IL NUOVO CIMENT() VOL. LVA, N. 4 21 Giugno 1968

Finite-Energy Sum Rules Ior Pion Photoproduction and the (,) Regge Pole Residue Functions.

P. DI VECCHIA and F. [)RAGO

Laboratori Naz iona l i del C N E N - Frascat i (Roma)

M. L. PACIELLO

Is t i tu to di _Fisica del l 'Universi t~ - Roma

(ricevuto il 16 Gennaio 1968)

Summary. - - We have studied the behaviour of the co residue func- tions in pion photoproduction. We show that they have no zeros at t _~-- 0.15 (GcV/c)2; this is in striking contradiction with the usual explana- tion of the cross-over phenomena. The behaviour of the (o trajectory near the wrong signature nonsense point ~ = 0 has also been investi- gated. The usual theory of the dip seems to work well and suggests that unlike the ,o. the (o trajectory is nonscnse-choosing.

1. - In troduc t ion and s t a t e m e n t of the problem.

During the last year the l~egge-pole theory has been applied successfully

to the description of high-energy e lementary particle interactions. However,

the large number of free parameters present in the theo ly would probably

make all the high-energy fits meaningless if one did not have the possibility

of connecting the various processes, extract ing in this way common features

of the various Regge poles. Such a possibility is provided, for the Regge-pole

residues, by the factorization theorem (1.2), which is assumed to hold in gen-

eral as a direct consequence of the uni tar i ty condition.

Although the factorization hypothesis gives generally self-consistent results,

(1) M. G]~LL-MANN: Phys . Rev. I~ett., 8, 263 (1962). (2) V. 1-~. GRIBOV and I. Y. PO~ERANCHUK: Phys . Rev. Lett., 8, 343 (1962).

8 1 0 P. DI VECCHIA, F. DR&GO and M. L. PACIELLO

i t has recent ly led to serious difficulties abou t the behaviour of the (o-pole residue functions (3.4). For the sake of clar i ty let us s ta te the prob lem f rom

the beginning. The problem arises in connection with the so-called cross-over phenom-

enon (*). I n fact i t is an exper imenta l fact , known for a long t ime (5), t h a t if

one considers elastic scat ter ing processes ~+p, K+p, and pp, and the processes

in which the incident part icle has been changed into the corresponding anti-

part icle, name ly ~-p, K - p , ~p, one finds t ha t the process which has the big-

gest cross-section a t t = 0 has also the biggest slope. Therefore the m o m e n t u m

t ransfer distr ibutions cross each other, i r respect ively of the value of the energy,

near a common value of m o m e n t u m transfer : t = t+ ~ - - 0 . 1 5 (GeV/c) 2. More

precisely s ta ted, if we define the quan t i ty

A - - d(~(AB) da(AB) dt dt '

we find t h a t i t changes sign near $ = tc a t all the explored energies (**). This

p rob lem has been discussed, f r J m the point of view of the Regge-pole theory, pa r t i cu la r ly in ref. (e.7). I n Regge theory the quan t i ty A has the following

expression:

$ A = 2 Re ~ E{a}O{~},

{~}

where (4} denotes the helicities in the crossed t-channel, E{~} is the sum of the contr ibut ions of all Regge-pole exchanges with even charge conjugation, and O{~} is the sum of those wi th odd charge conjugation.

There are now several exper imenta l facts which indicate t ha t in all the

processes under consideration, the helicity nonflip pa r t of E{a} is essential ly imag ina ry and large a t t = 0, while the helici ty flip pa r t of E~a} is much

smaller. Since the point where the cross-over occurs is not too far f rom t = 0,

the imag ina ry pa r t of the nonhel ici ty flip E{~} will still give the dominan t con-

t r ibu t ion to the full scat ter ing ampli tude, and will still be essentially imag-

(3) R. J. N. PHILLIPS: private communication and Heidelberg Con]erenee on Ele- mentary Particles Physics, October 1967.

(4) V. BARGER and L. DURAND I I I : Phys. Rev. Lett., 19, 1295 (1967). (*) We are indebted to Dr. R. J. N. PHILLIPS for pointing this out to us. (5) K. J. FOLEY, S. J. L I N D E N B A U M , W . A. LOVE+ S. 0 Z A K I , J. J. RUSELL and

L. C. L. YUAN: Phys. Rev. Lett., 10, 376 (1963); 11, 425, 503 (1963); |5, 45 (1965). (**) We are always interested in the high-energy range. (e) R. J. N. PHILLIPS and W. RARITA: Phys. Rev., 139, B 1336 (1965). (7) W. RARITA, R. J. RIDDELL, C. B. CHIU and R. J. N. PHILLIPS: University of

California Radiation Laboratory Report. UCRL-17523 (1967). Additional references are given in this paper.

FINITI~-ENI~I~GY SVM RULES FOR VIOl{ PYIOTOPRODUCTION ETC.

inary. We are thus led to the following approx imate expression:

811

A _~ 2 ~ Im E~*~ I m 0(a } . nonfliD

We see f rom this t ha t the quan t i ty A can change sign if and only if the imag-

inary pa r t s of the helicity nonflip ampli tudes change sign. Le t us see how this

a rgumen t works in various cases. Consider first the =• scat ter ing; in this process O{a } receives a contr ibut ion only f rom p-exchange ~*): a consistent

p ic ture is obta ined by requir ing t h a t the factorized helici ty nonflip A o ~ residue

change sign a t t = to, while the helicity flip residue mus t s tay finite in order

to explain the m a x i m u m of the m o m e n t u m t ransfer d is t r ibut ion in the reac- t ion ~ - + p - - > = O + n near t = to.

