# More about the Mass of the New Charmonium States

Authors: T. M. Aliev; H. Özşahin; M. Savcı
Journal: Advances in High Energy Physics
 (2015)

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## Abstract

IfX(3872) is described by the picture as a mixture of the charmonium and molecular D*D states; Y(3940) as a mixture of the χc0 and D*D* states; and Y(4260) as a mixture of the tetraquark and charmonium states, their orthogonal combinations should also exist. We estimate the mass and residues of the states within the QCD sum rules method. We find that the mass splitting among X,Y and their orthogonal states is at most 200 MeV. Experimental search of these new states can play critical role for establishing the nature of the new charmonium states.


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## Body

## 1. Introduction

Last decade was quite an exciting and productive period in particle physics. Starting from the first observation of theX(3872) meson by the Belle Collaboration [1] up to now more than twenty new charmonium states have been observed (for the experimental and theoretical reviews and the references to the original works on the charmonium spectroscopy, see, e.g., [2–8]). The main lesson from these discoveries is that these states (referred to as XYZ states) can not be described by the simple quark model, that is, as a quark-antiquark system, and they are believed to have more complex structures. Understanding the structure and dynamics of these states is one of the hot problems in particle physics. In investigating the properties of these states two pictures are widely used, namely, four-quark (tetraquark) or meson molecules (bound states of two mesons).The mass and some of the strong coupling constants of theXYZ mesons with light mesons are widely studied within the QCD sum rules method [9] in many works (see, e.g., [10, 11]). The first calculation of the mass of the X(3872) meson as a tetraquark state with the quantum numbers JPC=1++ is performed within QCD sum rules method in [10]. The X(3872) meson as a D*D molecular state with the interpolating current,
(1)jμmol=12q¯aγ5cac¯bγμqb-q¯aγμcac¯bγ5qb,
was investigated in [11]. The next step in analysis of the properties of X(3872) is that it is assumed to be a mixture of the charmonium and D*D molecules [12, 13], whose interpolating current has the form
(2)jμ(1)=cos⁡⁡θ1jμch+sin⁡θ1jμmol,
where
(3)jμch=162q¯qc¯γμγ5c
and jμmol is given in (1). In [12, 13], the mixing angle is calculated to have the value θ1=(9±4)0 in order to reproduce the experimental value of the mass of the X(3872) meson.As far as the the stateY(3940) is concerned, it is described by the mixture of the scalar χc0 and D*D* molecules [14]. In other words its interpolating current can be written as
(4)j(2)=cos⁡⁡θ2-q¯q2c¯c+sin⁡θ2q¯γμcc¯γμq.
In the same manner, the mixing angle is calculated by requiring that it should reproduce the mass of the Y(3940) state and is found to have the value θ2=(76±5)0.Finally, if one assumes thatY(4260) state is a mixture of the tetraquark and the charmonium current given as [15],
(5)jμ(3)=cos⁡⁡θ3jμ′ch+sin⁡θ3jμtet,
where
(6)jμ′ch=12q¯qc¯γμc,jμtet=12εabcεdecqaTCγ5cbq¯dγμγ5Cc¯eT+qaTCγμγ5cbq¯dγ5Cc¯eT,
and requires that it should reproduce the mass Y(4260) state, the mixing angle is calculated to be θ3=(53±5)0 [15]. It should be noted here that X(3872), Y(3940), and Y(4260), as well as other related states which are represented as the four-quark states within the QCD sum rules method, are studied in [16], [17], [18], and [19], respectively.If the correct pictures ofX(3872), Y(3940), and Y(4260) states were mixture of the molecular or tetraquark states with charmonium states, the orthogonal combinations of these states may also exist. In other words, these orthogonal X′, Y′, and Y′′ states should be described by the following interpolating currents:
