Double-gluon charmonium hybrid states with various (exotic) quantum numbers

We study the double-gluon charmonium hybrid states with various quantum numbers, each of which is composed of one valence charm quark and one valence charm antiquark as well as two valence gluons. We concentrate on the exotic quantum numbers $J^{PC} =0^{--}/0^{+-}/1^{-+}/2^{+-}/3^{-+}$ that the conventional $\bar q q$ mesons can not reach. We apply the QCD sum rule method to calculate their masses to be $7.28^{+0.38}_{-0.43}$ GeV, $5.19^{+0.36}_{-0.46}$ GeV, $5.46^{+0.41}_{-0.62}$ GeV, $4.48^{+0.25}_{-0.31}$ GeV, and $5.54^{+0.35}_{-0.43}$ GeV, respectively. We study their possible decay patterns and propose to search for the $J^{PC}=2^{+-}/3^{-+}$ states in the $D^*\bar D^{(*)}/D^{*}_s \bar D^{(*)}_s/\Sigma_c^* \bar \Sigma_c^{(*)}/\Xi_c^* \bar \Xi_c^{(\prime,*)}$ channels. Experimental investigations on these states and decay channels can be useful in classifying the nature of the hybrid state, thus serving as a direct test of QCD in the low energy sector.

In this letter we investigate the double-gluon charmonium hybrid states, each of which is composed of one valence charm quark and one valence charm antiquark as well as two valence gluons.We construct twenty doublegluon charmonium hybrid currents with various quantum numbers, and use them to perform QCD sum rule analyses.We refer to Ref. [27] for more QCD sum rule studies.Especially, these currents can reach the exotic quantum numbers J P C = 0 −− /0 +− /1 −+ /2 +− /3 −+ , whose masses are calculated to be 7.28 We further study their possible decay patterns from the two-/three-meson and two-baryon decay processes.Since these three processes are both at the O(α s ) order, the three-meson and two-baryon decay patterns are generally not suppressed severely compared to the two-meson decay pattern.Especially, we propose to search for the c channels directly at LHC, given that they may have relatively smaller widths due to their limited decay patterns.Experimental investigations on these states and decay channels can be useful in classifying the nature of the hybrid state, thus serving as a direct test of QCD in the low energy sector.
Double-gluon charmonium hybrid currents -As the first step, we combine the charm quark field c a (x), the charm antiquark field ca (x), the gluon field strength tensor G n µν (x), and the dual gluon field strength tensor Gn µν (x) = G n,ρσ (x) × ϵ µνρσ /2 to construct the doublegluon charmonium hybrid currents.Here a = 1 • • • 3 and n = 1 • • • 8 are color indices, and µ • • • σ are Lorentz indices.These currents can be generally constructed by combining the color-octet quark-antiquark fields and the color-octet double-gluon fields where d npq and f npq are the totally symmetric and antisymmetric SU (3) structure constants, respectively.
In the present study we shall investigate as many as twenty double-gluon charmonium hybrid currents with various quantum numbers J P C .We write them as and ca λ ab n γ µ c b , respectively: J α1β1,α2β2 J α1β1,α2β2 Here S represents the symmetrization and subtracting trace terms in the two sets The double-gluon hybrid currents with the light quarkantiquark fields qa λ ab n γ 5 q b and qa λ ab n σ µν q b (q = u, d, s) have been systematically investigated in Refs.[28][29][30], and in the present study we just need to replace the light quark fields by the charm quark fields.However, these currents can only reach the exotic quantum numbers J P C = 1 −+ /2 +− /3 −+ , and we need the other two currents J α 1 ±− C with the quark-antiquark field ca λ ab n γ µ c b in order to study the exotic quantum numbers J P C = 0 −− /0 +− , as discussed below.
QCD sum rule analyses -The QCD sum rule method has been widely applied in the study of hadron physics [31][32][33][34].In this letter we apply this method to study the double-gluon charmonium hybrid currents listed in Eqs.(3).We use the current J α 1 +− C as an example and calculate its two-point correlation function = (q α q β − q 2 g αβ ) Π 1 (q 2 ) + q α q β Π 0 (q 2 ) , at both the hadron and quark-gluon levels.The correlation functions Π 1 (q 2 ) and Π 0 (q 2 ) are respectively contributed by the J P C = 1 +− and 0 −− states through ⟨0|J α We concentrate on the exotic term Π 0 (q 2 ) and extract its spectral density ρ(s) ≡ ImΠ 0 (s)/π through the dispersion relation where s < = 4m 2 c is a kinematic limit, i.e., the square of the sum of the current charm quark masses of the hadron.
