Semileptonic weak Hamiltonian to O(α αs )

A systematic treatment of electromagnetic and strong corrections to the
semi-leptonic decays is needed in order to have a precise determination of
phenomenological parameters of the Standard Model (SM), such as CKM
matrix elements. Under the presence of QED, the matrix element associated
to the effective semi-leptonic operator on the lattice has to be renormalised,
thus requiring a matching to the continuum results. To this end, in
collaboration with Dr. M. Gorbahn, Dr. S. Jäger and Mr. E. van der Merwe, we
calculated the corresponding pertubative matching coefficients up to O(ααs ).
In our work, we emphasise the importance of appropriate choices of
renormalisation conditions on the lattice and show how these impact the
resulting perturbative matching. In particular, we have found that the
renormalization conditions as defined and used in the literature thus far lead
to extraneous and unnecessary QCD contributions that reflect in an artificial
dependence on the lattice matching scale. We suggest improvements to rectify
this problem and present the complete expression for the Leading-Log (LL) and
Next-to-Leading-Log (NLL) strong corrections to the electromagnetic
contributions of the low-scale Wilson Coefficient. Additional steps including
the matching to the full SM at the Electroweak scale and the 3-loop anomalous
dimensions of the semi-leptonic necessary to achieve the NLL result will also
be discussed.