With or without U(2)? Probing non-standard flavor and helicity structures in semileptonic B decays

Motivated by the recent hints of lepton flavor universality violation observed in semileptonic $B$ decays, we analyze how to test flavor and helicity structures of the corresponding amplitudes in view of future data. We show that the general assumption that such non-standard effects are controlled by a $U(2)_q \times U(2)_\ell$ flavor symmetry, minimally broken as in the Standard Model Yukawa sector, leads to stringent predictions on leptonic and semileptonic $B$ decays. Future measurements of $R_{D^{(*)}}$, $R_{K^{(*)}}$, ${\mathcal B}(\bar B_{c,u}\to \ell \bar \nu)$, ${\mathcal B}(\bar B \to \pi \ell \bar\nu)$, ${\mathcal B}(B \to \pi \ell \bar\ell)$, ${\mathcal B}(B_{s,d}\to\ell\bar\ell^{(\prime)})$, as well as various polarization asymmetries in $\bar B\to D^{(*)} \tau \bar \nu$ decays, will allow to prove or falsify this general hypothesis independently of its dynamical origin.


Introduction
Present data exhibit intriguing hints of violations of Lepton Flavor Universality (LFU) both in charged-current [1][2][3][4][5] and neutral-current [6][7][8][9][10][11] semileptonic B decays. These hints can be well described employing Effective Field Theory (EFT) approaches (see [12][13][14][15][16]f o r the early attempts), whose main ingredients are the assumptions that New Physics (NP) affects predominantly semileptonic operators, and that it couples in a non-universal way to different fermion species. In particular, NP should have dominant couplings to third generation fermions and smaller, but nonnegligible, couplings to second generation fermions. Interestingly enough, this non-trivial flavor structure resemble the hierarchies observed in the Standard Model (SM) Yukawa couplings, opening the possibility of a common explanation for the two phenomena.
An effective approach to address the question of a possible connection between the LFU anomalies and the SM Yukawa couplings and, more generally, to investigate the flavor structure of non-standard effects at low-energies, is the assumption of an appropriate flavor symmetry and a set of symmetry-breaking terms. The flavor symmetry does not need to be a fundamental property of the underlying theory, it could simply be an accidental lowenergy property. Still, its use in the EFT context provides a very powerful organizing principle for a bottom-up reconstruction of the underlying dynamics.
In the context of the recent anomalies, the flavor symmetry that emerges as particularly suitable to describe the observed data is U (2) q × U (2) ℓ , which is a subset of the larger U (2) 5 proposed in [17][18][19]a s a useful organizing principle to address the hierarchies in the SM Yukawa couplings and, at the same time, allow large NP effects in processes involving third-generation fermions (as expected in most attempts to address the electroweak hierarchy problem).
The scope of the present paper is a systematic investigation of the consequences of this symmetry hypothesis in (semi)leptonic B decays. Contrary to previous analyses, where this symmetry has been implemented in the context of specific new-physics models, our goal here is to investigate the consequences of this symmetry (and symmetry-breaking) ansatz in general terms, with a minimal set of additional assumptions about the dynamical origin of the anomalies. As we will show, in the case of charged-current interactions, the symmetry ansatz alone is sufficient to derive an interesting series of testable predictions. The predictive power is smaller for neutral-current transitions, but also in that case we can identify a few clean predictions which are direct consequences of the symmetry ansatz alone.

