Exchange degeneracy breaking in the scattering and particle regions

Volume 64B, number 2

PHYSICS LETTERS

EXCHANGE

DEGENERACY

SCATTERING

13 September 1976

BREAKING

AND PARTICLE

IN THE

REGIONS*

M. BISHARI Weizmann Institute o,f Science, Rehovot, Israel and Research Institute for Theoretical Physics *, University of Helsinki, Helsinki, Finland

Received 21 July 1976 We perform the analytic continuation, from the scattering (t < 0) to the particle (t > 0) region, of the non-planar diagrams (having torus topology) which generate a splitting between planar trajectories with non-zero iso-spin. Using heavily our previous work on the planar and cylinder amplitudes, we derive “asymptotic planarity” and estimate the rate of approach to it, also for the torus under rather general assumption on the Reggeon-Reggeon amplitude fixed pole residue.

Recently Chan et al. [ 1] have evaluated, at t = 0, the non-planar topology (torus) which separates between planar trajectories with non-zero iso-spin, such as p and A,, K* and K**. Although the fixed pole residue is not yet completely known, their result is nevertheless encouraging for the dual unitarization approach (or the “l/N expansion”) [2,3]. Namely, it can provide an explanation for the apparent exchange degeneracy breaking, in the scattering region (t < 0), between vector-tensor trajectories with iso-spin. A crucial question is how the torus contribution changes when passing from the scattering to the particle region (t > O)? The answer to this question will determine whether exchange degeneracy (EXD) improves in the particle region, as expected phenomenologically, or deteriorates. In other words, is there an “asymptotic planarity” [4,5] also for the torus contribution? It must be emphasized that the “asymptotic planarity” (AP) derived for the cylinder cannot be directly applied for the torus (see fig. 1). Indeed some of the Reggeons in the torus diagram cany a phase whereas in the cylinder the Reggeons exchanged between planar amplitudes carry no phase. Moreover, in the torus diagram, fig. 1, there appears a fixed pole residue of a Reggeon-Reggeon amplitude which may introduce an additional dependence on the various momentum transfers. The main goal in this note is to study the dependence of the EXD breaking (or the torus diagram, fig. 1) on both space and time-like t. This is done by analytically continuing the torus contribution from t < 0, where it is originally defined, to time-like t. Except for the fixed pole residue (the crossed clusters in fig. 1) all other physical parameters already appeared in the cylinder study [5] and will be taken from there. For the fixed pole residue we take a dependence on the various momentum transfers as in the simple dual model with, however, the involved parameters left free (see below). We consider it a rather general assumption on the structure of the fixed pole residue. Then employing our previous results [5] for the planar and the cylinder topologies, an AP is also obtained for the torus topology, and the rate of approach to it is estimated. We thus obtain the novel feature of the “l/N expansion” that EXD is no longer a property independent of the relevant Mandelstam variable: There may be a sizable breaking of EXD when t is a momentum transfer in one channel (t < 0) with this breaking dying out when t becomes an energy variable in the crossed channel (t > 0). Needless to say, our study is independent of the sign of the EXD breaking. The AP for the torus is particularly interesting since trajectories such as p and A,, K* and K** are relatively well understood in both the scattering and resonance regions. * Supported in part by the Israel Commission for Basic Research. * Summer visitor.

203

Volume

64B, number

2

PHYSICS

LETTERS

13 September

1976

t

6’

t1

+ t;

t2 + Fig. 1. Non-planar

corrections

to planar

trajectories

t

+

with non-zero

iso-spin.

For the quark

content

and other

details see ref.

[ 11.

Finally it should be remarked that the derivation of AP is fundamental for the approach since it entirely rests on the phase structure of untwisted and twisted lines. However, the rate of approach to AP depends, in general, on the detailed value of the involved parameters (see below). Thus, for example, with increasing time-like t, the cylinder and the torus may die out with a different rate. We now present explicitly the analytic continuation of the torus diagram to time-like t. Using the same notation as in ref. [5] , the insertion I(t) due to fig. 1, to a planar Reggeon with trajectory or(t), can be written as:

aqt)Ysdf9($2

I(t) =5%

‘yc,I

“c,2

V(t; 1) 2) 2 cos ncw’(tl - t;>.

