A General Method for Non-Perturbative Renormalization of Lattice Operators

Abstract

We propose a non-perturbative method for computing the renormalization constants of generic composite operators. This method is intended to reduce some systematic errors, which are present when one tries to obtain physical predictions from the matrix elements of lattice operators. We also present the results of a calculation of the renormalization constants of several two-fermion operators, obtained, with our method, by numerical simulation of $QCD$, on a $16^3 \times 32$ lattice, at $β=6.0$. The results of this simulation are encouraging, and further applications to four-fermion operators and to the heavy quark effective theory are proposed.

Figures (9)

• Figure 1: Diagram which mixes the four fermion operator in eq. (\ref{['eq:dt12']}) with the scalar and pseudoscalar densities. The mixing is induced by the chiral breaking term present in the Wilson and Clover actions.
• Figure 2: Vertex diagrams with the Clover action. The second and third diagrams refer to the contribution from the rotated part of the operator.
• Figure 3: Self energy diagram.
• Figure 4: $Z_{V^L}$ as a function of $\mu^2 a^2$. Since the lattice is not symmetric in space and time, the results obtained by using the time-component of the local current ($V_0$) and the space components, averaged in the three directions, ($V_k$) are shown separately. The dashed line is $Z^{{\rm BSPT}}_{V^L}$ from BSPT, and the curve is from BDPT, on a volume of size $16^3 \times 32$, see sec. \ref{['sec:per']}, see sec. \ref{['sec:per']}. The straight (continous) line is the result obtained using the Ward identities method wigribov.
• Figure 5: $Z_S$ as a function of $\mu^2 a^2$. We give the renormalization constant obtained by using the two possible definitions of the quark wave function renormalization, denoted by $Z_\psi$, eq. (\ref{['eq:zpsil']}), and $Z^\prime_\psi$, eq. (\ref{['eq:zpsi1']}).The dashed curve is $Z^{{\rm BSPT}}_S$, from boosted standard perturbation theory in the infinite volume limit, and the full curve is the result obtained in BDPT.