$|V_{ub}|$ and $B\to\eta^{(')}$ Form Factors in Covariant Light Front Approach

$B\to (\pi, \eta, \eta')$ transition form factors are investigated in the covariant light-front approach. With theoretical uncertainties, we find that $B\to (\pi, \eta, \eta')$ form factors at $q^2=0$ are $f^{(\pi, \eta, \eta')}_{+}(0)=(0.245^{+0.000}_{-0.001}\pm 0.011, 0.220 \pm 0.009\pm0.009, 0.180\pm 0.008^{+0.008}_{-0.007})$ for vector current and $f^{(\pi, \eta, \eta')}_{T}(0)=(0.239^{+0.002+0.020}_{-0.003-0.018}, 0.211\pm 0.009^{+0.017}_{-0.015}, 0.173\pm 0.007^{+0.014}_{-0.013})$ for tensor current, respectively. With the obtained $q^2$-dependent $f^{\pi}_{+}(q^2)$ and observed branching ratio (BR) for $\bar B_d\to \pi^+ \ell \bar \nu_{\ell}$, the $V_{ub}$ is found as $|V_{ub}|_{LF}= (3.99 \pm 0.13)\times 10^{-3}$. As a result, the predicted BRs for $\bar B\to (\eta, \eta') \ell \bar\nu_{\ell}$ decays with $\ell=e,\mu$ are given by $(0.49^{+0.02+0.10}_{-0.04- 0.07}, 0.24^{+0.01+0.04}_{-0.02-0.03})\times 10^{-4}$, while the BRs for $D^-\to (\eta,\eta')\ell\bar\nu_{\ell}$ are $(11.1^{+0.5+0.9}_{-0.6-0.9}, 1.79^{+0.07+0.12}_{-0.08-0.12})\times 10^{-4}$. In addition, we also study the integrated lepton angular asymmetries for $\bar B\to (\pi,\eta,\eta')\tau \bar\nu_{\tau}$:$(0.277^{+0.001+0.005}_{-0.001-0.007},0.290^{+0.002+0.003}_{-0.000-0.003},0.312^{+0.004+0.005}_{-0.000-0.006})$.

Compared with nonleptonic B decays, semileptonicB → η (′) ℓν ℓ andB s → η (′) ℓ + ℓ − decays are much cleaner and thus might be more helpful to explore the differences among various mechanisms. In particular, a sizable flavor-singlet component of η (′) predicts larger BRs for B → η ′ ℓν ℓ than the η modes, while the chiral symmetry breaking enhancement could give the reverse results [7]. Nevertheless, before one considers various possible novel effects on η (′) , it is necessary to understand the BRs forB → η (′) ℓν ℓ decays without these exotic effects. In our previous work [7], we used the perturbative QCD approach [8] to calculate the B → η (′) form factors at large recoil; then the same whole spectrum as a function of invariant mass of ℓν ℓ for the form factors is assumed with that in the light-cone sum rules (LCSRs). Despite the predicted results for various branching ratios are consistent with the experimental data, it is meaningful to examine the same processes in other parallel frameworks. This is helpful to reduce the dependence on the treatments of the dynamics in transition form factors. The motif of this work is to employ another method to deal with the form factors: the covariant light-front (LF) approach [9,10]. Since the predictions of B → π form factors in LF model match very well with those applied to the nonleptonic charmless B decays, it is worthy to understand what we can get the B → η (′) form factors by this approach.
At the quark level, theB → η (′) ℓν ℓ is induced by b → ulν transition which will inevitably involve theūu component of the η ( ′ ) meson. Then the convenient mechanism for the η − η ′ mixing would be the quark flavor mixing scheme, defined by [11,12]  where η q = (uū + dd)/ √ 2, η s = ss and angle φ is the mixing angle. By the definition of , the masses of η q,s can be expressed by Here, m qq and m ss are unknown parameters and their values can be obtained by fitting with the data. In terms of the quark-flavor basis, we see clearly that m qq and m ss are zero in the chiral limit. The advantage of the quark-flavor mixing scheme is: at the leading order in α s only the quark transition from the B meson into the η q component is necessary; while the other transitions like B → η s are suppressed by α s . The gluonic form factors (or referred to as flavor-singlet form factors) will be remarked later.
For calculating the transition form factors, we parameterize the hadronic effects as with P µ = (P ′ + P ′′ ) µ and q µ = (P ′ − P ′′ ) µ . Since the light quarks in B-meson are u-and d-quark, the meson P could stand for π and η q states.
