Masses and decay constants of the heavy tensor mesons with QCD sum rules
aa r X i v : . [ h e p - ph ] A ug Masses and decay constants of the heavy tensor mesons with QCDsum rules
Zhi-Gang Wang , Zun-Yan DiDepartment of Physics, North China Electric Power University, Baoding 071003, P. R. China Abstract
In this article, we calculate the contributions of the vacuum condensates up to dimension-6in the operator product expansion, study the masses and decay constants of the heavy tensormesons D ∗ (2460), D ∗ s (2573), B ∗ (5747), B ∗ s (5840) using the QCD sum rules. The predictedmasses are in excellent agreement with the experimental data, while the ratios of the decayconstants f D ∗ s f D ∗ ≈ f B ∗ s f B ∗ ≈ f Ds f D | exp , where the exp denotes the experimental value. PACS number: 13.20.Fc, 13.20.HeKey words: Decay constants, Tensor mesons, QCD sum rules
In recent years, the Babar, Belle, CLEO, CDF, D0, LHCb and BESIII collaborations have observed(or confirmed) many charmonium-like states and revitalized the interest in the spectroscopy of thecharmonium states [1]. There are also some progresses in the spectroscopy of the conventional heavymesons, several excited states have been observed, such as the D ∗ s (2700), D ∗ sJ (2860), D sJ (3040), D J (2580), D ∗ J (2650), D J (2740), D ∗ J (2760), D J (3000), D ∗ J (3000), B (5721), B ∗ (5747), B s (5830), B ∗ s (5840), B (5970), etc [2]. In 2007, the D0 collaboration firstly observed the B (5721) and B (5747) [3], later the CDF collaboration confirmed them [4]. Also in 2007, the CDF collaborationobserved the B s (5830) and B ∗ s (5840) [5]. The D0 collaboration confirmed the B ∗ s (5840) [6]. In2012, the LHCb collaboration updated the masses M B s = (5828 . ± . ± . ± .
41) MeV and M B ∗ s = (5839 . ± . ± . ± .
17) MeV [7]. The heavy-light mesons can be classified in doubletsaccording to the total angular momentum of the light antiquark ~s ℓ , ~s ℓ = ~s ¯ q + ~L , where the ~s ¯ q and ~L are the spin and orbital angular momentum of the light antiquark, respectively. Now the J Ps ℓ =(0 + , + ) doublets ( D ∗ (2400) , D ′ (2430)), ( D ∗ s (2317) , D ′ s (2460)), the J Ps ℓ = (1 + , + ) doublets( D (2420) , D ∗ (2460)), ( D s (2536) , D ∗ s (2573)), ( B (5721) , B ∗ (5747)), ( B s (5830) , B ∗ s (5840)) arecomplete [2].The QCD sum rules is a powerful nonperturbative theoretical tool in studying the ground statehadrons [8, 9, 10, 11]. In the QCD sum rules, the operator product expansion is used to expandthe time-ordered currents into a series of quark and gluon condensates which parameterize thelong distance properties. We can obtain copious information on the hadronic parameters at thephenomenological side by taking the quark-hadron duality [8, 9, 10, 11]. There have been manyworks on the J Ps ℓ = (0 − , − ) doublets with the full QCD sum rules [10, 11] (or Ref.[12] for recentworks), while the works on the J Ps ℓ = (0 + , + ) and (1 + , + ) doublets are few [13, 14, 15]. Forthe works on the QCD sum rules combined with the heavy quark effective theory, one can consultRefs.[16, 17]. In Ref.[14], H. Sundu et al study the masses and decay constants of the tensormesons D ∗ (2460) and D ∗ s (2573) with the QCD sum rules by taking into account the perturbativeterms and the mixed condensates in the operator product expansion, the contributions of thegluon condensate, three-gluon condensate and four-quark condensate are neglected. Neglectingthe vacuum condensates of dimension-4 and 6 impairs the predictive ability.In this article, we calculate the contributions of the vacuum condensates up to dimension-6 inthe operator product expansion consistently, study the masses and decay constants of the heavytensor mesons D ∗ (2460), D ∗ s (2573), B ∗ (5747), B ∗ s (5840) with the QCD sum rules. E-mail,[email protected].
