Ratio of Real to Imaginary for pp and pp Elastic Scatterings in QCD Inspired Model

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1 Coun. Theor. Phys. Beijing, China pp c International Acadeic Publishers Vol. 47, No. 3, March 15, 2007 Ratio of Real to Iaginary for pp and pp Elastic Scatterings in QCD Inspired Model LU Juan, 1,2 MA Wei-Xing, 1,3 and HE Xiao-Rong 2 1 Collaboration Group of Hadron Physics and Non-perturbative QCD Study, Guangxi University of Technology, Liuzhou , China 2 Institute of Physics Science and Engineering Technology, Guangxi University, Nanning , China 3 Institute of High Energy Physics, the Chinese Acadey of Sciences, Beijing , China Received April 28, 2006 Abstract We use the QCD inspired odel to analyze the ratio of the real to the iaginary for pp and pp elastic scatterings. A calculation for the ratio of the real to the iaginary is perfored in which the contributions fro gluongluon interaction, quark-quark interaction, quark-gluon interaction, and eikonal profile function are included. Our results show that the QCD inspired odel gives a good fit to the LHC experiental data. PACS nubers: p, Gg, Ve Key words: QCD inspired odel, eikonal profile function, ratio of real to iaginary 1 Introduction Measureents of ρ-values ratio of real to iaginary for pp and pp elastic scatterings have had a rich history. AGS and FNAL had been easured the value of, at 0 < < 40 GeV energy region. When the ISR was turned on in 1971, soe of the first experients done were an elastic scattering easureent of the ratio of the real to the iaginary ρ for pp and pp elastic scatterings at 30 GeV < s < 70 GeV energy region by CERN Roe group. Eagerly awaited high energy collision at the CERN large hadron collider LHC 1] will give access not only yet unexplored sall distances but also siultaneously to large distances that were neither explored. Now, the easureents of the ratio of the real to the iaginary ρ for pp and pp elastic scatterings have reached = 2 TeV at LHC. Recently soe odels with ulti-poeron structures were proposed, 2 4] soe of these 2,3] used Born aplitudes with two Poeron structures as single 2] or double poles. 3] The foral violation of the Froissart Martin bound in soe of these odels is considered as practically negligible, though in ters of particle-wave aplitudes unitarity violation is flagrant at present energies. Soe of these 5] used eikonal odel with three Poerons. The odel shows quite a good agreeent with LHC data. But the odel is polypoeron hypothesis what happens if one adits a fourth ect. Moreover, the origin and nature of the Poeron has not been known so far and the Poeron coupling to the proton has a vector for, γ µ, siilar to that of C = +1 isoscalar photon, which is in contradiction to the vacuu property of the Poeron. The principles of analyticity, unitarity and crossing syetry are truly fundaental to our understanding of particle physics. A requireent of analyticity is that the forward scattering aplitudes for elastic nuclear scattering coe fro the sae analytic function. Further, unitarity provides a relationship between the al cross section and the iaginary part of the forward scattering aplitude the optical theore. Cheng and Wu 9] proposed initially that eikonalization should properly unitarize odels. Now we used a QCD-inspired, and eikonalized odel to predict the ratio of the real to the iaginary at the energies of LHC. Using the ipact paraeter representation, the QCD-inspired odel predicts the experiental data by understanding of eleental scattering theory fro non relativistic quantu echanics. The eikonal odels that are capable of describing the data for non-zero transferred oenta are developed in Refs. 6] and 7], using an eikonal structure for the pp and pp scattering aplitude, Block and Kaidalov 8] have derived factorization on theore for pp, pp at high energies. In this paper, we will base on QCD inspired eikonal odel. Fro the conventional eikonal is suppleented with a QCD otivated part consisting of three ters. 7] A calculation for the ratio of the real to the iaginary ρ for pp and pp elastic scatterings is perfored in which the contributions fro gluon-gluon, quark-quark and gluonquark interactions are included. 2 Eikonal Model We introduce the eikonal forulis in the twodiensional transverse ipact paraeter space b, we use the coplex analytic eikonals, χ pp = χ even χ and χ pp = χ even +χ. even or under the transforation E E, E = k where E is the proton laboratory energy and is the proton ass. The data both for pp and pp are fitted using the al coplex analytic eikonal profile functions, i.e. phase transition, in ters of The project supported in part by National Natural Science Foundation of China under Grant Nos and and the Science Foundation of Guangxi Province of China under Grant Nos , , and

