Spectrum of Higher excitations of $B$ and $D$ mesons in the relativistic potential model

0.21 MB·English

by  Jing-Bin Liu

Spectrum of Higher excitations of B and D mesons in the relativistic potential model
Jing-Bin Liu and Mao-Zhi Yang
∗ †
School of Physics, Nankai University, Tianjin 300071, P.R. China
(Dated: January 20, 2015)
Westudythemassesandwavefunctionsofheavy-lightquark-antiquarksystemintherelativistic
potential model, where we focus on the higher excited states of B and D mesons. The resonances
of DJ(2740), DJ∗(2760), DJ∗(3000) and Ds∗J(2860) recently observed in experiments are considered.
Wefindthatthefourresonancescanbeidentifiedasthe|11,3D2i,|13D3i,|11,3F3iand|13D3istates
due to the prediction to the states with l = 2 and l = 3 in the potential-model calculation. The
masses of n=2 excited states are also calculated. All the observed mesons can be accommodated
successfully in thepotential model.
5
1 PACSnumbers: 12.39.Pn,14.40.Lb,14.40.Nd
0
2
I Introduction resultanddiscussion. InsectionIV wehavea briefsum-
n
mary.
a
J Heavy-light quark-antiquark system Qq¯ plays an im-

8 portant role in studying the strong interactions between
1 quark and antiquarks. Great progress was made in II The effective Hamiltonian and the solution of the
wave equation
the measurement of the spectrum of heavy-light quark-
]
h antiquark system in the last decade. In the charm sec-
In the potential model one assumes that hadrons can
p tor, several new excited meson states were discovered
- betreatedintermsofvalence-quarkconfigurations. Here
p in the previous several years [1–6]. For DJ mesons, we employ the relativistic potential model to describe
e the resonances of D(2740)0, D∗(2760)0,+ [5], D∗(2650)
h and D (3000) [6] were observed, while for D mesons, the heavy-lightB andD mesons. The effective potential
1 [ Dth+e (r2e7s∗o0n0a)n[3ce]sanDds+JD(2+63(320)4[01)],[4X] (w2e6r9e0)e,stDabs+lJiss(J2h8ed60.)T[2h]e, bgleutowne-eenxcthhaengqeua(rOkGaEn)ddaonmtiiqnuaanrtkatinshthoretmdiesstoannciessoanned-
sJ sJ with a linear confinement at long distances. Since there
v spectroscopy provides a powerful test of the theoretical
6 is a light quark in the heavy-light system, it requires to
predictions based on the quark model in the standard
6 include relativistic dynamics in the Hamiltonian. The
model.
2 relativisticversionofSchr¨odingerwaveequationdescrib-
4 The bound state of the heavy-light quark-antiquark
ing the heavy-light system is written as
0 system can be studied with the relativistic wave equa-
1. tion, where an effective potential compatible with QCD Hψ(~r)=Eψ(~r). (1)
0 is taken [7, 8]. The potential shows a linear confining
5 behavior at long distance and a Coulombic behavior at In the rest frame of the bound state system, the eigen-
1
short distance. The properties of B and D mesons have values of the Schr¨odinger equation will be the masses of
:
v been investigated in the relativistic potential model in mesons.
Xi Refs. [8–14]. In our recent work [15], the spectra for The effective Hamiltonian can be written as
lower orbital and radial excitations of B and D mesons
r
a were calculated, a good agreement with experiment [16] H =H0+H′, (2)
was obtained. Here we extend our previous work to in-
clude the higher excited states and compare the predic- with
tionwiththespectrummeasuredinexperimentinrecent
years. The excited states of the heavy-light system have H0 = p~21+m21+ p~22+m22+V(r), (3)
also been studied in Ref. [17] in the relativistic quark q q
model, which includes the leading order corrections in where m1, ~p1 are the mass and momentum of the heavy
1/mc,b expansion. The Hamiltonian used in Ref. [17] is quark,respectively,andm2,p~2 themassandmomentum
a kindof relativisticDirac Hamiltonian. Comparedwith ofthelightantiquark. V(r)isthefunnelpotential,which
Ref. [17], the method used in this work is different and includes only the spin-independent partofthe strongin-
the recently observed states are also carefully studied in teraction between the heavy and light quarks
this work.
4α (r)
s
The organizationofthis paper is asfollows. In section V(r)= +br+c. (4)
−3 r
II, we give the effective Hamiltonian for the heavy-light
quark-antiquark system and solve the relativistic wave The potential is taken as a combination of a Coulomb
equation theoretically. Section III is for the numerical term and a linear confining term, whose behavior is
2
compatible with QCD at both short- and long-distance The spin-dependent Hamiltonian H is the result of
′
[8,18,19]. Thefirsttermisobtainedfromthecalculation OGEdiagraminthenon-relativisticapproximation[7,8].
of one-gluon-exchange diagram in perturbative QCD. It Itisreasonablethattheremightbecontributionsofnon-
dominates at short-distance. The second term is the lin- perturbative dynamics in the bound state system and
ear confining term. The third term is a phenomenolog- relativistic corrections for the light quark. In this work,
ical constant, which can be adjusted to give the correct the δ(r) in the spin-spin contact hyperfine interaction
groundstateenergy-levelofthe quark-antiquarksystem. term is replaced by ( σ )3e σ2r2 as in Ref. [20]. The
√π −
The running coupling constant αs(r) in eq.(4) is ob- mass of the light quark m2’s in the denominators of eqs.
tained from the coupling constant in momentum space (7) and (10) are replaced by a set of new parameters
αs(Q2) after the Fourier transformation. It can be writ- m˜2i, i = a, b, c, d, which is assumned to include the
ten in a more convenient form [8] relativistic corrections and the bound-state effect in the
heavy meson [15]. In the original expression of the spin-
αs(r)= αi 2 γire−x2dx, (5) dependent interaction, the momenta of the quarks are
√π
i Z0 dropped. However, it is not reasonable to drop them
X
completely for the heavy-light system as the light quark
where α and γ are free parameters which can be fitted
i i should be highly relativistic. There may also be bound-
tomakethebehavioroftherunningcouplingconstantat
stateeffectinthespin-dependentinteractiontermswhich
short distance be consistent with the coupling constant
can not be described by the OGE potential. We assume
in momentum space predictedby QCD.In this work,we
that these effects can be included by replacing the light
take α = 0.15, α = 0.15, α = 0.20, and γ = 1/2,
1 2 3 1 quark masses with a set of new parameters, which can
γ =√10/2, γ =√1000/2 [14].
