How to Determine the Branch Points of Correlation Functions in Euclidean Space II: Three-Point Functions

Round 1

Reviewer 1 Report

The paper considers the analytic properties of specific three and two
point Feynman integrals. This is the continuation of a long term
research program, by the authors, to study the analytic properties of
Feynman integrals.  The results may help functional methods, such as
the Dyson-Schwinger equations, produce results at time-like momentum
in a new way. The paper does seem to be a more concise version
of reference [28]. Given the importance of extracting Minkowski
correlators from Euclidean correlators, I recommend it be published.

I have a couple of minor points.

In the abstract it says

Here we describe how to employ it for three-point 5 functions like
the three-gluon or quark-gluon vertices.

However, although the results will be useful for three-gluon or
quark-gluon vertices, there is no explicit use of them in the
paper, apart from being mentioned as future work in the conclusions.
It would be good if the above sentence was modified so potential

Some of the hyperlinks for older references refer to the old
Los Alamos server, so they no longer work.
For example reference [25] links to
https://www.lanl.gov/errors/system-notification.php

Author Response

We thank the reviewer for the positive review of our work. We changed the abstract as follows to make clear we do not treat the QCD vertices explicitly in this work:
"Here we describe how to employ it for three-point functions like the three-gluon or quark-gluon vertices." -> "Here we describe how to employ it for three-point functions. The basic mechanisms are discussed for a scalar theory, but they are the same for more complicated theories and are thus relevant, e.g., for the three-gluon or quark-gluon vertices of quantum chromodynamics."
As far as the links to the Los Alamos server are concerned, this problem was introduced by the journal style file. We changed the MDPI style file to fix this.

Reviewer 2 Report

In this work, the authors give an accessible presentation of a previous proposal (the contour deformation method) to determine the branch points of correlation functions in Euclidean spacetime. Here, the analytic structure of three-point functions are explained in detail. I find this approach interesting and general enough to serve as a bridge between calculations in Euclidean spacetime, scattering data, and the physical spectrum of Quantum Field Theories.

 

Author Response

We thank the reviewer for the positive review of our work.

Reviewer 3 Report

 

In the present manuscript, the authors employ the contour deformation method (CDM) as a tool for analyzing the analytic structure of one-loop

 integrals of two- and three-point functions in a scalar theory with cubic interaction. Although the authors are applying the CDM in a simplified context, all general mechanism for the emergence of thresholds is described there. Therefore, this study is an excellent toy model for more challenging applications, such as determining the analytic structure of the elementary QCD three-point functions. In addition, as stressed throughout the manuscript, an extended and more detailed version of this work is already available in the arXiv (see Ref. [28] ).

 The general idea and the manuscript results are quite interesting, and the present analysis may be faced as a starting point for a systematic exploration of the timelike momenta region. From the technical point of view, the paper is solid, the article organization is good, and the presentation of the material is coherent. For these reasons, in my opinion, the article merits publication in Symmetry.

 

Author Response

We thank the reviewer for the positive review of our work.