Models of Low Energy Effective Theory applied to Kaon Non-leptonic Decays and Other Matrix Elements

In this talk I describe work on computing non-leptonic matrix elements consistently with both long and short distance contributions included. On the simpler example of the ${\pi }^{+}$-${\pi }^0$ mass difference I explain in detail the matching procedure and the difference between various low-energy models. I then explain the new difficulties in non-leptonic Kaon decays and how the matching here can in principle be done in the same way when scheme dependences are correctly accounted for. In the end I summarize the results J. Prades and I obtain for the $\mathrm{\Delta }I=1\mathrm{/}2$ rule and $B_6$.