Decoding (pseudo)scalar operators in leptonic and semileptonic B decays
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Abstract
We consider leptonic \(B^\rightarrow \ell ^ {\bar{\nu }}_\ell \) and semileptonic \({\bar{B}} \rightarrow \pi \ell ^ {\bar{\nu }}_\ell \), \({\bar{B}} \rightarrow \rho \ell ^ {\bar{\nu }}_\ell \) decays and present a strategy to determine shortdistance coefficients of NewPhysics operators and the CKM element \(V_{ub}\). As the leptonic channels play a central role, we illustrate this method for (pseudo)scalar operators which may lift the helicity suppression of the corresponding transition amplitudes arising in the Standard Model. Utilising a new result by the Belle collaboration for the branching ratio of \(B^\rightarrow \mu ^ {\bar{\nu }}_\mu \), we explore theoretically clean constraints and correlations between New Physics coefficients for leptonic final states with \(\mu \) and \(\tau \) leptons. In order to obtain stronger bounds and to extract \(V_{ub}\), we employ semileptonic \({\bar{B}} \rightarrow \pi \ell ^ {\bar{\nu }}_\ell \) and \({\bar{B}} \rightarrow \rho \ell ^ {\bar{\nu }}_\ell \) decays as an additional ingredient, involving hadronic form factors which are determined through QCD sum rule and lattice calculations. In addition to a detailed analysis of the constraints on the New Physics contributions following from current data, we make predictions for yet unmeasured decay observables, compare them with experimental constraints and discuss the impact of CPviolating phases of the NewPhysics coefficients.
1 Introduction
In this paper, we propose a new strategy to probe NP effects by utilising leptonic B decays and the interplay with their semileptonic counterparts. Both decay classes are actually caused by the same lowenergy effective Hamiltonian. We will obtain constraints on shortdistance coefficients using the Belle result in Eq. (8) and address the question of how large the branching ratio of \(B^\rightarrow e^{\bar{\nu }}_e\) could be due to a lift of the helicity suppression through NP effects. We have addressed a similar question for the leptonic rare decays \(B^0_{s,d}\rightarrow e^+e^\), which could be enhanced to the level of the \(B^0_{s,d}\rightarrow \mu ^+\mu ^\) channels through new (pseudo)scalar interactions [16]. We shall also make predictions for various semileptonic decay ratios which will allow us to fully reveal the underlying decay dynamics, and extract \(V_{ub}\) while also allowing for NP contributions.
Contributions of NP to \(B^\rightarrow \ell ^{\bar{\nu }}_\ell \), \({\bar{B}}\rightarrow \pi \ell ^{\bar{\nu }}_\ell \) and \({\bar{B}}\rightarrow \rho \ell ^{\bar{\nu }}_\ell \) processes have also been addressed in, e.g., Refs. [23, 24, 25, 26, 27, 28].
The outline of this paper is as follows: after introducing briefly the theoretical framework in Sect. 2, we discuss the leptonic \(B^\rightarrow \ell ^{\bar{\nu }}_\ell \) decays in Sect. 3. The semileptonic \({\bar{B}}\rightarrow \rho \ell ^{\bar{\nu }}_\ell \) and \({\bar{B}}\rightarrow \pi \ell ^{\bar{\nu }}_\ell \) modes are analysed in Sect. 4, where we will also combine them with the leptonic constraints to obtain regions for shortdistance coefficients. The hadronic form factors, which are required for the study of experimental data, are discussed in Appendix A. In both Sects. 3 and 4 we will also address the impact of CPviolating phases on the regions for the shortdistance coefficients. Then, in Sect. 5 we determine \(V_{ub}\) in the presence of NP contributions. Finally, we give predictions for the not yet measured branching ratios \({\mathcal {B}}(B^\rightarrow \mu ^ \bar{\nu }_{\mu })\), \({\mathcal {B}}(\bar{B}\rightarrow \rho \tau ^ \bar{\nu }_{\tau })\), and \({\mathcal {B}}(\bar{B}\rightarrow \pi \tau ^ \bar{\nu }_{\tau })\) in Sect. 6. The conclusions are summarised in Sect. 7.
