I review the predictions of the total cross section for many models, and
point out that some of them lead to the conclusion that the standard
experimental analysis may lead to systematic errors much larger than expected.Comment: Contribution to the 13th International Conference on Elastic &
  Diffractive Scattering (13th "Blois Workshop"), CERN, 29th June - 3rd July
  200

Cudell, J. R.

English

arXiv

I review the predictions of the total cross section for many models, and point out that some of them lead to the conclusion that the standard experimental analysis may lead to systematic errors much larger than expected

Cudell, J R

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The Total Cross Section at the LHC:Models and Experimental ConsequencesJ.R. CudellIFPA, AGO Dept., Universite´ de Lie`ge, BelgiumI review the predictions of the total cross section for many models, and point out thatsome of them lead to the conclusion that the standard experimental analysis may lead tosystematic errors much larger than expected.The total cross section is a highly non-perturbative object that we cannot predict fromQCD. In fact, we have to rely on theoretical ideas that were developed before QCD, in thecontext of the analytic S matrix, such as analyticity, or the unitarity of partial waves.The natural place to discuss these ideas is the complex-j plane, where the singularities ofthe amplitudes determine their behaviour with s. But we do not know these singularities. Thesimplest ones, which correspond to the exchange of bound states, are simple poles. Accountingfor the exchange of meson trajectories is not too hard, as we know their spectrum. But as theircontribution falls with s, they will not matter at the LHC. One must thus model the pomeron,for which there is little spectroscopic guidance. Again, the simplest idea is to use a simple poleat j = 1 + ². But we know that if there are simple poles, there must be cuts, which correspondto multiple exchanges. We do not know how to calculate these: we only know general propertiesof the two-pomeron cut. This means that there are many possibilities, such as eikonal models,U -matrix unitarisation, extended eikonal/U -matrix models, or multi-channel eikonals. Thesecuts will be needed at the LHC to restore partial-wave unitarity. It is also possible that thepomeron is not an exchange of bound states, so that one should not start from a simple pole, butrather consider multiple poles (double or triple) at j = 1, which automatically obey unitarity.1 The COMPETE FitsThe multitude of models has been confronted with data, and the COMPETE collaboration [1]gathered and cleaned the datasets which are now available on the servers of the Particle datagroup [2], and which gather all the soft data measured at√s ≥ 4 GeV and t = 0, i.e. the totalcross sections and the ρ parameters when available, for pp, p¯p, pi±p, K±p, γp and γγ. Thesedata have some drawbacks. First of all, most cross sections have been measured at ISR energies,and there is a gap from√s ≈ 100 GeV to √s ≈ 500 GeV. This gap would have been filledby the pp2pp collaboration at RHIC, but it has unfortunately been stopped. Another problemcomes from the highest-energy data, where a 2σ disagreement exists between E710 and CDF.Finally, the compatibility of the various measurements of ρ at ISR energies is questionable, andnot enough information has been published to redo the analyses.Nevertheless, COMPETE considered several hundreds of possible parametrisations, basedon simple, double or triple poles, and kept only those which had a global χ2/point smaller42than 1, for√s ≥ 5 GeV. From these, one can predict the total cross section at the LHC,and estimate the error due to the extrapolation. Fig. 1 shows the result of this study. The0.080.120.11s  (TeV)10σ(b)Figure 1: The COMPETE prediction forthe total cross section in the LHC energyregion. The inner black band representsour best guess with statistical errors only,the outer band all the parametrisationswith χ2/point ≤ 1.extrapolation of the parametrisations whichpassed the χ2 test is given by the outer band. Itpredicts 84mb ≤ σ ≤ 112mb for √s = 10TeV,and 90–117mb for 14TeV.The inner band of Fig. 1 shows the best guess,and it is based on a series of statistical indica-tors explained in [4]. It corresponds to a universaltriple-pole (log2 s) parametrisation. One shouldnote that the multiple-pole fits have some