At the hadron level we parameterize Π 0 (q 2 ) using one-pole-dominance assumption for the possibly-existing ground state |X; 0 −− C ⟩ and the continuum At the quark-gluon level we calculate Π 0 (q 2 ) and extract the spectral density ρ(s) using the method of operator product expansion (OPE).The obtained results are given in the supplementary file "OPE.nb", with the integration parameters , and β max = 1 − α.The other spectral densities calculated in the present study are also summarized there.In the calculations we have taken into account the Feynman diagrams depicted in Fig. 1, and calculated ρ(s) up to the dimension eight (D = 8) condensates.We have calculated all the diagrams proportional to α 2 s × g 0 s and α 2 s × g 1 s , while we have partly calculated the diagrams proportional to α 2 s ×g n≥2 s .The gluon field strength tensor can be naturally separated into two parts: we use the single-gluon-line to describe the former two terms µ , and we use the double-gluon-line with a red vertex to describe the third term g s f npq A p,µ A q,ν , e.g., see the diagram depicted in Fig. 1(c-3).Eq. ( 9) indicates that there can exist a significant mixing among the single-/double-/triple-gluon hybrid states, and moreover, these FIG. hybrid states can also mix with the conventional mesons, tetraquark states, and glueballs, etc.It is still difficult to investigate this effect, so there is still a long long way to understand glueballs and hybrid states as well as the gluon degree of freedom.
After performing the Borel transformation to Eq. ( 7) at both the hadron and quark-gluon levels, we obtain where the continuum has been approximated as the OPE spectral density above the threshold value s 0 .Eq. ( 10) can be used to calculate the mass of |X; Numerical analyses -We perform numerical analyses using the following values for various QCD parameters at the QCD scale Λ QCD = 300 MeV and the renormalization scale 2 GeV [1,35,36]: As shown in Eq. ( 11), the mass of |X; 0 −− C ⟩ depends on two free parameters: the Borel mass M B and the threshold value s 0 .Firstly, we investigate the OPE convergence by requiring a) the α 2 s ×g n≥2 s terms to be less than 5%, b) the D = 8 terms to be less than 10%, and c) the D = 6 terms to be less than 20%: Secondly, we investigate the one-pole-dominance assumption by requiring the pole contribution (PC) to be larger than 40%: Altogether, we determine the Borel window to be 8.86 GeV 2 ≤ M 2 B ≤ 10.30 GeV 2 when setting s 0 = 64.0GeV 2 .We redo the same procedures and find that there exist the Borel windows as long as s 0 ≥ s min 0 = 58.3GeV 2 .Accordingly, we set s 0 to be slightly larger and determine the working regions to be 51.0GeV 2 ≤ s 0 ≤ 77.0 GeV 2 and 8.86 GeV 2 ≤ M 2 B ≤ 10.30 GeV 2 , where the mass of |X; 0 −− C ⟩ is calculated to be Its uncertainty comes from the threshold value s 0 , the Borel mass M B , and various QCD parameters listed in Eqs.(12).We show M |X;0 −− C ⟩ in Fig. 2 with respect to s 0 and M B .As shown in Fig. 2(a), we find a mass minimum around s 0 ∼ 45 GeV 2 , and the mass dependence on s 0 is moderate and acceptable inside the region 51.0 GeV 2 ≤ s 0 ≤ 77.0 GeV 2 .As shown in Fig. 2(b), the mass dependence on M B is weak inside the Borel window 8.86 GeV 2 < M 2 B < 10.30 GeV 2 .Similarly, we apply the QCD sum rule method to study the other nineteen double-gluon charmonium hybrid currents listed in Eqs.(3).The obtained results are summarized in Table I.
Decay analyses -As depicted in Fig. 3, the doublegluon charmonium hybrid states can decay after exciting two qq (q = u, d, s) pairs from two gluons, followed by Working Regions  recombining three color-octet cc/qq pairs into two/three color-singlet mesons or two color-singlet baryons: These three decay processes are both at the O(α s ) order, so the three-meson and two-baryon decay patterns are generally not suppressed severely compared to the twomeson decay patterns.Comparatively speaking, their decays into one charmonium meson and light mesons are at the O(α 2 s ) order and so suppressed, but these channels can be observed in experiments more easily, such as J/ψππ, J/ψπππ, and J/ψK K, etc.We list in Table II possible S-wave and P -wave as well as several D-wave decay patterns of the double-gluon charmonium hybrid states with the exotic quantum numbers J P C = 0 −− /0 +− /1 −+ /2 +− /3 −+ , separately for the two-/three-meson and two-baryon decay processes.
TABLE II: Possible S-wave (red) and P -wave (blue) as well as several D-wave (green) decay patterns of the double-gluon charmonium hybrid states with the exotic quantum numbers J P C = 0 −− /0 +− /1 −+ /2 +− /3 −+ , separately for the two-/three-meson and two-baryon decay processes.Some chargeconjugated decay patterns are omitted for simplicity.
The above mass values are accessible in the LHC experiments.
We propose to search for them experimentally in their possible decay channels D ( * ) D( * ) (π/η/η ′ /ρ/ω), D c channels directly at LHC. Experimental investigations on these states and decay channels can be useful in classifying the nature of the hybrid state, thus serving as a direct test of QCD in the low energy sector.