In this work, the mass and pole residue of resonance Yb is studied by using QCD sum rules approach at finite temperature. Resonance Yb is described by a diquark-antidiquark tetraquark current, and contributions to operator product expansion are calculated by including QCD condensates up to dimension six. Temperature dependencies of the mass mYb and the pole residue λYb are investigated. It is seen that near a critical temperature (Tc≃190MeV), the values of mYb and λYb decrease to 87% and to 44% of their values at vacuum.
Kocaeli UniversityBAP 2018/082Türkiye Bilimsel ve Teknolojik Arastirma KurumuBIDEP-22191. Introduction
Heavy quarkonia systems provide a unique laboratory to search the interplay between perturbative and nonperturbative effects of QCD. They are nonrelativistic systems in which low energy QCD can be investigated via their energy levels, widths, and transition amplitudes [1]. Among these heavy quarkonia states, vector charmonium and bottomonium sectors are experimentally studied very well, since they can be detected directly in e+e- annihilations. In the past decade, observation of a large number of bottomonium-like states in several experiments increased the interest in these structures [2–6]. However, these observed states could not be conveniently explained by the simple qq¯ picture of mesons. The presumption of hadrons containing quarks more than the standard quark content (qq¯ or qqq) is introduced by a perceptible model for diquarks plus antidiquarks, which was developed by Jaffe in 1976 [7]. Later Maiani, Polosa, and their collaborators proposed that the X, Y, Z mesons are tetraquark systems, in which the diquark-antidiquark pairs are bound together by the QCD color forces [8]. In this color configuration, diquarks can play a fundamental role in hadron spectroscopy. Thus, probing the multiquark matter has been an intensely intriguing research topic in the past twenty years and it may provide significant clues to understand the nonperturbative behavior of QCD.
In 2007, Belle reported the first evidence of e+e-→Υ(1S)π+π-,Υ(2S)π+π- and first observation for e+e-→Υ(3S)π+π-,Υ(1S)K+K- decays near the peak of the Υ(5S) state at s=10.87GeV [2]. Assigning these signals to Υ(5S), the partial widths of decays Υ(5S)→Υ(1S)π+π- and Υ(5S)→Υ(2S)π+π- were measured unusually larger (more than two orders of magnitude) than formerly measured decay widths of Υ(nS) states. Following these unusually large partial width measurement, Belle measured the cross sections of e+e-→Υ(1S)π+π-,Υ(2S)π+π- and Υ(1S)π+π- and reported that the resonance observed via these decays does not agree with conventional Υ(5S) line shape. These observations led to the proposal of existence of new exotic hidden-beauty state analogous to broad Y(4260) resonance in the charmonium sector, which is a Breit-Wigner shaped resonance with mass (10888.4-2.6+2.7±1.2) MeV/c2, and width (30.7-7.0+8.3±3.1) MeV/c2, and is called Yb(10890) [5]. In literature, there are several approaches to investigate the structure of exotic Yb resonance. In [9], Yb is considered as a ΛbΛ¯b bound state with a highly large binding energy. In [10, 11], Yb is interpreted as a tetraquark and its mass is estimated by using QCD sum rules at vacuum.
Moreover, it is likely that at very high temperatures within the first microseconds following the Big Bang, quarks and gluons existed freely in a homogenous medium called the quark-gluon plasma (QGP). One of the first quark-gluon plasma signals proposed in the literature is suppression of J/ψ particles [12]. In 2011, CMS collaboration reported that charmonium states ϕ(2S) and J/ψ melt or were suppressed due to interacting with the hot nuclear matter created in heavy-ion interactions [13, 14]. Following these observations in the charmonium sector, CMS collaboration also reported suppression of bottomonium states, Υ(2S) and Υ(3S) relative to the Υ(1S) ground state [15, 16]. The dissociation temperatures for the Υ states are expected to be related to their binding energies and are predicted to be 2Tc,1.2Tc and Tc for the Υ(1S),Υ(2S), and Υ(3S) mesons, respectively, where Tc is the critical temperature for deconfinement [16–18].
