Reanalysis of the $X(4140)$ as axialvector tetraquark state with QCD sum rules

In this article, we take the $X(4140)$ as the diquark-antidiquark type $cs\bar{c}\bar{s}$ tetraquark state with $J^{PC}=1^{++}$, and study the mass and pole residue with the QCD sum rules in details by constructing two types interpolating currents. The numerical results $M_{X_{L,+}}=3.95\pm0.09\,\rm{GeV}$ and $M_{X_{H,+}}=5.00\pm0.10\,\rm{GeV}$ disfavor assigning the $X(4140)$ to be the $J^{PC}=1^{++}$ diquark-antidiquark type $cs\bar{c}\bar{s}$ tetraquark state. Moreover, we obtain the masses of the $J^{PC}=1^{+-}$ diquark-antidiquark type $cs\bar{c}\bar{s}$ tetraquark states as a byproduct. The present predictions can be confronted to the experimental data in the future.


Introduction
In 2009, the CDF collaboration observed the X(4140) for the first time in the J/ψφ invariant mass distribution in the exclusive B + → J/ψ φK + decays in pp collisions at √ s = 1.96 TeV with a statistical significance more than 3.8σ [1]. In 2011, the CDF collaboration confirmed the X(4140) in the B ± → J/ψ φK ± decays with a statistical significance more than 5σ, and observed an evidence for the X(4274) in the J/ψφ invariant mass distribution with a statistical significance about 3.1σ [2]. In 2013, the CMS collaboration confirmed the X(4140) in the B ± → J/ψφK ± decays in pp collisions at √ s = 7 TeV collected with the CMS detector at the LHC with a statistical significance more than 5σ [3], the D0 collaboration confirmed the X(4140) in the B + → J/ψφK + decays with a statistical significance of 3.1σ based on the data sample corresponds to an integrated luminosity of 10.4 fb −1 of pp collisions at √ s = 1.96 TeV [4]. There have been several possible assignments for the X(4140) since its first observation by the CDF collaboration [1], such as a molecular state [5], a tetraquark state [6,7,8,9,10], a hybrid state [11,12] or a rescattering effect [13].
Recently, the LHCb collaboration performed the first full amplitude analysis of the decays B + → J/ψφK + with J/ψ → µ + µ − , φ → K + K − with a data sample corresponds to an integrated luminosity of 3fb −1 of pp collision data collected at √ s = 7 and 8 TeV with the LHCb detector, and observed that the data cannot be described by a model that contains only excited kaon states decaying into φK + [14,15]. The LHCb collaboration confirmed the two old particles X(4140) and X(4274) in the J/ψφ invariant mass distributions with statistical significances 8.4σ and 6.0σ, respectively, and determined the spin-parity-change-conjugation to be J P C = 1 ++ with statistical significances 5.7σ and 5.8σ, respectively [14,15]. Moreover, LHCb collaboration observed the two new particles X(4500) and X(4700) in the J/ψφ invariant mass distributions with statistical significances 6.1σ and 5.6σ, respectively, and determined the spin-parity-change-conjugation to be J P C = 0 ++ with statistical significances 4.0σ and 4.5σ, respectively [14,15]. The measured Breit-Wigner masses and widths are which are consistent with the experimental data. The inclusion of the first radial excited state beyond the ground state in the QCD sum rules leads to smaller ground state mass [9], which happens to lie in the same energy region of the X(3915). If the masses of the ground state diquark-antidiquark type 0 ++ and 2 ++ cscs tetraquark states are about 3.9 GeV and 4.1 GeV, respectively, we would expect that the ground state diquark-antidiquark type 1 ++ cscs tetraquark state has the mass about 3.9 − 4.1 GeV. In Ref. [6], F. Stancu calculates the mass spectrum of the ccss tetraquark states within a simple quark model with chromomagnetic interaction and effective quark masses extracted from meson and baryon spectra, and obtain the two lowest masses 4195 MeV and 4356 MeV of the tetraquark states with J P C = 1 ++ . The value 4195 MeV is consistent with the experimental data 4146.5 ± 4.5 +4.6 −2.8 MeV. In the simple chromomagnetic interaction model, there are no correlated quarks or diquarks [6].
