Vol.35 (Oct) 2025 | Article no.25 2025
In this work, we introduce an explicit P-wave to construct the diquarks \([qc]_{\widehat{V}}\), then construct the local four-quark currents to explore the hidden-charm tetraquark states with the \(J^{PC}=0^{++}\), \(1^{+-}\), and \(2^{++}\) in the framework of the QCD sum rules at length. Our calculations indicate that the light-flavor SU(3) breaking effects on the tetraquark masses are tiny. The predictions support assigning the X(4475) and X(4500) as the \([uc]_{\widehat{V}}[\overline{uc}]_{\widehat{V}}-[dc]_{\widehat{V}}[\overline{dc}]_{\widehat{V}}\) and \([sc]_{\widehat{V}}[\overline{sc}]_{\widehat{V}}\) tetraquark states with the \(J^{PC}=0^{++}\) respectively, and assigning the \(Z_{c}(4600)\) and \(Z_{\bar{c}\bar{s}}(4600)\) as the \([uc]_{\widehat{V}}[\overline{dc}]_{\widehat{V}}\) and \([qc]_{\widehat{V}}[\overline{sc}]_{\widehat{V}}\) tetraquark states with the \(J^{PC}=1^{+-}\) respectively. On the other hand, there is no room for the X(4710) and X(4700). Combined with previous works, the X(4475), X(4500), \(Z_{c}(4600)\), and \(Z_{\bar{c}\bar{s}}(4600)\) might have other important Fock components besides the \(\widehat{V}\widehat{V}\) type components.
In 2016, the LHCb collaboration performed the first full amplitude analysis of the decays \(B^+\rightarrow J/\psi \phi K^+\) with a data sample of \(3\,\textrm{fb}^{-1}\) of the pp collision data at \(\sqrt{s}=7\) and \(8\, \textrm{TeV}\) [1, 2]. And they observed four \(J/\psi \phi\) structures: two old particles X(4140) and X(4274) and two new particles X(4500) and X(4700). The statistical significances of the X(4140), X(4274), X(4500), and X(4700) are \(8.4\sigma\), \(6.0\sigma\), \(6.1\sigma\), and \(5.6\sigma\), respectively. While the statistical significances of their quantum numbers \(J^{PC}=1^{++}\), \(1^{++}\), \(0^{++}\), and \(0^{++}\) are \(5.7\sigma\), \(5.8\sigma\), \(4.0\sigma\), and \(4.5\sigma\), respectively [1, 2]. The LHCb collaboration determined the \(J^{PC}\) of the X(4140) to be \(1^{++}\), thus ruling out the \(0^{++}/2^{++}\) \(D_s^{*+}D_s^{*-}\) molecule assignments.
In 2021, the LHCb collaboration performed an improved full amplitude analysis of the exclusive \(B^+\rightarrow J/\psi \phi K^+\) decays using the pp collision data (6 times larger signal yield than previously analyzed) corresponding to a total integrated luminosity of \(9\textrm{fb}^{-1}\) at \(\sqrt{s}=7\), 8, and 13 \(\mathrm TeV\) [3]. They observed the \(Z_{cs}(4000)\) with the \(J^P=1^+\) in the \(J/\psi K^+\) mass spectrum with the statistical significance of \(15\sigma\), and the X(4685) (X(4630)) in the \(J/\psi \phi\) mass spectrum with the \(J^P=1^+\) (\(1^-\)) with the statistical significance of \(15\sigma\) (\(5.5\sigma\)). Furthermore, they confirmed the old particles X(4140), X(4274), X(4500), and X(4700). The measured Breit-Wigner masses and widths are,
If we adopt the scenario of the color \(\bar{\textbf{3}}\textbf{3}\)-type tetraquark states and consult the predictions based on the QCD sum rules, the X(4140) and X(4274) lie at the 1S region, the X(4500), X(4685), and X(4700) lie at the 2S region, the X(4630) lies at the 1P region, etc. [4,5,6].
In 2024, the LHCb collaboration performed the first full amplitude analysis of the decays \(B^+ \rightarrow \psi (2S) K^+ \pi ^+ \pi ^-\) using the pp collision data corresponding to an integrated luminosity of \(9\,\text {fb}^{-1}\), and they developed an amplitude model with 53 components comprising 11 hidden-charm exotic states: \(Z_c(4055)\), \(Z_c(4200)\), \(Z_c(4430)\), X(4475), X(4710), X(4650), X(4800), \(Z_{\bar{c}\bar{s}}(4000)\), \(Z_{\bar{c}\bar{s}}(4600)\), \(Z_{\bar{c}\bar{s}}(4900)\), \(Z_{\bar{c}\bar{s}}(5200)\) [7].
They confirmed the \(Z_c(4200)\) and \(Z_c(4430)\) in the \(\psi (2S) \pi ^+\) mass spectrum, and determined the spin-parity of the \(Z_c(4200)\) to be \(1^+\) for the first time with a significance exceeding \(5\sigma\), and observed the \(Z_{\bar{c}\bar{s}}(4600)\) and \(Z_{\bar{c}\bar{s}}(4900)\) in the \(\psi (2S)K^*(892)\) mass spectrum. The measured Breit-Wigner masses and widths are [7],
In 2019, the LHCb collaboration performed an angular analysis of the decays \(B^0\rightarrow J/\psi K^+\pi ^-\) using the pp collision data corresponding to an integrated luminosity of \(3\,\textrm{fb}^{-1}\), examined the \(m(J/\psi \pi ^-)\) versus the \(m(K^+\pi ^-)\) plane, and observed two structures in the vicinity of the energies \(m(J/\psi \pi ^-)=4200 \,\textrm{MeV}\) and \(4600\,\textrm{MeV}\), respectively [8]. However, the \(Z_c(4600)\) has not been confirmed yet.
Again, we adopt the scenario of the color \(\bar{\textbf{3}}\textbf{3}\)-type tetraquark states and consult the predictions based on the QCD sum rules, the \(Z_c(4430)\), \(Z_c(4600)\), and \(Z_{\bar{c}\bar{s}}(4600)\) lie at the 2S region, the \(Z_{\bar{c}\bar{s}}(4900)\) lies at the 3S region, while there is no room for the \(Z_c(4200)\) [6, 9,10,11].