2 Theoretical framework
3 Leptonic \(B^\rightarrow \ell ^\bar{\nu }_\ell \) decays
There is a subtlety related with Eqs. (17) and (18) when allowing for physics beyond the SM: the point is that the value of \(V_{ub}\) extracted from sophisticated analyses of semileptonic B decays (for an overview, see Ref. [3]) may include NP contributions, thereby precluding us from calculating the SM branching ratio. In order to deal with this issue, our analysis will be based on the study of ratios of branching fractions.
3.1 Constraints on pseudoscalar NP coefficients from leptonic decay observables
Let us first assume that the NP shortdistance coefficients \(C^{\ell }_P\) in Eq. (20) are real, and study the constraints we obtain from the leptonic decay ratios defined in Eq. (19). For the specific determination of \(R^{\ell _1}_{\ell _2}\), we consider the tau–muon and the electron–muon pairs, i.e. \(R^{\tau }_{\mu }\) and \(R^{e}_{\mu }\), respectively.
3.2 Implications of CPviolating phases
4 Semileptonic \(\bar{B} \rightarrow \rho \ell ^ \bar{\nu }_\ell \) and \(\bar{B} \rightarrow \pi \ell ^ \bar{\nu }_\ell \) decays
We may improve the constraints on the NP shortdistance contributions if in addition to the leptonic processes described in Sect. 3 we also include semileptonic decays caused by the transition \(b\rightarrow u\ell ^\bar{\nu }_{\ell }\). The relevant decays for our analysis are \({\bar{B}}\rightarrow \rho \ell ^ {\bar{\nu }}_\ell \) and \({\bar{B}}\rightarrow \pi \ell ^ {\bar{\nu }}_\ell \). The first mode depends only on \(C^{\ell }_P\) and therefore can be considered the counterpart of the leptonic channels. On the other hand, the process \({\bar{B}}\rightarrow \pi \ell ^ {\bar{\nu }}_\ell \) is sensitive to the short distance contribution \(C^{\ell }_S\).
The expressions for semileptonic decays have a more complicated structure than those for the leptonic modes due to the hadronic form factors used to calculate the transitions \({\bar{B}}\rightarrow \rho \) and \({\bar{B}}\rightarrow \pi \). The kinematical regimes for the semileptonic decays are described in terms of \(q^2 \equiv (p_B  p_{\rho ,\pi })^2\), where \(p_B\) and \(p_{\rho ,\pi }\) are the fourmomenta of the Bmeson and the \(\rho \) or \(\pi \), respectively. For low momentum transfer, i.e. \(q^2 \in [0,q^2_{\mathrm{{max}}}]\) where \(12~\mathrm{{GeV}}^2\le q^2_{\mathrm{{max}}}\le 16~\mathrm{{GeV}}^2\), the nonperturbative hadronic form factors are estimated using QCD sum rules. For higher \(q^2\) values, lattice determinations are available. Quark models were also used for the determination of the hadronic form factors, as discussed in Ref. [28].
The calculation of the nonperturbative contributions to \(\bar{B}\rightarrow \pi \) decays is well developed. As a matter of fact, the corresponding form factors are currently known with good precision. Here we use the parameterization that extrapolates from high to low \(q^2\) values introduced originally in Ref. [32] and discussed in more detail in Appendix A. In contrast, the form factors for the \(\bar{B}\rightarrow \rho \) transitions are less precisely known, and only determinations referring independently to either the low \(q^2\) or high \(q^2\) regimes are available in the literature. Moreover, high \(q^2\) calculations are more than one decade old [33] and have large uncertainties. Later in this section, we will argue on the importance of improving these results.