In addition to these studies, one of the first investigations on thermal properties of exotic mesons was done in [19], in which the authors investigated in medium properties of X(3872) under the hypothesis that it is a 1++ state or 2-+ state, but making no assumption on its structure. They estimated that the mass of the 1++ molecular state decreases with increasing temperature; the mass of charmonium or tetraquark state is almost stable.
Inspired by these findings and motivated by the aforementioned discussions, we focus on the Yb resonance and its thermal behavior. This paper is organized as follows. In Section 2, theoretical framework of Thermal QCD sum rules (TQCDSR) and its application to Yb are presented, and obtained analytical expressions of the mass and pole residue of Yb are given up to dimension six operators. Numerical analysis is performed and results are obtained in Section 3. Concluding remarks are discussed in Section 4. The explicit forms of the spectral densities are written in Appendix.
2. Finite Temperature Sum Rules for Tetraquark Assignment
QCD sum rules (QCDSR) approach is based on Wilson’s operator product expansion (OPE) which was adapted by Shifman, Vainshtein, and Zakharov [24] and applied with remarkable success to estimate a large variety of properties of all low-lying hadronic states [25–27]. Later, this model is extended to its thermal version that is firstly proposed by Bochkarev and Shaposnikov and led to many successful applications in QCD at T≠0 [28–35]. Very recently, in [36], the authors claimed that any QCDSR study on tetraquark states should contain O(αs2) contributions to OPE, which are unknown. It is a very important and strong argument for studying tetraquark states within QCDSR; however, those terms and their thermal behaviors are not known, and their calculation is beyond the scope of this work. Thus we follow the traditional sum rules to investigate the thermal behavior of hadronic parameters of Yb, which were used very successfully in predicting properties of tetraquark states as well. In this work, we proceed with traditional sum rules analysis by using a tetraquark current, following several successful applications to exotic hadrons [37–42].
In this section, the mass and pole residue of the exotic Yb resonance are studied by interpreting it as a bound bs[b¯s¯] tetraquark via TQCDSR technique which starts with the two point correlation function(1)Πμνq,T=i∫d4xeiq·xΨTημxην†0Ψ,where Ψ represents the hot medium state, ημ(x) is the interpolating current of the Yb state, and T denotes the time ordered product. The thermal average of any operator O^ in thermal equilibrium is given as(2)O^=Tre-βHO^Tre-βH,where H is the QCD Hamiltonian, and β=1/T is inverse of the temperature, and T is the temperature of the heat bath. Chosen current ημ(x) must contain all the information of the related meson, like quantum numbers, quark contents and so on. In the diquark-antidiquark picture, tetraquark current interpreting Yb can be chosen as [10](3)ημx=iϵϵ~2saTxCγ5Qbxs¯dxγμγ5CQ¯eTx+saTxCγ5γμQbxs¯dxγ5CQ¯eTx,where Q=b, C is the charge conjugation matrix and a,b,c,d,e are color indices. Shorthand notations ϵ=ϵabc and ϵ~=ϵdec are also employed in (3).