In addition, the LHCb collaboration observed that the \(\psi (2S) \pi ^+ \pi ^-\) mass spectrum are dominated by the \(X^0 \rightarrow \psi (2S) \rho ^0(770)\) decays with \(X^0=X(4475)\), X(4650), X(4710), and X(4800) [7], which are similar to the previously observed \(J/\psi \phi\) resonances X(4500), X(4685), X(4700), and X(4630), respectively [3]. The spin-parity of the X(4630) have not been unambiguously determined yet, the assignment \(J^P =1^-\) is favored over \(J^P =2^-\) with a significance of \(3\sigma\), and other assignments are disfavored by more than \(5\sigma\) [3]. As the strong decays conserve isospin, we can sort out those states according to the isospins of the final states \(\psi (2S)\rho ^0(770)\) and \(J/\psi \phi (1020)\),
The measured Breit-Wigner masses and widths are [7],
A possible explanation is that those states are genuinely different states, however, for example, if the X(4475) state is the \(c\bar{c}(u\bar{u} -d\bar{d})\) isospin partner of the X(4500) which interpreted as the \(c\bar{c}s\bar{s}\) state, we would generally expect a larger mass difference of \(M_{X(4500)}-M_{X(4475)}\approx 200\,\textrm{MeV}\) rather than several \(\textrm{MeV}\). It is interesting to explore the light-flavor SU(3) breaking effects.
On the other hand, we should bear in mind that the observation of the \(Z_c(4055)\), X(4800), and \(Z_{\bar{c}\bar{s}}(5200)\) should not be considered as confirmations of specific states but rather effective descriptions of the generic \(\psi (2S)\pi ^+\), \(\psi (2S)\rho ^0(770)\), and \(\psi (2S)[K^+\pi ^-]_S\) contributions respectively with the \(J^P=1^-\) [7].
Also in 2024, the LHCb collaboration accomplished the first investigation of the \(J/\psi \phi\) production in diffractive processes in the pp collisions, which is based on a data-set recorded at \(\sqrt{s}=13\,\textrm{TeV}\) corresponding to an integrated luminosity of \(5\,\textrm{fb}^{-1}\) [12]. The data are consistent with a resonant model including several resonant states observed previously in the \(B^+\rightarrow J/\psi \phi K^+\) decays. The X(4500) and X(4274) were observed with significances over \(5\,\sigma\) and \(4\,\sigma\), respectively.
Now we reach a short summary. The X(4140), X(4274), X(4500), X(4630), X(4685), and X(4700) observed in the \(J/\psi \phi\) mass spectrum have the symbolic valence quarks \(c\bar{c}s\bar{s}\), isospin \((I,I_3)=(0,0)\), and \(J^{PC}=0^{++}\), \(1^{++}\), \(2^{++}\) for the S-wave systems and \(0^{-+}\), \(1^{-+}\), \(2^{-+}\), \(3^{-+}\) for the P-wave systems. The X(4475), X(4650), X(4710), and X(4800) observed in the \(\psi (2S)\rho ^0(770)\) mass spectrum have the symbolic valence quarks \(c\bar{c}(u\bar{u}-d\bar{d})\), isospin \((I,I_3)=(1,0)\), and \(J^{PC}=0^{++}\), \(1^{++}\), \(2^{++}\) for the S-wave systems and \(0^{-+}\), \(1^{-+}\), \(2^{-+}\), \(3^{-+}\) for the P-wave systems. The \(Z_{\bar{c}\bar{s}}(4600)\) and \(Z_{\bar{c}\bar{s}}(4900)\) (\(Z_{\bar{c}\bar{s}}(4000)\)) observed in the \(\psi (2S)K^*(892)\) (\(\psi (2S)K\)) mass spectrum have the symbolic valence quarks \(c\bar{c}u\bar{s}\), isospin \((I,I_3)=(\frac{1}{2},\frac{1}{2})\), and \(J^{PC}=1^{+-}\) for the S-wave systems.
In the following, we would like to see from the perspective of the tetraquark picture for the X, Y, and Z states and resort to the QCD sum rules [6]. We often take the diquarks as the basic valence constituents to study the tetraquark states. The diquarks \(\varepsilon ^{ijk}q^{T}_j C\Gamma q^{\prime }_k\) have five spinor structures, where \(C\Gamma =C\gamma _5\), C, \(C\gamma _\mu \gamma _5\), \(C\gamma _\mu\), and \(C\sigma _{\mu \nu }\) (or \(C\sigma _{\mu \nu }\gamma _5\)) for the scalar (S), pseudoscalar (P), vector (V), axialvector (A), and tensor (T) diquarks, respectively, the i, j, k are color indexes. And the T-diquarks have both the \(J^P=1^+\) (\(\widetilde{A}\)) and \(1^-\) (\(\widetilde{V}\)) components.
In Ref. [13], we explored the energy scale dependence of the QCD sum rules for the X, Y, and Z states for the first time. Later, we suggested an energy scale formula,
via introducing the effective heavy quark masses \({\mathbb {M}}_Q\) to obtain the suitable energy scales of the QCD spectral densities for the hidden-charm (hidden-bottom) tetraquark states [14,15,16]. The formula can magnify the ground state contributions substantially and improve convergence of the operator product expansion substantially. If there exist valence s-quarks, we modify the energy scale formula,
to account for the light-flavor SU(3) breaking effects via introducing the effective s-quark mass \({\mathbb {M}}_s\), where the \(\kappa\) is the valence s-quark’s number. The modified energy scale formula plays an important role in exploring the light-flavor SU(3) breaking effects in the multiquark states, especially those X states [4, 5, 17,18,19,20,21,22,23,24].
We should bear in mind that there exist other theoretical schemes (irrespective the tetraquark picture [25,26,27,28,29,30], molecule picture [31,32,33], or charmonium picture [34,35,36,37]) and possible assignments for those X states. All the assignments can account for some experimental data to some extent. However, no definite conclusion can be obtained up to now, more theoretical and experimental works are still needed.
In the theoretical scheme of the QCD sum rules having our unique feature, we have performed comprehensive analysis of the exotic states [6]. For example, the hidden-charm tetraquark states with the \(J^{PC}=0^{++}\), \(0^{-\pm }\), \(1^{-\mp }\), \(1^{+\mp }\), \(2^{++}\) [4, 21, 24, 38, 39], hidden-bottom tetraquark states with the \(J^{PC}=0^{++}\), \(1^{+\mp }\), \(2^{++}\) [40], hidden-charm molecular states with the \(J^{PC}=0^{++}\), \(1^{+\mp }\), \(2^{++}\) [18], doubly-charm tetraquark (molecular) states with the \(J^{P}=0/1/2^{+}\) [41] ([20]), and hidden-charm pentaquark (molecular) states with the \(J^{P}={\frac{1}{2}}/{\frac{3}{2}}/{\frac{5}{2}}^{-}\) [42] ([43]).
