Possible 2S and 1D charmed and charmed-strange mesons
Bing Chen,* Ling Yuan, and Ailin Zhang
Department of Physics, Shanghai University, Shanghai 200444, China(Received 22 February 2011; published 10 June 2011)
Possible 2S and 1D excited D and Ds states are studied, the charmed states D25500, D2600,D27500, and D2760 newly observed by the BABAR Collaboration are analyzed. The masses of thesestates are explored within the Regge trajectory phenomenology, and the strong decay widths are computed
within the heavy-quark effective theory. Both the mass and the decay width indicate that D25500 isa good candidate for 21S0. The strong decay property of D
2600 and Ds12700 is described well bypure 23S1 states. If a mixing between 2
3S1 and 13D1 does exist, the mixing angle is not large and 2
predominant. D2760 and DsJ2860 are possibly the 13D3 D, and Ds, respectively. D27500 andD2760 seem two different states, and D27500 is very possibly the 1D2; 52 though the possibilityof 1D2; 32 has not been excluded. There may exist an unobserved meson DsJ2850 correspondingto DsJ2860.DOI: 10.1103/PhysRevD.83.114025 PACS numbers: 13.25.Ft, 11.30.Hv, 12.39.Hg
The properties of 2S and 1D Q q mesons have beenstudied for a long time. However, no such higher excitedQ q state has been established for lack of experimentaldata. In the past years, some higher excited charmed orcharmed-strange states were reported though most of themhave not yet been pinned down . It will be useful tostudy the possible 2S and 1D charmed and charmed-strange mesons systemically in time.
The first possible charmed radial excitation,D02640, was reported by DELPHI . This state isdifficult to be understood as a charmed radially excitedstate for the observed decaying channel D anddecay width
more models is required. In this paper, the method pre-sented by Eichten et al. (EHQs method)  is employedto study the strong decay of the heavy-light mesons. Wewill label them with the notation nLJP; jq in most cases,where n is the radial quantum number, L is the orbitalangular momentum, JP refers to the total angular momen-tum and parity, and jq is the total angular momentum of the
light degrees of freedom.The paper is organized as follows. In Sec. II, the spec-
trum of 2S and 1D Ds and D will be examined within theRegge trajectory phenomenology. In Sec. III, the two-bodystrong decay of these states will be explored with EHQsmethod. Finally, we present our conclusions and discus-sions in Sec. IV.
II. MASS SPECTRUM IN REGGE TRAJECTORIES
Linearity of Regge trajectories (RTs) is an importantobservation in particle physics . In the relativizedquark model , the RTs for normal mesons are linear.For Q q mesons, the approximately linear, parallel, andequidistant RTs were obtained both in J;M2 and innr;M2 planes in the framework of a QCD-motivatedrelativistic quark model .
However, when RTs are reconstructed with the experi-mental data, the linearity is always approximate. For orbi-tally excited states, Tang and Norbury plotted many RTsof mesons and indicated that the RTs are nonlinear andintersecting :
M2 aJ2 bJ c; (1)where the coefficients a, b, c are fixed by the experi-mental data, and jaj jbj . The coefficients areusually different for different RTs.
For radially excited light q q mesons, Anisovich et al.systematically studied the trajectories on the planes n;M2in the mass region up to M< 2400 MeV . The RTson n;M2 plots behave as
M2 M20 n 12; (2)where M0 is the mass of the basic meson, n is the radialquantum number, and 2 is the slope parameter of thetrajectory.
