Theoretical framework at low-q 2

The theoretical treatment used to describe the decay B → K^∗`<sup>+</sup>`⁻ at low squared invariant dilepton massesq² (where the most significant tensions with the SM are found) is based on QCDf supplemented by a sophisticated estimate of the power corrections of order Λ_QCD/m_B. This framework is calledimproved QCDf [113,147], where the term ”improved” stands for the fact that O(Λ_QCD/m_B) corrections that go beyond QCDf are also included as uncertainty estimates in our predictions.

The use of effective theories [24,29] allows one to relate the differentB →K^∗ form factors at leading order in ΛQCD/mB and ΛQCD/E, where E is the energy of the K^∗. As we mentioned in Section 2.1.4, this procedure reduces the required hadronic input from seven to two independent soft form factorsξ⊥, ξk, which in the region of lowq² can be calculated using LCSRs.

Two different types of LCSR form factors can be found in the literature, depending on the method adopted for their computation. On the one hand, there are form factors based on LCSRs withB-meson distribution amplitudes. Currently two form factor parametrisations of this kind are available: the calculation of form factors in [148] (KMPW), where up to leading-twist two-particle contributions were included, and the recent parametrisation presented in Ref. [149], where the leading-twist contributions are extended up to twist four. On the other hand, one can also use LCSRs withK^∗-meson distribution amplitudes, which lead to results with smaller uncertainties be-cause of the current better knowledge of light-meson LCDAs. The so-called BSZ form factors [150]

were computed within the aforementioned approach, with a prevalent application of equations of motion. For in-depth analyses of the implications of using these different form factor parametrisa-tions for the computation of relevant observables inB → K^∗`<sup>+</sup>`⁻ and related channels, we refer the interested reader to Refs. [113, 147]. Our default predictions rely on KMPW form factors which have larger uncertainties and thus lead to more conservative predictions for observables.

By construction, the choice of the set of form factors has a relatively low impact on optimized observables but it has a large impact on the error size of form-factor sensitive observables, such as the longitudinal polarisation F_L or the CP-averaged angular coefficientsS_i.

The large-recoil symmetry limit is enlightening as it allows us to understand the main be-haviour of optimized observables P_i⁽⁰⁾ [103, 112] (see also Section 5.1.2of this Thesis) in presence of New Physics in a form-factor independent way. However, for precise predictions of these ob-servables it has to be complemented with different kinds of corrections, separated in two classes:

factorisable and non-factorisable corrections. Improved QCDf provides a systematic formalism to include the different corrections as a decomposition of the amplitude in the following form [30]:

h`<sup>+</sup>`⁻K¯_i^∗|Heff|B¯i=X

a,±

Ci,aξa+ ΦB,±⊗Ti,a,±⊗ΦK^∗,a+O

Λ_QCD mB

, (6.1.1)

where i = 0,⊥,k and a =⊥,k. The equation above is formally related to the the factorisation ansatz of the hadronic form factors in Eq. (4.4.9) and all its elements have the same meaning as the ones defined there.

Factorisable power corrections

Factorisable corrections are the corrections that can be absorbed into the (full) form factors F by means of a redefinition at higher orders inαs and ΛQCD/mB:

6.1. An overview of the computation ofB →K^∗`<sup>+</sup>`⁻ observables 91

F(q²) = F^∞(ξ⊥(q²), ξ_k(q²)) + ∆F^αs(q²) + ∆F^Λ(q²). (6.1.2) The two types of corrections to the leading-order form factor F^∞(ξ⊥, ξ_k) are factorisable α_s -corrections ∆F^αs and factorisableO(ΛQCD/mB) corrections ∆F^Λ. While the former can be com-puted within QCDf and are related to the prefactors Ca,i of Eq. (6.1.1), the latter, representing part of theO(Λ_QCD/m_b) terms of Eq. (6.1.1) that QCDf cannot predict, can be parametrised as an expansion in q²/m²_B following [151]

∆F^Λ(q²) =a_F + b_F q²

m²_B + c_F q⁴

m⁴_B + . . . . (6.1.3) Whereas these corrections are expected to be small, one cannot simply neglect them as they break the symmetry relations that protect the optimised observables at leading order and reintroduce a form factor dependence at O(α_s,Λ_QCD/m_b) in them. Therefore it is of paramount importance to have these ∆F^Λ corrections under good theoretical control. Notice that the decomposition in Eq. (6.1.2) is not unique, due to the symmetry relations in Eqs. (2.1.27) and (2.1.28) (and Eq. (2.1.26) for pseudoscalars) one can always redefine ξ_⊥,k such that some of these corrections are partly absorbed. At the practical level, in order to unambiguously define the soft form factors one needs to fix a renormalisation scheme, i.e. a specific definition forξ_⊥,k in terms of full QCD form factors. Noa priory restrictions exist for the definition of a particular scheme, as long as it conforms with the symmetry relations among form factors at large recoil.

