J/ψ production in p+p and p+Pb collisions at ultrarelativistic energies

J/ψ production in p+p and p+Pb collisions at ultrarelativistic energies 1 corresp first-author Corresponding author, whmw@sina.com Corresponding author, whmw@sina.com whmw@sina.com 2 1 3 Physics Department, Academy of Army Armored Forces of PLA, Beijing 100072, China Physics Department, Academy of Army Armored Forces of PLA, Beijing 100072, China Physics Graduate School, Shijiazhuang Railway Institute, Shijiazhuang 050043, China Physics Graduate School, Shijiazhuang Railway Institute, Shijiazhuang 050043, China Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China The production of J/ψ mesons in p+p and p+Pb collisions is studied in the framework of Color Glass Condensate together with a simple color evaporation model. Considering the nuclear effects with the Glauber-Gribov approach, we calculate the cross section and the nuclear modification factor of forward J/ψ production in p+Pb collisions at <math id="M1"><mrow><msqrt><mi>s</mi></msqrt><mo>=</mo><mn>5.02</mn></mrow></math> TeV. Then, the backward J/ψ production in p+Pb collisions at <math id="M2"><mrow><msqrt><mi>s</mi></msqrt><mo>=</mo><mn>8.16</mn></mrow></math> TeV is also analyzed. In our calculation, the phenomenology KLR-AdS/CFT model and the rcBK approach, which are valid at a small x , are used to calculate the three point function. It is shown that the theoretical results fit well with the experimental data from ALICE and LHCb. Color Glass Condensate Forward J/ψ production Glauber-Gribov approach 1 Introduction 1 The production of forward J/ψ mesons in proton-proton (p+p) and proton-nucleus (p+A) collisions can provide valuable insight into the gluon saturation physics and strong color fields at a small x . Thus, this subject has received much attention on nuclear theory sides in recent years [ 1 1 - 5 5 1-5 ]. There are two stages for J/ψ production. The first stage is the J=ψ pair production in p+p and p+A collisions. The Color Glass Condensate (CGC) theory, which consider the p+p (A) as a dilute-dense system, is an effective method to study the J=ψ pair production. According to the CGC theory, the J=ψ pair production comes from two processes at leading order. One process is that the gluon from the incoming proton splits into a J=ψ pair before multiple scattering with the target, and the other process is that the incoming gluon multiply scatters with the target nucleus first and then splits into the J=ψ pair. The second stage is the J=ψ pair producing the J/ψ meson. This process can be described with the simple color evaporation model (CEM) [ 6 6 , 7 7 ]. In this model, the color of the J=ψ pair is assumed to be "evaporated" away in the form of a soft gluon, and then all the J=ψ pair are assumed to become the J/ψ mesons with a fixed fraction below the D -meson threshold. In the CGC framework, the dipole amplitude is the key ingredient to compute the cross section for J=ψ pair production. At Large Hadron Collider (LHC) energies, the value of Bjorken- x for J=ψ pair production in p+p (Pb) collisions will be down to 10 -6 . At this x domain, the common phenomenological saturation models, such as the CGC model [ 8 8 ], the Soyez model [ 9 9 ], and the Golec-Biernat and Wüsthoff (GBW) model [ 10 10 ], are not valid. Recently, it is more popular to use the running coupling Balitsky-Kovchegov (rcBK) equation for the dipole amplitude [ 1 1 - 3 3 1-3 ]. Unfortunately, there is a Fourier transform to obtain the unintegrated gluon distribution (UGD) from the dipole amplitude. Because the Fourier transform is an oscillatory integral [ 11 11 ], it is too difficult to ensure the accuracy of the theoretical results with the rcBK approach, especially at large p T . Here, we will introduce the phenomenological model proposed by Kovchegov, Lu, and Rezaeian (KLR) [ 12 12 , 13 13 ]. This model is depended on the Anti-de Sitter space/conformal field theory (AdS/CFT) approach, and proved very valid at a small Bjorken- x domain (10 -4 ∼10 -6 ). Furthermore, an analytic formula of the UGD can be obtained from the dipole amplitude given by the KLR-AdS/CFT model. Thus, the accuracy of the theoretical results can be ensured. In this paper, we will use the rcBK approach and the KLR-AdS/CFT model to investigate the J/ψ production in p+p and p+Pb collisions at LHC energies. In order to study the nuclear suppression of forward J/ψ production in p+A collisions, the nuclear effects should be considered. A simple method to consider nuclear effects is by relating the saturation scale of a heavy nucleus, <math id="M3"><mrow><msubsup><mi>Q</mi><mrow><mtext>sA</mtext></mrow><mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> , to that of a proton, <math id="M4"><mrow><msubsup><mi>Q</mi><mrow><mtext>sp</mtext></mrow><mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> , through formula <math id="M5"><mrow><msubsup><mi>Q</mi><mrow><mtext>sA</mtext></mrow><mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>A</mi><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup><msubsup><mi>Q</mi><mrow><mtext>sp</mtext></mrow><mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> [ 14 14 ]. In the frame of recently published papers [ 15 15 , 16 16 ], the authors consider impact parameter (b) dependence on the saturation scale using CGC approximation and investigate the nuclear configurational entropy. The results match the fitted experimental data at 1%. Besides, Refs. [ 17 17 - 20 20 17-20 ] offered a quantitative study of quarkonia radially excited S-wave states and paved a way into which bottomonium and charmonium were studied from