Measurements of the mass difference m(B⁰) − m(B⁺) and the energy dependence of the cross-section ratio at Belle and Belle II

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(B⁰) − m(B⁺) = (0.495 ± 0.024 ± 0.005) MeV/c². The results are based on a simultaneous fit to the variable, which is related to the B momentum, for B⁰ and B⁺ candidates; and to the energy dependence of, which is measured using changes in the average center-of-mass energy over the data taking periods. The phase-space hypothesis, 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 in a broader energy range than covered by the direct measurement and extract the energy dependence of in the range from the threshold up to 10.59 GeV. We interpret the results using a phenomenological model and constrain the parameters of the potential in the isovector channel.