Study on Decays of $D_{sJ}^{*}(2317)$ and $D_{sJ}(2460)$ in terms of the CQM Model

0.18 MB·English

by  Xiang Liu

D D
Study on Decays of (2317) and (2460) in terms of the
s∗J sJ
CQM Model
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February 2, 2008
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a Xiang Liu, Yan-Ming Yu, Shu-Min Zhao and Xue-Qian Li
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Department of Physics, Nankai University, Tianjin 300071, China

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1
Abstract
0
1 Basedonthe assumptionthat Ds∗J(2317)and DsJ(2460)are the (0+,1+) chiralpartners
0 ∗ ∗
6 ofDsandDs,weevaluatethestrongpionicandradiativedecaysofDsJ(2317)andDsJ(2460)
in the Constituent Quark Meson (CQM) model. Our numerical results of the relative ratios
0
/ of the decay widths are reasonably consistent with data.
h
p
PACS numbers: 13.25.Ft, 12.38.Lg, 12.39.Fe, 12.39Hg
-
p
e
h 1 Introduction
:
v
i The new discoveries of exotic particles D∗ (2317) and D (2460), which possess spin-parity
X sJ sJ
structures of 0+, 1+ respectively [1, 2, 3], attract great interests of both theorists and experi-
r
a mentalists of high energy physics. Some authors [4] suppose that D∗ (2317) and D (2460) are
sJ sJ
(0+,1+) chiral partners of D and D∗ i.e. p-wave excited states of D and D∗ [5]. Narison used
s s s s
the QCD spectral sum rules to calculate the masses of D∗ (2317) and D (2460) by assuming
sJ sJ
that they are quark-antiquark states and obtained results which are consistent with data within
a wide error range [6]. Beveren and Rupp also studied the mass spectra [7] and claimed that
their results support the cs¯ structures for D∗ (2317) and D (2460). Meanwhile, some other
sJ sJ
authors suggest that D∗ (2317) and D (2460) can possibly be four-quark states [8, 9, 10, 11].
sJ sJ
Thus, one needs to try various ways to understand the structures of D∗ (2317) and D (2460).
sJ sJ
Ingeneral, onecan take areasonabletheoretical approach toevaluate related physicalquantities
and then compare the results with data to extract useful information. One can determine the
structures of D∗ (2317) and D (2460) by studying the production rates of the exotic particles,
sJ sJ
and our recent work [12] is just about the production of D∗ (2317) in the decays of ψ(4415).
sJ
Recently, several groups have calculated the strong and radiative decay rates of D∗ (2317)
sJ
andD (2460) indifferenttheoretical approaches: theLightConeQCDSumRules, Constituent
sJ
Quark model, Vector Meson Dominant (VMD) ansatz, etc. [13, 14, 15, 16, 17, 18, 19]. For a
clear comparison, theresultsbydifferentgroupsarelisted inTables 2and3. TheauthorsofRef.
1
[9, 20] also calculated the rates based on the assumption that D∗ (2317) and D (2460) are in
sJ sJ
non-cs¯structures. Their predictions on the D∗ (2317) and D (2460) decay rates are obviously
sJ sJ
larger than that obtained by assuming the two-quark structure by orders. Thus studies on
the strong and radiative decays with other plausible models would be helpful. It can not only
deepen our understanding about the characters of these particles, but also test the reliability of
models which are applied to calculate the decays. Because D (2632) was only observed by the
sJ
SELEX collaboration[21], but not by Babar [22], Belle [23] and FOCUS[24], its existence is still
in dispute, so here we do not refer decays of D (2632).
sJ
In this work, we study the strong and radiative decays of D∗ (2317) and D (2460) in the
sJ sJ
framework of Constituent-Quark-Meson (CQM) model. CQM model was proposed by Polosa
et al. [25] and has been well developed later based on the works of Ebert et al. [26] (See the
Ref. [25] for a review). The model is based on an effective Lagrangian which incorporates the
flavor-spin symmetry for heavy quarks with the chiral symmetry for light quarks. Employing
the CQM model to study phenomenology of heavy meson physics, reasonable results have been
achieved [27, 28]. Therefore, we can believe that the model is applicable to our processes and
expect to get relatively reliable conclusion.
