P-Wave Charmed-Strange Mesons

0.25 MB·Indonesian

by  Yukiko Yamada

APS/123-QED
P-Wave Charmed-Strange Mesons
Yukiko Yamada,∗ Akira Suzuki, and Masashi Kazuyama†
Department of Physics, Tokyo University of Science, Shinjuku, Tokyo, Japan
Masahiro Kimura
Department of Electronics Engineering, Tokyo University of Science, Suwa, Nagano, Japan
(Dated: February 2, 2008)
We examine charmed-strange mesons within the framework of the constituent quark model, fo-
cusingonthestateswithL=1. Weareparticularlyinterestedinthemixingoftwospin-statesthat
areinvolvedinDs1(2536)andtherecentlydiscoveredDsJ(2460). Weassumethatthesetwomesons
formapairofstateswithJ =1. Thesespin-statesaremixedbyatypeofthespin-orbitinteraction
that violates the total-spin conservation. Without assuming explicit forms for the interactions as
functions of the interquark distance, we relate the matrix elements of all relevant spin-dependent
6
interactions to the mixing angle and the observed masses of the L = 1 quartet. We find that the
0
spin-spin interaction, among various types of the spin-dependent interactions, plays a particularly
0
2 interesting role in determining thespin structureof Ds1(2536) and DsJ(2460).

n
PACSnumbers: 12.38.Bx,12.39.Jh,12.39.Pn,14.40.Lb
a
J

5
2

I. INTRODUCTION
1
v
1 Recently a new charmed-strange meson, D∗ (2317), has been discovered by the BaBar collaboration [1]. It was
sJ
1 confirmed by the CLEO collaboration [2]. The CLEO reported another charmed-strange meson called D (2460).
sJ
2 Finally, thesemesons wereconfirmedby the Belle collaboration[3, 4]. The massesanddecaypropertiesofD∗ (2317)
sJ
1
and D (2460) have been investigated with particular structures assumed for them. Essentially there are two types
sJ
0 of structures assumed. The one is ordinary qQ¯ structure and the other an exotic structure such as KD molecule
6
[5, 6, 7,9]or tetra-quarkconfiguration[10, 11, 12, 13]. We workwith the formerstructure in this paper. Then, these
0
/ new entries together with Ds1(2536) and Ds2(2573) which were discovered earlier are expected to form a quartet
h with L=1 (P-states) of the cs¯(or sc¯) system. In this expectation, Godfrey studied various properties of D∗ (2317)
p sJ
and D (2460) [14, 15], following the works prior to the discoveries of these mesons [16, 17]. Also decay modes of
- sJ
p D∗ (2317) and D (2460) were analyzed by Colangero and De Fazio [18], Bardeen et al. [19], Mehen and Springer
sJ sJ
e [7] and Close and Swanson [8].
h With respect to the spin-structure of these mesons, there are four states, 1P , 3P , 3P and 3P , in terms of the
: 1 0 1 2
v (JLS) bases.[ ] While D∗ (2317) and D (2573) can probably be assigned to 3P and 3P , respectively, D (2536)
† sJ s2 0 2 s1
i and D (2460) are probably mixtures of 1P and 3P . The extent of the mixing can be parameterized by a mixing
X sJ 1 1
angle[14,15,16,17,20,21]. Inadditiontothemassesofthemesons,thebranchingfractionsforB D¯D followed
sJ
r →
a by the electromagnetic (EM) decays of DsJ have also been measured [3]. The mixing angle are closely related to the
EM decay rates of D [15].
sJ
The purpose of this paper is to examine the spin-structure of the four mesons. We use the constituent quark
model with the interquark interactions that arise from the nonrelativistic expansion of the QCD inspired Fermi-
Breit interaction. We have five types of interactions in the following sense. In addition to the spin-independent
interaction that consists of a confining potential and the color Coulomb interaction, we have four types of spin-
dependent interactions. They arethe spin-spin,tensorandtwotypes ofspin-orbitinteractionsonwhichweelaborate
in the next paragraph. The model is the same as the one used by Godfrey et al. [16, 17, 20] except that we do
not assume any explicit forms for the interactions as functions of the distance between the two quarks. We treat all
spin-dependent interactions perturbatively.
