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In 2003, Belle observed a new state known as
X(3872) , which definitely contained a charm-anticharm pair and could not be explained by the ordinary quark-antiquark model [1]. Since then, more new hadrons containing heavy quarks have been found and studied in numerous experiments [2]. These hadrons are known as XYZ states, which contain a heavy quark-antiquark pair and at least a light quark-antiquark pair; they are naturally exotics [3]. Many structures have been proposed to describe XYZ states that include molecular, tetraquark, and hybrid components [4–6]. Like the studies of other mesons with exotic quantum numbers, convincing explanations of the observed XYZ states remain an open question in phenomenological particle physics. Recently, a study ofX(3872) by LHCb argued that the compact component should be required in X states [7]; this result is more likely to support the tetraquark model of XYZ states but not exclude the molecular model of all exotic states. In this paper, we focus on the states in two simpleˉQqQˉq andˉQQˉqq combinations to study XYZ states (ˉQ represents a heavy c or b quark, while q represents a lightu,d or s quark). These two forms have been extensively studied previously [8, 9]. However,ˉQqQˉq andˉQQˉqq states are difficult to distinguish straightforwardly from the decay modes of XYZ states because XYZ states are usually observed to have bothˉQQ+ˉqq like decay modes andˉQq+Qˉq like decay modes. Many scenarios were studied to qualitatively distinguishˉQqQˉq andˉQQˉqq states [10–14]. Furthermore, the physical states are usually mixtures of different structures, and this makes the problem more complicated. In a previous study we have developed a method to estimate the mixing strength of different currents from a QCD sum rule (QCDSR) approach [15–18]; we use the same technique to study the mixing ofˉQqQˉq andˉQQˉqq XY states.In this study, we investigated three kinds of vector states with different quantum numbers
JPC=1++ ,1−− ,1−+ . These states have long been considered to beˉQqQˉq orˉQQˉqq molecular states in different studies [19–25]. However, since many of them have abundant decay modes, mixing scenarios should be taken into account. Besides, it is generally believed that there is a large background of two free mesons spectrum in the two points correlation function of four-quark currents. To avoid such a large uncertainty, it is especially important to estimate the mixing strength ofˉQqQˉq andˉQQˉqq currents to investigate the corresponding physical states. The calculations will show us whether the physical states prefer to beˉQqQˉq orˉQQˉqq molecular states, or whether they are strongly mixed states.As mentioned above, for the 1++ channel,
X(3872) has been extensively studied for a wide variety of structures [26, 27]. In molecular state schemes,X(3872) has been usually considered aD∗D molecular state [12–14]. However, although the pureD∗D molecular state was predicted to have a mass close toX(3872) , it had too large a decay width to agree with the experimental results [28, 29]. Moreover, theJ/ψρ andJ/ψω states have a similar mass, since the sum of the masses of their two constituent parts are close toX(3872) . Hence, the mixing of these two molecular states is naturally possible. Furthermore, in a recent observation ofX(3872) by LHCb [7], the compact component was found to be required. Hence, we will consider another stateˉcc in the mixing; this has been studied in [28].For the
1−− channel, many1−− states are found in the range of 4200–4700 MeV, permitting an abundance of possible pure or mixed molecular states. Some1−− states have very similar masses likeY(4220)/Y(4260) and Y(4360)/Y(4390) [30, 31]. Hence, it is interesting and meaningful to investigate the possible mixing of molecular states, which has not been previously studied.For 1–+ sector, no confirmed heavy hadrons with 1–+ quantum numbers have been observed. Some potential candidates include
X(3940), X(4160), X(4350) [2]. The constructions of 1–+ molecular states in theˉQqQˉq andˉQQˉqq scenarios are possible. As outlined below, we calculate the mass spectrum of these states and estimate their mixing strength in both theu,d and s quark cases to help guide searches for these states in the 1–+ sector.Our methodology is introduced in Section II. Then, we discuss the
1++ states,1−− states, and1−+ states in Sections III, IV, and V, respectively. We discuss the importance of non-perturbative terms in calculations that evaluate the mixing strength in Section VI. Finally, we present our summary and conclusions in the last section. -
In QCDSR, we normally construct a mixing current that combine two state interpretations. The two-point correlation function of the mixing currents can be written as
Π(q2)=i∫d4xeiqx⟨0|T{(ja(x)+cjb(x))(j+a(0)+cj+b(0))}|0⟩=Πa(q2)+2cΠab(q2)+c2Πb(q2),
(1) where
ja andjb have the same quantum numbers; c is a real parameter related to the mixing strength (not the mixing strength itself, since c may not been normalized), andΠa(q2)=i∫d4xeiqx⟨0|T(ja(x)j+a(0))|0⟩,
Πb(q2)=i∫d4xeiqx⟨0|T(jb(x)j+b(0))|0⟩,Πab(q2)=i2∫d4xeiqx⟨0|T(ja(x)j+b(0)+jb(x)j+a(0))|0⟩.
(2) Here, we consider the mixed correlator
Πab because it provides a signal that indicates which states couple to both currents. One can insert a complete set of particle eigenstates betweenja andjb , and the state with a relatively strong coupling to both these currents will be selected out through the QCDSR. By estimating the mass and coupling constants as well as taking experimental results into account, one can obtain insight into the constituent composition of the corresponding states. This method worked well in our previous study on vector and scalar meson states [15] and has been successfully applied in other systems [16–18].Πab usually can be decomposed into a different Lorentz structureΠab(q2)=∑nΠn(q2)An,
(3) where
n=1,2,3... ,Πn(q2) is the mixing state correlation function with specific quantum numbers, andAn is the corresponding Lorentz structure. The forms ofAn are related toja andjb , and we will define them in the sections below. For simplicity, we assume a specificΠH(q2) represents a mixed-state correlation function and is one of the possibleΠn(q2) . We also assume thatΠH(q2) obeys the dispersion relation [32]ΠH(q2)=∫∞smindsρH(s)s−q2−iϵ+…,
(4) where the spectral density
ρH(s)=1πImΠH(s) ,smin represents the physical threshold of the corresponding current, and the dots on the right hand side represent the polynomial subtraction terms to renderΠH(q2) finite. The spectral densityρH(s) can be calculated using the operator product expansion (OPE). In this paper, we calculate the spectral densityρH(s) up to dimension-six operators,ρ(OPE)H(s)=ρ(pert)H(s)+ρ⟨ˉqq⟩H(s)+ρ⟨G2⟩H(s)+ρ⟨ˉqGq⟩H(s)+ρ⟨ˉqq⟩2H(s)+…,
(5) then
Π(OPE)H(q2)=∫∞smindsρ(OPE)H(s)s−q2−iϵ+….
