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Mixing of X and Y states from QCD sum rules analysis

  • We study ˉQQˉqq and ˉQqQˉq states as mixed states in QCD sum rules. By calculating the two-point correlation functions of pure states of their corresponding currents, we review the mass and coupling constant predictions of JPC=1++, 1, and 1+ states. By calculating the two-point mixed correlation functions of ˉQQˉqq and ˉQqQˉq currents, we estimate the mass and coupling constants of the corresponding "physical state" that couples to both ˉQQˉqq and ˉQqQˉq currents. Our results suggest that for 1++ states, the ˉQQˉqq and ˉQqQˉq components are more likely to mix, while for 1 and 1+ states, there is less mixing between ˉQQˉqq and ˉQqQˉq. Our results suggest the Y series of states have more complicated components.
  • 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 [46]. 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 of X(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 simple ˉQqQˉq and ˉQQˉqq combinations to study XYZ states (ˉQ represents a heavy c or b quark, while q represents a light u,d or s quark). These two forms have been extensively studied previously [8, 9]. However, ˉQqQˉq and ˉQQˉqq states are difficult to distinguish straightforwardly from the decay modes of XYZ states because XYZ states are usually observed to have both ˉQQ+ˉqq like decay modes and ˉQq+Qˉq like decay modes. Many scenarios were studied to qualitatively distinguish ˉQqQˉq and ˉQQˉqq states [1014]. 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 [1518]; we use the same technique to study the mixing of ˉQqQˉq and ˉQQˉqq 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 be ˉQqQˉq or ˉQQˉqq molecular states in different studies [1925]. 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 of ˉQqQˉq and ˉQQˉqq currents to investigate the corresponding physical states. The calculations will show us whether the physical states prefer to be ˉQqQˉq or ˉQQˉqq 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 a DD molecular state [1214]. However, although the pure DD molecular state was predicted to have a mass close to X(3872), it had too large a decay width to agree with the experimental results [28, 29]. Moreover, the J/ψρ and J/ψω states have a similar mass, since the sum of the masses of their two constituent parts are close to X(3872). Hence, the mixing of these two molecular states is naturally possible. Furthermore, in a recent observation of X(3872) by LHCb [7], the compact component was found to be required. Hence, we will consider another state ˉcc in the mixing; this has been studied in [28].

    For the 1 channel, many 1 states are found in the range of 4200–4700 MeV, permitting an abundance of possible pure or mixed molecular states. Some 1 states have very similar masses like Y(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 the ˉQqQˉq and ˉQQˉqq scenarios are possible. As outlined below, we calculate the mass spectrum of these states and estimate their mixing strength in both the u,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, and 1+ 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)=id4xeiqx0|T{(ja(x)+cjb(x))(j+a(0)+cj+b(0))}|0=Πa(q2)+2cΠab(q2)+c2Πb(q2),

    (1)

    where ja and jb 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), and

    Πa(q2)=id4xeiqx0|T(ja(x)j+a(0))|0,

    Πb(q2)=id4xeiqx0|T(jb(x)j+b(0))|0,Πab(q2)=i2d4xeiqx0|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 between ja and jb, 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 [1618].

    Πab usually can be decomposed into a different Lorentz structure

    Πab(q2)=nΠn(q2)An,

    (3)

    where n=1,2,3..., Πn(q2) is the mixing state correlation function with specific quantum numbers, and An is the corresponding Lorentz structure. The forms of An are related to ja and jb, and we will define them in the sections below. For simplicity, we assume a specific ΠH(q2) represents a mixed-state correlation function and is one of the possible Πn(q2). We also assume that ΠH(q2) obeys the dispersion relation [32]

    ΠH(q2)=smindsρH(s)sq2iϵ+,

    (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 render ΠH(q2) finite. The spectral density ρH(s) can be calculated using the operator product expansion (OPE). In this paper, we calculate the spectral density ρH(s) up to dimension-six operators,

    ρ(OPE)H(s)=ρ(pert)H(s)+ρˉqqH(s)+ρG2H(s)+ρˉqGqH(s)+ρˉqq2H(s)+,

    (5)

    then

    Π(OPE)H(q2)=smindsρ(OPE)H(s)sq2iϵ+.

    (6)

    On the phenomenological side, by using the narrow resonance spectral density model,

    Π(phen)H(q2)=λaλb+λbλa21M2Hq2+s0dsρ(cont)H(s)sq2iϵ,

    (7)

    where λa and λb 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, and s0 is the continuum threshold. By using the QCDSR continuum spectral density assumptions

    ρ(cont)H(s)=ρ(OPE)H(s)Θ(ss0),

    (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)sq2iϵ+=λaλb+λbλa21M2Hq2.

    (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)esτ=λaλb+λaλb2eM2Hτ,

    (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 obtain

    M2H=sminds sρ(OPE)H(s)esτsminds ρ(OPE)H(s)esτ.

    (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)esτsminds ρ(OPE)H(s)esτ>40%,

    (12)

    while HDC (usually ˉqq2 in molecular systems) obeys the relation

    |sminds ρˉqq2H(s)esτ||sminds ρ(OPE)H(s)esτ|<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, with Δs0.5 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 parameter s0 separates the ground state and other excited states' contributions to spectral density. Hence, one can set s0 less than the first excitation threshold in the case of involving excited state contributions in the spectral density, and Δs 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 establish s0 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 around 0.5+0.10.1 GeV, and the fluctuations are all acceptable in the QCDSR approach.

    Table 1

    Table 1.  Charmed meson (c=±1) states, where q represents the u,d quark. The symbol indicates particles that have confirmed quantum numbers.
    PDG name Possible structure Ground state Possible 1st excited state Δs/MeV
    D ˉcγ5q D(1865) D(2550) ~685
    D1 ˉcγμγ5q D1(2420)
    D0 ˉcq D0(2300) DJ(2600) ~300
    D ˉcγμq D(2007) D(2640) ~633
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    Table 2

    Table 2.  Charmonium (possibly non-ˉcc states). The symbol indicates particles that have confirmed quantum numbers.
    Possible structure Ground state/MeV Possible 1st excited state/MeV Δs/MeV
    ˉcγ5c ηc(1S)/2984 ηc(2S)/3637 ~653
    ˉcγμγ5c χc1(1P)/3510 χc1(3872) ~362
    ˉcc χc0(1P)/3415 χc0(3860) ~445
    ˉcγμc J/ψ(1S)/3097 ψ(2S)/3686 ~589
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    In order to estimate the mixing strength of the physical state strongly coupled to both of the two different currents, we define

    N|λaλb+λaλb|2|λaλb|,

    (14)

    where λa and λb are coupling constants of the relevant current with a pure state (i.e., the coupling that emerges in the diagonal correlation functions Πa, Πb). Eq. (14) is analogous to the mixing parameter defined in Ref. [38]. By using appropriate factors of mass in the definitions of λa and λb, 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 that Ncosθsinθ, and N(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=m20ˉqq, m20=0.8 GeV2, αsG2=0.07±0.02 GeV4, andˉss=(0.8±0.2)ˉqq [40, 41]. In addition, the quark masses mc=1.27 GeV, mq=12(mu+md)=0.004 GeV, and ms=0.096 GeV, at the energy scale μ=2 GeV [2], are used.

