# Assignments of theX4140, X4500, X4630, and X4685 Based on the QCD Sum Rules

Authors: Zhi-Gang Wang
Journal: Advances in High Energy Physics
 (2021)

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## Abstract

In this article, we take into account our previous calculations based on the QCD sum rules, and tentatively assign theX4630 as the Ds∗D¯s1−Ds1D¯s∗ tetraquark molecular state or csPc¯s¯A+csAc¯s¯P tetraquark state with the JPC=1−+, and assign the X3915 and X4500 as the 1S and 2S csAc¯s¯A tetraquark states, respectively, with the JPC=0++. Then, we extend our previous works to investigate the LHCb’s new tetraquark candidate X4685 as the first radial excited state of the X4140 with the QCD sum rules and obtain the mass MX=4.70±0.12GeV, which is in very good agreement with the experimental value 4684±7−16+13MeV. Furthermore, we investigate the two-meson scattering state contributions in details and observe that the two-meson scattering states alone cannot saturate the QCD sum rules, the contributions of the tetraquark states play an unsubstitutable role, and we can saturate the QCD sum rules with or without the two-meson scattering states.


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## Body

## 1. Introduction

In 2009, the CDF collaboration observed an evidence for theX4140 in the J/ψϕ mass spectrum for the first time with a significance of larger than 3.8σ [1]. Subsequently, the existence of the X4140 was confirmed by the CDF, CMS, and D0 collaborations [2–5]. In 2016, the LHCb collaboration accomplished the first full amplitude analysis of the B+⟶J/ψϕK+ decays and acquired a good description of the experimental data in the 6D phase space and confirmed the X4140 and X4274 and determined the spin-parity-charge-conjugation JPC=1++ [6, 7]. Furthermore, the LHCb collaboration also observed two new exotic hadrons X4500 and X4700 in the J/ψϕ mass spectrum and determined the quantum numbers JPC=0++ [6, 7]. The Breit-Wigner masses and widths are
(1)X4140:M=4146.5±4.5−2.8+4.6MeV,Γ=83±21−14+21MeV,X4274:M=4273.3±8.3−3.6+17.2MeV,Γ=56±11−11+8MeV,X4500:M=4506±11−15+12MeV,Γ=92±21−20+21MeV,X4700:M=4704±10−24+14MeV,Γ=120±31−33+42MeV.Recently, the LHCb collaboration accomplished an improved full amplitude analysis of theB+⟶J/ψϕK+ decays using 6 times larger signal yields than previously analyzed and observed a hidden-charm and hidden-strange tetraquark candidate X4685 and (X4630) in the mass spectrum of the J/ψϕ with a significance of 15σ (5.5σ), the favored assignment of the spin parity is JP=1+(1−), and the Breit-Wigner mass and width are 4684±7−16+13MeV (4626±16−110+18MeV) and 126±15−41+37MeV (174±27−73+134MeV), respectively [8]. Furthermore, the LHCb collaboration also observed two new tetraquark (molecular) state candidates Zcs4000 and Zcs4220 in the mass spectrum of the J/ψK+ with the preferred spin parity JP=1+ and updated the experimental values of the masses and widths of the X4500 and X4700 [8]. The X4140, X4274, X4500, X4630, X4685, and X4700 were observed in the mass spectrum of the J/ψϕ, and their quantum numbers are JPC=0++, 1++, and 2++ for the S-wave couplings and 0−+, 1−+, 2−+, and 3−+ for the P-wave couplings. In the present work, we discuss the possible assignments of the X4140, X4500, X4630, and X4685 based on the QCD sum rules.The article is arranged as follows: in Section2, we discuss the possible assignments of the X4630 and X4500 based on the QCD sum rules; in Section 3, we get the QCD sum rules for the masses and pole residues of the tetraquark states X4140/X4685 with the JPC=1++; in Section 4, we obtain numerical results and give discussions, and Section 5 is aimed to get a conclusion.


