Scalar or Vector Tetraquark State Candidate: Z_c(4100)^*

The attractive interactions of one-gluon exchange favor formation of the diquarks in color antitriplet.^[1] The diquarks (or diquark operators) ${\varepsilon }^{abc}{q}_{b}^{T}C\Gamma {q}_{c}^{^{\prime} }$ in color antitriplet have five structures in Dirac spinor space, where CΓ = C, Cγ₅, Cγ_μ γ₅, Cγ_μ and Cσ_μν for the pseudoscalar (P), scalar (S), vector (V), axialvector (A) and tensor (T) diquarks, respectively, the a, b, c are color indexes. The pseudoscalar, scalar, vector and axialvector diquarks have been studied with the QCD sum rules in details, which indicate that the favored configurations are the scalar and axialvector diquark states.^[2–4] The scalar and axialvector diquark operators have been applied extensively to construct the tetraquark currents to study the lowest tetraquark states.^[5–12]

In 2018, the LHCb Collaboration performed a Dalitz plot analysis of B⁰ → η_cK⁺π⁻ decays and observed an evidence for an exotic η_cπ⁻ resonant state.^[13] The significance of this exotic resonance is more than three standard deviations, the measured mass and width are $4096\pm {20}_{-22}^{+18}{\rm{MeV}}$ and $152\pm {58}_{-35}^{+60}{\rm{MeV}}$, respectively. The spin-parity assignments J^P = 0⁺ and 1⁻ are both consistent with the experimental data.^[13] Is it a candidate for a scalar tetraquark state,^{[12, 14, 15]} hadrocharmonium,^[16] ${D}^{* }{\bar{D}}^{* }$ molecular state,^[17] or charge conjugation of the ${Z}_{c}^{+}(4050)$?^[18]

In Refs. [9] and [10], we study the ${[sc]}_{S}{[\bar{s}\bar{c}]}_{S}$-type, ${[sc]}_{A}{[\bar{s}\bar{c}]}_{A}$-type, ${[sc]}_{P}{[\bar{s}\bar{c}]}_{P}$-type and ${[sc]}_{V}{[\bar{s}\bar{c}]}_{V}$-type scalar tetraquark states with the QCD sum rules systematically, and obtain the ground state masses M_SS = 3.89 ± 0.05 GeV $({3.85}_{-0.17}^{+0.18}{\rm{GeV}})$, ${M}_{AA}={3.92}_{-0.18}^{+0.19}{\rm{GeV}}$, M_PP = 5.48 ± 0.10 GeV and ${M}_{VV}={4.70}_{-0.09}^{+0.08}{\rm{GeV}}$ for the SS, AA, PP and VV diquark-antidiquark type tetraquark states, respectively. The larger uncertainties ${}_{-0.17}^{+0.18}{\rm{GeV}}$ and ${}_{-0.18}^{+0.19}{\rm{GeV}}$ (compared to the uncertainties ± 0.05 GeV, ± 0.10 GeV and ${}_{-0.09}^{+0.08}{\rm{GeV}}$) in the ground state masses originate from the QCD sum rules, where both the ground state and the first radial excited state are taken into account at the hadron side. If only the ground states are taken into account in the QCD sum rules, the uncertainties |δ M| ≤ 0.10 GeV, so the uncertainties ${}_{-0.17}^{+0.18}{\rm{GeV}}$ and ${}_{-0.18}^{+0.19}{\rm{GeV}}$ (± 0.10 GeV) can be referred to as "larger uncertainties" ("smaller uncertainties").

In Ref. [11], we tentatively assign the ${X}^{* }(3860)$ to be the ${[qc]}_{S}{[\bar{q}\bar{c}]}_{S}$-type scalar tetraquark state, study its mass and width with the QCD sum rules in details, and obtain the mass M_SS = 3.86 ± 0.09 GeV. Now we can see that the SU(3) breaking effects of the masses of the $[sc][\bar{s}\bar{c}]$ and $[qc][\bar{q}\bar{c}]$ tetraquark states from the QCD sum rules are rather small, roughly speaking, they have degenerate masses. If we take the larger uncertainty, the predicted mass ${M}_{AA}={3.92}_{-0.18}^{+0.19}{\rm{GeV}}$ for the $[uc][\bar{d}\bar{c}]$ tetraquark state has overlap with the experimental data ${M}_{{Z}_{c}}=4096\pm {20}_{-22}^{+18}{\rm{MeV}}$ marginally,^[13] and favors assigning the Z_c(4100) to be the AA-type scalar tetraquark state.^[14] On the other hand, if we take the smaller uncertainty, the predicted mass M_AA = 3.92 ± 0.10 GeV for the $[uc][\bar{d}\bar{c}]$ tetraquark state has no overlap with the experimental data ${M}_{{Z}_{c}}=4096\pm {20}_{-22}^{+18}{\rm{MeV}}$, and disfavors assigning the Z_c(4100) to be the AA-type scalar tetraquark state. In a word, it is not robust assigning the Z_c(4100) to be the AA-type scalar tetraquark state.

In Ref. [12], Sundu, Agaev and Azizi obtain the mass M_SS = 4.08 ± 0.15 GeV, which differs from the prediction M_SS = 3.86 ± 0.09 GeV greatly.^[11] The differences originate from the different input parameters at the QCD side and different pole contributions at the hadron side. In Ref. [11], the pole contribution is about (46 − 70)%, which is much larger than the pole contribution in Ref. [12]. While in early works, we took the pole contributions >50% and chose the energy scales of the QCD spectral densities to be μ = 1 GeV, and obtained almost degenerate masses M_SS ≈ M_AA ≈ 4.4 GeV,^[6] which are much larger than the value M_SS = 4.08 ± 0.15 GeV.^[12] In Ref. [8], we observe that the energy scale μ = 1 GeV is not the optimal energy scale for the hidden-charm tetraquark states.

In summary, the QCD sum rules do not favor assigning the Z_c(4100) to be the ${[qc]}_{S}{[\bar{q}\bar{c}]}_{S}$-type, ${[qc]}_{A}{[\bar{q}\bar{c}]}_{A}$-type, ${[qc]}_{P}{[\bar{q}\bar{c}]}_{P}$-type and ${[qc]}_{V}{[\bar{q}\bar{c}]}_{V}$-type scalar tetraquark states.

In Refs. [14, 19], we take the scalar and axialvector diquark operators as basic constituents, introduce an explicit P-wave between the diquark and antidiquark operators to construct the vector tetraquark currents, and study the vector tetraquark states with the QCD sum rules systematically, and obtain the lowest vector tetraquark masses up to now, M_Y = 4.24 ± 0.10 GeV, 4.28 ± 0.10 GeV, 4.31 ± 0.10 GeV and 4.33 ± 0.10 GeV for the tetraquark states |0, 0; 0, 1; 1⟩, |1, 1; 0, 1; 1⟩, $\frac{1}{\sqrt{2}}(|1,0;1,1;1\rangle +|0,1;1,1;1\rangle )$ and |1, 1; 2, 1; 1⟩, respectively, where the tetraquark states are defined by $|{S}_{[qc]},{S}_{[\bar{q}\bar{c}]};S,L;J\rangle$, the S, L and J are the diquark spin, angular momentum and total angular momentum, respectively. For other QCD sum rules with the ${[qc]}_{S}{\partial }_{\mu }{[\bar{q}\bar{c}]}_{S}$-type interpolating currents, one can consult Ref. [20]. In fact, if we take the pseudoscalar and vector diquark operators as the basic constituents, even (or much) larger tetraquark masses are obtained.^[21–23] Up to now, it is obvious that the QCD sum rules do not favor assigning the Z_c(4100) to be the vector tetraquark state.

It is interesting to analyze the properties of the tensor diquark states, and take the tensor diquark operators as the basic constituents to construct the tetraquark currents to study the Z_c(4100).

