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. $G_E^v(Q^2)$ 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.

Figure 1

Left: Nucleon two-point (top) and three-point (bottom) functions. The solid black circles represent the nucleon source and sink, the black square in the three-point function represents the current insertion. The red line refers to the propagator which we use for computing the momentum derivatives of the correlators which carry therefore the derivative vertex (solid red circle). The right panel shows the representation of the derivative vertex for the simplified case of unsmeared propagators.

Figure 2

$C_2^{\prime }(0,t)$ (left) and $-C_2^{\prime \prime }(0,t{)}^{\mathrm{\Lambda },\mathrm{\Sigma }}/C_2(0,t)$ (right). The red and blue bands correspond to the combined fits of $C_2^{\prime \prime }(0,t{)}^{\mathrm{\Lambda },\mathrm{\Sigma }}$ and $C_2(0,t)$.

Figure 3

The derived values for $Z({\overrightarrow{p}}^2)$ from two-state fits of $C_2(\overrightarrow{p},t)$ (black points) followed by a linear fit (grey band) for extracting $Z(0)$ and $Z^{\prime \prime }(0)$.

Figure 4

Isovector magnetic moment (left) and isovector charge radius (right). For both ${\mu }^v$ and $(r_E^2{)}^v/a^2$, results from the ratio method are shown using source-sink separations $T/a\in \{10,13,16\}$, as well as the summation method.

Figure 5

Isovector electric (top row) and magnetic (bottom row) form factors using both the ratio method with $T=10a$ (left column) and the summation method (right column). The blue points show results from the standard method and the red bands show a $z$-expansion fit to those points. The green band (top) and point (bottom) show the slope and value of the respective form factor at $Q^2=0$, computed using the momentum derivative method. The black curves result from a phenomenological fit to experimental data by Kelly [36].

Figure 6

The induced pseudoscalar form factor at $Q^2=0$ (left) and nucleon axial radius (right). For both $G_P^v(0)$ and $(r_A^2{)}^v/a^2$, results from ratio method are shown using source-sink separations $T/a\in \{10,13,16\}$, as well as the summation method.

Figure 7

Nucleon axial (top row) and induced pseudoscalar (bottom row) form factors using both the ratio method for $T=10a$ (left column) and the summation method (right column). The blue points show results from the standard method and the red bands show a $z$-expansion fit to those points. The green band (top) and point (bottom) show the slope and value of the respective form factor at $Q^2=0$, computed using the momentum derivative method.