<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>

China

Rong Wang

Chengdong Han

Xurong Chen

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

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.

Wei Kou

Xiaopeng Wang

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

We present the gluon distribution functions of the pion and kaon in small-x and large-x regions, and compare them with the results obtained from lattice QCD and continuum Schwinger function methods. Whether in the small-x region or the large-x region, our gluon distribution of the pion is consistent with the results of lattice QCD and continuum Schwinger function methods. We also calculate the first four moments of gluon distributions of the pion and kaon at different  $$Q^{2}$$  <math> <msup> <mi>Q</mi> <mn>2</mn> </msup> </math>   scales. In addition, we present the mass decomposition of the pion and kaon with the dynamical parton distribution functions calculated by the DGLAP equation with parton–parton recombination corrections. The mass structures of the pion and kaon are completely different from that of the proton.

Gluon distribution and mass decomposition of the pion and kaon

In this study, we determine the mass radius of <math><mmultiscripts><mi>He</mi><mprescripts></mprescripts><none></none><mn>4</mn></mmultiscripts></math>, a very light nucleus, by examining the near-threshold <math><mi>ϕ</mi></math>-meson photoproduction data of the LEPS Collaboration. To assess the gravitational form factor of <math><mmultiscripts><mi>He</mi><mprescripts></mprescripts><none></none><mn>4</mn></mmultiscripts></math>, we employ multiple models for the mass distribution, including Yukawa-type, exponential, Gaussian, and uniform functions. The mass radius of <math><mmultiscripts><mi>He</mi><mprescripts></mprescripts><none></none><mn>4</mn></mmultiscripts></math> is measured to be <math><mrow><mn>1.70</mn><mo>±</mo><mn>0.14</mn></mrow></math> fm, which is approximately equal to the charge radius of <math><mmultiscripts><mi>He</mi><mprescripts></mprescripts><none></none><mn>4</mn></mmultiscripts></math>. Surprisingly, in contrast to the findings of the proton, no noticeable discrepancy between the charge radius and the mass radius is noted for the <math><mmultiscripts><mi>He</mi><mprescripts></mprescripts><none></none><mn>4</mn></mmultiscripts></math> nucleus. The proton and neutron distributions within <math><mmultiscripts><mi>He</mi><mprescripts></mprescripts><none></none><mn>4</mn></mmultiscripts></math> are likely to be identical, confirming its regular tetrahedral structure in a new way. We propose exploring the difference between charge and mass radii as a new approach to examine the nuclear structure.

Exploring the mass radius of <math><mmultiscripts><mi>He</mi><mprescripts></mprescripts><none></none><mn>4</mn></mmultiscripts></math> and implications for nuclear structure

We calculate the valence quark distribution functions of the π meson using the non-relativistic chiral constituent quark model. The π wave function is obtained by solving the two-body Schrödinger equation within the framework of constituent quark model. We transform the π wave function from the rest frame to the light cone or infinite momentum frame based on the Lorentz boost. The valence quark distributions at the initial evolution scale are obtained. The QCD evolution is given with the DGLAP equations with parton-parton recombination corrections. With tuning the valence up (down) quark mass to 70 MeV, the calculated valence up quark distributions at <math><msup><mrow><mi>Q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mn>20</mn></math> GeV$^{2}$ are in good agreement with the E615 experimental data. The structure functions <math><mrow><msubsup><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>π</mi></mrow></msubsup><mo>(</mo><mi>x</mi></mrow><mo>,</mo><mrow><msup><mrow><mi>Q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></math> of pion are also calculated which consist with the H1 experimental data. The proposed mechanisms here could be also used to study other hadrons.

Qian Wu

Canada

Di Qing

Fan Wang

Ju-Jun Xie

Pion parton distribution functions with the nonrelativistic constituent quark model