Under parity transform $\hat{P}$, the tensor diquark operators have the properties,

$$\begin{eqnarray}&&\begin{array}{l}\hat{P}{\varepsilon }^{abc}{q}^{Tb}(x)C{\sigma }_{\mu \nu }{\gamma }_{5}{Q}^{c}(x){\hat{P}}^{-1}={\varepsilon }^{abc}{q}^{Tb}(\mathop{x}\limits^{\sim })C{\gamma }^{0}{\sigma }_{\mu \nu }{\gamma }^{0}{\gamma }_{5}{Q}^{c}(\mathop{x}\limits^{\sim })={\varepsilon }^{abc}{q}^{Tb}(\mathop{x}\limits^{\sim })C{\sigma }^{\mu \nu }{\gamma }_{5}{Q}^{c}(\mathop{x}\limits^{\sim }),\\ \hat{P}{\varepsilon }^{abc}{q}^{Tb}(x)C{\sigma }_{\mu \nu }{Q}^{c}(x){\hat{P}}^{-1}=-{\varepsilon }^{abc}{q}^{Tb}(\mathop{x}\limits^{\sim })C{\gamma }^{0}{\sigma }_{\mu \nu }{\gamma }^{0}{Q}^{c}(\mathop{x}\limits^{\sim })=-{\varepsilon }^{abc}{q}^{Tb}(\mathop{x}\limits^{\sim })C{\sigma }^{\mu \nu }{Q}^{c}(\mathop{x}\limits^{\sim }),\end{array}\end{eqnarray} \tag{ 1 }$$

where the four vectors ${x}^{\mu }=(t,\overrightarrow{x})$ and ${\mathop{x}\limits^{\sim }}^{\mu }=(t,-\overrightarrow{x})$, and we have used the properties of the Dirac γ-matrixes, γ⁰ γ_μ γ⁰ = γ^μ and γ⁰σ_μν γ⁰ = σ^μν. The tensor diquark states have both J^P = 1⁺ and 1⁻ components,

$$\begin{eqnarray}&&\begin{array}{l}\hat{P}{\varepsilon }^{abc}{q}^{Tb}(x)C{\sigma }_{jk}{\gamma }_{5}{Q}^{c}(x){\hat{P}}^{-1}=+{\varepsilon }^{abc}{q}^{Tb}(\mathop{x}\limits^{\sim })C{\sigma }_{jk}{\gamma }_{5}{Q}^{c}(\mathop{x}\limits^{\sim }),\\ \hat{P}{\varepsilon }^{abc}{q}^{Tb}(x)C{\sigma }_{0j}{\gamma }_{5}{Q}^{c}(x){\hat{P}}^{-1}=-{\varepsilon }^{abc}{q}^{Tb}(\mathop{x}\limits^{\sim })C{\sigma }_{0j}{\gamma }_{5}{Q}^{c}(\mathop{x}\limits^{\sim }),\\ \hat{P}{\varepsilon }^{abc}{q}^{Tb}(x)C{\sigma }_{0j}{Q}^{c}(x){\hat{P}}^{-1}=+{\varepsilon }^{abc}{q}^{Tb}(\mathop{x}\limits^{\sim })C{\sigma }_{0j}{Q}^{c}(\mathop{x}\limits^{\sim }),\\ \hat{P}{\varepsilon }^{abc}{q}^{Tb}(x)C{\sigma }_{jk}{Q}^{c}(x){\hat{P}}^{-1}=-{\varepsilon }^{abc}{q}^{Tb}(\mathop{x}\limits^{\sim })C{\sigma }_{jk}{Q}^{c}(\mathop{x}\limits^{\sim }),\end{array}\end{eqnarray} \tag{ 2 }$$

where j, k = 1, 2, 3, and we have used the properties of the Dirac γ-matrixes, γ⁰ = γ₀, γ^j = −γ_j, σ^jk = σ_jk and σ^0j = −σ_0j. Now we introduce the four vector ${t}^{\mu }=(1,\overrightarrow{0})$ and project out the 1⁺ and 1⁻ components explicitly,

$$\begin{eqnarray}&&\begin{array}{l}\hat{P}{\varepsilon }^{abc}{q}^{Tb}(x)C{\sigma }_{\mu \nu }^{t}{\gamma }_{5}{Q}^{c}(x){\hat{P}}^{-1}=+{\varepsilon }^{abc}{q}^{Tb}(\mathop{x}\limits^{\sim })C{\sigma }_{\mu \nu }^{t}{\gamma }_{5}{Q}^{c}(\mathop{x}\limits^{\sim }),\\ \hat{P}{\varepsilon }^{abc}{q}^{Tb}(x)C{\sigma }_{\mu \nu }^{v}{\gamma }_{5}{Q}^{c}(x){\hat{P}}^{-1}=-{\varepsilon }^{abc}{q}^{Tb}(\mathop{x}\limits^{\sim })C{\sigma }_{\mu \nu }^{v}{\gamma }_{5}{Q}^{c}(\mathop{x}\limits^{\sim }),\\ \hat{P}{\varepsilon }^{abc}{q}^{Tb}(x)C{\sigma }_{\mu \nu }^{v}{Q}^{c}(x){\hat{P}}^{-1}=+{\varepsilon }^{abc}{q}^{Tb}(\mathop{x}\limits^{\sim })C{\sigma }_{\mu \nu }^{v}{Q}^{c}(\mathop{x}\limits^{\sim }),\\ \hat{P}{\varepsilon }^{abc}{q}^{Tb}(x)C{\sigma }_{\mu \nu }^{t}{Q}^{c}(x){\hat{P}}^{-1}=-{\varepsilon }^{abc}{q}^{Tb}(\mathop{x}\limits^{\sim })C{\sigma }_{\mu \nu }^{t}{Q}^{c}(\mathop{x}\limits^{\sim }),\end{array}\end{eqnarray} \tag{ 3 }$$

where

$$\begin{eqnarray}&&\begin{array}{l}{\sigma }_{\mu \nu }^{t}=\frac{{\rm{i}}}{2}\left[{\gamma }_{\mu }^{t},{\gamma }_{\nu }^{t}\right],\\ {\sigma }_{\mu \nu }^{v}=\frac{{\rm{i}}}{2}\left[{\gamma }_{\mu }^{v},{\gamma }_{\nu }^{t}\right],\\ {\gamma }_{\mu }^{v}=\gamma \cdot t{t}_{\mu },\\ {\gamma }_{\mu }^{t}={\gamma }_{\mu }-\gamma \cdot t{t}_{\mu }.\end{array}\end{eqnarray} \tag{ 4 }$$

In this article, we choose the axialvector diquark operator ${\varepsilon }^{abc}{q}^{Tb}(x)C{\sigma }_{\mu \nu }^{v}{Q}^{c}(x)$ and vector diquark operator ${\varepsilon }^{abc}{q}^{Tb}(x)C{\sigma }_{\mu \nu }^{t}{Q}^{c}(x)$ to construct the tetraquark operators. On the other hand, if we choose the axialvector diquark operator ${\varepsilon }^{abc}{q}^{Tb}(x)C{\sigma }_{\mu \nu }^{t}{\gamma }_{5}{Q}^{c}(x)$ and vector diquark operator ${\varepsilon }^{abc}{q}^{Tb}(x)C{\sigma }_{\mu \nu }^{v}{\gamma }_{5}{Q}^{c}(x)$ to construct the tetraquark operators, we can obtain the same predictions.

In this article, we separate the vector ($\mathop{V}\limits^{\sim }$) and axialvector ($\mathop{A}\limits^{\sim }$) components of the tensor diquark operators explicitly, construct the $\mathop{A}\limits^{\sim }\mathop{A}\limits^{\sim }$-type and $\mathop{V}\limits^{\sim }\mathop{V}\limits^{\sim }$-type scalar tetraquark currents, and the ST-type tensor tetraquark current to study the Z_c(4100) with the QCD sum rules, and try to assign the Z_c(4100) in the scenario of scalar and vector tetraquark states once more.