In Ref. [4], we take account of the light-flavor SU(3) breaking effects comprehensively, and revisit the assignments of the potential \(cs\bar{c}\bar{s}\) tetraquark candidates and supersede the old assignments [44,45,46,47]. The possible assignments of the X(4140), X(4274), X(4500), X(4685), and X(4700) in terms of the \(cs\bar{c}\bar{s}\) tetraquark states are given in Table 1. Compared with the \(cu\bar{c}\bar{d}\) or \(cu\bar{c}\bar{u}-cd\bar{c}\bar{d}\) tetraquark states [24], the light-flavor SU(3) breaking effects on the tetraquark masses are remarkable. If the X(4500), X(4685), X(4700), and X(4630) have the same Dirac spinor structures as the X(4475), X(4650), X(4710), and X(4800) respectively, but different isospin structures, the light-flavor SU(3) breaking effects on the hadron masses should be very small, we would like to explore the odd phenomenon.
| \(J^{PC}\) | \(X(1\textrm{S})\) | \(X(2\textrm{S})\) | |
|---|---|---|---|
| \([sc]_{A}[\overline{sc}]_{A}\) | \(0^{++}\) | ? X(4700) | |
| \([sc]_{\tilde{A}}[\overline{sc}]_{\tilde{A}}\) | \(0^{++}\) | ? X(4700) | |
| \([sc]_{S}[\overline{sc}]_{S}{}^*\) | \(0^{++}\) | ? X(3960) | ? X(4500) |
| \([sc]_{V}[\overline{sc}]_{V}{}^*\) | \(0^{++}\) | ? X(4700) | |
| \([sc]_S[\overline{sc}]_{A}+[sc]_{A}[\overline{sc}]_S\) | \(1^{++}\) | ? X(4140) | ? X(4685) |
| \([sc]_S[\overline{sc}]_{\widetilde{A}}+[sc]_{\widetilde{A}}[\overline{sc}]_S\) | \(1^{++}\) | ? X(4140) | ? X(4685) |
| \([sc]_{\widetilde{V}}[\overline{sc}]_{V}-[sc]_{V}[\overline{sc}]_{\widetilde{V}}\) | \(1^{++}\) | ? X(4274) |
In Ref. [48], we introduce an explicit P-wave to construct the doubly-charm diquarks \(\varepsilon ^{ijk} c^{T}_j C\gamma _5{\mathop {\partial }\limits ^{\leftrightarrow }}_\mu c_k\), then we take them as the basic constituents to study the fully-charm tetraquark states with the QCD sum rules, where the derivative \({\mathop {\partial }\limits ^{\leftrightarrow }}_\mu ={\mathop {\partial }\limits ^{\rightarrow }}_\mu -{\mathop {\partial }\limits ^{\leftarrow }}_\mu\) embodies the net P-wave effect. We can take the vector heavy-light diquarks \(\varepsilon ^{ijk} c^{T}_j C\gamma _5{\mathop {D}\limits ^{\leftrightarrow }}_\mu q_k\) with \(D_\mu =\partial _\mu -ig_sG_\mu\) as the elementary constituents to study the \(cq\bar{c}\bar{q}^\prime\) tetraquark states with the \(J^{PC}=0^{++}\), \(1^{+-}\), and \(2^{++}\), examine the light-flavor SU(3) breaking effects, and make reasonable assignments of the new X states. In the heavy quark limit, the c-quark is static, the diquarks \(\varepsilon ^{ijk} c^{T}_j C\gamma _5{\mathop {D}\limits ^{\leftrightarrow }}_\mu q_k\) are reduced to the form \(\varepsilon ^{ijk} c^{T}_j C\gamma _5D_\mu q_k\), and we would like to take the reduced operators and denote them as \(\widehat{V}\).
In Ref. [25], Chen et al. choose a D-wave diquark in the color \(\bar{\textbf{3}}\) (\(\textbf{6}\)) and a S-wave antidiquark in the color \(\textbf{3}\) (\(\bar{\textbf{6}}\)) to construct the four-quark current to study the tetraquark state with the \(J^P=0^+\), and obtain the tetraquark mass \(4.55^{+0.19}_{-0.13}\,\textrm{GeV}\) (\(4.66^{+0.20}_{-0.14}\,\textrm{GeV}\)), then assign the X(4500) and X(4700) tentatively. While in this work, we choose the \([cq]_{\widehat{V}}[\bar{c}\bar{q}^\prime ]_{\widehat{V}}\)-type currents, which have two P-waves (an effective D-wave) instead of a D-wave.
The article is arranged as follows: we obtain the QCD sum rules for the hidden-charm tetraquark states in Sect. 2; in Sect. 3, we present the numerical results and discussions; Sect. 4 is reserved for our conclusion.
At the beginning, we write down the two-point correlation functions \(\Pi (p)\) and \(\Pi _{\mu \nu \alpha \beta }(p)\),
where \(J_{\mu \nu }(x)=J^{\pm }_{\mu \nu }(x)\),
the i, j, k, m, n are color indexes, q, \(q^\prime =u\), d or s, the charge conjugation matrix \(C=i \gamma ^2 \gamma ^0\), and the superscripts ± denote the ± charge conjugation, respectively. We check the properties under parity \(\widehat{P}\) and charge-conjugation \(\widehat{C}\) transformations,
and
where the coordinates \(x^\mu =(t,\vec {x})\) and \(\tilde{x}^\mu =(t,-\vec {x})\).
In the isospin limit, the currents with the symbolic quark constituents \(\bar{c}c\bar{d}u\), \(\bar{c}c\bar{u}d\), \(\bar{c}c\frac{\bar{u}u-\bar{d}d}{\sqrt{2}}\), \(\bar{c}c\frac{\bar{u}u+\bar{d}d}{\sqrt{2}}\) couple potentially to the tetraquark states with degenerated masses. The currents with the isospins \(I=1\) and 0 lead to the same QCD sum rules, while the currents with the symbolic quark constituents \(\bar{c}c\bar{q}s\) and \(\bar{c}c\bar{s}q\) (with \(q=u\) and d) couple potentially to the tetraquark states with degenerated masses, and lead to the same QCD sum rules. In this work, we would like to choose the quark configurations \(\bar{c}c\bar{d}u\), \(\bar{c}c\bar{s}q\), and \(\bar{c}c\bar{s}s\) to explore the mass spectrum.
Now we insert a complete set of intermediate hadronic states with the same quantum numbers as the currents into the correlation functions to obtain the hadronic representation, then isolate the ground state contributions,
where \(\widetilde{g}_{\mu \nu }=g_{\mu \nu }-\frac{p_{\mu }p_{\nu }}{p^2}\). We add the superscripts ± in the \(\Pi ^{\pm }_{\mu \nu \alpha \beta }(p)\) to denote ± charge conjugation, respectively, and add the superscripts/subscripts ± in the \(X^{\pm }\)/\(\Pi _{\pm }(p^2)\)/\(\widetilde{\Pi }_{\pm }(p^2)\) to denote ± parity, respectively. The pole residues \(\lambda _{X^\pm }\) are defined by
where the \(\varepsilon _{\mu /\alpha }\) and \(\varepsilon _{\mu \nu }\) are the tetraquark polarization vectors.
