# Study of1D Strange Charmed Meson Family Using HQET

Authors: Pallavi Gupta; A. Upadhyay
Journal: Advances in High Energy Physics
 (2016)

---

## Abstract

Recently LHCb predicted spin 1 and spin 3 statesDs1⁎(2860) and Ds3⁎(2860) which are studied through their strong decays and are assigned to fit the 13D1 and 13D3 states in the charm spectroscopy. In this paper, using the heavy quark effective theory, we state that assigning Ds1⁎(2860) as the mixing of 13D1-23S1 states is rather a better justification to its observed experimental values than a pure state. We study its decay modes variation with hadronic coupling constant gxh and the mixing angle θ. We appoint spin 3 state Ds3⁎(2860) as the missing 1D3-JP state and also study its decay channel behavior with coupling constant gyh. To appreciate the above results, we check the variation of decay modes for their spin partners states, that is, 1D2 and 1D2′, with their masses and strong coupling constant, that is, gxh and gyh. Our calculation using HQET approach gives mixing angle of the 13D1-23S1 state for Ds1⁎(2860) to lie in the range (-1.6 radians ≤θ≤-1.2 radians). Our calculation for coupling constant values gives gxh to lie within value range of 0.17–0.20 and gyh to be 0.40. We expect from experiments to observe this mixing angle to verify our results.


