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-τ 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 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 fm. By comparing to pure gauge theory we find that charm quarks in the sea affect the hyperfine splitting at a level below 2%. The most significant effects are 5% in M c and 3% in the value of the charm quark content of the η 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 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.

In this work we investigate the effect of an external magnetic field B on the shape of flux tubes in QCD by means of lattice simulations, performed with N f = 2 + 1 flavors of stout improved dynamical staggered quarks with physical masses. After having discussed some difficulties in the practical definition of the flux tube at B = 0, we show that these ambiguities do not affect the determination of the flux tube modifications induced by the magnetic field. Different results are obtained depending on the relative orientations of the flux tube and of the magnetic field: they confirm that the magnetic field acts as transverse confinement catalyser and longitudinal confinement inhibitor; moreover, the flux tube itself loses its axial symmetry when it is not directed along the magnetic background. * Electronic address: claudio.bonati@df.unipi.it † Electronic address: scali@uni-wuppertal.de ‡ Electronic address: massimo.delia@unipi.it § Electronic address: michele.mesiti@swansea.ac.uk ¶ Electronic address: fnegro@pi.infn.it * * Electronic address: andrea.rucci@pi.infn.it † † Electronic address: francesco.sanfilippo@roma3.infn.it

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