A model, based on Gribov's Reggeon calculus, is proposed and applied to
processes of soft diffraction at high energies. It is shown that by accounting
for absorptive corrections for all legs of triple-Regge and loop diagrams a
good description of experimental data on inelastic soft diffraction can be
obtained. In this paper we give a brief description of the model and of its
predictions for LHC energies.Comment: Talk given at EDS'09 conferenc

Kaidalov, A. B.

Poghosyan, M. G.

English

arXiv

A model, based on Gribov's Reggeon calculus, is proposed and applied to processes of soft diffraction at high energies. It is shown that by accounting for absorptive corrections for all legs of triple-Regge and loop diagrams a good description of experimental data on inelastic soft diffraction can be obtained. In this paper we give a brief description of the model and of its predictions for LHC energies

Kaidalov, A B

Poghosyan, M G

CERN Document Server

Description of Soft Diffraction in the Frameworkof Reggeon Calculus. Predictions for LHCAlexei Kaidalov1, Martin Poghosyan21 Institute of Theoretical and Experimental Physics, 117526 Moscow, Russia2 Universita` di Torino and INFN, 10125 Torino, ItalyA model, based on Gribov’s Reggeon calculus, is proposed and applied to processes ofsoft diffraction at high energies. It is shown that by accounting for absorptive correctionsfor all legs of triple-Regge and loop diagrams a good description of experimental data oninelastic soft diffraction can be obtained. In this paper we give a brief description of themodel and of its predictions for LHC energies.1 IntroductionThe process of soft single- and double- diffraction dissociation are closely related to small angleelastic scattering in which each of the incoming hadrons may become a system which will thendecay into a number of stable final state particles. Regge-pole theory is the main method fordescription of high-energy soft processes. In this approach (see [1]), the inclusive cross-sectionof single and double diffraction dissociations is described by triple-Reggeon and loop diagrams,respectively. Triple-Reggeon description is in good agreement with the FNAL and ISR datafor soft diffraction dissociation [2]. However, the higher-energy data from SPS and Tevatrondo not show the increase of the cross section with energy expected from the simple fits and thecontribution of triple-Pomeron vertex (in the elastic scattering amplitude) violates unitarity. Anumber of different approaches have been proposed in order to be in agreement with the datafrom higher energy experiments: non-gaussian parameterisation for Reggeon-hadron vertex [3],renormalisation [4] or damping [5] of the Pomeron flux. A more realistic approach suggestedin [6] and [7] by the inclusion of initial state elastic scattering corrections to the triple-Reggeonvertices. However, the analysis done in [8] shows that this correction is not enough for restoringthe s-channel unitarity.Besides of its own role, the theoretical knowledge of soft diffraction is also important inanalysing hard diffraction data, which is an active area of study at HERA and Tevatron andwill continue to be interesting at LHC. The knowledge of the interaction between Pomerons(enhanced diagrams) is important for analysing data at very high energies, too. The contri-bution of these diagrams can be essential in hadron-nucleus and especially in nucleus-nucleuscollisions, where the thermalisation and the quark-gluon plasma formation strongly depend onthe strength of interactions between Pomerons [9].In this article we propose to describe data on soft diffraction dissociation in pp and pp¯interactions taking into account all possible non-enhanced absorptive corrections to triple-Reggevertices and loop diagrams. This approach describes available data on high-mass soft diffractionin the energy range from ISR, FNAL to Tevatron. The article is organised as