SciPost Submission Page
Color Confinement and Bose-Einstein Condensation
by Masanori Hanada, Hidehiko Shimada, Nico Wintergerst
This is not the current version.
|As Contributors:||Nico Wintergerst|
|Arxiv Link:||https://arxiv.org/abs/2001.10459v2 (pdf)|
|Date submitted:||2020-03-12 01:00|
|Submitted by:||Wintergerst, Nico|
|Submitted to:||SciPost Physics|
|Subject area:||High-Energy Physics - Theory|
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.
Submission & Refereeing History
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Reports on this Submission
Anonymous Report 2 on 2020-6-7 Invited Report
1. The paper is aimed at an interesting issue: color confinement remains an interesting problem, and a persuasive new connection between color confinement and Bose-Einstein condensation would be nice to understand.
I was originally excited to read this paper because it claimed to make
an interesting new connection between Bose-Einstein condensation and
confinement-deconfinement transitions in gauge theory. That's a big
claim, but it is not supported by a reading of the manuscript. I
expected the manuscript to start by explaining why the claimed
relation is somewhat surprising, and then explain why it works anyway
- or least to have a discussion along these lines somewhere in the
paper. But it isn't there.
Bose-Einstein condensation takes place *only* in the infinite volume
limit, and requires a small but non-vanishing repulsive coupling. The
most common examples feature a U(1) global symmetry (and indeed, the
non-relativistic example of Bose-Einstein condensation in the
manuscript has a U(1) particle number symmetry), which breaks
spontaneously. Then the condensed phase is gapless, with a
4d SU(N) YM theory has a Z_N center symmetry which breaks
spontaneously in the deconfined phase when the spatial volume is
infinite. Both the confined and deconfined phases are always gapped.
(In the deconfined phase, the electric sector is gapped thanks to a
Debye mechanism, while the magnetic sector is gapped
non-perturbatively.) When N is sent to infinity, the Z_N symmetry
does not act like a U(1) symmetry, and there is no gapless
Nambu-Goldstone boson in the deconfined phase. When N is strictly
infinite, the confinement-deconfinement phase can survive in finite
spatial volume - this is the regime studied in the manuscript. But as
soon as N is finite, this phase transition ceases to exist at finite
volume, and becomes a smooth crossover.
These are all completely obvious differences between confinement
physics and Bose-Einstein condensation physics, and any proposal to
connect the two pictures should have addressed why they don't matter
right on page one, metaphorically speaking. Yet they are not
discussed at all.
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!
Instead there are lots of arguments by analogy inspired by
manipulations of free theories and loosely worded claims. Just to
give two examples:
* Eq. 32 is claimed to be the "order parameter of the partial
deconfinement". But how precisely does its behavior distinguish
bona-fide distinct phases of matter? Do non-analyticities in the
quantity in Eq. 32 correspond to genuine phase transitions? The
paragraph below 32 gives a proposal for how it can be calculated
in general, and it appears to be a quite non-local quantity, so it
isn't obvious whether non-analyticities in it map to anything
sharp in terms of phase transitions. Is it at all meaningful at
finite N, given that Eq. 32 was obtained by studying a theory in
finite volume, where there are no phase transitions at all at
finite N? None of these questions are given a sharp answer, or
even acknowledged as serious concerns - at the end of the paper
there is some handwaving about how to extend the discussion to
finite N, but there is nothing sharp.
* Footnote 8 says "In the case of partial confinement, even if the
transition is not first order, the confined and deconfined phases
can coexist[14,11,15]. This happens for example if one introduce
fundamental matter." It's hard to say what this is supposed to
mean. When fundamental matter is introduced with N_f/N \ll 1,
there is still a first-order phase transition as a function of
temperature. When N_f/N \sim O(1), this phase transition
disappears, unless it is forced to persist by the behavior of e.g.
chiral symmetry. So what is phase transition referred to in that
To summarize, I do not think the manuscript makes a persuasive case
that its findings are sufficiently sharp and novel to be considered
for publication in SciPost.
Anonymous Report 1 on 2020-6-2 Invited Report
1. The paper points out a strong similarity between the physics of Bose Einstein condensation and the confinement/deconfinement phase transition in large N gauge theories. The connection between permutation symmetry and gauge symmetry is provocative (and new at least to the referee).
2. Evidence is presented of the correctness of this conjecture by analysis of models at weak coupling by formulating them on small spheres.
3. The analogy to BEC suggest that methods used for the latter may be useful for understanding confinement in gauge theories.
1. The models that can be treated this way require that the physics at weak coupling be continuously connected to strong coupling. While this may be true for certain supersymmetric theories it may not be true for other more physically relevant theories.
2. This analysis doesn't really lead to any new results that have not already been obtained using other methods. So the main point of the paper seems to be that it offers a new conceptual angle.
I think the paper offers some new and worthwhile observations and conclusions that are most likely relevant to the case of theories
exhibiting gauge/gravity duality. I would have liked to see more discussion of eq. 1 - what is $\mathcal U$ that appears there ? I understand that integrating over all gauge transformations produces a singlet but there seems to be more content than that in this eqn.
Can one extend this analysis to finite (small) N by exploiting large
N volume independence ? Eguchi-Kawai reduction should hold for
many of the theories that are potentially being considered eg N=4 SYM,
models with fermions in the adjoint representation etc.
It would be nice for the authors to also include a brief discussion of these issues.