Analysis of the strong decays D ∗ s3 (2860)→DK , D ∗ K with QCD sum rules
aa r X i v : . [ h e p - ph ] A ug Analysis of the strong decays D ∗ s (2860) → DK , D ∗ K with QCD sumrules Zhi-Gang Wang Department of Physics, North China Electric Power University, Baoding 071003, P. R. China
Abstract
In this article, we assign the D ∗ s (2860) to be a D-wave c ¯ s meson, study the hadronic cou-pling constants G D ∗ s (2860) DK and G D ∗ s (2860) D ∗ K with the three-point QCD sum rules, andcalculate the partial decay widths Γ ( D ∗ s (2860) → D ∗ K ) and Γ ( D ∗ s (2860) → DK ). The pre-dicted ratio R = Γ ( D ∗ s (2860) → D ∗ K ) / Γ ( D ∗ s (2860) → DK ) = 0 . ± .
38 cannot reproducethe experimental value R = Br ( D ∗ sJ (2860) → D ∗ K ) / Br ( D ∗ sJ (2860) → DK ) = 1 . ± . ± . PACS number: 14.40.Lb, 12.38.LgKey words: D ∗ s (2860), QCD sum rules In 2006, the BaBar collaboration observed the D ∗ sJ (2860) meson in decays to the final states D K + and D + K S , the measured mass and width are (2856 . ± . ± .
0) MeV and (48 ± ±
10) MeV,respectively [1]. In 2009, the BaBar collaboration confirmed the D ∗ sJ (2860) in the D ∗ K channel,and measured the ratio R among the branching fractions [2], R = Br ( D ∗ sJ (2860) → D ∗ K )Br ( D ∗ sJ (2860) → DK ) = 1 . ± . ± . . (1)The observation of the decays D ∗ sJ (2860) → D ∗ K rules out the J P = 0 + assignment, the possibleassignments are the 1 D c ¯ s meson [3, 4, 5, 6, 7, 8, 9, 10], the c ¯ s − cn ¯ s ¯ n mixing state [11], thedynamically generated D (2420) K bound state [12], etc.In 2014, the LHCb collaboration observed a structure at 2 .
86 GeV in the D K − mass distribu-tion in the Dalitz plot analysis of the decays B s → D K − π + , the structure contains both spin-1(the D ∗− s (2860)) and spin-3 (the D ∗− s (2860)) components [13, 14]. Furthermore, the LHCb col-laboration obtained the conclusion that the D ∗ sJ (2860) observed in the inclusive e + e − → D K − X production by the BaBar collaboration and in the pp → D K − X processes by the LHCb collabo-ration consists of at least two particles [2, 15].The QCD sum rules is a powerful theoretical tool in studying the ground state hadrons andhas given many successful descriptions of the masses, decay constants, form-factors and hadroniccoupling constants, etc [16, 17]. In Ref.[18], we assign the D ∗ s (2860) to be a D-wave c ¯ s meson,and study the mass and decay constant (or the current-meson coupling constant) of the D ∗ s (2860)with the QCD sum rules. The predicted mass M D ∗ s = (2 . ± .
10) GeV is in excellent agreementwith the experimental value M D ∗ s = (2860 . ± . ± . ± .
0) MeV from the LHCb collaboration[13, 14]. We obtain further support by reproducing the mass of the D ∗ s (2860) based on the QCDsum rules.If we assign the D ∗ sJ (2860) to be the 1 D state, the ratio R from the leading order heavymeson effective theory [3], the constituent quark model with quark-meson effective Lagrangians[5], the P model [6, 9, 19, 20, 21] and the relativized quark model [22] cannot reproduce theexperimental value R = 1 . ± . ± .
19 [2]. The values of the ratio R from different theoreticalmethods are shown explicitly in Table 1. From the table, we can see that even in the P modelthe predictions are quite different, as different harmonic oscillator wave-functions are chosen toapproximate the mesons’ wave-functions. E-mail,[email protected]. . ± . ± .
19 Experimental value from BaBar [2]0 .
39 Leading order heavy meson effective theory [3]0 .
40 Constituent quark model with effective Lagrangians [5]0 . P model [6]0 . P model [9]0 . − . P model [19]0 . P model [20]0 . P model [21]0 .
43 Pseudoscalar emission decay model [22]1 . ± . ± .
19 Heavy meson effective theory with chiral symmetry breaking corrections [24]Table 1: The values of the ratio R from different theoretical methods compared to the experimentaldata.The c ¯ q mesons can be sorted in doublets according to the total angular momentum of the lightantiquark ~s ℓ , ~s ℓ = ~s ¯ q + ~L , in the heavy quark limit, where the ~s ¯ q and ~L are the light antiquark’s spinand orbital angular momentum, respectively [23]. For the D-wave mesons, the doublets ( D ∗ s , D s )and ( D ′ s , D ∗ s ) have the spin-parity J Ps ℓ = (1 − , − ) and (2 − , − ) , respectively. The followingtwo-body strong decays can take place, D ∗ + s → D ∗ + K , D ∗ K + , D ∗ + s η , D + K , D K + , D + s η ,D + s → D ∗ + K , D ∗ K + , D ∗ + s η ,D ′ + s → D ∗ + K , D ∗ K + , D ∗ + s η ,D ∗ + s → D ∗ + K , D ∗ K + , D ∗ + s η , D + K , D K + , D + s η . (2)In Ref.[24], we assign the D ∗ s (2860) and D ∗ s (2860) to be the 1 D and 1 D c ¯ s states, respectively,study the strong decays with the heavy meson effective theory by taking into account the chiralsymmetry breaking corrections. We can reproduce the experimental value R = 1 . ± . ± . k Y and ¯ k X , which describe the chiral symmetry breakingcorrections. The coupling constant ¯ k X in the assignment D ∗ sJ (2860) = D ∗ s (2860) is much largerthan the coupling constant ¯ k Y in the assignment D ∗ sJ (2860) = D ∗ s (2860). Naively, we expectsmaller chiral symmetry breaking corrections, the assignment D ∗ sJ (2860) = D ∗ s (2860) is preferred[24]. On the other hand, if the chiral symmetry breaking effects are small enough to be neglected,we have to include some D + s (2860) and D ′ + s (2860) components, as they can only decay to the finalstates D ∗ K , which can enhance the ratio R efficiently.In the article, we take the mass and decay constant (or the current-meson coupling constant) ofthe D ∗ s (2860) from the QCD sum rules as input parameters [18], analyze the vertices D ∗ s (2860) DK and D ∗ s (2860) D ∗ K in details to select the pertinent tensor structures, and study the hadronic cou-pling constants G D ∗ s (2860) DK and G D ∗ s (2860) D ∗ K with the three-point QCD sum rules. Then we usethe G D ∗ s (2860) DK and G D ∗ s (2860) D ∗ K to calculate the partial decay widths Γ ( D ∗ s (2860) → D ∗ K )and Γ ( D ∗ s (2860) → DK ) and obtain the ratio R = Γ ( D ∗ s (2860) → D ∗ K ) / Γ ( D ∗ s (2860) → DK ),and try to reproduce the experimental value R = 1 . ± . ± .
