This study investigates the ultraviolet realization of semileptonic four-fermion operator $ O_{ledq}^{\alpha \beta 11} $ that incorporates Majorana dark matter (DM) in both lepton-flavor-conserving (LFC) and lepton-flavor-violating (LFV) scenarios at the one-loop level via box diagram, thereby effectively alleviating the lower bounds on the new physics scale. In addition, we investigate the interplay between model-independent constraints on the Wilson coefficients and DM direct detection, relic density, and collider searches in the context of fermion portal DM model with two mediators. We found that both the projected future constraint on the LFC Wilson coefficient $ C_{ledq}^{2211}/\Lambda^2 \lt $ $ (12.3\; \text{TeV})^{-2} $ from the measurements of neutrino non-standard interaction in the next-generation neutrino oscillation experiments, and the LFV constraint $ C_{ledq}^{1211}/\Lambda^2 \lt \left(2.2\times 10^3\; \text{TeV} \right)^{-2} $ from ongoing charged-lepton-flavor-violation searches, provide a complementary exploration of the parameter space encompassing the DM and scalar masses. With a colored mediator mass of approximately $ 2\; \text{TeV} $ , the sensitivity of the indirect constraints on the four-fermion operator can surpass those of collider searches and DM direct detection, in scenarios where the masses of the DM and scalar are close. By ensuring the correct DM relic density, however, we find that the collider searches and DM direct detection are more sensitive to the electroweak-scale DM and scalar than indirect constraints.

China

Yuxuan He

Gang Li

Jia Liu

Xiao-Ping Wang

Xiang Zhao

Lepton flavor of four-fermion operator and fermion portal dark matter * Gang Li and Xiang Zhao are supported by the National Natural Science Foundation of China (12347105, 12505127), the Guangdong Basic and Applied Basic Research Foundation (2024A1515012668). The work of Jia Liu is supported by the National Science Foundation of China (12235001, 12475103) and State Key Laboratory of Nuclear Physics and Technology (NPT2025ZX11). The work of Xiao-Ping Wang is supported by National Science Foundation of China (12375095) and the Fundamental Research Funds for the Central Universities

We propose a novel method for extracting non-singlet (NS) fragmentation functions (FFs) of light charged hadrons from charge asymmetries measured in hadron fragmentation using data from both single-inclusive electron-positron annihilation and semi-inclusive deep-inelastic scattering processes. We determine the NS FFs for pions and kaons at next-to-next-to-leading order in Quantum Chromodynamics through a comprehensive uncertainty analysis. The extracted FFs reveal a scaling index of approximately 0.7 at large momentum fractions and low energy scales, a strangeness suppression factor of approximately 0.5, and universality in fragmentation of light mesons. Our findings establish a valuable benchmark for testing non-perturbative QCD models and Monte Carlo event generators, and serve as crucial inputs for future electron-ion colliders.

Jun Gao

Chong-Yang Liu

Bin Zhou

Determination of fragmentation functions from charge asymmetries in hadron production * The work of Jun Gao is supported by the National Natural Science Foundation of China (NSFC) (12275173), the Shanghai Municipal Education Commission (2024AIZD007) and open fund of Key Laboratory of Atomic and Subatomic Structure and Quantum Control (Ministry of Education)

Large-momentum effective theory is a physics-guided systematic expansion to calculate light-cone parton distributions, including collinear (PDFs) and transverse-momentum-dependent ones, at any fixed momentum fraction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math> within a range of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">[</mo><msub><mi>x</mi><mi>min</mi></msub><mo>,</mo><msub><mi>x</mi><mi>max</mi></msub><mo stretchy="false">]</mo></math>. It theoretically solves the ill-posed inverse problem that afflicts other theoretical approaches to collinear PDFs, such as short-distance factorizations. Recently, Dutrieux et al. raised practical concerns about whether current or even future lattice data will have sufficient precision in the subasymptotic correlation region to support an error-controlled extrapolation—and if not, whether it becomes an inverse problem where the relevant uncertainties cannot be properly quantified. While we agree that not all current lattice data have the desired precision to qualify for an asymptotic extrapolation, some calculations do, and more are expected in the future. We comment on the analysis and results in Dutrieux et al. and argue that a physics-based systematic extrapolation still provides the most reliable error estimates, even when the data quality is not ideal. In contrast, reframing the long-distance asymptotic extrapolation as a data-driven-only inverse problem with ad hoc mathematical conditioning could lead to unnecessarily conservative errors.