## 2. Possible Assignments of theX4630 and X4500 Based on the QCD Sum Rules

In this section, we discuss the possible assignments of theX4630 and X4500 according to our previous calculations with the QCD sum rules.In Ref. [9], we construct the color-singlet-color-singlet type four-quark current Jμνx to investigate the Ds∗D¯s1−Ds1D¯s∗ molecular state:
(2)Jμνx=12s¯xγμcxc¯xγνγ5sx−s¯xγνγ5cxc¯xγμsx.The currentJμνx has definite charge conjugation C=1 but has not definite parity, and the components J0ix and Jijx have positive parity and negative parity, respectively, where the space indexes i, j=1, 2, 3. The neutral current Jμνx couples potentially to the Ds∗D¯s1−Ds1D¯s∗ two-meson scattering states or tetraquark molecular states XDs∗D¯s1−Ds1D¯s∗ with the quantum numbers JPC=1++ and 1−+, where we use the symbols Ds∗ and Ds1 to represent the color-neutral clusters with the same quantum numbers as the physical Ds∗ and Ds1 mesons, respectively. In the QCD sum rules, we choose the local currents, and it is better to call the XDs∗D¯s1−Ds1D¯s∗ as the color-singlet-color-singlet type tetraquark state than call it as the tetraquark molecular state. The traditional hidden-flavor mesons, such as the qq¯, cc¯, and bb¯ quarkonia, have the normal quantum numbers JPC=0−+, 0++, 1−−, 1+−, 1++, 2−−, 2−+, 2++, ⋯. The components J0ix and Jijx couple potentially to the JPC=1++ and 1−+ tetraquark molecular states, respectively. We construct projection operators to project out the contribution of the JPC=1−+ component unambiguously and explore the Ds∗D¯s1−Ds1D¯s∗ tetraquark molecular state with the exotic quantum numbers JPC=1−+ using the QCD sum rules and acquire the prediction [9],
(3)MX=4.67±0.08GeV,which happens to coincide with the mass of the X4630 from the LHCb collaboration, MX4630=4626±16−110+18MeV [8].The calculations based on the Bethe-Salpeter equation combined with the heavy meson effective Lagrangian also indicate that there exists such aDs∗D¯s1−Ds1D¯s∗ tetraquark molecular state with the exotic quantum numbers JPC=1−+ [10, 11]. The predictions in Refs. [9–11] were achieved before the discovery of the X4630. Whether or not the predictions of the QCD sum rules are reliable, the experimental data can reply. After the discovery of the X4630 by the LHCb collaboration, Yang et al. study the charmonium-like molecules with hidden-strange via the one-boson exchange mechanism and assign the X4630 to be the Ds∗D¯s1 molecular state with the quantum numbers JPC=1−+ [12].As long as the diquark-antidiquark type tetraquark states are concerned, we usually take the scalar (S), pseudoscalar (P), vector (V), axialvector (A), and tensor (T) diquark operators without introducing explicit P-waves as the elementary building blocks to construct the interpolating currents. The tensor currents have both vector and axialvector components, and we construct projection operators to project out the spin parity JP=1− and 1+ components explicitly and denote the corresponding operators as V~ and A~, respectively, to avoid ambiguity. In Ref. [13], we choose the diquark-antidiquark type vector currents Jμ−x and Jμ+x:
(4)Jμ−x=εijkεimn2sjxCckxs¯mxγμCc¯nx−sjxCγμckxs¯mxCc¯nx,Jμ+x=εijkεimn2sjxCckxs¯mxγμCc¯nx+sjxCγμckxs¯mxCc¯nx,to interpolate the csPc¯s¯A−csAc¯s¯P-type and csPc¯s¯A+csAc¯s¯P-type tetraquark states with the quantum numbers JPC=1−− and 1−+, respectively, and investigate their properties with the QCD sum rules, where the i, j, k, m, and n are color indexes. We acquire the predictions:
(5)MX4630=4.63−0.08+0.11GeVforJPC=1−+,MY4660=4.70−0.10+0.14GeVforJPC=1−−,which happen to coincide with the masses of the X4630 and Y4660, respectively [13], and support assigning the X4630 and Y4660 to be the tetraquark states with the symbolic quark constituents cc¯ss¯ and with the quantum numbers JPC=1−+ and 1−−, respectively. The prediction of the mass 4.63−0.08+0.11GeV was achieved long before the discovery of the Y4630.In Refs. [14, 15], we construct the diquark-antidiquark type currents to explore the csSc¯s¯S, csPc¯s¯P, csAc¯s¯A, and csVc¯s¯V tetraquark states with the quantum numbers JPC=0++ concordantly via the QCD sum rules, and the numerical results support assigning the X3915 and X4500 to be the ground state and first radial excited state of the csAc¯s¯A tetraquark states, respectively, with the quantum numbers JPC=0++ and assigning the X4700 to be the ground state csVc¯s¯V tetraquark states with the quantum numbers JPC=0++. Furthermore, we also obtain the potability that assigning the X3915 to be the ground state csSc¯s¯S tetraquark state with the JPC=0++ [15]. Our predictions,
(6)MX4500=4.50−0.09+0.08GeV,MX4700=4.70−0.09+0.08GeV,are in very good agreement with the LHCb improved measurement MX4500=4474±3±3MeV and MX4700=4694±4−3+16MeV [8]. Other assignments of the X4500 and X4700, such as the D-wave csc¯s¯ tetraquark states with the JP=0+, are also possible [16], and more theoretical and experimental works are still needed to obtain definite conclusion.In summary, according to the (possible) quantum numbers, decay modes, and energy gaps, we can assign theX3915 and X4500 as the ground state and first radial excited state of the hidden-charm tetraquark states with the JPC=0++ [14, 17], assign the Zc3900 and Zc4430 as the ground state and first radial excited state of the hidden-charm tetraquark states with the JPC=1+−, respectively [18–20], and assign the Zc4020 and Zc4600 as the ground state and first radial excited state of the hidden-charm tetraquark states with the JPC=1+−, respectively [21, 22]. If we assign the X4685 to be the first radial excited state of the X4140 tentatively, we can get the energy gap 566MeV, and it is reasonable, see Table 1.Table 1
The energy gaps between the ground states and first radial excited states of the hidden-charm tetraquark states with the possible assignments.
JPC1S2SEnergy gaps1+−Zc3900Zc4430591 MeV0++X3915X4500588 MeV1+−Zc4020Zc4600576 MeV1++X4140X4685566 MeVMoreover, in Ref. [17], Lebed and Polosa assign the X3915 and X4140 to be the JPC=0++ and 1++ diquark-antidiquark type hidden-charm tetraquark states csSc¯s¯S and csAc¯s¯S+csSc¯s¯A, respectively, based on the effective Hamiltonian with the spin-spin and spin-orbit interactions. In Ref. [23], we construct the scA~s¯c¯A+scAs¯c¯A~ type and scV~s¯c¯V−scVs¯c¯V~ type axialvector currents with the quantum numbers JPC=1++ to interpolate the X4140 and observe that only the scV~s¯c¯V−scVs¯c¯V~ type current can reproduce the mass and width of the Y4140 in a consistent way.