In the following, we write down the two-point correlation functions Π µναβ ( p ) in the QCD sumrules, Π µναβ ( p ) = i Z d xe ip · ( x − y ) h | T n J µν ( x ) J † αβ ( y ) o | i | y =0 , (1) J µν ( x ) = iQ ( x ) (cid:18) γ µ ↔ ∂ ν + γ ν ↔ ∂ µ − e g µν ↔ ∂ (cid:19) q ( x ) , (2) ↔ ∂ µ = → ∂ µ − ← ∂ µ , e g µν = g µν − p µ p ν p , where Q = c, b and q = u, d, s , the tensor currents J µν ( x ) interpolate the heavy tensor mesons D ∗ (2460), D ∗ s (2573), B ∗ (5747) and B ∗ s (5840), respectively. In Ref.[18], Aliev and Shifman takethe tensor currents η µν ( x ), η µν ( x ) = 12 iq ( x ) (cid:16) γ µ ↔ D ν + γ ν ↔ D µ (cid:17) q ( x ) , (3)with D µ = ∂ µ − ig s G µ and the G µ is the gluon field, to study the light tensor mesons. Later,Reinders, Yazaki, Rubinstein take the tensor currents η µν ( x ), η µν ( x ) = iq ( x ) (cid:18) γ µ ↔ ∂ ν + γ ν ↔ ∂ µ − e g µν ↔ ∂ (cid:19) q ( x ) , (4)to study the light tensor mesons [19]. In Ref.[20], Bagan and Narison restudy the light tensormesons with the tensor currents η µν ( x ). For recent works on the light tensor mesons with theQCD sum rules, see Ref.[21]. We have two choice, i.e. we can choose either the partial derivative ∂ µ or the covariant derivative D µ in constructing the interpolating currents. The currents η µν ( x )with the covariant derivative D µ are gauge invariant, but blur the physical interpretation of the ↔ D µ as the angular momentum; on the other hand, the currents η µν ( x ) with the partial derivative ∂ µ arenot gauge invariant, but manifest the physical interpretation of the ↔ ∂ µ as the angular momentum.In this article, we will present the results come from the currents with both the partial derivativeand the covariant derivative.We can insert a complete set of intermediate hadronic states with the same quantum numbersas the current operators J µν ( x ) into the correlation functions Π µναβ ( p ) to obtain the hadronicrepresentation [8, 9]. After isolating the ground state contributions from the heavy tensor mesons,we get the following result, Π µναβ ( p ) = f T M T M T − p P µναβ + · · · , = Π( p )P µναβ , (5)where the decay constants f T are defined by h | J µν (0) | T ( p ) i = f T M T ε µν , (6)2he ε µν are the polarization vectors of the tensor mesons with the following properties [22],P µναβ = X λ ε ∗ µν ( λ, p ) ε αβ ( λ, p ) = e g µα e g νβ + e g µβ e g να − e g µν e g αβ ,p µ P µναβ = e g µν P µναβ = e g αβ P µναβ = g µν P µναβ = g αβ P µναβ = 0 , µναβ P µναβ . (7)Now, we briefly outline the operator product expansion for the correlation functions Π µναβ ( p )in perturbative QCD. We contract the quark fields in the correlation functions Π µναβ ( p ) with Wicktheorem firstly,Π( p ) = 15 P µναβ Π µναβ ( p )= − i µναβ Z d xe ip · ( x − y ) T r n Γ µν S ij ( x − y )Γ αβ S Qji ( y − x ) o | y =0 , (8)where Γ µν = i γ µ ↔ ∂∂x ν + γ ν ↔ ∂∂x µ − e g µν γ τ ↔ ∂∂x τ , Γ αβ = i γ α ↔ ∂∂y β + γ β ↔ ∂∂y α − e g αβ γ τ ↔ ∂∂y τ , (9) S ij ( x ) = iδ ij x π x − δ ij m q π x − δ ij h ¯ qq i + iδ ij xm q h ¯ qq i − δ ij x h ¯ qg s σGq i
