We predict pp elastic differential cross sections at LHC at c.m. energy 14
TeV and momentum transfer range |t| = 0 - 10 GeV*2 in a nucleon-structure
model. In this model, the nucleon has an outer cloud of quark-antiquark
condensed ground state, an inner shell of topological baryonic charge (r ~
0.44F) probed by the vector meson omega, and a central quark-bag (r ~ 0.2F)
containing valence quarks. We also predict elastic differential cross section
in the Coulomb-hadronic interference region. Large |t| elastic scattering in
this model arises from valence quark-quark scattering, which is taken to be due
to the hard-pomeron (BFKL pomeron with next to leading order corrections). We
present results of taking into account multiple hard-pomeron exchanges, i.e.
unitarity corrections. Finally, we compare our prediction of pp elastic
differential cross section at LHC with the predictions of various other models.
Precise measurement of pp elastic differential cross section at LHC by the
TOTEM group in the |t| region 0 - 5 GeV*2 will be able to distinguish between
these models.Comment: To be published in the Proceedings of the 12th International
  Conference on Elastic and Diffractive Scattering, DESY, Hamburg. Presented by
  M. M. Islam, May 200

Islam, M. M.

Kaspar, J.

Luddy, R. J.

English

arXiv

We predict pp elastic differential cross sections at LHC at c.m. energy 14 TeV and momentum transfer range |t| = 0 - 10 GeV*2 in a nucleon-structure model. In this model, the nucleon has an outer cloud of quark-antiquark condensed ground state, an inner shell of topological baryonic charge (r ~ 0.44F) probed by the vector meson omega, and a central quark-bag (r ~ 0.2F) containing valence quarks. We also predict elastic differential cross section in the Coulomb-hadronic interference region. Large |t| elastic scattering in this model arises from valence quark-quark scattering, which is taken to be due to the hard-pomeron (BFKL pomeron with next to leading order corrections). We present results of taking into account multiple hard-pomeron exchanges, i.e. unitarity corrections. Finally, we compare our prediction of pp elastic differential cross section at LHC with the predictions of various other models. Precise measurement of pp elastic differential cross section at LHC by the TOTEM group in the |t| region 0 - 5 GeV*2 will be able to distinguish between these models

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$p p$ Elastic Scattering at LHC in a Nucleon-Structure Model

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Islam, M.M.

Luddy, R.J.