---

## Body

## 1. Introduction

Over the last decade many new heavy-light mesons(Qq¯) have been observed by various experimental collaborations. The state DsJ∗(2860) was first observed by the BaBar Collaboration in DsJ(2860)→D0K+,D+K0 with mass M=2856.6±1.5MeV and width Γ=48±7MeV [1]. It was supposed to have natural parity states, that is, 0+, 1-, 2+, 3-, and so forth. But the assignment of DsJ(2860) as the 0+ state was ruled out after the observation of DsJ(2860)→D∗K [2]. Along with the D∗K channel [2] BaBar also gives the ratio R measured as R=Br(DsJ∗(2860)→D∗K)/Br(DsJ∗(2860)→DK)=1.10±0.15±0.19. Along with this DsJ∗(2860), BaBar Collaboration also observed Ds1∗(2710) state in the DK invariant mass spectrum with mass =2688±4±3 MeV and decay width =112±736 [1]. In [2], BaBar Collaboration reported the branching ratio for this Ds1∗(2710) state as R=Br(Ds1∗(2710)→D∗K)/Br(Ds1∗(2710)→DK)=0.91±0.13±0.12. The DsJ(2860) and Ds1∗(2710) state had went through extensive discussions by various theoretical models, to find a place in strange charm spectrum. Various discussions suggest Ds1∗(2710) to be suitable as 13D3 state or as a radial excitation of S-wave, that is, 23S1 state. Zhang et al. have assigned DsJ∗(2860) as 23P0 or 13D3 states using the P03 model [3], Colangelo et al. assign DsJ∗(2860) to be 13D3 state using the heavy meson effective theory [4, 5], and Li et al. favor DsJ∗(2860) as the 23P0 or 13D3 state using Regge phenomenology [6]. All these different approaches calculated different value of the R ratio R=(Br(DsJ∗(2860)→D∗K)/Br(DsJ∗(2860)→DK)). Heavy quark effective theory predicts R to be ≈ 0.39 [4], while P03 model calculated it to be R=0.59; both of the predicted values of R are far from the experimental value R=1.10. All these references favored DsJ∗(2860) as 13Ds3 state due to observed narrow decay width, at the cost of mismatch of R with experiments.Recently LHCb Collaboration predicted a new resonance around 2.86 GeV in theD¯0K- invariant mass spectrum from decay channel Bs0→D¯0K-Π+, containing the mixture of spin 1 and spin 3 states components corresponding to Ds1∗(2860) and Ds3∗(2860) [7, 8] where the mass and width parameters are (1)MDs1∗2860=2859±12±6±23MeV,ΓDs1∗2860=159±23±27±72MeV,MDs3∗2860=2860.5±2.6±2.5±6.0MeV,ΓDs3∗2860=53±7±4±6MeV.Here the first error is statistical error, the second is the experimental systematic effects, and the last one is due to model variations. Thus LHCb observed two Ds∗(2860) states with spin 1 and spin 3. From the previous study it can be speculated that it is the spin 3 resonance of Ds∗(2860) that belongs to 13Ds3 state, with a narrow width Γ=53MeV. Theoretically, R value can be matched with the experimental value, considering its contribution coming from the spin 1 state of Ds1∗(2860) resonance. Comparing with earlier theoretical mass predictions, LHCb spin 1 resonance Ds1∗(2860) can be assumed to fit in 13Ds1 state of 1D family or can be a mixture of 13D1 and 23S1 states since both these states have the same orbital angular momentum. Assigning Ds1∗ as a mixing state of 13D1-23S1 may give a better justification than assigning it as a pure state, because the R value calculated by taking Ds1∗(2860) to be a mixed state of 13D1-23S1 now depends on the mixing angle of these mixed states. By choosing suitable mixing angle (θ), the calculated R value can be better justified with the experimental value. Li and Ma assign DsJ(2860) to be mixing state of 13D1-23S1 and Ds1∗(2700) to be its orthogonal partner [9] and obtained R=0.8, nearly close to the experimental value. Zhong and Zhao by chiral quark model [10, 11] studied the DsJ∗ state as the 13D3 state with some 13D2-11D2 mixing. Wang [12] tried to reproduce the experimental value R=1.10 with some suitable hadronic coupling constants, by including chiral symmetry breaking corrections in heavy quark effective theory. Besides these studies, Vijande et al. also assign DsJ(2860) to be the multiquark exotic state as cs¯-cns¯n¯ [13]. Godfrey and Jardine by adopting the pseudoscalar emission decay model [14] and Song et al. by adopting QPC model [15] studied DsJ∗ as 13D1-23S1. Various predictions are made to study the mixing effects in DsJ∗ state [16–19].In Particle Data Group [20] 1S and 1P strange charmed states are nicely described, but information for other states is still missing. The strange meson states with their JP states predicted by various theoretical model are gathered in Table 1. From the mass predicted by various theoretical models, that is, from second, third, and fourth columns of Table 1, Ds1∗(2860) and Ds3∗(2860) can be fitted as spin 1 and spin 3 state of 1D family.Table 1
Theoretically predicted masses.
J
P
(
2
s
+
1
L
j
)
GI [21]
PE [22]
EFG [23]
Experimental [20]
(MeV)
(MeV)
(MeV)
(MeV)
0
-
1
S
0
1979
1965
1969
1968
1
-
3
S
1
2129
2113
2111
2112
0
+
3
P
0
2484
2487
2509
2318
1
+
1
P
1
2459
2535
2536
2460
1
+
3
P
1
2556
2605
2574
2536
2
+
3
P
2
2592
2581
2571
2573
1
-
3
D
1
2899
2913
2913
2859
2
-
1
D
2
2900
2900
2931
—
2
-
3
D
2
′
2926
2953
2961
—
3
-
3
D
3
2917
2925
2871
2860In this paper, the four states of1D stranged charm meson family are analyzed by studying their decay widths and branching ratios. For this heavy quark effective theory is used and the importance of mixing of the two states is surveyed. In the past years, HQET has been successful in assigning suitable JP states to the observed D and B mesons using their decays widths in terms of coupling constants. We use HQET approach to study spin 1 resonance of LHCb Ds1∗(2860), by assigning it to be the first member of 1D stranged charm meson family. Properties of this state are examined in two ways, firstly by considering it as a pure spin 1 state of 13D1 and secondly by assuming this state as a mixture of 13D1 and 23S1 states. LHCb predicted spin 3 state Ds3∗(2860) is studied by assigning it the 13D3 position in 1D strange charm mesons. To complete the 1D family, we also try to study the behavior of their spin partners, that is, 1D2 and 1D2′, which are still missing experimentally.The paper is divided into the following sections. Section2 describes the heavy quark effective theory formalism used for the strong decays. Section 3 discusses the members of 1D family. In this section, all the four states with their decay modes in terms of their couplings are described in different subsections. To appreciate the experimental value of R, various mixing effects in terms of mixing angle theta are studied. We finally conclude our results in Section 4.


