In quantum field theories containing a 2π2\pi-periodic compact scalar φ\varphi and an independent Abelian global symmetry with current jj, Chen, Cherman, Choi and Neuzil (2024) identified a periodic 'Cheshire' coupling iθ^∫dDx ∂μφ2πjμi\hat{\theta}\int d^Dx\,\frac{\partial_\mu\varphi}{2\pi}j^\mu that is invisible in the particle spectrum and flat-space scattering amplitudes, yet induces Aharonov–Bohm phases eiθ^e^{i\hat{\theta}} for symmetry-charged particles encircling φ\varphi vortices. In QCD coupled to an axion such a term predicts model-independent Aharonov–Bohm scattering of baryons and leptons off axion strings and plasma drag that cuts off the axion-string stochastic gravitational-wave background above a frequency ffricf_{\rm fric}. The source paper explicitly leaves open five linked questions, which as of August 2026 remain unresolved: (i) which ultraviolet completions, such as current–current interactions, generate Cheshire θ^\hat{\theta} terms; (ii) whether the long-distance effective action of large-NcN_c QCD contains a quantized θ^=π\hat{\theta}=\pi term coupling the Skyrme (baryon-number) current to the η′\eta' meson; (iii) which previously discussed Aharonov–Bohm-like phases of global vortices are instances of a Cheshire θ^\hat{\theta}; (iv) whether distinct quantized values of θ^\hat{\theta} label distinct phases of matter; and (v) the quantitative shape of the friction-suppressed axion-string gravitational-wave spectrum.

Background

A θ\theta term in Euclidean quantum field theory is a periodic coupling iθ∫dDx q(x)i\theta\int d^Dx\,q(x) where ∫dDx q(x)∈Z\int d^Dx\,q(x)\in\mathbb{Z} is an integer topological charge; physics is 2π2\pi-periodic in θ\theta. The canonical example is the QCD θ\theta angle, whose small value is the strong-CP problem. Chen, Cherman, Choi and Neuzil (arXiv:2410.23355; published as Phys. Rev. D 112, 045012 (2025)) studied a different periodic parameter. Consider a theory in DD spacetime dimensions containing: a 2π2\pi-periodic compact scalar field φ\varphi (for example a pseudo-Nambu–Goldstone boson, i.e. a light field enjoying an approximate shift symmetry φ↦φ+const\varphi\mapsto\varphi+{\rm const}, such as an axion or the η′\eta' meson), whose winding number defines a global (D−2)(D-2)-form symmetry U(1)[D−2]U(1)^{[D-2]} — a higher-form symmetry whose charged objects are extended defects, namely the φ\varphi vortices (global strings); and an ordinary 0-form Abelian global symmetry G=U(1)G=U(1) with conserved current jj, assumed to have no 't Hooft anomaly (a quantum inconsistency that would obstruct gauging the symmetry) mixing it with the vortex symmetry. Then the effective action may contain the 'Cheshire θ^\hat{\theta} term' iθ^∫dDx ∂μφ2πjμi\hat{\theta}\int d^Dx\,\frac{\partial_\mu\varphi}{2\pi}j^\mu, with θ^≃θ^+2π\hat{\theta}\simeq\hat{\theta}+2\pi. Naively it vanishes by integrating by parts and using current conservation, so it is usually discarded; particle spectra and particle–particle scattering do not depend on it. Its only — but robust — effect is topological: a GG-charged particle transported around a unit-winding φ\varphi vortex acquires an Aharonov–Bohm (AB) phase eiθ^e^{i\hat{\theta}}, in the same way a charged particle encircling a magnetic flux tube acquires eiqΦe^{iq\Phi}, and this phase is protected by the vortex higher-form symmetry. The name echoes the 'Cheshire cat' idea that a property (here the AB flux) persists after the object carrying it (the vortex core) is resolved. Such particle–vortex AB phenomena have a long prehistory: AB interactions of particles with gauged cosmic strings (Alford and Wilczek, Phys. Rev. Lett. 62, 1071 (1989)); a global analog of the AB effect for cosmic textures (March-Russell, Preskill and Wilczek, Phys. Rev. Lett. 68, 2567 (1992)); 'AB defects' as order parameters for symmetry realizations (Chatterjee and Nitta, Phys. Rev. D 101, 085002 (2020)); anyonic particle–vortex statistics in dense quark matter (Cherman, Sen and Yaffe, Phys. Rev. D 100, 034015 (2019)); and matching of AB phases across Higgs–confinement transitions (Hayashi, Phys. Rev. Lett. 132, 221901 (2024)). The Cheshire θ^\hat{\theta} term gives these phenomena a single effective-field-theory parameter. Two applications frame the open questions. First, in QCD coupled to an axion (a pseudo-Goldstone field that arises when the Peccei–Quinn symmetry, an anomalous global symmetry introduced to solve the strong-CP problem, is spontaneously broken; the axion strings left by that breaking are global vortices), a Cheshire term with G=U(1)B−LG=U(1)_{B-L} (baryon number minus lepton number) makes baryons and leptons acquire AB phases around axion strings, yielding model-independent scattering, and the corresponding drag force ∣fdrag∣=βT3vˉ|\mathbf{f}_{\rm drag}|=\beta T^3\bar v on strings moving through the thermal plasma suppresses loop production above a temperature TfricT_{\rm fric} and hence cuts off the stochastic gravitational-wave background (SGWB) from axion strings at a frequency ffricf_{\rm fric}, on top of known frictionless spectra (Gouttenoire, Servant and Simakachorn, JCAP 07, 032 (2020)). Second, at large NcN_c (number of colors), where the η′\eta' meson becomes light in the chiral Lagrangian (the low-energy effective field theory of the pion fields), baryon number is realized solitonically and its current is the Skyrme current jB=−124π2⋆tr(U†dU)3j_B=-\frac{1}{24\pi^2}\star{\rm tr}(U^\dagger dU)^3 built from the pion field matrix UU; the chiral Lagrangian can in principle contain the Cheshire term iθ∫dη′2πfπ∧124π2tr(U†dU)3i\theta\int\frac{d\eta'}{2\pi f_\pi}\wedge\frac{1}{24\pi^2}{\rm tr}(U^\dagger dU)^3, which would make the solitonic baryons (Skyrmions) pick up AB phases around η′\eta' strings.

