Eradication of singularities in the next-to-leading order RG evolution for the \Delta S = 1 effective Hamiltonian with 3 quark flavours

Abstract

We consider the renormalization group evolution for the operators in the $\Delta S=1$ effective Hamiltonian with 3 active quark flavors, which is needed in the numerical analysis of data sets for ${ϵ}^{\prime }/ϵ$ calculated in lattice QCD. Singularities are present in the original solution of Buras et al. at next-to-leading order. We show how these can be eradicated through a method of analytic continuation to obtain the correct finite solution in this case. Furthermore, we trace the origin of the singularities to a breakdown of the approach of Buras et al. in the 3 flavour case, and show how it can be rectified so that singularitites are absent from the beginning.

Comments: 7 pages, presented at the XXV International Symposium on Lattice Field Theory, July 30 - August 4 2007, Regensburg