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}^2\rangle_{\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}^y\rangle_1^u $\end{document} and \begin{document}$ \langle b_{\bot}^y\rangle_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.