the point of view of the conifgurational entropy in the AdS/QCD correspondence as well as in the refinement of topological defects. Thus, for getting more reliable theoretical results, the impact parameter, b , dependence of the nuclear saturation scale should be considered. In this paper, the nuclear effects are considered in the Glauber-Gribov approach [ 21 21 , 22 22 ]. We assume that the dipole-nucleus scattering amplitude relate to the dipole-proton case by the formula <math id="M6"><mrow><msubsup><mi>Q</mi><mrow><mtext>sA</mtext></mrow><mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>N</mi><mrow><mtext>part,A</mtext></mrow></msub><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><msubsup><mi>Q</mi><mrow><mtext>sp</mtext></mrow><mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math> , where N part,A ( b ) is the number of participating nucleons in p+A collisions. This paper is organized as follows. In Sect. 2, we introduce the theoretical formalism for J=ψ pair production in the large N c limit. Then, we give the method of calculating the differential cross section for J/ψ production in Sect. 3. In Sect. 4, we give the theoretical results and compare them with the recent experimental data from ALICE and LHCb. 2 Quark Pair Production Cross Section in the Large N c Limit 1 In the CGC framework, the cross-section of quark pair production with the transverse momentum, p T (q T ), and rapidity, y p ( y q ) of the quark (anti-quark) can be written as [ 1 1 ] <math display="block" id="M7"><mtable columnalign="left"><mtr><mtd><mfrac><mrow><mtext>d</mtext><msub><mi>σ</mi><mrow><mtext>q</mtext><mover accent="true"><mtext>q</mtext><mo>¯</mo></mover></mrow></msub></mrow><mrow><msup><mtext>d</mtext><mtext>2</mtext></msup><msub><mtext>p</mtext><mtext>T</mtext></msub><msup><mtext>d</mtext><mtext>2</mtext></msup><msub><mtext>q</mtext><mtext>T</mtext></msub><mtext>d</mtext><msub><mi>y</mi><mtext>p</mtext></msub><mtext>d</mtext><msub><mi>y</mi><mtext>q</mtext></msub></mrow></mfrac><mo>=</mo><mfrac><mrow><msubsup><mi>α</mi><mtext>s</mtext><mn>2</mn></msubsup></mrow><mrow><mn>16</mn><msup><mi>π</mi><mn>2</mn></msup><msub><mi>C</mi><mtext>F</mtext></msub></mrow></mfrac><mstyle displaystyle="true"><mrow><mo>∫</mo><mrow><mfrac><mrow><msup><mtext>d</mtext><mtext>2</mtext></msup><msub><mtext>k</mtext><mtext>T</mtext></msub></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mo stretchy="false">)</mo></mrow><mn>4</mn></msup></mrow></mfrac><mfrac><mrow><msub><mi>Ξ</mi><mrow><mtext>coll</mtext></mrow></msub><mo stretchy="false">(</mo><msub><mtext>p</mtext><mtext>T</mtext></msub><msub><mrow><mtext>+q</mtext></mrow><mtext>T</mtext></msub><msub><mrow><mtext>,k</mtext></mrow><mtext>T</mtext></msub><mo stretchy="false">)</mo></mrow><mrow><msup><mrow><msub><mrow><mtext>(p</mtext></mrow><mrow><msub><mrow/><mtext>T</mtext></msub></mrow></msub><msub><mrow><mtext>+q</mtext></mrow><mrow><msub><mrow/><mtext>T</mtext></msub></mrow></msub><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></mfrac></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><msubsup><mi>ϕ</mi><mrow><mi>A</mi><mn>,</mn><msub><mi>y</mi><mn>2</mn></msub></mrow><mrow><mi>q</mi><mover accent="true"><mi>q</mi><mo>¯</mo></mover><mn>,</mn><mi>g</mi></mrow></msubsup><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mrow><msub><mrow/><mtext>T</mtext></msub></mrow></msub><mo>+</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>q</mi></mstyle><mrow><msub><mrow/><mtext>T</mtext></msub></mrow></msub><mn>,</mn><msub><mstyle mathvariant="bold" mathsize="normal"><mi>k</mi></mstyle><mrow><msub><mrow/><mtext>T</mtext></msub></mrow></msub><mo stretchy="false">)</mo><msub><mi>x</mi><mn>1</mn></msub><msub><mi>G</mi><mtext>p</mtext></msub><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mn>,</mn><msup><mi>Q</mi><mtable columnalign="left"><mtr><mtd><mrow/></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable></msup><mo stretchy="false">)</mo><mn>,</mn></mtd></mtr></mtable></math> where <math id="M8"><mrow><msub><mi>C</mi><mtext>F</mtext></msub><mo>=</mo><mo stretchy="false">(</mo><msubsup><mi>N</mi><mtext>c</mtext><mn>2</mn></msubsup><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>/</mo><mo stretchy="false">(</mo><mn>2</mn><msub><mi>N</mi><mtext>c</mtext></msub><mo stretchy="false">)</mo></mrow></math> with N c =3, and x 1 and x 2 are the longitudinal momentum fractions of the gluons from the projectile and target, respectively. Ξ coll is the "hard matrix element", which can be given as Ref. [ 23 23 ] <math display="block" id="M9"><mrow><msub><mi>Ξ</mi><mrow><mtext>coll</mtext></mrow></msub><mo>=</mo><msubsup><mi>Ξ</mi><mrow><mtext>coll</mtext></mrow><mrow><mtext>q</mtext><mover accent="true"><mtext>q</mtext><mo>¯</mo></mover><mn>,</mn><mtext>q</mtext><mover accent="true"><mtext>q</mtext><mo>¯</mo></mover></mrow></msubsup><mo>+</mo><msubsup><mi>Ξ</mi><mrow><mtext>coll</mtext></mrow><mrow><mtext>q</mtext><mover accent="true"><mtext>q</mtext><mo>¯</mo></mover><mn>,</mn><mtext>g</mtext></mrow></msubsup><mo>+</mo><msubsup><mi>Ξ</mi><mrow><mtext>coll</mtext></mrow><mrow><mtext>g,g</mtext></mrow></msubsup><mn>,</mn></mrow></math> where <math id="M10"><mrow><msubsup><mi>Ξ</mi><mrow><mtext>coll</mtext></mrow><mrow><mtext>q</mtext><mover accent="true"><mtext>q</mtext><mo>¯</mo></mover><mn>,</mn><mtext>q</mtext><mover accent="true"><mtext>q</mtext><mo>¯</mo></mover></mrow></msubsup></mrow></math> is the term of a quark pair scattering off the nuclei, <math id="M11"><mrow><msubsup><mi>Ξ</mi><mrow><mtext>coll</mtext></mrow><mrow><mtext>g,g</mtext></mrow></msubsup></mrow></math> is the term