Analysis of the mass and width of the $X(4140)$ as axialvector tetraquark state

In this article, we construct both the $[sc]_T[\bar{s}\bar{c}]_A+[sc]_A[\bar{s}\bar{c}]_T$ type and $[sc]_T[\bar{s}\bar{c}]_V-[sc]_V[\bar{s}\bar{c}]_T$ type axialvector currents with $J^{PC}=1^{++}$ to study the mass of the $X(4140)$ with the QCD sum rules. The predicted masses support assigning the $X(4140)$ to be the $[sc]_T[\bar{s}\bar{c}]_V-[sc]_V[\bar{s}\bar{c}]_T$ type axialvector tetraquark state. Then we study the hadronic coupling constant $g_{XJ/\psi\phi}$ with the QCD sum rules based on solid quark-hadron duality, and obtain the decay width $\Gamma(X(4140)\to J/\psi \phi)=86.9\pm22.6\,{\rm{MeV}}$, which is in excellent agreement with the experimental data $83\pm 21^{+21}_{-14} {\mbox{ MeV}}$ from the LHCb collaboration.


Introduction
In 2009, the CDF collaboration observed the X(4140) for the first time in the J/ψφ mass spectrum in the exclusive B + → J/ψ φK + decays in pp collisions with a statistical significance more than 3.8σ [1]. Then the X(4140) was confirmed by CDF, CMS, D0, LHCb collaborations [1,2,3,4,5,6,7]. The LHCb collaboration performed the first full amplitude analysis of the decays B + → J/ψφK + and confirmed the two old particles X(4140) and X(4274) in the J/ψφ mass spectrum with statistical significances 8.4σ and 6.0σ, respectively, and determined the spin-paritychange-conjugation to be J P C = 1 ++ with statistical significances 5.7σ and 5.8σ, respectively [6,7]. In Table 1, we present the mass, width, J P C of the X(4140) from the different experiments. Although the width from the LHCb collaboration [6,7] differs from other measurements greatly, the masses from different experiments are consistent with each other. The D * sD * s threshold is 4224.4 MeV from the Particle Data Group [8], which leads to the possible molecule assignment for the X(4140). The X(4140) was observed in the final state J/ψφ, its J P C = 0 ++ , 1 ++ , 2 ++ for the S-wave couplings, and 0 −+ , 1 −+ , 2 −+ , 3 −+ for the P-wave couplings. The most popular current to interpolate the D * s meson is J α (x) =s(x)γ α c(x), the most popular current to interpolate the D * sD * s molecular states is J αβ (x) =s(x)γ α c(x)c(x)γ β s(x). We can study the J P C = 0 ++ , 1 −+ , 2 ++ , 1 +− , 1 −− D * sD * s molecular states with the QCD sum rules by using the suitable projectors. The LHCb collaboration determined the quantum numbers of the X(4140) to be J P C = 1 ++ , which rules out the 0 ++ or 2 ++ D * sD * s molecule assignment, but does not rule out the existence of the 0 ++ or 2 ++ D * sD * s molecular states. The possible assignments for the X(4140) are tetraquark state [9,10,11,12,13], hybrid state [14,15] or rescattering effect [16], etc. In Ref. [9], F. Stancu calculates the mass spectrum of the ccss tetraquark states via a simple quark model with chromomagnetic interaction, and obtain two lowest masses 4195 MeV and 4356 MeV with J P C = 1 ++ . The value 4195 MeV is consistent with the LHCb data 4146.5 ± 4.5 +4.6 −2.8 MeV [6,7]. In the simple chromomagnetic interaction model, there are no correlated quarks or diquarks [9]. In Ref. [11], R. F. Lebed and A. D. Polosa assign the X(4140) to be the J P C = 1 ++ diquark-antidiquark state [cs] A [cs] S + [cs] S [cs] A based on the effective Hamiltonian with the spin-spin and spin-orbit interactions, then in Ref. [12], L. Maiani, A. D. Polosa and V. Riquer take the mass of the X(4140) as input parameter, and obtain the mass spectrum of the ccss tetraquark states with positive parity, however, they observe that there is no room for the X(4274), and suggest the X(4274) corresponds to two, almost degenerate, unresolved lines with J P C = 0 ++ and 2 ++ . In the QCD sum rules, we usually take the diquarks (or correlations) and antidiquarks (or correlations) as the basic constituents to construct the interpolating currents, the predictions can be compared to that based on the diquark-antidiquark model directly [11,12]. In the quantum field theory, the diquark operators (or diquarks) ε ijk q T j CΓq ′ k 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 (S), pseudoscalar (P ), vector (V ), axialvector (A) and tensor (T ) diquarks, respectively. The Cγ 5 and Cγ µ diquark states have the spin-parity J P = 0 + and 1 + , respectively, the C and Cγ µ γ 5 diquark states have the spin-parity J P = 0 − and 1 − , respectively, the Cσ µν and Cσ µν γ 5 diquark states (or operators) have both the J P = 1 + and 1 − components. The relevant diquark-antidiquark type scalar, axialvector and tensor tetraquark states scsc have been studied with the QCD sum rules [10,17,18,19,20,21,22,23], see Table 2.
