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

2. 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

3. 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

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

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

6. Hypergeometric functions

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

8. K+,p4
π0,p1
νµ,p3
µ+,p2

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

10. Feynman parameterization
K+,p4 k q h
Feynman parameterization

11. Full agreement with Cirigliano et al.

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

13. 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

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

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

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 l 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)