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

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

$$\begin{eqnarray}&&\begin{array}{l}\Pi (p)={\rm{i}}\displaystyle \int {{\rm{d}}}^{4}x{{\rm{e}}}^{{\rm{i}}p\cdot x}\left\langle 0\left|T\left\{J(x){J}^{\dagger }(0)\right\}\right|0\right\rangle,\\ {\Pi }_{\mu \nu \alpha \beta }(p)={\rm{i}}\displaystyle \int {{\rm{d}}}^{4}x{{\rm{e}}}^{{\rm{i}}p\cdot x}\left\langle 0\left|T\left\{{J}_{\mu \nu }(x){J}_{\alpha \beta }^{\dagger }(0)\right\}\right|0\right\rangle,\end{array}\end{eqnarray} \tag{ 5 }$$

where $J(x)={J}_{\mathop{A}\limits^{\sim }\mathop{A}\limits^{\sim }}(x)$, ${J}_{\mathop{V}\limits^{\sim }\mathop{V}\limits^{\sim }}(x)$,

$$\begin{eqnarray}\begin{array}{lll}{J}_{\mathop{A}\limits^{\sim }\mathop{A}\limits^{\sim }}(x) & = & {\varepsilon }^{ijk}{\varepsilon }^{imn}{u}^{Tj}(x)C{\sigma }_{\mu \nu }^{v}{c}^{k}(x){\bar{d}}^{m}(x){\sigma }_{v}^{\mu \nu }C{\bar{c}}^{Tn}(x),\\ {J}_{\mathop{V}\limits^{\sim }\mathop{V}\limits^{\sim }}(x) & = & {\varepsilon }^{ijk}{\varepsilon }^{imn}{u}^{Tj}(x)C{\sigma }_{\mu \nu }^{t}{c}^{k}(x){\bar{d}}^{m}(x){\sigma }_{t}^{\mu \nu }C{\bar{c}}^{Tn}(x),\\ {J}_{\mu \nu }(x) & = & \frac{{\varepsilon }^{ijk}{\varepsilon }^{imn}}{\sqrt{2}}\left[{u}^{Tj}(x)C{\gamma }_{5}{c}^{k}(x){\bar{d}}^{m}(x){\sigma }_{\mu \nu }C{\bar{c}}^{Tn}(x)\right.\\ & & \left.-{u}^{Tj}(x)C{\sigma }_{\mu \nu }{c}^{k}(x){\bar{d}}^{m}(x){\gamma }_{5}C{\bar{c}}^{Tn}(x)\right],\end{array}\end{eqnarray} \tag{ 6 }$$

the i, j, k, m, n are color indexes. We take the isospin limit by assuming the u and d quarks have degenerate masses. Under charge conjugation transform $\hat{C}$, the currents J(x) and J_μν(x) have the properties,

$$\begin{eqnarray}&&\begin{array}{l}\hat{C}J(x){\hat{C}}^{-1}=+J(x),\\ \hat{C}{J}_{\mu \nu }(x){\hat{C}}^{-1}=-{J}_{\mu \nu }(x),\end{array}\end{eqnarray} \tag{ 7 }$$

the currents have definite charge conjugation. The currents J(x) couple potentially to the scalar tetraquark states, while the current J_μν(x) couples potentially to both the J^PC = 1⁺⁻ and 1⁻⁻ tetraquark states,

$$\begin{eqnarray}&&\begin{array}{l}\left\langle 0\left|J(0)\right|{Z}_{c}^{+}(p)\right\rangle ={\lambda }_{{Z}^{+}},\\ \left\langle 0\left|{J}_{\mu \nu }(0)\right|{Z}_{c}^{-}(p)\right\rangle =\frac{{\lambda }_{{Z}^{-}}}{{M}_{{Z}^{-}}}{\varepsilon }_{\mu \nu \alpha \beta }{\varepsilon }^{\alpha }{p}^{\beta },\\ \left\langle 0\left|{J}_{\mu \nu }(0)\right|{Z}_{c}^{+}(p)\right\rangle =\frac{{\lambda }_{{Z}^{+}}}{{M}_{{Z}^{+}}}({\varepsilon }_{\mu }{p}_{\nu }-{\varepsilon }_{\nu }{p}_{\mu }),\end{array}\end{eqnarray} \tag{ 8 }$$

the ε_μ/α are the polarization vectors of the tetraquark states, the superscripts^± denote the positive parity and negative parity, respectively, the M_Z^± and λ_Z^± are the masses and pole residues, respectively.

We 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,^{[24, 25]} and isolate the ground state tetraquark contributions,

$$\begin{eqnarray}&&\begin{array}{l}\Pi (p)=\frac{{\lambda }_{{Z}^{+}}^{2}}{{M}_{{Z}^{+}}^{2}-{p}^{2}}+\cdots =\Pi ({p}^{2}),\\ \begin{array}{lll}{\Pi }_{\mu \nu \alpha \beta }(p) & = & \frac{{\lambda }_{{Z}^{-}}^{2}}{{M}_{{Z}^{-}}^{2}({M}_{{Z}^{-}}^{2}-{p}^{2})}({p}^{2}{g}_{\mu \alpha }{g}_{\nu \beta }-{p}^{2}{g}_{\mu \beta }{g}_{\nu \alpha }-{g}_{\mu \alpha }{p}_{\nu }{p}_{\beta }-{g}_{\nu \beta }{p}_{\mu }{p}_{\alpha }+{g}_{\mu \beta }{p}_{\nu }{p}_{\alpha }+{g}_{\nu \alpha }{p}_{\mu }{p}_{\beta })\\ & & +\frac{{\lambda }_{{Z}^{+}}^{2}}{{M}_{{Z}^{+}}^{2}({M}_{{Z}^{+}}^{2}-{p}^{2})}(-{g}_{\mu \alpha }{p}_{\nu }{p}_{\beta }-{g}_{\nu \beta }{p}_{\mu }{p}_{\alpha }+{g}_{\mu \beta }{p}_{\nu }{p}_{\alpha }+{g}_{\nu \alpha }{p}_{\mu }{p}_{\beta })+\cdots \\ & = & {\mathop{\Pi }\limits^{\sim }}_{-}({p}^{2})({p}^{2}{g}_{\mu \alpha }{g}_{\nu \beta }-{p}^{2}{g}_{\mu \beta }{g}_{\nu \alpha }-{g}_{\mu \alpha }{p}_{\nu }{p}_{\beta }-{g}_{\nu \beta }{p}_{\mu }{p}_{\alpha }+{g}_{\mu \beta }{p}_{\nu }{p}_{\alpha }+{g}_{\nu \alpha }{p}_{\mu }{p}_{\beta })\\ & & +{\mathop{\Pi }\limits^{\sim }}_{+}({p}^{2})(-{g}_{\mu \alpha }{p}_{\nu }{p}_{\beta }-{g}_{\nu \beta }{p}_{\mu }{p}_{\alpha }+{g}_{\mu \beta }{p}_{\nu }{p}_{\alpha }+{g}_{\nu \alpha }{p}_{\mu }{p}_{\beta })\end{array}\end{array}\end{eqnarray} \tag{ 9 }$$

We can project out the components Π_±(p²) explicitly by introducing the operators ${P}_{{Z}^{\pm }}^{\mu \nu \alpha \beta }$,

$$\begin{eqnarray}&&{\Pi }_{\pm }({p}^{2})={p}^{2}{\mathop{\Pi }\limits^{\sim }}_{\pm }({p}^{2})={P}_{{Z}^{\pm }}^{\mu \nu \alpha \beta }{\Pi }_{\mu \nu \alpha \beta }(p),\end{eqnarray} \tag{ 10 }$$

where

$$\begin{eqnarray}\begin{array}{lll}{P}_{{Z}^{-}}^{\mu \nu \alpha \beta } & = & \frac{1}{6}\left({g}^{\mu \alpha }-\frac{{p}^{\mu }{p}^{\alpha }}{{p}^{2}}\right)\left({g}^{\nu \beta }-\frac{{p}^{\nu }{p}^{\beta }}{{p}^{2}}\right),\\ {P}_{{Z}^{+}}^{\mu \nu \alpha \beta } & = & \frac{1}{6}\left({g}^{\mu \alpha }-\frac{{p}^{\mu }{p}^{\alpha }}{{p}^{2}}\right)\left({g}^{\nu \beta }-\frac{{p}^{\nu }{p}^{\beta }}{{p}^{2}}\right)-\frac{1}{6}{g}^{\mu \alpha }{g}^{\nu \beta }.\end{array}\end{eqnarray} \tag{ 11 }$$

In this article, we choose the correlation functions Π(p²), Π_-(p²) and Π₊(p²) to study the scalar, vector and axialvector tetraquark states, respectively.

