1 Introduction
###### Abstract

In this article, we distinguish the charge conjugations of the interpolating currents, calculate the contributions of the vacuum condensates up to dimension-10 in the operator product expansion, and study the masses and pole residues of the hidden charmed tetraquark states with the QCD sum rules. We suggest a formula with the effective mass to estimate the energy scales of the QCD spectral densities of the hidden charmed tetraquark states, which works very well. The numerical results disfavor assigning the , , as the diquark-antidiquark (with the Dirac spinor structure ) type vector tetraquark states, and favor assigning the , as the diquark-antidiquark type tetraquark states. While the masses of the tetraquark states with symbolic quark structures and favor assigning the as the diquark-antidiquark type tetraquark state, more experimental data are still needed to distinguish its quark constituents. There are no candidates for the positive charge conjugation vector tetraquark states, the predictions can be confronted with the experimental data in the future at the BESIII, LHCb and Belle-II.

Analysis of the , , and as vector tetraquark states with QCD sum rules

[2mm] Zhi-Gang Wang 111E-mail: zgwang@aliyun.com.

Department of Physics, North China Electric Power University, Baoding 071003, P. R. China

PACS number: 12.39.Mk, 12.38.Lg

Key words: Vector tetraquark state, QCD sum rules

## 1 Introduction

Recently, the BESIII collaboration studied the process at a center-of-mass energy of using a data sample obtained with the BESIII detector at the Beijing Electron Positron Collider, and observed a structure near the threshold in the recoil mass spectrum [1]. The measured mass and width of the are and , respectively [1]. Later, the BESIII collaboration studied the process at center-of-mass energies from to , and observed a distinct structure in the mass spectrum, the measured mass and width of the are and , respectively [2]. No significant signal of the was observed in the mass spectrum [2], the and maybe have different quantum numbers.

At first sight, the S-wave systems have the quantum numbers , , , while the S-wave systems have the quantum numbers , so the and are different particles. On the other hand, it is also possible for the P-wave () systems to have the quantum numbers (). We cannot exclude the possibility that the and are the same particle with the quantum numbers or . There have been several tentative assignments of the and , such as the re-scattering effects [3], molecular states [4], tetraquark states [5], etc. The and are charged charmonium-like states, their quark constituents must be or irrespective of the diquark-antidiquark type or meson-meson type substructures.

In 2013, the BESIII collaboration studied the process and observed the in the mass spectrum with the mass and width , respectively [6]. Later the was confirmed by the Belle and CLEO collaborations [7, 8]. Also in 2013, the BESIII collaboration studied the process and observed the in the mass spectrum with the mass and width , respectively [9]. The angular distribution of the system favors assigning the with [9]. We tentatively identify the and as the same particle according to the uncertainties of the masses and widths [10], one can consult Ref.[10] for more articles on the . The possible quantum numbers of the or are . There is a faint possibility that the and are the same axial-vector meson with according to the masses.

In 2007, the Belle collaboration measured the cross section for the process between threshold and using a data sample collected with the Belle detector at KEKB, and observed two structures and in the invariant mass distributions at with a width of   and with a width of   , respectively [11]. The quantum numbers of the and are , which are unambiguously listed in the Review of Particle Physics now [12]. In 2008, the Belle collaboration studied the exclusive process and observed a clear peak in the invariant mass distribution just above the threshold, and determined the mass and width to be and , respectively [13]. The and may be the same particle according to the uncertainties of the masses and widths (also the decay properties [14]). There have been several tentative assignments of the and , such as the conventional charmonium states [15], baryonium state [16], molecular states or hadro-charmonium states [17], tetraquark states [18, 19, 20], etc. One can consult Ref.[21] for more articles on the , and particles.

In this article, we study the diquark-antidiquark type vector tetraquark states in details with the QCD sum rules, and explore possible assignments of the , , and in the tetraquark scenario. In Ref.[10], we extend our previous works on the axial-vector tetraquark states [22], distinguish the charge conjugations of the interpolating currents, calculate the contributions of the vacuum condensates up to dimension-10 and discard the perturbative corrections in the operator product expansion, study the type axial-vector hidden charmed tetraquark states with the QCD sum rules. We explore the energy scale dependence of the charmed tetraquark states in details for the first time, and tentatively assign the and (or ) as the and tetraquark states, respectively [10]. In calculations, we observe that the tetraquark masses decrease monotonously with increase of the energy scales, the energy scale is the lowest energy scale to reproduce the experimental values of the masses of the and , and serves as an acceptable energy scale for the charmed mesons in the QCD sum rules [10].