The scattering amplitude for one-gluon exchange is proportional to where the T a is the generator of the SU c (3) gauge group, and the i, j and k, l are the color indexes of the two quarks in the incoming and outgoing channels respectively. The negative sign in front of the antisymmetric antitriplet indicates the interaction is attractive, which favors the formation of diquark states in the color antitriplet [18], so we usually take the diquarks in color antitriplet as the basic constituents in studying the baryon states, tetraquark states and pentaquark states. The diquarks ε ijk q T j CΓq ′ k in color antitriplet have five structures in Dirac spinor space, where the i, j and k are color indexes, CΓ = Cγ 5 , C, Cγ µ γ 5 , Cγ µ and Cσ µν for the scalar, pseudoscalar, vector, axialvector and tensor diquarks, respectively. The stable diquark configurations are the scalar (Cγ 5 ) and axialvector (Cγ µ ) diquark states from the QCD sum rules [19,20], we can construct the tetraquark states using the scalar or axialvector diquarks rather than the uncorrelated quarks to obtain the lowest masses.
In Ref. [21], we study the masses and pole residues of the axialvector hidden-charm tetraquark states in details with the QCD sum rules, and observe that the predictions M X(3872) = 3.87 +0.09 −0.09 GeV and M Zc(3900) = 3.91 +0. 11 −0.09 GeV support assigning the X(3872) and Z c (3900) to be the 1 ++ and 1 +− diquark-antidiquark type tetraquark states, respectively. If we take the X(4140) as the hiddenstrange cousin of the X(3872), then the mass difference M X(4140) −M X(3872) = 275 MeV, the SU (3) breaking effect is about m s − m q = 135 MeV, which is consistent with our naive expectation. In Ref. [22], Chen and Zhu obtain the value 4.07±0.10 GeV for the mass of the Cγ 5 ⊗γ µ C +Cγ µ ⊗γ 5 C type cscs tetraquark states based on the QCD sum rules, the theoretical value 4.07 ± 0.10 GeV overlaps with the experimental value 4146.5 ± 4.5 +4.6 −2.8 MeV, which supports assigning the X(4140) to be the axialvector tetraquark state [23]. Although the masses of the axialvector tetraquark states are calculated with the QCD sum rules, the routines are different [21,22]. In Ref. [21], we study the energy scale dependence of the QCD spectral densities for the first time, and in subsequent works [24,25,26], we suggest an empirical energy scale formula, with the effective heavy quark masses M Q to determine the ideal energy scales of the QCD spectral densities of the hidden-charm and the hidden-bottom tetraquark states in the QCD sum rules. Before the work [22], we performed a systematic study of the mass spectrum of the axialvector hidden-charm and hidden-bottom tetraquark states using the QCD sum rules, and obtained the ground state masses M cqcq = 4.32 ± 0.18 GeV and M cscs = 4.40 ± 0.16 GeV [27], the mass breaking effect M cscs − M cqcq = 80 MeV, which is much smaller than the experimental value M X(4140) − M X(3872) = 275 MeV. In Ref. [27], we extract the masses from the QCD spectral densities at the energy scale µ = 1 GeV, which is much smaller than the optimal energy scales determined by the empirical energy scale formula, and results in much larger mass M cqcq = 4.32 ± 0.18 GeV compared to the mass M X(3872)/Zc(3900) ≈ 3.9 GeV extracted at the optimal energy scales [21].
In Ref. [25], we study the masses and pole residues of the J P C = 1 −± hidden-charm tetraquark states at the optimal energy scales with the QCD sum rules. The predicted masses of the tetraquark states with symbolic quark structures ccss and cc(uū+dd)/ √ 2 support assigning the Y (4660) to be the 1 −− diquark-antidiquark type tetraquark state, the mass difference M ccss − M cc(uū+dd)/ √ 2 = 40 MeV is even smaller compared to the value 80 MeV obtained in Ref. [27]. Now we can draw the conclusion tentatively that the QCD sum rules support smaller SU (3) breaking effect than our naive expectation. It is interesting to perform detailed studies of the X(4140) as the axialvector cscs tetraquark state based on the QCD sum rules.