Bythetwotypesofthespin-orbitinteractionswemeantheonesthatarerespectivelysymmetricandantisymmetric
with respectto the interchangeofthe two quarks. We referto the formerasSLS andthe latter asASLSinteractions.
∗Presentaddress: Department ofPhysics,KyushuUniversity,Fukuoka, Japan.
†Presentaddress: DepartmentofPhysics,NagoyaUniversity,Nagoya,Japan.
[†]We use the ordinary spectroscopic notation 2S+1LJ that is used for a two-particle system where S, L and J are total spin, orbital
angularmomentum andtotal angularmomentumquantum numbers,respectively.
2
TABLE I: Summaryof observed charmed-strange mesons
Label Mass (MeV) Assignment (2S+1LJ) Yearof discovery
Ds± 1968.3 ± 0.5 1S0 1983 [28]
Ds∗± 2112.1 ± 0.7 Probably 3S1 1987 [29]
Ds∗J(2317)± 2317.4 ± 0.9 Probably 3P0 2003 [3]
DsJ(2460)± 2459.3 ± 1.3 ? 2003 [3]
Ds1(2536)± 2535.35 ± 0.34 ? 1989 [30]
Ds2(2573)± 2572.4 ± 1.5 Probably 3P2 1994 [31]
The SLS interaction commutes with the total spin of the two quarks whereas ASLS interaction does not. The ASLS
interactionviolates the conservationof the total spin. This is the agentthat induces the mixing of 1P and 3P . The
1 1
ASLS interactionis proportionalto the mass difference between the quarks. Hence its effect can be substantial when
the mass difference is large, leading to a specific amount of mixing in the heavy quark limit. This is indeed the case
with the cs¯(or sc¯) system as we will see. Historically effects of ASLS interaction were first examined for the Λ N
interaction and hypernuclei [22, 23, 24]. Regarding the particular role of the spin-orbit interactions in qQ¯ syste−ms,
we refer to a series of works by Schnitzer [25] and the work by Cahn and Jackson [26] in addition to those quoted
already [16, 17, 20].
As we said above we have five types of interactions. On the other hand there are five pieces of experimental data
now available, which are the masses of the four mesons and the branching ratio of the EM decays. (See Eq. (29).)
The matrixelementsofthe fiveinteractions(within theP-statesector)canbe determinedsuchthatthe fivepiecesof
the experimental data be reproduced. At the same time the spin structure of the four mesons can be determined. In
doing so we do not have to know the radial dependence of the interactions. As it turns out the spin-spin interaction,
among the four types of the spin-dependent interactions, plays a particularly interesting role in relation to the spin
structure of D (2536) and D (2460).
s1 sJ
We begin Sec. II by defining a nonrelativistic model Hamiltonian which incorporates relativistic corrections as
various spin-dependent interactions and proceed to determining the matrix elements of the interactions by using the
mass spectra of the L=1 quartetof charmed-strangemesons and the EM decay widths of D (2460). In Sec. III we
sJ
make remarkson the approximations that we use. Discussions and a summary are givenin the last section. In Table
I we list the observed charmed-strange mesons that we consider in this paper [27].
II. HAMILTONIAN AND MIXING ANGLE
We assume that the nonrelativistic scheme is appropriate for the system and relativistic corrections can be treated
as first order perturbation. The nonrelativistic expansion of the Fermi-Breit interaction gives us the Hamiltonian for
a charmed-strange meson in the form of
H = H +S S V (r)+S V (r)
0 s c S 12 T
·
+L SV(+)(r)+L (S S )V(−)(r) , (1)
· LS · s− c LS
where S is the spin operator of the strange quark when i = s and of the charmed quark when i = c, S = S +S ,
i s c
S the tensoroperatorand L the orbitalangularmomentumoperator. The lowest-orderterms in the nonrelativistic
12
expansion are all in H which also contains a phenomenological potential to confine the quarks. More explicitly H
0 0
reads as
p2 p2
H =m +m + s + c +V (r)+V (r) , (2)
0 s c C conf
2m 2m
s c
where m andp arethe massandmomentum ofquarki,respectively,V is the colorCoulombinteractionandV
i i C conf
theconfinementpotential. ThelasttwotermsofEq. (1)aretheSLSandASLSinteractions,respectively. Thespatial
functions attached to the operators in Eq. (1) can be expressed in terms of V and V [17, 32]. However, we do
C conf
not need such explicit expressions of these functions as it will become clear shortly.