192 + iδ ij x xm q h ¯ qg s σGq i − ig s G aαβ t aij ( xσ αβ + σ αβ x )32 π x + iδ ij x xg s h ¯ qγ µ t n q ¯ qγ µ t n q i − h ¯ q j σ µν q i i σ µν − h ¯ q j γ µ q i i γ µ + · · · , (10) S Qij ( x ) = i (2 π ) Z d ke − ik · x ( δ ij k − m Q − g s G nαβ t nij σ αβ ( k + m Q ) + ( k + m Q ) σ αβ ( k − m Q ) + g s D α G nβλ t nij ( f λβα + f λαβ )3( k − m Q ) − g s ( t a t b ) ij G aαβ G bµν ( f αβµν + f αµβν + f αµνβ )4( k − m Q ) + i h g s GGG i
48 ( k + m Q ) (cid:2) k ( k − m Q ) + 2 m Q (2 k − m Q ) (cid:3) ( k + m Q )( k − m Q ) + · · · ) ,f λαβ = ( k + m Q ) γ λ ( k + m Q ) γ α ( k + m Q ) γ β ( k + m Q ) ,f αβµν = ( k + m Q ) γ α ( k + m Q ) γ β ( k + m Q ) γ µ ( k + m Q ) γ ν ( k + m Q ) , (11) t n = λ n , the λ n is the Gell-Mann matrix, the i , j are color indexes(One can consult Refs.[9, 23] forthe technical details in deriving the full heavy quark and light quark propagators, respectively.);then compute the integrals both in the coordinate and momentum spaces; finally obtain the QCDspectral density through dispersion relation,Π( p ) = 1 π Z ∞ m Q ImΠ( s ) s − p = Z ∞ m Q ρ QCD ( s ) s − p . (12)3igure 1: The diagrams contribute to the mixed condensate h ¯ qg s σGq i .Figure 2: The diagrams contribute to the gluon condensate h α s GGπ i and three-gluon condensate h g s GGG i . Figure 3: The diagrams contribute to the four-quark condensate h ¯ qq i .4n Eq.(10), we retain the terms h ¯ q j σ µν q i i and h ¯ q j γ µ q i i originate from the Fierz re-ordering of the h q i ¯ q j i to absorb the gluons emitted from the heavy quark lines to form h ¯ q j g s G aαβ t amn σ µν q i i and h ¯ q j γ µ q i g s D ν G aαβ t amn i so as to extract the mixed condensate and four-quark condensate h ¯ qg s σGq i and g s h ¯ qq i , respectively. In Ref.[14], such contributions are neglected.We take quark-hadron duality below the continuum thresholds s and perform the Borel trans-form with respect to the variable P = − p to obtain the QCD sum rules, f T M T exp (cid:18) − M T M (cid:19) = 110 π Z s m Q ds ( s − m Q ) (3 s + 2 m Q ) + 10 m q m Q s ( s − m Q ) s exp (cid:16) − sM (cid:17) − m Q h ¯ qg s σGq i − m Q M ! + (cid:18) m q h ¯ qg s σGq i − g s h ¯ qq i (cid:19) m Q M ! exp − m Q M ! + m Q h α s GGπ i " − m Q M − M m Q ! exp − m Q M ! + m Q M + 29 m Q M ! Γ , m Q M ! + h g s GGG i π " − − m Q M ! exp − m Q M ! + m Q M + 163 m Q M ! Γ , m Q M ! , (13)Γ(0 , x ) = R ∞ dt t e − xt . In this article, we carry out the operator product expansion up to thevacuum condensates of dimension 6 in the leading order approximation. In Figs.1-3, we expressthe contributions of the mixed condensates, four-quark condensates, gluon condensates and three-gluon condensates in terms of Feynman diagrams, which are drawn up directly from Eqs.(8-11).In the Feynman diagrams, we use the solid and dashed lines to represent the light and heavyquark propagators, respectively. The perturbative contributions are consistent with that obtainedin Ref.[14] (also that in Refs.[18, 19, 20] in the limit m Q → m Q → − M h α s GGπ i exp − m Q M ! − h g s GGG i π