If we cannot project out the components Π_±(p²) explicitly, we have to choose the currents ${J}_{\mu \nu }^{\mathop{V}\limits^{\sim }}(x)$ and ${J}_{\mu \nu }^{\mathop{A}\limits^{\sim }}(x)$,

$$\begin{eqnarray}&&\begin{array}{l}{J}_{\mu \nu }^{\mathop{V}\limits^{\sim }}(x)=\frac{{\varepsilon }^{ijk}{\varepsilon }^{imn}}{\sqrt{2}}[{u}^{Tj}(x)C{\gamma }_{5}{c}^{k}(x){\bar{d}}^{m}(x){\sigma }_{\mu \nu }^{t}C{\bar{c}}^{Tn}(x)-{u}^{Tj}(x)C{\sigma }_{\mu \nu }^{t}{c}^{k}(x){\bar{d}}^{m}(x){\gamma }_{5}C{\bar{c}}^{Tn}(x)],\\ {J}_{\mu \nu }^{\mathop{A}\limits^{\sim }}(x)=\frac{{\varepsilon }^{ijk}{\varepsilon }^{imn}}{\sqrt{2}}[{u}^{Tj}(x)C{\gamma }_{5}{c}^{k}(x){\bar{d}}^{m}(x){\sigma }_{\mu \nu }^{v}C{\bar{c}}^{Tn}(x)-{u}^{Tj}(x)C{\sigma }_{\mu \nu }^{v}{c}^{k}(x){\bar{d}}^{m}(x){\gamma }_{5}C{\bar{c}}^{Tn}(x)],\end{array}\end{eqnarray} \tag{ 12 }$$

which couple to the J^P = 1⁻ and 1⁺ tetraquark states respectively,

$$\begin{eqnarray}&&\begin{array}{l}\left\langle 0\left|{J}_{\mu \nu }^{\mathop{V}\limits^{\sim }}(0)\right|{Z}_{c}^{-}(p)\right\rangle =\frac{{\mathop{\lambda }\limits^{\sim }}_{{Z}^{-}}}{{M}_{{Z}^{-}}}({\varepsilon }_{\mu }{p}_{\nu }-{\varepsilon }_{\nu }{p}_{\mu }),\\ \left\langle 0\left|{J}_{\mu \nu }^{\mathop{A}\limits^{\sim }}(0)\right|{Z}_{c}^{+}(p)\right\rangle =\frac{{\mathop{\lambda }\limits^{\sim }}_{{Z}^{+}}}{{M}_{{Z}^{+}}}({\varepsilon }_{\mu }{p}_{\nu }-{\varepsilon }_{\nu }{p}_{\mu }),\end{array}\end{eqnarray} \tag{ 13 }$$

as the current operators ${J}_{\mu \nu }^{\mathop{V}\limits^{\sim }}(x)$ and ${J}_{\mu \nu }^{\mathop{A}\limits^{\sim }}(x)$ have the properties,

$$\begin{eqnarray}&&\begin{array}{l}\hat{P}{J}_{\mu \nu }^{\mathop{V}\limits^{\sim }}(x){\hat{P}}^{-1}=-{J}_{\mu \nu }^{\mathop{V}\limits^{\sim }}(\mathop{x}\limits^{\sim }),\\ \hat{P}{J}_{\mu \nu }^{\mathop{A}\limits^{\sim }}(x){\hat{P}}^{-1}=+{J}_{\mu \nu }^{\mathop{A}\limits^{\sim }}(\mathop{x}\limits^{\sim }),\end{array}\end{eqnarray} \tag{ 14 }$$

under parity transform $\hat{P}$.

It is more easy to carry out the operator product expansion for the current J_μν(x) than for the currents ${J}_{\mu \nu }^{\mathop{V}\limits^{\sim }}(x)$ and ${J}_{\mu \nu }^{\mathop{A}\limits^{\sim }}(x)$. In this article, we prefer the current J_μν(x), and denote the corresponding vector and axialvector tetraquark states as $S\mathop{V}\limits^{\sim }$ type and $S\mathop{A}\limits^{\sim }$ type, respectively.

We carry out the operator product expansion for the correlation functions up to the vacuum condensates of dimension 10 in a consistent way, then obtain the QCD spectral densities through dispersion relation, take the quark-hadron duality below the continuum threshold s₀ and perform Borel transform with respect to P² = −p² to obtain the QCD sum rules:

$$\begin{eqnarray}&&{\lambda }_{Z}^{2}\exp \left(-\frac{{M}_{Z}^{2}}{{T}^{2}}\right)=\displaystyle {\int }_{4{m}_{c}^{2}}^{{s}_{0}}{\rm{d}}s\rho (s)\exp \left(-\frac{s}{{T}^{2}}\right),\end{eqnarray} \tag{ 15 }$$

the ρ(s) are the QCD spectral densities. The explicit expressions of the QCD spectral densities are available upon request by contacting me with E-mail. For the technical details, one can consult Refs. [8, 23]. In carrying out the operator product expansion for the correlation functions Π(p), we encounter the terms (p · t)², and set (p · t)² = p², just like in the QCD sum rules for the baryon states separating the contributions of the positive parity and negative parity, where we take the four vector ${p}^{\mu }=({p}_{0},\overrightarrow{0})$.^[26]

We derive Eq. (15) with respect to $\tau =\frac{1}{{T}^{2}}$, and obtain the QCD sum rules for the masses of the scalar, vector and axialvector tetraquark states Z_c through a ratio,

$$\begin{eqnarray}&&{M}_{Z}^{2}=-\frac{\displaystyle {\int }_{4{m}_{c}^{2}}^{{s}_{0}}{\rm{d}}s\frac{{\rm{d}}}{{\rm{d}}\tau }\rho (s)\exp (-\tau s)}{\displaystyle {\int }_{4{m}_{c}^{2}}^{{s}_{0}}{\rm{d}}s\rho (s)\exp (-\tau s)}.\end{eqnarray} \tag{ 16 }$$

We choose the standard values of the vacuum condensates $\langle \bar{q}q\rangle =-{(0.24\pm 0.01{\rm{GeV}})}^{3}$, $\langle \bar{q}{g}_{s}\sigma Gq\rangle ={m}_{0}^{2}\langle \bar{q}q\rangle$, ${m}_{0}^{2}=(0.8\pm 0.1){{\rm{GeV}}}^{2}$, $\langle \frac{{\alpha }_{s}GG}{\pi }\rangle ={(0.33{\rm{GeV}})}^{4}$ at the energy scale μ = 1 GeV,^[24,25,27] and choose the $\overline{MS}$ mass m_c(m_c) = (1.275 ± 0.025) GeV from the Particle Data Group,^[28] and set m_u = m_d = 0. Moreover, we take into account the energy-scale dependence of the input parameters at the QCD side,

