The fully-light vector tetraquark states with explicit P-wave via the QCD sum rules

In this paper, we apply the QCD sum rules to study the vector fully-light tetraquark states with an explicit P-wave between the diquark and antidiquark pair. We observed that the $C\gamma_\alpha\otimes\stackrel{\leftrightarrow}{\partial}_\mu\otimes\gamma^\alpha C$ (or $C\gamma_\alpha\otimes\stackrel{\leftrightarrow}D_\mu\otimes\gamma^\alpha C$) type current with fully-strange quarks couples potentially to a tetraquark state with the mass $2.16 \pm 0.14 \,\rm{GeV}$, which supports assigning the $Y/\phi(2175)$ as the diquark-antidiquark type tetraquark state with the $J^{PC}=1^{--}$. The $qs\bar{q}\bar{s}$ and $ss\bar{s}\bar{s}$ vector tetraquark states with the structure $C\gamma_\mu\otimes \stackrel{\leftrightarrow}{\partial}_\alpha \otimes\gamma^\alpha C + C\gamma^\alpha \otimes\stackrel{\leftrightarrow}{\partial}_\alpha \otimes\gamma_\mu$ (or $C\gamma_\mu\otimes \stackrel{\leftrightarrow}D_\alpha \otimes\gamma^\alpha C + C\gamma^\alpha \otimes\stackrel{\leftrightarrow}D_\alpha \otimes\gamma_\mu$) are consistent with the $X(2200)$ and $X(2400)$ respectively, which lie in the region $2.20$ to $2.40\,\rm{GeV}$. The central values of the masses of the fully-strange vector tetraquark states with an explicit P-wave are about $2.16-3.13\,\rm{GeV}$ (or $2.16-3.16\,\rm{GeV}$), and the predictions for other fully-light vector tetraquark states with and without hidden-strange are also presented.


Introduction
In 2006, the BaBar collaboration studied the cross section e + e − → φf 0 (980) and discovered a structure near threshold compatible with the quantum numbers J P C = 1 −− for the first time, the mass is M = 2.175 ± 0.010 ± 0.015 GeV and the decay width is Γ = 58 ± 16 ± 20 MeV [1], thereafter, it was named as the Y (2175) [2].It is a suitable fully-light tetraquark candidate [2] and the experimental results have attracted extensive interest as there also exist other possible interpretations beyond the tetraquark state [3].Later, the BESII collaboration confirmed the presence of the Y (2175) by providing the Breit-Wigner mass and width: M = 2.186 ± 0.010 ± 0.006 GeV and Γ = 65 ± 23 ± 17 MeV [4].And the Belle collaboration also conformed the Y (2175), in addition, there is a cluster of events near 2.4 GeV [5].Shen and Yuan studied the combined data from the Belle and BaBar collaborations and observed an evidence for the structure X(2400) with a mass 2.436 ± 0.026 GeV and a width 121 ± 35 MeV [6].
In previous works, the QCD sum rules have been used to explore the tetraquark states ssss with the quantum numbers J P = 0 ++ , 1 −− , 1 +− , 2 ++ , et al [2,15,16,17,18,30].In Refs.[2,15,16,18], the Y (2175)/φ(2170) is interpreted as the ssss vector tetraquark state (or as the sqsq state [19]) with an implicit P-wave (or with an explicit P-wave in the (anti)diquark constituent [17]) according to the calculations via the QCD sum rules.On the other hand, the mass spectrum and strong decays of the ssss states with the S-wave and P -wave are studied in the relativized quark model, the assignment of the Y (2175)/φ(2170) as a ssss candidate cannot be excluded [20,21].In the flux-tube model, the 3 P 0 model, the modified GI quark model and quark pair creation (QPC) model, the state Y (2175)/φ(2170) is assigned as the hidden strange ssss tetraquark state [22], the Λ Λ baryonium [23,24], and the 2 3 D 1 ss meson [25,26], respectively.For more details and literatures, one can consult the review [3].