Semileptonic decays are used for the exclusive determination of \(V_{ub}\), which is typically done using SM expressions. Therefore, a value of \(V_{ub}\) based on this approach may already be affected by NP contributions. Consequently, using this parameter as an input in other NP studies may lead to wrong conclusions. To avoid this problem, we propose a different method for the determination of \(V_{ub}\), which is described in more detail in Sect. 5. Our strategy is based on two key steps: we first obtain the NP shortdistance contributions \(C^{\ell }_S\) and \(C^{\ell }_P\) using only ratios of branching fractions of leptonic and semileptonic processes. Then, we substitute these results in the individual expressions for the branching fractions in order to extract the value of \(V_{ub}\).
4.1 \({\bar{B}}\rightarrow \rho \ell ^ {\bar{\nu }}_\ell \)
4.1.1 Constraints on pseudoscalar NP coefficients from \(\bar{B}\rightarrow \rho \ell ^ \bar{\nu }_{\ell }\)
Using the observable in Eq. (42) and making the assumption \(C^{e}_P=C^{\mu }_P\), we can derive further constraints on the regions shown in Fig. 1. In particular, the range for \(C_P^\mu \) following from \(\mathcal {R}^{\mu }_{\langle {e, \mu }\rangle ;\rho ~[q^2\le 12]~\mathrm {GeV}^2}\) yields the green vertical bands shown in Fig. 6. The combination with \(R_\mu ^\tau \) gives us then four allowed regions. Performing a \(\chi ^2\) fit to these two observables yields the \(1\sigma \) allowed regions given by the black contours. Since the Wilson coefficients \(C_P^\mu \) and \(C_P^e\) are correlated, we may in addition include the ratio \(R^{e}_{\mu }\) to obtain even stronger constraints. For \(C_P^e=C_P^\mu \), this observable yields the blue region in Fig. 6, selecting the right green band and excluding solutions 3 and 4 satisfying \(C_P^\mu < 0\).
4.2 \({\bar{B}}\rightarrow \pi \ell ^ {\bar{\nu }}_\ell \)
4.2.1 Constraints on (pseudo)scalar NP coefficients from \(\bar{B}\rightarrow \pi \ell ^ \bar{\nu }_{\ell }\)
4.3 Combining leptonic and semileptonic constraints
Experimental values of \(\mathcal {B}(\bar{B} \rightarrow \rho \ell ^ \bar{\nu }_\ell )\) in different \(q^2\) intervals [36]. The fourth column gives the isospin averages of the values in the second and third columns
\(\Delta q^2\) \((\mathrm {GeV}^2)\)  \(2\langle {\mathcal {B}(B^\rightarrow \rho ^0 \ell ^ \bar{\nu }_{\ell } )\rangle }_{[\ell =e, \mu ]}\)  \(\langle {\mathcal {B}(\bar{B}^0\rightarrow \rho ^+ \ell ^ \bar{\nu }_{\ell } )\rangle }_{[\ell =e, \mu ]}\)  \(\langle {\mathcal {B}(\bar{B}\rightarrow \rho \ell ^ \bar{\nu }_{\ell } )\rangle }_{[\ell =e, \mu ]}\) 

[0, 4]  \((5.54 \pm 0.92)\times 10^{5}\)  \((3.73 \pm 1.06) \times 10^{5}\)  \((4.76 \pm 0.69)\times 10^{5}\) 
[4, 8]  \((7.92 \pm 0.96)\times 10^{5}\)  \((7.18 \pm 1.16) \times 10^{5}\)  \((7.62 \pm 0.74)\times 10^{5}\) 
[8, 12]  \((6.84 \pm 0.89)\times 10^{5}\)  \((8.06 \pm 1.23) \times 10^{5}\)  \((7.26 \pm 0.72)\times 10^{5}\) 
Just as for the low\(q^2\) region, a small tension with the SM appears in the case of \(\beta =6.0\) with \(q^2=17~\mathrm {GeV}^2\). However, a more precise determination of the form factors in the high\(q^2\) regime is required in order to understand the origin of this discrepancy: it can certainly be triggered by the theoretical precision of the nonperturbative contributions. Indeed, the study presented in [33] was performed when the lattice calculations technology was in its early stages of development and an underestimation of the uncertainties cannot be discarded. A very interesting prospect would be the presence of NP; this possibility is quite exciting and is in principle allowed by the theoretical and experimental information available at the moment. In addition, an interpolation between the low and high\(q^2\) regimes for the \(B\rightarrow \rho \) transitions will allow a full use of the experimental determinations.