2 3 be determined by fitting the spectra of heavy mesons.
The second term H in eq.(2) is the spin-dependent
′ One can find in the next section that this assumption
part of the Hamiltonian
doeswork,allthemassesmeasuredinexperimentcanbe
accommodated well.
H′ =Hhyp+Hso, (6)
Inour previouswork[15], only the states ofB, D, B ,
s
where Hhyp is the spin-spin hyperfine-interaction term, Ds mesons with lower orbital quantum numbers l =0,1
areconsidered,themassesandwavefunctionsofthecor-
which is
responding states with JP = 0 , 1 , 0+, 1+, 2+ are
− −
32π
Hhyp = α (r)δ (r)~s ~s obtained. In this work we extend our previous work by
s σ 1 2
9m1m˜2a · studyingmoreexcitedstateswithhigherorbitalquantum
4 α (r) 1 3~s ~r~s ~r numbers. In the heavy quark limit the spin of the heavy
s 1 2
+ · · ~s ~s (7)
3m m˜ r3 r2 − 1· 2 quark decouples from the interaction [21–29], therefore
1 2b (cid:18) (cid:19)
the spin-dependent interaction can be treated perturba-
with tively. We choose the eigenstates of spin-independent
δ (r)=( σ )3e σ2r2, (8) Hamiltonian H0 as the basis of perturbative expansion.
σ √π − The eigenstates of H0 is denoted as JM,sl , where J
| i
is the quantum number of the total angular momentum,
wheretheparameterσ istakenasquarkmass-dependent M the magnetic quantum number, s the total spin, l
[8] the relevant orbital angular momentum. The δ (r) term
σ
of the hyperfine interaction Hhyp conserves all the four
4 2 quantum numbers, it gives the splitting of states with
1 1 4m m 2m m
σ = σ2 + 1 2 +s2 1 2 , different total spin. The tensor part of Hhyp does not
v 0 2 2 (m +m )2 0 m +m
u (cid:18) 1 2 (cid:19) ! (cid:18) 1 2(cid:19) conserve the orbital angular momentum, it causes mix-
u
t (9) ing betweenthe states withdifferent orbitalangularmo-
herTehσe0sapnind-osr0baitreinptheernaoctmioennotleorgmicaHl psaoracmonestiesrtss. of the dmoeenstano3tLJco↔ns3erLve′J,twhehitleottahlessppinin,-oitrbciatuisnetsermacitxiionngHbeso-
color-magnetic term and the Thomas-precessionterm tween the states with different total spin quantum num-
bers 1L 3 L .
4α (r) 1 1 ~s L~ ~s L~ J ↔ J
Hso = s + 1· + 2· The details of solving the wave equation have been
3 r3 (cid:18)m1 m˜2c(cid:19) m1 m˜2c ! given in [14, 15], which is omitted in the present paper
for brevity. The readers can refer to these references.
1 ∂V(r) ~s L~ ~s L~
1· + 2· , (10) The mass matrix is calculated perturbatively in the
− 2r ∂r m21 (m˜2d)2! basis of JM,sl , which can be written as
| i
whereL~ =~r P~ istherelativeorbitalangularmomentum H H
× H = 11 12 , (11)
between the quark and antiquark. H H
21 22
(cid:18) (cid:19)
3
whereH = ψ H ψ , withi,j =1,2,ψ ’s denotethe the mass matrix are
ij i j i,j
h | | i
eigenstates of the Hamiltonian H . By diagonalizingthe
0
3
mass matrix, we can get the masses and wave functions H = E(0) ψ(0)(r)f(r)ψ(0)(r) , (20)
ofthe mixing eigenstates. Theperturbativecontribution 11 l=3− 4h l=3 | | l=3 i
ofthe spin-dependentHamiltonianH′ tothe eigenvalues H12 = √3hψl(=0)3(r)|h1(r)−h2(r)|ψl(=0)3(r)i, (21)
of states with high orbital quantum numbers are given
H = H , (22)
21 1∗2
below (Contributions to states with lower orbital mo-
1 1
mentum have been given in our earlier work, Ref. [15]). H = E(0) + ψ(0)(r) f(r)+ g(r)
22 l=3 h l=3 |4 2
(1) The mass matrix of state JP =2− 1h (r) 1h (r)ψ(0)(r)) . (23)
−2 1 − 2 2 l=3 i
Both states with s = 0, l = 2 and s = 1, l = 2 can
constructthe JP =2 state. The basisforthe mixing is
− With eqs.(20) (23), the cases with more 1D and
ψ = 1D and ψ = 3D . The matrix elements of ∼ | 2i
t|h1eimas|s m2aitrix a|re2i | 2i |3D2i mixing states can be obtained.
(4) The mass matrix of state JP =4+
The JP = 4+ state is mixture of 3F and 3H states,
4 4
3
H = E(0) ψ(0)(r)f(r)ψ(0)(r) , (12) both states with s = 1, l = 3 and s = 1, l = 5 can
11 l=2− 4h l=2 | | l=2 i constructthe JP =4+ state. The basisfor the mixing is
3 ψ = 3F and ψ = 3H . The matrix elements of
H12 = 2hψl(=0)2(r)|h1(r)−h2(r)|ψl(=0)2(r)i, (13) |th1eimas|s m4aitrix a|re2i | 4i
r
H = H , (14)
21 1∗2 1 1
H22 = El(=0)2+hψl(=0)2(r)|41f(r)+ 12g(r) H11 = El(=0)3+hψl(=0)3(r)|[4f(r)− 6g(r)
3 3
−21h1(r)− 12h2(r)ψl(=0)2(r))i. (15) +2h1(r)+ 2h2(r)]|ψl(=0)3(r)i, (24)
√5
H = ψ(0)(r)g(r)ψ(0)(r) , (25)
12 3 h l=3 | | l=5 i
With eqs.(12) ∼ (15), the cases with more |1D2i and H21 = H1∗2, (26)
|3D2i mixing states can be obtained. H = E(0) + ψ(0)(r)[1f(r) 1g(r)
(2) The mass matrix of state JP =3 22 l=5 h l=5 | 4 − 3
−
3h (r) 3h (r)]ψ(0)(r) . (27)
The JP = 3− state is mixture of 3D3 and 3G3 states, − 1 − 2 | l=5 i
both states with s = 1, l = 2 and s = 1, l = 4 can
constructthe JP =3 state. The basisforthe mixing is With eqs.(24) (27), the cases with more 3F4 and
− ∼ | i
ψ = 3D and ψ = 3G . The matrix elements of 3H4 mixing states can be obtained.
