Regularization dependence of pion generalized parton distributions

Pion generalized parton distributions are calculated within the framework of the Nambu–Jona-Lasinio model using different regularization schemes, including the proper time regularization scheme, the three-dimensional (3D) momentum cutoff scheme, the four-dimensional momentum cutoff scheme, and the Pauli-Villars regularization scheme. Furthermore, we check the theoretical constraints of pion generalized parton distributions required by the symmetries of quantum chromodynamics in different regularization schemes. The diagrams of pion parton distribution functions are plotted, in addition, we evaluate the Mellin moments of generalized parton distributions, which are related to the electromagnetic and gravitational form factors of pion. Pion generalized parton distributions are continuous but not differential at begin{document}$ x=pm ,xi $end{document}, when considering the effect of the contact contribution term, generalized parton distributions become not continuous at begin{document}$ x=pm ,xi $end{document} in all the four regularization schemes. Generalized parton distributions in impact parameter space are considered, the width distribution of u quark in the pion and the mean-squared begin{document}$ langle {boldsymbol{b}}_{bot}^2rangle_{pi}^u $end{document} are calculated. The light-front transverse-spin distributions are studied when quark polarized in the light-front-transverse begin{document}$ +,x $end{document} direction, the transverse-spin density is no longer symmetric around begin{document}$ (b_x=0,b_y=0) $end{document}, the peaks shift to begin{document}$ (b_x=0,b_y>0) $end{document}, we compare the average transverse shift begin{document}$ langle b_{bot}^yrangle_1^u $end{document} and begin{document}$ langle b_{bot}^yrangle_2^u $end{document} in different regularization schemes. The light-cone energy radius begin{document}$ r_{E,LC} $end{document} and the light-cone charge radius begin{document}$ r_{c,LC} $end{document} are also evaluated, we found that in the proper time regularization scheme the values of these quantities were the largest, in the 3D momentum cutoff scheme they were the smallest.