We start with the eigenstates of H such that
0
H ψ (r)=E(0)ψ (r) , (3)
0 nJLS nL nJLS
3
where
J
ψ (r)=R (r) C M (θ,φ) . (4)
nJLS nL MYJLS
M=−J
X
Here C are constants such that C 2 = 1 and can be chosen as (2J +1)−1/2 since there is no preferable
M M| M|
direction. We concentrate on the P-states of n = 1 with no radial node. We denote each of the L = 1 states with
P
single index ν according to
1 1P
1
2 3P
ν = corresponding to  0 . (5)
 3 3P1
4 3P
2
Nextwecalculatethe matrixelementsofH intermsofthe basesdefinedbyEqs. (3)and(4). Nonvanishingmatrix
elements are
H =M 3v
11 0− 4 S
H =M + 1v 2v 4v
22 0 4 S − LS − T
H =M + 1v v +2v , (6)
33 0 4 S − LS T
H =M + 1v +v 2v
44 0 4 S LS − 5 T
H =H =√2∆
13 31
where
M = d3rψ∗ (r)H ψ (r)=E(0) (7)
0 J1S 0 J1S 1
Z
∞
v = drr2V (r)R2(r) (8)
S S 1
Z0
∞
v = drr2V(+)(r)R2(r) (9)
LS LS 1
Z0
∞
v = drr2V (r)R2(r) (10)
T T 1
Z0
∞
∆ = drr2V(−)(r)R2(r) . (11)
LS 1
Z0
We choose the phases of the wave functions involved in Eq.(11) such that ∆ is positive. Here we have suppressed
suffix n = 1 of the wave functions and the unperturbed P-state energy. We have ignored the tensor coupling of 3P
2
state to 3F state. We will remark on this point in the next section. Note that the ASLS interaction gives rise to
2
∆=0 that causes the mixing of 1P and 3P .
1 1
6
All of the matrix elements of the Hamiltonian that we need are parameterized in terms M , v , v , v and ∆.
0 S LS T
ThesefiveparameterscanbedeterminedbythefourobservedmassesandtheEMdecayratesofD (2460). Wehave
sJ
no other adjustable parameters. In this context we do not need explicit expressions of the radial wave function nor
the radial dependence of the potential functions.
The diagonalizationof H leads to four states whose masses are given by
1 1
= 2M v v +2v
+ 0 S LS T
M 2 − 2 −
(cid:20)
1/2
+ (v 2v v )2+8∆2 (12)
LS T S
− −
n o (cid:21)
1
=M + v 2v 4v (13)
2 0 S LS T
M 4 − −
1 1
= 2M v v +2v
− 0 S LS T
M 2 − 2 −
(cid:20)
1/2
(v 2v v )2+8∆2 (14)
LS T S
− − −
n o (cid:21)
1 2
=M + v +v v . (15)
4 0 S LS T
M 4 − 5
4
The second and fourth states with and are pure 3P and 3P states, respectively. We identify them with
2 4 0 2
D∗ (2317) and D (2573). Other twMo statesMwith and are composed of 1P and 3P states. We interpret
sJ s2 M+ M− 1 1
them as D (2536) and D (2460), respectively.
s1 sJ
Let us introduce a mixing angle θ that represents the extent of the mixing of 1P and 3P states in D (2536)and
1 1 s1
D (2460). Following Godfrey and Isgur [16], we define θ by
sJ
ψ (r)= ψ (r)sinθ+ψ (r)cosθ
+ − 110 111 (16)
ψ (r)=ψ (r)cosθ+ψ (r)sinθ
− 110 111
where ψ and ψ are the eigenstates that correspond to D (2536) and D (2460), respectively. The requirement
+ − s1 sJ
that the energy eigenvalues for ψ are leads to
± ±
M
2√2∆
tan(2θ)= . (17)
−v v +2v
S LS T
−
Itisunderstoodthatθ lies inthe intervalof π/2 θ 0 sothatitconformstothe signconventionusedinRef.[16].
− ≤ ≤
Since π/4 θ 0 (or π/2 θ π/4) if (v v +2v ) 0 (or 0), we have θ 0 (or π/2) as ∆ 0
S LS T
− ≤ ≤ − ≤ ≤− − ≥ ≤ → →− →
if (v v +2v ) 0 (or 0). In other words, when (v v +2v )>0 (or <0), D (2536) develops from the
S LS T S LS T s1
3P (o−r 1P ) state, w≥hile D ≤(2460) from the 1P (or 3P ) st−ate due to the ASLS interaction .
