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

In 2009, the CDF collaboration observed the X(4140) for the first time in the \(J/\psi \phi \) mass spectrum in the exclusive \(B^+ \rightarrow J/\psi \,\phi K^+\) decays in \(p{\bar{p}}\) collisions with a statistical significance more than \(3.8 \sigma \) [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 \(B^+\rightarrow J/\psi \phi K^+\) and confirmed the two old particles X(4140) and X(4274) in the \(J/\psi \phi \) mass spectrum with statistical significances \(8.4\sigma \) and \(6.0\sigma \), respectively, and determined the spin-parity-change-conjugation to be \(J^{PC} =1^{++}\) with statistical significances \(5.7\sigma \) and \(5.8\sigma \), respectively [6, 7]. In Table 1, we present the mass, width, \(J^{PC}\) 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 \(D_s^*{\bar{D}}_s^*\) threshold is \(4224.4\,\mathrm{MeV}\) 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 \(J/\psi \phi \), its \(J^{PC}=0^{++}\), \(1^{++}\), \(2^{++}\) 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 \(J_\alpha (x)={\bar{s}}(x)\gamma _\alpha c(x)\), the most popular current to interpolate the \(D_s^*{\bar{D}}_s^*\) molecular states is \(J_{\alpha \beta }(x)={\bar{s}}(x)\gamma _\alpha c(x){\bar{c}}(x)\gamma _\beta s(x)\). We can study the \(J^{PC}=0^{++}\), \(1^{-+}\), \(2^{++}\), \(1^{+-}\), \(1^{--}\) \(D_s^*{\bar{D}}_s^*\) 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 \(J^{PC}=1^{++}\), which rules out the \(0^{++}\) or \(2^{++}\) \(D_s^*{\bar{D}}_s^*\) molecule assignment, but does not rule out the existence of the \(0^{++}\) or \(2^{++}\) \(D_s^*{\bar{D}}_s^*\) molecular states.

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) \(\varepsilon ^{ijk}q^{T}_j C\Gamma q^{\prime }_k\) have five structures in Dirac spinor space, where the i, j and k are color indexes, \(C\Gamma =C\gamma _5\), C, \(C\gamma _\mu \gamma _5\), \(C\gamma _\mu \) and \(C\sigma _{\mu \nu }\) for the scalar (S), pseudoscalar (P), vector (V), axialvector (A) and tensor (T) diquarks, respectively. The \(C\gamma _5\) and \(C\gamma _\mu \) diquark states have the spin-parity \(J^P=0^+\) and \(1^+\), respectively, the C and \(C\gamma _\mu \gamma _5\) diquark states have the spin-parity \(J^P=0^-\) and \(1^-\), respectively, the \(C\sigma _{\mu \nu }\) and \(C\sigma _{\mu \nu }\gamma _5\) diquark states (or operators) have both the \(J^P=1^+\) and \(1^-\) components. The relevant diquark-antidiquark type scalar, axialvector and tensor tetraquark states \(sc{\bar{s}}{\bar{c}}\) have been studied with the QCD sum rules [10, 21, 22, 23, 24, 25, 26, 27], see Table 2.

From Table 2, we can see that the \([sc]_S[{\bar{s}}{\bar{c}}]_A+[sc]_A[{\bar{s}}{\bar{c}}]_S\) type axialvector current cannot reproduce the mass of the X(4140) if the pole dominance criterion is satisfied. If we take energy scale formula \(\mu =\sqrt{M^2_{X/Y/Z}-(2{{\mathbb {M}}}_c)^2}\) and choose the updated value \({{\mathbb {M}}}_c=1.82\,\mathrm{GeV}\) [35], we can obtain the optimal energy scales \(\mu =1.4\,\mathrm{GeV}\) and \(2.0\,\mathrm{GeV}\) for the QCD spectral densities in the QCD sum rules for the \(Z_c(3900)\) and X(4140), respectively. In Ref. [23], we observe that the mass of the X(4140) can be reproduced at the energy scale \(\mu =1.1\,\mathrm{GeV}\), a too low energy scale. The QCD sum rules do not support assigning the X(4140) to be the \([sc]_S[{\bar{s}}{\bar{c}}]_A+[sc]_A[{\bar{s}}{\bar{c}}]_S\) type axialvector tetraquark state.

