P H Y S I C A L REVIEW D VOLUME 6 , NUMBER 11 1 DECEMBER 1 9 7 2

Soft-Pion Emission in Proton-Antiproton Annihilation* Rein A . Uritam

Department of Phys ics , Boston College, Chestnut Hill, Massachusetts 02167 (Received 10 July 1972)

We calculate the amplitude for the annihilation processflp --K'K-T'T-, and the differen- tial rate for@ - KtK-atn-, normalized to thepp - K'K- rate, in the soft-pion limit, for annihilation in flight at vanishing momentum. The computation uses the formalism of cur- rent algebra and PCAC (partially conserved axial-vector current); of particular interest is the use of Low's soft-photon theorem to evaluate the isovector photon term. The final re- sult is expressed as thepp + K'K-r'r- rate, differential in all five independent kinematic variables, chosen for their symmetry and their utility in comparing with experimentally determined spectra. The integral of this expression yields the branching ratiopp - K'K-n'n-/flp - K'K- = 2.25, in reasonable agreement with experiment.

I. INTRODUCTION

It i s well known that one can, in principle, ex- p r e s s the amplitude of any process which involves emission of soft pions in t e rms of the amplitude of the same process without soft pions.' These theorems have been demonstrated a t one time or another in the las t few years within the framework of current algebra and PCAC (partially conserved axial-vector current),' chiral ~ y m m e t r y , ~ o r phe- nomenological ~ a g r a n g i a n s . ~ It has been shown that, whichever formalism one uses, the actual results of any computation a r e the same5; the lit- e ra ture i s rich in reformulations of the general computational schemes. Applications to explicit calculations of specific processes were begun in the important early work of Weinberg and Callan and Treiman7 on K-decay form factors and pion scattering lengths. More recent applications of the current-algebra-PCAC methods have included pion production by two photons and by electron- positron a n n i h i l a t i ~ n . ~

We justify the presentation of yet another soft- pion calculation on the following two grounds. We apply these methods to annihilation of a baryon- antibaryon pair, an intrinsically interesting and inadequately understood process . Secondly, we present our result, before integration, in a form fully differential in al l independent kinematic var i - ables. We adhere strongly to the point of view, s t ressed especially by van Hove recently,'' that, lacking a complete theory, one must present ex- perimental information in the most differential form available and compare i t with theoretical schemes that also predict an equally detailed dif- ferential form, rather than consider distributions with most, o r all, of the independent variables integrated over, whereby most of the complicated behavior of transition amplitudes is lost.

Specifically, we have calculated the amplitude of the annihilation process, Fp - K'K-n'r-, and from this the rate, differential in all five kine- matic variables, normalized to the g p -K+K- total rate, in the soft-pion l imit for anpihilation in flight a t vanishingly low momentum. The inte- gra l of this i s the branching rat io

pp - K'K-T'V- - Fp - K+K-

The calculation proceeds within the formalism of current algebra and PCAC. As i s well known, such a soft-pion calculation involves three steps - deriving a n identity that i s valid a t a l l four-mo- menta, taking the low four-momentum limit of this expression, and translating this unphysical result into a statement about pions with low phys- ical four-momenta on the mass shell. We begin with the matrix element of the time-ordered prod- uct of two axial-vector currents . Contracting this twice with pion four-momenta results in an identity in which appear three new matrix elements. One i s a o t e rm which we neglect. The second i s a commutator of two axial-vector currents which, evaluated by current algebra, yields the matrix element for the emission of an isovector photon; an early theorem of Low" re la tes this intermedi- a t e result to the process without pion emission. The third t e rm i s the amplitude for the emission of two soft pions. The original t e rm itself (matrix element of two axial-vector currents contracted with two pion momenta) vanishes in the soft-pion limit, except for those contributions which result from the attachment of the axial cur rents to ex- ternal baryon lines.

