The production of heavy-quark (HQ) jets is a new area that addresses the mass effect of jet quenching in heavy-ion physics. This paper presents a theoretical study of HQ jet yield suppression in Pb+Pb collisions at the Large Hadron Collider (LHC) and focuses on the energy loss of HQ jets produced by different mechanisms. The $\textit{p+p}$ baseline is provided by the generator simulation of high-energy reactions of particles (SHERPA), and the jet-medium interactions are described by the SHELL transport model, which considers the elastic and inelastic partonic energy loss in the quark-gluon plasma (QGP). In $\textit{p+p}$ collisions, our numerical results indicate that the HQ jets from gluon splitting ( $ g \rightarrow Q $ -jet) are the dominant contribution at high $ p_T $ , displaying more dispersive structures than the HQ-initiated ( $ Q \rightarrow Q $ -jet). In nucleus-nucleus collisions, our calculations were consistent with the inclusive and b-jet $ R_{AA} $ recently measured by the ATLAS collaboration, revealing a remarkable manifestation of the mass effect of jet energy loss. As a result of the dispersive substructure, the $ g \rightarrow Q $ -jet loses more energy than the $ Q \rightarrow Q $ -jet in the QGP. Due to the significant contribution of $ g \rightarrow c $ -jet, the $ R_{AA} $ of $\textit{c}$-jet is comparable or even smaller than that of inclusive jet. To experimentally distinguish the $ g \rightarrow Q $ -jet and $ Q \rightarrow Q $ -jet, we propose event selection strategies based on their topological features and test their performances. By isolating the $ c \rightarrow c $ -jet, $ b \rightarrow b $ -jet, and the jets initiated by heavy quarks, we predicted that the order of their $ R_{AA} $ are in line with the mass hierarchy of energy loss. Future measurements on the $ R_{AA} $ of $ Q \rightarrow Q $ -jet and $ g \rightarrow Q $ -jet will provide a unique opportunity for testing the flavor/mass dependence of energy loss at the jet level.

China

Sa Wang

Shuang Li

Yao Li

Ben-Wei Zhang

Enke Wang

Probing the mass effect of heavy quark jets in high-energy nuclear collisions * Supported by the MajorProject of Basic and Applied Basic Research of Guangdong Province, China (2020B0301030008, 2023A1515011460), and the National Natural Science Foundation of China (11935007, 12035007, 12247127, 12375137). Sa Wang is supported by the Open Foundation of Key Laboratory of Quark and Lepton Physics (MOE) (QLPL2023P01) and the Talent Scientific Star-up Foundation of the China Three Gorges University, China (CTGU) (2024RCKJ013)

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.

Yanbing Cai

Xiaopeng Wang

Xurong Chen

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

In this study we demonstrate the application of the maximum entropy method (MEM) to determine the valence quark distribution of exotic hadrons. Our investigation yields three key findings. Firstly, we observe a significant shift towards smaller Bjorken scale x in the peak position of the valence quark distribution for hadrons with an increasing number of valence quarks, consistent with previous results by Kawamura and Kumano. Secondly, assuming that the  $$Z_c(3900)$$  <math> <mrow> <msub> <mi>Z</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <mn>3900</mn> <mo>)</mo> </mrow> </mrow> </math>   initially consists of four valence quarks, we employ MEM to determine its initial valence quark distribution, estimating a radius of  $$r_c=1.276$$  <math> <mrow> <msub> <mi>r</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>1.276</mn> </mrow> </math>   fm at an extremely low resolution scale  $$Q^2$$  <math> <msup> <mi>Q</mi> <mn>2</mn> </msup> </math>  . Furthermore, we identify a notable discrepancy between our computed charge form factor  $$G_c(q)$$  <math> <mrow> <msub> <mi>G</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <mi>q</mi> <mo>)</mo> </mrow> </mrow> </math>   at leading order and the outcomes of hadron molecular state calculations. We propose that this form factor can be extracted from the QCD counting rule cross-section, which is grounded in generalized distribution amplitudes (GDA) linked to the multi-quark states.

