Non-planar corrections to the pomeron and reggeons in the dual unitarity scheme

Nuclear Physics B114 (1976) 137-146 © North-Holland Publishing Company

NON-PLANAR CORRECTIONS TO THE POMERON AND REGGEONS IN THE DUAL UNITARITY SCHEME J. USCHERSOHN Research Institute for Theoretical Physics, University of Helsinki

Received 13 April 1976 (Revised 14 June 1976)

We study a special class of non-planar diagrams, those in which the produced clusters are crossed. Insertion of these diagrams in the reggeon propagators accounts for exchange degeneracy breaking. The even-signaturereggeons get their intercept promoted to a larger value, while the odd-signature ones are left unchanged.

1. Introduction

A lot of attention has been devoted recently to iterative schemes for calculating hadronic amplitudes [1 ]. In one of the most successful attempts, Chan, Paton and Tsou [2] use the combined constraints of duality and unitarity to derive first bootstrap equations for the reggeons and the pomeron. Then, using the parameters determined from the bootstrap equations and some phenomenological input, the dual unitarity scheme was used in explaining a variety of aspects of strong interactions: energy dependence and t-slopes of two-body reaction amplitudes [2], clustering effects and correlations in multi-particle production [3], violations of the ZweigOkubo-Iizuka rule [4], the symmetry-breaking pattern of reggeon trajectories [5] and hadronic and electromagnetic decay rates of the new resonances [6]. A more rigorous and more ambitious approach is the topological expansion developed by Veneziano [7]. The central idea is that, instead of summing the diagrams according to the number of vertices (ordinary perturbation expansion in the coupring constant) one should sum according to the topological properties of the diagrams. The simplest term in this expansion, the sum of planar diagrams, corresponds to the bare reggeon, while the bare pomeron emerges after summing the diagrams containing crossed reggeon loops. More complicated topologies correspond to absorptive corrections, cuts, etc. The analyses of reggeons and the pomeron presented so far were restricted to the simplest loop diagrams, in which only the (space-like) reggeons in the loop were crossed. Recently, however, it has become more and more obvious that another 137

13~

.L Uschersohn / Corrections to the pomeron

class of diagrams plays an important role, namely those diagrams in which (timelike) produced clusters are crossed. Such diagrams, for example, are crucial in the calculation of ff decays [6] and, more general in understanding the violations of the Zweig-lizuka-Okubo rule. A detailed discussion of diagrams with crossed intermediate state particles and their connections to violations of the ZlO rule and exotic exchanges was recently given by Schmid, Webber and Sorensen [8]. In this paper we study the contribution of such diagrams to reggeons and the pomeron. In sect. 2 we discuss the diagrams and study their properties in terms of the exchanged quantum numbers. Sects. 3 and 4 contain an analysis of how the intercept of the reggeon trajectories and of the pomeron are modified when such diagrams are inserted. For the purpose of discussion we use a simple one-dimensional model. In sect. 5 we present our conclusions and summarize the results.

2. Properties of diagrams with crossed intermediate (time-like) clusters We shall consider here only diagrams where all clusters except the end ones are crossed. A typical example is shown in fig. 1a. For simplicity we shall always use the notation of fig. lb. At the ends of the diagram we have used semi-local duality to replace the average over the cluster contribution to particle-reggeon scattering through the reggeon contribution. Diagrams where even the end clusters are crossed will be considered separately, since they cannot be inserted as corrections to the leading reggeons. For instance, the diagram of fig. 2 has the leading/'-singularity at zero and not at c~ = ~. The reggeon loops can all be either crossed or uncrossed. The end loops must have only one of the reggeons crossed, while the "internal" loops can have either both reggeons crossed or both reggeons uncrossed. This follows from the fact that the number of twists in each loop must be even, in order to preserve the qq content of each reggeon or cluster. In what follows we shall distinguish b~tween two kinds of insertions: one type

$

Fig. la. Example of diagram with crossed clusters. The two uncrossed clusters at the ends represent in fact sums over uncrossed clusters.

