The problem of the relation between double pomeron exchange in inclusive and exclusive experiments

Abstract The pionization region of the inclusive single-particle spectrum is accounted for by double pomeron exchange in the absorptive part of a six-point amplitude. In this paper a multiperipheral model for the six-point amplitude with double pomeron exchange is used for continuation by crossing and analyticity to the physical region of the exclusive two particle → four particle production process. The cross section for π − p → π − ( π + π − )p in the double-Regge region is then calculated and compared with the experimental analysis of Lipes, Zweig and Robertson which sets an upper bound to the strength of the double pomeron exchange coupling. This upper bound, coupled with the model for continuation to the inclusive cross section, is shown to give too small a magnitude for the double pomeron exchange in the pionization region. Further avenues for investigation are discussed.


Introduction
One of the fundamental questions about the nature of the pomeron singularity is the strength of the coupling of two pomerons to a particle or particles.It is our purpose to investigate how current models relate the strength of this coupling in two of its appearances.One occurence is in the analysis of the central plateau or pionization region [1][2][3] of the inclusive pion single-particle spectrum a + b ~ c + X where the observed pion has c.m. momentum Iq ii1< O(s~).The optical-theorem approach of Mueller [4] succintly describes the single-particle spectrum as an absorptive part in missing mass M 2 = (pa+ Pb -q)2 of the forward three-particle ~ threeparticle amplitudes in the a + b + c channel.Pionization occurs at high energy by the exchange of two pomerons to two pions, fig.I.This coupling and the existence of the pionization region was indicated by the n-ac Ins growth of the average multiplicity [5] and has been confirmed directly by the recent ISR experiments [6].In a sense to be detailed in this paper, the coupling of two pomerons here is "strong." Another occurence of double pomeron exchange is in the exclusive two-particle -+ four particle production process 7r-p ~ 7r-(Tr+Tr-)p, fig.2, where the (Tr+rr -) pair is at low invariant mass and the subenergies s 12 and s34 are large.This experiment was analyzed exclusively in the double-Regge region by Lipes, Zweig and Robertson [7] (LZR) and no evidence for double pomeron exchange was found.
The three-particle ~ three-particle amplitude and the two-particle ~ four-particle amplitude are related by crossing to be analytic continuations of the same six-point function into the different physical regions.The program is to take current models for the six-point amplitude and continue them between the two-particle ~ four-particle production region and the forward three-particle -+ three-particle region where we take the M 2 = (Pa + Pb -Pc )2 discontinuity to obtain the pionization spectrum.Since both regions have large subenergies in which double pomeron exchange occurs, we will be able to relate the strengths of the double pomeron coupling in the two regions.In this paper we use a multiperipheral model for the six-point amplitude which has a dynamical structure in terms of intermediate states in the M 2 channel.The model is an analytic continuation [8] of a model for the single-particle spectrum with Regge behavior and exponential damping in momentum transfer previously studied by Silverman and Tan [9] and by Caneschi and Pignotti [10].The model incorporates double Regge behavior for the exchanged pomerons and possesses an analytic behavior for double Regge coupling which is consistent with the Steinmann relations [11][12][13].The amplitude can be explicitly continued between the production and pionization regions.The strength of the coupling in the pionization region is found by taking the absorptive part in M 2 and fitting to the experimental results on pionization.
The continued amplitude is then used to calculate the cross section for the production experiment and the results are compared with the analysis of LZR.We find that the coupling strength from pionization is much too large for the production region.By adjusting the coupling to be consistent with the upper bound on double pomeron exchange set by the LZR analysis, we find that this coupling strength is only about ~ of that observed in pionization.
In a subsequent paper [14] we will again carry out this continuation and comparison with experiment using a dual resonance model for the six-point amplitude with the pomeron exchanges included in a phenomenologically consistent way.
For comparison with the calculations with the continued amplitudes, we also calculate the production cross section using a simple pion-pole exchange in the production amplitude with the strength determined by factorization.The two-particle four-particle production cross section from the pion pole is below that obtained by LZR.
In sect. 2 we formulate the general six-point amplitude with double pomeron exchange and discuss the method of analytic continuation between the production and pioniza~ion region.In sect. 3 we present the multiperipheral model with exponential damping in momentum transfer and use it for the analytic continuation.In sect. 4 the absorptive part of the amplitude in M 2 is related to the single-particle spectrum in the pionization region using the Mueller optical theorem.The strength of the absorptive part is found by comparison with the pionization data.In sect.5 we present the method of performing the phase space integration to compute the 7r-p -+ n-Or+Tr-)p cross section from the continued amplitude.In sect.6 we present the calculated results for the production cross section.We find that a much smaller value for double pomeron coupling than that found in pionization is demanded by the non-observation of double pomeron exchange in the production experiment.Possible sources for this inconsistency are discussed as well as its relation to the proofs of pomeron decouplings.