The diquarks ε ijk q T j CΓq ′ k in the color antitriplet have five structures in Dirac spinor space, CΓ = Cγ 5 , C, Cγ µ γ 5 , Cγ µ and Cσ µν for the scalar, pseudoscalar, vector, axialvector and tensor diquarks, respectively, where the i, j and k are color indexes.The favored or stable configurations are the scalar ε ijk q T j Cγ 5 q ′ k and axialvector ε ijk q T j Cγ µ q ′ k diquark states based on the QCD sum rules [31], for the diquarks containing the same flavor, ε ijk q T j Cγ 5 q k = 0, we give priority to the axialvector diquarks ε ijk q T j Cγ µ q k as the basic constituents.However, the diquarks q T j Cγ 5 q k in the color sextet can exist indeed due to Fermi-Dirac statistics, although the attractive (repulsive) force from the one-gluon exchange favors (disfavors) formation of the diquarks in the color antitriplet (sextet) [32,33,34].
In case of the negative-parity is concerned, the relative P-waves in the diquarks can be implied 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 [35], or in the vector components in the Cσ µν and Cσ µν γ 5 type diquarks [36].In Refs.[15,16], Chen et al take the diquarks both in the color antitriplet and sextet as the basic constituents to construct two vector currents to interpolate the Y (2175), where the P-wave is implied in the diquarks, then take account of the mixing effects between the two currents, and obtain the lowest masses 2.41 ± 0.25 GeV and 2.34 ± 0.17 GeV for the vector ssss tetraquark states, the central values are larger than the experimental data.In Ref. [30], we choose the Cγ µ ⊗ γ ν C − Cγ ν ⊗ γ µ C-type current without introducing an explicit or implicit Pwave to interpolate the Y (2175), the P-wave is only implicitly embodied in the non-vanishing couplings to the vector tetraquark states.The calculations also lead to larger mass for the vector ssss tetraquark configuration having an implicit P-wave compared to the experimental mass of the Y (2175)/φ(2170).In Ref. [2], we choose the γ µ ⊗ γ 5 γ 5 -type current in the color octet-octet to interpolate the Y (2175), where the P-wave is implied in the underlined γ 5 , and reproduce the experimental mass of the Y (2175), however, the pole contribution is not large enough.
So we propose to introduce the explicit P-wave to study the fully-light vector tetraquark states, and construct the [37,38].The additional P-wave in the nonrelativistic quark model can alter the parity by adding a factor (−) L = −, where L = 1 is the angular momentum.
By comparing Ref. [38] with Ref. [39], we reveal that the masses of the type vector hidden-charm tetraquark states.The Y (2175)/φ(2170) could be viewed as a ss analogue of the Y (4260), or as a ssss state decays primarily to the φf 0 (980).Now we extend our previous work on the Y (4220/4260), Y (4320/4360), Y (4390) and Y (4660) carried out in Ref. [38], where we study the hidden-charm vector tetraquark states with an explicit P-wave between the diquark and antidiquark using the QCD sum rules in a systematic way, to explore the fully-light tetraquark states.More explicitly, we extend the doubly-heavy and doublylight quarks to the case where the four valence quarks are all light quarks, and study the ssss, sqsq, and qq q q tetraquark states with an explicit P-wave via the QCD sum rules, then compare the predictions to the corresponding ones without an explicit P-wave [15,16,30].
Generally speaking, we can choose either the partial derivative or covariant derivative to construct the interpolating currents.The currents with covariant derivative D µ are gauge invariant, but does not favor interpreting the as angular momentum.The currents with partial derivative ∂ µ are not gauge covariant, but favors interpreting the as angular momentum, furthermore, the covariant derivative D µ leads to some hybrid components in the hadron states due to the gluon field G µ .In this work, we present the results with both the partial derivatives ∂ µ and covariant derivatives D µ for completeness.
The paper is organized as follows: in section 2, we derive the QCD sum rules for the vector tetraquark states with explicit P-waves; in section 3, we show the numerical results and discussions; and in section 4, we draw conclusions.