56 + m Q M ! exp − m Q M ! + h g s GGG i π m Q M
116 + m Q M ! Γ , m Q M ! , (14)should be added to the right side of the Eq.(13). In the leading order approximation, the pertur-bative terms of the QCD spectral densities are not modified, we take into account the additional5igure 4: The additional diagrams contribute to the gluon condensate h α s GGπ i and three-gluoncondensate h g s GGG i from the covariant derivative.Figure 5: The additional diagram contributes to the mixed condensate h ¯ qg s σGq i from the covariantderivative.contributions by the modified vertexes,Γ µν = i γ µ ↔ ∂∂x ν + γ ν ↔ ∂∂x µ − e g µν γ τ ↔ ∂∂x τ + 2 g s (cid:18) γ µ G ν ( x ) + γ ν G µ ( x ) − e g µν γ τ G τ ( x ) (cid:19) , Γ αβ = i γ α ↔ ∂∂y β + γ β ↔ ∂∂y α − e g αβ γ τ ↔ ∂∂y τ + 2 g s (cid:18) γ α G β ( y ) + γ β G α ( y ) − e g αβ γ τ G τ ( y ) (cid:19) , (15)where G µ ( x ) = x θ G θµ (0) + · · · and G α ( y ) = y θ G θα (0) + · · · = 0 in the fixed point gauge. Thecontributions are shown explicitly by the Feynman diagrams in Fig.4. There are no additionalcontributions come from the mixed quark condensate, i.e. we calculate the Feynman diagram inFig.5 and observe that the contribution is zero.Differentiate Eq.(13) with respect to M , then eliminate the decay constants f T , we obtainthe QCD sum rules for the masses of the tensor mesons. In this article, we take into account thecontributions come from the covariant derivative. The masses of the tensor mesons listed in the Review of Particle Physics are M D ∗ (2460) ± = (2464 . ± .
6) MeV, M D ∗ (2460) = (2461 . ± .
7) MeV, M D ∗ s (2573) = (2571 . ± .
8) MeV, M B ∗ (5747) =65743 ±
5) MeV, M B ∗ s (5840) = (5839 . ± .
20) MeV [2]. We can take the threshold parametersas s D ∗ = (8 . ± .
5) GeV , s D ∗ s = (9 . ± .
5) GeV , s B ∗ = (39 ±
1) GeV , s B ∗ s = (41 ±
1) GeV tentatively to avoid the contaminations of the high resonances and continuum states, the energygaps p s T − M T = (0 . − .
6) GeV, the contributions of the ground states are fully included.The quark condensates and mixed condensates are taken to be the standard values h ¯ qq i = − (0 . ± .
01 GeV) , h ¯ ss i = (0 . ± . h ¯ qq i , h ¯ qg s σGq i = m h ¯ qq i , h ¯ sg s σGs i = m h ¯ ss i , m =(0 . ± .
1) GeV at the energy scale µ = 1 GeV [11]. The quark condensate and mixed quarkcondensate evolve with the renormalization group equation, h ¯ qq i ( µ ) = h ¯ qq i ( Q ) h α s ( Q ) α s ( µ ) i , h ¯ ss i ( µ ) = h ¯ ss i ( Q ) h α s ( Q ) α s ( µ ) i , h ¯ qg s σGq i ( µ ) = h ¯ qg s σGq i ( Q ) h α s ( Q ) α s ( µ ) i and h ¯ sg s σGs i ( µ ) = h ¯ sg s σGs i ( Q ) h α s ( Q ) α s ( µ ) i .The values of the gluon condensate and three-gluon condensate are also taken to be the standardvalues h α s GGπ i = 0 .
012 GeV and h g s GGG i = 0 .
045 GeV [11].In the article, we neglect the small u , d quark masses and take the M S masses m c ( m c ) =(1 . ± . m b ( m b ) = (4 . ± .
