The Quantum Yang–Mills Theory

Round 1

Reviewer 1 Report

The paper addresses an importante and still very updated issue. Quantum Yang-Mills Theories is a wide subject and still with many open questions to be clarified. The Introduction of the article gives clear motivations to justify the research question and places the contribution in the context of the field. The mathematical developments and technical aspects are treated with rigor. The results are discussed with good physical insights and the reference list is exhaustive. The quality of the text is very good and deserves publication in its present form.

Author Response

Thank you for your comments

Reviewer 2 Report

The manuscript highlights the critical aspect of the non-perturbative quantum Yang-Mills vacuum state as a mixed quantum state that it does not fulfil the cluster decomposition property. This aspect is discussed from the viewpoint on the axiomatic QFT, and the main aim of this work is to show that the quantum YM theory in the strong coupling regime does not admit a Lagrangian description. It appears that the basic purpose of this manuscript is to justify the results of the previous works by the same author [1,2] using the basics of axiomatic QFT.

While reading this manuscript, I have encountered a few doubts, questions and rather serious issues that need to be addressed before it can be considered for publication:

1. First, and foremost, the reference list is by far not complete. The key aspects of the ground state in quantum YM theories has been discussed since a long ago starting e.g. from fundamental works by Savvidy in '77 and by Pagels and Tomboulis in '78 (for a detailed discussion of the status of this research field, see for instance recent reviews 1910.00654, 2204.02950 and references therein). Specifically, the properties of the YM ground state discussed in these earlier references, their similarity or differences with the two vacuum structures discussed in the present manuscript, need to be properly elaborated upon, in order to actually highlight the novelty of the suggested approach.

2. On page 6 the author describes the properties of the two stable vacua, one with zero density, one with a positive one. It is not very clear how these states can be formed as the YM field gets cooled down and co-exist simultaneously in the ground state. Due to YM interactions, the state with positive energy must eventually decay into the state with zero energy, quantum-mechanically. I see no obvious way to isolate one state from another, other than to prove that the positive-energy state is actually a metastable (not stable!) state. This (meta)stability must to be clarified and explicitly proven before the approach and its results can be trusted.

3. It is very well-known that the effective YM theory accounting for the quantum effects of vacuum polarisation admits a stable chromo-magnetic solution, also known as Savvidy vacuum (see comprehensive discussions in the above references), which however has a net negative energy-density. This fact seems to be in apparent contradiction with the current proposal that must be resolved and properly understood before the suggested formulation can be considered as valid.

4. A way to stabilise the second, chromo-electric, vacuum in the framework of effective YM theory turning it into a meta-stable solution in real physical time has been discussed earlier in 1804.09826, also showing that both magnetic and electric condensates are cosmological attractors, thus leading to unavoidably vanishing of the Lorentz-invariant cosmological constant at large distances. In the present work, the cosmological constant is however always large and positive at any distances/physical times. Do these different solutions suggest that the ground-state solutions are actually not unique, and may be dependent on the basis/procedure one works in, due to YM interactions? If so, what guarantees that a given separation procedure (like the one present in the manuscript) yields truly unique and physical components of the vacuum? Again, a consistent proof, or at least a comprehensive discussion of this problem is needed to validate the suggested framework (and its advantages to more standard effective action approach).

I have not detected any significant issues with quality of English

Author Response

please see the attachment

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

The revised version addresses most of my concerns and questions indicated in the first report. I can now recommend it for publication in Universe.

I did not have any significant issues with the quality of English while reading the manuscript.