Sterile-Active Neutrino Mixing

• Sterile-active neutrino mixing is defined as the quantum-mechanical interplay between SM active flavors and hypothetical sterile states, altering oscillation probabilities.
• The phenomenon is analyzed via an extended PMNS framework that incorporates new mixing angles and mass-squared differences, explaining short-baseline anomalies.
• Experimental, cosmological, and astrophysical constraints tightly bound the mixing parameters, guiding theoretical models and future neutrino studies.

Sterile-active neutrino mixing denotes quantum-mechanical mixing between the three Standard Model (SM) “active” neutrino flavors—$\nu_e$, $\nu_\mu$, $\nu_\tau$—and one or more hypothetical “sterile” neutrino states, $\nu_s$, which lack SM gauge interactions. The possibility of such mixing arises both in theoretical frameworks extending the SM, such as seesaw models, and in phenomenological scenarios addressing anomalies in short-baseline oscillation data. Sterile-active mixing modifies oscillation probabilities, has critical implications for laboratory and astrophysical neutrino experiments, and is tightly constrained by cosmological and astrophysical observables.

1. Formalism of Sterile-Active Neutrino Mixing

A generalized oscillation framework extends the PMNS matrix to accommodate one or more sterile flavors. For one sterile, the flavor and mass eigenstates are related by a $4\times4$ unitary matrix: $$\nu_f = U\,\nu_m,\qquad \nu_f \equiv \begin{pmatrix} \nu_e \ \nu_\mu \ \nu_\tau \ \nu_s \end{pmatrix},\; \nu_m \equiv \begin{pmatrix} \nu_1 \ \nu_2 \ \nu_3 \ \nu_4 \end{pmatrix}$$ where $U$ can be decomposed into sequential rotations: the standard active-active mixing angles $(\theta_{12}, \theta_{13}, \theta_{23})$ and new active-sterile angles, e.g.\ $\theta_{14},\theta_{24},\theta_{34}$, plus additional CP-violating phases (Kisslinger, 2014, Majhi et al., 2019). For small mixing, $|U_{\alpha4}|^2 \simeq \sin^2\theta_{\alpha4}$ $(\alpha = e, \mu, \tau)$.

The corresponding mass-squared differences comprise the established solar ($\delta m_{12}^2$) and atmospheric ($\delta m_{13}^2$) scales, plus the new “sterile splitting” $\delta m_{j4}^2\;(\sim 0.1-1\;\text{eV}^2$ for light sterile neutrinos) (Kisslinger, 2014).

In models with several sterile states (e.g., 3+N), the mixing matrix expands accordingly, with parameterizations such as those in (Khruschov et al., 2018) for (3+3) schemes. Theoretical models invoking flavor symmetries, e.g.\ $A_4$ or $D_4$, impose relations between active-sterile and active-active angles, and frequently predict correlated values for $\theta_{13}$ and the sterile mixings (Merle et al., 2014, Singh et al., 2022).

2. Oscillation Phenomenology and Key Probabilities

The oscillation probability between active flavors is modified in the presence of sterile mixing. In vacuum and neglecting CP violation for simplicity, the $\nu_\mu \to \nu_e$ appearance probability reads: $$P(\nu_\mu\to\nu_e) = \sum_{i=1}^4 U_{ei}^2 U_{\mu i}^2 + 2\sum_{i<j} U_{ei} U_{\mu i} U_{ej} U_{\mu j} \cos (\Delta_{ij} L)$$ where $\Delta_{ij} = \delta m_{ij}^2/(2E)$ and $L$ is the baseline (Kisslinger, 2014, Majhi et al., 2019).

For short baselines with $\Delta m_{41}^2\sim 1\;\text{eV}^2$, the leading appearance and disappearance probabilities reduce to: $$P_{\text{SBL}}(\nu_\alpha\rightarrow\nu_\beta)\simeq 4|U_{\alpha4}|^2|U_{\beta4}|^2 \sin^2 \left(\frac{\Delta m^2_{41} L}{4E}\right)$$ Demonstrably, $|U_{e4}|^2 \sim 0.02$--$0.04$, $|U_{\mu4}|^2 \sim 0.01$--$0.03$ are required to explain MiniBooNE/LSND-type anomalies (Mirizzi et al., 2013, Khruschov et al., 2018). At longer baselines, interference with the standard oscillation phase and matter effects can induce degeneracies and suppress sensitivities to mass hierarchy and CP violation in forthcoming experiments (Majhi et al., 2019).

