D Wave Heavy Mesons Wei Wei, Xiang Liu, and Shi-Lin Zhu Department of Physics, Peking University, Beijing 100871, China We first extract the binding energy Λ¯ and decay constants of the D wave heavy meson doublets − − − − (1 ,2 )and(2 ,3 )withQCDsumruleintheleadingorderofheavyquarkeffectivetheory. Then we study their pionic (π,K,η) couplings using the light cone sum rule, from which the parameter Λ¯ can also be extracted. We then calculate the pionic decay widths of the strange/non-strange D waveheavyD/B mesonsanddiscussthepossiblecandidatesfortheDwavecharm-strangemesons. Furtherexperimentalinformation,suchastheratiobetweenDsηandDK modes,willbeveryuseful todistinguish various assignments for DsJ(2860,2715). PACSnumbers: 12.39.Mk,12.39.-x Keywords: Dwaveheavymeson,QCDsumrule 7 0 0 I. INTRODUCTION [11]. The decay property of the heavy mesons till L =2 2 was calculated using the 3P model in [12]. 0 n Recently BarBar reported two new D states, Light cone QCD sum rule (LCQSR) has proven very a s useful in extracting the hadronic form factors and cou- D (2860) and D (2690) in the DK channel. Their J sJ sJ pling constants in the past decade [13]. Unlike the tra- widths are Γ = 48 7 10 MeV and Γ = 112 7 36 9 ± ± ± ± ditional SVZ sum rule [14], it was based on the twist MeV respectively [1]. For D (2860) the significance of sJ expansion on the light cone. The strong couplings and signalis5σ intheD0K+ channeland2.8σ intheD+K0 3 s semileptonic decay form factors of the low lying heavy 6v JchPan=ne1l. BinelBle+obserDv¯e0dDanotheDr¯0sDta0tKe+Ds[2J](.27I1ts5)wwiditthh mesons have been calculated using this method both in − → sJ → full QCD and in HQET [15]. 6 is Γ = 115 20 MeV. No D K or D η mode has been 0 detected for±all of them. ∗ s Inthispaperwe firstextractthe massparametersand 2 decay constants of D wave doublets in section II. Then 1 The JP of DsJ(2860) and DsJ(2690) can be we study the strong couplings of the D wave heavy dou- 6 0+,1−,2+,3−, since they decay to two pseduscalar bletswithlightpseduscalarmesonsπ,K andηinsection ··· 0 mesons. DsJ(2860)was proposedas the first radialexci- III. We work in the framework of LCQSR in the lead- h/ tationofDsJ(2317)basedonacoupledchannelmodel[3] ing order of HQET. We present our numerical analysis p or an improvedpotential model [4]. Colangelo et al con- insectionIV. In sectionV wecalculate the strongdecay - sidered DsJ(2860)as the D wave 3− state [5]. The mass widths to light hadrons and discuss the possible D wave p e of DsJ(2715) or DsJ(2690) is consistent with the poten- charm-strange heavy meson candidates. The results are h tialmodelpredictionofthecs¯radiallyexcited23S1 state summarized in section VI. : [4,6]. Basedonchiralsymmetryconsideration,aDwave v 1 state with mass M = 2720 MeV is also predicted if i − X the D (2536) is taken as the P wave 1+ state [7]. The sJ II. TWO-POINT QCD SUM RULES strong decay widths for these states are discussed using r a the 3P model in [8]. 