19 based on the QCD sum rulesso as to obtain additional support for assigning the D ∗ sJ (2860) to be the D ∗ s (2860) [24].The article is arranged as follows: we derive the QCD sum rules for the hadronic couplingconstants G D ∗ s (2860) DK and G D ∗ s (2860) D ∗ K in Sect.2; in Sect.3, we present the numerical resultsand discussions; and Sect.4 is reserved for our conclusions.2 QCD sum rules for the hadronic coupling constants G D ∗ s (2860) DK and G D ∗ s (2860) D ∗ K In the following, we write down the three-point correlation functions Π µνρ ( p, p ′ ) and Π σµνρ ( p, p ′ )in the QCD sum rules,Π µνρ ( p, p ′ ) = i Z d xd ye ip ′ · x e i ( p − p ′ ) · ( y − z ) h | T (cid:8) J ( x ) J K ( y ) J † µνρ ( z ) (cid:9) | i | z =0 , (3)Π σµνρ ( p, p ′ ) = i Z d xd ye ip ′ · x e i ( p − p ′ ) · ( y − z ) h | T (cid:8) J σ ( x ) J K ( y ) J † µνρ ( z ) (cid:9) | i | z =0 , (4) J ( x ) = c ( x ) iγ d ( x ) ,J σ ( x ) = c ( x ) γ σ d ( x ) ,J K ( y ) = d ( y ) iγ s ( y ) ,J µνρ ( z ) = c ( z ) (cid:16) γ µ ↔ D ν ↔ D ρ + γ ν ↔ D ρ ↔ D µ + γ ρ ↔ D µ ↔ D ν (cid:17) s ( z ) , where the currents J ( x ), J σ ( x ), J K ( y ) and J µνρ ( z ) interpolate the mesons D , D ∗ , K and D ∗ s (2860),respectively, ↔ D µ = → ∂ µ − ig s G µ − ← ∂ µ − ig s G µ , the G µ is the gluon field.The current J µνρ (0) has negative parity, and couples potentially to the J P = 3 − ¯ cs meson D ∗ s (2860). Furthermore, the current J µνρ (0) also couples potentially to the J P = 2 + , 1 − , 0 + ¯ cs mesons. The current-meson coupling constants or the decay constants f D ∗ s , f D ∗ s , f D ∗ s and f D ∗ s are defined by h | J µνρ (0) | D ∗ s ( p ) i = f D ∗ s ε µνρ ( p, s ) , (5) h | J µνρ (0) | D ∗ s ( p ) i = f D ∗ s [ p µ ε νρ ( p, s ) + p ν ε ρµ ( p, s ) + p ρ ε µν ( p, s )] , h | J µνρ (0) | D ∗ s ( p ) i = f D ∗ s [ p µ p ν ε ρ ( p, s ) + p ν p ρ ε µ ( p, s ) + p ρ p µ ε ν ( p, s )] , h | J µνρ (0) | D ∗ s ( p ) i = f D ∗ s p µ p ν p ρ , (6)where the ε µνρ ( p, s ), ε µν ( p, s ) and ε µ ( p, s ) are the mesons’ polarization vectors with the followingproperties [25],P µνραβσ = X s ε ∗ µνρ ( p, s ) ε αβσ ( p, s )= 16 ( e g µα e g νβ e g ρσ + e g µα e g νσ e g ρβ + e g µβ e g να e g ρσ + e g µβ e g νσ e g ρα + e g µσ e g να e g ρβ + e g µσ e g νβ e g ρα ) −
115 ( e g µα e g νρ e g βσ + e g µβ e g νρ e g ασ + e g µσ e g νρ e g αβ + e g να e g µρ e g βσ + e g νβ e g µρ e g ασ + e g νσ e g µρ e g αβ + e g ρα e g µν e g βσ + e g ρβ e g µν e g ασ + e g ρσ e g µν e g αβ ) , (7)P µναβ = X s ε ∗ µν ( p, s ) ε αβ ( p, s ) = e g µα e g νβ + e g µβ e g να − e g µν e g αβ , (8) e g µν = X s ε ∗ µ ( p, s ) ε ν ( p, s ) = − g µν + p µ p ν p . (9)At the phenomenological side, we insert a complete set of intermediate hadronic states withthe same quantum numbers as the current operators J ( x ), J σ ( x ), J K ( y ) and J µνρ ( z ) into thecorrelation functions Π µνρ ( p, p ′ ) and Π σµνρ ( p, p ′ ) to obtain the hadronic representation [16, 17].3e isolate all the ground state contributions and write them down explicitly,Π µνρ ( p, p ′ ) = f D M D f K M K f D ∗ s G D ∗ s DK ( q )( m c + m d )( m d + m s ) ( M D − p ′ ) ( M K − q ) (cid:16) M D ∗ s − p (cid:17) h λ (cid:16) M D ∗ s , M D , q (cid:17) + 10 M D ∗ s M D i (cid:16) M D ∗ s + M D − q (cid:17) M D ∗ s p µ p ν p ρ + λ (cid:16) M D ∗ s , M D , q (cid:17) (cid:16) M D ∗ s + M D − q (cid:17) M D ∗ s ( p µ g νρ + p ν g µρ + p ρ g µν ) − λ (cid:16) M D ∗ s , M D , q (cid:17) M D ∗ s (cid:0) p ′ µ g νρ + p ′ ν g µρ + p ′ ρ g µν (cid:1) − λ (cid:16) M D ∗ s , M D , q (cid:17) + 5 M D ∗ s M D M D ∗ s (cid:0) p ′ µ p ν p ρ + p ′ ν p µ p ρ + p ′ ρ p µ p ν (cid:1) + M D ∗ s + M D − q M D ∗ s (cid:0) p ′ µ p ′ ν p ρ + p ′ ν p ′ ρ p µ + p ′ ρ p ′ µ p ν (cid:1) − p ′ µ p ′ ν p ′ ρ ) + f D M D f K M K f D ∗ s G D ∗ s DK ( q )( m c + m d )( m d + m s ) ( M D − p ′ ) ( M K − q ) (cid:16) M D ∗ s − p (cid:17) λ (cid:16) M D ∗ s , M D , q (cid:17) + 6 M D ∗ s M D M D ∗ s p µ p ν p ρ + λ (cid:16) M D ∗ s , M D , q (cid:17) M D ∗ s ( p µ g νρ + p ν g µρ + p ρ g µν )+ (cid:0) p ′ µ p ′ ν p ρ + p ′ ν p ′ ρ p µ + p ′ ρ p ′ µ p ν (cid:1) − M D ∗ s + M D − q M D ∗ s (cid:0) p ′ µ p ν p ρ + p ′ ν p µ p ρ + p ′ ρ p µ p ν (cid:1)) + f D M D f K M K f D ∗ s G D ∗ s DK ( q )( m c + m d )( m d + m s ) ( M D − p ′ ) ( M