In this article, we construct 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 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 Sects. 2 and 3 respectively; Sect. 4 is reserved for our conclusion.

Now we briefly outline the operator product expansion for the correlation functions \(\Pi _{\mu \mu ^\prime }(p)\). We contract the quark fields s and c in the correlation functions \(\Pi _{\mu \mu ^\prime }(p)\) with Wick theorem, and obtain the results,

$$\begin{aligned}&\Pi ^1_{\mu \mu ^{\prime }}(p) =-\frac{i}{2} \varepsilon ^{ijk}\varepsilon ^{imn}\varepsilon ^{i^{\prime }j^{\prime }k^{\prime }}\varepsilon ^{i^{\prime }m^{\prime }n^{\prime }} \int d^4x e^{ip \cdot x} \nonumber \\&\qquad \left\{ \mathrm{Tr}\left[ \sigma _{\mu \nu }\gamma _5 S_c^{kk^{\prime }}(x)\gamma _5\sigma _{\mu ^{\prime }\nu ^{\prime }} CS^{Tjj^{\prime }}(x)C\right] \right. \nonumber \\&\qquad \times \mathrm{Tr}\left[ \gamma ^{\nu ^\prime } S_c^{n^{\prime }n}(-x)\gamma ^\nu CS^{Tm^{\prime }m}(-x)C\right] \nonumber \\&\qquad +\mathrm{Tr}\left[ \gamma ^\nu S_c^{kk^{\prime }}(x)\gamma _5\sigma _{\mu ^{\prime }\nu ^{\prime }} CS^{Tjj^{\prime }}(x)C\right] \nonumber \\&\qquad \times \mathrm{Tr}\left[ \gamma ^{\nu ^{\prime }} S_c^{n^{\prime }n}(-x)\gamma _5 \sigma _{\mu \nu }CS^{Tm^{\prime }m}(-x)C\right] \nonumber \\&\qquad + \mathrm{Tr}\left[ \sigma _{\mu \nu }\gamma _5 S_c^{kk^{\prime }}(x)\gamma ^{\nu ^{\prime }} CS^{Tjj^{\prime }}(x)C\right] \nonumber \\&\qquad \times \mathrm{Tr}\left[ \sigma _{\mu ^{\prime }\nu ^{\prime }}\gamma _5 S_c^{n^{\prime }n}(-x)\gamma ^\nu CS^{Tm^{\prime }m}(-x)C\right] \nonumber \\&\qquad + \mathrm{Tr}\left[ \gamma ^{\nu } S_c^{kk^{\prime }}(x)\gamma ^{\nu ^{\prime }}CS^{Tjj^{\prime }}(x)C\right] \nonumber \\&\qquad \times \left. \mathrm{Tr}\left[ \sigma _{\mu ^{\prime }\nu ^{\prime }} \gamma _5 S_c^{n^{\prime }n}(-x)\gamma _5\sigma _{\mu \nu } CS^{Tm^{\prime }m}(-x)C\right] \right\} , \end{aligned}$$

(6)

