Probing the structure of the B meson with B\to\ell\ell\ell'ν
IINT-PUB-21-003
Probing the structure of the B meson with B → (cid:96)(cid:96)(cid:96) (cid:48) ν Aoife Bharucha , Bharti Kindra , Namit Mahajan Aix Marseille Univ, Universit´e de Toulon, CNRS, CPT, Marseille, France Indian Institute of Technology, Gandhinagar, India Physical Research Laboratory, Ahmedabad, India
Email: [email protected],[email protected],[email protected]
Abstract
We consider the decay B → (cid:96)(cid:96)(cid:96) (cid:48) ν , taking into account the leading 1 /m b and q corrections calculated in the QCD factorization framework as well as the softcorrections calculated employing dispersion relations and quark-hadron duality. Weextend the existing results for the radiative decay B → γ(cid:96)ν to the case of non-zero(but small) q , the invariant mass squared of the dilepton pair (cid:96) + (cid:96) − . This restrictsus to the case (cid:96) (cid:54) = (cid:96) (cid:48) as otherwise the same sign (cid:96) and (cid:96) (cid:48) cannot be distinguished. Wefurther study the sensitivity of the results to the leading moment of the B -mesondistribution amplitude and discuss the potential to extract this quantity at LHCband the Belle II experiment. Our understanding of the structure of the B meson lacks the precision needed to provideaccurate predictions of B -meson decays beyond naive factorization. The quantity mostvitally required for making these predictions, whether using QCD factorization or light-cone sum rules (LCSR), is the leading moment of the B -meson distribution amplitude, λ B , defined by (see e.g. Ref. [1]) λ − B ( µ ) = (cid:90) ∞ dkk φ + B ( k, µ ) , (1)where φ + B ( k, µ ) is the distribution amplitude of the B meson, depending on the scale µ . The theoretical uncertainty on λ B is large with estimates ranging from 200 MeVobtained using non-leptonic decays [2, 3] to 460 ±
110 MeV obtained from QCD sumrules [1].It was proposed in Refs. [4–6] to use the charged current decay of the B + meson, withan extra photon in the final state i.e., B + → (cid:96) + νγ to probe λ B in the kinematic limitwhen the energy of the photon is large in comparison to the scale of strong interactions(Λ QCD ). Note that the branching ratio is larger than for the purely leptonic final state,1 a r X i v : . [ h e p - ph ] F e b s the emission of the photon lifts the helicity suppression. On the experimental side,the most stringent limit from the BaBar collaboration was [7] BR ( B + → (cid:96) + νγ ) < . × − , (2)at 90% confidence level (C.L.), resulting in the lower limit λ B >
300 MeV [7]. The Bellecollaboration has provided an upper limit on the branching ratios for the electron andmuon final states [8]: BR ( B + → e + νγ ) < . × − , (3) BR ( B + → µ + νγ ) < . × − , (4)also at 90% C.L., providing a lower limit of λ B >
238 MeV. Note that the fact thatthe Belle lower limit lies below the BaBar limit, despite the limit on the branchingratio being more stringent, is due to both to a different choice of input parameters(particularly m b and | V ub | ) and the fact that in Ref. [8] the expression for the partialbranching fraction include state-of-the-art next-to-leading order corrections from Ref. [9]and the soft corrections calculated in Ref. [10].Updating this result is a priority at Belle II, and projections look very promising [11,12]. However, the measurement of B → γ(cid:96)ν at LHCb is challenging as the low energyphoton in combination with the neutrino in the final state provide complications for thetrigger. On the other hand, if the photon was off-shell and emitted a lepton pair, thenfor the final state consisting of three charged leptons and a neutrino the analysis wouldbe feasible. In fact, an upper limit at 95% C.L. has already been provided by LHCb inthe region where the lowest of the muon pair mass combination is below 980 MeV [13], BR ( B + → µ + µ − µ + ν ) < . × − . (5)In light of this measurement, it is of interest to assess whether B → (cid:96)(cid:96)(cid:96) (cid:48) ν could be a sourceof complementary information about the nature of the B -meson distribution amplitude.On the theoretical side the extension of the full existing formalism for B → γ(cid:96)ν to thiscase has not yet been attempted, although a first prediction of the branching ratio canbe found in Ref. [14, 15], based on the vector meson dominance approach. Here thebranching ratio was predicted to be ∼ × − (above the experimental limit fromLHCb) without an analysis of the associated uncertainties.Our aim is therefore to provide predictions for the differential decay spectrum andthe partial branching fraction for B → (cid:96)(cid:96)(cid:96) (cid:48) ν , which, with more data, could lead tothe measurement of the first moment of the B -meson distribution amplitude at LHCb.This would serve as an important cross-check for the results from Belle II. Here it isimportant to note that the framework in which our calculation is performed means thatthe results are only valid at low q , where q is the dilepton mass squared for the (cid:96) + (cid:96) − pair originating from the virtual photon. Since when (cid:96) = (cid:96) (cid:48) , this theoretical definitionof q cannot be measured experimentally, due to the ambiguity between the same