v. REGGE CUTS AND REGGEON CALCULUS

Transcription

1 v. REGGE CUTS AND REGGEON CALCULUS Presented by J. B. Bronzan Rutgers University New Bnmswick, New Jersey A. Introduction (R. D. I. Abarbanel) Bronzan prepared a very thorough review of "'M) rk on Regge cuts since the seminal work of Gribov, Pomeranchuk, and Ter-Martirosyan in i965. His main point was that with recent developments in the study of multiparticle amplitudes we are now able to make quite a number of quantitative statements about the magnitude of cuts in the 1 -plane -- the location has been known for ten years. Furthermore via the field theoretic machinery of the Reggeon calculus one is able to assess the importance of the collision of poles and many cuts which occurs whenever one takes the t = 0 intercept of the vacuum trajectory to be exactly one. Because the hour was quite late, discussion after Bronzan's presentation was severly limited. This was Wlfortunate because it is my opinion that the Reggeon calculus is a most powerful tool for the study of diffraction scattering. An example of its use is given in the written version of Gribov's talk which will appear in these proceedings. Because of this opinion I am including finally a large part of Bronzan's review of the Reggeon diagram techniques. B. Text of Report (J. B. Bronzan) I begin with relatively conservative statements about the t-channel, by which I mean the channel of the Hegge singularities. The existence of Regge cuts and the behavior of their discontinuities can be seen from t-channel unitarity. The requisite rewriting of the unitarity integrals was done by Gribov, Pomeranchuk and Ter-Martirosyan in the middle sixties. Consider the t-channel process The particles in parentheses are organized into pairs of energy, angular momenta and helicities \, 1 i and m and I am dealing with the t-channel partial amplitude of angular momentum j. Now i consider the contribution of such four particle intermediate states to the imaginary part of the - amplitude. If you assume that appropriate analytic continuations can be made in 1 i and m i the helicity and angular momentum sums can be converted into contour integrals, and then opened up. Hegge cuts develop because of Hegge am helicity poles at 1 i = m = a(t ). Call the i i residue of these poles T. (t) (t). This amplitude in turn has a fixed singularity analogous to J,a i,a the first nonsense pole in spinless scattering. It appears at the spin-promoted point j." a(tt) + a(t ) -i, so with appropriate normalization, T can be written Z -498

2 with N nonzero at the fixed singularity. Then the. -. amplitude has a two - Reggeon cut with its j-plane discontinuity related to N: DisC f(t. j) 'ITft:~)f Zp(t:tt' t z ) + j z Zi dtt tt/z Nj,a(tt),aCtz) Nj,O'(t:t.),a(t z) tz,(+) Here the superscripts ::I: indicate that the amplitude is to be evaluated above or below the cut. t (::I:) satisfy the equation z It is convenient to associate a Reggeon unitarity diagram with this equation, thus: Note that, because of the delta function, the Hegge cut discontinuity is proportional to the fixed singularity residue of the production amplitude T, a(tt), a(t)" If you write the Froissart-Gribov i partial wave amplitude relating T to the absorptive part of t~e Reggeon ~particle scattering amplitude, the fixed singularity residue can be calculated from inclusive reaction data. This rule was first given by Gribov and Migdal, and I shall return to it later. Recently Abarbanel and White have rederived the Reggeon unitarity relations in different ways. Abarbanel uses basically a multiperipheral framework, and obtains a result similar to this, but with the stipulation that the discontinuity across two-reggeon cuts in positive in the sense that the two-pomeron cut contributes positively to total cross sections. White has considered the matter from the t-channel point of view, but with a more elaborate treatment of the required analytic continuations, and of the problem of signature for the production amplitudes. White finds that for Pomerons the two-reggeon cut is negative in my sense of the term. Here we have the sign controversy about how cross sections approach their asymptotes. I haven1t been assigned this subject for discussion, so I merely point its intrusion at this point. Gribov, Pome ranchuk -499