1 3 2 3 | i
t|heimas|s maitrix ar|e i | i (5) The mass matrix of state JP =4−
Both states with s = 0, l = 4 and s = 1, l = 4 can
constructthe JP =4 state. The basis forthe mixing is
1 1 −
H = E(0) + ψ(0)(r)[ f(r) g(r) ψ = 1G and ψ = 3G . The matrix elements of
11 l=2 h l=2 | 4 − 7 | 1i | 4i | 2i | 4i
the mass matrix are
+h (r)+h (r)]ψ(0)(r) , (16)
1 2 | l=2 i
H12 = 37√3hψl(=0)2(r)|g(r)|ψl(=0)4(r)i, (17) H11 = El(=0)4− 34hψl(=0)4(r)|f(r)|ψl(=0)4(r)i, (28)
H21 = H1∗2, (18) H12 = √5hψl(=0)4(r)|h1(r)−h2(r)|ψl(=0)4(r)i, (29)
1 5
H22 = El(=0)4+hψl(=0)4(r)|[4f(r)− 14g(r) H21 = H1∗2, 1 1 (30)
(0) (0)
5h (r) 5h (r)]ψ(0)(r) . (19) H22 = El=4+hψl=4(r)|4f(r)+ 2g(r)
−2 1 − 2 2 | l=4 i 1 1
h (r) h (r)ψ(0)(r)) . (31)
−2 1 − 2 2 l=4 i
With eqs.(16) (19), the cases with more 3D and
3G mixing st∼ates can be obtained. | 3i With eqs.(28) (31), the cases with more 1G4 and
3 ∼ | i
| i 3G mixing states can be obtained.
4
(3) The mass matrix of state JP =3+ | (6)iThe mass matrix of state JP =5+
Both states with s = 0, l = 3 and s = 1, l = 3 can Both states with s = 0, l = 5 and s = 1, l = 5 can
constructthe JP =3+ state. The basisfor the mixing is constructthe JP =5+ state. The basisfor the mixing is
ψ = 1F and ψ = 3F . The matrix elements of ψ = 1H and ψ = 3H . The matrix elements of
1 3 2 3 1 5 2 5
| i | i | i | i | i | i | i | i
4
the mass matrix are some of the parameters. The ground states are not sen-
sitive to the value of the parameter b, which is slightly
3
H11 = El(=0)5− 4hψl(=0)5(r)|f(r)|ψl(=0)5(r)i, (32) increasedtogivethecorrectpredictionsofexcitedstates.
The values we obtain by fitting are
15
H = ψ(0)(r)h (r) h (r)ψ(0)(r) , (33)
12 r 2 h l=5 | 1 − 2 | l=5 i mb =4.99GeV, mc =1.58GeV,
H21 = H1∗2, (34) m =0.32GeV, m =m =0.06GeV,
s u d
1 1
H = E(0) + ψ(0)(r) f(r)+ g(r) b=0.175GeV2, c= 0.312GeV,
22 l=5 h l=5 |4 2 −
1h (r) 1h (r)ψ(0)(r)) . (35) σ0 =1.80GeV, s0 =1.55. (44)
−2 1 − 2 2 l=5 i
The values of m˜ , m˜ , m˜ and m˜ depend on the
2a 2b 2c 2d
With eqs.(32) (35), the cases with more 1H5 and quark-antiquarksystem, they can be written as
∼ | i
3H mixing states can be obtained.
5
| i
(7) The mass matrix of state JP =5− m˜2i =ǫim˜2, i=a,b,c,d. (45)
The JP = 5 state is mixture of 3G and 3I states,
− 5 5
both states with s = 1, l = 4 and s = 1, l = 6 can We find the values of ǫi’s and m˜2 are
construct the JP = 5 state. The basis for the mixing
−
(ǫ ,ǫ ,ǫ ,ǫ )=(1.10,1.30,1.30,1.32) (46)
is ψ = 3G and ψ = 3I . The matrix elements of a b c d
1 5 2 5
| i | i | i | i
the mass matrix are
for (bq¯) and (cq¯) systems, and
1 2
H = E(0) + ψ(0)(r)[ f(r) g(r)
11 l=4 h l=4 | 4 − 11 (ǫa,ǫb,ǫc,ǫd)=(1.10,1.10,1.10,1.31) (47)
+2h (r)+2h (r)]ψ(0)(r) , (36)
1 2 | l=4 i for (bs¯) and (cs¯) systems, and
3 15
H = ψ(0)(r)g(r)ψ(0)(r) , (37)
12 11 2 h l=4 | | l=6 i 0.562GeV for(bq¯)system,
r
H21 = H1∗2, (38) m˜2 = 00..461729GGeeVV ffoorr((cbqs¯¯))ssyysstteemm,, (48)
H22 = El(=0)6+hψl(=0)6(r)|[41f(r)− 272g(r)  0.488GeV for(cs¯)system,
7h (r) 7h (r)]ψ(0)(r) . (39) here q is the light quark u or d.
−2 1 − 2 2 | l=6 i
Numericalcalculationshowsthatthesolutionisstable
Witheqs.(36) (39),thecaseswithmore 3G and 3I when L > 5 fm, N > 50. Here we take L = 10 fm,
5 5
∼ | i | i
mixing states can be obtained. N = 100. The definition of L and N can be found in
In the above equations, the functions f(r), g(r), h (r) Refs. [14, 15].
1
and h (r) are defined as The radial wave functions ψ (r) for D meson in co-
2 nl
ordinate space are depicted in Fig.1 as an example.
