PARTICLES AND NUCLEI Volume 2, Part 1

PARTICLES AND NUCLEI

Volume 1, Part 1

Elastic Scattering of Protons by Nucleons in the Energy Range 1-70 Ge V V. A. Nikitin

Probability Description of High-Energy Scattering and the Smooth Quasi-potential A. A. Logunov and O. A. Khrustalev

Hadron Scattering at High Energies and the Quasi-potential Approach in Quantum Field Theory V. R. Garsevanishvili, V. A. Matveev, and L. A. Slepchenko

Interaction of Photons with Matter Samuel C. C. Ting

Short-Range Repulsion and Broken Chiral Symmetry in Low-Energy Scattering V. V. Serebryakov and D. V. Shirkov

CP Violation in Decays of Neutral K-Mesons S. M. Bilen'kii

Nonlocal Quantum Scalar-Field Theory G. V. Efimov

Volume 1, Part 2

The Model Hamiltonian in Superconductivity Theory N. N. Bogolyubov

The Self-Consistent-Field Method in Nuclear Theory D. V. Dzholos and V. G. Solov'ev

Collective Acceleration of Ions I. N. Ivanov, A. B. Kuznetsov, E. A. Perel'shtein, V. A. Preizendorf, K. A. Reshetnikov, N. B. Rubin, S. B. Rubin, and V. P. Sarantsev

Leptonic Hadron Decays E. I. Mal'tsev and I. V. Chuvilo

Three7Quasiparticle States in Deformed Nuclei with Numbers between 150 and 190 (E/T) K. Ya. Gromov, Z. A. Usmanova, S. I. Fedotov, and Kh. Shtrusnyi

Fundamental Electromagnetic Properties of the Neutron Yu. A. Aleksandrov

Volume 2, Part 1

Self-Similarity, Current Com;nutators, and Vector Dominance in Deep Inelastic Lepton-Hadron Inter­actions

V. A. Matveev, R. M. Muradyan, and A. N. Tavkhelidze Theory of Fields with Nonpolynomial Lagrangians

M. K. Volkov Dispersion Relationships and Form Factors of Elementary Particles

P. S. Isaev Two-Dimensional Expamions of Relativistic Amplitudes

M. A. Liberman, G. I. Kuznetsov, and Ya. A. Smorodinskii Meson Spectroscopy

K. Lanius Elastic and Inelastic Collisions of Nucleons at High Energies

K. D. Tolstov

PARTICLES AND NUCLEI

N. N. Bogolyubov Editor-in-Chief

Director, Laboratory for Theoretical Physics Joint Institute for Nuclear Research

Dubna, USSR

A Translation of Problemy Fiziki Elementarnykh Chastits i Atomnogo Yadra (Problems in the Physics of Elementary Particles and the Atomic Nucleus)

Volume 2, Part 1

® CONSULTANTS BUREAU' NEW YORK-LONDON • 1~72

Editorial Board Editor-in-Chief N. N. Bogolyubov

Associate Editors A. M. Baldin

Secretary 1. S. Isaev

K. Aleksander D. 1. Blokhintsev V. P. Dzhelepov G. N. Flerov 1. M. Frank V. G. Kadyshevskii Kh. Khristov A. Khrynkevich

Nguen Van Heu

N. Kroo R. M. Lebedev M. M. Lebedenko

V. G. Solov'ev

M. G. Meshcheryakov I. N. Mikhailov S. M. Polikanov Shch. Tsitseika A. A. Tyapkin

The original Russian text, published by Atomizdat in Moscow in 1971 for the Joint Institute for Nuclear Research in Dubna, has been revised and corrected for the present edition. This translation is published under an agreement with Mezhdunarodnaya Kniga, the Soviet book export

agency.

PROBLEMS IN THE PHYSICS OF ELEMENTARY PARTICLES AND THE ATOMIC NUCLEUS

PROBLEMY FIZIKI ELEMENTARNYKH CHASTITS I ATOMNOGO YADRA

n p06J1eMbi <pH3HKH 3J1eMeHTapHblX '1aCTJu.\ H aTOMHoro

library of Congress Catalog Card Number 72-835lO ISBN 978-1-4684-7558-6 ISBN 978-1-4684-7556-2 (eBook) DOl 10.1007/978-1-4684-7556-2

© 1972 Consultants Bureau, New York

Softcover reprint of the hardcover 1 st edition 1990

A Division of Plenum Publishing Corporation 227 West 17th Street, New York, N. Y. 10011

United Kingdom edition published by Consultants Bureau, London A Division of Plenum Publishing Company, Ltd.

Davis House (4th Floor), 8 Scrubs Lane, Harlesden, London NWI0 6SE, England

All rights reserved

No part of this pUblication may be reproduced in any form without written permission from the publisher

CONTENTS Volume 2, Part 1

Self-Similarity, Current Commutators, and Vector Dominance in Deep Inelastic Lepton-Hadron Interactions-V. A. Matveev, R. M. Muradyan, and A. N. Tavkhelidze ..............•.••.•....•................

Theory of Fields with Nonpolynomial Lagrangians- M. K. Volkov •........•...... Dispersion Relationships and Form Factors of Elementary Particles-Po S. Isaev ....• Two-Dimensional Expansions of Relativistic Amplitudes-M. A. Liberman,

G. I. Kuznetsov, and Ya. A. Smorodinskii ..........•..•...•.......... Meson Spectroscopy-K. Lanius •••.....•..••..••.•..••...••.•.•...... Elastic and Inelastic Collisions of Nucleons at High Energies- K. D. Tolstov •..••...

Engl.iRuss.

1 22 45

70 88

145

5 33 67

105 129 231

SELF-SIMILARITY, CURRENT COMMUTATORS,

AND VECTOR DOMINANCE IN DEEP INELASTIC

LEPTON-HADRON INTERACTIONS

V. A. Matveev, R. M. Muradyan, and A. N. Tavkhelidze

A general approach based on the principle of approximate self-similarity, current algebra, and vector dominance is developed for studying inelastic lepton-hadron interactions. Since the form factors for deep inelastic electromagnetic and weak interactions are self-similar, the number of independent variables in the asymptotic region can be reduced by one. Com­bined with current algebra, this circumstance leads to special sum rules which can in principle be used to solve the fundamental question of the structure of the electromagnetic or weak hadron current. It is shown that the mechanism for the violation of self-similarity or invariance is related to violation of conformal symmetry up to the Poincare symmetry group. The formation of a muon pair in a deep inelastic proton-proton collision, p + p -11 + + 11- + hadrons, is discus sed in detail.

INTRODUCTION

A basic approach in the theory of elementary particles is to study the behavior of electromagnetic and weak interactions at high energies. Figure 1 illustrates the most general case of an interaction of a lepton pair with a hadron system; this interaction factors into lepton and hadron parts:

(1.1)

The specific forms of the "coupling constant" c and the lepton part LI1 are well known; the hadron part pre­sents more difficulties. By analogy with electrodynamics, for which the local currents correctly describe the phenomena, we postulate that there exist operators corresponding to local hadron currents - the electro­magnetic operator J~m(x) and the weak operator J'ti (x). These operators have a definite experimental meaning: Their matrix elements are directly related to observables (aross sections, polarizations, etc.). These quantized currents are expressed most simply in the Bogolyubov formulation of field theory, where they arise as the response of a particle system to an unquantized external perturbation:

The factors in the matrix element can thus be written

( 4na emIT w c={ ;

lG/V2

Joint Institute for Nuclear Research, Dubna. Translated from Problemy Fiziki Elementarnykh Chastits i Atomnogo Yadra, Vol. 2, No.1, pp. 5-32, 1971.

«:11972 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, Nt'Z<' York, N. Y. 10011. All rights reserved. This article cannot be reproduced for allY purpose whatsol'l','r without permission of the publisher. A copy of this article is available from the publisher for $15.00.

(1.2)

(1.3)

1

Lepton pair

Hadrons

a c

Fig. 1 Fig. 2

Fig. 1. The matrix element Tfi> describing an arbitrary electromagnetic or weak interaction of a lepton pair and a hadron system.

Fig. 2. Matrix elements for processes a-c. a, c) q2 is time-like, q2> 0; b) q2 is space-like, q2 < O.

Although the explicit form of the hadron part of the matrix element is not yet known, we can find concrete information about the hadron part by using the requirements of the relativistic P, C, and T covariance and the selection rules which follow from the existence of internal SU(2) or SU(3) symmetry. We are left with the fundamental theoretical difficulty-the lack of a quantitative understanding of the dynamics of strong interactions. This leads to the appearance of unknown functions, the so-called structure functions or form factors, in the theory. A familiar example is presented by the electromagnetic form factors of the nucleon, which depend only on the Lorentz-invariant variable GE(q2) or GM(q2). In general, the form factors may depend on several Lorentz-invariant variables. The fundamental problem actually consists of a theoretical and experimental study of these form factors. Such studies will hopefully lead to a solution of such funda­mental problems as particle structure and the possible existence of hadron subparticles (quarks, partons, etc.).

The importance of studying deep inelastic processes has been emphasized in several papers [1-5]. General methods have been worked out [4] for studying deep inelastic strong interactions, and rigorous cal­culations have been carried out for the amplitUdes. These methods can also yield useful information in a study of the behavior of the form factors for deep inelastic lepton-hadron interactions. We can list some of the specific deep inelastic lepton-hadron interactions which can be studied experimentally. Depending on which particles in Fig. 1 are considered to be entering and which are considered to be leaving, these inter­actions can be divided into three groups: a) those involving annihilation of a lepton pair, b) those involving scattering of a lepton by a hadron, and c) those involving the formation of a lepton pair in the collision of two hadrons. The corresponding matrix elements are shown in Fig. 2.

The interactions studied most intenSively in recent years are electromagnetic and weak scattering, which correspond to the diagram in Fig. 2b. Deep inelastic scattering of electrons by protons,

e- + p -? e- + hadrons, (1.4)

has been studied experimentally at SLAC [6]. An extremely interesting "point" pattern for electron crea­tion has emerged. The differential cross section da/dq2 for large q2 has turned out to be large, roughly equal to the Mott cross section for scattering by a structureless nucleon. Accordingly, several theoretical concepts have been advanced and checked [1-5]. An analogous point pattern was observed at CERN in ex­periments involving deep inelastic scattering of a neutrino by a nucleon [7, 8]:

V I1 + N -? 11- + hadrons. (1.5)

The simplest explanation for these factors is based on the assumption that, as the number of channels in­creases, the net contribution of the channels to the form factors depends weakly on q2. For a point nucleon and for the case of neutrino creation, we have, on the basis of the simplest perturbation-theory diagram,

2

z k

y

b(p)

Fig. 3 Fig. 4

Fig. 3. Kinematics of the formation of a lepton pair.

Fig. 4. The c.m. system of the muon pair. The z axis lies along the momentum p, while the momentum pI lies in the xz creation plane. The normal to the creation plane lies along thc y axis.

where E is the laboratory neutrino energy (GeV). The CERN experiments yield

In principle, this interaction could be studied at neutrino energies of 50 GeV on the accelerator of the Insti­tute of High-Energy Physics.

Below we will discuss in detail the interaction corresponding to the diagram in Fig. 2c - the deep in­elastic formation of a muon pair in a hadron-hadron collision:

(1.6)

In §2 below we carry out a kinematic analysis of this interaction, and we treat three theoretical schemes, based on self-similarity (§3) current commutators (§4), and vector dominance (§5) in order to obtain dyna­mic information. This analysis is based on results recently obtained at Dubna [9-13]. An experimental study of interaction (1.6) is currently being carried out on the Brookhaven accelerator, and preliminary data have been reported [14].

There is considerable independent interest in the results obtained in a study of interaction (1.6); these results may be quite valuable in the search for the intermediate W meson, which is formed in strong interactions [15-19].

We note that the next step in the study of interactions (1.4) and (1.5) is to single out one of the hadrons in its final state. The interactions

e- -I- p ~ e- -+ p' + hadrons. (1. 7)

VI' + P ~ 11- + p' -1- hadrons (1.8)

were studied theoretically in [20].

2. KINEMATIC ANALYSIS

We consider the deep inelastic collision of two hadrons, a and b, which gives rise to a muon pair and some hadron system A:

(2.1)

3

In the lower-order approximation in terms of the electromagnetic interaction, this process involves the emission and decay of a virtual photon as depicted in Fig, 3, where the parentheses denote the 4-momenta of the particles. The corresponding matrix element of the T matrix is

Tji~ 4J12a j"(AoutIJ,,(O)lp,p', in)C, q (2.2)

where jlA. u (k) ylAv (k') is the electromagnetic current of the muon pair, J p,(x) is the operator correspond­ing to the electromagnetic hadron current, and Q! = e 2/47T = 1/137 is the fine-structure constant. The index "c" reminds us that we are to take into account only the coupled part of the current matrix element. If the colliding particles are not polarized, and if only the muon pair is detected in the final state, the cross sec­tion for this interaction can be expressed in terms of the following second-rank tensor:

PI-''' (p, p', q) ~~ ~ (2:n;)4 X 6(p -1- p' - q - PAl (p, p', in I J I-' (0) I A out) (A out I J" (0) I p, p', in)c. A

(2.3)

Because of conservation of electromagnetic current, this tensor must satisfy the condition for gradient in­variance, qlApl-''' = pl-'vq~ = 0; from the Hermiticityof PI''' = ptlA we see that the real part of the tensor must be symmetric, and the imaginary part must be anti symmetric , with respect to the interchange p, ~ v.

It is convenient to expand tensor p v in terms of structures [21, 9, 11, 22] corresponding to definite polarizations of the virtual photon. We Jetermine the directions of the three-dimensional polarization vec­tors e(T 1), e(T 2), e(L) in the rest system q = 0 of the virtual photon, i.e., in the c.m. system of the muon pair, as shown in Fig, 4. Then the corresponding 4-polarization vectors are

(2.4a)

(2.4b)

e(L) = 1 "" .. V "}"I-" ~ -:3"2 (2.4c)

where

(2.5)

It is not difficult to see that the polarization vectors are orthogonal to the virtual-photon momentum qP, and to each other; their norm is equal to -1:

(2.6)

In addition, the completeness condition holds:

(2.7)

We use these vectors to expand the tensor Pp,v in terms of the five independent structures:

4

The structure functions* (or form factors) PT l' PT 2' PL, p5;L and pS[L a re real functions which depend on four unknown Lorentz-invariant variables, which may be chosen to be, e.g., s = (P1 + P2)2, q2, V = pq, and ll2 = (p'- q)2 == m,2 + q2 - 2v. Other invariant variables could be selected, e.g., mk = (p + p' - q)2 (the square

of the effective mass of the hadron system) or {) == ~ p (p' - q) (the energy transfer in the p = 0 lab. system).

We note that in the q= 0 system there is a simple relationship between the spatial components of the tensor Pij and the form factors:

( pxx 0 PXZ)

II Pu II ~ 0 pyy O. =

PZX 0 (lzz

(2.9)

Within a normalization factor, this is the density matrix for the virtual photon, specified in a linear basis.t

The angular distribution summed over spins is by definition equal to the ratio of the pentuple differ­

ential cross section d50 (5, q2, ~2, V, fl, 'V) to the triple cross section da0 (5, q2, ~2, v) . --dq2 d~2 d~ dq2 d~2 dv .

W (8 ) W (0 2 1\" ) _ d50 (5, q2, ~2, v, (J, '1.) , cp = . , q', S, q, - ,V - !la0 (5, q2, ,\2, v)-

W (8, lp) = 1 v2 [PT, (I - v2 sin2 0 cos" (p) 4n (1- 3 ) fl

(2.13)

where P is given by Eq. (2.12), and V= I~I =V q2-24mt is the velocity of the muons in their c.m. system. L q

By stp..,qying this angular distribution we can determine PT l' PT 2' PL' and p!;L but not pfL. The form fac­tor PTi can be found by measuring the polarization of one of the muons along the normal to the creation plane (along the y axis in Fig. 2):

(2.14)

. (+) (-) *In the notation of [21], we have PT 1 = F 3, PT 2 = F 2, PL = F 1, PTL = F 4, PTL = F 5•

tTransforming from the linear basis to the helical basis,

(2.10)

we find, following Oakes [21], a relationship between the form factors and the orthonormal matrix elements of the density matrix in the helical basis:

1 plll) = p-1-1 = 2P (PT2 -\-PL);

1 pOO=-PL' P ,

1 p1-1= p-ll =2,) (PI', - PI,);

1 (2.11)

(110= pOh= _p-10= pO-1* = __ 1 ___ (p'll -\- ipji), V2p

where

(2.12)

5

q q

p/_--( p'

p

Fig. 5. Amplitude of the Compton effect with two hadrons in the forward direction.

Integrating (2.13) over d<p or over d cos e, we find the distributions only with re­spect to e or <p, respectively:

W (8) = ( 1 V2) [(PTl -+- (JT2) ( 1 - ~2 sin2 8) _i PI. (1- v2 cos2 G) ] ; 2p 1--3

(2.15a)

w (cp) C~ _--=-(_I_V2 [PTl ( 1 - fV2 cos2 rp) j- PT" ( 1 --- f v2 sin2 rp) : PI. ( 1- v; )j. (2.15b) 2Jl(l 1--- 3 ) -

The form factors PT ' •.. , pf"L have kinematic singularities. The form factors PI,

P2, P3' P4' and P5 can be determined from [9]

(2.16)

where if'1-' and iJ'v are defined above, in Eqs. (2.5). It is not difficult to find a relationship between these two sets of form factors:

( 6f':J") 2 (l ~'(l - (j>2p - -'-- P - 2(iJ>~V 0 . L I 2 :JD2 3 I -tt (2.17a)

(2.17b)

(2.17c)

(2.17d)

The triple differential cross section for interaction (2.1) for the case in which only the muon pair with definite q2, A2, and 0 is detected in the final state, and in which the summation has been carried out over all possible hadron states, is

d3 fJ(s,q2,"'2,Q) ___ .c::.( q2-4m~)I/q2_4m~ 1 os 2J\2fl dq2d",2db 8n2 1- 3q2 J' q2 -V s-(m+m')2 Vs-(m-m'\2 ' ( , q, ,), (2.18)

where a = e 2j 41T = 1/137; m, m I, and m/.L are the masses of hadron b, hadron a, and the muon, respectively; and

(2.19)

We note that tensor P/.LII(P, p', q) describes the contents of the hadron "black box" for the Compton effect with two hadrons, depicted in Fig. 5.

The distribution with respect to the square effective mass of the muon pair is found from (2.2) by in­tegration over dA2 and do within the physical region. Neglecting the muon mass, we find the following equation for the muon-pair mass spectrum:*

.l.2max 6 max dfJ a m 1 r r dq2 = -12Jt2' Vs-(m+m')2 Vs-(m-m')2(j2 } d/).2 6J d6p (s, q2, /).2,6) .

.l. min min

(2.20)

*The definition of the boundaries of the physical region is discussed in the Appendix.

6

For use of the vector-dominance hypothesis (see §5), it is convenient to write the mass spectrum as

da _..!!:.-.-. ~ 1'* ( 2) d 2 -- 3 2 (J S, q . q Jt q

(2.21)

where

(2.22)

is the total cross section for the creation of a virtual y* photon having mass q2 in the interaction

a -1- b ~ 1'* + hadrons. (2.23)

As the last item in this section, we note that there is an interesting kinematic analogy between the reaction considered here and inelastic neulrino creation: If we replace the square lepton mass mz by q2 in the Appendix of Adler's paper [2], and if we replace Adler's q2 by our .6,2, we are actually defining the boundaries of the physical region for interaction (2.1). The Appendix below takes up in detail the definition of the boundaries of the physical region. In order to obtain dynamic information about the form factors, we consider below three theoretical schemes, based on: 1) self-similarity or scale invariance, 2) vector dom­inance, and 3) current commutators.

3. THE SELF-SIMILARITY PRINCIPLE

As we mentioned above, the SLAC and CERN experiments indicate a point nature for deep inelastic interactions of leptons with hadrons. This behavior could be understood on the basis of the hypothesis of approximate self-similarity or scale invariance. We assume that in the description of deep inelastic lepton­hadron interactions, for which the energies and momentum transfers are large, no dimensional quantities, such as the mass, "elementary length," etc. can play important roles, so the form factors can depend only on the kinematic-invariant variables.

N. N. Bogolyubov drew our attention to the possible self-similar behavior of the form factors in these problems. He emphasized that this behavior of the form factors for inelastic weak and electromagnetic interactions might be extremely similar in nature to the so-called self-similar solutions for several prob­lems of classical hydrodynamics, e.g., the problem of a powerful point explosion [24, 25]. In the search for self-similar solutions for hydrodynamic problems, it turns out to be quite useful to use the methods of similarity and dimensionality theory in combination with certain qualitative arguments about the nature of the physical processes. It is well known that electromagnetic and weak interactions can be described quite successfully by means of local electromagnetic and weak currents. The strong interactions, on the other hand, are taken into account through the introduction of form factors. At low energies, the need to take into account the particle masses will presumably distort the strong-interaction picture, while at high ener­gies (and at high values of the other invariant variables), in which case the masses of the particles created can be neglected, the situation becomes much simpler and, in a certain sense, "hydrodynamic." This hy­pothesis is supported qualitatively by the fact that the prinCipal singularities of the Singular functions of field theory are mass-independent (see, e.g., [26 D.

We will attempt to discuss the principle of approximate self-similarity or scale invariance with re­gard to lepton-hadron interactions at high energies and high momentum transfers, and we will derive several consequences which can be checked experimentally. We will assume that the asymptotic behavior of the form factors for interactions involving leptons at high energies and high momentum transfers is governed by dimensionality considerations and by the requirement of approximate invariance under scale transforma­tions q - Aq and Pi - APi, where q is the momentum transferred from the lepton to the hadron, and Pi are the momenta of the hadrons participating in the reaction.

In essence, this assumption means that in the asymptotic limit in which q2 - 00 and qpi - 00, the form factors for lepton interactions are governed by functions of the dimensionless ratios Wi = q2jqPi and are approximately independent of the particle masses and of other dimensional parameters, such as the inter­action radius, etc. We emphasize that this principle does not hold for purely strong interactions, since in this case the processes apparently depend strongly on the constant dimensional quantities.

7

Below we derive several consequences of the principle of scale invariance for interactions involving the annihilation of electron-positron pairs into hadrons, electron creation, and the formation of lepton pairs in two-hadron collisions.

We first consider the simplest deep inelastic process involving leptons-the annihilation of a lepton pair resulting in the production of hadrons:

e+ + e- -7 hadrons.

In the one-photon approximation, this interaction corresponds to the diagram

and the total cross section is (for me == 0)

_~_ 8rr2(;t2 2 (Jtot -~ -q-2 - P (q ). (3.1)

All the information about the dynamics of the process is incorporated in the unknown spectral function (or form factor) p(q2), related by definition to the tensor p/1V(q2):

p",,, (q) =co J dx eiq " (0 I J fl (x) J" (0) I O)~ ~ (2n)4 & (q - PN) (0 I J", (0) I N) (N I J" (0) I 0)= ( - g",,,q2 -I- q)J.q,,) P (q2) (3.2) N

It is easy to calculate the dimensionality of the tensor* P/1v:

[P)J." (q)] = [m2] • (3.3)

We see that p(q2) is, as expected, dimensionless:

[p (q2)] = 1. (3.4)

For scale transformations of the momentum scale,

q-7Aq. (3.5)

it follows from an account of the self-similarity principle that

(3.6)

At large q2 the total cross section must thus asymptotically display "point" behavior, analogous to the case of the annihilation of an electron-positron pair resulting in the production a muon pair:

*We use a system of units here in which action and velocity are dimensionless and in which mass is a di­mensional quantity. We recall that in this system the current dimensionality is [J /1] = [m3], and In>, the partial state vector for relativistic invariant normalization, has the dimensionality

[I Plo PZ. P3 • ...• Pn»= [m-n ).

8

(3.7)

This behavior is the same as that predicted by quark-current algebra [27, 28]. Using an inverse Fourier transformation, we can construct the space-time picture, finding that the commutator of the elec­tromagnetic currents between the vacuum states is

(3.8)

where c = p(q2) = const; 0 is the d'Alembertian, and .') represents the principal value. In particular, it fol­lows that the simultaneous commutator between the time and space components is

I ic (0 I [Jo (x, 0), J i (O)J I 0) = lim 2'- \i8 (x).

't-} LJ T JL (3.9)

Le., it is equal to the Schwinger term with a numerical coefficient which diverges as c 2 [29].

In the one-photon approximation, electron creation is described by

p

and the cross section is expressed in the standard manner (see, e.g., [30]) in terms of the tensor

W ltv (p, q) = ~ (2n)' (p I J IL (0) IN)' (N I J v (0) I p)C 8 (p .. q ~ PN) N

(3.10)

It is easy to see that this tensor is dimensionless:

(3.11)

it follows that

(3.12)

It follows from self-similarity that for scale transformations

(3.13)

the form factors W 1 and W 2 must satisfy

WI (J...,2q2, J...,2pq) = Wit (q2, pq); } J...,ZWz ('AZq2, J...,2pq) = Wz (q2, pq).

(3.14)

9

These requirements can be satisfied by setting

(3.15)

This universal dependence on the single dimensionless variable q2/pq for the form factor W2 has actually been observed experimentally at SLAC [4]. Some theoretical arguments in favor of this dependence were advanced in [23]. We note that many attempts are currently being made to find a qualitative explanation for this "pOint" behavior of the form factors for electron creation for high momentum transfers through the construction of specific models (see, e.g., [31-33]).

We turn now to the formation of a muon pair in a strong interaction. The dimenSionality of the tensor Pp,v and thus of the form factors Pi is

(PILV] = [pd = [m-2 ]. (3.16)

Using the requirement of self-Similarity, and taking account of (3.2), we find

PILV(Ap, 'Ap', Aq)= 'A-2PILV(P, p', q); } Pi (1.25, ,,2q2, ,,2;\,.2, ,,28) ~ 'A-2Pi (5, q2, ;\,.2, 8),

(3.17)

from which it follows that the form factors are, for large invariants,

(3.18)

where a, {3, and ware dimensionless variables constructed from ratios of the invariants s, q2, to, and O.

The self-similarity principle can also be used to analyze the behavior of the form factors for weak interactions. There is considerable interest in an experimental check of the consequences of the principle of approximate self-similarity for deep inelastic scattering of neutrinos by nucleons. We note that the self-similar nature of the form factors for the electromagnetic and weak interactions permits the number of independent variables in the asymptotic region to be reduced by one; moreover, knowing the form factors for one set of invariants, we can predict them for another set if certain ratios remain fixed.

We believe it would be very interesting to experimentally check the behavior predicted by the self­similarity principle up to certain large values of the invariants. Deviations from these predictions could be interpreted as evidence that some dimensional factor, e.g., the "elementary length," etc. comes into play here and violates self-similarity over supersmall distances.

We have thus far treated the possible existence of "maximum self-similarity," i.e., self-similarity with respect to all variables. The possibility is not ruled out that "partial self-similarity" may occur, in which case the self-similarity would not hold for all variables, but only for some of them. In particular, it would of course be extremely interesting to understand the mechanism for the violation of the self-similarity principle and to work out methods for calculating the corrections to the self-similar approximations. This question is apparently closely associated with the concept of the spontaneous violation of conformal sym­metry up to the symmetry of the Poincare group.

Conformal symmetry is one of the possible generalizations of Poincare symmetry which is of physi­cal interest. We briefly review the basic information about the conformal group, which contains as one transformation the space-time scale transformation. This 15-parameter group includes the following trans­formations [34-37]:

1. Space-time translations (four parameters),

X'IL=x"+a". (3.19)

2. Homogeneous Lorentz transformations (six parameters),

(3.20)

10

3. Special conformal transformations (four parameters),

x'IJ..-;--- XI.t+~I.tX2

I +2~X+~2X2 • (3.21)

4. Scale transformations (one parameter),

(3.22)

According to the Noether theorem, local currents correspond to these transformations; in particular, the currents of the special conformal transformations C/.1v and scale transformations are expressed in the fol­lowing manner in terms of the moments of the energy-momentum tensor (the gravitational current):

(3.23)

(3.24)

The generators for the transformations are expressed in terms of the spatial integrals of the zeroth cur­rent components:

(3.25)

(3.26)

It can be shown [34, 38, 39] that the divergence fields of currents (3.23) and (3.24) are related in the follow­ing manner over a broad class of Lagrangian theories:

(3.27)

The vanishing of the current divergences corresponds to conservation of "charges" (3.25) and (3.26). Equa­tions (3.27) show that invariance with respect to the total conformal group follows in this case from scale invariance, so conformal symmetry is violated "minimally" because of the violation of scale invariance. If the Lagrangian is independent of the mass and other dimensional constants, we have e /.1 == 0, and we thus find, as suggested above, scale invariance. The question of the possible spontaneous viofation of this sym­metry is currently being discussed widely in the literature in connection with the violation of chiral sym­metry [35, 38-40].

4. CURRENT COMMUTATORS AND ASYMPTOTIC SUM RULES

We consider the Fourier transform of the matrix element of the electromagnetic-current commutator between the two-particle in-states [9, 11]:

(4.1)

We assume here that the enclosed particles are not polarized; the "c" indicates that we take that part of the matrix element which is coupled as a whole. Let us consider the quantity r/.1V in more detail. Using the completeness condition for the out-state vectors, we find

(4.2)

11

where the "c" on the summation symbol indicates that we take only the matrix elements of the product of two currents which are coupled as a whole. We extract from this sum the completely coupled part corres­ponding to the quantity*

{IJ.V(p, p', q)=pIJ.v(p, p', q)l-pIJ.v (p, p', q),

where P/lV denotes the contribution of 15 z diagrams. This separation can be depicted graphically as

+ Diagrams obtained from the symmetrization of the initial and final states.

From momentum conservation and the spectrality condition it follows that, for q2 > 0, we have

In the physical region the P/lv(p, p', q) contribution thus vanishes exactly.

(4.3)

(4.4)

(4.5)

In the derivation of the sum rules (see below), however, we use the entire region -both the physical and the nonphySical parts-so the z diagrams from the second part of the p(p, p', -q) commutator may give a nonvanishing contribution. We show below that, with the standard assumptions used in a current-algebra derivation of sum rules, the contribution of these diagrams tends toward zero as s - 00.

We will show that the problem of determining the behavior of the form factors for the creation of a muon pair at high energies of the colliding hadrons and at high energies and masses of the virtual photon, for which s, q2, v - 00 (Le., such that the ratios

(4.6)

*If the < A out I state contains particle p or pI ,the current matrix element < A out I J/l (0) I p, p' in> will contain uncoupled parts corresponding to free propagation of these particles. Graphically, the matrix element can be divided into coupled and uncoupled parts in the following manner:

p-~~-- p'--'!:._- 2Jo.

The first term here, the completely coupled part, is involved in the determination of the cross section for the physical process; the other three terms are uncoupled parts which lead to the appearance of so-called semi coupled z diagrams.

12

remain fixed), can be reduced to a study of the simultaneous commutation relations between the spatial components of the operator for the electromagnetic hadron current and for its time derivative.

The use of simultaneous commutation relations is much simpler in the c.m. system of the muon pair, where we have q = {qo, O}. In this system the expansion of the tensor Pij (p, p', qo), (i, j = x, y, z) becomes

(4.7)

We see that Rij' qj and Pij can be expanded in a similar manner in terms of the five structures:

(4.8a)

(4.8b)

(4.8c)

where

(4.9)

where e(qo) = ± 1, qo ~ 0; i = T 1, T 2' L, and TL<+);

Rj"l(p, p', qo) = rj"l(p, p', qo)+rj"l(p, p', -qo) =pj"l(p, p', IqoIH-pj"l(p, p', -Iqol)· (4.10)

We see that the quantities RT l' RT ' R L, and R~L are odd functions of qo, and R~L is an even function. In­tegrating Eq. (4.1) over dqo and qodho, we find several relations:

etc. Here we have

=

-21 r dqoRij (p, p', qo) =- iBij (p, p'), n J

00

21n I qodqoRij(p, p', qO)~Cij(p, p'),

Bij (P. p') = - i j dx (p, p' in I [J i (x, 0), J; (0)11 p, p', in)C;

Cij(p, p')= -i f dx (p, p', in I [ji (x, 0), Jj(O)11 p, p' in/.

Of these relations we retain only those which are nontrivial in terms of parity:

00 + .i dqoR~-:l (p, p', qo) = BXl (p, p') - Bzx (p, p'); o

00

~ I dq"qr,RTI (p, p', qo) C~ Cxx (p, p'); o

00

; i dqoqoRT2 (p, p', qo) = Cyy (p, p'); (I

(4.11)

(4.12)

(4.13)

(4.14)

(4.15a)

(4.15b)

(4.15c)

13

00

-k- ) dqoqoRdp, p', qo) = Czz (p, p'); (4.15d) o

00

~ ) dqoqoR~L (p, p', qo) = Cxz (p, p') + C zx (p, p'). (4.15e) o

We note that in the q = 0 system selected, the invariant variables on which the form factors depend are

s = m2 + m'2 + 2 (Pop~ - pp'); q2 = q~; V = Poqo; a ~-.£i Po

(4.16)

It follows that the variables sand Ct are fixed in the integration over dqo, while q2 = v2/P5; Le., in the (q2, v) plane, the integration in (4.15) is carried out along a parabola.

Such sum rules for arbitrary fixed momenta p and pI contain contributions from the spectral functions p of the corresponding z diagrams. As condition (4.5) shows, the contributions of the z diagrams in the limit s - 00 are governed by the intermediate states of hadrons A with infinitely heavy effective masses mAo Following the convention of the current-algebra method, we assume that the z-diagram contributions vanish as s - 00; this assumption is valid when the order of the integration and the transition to the s - 00 limit in Eqs. (4.15) can be changed. For sum rule, e.g., (4.15a), the z-diagram contribution is given by

00 ood 2

1 r (-) ( , 1 I) d 1 r mN (-) ( , 1 I) n J Pn p, P ,- qo qo = -n J 2EN PTL p, P , - qo . (4.17) o s

Taking the limit s - 00 under the integral for fixed mi, and using (4.5), we find that the contribution of the z diagrams to the sum rules vanishes in this limit.

In the c.m. system of the lepton pair, the limiting transition s - 00 is carried out under the conditions Po - 00, Po - 00. We assume that

a=.E.!!. Po

is fixed, ~ = P; pz is fixed.

(4.18) is fixed,

The constancy of {3 in the invariant form means that

is fixed. (4.19)

We now assume that there exist limits for fixed Ct and {3:

Bij(a.,~)= lim PoBij(p,p'); (4.20) Po, P~--+oo

Cij(a.,~) = lim Cij(p, p'), Po, P~--+oo

(4.21)

where the tensors on the left side are dimensionless.

We now take the limits s - 00, v - 00, q2 - 00 in the sum rules under the condition that Ct, {3, and ware constant. In this limit, as was mentioned above, the z-diagram contributions drop out, and the form factors Pi (s, q2, Ct, v) have the following self-similar behavior:

(4.22)

14

Converting to an integration over dw in Eq. (4.15), we find the following final sum rules relating the limiting self-similar form factors to the current matrix elements:

where

"'0

~ I' dwF~l(a,~, w)=Bxz(a, ~)~Bzx(a, ~); " J I)

"'0

"'0 -k- r dwwFTl(a,~, w)=Cxx(a, ~);

o

"'0

L r dwwFT2(a,~, (,»=Cyy(a, ~); .'t J

o

"'0

-k- i d(,)(,)F L (a, ~, w)= Czz (a, ~); o

*- ) dwwF(tl(a,~, w)=Cxz(a, ~H--Czx(a, ~), o

+ ' I () -~-~(1 ~l.~) , 0 -- 2po - 2 ,~ .

(4.23a)

(4.23b)

(4.23c)

(4,23d)

(4.23e)

(4.24)

The right sides of these equations depend on the specific model chosen for the current, so they may be used as a criterion for selecting some model or other.

In the model in which quarks interact by exchanging a neutral vector meson (the "gluon" model) and in the model of vector commutator fields, we have

where

{

~ & (x) tjJ+ (0) {i (aiaj + aA - 2nd&ij) - 2g (aiB j + ajB i ~ 2n8&ij) + 4M&d Q2tjJ (0) (quarks);

& (x) Cabn (0) J7 (0) + C - a number (fields).

(2/3 0 0)

Q= 0 -1/3 0 . o 0 -1/3

Using (4.27), we find from the sum rule for the polarization form factor that

"0 I' d F(--l ( P. ) _ { const (quark model); J w TL 0.,1'" W ~ o 0 (field algebra).

(4.25) (4.26)

(4.27) (4.28)

(4.29)

(4.30)

It can also be shown from the sum rules that the quark model predicts greater values for the trans­verse form factors FT 1 and FT 2 than for the longitudinal form factor FL.

15

b

a

Fig. 6. Diagram illustra­ting the vector-dominance model.

These sum rules can thus be used to select a model. Analogous sum rules were found in [41] (see also 22, 42) for the case of ordinary electron creation. The analogous sum rules were treated in [20] for electron creation with a single hadron distinguished in the final state.

5. VECTOR DOMINANCE AND THE MUON-PAIR MASS

SPECTRUM

According to the vector-dominance hypothesis (see, e.g., reviews [43, 44]) muon-pair formation involves the emission of a virtual vector meson, which converts into a virtual photon, which then decays into a muon pair, as shown in Fig. 6.

It can be shown that the vector-dominance hypothesis leads to a correct description of this process, since here q2 is time-like.

Let us determine the matrix density for the virtual vector meson V (V = Po, w, or ~ formed in the reaction

a + b ---0> V + hadrons, (5.1)

according to

W'IlV(P, p', q) = ~ (2n)4 6(p+p' -q- PA)(P, p', inl J(V)(O) IA out)" (A out I J~v) (0) I p, p' in)C, A Il

(5.2)

where Jf;') (x) = (02 - m~) V Il (x) is the density of the V-meson current. Using the current-field identity,

(5.3)

we find the following relation between the density matrices of the virtual photon and the vector mesons:

m2 2 1 PIlV (p, p', q) = ~ (-2 v ) (2 2)2 W~VJ (p, p', q) + interference terms.

Yv mv-q (5.4)

v

Equation (5.4) can be used to express the five form factors PT ' PT 2' PL> p!;i, which completely de­scribe the muon-pair formation in terms of the corresponding V-m~son form factors;!'

For use of the vector-dominance hypotheSiS it is convenient to write the equation for the mass spec­trum in the form [12]

~ _ ~ . ~ (1 _ q2 - 4m~) ... / q2 - 4m~ v* 2 dq2 - 2n q2 3q2 V q2 (J (S, q ), (5.5)

where

(5.6)

is the total cross section for the creation of a virtual/,* photon of mass q2 in the process

a + b ---0>')'* + hadrons. (5.7)

*We recall that only the form factors PTl' PT ' and PL contribute to the cross section; the form factor p!tl can be determined from the angular distributfon of the muon pair, while Ptt can be determined by measur­ing the polarization of one of the muons (see §2).

16

to~

10-32

10-33

> c3 1O-34

'-NE u~ 10cJ5

<:~ §. .:g 10-36

10-37 pp

Fig. 7. Mass spectra of the muon pair formed in pp collisions (with Plab. = 28.5 GeV /c and in 7r+ -p colli­sions (with Plab. = 8.5 GeV/c) predicted by the vector-dominance model [Eqs. (5.13) and (5.16)].

According to the vector-dominance hypothesis, this cross section is related to the total cross section for the formation of real vector mesons in process (5.1) by

v* 2 _ (,( [( m~ )2 4n P ,( m~ )2 4n (,) L ( m~ )2 a (s, q ) -""4 mZ _q2 'l'~ a (s) T m~-q2 'l'~ a (s)" m~-q2

~~ a lJl (s) J + interference terms. (5.8)

Substituting this approximate expression for aY* into Eq. (5.5), neglecting the muon mass (m/-t = 0), and assuming the contribution of the interference terms to be small, we find the following expression for the mass spectrum of the muon pair:

du (,(2 ~ (m;,)2 4n " ([(j2= 12n ~ m~_q2 'l'~ a (s).

v~po. Ul. IJl

(5.9)

The cp meson is weakly created in hadron-hadron collisions, so by retaining only the contributions of the pO and w mesons, and assuming mp ~ m." p~: y~ = I : 9, and YZ/4:n = 0.5, we convert Eq. (5.9) to (ml1l1 = Vif)

or, for large m /-t/-t '

du 2.10-6 I' em2

~ = m (m2 -06)2 [ aP (s) +gaUl(s) JGev ~ ~ ~ . .

du 2:10-6 [ 1 J em2 --= -.- aP(s)-I-aUl(s) -dml1l1 mlll1 9 GeV

(5.10)

(5.11)

We will use Eq. (5.10) [or (5.11)] to analyze muon-pair formation in specific hadron-hadron collisions.

a. Proton-Proton Collisions (a = b = p). The formation of the pO meson in a p + P _ P + P + pO reac­tion has not been observed at any energy up to Plab. = 28.5 GeV/c. In this range, the cross sections for w­meson formation in the p + p - p + p + W reaction are [45]

Plab .• GeV Ie 5 10 28.5

140;1::20 I 60 50±1O

This result is in agreement with the results of an analysis based on the double Regge-pole model [46]. Analysis of the six-ray reaction

pp -+- pp:n+:n+:n-JC

shows that about 24% of the events involve the formation of a pO meson; the corresponding cross section is 90 tffJ [47]. The cross section for the right-ray process pp - PP7T+7T+7T+7T-7T-7T- is 20 /-tb. Assuming that here also about one-fourth of the events involve the formation of the pO meson, we can estimate the corres­ponding cross section to be about 5 /-tb. We can thus assume that the total cross section for pO formation in pp collisions at Plab. = 28.5 GeV/c is equal to roughly 100 /-tb:

aPP-+P+, .. = 100 /lb. (5.12)

The w contribution in (5.10) can be neglected because of the factor of 1/9. If, on the other hand, we assume that ()'Ul~()'P = 100 /-tb, we find the following expression for the mass of the muon pair formed in pp colli­sions with Plab. = 28.5 GeV/c, from Eqs. (5.10) or (5.11):

17

p p

Fig. 8 Amplitude for the scattering of ha­dron a having 4-mo­mentum ~ and a non­physical mass ~2 by hadron b having mass p2 = m 2. In the lab. system (p = 0), the energy of the nonphys­ical hadron is () (11m) pD..

daPP dmfJ.fJ.

2.2.10-34 2 2.2.10-34 2 ( • -06) em /GeV ~--.- em /GeV.

mfJ.fJ. mfJ.fJ. . mfJ.fJ.

The corresponding curve is shown in Fig. 7.

(5.13)

b. Pion-Proton Collisions. We consider the case of 7r+P collisions (a = 11'+, b = p). We can conclude from the analysis in [48] that tlie cross section for pO meson formation in the reaction 11'+ + p - pO hadrons is greater than or approxi­mately equal to 1840 tLb at Plab. = 8.5 GeV/c,

o"+p ..... po+ ••• :;;,. 1840 /lb, (5.14)

and the cross section for w formation in the reaction 7r+P - w + hadrons is

(5.15)

From Eqs. (5.10) or (5.11) we thus find the following approximate (lower) estimate for the mass spectrum of the muon pair formed in 7r+P collisions with a momen­tum of Plab. = 8.5 GeV Ic:

3.7.10-33 2 3.7.10-33 2

( 2 -06)2 em /GeV :::::; --.- em /GeV. m~fJ. m~fJ. ' mfJ.fJ.

(5.16)

6. LOWER LIMIT FOR THE MASS SPECTRUM

To find an asymptotic expression for the muon-pair mass spectrum, we consider the hadron part of the matrix element for the formation of a muon pair as Ip'l- 00. With an accuracy to within terms 0(11 Ip'!), the matrix element is

p' e (A out 1 J!(0) 1 p, p', in)C I P.I ..... oo' ;. (A out 1 J 0 m(O) 1 p, p', in)" + 0 ( m) . (6.1)

This means that the muon-pair formation is governed primarily by the Jo(O) component of the electromag­netic field; Le., it is a "Coulomb" process.

USing the Bjorken limit, i.e., expanding the T product in a series of simultaneous commutators, and retaining only the first term of this asymptotic series, we find the following approximate relation with the matrix element for hadron-hadron scattering away from the energy surface:

(A out 1 J~(0) I p, p', in)C =" - i ) dxe iqx (A out! T (,fom(x) JV") (0) 1 p/

= ----+ VI _ r dxe- iqx (A out 1 [J~(X, 0), J(U) (0»)1 p/ q2-+oo q2 J

I = v- (A out 1 J(a) (0) 1 p/ + contribution of quasilocal terms, q2

where J(a) (x) is the current of the hadron carrying a 4-momentum ~.

Using Eqs. (6.1) and (6.2), we can estimate the form factor p governing muon-pair creaction:

( 2 L12 6 4m V62_~2 (6 L12) P S, q , ,) ~ q2 0ab, •

(6.2)

(6.3)

The quantity uab(6, ~2) which appears here is the analytic continuation of the total cross section for the in­teraction of a and b hadrons into the nonphysical region, where the square mass of hadron a is negative and equal to ~2; 6 is the lab. energy of the nonphysical hadron (Fig. 8). In this approximation we find the follow­ing triple differential cross section (neglecting m' and mJ:

(6.4)

18

The corresponding mass spectrum is

(6.5)

If axiomatic field theory or analytic S-matrix theory [22, 49, 50] imposes a restriction on uab(o, L).2) away from the mass. surface, (6.5) imposes a restriction on the mass spectrum. For the simpler case of the electromagnetic form factor F(t), field theory and S-matrix theory predict an exponential restriction for the lower limit of the form factor. By analogy, we can expect

(J (/5 ~2)'--(JPhe-aV-"'2 ab, ..-;:::::;- ab , (6.6)

where uab is the total cross section for the interaction of real particles, and a is some constant. Then we find the following lower estimate for the mass spectrum from (6.5) under the conditions s »q2 » 1/a2:

(6.7)

Another method for estimating the mass spectrum was discussed in [51].*

APPENDIX

Determination of the Boundaries of the Physical Region for Muon-Pair Formation. Conservation of 4-momentum is described by

(A.1)

Introducing the vector L). = p' - q, we obtain

(A.2)

Then we have L).2 = mk - m 2 - 2mo, where ° = l/m pL). = (e _qo).

The identity mN == m corresponds to elastic scattering. Then L).2 and ° are unambiguously related, Le., are not independent variables ° = - L).2/2m.

The Omin value is at a minimum, since qO here is at a maximum. We consider the case in which the virtual photon moves backward in the lab. system. For fixed invariants, this virtual photon clearly acquires a minimum energy (qo)min; this means that

IImax = e - (qo)min • (A.3)

We find (qo)min from

(A.4)

Setting m' == 0 and solving this equation, we find

q2-t.2 eq2 (qO)min=-4-e-+ q2-t.2 ; (A.5)

*By analogy with the dynamics of a planar explosion in hydrodynamics, we have formulated a principle for approximate self-similarity for high-energy hadron-hadron collisions [52].

19

Ilmax~e-(qo) . =e l---~---~ =e*+-~ (q2 q2_t..2) t..2 mm q2_t..2 4e e* '

(A.6)

where

In the physical region we thus have

(A.7)

We now find the physical region .602 for fixed sand q2, from the condition

(A.8)

The result is

(A.9)

where

We note that there is an interesting analogy between the reaction under consideration here and inelas­tic neutron formation: If we replace the square of the lepton mass in the Appendix of Adler's paper [2] by q2 and replace Adler's q2 by our .602, we essentially reduce Adler's problem to ours, and vice versa.

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20

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(1958) . 26. N. N. Bogolyubov and D. V. Shirkov, Introduction to Quantum Field Theory [in Russian], Gostekhizdat,

Moscow (1957). 27. J. D. Bjorken, Phys. Rev., !1§., 1467 (1966). 28. V. N. Gribov, B. L. Ioffe, and I. Ya. Pomeranchuk, Phys. Letters, 24B, 554 (1967). 29. I. Schwinger, Phys. Rev. Letters, ~, 296 (1959). 30. S. Drell and I. Walecka, Ann. Phys.,~, 18 (1964). 31. s. D. Drell, D. J. Levy, and T. M. Yan, SLAC Pub., 606, 645, 685 (1969). 32. J. D. Bjorken and E. A. Paschos, SLAC Pub., 572 (1969). 33. D. J. Gross and C. H. Llewellyn Smith, Preprint CERN TH-1043 (1969). 34. J. Wess, Nuovo Cimento, 18, 1086 (1960). 35. G. Mack and A. Salam, Ann. Phys., 53, 174 (1969). 36. D. J. Gross and J. Wess, Preprint CERN TH-1076, Geneva (1969). 37. c. J. Isham, A. Salam, and A. strathdee, Preprint IC/70/3, Trieste (1970). 38. Y. Nambu and G. Jona-Lasinio, Phys. Rev., 122,345 (1960). 39. B. A. Arbuzov, A. N. Tavkhelidze, and R. N. Faustov, Dokl. Akad. Nauk SSSR, 139, 345 (1961). 40. A. A. Logunov, V. A. Meshcheryakov, and A. N. Tavkhelidze, DokI. Akad. Nauk SSSR, 142,317 (1961). 41. C. Callan and D. Gross, Phys. Rev. Letters, 22, 156 (1969). --42. J. Cornwall and R. Norton, SLAC Pub., 458 (1968). 43. J. J. Sakurai, Currents and Mesons, University of Chicago Press, Chicago (1969). 44. D. Schildknecht, Preprint DESY 69/10. 45. F. Turkot, Topical Conference on High-Energy Collisions of Hadrons, Part I, CERN, Geneva (1968),

p. 316. 46. R. G. Roberts, Nuovo Cimento, 53A, 557 (1968). 47. I. R. Kenyon, NucI. Phys., ~, 255 (1969). 48. H. H. Kung, Preprint, Columbia University, Nevis 171 (1969). 49. M. A. Jaffe, Phys. Rev. Letters, 11, 661 (1966). 50. A. Martin, Nuovo Cimento, 37, 671 (1965). 51. s. M. Berman, D. J. Levy, and T. L. Neff, Preprint SLAC Pub., 681 (1969). 52. Preprint Joint Institute for Nuclear Research E2-5962, Dubna (1971).

21

THEORY OF FIELDS WITH NONPOLYNOMIAL

LAGRANGIANS

M. K. Volkov

This work contains a review of contemporary methods of constructing two-point Green's functions and particle-scattering amplitudes, applicable in the quantum theory of fields with rapidly increasing spectral functions. These methods must ensure that the theory is finite in all orders of modified perturbation theory and must be such that the S-matrix is unitary. The nonanalyticity in the coupling constants, arising in the use of these methods, indicates that the usual perturbation theory is not applicable here.

INTRODUCTION

Considerable progress has recently been achieved in the development of methods, applicable in quan­tum field theory, of describing unrenormalizable interactions between elementary particles. In all these methods, an attempt is made to avoid the usual perturbation-theory methods and their accompanying diffi­culties (the appearance of an infinite number of undetermined constants in unrenormalizable theories; the impossibility of describing the nonanalytic dependence of amplitudes and Green's functions on the coupling constants in these theories, although some solvable models indicate the presence of such a dependence [1-4], etc.). Many authors have succeeded in solving such problems. At the present, however, the develop­ment of such methods is far from complete, although they permit us to obtain descriptions of many unre­normalizable interactions in complete agreement with the basic requirements imposed on field theory by the causality principle and the principle that the S-matrix must be unitary. In the present work we review the contemporary state of all these methods.

Before beginning our review, we briefly sketch the history of the formulation and solution of the prob­lems discussed here.

The first article concerning the construction of two-point Green's functions in theories with Lagran­gian (1.1) or (1.3) was published in 1954 by S. Okubo [2]. The method proposed by Okubo is similar to the methods of Group 2 or Group 3 (cf. § 2). The technique used by Okubo and the formulas he obtained differ only slightly from contemporary technique and formulas. The only flaw in his work is the incorrect analy­tic continuation with respect to the coupling constants, which leads to a violation of the condition that the S­matrix must be unitary. Unfortunately this important work did not at first receive the attention it merited, and its value has only been appreciated recently.

The next article to appear was written by Arnowitt and Deser [5], and was published in 1955. The Lagrangian (1.1) was considered, but the method proposed by these authors for the construction of particle­scattering amplitudes and Green's functions was much cruder than Okubo's method. B. M. Barbashov and G. V. Efimov [6] showed in 1962 that the condition that the S-matrix be unitary is also violated in this work. Almost ten years passed before any important new work was published. In 1963, several authors returned simultaneously to the study of the theory of fields with nonpolynomially increasing spectral functions. Inde­pendent attempts were made by G. V. Efimov and E. S. Fradkin to construct a finite quantum theory for fields with nonpolynomial Lagrangians [30, 31]. In the same year G. Feinberg, in collaboration with A. Pais, proposed the peratization method, which was used in the theory of weak interactions for the construc-

Joint Institute for Nuclear Research, Dubna. Translated from Problemy Fiziki Elementarnykh Chastits i Atomnogo Yadra, Vol. 2, No.1, pp. 33-66, 1971.

22

C 1972 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without permission of the publisher. A copy of this article is available from the publisher for $15.00.

tion of particle-scattering amplitudes depending nonpolynomially on the amplitude [18]. After the appear­ance of this article, the number of papers published on work in this direction increased rapidly. An incom­plete list of the authors is as follows: Guttinger, M. K. Volkov, B. A. Arbuzov and A. T. Filippov, Fried, Delbourgo, Salam and Strathdee, Lee and Zumino, Lehmann and Pohlmeyer, Keck and Taylor, Budini and Calucci [4, 7-12].

1. EXAMPLES OF UNRENORMALIZABLE INTERACTIONS

We begin by giving several examples of unrenormalizable interactions of elementary particles which have been described by the methods under consideration.

