Decays of $D_{sj}^(2317)$ and $D_{sj}(2460)$ Mesons in the Quark Model

a r X i v :h e p -p h /0604090v 2 20 J u l 2006Decays of D ∗sj (2317)and D sj (2460)Mesons in the Quark ModelFeng-Lin Wang,Xiao-Lin Chen,Da-Hai Lu,Shi-Lin Zhu,and Wei-Zhen Deng ∗Department of Physics,Peking University,Beijing 100871,ChinaAbstractWe study the decay widths of the narrow resonances D ∗sj (2317)and D sj (2460)in the chiral quark model,together with the well-known D ∗and D ∗smesons.All the parameters in our calculation are taken from Godfrey and Isgur’s quark model except the π0−ηmixing angle which is fixed by theD ∗sdecay widths.The calculated electromagnetic decay widths agree with those from other groups and the experimental data available quite well.However,the pionic decay widths of D sj (2317)and D sj (2460)are too small to fit the experimental data.We suspect that the simple chiral quark pion axial-vector interaction Hamiltonian is not suitable for hadron strong decays of D sj (2317)and D sj (2460).PACS numbers:12.39.Pn,13.25.Ft,13.40.HqI.INTRODUCTIONThe discovery of narrow resonances D∗sj(2317)[1]and D sj(2460)[2]raises challenges to the quark model.Masses of these two states are about100MeV lower than the predictions of the potential quark model[3].Furthermore,the isospin conserving decay channels D(∗)sj→D(∗)Kare forbidden by kinematics.The observed pionic decays into D(∗)sπ0break the isospin symmetry.Several non-conventional schemes,such as molecules[4],tetraquark states[5,6,7,8], or the chiral partners of D s and D∗s[9,10]etc.,have been proposed(for a review,see Refs.11,12).But the conventional c¯s interpretation is still attractive if the experimental masses of D(∗)sj can be accommodated in the quark model[13,14,15].In the heavy quark limit,D sj(2317)and D sj(2460)naturally form a P-wave doublet J P=(0+,1+)with j l=1II.DECAYS OF D(∗)sj IN THE QUARK MODELThe electromagnetic interaction is standardLγ=e¯ΨγµΨAµ.(1) In the chiral quark model,the pions interact with the light up and down quarks throughthe axial-vector couplingLπq=g qA3m s−m d+m u2fη¯Ψγµλ8Ψ·∂µφη.(4)Thus,we have[34]Lπs=−g qA3fηθm¯sγµγ5s∂µπ0.(5)In our calculation,we do not make the non-relativistic reductions of the transition oper-ators.The quarkfields are kept in the form with Dirac spinorsΨ(x)= s d3p2p0 d†(p,s)v(p,s)e ip·x+b(p,s)u(p,s)e−ip·x ,(6) where the anti-commutation relation of quarks reads{d(p,s),d†(p′,s′)}={b(p,s),b†(p′,s′)}=(2π)3(2p0)δss′δ(p−p′).(7) Accordingly,the meson wave functions are expressed with the quark operators|M,P =1Nc d 3k 2TABLE I:The inner tensor structure matrix of D (∗)smesons.MesonJ PT M √√√√√2Here we treat the quark operators in a matrix formb †Q =b †Q,↑b †Q,↓T,d †¯q =b †¯q ,↑b †¯q ,↓.(9)The inner Q ¯q structure matrix T M of D s ,D ∗s ,D ∗sj (2317)and D sj (2460)mesons are takenfrom the quark model [3,23].They are listed in Table I.The spatial wave functions φ(k ,P )are normalized as2E =d 3k2.They form the (0+,1+)doublet of the P -wave orbital excitation.D sj (2460)is an ideal mixture of the 1P 1and 3P 1states in this limit|D sj (2460),P =3|D sj (1P 1),P +3|D sj (3P 1),P .(11)The spatial wave functions are related to the simple harmonic oscillator (SHO)wavefunctions in Ref.312Eφ(k ,P )=φ(k 2).(12)For the ground states,φ(k 2)= 2√β 3/2e −k 2/2β2.(13)For the P -wave states,φ(k 2)=√β 2√β3/2e −k 2/2β2.(14)In the rest frame of A,the decay width of a process A→B+C isΓ(A→B+C)=|p C|2M A.(17) The wave function in Eq.(12)is calculated in the rest frame,which is an approximation valid only for small|P|for mesons in motion.The calculation of the Lorentz invariant M matrix element should be done in a suitable frame in which the relativistic effect due to |P|is small.In our calculation,the C particle isπorγ,which is treated as an elementary particle.We calculate the invariant matrix element M in the frame P A+P B=0very like the Breit frame.If the heavy quark is the spectator,the relevant kinematics areP A=12p′C(19)p¯q,A=k+12p′C(21)p Q,A=−k(22) p Q,B=−k(23)k A=m Q p¯q,A−m q p Q,A2ηQ p′C(24)k B=m Q p¯q,B−m q p Q,B2ηQ p′C(25)whereηQ=m Q2M2A +2M2B−M2C.(28)The matrix elements are listed below.Following Ref.24,the formulae are all written in the way similar to the non-relativistic formulae except the overlapping integrals which approach unity in the non-relativistic limit m→∞and deviate from unity significantly for the light quarks.For the pionic decay of the D∗meson,we havePπ| L(0)|V4E V E P =ig qA√2fπ→−θm g qA3fη.