M1 Radiative Transitions in the $B_c$ System after the $B_c^{\ast }(1S)$ Observation: Radial-Node Cancellation and Quark-Line Filtering

Abstract

The recent ATLAS observation of the $B_c^{\ast }(1S)$ meson fixes the ground-state hyperfine splitting and thereby determines the phase space for $B_c^{\ast }(1S)\to B_c(1S)\gamma$. We study four $1S$ and $2S$ M1 radiative transitions using relativistic Salpeter wave functions calibrated to the measured ground-state masses, obtaining $\Gamma [B_c^{\ast }(1S)\to B_c(1S)\gamma ]=5.24\times 10^{-2}$~keV. The allowed transitions show no radial cancellation, whereas the hindered $2S\to 1S$ amplitudes are residuals of an approximately twenty-fold cancellation across the radial node. A quark-line-resolved analysis shows a strong asymmetry in the crossed hindered channel: the $b$-emission contribution undergoes an $\mathcal{O}(10^2)$ internal cancellation while the $\bar{c}$ contribution remains at $\mathcal{O}(20)$. The corresponding unsigned line strengths show that the small $b$-current is generated by enhanced cancellation rather than by a smaller unsigned contribution. The unmeasured $B_c^{\ast }(2S)$ mass remains the main limitation on the $2S\to 2S$ radiative width.