New hadron configuration: The double-gluon hybrid state

This is the first study on the double-gluon hybrid, which consists of one valence quark and one valence antiquark together with two valence gluons. We concentrate on the one with the exotic quantum number $J^{PC} = 2^{+-}$ that conventional $\bar q q$ mesons can not reach. We apply QCD sum rule method to evaluate its mass to be $2.26^{+0.20}_{-0.25}$ GeV, and study its possible decay patterns. Especially, its three-meson decay patterns are generally not suppressed severely compared to two-meson decay patterns, so the $S$-wave three-meson decay channels $f_1\omega\pi/f_1\rho\pi$ can be useful in identifying its nature, which is of particular importance to the direct test of QCD in the low energy sector.

Introduction.-A hybrid state consists of one valence quark and one valence antiquark together with some valence gluons. Its experimental confirmation is a direct test of Quantum Chromodynamics (QCD) in the low energy sector. Especially, the hybrid states with J P C = 0 −− /0 +− /1 −+ /2 +− / · · · are of particular interests, since these exotic quantum numbers arise from the manifest gluon degree of freedom and can not be accessed by conventionalqq mesons. In the past half century there have been a lot of experimental and theoretical investigations, but their nature remains elusive [1][2][3][4].
In this letter we further investigate the double-gluon hybrid, which consists of one quark-antiquark pair together with two valence gluons. This is motivated by the recent D0 and TOTEM experiments observing the evidence of a C-odd three-gluon glueball [30], given that two-gluon glueballs have been detailedly studied but still difficult to be identified unambiguously [1][2][3][4]. We construct twelve double-gluon hybrid currents and use them to perform QCD sum rule analyses. As the first study, we concentrate on the one with the exotic quantum number J P C = 2 +− , which makes it doubly interesting. Among the twelve currents, we find it to be the only one with the mass predicted to be smaller than 3.0 GeV, that is This mass value is accessible in the BESIII, GlueX, LHC, and PANDA experiments.
We study possible decay patterns of the doublegluon hybrids with I G J P C = 1 + 2 +− and 0 − 2 +− , separately for two-and three-meson final states. We propose to search for the one of I G J P C = 1 + 2 +− in its decay channels ρf 0 (980)/ωπ/K * K /f 1 ωπ/ρππ/ · · · , and the one of I G J P C = 0 − 2 +− in its decay channels ρa 0 (980)/ρπ/K * K /f 1 ρπ/ωππ/ · · · , both of which are worthy to be searched for in the decay process J/ψ → π/ππ/η + X(→ K * K * /K * K π/ρKK → KKππ). Especially, their three-meson decay patterns are generally not suppressed severely compared to two-meson decay patterns, since they are both at the O(α s ) order. Accordingly, the S-wave three-meson decay channels f 1 ωπ/f 1 ρπ can be useful in distinguishing their nature from the tetraquark picture, therefore, of particular importance to the direct test of QCD in the low energy sector.
Double-gluon hybrid currents.-As the first step, we use the light up/down quark field q a (x) and the gluon field strength tensor G n µν (x) to construct double-gluon hybrid currents. Here a = 1 · · · 3 and n = 1 · · · 8 are color indices; µ and ν are Lorentz indices. Besides, we need the antiquark fieldq a (x) and the dual gluon field strength tensorG n µν = G n,ρσ × ǫ µνρσ /2. Generally speaking, one can construct many doublegluon hybrid currents by combining the color-octet quark-antiquark fields, q a λ ab n γ µ q b ,q a λ ab n γ µ γ 5 q b ,q a λ ab n σ µν q b , and the relativistic color-octet double-gluon fields, with suitable Lorentz matrices Γ µν···αβγδ . Here d npq and f npq are totally symmetric and antisymmetric SU (3) structure constants, respectively.
As the first study on the double-gluon hybrid, in the present study we shall investigate the following doublegluon hybrid currents: J α1β1,α2β2 where S denotes symmetrization and subtracting trace terms in the two sets {α 1 α 2 } and {β 1 β 2 } simultaneously. The above double-gluon hybrid currents have very clear Lorentz structures, simply because the color-octet quark-antiquark fieldq a γ 5 λ ab n q b does not contain any surplus Lorentz index. Besides, this quark-antiquark pair has the S-wave spin-parity quantum number J P = 0 − , so these currents are capable of coupling to the lowestlying double-gluon hybrid states.
QCD sum rule analyses.-The method of QCD sum rules has been widely applied in the study of hadron phenomenology [31,32], and in this letter we apply it to study the double-gluon hybrid currents defined in Eqs. (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14). We find that only the double-gluon hybrid state coupled by the current J α1β1,α2β2 can be investigated at both hadron and quark-gluon levels. Here S ′ denotes anti-symmetrization in the four sets , and symmetrization and subtracting trace terms in the four sets {α 1 α 2 }, simultaneously. The spectral density ρ(s) ≡ ImΠ(s)/π can be extracted from Eq. (15) through the dispersion relation At the hadron level we parameterize it using one pole dominance for the possible ground state |X; 2 +− together with the continuum contribution: At the quark-gluon level we calculate Eq. (15) and extract ρ OPE (s) using the method of operator product expansion (OPE). After performing the Borel transformation to Eq. (16) at both hadron and quark-gluon levels, we obtain where we have approximated the continuum using the OPE spectral density above the threshold value s 0 .