before the gluon splitting into a quark pair, and <math id="M12"><mrow><msubsup><mi>Ξ</mi><mrow><mtext>coll</mtext></mrow><mrow><mtext>q</mtext><mover accent="true"><mtext>q</mtext><mo>¯</mo></mover><mn>,</mn><mtext>g</mtext></mrow></msubsup></mrow></math> corresponds to the interference between the quark and gluon. The explicit expressions are <math display="block" id="M13"><mrow><msubsup><mi>Ξ</mi><mrow><mtext>coll</mtext></mrow><mrow><mtext>q</mtext><mover accent="true"><mtext>q</mtext><mo>¯</mo></mover><mn>,</mn><mtext>q</mtext><mover accent="true"><mtext>q</mtext><mo>¯</mo></mover></mrow></msubsup><mo>=</mo><mfrac><mrow><mn>8</mn><msup><mi>p</mi><mo>+</mo></msup><msup><mi>q</mi><mo>+</mo></msup></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><msup><mi>p</mi><mo>+</mo></msup><mo>+</mo><msup><mi>q</mi><mo>+</mo></msup><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><msup><mrow><mo stretchy="false">(</mo><msubsup><mstyle mathvariant="bold" mathsize="normal"><mi>a</mi></mstyle><mrow><msub><mrow/><mtext>T</mtext></msub></mrow><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>m</mi><mtext>c</mtext><mn>2</mn></msubsup><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></mfrac><mo stretchy="false">[</mo><msubsup><mi>m</mi><mtext>c</mtext><mn>2</mn></msubsup><mo>+</mo><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><msup><mi>p</mi><mo>+</mo></msup><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><msup><mi>q</mi><mo>+</mo></msup><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><msup><mi>p</mi><mo>+</mo></msup><mo>+</mo><msup><mi>q</mi><mo>+</mo></msup><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></mfrac><msubsup><mstyle mathvariant="bold" mathsize="normal"><mi>a</mi></mstyle><mrow><msub><mrow/><mtext>T</mtext></msub></mrow><mn>2</mn></msubsup><mo stretchy="false">]</mo><mn>,</mn></mrow></math> <math display="block" id="M14"><mrow><msubsup><mi>Ξ</mi><mrow><mtext>coll</mtext></mrow><mrow><mtext>q</mtext><mover accent="true"><mtext>q</mtext><mo>¯</mo></mover><mn>,</mn><mtext>g</mtext></mrow></msubsup><mo>=</mo><mo>−</mo><mfrac><mrow><mn>16</mn></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mi>p</mi><mo>+</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo stretchy="false">(</mo><msubsup><mstyle mathvariant="bold" mathsize="normal"><mi>a</mi></mstyle><mrow><msub><mrow/><mtext>T</mtext></msub></mrow><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>m</mi><mtext>c</mtext><mn>2</mn></msubsup><mo stretchy="false">)</mo></mrow></mfrac><mo stretchy="false">[</mo><msubsup><mi>m</mi><mtext>c</mtext><mn>2</mn></msubsup><mo>+</mo><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><msup><mi>p</mi><mo>+</mo></msup><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><msup><mi>q</mi><mo>+</mo></msup><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><msup><mi>p</mi><mo>+</mo></msup><mo>+</mo><msup><mi>q</mi><mo>+</mo></msup><mo stretchy="false">)</mo></mrow><mn>3</mn></msup></mrow></mfrac><msub><mi>a</mi><mrow><msub><mrow/><mtext>T</mtext></msub></mrow></msub><mo>⋅</mo><mo stretchy="false">(</mo><msup><mi>p</mi><mo>+</mo></msup><msub><mstyle mathvariant="bold" mathsize="normal"><mi>q</mi></mstyle><mrow><msub><mrow/><mtext>T</mtext></msub></mrow></msub><mo>−</mo><msup><mi>q</mi><mo>+</mo></msup><msub><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mrow><msub><mrow/><mtext>T</mtext></msub></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mn>,</mn></mrow></math> <math display="block" id="M15"><mrow><msubsup><mi>Ξ</mi><mrow><mtext>coll</mtext></mrow><mrow><mtext>g,g</mtext></mrow></msubsup><mo>=</mo><mfrac><mn>8</mn><mrow><msup><mrow><mo stretchy="false">(</mo><mi>p</mi><mo>+</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><mn>4</mn></msup></mrow></mfrac><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mi>p</mi><mo>+</mo><mi>q</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>−</mo><mfrac><mn>2</mn><mrow><msup><mrow><mo stretchy="false">(</mo><msup><mi>p</mi><mo>+</mo></msup><mo>+</mo><msup><mi>q</mi><mo>+</mo></msup><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></mfrac><msup><mrow><mo stretchy="false">(</mo><msup><mi>p</mi><mo>+</mo></msup><msub><mstyle mathvariant="bold" mathsize="normal"><mi>q</mi></mstyle><mrow><msub><mrow/><mtext>T</mtext></msub></mrow></msub><mo>−</mo><msup><mi>q</mi><mo>+</mo></msup><msub><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mrow><msub><mrow/><mtext>T</mtext></msub></mrow></msub><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo stretchy="false">]</mo><mn>,</mn></mrow></math> with a T ≡q T -k T . x 1 G p ( x 1 , Q 2 ) is the collinear gluon distribution function of the proton. In this work we use the CTEQ6 LO parametrization [ 24 24 ]. The three-point function of the nucleus in Eq. (1) can be expressed as <math display="block" id="M16"><mrow><msubsup><mi>ϕ</mi><mrow><mtext>A</mtext><mn>,</mn><msub><mtext>y</mtext><mn>2</mn></msub></mrow><mrow><mtext>q</mtext><mover accent="true"><mtext>q</mtext><mo>¯</mo></mover><mn>,</mn><mtext>g</mtext></mrow></msubsup><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>l</mi></mstyle><mrow><msub><mrow/><mtext>T</mtext></msub></mrow></msub><mn>,</mn><msub><mstyle mathvariant="bold" mathsize="normal"><mi>k</mi></mstyle><mrow><msub><mrow/><mtext>T</mtext></msub></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><msup><mtext>d</mtext><mn>2</mn></msup><mstyle mathvariant="bold" mathsize="normal"><mi>b</mi></mstyle><mfrac><mrow><msub><mi>N</mi><mtext>c</mtext></msub><msubsup><mstyle