## 2. Framework

In the heavy quark limitmQ≫ΛQCD≫mq, Qq¯ system can be effectively studied using heavy quark effective theory. According to this theory, heavy quark acts like static color source with spin sQ, which, due to heavy flavor symmetry, interacts only with the light degree of freedom having spin sl through the exchange of soft gluons. This picture can be compared with that of hydrogen atom [24]. The basic idea is that, in a Qq¯ system, heavy quark plays the role of a nucleus and the light quark plays the role of an electron. This Qq¯ system can be categorized in doublets in relation to the total conserved angular momentum, that is, sl=sq¯+L, where sq¯ and L are the spin and orbital angular momentum of the light antiquark, respectively. For L=0 (S-wave), the doublet is represented by (D,D∗) with JslP=(0-,1-)1/2, in which for L=1 (P-wave), there are two doublets represented by (D0∗,D1) and (D1′,D2∗) with JslP=(0+,1+)1/2 and (1+,2+)3/2, respectively. Two doublets of L=2 (D-wave) are represented by (D1∗,D2) and (D2′,D3∗) belonging to JslP=(1-,2-)3/2 and (2-,3-)5/2, respectively. These doublets are described by the effective superfields Ha, Sa, Ta, Xa, Ya [25, 26], where field Ha describes the (D,D∗) doublet, that is, S-wave, and Sa and Ta fields represent the P-wave doublets (0+,1+)1/2 and (1+,2+)3/2, respectively. D-wave doublets are represented by the Xa and Ya fields. These fields are as follows:(2)Ha=1+v2Paμ∗γμ-Paγ5,Sa=1+v2P1aμγμγ5-P0a∗,Taμ=1+v2P2a∗μνγν-P1aν32γ5gμν-γνγμ-υμ3,Xaμ=1+v2P2aμνγ5γν-P1aν∗32gμν-γνγμ+γμ3,Yaμν=1+v2P3a∗μνσγσ-P2aαβ53γ5gαμgβν-gβνγαγμ-vμ5-gαμγβγν-vν5.The light pseudoscalar mesons are described by the fieldsξ=expiM/fπ. The pion octet is introduced by the vector and axial combinations Vμ=1/2ξ∂μξ†+ξ†∂μξ and Aμ=1/2ξ∂μξ†-ξ†∂μξ. We choose fπ=130MeV. Here, all traces are taken over Dirac spinor indices, light quark SU(3)V flavor indices a=u,d,s, and heavy quark flavor indices Q=c,b [25, 26]. The Dirac structure of chiral Lagrangian has been replaced by velocity vector v. At the leading approximation, the heavy meson chiral Lagrangian terms LHH, LSH, LTH, LXH, LXS, LXT, LYH, LYS, LYT for the two-body strong decays to light pseudoscalar mesons can be written as follows:(3)LHH=ghhTrH¯aHbγμγ5Abaμ,LSH=gshTrH¯aSbγμγ5Abaμ+h.c.,LTH=gthΛTrH¯aTbμiDμA+iDAμbaγ5+h.c.,LXH=gxhΛTrH¯aXbμiDμA+iDAμbaγ5+h.c.,LXS=gxsΛTrS¯aXbμiDμA+iDAμbaγ5+h.c.,LXT=1Λ2TrT¯aμXbνk1TDμ,DνAλ+k2TDμDλAν+DνDλAμbaγλγ5+h.c.,LYH=1Λ2TrH¯aYbμνk1HDμ,DνAλ+k2HDμDλAν+DνDλAμbaγλγ5+h.c.,LYS=1Λ2TrS¯aYbμνk1SDμ,DνAλ+k2SDμDλAν+DνDλAμbaγλγ5+h.c.,LYT=gytΛTrT¯aμXbμνiDνA+iDAνbaγ5+h.c.From the chiral Lagrangian terms LHH, LSH, LTH, LXH, LYH, the two-body strong decay of Qq¯ system to final state light pseudoscalar mesons M(π,η,K) can be described as(1-,2-)→(0-,1-)+M(4)Γ1-⟶0-=CM4gxh29πfπ2Λ2MfMipM3mM2+pM2,Γ1-⟶1-=CM2gxh29πfπ2Λ2MfMipM3mM2+pM2,Γ2-⟶1-=CM2gxh23πfπ2Λ2MfMipM3mM2+pM2.(2-,3-)→(0-,1-)+M(5)Γ2-⟶1-=CM4gyh215πfπ2Λ4MfMipM7,Γ3-⟶0-=CM4gyh235πfπ2Λ4MfMipM7,Γ3-⟶1-=CM16gyh2105πfπ2Λ4MfMipM7.(1-,2-)→(0+,1+)+M(6)Γ2-⟶1+=CM2gxs25πfπ2Λ2MfMipM5,Γ2-⟶0+=CM4gxs215πfπ2Λ2MfMipM5,Γ1-⟶1+=CM2gxs23πfπ2Λ2MfMipM5.(2-,3-)→(0+,1+)+M(7)Γ3-⟶1+=CM4gys215πfπ2Λ4MfMipM5mM2+pM2,Γ2-⟶1+=CM8gys275πfπ2Λ4MfMipM5mM2+pM2,Γ2-⟶0+=CM4gys225πfπ2Λ4MfMipM5mM2+pM2.In the above expressions of decay width, Mi, Mf stand for initial and final meson mass. Hadronic coupling constants gxh, gxs, gxt=k1T+k2T, gyh=k1H+k2H, gys=k1S+k2S, and gyt are dependent on the radial quantum number, Λ is the chiral symmetry breaking scale =1GeV, and pM and mM are the final momentum and mass of the emitted light pseudoscalar meson. The coefficients Cπ±, CK±, CK0, CK¯0=1, Cπ0=1/2, and Cη=2/3 or 1/6 [25, 26]. Different values of Cη correspond to the initial state being cu¯, cd¯, or cs¯, respectively.


