SciPost Phys. 17, 168 (2024) ·
published 13 December 2024
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We consider a 4d non-linear sigma model on the coset $(\text{SU}(N)_L × \text{SU}(N)_R × \text{SU}(2))/(\text{SU}(N)_{L+R}× \mathrm{U}(1))\cong \text{SU}(N) × S^2$, that features a topological Wess–Zumino–Witten (WZW) term whose curvature is $\frac{n}{24\pi^2}{Tr}(g^{-1}dg)^3 \wedge \mathrm{Vol}_{S^2}$ where $g$ is the $\text{SU}(N)$ 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 $\text{SU}(N)^2$ 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 $\text{SU}(N)_{L+R}$. This WZW term may even play a phenomenological role as a portal to a dark sector, that determines the relic abundance of dark matter.
SciPost Phys. 10, 074 (2021) ·
published 25 March 2021
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We give a general description of the interplay that can occur between local and global anomalies, in terms of (co)bordism. Mathematically, such an interplay is encoded in the non-canonical splitting of short exact sequences known to classify invertible field theories. We study various examples of the phenomenon in 2, 4, and 6 dimensions. We also describe how this understanding of anomaly interplay provides a rigorous bordism-based version of an old method for calculating global anomalies (starting from local anomalies in a related theory) due to Elitzur and Nair.
SciPost Phys. 8, 009 (2020) ·
published 23 January 2020
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What values of the Standard Model hypercharges result in a mathematically consistent quantum field theory? We show that the constraints imposed by the lack of gauge anomalies can be recast as the equation x^3 + y^3 = z^3. If hypercharge is quantised, then x, y and z must be integers. The trivial (and only) solutions, with x=0 or y=0, reproduce the hypercharge assignments seen in Nature. This argument does not rely on the mixed gauge-gravitational anomaly, which is automatically vanishing if hypercharge is quantised and the gauge anomalies vanish.
SciPost Phys. 7, 059 (2019) ·
published 8 November 2019
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We explore the phase structure of the Standard Model as the relative strengths of the SU(2) weak force and SU(3) strong force are varied. With a single generation of fermions, the structure of chiral symmetry breaking suggests that there is no phase transition as we interpolate between the SU(3)-confining phase and the SU(2)-confining phase. Remarkably, the massless left-handed neutrino, familiar in our world, morphs smoothly into a massless right-handed down-quark. With multiple generations, a similar metamorphosis occurs, but now proceeding via a phase transition. In the second half of the paper we introduce a two-parameter extension of the Standard Model, a chiral gauge theory with gauge group U(1) x Sp(r) x SU(N). We again explore the phase structure of the theory as the relative strengths of the Sp(r) and SU(N) gauge couplings vary.
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