Le t us now discuss the much more involved problem of the p-p and ~-p

cross-over. Empir ica l ly 0~} is found to receive the largest contr ibut ion f rom

co-exchange. In fact the coupling of the p- t ra jec tory to the nucleon-antinucleon sys tem is much weaker than the ~o coupling, as can be deduced f rom the small-

ness of the cross-section for the p-n charge-exchange react ion compared to the elastic scattering. The ~0 exchange is usual ly neglected since its coupling to the

nucleon-ant inucleon sys tem is appa ren t ly ve ry weak." I f this is the re~l situa-

tion, the cross-over in the p - p and p-p scat ter ing requires t h a t the helicity

nonflip-nonflip residue function much change sign a t t -= t~. This ~pp~rent ly

reasonable r equ i rement gives rise to one of the mos t serious difficulties for the

Regge-pole theory. In fact it is an easy m~t t e r to combine the factor izat ion

theorem with the ana ly t ic i ty and real i ty requi rements for the ampli tudes to

show tha t not only all the r factorized residue functions mus t vanish a t t = tc, but t ha t all the co residue functions mus t v~nish in ~ny react ion a t t ha t point . This possibility, ~lthough it explains nicely the cross-over in the K~=p scattering, is clearly unpleasant and in contradict ion with the infor- ma t ion t ha t can be deduced f rom other processes, as will be discussed below. We note ~t this point t h a t the behaviour assumed for the ~ residue is com- ple te ly different f rom the one assumed for the (o residue.

We emphasize t ha t the foregoing conclusions are a direct consequence of

the assumpt ion t ha t the ~o Regge pole gives the only significant C = - 1 con-

t r ibu t ion to the ]~p and pp elastic scattering' ampl i tudes and tha t the factor- izat ion t heo rem holds.

The difficulty arises when one looks a t the high-energy da ta on the reac-

t ion y+p__>~O+p. Dur ing the l'~st year this reaction has been studied b y

(*) The p'-exchange is neglected since it is not important for this kind of considerations.

8 1 2 P. DI VECCHIA, F. DRAGO ~ n d M. L. PACIELLO

several authors (s-~0) in the f r amework of the Regge-pole theory. The only

q u a n t u m numbers t ha t can be exchanged are those of the co, ~ and B mesons.

The p-exchange contr ibut ion is general ly neglected because of the re la t ively weak par3/ ' and pay coupling. I t can be shown (9) however t h a t the inclusion

of the p exchange does not change the results drastically. The B contribu- t ion however seems to be i m p o r t a n t (~o), a t least a t mode ra t e energies, despite

the fact t h a t its t r a j ec to ry is supposed to be low: in fact i t is needed in order

to explain the absence of a zero a t the nonsense point where the co t r a j ec to ry

goes to zero. Therefore if one looks only a t the photoproduct ion react ion,

disregarding comple te ly a n y other in format ion f rom other processes, one is

na tura l ly led to the following pic ture:

1) co and B exchange are bo th i m p o r t a n t in =o pho toproduc t ion ;

2) the k inemat ica l const ra in t a t t = 0 is satisfied b y evasion (~.~2) (see below);

3) the co t r a j ec to ry is similar to the well-known p t r a j ec to ry ; all the

residue functions can be app rox ima ted with constants .

In this way one can explain ve ry well the exper imenta l dip a t t-----0, and

the near ly forward m a x i m u m in the m o m e n t u m t rans fe r dis t r ibut ion a t

t _~ - - 0.1 (GeV/c) ~.

This model is however inconsis tent with the previous in te rp re ta t ion of the cross-over effect: in fact one has the problenl of explaining a m a x i m u m exact ly where one would expect a s t r iking min imum. I n order to explain this situa- tion, BARGEIr and DURAND (4) assumed tha t ex t ra co-type contr ibut ions mus t be present and denoted t hem b y ~ : they suggested t ha t the ~ could be a con-

spiring pole or a conspiring cut. Obviously in this case the cross-over in ~p and pp scat ter ing can be reproduced provided t ha t the helici ty nonflip com-

ponent of I m ( r changes sign a t t ~ t~. This does not give any restr ict ion

on the co-exchange residue. However , if this is the case, a fit to the high-energy

photoproduc t ion da ta would require a ve ry rapid and unusual behaviour of

the residue or of the discontinuities across the cut in the small ]t I region

(Itl ~ 1 (GeV/e) ~) and a dist inction between the various possibilities is b y no

means possible with the avai lable h igh-energy data .

In order to obta in fu r the r insight into this problem we have used the

(s) M. P. LOCHER and H. ROLLNIK: Phys. Lett., 22, 696 (1966). (9) 1 ). DI VECCHIA and F. D•A6o: Phys. Lett., 24B, 405 (1967).

(10) 1~. p. ADER, M. CAPDEVILLE and P~t. SALII~: CERN preprint, TH. 803 (1967). (11) M. B. HALPEI~N: Phys. Rev., 160, 1441 (1967). (1~) S. FRAUTSCtII and L. JONES: Phys. Bey., 163, 1820 (1967).

FINITE-ENERGY SUM RULES FOR PION PItOTOPRODUCTION ETC. 813

<~ finite-energy sum rule ~> (~3.~5) techniques by means of which the high-energy

paramete rs can be expressed us integrals over the low-energy ampli tudes.