peculiarproperties: in the triple-pole case, the contributionof the pomeron falls with s for energies smallerthan 6 GeV, whereas in the double-pole case, itactually becomes negative below 9.5GeV. Simplepoles seem to be excluded, but they can be con-sidered again if the lower cut-off on the energy israised to 10 GeV. The multiple-pole parametrisa-tions have been used successfully [5] to reproducelow-x data in deeply inelastic scattering (DIS),showing that universality does not seem to holdin general, as the coefficient of the triple pole de-pends on Q2. More recently, multiple poles havebeen used successfully [6] to reproduce the differential elastic cross sections [7].Hence multiple-pole fits seem to work, but it is hard to imagine how such a structure mayemerge from QCD. On the other hand, simple-pole fits have a simple phenomenology, a simpleinterpretation as the exchange of glueballs, and only a handful of parameters.2 The Simple-Pole Fit: Low EnergyClearly, one simple-pole pomeron has never been a reasonable choice: in their original soft-pomeron model [8], Donnachie and Landshoff included a two-pomeron cut, which lowers thecross section, and which enables one to go to ≈ 40 TeV before unitarity is broken. However,such a simple model seems to produce worse fits than multiple poles. Furthermore, HERA dataon DIS and on vector-meson production have lead to the introduction of a second singularity[9], taken also as a simple pole, and mimicking the BFKL cut. One can indeed follow thestandard transformation to the j-plane to convince oneself that hadronic singularities mustbe present in DIS [10]. This new singularity has a coupling which depends on Q2, or on thevector-meson mass, and, together with the soft pomeron, it enables one to reproduce all HERAmeasurements [11]. The question then is to know whether such a singularity might be presentin soft data.It actually came as a surprise that, indeed, the introduction of a hard pole brings the simple-pole pomeron fit on equal footing with multiple poles, as far as the χ2/point is concerned [12].This is true provided that one stops the fit at√s ≈ 200 GeV. Beyond that, the hard polemust be unitarised, as its contribution to the total cross section would be much too large.Nevertheless, it is remarkable that a global fit to soft data leads to a hard pomeron with anTHE TOTAL CROSS SECTION AT THE LHC: MODELS AND EXPERIMENTAL CONSEQUENCES43intercept ≈ 1.4, perfectly compatible with that obtained by Donnachie and Landshoff in DIS.It must be noted however that the coupling of the hard pomeron is tiny. It contributes atmost 7% to the total cross section for√s < 200 GeV, as shown in Fig. 2. Previous studies [1]considered only fits to the highest energy, in which case the hard pomeron coupling becomesextremely small. Even if the energy is limited to 200 GeV, the coupling remains of the orderof 1% of that of the soft pomeron.10 1000.000.501.00s (GeV)relative contributionf/aρ/ωsoft pom.hard pom.Figure 2: The contribution to the total ppcross section of the two pomerons and thetwo reggeons, divided by the average of thepp and p¯p total cross sections.The two-pomeron model was later applied tothe differential elastic cross section in [13], where itwas shown that the hard pomeron was compatiblewith data on the first cone (0.1 GeV2 < |t| ≤ 0.5GeV2), for energies smaller than 100 GeV, al-though in this case it did not significantly improvethe fit. Note also that the form factors of the var-ious exchanges were directly extracted from thedata, confirming the presence of a zero in the formfactor of the C = −1 trajectory, and suggestingone for the c = +1 meson trajectory.One has thus found an alternative to themultiple-pole fits: the two-pomeron model, whichcan reproduce DIS data, photoproduction data,soft forward data and differential elastic cross sec-tion, provided one stays at moderate energy andsmall |t|. As we shall now see, if this model isthe correct one, then it may have important con-sequences at the LHC.3 The Simple-Pole Fit: High EnergyIndeed, to go to higher energies, one must confront the question of unitarisation. At highenergy, the partial waves can be obtained by a Fourier transform of the amplitude A(s, t) toimpact-parameter space, as ` ≈ b√s. The unitarity constraint can then be written|A(s, b)|2 ≤ 2Im (A(s, b)) .It is easy to see that this means that the complex A(s, b) has to be in a circle of