In the covariant LF quark model, the transition form factors for B → P could be obtained by computing the lowest-order Feynman diagram depicted in Fig.1. Below we will adopt the same notation as Ref. [9] and light-cone coordinate system for involved momenta, in which the components of meson momentum are read by P ′ = (P ′− , P ′+ , P ′ ⊥ ) with P ′± = P ′0 ± P ′3 . The relationship between meson momentum and the momenta of its constitutent quarks is given by P ′ = p ′ 1 + p 2 and P ′′ = p ′′ 1 + p 2 with p 2 being the spectator quark of initial and final mesons. Additionally, one can also express the quark momenta in terms of the internal with x 1 + x 2 = 1. Here, the notation with tilde could represent all momenta in the initial and final mesons.
In order to formulate the results of Fig. 1, the quark-meson-antiquark vertex for incoming and outgoing mesons are respectively chosen to be where H ′ P is the covariant light-front wave function of the meson. Consequently, the amplitude for the loop diagram is straightforwardly written by where N c = 3 is the number of colors, N with the M ′ (M ′′ ) being the mass of the incoming (outgoing) meson. As usual, the loop integral could be performed by the contour method. Therefore, except some separate poles appearing in the denominator, if the covariant vertex functions are not singular, the integrand is analytic. Thus, when performing the integration, the transition amplitude will pick up the singularities from the anti-quark propagator so that the various pieces of integrand are led to be We work in the q + = 0 frame and the transverse momentum of the quark in the final meson is given as The new function of h ′ M for initial meson is given by with where e i can be interpreted as the energy of the quark or the antiquark, M ′ 0 can be regarded as the kinetic invariant mass of the meson system and ϕ ′ P is the LF momentum distribution amplitude for s-wave pseudoscalar mesons. The similar quantities associated with the outgoing meson can be defined by the same way.
After the contour integration, the valance antiquark is turned to be on mass-shell and the conventional LF model is recovered. The formulas of the form factors in the LF quark model shown in Eq. (7) would contain not only the terms proportional to P µ and q µ , but also the terms proportional to a null vectorω = (2, 0, 0 ⊥ ). This vector is spurious, because it does not appear in the standard definition of Eq.(3), and spoils the covariance. In the literature, it is argued that this spurious factor can be eliminated by including the so-called zero-mode contribution, and a proper way to resolve this problem has been proposed in Ref. [9]. In this method, one should obey a series of special rules when performing the p − integration.
A manifest covariant result can be given with this approach, which is physically reasonable.
Using Eqs. (7)-(9) and taking the advantage of the rules in Ref. [9,10], the B → P form factors are straightforwardly obtained by where the relation of f P − (q 2 ) to f P 0 (q 2 ) can be read by Clearly, one has f P + (0) = f P 0 (0). After we obtain the formulae for the B → P transition form factors, the direct application is the exclusive semileptonicB → P ℓν ℓ decays. The effective Hamiltonian for b → uℓν ℓ in the standard model (SM) is given by Although these decays are tree processes, however, if we can understand well the form factors, there still have the chance to probe the new physics in these semileptonic decays [14,15]. Hence, the decay amplitude forB → P ℓν ℓ is written as To calculate the differential decay rates, we choose the coordinates of various particles as follows q 2 = ( q 2 , 0, 0, 0), p B = (E B , 0, 0, |p P |), where . It is clear that θ is defined as the polar angle of the lepton momentum relative to the moving direction of the B-meson in the q 2 rest frame. With Eqs. (14) and (15), the differential decay rate forB → P ℓν ℓ as a function of q 2 and θ can be derived by Since the differential decay rate in Eq. (16) involves the polar angle of the lepton, we can define an angular asymmetry to be with z = cos θ. Explicitly, the asymmetry forB → P ℓν ℓ decay is Moreover, the integrated angular asymmetry can be defined bȳ The angular asymmetry is only associated with the ratio of form factors, which supposedly is insensitive to the hadronic parameters. Plausibly, this physical quantity could be the good candidate to explore the new physics such as charged Higgs [14], right-handed gauge boson [15], etc.
Before presenting the numerical results for the form factors and other related quantities, we will briefly discuss how to extract the input parameters for the η q in the presence of η −η ′ mixing. Following the divergences of the axial vector currents where G = G aµν are the gluonic field-strength andG =G aµν ≡ ǫ µναβ G a αβ , the mass matrix of η q,s becomes  with a 2 = 0|α s GG|η q /(4 √ 2πf q ) and y = f q /f s . Using the mixing matrix introduced in Eq.
(1), one can diagonalize the mass matrix and the eigenvalues are the physical mass of η and η ′ . Correspondingly, we have the relations [13] sin and m η (′) is the mass of η (′) . Once the parameters φ, y and a are determined by experiments, we can get the information for m qq,ss and f q,s . Then, they could be taken as the inputs in our calculations.
After formulating the necessary pieces, we now perform the numerical analysis for the form factors and the related physical quantities introduced earlier. For understanding how well the predictions of LF model are, we first analyze B → π form factors at q 2 = 0.
