Abstract
We present a precise calculation of the pion form factor using overlap fermions on seven ensembles of 2+1-flavor domain-wall configurations with pion masses varying from 139 to 340 MeV. Taking advantage of the fast Fourier transform and other techniques to access many combinations of source and sink momenta, we find the pion mean square charge radius to be $\langle {r_\pi^2} \rangle= 0.430(5)(13)\ {\rm{fm^2}}$, which agrees well with the experimental result, and includes the systematic uncertainties from chiral extrapolation, lattice spacing and finite-volume dependence. We also find that $\langle {r_\pi^2} \rangle$ depends on both the valence and sea quark masses strongly and predict the pion form factor up to $Q^2 = 1.0 \ {\rm{GeV^2}}$ which agrees with experiments very well.
Highlights
The spacelike pion electric form factor fππðQ2Þ is defined from the pionic matrix element and its slope at Q2 1⁄4 0 gives the mean square charge radius hπiðp0ÞjVjμð0ÞjπkðpÞi 1⁄4 iεijkðpμ þ p0μÞfππðQ2Þ; ð1Þ hr2π i ≡ −6 dfππ ðQ2 dQ2 Þ Q2 ; 1⁄40 ð2Þ where V j μ 1⁄4 ψ1 2 τjγμ ψ is the isovector vector current, τi are the Pauli matrices in flavor space, and jπii are the pion triplet states. hr2πi has been determined precisely based on the existing πe scattering data [1,2,3] and eþe− → πþπ−
We find that hr2πi depends on both the valence and sea quark masses strongly and predicts the pion form factor up to Q2 1⁄4 1.0 GeV2, which agrees with experiments very well
The colored bands show our prediction based on the global fit of hr2πi with χ2=d:o:f: 1⁄4 0.85; the inner gray band shows our prediction for the unitary case of equal pion mass in the valence and the sea in the continuum and infinite volume limits and the outer band includes the systematic uncertainties from excited-state contamination, z-expansion fit, chiral extrapolation, lattice spacing, and finite-volume dependence
Summary
The spacelike pion electric form factor fππðQ2Þ is defined from the pionic matrix element and its slope at Q2 1⁄4 0 gives the mean square charge radius hπiðp0ÞjVjμð0ÞjπkðpÞi 1⁄4 iεijkðpμ þ p0μÞfππðQ2Þ; ð1Þ hr2π i. Due to the multimass algorithm available for overlap fermions, we can effectively calculate several valence quark masses on each ensemble [25,26,27,28] and O (100) combinations of the initial and final pion momenta with little overhead with the use of the fast Fourier transform (FFT) algorithm [29] in the three-point function contraction This allows us to study both the sea and the valence quark mass dependence of hr2πi in terms of partially quenched chiral perturbative theory, besides giving an accurate result at the physical pion mass.
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