follows: In the next91two sections we briefly describe the Regge-pole approach, Gribovs’ Reggeon calculus and AGKcutting rules. Our proposed model is presented in Section 4 and its predictions are comparedwith data in Section 5.2 Single Regge-Pole ApproximationIn Regge theory, the simplest singularity in the j-plane is a moving pole α(t) in the t-channel(see the leftmost graph in Fig. 1) and in the small t-region the scattering amplitude, M(s, t),of the process a + b → c + d can be parameterised as:M(s, t) = γ(0)η(α(0))(s/s0)α(0)−1 exp(λ(s)t). (1)Here η(α(t)) is the signature factor, γ(t) ≡ gac(t)gbd(t) is the factorisation residue, and λ(s) =R2 + α′R ln(s/s0). The parameter R2 characterises the t-dependence of the product of residuefunction and of the signature factor. In our notations the normalisation of the scatteringamplitude is such that σtot = 8pi Im M(s, 0) and dσel/dt = 4pi|M(s, t)|2.Because the Pomeron’s intercept is larger than unity (which is required in order to guaranteethe growth of the total cross-section, σtot ∼ s∆ where ∆ ≡ αP (0) − 1), the correspondingcross-section grows as a power function of s and therefore the contribution of the Pomeron-polein the scattering amplitude violates unitarity. The easiest way to restore the unitarity is totake into account branch points which correspond to multi-Reggeon exchange. The calculationof the multi-Reggeon exchange amplitude is possible in eikonal (or eikonal-like) approximation,where only single particle intermediate states are taken into account.3 Eikonal Approximation and AGK Cutting RulesRegge poles are not the only singularities in the complex angular momentum plane. Exchangeof several Reggeons in t-channel leads to moving branch points in the j-plane (Fig. 1). ARegge pole exchange can be interpreted as corresponding to single scattering while Reggecuts correspond to multiple scatterings on constituents of hadrons. In case of the ‘super-critical’ Pomeron (∆ > 0) the contribution of n-Pomeron exchange in the scattering amplitude(M(n)P (s, 0) ∼ sn∆) is increasing with the increase of the energy and the entire series of n-Pomeron exchange should be summed. On the contrary, the contribution of the branch pointsFigure 1: Single pole and RP n cut contri-bution in the elastic scattering amplitude.R stands for secondary Reggeon and forPomeron.P + P P + P ÉP P+ R + R P + R ÉP Pconcerned with the exchange of several secondary Reggeons decreases very quickly with increas-ing collision energy and the contribution of such branch points can be neglected with respect tothe branch points due to the exchange of one secondary Reggeon and Pomerons that are neededfor properly matching low energy data. For instance, in eikonal approximation the amplitudeA KAIDALOV, AND M POGHOSYAN92of n-Pomeron exchange can be written in the following form [10]:M (n)(s, t) =(2i)n−1n!∫ n∏i=1[M (1)(s,q2i⊥)d2qi⊥2pi]δ(q⊥ −n∑i=1qi⊥). (2)Using the parameterisation (1) for the Regge pole contribution and performing the integrationover the transverse momenta of Reggeons in Eq. (2) it can be shown that the account of themulti-Pomeron exchanges results in the unitarisation of the scattering amplitude, which leadsto the Froissart behaviour of the total cross section for s À m2N : σtot ' 8piα′P ∆ ln2(s/s0).In the language of Regge poles the multiparticle production processes are related to cut-Reggeondiagrams. Abramovski, Kancheli and Gribov (AGK) proposed rules [11] for calculating thediscontinuity of the matrix element that represent the generalisation of the Optical Theoremfor the case of multi-Pomeron exchange. The basic results of AGK needed for the followingdiscussion are: a) There is one and only one cut-plane which separates the initial and final statesof the scattering. b) Each cut-Pomeron gives an extra factor of (−2) due to the discontinuityof the Pomeron amplitude. c) Each un-cut Pomeron obtains an extra factor of 2 since it canbe placed