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arXiv:0708.1156v1  [hep-ph]  8 Aug 2007Blois07/EDS07ProceedingsM. M. Islama, J. Kasparb and R. J. LuddycaDepartment of Physics, University of Connecticut, Storrs, CT 06269, USA, (islam@phys.uconn.edu)b TOTEM Collaboration, CERN, Geneva, Switzerland (Jan.Kaspar@cern.ch)Institute of Physics, Academy of Sciences of the Czech Republic, Praguec Department of Physics, University of Connecticut, Storrs, CT 06269, USA, (RJLuddy@phys.uconn.edu)pp Elastic Scattering at LHC in a Nucleon–Structure Model(Presented by M. M. Islam)AbstractWe predict pp elastic differential cross sections at LHC at c.m. en-ergy 14 TeV and momentum transfer range jtj= 0 – 10 GeV2 in anucleon-structure model. In this model, the nucleon has an outer cloudof quark-antiquark condensed ground state, an inner shell of topolog-ical baryonic charge (r ’ 0:44F ) probed by the vector meson !, anda central quark-bag (r ’ 0:2F ) containing valence quarks. We alsopredict d=dt in the Coulomb-hadronic interference region. Large jtjelastic scattering in this model arises from valence quark-quark scat-tering, which is taken to be due to the hard-pomeron (BFKL pomeronwith next to leading order corrections). We present results of takinginto account multiple hard-pomeron exchanges, i.e. unitarity correc-tions. Finally, we compare our prediction of pp elastic d=dtat LHCwith the predictions of various other models. Precise measurement ofpp d=dtat LHC by the TOTEM group in the jtjregion 0 – 5 GeV2will be able to distinguish between these models.High energy pp and pp elastic scattering have been at the forefront of accelerator researchsince the early seventies with the advent of CERN Intersecting Storage Rings (ISR) and mea-surement of pp elastic differential cross section in the c.m. energy rangeps= 23 – 62 GeV andmomentum transfer range jtj= 0.8 – 10 GeV2 [1]. This was followed by the Fermilab acceler-ator where pp elastic scattering was measured at c.m. energyps = 27.4 GeV in a fixed targetexperiment at large momentum transfers: jtj= 5.5 – 14 GeV2 [2]. Next came the CERN SPSCollider, where pp elastic scattering was measured at c.m. energies 546 GeV and 630 GeV – ajump of one order of magnitude in c.m. energy from ISR [3–5]. The Fermilab Tevatron followednext where pp elastic scattering was measured at c.m. energy 1.8 TeV, but in a rather small mo-mentum transfer range: jtj= 0 – 0.5 GeV2 [6, 7]. We are now at the threshold of a new periodof accelerator research with the LHC starting up soon and with the planned measurement of ppelastic scattering by the TOTEM group at c.m. energy 14 TeV and momentum transfer rangejtj’ 0 – 10 GeV2 [8, 9].My collaborators and I have been studying pp and pp elastic scattering since late seventies.From our phenomenological investigation, we have arrived at two results: 1) a physical pictureof the nucleon, 2) an effective field theory model underlying the physical picture [10]. Thephysical picture shows that the nucleon has an outer cloud, an inner shell of baryonic charge,and a central quark-bag containing the valence quarks (Fig. 1). The radius of the shell is about0.44 F and that of the quark-bag is 0.2 F. The underlying field theory model turns out to be agauged Gell–Mann–Levy linear -model with spontaneous breakdown of chiral symmetry andwith a Wess–Zumino–Witten (WZW) anomalous action. The model attributes the outer nu-cleon cloud to a quark–antiquark condensed ground state analogous to the BCS ground state insuperconductivity– an idea that was first proposed by Nambu and Jona-Lasinio. The WZW ac-tion indicates that the baryonic charge is geometrical or topological in nature, which is the basisof the Skyrmion model. The action further shows that the vector meson ! couples to this topo-logical charge like a gauge boson, i.e. like an elementary vector meson. As a consequence, onenucleon probes the baryonic charge of the other via !-exchange. In pp elastic scattering, in thesmall momentum transfer region, the outer cloud of one nucleon interacts with that of the othergiving rise to diffraction scattering. As the momentum transfer increases, one nucleon probes theother at intermediate distances and the !-exchange becomes dominant. At momentun transferseven larger, one nucleon scatters off the other via valence quark-quark scattering.Our calculated pp elastic d=dt at c.m. energy 14 TeV is shown in Fig. 2 by the solidline that includes all three processes: diffraction, !