Harari-Freund and other schemes for the pomeron in the topological expansion

Nuclear Physics B108 (1976) 285-292 © North-Holland Publishing Company

HARARI-FREUND AND OTHER SCHEMES FOR THE POMERON IN THE TOPOLOGICAL EXPANSION *

G. VENEZIANO )qeizmann Institute o f Science, Rehovot, Israel ** Research Institute for Fundamental Physics, Kyoto University, Kyoto, Japan

Received 24 February 1976

Consequences of t- and s-channel unitarity are derived for the bare (unabsorbed) reggeon and pomeron in the topological expansion. The "strict" Harari-Freund scheme is found to be inconsistent. Possible alternative schemes such as that of Chew and Rosenzweig are discussed. Further arguments in favour of "asymptotic planarity" are given.

1. Introduction

Chew and Rosenzweig (CR) have proposed [1 ] a scheme in which the bare pomeron is just the f trajectory, renormalized and mixed with the f' trajectory v/a the cylinder correction to the planar approximation, as defined in the topological expansion [2] (TE). At t = 0 the cylinder correction is important, but CR argue [1] that, as one goes to t ~ m 2, the cylinder coupling turns off and the S matrix becomes essentiaU.y planar. CR call this property "asymptotic planarity" (AP) and attribute to it the surprisingly good validity of exchange degeneracy, of ideal mixing and of the Okubo-Zweig-Iizuka (OZI) rule [3], in the time-like region. More recently, Chew and Rosenzweig, as well as Bishari, have given theoretical arguments [4] supporting the concept of AP. Although rather appealing, the CR scheme contradicts the rather well established Harari-Freund (HF) picture, in which the pomeron is a distinct, factorizable Regge trajectory and the f is still degenerate with p, A 2 and co and ideally mixed to the f' (we define this as the "strict" HF scheme). In this note we derive some general consequences of the TE, which have an important bearing on the previous issue. Our main results are: (i) Both t-channel and s-channel unitarity point at the sum of the plane and the

* Supported in part by the Isreal Commission for Basic Research. ** Permanent address. 285

286

G. Veneziano / Topological expansion

cylinder (called the "complete'cylinder" in the following) as the object exhibiting simple, factorized Regge poles. (ii) The "strict" HF scheme is not compatible with the TE. Previous problems with the extension of HF to inclusive cross sections are now easily understood. (iii) The CR scheme appears to be the simplest, but not the unique alternative to HF. (iv) Combining our results and previous results of dual perturbation theory, we arrive at a consistent scheme with distinct P and f trajectories. (v) Further arguments are given to support the idea of AP. We now proceed to give details on each of these points:

2. General results from unitarity in the TE l e t us denote * by P the planar 4-point function for ab -* cd and by C the cylinder contribution having no quantum number exchange on the t-channel (definedas ab -~ car). Since we work at the h = 0 level, unitarity equations are rather simple. We have, for instance: (a) t-channel unitarity below the inelastic threshold. For P we have "planar unitarity" [2] for each permutation of the external lines: - i Disct Pabcd = Pab12 ® P~21cd ,

(1)

and similarly for other permutations. For C, however, the equation is more complicated and reads - i Disc t Cab,c d = Cab, l 2 ® C~2,cd + P ® C* + C ® P* + P ® I f* ,

(2)

where the last term on the right-hand side has to be interpreted as Pabl2 ® P~2cd + +Pba21 ® P~lclc + .... Eq. (2) is depicted in fig. 1. It is quite clear graphically that, if we define a "complete cylinder" Cab,cd by the equation Cab, cd - Cab.cd + Pabcd + Pabac + Pbadc + Pbacd ,

(3)

we have for C, - i Disc t Cab, cd = Cab,12 ® ~ *1 2 , c a

•

(4)

The simple structure of eq. (4), as compared to eq. (2), is the important result. Before discussing its consequences we turn to s-channel unitarity. (b) s-channel unitarity at large s, freed t <~ O. The s-channel unitarity constraints * To be precise we should write P = Pabcd + Pbadc + Pabdc + Pbacd, corresponding to the (cyclic and anticyelic) permutations of the external particles giving t-channel intermediate states. We should also write C = Cab,cd = Cba,cd = Cab,dc ~ Cba,dc. We are occasionally leaving out these indices for simplicity.