(6) On the phenomenological side, by using the narrow resonance spectral density model,
Π(phen)H(q2)=λaλ∗b+λbλ∗a21M2H−q2+∫∞s0dsρ(cont)H(s)s−q2−iϵ,
(7) where
λa andλb are the respective couplings of the ground state to the corresponding currents;MH represents the mass of the mixed state, which has relatively strong coupling to the corresponding currents;ρ(cont)H represents continuum contributions to spectral density, ands0 is the continuum threshold. By using the QCDSR continuum spectral density assumptionsρ(cont)H(s)=ρ(OPE)H(s)Θ(s−s0),
(8) and equating the OPE side and the phenomenological side of the correlation function,
Π(phen)H(q2)=Π(OPE)H(q2) , we obtain the QCDSR master equation∫s0smindsρ(OPE)H(s)s−q2−iϵ+…=λaλ∗b+λbλ∗a21M2H−q2.
(9) After applying the Borel transformation operator
ˆB to both sides of the master equation, the subtraction terms are eliminated, and the master equation can be written as [32, 33]∫s0smindsρ(OPE)H(s)e−sτ=λaλ∗b+λ∗aλb2e−M2Hτ,
(10) where the Borel parameter
τ=1/M2 , and M is the Borel mass. The master equation (10) is the foundation of our analysis. By taking the τ logarithmic derivative of Eq. (10), we obtainM2H=∫∞sminds sρ(OPE)H(s)e−sτ∫∞sminds ρ(OPE)H(s)e−sτ.
(11) One can set
a=b in Eqs. (7), (9), and (10) to obtain the original pure state QCDSR.Eq. (10) is not valid for all values of τ because of the OPE truncation and the simplified assumption for the phenomenological spectral density; thus, the determination of the sum rule window, in which the validity of (10) can be established, is very important. In the literature, different methods are used in the determination of the τ sum rule window [34, 35]. In this paper, we follow similar previous studies to restrict the resonance and high dimension condensate contributions (HDC), i.e., the resonance part obeys the relation
∫s0sminds ρ(OPE)H(s)e−sτ∫∞sminds ρ(OPE)H(s)e−sτ>40%,
(12) while HDC (usually
⟨ˉqq⟩2 in molecular systems) obeys the relation|∫∞sminds ρ⟨ˉqq⟩2H(s)e−sτ||∫∞sminds ρ(OPE)H(s)e−sτ|<10%.
(13) Furthermore, the value of
s0 is also very important in QCDSR methods. It is often assumed that the threshold satisfies√s0=MH+Δs , withΔs≈0.5 GeV. This is especially the case in molecular state QCDSR calculations [36, 37].The approximation
√s0=MH+Δs can be understood in QCDSR because the parameters0 separates the ground state and other excited states' contributions to spectral density. Hence, one can sets0 less than the first excitation threshold in the case of involving excited state contributions in the spectral density, andΔs represents the approximate mass difference between the ground and first excited states. We assume that the first excited state is approximately equal to an excited constituent meson and another ground state constituent meson. Then, we can establishs0 by comparing the mass difference between the ground constituent meson and the first excited constituent meson of the corresponding state (like the charmonium and D meson family in our case). We have listed some experimental data for the charmonium and D meson families in Table 1 and Table 2. One can easily find that the mass difference between the ground state and first excited state are all around0.5+0.1−0.1 GeV, and the fluctuations are all acceptable in the QCDSR approach.PDG name Possible structure Ground state Possible 1st excited state Δs/ MeVD ˉcγ5q ∙D(1865) D(2550) ~685 D1 ˉcγμγ5q ∙D1(2420) − − D∗0 ˉcq ∙D∗0(2300) D∗J(2600) ~300 D∗ ˉcγμq ∙D∗(2007) D∗(2640) ~633 Table 1. Charmed meson (
c=±1 ) states, where q represents theu,d quark. The∙ symbol indicates particles that have confirmed quantum numbers.Possible structure Ground state/MeV Possible 1st excited state/MeV Δs/ MeVˉcγ5c ∙ηc(1S) /2984∙ηc(2S) /3637~653 ˉcγμγ5c ∙χc1(1P) /3510∙χc1(3872) ~362 ˉcc ∙χc0(1P) /3415χc0(3860) ~445 ˉcγμc ∙J/ψ(1S) /3097∙ψ(2S) /3686~589 Table 2. Charmonium (possibly non-
ˉcc states). The∙ symbol indicates particles that have confirmed quantum numbers.In order to estimate the mixing strength of the physical state strongly coupled to both of the two different currents, we define
N≡|λaλ∗b+λ∗aλb|2|λ′aλ′b|,
(14) where
λ′a andλ′b are coupling constants of the relevant current with a pure state (i.e., the coupling that emerges in the diagonal correlation functionsΠa ,Πb ). Eq. (14) is analogous to the mixing parameter defined in Ref. [38]. By using appropriate factors of mass in the definitions ofλ′a andλ′b , we can compare the magnitude of coupling constants and estimate the mixing strength self-consistently. The mixing strength depends on the definition and normalization of the mixed state. For example, in Ref. [39] the definition of the mixed state is|M⟩=cosθ|A⟩+sinθ|B⟩,
(15) where
|M⟩ is a mixed state composed of pure states|A⟩ and|B⟩ , and θ is a mixing angle. In this definition and normalization of the mixed state, we see thatN≈ cosθsinθ , andN∈(0,12) . We use Eq. (14) as a robust parameter to quantify mixing effects because of the different possible normalizations and mixed state definitions. Furthermore, because the behavior of N is not linear, we define˜N under the scenario of Eq. (15):˜N=sin2(arcsin(2N)2).