In the QCD sum rules for the hidden-charm (or hidden-bottom) tetraquark states and molecular states, the integrals s0 are sensitive to the energy scales µ, where the ρ QCD (s, µ) are the QCD spectral densities, the T 2 are the Borel parameters, the s 0 are the continuum thresholds parameters, the predicted masses depend heavily on the energy scales µ. In Refs. [24,25], we suggest an energy scale formula µ = M 2 X/Y /Z − (2M Q ) 2 with the effective Q-quark mass M Q to determine the ideal energy scales of the QCD spectral densities. The formula enhances the pole contributions remarkably, we obtain the pole contributions as large as (40 − 60)% in Refs. [10,17,18,19], otherwise, the pole contributions are about 40% [20] or 20% [22,23]. The energy scale formula also works well in the QCD sum rules for the hidden-charm pentaquark states [26].
From  [27], we can obtain the optimal energy scales µ = 1.4 GeV and 2.0 GeV for the QCD spectral densities in the QCD sum rules for the Z c (3900) and X(4140), respectively. In Ref. [19], we observe that the mass of the X(4140) can be reproduced at the energy scale µ = 1. T type axialvector currents to study the mass of the X(4140) as the axialvector tetraquark state with the QCD sum rules in details, then study the width of the X(4140) with the QCD sum rules based on the solid quark-hadron duality.
The article is arranged as follows: we derive the QCD sum rules for the mass and width of the X(4140) as axialvector tetraquark state in section 2 and in section 3 respectively; section 4 is reserved for our conclusion.   2 The mass of the X(4140) as the axialvector tetraquark state In the following, we write down the two-point correlation functions Π µµ ′ (p) in the QCD sum rules, where the i, j, k, m, n are color indexes. Under charge conjugation (parity) transform C ( P ), the currents J µ (x) have the properties, the four vectors T axialvector tetraquark states with J P C = 1 ++ , respectively. The tensor diquark operators have the properties, under parity transform, the tensor diquark operators couple potentially to both the J P = 1 + and 1 − diquark states. We should project out the 1 + or 1 − component by multiplying tensor diquark operators by the axialvector antidiquark operator ε imnsm (x)γ ν Cc T n (x) or vector antidiquark operator ε imnsm (x)γ 5 γ ν Cc T n (x). At the phenomenological side, we insert a complete set of intermediate hadronic states with the same quantum numbers as the current operators J µ (x) into the correlation functions Π µµ ′ (p) to obtain the hadronic representation [28,29], and isolate the ground state contributions, where the pole residues λ X are defined by 0|J µ (0)|X(p) = λ X ε µ , the ε µ are the polarization vectors of the axialvector tetraquark states X. Now we briefly outline the operator product expansion for the correlation functions Π µµ ′ (p). We contract the quark fields s and c in the correlation functions Π µµ ′ (p) with Wick theorem, and obtain the results, where and t n = λ n 2 [29], then compute the integrals both in the coordinate space and in the momentum space, and obtain the correlation functions Π µµ ′ (p) (i.e. Π 1 µµ ′ (p) and Π 2 µµ ′ (p)), therefore the QCD spectral densities through dispersion relation ρ(s) = lim ε→0 . For technical details, one can consult Ref. [30]. Now we take the quark-hadron duality below the continuum thresholds s 0 and perform Borel transform with respect to the variable P 2 = −p 2 to obtain the QCD sum rules: where ρ(s) = ρ T A (s) and ρ T V (s) for the [ the subscripts i in the QCD spectral densities ρ i (s) denote the dimensions of the vacuum condensates, the lengthy expressions of the QCD spectral densities are given in Appendix. We derive Eq.(11) with respect to τ = 1 T 2 , then eliminate the pole residues λ X to obtain the QCD sum rules for the masses, At the QCD side, we take the vacuum condensates to be the standard values qq = −(0.24 ± 0.01 GeV) 3  α s (µ) α s (2GeV) 12 33 , Λ = 210 MeV, 292 MeV and 332 MeV for the flavors n f = 5, 4 and 3, respectively [8,32], and evolve all the input parameters to the typical energy scales µ satisfying the energy scale formula µ = M 2 X/Y /Z − (2M c ) 2 to extract the masses of the axialvector tetraquark states.
We search for the optimal Borel parameters T 2 and 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, via try and error, and obtain the Borel windows T 2 , threshold parameters s 0 , optimal energy scales of the QCD spectral densities, and pole contributions of the ground states, see Table 3. Now we take a short digression to illustrate how to impose the four criteria to choose the Borel parameters T 2 and continuum threshold parameters s 0 . Firstly, we set M X = 3.9 GeV tentatively, and obtain the energy scale µ = 1.4 GeV according to the energy scale formula. Then we take the continuum threshold parameters to be √ s 0 = (3.9+0.5) GeV as the energy gap between the ground state and the first radial excited state is about (0.4 − 0.6) GeV, and obtain the predicted masses M X , pole contributions, and the contributions of the vacuum condensates of dimension 10. We observe that the predicted masses M X are much larger than 3.9 GeV and the pole contributions are much smaller than 50% in the regions where the Borel platforms appear, furthermore, the contributions of the vacuum condensates of dimension 10 are not small enough. Then we choose the masses M X > 3.9 GeV, say M X = 4.0 GeV, 4.1 GeV, · · · and reiterate the same procedure until obtain the optimal Borel parameters T 2 and continuum threshold parameters s 0 satisfying the four criteria.