(7)jμ(1)′=-sin⁡θ1jμch+cos⁡⁡θ1jμmol,(8)j(2)′=-sin⁡θ2-q¯q2c¯c+cos⁡⁡θ2q¯γμcc¯γμq,(9)jμ(3)′=-sin⁡θ3jμ′ch+cos⁡⁡θ3jμtet.By imposing the condition that the states defined by the currents in (2), (4), and (5) should be orthogonal to the states defined by the currents given in (7), (8), and (9), the corresponding mixing angles are estimated within the QCD sum rules method in [20], whose values are calculated to have the values θ1=(2.4±0.6)0, θ2=(20.0±2.0)0, and θ3=(20.0±3.0)0, respectively.The aim of this paper is to calculate the mass and residues of these states described by the interpolating currents (7), (8), and (9).Consider the following two-point correlation functions:(10)Πμν(1)=i∫d4xeiqx0Tjμ1′xj¯ν(1)′00=Π1q2gμν-qμqνq2+Π01qμqνq2,(11)Π(2)=i∫‍d4xeiqx0Tj(2)′xj¯(2)′00=Π(2)q2,(12)Πμν(3)=i∫‍d4xeiqx0Tjμ(3)′xj¯ν(3)′00=Π(3)q2gμν-qμqνq2+Π0(3)qμqνq2.According to the duality principle the phenomenological part of these correlation functions can be calculated in terms of quarks, gluons, and hadrons by inserting complete set of hadrons carrying the same quantum numbers as the interpolating currents themselves. Then isolating the ground states and performing summation over the spins (if ever exist) we get(13)Πμν(1)=0jμ(1)BqBqjν(1)0q2-mB12+⋯=λ(1)2mB12-q2-gμν+qμqνmB12+⋯Π(2)=λ(2)2mB22-q2+⋯Πμν(3)=λ(3)2mB32-q2-gμν+qμqνmB12+⋯,
where ⋯ means the contributions of the higher states and continuum, and we have used
(14)0jμ(1),(3)Bq=λ(1),(3)εμ(1),(3),0j(2)Bq=λ(2).
The continuum contribution to the spectral density is modeled as a spectral density from the operator product expansion (OPE) starting from some threshold s0; that is,
(15)ρconts=ρOPEθs-s0,
where θ(s-s0) is the Heaviside step function.In order to construct the sum rules for the mass and residues of new charmonium states the calculation of the correlation functions in terms of the quark and gluon degrees of freedom using the operator product expansion (OPE) are needed. On the other hand, to be able to calculate the correlation functions from QCD side the heavy and light quark propagators are needed. The expression for the propagator of the light quark in the coordinate space is given as(16)Sqx=ix2π2x4-mq4π2x2-q¯q121-imq4x-x2192m02q¯q1-imq6x-αsπx2x2335q¯q2+132π2gsGμνσμνx+xσμνx2-π2x42833q¯qgs2G2-αsmqπx42535q¯q2-mqx2x2732m02q¯q,
and that of the heavy quark is given as
(17)SQ=mQ24π2ix-x2K2mQ-x2+1-x2K1mQ-x2-mQgsGμν32π2iσμνx+xσμν-x2K1mQ-x2+2σμνK0mQ-x2iσμνx+xσμν-x2-gs2G22304π2imQx-6-x2K1mQ-x2+mQ-x2K2mQ-x2
(see [21] and [22], resp.).The theoretical part of the correlation function(s) can be written in terms of the dispersion relation as(18)Π(i)q2=∫4mc2∞dsρOPE(i)ss-q2.
Choosing the coefficient of the gμν structure for the correlators (10) and (12), for the spectral densities we get
(19)ρ1s=14096π6·∫αmin⁡αmax⁡‍dα∫βmin⁡βmax⁡‍dβ·16mcmq+α+βm023αβ1-α+β2μ14-6mcmqα+β1-α-β3-α+βμ13-48π2α+β1+α+βmc+2αβmq·q¯qμ12+24π216mcmq+α+βm02mcq¯qμ1·cos⁡2θ+11536π4·9mc-6α1-αmq36α1-αmqq¯qμ22cos⁡2θ-2m029mc-6α1-αmq+8π2α1-αq¯q9mc-6α1-αmq·q¯qμ2cos⁡2θ+43m02mq3mc2-α1-αs+2π210mc2-9mcmq+2α1-αsq¯q·q¯qcos⁡2θ-16π2mc2+α1-αμ2-s·q¯q2sinθsinθ+2cos⁡θ9mc-6α1-αmq,(20)ρ2s=1512π6·∫αmin⁡αmax⁡‍dα∫βmin⁡βmax⁡‍dβ·-3αβ1-α-βμ14+61-α-βα+βmcmqμ13+24π2α+βmcq¯qμ12-192π2mc2mqq¯qμ1cos⁡2θ+1384π4·∫αmin⁡αmax⁡‍dα·-36α1-αmqq¯qμ22cos⁡2θ+3m023mc-8α1-αmq+64π2α1-αq¯q·q¯qμ2cos⁡2θ+6m02mq-6mc2+2α1-αs+16π2·mc3mq-4mc-2α1-αsq¯q·q¯qcos⁡2θ+48π2mc2+α1-α2μ2-s·q¯q2sinθ3sinθ-22cos⁡θ,(21)ρ3s=13072π6·∫αmin⁡αmax⁡‍dα∫βmin⁡βmax⁡‍dβ·3αβ1-α+β2μ14-21-α-β3mc2μ13+192π2αβmqq¯qμ12-96π25-α-βmc2mqq¯qμ1-16π2m02mc2mqq¯qcos⁡2θ+1192π4·∫αmin⁡αmax⁡‍dα·q¯q2sinθ3sinθ-cos⁡θα1-αq¯qμ2·6mqμ2-m02mq+16π2q¯qcos⁡2θ-2m02mq3mc2+α1-αs+8π2mc2+α1-αm02-sq¯q·q¯qcos⁡2θ+24π2mc2-α1-αμ2-s·q¯q2sinθ3sinθ-cos⁡θ,
where
(22)μ1=mc2α+βαβ-s,μ2=μ1β⟶1-α,βmin⁡=αmc2αs-mc2,βmax⁡=1-α,αmin⁡=121-v,αmax⁡=121+v,v=1-4mc2s.For simplicity, in these expressions we do not present the terms proportional with the gluon condensategs2G2. It should be noted that these spectral densities are all calculated in [12–15]. Perturbative parts of our results coincide with those presented in [12–15], while there are some discrepancies in quark condensates and d=5 operators. These differences are due to the fact that some of the contributions corresponding to the quark condensates and d=5 operators are neglected in [12–15].Equating the spectral densities given in (19), (20), and (21) with the coefficients of Π(1), Π(2), and Π(3), respectively, and performing Borel transformation over -q2, we get
(23)λ(i)2e-m12/M2=∫4mc2s0‍dsρ(i)se-s/M2,
where i=1,2,3. In deriving (23) the quark-hadron duality ansatz has been used; that is, the contribution of the continuum and higher states are assumed to be the same as the perturbative ones starting on from the threshold.In order to obtain the mass sum rules for the mass, we take the derivative of (23) with respect to 1/M2 on both sides and divide the obtained result by itself, from which we obtain
(24)mi2=∫4mc2s0‍dssρise-s/M2∫4mc2s0‍dsρ(i)e-s/M2.