    We start from the following three forms of currents:

    jXAμν(x)=i2[ˉc(x)γμc(x)ˉq(x)γνq(x)ˉc(x)γνc(x)ˉq(x)γμq(x)],jXB1μ(x)=i2[ˉc(x)γμq(x)ˉq(x)γ5c(x)ˉq(x)γμc(x)ˉc(x)γ5q(x)],jXB2μ(x)=i2[ˉc(x)γμγ5q(x)ˉq(x)c(x)+ˉq(x)γμγ5c(x)ˉc(x)q(x)],jXCμ(x)=162ˉqqˉc(x)γμγ5c(x),

    (17)

    where X denotes the 1++ state, the subscript A of X denotes the ˉQQˉqq scenario, B denotes the ˉQqQˉq scenario, C denotes the ˉQQ 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 current jXAμν can be decomposed into J/ψ(1S)(3097) and ρ(770) currents; jXB1μ(x) can be decomposed into D(2007) and D(1865) currents. The sums of the masses of the two constituent mesons are both close to X(3872). Hence, we choose these two currents to study X(3872). The current jXB2μ has the same quantum numbers, and it cannot be excluded in 1++ mixing state structures. Besides, jXCμ(x) is normalized according to Ref. [28]. Since the mixing between jXCμ(x) and ˉQQˉqq is suppressed (ˉqq in ˉQQˉqq becomes a bubble and vanishes), we only consider the mixing between jXCμ(x) and jXB1μ(x) or jXB2μ(x).

    Table 3

    Table 3.  Summary of results for 1++ states. λ=λaλb+λaλb2 when mixed cases are involved, the same below.
    State Current structure Mass/GeV λ/104 GeV10 s0/GeV τ window/GeV−2
    XA J/ψρ 3.798+0.090.09 1.49+0.510.47 4.4 0.30 – 0.31
    XB1 DˉD 3.857+0.060.06 2.24+0.650.53 4.4 0.31 – 0.39
    XB2 D1ˉD0 5.310+0.040.04 69.0+15.014.0 5.8 0.20 – 0.29
    XC χc1 3.511+0.020.03 0.0229+0.00170.0018 4.5 0.29 – 0.31
    MX1 J/ψρ - DˉD 3.987+0.060.06 0.168+0.0490.042 GeV-1 4.4 0.30 – 0.32
    MX2 J/ψρD1ˉD0 4.945+0.080.06 0.760+0.290.20 GeV-1 5.45 0.22 – 0.24
    MC χc1 - DˉD 3.818+0.030.02 0.0282+0.00270.0024 4.5 0.28 – 0.30
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    To study the pure ˉQQˉqq, ˉQqQˉq, and ˉQQ states, the respective two-point correlation functions can be decomposed into different Lorentz structures.

    ΠXAμνρσ(q2)=ΠXAa(q2)1q2(q2gμρgνσq2gμσgνρqμqρgνσ+qμqσgνρqνqσgμρ+qνqρgμσ)+ΠXAb(q2)1q2(qμqρgνσ+qμqσgνρqνqσgμρ+qνqρgμσ),ΠXBk/Cμν(q2)=ΠXBk/C(1)(q2)(gμν+qμqνq2)+ΠXBk/C(0)(q2)(qμqνq2),

    (18)

    where k=1,2; ΠXAa and ΠXAb describe the pure molecular state contribution with respective quantum numbers 1++ and 1+; and ΠXBk/C(1) and ΠXBk/C(0) describe 1++ and 0+ state contributions, respectively. In the mixing scenario, we start from the off-diagonal mixed correlator described in the previous section, i.e.,

    ΠMX1μνσ(q2)=i2d4xeiqx0|T(jXAμν(x)jXB1+σ(0)+jXB1σ(x)jXA+μν(0))|0,ΠMX2μνσ(q2)=i2d4xeiqx0|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 structure

    ΠMXkμνσ(q2)=ΠMXk(q2)(qαϵαμνσ).

    (20)

    Furthermore, when we consider the two-quark states jXCμ(x), the mixed correlator and its Lorentz structure are

    ΠMCμν(q2)=i2d4xeiqx0|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) and ΠC(0) describe the 1++ and 0+ state contributions, respectively. Here, we just consider the state mixed with jXCμ and jXB1; this state is a candidate for X(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 specific s0, we use Eq. (11) for each state to plot the τ behavior of MH for the chosen s0 (see Appendix A for details). The mass prediction MH and s0 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 predicted MH and s0 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 threshold s0. 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 the Δs 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 states XA(3798) and XB1(3857) both have mass predictions close to X(3872). However, the large mass prediction of the XB2(5310) state is far beyond the D1+D0 threshold, and does not match the observed 1++ 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)=i2d4xeiqx0|T(jXAμν(x)jXBk+σ(0)+jXBkσ(x)jXA+μν(0))|0(λXAMHϵμναβεXAαqβ)(λXBkεXBkσ)+(λXAMHϵμναβεXAαqβ)(λXBkεXBkσ)+...=λXAλXBk+λXAλXBkMHqαϵμνσα+...,

    (22)

    where k=1,2;εXA/XBkα/σ is a polarization vector; MH represents the ground state mass of XA; 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 factor MnH with a suitable n. Hence, we define λXA/MH as the new coupling constant of the XA state. The mixing strength can be written as

    NMX1=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=0.0282GeV100.0229GeV5×2.24GeV5=0.125,˜NMC=sin2(arcsin(0.125×2)2)=1.6%,

    (23)

    The state MX1(3987) is a mixture of XA(3798) and XB1(3857), which have similar mass predictions close to X(3872); unsurprisingly, this state also has the same mass prediction. Due to X(3872), observed decays to π+πJ/ψ(1S), ωJ/ψ(1S) and ˉD0D0, MX1(3987) is a good candidate to describe X(3872) [2]. We can estimate the proportions of each constituent and decay width of the corresponding decay modes by using the parameter NMX1. Experimental results of X(3872) decay width Γ1 of the ˉQq+Qˉq like decay mode is >30%, while the decay width Γ2 of the ˉQQ+ˉqq like decay mode is >5%. By comparison, the parameter NMX1 shows that the proportions of the ˉQqQˉq and ˉQQˉqq parts of MX1(3987) are 86% and 14%, respectively. Considering the similar Lorentz-invariant phase-space of these two kinds of decay modes, we can roughly equate Γ1/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 of MC is jXCμ. However, we found that this result strongly depends on the normalization of jXCμ. Hence, a proper normalized current is essential in calculations.

    For the state MX2(4945), its mass prediction is larger than those for all observed 1++ states. However, our calculation suggests that it is relatively strongly mixed. The dominant part of MX2(4945) is more likely to be XB2(5310) by comparing mass predictions.

    In Ref. [19], the authors calculated the state XB1 with a similar method and obtained the following results: the mass mXB1=3.89+0.090.09 GeV and the decay constant λXB1=2.96+1.090.79×104 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)=i2[ˉc(x)γμq(x)ˉq(x)c(x)+ˉq(x)γμc(x)ˉc(x)q(x)],jYB2/YBs2μ(x)=i2[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 the ˉQQˉqq scenario, while B represents the ˉQqQˉq scenario. The additional subscript s represents the s quark case, and one can straightforwardly replace the q with the s quark when jYAs1μ, jYAs2μ, jYBs1μ, and jYBs2μ are involved. The Y(4230) was observed to decay to χc0ω, while Y(4660) was observed to have both the ψ(2S)π+π and D+sDs1(2536) decay modes [2]. Hence, we especially focus on the currents jYA1μ and jYAs2μ, which are consistent with the respective decay modes, to describe Y(4230) and Y(4660), respectively, and discuss the corresponding mixed states in both the u,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) and ΠYBk/YBsk(1) describe pure state contributions with quantum numbers 1; and ΠYAk/YAsk(0)and ΠYBk/YBsk(0) describe the pure state contribution with quantum numbers 0+.