$$\begin{eqnarray}&&\begin{array}{l}\left\langle \bar{q}\right\rangle (\mu )=\left\langle \bar{q}q\right\rangle (1{\rm{GeV}}){\left[\frac{{\alpha }_{s}(1{\rm{GeV}})}{{\alpha }_{s}(\mu )}\right]}^{\frac{12}{25}},\\ \left\langle \bar{q}{g}_{s}\sigma Gq\right\rangle (\mu )=\left\langle \bar{q}{g}_{s}\sigma Gq\right\rangle (1{\rm{GeV}}){\left[\frac{{\alpha }_{s}(1{\rm{GeV}})}{{\alpha }_{s}(\mu )}\right]}^{\frac{2}{25}},\\ {m}_{c}(\mu )={m}_{c}({m}_{c}){\left[\frac{{\alpha }_{s}(\mu )}{{\alpha }_{s}({m}_{c})}\right]}^{\frac{12}{25}},\\ {\alpha }_{s}(\mu )=\frac{1}{{b}_{0}t}\left[1-\frac{{b}_{1}}{{b}_{0}^{2}}\frac{\mathrm{log}t}{t}+\frac{{b}_{1}^{2}({\mathrm{log}}^{2}t-\mathrm{log}t-1)+{b}_{0}{b}_{2}}{{b}_{0}^{4}{t}^{2}}\right],\end{array}\end{eqnarray} \tag{ 17 }$$

where

$$\begin{eqnarray*}&&\begin{array}{l}t=\mathrm{log}\frac{{\mu }^{2}}{{\Lambda }^{2}},\,{b}_{0}=\frac{33-2{n}_{f}}{12\pi },\\ {b}_{1}=\frac{153-19{n}_{f}}{24{\pi }^{2}},\,{b}_{2}=\frac{2857-\frac{5033}{9}{n}_{f}+\frac{325}{27}{n}_{f}^{2}}{128{\pi }^{3}},\end{array}\end{eqnarray*}$$

Λ = 210 MeV, 292 MeV and 332 MeV for the flavors n_f = 5, 4 and 3, respectively,^{[28, 29]} and evolve all the input parameters to the ideal energy scales μ to extract the tetraquark masses. In the present work, we choose the flavor n_f = 4.

Now we search for the ideal Borel parameters T² and continuum threshold parameters s₀ to satisfy the four criteria:

• (a)  
  Pole dominance at the hadron side;
• (b)  
  Convergence of the operator product expansion;
• (c)  
  Appearance of the Borel platforms;
• (d)  
  Satisfying the energy scale formula,

via try and error. The resulting Borel parameters, continuum threshold parameters, energy scales of the QCD spectral densities and pole contributions are shown explicitly in Table 1. From the Table, we can see that the pole contributions are about (40 − 60)%, the pole dominance condition at the hadron side is well satisfied. In calculations, we observe that the contributions of the vacuum condensates of dimension 10 are ≪ 1%, the operator product expansion is well convergent.

We take into account the uncertainties of the input parameters and obtain the masses and pole residues of the tetraquark states, which are shown explicitly in Table 2 and in Figs. 1 and 2. From Tables 1 and 2, we can see that the energy scale formula $\mu =\sqrt{{M}_{X/Y/Z}^{2}-{(2{{\mathbb{M}}}_{c})}^{2}}$ is well satisfied, where we take the updated value of the effective c-quark mass ${{\mathbb{M}}}_{c}=1.82{\rm{GeV}}$.^[22] In Figs. 1 and 2, we plot the masses and pole residues of the tetraquark states with variations of the Borel parameters at much larger ranges than the Borel widows, the regions between the two perpendicular lines are the Borel windows. In the Borel windows, the uncertainties induced by the Borel parameters are ≪ 1% for the masses and ≤ 2% for the pole residues, there appear Borel platforms. Now the four criteria are all satisfied, we expect to make reliable predictions.

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Table 1. The Borel parameters, continuum threshold parameters, energy scales of the QCD spectral densities and pole contributions of the ground state tetraquark states, where the superscript^* denotes the energy scale formula is not satisfied.

Zc	T2 (GeV2)	$\sqrt{{s}_{0}}({\rm{GeV}})$	μ(GeV)	pole
${[uc]}_{\mathop{A}\limits^{\sim }}{[\bar{d}\bar{c}]}_{\mathop{A}\limits^{\sim }}$	3.1–3.5	4.55 ± 0.10	1.6	(42 − 62)%
${[uc]}_{\mathop{V}\limits^{\sim }}{[\bar{d}\bar{c}]}_{\mathop{V}\limits^{\sim }}$	4.9–5.5	5.90 ± 0.10	3.9	(43 − 61)%
${[uc]}_{S}{[\bar{d}\bar{c}]}_{\mathop{V}\limits^{\sim }}-{[uc]}_{\mathop{V}\limits^{\sim }}{[\bar{d}\bar{c}]}_{S}$	3.8–4.2	5.18 ± 0.10	2.8	(43 − 61)%
${[uc]}_{S}{[\bar{d}\bar{c}]}_{\mathop{A}\limits^{\sim }}-{[uc]}_{\mathop{A}\limits^{\sim }}{[\bar{d}\bar{c}]}_{S}$	3.1–3.5	4.56 ± 0.10	1.6	(42 − 62)%
${[uc]}_{\mathop{A}\limits^{\sim }}{[\bar{d}\bar{c}]}_{\mathop{A}\limits^{\sim }}^{* }$	3.3–3.7	4.70 ± 0.10	1.4	(41 − 61)%
${[uc]}_{\mathop{A}\limits^{\sim }}{[\bar{d}\bar{c}]}_{\mathop{A}\limits^{\sim }}\oplus {[uc]}_{\mathop{V}\limits^{\sim }}{[\bar{d}\bar{c}]}_{\mathop{V}\limits^{\sim }}$	3.1–3.5	4.70 ± 0.10	1.9	(42 − 62)%

Table 2. The masses and pole residues of the ground state tetraquark states, where the superscript^* denotes the energy scale formula is not satisfied.

Zc	MZ (GeV)	λZ (GeV5)
${[uc]}_{\mathop{A}\limits^{\sim }}{[\bar{d}\bar{c}]}_{\mathop{A}\limits^{\sim }}$	3.98 ± 0.08	(4.30 ± 0.63) × 10−2
${[uc]}_{\mathop{V}\limits^{\sim }}{[\bar{d}\bar{c}]}_{\mathop{V}\limits^{\sim }}$	5.35 ± 0.09	(4.86 ± 0.50) × 10−1
${[uc]}_{S}{[\bar{d}\bar{c}]}_{\mathop{V}\limits^{\sim }}-{[uc]}_{\mathop{V}\limits^{\sim }}{[\bar{d}\bar{c}]}_{S}$	4.61 ± 0.08	(6.15 ± 0.80) × 10−2
${[uc]}_{S}{[\bar{d}\bar{c}]}_{\mathop{A}\limits^{\sim }}-{[uc]}_{\mathop{A}\limits^{\sim }}{[\bar{d}\bar{c}]}_{S}$	3.99 ± 0.09	(2.73 ± 0.41) × 10−2
${[uc]}_{\mathop{A}\limits^{\sim }}{[\bar{d}\bar{c}]}_{\mathop{A}\limits^{\sim }}^{* }$	4.12 ± 0.08	(4.84 ± 0.71) × 10−2
${[uc]}_{\mathop{A}\limits^{\sim }}{[\bar{d}\bar{c}]}_{\mathop{A}\limits^{\sim }}\oplus {[uc]}_{\mathop{V}\limits^{\sim }}{[\bar{d}\bar{c}]}_{\mathop{V}\limits^{\sim }}$	4.10 ± 0.09	(1.43 ± 0.24) × 10−1

