Implications for models

As expected, C9µ^V and C9^U are highly anti-correlated, with its nominal value somewhat smaller than in [211]. Fit estimates of the parameters in scenario {C9µ^V = −C10µ^V ,C9^U = C10^U} are now slightly correlated, while before their correlation was negligible. Interestingly, however, we find the parameters of the new scenario{C9µ^V,C10^U}statistically independent to a large extent.

8.3 Implications for models

Our most updated model-independent fits to available b → s`` and b → sγ data in Ref. [197]

strongly favour several patterns of NP, with either purely LFUV or LFU and LFUV signatures.

As it was already identified in previous analyses [113,115, 136], the strongest signal of NP takes the form of an LFUV contribution to the coefficient C9 affecting mainly b → sµµ transitions.

However, more complex NP solutions with additional structures, involving either LFU types of NP or RHC, seem to emerge from the most recent data. This has important implications for some popular ultraviolet-complete models which we briefly discuss.

I LFUV: Given that leptoquarks (LQs) should posses very small couplings to electrons in order to avoid dangerous effects inµ→eγ, they naturally violate LFU. WhileZ⁰ models can easily accommodate LFUV data [220], variants based on the assumption of LFU [221,222]

are now disfavoured. The same is true if one aims at explaining P₅⁰ via NP in four-quark operators leading to a NP (q²-dependent) contribution from charm loops [223].

I C_9µ^NP: Z⁰ models with fundamental (gauge) couplings to leptons preferably yield C9µ^NP-like solutions in order to avoid gauge anomalies. In this context, L_µ−L_τ models [224–227]

are popular since they do not generate effects in electron channels. The new fit including R_K^∗ is also very favourable to models predicting C9µ^NP =−3C9e^NP [228]. Interestingly, such a symmetry pattern is in good agreement with the structure of the PMNS matrix. Concerning LQs, aC9µ^NP-like solution can only be generated by adding two scalar (anSU(2)L triplet and an SU(2)_L doublet with Y = 7/6) or two vector representations (an SU(2)_L singlet with Y = 2/3 and anSU(2)_L doublet withY = 5/6).

I C_9µ^NP =−C_10µ^NP: This pattern can be achieved inZ⁰models with loop-induced couplings [229]

or in Z⁰ models with heavy vector-like fermions [180, 230] which posses also LFUV. Con-cerning LQs, here a single representation (the scalar SU(2)L triplet or the vector SU(2)L

singlet with Y = 2/3) can generate a C9µ = −C10µ like solution [231–237] and this pat-tern can also be obtained in models with loop contributions from three heavy new scalars and fermions [238–240]. Composite Higgs models are also able to achieve this pattern of deviations [241].

I RHC:with anRK value closer to one, scenarios with right-handed currents, namelyC9µ^NP=

−C9⁰µ, (C9µ^NP,C9⁰µ) and (C9µ^NP,C10⁰µ), seem to emerge. The first two scenarios are naturally generated inZ⁰ models with certain assumptions on its couplings to right-handed and left-handed quarks, as it was shown in Ref. [224] within the context of a gaugedLµ−Lτ symmetry with vector-like quarks. One could also obtainC9µ^NP =−C9⁰µby adding a third Higgs doublet to the model of Ref. [226] with opposite U(1) charge. On the other hand, generating the

aforementioned contribution in LQ models requires one to add four scalar representations or three vector ones.

I C_9µ^V = −C^V_10µ & C₉^U: Scenario 8 of Ref. [211] can be realized via off-shell photon pen-guins [242] in a leptoquark model explaining also b→cτ ν data (we will return to this point below).

I C₁₀₍^U 0): The new scenarios 9–13 are characterized by aC₁₀₍^{U 0}₎ contribution. This arises natu-rally in models with modifiedZ couplings (to a good approximationC₉₍^U0₎ can be neglected).

The pattern of scenario 9 occurs in Two-Higgs-Doublet models where this flavour universal effect can be supplemented by aC9^V=−C10^V effect [213].

I More LFU-LFUV: In case of scenarios 11 to 13, one can invoke models with vector-like quarks where modified Z couplings are even induced at tree level. The LFU effect in C₁₀₍^{U 0}₎ can be accompanied by a C_9,10(^{V 0}₎ effect from Z⁰ exchanges [214]. Vector-like quarks with the quantum numbers of right-handed down quarks (left-handed quarks doublets) generate effects inC10^U and C9^V0 (C₁₀₍^{U 0}₎ and C9^V) for aZ⁰ boson with vector couplings to muons [214].

Concerning Hyps. 1 to 5 of Tab. 8.2, only two of them (2 and 4) can be explained within a Z⁰ model, while hypotheses 1 and 3 violate the relationship C9µ^NP× C10⁰µ = C10µ^NP × C9⁰µ [113] that minimalZ⁰models should obey. One would have to turn to other models (like LQs with a sufficient number of representations) to explain the hypothesis with the highest pull (Hyp. 1).

