The ground states and the first radially excited states of D-wave vector ρ and φ mesons
aa r X i v : . [ h e p - ph ] F e b The ground states and the first radially excited states of D-wave vector ρ and φ mesons Guo-Liang Yu , ∗ Zhi-Gang Wang , † Xiu-Wu Wang , and Hui-Juan Wang Department of Mathematics and Physics, North China Electric power university,Baoding 071003, People’s Republic of China (Dated: February 23, 2021)In this article, we firstly derive two QCD sum rules QCDSR I and QCDSR II which are respectivelyused to extract observable quantities of the ground states and the first radially excited states ofD-wave vector ρ and φ mesons. In our calculations, we consider the contributions of vacuumcondensates up to dimension-7 in the operator product expansion. The predicted masses for 1 D ρ meson and 2 D φ meson are consistent well with the experimental data of ρ (1700) and φ (2170).Besides, our analysis indicates that it is reliable to assign the recent reported Y (2040) state as the2 D ρ meson. Finally, we obtain the decay constants of these states with QCDSR I and QCDSR II.These predictions are helpful not only to reveal the structure of the newly observed Y (2040) statebut also to establish φ meson and ρ meson families. PACS numbers: 13.25.Ft; 14.40.Lb
Recently, BESIII Collaboration reported two resonant structures in measuring born cross sectionsfor the processes e + e − → ωη and e + e − → ωπ for center-of-mass energies between 2 .
00 and 3 . e + e − → ωη , a resonance with a mass of (2179 ± ± /c and a widthof (89 ± ± . σ . This structure is consistent with theproperties of φ (2170) resonance. Another structure with a mass of (2034 ± ± /c , widthof (234 ± ± )MeV was observed in the ωπ cross section. By analyzing its mass and decay width,BESIII Collaboration assigned this structure to be the ρ (2000) state, which had been suggested to bea 2 D ρ meson by Regge trajectory analysis[2, 3]. ρ (2000) was first observed in the pp → ππ process with a mass of 1988 MeV[4]. Later, its existencewas also confirmed in the processes pp → ωηπ and pp → ωπ [2, 5–7]. By analyzing its decay propertiesand Regge trajectory, scientists suggested ρ (2000) was the first radial excitation of ρ (1700)[2, 3].However, ρ (2150) was also predicted to be the 2 D state with 2 .
15 GeV by Godfrey and Isgur[8],which is contradictory with this above conclusion about ρ (2000). In ref.[9], Bugg speculated ρ (2000) ∗ Electronic address: [email protected] † Electronic address: [email protected] to be a mixed state with a significant component of D . In Ref.[10], the masses of ρ (1700) and ρ (2000) were predicted to be about 1 . . ∼
100 MeV. In order to clarify all of these questions, it is necessary to make a theoreticalanalysis about the ground state and the first radially excited state of ρ meson. φ (2170) state was also denoted as X (2170) or Y (2170) in some literatures and it was observed bythe BARBAR Collaboration in the process e + e − → γφf [11]. Later, its existence was confirmed byBelle[12], BESII[13] and BESIII[14, 15] Collaborations. Since the observation of φ (2170), scientistshave made considerable efforts to understand its natrue and have proposed many interpretations suchas a conventional 3 S or 2 D ss state[16–23], an ssg hybrid[24, 25], a tetraquark state[26–35], a ΛΛbound state[36–39], as well as φKK [40, 41] and φf [42] resonances. Recently, Particle Data Groupcategorized it as a vector ss meson with a mass of 2 .
