Radiative and pionic transitions from the $D_{s1}(2460)$ to the $D_{s0}^\ast(2317)$

0.16 MB·

by  Cheng-Jian Xiao

Radiativeand pionictransitions from the D (2460)to the D (2317)
s1 ∗s0
Cheng-Jian Xiao1,4, Dian-Yong Chen1,2 , and Yong-Liang Ma3
∗†
1Instituteof Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China
2ResearchCenterforHadronandCSRPhysics,LanzhouUniversity&InstituteofModernPhysicsofCAS,Lanzhou730000,China
3 CenterofTheoreticalPhysicsandCollegeofPhysics, JilinUniversity, Changchun, 130012, China
4 University of Chinese Academy of Sciences, Beijing 100049, China
(Dated:May16,2016)
WeestimatethepartialwidthsfortheradiativeandpionictransitionsfromtheD (2460) tothe D (2317)
s1 s0
inamolecularscenario,inwhichtheD (2460)andD (2317)areconsideredashadronicmolecularstatesof
s1 ∗s0
DKandD K,respectively. ThepartialwidthsfortheD (2460) D (2317)π0 andD (2460) D (2317)γ
∗ s1 → ∗s0 s1 → ∗s0
areevaluatedtobeabout 0.19–0.22and3.0–3.1keV, respectively. Inaddition, theratioof the D (2460)
6 s1 →
D (2317)γ and D (2460) D π0 is estimated to be about (6.6–10.6) 10 2, which is safely under the
1 s0 s1 → ∗s × −
0 measuredupperlimit.
2
PACSnumbers:14.40.Pq,13.20.Gd,12.39.Fe
y
a
M
I. INTRODUCTION atthe90%confidencelevel.
Theoretically,thequarkmodelpredictedthemassesofthe
3
1 In the last decade, great experimental progress in the D∗s0andDs1tobe2480MeVand2530MeV[2],respectively,
charmed-strange meson spectrum has been achieved. Some which are about 160 and 70 MeV, respectively, above the

] radially or orbitally excited charmed-strange mesons have experimental measured values. This disagreement between
h
beenobserved[1], andthese observationsnotonlymakethe thequarkmodelexpectationsandexperimentalmeasurements
p
charmed-strange meson family lengthy, but also raise some makes these two states unlike conventionalcharmed-strange
-
p challengestotheconventionalquarkmodel[2]. Amongthese mesons.
he newly observedcharmed-strangemesons, the D∗s0(2317)and The particular properties of the D∗s0(2317) and Ds1(2460)
[ Ds1(2460)aretwoparticularstates,sincetheirmassesarefar have stimulated the theorists’ interest in the nature of these
belowthequarkmodelexpectations[2]. two states. The coupledchannelestimates indicatedthat the

v2 oraTtihoenDin∗s0(t2h3e1D7)+πw0asinfivrasrtiarnetpomratesds bspyecthtreumBAoBfAtRheCoBlldabe-- mstraosnsegscoofupthlienDg∗os0f(2th3e1P7)-wanadveDcsh1a(2rm46e0d)-sctoraunldgeremsuelstofnrsomtotthhee
9 s
cay processand its mass was measured to be (2316.8 0.4) DK and D K, respectively [10, 11]. With some fine-tuning
9 ± ∗
MeV [3]. Later, the CLEO Collaboration confirmed the parameters,themassesoftheD (2317)andD (2460)could
3 ∗s0 s1
6 existence of this state and also reported another state, the bereproducedinarelativisticquarkmodel[12]. Thedecays
0 D (2460), in the D +π0 invariant mass distribution, which of the D (2317) and D (2460) were investigated in a con-
s1 ∗s ∗s0 s1
. is 351.2 1.7(stat.) 1.0(syst.) MeV heavier than the D ventionalcharmed-strangemesonsframewithdifferentmeth-
01 [4]. Besid±esthe D∗sπ0±mode,someotherdecaymodesofthe∗s ods,suchasthequarkpair-creationmodel[13,14],QCDsum
6 Ds1(2460)–like Dsγ, D∗sγ, Dsπ+π− and D∗s0γ–have also been rules[15–19], andchiraleffectivetheory[20]. However,the
1 measured[4]. large-N expansioncalculationsindicatedthat the D (2317)
c ∗s0
: AftertheobservationsfromtheBABARandCLEOcollab- couldnotbea standardquark-antiquarkmeson[21]. A cs¯qq¯
v
Xi orations[3,4],theexistenceofthe D∗s0(2317)andDs1(2460) tetraquarkinterpretationwasproposedtounderstandthemass
was confirmedbythe Belle Collaboration[5, 6] and BABAR anddecaybehavioroftheD (2317)[22–24]. TheQCDsum
∗s0
r Collaboration[7–9].Andnow,theParticleDataGroup(PDG) rule calculations also supported the idea that the D (2317)
a ∗s0
averagesofthemassesoftheD (2317)andD (2460)are[1] couldbeatetraquarkstate[25,26].
