Functional flows for complex effective actions

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This manuscript discusss the application of functional renormalization group approaches to complex effective actions. The manuscript is well written and the calculations are carefully laid out. The validity of the manuscript is high, although it is rather technical and really only of interest to specialists in this field. That being said it will be a valuable resource.

There are two points that should be commented on in some more detail.

1) The authors only consider complex actions due to complex external fields. They mention that the action itself could be complex (e.g. because of complex coupling constants), but do not elaborate on this. Different from the case of external fields the complex parameters would now couple to terms that are non-linear in the fields. Does the apporach still work?

2) Open quantum systems are often described using the Lindblad appraoch, which amounts to non-hermitean “Hamiltonians”. Can the authors comment on the applicability to this situation?

Some remarks along these lines would be helpful for readers who want to build on this work and possibly apply it to such models.

See report.

See report.

The authors show how the functional renormalization group approach (FRG) can be extended to complex effective actions. More precisely, they consider a $\phi^4$ theory with a complex external source that couples linearly to the field. Both the flow of the Wilsonian effective action and the 1PI effective action are discussed. There is a nice presentation of the derivation of the Wegner's flow, the Wilson-Polchinski flow and the flow of the 1PI effective action. The results are benchmarked by considering the zero-dimensional and four-dimensional theories and the location of the Lee-Yang zeros is also discussed. Although somewhat technical (many technical details are discussed in the appendices), the paper is clearly written and provides us with an interesting application of the FRG approach.

One of the main results of the paper is that the most efficient approach is the RG-adapted flow for the Wilsonian effective action whereas the 1PI flow is not very efficient. Since the latter is often the method of choice in practice to deal with many physical models, the authors' conclusion is somewhat disappointing. It would be interesting to discuss this issue further, for example in the conclusion. Figure 1 shows that the 1PI flow does not converge anymore way before the pole is approached, i.e. $J_y\lesssim 1$, while the RG-adapted flow converges up to $J_y\simeq 2.4$. Is there a simple way to understand why the 1PI flow does so badly?

The authors consider the case where the external source is complex but the action remains real. They point out that the bare action could also be complex (the mass or some coupling constants could be complex). In that case should we expect some complications in the FRG approach or is there no essential differences between a complex external source and a complex action?

Another (related) issue is whether the authors' results could be of interest for the study of out-of-equilibrium systems where the Hamiltonian may be non-Hermitian and the bare Wilsonian action as well as the effective action complex. In this respect, it would be interesting to discuss the possible connection with a recent paper of Grunwald {\it et al.} [SciPost Phys. 12, 179 (2022)] where a harmonic oscillator perturbed by a cubic term with imaginary coupling is studied within the FRG approach.

See remarks in the report.