16 GeV[43]. In addition, several collaborationshave predicted its mass as a 2 D ss meson[8, 10, 16, 20, 44]. However, the ground state of D-wavevector ss meson 1 D is still not established in experiments. Finally, the predicted masses of 1 D and2 D φ mesons by different collaborations were not consistent well with each other[8, 10, 16, 20, 44].In order to further confirm the inner structure of ρ (2000) and φ (2170), we calculate the masses andthe decay constants of 1 D and 2 D states of vector D-wave ρ and φ mesons based on QCD sumrules. QCD sum rules proved to be a most powerful non-perturbative method in studying the groundstate hadrons and it has been widely used to analyze the masses, decay constants, form factors andstrong coupling constants, etc[45, 46]. There have been many reports about its applications to theground state with spin-parity J P C = 0 ± , 1 ± , 2 + , 3 − mesons[47–54], while efforts on the excited statesare few[55, 56]. In this article, we assign the ρ (2000) and φ (2170) as the first radially excited statesof D-wave vector ρ and φ mesons, study their masses and decay constants with the full QCD sumrules in detail. In our calculations, we consider the contributions of the vacuum condensates up todimension-7 in the operator expansion.The layout of this paper is as follows: in Sec.2, we derive QCD sum rules I(QCDSR I) and II(QCDSRII) from two-point correlation function. QCDSR I and II are respectively used to analyze propertiesabout the ground states of 1 D ( ρ ), 1 D ( φ ) mesons and about the first radially excited states of2 D ( ρ ), 2 D ( φ ) mesons. Then we present the numerical results in Sec.3; and Sec.4 is reserved forour conclusions. ρ (1 D , D ) and φ (1 D , D ) In order to obtain the hadronic observables such as masses, decay constants, we begin our studywith the following two-point correlation function,Π µν ( p ) = i Z d D xe ip ( x − y ) D | T n J µ ( x ) J † ν ( y ) o | E(cid:12)(cid:12) y → = Π( p )( − g µν + p µ p ν p ) + Π ( p ) p µ p ν p (1)where T is the time ordered product and J is the interpolating current of vector ρ or φ meson. Theinterpolating current is a composite operator with the same quantum numbers as the studied hadrons.The current of D-wave vector mesons can be written as, J µ ( x ) = q ( x ) (cid:16) γ α ←→ D µ ←→ D α + γ µ ←→ D α ←→ D α + γ α ←→ D α ←→ D µ (cid:17) q ( x ) (2) J † ν ( y ) = q ( y ) (cid:16) γ β ←→ D µ ←→ D β + γ ν ←→ D β ←→ D β + γ β ←→ D β ←→ D ν (cid:17) q ( y ) (3)with ←→ D µ = ( −→ ∂ µ − ig s G µ ) − ( ←− ∂ µ + ig s G µ ), where D µ , ∂ µ are the covariant derivative, partial derivativeand G µ is the gluon field. This current can be decomposed into two parts, J µ ( x ) = η µ ( x ) + J Vµ ( x ) (4)where η µ ( x ) = q ( x ) (cid:16) γ α ←→ ∂ µ ←→ ∂ α + γ µ ←→ ∂ α ←→ ∂ α + γ α ←→ ∂ α ←→ ∂ µ (cid:17) q ( x ) (5) J Vµ ( x ) = − iq ( x ) h γ α (cid:16) g s G µ ←→ ∂ α + g s G α ←→ ∂ µ − ig s G µ G α (cid:17) + γ µ (cid:16) g s G α ←→ ∂ α + g s G α ←→ ∂ α − ig s G α G α (cid:17) + γ α (cid:16) g s G α ←→ ∂ µ + g s G µ ←→ ∂ α − ig s G α G µ (cid:17)i q ( x ) (6)and ←→ ∂ µ = −→ ∂ µ − ←− ∂ µ .This means we have two choices in contructing the interpolating currents of vector D-wave mesons,Eq.(5) with the partial derivative ∂ µ [57, 58], and Eq.(2) with the covariant derivative D µ [59]. FromRef.