Correlation with R D (∗) and R J/ψ

± 0.12 + 0.02C9^NP−0.04C10^NP+ 0.01C9⁰ −0.04C10⁰

+0.01C10^NPC10⁰ + 0.01C10^{NP 2}+ 0.08C10⁰²

, (9.2.7)

10⁷×Br B →K^∗τ⁺τ⁻[15,19]

= 0.98 + 0.38C9^NP−0.14C10^NP−0.30C9⁰ + 0.12C10⁰ −0.08C9^NPC9⁰

−0.03C10^NPC10⁰ + 0.05C9^{NP 2}+ 0.02C10^{NP 2}+ 0.05C9⁰²+ 0.02C⁰10²

± 0.09 + 0.03C9^NP−0.01C10^NP−0.03C9⁰ −0.01C9^NPC9⁰

−0.01C9⁰C10⁰ + 0.01C9⁰²−0.01C10⁰²

, (9.2.8)

10⁷×Br B_s→φτ⁺τ⁻[15,18.8]

= 0.86 + 0.34C9^NP−0.11C10^NP−0.28C9⁰ + 0.10C10⁰ −0.08C9^NPC9⁰

−0.02C10^NPC10⁰ + 0.05C9^{NP 2}+ 0.01C10^{NP 2}+ 0.05C9⁰²+ 0.01C⁰10²

± 0.06 + 0.02C9^NP−0.02C9⁰ + 0.02C10⁰²

(9.2.9)

As expected, there is a limited dependence of the uncertainties on the values of the Wilson coeffi-cients. In order to shorten the equations, we dropped the superscriptτ τ in the Wilson coefficients here. Comparing our results with Ref. [280], we find slightly lower central values for the SM (Eqs. (9.2.4)-(9.2.6)). On the other hand, we obtain the same dependence of the central values on the NP contributions to the Wilson coefficients (Eqs. (9.2.7)-(9.2.9)).

In this analysis we neglect the effects of scalar and tensor operators. This is justified since the current global analyses of b → s`<sup>+</sup>`⁻ anomalies do not favour such contributions [113, 153, 164, 166]. Moreover, the indirect bounds on the Wilson coefficients of scalar operators fromBs→τ⁺τ⁻ are much stronger than forC₉⁽⁰⁾ andC₁₀⁽⁰⁾ [279] and therefore they cannot lead to comparably large and observable effects in B →K^(∗)τ⁺τ⁻ orBs→φτ⁺τ⁻. We also neglect tensor operators since they are not generated at the dimension-6 level forb→s`<sup>+</sup>`⁻[286,287].

9.3 Correlation with R

_D^(∗)

and R

_J/ψ

It is interesting to correlate these results with the tree-levelb→cτ⁻ν¯τtransition. A solution of

9.3. Correlation with R_D(∗) and R_J/ψ 161 the∼3σ anomaly inR_D(∗) and R_J/ψ requires a NP contribution ofO(20%) to the branching ratio ofB →D^(∗)τ⁻ν¯_τ, which is rather large given that these decays are mediated in the SM already at tree level. In order to comply with theBclifetime [244] and theq² distribution ofR_D^(∗) [245–247], a contribution to the SM operator [¯cγ^µP_Lb][¯τ γ_µP_Lν_τ] is favoured such that there is interference with the SM. In principle, these constraints can be avoided with right-handed couplings (including possibly right-handed neutrinos [288]). However, no interference with the SM appears for such solutions, which require very large couplings close to the perturbativity limit, and we will not consider such solutions any further.

Since a NP contribution to the Wilson coefficient of the SMV −A operator amounts only to changing the normalisation of the Fermi constant forb →sτ⁺τ⁻ transitions, one predicts in this case:

R_J/ψ/R^SM_J/ψ =R_D/R^SM_D =R_D^∗/R^SM_D∗, (9.3.1) which agrees well with the current measurements.

If NP generates this contribution from a scale much larger than the electroweak symmetry breaking scale [250, 289], the semileptonic decays involving only left-handed quarks and leptons are described by the twoSU(2)_L-invariant operators

O⁽¹⁾_ijkl = [ ¯Q_iγ_µQ_j][ ¯L_kγ^µL_l],

O⁽³⁾_ijkl = [ ¯Qiγµσ^IQj][ ¯Lkγ^µσ^ILl], (9.3.2) where the Pauli matrices σ^I act on the weak-isospin components of the quark (lepton) doublets Q (L). Note that there are no further dimension-six operators involving only left-handed fields and that dimension-eight operators can be neglected for NP around the TeV scale. This approach has been used to correlate Wilson coefficients of the effective Hamiltonian for both charged- and neutral-current transitions in various broad classes of NP models (some examples are found in Refs. [234,235,251,290]).

