We derive the renormalization group equations for a generic nonrenormalizable theory. We show that the equations allow one to derive the structure of the leading divergences at any loop order in terms of one-loop diagrams only. In chiral perturbation theory, e.g., this means that one can obtain the series of leading chiral logs by calculating only one loop diagrams. We discuss also the renormalization group equations for the subleading divergences, and the crucial role of counterterms that vanish at the equations of motion. Finally, we show that the renormalization group equations obtained here apply equally well also to renormalizable theories.
We propose a new method to obtain the K → ππ amplitude from K → π which allows one to fully account for the effects of final state interactions. The method is based on a set of dispersion relations for the K → ππ amplitude in which the weak Hamiltonian carries momentum. The soft pion theorem, which relates this amplitude to the K → π amplitude, can be used to determine one of the two subtraction constants -the second constant is at present known only to leading order in chiral perturbation theory. We solve the dispersion relations numerically and express the result in terms of the unknown higher order corrections to this subtraction constant.
It has been recently suggested that it is possible to calculate the effect of final state interactions in K → ππ amplitudes by applying dispersive methods to the amplitude with the kaon off-shell. We critically reexamine the procedure, and point out the effects of the arbitrariness in the choice of the off-shell field for the kaon.
We calculate the leading divergences at NNLO for the octet part of the nonleptonic weak sector of chiral perturbation theory, using renormalization group methods. The role of counterterms which vanish at the equation of motion and their use to simplify the calculation is shown explicitly. The obtained counterterm Lagrangian can be employed to calculate the chiral double log contributions of quantities in this sector, most notably the K → ππ amplitude. The double log contribution of the latter is discussed in a separate paper.
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