## 2. Numerical Analysis

Here in this section we present the results of the numerical analysis on the mass and residues of the considered new charmonium states. For thec-quark masses we have used its MS¯ scheme masses, m¯c(m¯c)=(1.28±0.03)GeV [23], 〈q¯q〉=(1GeV)=-(0.246-19+28MeV)3 [24], m02=(0.8±0,2)GeV2, 〈s¯s〉=0.8〈q¯q〉 [25], and ms(2GeV)=(102±8)MeV [26].The sum rules for the mass and residue contain two auxiliary variables, namely, the Borel mass parameterM2 and the continuum threshold s0. Any physical quantity should be independent of them, and therefore our primary aim is to find the regions of M2 and s0, where the mass and residue are practically independent of them. It should be noted here that the continuum threshold itself is not a totally arbitrary parameter but related to the first excited state. In general, the quantity s0-m, where m is the mass of the ground state, corresponds to the energy of the first excited state with the same quantum number. The range of this quantity typically changes in the region 0.3 to 0.8GeV. The lower limit of M2 is determined by requiring that the operator product expansion series is convergent. In other words the contribution of the perturbative part must dominate over the nonperturbative one. The upper bound of M2 is decided from the condition that the contribution coming from the continuum constitutes about 1/3 of the contribution coming from the perturbative part, that is, the ratio
(25)R=∫s0∞‍dsρse-s/M2∫4mc2∞‍dsρse-s/M2<13,
which leads to the following “working regions” of the Borel mass parameter:
(26)2.4≤M2≤4.0GeV2forX′state,2.5≤M2≤4.0GeV2forY′state,2.5≤M2≤5.0GeV2forY′′state
In Figure 1, we present the dependence of the mass of X′ state on M2, at the fixed value of the continuum threshold s0=4.4GeV and of the mixing angle θ. We observe from these figures that, in the considered domains of M2, the results exhibit good stability with respect to the variation in M2 and seem to be practically insensitive to the variations in s0 and the mixing angle θ. As has already been noted, if X meson is represented as mixture of the charmonium and D*D molecular states, it is found in [20] the mixing angle is equal to θ=(2.4±0.3)0. In other words, in this picture X(3872) state can be said to be composed of pure charmonium and its orthogonal combination X′ which is described by the D*D molecular state. The study of the decay channels of X(3872) and X′ states can give unambiguous useful information about the “correct” pictures of these states. Our final result for the mass of X′ is that
(27)mX′=3.75±0.15GeV.Figure 1
The dependence of the mass of theX′ state on Borel mass square M2 for the mixing angle θ=2.40, at the fixed value of the continuum threshold s0=4.4GeV.In Figure2 we present the M2 dependence of the residue of X′ on M2 for the mixing angle θ=2.40, at the fixed value of the continuum threshold s0=4.4GeV. We observe that the residue is weakly dependent on M2, and we deduce from this figure that
(28)λX′=1.3×±0.2×10-2GeV3.
The dependencies of the mass and residue Y′ state on M2 for the mixing angle θ=200, at s0=4.4GeV, are presented in Figures 3 and 4, respectively. We observe from these figures that, again, the mass and residue of Y′ state seem to be practically insensitive with respect to the variation in M2, whose values are calculated to be