∗s0 s1
SincethemassesoftheD (2317)andD (2460)areabout
m (2317) = (2317.7 0.6)MeV, ∗s0 s1
D∗s0 ± 40 MeV below the thresholds of the DK and D∗K, respec-
mDs1(2460) = (2459.5±0.6)MeV. tively, a possible explanation of the structures of D∗s0(2317)
and D (2460) is that they are DK and D K hadronic
Inaddition,thedecaymodeofD (2460) D (2317)γwas s1 ∗
s1 → ∗s0 molecules, respectively. The calculations in the Bethe-
measuredbytheCLEOandBABARcollaborations[4,8]and
Salpeterapproach[27]andpotentialmodel[28]showedthat
theratioofΓ(D (2460) D (2317)γ)andΓ(D (2460)
D π0)wasreporst1edtobe→ ∗s0 s1 → the D∗s0(2317) could indeed be a DK hadronic molecule. In
∗ Ref. [29], the D (2317) and D (2460) were considered as
∗s0 s1
Γ(DΓs(1D(2s41(6204)6→0)D→∗s0D(2∗s3π107))γ)( <<00..5282,, CBALBEAOR[4[8],], (1) dkteeancosaniyviceblmyeohinlaevvcieuoslrteisgsaobtfoeduthniednbDthy∗se0s(tD2ro3Kn1g7a)snhdaonrDdt-rDKans1hg(ae2da4rt6otr0na)icctwimoenore.leTechxue--
∗
lar scenario [30–32]. The production of the D (2317) and
∗s0
D (2460) from the nonleptonic B decay were calculated in
s1
Ref.[33],inwhichD (2317)andD (2460)wereconsidered
∗Correspondingauthor ∗s0 s1
†Electronicaddress:[email protected] ashadronicmolecularstatesofDK andD∗K,respectively.
2
In thispaper,we study the radiativeandpionictransitions ofthecorrelationfunctionshoulddropfastenoughintheul-
from the D (2460) to the D (2317) in a hadronic molec- traviolet region. Here we choose the Fourier transformation
s1 ∗s0
ular scenario. With the assignment that the D (2317) and ofthecorrelationintheGaussianform,
∗s0
D (2460) are the hadronic molecules of DK and D K, re-
s1 ∗
spectively,onecouldfindthattheradiativeandpionictransi- Φ˜M(−p2,Λ2M)=exp(p2/Λ2M), M =(D∗s0, Ds1), (5)
tionsfromthe D (2460)tothe D (2317)occurviathesub-
s1 ∗s0 withΛ beingthesizeparameterwhichcharacterizesthedis-
processesD DγandD Dπ0,respectively. Asforthe M
∗ ∗
D (2460) →D (2317)π0,it→isanisospin-violatingprocess, tributionofcomponentsinsidethemolecule.
s1 → ∗s0
whichcouldresultfromthemass differencesof chargedand
neutralDandKmesonsandη π0mixing.Inaddition,thera- D D∗
−
tioofthepartialwidthsfortheD (2460) D (2317)γand
D (2460) D π0 wasmeasuresd1bytheC→LEO∗s0andBABAR D∗s0 D∗s0 Ds1 Ds1
s1 ∗
→
collaborations[4,8].Inthepresentwork,wecantesttheD K
∗
assignmentoftheD (2460)bycomparingtheestimatedratio K K
s1
of Γ(Ds1(2460) Ds0(2317)γ) and Γ(Ds1(2460) D∗π0) (a) (b)
→ →
withtheexperimentalmeasurements.
This work is organized as follows. The the hadronic FIG.1:MassoperatorsoftheD∗s0(2317)(a)andDs1(2460)(b).
molecular structures of the D (2317) and D (2460) are
∗s0 s1
dDi∗ss0c(u2s3s1e7d)πin0,SDec∗s.0(I2I3.1T7h)γeapnadrtDia∗slπw0iadrtehesstfiomraDtesd1(i2n4S6e0c).I→II. (3)ThceouclodupbleindgecteornmstiannetdsgbDy∗s0DthKeancodmgpDos1sDi∗tKenienssEqcso.n(d2i)tiaonnds
Thenumericalresultsare presentedin Sec.IV andSec. V is [30, 31, 34–36], where the renormalization constants of the
dedicatedtoashortsummary. compositeparticlesshouldbezero,i.e.,
Z 1 Σ (m2 )=0,
II. HADRONICMOLECULARSTRUCTURESOFTHE D∗s0 ≡ − ′D∗s0 D∗s0
D∗s0(2317)ANDDs1(2460) ZDs1 ≡ 1−Σ′Ds1(m2Ds1)=0, (6)
with Σ (m2 ) being the derivative of the mass operator of
DsI1n(24th6e0)haarderoansiscigmneodlecauslaSr-wscaevnearDioK, tahnedDD∗s∗0K(23h1a7d)roannidc the D∗s0′D(∗s20317D)∗s0. As forthe Ds1(2460),the massoperatorΣµDνs1
molecules, respectively. Here, we adoptthe followingeffec- presentedinFig. 1(b)canbedecomposedintothetransverse
tiveLagrangianstodescribetheinteractionsoftheD∗s0(2317) ΣDs1 andlongitudinalΣLDs1 componentsas
and D (2460)andtheir constituents. Theconcreteformsof
s1
theLagrangiansare[30,31] Σµν (p)=gµνΣ (p2)+ pµpνΣL (p2), (7)
Ds1 ⊥ Ds1 p2 Ds1
(x) = g D (x) dyΦ (y2)DT(x+w y)
LD∗s0 D∗s0DK ∗s0 Z D∗s0 KD with gµν = gµν pµpν/p2. The concrete forms of the mass
×K(x−wDKy)+H.c., (2) operato⊥rs of the−D∗s0(2317) and Ds1(2460) corresponding to
thediagramsinFig.1are
(x) = g Dµ (x) dyΦ (y2)D T(x+w y)
LDs1 Ds1D∗K s1 Z Ds1 ∗µ KD∗
d4q
×K(x−wD∗Ky)+H.c., (3) ΣD∗s0 = g2D∗s0DKZ (2π)4Φ˜2[−(q−wDKp)2,Λ2]
where 1 1
, (8)
K+ ×(p q)2 m2 q2 m2
D(∗)T =(D(∗)0,D(∗)+), K = K0 !. − −d4qK − D
Σµν = g2 Φ˜2[ (q w p)2,Λ2]
Thewij =mi/(mi+mj)iskinematicalparameterwithmibeing Ds1 Ds1D∗KZ (2π)4 − − D∗K
themassofthecorrespondingmeson. 1 gµν+qµqν/m2
ThecorrelationfunctionsΦ (y2)andΦ (y2),whichde- − D∗. (9)
D∗s0 Ds1 ×(p q)2 m2 q2 m2
pend only on the Jacobian coordinate y, are introduced to − − K − D∗
depict the distributions of the components in the hadronic
molecule.TheFouriertransformationofthecorrelationfunc-
III. RADIATIVEANDPIONICTRANSITIONSFROMTHE
tionis,
D (2460)TOTHED (2317)
s1 ∗s0
d4p
Φ (y2)= e ipyΦ˜ ( p2,Λ2 ), M =(D , D ).
