Nuclear Slope Parameter of pp and pp Elastic Scattering in QCD Inspired Model

Commun. Theor. Phys. (Beijing, China) 49 (28) pp. 456 46 c Chinese Physical Society Vol. 49, No. 2, Feruary 15, 28 Nuclear Slope Parameter of pp and pp Elastic Scattering in QCD Inspired Model LU Juan, 1,2 MA Wei-Xing, 1,3 and HE Xiao-Rong 2 1 Collaoration Group of Hadron Physics and Non-perturative QCD Study, Guangxi University of Technology, Liuzhou 5456, China 2 Department of Physics, Guangxi University, Nanning 534, China 3 Institute of High Energy Physics, the Chinese Academy of Sciences, P.O. Box 918, Beijing 149, China (Received January 19, 27) Astract Based on the quark-gluon structure of nucleon and the possile existence of Odderon in strong interaction process due to gluon self-interaction, the elastic scatterings of pp and pp at high energies are studied. The contriutions from individual terms of quark-quark, gluon-gluon interactions, quark-gluon interference, and the Odderon terms to the nuclear slope parameter B(s) are analyzed. Our results show that the QCD inspired model gives a good fit to the LHC experimental data of the nuclear slope parameter. PACS numers: 25.4.cm, 24.85.+p, 12.4.Gg, 25.4.Ve Key words: nuclear slope parameter, QCD inspired model, Odderon 1 Introduction A microscopic understanding of hadron-hadron elastic scattering remains an elusive goal of hadron physics, and it offers an opportunity to search for new physics and new particles, such as quest for H-particle, Higgs, σ-particle, glueall, and Odderon. The concept of Odderon first emerged in 1973 in the context of asymptotic theorems. [1] Seven years later, it was possily connected with 3-gluon exchange in perturative QCD (pqcd), [2] ut it took 27 years to firmly rediscover it in the context of pqcd. [3] The Odderon was also rediscovered recently in the Color Glass Condensate (CGC) approach [4] and in the dipole picture, [5] one can therefore assert that the Odderon is a crucial test of QCD. The forward hadronic amplitude of pp and pp elastic scattering is reflected in three experimentally determined parameters: the total cross section σ tot, the ratio of the real to the imaginary part of forward scattering amplitude ρ value, and the nuclear slope parameter B. Measurements of B-values of pp and pp elastic scattering have a rich history. AGS and FNAL measured the value of B at < s < 3 GeV energy region. When the ISR was turned on in 1971, measurements of B-values of pp and pp elastic scattering have reached s = 52.8 GeV. The measurements of B in the energy range 5 < s < 2 GeV do not form a smooth set in s, unlike the situation for σ tot and ρ. The recent SPS measurement for the nuclear slope parameter B has risen to s = 54 GeV, eagerly awaited high energy collision at the CERN large hadron collider (LHC) [6] will give access not only yet unexplored small distances ut also simultaneously to large distances that were neither explored. Now, the measurements of the nuclear slope parameters B of elastic scattering have reached s = 2 TeV. Given the importance of the hadron-hadron elastic scattering, it is not surprising that a numer of models have een developed to address the dynamics of hadronhadron interaction. The eikonal models that are capale of descriing the data for non-zero transformed momenta were developed in Refs. [7] and [8]. Using an eikonal structure for the pp and pp scattering amplitude, Block and Kaidalov [9] have derived factorization theorem for pp and pp at high energies. The principles of analyticity, unitarity and crossing symmetries are truly fundamental to our understanding of particle physics. A requirement of analyticity is that the forward scattering amplitudes of pp and pp elastic nuclear scattering come from the same analytic function. Further, unitarity provides a relationship etween the total cross section and the imaginary part of the forward scattering amplitude, which h is the optical theorem. Cheng and Wu [1] proposed initially that eikonalization should properly unitarized Regge models. Now we used a QCDinspired, and eikonalized model to predict the nuclear slope