Functional renormalization group and 2PI effective action formalism

Abstract

We combine two non-perturbative approaches, one based on the two-particle-irreducible (2PI) action, the other on the functional renormalization group (fRG), in an effort to develop new non-perturbative approximations for the field theoretical description of strongly coupled systems. In particular, we exploit the exact 2PI relations between the two-point and four-point functions in order to truncate the infinite hierarchy of equations of the functional renormalization group. The truncation is "exact" in two ways. First, the solution of the resulting flow equation is independent of the choice of the regulator. Second, this solution coincides with that of the 2PI equations for the two-point and the four-point functions, for any selection of two-skeleton diagrams characterizing a so-called Φ-derivable approximation. The transformation of the equations of the 2PI formalism into flow equations offers new ways to solve these equations in practice, and provides new insight on certain aspects of their renormalization. It also opens the possibility to develop approximation schemes going beyond the strict Φ-derivable ones, as well as new truncation schemes for the fRG hierarchy.

On overlapping Feynman (sub)graphs

“…Here we extend and refine the discussion of Ref. [5] to describe, on general grounds, how two subgraphs of a given Feynman graph can overlap. This allows us to derive a series of "non-overlap" theorems for one-particle-irreducible subgraphs with 2, 3 and 4 external legs.…”

“…( 5) is not a trivial application of the corresponding formula between the cardinals of the sets of external legs of Ḡ1 , Ḡ2 , Ḡ1 ∪ Ḡ2 and Ḡ1 ∩ Ḡ2 because the sets of external legs of Ḡ1 or Ḡ2 are not subsets of the set of external legs of Ḡ1 ∪ Ḡ2 and so the union of the sets of external legs of Ḡ1 and Ḡ2 is not the set of external legs of Ḡ1 ∪ Ḡ2 . Instead, the formula (5) needs to be seen as consequence of the overlapping structure depicted in Fig. 3.…”

“…Otherwise they always overlap, in a trivial manner. Moreover, 5 We could introduce overlapping submodes by considering all the possible ways one can distribute the given since the union of two overlapping 1PI subgraphs is also 1PI, it is necessarily contained in one of the 1PI components of the original graph G. Thus, without loss of generality, we can assume that the original graph is 1PI (and in particular connected).…”

“…One recent example of this situation is reported in Ref. [5] where the overlapping divergences that appear in the twoparticle-irreducible (2PI) formalism for the case of a scalar ϕ 4 theory are disentangled with the help of the functional renormalization group and classified into divergences of the twopoint function, divergences of the four-point function, and divergences of higher derivatives δ n Φ[G]/δG n (with n ≥ 3) of the so-called Luttinger-Ward functional Φ[G], a functional of…”

Emergent Hadrons and Diquarks

“…Related fRG for n-particle irreducible (nPI) actions can be found in e.g. [24,[115][116][117][118][119][120][121][122][123][124][125].…”