We se t up the above formalism in Sec. II. In Sec. I11 we deal with the isovector photon t e rm by Low's soft-photon theorem, and in Sec. lV we compute the pole t e rms in the matrix element of the two

3234 R E I N A . URITAM 6 -

axial-vector currents. When these separate parts a r e put together in Sec. V, we arrive a t a relation between the amplitude of the process pp -K+Kh'n- and of $p - K'K- . From this one obtains, after some kinematic manipulation, the final expression for the pp -K'K-n+n- rate, differential in five ap- propriate kinematic variables, normalized to the z p -1C'K- rate, in the soft-pion limit, for an- nihilation a t vanishing momentum. In Sec. VI we integrate over all kinematic variables and obtain the branching ratio

In addition to the philosophical preference men- tioned before, such an integration i s highly prob- lematic since it includes contributions from re - gions of high pion momentum, far from the "soft- pion region." Nevertheless, reasonable agree- ment with experiment i s achieved.

11. SOFT-PION FORMALISM

terms result: (i) The time-ordered product of two axial-vector

current divergences,

( f IT(a'q.4, aVA,a(y))li) . We replace the divergences of the axial-vector

currents by pion field operators through PCAC,

apA;(x) =c,#(x), (2.2)

where

M, i s the nucleon mass; IJ. i s the pion mass; g, is the rationalized, renormalized pion-nucleon cou- pling constant (g, '/4n - 14.6); KNN"(0) i s the pionic form factor of the nucleon [KNNn(-pZ) =I]; gA i s the renormalized axial-vector coupling constant (g*- 1.2).

(ii) The commutator of the time component of an axial-vector current with the divergence of the axial-vector current; this is the a term,

Consider the reaction 6 (xO -yo)(f l[A:(x), aUA!(y)]li) . i - f +n"(k1)+~8(k2) ,

where i and f a r e arbitrary multiparticle hadronic states, such a s gp or K'K-. The pions have mo- menta k, and k,, and carry isospin a and p .

Define the quantity

x ( f lT(A;(x), A,B(y))Ii) , (2.1)

where the operators A a r e axial-vector currents with the specified Lorentz and isospin indices.

If we contract M$ with kt and k,U, three kinds of

Following standard practice, we neglect this term.12 (iii) The commutator of two axial-vector cur-

rents,

6 (xo -Yo)(f I[A:(x), A,B(y)Ili) . This is evaluated by the commutation relations of SU(2) xSU(Z),

6 (x, -y0)[A:(x), A; (Y)] =ib4(x - Y ) E ~ ~ ~ V ~ ( X ) .

By repeating the contraction with the order of k, and k, reversed, combining with the original ex- pression, and using the three results above, we have the following identity:

We perform the trivial integration in the second term on the right-hand side and insert the Mein-Gordon operator in the first. Next we take the soft-pion limit of the expression, letting both k, and k, approach zero, and obtain the "soft-pion amplitude theorem,"

In this expression the f i rs t term on the right-hand side is, to within a numerical factor,13 the amplitude for the process i - f +na(kl) +aB(k2), and the second term i s similarly related to the amplitude for the pro- cess i- f + isovector photon; the left-hand side does not vanish since M$ has contributions from terms of order k-'. In the next two sections we see that both the isovector photon term and the term on the left-

6 - S O F T - P I O N EMISSION IN P R O T O N - A N T I P R O T O N . . . 3235

hand side can be expressed in terms of the i - f amplitude. As it stands, the entire expression i s valid in the unphysical region kf=O and k,U=O, i.e., with the pions off the mass shell. It is assumed that the same relation remains valid if we extrapolate from zero pion four-momenta to low momenta on the mass shell.14 Hence, we will arrive a t an expression for the amplitude for i- f +two soft pions in terms of the amplitude for i-f.

r --

111. LOW-ENERGY THEOREM FOR PHOTON M:' consists of al l diagrams not in M;, that is, EMISSION diagrams where the photon emerges from the in-

In the soft-pion limit, k, - 0 and k, - 0. The mo - mentum of the isovector photon, discussed above, is k = k, + k, ; therefore, in this limit also k - 0, and we may make use of any low-energy theorem for photon emission.