1. A neutral pseudo scalar theory, with pseudovector coupling between scalar and spinor fields [2, 5, 13]:*

L (x) = Lo (W (x), (jJ (x)) - ig : if (x) Y5Yv W (x) aV(jJ (x): (1.1)

By using Dyson's transformation [15]

W' (x) = exp { - gY5(jJ (x)} W (x) (1.2)

for the spinor field, we can get rid of the derivative in the interaction Lagrangian and write (1.1) as

L (x) == Lo (W' (x), (p (x)) - m : W' (x) (exp [ - 2gYb(jJ (x))- I) 0/' (x): (1.3)

(here the normal-product sign refers only to spinor fields).

The two-point Green's function, and also the scalar-particle-scattering amplitude, in the second order with respect to the dimensionless parameters (mg), can be expressed in terms of the function

F (x) = iC (m' g)2 Sp {SC (x) SC ( -x)} exp { - i (2g)2 N (x)},

where SC(x) is the propagator of the free spinor field; Llc(x) is the propagator of the free scalar field; m' =m exp {i 2g2L\' (O)}, and C is a constant.

(1.4)

The function F(x) has an essential singularity on the light cone, and its spectral function grows more rapidly than any polynomial in p2. This indicates that the usual field-theory methods are inapplicable, and we have a typical situation in which a qualitatively new method must be used to describe the interaction.

2. A weak interaction, not conserving parity, between a neutral vector meson and a spinor field [16]:

Linter (x) = G : W (x) yV (a -+ ibYb) 'l' (x) Wv (x): (1.5)

we use Stuekelberg's transformation [17]

w v (x) = (jJv (x) + mw-aa e (x), Xv

(1.6)

where () (x) is a scalar field and (/Iv (x) is a vector field with propagator

~(<P) ( ) _ g",v ",v P - 2 . mw_p2-18

(1. 7)

[the zero-spin part of the field (/Iv (x) has a negative metric], to separate the unrenormalizable part of the interaction. It is written

*In our notation we follow the monograph by N. N. Bogolyubov and D. V. Shirkov [14].

23

G - . v Linter(unre)(X) = mw : 'I' (x) y" (a + lby6) 'I' (x) a e (x):

and is similar to the interaction (1.1). Applying Dyson's transformation [15]

'1" (x) = exp { - iG (a + ibY6) ~<: } 'I' (x),

we again obtain an interaction of type (1.3) with a nonpolynomial dependence on e(x).

3. A weak four-Fermi interaction

Linter(x) '= GjA (x) j B (x),

(1.8)

(1.9)

(1.10)

where iA(x)= :'I'A (X)Y6'1'A (x):. For the case rnA = mB = 0 and s = (Pt + P2)2 = 0, Guttinger [7] found expres­sions for the Bethe-Salpeter scattering amplitude of Fermi particles in the "ladder" approximation. This amplitude is

r Va} { 11'0 } Fc(x)=aexp1--2-·- +bexp -2-'- , l x -1e x -Ie

(1.11)

and also differs only slightly from (1.4).

4. The scattering amplitude of spinor particles, found by Feinberg and Pais in the ladder approxima­tion for a weak interaction between vector mesons and Fermi mesons [18]:

Linter(x) = G:W (x) y" (1 + iY6) 'I' (x) W" (x): (1.12)

has a form qualitatively different from (1.4):

(1.13)

but it also leads to a spectral function growing more rapidly than any polynomial. In contrast to (1.4), the solution (1.13) does not satisfy the conditions of interaction localizability [19-22].

5. Chirally symmetric Lagrangians [8,23-27]. For a typical example we use Weinberg's Lagrangian for 7T-mesons [26]:

(1.14)

Here scattering-amplitude spectral functions also grow more rapidly than any polynomial, and the interac­tion has a nonlocal character.

There are examples of other interactions of a similar type (Einstein's gravitational Lagrangian, the Yang-Mills theory with massive fields, etc.). for example, in the work of Salam and his collaborators [8]. We confine ourselves to the five examples already described as being the most characteristic. For the demon­stration of various methods used in theories with rapidly grOwing spectral functions, we consider a certain Lagrangian of general form. For simplicity we assume that it depends only on a one-component scalar field, does not contain derivatives, and can be expressed as an infinite series in powers of cp(x):

00

L inter(x) = G ~ u ~~) :(<p (x»n: = G:U (<p (x»: (1.15) o

24

The coefficients u(n) are proportional to the second coupling constants gn, which are always present in such theories together with G. In the following, we construct the perturbation theory with respect to G. In each order with respect to G, all orders with respect to g are taken into consideration. We mainly consider second orders with respect to G, since this suffices for the description of characteristic features of the methods.

2. REMARKS CONCERNING ALL METHODS, AND CONDITIONS

IMPOSED ON TWO-POINT FUNCTIONS IN MOMENTUM SPACE

In theories with essentially nonlinear Lagrangian, the S-matrix is constructed as in the renormaliza­ble theories with renormalizable Lagrangians [14]:

where

00 on S = 1 + ~ -, S7" ,L.J n.

1

(2.1)

(2.2)

Elastic and inelastic scattering of scalar particles is possible for the Lagrangian (1.15) in the first order with respect to G. However,this order is not of interest since it is trivial (there is no divergence, and all amplitudes are expressed in terms of constants). We thus turn to the consideration of second-order pertur­bation theory:

where

and

00 00

S2 = - ) ) d4xl d4x2 ~ ~ FI.2ih2 (Xj - x2): o 0

(2.3)

(2.4)

(2.5)

is the propagator of a free scalar particle with mass m. We shall often consider the interaction of scalar particles with zero rest mass. In this case (2.5) has the simple form

t,.C (x) = _ i (2n)2 (x2 - ie) . (2.6)

All Green's functions and scattering amplitudes of scalar particles in the second order with respect to G will be expressed in terms of the two-point function (2.4).

Two difficulties arise in the study of this operator. The first is related to the appearance of ultra­violet divergence at the transition to the momentum space. The form of (2.4) shows that there is a pole of any order on the light cone. In localizable interactions, in particular, this causes the function F~2~ (x) to have an essential Singularity on the light cone. 2

25

Another difficulty is a specific feature of nonlocalizable interactions. It is that the series (2.4), for such interactions, either converges only in a bounded region of x2, or it does not converge in general for any region of x2 and is an asymptotic series. Hence, in addition to formerly known difficulties concerning divergent integrals, we are also confronted with the necessity of summing divergent series.

The principal requirement for the solution of these problems is that the 8-matrix of the final theory be finite and unitary. The microcausality prinCiple must be satisfied in theories with localizable interac­tions, and the macrocausality principle must be satisfied in theories with nonlocalizable interactions [21].

How shall we find the boundary between localizable and nonlocalizable interactions? This is most simply done by studying the asymptotic behavior of the operator (2.4) in momentum space, and in particular its imaginary part, which is Uniquely determined. For localizable interactions there is a definite limit to the rate of growth of Green's functions in momentum space, first discovered in somewhat different forms by Meiman [19] and Jaffe [20]. We start from the more general condition obtained by Meiman to the effect that for increasing p2 we have

(2.7)

for any a > 0 (cf. also [21, 22]).

In particular (2.7) must be satisfied by the imaginary part of the Fourier transform (2.4), whose be­havior for p2 - 00 is decisive for all functions F~2)k (p). Hence, it is natural to start with the determination of this imaginary part. 1 2

We also need the imaginary part of F~2)k (p) to verify that the S-matrix is unitary. What exactly is this condition? Consider the Fourier transf6rfn of the two-point function (2.4):

(2.8)

We require that the function F~2)k (p) have completely analytic properties. For definiteness, we consider the elastic scattering of two scJI~r particles. For p2 < 0 the function (2.8) must be real, for 0 < p2 < 4m2 there must be a simple pole for p2 = m 2, and at the roint p2 = 4m2 there is a branch point from which a cut extends to infinity. The discontinuity at the cut must be expressed in terms of the sum of the phase volumes of the scalar particles:

[ v:] • - u

1m F~~ (p) = nu!b (p2 -- m2) + n ~ ~r [lkm ) (p2), (2.9) h=~

where

(2.10)

If the rest mass of scalar particles is zero, the phase volume is simply a power function of p2:

Q(O) (2 4 -2 [ p2 ]k-2 a (p2) e (pO) ,. p) = (n) _ (4n)2 (k-l)! (k-2)1 ' (2.11)

and the upper limit of the sum in (2.9) is infinity. Concerning the analytic properties FW(P) must possess for mass-free particles, it must be real for p2 < 0, and it must have a pole and a branch point for p2 = O. There must be a logarithmic cut from the origin to plus infinity [ef. [11] and also formulas (6.26) and (0.32)].

Whatever method is used for constructing the finite function F~2k (P), it must possess the above prop-erties for the 8-matrix to be unitary. 1 2

26

We now consider the determination of the boundary between localizable and nonlocalizable interac­tions. We must find the asymptotic properties of ImFW(p) for p2 - 00. For mass-free particles, we use (2.11) for phase volumes. If the masses are not zero, we use an approximate expression for Q~m)(p2), valid for p2 »m2 [28]:

11-3 3k-5

P-2- (p _ km)-2-Q("m) (p2) ;;:::: m (p k) r (2k) 'Y"

(2.12)

where p = 0/, r (2k) is the gamma function, and <p(p, k) is a slowly varying function whose dependence on p can be neglected in the calculation of the asymptotic properties of Im'Fg>(p) for large p2. Results for vari­ous uk are given in Table 1.

By using the limitation on the growth of scattering amplitudes in localizable theories specified in (2.7), we easily obtain the corresponding conditions for the uk:

lim f Uk 11/1t =0 "-.00 r ( ~ )

(2.13)

or

lim ICm(k)II/"~O, "-'00

(2.14)

where

Cm (k) = (Uh;~)2 • (2.15)

Lagrangians for which the coefficients of the series expansion (1.15) satisfy conditions (2.13) or (2.14) de­scribe localizable interactions. If the above conditions are not satisfied, we have nonlocalizable interac­tions. There are two subclasses of the latter: the first in which the series (1.15) converges to a definite function of the field <p(x), and the second in which this series diverges. Only the first subclass is interest­ing from the physical point of view. In the second subclass, not only does the series (1.15) diverge, but in most cases the spectral function does not exist for mass-free particles (cf. Table 1).

An interesting difference between localizable and nonlocalizable interactions is the following. The asymptotic behavior of amplitudes for large momentum is identical in local theories for particles of zero and nonzero mass. For nonlocalizable interactions, the asymptotic behavior of scattering amplitudes for particles with mass and without mass is different, as can be seen from Table 1.* Hence, in approximate calculations for large energies in nonlocal theories, we cannot simply neglect the mass and use expres­sions from the case for mass-free particles. (This error is made in the paper by Salam and strathdee [9]; for large p2, they use for the propagator of a massive particle an approximate expreSSion coinciding with the expreSSion for a zero-mass particle.)

We thus give a brief description of problems to be discussed and claSSify the methods used for the solution of these problems.

We shall study a two-point function F(x), expressed as an infinite series in powers of the propagator of a free scalar particle:

00

F (x) = ~ C (n) [ .- iL1c (x)t. (2.16) 1

*We obtained this result recently, and it is published for the first time in the present article. For the def­inition of local and nonlocal theories, and also localizable and nonlocalizable interactions, see the footnote to Table 1.

27

28

TABLE 1. Asymptotic Properties of InrmFW<p2) for Large p2 --- ---

C lim I c (k) \1/k , In ImFW (p) In ImFg)(fi) U

.3 k--.oo "'~ U No. u (k) (uk)' U ('I') 1'-+00 r--~oo

.... c c (k) =~ -k-I- (mol- 0) (m=O) ~.g

<V SU (/)

N

I U (k) == 0 0 ~ U ~~) : CjJk (x) : 2(N-4) Inp 2(N-4) In p k>N 0

2 2 ~

2 - II' (yk)I(1,~O) 0 10 (V gCjJ (x) ).,,=1 canst p3+2 y canst p3+2 y "" '" I N

I I I I I -'" 3 - (const)k 0 exp (gCjJ (x)) const p'l3 canst p'/3 u 0

I -'

00 2 2

4 -I'(yk) 0 J dveVgq! (X)-v l / y const p3-2 y const p3-2 y I' < 1/. 0

I

5 - r ( ~ ) I

exp {+ g2rp2 (x)} canst p

I canst p -2

II 2 6 -1' (I'll) 00 exp (+ g2mCjJ2m (x)) const (21'-1) pin p canst p:J-2Y

1/2 < 1'< I m>2

( I' = I - dm ) I' < I <V

:;:; '" .;::; ';;l u

7 -I'(k) 00 gCjJ grp canst pin p const p2 ~

VI +g2CjJ2' I ± g2CjJ2 0 Z

2 const p3-2 y

00

8 - r(yk) ~ r ((1'-1) k) (gCjJ)k const (21'-1) pin p V<3/2

00

1'>1 0

III I Does not exist if

I 1'>3/2

00

9 k' -e 00 ~ en' (gCjJ)n const 2p2 Does not ex ist 0

*An interaction is localizable if it leads to a spectral function satisfying (2.7). If the spectral function grows more rapidly than (2.7) for large values of the momentum, then the interaction is called nonlocalizable.

Local theories are theories in which the microcausality principle holds. A theory is non­local if the microcausality principle is violated at small distances, and only the macrocausality principle holds (cf. [21]). Nonlocalizable interactions are always described by nonlocal theories, while localizable interactions can be described by both nonlocal and local theories, depending on the method used in constructing Green's functions and particle-scattering amplitudes [19-22].

We wish to construct the Fourier transform of F(x):

F (p) = i ~ d4xe ipxF (x), (2.17)

which must be a finite function of p2 and have completely analytic properties. The function pep) must have a spectral representation of the form [20, 21]

00

F(p)= m2-~~;-ie T I dx2 X2~~~2~ie V(X2, p2H-W(p2), (2.18) 4m2

where p('x.2); V(}t2, p2) and W(p2) are entire functions of}t2 and p2 in the corresponding complex planes and

(2.19)

The existing methods can be divided into the following four groups.

1. Determination of F(x).

2. Determination of F(p).

3. Determination of F(x) and F(p) by using a solution of the corresponding equations for Green's functions or scattering amplitudes.

4. Introduction of nonlocal form-factors.

Nonlocalizable interactions are usually investigated by the first type of method. Here (2.16) is an asymptotic series. We postulate that there is a "regular" function F(x) having no singularity for x 2 - 0, while AC(x) - 00. The series (2.16) is the asymptotic expansion of F(x) for x2 - 00, while A(x) - 0. The problem is to find this "regular" function.

Methods in the second group are close to the so-called analytic-regularization methods, which are applied in renormalizable theories. By using analytic continuation with respect to the propagator index n in (2.16), the originally divergent integral (2.17) is given a completely definite meaning. At the final stage of the calculation, this intermediate procedure is removed, and we obtain a finite expression for Pcp) with the required analytic properties.

In group 3, the equations for the Green's functions or the scattering amplitudes are solved in momen­tum space in a euclidean region of the momentum. Solutions are obtainec;1 for nonfixed values of the coupling constant and are then continued analytically to true values. The solutions of the equations can be used to find F(x) in x-space.

In the last group, form-factors are introduced which permit the avoidance of ultraviolet divergences, but which do not violate the condition that the S-matrix be unitary.

These methods have the following common features. Investigation begins in a euclidean region of the variables, and the resulting functions are continued analytically into the whole range of p2. Nonuniqueness arises in these methods only in the second order of perturbation theory.

Further nonuniqueness does not arise in higher orders.

3. CONDITIONS IMPOSED ON THE TWO-POINT FUNCTION

IN CONFIGURATION SPACE*

Before describing methods in group 1, we stress that all these methods apply to nonlocal theories. Some are theoretically applicable only to nonlocalizable interactions (cf. Secs. II and III of Table 1). Others can also be applied to localizable interactions, but we obtain only nonlocal theories.

*In §3 and 4, and in part of §2, we follow G. V. Efimov's review [29].

29

What are the conditions that the two-point function must satisfy, in order that transition to momentum space will yield a finite function with regular analytic properties, in conformity with the condition that the S-matrix be unitary? We shall indicate these general conditions for all methods of group 1.

Since (2.16) is an asymptotic series for nonlocalizable interactions, it can be associated with some function F(x) for x 2 - - 00 (~c(x) - 0). This function must satisfy the following two conditions:

1) the absence of ultraviolet divergence:

lim I F (x) I = 0; x2-+0

(3.1)

2) the reality of the amplitude in the nonphysical region of the variables x2 < 0 (or p2 < 0). The func­tion F(x) must be real and have no singularities in the interval

(3.2)

The second condition ensures the regular analytic properties of the scattering amplitude.

'" We now assume that conditions 1 and 2 are satisfied, and show that F(p) has the regularanalyticprop-erties corresponding to the unitary property of the S-matrix.

where

In the euclidean region p2 == - q2 < 0, the integral (2.17) can be written

N

F2 (p)= ~ d~xeiqxe(x2-a2) {F(x)- ~C(n)[-W(x)ln}; 1

N

F~ (p) .= I d:xeiqxe (x2_ a2) ~ C (n) [ - itic (x)]n ..

Here a is a nonzero real parameter, and N is any integer.

Integration over euclidean angles in (3.4) and (3.5) yields

(3.4)

(3.5)

(3.6)

(3.7)

(3.8)

where J 1 (x) and Kl (x) are Bessel and Macdonald functions. The integral (3.7) converges for any complex p2, and defines an entire function of order 1/2 in the p2 complex plane. The integral (3.8) defines an analytic function of p2 for

(3.9)

since, for large u, the expression in braces decreases like

30

~exp{-(N+l)mu}. (3.10)

The functions Fj(p) and F2(p) are real. The contribution from the imaginary part of F(p) yields only F3(p). Using the identity

00

[ - i~:'n (xW= - i J dx2Q~m) (X2) ~~ (x), (3.11) (nm)2

where A~ (x) is the causality function for a scalar particle with mass -x and ngn)(-x) is the phase volume of n particles of mass m, we reduce the integral (3.6) to the form

where

and

00

F. ( ).~ C (I) + r dx2 PN (X2) da (x2, p2) J P m2_p2-ie J x2_p2-ie'

4m2

N

PN (X2) ~~ ~ C (n) Qhm ) (X2), 2

The integral (3.12) converges because PN('rt2) '" -x 2N and da (x 2 , p2)~exp{-ax}. for 'rt- 00.

(3.12)

(3.13)

(3.14)

(3.15)

The function F3(P) has a simple pole at p2 = m 2, and a cut starting from the point p2 = 4m2. For 4m2< p2 < (N + 1)2m2, we have

[ ?J 1m F 3 (p) -= n ~ C (n) Qhm1 (p2), (3.16)

2

which is in agreement with the unitary condition.

Since N can be arbitrarily large, it may be chosen so that (3.9) always holds, and F(p) has a simple pole for p2 = m 2 and a cut starting from the point p2 = 4m2, across which the discontinuity is given by (3.16). There are no other singularities in the finite range of p2. Hence F(P) has a representation of the type (2.18), ensuring the regular unitary properties and the observance of the causality principle.

We now consider a question related to the nonuniqueness of these methods. Summation of asymptotic series does not yield a unique result. Different functions obtained by such summations will differ by a function with the following property:

lim ~ -At (~) = 0 (f (M = Fw (x) - F(2 ) (x». (3.17)

This function has an essential singularity for x 2 - - 00, and its contribution to the asymptotic series (2.16) is always zero. The Fourier transformf(A) is an entire analytic function [cf. W(p2) in (2.18)].

4. METHODS OF DETERMINING F(x)

1. The Efimov-Fradkin Method. This method was proposed independently by G. V. Efimov [30] and E. S. Fradkin [31] in 1963. It was the first attempt to construct a finite quantum theory with an essentially

31

nonlinear Lagrangian. The authors started from the position that a finite theory cannot be constructed for an arbitrary Lagrangian, and their object was to find the class of Lagrangians for which a finite theory was possible. Their method yields a finite theory for Lagrangians satisfying the following conditions.

a. U(a) is a continuous function with no singularities on the real axis, which can be expanded in the neighborhood of the origin in a Taylor series with a radius of convergence p:

00

U(a)= ~ U;~)ano

b. The integral of /U(CII)/2 must exist over any bounded region of the complex a-plane.

c. At infinity U(a) satisfies the condition

-10 - U (a) 0 lITI -2- = 0

I a \-+00 a

This method describes nonlocalizable interactions with a Lagrangian of type 7 (see Table 1).

(4.1)

(4.2)

We now describe this method. Using Vik's theorem [32], we write the symbol T for the product in (2.2) in the form

Relation (2.3) now becomes

Our problem is to find an expression of the type

Using the integral representation

exp { - f L1 aa~~aJ = * r~ dtl dt2 exp { - t~ - t~ + -V ~ [(tl + it2) a!1 + (tl-- it2) a~2]} -00

in (4.5) and noting that (4.6) contains a translation operator, we rewrite (4.5) as

The change of variables

yields

1 t2 = u2 - -V- (al- (2)'

2i~

(4.3)

(4.4)

(4.5)

(4.6)

(4.7)

(4.8)

S2 (L1, ajo (2) = * (r dU I dU2U (-V :i (U 1 + iU2») U (-V ~ (u 1 - iU2») exp { - [ u 1 - -Vl~~~r -[ U2 - CY-;i:2 r} 0 (4.9) -00

32

The expansion of this function in powers of 0'1 and 0'2 shows that F(x) can be expressed as a finite sum of functions

(4.10)

It is clear that for (3.1) to hold (the absence of ultraviolet divergence in the theory), it is necessary that

lim I U (z) 1=0. I z 1->00

(4.11)

Condition c follows. But condition b is necessary for the integral (4.10) to exist. This condition implies that U(z) has cuts of order y < 1 in the complex z-region, for example

(4.12)

We stress the fact that the intermediate stages of the calculation carried out here have no rigorous mathematical basis. For example, there is no mathematical justification for the conversion of (4.5) into (4.7) by the application of the translation operator to the argument of U (0') since, in the region of the trans­lation, this function may have singularities whose intersection can, in general, yield extra contributions to (4.7). It follows that the method does not possess the property of uniqueness.

The method is applicable in all orders of perturbation theory, and it leads to a finite theory with a unitary S-matrix.

2. The Lee-Zumino Method. A direct development from the Efimov-Fradkin method is a procedure described by Lee and Zumino [10]. While the former authors aimed to find a class of Lagrangians which could be used to construct a finite and unitary quantum field theory, Lee and Zumino had the object of de­scribing given Lagrangians obtained, for example, in the chirally symmetric theories [23-27]. They thus con­sidered a Lagrangian, differing slightly from chirally symmetric Lagrangians, but inconsistent with condi­tion b of the above method:

Here the function (2.16) is

00

lIXq:>(x) U(cp)= l-x<p2(x)

F (x) = ~ (2n + 1 )!x 2n+l [ - ii1C (X))2n+l. o

Borel summation is applied to the divergent series (4.14) [33]:

00 00 00

F (x) = r dte-I ~ x 2n+1 ( _ itN (x))2n+1 = _ ix i dt le-I!'!.c (x) J "'-I J 1 + [Ix!'!.c (x)j2 . o 0 0

For p2 < 0, the Fourier transform of F(x) is

00 00 (mKl (mr») ~ 4n2x~' -I i 2 T (V--2-) ~ F X2(P)=,;_ dtte drr. 1 -p r ( K ( »)2' v _p2 1-(2)(2 m 1 mr

o 0 ~~

(4.13)

(4.14)

(4.15)

(4.16)

The function F-x,2(P) is defined for all -x, 2 except for values on the positive real axis, where it has a discon­tinuity. But the physical value of -x,2 is on this cut. Hence the value of F(p) for -x, 2 > 0 must be determined from a combination of the values on the upper and lower sides of this cut, as in [11]:

33

(4.17)

where QI = 1/2 + i7J, with 7J a real parameter. This choice of QI ensures the regular unitary properties of F(P). The function F(p) is easily continued analytically from the region p2 < 0 to the whole range - 00 < p2 < 00. The nonuniqueness of the procedure is evinced by the undefined parameter 71. We do not discuss the construction of higher orders.

3. The Direct Summation Method. We name this method after its author, G. V. Efimov [21, 29, 34]. It is similar to those of group 4, except that the form-factor used is introduced in the definition of F(x) and not in the interaction Lagrangian or in the free-field propagator, as in group 4. This is related to the de­finition of the T-product.

We shall demonstrate how this is done. If the representation

00

U(cp)= ~ d~D(~)eitl<P(X) (4.18)

is used for the interaction Lagrangian, where

(4.19)

for any n, and the T-product is defined by (4.3), then the n-th order S-matrix is

00 n

Sn (XI' ... , xn)·~ ~ ... ~ II d~kD (~,,) eitl""'k I~t ei~iitlitlj, -00 1 1!S'L . .::}<:;:n

(4.20)

where (Xi = cp (Xi) and L1ij ~ L1G (Xi -Xj). We easily see that the resulting expression does not satisfy condi­tion (3.1) (the absence of ultraviolet divergence). To satisfy this condition we used the following regulariza­tion. We assume that the chronological ordering operation (4.3), acting under the integration sign in (4.2), yields

where

{ I (u>O) 8(u)= 0 (u<O)

(4.21)

(4.22)

and A is a positive parameter (here and everywhere in this section we are conSidering euclidean space, where -i.6.C (x) > 0). By using (4.21) and (4.20) we can write

00 n

Fk~) .. hn (Xl" . Xn) = J ... ~ II d~8D (~8) (i~8)": II el~iitlitlJe (1 + A.L1rj~lM)· (4.23) 1 l~i!Sj~n

This formula yields the usual expression for the coefficient functions in the case of renormalizable interac­tions of the type U(qJ) = qJ3(x) or U(qJ) = qJ4(x)j this is in agreement with the regularity of the procedure.

The procedure is nonunique. Not only is the parameter A undetermined, but the 8-function in (4.21) can be defined in various ways. Thus the 8-function can be e (1 - A (-iL1iit M) X e (1 - A. (-iL1ii)a~n or 8 (1 - A. (-iL1ijt~i~i)' where a is a new parameter. But the theory is finite and unitary for any of the indi­cated T-products.

34

The last method is close to methods of group 4. Since methods of this group have already been de­scribed in a review by G. V. Efimov, we shall only touch on them briefly.

One of these methods uses the introduction of nonlocal properties in the interaction Lagrangian [35 J. This is done by redefining the field in the interaction Lagrangian:

00

<D (x) = j d4yA (x- y) (P (y) = A (0) <p (x) = ] (;;)1 OnqJ (x), o

(4.24)

where A(x - y) is a nonlocal generalized function from the appropriate space of nonlocal distributions [21J.

In another method, a nonlocal form-factor V(-x2, p2) is used in (2.18):

V (x2 p2) = ~ (p2) , V(X2) ,

(4.25)

where V(p2) is an entire function with the same behavior when p2 - 00 or p2 - - 00 [35J. This method is appli­cable to any interaction Lagrangian in the table. The S-matrix is unitary.

5. DETERMINATION OF SCATTERING AMPLITUDES

BY SOLUTION OF THE EQUATIONS

1. The Peratization of Feinberg and Pais. A method differing only slightly from methods of the first group was proposed by Feinberg and Pais [18J. They also determine a function F(x) whose Fourier trans­form has no ultraviolet singularities, but do not start from an essentially nonlinear Lagrangian; they con­sider the interaction of lept<;ms through a vector boson. Solving the Bethe-Salpeter equation for lepton scat­tering amplitudes in the ladder approximation and retaining only terms involving higher powers of the variable of integration, they obtain an interative procedure yielding (1.13) for the amplitude in the space coordinate.

We note that the retention of only higher powers of the variable of integration in the equation is equiva­lent in the language of the Feinman diagram to the collapse, in the ladder diagram, of the ends of all boson lines into two points, one of which lies on one Fermi line and the other on the other. At the same time the propagators of boson lines are converted into propagators of ordinary scalar particles. Hence, in the itera­tion procedure, we obtain an infinite set of diagrams of the type corresponding to the series (2.16). By means of the equation, this series is summed to the expression (1.13).

Peratization is the specification of a definite method for constructing the Fourier transform from the amplitude (1.13). The first step of this program involves discarding an integral around a closed contour, and the retention of the integral along a radius from 0 to 00. The second step involves an intermediate reg­ularization (the Pauli-Villars regularization). It is assumed that the limiting transition M - 00 can be per­formed under the sign of integration with respect to r although, in general, the order of integration and re­gularization cannot be changed.

Another difficulty is encountered here which is not completely dealt with by Feinberg and Pais. The case in which the function F~) (x) in the integrand of the integral with respect to r has a pole (for q2 < 0) is avoided by Feinberg and Pais by assuming that the mass has a small imaginary part. It would be more correct to take the average of the circuits above and below (cf. [4, 11J, where a similar procedure is used with a cut, but not a pole). *

2. The Arbuzov-Filippov Method. This method is similar to methods of the second group, and has as its object the determination of F(P) [4J. Its authors consider an interaction Lagrangian of the type (1.1), but with a scalar field. Dyson's transformation [15 J is used to convert it to a free Lagrangian, and so the S­matrix is the unit matrix. However, the Green's function of the spinor field, the vertex function, and some

*A. Slavnovand A. Shabad [36J study in more detail the problem caused by the violation of the unitary condi­tion for the S-matrix in the Feinberg-Pais method.

35

other quantities,have a nontrivial form of the type (1.4). Hence,it is of interest to find how we can use the solution of the equations to determine the Fourier transform of the function (1.4).

For mass-free particles, the Green's spinor function is

(5.1)

Then starting from the Dyson-Swinger equation and using the Ward identity, we obtain the following equa­tion for f (p2 + i e) in the euclidean momentum region p2 < 0:

xSf"' (x) + 3x2f" (x) + ').}xf (x) = ° (5.2)

with the boundary conditions

xf (x) ---? 0; f (x) -;> canst. X-Hx:l X--+-O

(5.3)

Here x = - p2; 'A = g/2n. The boundary conditions (5.3) are necessary in order that the sign of 71.2 be nega­tive. This corresponds to a nonphysical imaginary value of the coupling constant g. Hence,we have the following variant for finding the actual function f(x).

First find f(x, 71.2) for negative values of 71.2• This function has a logarithmic cut with respect to 71.2 from 0 to 00. The actual function f(x) is expressed in terms of a combination of the functions on the upper and lower sides of the cut: *

(5.4)

where Ci = 1/2 + i17 and 17 is an arbitrary real parameter.

Here we first consider a method leading to the local theory [19]. We have

(5.5)

where G~~ (t.h 11, 0, -1) is Meyer's function [37]. For large x, this function increases faster than

(5.6)

i.e., condition (2.7) is satisfied.

The nonuniqueness of the method is shown by the involvement of the parameter 17.

In conclusion we make a few comments on the Feinberg-Pais method. At the beginning of the section we made a criticism concerning the construction of the Fourier transform of F(x) by the peratization meth­od which leads to the violation of the unitary condition for the S-matrix. This critism touches the part of Feinberg-Pais work which is especially close to the theme of the present work. However,we have another criticism referring to the derivation of the equation for F(x). As shown by B. A. Arbuzov and A. T. Filippov [4J, using the exact solution for a model, retention only of terms involving higher powers of the variable of integration in the Bethe-Salpeter equation causes a considerable change in the asymptotic behavior of the scattering amplitude for large p, as compared with the exact solution. In particular, this behavior is not in agreement with the local property of the theory [19, 21J, while the behavior of the exact solution is.

In support of the method we note that, if the Stuekelberg and Dyson transformations are applied to the Lagrangian (1.12) considered by Feinberg and Pais, and then the perturbation theory is constructed by our method (see below) for the unrenormalizable part of the Lagrangian with the form (1.3), an expression for

* Here we have departed from the description given by the authors of the method, who take Ci to be purely real and equal to 1/2 (17 = 0).

36

the scattering amplitude is obtained which satisfies the local property in each order of perturbation theory (with respect to mG).

6. DETERMINATION OF THE INTEGRAL F(p)

The methods of finding the function F(p) can be combined under the title "analytic regularization meth­ods." A characteristic feature of all these methods is the consideration of the degrees of particle prop­agators joining different vertices in the Feinman diagram (as complex numbers). Their real parts are chosen so that all integrals over intermediate momenta or coordinates converge, after which the actual values of the degrees of propagators are used by applying an analytic continuation procedure [38].

1. Speer's Analytic-Continuation Method. In considering the second group, we first describe a meth­od which does not, in general have a direct relation to theories with essentially nonlinear Lagrangians. This method was proposed by Speer for the study of theories with polynomial Lagrangians [39, 40]. In its main features, however, it is quite like methods of the second group.*

We describe the procedure briefly. Let G be a mathematical expression corresponding to any Fein­man diagram in momentum space. The corresponding regularized expression GR is constructed as follows.

1. For the k-th interior line of a particle with mass mi each propagator

(6.1)

is replaced by

;.,.A ( ) _ (mi)2A fi (1c) m. P il -- 2 2 1-' , ,

• (mi-Pk-is) T~ (6.2)

where A is an arbitrary, generally complex,number, and Ii (A) is any regular function of A such that Ii (0) = 1.

2. By taking A sufficiently large, we can arrange that no integrals with respect to intermediate-sized moments have ultraviolet divergences. We can use the integral representation

(6.3)

3. Finally the regularized expression GR is obtained by using the following analytic-continuation pro­cedure:

(6.4)

where the CQ(i) are circles of radius q with center the origin corresponding to the integration variable

i-I

ft> ~ fj (1 «i«L), 1

the summation being over all permutations of these circles.

In contrast to the well-known older Pauli-Villars regularization, introducing "ghost" states and violating the unitary condition for the S-matrix, only arbitrary but finite constants appear here.

*Speer's method, used in renormalizable field theory, does not yield essentially new results in comparison with the well known method due to N. N. Bogolyubov and O. S. Parasyuk [41]. These two methods differ only in the technique used.

87

2. Guttinger's Method. This method differs only slightly from the preceding [7]. There a transfor­mation of the type (6.2) is applied to the propagator, but in coordinate space.

'" Guttinger defines F(p) by the independent integration of each term of the sum (2.16). Consider the n-th term

[_iliC (x)t. (6.5)

On the light cone this term has a singularity of the type

1 '" (x2)n or

(the first for mass-free particles, the second for particles with mass). It is clear that the Fourier trans­form of this expression, starting with n = 2, is an integral which diverges more strongly for larger n.

To obtain a finite expression for the Fourier transform, Guttinger proposed the following method. Instead of (6.5), consider the quantity

(6.6)

where z is a complex number and an is an undetermined parameter with the dimensions of a squared length. Assume that Re z < 2 - n. Then the Fourier transform of (6.6) exists in the usual sense:

(6.7)

It is a function of z, regular for Re z < 2 - n, and having poles at the integers on the real axis for Re z ~ 2 -no

The regular functionfn(p) is defined as follows:

1 ,{:, dz f fn (p) = 2ni ';Y Z n (p, z), (6.8) c

where C is a circle with radius smaller than unity. The expression for f n (p) for mass-free particles is very simple:

(0) (-I) (p2+ie)n-t 1 .. { [ p2+ie _0 ] O} fn (P)=p2+ie (4n)2 (n-I)I(n-2)1- In an (4n)2 e "' --'I'(n)-'I'(n-I) • (6.9)

where \}I(n) is Euler's function.

By using (6.9), we obtain the expression

00

- C t 1 ~ C(n+l) (p2+ ie )n {[ p2+ie ° J } F(P)=-p2+ie- p2+ie ",,",nl(n-I)I (4n)2 In a n+1 (4n)2 e-ut -'I'(n)-'I'(n+I) •

(6.10)

for the case of mass-free particles. The regular imaginary part in the physical range p2 > 0 satisfies (2.9).

We now discuss the disadvantages of this method. First of all, since the Fourier transform of the function (6.6) is defined for Re z < 2 - n, it would appear that the contour of the integral (6.8) should contain at least a small part of this region. However, if C contains a part of the real axis from 0 to Re z < 2 - n, then the expreSSion (6.9) contains a further polynomial of degree n = 3 in p2. Hence, it must be prescribed that C may include a part of the region Re z < 2 - n, but only if this part contains no singularities. Sec­ondly, the method involves an infinite number of undetermined constants. Thirdly, the final expression has

38

no obvious nonanalytic dependence on the coupling constant, which is so characteristic of unrenormalizable theories. Finally, Guttinger attempts to apply the procedure developed for determining the Fourier trans­form of the propagator in a finite degree n to the construction of the spectral representation of F(p):

(6.11)

However, this formula does not define a finite expression, since the spectral function p(m 2) increases more rapidly than any polynomial in m 2, and no negative power of m 2 can compensate for this growth.

3. The Integral-Representation Method. This method, like the two foregoing methods, is based on the fact that the degrees n of the propagator in F(x) are converted into complex numbers z. For the series (2.16) there is an integral representation in which Re z < 2. The Fourier transform F(p) can be formed with no difficulties concerning ultraviolet divergences, after which we again return from the integral repre­sentation to a series [11, 42].

We rewrite (2.16) in a somewhat different form, separating out the coupling constant g2 from the co­efficients C (n):

00

F (x) = ~ a (n) [_ig211C (x))n. (6.12) I

There are two variants of this method, one of which is applicable to nonlocalizable interactions, and the other to localizable interactions. We start by considering the first.

a. Let the coefficients a(n) satisfy the condition

lim n-b [a (n) [lin = A, (6.13) n->oo

where A is a nonzero constant and the values of b satisfy

(6.14)

Theories in which the coefficients a(n) satisfy (6.13) describe almost all physically interesting cases of non­localizable interactions. In the table this includes all interactions of II and even those in III for which F(p) has an imaginary part for mass-free particles (y < 3/2).

Condition (6.13) can be written in a more rigorous form; namely. we can require that the entire func­tion X(z), which can be expressed in the form

00

X (z) = ] (-It ~2~)1 Zn,

1

(6.15)

be bounded in some sector I cp [-<. 8, 8 ~ 0, Z = reiq>. Condition (6.15) implies (6.13). We require that the latter condition hold.

Using (3.11), we can convert the expression for F(x) into

00

F' (x) = - i ~ (g2t+l a (n + 1) ) dx2Qh~l (X2) ll~ (x). (6.16) [(n+l) mJ2

Here we have discarded the first term in F(x), since for it the conversion to momentum space is trivial.

We consider further the case of mass-free particles, because it is simpler and better for explaining our procedure. The generalization to particles with mass is described in [43].

39

If m = 0, then (6.16) becomes

00 M2

H (x) = if- ~ ( __ l)n j ~~2 f (n, X2) M, (x), (6.17) 1 0

where

f( 2 _ [ (X )2Jn a(n+I) n,x)- f- 4n r(n)r(n+I) , (6.18)

Here A = - g2; r (n) is the gamma-function, and M2 < 00 is a cut-off introduced as an intermediate regulari­zero It will be used in the following. We assume that A > O. At the end of the calculation we return to physical values A < O.

We now use the values of fen, -x,2) at the sequence of points n = 1, 2, 3, ... to determine an analytic function fez, -x,2), regular in the right half-plane Re z < 0 and satisfying the following conditions (z = x + iy) [44]:

a) I fez, x 2) I <BeAlzl , Rez> 0; } b\ If(iy, x 2) I <Be(n-6) IYI, -00 <y<oo, 0> o. (6,19)

Conditions (6.19) ensure, on the one hand, the uniqueness of fez, -x, 2) , and on the other hand permits us to write the sum (6.17) as a Mellin-Burns (Sommerfeld-Watson) integral with a contour such that Re z < 1 in the whole range of integration:

a-ioo M2

F~ (ren' (x) = "2 r ~ \ dX22 f (z, X2) L1\'o (x),

b' J sm nz J x (6.20) a+ioo 0

with 0 < a < 1. This solves the problem stated at the beginning of this section. Conditions (6.19) are com­patible with the condition (6.13) imposed on the coefficients a(n) [11].

It is now simple to transfer to momentum space. Since all integrals involved in the application of the Fourier transform to (6.20) are absolutely convergent, this operation can be transferred directly to A~(x) with the intermediate regularization omitted:

,a-rioo d ({4~)2 ) z a (z + I) r (2 1

F~(p)=ii J sin~z' i'(z)r(z+l) J dX2x2~):=-ie' (6.21) a+ioo 0

"-

The resulting expression can be taken as the spectral representation of F~ (p). All integrals in (6.21) are easily calculated, and we have

00

F ( )=_"_~ (_' p2+ ie )n a(n+I) { [P2+ ie ] } .. P p2+ie ~ I\, (4n)2 n!(n-I)! In f- (4nJ2 e- i3t +(lna(n+l»'-1JI(n)-1JI(n+l) . (6.22) 1

"-

The ~nction Fi (p) has a logarith,.!llic cut for negative values of A. Hence, in order to obtain the actual func-tion F'(P), we use the values of F~(P) on both sides of the cut:

(6.23)

where a and f3 satisfy

(6.24)

40

The last condition is a consequence of the fact that the S-matrix is unitary. Hence,

(6.25)

where 7) is an arbitary parameter. The final result is

00

--'" (_g2) ~ ( 2 p2+ie)n a(n+l) { (P2+ie ) } F tp)= p"+ie ~ g (4;'"[) 2 n!(n-I)! In g2 (4n)2e-iJt+2Jtt] -Hlna(n+l»)'--'¥(n)-'¥(n+ 1) . j

(6.26)

The introduction of the parameter A was necessary to ensure the existence of the integral (6.21). For nega­tive A the integral with respect to z is divergent.

~ There is no spectral representation of the type (6.21) for F'(p), but for F'(p, y), which is equal to F' (p) for y = 1:

oo-ioo ( 4: ) 2< a (z+ I) F'( , ,)=.£ r dz cosnz. "' P I 2 ,J SIn,\,;[ Z ----;1"""' -c-( z7-) =r 7( z--1-"-, 7:1)-

a+ioo

Here y must be larger than 2 (more precisely y > 2 - b/2).

b. Now consider localizable interactions, for which

We confine ourselves to localizable interactions for which

limnkla(n)ll/n=A, A>O, O<k<2. n .... oo

(6.27)

(6.28)

(6.29)

In the nonphysical momentum range p2 < 0, we can convert to the euclidean metric in the integral (2.17) and take the integral with respect to the angles:

00 00

Hcreg > (p) =--ih ~ ( - (2~)2r a (n + 1) ) drr2nJI (I p 1 r); (I p 1 = 11 _p2). (6.30) 1 I

Here an intermediate regularization is introduced by truncating the integral with respect to r for r < 1. Using a procedure similar to the above, we rewrite (6.30) as

o::-ioo 00

F~(p).=i 2(~1 ) si~znz C2~)2ra(z+I») drr-2"J 1 (iplr). (6.31) IXTioo ()

Here the intermediate regularization has been removed, since the integral with respect to r converges at zero. The integral with respect to z is also absolutely convergent if 0 < k < 2. Integration with respect to rand z yields

00

H(p)= - 1 ;"12 ~ [I, (1:~2) r n~i~=:;! {In (A !In\:) +(Ina(n+ 1»)' - '¥(n)- '¥(n+ I)}. (6.32) !

This expression is easily continued into the physical region p2 > 0, and (6.23) yields the actual function F,(p).

The first term of the sum (2.16) is easily included in an integral representation of the type (6.27). ,..., Then the complete function F(p, y) is given by the formula

41

(J.-ioo .

- . 3 r e- IJtZ C (z) 2'. z-2 F(p,1')=1(2n) J dzctgnz siol'Jtz 'r(z-l)rTz)(P +lc) .

a+ioo

(6.33)

The value of y depends here on the behavior of the coefficients C (n) for large n. In the final result we must put y = 1.

The representation (6.33) can be used to show that there are no ultraviolet divergencies in any order of perturbation theory. In fact consider the n-th order. In it, n vertices are joined by n(n -1)/2 interior lines of the type (6.33). The common multiplicity of the integral with respect to interior moments is

n(n--I)/2

[2n(n -1)-4(n -1)]. The common degree of the momentum in the integrand is 2 ~ (Re z; -- 2). Thus the I

condition that there be no ultraviolet singularities can be written

n(n-I) -2-

2(n-I)(n-2)+2 ~ (Rezi-2)<O.

Hence, making all Re zi equal, we obtain

1

ex.= Rez<i.. n

(6.34)

(6.35)

But since 0 < a < 1 and a can be arbitrarily close to zero, condition (6.35) can always be satisfied [11].

A spectral representation of the form (6.27) ensures that the S-matrix is unitary in all orders of per­turbation theory.

In the study of particles with mass, we must keep in mind the considerable difference in the asympto­tic behavior of amplitudes for high m,.S>menta for the nonlocal theories for massive and mass-free particles. Moreover, the analytic properties of F(p) become much more complicated. In particular, the nonanalyticity with respect to the coupling constant is not of a simple logarithmic type, but of a more essential kind. The examinations of amplitudes we have carried out for low momenta show that, for massive particles, the function F(p) can be expressed as follows:

00

F (p) = !: ~ (gm)2n a (n -f-l) [fn (~: ) + In (gm) P::'!l (~: ) +(In (gm»2 P~~2 (~: ) + ... +(In (gm)t p~n) (~: ) J. (6.36) n=O

where the I1n)(p2/m2) are polynomials in p2/m2 of degree k, and the fn(p2/m2) are functions of p2/m2• The above series shows that the nonanalytic dependence of the amplitude on the coupling constant g is of a very involved nature.

We note in conclusion that Salam and Strathdee have recently published an interesting article devoted to a detailed description of the method described above [9]. They attempt to consider high orders in per­turbation theory and describe the case of massive particles. They study the problem of the S-matrix being unitary in high orders. In addition to the deficiency in this particle mentioned at the beginning of the pres­ent review, we note the not completely correct assumption made concerning the structure of F(p) in the case of massive particles. The authors assume that, for massive particles, F(p) has only a simple log­arithmic singularity with respect to the coupling constant (cf. p. 33 in [9 D.

C ONC L USIONS

Following the above compressed description of the many methods used in quantum field theory with spectral functions of rapid growth, we shall attempt to give an over-all estimate of the value of these meth­ods and to discuss briefly some questions arising concerning higher orders of modified pertUrbation theory, and also to indicate the main problems remaining open.

42

We first outline the main characteristics of contemporary methods. In our opinion, the Efimov-Frad­kin method is the most firmly based and interesting of the group-1 methods. This method modernized and improved in subsequent articles by G. V. Efimov [21], is applicable to a wide class of non polynomial Lagrangians. Efimov has shown that his method yields unitary S-matrices satisfying the macrocausality condition.

Of methods of the second group, we prefer the integral-representation method. This method, like the Efimov-Fradkin method, can be applied to a large class of nonpolynomial Lagrangians. It enables us to construct two-point Green's functions in momentum space with the correct analytic properties, completely compatible with the S-matrix being unitary. It also yields integral representations for two-point functions which greatly simplify the examination of higher orders of perturbation theory with respect to the "princi­pal" coupling constant.

Finally, from methods of the third group we choose the Arbuzov-Filippov method. This method can also be used for the description of many unrenormalizable interactions, and leads to expressions for Green's functions and scattering amplitudes such that the S-matrix is unitary. It is interesting that methods differ­ing as widely as our integral-representation methods and the Arbuzov-Fillipov method yield identical re­sults when applied to the same interactions [4, 11].

In this sense, a good method of checking a method is to apply it in the description of a model which can be solved exactly. Here the above methods yield compatible results and correctly describe the models investigated.

We now discuss some questions arising in the investigation of higher orders of perturbation theory with respect to the principal coupling constant. One of the main problems is that of ensuring that the S­matrix is unitary. To prove that this condition is satisfied in an arbitrary order of perturbation theory is rather difficult. However, if Kutovskii's theorem can be proved in any order of perturbation theory, then the S-matrix is unitary in this theory. G. V. Efimov [21], and also Salam and Strathdee [9] have proved that Kutkovskii's theorem is true in high orders of perturbation theory. In [11], we verified the unitary condition in the third order of perturbation theory and derived a spectral representation for two-point Green's functions which will, it is hoped, maintain the unitary property of the S-matrix in higher orders if these functions are used.

The following problem is connected with the finiteness of high-order perturbation theory with respect to the principal coupling constant. In [11] we show that the theory remains finite in any order, if the inte­gral-representation method is used to calculate the relevant quantities. The Efimov-Fradkin method also ensures that the theory is finite in any order of perturbation theory.

We call attention to the following unsolved problems.

1. Uniqueness of a Method. We believe that this problem must be solved by introducing further physi­cal principles. For example the Arbuzov-Filippov and integral-representation methods become unique if we require that

ReF (p) lim ~ O. p2--+oo 1m F (p)

This condition was first suggested by A. T. Filippov [4]. It has not yet any physical basis.

2. The Second Problem Concerning the Nonpolynomial Growth of Amplitudes in Every Order of Per­turbation Theory with Respect to the Coupling Constant. The methods described here can be used for cal­culation in low orders of perturbation theory with respect to the principal coupling constant only for small momenta. To draw conclusions concerning the asymptotic behavior of amplitudes for high energies in these theories, we must consider a series of perturbation theories with respect to the principal coupling constant (situations analogous to those in which properties are obtained for the usual renormalizable theories).

The above problems are difficult, but the author believes that they will soon be solved.

other problems arises in the description of charge fields and in taking account of the gradient invari­ance of theories. However, these questions have already been solved [12].

43

LITERATURE CITED

1. Transactions of the International Conference on Nonlocal Quantum Field Theory, Preprint Joint Insti-tute for Nuclear Research R2-3590, Dubna (1967).

2. S. Okubo, Progr. Theor. Phys., .!b 80 (1954). 3. T. D. Lee, Phys. Rev., 128, 899 (1962). 4. B. A. Arbuzov and A. T. Filippov, Zh. Eksperim. i Teor. Fiz., 49, 990 (1965); Nuovo Cimento, 38,

796 (1965); Yadernaya Fizika, .§" 365 (1968); A. T. Filippov, Preprint Joint Institute for Nuclear Re­search E2-4189 (1968).

5. R. Arnowitt and S. Deser, Phys., Rev., 100, 349 (1955). 6. B. M. Barbashov and G. V. Efimov, Zh. Eksperim. i Teor. Fiz., 43, 1057 (1962). 7. W. Guttinger, Fortsch. Phys., 14, 483 (1966). 8. R. Delbourgo, A. Salam, and J~trathdee, ICTP Trieste, Preprint IC/69/17. 9. A. Salam and J. Strathdee, Phys. Rev . .Q, b &3296 (1970).

10. B. W. Lee and B. Zumino, Nucl. Phys. B13, 671 (1969). 11. M. K. Volkov, Ann. Phys., ~, 202 (1968). 12. H. M. Fried, Nuovo Cimento, 52A, 1333 (1967); B. W. Keck and J. G. Taylor, Preprint, University of

Southampton, April (1970); P. Budini and G. Calucci, Preprint ICTP IC/70/37, Trieste (1970); H. Lehmann and K. Pohlmeyer, Commun. Math. Phys., 20, 101 (1971).

13. M. K. Volkov, Preprint ITF 69-5, Kiev (1969). 14. N. N. Bogolyubov and D. V. Shirkov, Introduction to Quantum Field Theory [in Russian], Moscow,

Gostekhizdat (1957). 15. F. J. Dyson, Phys. Rev., 73,929 (1948). 16. T. D. Lee, Nuovo Cimento, ~, 579 (1968). 17. E. C. Stueckelberg, Helv. Phys. Acta,.!b 225, 229 (1938). 18. G. Feinberg and A. Pais, Phys. Rev., 131,2724 (1963); 133B, 477 (1964). 19. N. N. Meiman, Zh. Eksperim. i Teor. fu., 47, 1966 (1964). 20. A. M. Jaffe, Ann. Phys., 32, 127 (1965); J. Math. and Phys., £" 1174 (1965); Phys. Rev. Lett.,.!I,

661 (1966). 21. G. V. Efimov, Preprint ITF-68-52 (1968); 68-54 (1968); 68-55 (1968); Commun. Math. Phys., 1, 138

(1968). 22. N. N. Bogolyubov, A. A. Logunov, and 1. T. Todorov, Elements of the Axiomatic Approach to Quantum

Field Theory [in Russian], Moscow, Nauka (1969). 23. F. Gursey, Nuovo Cimento, 1&, 230 (1960). 24. J. Schwinger, Phys. Rev. Lett., 24B, 473 (1967). 25. S. Coleman, 1. Wess, and B. Zumino, Phys. Rev., 177,2239 (1969); 177,2247 (1969). 26. S. Weinberg, Phys. Rev. Lett., ll, 188 (1967). 27. B. V. Struminskii, Preprint Joint Institute for Nuclear Research F2-3554, Dubne (1967). 28. v. A. Kolkunov, Zh. Eksperim. i Teor. Fiz., 43, 1448 (1962). 29. G. V. Efimov, Preprint CERN, Geneva, TH-1087 (1969). 30. G. V. Efimov, Zh. Eksperim. i Teor. Fiz., iio 2107 (1963); Nuovo Cimento, 32, 2046 (1964); Zh.

Eksperim. i Teor. Fiz., j!!, 596 (1965). 31. E. S. Fradkin, Nucl. Phys., 49, 624 (1963). 32. S. Hori, Progr. Theor. Phys., &" 578 (1952). 33. G. H. Hardy, Divergent Series [Russian translation], Izd-vo Inostr. Lit. (1951). 34. G. V. Efimov, Nucl. Phys., 74, 657 (1965). 35. G. V. Efimov, Preprint Joint Institute for Nuclear Research R2-4546, Dubna (1969); R2-4472 (1969). 36. A. Slavnov and A. Shabad, Yadernaya Fizika, 1. 721 (1965). 37. T. Bateman and A. Erdeli, Higher Trascendental Functions, Vol. 1 [Russian translation], Nauka,

Moscow (1965). 38. 1. M. Gel'fand and G. E. Shilov, Generalized Functions, Vol. 1 [in Russian], Fizmatgiz, Moscow

(1958). 39. E. J. Speer, Math. Phys.,~. 1404 (1968). 40. P. Breitenlohner and H. Mitter, Nucl. Phys., B7, 443 (1968). 41. N. N. Bogolyubov and O. S. Parasyuk, Acta Math., 97, 227 (1968). 42. M. K. Volkov, Yadernaya Fizika, &.. 2200 (1967); L 448 (1968). 43. M. K. Volkov, Commun. Math. Phys., l.2.. 69 (1969); Teor. i Mat. Fiz., ~, 197 (1970); &" 21 (1971). 44. M. A. Evgrafov, Asymptotic Estimates and Entire Functions [in Russian], Fizmatgiz, Moscow (1962).