1 1 sJ 1 1
We can express the five parameters, M , v , v , v and ∆, in terms of the four observed masses and the mixing
0 S LS T
angle such that
1 1 1 5
M = + + + (18)
0 + − 2 4
4M 4M 12M 12M
1 1
v = (1 2cos(2θ)) (1+2cos(2θ))
S + −
−3 − M − 3 M
1 5
+ + (19)
2 4
9M 9M
1 1
v = (1+cos(2θ)) (1 cos(2θ))
LS + −
−8 M − 8 − M
1 5
+ (20)
2 4
−6M 12M
5 5
v = (1+cos(2θ)) + (1 cos(2θ))
T + −
48 M 48 − M
5 5
(21)
2 4
−36M − 72M
1
∆ = (M M )sin(2θ) . (22)
+ −
−2√2 −
Equation(18)statesthefactthatthemassofthecenterofgravityofthel =1quartetisfreefromthespin-dependent
interactions involved in Eq.(1) in the lowest order perturbation scheme.
In order to determine the mixing angle, we consider EM decays of D (2460) to D and D∗. Generally the E1
sJ s s
decay width of a meson composed of quark 1 and anti-quark 2 is given by
4e2
Γ(i f +γ)= Q k3(2J +1) f r i 2S , (23)
f if
→ 27 |h | | i|
where e is the effective charge defined by
Q
m e m e
1 2 2 1
e = − , (24)
Q
m +m
1 2
k the momentum of the emitted photon
M2 M2
k = i − f , (25)
2M
i
and
1 for a transition between triplet states,
S = (26)
if
(3 for a transition between singlet states,
5
0
e) -20
e
r
g
e
d
(
(cid:84)
-40
-60
0 0.2 0.4 0.6 0.8
R
exp
FIG.1: VariationofthemixinganglewithRexp. Thedot-dashedlineshowsthevalueobtainedfromthecentralvalueofRexp,
and the vertical dotted lines indicate theupperand lower values of Rexp allowed within thestatistical errors.
is a statistical factor [33]. For the decays of D , we have
sJ
322.7MeV for the decay to D∗
k = s , (27)
( 442.0MeV for the decay to Ds
and (2J +1)S = 3 for both cases. Since only the 3P state in D undergoes the transition to D∗ and only the
f if 1 sJ s
1P state to D , the matrix element f r i is proportional to sinθ for the decay to D∗ and to cosθ for the decay to
1 s h | | i s
D [15]. Thus we obtain
s
Γ(D D∗γ) 322.7 3
sJ → s = tan2θ . (28)
Γ(D D γ) 442.0
sJ → s (cid:18) (cid:19)
The Belle collaboration made the first observation of B D¯D decays and reported the branching fractions for
sJ
B D¯D followedbytheEMdecaysofD [3]. Colangelo→et al. analyzedthedatatoextracttheratioofbranching
sJ sJ
fra→ctions for the EM decays of D (2460) to D and D∗ [34]. They obtained
sJ s s
Γ(D D∗γ)
R sJ → s =0.40 0.28 . (29)
exp
≡ Γ(D D γ) ±
(cid:20) sJ → s (cid:21)exp
The experimental value has the large statistical errorswhich results in a large uncertainty in determining the mixing
angle as can be seen in Fig. 1. The numerical value is
θ = 45.4◦ −7.5◦ , (30)
− +16.4◦
wheretheupperandlowerincrementsareduetothepositiveandnegativecorrectionsofthestatisticalerrorsinR ,
exp
respectively. This maybe comparedwith 38◦ obtainedby GodfreyandKokoski[17],and 54.7◦ thatemergesfrom
− −
sinθ = 2/3 in the heavy quark limit [15, 20].
−
We can calculate M , v , v , v and ∆ through Eqs. (18) to (22) by fitting the observed masses of Table I and
0 S LS T
p
the mixing angle of Eq. (30). Again these quantities are subject to uncertainties due to the statistical errors. Using
6
80
)
V
e
M
60 v
( LS
s
t
n
v
e
m 40 S
e
l (cid:39)
E

x
ri 20 v
t T
a
M
0
0 0.2 0.4 0.6 0.8
R
exp
FIG. 2: The matrix elements calculated by applying the experimental value of Rexp from Eq.(18) to Eq.(22) with Eq.(28).