In Ref.[23], we study the and type tetraquark states with the QCD sum rules by carrying out the operator product expansion to the vacuum condensates up to dimension-10 and setting the energy scale to be . In Refs.[5, 18, 24], the authors carry out the operator product expansion to the vacuum condensates up to dimension-8 to study the vector tetraquark states with the QCD sum rules, but do not show the energy scales or do not specify the energy scales at which the QCD spectral densities are calculated. In Refs.[5, 18, 23, 24], some higher dimension vacuum condensates involving the gluon condensate, mixed condensate and four-quark condensate are neglected, which maybe impair the predictive ability. The terms associate with , , in the QCD spectral densities manifest themselves at small values of the Borel parameter , we have to choose large values of the to warrant convergence of the operator product expansion and appearance of the Borel platforms. In the Borel windows, the higher dimension vacuum condensates play a less important role. In summary, the higher dimension vacuum condensates play an important role in determining the Borel windows therefore the ground state masses and pole residues, so we should take them into account consistently.

In this article, we extend our previous works [10] to study the vector tetraquark states, distinguish the charge conjugations of the interpolating currents, calculate the contributions of the vacuum condensates up to dimension-10 and discard the perturbative corrections, study the masses and pole residues of the type vector hidden charmed tetraquark states with the QCD sum rules. Furthermore, we explore the energy scale dependence in details so as to obtain some useful formulae, and make tentative assignments of the , , and as the or tetraquark states. The scalar and axial-vector heavy-light diquark states have almost degenerate masses from the QCD sum rules [25], the and type tetraquark states have degenerate (or slightly different) masses [23], as the pseudoscalar and vector heavy-light diquark states have slightly different masses.

The article is arranged as follows: we derive the QCD sum rules for the masses and pole residues of the vector tetraquark states in section 2; in section 3, we present the numerical results and discussions; section 4 is reserved for our conclusion.

## 2 QCD sum rules for the vector tetraquark states

In the following, we write down the two-point correlation functions in the QCD sum rules,

 Πμν(p) = i∫d4xeip⋅x⟨0|T{Jμ(x)J†ν(0)}|0⟩, (1) J1μ(x) = ϵijkϵimn√2{sj(x)Cck(x)¯sm(x)γμC¯cn(x)+tsj(x)Cγμck(x)¯sm(x)C¯cn(x)}, (2) J2μ(x) = ϵijkϵimn2{uj(x)Cck(x)¯um(x)γμC¯cn(x)+dj(x)Cck(x)¯dm(x)γμC¯cn(x) (3) +tuj(x)Cγμck(x)¯um(x)C¯cn(x)+tdj(x)Cγμck(x)¯dm(x)C¯cn(x)}, J3μ(x) = ϵijkϵimn√2{uj(x)Cck(x)¯dm(x)γμC¯cn(x)+tuj(x)Cγμck(x)¯dm(x)C¯cn(x)}, (4)

where , , the , , , , are color indexes, the is the charge conjugation matrix. Under charge conjugation transform , the currents have the properties,

 ˆCJ1/2μ(x)ˆC−1 = ±J1/2μ(x)fort=±1, ˆCJ3μ(x)ˆC−1 = ±J3μ(x)∣u↔dfort=±1, (5)

which originate from the charge conjugation properties of the pseudoscalar and axial-vector diquark states,

 ˆC[ϵijkqjCck]ˆC−1 = ϵijk¯qjC¯ck, ˆC[ϵijkqjCγμck]ˆC−1 = ϵijk¯qjγμC¯ck. (6)

We choose the neutral currents and with to interpolate the diquark-antidiquark type tetraquark states and , respectively. There are two structures in invariant mass distributions at about and in the mass spectrum, which maybe due to the scalar mesons and , respectively [11]. In the two-quark scenario, and in the ideal mixing limit, while in the tetraquark scenario, the and have the symbolic quark structures and , respectively. The couples to the current while the couples to the current . However, we cannot exclude the possibility that the has the symbolic quark structure , in that case the decay is Okubo-Zweig-Iizuka (OZI) allowed. We choose the charged vector current with to interpolate the and , the results for the scalar and tensor currents will be presented elsewhere. At present time, we cannot exclude the possibility that the and are the same vector particle.