radius 1 centredat A(s, b) = i/2, which we shall call the unitarity circle (Note that another normalisation isfrequently used, where the circle has radius 1 and is centred at i.)In our two-pomeron model, the amplitude A(s, 0) leaves the unitarity circle for energiesslightly below the Tevatron, and it is quite far out at the LHC, as can be seen in Fig. 3.The problem, of course, is that one does not know exactly how the amplitude will stay in theunitarity circle. It is known that multi-pomeron exchanges, and maybe multi-pomeron vertices,will restore unitarity. But how to implement these ingredients is far from unique [14]. Eventhe simplest two-pomeron cut is not fully calculable, let alone a full resummation.Mathematically, unitarisation schemes can be classified into three classes. The first onemaps the whole upper-half complex plane onto the unitarity circle. In this class, one finds theeikonal scheme, and the U -matrix scheme [16]. Note that the latter is the only one to providea one-to-one mapping between the half-plane and the circle, but the amplitude also has a newJ R CUDELL44pole at A(s, b) = −2i. The second class is a simple extension of the first one: instead of mappingthe upper half-plane to the full unitarity circle, one maps it to a smaller circle, still centred atA(s, b) = i/2. The third class, which we recently proposed [15], maps a given amplitude to thecircle, which is far less restrictive than mapping the half plane. In general, these schemes canhave very different properties [15]: they can lead to the Pumplin bound (σel < σtot/2 [17]) andto shadowing, but it is also possible to have anti-shadowing and σel ≈ σtot, both for eikonal-likeand U -matrix-like schemes. I shall not consider these more exotic possibilities here, but onehas to bear in mind that they cannot be a priori ruled out.LHCTevatronunitaritybound5 100012Im(A      )(s,b)(GeV   )b −1Figure 3: The imaginary part of the am-plitude A(s, b) at the Tevatron and at theLHC, before unitarisation.(TeV)sbare saturatedeikonal1010.16090150120180σtot(mb)Figure 4: Total cross section in a two-pomeron model without (bare) and withsimple unitarisation schemes.As we do not know what scheme to use, we have decided to be conservative, and to unitarisethe two-pomeron model using the simplest and most conservative approach. The minimalscheme assumes that something happens when the amplitude reaches the unitarity bound, andone simply cuts-off the amplitude at 1. We shall call it the saturation scheme. Note that to dothat while preserving analytic properties is not very simple, but nevertheless it can be done [18].The other scheme is the standard one-channel eikonal.Using either of these schemes produces an amplitude which respects unitarity, and thus canbe used to predict the total cross section at the LHC. The first scheme keeps by definition theresults of the fit below the Tevatron energy, and thus one can keep the parameters of the low-energy simple-pole fit. The eikonal on the other hand changes slightly the low-energy results,and thus necessitates some re- fitting. The result is shown in Fig. 4. One sees that, in this verysimple approach, the total cross section could be as large as 150 mb. It is interesting to notethat a similar number has been obtained by Landshoff, in a very different manner [14].4 The Measurement of σtotThe unitarised two-pomeron model has a consequence that has been overlooked so far. As theenergy grows, the protons become blacker, and edgier. This in turn changes the diffractionpattern, i.e. the exponential slope of the cross section varies quickly with t: the differentialcross section cannot be approximated any longer by eBt with B constant. In turn, the real partof the amplitude also develops a strong t dependence, and ρ(0) could be as large as 0.24. WeTHE TOTAL CROSS SECTION AT THE LHC: MODELS AND EXPERIMENTAL CONSEQUENCES45have checked that this is the case for the eikonal scheme, and for the saturation scheme [18],and show the results for the latter in Fig. 5.