[60], we can see that these two currents lead to little differences in the final results in heavymesons QCD sum rules. To study the effect on light meson, we will present the results coming fromthe currents with both the partial derivative and the covariant derivative.To extract hadronic observables, the correlation will be calculated in two different ways. On onehand, it is identified as as a hadronic propagator, called Phenomenological(Physical) side. On theother hand it is called the QCD side, where it is treated in the framework of the operator productexpansion(OPE), and the short and long distance quark-gluon interactions are separated. Then, theresult of the QCD calculation is matched, via dispersion relation, to a sum over hadronic states. Thesum rule obtained in this way allows to calculate not only observables of the hadronic ground statebut also those of excited states. We now describe the first step in the sum rule derivation how the Phenomenological side of thecorrelation is done. In this step, a complete set of intermediate hadronic states with the same quantumnumbers as the current operators J µ ( x ) is inserted into the correlation Π µν ( p )[45, 46]. It should benoticed that the interpolating currents J µ ( x ) couples not only to the ground states of hadrons, butalso to their excited states with the same quark contents and quantum numbers, h | J µ (0) | ρ/φ (1 D ) i = f M ε µ , h | J µ (0) | ρ/φ (2 D ) i = f M e ε µ (7)where f i and M i are the decay constants and the masses of 1 D and 2 D states of ρ or φ meson,and ε µ , e ε µ are their polarization vectors with the following properties, e g µν = X λ ε ∗ µ ( λ, p ) ε ν ( λ, p ) = − g µν + p µ p ν p (8)In correlation(1), the component Π( p ) denotes the contributions of the 1 D and 2 D state of vector ρ or φ meson, while Π ( p ) comes from the contributions of scalar mesons. Considering the properties p µ e g µν = p ν e g µν = 0, contributions from 1 D and 2 D states can be extracted out by the followingprojection method, Π( p ) = 13 e g µν Π µν ( p ) (9)After isolating the pole terms of the ground state and the first radially excited state, we obtain thefollowing results, Π H ( p ) = f M M − p + f M M − p · · · = Π ρ/φ (1 D ) ( p ) + Π ρ/φ (2 D ) ( p ) · · · (10)where · · · stands for the contributions of the higher resonances and continuum states, and the subscript H denotes the hadron side of the correlation functions. Then, we obtain the hadronic spectral densitiesfrom the imaginary part of correlation function, ρ H D ( s ) = lim ǫ → Im Π ρ/φ (1 D ) ( s + iǫ ) π = f M δ ( s − M ) ρ H D ( s ) = lim ǫ → Im Π ρ/φ (2 D ) ( s + iǫ ) π = f M δ ( s − M ) (11) We now turn to the next important step in sum rule derivation and describe how the QCD calculationof the correlation function (1) is done. The correlation function can be approximated at very large P = − p by contract all quark fields in Eq.(1) with Wick’s theorem. Combined with Eq.