The dot-dashed line shows the value obtained from the central value of Rexp, and the vertical dotted lines indicate the upper
and lower values of Rexp allowed within thestatistical errors.
the central values of the observed masses, we obtain
M =2513.6MeV (31)
0
−13.1
v =21.0 MeV (32)
S +27.5
+2.5
v =61.4 MeV (33)
LS −5.2
−2.1
v =19.7 MeV (34)
T +4.3
−1.0
∆=26.9 MeV . (35)
−4.1
InFig.2weshowhowthematrixelementsvarywhenR isvariedwithinthestatisticalerrors. Notethatthematrix
exp
elementofthespin-spininteractionisparticularlysensitivetothevariationofR . Sincethesignof(v v +2v )
exp S LS T
−
determines the main spin-states of , it is interesting to see the R dependence of this quantity shown in Fig. 3.
± exp
M
We see that (v v +2v ) changes its sign from positive to negative as R passes over 0.39. If R <0.39 the
S LS T exp exp
−
mainspin-statesofD (2536)andD (2460)arerespectively3P and1P . IfR exceeds0.39,thesetwospin-states
s1 sJ 1 1 exp
are interchanged.
In the nonrelativistic expansion of the Fermi-Breit interaction, the spin-spin interaction contains the derivative
of the color Coulomb interaction. If the color Coulomb interaction is of the form of 1/r, the spin-spin interaction
behaves like the delta function near the origin. In that case, the matrix element of the spin-spin interaction will
vanish in P-states because the P-state wave functions are strongly suppressed where the interaction acts. The real
situation, however,is not so simple. The singular spin-dependent interactions are smeared out due to the relativistic
corrections[16, 17] and the asymptotic freedom. The resultant spin-spininteraction will have a well-behavedformat
the origin. Consequently the matrix element of the spin-spin interaction can become sizable. Its magnitude depends
on the spatial form of the interaction which in turn depends on how one incorporates the relativistic corrections and
the asymptotic freedom. Equation (32) that we obtained above is a constraint that the spin-spin interaction has to
satisfy.
Earlierwehadexperimentalinformationontheeffectofthespin-spininteractiononP-statesofheavyquarksystems
only from the charmonia. In first order perturbation theory, we can estimate the matrix element by calculating the
difference between an weightedaverageof the masses of 3P-states and the mass of 1P-state. (See Eq.(6).) For the cc¯
system, if we canregardh (1P)as the 1P state [27], we obtain 0.85MeV forthis quantity. If one assumesthat the
c 0
−
spin-spin interaction is inversely proportional to the product of the quark masses and that the wave functions of the
cc¯systemand those ofthe charmed-strangemesons are the same, one obtains about -3 MeV for the charmed-strange
mesons. The value that emerged from our analysis is much larger in magnitude than this value.
7
80
)
V
e 60
M
(
40
)
T
v
2 20
+
S
L
v 0
-
S
v
( -20
-40
0 0.2 0.4 0.6 0.8
R
exp
FIG. 3: The matrix element (vS −vLS +2vT) versus Rexp. The value at which the matrix element changes its sign is 0.39.
The dot-dashed line shows the value obtained from the central value of Rexp, and the vertical dotted lines indicate the upper
and lower values of Rexp allowed within thestatistical errors.
Let us make remarks on the works of Godfrey and Isgur [16] and of Godfrey and Kokoski [17] in comparison
with the present work. They used basically the same Hamiltonian that we used and diagonalized it on the basis
of the harmonic oscillator eigenstates. They assumed explicit forms for the confinement potential and the color
Coulombinteractioninterms ofwhichthe spatialbehaviorofallspin-dependentinteractionscanbe expressed. They
accomplished the relativistic corrections by introducing a smearing function which softens the singular behavior of
the spin-dependent interactions at the origin. As a consequence a sizable contribution from the spin-spin interaction
to the matrix element for the P-state emerged. They fixed the parameters by fitting observed meson masses known
then and predicted unobserved meson masses. Although they worked out beyond the perturbation theory, it would
be interesting to estimate the matrix elements of the spin-dependent interactions perturbatively through Eqs.(18) to
(22) from the masses and the mixing angle that they obtained for charmed-strange mesons.