In TQCDSR, the correlation function given in (1) is calculated twice, as in two different regions corresponding two perspectives, namely the physical side (or phenomenological side) and the QCD side (or OPE side). By equating these two approaches, the sum rules for the hadronic properties of the exotic state under investigation are achieved. To derive mass and pole residue via TQCDSR, the correlation function is calculated in terms of hadronic degrees of freedoms in the physical side. A complete set of intermediate physical states possessing the same quantum number as the interpolating current are inserted into (1), and integral over x is handled. After these manipulations, the correlation function is obtained as(4)ΠμνPhysq,T=ΨημYbqTYbqην†ΨTmYb2T-q2+subtractedterms,here mYb(T) is the temperature-dependent mass of Yb meson. Temperature-dependent pole residue λYb(T) is defined in terms of matrix element as(5)ΨημYbqT=λYbTmYbTεμ,where εμ is the polarization vector of the Yb satisfying(6)εμεν∗=-gμν+qμqνmYb2T.After employing polarization relations, the correlation function is written in terms of Lorentz structures in the form(7)ΠμνPhysq,T=mYb2TλYb2TmYb2T-q2-gμν+qμqνmYb2T+…,where dots denote the contributions coming from the continuum and higher states. To obtain the sum rules, coefficient of any Lorentz structure can be used. In this work, coefficients of gμν are chosen to construct the sum rules and the standard Borel transformation with respect to q2 is applied to suppress the unwanted contributions. The final form of the physical side is obtained as(8)Bq2ΠPhysq,T=mYb2TλYb2Te-mYb2T/M2,here M2 is the Borel mass parameter. In the QCD side, ΠμνQCD(q,T) is calculated in terms of quark-gluon degrees of freedom and can be separated into two parts over the Lorentz structures as(9)ΠμνQCDq,T=ΠSQCDq2,Tqμqνq2+ΠVQCDq2,T-gμν+qμqνq2, where ΠSQCD(q2,T) and ΠVQCD(q2,T) are invariant functions connected with the scalar and vector currents, respectively. In the rest framework of Yb (q=0), ΠVQCD(q02,T) can be expressed as a dispersion integral,(10)ΠVQCDq02,T=∫4mb+ms2s0TρQCDs,Ts-q02ds+⋯, where corresponding spectral density is described as(11)ρQCDs,T=1πImΠVQCDs,T. The spectral density can be separated in terms of operator dimensions as(12)ρQCDs,T=ρpert.s+ρq¯qs,T+ρG2+Θ00s,T+ρq¯Gqs,T+ρq¯q2s,T.In order to obtain the expressions of these spectral density terms, the current expression given in (3) is inserted into the correlation function given in (1) and then the heavy and light quark fields are contracted, and the correlation function is written in terms of quark propagators as(13)ΠμνQCDq,T=-i2∫d4xeiq·xϵϵ~ϵ′ϵ~′Trγμγ5S~baa′-xγ5γνSsbb′-x×Trγ5S~sdd′xγ5Sbee′x+Trγ5S~baa′-xγ5Ssbb′-xγμ×Trγ5S~sdd′xγ5S~bee′xγνγ5Sbbb′x+Trγ5S~baa′-xγ5γν×Ssbb′-xTrγ5S~sdd′xγ5γμSbee′x+Trγ5S~baa′-xγ5×Ssbb′-xTrγ5S~sdd′-xγ5Sbee′xγνT, where Ss,bij(x) are the full quark propagators and S~s,bij(x)=CSs,bijT(x)C is used. The quark propagators are given in terms of the quark and gluon condensates [25]. At finite temperatures, additional operators arise due to the breaking of Lorentz invariance by the choice of thermal rest frame. Thus, the residual O(3) invariance brings additional operators to the quark propagator at finite temperature. The expected behavior of the thermal averages of these new operators is opposite of those of the Lorentz invariant old ones [43]. The heavy-quark propagator in coordinate space can be expressed as(14)Sbijx=i∫d4k2π4e-ik·xδijk+mbk2-mb2-gGijαβ4σαβk+mb+k+mbσαβk2-mb22+g212GαβAGAαβδijmbk2+mbkk2-mb24+…,and the thermal light quark propagator is chosen as(15)Ssijx=ix2π2x4δij-ms4π2x2δij-s¯s12δij-x2192m02s¯s1-ims6xδij+i3xms16s¯s-112uμΘμνfuν+13u·xuuμΘμνfuνδij-igsGijαβ32π2x2xσαβ+σαβx,where Gijαβ≡GAαβtijA is the external gluon field, tijA=λijA/2 with λijA Gell-Mann matrices, A=1,2,…,8 symbolizes color indices, ms implies the strange quark mass, uμ is the four-velocity of the heat bath, 〈q¯q〉 is the temperature-dependent light quark condensate, and Θμνf is the fermionic part of the energy-momentum tensor. Furthermore, the gluon condensate related to the gluonic part of the energy-momentum tensor Θαβg is expressed via relation [43]:(16)TrcGαβGλσT=gαλgβσ-gασgβλA-uαuλgβσ-uαuσgβλ-uβuλgασ+uβuσgαλB,where