# Three Flavoured neutrino oscillations and the Leggett Garg Inequality

## TL;DR

This paper explores how three-flavor neutrino oscillations violate the Leggett-Garg inequality, proposing an experimental setup to verify these quantum effects and examining how parameters like mixing angles and CP violation influence the violation.

## Contribution

It introduces an experimental proposal to test Leggett-Garg inequality violations in neutrino oscillations and analyzes the impact of various neutrino parameters on this violation.

## Key findings

- Maximum LGI violation is 2.17036 at specific neutrino path length.
- Presence of mixing angle θ13 increases LGI violation by 4.6%.
- Mass hierarchy parameter α increases LGI violation by 3.7%.

## Abstract

Three flavoured neutrino oscillations are investigated in the light of the Leggett-Garg inequality. The outline of an experimental proposal is suggested whereby the findings of this investigation may be verified. The results obtained are: (a) The maximum violation of the Leggett Garg Inequality (LGI) is $2.17036$ for neutrino path length $L_{1}=140.15 $ Km and $\Delta L=1255.7 $ Km.(b) Presence of the mixing angle $\theta_{13}$ enhances the maximum violation of LGI by $4.6\%$.(c) The currently known mass hierarchy parameter $\alpha = 0.0305$ increases the the maximum violation of LGI by $3.7\%$. (d)Presence of CP violating phase parameter enhances the maximum violation of LGI by $0.24\%$, thus providing an \textit{alternative indicator of CP violation} in 3-flavoured neutrino oscillations.