44

DISPERSION RELATIONSHIPS AND FORM FACTORS

OF ELEMENTARY PARTICLES

P. S. Isaev

The survey is devoted to the modern state-of-the-art in the field of the form factors of '/l"­mesons, K-mesons, and nucleons. The concept of form factors of elementary particles is presented, as well the description of the form factors using dispersion relations; an inter­pretation of experimental data on the scattering of electrons by pions and nucleons is given by introducing form-factor functions; the theoretical aspects of the link between the differ­ence in masses of charged and neutral K-mesons and the form factors of K-mesons are considered; the various representations of form-factor functions are treated.

INTRODUCTION*

The subject of the present review is connected with one of the fundamental problems in the physics of elementary particles which is being actively investigated at present - the problem of the structure of ele­mentary particles. The idea that the investigated objects - elementary particles - must have a definite structure is not new. This was discussed from the instant the electron was discovered. However,atpres­ent the notion of "structure of elementary particles" has a deeper content than the first concepts of the structure of an electron which were discussed in the beginning of the twentieth century in the papers by Abragam, Lorentz, Poincare, etc.

The review expounds in simple form the notion of the form factors of elementary particles, the link between form factors and the structure of particles, the description of form factors by means of dispersion relations, and the interpretation of experimental data on the scattering of electrons by pions and nucleons through the introduction of pion and nucleon form factors. There is an enormous number of papers devoted to all of these problems. Here I shall mention only a series of surveys [1-4] to which the reader should refer to obtain a more thorough familiarity with the history of the subject.

Before going over to the review itself, let me make several comments on the method of dispersion re­lations. At present the method of dispersion relations is one of the reliable methods in field theory and is being used with success to explain experimental data. Right now there is not a single case of contradiction between the dispersion relations and experimental data.

Regrettably, there are also no rigorous experimental proofs of the validity of the method. The exist­ence of dispersion relations was first proved theoretically by N. N. Bogolyubov for scattering of '/l"-mesons by nucleons in 1956 in his report to the International Convention of Theoretical Physicists in Seattle (see also [5]). Later the existence of dispersion relations was also proved for several other processes involving the interaction of elementary particles. Obtaining such proof (theoretically) is a very difficult matter. At present it is assumed that the dispersion relations are valid for any process. In the present review it is likewise assumed throughout that the use of the dispersion relations is valid for describing the form factors of elementary particles.

*The basis of the present review is a series of lectures read by the author in the International School on the Physics of Elementary Particles at Herzeg Novi (Yugoslavia) in 1966 and published in the book "Methods in Subnuclear Physics," Vol. II, Gordon and Breach, New York (1968). The lectures were published only in English. The text of the lectures was reexamined for the purpose of the present review and was considera­bly augmented with new material on the investigation of electromagnetic form factors of '/l"- and K-mesons and nucleons (published during the period from 1966 to 1970).

Joint Institute for Nuclear Research, Dubna. Translated from Problemy Fiziki Elementarnykh Chastits i Atomnogo Yadra, Vol. 2, No.1, pp. 67-104, 1971.

C! 1972 Consultants Bureau, a division o[ Plenum Publishing Corporation, 227 West 17th Street, Ne", York, N. Y. 10011. A II rights reserved. This article cannot be reproduced [or any purpose ",halsoet'cr without permission o[ the publisher. A copy o[ this article is available [rom the publisher [or $15.00.

45

b

1 .P ~c===

c

Fig. 1. Feynman diagrams of the simplest vertex functions.

A A A A A····· .. abc d e

Fig. 2. The different Feynman diagrams which make a con­tribution to the generalized vertex function displayed in Fig. 3.

Fig. 4

Fig. 3. Generalized vertex function.

Fig. 4. Generalized vertex function corres­ponding to the decay n - p + e + v.

1. THE FORM FACTORS OF ELEMENTARY

PARTIC LES

No unified definition of the notion of a form factor exists; in the majority of cases it is identified with the no­tion of a vertex function.

For interaction of an electromagnetic field with nu­cleons (or 1T-mesons) a vertex develops (Fig. 1a). Weak in­teractions have a vertex of the type depicted in Fig. lb. The interaction of an electromagnetic field with a p-meson is described by the vertex displayed in Fig. 1c. The interac­tion force is characterized by coupling constants. Let us

consider the three-particle form of interaction (Fig. 1a). We shall include only photons, 1T-mesons, and nu­cleons in the consideration. The experimental data indicates that the constant for coupling of a 1T-meson field with a nucleon field is G2/41T R:I 15, while the constant for coupling of an electromagnetic field with nu­cleons is e2/1ic == 1/137. Therefore, in describing the interaction of an electromagnetic field with a nucleon (7I'-meson) field it is necessary to remember strong coupling of nucleons (7I'-mesons) with the 1T-meson field (nucleon field). From this the vertex depicted in Fig. 1a must be augmented by a finite collection of dia­grams displayed in Fig. 2, in which the 1T-mesons are depicted by a dashed line. Symbolically this collec­tion of diagrams together with the diagram described in Fig. 1a may be depicted by a single diagram (Fig. 3) in which the hatched circle denotes the contribution of all possible diagrams allowed by the correspond­ing interaction Lagrangian. If one of the ends of the diagram (see Fig. 3) is a virtual y-quantum (q2 ;t 0), while the other two ends describe a free nucleon, then the vertex function F displayed in Fig. 3 will depend on a single variable q2: namely, F == F(p2 = M2, p,2 = M2, q2;r 0). Such a function F(M2, M2, q2) is called the electromagnetic form factor of a nucleon. Actually, the interaction of a proton or neutron with a photon can be described by means of two form-factor functions rather than by one, as is shown here for purposes of clarity. This will be considered in greater detail further on.

Methods of summing the infinite number of diagrams displayed in Fig. 3 do not exist, and the form factors are calculated only approximately.

The form factors are also introduced in the case of vertices which describe weak interactions (see Fig. 1b). For example, in the decay n - p + e + v (Fig. 4) the hatched circle denotes consideration of all corrections which occur due to strong interactions of a neutron and proton, while the functions F(M~, M~, q2) describing these interactions, where the variable quantity q2 is the difference between the four-dimen­sional proton and neutron momenta, are likewise called form factors, although in the case given all four particles (n, p, e, v) lie on the mass surface. The name form factor (the electromagnetic form factor of a 1r-meson or the 1T-meson form factor of bosons, etc.) is linked to the virtual particle according to whose

46

4-momentum the dependence of the form factor is investigated, and to those real particles with which the given virtual particle forms the vertex.

In the case of interaction of one virtual photon with a nucleon the electric charge may be included in the form-factor function in such a way that the "new charge" e' will be related to the conventional charge e by the equation

e' (q2) = F (q2); e' (0) = F (0) = e, (1)

l'1g. n. .ttepresemanon of the generalized Feyn­man diagram corres­ponding to the virtual Compton-effect. i.e., it is as if a theory were to be obtained with an interaction constant which

depends on the transfer of 4-momentum q2 = (p'_p)2. However, one cannot always represent the form factor function as a variable charge. This can easily be seen if one goes over to more complex diagrams. Considering, for example, a diagram with two virtual ends (Fig. 5), one can see that it may be described by a collection of functions Fi (qi, q~) which will depend in a complex manner on the two variables qi ;oe 0 and q~ ;oe 0, in connection with which neither of them may be interpreted in the spirit of Eq. (1). In the examples considered the collection of functions Fi (qr, q~) describes the amplitudes of the virtual Compton-effect. The form-factor function may not be interpreted as the real space distribu­tion of the electric charge of a nucleon (or of the actual nucleon matter). This derives from the fact that F(x) is a four-dimensional Fourier transform of the function F(q2):

and is not associated with the dimensions of the nucleon. It is only for a special selection of the coordinate system for small transfers q2 on the assumption that F(q2) is a smoothly varying function and for a corres­ponding definition of the distribution density of the electric charge that one can indicate a link between form factor function F(ql) and the mean-square radius of a nucleon < r2 > (see, for example, [2]):

F (q2) = F (0) [ I + i q2 «()2 + ... ] ' (2)

where q2 is the three-dimensional transfer (p'_p)2, and F(O) = e.

The form factors may be interpreted most accurately as the influence of the virtual particle clouds on the process amplitude of interest to us (or on the matrix elements of interest to us). Thus, the form factor depicted in Fig. 3 can be interpreted as the effect of the 7T-meson virtual cloud of a nucleon on the electromagnetic interaction of the nucleon.

The practical necessity of resorting to form factors to explain experimental data develops fairly fre­quently. They are also introduced to explain the electromagnetic interaction of particles with nucleons, and in describing weak interactions with allowance for electromagnetic and strong interactions.

In the present review the concept of a "form-factor function" will be used only for electromagnetic vertices (Fig. 1a) when one end of the vertex is a virtual y-quantum and the two other ends certainly lie on the mass surface (Le., the concept of a form factor will be used only for electromagnetic interactions of elementary particles).

The functions Fi (qr, q~) which characterize the amplitudes of the virtual Compton-effect (see Fig. 5) or the functions Fi(MA, Mb, q2) describing the process of neutron decay n - p + e + v (see Fig. 4) will be called relativistic structural coefficients.

In field theory form-factor functions appear for consideration of invariant properties of vertex func­tions. Thus, the electromagnetic form factor of a 7T-meson appears for consideration of a 7T-meson vertex (Fig. 6). It may be shown that the sole nonzero vector has the form

where t = (7T 2 - 7T 1)2; w2 = v + mj.; m7T is the 7T-meson mass; v is the square of the three-dimensional 7T­meson momentum; Vi = qr. w1,2 are the 7T-meson energies in the initial and final states, respectively; T3

47

Fig. 6. Generalized 'II'-meson vertex function.

is the third component of thp- isotopic 'II'-meson spin, while F'II'(t) is a scalar function of the invariant variable t. At the point t = 0 the function F 'II'(t) is normalized to the electric charge F'II'(O) = e. The expression on the right side of the equation given above is the most general one satisfy­ing the requirements of relativistic invariance. A nucleon vertex has a more complicated form (see Fig. 3), which is associated with the pres­ence of nucleon spin and isotopic spin. Taking account of invariance rel­ative to Lorentz transformations, gauge invariance, and the requirement that free nucleons must satisfy the Dirac equation, and simultaneously

considering the isotopic structure, we find that the most general expression for the matrix element of the nucleon current jr has the form

(p' I j~ I p) = V 1 (w (p') I (G (t) -I- F; (t) 'ta) 'YI. + i (F; (t) + F~ (t) 'ta) C11."k" I w (p», 4EfEI

(3)

where w(p') and w(P) are the nucleon spinors in the final and initial states; E j, Ei are the nucleon energies in the final and initial states; the symbols s and v denote the isotopically scalar part and the isotopically vector part of the form factor functions of the nucleon; (1l.v = 'YI.'Y" - 'Yv'Yl.. kv = (p' - p)". t = k~, and,

finally, 'ta = (~_~) .

The first term in Eq. (3)

(w (p') I (F~ (t) + F~ (t) 'ta) 'YI.I w (p» = (iii (p') I FI (t) 'YI.I w (p»

describes the electromagnetic interaction of a nucleon with a charge e and a magnetic moment equal to the Bohr magneton in the absence of a meson cloud. For t - 0 this term describes the motion of a free nonin­teracting proton, and therefore it is assumed equal to e (W (p') 'Y,,-w (p». whence it follows that for a proton

F IP (0) = F~ (0) + F~ (0) = e,

while for a neutron

F,n (0) = F~ (0) - F~ (0) = 0

or

F~ (0) = F~ (0) = f .

The second term in Eq. (3)

is associated with an anomalous nucleon magnetic moment. Sometimes the second term is called Paulian, since Pauli [6] showed that in describing particles having spin 1/2 the conventional Dirac equation

(n=c=l),

which considers the interaction with an electromagnetic field (the term -ienA;\), may have the term (ie-x./ 2M)y;\YvF;\v added to it, where F AV is the electromagnetic-field tensor, while -x. is a certain arbitrary con­stant which may be interpreted as the additional nucleon magnetic moment. For a transfer t - 0 the second term can be normalized as follows:

i (iii (p') I (F: (0) + F~ (0) 't3) C11."kv I w (p» = i 2~ (w (p') I(x. + Xv'ta) C11."k" I w (p».

48

whence it follows for a proton

while for a neutron

Since the anomalous proton magnetic moment 'Kp = 1.79, while the anomalous neutron magnetic moment 'Kn (it is the total neutron magnetic moment 141) is given by 'Kn = 141 = -1.91, it follows that

From this it is evident that the principal contribution to the proton and neutron form factors F2(q2) is made by the isotopic vector part.

The two 'IT-meson and nucleon vertices displayed above may be treated as parts of Feynman graphs describing the scattering processes

(4a)

(4b)

In the general case the total collection of relativistic structural coefficients for the scattering amplitudes (4) turns out to be larger than the number of form factors considered above. This is associated with the fact that the total amplitude of the scattering processes (4) includes an infinitely large collection of diagrams in addition to the diagrams displayed in Figs. 3 and 6; this collection includes, for example, exchange of two, three, etc., photons, which in general leads to an increase in the number of invariant functions or, stated differently, an increase in the number of relativistic structural coefficients. The total number of rel­ativistic structural coefficients is determined by the number of free ends on the Feynman diagram and the presence of spins and isotopic spins of the particles described by the free ends. In the case of a finite number of free ends on the Feynman diagram, the number of structural coefficients will be finite. For amplitudes of the type

the number of relativistic structural coefficients M (the isotopic structure of the amplitudes is not con­sidered here) can be determined by the following formulas [7].

1. If the spins of all particles participating in the reaction are integer spins, then

where ia is the spin of particle a; ~ = ~:~:, Xa is the internal parity of particle a.

2. In all remaining cases

M =i- (2ia + 1) (2i b + 1) (2ic + 1) (2id + 1).

49

The isotopic structure of the processes considered can easily be sought by means of the rule for the addi­tion of isotopic particle spins and the isotopic spin conservation laws. For elastic-scattering processes a + b - a + b invariance relative to time inversion reduces the number of structures. In particular, for the e + N - e + N process the number of relativistic structural coefficients decreases from eight to six. With allowance for istopic invariance, the number of structural coefficients in these reactions increases to 12.

The vertex parts displayed in Figs. 3 and 6 may also be treated as parts of Feynman graphs which describe annihilation processes:

In this case the expressions for the vertex parts will have the form:

(0 I h Inn) = (~-n-h F" (t), 4W+W_

t= (n++n-)2>4m~,

(Olj~INN)= V 1 (WN(PN)I(F~(t)+F~(t)T3)1').+ 4ENEN

+ i (F~ (t) + F~ (t) '3) o).vkv I WN (PN», t = (PN + P N)2 > 4M2.

(5a)

(5b)

(6)

Here the form-factor function of 1T-mesons and nucleons differ from those written out previously in (3) in their range of variation of the variable t.

In field theory it is stated that the form-factor function F 1T (t) must be the same function for both the scattering process (4a) and the annihilation process (5a). A similar statement is also made with regard to the form-factor functions of nucleons. The dispersion relations ensure the necessary link between form­factor functions stipulated in various t-domains.

2. THE ELECTROMAGNETIC FORM FACTOR OF A 1T-MESON

An investigation of the analytic properties of the vertex function of a 1T-meson [8, 2] leads to the state­ment that F1T (t) is an analytic function in the t plane having a cut 4rr;~ -< t -< 00, i.e., for the e+ + e- - 1T+ +

1T- annihilation process the form factor of the 1T-meson is a complex quantity, whereas in the domain t < 0 (Le., for the e + 1T - e + 1T scattering process) ImF1T (t) = 0 and the form factor is a real quantity. Thus, the dispersion relation has the form

F (t) = 2- r 1m F" (1') dt' . " n J t'-(t+ie)

C

(7)

The integration contour is depicted in Fig. 7. The imaginary correction ie to the value of t denotes that the form factor F1T (t) takes observable values as it approaches the real axis from above. In the subsequent ex­position the imaginary corrections ie will be dropped throughout. Performing the transition in the limit in the denominator of the integrand expression (7), we obtain*

(7a)

*The symbolic equation

t' -(t ± ie) t'~t ± in6(t'-t).

is used.

50

Ret

Fig. 7. Integration con­tour in the complex plane of the variable t.

Fig. 8. Graphical representation of the unitarity condition.

The behavior of 1m Frr(t) for t - 00 is unknown. But if it is assumed that lim 1m F" (t) c., const, then it follows that for convergence of the integral (7) it

i -+oo

is required to carry out one deduction. We shall carry out the subtraction at the point t = 0, where Frr(O) = e. Then the dispersion relation has the form

00

F,,(t)-F,,(O)=+ I 4m~

1m F" (I') dl' /' (/' -I) .

(8)

Relationship (8) is still an identity and may not be used for applications. In order to convert it into an equa­tion it is necessary to make use of additional information. Further work on calculating the function F7r(t) depends on the assumptions concerning the behavior of ImF7r (t) in the domain t ~ 4mir. For this purpose we use the unitarity condition

1m (0 I iv I Jl:+rn= ~ (0 I iv I a) (a I r+ I Jt+:rC), (9) a

where T+ denotes the Hermite-conjugate amplitude, while a takes all states allowed by the conservation laws (in particular, states with a total angular momentum J = 1, since a photon has a spin equal to 1). The lowest state is the state having two 7r-mesons; the 37r-state is forbidden, since (a I T+ I Jt+:rc) ~C. 0 according to a theorem similar to the Furry theorem Jn e~ctrodynamics. The next allowed state is the a-state with four 7r-mesons, etc, Among the a states KK, NN, NN, 'Tr1T, etc., pairs are possible. Theirproductionthresh­old with respect to t lies far from the beginning of the cut, which is frequently used in specific calcula­tions.

Equation (9) is displayed graphically in Fig. 8. Here it is appropriate to recall that the dispersion relations (7) or (8) together with the unitarity condition (9) in principle provide the possibility of carrying out the summation of an infinitely large number of the diagrams displayed in Fig. 3 or Fig. 6. For this pur­pose it is necessary to write out the dispersion relations for all amplitudes which are included in the uni­tarity condition (9) and to solve the infinite system of nonlinear singular integral equations. It is clear that this problem is unsolvable, and in practical calculations the practice is to restrict the analysis to the sim­plest cases.

Assume that we are interested in the region of small values of t. Whereas now there are no states in the unitarity condition for tf > t which make anomalously large contributions, then the high-energy re­gion will yield a small contribution which differs little from a constant, since the denominator of the inte­grand Eq. (8) may be written in the form

[' (t' -/)~· 1'2 (I _-{,) ~ ['2,

Thus, under the assumptions considered one may limit the analysis to the lowest states from a. In the con­sideration given let us limit ourselves to one state from a - the 27r-meson state. Then condition (9) takes the form

1m (0 I h I Jln) ~ (0 I h.1 :rUt) (JlJlI r+ I JtJl),

The vertex (0 I hi Jln) can be described by one p-wave. Therefore, the amplitude (1m! T+ I JlJl) must like­wise be described by one p-wave. Starting from the given parity condition for the matrix S, we obtain the following expression for the imaginary part of the form factor:

51

1m F" (t) = F" (t) e-i6,(t) sin OJ (t) (10)

where

ImF" (t) = ReF" (t) tg 6j (t), (11)

and 61 (t) is the phase of the 7I"7I"-interaction corresponding to the quantum numbers I = 1, J = 1. Having sub­stituted (11) into Eq. (8), we obtain

00

ReF~(t)=e+: g> (' ReF,,(t')tgi)dt')dt' .. "J t' (t' - I) .

4m~

Thus, the form-factor functions satisfy a linear singular integral equation which may be solved by the Muskhelishvili-Omnes method [9]. The general solution of Eq. (12) has a somewhat cumbersome form. Therefore, for simplicity we write out the solution of the integral equation without subtraction:

00

{ t (' i)j (1') dt' } F" (t) = e exp IT: g> J I' (I' _ t) •

4m~

(12)

(13)

Making various assumptions concerning the character of the behavior of the 61-phase of 7I"7I"-scattering, one can obtain various expressions for F7I"(t). The integration in (13) may be carried to completion by means of the theory of residues if the behavior of the phase 61 is chosen to be in the form of the ratio of polyno­mials P(k) and Q(k) [10]:

3 _ P (k) k ctg 61 (t) - Q (k) ,

where k is the three-dimensional momentum of a 7I"-meson in a coordinate system in which a pair of 71"­

mesons is produced. The invariant variable in this coordinate system is equal to the square of the total energy of two 7I"-mesons: t = 4(k2 + 1); m7l" = 1.

For the scattering of an electron by a 7I"-meson in the center of mass system in the case of back scattering the invariant variable t has the form

t= --:2v(1-cos1800)= -4v= -4(w2-1), w=ik.

After integrating the relationship (13) we obtain

where ki, j are the roots of the equations 1 + i tg 61 = 0, V = w2-1. For v = 0 the energy w = 1 and F7I"(0) = e. In particular, choosing the behavior of the phase 61 to be of the form aks ctg 61 (k) = k~ - k2; which corres­pond to the choice of the resonance Breit-Wigner equation

and assuming e = akr < 1, we obtain

52

iO, . .. _ ak3

e smUj - k2-k2_ iak3 T

kr+_l_ F ( ) _ e+k r

" v -e 002

kr+ ew+kr

(14)

(15)

If in Eq. (15) we carry out the expansion of the expression 1/ (£.w + kr) for the condition £.w/kr < 1, then the form factor (15) goes over into the expression

(16)

and coincides with the form factor obtained in [11] for a 1T-meson. At the point t = 0 the quantity k = i, and F(O) = e.*

The form factor (15) enters into the dispersion relation for the scattering of 1T-mesons by nucleons. Resonance interaction of p-mesons taken in the form (14) can be explained as exchange of a p-meson

(n + n -+ p -...n + n), whose mass is equal to mp = 750 MeV. For a value of the parameter £. = 0.2, one can provide a good explanation of the experimental data on the scattering of 1T-mesons by nucleons [10], which justifies the approximations used in the derivation of Eq. (15).

An original approach to a description of the electromagnetic form factor of a 1T-meson was proposed in [12, 13]. Its essence resides in the fact that the dispersion relations for the form-factor function in­cludes integrals within finite limits over that range of momentum transfers in which experimental data is available. The problem consists in finding the form-factor function of a 1T-meson and determining the boundary of the variation r1T of the electromagnetic radius of a 1T-meson. In particular, in [13] the author was able to obtain agreement between the experimental data and the analytic properties of a 1r-meson form factor and to estimate the upper boundary of the value r1r < 0.8 F.

An example of the nondispersive approach to a description of the form factor of 1r-meson may be found in [14]. The author uses the hypothesis of minimality of the electromagnetic interaction

i(V~ (x) = QeA!1 (x) J~ (x),

where Q = +, -, 0; J~ is the current which considers strong and electromagnetic interactions; Ap is the electromagnetic potential. In the absence of electromagnetic interaction the theory of strong interactions is isotopically invariant, and then the right side of the equation is equal to zero. The interaction which ap­pears in the right side of the expression leads to electromagnetic splitting of the masses of the isotopic multiplet (in the case given this multiplet is a 1T-meson triplet). If from this interaction we take the matrix element between one-pion states, then it can be seen that in the e 2-approximation it turns out to be linked to the electromagnetic form factor of a 1r-meson:

where T~s' are the isotopic matrices of the triplet; a is the fine-structure constant; F1T(1r21rl) is the form­factor function of a 1T-meson; G(1r 21r 1) is a function which is linked with the radiation correction to ,B-decay of a 1T-meson; s is the isotopic index of a 1T-meson. The form-factor function F1r(1r27T 1) satisfies the equation

with the additional condition F1r(m~) = 1.

In deriving this equation the one-particle approximation was used in the expansion of the product of the currents in the full collection of functions. This led to a situation in which it contained only one integral

*Having placed the value of a equal to zero and e = 1 in Eq. (16), we obtain the conventional expression for the form factor of a 1r-meson in the pole approximation (a p-pole):

I F,,(t)=--t-

1-­tp

[see Eq. (17a)].

53

b

Fig. 9 Fig. 10

Fig. 9. Process of electron (tt-meson) scattering by 7I"-mesons (a) and annihilation of electrons and positrons (tt±-mesons) into a pair of 1T-mesons, in which the electromagnetic form factor of a 1r-me­

son is measured (b).

Fig. 10. The various contributions to the amplitude of 1r-meson electroproduction.

FKr--------------------------------, Form factor of a 11"- meson

:: ~'~1:::.:::1:- -L - --, .... r __ ..... _- ------..... ---

0,6 p-dominance -7-----t~----f Fn - CE 2 Gev2

2 5 6 7

Fig. 11. Comparison of the experimental data with theoretical curves.

term and was a nonlinear integral equation. Considera­tion of the other states would lead to the addition of other integral terms of the type

in the right side of the equation considered; these terms contain products of more complex vertex functions which depend on many variables instead of the product of form­factor functions F(1r2q) F(q1rl)' It is very difficult to esti­mate the correction of the approximation used by the author. For the equation considered the approximate analytic solution is sought by expansion into a series in

eigenfunction of the motion group in Lobachevskii space in the domain of small momentum transfers. In this case the solution is simply related to the mean-square electromagnetic radius (see Eq. (2)). The ap­proximate solutions found in this manner have the form

F(q2)=[I-q: (1~2e+16E2)-:- ... J e,

where B=am,,/8t.m,,; a= 11137; m,,= 135 MeV; born = 4.6 MeV. The substitution of numerical values leads to the value (r) = 0.23 m;,l ~ 0.3 F.

The form factor of a 7I"-meson can be measured experimentally in all reactions which contain a vertex 'Y - 271". However, obtaining experimental data on the form factor of a 7I"-meson is an exceptionally difficult matter. Usually a number of other quantities which are determined from the same experiment is included in the investigated processes besides the form factor of a 1I"-meson. This complicates an unambiguous iso­lation of the form-factor function. Only in scattering of electrons or tt-mesons by 7I"-mesons (Fig. 9a) or in the e + e - 71" +71", tt + tt - 71" + 71" (Fig. 9b) can one measure the form factor of a 7I"-meson in the most efficient manner.

The 71" + e - 71" + e process was studied for the scattering of 1I"-mesons by the electrons of various nu­clei (see, for example, [15]). However, for low 1I"-meson energies the electron recoil is small, in connec­tion with which high requirements develop governing the accuracy of the experiment on the one hand, while on the other hand the form-factor function F7I"(t) is determined in a small domain of space-like momentum transfers in which F 1I"(t) differs little from unity.

In [16] the form factor of a 7I"-meson in the domain of space-like transfers was measured during the process of the electrical production of a 7I"-meson (Fig. 10):

e+p-.e+Ntn.

54

Fig. 12 Fig. 13

Fig. 12. Feynman diagrams describing e + 7r - + e + 7r scattering in the p-dominant approximation.

Fig. 13. Sum of the contributions to the form factor F7r: a) p-exchange diagram with allowance for rp; b) diagram which considers the contribu­tion of the interference p -w term.

The experimental points have rather large errors which are generated not only by the experimental errors but also by the estimates of the reality of the theoretical description. Figure 11 gives a comparison of ex­perimental data (the points) with two theoretical curves

(p is the dominant model, see Fig. 12);

1 Fn (t) C~ --1-1-­m2

p

F,,(t)-G1-,

(17a)

(17b)

where G~ is the isotopic vector part of the electrical nucleon form factor (see § 6 of the present review).

Equation (17a) derives trivially from Eq. (17) if instead of the imaginary part ImF7r (t) we substitute the expression Jlgr1"!.[n~(,c~ (I - m~) (Le., if we choose ImF7r (t) in the pole approximation having a zero decay width).* The quantities gpy and g.Tr7rP are the coupling constants of a p-meson with a y-quantum (see Fig. 1c) and of a p-meson with 7r-mesons, respectively. If in the p-dominant approximation the value of mp is chosen to equal (600±80) MeV, then the average value of the electromagnetic radius of a 7r-meson r7T [the determination of r7r is given in Eq. (2)] turns out to equal (0.80 ± 0.10) F.

With the launching of the Serpukhov accelerator (The Institute of High-Energy Physics) it will be pos­sible to obtain beams of 7r-mesons having a momentum :5 50 MeV/c which will ensure the investigation of the form factor of a 7r-meson up to transfers of q = V'-t ;p 200 MeV Ic. Under these conditions the mean­square radius V (r)" ;:::: 07 F can be measured with an error which is as small as 0.1 F [17]. The correct estimation of the radiation corrections is of important significance.

In [18] the radiation corrections to the process of elastic 7r - e scattering were calculated in a kine­matics that maximally approximates the kinematics of the experiment planned [17] at the Joint Institute for Nuclear Research, which will be carried out at Serpukhov (Institute of High-Energy Physics). Before [18] the radiation corrections to scattering were calculated in the Kahane paper [19] for the case when the ener­gies of the scattered particles are measured. However, in order to isolate the background processes, the

*In the domain t < 0 we have F7r (t) = Re F7r (t). In Eq. (7a) the denominator t' - t p! 0 for t< 0, and therefore the symbol .UP may be dropped:

The quantity {ZpvlZ;"" ~ 1. From this we obtain Eq. (17a) for the form factor of a 7r-meson which is normal­mp

ized to unity F(O) = 1 rather than to the charge e.

55

b

2'050 2'E, MeV

Fig. 14. Comparison of theoretical calculations with experi­mental data for ~ = 0 (a) and ~ = ~ (E. a) ~ 0 (b) [23].

Fig. 15. Electromagnetic cor­rection to the meson mass in the lowest order of perturbation theory •

planned experiments will measure the energies and emission angles of the final particles and also verify the coplanarity of the momenta. It is obvious that the measurements of additional parameters will affect the magnitude of the radiation corrections. In [18] the radiation cor­rections were calculated for the case of measuring three parameters (the energies of the final particles and the emission angle of the scat­tered ".-meson) out of the five enumerated above. The conditions and ranges of variation of the measured parameters within which the addi­tional measurement of the two remaining parameters does not alter the magnitude of the radiation corrections were indicated. An analysis is carried out of the contribution of the p-mesons in the diagram of two-photon exchange.

Recently the form factor of a ".-meson was investigated in the domain of time-like transfers t > 0, (i.e., inthe e+ + e- -".+ + ".-process) in a number of papers [20-23]. It can easily be seen (see Fig. 9b) that the cross section of the process will be proportional to the square of the modulus of the form factor F".. In [23] values of IF". 12 were obtained in the following range of energies E:

300<£ <500 MeV,

where E is the energy of an electron (or positron in the center of mass system of the reaction e + + e - -".+ + 11'-).

It is assumed that the principal contribution to F'/l' is made by the p-exchange diagram (Fig. 13a) with allowance for the width r p of p-meson decay. In order to co~sider the possible contribution of the interfer­ence p - w terms, a diagram (Fig. 13b) with the multiplier ~ela is added to the diagram shown in Fig. 13a. The full expression for the form factor FlI' is written in the form

(18)

where ~ and a are free parameters which describe the p - W interference, and d is chosen in such a way that the p-meson part of the form factor is normalized to unity for zero momentum transfer (d = 0.48);

P = V £2 - m~; Po = Y :~ - m~, s = 4£2. The results of comparing the theoretical calculations with the ex-

perimental data are displayed in Fig. 14. From this comparison it is evident that the case ~ ~ 0 is prefera­ble, although the case ~ = 0 cannot be excluded entirely.

3. THE ELECTROMAGNETIC MASS DIFFERENCE AND FORM

FACTORS OF K-MESONS [24]

The problems of the theoretical explanation of the observed mass difference and calculating the form factors of K-mesons are interrelated. This link depends on the assumptions made concerning strong inter­actions of K-mesons with other particles.

56

Independently of these assumptions a general formula relating the electromagnetic correction to the mass of a particle having a virtual Compton-effect amplitude T!-tv (see Fig. 5) is valid:

__ 1_ i T!LV (q2, v) g~tV 4

11m -- 8,,2 J q2+ie d q, (19)

where v = pq/m; q is the photon momentum; p is the particle momentum.

Riazzudin [25], restricting itself to the one-meson contribution to the imaginary part of the am­plitude of photon scattering by mesons, obtained an approximate equation for the corrections of the mass corresponding to the diagram displayed in Fig. 15:

(20)

If one uses a pole approximation of the type of Eq. (17a) for the form factor of a IT-meson, then Eq. (20) may be used to obtain the correct mass difference for 1T-mesons, but the pole approximation for the form factors of K-mesons and Eq. (20) yield the incorrect sign of the mass difference for K-mesons (instead of omK = mK+- mKo = -4 MeV the theory yields a positive value of omK).

There are two ways of affecting the calculations:

1) to reject Eq. (20) and go over to Eq. (19) with a more exact approximation of the amplitude T!-tv and

2) to modify the form factor F(q2) on the assumption that Eq. (20) is fairly accurate.

Tanaka [26], having chosen the first way, considered the contributions from the K*- and KA-resonance states in the amplitude of the virtual Compton-effect involving K-mesons. The terms which develop under these conditions in the expressions for the mass difference contain diverging integrals. The truncation con­stant must be chosen fairly far out; however, one can obtain the required mass difference. An unsuccessful choice of the truncation constant is evidently associated with the fact that a good asymptotic behavior has not been found for the amplitude.

G. Zinov' ev and B. Struminskii [27] obtained the correct difference of the K-meson masses after having improved the consideration of the asymptotic behavior of the amplitude T!-tv of the virtual Compton­effect in Eq. (19). They assumed that the amplitude T!-tv has a Regge asymptotic behavior ~va(O) which is determined by the A2-meson. The Regge residue ,8(q2) associated with the A2-meson was found by the authors using the sum rules for the final energy. The value of the Regge residue depends on the value of the coupling constant ghK of the decay K* - Ky. This constant was determined from the decay w - 1Ty using SU(3)-symmetry. y

Thus, the first way-rejection of Eq. (20) and refinement of the form T!-t v (q2, v)-leads to the correct value of the difference 0mK'

The second way-modification of the form factor FK(q2) -may likewise lead to a positive result.

Ogievetskii and Chou Huan-Chao [28] derived the correct expression for the mass difference using Eq. (20) and K-meson form factors of the form:

F'K = (4mk)2

( 1 - A) 4m2

! 'A ], ; } (4ml-q2)2 : 4mj(-q2 (21)

(4m}()2 41llJ( Fl( = (41ll}(_q2)2

(1 ), ) -f---~ 4mJ\.--q'2 '

where A is the parameter; A = 2. The form factors (21) have correct normalization, but the presence of double poles, the positions of the poles, and the residues at them regrettably have no physical substantia­tion.

One may attempt to find form factors F~' s having a more natural behavior at low energies:

57

1mt

a b

0

4m/r gm~.

c

~

Ret 411£

a) the form factor Fk must have a pole corresponding to a p-meson;

b) the form factor F~ must have poles corresponding to w- and cp-mesons;

Fig. 16. The beginning of the cuts in the unitarity condition (24) for states a con­taining two 7T-mesons (a), three 7T-me­sons (b), and a nucleon-antinucleon

c) F~ must be related by the two-particle unitarity con­dition to the amplitude of the 7T7T - KK process. An amplitude calculated according to the Venezziano model is chosen as the amplitude of 7T7T - KK;

d) the form factors must have the correct normaliza-tion;

pair (c). e) for describing the integral contribution of the high-

energy domain one may use the parametrization as-b which reminds one of the behavior of Regge amplitudes. The principal conclusion which is obtained from assump­tions a-e) is the following: the difference om = mK+ - mKo = -4 MeV can be obtained using the modified form factor Fk:

= F V = _) --c ~ ~ (~)-b

1( I 'em" )__ s., p m2 So

p

(22)

where C, b, and So are arbitrary parameters.

The form factor Fl( at low energies can be described well by the pole approximation, while at high energies it has a slowly decreasing ntail. n The quantity b may be small (10-3 ~ b ,.,; 10-2); 10-5 ,...:;; c ~ 10-2 ,

So is an arbitrary parameter beginning with which the contribution of the high-energy domain is considered; the dependence on the choice of So is weak.

4. DESCRIPTION OF NUCLEON FORM FACTORS BY MEANS

OF DISPERSION RELATIONS

The dispersion relations for nucleon form factors are written by complete analogy with the dispersion relations for the 7T-meson form factor. Let us agree to start by writing the dispersion relations for the isotopic form factors (the scalar FS and vector FV form factors) having one deduction:

v v I ~ ImF~, 2 (I') dl' F l , 2(t)=F I , 2(0)+n j 1'(1'-1)

4m; (23a)

F' _ sir 1m FL 2 (I') dl' 1,2 (t) - F l , 2(0) +n J I' (1'-1)

9m~

(23b)

In the last section of the review we shall consider dispersion relations with no deductions.

The imaginary parts ImF~\S for the nucleon current (6) are found from the unitarity condition for the 8-matrix (S = 1 + iT): '

1m (0 I it' v INN) = ~ (0 Ii;"" I a) (a I T+ INN). (24) a

The physical threshold of the process considered begins with an energy t~ 4 M2. However, the intermediate states a may contain a 27T-meson, a 37T-meson, and an infinite set of other states (Fig. 16). For a number of low intermediate states the quantities ImF~' i turn out to be nonvanishing in the nonphysical domain t:E 4M2. From an analysis of Eq. (24) it follo'ws (see, for example, [2]) that the isovector functions ImFi 2 are nonvanishing in the domain t ~ 4 mi and a contribution to them is made only by the states a having ~

58

,., I I I I

Fig. 17. Graphical repre­sentation of the unitarity condition.

even number of 1f'-mesons, while the isoscalar functions ImF~ 2 are non­vanishing in the domain t ~ -9 m~ and a contribution to them i~ made only by the states a having an odd number of 1f'-mesons. This can explain the differ­ent lower limits in Eqs. (23a) and (23b). The use of double Mandel'shtam dispersion relations allows us to obtain the analytic continuation of the unitarity condition from the physical domain t ~ 4 M2 into the domain of non­physical values 4m~ ~ t ~ 4 M2. Therefore, there are no difficulties in principle in carrying out the theoretical calculations. Let us proceed fur­ther by analogy with the procedure used in describing the form factor of a 1f'-meson. For low values of t we retain only one lowest intermediate state each in the unitarity condition (24) on the assumption that all of the remain­ing higher intermediate states make a negligibly small contribution. This means that for the isovector form factors only the 21f'-meson intermediate state remains, and the expressions for the function ImFj 2 take the form ,

1m p~. 2 = P" (t) (nn 1 n. 21 Nfl), (25)

where F rr (t) is the form factor of a 7r-meson, while (nn 1 T+ iN IV) are the parts of the amplitude of the anni­hilation process nn -+ N N, which make contributions to the Dirac or Pauli form factors, respectively. And for the imaginary parts of the isoscalar form factors only the three-meson intermediate state remains, and the expressions for them can be represented in the form of a product of two amplitudes:

1m Pt 2 (t) = (y 1 T 1 nnn) (nnn 1 Tt 21 Nfl), (26)

where (I' 1 T 1 mm) is the y - 3rr vertex, while (rmn 1 Tt. 2 1 N IV) are the parts of the amplitude of the 31f'­NN annihilation process which yield, by analogy with (25), contributions to the Dirac or Pauli form factors, respectively. Conditions (25). (26) are displayed graphically in Figs. 17a and 17b. Let us consider the case of an isovector form factor. Let us assume that the amplitudes (rm 1 Tt. 2 I N IV) can be described well by one resonance wave of the Eq. (14) type. In the limits of very narrow resonance we obtained the follow­ing results near resonance from the assumption that Re < nn 1 Tt, 21 NN > = 0 at the resonance point:

1 + - 1 + 1 - Ai (ak3)2 A k3 " * (nn Ti INN)~ 1m(nn Ti NN) ~ (tv-t)2+(ak3)2~ n: da )u(tv-t) , (27)

where Ai is a certain normalization constant; tv is the resonance position; a is the resonance width. The integration in (23a) can now be carried out in elementary fashion and leads to the result

pv (t) _ pv (0)' t const l • 2 1,2 - 1,2 -, tv- t • (28)

Expression (28) may be rewritten in the form of the well known Clementel-Villi form [29, 30]:

Ft 2 (t)=p~. dO) [l-a~. 2 + tv~t a~. 2]. (29)

If the presence of strong interaction in the 37r-meson intermediate state is allowed, then by complete analogy with the derivation of Eq. (29) one can obtain the Clementel-Villi form for isoscalar form factors as well:

ps (t) = ps (0) + t const l • 2 1.2 1,2 ts-t (30)

*In deriving (27) we used the relationship

6 (x-xo) =~ lim ( ~2+ 2' n a->O X-Xo a

59

or

F1. 2 (t) = F1, 2 (0) [ 1-bl, 2 + t8: t bt 2] • (31)

Using Eqs. (28) and (30), it is easy to obtain the expression for the charge form factor of a neutron:

The experimental data indicate the fact that for all values of t the function Fm(t) ~ 0, whence we obtain

bs aV __ 1 ____ 1_ ~ 0 ts-t tv-t .

Since this relationship is satisfied everywhere, it must also be valid near t :: 0, whence we obtain

Thus, in order to derive the form factors of nucleons one requires not six parameters (aI, ai, tv, b~, b~, t s ), as is evident from Eqs. (29), (31), but merely five: A, ai, tv, b~, ts. Moreover, the position of the resonan­ces ty and ts may be taken from experimental data, which reduces the number of independent parameters in the Clementel-Villi model to three.

The principal Singularity of the form factors (29) and (31) considered resides in the fact that with in­creasing transfer t all of them tend to constant limits. This may be interpreted as the presence of a core in the nucleons. It is precisely such an understanding of the problem which existed up to 1962 when experi­mental data on scattering of electrons by protons and neutrons were known for transfers q2 :: -t ~ 25-2 F and were interpreted by means of one vector p-meson and one scalar meson (see, for example, [31]). How­ever, with increasing energy of the incident electrons in an increase of the transfers q2 :: -t it turned out that the form factor function continued to decrease"'" 1/q2 [32] in such a way that the new data could not be reconciled with the existence of a core in the nucleons. Therefore, it was necessary to reexamine the theoretical interpretation of the results which had been obtained.

5. SURVEY OF EXPERIMENTAL DATA ON NUCLEON FORM

FACTORS AND THEIR THEORETICAL INTERPRETATION

Problems of the experimental determination of form factors, derivation of the Rosenbluth formula, and the validity of its use in the analysis of experimental data on the scattering of electrons by nucleons have been discussed in detail previously [33]. In this section we shall enumerate only the fundamental ex­perimental data and their interpretation from the modern viewpoint [34-37].

All of the conclusions concerning the behavior of the form factors as functions of the transfer tare based on the Rosenbluth formula [38] for the differential cross section of scattering of electrons by protons (q2 = -t):

where amo is the differential cross section of electron scattering by the Coulomb field, while () is the scattering angle of an electron in the laboratory coordinate system.

(32)

The data on the form factors of a neutron are obtained from experiments on the elastic and inelastic scattering of electrons by deuterons followed by subtraction of the effect of elastic scattering of electrons by protons. A description of this subtraction procedure may be found, for example, in [2, 33 J.

At present, as a rule, an analysis of the experimental data is carried out by certain combinations of the form factors Ff' v and F~' v rather than by means of the form factors individually:

60

TABLE 1. Experimental Information on the Functions GE p and GM Obtained for Various Assumptions Concerning the Behavio~ of GEp and GMp for Large Momentum Transfers q2

r"o' boo"''', w"h-in the limits of the 1/:!., Gli p GMp two standard errors

(GeV/c)2 Assumption --e GEp I o.1[p e

e e

4,81 GEp=O 0 I 0,056 0 0,076

G.1[p = 0 0,079

I 0 0,10 0

Gj[p = (1 +xp) G Ep 0,019 0,054 0,025 0,070

6,81 GEp=O I 0 0.038 I 0 0,052

GMp=O

I 0,070 0

I 0,097 0

GMp=(l I-xp) GEp 0,013 0,037 0.018 0,051 I

It can easily be verified that Eq. (32) may be written in especially simple form by means of (33):

yc~a+bx,

where

a=

i.e., in the x, y plane the Rosenbluth formula (34) may be represented by a straight line for q2 = const.

where

There exists still another frequently used notation for the Rosenbluth formula

~~ =A(8, t)G'i-!-B(8, f)Gk.

A (8, t)· Gmo t

1- 4M2

(33)

(34)

(35)

in which the absence of the "interference term" containing the newly determined quantities GE and GM [un­like the term [Fi (q2) + 2MF2 (q2)]" from Eq. (32)] is clearly emphasized; this leads to a reduction of the cor­relation errors in isolating the form factors from the experimental data. In the subsequent exposition we shall use only the new forlll-'factor functions G~ v and G~ v (or GEp, nand GMp, n)'

From Eqs. (33) it follows that:

GEp (0) = FiP (0) = e; GEn (0) = Fin (0) = 0;

GMp (0) = FiP(O) +2MF2P (0) =e(l +xp) =qtp, !!p 0_ 2.79;

G~!n (0) = FIn (0) + 2MF2n (0) = e!!n.

The form factor GE can be normalized to the charge, and therefore it is called the charge form factor, while GM can be normalized to the total magnetic moment of the nucleons and is called the magnetic form

61

c~ 0,4 ~j

\ t GN/J1.n

0,3 £ GN/J1.p T 'l' GEp

\1 0,2

0,1 t==d L-~--,----,:,-::----,--2x ,2"""x 50 fermis ,

o I,D 2,0 3,0 4,0 q~ (/J(J/C)2

Fig. 18. Comparison of the dipole model for the electromagnetic form factors of nucleons with the experimental data.

pGE/Gn

1,6

1,4

1,2 I 1,0

6,8

0,6

I T

n" H1 f I I

1,0 2,0 3,0 4,Oq~(GeV/C)2

Fig. 19. Check of the deviation from the linear law.

factor. In [34] experimental information on the func­tions GEp, n(q2) , GMp , n(q2) was obtained up to values q2 ~ 3(GeV Ic). Moreover, for the form-factor functions GEp and GMp the upper boundaries were obtained for q2 = 4.84 (GeV/c)2 and 6.81 (GeV/c)2, which could simultaneously serve as an estimate of the upper boundary of the core. Due to the smallness of the dif­ferential cross section for e + p - e + p scattering at

the two values indicated above, it was not possible simultaneously to determine the values of GEp and GMp from Eq. (35). Therefore, the analysis was carried out on the assumptions that:

~-A(8 2)G2 • b) if GMp = 0, then dQ -- ,q E p '

G GMp GMp th do !"A(e,q2) B(82-]G' c) if Ep=~= l+xp , en (jQ= _ (l+Xp)2 + ,q) MI'.

The results of the analysis in [34] are presented in Table 1.

For the form-factor functions GEn and GMn the upper boundary was obtained on the basis of assump­

tions a-b. For q2 = 6.81 (GeV /c)2 the upper boundary GMn = 0.024 ± g:g~~. For GEn the upper boundary

was obtained only for q2 = 3.89 (GeV.fc)2 and is equal to 0.081:1: 0.009.

The experimental data indicate the fact that the neutron form factor GMn has a behavior similar to the bp-havior of the form factors GEp and GMp. A check of the relationships

GMp GMn 1 GEp=~=~= ( q2 )2 =GD,

1 +if,7f

where q2 is measured in (GeV IC)2, substantiates this (Fig. 18, taken from [34]).

(36)

The linear relationship between the form factors GEp ' GMp/llp' and GMnllln, expressed by Eq. (36), is called the scaling law (Le., a linear law), while the dependence of these form factors on q2 is called the dipole fit (Le., the dipole model) and in the given review will be designated by the symbol Gd. Equation (36) indicates the possibility of describing form factors by means of the one-parameter model.

Recently [37] new data on the proton form factors GEp and GMp have been obtained from the q2 ~ 25 (GeV/c)2. In the range of transfers 1 (GeV/c)2 Z q2 Z 2 (GeV/c)2 a small systematic deviation from the linear law GEp = GMP/Ilp is observed. Figure 19 compares the experimental data with the ratio IJpGEp/

62

----- ----- ------------ ------.

1,1 f I TABLE 2. Experimental Information on the Func­tion (GEn)2

:~I'"-~J-~I i

___ -L-__ ~'_. ___ .l ___ .-1 ____ 1-_--"---_-'-, __ ----'--_---'-, _-,,_--,' W n m w ~ w m ~ ~ ~

q~ fermi

'1 2 , (GeV /C)2j (G!n r 0,389 +0,026 0,389 -0,034 0,623 +0,007 0,857 +0,007 1,17 -0,009

q2, (GeV /C)2j (G~n r 1,17 I -0,012 1,75 -0,003

2,92 0,102 3,89 0,0066

Fig. 20. Comparison of the magnetic form factor of a proton with the dipole model.

GMp == 1 from which it is evident that the deviation is approximately 10%. Figure 20 gives the compari­son between the magnetic form factor of a proton

and the dipole model. The deviation of the ratio GMp /llpGd from unity so far does not have a satisfactory theoretical explanation. Somewhat further on the four-pole model will be discussed, by means of which one can explain the experimental data.

The data on the electrical form factor of a neutron for large transfers are presented in Table 2 (taken from [34]), whence it is evident that GEn "'" O.

The experimental facts indicate that the form factors GEp' GMp ' GMn decrease with increasing trans­fer q2 no slower than 1/ q2. Consequently, if we wish to use the method of dispersion relations for describ­ing the form factors of nucleons, then the dispersion relations for the functions GE ,M should be taken with­out deductions. But this means that the dispersion relations must be taken without deductions for the form factors FP'i also, as is immediately evident from Eqs. (33) for GE M. * We see that the language of dis­persion reiations is convenient for the theoretical interpretation of ~xperimental data.

However, before writing the dispersion relations for the functions G let us note a certain fact. Having solved Eq. (33) for F j, we obtain:

0 1,(1)- 4~2 OM (I) F j (t) = ------cl--

1- 4M2 (37)

whence it is evident that at the point t == 4 M2 (Le., at the physical threshold of the annihilation channel) the

denominator 1 - 4:12 vanishes. Thus, the functions F j , 2 (t == 4 M2) go to infinity if it is not assumed that

GM(4M2) == GE(4M 2). We shall assume that such an equation exists, and we write the dispersion relations under these conditions.

Thus,

r 1m O~. M (I') , G~"M(t)=-n J 1'-1 dt, (38)

4m~

Analogous dispersion relations can be written for the functions G~ and G~, the only difference being that the lower limit begins at m~. The unitarity condition for ImG~ vM is graphically depicted just as it is in Fig. 17. Let us assume that the spectral functions ImG~' 'M may he represented in the form of a sum of 6-like terms (in the previous section the spectral functions 'ImF~:i were approximated by one (i-function):

*In [35] the authors showed that the dispersion approach is identically good for application to both the func­tions GE and GM and the functions F j , F 2•

63

TABLE 3. Values of the Parameters 01, (3i and the Core in Models b-d

0) 1=1

-0

I I core 0

al a2 ::?:

6 0,5 0 -8 0,525 0 -0,025 2 2,347 -1,847 -

1=0 1=1

a3 I "'4 I core ~1 I ~2 I core

1,477 -0,977 - 2,353 0 -1,090 -0,555 -0,035 2,471 0 -0,118 1,214 -0,714 - 8,268 -5,915 -

1m G'li 8 (t) = f Ctin:8 (t - til; }

1m GXi 8 (t) = ~~in:8(t-ti)' i

1=0

~3 I ~4 I core

I 1,612 -1,172 -1,060 -0,594 -0,031 1,093 -0,653 -

(39)

where ai, (3i are constants which must be calculated theoretically and must depend onthe interactionparam­eters 11' as other particles with nucleons, while ti = Mf are the positions of the isovector or isoscalar me­sons which yield the contribution to the form-factor function G by assumption. Having substituted Eqs. (39) into (38), we obtain:

G'li 8 (t)= ~ I ~i t ; 1 1 M~

GV,8(t)- ~ -~j- J M --.LJ t' j 1- M~

l

(40)

None of the form-factor functions (40) have a core, and for t - 00 all of them decrease according to the law '" 1/q2. If one of the meson masses (or several of them) turns out to be very large, then in this case the form- factor functions (40) can simulate the existence of a core.

Let us make use of the relations

GEp=GE+G~; GMp=GM+G~; GEn=G'E-G~; GMn=GM-G~.

and let us write the proton form factors in explicit form:

G - al I a2 a3 + a, + I Ep- 2 T 2 + 2 2'" 1+ q I' q 1+ q 1+ q M2 T M2 M2 M2

Iv 2v 3s ..

GMp =c ~I 2 -+ ~i 2 + ~3 2 + ~4 2 + ... 1 I+-q- I+-q- I+-q- l+-q-

M~v M~v M~s MZ.

(41)

In the notation (41) the contributions from isovector or isoscalar mesons are marked by the symbol v or s used as a subscript for the masses (Le., Ms or Mv); Le., the contribution from two isovector and two iso­scalar mesons are written out in explicit form. The neutron form factors differ from the notation (41) in that they have opposite signs which appear before the contributions,~rom isovector mesons.

In order to analyze the experimental data one may choose the one-pole, two-pole, three-pole, etc., approximations. At present three mesons are known: one isovector meson (a p-meson), and two isoscalar mesons (w and ~ each of which may make a contribution to the form factors of the nucleons in accordance with the unitarity conditions (25)- (27) and (39). In [34] an analysis is carried out for the following possible cases:

a) the dipole model in accordance with Eq. (36);

b) the three-pole approximation as the one which most completely considers the contribution from all known mesons and having no core;

64

c) the three-pole approximation - with a core;

d) the four-pole approximation if none of the previous ones provides satisfactory agreement with the experimental data.