After electroweak symmetry breaking, these operators contribute to semileptonic b → c(s) decays involving charged tau leptons and tau neutrinos. Working in the down basis when writing theSU(2) components of the operators in Eq. (9.3.2) (i.e., in the field basis with diagonal down quark mass matrices) we obtain

C⁽¹⁾O⁽¹⁾ → C23⁽¹⁾([¯s_Lγ_µb_L][¯τ_Lγ^µτ_L] + [¯s_Lγ_µb_L][¯ν_τγ^µν_τ]), (9.3.3) C⁽³⁾O⁽³⁾ → C23⁽³⁾(2V_cs[¯c_Lγ_µb_L][¯τ_Lγ^µν_τ] + [¯s_Lγ_µb_L][¯τ_Lγ^µτ_L]

−[¯s_Lγµb_L][¯ντγ^µντ]) +C33⁽³⁾(2V_cb[¯c_Lγµb_L][¯τ_Lγ^µντ]). (9.3.4) whereCij⁽ⁿ⁾ denote the Wilson coefficients for Oij⁽ⁿ⁾33.

We neglect the effect ofC13⁽³⁾ which would enterb→cτ ν_τ processes with a factor proportional toVcd. But it would contribute even more dominantly to b→ dτ τ and b→ uτ ντ processes such asB⁻→τ⁻ν¯_τ, where no deviation from the SM is observed [291,292]. We will thus not consider this contribution any more.

As a consequence, we see thatb→cτ ντ processes receive a NP contribution from C33⁽³⁾ also in scenarios with a flavour-diagonal alignment to the third generation, which would avoid any effects in down-quark FCNCs. However, due to the CKM suppression of this contribution, a solution

1.1 1.2 1.3 1.4 1.5 0

2 4 6 8 10

R_X/R_X^SM

Br×104 R

D^(*)&R_J/Ψ2σ R_D(*)&R_J/Ψ1σ Br[B_s→ττ]

Br[B→K^*ττ]

Br[B→Kττ]

Br[B_s→ϕττ]

Figure 9.1: Predictions of the branching ratios of theb→sτ⁺τ⁻processes (including uncertainties) as a function ofRX/R_X^SM.

of the R_D(∗) anomaly via this contribution requires a rather large C33⁽³⁾ coming into conflict with bounds from electroweak precision data [249] and direct LHC searches forτ⁺τ⁻final states [248].

TheR_D(∗) anomaly can thus only be solved viaC23^(1,3)which then must generate huge contribu-tions tob→ sτ τ and/or b→sντν¯τ processes. The severe bounds on NP fromB → K^(∗)νν¯ (e.g., Ref. [293]) rule out large effects in b→ sν¯ν and they can only be accommodated if the contribu-tion fromC23⁽³⁾ is approximately cancelled by the one fromC23⁽¹⁾, implyingC23⁽¹⁾≈C23⁽³⁾ [290]. Such a situation can for instance be realized by a vector leptoquark singlet [234, 235, 256, 257, 294]

or by combining two scalar leptoquarks [253]. Neglecting small CKM factors, the assumption C₂₃⁽¹⁾≈C₂₃⁽³⁾ implies that contributions tob→cτ⁻ν¯_τ andb→sτ⁺τ⁻ are generated together in the combination

[¯c_Lγµb_L][¯τ_Lγ^µντ] + [¯s_Lγµb_L][¯τ_Lγ^µτ_L]. (9.3.5) This correlation means that effects in b → sτ τ are of the same order as the ones required to explain R_D(∗), i.e., of the order of a tree-level SM process. We may neglect Cabibbo-suppressed contributions and assume that the NP contribution tob→cτ ντ is small compared to the SM one, so that we keep only the SM contribution and the SM-NP interference terms in b → cτ ν_τ decay rates. We find the relation

C₉₍₁₀₎^{τ τ} ≈ C^SM₉₍₁₀₎−(+)∆, (9.3.6)

with

∆ = 2π α

V_cb V_tbV_ts^∗

s R_X R^SM_X −1

!