0 The proper interpolating currents Jα1···αj for the The heavy quark effective theory (HQET) provides a j,P,jℓ states with the quantum number j, P, j in HQET were ℓ systematic expansion in terms of 1/m for hadrons con- Q given in [16], with j the total spin of the heavy meson, taining a single heavy quark, where m is the heavy Q P the parityandj theangularmomentumoflightcom- ℓ quark mass [9]. In HQET the heavy mesons can be ponents. In the m limit, the currents satisfy the Q grouped into doublets with definite jP since the angu- → ∞ ℓ following conditions lar monument of light components j is a good quan- ℓ tbulemt (n0um,1ber)iwnitthheormbiQta→l an∞gulliamrimt.onTuhmeyenatrLe 12=−0d,o1u+- h0|Jjα,P1·,·j·ℓαj(0)|j′,P′,jℓ′i=fPjlδjj′δPP′δjℓjℓ′ηα1···αj , doublet−(0−+,1+) and 32+ doublet (1+,2+) with L =21. ih0|T (cid:16)Jjα,P1·,·j·ℓαj(x)Jj†′β,P1·′·,·jβℓ′j′(0)(cid:17)|0i=δjj′δPP′δjℓjℓ′ For L=2 there are (1 ,2 ) and (2 ,3 ) doublets with − − − − ( 1)j gα1β1 gαjβj dtδ(x vt)Π (x), jP = 3−and 5− respectively. The states with the same × − S t ··· t Z − P,jℓ ℓ 2 2 JP, such as the two 1 and two 1+ states, can be dis- (1) − tinguished in the m limit, which is one of the ad- Q vantageofworkingin→HQ∞ET.The D waveheavymesons where ηα1···αj is the polarization tensor for the spin j ′ ′ ((B ,B ),(B ,B ))wereconsideredinthequarkmodel state. v denotes the velocity of the heavy quark. The 1∗ 2∗ 2∗ 3 [10]. The semileptonic decay of B meson to the D wave transverse metric tensor gαβ = gαβ vαvβ. denotes t − S doublets was calculatedusing three point QCDsum rule symmetrizingtheindicesandsubtractingthetraceterms 2 separatelyinthesets(α α )and(β β ). f isa get the following sum rules from eqs. (10) and (11) 1··· j 1··· j P,jℓ constant. Π is a function of x. Both of them depend only on P aPn,djℓjℓ. f2 exp 2Λ¯−,23 The interpolating currents are [16] −,32 h− T i 1 ωc 1 1 α 3 1 = 26π2 Z ω4e−ω/Tdω+ 16 m20hq¯qi− 25h πsG2iT, J1†,α−,23 =r4 h¯v(−i)(cid:18)Dtα− 3γtαD/t(cid:19)q , (2) f2 exp 0 2Λ¯−,25 J2†,α−1,,α232 =r12 Tα1,α2;β1,β2h¯v(−i) =−,52 1 h− ωcTω6ei−ω/Tdω+ 1 αsG2 T3 . (12) 2 γ / q , (3) 27·5π2 Z0 120h π i × (cid:18)Dtβ1Dtβ2 − 5Dtβ1 tβ2Dt(cid:19) Here m2 q¯q = q¯gσ Gµνq . Only terms of the low- 0h i h µν i 5 est order in α and operators with dimension less than J2†,α−1,,α522 =−r6 Tα1,α2;β1,β2h¯vγ5 six have beensincluded. For the 52− doublet there is no 2 mixing condensate due to the higher derivation. γ / q , (4) × (cid:18)Dtβ1Dtβ2 − 5Dtβ1 tβ2Dt(cid:19) We use the following values for the QCD parameters: q¯q = (0.24 GeV)3, α GG = 0.038 GeV4, m2 = J3†,α−,β,52,λ =−r12 Tα,β,λ;µ,ν,σh¯vγtµDtνDtσq , (5) h0.8RiGeqeuVir2−i.ngthatthe highh-osrderipowercorrectionsis0less J1†,α,1 =r12 h¯vγtαq , J0†, ,1 =r12 h¯vγ5q , (6) athnadnt3h0e%coonftrtihbeuptieorntuorfbtahteiopnotleertmerwmitihsoluartgtehretchuatnoff35ω%c − 2 − 2 ofthe continuumcontributiongivenby the perturbation 1 J1†,α+,12 =r2 h¯vγ5γtαq , (7) irnetgeiognraolfinthtehseumregriuolnesωω>=ωc3,.2we3a.r6rGiveeVa,tTth=e 0st.8abil1it.y0 c − − 3 1 GeV. J1†,α+,32 =r4 h¯vγ5(−i)(cid:18)Dtα− 3γtαD/t(cid:19)q , (8) The results for Λ¯’s are J2†,α+1,,3α2 =r12 h¯v(−2i) Λ¯−,23 = 1.42±0.08 GeV, (13) 2 Λ¯ = 1.38 0.09 GeV. (14) γα1 α2 +γα2 α1 2gα1α2 / q , (9) −,25 ± × (cid:18) t Dt t Dt − 3 t Dt(cid:19) The errors are due to the variation of T and the uncer- tainty in ω . In Fig. 1 and 2, we show the variations of c where hv is the heavy quark field in HQET and γtµ = masses with T for different ωc. γ v v/. ThedefinitionsofTα,β;µ,ν andTα,β,λ;µ,ν,σ are µ µ giv−en in Appendix A. 2 Wefirststudythetwopointsumrulesforthe(1 ,2 ) − − and (2 ,3 ) doublets. We consider the following corre- 1.8 − − lation functions: 1.6 s iZ d4xeikxhπ|T(J1α,−,23(x)J1β,−,23)|0i=−gtαβΠ−,23(ω), mas 1.4 (10) 1 i d4xeikxhπ|T(Jα1α2(x)Jβ1β2 )|0i= (gα1β1gα2β2 1.2 Z 2,−,5 2,−,5 2 t t 2 2 2 +gtα1β2gtα2β1 − 3gtα1α2gtβ1β2)Π−,52(ω), (11) 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 T where ω =2v k. · At the hadron level, FIG. 1: The variation of the mass parameter Λ¯−,3 with the f2 threshold ωc (in unit of GeV) for the 32− doublet.2The long- Π = P,jl + . dashed, short dashed and solid curvescorrespond to ωc=3.2, P,jl 2Λ¯ ω ··· 3.4, 3.6 GeV respectively. P,jl − At the quark-gluon level it can be calculated with the The masses of D wave mesons in the quark model are leading order lagrangian in HQET. Invoking quark- around 2.8 GeV for D meson and 6 GeV for B meson hadrondualityandmakingthe Boreltransformation,we [10]. 1/m correctionmaybequiteimportantforDwave Q 3 2 we will denote the light hadron as pion and discuss all 1.75 three cases. The strong decay amplitudes for D wave 1− and 3 states to ground doublet (0 ,1 ) are 1.5 − − − 1.25 M(B ′ Bπ)=Iǫµq g(B ′B), mass 0.715 M(B11∗∗′ →→B∗π)=Iiǫtµµνρσǫµ1∗ǫ′ν∗vρqtσg(B1∗′B∗), 1 0.5 M(B3 →Bπ)=I ǫαβλ(qtαqtβqtλ− 6qt2(gtαβqtλ 0.25 4 +g q + g q ))g(B B), tαλ tβ tβλ tα 3 3 0.4 0.6 0.8 1 1.2 M(B B π)=I iǫµνσαǫ ǫ v , T 3 → ∗ αβλ ∗µ σ 1 q qβ qλ q2 gβ qλ+gλqβ ×(cid:20) tν t t − 6 t(cid:18) tν t tν t tFhIrGes.h2o:ldTωhec (vianriuantiiotnofofGtehVe)mfoarssthpear25a−medtoeurbΛ¯le−t.,25Twhiethlonthge- +43gtβλqtν(cid:19)(cid:21)g(B3B∗), (17) dashed, short dashed and solid curves correspond to ωc=3.2, 3.4, 3.6 GeV respectively. where ǫαβλ, ǫµ, and ǫ′ are polarizations of the D wave ν 3 ,1 statesandground1 staterespectively. I =1, 1 − − − √2 heavymesons,whichwillbeinvestigatedinasubsequent for charged and neutral pion mesons respectively, while work. for the K and η mesons it equals one. g(B ′B) etc are 1∗ Inthefollowingsectionswealsoneedthevaluesoff’s: thecouplingconstantsinHQETandarerelatedtothose in full QCD by f = 0.39 0.03 GeV5/2 , (15) −,23 ± ′ ′ f−,52 = 0.33±0.04 GeV7/2 . (16) gfullQCD(B1∗ B)=pmB1∗′mB gHQET(B1∗ B). (18) Because of the heavy quark symmetry, the coupling III. SUM RULES FOR DECAY AMPLITUDES constants in eq. (17) satisfy ′ ′ Now let us consider the strong couplings of D wave g(B1∗ B) = g(B1∗ B∗), doublets 32− and 52− with light hadrons. For the light g(B3∗B) = g(B3∗B∗). (19) quark being a u (or d) quark, the D wave heavy mesons decay to pion, while for the light quark being a strange In order to derive the sum rules for the coupling con- quark,itcandecayeithertoBK orB η. Inthefollowing stants we consider the correlators s Z d4xeik·xhπ(q)|T (cid:16)J1α,−,32(x)J0†,−,12(0)(cid:17)|0i=qtαI G1(ω,ω′), (20) 1 Z d4xeik·xhπ(q)|T (cid:16)J1α,−,32(x)J1†,β+,21(0)(cid:17)|0i=(qtαqtβ − 3gtαβqt2)I G2(ω,ω′), (21) 1 Z d4xeik·xhπ(q)|T (cid:16)J1β,+,32(x)J1†,α−,23(0)(cid:17)|0i=(qtαqtβ − 3gtαβqt2)I Gd3(ω,ω′)+gtαβIGs3 , (22) 1 1 Z d4xeik·xhπ(q)|T (cid:16)J1α,−,32(x)J2†,α−1,α322(0)(cid:17)|0i=h2(gtαα1qtα2 +gtαα2qtα1)− 3gtα1α2qtαiI Gp4(ω,ω′) 1 4 +hqtαqtα1qtα2 − 6qt2(gtαα1qtα2 +gtαα2qtα1 + 3gtα1α2qtα)iI Gf4(ω,ω′), (23) for the jP = 3− doublet; ℓ 2 1 4 Z d4xeik·xhπ(q)|T (cid:16)J3α,β−λ,52(x)J0†,−,12(0)(cid:17)|0i=hqtαqtβqtλ− 6qt2(gtαβqtλ+gtαλqtβ + 3gtβλqtα)iI G5(ω,ω′), (24) 1 Z d4xeik·xhπ(q)|T (cid:16)J2α,β−,52(x)J0†,+,21(0)(cid:17)|0i=(qtαqtβ − 3gtαβqt2)I G6(ω,ω′), (25) 4 1 Z