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

Using data samples collected by the Belle and Belle II experiments at the $\mathrm{{\rm Y}}(4S)$ resonance with integrated luminosities of 571 fb${}^{-1}$ and 365 fb${}^{-1}$, respectively, we measure the pseudoscalar $B$-meson mass difference to be $m(B^0)-m(B^{+})=(0.495\pm 0.024\pm 0.005)$ 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 $\mathcal{R}=\sigma (e^{+}{e}^{-}\to B^0{\bar{B}}^0)\mathrm{/}\sigma (e^{+}{e}^{-}\to B^{+}{B}^{-})$, which is measured using changes in the average center-of-mass energy over the data taking periods. The phase-space hypothesis $\mathcal{R}=(p_{B^0}\mathrm{/}{p}_{B^{+}}{)}^3$, 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 $\mathcal{R}$ in a broader energy range than covered by the direct measurement and extract the energy dependence of $\mathcal{R}$ 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.