# Color Confinement and Bose-Einstein Condensation

### Submission summary

 As Contributors: Nico Wintergerst Arxiv Link: https://arxiv.org/abs/2001.10459v3 (pdf) Date submitted: 2020-07-22 22:32 Submitted by: Wintergerst, Nico Submitted to: SciPost Physics Academic field: Physics Specialties: Atomic, Molecular and Optical Physics - Theory High-Energy Physics - Theory High-Energy Physics - Phenomenology Approach: Theoretical

### Abstract

We propose a unified description of two important phenomena: color confinement in large-$N$ gauge theory, and Bose-Einstein condensation (BEC). We focus on the confinement/deconfinement transition characterized by the increase of the entropy from $N^0$ to $N^2$, which persists in the weak coupling region. Indistinguishability associated with the symmetry group -- SU($N$) or O($N$) in gauge theory, and S$_N$ permutations in the system of identical bosons -- is crucial for the formation of the condensed (confined) phase. We relate standard criteria, based on off-diagonal long range order (ODLRO) for BEC and the Polyakov loop for gauge theory. The constant offset of the distribution of the phases of the Polyakov loop corresponds to ODLRO, and gives the order parameter for the partially-(de)confined phase at finite coupling. This viewpoint may have implications for confinement at finite $N$, and for quantum gravity via gauge/gravity duality.

###### Current status:
Editor-in-charge assigned

### List of changes

- Expanded discussion after Eq.(1)
- Expanded discussion of large-N volume independence on pg. 22
- Typos corrected in Appendix A

### Submission & Refereeing History

Resubmission 2001.10459v3 on 22 July 2020
Submission 2001.10459v2 on 12 March 2020

## Reports on this Submission

### Report

I don't have very much to add to my original report. I tried to make
a constructive suggestion about relating deconfinement and BEC of a
scalar field by bringing up the Svetitsky-Yaffe results, but it sounds
like the authors don't agree that there's any relation to what they're
saying. That's fine. But I still don't understand how to make sense
of their claims.

In their reply, the authors seem to suggest that all they want to do
is draw an analogy between BEC transitions in the thermodynamic limit,
and the deconfimenent transition of infinite-N gauge theory in finite
volume. (The manuscript speculates about finite-N extensions of the
results...)

I am certainly aware that the large N limit is a thermodynamic limit.
But in large N gauge theory the deconfinment phase transition is
always very strongly first order. But BEC transitions could be
either first or second order. The fact *even the order* of the
transition is doesn't agree between deconfinement and BEC is a loud
signal that any analogy between the systems is being pushed too far.

Moreover, I have to insist that the essense of superfluidity is the
existence of a gapless Nambu-Goldstone boson excitation in the
superfluid phase. It arises because of the spontaneous breaking of
U(1) particle number symmetry. The NGB is responsible for almost all
of the interesting phenomenology of superfluids. There's no such NGB
in gauge theory, either in the confined or deconfined phases.
Confinement famously comes with a mass gap (except when continuous
global symmetries unconnected to confinement are spontaneously
broken.) This is again a loud signal that the analogy the authors try
to draw is being pushed way too far.

• validity: -
• significance: -
• originality: -
• clarity: -
• formatting: -
• grammar: -

Author Nico Wintergerst on 2020-08-02
(in reply to Report 1 on 2020-07-23)

We thank the referee for the very prompt reply. Again, however, we notice that the report is rather tangential to the main message of our work. As we have explained multiple times, we discuss fundamental common aspects of the BEC and the deconfinement/confinement transition that precede model dependent details of the theory, including the order of the phase transition and the U(1) symmetry breaking. These main points of our work are drawn into focus repeatedly throughout our manuscript, yet, remarkably, they remain without direct commentary from the referee.

Let us nonetheless address below the objections raised in the referee's second reply.