Taiwan

Jiunn-Wei Chen

United States

Xiang Gao

Jinchen He

Jun Hua

Xiangdong Ji

Germany

Andreas Schäfer

Yushan Su

Wei Wang

Yi-Bo Yang

Jian-Hui Zhang

Qi-An Zhang

Rui Zhang

Yong Zhao

Large-momentum effective theory’s asymptotic extrapolation vs the inverse problem

In a recent paper, “Nonlocal nucleon matrix elements in the rest frame” [1], it was observed that the next-to-leading order calculations of the renormalization factor can describe, to a few percent accuracy, the logarithm of the lattice quantum chromodynamics (QCD) rest frame matrix elements with separations up to distances of 0.6 fm on multiple lattice spacings. We argue that perturbative QCD breaks down at such a distance scale after resumming the associated large logarithms, while the Ansatz used in the analysis there is not justified in perturbation theory. Besides, we explain the observation in Ref. [1] and demonstrate that the Ansatz fails to describe the data for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>z</mi><mo>&gt;</mo><mn>0.3</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></mrow></math>, showing an opposite trend. Finally, although Ref. [1] proposes multiplying the Ansatz by a Gaussian correction model, which is shown to reduce the discrepancy with the data, this does not legitimize the use of perturbative QCD at such distance scales.

Comment on “Nonlocal nucleon matrix elements in the rest frame”

We present lattice QCD calculations of the odd Mellin moments of pion valence-quark generalized parton distribution up to fifth order <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">⟨</mo><msup><mi>x</mi><mn>4</mn></msup><mo stretchy="false">⟩</mo></math> and for the skewness range <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">[</mo><mo>−</mo><mn>0.33</mn><mo>,</mo><mn>0</mn><mo stretchy="false">]</mo></math> using operator product expansion of bilocal quark-bilinear operators. The calculations are performed on an ensemble with lattice spacing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>a</mi><mo>=</mo><mn>0.04</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></math> and valence pion mass 300 MeV, employing boosted pion states with momenta up to 2.428 GeV and momentum transfers reaching <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>2.748</mn><mtext> </mtext><mtext> </mtext><msup><mi>GeV</mi><mn>2</mn></msup></math>. We employ ratio-scheme renormalization and next-to-leading logarithmic resummed perturbative matching. At zero skewness, our results are consistent with previous lattice studies. By combining matrix elements at multiple values of skewness and momentum transfer, skewness-dependent moments are obtained through simultaneous polynomiality-constrained fits.

Swagato Mukherjee

Qi Shi

Fei Yao

Skewness-dependent moments of the pion GPD from nonlocal quark-bilinear correlators

We investigated the impact of a constant external magnetic field on the dressed propagators of up-, down-, and strange quarks. In the weak field limit, we derive a general momentum-space representation for the propagator and numerically solve the corresponding gap equation. Our analysis reveals that the vector term of the propagator can be decomposed into components parallel and perpendicular to the magnetic field, resulting in anisotropic effective masses, with the transverse mass consistently exceeding the longitudinal mass. This mass disparity exhibits a power law dependence on the magnetic field strength and is less pronounced for the strange quark compared to up and down quarks. Additionally, the magnetic field induces axial-vector and tensor terms, highlighting the Zeeman effect, resulting from quark interactions with the magnetic field. These findings have important implications for (inverse) magnetic catalysis and phenomena such as vector meson and pion condensations.

Minghui Ding

Fei Gao

Sebastian M. Schmidt

Anisotropic quark propagation and Zeeman effect in an external magnetic field * Fei Gao is supported by the National Science Foundation of China (12305134). Minghui Ding appreciates the support received from the Helmholtz-Zentrum Dresden-Rossendorf High Potential Programme

We present the first lattice QCD calculations of the isovector helicity transverse momentum-dependent parton distribution function (TMDPDF) and the flavor-dependent unpolarized TMDPDFs for up and down quarks in the proton. Our computations utilize domain-wall fermion discretization with physical quark masses. Employing Coulomb-gauge-fixed quark correlation functions within the large-momentum effective theory framework, we access nonperturbative transverse quark separations <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>b</mi><mi>T</mi></msub></math> up to approximately 1 fm, corresponding to transverse momenta as low as 200 MeV. Based on the quasi-TMD factorization theorem, we construct renormalization-group–invariant ratios that are equal to the corresponding light-cone TMDPDF ratios. At moderate <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math>, our results reveal that the isovector helicity and unpolarized TMDPDFs exhibit nearly identical transverse structure up to a normalization factor, and the unpolarized distributions display only mild flavor dependence. These findings not only support key trends observed in recent global analyses but also provide robust, nonperturbative constraints that can distinguish between different parameterizations. This Letter establishes a first-principles benchmark for TMDPDFs, offering valuable input for ongoing and future experimental efforts to map the proton’s three-dimensional structure.