3 and Ter-Martirosyan can accommodate either cut sign in their work. Further two-reggeon unitarity relations can be derived from Reggeon production andscattering amplitudes so that with the previous results we have a closed set of unitarity equations. Schematically, the additional equations are [Xl Disc. J )r= = DisC j [ftj::] ~ These unitarity equations can be solved near the two-reggeon threshold by exploiting the known threshold behavior of amplitudes. Multi-Reggeon states can be treated in the same way_ leading to a kind of Reggeon threshold representation for the elastic amplitude. -1. AZ(t,j) ( ) ( f(t,j)} = AO(taJ) + a (t) In j - az(t) A 3 (t, j) '" \ ( ) + a;(t) ~-a3(t)j ln~-a3(t) + an(t) = na(tl n Z) - n + 1 Note that all the Regge cuts appear in the denominator of f. In this representation, the Ai(t, j) are rneromorphic in the j-plane, or at least any cuts they have are weaker near multi-reggeon thresholds than the exhibited cuts. The nature of the threshold behavior of n-reggeon cuts is such that it vanishes as ~-a (tf-z log (j-a (t~for n '" 3, so the leading cootribution of an isolated n-reggeon cut is normally ~ a n(t) I (In S)n.!it for n ~ 3. The contribution can be abnormal if An(t, j) is not a constant at threshold. You can see that Reggeon unitarity alone has serious weaknesses as a dynamical tool. One is that the unlmown functions An(t, j) represent parameters in the theory corresponding roughly to the coupling of the two-particle state to the n-reggeon state. Another is that Pomeron cuts are not isolated; they coalesce at t = O. Nevertheless, a little more can be learned about the conventional Pomeron from Reggeon unitarityalone. Suppose we ignore cuts other than two-pomeron cuts and ask how l(t, j) can have a pole passing through j = 1 and t = 0, leading to constant total cross-section. One has to solve the functional equation for a(t): o = Aot, Az(t, a(tl) (. ) ( aft)) + t a' (t/4) In act) - Za (t/4) + 1 You immediately see that if A and A are taken to be constants, there is no solution with O Z a(o) ;: 1 and Q '(0) finite because of the singularity of the logarithm. Multiperipheralists will -500

4 recognize this as the deflection of the Pomeron pole by the two-pomeron cut. To obtain a conventional Pomeron, one must assume that A has a linear zero passing through J = f and t = o~ O which can be regarded as the Pomeron in the absence of Pomeron cuts, if you like. In addition, A z zero in the triple-pomeron vertex. Under these assumptions, the trajectory has a mild has a quadratic zero passing through J ={ and t = 0 on a different ray; this is the celebrated singularity a(t) = t + Att + A Zt In t +... What about neglecting the higher cuts, as I have done here? It is true that in the scattering region they lie above the pole and two-pomeron cuts. On the other hand, Reggeon unitarity indicates that they are successively weaker, and a simple estimate shows that what we have done is adequate lor the interval of t o ~ t ~ - (Const)/lns. In other words, the region shrinks with the diffraction peak. The existence of such a region is one of the most important predictions of Reggeon unitarity. since before there was no reason to think any finite number of cuts would ~r be adequate. At the same time it should remind us that models like the multi-regge model, with only two-reggeon cuts, have no reason to be valid at high s and fixed t. Several loose ends in this picture are clarified by the Gribov Reggeon calculus. I group the Gribov calculus with Reggeon unitarity as a t-channel theory because both neglect the constraints of s-channel unitarity. For example, in both formalisms the Froissart bound is respected only because one puts a(o) "" 1 by hand. The Reggeon calculus graphs are consistent with Reggeon unitarity in the sense that sums over all Gribov diagrams satisfy the unitarity equations. The relation of the two formalisms is analogous to that between ordinary field theory and unitarity. In the Reggeon calculus, the behavior of the Pom"eron pole in the presence of cuts is studied by looking at the Dyson equation for the full Pomeron propagator. The bare propagator is D 'w k ) =_i a", w +k Z w =j - 1, k - Att, k is two dimensional. spacelike. This bare propagator corresponds to the linear trajectory act) = f + A 1.t. The full propagator, D(w. k ). satisfies the Dyson equation -1-1 [ D(w, k )] = [ Do(w, k)] -L(w, k ), -50t