32π
f(r) = 9m m˜ αs(r)δσ(r), (40) |ψnl(r)|2r2 is the possibility density distributed along
1 2a the quark-antiquark distance r. We can use the Fourier
4 α (r) 1
s transformation of the wave function in the coordinate
g(r) = , (41)
3m1m˜2br3 space to get the momentum distribution, which is useful
4α (r) 1 1 for studying B and D decays.
s
h (r) = +
1 (cid:20)3 r3 (cid:18)m1 m˜2c(cid:19) In this work we calculate B, D, Bs, Ds bound states
withhigherorbitalquantumnumbers,thequantumnum-
1 ∂V(r) 1 1
, (42) ber JP of the new corresponding states are 2 , 3 , 3+,
−2r ∂r m m − −
1(cid:21) 1 4+, 4 . the numerical results for B, D, B , D bound
− s s
4α (r) 1 1 1
h (r) = s + states are presented up to l = 4 in four tables. The nu-
2 (cid:20)3 r3 (cid:18)m1 m˜2c(cid:19)m˜2c mericalresultsforD, Ds boundstatesaregiveninTable
1 ∂V(r) 1 I and II, the numerical results for B, B bound states
. (43) s
−2r ∂r (m˜ )2 are given in Table III and IV. For completeness, results
2d (cid:21)
for ground-states and lower orbital excitations are also
given. The mass predictions for lowerorbital excitations
III Numerical result and discussion have been calculated in Ref. [8] several decades before,
which are also presented in Tables I to IV for compari-
Theparametersusedinthisworkarethequarkmasses, son. The masses for most of the lower excitation states
the potential parameters b, c, m˜ , σ and s . Compar- obtained in this work are approximately consistent with
2i 0 0
ingwithourpreviouswork[15],wehaveslightlyadjusted themassesgiveninRef. [8]. Onlyforafewmesons,such
5
oration are listed in the following [6]:
) 8
r
(
ψ M(D (2420)0) = 2419.6 0.1 0.7MeV
1
± ±
6 M(D (2460)0) = 2460.4 0.4 1.2MeV
2∗ ± ±
1s M(D (2650)0) = 2649.2 3.5 3.5MeV
J∗ ± ±
4 M(D (2760)0) = 2761.1 5.1 6.5MeV
J∗ ± ±
M(D (2580)0) = 2579.5 3.4 5.5MeV
J
p ± ±
2 1 M(D (2740)0) = 2737.0 3.5 11.2MeV
1d 1f 1 g M(DJ(3000)0) = 2971.8±8.7±MeV
J
±
0.5 1.0 1.5 2.0 2.5 M(D (2460)0) = 2460.4 0.1 0.1MeV
r (fm) 2∗ ± ±
(a) M(DJ∗(2760)0) = 2760.1±1.1±0.1MeV
) M(D (3000)0) = 3008.1 4.0MeV
ψr( 8 M(D2∗J∗(2460)+) = 2463.1±±0.2±0.6MeV
M(D (2760)+) = 2771.7 1.7 3.8MeV
6 J∗ ± ±
2s M(DJ∗(3000)+) = 3008.1MeV.
4
The assignments of the above observed states are listed
2 2p in Table I and II. DJ(2740) can be identified as mixing
2d 2f 2 g state of|11D2iand |13D2i with JP =2−. DJ∗(2760)can
be identified as dominantly 13D state with JP = 3 ,
3 −
0.5 1.0 1.5 2.0 2.5 | i
which agrees with Ref. [30]. D (3000) can be identified
J∗
r (fm) as mixing state of 11F and 13F with JP =3+. The
(b) | 3i | 3i
other three observedstates are D (2580), D (2650)and
J J∗
D (3000), are identified as n=2 excited states in Table
J
FIG. 1: The radial wave function ψnl(r) of D meson in co- II. In Ref. [6], D (2580) is identified as 0 , D (2650)
ordinate space, l from 0 to 4. (a) is for n = 1, (b) is for J − J∗
is identified as 1 , but one can see that D (2580) is
n=2. − J
not fully compatible with the assignment 0 in our cal-
−
culation. In this work, we take a different assignment,
i.e., D (2580) is identified as dominantly 23S state
J 1
| i
with JP = 1 , D (2650) is identified as 23P state
− J∗ | 0i
with JP =0+, since our predictionfor 21S and 23S
0 1
| i | i
statearelowerthanthepredictionsofGodfreyandIsgur
[8]. The D (3000) resonance is identified as dominantly
J
23P state with JP =2+ in this work.
2
| i
Since 2004 many new D mesons have been observed,
s
asD1(2420),D∗s0(2317),D∗s1(2460)etc.,themasspre- such as DsJ(2632) [1], Ds∗J(2860) [2], Ds∗1(2710) [3],
dictions are improved, the results in this work are more DsJ(3040) [4]. One can see that our predicted mass
consistentwithexperiment. Mixingsbetweenstateswith spectrumforDs mesonsbelow2600MeVaccommodates
appropriatequantum numbers are consideredin our cal- the experimental data successfully [16], here we discuss
culation. FordoubletswithJP =1+,2−,3+,4−,mixing the assignment of the recently observed Ds meson reso-
occurs more strongly. The mixing from state of higher nances. The Ds∗J(2860) resonance parameters measured
radialquantumnumberwithn=2cannotbeneglected, in Ref. [4] are
itcanbeaslargeas10%. Themassesmeasuredinexper-
M(D (2860)+) = 2862 2+5 MeV
iments are given in the last column in Tables I and III, s∗J ± −2
fromwhichwecansee thatthe theoreticalcalculationin Γ = 48 3 6MeV
± ±
this work can accommodate the experimental data well.