## 3. TheX4140/X4685 as the 1S/2S Axialvector Tetraquark States

In this section, we extend our previous work [23] to investigate the X4685 as the first radial excitation of the X4140 with the QCD sum rules and discuss the possible assignment of the X4685 as the tetraquark state having the quantum numbers JPC=1++.Firstly, we write down the two-point correlation functionΠμμ′p in the QCD sum rules:
(7)Πμμ′p=i∫d4xeip·x0∣TJμxJμ′†0∣0,where
(8)Jμx=εijkεimn2sTjxCσμνckxs¯mxγ5γνCc¯Tnx−sTjxCγνγ5ckxs¯mxσμνCc¯Tnx.The currentJμx couples potentially to the scV~s¯c¯V−scVs¯c¯V~ tetraquark states with the JPC=1++. The tensor diquark operator εijksTjxCσμνckx has both the spin parity JP=1+ and 1− components, and we project out the 1− component via multiplying the tensor diquark operator by the vector antidiquark operator εimns¯mxγ5γνCc¯Tnx. In Ref. [23], we observe that the current Jμx can reproduce the mass and width of the Y4140 satisfactorily.At the hadron side, we isolate the ground state (X) and first radial excited state (X′) contributions, which are supposed to be the pole contributions from the X4140 and X4685, respectively,
(9)Πμμ′p=λX2MX2−p2+λX′2MX′2−p2+⋯−gμμ′+pμpμ′p2+⋯=Πp2−gμμ′+pμpμ′p2+⋯,where the pole residues or decay constants λX′ are defined by 0∣Jμ0∣X′p=λX′εμ, and the εμ is the polarization vectors of the axialvector tetraquark states X′.A hadron, such as the usually called quark-antiquark type meson, tree-quark type baryon, diquark-antidiquark type tetraquark state, and diquark-diquark-antiquark type pentaquark state, has definite quantum numbers and more than one Fock states. Any current operator with the same quantum numbers and same quark structure as a Fock state in the hadron couples potentially to this hadron; in other words, it has nonvanishing coupling to this hadron. Generally speaking, we can construct several current operators to interpolate a hadron or construct a current operator to interpolate several hadrons. Actually, a hadron has one or two main Fock states, and we call a hadron as a tetraquark state if its main Fock component is of the diquark-antidiquark type.In the present work, the diquark-antidiquark type local four-quark current operatorJμx having the quantum numbers JPC=1++ couples potentially to the diquark-antidiquark type tetraquark states with the same quantum numbers JPC=1++. On the other hand, this local current Jμx can be rearranged into a special superposition of a series of color-singlet-color-singlet type currents through the Fierz transformation both in the Dirac spinor space and color space:
(10)22Jμx=−c¯xσμνγ5cxs¯xγνsx+s¯xσμνγ5sxc¯xγνcx−3ic¯xγμγ5cxs¯xsx+3is¯xγμγ5sxc¯xcx−c¯xσμνsxs¯xγνγ5cx+s¯xσμνcxc¯xγνγ5sx−3c¯xγμsxs¯xiγ5cx+3s¯xγμcxc¯xiγ5sx,which couple potentially to the tetraquark molecular states or two-meson scattering states having the quantum numbers JPC=1++. The diquark-antidiquark type tetraquark states can be viewed as a special superposition of a series of color-singlet-color-singlet molecular states and embody the net effects, and vise versa.The diquark-antidiquark type tetraquark state can be plausibly described by two diquarks in a double well potential which are separated by a barrier [24, 25], the spatial distance between the diquark and antidiquark leads to smaller wave-function overlap between the quark and antiquark constituents, and the repulsive barrier or spatial distance frustrates the Fierz rearrangements or recombinations between the quarks and antiquarks and therefore suppresses hadronizing to the meson-meson pairs [24–27].If the color-singlet-color-singlet type components in Eq. (10), such as c¯xσμνγ5cxs¯xγνsx and s¯xσμνγ5sxc¯xγνcx, only couple potentially to the two-meson (TM) scattering states, we obtain the correlation function ΠTMp2 at the hadron side:
