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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/2209.02837v2 (pdf) |
| Date submitted: | 2023-02-02 14:41 |
| 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 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.
Author comments upon resubmission
List of changes
Report 2:
We thank the referee for carefully reading through our manuscript and for pointing out a number of misprints. We have corrected them accordingly. Additionally, we corrected a misidentification of $U_+$ and $U_-$ in parts of the text.
Report 1:
We thank the referee for their careful reading of our manuscript, especially for pointing out possible bound coincidences with earlier work.
Regarding said bound coincidences: for two bootstrap systems to have bound coincidences, the group theoretic dimensions of the externals must agree. For example, the bootstrap of a four point function of traceless symmetric two-index tensors of $O(n)$, $t_{ij}$, has bounds that coincide with the bootstrap of a four point function consisting of four $O(N)$ vectors $\phi_i$ with $N=(n-1)(n+2)/2$, i.e. the group theoretic dimension of $t_{ij}$. Similarly, a bootstrap system of $U(m)\times U(n)$ operators in the defining representation can enhance to $O(N)$ with $N=2mn$, i.e. the dimension of the defining representation in $U(m)\times U(n)$.
With these observations, we see that the $SU(6)$ adjoint bootstrap in the reference by Nakayama cannot directly coincide with our $U(3)\times U(3)$ bootstrap. This is because the group-theoretic dimension of the SU(6) adjoint is $6^2-1=35$, whereas the group-theoretic dimension of the external operator in our case is $3\cdot 3=9$.
Regarding the extraction of the extremal spectra, we did extract some sample spectra, however these do not seem to give conclusive results. For example, extracting the spectrum corresponding to $U_-$ for $U(2)\times U(20)$ (which should be unstable as a fixed point), we found one operator with dimension close to $\Delta_S=1$ and one operator with dimension close to $\Delta_S =3$, corresponding in the field theory language to "$\phi^2$" and "$\phi^6$". In other words the extremal spectrum misses the "$\phi^4$" singlet operator. We observed a similar situation when studying $O(m)\times O(n)$ symmetric theories in previous work. Hence we opted to not argue about stability based on extremal spectra.
Let us note that we also fixed a mislabeling of $U_+$ and $U_-$ in the text. $U_+$ (the stable fixed point) should be the fixed point with two Hubbard-Stratonovic fields at large-n, and $U_-$ the fixed point with one Hubbard-Stratonovich field at large-n.
Lastly, we added a footnote on page 25 pointing out that one would typically need a sufficiently large number of colors for the theory to be confining.
Current status:
Reports on this Submission
Report
In my opinion, conformal bootstrap is closer to experimental physics, and it does not seem scientifically sound that the authors report only results that are favorable to their interpretation while hiding the results that are not (i.e. the mismatch of the spectrum or stability condition in this case). I will eventually leave the editor on the decision, but if I were the authors, I would comment on the discrepancies in the paper.
Anonymous Report 2 on 2023-2-6 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2209.02837v2, delivered 2023-02-06, doi: 10.21468/SciPost.Report.6690
Strengths
This is an interesting paper exploring bootstrap and epsilon expansion bounds
which are claimed to have some possible applications to QCD at finite
temperature.
Apart from some important but fixable (at a cost of changes to e.g. the title
abstract, and introduction) issues to do with the desired application to QCD, the body of
the paper paper seems well-written and technically strong.
Weaknesses
I am confused by some statements in the introduction concerning U(1)_A symmetry.
As the authors note, due to the ABJ anomaly there is no such global symmetry in
QCD. As their references 1-4 show, in the limit T/\Lambda_{QCD} \to \infty, the
instanton density approaches to zero, the ABJ anomaly becomes negligible, and
U(1)_A becomes a good approximate symmetry of the resulting high-temperature
EFT. But I'm not aware of any plausible argument that the instanton density
drops to exactly zero at T_c, given that T_c/\Lambda_QCD \sim O(1). Without
such an argument, U(1)_A is not restored for any finite value of
T/\Lambda_{QCD}, and in particular it is not restored at T_c. So I don't
understand why G_{LRA} would be relevant for the thermal chiral phase transition
in three-color QCD. Can the authors supply an argument on why they think U(1)_A
should be restored at T_c?
As noted at the top of page 3 restoration of U(1)_A at T_c is crucial for there
to be an application of the results of this paper to QCD. Given that I think
there are persuasive reasons to think U(1)_A is NOT restored at T_c, is there
some reason the authors did not explore bootstrap founds on SU(N) \times SU(N)
3d CFTs? If there's some technical reason for this it should be discussed in
the paper.
Report
As explained above I don't think this paper can say something for three-color
QCD with any number of massless quarks. However, the way to rescue the
phenomenogical viability of the choice to consider U_1(A) is via the 1/N_c
expansion. That is, instanton effects are also suppressed by 1/N_c for any
temperature, and U(1)_A is expected to be restored in large-N_c QCD. Therefore
I think that the results of this paper help deduce constraints on N_f = 2 and
N_f = 3 QCD *in the large N_c limit*. If the authors agree, they should make this clear in all the places they refer to QCD, such as such as even the title. (I am very fond of large N
QCD, but large N QCD \neq QCD as such.).
Author: Stefanos Robert Kousvos on 2023-03-06 [id 3440]
(in reply to Report 2 on 2023-02-06)
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 of 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 close to 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.
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CFTs_with_U_m_xU_n__Global_Symmetry_in_3D_and_the_Chiral_Pha_SVoBklu.pdf
Author: Stefanos Robert Kousvos on 2023-03-06 [id 3439]
(in reply to Report 3 on 2023-02-16)We have added footnote 19 which comments on the absence 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).
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CFTs_with_U_m_xU_n__Global_Symmetry_in_3D_and_the_Chiral_Pha_VdwCqTT.pdf