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• If D$_{sJ}$ (3040) is ˜ D ′ s 1 ( s P ℓ = 1 2 $^{+}$, J P = 1 $^{+}$, n = 2), its spin partner is ˜ D ∗ s $_{0}$, a J P = 0 + state, the first radial excitation of D$_{sJ}$ (2317). This state can decay to DK and D$_{s}$η in s -wave; p -wave decays to D ′ $_{1}$K and D$_{1}$K are also allowed. 

• If D$_{sJ}$ (3040) is ˜ D$_{s}$$_{1}$ ( s P ℓ = 3 2 $^{+}$, J P = 1 $^{+}$, n = 2), its spin partner is ˜ D ∗ s 2 with J P = 2 $^{+}$. It is allowed to decay to DK , D$_{s}$η , D $^{∗}$K , D ∗ $_{s}$η , DK ∗ and D$_{s}$φ in d -wave, and to D ′ $_{1}$K , D$_{1}$K and D ∗ $_{2}$K in p -wave. 

• If D$_{sJ}$ (3040) is D$_{s}$$_{2}$ ( s P ℓ = 3 2 $^{−}$, J P = 2 $^{−}$, n = 1), its spin partner is the vector meson D ∗ s 1 with J P = 1 $^{−}$. It can decay to DK , D$_{s}$η , D $^{∗}$K , D ∗ $_{s}$η , DK ∗ and D$_{s}$φ in p -wave, to D ∗ $_{0}$K , D ∗ s $_{0}$η and D ′ $_{1}$K in d − wave and to D$_{1}$K in s − wave. The decay to D ∗ $_{2}$K is allowed at O ( 1 m$_{c}$ ) in d − wave. ∗′ P 5 P 

• If D$_{sJ}$ (3040) is D s 2 ( s ℓ = 2 $^{−}$, J = 2 $^{−}$, n = 1), its spin partner is D$_{s}$$_{3}$ with J P = 3 $^{−}$, decaying to DK , D$_{s}$η , D $^{∗}$K and D ∗ $_{s}$η , DK $^{∗}$, D$_{s}$φ in f − wave, and to D ′ $_{1}$K , D$_{1}$K and D ∗ $_{2}$K in d − wave. 

Since D$_{sJ}$ (3040) has a broad width, we expect that also its spin partner shares the same feature. Considering the previous list, we can argue that ˜ D ∗ s 0 is broad due to its s -wave decays into DK and D$_{s}$η . Also D ∗ s 1 has allowed s -wave decays, but only to D$_{1}$K which is suppressed by phase space effects. ′ 

The identification of D$_{sJ}$ (3040) with ˜ D s 1 is supported in Refs.[17, 18] on the basis of the c ¯ s mass spectrum [17] or of the decay widths computed in the $^{3}$P$_{0}$ model 

[18]. In the second case, the identification with ˜ D ′ s 1 and ˜ D$_{s}$$_{1}$ is discussed: the full widths of these two states are computed and compared to the experimental measure- ment of Γ( D$_{sJ}$ (3040)), concluding that for ˜ D ′ s 1 the ex- perimental width can be reproduced, with the predic- 

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