03) GeV and m s ( µ = 2 GeV) = (0 . ± . M S massesfrom the renormalization group equation, m s ( µ ) = m s (2GeV) (cid:20) α s ( µ ) α s (2GeV) (cid:21) ,m c ( µ ) = m c ( m c ) (cid:20) α s ( µ ) α s ( m c ) (cid:21) ,m b ( µ ) = m b ( m b ) (cid:20) α s ( µ ) α s ( m b ) (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) , (16)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 [2]. In calculations, we take n f = 4and µ c ( b ) = 1(3) GeV for the charmed (bottom) tensor mesons. We choose the energy scales µ c and µ b for the charmed mesons and bottom mesons respectively based on the crude estimation, µ c = p M D − m c ≈ √ . − . GeV ≈ µ b = p M B − m b ≈ √ . − . GeV ≈ M D and M B are the masses of the ground states (of the pseudoscalar mesons), the m c and m b are constituent quark masses. The strong coupling constant α s ( µ ) in itself is not aphysical observable, but rather a quantity defined in the context of perturbation theory, whichenters predictions for experimentally measurable observables. We can extract the value of the α s ( µ ) from the experimental data at a special energy scale µ , then fit the parameter Λ with theexpressions of the α s ( µ ) from one-loop, two-loop, three-loop, or four-loop renormalization groupequations. The values of the α s ( µ ) from three-loop renormalization group equation are alreadycompatible with that from different determination [2], we prefer the expression in Eq.(16), not thecrude one-loop approximation.We impose the two criteria (pole dominance and convergence of the operator product expan-sion) on the charmed (or bottom) tensor mesons, and search for the optimal values of the Borelparameters. The threshold parameters, Borel parameters, pole contributions and the resultingmasses and decay constants are shown explicitly in Table 1. The pole contributions are about(45 − − − − (GeV ) s (GeV ) pole M T (GeV) f T (MeV) D ∗ (2460) 1 . − . . ± . − . ± .
09 182 ± D ∗ s (2573) 1 . − . . ± . − . ± .
09 222 ± B ∗ (5747) 4 . − . ± − . ± .
06 110 ± B ∗ s (5840) 5 . − . ± − . ± .
06 134 ± f D ∗ = (0 . ± . f D ∗ s = (0 . ± . f D ∗ = (0 . ± . f D ∗ s = (0 . ± . D ∗ (2460) → D + π − , D ∗ + π − , D π , D ∗ π ,D ∗ s (2573) + → D K + , D ∗ K + , D + K , D ∗ + K , D + s π , D ∗ + s π ,B ∗ (5741) → B + π − , B ∗ + π − , B π , B ∗ π ,B ∗ s (5840) → B + K − , B ∗ + K − , B ¯ K , B ∗ ¯ K , B s π , B ∗ s π . (17)The central values f D ∗ s f D ∗ = f B ∗ s f B ∗ = 1 . , (18)the heavy quark symmetry works well. Furthermore, the SU (3) breaking effects are compatiblewith the experimental data [24], f D s f D = 1 . ± . , (19)the approximation f D ∗ s f D ∗ ≈ f B ∗ s f B ∗ ≈ f Ds f D is reasonable.If we use the non-covariant currents instead of the covariant currents, the gluon condensate andthe three-gluon condensate in Eq.(14) have no contributions, the masses and the decay constantschange as δM D ∗ = −
27 MeV , δf D ∗ = 3 MeV ,δM D ∗ s = −
20 MeV , δf D ∗ s = 2 MeV ,δM B ∗ = −
14 MeV , δf B ∗ ≈ ,δM B ∗ s = −
10 MeV , δf B ∗ s ≈ . (20)There are effective cancelations among the contributions of the three-gluon condensate h g s GGG i from different Feynman diagrams, and among the contributions of the three-gluon condensate h g s GGG i and the four quark condensate g s h ¯ qq i . The masses and the decay constants remain8 .5 1.6 1.7 1.8 1.9 2.0 2.11.61.82.02.22.42.62.83.03.23.43.6 (I) M T ( G e V ) M (GeV ) Central value; Upper bound; Lower bound. (II) M T ( G e V ) M (GeV ) Central value; Upper bound; Lower bound. (III) M T ( G e V ) M (GeV ) Central value; Upper bound; Lower bound. (IV) M T ( G e V ) M (GeV ) Central value; Upper bound; Lower bound.
