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Can perturbative QCD predict a substantial part of

arXiv:hep-ph/9209230v1 13 Sep 1992McGill/92–37September 1992Can perturbative QCD predict a substantial part ofdiffractive LHC/SSC physics?J.R. Cudell∗and B. MargolisPhysics Department, McGill UniversityMontr´eal, Qu´ebec H3A 2T8, CanadaAbstractWe examine a model of hadronic diffractive scattering which interpolates betweenperturbative QCD and non-perturbative fits.

We restrict the perturbative QCD re-summation to the large transverse momentum region, and use a simple Regge-poleparametrization in the infrared region. This picture allows us to account for existingdata, and to estimate the size of the perturbative contribution to future diffractivemeasurements.

At LHC and SSC energies, we find that a cut-offBFKL equation canlead to a measurable perturbative component in traditionally soft processes. In par-ticular, we show that the total pp cross section could become as large as 228 mb (160mb) and the ρ parameter as large as 0.23 (0.24) at the SSC (LHC).∗cudell@hep.physics.mcgill.ca

1IntroductionAs the energy of hadronic colliders increases, diffractive scattering will play an increas-ingly important role. On the discovery side, it will produce the highest mass statesaccessible at future colliders[1], and the physics of rapidity gaps might make their de-tection feasible[2].

On the background side, most interesting events will emerge fromthe small-x region, and will contain an appreciable “minijet” structure, so that softparton scattering and evolution has to be modelled to optimize the detection of newphysics.As lattice calculations can deal only with static problems, the only fundamentaltool that we have so far to deal with QCD scattering is perturbation theory, and re-summation techniques[3,4] have pushed the perturbative limit to the small-x region.However, it seems that, at present energies, these efforts have failed to reproduce softdata[5]. This failure has been ascribed to the intrinsically non-perturbative nature ofthe problem, and simple models have been proposed to extrapolate present measure-ments to higher energies[6,7,8].

However, the details of the process cannot be predictedthrough this approach.We thus want to address the following question: what fraction of events at futurecolliders can be understood by present perturbative techniques? The first step is to finda simple parametrization of the data, for which we use one of the existing models.

Weassume that this describes the infrared part of the QCD ladders, for which the gluontransverse momentum kT is smaller than some cutoffQ0. We then evolve this term viaperturbative resummation techniques[3], using gluons with kT > Q0, so that we aresure that perturbative QCD is valid.

This approach interpolates between the purelyperturbative ladders (Q0 = 0) and the purely non-perturbative models (Q0 = ∞).We limit ourselves to the most general features that one can expect from such anevolution, and do not attempt to make an explicit model. We simply assume that theinfrared region couples to the perturbative one through an unknown vertex.

For a givenQ0, present data constrain the size of this vertex and one can predict an upper boundon the perturbative contribution to the hadronic amplitude at higher energies. As theQCD equations are simpler at zero momentum transfer, we consider only the total crosssection and the ratio of real to imaginary part of the forward scattering amplitude, theρ parameter.

Even then, as the exchange will involve at least two gluons, it is possibleto demand that both have large transverse momenta, which add up to zero.Theperturbative evolution then can lead to a “gluon bomb” which remains dormant in thedata up to present energies, but which can bring large observable corrections at futurecolliders.In the next section, we give a simple model for soft physics at t = 0, which wecall the soft pomeron. We then briefly outline the BFKL equation [3] and mention itssolutions, which are very far from reproducing the data.

We then show how one canmake a very general model evolving soft physics to higher values of log s and constrainit using existing data for σtot and ρ. We then show that soft physics at the SSC and1

the LHC could have a substantial perturbative component.2Data: the soft pomeronAs explained above, we shall concentrate on the hadronic amplitude A(s, t = 0) de-scribing the elastic scattering of pp and p¯p with center-of-mass energy √s and squaredmomentum transfer t = 0. This amplitude is known experimentally: we normalize itso that its imaginary part is s times the total cross section; the ratio of its real andimaginary parts is by definition ρ.The most economical fit, inspired by Regge theory, is a sum of two simple Reggepoles:A(s, t)s= (a ± ib)sǫm+α′mt + C0sǫ0+α′t(1)with a, b, C0 constants independent of s.The phase of the amplitude is obtainedby the imposition of s to u crossing symmetry.