mathvariant="bold" mathsize="normal"><mi>l</mi></mstyle><mrow><msub><mrow/><mtext>T</mtext></msub></mrow><mn>2</mn></msubsup></mrow><mrow><mn>4</mn><msub><mi>α</mi><mtext>s</mtext></msub></mrow></mfrac><msub><mi>S</mi><mtext>Y</mtext></msub><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>k</mi></mstyle><mrow><msub><mrow/><mtext>T</mtext></msub></mrow></msub><mo stretchy="false">)</mo><msub><mi>S</mi><mtext>Y</mtext></msub><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>l</mi></mstyle><mrow><msub><mrow/><mtext>T</mtext></msub></mrow></msub><mo>−</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>k</mi></mstyle><mrow><msub><mrow/><mtext>T</mtext></msub></mrow></msub><mo stretchy="false">)</mo><mn>,</mn></mrow></math> where b is the impact parameter and l T =p T +q T . The UGD can be obtained by a Fourier transform [ 11 11 , 25 25 ] <math display="block" id="M17"><mrow><msub><mi>S</mi><mtext>Y</mtext></msub><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>k</mi></mstyle><mrow><msub><mrow/><mtext>T</mtext></msub></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><msup><mtext>d</mtext><mn>2</mn></msup><msub><mstyle mathvariant="bold" mathsize="normal"><mi>r</mi></mstyle><mrow><msub><mrow/><mtext>T</mtext></msub></mrow></msub><msup><mi>e</mi><mrow><mi>i</mi><msub><mstyle mathvariant="bold" mathsize="normal"><mi>k</mi></mstyle><mrow><msub><mrow/><mtext>T</mtext></msub></mrow></msub><mo>⋅</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>r</mi></mstyle><mrow><msub><mrow/><mtext>T</mtext></msub></mrow></msub></mrow></msup><msub><mi>S</mi><mtext>Y</mtext></msub><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>r</mi></mstyle><mrow><msub><mrow/><mtext>T</mtext></msub></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mfrac><mrow><mtext>d</mtext><msub><mi>r</mi><mrow><msub><mrow/><mtext>T</mtext></msub></mrow></msub></mrow><mrow><msub><mi>r</mi><mrow><msub><mrow/><mtext>T</mtext></msub></mrow></msub></mrow></mfrac><msub><mi>J</mi><mn>0</mn></msub><mo stretchy="false">(</mo><msub><mi>r</mi><mrow><msub><mrow/><mtext>T</mtext></msub></mrow></msub><mi>k</mi><mo stretchy="false">)</mo><msub><mi>S</mi><mtext>Y</mtext></msub><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>r</mi></mstyle><mrow><msub><mrow/><mtext>T</mtext></msub></mrow></msub><mo stretchy="false">)</mo><mn>,</mn></mrow></math> where S Y (r T ) is the dipole amplitude and J 0 is the spherical bessel function of the first kind. In this paper, the rcBK approach and the KLR-AdS/CFT model are used for S Y (r T ). In the rcBK approach, the dipole scattering amplitude can be given by Ref. [ 25 25 ] <math display="block" id="M18"><mrow><mfrac><mrow><mo>∂</mo><mi>S</mi><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>r</mi></mstyle><mrow><msub><mrow/><mtext>T</mtext></msub></mrow></msub><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><mi>Y</mi></mrow></mfrac><mo>=</mo><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mtext>d</mtext><msub><mstyle mathvariant="bold" mathsize="normal"><mi>r</mi></mstyle><mrow><msub><mrow/><mrow><mtext>1T</mtext></mrow></msub></mrow></msub><msup><mi>K</mi><mrow><mtext>Bal</mtext></mrow></msup><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>r</mi></mstyle><mrow><msub><mrow/><mtext>T</mtext></msub></mrow></msub><mn>,</mn><msub><mstyle mathvariant="bold" mathsize="normal"><mi>r</mi></mstyle><mrow><msub><mrow/><mrow><mtext>1T</mtext></mrow></msub></mrow></msub><mn>,</mn><msub><mstyle mathvariant="bold" mathsize="normal"><mi>r</mi></mstyle><mrow><msub><mrow/><mrow><mtext>2T</mtext></mrow></msub></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mi>S</mi><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>r</mi></mstyle><mrow><msub><mrow/><mrow><mtext>1T</mtext></mrow></msub></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><mi>S</mi><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>r</mi></mstyle><mrow><msub><mrow/><mrow><mtext>2T</mtext></mrow></msub></mrow></msub><mo stretchy="false">)</mo><mo>−</mo><mi>S</mi><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>r</mi></mstyle><mrow><msub><mrow/><mtext>T</mtext></msub></mrow></msub><mo stretchy="false">)</mo><mo>−</mo><mi>S</mi><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>r</mi></mstyle><mrow><msub><mrow/><mrow><mtext>1T</mtext></mrow></msub></mrow></msub><mo stretchy="false">)</mo><mi>S</mi><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>r</mi></mstyle><mrow><msub><mrow/><mrow><mtext>2T</mtext></mrow></msub></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mn>,</mn></mrow></math> where K Bal (r T ,r 1T ,r 2T ) is the kernel for the running term, and the GBW ansatz is used for the initial conditions of the dipole scattering amplitude <math display="block" id="M19"><mrow><msup><mi>S</mi><mrow><mtext>GBW</mtext></mrow></msup><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>r</mi></mstyle><mrow><msub><mrow/><mtext>T</mtext></msub></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mo>−</mo><mtext>exp</mtext><mo stretchy="false">[</mo><mo>−</mo><msup><mrow><mo stretchy="false">(</mo><mfrac><mrow><msubsup><mi>r</mi><mtext>T</mtext><mn>2</mn></msubsup><msubsup><mi>Q</mi><mrow><mi>s</mi><mn>0</mn></mrow><mn>2</mn></msubsup></mrow><mn>4</mn></mfrac><mo stretchy="false">)</mo></mrow><mi>γ</mi></msup><mo stretchy="false">]</mo></mrow></math> with γ =1 and <math id="M20"><mrow><msubsup><mi>Q</mi><mrow><mi>s</mi><mn>0</mn></mrow><mn>2</mn></msubsup><mo>=</mo><mn>0.24</mn></mrow></math> GeV 2 . The dipole-proton scattering amplitude in the KLR-AdS/CFT model can be written as Ref. [ 12 12 , 13 13 ] <math