(31)D∗sj(2317)can decay into D s through the emission of oneπ.The decay matrix element contains two termsPπ| L(0)|S4E S E P=−i 2g A m q F2(p2π,m q,ηQ)+iηQ 6g AβF1(p2π,m q,ηQ).(32) Its radiative decay matrix elements also contain two piecesVγ| Lγ(0)|S = Vγ| Lγ(0)|S E+ Vγ| Lγ(0)|S M,(33) Vγ| Lγ(0)|S E4E V E S= 3βǫ∗V·ǫγ µ¯q F4(p2γ,m q,ηQ)−µQ F4(p2γ,m Q,ηq) ,(34) Vγ| Lγ(0)|S M4E V E S= 6p2γ√32fπEπβ12fπp2π•A →P +γP γ| L γ(0)|A = P γ| L γ(0)|A E + P γ| L γ(0)|A M ,(37) P γ| L γ(0)|A E 4E P E A=3βǫA ·ǫγ µ¯q F 4(p 2γ,m q ,ηQ )−µQ F 4(p 2γ,m Q ,ηq ) (38)P γ| L γ(0)|A M4E P E A=6p 2γ√2√1βµ¯q ǫ∗V ×ǫA ·ǫγF 3(p 2γ,m q ,ηQ )−iηq3µQ 2p 2γ11·ǫA ×ǫγ.(42)•A →S +γSγ| L γ(0)|A = Sγ| L γ(0)|A E + Sγ| L γ(0)|A M ,(43)Sγ| L γ(0)|A E 4E A E S=i23ηq µQ F 4(p 2γ,m Q ,ηq )−µQ F 5(p 2γ,m Q ,ηq )p γ×ǫγ·ǫA(44)Sγ| L γ(0)|A M 4E A E S=ip 2γTABLE II:Decay widths of D∗→D+πin unit of MeV and the relevant F1values.V→P+πpπExp.[25]F1Present work Ref.24 D∗s→D s+π0490.807.4×10−3D∗sj(2317)→D s+π02980.700.521.9D sj(2460)→D∗s+π02970.700.521.9radiative decay widths are listed in Table IV together with some results from other groups. The relevant overlapping integrals are collected in Table V,where=F i(p2γ,m q,ηQ)(47)F qiF Q=F i(p2γ,m Q,ηq)(48)iFrom the table,we see clearly that the overlapping integrals,which are related to the relativistic effects,are very important for the light quarks.For the heavy quarks,the overlapping integrals always approach unity.TABLE IV:Radiative decay widths in unit of keV.Present work QM[26]QM[27]QM[28]QSR[29]LCSR[30]VMD[11]TABLE V:Overlapping integrals related to radiative decays.F q3F Q3F q4F Q4F q5F Q5D∗0→D0π0and D∗s→D sπ0D∗±→D±π0is nearly six times smaller than the experimental data. This may be partly attributed to the uncertainty of our naive SHO wave functions.In the V→P+γformula Eq.(30),there exists a strong cancellation between the light and heavy quark contributions.We haveµ¯d2m d≈−1600F Q3≈0.4,(50) i.e.,µ¯d F q3≈−µc F Q3which leads to the strong cancellation.The sensitivity of the overlapping integrals to the uncertainty of the wave function is amplified in this case.For example,if we change theβparameter toβ=300MeV,we have F q3≈0.48.Then the resulting ratioTABLE VI:Branching ratios between radiative and pionic decays.Exp.Present workD∗0→D0π00.620.35 D∗s→D sπ0D∗±→D±π00.0520.009 D∗sj(2317)→D∗sγD sj(2460)→D∗sπ00.313.3D sj(2460)→D∗sγD sj(2460)→D∗sπ0<0.220.094will increase to∼0.02.As can be seen in Table IV,the radiative decay widths of different channels in this work are comparable with those from other groups(see also ref.31).However,the pionic decay widths of D sj(2317)and D sj(2460)from the simple chiral quark model in Table III are ten times smaller than those from light-cone QCD sum rules approach[14]and the3P0decay model[20].Hence our calculated ratios between EM and pionic decay widths of D sj(2317) and D sj(2460)mesons are systematically larger than the experimental data by a factor of 10.Such a large systematic discrepancy cannot easily be ascribed to either the uncertainty of the meson wave function or the uncertainty of the value of theη−π0mixing amplitude [21,32].We tend to conclude that the simple strong decay mechanism based on the pion and chiral quark axial vector coupling is not realistic if D sj(2317)and D sj(2460)mesons are conventional c¯s states.In summary,we perform a systematic calculation of the decay widths of D∗s,D∗sj(2317), and D sj(2460).The EM radiative decay widths agree with the available experimental data and other model results reasonably well.But the isospin violating pionic decay widths ofD∗sj(2317)and D sj(2460)are too small tofit the experimental data.This disagreement cannot easily be resolved by changing the wave functions or theη−π0mixing amplitude in the chiral quark model.One may wonder whether other possible theoretical schemes such as the coupled-channel effects,hybrid meson,molecule state or tetraquark interpretations ofthese two resonances may resolve the above discrepancy.However,there is no clear evidence in favor of these exotic schemes from BABAR’s most recent extensive measurement[33]. 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