(11)ΠTMp2=1768π2∫mJ/ψϕ2∞ds1s−p2λ1/2s,mJ/ψ2,mϕ2sρ¯J/ψϕs+11536π2fϕ2mϕ2fhc2∫mhcϕ2∞ds1s−p2λ1/2s,mhc2,mϕ2sρ¯hcϕs+11536π2fJ/ψ2mJ/ψ2fhs2∫mhsJ/ψ2∞ds1s−p2λ1/2s,mhs2,mJ/ψ2sρ¯hsJ/ψs+3512π2ff02mf02fχc12mχc12∫mf0χc12∞ds1s−p2λ1/2s,mχc12,mf02sρ¯χc1f0s+3512π2ff12mf12fχc02mχc02∫mf1χc02∞ds1s−p2λ1/2s,mχc02,mf12sρ¯χc0f1s+3512π2ff02mf02fηc2∫mf0ηc2∞ds1s−p2λ3/2s,mηc2,mf02s2+3512π2fχc02mχc02fη2∫mχc0η2∞ds1s−p2λ3/2s,mχc02,mη2s2+1768π2fDs12mDs12fT,Ds∗2∫mDs1Ds∗2∞ds1s−p2λ1/2s,mDs12,mDs∗2sρ¯Ds1Ds∗s+1192π2∫mDsDs∗2∞ds1s−p2λ1/2s,mDs2,mDs∗2sρ¯DsDs∗s+3256π2fDs2mDs4fDs02mc2∫mDsDs02∞ds1s−p2λ3/2s,mDs2,mDs02s2+J/ψϕ⟶ψ′ϕ+J/ψϕ⟶ψ′′ϕ+hcϕ⟶hc′ϕ+⋯,where
(12)ρ¯J/ψϕs=fϕ2mϕ2fT,J/ψ2−s+8mJ/ψ2−mϕ2+mJ/ψ2−mϕ22s+s−mJ/ψ22mϕ2+2fJ/ψmJ/ψfϕmϕfT,ϕfT,J/ψ5s−4mJ/ψ2−4mϕ2−mJ/ψ2−mϕ22s+fJ/ψ2mJ/ψ2fT,ϕ2−s+8mϕ2−mJ/ψ2+mJ/ψ2−mϕ22s+s−mϕ22mJ/ψ2,ρ¯hcϕs=−2s−10mhc2−2mϕ2+2mϕ4−3mhc2mϕ2s+2s2−3smhc2mϕ2+mhc6smϕ2,ρ¯hsJ/ψs=−2s−10mhs2−2mJ/ψ2+2mJ/ψ4−3mhs2mJ/ψ2s+2s2−3smhs2mJ/ψ2+mhs6smJ/ψ2,ρ¯χc1f0s=10+mχc12−2mf02s+s−2mf02mχc12+mf04smχc12,ρ¯χc0f1s=10+mf12−2mχc02s+s−2mχc02mf12+mχc04smf12,ρ¯Ds1Ds∗s=−2s−10mDs∗2−2mDs12+2mDs14−3mDs∗2mDs12s+2s2−3smDs∗2mDs12+mDs∗6smDs12,ρ¯DsDs∗s=fDsfT,Ds∗s−mDs2−mDs∗24−3fDsmDs2fDs∗mDs∗2mc210+mDs∗2−2mDs2s+s−2mDs2mDs∗2+mDs4smDs∗2−fDs2fT,Ds∗2s−mDs2−mDs∗28−3fDs2mDs2fDs∗mDs∗fT,Ds∗4mcs+mDs∗2+2mDs2+3mDs∗2mDs2−mDs∗4−3mDs4s+3smDs2−s2−3mDs4mDs∗2+mDs6smDs∗2+fDs2fT,Ds∗24−s2−mDs∗4+mDs4+mDs∗2mDs2+smDs2+3smDs∗22+mDs84smDs∗2+6mDs∗2mDs4−4mDs∗4mDs2+mDs∗6−4mDs64s+s3−4s2mDs2+6smDs4−4mDs64mDs∗2,λa,b,c=a2+b2+c2−2ab−2ac−2bc,mJ/ψϕ=mJ/ψ+mϕ,mhcϕ=mhc+mϕ,mhsJ/ψ=mhs+mJ/ψ,mf0χc1=mf0+mχc1,mf1χc0=mf1+mχc0,mf0ηc=mf0+mηc,mχc0η=mχc0+mη,mDs1Ds∗=mDs1+mDs∗,mDsDs∗=mDs+mDs∗,mDsDs0=mDs+mDs0, and we have taken the standard definitions of the decay constants:
(13)0∣c¯0σμνc0∣J/ψp=ifJ/ψTεμpν−ενpμ,0∣s¯0σμνs0∣ϕp=ifϕTεμpν−ενpμ,0∣c¯0σμνs0∣Ds∗p=ifDs∗Tεμpν−ενpμ,0∣c¯0σμνc0∣hcp=ifhcεμναβεαpβ,0∣s¯0σμνs0∣hsp=ifhsεμναβεαpβ,0∣c¯0γμc0∣J/ψp=fJ/ψmJ/ψεμ,0∣s¯0γμs0∣ϕp=fϕmϕεμ,0∣c¯0γμs0∣Ds∗p=fDs∗mDs∗εμ,0∣c¯0γμγ5c0∣χc1p=fχc1mχc1εμ,0∣s¯0γμγ5s0∣f1p=ff1mf1εμ,0∣c¯0γμγ5s0∣Ds1p=fDs1mDs1εμ,0∣c¯0γμγ5c0∣ηcp=ifηcpμ,0∣s¯0γμγ5s0∣ηp=ifηpμ,0∣c¯0γμγ5s0∣Dsp=ifDspμ,0∣c¯0c0∣χc0p=fχc0mχc0,0∣s¯0s0∣f0p=ff0mf0,0∣c¯0iγ5s0∣Dsp=fDsmDs2mc,0c¯0γμs0Ds0p=fDs0pμ.The εμ is the polarization vectors of the vector and axialvector mesons.We accomplish the operator product expansion for the correlation functionΠμμ′p up to the vacuum condensates of dimension 10 consistently [28–30]. In calculations, we assume dominance of the intermediate vacuum state tacitly, just like in previous works [28–30], and insert the intermediate vacuum state alone in all the channels, and vacuum saturation works well in the large Nc limit [31]. Up to now, almost in all the QCD sum rules for the multiquark states, vacuum saturation is assumed for the higher dimensional vacuum condensates, except in some cases, the parameter ρ>1(14)0∣:q¯αiqβjq¯λmqτn:∣0=ρ16Nc2q¯q2δijδmnδαβδλτ−δinδjmδατδβλ,which parameterizes deviations from the factorization hypothesis, is introduced by hand for the sake of fine-tuning [32], where q=u, d, s, the i, j, m, and n are color indexes, and the α, β, λ and τ are Dirac spinor indexes.In the original works, Shifman, Vainshtein, and Zakharov took the factorization hypothesis based on two reasons [33, 34]. The first one is the rather