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

TL;DR

The paper presents the first lattice QCD calculation of the pion valence-quark TMDPDF within the LaMET framework using Coulomb-gauge fixed correlators. By computing the CG quasi-TMD beam function and quasi-TMDWF, it extracts the Collins-Soper kernel and intrinsic soft function, and performs NLL resummation in the matching to obtain the light-cone TMDPDF and TMDWF in both $b_\perp$ and $k_\perp$ spaces. The study demonstrates consistency with perturbative results at small transverse separations and finds reasonable agreement with global phenomenological fits for moderate $x$, while also providing a nonperturbative probe of the large-$b_\perp$ region. It further extends to a heavier pion mass to examine robustness and discusses systematic uncertainties due to power corrections and lattice artifacts. Overall, this work establishes a viable first-principles route to the transverse momentum structure of the pion and sets the stage for future refinements and extensions to other hadrons.

Abstract

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 $a = 0.06$ fm 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 ($b_\perp$) between the quarks. At larger $b_\perp$, 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 (NLL) factorization of the quasi-TMD beam function and the soft function, we obtain $x$-dependent pion valence-quark TMDPDF for transverse separations $b_\perp \gtrsim 1$ fm. Interestingly, we find that the $b_\perp$ dependence of the phenomenological parameterizations of TMDPDF for moderate values of $x$ are in reasonable agreement with our QCD determinations. In addition, we present results for the transverse-momentum-dependent wave function (TMDWF) for a heavier pion with 670 MeV mass.

Figures (23)

• Figure 1: Upper panel: effective mass plot of $C^{\rm SS}_{\text{2pt}}$ with various momenta. The dashed lines are the ground state energies of pion calculated using the dispersion relation $E_0 = \sqrt{m_0^2 + (P^z)^2}$, where the valence pion mass is $m_0 = 300$ MeV. Lower panel: the ground-state energies $E_0$ extracted from two-state fit of the two-point functions. The red line represents the exact dispersion relations with $m_0 = 300$ MeV.
• Figure 2: From left to right, the real part of the ratios $R_{\tilde{h}}$ between three-point and two-point functions for the pion quasi-TMD, corresponding to hadron momenta $P^z=1.83$, 2.43 and 3.04 GeV, are shown as functions of $t_{\rm sep}$ and $\tau$. The upper and lower panels are for the cases with $(b_\perp, z) = (1, 3)~a$ and $(b_\perp, z) = (3, 3)~a$, respectively. The colored bands are two-state fit results while the gray band is the estimated ground-state matrix element.
• Figure 3: The bare matrix elements of quasi-TMD in the coordinate space are plotted as the function of $b_\perp$ and $\lambda = z P^z$. These matrix elements are extracted from the two-state chained fit of $C_{\rm 2pt}$ and $R_{\tilde{h}}$. From left to right, the three panels correspond to hadron momenta of $P^z=1.83$, 2.43 and 3.04 GeV, respectively. It is observed that for all three hadron momenta, the quasi-TMD decays as the transverse separation $b_\perp$ increases and asymptotically approaches zero in the large $\lambda$ regime.
• Figure 4: The extrapolation of the renormalized quasi-TMD in coordinate space for various hadron momenta is shown in three panels. From left to right, they correspond to hadron momenta $P^z=1.83$, 2.43 and 3.04 GeV, respectively. The regions between the two red dashed lines indicate the extrapolation range using Eq. \ref{['eq:extrapolation_form']}. For different hadron momenta, the same starting point of $z = 0.78$ fm is chosen for the extrapolation.
• Figure 5: The ratio $\gamma^{\overline{\rm MS}}(b_\perp, P_1, P_2; \mu)$, as defined in Eq. \ref{['eq:cs_kernel_calc']}, of combinations of momentum $n_1^z/n_2^z$. The results at $\mu=2$ GeV for $b_\perp = 2a$ (upper panel) and $b_\perp = 8a$ (lower panel) are presented.