The strong isospin-breaking correction for the gluonic penguin contribution toε′/εat next-to-leading order in the chiral expansion

Abstract

The strong isospin-breaking correction ${\ensuremath{\Omega}}_{\mathrm{st}},$ which appears in estimates of the standard model value for the direct $\mathrm{CP}$-violating ratio ${\ensuremath{\epsilon}}^{\ensuremath{'}}/\ensuremath{\epsilon},$ is evaluated to next-to-leading order (NLO) in the chiral expansion using chiral perturbation theory. The relevant linear combinations of the unknown NLO $\mathrm{CP}$-odd weak low-energy constants (LEC's) which, in combination with one-loop and strong LEC contributions, are required for a complete determination at this order, are estimated using two different models. It is found that, to NLO, ${\ensuremath{\Omega}}_{\mathrm{st}}=0.08\ifmmode\pm\else\textpm\fi{}0.05,$ significantly reduced from the ``standard'' value, $0.25\ifmmode\pm\else\textpm\fi{}0.08,$ employed in recent analyses. The potentially significant numerical impact of this decrease on standard model predictions for ${\ensuremath{\epsilon}}^{\ensuremath{'}}/\ensuremath{\epsilon},$ associated with the decreased cancellation between gluonic penguin and electroweak penguin contributions, is also discussed.