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

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 $ρ(t) = {1/2} N(t) [\1 + \vecζ(t) \cdot \vecσ ] $, the norm of the state $N(t)$ is found to decrease monotonically from one to zero, while the magnitude of the Stokes vector $|\vecζ(t)|$ 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 $γ_L, γ_S$ and $Δm$ yields a bound on the CP-violating overlap $δ= \braket{K_L}{K_S}$, which is similar to, but weaker than, the known unitarity bound. A similar requirement on $|\vecζ(t)|$ yields a new bound, $δ^2 < {1/2} (\frac{Δγ}{Δm}) \sinh (\frac{3π}{4} \frac{Δγ}{Δm})$ which is particularly effective in limiting the CP-violating overlap in the $B^0$-$\bar{B^0}$ system. We obtain the Stokes parameter $ζ_3(t)$ which shows how the average strangeness of the beam evolves from zero to $δ$. The evolution of the Stokes vector from $|\vecζ| = 0$ to $|\vecζ| = 1$ has a resemblance to an order parameter of a system undergoing spontaneous symmetry breaking.