Computing the nucleon charge and axial radii directly at $Q^2=0$ in lattice QCD

We describe a procedure for extracting momentum derivatives of nucleon matrix elements on the lattice directly at $Q^2=0$. This is based on the Rome method for computing momentum derivatives of quark propagators. We apply this procedure to extract the nucleon isovector magnetic moment and charge radius as well as the isovector induced pseudoscalar form factor at $Q^2=0$ and the axial radius. For comparison, we also determine these quantities with the traditional approach of computing the corresponding form factors, i.e. $G^v_E(Q^2)$ and $G_M^v(Q^2)$ for the case of the vector current and $G_P^v(Q^2)$ and $G_A^v(Q^2)$ for the axial current, at multiple $Q^2$ values followed by $z$-expansion fits. We perform our calculations at the physical pion mass using a 2HEX-smeared Wilson-clover action. To control the effects of excited-state contamination, the calculations were done at three source-sink separations and the summation method was used. The derivative method produces results consistent with those from the traditional approach but with larger statistical uncertainties especially for the isovector charge and axial radii.