large value of the quark condensate q¯q, the second one is the duality between the quark and physical states, and they reproduce each other, counting both the quark and physical states (beyond the vacuum states) that maybe lead to a double counting [33, 34].In the QCD sum rules for theqq¯, qQ¯, and QQ¯ mesons, the q¯q2 are always companied with the fine-structure constant αs=gs2/4π and play a minor important (or tiny) role, and the deviation from ϱ=1, for example, ϱ=2~3, cannot make much difference in the numerical predictions; though in some cases, the values ϱ>1 can lead to better QCD sum rules [35, 36]. However, in the QCD sum rules for the multiquark states, the q¯q2 play an important role, large values, for example, if we take the value ϱ=2 in the present case, we can obtain the uncertainties δMX=+0.08GeV and δMX′=+0.09GeV, which are of the same order of the total uncertainties from other parameters. Sometimes, large values of the ρ can destroy the platforms in the QCD sum rules for the multiquark states [37].The true values of the higher dimensional vacuum condensates remain unknown or poorly known; if the true valuesϱ>1 or ≫1, the QCD sum rules for the multiquark states have considerably large systematic uncertainties and are less reliable than those of the conventional mesons and baryons [38]. We just make predictions for the multiquark masses with the QCD sum rules based on vacuum saturation and then confront them to the experimental data in the future to examine the theoretical calculations.After the analytical expression of the QCD spectral density was acquired, we take the quark-hadron duality below the continuum thresholdss0 and s0′ by including the contributions of the 1S state and 1S plus 2S states, respectively, and perform Borel transform in regard to the variable P2=−p2 to obtain the two QCD sum rules:
(15)λX2exp−MX2T2=∫4mc2s0dsρsexp−sT2=Πτ,λX2exp−MX2T2+λX′2exp−MX′2T2=∫4mc2s0′dsρsexp−sT2=Π′τ,where τ=1/T2. For the explicit expression of the spectral density ρs at the quark and gluon level, one can consult Ref. [23]. We define Dn=−d/dτn with n=0, 1, 2, ⋯ and then acquire the QCD sum rules for the masses:
(16)MX2=DΠτΠτ,(17)MX′2=b+b2−4c2,(18)λX′2=D−MX2ΠQCDτMX′2−MX2expτMX′2,where
(19)b=D3⊗D0−D2⊗DD2⊗D0−D⊗D,c=D3⊗D−D2⊗D2D2⊗D0−D⊗D,Dj⊗Dk=DjΠ′τDkΠ′τ,the indexes i=1,2 and j,k=0,1,2,3. For the technical details in obtaining the QCD sum rules in Eqs. (17) and (18), one can consult Refs. [14, 20, 39].On the other hand, if we saturate the hadron side of the QCD sum rules with the contributions of the two-meson scattering sates, we obtain the following two QCD sum rules:(20)ΠTMT2=1768π2∫mJ/ψϕ2s0′dsλ1/2s,mJ/ψ2,mϕ2sρ¯J/ψϕsexp−sT2+11536π2fϕ2mϕ2fhc2∫mhcϕ2s0′dsλ1/2s,mhc2,mϕ2sρ¯hcϕsexp−sT2+11536π2fJ/ψ2mJ/ψ2fhs2∫mhsJ/ψ2s0′dsλ1/2s,mhs2,mJ/ψ2sρ¯hsJ/ψsexp−sT2+3512π2ff02mf02fχc12mχc12∫mf0χc12s0′dsλ1/2s,mχc12,mf02sρ¯χc1f0sexp−sT2+3512π2ff12mf12fχc02mχc02∫mf1χc02s0′dsλ1/2s,mχc02,mf12sρ¯χc0f1sexp−sT2+3512π2ff02mf02fηc2∫mf0ηc2s0′dsλ3/2s,mηc2,mf02s2exp−sT2+3512π2fχc02mχc02fη2∫mχc0η2s0′dsλ3/2s,mχc02,mη2s2exp−sT2+1768π2fDs12mDs12fT,Ds∗2∫mDs1Ds∗2s0′dsλ1/2s,mDs12,mDs∗2sρ¯Ds1Ds∗sexp−sT2+1192π2∫mDsDs∗2s0′dsλ1/2s,mDs2,mDs∗2sρ¯DsDs∗sexp−sT2+3256π2fDs2mDs4fDs02mc2∫mDsDs02s0′dsλ3/2s,mDs2,mDs02s2exp−sT2+J/ψϕ⟶ψ′ϕ+J/ψϕ⟶ψ′′ϕ+hcϕ⟶hc′ϕ+⋯=κΠ′T2,(21)dd1/T2ΠTMT2=κdd1/T2Π′T2.In Eqs. (20) and (21), we introduce the parameter κ to measure the deviations from 1; if κ≈1, we can acquire the conclusion tentatively that the two-meson scattering states can saturate the QCD sum rules.