parameter B at the energies of experimental data. Using the impact parameter representation, the QCDinspired model predicts the experimental data y understanding of elemental scattering theory from nonrelativistic quantum mechanics. In this paper, we ase on QCD inspired eikonal model to study slope function B(s). The conventional eikonal model is supplemented with a QCD motivated part consisting of three terms. [8] A calculation of the nuclear slope parameter B of pp and pp elastic scattering is performed in which the contriutions from individual terms of quarkquark, gluon-gluon interactions, quark-gluon interference and the Odderon term are included. In addition to the The project supported in part y National Natural Science Foundation of China under Grant Nos. 15651 and 16472 and the Natural Science Foundation of Guangxi Province of China under Grant Nos. 5752, 54242, and 4813 and Guangxi University under Grant No. X511

No. 2 Nuclear Slope Parameter of pp and pp Elastic Scattering in QCD Inspired Model 457 leading quark-quark contriution, the Odderon contriution is quite important. In particular, the Odderon plays an essential role in fitting to data. 2 Theory and Formalism We start our study y considering elastic scattering process a + a +, (1) in the s-channel with amplitude F a (s, t). The related elastic scattering process in the u-channel a + a + (2) with its amplitude F a (s, t), can e otained from the former, Eq. (1), y crossing to the u-channel, Eq. (2), via the crossing relation F a (s, t, u) = F a (u, s, t), (3) where s, t, u are the Mandestaim variales. It is important to notice the order of variales s, t, and u in oth sides of Eq. (3). Let us now define two amplitudes F ± y definition of F ± (s, t) = 1 2 [F a (s, t) ± F a (s, t)]. (4) Under the crossing from the s-channel process, Eq. (1), to u-channel, Eq. (2), the amplitude F + (s, t) evidently remains unchanged whereas the amplitude F (s, t) changes its sign. Accordingly, they are called even under-crossing and odd under-crossing amplitudes. We oserved that the amplitude F + (s, t) is the same for particle-particle and particle-antiparticle scattering and thus corresponds to an exchange of even (or positive) C-parity, C=+1. The amplitude F (s, t) changes its sign when going from the particle-particle to the particleantiparticle scattering process and can hence e understood to have odd (or negative) C-parity, C = 1. The amplitude F + (s, t) hence has vacuum quantum numers and it is dominated at high energy y the Pomeron exchange, which identifies as tensor glueall ξ(223) with quantum numers I G J PC = + 2 ++, mass of 2.23 GeV, and decay width Γ ξ = 1 MeV. [11] The dominant contriution to the amplitude F (s, t) comes from the Odderon, which is made up of three Reggeized gluons and is a colorless ound state of the three gluons. Therefore, studying the contriution from amplitude F (s, t) is equivalently to study the Odderon contriution to hadron-hadron scattering cross section at high energies. As an example, we study pp and pp elastic scattering at high energies. From definition of Eq. (4), we arrive at F ± (s, t) = 1 2 [F pp(s, t) ± F pp (s, t)]. (5) In turn, from Eq. (5) we have the following expressions for the pp and pp elastic scattering amplitude F pp (s, t) = F + (s, t) + F (s, t), (6) F pp (s, t) = F + (s, t) F (s, t). (7) The cross sections, the ratio of the real to the imaginary part and the nuclear slope parameters of the forward scattering amplitude can e normalized such that σ tot (s) = 1 Im F(s, t = ), s (8) Re F(s, t = ) ρ(s) = Im F(s, t = ), (9) dσ(s, t) = 1 dt 16πs 2 F(s, t) 2, (1) B(s) = d [ ln dσ ] dt dt (s, t) t= Equations (8) (11) are the fundamental formulae of our present study of pp and pp elastic scattering. 