Proper-time functional renormalization in O(N) scalar models coupled to gravity

We focus on the use of the functional Wilsonian renormalization group framework characterized by a proper time regulator and test its use in the search of the scaling solutions and the critical properties of an O ( N )-invariant scalar field multiplet coupled to gravity in $$d=4$$ d = 4 and $$d=3$$ d = 3 dimensions. We employ the same background-fluctuation splitting and gauge fixing procedure, already adopted in a previous study based, instead, on the effective average action framework and a similar truncation of the effective action. Our main goal is to compare the results for the scaling solutions and some of the associated critical exponents. In this analysis, performed in a different framework, most of the picture previously uncovered is confirmed both at qualitative and quantitative level. There are, nevertheless, few differences both at finite N and in its large value limit, depending also on the schemes which in both frameworks are called “improved”.