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CFTs with $U(m)\times U(n)$ Global Symmetry in 3D and the Chiral Phase Transition of QCD
by Stefanos R. Kousvos, Andreas Stergiou
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Submission summary
| Authors (as registered SciPost users): | Stefanos Robert Kousvos · Andreas Stergiou |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2209.02837v1 (pdf) |
| Date submitted: | 2022-11-03 15:13 |
| Submitted by: | Stergiou, Andreas |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
Conformal field theories (CFTs) with $U(m)\times U(n)$ global symmetry in $d=3$ dimensions have been studied for years due to their potential relevance to the chiral phase transition of quantum chromodynamics (QCD). In this work such CFTs are analyzed in $d=4-\varepsilon$ and $d=3$. This includes perturbative computations in the $\varepsilon$ and large-$n$ expansions as well non-perturbative ones with the numerical conformal bootstrap. New perturbative results are presented and a variety of non-perturbative bootstrap bounds are obtained in $d=3$. Various features of the bounds obtained for large values of $n$ disappear for low values of $n$ (keeping $m<n$ fixed), a phenomenon which is attributed to a transition of the corresponding fixed points to the non-unitary regime. Numerous bootstrap bounds are found that are saturated by large-$n$ results, even in the absence of any features in the bounds. A double scaling limit is also observed, for $m$ and $n$ large with $m/n$ fixed, both in perturbation theory as well as in the numerical bootstrap. For the case of two-flavor massless QCD existing bootstrap evidence is reproduced that the chiral phase transition with effectively restored axial $U(1)$ symmetry may be second order, albeit associated to a universality class unrelated to the one usually discussed in the $\varepsilon$ expansion. Similar evidence is found for the case of three-flavor massless QCD, where we observe a pronounced kink.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2022-12-25 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2209.02837v1, delivered 2022-12-25, doi: 10.21468/SciPost.Report.6387
Report
This work revisits the prospect of bootstrapping chiral symmetry breaking for 3d nuclear matter at finite temperature. To a good approximation, this leads one to consider a scalar order parameter (which can be thought of as a quark bilinear) in the bifundamental of $U(m) \times U(n)$. The most physical cases are $m = n = 2, 3$ but the authors take a more general point of view, also studying regimes where one or both parameters are large. The perturbative sections of this paper, which assume the transition is reachable from a standard Landau-Ginzburg action, estimate the critical values of $m$ and $n$ at which it becomes first order. This is followed by a numerical bootstrap section which supports this basic picture and reveals a surprising tendency for certain fixed points to saturate the bounds even when there is no kink. For both types of calculations, the authors show how the necessary invariant theory is made more straightforward by embedding $U(m) \times U(n)$ into $O(2mn)$.
This is a well written paper which finds new results about an important phenomenological problem. I have only minor changes to recommend.
Requested changes
1. On page 10, it might be good to quickly say that the notation $RSSR$ for the sum will be used later.
2. The caption of Figure 11 should only mention circles and squares for $n = 10, 20$.
3. I think removing "number of" would make the first sentence of the last paragraph of section 7 sound better.
4. Appendix C's "plots are ran" should say "plots are run".
5. On page 6-7, I think it is confusing to say there are 9 rank four invariant tensors for $U(m) \times U(n)$ and 7 for indistinguishable $U(n)^2$. The counting here should match the number of projectors: 12 and 9. The reduction in number appears to come from treating index permutations of $\omega_{u, ijkl}$ as equivalent. But then it seems inconsistent to not also do this for $\delta_{ij} \delta_{kl}$.
Report #1 by Anonymous (Referee 1) on 2022-12-19 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2209.02837v1, delivered 2022-12-19, doi: 10.21468/SciPost.Report.6335
Report
The paper under review studies $SU(N) \times SU(M) \times U(1)$ symmetric CFTs from the conformal bootstrap approach. While the methods used in the paper are not brand new and the direct physical implication in nature is not immediately obvious (except for $N=M = 2$ case where there already have been other studies), the paper gives some addition to our knowledge of three-dimensional CFTs.
Before publication, I think several clarifications, as well as new additional information, are needed.
(1) On page 25, it was mentioned "QCD with 1000 massless flavors", but we usually call $SU(3)$ gauge theory QCD, and with 1000 flavors, it would not be confining. I can easily guess that what is meant here was larger color Yang-Mills theory with 1000 massless flavors, but more careful rewriting seems welcomed.
(2) The intrinsic difference between $U_+$ and $U_-$ is the stability (as reviewed in section 2 of the paper). It seems extremely important to study the operator contents at various "kinks" to see whether we have one or two relevant singlet scalar operator(s) in the spectrum. Since the authors seem to have an available method (i.e. extremal functional technique) from the previous studies, I strongly recommend mentioning the stability of these potential fixed points (kinks). In particular, any potential fixed points corresponding to QCD phase transition should have only one singlet relevant operator.
(3) As we know, some of the kinks can be interpreted as more symmetric CFTs (e.g. O(2NM) fixed point). I wonder if the kink found in $SU(3) \times SU(3) \times U(1)$ in this paper can be interpreted as $SU(L)$ or $U(L)$ fixed point. It seems that plot 12 of this paper looks very similar to plot 1 of
https://arxiv.org/abs/1705.02744 (where $SU(6)$ adjoint plot was shown).
If this is the case, the nature of the fixed point is not of $SU(3) \times SU(3) \times U(1)$ but it has the enhanced symmetry: then it has little to do with the QCD phase transition.