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Submitted on 1 Jan 1973
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CUTS IN THE COMPLEX ANGULAR MOMENTUM PLANE
I. Halliday
To cite this version:
I. Halliday. CUTS IN THE COMPLEX ANGULAR MOMENTUM PLANE. Journal de Physique
Colloques, 1973, 34 (C1), pp.C1-141-C1-144. �10.1051/jphyscol:1973113�. �jpa-00215193�
CUTS IN THE CCMPLEX ANGULAR MOMENTUM PLANE
CUTS IN THE COMPLEX ANGULAR MOMENTUM PLANE I. G
.
HALLIDAYImperial College
I have been asked to review the recent advances in the theory of cuts in the angular mamentum plane.
Only elastic scattering will be considered. In other words, we are considering the cuts associated with
the Pomeron. The topics I shall cover are:- (1) Hard cuts
(2) The quasi-stable Pomeron (3) Cut sign controversy
1.
-
HARD CUTS.-
In the literature there are several models which give rise to rising total cross- sections. I may mention the eikonal models of Cheng and Wu, or the s-channel unitary models of Auerbach, Aviv, Suger and Blankenbecler (1). These obtain their rising cross-sections by having cuts in the j-plane whose discontinuities do not vanish at their end points. These models are clearly of interest in view of the behaviour of a at the I.S.R. However,tot
these models do not manifestly satisfy the constraints imposed by t-channel unitarity, i.e. unitarity in the channel containing the Regge singularities. In par- ticular, we have the Bronzan and Jones (21 condition that the cut discontinuity should vanish at its end point. This was derived from t-channel elastic uni- rarity under plausible assumptions. In an interesting paper Creutz, Paige and Wang [ 3 : have given a "counter- example" which allows "hard" cuts and satisfies t-channel elastic unitarity. Unfortunately, it also has j-plane singularities which lie arbitrarily far to the right. Thus the consistency of these hard cut models with t-channel unitarity is still a not quite closed problem.
One should perhaps also mention at this point that Fried and Blankenbecler, and Fried alone (41 obtain large cancellations when graphs omitted in the eikonal method are included.
2.
-
THE QUASI-STABLE POMERON.-
Let me now turn to the model which to many has the greatest appeal. Thisis a factorizable pole plus the cut corrections which Mandelstam showed to be necessary 10 years ago.
The appeal of this model is that one can study the t-channel properties in a systematic manner. From my remarks on hard cut models it is clear that the choice of this model is a matter of taste rather than necessity at the present moment.
Thus given the existence of a Pomeron pole, one can study the properties of the cuts using t-channel unitarity. One considers the 4-particle discontinuity in the t-channel
This equation is projected into angular momentum states where the left and right pairs have angular momentum and helicities El, ml, L2, m2 respectively.
The equations are now continued to complex values of these variables and poles are inserted in L1, %2 etc. aorresponding to the Pomeron. The questions of signature and definitions of continuations
involved in this program, originally set out by Gribov, Pomeranchuk and Ter-Martirosyan [5], are formidable. White [6] has made a major contribution to unravelling this problem.
The major results of this technique are that the 2-Reggeon cut discontinuity is negative and the cut discontinuity may be expressed as an integral over fixed pole residue functions of the 2 particle- 2 Reggeon amplitude. The negative sign means that the I.S.R. result of a rising cross-section is very natural as the destructive cuts die away logarithmical-1y.- The fixed pole residues N are also
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1973113
C1-142 I.G. HALLIDAY
accessible from inclusive cross-sections using
Because of the possibility of Pomeron exchange in this amplitude, one has to be slightly more careful than this. The details have been set out by Muzinich.
Paige, Trueman and L.L. Wang : 7 ) , who estimate cuts of around 10% at s = 1000 G ~ v ~ . This is probably too small to give the rise in a at I.S.R.
tot
In several interesting papers, Bronzan f81 has studied the constraints imposed by studying the con- sistency of t-~hannel~unitarity with a(o) = 1. As Gribov et al. !5: showed, the unitarity equations for
particle
= Reggeon imply
where ac(t) = 2act/4)
-
1. The functions A,B only contain less singular terms (j-
a (t))" Ln(j-
ac(t)), n = 1, 2,....,
for the singularities at the end of the cut. B may also contain terms like2 n - 2 t 2
(j
-
na(t/n ) + n-
1) log (j - na( I n ) + n - 1) from higher order cuts. These are ignored. Bronzan finds that to obtain a constant total cross-section we need a double pole at t = 0 , j = 1 in A,B. If A,B are analytic he obtains a logarithmically dec- reasing total cross-section.In this notation h(j,t) = l/D(j,t) and clearly has a double zero. This gives gppp(0) = 0. The consistent trajectory is
where x,y depend only on A,B and z ie zero if the cut is omitted from D. This is quasi-stability, i.e. the trajectory is only slightly renormalized by the cuts
near t = 0.
The double pole in B, however, reduces the cut term in D effectively by t2, which means that the higher cut terms in B are no longer clearly negligible.
Collins and Williams (91 have exploited this to construct a self-consistent model including higher order cuts. If the 2 and 4 Reggeon cuts are allowed to have opposite signs they even manage to obtain a solution with g (0) # 0.
PPP
In the above, quasi-stability is fed in by brute force. In his second paper Bronzan (81 has shown an explicit mechanism leading to quasi-stability.
