Searching for the $N$-naturalness tower of neutrinos

Technical Summary: Searching for the N-naturalness Tower of Neutrinos

Problem and Motivation
The N-naturalness model proposes a solution to the hierarchy problem by positing the existence of $N$ distinct gauge sectors, each containing a Higgs field with a mass parameter $\mu_i$ distributed within the interval $[-\Lambda_H^2, \Lambda_H^2]$. In this framework, the lightness of the Standard Model (SM) Higgs is natural if the spacing between sector mass parameters is sufficiently small, scaling as $\sim \Lambda_H^2/N$. The model introduces a fine-tuning parameter $r$ to describe the specific placement of the SM sector within this distribution.

While originally motivated by the hierarchy problem, the N-naturalness framework also addresses the smallness of neutrino masses through the "many mixing partners" mechanism. In this scenario, the SM left-handed neutrino mixes with right-handed neutrinos (or Majorana states) across all $N$ sectors. This large number of degrees of freedom dilutes the effective coupling, suppressing the neutrino mass. The paper investigates whether terrestrial neutrino experiments can constrain the parameter space of this theory, specifically the number of sectors $N$ and the fine-tuning parameter $r$.

Methodology
The authors perform a global analysis of publicly available neutrino data to constrain the $(r, N)$ parameter space. The study distinguishes between two realizations of the model:

1. N-naturalness Dirac (NND): Neutrinos acquire mass via standard Dirac mass terms involving right-handed neutrinos in each sector.
2. N-naturalness Majorana (NNM): Neutrinos acquire mass via a Weinberg operator induced by a fermionic reheaton, leading to Majorana masses.

Theoretical Framework and Parametrization
The analysis utilizes the Newtrinos.jl framework to model the mass matrices for both Dirac and Majorana cases.

• Mass Matrices: The mass matrices are constructed based on Yukawa couplings $\lambda_{ij}$ in sector space, assumed to possess an approximate permutation symmetry. The diagonal elements are governed by parameter $a$, and off-diagonal elements by $b$. The authors adopt a natural expectation of democratic mixing ($a \approx b \sim 1/\sqrt{N}$) but treat them as independent parameters, defining $\tilde{a}$ and $\tilde{b}$ to simplify the matrix structure.
• Parametrization: To reduce the dimensionality of the fit, the authors fix several parameters based on theoretical assumptions and experimental constraints:
  • The absolute neutrino mass scale $m_0$ is fixed to $0.01$ eV (a conservative upper limit).
  • The ratio of couplings $\eta = \tilde{a}^2/\tilde{b}^2$ is set to $1 + 1/N$.
  • The overall mass scale is determined by oscillation data ($\Delta m^2_{21}$ and $\Delta m^2_{31}$) and the mass ordering (Normal or Inverted).
• Observables: The model predicts modifications to:
  • Oscillation Probabilities: The survival probability $P(\nu_l \to \nu_l)$ is modified by the mixing with heavy sterile states from other sectors, introducing fast oscillations and a suppression factor dependent on $N$.
  • Effective Masses: The effective electron neutrino mass $m_\nu$ (from $\beta$-decay) and the effective Majorana mass $m_{\beta\beta}$ (from neutrinoless double-beta decay, $0\nu\beta\beta$) are calculated by summing contributions from all $N$ sectors, weighted by the mixing matrix elements.

Experimental Data and Analysis
The analysis combines data from:

• Oscillation Experiments: Daya Bay (current data) and JUNO+TAO (projected 6-year sensitivity). These experiments measure electron antineutrino survival probabilities to constrain oscillation parameters and the presence of heavy sterile states.
• $0\nu\beta\beta$ Experiments: GERDA (current limit), LEGEND-200 (current limit), and LEGEND-1000 (projected). These experiments constrain the effective Majorana mass $m_{\beta\beta}$. The analysis accounts for nuclear matrix element uncertainties and the transition between light and heavy neutrino contributions.

A profile likelihood ratio test statistic is used to draw exclusion contours in the $(r, N)$ plane at 90% confidence level (C.L.). Nuisance parameters (flux normalization, energy resolution, etc.) are profiled over.

Key Results

1. Majorana Case (NNM):

   • Current Constraints: The combination of current GERDA and Daya Bay data excludes non-fine-tuned scenarios ($r=1$) up to $N \approx 10^4$.
   • Fine-Tuning Requirements: For smaller $N$ (e.g., $N \sim 100$), the model requires significant fine-tuning ($r \sim 10^{-6}$) to remain viable.
   • Future Sensitivity: Projected JUNO+TAO data will exclude all $N < 500$ for any value of $r$. In the range $500 \le N \le 10^4$, the theory remains viable only if $r \le 0.1$.
   • Benchmark Ruling Out: The specific benchmark scenario of N-naturalness with $N=10^4$ and no fine-tuning ($r=1$), which was proposed to solve the hierarchy problem while preserving gauge coupling unification, is ruled out by current data for both Normal and Inverted mass orderings.

2. Dirac Case (NND):

   • Constraints are significantly weaker because $0\nu\beta\beta$ experiments are not applicable.
   • Current Daya Bay data excludes small $N$ ($N \le 50$) for non-fine-tuned scenarios ($r \approx 1$).
   • Projected JUNO+TAO sensitivity extends the exclusion to $N > 100$, but the parameter $r$ remains less constrained compared to the Majorana case.

Significance and Claims
The paper claims to present the first experimental search for the N-naturalness tower of neutrinos using terrestrial data. The primary significance lies in the ability of current neutrino experiments to test and rule out specific theoretical benchmarks of the N-naturalness framework.

Specifically, the authors demonstrate that the "natural" solution to the hierarchy problem with $N=10^4$ sectors and no fine-tuning is incompatible with current neutrino oscillation and $0\nu\beta\beta$ data in the Majorana scenario. This result suggests that if N-naturalness is the correct solution to the hierarchy problem, it must either involve a much larger number of sectors ($N > 10^4$) or require a degree of fine-tuning ($r < 0.1$) that undermines the "naturalness" motivation of the model. The authors note that their results are robust across different assumptions for the absolute neutrino mass scale $m_0$.

The study highlights the complementarity of oscillation and mass-scale experiments in probing new physics with light sterile states and establishes a methodology for using global neutrino data to constrain theories with large numbers of hidden sectors.