By examining Eq. (11), we see that the main theoretical unknowns are the parameters of distribution amplitudes of mesons, masses of constitute quarks and the decay constants of mesons. As usual, we adopt the gaussian-type wave function for pseudoscalar mesons as with β ′ P characterizing the shape of the wave function. Other relevant values of parameters are taken as (in units of GeV) m B = 5.28 , m b = (4.8 ± 0.2) , m π = 0.14 , where m u,d are the constituent quark masses, the errors in them are from the combination of linear, harmonic oscillator and power law potential [16] and f P denotes the decay constant of P-meson. The shape parameters βs are determined by the relevant decay constants whose analytic expressions are given in Ref. [10]. Following the formulae derived in Eq. (11) and using the taken values of parameters, we immediately find where the first and second errors are from (i) β ′ B and β ′ ηq (ii) the quark masses m u and m b , respectively. From Eq. (11), one can see that the form factor f ηq + (q 2 ) does not depend on the mass m qq , while the dependence of m qq in f ηq T resides in the term M ′ +M ′′ (in this case m B + m qq ). The uncertainty of f T caused by the m qq is less than 2%. Furthermore, since the form factors are associated with mixing angle φ, the corresponding uncertainties for B → η (′) and BRs ofB → η (′) ℓν ℓ are expected to be 2.1% (1.4%) and 4.2% (2.8%), respectively. Despite different treatments of quarks' momenta, the results here are well consistent with that in light-cone quark model constructed in the effective field theory [18]: f ηq + (0) = 0.287 +0.059 −0.065 . Intriguingly, our results are also consistent with f η + (0)| LCSR = 0.231 +0.018 −0.020 and f η ′ + (0)| LCSR = 0.189 +0.015 −0.016 calculated by LCSRs [19]. In order to understand the behavior of whole q 2 , the form factors for B → P are parametrized by [17] where F i denotes any form factor among f +,0,T . The fitted values of a, b for B → (π, η, η ′ ) are displayed in Table I   In the quark flavor mixing mechanism, the η and η ′ meson receives additional coupling with two gluons, due to the axial anomaly. Thus to be self-consistent, in the study of the transition form factors, one also needs to include the so-called gluonic form factors which is induced by the transition from the two gluons into the η ( ′ ) . In our study, the gluonic form factors have been neglected and there are two reasons for this. In the light-front quark model, the leading order contribution to the form factor is of the order α 0 s while the gluonic form factor is suppressed by the α s , where the coupling constant is evaluated at the typical scale µ ∼ Λ QCD × m B (with Λ QCD hadronic scale). The inclusion of the gluonic form factors also requires the next-to-leading order studies for the quark content, which is beyond the scope of the present work. Secondly the factorization analysis of the gluonic form factors such as the perturbative QCD study in Ref. [22] reflects that there is no endpoint singularity in the gluonic form factors and the PQCD study shows that the gluonic form factors are negligibly small. This feature is also confirmed by the recent LCSR results [19]. For terms without endpoint singularity, different approaches usually obtain similar results. Thus our results of the semileptonic B → η (′) lν will not be sizably affected by the gluonic form factors, although they are not taken into account in the present analysis.
Besides the form factors could be the source of uncertainties, another uncertain quantity in exclusive b → uℓν ℓ decays is from the Cabibbo-Kobayashi-Maskawa (CKM) matrix element V ub ∼ λ 3 with λ being Wolfenstein parameter. Results for V ub determined by inclusive and exclusive decaying modes have some inconsistencies [15,20]. For a selfconsistent analysis, we take B → π form factors calculated by LF model and the data [20] as the inputs to determine the |V ub |. Neglecting the lepton mass, one gets the differential decaying rate forB → πℓ ′ν where only the f π + form factor involves. Accordingly, the value of V ub is found by With the obtained result of |V ub | LF , the form factors in the Table I, the predicted BRs for B − → (η, η ′ )ℓν ℓ , together with the experimental results measured by BaBar collaboration [21], are displayed in Table II. The predicted result for the BR of B − → ηℓν ℓ is about two times larger than that of B − → η ′ ℓν ℓ : the form factor of B − → η is larger than the form factor of B − → η ′ ; the phase space in B → η ′ ℓν ℓ is smaller. Branching ratios for decays with a tau lepton are naturally smaller than the relevant channels with a lighter lepton. Mode  Finally, we make some remark on the D decays. We find that the obtained information on η (′) can be directly applied to the semileptonic D + → η (′) ℓν ℓ decays. Since the associated respectively. It is found that B(D − → η ′ ℓν ℓ ) is almost one order of magnitude smaller than B(D − → ηℓν ℓ ). The reason for the resulted smallness is just phase space suppression. Our predictions are well consistent with the recent measurements by the CLEO collaboration [23]: B(D − → ηℓν ℓ ) = (1.33 ± 0.20 ± 0.06) × 10 −3 , This consistence is very encouraging. The D − → η ′ lν may be detected in the near future.
Our results are also consistent with the results given in Ref. [24].