The U (2) 5 symmetry in the SM
The U (2) 5 × U (1) B 3 × U (1) L 3 symmetry is the global symmetry that the SM Lagrangian exhibits in the limit where we ne- glect all entries in the Yukawa couplings but for third generation masses [17][18][19]. Under this symmetry, the first two SM fermion families transform as doublets of a given U (2) subgroup, while third-generation quarks (leptons) are only charged under . The largest breaking of this symmetry in the complete SM Lagrangian is controlled by the small parameter A minimal set of U (2) 5 breaking terms (spurions) which lets us reproduce all the observable SM flavor parameters (in the limit of vanishing neutrino masses), without tuning and with minimal size for the breaking terms, is In terms of these spurions, the 3 × 3Y u k a w a 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). 1 Note that u,d,e are 2 × 2 complex matrices, while The precise size of the spurions is not known; however, we can estimate it by the requirement of no tuning in the O(1) parameters. This implies |V q | = O (ǫ). In the limit of vanishing neutrino masses, the size of |V ℓ | cannot be unambiguously determined. As discussed below (see also [20]), a good fit of the anomalies in semileptonic B decays is obtained for which is perfectly consistent with: i) the estimate |V q | = O (ǫ); ii) the hypothesis of a common origin for the two leading U (2) 5 breaking terms in quark and lepton sectors. The entries in the 2 × 2 matrices u,d,e are significantly smaller than |V q,ℓ |, with a maximal size of O (10 −2 ) in the quark sector.
By appropriate field redefinitions and without loss of generality, one can remove unphysical parameters in the Yukawa matrices in (4)( s e e App. A). Working in the so-called interaction basis, where the second generation in U (2) q(ℓ) space is defined by the alignment of the leading spurions, one can bring the Yukawa matrices to the following form 1 In models with more than one Higgs doublet, the smallness of y b,τ can be justified in terms of approximate flavor-independent U (1) symmetries.
where ˆ u,d,e are 2 × 2 diagonal positive matrices, O u,e are 2 × 2 orthogonal matrices and U q is of the form with s d ≡ sin θ d and c d ≡ cos θ d .
The Yukawa matrices in (7)g e t diagonalized by means of appropriate unitary transformations: The most general form for these unitary transformations is with L u = L d V † CKM . Here we have taken advantage of the constraints imposed by fermions masses and CKM matrix elements to eliminate various parameters appearing in L f and R f . 2 These further imply that s d and α d are constrained by s d /c d =|V td /V ts | and α d = arg(V * td /V * ts ), and that s t = s b − V cb . The light-family leptonic mixing (s e ), appearing in O e , cannot be expressed in terms of measurable quantities. Two additional mixing angles which remain unconstrained are s b /c b =|x b | |V q | and s τ /c τ =|x τ | |V ℓ |. Finally, φ q is an unconstrained O (1) phase, that becomes unphysical in the limit s b → 0( o r , equivalently, x b → 0). This limit is phenomenologically required in models where F = 2o p e r a t o r s are generated at tree-level around the TeV scale: in such case one needs to impose a mild alignment to the down basis, i.e. |s b | 0.1 ǫ, to satisfy the constraints from B s,d -meson mixing [21][22][23].

Impact of U (2) 5 on the EFT for semileptonic B decays
Having defined the flavor symmetry and its symmetry breaking terms from the SM Yukawa sector, we are ready to analyze its implications beyond the SM. Assuming no new degrees of freedom below the electroweak scale, we can describe NP effects in full generality employing the so-called SMEFT. We limit the attention to dimension-six four-fermion operators bilinear in quark and lepton fields, 3 that we write generically as 2 The removal of unphysical parameters presented in App. A corrects a similar analysis presented in [21], where it was erroneously concluded that the parameter α d is unconstrained. 3 We neglect operators which modify the effective couplings of W and Z bosons.
These are highly constrained and cannot induce sizable LFU violating effects.
where v ≈ 246 GeV is the SM Higgs vev, {α, β} are lepton-flavor indices, and {i, j} are quark-flavor indices. The operators in the Warsaw basis [24]w i t h a non-vanishing tree-level matrix element in semileptonic B decays are where C V i ,S control the overall strength of the NP effects and V i ,S are tensors that parametrize the flavor structure. They are normalized by setting [3333] V i ,S = 1, which is the only term surviving in the exact U (2) 5 limit.
Let us consider first the structure of S , which is particularly simple. Neglecting U (2) d,e breaking spurions, it factorizes to where, in the interaction basis, Here x q,ℓ,qℓ are O (1) coefficients and we have neglected higherorder terms in V q,ℓ (that would simply redefine such coefficients).
Moving to the mass-eigenstate basis of down quarks and charged leptons, where (9)], with the new matrices assuming the following explicit form in 3 × The (complex) parameters x bτ qℓ , λ i q , λ α ℓ , and αi qℓ are a combination of the spurions in (14) and the rotation terms from L d,e , that satisfy On the r.h.s. of the first line of (16)w e have neglected tiny terms suppressed by more than two powers of |V q,ℓ | or s d,e .
If we consider at most one power of V q and one power of V ℓ , then also have the same structure as Ŵ L with, a priori, different O (1) coefficients for the spurions. Moving to the basis (15), Ŵ V i L assumes the same structure as Ŵ L in (16), with parameters which can differ by O(1) overall factors, but that obey the same flavor ratios as in (17). Corrections to the factorized structure in (18)a r i s e s only to second order in V q or V ℓ , generating terms which are either irrelevant or can be reabsorbed in a redefinition of the observable parameters in the processes we are interested in (see sect. 4).