d4xeik·xhπ(q)|T (cid:16)J2α,β−,52(x)J1†,γ+,32(0)(cid:17)|0i= 2i(ǫβγµνqtα+ǫαγµνqtβ)qtµvνI G7(ω,ω′), (26) 1 1 Z d4xeik·xhπ(q)|T (cid:16)J2α,β−,52(x)J1†,λ−,32(0)(cid:17)|0i=h2(gtαλqtβ +gtβλqtα)− 3gtαβqtλiI Gp8(ω,ω′) 1 4 + qαqβqλ q2(gαλqβ +gβλqα+ gαβqλ) I Gf(ω,ω ), (27) h t t t − 6 t t t t t 3 t t i 8 ′ Z d4xeik·xhπ(q)|T (cid:16)J2α,β−,25(x)J3†,µ−ν,σ52(0)(cid:17)|0i=TgI Gg9(ω,ω′)+TfI Gf9(ω,ω′)+Tp1I Gp91(ω,ω′) +Tp2I Gp2(ω,ω ), (28) 9 ′ for the jℓP = 52− doublet, where k′ = k+q, ω = 2v·k, quark-gluonlevel ω =2v k . NotethatthetwoPwavecouplingsbetween (2′,−,25)· a′nd (3,−,52) in eq. (28) are not independent G1(ω,ω′)= i√6fπ ∞dt 1duei(1−u)ω2teiuω2′tu and satisfy the relation gp2 = 1gp1. − 12 Z0 Z0 Firstletusconsiderthe9func−tio3n9G1(ω,ω′)ineq. (20). i[uϕ (u)] + 1 m2[uA(u)] + 1B(u) iu As a function of two variables, it has the following pole ×nt π ′ 16 π ′ 2 hq v · terms from the double dispersion relation 1 1 +µ uϕ (u)+ µ ϕ (u) + . (31) −(q v)2ti π p 6 π σ o ··· (2Λ¯ f−,ω12f)−(2,23Λ¯g1 ω) + 2Λ¯ c ω + 2Λ¯ c′ ω , Afterper·formingwickrotationanddoubleBoreltrans- −,12 − ′ −,23 − −,21 − ′ −,32 − formation with the variables ω and ω′ the single-pole terms in eq. (29) are eliminated and we arrive at the wheref denotesthedecayconstantdefinedineq. (1). Λ¯ =Pm,jℓ m . following result: P,jℓ P,jℓ − Q Wecalculatethecorrelator(20)onthelight-conetothe g f f 1 ,1 ,3 leading order of (1/m ). The expression for G (ω,ω ) − 2 − 2 reads O Q 1 ′ = √6fπexp Λ−,21 +Λ−,23 1[uϕπ(u)]′T2f1 ωc 6 h T i(cid:26)2 (cid:16)T (cid:17) √6i ∞dt dxeikxδ(x vt)Tr ( t 1γt /t) 1m2[uA(u)]′ +m2[G (u)+G (u)] − 8 Z0 Z − h Dα− 3 αD −8 π π 1 2 ×(1+v/)γ5hπ(q)|u(0)d¯(x)|0ii. (29) −µπ[uϕP(u)+ 61ϕσ(u)]Tf0(cid:16)ωTc(cid:17)(cid:27)(cid:12)(cid:12)u=u0 , (32) (cid:12) Thepion(orK/η)distributionamplitudesaredefined where u = T1 , T = T1T2 . T , T (cid:12)are the Borel pa- asthematrixelementsofnonlocaloperatorsbetweenthe 0 T1+T2 T1+T2 1 2 n vacuumandpionstate. Uptotwistfourtheyare[20,21]: rameters, and f (x) = 1 e x xk. The factor f is n − − k! n kP=0 π(q)d¯(x)γµγ5u(0)0 = ifπqµ 1dueiuqx ϕπ(u) utisoendotfotshuebtcroancttintuhuemin.teTghraelsRuω∞mc srnuele−sTswdeshaasvaecoobnttariinbeud- h | | i − Z0 h from the correlators (20)-(28) are collected in Appendix 1 i x 1 + m2x2A(u) f m2 µ dueiuqxB(u), C together with the definitions of G1 etc. 16 π i− 2 π πqx Z0 1 π(q)d¯(x)iγ5u(0)0 =fπµπ dueiuqxϕP(u), IV. NUMERICAL RESULTS h | | i Z 0 i π(q)d¯(x)σ γ u(0)0 = (q x q x )f µ For the ground states and P wave heavy mesons, we µν 5 µ ν ν µ π π h | | i 6 − will use [17, 18]: 1 dueiuqxϕ (u). (30) ×Z0 σ Λ¯ ,1 =0.5 GeV, f ,1 =0.25 GeV3/2 , − 2 − 2 Λ¯ =0.85 GeV, f =0.36 0.10 GeV1/2 , The expressions for the light cone wave functions ϕπ(u) +,21 +,12 ± etc are presented in Appendix B together with the rele- Λ¯ =0.95 GeV, f =0.26 0.06 GeV5/2 . vant parameters for π, K and η. +,23 +,32 ± Expressingeq. (29)withthelightconewavefunctions, ThemassparametersanddecayconstantsfortheDwave we get the expression for the correlator function in the doublets have been obtained in section II from the two 5 point sum rule: With the central values of f’s, we get the absolute values of the coupling constants: Λ¯ =1.42 GeV, f =0.39 0.03 GeV5/2 , −,23 −,32 ± g1π =(1.74 0.43) GeV−1 , Λ¯−,25 =1.38 GeV, f−,52 =0.33±0.04 GeV7/2 . g1K =(1.95±±0.63) GeV−1 , g1η =(2.87 0.65) GeV−1 . (35) We choose to work at the symmetric point T =T = ± 1 2 2T, i.e., u0 =1/2 as traditionally done in literature [15]. For the 5− doublet we have The working region for T can be obtained by requiring 2 that the higher twist contribution is less than 30% and g5π =(1.33 0.29) GeV−3 , thecontinuumcontributionislessthan40%ofthewhole ± sumrule,wethengetω =3.2 3.6GeVandtheworking g5K =(1.70 0.42) GeV−3 , c ± region 2.0<T <2.5 GeV for −eqs. (C2), (C6) and (C12) g5η =(1.45 0.30) GeV−3 . (36) ± in Appendix C and 1.2 < T < 2.0 GeV for others. The working regions for the first three sum rules are higher We do not include the uncertainties due to f’s here. than that for the others because there are zero points We can also extract the mass parameter from the between 1 and 2 GeV for them and stability develops strongcouplingformulasobtainedinthelastsection. By only for T above 2 GeV. From eq. (32) the coupling puttingtheexponentialfactorontheleftsideofeq. (C6) reads and differentiating it, one obtains g1πf−,12f−,23 =(0.17±0.04) GeV3 . (33) Λ¯−,32 = T22[ϕπ(du[0ϕ)Tπ(fu00()ωTTcf)0−(ωT41cm)−2πA41(mu02π)AT1(u−0)31T1µ]π/ϕdσT(u0)] . (37) We use the central values for the mass parameters and Withω =3.2 3.6andtheworkingregion2.0<T <2.5 the errorisduetothe variationofT andtheuncertainty c − GeV, we get ofω . ThecentralvaluecorrespondstoT =1.6GeVand c ω = 3.4 GeV. There is cancelation between the twist 2 c Λ¯ =1.36 1.56 GeV, (38) and twist 3 contributions in the sum rule. 1,−,32 − For D wave heavy mesons with a strange quark, the which is consistentwith the value obtained by two point couplings can be obtained by the same way. Notice that sum rule. We present the variation of mass with T and in the η case, f should be replaced by 2 f due to π −√6 η ωc in Fig.3. the quark components of η meson, where f =0.16 GeV η is the decay constant of η meson. From eq. (32) we can 2.4 get the couplings between the ground state doublet and D wave doublet with a strange quark, 2.2 2 g f f =(0.19 0.06) GeV3 , 1K −,21 −,32 ± s 1.8 g f f =(0.28 0.06) GeV3 . (34) as 1η −,21 −,32 ± m 1.6 The couplings between the 3− and 5− doublets and 1.4 2 2 other doublets are collected in TableI. We can see that 1.2 the SU(3) breaking effect is not very big here. f 1.6 1.8 2 2.2 2.4 2.6 2.8 3 T 3− 1− 1+ 3+ 3+ 3− 3− 2 2 2 2d 2s 2f 2p π 0.17 0.086 0.16 0.10 0.056 0.071 K 0.19 0.09 0.24 0.18 0.057 0.10 FIG. 3: The variation of the mass parameter Λ¯−,3 with η 0.28 0.046 0.22 0.11 0.030 0.078 the threshold ωc (in unit of GeV) for the 23− double2t from 5− 1− 1+ 3+ 3− 3− 5− 5− 5− LCQSR.Thelong-dashed,shortdashedandsolid curvescor- 2 2 2 2 2f 2p 2g 2f 2p π 0.11 0.36 0.072 0.13 0.12 0.015 0.05 0.01 respond to ωc=3.2, 3.4, 3.6 GeV respectively. K 0.14 0.48 0.083 0.11 0.16 0.015 0.09 0.02 η 0.12 0.42 0.074 0.10 0.14 0.008 0.08 0.01 V. STRONG DECAY WIDTHS FOR D WAVE TABLEI:Thepioniccouplingsbetween 3− and 5− doublets HEAVY MESONS 2 2 and other doublets. The values are the product of coupling constants and the decay constants of initial and final heavy Having calculated the coupling constants, one can ob- mesons. tain the pionic decay widths of D wave heavy mesons. 