Two- and three-gluon glueballs of $C=+$

We study two- and three-gluon glueballs of $C=+$ using the method of QCD sum rules. We systematically construct their interpolating currents, and find that all the spin-1 currents of $C=+$ vanish. This suggests that the ``ground-state'' spin-1 glueballs of $C=+$ do not exist within the relativistic framework. We calculate masses of the two-gluon glueballs with $J^{PC} = 0^{\pm+}/2^{\pm+}$ and the three-gluon glueballs with $J^{PC} = 0^{\pm+}/2^{\pm+}$. We propose to search for the $J^{PC} = 0^{-+}/2^{-\pm}/3^{\pm-}$ three-gluon glueballs in their three-meson decay channels in future BESIII, GlueX, LHC, and PANDA experiments.

Recently the D0 and TOTEM Collaborations studied pp and pp [31] cross sections, which are found to be different with a significance of 3.4σ [32]. Together with their previous result [33], this significance can be increased to 5.2σ-5.7σ. The above difference leads to the evidence of a t-channel exchanged odderon [34][35][36][37][38], that is predominantly a three-gluon glueball of C = −. We refer to Refs. [39][40][41][42][43][44][45][46][47] and the review [48] for more discussions. Due to these studies, interests in glueballs are reviving recently. Since the above odderon evidence is still indirect, it is crucial and important to directly study the glueball itself.
The lowest-lying two-, three-, and four-gluon glueballs have been systematically investigated in Ref. [49], where the authors constructed their corresponding nonrelativistic low-dimension operators. These operators have been successfully used in Lattice QCD calculations. In this paper we systematically study two-and threegluon glueballs of C = +. We shall construct their corresponding relativistic glueball currents, and calculate masses of these glueballs using the method of QCD sum rules. The same approach has been applied in Ref. [50] to study three-gluon glueballs of C = −, so a rather com-Accordingly, it couples to both positive-and negativeparity charmonia through 0|J αβ |h c (ǫ, p) = if T hc ǫ αβµν ǫ µ p ν , where f T hc and f T J/ψ are relevant decay constants. Given the Lorentz structures of J/ψ and h c to be totally different, they can be clearly separated from each other. For example, we can isolate h c at the hadron level by investigating the two-point correlation function containing since the correlation function of J/ψ dose not contain the above coefficient. It is not so easy to isolate J/ψ from J αβ at the hadron level. Instead, we can investigate its partner currentJ which couples to J/ψ and h c just in the opposite ways: Accordingly, we can use the two currents J αβ andJ αβ to study and well separate J/ψ and h c . We apply the above process to generally investigate the current J α1···αN ,β1···βN , which has 2N = 2J Lorentz indices with certain symmetries, e.g., the spin-2 current J α1α2,β1β2 has four Lorentz indices, satisfying (8) Its coupling can be written as: where X is the corresponding state having the same parity as J i1···iN ,j1···jN (i 1 · · · j N = 1, 2, 3); S denotes symmetrization and subtracting trace terms in the two sets {α 1 · · · α N } and {β 1 · · · β N } simultaneously, with Note that the current J α1···αN ,β1···βN can also couple to the other state X ′ having the parity opposite to X, but this state X ′ can not be easily isolated at the hadron level, so we do not consider it in the present study.

B. Two-gluon glueball currents
In this subsection we use the gluon field strength tensor G a µν to construct two-gluon glueball currents, with a the color index and µ, ν the Lorentz indices. We also needG a µν = G a,ρσ × ǫ µνρσ /2 to denote the dual gluon field strength tensor, and f abc to denote the totally antisymmetric SU (3) C structure constant. In the present study we only consider local glueball currents without explicit derivatives, although G a µν andG a µν contain covariant derivatives inside themselves.
In Ref. [49] the authors use the chromoelectric and chromomagnetic fields (i, j = 1, 2, 3), to write down all the non-relativistic low-dimension twogluon glueball operators: where S ′ denotes symmetrization and subtracting trace terms in the set {ij}.
