Probes of Lepton Flavor Universality in b → u Transitions

: Anomalies recently observed in semileptonic b → c (cid:96) − ¯ ν (cid:96) and b → s (cid:96) + (cid:96) − transitions point to violation of Lepton Flavour Universality. Strategies for new analyses of different modes are required, in particular for the modes induced by the b → u transition. We describe the purely leptonic B decay, the ¯ B → π (cid:96) − ¯ ν (cid:96) channel and the B semileptonic modes to ρ ( 770 ) and a 1 ( 1260 ) in extensions of the Standard Model involving Lepton Flavour Universality violating b → u operators. In particular, we review the observables in the four-dimensional angular ¯ B → ρ ( ππ ) (cid:96) − ¯ ν (cid:96) and ¯ B → a 1 ( ρπ ) (cid:96) − ¯ ν (cid:96) distributions, suitable to pin down deviations from the Standard Model. We discuss the complementarity among the various modes for New Physics searches.

Although the semileptonic b → u transitions are CKM suppressed with respect to the b → c ones, new precision measurements are foreseen by LHCb and Belle II Collaborations. At present, there is a tension between the inclusive determination of |V ub | and the exclusive measurement from theB → π −ν decay width. The B − purely leptonic and the semileptonic B → π mode have been analyzed in References [36][37][38][39][40][41][42][43], showing their sensitivity to BSM effects. Other decay modes can be exploited to pin down possible deviation from the Standard Model, in particular those involving several hadrons in the final state. This is the case of B to vector ρ(770) and axial-vector a 1 (1260) channels, for which the fully differential angular distributions, when ρ decays in two pions and a 1 decays into ρπ, produce a wealth of correlated observables, with coherent patterns within and beyond SM.
We extend the semileptonic b → u effective Hamiltonian with the inclusion of additional scalar (S), pseudoscalar (P), vector (V) and tensor (T) operators, and study the effects of such operators on B transitions to leptons and to π ν. Then, we provide the expressions of the fully differential decay distributions for theB → ρ(ππ) −ν andB → a 1 (ρπ) −ν modes (Section 4). In the Large Energy Limit for the light mesons, the distributions are expressed in terms of a small number of hadronic form factors. Observables inB → ρ(ππ) −ν are studied at a benchmark point in the parameter space of the new effective couplings, scrutinizing their sensitivity to the different operators (Section 5). For the a 1 (1260) mode, analyzed in Section 06, the uncertainties from the form factors are large; nevertheless, also in this channel observables particularly sensitive to New Physics (NP) can be identified [44].

Extended b → u −ν Effective Hamiltonian
Beyond the Standard Model (BSM) effects in beauty hadron decays can be analyzed using the Standard Model Effective Field Theory. For a NP scale Λ NP much larger than the EW scale the new massive degrees of freedom can be integrated out, recovering an effective Hamiltonian involving SM fields and invariant under the SM gauge group. This Hamiltonian contains additional operators with respect to SM, suppressed by powers of Λ NP : dimension-six four-fermion operators appear at O 1/Λ 2 NP [45]. We consider the general expression of the effective b → u Hamiltonian: consisting in the SM term and in NP operators with complex lepton-flavour dependent couplings V,S,P,T . V ub and V are independent parameters. A purely left-handed lepton current is assumed. Moreover, the quark right-handed vector current is excluded, since the only four-fermion operator of this type, invariant under the SM group, is non-linear in the Higgs field [46][47][48]. See References [37,38,40] for investigations of right-handed currents. The Hamiltonian (1) governs decays of B mesons, of B s (such as [50]), and so forth. Here we focus on B meson modes, for which precise measurements are available.
The couplings of the NP operators in (1) are constrained by the purely leptonic B − and semileptonicB → π −ν modes. The B − → −ν decay width obtained from Equation (1), with f B defined as 0|ūγ µ γ 5 b|B(p) = i f B p µ , shows that this mode is insensitive to the scalar and tensor operators. The pseudoscalar operator lifts the helicity suppression, which is effective for light leptons. This produces stringent bounds to e, µ P . TheB → π −ν decay distribution in the dilepton mass squared q 2 , obtained using (1) and adopting the definition of the form factors f i (q 2 ) = f B→π i (q 2 ) as in Reference [44], (with λ the triangular function), shows that the operator P does not contribute in this case. In Equations (2) and (3) the CKM matrix element V ub appears in the combination V ub (1 + V ). The lepton-flavour dependence of the effective couplings V would produce different values of an effective |V ub | when measured in channels involving different lepton generations.
In the B rest-frame the energy E of the emitted pion is obtained by the relation In the Large Energy Limit, E m B 2 , theB → π −ν distribution involves a single form factor ξ π [51,52]. The consequence is that, in this limit, the ratios dR(π) are free of hadronic uncertainties and only involve combinations of , V,S,T [44] .
For the modesB → ρ(→ ππ) −ν andB → a 1 (→ ρπ) −ν , the main sensitivity to the new operators in (1) is in the 4-dimensional differential decay distribution in (q 2 , θ, θ V , φ), with angles defined in Figure 1. For the ρ mode the distribution reads: . Other angular structures appear in the differential distribution if the quark right-handed vector operator is included in (1). The same expression holds for B s → K * + (Kπ) −ν . The relations of the angular coefficient functions in Equation (5) to the hadronic B → ρ matrix elements are collected in Reference [44], obtained in the narrow width approximation which factorizes the production and decay amplitude of the intermediate vector meson. A discussion on this procedure, together with references to studies of non-resonant contributions, can also be found in Reference [44].
For the a 1 (ρπ) channel with the final ρ transversely (ρ ⊥ ) or longitudinally (ρ ) polarized, the 4-dimensional distribution has the same expression as in (5) . The subscripts ⊥ and refer to the ρ polarizations.
The expressions of the coefficient functions in terms of B → a 1 hadronic matrix elements are in Reference [44]. The separation of the polarizations of the final ρ is justified by the different sensitivity of the distribution to the NP operators. The angular functions I ρ i (I a1 i ) in Equation (5) have the general expression with X = P in case of ρ, and X = S for a 1 , and functions I SM i , I NP i and I I NT i collected in Reference [44]. All angular coefficient functions, with the exception of I 7 , do not vanish in SM and are sensitive to V . Structures able to disentangle the other S, P and T operators can be identified. For B → ρ ν , the functions I 6c,⊥ are insensitive to the scalar operator. There are angular coefficient functions related only to the helicity amplitudes corresponding to the transversely polarized W, hence to transverse ρ (a 1 ) in B → ρ ν (B → a 1 ν ). Such observables depend on T , not on P (in the ρ channel) or S (in a 1 ).
In the Large Energy Limit of the light meson, the B → ρ(a 1 ) weak matrix elements can be written in terms of two form factors, ξ ρ ⊥ (ξ a 1 ⊥ ) and ξ ρ (ξ a 1 ) [44]. In this limit, several angular coefficients depend only on ξ ⊥ , others involve both ξ ⊥ and ξ . The coefficients depending only on ξ ρ,a 1 6c,⊥ (for transversely polarized ρ). When a single form factor is involved, ratios of coefficient functions are free of hadronic uncertainties.

