We investigated the effect of an isospin chemical potential ( $ \mu_{I} $ ) within the quark-meson model, which approximates quantum chromodynamics (QCD) by modeling low-energy phenomena such as chiral symmetry breaking and phase structure under varying conditions of temperature and chemical potential. Using the functional renormalization group (FRG) flow equations, we calculated the phase diagram in the chiral limit within the two-flavor quark-meson model in a finite $ \mu_{I} $ with $\textit{ρ}$ vector meson interactions. Fluctuation effects significantly decrease the critical chemical potential from the mean-field (MF) value $\mu_{I,{\rm MF}} \gt m_\rho$ to a lower value at which the $\textit{ρ}$ vector meson condensates alongside the chiral condensate once the isospin chemical potential exceeds the critical value $ \mu_{I}^{{\rm{crit}}} $ . This $\textit{ρ}$ condensation was investigated numerically for different meson coupling strengths. The $\textit{ρ}$ meson dominated region is delineated from other phases by a second-order phase transition at lower $ \mu_{I} $ and a first-order transition at slightly higher $ \mu_{I} $ .

China

Mohammed Osman

Defu Hou

Wentao Wang

United States

Hui Zhang

Functional renormalization group study of $\textit{ρ}$ meson condensate at a finite isospin chemical potential in the quark meson model * Supported in part by the National Key Research and Development Program of China (2022YFA1604900). This work is also partly supported by the National Natural Science Foundation of China (NSFC) (12435009, 12275104). Hui Zhang acknowledges the financial support from the Guangdong Major Project of Basic and Applied Basic Research (2020B0301030008) and the National Natural Science Foundation of China (12047523, 12105107)

The Collins-Soper (CS) kernel governs the rapidity evolution of transverse-momentum-dependent (TMD) parton distributions, a cornerstone for QCD factorization and linking nucleon structure data across scales. Its nonperturbative behavior at large transverse separations (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>b</mi><mo>⊥</mo></msub></math>) remains weakly constrained due to phenomenological model dependencies. We present a first-principles determination of the CS kernel at the continuum limit and physical pion mass from lattice QCD in the large-momentum effective theory framework. Using (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math>)-flavor configurations (lattice spacings <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>a</mi><mo>∈</mo><mo stretchy="false">[</mo><mn>0.052</mn><mo>,</mo><mn>0.105</mn><mo stretchy="false">]</mo><mtext> </mtext><mtext> </mtext><mi>fm</mi></math>, and pion mass <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>m</mi><mi>π</mi></msub><mo>≈</mo><mo stretchy="false">(</mo><mn>136</mn><mo>,</mo><mn>230</mn><mo>,</mo><mn>300</mn><mo>,</mo><mn>320</mn><mo stretchy="false">)</mo><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>), we simulating the nonlocal equal-time correlation function and extract the quasi-TMD wave functions. Taking into account systematic improvements including hypercubic smearing, nonperturbative renormalization, and a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>b</mi><mo>⊥</mo></msub></math>-unexpanded matching kernel, we obtain the CS kernel at the continuum, chiral, and infinite-momentum limits. Our results are determined up to <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>, with controllable uncertainties, and agree with perturbative QCD at small <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>b</mi><mo>⊥</mo></msub></math> and global TMD phenomenological extractions. We conduct a global analysis integrated with phenomenological fits and demonstrate the impact of our results on such fits. This work yields the most precise nonperturbative constraint on the CS kernel’s long-distance behavior from lattice QCD, which not only bridges lattice QCD, perturbation theory, and nucleon structure experiments for TMD studies, but also boosts the utility of our constraint for future global TMD analyses.

Jin-Xin Tan

Zhi-Chao Gong

Jun Hua

Xiangdong Ji

Xiangyu Jiang

Hang Liu

Germany

Andreas Schäfer

Yushan Su

Han-Zhang Wang

Wei Wang

Yi-Bo Yang

Jun Zeng

Jian-Hui Zhang

Jia-Lu Zhang

Qi-An Zhang

Lattice QCD determination of the Collins-Soper kernel in the continuum and physical mass limits

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

Xiang Gao

Jinchen He

Rui Zhang

Yong Zhao

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

It is well known that in the study of mixing between nonlocal gluon and quark bilinear operators there exists an ambiguity when relating coordinate space and momentum space results. In this work, we show that this ambiguity is due to the lack of a proper regularization prescription of the singularity that arises when the separation between the gluon/quark fields approaches zero. We then demonstrate that dimensional regularization resolves this issue and yields consistent results in both coordinate and momentum space. This prescription is also compatible with lattice extractions of parton distributions from nonlocal operators.

Yao Ji

Poland

Zhuoyi Pang

Fei Yao

Regularization prescription for the mixing between nonlocal gluon and quark operators

Light cone distribution amplitudes for a light baryon can be extracted through the simulation of the quasi-distribution amplitudes (quasi-DAs) on the lattice. We implement the hybrid renormalization for the quasi-DAs of light baryons. Lattice simulations are performed using <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi>f</mi></msub><mo>=</mo><mn>2</mn><mo>+</mo><mn>1</mn></math> stout-smeared clover fermions and a tree-level Symanzik-improved gauge action, with three lattice spacings of 0.105, 0.077, 0.052 fm. By analyzing zero-momentum matrix elements for different lattice spacings, we extract the linear divergence associated with the Wilson-line self-energy. Matching to perturbative matrix elements in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mover accent="true"><mi>MS</mi><mo stretchy="true">¯</mo></mover></math> scheme yields the residual self-renormalization factors. Using these factors, we renormalize the quasi-DAs within the hybrid scheme, which combines self-renormalization at large separations and the ratio scheme at short distances. The renormalized results demonstrate effective cancellation of linear divergences and yield smooth, continuumlike coordinate-space distributions suitable for subsequent Fourier transformation and perturbative matching. These results establish the viability of both self- and hybrid-renormalization frameworks for light baryon quasi-DAs, providing a robust foundation for LaMET-based determinations of light cone distribution amplitudes in future.