Using the technique developed by GriSov (The Reggeon Calculus) ';lo] one can calculate the diagrarcs shown below, where the l r v ~ denote some primitive model of the Pomeron and the higher diagrams denote corrections due to multi-Pomeron exchange.
t
+
(neglect higherterrns)Then the full propagator is of the form (- +
* ) . The zero of this object gives the pomeron.
The integral equation for
r
has a highly singular kernel and it turns out that quasi-stability and gppp(0) = 0 are built in even if r # 0.Thus for the first time one can see a dynamical mechanism for quasi-stability.
In one extremely interesting paper submitted to this Conference, Cardy and White (11) have extended this idea to show that this method of obtaining quasi-stability also allows one to avoid all the decoupling "theorems" (except gppp(0) = 0) which were so prevalent at the N.A.L. Conference. The sum-rule proofs (12: of the decoupling theorems are avoided by the presence of cuts which are just as large as the usual simple pole terms. This possibility has also been pointed out by Halliday and Sachrajda (131 for the proof of g
PPP= O.
3.
-
CUT SIGN.-
Abarbanel, supported by Chew, has argued that the cut sign should in fact be positive(14;. This argument may be sketched as follows.
CUTS IN THE C W L E X ANGULAR MOMENTUM PLANE C1-143
Any scattering event can be ordered
so that there are no large rapidity gaps between par- ticles in clumps but there are large gaps between pairs from different clumps. Then it is assumed that
large gaps imply Regge pole exchange. Thus we find
In the weak coupling approximation (only) the last term may be neglected. The first three terms contribute A, -4A, +2A respectively to the cut. The last three terms are clearly absent fromAbarbanelTs analysis. The second diagram is essentially an absorptive correction but with a proper fixed pole to couple the cut. The next term is interesting as it leads to large fluctuations (with higher order diagrams) in the multiplicity distributions. These have been investigated by Ter-Martirosyan ;171, who has estimated their effect in a model.
Unfortunately, this way of looking at the cut problem is somewhat hampered by the lack of a good model for the Pomeron.
and this leads to a linear integral equation. After some analysis one reaches the almost obviously posi- tive cut term in a
tot'
Many people (151 have suggested absorptive effects as being responsible forthe requisite change of sign to
obtain consistency with t-channel unitarity. Recently In conclusion, the quasi-stable pomeron now seems the Mandelstam diagram, which serves as a paradigm of to be in much better shape than it was one year ago the t-channel calculation, has been analysed in great at N.A.L. The technical problems of continuations detail, in the s-channel, in the weak coupling appro- in angular momenta are gradually being overcome and ximation (13,161. The s-channel unitarity equation is hence the continued t-channel unitarity equations
are being placed on a more solid footing. The decoupling "theorems" can now be avoided in the
As
(
=+
manner of Cardy and White. I should also mentionat this point Gribov's highly intuitive picture
n = k n-
< n,>
presented at the N.A.L. Conference, which alsoleads to the Pomeron coupling to total cross-sections and indeed total cross-sections being equal asymptotically.
n- 2<nL>
<
n,>=
ladder multiplicity References:1) CHENG (H.) and WU (T.T.), Phys. Rev. Lett. 24 (5) GRIBOV (V.N.)
,
POMERANCHUK (I .Ya) and(1970) 1456. TER-MARTIROSYAN (K.A.), Phys. Rev. 139B
AUERBACH (S.). AVIV (R.), SUGAR (R.L.) and (1965) 184.
BLANKENBECLER (R.). Phys. Rev. D6.(1972) 2216. ( 6 ) WHITE (A.), Nucl. Phys. B50 (1972) 93.
CHANG (S.J.) and YAN (T.M.), Phys. Rev. D4 (1971) WHITE (A.), Nucl. Phys. B50 (1972) 130.
537. :7) MUZINICH (I.J.), PAIGE (F.E.), TRUEMAN (T.L.) and
FINKELSTEIN ( J . ) and ZACHARIASEN (F.), Phys. Lett. WANG (L.L.), Phys. Rev. D6 (1972) 1048.
34B (1971) 631. (8) BRONZAN (J.B.), Phys. Rev. D4 (1971) 1097.
[21 BRONZAN (J.B.) and JONES (C.E.), Phys. Rev. 160, BRONZAN (J.B.), Phys. Rev. D7 (1973) 480.
(1967) 1494. (91 COLLINS (P.A.) and WILLIAMS (R.)
,
ICTP/72/28.[3] CREUTZ (M.), PAIGE (F.E.) and WANG (L.L.), Phys. [lo] GRIBOV (V.N.), JETP 26 (1968) 414.
Rev. Lett. 30, (1973) 343. [ll) CARDY (J.) and WHITE (A.R.), contributed paper (41 BLANKENBECLER (R.) and FRIED (H.M.), SLAC report no. 279.
1125 (1972). I121 JONES (C.E.), LOW (F.E.), TYE (S.H.), VENEZIANO FRIED (H.M.), Phys. Rev. 0 6 , (1972) 3562. (G.) and YOUNG (J.E.) Phys.Rev. D6 (1972) 1033.
C1-144 I . G
.
HALL IDAY( 1 3 ) HALLIDAY (I.G.) a n d SACHRAJDA (C.T.), ICTP/72/18 :16) GRIBOV (V.N.), i n P r o c . o f t h e XVl I n t e r n a t i o n a l :Phys. Rev. t o b e p u b l i s h e d ) . C o n f e r e n c e on High Energy P h y s i c s , B a t a v i a ( 1 4 ) ABARBANEL (H.D.I.), Phys. Rev. D6 (1972) 2788. 1972, Vol. 3, p. 491.
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