Electromagnetic form factor of the pion from twisted-mass lattice QCD at Nf=2

TL;DR

This work computes the electromagnetic form factor of the pion using $N_f=2$ twisted-mass lattice QCD with all-to-all propagators and twisted boundary conditions to access a broad $Q^2$ range. The form factor is extracted with high precision across ensembles and extrapolated to the physical point via continuum NNLO ChPT, incorporating finite-volume corrections and constraints from the pion scalar radius. The resulting pion charge radius, $\langle r^2\rangle^{phys}=0.456(30)(24)$ fm$^2$, and the $F_\pi(q^2)$ data up to $Q^2\approx0.8$ GeV$^2$ agree well with experimental measurements, while the analysis yields a robust set of low-energy constants, including $\bar{\ell}_4\approx4.4$ and $\bar{\ell}_6\approx14.9$. The study demonstrates the effectiveness of twisted BCs and all-to-all propagators in precision hadron form-factor calculations and provides a NNLO-consistent bridge between lattice results and phenomenology.

Abstract

We present a lattice calculation of the electromagnetic form factor of the pion obtained using the tree-level Symanzik improved gauge action with two flavors of dynamical twisted Wilson quarks. The simulated pion masses range approximately from 260 to 580 MeV and the lattice box sizes are chosen in order to guarantee that M L > 4. Accurate results for the form factor are obtained using all-to-all quark propagators evaluated by a stochastic procedure. The momentum dependence of the pion form factor is investigated up to values of the squared four-momentum transfer Q**2 ~ 0.8 GeV**2 and, thanks to the use of twisted boundary conditions, down to Q**2 ~ 0.05 GeV**2. Volume and discretization effects on the form factor appear to be within the statistical errors. Our results for the pion mass, decay constant and form factor are analyzed using (continuum) Chiral Perturbation Theory at next-to-next-to-leading order. The extrapolated value of the pion charge radius is <r**2>{phys} = 0.456 +/- 0.030(stat.) +/- 0.024(syst.) in nice agreement with the experimental result. The extrapolated values of the pion form factor agree very well with the experimental data up to Q**2 ~ 0.8 GeV**2 within uncertainties which become competitive with the experimental errors for Q**2 > 0.3 GeV**2. The relevant low-energy constants appearing in the chiral expansion of the pion form factor are extracted from our lattice data, which come essentially from a single lattice spacing, adding the experimental value of the pion scalar radius in the fitting procedure. Our findings are in nice agreement with the available results of ChPT analyses of pion-pion scattering data as well as with other analyses of our collaboration.

Figures (23)

• Figure 1: (a) Ratio of 2-point and 3-point correlation functions given by the r.h.s. of Eq. (\ref{['eq:ZV']}), evaluated for $t^\prime = T / 2$ at $\beta = 3.9$ and $V \cdot T = 24^3 \cdot 48 ~ a^4$, versus the (Euclidean) time $t$ in lattice units. (b) The vector renormalization constant $Z_V$ as obtained at different values of the bare quark mass in lattice units. Open dots correspond to the values extracted from the plateau region denoted by the vertical dotted lines in (a). Open squares are the results obtained from the WI using Eq. (\ref{['eq:ZV_WI']}) in Ref. renorm. The solid and dashed lines are simple linear interpolations of the lattice points and the full markers denote the corresponding values at the chiral point.
• Figure 2: Effective mass of the pion (\ref{['eq:M_eff']}) versus the (Euclidean) time distance in lattice units for $M_\pi \simeq 300~{\rm MeV}$ (a) and $M_\pi \simeq 440~{\rm MeV}$ (b) at $\beta = 3.9$ and $V \cdot T = 24^3 \cdot 48 ~ a^4$. The twisting angle $\vec{\theta}$ is chosen in the symmetric form $\vec{\theta} = ( \theta, \theta, \theta)$. The dots, squares, diamonds, triangles, the full dots and the full squares correspond to $\theta = \{ 0.0, 0.11, 0.19, 0.27, 0.35, 0.44 \}$, respectively. The dashed vertical line is drawn at $t / a = 10$, where the ground state starts to dominate.
• Figure 3: Squared pion energy $E_\pi^2(\vec{p})$ in lattice units, obtained from the time plateaux of the effective mass shown in Fig. \ref{['fig:Meff']} (by choosing the time interval $10 \leq t /a \leq 21$), versus the squared pion momentum $p^2 \equiv 3 (2 \pi \theta / L)^2$ in lattice units, for $M_\pi \simeq 300~{\rm MeV}$ (a) and $M_\pi \simeq 440~{\rm MeV}$ (b) at $\beta = 3.9$ and $V \cdot T = 24^3 \cdot 48 ~ a^4$. The solid line is the continuum-like dispersion relation $E_\pi^2(\vec{p}) = M_\pi^2(L) + |\vec{p}|^2$, while the dashed line in (a), which can be hardly distinguished from the solid one, represents the modified dispersion relation (\ref{['eq:p_renorm']}) predicted by partially twisted and partially quenched ChPT at NLO elaborated in Ref. JT07.
• Figure 4: Squared pion energy $E_\pi^2(\vec{p})$ in lattice units at $M_\pi \simeq 300~{\rm MeV}$ and $\beta = 3.9$ for the two runs $R_{2a}$ and $R_{2b}$, performed at the volumes $V \cdot T = 24^3 \cdot 48 ~ a^4$ (dots) and $V \cdot T = 32^3 \cdot 64 ~ a^4$ (squares). The values of the twisting angle $\theta$ are chosen in such a way that $\theta / L$ has the same values in the two runs. The solid and dashed lines represent the continuum-like dispersion relation $E_\pi^2(\vec{p}) = M_\pi^2(L) + |\vec{p}|^2$.
• Figure 5: Pion form factor $F_\pi(q^2)$ versus $q^2$ in lattice units for a simulated pion mass of $\simeq 300~{\rm MeV}$. The full dots are the results obtained using twisted BC's in the Breit frame and the one-end-trick procedure for calculating the all-to-all propagators for an ensemble of 80 gauge configurations taken from the run $R_{2b}$. The open squares correspond to the results of the standard procedure based on point-to-all propagators with fixed sources for 120 gauge configurations of the run $R_{2b}$. In this case spatially periodic BC's are applied in the frame where the final pion is at rest ($\vec{p}^{\,\prime} = 0$) and the momentum of the initial pion is given by $\vec{p} = 2\pi/L$$\{ (1,0,0), (1,1,0), (1,1,1), (2,0,0) \}$. At the two smallest values of $q^2$ and for the ensemble of gauge configurations considered, only the stochastic procedure provides time plateaux of enough good quality to allow the extraction of the pion form factor.