Weak Hamiltonian, CP Violation and Rare Decays
Andrzej J. Buras
TL;DR
The work develops a detailed framework for weak decays of mesons using the operator product expansion and renormalization-group methods, yielding a robust separation between short-distance Wilson coefficients $C_i(\mu)$ and long-distance matrix elements $\langle Q_i(\mu)\rangle$. It covers leading and next-to-leading order QCD corrections, including operator mixing, anomalous dimensions, and the treatment of gamma5 in D-dimensions and evanescent operators, culminating in explicit results for current-current, penguin, and box contributions. The text then applies this formalism to important phenomenological topics such as CKM parameters, CP violation, and rare decays like $B\to X_s\gamma$ and $K\to \pi\nu\bar{\nu}$, highlighting the predictive power and remaining nonperturbative uncertainties in hadronic matrix elements. The overall aim is to enable precise tests of the Standard Model flavor sector and to provide tools for probing new physics through weak decays, with careful attention to scale and scheme dependences and their cancellations in physical amplitudes.
Abstract
These lectures describe in detail the effective Hamiltonians for weak decays of mesons constructed by means of the operator product expansion and the renormalization group method. We calculate Wilson coeffcients of local operators, discuss mixing of operators under renormalization, the anomalous dimensions of operators and anomalous dimension matrices. We elaborate on the renormalzation scheme and renormalization scale dependences and their cancellations in physical amplitudes. In particular we discuss the issue of gamma-5 in D-dimensions and the role of evanescent operators in the calculation of two-loop anomalous dimensions. We present an explicit calculation of the 6 times 6 one-loop anomalous dimension matrix involving current-current and QCD-penguin operators and we give some hints how to properly calculate two-loop anomalous dimensions of these operators. In the phenonomenological part of these lectures we discuss in detail: CKM matrix, the unitarity triangle and its determination, two-body non-leptonic B-decays and the generalized factorization, the ratio epsilonprime/epsilon, B to X_s gamma, K^+ to pi^+ nu barnu, K_L to pi^0 nu barnu, B to X_s nu barnu, B_s to mu bar mu and some aspects of CP violation in B-decays.