(16) The quantity
˜N gives the approximate proportion of the pure part of the mixed state. A comparison of the the mixed state mass with the two relevant pure states suggests that the mixed state is dominated by the part whose pure state mass prediction is the closest to the mixed state mass. Different decay widths can also help us to distinguish the dominant part of the mixed state.We use the following numerical values of vacuum condensates consistent with other QCDSR analyses of XYZ states:
⟨ˉqq⟩= (−0.23±0.03)3 GeV3,⟨ˉqgsσGq⟩= m20⟨ˉqq⟩ ,m20=0.8 GeV2,⟨αsG2⟩=0.07±0.02 GeV4, and⟨ˉss⟩=(0.8±0.2)⟨ˉqq⟩ [40, 41]. In addition, the quark massesmc=1.27 GeV,mq=12(mu+md)=0.004 GeV, andms=0.096 GeV, at the energy scaleμ=2 GeV [2], are used. -
We start from the following three forms of currents:
jXAμν(x)=i√2[ˉc(x)γμc(x)ˉq(x)γνq(x)−ˉc(x)γνc(x)ˉq(x)γμq(x)],jXB1μ(x)=i√2[ˉc(x)γμq(x)ˉq(x)γ5c(x)−ˉq(x)γμc(x)ˉc(x)γ5q(x)],jXB2μ(x)=i√2[ˉc(x)γμγ5q(x)ˉq(x)c(x)+ˉq(x)γμγ5c(x)ˉc(x)q(x)],jXCμ(x)=16√2⟨ˉqq⟩ˉc(x)γμγ5c(x),
(17) where X denotes the 1++ state, the subscript A of X denotes the
ˉQQˉqq scenario, B denotes theˉQqQˉq scenario, C denotes theˉQQ scenario, and the corresponding mesonic structures of these currents are listed in Table 3. We note that the former two currents can be decomposed into two constituent meson currents, and the mass prediction of each corresponding pure state are usually close to the sum of the masses of these two constituent mesons. The currentjXAμν can be decomposed intoJ/ψ(1S)(3097) andρ(770) currents;jXB1μ(x) can be decomposed intoD∗(2007) andD(1865) currents. The sums of the masses of the two constituent mesons are both close toX(3872) . Hence, we choose these two currents to studyX(3872) . The currentjXB2μ has the same quantum numbers, and it cannot be excluded in1++ mixing state structures. Besides,jXCμ(x) is normalized according to Ref. [28]. Since the mixing betweenjXCμ(x) andˉQQˉqq is suppressed (ˉqq inˉQQˉqq becomes a bubble and vanishes), we only consider the mixing betweenjXCμ(x) andjXB1μ(x) orjXB2μ(x) .State Current structure Mass/GeV λ/ 10−4 GeV10√s0/GeV τ window/GeV−2 XA J/ψρ 3.798+0.09−0.09 1.49+0.51−0.47 4.4 0.30 – 0.31 XB1 D∗ˉD 3.857+0.06−0.06 2.24+0.65−0.53 4.4 0.31 – 0.39 XB2 D1ˉD∗0 5.310+0.04−0.04 69.0+15.0−14.0 5.8 0.20 – 0.29 XC χc1 3.511+0.02−0.03 0.0229+0.0017−0.0018 4.5 0.29 – 0.31 MX1 J/ψρ -D∗ˉD 3.987+0.06−0.06 0.168+0.049−0.042 GeV-14.4 0.30 – 0.32 MX2 J/ψρ –D1ˉD∗0 4.945+0.08−0.06 0.760+0.29−0.20 GeV-15.45 0.22 – 0.24 MC χc1 -D∗ˉD 3.818+0.03−0.02 0.0282+0.0027−0.0024 4.5 0.28 – 0.30 Table 3. Summary of results for
1++ states.λ=λaλ∗b+λ∗aλb2 when mixed cases are involved, the same below.To study the pure
ˉQQˉqq ,ˉQqQˉq , andˉQQ states, the respective two-point correlation functions can be decomposed into different Lorentz structures.ΠXAμνρσ(q2)=ΠXAa(q2)1q2(q2gμρgνσ−q2gμσgνρ−qμqρgνσ+qμqσgνρ−qνqσgμρ+qνqρgμσ)+ΠXAb(q2)1q2(−qμqρgνσ+qμqσgνρ−qνqσgμρ+qνqρgμσ),ΠXBk/Cμν(q2)=ΠXBk/C(1)(q2)(−gμν+qμqνq2)+ΠXBk/C(0)(q2)(qμqνq2),
(18) where
k=1,2 ;ΠXAa andΠXAb describe the pure molecular state contribution with respective quantum numbers1++ and1−+ ; andΠXBk/C(1) andΠXBk/C(0) describe1++ and0−+ state contributions, respectively. In the mixing scenario, we start from the off-diagonal mixed correlator described in the previous section, i.e.,ΠMX1μνσ(q2)=i2∫d4xeiq⋅x⟨0|T(jXAμν(x)jXB1+σ(0)+jXB1σ(x)jXA+μν(0))|0⟩,ΠMX2μνσ(q2)=i2∫d4xeiq⋅x⟨0|T(jXAμν(x)jXB2+σ(0)+jXB2σ(x)jXA+μν(0))|0⟩,
(19) where
MXk ,k=1,2 , are mixed states assumed to result from the corresponding currents. These mixed correlators have the Lorentz structureΠMXkμνσ(q2)=ΠMXk(q2)(qαϵαμνσ).