This analysis is in essential agreement with the s ta tements 1), 2) and 3) given

above and with an analysis of the reactions (~G) r:_+§ p~-kP and ~ - + p - - >

-+ p~ f rom which the co contr ibut ion can be ext rac ted easily.

I n Sect. 2 we give the photoproduct ion ampl i tudes ; in Sect. 3 we deduce

the sum rules; in Sect. 4 we discuss the general behaviour of the co residue funct ions; in Sect. 5 we discuss the problem of the dips for the r and ,z trajec-

tories using the sum-rule approach; the related problem of the sa turat ion of

the Schwarz supcrconvergencc reh~tions is also discussed.

2. - The photoproduction amplitudes.

We give in this Section the essential nmchinery of the photoproduct ion

reactions. Following the classical paper by CHEW, GOLDBERGER, LOW and :NAM]~U (17) the nmtr ix e lement for single-pion photoproduct ion can be writ-

ten as

(2.1)

(2.2)

with

H = ~. H(~)g r , J

H r -- ~ M,A~"(v, t). i

M, = @~(7"s)(r" k) ,

M2 = 2 i ) , ~ [ ( P . s ) ( q . k ) - ( P . k ) ( q . s ) ] ,

M3 = y s [ ( y ' e ) ( q , k) - - (7" k ) ( q . e ) ] ,

M4 -- 27~[(y" s)(P" k) - - (F" k ) ( P . s ) - - i~]/(7 �9 s)(y. ~)],

where k and q %re respect ively ttle photon and pion four -momenta , P = � 8 9 (Pl +P2),

Pl and P2 being tile initi~fl and final nucleon four -momenta , and s is the photon

(la) K. IGI and S. MATSUDA: Phys. Rev. Lett.. 18. 625 (1967). (la) A. A. LOGUNOV, L. D. SOLOVIEV and A. N. TAVKIIELIDZE: Phys. Lett., 24 B.

181 (1967); V. A. MESIICHERIAKOV, K. V. RERIEn. A. N. TAVKgLIDZE and V. I. ZHt "- RAREEV: Phys. Lett., 25 B, 341 (1967).

(15) R. DOLEN, D. HORN and C. SCHMID: Phys. Rev. Lett.. 19. 402 (1967); preprint CALT-68-143 (1967).

(16) A. P. CONTOGOI;RIS. J . TRAN TIIANH VAN and H. F. LUBATTI: Phys. Rev. L e t t . .

19, 1352 (1967). (17) G. •. CHEW, M. L. CTOLDBERGER, F. E. LOW T and Y. NAMBU: Phys. Rev.. 106.

1345 (1957).

814 P. DI VECCHIA, F. DRAGO and M. L. PACIELLO

polarization. We define, fur thermore

( qk ) ( P k ) W ~ - - M ~ v~ = 2 M ' v = M 2 M v~ , t = F ~ - 4 M v ~ , s = W ~ .

tt ~nd M are the pion and nucleon masses, and W is the c.m. energy. The superscript (j) refers as usual to the valnes (=1:, 0), and

g ' + ' ~ ~3 , = = g(O) =_ v u

where ~ is the isospin index of the outgoing pion. Using s tandard techniques, the inv~riant functions may be projected on

the helicity amplitudes in the crossed t-channel. One obtains, suppressing the isospin index,

(2.3)

(2.4)

(2.5)

(2.6)

where

]�89 = -]~ - - I~ ,~ / t (A1 - - 2 M A 4 ) , 2

- + - - ]~t /�89 - - ] + = ~ ( 2 M A ~ - - tA4) ,

- l OA-la-~l ]~, = (cos ,~)'~+~' [sin ~ - ) /~,,

and k t , P t , O, are respectively the boson momentum, nucleon momentum and scattering angle in the c .m system of the t-channel The ]a, are free from kinematical singularities in s (18) and satisfy an unsubtrac ted dispersion rela- tion at fixed t, ~s do the invariant functions (~7). Moreover one c~n show (18) tha t only natural (unnatural) par i ty poles contr ibute to the asymptot ic behav-

iour of the ]~, (];,). The 7:~ amplitudes are given by

~_ A {0) A~".' A~+' + -- , ,

the A (+~ amplitudes receive contributions from I----0- poles ( i .e . the co), and

(is) M. GELL-MANN, M. L. GOLDBERGER, 1 ~. E. Low, E. MARX and F. ZACItARIASEN : P h y s . Bey . , 133, B 145 (1964).

F I N I T E E N E R G Y SUM R U L E S F O R P I O N P H O T O P R O D U C T I O N E T C . 815

the A~ ~ from I = 1 + poles (i.e. p and B). In the following we shall be most ly

interested in the isospin (+ ) amplitudes so tha t we shall drop the super-

script (+ ) where this omission cannot give rise to ambiguity.

We remind the reader also of the kinematical constraint tha t has to be

satisfied ~t t = 0 (imp) (*)

( 2 . 7 ) ~/-t ];1 = i V-t -F.

I t is usually assumed tha t the co Regge pole satisfies this constraint by eva-

sion; tha t me,ms tha t the co helicity flip residue function must vanish at t = 0.

A similar evasive behaviour is assumed to hold also for the ? ~nd B trajec-

tories.

This proper ly of the residue function will play an impor tan t role in the fol-

lowing discussion.

3 . - T h e s u m r u l e s .

