Analytic formulae for factorisable power corrections to optimised observables 102

We have considered a particular observable and demonstrated numerically that the prediction for observables depends on the scheme chosen for the soft form factorsξ_⊥,k. In this section we illustrate this scheme dependence more explicitly by giving analytic formulae for the power corrections to the observablesP₅⁰, P₁ and P₂, both in the transversity and in the helicity basis. The two bases are related to each other via the relations given in Eq. (31) of Ref. [151]. In both cases we parametrise the power corrections according to Eq. (6.1.3). The formulae are given without fixing a particular scheme, i.e., before power corrections are partially absorbed into the non-perturbative input parametersξ⊥ and ξk.

In the helicity basis, the formula for P₅⁰ reads

P₅⁰ =P₅⁰|^∞ 1 +2aV−−2aT− We denote the large-recoil expression as P₅⁰|^∞ and leave aside non-local terms, corresponding to non-factorisable corrections. Our result agrees with Eq. (25) of Ref. [146] for the terms propor-tional toa_V−, a_T−, a_V0, a_T0, but we find an additional term proportional toa_V+. We would like to stress that precisely this term, which is hidden in “further terms” and not discussed in Ref. [146], dominates the power corrections in the anomalous region aroundq₀²∼6 GeV², as can be seen from the numerical evaluation of Eq. (6.2.13):

6.2. Anatomy of factorisable power corrections 103 This means that the discussion on the scheme dependence of P₅⁰ in Ref. [146] only takes into account numerically subleading contributions. Converted into the transversity basis, Eq. (6.2.13) becomes

with the dominant term being proportional to the combination a_A1 −a_V of power correction parameters. IfA1 orV is chosen as input forξ⊥, the corresponding parameter aA1 oraV vanishes identically. On the other hand, ifT₁is taken as input, botha_A1 ora_V survive and their independent variation leads to an increase of the errors associated to power corrections. This behaviour explains part of the inflated errors in Ref. [151] and it is analytically pinned down in Eqs. (6.2.13) and (6.2.19). The formulae support the numerical analysis reported in Fig. 2 of Ref. [147], where the binned predictions for P1, P2, P₄⁰, P₅⁰ were given in the two schemes with ξ⊥ defined from V or T₁, respectively. Without any further assumption on the correlations between the parametersa_F, Eqs. (6.2.13) and (6.2.19) manifest an explicit scheme dependence whose origin and interpretation was discussed in detail in Sec. 6.2.1.

For the observableP₁, which vanishes in the large-recoil limit, we find in the helicity basis

P1 = −2aV+

Our result, Eq. (6.2.17), fully agrees with Eq. (26) of Ref. [146]. The authors of Ref. [146] used this result to argue that P₁ should be much cleaner than P₅⁰ because it only involves one soft form factor and a lower number of power correction parametersaF. However, the total number of power correction parameters is not the relevant criterion to decide whether an observable is clean:

as seen before, in the case ofP₅⁰ the coefficients in front of the power correction parameters exhibit a strong hierarchy, so that in practice only one term becomes relevant. As a matter of fact, the leading power corrections for both P₅⁰ andP₁ stem from a_V+ and the respective coefficients are of the same size, as seen when comparing the evaluation of Eq. (6.2.17) for q²₀ = 6 GeV²,

with the corresponding one forP₅⁰ from Eq. (6.2.15). Therefore,P1 andP₅⁰ are on an equal footing with respect to power corrections, and all statements above, regarding the scheme dependence of P₅⁰, also apply to P1. Like P₅⁰, P1 suffers from an increase of power corrections when ξ⊥ is defined fromT₁ instead of fromV, as already demonstrated numerically in Fig. 2 of Ref. [147] and analytically in Eq.(6.2.18).

Turning finally to the observableP₂, we find in the helicity basis

P2 =P2|^∞ 1 +2a_V−−2a_T− in P2 involves both (axial-)vector and tensor power corrections, and at first sight it seems that there is no preference whether to defineξ⊥ fromV or fromT₁. Note, however, that the kinematic relationT1(0) =T2(0) impliesaT1 =aT2 and that a definition fromT1 hence absorbs bothaT1 and a_T2 and leads to smaller uncertainties from corrections. Again, this is confirmed by the numerical results in Fig. 2 of Ref. [147].

We see that the scheme dependence of the angular observables can be explicitly worked out by studying the analytic dependence on the power correction parameters. Our results agree with Ref. [146] for P1, but we have shown that the formula for P₅⁰ in Ref. [146] actually misses the dominant and manifestly scheme-dependent term. Our analytic formulae allow us to understand how different schemes can yield significantly different uncertainties if one treats power corrections as uncorrelated, in perfect agreement with the numerical discussion in Ref. [147]. We can spot the relevant form factor(s) whose power corrections are going to have the main impact on each observable, and thus identify appropriate schemes to compute each observable accurately.

6.3 Reassessing the reappraisal of long-distance charm loops

We now turn to the second main source of hadronic uncertainties: non-factorisable Λ_QCD/m_B corrections associated with non-perturbativec¯cloops. Notice that Eq. (6.1.11) suggests that these contributions formally enter C9 on the same footing as possible short-distance NP effects, so they could potentially mimic a NP contribution toC9. Hence, disentangling one contribution from the other is of utmost importance for NP searches in b→ s`` decays. While the latter would induce a q²-independent C⁹, universal for the three different transversities i =⊥,k,0, non-factorisable long-distance effects from c¯c loops in general introduce a q²- and transversity dependence that can be cast into effective coefficient functions C9^c¯ci(q²). A promising strategy thus consists in

6.3. Reassessing the reappraisal of long-distance charm loops 105 investigating whether the B → K^∗µ⁺µ⁻ data points towards a q²-dependent effect. To this end the authors of Refs. [144,145] performed a fit of the functionsC9^c¯ci(q²) to the data using a polynomial parametrisation. In Sec. 6.3.1we comment on the results, before presenting in Sec.6.3.2our own analysis based on a different, frequentist, statistical framework.