Observables inB → ρ(ππ) −ν
To give examples of the effects of NP operators in (1) inB → ρ −ν , we constrain the space of the effective couplings using a set of hadronic quantities and the available data. The couplings [53]. An upper bound B(B 0 → π + τ −ν τ ) has been established [55]. We use the B → π form factors in Appendix A and the B → ρ form factors in Reference [56]. The parameter space for the NP couplings, together with the chosen benchmark point, is shown in Figure 2 in the case of µ and τ. TheB → π µ −ν µ decay distribution, using the form factors in Appendix A, is depicted in Figure 3 for SM and NP at the benchmark point, compared to Belle [57] and BaBar measurements [58,59].
The plots in the third row show the allowed regions for the couplings τ P and τ T , with the stars indicating the benchmark points τ  The other coefficient functions are in Figure 5, and there is a zero in I ρ 6c (q 2 ) not present in SM. The function I ρ 7 in Figure 6 vanishes in SM, and is only sensitive to the imaginary part of the NP couplings. For the τ modes, the angular functions are in Figures 7 and 8, and also for this mode I Let us consider the ratios and R a 1 , 2s/1s = R a 1 , ⊥ 2c/1c . R ρ 2s/1s is form factor independent in SM. In NP it is still form factor independent in the Large Energy Limit, since I ρ 2s and I ρ 1s depend on ξ ρ ⊥ . As shown in Figure 9, the ratio (7) has a zero in NP, not in SM, whose position q 2 0, ρ depends only on | µ T | with a small form factor effect. In the Large Energy Limit we have Analogously, for the (a 1 ) channel (for (a 1 ) ⊥ considering R 2c/1c ) we have: The positions of the zeros in two modes are related, and their measurement would provide access to the tensor operator. The function I ρ 6c , shown for SM and NP in the right panel of Figure 5, is sensitive to V , P , T . Different possibilities can be envisaged [44]: (1) If there is no NP ( P = T = 0), I ρ 6c has no zero (right panel of Figure 5).  Figure 10).
(4) Finally, if there is NP with both P = 0 and T = 0, both real and imaginary parts of P and T are involved, and two zeroes are possible. Integrating the 4-dimensional distribution several observables can be constructed. The q 2 -dependent forward-backward (FB) lepton asymmetry is given by The transverse forward-backward (TFB) asymmetry, that is, the FB asymmetry for transversely polarized ρ, is given by A FB and A T FB are shown in Figure 11 for = µ and = τ. In NP the zero of A FB is shifted in the τ mode. A T FB is sensitive to the new operators, and in the case of τ it has a zero absent in SM. There are observables which depend on the ρ polarization, namely the differential branching ratio for longitudinally (L) and transversely (T) polarized ρ, as a function of q 2 or of one of the angles θ, θ V : dB L(T) /dq 2 , dB L(T) /dcosθ and dB L(T) /dcosθ V . They are shown in Figure 12 for = µ and = τ. Integrating the distributions we obtain for the ρ mode: As shown in Table 1, the ratios are modified by the NP operators in (1). The deviations are correlated, and large effects are possible varying the effective couplings in their allowed ranges ( Figure 13). Table 1. R π and R ρ in Equation (14) in SM and in NP at the benchmark point.