M Z (2π)4 − M − M ∗s0 s1 We estimatethe partialwidthsfortheradiativeandpionic
(4) transitionsfromtheD (2460)totheD (2317)inaneffective
s1 ∗s0
The introduced correlation function also plays the makes Lagrangianapproach. Theinteractionsofthe D (2317)and
∗s0
the Feynman diagrams finite in the ultraviolet region of Eu- D (2460)withtheircomponentsarepresentedinEqs.(2)and
s1
clideanspace,whichindicatesthattheFouriertransformation (3). Besides these effective Lagrangians, in our calculation,
3
weemploythefollowingphenomenologicalLagrangians[37– thebranchingratiosofthe D 0 D0γand D 0 D0π0 are
∗ ∗
→ →
40] measured. Here,wecanroughlyestimatethepartialwidthof
the D 0 D0π0 from that of the D + D+π0 via isospin
∗ ∗
LD∗Dπ =−ig√D∗2DPD†∂µ~π·~τD∗µ, sDy0mγm) aentrd→yΓ[4(D5,046]. WDi0thπ0t)h,ewmeecaasunreodbtra→aitniothoeftphaertΓia(lDw∗0id→th
∗
LD∗Dη =−igD∗DηD†∂µηD∗µ, ogf the Da∗s0g→ D0→γ=a2n.0dGtheeVco1r.respondingcoupling constant
LD∗sDK =igD∗DP(D∗s−µD∂µK†), DI∗0nD0γthe prDe∗0sDe0nγt work, the− decays of the Ds1(2460)
LD∗D∗π = 2√12gD∗D∗PǫµναβD∗†µ∂ν~π·~τ↔∂ αD∗β, pDr∗so0c(e2s3s1e7s),πw0haincdhDarse1(a2ls4o60co)n→triDbustπe0dafrreomthethiesoηspinπ-0vmioilxatiinn→gg.
−
Theη π0mixingschemeisintheform[47],
=g ǫ D µ∂νη↔∂ αD β, −
D D η D Dη µναβ ∗† ∗
L ∗ ∗ ∗ ∗ (m m )
LD∗sD∗K = 21gD∗D∗PǫµναβD∗s−µ∂νK† ↔∂ ∂αD∗β, Lηπ0 =µ d√−3 u π0η, (16)
wherem andm arethecurrentquarkmassesoftheuandd
=ig D ν ↔∂ µD K u d
LD∗sD∗K∗ D∗D∗V ∗s− ∗ν µ∗† quarks,respectively,andµisthecondensateparameter.
+4if D (∂µK ν ∂νK µ)D ,
D∗D∗V ∗sµ− ∗† − ∗† ∗ν
=ig K ~π ~τ↔∂ µK, (10) A. ThedecayofD (2460) D (2317)π0
LK∗Kπ K∗Kπ µ∗† · s1 → ∗s0
= ig K η↔∂ µK, (11)
LK∗Kη − K∗Kη µ∗†
where A ↔∂ B A(∂B) (∂A)B,~τisthePaulimatrix,~π rep- D+ D∗ π0 D+ D∗ η π0
resentsthepio≡ntriplets,−andK( ) andD( ) arethedoubletsof s1 D s1 D
∗ ∗
strangeandcharmedmesons,respectively, K D∗+ K D∗+
s0 s0
K( )+ D( )0 (a) (b)
K( ) = ∗ , D( ) = ∗ . (12)
∗ K(∗)0 ! ∗ D(∗)+ ! FIG. 2: Diagrams contributing to the pionic transition from the
D (2460) tothe D (2317). Diagram (a) isthedirect contribution
In the heavy quark-limit, the coupling constants g s1 ∗s0
DD()P anddiagram(b)isthecontributionfromη π0mixing.
∗ ∗
couldberelatedtothegaugecouplingconstantgvia −
The decay of the D (2460) D (2317)π0 occurs via a
2g 2g s1 → ∗s0
gD∗D∗P = fπ , gD∗DP = fπ √mD∗mD, (13) sthuebphraodcreosnsicD-∗le→velDdπes0cirniptthioenhaodfrtohnisicpmroocleescsuliasrppriecsteunrete,danind
where f =132MeVisthedecayconstantofthepionandthe Fig. 2(a). Since this decay is an isospin-violating process,
π
gaugecouplingg = 0.59is estimated fromthe experimental we also include the contribution from the η π0 mixing as
−
valueofthepartialwidthfortheD + D+π0. Theinvolved presented in Fig. 2(b). With the effective interactions listed
∗
couplingconstantsofK are[38], → above,wecangettheamplitudecorrespondingtoFig. 2(a)as
∗
d4q
gD∗sD∗K∗ = β√g2V, fD∗sD∗K∗ = λ√g2V √mD∗sD∗, (14) Ma = (i)3Z (2π)4(cid:2)gDs1D∗KǫDφs1Φ˜Ds1(−P212,Λ2Ds1)i
ig
g Φ˜ ( P2 ,Λ2 ) D∗DP( ipµ)
wherethegaugecouplingsβ=0.9,λ=0.56andgV =mρ/fπ. × D∗s0DK D∗s0 − 20 D∗s0 √2 − 3
h ih i
As for the coupling constants of g and g , we adopt
gK∗Kπ =3.21andgK∗Kη =4.47,whiKc∗hKaπreevaluKa∗KteηdbySU(3) −gφµ+pφ1pµ1/m21 1 1 , (17)
symmetry[41]. × p2 m2 p2 m2q2 m2
1− 1 2− 2 − q
Theinvolvedinteractionrelatedtothephotonfieldandthe
where P = (p w p w ) and P = qw p w .
