Fate of nonpolynomial interactions in scalar field theory

Abstract

We present an exact RG (renormalization group) analysis of O(N )-invariant scalar field theory about the Gaussian fixed point. We prove a series of statements that taken together show that the non-polynomial eigen-perturbations found in the LPA (local potential approximation) at the linearised level, do not lead to new interactions, i.e. enlarge the universality class, neither in the LPA or treated exactly. Non-perturbatively, their RG flow does not emanate from the fixed point. For the equivalent Wilsonian effective action they can be re-expressed in terms of the usual couplings to polynomial interactions, which can furthermore be tuned to be as small as desired for all finite RG time. For the infrared cutoff Legendre effective action, this can also be done for the infrared evolution. We explain why this is nevertheless consistent with the fact that the large field behaviour is fixed by these perturbations.arXiv:1605.06075v3 [hep-th]

Renormalization group properties in the conformal sector: towards perturbatively renormalizable quantum gravity

“…This was analysed in great detail in ref. [12], see also [10,11], however the focus there was different and model approximations were used (in particular the so-called Local Potential Approximation). Here, and in the rest of this paper, we make no approximations beyond the use of perturbation theory where it is legitimate to do so.…”

“…Their large field behaviour grows as ∼φ λ−5 exp(a 2φ2 ), so they lie outside L + . However it is not true that these solutions provide new continuum limits [10][11][12]. The reason is that for fixed , no matter how small, the linearised approximation, (2.5) or (2.22), is not valid for large field.…”

“…Non-polynomial perturbations with definite scaling dimension at finite field, do not scale correctly at large field. They do not emanate from the Gaussian fixed point and after RG evolution to the IR (infrared), they can be re-expanded in terms of the polynomial perturbations and thus do not lead to new continuum physics [10][11][12].…”

Momentum dependence of quantum critical Dirac systems

“…for perturbations around nontrivial solutions. In fact, the importance of a proper choice of the Hilbert space for spanning the fixed functions is also well studied for the analysis of fixed potentials in field space for Wilson-Fisher type fixed points 99,123,124 and beyond [125][126][127][128][129] . We observe a similar feature here for fixed functions in momentum space in the form of the maximum regularity criterion.…”

Fantasia of a Superfluid Universe — In Memory of Kerson Huang

“…In d = 4 the situation is complicated and a debate over the significance of the nonpolynomial solutions of (10) has continued for a long time (see e.g. a recent work 17 and references therein). To incorporate this scalar field theory into an inflation model, Huang et al suggested in ref.…”