The first term has a universal part(a) representing f and a2 exchange, and a part (b) changing sign between p and ¯pscattering, which comes from ρ and ω exchange. The second term (C0) is responsiblefor the rise in σtot and is referred to as the “soft pomeron”.

Its only obvious problem isthe eventual violation of the Froissart bound. Therefore, we also consider a unitarizedversion, for which we eikonalize the second term of Equation (1).We give the best fit values of the parameters and the χ2/d.o.f.

in Table 1.parameterpole fiteikonal fita124 mb141 mbb32.5 mb35.1 mbǫm-0.474-0.469C021.6 mb24.0 mbǫ00.0850.093α′-0.251χ2/d.o.f.1.031.08σtot at the SSC (LHC)125 (107) mb117 (102) mbρ at the SSC (LHC)0.131 (0.131)0.116 (0.113)Table 1: Values of the parameters of Equation (1) that result from a least-χ2 fit todata at t = 0.The pole fit is shown by the lower curve of Figure 1 and the eikonalized one by thelowest curve of Figure 2. Both fits reproduce the data [9], from √s = 10 Gev to 1800GeV with χ2/d.o.f.

very close to 1. The only failure is the UA4 value for ρ, which is notreproduced by most models, and for which further experimental confirmation seems tobe needed.

It is a curious fact that the eikonalized fit chooses the conventional valueof the pomeron slope α′ which is normally derived from other constraints[6]. Also,notice that unitarization does not make a big difference, and that even at the SSC, thedifference between the two fits is only 8 mb.2

Other parametrizations are possible, e.g. [7,8], and as shown by the proponentsof this one [6], multiple Regge exchanges are essential to describe the data at nonzerot.

However, as we limit ourselves here to the zero momentum transfer case for whichthe corrections are small, and as this simple form is particularly well suited for ourpurpose, we shall adopt it in the following as a starting point for the QCD evolution.3Theory: the hard pomeronIn order to describe total cross sections within the context of perturbative QCD, onecan try, for s →∞, to isolate the leading contributions and to resum them. Thisis made possible by the fact that perturbative QCD is infrared finite in the leadinglog s approximation and in the colour-singlet channel.

This suggests that very smallmomenta might not matter, and that one could use perturbation theory.Such a program has been developed by BFKL [3]. In a nutshell, one can show that,when considering gluon diagrams only, the amplitude is a sum of terms Tn of order(log s)n and that terms of order (log s)n are related to terms of order (log s)n−1 by anintegral operator that does not depend on n, and that we shall write ˆK:Tn+1(s, k2T) = ˆKTn(s′, k′2T )=3αSπ k2TZ ss0ds′s′Z dk′2Tk′2T[Tn(s′, k′2T ) −Tn(s′, k2T)|k2T −k′2T |+ Tn(s′, k2T)qk2T + 4k′2T].

(2)This leads to:T∞=XnTn = T0 + ˆKT∞(3)This is the BFKL equation at t = 0. Its extension to nonzero t is known, but toocomplicated to handle analytically.

We limit ourselves here to the zero momentumtransfer case.In this regime, the BFKL equation (3) possesses two classes of solutions. Firstof all, at fixed αS, the resummed amplitude is a Regge cut instead of a simple pole:T∞≈R dνsN(ν), with a leading behaviour given byNmax = 1 + 12 log 2παS(4)Even for a small αS, say of order of 0.2, this leads to a big intercept Nmax ≈1.5.

Asthis is much too big to accomodate the data, and as a cut rather than a pole leadsto problems with quark counting, subleading terms were added via the running of thecoupling constant. It was first claimed that such terms would discretize the cut andturn it into a series of poles [10], but further work has shown that the cut structureremains [5,11].

However, the leading singularity is slightly reduced, and one can derivethe bound [12]Nmax > 1 + 3.6π αS(5)3

Again, for values of αS of the order of 0.2, this leads to an intercept of the order of1.23.So, we reach a contradiction: on the one hand, the data demands that the am-plitude rises more slowly than s1+ǫ0, with ǫ0 < 0.1; on the other hand, perturbativeresummation leads to a power s1+ǫp, with ǫp > 0.23. The difference between the twois a factor 3 in the total cross section at the Tevatron.

