B-anomalies in U(2) flavor symmetry

We analyzed how to test flavor and helicity structures of the corresponding amplitudes in view of future data, motivated by the recent hints of lepton flavor universality violation observed in semileptonic B decays. The general assumption that non-standard effects are controlled by a U(2)5 flavor symmetry, minimally broken as in the Standard Model Yukawa sector, leads to stringent predictions on leptonic and semileptonic B decays. Future measurements will allow to prove or falsify this general hypothesis independently of its dynamical origin.


Introduction
The current data collected by LHCb, BaBar and Belle experiments exhibit intriguing hints of violations of Lepton Flavor Universality (LFU) both in charged-current [2][3][4][5][6] and neutral-current [7][8][9][10][11][12] semileptonic B decays, and it gets attention as B-anomalies. The features of the hypothetical NP should have dominant couplings to third generation fermions and smaller couplings to second generation fermions. This non-trivial flavor structure resembles the observed flavor hierarchies in the Standard Model (SM) Yukawa couplings, and the possibility of a common explanation for these phenomena is opened.
In the context of the recent anomalies, we adopt the general assumptions that the NP effects are controlled by U(2) 5 flavor symmetry, which is a useful organizing principle to address the flavor hierarchies in the SM [13][14][15]. The paradigm of U(2) 5 flavor symmetry turns out to be successful in addressing B-anomalies with satisfying all existing bounds (e.g. Ref [16]). It is a global symmetry that the SM Lagrangian satisfies in good approximation; in the limit where we neglect all entries in the Yukawa couplings but for third generation masse, and given as where the first two SM fermion families transform as doublets of the U(2). A minimal set of U(2) 5 breaking terms (spurions) which lets us reproduce all the observable SM flavor parameters without tuning and with minimal size for the breaking terms, is V q ∼ (2, 1, 1, 1, 1) , V ∼ (1, 2, 1, 1, 1) , ∆ u(d) ∼ 2, 1,2(1), 1(2), 1 , ∆ e ∼ 1, 2, 1, 1,2 .
(2) * This talk is based on the work with Javier Fuentes-Martín, Gino Isidori and Julie Pagès [1]. It is given at the Workshop "Flavour changing and conserving processes" 29-31 August '19 (Capri Island, Italy) e-mail: keiy@hiroshima-u.ac.jp In terms of these spurions, the 3 × 3 Yukawa matrices can be decomposed as where x t,b,τ and y t,b,τ are free complex parameters, expected to be of O(1). By the requirement of no tuning in the O(1) parameters, the order of spurion |V q | = O(10 −1 ) is implied, which shows a good fit of the anomalies in semileptonic B decays as discussed below. In this work, we present a systematic investigation of the consequences of this symmetry hypothesis in (semi)leptonic B decays, in model independent manner.

The EFT for semileptonic B decays based on the U(2) 5 flavor symmetry
Assuming no new degrees of freedom below the electroweak scale, we can describe NP effects in full generality employing the so-called SMEFT (SM effective lagrangian), and we write the Lagrangian as where v ≈ 246 GeV is the SM Higgs vev, {α, β} are leptonflavor indices, and {i, j} are quark-flavor indices. Under the U(2) flavor symmetry, the right handed light fermion operators are suppressed and this feature reduces the number of relevant semileptonic operators. Also, we do not consider O qe = (q i L γ µ q j L )(ē α R γ µ e β R ) for simplicity because it contributes at tree-level only to b → sττ, which is poorly constrained currently. Now, the relevant operators in the Warsaw basis [17] is given as and we are left with the following effective Lagrangian where C V i ,S control the overall strength of the NP effects and Λ V i ,S are tensors that parametrize the flavor structure. They are normalized by setting Λ [3333] V i ,S = 1, which is the only term surviving in the exact U(2) 5 limit. Note that the U(2) 5 assumption matches the U 1 vector leptoquark, which is the best solution for B-anomalies so far, transforming as (3, 1) 2/3 under the SM gauge group. The EFT in (6) nicely matches the structure generated by integrating out a U 1 vector leptoquark. The relation The flavor structure Λ S is factorizes to where, in the interaction basis, Here x q, ,q are O(1) coefficients and we have neglected higher-order terms in V q, . Moving to the mass-eigenstate basis of down quarks and charged leptons, where The new matriceŝ Γ L,R can be written aŝ The (complex) parameters x bτ q , λ i q , λ α , and ∆ αi q are a combination of the spurions in (8) and the rotation terms from L d,e , that satisfy The r.h.s. of the first line of (10) is at lowest order in the spurion (|V q, |) expansion. The structure of Λ [i jαβ] V is written in same manner.
From the flavor structure shown above, it is found that following special features are predicted by U(2) 5 .
• The NP effect in neutral-current b → sµµ is smaller than one in charged current b → cτν, which is compatible with the situation of B-anomalies.
• Scalar operators with light fermions suppressed by factor m s /m b and m µ /m τ .