. (9.3.7)

In our framework, ∆ is independent of the exclusiveb→c`<sup>−</sup>ν¯<sub>`</sub> channel chosenX, see Eq. (9.3.1).

Note that this prediction for the Wilson coefficientsC9^{τ τ} andC10^{τ τ} is model independent, in the sense that the only ingredients in the derivation are the assumptions that NP only affects left-handed

9.3. Correlation with R_D(∗) and R_J/ψ 163 quarks and leptons and that it couples significantly to the second generation in such a way that experimental constraints can be avoided.

We stress that the factor multiplying the bracket in Eq. (9.3.7) is very large (around 860).

Using the current values for R_D(∗), we obtain a positive (respectively negative) NP contribution to the Wilson coefficient C9^{τ τ} (respectively C10^{τ τ}) parametrised by ∆ = O(100) which overwhelms completely the SM contribution to these Wilson coefficients. Such large values of the Wilson coefficients are not in contradiction with the constraints obtained in Ref. [279] (when comparing with the results of this reference, one must be aware of the different normalisations of the operators in the effective Hamiltonian).

In view of these huge coefficients, we provide predictions for the relevant decay rates assuming that they are completely dominated by the NP contribution ∆, and thus neglecting both short-and long-distance SM contributions. We obtain the branching ratios of the various b → sτ⁺τ⁻ channels

Br B_s→τ⁺τ⁻

=

∆ C₁₀^SM

2

Br B_s →τ⁺τ⁻

SM (9.3.8)

Br B →Kτ⁺τ⁻

= (8.8±0.8)×10⁻⁹∆², (9.3.9) Br B →K^∗τ⁺τ⁻

= (10.1±0.8)×10⁻⁹∆², (9.3.10) Br Bs→φτ⁺τ⁻

= (9.1±0.5)×10⁻⁹∆², (9.3.11) where the last three branching ratios are considered over the whole kinematic range for the lepton pair invariant massq² (i.e., from 4m²_τ up to the low-recoil endpoint). We neglect the contributions only due to the SM. In the above expressions, the uncertainties quoted come from hadronic con-tributions multiplied by the short-distance NP contribution ∆. A naive estimate suggests that the contribution of theψ(2S) resonance to this branching ratio amounts to 2×10⁻⁶, which is negligible in the limit of very large NP contributions considered here. We thus may calculate the branching ratios for the whole kinematically allowedq² region, from the vicinity of theψ(2S) resonance up to the low-recoil endpoint, assuming that the result is completely dominated by the NP contribution.

Since we neglected all errors related to the SM contribution for the semileptonic processes, we should do the same forB_s→τ⁺τ⁻. For Br (B_s→τ⁺τ⁻)_SMin Eq. (9.3.8), we should only consider the uncertainties coming from theBsdecay constant and decay width as well as the different scales used to compute the Wilson coefficients here and in Ref. [139], leading to a relative uncertainty of 4.7% (to be compared with the larger 6.4% uncertainty in Eq. (9.1.3) that includes other sources of uncertainties irrelevant under our current assumptions).

In Fig.9.1, we indicate the corresponding predictions as a function ofR_X/R_X^SM(assumed to be independent of the b→c`ν` hadronic decay channelX in our approach). We have also indicated the current experimental range forR_X/R^SM_X , obtained by performing the weighted average ofR_D, R_D^∗ and R_J/ψ without taking into account correlations. We see that the branching ratios for semileptonic decays can easily reach 3×10⁻⁴, whereas Bs→τ⁺τ⁻ can be increased up to 10⁻³.

Up to now, we have discussed the correlation between NP inb →cτν¯_τ and b→ sτ τ under a limited set of assumptions that are fairly model independent. A comment is in order concerning the implications of these assumptions for b → sµµ. If we assume that the same mechanism is at work for muons and taus, we obtain also a correlation between b → sµµ and b → cµν_µ: the O(25%) shift needed inC9^µµ andC10^µµ to describeb→sµ⁺µ⁻data [136] translates into a very small

positive ∆ and a decrease ofb→cµνµ decay rates compared to the SM by a negligible amount of only a few per mille, so that there would be no measurable differences between electron and muon semileptonic decays.