The six-point amplitude with double pomeron exchange
The production process, fig.2, is a function of eight independent invariants which we choose as The analytic continuation to the pionization region consists of taking where q is the momentum of the observed pion in the single-particle spectrum.The variables describing the subenergies in pionization are chosen to be Ul = (Pa q)2, u2 = (Pb -q)2, which are related to M 2 = (Pa + Pb q)2 and s in the pionization region by where m is the pion mass.For the continuation to pionization for large s and large subenergies u I ~ O(sLXV), u2 ~ O(s 1 -ay) with 0 < 2xy < 1, the invariants are, to leading order in powers of s, Sl2~U l , s34-~ u 2, In constructing our model for the six-point amplitude we are concerned in this study mainly with the part which contains the discontinuity of T in M 2. Additional parts which we have, for example, a pion pole in t 2 are much smaller than the part with the discontinuity in M 2 and their effects will be discussed in sect 6. q'\ 1% (el qt q2 (d) Fig. 3.The four diagrams which produce the signature structure of the six-point amplitude.The complete amplitude also includes the diagrams with q] and q2 interchanged.
The models that we will use are put in the form of a double pomeron exchange amplitude with a function that gives the M 2 discontinuity observed in the pionization region.The double Regge form for T from the diagram, fig.3a, is   r(--~  The explicit occurence of GTr is for convenience in comparing the coupling strengths and does not mean to imply that the pomerons (P) in PP ~ lrTr couple to on-shell pions, gp is the intrinsic strength of the double pomeron coupling and will be determined in sect.4. The PP -+ lrTr coupling function F(r/) may also depend on s23, t 2, t 1 and t 3 but we will suppress this in the notation.
In an amplitude with double Regge exchange, the M 2 discontinuity occurs through the discontinuity in the variable r/ [8,12,[15][16][17] which involvesM 2 (2.6).In the models we will study the function F(r/) ~vill have a cut for r/~> O which occurs in the pionization region where r/---q2 + rn .r/is negative in the production region, eq.(2.6).
The signature factors for the Regge exchanges occur by adding the three diagrams with interchanges of(p a ~ --Pa), (Pb *-Pb) and (Pa '~-Pa, Pb ~' p~).These are each computed from the same function Tab, eq.(2.5), but with the appropriate subenergy variables resulting from interchanging the momenta in the definitions of slZ, S34andMZ, eq.( 2 The interchanges in M 2, however, produce invariants for different channels : Adding these four diagrams gives the signature factor in the production region, since all of the four r/s are approximately numerically equal.Since our choice of r/is not symmetric in ql and q2' we must add the same terms with ql and q2 interchanged.
The complete six-point amplitude in the production region is then Since r/is negative in the production region, it is not necessary to distinguish if it is above or below the r/cut in the various terms.
In taking the discontinuity in the channel M 2 = M2 b = (Pa + Pb -q2 )2 for pionization, only the first of the four terms, Tab, contributes since the other r/'s do not contain M 2, but variables which refer to other channels.In the four additional terms obtained from ql ~ q,, however, the term arising from T~'h', call it T,h,, contains M2'b' = (--P'a -Pb -ql )2 from exchangingq2 ¢~ ql in M~b" Now from four-momen.turn conservation M2b' = (Pa +Pb q2 )2 =M2 ' (2.]o) so this term gives an M 2 discontinuity identical to the first term Tab in the pionization region.