Fini te-energy sum rules are consistency conditions imposed by analy-

t icity (~a-~5): if a function ~'(v, t) satisfies a dispersion relation for fixed t and

can be expanded at high energy (v>>A) as a sum of Regge poles

• -- exp [ - - i~a~] l~_~, C*~ s i n ~ fi~(t) Wo]

then the following relations are equally valid (**)(***):

A

(3.1) ~,,+~ " I m F ( v , t) dv = �9 ~ + n ~-J A~'(*' '

o

where n is an even or an odd positive integer when F(~, t) is an odd or even

function of v. I t should be noted tha t it is irrelevant in the sum rule whether

a singularity is above or below a certain point, say a = - - 1 . In fact when

( a , + n ) < - - i one can obtain the relation (3.1) directly from the supereon-

vergence relation for the ampli tude v-F(v, t), while for ( a , + n ) > - - 1 one starts

by writing down a superconvergence relation for the ampli tude v~[F(v, t ) - -

(*) Note that both ]~-1 and ]+ have a singular behaviour ~(t)-~ for t ~ 0. (**) The integration is over tile right-hand cut in s and is always defined to include

the Born term even if the latter occurs at negative values of ,. (***) For simplicity of writing here and in the following we take Vo= 1 GeV,

all the energies being expressed in GeV.

8 1 6 p. DI VECCHIA~ F. DRAGO a n d M. L. PACIELLO

3 2 l . )

where

--Fa(v, t)], where -FR(V , t) contains the contr ibut ions of all the Regge poles such tha t ( a , + n ) > - - l ; f rom this superconvergence relat ion one again ob- tains the relat ion (3.1).

Sum rules of the kind (3.1) provide a relat ion between the high-energy l~egge parameters , t rajectories and residue functions, and the low-energy parameters , masses and widths, tha t characterize the s-channel resonance

spectrum. B y means of relations of the type (3.1) we shall tes t the compatibi l i ty of

the hypothesis made in the description of high-energy 7: ~ photoproduct ion with

the low-energy data, and we shM1 invest igate the s t ruc ture of the residue

functions fl,~(t). Star t ing from the dispersion relations for the ampli tudes A given in ref. (,7),

the following sum rules are easily obtained for the isospin ( § ampli tudes:

A

S~ = R(A~)~,B + ~ }v '~ ImA~(v, t) # siuz~,~

~ d v - - . . . . . . . g~ ( t ) A ~ ( t ) + ~

Yo

t --/~2 t --/~2 ~ - - 4M ' v 0 = j u + 4M-

1 R(A~ + ) = - ~ e / ,

R( ,.)=-- ~e/,

where #~ and #. are the rationalized anomalous nucleon magnetic moments and e and ] are the rationalized and renormatized electronic charge and pion nucleon coupling constant: e2/4~ = 1/137 and / ~ / 4 ~ 0.08. Taking into ac- count the crossing propert ies of the A + amplitudes (17) n must be an odd posi-

t ive integer for i = 1~ 2~ 4 and an even posit ive integer for i = 3. The func-

tions g~(t) are explicit ly given in the Appendix: they are proport ionM to some

combinations of the helici ty flip and helicity nonflip residue functions of the

Regge t ra jec tory . Relations (3.2) hold under the hypothesis t ha t no other singularities, in

the I = 0- channel beside the r Regge pole, are present in the complex J -p lane to the r ight of J - - - - - n. In the following, n will be equal to one or three for i = 1 , 2 , 4 and to zero or two for i = 3 . For A-->o% S~ turns into a well-

known supereonvergence relation (~").

(is) A. BIETTI. P. DI VECCHIA and F. DRAGO: NUOVO Cimento. 49A. 511 (1967). References to previous works carl be found in this papcr.

F I N I T E - E N E R G Y SUM I~ULES FOR PION P t I O T O P R O D U C T I O N ETC. 8 1 7

We sa tu ra te the sum rules (3.2) using a phenomenological mult ipole analysis

of pion pho toprodue t ion up to W = 1.8 GeV done by WALKER at Cal-

Teeh (2o). In this analysis the exper imenta l da ta are fi t ted by six resonances

(the A ~* J = ~ + at 1 .236GeV, the J = ~ - a t 1 .519GeV, the J = ~ - + ut 1.672 GeV, the J = ~ - at 1.652 GeV, the J = l + a t 1 .471GeV and the

J = �89 a t 1.561 GeV, respect ively with mult ipoles M~§ and El+, E,_ and

M2_ , E 3_ and M3_ , E~§ and M~+, M 1_ and Eo+), the Born ampl i tudes and a (~ background ~ in t roduced in order to have a be t t e r fit at certain energies. The resonant mult ipoles are given in Brei t -Wigner form with energy-dependent

width. Using the relat ions between the invar ian t ampl i tudes A~(v, t) and the

mult ipoles Mz~ and Ez~ given b y CGLN we can thus di rect ly sa tura te the

sum rules.

4 . - D i s c u s s i o n .

We give in Table I the values of the various S~ obta ined by numerical integrat ion. In Table I I we give the combinat ions /~o~ = [S]--2MS'~] and

T A B L E I .

t (GeV/c) 2

MS 1 (GeV) MS~ (GeV)3 MS~ (GeV) -1 MS~ (GeV) M81 MS] (GcV) 2

0.1

).02 ).00 ).14 ).18 ).34 ).17

0.0

0.03 0.01 0.13 0.15 0.30 0.13

--0.]