The constraint from the phase space of the final states forbids the processes D∗ (2317)
sJ →
D η(η′)andD (2460) D∗η(η′),sothattheonlyallowedstrongdecaymodesareD∗ (2317)
s sJ → s sJ →
D π0 and D (2460) D∗π0. In principle, D (2460) D∗ (2317) + π0 which is a 1+
s sJ → s sJ → sJ →
0+ +0− process, is allowed by the phase space. However, it is a p-wave reaction, and the total
rate is proportional to p 2, where p is the three-momentum of emitted pion in the center-of-
| |
mass frame of D (2460). In this case p is very small (about 28 MeV), so that this process
sJ
| | ∼
can only contribute to the total width a negligible fraction, in practice.
The aforementioned strong decay modes obviously violate isospin conservation. Therefore
the decay widths of D∗ (2317) D π0 and D (2460) D∗π0 must be highly suppressed.
sJ → s sJ → s
Moreover, direct emission of a pion is OZI suppressed [29].
Cho et al. suggested a mixing mechanism of η π0 where the isospin violation originates
−
from the mass splitting of u and d quarks [30]. In that scenario, D∗ (2317) and D (2460)
sJ sJ
firstly transit into D η and D∗η, and then η transits into π0 by the mixing. In the intermediate
s s
process, η obviously is off-shell. The mixing depends on the mass difference of η and π, and the
effects due to the mixing between η′ and π can be ignored.
Another sizable mode is the radiative decay. Even though, by general consideration, the
electromagnetic reaction should bemuch less important than the strong decay, it does not suffer
the suppression of isospin violation, therefore one may expect that it has a comparable size
to the strong processes described above. The relevant decay modes are D∗ (2317) D∗ +γ,
sJ → s
D (2460) D +γ and D (2460) D∗+γ and D (2460) D∗ (2317)+γ.
sJ → s sJ → s sJ → sJ
Currently the Babar and the Belle collaborations have completed precise measurements on
the ratio of Γ(D (2460) D γ) to Γ(D (2460) D∗π0) [3, 31, 32]. Now Babar and Belle
sJ → s sJ → s
collaborations begin to further measure other decays of D∗ (2317) and D (2460). We are
sJ sJ
expecting new results of Babar and Belle which can be applied to decisively determine the
structures of D∗ (2317) and D (2460).
sJ sJ
This paper is organized as follows, after the introduction, in Sect. 2, we formulate the strong
and radiative decays of D∗ (2317) and D (2460). In Sect.3, we present our numerical results
sJ sJ
along with all the input parameters. Finally, Sect. 4 is devoted to discussion and conclusion.
Some detailed expressions are collected in the Appendix.
2
2 Formulation
First, for readers’ convenience, we present a brief introduction of the Constituent-Quark-Meson
(CQM) model [25]. The model is relativistic and based on an effective Lagrangian which com-
bines the heavy quark effective theory (HQET) and the chiral symmetry for light quarks,
f2
L = χ¯[γ (i∂ + )]χ+χ¯γ γ χ m χ¯χ+ πTr[∂µΣ∂ Σ+]+h¯ (iv ∂)h
CQM 5 q µ v v
· V ·A − 8 ·
∂ 1 1
[χ¯(H¯ +S¯+iT¯µ µ)h +h.c.]+ Tr[(H¯ +S¯)(H S)]+ Tr[T¯µT ], (1)
v µ
− Λ 2G − 2G
3 4
where the fifth term is the kinetic term of heavy quarks with v/h = h ; H is the super-field
v v
corresponding to doublet (0−,1−) of negative parity and has an explicit matrix representation:
1+v/
H = (P∗γµ Pγ );
2 µ − 5
where P and P∗µ are the annihilation operators of pseudoscalar and vector mesons which are
normalized as
0P M(0−) = M , and 0P∗µ M(1−) = M ǫµ.
H H
h | | i h | | i
p p
S is the super-fields related to (0+,1+),
S = 1+v/[P∗′γµγ P ].
2 1µ 5− 0
iM
χ = ξq (q = u,d,s) is thelight quarkfieldand ξ = efπ , andM is theoctet pseudoscalar matrix.
We also have
1
µ = (ξ†∂µξ+ξ∂µξ†),
V 2
and
i
µ = − (ξ†∂µξ ξ∂µξ†).
A 2 −
Because the spin-parity of D∗ (2317) and D (2460) are 0+ and 1+, thus D∗ (2317) and
sJ sJ sJ
D (2460) can be embedded into the S-type doublet (0+,1+) [28], whereas D and D∗ belong
sJ s s
to the H type doublet (0−,1−). Then we can calculate the strong and radiative decay rates of
D∗ (2317) and D (2460) in the CQM model.
sJ sJ
2.1 The transition amplitude of D∗ (2317) D +π0 and D (2460) D∗ +π0
sJ → s sJ → s
strong decays in the CQM model.
As discussed above indirect D∗ (2317) D + π0 and D (2460) D∗ + π0 occur via two
sJ → s sJ → s
steps. InFig. 1,weshowtheFeynmandiagrams whichdepictthestrongprocessesD∗ (2317)
sJ →
D +η D +π0 and D (2460) D∗+η D∗+π0. According to chiral symmetry, η only
s → s sJ → s → s
couples with light quark, thus it can only be emitted from the light-quark leg in Fig.1.
The matrix elements of D∗ (2317) D +π0 and D (2460) D∗+π0 are written as
sJ → s sJ → s
i
[D∗ (2317) D +π0] = π0 η ηD D∗ (2317) , (2)
M sJ → s h |Lmixing| im2 m2h s|LCQM| sJ i
η − π
i
[D (2460)) D∗+π0]= π0 η ηD∗ D (2460) . (3)
M sJ → s h |Lmixing| im2 m2h s|LCQM| sJ i
η − π
3
π0 q
η
D∗ (2317) l l+q D
sJ s
v,p v,p′
l+p
Figure 1: The Feynman diagrams which depict the decays of D∗ (2317) D + π0 or
sJ → s
D (2460) D∗+π0. The double-line denotes the heavy quark (c quark) propagator.
sJ → s −
The mixing mechanism is described by the Lagrangian1
m2 m˜ m˜
= π u− dπ0η
mixing
L 2√3m˜u+m˜d
which originates from the mass term of the low energy Lagrangian for the pseudoscalar octet
[30]
m2f2
= π π Tr[ξm ξ+ξ†m ξ†], (4)
mass q q
L 4(m˜ +m˜ )
u d
where m is the light quark mass matrix. The matrix elements of ηD D∗ (2317) and
q h s|HCQM| sJ i
ηD∗ D (2460) will be calculated in the CQM model.
h s|HCQM| sJ i
m˜ (i = u,d,s) arethecurrentquarkmasses. Itis noted thatintheCQMmodelcalculations,
i
the quark masses (m , m ) which we denote as m , are constituent quark masses [25], whereas,
q c q
for the mixing, the concerned masses which we denote as m˜ are the current quark masses[30].
q
According to the CQM model [25], couplings of D∗ (2317) and D (2460) with light and
sJ sJ
heavy quarks are expressed as
1+v/
Z M ,
S 1
2
p
and
1+v/
Z M /ǫ γ ,
S 2 1 5
2
p
the couplings of D and D∗ to light and heavy quarks are
s s
1+v/
Z M γ
2 q S Ds 5
and
1+v/
ZSMD∗/ǫ2,
2 q s
where ǫ and ǫ denote the polarization vectors of D (2460) and D∗ respectively. M and M
1 2 sJ s 1 2
are respectively the masses of D∗ (2317) and D (2460). The concrete expressions of Z and
sJ sJ H
1Hereweignore themixingof η andη′,becausethemixingangle θ∼−11◦ is small anddoes notmuchaffect
our results. Therefore we simply assume that η is η8 and the contribution of η′, as discussed in the text, is
neglected in ourcalculations.