We choose the components \(\Pi _{+}(p^2)\) and \(p^2\widetilde{\Pi }_{+}(p^2)\) to explore the tetraquark states with the \(J^{PC}=0^{++}\), \(1^{+-}\), and \(2^{++}\) without contaminations from other states. Generally speaking, the quantum field theory does not forbid the currents J(x) and \(J_{\mu \nu }(x)\) coupling to the two-meson scattering states if they have the same quantum numbers, there might exist contaminations from the two-meson scattering states. In Refs. [49, 50], we illustrate that the two-meson scattering states play an unimportant role and cannot saturate the QCD sum rules by themselves, on the other hand, the tetraquark (molecular) states play an irreplaceable role. We can saturate the QCD sum rules with or without the two-particle scattering states, it is reliable to study the tetraquark (molecular) states. In other words, we choose the local four-quark currents, and the mesons have finite average radii \(\sqrt{\langle r^2\rangle }\), the net effects of the small overlapping of the wave-functions can be absorbed into the pole residues safely [51].
At the QCD side, we accomplish the operator product expansion up to the condensates of dimension 10 consistently based on our unique counting rules and adopt the truncation \({\mathcal {O}}(\alpha _s^k)\) with \(k\le 1\) for the quark-gluon operators so as to estimate reliability and feasibility [21, 24, 38, 39]. We compute the condensates \(\langle \bar{q}q\rangle\), \(\langle \frac{\alpha _{s}GG}{\pi }\rangle\), \(\langle \bar{q}g_{s}\sigma Gq\rangle\), \(\langle \bar{q}q\rangle ^2\), \(\langle \bar{q}q\rangle \langle \frac{\alpha _{s}GG}{\pi }\rangle\), \(\langle \bar{q}q\rangle \langle \bar{q}g_{s}\sigma Gq\rangle\), \(\langle \bar{q}g_{s}\sigma Gq\rangle ^2\), and \(\langle \bar{q}q\rangle ^2 \langle \frac{\alpha _{s}GG}{\pi }\rangle\) with \(q=u\), d, or s. Then, we obtain the QCD spectral densities \(\rho _{QCD}(s)\) through dispersion relation directly. In computations, we take the full quark propagators,
with \(Q=c\), \(t^n=\frac{\lambda ^n}{2}\), the \(\lambda ^n\) is the Gell-Mann matrix [13, 52, 53]. We introduce the new terms \(\langle \bar{q}_j\sigma _{\mu \nu }q_i \rangle\) in the full light-quark propagators through the Fierz re-arrangement [13], which absorb the gluons emitted from the other quark lines and result in the mixed condensate \(\langle \bar{q}g_s\sigma G q\rangle\).
We match the hadron side with the QCD side for the components \(\Pi _{+}(p^2)\) and \(p^2\widetilde{\Pi }_{+}(p^2)\) in the spectral representation below the continuum thresholds \(s_0\) and carry out the Borel transformation in regard to the \(P^2=-p^2\) to obtain the QCD sum rules:
where the \(T^2\) is the Borel parameter.
As last, we differentiate the QCD sum rules in Eq.(17) with respect to the variable \(\tau =\frac{1}{T^2}\), and obtain the QCD sum rules for the masses of the \(cq\overline{cq}\) tetraquark states \(X^+\) with the positive parity,
At first, we write down the energy-scale dependence of the input parameters from the re-normalization group equation with the lowest order approximation,
where the quarks \(q=u\), d, and s [54, 55], the strong fine-structure constant \(\alpha _s(\mu )\) is determined in most cases in the next-to-next-to-leading order approximation,
with \(t=\log \frac{\mu ^2}{\Lambda _{QCD}^2}\), \(b_0=\frac{33-2n_f}{12\pi }\), \(b_1=\frac{153-19n_f}{24\pi ^2}\), \(b_2=\frac{2857-\frac{5033}{9}n_f+\frac{325}{27}n_f^2}{128\pi ^3}\), \(\Lambda _{QCD}=210\,\textrm{MeV}\), \(292\,\textrm{MeV}\), and \(332\,\textrm{MeV}\) for the flavors \(n_f=5\), 4, and 3, respectively [54]. And we choose \(n_f=4\) in the present analysis.
At the initial points, we adopt the standard values \(\langle \bar{q}q \rangle =-(0.24\pm 0.01\, \textrm{GeV})^3\), \(\langle \bar{s} s \rangle =(0.8 \pm 0.1)\langle \bar{q}q \rangle\), \(\langle \bar{q}g_s\sigma G q \rangle =m_0^2\langle \bar{q}q \rangle\), \(\langle \bar{s}g_s\sigma G s \rangle =m_0^2\langle \bar{s}s \rangle\), \(m_0^2=(0.8 \pm 0.1)\,\textrm{GeV}^2\), \(\langle \frac{\alpha _s GG}{\pi }\rangle =(0.012\pm 0.004)\,\textrm{GeV}^4\) at the particular energy scale \(\mu =1\, \textrm{GeV}\) with \(q=u\) and d [52, 56, 57], and adopt the modified-minimal-subtraction masses \(m_{c}(m_c)=(1.275\pm 0.025)\,\textrm{GeV}\) and \(m_{s}(\textrm{2 GeV})=(0.095\pm 0.005)\,\textrm{GeV}\) from the Particle Data Group [54]. Moreover, we set \(m_u=m_d=0\) considering their tiny values.
In our previous works, we adopt the modified energy scale formula \(\mu =\sqrt{M^2_{X/Y/Z}-(2{\mathbb {M}}_c)^2}-\kappa \,{\mathbb {M}}_s\) to choose the suitable energy scales of the QCD spectral densities [4, 5, 15, 18,19,20,21,22], where the \({\mathbb {M}}_c\) and \({\mathbb {M}}_s\) are the effective c and s-quark masses respectively, and have common values. We take \({\mathbb {M}}_c=1.82\,\textrm{GeV}\) [58] and \({\mathbb {M}}_s=0.20\,\textrm{GeV}\) (\(0.12\,\textrm{GeV}\)) for the S-wave (P-wave) tetraquark states [4, 5, 18,19,20,21,22].
In the picture of tetraquark states, we can assign the (X(3960), X(4500)), (X(4140), X(4685)), (\(Z_c(3900)\), \(Z_c(4430)\)), and (\(Z_c(4020)\), \(Z_c(4600)\)) as the (1S, 2S) tetraquark states tentatively, see Table 2. We draw the conclusion tentatively that the mass gaps between the 1S and 2S tetraquark states are about \(0.55\sim 0.59 \,\textrm{GeV}\).