Problem Statement

Determine the microscopic status and physical consequences of Cheshire θ^\hat{\theta} couplings, along the five linked questions posed by Chen, Cherman, Choi and Neuzil (arXiv:2410.23355), which together span origin, realization, interpretation, and phenomenology: (i) UV origin — do well-motivated short-distance completions, in particular current–current interactions 1ΛUV2∫d4x jSμjμ\frac{1}{\Lambda_{\rm UV}^2}\int d^4x\,j^\mu_{\mathcal S}j_\mu where jSμj^\mu_{\mathcal S} is the ultraviolet current that flows to the φ\varphi shift-symmetry current, naturally produce the low-energy Cheshire θ^\hat{\theta} term in axion models, and are there other mechanisms that induce such terms? (ii) Large-NcN_c QCD — does the long-distance effective action of large-NcN_c QCD include a quantized θ^=π\hat{\theta}=\pi coupling between the Skyrme (baryon-number) current and the η′\eta' meson, i.e. the term iθ∫dη′2πfπ∧124π2tr(U†dU)3i\theta\int\frac{d\eta'}{2\pi f_\pi}\wedge\frac{1}{24\pi^2}{\rm tr}(U^\dagger dU)^3 with θ=π\theta=\pi, and if so with what consequences (e.g. an Aharonov–Bohm phase for Skyrmions encircling η′\eta' strings)? (iii) Unification — which of the previously discussed Aharonov–Bohm-like phases for U(1)U(1) global vortices can be re-expressed as instances of a Cheshire θ^\hat{\theta} parameter, and which cannot? (iv) Phases of matter — do effective field theories with distinct quantized values of θ^\hat{\theta} (for example values fixed by discrete symmetries or anomaly constraints) describe distinct phases of matter? (v) Gravitational waves — what is the quantitative shape of the axion-string stochastic gravitational-wave background in the friction-suppressed region f≳ffricf\gtrsim f_{\rm fric} caused by Cheshire-induced plasma drag on the strings?

References

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