on both sides of the cut-plane.4 The ModelWe propose to describe single- and double- diffraction processes by such diagrams where anynumber of Pomeron exchanges is taken into account together with each R of the triple-Reggeonand loop diagrams and as well as the screening corrections are considered, as shown in Fig. 2. Inthis figure the solid line accompanied with a dashed line corresponds to one Reggeon (Pomeronor secondary-Reggeon) exchange together with any number Pomeron exchange. The double-dashed lines stand for eikonal screening.The theory does not give any prediction on the structure of the vertices for n Pomeron to mPomeron transitions. The simplest approximation is to assume an eikonal-type structure. In thisapproximation the general approach of constructing elastic scattering amplitude with account ofenhanced diagrams has been proposed in [12]. Assuming pi-meson exchange dominance of multi-Pomeron interaction vertices, the authors summed high order enhanced diagrams iteratingFigure 2: Eikonalised triple-Reggeon and loop diagramswhich are proposed to de-scribe single- and double-diffractive processes inhadron-hadron collisions.multi-Pomeron vertices in both, s- and t- channels. In that article it was demonstrated that theinclusion of these diagrams in most of the cases leads to predictions that are very close to theresults of eikonal type models, where a Pomeron with suitably renormalised intercept is used.For calculating the diagrams shown in Fig. 2 we assume the mentioned pi-meson exchangedominance of multi-Pomeron interaction vertex corresponding to the transition of n PomeronsDESCRIPTION OF SOFT DIFFRACTION IN THE FRAMEWORK OF REGGEON CALCULUS. . . .93to m Pomerons:λ(n,m) = r3P gn+m−3pi exp(−R2pin+m∑i=1q2i). (3)As a secondary Regge pole we consider f -trajectory. The conservation laws allow us to assumethe same pion dominance at the same transition with participation of f -trajectory. In theseterms according to the AGK cutting rules, the cross-section corresponding to the cut dressedtriple-Reggeon graph for the process a + b → X + b has the form1:dσdζ=12∑i,j,k=P,RGijk∫db db1 Γibpi(ζ2,b2)Γjbpi(ζ2,b2)Γkapi(ζ,b1) exp{−2Ωab(ξ,b)} (4)Here we introduce the following notationsζ = ln(M2X/s0), ζ2 = ξ − ζ, b2 = b− b1, Ωαβ(ζ,b) =gαβλαβexp{∆ζ − b24λαβ}, (5)ΓPαβ(ζ,b) = 1− e−Ωαβ(ζ,b), ΓRαβ(ζ,b) =gRαβλRαβexp{(αR − 1)ζ − b24λRαβ− Ωαβ(ζ,b)}.Gijk stand for triple-Regge vertices strength. The expression of the cross-section in the (ζ, t)-space is rather long and we do not present it here.Analogously can be calculated the cross-section corresponding to the cut dressed loop dia-gram standing for the process a + b → X1 + X2, and it has the following form:dσdζ1dζ2=14∑i,j,k,l=P,RGijkGlik∫db db1 db2 Γipia(ζ1,b1)Γlpib(ζ2,b2)Γjpipi(ζ3,b3)Γkpipi(ζ3,b3)× exp{−2Ωab(ξ,b) − 2Ωapi(ξ − ζ1,b− b1)− 2Ωbpi(ξ − ζ2,b− b2)} (6)Here in addition to (5) we used the following notations:ζ1 = ln(M2X1/s0), ζ2 = ln(M2X2/s0), ζ3 = ξ − ζ1 − ζ2, b3 = b− b1 − b25 Extraction of the Parameters from Experimental DataBecause we do not consider the contribution of the enhanced diagrams in the elastic scatteringamplitude it allows us to differentiate data fitting procedure and realise it by two steps. At thefirst step we fix secondary-Reggeon and Pomeron parameters. The trajectories of secondary-Reggeons are fixed from fit to data on spin vs. mass for corresponding family of mesons, andthe following results are found: αf (t) = 0.7 + 0.8t, αω(t) = 0.4 + 0.9t, αρ(t) = 0.5 + 0.9t.Then the residues of secondary-Reggeons and the residues/trajectory of Pomeron are foundfrom fit to data on elastic scattering and total interaction cross-section. At the second step wefix triple-Reggeon interaction vertices’ constants from fit to data on high mass soft-diffractiondissociation, using the values of the parameters fixed in the first step as an input.We take into account P -, f - and ω- poles in pp and pp¯ elastic