(29)mY′=3.85±0.20GeV,λY′=1.9±0.4×10-2GeV-3.Figure 2
The dependence of the residue of theX′ state on Borel mass square M2 for the mixing angle θ=2.40, at the fixed value of the continuum threshold s0=4.4GeV.Figure 3
The dependence of the mass of theY′ state on Borel mass square M2 for the mixing angle θ=200, at the fixed value of the continuum threshold s0=4.4GeV.Figure 4
The dependence of the residue of theY′ state on Borel mass square M2 for the mixing angle θ=200, at the fixed value of the continuum threshold s0=4.4GeV.Performing similar approach for theY′′ state, we see from Figures 5 and 6 that
(30)mY′′=4.4±0.1GeV,λY′′=2.0±0.2×10-2GeV-3.Figure 5
The dependence of the mass of theY′′ state on Borel mass square M2 for the mixing angle θ=200, at the fixed value of the continuum threshold s0=4.6GeV.Figure 6
The dependence of the residue of theY′′ state on Borel mass square M2 for the mixing angle θ=200, at the fixed value of the continuum threshold s0=4.6GeV.We would like to note here that onlyY′′ state is sensitive to the change in the value of the mixing angle θ. With the increasing value of θ, the mass of the Y′′ state is decreasing; for example, at θ=400, mY′′=(4.0±0.1)GeV and at θ=500,  mY′′=(3.9±0.1)GeV.We also calculate the mass of theX(3872), Y(3940), and Y(4260) states at θ1=(2.4±0.6)0, θ2=(20±2)0, and θ3=(20±3)0, respectively, and obtain the following mass values: mX(3872)≡(3.65±0.10)GeV, mY(3940)≡(3.7±0.1)GeV, and mY(4260)≡(3.9±0.10)GeV. In other words, sum rules predictions for the masses are slightly lower compared to their experimental values.From these predictions of the masses ofX′, Y′, and Y′′ states and the experimental values of X(3872), Y(3940), and Y(4260) states we observe that the splittings among these states are at most 100MeV≤Δm≤200MeV. These results can be checked in future planned experiments after the discovery of the X′, Y′, and Y′′ states.The predicted results for the residues of theY′′ and Y(4260); Y′ and Y(3940) states, respectively, are very close to each other, while the result for the X′ state is three times larger compared to that of the X(3872) state.Few words about the decay channels of these states are in order. The main decay channels of theX(3872) and X′ states are Ψπ+π-, J/Ψρ, J/Ψω, J/Ψγ, J/Ψϕ, Ψ′γ, χcγ, DD¯*, and DD¯π; of the Y(4260) and Y′′ states are J/Ψη, Ψ′π+π-, DD¯, DD¯*, χcγ, and DD*π, and of the Y(3940) and Y′ states are J/Ψω, DD¯, D*D¯*, and D¯*+D-γ (see, e.g., [2–8] and the references therein). It should especially be stressed here that the measurement of the photon spectrum in the radiative decay channels can play critical role in testing the structure of these mesons.In conclusion, using the picture that assumesX(3872) as the mixture of the charmonium and the molecular D*D states; Y(3940) as the mixture of the χc0 and D*D* states; and Y(4260) as the mixture of the tetraquark and charmonium states, we calculate the mass and the residues of their orthogonal states within the QCD sum rules method. We obtain that the mass splittings among these states are at most 200MeV. Experimental search of new orthogonal states can be quite useful in establishing correct picture of these states. The study of the decay channels can also be very useful in this investigation.



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