In Ref. [30], we study the tensor-tensor type scalar hidden-charm tetraquark states with currents

$$\begin{eqnarray}&&\eta (x)={\varepsilon }^{ijk}{\varepsilon }^{imn}{q}^{Tj}(x)C{\sigma }_{\mu \nu }{c}^{k}(x){\bar{q}}^{m}(x){\sigma }^{\mu \nu }C{\bar{c}}^{Tn}(x),\end{eqnarray} \tag{ 18 }$$

via the QCD sum rules by taking into account both the ground state contributions and the first radial excited state contributions, where ${\sigma }_{\mu \nu }=\frac{{\rm{i}}}{2}[{\gamma }_{\mu },{\gamma }_{\nu }]={\sigma }_{\mu \nu }^{t}+{\sigma }_{\mu \nu }^{v}$, and obtain masses M_TT,1S = 3.82 ± 0.16 GeV and M_{TT, 2S} = 4.38 ± 0.09 GeV. We can rewrite the current η(x) as a linear superposition $\eta (x)={J}_{\mathop{V}\limits^{\sim }\mathop{V}\limits^{\sim }}(x)+2{J}_{\mathop{A}\limits^{\sim }\mathop{A}\limits^{\sim }}(x)$, the tensor-tensor type scalar hidden-charm tetraquark states have both the $\mathop{V}\limits^{\sim }\mathop{V}\limits^{\sim }$ and $\mathop{A}\limits^{\sim }\mathop{A}\limits^{\sim }$ components. Compared to the $\mathop{A}\limits^{\sim }\mathop{A}\limits^{\sim }$ ($\mathop{V}\limits^{\sim }\mathop{V}\limits^{\sim }$) type tetraquark mass ${M}_{\mathop{A}\limits^{\sim }\mathop{A}\limits^{\sim }}=3.98\pm 0.08{\rm{GeV}}$ (${M}_{\mathop{V}\limits^{\sim }\mathop{V}\limits^{\sim }}=5.35\pm 0.09{\rm{GeV}}$), the ground state mass M_TT,1S = 3.82 ± 0.16 GeV is (much) lower. In the QCD sum rules for the tensor-tensor type scalar tetraquark states, the terms ${m}_{c}\langle \bar{q}q\rangle$ and ${m}_{c}\langle \bar{q}{g}_{s}\sigma Gq\rangle$ disappear due to the special superposition $\eta (x)={J}_{\mathop{V}\limits^{\sim }\mathop{V}\limits^{\sim }}(x)+2{J}_{\mathop{A}\limits^{\sim }\mathop{A}\limits^{\sim }}(x)$, the contamination from the $\mathop{V}\limits^{\sim }\mathop{V}\limits^{\sim }$ component is large.

In Fig. 1, we also present the experimental values of the masses of the Z_c(4100) and Z_c(4020).^{[13,28,31,32]} From the figure, we can see that the predicted mass for the ${[uc]}_{S}{[\bar{d}\bar{c}]}_{\mathop{A}\limits^{\sim }}-{[uc]}_{\mathop{A}\limits^{\sim }}{[\bar{d}\bar{c}]}_{S}$ type axialvector tetraquark state is in excellent agreement with the experimental data in the Borel window, and favors assigning the Z_c(4020) to be the ${[uc]}_{S}{[\bar{d}\bar{c}]}_{\mathop{A}\limits^{\sim }}-{[uc]}_{\mathop{A}\limits^{\sim }}{[\bar{d}\bar{c}]}_{S}$ type axialvector tetraquark state, while the Z_c(4100) lies above the $\mathop{A}\limits^{\sim }\mathop{A}\limits^{\sim }$-type scalar tetraquark state, and much below the $\mathop{V}\limits^{\sim }\mathop{V}\limits^{\sim }$-type scalar tetraquark state and $S\mathop{V}\limits^{\sim }$-type vector tetraquark state in the Borel windows, the present QCD sum rules do not favor assigning the Z_c(4100) to be the scalar or vector tetraquark state. The masses of the scalar hidden-charm tetraquark states have the hierarchy,^[9,10,11]

$$\begin{eqnarray}&&\begin{array}{l}{M}_{SS}\le {M}_{AA}\le {M}_{\mathop{A}\limits^{\sim }\mathop{A}\limits^{\sim }}\lt {M}_{{Z}_{c}(4100)}\\ \ll {M}_{VV}\ll {M}_{\mathop{V}\limits^{\sim }\mathop{V}\limits^{\sim }}\le {M}_{PP},\end{array}\end{eqnarray} \tag{ 19 }$$

the QCD sum rules disfavor assigning the Z_c(4100) to be a scalar tetraquark state.

The masses extracted from the QCD sum rules depend on the Borel windows, different Borel windows lead to different predicted masses. From Fig. 1(a), we can see that if we choose larger Borel parameter for the $\mathop{A}\limits^{\sim }\mathop{A}\limits^{\sim }$-type scalar tetraquark state, for example, choose T² > 4.2 GeV² rather than choose T² = (3.1 − 3.5) GeV², we can obtain a mass about 4.1 GeV, which is compatible with the experimental value of the mass of the Z_c(4100). If we choose the Borel window T² = (4.9 − 5.3) GeV², the predicted mass ${M}_{\mathop{A}\limits^{\sim }\mathop{A}\limits^{\sim }}=4.09\pm 0.08{\rm{GeV}}$, which is in excellent agreement with the experimental data ${M}_{{Z}_{c}}=4096\pm {20}_{-22}^{+18}{\rm{MeV}}$.^[13] However, the pole contribution is about (14−24)%, the prediction is not robust.

The mass ${M}_{\mathop{A}\limits^{\sim }\mathop{A}\limits^{\sim }}$ extracted from the QCD sum rules decreases monotonously with increase of the energy scales of the QCD spectral density. If we choose μ = 1.4 GeV, T² = (3.3 − 3.7) GeV², $\sqrt{{s}_{0}}=4.70\pm 0.10{\rm{GeV}}$, the pole contribution is (41 − 61)% and the operator product expansion is well convergent, we obtain the tetraquark mass ${M}_{\mathop{A}\limits^{\sim }\mathop{A}\limits^{\sim }}=4.12\pm 0.08{\rm{GeV}}$, which is in excellent agreement with the experimental data ${M}_{{Z}_{c}}=4096\pm {20}_{-22}^{+18}{\rm{MeV}}$,^[13] see Fig. 3. In Fig. 3, we plot the mass ${M}_{\mathop{A}\limits^{\sim }\mathop{A}\limits^{\sim }}$ extracted at the energy scale μ = 1.4 GeV with variation of the Borel parameter, again the region between the two perpendicular lines is the Borel windows. However, the energy scale formula $\mu =\sqrt{{M}_{X/Y/Z}^{2}-{(2{{\mathbb{M}}}_{c})}^{2}}$ is not satisfied. In the QCD sum rules for the hidden-charm (or hidden-bottom) tetraquark states and molecular states, the integrals

$$\begin{eqnarray}&&\displaystyle {\int }_{4{m}_{Q}^{2}(\mu )}^{{s}_{0}}{\rm{d}}s{\rho }_{QCD}(s,\mu )\exp \left(-\frac{s}{{T}^{2}}\right),\end{eqnarray} \tag{ 20 }$$

are sensitive to the energy scales μ. We suggest an energy scale formula $\mu =\sqrt{{M}_{X/Y/Z}^{2}-{(2{{\mathbb{M}}}_{Q})}^{2}}$ with the effective Q-quark mass ${{\mathbb{M}}}_{Q}$ to determine the ideal energy scales of the QCD spectral densities in a consistent way.^[23]

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The energy scale formula works well for the tetraquark states,^{[8–11, 22, 23, 33, 34]} tetraquark molecular states,^[35] and even for the hidden-charm pentaquark states.^[36] For example, in Refs. [8,22,33] and the present work, we observe that there exist one axialvector tetraquark candidate ${[uc]}_{S}{[\bar{d}\bar{c}]}_{A}-{[uc]}_{A}{[\bar{d}\bar{c}]}_{S}$ for the Z_c(3900), three axialvector tetraquark candidates ${[uc]}_{A}{[\bar{d}\bar{c}]}_{A}$, ${[uc]}_{\mathop{A}\limits^{\sim }}{[\bar{d}\bar{c}]}_{A}-{[uc]}_{A}{[\bar{d}\bar{c}]}_{\mathop{A}\limits^{\sim }}$ and ${[uc]}_{S}{[\bar{d}\bar{c}]}_{\mathop{A}\limits^{\sim }}-{[uc]}_{\mathop{A}\limits^{\sim }}{[\bar{d}\bar{c}]}_{S}$ for the Z_c(4020), which is consistent with the almost degenerate scalar and axialvector heavy diquark masses from the QCD sum rules [3]. Furthermore, the Z_c(4430) can be assigned to be the first radial excited state of the Z_c(3900). If the Z_c(4100) is a diquark-antidiquark type scalar tetraquark state, the energy scale formula should be satisfied, as the Z_c(4100) has no reason to be an odd or special tetraquark state. The masses of the scalar tetraquark states SS, AA and $\mathop{A}\limits^{\sim }\mathop{A}\limits^{\sim }$ are estimated to be 3.9 – 4.0 GeV, if the spin-spin interactions are neglected, the QCD sum rules support this estimation.

