Lowest vector tetraquark states: $Y(4260/4220)$ or $Z_c(4100)$

In this article, we take the $Y(4260/4220)$ as the vector tetraquark state with $J^{PC}=1^{--}$, and construct the $C\gamma_5\otimes\stackrel{\leftrightarrow}{\partial}_\mu\otimes \gamma_5C$ type diquark-antidiquark current to study its mass and pole residue with the QCD sum rules in details by taking into account the vacuum condensates up to dimension 10 in a consistent way. The predicted mass $M_{Y}=4.24\pm0.10\,\rm{GeV}$ is in excellent agreement with experimental data and supports assigning the $Y(4260/4220)$ to be the $C\gamma_5\otimes\stackrel{\leftrightarrow}{\partial}_\mu\otimes \gamma_5C$ type vector tetraquark state, and disfavors assigning the $Z_c(4100)$ to be the $C\gamma_5\otimes\stackrel{\leftrightarrow}{\partial}_\mu\otimes \gamma_5C$ type vector tetraquark state. It is the first time that the QCD sum rules have reproduced the mass of the $Y(4260/4220)$ as a vector tetraquark state.

In Ref. [4], L. Maiani et al assign the Y (4260) to be the diquark-antidiquark type tetraquark state with the angular momentum L = 1 based on the effective Hamiltonian with the spin-spin and spinorbit interactions. In the type-II diquark-antidiquark model [5], where the spin-spin interactions between the quarks and antiquarks are neglected, L. Maiani et al interpret the Y (4008), Y (4260), Y (4290/4220) and Y (4630) as the four ground states with L = 1. By incorporating the dominant spin-spin, spin-orbit and tensor interactions, A. Ali et al observe that the preferred assignments of the ground state tetraquark states with L = 1 are the Y (4220), Y (4330), Y (4390), Y (4660) rather than the Y (4008), Y (4260), Y (4360), Y (4660) [6]. The QCD sum rules can reproduce the experimental values of the masses of the Y (4360) and Y (4660) in the scenario of the tetraquark states [8,9,10,11,24,25,26,27].
The diquarks ε ijk q T j CΓq ′ k have five structures in Dirac spinor space, where CΓ = Cγ 5 , C, Cγ µ γ 5 , Cγ µ and Cσ µν for the scalar, pseudoscalar, vector, axialvector and tensor diquarks, respectively, the i, j, k are color indexes. The attractive interactions of one-gluon exchange favor formation of the diquarks in color antitriplet, flavor antitriplet and spin singlet [28], while the favored configurations are the scalar (Cγ 5 ) and axialvector (Cγ µ ) diquark states based on the QCD sum rules [29,30,31,32]. We can take the Cγ 5 and Cγ µ diquark states as basic constituents to construct the scalar and axialvector tetraquark states [33,34]. In the non-relativistic quark models, we have to introduce additional P-waves explicitly to study the vector tetraquark states, while in the quantum field theory, we can also take other diquark states (C, Cγ µ γ 5 and Cσ µν ) as basic constituents without introducing the explicit P-waves to study the vector tetraquark states [8,9,10,11,24,25,35,36]. However, up to now, the QCD sum rules cannot reproduce the experimental value of the mass of the Y (4260/4220) in the scenario of the tetraquark state [8,9,10,11,24,25,26,27]. We often obtain much larger mass than the M Y (4260/4220) .
The net effects of the relative P-waves between the heavy (anti)quarks and light (anti)quarks in the heavy (anti)diquarks are embodied in the underlined γ 5 in the Cγ 5 γ 5 ⊗ γ µ C type and Cγ 5 ⊗ γ 5 γ µ C type currents or in the underlined γ α in the Cγ α γ α ⊗ γ µ C type currents [27]. If we introduce the relative P-waves between the heavy (anti)quarks and light (anti)quarks in obtain the hadronic representation [40,41]. After isolating the ground state contribution of the vector tetraquark state Y (4260/4220), we get the result, where the pole residue λ Y is defined by 0|J µ (0)|Y (p) = λ Y ε µ , the ε µ is the polarization vector of the vector tetraquark state Y (4260/4220). The vector and scalar tetraquark states contribute to the components Π(p 2 ) and Π 0 (p 2 ), respectively. In this article, we choose the tensor structure −g µν + pµpν p 2 for analysis, the scalar tetraquark states have no contaminations. Now we briefly outline the operator product expansion for the correlation function Π µν (p) in perturbative QCD. We contract the u, d and c quark fields in the correlation function Π µν (p) with Wick theorem, obtain the result: where the S ij (x) and C ij (x) are the full u/d and c quark propagators respectively, and t n = λ n 2 , the λ n is the Gell-Mann matrix [41,42]. In Eq. (6), we retain the term q j σ µν q i originate from the Fierz re-arrangement of the q iqj to absorb the gluons emitted from other quark lines to extract the mixed condensate qg s σGq [24,33].
It is very difficult (or cumbersome) to carry out the integrals both in the coordinate and momentum spaces directly due to appearance of the partial derives ∂ µ and ∂ ν . We perform integral by parts to exclude the terms proportional to the tensor structure pµpν p 2 , which only contributes to the scalar tetraquark states, and simplify the correlation function Π µν (p) greatly, Then we compute the integrals both in the coordinate and momentum spaces, and obtain the correlation function Π(p 2 ) therefore the spectral density at the level of quark-gluon degrees of freedom.
