Low-Scale Seesaw: RG & Threshold Effects

• Low-scale seesaw models are neutrino mass frameworks operating at the electroweak scale, characterized by successive heavy neutrino thresholds and altered RG evolution.
• They utilize tailored heavy mass assignments and flavor symmetries to induce significant threshold corrections that adjust neutrino mixing angles.
• The resulting phenomenology bridges high-scale flavor symmetry with observable low-energy neutrino oscillation data and potential collider signals.

A low-scale seesaw setup is a class of neutrino mass models in which the scale of new physics responsible for generating the smallness of active neutrino masses lies at or near the electroweak scale (typically TeV or below), as opposed to the canonical seesaw scenarios with right-handed neutrino masses at grand unification or Planckian scales. These frameworks have rich renormalization group (RG) and threshold effects; they allow for distinctive phenomenological signatures in mixing patterns and collider experiments. Their technical implementation often involves a tailored assignment of heavy masses, couplings, flavor symmetries, or extra field content to maintain phenomenological viability at the low scale.

1. Seesaw Threshold Effects and RG Matching

A defining feature in low-scale seesaw models is the presence of multiple closely spaced thresholds as the RG scale $\mu$ probes the masses of individual heavy neutrinos. Upon crossing each mass threshold $M_n$, a right-handed Majorana neutrino decouples, and the active neutrino mass matrix is immediately described by a low-energy effective operator structure that differs in its RG evolution from the full theory.

Specifically, immediately below a threshold at $\mu = M_{n-1}$, the light neutrino mass matrix is given by

$$m_\nu|_{M_{n-1}} \simeq b\,[\,m_\nu|_{M_n} + \epsilon\,v^2\,\kappa|_{M_n}\,]$$

with $b = \exp[\int (2\alpha_\nu)\;dt]$. The term proportional to $\epsilon\,v^2\,\kappa$ arises from a mismatch in the integrated (flavor-blind) beta-functions, with

$$\epsilon = \exp\left[\int (\alpha_\kappa - 2\alpha_\nu)dt\right] - 1$$

where $\kappa$ is the effective operator coefficient for light neutrino masses, and the $\alpha_i$ incorporate gauge and Higgs self-coupling contributions. The result is that threshold corrections—even for modest RG intervals—can yield sizable and nontrivial modifications to $m_\nu$ (Bergstrom et al., 2010).

2. Renormalization Group Evolution and Enhancement Factors

The RG running of neutrino parameters in a low-scale seesaw setup is uniquely affected by the sequence of heavy neutrino thresholds:

• Above the heaviest threshold, the full set of Yukawa and Majorana mass matrices run by coupled RGEs, i.e.,

$$16\pi^2 \frac{dY_e}{dt} = (\alpha_e + C_{ee}H_e + C_e^\nu H_\nu) Y_e$$

$$16\pi^2 \frac{dY_\nu}{dt} = (\alpha_\nu + C_{\nu\nu}H_\nu + C_\nu^e H_e) Y_\nu$$

• Below all thresholds, only the effective operator $\kappa$ runs, subject to

$$\frac{d\kappa}{dt} = \alpha_\kappa \kappa + C_e^\nu H_e \kappa + \kappa (C_e^\nu H_e)^T$$

where $H_e = Y_e Y_e^\dagger$, $H_\nu = Y_\nu Y_\nu^\dagger$.

Crucially, threshold corrections to mixing angles receive enhancement factors of the form $1/(m_i - m_j)$ when calculating the rotation for the mixing matrix: $$\Delta\theta_{12} \propto \epsilon\,\frac{\sqrt{m_1 m_2}}{m_2 - m_1}$$ For a quasi-degenerate light spectrum, these denominators become small, leading to possible enhancement factors $\mathcal{O}(10^2)$ or more, and potentially order-10${}^\circ$ RG-driven changes in $\theta_{12}$ even over a “short” RG interval between thresholds (Bergstrom et al., 2010).

3. Threshold-Induced Mixing Pattern Evolution

Threshold effects induce first-order corrections to the leptonic mixing matrix as

$$\widetilde{U}_{\alpha j} = U_{\alpha j} + \epsilon \sum_{k \neq j} \frac{\sqrt{m_j m_k}}{m_j - m_k} O_{kn} O_{jn} U_{\alpha k}$$

where $U$ diagonalizes $m_\nu$ at the high scale, $O_{jn}$ are elements from the Casas–Ibarra parameterization of the Dirac neutrino Yukawa couplings, and $\epsilon$ encodes RG threshold mismatch.

This analytic correction enables a high-energy flavor symmetric mixing pattern (bi-maximal, tri-bimaximal, etc.) to evolve across thresholds, morphing to observed values at low energy. For example, a high-scale pattern with $$(\theta_{12}, \theta_{23}, \theta_{13}) = (45^\circ, 45^\circ, 0)$$ can, via threshold corrections, yield $\theta_{12}$ at $34^\circ$ (within the experimental $3\sigma$ range), as shown in explicit numerical studies. The small denominator ($m_2-m_1$) ensures that $\theta_{12}$ receives the most pronounced radiative modification, while $\theta_{23}$ and $\theta_{13}$ are less sensitive due to larger mass splittings (Bergstrom et al., 2010).