charmedmesonsisintheform[42], 12 1 D∗K − 2 KD∗ 20 DK − 2 KD
TheamplituderelatedtoFig. 2(b)is,
g
LD∗Dγ =(cid:26) D∗4+D+γeǫµναβFµνD∗α+βD− Mb = (i)3Z (d2π4q)4 gDs1D∗KǫDφs1Φ˜Ds1(−P212,Λ2Ds1)
+ gD∗0D0γeǫµναβF D 0D¯0 +H.c., (15) (cid:2) ig i
4 µν ∗αβ (cid:27) × gD∗s0DKΦ˜D∗s0(−P220,Λ2D∗s0) √D∗2Dη(−ipµ3)
h ih i
wherethe field-strengthtensorsare definedas Fµν = ∂µAν − gφµ+pφpµ/m2 1 1
∂νAµ, D∗αβ = ∂αD∗β−∂βD∗α. ThecouplingconstantgD∗+D+γ = ×− p2 1m21 1 p2 m2q2 m2
0.5 GeV−1 is estimated from the partial width of D∗+ 1− 1 2− 2 − q
D−+γ: theminussignisadoptedaccordingtothelatticeQC→D µmd−mu 1 , (18)
andQCDsumrulecalculations[43,44]. AsforgD∗0D0γ,only × √3 m2π−m2η
4
wherem2π = (mu+md)µ, m2η = 32(m+2ms)µandm = (mu + K π0 D∗ π0 D∗ π0
m )/2.Theaboveamplitude canbereducedto D+ D+ D+
d Mb s1 K∗ s1 D s1 D∗
√3(m m ) D∗ D∗+ K D∗+ K D∗+
= d− u , (19) s s s
Mb Ma|π0→η 4 (ms m) (a) (b) (c)
−
K η π0 D∗ η π0 D∗ η π0
twhheerreelaMteda|πc0o→uηpilnindgiccaotenssttahnetsamofpπli0tuwdiethobthtaoisneeodfbηy.rTehpelatcoitnagl Ds+1 K∗ Ds+1 D Ds+1 D∗
amplitudeoftheDs1(2460)→D∗s0(2317)π0is D∗ Ds∗+ K Ds∗+ K Ds∗+
(d) (e) (f)
= + . (20)
MDs1→D∗s0π0 Ma Mb
FIG.4: DiagramscontributingtoprocessD+ D+π0. Diagrams
s1 → ∗s
(a),(b)and(c)aredirectprocesses,wheretheπ0 directlycouplesto
B. ThedecayofD (2460) D (2317)γ strange mesons or charmed mesons. Diagrams (d), (e) and (f) are
s1 → ∗s0 indirectprocesses, whereπ0 couplestostrangemesonsorcharmed
mesonsviaη π0mixing.
−
γ γ
D∗+ D∗0
Ds+1 D+ Ds+1 D0 C. ThedecayofDs1(2460)→D∗sπ0
K0 D∗+ K+ D∗+ WecanestimatethepartialwidthofD (2460) D π0and
s0 s0 s1 → ∗s
(a) (b) comparetheevaluatedratiooftheΓ(Ds1(2460)→ D∗s0γ)and
Γ(D (2460) D π0) to furthertest the hadronicmolecular
FIG. 3: Diagrams contributing to the radiative transition from the intersp1retation→softh∗seD (2460)andD (2317).Similartothe
s1 ∗s0
cDhsa1r(m24e6d0m) etosotnhseanDd∗s0((b2)3i1s7t)h.ec(oa)ntirsibtuhteiocnofnrotrmibuthtieonnefurtoramlcchhaarrmgeedd processDs1(2460)→D∗s0π0,thedecayofDs1(2460)→D∗sπ0
is also an isospin-violating process, which also arises from
mesons..
thedirectπ0 couplingandη π0 mixingasshowninFig. 4.