0.04 0.02 0.12 0.12 0.25 0.10

--0.2' . . . . 0.3 0.4

0.04 ] 0.04 0.04 o.o21o.o2 Io.o2 o.1110.lOl o.oo o lo oo8(oo6 0.21 0.17/0.13 0.08 0.06 [ 0.04

0.~

0.04 0.02 0.08 0.04 0.10 0.03

--0.(

0.04 0.02 0.07 0.03 0.06 0.01

--0.~

0.04 0.01 0.06 0.02 0.03 0.01

--0.8[ --0.9

0.03 0.03 0.00 / 0.00 0.06 0.05 0.01 0.00 0.00 --0.03 0.00 --0.01

--1.0

0.02 --0.01

0.05 --0.01 --0.06 --0.01

R[I = [2MS~--tsn4] which are propor t ional respect ively to tile helicity-nonflip

and helieity-flip ~o residue functions (see (2.3), (2.5) "rod Appendix) . The dis-

eussion of this da ta is complicated by the s imultaneous presence of various

aspects of the problem, as we shall see.

As a first step we mus t be sure t ha t only the r pole contr ibutes to the

isospin ( + ) ampli tudes . To this end we t ry to de te rmine the o) t ra jec tory

u~ing the me thod proposed in ref. (a~); in fact it is easy to show that , if one

(20) R. L. WALKER: private communication. This phenomenologieal fit has Mrcady been used in ref. (is).

53 - II Nuoro Cimenlo A.

818 P . D I V]gCCt t IA, F . D R A G O and M. L. P A C I E L L O

TABLE II .

t (GeV/c) ~ 0.1 0.0 - - 0 . 1 - - 0 . 2 - - 0 . 3 - - 0 . 4

1

2 ~ta

4 ~to

- -0 .01 1.73

- -0 .14

~ 0 . 5 5 1 . 4 5

- - 0 . 45

- -0 .29 1.11

- - 0.49

- -0 .17 0.79

- -0 .5 3

- -0 .13 0.49

- -0 .5 6

-o.16 0.19

- -0 .5 9

t (GeV/c) 2 - - 0.5 - - 0.6 - - 0.7 - - 0.8 - - 0.9 - - 1.0

1

2

4

- - 0 . 2 6 0.10

- -0 .63

- - 0 . 4 0 - -0 .37 - -0 .68

- -0 .60 - -0 .62 - -0 .77

- -0 .8 4 - -0 .8 4 - -3 .36

- -1 .1 3

- - 1 . 0 4

- -0 .51

- -1 .4 6 - -1 .21

- - 0.59

d e f i n e s X d t ) a 2 = S~/A S~ f o r i = 1, 2, 4 (*), t h e t r a j e c t o r y t h a t o n e o b t a i n s f r o m

t h e v a r i o u s s u m r u l e s is g i v e n b y

zcs = 3X~(t) - - 1

(4 .J ) 1 - x , ( t ) "

Using this formula one obtains ve ry bad results; for reference they are given in Table I I I . However it is ve ry easy to explain the complete failure of rela-

TABLE I I I .

t (GeV/c) 2 0.2 0.1 0.0 - - 0.1 - - 0.2

( s ~ - 2 Jgsi)2g (GeV)~ (2MS~ - - t S ~ ) M (GeV) 2 (2MS~ - - tS~) M (GeV) 4

- - 0.73 - - 0.41 - - 0 . 0 2 - - 0 . 0 7

- - 0.63 - - 0 . 3 2

0.01 - - 0 . 0 2

- - 0 . 5 3 - - 0 . 2 4

0.01 0.02

- - 0.44 0.18 0.10 0.05

- - 0 . 3 6 0.13 0.12 0.06

- - 0 . 3

0.28 0.09 0.13 0.06

- - 0 . 4

- - 0 . 2 1

- - 0.05 I 0.14 I 0.06

t (GeV/c) 2

(Si 2MS~)M (GeV) (S~ - - 2MS~) M (GeV) a

(2M8~ -- tS~)M (GeV)'

- - 0 . 5

- - 0 . 1 4

- - 0 . 0 3 0.13

0.05

- - 0 . 6

- - 0 . 0 8 - - 0 . 0 1

0.11 0.04

- - 0 . 7

- - 0.02 0.00 0.09 0.02

- - 0 . 8

q- 0.03 0.01 0.06 0.01

- - 0 . 9

0.08 0.01

0 . 0 2 - -0 .0 1

- - 1 . 0

0.13

0.01 - - 0 . 0 2

- - 0.03

(*) W e disregard in the following the sum rules ob ta ined f rom the ampl i tude A 3. This is because to u-pole en ters in th is ampl i tude only in n o n a s y m p t o t i c form (i.e. ~ v ~ - 2 ) and wi th a residue func t ion singular at t = 0, so t h a t daugh te r t rajec-

tories are necessary in order to res tore t he ana ly t ic i ty of t he full ampl i tude . The rela- t ion of t he daugh te r to the p a r e n t t ra jec tory , a l though known at t = 0, is general ly unknown, so t h a t the one-pole app rox ima t ion for the A 3 ampl i tude seems unlikely.

F I N I T E - E N E R G Y SUM R U L E S F O R P I O N P t t O T O P R O D U C ~ I O N E T C . 819

t ion (4.1)(*). In fact the value of ~(t) determined from (4.1) is extremely

sensitive to the value of X~ (for Xi = 0.3, ~ = - 0.14, for Xi----0.4 ~--~ 0.33,

and for Xl = 0.5, ~ = 1!) which in turns depends on the third momentum

smn rule. Now at W = 1.8 GeV the resonance spectrum is certMnly not ex-

hausted and it is clear tha t the contr ibution of the higher resonances will

affect much more strongly the third than the first momentum sum rule. This

is confirmed from the s tudy of the residue functions: in fact we find that the

residues determined from the third momentum sum rule are different from those

determined from the first momentum sum rule. They are smaller or big~'er

in such a way tha t the failure of relation (4.1) can be completely understood.