Haoyang Bai

Jian Liang

JiaLu Zhang

Mu-Hua Zhang

Hybrid renormalization for distribution amplitude of a light baryon in large momentum effective theory

We show that the traditional moments approach in lattice quantum chromodynamics based on operator product expansion can be realized such that it utilizes derivatives in momentum rather than in distance. This avoids power divergent mixings, and thus allows to extract moments order by order to all orders in principle. Further, by exploiting the symmetry of lattice matrix elements, we can determine the even and odd moments separately. As a demonstrative example, we determine the first three moments beyond the tensor charge $ g_T$ of the isovector quark transversity distribution in the nucleon.

Dian-Jun Zhao

Moments from momentum derivatives in lattice QCD * Supported in part by National Natural Science Foundation of China (12375080, 11975051, 12061131006), the Ministry of Science and Technology of China (2024YFA1611004) and CUHK-Shenzhen (UDF01002851)

We present the first lattice QCD calculation of the quark transverse spin-momentum correlation, i.e., the naive time-reversal-odd Boer-Mulders function, of the nucleon, using large-momentum effective theory (LaMET). The calculation is carried out on an ensemble with lattice spacing $\textit{a}$ = 0.098 fm and pion mass 338 MeV, at various proton momenta up to 2.11 GeV. We have implemented perturbative matching up to the next-to-next-to-leading order together with a renormalization-group resummation improvement. The result exhibits the expected decay behavior with increasing transverse separation $\textit{b}$ . We also compare the results in the nucleon and pion.

Lingquan Ma

Hai-Tao Shu

Peng Sun

Lisa Walter

Xiaonu Xiong

Quark transverse spin-momentum correlation of the nucleon from lattice QCD: the Boer-Mulders function

We study $ \cos2\phi $ azimuthal asymmetry in doubly longitudinally polarized proton-proton Drell-Yan collisions within the transverse momentum dependent factorization framework. The asymmetry arises from the convolution of the longitudinal transversity distribution $ h_{1L}^{\perp} $ for both protons. Using the Bacchetta-Delcarro-Pisano-Radici-Signori parameterization for the nonperturbative Sudakov form factor and the Wandzura-Wilczek approximation for collinear $ h_{1L}^{\perp} $ , we predict the double spin asymmetry $ A_{LL}^{\cos2\phi} $ at RHIC and NICA kinematics. Our results demonstrate sensitivity to sea quark distributions, with the asymmetry reaching up to 25% for maximal sea quark contributions. These predictions highlight the potential of polarized Drell-Yan measurements to probe sea quark dynamics and advance our understanding of nucleon structures.

Hui Li

Ting Hu

Liang-Liang Liu

Xiao-Yu Wang

Zhun Lu

Asymmetry ${{\boldsymbol A}_{\boldsymbol{LL}}^{\bf\cos2\boldsymbol\phi}} $ in the polarized proton-proton Drell-Yan process within TMD factorization * Hui Li is supported by Fundamental Research Program of Shanxi Province (202203021222224) and Natural Science Foundation of Shanxi Normal University (JCYJ2023021). Xiao-Yu Wang is supported by the Natural Science Foundation of Henan Province (242300421377, 232300421140)

We present the first lattice QCD calculation of the quark transverse spin-momentum correlation, i.e., the T-odd Boer-Mulders function of the pion, using large-momentum effective theory. The calculation is done at three lattice spacings <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>a</mi><mo>=</mo><mo stretchy="false">{</mo><mn>0.098</mn><mo>,</mo><mn>0.085</mn><mo>,</mo><mn>0.064</mn><mo stretchy="false">}</mo><mtext> </mtext><mtext> </mtext><mi>fm</mi></mrow></math> and pion masses <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo>∼</mo><mn>350</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>, with pion momenta up to 1.8 GeV. The matrix elements are renormalized in a state-of-the-art scheme and extrapolated to the continuum and infinite momentum limit. We have implemented the perturbative matching up to the next-to-next-to-leading order and carried out a renormalization-group resummation. Our results provide valuable input for phenomenological analyses of the Boer-Mulders single-spin asymmetry.

Sebastian Lahrtz

Hong Kong

Quark transverse spin-momentum correlation of the pion from lattice QCD: The Boer-Mulders function

We report a lattice QCD calculation of the parton distribution function (PDF) of a deuteronlike dibaryon system using large-momentum effective theory. The calculation is done on three Wilson-clover ensembles with a fixed lattice spacing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>a</mi><mo>=</mo><mn>0.105</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></mrow></math> and two pion masses. The lattice matrix elements are computed at proton momenta up to 2.46 GeV, with the signal of high-momentum modes being improved by applying the momentum smearing technique. The state-of-the-art renormalization, matching, and extrapolation are then applied to obtain the final result of the light-cone PDF. A comparison between the result of the dibaryon system and the sum of the proton and neutron PDFs is also given.