Towards establishing an abundant $B$ and $B_s$ spectrum up to the second orbital excitations

Stimulated by the exciting progress in experiments, we carry out a combined analysis of the masses, and strong and radiative decay properties of the $B$ and $B_s$-meson states up to the second orbital excitations. Based on our good descriptions of the mass and decay properties for the low-lying well-established states $B_1(5721)$, $B_2^*(5747)$, $B_{s1}(5830)$ and $B_{s2}^*(5840)$, we give a quark model classification for the high mass resonances observed in recent years. It is found that (i) the $B_{J}(5840)$ resonance may be explained as the low mass mixed state $B(|SDrangle_L)$ via $2^3S_1$-$1^3D_1$ mixing, or the pure $B(2^3S_1)$ state, or $B(2^1S_0)$. (ii) The $B_J(5970)$ resonance may be assigned as the $1^3D_3$ state in the $B$ meson family, although it as a pure $2^3S_1$ state cannot be excluded. (iii) The narrow structure around 6064 MeV observed in the $B^+K^-$ mass spectrum at LHCb may be mainly caused by the $B_{sJ}(6109)$ resonance decaying into $B^{*+}K^-$, and favors the assignment of the high mass $1D$-wave mixed state $B_s(1D_2)$ with $J^P=2^-$, although it as the $1^3D_3$ state cannot be excluded. (iv) The relatively broader $B_{sJ}(6114)$ structure observed at LHCb may be explained with the mixed state $B_s(|SDrangle_H)$ via $2^3S_1$-$1^3D_1$ mixing, or a pure $1^3D_1$ state. Most of the missing $1P$-, $1D$-, and $2S$-wave $B$- and $B_s$-meson states have a relatively narrow width, they are most likely to be observed in their dominant decay channels with a larger data sample at LHCb.