Two-loop virtual corrections to B→Xsℓ+ℓ− in the standard model

Abstract

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

B → Kℓ + ℓ − decay at large hadronic recoil

“…The definitions and nomenclature of the indices of the functions F (7,9) 1,2,8 are the same as in [16,17], where F (7,9) 1,2 are expressed as a double expansion inŝ = q 2 /m 2 b andm 2 c = m 2 c /m 2 b . It has been shown in [16,17] that keeping the terms up to the third power ofŝ andm 2 c provides a sufficient numerical accuracy in the region 0.05 ≤ŝ ≤ 0.25.…”

“…It has been shown in [16,17] that keeping the terms up to the third power ofŝ andm 2 c provides a sufficient numerical accuracy in the region 0.05 ≤ŝ ≤ 0.25. Here we use these expansions at q 2 < 0, restricting ourselves to |ŝ| < 0.25.…”

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].…”

Improved model-independent analysis of semileptonic and radiative rareBdecays

“…[16,17], recalling that the O(α s ) corrections to the matrix elements computed in Ref. [17] have been calculated only below the cc resonances.…”

“…In these one-loop contributions, m c has the meaning of the pole mass when using the expressions derived in Ref. [17]. Concerning the charm quark mass in the two-loop expressions, the definition m c is not fixed, like in B → X s γ.…”