We discuss the effect of infrared soft gluons on the asymptotic behaviour of the total cross-section. We use a singular but integrable expression for the strong coupling constant in the infrared limit and relate its behaviour to the satisfying of the condition of the Froissart bound, giving a specific phenomenological example.Raspravljamo učinak infracrvenih mekih gluona na asimptotsko ponašanje ukupnog udarnog presjeka. Primjenjujemo singularan ali integrabilan izraz za konstantu jakog vezanja u infracrvenoj granici i usklađujemo njeno ponašanje s Froissartovom granicom, dajući poseban fenomenološki primjer

Achilli, A.

Grau, A.

Pancheri, G.

Srivastava, N.

English

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Printed ISSN 1330–0016Online ISSN 1333–9133CD ISSN 1333–8390CODEN FIZBE7QCD CONTRIBUTIONS TO THE FROISSART BOUND FOR THE TOTALCROSS-SECTIONA. ACHILLIa, A. GRAUb, G. PANCHERIc and Y. N. SRIVASTAVAaaINFN and Physics Department University of Perugia, I-06123 Perugia, ItalyE-mail-addresses: achilli@fisica.unipg.it, Yogendra.Srivastava@pg.infn.itbDepartamento de Fisica Teórica y del Cosmos, Universidad de Granada,18071 Granada, Spain E-mail-address: igrau@ugr.escINFN Frascati National Laboratories, I-00044 Frascati, ItalyE-mail-address: pancheri@lnf.infn.itReceived 20 November 2007; Accepted 20 February 2008Online 11 July 2008We discuss the effect of infrared soft gluons on the asymptotic behaviour of thetotal cross-section. We use a singular but integrable expression for the strong cou-pling constant in the infrared limit and relate its behaviour to the satisfying of thecondition of the Froissart bound, giving a specific phenomenological example.PACS numbers: 11.55.Hx, 13.60.Hb, 25.20.Lj UDC 539.125Keywords: proton, Froissart bound, total cross-section1. IntroductionThe understanding of the energy behaviour of total cross-sections in hadron andphoton scattering is still an unresolved challenge. In our opinion, the major problemcan be summarized as follows. The simplest and the most successful parametrizationof the total cross-section for proton-proton and proton-antiproton scattering followsfrom the optical theorem and describes the initial fall through the exchange of Reggetrajectories, i.e. sαR(0)−1 with αR(0) < 1, and the rise as due to Pomeron exchange,i.e. sαP (0)−1 with αP (0) > 1. Although successful, this parametrization violatesthe Froissart bound on total cross-sections, established long time ago, namely thatσtot ≤ log2s as s → ∞. Other parametrizations have then been proposed, which arebased on purely phenomenological grounds, but as such do not really give an insightinto the mechanisms of the rise of total cross-sections. In another vein, there arethe eikonal models which use the impact parameter distribution of protons, oftenderived from the Fourier trasform of the proton form factor, and basic scatteringFIZIKA B (Zagreb) 17 (2008) 1, 143–150 143Achilli et al.: qcd contributions to the froissart bound for the total . . .cross-sections. Among them is the eikonal mini-jet model in which the term whichdrives the rise is a perturbatively calculated QCD jet cross-section. Our proposalis to use this eikonal mini-jet model with an impact parameter factor derived fromthe Fourier transform of the initial transverse momentum distribution of valencequarks in the proton, calculated through soft gluon resummation. This distributionis energy-dependent and produces a saturation mechanism which quenches the mini-jet rise in such a way that one restores the Froissart bound. In the following, we shallfirst describe the soft gluon transverse momentum distribution, then the eikonalmini-jet model and its application to the proton cross-sections. We shall then showhow this model in the very large asymptotic regime, and with a particular ansatzfor the zero momentum modes of the emitted soft gluons, satisfies the Froissartbound.2. The soft gluon transverse momentum distributionSoft gluon emission from the initial quarks and gluons introduces an acollinearitybetween the initial partons. Such acollinearity is energy dependent, and can beresponsible for reducing the fast rise of the mini-jet cross-section [1]. We havedescribed this effect in a number of papers. Here we first discuss in detail the maincharacteristics of the soft gluon resummed transverse momentum distribution.The emission of an infinite number of low-momentum massless quanta resultsin a four-momentum distributiond4P (K) =d4K(2π)4∫d4xei K·xe−h(x) , (1)withh(x) =∫d3n̄(k)[1 − e−i k·x] , (2)where d3n̄(k) is the average number of soft quanta, gluons or photons, emittedby a color or electrically charged source. Upon integrating over the energy andlongitudinal momentum variable, one obtains the well known transverse momentumdistributiond2P (K⊥) =d2K⊥(2π)2∫d2beiK⊥·be−h(b) , (3)withh(b) =∫d3n̄(k)[1 − e−ikt·b]. (4)The above transverse momentum distribution does not admit a closed-form solu-tion, unlike the energy distribution in QED, which easily integrates to a power lawfunction. This has led to various strategies to evaluate the integral in Eq. (3), noneof which is however fully satisfying, for reasons which we shall try to illustrate.144 FIZIKA B (Zagreb) 17 (2008) 1, 143–150Achilli et al.: qcd contributions to the froissart bound for the total . . .Indeed, while the importance of this function has been known for a long time, inour opinion such importance has not been fully exploited, and in this note we shalldiscuss this point and present our approach to the problem.The simplest case to examine is that of the low-momentum soft quanta in anAbelian gauge theory with a constant but not small coupling constant. It is thenpossible to make some simple approximations, as in Ref. [2], and obtain a closedform for the transverse momentum distribution, namelyd2P (K⊥) =β(2π)−1Γ(1 + β/2)d2K⊥2E2A(K⊥2E√A)β2 −1×K1−β/2(K⊥E√A), (5)where K1−β/2 is the modified Bessel function of the third kind, β is obtained byperforming the angular integration over d3n̄(k) and E is the maximum energyallowed to the single soft quantum emitted.Clearly, the above procedure fails in the non Abelian case, where the strongcoupling constant is large but not constant. This was included [3, 4, 5] in the sub-sequent QCD version of the transverse momentum distribution, which is presentlyused in many phenomenological applications. Using the asymptotic freedom expres-sion for αs, the relative transverse momentum distribution induced by soft gluonemission from a pair of, initially collinear, colliding partons is obtained, at LO, ash(b,E)=163E∫µαs(kt)πdktktln2Ekt≈ 3233−2Nf{ln(2EΛ)[ln(ln(EΛ))−ln(ln(µΛ))]−ln(Eµ)},(6)where the integration only extends down to a scale µ. This expression fails toreproduce the entire range of low-energy transverse momentum effects and an adhoc constant, intrinsic transverse momentum is introduced for phenomenologicalapplications.The problem we observe in the above use of this distribution is its limitedapplicability to minimum bias physics and its lack of connection to the questionof confinement. Our approach instead is to extend the integral over the soft gluonmomentum down to zero modes; to propose an ansatz for the behaviour of αs in theinfrared region, and compare its consequences on measurable quantities such as thetotal cross-section. In the next section, we shall present one such phenomenologicalapplication through the calculation of total cross-sections and in the last sectiondiscuss the consequences of our ansatz for the infrared soft gluon limit to thelimitations imposed by the Froissart bound on the total cross-section.3. A model for the total cross-sectionRecently, we obtained predictions for the total cross-section [7, 6, 8] at LHCenergies using a model based on hard component of scattering responsible for therise of the total cross-section and soft gluon emission from scattering particles,FIZIKA B (Zagreb) 17 (2008) 1, 143–150 145Achilli et al.: qcd contributions to the froissart bound for the total . . .which softens the rise and gives the impact parameter distribution. In this model,mini-jet events coming from parton-parton high-energy collisions are responsible forthe rise of the total cross-section. From perturbative QCD, we have the followingexpression for parton-parton cross section at high energy,σABjet (s, pt min) =√s/2∫pt mindpt1∫4p2t/sdx11∫4p2t/(x1s)dx2 ×∑i,j,k,lfi|A(x1, p2t )fj|B(x2, p2t )dσ̂klij (ŝ)dpt,(7)which depends on the minimum transverse momentum allowed to the scatteredpartons in the final state, pt min, ≈ 1−2 GeV. This parameter represents the energyscale which distinguishes hard processes from the soft ones. Evaluating the mini-jetcross-sections using different LO parametrizations available for the partonic densityfunctions fi|A [9], we obtain a growth of σjet with energy as a (small) power of s.Soft-gluon emission changes the parton collinearity and contributes in softening therise of the total cross-section. The average number of soft-gluon emissions increaseswith energy, and more emissions means more acollinearity between colliding partonsand less energy available for mini-jet production. These processes are best studiedthrough an impact parameter (b-)distribution. The overlap function which describesthe b-distribution of the incident partons in our model isABN (b, s) = N∫d2K⊥ e−iK⊥·bd2P (K⊥)d2K⊥=e−h(b,qmax)∫d2b e−h(b,qmax), (8)withh(b, qmax) =163qmax∫0αs(k2t )πdktktlog2qmaxkt[1 − J0(ktb)], (9)similar to the one obtained in the first section, but for the fact that the integralextends down to zero momentum and J0(ktb) needs