Matching to the U 1 leptoquark case
The EFT in (12), with factorized flavor couplings as in (13) and (18), nicely matches the structure generated by integrating out a U 1 vector leptoquark, transforming as (3, 1) 2/3 under the SM gauge group. As noted first in [16], this field provides indeed an excellent mediator to build in a natural, and sufficiently general way, an EFT for semileptonic B decays built on the U (2) 5 flavor symmetry broken only by the leading V q and V ℓ spurions (see [20][21][22][23][25][26][27][28][29][30]f o r other phenomenological analysis of the U 1 leptoquark in B physics).
Writing the interaction between the U 1 field and SM fermions in the basis of (15)a s [ 20] the flavor symmetry hypothesis imply a parametric structure for β jα L and β iα R identical to that of Ŵ iα L and Ŵ iα R in (16). Normalizing g U such that β bτ L = 1, and integrating out the U 1 field, leads to the following (tree-level) matching conditions for the parameters of L EFT and where β R ≡ β bτ R . Note that while (21)i s just a redefinition of the free parameters of the effective Lagrangian, (20)g i v e non-trivial constraints. We also stress that, beside the overall coupling (encoded in C V ) the four combinations of couplings in (21) indicate the helicity structure of the interactions (C S /C V ) and its alignment in quark and lepton flavor space (λ s q , λ μ ℓ and sμ qℓ ).
The condition C V 1 = C V 3 , arising naturally in the U 1 case, is important to evade the tight constraints on O (3) ℓq from b → sνν observables and electroweak precision tests [27]. In order to analyze charged currents as much independent as possible from other of redundant parameters as far as neutral currents are concerned.