signleptons, such that the existing limit from LHCb cannot be converted into a lower limiton λ B . We will therefore concentrate on the case (cid:96) (cid:54) = (cid:96) (cid:48) .In the following section we will introduce our theoretical framework, expressing theamplitude for the decay in terms of the form factors, and providing details of the variouscontributions to these form factors that we include. In Sec. 3, after having discussed the2inematics necessary to calculate the branching ratio and the parameters adopted, wewill present the results of our numerical analysis. Finally, a discussion of these resultsand our conclusions can be found in Sec. 4. The amplitude for the process B + ( p B ) → (cid:96) + ( q ) (cid:96) − ( q ) (cid:96) (cid:48) + ( p ) ν ( p ) can be written as iA = G F V ub √ ie q (¯ u (cid:96) γ µ v (cid:96) )[(¯ u ν γ ρ P L v (cid:96) (cid:48) ) T µρ − i (¯ u ν γ µ P L v (cid:96) (cid:48) ) f B ] (6)where G F is the Fermi constant, V ub is the CKM matrix element and e is the electriccharge. Here the first term describes the emission of the virtual photon from the B meson and the second term from the lepton. At leading order the second term cantrivially be written in terms of the B -meson decay constant f B , whereas the first termis more complicated and requires further study. On writing down this term in the mostgeneral form as possible, imposing the conservation of the electromagnetic current andapplying the equation of motion (see Appendix A), the tensor T µρ defined in Eq. (6)reduces to, T µρ = iF A ( g µρ ( p · q ) − p µ q ρ ) + F V (cid:15) ρµλσ p λ q σ , (7)where p = p + p and q = q + q . Therefore this contribution to the decay amplitudecan be expressed in terms of two form factors F V and F A , as in the case of B → γ(cid:96)ν .The amplitude can then be expressed as iA = G F V ub √ ie q (¯ (cid:96)γ µ (cid:96) )(¯ ν Γ ρ (cid:96) (cid:48) ) (cid:104) iF A ( g µρ p.q − p µ q ρ ) + F V (cid:15) ρµλσ p λ q σ (cid:105) , (8)Making predictions for these decays therefore comes down to obtaining expressions forthese form factors. Note that the mass of the leptons in the final state has been neglected,which results in the exact cancellation of the contact term with the contribution ofphoton emission from charged lepton. However, in case of tau leptons in the final state,additional form factors may appear and would need to be calculated. The form factors defined in Eq. (7) are functions of q . They have been calculated forthe case B → γ(cid:96)ν (i.e. q = 0) and therefore these results need to be extended. Wewill therefore first review the results for q = 0, before presenting our calculation atnon-zero q . 3 igure 1: (a): photon emission from u quark, (b): photon emission from b quark B → γ(cid:96)ν In the case of B → γ(cid:96)ν , from Ref. [16] we have F V = Q u f B m B E γ λ B ( µ ) R ( E γ µ ) + ξ ( E γ ) + ∆ ξ ( E γ ) F A = Q u f B m B E γ λ B ( µ ) R ( E γ , µ ) + ξ ( E γ ) − ∆ ξ ( E γ ) , (9)where the photon energy E γ = ( m B − p ) / (2 m B ), Q u is the charge of the u quarkand m B is the mass of the B meson. The first term is the leading contribution in theheavy quark expansion, which depends inversely on the quantity of interest, λ B , thefirst inverse moment of the B -meson distribution amplitude defined in Eq. (1). Notehere that the factor R ( E γ , µ ) contains the radiative corrections calculated in Ref. [9],and at tree level is equal to 1. The second term encodes the symmetry conservingsoft and O (1 /m b , /E γ ) corrections to the form factors which may be sizeable. Thesoft corrections, arising when the quark propagator between the electromagnetic andweak vertices becomes soft, have been calculated making use of dispersion relationsand quark-hadron duality, up to next-to-leading order (NLO) at leading twist and upto twist-6 at leading order [16–18]. The last term contains the symmetry breakingcorrections, which arise at O (1 /m b , /E γ ) in QCD factorization, as well as receivingcontributions from the soft corrections at twist-3 to 6 [16–18]. B → (cid:96)(cid:96)(cid:96) (cid:48) ν at leading order in α s We now wish to extend these calculations to the case of non-zero q . In B → γ(cid:96)ν ,factorization holds as long as the photon is energetic in the rest frame of the B meson,implying that the photon energy is ∼ m b . Similarly in our case a hard collinear virtualphoton coupling to the spectator u quark can be integrated out systematically, and thetotal amplitude is found to be factorized into a hard scattering kernel and a soft part,given by the distribution amplitude. In the same setting, the resulting contributionfrom the b quark is found to be power suppressed. A quick analysis of the diagramsfor the case of a soft emitted photon reveals that there are possibly power enhancedcontributions, coming both from the u -quark and b -quark legs. However, these are notexpected to be factorizable and will not be considered any further. We thus restrictour attention to the case of a hard collinear photon (defined below) for which the Note that the first correct calculation of this decay within the QCD factorization framework wasperformed in Ref. [27]. q to be not too far from zero. Thus at theleading order, only the left-hand diagram in Fig. 1 