(20) Furthermore, when we consider the two-quark states
jXCμ(x) , the mixed correlator and its Lorentz structure areΠMCμν(q2)=i2∫d4xeiq⋅x⟨0|T(jXCμ(x)jXB1+ν(0)+jXB1ν(x)jXC+μ(0))|0⟩,ΠMCμν(q2)=ΠMC(1)(q2)(−gμν+qμqνq2)+ΠMC(0)(q2)(qμqνq2),
(21) where
ΠC(1) andΠC(0) describe the1++ and0−+ state contributions, respectively. Here, we just consider the state mixed withjXCμ andjXB1 ; this state is a candidate forX(3872) .We follow the 40%–10% sum-rule window and
Δs methods mentioned in Section II. After establishing the τ window by the 40%–10% method at a specifics0 , we use Eq. (11) for each state to plot the τ behavior ofMH for the chosens0 (see Appendix A for details). The mass predictionMH ands0 are then compared with the constraint√s0=MH+Δs . Then,s0 is adjusted, and the analysis is repeated until we find the best(MH,s0) solutions that satisfy the relation√s0=MH+Δs . The coupling constants are naturally obtained through the predictedMH ands0 according to Eq. (10). The mass and coupling constant prediction and associated QCDSR parameters are presented in Table 3. All the parameters are the average values in the corresponding τ window.The uncertainties are mainly from the input parameters. For instance,
⟨αsG2⟩= 0.07±0.02 GeV4,⟨ˉss⟩= (0.8±0.2)⟨ˉqq⟩ , and⟨ˉqq⟩= (−0.23±0.03)3 GeV3. The quark masses and other parameters included in the calculations have uncertainties of less than 5% due to substantial numerical fittings by other researchers. There is also an uncertainty about the value of the thresholds0 . Analogous to the studies in Refs. [20, 42], the fluctuation of threshold is set to be±0.1 GeV(√s0 ).In the pure state calculations, a τ window of
XA (3798) state cannot be determined under 40%–10% and theΔs method, and we rearrange the limits of resonance and HDC to 35%–15% (one may naturally expect that the pure-state analysis requires such adjustments because of mixing). The statesXA (3798) andXB1 (3857) both have mass predictions close toX(3872) . However, the large mass prediction of theXB2 (5310) state is far beyond theD1 +D∗0 threshold, and does not match the observed1++ states.The mixing strength can then be estimated by computing the value of N via Eq. (14). Note that the coupling constants of the two mixed state correlators have the form
ΠMXkμνσ(q2)=i2∫d4xeiq⋅x⟨0|T(jXAμν(x)jXBk+σ(0)+jXBkσ(x)jXA+μν(0))|0⟩∼(λXAMHϵμναβεXAαqβ)(λ∗XBkε∗XBkσ)+(λ∗XAMHϵμναβε∗XAαqβ)(λXBkεXBkσ)+...=λXAλ∗XBk+λ∗XAλXBkMHqαϵμνσα+...,
(22) where
k=1,2 ;εXA/XBkα/σ is a polarization vector;MH represents the ground state mass ofXA ; and dots represent excited contributions to the spectral density and polynomial subtraction terms. In the definition of the mixing strength Eq. (14), we have omitted the Lorentz structures of corresponding currents. The dimension of the decay constant depends on the Lorentz structure we extract in the diagonal correlator. If the two currents have different Lorentz structures, we need to compensate the mass dimension of the decay constants, which are obtained from previous works, to make the mixing strength Eq. (14) dimensionless. The normal method is to make the Lorentz structures massless by multipling a factorMnH with a suitable n. Hence, we defineλXA/MH as the new coupling constant of theXA state. The mixing strength can be written asNMX1=0.168GeV9×MH(3.798GeV)√1.49GeV5×√2.24GeV5=0.349,˜NMX1=sin2(arcsin(0.349×2)2)=14%,NMX2=0.760GeV9×MH(3.798GeV)√1.49GeV5×√69.0GeV5=0.285,˜NMX2=sin2(arcsin(0.285×2)2)=9.0%,NMC=0.0282GeV10√0.0229GeV5×√2.24GeV5=0.125,˜NMC=sin2(arcsin(0.125×2)2)=1.6%,
(23) The state
MX1 (3987) is a mixture ofXA (3798) andXB1 (3857), which have similar mass predictions close toX(3872) ; unsurprisingly, this state also has the same mass prediction. Due toX(3872) , observed decays toπ+π−J/ψ(1S) ,ωJ/ψ(1S) andˉD∗0D0 ,MX1 (3987) is a good candidate to describeX(3872) [2]. We can estimate the proportions of each constituent and decay width of the corresponding decay modes by using the parameterNMX1 . Experimental results ofX(3872) decay widthΓ1 of theˉQq+Qˉq like decay mode is >30%, while the decay widthΓ2 of theˉQQ+ˉqq like decay mode is >5%. By comparison, the parameterNMX1 shows that the proportions of theˉQqQˉq andˉQQˉqq parts ofMX1 (3987) are 86% and 14%, respectively. Considering the similar Lorentz-invariant phase-space of these two kinds of decay modes, we can roughly equateΓ1/Γ2 to the ratio of these two parts, 86%/14% ~ 6, which is consistent with experimental results. It should be noted that our method can not determine definitely which constitute dominates the mixing state. We tend to the one whose pure mass is closer to the mixing state.When we consider the two-quark state
jXCμ , the corresponding mixing angle is arcsin (0.25)/2 = 7°, which is consistent with the result in Ref. [28], and the dominant part ofMC isjXCμ . However, we found that this result strongly depends on the normalization ofjXCμ . Hence, a proper normalized current is essential in calculations.For the state
