Semileptonic $B^{-}\to\pi^{+}\pi^{-}\ell^{-}\bar\nu_\ell$ decay over the full $\pi\pi$ invariant mass spectrum

Semileptonic b-hadron decays provide an ideal platform for testing heavy-to-light transition form factors [1–6]. Mediated by the quark-level $ b\to u\ell\bar\nu_\ell $ weak transition, they offer one of the most reliable avenues for determining the Cabibbo–Kobayashi–Maskawa (CKM) matrix element $ |V_{ub}| $ [7–15], a key parameter that governs the overall magnitude of CP violation in the Standard Model. Moreover, owing to their relatively small hadronic uncertainties, these decays provide a sensitive probe for potential contributions from physics beyond the Standard Model [16–18].

The four-body decay channel $ B\to\pi\pi\ell\bar\nu_\ell $ is among the semileptonic B decays that have been measured experimentally [19–23]. In this process, the quasi-two-body contribution $ B\to\rho\ell\bar\nu_\ell $, followed by the decay $ \rho\to\pi\pi $, plays a dominant role. Accordingly, this decay has been extensively studied within various theoretical frameworks, including factorization approaches [24–27], QCD light-cone sum rules [28–33], lattice QCD [34], model-independent parameterizations [35], and phenomenological analyses [36]. However, the nonresonant contribution to $ B\to\pi\pi\ell\bar\nu_\ell $ represents a genuine four-body decay topology but is often treated as a negligible background and has received comparatively little attention.

The Belle Collaboration has recently measured the branching fractions of $ B^{-}\to\pi^{+}\pi^{-}\ell^{-}\bar\nu_\ell $ over the full $ \pi\pi $ invariant-mass ($ M_{\pi\pi} $) spectrum, as reported in [23].

with $ \ell^-=e^- $ or $ \mu^- $. Here, the total branching fraction $ {\cal B}_{\text{T}} $ includes contributions from resonant processes of the form $ B^{-}\to R\ell^{-}\bar\nu_\ell $ followed by $ R\to\pi^{+}\pi^{-} $, with $ R=\rho^0 $ or $ R=f_2\equiv f_2(1270) $. Because the $ \pi\pi $ invariant-mass spectrum is measured over the full kinematic range, potential non-resonant contributions are also expected. However, a dedicated statistical analysis of the non-resonant component has not yet been performed. Assuming $ {\cal B}_{\text{T}}\simeq{\cal B}_\rho+ {\cal B}_{f_2}+{\cal B}_{\text{N}} $, where $ {\cal B}_{\rho}(B^{-}\to\rho^0\ell^{-}\bar\nu_\ell,\rho^0\to\pi^{+}\pi^{-}) =(15.8\pm 1.1)\times 10^{-5} $ [15] and $ {\cal B}_{f_2}(B^{-}\to f_2\ell^{-}\bar\nu_\ell,f_2\to\pi^{+}\pi^{-}) =(1.8\pm 0.9^{+0.2}_{-0.1})\times 10^{-5} $ [37], the non-resonant branching fraction can be estimated as $ {\cal B}_{\text{N}}(B^{-}\to\pi^{+}\pi^{-}\ell^{-}\bar\nu_\ell) \sim{\cal O}(10^{-5}) $. This indicates that the non-resonant contribution is comparable in magnitude to the resonant components and should not be regarded as a negligible background, thereby warranting dedicated investigation.

The non-resonant decay channel $ B^{-}\to\pi^{+}\pi^{-}\ell^{-}\bar\nu_\ell $ is governed by the $ B\to\pi\pi $ non-resonant form factors, denoted $ F_{\pi\pi} $. Notably, the non-leptonic B decay channel $ B^-\to\pi^-\pi^+\pi^- $ also requires $ F_{\pi\pi} $ to account for its sizable non-resonant contribution [15], although these form factors remain poorly determined. Currently, little information is available in the literature [7, 24–43]. This situation motivates us to extract $ F_{\pi\pi} $ from the measured $ \pi\pi $ invariant-mass distribution in $ B^{-}\to\pi^{+}\pi^{-}\ell^{-}\bar\nu_\ell $. To this end, we will separate the resonant contributions from the non-resonant component and perform a numerical analysis of the branching fractions and angular distributions. In addition, we will examine the sensitivity of the current data to the CKM matrix element $ |V_{ub}| $.