to be retained for infraredfiniteness. For αs we use a phenomenological expression, which coincides with theusual QCD limit for large kt, but is singular for kt → 0, namelyαs(k2t ) =12π33 − 2Nfpln[1 + p(kt/Λ)2p] , (10)with p < 1 for the integral over kt to exist and p ≥ 1/2 for the correct analyticityin the momentum transfer variable.The parameter qmax is linked by kinematics to the maximum transverse mo-mentum allowed to single soft-gluon emission in a given hard collision, and for this146 FIZIKA B (Zagreb) 17 (2008) 1, 143–150Achilli et al.: qcd contributions to the froissart bound for the total . . .purpose, we use an expression averaged over all parton densities [8],qmax(s) =√s2∑i,j∫(dx1/x1)∫(dx2/x2)1∫zmindzfi(x1)fj(x2)√x1x2(1 − z)∑i,j∫(dx1/x1)∫(dx2/x2)1∫zmindzfi(x1)fj(x2), (11)with zmin = 4p2t min/(sx1x2).The use of eikonal representation allows to consider multiple scattering. Ne-glecting the real part of the eikonal function, the expression for the total crosssection is given byσtot = 2∫d2b[1 − e−n(b,s)/2], (12)with n(b, s) = nsoft(b, s) + nhard(b, s) = nsoft(b, s) + ABN (b, s)σjet(s, pt min), denot-ing the average number of partonic collisions during the scattering. The hard term is40608010012014010 102103104√s ( GeV )σ tot(mb)G.G.P.S. model, PRD 72, 076001 (2005) using GRV and MRST P.D.F. G.G.P.S. model, using GRV P.D.F.σ0=48.0 mb p=0.75 ptmin=1.15Cudell et. al. hep-ph/0612046aLuna-Menon, hep-ph/0105076bc Block-Halzen PRD 73 054022 (2006)Donnachie-Landshoff, PRL B296 227 (1992)dDonnachie-Landshoff, PLB 595 393 (2004)DLhpbacdDLhpproton-antiprotonUA5UA1UA4CDFE710E811proton-protonFig. 1. Total cross-section obtained from our model [8] using different PDF’s [9],compared with data [10] and with other models [11].FIZIKA B (Zagreb) 17 (2008) 1, 143–150 147Achilli et al.: qcd contributions to the froissart bound for the total . . .evaluated using the expressions (7) and (8) for σjet and ABN , while the soft onehas a phenomenological parametrization,nsoft(b, s) = AsoftBN (b, s)σ0(1 + ǫ2√s), (13)where σ0 = constant is a parameter necessary to reproduce the right normalizationof the total cross-section, while ǫ = 0, 1 refer to pp and pp̄ scattering, while AsoftBN isparametrized as in Ref. [7].Figure 1 from Ref. [6] shows the range of values of the total cross-section ob-tained with this model using a set of phenomenological values for the parameterspt min, σ0 and p, and varying the parton densities.4. Soft gluons in the infrared limit and total cross-sectionAt very large energies, using the eikonal representation, the total cross-sectionin our model [7] readsσT (s) ≈ 2π∞∫0db2[1 − e−nhard(b, s)/2], (14)where nhard(b, s) = σjet(s)Ahard(b, s). We consider the asymptotic expression forσjet at high energies, which grows as a power of s: σjet(s) ≈ σ1(s/s0)ε. Ahard(b, s)is obtained through soft gluon resummation asAhard(b, s) ∝ e−h(b,s), (15)where h(b, s) is given by Eq. (4). If we take now the limit kt → 0, whered3n(k) ∝ αs(k2t ) and αs(kt) ≈ (Λ/kt)2p, with p < 1, the integral (4) givesh(b, s) ∝ (bΛ̄)2p. This implies Ahard(b) ∝ e−(bΛ̄)2pandnhard = 2C(s)e−(bΛ̄)2p , (16)where C(s) = A0(s/s0)εσ1. The resulting expression for σT isσT (s) ≈ 2π∞∫0db2[1 − e−C(s)e−(bΛ̄)2p](17)from which we deriveσT → [ε ln(s)](1/p) (18)The result shows explicitly that soft gluon resummation can restore the asymptoticgrowth of σT to lie within the Froissart bound. (Recall that 1/2 ≤ p < 1.)148 FIZIKA B (Zagreb) 17 (2008) 1, 143–150Achilli et al.: qcd contributions to the froissart bound for the total . . .5. ConclusionsWe have shown how soft gluon resummation acts as a saturation mechanism inthe rise of the total cross-section. This effect, embedded into the eikonal formalismwith QCD mini-jets to drive the rise, leads to asymptotic cross-sections whichsatisfy the Froissart bound.References[1] A. Corsetti, G. Pancheri, Y. N. Srivastava and A. Grau, Phys. Lett. 382 (1996) 282,[arXiv:hep-ph/9605314][2] G. Pancheri-Srivastava and Y. N. Srivastava, Phys. Rev. D 15 (1977) 2915.[3] Y. L. Dokshitzer, D. I. D’yakonov and S. I. Troyan, Phys. Lett. B 79 (1978) 269.[4] G. Parisi and R. Petronzio, Nucl. Phys. B 154 (1979) 427.[5] J. C. Collins and D. E. Soper, Phys. Rev. D 16 (1977) 2219.[6] A. Achilli, R. Hegde, R. M. Godbole, A. Grau, G. Pancheri and Y. Srivastava, Phys.Lett. B 659 (2008) 137; arXiv:0708.3626 [hep-ph].[7] R. M. Godbole, A. Grau, G. 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Primjenjujemo singularan ali integrabilan izraz za konstantujakog vezanja u infracrvenoj granici i usklad–ujemo njeno ponašanje s Froissartovomgranicom, dajući poseban fenomenološki primjer.150 FIZIKA B (Zagreb) 17 (2008) 1, 143–150

Doprinosi QCD-a Froissartovoj granici ukupnog udarnog presjeka

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