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Geometrical Constraints On Leptonic Unitarity Triangles

Mathieu Guigue, Lorenzo Restrepo

TL;DR

The paper investigates testing leptonic unitarity by framing the PMNS matrix in terms of leptonic unitarity triangles and deriving constraints from neutrino-oscillation amplitudes without assuming $U$ is unitary. It relates triangle apex coordinates $(\rho,\eta)$ and $(\rho',\eta')$ to combinations of disappearance and appearance amplitudes, showing that disappearance data yield circle constraints while appearance data constrain the apex slope. A Bayesian Stan framework with flat priors on matrix elements and a $5\%$ measurement precision demonstrates how combining both disappearance and appearance constraints can produce a tight, point-like overlap in the unitarity plane, consistent with unitarity in the illustrated scenario. The study discusses practical limitations, including degeneracies and experimental feasibility across $L/E$ regimes, and outlines strategies to enhance sensitivity by measuring multiple oscillation terms and leveraging diverse neutrino sources.

Abstract

The precision of the neutrino oscillation parameters measurements has improved and will continue to improve as the next-generation experiments become online. Beyond the more precise measurements of the mixing angles and phases used to parametrize the lepton mixing matrix, tests of its unitarity are of great interest. This paper studies how the amplitudes of the oscillation patterns can be used and combined to construct leptonic unitarity triangles.

Geometrical Constraints On Leptonic Unitarity Triangles

TL;DR

The paper investigates testing leptonic unitarity by framing the PMNS matrix in terms of leptonic unitarity triangles and deriving constraints from neutrino-oscillation amplitudes without assuming is unitary. It relates triangle apex coordinates and to combinations of disappearance and appearance amplitudes, showing that disappearance data yield circle constraints while appearance data constrain the apex slope. A Bayesian Stan framework with flat priors on matrix elements and a measurement precision demonstrates how combining both disappearance and appearance constraints can produce a tight, point-like overlap in the unitarity plane, consistent with unitarity in the illustrated scenario. The study discusses practical limitations, including degeneracies and experimental feasibility across regimes, and outlines strategies to enhance sensitivity by measuring multiple oscillation terms and leveraging diverse neutrino sources.

Abstract

The precision of the neutrino oscillation parameters measurements has improved and will continue to improve as the next-generation experiments become online. Beyond the more precise measurements of the mixing angles and phases used to parametrize the lepton mixing matrix, tests of its unitarity are of great interest. This paper studies how the amplitudes of the oscillation patterns can be used and combined to construct leptonic unitarity triangles.
Paper Structure (18 sections, 22 equations, 3 figures, 1 table)

This paper contains 18 sections, 22 equations, 3 figures, 1 table.

Table of Contents

  1. Unitarity triangles in the lepton sector
  2. Neutrino oscillation and oscillation amplitudes
  3. Neutrino oscillation probability
  4. Relevance of a generic three neutrino analysis framework
  5. Oscillation probability amplitudes
  6. Relationship between oscillation probability amplitudes and unitarity triangles
  7. Unitarity triangles construction
  8. Geometrical constraints from oscillation probability amplitudes
  9. Application: standard and generic unitary PMNS matrix
  10. Implementation of the constraints on a unitarity triangle
  11. Implementation framework
  12. Disappearance amplitudes constraints
  13. Appearance amplitudes constraints
  14. Combined constraints
  15. Discussion
  16. ...and 3 more sections

Figures (3)

  • Figure 1: Constraints on $(\rho_{e\mu},\eta_{e\mu})$ Eq.\ref{['eq:rho_eta_def']} (blue) and $(\rho'_{e\mu},\eta'_{e\mu})$ Eq. \ref{['eq:rho_eta_prime_def']} (orange) when considering only constraints on the disappearance amplitude measurements. Top: the amplitude values are computed using Table \ref{['tab:mean-unitarity']} and the error on each amplitude is set to $5~\%$. Bottom: Same, but where the error on $A_{\mu,23}$ is artificially increased by a factor of 5.
  • Figure 2: Future constraints on $(\rho_{e\mu},\eta_{e\mu})$ Eq. \ref{['eq:rho_eta_def']} (blue) and $(\rho'_{e\mu},\eta'_{e\mu})$ Eq. \ref{['eq:rho_eta_prime_def']} (orange) when considering constraints on the appearance amplitude measurements. Top: the amplitudes values are computed using Table \ref{['tab:mean-unitarity']} and the error on each amplitude is set to $5~\%$. Bottom: Same, but additional constraints on the $\left(\sum_{k}\vert U_{e k}\vert^2\right)^2$ and $\left(\sum_{k}\vert U_{\mu k}\vert^2\right)^2$ values are included as normal of mean value 1 and $5~\%$ error.
  • Figure 3: Future constraints on $(\rho_{e\mu},\eta_{e\mu})$ Eq. \ref{['eq:rho_eta_def']} (blue) and $(\rho'_{e\mu},\eta'_{e\mu})$ Eq. \ref{['eq:rho_eta_prime_def']} (orange) when considering all constraints on the appearance and disappearance amplitudes measurements. Top: the amplitudes values are computed using Table \ref{['tab:mean-unitarity']} and the error on each amplitude is set to $5~\%$. Bottom: Same, but a bias of 50 % is applied to $\mathcal{R}_{\mu e,13}$.