The CP - Violating Asymmetry in K L → π + π − e + e −

We update the theoretical analysis of the CP-violating asymmetry in the decay KL → ππee, relying on chiral perturbation theory and on the most recent phenomenological information. With the experimentally determined magnetic amplitude and branching ratio as input, the asymmetry can be calculated with good accuracy. The theoretical interpretation of the sign of the asymmetry is discussed. * Work supported in part by TMR, EC-Contract No. ERBFMRX-CT980169 (EURODAΦNE). 1. A large CP-violating asymmetry in the decay KL → ππee was originally predicted by Sehgal and Wanninger [1]. The effect is almost entirely due to indirect CP violation in K − K̄0 mixing. The predicted asymmetry in the angle between the ππ and the ee planes has been confirmed experimentally [2, 3]. The purpose of this note is to update the theoretical analysis of the asymmetry [1, 4, 5, 6, 7, 8] using the most recent phenomenological input available and employing the methods of chiral perturbation theory. The latter will mainly be invoked to estimate higher-order corrections but also to interpret the observed sign of the asymmetry. 2. The amplitude for the decay KL(p) → π(p1)π(p2)e(k+)e(k−) is expressed in terms of three invariant form factors E1, E2 (electric) and M (magnetic): A(KL → ππee) = e q2 u(k−)γ v(k+) Vμ Vμ = iE1p1μ + iE2p2μ +Mεμνρσp ν 1p ρ 2q σ (1) q = k+ + k− . We use the set of variables originally introduced by Cabibbo and Maksymovicz [9] for Ke4 decays. With this choice, the form factors E1, E2,M depend on sπ (invariant mass squared of the two pions), q (invariant mass squared of the leptons) and θπ (angle of the π in the ππ center-of-mass system with respect to the dipion line of flight in the kaon rest frame). The two remaining Dalitz variables are θl, the corresponding angle for the positron, and Φ, the angle between the dipion and dilepton planes in the kaon rest frame. After integration over four of these variables, the differential decay rate with respect to Φ assumes the general form [10, 11] dΓ dΦ = I1 + I2 cosΦ + I3 sinΦ + I4 cos 2Φ + I5 sin 2Φ . (2) Under a CP transformation cosΦ −→ cosΦ sin Φ −→ − sinΦ (3) so that non-zero I3, I5 signify CP violation. It turns out that I3 is very small in the standard model, being sensitive to direct CP violation only [4]. The quantity of interest here is I5 that is almost exclusively due to indirect CP violation [4]. A convenient measure of this term is the asymmetry ACP = 〈sgn(sin Φ cosΦ)〉 = 4I5 Γ(KL → π+π−e+e−) . (4) We use the conventions of Ref. [11], in particular ε0123 = 1 and the metric (+ −−−).