QCD sum rules for the fully-light vector tetraquark states
In the following, we write down the two-point correlation functions Π µν (p) and Π µναβ (p) in the QCD sum rules, where J µ (x) = J 1,2,3,4 µ,ssss/qsqs/qq q q(x), J 5 µ,qsqs (x), J µν (x) = J 6 µν,qsq s(x), where the i, j, k, m, n are color indexes, the C is the charge conjugation matrix, and We choose the tensor diquark operators ε ijk s T j (x)Cσ µν s k (x), ε ijk q T j (x)Cσ µν s k (x) and ε ijk q T j (x)Cσ µν q k (x) beyond the axialvector diquark operators to construct the four-quark currents, as they also exist due to Fermi-Dirac statistics.
We can take a simple replacement 3)-( 5) to acquire the corresponding gauge invariant currents, just like what we have done in Refs.[40,41].In this work, we choose the currents with the partial derivatives and covariant derivatives respectively to interpolate the vector tetraquark states.Under parity transform P and charge conjugation transform C, the currents J µ (x) and J µν (x) have the properties, where x µ = (t, x) and xµ = (t, − x).
On the hadron side, we acquire the hadronic representation by inserting a complete set of intermediate hadronic states with the same quantum numbers as the current operators J µ (x) and J µν (x) into the correlation functions Π µν (p) and Π µναβ (p) [42,43], and reach the following results by separating the ground state contributions of the vector tetraquark states, where the pole residues λ Y and λ Z are defined by the ε µ are the polarization vectors of the vector and axialvector tetraquark states Y and Z with the J P C = 1 −− and 1 +− , respectively.Now we project out the components Π Y (p 2 ) and Π Z (p 2 ) by introducing the operators where In this paper, we choose the components Π Y (p 2 ) to study the fully-light tetraquark states with the J P C = 1 −− , and acquire the hadronic spectral representation through dispersion relation, On the QCD side, if we take the current J µ (x) = J 1 µ,ssss (x) with the partial derivatives as an example, the correlation function Π µν (p) can be written as after performing the Wick's contractions, where the S ij (x) is the full s-quark propagator [43,44,45], and t n = λ n 2 , the λ n is the Gell-Mann matrix, we retain the term sj σ µν s i originates from the Fierz re-arrangement of the s i sj to absorb the gluons emitted from other quark lines to extract the mixed condensate sg s σGs [45].Then it is straightforward to carry out the integrals d 4 x in the coordinate space by setting d 4 x → d D x and using the basic integral, to obtain the correlation function Π µν (p), therefore we obtain the spectral representation at the quark-gluon level through dispersion relation, the QCD spectral density ρ QCD (s) = ImΠY (s) π , where we have used the formula, Then we take the simple replacement 2 in the vertexes in Eq.( 15) to take account of additional terms originating from the covariant derivatives, thus we obtain the corresponding correlation function for the gauge invariant current, the calculation is straightforward via the same procedure.The mass of the s-quark is taken as a small quantity and treated perturbatively in Eq.( 16), which requires that the lower bound of the integral ds should be zero.
When we acquire the analytical expressions of the QCD spectral densities ρ QCD (s), we take the quark-hadron duality below the continuum thresholds s 0 by setting, and implement the Borel transform in regard to the variable P 2 = −p 2 to get the QCD sum rules, the explicit expressions of the spectral densities ρ QCD (s) for the currents with both partial derivatives and covariant derivatives are all given in the Appendix.The vacuum condensates qq 3 make no contribution due to the special structures of the currents, where q = u, d or s.
