Pion Valence-Quark TMD from Continuum Schwinger Function Methods and Gaussian GTMD

TL;DR

The paper addresses whether a Gaussian intrinsic transverse momentum in the pion's valence-quark TMD can be derived from nonperturbative QCD dynamics. Using the continuum Schwinger function method in rainbow-ladder truncation, the authors compute seventeen generalized Mellin-transverse moments $\mathcal{M}^{n,m}$ and show that a factorized form with $\langle k_T^2\rangle=0.231~\text{GeV}^2$ reproduces the moments; the longitudinal shape $\mathfrak{q}(x)$ is extracted from a DF with $\rho_{\rm DF}=0.061$. They quantify Gaussianity via ratios $R_{n,m}$ and find near-Gaussian behavior for $m\le 3$, with mild non-Gaussian tails, and show that a Gaussian GTMD implies a Gaussian GPD and a Gaussian impact-parameter profile, yielding a pion electromagnetic form factor in agreement with data when parameterized by $B_0=4.699~\text{GeV}^{-2}$. The results provide a QCD-based justification for the Gaussian transverse structure at the hadron scale and furnish a baseline for future TMD evolution and large-$k_T$ tail analyses.

Abstract

We employ the continuum Schwinger function method to investigate the unpolarized valence-quark transverse-momentum-dependent parton distribution function (TMD) of the pion at the hadron scale. The first seventeen generalized Mellin-transverse moments, constructed from lightlike and transverse vectors, are computed and found to be well described by a factorized ansatz, in which the longitudinal component coincides with the distribution function (DF) and the transverse momentum follows a Gaussian form. The Gaussianity relation between the mean and mean-squared transverse momenta is satisfied with approximately $99\%$ accuracy in our numerical results, with the mean-squared transverse momentum equal to $0.231\,\text{GeV}^2$. Using the extracted TMD, we test the hypothesis that the quark's transverse spatial distribution also follows a Gaussian form and find that the resulting electromagnetic form factor is in good agreement with existing data. These results indicate that the intrinsic transverse-momentum and transverse-spatial distributions of valence quarks in the pion can be accurately approximated by a Gaussian ansatz, supporting its use in phenomenological analyses and experimental fits.

Figures (6)

• Figure 1: Pion valence-quark DF Mellin moments $\langle x^n \rangle$ ($n=0$-$4$) at the hadron scale $\mu_H$. Purple circles denote numerical results obtained from Eq. \ref{['eq:pdfmomentform']}, while the dashed curve shows the fit using Eq. \ref{['eq:pdffunc']} with $\rho_{\text{DF}}$ from Eq. \ref{['eq:rhopdf']}.
• Figure 2: Pion valence-quark DF $\mathfrak{q}(x)$ at the hadron scale $\mu_H$. The solid curve corresponds to Eq. \ref{['eq:pdffunc']} with $\rho_{\text{DF}}$ from Eq. \ref{['eq:rhopdf']}, while the dashed curve shows the scale-free reference form $\mathfrak{q}_{\text{sf}}(x)= 30\,x^2(1-x)^2$.
• Figure 3: Pion generalized Mellin-transverse TMD moments $\mathcal{M}^{n,m}$ at the hadron scale $\mu_H$. Symbols denote direct numerical results from Eq. \ref{['eq:tmdmomentform']}: circles ($m=0$), upward triangles ($m=1$), downward triangles ($m=2$), leftward triangles ($m=3$), and rightward triangles ($m=4$). Dashed curves show fits using Eq. \ref{['eq:tmdfuncmellin']} with parameters $\rho_m$ from Eq. \ref{['eq:rhotmdmellin']}, and solid curves represent Gaussian fits obtained from Eq. \ref{['eq:gaussianfunc']} with $\rho_{\text{TMD}}$ in Eq. \ref{['eq:rhotmd']}. Units of $\mathcal{M}^{n,m}$ are GeV$^{m}$.
• Figure 4: Pion valence-quark unpolarized TMD $f_1(x,k_T^2)$ at the hadron scale $\mu_H$, obtained from Eq. \ref{['eq:gaussianfunc']} with $\rho_{\text{TMD}}$ in Eq. \ref{['eq:rhotmd']}. The transverse dependence is Gaussian.
• Figure 5: Ratios $R_{n,1}$, $R_{n,3}$, and $R_{n,4}$. Symbols denote CSM results: circles ($R_{n,1}$), upward triangles ($R_{n,3}$), and downward triangles ($R_{n,4}$). The dotted line marks the Gaussian expectation $R_{n,m}=1$.