(9), thecorrelation function can be written as,Π QCD ( p ) = − i e g µν Z d xe ip. ( x − y ) × n S ji ( y − x )Γ µ ( x ) S ij ( x − y )Γ ν ( y ) o | y =0 (12)where Γ µ ( x ) and Γ ν ( y ) are the vertexes,Γ µ ( x ) = γ α ←→ D µ ←→ D α + γ µ ←→ D α ←→ D α + γ α ←→ D α ←→ D µ Γ ν ( y ) = γ β ←→ D µ ←→ D β + γ ν ←→ D β ←→ D β + γ β ←→ D β ←→ D ν and S are the quark propagators in the coordinate or momentum spaces which can be written as, S ijq ( x ) = i x/ π x δ ij − m q π x δ ij − h qq i (cid:16) − i m q x/ (cid:17) − x m h qq i (cid:16) − i m q x/ (cid:17) (13) − ig s λ ija G aθη π x h x/σ θη + σ θη x/ i + · · · ,S ijQ ( x ) = i (2 π ) Z d ke − ik.x n δ ij k/ − m Q − g s G cαβ t cij σ αβ ( k/ + m Q ) + ( k/ + m Q ) σ αβ ( k − m Q ) − g s ( t a t b ) ij G aαβ G bµν ( f αβµν + f αµβν + f αµνβ )4( k − m Q ) (14)+ 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 ) · · · o where 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 ) (15)In the fixed point gauge, G µ ( x ) = x θ G θµ (0) + · · · and G α ( y ) = y θ G θα (0) + · · · | y =0 = 0. Thus, forthe vertex Γ ν ( y ), we can get G α ( y ) = G µ ( y ) = 0, there are no gluon lines associated with the vertexat the point y = 0. After completing the integrals both in the coordinate and momentum spaces, weobtain the QCD spectral density through the imaginary part of the correlation, ρ QCD ( s ) = lim ǫ → Im Π QCD ( s + iǫ ) π (16)After lengthy derivation, we find there is no contribution of the condensate terms h qq i , h qg s σGq i and h qq i . For φ meson, its spectral densities of perturbative and non-perturbative terms are listed inEqs.(17) − (19), where ρ Vφ − nonpert ( s ) denotes the contributions from vertex and ρ ηφ − nonpert ( s ) representsthe other contributions in the non-perturbative terms. For ρ meson, because its spectral densities ofnon-perturbative terms are tediously lengthy, we only list its perturbative part in Eq.(20), ρ φ − pert ( s ) = 14 π × ( s − m ) ( s + 18 m ) √ s (17) ρ ηφ − nonpert ( s ) = h p s ( s − m )(162 m + 421 m s − s )36 s + m (320 m − m s + 241 s )10 s i ×h α s GGπ i + h − m + 15972 m s + 7804 m s − m s + 693 s π s (4 m − s ) p s ( s − m ) (18)+ 368 m − m s + 49 s π s i × h g s GGG i − h
179 + 87 m T + 7 m T + m T + m T i × m < qqGG > ρ Vφ − nonpert ( s ) = h − m + s (3 s − m )4 p s ( s − m ) − m s (5 s − m )9[ s ( s − m )] + 3 s ( s − m )4 p s ( s − m ) − m s ( s − m )3[ s ( s − m )] i × h α s GGπ i + 13 h s + 12 m p s ( s − m ) + m − s s + m − m s − s s − m ) i × h g s GGG i (19) ρ ρ − pert ( s ) = p [ s − ( m − m ) ][ s − ( m + m ) ]8 π s × h + 32 m m + 16 m ( m − s ) − m (4 m + 13 m s )+ m ( − m − m s + 17 s ) − m (4 m + 18 m s − m s ) (20)+ m (16 m − m s + 46 m s + s ) + 2 m (16 m − m s + 44 m s − m s )+16 m − m s + 17 m s + m s − s i Using dispersion relations, the correlation function can be written as the following form,Π
QCD ( p ) = Z ∞ ( m + m ) ρ QCD ( s ) s − p ds (21)We take quark-hadron duality and perform the Borel transformation to the Phenomenological side aswell as the QCD side to obtain the QCD sum rules, f M exp h − M T i = Z s ( m + m ) (cid:2) ρ H D ( s ) + ρ H D ( s ) (cid:3) exp h − sT i ds = Z s ( m + m ) ρ QCD ( s ) exp h − sT i ds (22) f M exp h − M T i + f M exp h − M T i = Z s ′ ( m + m ) (cid:2) ρ H D ( s ) + ρ H D ( s ) (cid:3) exp h − sT i ds = Z s ′ ( m + m ) ρ QCD ( s ) exp h − sT i ds (23)In Eq.(22), s is the continuum threshold parameter, which separates the contribution of the groundstate 1 D from those of the higher resonances and continuum states. While s ′ separates the contri-bution of 1 D plus 2 D states from the higher resonances and continuum states. By differentiatingEq.(22) with respect to T , and eliminating the decay constant f , we obtain, M = − dd (1 /T ) R s ( m + m ) ρ QCD ( s ) exp h − sT i ds R s ( m + m ) ρ QCD ( s ) exp h − sT i ds (24)After the mass M is obtained, it is treated as a input parameter to obtain the decay constant fromQCD sum rule from Eq.