Chengdong Han

Wei Kou

Maximum entropy method for valence quark distributions in exotic hadrons: a study of the  $$Z_c(3900)$$  <math> <mrow> <msub> <mi>Z</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <mn>3900</mn> <mo>)</mo> </mrow> </mrow> </math>   case

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

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  

Understanding the production mechanisms and utilizing them as probes to investigate the structure of exotic states represent some of the most actively studied research areas in particle physics. In this study, we present a theoretical analysis of the charmed meson <math><msubsup><mi>T</mi><mrow><mi>c</mi><mi>c</mi></mrow><mo>−</mo></msubsup></math> in the <math><mi>γ</mi><mi>p</mi><mo>→</mo><msup><mi>D</mi><mo>+</mo></msup><msubsup><mi>T</mi><mrow><mi>c</mi><mi>c</mi></mrow><mo>−</mo></msubsup><msubsup><mi>Λ</mi><mi>c</mi><mo>+</mo></msubsup></math> and <math><mi>γ</mi><mi>γ</mi><mo>→</mo><msup><mi>D</mi><mo>+</mo></msup><msubsup><mi>T</mi><mrow><mi>c</mi><mi>c</mi></mrow><mo>−</mo></msubsup><msup><mi>D</mi><mrow><mo>*</mo><mn>0</mn></mrow></msup></math> reactions, considering <math><msubsup><mi>T</mi><mrow><mi>c</mi><mi>c</mi></mrow><mo>+</mo></msubsup></math> as a <math><mi>D</mi><msup><mi>D</mi><mo>*</mo></msup></math> molecule. The differential cross section and total cross section for the photoproduction of <math><msubsup><mi>T</mi><mrow><mi>c</mi><mi>c</mi></mrow><mo>−</mo></msubsup></math> in the <math><mi>γ</mi><mi>p</mi><mo>→</mo><msup><mi>D</mi><mo>+</mo></msup><msubsup><mi>T</mi><mrow><mi>c</mi><mi>c</mi></mrow><mo>−</mo></msubsup><msubsup><mi>Λ</mi><mi>c</mi><mo>+</mo></msubsup></math> and <math><mi>γ</mi><mi>γ</mi><mo>→</mo><msup><mi>D</mi><mo>+</mo></msup><msubsup><mi>T</mi><mrow><mi>c</mi><mi>c</mi></mrow><mo>−</mo></msubsup><msup><mi>D</mi><mrow><mo>*</mo><mn>0</mn></mrow></msup></math> reactions are presented for nucleus-nucleus (nucleon-nucleon) ultraperipheral collisions at HL-LHC and RHIC, respectively. Taking into account the integrated luminosity per typical run and the luminosity of photons from the nucleus (nucleon), we observe a significant event count for <math><msubsup><mi>T</mi><mrow><mi>c</mi><mi>c</mi></mrow><mo>−</mo></msubsup></math> production in <math><mrow><mi>p</mi><mtext>−</mtext><mi>p</mi></mrow></math> ultraperipheral collisions at HL-LHC.

Yaping Xie

Yin Huang

Exclusive production of <math><msubsup><mi>T</mi><mrow><mi>c</mi><mi>c</mi></mrow><mo>−</mo></msubsup></math> in hadron-hadron ultraperipheral collisions

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 study the deep inelastic scattering (DIS) of a proton-targeted lepton in the presence of gluon condensation using gauge/gravity duality. We use a modified   ${\rm AdS}_5$   background where the modification parameter c corresponds to the gluon condensation in the boundary theory. First, by examining the electromagnetic field, we establish that a non-zero c can increase field magnitude. Our aim is to obtain the acceptable value of c for this scattering. Our method is based on setting the mass of the proton as an eigenvalue of the baryonic state equations of the DIS to find the acceptable value of the parameter c on the other side of the equations. Therefore, in the second step, we calculate wave function equations for the baryonic states where the mass of the proton target requires a value contribution of c as   ${\rm AdS}_5$  . Proceeding with the electromagnetic field and baryonic states, we derive the holographic interaction action related to the amplitude of the scattering. Finally, we compute the corresponding structure functions numerically as functions of x and q, which are Björken variables and the lepton momentum transfers, respectively. Comparing the Jlab Hall C data with our theoretical calculations, our results are acceptable.

Sara Tahery

Deep inelastic scattering with proton target in the presence of gluon condensation using holography  

The partons model reveals the dynamical structure of nucleons (protons and neutrons). Studies related to thermodynamic quantities of nucleons present interesting and topical questions. In this work, for the first time we apply Page’s theory of the studies of black holes to investigate the entropy of a proton system. Inspired by the quantum entanglement entropy in black hole information theories, we establish the proton entanglement with the similar way. Based our calculations, the proton entanglement entropy has the approximate form <math><mi>S</mi><mo>=</mo><mi>ln</mi><mi>m</mi><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn></math>, where <math><mi>m</mi></math> represents the partons density of proton (mainly gluon and sea quark contributions) at small Bjorken <math><mi>x</mi></math>. Our calculations using Page’s theory are well in agreement with the recent deep inelastic scattering measurements from the H1 Collaboration.