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WZW terms without anomalies: generalised symmetries in chiral Lagrangians
by Joe Davighi, Nakarin Lohitsiri
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Submission summary
| Authors (as registered SciPost users): | Joe Davighi · Nakarin Lohitsiri |
| Submission information | |
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| Preprint Link: | scipost_202411_00020v1 (pdf) |
| Date accepted: | 2024-11-19 |
| Date submitted: | 2024-11-10 17:48 |
| Submitted by: | Davighi, Joe |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We consider a 4d non-linear sigma model on the coset $(\mathrm{SU}(N)_L \times \mathrm{SU}(N)_R \times \mathrm{SU}(2))/(\mathrm{SU}(N)_{L+R}\times \mathrm{U}(1))\cong \mathrm{SU}(N) \times S^2$, that features a topological Wess-Zumino-Witten (WZW) term whose curvature is $\frac{n}{24\pi^2}\mathrm{Tr}(g^{-1}dg)^3 \wedge \mathrm{Vol}_{S^2}$ where $g$ is the $\mathrm{SU}(N)$ pion field. This WZW term, unlike its familiar cousin in QCD, does not match any chiral anomaly, so its microscopic origin is not obviously QCD-like. We find that generalised symmetries provide a key to unlocking a UV completion. The $S^2$ winding number bestows the theory with a 1-form symmetry, and the WZW term intertwines this with the $\mathrm{SU}(N)^2$ flavour symmetry into a 2-group global symmetry. Like a 't Hooft anomaly, the 2-group symmetry should match between UV and IR, precluding QCD-like completions that otherwise give the right pion manifold. We instead construct a weakly-coupled UV completion that matches the 2-group symmetry, in which an abelian gauge field connects the QCD baryon number current to the winding number current of a $\mathbb{C}P^1$ model, and explicitly show how the mixed WZW term arises upon flowing to the IR. The coefficient is fixed to be the number of QCD colours and, strikingly, this matching must be 'tree-level exact' to satisfy a quantization condition. We discuss generalisations, and elucidate the more intricate generalised symmetry structure that arises upon gauging an anomaly-free subgroup of $\mathrm{SU}(N)_{L+R}$. This WZW term may even play a phenomenological role as a portal to a dark sector, that determines the relic abundance of dark matter.
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Author comments upon resubmission
We are grateful to the first referee for their very constructive report. We have addressed the 5 requested changes in the new version of the manuscript. We summarise the changes below: 1. In the newly written Section 4.5, we sketch a `deeper UV' completion that is free of the Landau pole, whereby $SU(n_c)\times U(1)$ is embedded in $SU(n_c+1)$. As the referee anticipates, the no-go theorem requires that the quark flavour symmetry that participates in the 2-group symmetry is not present in the UV, but only emerges going towards the IR (along with the 1-form symmetry associated with the abelian gauge field) after breaking $SU(n_c+1)\to SU(n_c)\times U(1)$. We explain in the new manuscript how this plays out. 2. We include the citation to Freed's work here, as suggested by the referee. 3. We are very grateful for this correction concerning the discrete theta angle associated with the (pure torsion) non-vanishing fourth bordism group. We have amended and significantly extended the discussion here (section 2.1), adding also a more explicit account of how this $Z_2$ valued phase is fixed by anomaly matching. 4. We have followed the referee's suggestion to use the more precise integral cohomology notation for the Postnikov class - see eq. (2.19) of the revised manuscript. 5. We have completely rewritten the section 5.1 concerning the variation in which the scalar $\phi$ has non-minimal charge, taking into account the referee's comments.
We thank referee 2 for their positive and helpful report. Our revised manuscript addresses the specific comments in the report: 1. We now explicitly state (pg 2) that the mixed anomaly for $SU(N)$ at $\theta=\pi$ is present only for even $N$. 2. We have amended our citations concerning the introduction of higher-group symmetry in (T)QFTs, including the suggested reference of Kapustin and Thorngren, and also three references from the mathematical literature concerning higher gauge theory ([13-15] in the new version). 3. We have added an extensive discussion of the sign ambiguity (also known as a `discrete theta angle') and how it matches Witten's $SU(2)$ anomaly, in section 2.1 of the new version. We agree that this was lacking in our previous version.
List of changes
See author comments above for the main changes and additions to version 2. We further made small corrections to typos in Eqs. (2.15) and (5.9), as numbered according to the new version, and added a small number of additional references for completeness.
Published as SciPost Phys. 17, 168 (2024)
Reports on this Submission
Report
We thank the authors for the substantial effort in revising. The points I made are appropriately addressed, and therefore I recommend to publishing the manuscript.
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Publish (surpasses expectations and criteria for this Journal; among top 10%)