For the time being it is inexpedient to use more complex models to explain the nucleon form factors.

The dipole approximation is very attractive due to its exceptional simplicity. It may also be inter­preted as the two-pole approximation with close values of masses corresponding to the resonances and coupling constants having opposite signs. Comparisons with experimental data are presented in Fig. 18.

The three-pole approximation (with or without the core) likewise can provide good agreement with ex­perimental data. However, under these conditions it leads to the necessity of taking underestimated values of the p-meson mass.

In [35, 36] the proton and neutron form factors were measured and analyzed in the range q2 :s; 30 F-2,

In the three-pole model the authors obtained the following expressions for the isoscalar and isovector parts of the nucleon form factors:

G~=0.5e{2.I8±q~'06 _ 1.I1±q~.14 -0.07±0.l5}; 1+ 15.7 I-I 26.7

G~f = 0.44e { 2.42 ± ~.05 _ 1.35 ± ~.09_ 0.07 ± O. I5} ; I "q q

IT 15.7 1+ 26 .7

G~=0.5e{ 1.05±:2·07 -0.05±0.07}; 1+ (7.51 ± 0.32)

G~ = 2.353e { 1.05 ± q~.OI~ 0.05 ± 0.0 I}, 1+ (7.51 ± 0.32)

From Eqs. (42) it follows that in the range of small q2:

1) the constant terms are close to zero;

(42)

2) the p-meson mass is approximately 200 MeV below the value observed (Mp = 548 ± 24 MeV). In order to improve the agreement with respect to the p-meson mass, the authors proposed the existence of still another vector meson (a B-meson having a mass of 1220 MeV and a width of 100 MeV [39]) and used the four-pole approximation.

For higher transfers q2 the interpretation of the form factors remains the same. A check of Eq. (41) showed that if known masses of the isoscalar mesons qJ and ware specified, while the p-meson mass is made variable, then the best fit to the experimental data is achieved for the value Mp = 510 MeV in the b model, while in the c model it is achieved for the value Mp = 540 MeV. The best agreement with experi­mental data (according to the x2-estimate) can be achieved in the d model with two isovector and two iso­scalar mesons. The values of the masses of the p-, W-, and qJ-mesons are chosen from experiments. The value of the mass of the vector meson is the fit parameter. In Table 3 the parameters of approximations b-d) are written out. The parameter of the approximation a is indicated in Eq. (36). The mass of the sec­ond vector meson in the d model is Mp' = 875 MeV. Figures 21 to 24 demonstrate the comparison of the approximations a , c, d with the experimental data for GEp' GMp ' GMn, andGEn respectively (Figs, 21-24 taken from [34]).

In [40] the authors stated the purpose of using the four-pole model to describe the form-factor func­tions of a proton with allowance for deviation from the linear law governing the behavior of the form factor:

4

GEp (q2) = ~ M~~q2 ' i=1 t

(43)

where M1 is a fictitious pole which considers the contribution of the continuous spectrum which causes an abrupt drop at q2 = 0(M1 = 0.45 GeV); M2 is the mass of a p-meson; M3 is the mass of a hypothetical p'-

65

~p~--------------------------------------~

_ ___ a

_._.- b

-- c

o 1,0 2,0

Fig. 21. Comparison of the approximations (or models) a, c, and d for the function GEp with the experimental data.

~p~--------------------------------~

0,6

~ ,,~

____ a

- .- .. b

-_c

n' ~_

~L-~ ___ ~-~--=--L-~~~~-~~ o 1,0 2,0 3,0 1,0 5,0 5,Ot,(~V Id

Fig. 22. Comparison of the approximations (or models) a ,b, and d for the function GMp with experimental data.

~~.-------------------------------------,

1,0

""'-t .. __

o 1,0 2,0

---- a _._ .. b

--c

. - .. - ..:. -=:.:..,.. "'--=---=:: . ~ -=.::-.:...

1,0 4,0 5,0 6,0 q, (GeV Id

Fig. 23. Comparison of the approximations (or models) a, b, and d for the function GMn with experimental data.

C~.-------r-------.-------r-------r------'

0,-1

0,2

_____ a

- ._.- b ---c

--t----.-._._ . 2,0 ' -'-3,0

meson [41] which is equal to 1.31 GeV; M4 is the fictitious pole which considers the contribu­tion of the far singularities in the dispersion in­tegral and causes a drop of the form factor for large transfers (M4 = 2.456 GeV). For such a set of masses, the constants Ai take the values: At = 0.047, A2 = 0.834, A3 = -1.263, A = 0.446.

Figure 25 gives a comparison of the theo­retical curve (43) with the experimental data (GEP/GD = 1). Notwithstanding the clear physi­cal meaning of the representation (43), it never­theless cannot be considered satisfactory due to the large number of arbitrary parameters.

At the end of this section we shall discuss exponential models of form factors proposed in [42-45 ].

It is well known that the microcausality principle allows exponential growth of the ampli­tude at infinity; however, this growth must be slower than a linear exponential. In this case one may not write the dispersion relation with a finite number of deductions, as is done for the amplitudes having polynomial growth.

Logunov, Mestvirishvili, and Sillen [42] considered the class of functions which allows a nonpolynomial growth of the amplitude at in­finity and is convenient for describing the ex­perimental data. Assume the function G(t) is the form factor function of a nucleon having the cut O<a-<t-<+ 00. The t-plane is converted into a unit circle by means of the conformal trans­formation

~= Va-Va=f Va+Va-f'

so that G (t) ~ f (6)·

Assume f(~)EA, where A is the class of all functions which are analytic in the unit circle and satisfy the condition:

where

2"

lim j In+ If (pei9 ) I d8<:A < 00, p-.i 0

{ Ina a> 1 In+a= o a<l.

Fig. 24. Comparison of the approximations (or models) a, c, and d for the function GEn with experimental data. Such a representation allows a fairly

varied behavior at infinity for G(t). However, this is only a minimal extension in comparison with the polynomial-growth function. Functions from the class A may be represented in the form

66

1,1

1,0 -

0,9·

0,8

0,7 ~

I

f (~) = b (~) e<lJm,

where b(g) is a Bliascshke function, while ~(~) has the form

<D(~)=i<D(0).+-21. r W(Z~dz, Jtl J z-~

Izl=1

1 0,60

where w(z) is the discontinuity of the function on the 10 20 30 40 50 60 70 80 gO qJ(GeV Id cut.

Fig. 25. Comparison of the ratio of the function GEp obtained from Eq. (43) to the function GD with experimental data.

Returning now to the variable t, we obtain

b (~) = B (I) eg(t) and G (I) = B (t) e/1(tl+'P(t),

where cp(t) is the transform of ~(g) in the t plane, while g(t) is a function having a cut in the t plane t(a ::s t ::s 00). If it is assumed that B(t) has a finite number of zeros (we shall not consider the case of an infinite number here), then a representation in the form

~ ~

G(t)o H(t)expr*) Im~~~dt' +t~a) Im~~~dt' +!Jl(a)j o a

is obtained for the form-factor function G(t). Now one can find Img(t) and Imcp(t) by extrapolation of experi­mental data (i.e., the imaginary parts of the proton form factors GEp and GMp) in the time-like domain of the momentum t.

Let us suppose that B(t) = 1. This assumption can be justified by the fact that the function eg(tH'P(t)

provides a good description of the experimental data on the nucleon form factors in the domain of space­like momentum transfers. Let us also consider the additional information:

GEp (0)· I; GMp (0) ~.- 2.79;

GE " (4M2)c GMp (4M2); GEl' (- 00) = GMI , ( - 00)·' O.

Let us choose the following approximating expressions:

GEp = exp r t~ (; ·1 a1£·1 a2~21 ;

GMp = exp[ I ~(;~ + bl~ + b.2£2 J . After the coefficients ai and bi have been found according to the experimental data in the t < 0 domain, the expressions GEp and GMp may be continued analytically in the time-like domain of transfers t. In this do­main ImGEp and ImGMp oscill!te, and these oscillations may be detected experimentally for measurement of the polarization in the p -I- p -+ c+ (ft +) + e- (ft -) reaction.

Using the method of analytic continuation, one could hope to obtain the positions and widths of the well-known resonances p, w, cp from oscillations of ImGEp and ImGMp in the domain t > 4m~. The authors, regrettably, were not able to -obtain the correct values of the parameters of the resonances enumerated above. It should, however, be noted that in [42] not all the possibilities inherent in the model were used, since only the case B(t) = 1 was considered.

In [43] the authors started from experimental data on the scattering of protons by protons at high energies and high momentum transfers. The behavior of the differential cross sections of p + p - p + P can be described by the exponential Orear formula:

do -~ 1lQ(6,p.l)",Ce 0.15

67

I

1,2 \ I \

---f----------------.""".... ! "" ............ I .~ ............ [f& [M] ...........

• r44? ......... " L' 'J ...... " ... -

Fig. 26. Comparison of various experimental representations for the proton form factors witb experimental data.

where C is a certain constant, while P.L is the transverse momentum transfer. Wu and Yang made the natural as­sumption that in elastic e + p - e + p- scattering there must be a rapid drop of the differential cross section, which is of the exponential type, and such behavior may occur as a result of the exponential behavior of the pro­ton form factor functions:

(44)

where P.L should be replaced by qand Po is a certain constant which is chosen from experimental data. It is assumed that the factor B may be a constant or may de­pend weakly on q2.

Actually the form factors (44) provide a fairly good description of the experimental data for the value Po =

0.6 (GeV/c) and B = const. However, one should note two singularities of these form factors.

1. Their analytic properties are false, since the cross section of the functions G(q2) begins with the values t = 0 rather than with the values t = 4mi-, as derives from a field-theory consideration. This short­coming is easily corrected. It is sufficient to replace YQ2for the function V q2 + 4 m~:

2. As has already been said above in analyzing the paper by Logunov et al. [42], the exponent in the time-like domain q2 < 0 becomes imaginary and the form factor function (44) begins to oscillate so as to give equally spaced minima and maxima in G(q2). Such a behavior differs from the behavior of the four­pole approximation in the time-like intervals of q2 and provides the possibility of choosing one of the mod­els named here by comparison with experimental data on annihilation:

In [44] a form factor of the form

GMp [ ./-q J ~ = 27.8 exp - V 0.040 (45)

was proposed, while in [45] an exponential form factor with the asymptotic behavior

(46)

was proposed, where A and a are fitting constants.

A comparison of Eqs. (44)- (46) with experimental data is given in Fig. 26. It is evident that the ex­ponential form factors may describe experimental data only in a certain portion of the transfers q2.

LITERATURE CITED

1. R. Hofstadter, Nuclear and Nucleon Structure. A Collection of Reprints with an Introduction, W. A. Benjamin, New York (1963).

2. s. D. Drell and S. Zachariasen, Electromagnetic Structure of Nucleons, Oxford University Press, Oxford (1961) [Russian translation: Moscow, Izd. Inostr. Lit. (1962)] .

68

3. Michel Gourdin, Herzeg Novi Lectures, 1961, Vol. 2. Lectures on High-Energy Physics (Sixth Summer Meeting, Herzeg Novi) , Gordon and Breach (1965).

4. L. N. Hand, D. G. Miller, and R. Wilson, Rev. Mod. Phys., ~ 335 (1963). 5. H. J. Bremmerman,R.Oehme, and J. G. Taylor, Phys. Rev., 109,2178 (1968). 6. W. Paull, Rev. Mod. Phys., 11, 203 (1941). 7. s. M. Bil enky , et al., Nucl. Phys., 7.., 646 (1958). 8. Y. Nambu, Nuovo Cimento, &" 1064 (1957). 9. N. I Muskhelishvili, Singular Integral Equations, 2nd ed. [in Russian], Fizmatgiz, Moscow (1962);

R. Omnes, Nuovo Cimento, 8, 316 (1968). 10. P. S. Isaev and V. A. Meshcherakov, Zh. Eksperim. i Teor. Fiz., 43, 1339 (1962). p. S. Isaev, V. I.

Lend'el, and V. A. Meshcherakov, Zh. Eksperim. i Teor. Fiz., 45;294 (1963). 11. J. Bowcock, W. N. Cottingham, and D. Lurie, Nuovo Cimento, 16, 918 (1960). 12. L. A. Khalfin and Yu. B. Shcherbin, Pis'ma Zh. Eksperim. i Teor. Fiz., 8, 588 (1968); ~, 642 (1968). 13. Yu. P. Shcherbin, Pis'ma Zh. Eksperim. i Teor. Fiz., 10, 340 (1969). -14. A. V. Efremov, Zh. Eksperim. i Teor. Fiz., 53, 732 (1967). 15. E. G. Grishin, E. P. Kistenev, and Mu Tsyun~adernaya Fizika, ~, 886 (1965). 16. C. W. Akerlov, et al., Phys. Rev., 163, 1482 (1957). 17. D. Yu. Bardin, et al., Communications of the Joint Institute for Nuclear Research, El-4786, Dubna

(1969). 18. D. Yu. Bardin, V. B. Semikov, and N. M. Shumeiko, Yadernaya Fizika, .!Q (1969); Preprint of the

Joint Institute for Nuclear Research R-4-4532, Dubna (1969). 19. T. Kahane, Phys. Rev., 135, B975 (1964). 20. v. L. Auslender, G. I. Budker, et al., Phys. Lett., 25B, 433 (1967). 21. V. L. Auslender, G. I. Budker, et al., Yadernaya Fizika, ~, 114 (1969). 22. J. Augustin, J. Bizot, et al., Phys. Lett., 28B, 508 (1969). 23. R. Perez-y-Torba, Fourth International Symposium on Electron and Photon Interaction at High Ener­

gies, Liverpool (1968). 24. S. K. Smirnov, Diplomate Paper, Moscow State University, Technical PhysiCS Laboratory of the

Joint Institute for Nuclear Research (1969). 25. v. A. Riazzudin, Phys. Rev., 114, 1184 (1959). 26. Y. Tanaka, Nuovo Cimento, A60, 589 (1969). 27. G. M. Zinov'ev and B. V. Struminskii, Yadernaya Fizika, ~, 173 (1969). 28. v. I. Ogievetskii and Chou Huan-Chao, Zh. Eksperim. i Teor. Fiz., 37, 866 (1969). 29. E. Clementel and E. Villi, Nuovo Cimento, .1, 1207 (1956). 30. S. Bergia et al., Phys. Lett., &" 367 (1961). 31. R. Hofstadter and R. Hermann, Phys. Rev. Lett., &,,293 (1961). 32. K. W. Chen, et aI., Phys. Rev. Lett., 1.1 561 (1963). 33. A. Verganelakis, Electromagnetic Form Factors, Ecole Internationall de la Physique des Particles

Elementairer, Herzeg Novi (Yugoslavia) (1965). 34. L. H. Chan, et al., Phys. Rev., 141, 1298 (1966). 35. E. B. Hughes, et aI., Phys. Rev., 139, B458 (1965). 36. T. Janssens, et al., Phys. Rev., 142,922 (1966). 37. J. G. Rutherglen, Fourth International Symposium on Electron and Photon Interactions at High Ener­

gies, Liverpool. Daresbury Nuclear Physics Laboratory (1969). (This review paper contains refer­ences to experimental papers on the measurement of nucleonic form factors, which were performed on the DESY, Bonn, and SLAC accelerators in 1967-1969).

38. M. N. Rosenbluth, Phys. Rev., .TI!" 615 (1950). 39. M. Abolins et al., Phys. Rev. Lett., 11, 381 (1963). 40. Ch. Berger et al., Electromagnetic Form Factors of a Proton between 10 and 50 F- 2, Physikalisehes

Institute, University of Bonn, Preprint 1-075, July (1969). 41. C. Lovelace, Phys. Lett., 28B, 264 (1968). 42. A. A. Logunov, M. A. Mestvirishvili, and I. N. Silin, Preprint of the Joint Institute for Nuclear Re-

search R-2519, Dubna (1965). 43. T. T. Wu and C. N. Yang, Phys. Rev.,~, 708 (1965). 44. s. D. Drell, A. C. Finn, and M. H. Goldhaber, Phys. Rev., 157, 1402 (1967). 45. G. Mack, Phys. Rev., 159, 1615 (1967). 46. D. H. Coward, et aI., Phys. Rev. Lett., 20, 292 (1968).

69

TWO-DIMENSIONAL EXPANSIONS OF RELATIVISTIC

AMPLITUDES

M. A. Liberman, G. I. Kuznetsov, and Ya. A. Smorodinskii

The present paper is a review of research on double expansions of relativistic amplitudes. In the beginning of the review the general theory of the expansion of functions stipulated in both the time-like and space-like domains is presented; then the general theory of the ex­pansion of a scalar function and a function with spin is presented. After that various coordi­nate systems and matrices for the transition from certain basis functions to other ones are indicated. The review likewise considers certain corollaries deriving from the existence of relativistic expansions.

INTRODUCTION

Very many papers have been devoted to the analytic properties of relativistic amplitudes. Expres­sions in which the variables s, t, u are chosen as the variables are used as the original representation of the investigations. A natural first step turns out to be the expansion of the amplitudes into partial waves which represent a generalization of the expansions of nonrelativistic amplitudes into spherical ones. Re­cently, the attention paid to such expansions has increased, especially in connection with the papers by Toller et al. However, in most of the papers expansions in only one of the variables are used, and in this sense such expansions are not, strictly speaking, relativistic.

In 1964 N. Ya. Vilenkin and Ya. A. Smorodinskii produced the theory of two-dimensional expansions which are the relativistic analog of expansions into spherical waves. In this paper the relativistic ampli­tude is parametrized in such a way that it is determined on the upper fields of a two-cavity hyperboloid in the velocity space. It may be shown (Verdiev [20], Matthews and Feldman) that such a parametrization may also be arrived at by beginning the expansion with the determination and expansion of two-particle ampli­tudes.

In this survey we have brought together the equations which apply to the expansion of scattering ampli­tudes for the case of equal masses.

The dependence of the amplitude of the scattering of two particles on two variables (the energy and the scattering angle) allows us to determine the variables in terms of the coordinates of the points on a three-dimensional hyperboloid with allowance for the equation pi = mi, P1 + P2 - P3 + P4 [1] (Le., for the amplitudes of the four-tail it is sufficient to consider the coordinates of just one of the free ends). From this it follows that the scattering amplitude may be treated as a function stipulated on the upper field of a two-cavity hyperboloid in order to be specific. Then the shift operator on this surface (these operators ob­viously realize a proper Lorentz group) may be used to convert the scattering amplitudes from one value of the variables to another. We shall call the possibility of such a transition from one value of the scatter­ing amplitude to another by means of shift operators on a hyperboloid (or cone) the extended relativistic­invariance condition. Let us note that the conventional invariance condition consists only in the fact that the amplitUde depends on the invariant variables s, t. Such an extension of the definition of invariance,

Joint Institute for Nuclear Research, Dubna. Translated from Problemy Fiziki Elementarnykh Chastits i Atomnogo Yadra, Vol. 2, No.1, pp. 105-128, 1971.

70

C 1972 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without permission of the publisher. A copy of this article is available from the publisher for $15.00.

which is required for the introduction of the expansions, naturally leads to the conditions governing the be­havior of the functions at large values of the argument.

Thus, the problem consists in investigating expansions of the scattering amplitude in representations of the Lorentz group. Let us begin with an investigation of the realization of the representations of the Lorentz group.

1. THE FUNCTIONS ON A HYPERBOLOID

1. The Coordinate System and the Eigenfunctions of the Laplace

Operator on a Hyperboloid

From the principle of relativistic invariance and unitarity it follows that the scattering amplitude should be expanded into functions which realize unitary finite-dimensioned representations of the Lorentz group. These functions are solutions of the equations [2, 3]:

(1.1)

t...'f= -vpt, (1.2)

where Ll and Ll' are invariant Casimir operators of the Lorentz group which are constructed from infinites­imal operators of the space and hyperbolic rotations M and N and are respectively equal to:

3

t...= ~ (M~-M); i=1

3

t...'= ~ MiNi. i=1

(1.3)

(1.4)

Let us give detailed consideration to the expansion of the scalar function f(u) stipulated on the hyperboloid u 2 = p2/m2 = 1. In this case the operator Ll is equal to the Laplacian on the hyperboloid, while the operator Ll', which is associated with the spin, is identically equal to zero. Let us likewise indicate the modification of the expansions for the case of nonzero spin.

On the hyperboloid one may introduce various coordinate systems which we shall proceed to describe now.

1. The spherical coordinate system S:

Uo = ch a; U2 = sh a sin e cos qJ; 1 U3 = sh acose; u, = shasin e sin qJ; ! o < a < 00, 0 < e < n, 0 < qJ < 2n.

In the variables a, e, rp the Laplace operator takes the form

The eigenfunctions corresponding to this operator having the eigenvalue - (1 + p2) are equal to

((lIm I aeqJ) = (sh ar'/2 p=:;~+lp (ch a) Y lm (e, qJ).

2. The Lobachevskii coordinate system L:

Uo c~c ch a ch b ch c: U 3 ~ ch a ch b sh c;

U 2 = ch a sh b; } u j = sh a.

(1.5)

(1.6)

(1. 7)

(1.8)

71

The Laplacian is determined from the following equation in the variables a, b, c:

~ ~ .. -.-I-~ch2a~--l---I- (_I_~Shb~+_I_~) L - ch2 a 8a 8a ' ch2 a sh b 8b 8b sh2 b 8c2 '

and its eigenfunctions are equal to:

3. The hyperbolic coordinate system H:

Uo =ch achb; U 3 = ch a sh b cos cp;

U 2 = ch a sh b sin cp; } u l = sh a.

In this system the Laplace operator and the eigenfunctions corresponding to it have the form

4. The cylindrical coordinate system C:

110 = ch b ch a; 112 = sh b sin cp; }

u3 = sh b cos cp; ul = ch b sh a.

The Laplacian and the eigenfunctions in the C-system are written in the form

I 8 8 I 82 1 82 ~L = ch b sh b . ail ch b sh b ail + ch2 b • a£!2 -j- sh2 b . 8<p2 ;

5. The orispheric coordinate system 0:

uo= 1/2 [e-a+(r+ l)ea); uz=reacoscp;}

u3 = 1/2 [e-a-t (r-1) ea); UI = rea sin cpo

In the variables a, r, qJ the Laplacian is given by the expressions

and its eigenfunctions are equal to

(pkm I brqJ) = (kb) KiP (kb) Jm (kr) eimq>.

Here Ki (kb) and J m (kr) are respectively Macdonald and Bessel functions, while e-a = b.

2. The Method of Orispheres

(1.9)

(1.10)

(1.11)

(1.12)

(1.13)

(1.14)

(1.15)

(1.16)

(1.17)

(1.18)

(1.19)

Thus, we have constructed the various eigenfunctions. Nowit is necessary to orthonormalize these functions or, what amounts to the same thing, to find the equations for the inverse transformation. This

72

can easily be done using the Gel'fand-Oraev equations [4, 5]. Let feu) be a scalar function which is stipu­lated on a hyperboloid, while h(k) is a scalar function stipulated on the cone k 2 = O. If

then

for n = 2m + 1 and

for n = 2m.

h(k)= f(u)o«ku)-I)-, J dnu

uo

n---1

(_)-2- r dnk f (u) =c 2 (2,,;)n 1 J o,n-1> «uk) - I) h (k) ~

n

f ( ) = (_)2 I' (u) r (uk __ I )-n h (k) dnk u (2n)n j ko

(1.20)

(1.21)

(1. 22)

Here n is the dimensionality of the manifold, while dnu/uo and dnk/ko are invariant measures on the hyperboloid and cone.

Let us expand h(k) into homogeneous components:

6+ioo

h(k)= 2~i J <IJ(k, u)du*, (1. 23) 6 --ioa

where

00

CD(k, u)= J h(kt)t-a- 1 dt. (1.24) o

From Eqs. (1.20) and (1.24) it follows that

CD (k, u) -~ f (u) (ull) -. J cr dnu

uo (1.25)

Using Eqs. (1.21)-(1.23), we obtaint

n-l 6+ioo

f (u) = ~ r r (a+ n- I) \ (D (Il' u) (uk,)-a n. j dn Ik'du 21 (2n)n j r (a) J ' (1.26)

6-ioo r

for n = 2m + 1 and

'::'-1 61-100

f(u)= (_)2 r l'(a+n-l) ctg:n:u l' CD(k' u)(UIl,)-a-nl-1dn-1k'd 2i (2n)n J I' (a) J' fT.

(1.27) 6--ioo r

*The quantity <5 is chosen in such a way that the poles of the function 4? (k, u) lie outside the strip

tEquation (1.26) was obtained for the first time by I. S. Shapiro for n = 3 [6].

73

The integration contour r is arbitrary on a cone intersecting all generant cones; dn- 1k' is an element of the volume of this contour and is determined by the equation

d (tk') = tn - 3 dt dk' (O<t<oo).

In the unitary case the quantity 0 = _ n; 1 + ip.

3. Derivation of the Inversion Formulas

Let us derive the inversion formulas which are associated with the eigenfunctions of the Laplacian.

1. The S-System. The integration contour r in the S-system is a sphere (ko = 1). The function <I>(k, a) is stipulated on the sphere. Let us expand it into a series in spherical harmonics:

CD (k, 0) = ~ aim (0) Y lm (k/I k I). 1m .

(1.28)

Substituting (1.28) into (1.26) and performing integration with respect to drl, we obtain

__ I) 1 r(l-G) alm(G) -l-~ t(u)-. 312 ~(--) r( I I) -V- P 3(cha)ylm (e,cp)da.

21 (2n) 1m -G- - sh a -0-2 (1.29)

Taking account of Eqs. (1.28) and (1.25) for the coefficients aZm(a) , we find the following expression:

(1.30)

d3u Here - = shz ada sh e de dcp. Ua

Equations (1.29) and (1.30) are written for the conventional four-dimensional space (n = 3). The func­tions on multidimensional hyperboloids were investigated [7, 8].

2. The H.,.System. The expansion in the hyperbolic coordinate system is carried out by the same meth­od as that used in the spherical system. The integration contour r consists of two parts r + and r _ cor­responding to two cross sections of the cone produced by the planes k3 = ± 1. The calculations lead to the following results:

0+100 _+100

f (u) = _ (ch a)-l ~ I' I' r (<1+T+2) r (<1-T+ I) m 1m", 8 (2n)4 ~ J J r (<1) r (m-<1) T ctg nTP -1:- d ch b) e

m ~-ioo e-ioo

(1.31)

where

a± (T cr) = r (T) r (-<1-T-I) r (T-G) I' f (u) p O+1 (4= th a) pm ( h b) -1m", dBu , r(T-m+I)r(-<1) J" 1: C e Uo •

(1.32)

The unitary space corresponds to the quantities a = -1 + ip, T = - 1/2 + iq.

3. The O-System. In the orispheric coordinate system let us choose the cross section of the cone produced by the plane ko - k3 = 2.

ko = (1 + !lZ);

k3 = ( -1 + !lZ);

74

k2=2!lc~sa; } kl = 2!l SIn a.

(1.33)

Representing the function <I>(k, u) in the form

2" 00

<D (k, 0) == ) ) '¥ (x, e, 0) e1xIl cos (8-a)x dB dx o 0

and substituting this expression into (1.26), we obtain

6+ioo 2Jt 00

f(u)~, (2~)2 .i l';a) J ) '¥(x, e, 0) (; r+2 eiXIl cos (8-<I»K_,,_dbx)dxdedo 6-ioo 0 0

after performing the calculations. The coefficients '¥ (x, e, 0) are determined according to equation

\¥(x tJ 0)= __ 2_ (2)"+1 I f(u)e-ixrCos(8-<I»K (bx)~. d3u .. nr(-a) x j (1+1 b 2 Uo'

In Eqs. (1.35), (1.36) Kv(x) is a Macdonald function, c- a = b, d3u/uO = rdrdbd<D.

(1.34)

(1.35)

(1.36)

4. The C-System. Finally, let us consider the cylindrical coordinate system. The integration con­tour r in this system is the intersection of the hyperboloids with the cylinder kij - ki == 1. Let us parame­trize r as follows:

Using the Fourier-expansion of <I>(k, u)

k0 c~ ch c;

1<1 sh c;

00

sin a; 1f k~ -~ cos a.

<D (k, 0) = ~ j am (,r, 0) ei(ma, 'c) dT, m=-oo -00

we obtain the following representations for f(u) after calculation:

. 6+ioo 00

f(ll) ~ 1 ~ thmbe"ncp I ei,alam(T,a)I'(A)I'(B)(_2_)"+2F (A B m+I·th2b)d~d - 16:12i ~ r(m+l) j j r(a) chb 2 1 " , v T.

111 fl-ioo

(1.37,)

(1.38)

(1.39)

Here A c- ~ (m -i ()' + iT --i- 2); Bee -} (m +- 0 - iT + 2). The coefficients am (T, u) in this system have the following

form:

r(A')J'(B') I (2)" d3 a",(T, 0)=- 4:'1I'(m+l)r(-a) j th"'b chb - 2Fj(A', B', m+l, th2b)ei (m<I>+'ta)f(u)--d;. (1.40)

Thus, we have carried out the expansion of the scattering amplitude in four coordinate systems in functions which realize infinite-dimensioned unitary representations of a class-1 Lorentz group. A detailed derivation of Eqs. (1.29), (1.31), (1.35), and (1.39) was given in the paper by N. Ya. Vilenkin and Ya. A. Smorodinskii [11.

4. Expansion of the Functions Stipulated in the Space-like Domain

In order to carry out the expansion of a function stipulated in the space domain one should use the Gel'fand-Graev equation in the form [5):

75

~+Ioo 00

f(u)= 4~(;~)3 ~ a(a+ 1) ~ F(k, a) 1 (uk) 1-0 - 2 d2kda+ ;2 ~ 2n ~ F(k, u; 2n) 6 (uk)d2k. 6-100 r n=l r

(1.41)

Here f (u) is an even quadratically integrable function; u2 = u~ - u2 = - I; k2 = 0; r is the integration con­tour on a cone (a sphere for ko = 1). The numbers a and n are the weights of the representations of the Lorentz groups of the fundamental and discrete series, a = -1 + ip holding as previously in the unitary case.

In order to write the expansion for an odd function f (u) ~ -f (-u) (xo -+ -Xo, x -+ -x), it is re­quired to replace I (uk) 1-0 - 2 by I (uk) ,-0-2 sin (uk) in Eq. (1.41) and 2n by 2n-l in the second term (see [9]). If the expansion of the functions F(k, a) and F(k, u, 2n) is carried out, and, for example, the spherical co­ordinate system is introduced, we have

Uo =sha; ko = 1;

} U3 = ch a cos 8; k3 = cos 8'; u2 = ch a sin 8 cos Ij); kz = sin 8' cos CD; (1.42)

U1 = ch a sin 8 sin Ij); kl = sin 8' sin CD,

then, having carried out integration with respect to d2k = d cos 8' dCD, we obtain [9]

2 00 ( 2n 4n 1 (pfn (th a) ) +Jl2 ~ 2n-l) ~ 2/+1 elm cha p2n-l(tha) Y lm (8, Ij)h

n=l 1m I

(1.43)

p7+1 (th a) and Pf (th a) are associated Legendre functions; fe(u) andfo(u) are even and odd functions, re­spectively. The first term in this expression is similar to the expansion of the function on a two-cavity hyperboloid, which was obtained in [11; the second term is the expansion of the function in discrete repre­sentations of the Lorentz group [10]. Problems associated with the analytic continuation of the amplitudes into the space-like domain, and likewise the eigenfunctions of the Laplacian in the space-like domain, were considered in [11-13J.

5. The Expansion of Fields

We have considered the expansion of a scalar function in eigenfunctions of the Laplace operator on two-cavity and one-cavity hyperboloids. In order to expand a function having spin s and a projection a onto the direction of the momentum* (the spiral state; see [14]) it is necessary to modify the kernel of the inte­gral transformation-namely, one should write (uk)"t-ip in Eq. (1.26) instead of the kernel (Uk)-l- iP.fl5 S (R), where R is a certain rotation which considers the spin requantization; .fl5s (R) is a Wigner function. Thus, the expansion of the functions l/Jsa(u) having a stipulated spin on a hyperboloid in eigenfunctions ~pv (k), which are transformed according to representations (p, v) that are irreducible relative to the proper Lorentz group, is now determined by the following expression [15, 16, 17J:

00

qts" (u) = 2 (~n)3 ~ J dp(p2+ V2) J (Uk)-l-ip .fl5~v(R)CDvp(k)d2k, (1.44) V=-8 r

where r is the integration contour (a sphere for the cross section of the cone k 2 = 0 produced by the plane ko = 1); d~ = dO; R is a rotation in the (k, u) plane through the angle 7r - (J'. If the spherical coordinate

k system n = fkl = (cos 8, sin 8 cos Ij), sin e sin Ij), is introduced on the cone, then (see [15-17])

*In this section a is the spin projection and not at all the number characterizing the representation of the Lorentz group.

76

cos fJ' = _ Uo cos 8 -I u 1 . uo-(un)

Assume the function <l>pv(k) is represented in the form

(Dpv (k) = ~ an,! (p) (2J + I) 27J'frv (m ) . JM

(1.45)

(1.46)

Then, substituting Eq. (1.46) into Eq. (1.44) and carrying out integration with respect to d~, we obtain [14]

-1-00

Ifsa (u) = 2 (~n)3 ~ J dp [p2 i v2] ~ aiM (p) [~~.rM (u). (1.47) v--:-s .1M

Here

(1.48)

[ ,q(J () 2 ~ G G ( )r+i ( I' !,_ "r-- 2s) a l' (J ',s-[1 ; l) r ([1-1-1) rvJ a :n;..::::J J"vj savr - exp a - v - - Ip L. ]' (J '-"j=-2-)--r, j

(1.49)

G __ ,{f-V [(f-a)!U+o)!(J-a)! (JJ_V)!11/2 ./avj - 1 j! (J -a- il' (J +a- ill (a-v -\- ill

(1.50)

[t=ca-v+r+j;

.£lJif{f (tD. 0, - (D) is the matrix for the finite rotation of the rotation group (see [18 J); U (u ll , u) (eh a,

shacos8, shasin8costD shasin8sin(D).

The function (1.49) constitutes the matrix of a finite hyperbolic rotation of the Lorentz group [17, 19-21].

At the end of this chapter it should be noted that in order to construct functions which realize the rep­resentations of the Lorentz group (p, v) we have used the integral method. Other methods of constructing relativistic spherical functions have been expounded in [22-23].

6 . The T ran sit ion fro m C han n e Ito C han n elf 0 r S pi r al

Scattering Amplitudes

In modern elementary-particle theory the universality principle which allows the amplitudes of differ­ent channels to be associated with one and the same diagram plays an important role. When, for example, the contribution to the scattering amplitude in a given channel caused by the unitarity relation in the cross channel needs to be determined, it is necessary to make a transition from channel to channel. In the case of the scattering of spinless particles such a transition can be reduced to the substitution s = t or the sub­stitution of the variables u, t for s, t. However, in the case of the scattering of particles with spin it is necessary,in addition to performing the analytic continuation of the amplitude, to perform requantization of spins because of the transition from channel to channel.

The problem of the transition from channel to channel was investigated in [24-28] for the spin case. The simplest method of transition was proposed in [24]. However, in this paper the spins in the t-channel were quantized in different directions; this led to nonstandard equations and was corrected by E. L. Surkov [28]. In the present exposition we shall basically follow [28].

Let us consider a reaction of the type (s-channel)

I-I-2~3+4

77

Fig. 1. Kinematic diagram of the 1 + 2 - 3 + 4 process.

t

Fig. 2. Kinematic diagram of the 1 + '3 -2' + 4 process.

(the numbers denote particles). The law of energy-momentum conservation in the s-channel has the form

The spiral amplitude corresponding to such a transition is usually denoted by (P3A3; P4A4 I M I PtAt; P2A2)" Here A-i are the spiralities of particles which participate in the reaction. The spiral amplitudes of the reac­tion 1 + 2 - 3 + 4 are understood to mean the amplitudes defined in the center of inertia system. After this the quantization axes turn out to be rigidly fixed-they must always be directed from the center of in­ertia (point s in Fig. 1).

The spiral amplitudes depend on the invariants:

Figure 1 depicts the kinematic diagrams [24] corresponding to the reaction 1 + 2 - 3 + 4 in the velocity space. Points 1 and 2 are the velocities of the particles before scattering; points 3 and 4 are the velocities of the particles after scattering.

The pOint s is the center of inertia of the reaction and corresponds to the vector S = P1 + P2; the point t corresponds to the vector T = P1 - P3 (here we have depicted the case when the vector T, just as the vector S, is time-like, which is possible in the case of unequal particle masses).

Continuing the spiral amplitudes of the physical domain of the annihilation channel (the t-channel: 1 +"'3 - 2' + 4), we obtain amplitudes which are quantized as before relative to the point s (the spirality does not change under these conditions). The center of inertia vector of the t-channel now becomes the vector T = P1 + P3' while the point s turns out to be outside the kinematic quadrangle (P2 - - P2' P3 - - P';0 (Fig. 2). If we wish the spiral amplitudes to be quantized relative to the center of inertia in the t-channel itself, then we must carry out a Lorentz transformation from the point s to the point t'. Under these conditions the amplitudes are transformed by means of the matrices D~J.! (g) according to the equation

(1.51)

Let us note that gi == (0, ai, 0). Using the Lobachevskii trigonometry formulas it is very easy to determine the rotation angles~. From Fig. 2 we find that:

O _ ch (1,2) ch (I, 3)-ch (2,3) - (2-1-3)'1 ~t- -~ ,

sh (1, 2) sh (1, 3)

cos O2 = -co:(~24); cos 83 = -cos(l34); ~ cos04 =cos(342); J

( b ) _ ch (ab) ch (be) - ch (ae) cos a c - sh (ab) sh (be) .

(1.52)

78

Let us write out the explicit expression for cos 8 j :

We consider the case when the vectors T, S are time-like. But S may also be space-like (for example, in the case of equal masses). In this case the rotations can be determined similarly, since it is not the point s which is important to us but the direction to it.

II. FUNCTIONS ON A CONE

1. The Coordinate System, the Complete Sets of Quantum Numbers,

and the Basis Functions on a Cone

Let us consider the realization of representations of a Lorentz group using functions stipulated on a cone (k2 = 0) and let us calculate the matrices for a transformation between representations, which corres­pond to reduction into various subgroups [29]. It is obvious that the transition coefficients indicated do not depend on the method of realization of the representations, and therefore they may be used to obtain basis functions of the Lorentz group from the functions (1.48), which correspond to reduction into the subgroups 0(2.1), E(2),etc. The papers [30, 32] are likewise devoted to the calculation of the transformation coeffi­cients for a Lorentz group.

Let us choose the infinitesimal operators M and N to be in the form [33]:

where k = (ko, k) is a four-dimensional vector lying on a cone (Le., the momentum of a particle having zero rest mass). The basis functions of the Lorentz group are eigenfunctions of the Casimir operators .6. and .6.' defined by Eqs. (1.3) and (1.4) and of the two other commutating operators defined by choosing the sub­groups.* Let us introduce the coordinate systems on a cone and let us write out the corresponding diagonal operators and their eigenfunctions.

1. The S-System: the subgroup 0 (3) =:l 0 (2)

ko=e"; k2 =easinecoscp; } k j = ea sin e sin cp; k3 = ea cos e,

where - 00 < a < 00; 0"< 8 < n; 0 < cp < 2Jt. The diagonal operators are

M2_-M2 J1IA2 M2- ~ t 8~ __ I_~_j_- 2A ('~+A)' -- 1+ "2+ 3- - aU2 -c g as sin2S arp2 l+cosS 1 arp ,

having the eigenvalues J(J + 1) and M, respectively. The eigenfunctions are Wigner D-functions [18] D~A.(CP, e, - cp).

*These operators are the Casimir operator of the subgroup and the operator of its simple Abelian subgroup (here 0(2) throughout).

79

2. The H-System: the subgroup 0 (2, 1)::l 0 (2)

ko = ea ch ~; kz = ea sh ~ cos cp; } kl = ea sh ~ sin cp; k3 = eea,

where -oo<a<oo, -oo<~<oo, 0<cp<2Jt ,and

e= { +1 -1

for

for

The diagonal operators are:

and M3 = i a~ + A. having the eigenvalues (q2 + 1/4) and M, respectively (here q is real for unitary repre­

sentations). The eigenfunctions H2 and M3 are

where d~}~) (ch~) can be expressed as follows in terms of the Jacobi functions [34] ~~g (ch ~):

d\ln(ch~)=~~,"(ch~) for e=+I;

d~I," (ch~) = ~~Mdch~) for e = - 1.

3. The O-System: the subgroup E (2) ::l 0 (2)

where -oo<a<oo; O<r<oo; 0<cp<2Jt.

The diagonal operators are:

M3= i :'fJ + A..

The eigenvalues 0 2 and M3 are 1'(.2 and M, respectively, where I'(. is real. The eigenfunctions

Here J M +:.\ (I'(.r) are Bessel functions.

4. The C-System: the subgroup 0(2) 00(1, 1)

ko=ellch~; k2=eacoscp; kl = ea sin cp; k, = ea sh ~,

where - 00 < a < 00, - 00 < ~ < 00, 0 < cp < 2Jt. The diagonal operators are M3 and N 3 in thi s case and have the eigenvalues M and T. Their eigenfunctions are

80

Calculating the Casimir operators fl. and fl.' in all of the coordinate systems on the cone, we find that they have an identical form:

Solving Eqs. (1.1) and (1.2) with allowance for (2.1) and (2.2), we find that for unitary representation A == I),

while exp (-1 + ip)a is the portion of the basis functions depending on a.

2. The Matrix Elements

In various physics problems it is required to know the matrix elements of the transformation from the basis functions corresponding to the reduction of the Lorentz group to one chain of subgroups, to basis functions corresponding to a different reduction. In view of the fact that such matrix elements do not de­pend on the method of realizing the representations, they are simplest to calculate by means of the functions on a cone.

1. The Matrix Elements of the S-C Transition. Assume the basis functions in the S-system are (a8rp I pvJ M), while in the C-system they are (b~rp I pVT:M). The matrix elements of the transition from the S­system to the C-system are determined by the following integrals:

r d3k (pVT:M \ pvJ M') ~~ J Ii; (pvr:M I b~rp) (a8rp I pvJM).

Having expressed the variables in the C-system in terms of the variables in the S-system, eb = ea sin 8, th ~ = cos e, and having substituted the explicit form of the basis functions, we obtain

+00 n 2n (pVT:M \ pvJM') = N sNi:: ~ e2a da J sin e d8 J drp el-l-iP)bel-HiP)ae-i'\;e-ilV--M')'PeilV-M)'Ppitv (cos 8)

o 0

n

= N sNi:: ~ sin 8 d8 (sin 8)-HiP (tg ~ ) i, Pit" (cos 8), o

(2.3)

where NS and NC are the normalization factors of the functions (a1trp \ pvJM) and (b~rplpVT:M) respectively.

After calculating the integral (2.3) and substituting the values of NS and NC, we finally obtain

r C+M-~+iT:+iP) r C+M+~-i'+iP)

r (M-'O+I) r (I +M+ip) (pvT:M I pvJ M') = (j ,2ip.lilM-V)' / (2J + I) (J -'0)1 (J +M)!

MM V 2(J+'O)!(J-M)1

The matrix elements of the transition are calculated analogously [29] in the remaining cases; here we shall present only the final results.

2. The Matrix Elements S-O:

J-H (-)n(J+M+n)! (%2 )M+iP+n , iM- v ( (2J+I)(J-M)!(J+M)!)112 '"

(pvxMlpvJM)=(jMM'Jt2' x 2 (J+M)\(J+'O)I ~ (M-'O+n)!(J-M-n)!nlr(M+n+ip+1) KV-iP-n(X). n=O

Here Kv-iP-n (x) is a Macdonald function.

81

3. The Matrix Elements S-H:

(pvJM I pvqM') = fJMM , [J (v)+ (_1)2\J-M-V) J (-v)l,

where J (± v) denotes the following expression:

" " r(lj2-iq+v) J (± v) = :n:2iM 'f\> 2J+lq-l/2 I - I

r h-+ M-iq) r (J+M+I) r (J =+= v+l) r h-+iq-M)

I -./"(J+v)I(J+M)I, 1" r(J+M+n+l)r(J=+=v+n+l)r (}-M+iq+n')r ({+iq±v+n')

Xr (I +" )r( I +" +" )V (J±v)I(J-M)1 ~(- ) r(M+v+n+l)n!n!r(I-M±v+n')r(M+v+n+n') 2 lq±V 2 lq Ip nn'

f + M + iq -f- ip + n + n'; -1).

4. The Matrix Elements O-C:

5. The Matrix Elements H-C:

(1/2 M · 1+ . I+V_M_iP+i,) - -lq, 2 'V-lq, 2

X 3F2 M+""· 'V - M + 1, 1 - iq + v - 2 Ip -IT; 1

6. The Matrix Elements O-H:

(pvxM I pvqM') = bMM , 2:n: 1/;( IJ I (x) + J 2 (x)J,

where J 1 and J 2 are expressed as follows in terms of the Meyer G-function:

( I M+v x O~! ~2 0, I +. +.

I -2 lq Ip-n,

( /1 M. ) 020 ~ 2"+ +v-Iq-n

X 13 4 • M+v+n-ip, 0, M+v

82

Let us likewise note here that for unitary representations the matrix element of the inverse transformation is evidently defined by a Hermite-conjugate matrix; thus, for example, from the matrix elements (S I 0)

and (0 I C) one can obtain (oS I O):S I O),,~ (S I C) (C I 0) , etc.

III. THE CONSEQUENCES OF RELATIVISTICALLY

INVARIANT EXPANSIONS

1. Asymptotic Integral Representation of the Amplitude

and Quadratic Integrability

In order to obtain the asymptotic expansions we shall start by writing the amplitudes in the form

-I-~

f( E 0) f ~(_)l 2alm(P)1'(-ip)P-l/2~ie(Cha)y (e )d " (P ~2l2JT)" J L.J P J'(-ip-i) -Vsha m, <jl p,

- 00 1m

(3.1)

where E = ch a; e is the scattering angle; the angle cp has been written for completeness; p characterizes the unitary representation of the Lorentz group; the coefficient aZm (p) is determined by the expression

'1/2 . -1/2-1 (h ) J::::-)I_~~J.'J.'..r'2. r f( 0 (J) P-l/2+ip cay (0 ) h2 d dQ

21' (ip-I) J a, , I Vsha m, <jl S a a .

The function for which the direct and inverse transformations (3.1) and (3.2) are valid must satisfy the following conditions:

a) it must be quadratically integrable

where x is the ensemble of variables a, e, cp, and dx~' 5h2 adadQ is a volume element;

(3.2)

b) it must realize a regular representation Tg of the group of motions on a hyperboloid, which is iso­morphic in the Lorentz group [34]:

(3.4)

Assume E - 00 for fixed e. If the integral and the integrand function are uniformly converging functions, then the limit symbol may be carried over into the integrand (which is usually what is done; see [35, 36]). Using the asymptotic representation of the associated Legendre function P~ (z):

-1/2-1 1 {r(_iP)e-iPln2z r(iP)eiPln2Z} P-l/2-iP(Z) ~ -V2nz r(-ip+I+I) + r (ip+I+I) , (3.5)

we obtain

00

const i f (E e) ~ A (e, p) eip In 2E dp, asym ' ~ l;r -V y £ £2-1

(3.6)

where

~ 1 1 r (ip) 12 al (p) p2 A(e, p)= k.J (-) r(-ip-/)r(ip+I+1) PI (e), (3.7)

83

where A (8, -p) =I=- A* (8, p). It should be noted that in other coordinate systems a Fourier expansion of the type (3.6) is likewise obtained. Since In2E -- 00, for fixed 0, it follows that due to oscillations in the ex­ponent integration with respect to pin (3.6) must actually be limited to the range of small values of p. Taking this into account, as well as the cross-symmetry conditions in the form [37]:

7 (E + iO, 8) = 1 ( - E - iO, 8) = 1* (- E + iO, 8), (3.8)

it is not difficult to obtain [38, 29] a relationship between the scattering amplitudes f(E, 0) and f(E, 0) of a particle and antiparticle, respectively, on the same target: namely,

II(E, 8) lasym= 1 {(E, 8)lasym, (3.9)

which represents the contents of the Pomeranchuk theorem [35] in the case 0 = O. Quadratic integrability off(E, fJ) requires that /f(E, 0)/ decrease more rapidly than liE at infinity. If quadratic integrability of the amplitude f(E, 0) is rejected, then the partial amplitudes A(p, 0) become generalized functions (for example, 6-functions and their derivatives [38]).

2. The Relationship Between the Lorentz- and Regge­

Amplitudes

Following [39, 40], let us consider the expansion of the amplitude in representations of the Lorentz group which correspond to reduction to the subgroup 0(2.1). Previous papers [12, 41] have been devoted to analogous problems. This may be done most simply if the indicated expansions realized on basis functions of the Lorentz group, which are defined on a cone [42], are considered. In order to obtain the parametriza­tion corresponding to these expansions it is convenient to construct the isotropic vector:

(3.10)

where sh A (t) = (PiPO) while m2 '

Pi =m(cha ch~, cha sh~, 0, sh a)

if the momentum of one of the particles participating in the reaction (i = 1, 2, 3, 4); Po = m(O, 0, 0, 1) is the momentum of the origin of the Breit system. Having placed ch a = ea, we obtain

k iO = mea. (ch~, sh~, 0, I).

The corresponding equations for the expansion of the amplitude in irreducible representations of the Lorentz group have the form* [42]

~+ioo L+ioo

f(a,~)= J da(a+l) J A(a,1)e-(Ja.(21+1)ctgnlPl(ch~)dl. ~-ioo L-ioo

(3.11)

Let us recall that the unitary case corresponds to (J = -1 + ip and 1 = -1/2 + ia. The Mandelstam variables sand t are related to a and {3 in the following manner:

*Since the amplitude is scalar, it follows that the expansion occurs only in degenerate representations having v = O.

84

It may be shown [12, 42] that the Regge representation of the amplitude [12, 43] in the t channel is the analy­tic continuation of the portion of the expansion (3.11) corresponding to expansion in the subgroup 0(2.1) from the s-channel to the t-channel. Taking account of this, we find that the partial amplitude a(l, t) in the t-channel is related to the Lorentz-amplitude A(u, l) by a Laplace transform (a Fourier transform in the unitary case) [42]:

6+ioo

a(/, t)=2icosnl ~ da(a+ l)e-aaA(a, I). (3.12) 6-ioo

Going over to the unitary case (0 == -1, u == -1 + ip) in (3.12) and integrating zero to infinity with respect to p (see [5]), we find the form of the Lorentz-amplitude which generates a pole of the function aZ(t):

A(p, I)=+exp{± ipf(/)},

where/(l) is an arbitary functiont having Im/(l) ~ O.

3. Lorentz- and Regge-Poles

(3.13)

In the paper by 1. S. Shapiro [43] it was shown that the Regge asymptotic behavior of the amplitude sl(t) corresponds to the pole of the partial Lorentz-amplitude in the p-plane.

D. V. Volkov and V. N. Gribov [44] found whole families of Regge trajectories. Then Toller et al. [39] proved in the case t == 0 that the pole in the Lorentz-amplitude generates an entire series of equidis­tant Regge-poles (daughter trajectories). The case of an infinitely small transferred momentum was con­sidered by Salam [40]. Other papers [45] have likewise been devoted to an investigation of daughter trajec­tories; in these papers a classification of Regge poles is given in certain models according to the Lorentz­poles. The simplest method of showing the development of daughter trajectories consists in the following. Let us consider the scattering amplitude /(s, t) of two scalar particles having identical masses m. The variables s, t shall be defined as follows:

S=(Pl-+pZ)2=2m 2 (1+cha 12 ); ~

t=(PI -P3)2=2m2(1-cha j3 ), ) (3.14)

where aik is the distance between the points i, k lying on a hyperboloid having a radius m (Le., p2 == m2).

Hereafter we shall consider the case t == O.

Let PiE{c:P 4 : p~-p;-p~-pi=m2}. Since/(s, 0) is a Lorentz-invariant amplitude (in the sense Tgf(s, O)=f(g-ls, 0)= f(s, 0), we shall expand it in Lorentz-invariant functions, namely, in elementary spherical functions which are defined as

• ;- 2 • sin pal2 __ ~ '1'* (PJ.) 'I' (!!2) V :ct psha12-~ plm m plm m ' 1m

where lDplm ( ~) is the function (1.7). (Here the subscript i denotes the number of particles.)

Therefore, one may write the following expansion for /(s, 0) (see [22, 43]):

00 00 i ·V2 sinpa 2 - i sinpa f(s.O)= A(p) -'-h- P dp= a(p)-h-pdp. . :ct ps a s a o 0

(3.15)

(3.16)

Here a12 == a. On the other hand, it is well known that for scattering of spinless particles the spatial portion of the momentum is determined completely by the two components; in other words, it may be assumed that

tIn (3.13) A(p, Z) should be understood in the sense of the generalized function.

85

Pi C {@3 : p~- p~- pi =" m2} (this means that /(s, 0) should be expanded in a scalar product of functions which realize the representation of weight I of the 0(2.1) (Le., it should be expanded in Legendre functions):

Pz(cha)= ~ cDfm (~~) cDzm (~), (3.17) m

where cD zm ( ~ ) = p;n (ch ai) eim(l'i, while I is a complex number. Thus,

e+ioo

f (5, 0) = 8~i 1 dla (l) P -l-dch a) I ctg :rl, (3.18) £-ioc

where

00

a(l)= 1 f(s,O)Pz(cha)(sha)da . (3.19) o

for the unitary case I = - 1/2 + iq. Let us substitute Eq. (3.16) into (3.19), and then we obtain the following result after performing integration with respect to a:

=

a(l)=2~i 1 dpa(p)~(l, p), (3.20) o

where

(3.21)

If it is assumed that poles exist for the function a(p) then one may obtain

a(/)= - ~ Si;JtJtl r C+I/iPIt ) r C+I;-iPIt)r (_1+2iPIt ) r (_1-2iPIt ) + f dpa(p)~(l, p) It L

by shifting the contour. Thus, from Eq. (3.22) it is evident that each p-pole produces a series of equidis­tant poles in the I-plane as poles of r-functions.