3 Description of F(s, t) in QCD-Inspired Model We introduce the eikonal formalism in the twodimensional transverse impact parameter space. We use the complex analytic eikonal profile functions, χ pp = χ even +χ odd and χ pp = χ even χ odd, even or odd under the transformation E = k 2 + m 2, where E is the proton laoratory energy and m is the proton mass. The data oth for pp and pp are fitted using the total complex analytic eikonal profile functions, i.e., phase transition, in terms of the even and odd forward scattering amplitudes F + and F (even and odd under crossing), which is related to the even eikonal profile function χ even and odd eikonal profile functions χ odd, respectively. We write the center-of-mass pp and pp forward scattering amplitude, [12] F c.m (s, t) = k e i q A(, s)d 2, (12) π where k is the momentum in the center-of-mass system, t = 2k 2 (1 cos θ) is the invariant four-momentum transfer. θ is the center-of-mass system scattering angle. q is a two-dimensional vector in the impact parameter space such that q 2 = t. A(, s) is the scattering amplitude in the impact parameter space. d 2 = 2πd. Our complex eikonal, χ = χ R + iχ I is defined. Therefore, the complex forward scattering amplitude in the impact parameter space is given y A(, s) = i 2 (1 eiχ ) = i 2 (1 e χ I(,s)+iχ R (,s) ). (13) Sustituting Eq. (13) into Eq. (12), we arrive at F c.m (s, t) = k e i q i π 2 (1 e χ I(,s)+iχ R (,s) )d 2. (14) 4 Slope Parameters B Using the impact-parameter amplitude, we can otain a physical picture of the slope parameters, [8] B(s) = d [ ln dσ ] dt dt (s, t), (15) t= which we often evaluate at zero momentum transfer B(s) = B(s, t = ), (16)

458 LU Juan, MA Wei-Xing, and HE Xiao-Rong Vol. 49 eginning with F c.m d 2 e i q A(, s). (17) We expand e i q at aout q = to find [ F c.m 1 + i q 1 2 ( q ) ] 2 + A(, s)d 2 (18) with this expansion and the definition of B we can eventually write the general expression for B as B = Re { da(, s) da (, s)} 2 da(, s) 2. (19) If the phase of A(, s) is independent of (This is the case when we either have a factorizale eikonal or an eikonal with a constant phase), this expression reduces to the more tractale form B = d 3 A(, s) 2 da(, s) = 2 A(, s)d 2 2 A(, s)d 2. (2) We note that B measures the size of the proton, i.e., B is related to the average value of the square of the impact parameters, weighted y A(, s). Introducing the eikonal, we find B = 1 (1 e χ I (,s)+iχ R (,s) ) 2 d 2 2 (1 e χ I (,s)+iχ R (,s) )d 2. (21) Equation (21) is the starting point of our study. Now, let us consider the quarks and gluons degrees of freedom to contriute to the nuclear slope parameters for proton and antiproton elastic scattering processes in QCD. 5 QCD-Inspire Model 5.1 Even Eikonal Profile Function The even QCD inspired eikonal profile function χ even for nucleon-nucleon scattering is given y the sum of three contriutions from gluon-gluon, quark-gluon, quark-quark sectors. They are individually factorizale into a product of a cross section σ(s) times an impact parameter space distriution function W(; µ), i.e χ even = χ gg (, s) + χ qg (, s) + χ qq (, s) = i[σ gg (s)w(; µ gg ) + σ qg (s)w(; µ qq µ gg ) + σ qq (s)w(; µ qq )]. (22) The factor i is inserted in Eq. (22) since the high-energy eikonal profile function is largely imaginary. σ ij are the cross sections of the colliding partons. The impact parameter space distriution functions used in Eq. (22) is taken to e convolutions of two dipole form factors, i.e., we parameterize W(; µ) as the Fourier transform of two dipole form factor of the nucleon, [13] W(; µ) = µ2 96π (µ)3 K 3 (µ), (23) where K 3 (x) is the modified Bessel function of the second kind. It is normalized so that W(; µ)d 2 = 1. Studying W(; µ) indicates that the total cross sections are essentially independent of the choice of form factor shape. As a consequence of oth factorization and the normalization chosen for the W(; µ), the following identity should hold χ even (s, )d 2 = i[σgg (s) + σ qg (s) + σ qq (s)]. (24) In the QCD inspired model, it allows one to reformulate the Froissart ounding in axiomatic field theory. We found that the total cross section contriuted from the gluongluon interaction term can e given asymptotically [8] y ( ε σ gg = 2π µ gg ) 2 log 2 s s. (25) The quark-quark interaction is simulated y a constant cross section plus a Regge-even falling down cross section. It can e approximated y ( σ qq = Σ gg C + C even m ), (26) s Regge where m is the threshold