In this article, we take the X(4140) as the axialvector cscs tetraquark state, construct the diquark-antidiquark type axialvector currents, calculate the contributions of the vacuum condensates up to dimension 10 in the operator product expansion in a consistent way, use the empirical energy scale formula to determine the ideal energy scales of the QCD spectral densities, and study the ground state masses and pole residues in details with the QCD sum rules. We want to obtain additional support in assigning the X(4140) to be the 1 ++ cscs tetraquark state from the QCD sum rules.
The article is arranged as follows: we derive the QCD sum rules for the masses and pole residues of the axialvector cscs tetraquark states in section 2; in section 3, we present the numerical results and discussions; section 4 is reserved for our conclusion.

QCD sum rules for the axialvector cscs tetraquark states
In the following, we write down the two-point correlation functions Π ± µν (p) in the QCD sum rules, where the i, j, k, m, n are color indexes, the C is the charge conjunction matrix. We choose the currents J L/H,+ µ (x) to interpolate the J P C = 1 ++ diquark-antidiquark type hidden-charm tetraquark states.
Under charge conjunction transform C, the currents J L/H,± µ (x) have the properties, which originate from the charge conjunction properties of the scalar, pseudoscalar, axialvector and vector diquark states, where q = u, d, s. Naively, we expect that the currents J H,± µ (x) couple to the hidden-charm tetraquark states with higher masses than that of the currents J L,± µ (x), as the scalar (Cγ 5 ) and axialvector (Cγ µ ) diquark states are much stable compared to the corresponding pseudoscalar (C) and vector (Cγ µ γ 5 ) diquark states [19,20]. In this article, we study the J P C = 1 +− diquarkantidiquark type hidden-charm tetraquark states as a byproduct.
At the phenomenological side, we insert a complete set of intermediate hadronic states with the same quantum numbers as the current operators J L/H,± µ (x) into the correlation functions Π ± µν (p) to obtain the hadronic representation [28,29]. After isolating the ground state hidden-charm tetraquark states X L/H,± and X ′ L/H,± contributions from the pole terms, we get the following result, where the pole residues (or coupling constants) λ X L/H,± and λ X L/H,± are defined by the ε µ are the polarization vectors of the axialvector tetraquark states X L/H,± . In this article, we choose the tensor structure −g µν + pµpν p 2 for analysis, the pseudoscalar tetraquark states X ′ L/H,± have no contaminations. In the following, we briefly outline the operator product expansion for the correlation functions Π ± µν (p) in perturbative QCD. We contract the quark fields in the correlation functions Π ± µν (p) with Wick theorem firstly, and obtain the results: where t = ±, C ij (x) = γ 5 C ij (x)γ 5 , the S ij (x) and C ij (x) are the full s and c quark propagators, respectively [29,30], and t n = λ n 2 , the λ n is the Gell-Mann matrix, D α = ∂ α − ig s G n α t n [29], we add the superscripts L and H to denote which interpolating current is used. Then we compute the integrals both in the coordinate space and in the momentum space, and obtain the correlation functions Π L/H,± µν (p) at the quark level. The calculations are straightforward but tedious. Once the analytical expressions of the correlation functions Π L/H,± (p) are gotten, we can obtain the QCD spectral densities ρ L/H,± (s) through dispersion relation. In Eq. (14), we retain the terms s j σ µν s i and s j γ µ s i originate from the Fierz re-ordering of the s isj to absorb the gluons emitted from the heavy quark lines to form s j g s G a αβ t a mn σ µν s i and s j γ µ s i g s D ν G a αβ t a mn to extract the mixed condensate and four-quark condensates sg s σGs and g 2 s ss 2 , respectively. Once the explicit expressions of the QCD spectral densities ρ L/H,± (s) are obtained, we take the quark-hadron duality bellow the continuum thresholds s 0 and perform Borel transform with respect to the variable P 2 = −p 2 to obtain the following four QCD sum rules: where where the T 2 is the Borel parameter, y f = c appear. In this article, we carry out the operator product expansion for the vacuum condensates up to dimension 10, and assume vacuum saturation for the higher dimension vacuum condensates. The vacuum condensates are the vacuum expectations of the operators, we take the truncations n ≤ 10 and k ≤ 1 for the operators in a consistent way, and discard the operators of the orders O(α k s ) with k > 1. The terms of the orders O( 1 in the QCD spectral densities manifest themselves at small T 2 , we have to choose large T 2 to warrant convergence of the operator product expansion and appearance of the Borel platforms. The higher dimension vacuum condensates play an important role in determining the Borel windows, though they play a less important role in the Borel windows. We differentiate Eqs. (17)(18) with respect to 1 T 2 , then eliminate the pole residues λ XL,± and λ XH,± , and obtain the QCD sum rules for the masses of the X L,± and X H,± , respectively.