contributes to the form factors.For convenience, we choose to work in light-cone coordinates ( l ≡ ( l + , l − , l ⊥ )) where, l ± = l ± l , l ⊥ = ( l , l ) . (10)As the spectator quark is soft, its momentum will scale as k = ( k + , k − , k ⊥ ) ∼ ( λ, λ, λ ),while the momentum of the virtual photon scales as q = ( q + , q − , q ⊥ ) ∼ ( λ, , λ / ). Thisimplies that, ( q − k ) ∼ q − q − k + . . . (11)where we neglect terms that are suppressed by higher powers of λ , such that thepropagator can be expressed as /q − /k ( q − k ) = q − /n + q − q − k + − (cid:18) k + /n − q − q − k + + k − /n + q − q − k + + /k ⊥ q − q − k + (cid:19) . (12)Here the first term provides the leading contribution and the terms in brackets are sup-pressed by λ . Expressing the four-momentum of the virtual photon as q µ = ( E, q ⊥ , − q ),the components in light-cone coordinates, ( q + , q − , q ⊥ ), can be expressed as q + = 12 ( E − q ) q − = 12 ( E + q ) . (13)When the photon is hard-collinear, q − ∼ E at leading order. Thus, 2 q − k + (cid:39) E k + + O (1 /E ). Using this and following Ref. [10], the form factor at leading order in the heavyquark and the perturbative expansion, arising from the first term in Eq. (12), reads, F B → γ ∗ ( q , p ) = Q u f B (cid:90) ∞ dk + φ + B ( k + )2 E k + − q − i(cid:15) . (14)We now wish to include the soft contribution, i.e. the non-perturbative contribution tothe form factors which is not accesible via QCD factorization, calculated using dispersionrelations and quark-hadron duality in Ref. [1,16] using a technique similar to one appliedto the γ ∗ γπ form factor [19]. The idea is to relate via a dispersion relation the formfactor at the desired q to that at q = − m B Λ QCD , where the result can easily becalculated perturbatively to arbitrary precision in the expansion in 1 /m b , q , 1 /E . Thisrequires an assumption to be made about the hadronic spectral density as a function of q . Following Refs. [10, 16], one easily obtains the result for the two form factors F V and F A at leading order in the twist and perturbative expansion, F sym V/A = Q u f B (cid:18) (cid:90) ∞ s E dk + φ + B ( k + )2 E k + − q − i(cid:15) + (cid:90) s E dk + e − (2 Ek + − m ρ ) /M m ρ − q − i(cid:15) φ + B ( k + ) (cid:19) . (15)where s is the continuum threshold, m ρ is the mass of ρ meson, and M is the Borelparameter. Note that s corresponds to the value of q above which quark-hadronduality is expected to hold. The Borel parameter enters the result due to fact thata Borel transformation has been performed in order to reduce the sensitivity to this5ssumption. Note that here, on comparing Eqs. (14) and (15) the soft contribution atleading order to the form factor can be extracted, ξ soft = Q u f B (cid:90) s E dk + (cid:32) e − (2 Ek + − m ρ ) /M m ρ − q − i(cid:15) − E k + − q − i(cid:15) (cid:33) φ + B ( k + ) . (16) B → (cid:96)(cid:96)(cid:96) (cid:48) ν at NLL Comparing Eq. (15) to Eq. (9) we observe two differences. The first is that q nowappears in the denominator. The second is that we have not yet included the factor R ( E γ , µ ) containing the NLL corrections. The radiative corrections for B → (cid:96)νγ atNLL are significant and were seen in Refs. [9, 16] to affect the form factors by 20 − R ( E γ , µ ) introduced in Ref. via [9] can be adapted to our case, R ( p , q , µ ) = C ( p , q , µ h ) K − ( µ h ) U ( p , q , µ h , µ h , µ ) J ( p , q , µ ) . (17)Here C ( p , q , µ h ) is obtained from matching the QCD heavy-to-light current to thecorresponding SCET current at the hard scale µ h . The result at NLO was calculatedin Ref. [6, 20, 21] and can also be found in Ref. [9], and is applicable for both B → γ(cid:96)ν and B → (cid:96)(cid:96)(cid:96) (cid:48) ν . The same goes for the factor K ( µ h ), which accounts for the conversionfrom the static B -meson decay constant in the SCET current to the standard definitionin QCD, f B , used in our analysis, more details and the expression can be found inRef. [9]. Letting aside momentarily the renormalization group evolution (RGE) factor U ( p , q , µ h , µ h , µ ), let us first discuss the factor J ( p , q , µ ). This accounts for thehard-collinear radiative corrections, and was calculated for B → γ(cid:96)ν in Refs [6, 20]. For B → (cid:96)(cid:96)(cid:96) (cid:48) ν where q (cid:54) = 0 we can adopt the expression given in Ref. [17].Returning to the RGE factor U ( p , q , µ h , µ h , µ ), we see that for any choice of thescales µ h , µ h and µ , some large logarithms arising in the expressions for C ( p , q , µ h ), K ( µ h ) and J ( p , q , µ ) will remain. Therefore it makes sense to resum these logarithmsto all orders by solving a renormalization group equation [6]. This resummation resultsin the factor U ( p , q , µ h , µ h , µ ), which can be found in Ref. [9]. This factor dependson three scales: the hard scales µ h and µ h are taken to be 2 E and m b respectively,while the hard-collinear scale µ is set to ( m b Λ QCD ) / .Putting this together we find the result at leading order in the heavy quark expansion,and NLL in the perturbative expansion to be F NLL