-exchange, and qq scattering. The dottedcurve shows d=dt due to diffraction only. We see that diffraction dominates in the small jtjregion, but falls off rapidly. The dot-dashed curve shows d=dt due to !-exchange only andindicates that !-exchange dominates in the jtjregion 1.5 – 3.5 GeV2. Beyond that, the valencequark-quark scattering takes over. The dashed curve for jtj> 3.5 GeV2 represents d=dtwithsingle valence quark-quark scattering, whereas the solid curve represents d=dtwith all multiplevalence quark-quark scattering.Let us next examine how the three processes are described in our calculations [10]. Diffrac-tion is described by using the impact parameter representation and a phenomenological profilefunction:TD (s;t)= ipWR10bdb J0(bq) D (s;b); (1)q is the momentum transfer (q = p jtj) and  D (s;b)is the diffraction profile function, which isrelated to the eikonal function D (s;b):  D (s;b)= 1  exp(iD (s;b)). We take  D (s;b)to bean even Fermi profile function: D (s;b)= g(s)[11+ e(b  R )=a)+ 11+ e  (b+ R )=a  1]. (2)The parameters R and a are energy dependent: R = R 0 + R 1(lns  i2 ), a = a0 + a1(lns i2); g(s)is a complex crossing even energy-dependent coupling strength.The diffraction amplitude we obtain has the following asymptotic properties:1. tot(s) (a0 + a1lns)2 (Froissart-Martin bound)2. (s)’ a1a0+ a1 lns(derivative dispersion relation)3. TD (s;t) is ln2sf(jtjln2s) (Auberson-Kinoshita-Martin scaling)4. T ppD(s;t)= TppD(s;t) (crossing even)Incidentally, the profile function (2) has been used by Frankfurt et al. to estimate the ab-sorptive effect of soft hadronic interactions (gap survival probability) in the central production ofHiggs at LHC [11].The !-exchange amplitude in our model has the formT!(s;t)  exp[iD (s;0)]sF 2(t)m 2!  t. (3)where the factor s shows that ! is behaving like an elementary spin-1 boson. The two formfactors indicate that ! is probing two baryonic charge distributions – one for each nucleon. Thefactor exp[iD (s;0)]represents the absorptive effect due to soft hadronic interactions.We view large jtjelastic scattering as a hard collision of a valence quark from one protonwith a valence quark from the other proton (Fig. 3). Since this process involves high energyquark-quark scattering at large momentum transfer, one would expect that it should be describedby perturbative QCD. In fact, in perturbative QCD, the two quarks would interact via BFKLpomeron, that is, reggeized gluon ladders with rungs of gluons that lead to a fixed branch point inthe angular momentum plane at BFK L = 1 + !. The value of ! in next to leading order lies in therange 0.13 – 0.18 as argued by Brodsky et al. [12]. We refer to the BFKL pomeron with next toleading order corrections included as the “hard-pomeron”. In our calculations, we approximatethe hard-pomeron by a fixed pole and take the corresponding qq amplitude as [13]T^1(^s;t)= iqq s^s^ e  i2!1jtj+ r  20, (4)where s^ is the square of the c.m. energy of qq scattering.Our pp elastic d=dtcalculation at 14 TeV reported earlier [10,13] included only a singlehard-pomeron exchange in qq scattering. However, Eq. (4) shows that the hard-pomeron predictsa qq asymptotic total cross section ^tot(^s)/ s^! , i.e. ^tot(^s)grows like a power of s^and thereforeviolates unitarity and the Froissart-Martin bound. To restore unitarity in the qq channel, we usethe eikonal representation and write the full qq scattering amplitude asT^ (^s;t)= i^pW^R10bdb J0(bq)h1   ei^(^s;b)i. (5)Taking T^1(^s;t)in Eq. (4) as the Born or single-scattering amplitude, we obtain by inverting it^(^s;b)= 2 iqqs^ e  i2!K 0(br0). (6)Expanding the exponential in (5), we getT^ (^s;t)=   ip^ W^R10bdb J0(bq)hi^  ^22!  i^33!+^44!+ :::i. (7)The nth term in the series isT^n(^s;t)=   i(  1)n 2n  1n!nqq s^s^ e  i2n! R10bdb J0(bq)Kn0br0. (8)Now, s^’ xx0s, where x and x0are the longitudinal momentum fractions of the protons carried bythe valence quarks (Fig. 3). This leads to the following pp elastic amplitude due to qq scattering:Tqq(s;t)= T^1(s;t)F21(q? )+ T^2(s;t)F22(q? )+ :::+ T^n(s;t)F2n(q? ); (9)F1; F2 ;:::Fn are the structure factors that take into account momentum distributions of thevalence quarks inside the proton. Our earlier calculation kept only the first term in Eq. (9). Fig.2 shows that the effect of multiple hard-pomeron exchange in pp scattering is to decrease d=dtat large jtjcompared to d=dtdue to single hard-pomeron exchange (dashed line in Fig. 2). Forjtj< 3.5 GeV2, there is little effect due to multiple scattering, i.e. unitarization.Results of some of our quantitative calculations are shown in Figs. 4–7. The solid curve inFig. 