4
Z are given in Ref. [25] as
S
∂I (∆ )
Z−1 = (∆ +m ) 3 H +I (∆ ), (5)
H H s ∂∆ 3 S
H
∂I (∆ )
Z−1 = (∆ +m ) 3 S +I (∆ ), (6)
S S s ∂∆ 3 S
S
iN 1/µ2 dy
I (a) = c exp[ y(m2 a2)](1+erf(a√y)), (7)
3 16π4 Z1/Λ2 y3/2 − s −
where erf is the error function.
Now, we can write out the transition matrix elements as
ηD D∗ (2317)
h s|HCQM| sJ i
2 N reg d4l Tr[(/l+m )qµγ γ (/l+/q+m )γ (1+v/)]
= ( 1)i6 Z M Z M c s µ 5 s 5 ,
− q S 1 H Dsr32fπ Z (2π)4 (l2−m2s)[(l+q)2−m2s](v·l+∆S)
(8)
and
ηD∗ D (2460)
h s|HCQM| sJ i
2 N reg d4l Tr[(/l+m )qµγ γ (/l+/q+m )/ǫ (1+v/)γ /ǫ ]
= (−1)i6qZSM2ZHMDs∗r32fcπ Z (2π)4 (l2−sm2s)[(µl+5 q)2−m2s](sv·2l+∆S) 5 1 ,
(9)
where N = 3.
c
Omitting technical details in the text for saving space, we finally obtain
2
ηD D∗ (2317) = Z M Z M A , (10)
h s|HCQM| sJ i q S 1 H Ds(cid:16)−r3(cid:17)2fπ
2 (ǫ ǫ )
hηDs∗|HCQM|DsJ(2460)i = qZSM2ZHMDs∗(cid:16)−r3(cid:17) 12·fπ2 A, (11)
with
= 4(m2m ω m2m ) m2( +ω )+m ω(2 2 ), (12)
A s η − η s O− η O1 O2 η O3 −O4−O5− O6
where ω = v v′ = ∆S−∆H and the concrete expressions of and are listed in the appendix.
· 2mη O Oi
The decay widths of D∗ (2317) D +π0 and D (2460) D∗+π0 read as
sJ → s sJ → s
Γ[D∗ (2317) D +π0]
sJ → s
Z Z M p′ m˜ m˜ 2
= S H Ds| | u− d 2, (13)
1024πM f2 (cid:18)m˜ (m˜ +m˜ )/2(cid:19) |A|
1 π s− u d
Γ[D (2460) D∗+π0]
sJ → s
= ZSZHMDs|p′| m˜u−m˜d 2 2+ (M22+MD2s∗)2 2, (14)
3072πM f2 (cid:18)m˜ (m˜ +m˜ )/2(cid:19) (cid:20) M2M2 (cid:21)|A|
2 π s− u d 2 Ds∗
where the relations m2 = 2m˜B and m2 = 2(m˜ +2m˜ )B with m˜ = (m˜ +m˜ )/2 are employed
π 0 η 3 s 0 u d
toderiveisospinsuppressionfactor (m˜ m˜ )/[m˜ (m˜ +m˜ )/2] [35]. Theserelations arevalid
u d s u d
− −
at the leading order of the chiral theory, but the principle does not really apply for estimating
the mass of η′ (or η ) due to an extra contribution from the axial anomaly [33].
