1. The decays KS, L into four leptons
Rene Unterdorfer, Paul Scherrer Institut
1. Introduction
2. Basic facts
3. KS into 4 leptons
4. KL into 4 leptons
5. Conclusions

2. 1.Introduction
Decays of neutral kaons into four leptons are dominated by long-distance contributions with respect to the weak scale
They provide information on the coupling of kaons to two virtual photons
This information is needed to calculate the long-distance contributions to the decays KS,L + -, sensitive to possible new physics contributions

3. 2. Basic facts
General decomposition of amplitude of K0 into four leptons for distinguishable leptons (only terms that contribute):
KS (CP limit)
KL (CP limit)

4. In case of undistinguishable leptons one has to consider also a term A2, where p2 and p4 have been interchanged. This leads to the decay width:
with
For distinguishable leptons:

5. Four body decay  five phase space variables
In case of distinguishable leptons one can integrate over three of these variables analytically  only the invariant masses of the two leptons pair remain as variables.
1 is obtained by same method also for undistinguishable leptons. To get the interference term 2 (turns out to by a small contribution)  perform the whole phase space integration numerically.

6. Theoretical framework:
Chiral Perturbation Theory (CHPT), characterized by:
(Gasser, Leutwyler)
Chiral symmetry and its spontaneous breaking
Identification of the lightest pseudoscalar mesons as Goldstone bosons  asymptotic states are not quarks, but pions, kaons and the 
Effective low-energy expansion up to a certain order in the external momenta and masses  in the Lagrangian all terms that respect the symmetries
Large NC QCD in order to include resonance exchange, if high precision is needed.

7. 3. KS into four leptons
Up to fourth order in the external momenta the amplitude is given by loop graphs.

8. /MK2 8
/105
/MK2
Normalized differential decay width with respect to the invariant masses of the electron pair and the lepton pair

9. Decay widths and branching ratios
K S  e+e- e+e-
K S  e+e- +-
K S  +- +-
 (MeV)
7.4  10-22
6.7  10-23
1.1  10-25
1.5 10-14
1.0  10-10
9.0  10-12 Br
Errors about 25%

10. 4. KL into four leptons Loop graphs contain a weak O(p4) vertex
WZW anomaly graphs and O(p6) counter-term can be replaced by large NC expression

11. (Goity, Longzhe)
(DIP)

12. (K TeV)

13. /MK2 2
/104
/MK2
Normalized differential decay width with respect to the invariant masses of the electron pair and the lepton pair

14. Decay widths and branching ratios
K L  e+e- e+e-
K L  e+e- +-
K L  +- +-
5.1  0.7  10-22
3.4  0.5  10-23
1.3 0.2  10-26
 (MeV)
4.0  0.6  10-8
2.7  0.4  10-9
1.0  0.2  10-12 Br
4.2  0.3  10-8
2.7  0.3  10-9
Exp. Br
(K TeV)

15. 5. Conclusions
Decays of neutral kaons into four leptons provide information on the coupling of a kaon to two virtual photons needed for KS,L + -
CHPT and large NC QCD as theoretical framework
The theoretical results for the already measured channels K L  e+e- e+e- and K L  e+e- +- agree with experimental data