SM NP (Benchmark Point)
R π 0.60 ± 0.01 0.75 ± 0.02 R ρ 0.53 ± 0.02 0.49 ± 0.02 SM BP Figure 13. R ρ vs R π correlation when only the tensor operator is included in (1). The colors correspond to the different signs of Re( µ T ) and Re( τ T ) in the range of the parameter space. The red and brown points correspond to SM and NP at the benchmark point.

6.B → a 1 (1260) −ν
We analyze theB → a 1 (ρπ) −ν mode in SM and in the NP extension Equation (1) at the same benchmark points for V,S,T . Exclusive hadronic B decays into a 1 (1260) have been studied at the B factories considering the dominant a 1 → ρπ mode, and the measurements of the semileptonicB → a 1 mode are within the experimental reach, in particular at Belle II. The study ofB → a 1 −ν requires an assessment of the accuracy of the hadronic quantities. We use the set in Reference [60], for which an uncertainty of about 20% is quoted. The angular coefficient functions, for the µ and τ modes and for both the ρ polarizations are in Figure 14, Figure15 and in Figure 16, Figure 17, respectively. In general, the hadronic uncertainties obscure the effect of the NP operators. Nevertheless, there are coefficient functions in which deviations from SM can be observed, namely I a 1 2s, (q 2 ), I a 1 6c, (q 2 ) ( Figure 14) and I a 1 2c,⊥ (q 2 ) (Figure 16) for the µ channel, I a 1 1s, (q 2 ), I a 1 6s, (q 2 ) ( Figure 15) and I a 1 1c⊥ (q 2 ), I a 1 6c⊥ (q 2 ) (Figure 16) for the τ mode. The forward/backward lepton asymmetry shows sizable deviations from SM in the case of τ ( Figure 18). i (q 2 ) in SM and NP (at the benchmark point) forB → a 1 (ρ π) µ −ν µ using the form factors in Reference [60]. The band widths for SM and NP, with colors indicated in each plot, are due to the form factor uncertainty.  Figure 15. Angular coefficient functions forB → a 1 (ρ π) τ −ν τ (same notations as in Figure 14).  Figure 16. Angular coefficient functions forB → a 1 (ρ ⊥ π) µ −ν µ (same notations as in Figure 14).  Figure 17. Angular coefficient functions forB → a 1 (ρ ⊥ π) τ −ν τ (same notations as in Figure 14). For the ratio R a 1 = B(B → a 1 (1260)τ −ν τ ) B(B → a 1 (1260) −ν ) we obtain, in the SM and for NP at the benchmark point, R SM a 1 = 0.44 ± 0.07 , R NP a 1 = 0.67 ± 0.12 , with individual branching fractions in SM B(B → a 1 µ −ν µ ) = (3.0 ± 1.7) × 10 −4 and B(B → a 1 τ −ν τ ) = (1.3 ± 0.6) × 10 −4 [60]. Summarizing the complementarity between the various B modes to the NP search, we remark that the presence of the tensor operator in (1) can be established independently of the other operators, from deviations of the observables that depend only on T , for example, those involving transversely polarized ρ and a 1 . | T | is constrained looking at the zero of the ratios in Equations (7) and (8). A correlation between the position of the zero in the ρ and a 1 channels should be observed.
If the pseudoscalar operator is present, without other NP structures, deviations should be observed in leptonic B decays and in the semileptonic decay to ρ, not in π and a 1 modes. Determining the position of the zero in I

Outlook
The anomalies observed in b → c semileptonic modes also require new precision analyses of the CKM suppressed b → u processes. We have discussed the impact of an extended effective b → u Hamiltonian on purely leptonic B decays, onB → π −ν and, in details, onB → ρ(ππ) −ν and B → a 1 (ρπ) −ν , which involve more than one hadron in the final state. For these two modes we have studied the sensitivity of the 4-dimensional angular distributionst to different NP operators. The quantum numbers of the light mesons select the contributions of the NP terms, and complementary information can be gained on the new operators in the effective Hamiltonian. Some observables have zeroes absent in SM. Comparing the modes into µ and τ, LF universality in b → u weak transitions is probed. The error connected to the hadronic matrix elements represents a sizable uncertainty, in particular for the a 1 mode. The Large Energy Limit, in which the number of hadronic form factors is reduced, is therefore useful for the analyses.
There are exciting perspectives for NP searches in heavy hadron decays in the coming years, in particular investigating b → u induced semileptonic B modes.