Our implementation of these corrections makes particular emphasis in two aspects [147]:

I For a realistic estimation of the errors, one must consider all correlations among the parame-tersa_F,b_F andc_F in Eq. (6.1.3). This includes all kinematic constraints among form factors at maximum recoil (i.e. T₁(0) =T₂(0)) and correlations that naturally arise from the choice of scheme.

I It is relevant to choose anappropriate scheme. Fixing a scheme determines which corrections ofO(αs,ΛQCD/mb) are absorbed into the definition of the soft form factorsF^∞(ξ⊥(q²), ξk(q²)) and which remain as ∆F^αs(q²) and ∆F^Λ(q²) symmetry breaking corrections. The ∆F^αs(q²) can be computed within QCDf, but ∆F^Λ(q²) power corrections need to be determined through the decomposition in Eq. (6.1.2). Hence, this introduces a scheme dependence at O(Λ_QCD/m_b) in the computation of the observables. In Sec.6.2 we will further discuss the dependence of improved QCDf predictions on the scheme and we will argue that an appro-priate scheme must be such that it naturally minimizes the sensitivity to power corrections in the relevant observables likeP₅⁰. Additionally, we will also derive explicit formulae for the contribution from factorisable power corrections to the most important observables P₅⁰, P2

andP₁, which will allow us to confirm in an analytic way the numerical findings of Ref. [147].

In our approach we first determine the parametersa_F,b_F,c_F so that our decomposition reproduces the central values of the full QCD form factors. This has the important implication that central values of our predictions present no scheme dependence, only errors are affected. To this end, once a scheme is fixed, we fit the subsequent parametrisation to the full form factors and keep the (non-zero) correlated estimates of the parameters (ˆaF, ˆbF, ˆcF) as central values. Explicit

ˆ

aF ˆbF ˆcF

A0(KMPW) 0.002±0.000 0.590±0.125 1.473±0.251 A1(KMPW) −0.013±0.025 −0.056±0.018 0.158±0.021 A2(KMPW) −0.018±0.023 −0.105±0.022 0.192±0.028 T1(KMPW) −0.006±0.031 −0.012±0.054 −0.034±0.095 T2(KMPW) −0.005±0.031 0.153±0.043 0.544±0.061 T3(KMPW) −0.002±0.022 0.308±0.059 0.786±0.093

Table 6.1: Fit results for the power-correction parameters in the case of scheme 1 as defined in Section6.2. The label KMPW refers to LCSR input from ref. [148]. In this scheme,V receives no power corrections and therefore the corresponding parameters vanish.

values of these estimates obtained by fitting to the full form factors in the KMPW parametrisation can be found in Table 6.1. Interestingly, this procedure yields results of typically (5−10)%×F in size, as expected from naive power counting. We take an uncorrelated 10% error assignment to the factorisable power corrections ∆ˆa_F, ∆ˆb_F, ∆ˆc_F ∼ F ×O(Λ_QCD/m_b) ∼ 0.1F around the central values [147], which corresponds to an error of order 100% with respect to the central values obtained through the fit. Finally, for the computation of observables the parameters are varied within the range ˆa_F −∆ˆa_F ≤a_F ≤aˆ_F + ∆ˆa_F, and the same forb_F and c_F [147].

Notice that our assumption on the error size of power corrections is only based on dimensional arguments and, even though there is no rigorous way to validate this assignment, it was shown in Ref. [147] that this error estimation of 10% for power corrections is conservative with respect to the central values of KMPW form factors. In Sec.6.2we will perform the same analysis but now to extract the amount of power corrections (including errors) contained in the form factors from Ref. [150], finding them of the typical order of magnitude of 10%, in agreement with dimensional arguments.

Non-factorisable power corrections

Non-factorisable corrections refer to corrections that cannot be absorbed into the definition of the form factors due to their different structure. Thus, these corrections pose a conceptually different problem with respect to our previous discussion: even if all sources of factorisable corrections were precisely known, we would still have to deal with hadronic uncertainties of non-factorisable origin. One can identify two types of such corrections. On one side, non-factorisableα_s-corrections originating from hard-gluon exchange in diagrams with insertions of four-quark operators O¹⁻⁶ and the chromomagnetic operator O8. And, on the other side, there are non-factorisable power corrections involving long-distance c¯c loops. First we will describe the non-factorisable power corrections associated with hard-gluon exchanges and later we will address the charm loop.

As we saw in Section 4.4, non-factorisable contributions come from four-quark and chromo-magnetic operators where the dilepton is generated via the insertion of a virtual photon line. At smallq²and at leading order in ΛQCD/mb, the factorisation formula in Eq. (4.4.18) allows for their computation in terms of hard-vertex coefficients, hard-scattering kernels and the corresponding distribution amplitudes [30, 33]. However, at order Λ_QCD/m_b unknown non-perturbative power corrections appear. These are precisely the non-factorisable power corrections we mentioned above.

