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 $\overline{K^0}$. Denoting the density matrix by $\rho (t)=1/2N(t)[\text{1}+\vec{\zeta }(t)\cdot \vec{\sigma }]$, 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 $\delta =⟨K_L⟩K_S$, 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 $B^0$-$\overline{B^0}$ 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.

Comments: 13 pages, 6 figures. Inserted conon "." in title; minor change in text. To appear in Physical review D