MX2 (4945), its mass prediction is larger than those for all observed1++ states. However, our calculation suggests that it is relatively strongly mixed. The dominant part ofMX2 (4945) is more likely to beXB2 (5310) by comparing mass predictions.In Ref. [19], the authors calculated the state
XB1 with a similar method and obtained the following results: the massmXB1=3.89+0.09−0.09 GeV and the decay constantλXB1=2.96+1.09−0.79×10−4 GeV10 with√s0=4.41 GeV, which is consistent with our results. -
We start from two forms of currents as follows:
jYA1/YAs1μ(x)=ˉc(x)c(x)ˉq(x)γμq(x),jYA2/YAs2μ(x)=ˉc(x)γμc(x)ˉq(x)q(x),jYB1/YBs1μ(x)=i√2[ˉc(x)γμq(x)ˉq(x)c(x)+ˉq(x)γμc(x)ˉc(x)q(x)],jYB2/YBs2μ(x)=i√2[ˉc(x)γμγ5q(x)ˉq(x)γ5c(x)−ˉq(x)γμγ5c(x)ˉc(x)γ5q(x)],
(24) where Y denotes the
1−− state; the subscript A of Y represents theˉQQˉqq scenario, while B represents theˉQqQˉq scenario. The additional subscript s represents the s quark case, and one can straightforwardly replace the q with the s quark whenjYAs1μ ,jYAs2μ ,jYBs1μ , andjYBs2μ are involved. TheY(4230) was observed to decay toχc0ω , whileY(4660) was observed to have both theψ(2S)π+π− andD+sDs1(2536)− decay modes [2]. Hence, we especially focus on the currentsjYA1μ andjYAs2μ , which are consistent with the respective decay modes, to describeY(4230) andY(4660) , respectively, and discuss the corresponding mixed states in both theu,d and s quarks for simplicity.The two-point correlator functions of the pure states have the Lorentz structures
ΠYAk/YAskμν(q2)=ΠYAk/YAsk(1)(q2)(−gμν+qμqνq2)+ΠYAk/YAsk(0)(q2)(qμqνq2),ΠYBk/YBskμν(q2)=ΠYBk/YBsk(1)(q2)(−gμν+qμqνq2)+ΠYBk/YBsk(0)(q2)(qμqνq2),
(25) where
k=1,2 ;ΠYAk/YAsk(1) andΠYBk/YBsk(1) describe pure state contributions with quantum numbers1−− ; andΠYAk/YAsk(0) andΠYBk/YBsk(0) describe the pure state contribution with quantum numbers0+− .To study the mixed state, the off-diagonal mixing two-point correlation functions described in Section II are
ΠMY1μν(q2)=i2∫d4xeiq⋅x⟨0|T(jYA1μ(x)jYB1+ν(0)+jYB1ν(x)jYA1+μ(0))|0⟩,ΠMY2μν(q2)=i2∫d4xeiq⋅x⟨0|T(jYA1μ(x)jYB2+ν(0)+jYB2ν(x)jYA1+μ(0))|0⟩,ΠMYs1μν(q2)=i2∫d4xeiq⋅x⟨0|T(jYAs2μ(x)jYBs1+ν(0)+jYBs1ν(x)jYAs2+μ(0))|0⟩,ΠMYs2μν(q2)=i2∫d4xeiq⋅x⟨0|T(jYAs2μ(x)jYBs2+ν(0)+jYBs2ν(x)jYAs2+μ(0))|0⟩,
(26) where
MYk andMYsk , withk=1,2 , both represent mixed states coupled to their respective currents. These mixed correlators have the same Lorentz structures as the pure state cases,ΠMYk/MYskμν(q2)=ΠMYk/MYsk(1)(q2)(−gμν+qμqνq2)+ΠMYk/MYsk(0)(q2)(qμqνq2),
(27) where
k=1,2 , andΠMYk/MYsk(1) andΠMYk/MYsk(0) describe the mixed states with quantum numbers1−− and0+− , respectively.We follow same method mentioned in the
1++ channel. The mesonic structures, mass and coupling constant predictions, and the related QCDSR parameters are presented in Table 4.State Current structure Mass/GeV λ/ 10−4 GeV10√s0 /GeVτ window/GeV−2 YA1 χc0ω 4.207+0.08−0.09 1.64+0.63−0.49 4.8 0.27 – 0.28 YAs2 J/ψf(980) 4.621+0.05−0.06 21.3+5.2−4.7 5.1 0.25 – 0.32 YB1 D∗0ˉD∗ 4.922+0.04−0.04 34.5+6.7−6.0 5.4 0.21 – 0.34 YB2 D1ˉD 4.385+0.06−0.06 7.36+1.95−1.67 4.9 0.27 – 0.35 YBs1 D∗s0ˉD∗s 4.952+0.03−0.04 37.9+6.5−6.3 5.45 0.21 – 0.36 YBs2 Ds1ˉDs 4.494+0.08−0.05 9.60+3.7−2.1 5.0 0.26 – 0.39 MY1 χc0ω -D∗0ˉD∗ 4.770+0.07−0.06 1.16+0.37−0.28 5.3 0.24 – 0.25 MY2 χc0ω -D1ˉD 4.266+0.08−0.08 0.373+0.117−0.093 4.95 0.26 – 0.27 MYs1 J/ψf(980) -D∗s0ˉD∗s 4.610+0.05−0.06 2.56+0.67−0.56 5.1 0.24 – 0.33 MYs2 J/ψf(980) -Ds1ˉDs 4.450+0.05−0.06 1.64+0.43−0.22 4.95 0.26 – 0.33 Table 4. Summary of results for
1−− states.In the Y family of states,
Y(4160) ,Y(4260) ,Y(4415) ,Y(4660) are reported to have decay modes including an s quark in the final states, andY(4230) ,Y(4360) , andY(4390) have not been observed to have decay modes that include an s quark in the final states. Furthermore,Y(4260) only decays to the K meson whileY(4415) andY(4660) only decay to theDs meson when an s quark is directly involved in the final states. TheY(4160) has both decay modes, including the K andDs mesons in the final states. In contrast, all Y states have bothˉQQ+ˉqq like decay modes andˉQq+Qˉq like decay modes, with the exception ofY(4390) . The decay modeY(4390) toπ+π−hc was observed, but the other decay modes ofY(4390) have not yet been seen. We cannot exclude an s quark inY(4230) ,Y(4360) ,Y(4390) because the K meson may decay to the π meson and disappear in the final states [2]. Hence, we suggest thatY(4230) has candidatesYA1 (4207),MY2 (4266), andY(4360) ;Y(4390) has a candidateYB2 (4385);Y(4415) has candidatesYBs2 (4494) andMYs2 (4450); andY(4660) has candidatesYAs2 (4621) andMYs1 (4610). Although the remaining states are not compatible with known1−− states, they still possibly mix with other states, and their contributions can be estimated.For the
1−− states, the mixing strengths are given by the data in Table 4:NMY1=1.16GeV10√1.64GeV5×√34.5GeV5=0.15,NMY2=0.373GeV10√1.64GeV5×√7.36GeV5=0.11,NMYs1=2.56GeV10√21.3GeV5×√37.9GeV5=0.09,NMYs2=1.64GeV10√21.3GeV5×√9.60GeV5=0.11.
