Generalized distribution amplitudes (GDAs) offer valuable insights into the three-dimensional structure of hadrons, delineating the amplitudes associated with the transition from a quark–antiquark pair to a hadron pair. Currently, hadron GDAs can be probed in electron–positron collisions, with experimental feasibility demonstrated at facilities such as Belle and BESIII. In this study, we put forth the proposition that hadron GDAs can also be investigated in electron–hadron collisions at forthcoming Electron-Ion Colliders (EICs), specifically through the subprocess $$\gamma ^*\gamma \rightarrow h_1 h_2.$$ In this framework, a quasi-real photon, emitted by the ion, exhibits a photon flux proportional to the square of the ion’s electric charge. Consequently, we anticipate that the cross sections in EICs will be substantially larger than those in electron–positron collisions. We present numerical calculations pertaining to di- $$\pi ^0$$ production employing the equivalent photon approximation (EPA). Our findings suggest that, within the same kinematic region, electron-proton ($\textit{e}$-$\textit{p}$) collisions at the EIC could yield an event rate comparable to that of Belle II, while electron-gold ($\textit{e}$-Au) collisions are expected to generate an even greater number of events. This enhanced event rate facilitates a high-precision examination of di- $$\pi ^0$$ GDAs at the EIC.

China

Bing’ang Guo

Jing Han

Ya-Ping Xie

Xurong Chen

Di- $$\pi ^0$$ production and generalized distribution amplitudes at future electron-ion colliders

Understanding the interactions between quark–antiquark pairs is essential for elucidating quark confinement within the framework of quantum chromodynamics (QCD). This study investigates the field distribution patterns that arise between these pairs by employing advanced machine learning techniques, namely multilayer perceptrons (MLP) and Kolmogorov-Arnold networks (KAN), to analyze data obtained from lattice QCD simulations. The models developed through this training are then applied to calculate the string tension and width associated with chromo flux tubes, and these results are rigorously compared to those derived from lattice QCD. Moreover, we introduce a preliminary analytical expression that characterizes the field distribution as a function of quark separation, utilizing the KAN methodology. Our comprehensive quantitative analysis underscores the potential of integrating machine learning approaches into conventional QCD research.

Wei Kou

Machine learning insights into quark–antiquark interactions: probing field distributions and string tension in QCD

Investigation of decay and scattering processes of particles in a vortex state offers a novel and promising approach for probing particle structure. Our study reveals distinct properties of vortex neutron decay, which deviate from those of classical plane-wave neutron decay. We present the energy-angle distributions of the final-state electron and antineutrino in unpolarized vortex neutrons, as well as angle distributions integrated over their energies. Notably, we provide theoretical calculations of the decay behavior of neutrons with varying vortex cone angles and initial energies. We propose that identifying the vortex state of the initial neutron can be achieved by analyzing the angular and energy distributions of the final-state particles, introducing new degrees of freedom for studying weak interactions and neutron decay kinematics that have been previously overlooked in particle physics. This timely investigation takes advantage of recent advancements in vortex neutron preparation and analysis, opening up new avenues for exploring the fundamental properties of matter.

Bing'ang Guo

Unveiling the Secrets of Vortex Neutron Decay

We investigate a new class of transverse momentum-dependent functions (TMDs), known as aligned-spin (AS) functions. In the paper, we propose the most suitable proof of the existence of the AS function together with a demonstration of the preponderances if the framework of twisted quark states has been employed. The twisted state corresponds to the elementary particle (quark), which possesses nontrivial intrinsic orbital angular momentum owing to the swirling trajectory of motion. In turn, this leads to the cylindrical system being applied for consideration. In this connection, we reveal that the twisted (vortex) quark states serve as effective tools for the study of TMDs, thereby facilitating a comprehensive analysis of AS functions. In contrast to the previous studies, where the existence of new TMDs is related to the the corresponding interactions encoded in the correlators, we now focus on the leading order of interactions providing a simplified and robust method. This has been ensured by the twisted quark in the corresponding correlator because the essential transverse momentum dependence of quarks generated by interactions in the correlators can be alternatively described by twisted states. Using a cylindrical formulation for twisted states, we can combine the properties of plane-wave particles with a description stemming from spherical harmonics, resulting in well defined propagation directions accompanied by essential orbital angular momentum (OAM) projections. In particular, this innovative framework opens a new window for the direct investigation of AS functions, generating the unique angular <inline-formula><tex-math id='TM0001' notation='LaTeX'>$\phi$</tex-math></inline-formula> dependence of differential cross sections. It also points towards promising applications in experimental particle physics.

Russia

I V Anikin

TMD-like Functions through the Twisted Quark States

<p>A novel approach for describing the evolution of nucleon structure from low-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>Q</mi><mn>2</mn></msup></math> regime to high-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>Q</mi><mn>2</mn></msup></math> asymptotic region is proposed. This infrared-safe scheme modifies the parton distribution evolution equations to incorporate the corrections from emergent hadron mass mechanisms and parton-parton recombination at low <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>Q</mi><mn>2</mn></msup></math>. The effective quark mass, generated by dynamical chiral symmetry breaking, slows the evolution of parton distributions in the infrared region, causing the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi evolution to freeze when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>Q</mi><mn>2</mn></msup><mo>≪</mo><msub><mi>M</mi><mrow><mi mathvariant="normal">q</mi><mo>/</mo><mi mathvariant="normal">g</mi></mrow></msub></math>. Notably, this scheme renders the high-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>Q</mi><mn>2</mn></msup></math> parton distributions insensitive to the choice of input hadronic scale. The parton-parton recombination effect is crucial in suppressing the rapid growth of parton distributions at small <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math>, consistent with experimental data. When applied to three valence quark distributions derived from a quark model, our scheme yields parton distributions that agree well with deep-inelastic scattering data in both large-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math> and small-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math> regions, providing a unified description of nucleon structure across the entire <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>Q</mi><mn>2</mn></msup></math> range.</p>

Rong Wang

Chengdong Han

Unifying the landscape of nucleon structure: An infrared-safe evolution scheme

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

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

Exclusive photoproduction of heavy <math><mi>J</mi><mo>/</mo><mi>ψ</mi></math> meson is performed within the handbag factorization method. The general parton distributions (GPDs) for gluons, which are an essential ingredient here, are constructed using double distribution (DD) representation. The calculated cross section for few sets of gluon GPDs <math><msub><mi>H</mi><mi>g</mi></msub></math> describe fine <math><mi>J</mi><mo>/</mo><mi>ψ</mi></math> production experimental data from HERA to LHC energies. Based on this approach we estimate <math><mi>J</mi><mo>/</mo><mi>ψ</mi></math> spin asymmetries determined by <math><msub><mi>E</mi><mi>g</mi></msub></math> and <math><msub><mover><mi>H</mi><mo>˜</mo></mover><mi>g</mi></msub></math>. The GPDs factorization approach can be employed to analyze heavy vector meson production in electron-proton scattering. Investigations of this reactions at future electron-ion colliders in USA and China can give essential information on gluon GPDs.

JINR

S. V. Goloskokov

Study of gluon GPDs in exclusive <math><mrow><mi>J</mi><mo>/</mo><mi>ψ</mi></mrow></math> production in electron-proton scattering

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.

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.