b → cτν rates and polarization asymmetries
At fixed quark and lepton flavors, the effective Lagrangian in (12) depends only on two coefficients. 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 Ŵ V 3 L = Ŵ L , one would need to redefine C c V replacing λ s q with λ s q . Employing this hypothesis, as in the leptoquark case, the ratio C c blind and depends only on the helicity structure of the NP amplitude.
Using the results in [32,33]f o r the B → D ( * ) ℓν form factors and decay rates, and neglecting the tiny NP corrections in the ℓ = μ, e case (see below) leads to the following expression for the LFU where η S ≈ 1.7a r i s e s from the running of the scalar operator from the TeV scale down to m b [34]. Updated SM predictions for R D ( * ) can be found in [35]: (3), Current measurements of R D and R D * [1][2][3][4][5]l e a d to the constraints on C c S and C c V shown in Fig. 1 (dashed contour lines) where, for simplicity, we have assumed these couplings to be real. 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 are also indicated. Before discussing the impact of additional observables in constraining the same set of parameters (under additional assumptions), we stress that once C c S and C c V have been determined, all the other b → cτν observables are completely fixed by the U (2) 5 invariant structure of L EFT and can be used to test it. Particularly interesting in this respect are the polarization asymmetries, where the τ (±) denotes a τ with ±1/2 helicity. We find the following expressions for these observables 4 Taking C c V ,S real for simplicity, we obtain where [35] Since the effect of C c V is that of rescaling the SM amplitude, all the above ratios are largely insensitive to the value of C c V and become 1 in the limit C c This fact is clearly illustrated in Fig. 2, where we plot the deviations from unity of the polarization ratios vs. the difference on the two leading LFU ratios (which also vanishes in the limit C c S → 0), As can be seen, the predicted pattern of deviations is very precise and rather specific. At present, only P D * τ [3,38] and F D * L [39]h a v e been measured, still with large uncertainties. As shown in Fig. 2, it is not possible in our setup to reach the current experimental central value for F D * L (see [36,40]f o r a similar discussion).
The last b → cτν observable we take into account is B(B c → τν), that is particularly sensitive to the scalar amplitude. Despite it will be quite difficult to measure this branching ratio in the future, interesting bounds can be derived from the measurement of the B c lifetime [41,42]. The expression for this observable reads We find B(B c → τν) to be at most at the level of 10% for the best fit contours in Fig. 1, well below the B(B c → τν) 30% bound obtained in [41].
In principle, additional probes of the b → cτν amplitude are provided by the τ /μ LFU ratios in b → c ℓν [37] and in B c → ψℓν [43]d e c a y s . In both cases scalar amplitudes are subleading and, in our framework, one should expect an enhancement compared to the SM prediction similar to the one occurring in R D * .
However, measuring the LFU ratio in b → c ℓν -where we have a precise SM prediction [44]-is quite challenging, while in the B c → ψℓν case the current SM theory error is well above the 10% level [45].

b → uτν transitions
The analog of C c V (S) for b → u transitions are the effective cou- 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 minimallybroken U (2) 5 hypothesis. At present, the most significant constrains on b → uτν transitions are derived from B u → τν, whose branching ratio is  [47]( 1 σ range indicated by the purple band in Fig. 1).
In the future, very interesting constraints are expected from B → πτν. Using the hadronic parameters in [48,49], we find for where R SM π = 0.641 (16) [48,50,51]. In the limit where quadratic NP effects can be neglected, the following approximate relation holds which would allow a non-trivial test of the U (2) 5 structure of the interactions. In Fig. 3 where the difference among the two modes arises by sub-leading spectator mass effects in the chirality-enhancement factors χ c and χ u .