6 The widths for D wave states decaying to 0−, 1−, 1+ DK D∗K Dsη Ds∗η states are D∗′ → 8-28 10-48 8-22 8-20 s1 ∗ D → 4-16 3-7 s2 Γ(B1∗′ →B0π−)= 241πMMBB1∗′g12|p1|3 , Bs∗∗1′ → 1B2-K52 1B8∗-8K4 1B1-s2η7 1B6-s∗4η2 Γ(B ′ B 0π )= 1 MB∗ g2 p 3 , Bs2 → 6-28 6-14 1∗ → ∗ − 12πMB∗′ 1| 1| 1 Γ(B ′ B0π )= 1 MB1 g2 p 5 , TABLE III: The decay widths (in unit MeV) of the charm- 1∗ → 1 − 36πMB∗′ 2| 1| strange and bottom-strange D wave (1−,2−) to ground dou- 1 blets and K/η. Γ(B B 0π )= 1 MB∗g2 p 3 , 2∗ → ∗ − 36πM 1| 1| B2 1 M Γ(B3 →B0π−)= 140πMB g52|p1|7 , B3 VI. CONCLUSION Γ(B B 0π )= 1 MB∗g2 p 7 , (39) 3 → ∗ − 105πM 5| 1| B3 In this work we extract the mass and decay constants where p is the moment of final state π. Note that usingthetraditionaltwopointsumruleandcalculatethe 1 | | strong couplings of D wave heavy meson doublets with g(B B )= 2 g(B ′B )= 2 g . 2∗ ∗ 3 1∗ ∗ 3 1 light hadrons π, K and η using LCQSR in the leading q q orderofHQET.Wealsoextractthemassparameterfrom LCQSRforthecouplingwithinthesameDwavedoublet. The extractedmassparametersfromtwoapproachesare A. Nonstrange case consistentwitheachother. Wethencalculatethewidths of D wave heavy mesons decaying to light hadrons. Wetake2.8GeVand6.2GeVforthemassesofDwave charmed and bottomed mesons respectively. MD =1.87 We have not considered the 1/mQ correction and ra- GGqueeaVVr,,kMMmBDo∗∗de==l p52r..3e03d1iGcGteieoVVn,[1[M190]D].a1n=dAfMt2e.4Br21su=Gme5Vm.7,i5nMgGBoevV=erf5rto.h2m8e dfaoniradtBivneomtceossoorrnelsactrwigohene.sr.eHtoHhweeea1vv/eymr,qbtuchaoerrkr1e/ecmxtipocanncoissirorunencdtwieoornrkcoissnwtnreooltll charged and neutral modes, we get the results listed in so small for the charmed mesons. It will be desirable to Table II. consider both the 1/mQ and radiative corrections in the future investigation. ∗ ∗ Dπ D π D1π D π According to our present calculation, the ratios such D1∗′ → 9-27 13-39 0.2 D2∗ → 5-13 as Γ(DsJ(2860)→DK) are useful in distinguishing various Bπ B∗π B1π B∗π inteΓr(pDrseJt(a2t8i6o0n)→sDosfηD) (2860) and D (2715). Treating ∗′ ∗ sJ sJ B1 → 16-46 27-79 0.3 B2 → 9-27 DsJ(2860)as a D wave1− state, we find the aboveratio is 0.4 2.2. If it is the radial excitation of D , this ratio − s∗ is 0.09 [8]. TABLE II: The decay widths (in unit MeV) of the charmed − − and bottomed D wave (1 ,2 ) toground doubletsand π. The pionic widths of D wavestates are not very large. With a mass of 2.86 GeV, the partialdecay width of the 1 D wave D state into DK and Dη modes is 34 118 − s − MeV.Withamassof2.715GeVitspionicwidthis15 57 − MeV. Note that DK modes may be equally important. ∗ B. Strange case So detection of other decay channels, such as Dsη and D K modes, will be very helpful in the classification of ∗ these new states. We use MDs = 1.97 GeV, MDs∗ = 2.11 GeV, MBs = 5.37 GeV, MB∗ = 5.41 GeV [19]. Then for the charm- Acknowledgments: W. Wei thanks P. Z. Huang s strangesectorwehaveTableIII. Wedonotconsiderthe for discussions. This project is supported by the Na- DK∗ mode and three-body modes in the present work. tionalNaturalScienceFoundationofChinaunderGrants For the 2 , 3 states with j = 5, we find that the 10375003,10421503and10625521,MinistryofEducation − − ℓ 2 widths are quite smalland the branchingfractionis per- of China, FANEDD and Key Grant Project of Chinese haps more useful. In the charm-strange sector the ratio Ministry of Education (NO 305001). X.L. thanks the of widths (central values) for DK, D K, D η and D η supportfromtheChinaPostdoctoralScienceFoundation ∗ s ∗ modes is 1:0.4:0.1:0.02. (NO 20060400376). 