From Table 3, we can see that the pole dominance criterion is well satisfied. In calculations, we observe that the contributions of the vacuum condensates of dimension 10 are ≪ 1% (about 1%) in the QCD sum rules for the current J 1 µ (x)(J 2 µ (x)), the operator product expansion is well convergent. We take into account all uncertainties of the input parameters, and obtain the values of the masses and pole residues of the axialvector tetraquark states, see Table 3

The width of the X(4140) as the axialvector tetraquark state
We can study the two-body strong decay X(4140) → J/ψφ with the three-point correlation function Π αβµ (p, q), where the currents interpolate the mesons J/ψ, φ(1020) and X(4140) respectively, the f J/ψ and f φ are the decay constants, the ξ α and ζ β are polarization vectors of the mesons J/ψ and φ(1020), respectively. In this section, we will use the notation m X in stead of M X for the special case J µ (0) = J 2 µ (0). At the phenomenological side, we insert a complete set of intermediate hadronic states with the same quantum numbers as the current operators J J/ψ α (x), J φ β (y), J † µ (0) into the three-point correlation function Π αβµ (p, q) and isolate the ground state contributions to obtain the result, where p ′ = p + q, the g XJ/ψφ is the hadronic coupling constant defined by the four functions ρ Xφ ′ (p ′2 , p 2 , t), ρ Xψ ′ (p ′2 , t, q 2 ), ρ X ′ J/ψ (t ′ , p 2 , q 2 ) and ρ X ′ φ (t ′ , p 2 , q 2 ) have complex dependence on the transitions between the ground states and the higher resonances or the continuum states.
In this article, we choose the tensor structure ε αβµλ p λ to study the g XJ/ψφ to avoid the contaminations from the relevant scalar and pseudoscalar mesons according to the non-vanishing coupling constants, where the f χc0 , f f0 and λ X0 are the decay constants of the χ c0 (3414), f 0 (980) and X 0 (J P = 0 − ), respectively. We introduce the parameters C Xφ ′ , C Xψ ′ , C X ′ φ and C X ′ J/ψ to parameterize the net effects, Then the correlation function Π(p ′2 , p 2 , q 2 ) on the phenomenological side can be written as Now we carry out the operator product expansion up to the vacuum condensates of dimension 5 and neglect the tiny contributions of the gluon condensate. The correlation function Π QCD (p ′2 , p 2 , q 2 ) can be written as through dispersion relation, where the ρ QCD (p ′2 , s, u) is the QCD spectral density, we introduce the subscript QCD to denote the QCD side. We rewrite the correlation function Π H (p ′2 , p 2 , q 2 ) on the hadron side as through dispersion relation, where the ρ H (s ′ , s, u) is the hadronic spectral density, we introduce the subscript H to denote the hadron side. However, on the QCD side, the QCD spectral density ρ QCD (s ′ , s, u) does not exist, Im s ′ Im s Im u Π QCD (s ′ + iǫ 3 , s + iǫ 2 , u + iǫ 1 ) We math the hadron side of the correlation function with the QCD side of the correlation function, and carry out the integral over ds ′ firstly to obtain the solid duality [33], In calculations, we observe that there appears divergence due to the endpoint s = 4m 2 c , we can avoid the endpoint divergence with the simple replacement s GeV 2 by adding a small squared s-quark mass 4m 2 s . The hadronic parameters are taken as m φ = 1.019461 GeV, m J/ψ = 3.0969 GeV [8], f J/ψ = 0.418 GeV [34], f φ = 0.253 GeV, s 0 φ = 1.5 GeV [35], s 0 J/ψ = 3.6 GeV, M X = 4146.5 MeV [6,7], λ X = 4.30 × 10 −2 GeV 5 . At the QCD side, we take the energy scale of the QCD spectral density to be µ = 2 GeV, just like in the two-point QCD sum rules. Then we set the Borel parameters to be T 2 1 = T 2 2 = T 2 for simplicity. The unknown parameters are chosen as C X ′ J/ψ + C X ′ φ = −0.00261 GeV 7 to obtain platform in the Borel window T 2 = (3.6 − 4.6) GeV 2 .
In Fig.3, we plot the hadronic coupling constant g XJ/ψφ with variation of the Borel parameter T 2 . From the figure, we can see that there appears platform in the Borel window indeed. After taking into account the uncertainties of the input parameters, we obtain the hadronic coupling constant g XJ/ψφ , g XJ/ψφ = −(2.23 ± 0.29) .