display="block" id="M21"><mrow><msubsup><mi>S</mi><mtext>Y</mtext><mrow><mtext>AdS</mtext></mrow></msubsup><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>r</mi></mstyle><mrow><msub><mrow/><mtext>T</mtext></msub></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mo>−</mo><mi>e</mi><mi>x</mi><mi>p</mi><mo stretchy="false">[</mo><mo>−</mo><mi>r</mi><msubsup><mi>Q</mi><mrow><mtext>s,p</mtext></mrow><mrow><mi>A</mi><mi>d</mi><mi>S</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>/</mo><mo stretchy="false">(</mo><mn>2</mn><msqrt><mn>2</mn></msqrt><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mn>,</mn></mrow></math> where the saturation scale is given by <math display="block" id="M22"><mrow><msubsup><mi>Q</mi><mrow><mtext>s,p</mtext></mrow><mrow><mtext>AdS</mtext></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn>2</mn><msub><mi>A</mi><mn>0</mn></msub><mi>x</mi></mrow><mrow><msubsup><mi>M</mi><mn>0</mn><mn>2</mn></msubsup><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo><mi>π</mi></mrow></mfrac><mo stretchy="false">(</mo><mfrac><mn>1</mn><mrow><msubsup><mi>ρ</mi><mtext>m</mtext><mn>3</mn></msubsup></mrow></mfrac><mo>+</mo><mfrac><mn>2</mn><mrow><msub><mi>ρ</mi><mtext>m</mtext></msub></mrow></mfrac><mo>−</mo><mn>2</mn><msub><mi>M</mi><mn>0</mn></msub><msqrt><mrow><mfrac><mrow><mn>1</mn><mo>−</mo><mi>x</mi></mrow><mi>x</mi></mfrac></mrow></msqrt><mo stretchy="false">)</mo><mn>,</mn></mrow></math> With <math id="M23"><mtable columnalign="left"><mtr><mtd><msub><mi>ρ</mi><mi>m</mi></msub><mrow><mo>{</mo><mtable columnalign="left"><mtr><mtd><msup><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mrow><mn>3</mn><mi>m</mi></mrow></mfrac></mrow><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>4</mn></mrow></msup><msqrt><mrow><mn>2</mn><mi>cos</mi><mrow><mo>(</mo><mrow><mfrac><mi>θ</mi><mn>3</mn></mfrac></mrow><mo>)</mo></mrow></mrow></msqrt><mtext>   :</mtext><mi>m</mi><mo>≤</mo><mfrac><mn>4</mn><mrow><mn>27</mn></mrow></mfrac></mtd></mtr><mtr><mtd><msqrt><mrow><mfrac><mn>1</mn><mrow><mn>3</mn><mi>m</mi><mi>Δ</mi></mrow></mfrac><mo>+</mo><mi>Δ</mi></mrow></msqrt><mtext>  :</mtext><mi>m</mi><mo>></mo><mfrac><mn>4</mn><mrow><mn>27</mn></mrow></mfrac><mo>,</mo></mtd></mtr></mtable></mrow></mtd></mtr><mtr><mtd><mi>Δ</mi><mo>=</mo><mo stretchy="false">[</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>m</mi></mrow></mfrac><mo>−</mo><msqrt><mrow><mfrac><mn>1</mn><mrow><mn>4</mn><msup><mi>m</mi><mn>2</mn></msup></mrow></mfrac><mo>−</mo><mfrac><mn>1</mn><mrow><mn>27</mn><msup><mi>m</mi><mn>3</mn></msup></mrow></mfrac></mrow></msqrt><msup><mo stretchy="false">]</mo><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup><mn>,</mn></mtd></mtr><mtr><mtd><mi>m</mi><mo>=</mo><mfrac><mrow><msubsup><mi>M</mi><mn>0</mn><mn>4</mn></msubsup><msup><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mn>,</mn></mtd></mtr><mtr><mtd><mi>cos</mi><mi>θ</mi><mo>=</mo><msqrt><mrow><mfrac><mrow><mn>27</mn><mi>m</mi></mrow><mn>4</mn></mfrac></mrow></msqrt><mn>,</mn></mtd></mtr><mtr><mtd><msub><mi>A</mi><mn>0</mn></msub><mo>=</mo><msqrt><mrow><msub><mi>λ</mi><mrow><mtext>YM</mtext></mrow></msub></mrow></msqrt><mi/><mi>G</mi><mi>e</mi><mi>V</mi><mn>.</mn></mtd></mtr><mtr><mtd><mrow/></mtd></mtr></mtable></math> Here, we choose λ YM =20 and M 0 =6.16×10 -3 , which are obtained from a fit to the HERA data. For p+A collisions, the saturation scale of the nucleus can be given by Ref. [ 21 21 , 22 22 ] <math display="block" id="M24"><mrow><msubsup><mi>Q</mi><mrow><mtext>s,A</mtext></mrow><mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>N</mi><mrow><mtext>part</mtext><mn>,</mn><mtext>A</mtext></mrow></msub><msubsup><mi>Q</mi><mrow><mtext>s,p</mtext></mrow><mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mn>,</mn></mrow></math> where the participating nucleons in the collisions, N part,A , can be given by a convolution of the thickness function of the proton and nucleus [ 26 26 ]. For the thickness function of the nucleus, we use the Woods-Saxon form [ 27 27 ] <math display="block" id="M25"><mrow><msub><mi>T</mi><mtext>A</mtext></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold" mathsize="normal"><mi>b</mi></mstyle><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mtext>d</mtext><mi>z</mi><mfrac><mrow><msub><mi>ρ</mi><mn>0</mn></msub></mrow><mrow><mn>1</mn><mo>+</mo><mtext>exp</mtext><mo stretchy="false">(</mo><mfrac><mrow><mi>r</mi><mo>−</mo><mi>R</mi></mrow><mi>a</mi></mfrac><mo stretchy="false">)</mo></mrow></mfrac><mn>,</mn></mrow></math> where ρ 0 is the nucleon density in the center of the nucleus and R is the nuclear radius [ 26 26 ]. For the proton, the Gaussian form is used [ 28 28 ] <math display="block" id="M26"><mrow><msub><mi>T</mi><mtext>p</mtext></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold" mathsize="normal"><mi>b</mi></mstyle><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msup><mi>e</mi><mrow><mo>−</mo><msup><mi>b</mi><mn>2</mn></msup><mo>/</mo><mo stretchy="false">(</mo><mn>2</mn><mi>B</mi><mo stretchy="false">)</mo></mrow></msup></mrow><mrow><mn>2</mn><mi>π</mi><mi>B</mi></mrow></mfrac></mrow></math> with <math id="M27"><mrow><mi>B</mi><mo>=</mo><mfrac><mrow><msub><mi>σ</mi><mrow><mtext>in</mtext></mrow></msub></mrow><mrow><mn>14.30</mn></mrow></mfrac></mrow></math> fm 2 . 