There exists, conveniently, just such a low- energy theorem, due to Low,ll which relates the first two terms in the series expansion, in powers of the photon momentum, of the photon emission amplitude to the amplitude of the process without photon emission. We follow Low's procedure in a straightforward way, adapting it to the process we a r e discussing, the annihilation a t vanishing momentum, 3 p -Kt K-. (Until now we did not have to specify the initial and final particle states in the soft-pion amplitude theorem of Sec. 11, but must do so now since the details of Low's pro- cedure depend on the spins and charges of the particles.ll)

Consider the above annihilation accompanied by photon emission, j5p -K tK- +photon; the amplitude for this reaction can be divided into two parts:

M';' consists of the sum of a l l diagrams where the photon is emitted from an external line. In this reaction, photon emission can take place from the two charged K lines, but not from the initial proton-antiproton state, since this i s a neutral state of two antiparticles a t rest, with no net charge and no higher moments of the charge; hence, there is no net contribution to MY from $p to lowest order in photon momentum. See Fig. 1.

- - teraction "blob," rather than from the external lines. See Fig. 2.

As k - 0, we see below that M:) - k-I , a nd Mf ' - constant.

Calculating MA" from the diagrams of Fig. 1 (omitting temporarily the magnitude of the cou- pling constant), we have immediately

where the matrix elements of M a r e the invariant amplitudes of the process without photon emission with the momenta of K+ and K - a s shown in ( I ; P is the momentum of thepp system. Neglecting k2, and noting the transversality condition, kpc, = 0, we have

Writing the invariant amplitude M a s a function of relativistic invariants (in this case just the squares of the momenta),

I! -% M(-4MN2, -m2, -m2+2q2*k) . qzSk

(3.4)

Now M:' i s not gauge-invariant, but M, is; hence,

FIG. 1. Lowest-order diagrams contributing to MA^) . FIG. 2. Lowest-order diagram contributing to M L ~ ) .

3236 R E I N A . U R I T A M

k",, = 0 ,

and (3.5)

k"? :2' -k"(" . P

Now

kPM;' = a2M2q1 - k - a,M2q2. k ,

where 3,M and a,M a r e the derivatives of M with

Adding A[:' and M:', the derivative t e rms vanish, and one obtains

This i s the result we sought; it relates the in- variant amplitude13 of the process & -K+K- (pp a t res t ) to the invariant amplitude, M, , of the same process accompanied by emission of a pho- ton of vanishing momentum. We can insert the coupling constant now, noting that the coupling of the isovector cur rent i s to the isovector charges, which for Kf and K - a r e +$ and -*. Since Low's procedure refers to a physical, uncharged photon, we res t r ic t the isospin index of the vector current in the soft-pion-amplitude theorem of Sec. I1 to that of a neutral photon, i.e., the isospin index i s 3 . Having noted these two points, a s well a s keeping track of numerical factors, the result above can now be used to evaluate the isovector t e rm in the soft-pion-amplitude theorem, Eq. (2.4) of Sec. 11.

IV. POLE TERMS

We must now consider the left-hand side of Eq. (2.4), namely, k Y k g ~ 1 2 . This t e rm will also be related to the Jp -K'K- amplitude in our approxi- mation. Recalling the definition

since we a r e calculating the amplitudes to zeroth order in pion momenta, i t i s necessary to identify and retain only those par t s of ~vI,",B that go like for smal l k .

In the diagrams for I W ~ ! the axial-vector current vert ices may be attached (i) to an internal line, (ii) to a terminating external pion line, and (iii) to a nonterminating external line. Only the t e rms of this las t type a r e of order k-'. Furthermore, the insertion of A into a pseudoscalar-meson line i s

respect to i ts second and third arguments, r e - spectively. Since M:' no has no singularity a s k - 0, one can combine the las t two expressions and solve for M:',

M O ( ~ ) = - 2 q y a , ~ + 2 q g a , ~ . (3.6)

Expanding M:) itself to zeroth order in k , and letting k - 0 in the arguments of M , one finds

I

forbidden by parity. Thus, for 3p - KfK-, there a r e no insertions in the K lines, only in the p and p lines.

The diagrams to consider a r e shown in Fig. 3. We write down their contribution in a straightfor- ward way, using Feynman rules; p, i s the proton momentum, p, the antiproton momentum; k , and k , a r e the momenta carried by the axial-vector currents , and a and p their isospin indices. The central interaction we write out explicitly in t e rms of relativistic invariants, Zm =A + B y Q, where Q =ql - q 2 . Several points should be noted. Since we a r e dealing with annihilation a t vanishing mo- mentum, we retain only the la rge components of the Dirac spinors; for that reason the coefficient of A vanishes and only t e rms proportional to B r e -

FIG. 3 . Diagrams of order k - 2 in M $. For isospins corresponding to emission of T + and T-, only diagrams (a), (b), and (c) contribute.

