Finite volume effects on the extraction of form factors at zero momentum

Hadronic matrix elements that depend on momentum are required for numerous phenomenological applications. Probing the low-momentum regime is often problematic for lattice QCD computations on account of the restriction to periodic momentum modes. Recently a novel method has been proposed to compute matrix elements at zero momentum, for which straightforward evaluation of the matrix elements would otherwise yield a vanishing result. We clarify an assumption underlying this method, and thereby establish the theoretical framework required to address the associated finite volume effects. Using the pion electromagnetic form factor as an example, we show how the charge radius and two higher moments can be calculated at zero-momentum transfer and determine the corresponding finite volume effects. These computations are performed using chiral perturbation theory to account for modified infrared physics and can be generalized to ascertain finite volume effects for other hadronic matrix elements extracted at zero momentum.

Figure 1

Graphical depiction of the modified correlation function $\mathcal{C}$. The single line represents the spectator particle, while the double line represents the particle carrying momentum. The twist angles $\overrightarrow{\theta }$ allow the momentum to be varied continuously and justify the use of a Taylor series expansion in a fixed volume.

Figure 2

Quark-level contractions of the modified three-point function, ${\mathcal{C}}_{\mu }$. The single line represents the spectator quark, while the double lines represent the twisted initial- and final-state quarks. The twist angles $\overrightarrow{\theta }$ and ${\overrightarrow{\theta }}^{\prime }$ are devices that are required to derive the momentum expansion of the current correlation function on a lattice of fixed size. They are set to zero after differentiation.

Figure 3

Quark-level contractions required to obtain the pion charge radius at vanishing momentum. The solid circles represent the pion source and sink, the square represents the insertion of the external current, while the open circles are the momentum derivative vertices. Two higher moments of the charge distribution can be obtained by further insertions of such vertices on the initial- and final-state quark lines.

Figure 4

One-loop contributions to the pion current matrix element in partially quenched chiral perturbation theory. The top panel shows diagrams required to determine the pion wave function renormalization. Dashed lines represent mesons, while the cross represents the partially quenched hairpin interaction. The wiggly line shows the insertion of the vector current, which corresponds to a flavor-changing transition in the effective theory. The flavors differ only by their boundary conditions.

Figure 5

Finite volume effect on moments of the pion’s charge distribution determined at zero momentum. The relative differences $\mathrm{\Delta }{r}^2/⟨r^2⟩$, $\mathrm{\Delta }{r}^4/⟨r^4⟩$, and $\mathrm{\Delta }{r}^6/⟨r^6⟩$ are plotted as a function of the lattice size, $L$, and pion mass, $m_{\pi }$. The infinite volume value of the charge radius is taken from experiment, while values of the two higher moments are the one-loop predictions from chiral perturbation theory. In the first plot, the pion mass is fixed to its physical value, while in the second plot, the lattice size is fixed to 5.0 fm