Role of $\rho \text{−}\omega$ interference in semileptonic $B\to {\pi }^{+}{\pi }^{-}\ell {\overline{\nu }}_{\ell }$ decays

It is long known that interference effects play an important role in understanding the shape of the ${\pi }^{+}{\pi }^{-}$ spectrum of resonances near the threshold. In this manuscript, we investigate the role of the $\rho \text{−}\omega$ interference in the study of semileptonic $B\to {\pi }^{+}{\pi }^{-}\ell {\overline{\nu }}_{\ell }$ decays. We determine for the first time the strong phase difference between $B\to {\rho }^0\ell {\overline{\nu }}_{\ell }$ and $B\to \omega \ell {\overline{\nu }}_{\ell }$ from a recent Belle measurement of the $m_{\pi \pi }$ spectrum of $B\to {\pi }^{+}{\pi }^{-}\ell {\overline{\nu }}_{\ell }$. We find ${\varphi }_{\rho \text{−}\omega }=(-46_{-67}^{+155})°$ and extract the branching fraction of $\mathcal{B}(B\to {\rho }^0\ell {\overline{\nu }}_{\ell })=(1.41_{-0.38}^{+0.49})\times {10}^{-4}$. In addition, we set a limit on the $S$-wave component within an $m_{\pi \pi }$ window ranging from $2m_{\pi }$ to 1.02 GeV of $\mathrm{\Delta }\mathcal{B}(B\to [{\pi }^{+}{\pi }^{-}{]}_S\ell {\overline{\nu }}_{\ell })<0.51\times {10}^{-4}$ at 90% CL. We also determine the absolute value of the Cabibbo-Kobayashi-Maskawa matrix element of $|V_{ub}{|}_{\rho \text{−}\omega }=(3.03_{-0.44}^{+0.49})\times {10}^{-3}$, which takes into account the $\rho \text{−}\omega$ interference.

Figure 1

Measured $B\to {\pi }^{+}{\pi }^{-}\ell {\overline{\nu }}_{\ell }$ spectrum from Ref. [15].

Figure 2

Left: line shapes $|\mathcal{A}(s){|}^2$ for different parametrizations using the same of $M_0$ and ${\mathrm{\Gamma }}_0$ parameters. For details of the parametrizations see text. All curves were normalized to their respective modes. Right: the same comparison, when the parameters of the different parametrizations are chosen such that the parametrizations yield the same pole of $\sqrt{s_0}=(0.764-0.146i/2)\mathrm{GeV}$ in the complex $s$ plane.

Figure 3

Model for the $m_{\pi \pi }$ spectrum in $B\to \pi \pi \ell {\overline{\nu }}_{\ell }$ decays using the relativistic Breit-Wigner amplitude The blue curve shows the model for $R=0$, whereas the green curve for $R=5{\mathrm{GeV}}^{-1}$. The gray curve shows the line shape, i.e., the model without the phase-space and barrier factors applied for comparison. For details see text.

Figure 4

The impact of the phase space and barrier factor, that lead to the asymmetry of the line shape as a function of $m_{\pi \pi }$ are shown for $R=5{\mathrm{GeV}}^{-1}$ (plum) and $R=0$ (light plum). All curves are normalized to their mode.

Figure 5

The $m_{\pi \pi }$ spectrum from the interference is shown assuming $\rho$ and $\omega$ are produced with ${\varphi }_{\rho \text{−}\omega }=0$ (solid line), using the relativistic Breit-Wigner amplitude with mass and width chosen to possess a pole at $\sqrt{s_0}=(0.764-0.146i/2)\mathrm{GeV}$ and $R=5{\mathrm{GeV}}^{-1}$. The shaded light blue lines show the interference prediction using the other parametrizations which mass and width chosen to yield the same pole position. The dashed lines show the spectrum for the pure $\rho$ (blue) and $\omega$ (green) contributions.

Figure 6

The distortion of the $m_{\pi \pi }$ spectrum from interference is shown for four different values of ${\varphi }_{\rho \text{−}\omega }$.

Figure 7

The considered parametrizations for the $S$-wave contribution are shown.

Figure 8

Fit result using the relativistic Breit-Wigner to model the $\rho$ amplitude and using the shape of Ref. [50] to model the $S$-wave background. The blue line and shaded band show the total model curve and its uncertainty. The $S$-wave contribution is shown as a dash-dotted purple line. The dashed lines show the prediction of the pure $\rho$ (blue) and $\omega$ (green) contributions.

Figure 9

The $\mathrm{\Delta }{\chi }^2=1$ (38.3% CL, blue) and $\mathrm{\Delta }{\chi }^2=2.3$ (68.3% CL, black dashed) contours of $\mathcal{B}(B\to {\rho }^0\ell {\overline{\nu }}_{\ell })$ and ${\varphi }_{\rho \text{−}\omega }$ are shown for the $S$-wave described using Ref. [49]. The best fit point ($\mathrm{\Delta }{\chi }^2=0$) is indicated with a blue star. The red data point shows the result of a fit neglecting the interference effects between $\rho$ and $\omega$.

Figure 10

Fit results using the relativistic Breit-Wigner to describe the $\rho$ line shape with using phase space (left) or Refs. [52, 53] to model the $S$-wave component. The blue line show the full interference and background line shape. The $S$-wave contribution is shown as a dash-dotted purple line. The dashed lines show the prediction without interference for $\rho$ (blue) and $\omega$ (green).

Figure 11

The $\mathrm{\Delta }{\chi }^2=1$ (38.3% CL, blue) and $\mathrm{\Delta }{\chi }^2=2.3$ (68.3% CL, black dashed) contours of $\mathcal{B}(B\to {\rho }^0\ell {\overline{\nu }}_{\ell })$ and ${\varphi }_{\rho \text{−}\omega }$ are shown for the $S$-wave described using phase space (left) or the model of Refs. [52, 53] (right). The best fit point ($\mathrm{\Delta }{\chi }^2=0$) is indicated with a blue star. The red data point shows the result of a fit neglecting the interference effects between $\rho$ and $\omega$.

Figure 12

Left: the influence of changing the scale factor $R$ on the relativistic Breit-Wigner shape is seen, whose size is related to the strong potential.