As depicted in Figs. 1a and 1b, the decay $ B^{-}\to\pi^{+}\pi^{-}\ell^{-}\bar\nu_\ell $ proceeds via resonant and non-resonant $ B\to \pi\pi $ transitions, respectively, with the lepton pair produced by the emitted W boson. In the measured $ M_{\pi\pi} $ spectrum, a dominant peak and a small bump are identified as arising from the processes $ B^-\to \rho^0\ell^-\bar\nu_\ell $ and $ B^-\to f_2\ell^-\bar\nu_\ell $, respectively, where the vector meson $ \rho^0 $ and the tensor meson $ f_2 $ appear as intermediate resonances decaying into $ \pi^+\pi^- $. In addition, a non-resonant component is expected to contribute and can, in principle, be extracted from the data. For our analysis, we express the total amplitude of $ B^-\to \pi^+\pi^-\ell^-\bar\nu_\ell $ as [44]

Figure 1.  (color online) Semileptonic $ B^{-}\to\pi^{+}\pi^{-}\ell^{-}\bar\nu_\ell $ decay: (a) nonresonant contribution; (b) resonant contributions via the intermediate states $ \rho^0 $ and $ f_2 $.

where the subscripts T, N, and $ R=(\rho,f_2) $ denote the total, non-resonant, and resonant contributions, respectively. The hadronic matrix elements for the (non-)resonant $ B\to\pi\pi $ transitions are parameterized as in Refs. [1, 45]

where $ p=p_b+p_a $, $ q=p_B-p=p_\ell+p_\nu $, and $ (s,t)\equiv (q^2,p^2) $. The non-resonant form factors are denoted by $ F_{\pi\pi}=(h, r, w_{\pm}) $. The $ B\to \rho(f_2) $ transition matrix elements are parameterized as in Refs. [46–48]

where $ \epsilon^{\prime\mu}\equiv \epsilon^{\mu\nu}p_{B\nu}/m_B $. Here, $ \epsilon^\nu $ and $ \epsilon^{\mu\nu} $ denote the polarization vector and tensor, respectively, and $ F_{\rho(f_2)}=(V_1^{(\prime)}, A_{0,1,2}^{(\prime)}) $ collects the corresponding transition form factors. The strong decay vertices $ \langle \pi\pi|\rho,f_2\rangle $ are given in Refs. [10, 50, 51]

where $ g_{1,2} $ denote strong coupling constants. To perform the sums over the vector and tensor polarizations of the intermediate ρ and $ f_2 $ states, we follow Refs. [46–48].

with $ M_{\mu\mu'} = -g_{\mu\mu'} + p_\mu p_{\mu'}/p^2 $. The form factors in Eqs. (3) and (4) are momentum-dependent and are modeled using either single- or double-pole parametrizations [46–48]:

where $ F_{\rho} $ and $ F_{f_2} $ have been studied within QCD-based approaches [48, 49], while the parameters $ (a,b,F_{\pi\pi}(0)) $ are extracted from a global fit.

For the four-body decay channel $ B^-(p_B)\to \pi^+(p_a)\pi^-(p_b)\ell^-(p_\ell)\bar \nu_\ell(p_\nu) $, the phase-space integration is performed over the kinematic variables $ (s,t,\theta_M,\theta_L,\phi) $. As illustrated in Fig. 2, $ \theta_{M(L)} $ denotes the angle between the momenta of the $ \pi^+ $ and $ \pi^- $ (of the $ \ell^- $ and $ \bar \nu_\ell $) in the $ \pi^+\pi^- $ ($ \ell^-\bar \nu_\ell $) rest frame. In addition, the angle ϕ is defined as the angle between the decay planes of the $ \pi^+\pi^- $ and $ \ell^-\bar \nu_\ell $ systems, which are formed by $ \vec{p}_{a,b} $ and $ \vec{p}_{\ell,\bar \nu_\ell} $, respectively, in the B-meson rest frame. The differential decay width is given by [1, 3, 4]