b → sℓl (′) transitions
The b → s semileptonic transitions have a rich phenomenology and have been extensively discussed in the recent literature. Contrary to the charged-current case, here model-dependent assumptions, such as the constraints in (20), play a more important role. Rather than presenting a comprehensive analysis of the various observables accessible in these modes, our scope here is to focus on: i) model-independent predictions related to the minimally-broken U (2) 5 hypothesis; ii) clean observables controlling the size of the symmetry breaking terms.
b → sττ (νν) Under the assumption C V 1 = C V 3 , following from the hypothesis of a U 1 UV-completion, NP effects in b → sνν transitions are forbidden at tree level. On the other hand, NP contributions to b → sττ are almost as large as those in b → cτν for λ s q = O (0.1) (see e.g. [27,52]  However, the current experimental limit [53,54] B(B s → ττ ) B(B s → ττ ) SM is still well below the possible maximal enhancement. As a result, at present this observable does not put stringent constraints on the parameter space of the EFT: in Fig. 1 we show the 90% CL exclusion region in the (C c S , C c V ) plane for λ s q = 3 |V ts |.
As pointed out in [55], a possible large enhancement of the b → sττ amplitude can indirectly be tested via the one-loopinduced lepton-universal contributions to b → sℓl(ℓ = e, μ, τ ) in the O 9 direction (see below). This contribution is well compatible and even favored by current data [30,56].
b → sμμ(eē) FCNC decays to light leptons offer an excellent probe of the U (2) 5 breaking terms in the lepton sector. These transitions are commonly described in terms of the so-called weak effective Hamiltonian [57,58] H b→s with G F the Fermi constant, α the fine-structure constant and In the SM, C ℓ 9 ≈ 4.1, C ℓ 10 ≈−4.3 and C ℓ S = C ℓ P = 0. Matching to the Lagrangian in (12), we get ( while the corresponding tree-level effects in the electron sector are negligible. One of most relevant observables involving these transitions are the LFU ratios R K ( * ) = Ŵ(B → K ( * ) μμ)/Ŵ(B → K ( * ) eē), which are particular interesting due to their robust theoretical predictions: R SM K ( * ) = 1.00 ± 0.01 [59]. In our setup, one gets [60,61] 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. As observed first in [62], the relation R K ≈ R K * holds in any NP model where LFU contributions to b → sℓl decays are induced by a left-handed quark current: in our framework this is a direct consequence of the smallness of the flavor-symmetry breaking terms in the right-handed sector.
From the experimental point of view, this implies that all μ/e universality ratios in b → s transitions are expected to be the same, provided their SM contribution is dominated by C 9 and/or C 10 . In addition to (42), we thus expect 5 Current experimental data hint to sizable NP effects in R K and Assuming the NP effect to be the same in R K and R K * , the combined measurements imply R , as suggested by the R D ( * ) fit, the value of R K ( * ) implies λ μ ℓ = O (10 −1 ). 5 We define the universality ratios as R( assuming a region in m ℓℓ below the charmonium resonances and sufficiently above the di-muon threshold (m ℓℓ 1G e V ).
Within the UV leptoquark completion, the fact that R K ( * ) ≡ R K ( * ) − 1 < 0 allows us also to determine a non-trivial relative sign among the U (2) 5 breaking terms: according to (20)o n e has C V > 0, which implies sμ qℓ λ μ * ℓ < 0. Other data involving b → sμμ transitions, such as the measurements of P ′ 5 [63][64][65][66] and other differential distributions, also deviate from the SM predictions consistently with R K ( * ) , further supporting the hypothesis of λ s q , λ Expressing the deviations in the Wilson coefficients in terms of R K ( * ) , by means of (41) and (42), leads to Current experimental measurements [67][68][69][70][71] As in B(B s → ττ ), the large chiral enhancement of the scalar contribution make it an excellent probe of the helicity structure of the NP effects. Moreover, this observable provides a direct probe of ues of the other NP parameters, the expected branching fraction is about one order of magnitude smaller. The current experimental limit, B(B s → τ ± μ ∓ ) < 4.2 × 10 −5 (95% CL) [73], is close to the NP predictions when C S is sizable. Future improvements in this observable will therefore provide very significant constraints.

b → dℓl and other FCNCs
A key prediction of the minimally broken U (2) 5 framework is that NP effects in b → sℓl and b → dℓl transitions scale according to the corresponding CKM factors. More precisely, defining the effective hamiltonian of the leading b → d FCNC operators as , then it is easy to check that, because of (17), C ℓ 9,10 = C ℓ 9,10 ,C ℓ S,P = C ℓ S,P .
These relations lead to a series of accurate predictions which could be tested in various b → dℓl observables.
Leaving aside B decays, the effective Lagrangian (12) necessarily imply non-standard effects also in K and τ semileptonic decays.
Since NP effects in these processes are strongly constrained, it is important to check if these constraints limit the parameter space of the EFT. As far as τ decays are concerned, the most stringent constraint is obtained by the non-observation of τ → μγ . Only the chiral-enhanced contribution to τ → μγ , proportional to C S , can be reliably estimated in the EFT, yielding 7 Taking C S = O (10 −2 ) and λ μ ℓ = O (10 −1 ), we find B(τ → μγ ) = few × 10 −9 , which is below current bounds but within the expected Belle II sensitivity [77]. Finally, the constraints obtained from K decays do not yield significant bounds to our framework. As in the b → sνν case, NP effects in s → dνν transitions are forbidden at tree-level if we take On the other hand, K L → ℓl (′) decays receive strong spurion suppressions, resulting in bounds on C V that are significantly above the preferred values. These are shown in Table 1, together with the parametric spurion dependence of the corresponding ds → ℓl (′) transition. For comparison, we stress that the preferred value of C V emerging from the R D ( * ) fit in Fig. 1, assum-