In this study, we try to extract <math><mrow><mi>ω</mi><mi>n</mi></mrow></math>, <math><mrow><mi>ω</mi><mi>p</mi></mrow></math>, and <math><mrow><mi>ϕ</mi><mi>N</mi></mrow></math> scattering lengths from the differential cross-section data of near-threshold <math><mi>ω</mi></math> and <math><mi>ϕ</mi></math> photoproductions at threshold <math><msub><mi>t</mi><mi>thr</mi></msub></math>. The incoherent data of <math><mi>ω</mi></math> and <math><mi>ϕ</mi></math> photoproductions on deuterium target for scattering length extractions are provided by the CBELSA/TAPS Collaboration and LEPS Collaboration, respectively, where the deuteron is usually approximated as a weakly bound state of a quasifree neutron plus a quasifree proton. Under the vector meson dominance model and the assumption of energy independence of the differential cross section, we obtained the <math><mrow><mrow><mo>|</mo></mrow><msub><mi>α</mi><mrow><mi>ω</mi><msup><mi>n</mi><mo>*</mo></msup></mrow></msub><mrow><mo>|</mo></mrow></mrow></math> and <math><mrow><mrow><mo>|</mo></mrow><msub><mi>α</mi><mrow><mi>ω</mi><msup><mi>p</mi><mo>*</mo></msup></mrow></msub><mrow><mo>|</mo></mrow></mrow></math> of loosely bound neutron and proton in the deuteron to be 0.709 <math><mo>±</mo></math> 0.078 fm and 0.621 <math><mo>±</mo></math> 0.022 fm, respectively, at threshold <math><msub><mi>t</mi><mi>thr</mi></msub></math> for the first time by fitting the near-threshold data of <math><mrow><mi>γ</mi><mi>d</mi></mrow></math> <math><mo>→</mo></math> <math><mrow><mi>ω</mi><mi>n</mi><mo>(</mo><mi>p</mi><mo>)</mo></mrow></math> and <math><mrow><mi>γ</mi><mi>d</mi></mrow></math> <math><mo>→</mo></math> <math><mrow><mi>ω</mi><mi>p</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow></math>. For a comparison study, we also extracted the scattering length <math><mrow><mrow><mo>|</mo></mrow><msub><mi>α</mi><mrow><mi>ω</mi><mi>p</mi></mrow></msub><mrow><mo>|</mo></mrow></mrow></math> of the free proton from the <math><mi>ω</mi></math> photoproduction on the hydrogen target by the CBELSA/TAPS Collaboration. With the near-threshold and incoherent <math><mi>ϕ</mi></math> photoproduction data of <math><mrow><mi>γ</mi><mi>d</mi></mrow></math> <math><mo>→</mo></math> <math><mrow><mi>ϕ</mi><mi>p</mi><mi>n</mi></mrow></math> by the LEPS Collaboration, we obtained scattering length <math><mrow><mrow><mo>|</mo></mrow><msub><mi>α</mi><mrow><mi>ϕ</mi><mi>N</mi></mrow></msub><mrow><mo>|</mo></mrow></mrow></math> of the bound nucleon inside the deuteron to be 0.109 <math><mo>±</mo></math> 0.008 fm at threshold <math><msub><mi>t</mi><mi>thr</mi></msub></math>. The obtained <math><mrow><mrow><mo>|</mo></mrow><msub><mi>α</mi><mrow><mi>ω</mi><mi>p</mi></mrow></msub><mrow><mo>|</mo></mrow></mrow></math> and <math><mrow><mrow><mo>|</mo></mrow><msub><mi>α</mi><mrow><mi>ϕ</mi><mi>N</mi></mrow></msub><mrow><mo>|</mo></mrow></mrow></math> are basically in agreement with the experimental indications and the theoretical predictions within the current uncertainties.

Extraction of <math><mrow><mi>ω</mi><mi>n</mi></mrow></math>, <math><mrow><mi>ω</mi><mi>p</mi></mrow></math>, and <math><mrow><mi>ϕ</mi><mi>N</mi></mrow></math> scattering lengths from <math><mi>ω</mi></math> and <math><mi>ϕ</mi></math> differential photoproduction cross sections on a deuterium target

We present the polarized parton distribution functions from a QCD analysis of the worldwide polarized deep inelastic scattering data, based on the dynamical parton distribution model. All the sea quarks and gluons are dynamically generated from QCD radiations, with the nonperturbative input comprising only the polarized valence quark distributions. This approach leads to a simple parametrization, which has only seven free parameters. In the analysis, we apply the DGLAP equations with parton-parton recombination corrections. For the first step, the DGLAP equations, parton-parton recombination corrections, and experimental observable <math><msub><mrow><mi>g</mi></mrow><mrow><mn>1</mn></mrow></msub></math> are all evaluated at the leading order. The parameterized nonperturbative input at an extremely low <math><msubsup><mrow><mi>Q</mi></mrow><mrow><mn>0</mn></mrow><mrow><mn>2</mn></mrow></msubsup></math> reproduces well the spin-dependent structure functions measured at high <math><msup><mrow><mi>Q</mi></mrow><mrow><mn>2</mn></mrow></msup></math>. Comparisons with experimental observations and some other polarized parton distribution functions are also shown. Our results are in good agreement with the experimental data and consistent with some other parameterized models. Furthermore, our analysis presents the positive polarized gluon distribution and it suggests that the gluon distribution plays an important role to the proton spin content. The polarized antiquark distributions are non-zero at high <math><msup><mrow><mi>Q</mi></mrow><mrow><mn>2</mn></mrow></msup></math> but quite small compared to polarized valence quark distributions, based on this dynamical parton model analysis. This analysis gives smaller statistical uncertainties for the polarized sea quark and gluon distributions, thanks to the fewer free parameters used for the parametrization of the initial polarized PDFs.