3. Experimental and Observational Constraints

Laboratory Bounds

Disappearance and appearance channels in short-baseline oscillation experiments, such as reactor (Daya Bay, Double Chooz), accelerator (MINOS, NO$\nu$A, T2K), and source experiments, place direct upper bounds on $|U_{e4}|^2$, $|U_{\mu4}|^2$, and $|U_{\tau4}|^2$. Representative best-fit regions from global fits are: | Parameter | Best-fit Range | Source | |------------------------|-----------------------|------------------------------| | $|U_{e4}|^2$ | 0.02 – 0.04 | LSND/MiniBooNE, Reactors | | $|U_{\mu4}|^2$ | 0.01 – 0.03 | LSND/MiniBooNE, MINOS | | $\delta m^2_{41}$ | 0.5 – 2 eV$^2$ | Short-baseline anomalies |

NO$\nu$A's neutral-current analysis yields $|U_{\mu4}|^2 < 0.126$, $|U_{\tau4}|^2 < 0.268$ (90% CL, $\delta m^2_{41}\sim 0.05$–$0.5$ eV$^2$) (Collaboration et al., 2017), while correspondingly weaker limits are obtained for tau mixing.

Cosmological Constraints

Sterile neutrino production in the early universe is stringently constrained by measurements of the effective number of relativistic degrees of freedom, $N_{\rm eff}$, and by the neutrino mass contribution to the energy density, $\Omega_\nu h^2$. Planck data require (Mirizzi et al., 2013): $$N_{\rm eff} < 3.80,\qquad m_s^{\rm eff} < 0.42\,\textrm{eV}\ (95\%) \ |U_{e4}|^2,\ |U_{\mu4}|^2 \lesssim 10^{-3}\ \textrm{for}\ \delta m^2_{41}\sim 1\,\textrm{eV}^2$$ Thus, most laboratory-favored active-sterile mixing regions at the eV scale are excluded by cosmological constraints unless nonstandard cosmologies (e.g. large lepton asymmetry, secret interactions) suppress sterile production (Mirizzi et al., 2013, Hannestad et al., 2015, Alonso-Álvarez et al., 2022).

Astrophysical Bounds

Active-sterile mixing can be probed via its impact on supernovae, the cosmic microwave background, and high-energy astrophysical neutrino fluxes. Core-collapse supernovae are sensitive to keV–MeV sterile neutrinos via energy-loss arguments—mixing angles as small as $\sin^2 2\theta \lesssim 10^{-8}$ are excluded for $m_s\gtrsim 100\,$keV (Raffelt et al., 2011, Ray et al., 22 Apr 2024). For $m_s \lesssim 10\,$keV, matter suppression renders the constraints much weaker, making this mass window available for warm dark matter sterile scenarios.

The flavor composition of astrophysical neutrinos at Earth allows direct tests of non-unitarity in the active sector induced by sterile mixing. Measured flavor-ratio triangles constrain the non-unitarity parameter $\epsilon$ and are beginning to probe the favored ranges for $|U_{\alpha4}|^2$ (Argüelles et al., 2019). At present, constraints are at the level $\epsilon\lesssim 0.5$; next-generation telescopes will reach $\epsilon\sim 0.1$.

4. Theoretical Origin and Model Realizations

Seesaw Mechanisms

Sterile-active mixing is an inherent feature of seesaw-type extensions of the SM. In Type I seesaw, Yukawa couplings $Y_{\alpha I}$ generate a Dirac mass term, and the heavy Majorana mass suppresses active neutrino masses while inducing mixings: $$\theta_{\alpha i}\simeq \frac{(m_D)_{\alpha I}}{M_I}=\frac{v}{\sqrt{2}}\frac{Y_{\alpha I}}{M_I},\quad (\alpha=e,\mu,\tau)$$ For $M_I$ in the (sub-)GeV range, minimal values of the sum $$|U_{\alpha1}|^2+|U_{\alpha2}|^2 \gtrsim 10^{-11}(1\,\text{GeV}/M_I)$$ arise for two sterile states (Gorbunov et al., 2013); see Table: | Number below $D$-meson | Minimal $|U_{eI}|^2+|U_{\mu I}|^2$ | Scaling | |------------------------|-------------------------------------|-----------------| | 1 | 0 | — | | 2 | $\sim10^{-11}$ | $\propto 1/M$ | | 3 | Pairwise, as for 2 | — |

Incorporating CP-violating phases allows destructive interference, potentially suppressing the mixing to $10^{-20}$ in parts of parameter space (Krasnov et al., 2018).