We calculate the vacuum condensates with dimensions up to 11 in the operator product expansion, and consider the impact of the vacuum condensates which are vacuum expectations of the quark-gluon operators of the orders O(α k s ) with k ≤ 1, just like in our previous works [45,46,47].The vacuum condensates g 3 s f abc G a G b G c , αs π GG 2 and qg s σGq αs π GG are the vacuum expectations of the quark-gluon operators of the orders O(α s ), respectively, and are neglected, where q = u, d or s.Direct calculations indicate that those contributions are tiny in the QCD sum rules for the multiquark states [48].The vacuum condensates qq 4 are companied with the strong coupling constant g 2 s , also make tiny contributions and are neglected for simplicity.We differentiate Eq.( 21) with respect to τ = 1 T 2 , then eliminate the pole residues λ Y , and obtain the QCD sum rules for the masses of the fully-light vector tetraquark states, 3 Numerical results and discussions We take the standard values of the vacuum condensates qq = −(0.24± 0.01 GeV) 3 , qg s σGq = m 2 0 qq , m 2 0 = (0.8 ± 0.1) GeV 2 , ss = (0.8 ± 0.1) qq , sg s σGs = m 2 0 ss , αsGG π = (0.012 ± 0.004) GeV 4 at the energy scale µ = 1 GeV [42,43,49], and choose the M S mass m s (µ = 2 GeV) = 0.095 ± 0.005 GeV from the Particle Data Group [9].We choose the square, cube and Nth powers of the s quark mass m N s = 0 (N = 2, 3, 4...) and neglect the small u and d quark masses.We usually choose the energy scale µ = 1 GeV for the QCD spectral densities of the fully-light mesons and baryons [30,50,51], and evolve the s-quark mass to the energy scale µ = 1 GeV accordingly.
According to the experiential mass gaps between the ground states and first radial excited states of the mesons, we restrict the central values of the fully-light vector tetraquark masses M c and continuum threshold parameters √ s 0 c in the range about, in order to ensure uniformity and reliability of the results.For the conventional pseudoscalar and vector mesons with the valence quarks s, c and b, the mass gaps between the ground states and first radial excited states are about 0.55 ∼ 0.75 GeV from the Particle Data Group [9].We borrow some idea from the experimental data and add an uncertainty δ √ s 0 = ±0.10GeV, then the continuum threshold parameters are about √ s 0 = √ s 0 c ± 0.10 GeV = M c + 0.40 ∼ 0.70 GeV, such as a strict constraint is reasonable.We control the pole contributions (PC) range from 35% to 75% to ensure pole dominance at the phenomenological side whenever possible, and the PC are defined by The contributions of the vacuum condensates D(n) in the operator product expansion are defined by where the subscript n in the QCD spectral densities ρ n (s) denotes the dimension of the vacuum condensates, where q = u, d or s.We require the criterion D(11) ∼ 0% to judge the convergent behaviors.Then we get the suitable Borel parameters and continuum threshold parameters via trial and error.In Table 1, we show the numerical results for the fully-light vector tetraquark states in the case of the partial derivatives.
From the analytical expressions of the QCD spectral densities shown in the Appendix, we can see explicitly that the currents with the partial derivatives and covariant derivatives make no difference for the tetraquark states Y qq q q .On the other hand, for the tetraquark states qsqs and Y 4 qq q q , the currents with covariant derivatives lead to slightly/tinily different masses or pole residues comparing with the corresponding ones with partial derivatives.All the numerical results with the covariant derivatives would be presented in Table 2.
In calculations, taking the currents with the partial derivatives as an example, we observe that the perturbative terms D(0), D(3), D(6), D(8) and D(10) provide the most significant contributions, see Fig. 1.In general, although the contributions vary (remarkably) with the vacuum condensates of increasing dimensions, the contributions D(0) or D(8) serve as milestones, the higher dimensional vacuum condensates play a less (or decreasing) important role, and D(11) ∼ 0%.More explicitly, for the curves C, D, G, H, M and N , the dominant contributions come from the D(0); for the curves A, B, F and J, the largest contributions come from the D(0), the contributions from the D(8) are also large, then the higher vacuum condensate contributions decrease quickly to zero; for the curves E, I, K and L, the largest contributions come from the D(8), then the higher vacuum condensate contributions decrease quickly to zero.All in all, for the curves A, B, C, D, F , G, H, J, M and N , the operator product expansion converges very good; for the curves E, I, K and L, the operator product expansion converges.