    To study the mixed state, the off-diagonal mixing two-point correlation functions described in Section II are

    ΠMY1μν(q2)=i2d4xeiqx0|T(jYA1μ(x)jYB1+ν(0)+jYB1ν(x)jYA1+μ(0))|0,ΠMY2μν(q2)=i2d4xeiqx0|T(jYA1μ(x)jYB2+ν(0)+jYB2ν(x)jYA1+μ(0))|0,ΠMYs1μν(q2)=i2d4xeiqx0|T(jYAs2μ(x)jYBs1+ν(0)+jYBs1ν(x)jYAs2+μ(0))|0,ΠMYs2μν(q2)=i2d4xeiqx0|T(jYAs2μ(x)jYBs2+ν(0)+jYBs2ν(x)jYAs2+μ(0))|0,

    (26)

    where MYk and MYsk, with k=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, and ΠMYk/MYsk(1) and ΠMYk/MYsk(0) describe the mixed states with quantum numbers 1 and 0+, 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.

    Table 4

    Table 4.  Summary of results for 1 states.
    State Current structure Mass/GeV λ/104GeV10 s0/GeV τ window/GeV−2
    YA1 χc0ω 4.207+0.080.09 1.64+0.630.49 4.8 0.27 – 0.28
    YAs2 J/ψf(980) 4.621+0.050.06 21.3+5.24.7 5.1 0.25 – 0.32
    YB1 D0ˉD 4.922+0.040.04 34.5+6.76.0 5.4 0.21 – 0.34
    YB2 D1ˉD 4.385+0.060.06 7.36+1.951.67 4.9 0.27 – 0.35
    YBs1 Ds0ˉDs 4.952+0.030.04 37.9+6.56.3 5.45 0.21 – 0.36
    YBs2 Ds1ˉDs 4.494+0.080.05 9.60+3.72.1 5.0 0.26 – 0.39
    MY1 χc0ω - D0ˉD 4.770+0.070.06 1.16+0.370.28 5.3 0.24 – 0.25
    MY2 χc0ω - D1ˉD 4.266+0.080.08 0.373+0.1170.093 4.95 0.26 – 0.27
    MYs1 J/ψf(980) - Ds0ˉDs 4.610+0.050.06 2.56+0.670.56 5.1 0.24 – 0.33
    MYs2 J/ψf(980) - Ds1ˉDs 4.450+0.050.06 1.64+0.430.22 4.95 0.26 – 0.33
    DownLoad: CSV
    Show Table

    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, and Y(4230), Y(4360), and Y(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 while Y(4415) and Y(4660) only decay to the Ds meson when an s quark is directly involved in the final states. The Y(4160) has both decay modes, including the K and Ds mesons in the final states. In contrast, all Y states have both ˉQQ+ˉqq like decay modes and ˉQq+Qˉq like decay modes, with the exception of Y(4390). The decay mode Y(4390) to π+πhc was observed, but the other decay modes of Y(4390) have not yet been seen. We cannot exclude an s quark in Y(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 that Y(4230) has candidates YA1(4207), MY2(4266), and Y(4360); Y(4390) has a candidate YB2(4385); Y(4415) has candidates YBs2(4494) and MYs2(4450); and Y(4660) has candidates YAs2(4621) and MYs1(4610). Although the remaining states are not compatible with known 1 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=1.16GeV101.64GeV5×34.5GeV5=0.15,NMY2=0.373GeV101.64GeV5×7.36GeV5=0.11,NMYs1=2.56GeV1021.3GeV5×37.9GeV5=0.09,NMYs2=1.64GeV1021.3GeV5×9.60GeV5=0.11.

    (28)

    All mixed states have a much weaker mixing strength compared with the 1++ mixed states. We suggest that 1 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 numbers 1 are actually very pure. As mentioned above, MY1(4770), which contains no s quark, is close to Y(4660) and cannot be compatible with known 1 states. MY2(4266), which is a possible candidate for Y(4230), is a mixture of YA1(4207) and YB2(4385). By comparing the two mass predictions, MY2(4266) is closer to YA1(4207) rather than YB2(4385), and it is possibly dominated by ˉQQˉqq component. For the same reasons, MYs1(4610) is possibly dominated by a ˉQQˉqq component, while MYs2(4450) is possibly dominated by ˉQqQˉq. Hence, we suggest that Y(4230) and Y(4660) prefer a ˉQQˉqq state, and Y(4415) prefers a ˉQqQˉq state.

    In Ref. [30], authors have calculated the states YB1 and YB2 with a similar method and obtained the following results: mass mYB1=4.78+0.070.07 with s0=5.3 GeV-2, and mass mYB2=4.36+0.080.08 with s0=4.9 GeV-2; these numbers are consistent with our results. The small difference in the mass of YB1 is caused by the different values of s0. In addition, in Ref. [30], the authors have discussed different results of the similar states of YB1 and YB2 in previous papers. For instance, the authors in Ref. [42] did not distinguish between the charge conjugations and obtained mass of a YB1 like state mD0¯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: mass mYAs2=4.67+0.090.09 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)=i2[ˉc(x)γμc(x)ˉs(x)γνs(x)ˉc(x)γνc(x)ˉs(x)γμs(x)],jPB1/PBs1μ(x)=i2[ˉc(x)γμq(x)ˉq(x)c(x)ˉq(x)γμc(x)ˉc(x)q(x)],jPB2/PBs2μ(x)=i2[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 the ˉQQˉqq scenario, while B represents the ˉQqQˉq scenario. The additional subscript s represents the s quark case, and one can straightforwardly replace q with s when jPBs1μ and jPBs2μ are involved. The structures of these currents are similar to the 1++ and 1 cases, and it is interesting to compare the mass predictions of these currents to the 1++ and 1 states.

    To study the pure ˉQQˉqq and ˉQqQˉq states, the two-point correlation functions have the respective Lorentz structures,

    ΠPA/PAsμνρσ(q2)=ΠPA/PAsa1q2(q2gμρgνσq2gμσgνρqμqρgνσ+qμqσgνρqνqσgμρ+qνqρgμσ)+ΠPA/PAsb1q2(qμqρgνσ+qμqσgνρqνqσgμρ+qνqρgμσ),ΠPBk/PBskμν(q2)=ΠPBk/PBsk(1)(q2)(gμν+qμqνq2)+ΠPBk/PBsk(0)(q2)(qμqνq2),

    (31)

    where k=1,2; ΠPA/PAsaand ΠPA/PAsb describe the pure state contributions with quantum numbers 1++ and 1+, respectively; and ΠPBk/PBsk(1) and ΠPBk/PBsk(0) describe 1+ and 0++, respectively. In mixing scenarios, we start from the off-diagonal mixed correlator described in the previous sections, i.e.,

    ΠMP1/MPs1μνσ(q2)=i2d4xeiqx0|T(jPA/PAsμν(x)jPB1/PBs1+σ(0)+jPB1/PBs1σ(x)jPA/PAs+μν(0))|0,ΠMP2/MPs2μνσ(q2)=i2d4xeiqx0|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 structure

    ΠMPk/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.