Flavor Symmetry Approaches

Flavor models based on $A_4$ or $D_4$ symmetry can account for both reactor and sterile mixings as correlated effects of symmetry breaking and vacuum misalignment (Merle et al., 2014, Singh et al., 2022). In such frameworks, structure in the active–sterile sector, for instance, predicts that $\theta_{13}$ and $\theta_{14}$ are of similar magnitude and can be suppressed or enhanced together.

Alternative mixing schemes factor the full mixing matrix into a dominant “flavor symmetric” part and perturbations responsible for nonzero reactor and active–sterile angles, leading to analytic correlations among oscillation parameters (Dev et al., 2019).

Mirror Sector and Radiative Models

Models incorporating a “mirror” sector or multiple Higgs doublets naturally produce one light sterile state with eV-scale mass from Planck-suppressed dimension-5 operators, with $O(10\%)$ active-sterile mixing generated through loop effects or Higgs portal couplings (Zhang et al., 2013).

5. Oscillation Effects in Astrophysical and Cosmological Environments

Supernovae and Core-Collapse

In supernovae, active-sterile mixing induces additional MSW resonances, with matter effects governing adiabatic or nonadiabatic transitions. For $m_s\gtrsim0.1$–$10$ keV and sufficiently large mixing, $\bar\nu_\mu\to\bar\nu_s$ conversion followed by sterile escape enhances the muonization of the proto-neutron star and modifies the thermodynamics and lepton number transport (Ray et al., 22 Apr 2024, Raffelt et al., 2011). In addition, keV-scale sterile production is self-regulated via feedback on neutrino asymmetries, strongly quenching the emission rate at intermediate masses (Raffelt et al., 2011).

Early Universe: Quantum Kinetics and BBN

Quantum kinetic equations with momentum-dependent collision terms are required to accurately model sterile-state thermalization and its impact on $N_{\rm eff}$. Nearly complete thermalization ($\Delta N_{\rm eff}\approx 1$) occurs for $m_s\sim 1$ eV and mixing $\sin^2 2\theta \gtrsim 10^{-3}$ unless mechanisms such as lepton asymmetry or secret self-interactions block sterile production (Hannestad et al., 2015, Alonso-Álvarez et al., 2022). BBN sets upper bounds on the effective occupation factor of sterile states $(\zeta\lesssim0.65)$ and the active-sterile mixing angles (e.g., $\phi_{1,2} \lesssim 10^\circ$ for $\Delta m_{14}^2 = 1\,\text{eV}^2$ (Civitarese et al., 2016)).

Resonant active-sterile conversion around $T\sim10$ GeV can boost sterile production even for very small mixing, excluding angles as low as $\theta_s\sim 10^{-16}$ for MeV–GeV masses by combined BBN and CMB data (Alonso-Álvarez et al., 2022), thus closing many otherwise unconstrained regions of parameter space.

6. Model-Building and Phenomenological Implications

Sterile-active mixing alters both direct laboratory phenomenology and cosmology:

• Short-baseline anomalies can be explained with $m_4 \sim 1$ eV and $|U_{e4}|^2 \sim |U_{\mu4}|^2 \sim 0.01$.
• However, eV-scale sterile neutrinos are disfavored by cosmological datasets unless new physics suppresses their production.
• In extended seesaw or mirror models, a range of masses ($\sim$eV–GeV–keV) and mixings are natural, constrained by direct searches, BBN, and lepton flavor violating processes.
• High-precision supernova, reactor, and accelerator experiments—as well as cosmological probes such as future CMB surveys—will further constrain the parameter space, potentially uncovering new physics or conclusively ruling out most light sterile neutrino scenarios (Collaboration et al., 2017, Hannestad et al., 2015, Mirizzi et al., 2013).

Sterile-active mixing thus remains a central topic at the interface of particle physics, cosmology, and astrophysics, providing a unique window into both the origin of neutrino mass and potential portals to hidden sectors.