4.4 CP violation
Finally, we would like to study the implications of CPviolating phases once we combine the different leptonic and semileptonic constraints described at the beginning of this Section and in Sect. 3. Since the direct CP asymmetries defined in Eq. (10) would take essentially vanishing values for the (semi)leptonic decays, we follow the approach introduced in Sect. 3.2 for leptonic processes and explore the implications of new CPviolating phases in the short distance contributions, i.e. complex Wilson coefficients. Specifically, we analyse correlations between the norms and phases of the shortdistance contributions, as well as between norms of different coefficients.
Finally, we allow \(\phi ^{\mu }_P\) to change as well. Unlike the previous case, the evolution is now along the horizontal direction. By scanning \(\phi ^{\mu }_P\) and \(\phi ^{\tau }_P\) within the interval \([0^{\circ }, 180^{\circ }]\) we generate the smeared plot shown in Fig. 15. We have highlighted the steps corresponding to: \((\phi ^{\mu }_P=0^{\circ }, \phi ^{\tau }_P=0^{\circ })\), \((\phi ^{\mu }_P=0^{\circ }, \phi ^{\tau }_P=180^{\circ })\), \((\phi ^{\mu }_P=180^{\circ }, \phi ^{\tau }_P=0^{\circ })\) and \((\phi ^{\mu }_P=180^{\circ }, \phi ^{\tau }_P=180^{\circ })\).
5 Determination of \(V_{ub}\)
 1.
Perform a \(V_{ub}\)independent extraction of \(C^{\ell }_P\). This can be achieved using the ratios introduced in Sects. 3 and 4.
 2.
Substitute the ranges for \(C^{\ell }_P\) in any of the leptonic or semileptonic branching ratios available, i.e. \(\mathcal {B}(B^\rightarrow \mu ^\bar{\nu }_{\mu })\) or \(\mathcal {B}(\bar{B}\rightarrow \rho \ell ^\bar{\nu }_{\ell })\), and then solve for \(V_{ub}\).

Using the expressions for \(\mathcal {R}^{\mu }_{\langle {e, \mu }\rangle ;\rho }\) introduced in Eq. (42), we solve for \(C^{\mu }_P\). Since we are assuming universal NP contributions for electrons and muons, this ratio depends only on one single NP coefficient.

The previous step leads to the function \(C^{\mu }_P(\mathcal {R}^{\mu }_{\langle {e, \mu }\rangle ;\rho })\). There are two solutions satisfying independently \(C^{\mu }_P<0\) and \(0 < C^{\mu }_P\). Looking at Fig. 6, we see that only \(0 < C^{\mu }_P\) is consistent with all the available constraints.

Finally, we evaluate any of the individual branching fractions \(\mathcal {B}(B^\rightarrow \mu ^\bar{\nu }_{\mu })\) or \(\langle {\mathcal {B}(\bar{B}\rightarrow \rho \ell ^\bar{\nu }_{\ell })}\rangle _{[\ell =e, \mu ]}\) in the interval for \(C^{\mu }_P\) obtained above. From the resulting expression, we can determine the only unknown left: the value of \(V_{ub}\).
Up to now we have shown how it is possible to extract \(V_{ub}\) from observables involving \(C^{\mu }_P\). We can, however, incorporate also the constraints for \(C^{\tau }_P\). With this in mind, we consider \(R^{\tau }_{\mu }\) defined in Eq. (19), which depends both on \(C^{\mu }_P\) and on \(C^{\tau }_P\). We reduce the number of independent parameters by substituting \(C^{\mu }_P(\mathcal {R}^{\mu }_{\langle {e, \mu }\rangle ;\rho })\) in \(R^{\tau }_{\mu }\). The resulting expression will depend only on \(C^{\tau }_P\) and can be inverted to obtain this coefficient as a function of \(R^{\tau }_{\mu }\) and \(\mathcal {R}^{\mu }_{\langle {e, \mu }\rangle ;\rho }\), which can then be inserted into \(\mathcal {B}(B^\rightarrow \tau ^\bar{\nu }_{\tau })\) to extract \(V_{ub}\).