A and B coefficients are(17)A=124GαβaGaαβT+16uαΘαβguβT,B=13uαΘαβguβT.In order to remove contributions originating from higher states, the standard Borel transformation with respect to q02 is applied in the QCD side as well. By equating the coefficients of the selected structure gμν in both physical and QCD sides, and by employing the quark hadron duality ansatz up to a temperature-dependent continuum threshold s0(T), the final sum rules for Yb are derived as(18)mYb2TλYb2Te-mYb2T/M2=∫4mb+ms2s0TdsρQCDs,Te-s/M2.To find the mass via TQCDSR, one should expel the hadronic coupling constant from the sum rules. It is commonly done by dividing the derivative of the sum rule given in (18) with respect to (-M-2) to itself. Following these steps, the temperature-dependent mass is obtained as(19)mYb2T=∫4mb+ms2s0TdssρQCDs,Te-s/M2∫4mb+ms2s0TdsρQCDs,Te-s/M2,where the thermal continuum threshold s0(T) is related to continuum threshold s0 at vacuum via relation(20)s0T=s01-TTc8+4mb+ms2TTc8[44, 45]. For compactness, the explicit forms of spectral densities are presented in Appendix.
3. Phenomenological Analysis
In this section, the phenomenological analysis of sum rules obtained in (18) and (19) is presented. First, the input parameters and temperature dependance of relevant condensates are given. Following, the working regions of the obtained sum rules at vacuum are analyzed. The behavior of QCD sum rules at T=0 is used to test the reliability of our analysis.
3.1. Input Parameters
During the calculations, input parameters given in Table 1 are used. In addition to these input parameters, temperature-dependent quark and gluon condensates, and the energy density expressions are necessary. The thermal quark condensate is chosen as(21)q¯q=0q¯q01+exp18.100421.846921/GeV2T2+4.992161/GeVT-1,where 0q¯q0 is the light quark condensate at vacuum and which is credible up to a critical temperature Tc=190MeV. The expression given in (21) is obtained in [46, 47] from the Lattice QCD results given in [48, 49]. The temperature-dependent gluon condensate is parameterized via [46, 50](22)G2=0G201-1.65TTc8.735+0.04967TTc0.7211,where 0G20 is the gluon condensate in vacuum state and G2=GαβAGAαβ. Additionally, for the gluonic and fermionic parts of the energy density, the following parametrization is used [46](23)Θ00g=Θ00f=T4exp113.8671GeV2T2-12.1901GeVT-10.1411GeVT5,which is extracted from the Lattice QCD data in [51].
Input parameters [20–23].
ms=0.13±0.03MeV
mb=4.24±0.05GeV
m02=0.8±0.2GeV2
ss-=-0.8×0.24±0.013GeV3
01παsG20=0.022GeV4
3.2. Analysis of Sum Rules at <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M142"><mml:mi>T</mml:mi><mml:mo>=</mml:mo><mml:mn mathvariant="normal">0</mml:mn></mml:math></inline-formula>
In order to get reliable results, obtained sum rules should be tested at vacuum, and the working regions of the parameters s0 and M2 should be determined. Within the working regions of s0 and M2, convergence of OPE and dominance of pole contributions should be assured. In addition, the obtained physical results should be independent of small variations of these parameters. Convergence of the OPE is tested by the following criterion. The contribution of the highest order operator in the OPE should be very small compared to the total contribution. In Figure 1, the ratio of the sum of the terms up to the specified dimension to the total contribution is plotted to test the OPE convergence. It is seen that all higher order terms contribute less than the perturbative part for M2≥6 GeV2. On the other hand, dominance of the pole contribution is tested as follows. The contribution coming from the pole of the ground state should be greater than the contribution of the continuum. In this work, the aforementioned ratio is(24)PC=ΠMmax2,s0ΠMmax2,∞>0.50,when M2≤10GeV2 as can be seen in Figure 1. After checking these criteria, the working regions of the parameters M2 and s0 are determined as (25)6GeV2≤M2≤10GeV2;132GeV2≤s0≤134GeV2, which is also consistent with s0≃(mH+0.5 GeV)2 norm [10]. Within these working regions, the variations of the mass of Yb with respect to M2 and s0 are plotted in Figure 2. It is seen that the mass is stable with respect to variations of M2 and s0.