## 3. Numerical Results

OZI allowed two-body strong decays of1D strange charm family are calculated using the heavy quark effective approach as given in Section 2. In the present work, partial and total decay widths of these four 1D states are studied and compared with their experimental values. OZI allowed decay channels for Ds1∗(2860) and Ds3∗ states are DK, D∗K, Dsη, and Ds∗η, and for their spin partners 1Ds2 and 1Ds2′ states, they are D∗K, Ds∗η, D(2400)K, and Ds∗(2317)η. For this calculation, initial masses of Ds1∗ and Ds3∗ states, as given by the LHCb [7, 8], have been used as input parameters along with 2890 MeV and 2900 MeV for their spin partner states 1Ds2 and 1Ds2′, respectively. Heavy quark effective theory shows that decay widths also depend on the strong hadronic couplings gxh, gyh, gxs, and gys. The theoretical value of the strong coupling constants has been constrained within the range of 0 and 1 [27] though their experimental information is still missing. In the next subsections, we have calculated two of these coupling constants, that is, gxh and gyh, using the decay widths and available experimental data.


### 3.1.Ds1∗(2860)

D
s
1
∗
(
2860
) was first observed by BaBar Collaboration and in 2014 its spin, mass, and decay width were confirmed by LHCb. In this subsection, heavy quark effective theory is adopted to reproduce the experimental data given by these collaborations. Also the coupling constant gxh is determined by assigning Ds1∗(2860) state as the 1- member of the 1D charm family. Assuming it to be the pure 1D1- state, we calculated the total and partial decay widths of Ds1∗(2860) using the decay width formulae given in Section 2 in terms of their hadronic coupling constants. These partial decay widths and ratios are tabulated in Table 2. Along with the partial decay widths, we also studied the ratios such as(8)R=BrDs1∗2860⟶D∗KBrDs1∗2860⟶DK,R1=BrDs1∗2860⟶DsηBrDs1∗2860⟶DK.Table 2
Calculated partial and total decay widths ofDs1∗(2860) as pure (13D1).
Theory
D
K
D
∗
K
D
s
η
D
s
∗
η
Total
D
∗
K
/
D
K
D
s
η
/
D
K
Our
2865.45gxh2
693.135gxh2
508.189gxh2
85.70gxh2
4152.48gxh2
0.24
0.177
Experimental [8]
159
1.10It can be seen from Table2 that our calculated R value does not match with the experimental value 1.10. Ratios calculated in Table 2 have also been calculated by various other theoretical models as shown in Table 3.Table 3
Reference [15]
Reference [18]
Reference [5]
Γ
D
∗
K
/
D
K
0.46–0.70
12.5–7.6
0.06
Γ
D
s
η
/
D
K
0.10–0.14
0.30–0.14
0.23It can be seen thatR value, that is, Γ(D∗K/DK) calculated by our HQET approach and by other theoretical approaches [5, 15, 18], does not match with the experimental R value, that is, 1.10. As R is independent of couplings, to justify the experimental value of R, we include the mixing of the states. According to this scheme, state Ds1∗(2860) is assumed to be the mixture of 23S1 and 13D1 states with Ds(2700) to be its orthogonal partner satisfying the relation (9)Ds12SDs12860=cos⁡θsin⁡θ-sin⁡θcos⁡θ23S113D1,where θ is the mixing angle between these two mixed states. Effect of variation of total decay width of Ds1(2860) state with coupling constant gxh for different mixing angles is shown in Figures 1–4, which shows the variation for some typical values of mixing angle at θ=0°, θ=-30° and for θ=-60° and θ=-80° where θ=0° correspond to nonmixing, that is, pure 13D1 state.Figure 1
