The Decays K L → l + l − l ′ + l ′ − Revisited

The double lepton pair decay modes of the KL meson are analyzed including all contributions of order p in Chiral Perturbation Theory. The experimentally established eeee mode and the recently observed eeμμ mode are discussed in detail. Typeset using REVTEX 0 In this note we reconsider the double lepton pair decays KL → llll that were studied many years ago by Miyazaki and Takasugi [1]. The two modes of main interest are the KL → eeee and the KL → eeμμ decays. The first proceeds at a rate well determined experimentally [2] with a branching ratio equal to 4.1±0.8×10−8, and its interest resides partly in the possibility of determining the form factor associated with the virtual γ couplings to the KL as well as the interesting interference effect derived from the identical leptons in the final state. Both issues are analyzed here, and the main conclusions are that the form factor effect is small (4%), requiring a substantial experimental improvement over the current error of about 20%, and the interference effect is tiny (0.5%) and most likely beyond experimental access. The second decay has been recently observed for the first time by the E799 collaboration at Fermilab [3]. This experiment quotes a branching ratio of a few parts per billion, a remarkable improvement over the previous upper bound of 4.9×10−6 [2]. We discuss in this case the effects of a non-trivial form factor, that according to our result produces an increase of the branching ratio by 30%, which could therefore be tested in the future as the experimental accuracy in this mode improves. In the following we neglect a contribution to the amplitude due to CP violation. This contribution is of order p but suppressed by the CP violating mixing parameter ǫ which is of order 10 (Re(ǫ) = 1.6× 10, φǫ ≃ 43.5). Thus, the CP violating contributions to the widths considered here are suppressed by a factor approximately equal to | ǫ | Λ2χ/M2 K ∼ 1%, where Λχ = 4π Fπ. Thus, in the limit of CP conservation, the KL → γγ amplitude has the most general form: A(KL → γ∗γ′∗) = F(t, t′) ǫ ǫμkνǫ′ρk′ σ. (1) Here γ and γ are virtual photons with respective invariant mass squared t = k and t = k. F(t, t) is the form factor of order p in the chiral expansion that has a t and t independent contribution F1, due to the π , η and η poles [4], plus a t and t dependent contribution F2(t, t ) from one chiral loop plus counterterms. F1 is entirely fixed up to its sign by the KL → γγ decay, and F2(t, t) is given by [5]: 1 F2(t, t ′) = αemC8 192π3F 3 π {−(a2 + 2a4) D(t, t′, μ) + C(μ) (t+ t)} , (2) where the counterterm has eliminated an UV divergence proportional to (t + t), Fπ = 93 MeV is the pion decay constant, and D(t, t′, μ) = (t+ t′) [ 10 3 − (log M 2 K μ2 + log M π μ2 )] + 4 [ F (M π , t) + F (M 2 K , t) + F (M 2 π , t ′) + F (M K , t ′) ] , (3) with the chiral logarithms contained in the function F (m, t):