(22), f = R s ( m + m ) ρ QCD ( s ) exp h − sT i dsM exp h − M T i (25)In the following, we will refer to the QCD sum rules in Eq.(24) and Eq.(25) as QCDSR I.We introduce the notations τ = T , D n = ( − ddτ ) n and Π ′ QCD ( τ ) = R s ′ ( m + m ) ρ QCD ( s ) exp h − τ s i ds for simplicity. With these substitutions, Eq(23) can be written as, f M exp h − τ M i + f M exp h − τ M i = Π ′ QCD ( τ ) (26)Then, deriving the QCD sum rules in Eq.(26) with respect to τ , we obtain, f M exp h − τ M i + f M exp h − τ M i = D Π ′ QCD ( τ ) (27)From Eq.(26) and Eq.(27), we can obtain the following relation, f i exp ( − τ M i ) = ( D − M j )Π ′ QCD ( τ ) M i ( M i − M j ) (28)After deriving with respect to τ in Eq.(28), we obtain the following two relations, M i = ( D − M j D )Π ′ QCD ( τ )( D − M j )Π ′ QCD ( τ ) (29) M i = ( D − M j D )Π ′ QCD ( τ )( D − M j )Π ′ QCD ( τ ) (30)From these two relations, we can get a equation about the squared masses M i , M i − bM i + c = 0 (31)where b = D ⊗ D − D ⊗ DD ⊗ D − D ⊗ D , (32) c = D ⊗ D − D ⊗ D D ⊗ D − D ⊗ D , (33) D j ⊗ D k = D j Π ′ QCD ( τ ) D k Π ′ QCD ( τ ) , (34)the indexes i = 1 , j, k = 0 , , ,
3. By solving the equation in Eq.(31) analytically, we finallyobtain two solutions[61–63], M = b − √ b − c M = b + √ b − c M and M represent the masses of the ground state and the first radially excitedstate. Because the ground state mass in Eq.(35) suffers from additional uncertainties from the firstradially excited state 2 D , we neglect this result and still use QCDSR I to obtain the mass and decayconstant of ground state. In the following, we will refer to the QCD sum rules in Eq.(28) and Eq.(36)as the QCDSR II which is used to analyze the properties of the first radial excitation. In the QCD side, we take the the vacuum condensates to be the standard values h qq i = − (0 . ± . , h ss i = (0 . ± . h qq i , h α s GGπ i = (0 . ± . , h g s GGG i = 0 .
045 GeV [45, 46, 64].And the masses of quarks are taken to be m u = 2 . +0 . − . MeV, m d = 4 . +0 . − . MeV and m s = 93 +11 − MeV from the Particle Data Group[43]. The final results also depend on two parameters, the Borelmass parameter T and continuum threshold s ( s ′ ). In order to choose the working interval ofthe parameters T and s ( s ′ ), two criteria should be satisfied which are pole dominance and OPEconvergence. That is to say, the pole contribution should be as large as possible(larger than 40%)comparing with the contributions of the high resonances and continuum states. Meanwhile, we shouldalso find a plateau, which will ensure OPE convergence and the stability of the final results. Theplateau is often called Borel window. (GeV ) C on t r i bu t i on s ABCDEF
FIG. 1: The contributions of different condensateterms in the OPE with variations of the Borelparameters T for ρ (1 D ) meson, where A-Fdenote perturbative term, h α s GGπ i η µ , h α s GGπ i J Vµ , h g s GGG i η µ , h g s GGG i J Vµ , and h qqGG i . (GeV ) C on t r i bu t i on s ABCDEF
FIG. 2: The contributions of different condensateterms in the OPE with variations of the Borelparameters T for ρ (2 D ) meson, where A-Fdenote perturbative term, h α s GGπ i η µ , h α s GGπ i J Vµ , h g s GGG i η µ , h g s GGG i J Vµ , and h qqGG i . The contributions from different parts of spectra density are defined as ρ Di P ρ Di , where D i representsthe dimension of condensate terms. In Figs.1-4, we show the dependence of these condensate terms onBorel parameter T . From these figures, we can see good OPE convergence for both ρ and φ mesons. (GeV ) C on t r i bu t i on s ABCDEF