6.1. An overview of the computation ofB →K^∗`<sup>+</sup>`⁻ observables 93 These contributions are estimated by means of a parametrisation that originally was designed to jointly account for both factorisable and non-factorisable power corrections. This approach is based on a set of complex functions that multiply each transversity amplitude [111, 152], with characteristic absolute values of the order of 10%, which again follows from dimensional arguments.

Error estimates for the transversity amplitudes around singular points are usually underestimated within this framework, but it turns out to provide reasonable errors at the observable level due to interferences between the transversity amplitudes [147].

One possibility would be to use the same technique but only for the non-factorisable correc-tions, indeed factorisable corrections have already been taken into account with the procedure detailed above. However, this would clearly overestimate the effect as contributions that are not affected by non-factorisable power corrections, i.e. those generated by operators O⁽⁰⁾9,10, would be artificially inflated.

Our procedure is based on the same idea but applied to the hadronic form factorsTi(generically defined through the factorisation formula in Eq. (4.4.18)) that parametrise the matrix element hγ^∗K¯^∗| Heff|B¯i[30]. First, we single out the contributions of purely hadronic origin inTⁱby taking the limit Ti^had = Ti|_C⁽⁰⁾

7 →0 and then we multiply each of these amplitudes, that will serve as a normalisation factor, by a complex q²-dependent function

Ti^had →(1 +ri(q²))Ti^had, (6.1.4) where

r_i(q²) =r_i^ae^iφai +r_i^be^iφbi q²

m²_B

+r^c_ie^iφci q²

m²_B 2

. (6.1.5)

We define the central values as the ones withr_i(q²)≡0 and estimate the uncertainties from non-factorisable power corrections by varyingr^a,b,c_i ∈[0,0.1] andφ^a,b,c_i ∈[−π, π] independently, which corresponds to a∼10% error assignment with arbitrary phase.

On the other hand, power corrections that involve long-distance c¯c-loops correspond to one-loop contributions from the operator O², which can be recast as a contribution to C⁹ depending on the squared dilepton invariant massq², the transversity amplitudesA^L,R_j (j= 0,⊥,||) and the hadronic states (as opposed to a universal contribution from New Physics). These complicated objects are included as an additional source of uncertainty, estimated on the basis of the only existing computation [148] of soft-gluon emission from four-quark operators involvingc¯ccurrents.

The calculation in Ref. [148] was done in the framework of LCSRs with B-meson distribution amplitudes and makes use of an hadronic dispersion relation to obtain results in the whole large-recoil region. It is important to notice that, taken at face value, the resulting correction would increase the anomaly [153]. The phenomenological model used in Ref. [148] for estimating this effect includes the perturbative leading order contribution (also coming from an insertion of theO2

operator). Therefore, as we are only interested in the long distance charm-loop effect, we subtract the aforementioned contribution, and proceed to shift the value ofm_caccording to the instructions given in Ref. [148] in order to be consistent with the value used in our analyses. Then, we introduce two parametrisations for the contribution to the transverse amplitudes [113,147]

δC9^⊥(q²) = a^⊥+b^⊥q²(c^⊥−q²)

q²(c^⊥−q²) , δC9^k(q²) = a^||+b^||q²(c^||−q²)

q²(c^||−q²) , (6.1.6)

and similarly for the longitudinal amplitude (with no pole atq² = 0, as expected) δC9⁰(q²) = a⁰+b⁰(q²+s₀)(c⁰−q²)

(q²+s₀)(c⁰−q²) , (6.1.7) beings₀ = 1 GeV². These parameters are fixed in order to cover the results from Ref. [148] in the q²-region from 1 to 9 GeV². Following this approach, one obtains [113]

a^⊥, a^||= 9.25±2.25, a⁰ = 33±7, (6.1.8)

b^⊥, b^||=−0.5±0.3, b⁰ =−0.9±0.5, (6.1.9) c^⊥, c^||= 9.35±0.25, c⁰ = 10.35±0.55, (6.1.10) where all parameters are taken as uncorrelated. For the sake of being as conservative as possible in our estimate for this effect, since the sign of these contributions can be debated, for each correction to the three transversity amplitudes we introduce prefactors si that are scanned from −1 to +1.

Finally, the prescription used in our theory predictions to account for the long-distance charm loop reads

A^L,R_i : C9^eff(q²)→ C9^eff(q²) +C9^c¯ci(q²), i=⊥,k,0, (6.1.11) with the last piece defined as

C9^c¯ci =s_iδC9ⁱ. (6.1.12) It is interesting to note that our conservative approach typically leads to larger uncertainties for observables as compared to other estimates in the literature [150,151].