$$\begin{aligned}&\Pi ^2_{\mu \mu ^{\prime }}(p)=-\frac{i}{2} \varepsilon ^{ijk}\varepsilon ^{imn}\varepsilon ^{i^{\prime }j^{\prime }k^{\prime }}\varepsilon ^{i^{\prime }m^{\prime }n^{\prime }} \int d^4x e^{ip \cdot x} \nonumber \\&\qquad \left\{ \mathrm{Tr}\left[ \sigma _{\mu \nu }S_c^{kk^{\prime }}(x)\sigma _{\mu ^{\prime }\nu ^{\prime }} CS^{Tjj^{\prime }}(x)C\right] \right. \nonumber \\&\qquad \times \mathrm{Tr}\left[ \gamma ^{\nu ^\prime } \gamma _5 S_c^{n^{\prime }n}(-x)\gamma _5 \gamma ^\nu CS^{Tm^{\prime }m}(-x)C\right] \nonumber \\&\qquad +\mathrm{Tr}\left[ \gamma ^\nu \gamma _5 S_c^{kk^{\prime }}(x)\sigma _{\mu ^{\prime }\nu ^{\prime }} CS^{Tjj^{\prime }}(x)C\right] \nonumber \\&\qquad \times \mathrm{Tr}\left[ \gamma ^{\nu ^{\prime }} \gamma _5 S_c^{n^{\prime }n}(-x) \sigma _{\mu \nu }CS^{Tm^{\prime }m}(-x)C\right] \nonumber \\&\qquad + \mathrm{Tr}\left[ \sigma _{\mu \nu } S_c^{kk^{\prime }}(x)\gamma _5 \gamma ^{\nu ^{\prime }} CS^{Tjj^{\prime }}(x)C\right] \nonumber \\&\qquad \times \mathrm{Tr}\left[ \sigma _{\mu ^{\prime }\nu ^{\prime }} S_c^{n^{\prime }n}(-x)\gamma _5 \gamma ^\nu CS^{Tm^{\prime }m}(-x)C\right] \nonumber \\&\qquad + \mathrm{Tr}\left[ \gamma ^{\nu }\gamma _5 S_c^{kk^{\prime }}(x)\gamma _5 \gamma ^{\nu ^{\prime }}CS^{Tjj^{\prime }}(x)C\right] \mathrm{Tr}\nonumber \\&\qquad \left. \times \left[ \sigma _{\mu ^{\prime }\nu ^{\prime }} S_c^{n^{\prime }n}(-x)\sigma _{\mu \nu } CS^{Tm^{\prime }m}(-x)C\right] \right\} , \end{aligned}$$

(7)

where

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(8)

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(9)

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(10)

and \(t^n=\frac{\lambda ^n}{2}\) [38], then compute the integrals both in the coordinate space and in the momentum space, and obtain the correlation functions \(\Pi _{\mu \mu ^\prime }(p)\) (i.e. \(\Pi ^1_{\mu \mu ^\prime }(p)\) and \(\Pi ^2_{\mu \mu ^\prime }(p)\)), therefore the QCD spectral densities through dispersion relation \(\rho (s)=\mathrm{lim}_{\varepsilon \rightarrow 0}\frac{\mathrm{Im}\Pi (s+i\varepsilon )}{\pi }\). For technical details, one can consult Ref. [39].

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 \(M_X=3.9\,\mathrm{GeV}\) tentatively, and obtain the energy scale \(\mu =1.4\,\mathrm{GeV}\) according to the energy scale formula. Then we take the continuum threshold parameters to be \(\sqrt{s_0}=(3.9+0.5)\,\mathrm{GeV}\) as the energy gap between the ground state and the first radial excited state is about \((0.4-0.6)\,\mathrm{GeV}\), 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 \(3.9\,\mathrm{GeV}\) 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 \(M_X>3.9\,\mathrm{GeV}\), say \(M_X=4.0\,\mathrm{GeV}\), \(4.1\,\mathrm{GeV}\), \(\ldots \) 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 \(\ll 1\%\) (about \(1\%\)) in the QCD sum rules for the current \(J^1_\mu (x)\)(\(J^2_\mu (x)\)), 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 and Figs. 1, 2. From Table 3, we can see that the energy scale formula is well satisfied. From Figs. 1, 2 and Table 3, we can see that there appear platforms in the Borel windows. The four criteria are all satisfied, our predictions are reliable.

From Table 3, we can see the predicted mass \(M_X=4.14\pm 0.10\,\mathrm{GeV}\) for the \([sc]_T[{\bar{s}}{\bar{c}}]_V-[sc]_V[{\bar{s}}{\bar{c}}]_T\) axialvector tetraquark state is in excellent agreement with the experimental data \(4146.5\pm 4.5^{+ 4.6}_{-2.8}\,\mathrm{MeV}\) from the LHCb collaboration [6, 7], which supports assigning the X(4140) to be the \([sc]_T[{\bar{s}}{\bar{c}}]_V-[sc]_V[{\bar{s}}{\bar{c}}]_T\) tetraquark state with \(J^{PC}=1^{++}\). While the \([sc]_T[{\bar{s}}{\bar{c}}]_A+[sc]_V[{\bar{s}}{\bar{c}}]_A\) axialvector tetraquark state has a much larger mass than that of the X(4140).