Numerical results and discussions
In previous works, we described the hidden-charm and the hidden-bottom four-quark systems qq ′ QQ by a double-well potential [21,24,25,26]. In the four-quark system qq ′ QQ, the heavy quark Q serves as one static well potential and combines with the light quark q to form a heavy diquark D qQ in color antitriplet, while the heavy antiquarkQ serves as the other static well potential and combines with the light antiquarkq ′ to form a heavy antidiquark Dq′Q in color triplet. Then the D qQ and Dq′Q combine together to form a compact tetraquark state, the two heavy quarks Q and Q stabilize the tetraquark state [31]. The doubly-heavy tetraquark states are characterized by the effective heavy quark mass M Q and the virtuality It is natural to take the energy scale µ = V , the energy scale formula works well for the X(3872), Z c (3900), Z c (4020), Z c (4025), Z(4430), Y (4660), Z b (10610) and Z b (10650) in the scenario of tetraquark states [21,24,25,26,32,33]. In Refs. [21,25], we obtain the effective mass for the diquark-antidiquark type hidden-charm tetraquark states, M c = 1.8 GeV. Then we re-checked the numerical calculations and found that there exists a small error involving the mixed condensates. After the small error is corrected, the Borel windows are modified slightly and the numerical results are improved slightly, the conclusions survive. In this article, we choose the updated value M c = 1.82 GeV [33], and obtain the optimal energy scales µ = 1.4 GeV and 2.0 GeV for the QCD spectral densities of the QCD sum rules for the Z c (3900) and X(4140), respectively. Now we choose the input parameters at the QCD side of the QCD sum rules. We take the vacuum condensates to be the standard values qq = −(0.24 ± 0.01 GeV) 3 , ss = (0.8 ± 0.1) qq , qg s σGq = m 2 0 qq , sg s σGs = m 2 0 ss , m 2 0 = (0.8 ± 0.1) GeV 2 , αsGG π = (0.33 GeV) 4 at the energy scale µ = 1 GeV [28,29,34], and take the M S masses m c (m c ) = (1.275 ± 0.025) GeV and m s (µ = 2 GeV) = (0.095 ± 0.005) GeV from the Particle Data Group [17]. Moreover, we take into account the energy-scale dependence of the quark condensates, mixed quark condensates and M S masses from the renormalization group equation [35], 4 9 , ss (µ) = ss (Q) α s (Q) α s (µ) 4 9 , qg s σGq (µ) = qg s σGq (Q) α s (Q) α s (µ) 2 27 , sg s σGs (µ) = sg s σGs (Q) α s (Q) α s (µ) 2 27 , where t = log µ 2 , Λ = 213 MeV, 296 MeV and 339 MeV for the flavors n f = 5, 4 and 3, respectively [17]. In this article, we take the standard value of the quark condensate qq at the energy scale µ = 1 GeV from the Gell-Mann-Oakes-Renner relation [28,29,34,35,36]. The values of the quark condensates have been updated [37], however, we determine the effective heavy quark masses M Q with the standard values [21,24,25,26,32,33], so we choose the standard values in this article. In our next works, we will redetermine the M Q with the updated values, as the updated value qq (2GeV) = −(274 +15 −17 MeV) 3 differs from the standard value qq (2GeV) = −(257 ± 10 MeV) 3 considerably.
In this article, we have neglected the higher-order QCD corrections. Including the higher-order QCD corrections means refitting the effective c-quark mass M c . According to the energy scale formula µ = M 2 X/Y /Z − (2M c ) 2 , some uncertainties are introduced by neglecting the higherorder QCD corrections. In this article, we take the leading order approximations just as in the QCD sum rules for the X(3872), Z c (3900), Y (4660), some higher-order effects are embodied in the effective c-quark mass M c [21,25].
