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

Abstract

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 $b_T$ 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 $x$, 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 work establishes a first-principles benchmark for TMDPDFs, offering valuable input for ongoing and future experimental efforts to map the proton's three-dimensional structure.

Figures (14)

• Figure 1: Ratios $r(t_s,\tau;b_T,b_z,P_z=1.62~\mathrm{GeV})$ for unpolarized (top) and helicity (bottom) CG quasi-TMD beam function matrix elements at $b_T = 8a$ and $b_z = 0$. Dashed lines show results from two-state fits; bands indicate the extracted values from $\mathrm{Fit}_{\mathrm{st1}}$, $\mathrm{Fit}_{\mathrm{st2}}$, and SumFit.
• Figure 2: The $x$-dependent quasi-TMD beam functions are presented as functions of $x$ at several values of $b_T$. From left to right, the panels the isoscalar unpolarized, isovector unpolarized, and isovector helicity cases, respectively.
• Figure 3: Lattice QCD results (filled symbols with error bars) for the ratio $R^{u-d}_{g_{1L}/f_1}(x, b_T)$ of isovector helicity to isovector unpolarized TMDPDFs in Eq. \ref{['eq:HeliUnpol']}, shown as a function of $b_T$ at several values of $x$. The hatched bands indicate the corresponding global fits from MAP-Heli Bacchetta:2024yzl and TNTC Yang:2024drd.
• Figure 4: Lattice QCD results (filled symbols with error bars) for the ratio $R_{f_1}^{u/d}(x, b_T)$ of up- to down-quark unpolarized TMDPDFs in Eq. \ref{['eq:UnpolUD']}, shown as a function of $b_T$ at several values of $x$. The hatched bands denote the corresponding global fits from MAP24 Bacchetta:2024qre and ART25 Moos:2025sal.
• Figure 5: The effective mass obtained from two-point functions are shown. The lines are calculated from dispersion relation $E(P_z)=\sqrt{P_z^2+m_N^2}$ with $m_N=0.94$ GeV. The bands are constructed from the two-state fit results.