−
In our calculations, in addition to the diagrams considered
AsforthedecayofD (2460) D (2317)γ,itoccursvia
the subprocess D Ds1γ as sho→wn in∗s0Fig. 3. With the ef- in Ref. [31], we include the diagramsdue to the D∗D∗π and
fectiveLagrangian∗s→givenabove,wecanobtaintheamplitude D∗D∗ηinteractions.Theconcreteformsoftheamplitudescor-
respondingtoFigs. 4(a)–4(c)are
correspondingtoFig. 3(a)as
d4q
Ma = (i)3Z (d2π4q)4hgDs1D∗KǫDφs1Φ˜Ds1(−P212,Λ2Ds1)i Ma = (i)3igZ (2(πip)4ηh+gDisp1Dη)∗KǫigDφs1Φ˜Ds1ǫ(τ−gPτ21ρ2(,iΛpσ2Ds+1)iipσ)
××hǫgµνDα∗sβ0DǫγηK(Φi˜pDν3∗sg0(µ−ηP−22i0p,ν3Λg2Dνη∗s0))(iihpeα1ggDβ4∗τ+D−+iγpβ1gατ) +×h4ifKD∗∗KDπ∗VǫDτ1∗s(iqτg3ρσih−iDq∗ρDg∗Vτσ)Di∗sp21−1m221 4
×−gφτp+21−pφ1mp21τ1/m21 p22−1m22q2−1m2q. i (21) ×−gρφp+22−pρ2mp22φ2/m22−gησq+2−qηmq2qσ/m2q , (25)
As for the amplitude corresponding to Fig. 3(b), it can be d4q
= (i)3 g ǫφ Φ˜ ( P2 ,Λ2 )
obtained from the above amplitude by replacing the masses Mb Z (2π)4 Ds1D∗K Ds1 Ds1 − 12 Ds1
h i
andcouplingconstantswiththoseinFig. 3(b),i.e.,
ig
− D∗DP( ipµ) ig ǫν (ipν)
Mb =Ma(cid:12)mgDD∗∗++D→+γm→Dg∗0D,∗m0DD0+γ→mD0,mK0→mK+ . (22) ×h gφ√µ2+pµ−pφ/m3 2ih D1∗DP D∗s 12 i
(cid:12)(cid:12) − 1 1 1 , (26)
Thenthetotalamplit(cid:12)udeforD (2460) D (2317)γis × p2 m2 p2 m2q2 m2
s1 → ∗s0 1− 1 2− 2 − q
d4q
MDs1→D∗s0γ =Ma+Mb. (23) Mc = (i)3Z (2π)4 gDs1D∗KǫDφs1Φ˜Ds1(−P212,Λ2Ds1)
h i
Itshouldbenoticedthatafterperformingtheloopintegral,the 1 g ε ( ipτ)(ipρ+iqρ)
aboveamplitudecanbereducedtotheform, × 2√2 D∗D∗P ητρσ − 3 1
h i
1
MDs1→D∗s0γ =gDs1D∗s0γεµναβǫDµs1ǫγνpαγpβDs1, (24) ×h2gD∗D∗PεµναβǫDµ∗s(ipν2)(iqα+ipα4)i
whichisobviouslygaugeinvariantandthecouplingconstant −gσφ+pσ1pφ1/m21 1 −gηβ+qηqβ/m2q .(27)
g couldbeestimatedfromtheamplitudeinEq. (23). × p2 m2 p2 m2 q2 m2
Ds1D∗s0γ 1− 1 2− 2 − q
5
As for the contributionsfrom η π0 mixing, the amplitudes
− 13
correspondingtoFigs. 4(d)-4(f)by,
g ∗
Ds0DK
Md = Ma|π0→η √43((mmds−mmu)), 12 gDs1D∗K
− )
V
= √3(md−mu), Ge
Me Mb|π0→η 4 (ms−m) ng( 11
= √3(md−mu). (28) upli
Mf Mc|π0→η 4 (ms m) Co
−
ThetotalamplitudeofD (2460) D π0is 10
s1 → ∗s
f
MDs1→D∗sγ =Xn=aMn. (29) 91.0 1.2 1.4 1.6 1.8 2.0
Λ(GeV)

With the total amplitudes defined in Eqs. (20), (23) and
(29),onecanestimatethepartialwidthby, FIG. 5: The Λ dependence of the coupling constants g and
g ,whereΛ =Λ =Λ. D∗s0DK
1 1 ~p Ds1D∗K Ds1 D∗s0
Γ= | | 2, (30)
38πm2 |M|
Ds1
0.4
where ~p is the momentumof thefinalstate in the Ds1(2460) ΛD∗ =2.0 GeV
restframeandtheoverlineindicatessumoverpolarizationsof ΛDs∗0 =1.5 GeV
vectormesons. keV) ΛDs∗s00 =1.0 GeV
( 0.3
)
0
π
IV. NUMERICALRESULTS ∗+Ds0
→
+Ds1 0.2
TABLEI:ThemassesoftheinvolvedparticlesinunitsofGeV[1]. Γ(
State Mass State Mass State Mass State Mass
D0 1.8648 D 1.8696 D0 2.0069 D 2.0102
± ∗ ∗±
K0 0.4976 K 0.4936 K 0 0.8958 K 0.8916
± ∗ ∗±
D 2.1121 D 2.3177 D 2.4595 π0 0.1349 0.1
∗s± ∗s0± ±s1 1.0 1.2 1.4 1.6 1.8 2.0
η 0.5478 ΛD (GeV)
s1
AllthemassesoftheinvolvedparticlesarelistedinTableI. FIG.6: . TheΛDs1 dependenceofthedecaywidthforDs1(2460)→
BesidesthecouplingconstantsdiscussedinSec.III,thecou- D∗s0(2317)π0.
plingconstantsof D (2460)/D (2317)to their components .