(1)

I tried to make a constructive suggestion about relating deconfinement and BEC of a scalar field by bringing up the Svetitsky-Yaffe results, but it sounds like the authors don't agree that there's any relation to what they're saying. That's fine. But I still don't understand how to make sense of their claims.

The paragraph in the original report was:

Perhaps what the authors have in mind is drawing a connection between Bose-Einstein condensation in scalar models in a 3d spacetime Z_N symmetry, and the temperature-driven confinement-deconfinement transition in 4d gauge theory, all in infinite volume. Then their example in Section 2.3 is quite misleading, given that it has a U(1) particle-number symmetry! But if this is the goal, then they would be addressing a topic on which there is a huge amount of existing work starting with the famous paper by Svetitsky and Yaffe, http://old.inspirehep.net/search?p=recid:177233&of=hd . The Svetitsky-Yaffe connection between symmetry breaking in a scalar field theory and gauge theory is very direct, obviously works at finite N, etc - so how is the proposal of the present manuscript an improvement -- what does it add? But the Svetitsky-Yaffe paper isn't even cited, so there are no comments on this!

We are happy to learn this was a constructive suggestion rather than criticism. As is acknowledged by the referee now, we think that the main point of our work is unrelated to the results mentioned in this paragraph.

(2)

But in large N gauge theory the deconfinment phase transition is always very strongly first order. But BEC transitions could be either first or second order. The fact even the order of the transition is doesn't agree between deconfinement and BEC is a loud signal that any analogy between the systems is being pushed too far.

The criticism in this paragraph is unfounded in two ways.

First, it contains a simple factual error. The deconfinement phase transition is not always very strongly first order. For example, as the referee themself raised in the original report, with many fundamental fermions the transition is not of first order. For reference, see [15].

Second, as we have emphasized both in the manuscript itself and in our first reply, BEC and the deconfinement/confinement transition are characterized by fundamental properties which are independent of the order of the phase transitions. As we pointed out, the most fundamental example is the comparison between the lambda transition of superfluid He4, which is second order and the ideal Bose gas, which is third order. Would the referee challenge such an established idea as the analogy between the ideal BEC and the lambda transition of He4 based on the difference in the order of the transition?

(3)

Moreover, I have to insist that the essense of superfluidity is the existence of a gapless Nambu-Goldstone boson excitation in the superfluid phase. It arises because of the spontaneous breaking of U(1) particle number symmetry. The NGB is responsible for almost all of the interesting phenomenology of superfluids. There's no such NGB in gauge theory, either in the confined or deconfined phases. Confinement famously comes with a mass gap (except when continuous global symmetries unconnected to confinement are spontaneously broken.) This is again a loud signal that the analogy the authors try to draw is being pushed way too far.

Again, we are addressing fundamental aspects of BEC and the deconfinement/confinement transition that can be phrased independently of Nambu-Goldstone bosons. In fact, the language of spontaneous symmetry breaking is not particularly useful here. Let us remind the referee that on one side of the analogy, N is played by the number of particles, whereas on the other it corresponds to the rank of the gauge group. Consequently, we are forced to consider BEC in particle number eigenstates with fixed N and thus adopt the more useful framework of first quantization. As pointed out in the first reply, this framework allows one to characterize all of the condensation phenomena via the permutation symmetry, or ODLRO, and can arguably be considered more fundamental than the U(1) symmetry breaking. We may add here that, for example, a modern standard textbook on BEC, "Quantum Liquids: Bose condensation and Cooper pairing in condensed-matter systems" by A. Leggett employs this point of view throughout, and discusses all properties of BEC solely based on the ODLRO and the permutation symmetry.

We also wish to add that in gauge theories, whether a mass gap is formed is a model-dependent property (such as the matter content), as also acknowledged by the referee. Indeed, for a YM theory with gauge/gravity duality, gapless gravitons should exist. In the last section of our paper, we have in fact spelled out a possible analogy between the gapless mode dual to the graviton in YM/gravity theory and the NG boson (the phonon) in BEC.