Natural neutrino mass hierarchy in a theory of gauge flavour deconstruction

1. Problem Statement

The flavour problem in the Standard Model (SM) concerns the origin of the hierarchical fermion masses and the distinct mixing patterns between quarks (small mixing, CKM matrix) and leptons (large mixing, PMNS matrix).

• Flavour Deconstruction: This framework proposes that the SM gauge group is extended to include separate gauge groups for each fermion family (e.g., $U(1)_{Y_1} \times U(1)_{Y_2} \times U(1)_{Y_3}$). While successful in generating quark hierarchies and small mixing, standard implementations (like Tri-Hypercharge) typically predict anarchic neutrino sectors.
• The Anarchy Issue: In minimal Tri-Hypercharge, right-handed neutrinos are indistinguishable singlets, leading to random $O(1)$ coefficients in the effective neutrino mass matrix. This fails to explain the observed natural neutrino mass hierarchy ($m_3 \gg m_2 \gg m_1$) and the specific structure of large mixing angles without fine-tuning.
• Goal: The authors aim to demonstrate that a natural neutrino mass hierarchy with large mixing angles can emerge naturally from gauge flavour deconstruction by moving beyond the "anarchy" hypothesis and implementing Sequential Dominance (SD).

2. Methodology

The authors extend the minimal Tri-Hypercharge (TH) theory into a larger Ultraviolet (UV) gauge group to distinguish between right-handed neutrinos and enforce Sequential Dominance.

A. Gauge Symmetry Extension

Instead of the standard TH group $G_{TH} = U(1)_{Y_1} \times U(1)_{Y_2} \times U(1)_{Y_3}$, they propose a UV extension:
$$G_{UV} = U(1)_{Y_1} \times U(1)_{R_2} \times U(1)_{(B-L)_2/2} \times U(1)_{R_3} \times U(1)_{(B-L)_3/2}$$

• Decomposition: The hypercharges for the 2nd and 3rd families are decomposed into Right-handed ($R$) and Baryon-minus-Lepton ($B-L$) components.
• Symmetry Breaking Chain:
  1. High Scale: Scalars $\chi_2, \chi_3$ acquire Vacuum Expectation Values (VEVs), breaking $G_{UV} \to G_{TH}$. This generates large Majorana masses for the right-handed neutrinos $\nu^c_2, \nu^c_3$ (proportional to $\langle \chi_{2,3} \rangle$) while keeping $\nu^c_1$ effectively decoupled (as $Y_1$ is not decomposed at this stage).
  2. Intermediate Scale: Scalars $\phi^{R,L}_{12, 23}$ (hyperons) acquire VEVs, breaking $G_{TH} \to U(1)_Y$ (SM hypercharge).

B. Sequential Dominance Implementation

The model is constructed such that the right-handed neutrinos contribute sequentially to the Type-I Seesaw mechanism:

1. Dominant: $\nu^c_3$ (mass $\sim \langle \chi_3 \rangle$) dominates the heaviest neutrino mass ($m_3$) and atmospheric mixing.
2. Subdominant: $\nu^c_2$ (mass $\sim \langle \chi_2 \rangle$) dominates the second heaviest mass ($m_2$) and solar mixing.
3. Decoupled: $\nu^c_1$ is effectively decoupled, resulting in a massless lightest neutrino ($m_1 \approx 0$).

C. Analytical Framework

The authors derive model-independent formulas for Sequential Dominance that account for contributions from both the neutrino and charged lepton sectors. This is a crucial methodological advancement, as previous SD literature often assumed mixing originated solely from the neutrino sector. They utilize a specific parametrization of the PMNS matrix ($U_{PMNS} = U_e U_\nu^\dagger$) to derive analytical expressions for mixing angles and masses.