G. Veneziano / Topologicalexpansion b

c

Ira.

b

287

I

c

= o

d

+ o

1 +

+ ........ +

o

2

d

......

o

2

Fig. 1. t-channel unitarity for the cylinder below the inelastic threshold (eq. 2). Planar amplitudes, denoted by a solid bubble, have an orientation. The dotted line indicates the cylinder shape.

f o r P and C at large s can be converted, by standard techniques, into multi-Regge integral equations, provided a multiperipheral assumption is made. For P the integral equation takes the form [5] Discs P(s) = Discs Po(s) +

fd¢ Kp @Discs, P(s'),

(5)

with d$ representing longitudinal and transverse momentum phase space;P0(s ) is the low-energy (single cluster) inhomogeneous term, and Kp is the planar kernel. In the J plane one gets [6] P(J, q) = P0(J, q) +f dq'Kp(J, q, q')P(J, q').

(6)

Here q, q' are transverse momenta and PO(J, q) has no singularity in J. Under the same assumption one obtains for C,

C(J,q)=Co(J,q) +/dq'Kc(J,q,q')C(J,q') + f dq'Kc(J,q,q')P(J,q') . (7) The (crucial) last term on the right-hand side of eq. (7) comes from the possibility of generating C by joining a planar (or non-planar) kernel and a planar amplitude through a pair of twisted reggeon links [1,5]. As a result, the equation for C(J) is more complicated than the one for P(J). On the other hand, for C = C + P, we easily find C(J, q) = C0(J, q)

+f dq'Ke(J,q, q')C-'(J, q') .

(8)

Hence, for C, the integral equation is again a simple one. (c) Consequences of the unitarity equations. The general lesson to learn either from t-channel or from s-channel unitarity is that simple, factorized Regge poles can be expected to occur in P and in C = C + P, but not in C itself. For instance, eqs. (1) and (4) can be used in a standard way [7] to prove that Regge poles are simple poles and have factorized residues, but eq. (2) cannot be used for that purpose. From the s-channel point of view, integral equations, such as eqs. (6) and (8), give rise to Regge

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G. Veneziano/ Topological expansion

pole solutions under standard assumptions on the kernels. Again, this is not true for eq. (7). In conclusion it is on C (and not on C) that we have to make our assumptions on the spectrum of Regge poles in the t-channel. In the presence of various channels we will have to distinguish those to which C does not contribute (e.g. charge or strangeness different from zero, I ~ 0 if isospin is exact) and for which C = P. Other channels, which couple to C get mixed, even if they did not at the planar level, and we have to face a diagonalization problem. In this way violations of the ZOI rule are produced [1 ].

3. The HF scheme in the TE In order to arrive at the HF results in our framework, we have to set c(J)

= j _

1

- j

1

(9)

i.e. we have to assume that C has just the pomeron pole and P has a set of exchange degenerate Regge poles. Inserting eqs. (9) into eq. (2) we arrive at an inconsistency since there is nothing to compensate for the poles at J = ctR on the right-hand side. A similar result follows from eq. (7): The last term on the right-hand side gives a pole at J = a R, and, again, there is nothing to cancel it if we assume eqs. (9). We conclude that the HF scheme is incompatible with the TE. It is worthwhile to mention that the problem found here with a strict HF scheme is closely related to the difficulties found some time ago [8] in extending the HF scheme to inclusive cross sections. In a strict HF scheme (called strong HF in ref. [8]) it was impossible to satisfy certain inclusive sum rules. Charge conservation gave, for instance, Discs Cab,ab = ~ Qc DiscM2 Pacbbca + (cylinder terms).

