1 Introduction

## Abstract

In this article, we construct the axialvector-diquark-scalar-antidiquark type currents to interpolate the axialvector doubly heavy tetraquark states, and study them with the QCD sum rules in details by carrying out the operator product expansion up to the vacuum condensates of dimension 10.

Analysis of the axialvector doubly heavy tetraquark states with QCD sum rules

[2mm] Zhi-Gang Wang 1

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

PACS number: 12.39.Mk, 12.38.Lg

Key words: Tetraquark state, QCD sum rules

## 1 Introduction

The scattering amplitude for one-gluon exchange is proportional to

 taijtakl = −13(δijδkl−δilδkj)+16(δijδkl+δilδkj), (1)

where , the is the Gell-Mann matrix. The negative sign in front of the antisymmetric antitriplet indicates the interaction is attractive while the positive sign in front of the symmetric sextet indicates the interaction is repulsive, the attractive interaction favors formation of the diquarks in color antitriplet [1]. The color antitriplet diquarks with or only have two structures in Dirac spinor space, where and for the axialvector and tensor diquarks, respectively. The axialvector diquarks are more stable than the tensor diquarks . Recently, the LHCb collaboration observed the doubly charmed baryon state in the mass spectrum in a data sample collected by LHCb at with a signal yield of , and measured the mass, but did not determine the spin [2]. The maybe have the spin or , we can take the diquark as basic constituent to construct the current

 JΞcc(x) = εijkcTi(x)Cγμcj(x)γ5γμuk(x), (2)

or

 JμΞcc(x) = εijkcTi(x)Cγμcj(x)uk(x), (3)

to study it with the QCD sum rules [3].

In this article, we choose the axialvector diquarks to construct the currents to interpolate the doubly heavy tetraquark states. Up to now, no experimental candidates for the quark configurations or have been observed. There have been several works on the doubly heavy tetraquark states, such as potential quark models or simple quark models [4, 5], QCD sum rules [6, 7, 8], heavy quark symmetry [9, 10], lattice QCD [11, 12], etc. Although the doubly heavy tetraquark states have been studied with the QCD sum rules, the energy scale dependence of the QCD sum rules has not been studied yet. In Refs.[13, 14, 15, 16, 17], we observe that in the QCD sum rules for the hidden charm (or bottom) tetraquark states and molecular states, the integrals

 ∫s04m2Q(μ)dsρQCD(s,μ)exp(−sT2), (4)

are sensitive to the heavy quark masses , where the denotes the QCD spectral densities and the denotes the Borel parameters. Variations of the heavy quark masses or the energy scales lead to changes of integral ranges of the variable besides the QCD spectral densities , therefore changes of the Borel windows and predicted masses and pole residues. In this article, we revisit the QCD sum rules for the axialvector doubly heavy tetraquark states and choose the optimal energy scales to extract the masses.

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

## 2 The QCD sum rules for the axialvector doubly heavy tetraquark states

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

 ΠJ/ημν(p) = i∫d4xeip⋅x⟨0|T{J/ημ(x)J/η†ν(0)}|0⟩, (5)

where

 Jμ(x) = εijkεimnQTi(x)CγμQj(x)¯um(x)γ5C¯sTn(x), (6) ημ(x) = εijkεimnQTi(x)CγμQj(x)¯um(x)γ5C¯dTn(x), (7)

, the , , , , are color indexes, the is the charge conjugation matrix.

On the phenomenological side, we insert a complete set of intermediate hadronic states with the same quantum numbers as the current operators and into the correlation functions and respectively to obtain the hadronic representation [18, 19], and isolate the ground state contributions,

 ΠJ/ημν(p) = λ2ZM2Z−p2(−gμν+pμpνp2)+⋯ (8) = ΠJ/η(p2)(−gμν+pμpνp2)+⋯,

where the pole residues are defined by , the are the polarization vectors of the axialvector tetraquark states .

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

 ΠJμν(p) = −2iεijkεimnεi′j′k′εi′m′n′∫d4xeip⋅x (9) Tr[γμSkk′Q(x)γνCSTjj′Q(x)C]Tr[γ5Um′m(−x)γ5CSTn′n(−x)C], Πημν(p) = −2iεijkεimnεi′j′k′εi′m′n′∫d4xeip⋅x (10) Tr[γμSkk′Q(x)γνCSTjj′Q(x)C]Tr[γ5Um′m(−x)γ5CDTn′n(−x)C],