K − q ) (cid:16) M D ∗ s − p (cid:17) (cid:16) M D ∗ s + M D − q (cid:17) M D ∗ s p µ p ν p ρ − (cid:0) p ′ µ p ν p ρ + p ′ ν p µ p ρ + p ′ ρ p µ p ν (cid:1) + f D M D f K M K f D ∗ s G D ∗ s DK ( q )( m c + m d )( m d + m s ) ( M D − p ′ ) ( M K − q ) (cid:16) M D ∗ s − p (cid:17) p µ p ν p ρ + · · · , (10)4 σµνρ ( p, p ′ ) = f D ∗ M D ∗ f K M K f D ∗ s G D ∗ s D ∗ K ( q )( m d + m s ) ( M D ∗ − p ′ ) ( M K − q ) (cid:16) M D ∗ s − p (cid:17) λ (cid:16) M D ∗ s , M D ∗ , q (cid:17) M D ∗ s (cid:0) g µν ε σρλτ p λ p ′ τ + g µρ ε σνλτ p λ p ′ τ + g νρ ε σµλτ p λ p ′ τ (cid:1) + λ (cid:16) M D ∗ s , M D ∗ , q (cid:17) + 5 M D ∗ s M D ∗ M D ∗ s (cid:0) ε σρλτ p µ p ν p λ p ′ τ + ε σνλτ p µ p ρ p λ p ′ τ + ε σµλτ p ν p ρ p λ p ′ τ (cid:1) − M D ∗ s + M D ∗ − q M D ∗ s (cid:0) ε σρλτ p ′ µ p ν p λ p ′ τ + ε σρλτ p µ p ′ ν p λ p ′ τ + ε σνλτ p ′ µ p ρ p λ p ′ τ + ε σνλτ p µ p ′ ρ p λ p ′ τ + ε σµλτ p ′ ν p ρ p λ p ′ τ + ε σµλτ p ν p ′ ρ p λ p ′ τ (cid:1) + 13 (cid:0) ε σρλτ p ′ µ p ′ ν p λ p ′ τ + ε σνλτ p ′ µ p ′ ρ p λ p ′ τ + ε σµλτ p ′ ν p ′ ρ p λ p ′ τ (cid:1)(cid:27) + f D ∗ M D ∗ f K M K f D ∗ s G D ∗ s D ∗ K ( q )( m d + m s ) ( M D ∗ − p ′ ) ( M K − q ) (cid:16) M D ∗ s − p (cid:17)(cid:26) − (cid:0) ε σρλτ p ′ µ p ν p λ p ′ τ + ε σρλτ p µ p ′ ν p λ p ′ τ + ε σνλτ p ′ µ p ρ p λ p ′ τ + ε σνλτ p µ p ′ ρ p λ p ′ τ + ε σµλτ p ′ ν p ρ p λ p ′ τ + ε σµλτ p ν p ′ ρ p λ p ′ τ (cid:1) + M D ∗ s + M D ∗ − q M D ∗ s (cid:0) ε σρλτ p µ p ν p λ p ′ τ + ε σνλτ p µ p ρ p λ p ′ τ + ε σµλτ p ν p ρ p λ p ′ τ (cid:1)) + f D ∗ M D ∗ f K M K f D ∗ s G D ∗ s D ∗ K ( q )( m d + m s ) ( M D ∗ − p ′ ) ( M K − q ) (cid:16) M D ∗ s − p (cid:17)(cid:0) ε σρλτ p µ p ν p λ p ′ τ + ε σνλτ p µ p ρ p λ p ′ τ + ε σµλτ p ν p ρ p λ p ′ τ (cid:1) + · · · , (11)where the · · · denotes the contributions come from the higher resonances and continuum states, λ ( a, b, c ) = a + b + c − ab − bc − ca , the decay constants f D , f D ∗ , f K and the hadronic couplingconstants G D ∗ s DK , G D ∗ s DK , G D ∗ s DK , G D ∗ s DK , G D ∗ s D ∗ K , G D ∗ s D ∗ K , G D ∗ s D ∗ K are defined by h | J (0) | D ( p ′ ) i = f D M D m c + m d , h | J σ (0) | D ∗ ( p ′ ) i = f D ∗ M D ∗ ε σ ( p ′ , s ) , h | J K (0) | K ( q ) i = f K M K m s + m d , (12) h D ( p ′ ) K ( q ) | D ∗ s ( p ) i = G D ∗ s DK ε αβγ ( p, s ) p ′ α p ′ β p ′ γ , h D ( p ′ ) K ( q ) | D ∗ s ( p ) i = G D ∗ s DK ε αβ ( p, s ) p ′ α p ′ β , h D ( p ′ ) K ( q ) | D ∗ s ( p ) i = G D ∗ s DK ε α ( p, s ) p ′ α , h D ( p ′ ) K ( q ) | D ∗ s ( p ) i = G D ∗ s DK , (13)5 D ∗ ( p ′ ) K ( q ) | D ∗ s ( p ) i = G D ∗ s D ∗ K ε αβλτ ε ∗ α ( p ′ , s ′ ) ε βωθ ( p, s ) p λ p ′ τ p ′ ω p ′ θ , h D ∗ ( p ′ ) K ( q ) | D ∗ s ( p ) i = G D ∗ s D ∗ K ε αβλτ ε ∗ α ( p ′ , s ′ ) ε βω ( p, s ) p λ p ′ τ p ′ ω , h D ∗ ( p ′ ) K ( q ) | D ∗ s ( p ) i = G D ∗ s D ∗ K ε αβλτ ε ∗ α ( p ′ , s ′ ) ε β ( p, s ) p λ p ′ τ , (14)the ε µνρ ( p, s ), ε µν ( p, s ) and ε µ ( p, s ) are the mesons’ polarization vectors.Now we rewrite the correlation functions Π µνρ ( p, p ′ ) and Π σµνρ ( p, p ′ ) at the phenomenologicalside into the following form,Π µνρ ( p, p ′ ) = Π DK, ( p , p ′ ) p ′ µ p ′ ν p ′ ρ + e Π DK, ( p , p ′ ) (cid:0) p ′ µ g νρ + p ′ ν g µρ + p ′ ρ g µν (cid:1) +Π DK, / / / ( p , p ′ ) p µ p ν p ρ + Π DK, / ( p , p ′ ) ( p µ g νρ + p ν g µρ + p ρ g µν )+Π DK, / / ( p , p ′ ) (cid:0) p ′ µ p ν p ρ + p ′ ν p µ p ρ + p ′ ρ p µ p ν (cid:1) +Π DK, / ( p , p ′ ) (cid:0) p ′ µ p ′ ν p ρ + p ′ ν p ′ ρ p µ + p ′ ρ p ′ µ p ν (cid:1) , (15)Π σµνρ ( p, p ′ ) = Π D ∗ K, ( p , p ′ ) 13 (cid:0) ε σρλτ p ′ µ p ′ ν p λ p ′ τ + ε σνλτ p ′ µ p ′ ρ p λ p ′ τ + ε σµλτ p ′ ν p ′ ρ p λ p ′ τ (cid:1) + e Π D ∗ K, ( p , p ′ ) (cid:0) g µν ε σρλτ p λ p ′ τ + g µρ ε σνλτ p λ p ′ τ + g νρ ε σµλτ p λ p ′ τ (cid:1) +Π D ∗ K, / / ( p , p ′ ) (cid:0) ε σρλτ p µ p ν p λ p ′ τ + ε σνλτ p µ p ρ p λ p ′ τ + ε σµλτ p ν p ρ p λ p ′ τ (cid:1) +Π D ∗ K, / ( p , p ′ ) (cid:0) ε σρλτ p ′ µ p ν p λ p ′ τ + ε σρλτ p µ p ′ ν p λ p ′ τ + ε σνλτ p ′ µ p ρ p λ p ′ τ + ε σνλτ p µ p ′ ρ p λ p ′ τ + ε σµλτ p ′ ν p ρ p λ p ′ τ + ε σµλτ p ν p ′ ρ p λ p ′ τ (cid:1) , (16)so as to isolate the components associated with the special tensor structures which only receivecontributions come from the spin-3 meson D ∗ s (2860), where the contributions come from thehigher resonances and continuum states are neglected, the subscripts 3, 2, 1 and 0 denote thatthere are contributions come from the J P = 3 − , 2 + , 1 − and 0 + c ¯ s mesons, respectively. FromEqs.