(10)

¢

For ab exotic the left-hand side goes like s ~P, but the term shown on the right-hand side goes like s aR log s and, in HF, cannot be cancelled. Eq. (19) can be derived in the TE and this, together with the factorization constraints we obtained, appears to rule out the "strong" HF scheme discussed in ref. [8]. In other words, the TE makes the problem one of a general nature, but also suggests in which direction possible alternative schemes can be looked for.

4. The CR scheme In the CR scheme [1 ] the object which has a simple structure of Regge poles is and, therefore, that scheme is consistent with (and indeed has been obtained within)

G. Veneziano / Topological expansion

289

the TE. The CR scheme has the further property that the number of leading Regge poles in P is the same as that in C; in other words the effect of C is simply that of renormalizing and mixing those poles of P which can couple through C. Since, however, the cylinder correction is seen essentially as a perturbation, the number of levels (Regge poles) is not changed. Of course, this feature of the CR scheme is not a consequence of our equations. The CR scheme has several attractive features; it is a very constrained and predictive scheme. It is not clear, however, that it can pass all the experimental checks, even if some of them have been successful [9]. Theoretically the CR scheme contradicts the perturbative results of dual model calculations, to which we now turn our attention.

5. Relation to the conventional dual scheme In the dual model [10] one has, to lowest order in the coupling constant, a leading reggeon at a(t) = 1 + a't and a leading pomeron at Otp(t) = 2 + 1/2dt. The situation in the TE has to be compared to that of dual model calculations only after all planar loops are included. Hanar loops are expected to renormalize a(t) and ap(t) (making them non-linear) but also to mix these two types of regge poles. Reggeonpomeron transition amplitudes have actually been computed [10]. Consider the situation in SU(2). There will be a direct coupling of the f to the pomeron gRC. One finds for C, C~(J, t) = j _ V=t~t) + ~

~CR J _ ~p(t-----~+ "'"

j - ~e(t) =

(11)

(J - ~(t)) (J - ~p(t)) - g2CR ' where ~(t) and ~p(t) are the (planar) renormalized, but non-mixed reggeon and pomeron trajectory respectively. C will have two Regge poles at J = ~ with 2tx± = (~ + ~p) -+X/(~p - ~--')2 + 4g2 R .

(12)

I f ~ and ~p do not differ too much qualitatively from t~ and rVp we find that ct+ ~ ~p for ~p -- ~ > > IgRC] i.e. t large and negative and o% ~ ~ for ~ - ~p > > IgRCl i.e. t large and positive. As shown in fig. 2, the new eigenstates will not cross and the leading trajectory is like a pomeron (reggeon) for ct't < < - 1 (t~t > > + 1). Probably, this turning of the pomeron into the reggeon at large positive t is related to the idea of AP [1,4]. We see that, in this scheme, we do end up with two trajectories (three in SU(3)). Still, the object exhibiting factorizable poles is C and not C. A possible way in which a new singularity can occur in C within an integral equa-

290

G. Veneziano / Topological expansion

J

oa~l

Fig. 2. Possible form of f and pomeron trajectories before (dotted line) and after (solid line) mixing.

tion such as eq. (8), is by having Ke become a kernel of rank two i.e. by having a new factorizable term in K~-, corresponding to low-energy resonance plus background. Treating this problem as in ref. [6], one finds solutions with two output self-consistent poles. The problem clearly deserves further study, but, again the CR scheme does not appear as a necessary consequence of the TE.

