P H Y S I C A L R E V I F , W D V O L U M E 2 0 , N U M B E R 9 1 N O V E M B E R 1 9 7 9

Behavior of the A, trajectory in Reggeon field theory

Vera Lucia Vieira Baltar Departamento de ~ i s i c a , ~ont$cia Uniuersidade Cathlica, Cx.P. 38071, Rio de Janeiro, RJ , Brasil

(Received 16 May 1979)

Comparing experimental data for the reaction r - p -7n with results obtained in the context of Reggeon field theory, strong indications are found that at high energies the trajectories of the A, pole and all its associated branch points overlap. This explains why simple Regge-pole models for processes mediated by the A , pole yield results in good agreement with experimental data, while in diffractive reactions all the branch points must be explicitly taken into account.

I. INTRODUCTION

Reggeon field theory was introduced by Gribov' a s a means of systematically analyzing the ex- change of Regge poles and associated branch points in hadronic p rocesses a t high energies. Together with renormalization-group technique^,^.^ th i s theory w a s found capable of consistently describing diffractive react ions, viewed a s mediated p r i m a r - ily by Pomerons.

This w a s the motivation for using Reggeon field theory in the study of charge-exchange processes , in which besides Ponlerons a t l eas t one secondary Reggeon, carrying the exchanged quantum number, must b e taken into account. This study showed that th ree s table fixed points govern the infrared behavior of the theo~-y.4 Each one of them leads to a different expression for the scat ter ing ampli- tude. Comparing with experimental resu l t s the differential c r o s s section for the react ion a-fi -non obtained f rom each of these al ternat ive expres - sions, s t rong indications were found that only one of them is physically acceptable. T h i s selected solution corresponds t o a fixed point which leads to a superposition of the Regge pole and a l l i t s associated branch points implied by t-channel unitarity. The superposition occurs not only for ze ro center-of-mass t r a n s v e r s e momentum t rans - f e r 5 =0, where we already expected the collision of all singularities, but a l so f o r $#O. It is thus possible t o understand why simple Regge-pole models in which the p t ra jec tory i s exchanged a r e in good agreement with experimental resu l t s , a s in the reaction a-fi-7r0n, while in the c a s e of dif- f ract ive processes it is necessary t o explicitly take into account a l l the branch point^.^

I

The react ion a-p-qn is a l s o known to be well described by a s imple Regge-pole model with ex- change of t h e A , t rajectory. Therefore, i t would be interesting t o analyze the q production in the s a m e way a s was done for the n charge exchange, in o rder to find out if the t ra jec tor ies of the A, pole and a l l i t s associated branch points overlap a t high energies , a s we believe t o happen in the p - exchange case.

In o r d e r t o render the paper self-contained, we present , in the next section, a brief summary of the techniques used t o calculate the scat ter ing amplitude F (Y, t) , which a r e fully described in Ref. 5. Section I11 contains t h e analysis of the reaction n'p -On, our resu l t s , and conclusions.

11. CALCULATION OF THE SCATTERING AMPLITUDE

We proceed t o very briefly summar ize the tech- niques used in Ref. 5. They a r e especially de- signed f o r the problem of secondary t ra jec tor ies , where the presence of a l a r g e r number of dimen- s ionless quantities a s compared to the pure Pom- eron case makes the situation more complicated.

Linear t ra jec tor ies a r e considered both for the Pomeron and the Reggeon:

ao( t ) =ao +a;t , (2.1)

~ o ~ ( t ) = a o ~ + a h ~ t (2.2)

s o that, fo r t - -Ej2 we have

~ ( p ) + A ~ , a, = l - ao (2.3)

ER(P) =a;&' +AOR, A o R = l -aOR. (2.4)

The Lagrangian i s , a s usual,

@ 1979 The American Physical Society

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with Y, and Xo real.' It i s the infrared limit of this theory that will be relevant to the analysis of high- energy scattering processes. Such limit is studied by means of renormalization-group techniques.

The Pomeron field +, and the Reggeon field X , a r e defined in D space dimensions. Physics takes place at D =2. Effecting the phase change x,- eiAo~*x0, the Reggeon energy E, = 1 - J i s r e - placed by E, - A,, =a,, - J = &.