0
5
2.2 The transition amplitude of D∗ (2317) and D (2460) radiative decays in
sJ sJ
the CQM model.
the Feynman diagrams of D∗ (2317) D∗+γ, D (2460) D +γ, D (2460) D∗+γ and
sJ → s sJ → s sJ → s
D (2460) D∗ (2317)+γ are presented in Fig. 2. We will evaluate these radiative decays in
sJ → sJ
γ q
D∗ (2317) l D∗
sJ s
D∗ (2317) l l+q D∗ v,p l+p l+p−q v,p′
sJ s
v,p v,p′
l+p γ q
(a) (b)
Figure 2: The Feynman diagrams depicts the radiative decays of D∗ (2317) and D (2460).
sJ sJ
the CQM model.
(a) D∗ (2317) D∗+γ.
sJ → s
In contrast with the case in Fig.1, the heavy quark also couples to γ. Thus the transition
matrix element of D∗ (2317) D∗+γ is written as
sJ → s
[D∗ (2317) D∗+γ]
M sJ → s
e reg d4l Tr[(/l+m )/ǫ (/l+/q+m )/ǫ (1+v/)]
= (−1)i6qZSM1ZHMDs∗Nc(cid:26)(cid:18)−3 (cid:19)Z (2π)4 (l2−m2s)[(sl+γq)2−m2s](sv·2l+2∆S)
2e reg d4l Tr[(/l+m )/ǫ (1+v/)/ǫ (1+v/)]
+ s 2 2 γ 2
3 Z (2π)4(l2 m2)(v l+∆ )(v l+∆ )(cid:27)
− s · H · S
2e (p′ q)(m + +2 )
= qZSM1ZHMDs∗(cid:16) 3 (cid:17)n(ǫγ ·ǫ2)hm2sR−R3−2R6− · sRMD∗R1 R5 i
s
m + +2
+(ǫ p′)(ǫ q) sR R1 R5 . (15)
γ 2
· · h MD∗ io
s
(b) D (2460) D +γ.
sJ s
→
The transition matrix element of D (2460) D +γ is
sJ s
→
[D (2460) D +γ]
sJ s
M →
e reg d4l Tr[(/l+m )/ǫ (/l+/q+m +iǫ)γ (1+v/)γ /ǫ ]
= ( 1)i6 Z M Z M N − s γ s 5 2 5 1
− q S 2 H Ds c(cid:26)(cid:18) 3 (cid:19)Z (2π)4 (l2−m2s)[(l+q)2−m2s](v·l+∆S)
2e reg d4l Tr[(/l+m )γ (1+v/)/ǫ (1+v/)γ /ǫ ]
+ s 5 2 γ 2 5 1
3 Z (2π)4(l2 m2)(v l+∆ )(v l+∆ )(cid:27)
− s · H · S
2e (p′ q)(m + +2 )
= Z M Z M (ǫ ǫ ) m2 2 · sR R1 R5
q S 1 H Ds(cid:16) 3 (cid:17)n γ · 1 h sR−R3− R6− MDs i
m + +2
+(ǫ p′)(ǫ q) sR R1 R5 . (16)
γ 1
· · h MDs io
6
(c) D (2460) D∗+γ.
sJ → s
The transition matrix element of D (2460) D∗+γ is
sJ → s
[D (2460) D∗+γ]
M sJ → s
e reg d4l Tr[(/l+m )/ǫ (/l+/q+m )/ǫ (1+v/)γ /ǫ ]
= (−1)i6qZSM2ZHMDs∗Nc(cid:26)(cid:18)−3 (cid:19)Z (2π)4 (l2−m2ss)[(lγ+q)2−m2ss](v2·l2+∆5S)1
2e reg d4l Tr[(/l+m )/ǫ (1+v/)/ǫ (1+v/)γ /ǫ ]
+ s 2 2 γ 2 5 1
3 Z (2π)4 (l2 m2)(v l+∆ )(v l+∆ )(cid:27)
− s · H · S
2e 2 (p′ q) m2 2
= qZSM2ZHMDs∗(cid:18) 3 (cid:19)εαβρσqαǫβγǫρ1ǫσ2(cid:20)Rms−R1+ RM2 D∗· + R s −MRD3∗− R6
s s
(2 +2 )(p′ q)
5 1
+ R R · . (17)
M2 (cid:21)
D∗
s
(d) D (2460) D∗ (2317)+γ.
