Models for the Electric Dipole Moment and Anomalous Magnetic Moment of the Tau Lepton

The Belle II experiment and other ongoing and projected lepton facilities are expected to greatly enhance the sensitivity to the electric dipole moment (EDM) and anomalous magnetic moment ($g-2$) for the tau lepton, making it timely to explore models that predict these observables. We present a class of models that generate a sizable EDM and $g-2$ of the tau lepton via radiative tau mass generation. Two benchmark models with different hypercharge assignments are investigated. The first model contains neutral fermions and charged scalars. We find that the model can predict a large signal of the tau EDM, $d_τ = \mathcal{O}(10^{-19}) , e , {\rm cm}$, and $g-2$, $a_τ = \mathcal{O}(10^{-5})$, which are within the reach of future updates of their measurements. In contrast, the second model, containing a charged fermion and neutral scalars, yields a similar magnitude for the $g-2$ but predicts a comparatively smaller EDM signal. Our models serve as benchmarks for new physics generating sizable EDM and $g-2$ of the tau lepton.

💡 Research Summary

The paper addresses the imminent improvement in experimental sensitivity to the tau lepton’s electric dipole moment (EDM) and anomalous magnetic moment (g‑2) at forthcoming facilities such as Belle II, BEPC II, and CEPC. Motivated by the possibility of observing sizable deviations from the Standard Model (SM) in these observables, the authors construct a class of models in which the tau mass itself is generated radiatively rather than through the usual Higgs Yukawa coupling. This radiative mass generation naturally enhances loop contributions to the dipole form factors because the loop factor that would normally suppress the EDM and g‑2 is absorbed into the expression for the tau mass.

Two benchmark realizations are studied, distinguished by the hypercharge assignments of the new exotic fields:

1. Majorana Fermion Model (MF Model) – The new fermions are SM‑singlet Majorana particles (hypercharge Yψ = 0) and the new scalars are electrically charged. The Lagrangian contains Yukawa interactions yϕ Lτ ϕ† ψR and yη ψL η τR, together with a scalar mixing term a H η† ϕ that induces mixing between the charged scalars ϕ and η. A Dirac mass term mD and Majorana masses mLL, mRR for the ψ fields introduce a physical CP‑violating phase θMFphys = ½ arg(mLL mRR mD²). After diagonalising the fermion and scalar mass matrices, the radiatively generated tau mass reads
   mradτ = (yϕ yη)/(16π²) FMF,
   where FMF is a complex loop function proportional to the masses of ψ and the charged scalars and to the mixing angles α (fermion) and θ (scalar). The complex phase of FMF, θτ, feeds directly into the dipole operators. The authors compute the coefficients CT(0) and CT′(0) governing the magnetic and electric dipole form factors, respectively, using Passarino‑Veltman functions B0, C0, C1. In the limit where the exotic masses are much larger than the tau mass, simple analytic expressions are obtained. Numerical scans show that for exotic masses around 100–500 GeV and sizable mixing (sin²α ≈ sin²θ ≈ 1), the product yϕ yη must be of order ten to reproduce the observed tau mass, which is compatible with perturbativity (yϕ yη < 4π). In this regime the model predicts
   aτ ≈ 2 × 10⁻⁵ and dτ ≈ 1 × 10⁻¹⁹ e·cm,
   both within the projected reach of Belle II and CEPC.

2. Real Scalar Model (RS Model) – Here the exotic fermion carries hypercharge Yψ = −1 (i.e., it is electrically charged) while the new scalars are neutral. The structure of the Lagrangian mirrors that of the MF model, but the CP‑violating phase now resides mainly in the fermion mass terms. The radiative mass formula is analogous, with a loop function FRS that depends on the neutral scalar masses and the charged fermion mass. Because the neutral scalars do not carry electric charge, the EDM contribution is suppressed relative to the MF model, while the g‑2 remains of comparable magnitude. The authors find typical predictions aτ ≈ 10⁻⁵ and dτ ≈ (0.2–0.5) × 10⁻¹⁹ e·cm for viable parameter choices.

Both constructions employ three Z₂ symmetries (Lτ, X, Sa) to forbid tree‑level τ Yukawa couplings, prevent lepton‑flavour‑violating decays (τ → ℓγ, τ → 3ℓ), and ensure that the exotic sector couples only to the third‑generation lepton doublet. The symmetries also allow a soft breaking term a H η† ϕ that is essential for generating the scalar mixing.

The paper proceeds to discuss experimental and theoretical constraints. Direct searches at the LHC and LEP set lower bounds on the masses of charged scalars and fermions (≈ 100 GeV). Electroweak precision observables, Higgs signal strengths, and vacuum stability impose restrictions on the scalar quartic couplings and the mixing parameter a. Indirect limits from the electron EDM, via three‑loop light‑by‑light diagrams, translate into a bound |dτ| ≲ 10⁻¹⁸ e·cm, which is still above the predicted values, leaving the models viable. Perturbativity of the Yukawa couplings (yϕ yη < 4π) further narrows the allowed region, especially because the heavy tau mass forces yϕ yη to be O(10).

A comprehensive numerical analysis scans the parameter space, varying exotic masses, mixing angles, and the soft term a, while imposing all constraints. The authors present contour plots showing regions where the predicted aτ and dτ fall within the anticipated experimental sensitivities. The MF model yields a larger EDM signal due to the presence of charged scalars in the loop, whereas the RS model predicts a smaller EDM but a similar g‑2.

In conclusion, the authors demonstrate that radiative tau‑mass generation provides a natural mechanism to obtain simultaneously large EDM and g‑2 contributions without excessive fine‑tuning. The two benchmark models serve as concrete targets for upcoming precision measurements of tau dipole moments. Observation of aτ ∼ 10⁻⁵ or dτ ∼ 10⁻¹⁹ e·cm would constitute strong evidence for new physics of the type described, and would complement analogous studies in the electron and muon sectors, offering a unique probe of lepton‑mass‑scaled CP‑violating interactions.