4 TABLE II: Features of the decay modes of D$_{sJ}$ (3040) and of its spin partner at leading order in 1 m$_{c}$ expansion for the four proposed assignments. decay modes ˜ D ′ s 1 ( n = 2 , J P s$_{ℓ}$ = 1 + 1 / $_{2}$) ˜ D$_{s}$ 1 ( n = 2 , J P s$_{ℓ}$ = 1 + 3 / $_{2}$) D$_{s}$ 2 ( n = 1 , J P s$_{ℓ}$ = 2 − 3 / $_{2}$) D ∗′ s 2 ( n = 1 , J P s$_{ℓ}$ = 2 − 5 / $_{2}$) D $^{∗}$K , D ∗ $_{s}$η s − wave d − wave p − wave f − wave R 1 0.34 0.20 0.245 0.143 D ∗ $_{0}$K , D ∗ s $_{0}$η , D ′ $_{1}$K p − wave p − wave d − wave d − wave D 1 K p − wave p − wave - d − wave D ∗ $_{2}$K p − wave p − wave s − wave d − wave DK $^{∗}$, D$_{s}$φ s − wave s − wave p − wave p − wave Γ ≃ 140 MeV Γ ≃ 20 MeV negligible negligible spin partner ˜ D ∗ s 0 ( n = 2 , J P s$_{ℓ}$ = 0 + 1 / $_{2}$) ˜ D ∗ s 2 ( n = 2 , J P s$_{ℓ}$ = 2 + 3 / $_{2}$) D ∗ s 1 ( n = 1 , J P s$_{ℓ}$ = 1 − 3 / $_{2}$) D$_{s}$ 3 ( n = 1 , J P s$_{ℓ}$ = 3 − 5 / $_{2}$) DK , D$_{s}$η s − wave d − wave p − wave f − wave D $^{∗}$K , D ∗ $_{s}$η - d − wave p − wave f − wave D ∗ $_{0}$K , D ∗ s $_{0}$η - - d − wave - D ′ $_{1}$K p − wave p − wave d − wave d − wave D 1 K p − wave p − wave s − wave d − wave D ∗ $_{2}$K - p − wave - d − wave [18]. In the second case, the identification with ˜ D ′ s 1 and ˜ D$_{s}$$_{1}$ is discussed: the full widths of these two states are computed and compared to the experimental measure- ment of Γ( D$_{sJ}$ (3040)), concluding that for ˜ D ′ s 1 the ex- perimental width can be reproduced, with the predic- • If D$_{sJ}$ (3040) is ˜ D ′ s 1 ( s P ℓ = 1 2 $^{+}$, J P = 1 $^{+}$, n = 2), its spin partner is ˜ D ∗ s $_{0}$, a J P = 0 + state, the first radial excitation of D$_{sJ}$ (2317). This state can decay to DK and D$_{s}$η in s -wave; p -wave decays to D ′ $_{1}$K and D$_{1}$K are also allowed. • If D$_{sJ}$ (3040) is ˜ D$_{s}$$_{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}$$_{2}$ ( s P ℓ = 3 2 $^{−}$, J P = 2 $^{−}$, n = 1), its spin partner is the vector meson D ∗ s 1 with J P = 1 $^{−}$. It can decay to DK , D$_{s}$η , D $^{∗}$K , D ∗ $_{s}$η , DK ∗ and D$_{s}$φ in p -wave, to D ∗ $_{0}$K , D ∗ s $_{0}$η and D ′ $_{1}$K in d − wave and to D$_{1}$K in s − wave. The decay to D ∗ $_{2}$K is allowed at O ( 1 m$_{c}$ ) in d − wave. • If D$_{sJ}$ (3040) is D ∗′ s 2 ( s P ℓ = 5 2 $^{−}$, J P = 2 $^{−}$, n = 1), its spin partner is D$_{s}$$_{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 FIG. 1: Spectrum of the c ¯ s system. All observed D$_{sJ}$ states, with mass indicated on the y axis, are assigned to a level with J P and proper name. The four assignments discussed for D$_{sJ}$ (3040) are shown in correspondence to the mass value M = 3040 MeV. D$_{s}$ D$_{s}$ GLYPH<Star> D$_{s0}$ GLYPH<Star> D$_{s1}$ ' D$_{s1}$ D$_{s2}$ GLYPH<Star> D$_{s2}$ D$_{s2}$ ' GLYPH<Star> D$_{s3}$ D GLYPH<OverTilde> s GLYPH<Star> D GLYPH<OverTilde> s1 ' D GLYPH<OverTilde> s1 0 GLYPH<Minus> 1 GLYPH<Minus> 0 GLYPH<Plus> 1 GLYPH<Plus> 1 GLYPH<Plus> 2 GLYPH<Plus> 1 GLYPH<Minus> 2 GLYPH<Minus> 2 GLYPH<Minus> 3 GLYPH<Minus> 1969 2112 2317 2460 2536 2573 2710 2860 3040 J P mass GLYPH<LParen1> MeV GLYPH<RParen1>