Extending the $S$-duality soliton computation of $\theta$-term-induced electric dipole moments beyond $N=2$ supersymmetry

In $N=2$ supersymmetric $SU(2)$ Yang-Mills theory, the Seiberg-Witten solution and electromagnetic $S$-duality make the effect of the topological $\theta$-term non-perturbatively computable: supersymmetry transformations of the classical Julia-Zee dyon solution give exact electric and magnetic dipole moments for the spin-$1/2$ monopole and dyon fermions, and an $S$-duality transformation maps them to an explicit prediction, $d=-(e^2\theta/8\pi^2)\cdot(e/m)J$, for the electric dipole moment (EDM) of a charged spin-$1/2$ fermion in the dual theory (Kobayashi, Kugo and Tokunaga, 2007; reviewed in Kobayashi's 2012 talk 'Dipole Moment of Solitons'). The source authors state that this computation 'heavily depends on the existence of $N=2$ supersymmetry', that it is unknown how much it can be generalized to systems with $N=1$ or no supersymmetry, and that it is 'completely unclear' how the Standard Model with quarks and leptons could be reconstructed or embedded in such a dual world. The open problem is whether this exact soliton/duality-based non-perturbative computation can be generalized to systems with $N=1$ or no supersymmetry; the physically most interesting applications — such as the $\theta$-term contribution to the neutron EDM, the key strong-CP observable, in non-supersymmetric confining $SU(3)$ QCD — lie beyond the reach of the present $N=2$ method. This generalization remains open.

Background

The observable. An electric dipole moment (EDM) of a spin-$J$ particle measures the separation of its positive and negative charge distributions along the spin axis; it violates parity $P$ and time reversal $T$, and hence (by the CPT theorem) CP. Gauge theories admit a topological, CP-violating addition to the action, the $\theta$-term $\frac{\theta g^2}{32\pi^2}\int F^a_{\mu\nu}\tilde F^{a\mu\nu}$ (with $\tilde F$ the dual field strength), which in QCD induces an EDM on the neutron, $d_n(\theta)$. This is the central observable of the strong CP problem: the experimental bound of order $10^{-26}\ e\cdot\mathrm{cm}$ (Abel et al., 2020) forces the effective parameter $\bar\theta$ (the combination of the $\theta$ angle and the CP-violating phase of the quark mass matrix) below about $10^{-10}$, so either $\bar\theta$ is set to nearly zero by some mechanism such as an axion, or the smallness of $d_n$ awaits an explanation. Because the $\theta$-term is a total derivative, it changes neither the classical equations of motion nor the perturbative Feynman rules; no expansion in $\theta$ can therefore determine $d_n$. The conventional analytic route rotates $\theta$ into a complex quark-mass phase via the axial (chiral) anomaly and works within chiral effective field theory, giving the classic estimate that $d_n$ is of order $10^{-16}\ \bar\theta\ e\cdot\mathrm{cm}$ (Crewther, Di Vecchia, Veneziano and Witten, 1979); otherwise, first-principles values come only from large-scale numerical simulations of lattice QCD (QCD regulated on a discrete space-time grid).

The exactly solvable supersymmetric laboratory. A very different, exact laboratory exists in $N=2$ supersymmetric Yang-Mills theory ($N=2$ SYM), a gauge theory constrained by two independent supersymmetries relating bosons and fermions. For gauge group $SU(2)$ broken to $U(1)$, the Seiberg-Witten solution determines the exact low-energy effective action (Seiberg and Witten, 1994), and the theory has electromagnetic $S$-duality: the complexified coupling $\tau=\frac{\theta}{2\pi}+\frac{4\pi i}{e^2}$ transforms under $\mathrm{SL}(2,\mathbb{Z})$, exchanging electric and magnetic variables. The theory contains BPS monopoles — classically stable solitons carrying magnetic charge and saturating a Bogomol'nyi energy bound — which become dyons (carrying both magnetic and electric charge) in the presence of $\theta$: by the Witten effect, a $\theta$-term endows a monopole with the electric charge $e\theta/2\pi$ (Witten, 1979). Fermionic zero modes in the monopole background convert the soliton multiplet into a spinning, spin-$1/2$ 'monopole fermion' (and dyon fermions), so the soliton is exactly the kind of object that can carry an EDM. Kastor and Na (1999) computed the electric dipole moment of the BPS monopole by applying finite supersymmetry transformations to the classical Julia-Zee dyon solution (the classical field configuration carrying both electric and magnetic charge); Kobayashi (2007) computed the electric and magnetic dipole moments of the monopole fermion, the charge-distribution contribution to the electric analogue of the gyromagnetic ratio involving $\zeta(3)$; and Kobayashi, Kugo and Tokunaga (2007) computed the dipole moments of the dyon fermions at $\theta\neq 0$ and then applied an $S$-duality transformation, obtaining for a charged spin-$1/2$ fermion ('electron') in the dual world the exact EDM $d=-\frac{e^2\theta}{8\pi^2}\cdot\frac{e}{m}J$, i.e. $-(\alpha/2\pi)\theta$ times the Dirac magnetic moment, with $\alpha=e^2/4\pi$ and spin $J=1/2$. They stressed that this provides a novel non-perturbative method for computing $\theta$-induced EDMs, because the $\theta$-dependence enters through the exact duality structure rather than through perturbation theory.

