Electromagnetic Leptogenesis -- an EFT-Consistent Analysis via Wilson Coefficients I: Low-Scale, Non-Resonant Regime (2509.07698v1)

Abstract: We analyse electromagnetic leptogenesis within the framework of an effective field theory, where the dynamics is governed by the gauge-invariant dipole operator $O_{NB}$. The Wilson coefficient $C_{NB}$ is matched at one loop and renormalisation-group (RG) evolved to the electroweak scale. After electroweak symmetry breaking we compute flavour-dependent two-body decay widths and CP asymmetries for $N\to\nu+\gamma/Z$, and solve the fully flavoured Boltzmann equations. In the $N_1$-dominated regime the freeze-out baryon asymmetry is $Y_B^{\rm FO}\lesssim 10^{-17}$, far below the observed value $Y_B^{\rm obs}\simeq 8.7\times 10^{-11}$. The suppression is structural: gauge invariance forces a Higgs insertion; therefore dipole couplings $\mu\propto v/M_{\Psi}^2$ while the matched coefficient $C_{NB}$ is loop-generated and further reduced by RG running. We note that in the quasi-degenerate limit the self-energy resonance can be operative and suggest a plausible path to $Y_B^{\rm obs}$.

• The paper presents an EFT-consistent analysis demonstrating that electromagnetic leptogenesis in the low-scale non-resonant regime is structurally suppressed by gauge invariance and loop factors.
• It uses detailed matching at one loop, renormalization group evolution, and full-flavour Boltzmann equations to quantify CP asymmetries and decay widths.
• Numerical benchmarks reveal a baryon asymmetry seven to eight orders of magnitude below the observed value, highlighting the need for resonant enhancement.

Electromagnetic Leptogenesis in the EFT Framework: Suppression Mechanisms and Phenomenological Implications

Introduction and Motivation

This work presents a comprehensive effective field theory (EFT) analysis of electromagnetic leptogenesis (EMLG), focusing on the low-scale, non-resonant regime. The baryon asymmetry of the Universe (BAU) is addressed via the out-of-equilibrium, CP-violating decays of heavy right-handed neutrinos, with the CP source arising from the gauge-invariant electromagnetic dipole operator $O=(\bar{L}\sigma^{\mu\nu}N)\tilde{H}B_{\mu\nu}$. The analysis is performed at the level of Wilson coefficients, matched at one loop from a renormalizable UV completion and evolved via the renormalization group (RG) to the electroweak scale. The paper includes a full treatment of flavour-dependent decay widths, CP asymmetries, and the solution of the three-flavour Boltzmann equations, incorporating spectator effects and sphaleron transitions.

EFT Matching and Wilson Coefficient Structure

The UV model introduces right-handed neutrinos $N_i$, a vector-like fermion $\Psi$, and a charged scalar $S$, all as $\mathrm{SU}(2)_L$ singlets with appropriate hypercharges to ensure gauge invariance. Integrating out $\Psi$ and $S$ at one loop generates the dimension-six dipole operator $O$, with the Wilson coefficient $NB$ given by

$$NB \sim \frac{1}{16\pi^2} f(r, \xi_j) \times (\text{couplings}),$$

where $r = M_S^2/M_\Psi^2$ and $\xi_j = M_\Psi/M_j$ are heavy-mass ratios, and $f(r, \xi_j)$ is an $O(1)$ loop function. The flavour structure is antisymmetric in the right-handed neutrino indices, requiring at least two generations for a nonzero effect. The matching calculation isolates the $\mathrm{U}(1)_Y$ dipole operator, with no one-loop mixing into $O_{NW}$ due to the absence of $B$--$W^3$ kinetic mixing.

Renormalization Group Evolution

The Wilson coefficient $NB$ is evolved from the matching scale $M_\Psi$ down to the electroweak reference scale $\mu_{\rm ref}$ using the one-loop anomalous dimension,

$$\gamma_{NB}^{(1)} = \frac{1}{32\pi^2} \left( \frac{91}{12}g_1^2 + \frac{9}{4}g_2^2 - 3y_t^2 \right),$$

where $g_1$, $g_2$, and $y_t$ are the running gauge and top Yukawa couplings. The RG running modifies $|NB|$ at the $O(10\%)$ level for typical parameter choices, with the evolution controlled by the coupled running of the gauge and Yukawa couplings.