## 4. Numerical Results and Discussions

At the QCD side, we choose the conventional values of the vacuum condensatesq¯q=−0.24±0.01GeV3, s¯s=0.8±0.1q¯q, s¯gsσGs=m02s¯s, m02=0.8±0.1GeV2, and αsGG/π=0.33GeV4 at the energy scale μ=1GeV [33, 34, 40, 41] and prefer the modified minimal subtracted masses mcmc=1.275±0.025GeV and msμ=2GeV=0.095±0.005GeV from the Particle Data Group [42]. In addition, we consider the energy-scale dependence of the input parameters:
(22)s¯sμ=s¯s1GeVαs1GeVαsμ1233−2nf,s¯gsσGsμ=s¯gsσGs1GeVαs1GeVαsμ233−2nf,mcμ=mcmcαsμαsmc1233−2nf,msμ=ms2GeVαsμαs2GeV1233−2nf,αsμ=1b0t1−b1b02logtt+b12log2t−logt−1+b0b2b04t2,where t=logμ2/Λ2, b0=33−2nf/12π, b1=153−19nf/24π2, b2=2857−5033/9nf+325/27nf2/128π3, Λ=210MeV, 292MeV, and 332MeV for the flavors nf=5, 4, and 3, respectively [42, 43], and evolve all the input parameters to the typical energy scales μ with the flavor number nf=4, which satisfy the energy scale formula μ=MX/Y/Z2−2Mc2 with the updated value of the effective charmed quark mass Mc=1.82GeV [44, 45], to extract the masses of the hidden-charm tetraquark states. We tentatively assign the X4140 and X4685 to be the ground state and first radial excited state of the hidden-charm tetraquark states, respectively, and the corresponding pertinent energy scales of the spectral densities at the quark-gluon level are μ=2.0GeV and 3.0GeV, respectively.In Ref. [23], we obtain the Borel window T2=2.7−3.3GeV2, continuum threshold parameter s0=4.7±0.1GeV, and pole contribution 41−69% for the X4140, then acquire the mass and pole residue MX=4.14±0.10GeV and λX=4.30±0.85×10−2GeV5, which are all shown plainly in Table 2. In the present work, we choose the same Borel parameter T2=2.7−3.3GeV2 as in Ref. [23] and assume that the energy gap between the first and second radial excited states is about 0.3~0.5GeV, take the continuum threshold parameter as s0′=5.1±0.1GeV, and then obtain the pole contribution 69−90%, and it is large enough to extract the mass of the first radial excited state. Moreover, the convergent behaviors of the operator product expansion are very good, and the contributions from the vacuum condensates of dimension 10 in the QCD sum rules for the X4140 and X4685 are <1% and ≪1%, respectively.Table 2
The Borel windows, continuum threshold parameters, and ideal energy scales of the spectral densities, pole contributions, masses and pole residues for the axialvector tetraquark states.
T2GeV2s0GeVμGeVPoleMGeVλGeV5X41402.7−3.34.7±0.12.041−69%4.14±0.104.30±0.85×10−2X46852.7−3.35.1±0.13.069−90%4.70±0.121.08±0.17×10−1Finally, we take into account all the uncertainties of the input parameters and get the mass and pole residue of the first radial excited stateX′, which are shown plainly in Table 2 and Figure 1. From the table, we observe that the predicted mass MX′=4.70±0.12GeV is in very good agreement with the experimental value MX4685=4684±7−16+13MeV from the LHCb collaboration [8].Figure 1
The masses of theX and X′ with variations of the Borel parameter T2, where the expt stands for the experimental values of the masses of the X4140 and X4685, respectively.In Figure1, we plot the predicted masses MX and MX′ with variations of the Borel parameter T2; from the figure, we can see clearly that there appear rather flat platforms both for the ground state and first radial excited state, and we are confidential to obtain reliable predictions. In addition, we present the experimental values of the masses of the X4140 and X4685, which happen to lie in the center regions of the predicted values.If the masses of the ground stateX4140, first radial excited state X4685, second radial excited state X′′, etc, satisfy the Regge trajectory,
(23)Mn2=αn−1+α0,where the α and α0 are some constants to be fitted experimentally, and the n is the radial quantum number. We take the masses of the ground state and first radial excited state, MX4140=4118MeV and MX4685=4684MeV [8], as input parameters to fit the parameters α and α0, and obtain the mass of the second radial excited state, MX′′=5.19±0.10GeV, which is consistent with the continuum threshold parameter s0′=5.1±0.1GeV, and the contamination from the second radial excited state is avoided; here, we add an uncertainty δ=±0.1GeV to the mass MX′′ according to Table 2. Now, we reach the conclusion tentatively that the calculations are self-consistent.The valuesMX′′=5.09GeV, 5.19GeV and 5.29GeV correspond to the continuum threshold parameters s0′=5.0GeV, 5.1GeV, and 5.2GeV, respectively, and have the relation MX′′>s0′, and the contamination from the second radial excited state can be neglected. At the beginning, we assume that the energy gap between the first and second radial excited states is about 0.3~0.5GeV and tentatively take the continuum threshold parameter as s0′=5.1±0.1GeV to obtain the mass of the X′, then resort to the Regge trajectory to check whether or not such a choice is self-consistent. Fortunately, such a choice happens to be satisfactory. On the other hand, if it is not self-consistent, we can choose another value of the s0′, then repeat the same routine to obtain self-consistent MX′, MX′′ and s0′ via trial and error. In all the calculations, we should obtain flat Borel platforms to suppress the dependence on the Borel parameters.In Refs. [9, 13, 14, 23], we investigate the hidden-charm tetraquark (molecular) states via the QCD sum rules using the energy scale formula μ=MX/Y/Z2−2Mc2 to choose the pertinent energy scales of the spectral densities at the quark and gluon level, which can enhance the pole contributions remarkably and improve the convergent behaviors of the operator product expansion remarkably. It is a unique feature of our works. The predictions MDs∗D¯s1−Ds1D¯s∗=4.67±0.08GeV [9], McsPc¯s¯A+csAc¯s¯P=4.63−0.08+0.11GeV [13], McsAc¯s¯A,1S=3.92−0.18+0.19GeV, McsAc¯s¯A,2S=4.50−0.09+0.08GeV [14], MscV~s¯c¯V−scVs¯c¯V~,1S=4.14±0.10GeV [23], and MscV~s¯c¯V−scVs¯c¯V~,2S=4.70±0.12GeV based on the QCD sum rules support assigning the X4630 as the Ds∗D¯s1−Ds1D¯s∗ tetraquark molecular state [9] or csPc¯s¯A+csAc¯s¯P tetraquark state with the quantum numbers JPC=1−+ [13], assigning the X3915 and X4500 as the 1S and 2S csAc¯s¯A tetraquark states, respectively, with the quantum numbers JPC=0++, and assigning the X4140 and X4685 as the 1S and 2S scV~s¯c¯V−scVs¯c¯V~ tetraquark states, respectively, with the quantum numbers JPC=1++ [23]. The predictions with the possible assignments are given plainly in Table 3. We should bear in mind that other assignments of the X4500 and X4700, such as the D-wave csc¯s¯ tetraquark states with the JP=0+, are also possible [16], and more theoretical and experimental works are still needed to obtain definite conclusion.Table 3
