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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
This Submission thread is now published as
Submission summary
| Authors (as registered SciPost users): | Stefanos Robert Kousvos · Andreas Stergiou |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2209.02837v3 (pdf) |
| Date accepted: | 2023-06-06 |
| Date submitted: | 2023-04-04 15:37 |
| Submitted by: | Kousvos, Stefanos Robert |
| 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 as 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 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.
List of changes
Below we list the changes made to address the issues raised by each report.
Report 3:
We have added footnote 19 which comments on the absense of the "$\phi^4$" singlet from the spectrum. The footnote also gives examples of other work where operators have been found to be missing from the spectrum ($O(m)\times O(n)$ symmetric theories and the Ising model).
Report 2:
We thank the referee for a careful reading of our manuscript, as well as pointing out possible issues with our discussion in the introduction.
Let us note that our numerical bounds, due to the conformal bootstrap, do indeed also apply to $SU(n) \times SU(n)$
3d CFTs. This is because the sum rules for $U(n) \times U(n)$ and $SU(n) \times SU(n)$ resulting from a four point function of operators in the defining representation are identical (with one particular exception, n=4, now mentioned in the introduction). Thus, our numerical results do indeed apply even if the axial symmetry is not restored exactly.
On the other hand, our perturbative results only apply to the case where the axial symmetry is restored. This is because the term in the Lagrangian corresponding to the breaking of $U(1)_A$, namely $\delta \mathcal{L}\sim g(det(\Phi)+ det(\Phi^\dagger))$ ($g$ being a coupling). The perturbative fixed points in our work are reached under the necessary condition $g=0$.
To clarify the possible applicability of our results to QCD, we have added a new paragraph on the top of page 3. Additionally, we made a number of adjustments to phrases in the introduction as well as the numerical bootstrap results section to reflect this. We would be happy to clarify further if the referee believes there are additional issues left un-adressed.
Published as SciPost Phys. 15, 075 (2023)
Reports on this Submission
Report
As for my request, the revision seems satisfactory and I suggest the manuscript be accepted.
As for the comment by the other referee, I think that the large N QCD may not be an ideal place to realize the fixed points studied in this manuscript. I believe that the large N QCD shows the first-order phase transition rather than the second-order phase transition due to the jump of free energy (order N^2 versus order N^0), which is also suggested by holographic constructions.
Report
This paper looks like it's ready for publication.
I am not bothered by the singlets which could not be found using the extremal functional. It has been known for awhile that this method, with achievable computing power, generically misses several operators even at low scaling dimension.