We construct their corresponding relativistic currents in order to perform QCD sum rule analyses: We shall explicitly prove in Appendix A that the third current J αβ 1 vanishes, suggesting that the "ground-state" two-gluon glueball of J P C = 1 −+ does not exist within the relativistic framework.
The former two currents J 0 of J P C = 0 ++ andJ 0 of J P C = 0 −+ couple to the J P C = 0 ++ and 0 −+ two-gluon glueballs |GG; J P C , respectively: where f 0 ++ and f 0 −+ are decay constants. Besides, the current J 0 has a partner, whose sum rule result is the same as that of J 0 . The latter two currents J α1α2,β1β2 2 andJ α1α2,β1β2 2 couple to the J P C = 2 ++ and 2 −+ glueballs through: The current J α1α2,β1β2 2 also has a partner, whose sum rule result is the same as that of J α1α2,β1β2 2 .
C. Three-gluon glueball currents of C = + In this subsection we use G a µν andG a µν to construct three-gluon glueball currents of C = +. Some of their corresponding non-relativistic operators have been constructed in Ref. [49]: We further construct their corresponding relativistic currents as follows: η αβ η αβ We shall explicitly prove in Appendix A that the third and fourth currents η αβ 1 andη αβ 1 both vanish, suggesting that the "ground-state" three-gluon glueballs of J P C = 1 ++ and 1 −+ do not exist within the relativistic framework.
We take the currentJ 0 defined in Eq. (14) as an example, and calculate its two-point correlation function separately at hadron and quark-gluon levels.
At the hadron level we express Eq. (35) using the dispersion relation as with ρ(s) = ImΠ(s)/π the spectral density. It is parameterized using one pole dominance for the ground state X as well as the continuum contribution, At the quark-gluon level we insert Eq. (14) into Eq. (35), and calculate it using the method of operator product expansion (OPE). After performing the Borel transformation to Eq. (36) at both hadron and quarkgluon levels, we approximate the continuum using the spectral density above a threshold value s 0 , and obtain This equation can be used to further calculate the mass of X through Since the gluon field strength tensor G a µν is defined as it can be naturally separated into two parts. As shown in Fig. 1, we depict the former two terms using the single-gluon-line, and the latter one term using the double-gluon-line with a red vertex (see also the diagram FIG. 1: The gluon field strength tensor G a µν = ∂µA a ν −∂νA a µ + gsf abc A b,µ Ac,ν, naturally separated into two parts (a) and (b). Fig. 2(c − 3)). Here A a µ is the gluon field, whose propagator is [57]: We work in the fixed-point gauge so that In the present study we consider the Feynman diagrams depicted in Fig. 2 (for three-gluon glueballs), and calculate OPEs up to the dimension eight (D = 8) condensates. We take into account the perturbative term, the two-gluon condensate g 2 s GG , the three-gluon condensate g 3 s G 3 , and the D = 8 condensate g 2 s GG 2 : In the calculations we have considered all the diagrams proportional to α n s × g 0 s and α n s × g 1 s (n = 2 for two-gluon glueballs and n = 3 for three-gluon glueballs); however, there are so many diagrams proportional to α n s × g 2 s , so we have only taken into account one of them. Especially, we find all the D = 8 terms proportional to g 2 s GG 2 vanish, so the convergence of the above OPE series are quite good.
In Ref. [22] the authors studied J P C = 0 ++ threegluon glueballs using the current η 0 defined in Eq. (25), where they calculated the Feynman diagrams depicted in Figs. 2(a, b − i, c − 1, c − 2). In Ref. [24] the authors studied J P C = 0 −+ three-gluon glueballs using the currentη 0 defined in Eq. (26), where they calculated the diagrams depicted in Figs. 2(a, b − i, c − i). Their calculations are done (mainly) by hand. In the present study we use the software Mathematica with the package Feyn-Calc, and we can obtain exactly the same results for these diagrams. In Refs. [16,21,23] the authors studied J P C = 0 ++ and 0 −+ two-gluon glueballs using the currents J 0 andJ 0 defined in Eqs. (13) and (14), where they calculated more diagrams than those calculated in the present study. However, such calculations are too complicated to be applied to three-gluon glueballs, and we still calculate similar diagrams as those depicted in Fig. 2 for two-gluon glueballs to make the present study unified as a whole.