sJ → sJ
The transition matrix element of D (2460) D∗ (2317)+γ is
sJ → sJ
[D (2460) D∗ (2317)+γ]
M sJ → sJ
e reg d4l Tr[(/l+m )/ǫ (/l+/q+m )(1+v/)γ /ǫ ]
= ( 1)i6 Z M Z M N − s γ s 2 5 1
− S 1 S 2 c(cid:26)(cid:18) 3 (cid:19)Z (2π)4 (l2 m2)[(l+q)2 m2](v l+∆ )
p − s − s · S
2e reg d4l Tr[(/l+m )(1+v/)/ǫ (1+v/)γ /ǫ ]
+ s 2 γ 2 5 1
6 Z (2π)4(l2 m2)(v l+∆ )(v l+∆ )(cid:27)
− s · S · S
2em
= Z M Z M sRε qαp′βǫρǫσ. (18)
S 1 S 2 3 M αβρσ γ 1
p 1
The concrete expressions of and are listed in the appendix.
i
R R
3 Numerical results
With the formulation we derived in last section, we can numerically evaluate the corresponding
decay rates. Besides, we need several input parameters for the numerical computations. They
include: f = 132 MeV, m = 0.5 GeV, Λ = 1.25 GeV, the infrared cutoff µ = 0.51 GeV
π s
and ∆ ∆ = 335 35 MeV [28]. m = 547.45 MeV, M = 2317 MeV, M = 2460 MeV,
S H η 1 2
− ±
MDs = 1968 MeV and MDs∗ = 2112 MeV [34]. The suppression parameter was estimated in
ref.[35] as
m˜ m˜ 1
u d
− .
m˜ (m˜ +m˜ )/2 ∼ 43.7
s u d
−
(a) We present the decay widths of D∗ (2317) D +π0 and D (2460) D∗+π0 in Table
sJ → s sJ → s
1.
For a comparison, we also list the values of the decay widths of D∗ (2317) D +π0 and
sJ → s
D (2460) D∗+π0 which are calculated by other groups, in Table 2.
sJ → s
7
∆ (GeV) ∆ (GeV) Z (GeV)−1 Z (GeV)−1 Γ (keV) Γ (keV)
H S H S 1 2
0.5 0.86 3.99 2.02 3.68 1.86
0.6 0.91 2.69 1.47 5.36 2.72
0.7 0.97 1.74 0.98 8.71 4.42
Table1: Thevaluesof∆ and∆ aretaken fromRef. [28]. Accordingeqs. (5)and(6), onegets
S H
thevaluesofZ andZ . Γ andΓ denoterespectivelythedecaywidthsofD∗ (2317) D +π0
S H 1 2 sJ → s
and D (2460) D∗+π0.
sJ → s
CQM model [13] [14] [15] [16] [17] [18]
Γ (keV) 3.68 8.71 6 2 34 44 7 1 21.5 10 16
1
∼ ± ∼ ± ∼
Γ (keV) 1.86 4.42 - 35 51 7 1 21.5 10 32
2
∼ ∼ ± ∼
Table 2: In this table, we list our calculation of D∗ (2317) D +π0 and D (2460) D∗+π0
sJ → s sJ → s
and that obtained by other groups. Where Γ and Γ are respectively denote the decay widths
1 2
of D∗ (2317) D +π0 and D (2460) D∗+π0.
sJ → s sJ → s
Our values depend on the model parameters, but qualitatively, the order of magnitude is
unchanged when the parameters vary within a reasonable region. All the values in table 2 are
somehow consistent with each other for order of magnitude, even though there is obvious differ-
ence in numbers.