(15-16), we can see that the components Π DK, ( p , p ′ ), e Π DK, ( p , p ′ ), Π D ∗ K, ( p , p ′ ) and e Π D ∗ K, ( p , p ′ ) only receive contributions come from the spin-3 meson D ∗ s (2860). The polarizationvector ε µνρ ( p, s ) satisfies the relation g µν ε µνρ ( p, s ) = g µρ ε µνρ ( p, s ) = g νρ ε µνρ ( p, s ) = 0. If wemultiply both sides of Eq.(5) by g µν , we can obtain g µν h | J µνρ (0) | D ∗ s ( p ) i 6 = f D ∗ s g µν ε µνρ ( p, s ) = 0 , (17)the equation does not survive. We have to introduce the traceless current J µνρ by taking thefollowing replacement, J µνρ → J µνρ = J µνρ − g µν g αβ J αβρ − g µρ g αβ J ανβ − g νρ g αβ J µαβ , (18)then the traceless current J µνρ satisfies the relations g µν J µνρ = g µρ J µνρ = g νρ J µνρ = 0, and h | J µνρ (0) | D ∗ s ( p ) i = f D ∗ s ε µνρ ( p, s ) . (19)According to Eq.(5) and Eq.(19), we can choose either the current J µνρ ( x ) or the current J µνρ ( x )to interpolate the D ∗ s (2860), as the components Π DK, ( p , p ′ ), e Π DK, ( p , p ′ ), Π D ∗ K, ( p , p ′ )and e Π D ∗ K, ( p , p ′ ) at the phenomenological side are not changed. At the QCD side, if thecurrent J µνρ ( x ) is chosen, the components Π DK, ( p , p ′ ) and Π D ∗ K, ( p , p ′ ) are not modified,but the components e Π DK, ( p , p ′ ) and e Π D ∗ K, ( p , p ′ ) are modified remarkably. In calcula-tions, we observe that the components e Π DK, ( p , p ′ ) and e Π D ∗ K, ( p , p ′ ) cannot lead to reli-able QCD sum rules and they are discarded. The pertinent tensor structures are p ′ µ p ′ ν p ′ ρ and ε σρλτ p ′ µ p ′ ν p λ p ′ τ + ε σνλτ p ′ µ p ′ ρ p λ p ′ τ + ε σµλτ p ′ ν p ′ ρ p λ p ′ τ , we choose the two components Π DK, ( p , p ′ )and Π D ∗ K, ( p , p ′ ) to study the hadronic coupling constants G D ∗ s DK and G D ∗ s D ∗ K , respectively.6ow, we briefly outline the operator product expansion for the correlation functions Π µνρ ( p, p ′ )and Π σµνρ ( p, p ′ ) in perturbative QCD. We contract the quark fields in the correlation functionsΠ µνρ ( p, p ′ ) and Π σµνρ ( p, p ′ ) with Wick theorem firstly,Π µνρ ( p, p ′ ) = Z d xd ye ip ′ · x e i ( p − p ′ ) · ( y − z ) Tr { iγ U ij ( x − y ) iγ S jk ( y − z )Γ µνρ C ki ( z − x ) } | z =0 , Π σµνρ ( p, p ′ ) = Z d xd ye ip ′ · x e i ( p − p ′ ) · ( y − z ) Tr { γ σ U ij ( x − y ) iγ S jk ( y − z )Γ µνρ C ki ( z − x ) } | z =0 , (20)where Γ µνρ = γ µ ↔ ∂∂z ν ↔ ∂∂z ρ + γ ν ↔ ∂∂z µ ↔ ∂∂z ρ + γ ρ ↔ ∂∂z µ ↔ ∂∂z ν , (21) C ij ( x ) = i (2 π ) Z d ke − ik · x (cid:26) δ ij k − m c − g s G nαβ t nij σ αβ ( k + m c ) + ( k + m c ) σ αβ ( k − m c ) + ig s GGδ ij m c k + m c k ( k − m c ) + · · · (cid:27) , (22) t n = λ n , the λ n is the Gell-Mann matrix, the i , j , k are color indexes [17]. We usually choose thefull light quark propagators in the coordinate space. In the present case, the quark condensatesand mixed condensates have no contributions, so we can take a simple replacement c → d/s toobtain the full d/s quark propagators. We compute all the integrals, then obtain the QCD spectraldensity through dispersion relation.The leading-order contributions Π µνρ ( p, p ′ ) and Π σµνρ ( p, p ′ ) can be written asΠ µνρ ( p, p ′ ) = 3 i (2 π ) Z d k Tr { γ [ k + m d ] γ [ k + p − 6 p ′ + m s ] Γ µνρ [ k − 6 p ′ + m c ] } [ k − m d ] [( k + p − p ′ ) − m s ] [( k − p ′ ) − m c ] , = Z dsdu ρ µνρ ( s, u )( s − p )( u − p ′ ) , (23)Π σµνρ ( p, p ′ ) = 3(2 π ) Z d k Tr { γ σ [ k + m d ] γ [ k + p − 6 p ′ + m s ] Γ µνρ [ k − 6 p ′ + m c ] } [ k − m d ] [( k + p − p ′ ) − m s ] [( k − p ′ ) − m c ] , = Z dsdu ρ σµνρ ( s, u )( s − p )( u − p ′ ) , (24)where Γ µνρ = − γ µ ( p − k − p ′ ) ν ( p − k − p ′ ) ρ − γ ν ( p − k − p ′ ) µ ( p − k − p ′ ) ρ − γ ρ ( p − k − p ′ ) µ ( p − k − p ′ ) ν . (25)The gluon field G µ ( z ) in the covariant derivative has no contributions as G µ ( z ) = z λ G λµ (0)+ · · · =0. We put all the quark lines on mass-shell by using the Cutkosky’s rules, see Fig.1, and obtainthe leading-order QCD spectral densities ρ µνρ ( s, u ) and ρ σµνρ ( s, u ), ρ µνρ ( s, u ) = 3(2 π ) Z d k δ (cid:2) k − m d (cid:3) δ (cid:2) ( k + p − p ′ ) − m s (cid:3) δ (cid:2) ( k − p ′ ) − m c (cid:3) Tr { γ [ k + m d ] γ [ k + p − 6 p ′ + m s ] Γ µνρ [ k − 6 p ′ + m c ] } , (26) ρ σµνρ ( s, u ) = − i (2 π ) Z d k δ (cid:2) k − m d (cid:3) δ (cid:2) ( k + p − p ′ ) − m s (cid:3) δ (cid:2) ( k − p ′ ) − m c (cid:3) Tr { γ σ [ k + m d ] γ [ k + p − 6 p ′ + m s ] Γ µνρ [ k − 6 p ′ + m c ] } . (27)7t is straightforward to compute the integrals , some useful identities are given explicitly in theappendix. The contributions of the gluon condensates shown in Fig.2 are calculated in the sameway.Once the analytical expressions of the QCD spectral densities are obtained, we can take quark-hadron duality below the continuum thresholds s and u respectively, and perform the doubleBorel transform with respect to the variables P = − p and P ′ = − p ′ to obtain the QCD sumrules, Π DK, ( M , M ) = − f D M D f K M K f D ∗ s G D ∗ s DK ( q )( m c + m d )( m d + m s ) ( M K − q ) exp − M D ∗ s M − M D M ! = Z dsdu exp (cid:18) − sM − uM (cid:19) π p λ ( s, u, q ) ρ DK , (28)Π D ∗ K, ( M , M ) = f D ∗ M D ∗ f K M K f D ∗ s G D ∗ s D ∗ K ( q )( m d + m s ) ( M K − q ) exp − M D ∗ s M − M D ∗ M ! = Z dsdu exp (cid:18) − sM − uM (cid:19) π p λ ( s, u, q ) ρ D ∗ K , (29)where Z dsdu = Z s m c ds Z u m c du | − ≤ cos θ ≤ , cos θ = (cid:0) u − q − m c (cid:1) (cid:0) s + u − q (cid:1) − s (cid:0) u − m c (cid:1) | u − q − m c | p λ ( u, s, q ) , (30) We choose the four-vectors as p = ( √ s, p ′ = ( p ′ , ~p ′ ), k = ( k , ~k ), and obtain the following solutions k = u − q + m s − m c √ s , | ~k | = s(cid:18) u − q + m s − m c √ s (cid:19) − m d ,p ′ = s + u − q √ s , | ~p ′ | = p λ ( s, u, q )2 √ s , from the three Dirac δ -functions in Eq.(26) or Eq.(27). Then we obtain cos θ cos θ = ( u − q + m s − m c )( s + u − q ) − s ( u + m d − m c ) q ( u − q + m s − m c ) − sm d p λ ( s, u, q ) , from the identity ( k − p ′ ) − m c = m d + u − k p ′ + 2 | ~k || ~p ′ | cos θ − m c = 0 , where we have used ~k · ~p ′ = | ~k || ~p ′ | cos θ . If we take the approximation m d ≈ m s ≈
0, then we obtain the constraintin Eq.(30). ρ DK = − m c + 4 m d m c + 2 u + 2 q + b (cid:0) m c − m d m c + 4 m s m c + 2 s − u − q (cid:1) + b (cid:0) − m c + 12 m d m c − m s m c − s + 6 u + 6 q (cid:1) + f (cid:0) − m d m c + 4 m s m c + 2 s − u − q (cid:1) + π h α s GGπ i (cid:26) ∂b ∂m B + 16 ∂b ∂m A + 16 ∂b ∂m c − ∂b ∂m B − ∂b ∂m A − ∂b ∂m c +6 ∂f ∂m B + 4 ∂f ∂m A + 6 ∂f ∂m c + (cid:0) u − m c − s + 9 q (cid:1) ∂ b ∂m A ∂m B + (cid:0) s − u − q (cid:1) ∂ f ∂m A ∂m B + (cid:0) u − m c − s + 3 q (cid:1) ∂ b ∂m A ∂m c + (cid:0) m c + s − u − q (cid:1) ∂ f ∂m A ∂m c + (cid:0) s + 3 u − m c + 3 q (cid:1) ∂ b ∂m B ∂m c + (cid:0) m c − s − u − q (cid:1) ∂ f ∂m B ∂m c − m c ∂ b ∂ ( m c ) + m c (cid:0) s − u − q (cid:1) ∂ f ∂ ( m c ) (cid:27) , (31) ρ D ∗ K = − m d + 4 ( m s − m c ) a − m c − m d ) b + 4 (2 m c − m d ) b + 8 ( m c − m s ) c +4 ( m s − m c ) e + 4 ( m d − m c ) f + 2 π m c h α s GGπ i (cid:26) ∂ b ∂m A ∂m B + 2 ∂ c ∂m A ∂m B − ∂ e ∂m A ∂m B − ∂ f ∂m A ∂m B − ∂ b ∂m A ∂m c − ∂ c ∂m A ∂m c + ∂ e ∂m A ∂m c + ∂ f ∂m A ∂m c − ∂ b ∂m B ∂m c − ∂ c ∂m B ∂m c + ∂ e ∂m B ∂m c + ∂ f ∂m B ∂m c +2 ∂ b ∂ ( m c ) + 2 ∂ c ∂ ( m c ) − ∂ e ∂ ( m c ) − ∂ f ∂ ( m c ) − m c ∂ e ∂ ( m c ) − m c ∂ f ∂ ( m c ) (cid:27) , (32)the explicit expressions of the coefficients a , b , b , c , e , f are given in the appendix.9igure 2: The gluon condensate contributions. The value of the gluon condensate is taken to be the standard value h α s GGπ i = 0 .
012 GeV [16, 17].In the article, we take the M S masses m c ( m c ) = (1 . ± . m s ( µ = 2 GeV) =(0 . ± . M S masses from the renormalization group equation, m s ( µ ) = m s (2GeV) (cid:20) α s ( µ ) α s (2GeV) (cid:21) ,m d ( µ ) = m d (1GeV) (cid:20) α s ( µ ) α s (1GeV) (cid:21) ,m c ( µ ) = m c ( m c ) (cid:20) α s ( µ ) α s ( m c ) (cid:21) ,α s ( µ ) = 1 b t (cid:20) − b b log tt + b (log t − log t −
1) + b b b t (cid:21) , (33)where t = log µ Λ , b = − n f π , b = − n f π , b = − n f + n f π , Λ = 213 MeV, 296 MeVand 339 MeV for the flavors n f = 5, 4 and 3, respectively [26]. Furthermore, we obtain the values m u = m d = 6 MeV from the Gell-Mann-Oakes-Renner relation at the energy scale µ = 1 GeV. Incalculations, we take n f = 4 and µ = µ D ∗ s = 2 . M D = 1 .