3 Differential Cross Section in Terms of the J/ψ Variate 1 In order to compare with the experimental data, we should express the cross section in Eq. (1) in terms of the variate of J/ψ , which are the invariant mass, M 2 , the transverse momentum, P T , and rapidity, Y . The relation equation can be written as Ref. [ 29 29 ] <math display="block" id="M28"><mrow><mfrac><mrow><mtext>d</mtext><mi>σ</mi></mrow><mrow><mtext>d</mtext><msup><mi>M</mi><mn>2</mn></msup><msup><mtext>d</mtext><mn>2</mn></msup><msub><mstyle mathvariant="bold" mathsize="normal"><mi>P</mi></mstyle><mrow><msub><mrow/><mtext>T</mtext></msub></mrow></msub><mtext>d</mtext><mi>Y</mi></mrow></mfrac><mo>=</mo><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mrow><msqrt><mrow><msup><mi>M</mi><mn>2</mn></msup><mo>/</mo><mn>4</mn><mo>−</mo><msubsup><mi>m</mi><mtext>c</mtext><mn>2</mn></msubsup></mrow></msqrt></mrow></msubsup></mrow></mstyle><mtext>d</mtext><mover accent="true"><mi>q</mi><mo>˜</mo></mover><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mrow><mn>2</mn><mi>π</mi></mrow></msubsup></mrow></mstyle><mtext>d</mtext><mi>ϕ</mi><mfrac><mrow><mover accent="true"><mi>q</mi><mo>˜</mo></mover><msub><mi>γ</mi><mi>x</mi></msub></mrow><mrow><msub><mi>ω</mi><mtext>p</mtext></msub><msub><mi>ω</mi><mtext>q</mtext></msub><mo>|</mo><mtext>sinh</mtext><mo stretchy="false">(</mo><msub><mi>y</mi><mtext>p</mtext></msub><mo>−</mo><msub><mi>y</mi><mtext>q</mtext></msub><mo stretchy="false">)</mo><mo>|</mo></mrow></mfrac><mfrac><mrow><mtext>d</mtext><mi>σ</mi></mrow><mrow><msup><mtext>d</mtext><mn>2</mn></msup><msub><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mrow><msub><mrow/><mtext>T</mtext></msub></mrow></msub><msup><mtext>d</mtext><mn>2</mn></msup><msub><mstyle mathvariant="bold" mathsize="normal"><mi>q</mi></mstyle><mrow><msub><mrow/><mtext>T</mtext></msub></mrow></msub><mtext>d</mtext><msub><mi>y</mi><mtext>p</mtext></msub><mtext>d</mtext><msub><mi>y</mi><mtext>q</mtext></msub></mrow></mfrac><mn>,</mn></mrow></math> where <math id="M29"><mrow><msub><mi>ω</mi><mtext>q</mtext></msub><mo>=</mo><msqrt><mrow><msubsup><mstyle mathvariant="bold" mathsize="normal"><mi>q</mi></mstyle><mrow><msub><mrow/><mtext>T</mtext></msub></mrow><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>m</mi><mtext>c</mtext><mn>2</mn></msubsup></mrow></msqrt></mrow></math> ( <math id="M30"><mrow><msub><mi>ω</mi><mtext>p</mtext></msub><mo>=</mo><msqrt><mrow><msubsup><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mrow><msub><mrow/><mtext>T</mtext></msub></mrow><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>m</mi><mtext>c</mtext><mn>2</mn></msubsup></mrow></msqrt></mrow></math> ) is the transverse masses of the quark (anti-quark). <math id="M31"><mstyle mathvariant="bold" mathsize="normal"><mover accent="true"><mi>q</mi><mo>˜</mo></mover></mstyle></math> is the quark (or anti-quark) transverse momentum in the center of mass system of the quark pair and ϕ is the angle between <math id="M32"><mstyle mathvariant="bold" mathsize="normal"><mover accent="true"><mi>q</mi><mo>˜</mo></mover></mstyle></math> and the x -axis. In Eq. (15), the momenta of the quark and anti-quark are given by <math display="block" id="M33"><mtable columnalign="left"><mtr><mtd><msup><mi>q</mi><mi>μ</mi></msup><mo>=</mo><msub><mi>L</mi><mi>z</mi></msub><msubsup><mrow><mo>(</mo><mrow><mi>β</mi><mi>z</mi></mrow><mo>)</mo></mrow><mi>ν</mi><mrow><msup><mrow/><mi>μ</mi></msup></mrow></msubsup><mi>L</mi><mi>x</mi><msubsup><mrow><mo>(</mo><mrow><mi>β</mi><mi>x</mi></mrow><mo>)</mo></mrow><mi>σ</mi><mi>ν</mi></msubsup><msubsup><mi>q</mi><mrow><mi>c</mi><mi>m</mi></mrow><mi>σ</mi></msubsup></mtd></mtr><mtr><mtd><msup><mi>p</mi><mi>μ</mi></msup><mo>=</mo><msub><mi>L</mi><mi>z</mi></msub><msubsup><mrow><mo>(</mo><mrow><mi>β</mi><mi>z</mi></mrow><mo>)</mo></mrow><mi>ν</mi><mrow><msup><mrow/><mi>μ</mi></msup></mrow></msubsup><mi>L</mi><mi>x</mi><msubsup><mrow><mo>(</mo><mrow><mi>β</mi><mi>x</mi></mrow><mo>)</mo></mrow><mi>σ</mi><mi>ν</mi></msubsup><msubsup><mi>p</mi><mrow><mi>c</mi><mi>m</mi></mrow><mi>σ</mi></msubsup><mo>,</mo></mtd></mtr></mtable></math> where the momenta in the center of mass system of the quark pair <math id="M34"><mrow><msubsup><mi>q</mi><mrow><mtext>cm</mtext></mrow><mi>σ</mi></msubsup></mrow></math> and <math id="M35"><mrow><msubsup><mi>p</mi><mrow><mtext>cm</mtext></mrow><mi>σ</mi></msubsup></mrow></math> can be given as <math id="M36"><mrow><msubsup><mi>q</mi><mrow><mtext>cm</mtext></mrow><mi>σ</mi></msubsup><mo>=</mo><mo stretchy="false">(</mo><mfrac><mi>M</mi><mn>2</mn></mfrac><mn>,</mn><mover accent="true"><mi>q</mi><mo>˜</mo></mover><mtext>cos</mtext><mi>ϕ</mi><mn>,</mn><mover accent="true"><mi>q</mi><mo>˜</mo></mover><mtext>sin</mtext><mi>ϕ</mi><mn>,</mn><msqrt><mrow><msup><mi>M</mi><mn>2</mn></msup><mo>/</mo><mn>4</mn><mo>−</mo><msubsup><mi>m</mi><mtext>c</mtext><mn>2</mn></msubsup><mo>−</mo><msup><mover accent="true"><mi>q</mi><mo>˜</mo></mover><mn>2</mn></msup></mrow></msqrt><mo stretchy="false">)</mo></mrow></math> and <math display="block" id="M37"><mrow><msubsup><mi>p</mi><mrow><mtext>cm</mtext></mrow><mi>σ</mi></msubsup><mo>=</mo><mo stretchy="false">(</mo><mfrac><mi>M</mi><mn>2</mn></mfrac><mn>,</mn><mo>−</mo><mover accent="true"><mi>q</mi><mo>˜</mo></mover><mtext>cos</mtext><mi>ϕ</mi><mn>,</mn><mo>−</mo><mover accent="true"><mi>q</mi><mo>˜</mo></mover><mtext>sin</mtext><mi>ϕ</mi><mn>,</mn><mo>−</mo><msqrt><mrow><msup><mi>M</mi><mn>2</mn></msup><mo>/</mo><mn>4</mn><mo>−</mo><msubsup><mi>m</mi><mtext>c</mtext><mn>2</mn></msubsup><mo>−</mo><msup><mover accent="true"><mi>q</mi><mo>˜</mo></mover><mn>2</mn></msup></mrow></msqrt><mo stretchy="false">)</mo><mn>.