Dennis Bollweg

Lattice QCD Benchmark of Proton Helicity and Flavor-Dependent Unpolarized Transverse Momentum-Dependent Parton Distribution Functions at Physical Quark Masses

We present the first lattice quantum chromodynamics (QCD) calculation of the pion valence-quark transverse-momentum-dependent parton distribution function (TMDPDF) within the framework of large-momentum effective theory (LaMET). Using correlators fixed in the Coulomb gauge (CG), we computed the quasi-TMD beam function for a pion with a mass of 300 MeV, a fine lattice spacing of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>a</mi><mo>=</mo><mn>0.06</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></math>, and multiple large momenta up to 3 GeV. The intrinsic soft functions in the CG approach are extracted from form factors with large momentum transfer, and as a byproduct, we also obtain the corresponding Collins-Soper (CS) kernel. Our determinations of both the soft function and the CS kernel agree with perturbation theory at small transverse separations (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>b</mi><mo>⊥</mo></msub></math>) between the quarks. At larger <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>b</mi><mo>⊥</mo></msub></math>, the CS kernel remains consistent with recent results obtained using both CG and gauge-invariant TMD correlators in the literature. By combining next-to-leading logarithmic factorization of the quasi-TMD beam function and the soft function, we obtain an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math>-dependent pion valence-quark TMDPDF for transverse separations <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>b</mi><mo>⊥</mo></msub><mo>≳</mo><mn>1</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></math>. Interestingly, we find that the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>b</mi><mo>⊥</mo></msub></math> dependence of the phenomenological parametrizations of TMDPDF for moderate values of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math> are in reasonable agreement with our QCD determinations. In addition, we present results for the transverse-momentum-dependent wave function for a heavier pion with 670 MeV mass.

Transverse-momentum-dependent pion structures from lattice QCD: Collins-Soper kernel, soft factor, TMDWF, and TMDPDF

We study the transverse momentum ( $p_T$ ) differential anisotropic flow and flow fluctuation in Pb+Pb collisions at $\sqrt{s_{_{NN}}}$ =5.02 TeV at the LHC. A (3+1)-dimensional CLVisc hydrodynamics framework with fluctuating TRENTO (or AMPT) initial conditions is utilized to simulate the space-time evolution of the quark-gluon plasma (QGP) medium. The effects of shear viscosity and the sub-nucleon structure on anisotropic flow and flow fluctuation are analyzed. Our result shows that shear viscosity tends to suppress both flow coefficients ( ${v_2\{2}\}$ , ${v_2\{4\}}$ , and ${\langle v_2\rangle}$ ) and flow fluctuation ( ${\sigma_{v_2}}$ ) owing to its smearing effect on local density fluctuation. The flow coefficients appear to be insensitive to the sub-nucleon structure, whereas the flow fluctuation ${\sigma_{v_2}}$ tends to be suppressed by the sub-nucleon structure in central collisions but enhanced in peripheral collisions. After taking into account the sub-nucleon structure effect, our numerical result can qualitatively describe the relative flow fluctuations ( ${v_2\{4\}/v_2\{2\}}$ , $F({v_2})$ ) measured by the ALICE Collaboration at the LHC. We further investigate the effects of shear viscosity, sub-nucleon structure, and initial condition model on the flow angle and flow magnitude decorrelations ( ${A_2^f}$ , ${M_2^f}$ ) using the four-particle correlation method. We find that the flow decorrelation effect is typically stronger in central collisions than in peripheral collisions. The flow angle decorrelation is found to be insensitive to the shear viscosity and sub-nucleon structure, whereas the flow magnitude decorrelation shows quite different behaviors when using the TRENTO and AMPT initial condition models. Our study sheds light on the anisotropic flow, transport properties, and initial structure of the QGP created in high-energy nuclear collisions.

Jie Zhu

Canada

Xiang-Yu Wu

Guang-You Qin

Anisotropic flow, flow fluctuation, and flow decorrelation in relativistic heavy-ion collisions: the roles of sub-nucleon structure and shear viscosity * Supported in part by the Natural Science Foundation of China (12225503, 11890710, 11890711, 11935007). Zhu Jie is supported in part by the China Scholarship Council (202306770009). Wu Xiang-Yu is supported in part by the National Science Foundation within the framework of the JETSCAPE collaboration (OAC-2004571) and in part by the Natural Sciences and Engineering Research Council of Canada. Some of the calculations were performed in the Nuclear Science Computing Center at Central China Normal University (NSC$^{3}$), Wuhan, Hubei, China.

In this work, we report a lattice calculation of $\textit{x}$-dependent valence pion generalized parton distributions (GPDs) at zero skewness with multiple values of the momentum transfer −$\textit{t}$. The calculations are based on an $\textit{N}$ = 2 + 1 gauge ensemble of highly improved staggered quarks with Wilson-Clover valence fermion. The lattice spacing is 0.04 fm, and the pion valence mass is tuned to be 300 MeV. We determine the Lorentz-invariant amplitudes of the quasi-GPD matrix elements for both symmetric and asymmetric momenta transfers with similar values and show the equivalence of both frames. Then, focusing on the asymmetric frame, we utilize a hybrid scheme to renormalize the quasi-GPD matrix elements obtained from the lattice calculations. After the Fourier transforms, the quasi-GPDs are then matched to the light-cone GPDs within the framework of large momentum effective theory with improved matching, including the next-to-next-to-leading order perturbative corrections, and leading renormalon and renormalization group resummations. We also present the 3-dimensional image of the pion in impact-parameter space through the Fourier transform of the momentum transfer −$\textit{t}$.

Heng-Tong Ding

Peter Petreczky

Sergey Syritsyn

Three-dimensional imaging of pion using lattice QCD: generalized parton distributions