On the contribution of electromagnetic penguins to ϵ′

Abstract

I discuss the importance of the electromagnetic penguin (EMP) contribution to ϵ′. I confirm the corrections to earlier calculations found by Buras and Gerard (BG). Incorporating these corrections, I calculate the Wilson coefficients of the EMP operators using the full anomalous dimension matrices, and also using the large- N (number of colors) approach of BG. I disagree with BG on the coefficient of the EMP operator dominant at large N: my result is of opposite sign and smaller in magnitude. This means that for large N the EMP contribution increases ϵ′, but by less than 1%. I also find that this Wilson coefficient is poorly estimated in the large- N approach. I agree with the BG result for the coefficient of the EMP operator subdominant at large N. This coefficient is large, and is well estimated using the large- N anomalous dimension matrices. I emphasize that the crucial factor determining the EMP contribution to ϵ′ is the size of the hadronic matrix elements of the subdominant EMP operator. Though suppressed at large N, it increases ϵ′ by 5–20% if one assumes vacuum insertion approximation, and by up to 100% if one uses the results from a recent lattice calculation.