Lattice simulations with G -parity boundary conditions

Abstract

We discuss G-parity lattice boundary conditions as a means to impose momentum on the pion ground state without breaking isospin symmetry. This technique is expected to be critical for the precision measurement of K → (ππ) I=0 matrix elements where physical kinematics demands moving pions in the final state and the statistical noise caused by disconnected contributions will make it difficult to use multi-exponential fits to isolate this as an excited state. We present a formalism for computing hadronic Green's functions with G-parity boundary conditions, derive the discretized action and its symmetries, discuss how the strange quark can be introduced and detail techniques for the numerical implementation of these boundary conditions. We demonstrate and test these methods using several 16 3 × 32 dynamical domain wall ensembles with a 420 MeV pion mass and G-parity boundary conditions in one and two spatial directions.

Direct CP violation and the ΔI=1/2 rule in K→<

“…In Ref. [10] we introduced a notation whereby the quark field and its G-parity partner are placed in a two-component vector,…”

“…With the introduction of this extra quark flavor a square root of the s=s 0 determinant is required in order to generate a 2 þ 1 flavor ensemble [10].…”

“…We would therefore need to resort to multistate fits to rather noisy data in order to obtain the physical amplitudes. The change in the finite-volume corrections induced by modifying the boundary conditions is exponentially small [9,10] or else accounted for by the Lüscher and Lellouch-Lüscher [11,12] prescriptions with minor alterations [13].…”

“…While more general than the APBC approach, the use of GPBC introduces a number of technical and computational difficulties that we discuss in Ref. [10] and below.…”

“…The first of these is Ref. [10] which gives a detailed discussion of the implementation and properties of lattice calculations which impose G-parity boundary conditions. The second paper is Ref.…”