Fig. lb. Simplified representation of the above diagram.

Jr. Uschersohn / Corrections to the pomeron

139

Fig. 2. Diagram with crossed end clusters.

will leave the orientation of the @[ pair unchanged while the other will reverse the orientation. We call the former a II-type diagrams and the latter a X-type diagram. They are exemplified in fig. 3. The simplest diagrams with crossed clusters are shown in fig. 4. The diagrams 4a and d are of ll-type, while 4b and c are of X-type. Each diagram separately is not necessarily real. However, the sum of 4a and d yields a real quantity. Same is true for 4b and c. Consider now a general diagram, with n loops. There are 2 n different diagrams with n loops. How can one tell which diagrams are of If-type and which are of X-type? Comparing the two diagrams in fig. 5, we see that each loop in which the reggeons are crossed changes the orientation of the q~ pairs while the loops in which the reggeons are uncrossed leave the orientation unchanged. Thus we can abstract the following general rule: when the number of crossed reggeons on each side of the diagram is even (odd), the diagram is of II-type (X-type). Thus, the diagram in fig. 1 is of If-type.' The next hnportant question is to find out which diagrams allow exchange of quantum numbers and which diagrams forbid it. Fig. 6 represents the quark content of a general, n-loop diagram with crossed clusters, inserted into the reggeon propagator. The diagrams in which a and b are identical to 1 and 2 will allow quantum numbers exchange, and such diagrams will contribute to the reggeon propagator. If a and b are identical to 3 and 4, the diagram will contribute to the pomeron. From fig. 6 one can see that if the number of loops is even, a and b cannot be identical to

Fig. 3. II-type and X-type diagrams.

--¢XS-a)

b)

-AL>c)

d)

Fig. 4. Diagrams with two loops.

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J. Uschersohn / Corrections to the p o m e r o n

722

Fig. 5. Quark diagrams for loops with uncrossed and crossed reggeons.

3 and 4. hr other words, diagrams with an even number of loops will always allow exchange of quantum numbers. Consider now a diagram with an odd number of loops. If a and b are identical to 1 and 2 the diagram will allow quantum numbers exchange. However, if instead of crossing c and a in the last loop, one crosses b and d, then the quark content of the external reggeon will be 3 and 4. Such diagrams will contribute to the pomeron. Note also that when a and b are identical to 1 and 2 the quarks 3 and 4 will make a closed quark line. In models with SU(N) symmetry, this means that the summation over internal quantum numbers yields a factor N. We summarize this discussion with the following rule: Diagrams with an even number of loops allow quantum numbers exchange. To each type II diagram there is a type X diagram, obtained by interchanging the crossed and uncrossed reggeons in one of the end loops. Diagrams with an odd number of loops contribute both to quantum numbers exchange and to vacuum exchange processes. For each diagram which allows the exchange of quantum numbers there is a diagram which forbids it, obtained by interchanging the crossed and uncrossed reggeons in one of the end loops. For diagrams with a fixed, odd number of loops, the following is true: among the diagrams which allow quantum numbers exchange, half are of If-type, and half of X-type. The same is true for the diagrams which forbid quantum numbers exchange. The proof of these statements is rather lengthy, but does not pose any serious difficulties. In order to calculate the effect of these diagrams on the reggeon and pomeron parameters we shall use a very simple one-dimensional model [91. This model has been widely used recently [~0] in connection with different aspects of production of clusters separated by reggeon exchanges. In this model, the contribution of a diagram with n loops can be written as [ 11 ] n-1

D

=g2(g'2)n-lfdxo

n

e(~-l)Xo

n

X [I1 dz/e2(c~-l)zJ 6 ( Y -

fI

1

n

dx. e (~E-l)xi dx n e (~-l)xn t

n

~0 xi - ~1 zl")'

(1)

where x i is the rapidity interval occupied by the ith cluster

s. = rn 2 e xi ,

(2)

z i is the rapidity gap between clusters Zi

=

1 1 QVi+I -- 2 X i + l ) -- ( Y i + ~ X i ) '

(3)

J. Uschersohn / Corrections to the pomeron 3 34

~4 3

1

c

d

c

141

a

~

2

Fig. 6. Quark content of a general, n-loop diagram.

t

Fig. 7. Amplitude with an intermediate crossed cluster.