Following any of the two methods described above leads to consistent results. This is actually not surprising since by adding \(\mathcal {B}(B^ \rightarrow \tau ^ \bar{\nu }_\tau )\) to our set of observables we are also including an additional coefficient \(C_P^\tau \). The result will be the same if we consider ratios containing \(\langle {\mathcal {B}(\bar{B} \rightarrow \pi \ell ^ \bar{\nu }_\ell )}\rangle _{[\ell =e, \mu ]}\), which bring \(C_S^\mu \) as an extra parameter into the analysis.
 1.\(C^{e}_P\ll C^{\mu }_P\); in particular, we explore$$\begin{aligned} C^{e}_P=(1/10)C^{\mu }_P. \end{aligned}$$(73)
 2.\(C^{\mu }_P \ll C^{e}_P\); we focus on$$\begin{aligned} C^{e}_P= 10C^{\mu }_P. \end{aligned}$$(74)
 3.The 2HDM, where according to Eq. (16), we have$$\begin{aligned} C^e_P=\frac{m_e}{m_{\mu }}C^{\mu }_P, \quad \quad \quad C^{\tau }_P=\frac{m_{\tau }}{m_{\mu }}C^{\mu }_P. \end{aligned}$$(75)
 4.NP entering only through the third generation:$$\begin{aligned} C^{\tau }_P\ne 0, \quad \quad \quad C^{e}_P=C^{\mu }_P=0. \end{aligned}$$(76)
Summary of the determination of \(V_{ub}\) and the predictions for \(\mathcal {B}(B^\rightarrow e^ \bar{\nu }_e)\) and \(\langle {{{\mathcal {B}}}({\bar{B}}\rightarrow \rho \tau ^ \bar{\nu }_{\tau })}\rangle \) in the different scenarios discussed in the text
Scenario  \(V_{ub}\)  \(\mathcal {B}(B^\rightarrow e^ \bar{\nu }_e)\)  \(\langle {{{\mathcal {B}}}(\bar{B}\rightarrow \rho \tau ^ \bar{\nu }_{\tau })}\rangle \)  

\(C_P^e = C_P^\mu \)  1  \((3.31 \pm 0.32) \times 10^{3}\)  \((6.7_{6.7}^{+9.3})\times 10^{8}\)  \((7.81\pm 0.66)\times 10^{5}\) 
2  \((6.30\pm 0.45)\times 10^{5}\)  
3  –  –  –  
4  
\(C_P^e = (1/10)C_P^\mu \)  1  \((3.31 \pm 0.32) \times 10^{3}\)  \((8.0_{8.0}^{+10.1})\times 10^{10}\)  \((7.81\pm 0.66)\times 10^{5}\) 
2  \((6.30\pm 0.45)\times 10^{5}\)  
3  \((3.31 \pm 0.32) \times 10^{3}\)  \((1.76\pm 0.47)\times 10^{8}\)  \((6.30\pm 0.45)\times 10^{5}\)  
4  \((7.82\pm 0.66)\times 10^{5}\)  
\(C_P^e = 10 C_P^\mu \)  1  \((3.31 \pm 0.32) \times 10^{3}\)  \((6.6_{6.6}^{+9.2})\times 10^{6}\)  \((7.81\pm 0.66)\times 10^{5}\) 
2  \((6.29\pm 0.45)\times 10^{5}\)  
3  –  –  –  
4  
2HDM  1  \((3.31 \pm 0.32) \times 10^{3}\)  \((1.15\pm 0.25)\times 10^{11}\)  \((6.26\pm 0.45) \times 10^{5}\) 
2  \((3.31 \pm 0.32) \times 10^{3}\)  \((1.15\pm 0.25)\times 10^{11}\)  \((8.00\pm 0.74)\times 10^{5}\)  
\(C_P^e = C_P^\mu =0\)  1  \((4.85 \pm 1.03) \times 10^{3}\)  \((1.51\pm 0.64)\times 10^{11}\)  \((6.42\pm 0.45)\times 10^{5}\) 
2  \((4.85 \pm 1.03)\times 10^{3}\)  \((7.45\pm 0.66)\times 10^{5}\) 
If we consider the correlation \(C^e_P=10 C^{\mu }_P\), we obtain the right plot in Fig. 17, where the observable \(R^e_{\mu }\) selects two narrow vertical sections inside the two ellipses on the right. As for \(C^e_P=(1/10)C^{\mu }_P\), the numerical results are summarized in Table 2.
