Three-quark force in p–p elastic scattering

Eur. Phys. J. C (2018) 78:1029 https://doi.org/10.1140/epjc/s10052-018-6435-3 Regular Article - Theoretical Physics Three-quark force in p–p elastic scattering M. A. Hassan1,a , A. A. E. Hefny2 , T. N. E. Salama1 1 Mathematics Department, Faculty of Science, Ain Shams University, Cairo, Egypt 2 Basic Science Department, Faculty of Computer and Information Sciences, Ain Shams University, Cairo, Egypt Received: 5 July 2018 / Accepted: 10 November 2018 © The Author(s) 2018 Abstract In the framework of Glauber optical limit approximation, considering the proton has an outer pion cloud of radius ∼ 0.87 f m and an inner core of radius ∼ 0.44 f m where the valence three quarks are confined, and including two-gluon exchange three-quark force, a good fit with the experimental data of p–p elastic scattering differential cross section up to q 2 ≈ 3 (GeV/c)2 , total cross section and the ratio of real to imaginary parts of elastic scattering amplitude in the forward direction is obtained at laboratory momenta 200, 290, 500, 1070 and 1500 GeV/c. The radii of two-quark interaction rt and three-quark force rth are calculated. The quant energy representing the gluon E g is evaluated. 1 Introduction Proton–proton scattering at very high energy taking into account the quark structure of proton located is becoming in the centre of the attention of researchers, especially with many measurements of proton–proton scattering cross sections. Different theoretical models are suggested to study different properties of proton structure and of proton–proton scattering mechanism. In most cases the phenomenological models with some physical picture of proton structure are used. Despite the availability of LHC data at very high energy [1–4], we will focus attention on the relatively low energy of FNAL and CERN-ISR data, where the centre of mass energy √ s = 19.42 − 53 GeV ( p L = 200 − 1500 GeV/c) [5–9]. One of the most prevalent models of proton–proton elastic scattering calculations at high energy is the Regge-pole theory with the Pomeron exchange. In general, in the framework of this theory, a good agreement with the proton–proton elastic scattering differential cross section, total cross section σt and the ratio ρ of real to imaginary parts of proton– proton elastic scattering amplitude in the forward direction was obtained at FNAL and CERN-ISR energy [10–14]. A similar fit with the experimental data of the same proton– proton quantities at the same energy was obtained by using many different other approaches, for example, the eikonalization approach [15], Bialas-Bzdak model [16], axiomatic quantum field theory [17] and others [18–22]. At higher energies of CDF, HERA and LHC the picture of the theoretical results of proton–proton elastic scattering may be a little different and we may need a new physics [23]. With Regge-pole theory, considering a parametrization of the total cross section σt and the ratio ρ with two Reggeons and four Pomeron contributions the authors in [24] cannot obtain √ a good agreement with the data at s = 13 TeV. However, in the framework of constituent quark model with pomeron exchange [25] a fit with the CDF and HERA data of proton– proton total cross section σt and elastic cross section σe was improved including the triple pomeron vertex and with the double pomeron exchange and the authors in [25] show that the radius of the constituent quark Rq ≈ 0.0624−0.0882 fm. With the same model including all possible quark interactions and taking quark-quark scattering amplitude in eikonal approximation a good fit with experimental data of proton– proton total cross section σt and elastic cross section σe at √ s = 23−1855 GeV was obtained [23]. The used quark radius to obtain the fit is Rq ≈ 0.0789 fm. At the same time, √ at s = 18000 GeV the authors found Rq ≈ 0.3−0.4 fm. In additive quark model with pomeron exchange approach the total cross section σt , differential cross section dσ dt [26] and the ratio ρ [27] for proton–proton elastic scattering at LHC energies are calculated. In this model with only four order of √ s = 7 TeV was interaction a good fit with data of dσ dt at obtained up to q 2 = 2.5 (GeV/c)2 with the quark radius Rq ≈ 0.44 fm. However, with the all order of quark-quark interactions the agreement with the data was observed up to q 2 = 0.4 (GeV/c)2 only. Something must be discussed with increasing of q 2 . At LHC energy, for σt a good fit was obtained [26], while the values of the ratio ρ were little below the data [27]. a e-mail: 0123456789().: V,-vol 123 1029 Page 2 of 8 Eur. Phys. J. C One of the successful models in the study of proton–proton scattering as composite systems is the multiple scattering theory of Glauber [28]. Using Glauber theory [29–31] the authors show that the multiple scattering mechanism with the quark degree of freedom describes the diffraction pattern of particle-particle elastic scattering and quarks having spatial dimensions which are small compared to the corresponding hadrons and a good agreement with the experimental data of high energy was obtained. With simple Gaussian form of proton wave function, by introducing the time-ordering effect in quark-quark multi-scattering the good agreement with the proton–proton data was obtained at CERN-ISR energies [35]. Different wave functions were used in the framework of Glauber model to describe the quark distribution in the proton and obtained a good fit with the experimental data at the CERN-ISR energies [31,32]. In [32], by introducing quark-quark short rang correlations in the wave function obtained the quark radius Rq ≈ 0.17 fm. Using the geometrical impact parameter representation [33] obtained Rq ≈ 0.15.5 fm. On the other hand, evaluations of the quark radius in the framework of Glauber theory at CERN-ISR energies are of order Rq ≈ 0.4 fm [34]. Thus, approximately same value of quark radius Rq at different energies, (CERN-ISR and LHC) may be related to different models and then different parametrization of quarkquark interaction. This interpret, also, the similar results of proton–proton cross sections in the framework of different models. The optical limit approximation in the framework of Glauber theory is not tested for proton–proton elastic scattering. Therefore, in this work, using this approximation, we consider the physical picture of proton as used in Ref. [36], where the proton has an outer pion cloud of radius ∼ 0.87 f m and an inner core of radius ∼ 0.44 f m where the valence three quarks are confined. With this picture of the proton, the proton–proton elastic scattering amplitude can be written as [36] T (q) = T0 (q) + F(q), (1) where T0 (q) represents the cloud effect, i.e., the interaction of incident proton pion cloud with the target proton pion cloud (soft Pomeron), and F(q) represents the contributions of quark–quark interaction in terms of quark-quark elastic scattering amplitudes (hard Pomeron). The expressions between brackets may explain the comm (...truncated)