The OPE convergence of the sum rules: the ratio of the sum of the contributions up to specified dimension to the total contribution is plotted with respect to M2 at s0=134GeV2,T=0 (left). Pole dominance of the sum rules: relative contributions of the pole (blue) and continuum (red-dashed) versus to the Borel parameter M2 at s0=134GeV2,T=0 (right).
Mass of Yb as a function of M2 (left) and s0 (right).
4. Results and Discussions
Following the analysis presented in previous section, the mass of the ground state estimated by the tetraquark current given in (3) at T=0 is presented together with other results from literature in Table 2. It is seen that our results agree with other theoretical estimates and also with the experimental data on Yb(10890) [10, 11, 20]. Thus, the broad resonance Yb can be described by the tetraquark current given in (3), and our analysis can be extended to finite temperatures.
Results obtained in this work for the mass of Yb at T=0, in comparison with literature.
mYb(MeV)
Present Work
10735-107+122
Experiment
10889.9-2.6+3.2 [20]
QCDSR
10880±130 [11]
10910±70 [10]
To analyze the thermal properties of Yb(10890) resonance, the temperature dependencies of the mass and the pole residue of Yb are plotted in Figure 3. It is seen that the mass and the pole residue of Yb stay monotonous until T≅0.12GeV. However, after this point, they begin to decrease promptly with increasing temperature. At the vicinity of the critical (or so called deconfinement) temperature, the mass reaches nearly 87% of its vacuum value. On the other hand, the pole residue decreased to 44% of its value at vacuum as shown in Figure 3.
The mass (left) and pole residue (right) of Yb as a function of temperature.
Our predictions presented in Figure 3 are in good agreement with other QCD sum rules analysis on thermal behaviors of conventional or exotic hadrons [35–38]. However in [19], authors predicted a decrease of 5% in the mass of molecular 1++ state, and no change in the mass of charmonium 2-+ state, even beyond Hagedorn temperature TH~177 MeV. Since the decay properties also depend on temperature, and while the mass and pole residue diminish, the decay width might increase with increasing temperature [52] and decay widths at finite temperature should also be investigated. However the current status of Yb resonance is very complicated, since it is very close to Υ(5S) state. Thus studying its decays requires establishment of a good model in the hidden-beauty sector.
Finally, we would like to highlight the following remarks:
We observed that the mass (the pole residue) of exotic Yb(10890) state starts to decrease near T/Tc≃0.7.
Both quantities tend to diminish with increasing temperature up to critical temperature Tc.
Even though the sum rules at T=0 estimates the mass of Yb consistent with experimental data, more theoretical efforts are required to discriminate Yb and Υ(5S).
In order to get more reliable results on tetraquarks from QCD sum rules, O(αs2) contributions suggested in [36] should be investigated.
In summary, we revisited the hidden-beauty exotic state Yb and studied its properties at vacuum and finite temperatures. To describe the hot medium effects to the hadronic parameters of the resonance Yb, TQCDSR method is used considering contributions of condensates up to dimension six. Our results for T=0 are in reasonable agreement with the available experimental data and other QCD studies in the literature. Numerical findings show that Yb can be well described by a scalar-vector tetraquark current. In the literature, remarkable drop in the values of the mass and the pole residue in hot medium was regarded as the signal of the QGP, which is called the fifth state of matter, phase transition. We hope that precise spectroscopic measurements in the exotic bottomonium sector can be done at Super-B factories, and this might provide conclusive answers on the nature and thermal behaviors of the exotic states.