87 GeV, M D ∗ = 2 .
01 GeV, f D = 208 MeV, f D ∗ = 263 MeV, M ( D ) = (1 . − .
8) GeV , M ( D ∗ ) = (1 . − .
5) GeV , u D = (6 . ± .
5) GeV , u D ∗ = (6 . ± .
5) GeV determinedin the two-point QCD sum rules [27]. In Ref.[18], we assign the D ∗ s (2860) to be a D-wave c ¯ s meson,and study the mass and decay constant (or current-meson coupling constant) of the D ∗ s (2860) withthe QCD sum rules by calculating the contributions of the vacuum condensates up to dimension-6in the operator product expansion. In this article, we take the values M D ∗ s = 2 .
86 GeV, f D ∗ s =10 .9 2.0 2.1 2.2 2.3 2.4 2.505101520253035404550 (I) M =1.5GeV | G | ( G e V - ) M (GeV ) Q =2GeV Q =3GeV Q =4GeV (I) M =2.2GeV | G | ( G e V - ) M (GeV ) Q =2GeV Q =3GeV Q =4GeV (II) M =2.2GeV | G | ( G e V - ) M (GeV ) Q =12GeV Q =13GeV Q =14GeV (II) M =2.2GeV | G | ( G e V - ) M (GeV ) Q =12GeV Q =13GeV Q =14GeV Figure 3: The hadronic coupling constants G D ∗ s DK ( Q ) (I) and G D ∗ s D ∗ K ( Q ) (II) with variationsof the Borel parameters M and M , respectively.6 .
02 GeV , M ( D ∗ s ) = (1 . − .
5) GeV , s D ∗ s = (11 . ± .
7) GeV determined in the two-pointQCD sum rules [18]. Furthermore, we take the values M K = 0 .
495 GeV and f K = 0 .
160 GeV fromthe Particle Data Group [26]In the following, we write down the definitions for the pole contributions of the D ∗ s (2860), D and D ∗ in the QCD sum rules,pole D ∗ s = R s m c ds R ∞ m c du ρ QCD ( s, u ) | − ≤ cos θ ≤ exp (cid:16) − sM − uM (cid:17)R ∞ m c ds R ∞ m c du ρ QCD ( s, u ) | − ≤ cos θ ≤ exp (cid:16) − sM − uM (cid:17) , pole D/D ∗ = R ∞ m c ds R u m c du ρ QCD ( s, u ) | − ≤ cos θ ≤ exp (cid:16) − sM − uM (cid:17)R ∞ m c ds R ∞ m c du ρ QCD ( s, u ) | − ≤ cos θ ≤ exp (cid:16) − sM − uM (cid:17) , (34)where the ρ QCD ( s, u ) denotes the spectral densities at the QCD side. If we choose the Borel win-dows determined by the two-point QCD sum rules [18, 27], the pole contributions pole D ∗ s /D/D ∗ ≫ D ∗ s = (92 − M = 1 . and q = − ; pole D = (86 − M = 2 . and q = − . The pole dominance is well satisfied. Moreover, in theBorel windows, the contributions come from the gluon condensate are of percent level, the operatorproduct expansion is well convergent. The Borel windows determined by the two-point QCD sumrules still work in the three-point QCD sum rules, and we expect to make reasonable predictions.In Fig.3, we plot the hadronic coupling constants G D ∗ s DK ( Q ) and G D ∗ s D ∗ K ( Q ) with varia-tions of the Borel parameters M and M , where Q = − q . From the figure, we can see that thevalues are not very stable with variations of the Borel parameters M and M . From the QCD sum11ules in Eqs.(28-32) or the explicit expressions of the ρ DK and ρ D ∗ K , we can see that there are nocontributions come from the quark condensates and mixed condensates, and no terms of the orders O (cid:16) M (cid:17) , O (cid:16) M (cid:17) , O (cid:16) M (cid:17) , O (cid:16) M (cid:17) , · · · , which are needed to stabilize the QCD sum rules so asto warrant a platform. The uncertainties originate from the Borel parameters are rather large, wetake them into account. In calculations, we observe that the values of the | G D ∗ s DK ( Q ) | at theregion Q > decrease monotonously with increase of the Q , while the values G D ∗ s D ∗ K ( Q )change sign at the region Q = (1 −
2) GeV , we have to postpone the Q to large values.Now we fit the central values of the hadronic coupling constants G D ∗ s DK ( Q ) at Q = (2 −
4) GeV and G D ∗ s D ∗ K ( Q ) at Q = (12 −
14) GeV into the functions of the form A + BQ , | G D ∗ s DK ( Q ) | = 22 .
88 GeV − − . Q GeV − , (35) | G D ∗ s D ∗ K ( Q ) | = 10 .
61 GeV − − . Q GeV − , (36)then we extend the values to the physical region Q = − M K , and obtain | G D ∗ s DK ( Q = − M K ) | = 23 . − , (37) | G D ∗ s D ∗ K ( Q = − M K ) | = 10 . − , (38)the uncertainties of the G D ∗ s DK ( Q = − M K ) and G D ∗ s D ∗ K ( Q = − M K ) are about 18% and 28%,respectively.We can take the physical values of the hadronic coupling constants G D ∗ s DK and G D ∗ s D ∗ K asinput parameters and study the two-body strong decays, which take place through relative F-wave,Γ (cid:0) D ∗ s (2860) → D + K + D K + (cid:1) = 1140 πM D ∗ s G D ∗ s DK p × , = 28 . ± . , (39)Γ (cid:0) D ∗ s (2860) → D ∗ + K + D ∗ K + (cid:1) = 1105 π G D ∗ s D ∗ K p ′ × , = 16 . ± . , (40)where p = r λ (cid:16) M D ∗ s , M D , M K (cid:17) M D ∗ s = 709 MeV ,p ′ = r λ (cid:16) M D ∗ s , M D ∗ , M K (cid:17) M D ∗ s = 585 MeV . (41)If we saturate the decay width of the D ∗ s (2860) with the strong decays to the final states D + K , D K + , D ∗ + K , D ∗ K + , the total decay width is 44 . ± . ± . D ∗ s = (53 ± ± ±
6) MeV from the LHCb collaboration [13, 14]. The predicted ratio
R R = Γ ( D ∗ s (2860) → D ∗ K )Γ ( D ∗ s (2860) → DK ) = 0 . ± . , (42)which has minor overlap with the experimental value, R = Br ( D ∗ sJ (2860) → D ∗ K )Br ( D ∗ sJ (2860) → DK ) = 1 . ± . ± . , (43)12rom the BaBar collaboration [2] due to the uncertainties, while the central value is much smallerthan the experimental value. If we assign the D ∗ sJ (2860) to be the D ∗ s (2860), the theoreticalvalues R from the leading order heavy meson effective theory [3], the constituent quark modelwith quark-meson effective Lagrangians [5], the P model [6, 9, 19, 20, 21] and the pseudoscalaremission decay model [22] are much smaller than the experimental value, see Table 1. If we takeinto account the chiral symmetry breaking corrections, the experimental value can be reproducedwith suitable parameters in heavy meson effective theory [24]. In the present work, we cannotreproduce the experimental value R = 1 . ± . ± .