(b)Withthesameparameters,weobtaintheradiativedecayratesofD∗ (2317)andD (2460).
sJ sJ
The results are listed in Table. 3
CQM model [19] [15] [16] [17]
Γ(D∗ (2317) D +γ)(keV) 1.1 4 6 0.85 1.74 1.9
sJ → s ∼ ∼
Γ(D (2460) D +γ)(keV) 0.6 2.9 19 29 3.3 5.08 6.2
sJ s
→ ∼ ∼
Γ(D (2460) D∗+γ)(keV) 0.54 1.4 0.6 1.1 1.5 4.66 5.5
sJ → s ∼ ∼
Γ(D (2460) D∗ (2317)+γ)(keV) 0.13 0.22 0.5 0.8 - 2.74 0.012
sJ → sJ ∼ ∼
Table 3: The rates of D∗ (2317) D +γ, D (2460) D +γ, D (2460) D∗ +γ and
sJ → s sJ → s sJ → s
D (2460) D∗ (2317)+γ. Meanwhile we also list the results obtained by other group.
sJ → sJ
For convenience, we define the relevant ratios as
R = Γ(D∗ (2317) D +γ) : Γ(D∗ (2317) D +π0),
1 sJ → s sJ → s
R = Γ(D (2460) D +γ) :Γ(D (2460) D∗+π0),
2 sJ → s sJ → s
R = Γ(D (2460) D∗+γ) : Γ(D (2460) D∗+π0),
3 sJ → s sJ → s
R = Γ(D (2460) D∗ (2317)+γ) : Γ(D (2460) D∗+π0).
4 sJ → sJ sJ → s
Then we calculate the R ’s and tabulate the results below.
i
8
Belle Babar CLEO [2] CQM model
R < 0.18[31] - < 0.059 0.12 0.30
1
∼
R 0.55 0.13 0.08 [31] 0.375 0.054 0.057 [32] < 0.49 0.32 0.66
2
± ± ± ± ∼
R <0.31 [31] - < 0.16 0.29 0.32
3
∼
R - < 0.23 [32] < 0.58 0.05 0.07
4
∼
Table 4: The ratios of radiative decay widths to strong decay widths for D∗ (2317) and
sJ
D (2460). The first three columns are the experimental data and the fourth column is our
sJ
result calculated in the CQM model.
4 Conclusion and Discussion
In this work, based on the assumption that D∗ (2317) and D (2460) are chiral parters of D
sJ sJ s
andD∗,wecalculatetheratesofD∗ (2317) D π0 andD (2460) D∗π0 intheConstituent-
s sJ → s sJ → s
Quark-Meson (CQM) model and take into account the η π0 mixing mechanism [30]. We also
−
estimate the rates of D∗ (2317) D∗ + γ, D (2460) D + γ, D (2460) D∗ +γ and
sJ → s sJ → s sJ → s
D (2460) D∗ (2317)+γ in the same model.
sJ → sJ
Comparing our results about the strong decay rates with that obtained by other groups, we
find that our results are reasonably consistent with the values listed in table 2.
For the radiative decay, our results generally coincide with that obtained by other groups
and especially these results of the QCD sum rules.
Theratio R hasbeenmeasuredwithrelatively highprecision [3,31,32], however forR , R
2 1 3
andR ,thereonlyareupperlimitsgivenbyBabar,Belle, andCLEOcollaborations[2,31,32]. It
4
seems that our results on theratios well coincide with the experimental values. This consistency
somehow implies that the assumption of D∗ (2317), D (2460 being p-wave chiral partners of
sJ sJ
D , D∗ does not contradict to the data with the present experimental accuracy.