In Ref. [32], we observed that the Z c (3900) and Z(4430) can be assigned to be the ground state and the first radial excited state of the axialvector tetraquark states with J P C = 1 +− , respectively based on the QCD sum rules. We expect the energy gap between the ground state and the first radial excited state of the hidden-charm tetraquark states is about 0.6 GeV according to the mass difference M Z(4430) − M Zc(3900) = 576 MeV. In this article, we assume X(4140) = X L,+ , then the threshold parameters can be taken as √ s 0 = (4.6 − 4.8) GeV. If we choose the energy scale determined by the empirical energy scale formula, then µ = 2.0 GeV. In calculations, we observe that it is impossible to reproduce the experimental value M X(4140) = 4146.5 ± 4.5 +4.6 −2.8 MeV. Now we explore the energy scale dependence of the predicted mass of the X L,+ . In Fig.1 which are shown in Fig.2 at a large interval of the Borel parameter. The predicted mass M XL,+ = (4.15 ± 0.09) GeV is in excellent agreement with the experimental value M X(4140) = 4146.5 ± 4.5 +4.6 −2.8 MeV, which favors assigning the X(4140) to be the 1 ++ diquark-antidiquark type cscs tetraquark state. However, we reproduce the experimental value M Zc(3900) at the energy scale µ = 1.4 GeV of the QCD spectral density, while we reproduce the experimental value M X(4140) at the energy scale µ = 1.1 GeV of the QCD spectral density. The empirical energy scale formula can be re-written as which puts a strong constraint on the masses of the hidden-charm and the hidden-bottom tetraquark states. If the two heavy quarks Q andQ serve as a double-well potential and stabilize the tetraquark states, the X(4140) should correspond to a larger energy scale than that of the Z c (3900), i.e. µ X(4140) > µ Zc(3900) . Moreover, in previous works, we used the empirical energy scale formula and reproduced the experimental values of the masses of the X(3872), Z c (3900), Z c (4020), Z c (4025), Z(4430), Y (4660), Z b (10610) and Z b (10650) in the scenario of tetraquark states [21,24,25,26,32,33]. It is odd that the QCD spectral density of the QCD sum rules for the X(4140) does not obey the empirical energy scale formula. Now we search for the Borel parameters T 2 and continuum threshold parameters 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, to obtain the ground state masses of the X L,± and X H,± .
The resulting Borel parameters, continuum threshold parameters, energy scales, pole contributions, contributions of the vacuum condensates of dimension 8 and 10 are shown explicitly in Table 1, where the vacuum condensate contributions D 8 and D 10 correspond to the central values of the threshold parameters. From the Table, we can see that the first two criteria are satisfied.
We take into account all uncertainties of the input parameters, and obtain the values of the ground state masses and pole residues, which are shown explicitly in Table 2 and Figs.3-4. From Table 1 and Table 2, we can see that the empirical energy scale formula is satisfied. From Figs.3-4, we can see that in the Borel windows, the masses and pole residues are rather stable with variations of the Borel parameters. The four criteria are all satisfied, we expect to make reliable predictions. From Fig.3, we can see that the upper error bound of the theoretical value M XL,+ lies below the experimental value M X(4140) , the present prediction disfavors assigning the X(4140) to be diquark-antidiquark type cscs tetraquark state with the J P C = 1 ++ . The present predictions of the masses of the axialvector cscs tetraquark states can be confronted to the experimental data in the future. Now we perform Fierz re-arrangement to the currents J L/H,± µ both in the color space and        to dimension 10 in the operator product expansion in a consistent way, use the empirical energy scale formula to determine the ideal energy scales of the QCD spectral densities, and study the ground state masses and pole residues with the QCD sum rules. The numerical results M XL,+ = 3.95±0.09 GeV and M XH,+ = 5.00±0.10 GeV disfavor assigning the X(4140) to be the J P C = 1 ++ diquark-antidiquark type tetraquark states. Moreover, we obtain the masses of the J P C = 1 +− diquark-antidiquark type cscs tetraquark states as a byproduct. The present predictions of the masses of the axialvector cscs tetraquark states can be confronted to the experimental data in the future.