Radiative corrections in K l3 decays

1. Radiative corrections in Kl3 decays Andrea MarroccoPh.D. student, University “Roma Tre” [Ph.D. thesis supervisor: G. Isidori]

2. Improve the theoretical indetermination of this parameter Independent determination of |Vus| The rates ratio of the electronic and the muonic channel is |Vus| independent It is possible to test the accuracy required in the CHPT expansion Thesis Decay rate calculation including radiative corrections Dimensionalregularization for both UV & IR divergences

3. q π0,p1 K+,p4 π0,p1 K+,p4 π0,p1 K+,p4 + + + h µ+,p2 νµ,p3 µ+,p2 νµ,p3 νµ,p3 Virtual photon exchange µ+,p2 K+,p4 π0,p1 K+,p4 π0,p1 + + + + νµ,p3 νµ,p3 µ+,p2 K+,p4 π0,p1 µ+,p2 h 2 q µ+,p2 νµ,p3 π0,p1 K+,p4 k π0,p1 K+,p4 + µ+,p2 νµ,p3 π0,p1 µ+,p2 νµ,p3 K+,p4 π0,p1 µ+,p2 νµ,p3 K+,p4 γ,q 2 π0,p1 K+,p4 π0,p1 K+,p4 π0,p1 K+,p4 µ+,p2 νµ,p3 Real photon emission γ,q µ+,p2 νµ,p3 µ+,p2 νµ,p3 µ+,p2 νµ,p3 γ,q The decay amplitude

4. “A” particle self energy A,p A,p Wave function renormalization Vertex modification + Self energy

5. K+,p4 K+,p4 K+,p4 K+,p4 + + q q K+,p4 K+,p4 K+,p4 K+,p4 h h Standard Feynman parameterization Ki= coefficients of the O(e2p2) mesonic Lagrangian [Urech, ‘95]

6. Hypergeometric functions

7. μ+,p2 μ+,p2 q μ+,p2 μ+,p2 + h Xi= coeff. of the O(e2p2) leptonic Lagrangian [Neufeld & Rupertsberger, ’95-96]

8. K+,p4 π0,p1 νµ,p3 µ+,p2

9. π0,p1 K+,p4 µ+,p2 νµ,p3 Contributes only to the f- function

10. K+,p4 π0,p1 h q k νµ,p3 µ+,p2 Feynman parameterization

11. Full agreement with Cirigliano et al.

12. γ,q 2 π0,p1 K+,p4 π0,p1 K+,p4 π0,p1 K+,p4 γ,q µ+,p2 νµ,p3 µ+,p2 νµ,p3 µ+,p2 νµ,p3 γ,q Real photon emission process

13. Decay rate In K+ rest frame Phase space separation This formula is valid in n dimensions and the result is based on Lorentz-covariance considerations

14. The infrared divergence is hidden in this factor It has no divergent terms and can be analytically expressed using hypergeometric functions

15. All other factors are in this function result of the integration on the photon variables Coordinate transformation Strategy to isolate the divergences

16. Conclusions & Outlook • Full analytical agreement with Cirigliano et al. in the virtual corrections • Differential decay rate calculation with real emission completed -> explicit check of the cancellation of infrared divergences • The IR-safe observable differential rate depends on z and l2. For each bin of l2 we are numerically calculating the O(α)corrections to the decay rate (numerical results in progress..). • Calculation performed for both K+ and K0 decays • For each channel we expect to reach the same accuracy of Cirigliano et al. (counterterms error ~ few x 0.001) • For the ratio between the decay rates of electronic channel and muonic channel we expect a better accuracy because many counterterms cancel (~ 0.001)