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Using extended Khuri-Treiman equations, we evaluate the final state interactions due to two-pion rescatterings to the decays η → π 0 π + π − and η → π 0 π 0 π 0 . As subtraction to the dispersion relation we take the one-loop chiral perturbation theory result of Gasser and Leutwyler. The calculated corrections are moderate and amount to about 14% in the amplitude at the center of the decay region. A careful analysis of the errors inherent to our approach is given. As a consequence, the experimental rate of the decay can only be reproduced if the double quark mass ratio Q −2 ≡ m d −mu ms−m · m d +mu ms+m is increased from the usual value of 1/(24.1) 2 to 1/(22.4 ± 0.9) 2 . We have also calculated the ratio of the rates of the two decays and various Dalitz Plot parameters. In particular, the linear slope a in the charged decay is different from the one-loop value and agrees better with experiment.

We present in detail the calculation of the O(α s ) virtual corrections to the matrix element for b → sγ. Besides the one-loop virtual corrections of the electromagnetic and color dipole operators O 7 and O 8 , we include the important two-loop contribution of the four-Fermi operator O 2 . By applying the Mellin-Barnes representation to certain internal propagators, the result of the two-loop diagrams is obtained analytically as an expansion in m c /m b . These results are then combined with existing O(α s ) Bremsstrahlung corrections in order to obtain the inclusive rate for B → X s γ. The new contributions drastically reduce the large renormalization scale dependence of the leading logarithmic result. Thus a very precise Standard Model prediction for this inclusive process will become possible once also the corrections to the Wilson coefficients are available.