Figure 2.  (color online) The angular variables $ (\theta_{M}, \theta_{L}, \phi) $ in the four-body $ B^-\to\pi^+\pi^-\ell^-\bar \nu_\ell $ decay.

where X, $ \alpha_{M} $, and $ \alpha_{L} $ are defined as in [43, 52, 53]

With $ \lambda(a,b,c)=a^2+b^2+c^2-2ab-2bc-2ca $. The allowed ranges for $ (s,t) $ and the angular variables ($ \theta_M,\theta_L,\phi $) are:

With $ m_{\ell}+m_{\bar \nu_\ell}\simeq 0 $, the angular asymmetry is defined from Eq. (8), as in Refs. [54–56].

where $ d\Gamma/d\cos\theta_M $ represents the angular distribution.

In the numerical analysis, we perform a minimum $ \chi^2 $ fit to extract, as free parameters, $ |V_{ub}| $, the nonresonant form factors $ F_{\pi\pi} $, and $ \delta_{1,2} $, where $ \delta_{1(2)} $ denotes the relative phase associated with $ {\cal M}_{\rho(f_2)} $. The $ \chi^2 $ function is defined as

where $ d{\cal B}/dM_{\pi\pi} $ denotes the differential branching fraction, and $ \sigma_{ex} $ ($ \delta F_{th} $) represents the experimental (theoretical form-factor) uncertainty. The theoretical inputs $ {\cal B}_{\rho(f_2)\,th} $ and $ d{\cal B}_{th}/dM_{\pi\pi} $ are obtained from the amplitudes in Eq. (2), whereas the experimental inputs are taken from Eq. (1) and Fig. 3. We use $ F_\rho^{(j)}=(V_1(0),A_1(0),A_2(0)) $ and $ F_{f_2}^{(k)}=(V'_1(0),A'_1(0),A'_2(0)) $ from Table 1 as input values for Eq. (12), together with $ |g_1|=5.98 $ and $ |g_2|=18.56 $ in units of GeV^-1 [51, 57].

Figure 3.  (color online) The $ \pi\pi $ invariant-mass spectrum. The solid curve includes all contributions and reproduces the data points from Belle [23]. The dashed (dotted) curve represents the contributions from $ B^-\to\rho(f_2)\ell\bar \nu $, with $ \rho(f_2)\to \pi^+\pi^- $, while the dot-dashed curve denotes the nonresonant contribution.

From the fit, we obtain

with $ n.d.f.=7 $ denoting the number of degrees of freedom. We note that the form factor r in Eq. (3) and $ A_0^{(\prime)} $ in Eq. (4) do not contribute to the fit, as their contributions to the amplitudes vanish since $ q_\mu\bar u_\ell\gamma^\mu(1-\gamma_5)v_\nu=0 $ for nearly massless leptons.

Using the fitted parameters in Eq. (13), we obtain

where the first uncertainty arises from $ |V_{ub}| $, the second from the form factors, and the third (for $ {\cal B}_T $) from the phase $ \delta_1 $. The partial branching fractions as functions of $ M_{\pi\pi} $ and $ \cos\theta_M $ are shown in Fig. 3 and Fig. 4, respectively. We further evaluate the angular asymmetries using Eq. (11) and obtain

Figure 4.  (color online) Angular distributions for the decay $ B^{-}\to\pi^{+}\pi^{-}\ell^{-}\bar\nu_\ell $. The solid, dashed, dotted, and dot-dashed curves represent the same contributions as in Fig. 3

where the first set of uncertainties arises from the form factors, and the second (for $ {A}_{\theta_M,\text{T}} $) arises from the phase $ \delta_1 $.

