Lattice Perturbation Theory
  The consideration of quantum fields defined on a spacetime lattice provides computational techniques which are invaluable for studying gauge theories nonperturbatively from first principles. Perturbation theory is an essential aspect of computations on the lattice, especially for investigating the behavior of lattice theories near the continuum limit. Particularly important is its role in connecting the outcome of Monte Carlo simulations to continuum physical results. For these matchings the calculation of the renormalization factors of lattice matrix elements is required. In this review we explain the main methods and techniques of lattice perturbation theory, focusing on the cases of Wilson and Ginsparg-Wilson fermions. We will illustrate, among other topics, the peculiarities of perturbative techniques on the lattice, the use of computer codes for the analytic calculations and the computation of lattice integrals. Discussed are also methods for the computation of 1-loop integrals with very high precision. The review presents in a pedagogical fashion also some of the recent developments in this kind of calculations. The coordinate method of L\"uscher and Weisz is explained in detail. Also discussed are the novelties that Ginsparg-Wilson fermions have brought from the point of view of perturbation theory. Particular emphasis is given throughout the paper to the role of chiral symmetry on the lattice and to the mixing of lattice operators under renormalization. The construction of chiral gauge theories regularized on the lattice, made possible by the recent advances in the understanding of chiral symmetry, is also discussed. Finally, a few detailed examples of lattice perturbative calculations are presented. 
Stefano Capitani