Here “c,i = o(ti) + 4ti) - 1, gi z g(t, ti, ti) (the triple Reggeon coupling), related to a Reggeon-Reggeon amplitude fixed pole residue) and

(1)

V(t; 1, 2) z V(t; tl , t;, t2, ti) (closely

dti dti @(-A) d~i =

[_A(& &, &)I i/2 169

’

with ’ = ” 2*

Note that here, in contrast with ref. [5] , the number of quarks N does not appear since there is no free quark loop in fig. 1 (see ref. [l]). The triple Reggeon coupling is parametrized as in ref. [5] , namely gi zg(t,

(” ) exp (” +tl .+ fJ!)

tip t::>=g[cr(t) - Q-c,i]exp Tff

(i= 1,2)

(2)

with the various parameters being constrained by the planar bootstrap (eqs. (5), (5’) and (5”) of ref. [5]). We must discuss now the vertex V(t, 1, 2). As a guide one may take the simple Veneziano (s, u) term, namely

A(s,u;t,tl,t;,t2,t’2)=g 2 U-4sN r(-4u))

W4s) -4uN ’

with s + L(+ t = tl t t; t t2 + t> and evaluate the fixed pole residue &t;

1, 2) Ei

jdol(s) so

Disc,(,) A(s, u; t, ...)

(3)

tt;tt2+tl2,--t)a’ln2].

(4)

to obtain v(t;1,2)=g2exp[(tl

Our V(t; 1,2) is derived from eq. (3) by introducing the Mellin transform factor (4~))-4~)-~. With this factor eq. (4) w-ill be multiplied by g1 exp(-A’t) where gl and A’ > 0 can be calculated from the simple Veneziano (s, U) term. For our purpose it is sufficient only to know that A’ > 0 which is just a result of the Mellin transform factor. Thus in order not to be bound to the parameters of the simple Veneziano model we take V(t; 1,2) = G2 exp 204

(tl + ti + t2 + tb)

)

exp

-_z t exp(-A’t) (“)

(5)

Volume 64B, number 2

PHYSICS LETTERS

13 September 1976

with A,A’>

0.

(5’)

The positivity of A is also model independent, and ensures the convergence of integrals involved in the study of the Reggeon-Reggeon cut in the J-plane. Inserting eqs. (2) and (5) into eq. (1) one obtains: Z(t) = g2G2 exp(ut) exp

exp(-A’t)

JdGl

exp

(6)

Eq. (6) is defined for t < 0 because of [-A(t, ti, ti)] l/2 which appears in d~i. One may directly analytically continue (-A)1/2 and the other factors in eq. (6) into t > 0 and then integrate. However, it is much simpler to integrate first for t < 0 and then trivially analytically continue into t > 0. One thus obtains from eq. (6) 1(t) = 2 $

$-n exp(at) exp

](/, +b)2 +o’)z] ‘(7)

As it stands eq. (7) cannot tell us whether Z(t) increases or decreases with increasing time-like t. However, using the planar bootstrap constraints (eqs. (5), (5’) and (5”) of ref. [5]), namely

g2N12b = 1

16n’

one can write eq. (7) as: (9) Eq. (9) now shows explicitly the occurrence of AP for the torus, irrespective of A’ (we now take, for simplicity, A’ = 0). Thus EXD breaking is expected to improve with increasing time-like t and deteriorate in the space-like region with increasing I tl . Obviously, because of the parametrization of the involved vertices, our analysis is not reliable for extremely large I tl , but is expected to be valid for -1 GeV2
quenching 2b

tT

1 GeV2. of the torus, in the time-like region, already occurs for

(A + b)2 + (na’)2

c-----(~a’)~ A 2 + Ab + (~a’)~ ’

(10)

Similarly the torus will be significantly enhanced in the space-like region at -tT. Note that for the cylinder the analogous quantity to tT is given by [4, 51 t, = 2b/(na’)2.