(GeV  )2−t(GeV   )−1B500 GeV500 GeV5 TeV5 TeV14 TeV14 TeV0 0.1 0.200.10.230252015ρFigure 5: The slope B and the ration of thereal to the imaginary parts of the ampli-tude for pp scattering, for various energies,in the saturation scheme.A hadronic slope B almost constant with tfor |t| < 0.25 GeV2 and a small ρ parameterare standard assumptions for the planned mea-surement of the total cross section [19]. If thetwo-pomeron model is correct, then these as-sumptions will be wrong, and we can evaluatethe systematic uncertainty that they will gen-erate. So, we have used our unitarised two-pomeron model [20] to simulate data, using binsand errors similar to those of the UA4/2 exper-iment [21], and including the Coulomb-nucleoninterference [22]. We then performed the stan-dard analysis on these simulated data, and ex-tracted a measurement of the total cross sec-tion, which we could compare to the input valuefrom our model. The result of this analysis isthat the extracted value of the total cross sectionwill systematically overshoot the model value byabout 10 mb in a luminosity-dependent method,and about 15 mb in a luminosity-independentone. Hence, an additional study of the t depen-dence of the slope and of the ρ parameter willbe needed before one reaches the 1 % precisionlevel.5 ConclusionThe measurement of the total cross section at the LHC will tell us a lot about the analyticstructure of the amplitude, as there is a variety of predictions that span the region from 90 to230 mb:• σtot > 200 mb: the only unitarisation scheme able to accommodate such a large number isthe U matrix [16]. It basically predicts the same inelastic cross section as more standardschemes, but the elastic cross section is much larger, and accounts for the difference.• 120 mb < σtot < 160 mb: this would be a clear signal for a two-pomeron model, andwould also tell us about the unitarisation scheme.• σtot ≈ 110 mb: this is the standard prediction not only of the COMPETE fits, but alsoof many models based on a simple eikonal and only one pomeron pole.• σtot < 100 mb: this would indicate either the validity of double-pole parametrisations, orthat of unitarisation schemes in which multiple-pomeron vertices are important [23, 24].J R CUDELL46And finally, if one really enters a new regime at the LHC, the study of the t dependence ofthe elastic cross section, and of the real part of the amplitude, may be of great importance tounravel the underlying dynamics and to improve the experimental measurement of σtot.AcknowledgementsI acknowledge the contribution of my collaborators of the COMPETE collaboration to theinvestigations summarised here, and in particular of A. Lengyel, E. Martynov and O.V. Selyugin.I also acknowledge discussions with A. Donnachie, P.V. Landshoff, U. Maor and S. Troshin,and thank Karine Gilson for a careful proofreading.References[1] J. R. Cudell, V. Ezhela, K. Kang, S. Lugovsky and N. Tkachenko, Phys. Rev. D 61 (2000) 034019 [Erratum-ibid. D 63 (2001) 059901] [arXiv:hep-ph/9908218]; J. R. Cudell et al., Phys. Rev. D 65 (2002) 074024[arXiv:hep-ph/0107219].[2] See http://pdg.lbl.gov/2009/hadronic-xsections/hadron.html[3] J. R. Cudell et al. [COMPETE Collaboration], Phys. Rev. Lett. 89 (2002) 201801 [arXiv:hep-ph/0206172].[4] J. R. Cudell et al. [COMPETE collaboration], arXiv:hep-ph/0111025.[5] J. R. Cudell and G. Soyez, Phys. Lett. B 516 (2001) 77 [arXiv:hep-ph/0106307].[6] E. Martynov and B. Nicolescu, Eur. Phys. J. C 56 (2008) 57 [arXiv:0712.1685 [hep-ph]]; E. Martynov,Phys. Rev. D 76 (2007) 074030 [arXiv:hep-ph/0703248].[7] A complete dataset for differential elastic cross sections is available athttp://www.theo.phys.ulg.ac.be/∼cudell/data/[8] A. Donnachie and P. V. Landshoff, Nucl. Phys. B 231 (1984) 189; Nucl. Phys. B 267 (1986) 690.[9] A. Donnachie and P. V. Landshoff, Phys. Lett. B 437 (1998) 408 [arXiv:hep-ph/9806344].[10] J. R. Cudell, A. Donnachie and P. V. Landshoff, Phys. Lett. B 448 (1999) 281 [Erratum-ibid. B 526 (2002)413] [arXiv:hep-ph/9901222].[11] A. Donnachie and P. V. Landshoff, arXiv:0803.0686 [hep-ph]; Phys. Lett. B 550 (2002) 160 [arXiv:hep-ph/0204165]; A. Donnachie and P. V. Landshoff, Phys. Lett. B 518 (2001) 63 [arXiv:hep-ph/0105088].[12] J. R. Cudell, E. Martynov, O. V. Selyugin and A. Lengyel, Phys. Lett. B 587 (2004) 78 [arXiv:hep-ph/0310198].[13] J. R. Cudell, A. Lengyel and E. Martynov, Phys. Rev. D 73 (2006) 034008 [arXiv:hep-ph/0511073].[14] P. V. Landshoff, arXiv:0903.1523 [hep-ph].[15] J. R. Cudell, E. Predazzi and O. V. 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C 60 (2009) 249 [arXiv:0812.2407 [hep-ph]].THE TOTAL CROSS SECTION AT THE LHC: MODELS AND EXPERIMENTAL CONSEQUENCES47

The Total Cross Section at the LHC: Models and Experimental Consequences

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