Gauge Symmetry Beyond Perturbation Theory: BRST and Anti-BRST Structure, Background Fields, and Infrared Dynamics of Yang–Mills Theory

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The manuscript provides a comprehensive and pedagogical review of the non-perturbative gauge theory. The author systematically describe the BRST and anti-BRST formalisms, and demonstrates how these symmetries leads powerful functional identities. The process independent QCD effective charge is presented, which is supported by lattice QCD results and Dyson-Schwinger equation analyses. The manuscript is well written, high-quality review in QCD and non-Abelian gauge theory.

Prior to acceptance, there are following minor issues must be addressed :

1. In the paragraph surrounding Eq. (108), there is an incomplete equation reference marked as “Eq. (??)”. The author should cite the equation properly.

2. In the paragraph containing Eq. (196), the text states “whereas the WI Eq. (4.2) gives”. No equation is numbered (4.2), please check the numbering of the equation.

3. In the first paragraph of Section 6.1, the statement “This picture is robust under changes of dimension, lattice spacing, and inclusion of dynamical quarks, and extends smoothly from Landau gauge to small values of the gauge parameter in linear covariant gauges.” lacks appropriate supporting citations.

To conclude, this is a highly insightful review that makes a meaningful contribution to the field. The manuscript will be suitable for publication in Particles once the issues outlined above are satisfactorily addressed.

Author Response

We thank the reviewer for her/his careful reading of our manuscript. 
Our answers to her/his comments are given below

Comment 1: In the paragraph surrounding Eq. (108), there is an incomplete equation reference marked as “Eq. (??)”. The author should cite the equation properly.
Response 1: The missing equation reference has now been resolved

Comment 2: In the paragraph containing Eq. (196), the text states “whereas the WI Eq. (4.2) gives”. No equation is numbered (4.2), please check the numbering of the equation.
Response 2: The number of the equation referenced has been changed to the correct one

Comment 3: In the first paragraph of Section 6.1, the statement “This picture is robust under changes of dimension, lattice spacing, and inclusion of dynamical quarks, and extends smoothly from Landau gauge to small values of the gauge parameter in linear covariant gauges.” lacks appropriate supporting citation.
Response 3: The sentence has been changed and now reads: "This picture is~[53-71]: robust under changes of dimension, lattice spacing, and inclusion of dynamical quarks; and extends smoothly from Landau gauge to small values of the gauge parameter in linear covariant gauges."

We hope that after these corrections the manuscript can be published in Particles.

Reviewer 2 Report

Comments and Suggestions for Authors

The review is very well written and can be a useful reference
for people interested in the subject.
Here are some comments that the Author may wish to consider:
- at page 12 please explain in more detail the relation
between s-closed and s-exact functionals and gauge-invariant 
and pure gauge observables. In this connection a slightly more expanded presentation of cohomological tools might be beneficial to the reader;
- In Eq.(75) an integral is missing;
- In Eq.(76) there is likely a mismatch on the symbols \Gamma and \Gamma_min;
- Just above Eq.(108) one equation reference is missing.

Author Response

We thank the reviewer for her/his careful reading of the manuscript.
Our response to her/his queries are detailed below.

Comment 1: at page 12 please explain in more detail the relation between s-closed and s-exact functionals and gauge-invariant and pure gauge observables. In this connection a slightly more expanded presentation of cohomological tools might be beneficial to the reader.
Response 1: We have expanded the s-closed and s-exact section. However we did not add more details about cohomology, due to their limited use in the paper.

Comment 2: In Eq.(75) an integral is missing
Response 2: The integral has been added

Comment 3: In Eq.(76) there is likely a mismatch on the symbols \Gamma and \Gamma_min;
Response 3: The mismatch has now been resolved

Comment 4: Just above Eq.(108) one equation reference is missing
Response 4: The missing equation number has been added

We hope that the new version of the manuscript is now deemed  suitable for publication in Particles