FIG. 3: The contributions of different condensateterms in the OPE with variations of the Borelparameters T for φ (1 D ) meson, where A-Fdenote perturbative term, h α s GGπ i η µ , h α s GGπ i J Vµ , h g s GGG i η µ , h g s GGG i J Vµ , and h qqGG i . (GeV ) C on t r i bu t i on s ABCDEF
FIG. 4: The contributions of different condensateterms in the OPE with variations of the Borelparameters T for φ (2 D ) meson, where A-Fdenote perturbative term, h α s GGπ i η µ , h α s GGπ i J Vµ , h g s GGG i η µ , h g s GGG i J Vµ , and h qqGG i . When the Borel parameters are larger than 1 . for ρ (1 D ), ρ (2 D ) and φ (1 D ) and largerthan 1 . for φ (2 D ), the results are rather stable with variations of the Borel parameters andthe contributions from perturbative term is larger than 60%(see Table I).The threshold parameters s and s ′ are used to include the ground state and the ground stateplus the first radially excited state, respectively. Commonly, these parameters are chosen to be √ s = M + ∆ and p s ′ = M + ∆ . In order to include contributions from the ground state andto exclude the contaminations from the first radial excitation, the value of ∆ should be little thanthe gap between 1 D state and 2 D state. For ρ meson as an example, the masses of 1 D stateand 2 D state are predicted to be about 1 . . is ∼ . D state which waspredicted to be (234 ± ±
25) MeV in reference[1]. Thus, the value of ∆ for ρ meson should belittle than 0 . D ρ meson on Borel parameters T in different values of ∆ , where √ s = M + ∆ = 1 . . . . . . . . . . , we can see from Fig.5that the curve with √ s = M + ∆ = 1 . . . √ s = 1 . √ s = 2 . √ s is taken to be 1 . . . D . Combined with these above considerations, the working regionfor the Borel parameter and continuum threshold for ρ (1 D ) meson are determined to be 1 . − . and √ s = 1 . (GeV ) M a ss ( G e V ) s =1.8 GeV s =1.9 GeV s =2.0 GeV FIG. 5: The results for ρ (1 D ) meson on Borelparameter T , with different values of s inQCDSR I. (GeV ) M a ss ( G e V ) s’ =2.0 GeV s’ =2.1 GeV s’ =2.2 GeV FIG. 6: The results for ρ (2 D ) meson on Borelparameter T , with different values of s ′ inQCDSR II. In reference[3, 65], the mass and decay width of the second radial excitation ρ (3 D ) were predictedto be ∼ .
27 GeV and ∼
300 MeV. The energy gap between the first and the second radially excitedstates is about 250 MeV which suggests the continuum threshold parameter for ρ (2 D ) in QCDSRII is p s ′ ≈ . .
25 = 2 .
25 GeV. Considering the width of the second radially excited state, thevalue of p s ′ in QCDSR II should be little than 2 .
25 GeV. From Fig.6, we can see that the resultsare more stable with continuum parameter p s ′ = 2 . p s ′ = 2 . p s ′ = 2 . s ′ will lead to contaminationsfrom the second radially excited state and too small values can not totally include contribution ofthe first radially excitation. This above conclusion about ρ meson is also applicable to φ meson. Inreference[8, 10, 16, 44], we know that the masses of the ground state, the first and the second radiallyexcited states of vector φ meson are ∼ .
