The longitudinal structure function is considered as the next-to-leading order approximation using the expansion method, defined by Ducati and Goncalves [Phys. Lett. B 390 , 401 (1997)] and further developed by Chen $\textit{et al}$. [Chin. Phys. C 48 , 063104 (2024)]. This method provides results for a wide range of $\textit{x}$ and $ Q^2$ values. The longitudinal structure function has been observed to depend on the fractional momentum carried by gluons at low $\textit{x}$. The extracted longitudinal structure functions $ F_{L}(x,Q^2)$ were in line with the data from the H1 Collaboration [Andreev $\textit{et al}$. (H1 Collaboration), Eur. Phys. J. C 74 , 2814 (2014)] and the CT18 parametrization method [Hou $\textit{et al}$. Phys. Rev. D 103 , 014013 (2021)].

Iran

China

G.R. Boroun

Yanbing Cai

An analysis of the longitudinal structure function at next-to-leading order approximation at small-x * G. R. Boroun Supported by Razi University

The GLR-MQ equation is a nonlinear evolution equation that takes into account the shadowing effect, which tames the growth of the gluon at small-x. In this study, we analytically solve for the first time the nonlinear GLR-MQ equation using the homogeneous balance method. The definite solution of the GLR-MQ equation is obtained by fitting the MSTW2008LO gluon distribution data. We find that the geometric scaling is an inherent property of our analytical solution. Furthermore, the gluon distribution functions derived from our solution are capable of accurately reproducing the MSTW2008LO data across most regions. These results indicate that our analytical solution from the homogeneous balance method is valid to describe the gluon behavior at small-x and moderate <math><msup><mrow><mi>Q</mi></mrow><mrow><mn>2</mn></mrow></msup></math> regions. Moreover, the saturation scale <math><msub><mrow><mi>Q</mi></mrow><mrow><mi>s</mi></mrow></msub></math> has been extracted from our analytical solution, we find that the energy-dependent saturation scale obeys the exponential law <math><msubsup><mrow><mi>Q</mi></mrow><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mspace width="0.2em"></mspace><mo>∝</mo><mspace width="0.2em"></mspace><msubsup><mrow><mi>Q</mi></mrow><mrow><mn>0</mn></mrow><mrow><mn>2</mn></mrow></msubsup><msup><mrow><mi>e</mi></mrow><mrow><mi>λ</mi><mi>Y</mi></mrow></msup></math>. We also perform a numerical simulation using the analytical results at <math><msup><mrow><mi>Q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mn>5</mn><mspace width="0.25em"></mspace><msup><mrow><mtext>GeV</mtext></mrow><mrow><mn>2</mn></mrow></msup></math> as the initial condition. We find that our analytical results are consistent with the numerical results in small-x.

Xiaopeng Wang

Xurong Chen

New analytical solution of the nonlinear Gribov-Levin-Ryskin-Mueller-Qiu equation

The Abelian decomposition of QCD reveals two types of gluons: color-neutral “neurons” and color-carrying “chromons”. This classification does not alter the overall properties of QCD, but the investigation of different types of gluon dynamics is necessary. By employing the Cho–Duan–Ge decomposition theorem, we have derived dynamic evolution equations for two types of gluons by using the time-ordered perturbation theory. We propose that the new equations are compatible with the DGLAP equations, requiring only the separate contributions of neurons and chromons to be summed. Surprisingly, with the evolution to high  $$Q^2$$  <math> <msup> <mi>Q</mi> <mn>2</mn> </msup> </math>  , the ratio of the number of chromons to neurons is approximately 3:1 in small-x region regardless of the inputs at evolution starting point. The new gluon dynamic equations reevaluate the gluon distribution functions and allow for a elaborate investigation of the distinct contributions of gluons in high-energy collisions.