Charged-current transitions to light leptons
The U (2) 5 breaking in the lepton sector could in principle be tested also in charged-current decays to light leptons, both in b → c and b → u transitions. The most relevant observables in each and B(B u → μν), whose experimental measurements will be improved at Belle II [77]. In contrast to R D ( * ) , scalar contributions to R μe D ( * ) are extremely suppressed due to the U (2) 5 flavor symmetry [see (16)].
The μ/e LFU ratios can be expressed as where the first term corresponds to the mode with ν = ν μ and the second one with ν = ν τ and where, similarly to C c V , we have defined 7 The full branching fraction, including both vector and non-chiral-enhanced scalar contributions, was computed in [20]i n a specific U 1 UV-completion. There, it was found that the additional contributions are much smaller than the ones quoted , we find that NP corrections to these observables are at most at the per-mille level, hence beyond the near-future experimental sensitivity. This is quite different from what is expected in other NP models addressing the anomalies, such as the scalar leptoquark models considered in [79,80].
It is also worth stressing that the phenomenological condition sμ qℓ λ μ * ℓ < 0, required to accommodate R K ( * ) with C V > 0, yields a partial cancellation between the two terms in C cμ V . As a result of this cancellation, NP effects are typically at the sub per-mille level, hence beyond any realistic sensitivity even in a long-term perspective. Similarly, we find possible NP contributions to B(B u → μν) to be at or below the per-mille level, very far from the experimental reach.

Conclusions
The hints of LFU violations observed in B meson decays have shaken many prejudices about physics beyond the SM, opening new directions in model building. One of the most intriguing possibilities is the existence of a link between the (non-standard) dynamics responsible for these anomalies and that responsible for the fermion mass hierarchies. A specific realization of this idea is the hypothesis that, at low energies, the new dynamics manifests via an EFT controlled by an approximate U (2) 5 symmetry, with leading breaking in specific directions in the U (2) q and U (2) ℓ subgroups.
In this paper we have explored in generality the consequences of this symmetry and symmetry-breaking hypothesis in (semi)leptonic B decays, trying to avoid making additional dynamical assumptions about the origin of the anomalies. As we have shown, the symmetry hypothesis alone leads to a significant reduction in the number of free parameters of the EFT which, in turn, can be translated into stringent predictions on various low-energy observables. The situation is particularly simple in the case of charged currents, where all relevant processes are controlled by two independent combinations of effective couplings. The latter can be determined for instance from R D and R D * , leading to a series of unambiguous predictions for B(B c,u → ℓν), B(B → πℓν), polarization asymmetries in B → D ( * ) τν, as well as other processes. As shown in Fig. 1, the available data on B(B → τν) perfectly support the initial hypothesis.
In neutral currents, additional combinations of effective couplings appear, but also in this case a series of stringent predictions, which are genuine consequences of the symmetry and symmetrybreaking hypothesis alone, can be derived. The two most notable ones are: i) the (approximate) universality of the deviations from 1 in μ/e ratios in short-distance dominated b → sℓℓ transitions, leading to (43), and ii) the SM-like CKM scaling of NP effects in b → s and b → d transitions, which leads to the relations (50) and (51). If the significance of the current anomalies will increase, the experimental tests of these relations, which are within the reach of current facilities, will provide an invaluable help in clarifying the origin of this intriguing phenomenon.