7 APPENDIX A φ (u) = 6u(1 u) 1+ 5η 1η ω 27ρ2 σ − (cid:26) (cid:18) 3− 2 3 3− 20 η The Tα,β;µ,ν and Tα,β,λ;µ,ν,σ are defined as 3 ρ2a C3/2(ζ), −5 η 2(cid:19)(cid:27) 2 1 1 Tα,β;µ,ν = 2(gαµgβν +gα,νgβ,µ)− 3gtαβgtµν , g (u) = 1+ 1+ 18a +60η + 20η C1/2(ζ) Tα,β,λ;µ,ν,σ = 1(gαµgβνgλσ+gαµgβσgλβ +gανgβµgλσ π h 9 7 2 3 3 4i 2 6 + a 6η ω C1/2(ζ), + gανgβσgλµ+gασgβµgλν +gασgβνgλµ) 1h6− 2284 2− 3 3i 240 1 1 A(u) = 6uu¯ + a +20η + η + + 1(gαβgµνgλσ+gαλgµνgβσ+gβλgµνgασ). (cid:26)15 35 2 3 9 4 h− 15 16 − 9 t t t t t t t t t 7 10 11 η ω η C3/2(ζ)+ a ThetensorstructuresforGwave,FwaveandtwoPwave −27 3 3− 27 4i 2 h− 210 2 decays in eq. (28) for the coupling between the two 5− 4 η ω C3/2(ζ) + 18a +21η ω state are 2 −135 3 3i 4 (cid:27) (cid:16)− 5 2 4 4(cid:17) Tg =Tµ,ν,σ;µ1,ν1,σ1Tα,β;α1,β1q q q q q , 2u3(10 15u+6u2)lnu+2u¯3(10 15u¯ µ1 ν1 σ1 α1 β1 (cid:26) − − 1 1 Tf = qµqαqβ q2(gµαqβ +gµβqα 3nh t t t − 6 t t t t t +6u¯2)lnu¯+uu¯(2+13uu¯)(cid:27), (B1) 4 + gαβqµ) gνσ+(µ,ν,σ) , 3 t t t i o whereu¯ 1 u, ζ 2u 1. C3/2,1/2(ζ)areGegenbauer Tp1 = 61hgtµαgtνσqtβ +gtµβgtνσqtα+(µ,ν,σ)i, polynom≡ials.−Hereg≡π(u)−=B(u1),2+ϕπ(u). aπ1,η =0,aK1 = 1 0.06, aπ,K,η = 0.25, ηπ,K = 0.015, ηη = 0.013, ωπ,K,η = Tp2 = 3(qtµgtνσ+qtνgtµσ+qtσgtµν)gtαβ . −3, η4π2= 10, η4K =30.6, η4η = 0.53, ω4π,K,η = 03.2. ρ2π etc give the mass corrections and are defined as ρ2 = π (mu+md)2, ρ2 = m2s , ρ2 = m2s. m = 0.125 GeV. µ = APPENDIX B m2 K m2 η m2 s π π K η m2π (1 ρ2), µ = m2K,η(1 ρ2 ). f =0.13 GeV, mu+md − π K,η ms − K,η π The distribution amplitudes ϕ etc can be parameter- f =0.16 GeV, f = 0.156 GeV. All of them are scaled π K η ized as [20, 21] at µ=1 GeV. ϕ (u) = 6uu¯ 1+a C3/2(ζ)+a C3/2(ζ) , π (cid:20) 1 1 2 2 (cid:21) APPENDIX C 5 φ (u) = 1+ 30η ρ2 C1/2(ζ)+ 3η ω p (cid:20) 3− 2 η(cid:21) 2 (cid:20)− 3 3 In this appendix we collect the sum rules we have ob- tainedforthestrongcouplingsofDwaveheavydoublets 27 81 ρ2 ρ2a C1/2(ζ), with light hadrons. −20 η− 10 η 2(cid:21) 4 g1f−,12f−,32 = √66fπexp(cid:20)Λ−,21 +T Λ−,23(cid:21)(cid:26)12[uϕπ(u)]′T2f1(cid:16)ωTc(cid:17)− 18m2π[uA(u)]′ +m2π[G1(u)+G2(u)] 1 ω c µ uϕ (u)+ ϕ (u) Tf , (C1) − πh P 6 σ i 0(cid:16)T (cid:17)(cid:27)(cid:12)(cid:12)u=u0 (cid:12) (cid:12) g2f+,21f−,32 = √46fπexp(cid:20)Λ+,21 +T Λ−,32(cid:21)u(cid:26)ϕπ(u)Tf0(cid:16)ωTc(cid:17)− 41m2πA(u)T1 − 31µπϕσ(u)(cid:27)(cid:12)(cid:12)u=u0 , (C2) (cid:12) (cid:12) g3df+,32f−,32 = 18fπexp(cid:20)Λ+,23 +T Λ−,23(cid:21)(cid:26)[(u(1−u)ϕπ(u)]′T2f1(cid:16)ωTc(cid:17)− 14m2π[(u(1−u)A(u)]′ 1 ω +2m2 G (u)+G (u)+2G (u) +2µ u(1 u)ϕ (u)+ ϕ (u) Tf c , (C3) π(cid:2) 3 4 5 (cid:3) π(cid:2) − p 6 σ (cid:3) 0(cid:16)T (cid:17)(cid:27)(cid:12)(cid:12)u=u0 (cid:12) (cid:12) 8 g3sf+,23f−,32 = −418fπexp(cid:20)Λ+,23 +T Λ−,23(cid:21)(cid:26)[(u(1−u)ϕπ(u)]′′′T4f3(cid:16)ωTc(cid:17)+2µπ(cid:20)(cid:16)u(1−u)ϕp(u)(cid:17)′′ +1ϕ (u)′′ T3f ωc m2 4 u(1 u)ϕ (u) ′ + 1 u(1 u)A(u) ′′′ 6 σ (cid:21) 2(cid:16)T (cid:17)− πh (cid:0) − π (cid:1) 4(cid:0) − (cid:1) +3A(u)′ 2(1 2u)B(u) 2 u(1 u)B(u) ′ 4G (u) T2f ωc 6 1 2 − − − − − i (cid:16)T (cid:17) (cid:0) (cid:1) 1 ω 8m2µ u(1 u)ϕ (u)+ ϕ (u) Tf c , (C4) − π π h − p 6 σ i 0(cid:16)T (cid:17)(cid:27)(cid:12)(cid:12)u=u0 (cid:12) (cid:12) g4pf−,32f−,32 = √966fπexp(cid:20)Λ−,32 +T Λ−,23(cid:21)(cid:26)m2πA(u)Tf0(cid:16)ωTc(cid:17)− 