| \(J^{PC}\) | \(X(1\textrm{S})\) | \(X(2\textrm{S})\) | References | |
|---|---|---|---|---|
| \([sc]_{S}[\overline{sc}]_{S}{}^*\) | \(0^{++}\) | ? X(3960) | ? X(4500) | [4] |
| \([sc]_S[\overline{sc}]_{A}+[sc]_{A}[\overline{sc}]_S\) | \(1^{++}\) | ? X(4140) | ? X(4685) | [4] |
| \([sc]_S[\overline{sc}]_{\widetilde{A}}+[sc]_{\widetilde{A}}[\overline{sc}]_S\) | \(1^{++}\) | ? X(4140) | ? X(4685) | [4] |
| \([uc]_S[\bar{d}\bar{c}]_A- [uc]_A[\bar{d}\bar{c}]_S\) | \(1^{+-}\) | ? \(Z_c(3900)\) | ? \(Z_c(4430)\) | [9, 59, 60] |
| \([uc]_{A}[\overline{dc}]_{A}\) | \(1^{+-}\) | ? \(Z_c(4020)\) | ? \(Z_c(4600)\) | [10, 11] |
| \([uc]_S[\overline{dc}]_{\widetilde{A}}-[uc]_{\widetilde{A}}[\overline{dc}]_S\) | \(1^{+-}\) | ? \(Z_c(4020)\) | ? \(Z_c(4600)\) | [10, 11] |
| \([uc]_{\widetilde{A}}[\overline{dc}]_{A}-[uc]_{A}[\overline{dc}]_{\widetilde{A}}\) | \(1^{+-}\) | ? \(Z_c(4020)\) | ? \(Z_c(4600)\) | [10, 11] |
In the present analysis, we would like to set the continuum threshold parameters as \(\sqrt{s_0}=M_X+0.60\pm 0.10\,\textrm{GeV}\), then vary the continuum threshold parameters and Borel parameters to meet with the four criteria:
Pole dominance at the hadron side;
Convergence of the operator product expansion;
Appearance of the enough flat Borel platforms;
Fulfillment of the modified energy scale formula, via trial and error. We define the pole contributions (PC),
and the contributions of the vacuum condensates D(n) of dimension n,
in the same way as in our previous works.
Now we add the subscripts 0, 1, and 2 to represent the spins of the tetraquark states. In the light-flavor SU(3) symmetry limit, the QCD spectral densities,
where \(q=u\), d, or s, the Pert denotes the perturbative terms. All the condensates are not companied with inverse powers of the Borel parameter \(\frac{1}{T^2}\), \(\frac{1}{T^4}\) \(\cdots\), thus they cannot manifest themselves at small values of \(T^2\) to result in very flat platforms. In the QCD sum rules for the \(Q\bar{Q}q\bar{q}\) tetraquark states, the gluon condensate always play an unimportant role [6]. Furthermore, the vacuum condensates companied with the light quark mass \(m_q\) play a tiny role due to its small value. We can estimate the convergent behavior of the operator product expansion by considering the important terms,
If we require the contributions |D(10)| to be about \(1\%\), the dominant contributions would come from the perturbative terms plus \(m_c\langle \bar{q}g_s \sigma G q\rangle\), the condensates \(m_c^2\langle \bar{q}g_s \sigma G q\rangle ^2\) with \(q=u\), d, or s almost make no difference. As there exists an additional term \(m_c\langle \bar{q}g_s \sigma G q\rangle\) in the spectral densities \(\rho _{QCD}^{1/2}(s)\), the light-flavor SU(3) breaking effects on the masses of the tetraquark states with the \(J^{PC}=1^{+-}\) and \(2^{++}\) would be slightly larger.
In the QCD sum rules from the four-quark currents without explicit P-waves [4, 18, 20, 21, 24], there exist the terms \(m_c\langle \bar{q}q\rangle\), \(m_c\langle \bar{q}g_s \sigma G q\rangle\), \(m_c^2\langle \bar{q}q\rangle ^2\), \(m_c^2\langle \bar{q}q\rangle \langle \bar{q}g_s \sigma G q\rangle\), and \(m_c^2\langle \bar{q}g_s \sigma G q\rangle ^2\), some are companied with inverse powers of the Borel parameter \(\frac{1}{T^2}\), \(\frac{1}{T^4}\) \(\cdots\), the light-flavor SU(3) breaking effects on the tetraquark masses are normal. In the present case, due to the tiny light-flavor SU(3) breaking effects, the modified energy scale formula \(\mu =\sqrt{M^2_{X/Y/Z}-(2{\mathbb {M}}_c)^2}-\kappa \,{\mathbb {M}}_s\) with \(\kappa =0\), 1, and 2 cannot be satisfied, we have to set the effective s-quark mass \({\mathbb {M}}_s\) to be zero, in other words, we have to resort to the energy scale formula \(\mu =\sqrt{M^2_{X/Y/Z}-(2{\mathbb {M}}_c)^2}\) to restrict the QCD sum rules.
After tedious trial and error, at last, we obtain the Borel windows, continuum threshold parameters, energy scales of the spectral densities, pole contributions, and contributions of the condensates of dimension 10, and show them explicitly in Table 3. From Table 3, we can see explicitly that the ground state contributions are about \((37-66)\%\) for the tetraquark states with the \(J^{PC}=0^{++}\) and \((33-57)\%\) for the tetraquark states with the \(J^{PC}=1^{+-}\) and \(2^{++}\). For the tetraquark states with the \(J^{PC}=0^{++}\), the pole dominance is satisfied certainly, while for the tetraquark states with the \(J^{PC}=1^{+-}\) and \(2^{++}\), the pole dominance is only satisfied marginally. As the largest power of the QCD spectral densities \(\rho (s)\sim s^6\) (for the tetraquark states with explicit two P-waves or a D-wave) instead of \(s^4\) (for the tetraquark states without explicit P/D-waves), the pole dominance criterion is difficult to satisfy. In Ref. [25], the pole contributions only \(\ge 20\%\) in the QCD sum rules for the X(4500) and X(4700). On the other hand, the contributions of the vacuum condensates \(D(0)\sim 100\%\) for the tetraquark states with the \(J^{PC}=0^{++}\) and \(D(0)+D(3)\sim 100\%\) for the tetraquark states with the \(J^{PC}=1^{+-}\) and \(2^{++}\), the operator product expansion is convergent in all the QCD sum rules.
We take account of all uncertainties of the parameters and obtain the masses and pole residues of the hidden-charm tetraquark states with the \(J^{PC}=0^{++}\), \(1^{+-}\), and \(2^{++}\), and show them explicitly in Table 4. The uncertainties of the masses and pole residues \(\Delta f\) are approximately estimated via,
where the f(x) stands for the analytical expressions, the \(\bar{x}_i\) stands for the central values of the relevant parameters, the \(\Delta x_i\) are their uncertainties, and \(|f(\bar{x}_i+ \Delta x_i)-f(\bar{x}_i)|\approx |f(\bar{x}_i- \Delta x_i)-f(\bar{x}_i)|\) as the large energy scales of the QCD spectral densities weaken the sensitivity to the \(x_i\).