Possible 1S and 2S D and Ds states are listed in Table I,where yD0s2635 is the predicted mass of 2S1; 12 Dsmeson. It is easy to notice that these candidates for 1S and2S meet well with the trajectories on the n;M2 plotaccording to Eq. (2). The narrow charmed-strange state
DsJ2632 is located around the mass region of2S, Ds. However, the exotic relative branching ratioD0K=Ds 0:14 0:06 excludes its 2S1; 12possibility. Therefore, we denote the 2S1; 12 Ds mesonwith yD0s2635. As indicated in Ref. , the 2P candi-date DsJ3040 meets well with the trajectory on then;M2 plot.The measured masses of D27500, D2760, and
DsJ2860 seem a little lower than most theoretical pre-dictions of the 1D states [24,25,28]. In Fig. 1, nonlinearRTs of D and Ds states consisting of 1
3S11, 13P22,and 13D33 were reconstructed, where the polynomialfits indicate jaj jbj. In a relativistic flux tube model, aratio bhl=bll 2 was obtained at the lowest order ,where bhl is the coefficient for the heavy-light mesonand bll is the coefficient for the light-light meson inEq. (1). The bll (about 0.701.60) has been obtained inRef. . The fitted bhl of D and Ds in Fig. 1 is about 2.74and 3.03, respectively. Obviously, the fitted ratio is con-sistent with the theoretical prediction.Through the analysis of the spectrum only, D25500,
D2600, and Ds12700 are very likely the first radiallyexcited D and Ds states, and D
2760 and DsJ2860 arelikely the 13D3 states.However, as is well known, the RTs can only give a
preliminary analysis of the observed states, the investi-gation of the decay widths and the ratios of branchingfractions will be more useful to shed light on the under-lying properties of these states.
III. DECAY WIDTH IN EHQS FORMULA
As is well known, in the heavy-quark symmetrytheory, the heavy-light mesons degenerate in jPq , i.e.,
two orbital ground states form a spin doublet 1S0; 1with jPq 12 , and the decay amplitude satisfies certainsymmetry relations due to the heavy-quark symmetry .
TABLE I. 1S and 2S D and Ds mesons are shown.
States 0; 12 1; 12 0; 12 1; 122S D25500 D12600 yD0s2635 Ds127001S D1869 D20070 Ds1968 Ds21122 (GeV2) 2.97 2.78 3.07 2.88
1.0 1.5 2.0 2.5 3.0
FIG. 1. Nonlinear RTs of the D and Ds triplet with N, S 1.The polynomial fits are M2 0:23J2 2:74J 1:53 GeV2and M2 0:29J2 3:03J 1:72 GeV2, respectively.
BING CHEN, LING YUAN, AND AILIN ZHANG PHYSICAL REVIEW D 83, 114025 (2011)
The decay properties of heavy-light mesons have beenstudied in detail in the heavy-quark effective theory.When 1=mQ corrections to heavy-quark symmetry predic-tions for strong decay are ignored, the decays of the twomesons in one doublet are governed by the same transitionstrength [4,22,30,31]. As mentioned above, the concisemethod presented by Eichten et al.  is employed tostudy the decays of D and Ds mesons.
In the decay of an excited heavy-light meson H, char-acterized by nLJP; jq, to a heavy-light meson H0[n0L0J0P0 ; j0q] and a light hadron hwith spin sh and orbitalangular momentum l relative to H0, the two-body strongdecay width is written as [4,22]
H!H0h CsQ;j0q;J0jh;jq;J 2Fjq;j
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2J0 12jq 1
qf sQ j
0jh J jq
and ~jh ~sh ~l. F jq;j0q
jh;l0 is the transition strength, and p
is the momentum of decay products in the rest frame of H.The coefficients C depend only upon the total angularmomentum jh of the light hadron, and not separately onits spin sh and the orbital angular momentum l of the decay.The 6 j symbols of the coefficients C exhibit the heavy-quark symmetry in the strong decays of heavy-light me-sons [4,32]. The flavor factor for different decay channelscan be found in Ref. .