5 where the self-energy Z (w, k ) satisfies the graphical equation G ribov and Migdal found that when the two-pomeron bubble is evaluated using bare propagators and a constant triple-pomeron couplings exactly the same inconsistency arises as when we took A and A to be constants in Reggeon unitarity, namely the renormalized trajectory, i. e., the o Z Pomeronin the presence of cuts, can no longer pass through a(o) = i with a' (0) finite. Gribov and Migdal then argued that one of two thinga must happen. One possibility is that D and Do are very different, or that the cuts change the Pomeron trajectory in an essential way. This they called "strong coupling". The other is that the triple-pomeron coupling vanishes when (,I), k and other appropriate variables are zero. This they called weak coupling or quasistability. Now the Reggeon calculus as a field theory has more predictive power than Reggeon unitarity alone, so one can hope to learn which possibility the theory chooses. The Soviet group was able to argue fairly convincingly that the triple-pomeron coupling vanishes, giving a Pomeron trajectory like that given by unitarity. Their arguments are threefold. 1) A vanishing triple-pomeron coupling is consistent with the Schwinger-Dyson equations of the Reggeon calculus; ) No satisfactory strong coupling solution could be constructed; 3) In solvable but unphysical models, no possibili ties are found for D(w, k ) beyond those surveyed in their general reasoning. As I mentioned, the quadratic zero in A in my unitarity argument corresponds to a linear zero in the triplez Pomeron vertex. Incidentally, it is curious that the vanishing of the triple-pomeron vertex can be discovered by either t-channel arguments alone involving pole-cut collisions, as in the Gribov calculus, or by s-channel considerations alone, as in the four-momentum sum rules for inclusive reactions. The real utility of the Reggeon calculus stems from the fact that it tells us how to reduce the arbitrary constants we found in the unitarity relations to a manageable few. Obviously, Hegge cuts will be a hopeless mess until we can do this, since the asymptotic behavior at fixed t is governed by a sum over all the cuts. At first glance it seems that the Gribov calculus contains as many parameters as did Reggeon unitarity since the bare coupling of two particles to n-reggeons is a parameter. The only way I know to reduce the number of such parameters is to use Lovelace's dual Reggeon calculus. However, Gribov and Migdal argue that these parameters are really not so important because of the phenomenon of enhancement. For example, consider the four Pomeron intermediate state. Itnas a finite coupling to particles that is of course a parameter. but there is also an enhanced graph, given by the first multi-pomeron coupling that does not vanish in the forward direction. -50

6 New Parameter '* Old Parameters >6{ Gribov and Migdal argue that the enhanced coupling dominates near w = k = 0 because of the Pomeron intermediate state. In the same way, the coupling of two particles to any number of Pomerons is dominated by a tree connection in which the only couplings are {3, the ordinary Hegge residue, and the lowest non-vanishing renormalized multi-pomeron vertex. Note that in this formalism cuts connect to the particles through poles. so they factorize near t ::;; o. An entirely different determination of the magnitude of Regge cuts is given by the Gribov Migdal rule. It connects the two Pomeron cut and the absorptive part of particle-pomeron scattering. It really is nothing more than the statement that N is a fixed singularity residue and the Froissart-Gribov partial wave projection N(t. t i t z } = f:s Discs Ta(tt)'.>(tz) (s, t) o ~~~--t is provided by the single particle interm:~* 1 1 Gribov and Migdal originally argued that since Discs TaCt ), act )(S" 0) is positive. a lower bound This result would of course say that the absorption model provides a lower bound to two-reggeon cut effects. However. Landshoff. Lovelace and the Brookhaven theory group have pointed out that conventional Hegge theory predicts that the absorptive part has the asymptotic behavior sa(t) - a(tt) - a (tz) so that the integral diverges when t =tt = t = O. The vanishing of the triple Pomeron vertex does not save the situation because Pomeron cuts also lead to a divergent integral. The Brookhaven group handle the Regge tail in a manner like that of finite energy sum rules and find that the Hegge term is negative. Therefore, absorption cuts need not be weaker than the full cuts. For proton-proton scattering. the Brookhaven group estimate that the Regge term reduces the proton pole term by 40 0/