s1 ∗s0
couldbeestimatedbythecompositenessconditionsgivenby
Eq.(6). ThephenomenologicalparametersΛ andΛ are
ofdsrefeonnomtceredd1seirontof1FtG2ihgee.GV5ce.o.VuHTphe[lr3eiens0,eg,wtc3weo1on]v.sactroaTynuhtptsehliegnΛDgpDsa1csDr1oa∗nKm=setaatnΛendrtDsDs∗gs0smΛ1D∗s=Do0Dns1KoΛatoandrndDeeo∗s0ΛpupesrDnel∗sy0-- ppfinraaogrrmtaiomafl0e.Λtw2e5rDidsst1tohΛo0Dfr.os21Λr1aDtnhk∗s0dee.VΛDIwDns1∗si0(tt,2hh4eaΛ6nc0dDa)ss1de→eincocrferDeaΛas∗se0sDi(s∗sn20g3w=1ift7rho)1πm.t00he1dG.e0iencVrcteor,ae2tsahe.s0es-
decrease with the increasingof the parameter Λ. In particu- GeV. In the considered parameter region, the partial width
l1a1r.,7t3hetoc1o0u.p2l5inGgecVonasntdanftrsomgD1s11D.∗2K0atnod9.g8D5∗s0GDKeVd,ercersepaescetifvreolmy, f0o.1r9the0D.2s51(k2e4V6.0) → D∗s0(2317)π0 is predicted to be about
∼
whenΛincreasesfrom1to2GeV. The Λ dependence of the partial width for the
Ds1
Thepartialwidthofthe D (2460) D (2317)π0ispre- D (2460) D (2317)γis presented in Fig. 7. Similar to
s1 → ∗s0 s1 → ∗s0
sentedinFig.6. Inthepresentcalculation,wevarytheΛ thepionictransitionfromtheD (2460)totheD (2317),the
Ds1 s1 ∗s0
from1.0to2.0GeVandtaketypicalvaluesofΛ =1.0,1.5 partial width for the D (2460) D (2317)γ also weakly
D∗s0 s1 → ∗s0
and 2.0GeV. Our calculations indicate that the partial width depends on the parameters Λ and Λ . In the consid-
oftheD (2460) D (2317)π0isoforder0.1keV,whichis eredparameterregion,thepartDias1lwidthfDo∗sr0the D (2460)
s1 → ∗s0 s1 →
rathersmallsincethephasespaceofthisprocessisverylim- D (2317)γvaries from 2.96 to 3.13 keV. The PDG average
∗s0
ited. In addition, this partial width weakly depends on the of the branching ratio of the D (2460) D (2317)γ is
s1 → ∗s0
6
D (2460) D (2317)γand D (2460) D π0 havebeen
4.0 s1 → ∗s0 s1 → ∗s
estimatedinthepresentwork,andtheratioofΓ(D (2460)
V) ΛΛΛDDD∗s∗s∗s000 ===211...050 GGGeeeVVV (sD6a∗sf.06e(l−2y31u10n7.2d))eγ×r)t1ha0en−du2piΓnp(etDhrsel1imc(2oi4nt6srei0dp)eorr→etdedpDbary∗saπtmh0)eetiCesrLerEesOtgiimoas1nant,dewdBhAtiocBhA→biRes
e
k
( 3.5 collaborations[4,8].
)
γ
∗+Ds0
→ TABLEII:Acomparisonofthethepartialwidths(inunitsofkeV)
+Ds1 3.0 fromChdainffneerlentmoPdreelsse.nt Ref.[48] Ref.[18] Ref.[49] Ref.[50]
(
Γ D D γ 3.0 3.1 2.74 0.5 0.8 0.012
s1 → ∗s0 ∼ ∼ ···
D D π0 0.19 0.22 0.0079
s1 → ∗s0 ∼ ··· ··· ···
D D π0 31.3 45.2 21.5 10 11.9
s1 → ∗s ∼ ··· ∼
2.5
1.0 1.2 1.4 1.6 1.8 2.0 In Table II, we collect our estimates of the partial widths
ΛDs1(G eV) oftheDs1(2460)→ D∗s0(2317)γ, D∗s0(2317)π0,andD∗sπ0and
compare with the results evaluated in the P-wave charmed-
FIG.7: ThesameasFig. 6butforDs1(2460) → D∗s0(2317)γpro- strange meson scheme. In Ref. [48], the decays of the
cess. D (2460)wereestimatedin a fullchiraltheoryandthepar-
s1
. tialwidthsfortheD (2460) D γandD (2460) D π0
s1 → ∗s0 s1 → ∗
are very similar to the present results obtained in a molec-
ular scenario, but for the D (2460) D (2317)π0 mode,
3.7+52..04%. However,thewidthof Ds1(2460)isnotwelldeter- the resultsfromRef. [48] arse1 muchs→malle∗sr0thanthe present
min−ed,asonecannotcomparethetheoreticalvalueofthepar-
one. The light-cone sum rule calculation for D (2460)
tialwidthwiththeexperimentalmeasurement. Here,wealso s1 →
D (2317)γis about20%ofthatobtainedin thepresentcal-
noticethatbothwidthsfortheD (2460) D (2317)π0and ∗s0
s1 → ∗s0 culation[18]. Theestimationsintherelativisticquarkmodel
D (2460) D (2317)γ weakly depend on the model pa-
rams1eters, an→d the∗s0former one is about 1 order smaller than indicated that the partial widths of Ds1(2460) → D∗s0γ and
D (2460) D π0 were 0.012 and about 10 keV, respec-
the latter one, which indicates that the branching ratio of s1 → ∗s
tively [49, 50], which are rather different with the results in
D (2460) D (2317)π0shouldbeoforder10 3.
s1 → ∗s0 − thepresentwork.