87 GeV, ∼ . ∼ . ∼
400 MeV, ∼
200 MeV and ∼
200 MeV respectively. Combining with results obtainedin different values of s ( s ′ ) which are shown in Figs.7-8, we determine the working regions of Borelparameter T and thresholds s ( s ′ ) and also present them in Table I. TABLE I: The Borel parameters T and continuum threshold parameters s ( s ′ ) for ρ and φ meson, where thepole stands for the pole contributions from the ground states or the ground states plus the first radially excitedstates, and the perturbative stands for the absolute value of contributions from the perturbative terms.State T (GeV ) √ s (GeV) pole perturbative ρ (1 D ) 1 . − . . −
44% 61 − ρ (2 D ) 1 . − . . −
50% 67 − φ (1 D ) 1 . − . . −
46% 62 − φ (2 D ) 1 . − . . −
50% 69 − (GeV ) M a ss ( G e V ) s =2.0 GeV s =2.1 GeV s =2.2 GeV FIG. 7: The results for φ (1 D ) meson on Borelparameter T , with different values of s inQCDSR I. (GeV ) M a ss ( G e V ) s’ =2.3 GeV s’ =2.4 GeV s’ =2.5 GeV FIG. 8: The results for φ (2 D ) meson on Borelparameter T , with different values of s ′ inQCDSR II. After two criteria of QCD sum rules are both satisfied, we can easily extract physical quantities.Before these extraction, we give a simple discussion about effects of different currents defined byEqs.(2) and (5) on the final results. It can be seen from Figs.9-12 that the contribution of vertex J Vµ in J µ has a significant influence on the masses. These figures clearly show that the values coming fromcurrent J µ = η µ + J Vµ are more stable in the Borel window than those coming from η µ . That is, we cannot obtain ideal Borel window if contribution coming from vertex J Vµ was not considered. This effectcan easily be explained according to Figs.1-4, which show that the contributions of vertex J Vµ mainlyoriginate from the condensate term h α s π GG i J Vµ . This condensate term from vertex is at the sameorder of magnitude with h α s π GG i and its contribution makes up 5% −
8% of the total contributions.Thus, its influence on the final results is obvious. In addition, contribution of h g s GGG i J Vµ is oneorder of magnitude lower than that of h α s π GG i J Vµ , but its contribution is comparable with that of h g s GGG i . Thus, all these condensate terms coming from vertex play an important role and shouldnot be neglected in light meson QCD sum rules. TABLE II: The masses of ground state and the first radial excited state of vector D-wave ρ and φ mesons.All values in units of GeV.State This work GI Ebert MGI Bug ERT Exp φ (1 D ) 1 . ± .
095 1 . . . − . − φ (2 D ) 2 . ± .
075 2 . . . − . . ± . ρ (1 D ) 1 . ± .
040 1 . . − . − . ± . ρ (2 D ) 2 . ± .
044 2 . . − . − . ± . Finally, we can easily obtain the values of the masses for ρ (1 D ), ρ (2 D ), φ (1 D ) and φ (2 D )states, which are shown in Figs.13-16 and Table II. After taking into account the uncertainties of the2 (GeV ) M a ss ( G e V ) J µ = η µ +J V µ (1 D ) η µ (1 D ) FIG. 9: The mass of ρ (1 D ) state for differentvalues of the Borel parameter T , where the cur-rents J µ and η µ are considered respectively. (GeV ) M a ss ( G e V ) J µ = η µ +J V µ (2 D ) η µ (2 D ) FIG. 10: The mass of ρ (2 D ) state for differ-ent values of the Borel parameter T , where thecurrents J µ and η µ are considered respectively. (GeV ) M a ss ( G e V ) J µ = η µ +J V µ (1 D ) η µ (1 D ) FIG. 11: The mass of φ (1 D ) state for differ-ent values of the Borel parameter T , where thecurrents J µ and η µ are considered respectively.. (GeV ) M a ss ( G e V ) J µ = η µ +J V µ (2 D ) η µ (2 D ) FIG. 12: The mass of φ (2 D ) state for differ-ent values of the Borel parameter T , where thecurrents J µ and η µ are considered respectively. input parameters, the uncertainties of the masses are also presented in Figs.13-16 which are markedas the Upper bound and Lower bound. Using these predicted masses as input parameters, we alsoobtain the decay constants from Eq.(22) and Eq.(28), which are presented in Figs.17-20 and Eq.(37).These estimated decay constants can be used to study the strong decay properties involving ρ (1 D ), ρ (2 D ), φ (1 D ) and φ (2 D ) with three-point QCD sum rules or light-core QCD sum rules.There is no doubt that ρ (1700) is a good candidate of 1 D ρ meson[43]. On this base, scientistsestimated the mass of 2 D ρ meson to be ∼ .