LITERATURE CITED

1. N. Ya. Vilenkin and Ya. A. Smorodinskii, Zh. Eksperim. i Teor. Fiz., 46, 793 (1964). 2. 1. M. Gel'fand, R. A. Minlos, and Z. Ya. Shapiro, Representation of the Rotation Group and the

Lorentz Group [in Russian], Fizmatgiz, Moscow (1958). 3. M. A. Naimark, Linear Representations of the Lorentz Group [in Russian], Fizmatgiz, Moscow (1968). 4. 1. M. Gel'fand, and M. 1. Graev, Transactions of the Moscow Mathematical Society [in Russian],

Vol. 11 (1962), p. 243. 5. 1. M. Gel'fand, M. 1. Graev, and N. Ya. Vilenkin, Integral Geometry and Problems of Representation

Theory Associated with It [in Russian], Fizmatgiz, Moscow (1962). 6. 1. S. Shapiro, Dokl. Akad. Nauk SSSR, 106, 647 (1956). 7. N. Ya. Vilenkin, Matern. Sbornik, 68 (100), No.3, 432 (1965). 8. G. 1. Kuznetsov, Zh. Eksperim. i Thor. Fiz., .21., 216 (1966). 9. G.1. Kuznetsov, Zh Eksperim. i Teor. Fiz., 54, 1756 (1968).

10. V. L. Ginzburg and 1. E. Tamm, Zh. Eksperim. i Teor. Fiz., 17, 227 (1947). 11. G.1. Kuznetsov and Ya. A. Smorodinskii, Yadernaiya Fizika,1!" 383 (1966). 12. J. F. Boyce, J. Math and Phys., .§" 675 (1967).

86

13. A. K. Agamaliev, N. M. Atakishev, and I. A. Verdiev, Yadernaya Fizika, 10, 187 (1969). 14. M. Jacob and G. C. Wick, Ann. Phys., 7,404 (1959). 15. Chou Huan-Chao and L. B. Zastavenko-;- Zh. Eksperim. i Teor. Fiz., 35, 1417 (1968). 16. V. S. Popov, Zh. Eksperim. i Teor. Fiz., 37, 116 (1969). 17. G. I. Kuznetsov, et aI., Yadernaya Fizika, lQ, 641 (1969). 18. A. R. Edmonds, Angular Momenta in Quantum Mechanics, Princeton (1957). 19. Nguyen Van Hieu and Dau Vong Dyk, Dok1. Akad. Nauk SSSR, 137, 1281 (1967). 20. I. A. Verdiev, Zh. Eksperim. i Teor. Fiz., 55, 1173 (1968); I. A. Verdiev and L. A. Dadashev,

Yadernaya Fizika, ,2., 1094 (1967). 21. S. Strom, Arkiv Fys., 29,467 (1965). 22. A. Z. Dolginov, Zh. Eksperim. i Teor. Fiz., 30, 746 (1956). A. Z. Dolginov and I. N. Toptygin,

Zh. Eksperim. i Teor. Fiz., 35, 794 (1968); 37, 1441 (1969). A. Z. Dolginov and A. N. Moskalev, Zh. Eksperim. i Teor. Fiz., 37, 1697 (1969).

23. M. A. Liberman, Ya. A. Smorodinskii, and M. B. Sheftel', Yadernaya Fizika, 1, 202 (1967). 24. Ya. A. Smorodinskii, Zh. Eksperim. i Teor. Fiz., 45, 604 (1963). 25. M. S. Marinov and V. I. ROginskii, Nucl. Phys., 49, 251 (1963). 26. T. Z. Trueman and G. C. Wick, Ann. Phys., 26, 322 (1964). 27. o. A. Atkinson, Crossing Symmetry for Helicity Amplitudes, Preprint RI 19-20 (1964). 28. E. L. Surkov, Yadernaya Fizika, 1., 1113 (1965). 29. N. A. Liberman and A. A. Makarov, Yadernaya Fizika, ~, 1314 (1959). 30. S. Storm, Arkiv Fys., 34, 295 (1967). 31. R. L. Delburgo, K. Koller, and P. Mahanta, Nuovo Cimento, 52A, 1254 (1967). 32. D. A. Akyeampong, J. F. Boyce, and M. A. Rushid, Nuovo Cimento, L1ll, 737 (1968). 33. J. S. Lomont and M. E. Moses, J. Math. and Phys.,,!!, 405 (1963). 34. N. Ya. Vilenkin, Special Functions and Theory of Group Representations [in Russian], Nauka, Moscow

(1966). 35. I. Ya. Pomeranchuk, Zh. Eksperim. i Teor. Fiz., 34, 725 (1968). 36. N. N. Meiman, in: Problems in Elementary-Particle Physics, Vol. 4 [Russian translation], Izd.

Akad. Nauk Arm. SSR, Erevan (1964), p. 258. 37. V. B. Barestetskii, Uspekhi Fiz. Nauk, 26, 25 (1962). 38. G. I. Kuznetsov and Ya. A. Smorodinskii, Yadernaya Fizika,,2., 1308 (1967). 39. M. Toller, Nuovo Cimento, 37, 631 (1965). A. Sciarrini and M. J. Toller, Math. and Phys., 8, 1252

(1967). M. Toller and I. Sertorio, Nuovo Cimento, 33,413 (1964). M. Toller,Nuovo Cimento, 52A, 671 (1968). 40. R. Delburgo, A. Salam, and J. Strathdee, J. Phys. Lett., 25B, 230 (1957). 41. T. Winternitz, Ya. A. Smorodinskii, and M. B. Sheftel', Yadernaya Fizika,1, 1 (1968). 42. M. A. Liberman, Yadernaya Fizika, 10, 882 (1968). 43. I. S. Shapiro, Zh. Eksperim. i Teor. Ftz., 43, 1727 (1962). 44. D. V. Volkov and V. N. Gribov, Zh. Eksperim. i Teor. Fiz., 44, 1068 (1963). 45. M. L. Goldberger and G. E. Jones, Phys. Rev., 150, 1260 (1966); D. Freedman and J. Wang, Phys.

Rev., 153, 1596 (1967); D. Freedman, C. E. Jones, and J. Wang, Phys. Rev., 155, 1645 (1967); G. Domocos, Phys. Rev., 159, 1387 (1967); D. Freedman and J. Wang, Phys. Rev. Lett., ~ 863 (1967); G. Domocos, Phys. Lett., 24B 293 (1967); L. Durand, Phys. Rev., 154,1537 (1967); J. C. Taylor, Preprint 19-16, Oxford (1967); M. A. Liberman, Yadernaya Fizika,~, 665 (1969).

87

MESON SPEC TROSC OPY

K. Lanius

A review of the present contradictory state of experimental research into meson reson­ances is presented. The simple quark model is used as a scheme of classification. Reli­ably established mesons are in general not considered. Problems particularly important for the further development of meson spectroscopy are discussed.

INTRODUCTION

Following the general classification, a distinction may be drawn between particles and resonances [1]. A state is called a "particle" if it is stable with respect to decay arising from nuclear forces. The decay of a particle may occur as a result of electromagnetic or weak interaction. A state is called a "resonance" if open channels of decay exist in the presence of strong interaction. The question "What is a resonance" has been theoretically discussed on various occasions. At the present time, this question is again becoming vital in view of the introduction of the concept of dualism [2]; however, it will not be dis­cussed in the present experimental review. Associated theoretical aspects are treated, for example, in a review presented to the Lund Conference on Elementary Particles by Greenberg [3].

Experimentally, resonances appear: a) as a structure (a maximum or shoulder) on the curve relating the interaction cross section to the energy; b) when analyzing the phase shift of the scattering process; c) as a peak in the effective-mass distribution in many-particle final states.

In accordance with the laws of conservation governing strong interactions, resonances obey the law of conservation of quantum numbers.

In this review we shall use the following notation for quantum numbers: the barion number B (for mesons B = 0), the hypercharge Y = B + S, where S is the singularity (for mesons Y = S), and the isospin I. The third isospin component Iz is related to the charge by the Gell-Mann-Nishijimarelation Q = IZ+Y/2. Other notation includes the spin J, the inner parity P, the charge parity C, and the G-parity G (only for mesons with Y = S = 0).

The mass M and half width r are specific properties of a resonance; they should not depend on the reaction channel in which the resonance appears. Resonances may appear both in formation and in gener­ation experiments. Meson states may be studied in formation experiments involving pp and pn interactions by using the antiproton beams now available in large acceterators with nucleons as targets. Thus resonances with S = 0 and M > 2mp = 1876 Me V /c2 may be found in this way. Another possibility for studying meson states in formation experiments lies in the obtaining of vector mesons in electron-positron collisions (see, far example, the Novosibirsk experiment e+ + e--- pO __ 1T+1T- [4]). The majority of all known meson resonances have been identified in generation experiments. Great difficulties have arisen in analyzing the many-particle final states, owing to their severe complexity.

In this experimental review, firmly established mesons [1] are as a rule not considered. Chief dis .. cussion is centered around certain problems which are important for the future development of meson spectroscopy.

Institute of High-Energy Physics, German Academy of Sciences,Berlin-Zeutchen,German Democratic Republic. Translated from Problemy Fiziki Elementarnykh Chastits i Atomnogo Yadra, Vol. 2, No.1, pp. 129-230, 1971.

88

C 1972 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011. A II rights reserved. This article cannot be reproduced for any purpose whatsoever without permission of the publisher. A copy of this article is available from the publisher for $15.00.

TABLE 1. Properties of Quarks

Quark

ql =cp q2=n Q3="

Q

+2/3 -1/3 -1/3

+1/2 -1/2 o

B

+1/3 +1/3 -1-1/3

v

+1/3 -1-1/3 -2/3

1. UNITARY SYMMETRY AND MODEL

OF QUARKS

All the barions and mesons so far reliably identified support the validity of the SU(3) classifi­cation. Hadrons may be disposed in singlets, octets, decuplets, antidecuplets, and 27-plets. The state of one particular multiplet has the same spin and the same parity. If unitary symmetry were exactly

satisfied, the states of anyone multiplet would have the same masses. Starting from certain assumptions as to the disruption of SU(3) symmetry, we may derive a formula for the mass of the mesons in one octet (Ge 11- Mann - Okubo formula):

(1)

The indices relate to the isospins. Since there are singlets and octets with equal spins and the same parity, the I = Y = 0 components of the octet \fI8 may interfere with the singlet \fI1. This mixing may explain the deviations from the Ge11-Mann-Okubo formula for the mass in known meson octets. Starting from the requirements of normalization, the observed physical particles >¥ and -¥, are expressed as

1 'I") = cos 81 'I'I) + sin 81 'I's); }

1 'I') = - sin 81 'I'I) + cos 0 1 'I's). (2)

Here () is the mixing angle. The following relationships exists between the masses of the physical particles M and M' and the masses M1 and M8 of the states without mixing

M2+M'2=M~+M:;

2 M'2_M; tg e= M2_M2 •

8

(3a)

(3b)

It follows from this that the sign of the mixing angles cannot be established simply from a determination of the mass. In addition to the classification SU(3) we may, for example, also draw certain conclusions regarding the branching coefficient in the decay of the states belonging to one multiplet. We find that the form of decay of a meson in anyone nonet is a function of the angle of mixing.

The simple model of quarks [5] has been very successful in the classification of mesons and barions. Following the hypothesis of Gell-Mann and Zweig, a basic triplet of quarks (q1, q2, q3) is considered to exist; the properties of these are indicated in Table 1.

In the simplified model of quarks, it is considered that meson states consists of strongly coupled qq systems, similar to small-particle diatomic molecules. We may therefore expect that these quasimolecular systems will have forms of excitations similar to those employed in describing the lower states of diatomic molecules. The quantum rotational excitation is characterized by the orbital moment L of the qq system; the quantized radial excitation is characterized by the vibrational quantum number n. To this we add the total spin S = 0 or 1 for a qq system consisting of two fermions. For specified values of n, L, S, together with J = L + S, we obtain in the general spectroscopic formulation (n2S + 1Lj), i.e., nine meson states 3 x 3 = 1 + 8. This corresponds to one meson singlet and one meson octet. In contradistinction to the SU(3) group, the simple quark model also reduces to the representations (1), (8), and (10), the latter for barions which may be constructed from the 3q systems. So far it has never been satisfactorily proved that the exotic representations (10) and (27) should be necessary for the description of the observed hadron states (see Ch. 6)

Since quarks are fermions, the parity of the mesons in the nonet is given by

p=(_I)L+l.

The charge parity of the I = Y = 0 state is given by

C=( _1/+8 .

For the Y = 0 state the G-parity equals

(4)

(5)

(6)

89

LHI I I I I Y?r4~5"5'7i++ I I I I

5, I I i I I I I I I I I I G I I r'4--.-'S-- I JI'C

I'?II 41 I I I I I I 2++3''3'-,'' I I I I

F iT' II I 3r I I I I I I I I I I I I I I I I I l--rr'r- I I I I I

o I I tTT;n1 II II I 2, i I~II : I II I

I 0"1"1'-2'.' I II I I : II I I I I I I

P I I I I I I 1[ I I I I I I

0_, I I I I II I I I I I I

OS.-L~~-L __ ~ ____ L-__ -L. -L1 ____ Li ~ ____ ~~I~~I o 2 J 4 S M: (GeV/c 2)2

Fig. 1. Schematic representation of supermultiplets.

The confi~uration of the ground state with n = 1 and L = 0 leads us to expect, two nonets, apseud~~calar nonet 11So with JPC = 0-+ in accordance with Eqs. (4) and (5), and a vector nonet 13S1 with J C= 1 • For an orbital moment L ;c 0 there may be four different nonets; these are indicated in Table 2.

For the case in which there is no radial excitation (n= 1) and the orbital moment of the qq system L = 1 we should have the four nonets as shown in Table 3.

With increasing value of L, we may expect a corresponding sequence of such supermultiplets, in each case consisting of four nonets. Starting from the corresponding wave equation for a nonrelativistic qq system, Dalitz took a suitable harmonic-oscillator potential [5a) and obtained a linear relationship between the eigenvalue of the energy of the system E 2 ex: M2 and its orbital moment L. If this were the only effect leading to the splitting of the mass, we might expect that the four meson nonets belonging to each value of L would be degenerate. Figure 1 gives a schematic representation of this interrelationship between Land M2 for a sequence of supermultiplets [6).

Since experiments do in fact reveal a certain splitting of the mass within the supermultiplet, a cor­responding assumption may be made, within the framework of the quark model, as to the existence of an interaction which disrupts the symmetry of the system [5). If, for example, we consider a super multiplet with L = 1 and the states of the four associated nonets with 1= 1 (Table 3), we may expect to find an approxi­mately equidistant splitting of the mass (D.M)2 Rl 0.3 (GeV/c2)2 between them experimentally. This splitting is ascribed to a spin-orbital interaction, so that the splitting of the masses is characterized by a coefficient

LS =i[J (J + 1)-L(L+ 1)-S(S+ 1)],

which takes the following values for the corresponding states within the nonet:

For L = 1 this corresponds to equidistantsplitting. With increasing L the relative distance between the two states with J = L becomes smaller by comparison with their distance from the states with J = L + 1.

Tensor forces would give a different splitting of the masses. With L = 0, for nonets 1So and 3S h the considerable splitting of the masses between the 7r and p mesons may be ascribed to spin-spin interaction.

The states of anyone nonet may be constructed from a quark-antiquark pair in the following manner:

TABLE 2. Properties of Nonets with L ;c 0

Quantum number

90

J

P C

S=1

- I - - -isotriplet: pn = n+; -V- (pp - nn) = nO; np = n-; 2 _ _

doublets with singularity: { P~ = K+; ~A = ~o; Ap=K-; An=Ko;

I - - -lsoscalar of octet: -Vii (pp + nn + 1..1..) = 'Ils;

I - - -isoscalar of singlet: va(pP +nn+2AA) = 'Ill •

In the abbreviations the usual symbolism has been used [1].

TABLE 3. Properties of Nonets with L = 1 As already mentioned, interference may occur between the two isoscalar states. Since the mixing angle is expressed in theformtan8 = tg8 0 = 1//2, with the he lp of Eq. (2) we obtain

Quantum number

0++

:1/)1

I" 1+-1-- -

lJ' = V2 (pp f- nn); 11 ~= - 'A'A,

i.e., an isosinglet which consists solely of pp or nn quarks and an isosinglet which consists solely of lI.'A quarks. The angle 00 = 35.3° is called the ideal mixing angle. If there is no disruption of symmetry, all the states within anyone nonet should have equal masses. The mass of the octet Ms should also be equal to the mass Ml of the singlet. If we allow for the disruption of symmetry within the nonet, then we may suppose that quarks with singularity have a constant excess mass over and above that of the p and n quarks:

Hence

and

However, the last formula is the Gell-Mann-Okubo mass formula, Eq. (1). In the case of ideal mixing we shall have

MTl,=M" and M~=M~+2~.

If each nonet is characterized by the mass Ml of the unitary pure state of the singlet, by Ms, the mass of the unitary pure state of the octet, and by .6., the universal excess mass, then the masses of the nine states should satisfy the Schwinger mass formula on eliminating these three parameters:

(7)

Here M'o and M'b are the masses of the physical 1= Y = 0 state, Ml is the mass of the state with 1= 1, Y = 0, and Mil is the mass of the state with 1= 1/2, Y = ±1, while, I'is the overlap integral between the singlet and octel states. For M j = Ms, I' = 1.

Subsequent experimental data relating to meson states will be considered within the framework of simple mode I of quarks. Despite its simplicity, this model has re mained re markably successful in describing hadron states up to the present time.

Pseudoscalar Vector nonet nonet

JPc~o-+ JPc-r-

ISO Iqq) "SI(qq)

2,0 ---[°(1420) I I I I I

:1-1,5 I I I U I .......

> I Q) I

Sl,O I -1/1

-Xoor I

Ti' I

"'- I =/(* ::t I = I =w I - -:P I~ 0,5 I

Ti I ~I II

---I( 1~~: '"

0 :It ~I -1 0 -1 0 Jz

Fig. 2. Schematic representation of the two L = 0 nonets.

II. MESON STATES WITH L 0

2 .1. pseudoscalar Nonet

The ascription of a 'Ir- meson triplet, a K doublet, and an rt meson to the JP = 0- nonet has been known for a long time. For the ninth isoscalar meson, however, there are two candidates: an X" meson and an EO meson. The pseudoscalar nonet is shown schematically in Fig. 2.

a. XO Meson. The mass of this equals M= (985 ± 1) MeV/c2, r < 4 MeV/c2• and it decays mainly into 7J 7r7r and 'IT''IT''Y.

The once-controversial question as to whether, owing to the equal masses of the XO and 0:, the XO meson belonged to the isospin triplet has been unambiguously resolved by experi­ment. The XO meson has 1= 0 [7].

The determination of JP so far made admits no unique identification.

The predominant form of decay of the XO meson is XO -. 7J 7r7r • For this form of decay the Dalitz diagram gives an

91

rI ·a ;::l

oS '"

"0 c

a

.;! c ;::l

.d :;

b

., e .,

N~ ~4-+-~~------~ ~

Ol >< ·s !

·-1,0

~ Ol >< ·s ~ ~ __ ~ __ ~~~~ __ ~L-

1,0 ~ -1,0

Fig. 3. IT/relationship: a) 0+0- uniform (A) and linear (B) quadratic matrix elements; b) simplest quadratic matrix element for the other JP ascriptions considered.

approximately uniform density distribution. It follows from this that the spin and parity should lie in the abnormal series JP = 0-, I+, 2- • A number of old references are cited in [1]. Dufey and others [8] pub­lished some brief results of experiments with a spark chamber of 1.5 GeV/c relating to the reaction 'II"-p -nX° -nT/°'ll" +'11"-. The author ascribed some 75% of the 392 observed events to this reaction. Analysis of the Dalitz diagram gives a probability of over 70% for JP = 0- on the assumption of a linear dependence of the matrix element on the kinetic enegry T of the TT/T/-mesons. The ascription JP = 2- also cannot be excluded. The TT/ relationship, however, eRables us to eliminate JP = 1+ (Fig. 3). The determination of the spin and parity in the electromagnetic decay XO_p0'Y - 'II"+7!"-'Y follows from the angular distribution of 'Y. For JP = 0- or 2- we should expect that this angular distribution would be proportional to sin29 of (6 + sin29). Here 9 is the angle in the pO system between the 'II"+mesonansy. Existing experimentally data in no way contradict the two cases [9]; in addition to this, they allow for the possibility of JP = 1+, as well as the lower electric or magnetic multipole transitions.

The decay of a particle with spin 1 into two photons is forbidden. Bollini et al. [1] probably observed a XO- 'Y'Y decay when studying the 7!"-P -n + MM reaction at 1.93 GeV /c. Figure 4 illustrates the result. The upper distribution is the effective-mass distribution of all the 'YY candidates. The lower mass dis­tribution contains only those events which are coplanar, and in which the determinations of the mass from the recoil neutron and from the angle 'Y agree with one another. Five events were observed on a back-

~ r---------------~

8

IQ 0 ........ .:.=:...L...'--__ -'-__ ----L...L.....J

~ ~ 6.-------~-------. t 01 eVeht

'0 4

~ 2

°800

Fig.4. The M('Y'Y) of the 'II" -p - n'YY reactions at 1.93 GeV/c. The upper distribution shows all the events, the lower distribu­tion simply those for which the selection rules are satisfied (see text).

92

ground of one event. The probability that the peak might simply be a statis­tical background fluctuation is under 1%.

Thus experimentally the xO meson may have JP = 0- or 2-. Owing to the relatively small mass, the first of these is the more like ly. We may therefore accept that XO = T/' is the isoscalar of the 0- nonet and thus obtain a value of 9 = -10.4 ± 0.20 for the mixing angle 9 from the quadratic Gell­Mann-Okubo formula for the mass (1). The negative Sign follows from the quark model. The overlap function of the Schwinger mass formula (7) I'RlO.7.

An independent determination of this mixing angle may be carried out by comparing the production of XO and T/ in 7!" interactions [11]. Here it is considered that the quark and anti quarks interact independently of each other, i.e., their scattering amplitudes add. Since the 'II" system consists exclusively of quarks without singularity, we may expect that only such xO and T/ com­binations will arise which contain no quarks with Singularity. Using Eq. (2) we obtain

(8)

As proportionality coefficient we use a correcting factor, which is only determined approximately. Butler et al. [12] recently gave a new determination of R2; they furthermore obtained a mixing angle of 9 (+VR) = -29.00 ± 3.30. This fails to agree with the value of 191 = 10.4°, obtained

'0 o z

fOO

Fig. 5. Effective-mass ( KOK±rt'f ) spectrum in pp annihilation at 0.7 GeV/c. The continu­ous curve was obtained by the fitting method, without including the formation of resonances.

from the quadratic Gell-Mann-okubo mass formula (1). The linear mass formula, however, would give a better approximation 10 I = 23.7 0 ± 0.3 0 • At the Thirteenth C on­ference on High-Energy Physics in Berkeley, Dalitz [13] gave a determination of R which leads to a mixing angle agreeing with the Butler value. However, this once again indicated the dependence of this method of determining the mixing angle on the quark model.

b. EO Meson. This has a mass of M = (1424 ± 6) MeV/c 2 J r = (71 ± 10) MeV'/c 2 and decays chiefly into K*K- and KK 7r. The formation of EO in pp annihilation at rest leads to I = O. Lorstadt et al. [14] independently mentioned obtaining EOina pp annihilation at 0.7 GeV/c (Fig. 5). In addition to this, the absence of a peak at 1420 MeV/c 2 in the effective mass distribution of the charged KK7r system in this experiment confirms that 1= O.

It follows from the absence of an EO - K~ K~ 1'1"0 decay on the one hand, and from the observations of an EO - K~K~1'I"° decay on the other, that C = +1. Baillon et al. [15] fitted the experimental distribution in order to determine the spin and parity of the EO meson by taking the corresponding matrix elements for the decomposition K'*K; K'*K and (KK)7r; they obtained probabilities of 2 and 0.2% for pp - EOn- +1'1"- and 30 and 5% for pp - EO", ° for JP and 1+ respective ly.

If the formation of the EO meson takes place in different three-particle reactions during annihilation at rest, then the initial state pp has certain specific quantum numbers eSo or 3S 1). Analysis of the Dalitz diagram for the formation process (or analysis of the specific angular distribution) enables us, in any case, to determine the JP of the EOmeson. The result of the corresponding matching process gave approximately the same probabilities for JP = 0- and 1+ [15]. However, in the latter case it would be necessary to take an unacceptably large D-wave component of the 1'1"7r system (the energy of the particle is limited to 500 MeV). For pp annihilation at 0.7 GeV /c [14], analysis of the decay properties also indicated the preferability of JP = 0-. However, JP = 1+ cannot be excluded even here (Fig. 5).

If the EO meson is aSSigned to the pseudoscalar nonet, the quadratic Gell-Mann- Okubo mass formula (11) gives a mixing angle 10 I = 6.2 ± O.P.

Another possible interpretation for the tenth pseudoscalar meson would be its identification with the first radially excited state (n = 2) of the l S0 nonet. With Dalitz' proposed [5] harmonic-oscillator potential, we might expect an S state with n = 2 in the mass range 1600-1700 MeV. The corresponding states have never yet been identified. The quadratic mass excess of quarks with Singularity for the pseudoscalar nonet is equal to !::. = 0.23 Ge V2•

2.2. The 1- Nonet

Nine vector mesons, a p triplet, a K * (890) doublet, an wand a cP meson, have been reliably identified. The r nonet is illustrated schematically in Fig. 2.

Electromagnetic decay with the disruption of G-parity w- 1'1"+1'1"- may lead to interference with the pO _ 7r+1'1"- decay. Depending on acceptance of this interference, the percentage r:'ontent of w -'11'+'11'­

decays so far observed varies between 1 and 10% [1].

Goldhaber et al. [16] studied the reaction 1'1" +p - N * ++1'1" +11" - (6634 events) for momenta between 3.7 and4 GeV/c; they observed a fall in the effective value of the w meson (Fig. 6). The effect amounted to some four standard deviations. For the forward-emitted two-ion system (cos 0*> 0.95) the angular distri­bution of?!' + - 7r- is isotropic in the mass range 780-790 MeV! c 2, whereas in other mass ranges it is anisotropic. The authors in question explained these observations as being due to a destructive interference between w - 1'1"+1'1"- (2.7~~)% and p - 1'1"+11"- .

93

The quadratic Gell-Mann-Okubo mass formula gives a mixing angle of lei = 39.9 ± loP for vector mesons. This is close to the ideal mixing angle of £I 0 = 35.3 ± 1.10. Hence the <I> me son should consists soley of M quarks, and the w meson mainly of <pp + nn) quarks. The approximately ideal mixing is also indicated by the fact that Mw "" Mp and the overlap integral of the Schwinger mas~ formula (7) I' RI 1. This also explains the observation that the formation of <I> meson takes place preferentially in K-p reactions. Dahl et al. [17]compared the formation of <I> and w mesons in 7r interactions for momenta between 3 and 4 GeV/c. For the interaction cross sections of the reactions 7r-P -ncl> and 71'+n -pw a ratio of 1:50 was obtained. This also simply explained the observation that the decay of the cl>meson in 71" + 7r ~7r0(p7r) was strongly suppressed [1]. Among the equally valid, model-dependent hypotheses used for the derivation of Eq. (8), we may also mention the following relation:

(9)

Butler et ale [12] also made a new determination of this ratio of cross sections, obtaining values of e (+VR) = 33.1:!::~:~: and e (-VR) = 37.5±~:~:. These values yield no distinction between the quadratic and linear mass formulas (1), since the latter leads to a mixing angle of £I = 37.1 ± 1.1". The more accurate value of Butler et al. in no way contradicts the earlier measurement of R2 [13].

The same review [13] inllicates two new independent tests for determining the mixing angle. The first test is based on the measurement of the width of the decays r(cI> - P7l") and r(w - 371") [18], and leads to a mixing angle of £I = 390 ± 1, in excellent agreement with the mixing angle determined from the quadratic mass formula. The second test is based on measuring the width of the decays r(w - e+e-) and r(cI> -e+s-) [19]. This tests leads to a mixing angle equal to I e I = 35,2±U= !20]. The quadratic mass excess for quarks with singularity in the nonet of vector mesons is .t- = 0.21 GeV/ c Z• This value agrees closely with the .t-of the 0- nonet, in accordance with the requirements of the simple quark model

III. SUPERMULTIPLET WITH L = 1

To this multiplet belong the tensor nonet 3P2, with J PC = 2++, both axially vectorial nonets, 1P1 with J PC = I-+- and 3P t with J PC= 1++, and a scalar nonet 3pO with J PC = 0++.

3.1. Tensor Nonet

To the J PC = 2++ nonet we ascribe the A2 triplet, the K*(1420) doublets and the two isospin singlets, the f meson and the f' meson. The tensor nonet is illustrated schematically in Fig. 7.

The A2 (1300) meson was for a long time regarded as a reliably identified resonance, which decays into p7l', rrrr and KK and possesses the quantum numbers fiJPC = r2++. With the discovery of the splitting of the peak into two [21], a new discussion started as to the properties of A2• Thequestionas to the properties of the double peak of the A2 meson and the reasons for the structure became a central problem in meson spectroscopy. For this reason we shall devote some detailed consideration to the A2 meson.

Different forms of decay of the A2 meson have been studied in many experiments. By way of example, Fig. 8 shows the result of the study of the A2-meson decay channel carried out by the Aachen-Bertin-CERN group [22]. In agreement with other experiments, we see that, together with the principal decay A2 - P7r, there are also other decays A2 - 1/ 7r and A2 - KK. It follows first of all from studying the different channels of the decay of the A2 meson that the isospin should be 1= 1. It then follows from the A2 - p7r -

371" decay that the G-parity equals -1. The spin J and parity P of the meson may easily be determined from the existence of the decay A2 - KK. For the boson-antiboson pair KK the following relations between the quantum numbers are operative: G = (_I)J + I and P = (-I)J. It follows from the first relation that for the A2 meson J should be even. If we further consider the second relation, then we obtain the following possible quantum numbers for the A2 meson from the KK decay: JP = O+, 2+, 4 +, •.•• From the A2 -- p1r decay we have P A = P P P7r (-I)L = ( -1)L. Here L is the relative orbital moment of the system P7r. If we suppose that JA2=2(), then it follows that L can only be equal to 1, i.e., PA2 = -1. However, this means that JA2 = 0+ is forbidden, owing to the presence of the p7r decay. If we make the usual assumption that the lowest permitted spin is correct, then it follows that the A2 meson has JP = 2+.

In order to make sure that, in the A2 -KK decay, which is a fairly rare form of the decay in question (about 4%), the same A2 mesons are formed as in the :main channel of the decay A2 - P 71", it is natural to attempt a further determination of the spin and parity. Figure 9 shows the expected density distributions

94

200

N 150 o ~ oJ ;E o .... ...... '/3 100 c oJ > oJ

'0 o z

50

600 BOO

1f+P -- tJ" ~ n+ 1[ -

6634 even ts

1000 1200 11, Me VIr!-

Fig. 6. Effective-mass spectrum of (7r+1("-) from the re­action 7r+P -- N*++1("+1("- at 3.7 GeV/c.

on the Dalitz diagram for various values of JP. Since during the Ai -- p01("+ -- 1f+1f+')!"- process the two 11"+ mesons, together with the 7r- meson, may form a p meson, an interference effect may be visible in the range of overlapping of the p bands. We see from Fig. 9a that for JP = r destructive interference occurs in the overlap region, and the density at the edges of the Dalitz diagram vanishes. For JP = l+,JP = 2+ and JP = 2-, constructive interference occurs in the overlap region, the density at the edges only vanishing in the case of JP = 2+. Despite this last circumstance, the density distributions in Fig. 9b-d are similar. Figure 10 shows the experimental distribution on the Dalitz diagram obtained from a 1("+ experiment at 8 GeV/c. The p bands are superimposed on the Dalitz diagram.

In order to compare the experimental and theoretical distributions, we may project the p bands and compare the experimental values with those expected theoretically. Figure 11 illustrates this comparison. The application of the X2 test shows that the best agreement occurs for JP = 2- P, but JP = l+S and JP = 2+ are also possible. If we deduce from this that JP = 2- is the spin-parity for the A2 meson, then this deduc­tion will be inconclusive, since in comparing the theoretical and experimental distributions we have not allowed for the presence of a strong background in the A2 region. The two following illustrations should indicate the extremely strong effect of the background on the determination of JP. For different values of spin parity, a background component varying between 0 and 100% contributes to the expected theoretical distribution of the projections of the p bands. For every theoretical distribution thus obtained a comparison was made with the experimental distribution of the projections of the p bands by using the X 2 test.

The different values of X 2 so obtained are shown together with the corresponding probabilities for different JP states in Fig. 12a and b. In Fig. 12a it is considered that the background behaves as a phase space, Le., it has a uniform density over the whole Dalitz diagram, while in Fig. 12b it is considered that the background may be expressed by a one-pion exchange model. We see from Fig. 8 that the proportion of background in the A2 region is about 50%. However, we see from Fig. 12a that for 50% background of

2,5

~5 -1

-('

= K'(I4Z0) T

-A -($

o 1

L1z-D,29 J .

IZ

Fig. 7. Schematic represen­tation of the tensor nonet JPC = 2++3P 2(qq).

JP = 1+, JP = 2+ and JP = 2- would constitute the quantum numbers of the A2 meson with approximately equal probability, whereas for a background calculated by the one-pion exchange model in Fig. 12b, JP = 2- and JP = 1+ are the only possible quantum numbers. Thus in order to be able to determine the spin and parity of the A2 meson from the Dalitz diagram we must allow for the behavior of the background.

In a combined operation carried out in Berlin, Aachen, and CERN [23], a method differing from those just described was employed in order to determine the spin and parity of the A2 meson. This method may best be demonstrated in relation to an attempt at determining the JP of the A2 meson obtained in 1f+ interactions at 8 GeV/c [24].

95

b

c

;1(+11 (mass

5 ~ ~

d

fa ~ K+Kll. mass

Fig. 8. Spectra of the effective masses in the '" + interaction at 8 Ge V /c: a) '" +p -­P'" +p 0 - {11r +",+ 1'"'; N*++ excluded; 1685 events; curve calculated from the model of one-pion exchange; b) ",+p - P"'+1j-­P'" +",+. 'If" .,p; N*++ excluded; c) '" +p -p"'+1j' - p",+",-Zo, N*++ excluded; ZO in the region 1]; d) '" -Ip -- P K~o •

It is considered that this resonance is formed in the A2+ (1.22-1.38) GeV/c2 mass range with a density distribution of W(JP) in accordance with the specified value of JP (Fig. 9). The background should be fromed by the incoherent combination of the following processes: ",+p- p ",+",+",- and ",+p _PP 0",+; it is here considered that the first of these two processes gives a uniform density distribution W(p 0) (isotropic) in the Dalitz diagram, while for the second process the density distribution is given by two overlapping Breit-Wigner P bands in the Dalitz diagram. In accordance with the foregoing assump­tion, the following probability functions were calculated:

N

= ~ Jg [A W (JP) + B W(isotr.) -l C W (pO) -j i= 1 - S W (JP) dF .I W (isotr.) dF - S W (pO) dF _ ' (10)

where A, B, and C = 1 - (A + B) are the contributions of the corresponding components. The summation extends over all particles in the Dalitz diagram. Figure 13 shows the results of calculations relating to the probability function Wobtained by varying the value of A, B, and C for three different values of JP. The numbers in the corners of this isosceles triangle give the values of the probability function obtained on the assumption of 100% formation of resonances (top corner), 100% isotropic background cp (right-hand corner). and 100% overlapping Breit-Wigner PP bands (left-hand corner), re­specti vely. In the triangle we indicate the regions in which the probability function reaches its maximum value. If we compare the different maximum probabilities with one another, we see that, although JP = 2- is the preferred solution, the other two values of JP cannot be excluded, since the difference in probabilities is insignificant. The reason for this indeterminacy of the results may lie in the fact that our assumption regarding the background, particulary as regards the incoherent addition of the different components, is invalid.

The background is quite different if we consider the reaction ",+p _p",+",+",-",o, ",- in which theA2 is formed in com­bination, not with a proton, but with a proton and a positive '" meson, which in the majority of cases form N+++. Figure 14a illustrates the effective-mass distribution for the", +p

experiment at8 GeV/c. At the mass of the A~ meson a peak is clearly visible. If we sketch the smoothed background curve (shown by a broken line in Fig. 12a) and fit the peak to the Breit"Wigner distribution, we obtained the mass value MAO = (1317 :I: 8) MeV/c 2 and the half width r = (62 :I: 20) MeV/c2. For the mass range (1.28-1.36) GeV/c2 (rJoge 2) 129 events fall in the background and 72 in the A2 peak over the back­ground.

Together with the mass range 2, we also consider ranges 1 (1.12-1.22) and 3 (1.42-1.50) GeV/c2 as a control region. The Dalitz diagram for events of range 2 is given in Fig. 14. Analogous diagrams were also constructed for ranges 1 and 3. The method of determining the spin parity by calculating the maximum probability function earlier described for the case of A2+ was applied to the three ranges. The results obtained by this method [23] for three different hypotheses of ~in parity in range 2 are collected together in Table 4. From Table 4 we see an obvious preference for J = 2+, with a 33% proportion of resonances corresponding to 67 events in the range (1.28-1.38) GeV/c 2. The total number of events in this range is 201, so that 134 events belong to the background contribution. Table 5 contains the corresponding results for the two control regions. If we combine the two control ranges and make an interpolation into the

96

b ' . . ". "._ .... .. . "._._ .. .... , , .... __ ..... ..

' .. . __ n ..... . ' .. I_·_ .... uu~ .

• ·· .. - ·- '· ... I".· .. ~ ... .... _._, .......... . • ,..... . .. . 1 .. UI ___ .... Hn .. ..

·.11.11 . 111 .. • .. · · " · · .... .. _· ..... _ .. _ ... 10 .. · •• ·' .. • .. 111 •• 11 . .. ' ....... ,._ •. _ " ...... ", .. .. ···,, ' .. • ..... II1 ... ..... U-. .• _.'._un .... . · 1 101' .. • .. ' .. • .. ' .. . ·1, .. ' ... ' __ ,· .... · ........ ..

::! :: ~::~ ::;:::::;::~::::::=;:::.:;:: ::;:!::::!::. .lnu .. II .... ' ... II' .. "".,._u,.u ..... ... " ....... t .. ".

: ::::::;~~;!;~!;!::::!!::L~!=.~:ffia:!!i!:!!:'1:E. , ..................... _,. .. H._ .... ' ........... u_ ==-~;;;:::::~~::.~~::::=:~::~:::~:::::===-~:!:~~!~:~~::::~::::!:;!:::::::;::~::::=:=-... • ..... ".OI ......... ~n .. '".IIo.~ .... 'd ........ . . , ................... .. . )1· .... ·"·· .. ""'···· ... , .. _ .. ' .. " ....... .,""""'1111" ...... ..

1. 'Io' .. . . UU ...................... . . " .. 'lItI •• U ..... ' .. ' ... II' .. ... ' .. .

" """'"iii[,~r~'~ii:ii::'' ' '.

d

.. '""' .... _. .. , ..................... . ...... _._ ...... . .. .... _._ ... ..... . ...... _-, ........ . ' .. '1_·_ · .......... , . .. , ... --, ....... ~ ... ,.

:: ::::==:~:::::!:m: : ..

1111,"i,~t' ·'''i : il~lr~1,~I,i::,

""""" """11'" .... .. . 1601 ... 1 •• · .... ' • ",I •• #lIIIII·· ·ot,,·U., •. I"'u .... I··~ ..... ··OI •. . ... , ....... .. " .......... , . ""."U.U,·"'._J_'~ .• ,, . .. '1""'11 ...... ·._·., .. . . ................ · ..... U_ ... , . . .. ,,, ............ _oo:_n ... .

• ... '''.111 .. _ ... _._1'9 ... . 111.'U ... '.· ... . .. _I_ ...... ·'h , ............................ _ ... ..

• .. "OIU .. • .... ,.._.···_ .. N •• 111 "" .... • ... • .. ·"' __ ". ..... . . ...... U.H ...... ..... _.·_U .... .. .. ..... U.n ... n ••. __ f.·· ... . ...IIIII· ...... . .... w.· .. _ •. '."'. ,. , ......... _ ...... _ ...... ". ... ,.

• , ••.•• 1 .. . .... ' .. ..... - . ...... . .... , • . ,IIIII ... ' ,.. ' ... _ .. • ...... .. f .. oI. • . ......... H., .... .r .. . .... .. __ . 11 - .1."··· .... ·" .. ·_ .. · ..... ···_.·" • • ., ....... • .... _ .. • •• .. •• ... 1 ....

. 01 •. · •• ,.. •• ••• • "' .. , .... .. .. _.. .. ."'UIUU"IU ... . • .. '1_ · .. ·_ ... • .......... _ · · ... "III •• ••••• · . ... .. , ... . · , . .. w .......... . ... ,. . . ..... _.. • .. ...... . . .................. . . .. .. H / t • • M ...... _h} . . .. •• , •• ~ ...... U •••••• • • • t., ... . .." .... ... _ I." 'IOIt._eo\IU I '.'''W .......... ... " ......... . " •. • '11 . .. . . ........ _" ...... " , ... " . ... ___ .,01" 1 ..... ' ••• ". · I'.'.n_n o .. .. '"....... .. H· •..•.••• .. _· ... h .. · .• · .. ·" •.. · ........................ ,' ...... _·· .... ~_·jlotlU·.··.'.M·H.· · ,,, .. ,,,.U.I.""'11 • __ .... . _ ... _ _ .. ,.,. .... _ ....... ........ 01 ... . .. ' ,._ . ..... . _ • ...-..... ....... , _ .. " , . " :,~;I_t~,~~~:i

c

.. .......... . .... ,111 .. ' .. '

. ... .......... ". ".100) ........... ..

.. '"11.· ........ . .... 11' .. _ ...... , · ... 11.·_ ... · ..... , .' .... ,._' ......... ..

. , ::: :::H:E!:;;l::::;m~::: .. ... .. ... , ........................... ,. ,., .. ::: :: ::: : ::!:!:::::~:t.-=:::!:::~ :~: :::::i::: :.. . .

: :i;;mm~mg~ili~.:F.L'§gg~fdmg~mmm:t:::·· . :: 1~~gigmg~m~~~~~~i~~m~iliggll~ifi if;f1;i1:

· ·: :: ::::;:::i:~:::=-~:;:;::::::::::: : ::::::: : ::::·" .::::::H!::::=::::::::::::::: ..... , .... ,

·::::::::m::?::;::::::· .. .. . ' .. 10 ... · ...... ..

":::::::::Y'

a

Fig. 9. Expected density distribution on the Dalitz diagram for the Az -+ p7r decay making various assumptions regarding spin parity: a) J P = 1- ; b) JP = 1+ (I = 0); c) JP = 2+; d) JP = 2- (I= 1).

TABLE 4. Results of Fitting the Maximum Probability Function for Different JP Hypotheses in the Resonance Range (2)

Proponion Spin parity

I I resonance p+ pO

I uniform background

Probability function

1+ 0,09 0,22 0,17 0,52 3,34 2- 0,25 0,12 0,19 0,45 5.61 2+ 0,33 0,04 0,14 0,49 13,87

97

~r-----------------------------~

J

::-. 1,4- 2 u

>-Q)

2 A; 408 events

0 0,5 1,0 1,5

~

::: 5 c :> --0-

-E ~ (' ----- 2~ co I I - ·- ·-1-~ I I

-= I I . 0> J ~~ . > Q) ... I \ .... . . 0 ,.,..- ......... . '\ o 2 4 ,. . . ..... . .. . . Z // '\~. . . /

42 a5 1.0 ',4 Square ci the effective mass ('11"+'11" - ) . (GeV/ cz)t

o ~ 1,0 1,S Square of the effective mass ('11"+'11" - ),(GeV/c2)z

Fig. 10 Fig. 11

Fig. 10. Experimental distribution on the Dalitz diagram for the 1("+P experiment at 8 GeV/c.

Fig. 11. Comparison between the experiment data of the 1(" +p experiment and different JP hypotheses at 8 GeV/c.

98

80 \ \

20

o

a

V 2~

\ ~

\ I' ~. I I ......... /,/;1

.......

1 ?;­a

10 ~

50-8 c: o,s 1,0

Proportion of back wound

120

e-.g 100 ~

.::: 'S

Az b /l

, .... 2' / 1.·· .. · ....

~ ~~6c:-.s l' § '"C 10 ~ .. e rp __ so -8

x ~"! st o o,s 1,0 Proportion of background

Fig. 12. Values of X 2 and corresponding probabilities for different background contributions a) Background in the form of a phase space; b) background corresponding to the one­pion exchange model.

TABLE 5. Results of Fitting the Maximum Probability Func­tion for Different JP Hypotheses in the Two Control Ranges (Ranges 1 and 3)

Proportion

Spin parity

I I I uniform Probabili ty resonance p+ pO background function

1+ 0,20 0,07 0,14 0,60 79,0 2- 0,24 0,05 0,12 0,59 79,4 2+ 0 , 10 0, 18 0,10 0,62 78 ,0

1+ 0,0 0, 15 0,08 0,77 -61,6 2- 0,01 0,14 0,08 0,76 -61,6 2+ 0,07 0,10 0,08 0,75 - 61,2

resonance range, we obtain the expected value for the events in the background as 117. This value may be compared with the fitting result in the resonance range, for which there are 134 events in the background; both figures are comparable with the result obtained by fitting to the Breit-Wigner distribution, which gave 129 background events.

Y. 100

80

60

~O

1+S ~f, r

• s16 '~6 62,1 B2,J 90,9 • • • 68f 8f ~5 9;g

f'5; 91.8- maximum val. of probability

7~4 6f gS,4 !5,6

'f 9f ~, gi l

1P( 9~O • W ~ 'i'z 81~ 921 *'! 9 ~ 81t ~f

5070 f +S Z07.PP JOy. rp

81( Vf~ 95f ~ 8~ 't' ~4 si S 81, ~s ~6 9~f '!4 'f" '~ « ... 5" S~O

o 16 '¥ " If 151/ "It ro.: 59, 4 4Sf 2~ 6, pp tp

20

In order to verify the fact that the maximum probability value obtained for the hypothesis JP = 2+ (Table 4) is preferable to the maximum probability obtained for the other hypotheses, a technique based on the fundamental principles of Swanson [25] was used. The application of this method showed. that for all the various pro~oSitions made regarding the behavior of the background, the value of J = 2+ for the A~ meson may be regarded as the most reliable [23].

Another method of determining the spin and parity of the A2 meson was proposed by Morrison [26]. For the total cross section of all the two-particle and quasi-two-particle reactions, the following relation is 'approximately valid:

(J ~o const(potn.

Here Po is the momentum of the incident particle in the laboratory syste m and n is an exponent depending on the nature of the reaction • In particular it appeared that, in all the reactions capable of taking place with the exchange of a meson with singularity S = 0, the exponent n ~ 1.5.

In the reactions 11' +p - N ++ A~ and 1I'-p - nA~ exchange by a Pomeranchuk pole is impossible. Figure 15 shows the cross section for the formation of A~. In accordance with the mode of variation of the cross section in the presence of a meson-exchange process, we find n = 1.65 % 0.35. For the cross section of formation of a charged A2 meson in the reactions 11' +p - pAt and 11' -p - pA'2 we obtain an exponent value of n = 0.51 ± 0.20. These results suggest by way of interpretation that there are in fact two A2 mesons. One has spin­parity JP = 2+ and the other JP = 2- (or 1+). The latter resonanoe is formed with a constant cross section in the reactions in which exchange by a Pomeranchuk pole is allowed. The A2 meson with JP = 2+, on the other hand, cannot be formed in reactions involving exchange by a Pomeranchuk pole, but it may be formed in reactions in which meson exchange is permitted. A decisive proof of the existence of the second A2 meson was obtained by means of a missing­mass spectrometer in CERN [21]. This spectrometer was used to study reactions of the 1I'-p _pX- type. For this process, it follows directly from the laws of conservation of energy and momentum that:

(11)

99

100

.. 0

>-~ 50 co ~ 0

?r 1/ c t <) ;:. 0) ... 0

a

Pit; enter Into region N 0++

Ai ~ ~ ~ mr------------------,

5

2

~o ~4 1,8

11(1f;1fR~, GeV/ e2

b

1,28<I1(Ai)<~'!5 GeV If!

2 5 10 20 Beam momentum. GeV Ie

Fig. 14 Fig. 15

Fig. 14. Effective-mass distribution (71"+71"-71"0) from the 71"; experiment at 8 GeV/c (a) and Dalitz diagram in the A region (b).

Fig. 15. Cross section for the formation of neutral resonances in a quasi­two-particle process as a function of the momentum of the beam.

This means tliat MX is determined directly by measuring the momentum P3 and the direction (J of the secondary proton in the laboratory system for a primary momentum Po and a primary energy Eo. The spectrometer operated in region I (region of the Jacobian peak) in Fig. 16. Here dMX/dP3 R: o. It is only necessary to make an accurate measurement of the scattering angle (J. In 1965-1967 further measurements were made of the mass spectra in the region A2 for various initial momenta Po. with certain changes in the spectrometer. The transferred 4-momentum was varied over the range 0.21 :s It I :s 0.39 GeV/c 2: The mass resolution achieved r depends sharply on the P3 range chosen in the Jacobian peak method. The best resolution achieved lies at rexp = (16-18) MeV/c2. Figure 17 combines the data obta.ined with the best resolution in each case [27]. After this came the installation of a new device, the so-called CERN boson spectrometer, with a simultaneous change in working conditions. The boson spectrometer worked in region II as indicated in Fig. 16. The scattering angle (J R: 0° and dMx/d(J R: O. An exact measurement of the momentum P3 of the recoil proton was required.

In 1968 a number of irradiations were carried out with different initial momenta. The four-dimensional momentum transferred equalled It I R: 0.22 (GeV/c)2. The mass resolution remained at r R: 10 MeV for the most accurate measurements. Figure 18 shows the combined results [28]. An analogous picture is obtained for the mass distribution constituting a combination of the results obtained from the missing-mass and boson

100

TABLE 6. Results of Fitting the Maximum Probability Func­tions for Various JP Hypotheses in the Resonance Range [29]

Mass range, MeV/cz

I 1260-1360

I 1254-1307

1307-1360

J P

2+ 1-2-1+ 1+

resonance

0, 40 0 , :)0 0, 30 0 ,40 0 ,30

0 ,40 0 ,30

0, 40 0, 30

I

Contribution

II

0,20 0,40 0,40 0 ,30 0 ,40

0,10 0,40

0,0 0,40

I uniform background

I 0,40 0,30 0 , :)0 0,30

I 0 ,:30

0 ,50 0 ,30

0,60 0,30

I , (x')

:,8 0,1 0, I 0, I 0 , I

38 0, :)

54 10

spectrometers. Despite the large background, splitting occurs in this case. Assuming that both peaks may be ascribed to the same values of JP, we obtain an excellent fitting of the experimental mass dis­tribution on the following three hypotheses:

a. Two neighboring Breit-Wigner resonance occur, AIz with M = 1289 MeV/c 2, and A¥ with M = 1309 MeV/c2; in both r = 22 MeV/c 2.

b. As a special case of "a", the two-pole resonance is given by

N(M)<x (M-Mo)2t'2 _ 2

[(M-Mo)2+ T \'2] with Mo = 1289 MeV/c 2 and r = 28 MeV/c 2.

c. Destructive interference occurs between a wide A2 resonance with r = 90 MeV/c 2 and M = 1298 MeV/c2 and a narrow A2 resonance with the same mass and r = 12 MeV/c 2•

An experiment to determine JP was also carried out with the CERN missing-mass spectrometer [29a]. Using a large wire spark chamber, the decay angle of pions from the A2 decay was measured. Some 675 events were observed in the A2 range. A small selective sampling showed no splitting. Owing to the indis­tinguishability of the charges on the pions, the six-fold composite Dalitz diagram was analyzed by the method described earlier for determining the JP of the A2 meson. Table 6 gives the results of the fitting. We see that in both regions (AtandA~)JP = 2+ is preferable. This result was confirmed by a new inves-

o Recoil momentum PI

Fig. 16. Relation between the momentum of a recoil proton and the cosine of the angle at which it was formed in the laboratory system of coordi­nates for the reaction 1!"-P -­pX-.

tigation involving 3325 events in the A2 range [29b].

The first proof of the splitting of the A2 peak was obtained in an experiment with a bubble chamber by Crennell et al. [30], who studied some 10,000 events of the reaction 1!"-P -P1!"- + (MM)o at 6 GeV/c. Using the same kinematic conditions as in the missing-mass spectrometer, events were chosen in the region of the Jacobian peak and also in two neighboring regions. The resolving power at the Jacobian peak was r f::: 10 MeV/c 2• The 4-momentum transferred lay in the range 0.22 ::s It f ::s 0.40 (GeV/c)2. For events in this range the mass distribution illustrated in Fig. 19a shows a clear (though statistically not assured) fall. The neighboring regions show a weak A2 signal.