    Table 5

    Table 5.  Summary of results for 1+ molecular states.
    State Current structure Mass/GeV λ/104GeV10 s0/GeV τ window/GeV−2
    PA J/ψρ 4.658+0.050.06 13.1+3.23.1 5.15 0.24 – 0.29
    PAs J/ψf(980) 4.694+0.050.05 14.0+3.52.9 5.2 0.24 – 0.32
    PB1 D0ˉD 4.927+0.050.04 34.1+7.75.7 5.4 0.21 – 0.30
    PB2 D1ˉD 4.528+0.060.05 9.7+2.92.1 5.05 0.26 – 0.31
    PBs1 Ds0ˉDs 4.999+0.040.03 42.8+8.26.8 5.5 0.21 – 0.33
    PBs2 Ds1ˉDs 4.642+0.050.05 13.0+3.42.8 5.15 0.25 – 0.34
    MP1 J/ψρ - D0ˉD 4.505+0.060.04 0.401+0.0960.073 GeV-1 5.05 0.21 – 0.30
    MP2 J/ψρ - D1ˉD 4.494+0.060.06 0.240+0.0600.052 GeV-1 5.05 0.23 – 0.27
    MPs1 J/ψf(980) - Ds0ˉDs 4.544+0.050.05 0.405+0.0850.077 GeV-1 5.1 0.21 – 0.33
    MPs2 J/ψf(980) - Ds1ˉDs 4.536+0.060.05 0.269+0.0670.055 GeV-1 5.1 0.22 – 0.31
    DownLoad: CSV
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    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)=i2d4xeiqx0|T(jPAμν(x)jPBk+σ(0)+jPBkσ(x)jPA+μν(0))|0λPAMH(εPAμqνεPAνqμ)(λPBkεPBkσ)+λPAMH(εPAμqνεPAνqμ)(λPBkεPBkσ)+...=λPAλPBk+λPAλPBkMH(gμσqν+gνσqμ)+...,

    (34)

    where k=1,2; εPA and εPBk are polarization vectors; and MH represents the ground state mass of PA. Analogous to the 1++ channel, the values of N obtained from Table 5 can be written as

    NMP1=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 corresponding ˉQQˉqq ground state mass. Like the 1 channel, all the mixed states that have quantum numbers 1+ 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) and MPs1(4494) are dominated by the ˉQQˉqq components; MP2(4544) and MPs2(4536) are more likely dominated by the ˉQqQˉq components.

    In Ref. [30], authors have calculated the states PB1 and PB2 with a similar method and obtained the following results: mass mPB1=4.73+0.070.07 with s0=5.2 GeV-2, and mass mPB2=4.60+0.080.08 with s0=5.1 GeV-2; these numbers are consistent with our results. The small difference in the mass of PB1 is caused by the different values of s0. Moreover, the authors in Ref. [43] obtained the mass of state PB1, mPB1=4.19 GeV. Our results are more supportive of the results in Ref. [30].

    We can convert the ˉQQˉqq and ˉQqQˉq 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, and λa are Gell-Mann matrices. That is, ˉQQˉqq currents can be decomposed into a series of ˉQqQˉq currents and a series of ˉQqQˉq color-octet currents, and vice versa. In this study, we have computed two-point correlation functions of ˉQQˉqq and ˉQqQˉq 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)=i22i2[(ˉcγμq)(ˉqc)+(ˉqγμc)(ˉcq)]+i22i2[(ˉcγμγ5q)(ˉqγ5c)(ˉqγμγ5c)(ˉcγ5q)]+...=i22jYB1μ+i22jYB2μ+...,

    (38)

    where jYA1μand jYB1/B2μ are defined in Section IV. When we compute two-point correlation functions of jYA1μ and jYB1/B2μ, it seems that the result may highlight states YB1/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 and YB2) will be suppressed, QCDSR calculations are sensitive to the changes of borel window and threshold s0, 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 two ˉQQˉqq and ˉQqQˉq currents and their mixed state,

    jZAμ(x)=ˉc(x)γμc(x)ˉq(x)γ5q(x)jZBμν(x)=i2[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 the ˉQQˉqq scenario, while B represents the ˉQqQˉq scenario. The mixed state is described by

    ΠMZμνσ(q2)=i2d4xeiqx0|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.

    Table 6

    Table 6.  Summary of results for 1+ states.
    State Current structure Mass/GeV λ/104GeV10 s0/GeV τ window/GeV−2
    ZA J/ψη 3.578+0.080.08 1.04+0.360.27 4.2 0.33 – 0.34
    ZB DˉD 4.018+0.060.06 2.90+0.880.67 4.55 0.29 – 0.36
    MZ J/ψη - DˉD 3.563+0.070.06 0.054+0.0170.012 GeV−1 4.1 0.31 – 0.35
    DownLoad: CSV
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    Compared to the 1++ currents jXA/B1μ and their mixed two-point correlator ΠMX1μνσ(q2), which are given in Eq. (17) and Eq. (19), jZA/Bμ and ΠMZμνσ(q2) have similar structures. According to our previous calculations in Section II, MX1 is relatively strongly mixed with different components, and MZ is supposed to have similar properties. However, the resulting mixing strength of MZ is

    NMZ=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 state MZ 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, threshold s0, 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 and ˉQqQˉq states. Such states strongly couple to ˉQQˉqq or ˉQqQˉq currents. Therefore, state components of ˉQQˉqq and ˉQqQˉq can be mixed with each other. Such mixing can be studied via the mixed correlators of ˉQQˉqq and ˉQqQˉq 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.

    Table 7

    Table 7.  Summary of mixed state results.
    Mixed state Mass/GeV N ˜N Dominant part Possible Candidate
    MX1 3.987+0.060.06 0.349 14% ˉQqQˉq X(3872)
    MX2 4.945+0.080.06 0.285 9.0% ˉQqQˉq
    MC 3.818+0.030.02 0.125 1.6% ˉQQ X(3872)
    MY1 4.770+0.070.06 0.15 2.3% ˉQqQˉq
    MY2 4.266+0.080.08 0.11 1.2% ˉQQˉqq Y(4230)
    MYs1 4.610+0.050.06 0.06 < 1% ˉQQˉqq Y(4660)
    MYs2 4.450+0.050.06 0.11 1.2% ˉQqQˉq Y(4415)
    MP1 4.505+0.060.04 0.09 < 1% ˉQQˉqq
    MP2 4.494+0.060.06 0.10 1.0% ˉQqQˉq
    MPs1 4.544+0.050.05 0.08 <1% ˉQQˉqq
    MPs2 4.536+0.060.05 0.09 <1% ˉQqQˉq
    DownLoad: CSV
    Show Table

    For the 1++ channel, we find that the two states MX1(3987) and MX2(4945) are relatively strongly mixed with the ˉQQˉqq and ˉQqQˉq components. Furthermore, we estimate the ratio of decay width of two kinds of decay modes of MX1(3987); this ratio is roughly consistent with the experimental results for X(3872). When we consider the mixing state combined with ˉQQ and ˉQqQˉq, we revisit the result in Ref. [28] with the new technique in Ref. [15]. The result argues that ˉQQ is the dominant part of X(3872), which can explain the latest observation to X(3872) of LHCb [7]. Our calculations just support that these two components can relatively strongly mix with each other in quantum numbers 1++.

    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) and YAs2(4610) configurations are good candidates for Y(4230) and Y(4660), respectively. However, by checking the assumed mixed states that mix with ˉQQˉqq and ˉQqQˉq molecular states, we find that these candidates have small components of ˉQqQˉq; this is inconsistent with the fact that Y(4230) and Y(4660) have more abundant decay modes. Thus, we can establish the dominant part of Y(4415). Our result suggests that Y(4415) is dominated by ˉQqQˉq and agrees with the absence of the K meson in observed decay final states. However, Y(4415) still has a small component of ˉQQˉqq. 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 and ˉQqQˉq 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.

    Figure A1

    Figure A1.  (color online) M2H behaviors on τ for 1++ mixed states.

    Figure A2

    Figure A2.  (color online) M2H behaviors on τ for 1 mixed states.