We investigate the four-body decay channel $ B^-\to\pi^+\pi^-\ell^-\bar \nu_\ell $ by performing a global fit to the $ M_{\pi\pi} $ invariant mass spectrum shown in Fig. 3. The resulting $ \chi^2/n.d.f.\simeq 1 $ indicates that our model provides a statistically robust description of the data. By employing the form factors extracted from the global fit (as detailed in Eq. (13)), we present the differential branching fractions $ d{\cal B}_{(T,\rho,f_2,N)}/dM_{\pi\pi} $ in Fig. 3, illustrating the individual and total contributions to the mass spectrum. The solid curve representing $ d{\cal B}_T/dM_{\pi\pi} $ is in good agreement with the experimental data. Notably, the non-resonant contribution ($ d{\cal B}_N/dM_{\pi\pi} $), shown by the dot-dashed curve, plays a critical role in shaping the overall distribution, particularly in the low $ M_{\pi\pi} $ region. Our analysis yields the first estimate of the non-resonant branching fraction, $ {\cal B}_{\text{N}}=(3.5\pm 1.4^{+4.3}_{-2.4})\times 10^{-5} $. This result indicates that $ {\cal B}_{\text{N}} $ can reach the $ 10^{-5} $ level and, therefore, should no longer be neglected in precision studies.

The fitted relative phase $ \delta_1 $ leads to destructive interference involving the resonant process $ B^-\to(\rho^0\to)\pi^+\pi^-\ell^-\bar \nu_\ell $, an effect that is most prominent in the $ M_{\pi\pi}\simeq m_\rho $ region. In contrast, $ \delta_2 $ is found to be consistent with zero. This reflects the suppression of the non-resonant contribution for $ M_{\pi\pi}>1 $ GeV, leading to negligible interference with the $ B^-\to (f_2\to)\pi^+\pi^- \ell\bar \nu $ channel. From the fit results in Eq. (13), we obtain $ |V_{ub}|=(3.31\pm 0.61)\times 10^{-3} $, which is consistent with current determinations. Although the relatively large uncertainty renders this result less competitive than existing extractions reported by the PDG [15], it nevertheless demonstrates that the four-body semileptonic decay $ B\to\pi\pi\ell\bar \nu $ is sensitive to $ |V_{ub}| $. This provides a novel and independent avenue for its determination. Future high-precision measurements will be essential to reduce the experimental uncertainty and fully exploit the potential of this decay channel.

The angular asymmetries provide stringent constraints on the underlying form factors [54–56]. Our analysis, as summarized in Eq. (15), shows that $ {A}_{\theta_M,\rho} $ and $ {A}_{\theta_M,f_2} $ are both approximately zero. Specifically, $ {A}_{\theta_M,\rho} $ exhibits a symmetric distribution, as shown in Fig. 4, while $ {A}_{\theta_M,f_2} $ remains nearly flat. By contrast, $ {A}_{\theta_M,\text{N}}\sim -40\% $ displays a clear, decreasing dependence on $ \cos\theta_M $. This behavior is dominated by the form factor associated with the structure $ w_-(p_b-p_a) $. In the $ \pi^+(p_a)\pi^-(p_b) $ rest frame (see Fig. 2), the term $ p_b-p_a=(0,2\vec{p}_b) $ from the $ w_- $ contribution projects onto the four-momentum of the lepton-pair system, inducing a pronounced $ \cos\theta_M $ dependence. Consequently, future experimental measurements of these angular asymmetries will serve as a powerful tool to test the existence and magnitude of the non-resonant contribution.

In summary, we have investigated the four-body semileptonic decay $ B^-\to\pi^+\pi^-\ell\bar \nu $. By analyzing the full $ \pi\pi $ invariant mass spectrum measured by the Belle Collaboration, we have determined $ |V_{ub}|=(3.31\pm 0.61)\times 10^{-3} $, a value consistent with current determinations. Furthermore, we have extracted the non-resonant $ B\to\pi\pi $ transition form factors and predicted the non-resonant branching fraction to be $ {\cal B}_{\text{N}}(B^{-}\to\pi^{+}\pi^{-}\ell^{-}\bar\nu_\ell) = (3.5\pm 1.4^{+4.3}_{-2.4})\times 10^{-5} $. We have also presented the non-resonant angular asymmetry $ {A}_{\theta_M,\text{N}}(B^{-}\to\pi^{+}\pi^{-}\ell^{-}\bar\nu_\ell)=(-43.0\pm 22.3){\text{%}} $. These predictions offer a clear signature of the non-resonant component and can be rigorously tested in future high-precision measurements at Belle II and LHCb.