The resolution of this problemis far from clear, and one can envisage the implementation of some non-perturbativeeffects within the BFKL equation [5]. Rather than trying to understand ǫ0, we shallhere take a much simpler approach, i.e.

assume a low-kT, low-s behaviour consistentwith the data, and see what general features its perturbative evolution might exhibit.The idea is to cut offEquation (3) by imposing k2T > Q20, with Q0 big enough forperturbation theory to apply, so that one uses the perturbative resummation only atshort distances. Furthermore, one takes T0 ∼s1+ǫ0 as the non-perturbative drivingterm, valid for k2T < Q20.

This cut-offequation has been recently solved by Collinsand Landshoff[4] in the case of deep inelastic scattering. Most of their results andapproximations can be carried over to the hadron-hadron scattering case, and we shallgive here the basic features of the solution in this case.First of all, the hadronic amplitude can be thought of as the convolution of two formfactors times a resummed QCD gluonic amplitude obeying a cut-offBFKL equation.A(s, t) =Z √sQ0dk1V (k1)k41Z √sQ0dk2V (k2)k42T(k1, k2; s)(6)k1 and k2 are the momenta entering the gluon ladder from either hadron, √s is thetotal energy, the two form factors V (ki), i=1,2, represent the coupling of the protonto the perturbative ladder via a non-perturbative exchange, and the 1/k4i come fromthe propagators of the external legs.

T(k1, k2; s) will obey the BFKL equation bothfor k1 and k2, and the two independent evolutions will be related by the driving termT0 representing the 2-gluon exchange contribution and thus proportional to δ(k1 −k2)s1+ǫ0. The next terms Tn will be given by Equation (2) but cut offat small k:Tn(kT, k2; s) = θ(√s > kT > Q0) ˆKTn−1(k′T, k2; s′)(7)Under these assumptions, and working at fixed αs, one can show that the amplitude(6) conserves the structure found in [4]:As = C0sǫ0 +∞Xn=1Cn(s)sǫn(s)(8)This solution reduces to the usual solution of the BFKL equation when s →∞andQ0 →0.

The coefficients Cn depend on the model assumed for the coupling V (k)between the non-perturbative and the perturbative physics and their s dependence isa threshold effect coming from the integration in (6). Their only general property isthat they are positive.

On the other hand, the powers ǫn(s) are universal functionsthat depend only on αS and √s/Q0.4

4Interplay between soft and hard QCD: a modelAs the coefficients of the series (8) are model-dependent, we do not attempt to calculatethem, but rather try to assess the constraints that present data place on them. Weshall then be able to decide whether such perturbative effects could play a substantialrole in soft physics at future colliders.

As all the Cn are positive, the behaviour of theseries (8) will not be very different from that of its leading term, and so we truncate it.We also make an educated guess for the threshold function contained in C1(s). Thisdoes not affect our results for the values of Q0 shown here.

We finally impose crossingsymmetry to get the real part of the amplitude. This gives˜As=C0sǫ0 + [c1(1 −Q0√s)2 θ(√s −Q0)] sǫ1(s)(9)A(s)=˜A(s) + ˜A(se−iπ)(10)with c1 a positive constant.

To calculate ǫ1(s) we assume that Q0 is the scale of αSand take ΛQCD = 200 MeV. Using the results of reference [4], we calculate the curvesof Figure 3, for various values of the cutoffQ0 and thus of αS.

One sees that theeffective power is much smaller than its purely perturbative counterpart (4), e.g. forQ0=2 GeV, the usual estimate (4) gives ǫ=0.8, whereas a cut-offequation gives valueshalf as big at accessible energies.Again, we consider both a pole fit and an eikonalized one.

Note that the use ofsuch an eikonal formalism [13] is not derived from QCD. In fact, the BFKL equationin principle sums multi-gluon ladders in the s and t channels, so that in the purelyperturbative case the eikonal formalism is probably too na¨ıve.

However, in this case, itcan be thought of as an expansion in the number of form factors V (k1)V (k2). This isdefinitely not included in the BFKL equation.

We further add the meson trajectoriesof (1) to the amplitude (9), and proceed to fit the data.The first obvious observation is that the extra perturbative terms do not help thefit: due to the positivity of the Cn, they cannot produce a bump in ρ that wouldexplain the UA4 measurement. So, one gets the best fit when the new QCD terms areactually turned off.

We want to examine here what constraints are placed on themby present data, and so we proceed as follows: we choose two values of Q0, 2 and 10GeV, where one could imagine to cut-offperturbative QCD. We then proceed to fitthe data, for increasing values of c1, letting all other parameters free.