Once analytical expressions of the QCD spectral density are obtained, we can take the quarkhadron duality below the continuum threshold s 0 and perform Borel transform with respect to the variable P 2 = −p 2 to obtain the QCD sum rules: where where dydz = s ), respectively, and are discarded [24,33]. We derive Eq.(9) with respect to τ = 1 T 2 , then eliminate the pole residue λ Y , and obtain the QCD sum rules for the mass of the vector tetraquark state Y (4260/4220),
In this article, we search for the ideal Borel parameter T 2 and continuum threshold parameter s 0 to satisfy the following four criteria: 1. Pole dominance at the phenomenological side; 2. Convergence of the operator product expansion; 3. Appearance of the Borel platforms; 4. Satisfying the energy scale formula, using try and error.
In the four-quark system qq ′ QQ, the Q-quark serves as a static well potential and combines with the light quark q to form a heavy diquark D in color antitriplet or combines with the light antiquarkq ′ to form a heavy meson-like state or correlation (not a physical meson) in color singlet, while theQ-quark serves as another static well potential and combines with the light antiquarkq ′ to form a heavy antidiquarkD in color triplet or combines with the light quark state q to form another heavy meson-like state or correlation (not a physical meson) in color singlet [24,34,38]. Then the D andD combine with together to form a compact tetraquark state, the two meson-like states (not two physical mesons) combine together to form a physical molecular state [24,34,38], the two heavy quarks Q andQ stabilize the tetraquark state [7]. The tetraquark states DD are characterized by the effective heavy quark masses M Q and the virtuality [24,34,38]. We cannot obtain energy scale independent QCD sum rules, but we have an energy scale formula to determine the energy scales consistently, which works well even for the hidden-charm pentaquark states [46], the updated value M c = 1.82 GeV [11].
In Refs. [24,33,34], we study the hidden-charm or hidden-bottom tetraquark states, the heavy diquarks and heavy antidiquarks are in relative S-wave, if there exist relative P-waves, the Pwaves lie in between the heavy (anti)quark and light (anti)quark in the heavy (anti)diquark. In the present work, we study the vector tetraquark state which has a relative P-wave between the charmed diquark and charmed antidiquark. If a relative P-wave costs about 0.5 GeV, then the energy scale formula is modified to be In calculations, we observe that if we take the continuum threshold parameter √ s 0 = 4.8±0.1 GeV, Borel parameter T 2 = (2.2 − 2.8) GeV 2 , energy scale µ = 1.1 GeV, the pole contribution of the ground state vector tetraquark state Y (4260/4220) is about (49 − 81)%, the predicted mass is about M Y = 4.24 GeV, the modified energy scale formula is well satisfied.
In Fig.1, we plot the pole contribution with variation of the Borel parameter, from the figure, we can see that the pole contribution decreases monotonously with increase of the Borel parameter, the pole contribution reaches about 50% at the point T 2 = 2.8 GeV 2 and √ s 0 = 4.7 GeV, we can obtain the upper bound T 2 max = 2.8 GeV 2 . In Fig.2, we plot the contributions of the vacuum condensates of dimension n in the operator product expansion, which are defined by From the figure, we can see that the contributions of the vacuum condensates of dimensions 3, 5, 6 and 8 are very large, and change quickly with variation of the Borel parameter T 2 at the region T 2 < 2.2 GeV 2 , the operator product expansion is not convergent, we can obtain the lower bound  [44], which supports assigning the Y (4260/4220) to be the Cγ 5 ⊗ ↔ ∂ µ ⊗γ 5 C type vector tetraquark state. The average value of the width of the Y (4260) is 55 ± 19 MeV, the relative P-wave between the diquark and antidiquark disfavors rearrangement of the quarks to form meson pairs, which can account for the small width.

Conclusion
In this article, we take the Y (4260/4220) as the vector tetraquark state with J P C = 1 −− , and construct the Cγ 5 ⊗ ↔ ∂ µ ⊗γ 5 C type current to study its mass and pole residue with the QCD sum rules in details by taking into account the vacuum condensates up to dimension 10 in a consistent way in the operator product expansion, and use the modified energy scale formula µ = M 2 X/Y /Z − (2M c + 0.5GeV) 2 with the effective c-quark mass M c to determine the optimal energy scale of the QCD spectral density. The predicted mass M Y = 4.24 ± 0.10 GeV is in excellent agreement with the experimental value M Y (4220) = 4222.0 ± 3.1 ± 1.4 MeV from the BESIII collaboration or the experimental value M Y (4260) = 4230.0 ± 8.0 MeV from Particle Data Group, and supports assigning the Y (4260/4220) to be the Cγ 5 ⊗ ↔ ∂ µ ⊗γ 5 C type vector tetraquark state, and disfavors assigning the Z c (4100) to be the Cγ 5 ⊗ ↔ ∂ µ ⊗γ 5 C type vector tetraquark state. It is the first time that the QCD sum rules have reproduced the mass of the Y (4260/4220) as a vector tetraquark state.