Conclusions

Over the last years, a very interesting pattern of deviations has emerged in b → s`` transitions.

After the initialP₅⁰ anomaly identified inB →K^∗µµby the LHCb experiment, several systematic deviations have been observed in various branching ratios. At the same time, new observables comparing electron and muon modes have been measured at LHCb (RK) and Belle (Q4,5) hinting at a violation of lepton flavour universality. Global analyses of all these deviations find a preference for NP solutions with respect to the SM with high significances (above 5σ) with distinctive features:

i) the dominant NP contribution enters the semileptonic operator O^9µ, ii) NP affects b → sµµ transitions much more noticeably than b → see ones and iii) strong consistency between the pattern of deviations in b→sµµ and LFUV observables.

In PartI we have discussed the basics needed for performing global analyses of b→s`` data.

These include the use of an effective Hamiltonian for the description of the underlyingb→squark level transition, the leading order parametrisation of matrix elements in terms of form factors, the development of a factorisation formula for the computation ofO(αs) corrections to the amplitude in a systematic way (which emerges from the simplified dynamics one obtains at the heavy quark and large energy limits) and the construction of a basis of optimised observables where most of hadronic information cancels at leading order.

On the theoretical side, we have seen in Chapter 6 that hadronic uncertainties conform to theoretical expectations and unexpectedly large effects (power corrections to form factors, charm-loop contributions) are disfavoured, in particular by the significant amount of LFUV observed.

However, it would be very useful to have more determinations of the form factors involved, both at low and large meson recoils, as well as refined estimates of charm-loop contributions, in order to improve the accuracy of theoretical predictions.

In Chapter 7 we have discussed how angular analyses of B → K^∗ee and B → K^∗µµ decay modes can be combined to understand better the pattern of anomalies observed and to get a solid handle on the size of some SM long-distance contributions.We have proposed different sets of observables comparing B → K^∗ee and B → K^∗µµ, discussing their respective merits. A first set of observables is obtained directly from the observables that have been introduced forB →K^∗µµ, namely Q_i (related to the optimised observables P_i), T_i (related to the angular averages S_i) and Bi (related to the angular coefficients Ji), measuring in each case the differences between muon and electron modes.

As the analysis in Chapter8demonstrates, recent experimental updates (R_K,R_K^∗andB(B_s→ µ⁺µ⁻)) yield a very similar picture to the one previously found in Refs. [136,211] for the various NP scenarios of interest with some important peculiarities. In presence of LFUV NP contributions only, the 1D fits to “All” observables remain basically unchanged showing the preference for C9µ^NP

scenario overC9µ^NP=−C10µ^NP. If only LFUV observables are considered the situation is reversed, as 165

already found in Ref. [136], but now with an increased gap between the significances. This difference between the preferred hypotheses, depending on the data set used, can be solved introducing LFU NP contributions [211].

The main differences arise for the 2D scenarios: the cases including RHC, (C9µ^NP,C10⁰µ), (C9µ^NP,C9⁰µ) or (C9µ^NP,C9⁰µ =−C10⁰µ), can accommodate better the recent updates, which enhances the signifi-cance of these scenarios compared to Ref. [136], pointing to new patterns including RHC. A more precise experimental measurement of the observable P1[83,103] would be very useful to confirm or not the presence of RHC NP encoded inC9⁰µ and C10⁰µ.

We also observe interesting changes in the 2D fits in the presence of LFU NP, where new scenarios (not considered in Ref. [211]) give a good fit to data with C₁₀^U(0) and additional LFUV contributions. For example scenario 11 (C9µ^V,C10⁰µ) can accommodate b→ s`<sup>+</sup>`⁻ data very well, at the same level as scenario 8. Scenarios including LFU NP in left-handed currents (discussed in Ref. [211]) stay practically unchanged but with some preference for scenarios 6 and 8, which have a (V−A) structure for the LFUV-NP and aV or (V+A) structure for the LFU-NP. Furthermore, we have included additional scenarios 9 and 10 that exhibit a significance of 5.0σand 5.5σrespectively.

We note that the amount of LFU NP is sensitive to the structure of the LFUV component.

For instance, in scenario 7 (C9µ^V and C9^U) the LFU component is negligible at its best fit point.