    Figure A3

    Figure A3.  (color online) M2H behaviors on τ for 1+ mixed states.
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1. Güngör, E., Sundu, H., Y. Süngü, J. et al. Possible Molecular Explanation for the Resonance Y (4500)[J]. Few-Body Systems, 2023, 64(3): 53. doi: 10.1007/s00601-023-01807-y
2. Wang, Z.-G.. Three-body strong decays of the Y ð4230Þ via the light-cone QCD sum rules[J]. International Journal of Modern Physics A, 2023. doi: 10.1142/S0217751X23501750

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Ze-Sheng Chen, Zhuo-Ran Huang, Hong-Ying Jin, T.G. Steele and Zhu-Feng Zhang. Mixing of X and Y states from QCD Sum Rules analysis[J]. Chinese Physics C. doi: 10.1088/1674-1137/ac531a
Ze-Sheng Chen, Zhuo-Ran Huang, Hong-Ying Jin, T.G. Steele and Zhu-Feng Zhang. Mixing of X and Y states from QCD Sum Rules analysis[J]. Chinese Physics C.  doi: 10.1088/1674-1137/ac531a shu
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Mixing of X and Y states from QCD sum rules analysis

  • 1. Institute of Modern Physics, Department of Physics, Zhejiang University, Hangzhou 310027,China
  • 2. Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China
  • 3. Department of Physics and Engineering Physics, University of Saskatchewan, Saskatoon, Saskatchewan, S7N 5E2, Canada
  • 4. Department of Physics, Ningbo University, Ningbo 315211, China

Abstract: We study ˉQQˉqq and ˉQqQˉq states as mixed states in QCD sum rules. By calculating the two-point correlation functions of pure states of their corresponding currents, we review the mass and coupling constant predictions of JPC=1++, 1, and 1+ states. By calculating the two-point mixed correlation functions of ˉQQˉqq and ˉQqQˉq currents, we estimate the mass and coupling constants of the corresponding "physical state" that couples to both ˉQQˉqq and ˉQqQˉq currents. Our results suggest that for 1++ states, the ˉQQˉqq and ˉQqQˉq components are more likely to mix, while for 1 and 1+ states, there is less mixing between ˉQQˉqq and ˉQqQˉq. Our results suggest the Y series of states have more complicated components.

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    I.   INTRODUCTION
    • 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 [46]. 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 of X(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 simple ˉQqQˉq and ˉQQˉqq combinations to study XYZ states (ˉQ represents a heavy c or b quark, while q represents a light u,d or s quark). These two forms have been extensively studied previously [8, 9]. However, ˉQqQˉq and ˉQQˉqq states are difficult to distinguish straightforwardly from the decay modes of XYZ states because XYZ states are usually observed to have both ˉQQ+ˉqq like decay modes and ˉQq+Qˉq like decay modes. Many scenarios were studied to qualitatively distinguish ˉQqQˉq and ˉQQˉqq states [1014]. 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 [1518]; we use the same technique to study the mixing of ˉQqQˉq and ˉQQˉqq 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 be ˉQqQˉq or ˉQQˉqq molecular states in different studies [1925]. 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 of ˉQqQˉq and ˉQQˉqq currents to investigate the corresponding physical states. The calculations will show us whether the physical states prefer to be ˉQqQˉq or ˉQQˉqq 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 a DD molecular state [1214]. However, although the pure DD molecular state was predicted to have a mass close to X(3872), it had too large a decay width to agree with the experimental results [28, 29]. Moreover, the J/ψρ and J/ψω states have a similar mass, since the sum of the masses of their two constituent parts are close to X(3872). Hence, the mixing of these two molecular states is naturally possible. Furthermore, in a recent observation of X(3872) by LHCb [7], the compact component was found to be required. Hence, we will consider another state ˉcc in the mixing; this has been studied in [28].

      For the 1 channel, many 1 states are found in the range of 4200–4700 MeV, permitting an abundance of possible pure or mixed molecular states. Some 1 states have very similar masses like Y(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 the ˉQqQˉq and ˉQQˉqq scenarios are possible. As outlined below, we calculate the mass spectrum of these states and estimate their mixing strength in both the u,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, and 1+ 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.

    II.   QCDSR APPROACH AND MIXING STRENGTH
    • 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)=id4xeiqx0|T{(ja(x)+cjb(x))(j+a(0)+cj+b(0))}|0=Πa(q2)+2cΠab(q2)+c2Πb(q2),

      (1)

      where ja and jb 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), and

      Πa(q2)=id4xeiqx0|T(ja(x)j+a(0))|0,

      Πb(q2)=id4xeiqx0|T(jb(x)j+b(0))|0,Πab(q2)=i2d4xeiqx0|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 between ja and jb, 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 [1618].

      Πab usually can be decomposed into a different Lorentz structure

      Πab(q2)=nΠn(q2)An,

      (3)

      where n=1,2,3..., Πn(q2) is the mixing state correlation function with specific quantum numbers, and An is the corresponding Lorentz structure. The forms of An are related to ja and jb, and we will define them in the sections below. For simplicity, we assume a specific ΠH(q2) represents a mixed-state correlation function and is one of the possible Πn(q2). We also assume that ΠH(q2) obeys the dispersion relation [32]

      ΠH(q2)=smindsρH(s)sq2iϵ+,

      (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 render ΠH(q2) finite. The spectral density ρH(s) can be calculated using the operator product expansion (OPE). In this paper, we calculate the spectral density ρH(s) up to dimension-six operators,

      ρ(OPE)H(s)=ρ(pert)H(s)+ρˉqqH(s)+ρG2H(s)+ρˉqGqH(s)+ρˉqq2H(s)+,

      (5)

      then

      Π(OPE)H(q2)=smindsρ(OPE)H(s)sq2iϵ+.

      (6)

      On the phenomenological side, by using the narrow resonance spectral density model,

      Π(phen)H(q2)=λaλb+λbλa21M2Hq2+s0dsρ(cont)H(s)sq2iϵ,

      (7)

      where λa and λb 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, and s0 is the continuum threshold. By using the QCDSR continuum spectral density assumptions

      ρ(cont)H(s)=ρ(OPE)H(s)Θ(ss0),

      (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)sq2iϵ+=λaλb+λbλa21M2Hq2.

      (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)esτ=λaλb+λaλb2eM2Hτ,

      (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 obtain

      M2H=sminds sρ(OPE)H(s)esτsminds ρ(OPE)H(s)esτ.

      (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)esτsminds ρ(OPE)H(s)esτ>40%,

      (12)

      while HDC (usually ˉqq2 in molecular systems) obeys the relation

      |sminds ρˉqq2H(s)esτ||sminds ρ(OPE)H(s)esτ|<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, with Δs0.5 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 parameter s0 separates the ground state and other excited states' contributions to spectral density. Hence, one can set s0 less than the first excitation threshold in the case of involving excited state contributions in the spectral density, and Δs 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 establish s0 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 around 0.5+0.10.1 GeV, and the fluctuations are all acceptable in the QCDSR approach.

      PDG name Possible structure Ground state Possible 1st excited state Δs/MeV
      D ˉcγ5q D(1865) D(2550) ~685
      D1 ˉcγμγ5q D1(2420)
      D0 ˉcq D0(2300) DJ(2600) ~300
      D ˉcγμq D(2007) D(2640) ~633

      Table 1.  Charmed meson (c=±1) states, where q represents the u,d quark. The symbol indicates particles that have confirmed quantum numbers.

      Possible structure Ground state/MeV Possible 1st excited state/MeV Δs/MeV
      ˉcγ5c ηc(1S)/2984 ηc(2S)/3637 ~653
      ˉcγμγ5c χc1(1P)/3510 χc1(3872) ~362
      ˉcc χc0(1P)/3415 χc0(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|λaλb+λaλb|2|λaλb|,

      (14)

      where λa and λb are coupling constants of the relevant current with a pure state (i.e., the coupling that emerges in the diagonal correlation functions Πa, Πb). Eq. (14) is analogous to the mixing parameter defined in Ref. [38]. By using appropriate factors of mass in the definitions of λa and λb, 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 that Ncosθsinθ, and N(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=m20ˉqq, m20=0.8 GeV2, αsG2=0.07±0.02 GeV4, andˉss=(0.8±0.2)ˉqq [40, 41]. In addition, the quark masses mc=1.27 GeV, mq=12(mu+md)=0.004 GeV, and ms=0.096 GeV, at the energy scale μ=2 GeV [2], are used.