V/A = Q u m B f B E λ B C ( p , q , µ h ) K − ( µ h ) U ( p , q , µ h , µ h , µ ) J ( p , q , µ ) , (18)where we remind the reader that these are symmetric contributions. As mentionedearlier, including these NLL corrections has the consequence that the expression forsoft contribution to the form factors given in Eq. (15) is no longer valid. This can beunderstood in terms of the following relation [9], ξ soft = 1 π (cid:90) s d s (cid:32) e − ( s − m ρ ) /M m ρ − q − iε − s − q − iε (cid:33) Im F QCD ( E, s ) . (19)6dopting F QCD ( E, s ) = F NLL
V/A from Eq. (18), setting ω (cid:48) = s/ (2 E ), and with the help ofAppendix A of Ref. [17] we therefore obtain ξ NLLsoft = Q u m B f B C ( p , q , µ h ) K − ( µ h ) U ( p , q , µ h , µ h , µ ) (cid:90) s E d ω (cid:48) (cid:32) e − (2 E ω (cid:48) − m ρ ) /M m ρ − q − iε − E ω (cid:48) − q − iε (cid:33) φ eff+ ( ω (cid:48) , µ ) (20)where an expression for φ eff+ ( ω (cid:48) , µ ) can be found in Ref. [16]. Note that, as was observedin Ref. [16] for the radiative mode, we find that on including NLL contributions theseparation between the hard and soft parts, F NLL
V/A and ξ NLLsoft respectively, becomescomplicated, and henceforth we will consider the finite quantity F NLL
V/A + ξ NLLsoft = Q u m B f B C ( p , q , µ h ) K − ( µ h ) U ( p , q , µ h , µ h , µ ) (cid:32)(cid:90) s E d ω (cid:48) e − (2 E ω (cid:48) − m ρ ) /M m ρ − q − iε φ eff+ ( ω (cid:48) , µ ) − (cid:90) ∞ s E d ω (cid:48) φ eff+ ( ω (cid:48) , µ )2 E ω (cid:48) − q − iε (cid:33) . (21) Having obtained the expression for the symmetry-conserving contributions to the formfactors, we now consider the symmetry-breaking correction terms. Considering theleading power suppressed term in the u -quark propagator, we find∆ F ( u ) V = − ∆ F ( u ) A = Q u f B (cid:90) ∞ dk + φ + B E (cid:18) q E k + − q + 1 (cid:19) = 14 E (cid:16) Q u f B + q F sym V/A (cid:17) (22)The contribution arising due to the virtual photon emission from the b -quark leg is alsopower suppressed and has been neglected while computing the symmetry preserving,leading order contribution to the form factors. Similar to the u -quark contribution, theemission from the b -quark leg yields symmetry breaking correction which is convenientlywritten as ∆ F ( b ) V = − ∆ F ( b ) A = Q b f B (cid:90) ∞ dk + φ + B m b E (cid:18) q E k + − q + 1 (cid:19) = Q b m b E (cid:18) f B + q Q u F sym V/A (cid:19) (23)
Combining Eqs. (21), (22) and (23), we obtain the final results for our form factors forthe decay B → (cid:96)(cid:96)(cid:96) (cid:48) ν , F V = F NLL V + ξ NLLsoft + ∆ F ( u ) V + ∆ F ( b ) V F A = F NLL A + ξ NLLsoft + ∆ F ( u ) A + ∆ F ( b ) A , (24)In order to compare the different contributions to the form factors we plot the individualterms as a function of q in Fig. 2 for p = 10 GeV . The dotted blue curve shows the7 .2 0.4 0.6 0.8 1.0 - q ( GeV ) F NLLV / A F sym V / A
10 × F ( u ) V
10 × F ( b ) V + ξ NLL soft
Figure 2:
Different contributions to form factors are shown as a function of q at a fixedvalue of p = 10 GeV . The dotted (solid) blue curve shows the symmetric leading order (NLL)contribution F sym V/A ( F NLL
V/A + ξ NLLsoft ). The red and green dotted curves show the symmetry breakingcontributions of u quark (∆ F ( u ) V ) and b quark (∆ F ( b ) V ) respectively, scaled by a factor of 10. form factors at leading order F sym V/A , as defined in Eq. (15). The solid blue curve showsthe form factors after including the radiative and soft corrections, i.e. F NLL
V/A + ξ NLLsoft asgiven in Eq. (21). Note that the contribution of the symmetry-breaking terms is at most10% of the symmetry-preserving terms, and therefore we have scaled them by a factor of10 in the plot. The red and green dotted curves show these scaled symmetry breakingterms, ∆ F ( u ) V from Eq. (22) and ∆ F ( b ) V from Eq. (23) respectively. The contributionof photon emission from the b quark is always negative due to the negative charge ofthe b quark. In Fig. 2 we see that the contribution of ∆ F ( b ) V is highly suppressed incomparison to ∆ F ( u ) V near q = 1 GeV (∆ F ( u ) V (1 GeV ) ∼ F ( b ) V (1 GeV )) but thetwo quantities are comparable at low q . We further observe that radiative correctionsreduce the form factor at leading order by a factor of 30 − q range.The central values (for λ B = 350 MeV) of the symmetry-preserving and symmetry-breaking terms contributions to the form factors, as well as the uncertainty bands, areshown in Fig. 3. The input parameters and the related uncertainties used to calculatethis uncertainty band are given in Table 1. On comparing the uncertainty due to thehard-collinear factorization scale µ and the total uncertainty, it is clear that the scaleprovides the dominant contribution to the uncertainty on the form factors. We furthershow | F A ± F V | / λ B = 200 and 500 MeV, and observe a strong dependence of theform factors on λ B . This is in accordance with the results for the branching ratio, whichwill be discussed in detail in Sec. 3.Note that there are several contributions which have been calculated in the state-of-the-art B → γ(cid:96)ν analysis which have been neglected here, namely the O ( α s ) and thehigher twist contributions. These contributions were seen to be relatively small, we willcomment more on the uncertainty due to missing higher-twist and higher O ( α s ) termsin the following section. 8arameter Value Ref. Parameter Value Ref. m B .