For most of these studies, the values of \(V_{ub}\) coincide with one another at the level of the significant digits. However, in the scenario where the NP enters only through the third generation, our numerical result for \(V_{ub}\) is higher in comparison with the other cases. In this respect it agrees with the inclusive \(V_{ub}\) determinations. This is certainly an interesting observation, although the uncertainty is still too large to draw any further conclusions.
6 Predictions of branching ratios
 1.With the values of \(C_P^\mu \) and \(C_P^\tau \) calculated as in Sect. 5, we determine \(R^{e}_{\tau }\). In the case of universal Wilson coefficients for the light leptons, we obtain$$\begin{aligned} R^{e}_{\tau } = (5.8_{5.8}^{+8.2})\times 10^3. \end{aligned}$$(80)
 2.In order to obtain \(\mathcal {B}(B^\rightarrow e^ \bar{\nu }_{e})\), we multiply the theoretical determination of \(R^{e}_{\tau }\) with the experimental value of \(\mathcal {B}(B^\rightarrow \tau ^ \bar{\nu }_{\tau })\) and the relevant mass factors (see Eq. 19). We employ the experimental value in Eq. (7) which yieldsConsequently, the branching ratio for the process \(B^ \rightarrow e^ \bar{\nu }_e\) could be enhanced by up to four orders of magnitude with respect to the SM value given in Eq. (6). Interestingly, our determination in Eq. (81) is only one order of magnitude below the current experimental bound in Eq. (9).$$\begin{aligned} \mathcal {B}(B^\rightarrow e^ \bar{\nu }_e) = (6.7_{6.7}^{+9.3})\times 10^{8}. \end{aligned}$$(81)
 1.Substitute the results for \(C^{\mu }_P(\mathcal {R}^{\mu }_{\langle {e, \mu }\rangle ; \rho }, \)) and \(C^{\tau }_P(\mathcal {R}^{\mu }_{\langle {e, \mu }\rangle ; \rho }, R^{\tau }_{\mu })\) obtained in Sect. 5 inside the ratioconstructed in analogy with \(\mathcal {R}^{\langle {e, \mu }\rangle ;\rho ~[0 \le q^2 \le 12]~\mathrm {GeV}^2}_{\langle {e, \mu }\rangle ;\pi }\) as given by Eq. (64). In the case of universality for the light leptons our theoretical predictions are$$\begin{aligned}&\mathcal {R}^{\tau ;\rho ~[m^2_{\tau } \le q^2 \le 12]~\mathrm {GeV}^2}_{\langle {e, \mu }\rangle ;\rho }\nonumber \\&\equiv \langle { \mathcal {B}(\bar{B}\rightarrow \rho \tau ^ \bar{\nu }_{\tau })}\rangle \Bigl ^{12~\mathrm {GeV}^2}_{m^2_{\tau }}/ \langle {\mathcal {B}(\bar{B}\rightarrow \rho \ell ^ \bar{\nu }_{\ell })}\rangle _{[\ell =e, \mu ]}, \end{aligned}$$(82)Note that we have two solutions, corresponding to the two allowed regions in Fig. 6, which have the same \(C_P^\mu \) but different values of \(C_P^\tau \).$$\begin{aligned} \mathcal {R}^{\tau ;\rho ~[m^2_{\tau } \le q^2 \le 12]~\mathrm {GeV}^2}_{\langle {e, \mu }\rangle ;\rho }= & {} 0.395 \pm 0.025, \nonumber \\ \mathcal {R}^{\tau ;\rho ~[m^2_{\tau } \le q^2 \le 12]~\mathrm {GeV}^2}_{\langle {e, \mu }\rangle ;\rho }= & {} 0.318 \pm 0.011 \end{aligned}$$(83)