It is straightforward to obtain the masses and pole residues of the vector tetraquark states, which are shown explicitly in Tables 1-2, where we calculate the uncertainties δ with the formula, where the f denote the tetraquark masses M Y and pole residues λ Y , the x i denote all the input parameters m s , qq , ss , • • • .As the partial derivatives ∂f ∂xi are difficult to carry out analytically, we take the approximation ∂f ∂xi 2 In calculations, we observe that if we choose the continuum threshold parameter √ s 0 = 2.75 ± 0.10 GeV and Borel parameter T 2 = (1.3−1.6)GeV 2 , then the pole contribution is about (37−70)% for the vector tetraquark state Y 1 ssss , the predicted mass M = 2.16 ± 0.14 GeV, see Tables 1-2,  Table 2: The Borel parameters, continuum threshold parameters, pole contributions, masses and pole residues of the P-wave (with covariant derivatives) fully-light vector tetraquark states.matches the experimental value M Y = 2.175 ± 0.010 ± 0.015 GeV for the Y (2175)/φ(2170) [1], and supports assigning the Y (2175)/φ(2170) as a vector ssss tetraquark state.
In this paper, we choose the pole contributions about (35 − 75)% consistently, the largest pole contributions up to know, just like in our previous work [30], where the S-wave diquark-antidiquark type ssss and qq q q states are explored with the QCD sum rules.In the QCD sum rules, we extract the masses in the Borel windows, which depend on the pole contributions and convergent behaviors of the operator product expansion, the predicted masses change with variations of the Borel windows.There are two choices in defining the contributions of the vacuum condensates D(n), the definition in Eq.( 25) is chosen in the present work and Refs.[15,16,30], while in Ref. [17], the D(n) is defined by setting s 0 → ∞.It is not odd that the existing predictions are compatible or slightly different.In Refs.[15,16], Chen et al obtain the masses 2.41±0.25 GeV and 2.34±0.17GeV for the two lowest vector tetraquark states ssss, the central values are larger than that of the present calculation 2.16 ± 0.14 GeV.Furthermore, we should bear in mind, the interpolating currents have the same quantum numbers J P C but different structures, which correspond to several tetraquark states or a tetraquark state with several Fock components, it is not odd that the predictions are also different.
Another noteworthy point is that the masses of the Y 2 qsqs and Y 2 ssss , see Tables 1-2, are M = 2.24 ± 0.17 GeV and 2.35 ± 0.17 GeV respectively, which happen to fit the experimentally observed states X(2240) and X(2400) with the masses in the range (2.20 − 2.40) GeV [7,8,10,11].We think it is possible that the X(2240) and X(2400) are the qsqs and ssss states respectively with the The assignments are consistent with the observation of the X(2240) in the K + K − invariant mass spectrum [7], as the decay Y 2 qsqs → K + K − can take place through the Okubo-Zweig-Iizuka super-allowed fall-apart mechanism.While the decay Y 2 ssss → Λ Λ could take place through annihilating an ss pair and creating a uū pair and a d d pair, which is not Okubo-Zweig-Iizuka favored, we can search for the two-body strong decays Y 2 ssss → φf 0 (980), φη ′ to diagnose the nature of the X(2400).In our previous work [30], the predicted mass of the vector tetraquark state without an explicit P-wave between the diquark and antidiquark pair, M X = 3.08 ± 0.11 GeV, is much larger than the experimental value of the mass of the Y (2175)/φ(2170), and opposes assigning the Y (2175)/φ(2170) as a hidden-strange partner of the Y states without an explicit P-wave.Most of the vector tetraquark states with an explicit P-wave have lower masses than the corresponding vector tetraquark states with an implicit P-wave we studied earlier [30], see Tables 1-2.If we release the pole contributions (37 − 70)% to smaller values, we can obtain even smaller masses than those presented in Tables 1-2, for example, the values obtained in Refs.[2,51].
The predicted masses are stable with variations of the Borel parameters, see Figs. 2-4 for the currents with partial derivatives as an example, the uncertainties come from the Borel parameters are rather small, the predictions are robust.Furthermore, from Tables 1-2, we can see clearly that the currents with the covariant derivatives and partial derivatives only lead to slight/tiny difference, we prefer the covariant derivatives in constructing the interpolating currents if gauge invariance is emphasized.On the other hand, if only final numerical results are concerned, we can choose either covariant derivatives or partial derivatives.