On the contrary, if the LFUV-NP has a (V −A) structure, the LFU-NP component (C9^U) is large, as illustrated by scenarios 6, 8 and 9. Scenarios with NP in RHC (either LFU or LFUV) prefer such contributions at the 2σ level (see scenarios 11 and 13) with the exception of scenario 12 with negligibleC₉^V0_µ. The new values ofRK andRK^∗ seem thus to open a window for RHC contributions while the new B(B_s→ µµ) update (theory and experiment) helps only marginally scenarios with C10µ^NP.

Then, we have discussed the impact of forthcoming measurements ofhR_Ki[1.1,6]andhQ₅i[1.1,6]

to disentangle NP hypotheses. We considered various central values for these two measurements and we assumed some reduction in the experimental uncertainties in order to study how pulls w.r.t. SM and best-fit points would evolve. hR_Ki[1.1,6] alone proves to have only a limited ability to separate the various NP hypotheses: C9µ^V =−C₉^V0_µ is the only hypothesis strongly affected. On the other hand, the combination ofhR_Ki[1.1,6]andhQ₅i[1.1,6]proves much more efficient to separate various favoured hypotheses, either with only LFUV-NP contributions or with both LFUV and LFU contributions.

Finally, in Chapter 9 we have also analysed the correlation between NP contributions to b→sτ⁺τ⁻andb→cτ⁻ν¯_τ under general assumptions in agreement with experimental indications:

the deviations inb→cτ⁻¯ν_τ decays come from a NP contribution to the left-handed four-fermion vector operator, this NP contribution is due to physics coming from a scale significantly larger than the electroweak scale, and the resulting contribution to b → sν_τ¯ν_τ is suppressed. Under these assumptions, an explanation of R_D^(∗) requires an enhancement of all b → sτ⁺τ⁻ processes by approximately three orders of magnitude compared to the SM, confirming the potential of b → sτ⁺τ⁻ decays to look for NP in the context of the measurements searching for violation of LFU in semileptonicb-decays.

Therefore, we consider our analyses in solid grounds, both from a theoretical perspective and from the point of view of our data analysis strategies, being our approach characterised for always using the most conservative estimation of errors. Hence, if new data from LHCb and, more importantly, results from a completely independent experiment such as Belle II, confirm the same

9.3. Correlation with R_D(∗) and R_J/ψ 167 picture we have already observed, there are arguments enough for taking theB-anomalies as the new guideline to follow in the short- and mid-term future in the research field of fundamental Particle Physics. In this context, theB-anomalies can potentially play the role of the Higgs boson during the LEP to LHC transition for the new experiments to come.

Some kinematics for B → K ^∗ ` <sup>+</sup> ` ⁻

This appendix is a compilation of the most relevant definitions and results concerning the kine-matics of four-body decays. Our notation is going to be general (X→Y(→ab)Z(→cd)), so that it can be applied to any decay, but at the end of the appendix you can also find a translation table between the notation used here and the one in Chapter 3 for theB →K^∗(→Kπ)`<sup>+</sup>`⁻ mode.

The dimension of the four-body phase space is 4×4. However, because of the on-shell and 4-momentum conservation conditions (four constraints each), the dimension is reduced to 4×2.

Moreover, exploiting isotropic symmetry, one can fix three Euler angles and ends up with 5 physical degrees of freedom. Following [295], the typical five independent kinematical degrees of freedom are portrayed by

I m²_ab, the effective mass squared of the absystem, ma+mb < mab < mX−mc−md. I m²_cd, the effective mass squared of the cdsystem, m_c+m_d< m_ab < m_X −m_a−m_b.

I θY, the angle of the a particle in the C.M. system of the particles a and b with respect to the direction of flight of (a,b) in theX rest system (0< θ_Y < π).

I θZ, the angle of thecparticle in the C.M. system of the particlescanddwith respect to the direction of flight of (c,d) in the X rest system (0< θ_Y < π).

I θX, the angle between the plane formed by the decay products (a,b) and the corresponding plane of (c,d) in theX rest frame (−π < θ_X < π).