Yirui Yang

Wei Kou

Deriving the QCD evolution equations under the Abelian decomposition scheme

This paper presents a comprehensive analysis of the MD-BFKL equation, considering both shadowing and anti-shadowing effects in gluon recombination processes. By deriving analytical expressions for unintegrated gluon distributions through the solution of the MD-BFKL equation, with and without the incorporation of the anti-shadowing effect, we offer new insights into the influence of these effects on the behavior of unintegrated gluon distributions. Our results, when compared to those from the CT18NLO gluon distribution function, demonstrate that the anti-shadowing effect has a notably stronger impact on the characteristics of unintegrated gluon distributions, particularly in regions of high rapidity and momentum. This work significantly contributes to the understanding of gluon recombination mechanisms and their implications in high energy physics.

Exploring the impact of anti-shadowing effect on unintegrated gluon distributions in the MD-BFKL equation  

An approximated solution for the gluon distribution from DGLAP evolution equations with the NLO splitting function in the small-x limit is presented. We first obtain simplified forms of the LO and NLO splitting functions in the small-x limit. With these approximated splitting functions, we obtain the analytical gluon distribution using the Mellin transform. The free parameters in the boundary conditions are obtained by fitting the CJ15 gluon distribution data. We find that the asymptotic behavior of the gluon distribution is consistent with the CJ15 data; however, the NLO results considering the "ladder" structure of gluon emission are slightly better than the LO results. These results indicate that the corrections from NLO have a significant influence on the behavior of the gluon distribution in the small-x region. In addition, we investigate the DGLAP evolution of the proton structure function using the analytical solution of the gluon distribution. The differential structure function reveals that our results have a similar tendency to the CJ15 data at small-x.

Jingxuan Chen

Qian Wang

Analysis of the gluon distribution with next-to-leading order splitting function at small-x  

In this study, we employ the homogeneous balance method to obtain an analytical solution to the Balitsky-Kovchegov equation with running coupling constant. We utilize two distinct prescriptions of the running coupling scale, namely the saturation scale dependent running coupling and the dipole momentum dependent running coupling. By fitting the proton structure function experimental data, we determine the free parameters in the analytical solution. The resulting <math><msup><mi>χ</mi><mn>2</mn></msup><mo>/</mo><mrow><mi>d</mi><mo>.</mo><mi>o</mi><mo>.</mo><mi>f</mi><mo>.</mo></mrow></math> values are determined to be 1.07 and 1.43, respectively. With these definitive solutions, we are able to predict exclusive <math><mi>J</mi><mo>/</mo><mi>ψ</mi></math> production and demonstrate that analytical solutions with running coupling are in excellent agreement with <math><mi>J</mi><mo>/</mo><mi>ψ</mi></math> differential and total cross section. Furthermore, our numerical results indicate that the analytical solution of the BK equation with running coupling constant can provide a reliable description for both the proton structure function and exclusive vector meson production.

Analytic solution of the Balitsky-Kovchegov equation with a running coupling constant using the homogeneous balance method

We calculate the centrality dependence for coherent photoproduction of very low-<math id="M2"><msub><mrow><mi>p</mi></mrow><mrow><mi>T</mi></mrow></msub></math><math id="M3"><mi>J</mi><mo>/</mo><mi>ψ</mi></math> at Relativistic Heavy Ion Collider (RHIC) and Large Hadron Collider (LHC) energies within the impact parameter-dependent saturated color dipole model. By using the large equivalent photon fluxes, we present the differential cross-section of very low-<math id="M4"><msub><mrow><mi>p</mi></mrow><mrow><mi>T</mi></mrow></msub></math><math id="M5"><mi>J</mi><mo>/</mo><mi>ψ</mi></math> produced by coherent photonuclear in peripheral heavy ion collisions. The numerical results demonstrate that our calculation agrees with <math id="M6"><mi>J</mi><mo>/</mo><mi>ψ</mi></math> data in peripheral heavy ion collisions at Relativistic Heavy Ion Collider (RHIC) energies.