(28) All mixed states have a much weaker mixing strength compared with the
1++ mixed states. We suggest that1−− states are preferred to be pure and weakly mixed with other states. This becomes more clear when we convert N to˜N ,˜NMY1=sin2(arcsin(0.15×2)2)=2.3%,˜NMY2=sin2(arcsin(0.11×2)2)=1.2%,˜NMYs1=sin2(arcsin(0.09×2)2)=0.82%,˜NMYs2=sin2(arcsin(0.11×2)2)=1.2%,
(29) where the values of
˜N suggest that the assumed mixed states with quantum numbers1−− are actually very pure. As mentioned above,MY1 (4770), which contains no s quark, is close toY(4660) and cannot be compatible with known1−− states.MY2 (4266), which is a possible candidate forY(4230) , is a mixture ofYA1 (4207) andYB2 (4385). By comparing the two mass predictions,MY2 (4266) is closer toYA1 (4207) rather thanYB2 (4385), and it is possibly dominated byˉQQˉqq component. For the same reasons,MYs1(4610) is possibly dominated by aˉQQˉqq component, whileMYs2 (4450) is possibly dominated byˉQqQˉq . Hence, we suggest thatY(4230) andY(4660) prefer aˉQQˉqq state, andY(4415) prefers aˉQqQˉq state.In Ref. [30], authors have calculated the states
YB1 andYB2 with a similar method and obtained the following results: massmYB1=4.78+0.07−0.07 with√s0=5.3 GeV-2, and massmYB2=4.36+0.08−0.08 with√s0=4.9 GeV-2; these numbers are consistent with our results. The small difference in the mass ofYB1 is caused by the different values of√s0 . In addition, in Ref. [30], the authors have discussed different results of the similar states ofYB1 andYB2 in previous papers. For instance, the authors in Ref. [42] did not distinguish between the charge conjugations and obtained mass of aYB1 like statemD∗0¯D∗=4.26 GeV. Our results are more supportive of the results in Ref. [30].In Ref. [8], the authors have calculated the states
YAs2 with a similar method and obtained the following result: massmYAs2=4.67+0.09−0.09 with√s0=5.1 GeV-2; this is consistent with our result. -
We start from two forms of currents as follows:
jPAμν(x)=jXAμν(x),jPAsμν(x)=i√2[ˉc(x)γμc(x)ˉs(x)γνs(x)−ˉc(x)γνc(x)ˉs(x)γμs(x)],jPB1/PBs1μ(x)=i√2[ˉc(x)γμq(x)ˉq(x)c(x)−ˉq(x)γμc(x)ˉc(x)q(x)],jPB2/PBs2μ(x)=i√2[ˉc(x)γμγ5q(x)ˉq(x)γ5c(x)+ˉq(x)γμγ5c(x)ˉc(x)γ5q(x)],
(30) where P denotes the
1−+ state; the subscript A of P represents theˉQQˉqq scenario, while B represents theˉQqQˉq scenario. The additional subscript s represents the s quark case, and one can straightforwardly replace q with s whenjPBs1μ andjPBs2μ are involved. The structures of these currents are similar to the1++ and1−− cases, and it is interesting to compare the mass predictions of these currents to the1++ and1−− states.To study the pure
ˉQQˉqq andˉQqQˉq states, the two-point correlation functions have the respective Lorentz structures,ΠPA/PAsμνρσ(q2)=ΠPA/PAsa1q2(q2gμρgνσ−q2gμσgνρ−qμqρgνσ+qμqσgνρ−qνqσgμρ+qνqρgμσ)+ΠPA/PAsb1q2(−qμqρgνσ+qμqσgνρ−qνqσgμρ+qνqρgμσ),ΠPBk/PBskμν(q2)=ΠPBk/PBsk(1)(q2)(−gμν+qμqνq2)+ΠPBk/PBsk(0)(q2)(qμqνq2),
(31) where
k=1,2 ;ΠPA/PAsa andΠPA/PAsb describe the pure state contributions with quantum numbers1++ and1−+ , respectively; andΠPBk/PBsk(1) andΠPBk/PBsk(0) describe1−+ and0++ , respectively. In mixing scenarios, we start from the off-diagonal mixed correlator described in the previous sections, i.e.,ΠMP1/MPs1μνσ(q2)=i2∫d4xeiq⋅x⟨0|T(jPA/PAsμν(x)jPB1/PBs1+σ(0)+jPB1/PBs1σ(x)jPA/PAs+μν(0))|0⟩,ΠMP2/MPs2μνσ(q2)=i2∫d4xeiq⋅x⟨0|T(jPA/PAsμν(x)jPB2/PBs2+σ(0)+jPB2/PBs2σ(x)jPA/PAs+μν(0))|0⟩,
(32) where
MPk/MPsk , withk=1,2 , are mixed states assumed to result from the corresponding currents. The correlators have the Lorentz structureΠMPk/MPskμνσ(q2)=ΠMPk/MPsk(q2)(−gμσqν+gνσqμ).
(33) We follow same method used in previous sections. The mesonic structures, mass and coupling constant predictions, and related QCDSR parameters are presented in Table 5.