The multiperipheral model with exponential damping in momentum transfer
The model we take for the functional form ofF(??) arises from an analytic continuation [8] of the Caneschi-Pignotti [10] model for the single-particle spectrum which was studied analytically by Silverman and Tan [9].It views production of a particle in the central region by the peripheral form, fig.4, where any number of particles may be emitted to the left or right of the observed particle, it is assumed that the sums of these inclusive particles behave like total cross sections which are Regge behaved and pomeron dominated at large s t, s r (fig.4).In order to obtain exponential damping in q2, exponential damping is assumed in the momentum transfers t 1 and t r.Using the described form for the amplitude in fig. 4 where J is the Jacobian of the phase space integrals and c is a constant.This has been computed for o~ 1 = o~ 3 = 1 in ref. [9], and for arbitrary Ul' c~3 in the three-particle -+ three-particle scattering region in ref. [8].In the pionization region the result is a function only ofK = kr/where k = 2~1~2r/(~21 + f2r).The result is also the absorptive part in K of the six-point amplitude in the pionization region.It may be extended to an anlytic function in K in a way consistent with the Steinmann relations (discussed below).We use the result of the continuation in K (ref.[8] ): where U is the confluent hypergeometric function [18] (also called qz [ 19] ), as the basic model for the K continuation in this paper.It should be noted that the entire amplitude is not analytically continued to the production region, but only the K dependence is abstracted from the forward three-particle --> three-particle amplitude.
If the entire amplitude were continued to the production region some of the internal momentum transfers can become positive, and the increasing exponential dependence would be a severe overestimate.
It is seen that F(K) has a cut for K ~> 0 which gives the absorptive part in the pionization region.The same function arises in the six-point function in the dual resonance model [14,15] where K is replaced by x = e/q/z(1 -z) in the pionization region and an integration over z is performed.This significantly alters the continuation and will be discussed in the paper on the dual resonance model [14].The analytic structure is directly expressed [18] in terms of the entire function M (also called ¢, [19]): The Steinmann relations state that the double discontinuity of two overlapping variables must vanish in the physical region.Considering the three-particle --> threeparticle scattering in the physical region of forward a + b + ~---> a + b + g we have u 1 > 0, u 2 > 0 and the double Regge form (--Ul)% (-u2)~F(K) (3.4) from one of the terms in (2.9).Since u 1 and u 2 are overlapping variables, the Steinmann relations state that we cannot have simultaneous singularities in these.So the form ofF(K) contains (-K)-% or (-K) % which gives respectively (--Ul)% (--u2)°% (-K)-% oc(-u2)%-% (-M2) °q , (-Ul)% (--u2)% (-g)-&3 ac(-Ul)%-c% (-M2)% .(3.5)The l/F(-a3) in the coefficient of(-K) -% prevents simultaneous poles in the amplitude in ot 3 and u 2 since they are overlapping variables, and similarly for the 1/F(-a 1) term in the coefficient of (-K)-%.The function F(K) does not in fact contain poles at o~ 1 = ot 3.
We now perform the continuation to the pionization region eqs.(2.2) and (2.3).We recall that the M 2 or r/= r/ab discontinuity is only present in the term Tab in (2.8) and the term Ta, b, with ql oq2' Then in the pionization region we have In taking the discontinuity of Tab it is important to keep c~ 1 and o~ 3 not equal to 1.
Tab has a pole at these points and since M 2 is an overlapping variable, T b could not have an M 2 discontinuity in a residue of these poles.It is perhaps simplest for analytic purposes to consider the pomeron intercept as slightly less than one, and in the later numerical evaluations take a 1 = 1 and a 3 = 1 as a good numerical approximation.
Evaluating lm F(K) is simplest from the form (3.3) where the discontinuity r/ arises from the terms (-K)-% and (-K)-%.After using the Kummer transformation [18] M(a, b, z) = e z M(b -a, b, -z) , ( and rearranging gamma functions we can recombine terms to obtain for K ) 0