(Yi are the cluster rapidities), g is the ordinary triple-reggeon vertex coupling and is the intercept of the ordinary reggeon trajectory (a = ½). The limits of integration are

x i>~ O,

z~ >10.

(4)

We have assumed that the energy dependence of the crossed cluster production amplitude (fig. 7) is see where aE is presumably low and will be determined later from the data. In fact, since the coupling constant g'2 is also unknown we can only get constraints on both of them. These constraints allow the calculation of either of them, once the other is known. Although not necessary in this work, one can consider that the contribution in fig. 7 is dual to an exotic (qq7:t~ crossed exchange and g'2 is the product of the two triple-reggeon couplings involving the exotic trajectory and the two loop trajectories. Transforming to J space, we get

f dYe _

(J-1)Y Dn(Y ) g2

1

"(

(j_o02 (S_2ot+l)

g,2

)n-1

~_OeE)(J_2o<+I )

.

(5)

We note that all diagrams with the same number of loops give the same contribution. Because of the one-dimensional approximation, crossed and uncrossed reggeon loops contribute equally. This fact will have important consequences later, We close this section with a comment on diagrams where even the end clusters are twisted. The contribution of such a diagram with n loops is g,2

~n+l

( S - 2c~ + 1) ( a _ e e E ) ( J _ 2 e e + l ) ]

.

(6)

They cannot be inserted in reggeon (or pomeron) propagators but they contribute to total cross sections. Such diagrams may be responsible for the fact that the factorization of reggeon contributions to total cross sections is not exact.

J. Uschersohn / Corrections to the pomeron

142

3. Corrections to reggeon propagators

We study in this section the effect of inserting diagrams with crossed clusters in the reggeon propagators. For every given number of loops n, there are 2 n different diagrams, obtained by taking all possible combinations of crossed and uncrossed reggeon loops. If n is even, all 2 n diagrams allow quantum numbers exchange, while for n odd only 2 n-1 diagrams can contribute to reggeons. For every fixed number of loops, half the diagrams are of II-type and the other half are of X-type. Contributions to even signature reggeons are obtained by taking the sum of II-type and X-type diagrams, while odd signature reggeons get contributions from the difference of II-type and X-type diagrams. Since in our one-dimensional model all diagrams give the same contribution, the difference of II-type and X-type diagrams will be zero. Thus, odd signature reggeons (p, co etc.) are not affected by the insertion of diagrams with crossed clusters. For the even signature reggeons we have, however, 2 n contributions for n even and 2 n -1 contributions for n odd. For diagrams with an odd number of loops there is also an extra factor N in SU(N) symmetric theories, as shown in the previous section. We sum first over n, assuming that each diagram is inserted only once in the reggeon propagator, as in fig. 8. Cutting off the external reggeons, and denoting the sum by S, we have J-2~+l

n odd n>~3

2ng2 n even n>2

U-~E)(J-2a+

1)

1 ( g,2 )n-I J - 2 o ~ + 1 (-~- O~E)~ - - 2~ + 1)

4927 [(J - 2c~ + 1) + NT] - ( J - 2o~ + 1) [(Jr- 2~+ 1) 2 '-43'21 '

(7)

where g,2 T-j_

aE.

Next, to calculate the correction to the reggeon propagator, we sum over all possible insertions of S in the reggeon propagator. The sum is shown in fig. 9, where R

22• ~

+22N- ~ +2~" ~

+ +.'.

Fig. 8. Corrections to the reggeon propagator. Each diagram is inserted only once.

