Regularization dependence of pion generalised parton distributions

Pion generalised 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 momentum cutoff scheme, the four dimensional momentum cutoff scheme, and the Pauli-Villars regularization scheme. Furthermore, we check the theoretical constraints of pion generalised 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 generalised parton distributions, which related to the electromagnetic and gravitational form factors of pion. Pion generalised parton distributions are continuous but not differential at $x=\pm \,\xi$, when considering the effect of D-term, generalised parton distributions become not continuous at $x=\pm \,\xi$ in all the four regularization schemes. Generalised parton distributions in impact parameter space are considered, the width distribution of $u$ quark in the pion and the mean-squared $\langle \mathbf{b}_{\bot}^2\rangle_{\pi}^u$ are calculated. The light-front transverse-spin distributions are studied, when quark polarized in the light-front-transverse $+\,x$ direction, the transverse-spin density is no longer symmetric around $(b_x=0,b_y=0)$, the peaks shift to $(b_x=0,b_y>0)$, we compare the average transverse shift $\langle b_{\bot}^y\rangle_1^u$ and $\langle b_{\bot}^y\rangle_2^u$ in different regularization schemes. The light-cone energy radius $r_{E,LC}$ and the light-cone charge radius $r_{c,LC}$ are also evaluated, we find that in the proper time regularization scheme the values of these quantities are the largest, in the three dimensional momentum cutoff scheme they are the smallest.