Gang Xie

An analysis of polarized parton distribution functions with nonlinear QCD evolution equations

Measuring the kaon structure beyond proton and pion structures is a prominent topic in hadron physics, as it is one way to understand the nature of the Nambu-Goldstone boson of QCD and observe the interplay between the EHM and HB mechanisms for hadron mass generation. In this study, we present a simulation of the leading Λ baryon tagged deep inelastic scattering experiment at EicC (Electron-ion collider in China), which is engaged to unveil the internal structure of kaon via the Sullivan process. According to our simulation results, the suggested experiment will cover the kinematical domain of   $ 0.05\lesssim x_{\rm K} \lesssim 0.85 $   and   $ 0.05\lesssim x_{\rm K} \lesssim 0.85 $   up to 50 GeV   $ 0.05\lesssim x_{\rm K} \lesssim 0.85 $  , with the acceptable statistical uncertainties. In the relatively low-   $ 0.05\lesssim x_{\rm K} \lesssim 0.85 $   region (   $ 0.05\lesssim x_{\rm K} \lesssim 0.85 $   GeV   $ 0.05\lesssim x_{\rm K} \lesssim 0.85 $  ), the Monte-Carlo simulation shows a good statistical precision (   $ 0.05\lesssim x_{\rm K} \lesssim 0.85 $  %) for the measurement of the kaon structure function   $ 0.05\lesssim x_{\rm K} \lesssim 0.85 $  . In the high-   $ 0.05\lesssim x_{\rm K} \lesssim 0.85 $   region (up to 50 GeV   $ 0.05\lesssim x_{\rm K} \lesssim 0.85 $  ), the statistical uncertainty of   $ 0.05\lesssim x_{\rm K} \lesssim 0.85 $   is also acceptable (   $ 0.05\lesssim x_{\rm K} \lesssim 0.85 $  %) for the data at   $ 0.05\lesssim x_{\rm K} \lesssim 0.85 $  . To perform such an experiment at an electron-ion collider, a high-performance zero-degree calorimeter is suggested. The magnitude of the background process and the assumed detector capabilities are also discussed and illustrated in the paper.

Tackling the kaon structure function at EicC  

We analyze the exclusive <math><mi>ϕ</mi></math>-meson photoproduction on both hydrogen and deuterium targets based on the published data of CLAS, SAPHIR and LEPS collaborations. A dipole-form scalar gravitational form factor is applied to describe the <math><mo>|</mo><mi>t</mi><mo>|</mo></math>-dependence of the differential cross section. Based on the average of all the near-threshold data from CLAS, SAPHIR and LEPS, we find that the proton and deuteron mass radii are <math><mn>0.75</mn><mo>±</mo><mn>0.02</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></math> and <math><mn>1.95</mn><mo>±</mo><mn>0.19</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></math> respectively. The coherent and near-threshold quarkonium photoproduction seems to be sensitive to the radius of the hadronic system. The vector-meson-dominance model together with a low-energy QCD theorem well describe the data of near-threshold <math><mi>ϕ</mi></math> photoproduction on the hadronic systems.

Jarah Evslin

Proton and deuteron mass radii from near-threshold <math><mi>ϕ</mi></math>-meson photoproduction

We present pion and kaon parton distribution functions from a global QCD analysis of the experimental data within the framework of dynamical parton model. We use the DGLAP equations with parton–parton recombination corrections and the valence input of uniform distribution which maximizes the information entropy. At our input scale  $$Q_0^2$$  <math> <msubsup> <mi>Q</mi> <mn>0</mn> <mn>2</mn> </msubsup> </math>  , there are no sea quark and gluon distributions. All the sea quarks and gluons of the pion and the kaon are completely generated from the parton splitting processes. The mass-dependent parton splitting kernel is applied for the strange quark distribution in the kaon. The obtained valence quark and sea quark distributions at high  $$Q^{2}$$  <math> <msup> <mi>Q</mi> <mn>2</mn> </msup> </math>   ( $$Q^2&gt;5$$  <math> <mrow> <msup> <mi>Q</mi> <mn>2</mn> </msup> <mo>&gt;</mo> <mn>5</mn> </mrow> </math>   GeV $$^2$$  <math> <msup> <mrow></mrow> <mn>2</mn> </msup> </math>  ) are compatible with the existed experimental measurements. Furthermore, the asymptotic behaviours of parton distribution functions at small and large x have been studied for both the pion and the kaon. Lastly, the first three moments of parton distributions at high  $$Q^{2}$$  <math> <msup> <mi>Q</mi> <mn>2</mn> </msup> </math>   scale are calculated, which are consistent with other theoretical predictions.