J. Uschersohn / Corrections to the pomeron

143

Iqg. 9. Corrections to the reggeon propagator.

stands for a bare reggeon constructed only from planar diagrams. R R + RSR + RSRSR

+ ...

1-

RS

'

(8)

where S is given by formula (7) and R = (J - c0 -1. The new poles of the reggeon propagator are the solutions of the following equation:

J- ~ :

4g23` (J-2c~+l)

[ ( J - 2a + l) + N3`] [(j 2 ~ + 1 ) 2 - 4 3 , 2 ] "

(9)

This relation can be used to constrain g2, g'2, N and aE, once we know the intercept of the leading even signatured reggeon. For g'2 small enough the r.h.s, of eq. (9) is small and positive, so that the leading solution can be imposed at a value slightly larger than c~. The most general constraints on g'2 and aE arise from the requirement that the r.h.s, of (9) is positive but less than ½ for the leading J. To conclude, after the insertion of diagrams with crossed clusters, the odd signature reggeons are left unchanged, while the even signature reggeons have their intercept promoted to a larger value. Diagrams with crossed clusters provide thus a natural explanation for exchange degeneracy breaking. Line reversal invariance holds only at the planar level.

4. Corrections to the pomeron There are two kinds of insertions which can contribute to the pomeron. One can either insert a diagram with crossed clusters with an odd number of loops which forbids quantum numbers exchange into the reggeon propagator or one can insert a diagram with an even number of loops into the pomeron propagator. Using an argument identical to the one presented in the previous section, if one separates the crossing even part and the crossing odd part of the pomeron, only the crossing even part will be affected by diagrams with crossed clusters. (We disregard here the problem of whether a crossing odd part exists at all. In some models, delicate cancellations occur and the crossing odd part of the pomeron vanishes, while in other models the crossing odd part decreases very fast with increasing energy.) We calculate first the effect of inserting diagrams with odd number of loops which forbid the exchange of quantum numbers into the reggeon propagator. Summing over the number of loops, with each diagram inserted only once, we get, denoting

144

J. Uschersohn / Corrections to the pomeron

the sum by G: 2n-lg2

(

g'2

)n 1

n~>3 4 g2 -

,),2

(J - 2a + 1) [(i _ 2a + 1) 2 - 472 ]

(10)

Again we have cut off the external reggeon propagators. The most general form for the pomeron propagator can be obtained by taking all possible combinations of insertions of G and S, separated by reggeon (R) or pomeron (1P) propagators such that only vacuum quantum numbers can be exchanged. By IP we have denoted here the pomeron propagator constructed only from diagrams with crossed reggeon loops separated by ordinary regge exchanges. The expression of IP in the one-dimensional model has already been calculated by Sakai [10]. In our simple one-dimensional model, where all quantities commute, the general expression can be simply written as oo

n=0

{(G + s ) n

+ R)

- SnR

IP+R

)

R

= 1-(1P+R)(G+S)

(11)

1 -SR"

The new poles of the crossing even part of the pomeron can be obtained from the condition 1 - (G + S)(IP + R) = 0.

(12)

The expression for IP (cf. ref. [11]) is

1

(13)

IP - (j _ ~) [(S - ~)(J - 2~ + 1) - Ng2l Introducing relations (7), (10), (13) and R = (J - c0 -1 in eq. (12) one obtains a rather complicated equation for the pomeron intercept. Imposing the leading solution at 1 one gets a constraint relation for g2, g'2, N and ~E" The constraint for the leading J reads 4g27 (J-2~+1) =1.

~(J - 2ol+ 1) + (N+ 1)T']F ( J - o O ( J - 2 o ~ + l ) - ( N - 1 ) g [_ U - 2 d ~ l ) 2 - _ - 4 5

2

Jl_(j_--~)[(j-_-~j~2~-+l~--NggZ]] (14)

An interesting observation is that the pomeron has a piece which exactly cancels the reggeon part (compare eqs. (11) and (8)). This effect has been found by many authors. It is not clear if this is due to the very simplified nature of the one-dimensional model or whether it is a general, model-independent feature.

