Finite-volume partially-quenched two-pion amplitudes in the I=0 channel

C.-J.D Lin,G Martinelli,E Pallante,C.T Sachrajda,G Villadoro

doi:10.1016/j.physletb.2003.12.019

Abstract

We present a study of the finite-volume two-pion matrix elements and correlation functions of the I=0 scalar operator, in full and partially quenched QCD, at one-loop order in chiral perturbation theory. In partially quenched QCD, when the sea and valence light quark masses are not equal, the lack of unitarity leads to the same inconsistencies as in quenched QCD and the matrix elements cannot be determined. It is possible, however, to overcome this problem by requiring the masses of the valence and sea quarks to be equal for the u and d quarks while keeping the strange quark (s) quenched (or partially quenched), but only in the kinematic region where the two-pion energy is below the two-kaon threshold. Although our results are obtained at NLO in chiral perturbation theory, they are more general and are also valid for non-leptonic kaon decays (we also study the matrix elements of (8,1) operators, such as the QCD penguin operator Q6). We point out that even in full QCD, where any problems caused by the lack of unitarity are clearly absent, there are practical difficulties in general, caused by the fact that finite-volume energy eigenstates are linear combination of two-pion, two-kaon and two-η states. Our Letter implies that extracting ΔI=1/2, K→ππ decay amplitudes from simulations with ms=md,u is not possible in partially quenched QCD (and is very difficult in full QCD).

Highlights

Several methods have been proposed to compute non-leptonic kaon decay amplitudes in lattice QCD
To overcome this problem by requiring the masses of the valence and sea quarks to be equal for the u and d quarks while keeping the strange quark (s) quenched, but only in the kinematic region where the two-pion energy is below the two-kaon threshold
Our results are obtained at next-to-leading order (NLO) in chiral perturbation theory, they are more general and are valid for non-leptonic kaon decays (we study the matrix elements of (8, 1) operators, such as the QCD penguin operator Q6)

Summary

Introduction

Several methods have been proposed to compute non-leptonic kaon decay amplitudes in lattice QCD. [7], the terms proportional to θ (E − mK ) and θ (E − mη) were omitted in Eq (A.1) of that Letter, but here we want to study the case mK = mπ = mη E, for which P (wK ) and P (wη) have vanishing denominators, so that the energy shift and the finite-volume corrections to the matrix element are modified This result is in agreement with the Lüscher quantization condition [10], because the FSI do depend on whether we are above or below the twokaon (and two-η) threshold. The procedure appears to be very complicated to implement in practice

Two-pion matrix elements in partially quenched QCD

Conclusions