Measurements of the mass difference $m(B^0)-m(B^+)$ and the energy dependence of the cross-section ratio $σ(e^+e^-\to B^0\bar{B}^0) / σ(e^+e^-\to B^+B^-)$ at Belle and Belle II

TL;DR

This study measures the $B^0$-$B^+$ mass difference $\Delta m$ and the energy dependence of the cross-section ratio ${\cal R} = \sigma(e^+e^-\to B^0\bar{B}^0)/\sigma(e^+e^-\to B^+B^-)$ using Belle and Belle II data collected near the $\Upsilon(4S)$. A simultaneous fit to the beam-constrained mass variable $\widetilde{M}_{\rm bc}$ and the energy dependence of ${\cal R}$, including $E_{\rm cm}$ spread and ISR effects, yields $\Delta m = (0.495\pm0.024\pm0.005)$ MeV/$c^2$ and shows the phase-space form ${\cal R}=(p_{B^0}/p_{B^+})^3$ is strongly disfavored. The analysis constrains ${\cal R}(E_{\rm cm})$ over the $B\bar{B}$ threshold up to $10.59$ GeV and interprets the results with a phenomenological isovector $B\bar{B}$ potential, finding two solutions with a hint of an attractive potential at about the 2-sigma level. These results improve the precision on $\Delta m$, challenge previous BaBar conclusions tied to the phase-space assumption, and provide new insights into near-threshold $B$-meson interactions with implications for molecular-state scenarios.

Abstract

Using data samples collected by the Belle and Belle II experiments at the $Υ(4S)$ resonance with integrated luminosities of 571 fb$^{-1}$ and 365 fb$^{-1}$, respectively, we measure the pseudoscalar $B$-meson mass difference to be $m(B^0)-m(B^+) = (0.495\pm0.024\pm0.005)$ MeV/c$^2$. The results are based on a simultaneous fit to the variable $\tilde{M}_{bc}$, which is related to the $B$ momentum, for $B^0$ and $B^+$ candidates; and to the energy dependence of ${\cal R}=σ(e^+e^-\to B^0\bar{B}^0) / σ(e^+e^-\to B^+B^-)$, which is measured using changes in the average center-of-mass energy over the data taking periods. The phase-space hypothesis ${\cal R}=(p_{B^0}/p_{B^+})^3$, upon which previous measurements rely, is strongly disfavored by our fit; the measured mass-difference value for the phase-space hypothesis also differs significantly from our measurement. We constrain ${\cal R}$ in a broader energy range than covered by the direct measurement and extract the energy dependence of ${\cal R}$ in the range from the $B\bar{B}$ threshold up to 10.59 GeV. We interpret the results using a phenomenological model and constrain the parameters of the $B\bar{B}$ potential in the isovector channel.

Figures (12)

• Figure 1: The $\tilde{M}_{\rm bc}$ distributions in the Belle $\Upsilon(4S)$ data for the $B^0$ (left) and $B^+$ (right) candidates. The top and bottom parts of the figure correspond to the $\Delta E'$ signal region and sideband, respectively. The red solid histogram is the result of the combined fit described in section \ref{['sec:results']}. The red dashed histogram shows the background, and the black dotted histogram shows the sum of the background and the broken signal (for the $\Delta E'$ sideband, this coincides with the total fit, and is thus not visible). The bottom panels show pulls (deviations of the data points from the fit function divided by the uncertainties on the data).
• Figure 2: The $\tilde{M}_{\rm bc}$ distributions in the Belle II $\Upsilon(4S)$ data for the $B^0$ (left) and $B^+$ (right) candidates. The panels and the curves have the same meaning as in figure \ref{['mbc_y4s_bp_belle_280324']}.
• Figure 3: The distribution of $B$-meson candidates in the calibrated energy of the $B\bar{B}$ pair in Belle (left) and Belle II data (right).
• Figure 4: Energy of the colliding beams as a function of the energy of the $B\bar{B}$ pair. The black and red dots correspond to the Belle and Belle II data, respectively. Horizontal error bars indicate $\bar{E}_{B\bar{B}}$ bins, while vertical error bars show the statistical uncertainty in $\bar{E}_{\rm cm}$.
• Figure 5: Energy dependence of the ratio of visible cross sections $\sigma_{\mathrm{v}}(e^+e^-\to B^0\bar{B}^0)\,/\,\sigma_{\mathrm{v}}(e^+e^-\to B^+B^-)$. The dots with error bars show the direct measurements performed using Belle (left) and Belle II (right) data; the vertical inner and outer error bars show statistical and total uncorrelated uncertainties, respectively; the horizontal error bars show statistical uncertainty. The curve is the result of the combined fit described in section \ref{['sec:results']}.