Observables in Charged-current
In this section, we discuss the NP effects on the chargedcurrent. In the b → cτν case, we conveniently re-define them as where, in the last line, we have used CKM unitarity. When defining C c V(S ) , we have factorized the CKM factor V cb , such the that the left-handed part of the interactions is modified as In the absence of the simplifying hypothesis Γ V 3 L = Γ L , one would need to redefine C c V replacing λ s q withλ s q . Employing this hypothesis, as in the leptoquark case, the ratio C c S /C c V = C S /C V is flavor blind and depends only on the helicity structure of the NP amplitude.
Current measurements of the LFU ratios R D and R D * , where R H = Γ(B → Hτν)/Γ(B → H ν), lead to the constraints on C c S and C c V . In fig.1, chi-square fit results (dashed contour lines) for b → cτν τ together with b → uτν τ are shown. Here, we use the results in [18,19] for theB → D ( * ) ν form factors and decay rates. For comparison, the directions corresponding to a pure left-handed (β R = 0) or a vector-like interaction (β R = −1) for the U 1 leptoquark, where β R is right-handed d R -e R coupling mediated by U 1 , are also indicated. It is found that the fit results taking into account only the information from b → cτν τ transitions (R D and R D * ) are deviated from 3σ from the SM predictions (zero point), and the U(2) prediction for b → u transition (B → τν) is compatible with them.
The predictiveness of U(2) 5 is also found in the predictions for the polarizations in B → D ( * ) τν τ . The τ polarizations P D τ , P D * τ and the D * polarization F D * L are sensitive to the NP models (e.g. Ref [21]) and expected to be measured at the Belle II experiment in the near future. In fig.2, the sharp correlations between R D , R D * , and the τ polarizations P D τ , P D * τ and the D * polarization F D * L are shown, which can be tested by Belle II in the foreseeable future. The predictions are obtained using the fit in Fig. 1 (continuous  lines). In gray, the experimental value of ∆F D * L at 1σ and 2σ.
Next, we discuss the b → cτν transitions. The analog of C c V(S ) for b → u transitions are the effective couplings where the result in the second line follows from CKM unitarity. The prediction of same size NP effects, relative to the SM, in b → u and b → c transitions is a distinctive feature of the minimally-broken U(2) 5 hypothesis. It predicts where the difference among the two modes arises by subleading spectator mass effects in the chirality-enhacement factors. Also, in the future, very interesting constraints are expected fromB → πτν. This process also has specific relation withB → D ( * ) τν, and we get following approximate relation where R π ≡ B(B → πτν)/B(B → π ν) and we use the hadronic parameters in [22,23]. This relation would allow a non-trivial test of the U(2) 5 structure of the interactions. In Fig. 3

Observables in Neutral current
The b → s semileptonic transitions have a rich phenomenology and have been extensively discussed in the One of most relevant observables are the LFU ratios R K ( * ) = Γ(B → K ( * ) µμ)/Γ(B → K ( * ) eē), which are particular interesting due to their robust theoretical predictions. In our setup, one gets [24,25] The prediction R K ≈ R K * , is a direct consequence of our flavor symmetry assumptions and is independent of the initial set of dimension-six SMEFT operators. In addition to (19), we expect (20) Current experimental data hint to sizable NP effects in R K and R K * consistent with R K ≈ R K * . This numerical value R K and R K * provides an important constraint on the size of the leptonic spurion (λ µ ): since ∆ sµ q = O(λ s q λ µ ), setting λ s q = O(10 −1 ) and C V = O(10 −2 ), as suggested by the R D ( * ) fit, the value of R K ( * ) implies λ µ = O(10 −1 ).
Among b → sµμ transitions, a special role is played by B s → µμ, where the chiral enhancement of the scalar amplitude allows us to probe the helicity structure of the NP interaction. In Fig. 4, we show the predictions for this observable as a function of ∆R K ( * ) for s τ = 0 (purple band), where s τ is mixing parameter in the rotation matrix, and for s τ = −0.1 λ µ setting C S /C V = 2 (green band). As can be seen, the current experimental values show tension with the SM [26] and are consistent with U(2) 5 prediction.

Conclusions
Motivated by the recent hints of lepton flavor universality violation observed in semileptonic B decays (Banomalies), we adopt the general assumptions that the NP effects are controlled by a U(2) 5 flavor symmetry, minimally broken as in the SM Yukawa sector, and anlyzed how to test flavor and helicity structures of the corresponding amplitudes in view of future data. It is found that stringent predictions on leptonic and semileptonic B decays in charged-current and neutral-current are led, and the current data consistent with a U(2) 5 flavour symmetry. A U(2) 5 flavor symmetry is very predictive, and future measurements by LHCb and Belle II will provide an invaluable help in clarifying the origin of this intriguing phenomenon.