Figure 19b and c shows the observed mass distribution K~ K~ and k-ko1 in the same experiment [30]. The mass distribution k -ko1 shows a weak A2 signal. In the effective-mass distribution K~K~ there is a narrow peak in the region of A¥. Matching gives M(K~K~) = 1311 ± 5 MeV/c2 and r = (21 ± JO)2 MeV/c 2• The authors conclude from this that A¥ has a spin and parity 2+, while in the case of At we are dealing with another meson: JP = 1-, 3-, .... In his review contribution on meson resonances presented to the International Conference on High-Energy Physics (Vienna, 1968) [31], French gave the effective-mass distribution

101

N (J

~

500

280

260

240

~ 220

~ <:: ~ 200 III

'Ci o z

180

160

Best Breit-Wigner fit

/'1-1297 MeV/c2

r-U6MeV/c 2

r exp

Fig. 17

1300 MeV a

~ 500 p, - 2,55j2,60j2,65 GeV le!tp

400

b

500 p,-5,!J8jl,OGeV tc!1Cp

~ ~ 400 ::E 0 ... ....... ~ C III >-III

'Ci &1000 z

gOO

800

1200 1300 HOD Missing mass Mx. MeV/cz

Fig. 18

Fig. 17. Splitting of the missing-mass spectrum in the A2 region in different experiments.

Fig. 18. Combination of different missing-mass spectra in the A2 region: a) CERN boson spectrometer (1968); b) CERN missing-mass spectrometer (1965-1967); c) total of a + b.

of M(K~K~), M(K~K2D and (1/11"~; these are reproduced in Fig. 20. The distribution incorporates many different experiments with a bubble chamber. The three mass distributions show a clear signal in the A2 region. The width r ~ 100 MeV/c2. The mass in the M(K~K-) distribution is shifted to a slightly higher value.

In a further important experimental contribution to the study of the A2 region, the pp - K~ K~'f reactions were studied by Aguilar-Benitez et al. [32] at 0, 0.7, and 1.2 GeV/c. These data are presented in Fig. 21 for 3217 selected events. The mass resolving power is rexp = (7-10) MeV/c 2• The agreement between these results and two incoherent Breit-Wigner distributions [ML = (1281 * 3) MeV/c2 and MH = (1325 ± 3) Mev/c2 rL H = (22~1~) MeV/c2] is characterized by a probability of 28%. Matching with a two­pole resonance with Mo = (1303 ± 2) MeV/c2 and r= 211:4 MeV/c2 has a probability of 65%, while matching with a simple Breit-Wigner distribution has a probability of only 4%. This result, which contradicts that of Crennell et al. [30], again supports the ascription of JP = 2+ to At and AV .

Aguilar-Benitez et al. [33] recently published an analysis of the mass spectrum of K~K~ in the reaction pI> -K~K~'1I'+1I"- at 0.7 and 1.2 GeV/c. Here also two peaks appear (Fig. 22). However, the upper peak lies below 1300 MeV/c2, which contradicts the data of Crennell et al. [30]. The lower peak may be due to an f meson, which also has JP = 2+ and possesses a K~K~ decay. If we consider the great width (r = 145 MeV/c2)

102

a

25 "0 ·20 >' ~ 15 o .... ......

5 ~

1297 d

'Cl 0 '-,--'-:--'---' 7,2 7,3 1,', o

Z /'1, GeV lez

O'---~~~~_~~_~~-L-7~-L~WL~ 7,1 7,2 I,J I,It 1,5

o ..... ......

75

~ 5 5 ~

'Cl .8 0 ::> ::> Z

5

11(1<-+/'111), GeV lez

b ~ 'f.-1< -p-nK:K:

~ I1 -tr -p-X: K:~ neutrals

Fig. 19

1,.10

t 6

50 (~1()() -=~-

40

"0

JO

20

fO

200

>- 150

~ 0 100 C'I

~ 5 50 ~

1,JO

~

(-120

a

b

'Cl o '----"-L......I--.....I:.~-~-'-o o,s 1,0 }5 2,0 1,5 1,0 Z 11 (K, Kf), GeV/e I

100 1,10 e

75

50

25

1,0

Fig. 20

Fig. 19. Comparison of different effective-mass spectra in the A2 region: a) For effective mass 7f- + MM; b) for effective mass K~K~; c) for effective mass K-K~; d) combination of the effective-mass distributions M(K~K~) in Fig. 19b and c with the corresponding distribu­tion from 7r-P interactions at 3.9 GeV/c.

Fig. 20. Combination of different effecti ve-mass spectra in the A region; a) Total effecti ve­mass spectrum 7)0?r±; b) total effective-mass spectrum K~ K~; c) total effective-mass spectrum K~K±.

of the f resonance, we see that there is a possibility of interference with the A2 peak. There may therefore be a considerable distortion of the mass distribution K~ K~. The re suUs of an experiment with a bubble chamber carried out simultaneously by laboratories in Bonn, llirham, Nijmegen, Paris (E. P.), and Turin [34] were presented to the International Conference on Elementary Particles (Lund, 1969). In the ?r +p -p?r +7f +7r- reaction, the authors observed the mass distribution p~+ shown in Fig. 23 at 5 GeV/c. The mass resolving power was rexp = 5-10 MeV/c 2. The 4-momentum transferred lies in the range 0.1 < I t 1< 1.0 (GeV/c)2. Fitting to two incoherent Breit-Wignerdistriuutions [ML = (1275 ± 6) MeV/c 2, rL = (27 ± 13)

103

104

~Or-------------------------------------~

pp-KfK~1f1' at 0i0,7,,1,2 GeV/c

3217 events (K· excluded at 0.7 and 1.2)

~ O~~------~=-------~------~~~~~~~ 1,70 2,JO J,JO

Hl(KfK~), (GeV/c~)z

Fig. 21. Effective-mass spectrum of (KoK±).

~ 20

~ > 61 ... 0

0 10 z

0 ~10 1,70

/'fl(KfKf), (GeV/cZ) 2

Fig. 22. Effective-mass distribution of (K~K~) from the reaction PI> -- K~K~1r+1r- at 0.7 and 1.2 GeV/c.

2,0

1,0 1,1

J(p-pp(JJ(+--p~ff'tr 41</tl <1,0 (Gev/c)z

Fig. 23. Effective-mass distribution of (p01r ~ from the 7f+P -­

P1f+7f+1r- reaction at 5 GeV/c.

q 0 n

~2 1,J 1,4 1,5

Meff' GeV feZ

Fig. 24. Effective-mass distributions of (r}1r ~ and (K~1C) from the lI"+P experiment at 5 GeV/c.

Me V /c2 and MH == (1338 ± 4) MeV/c2, rH == (17 ± 5) MeV/c2] has a 70% probability. Fitting to a two-pole resonance with Ma == (1306 ± 4) MeV/c 2 and r == (41 ± 5) MeV/c2 has a 63% probability, while fitting to a simple Breit-Wigner distribution has a probability of 20%. Figure 24 shows the effective-mass distribution of 'f/7r+ and K+J<- from the reactions n+o-+ O'Y]n+and n+p-+ N*++J(+K- for the same experiment. This M(K~-) distribution, like Fig. 19, shows no asymmetry. New resul ts of Crennell et al. [35] were also presented to the Lund conference. These authors studied the K-n-+ ;\X- reactions at 3.9 GeV/c. Figure 25 a shows the effective-mass distribution of 746 selected events with :x- == pOll". Fitting to a Breit-Wigner resonance gave M (1::A;») = 1289 ± 10 MeV/c 2 and r ((A~») -< 40 MeV/c 2 • Figure 25 band c shows the mass distributions for 'f/ 7r - and K~K-, from which only the upper limiting value for the corresponding A2 decay may be estimated. Using the foregoing method of determining JP from the Dalitz diagram for the p7l" decay, these authors obtained JP == 1- with a 25% probability, and JP == 2+ with a 2% probability, for the (Ai') peak. Blumenfeld et al. [36], studying the formation of A2 in 71"- interactions at 3.9 GeV/c, did not find any narrow peak at 1300 MeV /c 2 in the M(K~K~) effective-mass distribution. The experimental mass resolution was rexp == 4 MeV/c 2. Figure 19 d combines the experimental M(K~K~) distribution of both investigations [36, 30] in the A2 region. The sharp peak at the mass value of A¥ observed by Crennell et al. [30] starts blurring on simply doubling the experimental material.

The question as to the splitting of the (KK) mass distribution in the A2 region was studied further in new experiments with the CERN boson spectrometer [37]. Using this spectrometer, 251 events of the 7I"-P -pAi' -pK-K~ -pK-7r+1I"- type were identified at 7 GeV/c. The mass resolution in the A2 region was rexp == 10 MeV/c 2• In the M(K-K~) mass distribution at (1300 ± 5) MeV/c 2 a sharp fall may clearly be seen (Fig. 26). Here there is a very slight background (~ 10 %), in the A2 region so that the possibility of an interference effect with the background may be excluded. From the new data regarding the splitting of the KK mass spectrum we may conclude that At and A¥ have the same spin and parity, JP == 2+.

Blumenfeld et al. [36] also studied the effective-mass spectrum M(7I" +71" -71" -) of the 1I"-P - p7l" +11" -71"­reaction for different ranges of 4-momentum transferred. By restricting consideration to events with a reliably identified proton (pp < 800 MeV/c), a high experimental resolving power was achieved in the A2 region (rexp == 3.5 MeV/c 2). Figure 27 gives the M(1I"+1I"-1I"-) mass distribution for three ranges of the 4-momentum transferred. A statistically justified structure is revealed (over 3-4 standard deviations). It is an interesting fact that the peak mass in the A2 region changes its position according to the t range in question. For 0.08 < I t I < 21 (GeV/c)2 the fall occurs in the A2 region at 1290 MeV /c 2, whereas for 0.21 sit I s 0.52 (GeV/c)2 a peak appears at approximately the same energy.

A more precise discussion will only become possible when bubble-chamber experiments with a high resolving power and simultaneous good statistics have been carried out.

A further proof of the dependence of the A2 effect on the 4-momentum transferred appears in a paper by Anderson et at. [38]. These authors used a miSSing-mass spectrometer at 16 Ge V /c to study the for­mation of A2 at an angle close to 1800, i.e., for small values of u. Despite the high mass resolving power rexp < 9 MeV,k2, no splitting was observed, but a peak appeared at 1295 MeV/c 2, i.e., at the mass value which exhibited a fall in the boson-spectrometer experiment [28].

105

30

20

~ 10 > ~ '" o o

.......

a 1289

~ r -pDJr­

(746 events)

1,JOO

~ 16

15

~ ~ g Or---~--~--~~~----L---llL--~

Q)

,x-·,,(550)'r ~ 10 > Q) b

LJr'''-{r 10 .;

....... ....... fI

(156 events) c Q)

o Z 10

> Q)

'0 d Z5

c ,x-·x:r

5 ....... (42 events)

0 (j.J events fi1 events

01100 O,G 0,8 I,D 1,2 1,1f 1,G 1,8 2,0 1200 1,J00 TWO H(KfKJ, MeV/c2

106

/tI, GeV/c2

Fig. 25 Fig. 26

Fig. 25. Effective-mass distribution: a) For P°'lf -. Events in the ~(1385+-) region are removed. The shaded region indicated the experimental resolving power in the A2 region. b) For 17 'If -.

Events in the ~(1385) region are removed. c) For K-K~. In graphs b) and c) the continuous curves give the phase-space distribution, in graph a) the phase space and two resonances.

Fig. 26. Effective-mass distribution of (K~K-).

a b c ~o

~O #J

"0

?> JO ,JO ~ 30

'8 ~ 20

20 20 t:J c ~ ! 10 0

0 Z

O/FOO 1,000 1,JOO 1,006 1,J06 1,606 1,000 1,,J00 1,600 H(7f+100, GeV/c2 H(I(+10t], Gev/c2 1'1(1(+1(-,0, GeV/c2

Fig. 27. Effective-mass distribution of ('If +11'-11'-) for various t ranges of the 1T-P - P1l'-1T-1T+ reaction at 3.9 GeV/c: a) I t I :s; 0.08 (GeV/ C)2. 944 events; b) 0.08 < I tl < 0.21. 1487 events; c) 0.21 < I t 1< 0.52. 1044 events.

L-1500

~ 800 ........

~ ~ .., 400 ... o o z

a b

"u120 ~ .., ~

o " ....... VV' Z

800 - - -"-12~(r··- 16jjj °1'200 UOO 1600 N(K~1T -}; Mev/ cz H(K~7T-), MeV/c2

"u JOO ~ .., ~

~200 C .., > ..,

'0100

o z

c

Fig. 28. Effective-mass distribution of (K+1f-) in the K+p-+ K+1f-1I"+1rP reaction at 12 GeV/c: a) All 27,000 events; b) K*(1420) region; c) K*(890) region.

If we consider the fact that investigations into other states of the JP = 2+ nonet have so far never provided any proof of splitting [39), the most probable explanation for the splitting of the A2 is a chance degeneracy of two independent states (I = 1 with JP = 2+), which interfere with each other. All observations lead to the conclusion that the two peaks usually exhibit the same intensity for different formation and decay processes.

Within the framework of the simple quark model, interference may arise between an A2 meson cor­responding to the 3P2 configuration and narrow 1= 1 qq states of the 3F 2 nonet for equal mass values [40). Such degeneracies require that the spin-orbital splitting should increase with rising L. Arnold et al. [41) considered the possibility that the narrow resonance interfering with the A2 meson corresponded to an exotic state JP = 2+, in which the I and (or) G differed from those of the A2• So far there has never been any proof as to the existence of the exotic meson (Chapter 6).

These and other attempts at explaining the effect of the splitting of the A2 meson are of a preliminary character. Further experimental data relating to the splitting phenomenon will have a decisive significance.

A recent extremely important experiment is that of Davis et al. [39). who studied the splitting of the K*(890) and K*(1420) mesons. These authors identified 27,000 events in the K+p -+pK+7r+7r- reaction at 12 GeV/c. The mass resolving power in the K*(890) and K*(1420) regions was determined as 5.1 and 6.5 MeV /c2. The effective-mass distribution of (K+1r-) shown in Fig. 28 does not show any characteristic peak structure. Division into different t ranges also fails to provide a single region with characteristic fine structure. The distribution was compared with the Breit-Wigner function and also with a double pole; the results are shown in Table 7.

The authors came to the conclusion that the K* (890) and K* (1420) mesons had neither a two-pole structure like the A2 mesons nor any Significant fine structure. The latest investigations into the spin and parity of the K*(1420) meson strongly suggest that JP = r [42); however, apart from the probable ascrip­tion J P = 2+, the ascription JP = 3- still re mains entire ly possible.

Two other states of the tensor nonet have been firmly identified. For the f meson we may mention simply the two la test investigations into its decay channels.

Ascoli et at. [43) studied the 1f-P -+ n1r+1T+1T-1T- reaction at 5 GeV/c and observed a peak in the 41T combination effective-mass distribution at M = (1270 ± 10) MeV/c2 with r = (90 ± 30) MeV/c2. If this peak is ascribed to the f meson, we may obtain an estimate for the ratio of the decay branches ([°-+271"+271"-): (/ .... 71"+71"-)= 10%.

Aderholz et al. [44) studied the 11" +p -+ 11" +pK~- reaction at 8 Ge V /c and observed a peak at the mass of f in the M(K~-) spectrum (Fig. 29). This leads to the cross section

(J (n+p -~ pn+fo ~ pn+K+K-) = (5 ± 2) tIb •

If we take a central value of the mass for all the resonances of the 2+ nonet, in the same way as for the A2 meson, we obtain () = 27.6 ± 2.3 0 for the mixing angle. This value is also not very far from the ideal mixing angle. The approximately ideal nature of the mixing also appears in the fact that the mass of the f meson differs very little from the central mass of the A2 meson (1297 Me V /c 2), and that the overlap integral of the Schwinger mass formu"la (7) takes a value of I' RI 0.92.

107

TABLE 7. Results of the Fitting of the Resonances

Breit-Wigger fitting Double- ~ole fi tting Resonance

M ,Mev/c21 MeV/c2 1 p (,,2), % M, Mev/czl MeV/c2 j p (,,2), %

K* (890) 895,7±0,5 53,2±1,6 42 I 16,3±0,3 10-1 0 I 892,3±0,5 K* (1420) 1421,1±2,6 101±1O 47 1421,7±1,3 27 ,6±1,6 1

Bassano et al. [45] compared the SU(3) predictions as to the decay branch ratio in the two-particle decay of the JP = 2+ state with the corresponding experimental values. For both types of decay 2+ -1- + 0- and 2+ -0- + 0+ matching gives excellent agreement with the measured values if we assume a wide A2 and a wide K* (1400). The quadratic mass excess of a quark with singularity is Do = 0.34 (Ge V /c)2 for the nonet of tensor mesons.

3.2. The Two 1+ Nonets

Within the framework of the quark model we may expect two axially vectorial nonets with J PC = 1+­and JPC = 1 ++. The total quantum numbers are not guaranteed experimentally for a single one of the possible candidates of the two nonets. In certain cases even the question as to the existence of the reson­ances in question remains open.

a. The Ai Meson. In the effective-mass distribution M(p1J") there is one peak at 1070 MeV/c 2• Tn some experiments a sharp peak appears. while in others the effective-mass distribution only shows a shoulder. In the generalized mass spectra of the three pions from the 1f±P reactions with various primary momenta so far encountered [31]. only one maximum projects above the strong background.

For low values of M(p7l') the maximum can only be repoduced from the kinematic effects of the periph­eral mechanism of formation.

Dekk suggested that the virtual exchange pion experienced diffraction scattering at the proton in the baryon point; he obtained a maximum at 110 MeV. However, allowance for a single p exchange with scat­tering at the baryon point and diffractive dissociation [46] does not lead to a quantitative reproduction of the experimental distributions of all the kinematic parameters in the Ai region.

If we use the Regge amplitude for the exchange diagram [47] we may describe some of the observations. For example, Fig. 30 shows the M(p\ -) distribution for the reactions 7!'-p - P p07!' - at 19 and 20 Ge V /c [48]. The continuous curve shows the prediction of the two-pole Regge model, normalized to the number of events.

From this we may tentatively conclude that the earlier idea of there being no resonance in the Ai region is probably incorrect [49]. Starting from the concept of dualis m, the behavior of the Regge ampli­tude in the t channel, i.e., the diffractive scattering of the resonance components, completely agrees with