19 based on the QCD sum rules, and fail toobtain additional support for assigning the D ∗ sJ (2860) to be the D ∗ s (2860).We have two choices to reproduce the experimental value R = 1 . ± . ± .
19, one choiceis taking into account the chiral symmetry breaking corrections by fitting the revelent parametersin the heavy meson effective Lagrangians [24]; the other choice is introducing some D s (2860)and D ′ s (2860) components in the D ∗ sJ (2860) beyond the D ∗ s (2860) and the D ∗ s (2860). The J P = 2 − mesons D s (2860) and D ′ s (2860) decay only to the final states D ∗ K . If the D ∗ sJ (2860)consists of at least four resonances D ∗ s (2860), D s (2860), D ′ s (2860), D ∗ s (2860), the large ratio R = 1 . ± . ± .
19 is easy to account for, as the components D s (2860) and D ′ s (2860) canenhance the branching fraction Br ( D ∗ sJ (2860) → D ∗ K ) efficaciously. In this article, we assign the D ∗ s (2860) to be a D-wave c ¯ s meson, study the vertices D ∗ s (2860) DK and D ∗ s (2860) D ∗ K in details to select the pertinent tensor structures, then calculate the hadroniccoupling constants G D ∗ s (2860) DK and G D ∗ s (2860) D ∗ K with the three-point QCD sum rules. Finallywe obtain the partial decay widths Γ ( D ∗ s (2860) → D ∗ K ) and Γ ( D ∗ s (2860) → DK ), and the ratio R = Γ ( D ∗ s (2860) → D ∗ K ) / Γ ( D ∗ s (2860) → DK ) = 0 . ± .
38. The predicted ratio R = 0 . ± .
38 cannot reproduce the experimental value R = 1 . ± . ± .
19, although the theoretical andexperimental values overlap slightly with each other due to the uncertainties. Some components D s (2860) and D ′ s (2860) are needed to reproduce the experimental value, if one would like not toresort to the chiral symmetry breaking corrections to dispel the discrepancy. Appendix
The explicit expressions of the coefficients a , b , a , b , c , d , a , b , c , d , e , f and ∂∂m i f . = ∂∂m i f ( m A , m B , m c ) | m A =0; m B =0 ,∂ ∂m i ∂m j f . = ∂ ∂m i ∂m j f ( m A , m B , m c ) | m A =0; m B =0 ,∂ ∂m i ∂m j ∂m k f . = ∂ ∂m i ∂m j ∂m k f ( m A , m B , m c ) | m A =0; m B =0 , (44)with f ( m A , m B , m c ) = a ( m A , m B , m c ), b ( m A , m B , m c ), a ( m A , m B , m c ), b ( m A , m B , m c ), · · · , m i , m j , m k = m A , m B , m c . 13 d k δ = π p λ ( s, u, q ) , Z d k δ k µ = π p λ ( s, u, q ) (cid:2) a ( m A , m B , m c ) p µ + b ( m A , m B , m c ) p ′ µ (cid:3) , Z d k δ k µ k ν = π p λ ( s, u, q ) (cid:2) a ( m A , m B , m c ) p µ p ν + b ( m A , m B , m c ) p ′ µ p ′ ν + c ( m A , m B , m c ) (cid:0) p µ p ′ ν + p ′ µ p ν (cid:1) + d ( m A , m B , m c ) g µν (cid:3) , Z d k δ k µ k ν k ρ = π p λ ( s, u, q ) [ a ( m A , m B , m c ) p µ p ν p ρ + b ( m A , m B , m c ) ( p µ g νρ + p ν g µρ + p ρ g µν )+ c ( m A , m B , m c ) (cid:0) p ′ µ g νρ + p ′ ν g µρ + p ′ ρ g µν (cid:1) + d ( m A , m B , m c ) (cid:0) p ′ µ p ν p ρ + p ′ ν p µ p ρ + p ′ ρ p µ p ν (cid:1) + e ( m A , m B , m c ) (cid:0) p ′ µ p ′ ν p ρ + p ′ ν p ′ ρ p µ + p ′ ρ p ′ µ p ν (cid:1) + f ( m A , m B , m c ) p ′ µ p ′ ν p ′ ρ (cid:3) , (45) δ = δ (cid:2) k − m A (cid:3) δ (cid:2) ( k + p − p ′ ) − m B (cid:3) δ (cid:2) ( k − p ′ ) − m c (cid:3) , (46) a ( m A , m B , m c ) = 1 λ ( s, u, q ) (cid:2) m c ( u − s + q ) + u ( s − u + q ) − um B + m A ( u + s − q ) (cid:3) ,b ( m A , m B , m c ) = 1 λ ( s, u, q ) (cid:2) m c ( s − u + q ) + u ( u − s − q ) + q ( q − s ) − sm A + m B ( u + s − q ) (cid:3) , (47)14 ( m A , m B , m c ) = 1 λ ( s, u, q ) (cid:2) ( u − m c ) − m A ( u + m c ) (cid:3) + 6 uλ ( s, u, q ) (cid:8) q (cid:2) m c − ( u + s − q ) m c + su (cid:3) + m A m B ( q − u − s ) − m A (cid:2) s ( u − s + q ) + m c ( s − u + q ) (cid:3) − m B (cid:2) u ( s − u + q ) + m c ( u − s + q ) (cid:3)(cid:9) ,b ( m A , m B , m c ) = 1 λ ( s, u, q ) (cid:2) ( u − q − m c ) + 2 m B ( u − q − m c ) − sm A (cid:3) + 6 sλ ( s, u, q ) (cid:8) q (cid:2) m c − ( u + s − q ) m c + su (cid:3) + m A m B ( q − u − s )+ m A (cid:2) s ( s − u − q ) + m c ( u − s − q ) (cid:3) + m B (cid:2) u ( u − s − q ) + m c ( s − u − q ) (cid:3)(cid:9) ,c ( m A , m B , m c ) = 1 λ ( s, u, q ) (cid:2) ( u − m c )( m c + q − u ) + m B ( m c − u )+ m A ( m c − q − m B + 2 s + u ) (cid:3) − u + s − q ) λ ( s, u, q ) (cid:8) q (cid:2) m c − ( u + s − q ) m c + su (cid:3) + m A m B ( q − u − s ) − m B (cid:2) m c ( u − s + q ) + u ( s − u + q ) (cid:3) − m A (cid:2) m c ( s − u + q ) + s ( u − s + q ) (cid:3)(cid:9) ,d ( m A , m B , m c ) = 12 λ ( s, u, q ) (cid:8) q (cid:2) m c − ( u + s − q ) m c + su (cid:3) + m A m B ( q − u − s )+ m A (cid:2) s ( s − u − q ) + m c ( u − s − q ) (cid:3) + m B (cid:2) u ( u − s − q ) + m c ( s − u − q ) (cid:3)(cid:9) , (48) a (0 , , m c ) = 1 λ ( s, u, q ) (cid:8) ( m c − u ) ( u − s ) + 3( m c − u ) ( u − s )( u + 3 um c − us − sm c ) q − m c − u ) (cid:2) m c ( s − u ) + 6 um c ( s − u ) + u (3 s − u ) (cid:3) q + (cid:0) m c + 9 um c + 9 u m c + u (cid:1) q (cid:9) ,b (0 , , m c ) = 12 λ ( s, u, q ) (cid:8) ( s − m c )( m c − u ) ( s − u ) q + ( m c − u )( m c − sm c + 2 um c − su ) q + m c ( m c + u ) q (cid:9) ,c (0 , , m c ) = 12 λ ( s, u, q ) (cid:8) ( m c − s )( m c − u ) ( s − u ) q + ( m c − u ) (cid:2) m c − ( s + 3 u ) m c + s ( s + 2 u )] q + (cid:0) m c − sm c − um c + su (cid:1) q + m c q (cid:9) , (49) d (0 , , m c ) = 1 λ ( s, u, q ) (cid:8) ( m c − u ) ( s − u ) + ( m c − u ) ( s − u ) (cid:2) u + m c ( s + 5 u ) − us − s (cid:3) q + ( m c − u ) (cid:2) m c (3 u − s ) + m c (9 s − us − u ) + u (3 s + 13 us − u ) (cid:3) q + (cid:2) m c + m c (6 u − s ) − um c (2 s + 5 u ) + u (5 s − u ) (cid:3) q + (cid:0) m c + 6 um c + u (cid:1) q (cid:9) , (50)15 (0 , , m c ) = 1 λ ( s, u, q ) (cid:8) ( m c − u ) ( u − s ) + ( m c − u ) ( u − s ) (cid:2) u + m c ( u + 5 s ) − us − s (cid:3) q + (cid:2) m c (3 s − u ) + m c (9 u + 15 us − s ) − m c ( s + us + 10 u s − u )+ u (3 s + 9 us + 12 u s − u ) (cid:3) q + (cid:2) m c − m c (5 u + 2 s ) + 3 m c (3 s + 8 us + 2 u ) + u (10 u − us − s ) (cid:3) q + (cid:2) m c − m c ( u + s ) + u ( s − u ) (cid:3) q + ( u + 3 m c ) q (cid:9) , (51) f (0 , , m c ) = 1 λ ( s, u, q ) (cid:8) ( m c − u ) ( s − u ) + 3( m c − u ) ( u − s ) (cid:2) s + us − u + m c ( u − s ) (cid:3) q + 3 (cid:2) m c (3 s − u ) + m c (6 u − us − s ) + m c (3 s + 9 us + 12 u s − u )+ u (5 u − u s − us − s ) (cid:3) q + (cid:2) m c + 3 m c (5 s − u ) − m c (5 s + 6 us − u ) + 6 s u − su − u − s (cid:3) q +3 (cid:2) m c + m c ( s − u ) + s + 3 su + 5 u (cid:3) q + 3( m c − s − u ) q + q (cid:9) , (52) ∂f ∂m A = 6 sλ ( s, u, q ) (cid:8) ( m c − u )( s − u ) ( u + s − m c ) − (cid:2) s + 4 us + u s − u + m c (6 s − u )+ m c (9 u − s − us ) (cid:3) q + (cid:2) um c − m c − sm c + s − u − us (cid:3) q + (cid:0) s + 4 u − m c (cid:1) q − q (cid:9) ,∂f ∂m B = 3 λ ( s, u, q ) (cid:8) ( m c − u ) ( s − u ) (3 s + u ) + (cid:2) m c (3 s + 4 us − u )+ m c (2 u s − s − us + 8 u ) + u (6 s + 9 us − u s − u ) (cid:3) q + (cid:2) s − us + 12 u s + 10 u + m c (3 u − s ) + 2 m c (5 s + us − u ) (cid:3) q − (cid:2) m c + 2 m c ( s − u ) + 3 s + 10 u + 10 us (cid:3) q + (3 s + 5 u − m c ) q − q (cid:9) , (53) ∂ e ∂m A ∂m B = 2 λ ( s, u, q ) (cid:8) ( u − s ) (cid:2) s + 12 us + 15 u s + 2 u − m c (3 s + 6 us + u ) (cid:3) + (cid:2) s − us − u s − u + m c (9 u + 12 us − s ) (cid:3) q + (cid:2) s + 23 us + 12 u + 3 m c ( s − u ) (cid:3) q + (3 m c − s − u ) q + 2 q (cid:9) ,∂ e ∂m A ∂m c = 2 λ ( s, u, q ) (cid:8) − ( s − u ) (cid:2) s + 4 us + u ) − m c ( u + 3 s ) (cid:3) + (cid:2) s m c − s − us + 4 u s + 12 usm c + 8 u − u m c (cid:3) q + (cid:2) s − sm c + 4 us − u + 9 um c (cid:3) q + (8 u − s − m c ) q − q (cid:9) ,∂ e ∂m B ∂m c = 6 λ ( s, u, q ) (cid:8) u − m c )( s − u ) ( u + s )+ (cid:2) s + 3 us + 5 u s − u + 2 m c ( s − us + u ) (cid:3) q + (cid:2) m c ( u + s ) − s + 2 us − u ) (cid:3) q + ( u + 3 s − m c ) q − q (cid:9) , (54)16 f ∂m A ∂m B = 6 sλ ( s, u, q ) (cid:8) ( s − u ) (cid:2) s − sm c + 6 us + 3 u − um c (cid:3) + (cid:2) s + 8 us + 9 u + 2 m c ( s − u ) (cid:3) q + (cid:0) m c − s − u (cid:1) q + 3 q (cid:9) ,∂ f ∂m A ∂m c = 6 sλ ( s, u, q ) (cid:8) ( s − u ) ( s + 3 u − m c ) + (cid:0) s − sm c + 8 us − u + 8 um c (cid:1) q + (cid:0) u − s − m c (cid:1) q − q (cid:9) ,∂ f ∂m B ∂m c = 6 λ ( s, u, q ) (cid:8) ( m c − u )( s − u ) (3 s + u )+ (cid:2) u s − s − us + 4 u + m c (3 s + 4 us − u ) (cid:3) q + (cid:0) s − sm c + us − u + 3 um c (cid:1) q + (cid:0) u − s − m c (cid:1) q − q (cid:9) , (55)here we have neglected the terms m A and m B in the a , b , c and d as they are irreverent inpresent calculations. Acknowledgements
This work is supported by National Natural Science Foundation, Grant Numbers 11375063, andNatural Science Foundation of Hebei province, Grant Number A2014502017.
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