The ΔS = 1 effective hamiltonian including next-to-leading order QCD and QED corrections

Abstract

In this paper we present a calculation of the ∆S = 1 effective weak Hamiltonian including next-to-leading order QCD and QED corrections. At a scale µ of the order of few GeV, the Wilson coefficients of the operators are given in terms of the renormalization group evolution matrix and of the coefficients computed at a large scale ∼ M W . The expression of the evolution matrix is derived from the two-loop anomalous dimension matrix which governs the mixing of the relevant current-current and penguin operators, renormalized in some given regularization scheme. We have computed the anomalous dimension matrix up to and including order α 2 s and α e α s in two different renormalization schemes, NDR and HV, with consistent results. We give many details on the calculation of the anomalous dimension matrix at two loops, on the determination of the Wilson coefficients at the scale M W and of their evolution from M W to µ. We also discuss the dependence of the Wilson coefficients/operators on the regularization scheme.

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“…(10). In (17) we have indicated the quark operators from which the G j 's the get their contrubution.…”

“…Within this basis the Wilson coefficients are calculated to the order α 2 s and α s α em [9,10], and it is now a basic element used by all groups estimating ε ′ /ε .…”

A status report concerning theoretical predictions for various kaon decays

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“…The operator product expansion (OPE) let us to compute nonleptonic |∆S| = 1 transitions at low energies (µ ≪ M W ) by means of an effective four-fermion Lagrangian [18,19]:…”

A 1996 analysis of the CP violating ratio ɛ′/ɛ

“…First of all the complete next-to-leading order (NLO) effective hamiltonians for ∆S = 1 [6][7][8], ∆S = 2 [9][10][11] and ∆B = 2 [9] are now available so that a complete NLO analysis of ε ′ /ε including constraints from the observed indirect CP violation (ε K ) and the…”