3. Key Contributions

1. Natural Hierarchy from Gauge Structure: The paper proves that the decomposition of family hypercharges into $B-L$ groups naturally enforces the conditions for Sequential Dominance. The hierarchical structure of the charged lepton masses (driven by $\epsilon^R_{ij}$) combined with the specific gauge charges of the right-handed neutrinos creates the necessary texture for SD without fine-tuning dimensionless coefficients.
2. Dual Origin of PMNS Mixing: A novel finding is that large PMNS mixing angles arise from the interference of mixing in both the charged lepton and neutrino sectors:
   • Atmospheric Angle ($\theta_{23}$): Generated by the sum of 23-mixing in both sectors.
   • Reactor Angle ($\theta_{13}$): Dominated by 12-mixing in the charged lepton sector, naturally linking it to the Cabibbo angle ($\theta_C \sim 0.2$).
   • Solar Angle ($\theta_{12}$): Dominated by 12-mixing in the neutrino sector.
3. New Analytical Formulas: The authors provide general, model-independent formulas for PMNS angles (Appendix C) and charged lepton masses/mixing (Appendix B.2) within the SD framework, allowing for simultaneous contributions from $U_e$ and $U_\nu$.
4. Quark Sector Consistency: The model retains the successful prediction of natural quark mass hierarchies and small CKM mixing, consistent with the original Tri-Hypercharge framework.

4. Results

A. Neutrino Sector

• Mass Spectrum: Predicts a normal hierarchy with a massless lightest neutrino ($m_1 = 0$).
  • $m_3 \approx \frac{|a^\nu_{33}|^2 + |a^\nu_{23}\epsilon^L_{23}|^2}{\langle \chi_3 \rangle} \langle H_u \rangle^2$
  • $m_2 \approx \frac{|a^\nu_{12}\epsilon^L_{12}\epsilon^R_{23}|^2}{\langle \chi_2 \rangle (s^\nu_{12})^2} \langle H_u \rangle^2$
• Mixing Angles:
  • $\tan \theta^\nu_{23} \approx |a^\nu_{23}|/|a^\nu_{33}| \epsilon^L_{23}$
  • $\theta_{13} \sim \epsilon^L_{12} \sim 0.1$ (driven by charged lepton mixing).
• Scale Estimates: To reproduce observed neutrino masses with $O(1)$ couplings, the symmetry breaking scales are estimated at $\langle \chi_2 \rangle \sim 10^{13}$ GeV and $\langle \chi_3 \rangle \sim 10^{14}$ GeV.

B. Charged Lepton Sector

• Mass Hierarchy: The model naturally reproduces $m_e \ll m_\mu \ll m_\tau$ via the parameters $\epsilon^R_{12} \sim m_e/m_\mu \approx 0.005$ and $\epsilon^R_{23} \sim m_\mu/m_\tau \approx 0.06$.
• Mixing: The charged lepton mixing angles are suppressed but non-negligible, specifically $\theta^e_{12} \sim 0.1$, which is crucial for generating the correct $\theta_{13}$.

C. Quark Sector

• The model reproduces the CKM hierarchy:
  • $V_{cb} \sim \epsilon^q_{23} \approx 0.04$
  • $V_{us} \sim \epsilon^q_{12} \approx 0.2$
  • $V_{ub} \sim \epsilon^q_{23}\epsilon^q_{12}$

5. Significance

• Beyond Anarchy: This work demonstrates that flavour deconstruction does not inevitably lead to anarchy. By extending the gauge group to include $B-L$ symmetries, the framework can naturally select a specific dynamical mechanism (Sequential Dominance) that explains the observed neutrino data.
• Predictivity: Unlike anarchic models where $O(1)$ coefficients are fitted to data, this model offers predictivity. The mixing angles are determined by ratios of Yukawa couplings and symmetry breaking scales, and the lightest neutrino is predicted to be massless.
• Unified Origin of Mixing: The result that $\theta_{13}$ originates from the charged lepton sector (similar to the Cabibbo angle) provides a compelling link between quark and lepton mixing patterns, suggesting a unified origin for the disparate mixing structures in the SM.
• Theoretical Robustness: The inclusion of novel, general formulas for SD that account for both charged lepton and neutrino mixing fills a gap in the literature, making the framework applicable to a broader class of models beyond the specific Tri-Hypercharge example.

In summary, the paper successfully bridges the gap between gauge flavour deconstruction and the observed neutrino phenomenology, offering a natural, non-anarchic explanation for the neutrino mass hierarchy and large mixing angles.