where the , , and are the full , , and quark propagators, respectively [19, 20],

 U/Dij(x) = iδij⧸x2π2x4−δij⟨¯qq⟩12−δijx2⟨¯qgsσGq⟩192−igsGaαβtaij(⧸xσαβ+σαβ⧸x)32π2x2 (11) −18⟨¯qjσμνqi⟩σμν+⋯,
 Sij(x) = iδij⧸x2π2x4−δijms4π2x2−δij⟨¯ss⟩12+iδij⧸xms⟨¯ss⟩48−δijx2⟨¯sgsσGs⟩192+iδijx2⧸xms⟨¯sgsσGs⟩1152 (12) −igsGaαβtaij(⧸xσαβ+σαβ⧸x)32π2x2−18⟨¯sjσμνsi⟩σμν+⋯,
 SijQ(x) = i(2π)4∫d4ke−ik⋅x{δij⧸k−mQ−gsGnαβtnij4σαβ(⧸k+mQ)+(⧸k+mQ)σαβ(k2−m2Q)2 (13) −g2s(tatb)ijGaαβGbμν(fαβμν+fαμβν+fαμνβ)4(k2−m2Q)5+⋯⎫⎬⎭,
 fλαβ = (⧸k+mQ)γλ(⧸k+mQ)γα(⧸k+mQ)γβ(⧸k+mQ), fαβμν = (⧸k+mQ)γα(⧸k+mQ)γβ(⧸k+mQ)γμ(⧸k+mQ)γν(⧸k+mQ). (14)

Then we compute the integrals both in coordinate space and in momentum space, and obtain the correlation functions at the quark level, therefore the QCD spectral densities through dispersion relation.

 limϵ→0ImΠJ/η(s+iϵ)π = ρJ/η(s). (15)

In Eqs.(11-12), we retain the terms and come from the Fierz re-ordering of the and to absorb the gluons emitted from other quark lines to form and to extract the mixed condensates and , respectively. In this article, we carry out the operator product expansion to the vacuum condensates up to dimension-10, and take into account the vacuum condensates which are vacuum expectations of the operators of the orders with in a consistent way [13, 14, 15, 16, 17].

Once the analytical expressions of the QCD spectral densities are obtained, we can take the quark-hadron duality below the continuum thresholds and perform Borel transform with respect to the variable to obtain the following QCD sum rules,

 λ2Zexp(−M2ZT2)=∫s04m2QdsρJ/η(s)exp(−sT2), (16)

where

 ρJ(s) = ρ0(s)+ρ3(s)+ρ4(s)+ρ5(s)+ρ6(s)+ρ8(s)+ρ10(s), (17) ρη(s) = ρJ(s)∣ms→0,⟨¯ss⟩→⟨¯qq⟩,⟨¯sgsσGs⟩→⟨¯qgsσGq⟩, (18)
 ρ0(s) = 1512π6∫yfyidy∫1−yzidzyz(1−y−z)2(s−¯¯¯¯¯m2Q)3(5s−¯¯¯¯¯m2Q) (19) +m2Q128π6∫yfyidy∫1−yzidz(1−y−z)2(s−¯¯¯¯¯m2Q)3,
 ρ3(s) = ms[⟨¯ss⟩−2⟨¯qq⟩]32π4∫yfyidy∫1−yzidzyz(s−¯¯¯¯¯m2Q)(3s−¯¯¯¯¯m2Q) (20) +msm2Q[⟨¯ss⟩−2⟨¯qq⟩]16π4∫yfyidy∫1−yzidz(s−¯¯¯¯¯m2Q),
 ρ4(s) = −m2Q384π4⟨αsGGπ⟩∫yfyidy∫1−yzidz(zy2+yz2)(1−y−z)2(2s−¯¯¯¯¯m2Q) (21) −m4Q384π4⟨αsGGπ⟩∫yfyidy∫1−yzidz(1y3+1z3)(1−y−z)2 +m2Q128π4⟨αsGGπ⟩∫yfyidy∫1−yzidz(1y2+1z2)(1−y−z)2(s−¯¯¯¯¯m2Q) −11536π4⟨αsGGπ⟩∫yfyidy∫1−yzidz(1−y−z)2(s−¯¯¯¯¯m2Q)(5s−3¯¯¯¯¯m2Q) +1256π4⟨αsGGπ⟩∫yfyidy∫1−yzidzyz(s−¯¯¯¯¯m2Q)(3s−¯¯¯¯¯m2Q) +m2Q128π4⟨αsGGπ⟩∫yfyidy∫1−yzidz(s−¯¯¯¯¯m2Q),
 ρ5(s) = ms[3⟨¯qgsσGq⟩−⟨¯sgsσGs⟩]48π4∫yfyidyy(1−y)s, (22)
 ρ6(s) = ⟨¯qq⟩⟨¯ss⟩3π2∫yfyidyy(1−y)s, (23)
 ρ8(s) = Missing or unrecognized delimiter for \left
 ρ10(s) = ⟨¯qgsσGq⟩⟨¯sgsσGs⟩48π2∫yfyidyy(1−y)(sT2+2s2T4+s3T6)δ(s−˜m2Q) (25) −11⟨¯qgsσGq⟩⟨¯sgsσGs⟩6912π2∫yfyidy(1+s2T2)δ(s−˜m2Q),

, , , , , , when the functions and appear.