The previous discussion leads to the conclusion tha t relations of the type

(4.1) are generally not reliable, so that , for example, the need of additional

trajectories derived from the failure of this kind of relations is not to be taken

too seriously in our opinion.

I n order to find the t ra jec tory we have therefore used the following meth-

od. We have evaluated the first momentum sum rules at various values of t

( --0.5(GeV/c)2< t < 0) with different upper limits A, and then we have fitted the

r ight-hand side of eq. (3.2) to the values so obtained. The best fit is obtained

with constant residue functions (once the evasion of the spin flip residue has been

taken in account, see below) and a linear t ra jec tory given by

(4.2) ~,~(t) = 0.61 -~ 0.75t .

The co t ra jec tory is not so well known from other processes, different values

being given by the various authors. I n our opinion this t ra jec tory is per-

fectly acceptable for the r exchange(**), so tha t no other contribution besides

the r pole is necessary in order to saturate the sum rules (3.2). This conclu-

sion will be strengthened by the following discussion: in fact we shall assume

from now on tha t only the o~-exchange contribution is present and we shall n see tha t in this way the behaviour of Rol and RI~ can be understood in a

sat isfactory way.

Let us first look at the sign change of /~1 and /~1 near t = 0.1 (GeV/c) 2.

This zero is expected from the evasive behaviour at t = 0 of the pole; of course

one cannot expect tha t the sum rules reproduce it exactly at t = 0. One may

ask if it is not possible tha t the o) helicity flip residue satisfies the con-

s traint (2.7) going t~ zero at t = 0 like ~n even power of t, say like t 2, so

tha t the sign change in RI~ and R[~ would be caused by the sign change of the

residue at t = tc ~ - - 0 . 1 3 (GeV/c) 2. This possibility does not appear very

(*) We are indebted to Prof. A. BIETTI for a discussion about this. (**) It is interesting to note that the value ~ ( t ) = 0.60~-0.75t has been obtained

independently from a fit to the high-energy T:0 photoproduction (8).

8 2 0 P . D I V E C C I t I A , F . Dt tAGO and M. L. P A C I E L L O

l ikely; moreover if one believes in the faetorizat ion theorem it can easily be shown tha~ the behaviour ]+~--t 2 near t ~ 0 would imply tha t the ampli- tude ]+l,~t and the (o contr ibut ion to the nucleon-nucleon scattering ampli-

+ t 2 ~+ ~ ~3. tude ]�89189189189 goes like or t ha t bo th these possibilities appear to 1�89189189 ~ ,

be very unlikely. Moreover in the isospin (0) amplitudes the same sign change around t = 0 is reproduced by the low-energy data, via finite-energy sum rules, in the ? residue function of the ampli tude ]+1 and in the B residue func- t ion of the ampli tude ]~.

We conclude tha t almost cer ta inly the co helicity flip residue functions do

not change sign at t =-t~ and tha t cer ta inly the helieity nonflip residue does not change sign at t~. In fact if the helici ty nonfiip residue changes sign at t~

the only possible way not to have an overall sign change is tha t it has another zero near t ---- 0. However, if this is the case we should expect t h a t / ~ and )?]~

are strongly depressed in the small t region, and this does not happen, as can be seen from Table II .

5. - The prob lem of the dips.

As a by-product of our invest igat ion we can s tudy the behaviour of the

co exchange at the wrong signature nonsense point ~ = 0. The usual theory of the dips, neglecting for the moment the th i rd double spectral function effects, gives, according to the various mechanisms, the following behaviour near

---- 0 for the amplitudes ]o+1 (sense-nonsense transition) and /+1 (nonsense- nonsense transition) :

1) Sense choosing:

2) l~onsense choosing:

3) Chew mechanism;

4) l~oncompcnsating mechanism:

- + 2 ] : 1 ~ , / 1 1 ~ �9

10+1- 2, ltl- 3

11 ~ ~ "

W e see from Table I I tha t ]+ changes sign near t = - 0.7 (GeV/e) 2 and ]+1

between - - 0.8 and - - 1 (GeV/c) 2. A similar investigation has been carried out for the p-exchange contribu-

t ion; we found in this ease tha t ]+ changes sign near t = - -0 .4 (GeV/c) 2 and ]+1 does not change sign; nor does i t go to zero. Moreover the p residue functions are found, as expected, to be smaller than tha t for co by something more than an order of magnitude. If these data can be taken seriously, various inter-

pretat ions are possible. The fact t ha t for the p-exchange contr ibut ion ]+ changes sign and ]+ does not can be unders tood assuming tha t the p-trajectory

F I N I T E - E N E R G Y SUM RULES FOR PION PI tOTOPRODUCTION ETC. 821

(( chooses sense ~): this is the expected behaviour for this pole (~1). The displace- merit of the zero near t ~ - - 0 . 4 (GeV/c) 2 can p robab ly be understood, since of course we cannot expect t h a t the low-energy fit reproduces, via sum rules,

a zero exac t ly where it is expected. The d isplacement also m a y be due to the th i rd double spectral funct ion effects. The nonvanishing of ]+ a t ~p = 0

can p robab ly be unders tood since this ampl i tude has to vanish a t t = 0, since

i t is assumed to sat isfy the constraints (2.7) b y evasion, and again a t % = 0,

where i t is expected to have a ve ry nar row dip owing to the ~ behaviour .

