In order to study the running coupling in four-flavour QCD, we review the setup of the Schrödinger functional (SF) with staggered quarks. Staggered quarks require lattices which, in the usual counting, have even spatial lattice extent L/a while the time extent T /a must be odd. Setting T = L is therefore only possible up to O(a), which introduces different cutoff effects already in the pure gauge theory. We redefine the SF such as to cope with this situation and determine the corresponding classical background field. A perturbative calculation yields the coefficient of the pure gauge O(a) boundary counterterm to one-loop order.
We calculate the disconnected contribution to the form factor for the semileptonic decay of a D-meson into a final state, containing a flavor singlet eta meson. We use QCDSF n f = 2 + 1 configurations at the flavor symmetric point m u = m d = m s and the partially quenched approximation for the relativistic charm quark. Several acceleration and noise reduction techniques for the stochastic estimation of the disconnected loop are tested.
Both cases have previously been investigated in the pure gauge theory. Here we define fermionic correlation functions and study their approach to the continuum limit at tree-level of perturbation theory.
The set-up of the QCD Schrödinger functional (SF) on the lattice with staggered quarks requires an even number of points L/a in the spatial directions, while the Euclidean time extent of the lattice, T /a, must be odd. Identifying a unique renormalisation scale, L = T , is then only possible up to O(a) lattice artefacts. In this article we study such lattices in the pure SU(3) gauge theory, where we can also compare to the standard set-up. We consider the SF coupling as obtained from the variation of an SU(3) Abelian and spatially constant background field. The O(a) lattice artefacts can be cancelled by the existing O(a) boundary counterterm. However, its coefficient, c t , differs at the tree-level from its standard value, so that one first needs to re-determine the induced background gauge field. The perturbative one-loop correction to the coupling allows to determine c t to one-loop order. A few numerical simulations serve to demonstrate that residual cutoff effects in the step scaling function are small in both cases, T = L ± a and comparable to the standard case with T = L.
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