Finally, we calculate the mass of |X; 2 +− through In the present study we take into account the Feynman diagrams depicted in Fig. 1, and calculate ρ OPE (s) up to the dimension eight (D = 8) condensates. The gluon field strength tensor G n µν is defined as so it can be naturally separated into two parts. We depict the former two terms using the single-gluon-line, and the third term using the double-gluon-line with a red vertex, e.g., the diagram depicted in Fig. 1(c-3). We calculate the spectral density from the current J α1β1,α2β2 where we have taken into account all the diagrams proportional to α 2 s × g 0 s and α 2 s × g 1 s ; while there are so many FIG. 1: Feynman diagrams for the double-gluon hybrid, including the perturbative term, the quark condensate qq , the quark-gluon mixed condensate ḡsqσGq , the two-gluon condensate g 2 s GG , the three-gluon condensate g 3 s G 3 , and their combinations. The diagrams (a) and (b-i) are proportional to α 2 s × g 0 s , the diagrams (c-i) and (d-i) are proportional to α 2 s × g 1 s , and the diagrams (e-i) are proportional to α 2 s × g 2 s .
diagrams proportional to α 2 s × g 2 s , and we have kept only three of them, as depicted in Fig. 1(e-i).
Numerical analyses.-We study the sum rules given in Eq. (21) numerically using the following values for various QCD parameters at the renormalization scale 2 GeV and the QCD scale Λ QCD = 300 MeV [1,[33][34][35][36][37]: , qq = −(0.240 ± 0.010) 3 GeV 3 , As shown in Eq. (19), the mass of |X; 2 +− depends on the Borel mass M B and the threshold value s 0 . To insure the convergence of Eq. (21), we require a) the α 2 s × g 2 s terms to be less than 5%, and b) the D = 8 terms to be less than 10%: To insure the one-pole-dominance assumption, we require the pole contribution (PC) to be larger than 40%: Altogether we determine the Borel window to be 1.61 GeV 2 ≤ M 2 B ≤ 1.78 GeV 2 when setting s 0 = 7.0 GeV 2 .
We redo the same procedures by changing s 0 , and find that there are non-vanishing Borel windows as long as s 0 ≥ s min 0 = 6.3 GeV 2 . Accordingly, we set s 0 to be about 10% larger, and determine our working regions to be 5.0 GeV 2 ≤ s 0 ≤ 9.0 GeV 2 and 1.61 GeV 2 ≤ M 2 B ≤ 1.78 GeV 2 . The mass of |X; 2 +− is evaluated to be whose uncertainty is due to the threshold value s 0 , Borel mass M B , and various quark and gluon parameters listed in Eqs. (22), respectively. We show it in Fig. 2 as a function of the Borel mass M B and the threshold value s 0 . This mass value is obtained for both isoscalar and isovector states, so actually we can not differentiate them in the present QCD sum rule study. For completeness, we also use the other eleven hybrid currents defined in Eqs. (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13) to perform QCD sum rule analyses. We explicitly prove the four currents J ··· 0 −− /0 +− /1 −+ /1 ++ to be zero, while masses extracted from the seven currents J ··· 0 −+ /0 ++ /1 −− /1 +− /2 −− /2 −+ /2 ++ are all larger than 3.0 GeV. We leave their detailed discussions for our future studies.
Decay analyses.-The double-gluon hybrid can decay after exciting twoqq/ss (q = u/d) pairs from two gluons, followed by recombining three color-octetqq/ss pairs into two color-singlet mesons or three mesons, as depicted in Fig. 3. These two possible decay processes are both at the O(α s ) order, so three-meson decay patterns are generally not suppressed severely compared to two-meson decay patterns, or even enhanced due to the quark-antiquark annihilation during the two-meson decay process. This behavior can be useful in identifying the nature of the double-gluon hybrid. To investigate decay properties of the double-gluon hybrid, we assume its final quark content to be either Accordingly, we list some possible decay patterns of the double-gluon hybrids with the exotic quantum numbers I G J P C = 1 + 2 +− and 0 − 2 +− in Table I, separately for two-and three-meson decay processes. The one of I G J P C = 1 + 2 +− may be observed in its two-meson decay channels ρf 0 (980)/ωπ/K * K / · · · and three-meson decay channels f 1 ωπ/ρππ/ · · · ; the one of I G J P C = 0 − 2 +− may be observed in its two-meson decay channels ρa 0 (980)/ρπ/K * K / · · · and three-meson decay channels f 1 ρπ/ωππ/ · · · . Especially, both of them are worthy to be searched for in the decay process J/ψ → π/ππ/η + X(→ K * K * /K * K π/ρKK → KKππ), and the S-wave three-meson decay channels f 1 ωπ/f 1 ρπ can be useful in identifying their nature.
Summary.-As the first study on the double-gluon hybrid, we systematically construct twelve double-gluon hybrid currents and use them to perform QCD sum rule analyses. These currents are constructed by using the S-wave color-octet quark-antiquark fieldq a γ 5 λ ab n q b has the mass smaller than 3.0 GeV, that is which is accessible in the BESIII, GlueX, LHC, and PANDA experiments. Moreover, this state has the exotic quantum number J P C = 2 +− that conventionalqq mesons can not reach, making it doubly interesting. We study its possible decay patterns separately for two-and three-meson final states. We propose to search for the one of I G J P C = 1 + 2 +− in its decay channels ρf 0 (980)/ωπ/K * K /f 1 ωπ/ρππ/ · · · , and the one of I G J P C = 0 − 2 +− in its decay channels ρa 0 (980)/ρπ/K * K /f 1 ρπ/ωππ/ · · · , both of which are worthy to be searched for in the decay process J/ψ → π/ππ/η + X(→ K * K * /K * K π/ρKK → KKππ). Especially, their three-meson decay patterns are generally not suppressed severely compared to two-meson decay patterns, since they are both at the O(α s ) order. Accordingly, the S-wave three-meson decay channels f 1 ωπ/f 1 ρπ can be useful in identifying their nature, therefore, of particular importance to the direct test of QCD in the low energy sector.