We feel t h a t it is unl ikely t ha t the low-energy data , a t least in the fo rm in

which we know t h e m now, could reproduce such a narrow dip.

The behaviour of the r t r a jec to ry however does not seem to agree wi th the usual naive s y m m e t r y a rguments : in fact in t e rms of the possibilities 1) ... 4)

given above, we see t h a t one is forced to ~ssume tha t , unlike t h a t of the p, the r t r a j ec to ry is nonsense-choosing.

The previous discussion has ignored the th i rd spectral function effects, or

in other words, the possible presence of fixed poles a t nonsense-wrong s ignature

points. In fact i t has been shown b y MA~D~,LSTAM and WAN~ (2~) tha t , if the

th i rd spectra l funct ion effects are impor tan t , a precise dist inction between

t ra jector ies which choose sense and those which choose nonsense a t an integer

of the wrong s ignature is no longer possible and all the usual dip-analysis

breaks down. Since in our case this analysis seems to work well if one assumes

t ha t the small d isplacement of the zero is due to the fit, one is led to the con-

elusion t h a t the th i rd double spectral funct ion effects are negligible. However

if this is the case, one can derive an observable consequence of the absence of fixed poles, name ly a Schwarz (2~) superconvergence relation. SC~WRZ has shown tha t , if no fixed poles are present a t the nonsense points of the

wrong signature, superconvergence relat ions can be derived also for the ampli-

tudes which have the wrong crossing prnper t ies (the condition t ha t no fixed poles are present a t the r ight s ignature-nonsense points gives the ordinary superconvergence relations).

In the photoprodnct ion case we can derive Schwarz superconvergence (-~ [A(I+~tA(+)I ~o~ (o) re la t ions for the ampl i tudes A a , ~ j, [A~ +tA~ ]. The ampl i tude

[A~~176 ] receives contr ibut ions f rom the B exchange and the A~ -) eventual ly

only f rom the A 1 t ra jec tory , which does not seem to p lay an i m p o r t a n t role in pion photoproduct ion . The ampl i tude c+) (+~ [A~ [-tA~ ] can receive a contri- but ion only f rom the exchange of unna tu ra l pa r i t y I a : 0- t ra jector ies: none

of the k n o ~ l t ra jector ies have these qua n tum numbers , so t ha t the super-

(21) A. AI4BAB and C. B. Cmu: Phys. Rev., 147, 1045 (1966); G. HOLER, J. BAACKE. H. SCHLAILE and P. SONDERE(~GER: Phys. Lett., 20, 79 (1966).

(22) S. MANDELSTAM ~nd L. L. WANG: Phys. Rev., 160, 1490 (1967). (23) j . it. SCHWARZ: Phys. Rev.. 159, 1269 (1967).

8 2 2 P . D I V E C C t t I A , F . D R A G O and M. L. P A C I E L L O

convergence relation for this ampli tude is expected to converge very rapidly, if i t converges at all. In fact the ordinary superconvergence relat ion for ~,[A+~-tA +] converges ve ry rapidly: from Table I we can see the cancellation

which is effective between $1 and tS~. The Schwarz relations for the amplitudes in discussion are

(5.1)

(5.2)

where

co

{R[A~o +) ] § tR[A~o+)]} 4- f*m t) § tA~o-)(v, t)} dv --~ O,

+ z f , = o, vo

R[A<~ ~ ---- R [ A ? ' ] ,

R[A~ ->] = R[A<4+)].

All these relations do not seem to converge, as can be seen from Table IV. This fact can be understood, par t icular ly for the isospin ( + ) amplitudes, only if one assumes tha t fixed poles with nonnegligible residues are present at J----0 in the wrong signature part ia l-wave amplitudes.

The )revious analysis seems to show tha t a simple direct connection be-

TABLE IV a.

t

(GeV/c) 2

0.0

--0.1

--0.2

--0.3

--0.4

M (Born term)

--0.142 {

0.098 (

0.117 {

0.126 {

0.130 (

Wm [{R[A~] + tRIAd~ + {[R(A +) + tR(A;)] +

~m Ym

+(/~/~) f Im[A~ + tA~ dv]M +(tt/z~ Im[Al + + tA +] dv}M % i

1.60 1.80

1.60 1.80

1.60 1.80

1.60 1.80

1.60 1.80

--0.08 --0.07

0.18 0.18

0.21 0.21

0.23 0.23

0.24 0.24

--0.15 --0.15

0.09 0.09

0.12 0.11

0.12 0.12

0.13 0.13

14~m is t h e c . m . e n e r g y u p to w h i c h t h e i n t e g r a l h a s b e e n e v a l u a t e d . A l l t h e n u m b e r s

g i v e n , e x c e p t t, a r e i n G e V .

FINITE-ENERGY SUM RULES FOR PION PItOTOPRODUCTION ETC.

T A B L E I V b.

823

t (GoV/c) ~ W,,~ l~ {/~(AT) + (tt/~)'flm,,. AT(v,

I

o [ I

--0.1 /

--0.2 {

--0.3

--0.4 {

i

1,60 1.80

1.60 1 .80

1.60 1.80

1 .60 1 .80

1 .60 1 .80

0.04 0.03

0.05 0.03

0.05 0.04

0.06 0.05

0.07 0.06

t) d~

IVan is the c .m . ene rgy (in GcV) up to which the intr hus been eva lua t ed . The Born t e r m I~fR(_4~)=--0.0098.

tween the noneonvergence of the Schwarz relat ions and the breakdown of the

dip analysis cannot general ly be inferred. A simi]ar si tuation has been me t in ref. (~s) for the ~-A p system.

