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Two-loop renormalization of vector, axial-vector and tensor fermion bilinears on the lattice

A. Skouroupathis, H. Panagopoulos

Abstract

We compute the two-loop renormalization functions, in the RI' scheme, of local bilinear quark operators $\barψΓψ$, where $Γ$ corresponds to the Vector, Axial-Vector and Tensor Dirac operators, in the lattice formulation of QCD. We consider both the flavor nonsinglet and singlet operators. We use the clover action for fermions and the Wilson action for gluons. Our results are given as a polynomial in $c_{SW}$, in terms of both the renormalized and bare coupling constant, in the renormalized Feynman gauge. Finally, we present our results in the MSbar scheme, for easier comparison with calculations in the continuum. The corresponding results, for fermions in an arbitrary representation, together with some special features of superficially divergent integrals, are included in the Appendices.

Two-loop renormalization of vector, axial-vector and tensor fermion bilinears on the lattice

Abstract

We compute the two-loop renormalization functions, in the RI' scheme, of local bilinear quark operators , where corresponds to the Vector, Axial-Vector and Tensor Dirac operators, in the lattice formulation of QCD. We consider both the flavor nonsinglet and singlet operators. We use the clover action for fermions and the Wilson action for gluons. Our results are given as a polynomial in , in terms of both the renormalized and bare coupling constant, in the renormalized Feynman gauge. Finally, we present our results in the MSbar scheme, for easier comparison with calculations in the continuum. The corresponding results, for fermions in an arbitrary representation, together with some special features of superficially divergent integrals, are included in the Appendices.

Paper Structure

This paper contains 13 sections, 69 equations, 10 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Formulation of the problem
  3. Lattice action
  4. Definition of renormalized quantities
  5. Conversion to the $\mathbf{\overline{MS}}$ scheme
  6. Renormalization of fermion bilinears
  7. Computation and Results
  8. One-loop results
  9. Two-loop results
  10. Discussion
  11. Fermions in an Arbitrary Representation
  12. Manipulation of superficially divergent and subdivergent terms
  13. Per Diagram Results

Figures (10)

  • Figure 1: One-loop diagram contributing to $Z_V$, $Z_{AV}$ and $Z_T$. A wavy (solid) line represents gluons (fermions). A cross denotes the Dirac matrices $\gamma_\mu$ (vector), $\gamma_5\gamma_\mu$ (axial vector) and $\gamma_{5}\sigma_{\mu\nu}$ (tensor).
  • Figure 2: Two-loop diagrams contributing to $Z_V$, $Z_{AV}$ and $Z_T$. Wavy (solid, dotted) lines represent gluons (fermions, ghosts). A solid box denotes a vertex from the measure part of the action; a solid circle is a mass counterterm; crosses denote the matrices $\gamma_\mu$ (vector), $\gamma_5\gamma_\mu$ (axial-vector) and $\gamma_{5}\sigma_{\mu\nu}$ (tensor).
  • Figure 3: Extra two-loop diagrams contributing to $Z_{AV,\,singlet}$. A cross denotes an insertion of a flavor singlet operator. Wavy (solid) lines represent gluons (fermions).
  • Figure 4: $Z_V^{L,RI'}(a_{_{\rm L}}\bar{\mu})=Z_V^{L,\overline{MS}}(a_{_{\rm L}}\bar{\mu})$ versus $c_{{\rm SW}}$ ($N_c=3$, $\bar{\mu}=1/a_{_{\rm L}}$, $\beta_{\rm o}=6.0$). Results up to 2 loops are shown for $N_f=0$ (solid line) and $N_f=2$ (dashed line); one-loop results are plotted with a dotted line.
  • Figure 5: $Z_{AV}^{L,RI'}(a_{_{\rm L}}\bar{\mu})$ versus $c_{{\rm SW}}$ ($N_c=3$, $\bar{\mu}=1/a_{_{\rm L}}$, $\beta_{\rm o}=6.0$). Results up to 2 loops, for the flavor nonsinglet operator, are shown for $N_f=0$ (solid line) and $N_f=2$ (dashed line); 2-loop results for the flavor singlet operator, for $N_f=2$, are plotted with a dash-dotted line; one-loop results are plotted with a dotted line.
  • ...and 5 more figures