J. Uschersohn / Corrections to the porneron

145

Conclusions The purpose of this paper was to study the contributions to reggeons and the pomeron coming from a certain class of non-planar diagrams. We have studied the effect of inserting diagrams with crossed clusters in the reggeon propagators and in the pomeron propagator. Only the even signature reggeons and the crossing-even part of the pomeron are affected. The intercept o f the leading even signature reggeon was shown to be larger than the intercept of the odd signature one. The fact that the intercept of the odd signature reggeons is left unchanged is presumably due to our simplified model. In more realistic models, which include the t-dependence of the reggeon propagators and the triple-regge vertices, the contributions coming from loops with crossed reggeons are expected to be larger than the contributions of loops with uncrossed reggeons. Then, we do not find any longer cancellations of diagrams contributing to the odd signature reggeons and the crossing odd part of the pomeron. Another important effect which may change the quantitative results is the symmetry breaking o f reggeon trajectories. This effect is now under study. All the considerations in this paper were valid only for mesons. Baryons and baryon trajectories were neglected. Preliminary investigations show that the introduction of baryon trajectories shifts all intercepts to larger values [12]. The class o f diagrams studied here can be considered as a particular case o f a more general scheme proposed b y C.-I. Tan [I 3]. Finally, we would like to mention that the two-loop diagrams discussed here were used previously to explain the o-A 2 exchange degeneracy breaking [ 141. I would like to thank Dr. Hong-Mo Chan for explaining to me how these diagrams appear in the dual unitarity scheme and for showing me his results prior to publication. I thank Dr. C. Montonen for discussions and comments on the manuscript.

References [1] Proc. 6th Int. Coll. on multiparticle reactions, Oxford, 1975, RL-75-143, ed. tf.M. Chan, R.J.N. Phillips and R.G. Roberts. [2] H.M. Chan, J.E. Paton and S.T. Tsou, Nucl. Phys. B86 (1975) 479; H.M. Chan, J.E. Paton, S.T. Tsou and S.W. Ng, Nucl. Phys. B92 (1975) 13. [3] A. Gu/a, Rutherford Lab. preprints RL-75-030, RL-75-080 (1975). [41 C.F. Chew and C. Rosenzwcig, Phys. Letters 58B (1975) 93; Phys. Rev. D12 (1975) 3907. [5] C. Schmid and C. Sorensen, Nucl. Phys. B96 (1975) 209; N. Papadopoulos, C. Schmid, C. Sorensen and D.M. Webber, Nucl. Phys. B10I (1975) 189; J. Dias de Deus and J. Uschersohn, Rutherford Lab preprint RL-75-042, to be published. [6] H.M. Chart, J. Kwiecinski and R.G. Roberts, Phys. Letters 60B (1976) 367; H.M. Chan, K. Konishi, J. Kwiecinski and R.G. Roberts, Phys. Letters 60B (1976) 469. [7] G. Veneziano, Phys. Letters 52B (1974) 220; Nucl. Phys. B74 (1974) 365. [8] C. Schmid, D.M. Webber and C. Sorensen, ETH preprint (January, 1976).

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J. Uschersohn / Corrections to the pomeron

[9] C.F, Chew and A. Pignotti, Phys. Rev. 176 (1968) 2112; C.E. DeTar, Phys. Rev. D3 (1971) 128. [10] N. Sakai, Nucl. Phys. B99 (1975) 167; S. Matsuda, K. Sasaki and T. Uematsu, Kyoto preprint KUNS 294 (1974); S. Jadach, Nucl. Phys. B99 (1975) 514; J. Finkelstein and J. Koplik, Columbia preprint CO-2271-72 (1975). [11] See ref. [10[. [12] H.M. Chan. private communication. [13] C.-I. Tan, Preprint BNL-20254 (1975). [14] H.M. Chan, K. Konishi, J. Kwiecinski and R.G. Roberts, work in preparation.