(11)

As one sees from eqs. (9) and (lo), as in the cylinder [4,5] case, also for the torus, AP is a direct result of the planarity factor; exp(-irrol) for untwisted lines. Indeed in both ref. [5] and here the only trace of the planarity phases is through the slope cw’;taking or + 0 there is no asymptotic planarity. However, the rate of quenching, for t > 0, for the cylinder and the torus may be different and depends on the details of the involved parameters (see eqs. (10) and (11)). It is clear that the derivation of AP in eq. (9) rests on a rather firm basis and does not suffer from the ambiguities present in a numerical evaluation of Z(t) because of the unknown fixed pole residue of a Reggeon-Reggeon amplitude. Of course, in a complete theory, yet lacking, the planar amplitude should uniquely specify all higher 205

Volume 64B, number 2

PHYSICS LETTERS

13 September 1976

topologies. However, a numerical evaluation of the insertion, such as in ref. [l] , is vital for the approach and may be highly instructive. We present here the complementary numerical study to ref. [l] , namely the evaluation of the torus quenching parameter tT. Assuming that v(t; 1, 2) has the same slope as gf (eqs. (5) and (2)), that is A = 2b, one obtains: 9b2 + (n+

~ j& tT

(~0’)~ 6b2 t (na’)2 *

Now from the study in ref. [S] of the bare pomeron intercept

compared with the cylinder quenching

2bFt:-

te= (&)2

0.45 a’ ’

and slope, b m 2.2 (IL’,leading to

parameter

(a’ = 0.9 GeV2).

Obviously these are only estimates, but nevertheless may indicate that the torus dies out, for t > 0, at a rate similar to the dying out of the cylinder (Note that A’ in eq. (9) is in practice non-zero and a reasonable value of A’ N 1 GeV2 will make tT even closer to tc.) In summary we have studied the contribution of fig. 1 in both the scattering and particle regions. In ref. [l] it has been shown that fig. 1 represents the most important mechanism for breaking EXD between planar trajectories with non-zero iso-spin, for example p and A,, K* and K**. The analytic continuation of the insertion Z(t), eq. (I), to the particle domain, indicates the existence of AP also for the torus. Namely, EXD is recovered with increasing time-like t, whereas sizable breaking may occur in the scattering domain with increasing 1tJ . The present analysis is particularly suitable for the p-A2 and K*-K** problem since the involved particles do not have high masses, in contrast to the case of the J/$ family. A rough numerical estimate presented above suggests that the torus quenching is not too different from the cylinder quenching. That is, already at the p mass, EXD breaking will be very small. The work in ref. [l] , in which the evaluation was carried out for t = 0, and the present work lend a further support to the “1 /N expansion”, or the “dual unitarization” approach, as a model for understanding hadron dynamics. This work has been carried out during a summer visit to the Research Institute for Theoretical Physics, University of Helsinki, where the board of the Institute is gratefully acknowledged for the pleasant hospitality. The author also thanks J. Uschersohn for showing him the work in ref. [l] .

References [l] Chan Hong-MO, Ken-ichi Konishi, J. Kwiecinski and R.G. Roberts, Rutherford Lab. preprint, RL-76-056, T.161. [2] G. Veneziano, Phys. Lett. 52B (1974) 220; Nucl. Physics B74 (1974) 365; M. Ciafaloni, G. Marchesini and G. Veneziano, Nucl. Phys. B98 (197.5) 472,493. [3] Chan Hong-MO, J.E. Paton and Tsou Sheung Tsun, Nucl. Phys. B86 (1975) 479; Chan Hong-MO, J.E. Paton, Tsou Sheung Tsun and Ng Sing Wai, Nucl. Physics B92 (1975) 13. [4] G.F. Chew and C. Rosenzweig, Phys. Lett. B58 (1975) 93; Phys. Rev. D12 (1975) 3907; Nucl. Phys. B104 (1976) [5] M. Bishari,

206

Phys. Lett. 59B (1975)

461.

290.