Decay widths ofDs1(2860) for θ=0°.Figure 2
Decay widths ofDs1(2860) for θ=-30°.Figure 3
Decay widths ofDs1(2860) for θ=-60°.Figure 4
Decay widths ofDs1(2860) for θ=-80°.Figures1, 2, 3, and 4 show that DK is the main decay channel of 13D1 state. Apart from DK, D∗K and Dsη are also important decay channels of 13D1, whereas the calculated decay width for Ds∗η is found to be small. Dominance of Ds∗η decay channel increases with large value of mixing angle. R ratio defined in Section 1 now depends on both the mixing angle and strong coupling constants gxh and ghh. Variation of R value with the mixing angle, by fixing ghh=0.17 [25], is shown in Figure 5. This figure shows that R≃8.5 corresponding to the mixing angle of range -1.6≤θ≤-1.2 radians. This obtained R value is near to the experimental R value, R=1.10±0.15±0.19. For this range of mixing angle our hadronic coupling constant comes out to be within 0.17≤gxh≤0.20. This variation of gxh hadronic coupling with the mixing angle has been shown in Figure 6.Figure 5
Variation ofR value with mixing angle.Figure 6
Variation of coupling constantgxh with mixing angle.For these calculated values of mixing angle and coupling constant, partial and total decay widths are again studied. Total widthΓ comes out to be 159 MeV, which matches very well with the experimental data. Other partial decay widths are listed in Table 4. The calculated R value from our approach is matching well with the experimental observed value 1.10±0.15±0.19.Table 4
Calculated partial and total decay widths ofDs1∗(2860) as a mixture of (13D1) and 23S1.
Theory
D
K
D
∗
K
D
s
η
D
s
∗
η
Total
D
∗
K
/
D
K
D
s
η
/
D
K
(MeV)
(MeV)
(MeV)
(MeV)
(MeV)
Our
60.61
55.26
11.85
18.66
146.39
0.91
0.19
Experimental
159
1.10


### 3.2.Ds3∗(2860)

Considering spin 3 resonanceDs3∗(2860) of LHCb, as the 13Ds3 state, decay channels and partial decay widths are presented in Table 5. Figure 7 shows the variation of the partial and total decay width with coupling constant gyh.Table 5
Calculated partial and total decay widths ofDs3∗(2860) as 13Ds3 state in terms of hadronic coupling gyh.
Theory
D
K
D
∗
K
D
s
η
D
s
∗
η
Total
(MeV)
(MeV)
(MeV)
(MeV)
(MeV)
Our
249.18gyh2
96.39gyh2
24.72gyh2
4.54gyh2
374.846gyh2
Experimental
53Figure 7
The variation of partial decay widths ofDs3∗(2860) as the 13Ds3 state with hadronic coupling.Figure7 clearly shows that DK is the dominant decay mode of Ds3∗(2860). Other important decay channels are D∗K, Dsη with Ds∗η contributing least. Computing it with the experimental value of total decay width Γ=53MeV, coupling constant gyh comes out to be 0.40. These partial decay widths can be used to calculate the ratio R:(10)R=ΓDs3∗2860⟶D∗KΓDs3∗2860⟶DK=0.38R1=ΓDs3∗2860⟶DsηΓDs3∗2860⟶DK=0.03.These ratios are compared with predictions made by various other theoretical models as shown in Table 6.Table 6
Reference [5]
Reference [10]
Reference [7]
Γ
D
∗
K
/
D
K
0.39
0.43
0.8
Γ
D
s
η
/
D
K
0.13
0.11
0.05