lmKF(-o~I)F(-o~3)F(K)= 7re-KK-% U(ol3+l,-al +OC3+l,K).
(3.8) The continuation can also be performed and the absorptive part taken when a 1 = a 3 as occurs at t 1 = t 3 = 0, by using the appropriate form of the hypergeometric function for this case [18,19].The results are the same as obtained from (
To complete the specification of our amplitude in the production region, we recall that the general pomeron-pomeron to 7rTr amplitude F in (2.5) can also have additional dependence on s23, t 2, t 1 and t 3. Lacking further dynamical knowledge, we assume the smoothest behavior possible.First, we assume that the t 1 and t 3 dependence has been largely removed in the elastic scattering residue factors /3a~r(tl),/3b~r(t3) which will be taken to be /3a~(t ) = ~3bTr(t ) = e~qt , (3.14) with g2 = 5 GeV 2 as given by elastic scattering data.This magnitude of damping is consistent with that observed in the production experiment for a single pomeron exchange [7].Second, we assume no explicit dependence in s23 in continuing from pionization, s23 = 0, to production, s23 ) 4m 2. Resonance structure would only enhance the production amplitude, which is not needed.In the dual resonance model [14] we include the effects of resonances in s23.Finally, since all hadron ampli-Fig.5.The exponentially damped multiperipheral model as a six-point amplitude.
tudes appear to be damped in momentum transfer, we will estimate the effects of possible damping in the variable t 2 by including in the amplitude the form e a(t2 -m)2 (3.15)This does not affect pionization where t 2 = rn 2.
In this paper we have abstracted only the K dependence in the pionization region from the multiperipheral model with exponential damping.We have also considered the subenergy dependence of the model in the three-particle ~ three-particle nonforward region as indicated in fig. 5 ).We also computed the cross section in the production region with these variables.The results did not differ significantly from those obtained from (2.9) with K = k~.

Comparison of the absorptive part with pionization
We now relate ImM2 T to the single particle inclusive cross section and evaluate gp and k by comparison with the pionization experiment.We use covariant normalization in which and K = k(q 2 + m2).
The single particle spectrum for pp ~ n -+ + X has been obtained in the pionization region (x = 0) at values of s up to 2820 GeV 2 at ISR by the Saclay/Strasbourg collaboration for the range 0.2 GeV ~< q± ~< 0.8 GeV [6].The form (4.7) gives an excellent fit to the data with the parameters [8] do mb -50 To evaluate gp we now need only Gp and G r, the pomeron residues at zero momentum transfer.With the pomeron exchange contributing Considering the definition ofgp as in (2.9), we see that it is the enhancement of the pomeron-pomeron to 7rn interaction over the on-shell coupling of pomerons to pions G2.In this quantitative sense the double pomeron exchange in pionization is very strong.

Cross section for two-pion production via double pomeron exchange
In this section we will evaluate the cross section for two pion production via double pomeron exchange using the model for T in sect.3 with the values of the coupling obtained in sect.4. In order to carry out the phase space integration for the two-particle -+ four-particle cross section and to compare with the experimental analysis of LZR [7], we convert to the subenergies t St =(P'a +ql +q2 )2' s2=(Pb+ql +q2 )2" ( Then by considering the two produced pions as a body of mass-squared s23 = (ql + q2 )2 we can consider the final state as a pseudo-three-body final state with subenergies Sl, s 2 and momentum transfer tl, t 3.In the limit t 1 ~_ 0, t 3 _~ 0 enforced by damping of pomeron exchange the amplitude and phase space will factorize into a function of(s 1, s2, tl, t3) times a function of (s23, t2).This approximation will be used to simplify the calculation.Because of the sharp damping in t I and t 3 (3.14),we calculate the pomeron trajectories effectively at cq = 1, c~ 3 = 1.We then obtain for F(K) in (3.2) the simple form [18] 1 !
~ (pa+Pb+ql +q2-Pa-Pb) 2s i=1 (s.9) Converting to the d4Q integration and using T24 in (4.7)  (pa+Pb+Q-pa-Pb) 12 is the two-body phase-space integral for pomeron + pomeron ~ 7rTr : (5.12) In order to complete the factorization of phase space we have deleted the dependence on t I and t 3 in the limits of integration ofI 2 by making the effective approximation oft 1 = 0, t 3 = 0.12 is then a function only ofs23 : (5.18) The boundary of integration is given by the Kibble function (5.20) The boundary is also subject to the additional constraints imposed on the data by LZR [7] to isolate the multi-Regge region Sl ~> 2GeV2 ' s2 >/4GeV2 ' s3 = (P'a + p;)2 >/4GeV 2 " (5.21) LZR also impose cutoffs on the momentum transfers t 1 and t 3 which we have neglected since they are effectively included by the damping in t 1 and t 3.