The possible assignments of the LHCb’sX states based on the predictions from the QCD sum rules, where the superscript ∗ denotes the mass of the X4140 taken from Ref. [23].
JPC1S [mass (GeV)]2S [mass (GeV)]ReferencesDs∗D¯s1−Ds1D¯s∗1−+X4630 [4.67±0.08][9]csPc¯s¯A+csAc¯s¯P1−+X4630 [4.63−0.08+0.11][13]csAc¯s¯A0++X3915 [3.92−0.18+0.19]X4500 [4.50−0.09+0.08][14]scV~s¯c¯V−scVs¯c¯V~1++X4140 [4.14±0.10]X4685 [4.70±0.12][23]∗Now, we explore the outcome in the case of saturating the hadron side of the QCD sum rules with the two-meson scattering states. At the hadron side of the QCD sum rules in Eqs. (20) and (21), we choose the parameters mJ/ψ=3.0969GeV, mηc=2.9839GeV, mhc=3.52538GeV, mχc0=3.41471GeV, mχc1=3.51067GeV, mϕ=1.019461GeV, mh1=1.416GeV, mf1=1.4263GeV, mf0=1.506GeV, mη=0.547862GeV, mDs=1.969GeV, mDs∗=2.1122GeV, mDs0=2.318GeV, mDs1=2.4596GeV, mψ′=3.6861GeV, and mψ′′=4.0396GeV from the Particle Data Group [42]; mhc′=3.9560GeV from the Godfrey-Isgur model [46]; fJ/ψ=0.418GeV, fηc=0.387GeV, fJ/ψT=0.410GeV, and fhc=0.235GeV from the lattice QCD [47]; fχc1=0.338GeV, fχc0=0.359GeV [48], fϕ=0.231GeV, fϕT=0.200GeV [49, 50], fη=1.34fπ [51], fh1=0.183GeV, ff1=0.211GeV [52], ff0=0.490GeV [53], fDs=0.240GeV, fDs∗=0.308GeV, fDs0=0.333GeVmc/mDs0, and fDs1=0.345GeV [54] from the QCD sum rules; fπ=0.130GeV, fψ′=0.295GeV, and fψ′′=0.187GeV extracted from the experimental data [42]; and fDs∗T=fDs∗, fψ′T=fψ′, fψ′′T=fψ′′, and fhc′=fhcfψ′/fJ/ψ=0.166GeV estimated in the present work (also in Ref. [27]).In Figure2, we plot the values of the κ with variations of the Borel parameter T2 for the central values of the input parameters. From Figure 2, we can see that the values of the κ increase monotonically and quickly with the increase of the Borel parameter T2, no platform appears, which indicates that the QCD sum rules in Eqs. (20) and (21) are not satisfactory, and the two-meson scattering states alone cannot saturate the QCD sum rules at the hadron side.Figure 2
The values of theκ with variations of the Borel parameter T2, where the A and B come from the QCD sum rules in Eqs. (20) and (21), respectively.In Ref. [27], we investigate the Zc3900 with the QCD sum rules in details by including all the two-meson scattering state contributions and nonlocal effects between the diquark and antidiquark constituents. We observe that the two-meson scattering states alone cannot saturate the QCD sum rules at the hadron side, and just like in the present case, the contribution of the Zc3900 (or pole term) plays an unsubstitutable role, and we can saturate the QCD sum rules with or without the two-meson scattering state contributions. We expect the conclusion is also applicable in the present case.Now, we explore the two-meson scattering state contributions; besides, the tetraquark statesXc and Xc′, and take account of all the contributions:
(24)Πμνp=−λ^X2p2−M^X2+ΣJ/ψϕp2+⋯gμν−λ^X′2p2−M^X′2+ΣJ/ψϕp2+⋯gμν+⋯.We choose the bare masses and pole residues M^X, M^X′, λ^X, and λ^X′ to absorb the divergent terms in the self-energies ΣJ/ψϕp2, ⋯. The renormalized self-energies satisfy the relations p2−MX,R2+Σ¯J/ψϕp2+⋯=0 and p2−MX′,R2+Σ¯J/ψϕp2+⋯=0, where the subscripts R represent the MS¯ masses, and the overlines above the self-energies represent that the divergent terms have been subtracted. The tetraquark states Xc and Xc′ have finite widths and are unstable particles, and the relations should be modified, p2−MX,R2+ReΣ¯J/ψϕp2+⋯=0, p2−MX′,R2+ReΣ¯J/ψϕp2+⋯=0, ImΣ¯J/ψϕp2+⋯=p2ΓXp2, and ImΣ¯J/ψϕp2+⋯=p2ΓX′p2. The renormalized self-energies contribute a finite imaginary part to modify the dispersion relation:
(25)Πμνp=−λX2p2−MX2+ip2ΓXp2gμν−λX′2p2−MX′2+ip2ΓX′p2gμν+⋯,where MX′2=MX′,R2+Σ¯J/ψϕMX′2.We can take account of the finite width effects by the simple replacements of the hadronic spectral densities:(26)λX′2δs−MX′2⟶λX′21πMX′ΓX′ss−MX′22+MX′2ΓX′2s,where
(27)ΓX′s=ΓX′MX′ss−mJ/ψ+mϕ2MX′2−mJ/ψ+mϕ2.The ΓX and ΓX′ are the physical decay widths. Then, the hadron sides of the QCD sum rules undergo the following changes:
(28)λX′2exp−MX′2T2⟶λX′2∫mJ/ψ+mϕ2s0′ds1πMX′ΓX′ss−MX′22+MX′2ΓX′2sexp−sT2,=0.81~0.82λX20.97~0.98λX′2exp−MX′2T2.λX′2MX′2exp−MX′2T2⟶λX′2∫mJ/ψ+mϕ2s0′dss1πMX′ΓX′ss−MX′22+MX′2ΓX′2sexp−sT2,=0.82~0.83λX20.96~0.97λX′2MX′2exp−MX′2T2.We can absorb the numerical factors0.81~0.82, 0.82~0.83, 0.97~0.98, and 0.96~0.97 into the pole residues safely, and the two-meson scattering states cannot affect the masses MX and MX′ significantly [55]. Again, we obtain the conclusion, the pole terms or tetraquark states play an unsubstitutable role, we can saturate the QCD sum rules with or without the two-particle scattering state contributions, and the two-particle scattering states can only modify the pole residues [27].In the present work, we choose the local four-quark currentJμx, while the traditional mesons are spatial extended objects and have average spatial sizes r2≠0, for example, r2=0.41fm (0.42fm) for the J/ψ ([56, 57]) and r2=0.63fm for the ϕ1020 [58]. On the other hand, the diquark-antidiquark type tetraquark states have the average spatial sizes r=0.5~0.7fm [59]. TheJ/ψ,ϕ1020,X4140, andX4685have average spatial sizes of the same order, and the couplings to the continuum statesJ/ψϕ, etc. can be neglected, as the overlappings of the wave-functions are small enough.