State Current structure Mass/GeV λ/ 10−4 GeV10√s0 /GeVτ window/GeV−2 PA J/ψρ 4.658+0.05−0.06 13.1+3.2−3.1 5.15 0.24 – 0.29 PAs J/ψf(980) 4.694+0.05−0.05 14.0+3.5−2.9 5.2 0.24 – 0.32 PB1 D∗0ˉD∗ 4.927+0.05−0.04 34.1+7.7−5.7 5.4 0.21 – 0.30 PB2 D1ˉD 4.528+0.06−0.05 9.7+2.9−2.1 5.05 0.26 – 0.31 PBs1 D∗s0ˉD∗s 4.999+0.04−0.03 42.8+8.2−6.8 5.5 0.21 – 0.33 PBs2 Ds1ˉDs 4.642+0.05−0.05 13.0+3.4−2.8 5.15 0.25 – 0.34 MP1 J/ψρ -D∗0ˉD∗ 4.505+0.06−0.04 0.401+0.096−0.073 GeV-15.05 0.21 – 0.30 MP2 J/ψρ -D1ˉD 4.494+0.06−0.06 0.240+0.060−0.052 GeV-15.05 0.23 – 0.27 MPs1 J/ψf(980) -D∗s0ˉD∗s 4.544+0.05−0.05 0.405+0.085−0.077 GeV-15.1 0.21 – 0.33 MPs2 J/ψf(980) -Ds1ˉDs 4.536+0.06−0.05 0.269+0.067−0.055 GeV-15.1 0.22 – 0.31 Table 5. Summary of results for
1−+ molecular states.All pure states have mass predictions of over 4.5 GeV and cannot be compatible with those known states, which are probably
1−+ candidates [2].The mixing strength can then be estimated by computing the value of N. Note that the coupling constants of the mixed state correlator has the form
ΠMPkμνσ(q2)=i2∫d4xeiq⋅x⟨0|T(jPAμν(x)jPBk+σ(0)+jPBkσ(x)jPA+μν(0))|0⟩∼λPAMH(εPAμqν−εPAνqμ)(λ∗PBkε∗PBkσ)+λ∗PAMH(ε∗PAμqν−ε∗PAνqμ)(λPBkεPBkσ)+...=λPAλ∗PBk+λ∗PAλPBkMH(−gμσqν+gνσqμ)+...,
(34) where
k=1,2 ;εPA andεPBk are polarization vectors; andMH represents the ground state mass ofPA . Analogous to the1++ channel, the values of N obtained from Table 5 can be written asNMP1=0.401GeV9×MH(4.658GeV)√13.1GeV5×√34.1GeV5=0.09,NMP2=0.240GeV9×MH(4.658GeV)√13.1GeV5×√9.7GeV5=0.10,
NMPs1=0.405GeV9×MH(4.694GeV)√14.0GeV5×√42.8GeV5=0.08,NMPs2=0.269GeV9×MH(4.694GeV)√14.0GeV5×√13.0GeV5=0.09,
(35) where
MH represents the correspondingˉQQˉqq ground state mass. Like the1−− channel, all the mixed states that have quantum numbers1−+ are weakly mixed with corresponding currents; this becomes clearer when we convert N to˜N ,˜NMP1=sin2(arcsin(0.09×2)2)=0.82%,˜NMP2=sin2(arcsin(0.10×2)2)=1.0%,˜NMPs1=sin2(arcsin(0.08×2)2)=0.64%,˜NMPs2=sin2(arcsin(0.09×2)2)=0.82%.
(36) For the same reasons mentioned in the
1−− channel,MP1 (4505) andMPs1 (4494) are dominated by theˉQQˉqq components;MP2 (4544) andMPs2 (4536) are more likely dominated by theˉQqQˉq components.In Ref. [30], authors have calculated the states
PB1 andPB2 with a similar method and obtained the following results: massmPB1=4.73+0.07−0.07 with√s0=5.2 GeV-2, and massmPB2=4.60+0.08−0.08 with√s0=5.1 GeV-2; these numbers are consistent with our results. The small difference in the mass ofPB1 is caused by the different values of√s0 . Moreover, the authors in Ref. [43] obtained the mass of statePB1 ,mPB1=4.19 GeV. Our results are more supportive of the results in Ref. [30]. -
We can convert the
ˉQQˉqq andˉQqQˉq states to each other through the Fierz transformation. Generally,(ˉQΓ1Q)(ˉqΓ2q)=∑ijkCi(ˉQΓjq)(ˉqΓkQ)+∑lmnCl(ˉQΓmλaq)(ˉqΓnλaQ),(ˉQΓ1q)(ˉqΓ2Q)=∑ijkCi(ˉQΓjQ)(ˉqΓkq)+∑lmnCl(ˉQΓmλaQ)(ˉqΓnλaq),
(37) where
Γi are gamma matrices,Ci are the parameters corresponding to the related currents, andλa are Gell-Mann matrices. That is,ˉQQˉqq currents can be decomposed into a series ofˉQqQˉq currents and a series ofˉQqQˉq color-octet currents, and vice versa. In this study, we have computed two-point correlation functions ofˉQQˉqq andˉQqQˉq currents. One can convert one current to a series of other kinds of currents and make calculations analogous to a series of calculations of pure currents. For instance,jYA1μ=(ˉcc)(ˉqγμq)=−i2√2i√2[(ˉcγμq)(ˉqc)+(ˉqγμc)(ˉcq)]+−i2√2i√2[(ˉcγμγ5q)(ˉqγ5c)−(ˉqγμγ5c)(ˉcγ5q)]+...=−i2√2jYB1μ+−i2√2jYB2μ+...,
(38) where
jYA1μ andjYB1/B2μ are defined in Section IV. When we compute two-point correlation functions ofjYA1μ andjYB1/B2μ , it seems that the result may highlight statesYB1 /YB2 , and the parameters of the current decomposition are likely to be directly related to mixing strength. However, our calculations show different results. Although the contributions in perturbative terms from different currents (e.g.,YB1 andYB2 ) will be suppressed, QCDSR calculations are sensitive to the changes of borel window and thresholds0 , which depend on the contributions of non-perturbative terms. Moreover, the mixing strength is related to both decay constants and parameters of the corresponding currents from the Fierz transformation, and the decay constants are also sensitive to the Borel window, which again depends on non-perturbative terms. To clarify this, we have computed another twoˉQQˉqq andˉQqQˉq currents and their mixed state,jZAμ(x)=ˉc(x)γμc(x)ˉq(x)γ5q(x)jZBμν(x)=i√2[ˉc(x)γμq(x)ˉq(x)γνc(x)−ˉq(x)γμc(x)ˉc(x)γνq(x)],
(39) where Z denotes the
1+− state, and the subscript A of Z represents theˉQQˉqq scenario, while B represents theˉQqQˉq scenario. The mixed state is described byΠMZμνσ(q2)=i2∫d4xeiq⋅x⟨0|T(jZAσ(x)jZB+μν(0)+jZBμν(x)jPA+σ(0))|0⟩,
(40) where
MZ is assumed to be mixed from corresponding currents. The hadronic structures along with results of mass, coupling constant, and mixing strength predictions are presented in Table 6.State Current structure Mass/GeV λ/ 10−4 GeV10√s0 /GeVτ window/GeV−2 ZA J/ψη 3.578+0.08−0.08 1.04+0.36−0.27 4.2 0.33 – 0.34 ZB D∗ˉD∗ 4.018+0.06−0.06 2.90+0.88−0.67 4.55 0.29 – 0.36 MZ J/ψη -D∗ˉD∗ 3.563+0.07−0.06 0.054+0.017−0.012 GeV−14.1 0.31 – 0.35 Table 6. Summary of results for
1+− states.Compared to the
1++ currentsjXA/B1μ and their mixed two-point correlatorΠMX1μνσ(q2) , which are given in Eq. (17) and Eq. (19),jZA/Bμ andΠMZμνσ(q2) have similar structures. According to our previous calculations in Section II,MX1 is relatively strongly mixed with different components, andMZ is supposed to have similar properties. However, the resulting mixing strength ofMZ isNMZ=0.054GeV9×MH(4.018GeV)√1.04GeV5×√2.90GeV5=0.125,˜NMZ=sin2(arcsin(0.125×2)2)=1.6%.