000 GeV[3, 65]. From Table II, we can see that thepredicted masses in this work for ρ (1 D ) and ρ (2 D ) are 1 . ± .
040 GeV and 2 . ± . ωπ cross section by BESIII collaboration which wasdenoted as Y (2040)[1]. This structure was predicted to be the 2 D ρ meson by its mass (2034 ± ± /c and decay width (234 ± ± )MeV. Our predicted mass about this state is consistent well3 (GeV ) M a ss ( G e V ) Lower boundCentral valueUpper bound
FIG. 13: The mass of ρ (1 D ) state with varia-tions of the Borel parameter T . (GeV ) M a ss ( G e V ) Lower boundCentral valueUpper bound
FIG. 14: The mass of ρ (2 D ) state with varia-tions of the Borel parameter T . (GeV ) M a ss ( G e V ) Lower boundCentral valueUpper bound
FIG. 15: The mass of φ (1 D ) state with varia-tions of the Borel parameter T . (GeV ) M a ss ( G e V ) Upper boundCentral valueLower bound
FIG. 16: The mass of φ (2 D ) state with varia-tions of the Borel parameter T . with the experimental data, which means it is reasonable to assign Y (2040) as the 2 D ρ meson.Although the ground state of D-wave vector ss meson is still not observed in experiment, recentlyParticle Data Group assigned φ (2170) to be the 2 D φ meson with a mass of 2 .
16 GeV[43]. Inaddition, BESIII collaboration reported the mass of φ (2170) to be (2179 ± ± c in the decayprocess ωη [1] From Table II, we can see that the predicted masses of 2 D φ meson with differenttheoretical methods are not agreement well with each other. Our predicted mass for 2 D φ mesonis 2 . ± .
075 GeV which agrees well with experimental data. In addition, we predict the mass of1 D φ meson to be 1 . ± .
095 GeV which is roughly compatible with those of other collaborations.4 (GeV ) f ( G e V ) Lower boundCentral valueUpper bound
FIG. 17: The decay constant ρ (1 D ) with varia-tions of the Borel parameter T . (GeV ) f ( G e V ) Upper boundCentral valueLower bound
FIG. 18: The decay constant ρ (2 D ) with varia-tions of the Borel parameter T . (GeV ) f ( G e V ) Lower boundCentral valueUpper bound
FIG. 19: The decay constant φ (1 D ) with vari-ations of the Borel parameter T . (GeV ) f ( G e V ) Central valueLower boundUpper bound
FIG. 20: The decay constant φ (2 D ) with vari-ations of the Borel parameter T . This prediction is helpful in the future to search for this missing ground state in experiment. f ρ (1 D ) = 0 . ± . GeVf ρ (2 D ) = 0 . ± . GeVf φ (1 D ) = 0 . ± . GeV (37) f φ (2 D ) = 0 . ± . GeV
In the past decades, more and more ρ and φ states have been observed in experiments. How tocategorize these states into the meson family is a interesting topic, which can improve our knowledgeof light meson spectrum. In this work, we study the masses and decay constants of the groundstate and the first radially excited state of D-wave vector ρ and φ mesons with the QCD sum rules.Our calculation successfully reproduce the experimental data of 1 D ρ meson and 2 D φ meson,5which indicates our analysis is reliable. We predict the mass of 2 D ρ meson to be 2 . ± . Y (2040) state[1]. This result supportsassigning Y (2040) resonance to be the 2 D state. We also predict the mass of 1 D φ meson to be1 . ± .
095 GeV. This result are roughly compatible with the values of other collaborations. Usingthe obtained masses as input parameters in QCDSR I and QCDSR II, we finally predict the decayconstants for these meson states. The theoretical analysis in this work is not only helpful to confirmthe underlying properties of these light mesons, but also serve further experimental investigation.
Acknowledgment
This work has been supported by the Fundamental Research Funds for the Central Universi-ties, Grant Number 2016
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