Figure 6: The masses of the tensor mesons with variations of the Borel parameters M , wherethe (I), (II), (III), (IV) denote the D ∗ (2460), D ∗ s (2573), B ∗ (5747), B ∗ s (5840), respectively.9 .5 1.6 1.7 1.8 1.9 2.0 2.104080120160200240280320360400 (I) f T ( M e V ) M (GeV ) Central value; Upper bound; Lower bound. (II) f T ( M e V ) M (GeV ) Central value; Upper bound; Lower bound. (III) f T ( M e V ) M (GeV ) Central value; Upper bound; Lower bound. (IV) f T ( M e V ) M (GeV ) Central value; Upper bound; Lower bound.
Figure 7: The decay constants of the tensor mesons with variations of the Borel parameters M ,where the (I), (II), (III), (IV) denote the D ∗ (2460), D ∗ s (2573), B ∗ (5747), B ∗ s (5840), respectively.10lmost unchanged if the dimension-6 vacuum condensates are neglected, δM D ∗ ≈ , δf D ∗ ≈ ,δM D ∗ s ≈ , δf D ∗ s ≈ ,δM B ∗ ≈ , δf B ∗ ≈ ,δM B ∗ s ≈ , δf B ∗ s ≈ . (21)In this article, we neglect the perturbative α s corrections. In the massless limit, taking intoaccounting the perturbative α s corrections amounts to multiplying the perturbative terms by afactor (cid:0) − α s π (cid:1) [19]. Now, we estimate the perturbative α s contributions by multiplying theperturbative terms by the factor (cid:0) − α s π (cid:1) , which leads to the following changes, δM D ∗ ≈ , δf D ∗ = −
15 MeV ,δM D ∗ s ≈ , δf D ∗ s = −
18 MeV ,δM B ∗ ≈ , δf B ∗ = − ,δM B ∗ s ≈ , δf B ∗ s = − . (22)In calculations, we observe that the masses decrease monotonously with increase of the energyscales while the decay constants increase monotonously with increase of the energy scales. If weenlarge the energy scales by µ c ( b ) → µ c ( b ) + 300 MeV, then δM D ∗ = −
44 MeV , δf D ∗ = 27 MeV ,δM D ∗ s = −
46 MeV , δf D ∗ s = 27 MeV ,δM B ∗ = −
37 MeV , δf B ∗ = 11 MeV ,δM B ∗ s = −
36 MeV , δf B ∗ s = 12 MeV , (23)the changes are sizeable, but they are small compared to the energy scale augment 300 MeV. Thecorrelation functions Π( p ) can be written asΠ( p ) = Z s m Q ( µ ) ds ρ QCD ( s, µ ) s − p + Z ∞ s ds ρ QCD ( s, µ ) s − p , (24)through dispersion relation at the QCD side, and they are scale independent, ddµ Π( p ) = 0, whichdoes not amount to ddµ Z s m Q ( µ ) ds ρ QCD ( s, µ ) s − p → , (25)as the perturbative corrections to all orders are neglected and truncations s set in. The correlationbetween the threshold m Q ( µ ) and continuum threshold s is unknown. We cannot obtain energyscale independent QCD sum rules, but we can choose the reasonable energy scales based on sometheoretical analysis. In this article, we calculate the contributions of the vacuum condensates up to dimension-6 in theoperator product expansion, study the masses and decay constants of the heavy tensor mesonsusing the QCD sum rules. The predicted masses of the D ∗ (2460), D ∗ s (2573), B ∗ (5747), B ∗ s (5840)are in excellent agreement with the experimental data, while the ratios of the decay constants f D ∗ s f D ∗ ≈ f B ∗ s f B ∗ ≈ f Ds f D | exp , where the exp denotes the experimental value. The decay constants canbe taken as basic input parameters in studying the strong decays with the three-point QCD sumrules or the light-cone QCD sum rules. 11 cknowledgements This work is supported by National Natural Science Foundation, Grant Numbers 11375063, theFundamental Research Funds for the Central Universities, and Natural Science Foundation of Hebeiprovince, Grant Number A2014502017.
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