8.3.1 Model-independent connection to b→c`ν

We close our discussion of models by commenting on the model-independent connection between the anomalies inb→s``neutral currents and those inb→cτ νcharged currents (i.e. R_D andR_D^∗), which are now at the 3.1σ level [243]. Such a connection, however, requires further hypotheses.

A solution of the R_D(∗) anomaly can naturally be achieved with a NP contribution to the SM operator (¯cγ^µP_Lb) (¯τ γ_µP_Lν), as it complies with the B_c lifetime [244] and q² distributions [245–

247]. Assuming SU(2) invariance, the effect inR_D(∗) is correlated tob→ s`` and/or to b→ sν¯ν, following the pattern C9µ = −C10µ. From model-independent arguments, b → sτ τ must then be significantly enhanced, as we will discuss at length in Chapter 9. Indeed, since b → c`ν processes are mediated already at tree level in the SM, one needs large NP contributions in order to explain the anomalies in R_D and R_D^∗. In principle, these large NP effects would also generate large contributions to b→ sν¯ν processes, due to SU(2) invariance, however contributions to this channel are strongly constrained by B → K^(∗)νν¯. A possible way to bypass this problem is to impose a coupling structure that is mainly aligned to the third generation, but this disagrees with direct LHC searches [248] and electroweak precision observables [249]. Another alternative, which yields no effects inb→sν¯ν processes, arises from the Standard Model Effective Theory (SMEFT) scenario whereC⁽¹⁾ =C⁽³⁾expressed in terms of gauge-invariant dimension-6 operators [3,250,251].

The operator involving-third generation leptons explains R_D(∗) and the one involving the second generation gives a LFUV effect in b → sµµ processes. Form a model-building perspective, this scenario stems naturally from models with an SU(2) singlet vector LQ [234, 235, 252] or with a combination of two scalar LQs [253]. Both the two aforementioned models are predicted to induce large effects in b→sτ τ (of the order of 10⁻³ forBs→τ⁺τ⁻) [253,254].

8.3. Implications for models 145

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Figure 8.5: Preferred regions at the 1, 2 and 3σ level (green) in the (C9µ^V =−C10µ^V ,C9^U) plane from b→ s`<sup>+</sup>`⁻ data. The red contour lines show the corresponding regions once R_D(∗) is included in the fit (for Λ = 2 TeV). The horizontal blue (vertical yellow) band is consistent with R_D(∗) (RK) at the 2σ level and the contour lines show the predicted values for these ratios.

Assuming that the coupling to the second generation is sizeable in order to avoid the bounds from direct LHC searches and electroweak precision observables one finds

C9(10)τ ≈ C9(10)^SM −(+)2π α

V_cb V_ts^∗

sR_D(∗)

R^SM

D^(∗)

−1

!

. (8.3.1)

Notice that our discussion above, implicitly assumes LFUV-NP contributions to b→sµµ, as we only consider effects modifying the muonic Wilson coefficients through C9µ = −C10µ. However, the same SMEFT C⁽¹⁾ =C⁽³⁾ scenario also provides an interpretation of Scenario 8 in Table 8.4, based on both LFU and LFUV types of NP, that jointly accounts for the anomalies in b → sµµ andb→cτ ν. As we mentioned, the constraint fromb→cτ ν andSU(2) invariance generally leads to large contributions to the operator (¯sγ^µP_Lb) (¯τ γ_µP_Lτ), which enhancesb→sτ τ processes (see Chapter 9), but also mixes into O9 and generates C9^U atµ= mb [242]. Note that not all models addressing the charged and neutral current anomalies simultaneously have an anarchic flavour structure. In fact, in the case of alignment in the down-sector [255, 256] one does not find large effects in b→sτ τ orC9^U.

Therefore, Scenario 8 is reproduced in this setup with an additional correlation betweenC9^Uand R_D(∗). Assuming a generic flavour structure so that small CKM elements can be neglected [3,242], we get

C9^U≈7.5 1−

s R_D(∗)

R_D^(∗)_SM

!

1 +log(Λ²/(1TeV²)) 10.5

. (8.3.2)

Realizations of this scenario in specific NP models also usually yield an effect inC⁷ [242]. However, since this effect is model dependent (and in fact small in some UV complete models [257,258]), we neglect it here, leading to the plot in Fig.8.5, where we include the recent update of Ref. [259] to

draw the band for R_D(∗). Note that this scenario has a pull of 7.0σ due to the inclusion of R_D(∗), which increases our ∆χ² by∼20.

In addition to the above-mentioned effects, in LQ models able to generate the effects described one expects sizeable branching ratios forb→sτ µprocesses, reaching the level of 10⁻⁵ [253].