How much do charm sea quarks affect the charmonium spectrum?

Abstract

The properties of charmonium states are or will be intensively studied by the B-factories Belle II and BESIII, the LHCb and PANDA experiments and at a future Super-c-$$\tau$$ Factory. Precise lattice calculations provide valuable input and several results have been obtained by simulating up, down and strange quarks in the sea. We investigate the impact of a charm quark in the sea on the charmonium spectrum, the renormalization group invariant charm–quark mass $$M_{\mathrm{c}}$$ and the scalar charm–quark content of charmonium. The latter is obtained by the direct computation of the mass-derivatives of the charmonium masses. We do this investigation in a model, QCD with two degenerate charm quarks. The absence of light quarks allows us to reach very small lattice spacings down to $$0.023~\hbox {fm}$$. By comparing to pure gauge theory we find that charm quarks in the sea affect the hyperfine splitting at a level around 2%. The most significant effects are 5% in $$M_c$$ and 3% in the value of the charm quark content of the $$\eta _c$$ meson. Given that we simulate two charm quarks these estimates are upper bounds for the contribution of a single charm quark. We show that lattice spacings $$<0.06~\hbox {fm}$$ are needed for safe continuum extrapolations of the charmonium spectrum with O(a) improved Wilson quarks. A useful relation for the projection to the desired parity of operators in two-point functions computed with twisted mass fermions is proven