Renormalization of Feynman integrals on the lattice

Abstract

A perturbative renormalization procedure is proposed which applies to massive field theories on a space-time lattice and is analogous to the BPHZ finite part prescription for continuum Feynman integrals. The renormalized perturbation theory is shown to be universal, i.e. under very natural assumptions the continuum limit exists and is independent of the details of the lattice action.

Wegner-Houghton equation in low dimensions

“…This is sufficient to establish convergent physics at finite scales when a → 0 [14], a claim to be verified in this Section numerically by means of the implementation of the Wegner-Houghton scheme on the lattice. But this convergence can not rule out a modification of the scaling laws in the asymptotical UV regime.…”

Power-counting theorem for staggered fermions

“…It is of great practical importance that his power-counting theorem for generalized continuum Feynman integrals ( §5 of [10]) continues to be applicable. One thing that remains to be done is to use the staggered fermion power-counting theorem to prove perturbative renormalizability, following [7,8]. Also, applications of the theorem to higher orders in perturbation theory should be explored in more detail.…”

“…By way of analogy, renormalizability of SU(N) Yang-Mills coupled to Wilson fermions has been proven some time ago by Reisz [7]. This result was based on his earlier work on BPHZ-like renormalization theory on the lattice [8,9]. That work rested crucially on his lattice power-counting theorem [10,11].…”

Renormalization Group in Quantum Mechanics

“…Each time derivative corresponds to the multiplication by 1 ∆t (e iω∆t − 1), which has 1 degree of ultraviolet divergence in agreement with the general construction of Ref. [11]. To find the superficial degree of divergence of the loop integrals we can follow the usual dimensional argument after separating the contribution of the frequency independent Planck constanth.…”

“…The answer is not trivial because the genuine lattice vertices are non-renormalizable and may generate new ultraviolet divergences which can compensate the tree level suppression of the vertices. Nevertheless one can prove that for certain class of models the lattice perturbation expansion converges to the result of the continuum regularization and the lattice artifacts are suppressed in the continuum limit [11]. The proof starts with the properly substracted theory where all loop integrals are made finite by the help of the counterterms.…”