6. Further arguments for AP We conclude with two arguments which, besides those of ref. [4], support the conjecture [ 1 ] of AP. (a) The first argument is a qualitative one; if we are interested at a particular resonating partial waves in the t-channel Pl(t) or Cl(t), at large t and positive, we have to look at the behavior of P and C for t -* +co and s(u) fixed and near zero. For Pt we get the usual behavior, Pl ~ (ct't)al(O)-l/l°g t, where otI is the leading trajectory in the s (or u) channel. For C the leading singularity in s and u is a reggeon-reggeon cut, giving Cl(t ) ~ (a't) ~ (0)÷~2(0)-2/log2t, where '~1 and a 2 are the two trajectories

G. Veneziano / Topological expansion

291

making the cut. One thus finds

Cl(t) + < (a,t)a2(O)- 1/log t , PI(t) t-*+-

(13)

I fixed where a 2 is the trajectory appearing in the t channel of P. Thus, for violation of the ZOI rule for ~, quarks (¢ ~ ~oTr),we find ClIP! <~(a't) ~ - 1 ~ l/a't and, for c quarks (~b -+ On), CI/P1 <~(ct t) ~° - " < < 1[a't. The main question to investigate here is the meaning of being asymptotic in t, when large masses appear in the t-channel. (b) Our second argument for AP uses a simple no-correlation model for computing ap(t) - a(t), which is a measure of non-planar corrections [ 1 ]. One finds (see e.g. ref. [ 11 ]), tip(t) - ct(t) = lim

f(d3ec/2Ec)DiSCM2 . . . .

aP~_/log

s

• .r.~. ., . , A aplanar bba

"

(14)

S---w.

An estimate of the right-hand side of eq. (14) using Born amplitudes gives ap(t) - or(t) ~ const. (a't)#. 4 -a't , t--,**

(15)

where ~ seems to depend on the particular model (fl = - 4 in the ordinary model) t 2 whereas the factor 4 -~'t, having an origin similar to that of the e - ,lap ± cutoff of DiSCM2Aacbbca, appears to be a rather model-independent feature. We see clearly here that AP does not work in the negative t region. Again, a more realistic model is needed, but the results obtained look quite encouraging for asymptotic planarity. I wish to thank J.D. Jackson, Y. Yamaguchi and Z. Maki for the kind hospitality I enjoyed at the Lawrence Berkeley laboratory, at Tokyo University and at the Research Institute for Fundamental Physics in Kyoto, respectively, while this work was done. Interesting exchanges with G.F. Chew, C. Rosenzweig, M. Bishari and K. lgi are also gratefully acknowledged.

References

[ 1 ] G.F. Chew and C. Rosenzweig, Phys. Letters 58B (1975) 93, Phys. Rev. DI2 (1975) 3907; C. Schmid and C. Sorensen, Nucl. Phys. B96 (1975) 209. [2] G. Veneziano, Nucl. Phys. B74 (1974) 365; Phys. Letters 52B (1974) 220. [3] G.Zweig, CERN preprint 8419/TH 412 (1964), unpublished; S. Okubo, Phys. Letters 5 (1963) 165; I. lizuka, K. Okado and O. Shito, Progr. Theor. Phys. 35 (1966) 1061. [4] G.F. Chew and C. Rosenzweig, Nucl. Phys. B104 (1976) 290; M. Bishari, Paris preprint, to be published.

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G. Veneziano / Topological expansion

[5] Chan Hong Me, J.E. Paten and S.T. Tsou, Nucl. Phys. B86 (1974) 479; Chan Hong Me, J.E. Paten, S.T. Tsou and S.W. Ng, Nucl. Phys. B92 (1975) 13. [6] M. Bishari and G. Veneziano, Phys. Letters 58B 0975) 445. [7] E.J. Squires, Complex angular momenta and particle physics (Benjamin, 1963). p. 62. [8] S.-H. H. Tye and G. Veneziano, Nuovo Cimento 14A 0973) 711. [9] J. Dash and J. Koplik, Columbia preprint CO 2271-45 (1975); P. Stevens, G.F. Chew and C. Rosenzweig, Calteeh preI~rint 68-541 (1976). [10] Dual theory, collection of Physics Reports, ed. M. Jacob (North-HoUand, Amsterdam, 1974). [11] M. Ciafaloni, G. Marchesini and G. Veneziano, Nucl. Phys. B98 (1975) 472.