For completeness, we also investigate the following three-gluon glueball currents of C = −: FIG. 2: Feynman diagrams for three-gluon glueball currents, including the perturbative term, the two-gluon condensate g 2 s GG , the three-gluon condensate g 3 s G 3 , and the D = 8 condensate g 2 s GG 2 . The diagrams (a) and (b − i) are proportional to where d abc is the totally symmetric SU (3) C structure constant. Their sum rule equations are: s e −s/M 2 B ds, s e −s/M 2 B ds.
The above three-gluon glueball currents of C = − have been systematically studied in Ref. [50], but there we did not calculate the Feynman diagrams depicted in Figs. 2(c − 3, c − 4, c − 5). Similar to Eqs. (43)-(50), we find all the D = 8 terms proportional to g 2 s GG 2 vanish, so the convergence of the above OPE series are also quite good.
We shall use the above sum rule equations to perform numerical analyses in the next section.

IV. NUMERICAL ANALYSES
In this section we perform numerical analyses using the sum rules given in Eqs. (43)-(50) and Eqs. (57)-(62). The glueball mass M X depends significantly on the gluon condensates g 2 s GG and g 3 s G 3 , both of which are still not precisely known. In the present study we use the following values for these parameters [58,59]: Besides, we use the following value for the strong coupling constant at the QCD scale Λ QCD = 300 MeV [60]:  We still take the currentJ 0 as an example, and use Eq. (39) to calculate the mass of |GG; 0 −+ . It depends on two free parameters, the Borel mass M B and the threshold value s 0 . We use two criteria to determine the Borel window. The first criterion is to insure the convergence of OPE by requiring a) the α 2 s × g 2 s term α 2 s g 2 s GG to be less than 5%, and b) the D = 6 term α s g 3 s G 3 to be less than 10%: As shown in Fig. 3 using the dashed curves, we determine the lower limit of M B to be M 2 B ≥ 3.28 GeV 2 when setting s 0 = 9.0 GeV 2 .
The above condition is the cornerstone of a reliable sum rule analysis, where we have taken into account two terms because the OPE is expanded in two directions: the dimension of condensates and the coupling constant g s . Eqs. (65) and (66) are for two-gluon glueball currents, and the conditions for three-gluon glueball currents are The second criterion is to insure the one-poledominance assumption by requiring the pole contribution (PC) to be larger than 40%: As shown in Fig. 3 using the solid curve, we determine the upper limit of M B to be M 2 B ≤ 3.70 GeV 2 when setting s 0 = 9.0 GeV 2 .
Altogether we determine the Borel window to be 3.28 GeV 2 ≤ M 2 B ≤ 3.70 GeV 2 for the fixed threshold value s 0 = 9.0 GeV 2 . Then we redo the same procedures by changing s 0 , and find that there exist non-vanishing Borel windows as long as s 0 ≥ s min 0 = 8.2 GeV 2 . Accordingly, we choose s 0 to be slightly larger, and determine our working regions to be 8.0 GeV 2 ≤ s 0 ≤ 10.0 GeV 2 and 3.28 GeV 2 ≤ M 2 B ≤ 3.70 GeV 2 , where we calculate the mass of |GG; 0 −+ to be Its central value corresponds to M 2 B = 3.49 GeV 2 and s 0 = 9.0 GeV 2 , and its uncertainty comes from the threshold value s 0 , the Borel mass M B , and the gluon condensates listed in Eqs. (63).
