Exact Matching Condition for Matrix Elements in Lattice and Ms Schemes

The exact matching condition is given for hadron matrix elements calculated in any two different schemes, in particular, in the lattice and dimensional regularization, (modified) minimal subtraction MS schemes. The result provides insight into and permits to go beyond Lepage and Mackenzie’s mean field theory of removing tadpole contributions in lattice operators. Typeset using REVTEX This work is supported in part by funds provided by the U.S. Department of Energy (D.O.E.) under cooperative agreement #DF-FC02-94ER40818. 1 Presently, lattice QCD provides the unique method with controlled approximation to compute hadron properties directly from the QCD lagrangian. In the last few years, a number of groups have calculated on the lattice an impressive list of hadron matrix elements, ranging from the axial and scalar charges of the nucleon to lower-order moments of deep-inelastic structure functions [1–3]. Note, however, that most of the hadron matrix elements are not directly physical observables. In field theory, apart from the S-matrix, physical observables are related to symmetry generators of the lagrangian, such as the vector and axial-vector currents or hadron masses. Nonetheless, hadron matrix elements are useful intermediate quantities to express physical observables. Being intermediate, they often depend on specific definitions in particular context. Or in field theory jargon, they are scheme-dependent. Since schemes are generally introduced to eliminate ultraviolet divergences in composite operators, the scheme dependence of a matrix element is in fact perturbative in asymptotically-free QCD. Understanding scheme dependence has important practical values. In calculating hadron matrix elements on a lattice, one is automatically limited to the lattice scheme. On the other hand, hadron matrix elements entering physical cross sections are often defined in connection with perturbation theory. The best scheme for doing perturbation theory is not the lattice QCD, because the lattice has complicated Feynman rules and accommodates only Euclidean Green’s functions. The most popular scheme for perturbative calculations is the dimensional regularization introduced by t’ Hooft and Veltman more than two decades ago, followed by the (modified) minimal subtraction (MS). A popular practice currently adopted in the literature for matching the matrix elements in the lattice and MS schemes goes like this [1,4]. Consider, for instance, a quark operator O. First, the one-loop matrix element of O in a single quark state |k〉 is calculated on the lattice,