The value of parameter is important to the decaywidth. In Ref. , the momentum scale was assumeduniversally 1 GeV, which implies 0:41 GeV. Inthis work, the optimum value of is taken as 0.38 GeV.It is consistent with the harmonic oscillator parameter(0.350.50 GeV) which usually appears in thepseudoscalar-meson emission model , the chiral quarkmodel [14,33,34], and the 3P0 model .Because of lack of measurements of partial widths in
the charmed states, the decay width of K mesons [i.e.K11270 ! K] was used to fix the transition strengthin Ref. . c and b quarks are much heavier than u, d, ands quarks, so the open charm or bottom mesons providebetter place to test EHQs formula. Systematic studiesof S- and P-wave heavy-light meons (D, B, Ds, and Bsmesons) by EHQs formula have been presented inRef. .The EHQs formula is also obtained by the 3P0 model
where a unitary rotation between the bases of Q q mesonsJ2; j2q; s2Q; Jz and q q mesons J2; L2; S2; Jz has been per-formed . In this way, the transition strength F
obtained in the 3P0 model includes only two parameters:the dimensionless parameter and the harmonic oscillatorparameter . In fact, the nodal Gaussian form factorobtained by the 3P0 model has been used for the transition
jh;l0 to interpret D02640 in terms of EHQs
The relevant transition strengths Fjq;j
jh;l0 used in this
paper are given in Table II. Some expressions in the tablecan be found in Refs. , and others are obtained in
TABLE II. The transition strength Fjq;j
jh;l0, where the sign P denotes a light pseudoscalar-
meson or a light vector meson is shown.
nLjPq ! nLjPq P F jq;j0q
jh;l0 Polynomial of p=
2S12 ! 1S12 0 F 1=2;1=21;1 0 52
34121 215 p
2S12 ! 1P12 0 F 1=2;1=20;0 0 1233 1 79 p2
2 227 p
2S12 ! 1P32 0 F 1=2;3=22;2 0 132
37141 239 p
1D32 ! 1S12 0 F 3=2;1=21;1 0 5234 12 1 215 p2
1D32 ! 1S12 1 F 3=2;1=21;1 0 22
34121 215 p
1D32 ! 1P12 0 F 3=2;1=22;2 0 537 14 1 215 p2
1D32 ! 1P32 0 F 3=2;3=20;0 0 22533
1 518 p2
2 1135 p
F 3=2;3=22;2 0 132375 14 1 239 p2
1D52 ! 1S12 0 F 5=2;1=23;3 0 23
1D52 ! 1S12 1 F 5=2;1=23;3 0 25
F 5=2;1=22;1 0 2434 12 1 215 p2
1D52 ! 1P12 0 F 5=2;1=22;2 0 22537
141 115 p
1D52 ! 1P32 0 F 5=2;3=22;2 0 257375
141 142 p
F 5=2;3=24;4 0 243857 18
POSSIBLE 2S AND 1D CHARMED AND CHARMED- . . . PHYSICAL REVIEW D 83, 114025 (2011)
the 3P0 model in detail in Ref. . For these transitionstrengths, a constant
G 1=22 210
was omitted. Here the phase space normalization ofKokoski and Isgur is employed [24,38]. ~MA, ~MB, ~MCare the mock-meson masses of A, B, C, respectively.The constant G absorbs the dimensionless parameter inthe 3P0 model. The variation of the constant G with themock-meson masses ~Mi is slow.
In the analysis that follows, the decay widths of possible2S and 1D D and Ds states are computed in terms ofEq. (3).
A. 21S0 or [2S0; 12]D25500 observed in the decay channel D is a
good candidate for a 21S0 charmed meson. Following theprocedure in Ref. , we take the decay width ofD224600 as an input and obtain the d-wave transitionstrength F 3=2;1=22;2 0 0:964 GeV4, where
F 3=2;1=22;2 0 G22
All the other transition strengths Fjq;j
jh;l0 in Table II
could be fixed easily once the mock-meson masses ~Mieffect has been taken into account. According to ourcomputation , the total decay width of D25500 isabout 124.1 MeV. The dominating decay mode is the Dchannel with D 121:0 MeV, and the decaywidth of another allowed D02400 channel is 3.1 MeV[the mass of D02400 is taken as 2318 MeV ].