28 GeV [22] f B . ± . | V ub | excl (3 . ± . × − [22] G F . × − GeV − [22] m µ .
105 GeV [22] τ B (1 . ± . × − s [22] M ρ .
775 GeV [22] m e . × − GeV [22] α em /
137 [22] λ B [200 − s . ± . [16] M . ± .
25 GeV [16] Table 1:
Numerical values of parameters adopted in our analysis
In this section, we will first outline the kinematics required to describe the differentialdecay distribution of the four leptonic decay of charged B meson. This will be followedby a discussion of the numerical parameters implemented in our analysis. We will thenpresent the results, where numerical predictions for the decay process under study areprovided in specific q bins, along with a thorough analysis of the uncertainties. We follow the definitions in [23] for the kinematics. In order to examine the partialdecay rate of B + ( p B ) → (cid:96) + ( q ) (cid:96) − ( q ) (cid:96) (cid:48) + ( p ) ν ( p ), it is useful to introduce the followingcombinations of the final state particles’ four-momenta: q = q + q ; Q = q − q p = p + p P = p − p . (25)This four-body partial decay rate can then be described via five independent variables: • the effective mass squared, p and q , of the (cid:96) (cid:48) ν and (cid:96) + (cid:96) − system respectively, • the angles θ γ of the (cid:96) + in the (cid:96) + (cid:96) − center-of-mass system with respect to the (cid:96) + (cid:96) − line of flight in B rest frame, and θ W of the (cid:96) (cid:48) in the (cid:96) (cid:48) ν center-of-mass systemwith respect to the (cid:96) (cid:48) ν line of flight in B rest frame, wherecos θ γ = − (cid:126)Q . (cid:126)p | (cid:126)Q | | (cid:126)p | , and cos θ W = − (cid:126)q . (cid:126)P | (cid:126)q | | (cid:126)P | , (26) • the angle φ between the planes of the two lepton pairs,sin φ = ( (cid:126)q × (cid:126)P ) × ( (cid:126)p × (cid:126)Q ) | (cid:126)q × (cid:126)P | | (cid:126)p × (cid:126)Q | . (27)In terms of these five variables, the partial decay distribution is given by, d Γ = πλ ( m B , q , p ) / (4 π ) m B | A | d Φ (28)9here d Φ denotes an element of the four-body phase space, and is defined via d Φ = dq dp d cos θ γ d cos θ W dφ, (29) A is the amplitude of the process defined in Eq. (8) and the K¨all´en function is given by λ ( a, b, c ) = a + b + c − a b + b c + c a ) . (30)Note that the decay distribution in Eq. (28) corresponds to the case when (cid:96) (cid:54) = (cid:96) (cid:48) . If theleptons in the final state are all of the same flavour, there is an additional contributionto the amplitude due to the possible exchange of same-sign leptons. The expression forthe partial decay rate given in Eq. (28), is then modified as A → M ≡ A − A (cid:48) , resultingin, |M| = 12 (cid:104) | A | d Φ + (cid:12)(cid:12) A (cid:48) (cid:12)(cid:12) d Φ (cid:48) − ( AA (cid:48)† + A † A (cid:48) ) d Φ (cid:105) , (31)where A (cid:48) and Φ (cid:48) are obtained from A and Φ respectively by substituting p ↔ k .To obtain the results for the binned branching ratios, we first integrate over allvariables except q in the ranges m (cid:96) (cid:48) ≤ p ≤ ( m B − (cid:112) q ) ≤ θ γ , θ W ≤ π, ≤ φ ≤ π. (32)While the physical range of q is given by 4 m (cid:96) ≤ q ≤ ( m B − m (cid:96) (cid:48) ) , since the factor-ization is only valid for low q , we will present binned branching ratios in the q bins,[4 m µ , ] and [0 . ,
1] GeV . For the case where (cid:96) = e , we additionally considerthe bin [0 . ,
1] GeV where the lower limit is chosen to avoid the very low detectorefficiency below 0 .