## 5. Conclusion

At the first step, we take into account our previous calculations based on the QCD sum rules and make possible assignments of the LHCb’s new particlesX4630 and X4500. We tentatively assign the X4630 to be the Ds∗D¯s1−Ds1D¯s∗ tetraquark molecular state or csPc¯s¯A+csAc¯s¯P tetraquark state with the quantum numbers JPC=1−+ and assign the X3915 and X4500 to be the 1S and 2S csAc¯s¯A tetraquark states, respectively, with the quantum numbers JPC=0++ according to the predicted masses.Then, we extend our previous works to explore theX4685 as the first radial excited state of the X4140 with the QCD sum rules and obtain the value of the mass MX4685=4.70±0.12GeV, which is in very good agreement with the experimental value MX4685=4684±7−16+13MeV from the LHCb collaboration, and support assigning the X4140 and X4685 as the 1S and 2S scV~s¯c¯V−scVs¯c¯V~ tetraquark states, respectively, with the quantum numbers JPC=1++. Furthermore, we investigate the two-meson scattering state contributions in details and observe that the two-meson scattering state contributions alone cannot saturate the QCD sum rules at the hadron side, the contributions of the tetraquark states (or pole terms) play an unsubstitutable role, we can saturate the QCD sum rules with or without the two-meson scattering state contributions, the two-meson scattering state contributions can only modify the pole residues, and the predictions of the tetraquark masses are robust.



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