We can insert a complete set of intermediate hadronic states with the same quantum numbers as the current operators into the correlation functions to obtain the hadronic representation [26, 27]. After isolating the ground state contributions of the vector tetraquark states, we get the following results,

 Πμν(p) = λ2Y/ZM2Y/Z−p2(−gμν+pμpνp2)+⋯, (7)

where the pole residues are defined by

 ⟨0|Jμ(0)|Y/Z(p)⟩=λY/Zεμ, (8)

the are the polarization vectors of the vector tetraquark states , , , , etc.

In the following, we take the current as an example and briefly outline the operator product expansion for the correlation functions in perturbative QCD. We contract the and quark fields in the correlation functions with Wick theorem, obtain the results:

 Πμν(p) = iϵijkϵimnϵi′j′k′ϵi′m′n′2∫d4xeip⋅x (9) {Tr[Ckk′(x)CSjj′T(x)C]Tr[γνCn′n(−x)γμCSm′mT(−x)C] +Tr[γμCkk′(x)γνCSjj′T(x)C]Tr[Cn′n(−x)CSm′mT(−x)C] ∓Tr[γμCkk′(x)CSjj′T(x)C]Tr[γνCn′n(−x)CSm′mT(−x)C] ∓Tr[Ckk′(x)γνCSjj′T(x)C]Tr[Cn′n(−x)γμCSm′mT(−x)C]},

where the correspond to respectively, the and are the full and quark propagators respectively,

 Sij(x) = iδij⧸x2π2x4−δijms4π2x2−δij⟨¯ss⟩12+iδij⧸xms⟨¯ss⟩48−δijx2⟨¯sgsσGs⟩192+iδijx2⧸xms⟨¯sgsσGs⟩1152 (10) −igsGaαβtaij(⧸xσαβ+σαβ⧸x)32π2x2−iδijx2⧸xg2s⟨¯ss⟩27776−δijx4⟨¯ss⟩⟨g2sGG⟩27648−18⟨¯sjσμνsi⟩σμν −14⟨¯sjγμsi⟩γμ+⋯,
 Cij(x) = i(2π)4∫d4ke−ik⋅x{δij⧸k−mc−gsGnαβtnij4σαβ(⧸k+mc)+(⧸k+mc)σαβ(k2−m2c)2 +gsDαGnβλtnij(fλβα+fλαβ)3(k2−m2c)4−g2s(tatb)ijGaαβGbμν(fαβμν+fαμβν+fαμνβ)4(k2−m2c)5+⋯⎫⎬⎭, fλαβ = (⧸k+mc)γλ(⧸k+mc)γα(⧸k+mc)γβ(⧸k+mc), fαβμν = (⧸k+mc)γα(⧸k+mc)γβ(⧸k+mc)γμ(⧸k+mc)γν(⧸k+mc), (11)

and , the is the Gell-Mann matrix, [27], then compute the integrals both in the coordinate and momentum spaces, and obtain the correlation functions therefore the spectral densities at the level of quark-gluon degrees of freedom. In Eq.(10), we retain the terms and originate from the Fierz re-arrangement of the to absorb the gluons emitted from the heavy quark lines to form and so as to extract the mixed condensate and four-quark condensates and , respectively. One can consult Ref.[10] for some technical details in the operator product expansion.

Once analytical results are obtained, we can take the quark-hadron duality below the continuum threshold and perform Borel transform with respect to the variable to obtain the following QCD sum rules:

 λ2Y/Zexp⎛⎝−M2Y/ZT2⎞⎠=∫s04m2cdsρ(s)exp(−sT2), (12)

where

 ρ(s) = ρ0(s)+ρ3(s)+ρ4(s)+ρ5(s)+ρ6(s)+ρ7(s)+ρ8(s)+ρ10(s), (13)

the 0, 3, 4, 5, 6, 7, 8, 10 denote the dimensions of the vacuum condensates, the explicit expressions of the spectral densities are presented in the Appendix. In this article, we carry out the operator product expansion to the vacuum condensates up to dimension-10 and discard the perturbative corrections, and assume vacuum saturation for the higher dimension vacuum condensates. The higher dimension vacuum condensates are always factorized to lower condensates with vacuum saturation in the QCD sum rules, factorization works well in large limit. In reality, , some (not much) ambiguities maybe come from the vacuum saturation assumption. The condensates , , , and are the vacuum expectations of the operators of the order . The four-quark condensate comes from the terms , and , rather than comes from the perturbative corrections of (see Ref.[10] for the technical details). The condensates , , have the dimensions 6, 8, 9 respectively, but they are the vacuum expectations of the operators of the order , , respectively, and discarded. We take the truncations and in a consistent way, the operators of the orders with are discarded. Furthermore, the numerical values of the condensates , , are very small, and they are neglected safely.