More than the individual momenta of each of the particles, it is usual to work with the following combinations

P_ab =p_a+p_b, Q_ab=p_a−p_b, P_cd =pc+p_d, Q_cd=pc−p_d, Then, the diparticle masses read

P_ab² =m²_ab P_cd² =m²_cd (A.0.1) 168

169 The three kinematical angles introduced before can be expressed in terms of momenta in the following way

cosθY =−Qab·Pcd

|Q_ab||P_cd|, cosθZ=−Qcd·Pab

|Q_cd||P_ab| (A.1)

sinθ_X = (P_ab×Q_ab)×(P_cd×Q_cd)

|P_ab×Q_ab||P_cd×Q_cd| (A.2)

where all these quantities must be evaluated, respectively, at theY,Z and X rest frames.

Using the five independent variables θ_X, θ_Y, θ_Z,m²_ab and m²_cd, the invariant products of the vectorsP_ab,P_cd,Q_ab and Q_cd write as [31]

PabQab=m²_a−m²_b, P_abP_cd= ¯p,

P_abQ_cd= m²_c−m²_d m²_cd p¯+ 2

m²_cdσσ_cdcosθZ, Q_abQ_ab= 1

m²_abm²_cd[(m²_a−m²_b)(m²_c−m²_d)¯p + 2σσ_ab(m²_c−m²_d) cosθ_Y

+ 2σσ_cd(m²_a−m²_b) cosθ_Z + 4σ_abσ_cdp¯cosθY cosθZ

+ 4σabσcdmabmcdsinθY sinθZcosθX], αβγδP_ab^αQ^β_abP_cd^γQ^δ_cd=−4σσabσcd

m_abm_cd sinθY sinθZsinθX

with the quantities

¯ p= 1

2(M_X² −m²_ab−m²_cd), σ=

q

¯

p²−m²_abm²_cd,

¯ p_ab= 1

2(m²_ab−m²_a−m²_b) σab=

q

¯

p²_ab−m²_a−m²_b

General B →K^∗(→Kπ)`<sup>+</sup>`⁻

(ab) (Kπ)

(cd) (`<sup>+</sup>`⁻)

m_ab m_K^∗

m_cd p

q²

σ_X φ

σ_Y θ_K^∗

σ_Z θ_`

σ_ab² m⁴_K∗β²/4 σ_cd² q⁴β_`²/4

σ² λ/4

¯

p (p_K^∗·q)

Table A.1: Translation table between the general X → Y(→ ab)Z(→ cd) variables and the particular notation used for the decayB →K^∗(→Kπ)`<sup>+</sup>`⁻ in chapter 3.

Appendix B

Large-recoil expressions for M and M f

Under the notation and hypotheses in Section7.1.2, we can separate the charm contributions from the rest of theMfobservable

Mf=Mf₀+A⁰δC10∆C9+B⁰δC10² ∆C9² (B.0.1) with

Mf₀ = (2C7δC10mˆ_b+δC9C10ˆs+δC10(C9+δC9)ˆs)

C7δC9C10(C10+δC10) ˆm_b(ˆs−1)ˆs (B.0.2)

×(C7δC10mˆb+δC9C10sˆ+δC10(C7mˆb+C9+δC9)ˆs) A⁰ = 2δC9C10sˆ+δC10(2(C9+δC9)ˆs+C7mˆ_b(3 + ˆs))

C7δC9C10(C10+δC10) ˆm_b(ˆs−1) (B.0.3)

B⁰ = sˆ

C7δC9C10(C10+δC10) ˆm_b(ˆs−1) (B.0.4) M can be expressed in terms of Mf and considering all the lepton mass effects coming from β_` =

q

1−4m²_`/sin the large recoil limit and up to leading order

M =Mf+ ∆M+A∆C9+B∆C9² (B.0.5)

∆M = −β_e²−β_µ² β_e²β_µ²

1

C7δC9C10(C10+δC10) ˆm_b(ˆs−1)ˆs

×h

− C10² (2C7mˆ_b+C9s)(ˆ C9ˆs+C7mˆ_b(1 + ˆs))β_e² (B.0.6) +(C¹⁰+δC¹⁰)²(2C⁷mˆ_b+ (C⁹+δC⁹)ˆs) ((C⁹+δC⁹)ˆs+C⁷mˆ_b(1 + ˆs))β²_µ

i

A = β_e²−β²_µ β_e²β²_µ

1

C7δC9C10(C10+δC10) ˆm_b(ˆs−1) (B.0.7)

×

C10² (2C⁹ˆs+C⁷mˆ_b(3 + ˆs))β²_e−(C¹⁰+δC¹⁰)²(2(C⁹+δC⁹)ˆs+C⁷mˆ_b(3 + ˆs))β²_µ B = β_e²−β²_µ