S O F T - P I O N E M I S S I O N IN P R O T O N - A N T I P R O T O N .

main. We retain only the lowest-order terms in diagrams of Figs. 3 (a), 3 (b), and 3 (c) contribute. pion momenta in kfk:M$, that is, zeroth-order Evaluating in the 3p center-of-mass frame, for terms. We specialize to isospins corresponding definite spin states s and Y, the sum of these to nt and .rr- emission; for that case only the three three terms of Fig. 3 is

This expression i s proportional to B. Now, using EX =A +By. Q, the invariant amplitude for the annihila- tion pp - K'K- a t res t is itself also proportional to B,

V. fip + K + K ' r + n - AMPLITUDE AND RATE

We have evaluated separately all the terms in the soft-pion-amplitude theorem, Eq. (2.4). Putting to- gether all three terms, and keeping track of numerical factors in the various amplitudes,13 we have the invariant amplitude for soft-pion emission in pp - KcK-.rrta- a t vanishing momentum,

where B is the constant in the corresponding invariant amplitude for j$-K'K- in Eq. (4.3), which has been explicitly used above. Equation (5.1) i s an important intermediate result of our calculation, and forms the basis of all subsequent results.

Since we a r e dealing with the proton and antiproton both at vanishing momentum, we can introduce non- relativistic Pauli spinor notation; in that notation we have

where

From Eq. (5.2) we compute the absolute square of the amplitude, (MI2, averaged over initial and summed over final spin states,

+ (kg)' ( tk l . Q)'E 22 - 2[ (k:)' + (hi)' + k%;] (f)k Hz) (G1 - 6 ) (2 6)) +4qf (k:+kg)[(tk,.tk,)G2- ( ~ l . Q ) ( ) k 2 . Q ) ] + ~ f 2 G 2 ) .

The differential rate i s given in terms of ((MI2),, by the usual expression,

Of the twelve kinematic variables for a four-particle final state, energy-momentum conservation elimi- nates four; the arbitrary direction of one vector and the arbitrary orientation of the entire system about this vector provide three additional trivial integrations. Hence, five independent variables remain. We choose the following particular se t of five such variables that a r e useful for comparing with experimental-

R E I N A . URITAM

ly determined spectra:

where mKK2 = -R2 is the invariant mass of the dikaon system (R =q, +q,); nzVn2 = -K2 is the invariant mass of the dipion system (K =k, +k,); 6, is the angle in the dipion res t frame between pion 1 and the negative of the total momentum; 6, is the angle in the dikaon res t frame between kaon 1 and the negative of the total momentum; $J is the relative azimuthal angle between the decay planes of the two-pion system and the two- kaon system. This set of variables possesses attractive symmetry, and makes calculation easier since the variables a r e not kinematically restricted, except in a simple way, i.e.,

A straightforward calculation yields the expression for the rate, differential in the new set of variables,

In order to normalize our final answer to the jip - KfK- rate, we must first calculate the pp - KfK- rate. This is done easily; since

M, = rfl(MN)yS Qur (MN),

one finds BZ (M 2 -m 2)3/2

W ( ~ ~ P - Z ~ * K - ) = ~ MN

(5.6)

Thus, one obtains the final result, the differential rate of pp -KfK-nf a-, normalized to the total 5p -K+K- rate:

d5w(Fp -KfK-s*a-) ~ ( 5 p - K + K - )

= dmTT2 dmKK2 d (coe.8,) d (cost?,) d$J

where /'I2 is the bracketed part of ((M/~),, in Eq. (5.3); explicitly

1 f i ( 2 = 3 2 1 7 2 { d t . ~ + k ~ ) 2 ( ( i ; l ~ t k 2 - ~ ) 2 + ( k : + k ~ ) 2 ( ~ 1 . t k 2 ) 2 ~ 2 + (k:)2(s2.~)2c12