2 No. 3 Ratio of Real to Iaginary for pp and pp Elastic Scatterings in QCD Inspired Model 551 the even and forward scattering aplitudes f + and f even and under crossing], which is related to the even eikonal profile function χ even and eikonal profile functions χ respectively. We write the center-of-ass pp and pp forward scattering aplitude, 10] f c. s, t = k e i q b ab, sd 2 b, 1 π where k is the oentu in center-of-ass syste, t = 2k 2 1 cos θ is the invariant four-oentu transfer. θ is the center-of- ass syste scattering angle. q is a twodiensional vector in the ipact paraeter space b such that q 2 = t, where ab, s is the scattering aplitude in ipact paraeter space, d 2 b = 2πbdb. Our coplex eikonal, χ = χ R + iχ I, is defined. So that the coplex forward scattering aplitude in ipact paraeter space b is given by ab, s = i 2 1 e iχ = i 2 1 e χ I b,s+iχ R b,s. 2 Substituting Eq. 2 into Eq. 1, we arrive at f c. s, t = k e i q b i π 2 1 e χ I b,s+iχ R b,s d 2 b, 3 f c. s, t = 0 = k i π 2 1 e χ I b,s+iχ R b,s d 2 b. 4 To calculate ρ, the ratio of the real to the iaginary part of the forward pp and pp scattering aplitude, we write ρs = Re {f c.s, 0} I {f c. s, 0} = Re {i 1 e χ I b,s+iχ R b,s d 2 b} I {i 1 e χ I b,s+iχ R b,s d 2 b}. 5 Fro Eq. 5, 1 e iχ iχ, so that equation 5 becoes = Re { χ R b, s + iχ I b, sd 2 b} I { χ R b, s + iχ I b, sd 2 b}. 6 Because of χb, s = χ R b, s + iχ I b, s, so that ρ = Re { χb, sd 2 b} I { χb, sd 2 b}. 7 The al eikonal profile functions for pp and pp elastic scattering are defined as χ pp = χ even χ and χ pp = χ even + χ. The ratio of the real to the iaginary for pp and pp elastic scattering can be written, = Re { χ even b, s χ b, sd 2 b} I { χ even b, s χ b, sd 2 b}, 8 = Re { χ even b, s + χ b, sd 2 b} I { χ even b, s + χ b, sd 2 b}. 9 Using the optical theore, we can write the al cross section σ in the center-of-ass syste as σ = 4π k I {f c.s, t = 0}. 10 Substituting Eq. 4 into Eq. 10, we arrive at { } σ = 2 I χ even b, s + χ b, sd 2 b. 11 Substituting Eq. 11 into Eqs. 8 and 9, the ratios of the real to the iaginary for pp and pp elastic scattering becoe = 2 Re { χ even b, s χ b, sd 2 b} = 2 Re { χ even b, s + χ b, sd 2 b}, Equations 12 and 13 are starting point of our study. Now, let us consider the quarks and gluons degrees of freedo to contribute to the ratios of the real to the iaginary for proton and antiproton elastic scattering processes in QCD. 3 QCD Inspire Model 3.1 Even Eikonal Profile Function The even QCD inspired eikonal profile function χ even for nucleon-nucleon scattering is given by the su of three contributions fro gluon-gluon, quark-gluon, quark-quark sectors. They are individually factorizable into a product of a cross section σs ties an ipact paraeter space distribution function W b; µ, i.e., χ even = χ gg b, s + χ qg b, s + χ qq b, s = iσ gg sw b; µ gg + σ qg sw b; µ qq µ gg + σ qq sw b; µ qq ]. 14 The factor i is inserted in Eq. 14 since the high energy eikonal profile function is largely iaginary. σ ij are the cross section of the colliding partons. The ipact paraeter space distribution functions used in Eq. 14 were taken to be convolutions of two dipole for factors, i.e. we paraeterize W b; µ as the Fourier transfor of two dipole for factors of the nucleon, 11] W b; µ = µ2 96π bµ3 K 3 bµ, 15 where K 3 χ is the odified Bessel function of the second kind. It is noralized so that W b; µd 2 b = 1. Studying wb; µ indicates that the al cross sections are essentially independent of the choice of for factor shape. As a consequence of both factorization and the noralization chosen for the W b; µ, the following identity should be held χ even s, bd 2 b = iσgg s + σ qg s + σ qq s]. 16 In the QCD inspired odel, it allows one to reforulate the Froissart bounding in axioatic field theory. We found that the al cross section contributed fro the gluongluon interaction ter can be given asypically 7] by ε σ gg = 2π µ gg 2 log 2 s s 0. 17