 2.Multiply the theoretical determination of \(\mathcal {R}^{\tau ;\rho ~[m^2_{\tau } \le q^2 \le 12]~\mathrm {GeV}^2}_{\langle {e, \mu }\rangle ;\rho }\) by the experimental value of the branching fraction \(\langle {\mathcal {B}(\bar{B}\rightarrow \rho \ell ^ \bar{\nu }_{\ell })}\rangle _{[\ell =e, \mu ]}\). The resulting value is precisely \(\langle { \mathcal {B}(\bar{B}\rightarrow \rho \tau ^ \bar{\nu }_{\tau })}\rangle \Bigl ^{12~\mathrm {GeV}^2}_{m^2_{\tau }}\). In the universal case for light leptons, we obtain$$\begin{aligned} \langle { \mathcal {B}(\bar{B}\rightarrow \rho \tau ^ \bar{\nu }_{\tau })}\rangle \Bigl ^{12~\mathrm {GeV}^2}_{m^2_{\tau }}= & {} (7.81\pm 0.66)\times 10^{5}, \nonumber \\ \langle { \mathcal {B}(\bar{B}\rightarrow \rho \tau ^ \bar{\nu }_{\tau })}\rangle \Bigl ^{12~\mathrm {GeV}^2}_{m^2_{\tau }}= & {} (6.30\pm 0.45)\times 10^{5}. \end{aligned}$$(84)
7 Conclusions
We have presented a detailed analysis of leptonic \(B^\rightarrow \ell ^ {\bar{\nu }}_\ell \) decays and their semileptonic counterparts \({\bar{B}} \rightarrow \rho \ell ^ {\bar{\nu }}_\ell \) and \({\bar{B}} \rightarrow \pi \ell ^ {\bar{\nu }}_\ell \), aiming at tests of lepton flavour universality in processes caused by \(b\rightarrow u \ell ^ {\bar{\nu }}_\ell \) transitions. A key requirement to constrain the shortdistance coefficients of NP operators is to consider only quantities which do not depend on \(V_{ub}\). The point is that the values of this CKM parameter extracted from semileptonic decays assume the SM while we allow for NP contributions to these processes. Since the leptonic decays, which exhibit helicity suppression in the SM, play a key role in this endeavour, we focused on new (pseudo)scalar operators which may lift the helicity suppression, thereby having a potentially dramatic impact on these modes.
The \(B^\rightarrow \ell ^ {\bar{\nu }}_\ell \) decays involve actually the pseudoscalar coefficient \(C_P^\ell \). Using a recent Belle result for the \(B^\rightarrow \mu ^ {\bar{\nu }}_\mu \) branching ratio in combination with the measured \({{\mathcal {B}}}(B^\rightarrow \tau ^ {\bar{\nu }}_\tau )\), we obtained theoretically clean constraints in the \(C_P^\mu \)–\(C_P^\tau \) plane. One branch of the solutions is consistent with the SM picture within the current uncertainties. Thanks to the lift of the helicity suppression, we obtain a remarkably constrained picture despite the significant experimental uncertainty for the \(B^\rightarrow \mu ^ {\bar{\nu }}_\mu \) mode.
