$K\to \pi \pi$ Decays in a Finite Volume

Abstract

We discuss finite-volume computations of two-body hadronic decays below the inelastic threshold (e.g. $K\to \pi \pi$ decays). The relation between finite-volume matrix elements and physical amplitudes, recently derived by Lellouch and Lüscher, is extended to all elastic states under the inelastic threshold. We present a detailed comparison of our approach with that of Lellouch and Lüscher and discuss the possible limitations of the method which could arise due to the presence of inelastic thresholds. We also examine a standard alternative method which can be used to extract the real part of the decay amplitude from correlators of the form $<0|T[\pi \pi {\mathcal{H}}_{W}K]|0>$. We show that in this case there are finite-volume corrections which vanish as inverse powers of the volume, which cannot be removed by a multiplicative factor.

arXiv comment36 Pages, Expanded version accepted for publication in Nuclear Physics B. More detailed explanations, same conclusions

References in corpus (3)

• Weak transition matrix elements from finite-volume correlation functions
• New lattice approaches to the $\Delta I=1/2$ rule
• Two-body non-leptonic decays on the lattice

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