mass which is determined y experiment and it takes the value of m =.6 GeV. The contriution from quark-gluon interaction is also simulated y σ qg (s) = Σ gg C log qg log s s. (27) 5.2 Odd Eikonal Profile Function Similarly, the odd eikonal profile function is parameterized as χ odd (s) = iσ odd (s)w(, µ), (28) which accounts for difference etween pp and pp elastic scattering, and must vanish at high energies. A Regge ehaving analytic odd eikonal profile function can e parameterized as χ odd (, s) = σ odd W(; µ odd ) = C odd Σ gg m s W(; µ odd ), (29) where W(; µ odd ) is determined y experiment and the normalized constant C odd is to e fitted. The parameters and functions in the aove equations, Eqs. (25) (29), are given in Tale 1. However, the odd eikonal profile function is completely different from the even one y their physical meaning. Tale 1 Values of the prameters used in the fit. Fitted Fixed m =.6 GeV C = 5.65 ±.14 ɛ =.3 C log qg =.167 ±.37 µ qq =.89 GeV Σ gg = 9πα 2 s/m 2 µ gg =.73 GeV CRegge even = 25.3 ± 2. µ odd =.53 GeV C odd = 7.62 ±.28 α s =.5 s = 16.9 ± 4.9 GeV 2

No. 2 Nuclear Slope Parameter of pp and pp Elastic Scattering in QCD Inspired Model 459 6 Quark-Quark, Gluon-Gluon, Quark-Gluon Interference Term and Odderon Contriutions to Nuclear Slope Parameters From Eq. (22), we expand the factor of (1 e iχ ), 1 e iχ = iχ + 1 2 χ2 + 1 3 iχ3 + (3) Using Eq. (22), equations (23) (3) lead to B = 1 χ 2 d 2, (31) 2 χd 2 χ pp pp = χ even ± χ odd. (32) The eikonal profile function χ is given y four terms: gluon-gluon, quark-quark, quark-gluon interference and Odderon terms. So that the nuclear slope parameter B of pp and pp elastic scattering is divided into four terms. B pp pp = 1 [ 2 iχd 2 σ gg W(, µ gg ) 2 d 2 + σ qq W(, µ qq ) 2 d 2 + σ qg W(, µ gg µ qq ) 2 d 2 ± σ odd W(, µ odd ) 2 d 2 ]. (33) The nuclear slope parameters B from gluon-gluon contriution can e rewritten as σgg W(, µ gg ) 2 d 2 B gg = 2 iχd 2. (34) and from quark-quark contriution can e rewritten as σqq W(, µ qq ) 2 d 2 B qq = 2 iχd 2. (35) Similarly quark-gluon contriution is given y σqg W(, µ gg µ qq ) 2 d 2 B qg = 2 iχd 2. (36) Finally, Odderon contriution is σodd W(, µ odd ) 2 d 2 B odd = 2 iχd 2. (37) The Odderon (under crossing) forward scattering amplitude accounts for the difference etween pp and pp elastic scattering. It should e noticed that all the parameters in the aove equations given in Tale 1 are fixed y fitting experiments. From Eqs. (24) (28), 2 iχd 2 will e studied, 2 iχd 2 = 2(σgg + σ qq + σ qg ± σ odd ). (38) The sign + stands for the pp elastic scattering, and sign is for pp elastic scattering, respectively. 7 Numerical Calculations and Results Using the aove discussions and formulae, we can study the contriutions from quark-quark, gluon-gluon, quark-gluon, and Odderon terms to the nuclear slope parameters of the pp and pp elastic scattering. In this section we give our theoretical predictions and comparisons with experimental data availale at present. With Eq. (33), the numerical calculations of the total nuclear slope parameter are performed. The predictions are shown in Fig. 1. By the solid curve in Fig. 1, we plot the total nuclear slope parameter B of pp elastic scattering vs. energy s. The dotted curve in Fig. 1 is the result for pp elastic scattering. Clearly, we have gotten excellent fits to experimental data oth for pp and pp elastic scattering at high energies. Fig. 1 The total nuclear slope parameter B of pp and pp elastic scattering against s the center-of-mass energy in units of GeV. The solid line and up-triangle points stand for pp scattering and the dotted line and lack points are for pp scattering. Experimental data of total nuclear slope parameter are also shown with up-triangle points for pp elastic scattering while lack points are for pp elastic scattering. Fig. 2 The nuclear slope parameter B of pp elastic scattering. The solid curve is gluon-gluon contriution, the dished curve is quark-quark contriution, the dotted curve is quark-gluon interference term contriution, and Odderon contriution is shown y the up-triangle curve, which is negative.

