The CP Conserving Contribution to $K_L\to\pi^0\nu\bar\nu$
  The rare decay $K_L\to\pi^0\nu\bar\nu$ is dominated by direct CP violation and can be computed with extraordinarily high precision. In principle also a CP conserving contribution to this process can arise within the Standard Model. We clarify the structure of the CP conserving mechanism, analyzing both its short-distance and long-distance components. It is pointed out that the calculation of the CP conserving amplitude, although sensitive in part to nonperturbative physics, is rather well under control. The resulting CP conserving contribution to the rate for $K_L\to\pi^0\nu\bar\nu$ turns out to be very strongly suppressed due to several factors, which we discuss in detail. 
Gerhard Buchalla and Gino Isidori