6 . - C o n c l u s i o n s .

Using finite-energy sum rules we have shown tha t the photoproduct ion isospin (~-) ~mpli tudes are dominated by the co-exchange and tha t there is no need of addi t ional ~ contributions, in the form of a new p o l e or of a cut;

moreover the co residue functions do not change sign in contradict ion with the

usual in te rpre ta t ion of the cross-over phenomenon. I f one cannot show with some ce r ta in ty t ha t addi t ional ~ contr ibut ions are present in K-+p, ~-p and

p-p sca t ter ing (*) and if no other explanat ion of the cross-over phenomena

can be found in the f r amework of the Regge-pole theory, one is forced to admi t t h a t the factor izat ion theorem is not of general validity.

(*) Finite-energy sum rules at t--~ 0 have been applied to the K-z~ system (24). The sum rules have been saturated with the experimental cross-section, via the optical theorem, and quite good agreement has been found with the high-energy model of PHILLIPS and RARITA (6). This result however cannot be considered conclusive since one can certainly conceive of a model in which, for example, a conspiring pole does not contribute at t---- 0 (a model of this kind has been discussed for the p' in ref. (25)).

(24) A. BORGESE, M. COLOCCI, M. LUSIGNOLI, M. RESTIGNOLI and G. VIOLINI: University of Rome preprint (1967).

(25) L. SERTORIO a.nd M. TOTJLER: Phys. Rev. Lett., 19, 1146 (1967).

824 P. DI VECCt I IA , F. DIg~GO lMld M. L. P A C I E L L O

We have also found some evidence tha t the r t ra jec tory is nonsense-choos- ing at g~ = O, while i t is generally assumed on symmet ry grounds tha t i t is sense-choosing, like t ha t of the p.

W e are part icular ly grateful to Prof. L. BERTOCCHI for many useful dis-

cussions and for critical reading of the manuscript . We also thank Prof. A. BIETTI for valuable comments.

APPENDIX

We give here some details about the Reggeization of the photoproduct ion amplitudes.

The relation between the invar iant amplitudes and the par i ty-conserving helicity amplitudes free from .~ and t kinematical singularities (56) are given by (*)

/o + = A 1 - 2 M A 4 ,

~ = (t -- tt~)[A1 + t A x i ,

~+1 = 2 M A ~ - - tA4 ,

We are only interested in the asymptot ic o-exchange contr ibut ion to these amplitudes. Using the s tandard Reggeization rules (is) we find

where

/o+1 = 2h(t) ( t ) ( - 2z ) ~-1,

/~ = - 2h( t ) o~ ~ ( t ) ( - '2Z) ~-~,

/ ~ = O ,

y~ = o(Z~-i) ,

s inna /~(c~ + ] ) 2 and Z = cos Ot,

0t being the scattering angle in the c.m. system of the t-channel.

(28) L. L. WANG: _Phys. Rev. , 142, 1187 (1966); 153, 1664 (1967). (*) The relation between the T+u and the ]~:u defined in Sect. 2 is given by

]~u = K• , where K• contains all the kinematical singularities in t of the ampli- tude, and can be evaluated using the rules given in ref. (26).

FINITE-ENERGY SUM RULES FOE PION PKOTOPI~ODUCTION ETC. 8 2 5

F r o m these expressions we ob ta in

\Vo/

-

/+ = -- 2h(t)./~h[~Jy+(t) \Vol

where the r educed residues y(t) are a s sumed to be ana ly t i c funct ions of t for t < 0 and ~ ( ~ ) and ~1(~) are some func t ions of ~(t) which depend on the pa r t i cu la r m e c h a n i s m t h a t the t r a j e c t o r y chooses a t the nonsense points . The

behav iou r nea r ~ = 0 of 5 + and [+ has been given in Sect. 5. The t in ]+ comes f rom the a s sumpt ion t h a t the k inemat ica l cons t ra in t (2.7) is satisfied by evasion.

F r o m the relat ions

A2 = -- 1 A~, t

A~= O(v ~-~) ,

1 [/~+ _ 2Mr+ ] A 4 - 4M2 t

the funot ions g~(t) defined in Sect. 3 are easily ob ta ined :

g g t ) - 4 M 2 - t s i n ~ F ( ~ I )

1 g2(t) = -- 7 g~(t) ,

~* 2~ + 1 F(~ + �89 g4(t) - - 4 i ~ - - t sinze~ l ' ( ~ - - l ) ~[~tlvi(~)y+I+ 2 M ~ ( ~ ) y ~ I ] "

R I A S S U N T O

Si ~ studiato il comportamento dei residui dell'omega nella fotoproduzione di pioni. Si mostra che questi non hanno zeri a t ~ ' - 0.15 (GeV/c)2; cib ~ in contraddizione con l'usuale interpretazione del fenomeno del cross-over. Si 6 inoltre studiato fl comporta- mento della traiettoria dell 'o intorno al punto ~ = 0. Viene suggerito che, a diffe- renza di queUa del o, la traiettoria dell'~ sceglie nonscnso ad ~ = 0.

826 1". DI VECCttIA~ F. DR~GO & n d M. L. PACIELLO

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H Co OeTaTOqHble ~yHKIIHH ~Ylg no~lOeOB Pe~ace.

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HerIpaBHJIbHO~ c n r H a T y p o ~ c~=-0. Ka>KeTcn, qTO o6biq~Ian TeopnR ~< HaKJIoaa ~ x o p o m o

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