We try to choose uniform continuum threshold parameters and energy scales for the tetraquark states with the \(J^{PC}=0^{++}\), \(1^{+-}\), and \(2^{++}\) respectively to scrutinize the light-flavor SU(3) breaking effects. From Table 4, we can obtain the central values \(\mu =2.64583\,\textrm{GeV}\) for the tetraquark states with the \(J^{PC}=0^{++}\), \(\mu =2.79616\,\textrm{GeV}\), \(2.82887\,\textrm{GeV}\), and \(2.86135\,\textrm{GeV}\) for the tetraquark states with the \(J^{PC}=1^{+-}\), and \(\mu =2.8936\,\textrm{GeV}\), \(2.92563\,\textrm{GeV}\), and \(2.95745\,\textrm{GeV}\) for the tetraquark states with the \(J^{PC}=2^{++}\) from the energy scale formula \(\mu =\sqrt{M^2_{X/Y/Z}-(2{\mathbb {M}}_c)^2}\), which are consistent with the values shown in Table 3. We combine Table 3 with Table 4, and obtain the conclusion tentatively that the light-flavor SU(3) breaking effects on the tetraquark masses are tiny.
In Fig.1, we plot the masses of the \([uc]_{\widehat{V}}[\overline{dc}]_{\widehat{V}}\) and \([sc]_{\widehat{V}}[\overline{sc}]_{\widehat{V}}\) tetraquark states with the \(J^{PC}=0^{++}\) according to variations of the Borel parameters at much larger regions than the Borel windows, which are characterized by two short vertical lines. For comparison, we present the experimental values of the masses of the X(4475) and X(4500) from the LHCb collaboration [3, 7]. The predicted masses increase monotonically and quickly with increase of the Borel parameters, at the point \(T^2=2.4\,\textrm{GeV}^2\), the beginning of the Borel windows, the masses begin to increase slowly, it is feasible to fix the lower bounds of the Borel windows. On the other hand, the PC (D(10)) decreases monotonically and steadily (quickly) with increase of the Borel parameters, for example, the lowest value \(37\%\) (\(3\%\)) determines the upper (lower) bound of the Borel window for the \([uc]_{\widehat{V}}[\overline{dc}]_{\widehat{V}}\) tetraquark state, where the predicted mass happens to vary slowly. Other Borel windows shown in Table 3 are fixed in the same way by considering the four criteria of the QCD sum rules. The energy gaps \(\sqrt{s_0}-M_X\sim 0.55\,\textrm{GeV}\) for the tetraquark states with the \(J^{PC}=0^{++}\) and \(\sim 0.65\,\textrm{GeV}\) for the tetraquark states with the \(J^{PC}=1^{+-}\) and \(2^{++}\), although the value \(0.65\,\textrm{GeV}\) is somewhat large [4, 9,10,11, 59, 60].
The masses of the \([uc]_{\widehat{V}}[\overline{dc}]_{\widehat{V}}\) (A) and \([sc]_{\widehat{V}}[\overline{sc}]_{\widehat{V}}\) (B) tetraquark states with the \(J^{PC}=0^{++}\) via variations of the Borel parameter \(T^2\), where the isospin limit is taken
In Table 4, we present the possible assignments of the LHCb’s new tetraquark candidates. The predictions \(M_X=4.50\pm 0.12\,\textrm{GeV}\) and \(4.50\pm 0.12\,\textrm{GeV}\) support assigning the X(4475) and X(4500) as the \([uc]_{\widehat{V}}[\overline{uc}]_{\widehat{V}} -[dc]_{\widehat{V}}[\overline{dc}]_{\widehat{V}}\) and \([sc]_{\widehat{V}}[\overline{sc}]_{\widehat{V}}\) tetraquark states with the \(J^{PC}=0^{++}\) respectively. The central values of the masses of the \([uc]_{\widehat{V}}[\overline{dc}]_{\widehat{V}}\), \([qc]_{\widehat{V}}[\overline{sc}]_{\widehat{V}}\), and \([sc]_{\widehat{V}}[\overline{sc}]_{\widehat{V}}\) tetraquark states with the \(J^{PC}=0^{++}\) are \(4.5001\,\textrm{GeV}\), \(4.5013\,\textrm{GeV}\), and \(4.5014\,\textrm{GeV}\), respectively, the light-flavor SU(3) breaking effects on the tetraquark masses are extremely tiny.
In Ref. [4], we tentatively assign the X(3960) and X(4500) as the 1S and 2S \([sc]_{S}[\overline{sc}]_{S}{}^*\) tetraquark states with the \(J^{PC}=0^{++}\) respectively based on the QCD sum rules, where the light-flavor SU(3) breaking effect \(M_{X(4140)}-M_{X(3872)}=275\,\textrm{MeV}\) is consistent with the mass difference \(m_s-m_q=135\,\textrm{MeV}\) and is normal [54]. So the X(4500) might have two significant Fock components at least. On the other hand, it is also possible to assign the X(3960) as the \(D_s\bar{D}_s\) molecular state based on the QCD sum rules [61].
Also in Ref. [4], we tentatively assign the X(4700) as the 2S \([sc]_{A}[\overline{sc}]_{A}\)/\([sc]_{\tilde{A}}[\overline{sc}]_{\tilde{A}}\) or 1S \([sc]_{V}[\overline{sc}]_{V}{}^*\) tetraquark state with the \(J^{PC}=0^{++}\), see Table 1. In case of the constituent diquarks without an explicit P-wave, the light-flavor SU(3) breaking effects are normal, the tetraquark states with the symbolic valence structures \(c\bar{c}(u\bar{u}-d\bar{d})\) lie about \(200\,\textrm{MeV}\) below the corresponding \(c\bar{c}s\bar{s}\) tetraquark states, there is no room for the X(4710) [24]. In the present case, there is also no room for the X(4710), see Table 4. We should resort to other diquarks with an explicit P-wave to construct the four-quark currents to interpolate the X(4710) and X(4700) consistently.
