# CP Violation and Arrows of Time Evolution of a Neutral $K$ or $B$ Meson from an Incoherent to a Coherent State

## TL;DR

This paper investigates the time evolution of neutral mesons, revealing how their state properties serve as arrows of time and establishing new bounds on CP violation parameters, with implications for understanding symmetry breaking.

## Contribution

It introduces a novel analysis of meson state evolution using Stokes parameters, deriving new bounds on CP violation that are especially relevant for B meson systems.

## Key findings

- Monotonic decrease of state norm from 1 to 0
- Monotonic increase of Stokes vector magnitude from 0 to 1
- New bound on CP-violating overlap in B meson system

## Abstract

We study the evolution of a neutral $K$ meson prepared as an incoherent equal mixture of $K^0$ and $\bar{K^0}$. Denoting the density matrix by $\rho(t) = {1/2} N(t) [\1 + \vec{\zeta}(t) \cdot \vec{\sigma} ] $, the norm of the state $N(t)$ is found to decrease monotonically from one to zero, while the magnitude of the Stokes vector $|\vec{\zeta}(t)|$ increases monotonically from zero to one. This property qualifies these observables as arrows of time. Requiring monotonic behaviour of $N(t)$ for arbitrary values of $\gamma_L, \gamma_S$ and $\Delta m$ yields a bound on the CP-violating overlap $\delta = \braket{K_L}{K_S}$, which is similar to, but weaker than, the known unitarity bound. A similar requirement on $|\vec{\zeta}(t)|$ yields a new bound, $\delta^2 < {1/2} (\frac{\Delta \gamma}{\Delta m}) \sinh (\frac{3\pi}{4} \frac{\Delta \gamma}{\Delta m})$ which is particularly effective in limiting the CP-violating overlap in the $B^0$-$\bar{B^0}$ system. We obtain the Stokes parameter $\zeta_3(t)$ which shows how the average strangeness of the beam evolves from zero to $\delta$. The evolution of the Stokes vector from $|\vec{\zeta}| = 0$ to $|\vec{\zeta}| = 1$ has a resemblance to an order parameter of a system undergoing spontaneous symmetry breaking.