Differentiate Eq.(12) with respect to , then eliminate the pole residues , we obtain the QCD sum rules for the masses of the vector tetraquark states,

 M2Y/Z=∫s04m2cdsdd(−1/T2)ρ(s)exp(−sT2)∫s04m2cdsρ(s)exp(−sT2). (14)

We can obtain the QCD sum rules for the vector tetraquark states and with the simple replacements,

 ms → 0, ⟨¯ss⟩ → ⟨¯qq⟩, ⟨¯sgsσGs⟩ → ⟨¯qgsσGq⟩, (15)

the QCD sum rules for the and degenerate in the isospin limit.

## 3 Numerical results and discussions

The vacuum condensates are taken to be the standard values , , , , , at the energy scale [26, 27, 28]. The quark condensate and mixed quark condensate evolve with the renormalization group equation, , , and .

In the article, we take the masses and from the Particle Data Group [12], and take into account the energy-scale dependence of the masses from the renormalization group equation,

 mc(μ) = mc(mc)[αs(μ)αs(mc)]1225, ms(μ) = ms(2GeV)[αs(μ)αs(2GeV)]49, αs(μ) = 1b0t[1−b1b20logtt+b21(log2t−logt−1)+b0b2b40t2], (16)

where , , , , , and for the flavors , and , respectively [12].

In Ref.[10], we observe that the energy scale is an acceptable energy scale of the QCD spectral densities in the QCD sum rules for the hidden and open charmed mesons, as it can reproduce the experimental values and with suitable Borel parameters. However, such energy scale and truncation in the operator product expansion cannot reproduce the experimental values of the decay constants and . In calculation, we observe that the masses of the axial-vector tetraquark states decrease monotonously with increase of the energy scales of the QCD spectral densities, the energy scale is the lowest energy scale to reproduce the experimental values of the masses of the and (or ), and serves as an acceptable energy scale (not the universal energy scale) for the tetraquark states [10]. On the other hand, it is hard to obtain the true values of the pole residues of the tetraquark states, so we focus on the masses to study the tetraquark states, and the predictions of the pole residues maybe not as robust. If the and are the vector tetraquark states, we can choose the threshold parameters and energy scales tentatively, and search for the ideal parameters, such as the threshold parameters, energy scales and Borel parameters.

In Fig.1, the masses of the vector tetraquark states are plotted with variations of the Borel parameters , energy scales , and continuum threshold parameters . From the figure, we can see that the masses decrease monotonously with increase of the energy scales, the parameters and can be excluded, as the predicted masses for the values of the Borel parameters at a large interval. We have to choose larger threshold parameters or (and) energy scales, the resulting masses are larger than for the parameters and . The predictions based on the QCD sum rules disfavor assigning the and as the diquark-antidiquark type vector tetraquark states. We cannot satisfy the relation with reasonable compared to the experimental data.

The BESIII collaboration observed the and in the following processes [1, 2],

 e+e− → Z±c(4025)π∓→(D∗¯D∗)±(0++,1+−,2++,0−+,1−−,2−+,3−−)π∓, e+e− → Z±c(4020)π∓→(hcπ)±(1−−,0++,1+−,2++)π∓, (17)

where we present the possible quantum numbers of the and systems in the brackets. If the and are the same particle, the quantum numbers are , , , . On the other hand, the and systems have the quantum numbers , then the survived quantum numbers of the and are , and . The predictions based on the QCD sum rules reduce the possible quantum numbers of the and to and .

The strong decays

 Y(4260)/γ∗(4260) → Z±c(4025/4020)(2++)π∓, (18)

take place through relative D-wave, and are kinematically suppressed in the phase-space. The assignment is disfavored, but not excluded.

In the following, we list out the possible strong decays of the and in the case of the assignment.

 Z±c(4025)(1+−) → hc(1P)π±,J/ψπ±,ηcρ±,ηc(ππ)±P,χc1(ππ)±P,(D¯D∗)±,(D∗¯D∗)±, Z±c(3900)(1+−) → hc(1P)π±,J/ψπ±,ηcρ±,ηc(ππ)±P,χc1(ππ)±P, (19)

where the denotes the P-wave systems have the same quantum numbers of the . We take the and as the same particle in the assignment, and will denote them as . In Ref.[10], we observe that the couples to the axial-vector current . Now we perform Fierz re-arrangement both in the color and Dirac-spinor spaces and obtain the following result,

 Jμ1+− = <