### 3.3.1Ds2 and 1Ds2′

1
D
s
2 is the spin partner of Ds1∗(2860) belonging to JP as 23/2- state, and 1Ds2′ state belongs to JslP to 25/2-. Both these states are still unknown in the charm meson spectrum. As shown in Table 1, their masses have been already predicted by various theoretical models [21–23]. Taking their masses to be within the allowed range 2800 MeV to 3000 MeV, variation of their total OZI allowed two-body strong decay width has been plotted with respect to their mass and their corresponding coupling constant, in Figures 8 and 9, respectively.Figure 8
Variation of1Ds2 with its mass and coupling constant gxh.Figure 9
Variation of1Ds2′ with its mass and coupling constant.Using the hadronic couplings obtained in Sections3.1 and 3.2, gxh=0.20 and gyh=0.40, partial and total decay widths of these states are listed in first column of Table 7. Also, these two states can mix through spin-orbit interaction or by some other mechanism and physically Ds2′ and Ds2 can be represented as the linear combination of D23 and D21 states as (11)1D2-1D′2-=cos⁡θ1Dsin⁡θ1D-sin⁡θ1Dcos⁡θ1D13D211D2,where θ1D is the mixing angle between the two D23 and D21 states. In general the mixing angle between Ll3 and Ll1 in heavy quark limit is given by θ1D=arctan⁡L/(L+1). For this case the mixing angle corresponds to L=2 and comes out to be 39.2°~39°. In Table 7, the last column gives partial decay widths by taking this mixing into account.Table 7
Calculated partial and total decay widths of1Ds2 and 1Ds2′. First section is calculated by taking them as pure states and the second section includes mixing scheme into account.
Decay channel
As pure states
As mixed states
(θ=0°)
(θ=-39°)
1
D
2
1
D
2
′
1
D
2
1
D
2
′
(MeV)
(MeV)
(MeV)
(MeV)
D
∗
K
61.37
20.51
36.02
56.95
D
s
∗
η
16.69
2.51
11.62
13.42
D
(
2
400
)
K
—
7.5
×
1
0
-
5
—
1.5
×
1
0
-
4
D
s
∗
(
2317
)
η
0.0037
0.005
0.0014
0.01
Total
78.06
21.04
47.65
70.38


## 3.1.Ds1∗(2860)

D
s
1
∗
(
2860
) was first observed by BaBar Collaboration and in 2014 its spin, mass, and decay width were confirmed by LHCb. In this subsection, heavy quark effective theory is adopted to reproduce the experimental data given by these collaborations. Also the coupling constant gxh is determined by assigning Ds1∗(2860) state as the 1- member of the 1D charm family. Assuming it to be the pure 1D1- state, we calculated the total and partial decay widths of Ds1∗(2860) using the decay width formulae given in Section 2 in terms of their hadronic coupling constants. These partial decay widths and ratios are tabulated in Table 2. Along with the partial decay widths, we also studied the ratios such as(8)R=BrDs1∗2860⟶D∗KBrDs1∗2860⟶DK,R1=BrDs1∗2860⟶DsηBrDs1∗2860⟶DK.Table 2
Calculated partial and total decay widths ofDs1∗(2860) as pure (13D1).
Theory
D
K
D
∗
K
D
s
η
D
s
∗
η
Total
D
∗
K
/
D
K
D
s
η
/
D
K
Our
2865.45gxh2
693.135gxh2
508.189gxh2
85.70gxh2
4152.48gxh2
0.24
0.177
Experimental [8]
159
1.10It can be seen from Table2 that our calculated R value does not match with the experimental value 1.10. Ratios calculated in Table 2 have also been calculated by various other theoretical models as shown in Table 3.Table 3
Reference [15]
Reference [18]
Reference [5]
Γ
D
∗
K
/
D
K
0.46–0.70
12.5–7.6
0.06
Γ
D
s
η
/
D
K
0.10–0.14
0.30–0.14
0.23It can be seen thatR value, that is, Γ(D∗K/DK) calculated by our HQET approach and by other theoretical approaches [5, 15, 18], does not match with the experimental R value, that is, 1.10. As R is independent of couplings, to justify the experimental value of R, we include the mixing of the states. According to this scheme, state Ds1∗(2860) is assumed to be the mixture of 23S1 and 13D1 states with Ds(2700) to be its orthogonal partner satisfying the relation (9)Ds12SDs12860=cos⁡θsin⁡θ-sin⁡θcos⁡θ23S113D1,where θ is the mixing angle between these two mixed states. Effect of variation of total decay width of Ds1(2860) state with coupling constant gxh for different mixing angles is shown in Figures 1–4, which shows the variation for some typical values of mixing angle at θ=0°, θ=-30° and for θ=-60° and θ=-80° where θ=0° correspond to nonmixing, that is, pure 13D1 state.Figure 1
Decay widths ofDs1(2860) for θ=0°.Figure 2
Decay widths ofDs1(2860) for θ=-30°.Figure 3
Decay widths ofDs1(2860) for θ=-60°.Figure 4
Decay widths ofDs1(2860) for θ=-80°.Figures1, 2, 3, and 4 show that DK is the main decay channel of 13D1 state. Apart from DK, D∗K and Dsη are also important decay channels of 13D1, whereas the calculated decay width for Ds∗η is found to be small. Dominance of Ds∗η decay channel increases with large value of mixing angle. R ratio defined in Section 1 now depends on both the mixing angle and strong coupling constants gxh and ghh. Variation of R value with the mixing angle, by fixing ghh=0.17 [25], is shown in Figure 5. This figure shows that R≃8.5 corresponding to the mixing angle of range -1.6≤θ≤-1.2 radians. This obtained R value is near to the experimental R value, R=1.10±0.15±0.19. For this range of mixing angle our hadronic coupling constant comes out to be within 0.17≤gxh≤0.20. This variation of gxh hadronic coupling with the mixing angle has been shown in Figure 6.Figure 5
Variation ofR value with mixing angle.Figure 6
Variation of coupling constantgxh with mixing angle.For these calculated values of mixing angle and coupling constant, partial and total decay widths are again studied. Total widthΓ comes out to be 159 MeV, which matches very well with the experimental data. Other partial decay widths are listed in Table 4. The calculated R value from our approach is matching well with the experimental observed value 1.10±0.15±0.19.Table 4
Calculated partial and total decay widths ofDs1∗(2860) as a mixture of (13D1) and 23S1.
Theory
D
K
D
∗
K
D
s
η
D
s
∗
η
Total
D
∗
K
/
D
K
D
s
η
/
D
K
(MeV)
(MeV)
(MeV)
(MeV)
(MeV)
Our
60.61
55.26
11.85
18.66
146.39
0.91
0.19
Experimental
159
1.10