</mn></mrow></math> In Eq. (16), L x ( β x ) and L z ( β z ) are the Lorentz boosts in the x -axis and z -axis, respectively. <math id="M38"><mrow><msub><mi>β</mi><mtext>x</mtext></msub><mo>=</mo><mo>|</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>P</mi></mstyle><mrow><msub><mrow/><mtext>T</mtext></msub></mrow></msub><mo>|</mo><mo>/</mo><msqrt><mrow><msup><mi>M</mi><mn>2</mn></msup><mo>+</mo><msubsup><mstyle mathvariant="bold" mathsize="normal"><mi>P</mi></mstyle><mrow><msub><mrow/><mtext>T</mtext></msub></mrow><mn>2</mn></msubsup></mrow></msqrt></mrow></math> and β z =tanh( Y ). To calculate the cross section of J/ψ , we will use the simple color evaporation model [ 6 6 , 7 7 ]. This model assumes that all <math id="M39"><mrow><mi>c</mi><mover accent="true"><mi>c</mi><mo>¯</mo></mover></mrow></math> pairs produced become J/ψ mesons below the D-meson threshold. This can be expressed as <math display="block" id="M40"><mrow><mfrac><mrow><mtext>d</mtext><msub><mi>σ</mi><mrow><mi>J</mi><mo>/</mo><mi>ψ</mi></mrow></msub></mrow><mrow><mtext>d</mtext><mi>Y</mi><msup><mtext>d</mtext><mn>2</mn></msup><msub><mstyle mathvariant="bold" mathsize="normal"><mi>P</mi></mstyle><mrow><msub><mrow/><mtext>T</mtext></msub></mrow></msub></mrow></mfrac><mo>=</mo><msub><mi>F</mi><mrow><mi>J</mi><mo>/</mo><mi>ψ</mi></mrow></msub><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mrow><mn>4</mn><msubsup><mi>m</mi><mtext>c</mtext><mn>2</mn></msubsup></mrow><mrow><mn>4</mn><msubsup><mi>m</mi><mtext>D</mtext><mn>2</mn></msubsup></mrow></msubsup></mrow></mstyle><mtext>d</mtext><msup><mi>M</mi><mn>2</mn></msup><mfrac><mrow><mtext>d</mtext><mi>σ</mi></mrow><mrow><mtext>d</mtext><msup><mi>M</mi><mn>2</mn></msup><msup><mtext>d</mtext><mn>2</mn></msup><msub><mstyle mathvariant="bold" mathsize="normal"><mi>P</mi></mstyle><mrow><msub><mrow/><mtext>T</mtext></msub></mrow></msub><mtext>d</mtext><mi>Y</mi></mrow></mfrac><mn>,</mn></mrow></math> where F J/ψ (=0.01-0.05) is a nonperturbative quantity representing the probability that a J=ψ pair will become a J/ψ [ 23 23 ]. 4 Results and Discussion 1 For the UGD of the KLR-AdS/CFT model, an analytic form can be obtained from Eq. (7) <math display="block" id="M41"><mrow><msub><mi>S</mi><mtext>Y</mtext></msub><mo stretchy="false">(</mo><mi>x</mi><mn>,</mn><msub><mstyle mathvariant="bold" mathsize="normal"><mi>k</mi></mstyle><mtext>T</mtext></msub><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn>32</mn><mi>π</mi></mrow><mrow><msup><mrow><mo stretchy="false">[</mo><msubsup><mi>Q</mi><mrow><mtext>s,p</mtext></mrow><mrow><mtext>AdS</mtext></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><mn>2</mn></msup></mrow></mfrac><mfrac><mn>1</mn><mrow><msup><mrow><mo>{</mo><mn>1</mn><mo>+</mo><mn>16</mn><msubsup><mi>k</mi><mtext>T</mtext><mn>2</mn></msubsup><mo>/</mo><mo stretchy="false">[</mo><msubsup><mi>Q</mi><mrow><mtext>s,p</mtext></mrow><mrow><mtext>AdS</mtext></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><msup><mrow><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><mn>2</mn></msup><mo>}</mo></mrow><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></mfrac><mn>.</mn></mrow></math> The UGD S Y ( x ,k T ) versus x at k T = 1 GeV (a), 2 GeV (b), 4 GeV (c), and 6 GeV (d) are shown in Fig. 1. Fig. 1. The solid and dashed curves are the results of the rcBK approach and the KLR-AdS/CFT model, respectively. It is shown that the results of the KLR-AdS/CFT model are larger than that of the rcBK approach at small k T , but the results are just the opposite at large k T . It is also shown that the results of the KLR-AdS/CFT model are obviously independent of Bjorken- x when x is small. In Fig. 2 Fig. 2 , we give the results for the production cross section of J/ψ in p+p collisions versus Y (a,c) and P T (b,d) at a center of mass energy <math id="M42"><mrow><msqrt><mi>s</mi></msqrt><mo>=</mo><mn>7</mn></mrow></math> TeV. Fig. 2 Fig. 2 (a,b) and (c,d) are the results of the KLR-AdS/CFT model and the rcBK approach, respectively. The uncertainty band includes the variation of the factorization scale between 2 M T and M T /2 with <math id="M43"><mrow><msub><mi>M</mi><mrow><msub><mrow/><mtext>T</mtext></msub></mrow></msub><mo>=</mo><msqrt><mrow><msup><mi>M</mi><mn>2</mn></msup><mo>+</mo><msubsup><mi>P</mi><mrow><msub><mrow/><mtext>T</mtext></msub></mrow><mn>2</mn></msubsup></mrow></msqrt></mrow></math> . The charm quark mass is taken as 1.2 GeV. The data comes from ALICE [ 30 30 ]. The integral range in Fig. 2(b,d) Fig. 2(b,d) is 2.5 < Y < 4. For the KLR-AdS/CFT model, as shown in Fig. 2(a,b) Fig. 2(a,b) , the theoretical results fit well with the experimental data except at a very large transverse momentum of J/ψ . The reason is that the CGC theory is not very accurate at a very large P T . In Fig. 2(c,d) Fig. 2(c,d) , it is shown that the normalization uncertainty for the results of the rcBK approach are larger than those of the KLR-AdS/CFT model. It is also shown that the curve in Fig. 2(d) Fig. 2(d) is not smooth enough. This is because there is a Fourier transform in obtaining the UGD, as shown in Eq. (7). Because the Fourier transform is an oscillatory integral, it is very difficult to ensure the correctness of the results for the rcBK approach, especially at a large P T . Thus, we will only give the results of the KLR-AdS/CFT model in p+Pb collisions. Using Eq. (12), we show the differential cross section of J/ψ production versus Y (a) and P T (b) in p+Pb collisions at a center of mass energy of <math id="M45"><mrow><msqrt><mi>s</mi></msqrt><mo>=</mo><mn>5.02</mn></mrow></math> TeV in Fig. 3 Fig. 3 . The theory uncertainty results are obtained by the same method as in Fig. 2 Fig. 2 and the data come from ALICE [ 31 31 ] and LHCb [ 32 32 ]. The integral range in Fig. 3(b) Fig. 3(b) is 1.5 < Y < 4. It is shown that the theoretical results fit well with the experimental data. Here, there is only one experimental data at large P T (=8.5-14) GeV. If there