Scalar size of the pion from lattice QCD

We present our recent lattice QCD study of the pion scalar form factor and the associated scalar radii, with fully controlled systematic uncertainties. The results are obtained from a set of 17 gauge ensembles with $N_f=2+1$ Clover-improved Wilson sea quarks. These ensembles span four lattice spacings between $a=0.049\mathrm{fm}$ and $a=0.086\mathrm{fm}$, pion masses in the range $130 - 350\mathrm{MeV}$, and various physical volumes. A precise evaluation of the computationally demanding quark-disconnected contributions enables unprecedented momentum resolution, particularly on large and fine ensembles near the physical quark masses.

A reliable extraction of the required ground-state matrix elements at both zero and nonzero momentum transfer is achieved by employing a broad range of source–sink separations in the relevant three-point functions, $1.0\ \mathrm{fm} \lesssim  \lesssim 3.25\ \mathrm{fm}$. For the first time, the scalar radii are determined using a $z$-expansion parametrization of the $Q^2$-dependence of the form factor, rather than relying on a simple linear approximation at low momentum transfer.

The physical extrapolation is guided by the next-to-leading order expressions for the scalar radii from chiral perturbation theory. The relevant low-energy constants are fitted directly from the lattice data, leading to the first lattice determination of $L_4^r$ with a result that is not compatible with zero. Systematic uncertainties associated with ground-state extraction, form-factor parametrization, and physical extrapolation are rigorously quantified using model averaging based on the Akaike Information Criterion (AIC)