The results are summarized in Table II, compared with preceding works by Godfrey and Kokoski [17] and Lucha
andScho¨berl[21]. The lastfour numbers under the mesonsymbols in the firstrownamed this workare the observed
masses that we used to evaluate the matrix elements in our analysis. The last four numbers in the other rows are
the predicted masses. The values in parenthesesin the thirdrow arethe matrix elements obtainednonperturbatively
in Ref. [17]. Note that these values are very close to the corresponding ones of the second row, showing that our
perturbative treatment is adequate.
The masses of D∗ predicted in Refs. [17] and [21] are much larger than the experimental value. The matrix
sJ
elements of the SLS and tensor interactions come into the masses of D∗ with negative sign as seen in Eq.(13).
sJ
In Refs. [17] and [21] the magnitudes of these matrix elements are very small compared with the ones that the
experiments require. This is why they had approximately 110 to 120MeV larger masses for D∗ compared with the
sJ
experimental value even when one corrects the overestimateof the center of gravity(M ) for the P-state masses. On
0
the other hand,the matrix elements of the SLS andtensor interactionscome with opposite signsfor the massof D .
s2
This moderates the overestimate of the mass of D . The mass differences between D and D in Refs. [17] and
s2 s1 sJ
[21] are very small as compared with 76MeV of the experimental value. This is simply due to the feature that the
values of ∆ of Refs. [17] and [21] are much smaller than the one that implied by experiments.
III. VALIDITY OF THE APPROXIMATIONS USED
We give remarks on the approximations that we have used in Sec. II. First, we have regarded all spin-dependent
interactions as perturbation and obtained their matrix elements as given in Eqs.(32)-(35). A typical mass difference
∆M betweentwoconsecutiveprincipalstatesthatemergesfromH isprobablylike400MeVto500MeV.Thevalues
0 0
of v , v , v and ∆ are much smaller than ∆M . This justifies our perturbative treatment of the spin-dependent
S LS T 0
interactions.
8
TABLE II: Matrix elements, the mixing angle θ and the masses of the P-state charmed-strange mesons in the columns with
thecorrespondingmesonsymbols. TheyaregiveninMeVexceptθ. Inthefirstrownamedthiswork,thecentralvaluesofthe
masses reported by Particle Data Group [27] were listed and we used them to obtain the matrix elements. The mixing angle
was given byEq.(30) with thestatistical errors suppressed. The valuesof themasses and themixing angles in thesecond row
arepredictionsbytheindicatedauthors. Thematrixelementsineachrowwerecalculatedbysubstitutingthesequantitiesinto
Eqs.(18)-(22). Thenumbersin theparentheses in thethird row are thematrix elements obtained in Ref. [17].
M0 vs vLS vT ∆ θ Ds∗J DsJ Ds1 Ds2
This work 2513.6 21.0 61.4 19.7 26.9 −45.4◦ 2317.4 2459.3 2535.35 2572.4
Godfrey and Kokoski[17] 2563 13 27 8 3 −38◦ 2480 2550 2560 2590
(2564) (15) (27) (7) (3)
Luchaand Sch¨oberl [21] 2531 14 29 8 4 −44.7◦ 2446 2515 2517 2561
Secondly, we have ignored the tensor coupling of 3P state to 3F state. The nonvanishing matrix element of the
2 2
tensor interactionbetween these states gives rise to an additive correctionto H in Eq.(6) through the second order
44
perturbation. Let us have an estimate of the second order correction. Note that a quark in a state with L 1 feels
≥
the color Coulomb interaction much less than a quark in an S state. This is because the wave function of the former
is much suppressed near the origin as compared with the wave function of the latter. Therefore, the P- and F-state
wave functions are not very different from those emerging from the confinement potential alone. Let us ignore the
color Coulomb interaction in obtaining the P- and F-state wave functions and use the harmonic oscillator potential
for V . Then the radial parts of nodeless P- and F-state wave functions are given by
conf
R (r) = 8 (µω)5 1/4re−µωr2/2 (36)
1
3 π
r (cid:20) (cid:21)
R (r) = 32 (µω)9 1/4r3e−µωr2/2 , (37)
3
105 π
r (cid:20) (cid:21)