    III.   MIXED STATE IN 1++ CHANNEL
    • We start from the following three forms of currents:

      jXAμν(x)=i2[ˉc(x)γμc(x)ˉq(x)γνq(x)ˉc(x)γνc(x)ˉq(x)γμq(x)],jXB1μ(x)=i2[ˉc(x)γμq(x)ˉq(x)γ5c(x)ˉq(x)γμc(x)ˉc(x)γ5q(x)],jXB2μ(x)=i2[ˉc(x)γμγ5q(x)ˉq(x)c(x)+ˉq(x)γμγ5c(x)ˉc(x)q(x)],jXCμ(x)=162ˉqqˉc(x)γμγ5c(x),

      (17)

      where X denotes the 1++ state, the subscript A of X denotes the ˉQQˉqq scenario, B denotes the ˉQqQˉq scenario, C denotes the ˉQQ 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 current jXAμν can be decomposed into J/ψ(1S)(3097) and ρ(770) currents; jXB1μ(x) can be decomposed into D(2007) and D(1865) currents. The sums of the masses of the two constituent mesons are both close to X(3872). Hence, we choose these two currents to study X(3872). The current jXB2μ has the same quantum numbers, and it cannot be excluded in 1++ mixing state structures. Besides, jXCμ(x) is normalized according to Ref. [28]. Since the mixing between jXCμ(x) and ˉQQˉqq is suppressed (ˉqq in ˉQQˉqq becomes a bubble and vanishes), we only consider the mixing between jXCμ(x) and jXB1μ(x) or jXB2μ(x).

      State Current structure Mass/GeV λ/104 GeV10 s0/GeV τ window/GeV−2
      XA J/ψρ 3.798+0.090.09 1.49+0.510.47 4.4 0.30 – 0.31
      XB1 DˉD 3.857+0.060.06 2.24+0.650.53 4.4 0.31 – 0.39
      XB2 D1ˉD0 5.310+0.040.04 69.0+15.014.0 5.8 0.20 – 0.29
      XC χc1 3.511+0.020.03 0.0229+0.00170.0018 4.5 0.29 – 0.31
      MX1 J/ψρ - DˉD 3.987+0.060.06 0.168+0.0490.042 GeV-1 4.4 0.30 – 0.32
      MX2 J/ψρD1ˉD0 4.945+0.080.06 0.760+0.290.20 GeV-1 5.45 0.22 – 0.24
      MC χc1 - DˉD 3.818+0.030.02 0.0282+0.00270.0024 4.5 0.28 – 0.30

      Table 3.  Summary of results for 1++ states. λ=λaλb+λaλb2 when mixed cases are involved, the same below.

      To study the pure ˉQQˉqq, ˉQqQˉq, and ˉQQ states, the respective two-point correlation functions can be decomposed into different Lorentz structures.

      ΠXAμνρσ(q2)=ΠXAa(q2)1q2(q2gμρgνσq2gμσgνρqμqρgνσ+qμqσgνρqνqσgμρ+qνqρgμσ)+ΠXAb(q2)1q2(qμqρgνσ+qμqσgνρqνqσgμρ+qνqρgμσ),ΠXBk/Cμν(q2)=ΠXBk/C(1)(q2)(gμν+qμqνq2)+ΠXBk/C(0)(q2)(qμqνq2),

      (18)

      where k=1,2; ΠXAa and ΠXAb describe the pure molecular state contribution with respective quantum numbers 1++ and 1+; and ΠXBk/C(1) and ΠXBk/C(0) describe 1++ and 0+ state contributions, respectively. In the mixing scenario, we start from the off-diagonal mixed correlator described in the previous section, i.e.,

      ΠMX1μνσ(q2)=i2d4xeiqx0|T(jXAμν(x)jXB1+σ(0)+jXB1σ(x)jXA+μν(0))|0,ΠMX2μνσ(q2)=i2d4xeiqx0|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 structure

      ΠMXkμνσ(q2)=ΠMXk(q2)(qαϵαμνσ).

      (20)

      Furthermore, when we consider the two-quark states jXCμ(x), the mixed correlator and its Lorentz structure are

      ΠMCμν(q2)=i2d4xeiqx0|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) and ΠC(0) describe the 1++ and 0+ state contributions, respectively. Here, we just consider the state mixed with jXCμ and jXB1; this state is a candidate for X(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 specific s0, we use Eq. (11) for each state to plot the τ behavior of MH for the chosen s0 (see Appendix A for details). The mass prediction MH and s0 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 predicted MH and s0 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 threshold s0. 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 the Δs 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 states XA(3798) and XB1(3857) both have mass predictions close to X(3872). However, the large mass prediction of the XB2(5310) state is far beyond the D1+D0 threshold, and does not match the observed 1++ 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)=i2d4xeiqx0|T(jXAμν(x)jXBk+σ(0)+jXBkσ(x)jXA+μν(0))|0(λXAMHϵμναβεXAαqβ)(λXBkεXBkσ)+(λXAMHϵμναβεXAαqβ)(λXBkεXBkσ)+...=λXAλXBk+λXAλXBkMHqαϵμνσα+...,

      (22)

      where k=1,2;εXA/XBkα/σ is a polarization vector; MH represents the ground state mass of XA; 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 factor MnH with a suitable n. Hence, we define λXA/MH as the new coupling constant of the XA state. The mixing strength can be written as

      NMX1=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=0.0282GeV100.0229GeV5×2.24GeV5=0.125,˜NMC=sin2(arcsin(0.125×2)2)=1.6%,

      (23)

      The state MX1(3987) is a mixture of XA(3798) and XB1(3857), which have similar mass predictions close to X(3872); unsurprisingly, this state also has the same mass prediction. Due to X(3872), observed decays to π+πJ/ψ(1S), ωJ/ψ(1S) and ˉD0D0, MX1(3987) is a good candidate to describe X(3872) [2]. We can estimate the proportions of each constituent and decay width of the corresponding decay modes by using the parameter NMX1. Experimental results of X(3872) decay width Γ1 of the ˉQq+Qˉq like decay mode is >30%, while the decay width Γ2 of the ˉQQ+ˉqq like decay mode is >5%. By comparison, the parameter NMX1 shows that the proportions of the ˉQqQˉq and ˉQQˉqq parts of MX1(3987) are 86% and 14%, respectively. Considering the similar Lorentz-invariant phase-space of these two kinds of decay modes, we can roughly equate Γ1/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 of MC is jXCμ. However, we found that this result strongly depends on the normalization of jXCμ. Hence, a proper normalized current is essential in calculations.

      For the state MX2(4945), its mass prediction is larger than those for all observed 1++ states. However, our calculation suggests that it is relatively strongly mixed. The dominant part of MX2(4945) is more likely to be XB2(5310) by comparing mass predictions.

      In Ref. [19], the authors calculated the state XB1 with a similar method and obtained the following results: the mass mXB1=3.89+0.090.09 GeV and the decay constant λXB1=2.96+1.090.79×104 GeV10 with s0=4.41 GeV, which is consistent with our results.

    IV.   MIXED STATE IN 1 CHANNEL
    • 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)=i2[ˉc(x)γμq(x)ˉq(x)c(x)+ˉq(x)γμc(x)ˉc(x)q(x)],jYB2/YBs2μ(x)=i2[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 the ˉQQˉqq scenario, while B represents the ˉQqQˉq scenario. The additional subscript s represents the s quark case, and one can straightforwardly replace the q with the s quark when jYAs1μ, jYAs2μ, jYBs1μ, and jYBs2μ are involved. The Y(4230) was observed to decay to χc0ω, while Y(4660) was observed to have both the ψ(2S)π+π and D+sDs1(2536) decay modes [2]. Hence, we especially focus on the currents jYA1μ and jYAs2μ, which are consistent with the respective decay modes, to describe Y(4230) and Y(4660), respectively, and discuss the corresponding mixed states in both the u,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) and ΠYBk/YBsk(1) describe pure state contributions with quantum numbers 1; and ΠYAk/YAsk(0)and ΠYBk/YBsk(0) describe the pure state contribution with quantum numbers 0+.

