2 TABLE I: Known c ¯ s mesons organized according to s P ℓ and J $^{P}$; the measured masses are indicated. Allowed possibilities for D$_{sJ}$ (3040) are displayed as D$_{sJ}$ ( 3040 )?. s P ℓ 1 2 − 1 2 + 3 2 + 3 2 − 5 2 − ( n = 1) J P = s P ℓ − 1 2 D$_{s}$ (1965) (0 $^{−}$) D$_{sJ}$ (2317) (0 $^{+}$) D$_{s}$ 1 (2536) (1 $^{+}$) (1 $^{−}$) D$_{sJ}$ ( 3040 )?(2 $^{−}$) J P = s P ℓ + 1 2 D ∗ $_{s}$(2112) (1 $^{−}$) D$_{sJ}$ (2460) (1 $^{+}$) D ∗ s $_{2}$(2573) (2$^{+}$) D$_{sJ}$ ( 3040 )?(2 $^{−}$) D$_{sJ}$ (2860)?(3 $^{−}$) ( n = 2) J P = s P ℓ − 1 2 (0 $^{−}$) (0 $^{+}$) D$_{sJ}$ ( 3040 )?(1 $^{+}$) (1 $^{−}$) (2 $^{−}$) J P = s P ℓ + 1 2 D$_{sJ}$ (2710) (1 $^{−}$) D$_{sJ}$ ( 3040 )?(1 $^{+}$) (2 $^{+}$) (2 $^{−}$) (3 $^{−}$) ( D$_{sJ}$ (2860) has been assigned to the s$_{ℓ}$ = 5 2 doublet, with a question mark since confirmation is needed). Some indications about the masses of these states come from potential model calculations. For example, in Ref.[11] the spectrum of heavy-light mesons is computed in the framework of a relativistic quark model (RQM), with results: Notice that, if the identification of D$_{sJ}$ (2860) as the J P s$_{ℓ}$ = 3 − 5 / 2 state were experimentally confirmed, this would disfavor the assignment of D$_{sJ}$ (3040) to its spin partner D ∗′ s 2 with J P s$_{ℓ}$ = 2 − 5 / 2 , since a mass inversion in a spin doublet seems unlikely. For a similar reason, one would also disfavor the identification of D$_{sJ}$ (3040) with D$_{s}$$_{2}$ , although in that case the two mesons would belong to different doublets. with the various operators annihilating mesons of four- velocity v which is conserved in strong interaction pro- cesses (the heavy field operators contain a factor √ m$_{P}$ and have dimension 3 / 2). Let us consider decays with the emission of a light pseudoscalar meson. The octet of light pseudoscalar mesons is introduced considering the fields: ξ = e i M f π , Σ = ξ $^{2}$, and the matrix M containing π, K and η fields ( f$_{π}$ = 132 MeV): M ( ˜ D$_{s}$$_{1}$ ) ( RQM ) = 3114 MeV M ( ˜ D ′ s $_{1}$) ( RQM ) = 3165 MeV M ( D$_{s}$$_{2}$ ) ( RQM ) = 2953 MeV (3) M ( D ∗′ s $_{2}$) ( RQM ) = 2900 MeV . M =    √ 1 $_{2}$π 0 + √ 1 $_{6}$η π + K + π − − √ 1 $_{2}$π 0 + √ 1 $_{6}$η K 0 K − ¯ K 0 − √ 2 $_{3}$η    (5) At the leading order in the heavy quark mass and light meson momentum expansion, the decays F → HM ( F = H, S, T, X, X ′ and M a light pseudoscalar meson) are described by the Lagrangian interaction terms [12]: The four classifications for D$_{sJ}$ (3040) in Table I can be discussed computing the allowed strong decays. To this purpose, we work in the heavy quark limit in which the various spin doublets are described by effective fields: H$_{a}$ for s P ℓ = 1 2 − ( a = u, d, s is a light flavour index); S$_{a}$ and T$_{a}$ for s P ℓ = 1 2 + and s P ℓ = 3 2 $^{+}$, respectively; X$_{a}$ and X ′ a for the doublets corresponding to orbital angular momentum ℓ = 2, i.e. s P ℓ = 3 2 − and s P ℓ = 5 2 $^{−}$: H$_{a}$ = 1 + v / 2 [ P ∗ $_{aµ}$γ µ − P$_{a}$γ$_{5}$ ] S$_{a}$ = 1 + v / 2 [ P ′ µ 1 $_{a}$γ$_{µ}$ γ$_{5}$ − P ∗ 0 a ] T µ a = 1 + v / 2 { P µν 2 a γ$_{ν}$ − P$_{1}$$_{aν}$ √ 3 2 γ$_{5}$ [ g µν − 1 3 γ $^{ν}$( γ µ − v $^{µ}$) ] } (4) X µ a = 1 + v / 2 { P ∗ µν 2 a γ$_{5}$γ$_{ν}$ − P ∗′ 1 aν √ 3 2 [ g µν − 1 3 γ $^{ν}$( γ µ − v $^{µ}$) ] } X ′ µν a = 1 + v / 2 { P µνσ 3 a γ$_{σ}$ − P ∗ $^{′}$αβ 2 a √ 5 3 γ$_{5}$ [ g µ $_{α}$g ν β − 1 5 γ$_{α}$g ν $_{β}$( γ µ − v $^{µ}$) − 1 5 γ$_{β}$g µ $_{α}$( γ ν − v $^{ν}$) ]} L H = g T r [ ¯ H$_{a}$H$_{b}$γ$_{µ}$γ$_{5}$ A µ ba ] L S = h T r ¯ H$_{a}$S$_{b}$γ$_{µ}$γ$_{5}$ A µ ba + h.c. L T = h ′ Λ$_{χ}$ T r [ ¯ H$_{a}$T µ b ( iD$_{µ}$ ̸ A + i D A $_{µ}$)$_{ba}$ γ$_{5}$ ] + h.c. L X = k ′ Λ$_{χ}$ T r ¯ H$_{a}$X µ b ( iD$_{µ}$ ̸ A + i D A $_{µ}$)$_{ba}$ γ$_{5}$ + h.c. (6) L X ′ = 1 Λ$_{χ}$ $_{2}$T r [ ¯ H$_{a}$X ′ µν b [ k$_{1}$ { D$_{µ}$, D$_{ν}$ }A λ + k$_{2}$ ( D$_{µ}$D$_{λ}$ A ν + D$_{ν}$D$_{λ}$ A $_{µ}$) ] $_{ba}$γ$^{λ}$γ$_{5}$ ] + h.c.