Running of neutrino mass parameters in the Zee model

We analyse the size of quantum corrections in the Zee model using effective field theory techniques. We derive the relevant 1-loop matching conditions and use them together with the existing renormalisation group equations in the two Higgs doublet model to calculate quantum corrections to the neutrino mass squared differences, mixing angles, and phases. Using four benchmark scenarios, we demonstrate when quantum corrections have to be included in studies of neutrino mass parameters in the Zee model.

💡 Research Summary

The paper presents a comprehensive study of quantum corrections to neutrino mass parameters within the Zee model, employing effective field theory (EFT) techniques to connect the high‑scale physics of the model to low‑energy observables. The Zee model extends the Standard Model (SM) by adding a second Higgs doublet (H₂) and a singly‑charged scalar (h), which together generate Majorana neutrino masses at one‑loop order via lepton‑number‑violating interactions.

The authors first construct the full Lagrangian, detailing the scalar potential, Yukawa sector, and the antisymmetric coupling f that mediates lepton‑flavour violation. They discuss perturbativity and vacuum‑stability constraints on the quartic couplings λ_i and outline the hierarchy of mass scales: the SM Higgs mass (≈125 GeV), the heavy neutral scalars (H, A), and the charged scalars (h⁺₁, h⁺₂). By working in the Higgs basis (β → 0) and assuming alignment (c_{β‑α} ≪ 1), they ensure a clear separation between the electroweak scale and the new heavy states, which is essential for a reliable EFT expansion.

The analysis proceeds in two matching steps. In the first step, the full Zee model is matched onto a Two‑Higgs‑Doublet Model (2HDM) EFT at the scale of the heavy scalars. The authors compute the one‑loop matching conditions for the dimension‑5 Weinberg operator, expressing its Wilson coefficient C₅ in terms of the antisymmetric coupling f, the Yukawa matrices Y₁ᵉ, Y₂ᵉ, and the scalar masses. The second step matches the 2HDM EFT onto the Standard Model EFT (SMEFT) below the electroweak scale, using existing one‑loop renormalisation‑group equations (RGEs) for the 2HDM. By solving the coupled RGEs for C₅, the quartic couplings, and the Yukawa matrices, they evolve the neutrino mass matrix from the high scale down to the experimental scale.

Four benchmark scenarios are investigated to illustrate the impact of quantum corrections:

1. Minimally constrained case (Y₁ᵉ ≲ Y₂ᵉ, α = 1): Here the alignment parameter α is close to the SM limit, and the Yukawa hierarchy is mild. Corrections to Δm²₁₂, Δm²₃₁, and the mixing angles are below 1 %, indicating that tree‑level predictions are sufficient.

2. Variations of the Yukawa couplings in the UV: By allowing sizable Y₂ᵉ entries, the RG flow amplifies the Weinberg operator. Corrections to the solar mass‑splitting Δm²₁₂ and the solar mixing angle θ₁₂ reach 5–15 %, a level detectable by upcoming experiments such as JUNO.

3. α = 0 (exact alignment limit): In this limit the mixing between the Higgs basis and the mass basis vanishes, enhancing the sensitivity of the neutrino parameters to the high‑scale couplings. Similar to scenario 2, sizable shifts (up to ~10 %) appear in Δm²₁₂ and θ₁₂, while the atmospheric parameters remain relatively stable.

4. Heavy‑charged‑scalar hierarchy (m_h ≫ m_{H₂}): When the singly‑charged scalar is much heavier than the second doublet, its contribution to the matching is suppressed, leading to negligible RG effects (≲1 %).

The numerical analysis employs a dedicated pipeline (e.g., SARAH/SPheno) to perform the matching, solve the RGEs, and extract the low‑energy neutrino observables. The authors also discuss phenomenological constraints from charged‑lepton flavour violation (μ → eγ, τ → μγ) and electroweak precision data, which restrict the allowed ranges of Y₂ᵉ and the scalar masses.

Key insights from the study are:

• The size of RG corrections is highly dependent on the hierarchy between the charged scalar and the second Higgs doublet, as well as on the magnitude of the Yukawa couplings Y₂ᵉ.
• In regions of parameter space where the antisymmetric coupling f and Y₂ᵉ are large, quantum corrections can no longer be ignored and must be included for accurate predictions.
• The alignment parameter α (or equivalently β‑α) plays a crucial role: approaching the exact alignment limit amplifies the running effects.
• The corrections to the solar sector (Δm²₁₂, θ₁₂) are generally larger than those to the atmospheric sector, making solar‑neutrino experiments a sensitive probe of the Zee model’s high‑scale structure.

The paper concludes that a consistent EFT treatment, including one‑loop matching and full RG evolution, is essential for confronting the Zee model with the forthcoming high‑precision neutrino data. The authors provide extensive appendices with explicit matching formulas, RG equations for both the 2HDM EFT and SMEFT, and detailed tables of input parameters, facilitating future studies and model‑building efforts.