46 LU Juan, MA Wei-Xing, and HE Xiao-Rong Vol. 49 We consider contriution from quark and gluon degree of freedom of QCD to the nuclear slope parameter of pp and pp elastic scattering. Because the eikonal profile function is consisting of four terms, the nuclear slope parameter B then correspondently contains four terms. With the Eqs. (34) (37), we get the four terms contriuting to the nuclear slope parameter B. The results are given in Figs. 1 3. Figure 1 shows our theoretical predictions of the energy dependence of B(s) and its comparison with experimental data. As is seen from Fig. 1, the agreement is good. Figure 2 shows is contriutions from quark-quark, gluon-gluon, quark-gluon, and Odderon terms to the nuclear slope parameter of pp elastic scattering. Fig. 3 The nuclear slope parameter B of pp elastic scattering. The solid curve is gluon-gluon contriution, the dished line is quark-quark contriution, the dotted line is quark-gluon interference term contriution, Odderon contriution is represented y the up-triangle curve which is positive. Figure 3 is contriutions from quark-quark, gluongluon, quark-gluon, and Odderon terms to the nuclear slope parameter of pp elastic scattering. Figure 3 shows that the gluon-gluon interaction term B gg rises quickly as energy increases. It makes dominant contriution to the total nuclear slope parameter B. The quark-quark term decreases as energy increasing and eventually ecomes zero. The quark-gluon interference term is negligile. For pp elastic scattering, the Odderon contriution is negative, and finally it ecomes zero at the extra high energy region. For pp elastic scattering, the Odderon contriution is positive, also decreases as energy increasing and eventually ecomes zero. It should e emphasized that since the Odderon, corresponding to the χ odd term, makes a great contriution to the total nuclear slope parameter B at low energies and causes the difference etween pp and pp elastic scattering, it should e considered in any calculations of scattering processes. 8 Summary and Concluding Remarks In this paper, we use QCD-inspired eikonal model to analyze the nuclear slope parameter B of pp and pp elastic scattering. The model simulates the fundamental theory of strong interaction QCD. In conclusion, as we have concluded in the calculations of the total cross section, we claim that the gluon-gluon interaction makes a dominant contriution to the protonproton and antiproton-proton elastic scatterings comparing with the quark-quark interaction and the quark-gluon interference contriution at high energies. Therefore, it must e considered in any theoretical understanding of hadron-hadron interacting processes at very high energy region. However, the most theoretical calculations did not consider the gluon-gluon contriutions. We also found that the Odderon plays a significant role in description difference etween pp and pp elastic scattering. We may quest for the existence of the Odderon in pp and pp elastic scattering processes. We will investigate this prolem in our coming work. References [1] L. Lukaszuk and B. Nicolescu, Nuovo Cim. Lett. 8 (1973) 45; J. Bartels, Nucl. Phys. Lett. B 175 (198) 365. [2] T. Jaroszewicz, Acta Phys. Polon. B 11 (198) 965; J. Kwiecinski and M. Praszalowicz, Phys. Lett. B 94 (198) 413. [3] J. Bartels, L.N. Lipatov, and G.P. Vaaca, Phys. Lett. B 477 (2) 178. [4] Y. Hatta, E. Iancu, K. Itakura, and L. Mclerran, Nucl. Phys. A 76 (25) 172, hep-ph/51171; S. Jeon and R. Venugopalan, Phys. Rev. D 7 (25) 1253, hepph/53219. [5] Y.V. Kovehegov, L. Szymanowski, and S. Wallon, Phys. Lett. B 589 (24) 267, hep-ph/39218. [6] A. Faus-Golfe, J. Velasco, and M. Haguenauer, hepex/1211. [7] P. Degrolard, M. Giffon, E. Martynov, and E. Predazzi, Eur. Phys. J. C 16 (2) 499. [8] M.M. Block, E.M. Gregores, F. Halzen, and G. Pancheri, Phys. Rev. Lett. D 6 (1999) 5424. [9] M.M. Block, F. Halzen, and G. Pancheri, hepph/11146. [1] H. Cheng and T.T. Wu, Phys. Rev. Lett. 24 (197) 1456. [11] Zhou Li-Juan, WU Qing, Ma Wei-Xing, and Gu Yun- Ting, Commun. Theor. Phys. (Beijing, China) 46 (26) 287. [12] M. Froissart, Phys. Rev. 123 (1961) 153. [13] M.M. Block and R.N. Cahn, Rev. Mod. Phys. 57 (1985) 563.