The renormalization conditions imposed a r e

where r'n*m;k' i s the renormalized connected prop- e r vertex function corresponding to n Pomerons + k Reggeons -m Pomerons + k Reggeons.

Renormalized quantities a r e related to bare ones by

Four dimensionless quantities a r e present in the problem:

Renormalization-group functions for the Reggeon a r e defined by

P R =EN-gR a 1 (2.23) a ro, Ao, a:, abfi P, fixed '

(2.24) ~o .Ao ,~o ,~&~PR fixed '

, (2.25) ? O * X O * ~ $ ~ ~ & , P R f i e d

a YPRR = -1nZ3, (2.26)

a p ~ 1 r o * h O s d t & p E N fixed ' a rER = EN-lnZ,Z,

BEN 9

rotho,&& PR fixed (2.27)

In order to extract the maximum amount of in- formation from our renormalization-group analy - sis, we introduce the scale factor t' = ln(-&/EN), and study the infrared limit by allowing the mo- mentum to go to zero with the energy in such a way that pR remains constant throughout the process. Using perturbation theory to obtain the renormali- zation-group functions, three infrared-stable fixed points a re found, namely:

where E = 4- 0 and Q = $ r ( 3 - * ~ ) / ( 8 ? r ) ~ / ~ . For small values of E , the effective coupling constants gl and gRl a re small; perturbation theory can then be used to obtain the renormalization-group func- tions in the infrared limit, and our calculations a re self-consistent. So we use perturbation theory and expand our results around D = 4, in a power ser ies in €. At the end of the calculations, we set € = 2.

The t J dependence of Z,(t',p,) can be obtained l t ~ f , ~ ( p ~ ) = JypRRdpR + l n c from the renormalization-group function y,, :

- -4Qsn12tJ+ lnFm(pn) , (2'29) where the infrared limit of yo,,, calculated to where the infrared limit of y,,, calculated to f i r s t order in the E expansion, has been used. f i r s t order in the E expansion, has been used. Analogously, from T,, and rPRP the infrared be- The p, dependence of Z3,(tJ,pR) can be obtained havior of Z,, 2,-' can be obtained. We may now from y,,,: integrate

2LirAo,o;~ a & PR t ixed

to get

where A i s a constant, and

This expression i s valid for all values of p,, and gives us the explicit energy -momentum depend- ence of the Reggeon propagator in the infrared li- mit. The full scattering amplitude i s now obtained by means of a Sommerfeld-Watson transform:

where Y = lns/so; the t dependence of the signature factor ( i s not taken into account, since i t i s negli- gible in our scheme of approximations.

Let u s now consider the f i r s t fixed point: gl= ( ~ / 2 4 ~ ) ' / ~ , g R l = i g l , vl= 0. It leads to the superposition, in the complex J plane, of all branch point t rajectories & = [(a;,at)/(ff' + ~ f f & ) ] k ' , and the pole trajectory & = ff;k2, since in the limit v - 0, these two expressions coincide. Indeed, ex- amining the behavior of the Reggeon propagator G(OtO;* corresponding to the f i r s t fixed point, we find that i t s singularities a r e a pole and a branch point which overlap at & = C'a&k2.

Performing the Sommerfeld-Watson trarisform we get

Considering now the other two fixed points, we have

' + 4Qgb2 C R O ) Y ~ ( Y E ~ ) Q ~ * ~ ~ ; ( Y , F*(Y,~)=- A

where

111. THE REACTION a-p + qn

Let us now examine explicitly the reaction 7-p - qn. Two amplitudes-spin flip and spin nonflip- must be used to describe it. The spin-flip ampli- tude contains a multiplicative factor a c r ( t ) , found by performing the usual analysis in the com- plex angular momentum plane. The spin-nonflip amplitude has a ghost-eliminating multiplicative factor c ~ ( t ) . ~ Therefore, the contribution of the in- verse Reggeon propagator riO'O" to the elastic angular distribution is

Within the range of values of 1 considered in the comparisorl of our results with experimental data, the t dependence of n++(l), n+-(t), and ff2(t) / t; l 2 will be neglected.