50
V. SUMMARY
) In the present work, we estimated the partial widths for
V
45
(ke theradiativeandpionictransitionsfromtheDs1(2460)tothe
0π) D∗s0(2317) in a molecular scenario, in which the Ds1(2460)
+ andthe D (2317)areassignedasa DK anda D K hadronic
∗Ds 40 molecule,∗sr0espectively. Tofurthertestthemolecu∗larinterpre-
→
tationsoftheD (2460)andtheD (2317),wealsocalculated
+Ds1 thepartialwidths1forD (2460) ∗sD0 π0.Intheconsideredpa-
( s1 → ∗s
Γ 35 rameterregion,thepartialwidthsareevaluatedtobe
Γ(D (2460) D (2317)π0) = 0.19 0.22keV,
s1 → ∗s0 ∼
Γ(D (2460) D (2317)γ) = 3.0 3.1keV,
30 s1 → ∗s0 ∼
1.0 1.2 1.4 1.6 1.8 2.0 Γ(D (2460) D π0) = 31.3 45.2keV. (31)
ΛD (GeV) s1 → ∗s ∼
s1
Our estimates indicate that the partial width for the
FDI∗sGπ.0.8: TheΛDs1 dependenceofthepartialwidthforDs1(2460) → tDhsa1t(2o4f6D0)s1(→2460D)∗s0→(23D17∗s)0π(203i1s7)aγb.ouTth1e borradnecrhsimngalrlaetriothfaonr
. Ds1(2460) → D∗s0(2317)γ is measured to be 3.7+52..04% [1],
and thus the branching ratio for D (2460) D−(2317)π0
widInthFfiogr.th8e,Dwe(p2r4e6s0e)nt thDe ΛπD0sw1 hdiecphenindcernecaeseosfwtihtehtphaertiina-l is roughly determined to be of ordse1r 10−3. →In ad∗sd0ition, we
creasing of Λ s1. In par→ticula∗sr, the partial width varies from furtherestimatetheratioofΓ(Ds1(2460)→D∗s0(2317)γ)and
32to46keVwDsi1thΛDs1 increasingfrom1.0to2.0GeV,which Γ(Ds1(2460)→D∗s+π0)tobe
ismuchlargerthanthepartialwidthsfortheD (2317)γand Γ(D (2460) D (2317)γ)
∗s0 s1 → ∗s0 =(6.6 10.6) 10 2, (32)
D∗s0(2317)π0 modes. In addition, the partial widths for the Γ(Ds1(2460)→D∗s+π0) − × −
7
whichisconsistentwiththeexperimentalmeasurementsfrom L.M.issupportedinpartbytheNationalScienceFoundation
theCLEOandBABARcollaborations[4,8]. ofChina(NSFC)underGrantNo. 11475071,11547308and
Atpresent,theexperimentalinformationontheD (2460) theSeedsFundingofJilinUniversity.
s1
and D (2317)isstill notabundant. Inparticular,the widths
s0
ofthesestatesarenotwelldetermined. Themeasurementsof
their decay behaviors at LHCb and the forthcomingBelle II
couldprovideafurthertesttotheresultsinthepresentwork.
Acknowledgements
TheworkofD.-Y.C.issupportedbytheNationalNatural
ScienceFoundationofChinaunderGrantNo. 11375240. Y.-
[1] K. A. Olive et al. [Particle Data Group], Review of particle Lett.B570,180(2003).
physics,Chin.Phys.C38,090001(2014). [18] P.Colangelo,F.DeFazioandA.Ozpineci,Radiativetransitions
[2] S.GodfreyandN.Isgur,Mesonsinarelativizedquarkmodel ofD (2317)andD (2460),Phys.Rev.D72,074004(2005).
∗sJ sJ
withchromodynamics,Phys.Rev.D32,189(1985). [19] P. Colangelo, F. De Fazio, F. Giannuzzi and S. Nicotri, New
[3] B.Aubertetal.[BABARCollaboration],Observationofanar- mesonspectroscopywithopencharmandbeauty,Phys.Rev.D
rowmesondecayingtoD+π0 atamassof2.32GeV/c2,Phys. 86,054024(2012).
s
Rev.Lett.90,242001(2003). [20] S. Fajfer and A. P.Brdnik, Chiral loops in the isospin violat-
[4] D.Bessonetal.[CLEOCollaboration], Observationofanar- ing decays of D (2460)+ and D (2317)+, Phys. Rev. D 92 ,
s1 ∗s0
row resonance of mass 2.46 GeV/c2 decaying to D+π0 and 074047(2015).
∗s
confirmationofthe D (2317) state, Phys.Rev. D68, 032002 [21] Z.H.Guo,U.G.Meiner,andD.L.Yao,Newinsightsintothe
∗sJ
(2003). D (2317)andothercharmscalarmesons,Phys.Rev.D92,no.
∗s0
[5] K.Abeetal.[BelleCollaboration], Measurements of the D 9,094008(2015).
sJ
resonanceproperties,Phys.Rev.Lett.92,012002(2004). [22] M.Nielsen,R.D.Matheus,andF.S.Navarra,Tetraquarkstates,
[6] P. Krokovny et al. [Belle Collaboration], Observation of the Nucl.Phys.B, Proc.Suppl.174,138(2007).
D (2317) and D (2457) in B decays, Phys. Rev. Lett. 91, [23] M.Nielsen,D+(2317) D+π0decaywidth,Phys.Lett.B634,
sJ sJ sJ → s
262002(2003). 35(2006).
[7] B. Aubert et al. [BABAR Collaboration], Study of B [24] K. Terasaki, BABAR resonance as a new window of hadron
D(s∗J)+D¯(∗)decays,Phys.Rev.Lett.93,181801(2004). → physics,Phys.Rev.D68,011501(2003).
[8] B.Aubertetal.[BABARCollaboration],Observationofanar- [25] Z.G.WangandS.L.Wan,D (2317)asatetraquarkstatewith
s
rowmesondecayingtoD+π0γatamassof2.458-GeV/c2,Phys. QCD sum rules in heavy quark limit, Nucl. Phys. A778, 22
s
Rev.D69,031101(2004). (2006).
[9] B. Aubert et al. [BABAR Collaboration], A study of the [26] M. E. Bracco, A. Lozea, R. D. Matheus, F. S. Navarra and
D (2317) and D (2460) mesons in inclusive cc¯ production M. Nielsen, Disentangling two- and four-quark state pictures
∗sJ sJ
near √s=10.6GeV,Phys.Rev.D74,032007(2006). ofthecharmedscalarmesons,Phys.Lett.B624,217(2005).
[10] M.F.M.LutzandM.Soyeur, Open-charm mesonsystemsin [27] Z.X.Xie,G.Q.Feng,andX.H.Guo,AnalyzingD (2317)+in
∗s0
the hadrogenesis conjecture, Prog. Part. Nucl. Phys. 61, 155 the DK moleculepictureintheBeth-Salpeterapproach, Phys.
