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

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.