The unresolved target. The same authors explicitly flag the limits: the computation 'heavily depends on the existence of $N=2$ supersymmetry', it is unknown how much it can be generalized to systems with $N=1$ or no supersymmetry, and it is 'completely unclear' how the usual Standard Model with quarks and leptons could be reconstructed or embedded in such a dual world in which the real electron is identified with an object dual to the monopole fermion. Kobayashi's 2012 talk 'Dipole Moment of Solitons' (Shoichi Sakata Centennial Symposium) reviews this program; its abstract states that electric and magnetic dipole moments of spinning solitons are discussed and that the effects of the topological term can be calculated non-perturbatively in the Seiberg-Witten model. The physically interesting cases lie beyond the reach of the present method: for instance the neutron, a spin-$1/2$ bound state in non-supersymmetric, confining $SU(3)$ QCD, whose $\theta$-induced EDM the $N=2$ computation cannot reach. How much the computation can be generalized to systems with $N=1$ or no supersymmetry is the open question; how the Standard Model could be embedded in the dual world is a companion open point the same authors raise.

Problem Statement

Can the non-perturbative computation of Kobayashi, Kugo and Tokunaga — based on $S$-duality ($\mathrm{SL}(2,\mathbb{Z})$ electric-magnetic duality) and classical soliton solutions, which in $N=2$ supersymmetric $SU(2)$ Yang-Mills theory computes the $\theta$-term-induced electric dipole moments of spinning solitons and, via the duality map, yields the exact EDM $d=-\frac{e^2\theta}{8\pi^2}\cdot\frac{e}{m}J$ of a charged spin-$1/2$ fermion in the dual theory — be generalized to systems with $N=1$ or no supersymmetry? The companion open point stated by the same authors is how the usual Standard Model with quarks and leptons can be reconstructed or embedded in such a dual world; physically interesting targets such as the $\theta$-term-induced neutron EDM $d_n(\theta)$ in non-supersymmetric confining $SU(3)$ QCD lie beyond that generalization.

The question is settled by either a carried-out generalization or a demonstration of obstruction. A carried-out generalization is a controlled analytic, non-perturbative computation — genuinely extending the exact Seiberg-Witten soliton/duality computation — of $\theta$-term-induced electric dipole moments of spinning solitons or spin-$1/2$ bound states in a system with $N=1$ or no supersymmetry, with the assumptions and the degree of control (exactness, controlled approximation, or quantified error) stated explicitly and with which structural ingredients of the $N=2$ computation (exact duality, BPS-type solitons with fermionic zero modes, supersymmetric control of the soliton's fields) survive or are replaced. A demonstration of obstruction is an argument or proof that no such controlled generalization exists, identifying which structural ingredients of the $N=2$ computation (exact duality, BPS-type solitons with fermionic zero modes, supersymmetric control of the soliton's fields) fail in systems with $N=1$ or no supersymmetry. Reconstructing or embedding the Standard Model in the dual world is a companion point of context, not required for settling this question.

Numerical lattice QCD determinations of $d_n(\theta)$ and effective-model computations (Skyrme or holographic solitons) are valuable benchmarks and context, but by themselves they do not answer this question: they replace the exact control of the $N=2$ soliton computation with numerical simulation or model assumptions rather than generalizing that computation to systems with $N=1$ or no supersymmetry.

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