Symmetries and the K S → μ+μ− branching ratio

Abstract

Implications of the low experimental K L → μ + μ − decay rate, and of CP, T and CPT invariances, for the K S → μ + μ − branching ratio are considered. In the spirit of economy, physical (energy-conserving) kaon decay amplitudes for the strongly coupled channels ππ and γγ are taken to be CP, T and CPT invariant, but those for the very feeble channel μ + μ − are allowed to contain non-invariant pieces in only their dispersive parts. Then, the lower bound on the K S → μ + μ − branching ratio is 10 −6, zero and 10 −6 respectively for the case of only T-invariance, of only CP invariance, and of only CPT invariance for the K → μ + μ − dispersive amplitudes. The last value was obtained previously under somewhat different assumptions. Lower bounds on the K S → μ + μ − branching ratio were previously derived in a CPT invariant description, without assuming CP invariance. We give a systematic discussion of the modifications in all these bounds if possible CPT non-invariant contributions were present, either in the K o − K o mass matrix, or in kaon decay amplitudes as restricted in the above economical scheme. Some of the previous bounds are lost, some get only quantitatively changed, and others are unaffected. In particular, with the K → ππ, γγ assumption above, the bound is 5 × 10 −7, allowing the most general K → μ + μ − amplitudes, if the imaginary part of the CP conserving total K → μ + μ − ( 1S 0) amplitude is determined by the unitarity contribution of the γγ intermediate state. Beyond the economical scheme, we also consider weakenings of all the above bounds (ours and the existing ones) if the mass-matrix parameter needed to derive these bounds is taken directly from experiment, rather than being determined through a theoretical assumption. In particular, we give the weakenings of the existing CPT invariant bounds if we take the mass-matrix parameters from data, rather than assume CPT invariance. As an illustrative example, the lowest existing bound 2 × 10 −7 is weakened to 0.08 × 10 −7.