Gongming Yu

Yongping Fu

Haitao Yang

Quangui Gao

Qiang Hu

Liyuan Hu

Wei Li

Yushou Song

Coherent Photoproduction of Low- Charmonium in Peripheral Heavy Ion Collisions within the Color Dipole Model

To test the contributions of the quark subprocesses to the inclusive diffractive heavy quarkonium photoproduction, we first investigate the inclusive nondiffractive direct and resolved photoproduction of <math><mrow><mi>J</mi><mo>/</mo><mi>Ψ</mi></mrow></math> in the framework of nonrelativistic quantum chromodynamics (NRQCD). We find that the theoretical total cross section of heavy quarkonium productions are in good agreement with the data available at HERA, once the <math><mrow><mi>γ</mi><mi>q</mi></mrow></math>, <math><mrow><mi>q</mi><mi>g</mi></mrow></math>, and <math><mrow><mi>q</mi><mi>q</mi></mrow></math> subprocesses in the heavy quark pair productions are taken into account. Then we predict the inclusive diffractive rapidity and transverse momentum distributions of <math><mrow><mi>J</mi><mo>/</mo><mi>Ψ</mi></mrow></math>, <math><mrow><mi>Ψ</mi><mo>(</mo><mn>2</mn><mi>S</mi><mo>)</mo></mrow></math>, and <math><mrow><mi>Υ</mi><mo>(</mo><mn>1</mn><mi>S</mi><mo>)</mo></mrow></math> in <math><mrow><mi>p</mi><mi>p</mi></mrow></math>, <math><mrow><mi>p</mi><mi>Pb</mi></mrow></math>, and <math><mi>PbPb</mi></math> collisions at the CERN Large Hadron Collider by using our quark improved NRQCD model. We find that the contributions from the quark involved subprocesses to quarkonium photoproduction are about <math><mrow><mn>6</mn><mo>%</mo><mtext>–</mtext><mn>15</mn><mo>%</mo></mrow></math>. The numerical results show that the contributions from quark involved subprocesses are significant modification in inclusive diffractive heavy quarkonium photoproduction.

Jiayu Wu

Wenchang Xiang

Inclusive diffractive heavy quarkonium photoproduction including quark subprocesses

An extended collinearly improved Balitsky-Kovchegov evolution equation in the target rapidity representation is derived by including the running coupling corrections during the expansion of the “real” <math><mi>S</mi></math>-matrix. We find that the running coupling brings important modifications to the evolution equation, as one can see that there are extra contributions to the evolution kernel once the running coupling is included. To identify the significance of the modifications, we numerically solve the evolution equation with and without the running coupling contributions during the <math><mi>S</mi></math>-matrix expansion. The numerical results show that the scattering amplitude is largely suppressed by the running coupling corrections, which indicate that one needs to consider the running coupling contributions during the derivation of the nonlinear evolution equation in the target rapidity representation.

Mengliang Wang

Daicui Zhou

Extended collinearly-improved Balitsky-Kovchegov evolution equation in target rapidity

The exclusive photoproduction of vector mesons (
					 and 
					) is investigated by considering the next-to-leading order corrections in the framework of the color glass condensate. We compare the next-to-leading order modified dipole amplitude with the HERA data, finding a good agreement. Our studies show that the 
					 from the leading order, running coupling, and collinearly improved next-to-leading order dipole amplitudes are 2.159, 1.097, and 0.932 for the elastic cross-section, and 2.056, 1.449, and 1.357 for the differential cross-section, respectively. The results indicate that the higher-order corrections contribute significantly to the vector meson productions, and the description of the experimental data is dramatically improved once the higher order corrections are included. We extend the next-to-leading order exclusive vector meson production model to LHC energies using the same parameters obtained from HERA. We find that our model provides a rather good description of the 
					 and 
					 data in proton-proton collisions at 7 TeV and 13 TeV in LHCb experiments.
				

Exclusive photoproduction of vector meson at next-to-leading order from color glass condensate Supported by the National Natural Science Foundation of China (11765005, 11947119, 11305040, 11847152, 11775097), the Fund of Science and Technology Department of Guizhou Province ([2018]1023, [2019]5653), the Education Department of Guizhou Province (KY[2017]004), Qian Kehe Platform Talents ([2017]5736-027), the National key research and development program of China (2018YFE0104700, CCNU18ZDPY04), and the 2018 scientific research startup foundation for the introduced talent of Guizhou University of Finance and Economics (2018YJ60)