TheJP =0+assignmentisproposedinRef. [31],butthe
observation of D (2860)+ D K decay rules out the
s∗J → ∗
JP =0+ assignment[4]. Our mass prediction for 13D
3
| i
state is 2853 MeV, very close to the mass of D (2860),
s∗J
Manynewresonanceswithopencharmflavorwereob- thus we suggest that D (2860)is JP =3 state, which
s∗J −
served in experiments recently [5, 6]. Our mass predic- is also proposed in Ref. [32]. For the D (2710) state, it
s∗1
tions for these resonancesareconsistentwiththe experi- ismeasuredtobe 2710 2+12MeVinRef. [6], theJP =
mental data. The masses measuredby the LHCb collab- 1 assignment is propo±sed−7in Refs. [33, 34], comparing
−
6
the experimentalmassofD (2710)withourcalculation and JP =(1+,2+) , we get
s∗1 sl 3/2
for 23S state in Table II, our prediction favors this
1
assig|nmenit. The meson DsJ(2632) was firstly observed |0+1/2i = |3P0i, (51)
inRef. [1]atamassof2632.5 1.7MeV,inourpredicted
± 1 2
mass spectrum for D mesons we assign it as the 23P 1+ = 1P + 3P , (52)
state with JP =0 , ist is the closeststates to 2632|MeV0i, | 1/2i r3| 1i r3| 1i
−
although 50 MeV lower. The DsJ(3040) resonance is 1+ = 2 1P + 1 3P , (53)
observedintheD K massspectrumatamassof3044 | 3/2i − 3| 1i 3| 1i
8+30 in Ref. [4].∗The assignment of this resonance ±is 2+ = 3Pr . r (54)
d−is5cussed in detail in Ref. [35]. We identify it as 21P | 3/2i | 2i
1
| i
with JP =1+, since its predicted mass 3012 MeV is the (3) For l = 2 states in the doublets JP = (1 ,2 )
sl − − 3/2
closest to the mass of the DsJ(3040) resonance. and JsPl =(2−,3−)5/2, we get
For b-flavoredmeson sector, one can see the predicted
mass spectra in Table III and IV are in good agreement |1−3/2i = |3D1i, (55)
with experimental results. Although experimental in-
2 3
fmoremasautrieodnJfoPr=b-0fl−av,o1r−e,d1m+easnodn2s+pescttartuemsciasnlbimeiatende,xtthrae |2−3/2i = r5|1D2i+r5|3D2i, (56)
test to our model besides the charmed meson sector. 3 2
Next we would like to discuss the properties of the |2−5/2i = − 5|1D2i+ 5|3D2i, (57)
r r
heavy-lightquark-antiquarkboundstatesfromthe point |3−5/2i = |3D3i. (58)
view of heavy quark symmetry. As discussed in Refs.
[15, 30], the spin of the heavy qurk decouples from the (4) For l = 3 states in the doublets JP = (2+,3+)
sl 5/2
light degrees of freedom in the heavy quark limit. The and JP =(3+,4+) , we get
totalangularmomentumofthe lightquarkis~s =~s +~l, sl 7/2
l q
where ~sq is the spin of the light quark,~l the orbital an- |2+5/2i = |3F2i, (59)
gular momentum of the light quark. The total angular
3 1
momentum of the heavy light meson is J~ = ~s +~s , 3+ = 1F +2 3F , (60)
l Q | 5/2i 7| 3i 7| 3i
where~s is the spin of the heavyquark. Since the prop- r r
Q
erties of the hadronic states do not depend on the spin 3+ = 2 1 1F + 3 3F , (61)
~s and the flavor of the heavy quark due to the heavy | 7/2i − 7| 3i 7| 3i
Q r r
quark symmetry, the heavy-light meson states only de- 4+ = 3F . (62)
| 7/2i | 4i
pend on the quantum numbers s and l, thus we can
l
classify the heavy-light mesons by these two quantum (5) For l = 4 states in the doublets JP = (3 ,4 )
sl − − 7/2
numbers. Heavy-light meson states can be divided into and JP =(4 ,5 ) , we get
sl − − 9/2
differentdoubletsaccordingtothis classification. Adou-
blet with JsPl = (0−,1−)1/2 exists for the case l = 0, for |3−7/2i = |3G3i, (63)
example. Mesons within the same doublet degenerate in
2 √5
tPhe=he(av1y)lq+u1a,rtkhleimpiatr.itSyinPcectahnedpeanroittyeotfhethoerbmiteasolnanis- |4−7/2i = 3|1G4i+ 3 |3G4i, (64)
mguelsaornm−sotamteenwtuitmh slpoefcitfihceqmueasnotnuminnaumdobuebrslect.lasFsoifireedacinh |4−9/2i = −√35|1G4i+ 32|3G4i, (65)
the heavy quark limit, we can denote them by |JsPli. |5−9/2i = |3G5i. (66)
By analyzing angular momentum addition with the
help of Clebsch-Gordan coefficients, we can decompose Particularly, for P = ( 1)J+1 unnatural parity states ,
−
the basis states JP in the heavy quark symmetry into they satisfy the relations
| sli
combination of states with definite quantum numbers l,
S and J, i.e. the state 2S+1LJ . The decompositions of lP = l 1l + l+1 3l , (67)
all the doublets involve|d in thisiwork are as follows: | l−1/2i r2l+1| li r2l+1| li
(1) For l = 0 states in the doublet JP = (0 ,1 ) , l+1 l
sl − − 1/2 lP = 1l + 3l . (68)
we get | l+1/2i − 2l+1| li 2l+1| li
r r
|0−1/2i = |1S0i, (49) ityCaorneseliqnueeanrtclyo,mthbeinpahtiyosniscaolfst1altesawndith3uln,nathtuermalixpianrg-
l l
|1−1/2i = |3S1i. (50) can be described by a mixing a|ngile. We|miust note here
that the sign of the mixing angle is fraught with ambi-
(2) For l = 1 states in the doublets JP = (0+,1+) guities [36, 37]. It depends on the coupling order of the
sl 1/2
7
angular momentum and spins, ~l, ~s and ~s . The sign consistent with Belle’s measurement.
1 2
of the mixing angle will flip if the coupling order of the Finally the wave function of each bound state can be
quark spins changes, for example. obtainedsimultaneouslywhensolvingthewaveequation.
In this work, the properties of the heavy flavored Figure 1 is the radial wave functions for D meson as an
mesons implied by the heavy quark limit are approx- example. It is easy to get all the wave functions when
imately maintained in the calculation in the relativis- they are needed.
tic potential model. The mixing angle in the heavy
quark limit should be ArcSin l/2l+1. For l = 1,
− IV Summary
we get ArcSin 1/3 = 0.615 rad, for l = 2 we have
− − p
ArcSin 2/5 = 0.685 rad, etc. The mixing angle
− p− The bound states of heavy-light quark and antiquark
increases as l increases. Comparing with the column
p system are studied in the relativistic potential model.
