Fixed-point actions in 1-loop perturbation theory

Abstract

It has been pointed out in recent papers that the example considered earlier in the O(N ) σ-model to test whether fixed-point actions are 1-loop perfect actually checked classical perfection only. To clarify the issue we constructed the renormalized trajectory explicitly in 1-loop perturbation theory. We found that the fixed-point action is not exactly 1-loop perfect. The cut-off effects are, however, strongly reduced also on the 1-loop level relative to those of the standard and tree level improved Symanzik actions. Some points on off-and on-shell improvement, Symanzik's program and fixed-point actions are also discussed.

Fixed point action for the massless lattice Schwinger model

“…The lattice actions obtained in this way have the remarkable property that they are classically perfect, in the sense that the solutions of the equations of motion are related to their continuum counterparts. Furthermore the FPaction is tree-level Symanzik-improved to all orders in the lattice spacing a [10] and there is evidence, that cut-off effects are strongly reduced also at the 1-loop-level [11,12,13].…”

Testing the efficiency of different improvement programs

“…Extending the perturbative calculation of Ref. [10] we will see that this interpretation is unlikely. Indeed with a different choice of the parameter κ one can define perfect actions that are local but that are not expected to have such a good behaviour.…”

“…By definition classically perfect actions do not show any lattice effect at tree level. In the standard language they are Symanzik tree-level on-shell improved to all orders of a [10].…”

Fixed point actions and on-shell tree-level Symanzik improvement

“…2 of Ref. [9] where the relation between FP actions and on-shell tree-level Symanzik improvement was already clearly evidenced.…”

“…In many cases, it is sufficient to check that there is convergence for the free theory 7. The notation is that of Ref [9]…”