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

Thesis Decay rate calculation including radiative corrections Dimensional regularization for both UV & IR divergences Independent determination of |Vus| Improve the theoretical indetermination of this parameter 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

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

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

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

Hypergeometric functions

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

K+,p4 π0,p1 νµ,p3 µ+,p2

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

Feynman parameterization K+,p4 k q h Feynman parameterization

Full agreement with Cirigliano et al.

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

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

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

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

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)