+ (k;)2((l . ~ ) ~ c ~ 2 -2[(k;l2+ ( k ; ) % k : k ; ~ ( ~ ~ . ~ ~ ) ( ~ , - G ) ( ( i ; ~ * a ) )

+ 4 ~ f (k :+k~) [ (~ , .~ , )Q2 - (tk, .Q)((i;,.G)] + +f2G2. 1$12 must of course be expressed in terms of the preferred set of variables, mKK2, mTa2, cost?,, cosf?,, and @. We express the variables appearing in Eq. (5.8) in terms of Lorentz-invariant variables, and then write these in turn in terms of the preferred set. These conversion formulas a r e tabulated in Eqs. (A3)- (A21) in the Appendix.

Our final expression, Eq. (5.7), is differential in all five independent kinematic variables. Although this is a relatively complicated expression, we feel i t is important to work, if possible, with such fully differ- ential expressions, before integrating to obtain the branching ratio.

i

VI. REMARKS which occurs in the Low procedure and subse- Certain aspects of our result deserve comment. quently in Eq. (5 .I), which expresses the pp

The first specific point is the presence of the fac- - K " ~ - n " n - amplitude. This term is zeroth order tor in k, and k,; yet i t nonetheless depends on k, and

k,, i.e., on which pion is soft. This is reminis- (41. 92 ' @ 2 (k -K+K- cent of a similar term in Weinberg's result for 41' (k +k,) -9,. (k2 +k,) the K,, form factor,' which is present because both

6 - S O F T - P I O N EMISSION IN P R O T O N - A N T I P R O T O N . . . pions a r e simultaneously taken off the mass shell in the calculation.

As we emphasized in the Introduction, although the resulting expressions a r e cumbersome, there is merit in computing theoretical expressions, and presenting experimental data, in fully differential form. Our final result, in its present form, in- cludes a large quantity of information about the reaction, Pp - K'K-a'n-, and predicts a spectrum in the five independent kinematic variables. The process of pp annihilation has had increasing ex- perimental study,15 and sufficient data a r e begin- ning to emerge to allow detailed comparisons of spectra, although a t present one cannot yet make a full comparison in the five kinematic variables.

The simplest comparison of course is with the branching ratio; if one integrates over a l l five ki- nematic variables one obtains the branching ratio (B.R.)

for annihilation a t vanishing momentum. This in- tegration is highly problematic, since it includes all allowed pion momenta, including values up to 735 M ~ V / C , whereas the theory applies only for low momenta, ideally for unphysical zero four- momenta. Nevertheless, we obtain

a r e

We expect that B.R.(pp -pK-n+n- ) i s comparable to the last two numbers listed. For comparison, from the above numbers, one finds

B.R. Pp-KY~in'a- =4. $P-K;K;

The proton-antiproton annihilation reaction has lent itself to a calculation of soft-pion emission since even for annihilation a t vanishing mamenturn there is enough energy to produce the desired final states; a t the same time, in this limit useful computational simplifications occur.

The annihilation process itself has attracted rel- atively little theoretical interest, due to its com- plexity and lack of detailed experimental results. Some proposed models have included production and decay of two or three vector mesons,lg rear- rangement of three quarks and three antiquarks," and direct-channel interference of massive bo- sons.21 We have confined ourselves to a relatively straightforward application of soft-pion theorems to j?p annihilation; a similar procedure i s being applied to soft-pion emission in 3p -KEKE.~~

for annihilation a t vanishing momentum,16 in rea- ACKNOWLEDGMENTS sonable agreement with experiment.

Experimentally, KE(mn) final states a re com- I a m grateful to Professor S. B. Treiman, who monly observed for those events in which a t least brought this problem to my attention and provided one kaon i s neutral17; thus data a re lacking for valuable advice in its solution. I have also had Pp -KfK-n+n-. Typical branching ratios of spe- useful discussions with P. Nuthakki and J. Roche, cific final states, compared to all final states, who also assisted with the computer calculations.