“Multiplet” in the Table I IV for each meson, one can
∼ The predictions of heavy-light charmed and b-flavored
find that this property is maintained in the numerical
meson states are presented. The low-lying meson states
results. As discussed in our previous work [15], only
the mixing between 11P and 13P in 1+ state of are in good agreement with experiments. The higher
1 1
| i | i excitation states are particularly focused on. All the ob-
cq¯ is very small, which seriously deviates from the re-
served mesons can be successfully accommodated in our
sult in the heavy quark limit 0.615 rad, the reason
− calculation. The wave functions of each bound state can
has been explained in our previous work [15]. The
be obtainedby solvingthe waveequationandbe usedin
Belle collaboration determined this mixing angle,which
the study of B and D decays.
is θ = 0.10 0.03 0.02 0.02 rad [38]. Our predic-
− ± ± ±
tion with small mixing angle for 1+ state of cq¯ is well
Acknowledgments [9] P. Cea, P. Colangelo, G. Nardulli, G. Paiano, and G.
Preparata, Phys.Rev.D26, 1157 (1982).
[10] P.Cea,G.Nardulli,andG.Paiano,Phys.Rev.D28,2291
This work is supported in part by the National Nat-
(1983).
ural Science Foundation of China under contracts Nos. [11] P. Cea, P. Colangelo, L. Cosmai and G. Nardulli, Phys.
11375088,10975077,10735080,11125525. Lett. B206, 691 (1988).
[12] P. Colangelo, G. Nardulli, M. Pietroni, Phys. Rev. D43,
3002 (1991).
[13] P. Colangelo, F. De Fazio, M. Ladisa, G. Nardulli, P.
Santorelli, A. Tricarico, Eur. Phys.J. C8, 81 (1999).
∗ Electronic address: [email protected] [14] M.Z. Yang, Euro. Phys.J. C72, 1880 (2012).
† Electronic address: [email protected] [15] J.B. Liu,M.Z. Yang, JHEP 1407, 106 (2014).
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[3] J.Brodzickaetal.(BelleCollaboration),Phys.Rev.Lett. [18] E. Eichten, K. Gottfried, T. Kinoshita, K.D. Lane, and
100, 092001 (2008). T.-M.Yan,Phys.Rev.D17,3090(1978)[Erratum: Phys.
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8
TABLE I: Theoretical spectrum of (cq¯) and (cs¯) bound states mainly with the radial quantum number n=1. The results in
thecolumn labeled “GI” are theoretical masses from Ref. [8].
Meson JP Multiplet Mass (GeV) GI (GeV) Exp. (MeV) [6, 16]
D 0− |11S0i 1.867 1.88 1869.62±0.15
D∗ 1− 0.997|13S1i−0.035|13D1i+0.051|23S1i+0.033|23D1i 2.004 2.04 2010.28±0.13
−0.026|33S1i−0.031|33D1i
0.046|13S1i+0.979|13D1i−0.090|23S1i−0.148|23D1i 2.785 2.82
−0.041|33S1i+0.093|33D1i
D0∗(2400)0 0+ |13P0i 2.302 2.40 2318±29
1+ 0.017|11P1i+0.990|13P1i−0.076|21P1i−0.117|23P1i 2.415 2.44
D1(2420) 0.996|11P1i−0.028|13P1i−0.028|21P1i−0.076|23P1i 2.429 2.49 2421.3±0.6
D∗(2460) 2+ 0.988|13P2i−0.014|13F2i+0.135|23P2i+0.011|23F2i 2.468 2.50 2464.4±1.9
2 −0.071|33P2i−0.009|33F2i
0.017|13P2i+0.997|13F2i−0.024|23P2i−0.047|23F2i 3.123
−0.021|33P2i+0.039|33F2i
DJ(2740)0 2− 0.769|11D2i−−0.06.3090|71|33D1D2i2+i+0.00.000032||2313DD22ii+0.007|23D2i 2.737 2737.0±3.5±11.2
0.639|11D2i+0.768|13D2i−0.005|21D2i−0.021|23D2i 2.834
0.013|31D2i+0.022|33D2i
(cq¯) D∗(2760)0 3− 0.997|13D3i−0.008|13G3i+0.063|23D3i+0.006|23G3i 2.754 2760.1±1.1±3.7
J −0.039|33D3i+0.005|33G3i
0.008|13D3i+0.9997|13G3i−0.007|23D3i+0.002|23G3i 3.397
−0.009|33D3i+0.018|33G3i
D∗(3000)0 3+ 0.759|11F3i−0.650|13F3i−0.016|21F3i+0.017|23F3i 2.997 3008.1±4.0
J −0.001|31F3i−0.001|33F3i
0.650|11F3i+0.759|13F3i+0.020|21F3i+0.016|23F3i 3.138
+0.002|31F3i+0.006|33F3i
4+ 0.9995|13F4i−0.005|13H4i+0.021|23F4i+0.004|23H4i 3.003
−0.022|33F4i−0.003|33H4i
0.005|13F4i+0.9994|13H4i−0.003|23F4i+0.031|23H4i 3.638
−0.007|33F4i+0.008|33H4i
4− 0.750|11G4i−0.660|13G4i−0.028|21G4i+0.026|23G4i 3.229
+0.003|31G4i−0.003|33G4i
0.660|11G4i+0.750|13G4i+0.034|21G4i+0.032|23G4i 3.398
−0.002|31G4i+0.0002|33G4i
Ds± 0− |11S0i 1.963 1.98 1968.49±0.32
D∗± 1− 0.996|13S1i−0.045|13D1i+0.052|23S1i+0.041|23D1i 2.103 2.13 2112.3±0.5
s −0.028|33S1i−0.036|33D1i
0.064|13S1i+0.950|13D1i−0.191|23S1i−0.201|23D1i 2.803 2.90
−0.047|33S1i+0.117|33D1i