Decay Widths and CP Asymmetries

After electroweak symmetry breaking, the operator $O$ matches onto a dimension-five dipole operator, with the dipole coupling

$$\mu_{\alpha i} = \frac{v \cos\theta_W}{\sqrt{2} M_\Psi^2} C_{NB,\alpha i}(\mu_{\rm ref}),$$

where $v$ is the Higgs VEV and $\theta_W$ is the weak mixing angle. The two-body decay channels $N \to \nu + \gamma$ and $N \to \nu + Z$ are open, with the $Z$ mode contributing non-negligibly to the total width. The partial widths scale as $M_i^3 |\mu_{\alpha i}|^2$, and the CP asymmetries are computed from the interference of tree and one-loop diagrams, with both vertex and self-energy contributions included.

Figure 1: Loop functions $f_{V_a}(x)$, $f_{S_a}(x)$, $f_{S_b}(x)$ and their sum $F(x)$, relevant for the CP asymmetry calculation.

The CP asymmetries inherit the same Lorentz structure as the tree-level widths, and the ratio of $Z$-to-photon-channel asymmetries is fixed by the phase-space and mixing-angle factors. The loop functions exhibit resonant behaviour in the quasi-degenerate limit, but are $O(1)$ in the hierarchical regime.

Flavour Dynamics and Boltzmann Evolution

The kinetic evolution is performed in the fully-flavoured regime, with three independent CP asymmetries $$\{\varepsilon_{e1}, \varepsilon_{\mu 1}, \varepsilon_{\tau 1}\}$$ tracked via Boltzmann equations. Spectator effects are encoded in a flavour-coupling matrix, and the final baryon asymmetry is obtained from the frozen $B-L$ charge at sphaleron decoupling,

$$Y_B^{\rm FO} = c_{\rm sph} Y_{B-L}(T_{\rm sph}), \qquad c_{\rm sph} = \frac{12}{37}.$$

Numerical Results: Suppression of the Baryon Asymmetry

Numerical solutions of the Boltzmann system are presented for benchmark masses $$(M_1, M_2, M_S, M_\Psi) = (0.5, 1.5, 8, 10)~\text{TeV}$$, with all inputs specified at $\mu_{\rm ref} = 150~\text{GeV}$. The time evolution of the $N_1$ abundance and the flavoured lepton asymmetries is shown, with the asymmetries sourced as $N_1$ departs from equilibrium and freezing out at sphaleron decoupling.

Figure 2: Time evolution of the $N_1$ abundance and of the flavoured lepton asymmetries in the fully-flavoured regime.

The freeze-out baryon asymmetry $Y_B^{\rm FO}$ is found to be

$Y_B^{\rm FO} \lesssim 10^{-17},$

which is seven to eight orders of magnitude below the observed value $Y_B^{\rm obs} \simeq 8.7 \times 10^{-11}$. This suppression is structural: both the dipole couplings and the CP source scale as $v^2/M_\Psi^4$, with an additional loop factor $1/(16\pi^2)$ and RG suppression.

Figure 3: Freeze-out baryon asymmetry $Y_B^{\rm FO}$ as a function of $\tilde{m}_1^{\rm EM}$, compared to the observed value and the neutrino mass scale.

Increasing the dipole interaction strength enhances both the production and washout in lockstep, leaving the final asymmetry strongly suppressed in the non-resonant regime.

Theoretical and Phenomenological Implications

The analysis demonstrates that gauge invariance enforces a Higgs insertion in the dipole operator, leading to a parametric suppression of both the decay widths and CP asymmetries. This suppression persists in both low-scale and high-scale scenarios, with the latter suffering an additional $(M_i/8\pi\Lambda)^2$ factor. The results question the viability of EMLG in the hierarchical regime, unless a resonant enhancement is operative.

The paper also highlights the importance of flavour resolution: in the quasi-degenerate limit, the self-energy contributions can resonate, and the $f_{S_b}(x)$ term survives without flavour summation, suggesting a plausible path to achieving the observed BAU via resonant EMLG.

Future Directions

The findings motivate several avenues for further research:

• A controlled resonant analysis with flavour-resolved kinetics and self-energy resummation to exploit the resonant enhancement in the quasi-degenerate regime.
• Exploration of UV completions that mitigate the Higgs-insertion suppression while maintaining gauge invariance.
• Systematic inclusion of finite-temperature corrections and $\Delta L=1,2$ scatterings in the kinetic framework.
• Direct experimental searches for dipole-induced heavy neutral lepton interactions at colliders and intensity-frontier experiments.

Conclusion

This EFT-consistent analysis of electromagnetic leptogenesis establishes that, in the low-scale, non-resonant regime, the baryon asymmetry is structurally suppressed by gauge invariance and loop effects, rendering the mechanism insufficient to account for the observed BAU. The results sharpen the theoretical understanding of EMLG and delineate the conditions under which resonant enhancement may restore viability. Future work should focus on resonant scenarios and UV model-building to overcome the identified suppression mechanisms.