Emergence of heavy quark spin symmetry breaking in heavy quarkonium decays

Heavy-quark spin symmetry (HQSS) implies that in the direct decay of a heavy quarkonium with spin $S$, only lower lying heavy quarkonia with the same spin $S$ can be produced. However, this selection rule, expected to work very well in the $b$-quark sector, can be overcome if multiquark intermediate states are involved in the decay chain, allowing for transitions to the final-state heavy quarkonia with a different spin $S^{\prime }$. In particular, the measured decays $\mathrm{\Upsilon }(10860)\to \pi Z_b^{(\prime )}\to \pi \pi \mathrm{\Upsilon }(nS)$ ($n=1$, 2, 3) and $\mathrm{\Upsilon }(10860)\to \pi Z_b^{(\prime )}\to \pi \pi h_b(mP)$ ($m=1$, 2) appear to have nearly equal strengths, which is conventionally explained by a simultaneous presence of both $S_{b\overline{b}}=0$ and $S_{b\overline{b}}=1$ components in the wave functions of the $Z_b\mathrm{s}$ in equal shares. Meanwhile, the destructive interference between the contributions of the $Z_b$ and $Z_b^{\prime }$ to the decay amplitude for a $\pi \pi h_b$ final state kills the signal to zero in the strict HQSS limit. In this paper, we discuss how the HQSS violation needs to be balanced by the narrowness of the $Z_b^{(\prime )}$ states in the physical case to allow for equal transition strengths into final states with different total heavy quark spins, and how spin symmetry is restored as a result of a subtle interplay of the scales involved, when the mass of a heavy quark becomes infinite. Moreover, we demonstrate how similar branching fractions of the decays into $\pi \pi h_b$ and $\pi \pi \mathrm{\Upsilon }$ can be obtained and how the mentioned HQSS breaking can be reconciled with the dispersive approach to the $\pi \pi /K\overline{K}$ interaction in the final state and matched with the low-energy chiral dynamics in both final states.