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The hadronic parameters are taken as \(m_{\phi }=1.019461\,\mathrm{GeV}\), \(m_{J/\psi }=3.0969\,\mathrm{GeV}\) [8], \(f_{J/\psi }=0.418 \,\mathrm{GeV}\) [43], \(f_{\phi }=0.253\,\mathrm{GeV}\), \(\sqrt{s^0_{\phi }}=1.5\,\mathrm{GeV}\) [44], \(\sqrt{s^0_{J/\psi }}=3.6\,\mathrm{GeV}\), \(M_X=4146.5\,\mathrm{MeV}\) [6, 7], \(\lambda _{X}=4.30\times 10^{-2}\,\mathrm{GeV}^5\). At the QCD side, we take the energy scale of the QCD spectral density to be \(\mu =2\,\mathrm{GeV}\), just like in the two-point QCD sum rules. Then we set the Borel parameters to be \(T_1^2=T_2^2=T^2\) for simplicity. The unknown parameters are chosen as \(C_{X^{\prime }J/\psi }+C_{X^{\prime }\phi }=-0.00261\,\mathrm{GeV}^7 \) to obtain platform in the Borel window \(T^2=(3.6-4.6)\,\mathrm{GeV}^2\).

In Fig. 3, we plot the hadronic coupling constant \(g_{XJ/\psi \phi }\) 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 \(g_{XJ/\psi \phi }\),

$$\begin{aligned} g_{XJ/\psi \phi }= & {} -(2.23\pm 0.29). \end{aligned}$$

(33)

Now it is easy to obtain the decay width,

$$\begin{aligned}&\Gamma (X(4140)\rightarrow J/\psi \phi )\nonumber \\&\quad = \frac{p\left( m_{X},m_{J/\psi },m_{\phi }\right) }{24\pi m_{X}^2}g_{XJ/\psi \phi }^2\left\{ \frac{\left( m_{X}^2-m_{\phi }^2\right) ^2}{2m_{J/\psi }^2}\right. \nonumber \\&\left. \qquad +\frac{\left( m_{X}^2-m_{J/\psi }^2\right) ^2}{2m_{\phi }^2} +4m_X^2-\frac{m_{J/\psi }^2+m_\phi ^2}{2}\right\} \nonumber \\&\quad =86.9\pm 22.6\,{\mathrm{MeV}}, \end{aligned}$$

(34)

where \(p(a,b,c)=\frac{\sqrt{[a^2-(b+c)^2][a^2-(b-c)^2]}}{2a}\). The width \(\Gamma (X(4140)\rightarrow J/\psi \phi )=86.9\pm 22.6\,{\mathrm{MeV}}\) is in excellent agreement with the experimental data \(83\pm 21^{+21}_{-14} { \text{ MeV }}\) from the LHCb collaboration [6, 7]. The present work supports assigning the X(4140) to be the \([sc]_T[{\bar{s}}{\bar{c}}]_V-[sc]_V[{\bar{s}}{\bar{c}}]_T\) tetraquark state with \(J^{PC}=1^{++}\).

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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 by carrying out the operator product expansion up to the vacuum condensates of dimension 10 and take the energy scale formula \(\mu =\sqrt{M^2_{X/Y/Z}-(2{{\mathbb {M}}}_c)^2}\) to determine the ideal energy scales of the QCD spectral densities. 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 calculate the hadronic coupling constant \(g_{XJ/\psi \phi }\) with the QCD sum rules based on the solid quark-hadron duality, and obtain the decay width \(\Gamma (X(4140)\rightarrow J/\psi \phi )=86.9\pm 22.6\,{\mathrm{MeV}}\), which is in excellent agreement with the experimental data \(83\pm 21^{+21}_{-14} { \text{ MeV }}\) from the LHCb collaboration. In summary, the present work supports assigning the X(4140) to be the \([sc]_T[{\bar{s}}{\bar{c}}]_V-[sc]_V[{\bar{s}}{\bar{c}}]_T\) type tetraquark state with \(J^{PC}=1^{++}\).