| X(Z) | \(J^{PC}\) | \(T^2 (\textrm{GeV}^2)\) | \(\sqrt{s_0}(\mathrm GeV)\) | \(\mu (\textrm{GeV})\) | pole | |D(10)| |
|---|---|---|---|---|---|---|
| \([uc]_{\widehat{V}}[\overline{dc}]_{\widehat{V}}\) | \(0^{++}\) | \(2.4-2.8\) | \(5.05\pm 0.10\) | 2.6 | \((37-65)\%\) | \((1\sim 3)\%\) |
| \([qc]_{\widehat{V}}[\overline{sc}]_{\widehat{V}}\) | \(0^{++}\) | \(2.4-2.8\) | \(5.05\pm 0.10\) | 2.6 | \((38-66)\%\) | \((1\sim 2)\%\) |
| \([sc]_{\widehat{V}}[\overline{sc}]_{\widehat{V}}\) | \(0^{++}\) | \(2.4-2.8\) | \(5.05\pm 0.10\) | 2.6 | \((37-65)\%\) | \((1\sim 1)\%\) |
| \([uc]_{\widehat{V}}[\overline{dc}]_{\widehat{V}}\) | \(1^{+-}\) | \(3.2-3.6\) | \(5.25\pm 0.10\) | 2.8 | \((35-57)\%\) | \((1\sim 1)\%\) |
| \([qc]_{\widehat{V}}[\overline{sc}]_{\widehat{V}}\) | \(1^{+-}\) | \(3.2-3.6\) | \(5.25\pm 0.10\) | 2.8 | \((35-56)\%\) | \((1\sim 1)\%\) |
| \([sc]_{\widehat{V}}[\overline{sc}]_{\widehat{V}}\) | \(1^{+-}\) | \(3.2-3.6\) | \(5.25\pm 0.10\) | 2.9 | \((34-56)\%\) | \((0\sim 1)\%\) |
| \([uc]_{\widehat{V}}[\overline{dc}]_{\widehat{V}}\) | \(2^{++}\) | \(3.3-3.7\) | \(5.30\pm 0.10\) | 2.9 | \((34-55)\%\) | \((1\sim 1)\%\) |
| \([qc]_{\widehat{V}}[\overline{sc}]_{\widehat{V}}\) | \(2^{++}\) | \(3.3-3.7\) | \(5.30\pm 0.10\) | 2.9 | \((33-54)\%\) | \((1\sim 1)\%\) |
| \([sc]_{\widehat{V}}[\overline{sc}]_{\widehat{V}}\) | \(2^{++}\) | \(3.3-3.7\) | \(5.30\pm 0.10\) | 3.0 | \((33-54)\%\) | \((0\sim 1)\%\) |
The predictions \(M_Z=4.59\pm 0.11\,\textrm{GeV}\) and \(4.61\pm 0.11\,\textrm{GeV}\) support assigning the \(Z_{c}(4600)\) and \(Z_{\bar{c}\bar{s}}(4600)\) as the \([uc]_{\widehat{V}}[\overline{dc}]_{\widehat{V}}\) and \([qc]_{\widehat{V}}[\overline{sc}]_{\widehat{V}}\) tetraquark states with the \(J^{PC}=1^{+-}\) respectively. Again, the light-flavor SU(3) breaking effects on the tetraquark masses are tiny. In Refs. [10, 11, 24], the \(Z_c(4020)\) and \(Z_c(4600)\) are assigned as the 1S and 2S \([uc]_{A}[\overline{dc}]_{A}\), \([uc]_S[\overline{dc}]_{\widetilde{A}}-[uc]_{\widetilde{A}}[\overline{dc}]_S\), or \([uc]_{\widetilde{A}}[\overline{dc}]_{A}-[uc]_{A}[\overline{dc}]_{\widetilde{A}}\) tetraquark states with the \(J^{PC}=1^{+-}\) respectively, see Table 2. In Ref. [4], the X(4140) and X(4685) are assigned as the 1S and 2S \([sc]_S[\overline{sc}]_{A}+[sc]_{A}[\overline{sc}]_S\) or \([sc]_S[\overline{sc}]_{\widetilde{A}}+[sc]_{\widetilde{A}}[\overline{sc}]_S\) tetraquark states with the \(J^{PC}=1^{++}\) respectively, also see Table 2. In Table 2, the light-flavor SU(3) breaking effects on the tetraquark masses are normal. On the other hand, if the \(Z_{cs}(3985/4000)\) is assigned as the 1S \([sc]_A[\overline{qc}]_{A}\) tetraquark state with the \(J^{PC}=1^{+-}\) [22], then the \(Z_{\bar{c}\bar{s}}(4600)\) and \(Z_{\bar{c}\bar{s}}(4900)\) can be assigned as the 2S and 3S \([qc]_A[\overline{sc}]_{A}\) tetraquark states with the \(J^{PC}=1^{+-}\), respectively, such a possibility also exists. The \(Z_c(4600)\) and \(Z_{\bar{c}\bar{s}}(4600)\) might have several significant Fock components.
We can take the pole residues \(\lambda _X\) in Table 4 as input parameters to explore the strong decays of those tetraquark states with the (light-cone) QCD sum rules, and acquire the partial decay widths and branching fractions to diagnose the nature of those exotic states [6].
| X(Z) | \(J^{PC}\) | \(M_X (\textrm{GeV})\) | \(\lambda _X (\textrm{GeV}^7)\) | Assignments |
|---|---|---|---|---|
| \([uc]_{\widehat{V}}[\overline{dc}]_{\widehat{V}}\) | \(0^{++}\) | \(4.50\pm 0.12\) | \((4.96\pm 1.39)\times 10^{-2}\) | ? X(4475) |
| \([qc]_{\widehat{V}}[\overline{sc}]_{\widehat{V}}\) | \(0^{++}\) | \(4.50\pm 0.12\) | \((5.18\pm 1.41)\times 10^{-2}\) | |
| \([sc]_{\widehat{V}}[\overline{sc}]_{\widehat{V}}\) | \(0^{++}\) | \(4.50\pm 0.12\) | \((5.38\pm 1.42)\times 10^{-2}\) | ? X(4500) |
| \([uc]_{\widehat{V}}[\overline{dc}]_{\widehat{V}}\) | \(1^{+-}\) | \(4.59\pm 0.11\) | \((2.15\pm 0.44)\times 10^{-2}\) | ? \(Z_{c}(4600)\) |
| \([qc]_{\widehat{V}}[\overline{sc}]_{\widehat{V}}\) | \(1^{+-}\) | \(4.61\pm 0.11\) | \((2.20\pm 0.45)\times 10^{-2}\) | ? \(Z_{\bar{c}\bar{s}}(4600)\) |
| \([sc]_{\widehat{V}}[\overline{sc}]_{\widehat{V}}\) | \(1^{+-}\) | \(4.63\pm 0.11\) | \((2.28\pm 0.46)\times 10^{-2}\) | |
| \([uc]_{\widehat{V}}[\overline{dc}]_{\widehat{V}}\) | \(2^{++}\) | \(4.65\pm 0.11\) | \((3.52\pm 0.71)\times 10^{-2}\) | |
| \([qc]_{\widehat{V}}[\overline{sc}]_{\widehat{V}}\) | \(2^{++}\) | \(4.67\pm 0.11\) | \((3.61\pm 0.73)\times 10^{-2}\) | |
| \([sc]_{\widehat{V}}[\overline{sc}]_{\widehat{V}}\) | \(2^{++}\) | \(4.69\pm 0.11\) | \((3.75\pm 0.75)\times 10^{-2}\) |
In this work, we introduce an explicit P-wave to construct the diquarks, then construct the local four-quark currents to explore the hidden-charm tetraquark states with the \(J^{PC}=0^{++}\), \(1^{+-}\), and \(2^{++}\) in the framework of the QCD sum rules at length. We carry out the operator product expansion up to the condensates of dimension 10 in a consistent way. Direct calculations indicate tiny light-flavor SU(3) breaking effects on the tetraquark masses and the modified energy scale formula \(\mu =\sqrt{M^2_{X/Y/Z}-(2{\mathbb {M}}_c)^2}-\kappa \,{\mathbb {M}}_s\) can be satisfied only by setting \({\mathbb {M}}_s=0\). The present calculations support assigning the X(4475) and X(4500) as the \([uc]_{\widehat{V}}[\overline{uc}]_{\widehat{V}}-[dc]_{\widehat{V}}[\overline{dc}]_{\widehat{V}}\) and \([sc]_{\widehat{V}}[\overline{sc}]_{\widehat{V}}\) tetraquark states with the \(J^{PC}=0^{++}\), respectively, support assigning the \(Z_{c}(4600)\) and \(Z_{\bar{c}\bar{s}}(4600)\) as the \([uc]_{\widehat{V}}[\overline{dc}]_{\widehat{V}}\) and \([qc]_{\widehat{V}}[\overline{sc}]_{\widehat{V}}\) tetraquark states with the \(J^{PC}=1^{+-}\), respectively. On the other hand, there is no room for the X(4710) and X(4700). Considering previous works, we can obtain the conclusion tentatively that the X(4475), X(4500), \(Z_{c}(4600)\), and \(Z_{\bar{c}\bar{s}}(4600)\) might have other important Fock components besides the \(\widehat{V}\widehat{V}\) type components, which could account for the experimental data from the LHCb collaboration. Other predictions can be confronted to the experimental data in the future to examine the exotic states.