Ds∗0(2317)0 0+ |13P0i 2.317 2.48 2317.8±0.6
Ds1(2460) 1+ 0.475|11P1i+0.851|13P1i−0.117|21P1i−0.193|23P1i 2.436 2.53 2459.6±0.6
Ds1(2536) 0.871|11P1i−0.488|13P1i+0.040|21P1i−0.033|23P1i 2.533 2.57 2535.12±0.13
D∗ (2573) 2+ 0.983|13P2i−0.018|13F2i+0.164|23P2i+0.013|23F2i 2.573 2.59 2571.9±0.8
s2 −0.080|33P2i−0.010|33F2i
0.027|13P2i+0.991|13F2i−0.063|23P2i−0.097|23F2i 3.145
−0.036|33P2i+0.058|33F2i
2− 0.740|11D2i−0.672|13D2i+0.027|21D2i−0.007|23D2i 2.835
−0.017|31D2i+0.005|33D2i
0.671|11D2i+0.737|13D2i−0.037|21D2i−0.063|23D2i 2.864
+0.025|31D2i+0.039|33D2i
(cs¯) D∗ (2860) 3− 0.994|13D3i−0.009|13G3i+0.093|23D3i+0.007|23G3i 2.853 2862±2+5 [4]
sJ −0.047|33D3i−0.005|33G3i −2
0.011|13D3i+0.998|13G3i−0.014|23D3i−0.043|23G3i 3.416
−0.010|33D3i+0.031|33G3i
3+ 0.762|11F3i−0.648|13F3i+0.006|21F3i−0.002|23F3i 3.092
−0.008|31F3i+0.005|33F3i
0.648|11F3i+0.761|13F3i−0.008|21F3i−0.016|23F3i 3.165
+0.010|31F3i+0.015|33F3i
4+ 0.998|13F4i−0.006|13H4i+0.051|23F4i+0.004|23H4i 3.097
−0.030|33F4i−0.003|33H4i
0.006|13F4i+0.9997|13H4i−0.006|23F4i−0.010|23H4i 3.653
−0.005|33F4i+0.018|33H4i
4− 0.754|11G4i−0.657|13G4i−0.005|21G4i+0.006|23G4i 3.320
−0.004|31G4i+0.002|33G4i
0.657|11G4i+0.754|13G4i+0.007|21G4i+0.004|23G4i 3.420
+0.004|31G4i+0.007|33G4i
9
TABLE II: Theoretical spectrum of (cq¯) and (cs¯) bound states mainly with the radial quantumnumbern=2.
JP Multiplet Mass (GeV) GI (GeV) Meson Exp. (MeV)
0− |21S0i 2.483 2.58
1− −0.043|13S1i++00..008635||1333DS11ii++00..096971||3233DS11ii−0.055|23D1i 2.613 2.64 DJ(2580)0 2579.5±3.4±5.5[6]
−0.040|13S1i+0.161|13D1i+0.058|23S1i+0.951|23D1i 3.155
−0.180|33S1i−0.180|33D1i
0+ |23P0i 2.685 DJ∗(2650)0 2649.2±3.5±3.5 [6]
1+ 0.083|11P1i+0.137|13P1i+0.4755|21P1i+0.867|23P1i 2.865
−0.010|11P1i+0.014|13P1i+0.887|21P1i−0.462|23P1i 2.902
2+ −0.111|13P2i++00..022598||1333FP22ii++00..905196||2333PF22ii−0.021|23F2i 2.969 DJ(3000)0 2971.8±8.7 [6]
−0.057|13P2i+0.040|13F2i+0.124|23P2i+0.898|23F2i 3.458
−0.412|33P2i−0.059|33F2i
(cq¯) 2− −0.003|11D2i−0.011|13D2i+0.757|21D2i−0.653|23D2i 3.144
+0.008|31D2i+0.005|33D2i
0.009|11D2i+0.020|13D2i+0.653|21D2i+0.755|23D2i 3.211
−0.021|31D2i−0.044|33D2i
3− −0.057|13D3i+0.009|13G3i+0.990|23D3i−0.010|23G3i 3.175
+0.124|33D3i+0.008|33G3i
−0.007|13D3i−0.002|13G3i+0.012|23D3i+0.9996|23G3i 3.704
−0.018|33D3i−0.015|33G3i
3+ 0.145|11F3i−0.020|13F3i+0.755|21F3i−0.654|23F3i 3.360
−0.017|31F3i+0.020|33F3i
−0.018|11F3i−0.018|13F3i+0.654|21F3i+0.755|23F3i 3.480
+0.020|31F3i+0.011|33F3i
4+ −0.020|13F4i+0.004|13H4i+0.998|23F4i−0.006|23H4i 3.373
+0.050|33F4i+0.005|33H4i
−0.004|13F4i−0.032|13H4i+0.006|23F4i+0.999|23H4i 3.926
−0.006|33F4i+0.029|33H4i
0− |21S0i 2.582 2.67 DsJ(2632) 2632.5±1.7 [1]
1− −0.035|13S1i+0.175|13D1i+0.976|23S1i−0.087|23D1i 2.709 2.73 D∗ (2710) 2708±9+11[3]
+0.063|33S1i+0.065|33D1i s1 10
−0.050|13S1i+0.190|13D1i+0.082|23S1i+0.816|23D1i 3.191
−0.507|33S1i−0.180|33D1i
0+ |23P0i 2.707
1+ 0.117|11P1i+0.193|13P1i+0.498|21P1i+0.837|23P1i 2.911
−0.044|11P1i+0.026|13P1i+0.858|21P1i−0.510|23P1i 3.012 DsJ(3040) 3044±8+−350[4]
2+ −0.130|13P2i+0.070|13F2i+0.941|23P2i−0.030|23F2i 3.070
+0.301|33P2i+0.022|33F2i
−0.001|13P2i+0.108|13F2i−0.011|23P2i+0.976|23F2i 3.477
+0.119|33P2i−0.143|33F2i
(cs¯) 2− 0.048|11D2i+0.041|13D2i−0.087|21D2i+0.990|23D2i 3.240
−0.072|31D2i−0.051|33D2i
0.011|11D2i+0.051|13D2i+0.992|21D2i+0.080|23D2i 3.246
−0.009|31D2i−0.080|33D2i
3− −0.083|13D3i+0.015|13G3i+0.982|23D3i−0.013|23G3i 3.272
+0.171|33D3i+0.010|33G3i
−0.011|13D3i+0.044|13G3i+0.021|23D3i+0.995|23G3i 3.724
−0.051|33D3i−0.071|33G3i
3+ −0.007|11F3i+0.0002|13F3i+0.754|21F3i−0.657|23F3i 3.454
+0.016|31F3i−0.007|33F3i
0.010|11F3i+0.016|13F3i+0.657|21F3i+0.753|23F3i 3.507
−0.019|31F3i−0.033|33F3i
4+ −0.048|13F4i+0.006|13H4i+0.994|23F4i−0.007|23H4i 3.467
+0.096|33F4i+0.006|33H4i
−0.005|13F4i+0.011|13H4i+0.008|23F4i+0.9994|23H4i 3.941
−0.012|33F4i−0.027|33H4i
10
TABLE III:Theoretical spectrum of (bq¯) and (bs¯) bound states mainly with theradial quantumnumbern=1.
