Volume 193,number 1 PHYSICS LETTERS B 9 July 1987 WEAK MATRIX ELEMENT CALCULATIONS ON THE LATFICE USING STAGGERED FERMIONS David DANIEL, Simon HANDS 1, T.D. KIEU and Stephen N. SHEARD Department of Physics, Universityof Edinburgh, MayfieldRoad, EdinburghEH9 3JZ, Scotland, UK Received24 February 1987;revisedmanuscriptreceived 16 April 1987 Weoutline a latticemethod, usingstaggered fermions,for the evaluationof matrix elements relevantfor weakdecays of strange mesons. We require three inversionsper configuration. 1. Introduction. The recent interest in the numerical lattice evaluation of matrix elements relevant to low- energy weak hadronic interactions has focused on formulating the problem in terms of Wilson fermions [ 1-3]. However, the Wilson fermion formulation does not respect the chiral symmetry which is expected to be approx- imately valid for the continuum limit, and there are consequent complications [4]. Therefore one might prefer to consider the problem in terms of staggered fermions, where sufficient chiral symmetry is retained to ensure that the matrix elements have chiral behaviour analogous to that in the continuum [ 5 ]. Unfortunately, staggered fermions have problems of their own. Firstly, their use entails the use of multi-link operators and, ideally, the generation of propagators from many sources. Both of these points tend to increase the computational effort required. Secondly, in the continuum limit, one staggered fermion field corresponds to four "flavours" of Dirac fermion [6]. However, at non-zero lattice spacing, these "Susskind flavours" mix, and even with degenerate masses we have only a discrete flavour symmetry [ 5 ]. Thus, if we identify Susskind flavour with continuum flavour, the formulation of weak hadronic transitions, whose characteristic feature is flavour non-conservation, somewhat non-intuitive. One way to avoid the second difficulty, as proposed in.ref. [ 5 ], is the use of three species of staggered fermion field, one for each of the (light) continuum flavours. This leads to a direct correspondence between the chiral symmetry in the continuum and on the lattice. The apparent disadvantage of having four lattice fermions for each continuum flavour should result in nothing more serious than an overcounting, which can be kept track of provided that we are working within the quenched approximation. The approach outlined in ref. [ 5 ] was motivated by the desire to have lattice matrix elements satisfying Ward identities analogous to those in the continuum. In this letter we propose a method for weak matrix elements, again using three species of staggered fermion, in which our prime concern is to use propagators from as few sites as possible (one, with the requirement of two extra inversions), and thus make progress possible with little more effort than is required for hadron mass calculations. 2. Review of aims. To be specific, we consider K°~n+n-, AS= 1 transitions, for which the effective ham- iltonian [7] is an expansion over six four-quark operators, ¢, = ( 3y,,x. s) ( a~,,,~ u) - ( 3~,~,Lu) ( a~,,,Ls) , 02 = ( 3YuLS)( ayuL U) + (3Y~LU)(ay~LS) + 2( dYuLS) [ ( dyuLd) + ( YyuLS)] , Present address: Department of Theoretical Physics,OxfordUniversity, 1 KebleRoad, OxfordOX1 3NP, UK. (l) 0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) 85