Abstract
Hadron masses are subject to few MeV corrections arising from QED interactions, almost entirely arising from the electric charge of the valence quarks. The QED effects include both self-energy contributions and interactions between the valence quarks/antiquarks. By combining results from different signs of the valence quark electric charge we are able to isolate the interaction term which is dominated by the Coulomb piece, $⟨{\ensuremath{\alpha}}_{\mathrm{QED}}{e}_{{q}_{1}}{e}_{{\overline{q}}_{2}}/r⟩$, in the nonrelativistic limit. We study this for ${D}_{s}$, ${\ensuremath{\eta}}_{c}$ and $J/\ensuremath{\psi}$ mesons, working in lattice QCD plus quenched QED. We use gluon field configurations that include up, down, strange and charm quarks in the sea at multiple values of the lattice spacing. Our results, including also values for mesons with quarks heavier than charm, can be used to improve phenomenological models for the QED contributions. The QED interaction term carries information about meson structure; we derive effective sizes $⟨1/{r}_{\mathrm{eff}}{⟩}^{\ensuremath{-}1}$ for ${\ensuremath{\eta}}_{c}$, $J/\ensuremath{\psi}$ and ${D}_{s}$ of 0.206(8) fm, 0.321(14) fm and 0.307(31) fm respectively.
Highlights
Lattice QCD calculations can achieve a very high level of accuracy for ground-state meson masses
A recent calculation of the mass splitting between the J=ψ and ηc achieved an accuracy of 1 MeV [1]. This precision requires that QED effects arising from the electric charge of the quarks be included in the calculation and this is being widely done, with a variety of approaches [1,2,3,4,5,6]
To OðαQEDÞ we expect the impact of QED on a meson made of quark q1 and antiquark q 2 to take the form
Summary
Lattice QCD calculations can achieve a very high level of accuracy for ground-state meson masses. The first term, with coefficient A, is physical, It is dominated, for nonrelativistic quarks, by the Coulomb interaction between the valence quark and antiquark in the meson. We will use lattice QCD calculations to which we add the effect of QED on the valence quarks in an approach known as “quenched QED” [7] This is achieved by generating a random photon field in momentum space and packaging the field in position space into a compact U (1) variable that can be multiplied into the gluon field as the Dirac equation is solved for each quark propagator. Since QCD is responsible for binding the quark and antiquark into the meson and the effect of QED is a perturbation to the meson mass, the QED interaction term Aeq1eq in Eq (1) can be isolated by comparing results from lattice calculations in which we flip the sign of the electric charge for one of the quarks.
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