      To study the mixed state, the off-diagonal mixing two-point correlation functions described in Section II are

      ΠMY1μν(q2)=i2d4xeiqx0|T(jYA1μ(x)jYB1+ν(0)+jYB1ν(x)jYA1+μ(0))|0,ΠMY2μν(q2)=i2d4xeiqx0|T(jYA1μ(x)jYB2+ν(0)+jYB2ν(x)jYA1+μ(0))|0,ΠMYs1μν(q2)=i2d4xeiqx0|T(jYAs2μ(x)jYBs1+ν(0)+jYBs1ν(x)jYAs2+μ(0))|0,ΠMYs2μν(q2)=i2d4xeiqx0|T(jYAs2μ(x)jYBs2+ν(0)+jYBs2ν(x)jYAs2+μ(0))|0,

      (26)

      where MYk and MYsk, with k=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, and ΠMYk/MYsk(1) and ΠMYk/MYsk(0) describe the mixed states with quantum numbers 1 and 0+, 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/104GeV10 s0/GeV τ window/GeV−2
      YA1 χc0ω 4.207+0.080.09 1.64+0.630.49 4.8 0.27 – 0.28
      YAs2 J/ψf(980) 4.621+0.050.06 21.3+5.24.7 5.1 0.25 – 0.32
      YB1 D0ˉD 4.922+0.040.04 34.5+6.76.0 5.4 0.21 – 0.34
      YB2 D1ˉD 4.385+0.060.06 7.36+1.951.67 4.9 0.27 – 0.35
      YBs1 Ds0ˉDs 4.952+0.030.04 37.9+6.56.3 5.45 0.21 – 0.36
      YBs2 Ds1ˉDs 4.494+0.080.05 9.60+3.72.1 5.0 0.26 – 0.39
      MY1 χc0ω - D0ˉD 4.770+0.070.06 1.16+0.370.28 5.3 0.24 – 0.25
      MY2 χc0ω - D1ˉD 4.266+0.080.08 0.373+0.1170.093 4.95 0.26 – 0.27
      MYs1 J/ψf(980) - Ds0ˉDs 4.610+0.050.06 2.56+0.670.56 5.1 0.24 – 0.33
      MYs2 J/ψf(980) - Ds1ˉDs 4.450+0.050.06 1.64+0.430.22 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, and Y(4230), Y(4360), and Y(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 while Y(4415) and Y(4660) only decay to the Ds meson when an s quark is directly involved in the final states. The Y(4160) has both decay modes, including the K and Ds mesons in the final states. In contrast, all Y states have both ˉQQ+ˉqq like decay modes and ˉQq+Qˉq like decay modes, with the exception of Y(4390). The decay mode Y(4390) to π+πhc was observed, but the other decay modes of Y(4390) have not yet been seen. We cannot exclude an s quark in Y(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 that Y(4230) has candidates YA1(4207), MY2(4266), and Y(4360); Y(4390) has a candidate YB2(4385); Y(4415) has candidates YBs2(4494) and MYs2(4450); and Y(4660) has candidates YAs2(4621) and MYs1(4610). Although the remaining states are not compatible with known 1 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=1.16GeV101.64GeV5×34.5GeV5=0.15,NMY2=0.373GeV101.64GeV5×7.36GeV5=0.11,NMYs1=2.56GeV1021.3GeV5×37.9GeV5=0.09,NMYs2=1.64GeV1021.3GeV5×9.60GeV5=0.11.

      (28)

      All mixed states have a much weaker mixing strength compared with the 1++ mixed states. We suggest that 1 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 numbers 1 are actually very pure. As mentioned above, MY1(4770), which contains no s quark, is close to Y(4660) and cannot be compatible with known 1 states. MY2(4266), which is a possible candidate for Y(4230), is a mixture of YA1(4207) and YB2(4385). By comparing the two mass predictions, MY2(4266) is closer to YA1(4207) rather than YB2(4385), and it is possibly dominated by ˉQQˉqq component. For the same reasons, MYs1(4610) is possibly dominated by a ˉQQˉqq component, while MYs2(4450) is possibly dominated by ˉQqQˉq. Hence, we suggest that Y(4230) and Y(4660) prefer a ˉQQˉqq state, and Y(4415) prefers a ˉQqQˉq state.

      In Ref. [30], authors have calculated the states YB1 and YB2 with a similar method and obtained the following results: mass mYB1=4.78+0.070.07 with s0=5.3 GeV-2, and mass mYB2=4.36+0.080.08 with s0=4.9 GeV-2; these numbers are consistent with our results. The small difference in the mass of YB1 is caused by the different values of s0. In addition, in Ref. [30], the authors have discussed different results of the similar states of YB1 and YB2 in previous papers. For instance, the authors in Ref. [42] did not distinguish between the charge conjugations and obtained mass of a YB1 like state mD0¯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: mass mYAs2=4.67+0.090.09 with s0=5.1 GeV-2; this is consistent with our result.

    V.   MIXED STATE IN 1+ CHANNEL
    • We start from two forms of currents as follows:

      jPAμν(x)=jXAμν(x),jPAsμν(x)=i2[ˉc(x)γμc(x)ˉs(x)γνs(x)ˉc(x)γνc(x)ˉs(x)γμs(x)],jPB1/PBs1μ(x)=i2[ˉc(x)γμq(x)ˉq(x)c(x)ˉq(x)γμc(x)ˉc(x)q(x)],jPB2/PBs2μ(x)=i2[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 the ˉQQˉqq scenario, while B represents the ˉQqQˉq scenario. The additional subscript s represents the s quark case, and one can straightforwardly replace q with s when jPBs1μ and jPBs2μ are involved. The structures of these currents are similar to the 1++ and 1 cases, and it is interesting to compare the mass predictions of these currents to the 1++ and 1 states.

      To study the pure ˉQQˉqq and ˉQqQˉq states, the two-point correlation functions have the respective Lorentz structures,

      ΠPA/PAsμνρσ(q2)=ΠPA/PAsa1q2(q2gμρgνσq2gμσgνρqμqρgνσ+qμqσgνρqνqσgμρ+qνqρgμσ)+ΠPA/PAsb1q2(qμqρgνσ+qμqσgνρqνqσgμρ+qνqρgμσ),ΠPBk/PBskμν(q2)=ΠPBk/PBsk(1)(q2)(gμν+qμqνq2)+ΠPBk/PBsk(0)(q2)(qμqνq2),

      (31)

      where k=1,2; ΠPA/PAsaand ΠPA/PAsb describe the pure state contributions with quantum numbers 1++ and 1+, respectively; and ΠPBk/PBsk(1) and ΠPBk/PBsk(0) describe 1+ and 0++, respectively. In mixing scenarios, we start from the off-diagonal mixed correlator described in the previous sections, i.e.,

      ΠMP1/MPs1μνσ(q2)=i2d4xeiqx0|T(jPA/PAsμν(x)jPB1/PBs1+σ(0)+jPB1/PBs1σ(x)jPA/PAs+μν(0))|0,ΠMP2/MPs2μνσ(q2)=i2d4xeiqx0|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 structure

