Predictive power of renormalisation group flows: a comparison

Abstract

We study a proper-time renormalisation group, which is based on an operator cut-off regularisation of the one-loop effective action. The predictive power of this approach is constrained because the flow is not an exact one. We compare it to the Exact Renormalisation Group, which is based on a momentum regulator in the Wilsonian sense. In contrast to the former, the latter provides an exact flow. To leading order in a derivative expansion, an explicit map from the exact to the proper-time renormalisation group is established. The opposite map does not exist in general. We discuss various implications of these findings, in particular in view of the predictive power of the proper-time renormalisation group. As an application, we compute critical exponents for O(N )-symmetric scalar theories at the Wilson-Fisher fixed point in 3d from both formalisms.

Derivative expansion and renormalisation group flows

“…The bounds on field theoretical methods, reviewed in [33], are much tighter. They agree surprisingly well with the findings of [29] (within less than 1% for ν and around 5% for ω, for all N = 0, · · · , 4), and are represented by the short-dashed line in Figs. 4 and 5.…”

“…While these results confirm our reasoning based on the arguments explained in sections 4 and 5, the numerical agreement with results obtained by other methods, or experiment, remains unclear. Currently, it is not understood why the derivative expansion of (33) should converge towards the physical scaling solution, bearing in mind that (33) is not an exact flow [29]. An answer to this question, however, is outside the range of the present study.…”

“…We have considered the radius of convergence for amplitude expansions of the specific PT flow (33), which is found to be larger than for exact RG flows. This qualitative difference between exact and PT flows is linked to their fundamental inequivalence, even to this order of the approximation [29]. Still, the large effective radius of convergence explains why the derivative expansion for the PT flow (33) should have even better convergence properties.…”

“…To leading order in the derivative expansion, (33) turns into (19) with ℓ PT (ω) = Γ( d 2 ) exp(−ω) [29,30].…”

Aspects of the functional renormalisation group

“…Eq. (3.78) also covers the interesting class of propertime flows [50][51][52][53][54], whereΓ comprises a full non-trivial quantum theory [45][46][47][48]. A detailed discussion of the general situation will be given elsewhere.…”