Results and conclusions
Our most striking result is the enormous value of the cross section o2~ 4 in the LZR region (5.21) computed using the continuation function (3.2) with the large double pomeron exchange strength gp = 70.Computing 12 and 13 numerically and using (5.10) we find for the cross section in the LZR region of phase space This is even larger than the total o2__.4 _~ 5mb and very much larger than the cross section in the LZR region of phase space _LZR 02__.4 ~ 90 ktb.This is a reflection of the occurence of the large double pomeron exchange strength gp linearly in the pionization region but quadratically in the two-particle -~ two-particle production cross a 2 (t 2 m ) section.Even exponential damping in t 2 like e -in the continuation only reduces o2~ 4 by order 1/a.
An alternative and perhaps more reasonable way of presenting this result is to find what is the largest value ofgp consistent with the LZR analysis and comparing this with the coupling strength found in pionization.The LZR analysis find for 02 4 in the double Regge region (5.21) integrating over s23 from 4m 2 to 2.25 GeV 2 -~ the value 02__+ 4 = 90~tb.From studying the distributions in s 1 and s 2 in LZR, which show that the cross section comes mainly from (P, P') and (P,o) exchange, we can estimate that the double-pomeron contribution is at most a tenth of this or 10/ab.To reduce (5.1)The result for o~a~lc 4 are about 50% higher for this form than for (5.8), which is not a significant difference.
To check that the problem of reconciling the pionization and production experiments lies in the continuation model and not in the double Regge structure, we have calculated the production cross section in the LZR region from pion pole exchange in the coupling pomeron + pomeron -+ 7r + + 7r-.Since the on-shell pomeron -rr -rr coupling has already been exhibited in (2.9), we simply replace gp by 1 and replace 1/SoFby the pion propagator l/(t 2 -m 2) to obtain the pion-pole dominated two-particle -+ two-particle amplitude.With the term ql ~ q2 added, the calculation is carried out as before but with qs in (5.8) replaced by --1 'I, po ~ , (6.4) The resulting cross section for the LZR region (5.21) is o~°~ = 50/~b, which is below the 90/lb observed by LZR.
In addition to the model for F(K) studied in this paper, the formalism presented above for calculating the magnitude of the six-point amplitude from pionization and for computing the two-particle ~ four-particle cross section can be applied to other models of the continuation from pionization to production.In another paper we carry out the continuation with the dual resonance model using the formalism of this paper.The results of the dual resonance model also give too large a value for o2~ 4 using the magnitude ofgp from pionization.
The importance of the inconsistency presented here is that it uses current dynamic al models of inclusive hadron production which successfully predict the qualitative behavior of those experiments, but it demonstrates that they are still quantitatively unsuccessful.Some calculations on the multiperipheral model used in this paper by inserting physical cross sections are quantitatively below the observed magnitude of pionization by about an order of magnitude.This is reflective of our result of accounting for only 1 of the observed pionization using gp ~< 1 consistent with LZR.A related problem is that using only the physical coupling strength in the multiperipheral model will not give an output trajectory near 1.
Another source of the inconsistency may be the analyticity structure in K used to satisfy the Steinmann relations.Dynamically it was the (-K) -c~ -+ -1/K term in F(K) that led to the numerically large result for o2~ 4.This analyticity structure has been verified for double Regge coupling only in planar graphs and in dual resonance models, which are just the type that lead to the inconsistency observed above.Recently, Brower and Weis [21 ] have shown that this type of analytic form for double Regge coupling would cause a pomeron of intercept 1 to decouple from total cross sections.Further investigations of the analyticity structure in K are suggested both by the Brower and Weis result and by the numerical inconsistency demonstrated in this paper.

Fig. 4 .
Fig. 4. Diagram for peripheral production of a particle in the central region.
3.8) by setting 0¢ 1 = o¢ 3. Finally, after taking the discontinuity we set o¢ 1 = 1, e¢ 3 = 1 at t I = O, t 3 = O.From (3.8) |m K I'(Otl) F(-a3)F(K ) 77 lm K P(-al) P(-a3)F(K) -+ ~ [(1 +K)El(K) e -K] .(3.12) Finally, combining with (3.6) and usingK = ku 1 u2/M2 we find for the M 2 absorptive part in the pionization region . The diagram is explicitly symmetric under (a ~" g), (b ~ b) and (c "~ ~-), and the result of the calculation is the same function F(K) used in (3.2) but with K replaced by the symmetric combination K' = k(Sa~ + s~c ) (sb~-+ sffc ) forK' and the Regge powers we need to keep only the four signature generating interchanges of a ~' a and b "~ b since the sum is then already symmetric under c ¢~ c.The amplitude in the pionization region is identical to (3.13
inner integrals for three-body phase space with exponential momentum transfer damping have been performed exactly by Chan, Kajantie and Ranft[20]