scattering amplitude. Sincewe assume pion exchange dominance at the coupling of Reggeons, we fix the parameters of1If the transferred momentum is very low and the mass of the diffracted system is high the pi-meson exchangeplays an important role. This we take into account based on the OPER model [13].A KAIDALOV, AND M POGHOSYAN94secondary Reggeon- and Pomeron- pion coupling as well. In pi±p elastic scattering amplitude,we take into account P -, f - and ρ- poles. Thus, we assume M = MP + Mf ±Mω for pp andpp¯ collisions and M = MP + Mf ±Mρ for pi+p and pi−p collisions, respectively. For Pomeron-trajectory we have found the following parameterisation: αP (t) = 1.117 ± 0.252t, and otherparameters are listed in the Tables 1 and 2.Table 1: p+p(p¯) interaction pa-rameters in GeV−2 units.gN = 1.366± 0.004R2N = 1.428± 0.006gfN = 2.871± 0.008Rf2N = 0.918± 0.023gωN = 2.241± 0.074Rω2N = 0.945± 0.026Table 2: pi±p interaction pa-rameters in GeV−2 units.gpi = 0.85± 0.0004R2pi = 0.5± 0.002gfpiN = 3.524± 0.001Rf2piN = 1.± 0.001gρpiN = 1.12± 0.017Rρ2piN = 9.19± 0.837Table 3: Found valuesof Gijk in GeV−2 units.GPPP =0.0098±0.0005GPPR=0.03 ±0.004GRRP =0.005 ±0.001GRRR=0.05 ±0.002GPRP =0.013 ±0.001GPRR=0.033 ±0.005Next we fix the triple-Reggeon vertices strengths (Gijk) from fit to data on soft single-diffraction dissociation in pp and pp¯ interactions. We used the available data on spectra of non-diffracted proton from fixed-target experiments [14] and [15], from ISR [16] and from CDF [4].Being interested in soft diffraction, we have chosen measurements done for d2σ/dζdt within thediffractive cone (-t ≤ 0.2GeV2). The values of Gijk found are reported in Table 3 and the fitresult is compared with data in Figs. 3-5.Fx-210 -110-2dt mb GeVF/dxσd 210310410510 = 65 GeV/clab) P2 10×( = 154 GeV/clab 10) P×( = 372 GeV/clabP2t = -0.05 GeVζ-210 -110-2dt mb GeVζ/dσd 210310410510610710 = 14 GeVs) 3 10×( = 20 GeVs) 2 10×( = 546 GeVs 10)  ×( = 1800 GeVs2t = -0.05 GeVFigure 3: Double differential cross-section d2σ/dζdt for p(p¯) + p → p(p¯) + X measured atFermilab at various√s and fixed t. The data are taken from [4, 15].In Fig. 6 we compare predictions of the model on single-diffractive integrated cross-sectionswith experimental data [17]. In each case the integration is done in accordance with the corre-sponding measurement as they are indicated in the figure.In Fig. 7 we compare predictions of the model on double-diffractive integrated cross-sectionswith experimental data [18, 19]. The data at√s >100GeV correspond to the cross-sectionfor minimum 3 units of rapidity gap between two produced clusters and are taken from [18].The rest of data (at√s <100GeV) are taken from [19] where exclusively and semi-inclusivelymeasured data are reduced to totally inclusive cross-section. The theoretical curve is calculatedusing Eq. (6) and requiring minimum 3 units of rapidity gap between two diffracted clusters.DESCRIPTION OF SOFT DIFFRACTION IN THE FRAMEWORK OF REGGEON CALCULUS. . . .95ζ-210 -110-2dt mb GeVζ/dσd 210310410510610710 )310× (2t = -0.036 GeV )210× (2t = -0.075 GeV10)× (2t = -0.131 GeV2t = -0.197 GeV 2s = 262 GeVζ-210 -110-2dt mb GeVζ/dσd 210310410510610710 )310× (2t = -0.036 GeV )210× (2t = -0.075 GeV10)× (2t = -0.131 GeV2t = -0.197 GeV 2s = 366 GeVζ-210 -110-2dt mb GeVζ/dσd 210310410510610710 )310× (2t = -0.036 GeV)210× (2t = -0.075 GeV10)× (2t = -0.131 GeV2t = -0.197 GeV 2s = 565 GeVζ-210 -110-2dt mb GeVζ/dσd 210310410510610710 )310× (2t = -0.036 GeV)210× (2t = -0.075 GeV10)× (2t = -0.131 GeV2t = -0.197 GeV 2s = 741 GeVFigure 4: Double differential cross-section d2σ/dζdt for pp → pX measured at Fermilab atvarious√s and t. The data are taken from [14].