formation of a resonance. Within the bounds of this

a b I

~ 20

~~~~~ oua~ruu=-~~~ 1,5 2,0 2,5 1.0 1,5

M(K+n,GeV/c2

Fig. 29. Effective-mass distribution of (K~-): a) 1J"+p -1J"+pK~'" reactions at 8 GeV/c; shaded region indicates events with M(p1J" +) in the region N*++; b) K-p reactions at 10 GeV/c; c) shaded region indicates events with M(pK-) in the A 1520) region.

108

representation, the Dekk effect and the formation of a resonance support one another.

These investigations into the At meson also emphaSize the importance of analyzing the background and determining its properties.

The best proof of the existence of an Ai meson is its observation in reactions in which a dominant kine­matic effect is least likely. An A1 peak was observed in two experiments on pI> annihilation [50] (Figs. 31 and 32). In the K+p reaction at 9 GeV /c, with five and six particles in the final state, Alexander et al. [51] observed a peak at. M = (1060 ± 20) MeV/c2, rising over a large background (see Fig. 33). Berlinghieri et al. [52] studied the reaction K-+P-K~p7!'+7l'-1J"Oat 12.7 GeV/c and found both At and A~ with M = (1030 ± 20) Me V /c2

(Fig. 34). Figure 35 shows the distribution of M(1J"+1J"-1J"-) in the 7!'-p - P7l'+7l'+7l'-7l'-1I"- reaction at 16 GeV /c [53].

50 b a '-J:P ~ ,,-

~ p p

~ JO JO

U) p-1JGeV!c p-20GeV/c

0 .:;

.......

~.zo 20

ii 't5

~ 10 10

Fig. 30. Effective-mass distribution (,rr-po) of the reaction n-p - p p07r-. The continuous curves correspond to the two­pole Regge model, normalized to the number of events.

Here in addition to the AI peak at M = (1055 ± 6) MeV/c 2 we also have the so-caned AI.5 peak at M = (1177 ± 8) MeV/c 2• The next example of a 1Cp experiment in which both an AI and an AI.5 peak may be observed in the many-particle final state is that of Ascoli et aL [54) at 5 GeV/c. Figure 36 shows the effective-ma ss distribution M(o°7r-) for reactions with four and five particles in the final state. The fact that an AI. 5 peak arises at around 1200 MeV in channels in which the Dekk effect is capable of taking place is demonstrated by one of the latest experiments (Fig. 27) [36). It is well known that the diffraction peak of quasi-two­particle reactions may, to a first approximation, be described by du/dt = K exp (At). For the formation of AI' the coefficient has a value of about 10 for I ti s 0.5 Ge V /c 2 [55). It is found, in particular, that the differential cross section of the many-particle channel near the Aj mass range exhibits the same beha vior, with approximate ly the sa me value of A as in the resonance region [56).

m.-----~.-------------~ .. ~ > 60 41 Cl If)

~ ~o d .......

~ 40 .. .9 1l JO o u ... 0 20 d Z

10

Fig. 31. Effective-mass distri­bution of (o07r±) from a pp anni­hilation at 3 and 3.6 Ge V / c. The lower histogram corresponds to the case in which 7r0 does not enter into the 7r± 7r0 combination having the mass of a p meson.

Particularly interesting in this respect is the result of the earlier-mentioned experiment relating to missing masses [38]. In the reaction 7r -p - pA i the formation of resonance s was studied at approximately 180°, i.e., for small values of u back scattering. The dependence of the differential cross section on u m ay be also described in terms of da/du = K exp (Bu). In contradistinction to the case of forward scattering, we obtain B ~ 17 for the AI region, whereas for the background B~ 3 •

In all the observations so far made the existence of an Ai meson is very probable. It follows from the observed p7r decay of the AI meson that :rG = 1- and C = +1. Direct experiments aimed at deter­mining the spin a nd parity of the Aj me son give J P = 1 +; although 2-also cannot be excluded [1]. Ascoli [57] concluded from the angular distribution of the decay Ai- p 07r- that, for JP = 1+, the Aj meson ha d a large d-wave component, in opposition to the usual idea according to which the lowest states of orbital moment predomina te in the decay. If the Aj and Aj s meson and the coherent background have the same spin and parity JP = 1+, then the deviation observed in different experiments may be at least qualitatively explained by interference .

b. The B Meson. The B meson has a mass of M = (1221 ± 6) MeV/c 2, a width r = (1221 ± 16) MeV/c 2, and decomposes into W 7r. Hence P = 1+; since G = C(-l), C = -1.

109

110

240

200

160

HO

>: 'II

I.:>

-g,120 ~ ~

5 > <II

~ <580 Z

40

,4,(1080)

150

5J8 events 22tH combinations

1,0 1,5 2jJ Heff (pDrJ, GeV/c2

Fig. 32. Effective-mass distribution of (p07r::l:) in pp anni­hilation at 5.7 Ge V/c.

KOptr+tr+tr­

KDp tr+tr+tr"tr0

1,2 2,0 f1(1l1l1l), GeV/Cl

Fig. 33

2,8

.. ~ :> Q)

::s 0 300 11:1

'-::l c:: Q)

> Q)

~ <5 z 200

100

1500 2000 7.,00 JOOO f1(1l+f(+f(J, MeV/c2

(,)0

AU 1

t J4!l7 events

tpp "'-1,O GeV2

1000 1500 2000 2500 JOOO H(1l+:;()(o), MeV/cz

Fig. 34

Fig. 33. Effective-mass distribution of three pions in K+p interactions at 9 GeV/c; 1) From the six-particle reaction (7r+7r+7l"-); 2) from the six-particle reaction (7r+7l"+7l"-) and the six­particle reaction (7l"+7l"-7l"0), two combinations/events.

Fig. 34. Effective-mass distribution of three pions in the reactions: a) K+p -pK°7r+7l"+7l"-; b) K+p -pK +7l"+7r-7l"0.

100

50

.. ()

"> ~ ." 0 COl HJ 0

~ S '0 «I

At 16GeV/c%

a

b

(1(~1() into p

Ascoli et al. [57] studied the reaction n-p-pB- _ pu)1t- at 5 GeV/c. Analysis of the spin and parity of the J P decay, allowing for the polarization of the B meson, favors JP = 1+. Also possible, how­ever, are JP = 2+, 3-, .... On account of the absence of the KK decay, the B meson Rhould have JP.., r.

Bizarri et al. [58] studied the formation of the B meson in a pp annihilation at rest. The initial con­dition was the 1S0 or 3S1 state. The best fitting of the Dalitz diagram was obtained for a B meson with JP = 1 + or 1. Since existing observations exclude J P = 1-, the ascription of J PC = 1+- to the B meson is assured.

~ 50

c. The H Meson. As indicated earlier, the two isovectors with J PC = 1+- and J PC = 1++ may be con­sidered as assured. As regards the 1= Y = 0 state of the two nonets, the experimental situation is far less clear. A hint of an isoscalar state with M = 990 MeV/c 2

and JP = I+, 2-, •.. , which decomposes preferentially from P1f to 1T+1f -1f0, was obtained in 1T+P- and 1T +n experi­ments with primary momenta between 3 and 4 Ge V /c [59]. Other investigations, such as, for example, the reactions 1T+P -N*++1T+1T-1TO in the same experimental

8 0 ()

'()

0 z

0 \-uL...I...Jc.Li..LL...I...JL...I...JL...I...JL...I....L.l..~~~t:.~~~~~.a.L..Lj range[60] or the 1T+n reactions at 5.1 GeV/c [61], con­

(tr; 10 into p (ptr;) into N"H

20 ·

c tained no hint of an H peak.

Fung et al. [60] found that one p selection with a statistically formed phase space in the M (<<p»n) mass distribution led to a peak at 1 GeV/c 2•

Barbaro-Galtieri and Soding [62], in a critical review containing complete data relating to the H peak, considered that the decay Xo- 1T+1T-Y, which introduces

oLLLutLLLLLL~W:~;:Ellliili!fg~~~EH~.J a slight perturbation into the channel with 1fo, might 2 J -1 also contribute to the wide peak at 990 MeV/c 2• These

10

l1(tr+tr-'O, GeV/cz authors came to the conclusion that observations made

Fig. 35. Effective-mass distribution (1T+1T-1T-). up to the present time failed to justify the existence of

The continuous curves correspond to the fitting the H meson. Goldhaber et al. [63] at the Lund Con­

of the masses to the Breit-Wigner distribution, ference described their study of the H effect in approxi-

the background being drawn in by hand: a) With- mately 17,000 events of the reaction 1T+P _ p1T +1T+1T-1T0 out limitations; b) M(1T+1T-) in the region of the

at 3.7 GeV/c. Without p selection, but with a quanti-p meson; c) M(1T 131T-) in the region of p and tative consideration of the XO -1T+1Ty decay, these M(P1l'A+) in the region of N* ++.

authors analyzed the M(1r+1T -1T0) distribution and obtained a clear peak at M = 1000 MeV/c 2, r = 50 MeV/c 2: The effect lay outside the N*++ band. Figure 37 shows the effective mass distribution M(1T+1T-1TO). In order to reduce the background, the following events were here removed: a) Events involving the w meson; b) M(1T + 1T-1T 0) combinations, which correspond to the reflec­tion of an T/ meson and N*++ baryon.

The strong background prevents the determination of JP for the H meson.

d. The D Meson. The D meson has a mass of M = (1285 ± 4) MeV/c 2 and a half width of r= 34 ± 4 Mev/c2; it has been reliably verified in the decay channel D - (Kir1T)O (See Fig. 5). It follows from the decays observed that 1=0 and C = + 1. There are also some unreliable hints as to a DO ..... 6± 11'~ decay [64]. This unreliability lies in the questionable identification of the 0:1: meson (see Sec. 3.3). None of the experi-ments so for carried out has enabled JP to be determined with any certainty [I]. Lorstadt et al. [14] stud­ied the formation of a D meson near the reaction threshold in the pp"'" DOpo process. The angular dis­tribution of the decay of the pO meson in its rest system relative to the flight direction of the primary p depended on the spin and parity of the D meson. By analyzing the angular distribution of the decay and assuming a 50% phase-space backgro\lnd, these authors found a slight preference for JP = 1+ over JP = 0-. Even now the possibility of JP = 2- cannot be excluded for the D meson.

111

Fig. 36. Effective-mass distribution (p01T-) for the reactions 1T-P- P1r+1T-1T-- p1T +1T-1T-1r ° at 5 GeV/cj 62 > 0.1.

Fig. 37. Effective-mass distribution (1T+1T-1TO) for the reaction 1T+P --1T+P1T+1T-1TO at 3.7 GeV/cj W-. rr. 6** reflections removed.

f,320

150

100

o

J159 events

ImJ Itl.c:46(Gevld

1,780

Fig. 38

N*H excluded 1922 events

... o

Fig. 39

812 events

Fig. 38. Effective-mass distribution of (K1T1T)- for the K-p ---- pK-1T+1r- reaction at 10 GeV/c. The separ­ated Breit-Wigner distribution corresponds to the calculated proportion of the K*(1420) resonance in the region.

Fig. 39 Effective-mass distribution of (K+1T-1T+) for the reaction K*01T+""" K*1T-1T+ at 5.5 GeV/c. The con­tinuous curve corresponds to the model of diffractive dissociation.

112

::I ~ 20 > II)

'c;

o 0 z

1800 I'1(K·7r),MeV/c2

b

gO 180 ¢, deg

2600

... ~ 0 ~ JO c

II)

C>

;:; 20 ....... ~ c II) 10 > 0)

'S 0 z 0

Fig.40. The K-p reaction at 12.6 GeV/c. The con­tinuous curves, corresponding to the predictions of the two-pole Regge model, are normalized to the number of events: a) Effective-rnass distribution of (K*tr) for the K-p -K*op7f- reaction; b) angular distribution of the Treiman-Yang angles; c) dis­tribution with respect to the 4-momentum trans­ferred.

e. The Q Region. In the K+- p(K7ftr)± reaction, with a primary momentum between 3 and 13 GeV/c, the effective-mass distribution of (Ktr7f)± exhibited a wide maximum between noo and 1450 MeV/c 2• Inves­tigations showed that the possible resonance in U~is so-called Q region decayed into [K*(890)7f] and (Kp).

The contribution of the two axially vectorial mesons to the Q decay is hard to estimate, since strong inter­ference occurs between the two forms of decay. The Q maximum has' been interpreted as a) a kinematic effect, b) a wide resonance, and c) many resonances.

A general review of experimental data up to the beginning of the summer of 1968 may be found in work by Goldhaber [65] and French [31].

The K*(1420) meson decays with an approximately equal probability into K7f and K7f7f. In experiments with primary momenta below 5 GeV/c, the K*(1420) meson appears in the M(K7f7f)± distributions as a single peak. For higher primary momenta the meson is not distinguished any more clearly in the effective-mass distribution. Figure 38 shows a typical picture of such a mass distribution [66]. On the assumption of a branching coefficient of K7f/K7f7f ;:;;: 1 for the K*(1420) decay, we may easily estimate the yield of the K*(1420) from the (K7f7f) mass distribution. The cross section for the formation of the Q maximum remains almost constant as the primary momentum increases. The cross section for the formation of the K*(1420) is proportional to the P02. Hence the proportion of the K*(1420) meson in the Q region diminishes with increas­ing primary momentum.

The experimentally established properties of the Q maximum, such as its strong dependence on t, or the presence of the maximum in reactions involving charge exchange, emphasizes the similarity with the A j (A 1•5) problem.

Various authors have attempted to describe the Q maximum by means of a model of diffractive dissoc­iation in the same sense as the Dekk effect. Allowance was made in these cases for the elastic scattering of both virtualtr-(p) mesons and virtual K-[K*(890)] meson at a proton point. It was always hard to explain

113

0

~ 1,22 t 1,32 .... b c t <Il

30 > <Il ..... 0

0 Z

fO

f,8 /'I (K~1l'~JOJGeV/c2

Fig.41. Effective (K+1T+1T-) mass spectra for equatorial (a) and polar (b) decays in the K+p interaction at 5 GeV /c (upper histograms) and 5.5 GeV/c (lower histograms): 1:12pp > 0.1 GeV /c 2; I cos 0KK I < 0.6.

the observed peak completely by the Dekk mechanism. Figure 39 shows (by way of example) a comparison between the K*o(890)r+ mass distributions from the K+p --K+p1T+r reaction at 5.5 GeV/c [67] and the diffractive-dissociation model of Ross and Yam [46]. Andrews et al. [68] tried to interpret the Q region obtained from the K-p-- K"'o(890) P1T- --K-P1T+1T- reaction at 12.6 GeV/c by a two-pole Regge model [47]. Figure 40 shows a comparison of the model with experiment. A similar good agreement between the two­pole Regge model and the coherently formed (K-1T +1T-) system was found by Werner et al. [69] when studying the K-d --K-d1T+1T- reaction at 5.5 GeV/c.

The obvious success of this model does not exclude the possibility, within the framework of the hypoth­esis of dualisms, that one or more resonances may occur in the Q region. It is very difficult to draw any conclusion as to the existence of one or several resonances from the observed M(K1T1T)± distribution. There are certain indications that the structure of the Q region depends on the initial momentum, the amount of 4-momentum transferred, and the orientation of the K*(890) obtained from the Q decay. A comparison of the (K+1T+"" ) mass spectra of the K+p --K+P1T+1T - reaction at 5 and 5.6 GeV/c [31] (Fig. 41) shows how strong these dependences may be. The position of the peak changes, on the one hand, when the events are divided into eventswithanequatorial K*(890) decay (cos 0KK < 0.6) and a polar decay; on the other band it

114

.. -0 ...... > ~ c... o o ';;;

5 > ., ... o • eo o z

600

~ .1++ removed (7321 events)

2,0 1,0 4,0 I1(K1l7r) ,GeVjC2

Fig. 42. Effective (K7r7r) mass distribution in the K+p -pK+7r+1f- reaction at 12 GeV/c: 14,310 events altogether.

200

160

40 1, 2 1,4 1,6 11 2,0 N(K7m),GeV/c

~ 400

~ ~ . J60 o ...... E i20 ., > ., 't:i 280 <5 Z

240

200

160

120

e

"~1,~2---~~,4~~1,6----~Le---2~,o~ M(Km1),GeV/c2

Fig. 43. Effective-mass distribution of (K7r7r): a) in reactions K+p-K+p7r+7r- and K+p-Kop7r+7r0 at 9 GeV/c; the shaded region indicates events with the exclusion of N1 ~n; b) in the same re­actions with the exclusion of N1!t8+ and the inclusion of K~o or p c) in the reaction K+p- KOn7r+7r+ ; d) in the same reactions as a) b) but at 10 GeV/c; e) total distribution for a), b) and d).

115

Fig. 44. Dependence of the (K7l"7l") effective-mass spectrum on the phase angle for the coherent addition of two reson­ances in the Q region.

also changes when there is even a very slight change of initial energy. If we suppose that the reason for these considerable variations lies in an interference effect, then this must be extremely dependent upon the energy, so that the combination of distributions from different experiments with different initial momenta is not a very reasonable procedure.

Of the many experimental investigations which have been made into the Q region, we may take as examples two recent experiments with high statistics. Barbaro-Galtieri et al. [70] used a momentum of 12 Ge V /c to study 14,310 events of the K+p -- K+p7l"+7l"- reaction; they found no structure in the Q region, and only found one peak at 1300 MeV/c with a width of about 250 MeV/c2 (Fig. 42). Alexander et al. [71] studied 7577 events of the K+p-- K+p7l"+7l"- and 2272 events of the reaction K+p --Kop7l"+7r0 at 9 GeV/c. As indicated by Fig. 43a, the effective-mass distribution (K7l"7l") contains a statistically confirmed dip in the Q region. The two peaks have the following masses and widths M = (1260 ± 20) MeV/c2, r = (40 ± 10) MeV/c2 and M = (1380 ± 20) MeV/c2, r = (120 ± 20) MeV /c~ In combined investigations by laboratories in Birmingham, Glasgow, and Oxford, the same reactions were studied at 10 GeV/c; Fig. 43d gives the mass distribution. Figure 43e combines the two experiments and emphasizes the character of the structure in Q region.

For different initial momenta, the spin parity has, as a rule, been analyzed for different ranges of mass in the Q region. On studying the density distribution of the Dalitz diagram for the (K7r7r) system and analyzing the angular distribution of the (K*7l") decay we always obtain JP = 1+ for all regions of the Q maximum. In certain experiments the possibility of JP = 2- cannot be completely excluded.

If we suppose that twosingular axially vectorial mesons contribute to the Q region, then these may interfere with one antoher. The mesons be long to two octets which only differ in their charge parity. Goldhaber [72] studied the coherent combinationof two axially vectorial mesons with the inclusion of an K*(1420) contribution. Figure 44 shows the change in the shape of the mass distribution on changing the phase angle 4> between the two amplitudes. Only for a proper choice of 4> can we obtain a mass spectrum which approximately corresponds to the varying experimental data. It must here be remembered that the possible existence of a coherent background was not taken into account.

116

a 250 250

.200 ZOO

150 150

"'~ 100 TOO '" u '>-'" l)

50 50 'i' 0

0 '-

Cl 0 0 1,0 1,~ I.B '0 <II MZ(KIninO), (GeV/c2)2 .5 ..0 S b d 0 (J

4-0 200 ZOO 0

0 z 150

:00

50 50

Fig. 45. Effective-mass distribution of (K7T7r) for the reaction: a) pp -- K~ K~7r+7r-; b) pp -- K~ K~7r+7r- ; c) pp -­K~K:t:". 'f".o neutral (K7r7r) combination; d) pp -- K~K ±", :f'1TO charged (K7r7r) combinations. The smooth curves cor­respond to phase space.

The following result of Alexander et al. [71] is also quite interesting: when observing the Q -- (K+7r+ 7/')

and Q -- (K°7r+7r°) decays a correct description can only be achieved on the following assumptions.

a) Interference takes place between the [K*(890) 7r] and (K, p) decays of the two mesons in the Q region.

b) In addition to these decays, there is also one (K+€o) decay. Here €0(720) is an 1= 0 -s wave (7r7r)0 system (see s 3.3). Direct grounds for the latter proposition are provided by the (7r7r) mass distribution for events from the middle of the Q region. In the final state K+p7r+7r- the (7r+7r-) mass spectrum gives a peak at M ~ 720 MeV/c2 and r ~ 180 MeV/c 2• In contradistinction to this, the corresponding distribution of the final state KOp7r+7r0 exhibits a peak at M ~ 760 MeV/c 2 and r ~ 80 MeV/c 2•

In addition to investigations relating to the channels capable of exhibiting diffracti ve dissociation, a resonance I =1/2, described as a c meson, was observed in pp -- KK7r7r reactions. The annihilation took place in a state of rest. A more detailed analysis of the extensive experimental material was recently given by Astieretal. [73]. Figure 45 shows the (K7r7r)o mass distribution for various final states. In all cases a peak appears at 1250 MeV/c 2• A shoulder appears at 1320 MeV/c 2• In the (K7m)± mass distribution there is also a peak at 1250Me V;c 2• The fitting of all the experimental data to the simple model by the method of maximum probability only yields good agreement on the assumption of a (Krr7r) resonance with M = (1242~~o) MeV/c 2, r = (127:t-~ 5) MeV/c 2 and LJP = 1/2 1+. The authors further showed tha t the two decay amplitudes (K*7r) and (Kp) were approximately in phase. In the absence of mixing between the octets, we may conclude from this that the c meson belongs to the 3P t nonet with J PC = 1++. As before, the generalized experimenta l situations are contradictory. The Simplest interpretation evidently lies in the assumption that two JP = 1+ c mesons with C = ±1 and masses of 1240 and 1350 MeV/c 2 exist. Further experime ntal investigations are very deSirable, in particular in reaction channels without diffractive dissociation. In addition to this,

117

150

100

50

'0

60

20

60

150

100

50 II I', I I I I

r l : : L .... ,

,r' L

3,0 GeV/c II ~ excluded

3,0 GeV/c N" excluded

Combined data 11* excluded

0,8 1,0 1,2 1,4 1,6 1,8 H(Kftr+), Gev/c2

Fig.46. Separate and combined (K~7!'"1 effective­mass spectra of different K+p -K~1r+p experi­ments.

contradictory results are also obtained in regard to the existence of resonances with natural parity. Dodd et al. [74] carried out a generalization of four investigations into the K+p -+ pK~ '/!' + reaction with initial momenta between 3.0 and 3.5 GeV/c. Figure 46 shows the (K~7!'"1 effective-mass distribution. A peak appears at 1260 MeV/c2• Crennell et al, [75], studying the K-n -nK~7!'- reaction at 3.9 GeV/c with approxi­mately comparable statistics, observed a peak at 1160 MeV/c2, but found no effect at 1250 MeV/c 2 (Fig . 47).

If we suppose that the A1 meson is the isovector of the 1++ nonet and the B meson the isovector of the 1+- nonet, and that the quadratic mass excess of the two axially vectorial nonets has the same value of t:. = 0.34 (Ge V /C}2 as in the nonet of tensor meson, then, as the expected mass of the K* doublet, we obtain M R! 1220 MeV/c2 for the 3P1 nonet and M R! 1340 MeV/c2 for the 1P1 nonet. These values are close to the masses of the experimental candidates . It follows from this that any possible SU(3) mixing in the two K* states can only be slight.

From the quadratic Gell-Mann-Okubo mass formula (I), for the isoscalar states of the 1++ octet, the expected value is M R! 1290 MeV/c2• This almost agrees with the mass of the D meson. The D' meson, i.e., the second 1= Y = 0 state of the 1++ nonet, has so far never been observed. The allowed channe Is for the decay of the D' meson are KK7!', TJ7!'7!' and 471' [5a] .

118

Z7°f .L

180 JOH events

700

NU ,'1.0 ~

~ 7't20 ~ !O l ~ 0 ..... ~ 100 1100

t c Q)

> Q) ... 0

0 z

60 ....

GO

'to

o,ti 7,0 7,¥ 1,8 N(Krtr)1GeV/c2

aJ

00

'to

O,G 1,0 7,It 7,8 2,0 N(K: tr),GeV/c 2

Fig. 47. Effective-mass spectrum of (K~ 7r-) from the reaction K-n -K~7r-n at 3.9GeV/c (a); mass spectra for the half of the Dalitz diagram in which K~ is emit­ted forward (b) and backward (c) with respect to the K7r system.

TABLE 8. Investigations into the {j Meson

st""ti"1 Litera

M, 2 r, 2 Notes ture Reaction Decay MeV/c MeV/c assurance

I n-p -> pM- - 962±51 <51 4,7u 1 = I or 2 I [77j pp- -> dM+ - 966±8 <10 2,5u 1=1 [78J n-p -" pM- - 1)- not observed 1=1 (GeV/c)2 [79] pp--> dM+ - 1)+ not observed : [SO] n-p - --" pn-no nOno 965±101 10 I 2,5u t> [81]

> 0,5 (GeV/c/ n-p --;. pn-no '11-'110 0- not observed t-> [82]

> 0,4(GeV/c)2 K-p - -> ~±n'F n+n-no n+n':f n - I) ± not 0 bserved [83J K-p -> pK-n+n-no n+n-nO 948±1O <;: 40 2,5u Assoc. with [841

N>++ (1236\ K-p - c' An+n- MM «1l»n;- 980±1O 80±30 3,5u lJnreliable [851

'I]~ neu-tral

K-n - c· An-MM «l1»n- 980±10 60± 30 3,5u Unreliable [861 'I]~neu-

tral A -n ----? I.n-'I]c 'l]n- 0- not observed [871

-> An-MM «ll»n- ,,~n) as a kinematic effect -K-p -o> «r) >>rr:- 970± 15 <;: 50 4u on allowing [88]

----? L + (1385) n-MM for the kine-pp --> 3n-'3n-nO llcn ± 970 ~25 3u matic effect [891

119

~Or-------------~--------------·--------'

qL-~~~--~~~O----~1,~~,,~----~~71 ------l~,S~----~~~ I1'(KfKr), (GeV/C2)2

Fig. 48. Effective (K~K±) mass spectrum obtained from the annihilation reaction pp - KfK:!: '" at rest. The con­tinuous curve corresponds to fitting based on the coherent formation of K+A2 and K7T-S waves. The broadening of the curve reflects the variation of the K7T scattering length between -0.4 and 0.4 F.

The situation is also unsatisfactory in regard to the 1+- nonet. There is insufficient confirmation of the 1= Y = 0 state to support the unreliably predicted H meson. The observation of the 1= Y = 0 state is experimentally very difficult. On the one hand, there are twice as many of these states as there are of the corresponding I = I, Y = 0 resonances, on the other hand, recognition of the 13 = 0 state of the isovector is still awaited.

3 .. 3. Scalar Nonet

Within the framework of the quark model, the J PC = 0++ nonet also belongs to the L= 1 supermultiplet. With all its quantum numbers, not one of the states of this sPo nonet can be regarded as experimentally established. Observations are made difficult by the fact that a state with J PC = 0++ has very few strong allowed decay channels.

a. Isovector. Allowed strong decay channels for P = 1- include KK and r/1r. When studying pp annihilation at rest, Astier et al. [76] observed an S wave threshold effect in the KK system for the pp -K~Koi: ... + reaction. These authors showed that the threshold effect depicted in Fig. 48 could be explained neither by interference nor by the reflection of known resonances; tliey later showed that it could be described either by a resonance with M = (1016 ± 10) MeV/c 2 at JP = 0+, by a positive real scattering length of (2.5 ± 1) F, or else by a complex scattering length a + ib. The duly matched values of a and b indicate the existence of a narrow resonance slightly below the threshold, with M = (975 ± ::l~) MeV/c 2• If the latter proPQsition is correct, we may well be able to identify this resonance with the 0 meson; we should then have J PC = 0++. The 0 meson was initially observed in the missing-mass spectrometer as a narrow peak at (962 ± 5 MeV/c 2

[77]. Research recently carried out with spectrometers and bubble chambers has been characterized by contradictory results. Reference is frequently made to an investigation constituting an experimental con­firmation of the existence of the 0 meson, although the authors did not in fact take this meson into account. Table 8 gives a review of the results obtained.

This table is incomplete; other papers are mentioned in the earlier review [1]. In the interactions [85] and [86], a neutrally decaying 1/ meson was "identified" by cutting out a corresponding .r.ange of the mass spectrum of unobserved neutral particles. Crennell et al. [87] used a similar method of identification and also found a peak due to the neutrally decaying 1/ meson at 980 MeV /c 2• Events with a reliably identified 1/ - 1f+1f- (7TO or 1'), however, did not give any peak at 980 MeV/c2 in the (&7",-) mass distribution (Fig. 49). The authors showed that the «(1/» 1f-) peak might be attributed to the intrusion of (P7TO_ 7T-7T07TO) events at 1 GeV/c2•

Barnes et al. [88] allowed for this kinematic effect in studying the K-p - ~+ (1385)1f-MM reaction and came to the different conclusion that an effect of the 0-- 1/"'- type was taking place. Investigations

120

20

~ 200 ~ ... 8 ~ 150 1 5 ~ ... o o z

a

o

K"d-psA1r-"c ~ Ps visible 47 events

K"d~ps"7(- .. neutral

c

10

0 .. d ~ ~¥O Cl

co 0 0 Ii 30 5 > ... ~ 0 20 z

70

0 0,2 0,'< 0,0 1,0 1,2

/1 2, (GeV/ c2l 11, GeV/cz

1 • . , f

173 events

879 events

1,0 7,8

Fig. 49. Effective-mass distribution of (7r + 7r - (7r0 o!: ')I» in th~ K-d - PSA7r - TJc (a); effective- mass distribution of neutral particles for the reaction K d -+- PSA7r + neutral (b); effective-mass dis­tribution of (7r-TJd, the continuous curve corresponding to phase space (c); effective-mass dis­tribution of (7r- «TJ»N) for the second reaction, the broken curve corresponding to the estimated background (d).

carried out by Defoix et al. [89] indica ted not only the existence of the 0 meson but also the possible existence of a decay chain:

The difficulty in interpreting the foregoing experiments lies in the different widths of the maximum ob-

0,1

0

<r;>

42

41

0

-41

+ +

+ ++t 0,8 +t+1,i 1,4't

K+p-K°7(°N*

+

tt +

48

t+++t

a

I I I 1,6 1,8 11 (K+,'-J, Ge V /c~

5rGev. c

+ b

Fig. 50. Course of the spherical func­tion of moment (Y~) : a) For K+ 7r­scattering; t - intervals weighted on the C hew- Low principle; t - interva l = 10 tlm2•

tained in measurements with spectrometers and bubble chambers, The most attractive explanation would be that an uP = 1 0+ me son with M:=:::i 975 MeV/c 2 and r = 50 MeV/c 2 existed, this having both a strong KK decay and a decay of the TJ1r type. It re mains to be proved whether this meson can be identified with the 0 meson.

h. K* doublet. The search for a scalar K* meson decaying intensively into K7r has so far yielded contradictory results. In discussing the Q region we already mentioned observing (K~1r*) in the mass spectrum. Dodd et al. [74] found indications of a peak at :=:::i 1080 MeV/c 2 in their review, as illustrated in Fig. 4. Crennell et al. [75}, on the other hand, observed a peak at :=:::i 1160 MeV/c 2, but found no effect at 1080 MeV/c2 (Fig. 47). A di:J;-ect determination of the spin parity was impossible in both cases, so that JP might be O+, r, 2+, ...•

Trippe et al. [90} studied K7r scattering in the reactions K+p -K+7r-N*++ and K+p -K°7r°N*++ at 7.3 GeV/c. Using pole extrapolation, these authors found that, in the part of the (K7r) mass spectrum between 1.1 and 1.2 Ge V /c 2• after a gradual rise. the phase shift o! = Of2 reached :=:::i 90°. If this effect corresponded to a resonance, it would have a width r :=:::i (200 to 400) MeV/c 2•

At the conference on 7r7r and K7r interactions held in Argonne National Laboratory in May 1969, Henri et al. [91} mentioned a corresponding analysis of K7r scattering in the reaction

121

120

100

80

,t:J60 8

2D

I I I , , , , \

I \ t I \

/ \ t 2

A''') \, .... ~+; + + o IV ... 0,8 o,g 1,0-- 1.1 -1,2 1,3

H (K~J(J, GeV/c2

Fig.51. Course of the total interaction cross section for (K+7r-): 1) Unitary p wave 4/9 12'lr>..2; 2) unitary s wave.

K+p -Kp7r7r at 5 GeV/c. Figure 50 illustrates the different behavior of the S wave in the mass range between 1.1 and 1.2 Gev/c2 by comparing K+7r- and K°7r° scattering. However, a T = 1/2 S wave resonance should lead to the same type of angular distribution of K°7r° as in K + 7r - scattering.

Also in dis~reement with the foregoing discussion [90] is the conclusion drawn in a subsequent investigation of K 7r- scattering [92]. The form of the total K+7r- cross section hardly shows any sign of resonance in the range (1.1-1.2) GeV/c2 (Fig. 51), and, although oJ/2 gradually reaches 90° at 1.1 GeV/c2,

it rises very little for higher mass values. The authors found it hard to explain this behavior as an indica­tion of resonance.

Finally we may mention recent work of Goldhaber et al. [93], who studied the angular decay distribution for the reactions K-p -nK-7r+ and K-n - N*-7r ~-. In both reactions the behavior of the spherical functions of moment (X~) agrees with the corresponding distribution of the K+7r- decay in the earlier papers [90, 91]. This independence of the decay symmetry relative to the mechanism of formation may serve as a strong argument in favor of the S-wave effect of K7r scattering.

Proof of the existence of scalar K7r resonance, however, will require a more detailed analysis with better statistics.

c. Isoscalars. A great deal of work has been done on the existence of one, or possibly two, mesons with ]G = o+, which decay into 7r+7r- and 7r07r0, and the mass of which lies approximately in the region between 300 and 900 MeV /c2 [1]. We may first of all mention two new investigations into the existence of the narrow (1 meson. Maglic [94] mentioned measuring the missing-mass spectrum in the dp - He3Xo reaction at the Lund Conference. A maximum was found at M:r: = (450 ± 10) Me V /c2 for large values of the transferred 4-momentum 1.2 :s I t(d/He3) I :s 2.0 (GeV/c) . ° The poor mass resolution (1':::1 80 MeV/c.2) prevented any determination of the peak width. In a generalization of the (7r+r) mass spectrum for approximately 13,000 events of the reactions 7r-P -n7r +7r- with I t I < 0.2 (GeV/c)2, l)].bal and Roos [95] found a maximum at M = (482 ± 3) MeV/c2; r :s 25 MeV/c 2• An effect amounting to 4-5 standard deviation was involved. Inves­tigations into the existence of the eO(720) meson are based, on the one hand, on an analysis of the phase displacement of the 7r+7r- scattering, and on the other on the search for a peak in the (7r07r0) mass distribution.

The following difficulties are encountered in determining the 7r7r phase shift 61 from measured data relating to the 7rN -7r7rN and 7rN -7r7rN* reactions:a)theseparationof'lr'lr scattering from inelastic processes; b) the fact that the angular distribution of the pions in the final state depends not only on the effective mass of the two-pion system but also on the amount of 4-momentum transferred t.

Thus, in order to separate the S-wave component of the background from the absorption effect, we require statistical sets exceeding the number of events so far obtained in experiments by an order of magnitude. It is therefore hardly surprising that the various kinds of analysis so far attempted, based as they are on approximating assumptions, lead to different results for o~, althougl.l in part using exactly the same experimental material.

122

180

160

HO

120

100

80

60 I 1=6' '>0 40 • -::±t~ " cit .g 20 "- i=- ' ~L r

.. " f 4:;:,~ • •

o a48 0,5211,56 0,60 0,64 o,M tV2 0,76 o,ao 4~ o,g2 ~ t,fH ys GeV -20 t::m~ . + ; Set I '

-40 , .tr. ~ • t - 6() "f -+fj" -80 pf.Xj -100 7dec~ I f

---o..--,r,-12rJ!l0 r:-" i ' \ ", I

--160

Fig. 52. Results of the determination of o~ in eleven dif­ferent experiments.

Figure 52, taken from Gutay's review [96], shows the phase shift o~ obtained from eleven different publications. The indeterminacy of the solution arises from the following causes, Owing to the invariance of the amplitude, in the transformation 01- o[ + nn a solution l' is obtained in addition to the solution 1. If we neglect the term responsible for the isotropy in the differential angular cross section of the nn scat­tering, then the analysis is based on terms which depend linearly and quadratically on the cosine of the scattering angle (). These terms, on the other hand, are invariant with respect to the transformation 00 - o'd = n/2 - (o~ - 0]). Hence a solution 2(0'd) is also obtained.

In order to discover which of the three solutions is correct, repeated analyses of the (nono) mass spectrum have been caried out. An advantage of the (nono) system is the absence of p resonance with I = 1. Hagopian et al. [97], discussing certain data relating to the (nono) scattering, came to the conclusion that these data were consistent with solution 1. This corresponds to the result obtained by the extrapol­ation method, i.e., an [0 meson with M ~ 720 MeV/c 2 and r ~ 140 MeV/c 2• Walker [98], on the other hand, considering all the (nono) data so far obtained, came to the conclusion that the results supported solutions 2, i.e., a very wide maximum of £0 between 700 and 900 MeV/c 2•

A very large number of matching observations debcribing the maximum in the I = 0 = (KK) mass distribution exists. Figure 53 constitutes a generalization of the experimental mass distributions prepared by Butterworth [20]. This J PC = 0++ threshold effect may be described either as a resonance S*, with M = (1062 ± 5) MeV/c 2 and r = (97 ± 11) MeV/c2 as weighted mean values, or else as a complex scattering length [99]. The difference between these interpretations cannot be resolved on the basis of the divergent experimental data so far obtained.

If the S* resonance were scalar, we might expect a more or less pure nn decay, depending on the mixing angle. On studying the n-p- nn+n- reaction, certain experiments gave an I= 0 peak at M ~ 1060 MeV/c2, The angular decay distribution of the corresponding events, however, was in no way opposed to the value J = 2 [20]. The (nn) maximum cannot therefore yet be considered as a confirmation of the S* (1070) resonance.

By correlating the results, we find that none of the candidates so far proposed for the scalar nonet can be regarded as assured. The quadratic mass excess obtained form the 0 (960) and K* (1080) is equal to A ~ 0.25 (GeV/c)2, i.e., it has the expected order of magnitude. On the other hand, the quadratic Gell­Mann -Okubo mass formula (1) gives M ~ (1120) MeV /c 2 for the mass of the isosinglet in the octet. This value exceeds both the mass of the £0 and that of S*. The mixing angle cannot therefore be specified. If

* we consider S (1070) as the isoscalar state of the nonet, then the Schwinger mass formula (7) with I' = 1

123

'" 5 iii

40

20

50

100

"0 10 o z

5

40

20

1,0

1,0

:Jfp- T/K7 Kf Hess et ai, 1,6-4,5 GeV/c

1,2

Huang et ai, 4 GeV/c

1,1 1,2

Huang et al. 5 GeV/c

1,0 1,1 1,2 /'1,GeV /c2

40

20

pp-KfKf f('Jt-

1,0

5

1,0

1,1 1,2

1,1 1,2

rN-+KtKf /'1 Ali tti et ai,

3,6-5GeV/c

Fig,53, Comparison of the (K~K~) effective­mass spectra in different experiments,

gives the expected value of M R:I 1320 MeV/c2 for the mass of the expected scalar S (-+~:) meson, On the

other hand, if we consider £0 as the isoscalar state of the nonet, then on the assumption of ideal mixing, we obtain M R:I (1280 Me V /c2) as the expected value of the S* (- KK) state.

The foregoing review of the four nonets of the L == 1 supermultiplet shows that, even for the mass range M ~ 1550 MeV!c 2, we need better experimental data as regards mass resolution and statistics before we can secure a satisfactory classification.

IV. STATES WITH A MASS BETWEEN 1550 AND 1876 MeV/c 2

A large number of investigations contain certain indications as to the existence of states with masses higher than those of the resonances hitherto discussed, Hardly any of these states have yet been reliably identified with all their properties,

On considering the various possible mechanisms of formation and the various decay channels open, we find it convenient to divide the states up in accordance with their masses M ~ 2Mp == 1876 MeV/c,

124

TABLE 9. Peaks in the R Region

Region

Rl (1630) >- 1 R2 (1700) :;;' 1 Ra (1750) 1

M,Mev/c2 j r, Mev/c2 j Decay into charged particles

1630± 15 1700± 15 1748± 15

< 21 < 30 <;: 38

1 : 3 : > 3 "'" 0,37 : 0,59 : 0,04 1 : 3 : > 3 "'" 0,43 : ° ,56 : 0 , 01 I: 3: > 3 "'" 0, 14: 0 ,80: 0 , IS

First of all we reca ll the r e sults of experiments with the missing-mass spectrometer in CE RN. For a very strong background (Fig. 54) three sta tistically assured narrow peaks with the propertie s indicated in Table 9 [100] are obtained.

To these we may add one further, le ss obvious peak at M = (1830 ± 15) MeV. The 4-momentum transferred equals 0.2 < I t I < 0.3 (GeV /c)2. In the earlier-mentioned experiment of Anderson et aL [38] relating to the study of missing masses, the formation of resonance s was studied at an angle of approxima tely 180°, Le., for small values of u. The authors also observed a peak at (1700 ± 47) MeV/c 2, although this had a width of r ~ 195 MeV/c 2•

The description of the results obtained with bubble chambers may be divided into state s with positive and negative G parity. The latest review may be found, for example, in [31] and [101].

.. 1100 ~ ~ ~1000 o

'S 0800 z

a

Combined data

7,O+'~5"'2,0 GeV/c

4 . 1 States with G = +1

a. The pN (1650) or g Meson. In the reactions 1f± - p1f± 11 ° and 11 - P -n11+11-, the two pion mass spectrum exhibits a peak at M = (1650 ± 20) MeV/c 2

with r = 120 ± 30 MeV/c 2• This is the so-called g meson, an isovector of the series JP = 1-, 3-, 5-, ... Figure 55 presents some condensed data regarding the mass and width of the g± and gO mesons obtained from various experiments [20]. It appears that the gO meson is slightly heavier than the g± meson. This may be expla ined by the presence of an isoscalar state in the gO meson mass range . There are experimental indications of a g - KK decay [102]. There are several methods of determining the spin and parity of the g meson. Thus Crennell et al. [102] analyzed the 71"-110 angular distribution by uSin~ a Legendre polynomial. The results are not entirely unambiguous, although J = 3- is preferred.

b. The p (1700) Meson. Figure 56 gives the total (411)± mass distribution from several experimental investigations into the 71"±P - P11±11 +71"- 11"0 reaction

700 [31]. For M = (1700 ± 20) MeV!c 2 andr = (110 ± 25) MeV/c 2 a maximum is § clearly to be seen. Various investigations into the decay properties of this ~600 isovector show that, in addition to the 471" decay, cJ>71"± and p ±p ° decays of the ~ 1,5 1,5 l~,~/,gN GeV/cZ b p(1700) state are also encountered. As regards the A~11"± decay, observations ~ Rz(17O!J t I Typical errOl in are contradictory. A study of the (411"°) mass distribution reveals no sharp ~ 200 R,(f5JO} I ~i&t~~~c2 structure at 1700 MeV/c 2• This may be understood if we suppose that the ~ t isovector decays preferentially to 11"W and pp. Danysz et al. [103] studied the ~ (471"°) mass distribution and also found a peak at M = (1717 ± 7) Me V /c 2 with a 0 100 width r = (40 ± 17) MeV/c 2 over a strong background (Fig. 57). These authors ~ stUdied the p p -311"+311"- reaction at 2.5 and 3 GeV/c. The peak may clearly ~ a be associated with the p011" +11"- decay. There may also be a certain link with the ; pOp ° decay, although this latter would mean that the state was isoscalar.

o · , !

z 15{)() 1700 1900 MeV/c2

Fig. 54. Missing.,.mass spectrum in the R region (a), and the same after substracting the hand­sketched background (b) •

The simplest assumption would be to the effect that the p (1700) is a 411" decay of PN(1650). However, owing to the difference in mass, it would at present appear more reasonable to assume the existence of two isovector states.

c. other Possible States withG = ± 1. Davier et al. [104] recently studied the yP -P71"+11"- and YP -P71"\+11"-11"-reactions from 4.5 to 18 GeV/c. In the A j 11" mass distribution, these authors observed a peak at M = (1550 ± 40) MeV/c 2

with r = (260 ± 110) MeV/c 2 (Fig. 58), and in the 11"+11- mass distribution a peak with M= (1540 ± 20) Me V!c 2 and r= (240 ± 80) MeV/c 2• In the paper mentioned

125

I !

a 100 200 300 1500

Johnston et al. (?-/"kp)

Crennell et al. (6\f-'kp)

Boesebeck et al.(8"f-'*p)

t----<>--i Armenise et al. (flC: 'it d)

f--<>-<

'-<>---<

>--<>--< I

1700

Armenise et al(5J.f17(+d)

Johnston et al. (7~¥rp)

Crennell et al. (6?(Ycp)

Purier et al. (f!-:V7(p)

GeV Knops et al. (8c 7(+d)

I

1800

Fig. 55. Masses and widths of the g:i: and gO mesons obtained in different experiments.

"()

~ C>I o

~ ::I 5

200

i,j 100 ~ o z

OL-----~~----L---~2~O----~----J.~.0~---"

H(-f1£il,Gev/cz

Fig. 56. Combination of the effective-mass distri­butions of four pions in the reaction 1I"::I: p _ p1l" ±,r+1I"-1I" ° obtained from a number of experiments.

earlier [103], the (411")° mass distribution also exhibited a peak at M = 1832 ± 6 Me V /c2 with r = 42 ± 11 MeV/c2 (Fig. 57). This decay has the same properties as the decay at 1717 MeV/c2•

4.2. States with G = - 1

a. The 11" A (1640) or A3 Meson. The (311"+) mass distribution of the 1I"±P - P1l"±1I" - 7r - reaction has a maximum at M = (1633 ± 9) Me V/ c 2 with r= (93 ± 24) Me V /c2• Figure 59 reproduces a generalization of various experiments with initial momenta between 5 and 20 GeV/c [101].

Investigations into the branch ratio of the decay of the A2 meson showed that, in addition to the preferential 311" decay, some (35-20)% decay to .,;j. It also follows from this that the whole peak cannot be ascribed to a single kinematic 7rjthreshold effect.

In order to determine the JP of the A3 meson, Bartsch et ale [105] analyzed the Dalitz diagram for the decay; they found that the A3 was probably a state of the series JP = 0-. I+, 2-, 3+, ••• At the lllnd Confer­ence. mention was made of a further analysis of the spin parity of the A3 me son in the 11'+ d - nsP1I' +11'- 11'-

126

15

12

6

't

~ 0 Q)

> Q)

'5 15 <> z 72

8

't -

0

JOO

200

.... 0 700

"> Q)

~ II)

c-I 0 0

0 ~ 100 0: .2 80 .... <II 0: 50 :E E 'to 0 0 ... 20 0

<> 0 z 720

700

80

50

'to

20

0 0,8

Total di~tribution

pp- JX'JJ{-

3v~~/~t I CERN, 302

I Liverpool, 360 events at 2.5 GeV/c

a

c

1,0 1,2 7,+ 1,. 1,6 2,0 2,2 2,+ 1'1 (;r ' 1T'1T -;r-),G eV/c2

2,5

11, ·m7:t7 r; - fO:t12

I1r-18J2:tG fZ -It2:t71

b

d

Fig . 57. Effective-mass spectrum of four pions, (The ma ss di s tribution of the pp combinations is given by the lower his togr a ms in b and d).

At :;r!p_p:;ri1(+1( -

/).+ + excluded

6.+ + excluded A, present

125 events

2800 J200

.... 0

"> Q)

::E II) 0

0 ...... ~ 5 > Q) ... 0

<> z

800

500

'tOO

7000

t Incident momentum 5 GeV/c

1((75"0)

l

1500

Background and two Breit - W igner distributions

2000

Fig. 58. Effective-mass distribution of four pio 11 (:;r !:;r ',r) , MeV/c2

Fig. 59. Combinations of the effective-mass distri­butions of three pions obta ined from different experi-ments.

127

80

a

~ GO > <l \!)

GO "" d ~O

II!! .....

50 ::l 535 events 0:

<l

I;() > :.. 20 0

30

I II 11111111 0 z

20 0

I jlllj!! Iii Ij 1,2 7,5 2,0 2,~

10 /1(J(P)o, GeV /e2

0 ......... i JO

"0

"> ~O

<l \!)

J8~ events ~ 20 30 0 .....

::l 10

c:: <l ~ 10

10 .... 0

0 0 Z 0

1100 7300 1500 1700 2100 0,8 7,2 1,5 2,0 2,ft 2,8 I1(W7[ '7[-; , l1eWe l 11 (7r'p0) , GeV f e2

Fig. 60 Fig. 61

Fig. 60. Effective-mass distribution of (w 7r+7!'-) obtained from the reaction K-p -+ AOW7r+7!'- at 4.6 GeV/c. >1<+-

In the lower distribution the Y (1385) events are removed.

Fig. 61. Effective-mass distribution of (7rp)o after the removal of the N *+ events from ~'f 7!'+d - PP7!' +71'-71'0 reaction at 8 GeV/c (a); effective-mass distribution of (7!''J,0), also after removing the N events. The continuous curve corresponds to the expected background (b). The mass spectrum is shaded.

reaction at 5 GeV/c [94]. The result was JP(As) = 2- or 3+. Barnes et al. [106] spoke in support of the existence of a hypothetical As meson in the K-p -+ A°7r+7!'+ 71'- -ff7!'0 reaction at 4.5 GeV/c (Fig. 60). In the W7!'7r mass distribution an 19 = 1- peak appeared at M = (1695 ± 20) MeV/c 2 with r= (90 ± 20) MeV/c2•

b. other Possible States with G = -1. Armenise et al. [107] mentioned observing (in all probability) an isoscalar (p7!') peak at 1640 ± 20 Me V/ c 2 with r = (112 ± 60) Me V /c 2 in the 1f+d -+ PSP1f+7!'-7!'° reaction at 5 GeV/c. This result is now supported by two auxiliary experiments.

Kenyon et al. [108] studied the same reaction at 8 GeV/c; they found for the state denoted as cp (1670)

a peak at M = (1670 ± 20) MeV/c2with r=(100 ± 40) MeV/c 2 and a branch ratio of the 7!'P decay (p+7!'- + P-7r~: (p07!'0) = 58 ± 15:(30 ± 10) (Fig. 61), leading to the result J.G = 0-. An attempt at determining the spin gave J ~ 1.

In studying the 7!'+d reaction at 9 GeV/c, an isoscalar state cp(1670) was also observed [94]. Figure 62a gives the (37r) mass distribution from the 7!'-p -+P7!'+7!'-7!'- reaction at 13 and 20 GeV/c [31]. In addition to the As meson, we may distinguish a peak (of the order of four standard deviations) at 1840 MeV/c 2 with r R1 80 Mev/c2• On adding the corresponding mass distributions from 279geventsat16GeV/c, the relatively sharp peak vanishes, as indicated in Fig. 62b. Danysz et al. [50a], studying the pp -+ 31f +31f -71'0 reaction and the (p°cJ - 51f) combination of masses, observed a peak at approximately the same mass value with rR1 70 MeV/c2•

128

a 80

All events (2317)

2,0 J,O N(tr''ff -1(-) ,GeV /c2

1Go b

51fG events

Fig. 62. Combination of the effective-mass distribution of (rr+7f -rr -) for the rr -p interaction: a) At 13 and' 20 GeV/c; b) at 13, 16, and 20 GeV/c.

4.3. States of Unknown G-Parity

When studying the pp - 0K>rr + rr - reaction at 0.7 Ge V /c, Aguilar- Benitez et al. [109] found a sub­stantial peak of five or six standard deviations at M = (1540 ± 5) MeV/c 2 with r= (40 ± 15) MeV/c 2 in the KKrr mass distribution (Fig. 63). It follows from the decays observed, denoted as F i - K*K' and F i - KK*, that 1= 1. Analysis of the angular distribution of the decay favors a spin-parity combination of .IP = 2+, I+, and, less probably, JP = 0+.

In this connection, an observation of Aderholz et al. [110] is also of interest. When studying the rr+p -p(KKrr)+ reaction at 8 GeV/c these authors observed the (KKx-)+ mass distribution illustrated in Fig. 64. The first peak has a maximumat M = (1490 ± 20) MeV/c 2 and a width r = 85 ± 39 MeV/c 2, and cor­responds to an effect equivalent to four standard deviations. On allowing for the limits of experimental error this state may be identified with the Pi meson.

The second peak has a mass of M = (1690 ± 16) MeV/c 2 and a width of r= (112 ± 60) MeV/c2; it corresponds to an effect of 3.5 standard deviations. Table 10 correlates those states of the region under consideration which, on the basis of existing bubble-chamber experiments, may be regarded as fairly reliably established.

129

130

~ o.

50

o 30 "-... .9 ~ 20 A e o u

'IS o z

TABLE 10. States in the R Region

State I IG I M,MeV/c2

PI (1540) I? 1540±5 g (l650~ 1+ 1650±20 p (1700 .. 1+ 1700±20 As (1640) 1- 1633±9 <p (1670) 0- 1670±20

*May be identical with g(1650).

2,22 J,02 111(KN!(D):Jr %), (Gf!{cZjZ

Fig. 63

~ C) II) o

I

30

~ ~ <:I OJ 10 ~

'IS o z

o 1

r, Me~c2 IP

40±15 2-, 1+, 0+ 120±30 3-, (1-, a-I 110±25 93±24 2-, 3-

1oo±20 I

F}2J (K7f) in the region r(I9Dj

WJD

7,5 2,0 2,5 J,(J I1(KiJ[) ,Gev/c t

Fig. 64

Fig. 63. Effective-mass distribution of (K°K°7r)2: in the pp -K~(Ko)7r+7r- reaction at 0.7 Ge V /c for 792 combinations. The continuous curve corresponds to fitting with the F 1 meson for JP = 2-. The broken curve indicates fitting without the F 1 meson.

Fig. 64. Effective-mass distribution of (KK7r)+ at 8 GeV/c. The curve corresponds to a fitting with two Breit-Wigner functions and a phase-space background.

J

J

z

lJ

Fig.65. Dependence of J on the square of the mass for the A2 and g mesons.

NO

~ "- a ~ SOO ~ "JOO e (,!) t:

70U8 events ." :::;, +00 0 , , ~ 200 ~ 300 .. c ~ 7780 Gl Gl ~ 200 >

~ Gl 700 ... 0

ci z 0

..... 1,0 2,0 3,0

H(K-(890)1T). GeV lez

N b 0 d "- ~ JO

~ :;; too • X-(7't20) events t: 0 ~ 20 0 "-..

~ C 700 c Gl ., > > ., ., 70 '0 ...

0

ci ci z 0 z 0 1,0 2,0 3,0 It,S H(K7t1r) , GeV/ e2

Fig. 66. Effective-mass distributions: a) The (K* (890) 1f) distribution in the K+p - K+p1f + 1f - reaction; 3234 events; ~++ removed; b) the (K1f1f) in the same reaction, when M(1f1f) is in the region of the pmeson; cases of * K (1420) falling in the region of the p meson are shown

in black; p is included,~++ removed; 2201 events; c) (1f1r7i) for the K+ p - K+ p1f +1f-1fo reaction; 10,288 events; d) (Kw) for the same reaction, 691 events.

A comparison of Tables 9 and 10 shows that the ascription of the narrow peak in the missing-mass spectrum to the state identified by the bubble-chamber method is at present impossible. The great dif­ference in the values of r cannot be explained by poor resolving power of the bubble-chamber experiments. For example, for the 7r±P- p1f±1f+1f - reaction channel the resolving power rexp < 20 MeV/c 2• On the other hand, the choice of a specific form of decay for the resonances is quite impossible with the missing­mass spectrometer.

Owing to the extremely strong background, this method of observation is principally sensitive with respect to narrow peaks, which are clearly distinguished on the background. Further investigations are required, both with the missing-mass spectrometer (for various ranges of 4-momentum transferred), and also with a bubble chamber, characterized by a high resolution and better statistics, before proceeding to identify the state of the R region.

The g meson probably has JP = 3-. In Fig. 65 the spins of the p, A2 and g mesons are shown as functions of the square of the masses of these particles.

We see that the g meson lies accurately on the p, A2 trajectory. This may easily be explained with t'ircfe1p _of the Simple model of quarks. In conformity with this model, the g meson should be a state of the J = 3 - nonet (Table 2). Reliable identification of the other states of the R region in the L-2 super-multiplet so far observed requires further experimental investigation.

4.4. Mesons with Singularity in the R Region

In the (K1f1f)± mass spectrum of the K±p - pK±1f+1f- and ~p - pK°7r±7r ° reactions a characteristic maximum was observed at M ~ 1780 Me V /c2, with r ~ 80 Me VI c 2• Example of this peak (defined as an L meson) may be found in Fig. 39 [661, Fig. 42 [701, Fig. 43 [711, and elsewhere [1111.

131

TABLE 11. Peaks above the R Region

S (1930) 1929±141 < 35 0,22- X- (3070) a075 ~25 0,23-0,36 0,44

T (2200) 2195±15 < 13 0,22- X- (3140) 3145 <10 0,16-0,36 0,52

u (2380) 2382±24 30 0,28- X- (3470) 3475 ~30 0,27-0,36 0,37

X- (2620) 2620±20 85±30 0,29 X- (3530) 3535 ~30 0,30-X- (2880) 288O±20 <15 0,64 0,41 X- (3020) 3025 ~ 25 0,21-

0,39

7J20 ~ ~ , .... ~ ""' 720 t t 7000

90 a

.. ~ SOD > ~ ~ > Q) .-< \!) 0

g, GQ ~ b d ~

';;; d C Q)

7790 >

Q) Q)

&'i

1 .... 0

'S <> JOO o JO Z Z

! ,

3,5 J, 5 3 ,7 t1,GeV / c2

J,8

Fig. 67 Fig. 68

Fig. 67. Effective-mass distribution of (K*(890)1T) for the K-p --pK-1T+7I"- reaction at 10 GeV/c: K-p - pK°7l" -1TO; N *++ excluded; 1303 events.

Fig. 68. Missing-mass spectrum in the range 3.4 :5 MX:5 3.8 GeV/c2 for 1T-P --pX-: a) Small set (:5 3 secondary charged particles); b) large set.

For the decay of the L meson, these authors obtained the following branching ratios:

Decay Kp I K* (890) it I K* (1420) it i Kw

Proportion of de - I cay along channels, "/0

28±13 11±9 I 34±12 I 19±15 I 8±5

For the isospin of the L meson, these observations gave 1= 1/2. This result is also supported by the coherent formation of the L meson in the K-d -- dL--- dK-1T +71"- reaction at 12 GeV/c [112].

132

170

89

79

1,0 2,0 J,O Momentum of antiproton in lab. coordinate system, GeV Ic

I I I ! ! I ! 1 ,

2100 2JOO 2500 2700 291JO

Total energy in center of mass system, MeV

Fig. 69. Total antinucleon (I == 0, 1== 1) inter­action cross section.

The diffractive formation of the L meson takes place directly above the K* (1420)7r threshold. Barbaro­Gatlieri et al. [70] studied 14,310 events of the K+p - K-p7r+7r - reaction at 12 GeV Ic; they came to the con­clusion that the L meson might be described as a threshold effect in the K* (1420) 7r system. The authors observed similar peaks in the (K7r7r) mass distribution for various K7r mass intervals, independently of the formation of a K7r resonance. On the other hand, no L formation was observed in the K* (890)7r, Kp and Kw channels, as indicated in Fig. 66. These observations contradict the results of other experiments, as indicated, for example, in Fig. 43 [71] and Fig. 67 [66].

The reasons for these discrepancies are not as yet clear. We shall have to wait for further results to improve the statistics, particularly in the K-p experiments, since here N*++ formation is much less marked than in the K+p reaction. As yet there is no reliable proof as to the existence of other mesons with singularity in the R region.

V. STATES WITH MASSES OF OVER 1876 MeV / c 2

A large number of statistically assured peaks (more than four standard deviations) with 12>: 1 have been observed above a very high background [113] in the CERN missing-mass and boson spectrometers . Table 11 presents the collected results.

Figure 68 illustrates part of the observed mass spectrum. It is well known that X-(3470) and X­(3530) always decay into three or less charged particles. Inthe earlier-mentioned experiment on missing masses, Anderson et at. [38] studied the formation of resonances at approximately 180 0 , i.e., for small values of u; they observed the following peaks:

M, M eV/e2 i2086± :l8\ 2050± 18 2370± 17 2500cL32

1', MeV le2

\ "" 150

\

"". 25 "" 57 ,"" 87

In the formation experiments, mesons with a mass of over 1876 MeV/c 2 are formed in nucleon-antinucleon interactions.

133

2,5

2,0 -0,05< cos 8<+0,05

7,5

1,0

0,5 1,25

7,00 -0,6"'COoffj<-0,'r

JOO 'tOO 500 700

-0,2 < COof fj <0

tl I/) JOO 700 ~ ....

7,25 .0 ... 8 0 -0,8<cosfj< -O,6 VI :;; lOO 'o~c;!

7,0 1/)'

'= .o~ ~ E .... 0,75 .5 0,5 ...

~O .. \0 .; 0,50 0 ...

JOO 'tOO 600 700 ~ 0,25

7,75 .5 5 ..

5 0 JOO 'too 500 600 700 ... 7,50 -0,+ < COS 9< -0,2

E 0 8 7,25 -T, ° <cos fj< - 0,8 cu

!if ... 'a

0,75 !'l '60 ~ 'f

t---l---i f-l----J....

JOO 400 500 500 700 JOO 'tOO 500 600 700 12007500

Plab' MeV/ c Plab, MeV/c

Fig. 70. Differential cross section of pp as a function of energy for various ranges of cos (J •

On making more accurate measure ments of the course of the total pp and pd cross sections at energie s of over 1 Ge V, Abrams et al. [114] found a characteristic structure in the cross section for the isospin states 1= 0 and 1= 1 (Fig. 69). If these structures are considered as meson resonances, they have the following parameters

M, MeV/c2 2190±5 2345± 10 2380±10

r, MeV/c2 85 140 140

o

We see that the masses of the two 1= 1 states approximately agree with T(2200) and u(2380), whereas the values of r differ considerably.

The large number of open reaction channels greatly impedes any proof of the direct S-channel effect in the curve representing the total cross section. The same applies to the forward elastic scattering, in which diffractive scattering tends to predominate. On the other hand, the cross section of elastic pp scat­tering in the backward direction, expressed as a function of Is, should exhibit a corresponding structure in the presence of resonances. Owing to the presumed high spin of the heavy resonances a narrow peak

134

Energy in CI~I1ter-of-mass system, GeV

0,05 2,0 2.2 2,'f 2,& 2,115 2,1f9 2,181, 2,219 2,25f I1(PP),Gell c2 I I , I I I

f 2190 - --, <>

f>. K:K:W'

t f1

t ' , . II'II-w· '. 0,02 80 ' , , ,

l<; + I "" 0 , ~ -----.... - -j---j--r I """ -0,02

, , b

, , 50

;1 -0,00 ,

t " '" " • This experiment 5 " " 0 Combs et al. " ii 12 , Elliot et aI. " " '" 'Cl ~O , ....

1- '

f1"'r-.... "

c Armenteros et al • e / T " .......... Z .c " f' E 8 '-'0:> --

--4

20

a

0 0,& 2 J

PIab, GeV/c 0 1,2 l,t p, GeV/c

Fig. 71 Fig. 72

Fig.71. Total pp -nn cross section between 1 and 3.6 GeV/c. The points in b) were obtained after subtracting the continuous line in a).

Fig. 72. Cross section of the pp - KKwO reaction between 1.1 and 1.6 Ge V /c.

should appear in the backward direction. The elastic pp back scattering has been studied with counters and bubble chambers for various energy ranges. As an example of this kind of investigation, in the range -1.0 < cos I) < - 0.8 (Fig. 70)we see two narrow peaks with 1= 0 or 1 at M = (1925) Me V/ c 2 with r f'::I 10 MeV and at M = 1945 MeV /c 2 with r f'::I 22 MeV/c 2 [115].

The experimental data relating to elastic pp back scattering so far obtained are compatible with the existence of a narrow resonance in the S region (1930) and a wide resonance (r f'::I 200 MeV/c 2) in the T region [116]. The width of the peak in the backward direction shows that the spin of this state has a value between 3 and 6.

Recently the cross section of the charge-exchange reaction pp - IDl was measured to a high accuracy for the first time between 1 and 3 Ge V /c (Fig. 71) [117]. The mode of variation of the cross section indicates an upper limiting value for the structure observed by Abrams et al. [114]. Further measurements must be made before any specific conclusion can be drawn.

Another possibility for proving the existence of the heavy meson resonances lies in studying selected annihilation channels, such as, for example pp - 7[+7[-,P'P -K~-. The experiment so far carried out show that the pp - 7[+7[- channel has a considerable effect on one or two wide resonances around 2200 MeV/c 2•

The properties of these states have never yet been reliably determined. The K~- channel does not show such a strong connection with these resonances as the rr+rr - system. It would be highly desirable to study other annihilation channels as well. Kalbfleisch et al. [118] carried out a linear interpolation between 1.11 and 1.52 GeV/c and found a rise (like a maximum) for an annihilation involving five pions at 1.33 GeV/c. The authors ascribed this rise at least partially to a pOporro state with M = 2190 MeV/c 2, r = 80 MeV/c 2,

and P = 1-. Figure 72 shows the number of pp - KKcJl annihilations as a function of the initial momentum [116]. Fitting of the K1K~ data leads to a resonance with M = (2176 ± 5) MeV/c 2, r = (20~~6) MeV/c 2, and IG = 0- or 1+ [119]. Ring et al. [120], in a similar experiment, found a rise over the background in the case of K~K1 events; this indicates the existence of another C = -1 resonance at M = 2370 MeV/c 2 with r = 20 Me V /c 2• Other information favoring the existence of this state was obtained by Ming Ma et al. [121] in a systematic analysis of pp interaction between 1.5 and 2.0 Ge V (Fig. 73).

135

136

GO .. ~ >

a ., Cl $~o d ...... rl c: ~ 20 ., ... 0

o

/'I~2J70 ~ 10 MeV Ic 2

r~ JUnO MeV Ic2 a

O,!' r-

JM\ t ~ -f--1 ''l-t

0,2

2J50 fJ90M eV/c2

o I i

O,J b

2 f ? ?{

7f- t !~ t 0,

0,

1 I I I I I I

t c

:~f'-~~ 1,

1,

0, 8 o State Universi ry of Michigan

• Michigan Universiry

0 I I I I I I I J 1,9 1,6 7,8 f ,O Pi' Gev/c

Fig. 73. Cross section of the following reactions in the range 1.5-2.2 GeV/c: a) K~K~7r+7r-7r°; K~K~(n7r°);b) K~K~w; K~K~ (n7r°);c) (K~K~7r-t7r -7r0) x 2; K~K±7r+7r-; K+K-7r+7r-7r°, K~K~(n7r°).

b c

Z O.~~~~~~~~~~~--~~-L~~~~~~~~ 1,~' 2,Z J,O 7,1; 2,2 J,o ~*

t1{X"JroJr 0Jr -Jr'), Ge V Ie Z

Fig. 74. Distribution of the effective masses of (K°7r+7r+7r-7r°) in the K+p interaction at 9 GeV/c: a) All combinations; b) only those with M(37r) in the region of the A mesons; c) only combinations with M(K7r) in the region K8~O or M(311") in the region of the A mesons.

7280

J~,

"u 20 "> t3 II";) o

c Brookhaven

~ O~-L __ L-~ __ L-~~~-L __ _

5 ~

'Ci o ~o z

20

O~~~ __ ~-L __ ~~~~

1,2 1,* 7,5 I1(KK) , GeV/c2

(KK;+T or (KKr, for secondary particles

Fig. 75

'J2Q

50 ~

50

'(J on C .., ~ J(J '-0

#. 20

10

Fig. 76

Fig. 75. Effective-mass distribution of (KK)++ obtained from different experiments: a) K+p -K~+i\, K+K~o, K~o~+ at 3.0 and 3.5 GeV/c, 136 events; b) K+p - K+K+i\, K~+~o at 3.5 GeV/c, 105 events; c) K+p -K~+i\,K+K~o, K+Ko~+at 3.0 GeV/c; d; three second­ary particles, 397 events in all.

Fig. 76. Effective-mass distribution of (p-7r-) obtained from various experiments: A) Seidlitz et al., p = 3.2 GeV/ c 6,.2 < 0.7 (GeV/c)2, N* excluded, 165 events; B) Abolins et al., p = 3.7 GeV/ c, all d, N* excluded, 342 events; C) Vanderhagen et al., p = 5 GeV/c,

2 * all is. , N excluded, 892 events.

Generalizing the present situation, we may say that a variety of pp formation experiments have proved the existence of both narrow and wide resonances above 1900 Mev/c 2• In order to determine the properties of the resonances reliably, we require further investigations with higher statistics and greater accuracy. The contradictory results so far achieved may possibly be explained by the high density of states in the mass region under consideration.

At the present time there are several unreliable indications of the existence of heavy Singular mesons. Figure 74 showfl the mass distribution of five mesons from the reaction K+p - KOp7r + 1T +1TO at 9 Ge V /c [51]. At M = 2460 MeV/c 2 we observe a peak of four standard deviations with r~ 80 MeV/c 2• In similar experi­ments. Alexander et al. [122] studied the K+p - 7jNN reaction; in the X"N mass distribution they found a peak at M = (2240 ± 20) MeV/c2 with rRJ 70 MeV/c 2•

It is obvious that the few and unreliable data relating to heavy resonances are insufficient to identify these with the states of the quark model. We must also remember that, with increasing orbital moment, the increasing overlap of the supermultiplet introduces serious complications into the identification of mesons.

137

Peak directed forward ~ Meson exchange Peak directed forward ~ Meson exchange

\ ~K~ \

Meson \ exchange

B~on ex­change

I

J71' I

Wisconsin data 7['p-J(~X+

JOO J,2GeV/c

250

';;; 2/J0 "-~ 150

~ 100 ~ ~ 50

t.J,., ~ 7,0 0 -1,0

\i' \

Meson \ exchange I

I

Ix-I

Baryon ex­change

cos 8 c.m.s. (=center of mass system)

Sac1ay data

7,0 0 -1,0 cos Bc•m•s•

Peak directed backward Lawrence

~ Baryon exchange Peak directed backward ~ Baryonexchange

Laboratory data

,--Jr--,--P_L--:---K.,· 70 3 GeV/c

1,0 0 -7,0 cos 8 c _m. s•

~ ~ '5

\

\ X' \

\

I

Iff" I

Saclay data

Baryon ex­change

Me~n exc ange

Kp-Z-K+ r---'-----,10

3 GeV/c

1,0 0 -1,0 cos ec•m•s•

Baryon ex­change

\~ \ K+ Z -,

\ Meson / 27 exchange

/1(" ? P I

Fig. 77. Proofs as to the existence and suppression of various exchange processes.

VI. EXOTIC RESONANCES

As indicated in Chapter I, the simple model of quarks requires the formation of a meson from a quarlt­antiquark pair and that of a baryon from three quarks. The so-called exotic meson resonances of the first kind with T ~ 2/3 and 1 sl ~ 2 and exotic meson resonances of the second kind with values of JPSnotencoun­tered in the qq system, such as 0- - or 1- +, should therefore be forbidden.

Exotic mesons of the second kind might, however, appear in the Regge model as daughter trajectories.

As indicated in Chapters 2 and 3, there are no particularly Significant indications at the present time as to the existence of exotic mesons of the second kind. The simplest exotic state of the second kind is an isovector with J PC = 1- + and a p7r decay but no strong KK decay. A review listing direct proofs as to the existence of exotic resonances of the first kind was presented by Rosenfeld [123], who showed that the exotic peaks observed in the corresponding mass spectra could not be considered as statistically assured, and that on improving the statistics the corresponding peaks would again vanish. Figure 75 shows how the S = 2K~+ peak at 1280 MeV /c2 vanishes again on combining several experiments with tripled statistics [74]. When studying the 7r-d -PsP7r-7r-7r° reaction at 5 GeV/c, Vanderhagen et al. [124] observed an 1= 2p-7r - peak at 1320 MeV /c 2 ('" 3 standard deviations). The mass distribution from this investigation, combined with that of two later experiments, is shown in Fig. 76. In evaluating the maximum at 1320 Me v/c 2, we must remember that, even for a pure phase-space distribution, the p selection leads to a maximum at approximately 1300 MeV/c2 [60].

An indirect proof of the absence, or at least strong suppreSSion, of exotic resonances of the first kind is obtained on studying peripheral interactions. If a two-particle process takes place with meson exchange, then a forward peak appears in the angular distributions. If the process takes place with baryon exchange,

138

2 GeV/c 10

8. 3 GeV/c 4 GeV/c

~ 6 ~~

~ 2

~ O

+ -t--t-+ r-+- -+-

l,·'-::(I.J........L-'---'--JO-'-·-'--'-'-1...J.O -1.0 0 I,D -1,0

cos off. i )

f--....-__ =::1"= _ _

illC esc ' fJ

Fig. 78. Differential cross section as a function of cos e A A + * 0 *+ (11'incKesc) for the 11'-p - K Y - and 11'+n - K Y reactions.

~ J[ '

J + J[ -

J ill!! ! j j . Pre lim inary

t j ! !

i 1.. + + 2

ltHt T ,

/mn o 0,2 0,'- 0,8 7.0 7,2 rFt,GeV/c

Fig. 79. Ratio of photo-formation for 11' +N* and 11'-N*. N o~ yD-+ n±t;Ns

ylf-+n't;

1,0

then a backward peak appears. Reactions which require exchange by an exotic resonance of the first kind (as indicated, for example, in Fig. 77), have a cross section at least two orders lower than in the case of exchange by known mesons [125]. There are some new experimental results which, on interpretation by the one-meson exchange model, indirectly indicate the existence of exotic resonances.

Abolins et al. [126] used momenta between 2 and 4 GeV/c in studying the 11'-p -K+Y*- and 11'+n _KOy*+ reactions. The distributions of the cosine of the angle between the incident pion and the kaon so formed indicates a statistically assured forward peak (Fig. 78). The cross section in the forward direction equals about 7 /Lb/sr at 2 GeV/c and is of the order of 2/Lb/sr at 4 Gev/c. If we interpret the forward peak on the basis of exchange by one meson, then it should have the quantum numbers 1== 3/2 and S == 1. However, the separation of the Y* peak from a possible kinematic effect would appear dubious.

For the K-p -+ K+S*- reaction, in which the baryon resonance is adecuplet JP == 3/2+ state, a forward peak of approximately the same magnitude has also been observed [127].

In an experiment relating to the photo-formation of singular particles at 11 Ge V, in which the 4-momen­tum transferred equa lled 0.025 :so I tl :so 0.46 (GeV/ c)2, Boyarskiiet al. [128] obtained the following ratio of cross sections:

139

On the assumption of exchange by one 1== 1/2 meson in the t channel, this ratio should have a value of 2. The same authors [128] used an energy of 16 Ge V in the 0 ~ I t I ~ 2 (Ge V /c)2 region to study the ratio of the following cross sections:

R (rc) =_ cr (yp -+ n-N*++)+cr (yn -+ n-N*+) . cr (l'P -+ n N*++) ,

R (n+) = cr (yp -+ n+N*O)+cr (yn -+ n+N*-) cr (yp -+ n+N*O)

As indicated in Fig. 79, the ratio R(,rr-) agrees with the value of 4/3 expected for exchange by an 1== 1 meson. The R(,rr +) ratio, on the other hand, deviates seriously from the expected value of 4. This indicates the participation of 1== 2 exchange. It would appear vital to carry out similar measurements for hadron reactions and to confirm the results of Boyarski et al.

C ONC L USION

In evaluating the many mass-distribution peaks listed in the foregoing chapters, as well as many others which we have not mentioned, the question of statistical assurance is of decisive importance. In his contribution relating to exotic resonances in 1967, Rosenfeld analyzed events recorded in photographs from bubble chambers, and estimated the peaks owing their origin to statistical fluctuations to be expected in the effective-mass spectra. In several hundred observations of the 30" effect, a few 40" peaks were expected. If allowance is made for the annual increase in the number of events measured in photographs from bubble chambers, in 1970 we should expect about 100 40" peaks and one 50" peak due to statistical de viations.

