A precision study of the semileptonic effective weak hamiltonian

The Standard Model has many hurdles it needs to pass, such as CKM unitarity. Unitarity can be tested through semileptonic decays of mesons which can be studied by utilising the Effective Weak Hamiltonian. This is a convenient tool for studying the short-distance physics of meson and nuclear beta decays. The decays in question can be studied both perturbatively, or non-perturbatively on the lattice. Results found utilising either of these methods are not a priori compatible because different renormalization methods must be used on the lattice to account for spacetime discretization. To aid in connecting results found using these two methods, we calculate Matching Coefficients for the semileptonic effective weak operator, between the MS scheme, and momentum subtraction schemes RI′-MOM and RI-SMOM to O(ααS), as well as the new RI-MOM and RI-SMOM schemes defined in Gorbahn et al. [1] and this thesis. We have devised the new schemes to maintain the Ward Identity. Furthermore, the Effective Weak Hamiltonian must be matched to the Standard Model via Wilson Coefficients in order for calculations made using the effective theory to be pertinent to the study of Standard Model processes. The Wilson Coefficient for the semileptonic effective operator is calculated by matching S-matrix elements for semileptonic weak decay using the Standard Model and the Effective Weak Hamiltonian respectively at the weak scale (∼ MZ). We present these results in this thesis at O(ααS). Once a Wilson Coefficient is calculated, it is only used correctly at the scale at which it was calculated. In order to use it at another scale in a different calculation, the Wilson Coefficient needs to be run to the scale at which the given calculation is performed. In the case of the lattice, or a corresponding perturbative calculation, this is ∼ 1 GeV. In order to run the Wilson Coefficient down to this scale, we must use the Renormalization Group Equation, which requires the semileptonic effective weak operator’s Anomalous Dimension. In this thesis, work contributing to the O(αα2S ) Anomalous Dimension calculation for the semileptonic operator is described. This work is still in progress, so final results are still to be published. All code for this PhD will be made available online in a GitHub repository. Or you can visit the repository manually at https://github.com/evandermerwe/PhD-Code.git.