ζ -110-2dt mb GeVζ/dσd 210310410510610710810 )310× (2t = -0.0425 GeV)210× (2t = -0.0625 GeV10)× (2t = -0.095 GeV 2t = -0.135 GeV 2s=1050 GeVζ -110-2dt mb GeVζ/dσd 210310410510610710810 )310× (2t = -0.0475 GeV)210× (2t = -0.0675 GeV10)× (2t = -0.105 GeV 2t = -0.145 GeV 2s=1050 GeVζ -110-2dt mb GeVζ/dσd 210310410510610710810)310× (2t = -0.0525 GeV)210× (2t = -0.075 GeV)110× (2t = -0.115 GeV 2t = -0.15 GeV 2s=1050 GeVζ -110-2dt mb GeVζ/dσd 210310410510610710810)310× (2t = -0.0575 GeV)210× (2t = -0.085 GeV10)× (2t = -0.125 GeV 2t = -0.19 GeV 2s=1050 GeVFigure 5: Double differential cross-section d2σ/dζdt for pp → pX measured at ISR at various√s and t. The data are taken from [16].A KAIDALOV, AND M POGHOSYAN962s GeV210 310mb0.511.522.532dtdMσ2d2dM2s+mηth∫dt-0.024-0.234∫ = 0.01η = 0.02η = 0.04η = 0.06η = 0.1η GeVs210 310 410 mbSDσ2468101214161820/s < 0.152M/s < 0.062M/s < 0.12M/s < 0.052MFigure 6: Integrated single-diffractive cross-section as a function of√s. The integrations aredone in accordance with corresponding measurement as they are indicated in the plots. GeVs210 310 410 mbDDσ12345678910 > 3η∆Figure 7: Double-diffractive cross-sectionas a function of√s . The theoretical curveis calculated requiring minimum 3 units ofrapidity gap between two diffracted clusters.6 Summary and Predictions for LHCIn this article we report the results of calculations of all non-enhanced absorptive correctionsto triple-Regge vertices and loop diagrams in eikonal approximation using Gribov’s Reggeoncalculus. Numerically evaluating the model we have found a good description of data on high-mass soft diffraction dissociation in the energy range from ISR, FNAL to Tevatron (from Plab =65GeV/c to√s = 1800GeV). It is worth to emphasise that such a detailed description ofinclusive diffraction in this broad region of energies is achieved for the first time. In Table 4 wepresent the predictions of the model on single- and double- diffractions cross-section for differentenergies of LHC. The single-diffractive cross-section is obtained integrating over masses up toM2/s = 0.05, and the double-diffractive cross-section is obtained requiring minimum 3 units ofrapidity gap between two diffracted clusters.√s TeV σSD mb σDD mb0.9 8.2 5.77 11.6 6.110 12 6.214 13 6.4Table 4: Predictions for LHC.DESCRIPTION OF SOFT DIFFRACTION IN THE FRAMEWORK OF REGGEON CALCULUS. . . .97AcknowledgementsWe thank A. Grigiryan, J.-P. Revol and K. Safarik for their interest in this work.References[1] A.B. Kaidalov, Phys. Rep. 50 157 (1979).[2] A.B. Kaidalov et al., Phys. Lett. B45 493 (1973); JETP Lett. 17 440 (1973).[3] A. Donnachie and P.V. Landshoff, arXiv:hep-ph/0305246;R. Fiore et al., Phys.Rev. D61 034004 (2000).[4] K. Goulianos and J. Montanha, Phys. Rev. D59 114017 (1999).[5] E. Erhan and P. Schlein, Phys. Lett. B427 389 (1998).[6] E. Gotsman et al., Phys. Rev. D49 R4321 (1994).[7] E.G.S. Luna et al., Eur. Phys. J. C59 1 (2009).[8] E.S. Martynov and B.V. Struminsky, Yad.Fiz. 59 1817 (1996) [Physics of Atomic Nuclei 59 1755 (1996)].[9] A.B. Kaidalov, Nucl. Phys. A525 39c (1991).[10] K.A. Ter-Martirosyan, Preprints ITEP No. 70, 71 (1975); No. 7, 11, 133, 134, 135, 158 (1976).[11] V.A. Abramovski, O.V. Kanceli and V.N. Gribov, Sov. J. Nucl. Phys. 18 308 (1974).[12] A.B. Kaidalov, L.A. Ponomarev and K.A. Ter-Martirosyan, Sov. J. Nucl. Phys. 44 483 (1986).[13] K.G. Boreskov et al., Sov. J. Nucl. Phys. 15 203 (1972); 17 669 (1973); 19 565 (1974).[14] R.D. Schamberger et al., Phys. Rev. D17 1268 (1978).[15] Y. Akimov et al., Phys. Rev. Lett. 39 1432 (1977).[16] J.C.M. Armitage et al., Nucl. Phys. B194 365 (1982).[17] R.E. Ansorge el al., Z. Phys. C33 175 (1986); G.J. Alner et al., Phys. Rept. 154 247 (1987); F. Abe et al.,Phys. Rev. D50 5535 (1994); N. Amos et al., Phys. Lett. B301 313 (1993).[18] T. Affolder et al., Phys. Rev. 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Description of soft diffraction in the framework of reggeon calculus. Predictions for LHC

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Description of soft diffraction in the framework of reggeon calculus. Predictions for LHC

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