When we reachthe 90% confidence letter (C.L.) as defined by our χ2/d.o.f., we have found the highestperturbative contribution permissible.

For Q0 = 2 GeV, this gives c1/C0 ≤6 × 10−4for the pole fit, and 2 × 10−3 for the eikonal one. For Q0 = 10 GeV, the correspondingvalues are 8 × 10−3 and 11 × 10−3.

We plot the resulting curves in Figures 1 and 2.The parameters of the non-perturbative component are modified from those ofTable 1 by a few percent only. The coupling C0 is increased a little while the power ǫ0goes down.

This maximizes the perturbative contribution, which is mainly constrainedby the Tevatron measurement of the total cross section.One sees that the small5

perturbative coupling can lead to quite dramatic consequences at the SSC and LHC.We show in Table 2 the 90% C.L. on the total cross section and the ρ parameter atfuture colliders.

As the perturbative contribution can become very large, the effectof unitarization is non negligible, and depending on the value of Q0, the total crosssection could reach values as high as 230 mb at the SSC, with half of it coming fromperturbative resummation.collider/pole fiteikonal fitpole fiteikonal fitquantityQ0 = 2 GeVQ0 = 2 GeVQ0 = 10 GeVQ0 = 10 GeVSSC σtot(mb)565 (459)228 (133)180 (71)139 (34)LHC σtot(mb)231 (138)160 (74)130 (35)113 (19)Tevatron σtot(mb)77 (9)77 (10)75 (6)74 (5)SSC ρ0.900.230.270.26LHC ρ0.660.240.210.20Tevatron ρ0.200.160.140.12Table 2: Allowed values of the cross section and the ρ parameter. The first two columnscorrespond to an infrared cutoffQ0 = 2 GeV and the two last ones to Q0 = 10 GeV,see Equation (9).

The number in parenthesis next to the cross section is the value ofthe perturbative component.We emphasize that the estimates of Table 1 are conservative, as they correspondto cutoffs of 2 and 10 GeV. Cutting offthe evolution when αS ≈1 would give a totalcross section of at least 3 b at the SSC, and be consistent with all available colliderand fixed target data!5ConclusionWe have shown that the BFKL equation can be used to evolve the soft pomeron tohigher s, and that perturbative effects could become measurable at the SSC/LHC.These effects are cutoffdependent, and perturbative physics seems to couple veryweakly to the proton in the diffractive region, its coupling strength being a few percentof that of the soft pomeron.

However, even a very weak coupling turning on at anenergy of a few GeV can lead to measurable effects at sufficiently large energy. Itis known that the pomeron couples to quarks, and quarks to gluons.

Therefore, thecoupling to the BFKL ladder cannot be zero, and specific models can be built for it[14].This contribution is genuinely new and comes entirely from a QCD analysis. Oneshould not be misled by previous parton models [8] which, while using a partonicpicture, keep it mostly non-perturbative, replacing the small power sǫ0 of (1) by asmall power x−ǫ0 in the gluon structure function xg(x).

In the present model, xg(x)will contain the same powers ǫi(Q0/x) as the total cross section, but their coefficients6

will in general be different from those entering the total cross section, and the relationbetween them will be model dependent.The existence of such possibilities, and the fact that very large total cross sectionsare expected from the same kind of arguments that lead one to predict a rising crosssection [13,15], shows that small momentum physics contains a wealth of open possibil-ities worth exploring experimentally. It also suggests that a large proportion of eventscould become calculable at very high energy, and so could be used for the detection ofnew physics.AcknowledgmentsThis work was supported in part by NSERC (Canada) and les fonds FCAR (Qu´ebec).7

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Figure CaptionsFigure 1: Pole fits (1) and (10) at zero momentum transfer, for pp (squares) and ¯pp(crosses) scattering. The lowest curve is the best (non-perturbative) fit, while the twoupper curves are allowed by a cut-offBFKL equation at the 90% C.L., for an infraredcutoffQ0=2 GeV or 10 GeV, as indicated.

(a) shows the total cross section and (b)the ratio of the real-to-imaginary parts of the amplitude. The data are from reference[9].Figure 2: Same as Figure 1, but after eikonalization.Figure 3: The effective power of s of Equation (8) that results from a cut-offBFKLequation, for various values of the infrared cutoffQ0, as indicated next to the curves.9

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