From (2.35) and (3.1) we see that the f i r s t fixed point leads to an expression for the differential c ros s section whose t dependence i s the same a s the one obtained f rom simple Regge-pole models. This i s no surpr ise , since in this case, the t r a - jectories of all singularities in the complex J plane coincide. The energy dependence of do/dt

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FIG. 1. Comparison between experimental data (Ref. 7) for differential c ros s sections of the reaction ?r-p -7n and results obtained from Eq. (3.1) for the fixed point (1). The values chosen fo r the parameters a r e C' cutR = 1 ( G ~ V / C ) ' ~ and n+-/n++ = 3 .

differs from the one obtained from a Regge pole by a factor of (lns)'I6, which i s too small to be de tected even at the highest available energies.

Comparison of our results with experimental data7 i s shown in Fig. 1, for incident 71 momenta P = 100.7 ~ e V / c and P = 199.3 G ~ V / C . The c ros s sections a r e normalized to one at the origin, and the data have been corrected for the effects of ex- perimental t resolution and finite bin widths, a s

0.005 0.05 0.25 0.5 0.75 0.05 0.25 0.5 0.75 1.0

- t [(G~v/c)']

FIG. 2. Graphs of the function Fi ( Y , t ) defined by Eq. (2.37). The value chosen for C 1 a B R is 0.63 ( G ~ v / c)' 2 .

FIG. 3. Comparison between experimental data (Ref. 7) for differential c ros s sections of the reaction n'p -?In and results obtained from Eq. (3.1) for the fixed point (2). The values chosen for the parameters a r e C' f f bR = 0.63 (G~v/c)-' and n,, /n++ = 26.

explained in Ref. 7. The values chosen for the parameters a r e C'a& = 1 ( G e ~ / c ) - ~ and n+_/n, = 31.

Let u s now consider the second fixed point: gl= ( ~ / 2 4 & ) ~ / ~ , gR1=g+=g l ( JT-k ) , v ~ = ~ . The dif- ferential c ros s section i s obtained from (2.36) and (3.1). Graphs of F :(Y , t ) a r e shown in Fig. 2 , which exhibits the presence of an energy-dependent ze ro in this function. The value chosen for Cia& i s 0.63 ( G e ~ / c ) - ~ .

Comparison with experimental data of the c r o s s sections obtained in this case i s shown in Fig. 3. The value chosen for n+- /n, is 26. We notice that up to -t- 0.3 Gev2/c2 the f i t i s such that we cannot discard a s physically unacceptable the ex- pression for da/dt obtained from the second fixed point. But the behavior of F: (Y , t ) leads to a dif- ferential c ros s section with an energy-dependent dip, which occurs in a range of values of t where

we expect our theory to be at least qualitatively reliable. The presence of this dip i s not in agree- ment with experimental results. The situation i s entirely analogous in the case of the third fixed point.

We a re therefore led to the conclusion that, a l - though the three fixed points correspond to valid mathematical solutions for our problem, only one of them i s physically acceptable, namely the one corresponding to a linear dispersion relation. This means that, at high energies, the trajector- i e s of the A, pole and all its associated branch points do indeed overlap.

We must, however, remember that there a re lower-order contributions to the amplitude F ( Y , t ) which were neglected in our calculations. The t dependence of n++(t), n+-(t), a2(t) / 5 1' was also neglected. It is not impossible that all these cor-

rections combine in such a way that the resulting expressions for du/dt obtained from the second and third fixed points turn out to be in good agree- ment with experimental results. But this possi- bility looks extremely unlikely. We therefore be- lieve that the overlapping of singularities in the complex J plane at high energies does occur, which explains why simple Regge-pole models for processes mediated by the A, trajectory agree with experimental data, while for diffractive pro- cesses all the branch points must be explicitly taken into account. As expected, the situation is analogous to the one studied in Ref. 5 , in connec- tion with the p trajectory.

ACKNOWLEDGMENTS

The author i s grateful to E. Ferre i ra and N. Zagury for their interest and comments.

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