We show M |GG;0 −+ in the left panel of Fig. 4 as a function of the Borel mass M B , and find it quite stable inside the Borel window 3.28 GeV 2 ≤ M 2 B ≤ 3.70 GeV 2 . We also show it in the right panel of Fig. 4 as a function of the threshold value s 0 . We find there exists a mass minimum around s 0 ∼ 5 GeV 2 , and the s 0 dependence is weak and acceptable inside the working region 8.0 GeV 2 ≤ s 0 ≤ 10.0 GeV 2 .
Similarly, we use the sum rules given in Eqs.    Table I, where we choose threshold values s 0 for two-gluon glueballs to be around 9 ∼ 10 GeV 2 , and those for threegluon glueballs to be around 33 ∼ 38 GeV 2 .

V. SUMMARY AND DISCUSSIONS
In this paper we study two-and three-gluon glueballs of C = + using the method of QCD sum rules, including • the two-gluon glueballs with the quantum numbers J P C = 0 ±+ , 1 −+ , and 2 ±+ ; • the three-gluon glueballs with the quantum numbers J P C = 0 ±+ , 1 ±+ , and 2 ±+ .
We systematically construct their interpolating currents, and find that all the spin-1 currents of C = + vanish, suggesting that the "ground-state" spin-1 glueballs of C = + do not exist within the relativistic framework. This behavior is consistent with Lattice QCD calculations [11][12][13][14].
We use spin-0 and spin-2 glueball currents to perform QCD sum rule analyses, and calculate masses of their corresponding spin-0 and spin-2 glueballs. All these spin-2 currents have four Lorentz indices with certain symmetries, so that they couple to both positive-and negativeparity glueballs, which need to be further separated at the hadron level. We refer to Ref. [50] for detailed discussions.
We summarize the obtained results in Table II, which are compared with the Lattice QCD results obtained using non-relativistic glueball operators [11][12][13][14]. For completeness, we also reanalysis the results of C = − threegluon glueballs (also called as odderons), which have been previously studied in Ref. [50]. Therefore, a rather complete QCD sum rule study have been done on the lowestlying glueballs composed of two or three valence gluons. We find that our QCD sum rule results are generally consistent with unquenched Lattice QCD results [14].
To end this paper, we briefly discuss possible decay patterns of two-and three-gluon glueballs. The two-gluon glueballs can decay after exciting two quarkantiquark pairs, and recombine into two mesons. However, it is rather difficult to differentiate them from standard qq states. The three-gluon glueballs can decay after exciting three quark-antiquark pairs, and recombine into three mesons. Their possible decay patterns are: where P and V denote light vector and pseudoscalar mesons, respectively. Considering their limited decay patterns, the J P C = 0 −+ /2 −± /3 ±− three-gluon glueballs may have relatively smaller widths, and we propose to search for them in their V V V and V V P decay channels in future BESIII, GlueX, LHC, and PANDA experiments. 1 all vanish. Their definitions are given in Eqs. (15), (27), and (28), respectively. For simplicity, we shall not differentiate the superscript and subscript in the following calculations.
Firstly, we investigate the current J αβ 1 . Due to the Lorentz invariant, we simply assume α = 0 and β = 1 · · · 3; besides, we need the Lorentz indices µ = 0/i, ρ = 0/k, and σ = 0/l, with i/k/l = 1 · · · 3. We obtain: After interchanging i ↔ k, the first term turns out to be zero: So does the second term. The third term is non-zero when β = k or β = l. However, for the case β = k, we can interchange i ↔ l and obtain (not sum over β here): So does the case β = l. Therefore, the third term is also zero, and the current J αβ 1 vanishes. Secondly, we investigate the current η αβ 1 : In the above expressions, we have consequently interchanged µν ↔ ρσ and a ↔ b. Similarly, we can prove the currentη αβ 1 to be zero. One can construct more spin-1 three-gluon glueball currents of C = +, such as: η ′αβ It is straightforward to prove the former current η ′αβ 1 to be zero: II: Masses of two-and three-gluon glueballs, in units of GeV. Our QCD sum rule results are listed in the 2nd column. Lattice QCD results are listed in the 3rd-6th columns, taken from Refs. [11][12][13] (quenched) and Ref. [14] (unquenched).