05 GeV [24, 25].
Before coming to the results, a discussion of our choices for the numerical inputparameters is in order. The critical hadronic parameters for our analysis include thelight-cone distribution amplitude φ + B , the continuum threshold s , Borel Parameter M ,and the decay constant of B meson f B . In addition, the hard-collinear scale µ and theCKM matrix element V ub play an important role. Starting with φ + B , there exist severalpossible choices for the parametrisation in the literature (see for example: [10, 26, 27]).We choose the form of B -meson distribution amplitude (DA) to be (see e.g. Ref. [10]), φ + B ( k, µ ) = kλ B ( µ ) e − k/λ B ( µ ) (33)where, λ B is the first inverse moment of the B -meson DA defined in Eq. (1). The scaledependence of the B -meson LCDA is given in [28]. It is to be noted that at the leadingorder, only φ + B contributes and one does not have to worry about the other B -mesonDAs, namely φ − B , which does appear in the definition of the matrix element of thenon-local quark-quark operator sandwiched between the B -meson state and the vacuumwhich defines the DAs (see for example [4, 27]). This means that our results depend onthe choice of λ B , for which we adopt the range given in Table 1, following Ref. [16].For the continuum threshold and the Borel parameter we take the standard valuesas advocated in Refs. [10, 16]. The final hadronic parameter is the B -meson decay10 B = 0.5 GeV λ B = 0.2 GeV λ B = 0.35 GeV - q ( GeV ) | F A + F V | / q ( GeV ) | F A - F V | / λ B = 0.5 GeV λ B = 0.2 GeV λ B = 0.35 GeV
Figure 3: | F V + F A | / | F A − F V | / q for p = 10 GeV .The black curve corresponds to central values of the input parameters, the dotted blue curvesshow the variation due to the hard-collinear scale, µ varied between [1 −
2] GeV, the solid bluecurves show the variation due to the input parameters: m b , s , M , and f B , where the errorshave been added in quadrature. The green and red curves correspond to λ B = 0.2 and 0.5 GeVrespectively. constant f B , for which we adopt the average found in Ref. [22]. V ub is responsible forthe dominant contribution to the uncertainty after λ B . As this is an exclusive decay, wefeel that it is appropriate to make use of the exclusive average as an input parameter, asfound in Ref. [22]. While the b -quark mass does not have a large impact on our results,we choose to use the pole mass as input, adopting conservative errors 4 . ± . We will now present our predictions for the branching ratios for B → (cid:96)(cid:96)(cid:96) (cid:48) ν in the q bins: [4 m µ , ] and [0 . ,
1] GeV as well as [0 . ,
1] GeV when (cid:96) = e . Theseresults can be found in Table 2, where the final states considered are µµµν and eeµν . Itis evident that the effect of interference for the case (cid:96) = (cid:96) (cid:48) enhances the branching ratioby ∼ λ B = 350 MeV):10 BR ( B → eeµν ) (cid:12)(cid:12) [ 0 . , = 0 .
411 + (cid:18) +0 . − . (cid:19) µ + (cid:18) +0 . − . (cid:19) | V ub | + (cid:18) +0 . − . (cid:19) m b + (cid:18) +0 . − . (cid:19) M + (cid:18) +0 . − . (cid:19) s + (cid:18) +0 . − . (cid:19) f B . (34)We see that the largest contribution to the uncertainty, of around 20 − µ . This is in contrast with the radiative B + decay where the dependence on the scale µ nearly vanishes at NNLO. Another majorsource of uncertainty is the CKM matrix element | V ub | , contributing at the 20% level.11ode q bin (GeV ) BR × (10 ) B → µµµν [4 m µ ,
1] 0 . +0 . − . [0 . ,
1] 0 . +0 . − . B → eeµν [0 . ,
1] 0 . +0 . − . [4 m µ ,
1] 0 . +0 . − . [0 . ,
1] 0 . +0 . − . Table 2:
Branching ratios of B → µµµν and B → eeµν Since | V ub | enters as an overall factor in the branching ratio, the resulting uncertainty isindependent of the phase space. The mass of the b -quark m b , the decay constant of the B meson f B , and the Borel parameter M each provide an uncertainty of ∼ s results in an error of only 1 − BR ( B → eeeν ) (cid:12)(cid:12) [ q ,q ] = BR ( B → µµµν ) (cid:12)(cid:12) [ q ,q ] BR ( B → µµeν ) (cid:12)(cid:12) [ q ,q ] = BR ( B → eeµν ) (cid:12)(cid:12) [ q ,q ] (35)In this study, we found that the relation holds upto the accuracy of the results given inTable 2. Hence, we give predictions for two decay processes, B → eeµν and B → µµµν .The branching ratios of other processes follow from Eq.(35).In the above discussion of the uncertainties, we have not yet mentioned the inputparameter λ B . The reason is that this is the very parameter we propose to measure orconstrain via measurements of the branching ratios give in Table 2. Such a measurementrelies crucially on the dependence of the integrated branching ratio for B → (cid:96)(cid:96)(cid:96) (cid:48) ν inthe q bin [4 m µ , ] on λ B , we show the case of B → µµeν in Fig. 4. Here onesees the branching ratio in the bin [4 m µ , ] as a function of λ B , where the centralvalue is shown in black, the total uncertainty is indicated by the solid blue lines and theuncertainty band due to the hard-collinear scale is shown by the dotted blue lines. Wefurther show the lower bound λ B >
238 GeV obtained by Belle mentioned in Sec. 3 [8].There is clearly a strong dependence of this binned branching ratio on λ B , far beyond theuncertainties arising from the remaining parameters. The final question to be answeredis therefore whether the experiments LHCb and Belle II could possibly measure thesebinned branching ratios, and if so with what accuracy.Here we stress again that the theoretical definition of q , i.e. the dilepton masssquared of the leptons originating from the virtual photon, can only be determinedexperimentally for the case (cid:96) (cid:54) = (cid:96) (cid:48) . Therefore the measurement of the integratedbranching ratio in the q bin we advocate can only be performed for this case. Asmentioned in Sec. 1, so far the only limit on B → (cid:96)(cid:96)(cid:96) (cid:48) ν decays available is from theLHCb experiment for the case (cid:96) = (cid:96) (cid:48) = µ , where with 4.7 fb − they obtain an upper12 .20 0.25 0.30 0.35 0.40 0.45 0.5005. × - × - × - × - × - λ B ( GeV ) BR ( B → μμ e ν ) (cid:1) m μ , (cid:2) B e ll e Figure 4:
