The Scalar Size of the Pion from Lattice QCD

We present a lattice QCD calculation of the pion scalar form factor and associated radii with fully controlled systematics. Lattice results are computed on a large set of 17 gauge ensembles with $N_f=2+1$ Wilson Clover-improved sea quarks. These ensembles cover four values of the lattice spacing between $a=0.049\mathrm{fm}$ and $a=0.086\mathrm{fm}$, a pion mass range of $130 - 350\mathrm{MeV}$ and various physical volumes. A precise determination of the notorious quark-disconnected contributions facilitates an unprecedented momentum resolution for the form factor, particularly on large and fine ensembles in the vicinity of physical quark mass. A large range of source-sink separations $1.0\mathrm{fm} \lesssim t_\mathrm{sep} \lesssim 3.25\mathrm{fm}$ is used to reliably extract the relevant ground state matrix elements at vanishing and non-vanishing momentum transfer. This allows us for the first time to obtain the scalar radii from a $z$-expansion parametrization of the $Q^2$-dependence of the resulting form factors rather than a simple, linear approximation at small momentum transfer. The physical extrapolation for the radii is carried out using three-flavor NLO chiral perturbation theory to parametrize the quark mass dependence in terms of three low-energy constants, including the first lattice determination of $L_4^r$. Systematic uncertainties on the final results related to the ground state extraction, form factor parametrization as well as the physical extrapolation are accounted for via model averages based on the Akaike Information Criterion.