are more data, the deviation of the theory results and the experimental data at large P T may be the same as in Fig. 2. Fig. 2. In order to compare with experimental data, we introduce the nuclear modification factor <math display="block" id="M47"><mrow><msub><mi>R</mi><mrow><mtext>pA</mtext></mrow></msub><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mi>A</mi></mfrac><mfrac><mrow><mo>〈</mo><mtext>d</mtext><mi>σ</mi><mo>/</mo><mtext>d</mtext><mi>Y</mi><mo>〉</mo><msub><mo>|</mo><mrow><mtext>pA</mtext></mrow></msub></mrow><mrow><mo>〈</mo><mtext>d</mtext><mi>σ</mi><mo>/</mo><mtext>d</mtext><mi>Y</mi><mo>〉</mo><msub><mo>|</mo><mrow><mtext>pp</mtext></mrow></msub></mrow></mfrac><mn>,</mn></mrow></math> and <math display="block" id="M48"><mrow><msub><mi>R</mi><mrow><mtext>pA</mtext></mrow></msub><mo stretchy="false">(</mo><msub><mi>P</mi><mtext>T</mtext></msub><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mi>A</mi></mfrac><mfrac><mrow><mo>〈</mo><mtext>d</mtext><mi>σ</mi><mo>/</mo><msup><mtext>d</mtext><mn>2</mn></msup><msub><mstyle mathvariant="bold" mathsize="normal"><mi>P</mi></mstyle><mrow><msub><mrow/><mtext>T</mtext></msub></mrow></msub><mo>〉</mo><msub><mo>|</mo><mrow><mtext>pA</mtext></mrow></msub></mrow><mrow><mo>〈</mo><mtext>d</mtext><mi>σ</mi><mo>/</mo><msup><mtext>d</mtext><mn>2</mn></msup><msub><mstyle mathvariant="bold" mathsize="normal"><mi>P</mi></mstyle><mrow><msub><mrow/><mtext>T</mtext></msub></mrow></msub><mo>〉</mo><msub><mo>|</mo><mrow><mtext>pp</mtext></mrow></msub></mrow></mfrac><mn>,</mn></mrow></math> where <math id="M49"><mrow><mo>〈</mo><mtext>d</mtext><mi>σ</mi><mo>/</mo><mtext>d</mtext><mi>Y</mi><mo>〉</mo><mo>=</mo><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><msup><mtext>d</mtext><mn>2</mn></msup><msub><mstyle mathvariant="bold" mathsize="normal"><mi>P</mi></mstyle><mrow><msub><mrow/><mtext>T</mtext></msub></mrow></msub><mfrac><mrow><mtext>d</mtext><mi>σ</mi></mrow><mrow><msup><mtext>d</mtext><mn>2</mn></msup><msub><mstyle mathvariant="bold" mathsize="normal"><mi>P</mi></mstyle><mrow><msub><mrow/><mtext>T</mtext></msub></mrow></msub><mtext>d</mtext><mi>Y</mi></mrow></mfrac></mrow></math> and <math id="M50"><mrow><mo>〈</mo><mtext>d</mtext><mi>σ</mi><mo>/</mo><msup><mtext>d</mtext><mn>2</mn></msup><msub><mstyle mathvariant="bold" mathsize="normal"><mi>P</mi></mstyle><mrow><msub><mrow/><mtext>T</mtext></msub></mrow></msub><mo>〉</mo><mo>=</mo><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mtext>d</mtext><mi>Y</mi><mfrac><mrow><mtext>d</mtext><mi>σ</mi></mrow><mrow><msup><mtext>d</mtext><mn>2</mn></msup><msub><mstyle mathvariant="bold" mathsize="normal"><mi>P</mi></mstyle><mrow><msub><mrow/><mtext>T</mtext></msub></mrow></msub><mtext>d</mtext><mi>Y</mi></mrow></mfrac></mrow></math> . In Fig. 4 Fig. 4 , we show the nuclear modification factor versus Y (a) andP T (b) at <math id="M51"><mrow><msqrt><mi>s</mi></msqrt><mo>=</mo><mn>5.02</mn></mrow></math> TeV. The experimental data come from ALICE [ 31 31 , 33 33 ]. In Fig. 2 Fig. 2 and Fig. 3 Fig. 3 , it is shown that the theoretical results have a large uncertain region. Thus, although the theoretical results can describe the J/ψ cross-section versus Y in p+p and p+Pb collisions, as shown in Fig. 4(a) Fig. 4(a) , their ratios can’t fit well with the data of R pPb ( Y ). In Fig. 4(b) Fig. 4(b) , it is shown that the theoretical results fit the data well except at very low P T . In Fig. 5 Fig. 5 , we analyze the J/ψ production in p+Pb collisions at forward (a) and backward (b) rapidity at the center of mass energy of <math id="M53"><mrow><msqrt><mi>s</mi></msqrt><mo>=</mo><mn>8.16</mn></mrow></math> TeV. The integral ranges of rapidity are 1.5 < Y < 4 and -5 < Y < -2.5 for Fig. 5(a) and (b) Fig. 5(a) and (b) , respectively. The theory uncertainty results are the same as in Fig. 2 Fig. 2 and the data come from the recent results of LHCb [ 34 34 ]. It is shown that the theoretical results at backward rapidity do not fit well with the data, especially at large P T . Because the Bjorken- x is very large at backward rapidity, the CGC theory is not very effective in this rapidity range. The commonly called cold nuclear effect is an effective method to study J/ψ production at backward rapidity [ 6 6 ]. In summary, we have investigated forward J/ψ production in p+p and p+Pb collisions at LHC energies in the CGC framework together with the CEM. Considering the transverse distribution of the proton and nucleus, the theoretical results are in good agreement with the data from ALICE and LHCb with the analytic UGD from the KLR-AdS/CFT model. 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Accessed 15 Sep 2014 Charmonium Production at forward rapidity in pp, p-Pb and Pb-Pb collisions, with ALICE LHCb Collaboration , Prompt and nonprompt J/ψ production and nuclear modification in pPb collisions at <math id="M58"><mrow><msqrt><mrow><msub><mi>s</mi><mrow><mtext>NN</mtext></mrow></msub></mrow></msqrt></mrow></math> =8.16 TeV . Phys. Lett. B 774 , 159 ( 2017 ). doi: 10.1016/j.physletb.2017.09.058 http://doi.org/10.1016/j.physletb.2017.09.058 Prompt and nonprompt J/ψ production and nuclear modification in pPb collisions at sNN =8.16 TeV 10.1007/s41365-018-0452-7 This work was supported by the Natural Science Foundation of Hebei Province (A2012210043). 2018-02-22 2018-04-14 2018-04-16 2018-07-04 2018-08-01 2021-04-11 © China Science Publishing & Media Ltd. (Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2018 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-018-0452-7 http://dx.doi.org/10.1007/s41365-018-0452-7 . 2018 Nuclear Science and Techniques 08 29