The data are available via contacting the corresponding author upon request.
R. Aaij et al., Physical Review Letters 118, 022003 (2017)
R. Aaij et al., Physical Review D 95, 012002 (2017)
R. Aaij et al., Physical Review Letters 127, 082001 (2021)
Z.G. Wang, Nuclear Physics B 1007, 116661 (2024)
Z.G. Wang, Nuclear Physics B 1002, 116514 (2024)
Z. G. Wang, arXiv:2502.11351 [hep-ph]
R. Aaij et al., Journal of High Energy Physics 01, 054 (2025)
R. Aaij et al., Physical Review Letters 122, 152002 (2019)
Z.G. Wang, Communications in Theoretical Physics 63, 325 (2015)
H.X. Chen, W. Chen, Physical Review D 99, 074022 (2019)
Z.G. Wang, Chinese Physics C 44, 063105 (2020)
R. Aaij et al., Physical Review Letters 134, 031902 (2025)
Z.G. Wang, T. Huang, Physical Review D 89, 054019 (2014)
Z.G. Wang, Communications in Theoretical Physics 63, 466 (2015)
Z.G. Wang, The European Physical Journal C 74, 2874 (2014)
Z.G. Wang, T. Huang, Nuclear Physics A 930, 63 (2014)
Z.G. Wang, International Journal of Modern Physics A 36, 2150071 (2021)
Z.G. Wang, International Journal of Modern Physics A 36, 2150107 (2021)
Z.G. Wang, Q. Xin, Chinese Physics C 45, 123105 (2021)
Q. Xin, Z.G. Wang, The European Physical Journal A 58, 110 (2022)
Z.G. Wang, Q. Xin, Nuclear Physics B 978, 115761 (2022)
Z.G. Wang, Chinese Physics C 46, 123106 (2022)
Z.G. Wang, Nuclear Physics B 973, 115592 (2021)
Z.G. Wang, Physical Review D 102, 014018 (2020)
H.X. Chen, E.L. Cui, W. Chen, X. Liu, S.L. Zhu, The European Physical Journal C 77, 160 (2017)
Q.F. Lu, Y.B. Dong, Physical Review D 94, 074007 (2016)
P.P. Shi, F. Huang, W.L. Wang, Physical Review D 103, 094038 (2021)
J. Wu, Y.R. Liu, K. Chen, X. Liu, S.L. Zhu, Physical Review D 94, 094031 (2016)
M.N. Anwar, J. Ferretti, E. Santopinto, Physical Review D 98, 094015 (2018)
X. Liu, H.X. Huang, J.L. Ping, D.Y. Chen, X.M. Zhu, The European Physical Journal C 81, 950 (2021)
F.Z. Peng, M.J. Yan, M. Sanchez, M.P. Valderrama, Physical Review D 107, 016001 (2023)
E. Wang, J.J. Xie, L.S. Geng, E. Oset, Physical Review D 97, 014017 (2018)
X.K. Dong, F.K. Guo, B.S. Zou, Progress in Physics 41, 65 (2021)
Q. Deng, R.H. Ni, Q. Li, X.H. Zhong, Physical Review D 110, 056034 (2024)
D.Y. Chen, The European Physical Journal C 76, 671 (2016)
X.H. Liu, Physics Letters B 766, 117 (2017)
T. Barnes, S. Godfrey, E.S. Swanson, Physical Review D 72, 054026 (2005)
Z.G. Wang, The European Physical Journal C 79, 29 (2019)
Z.G. Wang, Chinese Physics C 48, 103103 (2024)
Z.G. Wang, The European Physical Journal C 79, 489 (2019)
Z.G. Wang, Z.H. Yan, The European Physical Journal C 78, 19 (2018)
Z.G. Wang, International Journal of Modern Physics A 35, 2050003 (2020)
X.W. Wang, Z.G. Wang, G.L. Yu, Q. Xin, Science China Physics, Mechanics & Astronomy 65, 291011 (2022)
Z.G. Wang, Z.Y. Di, The European Physical Journal C 79, 72 (2019)
Z.G. Wang, Advances in High Energy Physics 2021, 4426163 (2021)
Z.G. Wang, The European Physical Journal A 53, 19 (2017)
Z.G. Wang, The European Physical Journal C 77, 78 (2017)
Z.G. Wang, International Journal of Modern Physics A 36, 2150014 (2021)
Z.G. Wang, Physical Review D 101, 074011 (2020)
Z.G. Wang, International Journal of Modern Physics A 35, 2050138 (2020)
Z. G. Wang, arXiv: 2005.12735 [hep-ph]
L.J. Reinders, H. Rubinstein, S. Yazaki, Physics Reports 127, 1 (1985)
P. Pascual, R. Tarrach, QCD: Renormalization for the Practitioner (Springer, Berlin Heidelberg, 1984)
S. Navas et al., Physical Review D 110, 030001 (2024)
S. Narison, R. Tarrach, Physics Letters B 125, 217 (1983)
M. A. Shifman, A. I. Vainshtein, V. I. Zakharov, Nuclear Physics B147, 385 (1979). Nuclear Physics B147, 448 (1979)
P. Colangelo, A. Khodjamirian, arXiv: hep-ph/0010175
Z.G. Wang, The European Physical Journal C 76, 387 (2016)
L. Maiani, F. Piccinini, A.D. Polosa, V. Riquer, Physical Review D 89, 114010 (2014)
M. Nielsen, F.S. Navarra, Modern Physics Letters A 29, 1430005 (2014)
Q. Xin, Z.G. Wang, X.S. Yang, AAPPS Bulletin 32, 37 (2022)
This work is supported by the National Natural Science Foundation, Grant Number 12175068.
This research was supported by the National Natural Science Foundation of China through Grant No. 12175068.
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