3 552 LU Juan, MA Wei-Xing, and HE Xiao-Rong Vol. 47 The quark-quark interaction is siulated by a constant cross section plus a -even falling down cross section. It can be approxiated by σ qq = Σ gg C + C even s is the threshold ass, which is deterined by experient and takes the value of = 0.6 GeV. The contribution fro quark-gluon interaction is also siulated by σ qg s = Σ gg C log qg log s s For large s, the even aplitude in Eq. 14 is ade analytic by the substitution s s e iπ/2. So that gluongluon, quark-gluon, quark-quark, eikonal profile function contributions for the al cross section and the ratio of the real to iaginary can be rewritten as ε 2 σ gg = 2π log 2 s 4 iπ 2 ε 2 s log, 20 σ qq = Σ gg C + C even ] + iσ gg C even sin, 21 s 4 σ qg s = Σ gg Cqg log log s iσ gg C log π qg s The values of all paraeters are given in Table 1 and are used coonly. Table 1 Fitted Values of the paraeters used in the fit. Fixed = 0.6 GeV C = 5.65 ± 0.14 ɛ = 0.3 Cqg log = ± µ qq = 0.89 GeV Σ gg = 9πα 2 s /2 0 µ gg = 0.73 GeV C even = 2.53 ± 0.2 µ = 0.53 GeV C = 7.62 ± 0.28 α s = 0.5 s 0 = 1.0 GeV Odd Eikonal Profile Function The eikonal profile function, χ b, s = σ W b; µ, accounts for difference between pp and pp elastic scatterings, and ust vanish at high energies. A behaved analytic eikonal profile function can be paraetrized as follows: χ b, s = σ W b; µ = C Σ gg W b; µ, 23 the aplitude in Eq. 23 is ade analytic by the substitution s s e iπ/2. So that χ b, s = C Σ gg + ic Σ gg in. 24 s 4 The paraeters and functions in the above equations, equations 20 24, are given in Table 1, and explained in the sae way as before. However, the eikonal profile function is copletely different fro the even one. 4 gg, qq, qg and Odd Profile Function Contributions for Ratio of Real to Iaginary The eikonal profile function is given by four ters: gluon-gluon ter, quark-quark ters, quark-gluon ters and eikonal profile function ter. So that we consider the ratio of the real to the iaginary for pp and pp elastic scatterings is divided into four ters, = 2σ gg + σ qg + σ qq σ ε 2 = 4π log 2 s 4 + 2Σ gg C + C even ] + 2Σ gg Cqg log log s 2C Σ gg in, 25 s 0 s 4 = 2σ gg + σ qg + σ qq + σ ε 2 = 4π log 2 s 4 + 2Σ gg C + C even ] + 2Σ gg Cqg log log s + 2C Σ gg in. 26 s 0 s 4 The ratio of the real to the iaginary fro gluon-gluon contribution can be rewritten, gg = 2π2 ε/µ gg 2 logs/s 0, 27 gg = 2π2 ε/µ gg 2 logs/s The ratio of the real to the iaginary fro quark-quark qq = 2Σ ggc even / sinπ/4, 29 qq = 2Σ ggc even / sinπ/4. 30 The ratio of the real to the iaginary fro quark-gluon qg π/2 qg = 2Σ ggc log, 31 qg = 2Σ ggcqg log π/2. 32 The ratio of the real to the iaginary fro eikonal = 2C Σ gg / cosπ/4, 33

4 No. 3 Ratio of Real to Iaginary for pp and pp Elastic Scatterings in QCD Inspired Model 553 = 2C Σ gg / cosπ/4. 34 The under crossing forward scattering aplitude accounts for the difference between pp and pp elastic scatterings, respectively. The ratio of the real to iaginary for pp and pp elastic scatterings fro eikonal contribution also accounts for difference of pp and pp. It should be noticed that all the paraeters in the above equations given in Table 1 are fixed by fitting experients. Now, we can plot gg, qq, qg, against the energy, the theoretical curves are given in Fig. 1, respectively, for proton-proton elastic scattering. Fro Fig. 1, we see that the gluon-gluon interaction contribution asypically grows as logs/s 0, when we go up in energy, we can see that the gluon-gluon interaction ter doinates the other three ter contributions. Both quark-quark interaction contribution and eikonal profile function contribution contained factor of 1/, because of the different fitting paraeters, we can see that the eikonal profile function contribution goes up ore quickly than the quark-quark interaction contribution. Finally they becoe zero at the high energies region. The quark-gluon interaction contribution actually is a constant ters C, and alost equal to zero at the whole energy region. quark-gluon interaction contribution. On the contrary, eikonal profile function contribution to pp is positive. The forward scattering aplitude accounts for the difference between the pp and pp elastic scattering. Fig. 2 The ratio of the real to iaginary ρ for pp elastic scattering. The solid curve is gluon-gluon contribution, the dotted-dashed curve is quark-quark contribution, the dashed curve is quark-gluon contribution and eikonal profile function contribution is the dotted curve and it is positive. All the behaviors of individual ters tell