APPENDIX

In Eq. (5.8) the term is given by the expression

1 l t i l ~ 3 2 ~ ~ { ( k : + k ~ ) ~ ( i ; ~ ~ H ~ . ~ ) ~ + ( k ~ , + k g ) ~ ( g ~ . g ~ ) ~ G ~ + (k01)2(~2 .6 )2~12

+(k;)L($1*~)2~22-2[(k:)2+(k;)2+k~kg](~l.~2)(z1.6)(~2.G)} 1 2 - 2 +4~f(k~+~)[(~l.~2)82-(~l.6)(~2-6)l+~f Q ,

where

We express the variables occurring in l@fI2 f irst in terms of Lorentz-invariant quantities, and finally in terms of the preferred set, mKK2, mXx2, cose,, case,, @, defined in Sec. V. Define the following variables:

and form the invariant quantities, Q .K, Q2, Q .A, R2, R .K, K 2 , R .A, h2. In terms of these, the vari- ables in lfi l 2 are given by

R E I N A . U R I T A M

In turn, the invariant quantities a re expressed in terms of the preferred set through the following relations:

K 2 = -mTR2, (A131

R~ = -mKK2, (A14)

R . IC = (rnRV2 tmKK2- M ~ $ ' ) , @15)

Q~ = mKK2 - 4 mK2, (A16)

~ ~ = m ~ ~ ~ -4p2 , (A17)

coso h R = -/ ( rvzTT2 - 4 / ~ ~ ) ~ / ~ [ ( n z , ~ + mKK2 -Mpd2)2 - 4mTR2mKK2]1/2,

2 fir, , (A181

*Work begun a t Princeton University a s part of the author's doctoral thesis; R. A . Uritam, Ph. D. thesis, Princeton (unpublished) Supported in part by thc U. S. Air Force Office of Scientific Research under Contract NO. AF49 (638)-1545

IS. Weinberg, Phys. Rev. Letters s, 879 (1966); Y. Nambu and D. Lurig, Phys. Rev. 125, 1429 (1962); Y. Nambu and E. Shrauner, i h d . 128, 862 (1962).

2 ~ . Weiaberg, Ref. 1. 3~ Nalllbu et nl . , Ref. 1; R . Dashen and M. Weinstein,

Phys. Rev. 183, 1261 (1968). 4 ~ . Weinberg, Phys. Rev. Letters 2, 188 (1967). 5 ~ . Dashen and M. Weinstein, Ref. 3 . 's. Weillberg, Phys. Rev. Letters 17, 336 (1966);

17, 616 (1966). - ' c . G. Callan and S. B. Treiman, Phys. Rev. Letters

16 , 153 (1966). - 's. L. Adler, B. W. Lee, S. B. Treiman, and A. Zee,

Phys. Rev. D 4, 3497 (1971); H. Terazawa, Phys. Rev. Letters 26, 1207 (1971).

6 - S O F T - P I O N E M I S S I O N I N P R O T O N - A N T I P R O T O N . . . 'A. Pais and S. B. Treiman, Phys. Rev. Letters 25,

975 (1970). 'OL. van Hove, Phys. Reports s, 347 (1971). "F. E. Low, Phys. Rev. 9, 974 (1958); H . Chew,

ibid. 2, 377 (1961). ''5. Weinberg, Phys. Rev Letters 17, 336 (1966);

H. D. I. Abarbanel, Phys. Rev. 153, 1547 (1966). I3In our notation, the Lorentz-invariant amplitude, & I f i ,

i s defined by

w h e r e N 2 = n i N i 2 ; N ~ ~ = ~ E forboson, N i 2 = ~ / r n for fermions. The integrals in Eq. (2.4) a re related to S f i through the LSZ reduction formulas. In this way Eq. (2.4) can be expressed in t e rms of the Lorentz- invariant amplitudes, which a re henceforth utilized. ' 4 ~ s i s customary, PCAC i s used in the role of a