β_e²β²_µ ˆ

s C10² β_e²−(C10+δC10)²β_µ²

C7δC9C10(C10+δC10) ˆm_b(ˆs−1) (B.0.8)

171

Definition of binned observables

The binned observables are defined following the same rules as in Ref. [112]:

hQii = hP_i^µi − hP_i^ei hQˆii=hPˆ_i^µi − hPˆ_i^ei hTii= hS_i^µi − hS_i^ei

hS_i^µi+hS_i^ei (C.0.1) hB_ii = hJ_i^µi

hJ_i^ei −1 hBe_ii= hJ_i^µ/β_µ²i

hJ_i^e/β_e²i −1 (C.0.2)

hMi = (hJ₅^µi − hJ₅^ei) (hJ_6s^µi − hJ_6s^ei)

hJ_6s^µihJ₅^ei − hJ_6s^e ihJ₅^µi (C.0.3) hMfi = hJ₅^µ/β²_µi − hJ₅^e/β_e²i

hJ_6s^µ/β_µ²i − hJ_6s^e /β_e²i

hJ_6s^µ/β_µ²ihJ₅^e/β_e²i − hJ_6s^e/β²_eihJ₅^µ/β_µ²i (C.0.4) wherehP_i^{`</sup>iandhS<sub>i</sub><sup>`}i correspond to the observables defined in Ref. [112] with `=eorµ. Similarly, thehPˆ<sub>i</sub><sup>`</sup>i are obtained from Eqs. (7.1.2)-(7.1.6), substituting J_i^{`</sup> → hJ<sub>i</sub><sup>`}i.

172

Appendix D

Predictions for the observables in various benchmark scenarios

Our predictions are obtained following Ref. [113]. We quote two uncertainties, the second corre-sponding to the charm contributions, the first to all other sources of uncertainties. Bars denote predictions affected by a very large uncertainty (presence of a pole).

D.1 SM

Bin QF_L Q1 Q2 Q3

[0.1,0.98] −0.041±0.044±0.010 −0.001±0.001±0.001 0.019±0.003±0.001 0.000±0.000±0.000 [1.1,2.5] −0.027±0.014±0.001 −0.000±0.000±0.000 0.007±0.000±0.000 0.000±0.000±0.000 [2.5,4.] −0.016±0.009±0.000 0.000±0.000±0.000 0.001±0.001±0.000 0.000±0.000±0.000 [4.,6.] −0.010±0.008±0.000 0.000±0.000±0.000 −0.001±0.000±0.000 0.000±0.000±0.000 [6.,8.] −0.006±0.006±0.000 0.000±0.000±0.000 −0.001±0.000±0.000 0.000±0.000±0.000 [15.,19.] −0.001±0.000±0.000 −0.000±0.000±0.000 −0.000±0.000±0.000 0.000±0.000±0.000

Bin Q4 Q5 Q6 Q8

[0.1,0.98] 0.005±0.002±0.004 0.047±0.003±0.008 −0.005±0.002±0.001 0.001±0.000±0.000 [1.1,2.5] 0.002±0.000±0.000 0.001±0.002±0.001 −0.001±0.000±0.000 0.000±0.000±0.000 [2.5,4.] 0.000±0.000±0.000 −0.004±0.001±0.000 −0.000±0.000±0.000 −0.000±0.000±0.000 [4.,6.] 0.000±0.000±0.000 −0.004±0.000±0.000 −0.000±0.000±0.000 0.000±0.000±0.000 [6.,8.] 0.000±0.000±0.000 −0.003±0.000±0.000 −0.000±0.000±0.000 0.000±0.000±0.000 [15.,19.] 0.000±0.000±0.000 −0.001±0.000±0.000 0.000±0.000±0.000 0.000±0.000±0.000