(41) Compared to
MX1 (NMX1 = 0.349,˜NMX1 = 14%), the mass predictions of two parts of the mixed stateMZ differ, and although the contributions of the perturbative terms in two-point correlator functions are similar, the mixing strength of the two states are quite different. Hence, we suggest that the mixing strength is considerably sensitive to the Borel window, thresholds0 , mass prediction, and decay constant, which are all influenced by non-perturbative terms in QCDSR calculations. -
In this study, we used QCD sum-rules to calculate the mass spectrum of
ˉQQˉqq andˉQqQˉq states. Such states strongly couple toˉQQˉqq orˉQqQˉq currents. Therefore, state components ofˉQQˉqq andˉQqQˉq can be mixed with each other. Such mixing can be studied via the mixed correlators ofˉQQˉqq andˉQqQˉq currents. Our studies focus on the mixing strength, which may determine whether the mixing picture accommodates candidates that have more than single dominant decay modes.We list all the mixed state results in Table 7. The uncertainties of the masses are less than 5%, and the uncertainties of the coupling constants are approximately 25%, which are induced by the uncertainties of the input parameters and threshold
s0 . The relations√s0=MH+Δs and 40%–10% are required to determine the window of τ. These two conditions are not always satisfied well. In some cases, the windows of τ are very narrow. If higher dimension condensates are considered, we may reconsider the constraint of 40%–10%, and the situation may change.Mixed state Mass/GeV N ˜N Dominant part Possible Candidate MX1 3.987+0.06−0.06 0.349 14% ˉQqQˉq X(3872) MX2 4.945+0.08−0.06 0.285 9.0% ˉQqQˉq − MC 3.818+0.03−0.02 0.125 1.6% ˉQQ X(3872) MY1 4.770+0.07−0.06 0.15 2.3% ˉQqQˉq − MY2 4.266+0.08−0.08 0.11 1.2% ˉQQˉqq Y(4230) MYs1 4.610+0.05−0.06 0.06 < 1% ˉQQˉqq Y(4660) MYs2 4.450+0.05−0.06 0.11 1.2% ˉQqQˉq Y(4415) MP1 4.505+0.06−0.04 0.09 < 1% ˉQQˉqq − MP2 4.494+0.06−0.06 0.10 1.0% ˉQqQˉq − MPs1 4.544+0.05−0.05 0.08 <1% ˉQQˉqq − MPs2 4.536+0.06−0.05 0.09 <1% ˉQqQˉq − Table 7. Summary of mixed state results.
For the
1++ channel, we find that the two statesMX1 (3987) andMX2 (4945) are relatively strongly mixed with theˉQQˉqq andˉQqQˉq components. Furthermore, we estimate the ratio of decay width of two kinds of decay modes ofMX1 (3987); this ratio is roughly consistent with the experimental results forX(3872) . When we consider the mixing state combined withˉQQ andˉQqQˉq , we revisit the result in Ref. [28] with the new technique in Ref. [15]. The result argues thatˉQQ is the dominant part ofX(3872) , which can explain the latest observation toX(3872) of LHCb [7]. Our calculations just support that these two components can relatively strongly mix with each other in quantum numbers1++ .In other quantum number channels, states are found to be weakly mixed. However, the calculations of these states is still meaningful to help us establish the physical structure of a corresponding state. For instance, pure
ˉQQˉqq YA1 (4207) andYAs2 (4610) configurations are good candidates forY(4230) andY(4660) , respectively. However, by checking the assumed mixed states that mix withˉQQˉqq andˉQqQˉq molecular states, we find that these candidates have small components ofˉQqQˉq ; this is inconsistent with the fact thatY(4230) andY(4660) have more abundant decay modes. Thus, we can establish the dominant part ofY(4415) . Our result suggests thatY(4415) is dominated byˉQqQˉq and agrees with the absence of the K meson in observed decay final states. However,Y(4415) still has a small component ofˉQQˉqq . These states may therefore have a more complicated construction; for instance,ˉqq could be a color-octet state. Other models, such as the tetraquark model, could be valuable. By using the Fierz transformations, tetraquark currents can be decomposed into various molecular currents and color-octet currents, to show more mixed effects of different possible states [44]. Since the mixing effects are normally small, the studies via tetraquark currents cannot distinguish the details of mixing between the different currents and only give the average of those currents. As such, the tetraquark model is not a self-verifying because it cannot show which parts (via the Fierz transformations) interact with each other strongly and which ones do not. It may also address challenges in the quantitative descriptions of XYZ states.The calculations based on pure molecular currents have been criticized because there is a large background of the two free mesons spectrum. If the states indeed have an absolutely dominate decay mode [45], there is no problem (actually, the mass of the molecule state is close to that of two free mesons). Otherwise, the mixing pattern must be taken into account. The mixing of the typical molecular currents
ˉQQˉqq andˉQqQˉq are suppressed (perturbatively) by the small coefficients of Fierz transformations, so the background of the two free mesons spectrum is also suppressed. Non-perturbative corrections play more important roles in the mixing correlator, which can distinguish the real four-quark resonance from the two free mesons. It should be the essential feature of the mixing pattern. Our calculations show that the mixing pattern is consistent with some of the XYZ states but fails for many others. Since the mixing correlator is normalized by the two diagonal correlators, which may be affected by a large background of two free meson spectrum, the real mixture may be larger than our estimate. How to remove the background of the two free meson spectrum is still a major problem. -
Here, we show the τ dependence of
M2H defined in Eq. (11) for all mixed states.
PDG name | Possible structure | Ground state | Possible 1st excited state | ![]() |
D | ![]() | ![]() | ![]() | ~685 |
![]() | ![]() | ![]() | − | − |
![]() | ![]() | ![]() | ![]() | ~300 |
![]() | ![]() | ![]() | ![]() | ~633 |