"smoothness condition" for the extrapolation matrix elements off the pion mass shell; the dependence of the matrix elements on pion momenta i s "gentle" when aAp/ax,, i s employed a s the pion field. S. B. Treiman, Comments Nucl. Particle Phys. L, 13 (1967); S. Wein-

berg, Refs. 2,G. 1 5 ~ . Barash, Ph.D. thesis, Columbia University (unpub-

lished); C. Baltay et a1 ., Phys. Rev. Letters l5, 532 (1965); N. Barash et a l ., Phys. Rev. 145, 1095 (1966); S. Devons et a2 ., Phys. Rev L e t t e r s x , 1614 (1971); D. Cline, J. English, and D. D. Reeder, i b i d . x , 7 1 (1971); P . Espegat et a1 ., Nucl. Phys. E, 93 (1972); R . Armenteros and B. French, in High Energy Physics, edited by E. H. S. Burhop (Academic, New York, 1969). '%reliminary results for the branching ratio were

reported in J. Roche and R. A. Uritam, Bull. Am. Phys. SOC. 17, 777 (1972). 1 7 ~ . Armenteros and B. French, Ref. 15. At most

energies, charged kaons cannot be distinguished from pions by ionization measurements. I*N. Barash, Ref. 15. "J. J. Sakurai, Ann. Phys. (N.Y.) 11, 1 (1960). 'OH. R. Rubinstein and H. Stern, Phys. Letters 2l, 447

(1966). "D. Cline and R . Rutz, Phys. Rev. D 5 778 (1971). 2 2 ~ . Nuthakki and R . A. Uritanl (unpublished).

P H Y S I C A L R E V I E W D V O L U M E 6 . N U M B E R 11 1 D E C E M B E R 1 9 7 2

Absorption Picture as a Coilsequence of Multi - Regge Iteration of Low -Energy Peripheral Resonances*

Bipin R. Desa i Department of Physics, University of Calgo fornia, Riverside, California 92502

(Received 17 May 1972)

Using the empirical fact that the dominant low -energy resonances a re peripheral (i.e., they satisfy the relation 3, + 4 = kR = ~RV%) , tlle imaginary part, Ao(s, t ) , of the low-energy amplitude is written in the factorized form F ( S ) J ~ ( R ~ ) . The resonance contribution to B(s) near the resonance position i s " ( 2 ~ ~ + 1 ) , which i s " s ' /~ ; and thus the well-known value for the vector-tensor trajectory intercept of =; follows simply from peripherality. From finite-energy sum rules the residue of the trajectory i s = J , , ( R G ) . The quantity Ao(s, t ) , which in the t channel corresponds to a fixed pole at j , = $, through ale multi-Regge i tera- tion gives r i s e to a moving pole; and the imaginary par t of the total nn amplitude A(s, t ) with I, = 1 satisfies the dual absorption result, S ~ J ~ ( R ~ ) . A crude estimate gives a(0) - + ( r p / m p ) (= 0.6) and a'(0) ;= ~ ( r p / m p ) ~ 2 (= 1.0 B~v- ' for R = 1 F) . Thus, for non- diffractive processes, the multi-Regge (or multiperipheral) model i s consistent with the absorption picture and a s a consequence an approximate bootstrap solution is obtained. The physical interpretation in t e rms of j -plane cuts i s discussed.

I. INTRODUCTION

It i s a n e m p i r i c a l f a c t that the dominant low-en- e r g y r e sonances are pe r iphe ra l . T h i s i s t r u e of m e s o n as wel l as baryon res0nances . I By pe r iph - e r a l w e m e a n that the angu la r momentum j, and the 'energy of the r e s o n a n c e in the s -channel sat- isfy the r e l a t ion

j s + $ = k ~ = $ ~ G , (1)

w h e r e k is the c.m. momentum and R i s the r ad ius parameter which t u r n s out to be of the o r d e r of

the r ad ius of in teract ion, e.g., 1 F. T h e i m p o r - t an t consequence of th i s i s that the Legendre poly- nomial, Pf(cosO), a s soc ia t ed with individual reso- nances can b e e x p r e s s e d as follows:

Pj,(cos 8) = J0((2j, + 1 ) sink61

= ~ ~ ( ~ 4 3 ) ~ (2)

w h e r e t [= -2k2(1 - cosO)] i s the momentum t r a n s - f e r . The contr ibut ions of t he d i f ferent r e s o n a n c e s will , t he re fo re , vanish at the same va lue of t. Of cour se , t he lowest r e s o n a n c e (e.g., p meson) wi l l have only the f i r s t zero; the next one, pe rhaps ,