In Ref. [12], Sundu, Agaev and Azizi choose the SS type scalar current to study the mass and width of the Z_c(4100), and obtain the mass M_SS = 4.08 ± 0.15 GeV with the pole contribution (19 − 54)%. In Ref. [11], we tentatively assign the X^*(3860) to be the ${[qc]}_{S}{[\bar{q}\bar{c}]}_{S}$-type scalar tetraquark state, study its mass and width with the QCD sum rules, and obtain the mass M_SS = 3.86 ± 0.09 GeV with the pole contribution (46 − 70)%, which is much larger than the pole contribution (19 − 54)% in Ref. [12]. In Ref. [11], just like in the present work, we use the energy scale formula $\mu =\sqrt{{M}_{X/Y/Z}^{2}-{(2{{\mathbb{M}}}_{c})}^{2}}$ to enhance the pole contribution. In the QCD sum rules, we prefer larger pole contributions to obtain more robust predictions.

Now we assume that the Z_c(4100) is a $\mathop{A}\limits^{\sim }\mathop{A}\limits^{\sim }\oplus \mathop{V}\limits^{\sim }\mathop{V}\limits^{\sim }$ type mixing scalar tetraquark state, and study it with the current J(x),

$$\begin{eqnarray}&&J(x)=2{J}_{\mathop{A}\limits^{\sim }\mathop{A}\limits^{\sim }}(x)\cos \theta +{J}_{\mathop{V}\limits^{\sim }\mathop{V}\limits^{\sim }}(x)\sin \theta,\end{eqnarray} \tag{ 21 }$$

where we introduce the factor 2 due to the relation ${\sigma }_{\mu \nu }={\sigma }_{\mu \nu }^{t}+{\sigma }_{\mu \nu }^{v}$. If we choose the mixing angle θ = 26.4°, the energy scale μ = 1.9 GeV, the Borel parameter T² = (3.1 − 3.5) GeV², the continuum threshold parameter $\sqrt{{s}_{0}}=4.70\pm 0.10{\rm{GeV}}$, the pole contribution is (42 − 62)%, the operator product expansion is well convergent, the energy scale formula is also satisfied. We obtain the tetraquark mass ${M}_{\mathop{A}\limits^{\sim }\mathop{A}\limits^{\sim }\oplus \mathop{V}\limits^{\sim }\mathop{V}\limits^{\sim }}=4.10\pm 0.09{\rm{GeV}}$, which is in excellent agreement with the experimental data ${M}_{{Z}_{c}}=4096\pm {20}_{-22}^{+18}{\rm{MeV}}$,^[13] see Fig. 4. If the Z_c(4100) is a diquark-antidiquark type tetraquark state, it may be a $\mathop{A}\limits^{\sim }\mathop{A}\limits^{\sim }\oplus \mathop{V}\limits^{\sim }\mathop{V}\limits^{\sim }$ type mixing scalar tetraquark state.

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We can also introduce more parameters and write down the most general scalar current J(x),

$$\begin{eqnarray}\begin{array}{lll}J(x) & = & 2{J}_{\mathop{A}\limits^{\sim }\mathop{A}\limits^{\sim }}(x){t}_{1}+{J}_{\mathop{V}\limits^{\sim }\mathop{V}\limits^{\sim }}(x){t}_{2}+{J}_{SS}(x){t}_{3}\\ & & +{J}_{PP}(x){t}_{4}+{J}_{AA}(x){t}_{5}+{J}_{VV}(x){t}_{6},\end{array}\end{eqnarray} \tag{ 22 }$$

where

$$\begin{eqnarray}\begin{array}{lll}{J}_{SS}(x) & = & {\varepsilon }^{ijk}{\varepsilon }^{imn}{u}_{j}^{T}(x)C{\gamma }_{5}{c}_{k}(x){\bar{d}}_{m}(x){\gamma }_{5}C{\bar{c}}_{n}^{T}(x),\,{J}_{PP}(x)={\varepsilon }^{ijk}{\varepsilon }^{imn}{u}_{j}^{T}(x)C{c}_{k}(x){\bar{d}}_{m}(x)C{\bar{c}}_{n}^{T}(x),\\ {J}_{AA}(x) & = & {\varepsilon }^{ijk}{\varepsilon }^{imn}{u}_{j}^{T}(x)C{\gamma }_{\mu }{c}_{k}(x){\bar{d}}_{m}(x){\gamma }^{\mu }C{\bar{c}}_{n}^{T}(x),\,{J}_{VV}(x)={\varepsilon }^{ijk}{\varepsilon }^{imn}{u}_{j}^{T}(x)C{\gamma }_{\mu }{\gamma }_{5}{c}_{k}(x){\bar{d}}_{m}(x){\gamma }_{5}{\gamma }^{\mu }C{\bar{c}}_{n}^{T}(x),\end{array}\end{eqnarray} \tag{ 23 }$$

the t_i with i = 1, 2, 3, ··· are arbitrary parameters. We can obtain any value between the largest mass M_PP and the smallest mass M_SS by fine tuning the parameters t_i, see Eq. (19). In fact, we cannot assign a tetraquark state unambiguously with the mass alone, we have to study the decays exclusively to obtain the partial decay widths and confront the predictions to experimental data in the future. The cumbersome calculations may be our next work. In the present work, we can obtain the conclusion tentatively that we cannot reproduce the experimental value of the mass of Z_c(4100) without introducing mixing effects.

In this article, we take the zero width approximation. In fact, we can take into account the finite width effect with the simple replacement of the hadronic spectral density,

$$\begin{eqnarray}&&{\lambda }_{{Z}_{c}}^{2}\delta (s-{M}_{{Z}_{c}}^{2})\to {\lambda }_{{Z}_{c}}^{2}\frac{1}{\pi }\frac{{M}_{{Z}_{c}}{\Gamma }_{{Z}_{c}}(s)}{{(s-{M}_{{Z}_{c}}^{2})}^{2}+{M}_{{Z}_{c}}^{2}{\Gamma }_{{Z}_{c}}^{2}(s)},\end{eqnarray} \tag{ 24 }$$

where

$$\begin{eqnarray}&&{\Gamma }_{{Z}_{c}}(s)={\Gamma }_{{Z}_{c}}\frac{{M}_{{Z}_{c}}}{\sqrt{s}}\sqrt{\frac{[s-{({M}_{{\eta }_{c}}+{M}_{\pi })}^{2}][s-{({M}_{{\eta }_{c}}-{M}_{\pi })}^{2}]}{[{M}_{{Z}_{c}}^{2}-{({M}_{{\eta }_{c}}+{M}_{\pi })}^{2}][{M}_{{Z}_{c}}^{2}-{({M}_{{\eta }_{c}}-{M}_{\pi })}^{2}]}}.\end{eqnarray} \tag{ 25 }$$