The currents J α with the same quantum numbers could mix with each other under renormalization, we should introduce the mixing matrixes U to obtain the diagonal currents, J ′ α = U J α , which couple potentially to (more) physical states, as the physical states have several Fock components.The matrixes U can be determined by direct calculating anomalous dimensions of all the currents, it is difficult to obtain a diagonal current, which is an special superposition of several non-trial currents to match with all the considerable Fock states.For the vector tetraquark state ssss, if we want to obtain a more physical state, we have to take account of the mixing effects of the currents J 1 µ,ssss , J 2 µ,ssss , J 3 µ,ssss and J 4 µ,ssss at least, and introduce three mixing angles, θ, θ 12 and θ 34 , J µ,ssss = cos θ cos θ 12 J 1 µ,ssss + sin θ 12 J 2 µ,ssss + sin θ cos θ 34 J 3 µ,ssss + sin θ 34 J 4 µ,ssss , (28) it is a difficult work.The currents J 1 µ,ssss , J 2 µ,ssss , J 3 µ,ssss and J 4 µ,ssss maybe couple potentially  qq q q , Y 2 qq q q , Y 3 qq q q and Y 4 qq q q, respectively.
to four different tetraquark states or to a tetraquark state with four different Fock components.Without exploring the mixing effects and strong decays in details combined with precise experimental data, we cannot assign the Y (2175)/φ(2170), X(2240), X(2400) in a rigorous way.At the present time, we just assign those exotic states tentatively and roughly.According to the BESIII and BaBar data, the Y (2175)/φ(2170) and X(2240) are two different states [7,10], their valence quark constituents are favored to be ssss and qsqs, respectively, the mixing effects between the currents J 1 µ,ssss and J 2 µ,qsqs are expected to be small, the present assignments make sense.We hope those vector tetraquark states can be experimentally verified in the future, and confront the predictions to the experimental data in the future at the BESIII, LHCb, Belle II et al.

Conclusion
In this paper, we construct the fully-light vector/tensor four-quark currents and introduce an explicit P-wave between the diquark and antidiquark pair.We take account of the contributions of the vacuum condensates up to dimension 11 in the operator product expansion and obtain the The masses with variations of the Borel parameters T 2 , where the K, L, M and N denote the Y 1 qq q q , Y 2 qq q q, Y 3 qq q q and Y 4 qq q q P-wave (with partial derivatives) fully-light vector tetraquark states, respectively.QCD sum rules for the masses and pole resides of the vector tetraquark states.For one of the ssss structures, the predicted mass M = 2.16 ± 0.  2400) respectively in the region 2.20 to 2.40 GeV from the BaBar/BESIII collaborations.All in all, we predict that the central values of the masses of the vector ssss tetraquark states with an explicit P-wave lie at the region 2.16 − 3.13 GeV (or 2.16 − 3.16 GeV).We also obtain masses and pole resides of the sqsq and qq q q vector tetraquark states and hope those fully-light vector tetraquark states can be observed in the future.

Figure 1 :
Figure 1: The absolute contributions of the vacuum condensates of dimension n for the central values of the input parameters for the ssss and qsqs P-wave (with partial derivatives) fully-light vector tetraquark states, where the A, B, C and D denote the Y 1 ssss , Y 2 ssss , Y 3 ssss and Y 4 ssss , the E, F , G, H, I and J denote the Y 1 qsqs , Y 2 qsqs , Y 3 qsq s, Y 4 qsqs , Y 5 qsq s and Y 6 qsqs , the K, L, M and N denote the Y 1qq q q , Y 2 qq q q , Y 3 qq q q and Y 4 qq q q, respectively.

Table 1 :
The Borel parameters, continuum threshold parameters, pole contributions, masses and pole residues of the P-wave (with partial derivatives) fully-light vector tetraquark states.
The masses with variations of the Borel parameters T 2 , where the E, F , G, H, I and J denote the Y 1 qsq s, Y 2 qsqs , Y 3 qsqs , Y 4 ssss ,Y 5 qsqs and Y 6 qsqs P-wave (with partial derivatives) fully-light vector tetraquark states, respectively.