32µπ(cid:2)(1−u)ϕσ(u)(cid:3)′T2f1(cid:16)ωTc(cid:17)(cid:27)(cid:12)(cid:12)u=u0 , (C5) (cid:12) (cid:12) g4ff−,32f−,32 = √46fπexp(cid:20)Λ−,23 +T Λ−,23(cid:21)u(1−u)(cid:26)ϕπ(u)Tf0(cid:16)ωTc(cid:17)− 41Tm2πA(u)− 31µπϕσ(u)(cid:27)(cid:12)(cid:12)u=u0 , (C6) (cid:12) (cid:12) g5f−,21f−,52 = 12fπexp(cid:20)Λ−,32 +T Λ−,23(cid:21)u2(cid:26)ϕπ(u)Tf0(cid:16)ωTc(cid:17)− 41Tm2πA(u)+ 31µπϕσ(u)(cid:27)(cid:12)(cid:12)u=u0 , (C7) (cid:12) (cid:12) g6f+,12f−,25 = −√2105fπexp(cid:20)Λ+,12 +T Λ−,25(cid:21)(cid:26)(cid:2)u2ϕπ(u)(cid:3)′T2f1(cid:16)ωTc(cid:17)− 14m2π(cid:2)u2A(u)(cid:3)′ 1 1 ω m2 G (u)+2G (u)+2G (u) +2µ u uϕ (u)+ ϕ (u) Tf c , (C8) −2 π(cid:2) 7 8 5 (cid:3) π h p 3 σ i 0(cid:16)T (cid:17)(cid:27)(cid:12)(cid:12)u=u0 (cid:12) (cid:12) g7f+,23f−,52 = −√3100fπexphΛ+,23 +T Λ−,25i(cid:26)− 14(cid:2)u2(1−u)ϕπ(u)(cid:3)′′T3f2(cid:16)ωTc(cid:17)− 112µπh7(cid:0)u(1− 72u)ϕσ(u)(cid:1)′ ′′ ω 7 + u2(1 u)ϕ (u) T2f c +m2 u2(1 u)ϕ (u)+ uA(u) (cid:0) − σ (cid:1) i 1(cid:16)T (cid:17) πh − π 8 1 ′′ ω 1 + u2(1 u)A(u) Tf c + m2µ u2(1 u)ϕ (u) , (C9) 16(cid:0) − (cid:1) i 0(cid:16)T (cid:17) 3 π π − σ (cid:27)(cid:12)(cid:12)u=u0 (cid:12) (cid:12) g8pf−,32f−,25 = √1180fπexphΛ−,32 +T Λ−,25i(cid:26)81(cid:2)u2(1−u)ϕπ(u)(cid:3)′′′T4f3(cid:16)ωTc(cid:17)+ 41µπh(cid:0)u2(1−u)ϕp(u)(cid:1)′′ 1 5 ′′ ω 1 ′ 27 ′ + u(1 u)ϕ (u) T3f c m2 u2(1 u)ϕ (u) + uA(u) 3(cid:0) − 2 σ (cid:1) i 2(cid:16)T (cid:17)− 2 πh(cid:0) − π (cid:1) 40(cid:0) (cid:1) 1 ′′′ 1 3 1 ′ + u2(1 u)A(u) u 1 u B(u) u2(1 u)B(u) G (u) 9 16 − − 5 − 2 − 2 − − (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) 9 ω 1 5 ω + G (u) T2f c m2µ u2(1 u)ϕ (u)+ 1 u ϕ (u) Tf c , (C10) 5 2 i 1(cid:16)T (cid:17)− π πh − p 3(cid:0) − 2 (cid:1) σ i 0(cid:16)T (cid:17)(cid:27)(cid:12)(cid:12)u=u0 (cid:12) (cid:12) g8ff−,23f−,52 = √1150fπexphΛ1,−,23 +T Λ2,−,52i(cid:26)− 21(cid:2)u2(1−u)ϕπ(u)(cid:3)′T2f1(cid:16)ωTc(cid:17)+ 81m2π(cid:2)u2(1−u)A(u)(cid:3)′ +m2 G (u) 2G (u)+2G (u) 6G (u) π 10 − 11 12 − 13 (cid:2) 1 ω (cid:3) c µ u(1 u) uϕ (u)+ ϕ (u) Tf , (C11) − π − h p 12 σ i 0(cid:16)T (cid:17)(cid:27)(cid:12)(cid:12)u=u0 (cid:12) (cid:12) 9 g9gf−2,52 = √615fπexphΛ−,52 +T Λ−,52iu2(1−u)2(cid:26)ϕπ(u)Tf0(cid:16)ωTc(cid:17)− 41m2πA(u)T1 − 31µπϕσ(u)(cid:27)(cid:12)(cid:12)u=u0 , (cid:12) (cid:12) (C12) g9ff−2,52 = √4155fπexphΛ−,52 +T Λ−,25i(cid:26)− 14(cid:2)u2(1−u)2ϕπ(u)(cid:3)′′T3f2(cid:16)ωTc(cid:17)+ 112µπh(cid:0)u2(1−u)2ϕσ(u)(cid:1)′′ , 3 ′ ω 1 u(1 u)2ϕ (u) T2f c +m2 u2(1 u)2ϕ (u) u(3+7u)A(u) −8(cid:0) − σ (cid:1) i 1(cid:16)T (cid:17) πh − π − 8 ω 1 3 G (u)+2G (u)+2G (u) 6G (u) Tf c m2µ u2(1 u)2ϕ (u) , (C13) − (cid:0) 10 11 12 − 13 (cid:1)i 0(cid:16)T (cid:17)− 3 π π − σ (cid:27)(cid:12)(cid:12)u=u0 (cid:12) (cid:12) g9p1f−2,25 = −√4155fπexphΛ2,−,52 +T Λ3,−,25i(cid:26)214µπ(cid:2)u2(1−u)ϕσ(u)(cid:3)′′′T4f3(cid:16)ωTc(cid:17) 1m2 3(u(1 2u)A(u))′′ +u(2 3u)B(u)+ u2(1 u)B(u) ′ −8 πh2 − − − (cid:0) (cid:1) ω 1 ′ ω +2G (u) 6G (u) T3f c m2µ u2(1 u)2ϕ (u) T2f c , (C14) 9 − 2 i 2(cid:16)T (cid:17)− 6 π π(cid:2) − σ (cid:3) 1(cid:16)T (cid:17)(cid:27)(cid:12)(cid:12)u=u0 (cid:12) (cid:12) The G’s are defined as integrals of light cone wave function B(u) u u x G (u) tB(t)dt, G (u) dx B(t)dt, 1 2 ≡Z ≡Z Z 0 0 0 u u x G (u) t(1 t)B(t)dt, G (u) dx (1 2t)B(t)dt, 3 4 ≡Z − ≡Z Z − 0 0 0 u x y u G (u) dx dy B(t)dt, G (u) B(t)dt, 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