## 3.2.Ds3∗(2860)

Considering spin 3 resonanceDs3∗(2860) of LHCb, as the 13Ds3 state, decay channels and partial decay widths are presented in Table 5. Figure 7 shows the variation of the partial and total decay width with coupling constant gyh.Table 5
Calculated partial and total decay widths ofDs3∗(2860) as 13Ds3 state in terms of hadronic coupling gyh.
Theory
D
K
D
∗
K
D
s
η
D
s
∗
η
Total
(MeV)
(MeV)
(MeV)
(MeV)
(MeV)
Our
249.18gyh2
96.39gyh2
24.72gyh2
4.54gyh2
374.846gyh2
Experimental
53Figure 7
The variation of partial decay widths ofDs3∗(2860) as the 13Ds3 state with hadronic coupling.Figure7 clearly shows that DK is the dominant decay mode of Ds3∗(2860). Other important decay channels are D∗K, Dsη with Ds∗η contributing least. Computing it with the experimental value of total decay width Γ=53MeV, coupling constant gyh comes out to be 0.40. These partial decay widths can be used to calculate the ratio R:(10)R=ΓDs3∗2860⟶D∗KΓDs3∗2860⟶DK=0.38R1=ΓDs3∗2860⟶DsηΓDs3∗2860⟶DK=0.03.These ratios are compared with predictions made by various other theoretical models as shown in Table 6.Table 6
Reference [5]
Reference [10]
Reference [7]
Γ
D
∗
K
/
D
K
0.39
0.43
0.8
Γ
D
s
η
/
D
K
0.13
0.11
0.05