Then the hadron sides of the QCD sum rules in Eqs. (15)–(16) undergo the changes,

$$\begin{eqnarray}\begin{array}{lll}{\lambda }_{{Z}_{c}}^{2}\exp \left(-\frac{{M}_{{Z}_{c}}^{2}}{{T}^{2}}\right)\, & \to & {\lambda }_{{Z}_{c}}^{2}\displaystyle {\int }_{{({M}_{{\eta }_{c}}+{M}_{\pi })}^{2}}^{{s}_{0}}{\rm{d}}s\frac{1}{\pi }\frac{{M}_{{Z}_{c}}{\Gamma }_{{Z}_{c}}(s)}{{(s-{M}_{{Z}_{c}}^{2})}^{2}+{M}_{{Z}_{c}}^{2}{\Gamma }_{{Z}_{c}}^{2}(s)}\exp \left(-\frac{s}{{T}^{2}}\right),\\ & = & 0.96{\lambda }_{{Z}_{c}}^{2}\exp \left(-\frac{{M}_{{Z}_{c}}^{2}}{{T}^{2}}\right),\end{array}\end{eqnarray} \tag{ 26 }$$

$$\begin{eqnarray}\begin{array}{lll}{\lambda }_{{Z}_{c}}^{2}{M}_{{Z}_{c}}^{2}\exp \left(-\frac{{M}_{{Z}_{c}}^{2}}{{T}^{2}}\right) & \to & {\lambda }_{{Z}_{c}}^{2}\displaystyle {\int }_{{({M}_{{\eta }_{c}}+{M}_{\pi })}^{2}}^{{s}_{0}}{\rm{d}}ss\frac{1}{\pi }\frac{{M}_{{Z}_{c}}{\Gamma }_{{Z}_{c}}(s)}{{(s-{M}_{{Z}_{c}}^{2})}^{2}+{M}_{{Z}_{c}}^{2}{\Gamma }_{{Z}_{c}}^{2}(s)}\exp \left(-\frac{s}{{T}^{2}}\right),\\ & = & 0.94{\lambda }_{{Z}_{c}}^{2}{M}_{{Z}_{c}}^{2}\exp \left(-\frac{{M}_{{Z}_{c}}^{2}}{{T}^{2}}\right),\end{array}\end{eqnarray} \tag{ 27 }$$

where we have used the central values of the input parameters for the ${[uc]}_{\mathop{A}\limits^{\sim }}{[\bar{d}\bar{c}]}_{\mathop{A}\limits^{\sim }}\oplus {[uc]}_{\mathop{V}\limits^{\sim }}{[\bar{d}\bar{c}]}_{\mathop{V}\limits^{\sim }}$ type scalar tetraquark state shown in Table 1 and the physical values of the mass and width of the Z_c(4100). We can absorb the numerical factors 0.96 and 0.94 into the pole residue λ_Zc safely with the simple replacement λ_Zc → (0.97 − 0.98) λ_Zc, the zero width approximation works well.

The predicted mass of the ${[uc]}_{S}{[\bar{d}\bar{c}]}_{\mathop{V}\limits^{\sim }}-{[uc]}_{\mathop{V}\limits^{\sim }}{[\bar{d}\bar{c}]}_{S}$ type vector tetraquark state ${M}_{S\mathop{V}\limits^{\sim }}=4.61\pm 0.08{\rm{GeV}}$ is much larger than the mass of the Z_c(4100), see Table 2. Furthermore, the mass of the Z_c(4100) is even smaller than the lowest vector hidden-charm tetraquark mass from the QCD sum rules,^[14,19]

$$\begin{eqnarray}&&\begin{array}{l}{M}_{{Z}_{c}(4100)}\lt {M}_{Y(4260/4220)}=4.24\pm 0.10{\rm{GeV}}\\ \ll {M}_{S\mathop{V}\limits^{\sim }}=4.61\pm 0.08{\rm{GeV}},\end{array}\end{eqnarray} \tag{ 28 }$$

the QCD sum rules also disfavor assigning the Z_c(4100) to be a vector tetraquark state.

If the Z_c(4100) is a SS, AA or $\mathop{A}\limits^{\sim }\mathop{A}\limits^{\sim }$ type scalar hidden-charm tetraquark state without mixing effects, it should have a mass about 3.9 GeV or 4.0 GeV rather than 4.1 GeV; on the other hand, if the Z_c(4100) is a vector hidden-charm tetraquark state, it should have a mass about 4.2 GeV rather than 4.1 GeV. However, a $\mathop{A}\limits^{\sim }\mathop{A}\limits^{\sim }\oplus \mathop{V}\limits^{\sim }\mathop{V}\limits^{\sim }$ type mixing scalar tetraquark state can have a mass about 4.1 GeV and reproduce the experimental value of the mass of the Z_c(4100).^[13]

The ${Z}_{c}^{-}(4100)$ is observed in the η_c π⁻ mass spectrum, the spin-parity assignments J^P = 0⁺ and 1⁻ are both consistent with the experimental data. We can search for its charge zero partner ${Z}_{c}^{0}(4100)$ in the ${\eta }_{c}{\pi }^{0}$ mass spectrum, if the η_c π⁰ is in relative S-wave, the ${Z}_{c}^{0}(4100)$ has the J^PC = 0⁺⁺, on the other hand, if the η_c π⁰ is in relative P-wave, the ${Z}_{c}^{0}(4100)$ has the ${J}^{PC}={1}^{-+}$. The quantum numbers ${J}^{PC}={1}^{-+}$ is exotic, we can also search for the ${Z}_{c}^{0}(4100)$ in the J/ψ ρ⁰ mass spectrum, which maybe shed light on the nature of the Z_c(4100). In Ref. [23], we observe that the diquark-antidiquark type vector $cu\bar{c}\bar{d}$ (or $cd\bar{c}\bar{u}$) tetraquark state with J^PC = 1⁻⁺ has smaller mass than the corresponding vector tetraquark state with J^PC = 1^−-, about 0.09 GeV. In the present case, we can choose the current η_μν(x),

$$\begin{eqnarray}\begin{array}{lll}{\eta }_{\mu \nu }(x) & = & \frac{{\varepsilon }^{ijk}{\varepsilon }^{imn}}{\sqrt{2}}\left[{u}^{Tj}(x)C{\gamma }_{5}{c}^{k}(x){\bar{d}}^{m}(x){\sigma }_{\mu \nu }C{\bar{c}}^{Tn}(x)\right.\\ & & \left.+{u}^{Tj}(x)C{\sigma }_{\mu \nu }{c}^{k}(x){\bar{d}}^{m}(x){\gamma }_{5}C{\bar{c}}^{Tn}(x)\right],\end{array}\end{eqnarray} \tag{ 29 }$$

which couples potentially to the vector tetraquark state with J^PC = 1⁻⁺, the mass of the vector tetraquark state with J^PC = 1⁻⁺ is estimated to be 4.52 ± 0.08 GeV, which is much larger than the experimental value of the mass of the ${Z}_{c}^{-}(4100)$, $4096\pm {20}_{-22}^{+18}{\rm{MeV}}$.

In this article, we separate the vector and axialvector components of the tensor diquark operators explicitly, construct the axialvector-axialvector-type and vector-vector type scalar tetraquark currents and scalar-tensor type tensor tetraquark current to study the scalar, vector and axialvector tetraquark states with the QCD sum rules by carrying out the operator product expansion up to vacuum condensates of dimension 10 in a consistent way. In calculation, we use the energy scale formula to determine the ideal energy scales of the QCD spectral densities to extract the masses from the QCD sum rules with the pole contributions about (40 − 60)%. The present calculations do not favor assigning the Z_c(4100) to be the scalar or vector tetraquark state. If the Z_c(4100) is a scalar tetraquark state without mixing effects, it should have a mass about 3.9 GeV or 4.0 GeV rather than 4.1 GeV; on the other hand, if the Z_c(4100) is a vector tetraquark state, it should have a mass about 4.2 GeV rather than 4.1 GeV. If we introduce mixing effects, a $\mathop{A}\limits^{\sim }\mathop{A}\limits^{\sim }\oplus \mathop{V}\limits^{\sim }\mathop{V}\limits^{\sim }$ type mixing scalar tetraquark state can have a mass about 4.1 GeV. More precise measurements of the mass, width and quantum numbers are still needed. As a byproduct, we obtain an axialvector tetraquark candidate for the Z_c(4020).