      ΠMPk/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/104GeV10 s0/GeV τ window/GeV−2
      PA J/ψρ 4.658+0.050.06 13.1+3.23.1 5.15 0.24 – 0.29
      PAs J/ψf(980) 4.694+0.050.05 14.0+3.52.9 5.2 0.24 – 0.32
      PB1 D0ˉD 4.927+0.050.04 34.1+7.75.7 5.4 0.21 – 0.30
      PB2 D1ˉD 4.528+0.060.05 9.7+2.92.1 5.05 0.26 – 0.31
      PBs1 Ds0ˉDs 4.999+0.040.03 42.8+8.26.8 5.5 0.21 – 0.33
      PBs2 Ds1ˉDs 4.642+0.050.05 13.0+3.42.8 5.15 0.25 – 0.34
      MP1 J/ψρ - D0ˉD 4.505+0.060.04 0.401+0.0960.073 GeV-1 5.05 0.21 – 0.30
      MP2 J/ψρ - D1ˉD 4.494+0.060.06 0.240+0.0600.052 GeV-1 5.05 0.23 – 0.27
      MPs1 J/ψf(980) - Ds0ˉDs 4.544+0.050.05 0.405+0.0850.077 GeV-1 5.1 0.21 – 0.33
      MPs2 J/ψf(980) - Ds1ˉDs 4.536+0.060.05 0.269+0.0670.055 GeV-1 5.1 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)=i2d4xeiqx0|T(jPAμν(x)jPBk+σ(0)+jPBkσ(x)jPA+μν(0))|0λPAMH(εPAμqνεPAνqμ)(λPBkεPBkσ)+λPAMH(εPAμqνεPAνqμ)(λPBkεPBkσ)+...=λPAλPBk+λPAλPBkMH(gμσqν+gνσqμ)+...,

      (34)

      where k=1,2; εPA and εPBk are polarization vectors; and MH represents the ground state mass of PA. Analogous to the 1++ channel, the values of N obtained from Table 5 can be written as

      NMP1=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 corresponding ˉQQˉqq ground state mass. Like the 1 channel, all the mixed states that have quantum numbers 1+ 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) and MPs1(4494) are dominated by the ˉQQˉqq components; MP2(4544) and MPs2(4536) are more likely dominated by the ˉQqQˉq components.

      In Ref. [30], authors have calculated the states PB1 and PB2 with a similar method and obtained the following results: mass mPB1=4.73+0.070.07 with s0=5.2 GeV-2, and mass mPB2=4.60+0.080.08 with s0=5.1 GeV-2; these numbers are consistent with our results. The small difference in the mass of PB1 is caused by the different values of s0. Moreover, the authors in Ref. [43] obtained the mass of state PB1, mPB1=4.19 GeV. Our results are more supportive of the results in Ref. [30].

    VI.   NON-PERTURBATIVE EFFECTS OF MIXING STRENGTH
    • We can convert the ˉQQˉqq and ˉQqQˉq 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, and λa are Gell-Mann matrices. That is, ˉQQˉqq currents can be decomposed into a series of ˉQqQˉq currents and a series of ˉQqQˉq color-octet currents, and vice versa. In this study, we have computed two-point correlation functions of ˉQQˉqq and ˉQqQˉq 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)=i22i2[(ˉcγμq)(ˉqc)+(ˉqγμc)(ˉcq)]+i22i2[(ˉcγμγ5q)(ˉqγ5c)(ˉqγμγ5c)(ˉcγ5q)]+...=i22jYB1μ+i22jYB2μ+...,

      (38)

      where jYA1μand jYB1/B2μ are defined in Section IV. When we compute two-point correlation functions of jYA1μ and jYB1/B2μ, it seems that the result may highlight states YB1/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 and YB2) will be suppressed, QCDSR calculations are sensitive to the changes of borel window and threshold s0, 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 two ˉQQˉqq and ˉQqQˉq currents and their mixed state,

      jZAμ(x)=ˉc(x)γμc(x)ˉq(x)γ5q(x)jZBμν(x)=i2[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 the ˉQQˉqq scenario, while B represents the ˉQqQˉq scenario. The mixed state is described by

      ΠMZμνσ(q2)=i2d4xeiqx0|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/104GeV10 s0/GeV τ window/GeV−2
      ZA J/ψη 3.578+0.080.08 1.04+0.360.27 4.2 0.33 – 0.34
      ZB DˉD 4.018+0.060.06 2.90+0.880.67 4.55 0.29 – 0.36
      MZ J/ψη - DˉD 3.563+0.070.06 0.054+0.0170.012 GeV−1 4.1 0.31 – 0.35

      Table 6.  Summary of results for 1+ states.

      Compared to the 1++ currents jXA/B1μ and their mixed two-point correlator ΠMX1μνσ(q2), which are given in Eq. (17) and Eq. (19), jZA/Bμ and ΠMZμνσ(q2) have similar structures. According to our previous calculations in Section II, MX1 is relatively strongly mixed with different components, and MZ is supposed to have similar properties. However, the resulting mixing strength of MZ is

      NMZ=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 state MZ 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, threshold s0, mass prediction, and decay constant, which are all influenced by non-perturbative terms in QCDSR calculations.

    VII.   SUMMARY
    • In this study, we used QCD sum-rules to calculate the mass spectrum of ˉQQˉqq and ˉQqQˉq states. Such states strongly couple to ˉQQˉqq or ˉQqQˉq currents. Therefore, state components of ˉQQˉqq and ˉQqQˉq can be mixed with each other. Such mixing can be studied via the mixed correlators of ˉQQˉqq and ˉQqQˉq 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+0.060.06 0.349 14% ˉQqQˉq X(3872)
      MX2 4.945+0.080.06 0.285 9.0% ˉQqQˉq
      MC 3.818+0.030.02 0.125 1.6% ˉQQ X(3872)
      MY1 4.770+0.070.06 0.15 2.3% ˉQqQˉq
      MY2 4.266+0.080.08 0.11 1.2% ˉQQˉqq Y(4230)
      MYs1 4.610+0.050.06 0.06 < 1% ˉQQˉqq Y(4660)
      MYs2 4.450+0.050.06 0.11 1.2% ˉQqQˉq Y(4415)
      MP1 4.505+0.060.04 0.09 < 1% ˉQQˉqq
      MP2 4.494+0.060.06 0.10 1.0% ˉQqQˉq
      MPs1 4.544+0.050.05 0.08 <1% ˉQQˉqq
      MPs2 4.536+0.060.05 0.09 <1% ˉQqQˉq

      Table 7.  Summary of mixed state results.

      For the 1++ channel, we find that the two states MX1(3987) and MX2(4945) are relatively strongly mixed with the ˉQQˉqq and ˉQqQˉq components. Furthermore, we estimate the ratio of decay width of two kinds of decay modes of MX1(3987); this ratio is roughly consistent with the experimental results for X(3872). When we consider the mixing state combined with ˉQQ and ˉQqQˉq, we revisit the result in Ref. [28] with the new technique in Ref. [15]. The result argues that ˉQQ is the dominant part of X(3872), which can explain the latest observation to X(3872) of LHCb [7]. Our calculations just support that these two components can relatively strongly mix with each other in quantum numbers 1++.

      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) and YAs2(4610) configurations are good candidates for Y(4230) and Y(4660), respectively. However, by checking the assumed mixed states that mix with ˉQQˉqq and ˉQqQˉq molecular states, we find that these candidates have small components of ˉQqQˉq; this is inconsistent with the fact that Y(4230) and Y(4660) have more abundant decay modes. Thus, we can establish the dominant part of Y(4415). Our result suggests that Y(4415) is dominated by ˉQqQˉq and agrees with the absence of the K meson in observed decay final states. However, Y(4415) still has a small component of ˉQQˉqq. 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 and ˉQqQˉq 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.

    APPENDIX A: QCDSR ANALYSIS RESULTS
    • Here, we show the τ dependence of M2H defined in Eq. (11) for all mixed states.

      Figure A1.  (color online) M2H behaviors on τ for 1++ mixed states.

      Figure A2.  (color online) M2H behaviors on τ for 1 mixed states.

      Figure A3.  (color online) M2H behaviors on τ for 1+ mixed states.

Reference (45)

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