4. Experimental Phenomenology and Collider Accessibility

Low-scale seesaw models with right-handed neutrino masses in the range 100 GeV–1 TeV have significant experimental implications:

• Oscillation parameters: RG and threshold-induced corrections can shift mixing angles (especially $\theta_{12}$) by up to $10^\circ$ or more, potentially reconciling flavor symmetric high-energy scenarios with observed neutrino data.
• Collider production: Should right-handed neutrinos lie below a TeV, they may be produced directly at the LHC. Although the neutrino Yukawa couplings must remain small to yield sub-eV active masses, new gauge sectors or additional interactions may exist that enhance production rates, depending on model specifics.
• Interplay with flavor models: The fact that threshold effects in low-scale seesaw models can “bridge” theoretical high-scale flavor patterns with measured low-energy parameters offers a testable realization of high-energy flavor symmetry breaking.

Table: Illustrative Impact of Threshold Corrections

Scenario	Scale (TeV)	High-Scale Pattern	RG/Threshold Effect	Low-Energy Result
Bi-maximal	1	$\theta_{12}=45^\circ$	$\Delta\theta_{12}\sim -10^\circ$	$\theta_{12}\sim35^\circ$
Tri-bimaximal	10	$\theta_{12}=35.3^\circ$	$\Delta\theta_{12}\sim -2-5^\circ$	$\theta_{12}\sim33^\circ$

Even with a relatively short RG interval between 1 TeV and $M_Z$, sizable corrections are possible, especially in the quasi-degenerate regime.

5. Mathematical Structure and RG Formalism

The technical core of the analysis is the careful separation of RG evolution above and below each seesaw threshold, leading to the formula

$$m_\nu|_{M_{n-1}} \simeq b\,[\,m_\nu|_{M_n} + \epsilon\,v^2\,\kappa|_{M_n}\,]$$

with

$$\epsilon = \exp\left[\int_{t_{n}}^{t_{n-1}} (\alpha_\kappa - 2 \alpha_\nu) dt\right] - 1$$

and $a \equiv \exp[\int \alpha_\kappa\,dt]$, $b \equiv \exp[\int 2\alpha_\nu\,dt]$. These are flavor-blind integrals, and the mismatch drives threshold corrections.

The analytic correction to the mixing matrix,

$$\widetilde{U}_{\alpha j} = U_{\alpha j} + \epsilon \sum_{k \neq j} \frac{\sqrt{m_j m_k}}{m_j - m_k} O_{kn} O_{jn} U_{\alpha k}$$

shows that the size of the correction is controlled by the small splitting $(m_j-m_k)$ and the magnitude of $\epsilon$, which depends logarithmically on scale ratios and couplings.

6. Model-Building and Theoretical Significance

The insights from threshold effects and RG running in low-scale seesaw scenarios highlight several important points for model-building:

• The possibility of significant RG-induced corrections means that the requirement for high-scale flavor symmetries to exactly match low-scale neutrino mixing is softened; radiative effects induced via thresholds can account for observed deviations from symmetric patterns.
• Low energy neutrino properties may ultimately reflect physics at scales near the electroweak or TeV range, placing experimental probes (oscillation measurements, direct collider searches) within reach.
• Parameter “tuning” (through choices in the Casas–Ibarra $O$ matrix) can exploit threshold enhancements to precisely dial low-energy mixing parameters to match observation, even starting from highly constrained high-scale textures (Bergstrom et al., 2010).

7. Outlook and Experimental Probes

The interplay between low seesaw scales, threshold-generated RG running, and experimental observables fosters strong connections between theoretical construction and testability:

• Precision oscillation experiments: Future improved measurements of $\theta_{12}$ and the other leptonic mixing angles will further constrain or support low-scale threshold effects as a source of deviation from symmetric high-scale models.
• Direct search for heavy neutrinos: Collider experiments (e.g., LHC) can probe the parameter space where right-handed neutrinos are accessible, through lepton-number violating signals or deviations from Standard Model predictions.
• Further model extensions: Models with enlarged gauge sectors (such as left–right symmetry) or specific flavor symmetries may be constructed to maximize the phenomenological utility of low-scale threshold-induced enhancements.

In conclusion, low-scale seesaw setups fundamentally alter the renormalization group evolution of neutrino masses and mixing due to threshold effects from successive decoupling of right-handed neutrinos. The radiative corrections mediated by thresholds can substantially reshape mixing patterns, accommodate flavor symmetric scenarios at low energy, and yield potentially observable consequences at both oscillation experiments and high energy colliders. These features render low-scale seesaw models uniquely positioned among neutrino mass generation mechanisms for theoretical flexibility and empirical accessibility (Bergstrom et al., 2010).