Meson JP Multiplet Mass (GeV) GI (GeV) Exp. (MeV) [16]
B 0− |11S0i 5.271 5.31 5279.25±0.17
B∗ 1− 0.9996|13S1i−0.012|13D1i+0.019|23S1i+0.012|23D1i 5.317 5.32 5325.2±0.4
−0.009|33S1i−0.011|33D1i
0.014|13S1i+0.995|13D1i−0.020|23S1i−0.080|23D1i 6.087
−0.011|33S1i+0.051|33D1i
0+ |13P0i 5.696
B1(5721) 1+ 0.511|11P1i+0.847|13P1i−0.082|21P1i−0.117|23P1i 5.732 5723.5±2.0
0.855|11P1i−0.517|13P1i+0.031|21P1i−0.031|23P1i 5.759
B∗(5747) 2+ 0.995|13P2i−0.005|13F2i+0.090|23P2i−0.004|23F2i 5.775 5.8 5743±5
2 −0.049|33P2i−0.004|33F2i
0.006|13P2i+0.9995|13F2i−0.008|23P2i−0.021|23F2i 6.374
−0.006|33P2i+0.021|33F2i
2− 0.764|11D2i−0.645|13D2i+0.010|21D2i−0.005|23D2i 6.051
−0.010|31D2i+0.007|33D2i
0.644|11D2i+0.763|13D2i−0.017|21D2i−0.025|23D2i 6.105
0.016|31D2i+0.021|33D2i
(bq¯) 3− 0.999|13D3i−0.003|13G3i+0.038|23D3i+0.002|23G3i 6.061
−0.025|33D3i−0.002|33G3i
0.003|13D3i+0.9999|13G3i−0.003|23D3i+0.007|23G3i 6.614
−0.002|33D3i+0.010|33G3i
3+ 0.753|11F3i−0.658|13F3i−0.006|21F3i+0.007|23F3i 6.296
−0.004|31F3i+0.003|33F3i
0.658|11F3i+0.753|13F3i+0.007|21F3i+0.005|23F3i 6.383
+0.005|31F3i+0.007|33F3i
4+ 0.9999|13F4i−0.002|13H4i+0.011|23F4i+0.001|23H4i 6.302
−0.013|33F4i−0.001|33H4i
0.002|13F4i+0.9997|13H4i−0.002|23F4i+0.024|23H4i 6.826
−0.001|33F4i+0.004|33H4i
4− 0.744|11G4i−0.667|13G4i−0.017|21G4i+0.016|23G4i 6.511
−0.0004|31G4i−0.0001|33G4i
0.667|11G4i+0.744|13G4i+0.020|21G4i+0.020|23G4i 6.619
+0.001|31G4i+0.002|33G4i
Bs 0− |11S0i 5.360 5.39 5366.77±0.24
B∗ 1− 0.9995|13S1i−0.014|13D1i+0.020|23S1i+0.013|23D1i 5.408 5.45 5415.4+2.4
s −0.010|33S1i−0.012|33D1i −2.1
0.017|13S1i+0.989|13D1i−0.030|23S1i−0.122|23D1i 6.120
−0.010|33S1i−0.069|33D1i
0+ |13P0i 5.731
1+ 0.537|11P1i+0.822|13P1i−0.108|21P1i−0.159|23P1i 5.765
Bs1(5830) 0.835|11P1i−0.546|13P1i+0.056|21P1i−0.042|23P1i 5.850 5829.4±0.7
B∗ (5840) 2+ 0.992|13P2i−0.006|13F2i+0.112|23P2i+0.005|23F2i 5.866 5.88 5839.7±0.6
s2 −0.056|33P2i−0.004|33F2i
0.008|13P2i+0.998|13F2i−0.015|23P2i−0.056|23F2i 6.405
−0.008|33P2i+0.033|33F2i
2− −0.684|11D2i+0.728|13D2i−0.033|21D2i+0.011|23D2i 6.134
+0.019|31D2i−0.007|33D2i
0.727|11D2i+0.682|13D2i−0.037|21D2i−0.058|23D2i 6.142
+0.023|31D2i+0.034|33D2i
(bs¯) 3− 0.998|13D3i−0.004|13G3i+0.061|23D3i+0.003|23G3i 6.144
−0.031|33D3i−0.002|33G3i
0.004|13D3i+0.9995|13G3i−0.005|23D3i−0.024|23G3i 6.641
−0.003|33D3i+0.018|33G3i
3+ 0.749|11F3i−0.662|13F3i+0.011|21F3i−0.007|23F3i 6.374
−0.009|31F3i+0.007|33F3i
0.662|11F3i+0.749|13F3i−0.014|21F3i−0.019|23F3i 6.416
+0.011|31F3i+0.014|33F3i
4+ 0.9992|13F4i−0.002|13H4i+0.033|23F4i+0.002|23H4i 6.380
−0.019|33F4i−0.001|33H4i
0.002|13F4i+0.9999|13H4i−0.002|23F4i−0.006|23H4i 6.849
−0.001|33F4i+0.010|33H4i
4− 0.744|11G4i−0.668|13G4i+0.001|21G4i+0.001|23G4i 6.585
−0.005|31G4i+0.004|33G4i
0.668|11G4i+0.744|13G4i−0.001|21G4i−0.003|23G4i 6.647
+0.006|31G4i+0.007|33G4i