Effective Hamiltonian for non-leptonic decays at NNLO in QCD

Abstract

We compute the effective hamiltonian for non-leptonic |∆F | = 1 decays in the standard model including next-to-next-to-leading order QCD corrections. In particular, we present the complete three-loop anomalous dimension matrix describing the mixing of currentcurrent and QCD penguin operators. The calculation is performed in an operator basis which allows to consistently use fully anticommuting γ 5 in dimensional regularization at an arbitrary number of loops. The renormalization scheme dependences and their cancellation in physical quantities is discussed in detail. Furthermore, we demonstrate how our results are transformed to a different basis of effective operators which is frequently adopted in phenomenological applications. We give all necessary two-loop constant terms which allow to obtain the three-loop anomalous dimensions and the corresponding initial conditions of the two-loop Wilson coefficients in the latter scheme. Finally, we solve the renormalization group equation and give the analytic expressions for the low-energy Wilson coefficients relevant for non-leptonic B meson decays beyond next-to-leading order in both renormalization schemes.

On the LHCb anomaly in B → K * ℓ + ℓ −

“…This inclusive decay is theoretically wellexplored. The NNLL QCD calculations for the branching ratio [58][59][60][61][62][63][64] and the forwardbackward asymmetry [65][66][67][68][69][70] have been finalised some time ago by an effort of several research groups. Even electromagnetic corrections have been already calculated [69,71,72].…”

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“…It is known to NLO in QCD and electroweak interactions [36,38,42,44]. The parameter P c describes the short-distance contribution of the charm quark and has been calculated including NNLO QCD [45][46][47] and NLO electroweak corrections [48]. Other long-distance contributions are contained in the parameter δP c,u [43].…”

$$B\rightarrow K^*\mu ^+\mu ^-$$ B → K ∗ μ + μ - optimised observables in the MSSM

“…We have neglected the tensor operators as they are highly suppressed both in the SM and in the Minimal Supersymmetric Standard Model (MSSM). The corresponding Wilson coefficients are well known in the SM [18][19][20][21][22][23][24], where C SM [25][26][27][28], specially when the latest B → K * μ + μ − data is included [1][2][3][4].…”