In this connection an estimation of the background is of fundamental importance. By varying the position of the boundary between the peak and the background, a 30" peak may easily be transformed into a 40" or even a 50" peak.

In determining the quantum numbers of a resonance, the properties of the background are of decisive importance. As indicated earlier for the case of the A2 meson, a comparison of the value of X2 for the spin­parity hypotheses under consideration can only provide useful information if the correct assumption has been made regarding the properties of the background. In Chapter 3 we devoted some detailed consideration to the question of explaining a maximum by analyzing the kinematic effect of the peripheral mechanism of formation. In Fig. 30 the continuous curves show, for example, a calculation of the background based on the two-pole Regge model. The question of substracting a background which depends on the model employed in the formation of resonances is problematical. On the basis of the dualism consept, the diffractive descrip­tion of the amplitude of the resonance components in the t channel in terms of the Regge model (in the sense of the "boot-strap" representation) itself leads to the formation of a resonance. In discussing experimental effective-mass spectra, it therefore appears rather dubious to attempt any proof of the existence of reson­ances by subtracting a background which depends on the nature of the model.

The experimental data so far obtained in the range above 1550 MeV are insufficient to permit a classification of the meson states. In the region of the L == 1 supermultiplet, in addition to the absence of any proof regarding, in particular, several 1== Y == 0 states, other vital problems (such as the splitting of the A2) have not yet been elucidated.

A decisive factor in solving the remaining problems is the introduction of a new phase in meson spectroscopy, that of fine-structure analysis. However, this demands a substantial improvement in the techniques of observation, particularly in the following aspects:

1. An increase in the resolving power in the region of heavy resonances, so as to achieve a value of a few MeV.

2. An increase in the number of events in the reaction channels studied, so that the order of these may be at least high enough to permit an accurate analysis of the states of the decay products over a wide range of the 4 momentum transferred.

These two requirements are closely linked. A high resolution without extensive statistics is useless. The same applies in reverse. Various projects now in preperation promise to satisfy these requirements.

140

Examples. The use of superconducting magnets in the chambers; the use of controlled spark chambers in spectrometers and controlled streamer chambers with a fast cycle. A further important step for improv­ing experimental conditions would be the use of polarized beams of particles and targets. This review may we 11 be brought to a close by a citation from Dalitz I contribution [52] to the Conference on Meson Spectro­scopy held in Philadelphia in April, 1968, which even now retains its importance.

"It is vital to emphasize the extent and value of this work. It is not true to say that the problem lies in collecting new resonances and classifying these in a zoological spirit; in order to explore these meson states and the phenomena involving their interaction and establish them on a firm basis, it is essential to know the complete picture. Together with corresponding work on barions, these empirical investigations open the clearest way to an understanding of the internal structure ofhadrons and the nature of the extremely strong interactions leading to them. At the present time there are few data regarding meson resonances which do not agree specifically with the simple quark model. This in no way assures us that the model is sufficiently flexible to explain all the data. Actually the situation tends in the opposite direction; if, in the determination of spin parity, a reliable deviation from the expected properties were established, this would remove the quark model from the sphere of working hypotheses. It would be a very trivial matter that all the hadron data could be explained on the basis of the quark model, and it is therefore extr3mely important to study the whole range of meson phenomena, since these should provide a rigorous basis for the confirmation of this concept."

In conclusion, the author wishes to thank Dr. K. Grote, Dr. I. Kundt, and Dr. S. Nowak for critical comments in the formulation of this communication; he also wishes to thank his wife for her coopera­tion

LITERATURE CITED

1. Particle Data Group, Review of Particle Properties, UCRL 8030. Pt. I (1969). 2. R. Dolen, D. Horn, and C. Schmid, Phys. Rev., 166, 1768 (1968). 3. O. W. Greenberg, Resonance Models, Technical Report No. 70-017, University of Maryland (1969). 4. V. L. Auslander et aI., Phys. Lett., 25B, 433 (1967).

5a. R. H. Dalitz, in: Meson Spectroscopy, edited by C. Baltay and A. H. Rosenfeld, Benjamin, New York (1968), p. 497.

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14. B. Lorstadt et aI., Nucl. Phys., B14, 63 (1969). 15. P. Baillon et al., Nuovo Cimento, 50A, 393 (1967). 16. G. Goldhaber et aI., Preprint UCRL-8, 894 (1969). 17. O. I. Dahl et aI., Phys. Rev., 163, 1377 (1967). 18. S. Glashow and R. Socolow, Phys. Rev. Lett., 15, 329 (1964). 19. R. Dashen and D. Sharp, Phys. Rev., 133, 1585 (1964). 20. I. Butterworth, Boson Resonances, Preprint (1969). 21. G. E. Chikovani et aI., Phys. Lett., 25B, 44 (1967). 22. K. Boesebeck et al., Nucl. Phys., B4, 501 (1968). 23. J. Bartsch et aI., Phys. Lett., 25B, 48 (1967). 24. S. Nowak, Berlin-Zeuthen, Dissertation (1968).

141

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144

ELASTIC AND INELASTIC COLLISIONS OF NUCLEONS

AT HIGH ENERGIES

K. D. Tolstov

Experimental data obtained in proton-proton and proton-neutron collisions at high energy are reviewed, including elastic scatterings, generation of particles and resonances, and momentum spectra of secondary particles. The results of experiments are compared with the theoretical models.

INTRODUCTION

In formulating a theory of strong interaction the investigation of elastic and inelastic collisions between nucleons occupies a central position. From the experimental point of view we are dealing with the most intense primary proton beams which exceed the intensity of secondary particles by many orders, and by using hydrogen targets we can investigate interactions between identical particles. This has allowed, for example, the achievement of differential cross sections ~ 10-36 cm 2 for elastic pp-scattering.

OJ 16 a [ t'/ .... <.J /;

<=1

[12 + .... 7

"0 " ..... .£-' .... <ll b.O 3 ~

2 I ; .... tl 8

'c>

.8 " 1 I ..... '.;r; .......... § I £- ..... A; <= ..... <ll 0 I

rf 7 10 100 1000 E, GeV <ll

~

n~ b II' •

ft-

o I J "[I •

I 1 il 3 •

20 5 70 75 20 25 Po, GeV Ie

Fig. 1. Dependence of the average number of charged particles on photon energy (a) according to the data obtained at: 1) Berkeley; 2) Dubna; 3) CERN; 4) Physics Institute of the Academy of Sciences of the USSR (Moscow); 5) Tokyo (photoe mulsion data); 6) pre­dictions for the CERN colliding-beam accelerator; 7) Chicago (photoemulsion data). The experimental and calculated values of the average number of charged particles in pp-collisions as a function of energy (b): 0) are the experimental data; .) are the calculated re suIts.

Joint Institute for Nuclear Research, Dubna. Translated from Problemy Fiziki Elementarnykh Chastits i Atomnogo Yadra, Vol. 2, No.1, pp. 231-278, 1971.

© 1972 Consultants Bureau, a division o{ Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011. All rights reserved. This article cannot be reproduced {or any purpose whatsoever without permission o{ the publisher. A copy o{ this article is available {rom the publisher {or $15.00.

145

6, ~--------

.... --- __ .... __ ..... _--... __ 4 charged 10 --.. particles

_...--- .... -----e __ ..-- 6 ... ---

1,0

0,1

0,07

o~_~~_~~~_~_~~_~! __ ~ 10 20 Pna5{;eV/c

Fig. 2. Dependence of the production cross section of various numbers of charged particles on energy (from data obtained on the SO-in. hydrogen bubble chamber at Berkeley).

The identity of the colliding particles obviously simplifies the theoretical analysis.

The ranges of high energies, besides being of great significance in connection with the asymptotic behavior of the processes, is also promising in that the interaction is less associated with resonance phenomena. The latter may be more substantial in a comparatively narrow energy range and thus may not reflect the basic features of the processes at high energies.

The interaction of colliding nucleons may be subdivided into elastic scattering, scattering with charge exchange, quasiparticle reactions,and multiple production, these processes being linked by relationships which derive from the unitarity properties of the S-matrix.

For specific colliding particles the S-matrix depends on the energy and angular momentum, and its unitarity also relates the elastic scattering to aU of the remaining processes for a fixed value of angular momentum. If data were to be available on the dependence of the interactions on the angular momentum (Le., on the impact parameter), then it would be possible to obtain many important conclusions, including conclusions concerning the so-called central and peripheral collisions (which for the time being have an arbitrary character) and consequently concerning the structural features of nucleons in strong interactions.

1. INELASTIC pp-COLLISIONS

1.1. The Multiplicity of the Particles

The dependence of the number of charged particles in pp-collisions on energy n(E) in the laboratory system is displayed in Fig. 1. From Fig. la it follows that in the 100 to 1000 GeV range where there are more points and the error is smaller, we have n(E) ,..., aE1/ 3 •

146

-T·~· -- -~·r-- ··-I-~

a, mb

0,,1

1 0,2 III f f 0,7

f [7 5 J[7 Hi

f

2[7 Plab'

2

GeV/c

Fig. 3. The dependence of the total w-meson production cross section in pp collisions on proton momentum.

In accordance with Fig. 1b we have n(E)~ aEo. 4 in the 10 to 30 Ge V range. Thus, the growth of n(E) occurs substantially faster than the Et/ 4 which was obtained in the Fermi statistical theory and the Landau hydrodynamic theory, or than the logarithmic growth n(E) ~ 10g(E) obtained according to the multiperipheral model.

In this connection one should dwell on the recent paper by P. Rotelly [1) in which the author does not use the model involving the formation of one or several fireballs in considering statistically multiple pro­duction of particles. Rotelly notes that previously no statistical consideration of multiple production had been carried out which was not based on the concept of the merging of colliding particles. In reality a pro­cess without merging is more natural, since we do not have an equilibrium state during collision but more likely an explosion.

Rotelly bases himself on the classical graincanonical distributions, postulating that the entropy does not depend on energy in the center of mass system for a fairly high energy of the primary particles. Further he starts from a constant inelasticity coefficient which is equal to 0.4 in the center of mass system (Le., the secondary particles carry away on the average a constant energy fraction equal to -OAEc )' This leads to the following formulas for the number of generated rr- and K-mesons:

(1)

(2)

where Err' EK are the energies of rr - and K-mesons in the center of mass syste m, while mrr , mK are their masses. Consequently the dependence of the number of particles on energy in the laboratory coordinate system can be expressed by the law El/3.

In explaining multiple production R. Hagedorn [2) starts from the fact that thermodynamic equilibrium and collecti ve motion along the collision axis holds in collisions at high energies. The simplest method of separating these processes is to consider quantities which do not depend on the collective motion - the multiplicity of the particles is among them.

According to Hagedorn, the multiplicity follows the law ~xp (-miT), where the temperature T - To, To increasing with energy so that T ° ~ 160 Me V at high energies. The results of the calculations carried out by

Fig. 4. Distribution of the effective masses of the 7T+rr-rro system in the pp -pprr+rr-rro reaction at 19 GeV/c.

Hagedorn and Ranft are plotted in Fig. 1b (the filled circles). Let us note that according to Hagedorn the onset of satura­tion of the number of produced particles must occur at high energies, which does not derive from the data shown in Fig. 1a. The energy dependence of the production cross sections of various numbers of charged particles is shown in Fig. 2.

1. 2. Production of Strange Particles

Calculations of the production of strange particles have usually led to a small fraction of them as compared with rr-mesons. However, Rotelly in [1), USing Eqs. (1) and (2) and assuming that at high energies the equation E7T = EK = 1/3Ec holds asymptotically, obtained NK/N7T '" 0.43. Such a high fraction of K-mesons is a new result which may serve as a check of the Rotelly concept.

147

(J,t,.

(J,2

(J

(J,S

° 0,8

(J (J,t,.

0,2

°

r t

!

t

t

5 10

p.p-p+N*O,2t,.J

! p+p-p.N*(I,t,.O)

f ! t

P'p-p+N"(1,52)

! ! t 1 p+p-p+N*(1,o9) ~

! ! !

p.p-p.N *(2, 19)

! ! 15 20 25 p,GeV/c

Fig. 5. Dependence of the total cross section of isobar production in the pp - p + N* re­action on proton momentum.

The experimental data on the total cross sections of the production of strange particles in pp-col-lis ions are as yet not numerous. At 8~ GeV Firebugh et al. [3] obtained CTtot = 1.8 mb; Holmgren et al. [4] obtained CTtot = 2.05 ± 0.14 mb at 10 GeV, assuming equality of the production cross sections of YK+ and YKo; Bartke et al. [5] obtained the production cross section YK 3.0 ± 0.3 mb for the production of YK and and a cross section KK 1.2 ± 0.3 mb for a pair production of KK at 24.5 GeV (Le., CTtot = 4.2 ± 0.4 mb, which is ~ 0.15 of the total inelastic cross section). Consequently, a substantial growth of the cross section with increasing photon energy is observed.

At cosmic energies the ratio between the number of K-mesons and the number of 1I"-mesons is esti­mated to be 0.2 to 0.4. Thus, the experimental data do not contradict the Rotelly calculations. (Let us note likewise that in 7rp-collisions at 25 GeV the production cross section of K-mesons is d 4 mb according to the data cited by Erwin [6]; Le., it is more than 0.2 of the inelastic cross section.)

t,.o~----~---------------------, 1,23 1,'f7 7,70

t t ~

Fig. 6. Effective mass distribution of the P1l'- system in the case when a Do++ isobar is formed in the pp -PP1l"+1I"­reaction at 16 GeV/c.

148

1.3. Production of Boson Resonance

Boson resonances yield a relatively small contribution to the total cross section in pp-collisions at high energies, especially in events having a large multiplicity of particles. The largest contribution is made by production of a wo meson which Figs. 3 and 4 from [7], for example, illustrate. Figure 3 shows the de­pendence of the w-meson production cross section on proton energy. The effective mass distribution of the 11'+11"-11"0 system in the pp­pP1l" +11"-11"0 reaction at 19 GeV Ic is shown in Fig. 4.

The resultant cross section of p-meson production in pp -.t.pN reactions at 19 GeV/c was found to equal 0.57 ± 0.07 mb.

1.4. Produc tion of Baryon Re sonance s

Experimental data show that at high energies a relatively small fraction of baryon resonances isformed. Similarly to boson

1000

o 500 7000 1500

T, MeV 750 7't1l 720

-=::...-_--- 700

2000 m,MeV

Fig. 7. Dependence of the average transverse momentum of the secondary particles on their mass and on the temperature of the excited system: 0) is for 12 to 30 Ge vic (different ex­periments); .6.) is for 10 3 to 10 5 GeV/c (Kranov); e) is for 3'10 3

GeV/c (Brazilian-Japanese collaboration).

resonances. Consequently, it is also true that the possible cascade decay of baryon resonances will not make a significant contribution to the multiple production of particles.

Figure 5 shows the dependence of the total cross section of the production of baryon resonances having different masses on the proton momentum according to the data published by Anderson et al. [8]. At 30 GeV/c the sum of the total cross sections is ~Utot(M) = 1.7 ± 0.4 mb. In reactions with production of two mesons, as Bertocchi notes [9], the probability of the transition of both colliding protons into a resonance state is low, since the charge conservation prevents the formation of two .6.++ isobars and the isotopic spin relationships reduce the probability of the formation of the .6.0 isobar. However, in the paper by Rushbrooke [10] there isanindicationof double production of .6.++,.6.0 isobars. This is evident from Fig. 6, where the dependence of the number of events on the mass of the P7r- system is given in cases when .6.++ isobars are formed.

1.5. The Momentum Spectra

At presen.t the general regularities governing the momentum spectra and the emission-angle dis­tribution of secondary protons and particles produced in inelastic collisions are well known. A strong

collimation of protons in the center of mass system

.. 70 l~tI l,O~ f >

'. It~ S ., l i t f fl • ([ ......

~ .., ~ 1,0 i H E

ii 0,7 • E . " . "

f t'~ %

{} ~ . ~ y, {} y;. 0,7 Gl 0,01'''''

""

0,010 O,'t 0,8 7,2 7,5 2,0 ~4 p~,GeV/c

Fig. 8. Spectrum of secondary protons as a func­tion of P.L in pp collisions at momenta of 10, 20, 30 GeV/c: a) Po = 30 GeV/c; b) 0 is for Po = 10 GeV/c; e is for Po = 20 GeV/c; the solid line is for the fictitious curve at 30 Ge V / c.

is observed in the direction of the primary momentum.

The average inelasticity coefficient k and the average transverse momentum of protons are practi­cally constant over a wide energy range and are equal to (k) "'" 0.4-0.5; (p.L) '" 0.4 Ge vic. Predominant emission of 7r± and K± mesons in the direction of primary protons and a more pronounced emission for positive particles are observed in the center of mass system. With increasing multiplicity of the secondary particles this directivity decreases.

The momentum spectra and the dependence of the cross sections on the momenta are very essential for clarifying the mechanism of ine lastic collisions. Hagedorn, for example, assumed that the distribution over the primary momenta N(Pl.) of the produced particles does not depend on their collective motion along the collision axis and follows a Boltzmannian distribution (i.e., N(pl.) "" exp(-(pl.)/(T)). This de­pendence is analogous to the previously considered distribution for multiple production of particles accord­ing to the Hagedorn scheme. At 30 GeV the tempera-

149

150

a

e e e e e

0,1,1

Fig. 9. Dependence of the differential cross section for the production of 1T+ mesons (a) and 1T­

measons (b) for fixed p 1. on the longitudinal momentum in pp collisions at 12.2 Ge Vic in the center of mass system .

.. o "> ~',O .0 E

~mL-~~ __ ~_L-I~~ __ ~~ __ ~~~ __ ~~~ o 0," 0,8 1,2 1,6 2,0 2," 2,1

p,~,GeV/c

Fig. 10. Dependence of the differential cross section of the emission of secondary protons for p: = 0.18 ± 0.02 GeV/c (e); for PI == 1.0 ± 0.04 GeV/c (0) and 1T­

for P;= 0.18 ± 0.02 GeV/c (.6) on PIT.

ture T ~ 120 MeV, and at higher (cosmic) energies T ~ 160 MeV. Hagedorn applies statistical thermo­dynamics, assuming the formation of highly excited blobs of hadron matter which decay in ~ lO-23 sec.

In [11] Hagedorn derived the following equations for the average magnitudes of the transverse momen­tum of the secondary particles as a function of their mass and temperature:

(3)

where K 5/2, K2 is the "K" function.

Figure 7 shows the experimental data and the values of (Pi. (m, T), calculated according to Eqs. (3). In the paper by Anderson [12] the dependence of the number of secondary protons on the transverse and longitudinal momentum p~. is obtained in the center of mass system. This dependence is expressed by the differential cross sections d2u/dPJdpn, which are shown in Fig. 8 for three values of proton momentum: 10,20, and 30 GeV/co It turned out that the cross sections baSically do not depend on p~, whose values varied in the range from 0.2 to 3 Ge V /c. The experimental pOints in Fig. 8 may be described by empirical curves corresponding to the equation

... I

U ~

0,+

t'5 0,2 ... I

:;t

~ , e 0.1 ' ~ , ,

.. " I , r

1[ ' " 78,8(£V/c • 23, 1

o 0,5 1,0 1,5 p",'GeV/c

Figo 110· Dependence of the differential cross section of 1['± mesons on the longitudinal momentum in the center of mass system in the case when the mesons were emitted at the angle 0° in the laboratory sys­tem (the dashed line is the statistical model) 0

d2a ( P J..) rob --=610'PJ exp ._-- . dp jdpn . 0,166 (GeV lei (4)

Equation (4), after transformation to the variables p, () (the momentum and angle in the laboratory system), takes the for m

_~~_ ~c p2 • l'e (!:'-~c cos 8) 610 ex (_ p .L) . rob • dp,h! 2,1 I: PJ p 0,166 (GeV/c)sr (5)

which is similar to the dependence obtained by Hagedorn in statistical theory for central collisions. Figure 9 shows the dependence of the differential cross sections d2u/dpL dp ~ for production of pions from p ~ for fixed values of p.L in pp - 7T± M reactions, where M is the defect mass at 1205 Ge V /c. Figure 10 shows the dependence of the cross sections d2u/dp.L· d!)li on Pi, for fixed pJ... It follows from these reactions that for protons a strong dependence of the cross sections on p.L and practi­cally constant cross sections as a function of p~ are observed. For pions the cross sections drop almost by three orders when pn varies in the 0.8:s p~:s 2.8 GeV range. Assuming that the uncertainty principle is valid with the equality sign in I13] in inelastic collisions of fast particles, the autp.or derives the equation for the ratio between the dispersions of the longitudinal (i\2~)'!2 and transverse momentum «i\Pl> = (p'i») :

(6)

where 'Yc is the Lorentz factor of the center of mass system.

The use of Eq. (6) for protons and 11. hyperons in 7T p collisions of of 7 GeV/c in the present paper, and likewise in [14], shows that Eq. (6) is valid within the limits of the experimental accuracy of '" 10%. (Data on the dispersions PI! of the momenta in pp-collisions are unknown to the author). If Eq. (6) is valid, then inelastic collisions occur in such a

151

o

AAAA

f; f;f; "'A. 'bLl. ..

'b ••

'*' o

2

• • •

o

• •

o •

p,GeV/c

Fig. 12. Dependence of the production cross section for 11"* and K* mesons on their momen­tum if they are emitted at an angle 6°40' in the laboratory coordinate system in pp collisions at 19 GeV/c: ,6.e) 11"+; t:.O)7f"; .) K+; D) K-.

way and in such a time that the Lorentz contradiction of the nucleons in the longitudinal direction does not change. The momentum spectrum of 11'± mesons emitted at the angle 0° in the laboratory system in pp­collisions at 19 and 23 Ge vic was measured in the paper by Dekkers [15]. The differential cross sections

P",·GeV/c o 7(' 0,60 • 7(- 0,60 .7(- o,t,.o o K' 0,50 • r 0,50 ". P (1,6'0

o (1,2 o,f ~5 ~8 ~D ~2 ~f ~5 P:, (GeV/c)2

Fig. 13. Differential cross sections for various particles as a function ofP.L for fixed values of the longitudinal momentum in the center of mass system PO = 0.4 or 0.6 GeV/c.

152

d2a/ dp dn are shown in Fig. 11 as a function of the longitudi­nal momentum in the center of mass system. It follows from this figure that statistical theory does not describe the spectra of 11' * mesons, and a specially large difference is observed for the 11"+ meson. Figure 12 from the paper by A. D. Krisch [16] shows the dependence of the production cross sections of 11"± and K± mesons on the momenta for pp-collisions at 19 Ge Vic when the mesons were emitted at an angle 6°40' in the laboratory system. It follows from the figure that throughout the entire momentum range the production and rotation of positive particles is larger than the production and cross section of negative particles.

The constancy of (p 1-) for secondary protons allows the assumption to be made that this is possibly associated with the stability of the interaction region during a change in energy. The interaction region may be estimated quantita­tively on the basis of the uncertainty principle if it is valid with the equality sign. According to W. Heisenberg [17] this is fulfilled for a Gaussian distribution of the components of the momenta and coordinates when the product .6.p~x is minimized. If the momentum distribution over the coordinate axes x, y in a plane perpendicular to the trajectory of the impinging protons is Gaussian, then the distribution has the following form as a function of p.L:

( Pi) dN (p 1.) = cp 1- exp - (P}) dp 1-.

If p ~ is fixed, then dp dn = 211. sec 8 d8 dp 1- and Eq. (7) is transformed to

d2(1 ( pi ) dpd>1=ccos8P1-exp -(pi) •

(7)

(8)

6

7

200

_ p; sine " fl = --o,f

o 0,1 0,2 0,3 O,ft o,s D,B 0,7 k J,O Pl , GeV/ c

Fig. 14 Fig. 15

Fig. 14. Dependence of the distribution of pp -+ ~ PPJt2 +~-2(~)( ) events on the inelasticity ~pp It It - --

factor k in coLLisions at 28.5 GeV/c.

Fig. 15. Dependence of the average number of pions on the transverse momentum of the secondary protons.

Figure 13 shows the functions d2u/dp dQ for various secondary particles for fixed values of p ~ which are equal to 0.6 and 0.4 GeV/c. The variation of p1. takes place in the limit 0.45 ::os p1. ::os 1.3 GeV/c, i.e., the factor ~1. in Eq. (8) varies by approximately two orders ,whereas d2u/dpdQ in Fig. 13 is reduced by three orders. Consequently, in accordance with Eq. (8) the linearity in the logarithmic scale of the function d2u/dpdQ in Fig. 13 corresponds to a Gaussian dependence of this function on the variable Pi-

For different secondary particles the functions d2u/dp dQ have a closely similar slope: d2u/dpd~ ~ 2

e -3,3p 1., i.e., the parameter (Pl) in Eq. (8) is the same. In accordance with the unce rtainty principle this indicates that iJ-LL the particles are emitted from one region. On this subject, A. D. Krisch (16) comments: this is strange, since K± and p have a large mass, but it is evidently true.

500

"00

t JOO ?

f

Let us note that this regularity is in accord with that con­sidered inSecs.1.1 and 1.2 of the paper by P. RoteLLy.

Purposeful work on the investigation of cross sections and momenta was carried out at the Argonne Laboratory. In the paper by A. D. Krisch (16) the dependence d2u/dpdQ (p 1) was investigated for secondary protons by registering their total momentum at the center of mass system (i.e., for a condition analogous to elastic scattering). It was found that at large angles Pc = 2.1 Ge V /c the elastic and ine lastic cross sections have an identical slope: du/d(-t)cx:dtr,.{lpdQ ~ e-3pl. Consequently, the break in the slope of the graph for the inelastic cross sections (i.e., the growth of the parameter (Pi> in Eqs. (7) and (8)) takes place by analogy with the case of elastic scattering (see Ch. 2).

In the paper by Ashbury (18) the production of 7r±-mesons 200 L-----'-- --....L..---7'----! in pp collisions at 12.5 GeV/c were investigated. It was found that

o 2 6 8 for a longitudinal momentum pM = 0.6 GeV/c which is fixed in the center Number of pions of mass system the cross sections d2u/dpdQ (pi) have a break and

Fig. 16. Dependence of (p 1.> for pro­tons (e) and 7r mesons (0) on the num­ber of produced mesons in collisions at 28.5 GeV/c.

can be described by two or more Gaussian functions. These reg­ularities are possibly associated with the change in the interaction region when the transition is made to higher values of pL. This is

153

a

p+p--p+ all

• DB} 72,S {96/c • [Z2} 72,¥

a

Elastic circle

o 1,0

III c.m.s. Ge V Inelasticity factor

.. 1.¥7 31t7-• 1,37 877-v 7,10 707-~ 0,95 fi170 [2Z] .0,77 ¥37. o O,H 2570 )( 0,29 797-00,13 6Y. • 0,83 537- as) • 0,22 1¥7-

• • •

•

Fig. 17. The measurement conditions (momenta and angles in the center of mass system) for fixed inelasticity factors (a) and the dependence of the cross section d2a/dp dn on pi for various values of the inelasticity factor (b).

0,25< -tp <0,80

b

> ~ ..... d ....... 0

~ 100 20

+t ~ 0,02<-tp <0,70 '5 0 80 15 z

1, 7<11(1l+n) <2,7 ., 50 8- 10 ...

<I)

70

5

¥O 5 -+-t-

2

lL-L-~~-LLL~-L~~~L-~~~ o 0,2 o,lt O,G 0 0,2

Itp l , (Gev/cf

20

0 1,0 1,5 2,0

° 7,0 1,It 1,8 11 (1l +n)

2,5 J,O H(tr+nJ, GeV

Fig. 18. Distribution of events for various intervals of the nl!'+ masses as a function of the square of the transferred 4-momentum I tp I at 28.5 GeV/c in the pp -pnl!'+ reaction (a), and mass distribution in the pp - pnl!'+ reaction (b).

154

... 't:! 10 ~ 0)

5 ... 0

0 z 2

1 0

20

15

10 8-0 ... V) 5

I,J~I1(I/ '1(-)<I,!i

GeV

+ N 0

o 10

+1 !i

--f- ~ 0 2 z

Fig. 19. Dependence of the distribution of events for various mass intervals of the ~++'Tr- system on I tp I in the pp -p~++'Tr- reaction at 2S.5 GeV/ c.

likewise indicated by the relationships between the inelasticity coefficients, the multiplicity of 'Tr-mesons and p..L. Figure 14 shows the result obtained by P. Connally reported in [19) which indicates a growth of almost a factor of two in the averaged inelasticity factor for the number of'Tr±- mesons in the pp -PP'Tr+'Tr- and pp -pp'Tr+'Tr+'Tr-'Tr- directions at 2S.5 GeV. In the paper by E. V. Anderson [20) processes characterized by a high inelasticity factor are considered at three values of primary energy, 10, 20, and 30 GeV, and a growth of the average number of pions with increasing p..L of the secondary protons is indicated. The results are displayed in Fig. 15, and the authors explain them in the spirit of the "bremsstrahlung" model of Lewis, Oppenheimer, and Votazin. In this model, and likewise in the paper by Kastrup [21], a relationship was found between the emission angle of the proton and the multiplicity of the mesons.

In Fig. 16 from [19) the dependence of (p~) for protons and mesons on the multiplicity of 'Tr-mesons is shown for pp­collisions at 28.5 Ge V, and a substantial growth of (p~ ) is observed for protons with an increase in the number of mesons.

In aggregate the results enumerated are in agreement with the proposition that the average inelasticity factor increases (and, consequently, the number of secondary particles likewise increases) for an increase in the average transverse momentum (i.e., for a decrease of the "impact parameter If).

However, the relationship between the average magnitude of the transverse momentum and the inelasti­city factor K does not convey the details of the interaction. Actually, the investigations performed in the work by Day [22) indicate a more complex link between K and p..L. In that work the differential cross sections were measured in a wide range of sca ttering angles for pp-collisions at 12.4 GeV Ic while registering the values of K in individual series of measurements; these values, in turn, were varied within the limits 0.08 s K s 0.94. The conditions for these experiments are illustrated by Fig. 17a, while Fig. 17b shows the momenta of the secondary protons corresponding to various K.

It was found that in the range 0.08 s K s 0.62 the absolute values of the cross sections-d2aldpdQ in the center of mass system are close to being constant for fixed values of p..L. This can be seen from Fig. 17b where for various K the points showing the cross sections d2a I dp dQ are superimposed on each other. Consequently, in the range O.OS s K s O .62 the quantity p..L does not.depend on K. In contrast to this the the cross sections d2aldp dQ decreases with an increase in K in the range 0.7 s K :5 0.94.

Further on it is clarified that for various K the dependence of the cross sections on p..L have the form

i.e., in accordance with Fig. 13 it is close to the dependence of analogous cross sections for various secondary particles. Consequently, similar regularities are manifested in the spectra of the protons and the produced particles.

1.6. Quasi-two-particle Processes

Quasi-two-particle processes are more probable at a high multiplicity of the secondary particles.

Chan Hong Mo [23) considers that the separation between two-particle and multiparticle processes is basically artificial and is merely a historical relic. This conclusion is based on the fact that without knowing the dynamics of hadron collisions one cannot subdivide events according to multiplicity, by analogy with the fact that the subdivision of interactions into peripheral and central interactions was never assured (in the experimental sense) by its original content. However, the singling out of quasi-two-particle pro­cesses is justified from the experimental standpoint by the fact that for the time being more complete data

155

156

70

10 100 Plab' GeV/c

Fig. 20. Dependence of the total cross section of the pp-+ pm!" + reaction on the momentum of the protons.

0 z

0 30

"if G,O ~ ..c S 5,0

';' o X ~,O

~ 8 0 u JO

1& Gev!c

50 90 120

G,oGeV/c

00 90 120 ¢,deg

28,S GeV/c

160

",/20 0) "0 0 ..... ';;;

5 > 0)

'to .... 0

0 z I/J, deg 0 50 12011, deg

Fig. 21. Distribution of events over the Treiman-Yang angles in pp -+ ,A++P7r- reactions (the solid line corre­sponds to the Regge model; the dashed line corresponds to the one-pion model).

TABLE 1. Dependence of the Parameters a, b, c on the Prim-ary Momentum

GeV/c; 4-momen-Prim. momentum'j

tum t. (GeV /c)2 b, (Gev/cf2 j c, (Gev/cf4 j Literature

15, I I 4,08±O,14 1 7,89±O,59

I

-O,43±O,59

I [261

0,22 < ~- 1 < 0,78

18,4 \ 4,18±O,O81 8,58±O,24

I I [271

0,2 < ~~I < 0,5

19,84 14, 19±O, 151 8,68±0,79

I 0,70±0,92

I [281

0,2<-1<0,8

20,0 1 4 ,23±0,1O I 9,15±0,45

I

0,72±0,45

j [261

0,21 <~-I<O,8

24,63 1 4 ,09±0,30 I 7 ,97±1 ,56

I 0,82±1,83

I 1281

0,25 < -I < 0,75 ------_._--

\ 3,76ctO,12I I I

29,7 8,O2±O,60 -O,64±O,65 [261 0,21 <-I < 0,73

can be obtained at a lower multiplicity. Experiments show that an analogy with elastic scattering is ob­served - each of the colliding particles has a tendency to transfer the state of its motion to the corres­ponding final particle. Further, the excitation energy is small, as well as the 4-momentum transfer. Therefore, at high energies reactions of this type are likewise called quasielastic scattering.

In the paper by Bertocchi [9] it is noted that the dynamic causes of this similarity are completely different. In elastic scattering this is basically a consequence of the diffraction pattern. In quasielastic scattering there is no diffraction, and the observed predominance of 4-momentum transfers having a small absolute magnitude is of dynamic origin as a consequence of another interaction mechanism.

For the pp -- pll1T + reaction at 28.5 GeV the distribution of events as a function of the transferred 4-momentum -t of a proton is displayed in Fig. 18a for various mass intervals of the 1l1/ system. In the first two mass i~tervals the distribution with respect to tp may be represented by '" exp (18tp)' breaks being observed at -tp '" 0.3 (GeV/c)2.

Figure 18b gives the distribution over the effective masses of n1T+ and the slopes for small I t I as a function of the mass of n1T +.

Figure 19 displays analogous distributions for the pp -- p.6. ++1T- reaction at 28.5 GeV/c also, as well as the slope of the distribution with respect to tp as a function of the mass of .6.++1T-. It follows from Figs. _ 18 and 19 that the slope decreases when the transition is made to large masses of the system n1[- and b.++1T •

9 Baroni. et al . " Taylor o BeUetini '" Kirillova

p-p

o Lorman S • Lindenbaum ystematic Or-____________________ ~--~eIT~o-[ ----~

-0,4 __ -i--- 'I'

-O'6~----~----~----~L-----~--~~, , S 70 75 20 25 p, GeV/c

Fig. 22. Dependence of the ratio between the real part of the scattering amplitude and the imaginary part on momentum.

For small values of I t I it may be assumed that -t '" P 1 ' and a comparison with Eq. (7) shows that the reduction of the slope with increasing mass corresp9nds to an increase of (pi> (i.e' J a reduction of the "impact parameter"). In order to substantiate this conclusion it is necessary to use the data on the dependence of the processes described on the angular momenta.

Events having a low multiplicity of secondary particles in the low-energy range have been explained successfully by the mode 1 with exchange of one pion (OPM). In the energy range which is fairly far from the threshold of baryon resonances the Satz paper [24] carries out a cal­culation of the cross sections of the pp -- pn1T+ reaction on the basis of theOPM;however, it uses the experimental values of the transferred momentum.

Figure 20 presents the results of this calculation. A difference from experimental data is observed at low proton momenta and in the'" 30 GeV/c region where the reaction cross section is already less than 1 mb.

157

dN/dP~

b. (CieV: cr

~t\ 71

10

9

8

7 2

50

HtHt~t

Jl\\1 !

M

i JO

20

! 70

+ o 6 10 20 JO 50 [lab' GeV 0 0,25 0,5

Fig. 23 Fig. 24

Fig. 23. Dependence of the slope parameters b on proton energy.

Fig. 24. dN/dPJ.(p.l) in elastic pp collisions at 11 GeV/c.

Pi' GeV/c

In describing a number of phenomena good agreement with experiment is obtained on the basis of the Regge model. This is evident, for example, in Fig. 21 where histograms of the distributions with respect to the Treiman-Yang angles and calculations according to the Regge and OPM models are presented for the pp -+ ~++P1T - reaction at 6.6, 16, and 28.5 Ge V /c.

II. ELASTIC pp-SCATTERING

The general case is complexity of scattering matrix T = D + iA. For elastic scattering of fermions the differential cross section is related to T by the equation

10.27.--_ _ _ ___ .,....-_____ _ -,

g,G

Z,O

1,0

o

70'

I I

I I , ,

I I I I I

-'

5 70 15 P" GeV:

w·u~---~----~~----~-~7'_~~ o 10 20 JO s sinO, GeV2

Fig. 25. du/dt (s·sin 8).

158

(9)

where S = (PI + P2)2 is the square of the total energy in the center of mass system. For zero-angle scattering:

(10)

If the real part D of the scattering amplitude is zero, then we obtain the equation

(~) = (PCOtOL) 2 dQ 9=0 4n '

(11)

which relates the total cross section Utot to the minimal differential cross section at 0° which is called the optical value. It is convenient to consider the relativistic invariant du/ d(-t), where t = (Pl-pl)2 = (P2-P;)2 where p~, P2 and p~, p~ are the initial and final momenta.

The following relationships hold:

(12

t= -2p~(l-cose). (13)

For the subsequent considerations it is convenient to separate the ranges of small and large scattering angles, and likewise to isolate the angles'" 90°,

10-n

1//"16

10'11

"'~ 70-1/

0

~ 10-Z3 0) \!)

1::::'. .os 10.JO

0 ... ~ -s:. ~

IO"JI

!f!"' 10·JJ

1~,25 10-

19,2; 19,3

10-J1 19,3 21,3

10-Jl 0 2

80· 90 ~ (j 8 10 12 It, 15 Itl, (GeV/c)1

Fig. 26. du / dt (I t I ) in elastic pp scattering.

2 . 1 . Small-Angle Scattering

A comparison of the experimental values of the differential cross sections near OOwith those cal­culated according to Eq. (11) allows a conclusion to be drawn concerning the real part of the scattering amplitude. The first indications of the presence of a real part were obtained on the synchrophasotron at the High-Energy Labora tory [25]. For extrapolation of the differential cross sections to a zero angle an excess above the optical cross section was obtained which corresponded to a ratio'" 0.7 ± 0.2 between the real part of the potential and the imaginary part. Currently known values of the magnitude of the ratios

a(S) of the real part of the scattering amplitude to the imaginary part

",27,9 "'21,9

.. '" 1,8 JO,9

are given in Fig. 22.

If the scattering angles are still not very small (i.e., interference between the Coulomb scattering amplitude and the real part of the nuclear­scattering amplitude is negligible), then from Eq. (10) and Fig. 22 it follows that at energies'" 10 Ge V the contribution from the real portion of the amplitude to the cross section is '" O.l.

The cross sections du/d(-t) (t) in the range of small values of I t I may be described by the phenomenological equation:

d t::.. t) = ex p (a + bt + ct2) . (14)

In [26 -28] values of the parameters band c were obtained which are given in Table l.

Measurements of the parameter b were carried out by G. G. Beznogikh [29] on the Serpukhov accelerator in the energy range from 12 to 70 Ge V at very high values of I t I :

0,01<:lt\<0.12 (GeV/c)2. 10.,°L-_-"-:_--'-_----''-:--;=::-...1.-...J

1,0 2,0 J,O ~,O viti, GeV/c The dependence of b on the energy is expressed by the author using the

Fig. 27. du /dn (\ t \) in elastic pp scattering.

formula:

b = r (6.8 ± 0.3) + (0.94 ± 0.18) In :0] (GeV/d. (15)

159

10-15

10-1•

1;-11T11

~ .., <.::> 10-r. .:.:::

"'E 10-11 <J

;::;

~ 11)"30 ~

10-n

llr'l

10-JJ 0 0,5 1,5 2 A, GeV/c

Fig. 28. da/dl t 1 (I t I) in elastic pp scattering at 19.2 GeV/c. The curve was calculated according to Eq. (24).

Figure 23 from that paper ahows the results obtained along with previous results.*

Let us note that these values of b are somewhat higher (within the limits of two experimental errors) than those previously known. Possibly, this is associated with the fact that the measurements were carried out at very small values of 1 t 1 • The growth of the parameter b with energy causes a narrowing of the diffraction peak, since t is negative and c« b. In optical theory, if the scattering is produced by a spherical nucleus having a radius R and a transparency coefficient a for r < R, then for differential cross sections of scattering at an angle () we have

da (8 R2 (I 2J2 112 dQ ) = sin2 e - a) 1 [( - t) R). (16)

The expansion of the Bessel function J 1 into a series and the comparison of Eqs. (14) and (16) yields the relationship

c= -b2/12

between the parameters.

It follows from Table 1 that this is untrue not only with respect to the magnitude but also with respect to sign; this, combined with the presence of the real part of the scattering amplitude, demonstrates the imperfection of the optical model.

In papers by the author [30, 31] Eq. (7) was used to describe the differential cross sections of elastic scattering in the region of the diffraction ~ak. Figure 24 shows the curve calculated according to Eq. (7) for ppscattering at '" 11 Ge vic when < p 1. > 1/2 = 0.35 Ge V /c. The experimental points in Fig. 24 have been plotted according to the data given in the papers by L. S. Kirillova [32) and Foley [33]. Using the relationships

Eq. (7) may be transformed to

12 Pl. = -t+ 4p~ , (17)

(18)

At small 1 tithe factor 1 + (t/2p~) is negligible, and Eqs. (14) and (18) practically coincide, whence we obtain the relationships

between the parameters.

b c= 4p~

Thus, assuming a Gaussian spectrum of the momenta along the coordinate axes (Le., the momentum distribution corresponding to the law for the case which was the basis of the derivation of Eq. (7) or its analog (18» it is possible to describe the experimental data in the region of the diffraction peak.

2.2. Large-Angle Scattering

Elastic scattering must accompany the inelastic processes just like a shadow throughout the entire range of angles. In the range of large scattering angles the so-called universal Orear distribution was initially used extenSively:

da (P 1. ) --=Aexp -_. d(-I) a '

.. A comparisonwith calculations based on the theory of complex angular momenta is given in the paper by K. A. Ter-Martirosyan [30].

160

(19)

I I

10-2 ~ ~ II II I'

~ 70-+ \\ :;:; \~7J,Q 't; , 110 ~ 16,9~, ih 11 GeV Ie ~ 10-6 '\\ 113,111 1 / :::::; Zf;,3t·., .... ~ l./~_ II" I~ - 181 ' « t ,7 " ~ ' •. '~':./f I2,9 1,,'7/ ~ 10-''r- 21'~{'~1<1;~7-t!5,0 ~ Jl,5 , ~~ -- 20

25, Ot";D)J " 1 19,~ t 16;0 L 28,7t!J,8t-·ili~;-.-- t 21,9

t~:! t Z5;S JO ......... ~ __ 125,2 L

10-12 ----~m- -,pu,,r o 2 " 6 8 70 72 1* 75 78 20 It I ,.(Gevlcr

Fig. 29. (da /dl tl )/(da/dl t 1)0 as a function I t I for elastic pp scattering at various energies.

in which a '" 160 MeV/c. It was likewise used to describe two-particle reactions. With a refinement of the experimental data breaks in the distribution of da/ d n (Akerlof [34]) appeared on the straight line in the logarithmic scale, and the equation

da (SSinO) d(-t) ~exp ---g- (20)

was used to describe them. Figure 25 shows the application of Eq. (20) to the description of the data from the papers by Ankenbrandt [35] and Allaby [36]. However. as the experimental data became more refined a "finer structure" was revealed as a function of[da/d(-t)](t,s),which cannot be described by Eqs. (19) or (20) or their modifications. Figure 26 shows the results of experiments of many groups of authors which were performed over wide ranges of t and s, and taken from the Allaby paper [37]; here the dependence of da/d(-t) is expressed in the variables pl. and tl..

In order to explain large-angle s(,3.ttering a number of models have been advanced. In the paper by Wu [38] a decrease of the cross sections with increaSing p, as well as the presence of excitation models of a nucleon having an energy above 300 Me V, is associated with the difficulty of imparting a large transverse momentum to the nucleon without "breaking it." Kinoshita [39] relates large-angle scattering to the con­dition requiring minimal scattering amplitude in accordance with the analyticity requirement. This treat­ment was continued by Martin [40]. He showed that calculations based on the nucleon form factor as pro­posed by Wu and Yang are in agreement at large I t I with the fastest asymptotic reduction allowed by analyticity and the boundary conditions.

In papers by V. R.Garsevanishvili [41] a relativistic model of scattering events at high energies is considered which is based on the A. A. Logunov and A. N. Tavkhelidze quasipotential 3quation [42] for the scattering amplitude in quantum field theory. In this model the scattering of hadrons may be described by a smooth complex potential V(r. E), which depends on energy and is a nonsingular function of the relative coordinate r. The imaginary part of this potential is caused by inelastic processes. This assumption means that the scattering may be treated as an interaction between two "loose" systems.

Let us introduce the simplest nonsingular quasipotential of the Gaussian type:

- (n 3/2 -.!-V (5, r) = iSgo a) e 4a,

in which go > 0, while the parameter a determines the effective interaction region and in the general case cannot increase more rapidly than la I :5 In s, s - 00 with increaSing energy. This potential ensures a constant total cross section at high energies.

The calculations were carried out for the range of small angles on the assumption that I tis I « 1, as »1. For large fixed angles the condition I tis I "'" sin 2( 8/2) was used. The equation

(21)

was derived in which q= vm. 161

162

25 GeVic

(d6/dlt )/(d6/d lt l),

, " \~~ .... , \' '- I a ,', ------11 GeV c , .... , , , .......

.... --" ----'::-~.... -----1. ,,"" ..... .....

'~ ..... .:---' ..... , ................. '-___ l!_ ... ~ ......... ...... ..... ...... .... 25 ----

.... ..... ...... ..... b ............ _ 30 ---_

... --- ---o + 8 12 Itl. (GeV/c)'

Fig. 30 Fig. 31

Fig. 30. M/dl t I as a function of I t I , calculated according to the hybrid model [50].

Fig. 31. M/dl t 1/ (da/dl t 1)0 as a function of I t I according to the data of [51] (a) and [52] (b).

35

• p-p t p-n

o p-p c p-n v p-p

i (p~n 1>p ~ •

30~ ________ ~ ________ ~ ________ ~~ ________ ~ ________ ~ __ ~

o 5 10 a 15 20 P" GeV c

.,. rob "tot'

1,5

H

1,2

1,0 0

38

36

o

o (pd)

c (pd) A (pd)

c2 ~ 1. ! (np) ~'I' I 1. ~ .,. (np) t t-'I'-'I'f~J!if~{\2-f}- i

5 10 15

b

20 P, ,GeV/c

Fig. 32. Dependence of the total cross section of pp scattering (a) and np scattering (b) on the momentum.

Fig. 33. Dependence of the cross section for the production of three and five charged particles in collisions on the primary momentum.

A peculiarity of Eq. (21) resides in the fact that for fixed momentum transfers at large scattering angles dojdn depends weakly on energy and only via the parameter a. This parameter is related to the width of the diffraction peak at small angles. Figure 27 shows the experimental data and the curves calculated according to Eq. (21) for a =3.0 (GeV/c)2, the authors noting that Eq. (21) is not applicable to angles ,...., 90°.

ill the papers by Ansel'm [43] large- angle scattering of protons was considered on the basis of the theory of complex angular momenta, and the resulting equation shows a drop of the cross section which is close to the Orear empirical dep­endence: da/dl t I ,...., e-plja , where the parameter a ~ 140 Me V if V In (s/4m2) "'"' 2 • Further on it is shown that for scattering angles 8 < 60- 65° the experimental data at different energies may be described approximately by the linear depen­dence

-- In r d (Il~ I I ( d (i~ I ) 0 J ~~ 3.G (1 -I- I /~),

where ~ = [n(s/4 GeV)2, T =-t/lGeV2. For stipulated T the values of-ln[da/dl til (dal dl t I )01 depend fairly essentially on ~, which qualitatively substantiates the theoretically predicted logarithmic energy dependence (,...., {~ ) of the exponent of the exponential drop of the cross sections with respect to h.

Then we find that the equation for dal d I t I includes oscillation terms containing (T, O. Oscillations in the differential cross sections were likewise obtained in the paper by I. V. Andreev [44] in which the authors started from the shape of the diffraction peak and the unitarity condition for the scattering amplitude, which is taken in the form A(p, 8) = I2 + F(p, 8), in deriving the equation for large-angle scattering. Here

/, _1_ , J dO dO sin 01 sin O2 A (p, ell A* (p, 02) ,.) 1 2, 1 ~ , ,l2.~". {[cosO--cos(OJ i 0,)1 [cos (OI-82)-cosOli /

(22)

where p, 8 are the momentum and scattering angle in the center of mass system; the integration domain is 18 1-821 < 8; 8 < (8 1 + ( 2) < 27[-8.

The function F (p, 8) describes the contribution of ine lastic processes to the elastic-scattering amplitude. The authors use the approximation in which the scattering amplitude is considered to be purely imaginary throughout the entire 'region of the diffraction cone ·8 < 8 d, while the amplitude normalization corresponds to the optical theorem:

1m A (p, 0)0-0 = 4p"ot.

Assuming for () < 8d that A (p, 0) ~ 4ip2atot e-a P282/2 and substituting this expression into (22), one may confirm the fact that the two-particle contribution I2 has a less abrupt angular dependence than does 1111 A «(7, O)fI, ~ cxp (-a"p2()2!4J. This means that for 8 < 8d the principal contribution to ImA(p, 8) is made by the function F(p, 8). Consequently, the presence of a two-particle contribution in the unitarity condition must lead to a weakerfall-offofthe amplitude with increaSing angles, which is what is observed.

ill the ranges of angles 8 »8d it is assumed that ImA(p, 8) »F(p,8), but that in a certain range of angles one still may not neglect F(p, 8). As a result the authors obtained the following equations for the logarithm of the differential cross section:

In (-ff~1 c) ~ --21n~(1-111 (4na/otot \pe +- 2 ~: exp l- (V2na-V2a In (4nalatot ») pElI cos (V2na pEl-rp).

The authors note good agreement of the calculations performed according to this equation with the experimental results given in [36] at energies of 9.2, 10.1, 11.1, and 12.1 GeV. However, it is more difficult to match the calculations with more exact data from [37] in which detailed measurements were carried out in the range I t I,...., 1 (GeV/c)2. A description of the results of [37] is given in the paper by the author [45] on the basis of the generalized equation (18).

163

12

20

18

12

8

#' np7(+J[-[mff'j 'l#' m=l,Z ..•

f ~

d,mb

12

8

4

6,mb

nn7(+7(+7(-[mf(°1 2 m= 0, "2 ...

ppr

2

1

2

2

2

nnJ[+ff+f(+f(-f(-[mJ["J m~0,1,2 ...

o 2 If 881012Po,GeV/c 0 2 4;; 8 1012 Po, GeV/c U 2 t,. 6~ 8 10 i2 Po' GeV/c

Fig. 34. Dependence of the cross sections of the partial reaction in np collisions on the primary momentum.

It is assumed that scattering occurs in three discrete interaction regions (Le., quantization of the impact parameters or of the mean-square momenta occurs):

<Pl>i/2:(pl>~/2:(Pl)~12= 1:2: 3. (23)

Further, summation of the partial amplitudes corresponding to the values (p 1> 1~2 is assumed with allow­ance for the relative phase shift (/Ji ~ On the basis of this the following equation is obtained for the differ­ential cross sections:

70 <:;-'.--------------,

p2 ) ( PO) +2 (CjC2)1/2COS (jl2exP (-~ +2 (C jC3) 1/2 COS(jl3 exp -~ s(P]) s(Pl>

1/2 (Pl )] ( t) + 2 (C2C3) cos «jl2 - (jl3) exp - -72-- 1 + 2"2 . _ (po> Pc 13 1-

(24)

Figure 28 displays the experimental data for scattering at 19.2 GeV/c from [37] and the curve calculated according to Eq. (24) when the para­meters were assumed equal to (pI> 1/2 = 0.35 GeV /c; C10 c2, C3 in units 10-27 cm2/ (GeV/c)2 were equal to 88, 0.15, 0.001;(/J2 = 150°, (/J3 = 0 0

• Equation (24) describes well-known experimental data for other values of high energy as well. the prinCipal parameter (p 1>1/2 being constant within the limits of the accuracy with which they can be determined from the experimental errors: (p3//2 = 0.35 % 0.01 GeV /c. The central point in the model proposed is, however, not the constancy of the parameter (pI>, which may be dynamically variable as a function of energy, but the condition (23).

If in Eq. (23) we do not restrict the analysis to three values and introduce (pD. Ch C2. C3, then for do/d (-t) we have the following

D,0~L.,2..L--'o,Lt,.--'--:D,Lfi--'--/J,L8--'--l,Lo"--JtLI,L(~G~ev/c'f equation instead of (24):

Fig. 35. Differential cross sec­tions for elastic np scattering for 8 < Ekin < 10 GeV.

164

n 2 n 2

dt~t)= {~Ckexp( -k2~~2 J +2 ~ (CjCk)!/2coS(jlkexp[ -2~~ > (1 + ;2)] k=! 1- k=2 1-

k+m=n 2

+2 ~ (CkCh+m)!/2 COS «jlh-(jlh+m)exp[ -2~P~>(kI2 +(k~m)2)]) (1+-A-).(25) _ 1- ~

m=!

2 5 I 10 12 75 [kin' GeV

Fig. 36. Dependence of the slope parameter b of elastic np and pp scattering on Ekin of the primary nucleon.

From the values of cr, C2' C3 given earlier it follows that the convergence of the sums in (25) is rapid. Equations (24) and (25) yield do"/d(-t) --0 for a scattering angle () -- 90°. Consequently, it is necessary to introduce a small additional term which is essential for (j -- 90° and makes practically no contribution to the total cross section. Integrating Eq. (24), we obtain the total elastic scattering cross section:

(26)

The values of O"ela determined from Eqs. (26) for chosen values of - (pi), Cj, C2, C3 are in agreement with experimental data which for the time being, regrettably, are known with an error of at least 10%.

2.3. Scattering in the Range of Angles near 90°

An experimental peculiarity of scattering events at small angles when the energy is fixed lies in the fact that the differential cross section tends to a constant value for (j -- 90°. With increasing energy, and likewise for fixed I t I an abrupt decrease of the cross sections near the limiting values occurs (see Fig. 29). An explanation of these phenomena is advanced on the basis of opposite models. In the first model scattering at (j -- 90° is caused by the processes which occur at smaller angles. In the second model, which is based on statistical theory, the formation of an excited system occurs and in a particular case its decay into two primary particles. It is evident that this process must yield an isotropic angular distribution in the center of mass system and consequently also a contribution to the differential cross sections at all angles. In [46] the central collisions are assumed to be those for which thermal equilibrium is established; i.e., the collision time is sufficient for the propagation of a shock wave forward and back in the portion of

Wlr----r---,----~---r----~---r--~ '1 , • np-np

opp-pp

2J,4:!2 Gev/c 2~,6 Gev/c

I Fig. 37. Differential cross sections for elastic pp and np scattering in the momen­tum range'" 21-25 GeV/c.

nucleon which does not intersect another. By determining the probability of central collisions and the statistical weight of the elastic channel for the decay of an excited system, Hagedorn obtained the following equation for the differential elastic-scattering cross section:

dcr cr· 4m2 (--) 1 -=--.!!!....-exp [-3.25 V s-2 dQ 2n s ' (27)

where O"in is the total cross section for inelastic scattering, a/s is taken in nucleon-mass units. This equation ensures the abrupt decrease of the cross section with increasing energy and shows agreement with experiments.

It is of interest to carry out an analogous calcul­ation according to the P. Rotelly model [1] expounded in Sec. 1.1, which was in accord with experimental data on the dependence of the number of produced particles on energy and likewise on the ratio K±/~+; consequently, one can calculate the probability of elastic nondiffractional scattering. In [47-49] scattering at high energies is considered as a random process by means of which the angular distribution of the elastic scattering is also explained. In these papers the conditions were obtained

165

lol 2,0 It.O J,OGeV/c

0,01.,-:-,--:------,---:-, 1,0 0 -1,0

o 100

10

~,O 8,0

S,I Gev/c

It I ! (GeV/c Y-o 2,0 ~,O 0 2.0 ~O 5. 0 0 2,0 It,O 0.0 • d . ~ d

, , J,5 GeVtc It,GGfMc ¥,l GeV/c

10 ~ t

t 1 f

\ 0,1 ~

10

0,1

. " ~ t

\ 10 •

~ .. 1 •

I "'l 0,01<-:---<----'-........... o -1,0 1,0 0 -1,0 o,Ol.,-:-,--'--'--.J 0,0

1,0 ° -7,0 1,0 cos 8'

Itl , (GeV/C)2 o

100 ~O 8.0 ° ~O 8,(} 0

100 100 , , ,

S,5GeV/c 5,i Gev/c

10 ~ 10 ~ 70

~

~ + ,

1 \ t

I )rI

~ I

'n. 0, 0,

0,01 0,0

~, o 8,0

5,8 GeV/c

I I 0,001 0,00 0,00 1,0 0 -1,0 1,0 0 -1,0 1,0

y O 0 0, 001 L......L.""",-"--'

-I, 1,0 0 -1,0 cos rr

Fig. 38. Dependence of the differential cross section for elastic np scattering on cos 0*.

for the correlation of the longitudinal and transverse components of the momentum. and likewise for the dispersions; on the basis of this it was concluded that a sudden "emergence onto a plateau" is possible for the elastic-scattering differential cross sections at transverse momenta which are comparable with the energy of the scattering particles.

In the hybrid model developed by Chiu [50] the scattering amplitude is taken in the form of the sum of the optical diffraction part and the part which develops into exchange of an "absorbed" Regge pole. This model predicts. in particular. that the differential cross sections for elastic and inelastic processes must have an identical dependence on t at large I t I regardless of the behavior of these cross sections at small I t I . Figure 30 shows the result of the calculations according to the hybrid model.

In the papers by Yang [51. 52] a model is proposed in which the colliding particles are treated as finite extended objects which penetrate each other. Under these conditions a damping of the wave function occurs. the result of which is elastic scattering. Figure 31 shows the curves calculated according to this model using the dipole electromagnetic proton form factor. Curve a was calculated for a purely imaginary scattering amplitude; curve b was calculated with inclusion of the small real part of the amplitude. The dashed lines illustrate the experimental results.

It follows from Fig. 31, that although the slope of the calculated curve decreases at angles (} - 90°. they nevertheless yield a more rapidfall-offofthe cross section.

III. NEUTRON - PROTON COLLISIONS

From general propositions on strong interactions it should be expected that in the high-energy range. neutron-proton interactions have properties similar to proton interactions. These reactions allow the

166

15(1), mb/(GeV/c)2

Q, ~,~~~~~~~~~~ 7,0 0,8 0,0 0,9 0,2 ells 0

Fig. 39. Dependence of the differential cross section for elastic np scattering at 5 Ge Vic for states with iso­topic spin 0 and 1 on cos ().

details of the interaction to be revealed from the asymmetry of the col­liding particles in the center of mass system and information on the role of identity to be given for purposes of theory. In np collisions an additional process - elastic charge exchange - occurs also.

Data on neutron-proton interactions are less detailed and exact as a consequence of the fact that experimentally the study of np collisions present substantially greater difficulties - there are no primary intense monochromatic neutron beams and no pure neutron targets. In investi­gating np collisions the simplest way is to perform experiments in proton beams using nuclear targets (deuterium targets are the best version in electron experiments) and likewise experiments in filling bubble chambers with deuterium. However, the nuclear binding of a neutron complicates an analYSis of the experiment. The second way is to perform experiments using hydrogen targets in neutron beams which are obtained as a result of the interaction of primary particles. In experiments on elastic np scat­tering the detection of a scattered neutron and a recoil proton allows the kinematic relationships to be used for determining the energy of the primary neutron.

3.1. The Total Cross Sections

Figure 32 shows the energy dependence of the total cross sections for np- pp-collisions. As is evident in the figure, it should be assumed that at the energy which is the largest of those achieved the cross sections are identical within the limits of the experimental errors. This is in agree­ment with the papers by I. Ya. Pomeranchuk and L. B. Okun' [53,54] on

the equality of the asymptotic limit of the total cross sections for interaction of particles belonging to one multiplet with a stipulated target.

3.2. Production of Particles and Resonances

The energy dependence of the average number of particles produced in np collisions is close to the analogous dependence for pp colliSions, but the lower accuracy of the experiments and the narrower energy range in which they are conducted for the time being does not allow the exponents to be estimated in the equation n(E) = EX.

In a number of papers data have been obtained on the energy dependence of the partial cross sections and effective masses in np reactions. The results of investigations carried out in papers by V. I. Moroz [55,56, 57] involving the irradiation of a propane bubble chamber with neutrons are shown in Fig. 33. This figure presents the dependence of the production cross sections of events having three or five charged particles, as well as the ratio of these cross sections, as a function of momentum.

Figure 34 presents the cross sections of various reactions as a function of energy.

In these experiments effective-mass distribution were obtained for the systems: p7r + 7r + • p7r - 7r - and p7r +7r- which were formed in the following reactions, respectively:

These distributions, according to the conclusions of the authors of [57]. can be explained if the mass of the isobars t:l++ and [ is equal to 1236 MeV, while in the mass range 1400 to 1700 MeV, the upper bound­aries for the formation of the t:l+++ isobars having isotopic spin 5/2 and decaying into p7r + 7r + amounts to '" 30 lib.

In the experiments by B. A. Shakhbazian [58] which were likewise carried out by irradiating a propane bubble chamber with neutrons having an average momentum of 7.5 GeV/c, resonance phenomena in two­baryon systems were discovered for the first time. It was determined that near the sum of the masses

167

10D"l-----------, , , , , , , , , , ,

a

, , , , , ' "

10-7 10-2 L-~_:__-!-::____7_:-_;!_;___;:' 0~--D.-:",O':-::'5----:O':":-,10:--"f-D.~,lS 0 0,1 0,2 0,3 o,y 0,5

Itl, (GeV/c)2

Fig. 40. Dependence of the differential cross section for elastic np-charge exchange on I t I at Blab == 0 to 45 mrad (a) and 0 to 90 mrad (b).

A, P the spectrum may be explained by resonance at the virtual level of the AP system having an energy Q == (4.4 ± 0.1) MeV. This result has been substantiated in pp and KD collisions in [59) and [60).

3.3. Elastic Scattering

In the range of small scattering angles the differential cross sectionsda/d( - t) of elastic np and pp scattering have an identical slope for equal primary energies, and the diffraction peaks are compressed with increasing energy. In the 3 to 7 GeV range this is shown, for example, by the results of [61) where da/d( -t) ,... A exp (bt) was obtained for I tIs 0.4 (Ge V /c)2. On the basis of this the authors of [61) express support of the idea that the distributions of hadron matter in a neutron and photon are very similar.

In [62) the differential cross sections were determined for neutrons in the kinetic energy range 8 to 10 GeV. They are shown in Fig. 35, where for t == 0 they are normalized to the value of the cross section based on the optical theory, and the straight line in the figure corresponds to .... exp (A + bt).

The aggregate results of experiments performed by many groups of authors in the region of the dif­fraction peaks are displayed in Fig . 36. It follows from this figure that in the energy range from 10 to 20 GeV the parameter b is close to 8 (GeV/c)-2; i.e., it is identical to its average value in the case of pp ~cattering according to Table 1. This indicates coincidence of the effective interaction radii in elastic pp and np collisions. Note likewise that the value b == 8 (Ge V /cr2 is close to 1/ (p 1> == 8.1 (Ge V /c)-2 from Eqs. (7), (18) for the value (pl )l/2 == 0.35 GeV/c which was used.

For larger scattering angles, including"" 90°, it is not clear a priori what the behavior of the np­scattering cross section is in comparison with the pp-scattering cross section. In [38], for example, the notion was stated that for 90° at high energies the cross section of elastic np scattering may amount to .... 1/2 elastic pp-scattering cross sections. The difficulty in predicting the behavior of the cross sections at large scattering angles, as noted in [61), is associated with the absence of a simple model and a large number of independent scattering amplitudes. For each isotopic spin state J == 1 and J == 0 there are five independent scattering amplitudes, each having its own symmetry property near 90° in accordance with the generalized Pauli principle.

Figure 37 displays the cross sections of np and pp scatterings in the It I == 0.2 to 1.2 (GeV/c)2 range according to the data of [63, 64) for energies ,.., 24-25 GeV. As is evident from the figure, the cross sections have closely similar values.

168

In [6) it was found that in the 4 to 7 Ge V range the ratio between the cross sections is

do I do d(-I) (np) d ( - I) (pp) = 1.1 ± 0.1.

Thus, it should be concluded that closeness of the cross sections for elastic np and pp scattering holds throughout the entire range of angles. Note, however, that as yet no details or precise data exist for np scattering for I t I ~ 1 (GeV/c)2 where a fine structure of the cross sections of elastic pp scattering was manifested.

Let us dwell on the dependences of the cross sections on the isotopic spin. Figure 38 from [65] shows the differential cross sections of elastic np scattering in the energy range from 3 to 6.8 GeV, including scattering angles () larger than 90° in the center of mass system. According to the figure, symmetry of the cross sections as a function of () is observed when I cos () I s: 0.3. This symmetry indicates the small­ness of the interference term for the scattering amplitudes in states having isotopic spins 1 and O.

Figure 39 from [61] shows the dependence of the cross sections a1 of elastic np scattering in the isotopic spin state J = 1 at an energy of 5 Ge V on cos () (a 1 are identical with the cross sections of pp scattering), as we 11 as the same dependence of the cross sections aO for J = O. It follows from the figure that aO is three time s as large as a 1•

3.4. Elastic Charge Exchange

Elastic charge exchange is one of the cases of exchange scattering (Le., a process in which the particles exchange quantum numbers). In the case of np collisions an exchange of electric charge occurs. The charge exchange, on the other hand, may be treated as a case of elastic scattering, since the initial and final particles are identical, while on the other hand it may be treated as an inelastic process for an indi vidual particle.

Experiments at various energies showed that np-charge exchange has, similarly to elastic scattering, a sharp diffraction peak. This is evident, for example, in Fig. 40 of [66] at 8 GeV/c. The dashed lines in the figure are processed data on elastic np scattering at 8.9 GeV/c from [33]. The abrupt growth of the differential cross section for charge exchange can be understood by starting from the fact that the transfer of charge between a neutron and a proton is associated with a small energy transfer. In [67] the authors explain exchange and elastic scattering as coherent excitation of particles during passage through an absorptive medium.

CONCLUSION

In aggregate, the results of the experiment which have been considered show that their explanation cannot be achieved using any of the existing models of nucleon-nucleon interaction. , Each of the models is better fitted to explain specific phenomena, and even in this case the substantiation or invalidation of the model is limited by the low experimental accuracy (for example, the total cross sections for elastic pp scattering are known with an error of at least 10%). A substantial improvement of the known experimental data is required, as well as a discovery of new regularities. In this respect the results of experiments carried out on the Serpukhov accelerator, as well as on the colliding-beam accelerator at CERN, will have great significance. In the latter experiments an extensive program of research on elastic and inelastic pp collisions has been planned for a high energy equivalent in the laboratory system (1650 Ge V).

However, obtaining precision data in the energy range ~ 10 Ge V is also no less important.

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