## 3.3.1Ds2 and 1Ds2′

1
D
s
2 is the spin partner of Ds1∗(2860) belonging to JP as 23/2- state, and 1Ds2′ state belongs to JslP to 25/2-. Both these states are still unknown in the charm meson spectrum. As shown in Table 1, their masses have been already predicted by various theoretical models [21–23]. Taking their masses to be within the allowed range 2800 MeV to 3000 MeV, variation of their total OZI allowed two-body strong decay width has been plotted with respect to their mass and their corresponding coupling constant, in Figures 8 and 9, respectively.Figure 8
Variation of1Ds2 with its mass and coupling constant gxh.Figure 9
Variation of1Ds2′ with its mass and coupling constant.Using the hadronic couplings obtained in Sections3.1 and 3.2, gxh=0.20 and gyh=0.40, partial and total decay widths of these states are listed in first column of Table 7. Also, these two states can mix through spin-orbit interaction or by some other mechanism and physically Ds2′ and Ds2 can be represented as the linear combination of D23 and D21 states as (11)1D2-1D′2-=cos⁡θ1Dsin⁡θ1D-sin⁡θ1Dcos⁡θ1D13D211D2,where θ1D is the mixing angle between the two D23 and D21 states. In general the mixing angle between Ll3 and Ll1 in heavy quark limit is given by θ1D=arctan⁡L/(L+1). For this case the mixing angle corresponds to L=2 and comes out to be 39.2°~39°. In Table 7, the last column gives partial decay widths by taking this mixing into account.Table 7
Calculated partial and total decay widths of1Ds2 and 1Ds2′. First section is calculated by taking them as pure states and the second section includes mixing scheme into account.
Decay channel
As pure states
As mixed states
(θ=0°)
(θ=-39°)
1
D
2
1
D
2
′
1
D
2
1
D
2
′
(MeV)
(MeV)
(MeV)
(MeV)
D
∗
K
61.37
20.51
36.02
56.95
D
s
∗
η
16.69
2.51
11.62
13.42
D
(
2
400
)
K
—
7.5
×
1
0
-
5
—
1.5
×
1
0
-
4
D
s
∗
(
2317
)
η
0.0037
0.005
0.0014
0.01
Total
78.06
21.04
47.65
70.38


## 4. Conclusion

Due to advancement in high energy accelerators, large amount of information is available on heavy-light charm and bottom mesons. This information motivates theorists to explore more about these heavy-light mesons. TheseD and B meson states are studied by observing their decaying behavior, masses, their JP states, coupling constants, branching ratios, and so forth. Many models like heavy quark effective theory, quark pair creation model, potential models, and so forth, are framed to study these heavy-light mesons. Recently, LHCb predicted spin 1 and spin 3 strange charm mesons. In this paper, we use the heavy quark effective approach to study the recently observed spin 1 and spin 3 strange charm states. This theory treats the heavy quark as static and provides Lagrangian and decay widths formulas to the available states. This theory has adequately studied the previously determined experimental states and successfully allotted their positions in the charm and bottom spectroscopy.Observation of spin 1 and spin 3 resonances ofDs∗(2860) by LHCb has clearly indicated that there are two different states of DsJ∗(2860). In the last 5 years, various theoretical models [5–7, 10–19], which studied DsJ∗(2860), favored it as 13D3 state with narrow decay width. From the LHCb data, Ds3∗(2860) state with Γ=53MeV can be correlated with this DsJ∗(2860) state. We too studied the decay behavior of Ds3∗(2860) assuming it to be in the 13D3 state and calculated the hadronic coupling constant gyh=0.40. This value can be compared with the one obtained by Wang gyh=0.52 [25].We also studied the remaining spin 1 observed state by LHCbDs1∗(2860), assuming it to be pure 13D1s state and to be a mixture of 13D1 and 23S1 state. We study its decay channels (D∗K,DK,Ds∗η,Dsη) and R value (D∗K/DK) calculated for the pure state (R=0.48) which does not lie within the given experimental data (R=1.10). So we adopted it to be as a mixture of radially excited 23S1 and orbitally excited 13D1. Using this interpretation, decay widths and R value depend on mixing angle (θ) and coupling constant gxh. We studied the variation of partial widths with coupling constant gxh for some fixed values of mixing angle (θ) which shows DK is the dominant decay channel. In the variation of R value with mixing angle (θ), experimental R value favors the large mixing angle. This large mixing angle implies the predominance of 23S1 state for Ds1∗(2860). We obtained R=0.85 corresponding to the mixing angle -1.6≤θ≤-1.2. Along with this mixing angle, we constrain the coupling constant gxh to be lying in the range 0.17≤gxh≤0.20. This obtained coupling value is close to the value given by Wang gxh=0.19 in [25].Using these coupling constants, we also calculated the decay behavior of the spin partners of these states1D2 and 1D2s′. These states are studied using two ways, first by considering them as pure states and secondly by taking their mixing into account. In both cases D∗K is the dominating decay channel. Decay width for 1D2s′ as a pure state comes to be small indicating the presence of other decay modes. As we have only considered the decays to pseudoscalar mesons, there may be a possibility that decays to light vectors mesons may also be present for this state.



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