Symanzik Improvement Research Articles

Perturbative construction and rigorous proof of locality of effective lattice actions for continuum theories

We construct an effective lattice action for a continuum theory, by fixing a set of collective coordinates which play the role of lattice variables. As opposed to Symanzik's improvement program our method involves no expansion in powers of the lattice spacing; in other words it simultaneously yields all “irrelevant” operators generated by the renormalization group to a given order in the continuum coupling constant. We are thus able to rigorously establish that the effective lattice action, for both smooth and singular collective coordinates, is local in the sense that long-range couplings decay exponentially over a distance independent of the mass gap of the theory; for asymtotically free theories this is interpreted as an existence proof of Wilson's infrared stable trajectories. Our methods are for convenience described in the context of dimensional φ 4, but can be easily extended to any theory with a set of collective coordinates which (i) are renormalizable and (ii) provide an infrared cutoff. Application to the 2-dimensional O( N) σ-model is, in particular, discussed; the technical problems of renormalization posed by gauge invariance are, on the other hand, not dealt with in this paper, although our treatment of singular coordinates is meant as a prelude to them. A by-product of our proofs is the derivation of an interesting factorization property of Zimmermann-subtracted diagrams.

Measurement of the string tension in Su(2) lattice gauge theory with action including six-link loops

A Monte Carlo measurement of the string tension has been carried out in SU(2) lattice gauge theory with the action including up to six-link loops. With the Wilson coefficients for the six-link loops, the approach to scaling is smoother than with the standard single-plaquette action, but the correlation length at the onset of scaling ( 1 √σa = 1.4−1.5 ) is roughly the same as in the standard action. We found that this value could be reduced (1/√ σa ≃ 1.2) with a set of coefficients satisfying Symanzik's improvement programme. For the Wilson coefficients we obtained the scale parameter Λ W √σ = 0.32 ± 0.01 .