In this appendix, the explicit forms of the spectral densities obtained in this work are presented. The expressions for ρpert.(s) and ρnonpert.(s,T) are shown below as integrals over the Feynman parameters z and w, where θ is the step function, (A.1)ρpert.s=13072π6∫01dz∫01-zdw1κ8ξ2-κmb2z+w+szwξ2κ2mb4zwz+ww2+-1+ww+z-1+4w-2κmb26ms2Φ27-1+zz-7+8zw+7w2+sz2w212-1+zz+-12+25zw+12w2ξ+szw12ms2Φ2+35sz2w2ξ3θLs,w,z,(A.2)ρss¯s,T=s¯s128π4∫01dz∫01-zdw1κ6κ3mb5wz+w2+κ2mb4msz+w19z4+19-1+w2w2+2z-1+ww-19+25w+z3-38+50w+z219+w-88+81w-κ2mb3zz+wms2Φ+2sw2ξ-κmb2msszw22z4+22-1+w2w2+z3-44+111w+z-1+ww-44+111w+z222+w-155+197wξ+κmbsz2wms2Φ+sw2ξ2+3mss2z2w2z2+-1+ww+z-1+21wξ3θLs,w,z,(A.3)ρG2+Θ00s,T=14608π2∫01dz∫01-zdw1κ6ξ2192π2Θ00fzwξ2mb4Φ2z+w×3-1+zz+-3+5zw+3w2+mb2Φszw-27-1+zz+27w-53zw-27w2ξ+30s2z2w2ξ3-gs2Θ00g3mb4Φ2zwz+w2-1+z2z2+-1+zz-4+3zw+-2+z-1+3zw2+-4+3zw3+2w4-mb2Φsz2w224-1+z2z2+-1+zz-48+85zw+24+z-133+121zw2+-48+85zw3+24w4ξ-12mb3msΦ3z+w3ξ2+12mbmsΦ2szzz2+10zw+w2ξ3+30s2z3w3z+wξ4+αsG2ππ2κmb2-6ms2Φ25-1+zz3+z3w+-5+zw3+5w4+sz2w22-1+zz2-18+11z+z72+z-221+133zw+36+z-221+231zw2-58+133zw3+22w4ξ-36κ3mb3msz+wz2-6zw+w2ξ2-60s2z3w3z+wξ4+ξ2mbzwmb3z+w4-1+zz2-3+4z+6z4+z-11+8zw+12+11z-6+5zw24-7+12z3+16w4+12mss3z2-26zw+3w2ξ3θLs,w,z(A.4)ρs¯Gss,T=msm02s¯s64π4∫01dz3mb2+sz-1zθL′s,z+∫01dz∫01-zdw13κ5×zwξκmb25z2+5w-1w+z11w-5-16szwξ2θLs,w,z,(A.5)ρs¯s2s,T=s¯s25184π4∫01dz648mb2π2+gs2mbmsz+54π25ms2+4sz-1zθL′s,z+gs2∫01dz∫01-zdwzwξκ5×3κmb27z2+7w-1w+z-7+15w-64szwξ2θLs,w,z,where explicit expressions of the functions L(s,w,z) and L′(s,z) are (A.6)Ls,w,z=κ-2-1+w-1+ww2+2-1+wwz+-1+2wz2-swzξ+z3mb2,(A.7)L′s,z=sz1-z-mb2. The below definitions are used for simplicity:(A.8)κ=z2+zw-1+w-1w,Φ=z-1w+z-1w+w2,ξ=z+w-1.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
J. Y. Süngü, A. Türkan, and E. Veli Veliev thank Kocaeli University for the partial financial support through the grant BAP 2018/082. H. Dag˘ acknowledges support through the Scientific and Technological Research Council of Turkey (TUBITAK) BIDEP-2219 grant.
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