Decay Amplitudes: Effective Hamiltonian

Abstract

Abstract The diagrammatic approach discussed in the previous chapter is useful to get an order-of-magnitude estimate of the importance of the various contributions to each amplitude, and to find approximate relations among the amplitudes for different decays. However, rigorous computations require a correct treatment of the QCD corrections, and this is best accomplished using the operator product expansion (OPE) already discussed in the context of kaon decays-see Chapter 20. The OPE allows the construction of a low-energy effective Hamiltonian as a sum of local operators multiplied by Wilson coefficients. The effective Hamiltonian is the sum of pieces, each of them of the type in the Wilson coefficients, from the low-energy non-perturbative confinement effects implicitly contained in the operator matrix elements. However, upon change ofµ or of the renormalization scheme, contributions previously included in the matrix elements of some operator may move into the Wilson coefficients or into the matrix elements of other operators.