us that the gluon-gluon interaction contribution doinates quarkquark interaction ter, quark-gluon interference ter and eikonal profile function contributions. Fig. 1 The ratio of the real to iaginary ρ for pp elastic scattering. The solid curve is gluon-gluon contribution, the dotted-dished curve is quark-quark contribution, the dashed curve is quark-gluon contribution, and eikonal profile function contribution is the dotted curve and it is negative. We can plot gg, qq, against the energy, the theoretical curves are given in Fig. 2, respectively, for antiproton-proton elastic scattering. We see that the ratio of the real to the iaginary for pp elastic scattering fro the gluon-gluon interaction contribution, quark-quark interaction contribution, quark-gluon interaction contribution is asypically equal to the ratio of the real to the iaginary fro the gluon-gluon interaction contribution, quark-quark interaction contribution, qg, 5 Predictions for Total Ratio of Real to Iaginary Our theoretical predictions for the al ratio of the real to iaginary of pp and pp elastic scattering at high energies are given by the following forulae: = gg + = 1 qq + qg 2π 2 ε 2 s log 2Σ gg C even + πσ gg C log qg 2C Σ gg cos = gg + = 1 qq + qg 2π 2 ε 2 s log 2Σ gg C even + πσ gg C log qg + 2C Σ gg cos in s 4 π ], 35 4 π in s 4 π ] With Eqs. 35 and 36, the nuerical calculations of al ratio of the real to the iaginary are perfored. The predictions are shown in Fig. 3. By the lower curve of Fig. 3, we plot the al ratio of the real to the iaginary of pp elastic scattering vs. energy s. The up curve in Fig. 3 is the result for pp elastic scattering. Clearly, we

5 554 LU Juan, MA Wei-Xing, and HE Xiao-Rong Vol. 47 have gotten excellent fits to experiental data both for pp and pp elastic scatterings at high energies. Fig. 3 The al ratio of the real to iaginary for pp and pp scattering against the center-of-ass energy in GeV. The solid line and circles points are for pp scattering and the dotted-dashed line and square black points are for pp scattering. The points on the figure are experiental data of al ratio of the real to iaginary at FNAL and LHC. 6 Conclusion In this paper, we used QCD-inspired eikonal odel to analyze the ratio of the real to the iaginary for pp and pp elastic scatterings, which is based on fundaental theory of strong interaction QCD. We consider contribution fro quark and gluon degrees of freedo of QCD to the ratio of the real to the iaginary for pp and pp elastic scatterings. Because the eikonal is consisting of four ters, the ratio of the real to the iaginary ρ includes four ters. The gluon-gluon ter ρ gg, which rises as logs/s 0 s 0 is an energy scale paraeters doinates the other ters at the whole energies. It akes doinant contribution to the al ratio of the real to the iaginary ρ. The quark-quark ter contains a factor 1/, it goes up slowly with energy increasing and eventually becoes zero. The quark-gluon ter is equal to zero. In other words, it is no contribution to the al ratio of the real to the iaginary ρ. The eikonal profile function ter also contains a factor 1/, but it is negative for the ratio of the real to the iaginary, it rises slowly with energy increasing. The eikonal profile function ter for the ratio of the real to the iaginary is positive and eventually becoes zero. In conclusion, we clais that gluon-gluon interaction akes doinant contributions to the al ratio of the real to the iaginary ρ for pp and pp elastic scattering, as we have done to the al cross section. References 1] A. Faus-Golfe, J. Velasco, and M. Haguenauer, hepex/ ] P. Gayron and B. Nicolescu, Phys. Lett. B , hep-ph/ ] K. Kontros, A. Lengyel, and Z. Tarics, hep-ph/ ] A. Donnachie and P.V. Landshoff, Phys. Lett. B ] V.A. Petrov and A.V. Prokudin, Eur. Phys. J. C ] P. Degrolard, M. Giffon, E. Martynov, and E. Predazzi, Eur. Phys. J. C ] M.M. Block, E.M. Gregores, F. Halzen, and G. Pancheri, Phys. Rev. Lett. D ] M.M. Block, F. Halzen, and G. Pancheri, hepph/ ] H. Cheng and T.T. Wu, Phys. Rev. Lett ] M.M. Block and R.N. Cahn, Rev. Mod. Phys ] D. Cline, F. Halzen, and J. Luthe, Phys. Rev. Lett ; P. L Heureux, B. Margolis, and P. Valin, Phys. Rev. D ; L. Durand and H. Pi, Phys. Rev. Lett ; Phys. Rev. D ; V. Innocente, A. Capella, and J.T.T. Van, Phys. Rev. B ; B. Margolis, et al., ibid ; B.Z. Kopeliovich, N.N. Nikolaev, and I.K. Potashnikova, Phys. Rev. D ; J.C. Collins and G.A. Ladinsky, ibid