These results agree well with the experiments. It ex-plains the fact that D25500 was first observed in D. In Fig. 2, the variation of the decay width with is plotted. Obviously, the observed decay width of
D25500 is well obtained in the reasonable region of (0.350.42 GeV).In Ds states, the mass of the 2
1S0 state is predictedaround 2635 20 MeV [a little smaller than the thresholdof D and D02400K], and DK is the only two-bodystrong decay channel. Our result for this decay channel isDK 82:2 15:1 MeV, so it is impossible that theobserved DsJ2632 is 21S0.
B. Mixing states of 23S1 and 13D1
The predicted masses of 23S1, D are almost about26002640 MeV, and the masses of 23S1, Ds are almostabout 27102730 MeV (Table III) [24,25,4042]. Thespectrum and the helicity-angle distributions suggest thatD2600 is the 23S1 . In our analysis, it is possible toexplain both D2600 and Ds12700 as the pure 23S1states. In this case, the variations of the branching fractionsand decay widths with are given in Figs. 3 and 4,respectively. Obviously, theoretical decay widths and ra-tios in the reasonable region of are consistent with theexperimental data.In the charmonium system, c 2S and c 3770 are
two orthogonal partners of mixtures of 23S1 and 13D1
with JPC 1 . This mixing scheme has also beenemployed to explain the decay width and the ratio ofbranching fractions of Ds12700 and DsJ2860 .If this mixing does exist, there are two orthogonal partners(JP 1) of D and Ds. They can be denoted as
D 2550 0
D 2550 0 : 130 12stat 13syst
0.35 0.36 0.37 0.38 0.39 0.40 0.41 0.42
FIG. 2 (color online). The decay width versus , whereD25500 is taken as a pure 21S0 (green line) state. Thedashed line refers to central values of the decay width givenby experiment.
TABLE III. D, Ds masses of the states 23S1 and 1
3D1predicted in different models are shown (MeV).
States Ref.  Ref.  Ref.  Ref.  Ref. 
D123S1 2640 2632 2620 2636D01 13D1 2820 2788 2710 2740Ds123S1 2711 2730 2731 2730 2714D0s113D1 2784 2900 2913 2820 2804
D K DK
Ds1 2700 : 0.91 0.13stat 0.12syst
D 2600 0 : 0.32 0.02stat 0.09syst
0.35 0.36 0.37 0.38 0.39 0.40 0.41 0.420.0
FIG. 3 (color online). Branching ratios of D26000 andDs12700 with . The dashed lines refer to central values ofdecay width given by experiment.
BING CHEN, LING YUAN, AND AILIN ZHANG PHYSICAL REVIEW D 83, 114025 (2011)
jSD1iL cosj23S1i sinj13D1i;jSD1iR sinj23S1i cosj13D1i: (5)
Details for the estimate of the decay width are given inthe Appendix where D2600 is identified with thejSD1iL of D.
To proceed our analysis, the masses of pure 23S1 and13D1 obtained in Refs. [24,41] are used. For 2
3S1, themasses from these two groups are almost the same. For13D1, the mass given by Ref.  is much larger than thatin Ref.  (Table III).
When mixing angles are treated as free variables,the decay widths and ratios of D26000 and Ds12700dependence on them are presented in Figs. 5 and 6,respectively. In the figures, the red lines and the blue lines
result from the predicted masses of the pure 23S1 and 13D1
in Ref.  and in Ref. , respectively. However, whenD26000 and Ds12700 are identified with the jSD1iLof D and Ds, respectively, the mixing angles are fixed(Table IV). The mixing angle does not seem to bestrongly dependent on the masses input of 13D1. Themixing angles determined from two different massesinput are used as the reasonable boundaries of the vari-ables. Obviously, the ratio D=D of D26000 andDK=DK of Ds12700 in the reasonable region agreewell with experiments. The decay widths are a little largerthan the experimental data.In summary, both D26000 and Ds12700 can be
explained as the pure 23S1 states. If the mixing between23S1 and 1
3D1 exists, the mixing ang...