173

Bin Qˆ_FL Qˆ1 Qˆ2 Qˆ3

[0.1,0.98] 0.018±0.017±0.004 −0.007±0.006±0.018 −0.008±0.004±0.001 0.000±0.001±0.001 [1.1,2.5] 0.014±0.002±0.000 −0.000±0.003±0.000 0.013±0.032±0.002 0.000±0.000±0.000 [2.5,4.] 0.010±0.002±0.000 0.000±0.003±0.000 0.010±0.025±0.001 0.000±0.001±0.000 [4.,6.] 0.008±0.001±0.000 0.001±0.006±0.000 −0.004±0.005±0.000 0.000±0.000±0.000 [6.,8.] 0.006±0.002±0.000 0.000±0.003±0.000 −0.004±0.007±0.000 0.000±0.000±0.000 [15.,19.] 0.001±0.000±0.000 0.001±0.000±0.000 −0.000±0.000±0.000 0.000±0.000±0.000

Bin Qˆ4 Qˆ5 Qˆ6 Qˆ8

[0.1,0.98] 0.111±0.007±0.037 −0.097±0.013±0.019 0.008±0.003±0.001 −0.004±0.004±0.003 [1.1,2.5] 0.003±0.005±0.002 −0.003±0.007±0.001 0.001±0.003±0.000 −0.001±0.002±0.000 [2.5,4.] 0.001±0.016±0.001 −0.005±0.017±0.001 −0.001±0.003±0.000 0.000±0.002±0.000 [4.,6.] −0.002±0.015±0.000 −0.002±0.017±0.000 −0.000±0.001±0.000 −0.000±0.001±0.000 [6.,8.] −0.005±0.009±0.001 0.002±0.010±0.000 0.000±0.000±0.000 −0.000±0.000±0.000 [15.,19.] −0.003±0.000±0.000 0.001±0.000±0.000 0.000±0.000±0.000 0.000±0.000±0.000

Bin T3 T4 T5

[0.1,0.98] −− −0.116±0.002±0.005 −0.075±0.003±0.001

[1.1,2.5] −− −− −0.017±0.004±0.001

[2.5,4.] −− −0.010±0.003±0.000 −0.006±0.003±0.000 [4.,6.] −0.007±0.006±0.000 −0.007±0.003±0.000 −0.004±0.003±0.000 [6.,8.] −0.005±0.004±0.060 −0.005±0.002±0.000 −0.003±0.002±0.000 [15.,19.] −0.001±0.000±0.000 −0.001±0.000±0.000 −0.000±0.000±0.000

Bin T7 T8 T9

[0.1,0.98] −0.067±0.003±0.000 −0.081±0.025±0.051 −−

[1.1,2.5] −0.013±0.003±0.000 −0.020±0.003±0.000 −−

[2.5,4.] −0.007±0.003±0.000 −0.010±0.003±0.000 −0.010±0.027±0.000 [4.,6.] −0.005±0.003±0.000 −0.007±0.003±0.000 −0.007±0.003±0.000 [6.,8.] −0.003±0.002±0.000 −0.005±0.002±0.000 −0.005±0.004±0.000 [15.,19.] −0.000±0.000±0.000 −0.001±0.001±0.004 −0.001±0.002±0.001

Bin B5 B6s M

[0.1,0.98] −0.155±0.002±0.002 −0.121±0.001±0.000 0.548±0.021±0.024 [1.1,2.5] −0.034±0.005±0.002 −0.027±0.000±0.000 0.150±0.071±0.037 [2.5,4.] −0.013±0.000±0.000 −0.015±0.001±0.000 −0.095±0.033±0.007 [4.,6.] −0.009±0.000±0.000 −0.008±0.021±0.000 0.149±0.122±0.019 [6.,8.] −0.006±0.000±0.000 −0.006±0.000±0.000 0.617±0.253±0.204 [15.,19.] −0.003±0.000±0.000 −0.003±0.000±0.000 −−

Bin Be5 Be6s Mf

[0.1,0.98] 0.000±0.000±0.000 0.000±0.000±0.000 0.000±0.000±0.000 [1.1,2.5] 0.000±0.000±0.000 0.000±0.000±0.000 0.000±0.000±0.000 [2.5,4.] 0.000±0.000±0.000 0.000±0.000±0.000 0.000±0.000±0.000 [4.,6.] 0.000±0.000±0.000 0.000±0.000±0.000 0.000±0.000±0.000 [6.,8.] 0.000±0.000±0.000 0.000±0.000±0.000 0.000±0.000±0.000 [15.,19.] 0.000±0.000±0.000 0.000±0.000±0.000 0.000±0.000±0.000

Dans le document Thesis submitted for the degree of Philosophae Doctor (PhD) Departament de F´ısica (Page 170-185)