In order to further constrain the pseudoscalar NP coefficients, we employ the semileptonic \({\bar{B}} \rightarrow \rho \ell ^ {\bar{\nu }}_\ell \) modes which involve \(C_P^\ell \) as well. While the leptonic decays depend on the \(B^\) decay constant as the only nonperturbative parameter, the semileptonic decay requires a variety of hadronic form factors which can be determined by means of QCD sum rule and lattice calculations. Using results available in the literature, we have performed a comprehensive study of the available data. Interestingly, to the best of our knowledge, measurements of differential decay rates of \({\bar{B}} \rightarrow \rho \ell ^ {\bar{\nu }}_\ell \) for \(\ell =\mu \) and \(\ell =e\) are not available. It would be important for probing violations of lepton flavour universality if experimental collaborations would report such analyses. We obtain a picture which is consistent with the SM at the \(1\,\sigma \) level, taking both experimental and theoretical uncertainties into account.
The general lowenergy effective Hamiltonian including NP effects has also a scalar operator which does not contribute to the \(B^\rightarrow \ell ^ {\bar{\nu }}_\ell \) and \({\bar{B}} \rightarrow \rho \ell ^ {\bar{\nu }}_\ell \) modes but has an impact on the semileptonic \({\bar{B}} \rightarrow \pi \ell ^ {\bar{\nu }}_\ell \) channels. A comment similar to the one for the \({\bar{B}} \rightarrow \rho \ell ^ {\bar{\nu }}_\ell \) modes applies also in this case, i.e. it would be very useful to have experimental results for electrons and muons in the final states. Making a simultaneous analysis of the leptonic \(B^\rightarrow \mu ^ {\bar{\nu }}_\mu \) and semileptonic \(\langle {\bar{B}} \rightarrow \pi \ell ^ {\bar{\nu }}_\ell \rangle _{[\ell =e,\mu ]}\) decays, we derived a constraint in the \(C_P^\mu \)–\(C_S^\mu \) plane, showing one solution in agreement with the SM. Yet another constraint follows from the ratio of the differential \(\langle {\bar{B}} \rightarrow \rho \ell ^ {\bar{\nu }}_\ell \rangle _{[\ell =e,\mu ]}\) and \(\langle {\bar{B}} \rightarrow \pi \ell ^ {\bar{\nu }}_\ell \rangle _{[\ell =e,\mu ]}\) rates, which we discussed for various values of the momentum transfer \(q^2\). Interestingly, for certain values, we obtain tension with the SM at the \(1\sigma \) level which will be interesting to monitor in the future. It would be very desirable to have more sophisticated nonperturbative analyses of the form factors available, in particular for the semileptonic \({\bar{B}} \rightarrow \rho \) transitions. In our study, we have also explored the impact of CPviolating phases of the NP coefficients.
Using the NP constraints, we could make corresponding predictions for decay observables which have not yet been measured. In particular, we find a potentially huge enhancement of the \(B^\rightarrow e^ {\bar{\nu }}_e\) branching ratio, lifting it up to the regime of the experimental upper bound. Moreover, we determined the CKM element \(V_{ub}\), obtaining values in agreement with other analyses in the literature although having larger uncertainties.
The method which we proposed and explored for decays caused by \(b\rightarrow u \ell ^ {\bar{\nu }}_\ell \) quarklevel processes can actually also be implemented for exclusive \({\bar{B}}\) decays originating from \(b\rightarrow c \ell ^ {\bar{\nu }}_\ell \) modes. In this case, the leptonic decay \(B_c^\rightarrow \ell ^ {\bar{\nu }}_\ell \) channels are key ingredients. Unfortunately, these decays are very challenging from an experimental point of view and no measurements are currently available, despite the fact that many \(B_c\) mesons are produced at the LHC. Hopefully, in the future, innovative ways will be found to get a handle on the leptonic \(B_c\) modes.
It will be very interesting to apply the strategy presented in this paper in the future highprecision era of B physics, thereby shedding more light on contributions of new (pseudo)scalar operators and probing lepton flavour universality in yet another territory of the flavour physics landscape.
Notes
Acknowledgements
This research has been supported by the Netherlands Foundation for Fundamental Research of Matter (FOM) Programme 156, “Higgs as Probe and Portal”, and by the National Organisation for Scientific Research (NWO). G.B. acknowledges the support through a Volkert van der Willigen grant of the University of Amsterdam.
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