We present the calculation of the virtual-and bremsstrahlung corrections of O(α s ) to the matrix elements d ℓ + ℓ − |O i |b . This is the missing piece in the full next-to-next-to-leading logarithmic (NNLL) results for various observables associated with the process B → X d ℓ + ℓ − , like the branching ratio, the CP-rate asymmetry and the forward-backward asymmetry. This paper is an extension of analogous calculations done by some of us for the process B → X s ℓ + ℓ − . As the contributions of the diagrams induced by the four-quark operators O u 1 and O u 2 with a u-quark running in the quark loop are strongly CKM suppressed, they were omitted in the analysis of B → X s ℓ + ℓ − . This is no longer possible for B → X d ℓ + ℓ − , as the corresponding contributions are not suppressed. The main new work therefore consists of calculating the O(α s ) corrections to d ℓ + ℓ − |O u 1,2 |b . In this paper we restrict ourselves to the range 0.05 ≤ s/m 2 b ≤ 0.25 (s is the invariant mass of the lepton pair), which lies above the ρ-and ω-resonances and below the J/ψ-resonance. We present the analytic results for the mentioned observables related to the process B → X d ℓ + ℓ − as expansions in the small parametersŝ = s/m 2 b , z = m 2 c /m 2 b and s/(4 m 2 c). In the phenomenological analysis at the end of the paper we discuss the impact of the NNLL corrections on the observables mentioned above. * Electronic address: hrachia@jerewan1.yerphi.am † Electronic address: bierik@itp.unibe.ch ‡ Electronic address: greub@itp.unibe.ch § Electronic address: walker@itp.unibe.ch,mwalker@bioc.unizh.ch However, restricting √ s to a region below thecc resonances, the long distance effects in B → X s ℓ + ℓ − are under control. The same is true for B → X d ℓ + ℓ − when choosing a region of √ s which is below the J/ψ-and above the ρ, ω-resonance regions. It turns out that in those ranges of √ s the corrections to the pure perturbative picture can be analyzed within the heavy quark effective theory (HQET). In particular, all available studies indicate that for the region 0.05 <ŝ = s/m 2 b < 0.25 the non-perturbative effects are below 10% [ [12][13][14][15][16][17]. Consequently, observables like differential decay rates, forward-backward asymmetries and CP-rate asymmetries for B → X s,d ℓ + ℓ − can be precisely predicted in this region of √ s using renormalization group improved perturbation theory. It was pointed out in the literature

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In a recent paper [1], we presented the calculation of the O(α s ) virtual corrections to b → s + − and of those bremsstrahlung terms which are needed to cancel the infrared divergences. In the present paper we work out the remaining O(α s ) bremsstrahlung corrections to b → s + − , which do not suffer from infrared and collinear singularities. These new contributions turn out to be small numerically. In addition, we also investigate the impact of the definition of m c on the numerical results. * Work partially supported by Schweizerischer Nationalfonds and SCOPES program

In a recent paper we presented a calculation of NNLL virtual corrections to the forwardbackward asymmetries in b → X s ℓ + ℓ − decay. That result does not include bremsstrahlung corrections which are free from infrared and collinear singularities. In the present paper we include the remaining O(α s ) bremsstrahlung corrections to the forward-backward asymmetries in b → X s ℓ + ℓ − decay. The numerical effect of the calculated contributions is found to be below 1%.

We present new contributions to the decay matrix element Γ q 12 of the B q −B q mixing complex, where q ¼ d or s. Our new results constitute the order α 2 s N f corrections to the penguin contributions to the Wilson coefficients entering Γ q 12 with full dependence on the charm quark mass. This is the first step toward the prediction of the CP asymmetry a q fs quantifying CP violation in mixing at next-to-next-to-leading logarithmic order (NNLO) in quantum chromodynamics (QCD) and further improves the prediction of the width difference ΔΓ q between the two neutral-meson eigenstates. We find a sizable effect from the nonzero charm mass and our partial NNLO result decreases the NLO penguin corrections to a q fs by 37% and those to ΔΓ q by 16%. We further update the Standard-Model NLO predictions for a q fs and the ratio of the width and mass differences of the B q eigenstates: If we express the results in terms of the pole mass of the bottom quark, we find a s fs ¼ ð2.07 AE 0.10Þ × 10 −5 , a d fs ¼ ð−4.71 AE 0.24Þ × 10 −4 , ΔΓ s =ΔM s ¼ ð4.33 AE 1.26Þ × 10 −3 , and ΔΓ d =ΔM d ¼ ð4.48 AE 1.19Þ × 10 −3. In the MS scheme these numbers read a s fs ¼ ð2.04 AE 0.11Þ × 10 −5 , a d fs ¼ ð−4.64 AE 0.25Þ × 10 −4 , ΔΓ s =ΔM s ¼ ð4.97 AE 1.02Þ × 10 −3 , and ΔΓ d =ΔM d ¼ ð5.07 AE 0.96Þ × 10 −3 .