s s
The experimental upper bounds on the total widths are Γ(D∗ (2317)) < 4.6 MeV and
sJ
Γ(D (2460)) < 5.5 MeV [34]. Obviously, the overwhelming decay modes of D∗ (2317) and
sJ sJ
D (2460) are their strong and radiative decays, therefore we can roughly take sums of these
sJ
widths as the total widths of D∗ (2317) and D (2460). However, our numerical results as well
sJ sJ
asthatgivenbyothergroupsareinorderoftensofkeV,muchsmallerthantheupperboundsset
by recent experiments. The reason is obvious that the aforementioned reactions violate isospin
conservation, there is a large suppression factor of about (1/43.7)2, which reduces the widths
by 3 orders. The calculations which are based on the assumption that the newly discovered
D∗ (2317) and D (2460) are p-wave excited states of D and D∗ predict their widths to be at
sJ sJ s s
orderofafewtotensofkeV.Bycontrary, iftheyarefour-quarkstates, ormolecularstates, there
may be more decay channels available, i.e. some modes are not constrained by the OZI rule,
thus much larger total widths might be expected in that scenario. The authors of refs.[9, 20],
for example, suggested that D∗ (2317) and D (2460) are in non-cs¯structure (four-quark states
sJ sJ
etc.), and obtained much larger rates, even though still obviously smaller than the experimental
upper bounds. So far, it is hard to conclude if they are p-wave chiral partners of D and D∗ or
s s
four-quark states yet.
We hope that the further more precise measurements of Babar, Belle and CLEO may offer
more information by which we may determine the structure of the newly discovered mesons.
9
Acknowledgment:
We would like to thank Dr. Xin-Heng Guo for helpful discussions. This work is supported
by the National Natural Science Foundation of China (NNSFC).
Appendix
I (∆ ,m /2,ω) I (∆ , m /2,ω)
5 S η 5 H η
= − − , (19)
O 2m
η
I ( m /2) I (m /2)+ω[I (∆ ) I (∆ )] [∆ ωm /2]
3 η 3 η 3 S 3 H S η
= − − − O − , (20)
O1 2m (1 ω2) − 1 ω2
η
− −
I (∆ )+I (∆ ) ω[I ( m /2) I (m /2)] [m /2 ∆ ω]
3 S 3 H 3 η 3 η η S
= − − − − O − , (21)
O2 2m (1 ω2) − 1 ω2
η
− −
2ω
1 4 2 3
= B + B −B −B , (22)
O3 2 2(1 ω2)2
−
3 6ω + (2ω2+1)
1 2 4 3
= −B + B − B B , (23)
O4 2(1 ω2) 2(1 ω2)2
− −
3 6ω + (2ω2+1)
1 3 4 2
= −B + B − B B , (24)
O5 2(1 ω2) 2(1 ω2)2
− −
ω 2 (2ω2+1) 3ω( + )
1 4 2 3
= B + B − B B , (25)
O6 2(1 ω2) 2(1 ω2)2
− −
= m2 I (∆ ), (26)
B1 sO− 3 H
I (m /2) I ( m /2)
= ∆2 3 η − 3 − η (ωm +2∆ ), (27)
B2 SO− 4m η S
η
m2 I (∆ ) 3I (∆ ) ω
= ηO + 3 S − 3 H + [∆ I (∆ ) ∆ I (∆ )], (28)
3 S 3 S H 3 H
B 4 4 4 −
m ∆ ∆ [I (∆ ) I (∆ ) I (m /2) I ( m /2)
η S S 3 S 3 H 3 η 3 η
= O + − + − − , (29)
4
B 2 2m 4
η
1 1 6 1/µ2 dy
I (a ,a ,a ) = dx ξexp[ y(m2 ξ2)]
5 2 3 4 Z0 1+2x2(1 a4)+2x(a4 1)(cid:26)16π3/2 Z1/Λ2 √y − s −
− −
6 1/µ2 dy
[1+erf(ξ√y)]+ exp[ y(m2 2ξ2)] , (30)
× 16π2 Z1/Λ2 y − s − (cid:27)
a (1 x)+a x
2 3
ξ = = − . (31)
1+2(a 1)x+2(1 a )x2
4 4
− −
p
3 1/µ2 exp[ sm2] 1
= ds − s dxexp[s∆2(x)][1+erf(∆(x)√s)], (32)
R 16π3/2 Z1/Λ2 s1/2 Z0
n
2
= , (33)
R1 q
| |
n
1 1
= −R , (34)
R2 q
| |
10