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The hadronic light-by-light contribution to the muon anomalous magnetic moment depends on an integration over three off-shell momenta squared (Q 2 i) of the correlator of four electromagnetic currents and the fourth leg at zero momentum. We derive the short-distance expansion of this correlator in the limit where all three Q 2 i are large and in the Euclidean domain in QCD. This is done via a systematic operator product expansion (OPE) in a background field which we construct. The leading order term in the expansion is the massless quark loop. We also compute the non-perturbative part of the next-to-leading contribution, which is suppressed by quark masses, and the chiral limit part of the next-to-next-to leading contributions to the OPE. We build a renormalisation program for the OPE. The numerical role of the higher-order contributions is estimated and found to be small.

The known isospin-breaking contributions to the K → ππ amplitudes are reanalyzed, taking into account our current understanding of the quark masses and the relevant non-perturbative inputs. We present a complete numerical reappraisal of the direct CP-violating ratio / , where these corrections play a quite significant role. We obtain the Standard Model prediction Re ( / ) = (14 ± 5) · 10 −4 , which is in very good agreement with the measured ratio. The uncertainty, which has been estimated conservatively, is dominated by our current ignorance about 1/N C -suppressed contributions to some relevant chiral-perturbation-theory low-energy constants.

The hadronic light-by-light contribution to the muon anomalous magnetic moment depends on an integration over three off-shell momenta squared ($$ {Q}_i^2 $$ Q i 2 ) of the correlator of four electromagnetic currents and the fourth leg at zero momentum. We derive the short-distance expansion of this correlator in the limit where all three $$ {Q}_i^2 $$ Q i 2 are large and in the Euclidean domain in QCD. This is done via a systematic operator product expansion (OPE) in a background field which we construct. The leading order term in the expansion is the massless quark loop. We also compute the non-perturbative part of the next-to-leading contribution, which is suppressed by quark masses, and the chiral limit part of the next-to-next-to leading contributions to the OPE. We build a renormalisation program for the OPE. The numerical role of the higher-order contributions is estimated and found to be small.