(2008). Rev.D81,036014(2010).
[11] D.S.HwangandD.W.Kim,Massof D (2317)andcoupled [28] Y.J.Zhang,H.C.Chiang,P.N.ShenandB.S.Zou,Possible
∗sJ
channeleffect,Phys.Lett.B601,137(2004). S-wavebound-states of two pseudoscalar mesons, Phys. Rev.
[12] J. B. Liu and M. Z. Yang, Spectrum of the charmed and b- D74,014013(2006)
flavoredmesonsintherelativisticpotentialmodel,J.HighEn- [29] P. Bicudo, The Family of strange multiquarks as kaonic
ergyPhys.07(2014)106. moleculesboundbyhardcoreattraction,Nucl.Phys.A748,537
[13] X.Liu,Y.M.Yu, S.M.Zhao,andX.Q.Li,Studyondecays (2005).
ofD (2317)andD (2460)intermsoftheCQMmodel,Eur. [30] A. Faessler, T. Gutsche, V. E. Lyubovitskij and Y. L. Ma,
∗sJ sJ
Phys.J.C47,445(2006). StrongandradiativedecaysoftheD (2317)mesonintheDK-
∗s0
[14] J.Lu,X.L.Chen,W.Z.DengandS.L.Zhu,Pionicdecaysof moleculepicture,Phys.Rev.D76,014005(2007).
D (2317), D (2460)and B (5718), B (5765),Phys.Rev.D [31] A.Faessler,T.Gutsche,V.E.LyubovitskijandY.L.Ma,D K
sJ sJ sJ sJ ∗
73,054012(2006). molecularstructureoftheD (2460)meson, Phys.Rev.D76,
s1
[15] Z.G.Wang,RadiativedecaysoftheD (2317),D (2460)and 114008(2007).
s0 s1
therelatedstrongcouplingconstants,Phys.Rev.D75,034013 [32] M. Cleven, H. W. Griehammer, F. K. Guo, C. Hanhart and
(2007). U. G. Mei ner, Strong and radiative decays of the D (2317)
∗s0
[16] Y.B.Dai,C.S.Huang,C.LiuandS.L.Zhu,Understandingthe andD (2460),Eur.Phys.J.A50,149(2014).
s1
D+(2317)andD+(2460)withsumrulesinHQET,Phys.Rev. [33] A. Datta and P. J. O’donnell, Understanding the nature of
sJ sJ
D68,114011(2003). D (2317) and D (2460) through nonleptonic B decays, Phys.
s s
[17] P.ColangeloandF.DeFazio,UnderstandingD (2317),Phys. Lett.B572,164(2003).
sJ
8
[34] S.Weinberg,Elementaryparticletheoryofcompositeparticles, [43] D. Becirevic and B. Haas, D Dπ and D Dγ decays:
∗ ∗
→ →
Phys.Rev.130,776(1963). axialcouplingandmagneticmomentof D meson,Eur.Phys.
∗
[35] A.Salam,Lagrangiantheoryofcompositeparticles,NuovoCi- J.C71,1734(2011).
mento25,224(1962). [44] S. L. Zhu, W. Y. P. Hwang, and Z. S. Yang, D Dγ and
∗
→
[36] K.Hayashi, M.Hirayama, T.Muta, N.Seto, andT.Shirafuji, B BγasderivedfromQCDsumrules,Mod.Phys.Lett.A
∗
→
Compositenesscriteriaofparticlesinquantumfieldtheoryand 12,3027(1997).
S-matrixtheory,Fortschr.Phys.15,625(1967). [45] Y. B. Dong, A. Faessler, T. Gutsche, and V. E. Lyubovitskij,
[37] O. Kaymakcalan, S. Rajeev and J. Schechter, Nonabelian Estimateforthe X(3872) γJ/ψdecaywidth, Phys.Rev.D
→
anomaly and vector meson decays, Phys. Rev. D 30, 594 77,094013(2008).
(1984). [46] D. Y. Chen and Y. B. Dong, Radiative decays of the neutral
[38] Y. S. Oh, T. Song and S. H. Lee, J/ψ absorption by π and ρ Z (3900),Phys.Rev.D93,014003(2016).
c
mesonsinmesonexchangemodelwithanomalousparityinter- [47] J.GasserandH.Leutwyler,Chiralperturbationtheory: expan-
actions,Phys.Rev.C63,034901(2001). sionsinthemassofthestrangequark, Nucl.Phys.B250,465
[39] R. Casalbuoni, A. Deandrea, N. Di Bartolomeo, R. Gatto, (1985).
F. Feruglio and G. Nardulli, Phenomenology of heavy meson [48] W.A.Bardeen,E.J.EichtenandC.T.Hill,Chiralmultipletsof
chiralLagrangians,Phys.Rep.281,145(1997). heavy-lightmesons,Phys.Rev.D68(2003)054024.
[40] P.Colangelo,F.DeFazioandT.N.Pham,B K χ decay [49] S. Godfrey, Testing the nature of the D (2317)+ and
fromcharmedmesonrescattering,Phys.Lett.−B→542,−71c0(2002). D (2463)+ statesusingradiativetransitions,Ph∗syJs.Lett.B568
sJ
[41] W. Liu, C. M. Ko and L. W. Chen, η absorption by mesons, (2003)254.
Nucl.Phys.A765,401(2006). [50] Q. T. Song, D. Y. Chen, X. Liu and T. Matsuki, Charmed-
[42] D.Y.Chen,Y.B.Dong,andX.Liu,Long-distantcontribution strangemesonsrevisited:massspectraandstrongdecays,Phys.
andχ radiativedecaystolightvectormeson,Eur.Phys.J.C Rev.D91,054031(2015).
c1
70,177(2010).