We derive Eq.(16) with respect to , then eliminate the pole residues to obtain the QCD sum rules for the masses,

 M2Z=−ddτ∫s04m2QdsρJ/η(s)e−τs∫s04m2QdsρJ/η(s)e−τs. (26)

## 3 Numerical results and discussions

We take the standard values of the vacuum condensates , , , , , at the energy scale [18, 19, 21], and choose the masses , , from the Particle Data Group [22]. Furthermore, we take into account the energy-scale dependence of the input parameters,

 ⟨¯qq⟩(μ) = ⟨¯qq⟩(Q)[αs(Q)αs(μ)]49, ⟨¯ss⟩(μ) = ⟨¯ss⟩(Q)[αs(Q)αs(μ)]49, ⟨¯qgsσGq⟩(μ) = ⟨¯qgsσGq⟩(Q)[αs(Q)αs(μ)]227, ⟨¯sgsσGs⟩(μ) = ⟨¯sgsσGs⟩(Q)[αs(Q)αs(μ)]227, mc(μ) = mc(mc)[αs(μ)αs(mc)]1225, mb(μ) = mb(mb)[αs(μ)αs(mb)]1223, ms(μ) = ms(2GeV)[αs(μ)αs(2GeV)]49, αs(μ) = 1b0t[1−b1b20logtt+b21(log2t−logt−1)+b0b2b40t2], (27)

where , , , , , and for the flavors , and , respectively [22], and evolve all the input parameters to the optimal energy scales to extract the masses of the .

In Refs.[13, 14, 15, 16, 17], we study the acceptable energy scales of the QCD spectral densities for the hidden-charm (hidden-bottom) tetraquark states and molecular states in the QCD sum rules in details for the first time, and suggest an energy scale formula to determine the optimal energy scales, which enhances the pole contributions remarkably and works well. The energy scale formula also works well in studying the hidden-charm pentaquark states [23]. We can assign the and to be the axialvector tetraquark states with the quark constituents and respectively, and choose the currents,

 JQ¯Qμ(x) = εijkεimn√2{uTj(x)Cγ5Qk(x)¯dm(x)γμC¯QTn(x)−uTj(x)CγμQk(x)¯dm(x)γ5C¯QTn(x)},

with to study them with the QCD sum rules [13, 16]. If we take the updated values of the effective heavy quark masses and [24], the optimal energy scales of the QCD spectral densities of the and are and , respectively.

There are no experimental candidates for the doubly heavy tetraquark states. Firstly, we suppose that the ground state type axialvector tetraquark states and have degenerate masses, and study the masses of the ground state axialvector tetraquark states at the same energy scales of the QCD spectral densities as the ones for the ground state axialvector tetraquark states . In Fig.1, we plot the predicted masses of the () and () with variations of the Borel parameter for the continuum threshold parameter () and the energy scale () [13, 16, 24]. From the figure, we can see that the experimental values of the masses of the and can be well reproduced, there appear platforms for the masses of the tetraquark states, which lie slightly below the corresponding masses of the and , respectively. If we choose the Borel windows as and for the tetraquark states and , respectively, the pole contributions are and , respectively, it is reliable to extract the masses. Furthermore, the continuum threshold parameters satisfy the relation and , respectively, which are consistent with our naive expectation that the mass gaps of the ground states and the first radial excited states of the tetraquark states are about [25, 26]. The energy scales and work well.

In Ref.[5], Karliner and Rosner obtain the masses and for the type axialvector tetraquark states and respectively based on a simple potential quark model, which can reproduce the mass of the doubly charmed baryon state . In Ref.[10], Eichten and Quigg obtain the masses and for the type axialvector tetraquark states and respectively based on the heavy quark symmetry, where the mass of the doubly charmed baryon state is taken as input parameter in the charm sector, while in the bottom sector, there are no experimental candidates for the baryon states and . From Fig.1, we can see that if we take the same parameters, such as the energy scales, continuum threshold parameters, etc, in the charm sector, the predicted mass is slightly smaller than the value from a simple potential quark model [5] and much smaller than the value from the heavy quark symmetry [10], in the bottom sector, the predicted mass is much larger than the value from a simple potential quark model [5] and slightly larger than the value from the heavy quark symmetry [10].

Now we revisit the subject of how to choose the energy scales of the QCD spectral densities. In calculation, we neglect the perturbative corrections to the currents , which can be taken into account in the leading logarithmic approximation through an anomalous dimension factor, , the are the anomalous dimension of the interpolating currents ,

 ⟨0|J/ηα(0;μ)|ZQQ(p)⟩ = [αs(μ0)αs(μ)]γJ⟨0|J/ηα(0;μ0)|ZQQ(p)⟩ (29) = [αs(μ0)αs(μ)]γJλZ(μ0)εα=λZ(μ)εα.

The pole residues are energy scale dependent quantities, at the leading order approximation, we can set .

At the QCD side, the correlation functions can be written as

 Π(p2) = ∫s04m2Q(μ)dsρJ/η(s,μ)s−p2+∫∞s0dsρJ/η(s,μ)s−p2, (30)

through dispersion relation, and they are energy scale independent according to the approximation or ,

 ddμΠ(p2) = 0, (31)

which does not mean the pole contributions are energy scale independent,

 ddμ∫s04m2Q(μ)dsρJ/η(s,μ)s−p