Rare Kaon Decays: Matching Long And Short Distance Physics In K - > Pi e+ e-

In this thesis we try to understand the non-perturbative regime of QCD through long and short distance matching of the decay amplitude K->pi e^+ e^-. The first Chapter contains brief introduction and motivation then in the end we introduce the notations and transformation properties of various quantities and tables of phenomenological parameters. In the second chapter we discuss the Chiral Symmetry and its breaking and the construction of Delta S = 0,1 Lagrangians. Then we apply chiral Lagrangian to calculate the amplitude of K^+ -> pi^+ pi^- decay at one loop in section of the second chapter, the approach is a little bit different than that of Ecker, Pick and Raffael but a few tricks introduced by them were used. In section 2.7.3 we introduce the beyond leading order dispersive calculation of the same decay by D'Ambrosio et al. where the phenomenological parameters a_i and b_i (that completely fixes the form factor of the decay under study) were introduced and were predicted in the last chapter. In chapter 3 we start with a brief discussion of long and short distance matching of QCD and the calculation of Wilson coefficients. Then we introduce in reasonable detail the Bardeen, Buras and G`erard (BBG) scheme of matching which plays the central role in our work. In the last chapter we apply BBG scheme to calculate the form factor of the deacy K -> pi e^+ e^-, first in section 4.0.6 without vector meson resonances and find values which are extremely small compared to the experimental values then in section 4.1 we introduce the resonances through Hidden Local Symmetry (HLS) and construct the weak chiral Lagrangian containing vector coupling based on the Gilman-Wise Delta S = 1 Hamiltonian, we then use it to calculate the appropriate extension of the BBG long distane evolution operator introduced in chapter 3 and calculate the a_i and b_i parameters. Vector inclusion shows huge enhancements in both parameters. We provide detailed evaluations of the loop integrals in Appendix B, Feynman rules and other conventions are presented in Appendix A and in Appendix C we present the large N structure of relevant Wilson coefficients. Notations, symbols and transformation properties of quantities along with various phenomenological parameters and their values are provided in the end of the introductory chapter.