Integrated branching ratio of B → µµeν in the q bin [4 m µ , ] as a function of λ B . The black curve shows the central value, dotted blue curves show the variation due to thehard-collinear scale, µ varied between [1 −
2] GeV, solid blue curves show the variation due toall input parameters, where the errors have been added in quadrature. The red line correspondsto the lower limit on λ B at 90% C.L. measured by Belle [8]. limit on the branching ratio of 1 . × − at 95% C.L., in the region where the lowestof the two µ + µ − mass combinations is below 0.98 GeV. As our formalism is only validfor low q , we cannot compare our results to this prediction. The sensitivity of LHCbfor the case (cid:96) (cid:54) = (cid:96) (cid:48) has not yet been studied, but given the existing limit for (cid:96) = (cid:96) (cid:48) = µ ,and taking a conservative guess that the yield would diminish by a factor of 3-4, theprospects for this channel with the the 50 ab − expected by the end of Upgrade I, letalone the 300 ab − at the end of Upgrade II, look very promising. For Belle II, themeasurement of B → γ(cid:96)ν would probably provide a more precise measurement of λ B ,given the branching ratio is O (10) times larger [12]. With the full Belle II dataset of50 ab − expected by 2025 from Belle II, a factor O (10) reduction in the statisticaluncertainty should be possible, more details can be found in Ref. [12]. However, ameasurement of the partial branching fraction for B → eeµν and B → µµeν , in the low q bins [0 . ,
1] and [0 . ,
1] GeV respectively, could provide additional interestinginformation. In this paper we have studied the purely leptonic decay modes B → (cid:96)(cid:96)(cid:96) (cid:48) ν for (cid:96), (cid:96) (cid:48) = e, µ in the low q region at NLL, including the leading 1 /m b and q corrections, as well asthe soft corrections at NLL. This work is motivated by the possibility to measure λ B at LHCb, since a systematic theoretical study with uncertainties for these four-leptonmodes was lacking, and in light of the recent results from LHCb [13], here we havetaken a first step towards achieving this goal. We stress that our result agree withthose of Ref. [16] in the q = 0 limit. We have provided a numerical comparison of13he various contributions to the form factors F V and F A in Fig. 3, where we see theimportance of calculating to NLL. Note that certain higher twist contributions whichwere calculated in the state-of-the-art B → γ(cid:96)ν analysis have been neglected here,i.e. the 1 /m b and 1 /E γ higher twist corrections and the twist 3 to 6 contributions tothe soft correction. The effect of these missing pieces is conservatively estimated toadd (cid:46)
5% to the uncertainty, negligible compared to the uncertainty coming from | V ub | .However this is only suggestive at this stage and a proper evaluation of these correctionsis called for. Further, the calculation of the additional form factors which may be neededin order to account for massive leptons would be desirable.We advocate the measurement of the partial branching fractions for B → eeµν and B → µµeν , in the low q bins [0 . ,
1] and [0 . ,
1] GeV respectively. Our results forthese quantities can be found in Table 2 for λ B = 350 MeV and central values of theremaining parameters, where the uncertainty is at the 30% level, dominant contributionscoming from | V ub | and the variation of the scale. We further show the dependence ofthe partial branching ratio on λ B in Fig. 4, and find that the dependence far outweighsthe remaining uncertainties, suggesting that given a value of the partial branching ratioa measurement of λ B should be feasible. While there are no official projections for thesechannels at LHCb and Belle II, naive estimates show that the prospects to measure thepartial branching ratio is promising. We therefore look forward to these results and tothe potential measurement of λ B , complementary to that of B ( B → γ(cid:96)ν ) at Belle II. Acknowledgements
We thank J´erˆome Charles for a careful reading of the manuscrip, Martin Beneke andYao Ji for useful discussions and Racha Cheaib, Francesco Polci, Justine Serrano, andWilliam Sutcliffe for important input concerning the potential experimental sensitivities.AB and BK are further grateful for their time spent at the Institute of Nuclear Theory,Seattle, attending the Heavy-Quark Physics and Fundamental Symmetries program(INT-19-2b), during which important progress on the project was made.
A Hadronic matrix element
Let us define, T µρ ( p, q ) = i (cid:90) d xe iqx (cid:10) | T { j emµ ( x )¯ u Γ ρ b (0) }| B ( p + q ) (cid:11) . (36)The most general decomposition of the hadronic matrix element defined in Eq. (36) isgiven by, T µρ = ( a q + b p.q ) g µρ + c p µ q ρ + d (cid:15) ρµλσ p λ q σ . (37)The terms containing p ρ have been neglected since their contribution will always beproportional to the mass of lepton. Without any loss of generality, Eq. (37) can berewritten as, T µρ = iF A ( g µρ p.q − p µ q ρ ) + α g µρ p.q + β p µ q ρ + F e q g µρ + F V (cid:15) ρµλσ p λ q σ (38)14onstraints on T µρ can be obtained using the current conservation of em current i.e, ∂ µ j emµ = 0. This is done by differentiating the correlation function in the definition of T µρ , which gives, q µ T µρ = i ( p + q ) ρ f B (39)Using the definition in Eq. (37), this implies F e q = if B − ( α + β ) p.q (40)Using this condition,, T µρ is reduced to, T µρ = ( iF A ) ( g µρ p.q − p µ q ρ ) + F V (cid:15) ρµλσ p λ q σ + if B g µρ , (41)where, iF A has been redefined as iF A + β for further simplification. References [1] V. M. Braun, D. Yu. Ivanov, and G. P. Korchemsky. The B meson distributionamplitude in QCD.