where ω is an oscillator constant and µ = (1/m +1/m )−1 the reduced mass. Remember that ω is related to the
s c
mass difference between two consecutive principal states and, hence, ω ∆M . Since the tensor interaction can be
0
≈
expressed as
V′(r) rV′′(r)
V (r)= C − C (38)
T
12m m r
s c
in terms of the color Coulomb interaction, the needed diagonal and off-diagonalmatrix elements are given by
8 (µω)3 α
3P S V (r) 3P = (39)
2 12 T 2
| | −45 π m m
r s c
(cid:10) (cid:11) 16 6 (µω)3 α
3P S V (r) 3F = , (40)
2 12 T 2
| | 15 35 π m m
r r s c
where we have used (cid:10) (cid:11)
4α
V (r)= (41)
C
−3 r
with the strong coupling constant α. Thus we obtain
3P S V (r) 3F 6
2 12 T 2
| | = 6 2.5 . (42)
3P S V (r) 3P − 35 ≈−
(cid:10)h 2| 12 T | 2i(cid:11) r
If we equate the denominator to the matrix element of the tensor operator times the quantity given in Eq.(34), that
is,
3P S V (r) 3P 8MeV , (43)
2 12 T 2
| | ≈−
an approximate magnitude of the off-diago(cid:10)nal element become(cid:11)s
3P S V (r) 3F 20MeV . (44)
2 12 T 2
| | ≈
Since the energy difference between the(cid:10)3P2 and 3F2 state(cid:11)s is approximately 2 ω 1 GeV, the second order
correctionwill be about 0.4MeV, that is, about5% of the diagonalelement for the×3P s≈tate. Thus we conclude that
2
the tensor coupling to 3F state will not change our result appreciably.
2
9
IV. DISCUSSIONS AND SUMMARY
We haveexaminedthe P-statecharmed-strangemesons,focusingonthe mixingof1P and3P statesinD (2536)
1 1 s1
andD (2460)thatiscausedbytheantisymmetricspin-orbit(ASLS)interaction. Wehavetreatedthespin-dependent
sJ
interactions that arise from the nonrelativistic expansion of the Fermi-Breit interaction perturbatively. We have not
assumedanyexplicitformsforthe interactionsasfunctions ofthe interquarkdistance. We haveexpressedthe matrix
elementsoftheseinteractionsintermsoftheobservedmassesoftheP-statequartetandthemixingangledetermined
from the EM decay rates of D (2460).
sJ
The EM decay rates have large statistical errors. If we vary the decay rates within the errors, the mixing angle
varies widely. The matrix elements of the spin-dependent interactions also vary accordingly. The matrix elements of
theSLS,tensorandASLSinteractionsarerelativelystablewiththevariationofthemixingangle,varyingonlywithin
20%. On the other hand, the matrix element of the spin-spin interaction varies from 48.5MeV to 7.9MeV when the
mixing angle varies from one end to the other determined from the EM decay rates with the statistical errors. Note
that Godfrey and Kokoski obtained the matrix element of the spin-spin interaction 15MeV that lies in this interval
[17].
Thematrixelementofthespin-spininteractionisparticularlysensitivetothemixingangleandofcrucialimportance
in determining the dominant states of D (2536) and D (2460). With the large variation of the mixing angle, the
s1 sJ
dominant state of D (2536) is transferred from 3P to 1P state and that of D (2460) from 1P to 3P state.
s1 1 1 sJ 1 1
This implies that the spin-spin interaction is the most important among the spin-dependent interactions for the
determination of the dominant states in D (2536) and D (2460). It will be crucial for their assignments, provided
s1 sJ
other mechanisms for the mixing such as the coupling to decay channels are less significant than what we have
discussed [35, 36, 37].
Ouranalysisisbasedonthebranchingfractionsthatwereobtainedfromthe firstobservationofB D¯D decays
sJ
→
by the Belle collaboration. The analyses are accompanied by large statistical errors,and so be the mixing angle that
isextractedfromthe branchingfractions. Forfurtherdiscussionofthe relationshipbetweenthe mixingangleandthe
spin-dependent interactions, we need more refined data of the branching fractions by the experimental groups.
Acknowledgments
We are grateful to Yuki Nogami for helpful comments on the manuscript. A. S. would like to thank Professor T.
Udagawa for the warm hospitality extended to him at the University of Texas at Austin where the last part of this
work was done.
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