4 represents our calculated total cross section as a function ofps. Dotted curves representthe error band given by Cudell et al. [14]. In Fig. 5, solid and dashed curves represent ourcalculated pp and pp respectively ( = Re T(s;0)=Im T(s;0)). Dotted curves, as before,represent the error band given by Cudell et al. Atps=14 TeV, our values of tot and pp are109.4 mb and 0.12 respectively. Fig. 6 shows our calculated d=dt for pp elastic scatteringatps = 541 GeV in the Coulomb–hadronic interference region using the Kundra´t–Lokajı´cekformulation (upper curve) and West–Yennie formulation (lower curve). Experimental data arefrom Augier et al. [15]. Fig. 7 shows our predicted d=dt for pp elastic scattering at p s = 14TeV in the Coulomb–hadronic interference region. Finally, in Fig. 8, we compare our predictedpp elastic d=dtat LHC with the predictions of other models proposed by various groups: Avilaet al. [16], Block et al. [17], Bourrely et al. [18], Desgrolard et al. [19], and Petrov et al. (threepomeron) [20].Conclusions1. Precision measurement of pp elastic d=dtat LHC by the TOTEM group in the region jtj= 0– 5 GeV2 will be able to distinguish between various proposed models (see Fig. 8).2. In our nucleon-structure model, the qualitative saturation of the Froissart-Martin bound is dueto soft hadronic interactions.3. Large jtjelastic scattering in our model is due to valence quark-quark scattering. This hasbeen described by us as due to the exchange of a hard-pomeron (BFKL pomeron plus next toleading order corrections).4. Unitarization of the hard-pomeron exchange leads to a decrease of d=dtat large jtj, but haslittle effect on forward d=dt.5. The nucleon structure that we find embodies salient features of many leading models– suchas Nambu-Jona-Lasinio model, Skyrmion model, nonlinear -model, chiral-bag model– but, atthe end, it presents a unique description of the nucleon.References[1] E. Nagy et al., Nucl. Phys. B 150, 221 (1979).[2] W. Faissler et al., Phys. Rev. D 23, 33 (1981).[3] UA4 Collaboration, M. Bozzo et al., Phys. Lett. B 147, 385 (1984).[4] UA4 Collaboration, M. Bozzo et al., Phys. Lett. B 155, 197 (1985).[5] D. Bernard et al., Phys. Lett. B 171, 142 (1986).[6] N. Amos et al., Phys. Lett. B 247, 127 (1990).[7] F. Abe et al., Phys. Rev. D 50, 5518 (1994).[8] TOTEM Collaboration, Technical Design Report CERN-LHCC-2004-002.[9] TOTEM Collaboration, CERN-LHCC-2007-013/G130.[10] M. M. Islam, R. J. Luddy, and A. V. Prokudin, Int. J. Mod. Phys. A 21, 1 (2006).[11] L. Frankfurt, M. Strickman, C. Weiss, and M. Zhalov, Czech. J. Phys. 55, B675 (2005).[12] S. Brodsky, V. S. Fadin, V. T. Kim, L. N. Lipatov, and G. B. Pivovarov, JETP Lett. 70, 155 (1999).[13] M. M. Islam, R. J. Luddy, and A. V. Prokudin, Phys. Lett. B 605, 115 (2005).[14] J. R. Cudell et al., Phys. Rev. Lett. 89, 201801 (2002).[15] C. Augier et al., Phys. Lett. B 316, 498 (1993).[16] R. F. Avila, S. D. Campos, M. J. Menon, and J. Montanha, Eur.Phys.J. C 47, 171 (2006).[17] M. M. Block, E. M. Gregores, F. Halzen, and G. Pancheri, Phys. Rev. D 60, 054024 (1999).[18] C. Bourrely, J. Soffer, and T. T. Wu, Eur. Phys. J. C 28, 97 (2003).[19] P. Desgrolard, M. Giffon, E. Martynov, and E. Predazzi, Eur. Phys. J. C 16, 499 (2000).[20] V. Petrov, E. Predazzi, and A. V. Prokudin, Eur. Phys. J. C 28, 525 (2003).10-1110-1010-910-810-710-610-510-410-310-210-111010 210 30 1 2 3 4 5 6 7 8 9 10|t| (GeV2)Differential cross section ds/dt (pp) (mb/GeV2 )diffraction onlyw  amplitude onlyquark-quark (single)quark-quark (unitarized)pp     14 TeVdiffraction onlyw  onlysingleunitarizedFig. 1 Fig. 2Fig. 302040608010012014016010 102 103 104 105√s— (GeV)Total cross sections s(pp and pp–) (mb)pp–pp-0.4-0.200.20.410 102 103 104 105√s— (GeV)rpp–pppp–ppFig. 4 Fig. 5)2-t   (GeV0 0.005 0.01 0.015 0.02  2GeVmb    dtsd210·2210·3210·4210·5)2-t   (GeV0 0.002 0.004 0.006 0.008 0.01  2GeVmb    dtsd210·5210·6210·7210·8210·9310Fig. 6 Fig. 710-1310-1210-1110-1010-910-810-710-610-510-410-310-210-111010 210 30 1 2 3 4 5 6 7 8 9 10|t| (GeV2)Differential cross sections ds/dt (pp) (mb/GeV2 )Avila et al.Block et al.Bourrely et al.Desgrolard et al.Islam et al.Petrov et al.pp   14 TeVFig. 8

pp Elastic Scattering at LHC in a Nucleon-Structure Model

pp Elastic Scattering at LHC in a Nucleon-Structure Model

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