Phys. Rev. , D69:034014, 2004.[2] M. Beneke, T. Huber, and Xin-Qiang Li. NNLO vertex corrections to non-leptonicB decays: Tree amplitudes.
Nucl. Phys. , B832:109–151, 2010.[3] Martin Beneke and Matthias Neubert. QCD factorization for B → P P and B → P V decays.
Nucl. Phys. , B675:333–415, 2003.[4] M. Beneke and T. Feldmann. Symmetry breaking corrections to heavy to light Bmeson form-factors at large recoil.
Nucl. Phys. , B592:3–34, 2001.[5] A. G. Grozin and M. Neubert. Asymptotics of heavy meson form-factors.
Phys.Rev. , D55:272–290, 1997.[6] S. W. Bosch, R. J. Hill, B. O. Lange, and M. Neubert. Factorization and Sudakovresummation in leptonic radiative B decay.
Phys. Rev. , D67:094014, 2003.[7] Bernard Aubert et al. A Model-independent search for the decay B + → l + ν l γ . Phys. Rev. , D80:111105, 2009.[8] A. Heller et al. Search for B + → (cid:96) + ν (cid:96) γ decays with hadronic tagging using the fullBelle data sample. Phys. Rev. , D91(11):112009, 2015.[9] M. Beneke and J. Rohrwild. B meson distribution amplitude from B → γlν . Eur.Phys. J. C , 71:1818, 2011.[10] V. M. Braun and A. Khodjamirian. Soft contribution to B → γ(cid:96)ν (cid:96) and the B -mesondistribution amplitude. Phys. Lett. , B718:1014–1019, 2013.[11] M. Gelb et al. Search for the rare decay of B + → (cid:96) + ν (cid:96) γ with improved hadronictagging. Phys. Rev. D , 98(11):112016, 2018.[12] W. Altmannshofer et al. The Belle II Physics Book.
PTEP , 2019(12):123C01, 2019.[Erratum: PTEP 2020, 029201 (2020)].1513] Roel Aaij et al. Search for the rare decay B + → µ + µ − µ + ν µ . Submitted to: Eur.Phys. J. , 2018.[14] A. V. Danilina and N. V. Nikitin. Four-Leptonic Decays of Charged and Neutral B Mesons within the Standard Model.
Phys. Atom. Nucl. , 81(3):347–359, 2018.[Yad. Fiz.81,no.3,331(2018)].[15] A. Danilina, N. Nikitin, and K. Toms. Decays of charged B -mesons into threecharged leptons and a neutrino. Phys. Rev. D , 101(9):096007, 2020.[16] M. Beneke, V.M. Braun, Yao Ji, and Yan-Bing Wei. Radiative leptonic decay B → γ(cid:96)ν (cid:96) with subleading power corrections. JHEP , 07:154, 2018.[17] Yu-Ming Wang. Factorization and dispersion relations for radiative leptonic B decay. JHEP , 09:159, 2016.[18] Yu-Ming Wang and Yue-Long Shen. Subleading-power corrections to the radiativeleptonic B → γ(cid:96)ν decay in QCD. JHEP , 05:184, 2018.[19] Alexander Khodjamirian. Form-factors of γ ∗ ρ → π and γ ∗ γ → π transitions andlight cone sum rules. Eur. Phys. J. C , 6:477–484, 1999.[20] Enrico Lunghi, Dan Pirjol, and Daniel Wyler. Factorization in leptonic radiative B → γeν decays. Nucl. Phys. , B649:349–364, 2003.[21] Christian W. Bauer, Sean Fleming, Dan Pirjol, and Iain W. Stewart. An Effectivefield theory for collinear and soft gluons: Heavy to light decays.
Phys. Rev. ,D63:114020, 2001.[22] M. et al Tanabashi. Review of particle physics.
Phys. Rev. D , 98:030001, Aug 2018.[23] A. Pais and S. B. Treiman. Pion Phase-Shift Information from K (cid:96) Decays.
Phys.Rev. , 168:1858–1865, 1968.[24] R Aaij et al. Measurement of the B → K ∗ e + e − branching fraction at low dileptonmass. JHEP , 05:159, 2013.[25] Roel Aaij et al. Angular analysis of the B → K ∗ e + e − decay in the low- q region. JHEP , 04:064, 2015.[26] S. Descotes-Genon and Christopher T. Sachrajda. Sudakov effects in B → πlν ( l )form-factors. Nucl. Phys. B , 625:239–278, 2002.[27] S. Descotes-Genon and C.T. Sachrajda. Factorization, the light cone distributionamplitude of the B meson and the radiative decay B → γ(cid:96)ν ( l ). Nucl. Phys. B ,650:356–390, 2003.[28] Bjorn O. Lange and Matthias Neubert. Renormalization group evolution of the Bmeson light cone distribution amplitude.