# Hybrid Meson Interpretation of the Exotic Resonanceπ11600

Authors: Azzeddine Benhamida; Lahouari Semlala
Journal: Advances in High Energy Physics
 (2020)

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## Abstract

The exoticJPC=1−+resonance π11600 is examined in the framework of the Quark Model with Constituent Gluon (QMCG). We report the possibility of interpreting that resonance as qq¯g meson, with a masse ≃1.65−0.04+0.05 GeV, and a decay width to ρπ≃0.28−0.09+0.14 GeV.


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## Body

## 1. Introduction

Over the last four decades, intensive experimental activity has been carried out seeking to detect new hadrons beyond the quark model: glueballs or gluonium, hybrids, diquonia, and tetraquarks. These “exotic” species are most likely the new hadrons allowed by the QCD and are the subject of numerous researches, both theoretical and experimental.Hybrid mesons (quark-antiquark-gluon) can haveJPC quantum numbers which are not allowed by the naive quark model, like 0−−, 0+−, 1−+, and 2+−, then they cannot mix with the standard mesons and hence can facilitate their observation. These “exotic” objects are the most promising new species of hadrons allowed by QCD and subject of lot of works both in the theoretical and experimental levels. In fact, several JPC=1−+ exotic resonances have been claimed to be identified, especially π11600 and π11400 have received great interest, but some doubts are raised about the last one (for a review, see Ref [1]).In the theoretical framework, these hybrid mesons were studied from different models: lattice QCD [2–7], flux tube model [8–11], bag model [12, 13], QCD sum rules [14–18], constituent gluon models [19–26], and the effective Hamiltonian model [11, 27, 28]. Some of these models can perform both estimations of masses and decay widths, they predicted that the lightest hybrid mesons will be in 1.4-2.1 GeV mass range which is consistent with the confirmed 1−+ candidates.Theπ11600 was observed decaying into b1π, f1π, η′π, and ρπ. But although the first three modes have been confirmed for a long time, the ρπ mode has been incorporated only recently in PDG since 2018 [29]. Indeed, this mode is forbidden due to the “standard” flux tube predictions in a symmetry limit where the ρ and π have the same size and in the case where the decay is triggered by breaking the flux tube [8–11], although a value of 57 MeV was calculated beyond this limit [30]. This remains quite far from the very recent measurements made by COMPASS experiment [31] (see also PDG-2018 [29]).In this work, we focus our attention on the1−+ hybrid meson in the context of the Quark Model with Constituent Gluon (QMCG), and we shall see that this constituent glue model gives values of the mass and the ρπ decay width of the lightest 1−+qq¯g quite compatible with the observed exotic candidate π11600.This paper is organized as follows. In Section2, we briefly present the experimental situation of the exotic JPC=1−+π11600. We give predictions of the model QMGC in Section 3, and we conclude in Section 4.


## 2. The Experiemental Status ofπ11600

We consider here only the status of the resonanceπ11600 (for a review of the experimental situation on the exotic hybrid mesons, see the Ref [1] that we have mainly used when preparing this section).VES Collaboration [32] observed a broad peak at a mass value of ~1.6 GeV in the η′π, f11235π, and b11235π systems, interpreted as an exotic resonance of width about 300 MeV. Actually, they are unable to make a definitive conclusion on the resonance nature of it. For the ρπ final state, they are unable to conclude that the π11600 is present, while the following experimental relationship between the branching fractions of the π11600 decays is obtained (and therefore a limit on the branching fraction of ρπ):
(1)b1π:f1π:ρπ:η′π=1.0±0.3:1.1±0.3:<0.3:1.E852 Collaboration at BNL reported an evidence for the1−+π11600 resonance decaying into ρπ [33, 34], η′π [35], f1π [36], and b1π [37], regarding the ρπ channel, in the earlier E852 analysis [33, 34] of 250 K π−π−π+ events showed a possible evidence for a 1−+ exotic meson with a mass of ~1.6 GeV and width ~168 MeV, this state have been excluded by a more refined analysis [38], with 2.6 M π−π−π+ and 3 M π−π0π0 events.The exoticπ11600 is observed decaying to b1π from the Crystal Barrel data, and only results with mass and width fixed to the PDG values were reported [39].CLEO Collaboration found evidence for an exotic P-waveη′π amplitude, which, if interpreted as a resonance, would have parameters consistent with the π11600 state with a mass of 1670±30±20 MeV and a width of 240±50±60 MeV [40].A search for exotic mesons in theπ+π+π− system photoproduced by the charge exchange reaction γp⟶π+π+π−n was carried out by the CLAS Collaboration at Jefferson Lab., and no evidence is shown of the exotic π11600 decaying to three charged pions [41, 42].COMPASS collaboration observed the spin-exoticπ11600 in their partial-wave analysis of the 3π final state. They reported the observation of the π11600 in the ρπ decay mode initially with a mass 1660±10−64+0 MeV and width 269±21−64+42 MeV [43], superseded by a mass 1.600−0.060+0.100 GeV and width 0.580−0.230+0.100 GeV in their recent analysis [31]. COMPASS collaboration has also examined the exclusive production of ηπ and η′π and reported that odd partial waves, which carry non-qq¯ quantum numbers, are suppressed in the ηπ system relative to the η′π system. Even though they saw the exotic 1−+ wave in η′π as the dominant wave, they were unable to confirm the resonant nature of the signal [44]. This has recently been improved by the JPAC collaboration [45] which performed the first coupled-channel analysis of the P-wave in the η′π system measured at COMPASS [44] and reported a single exotic π1 with mass and width determined to be 1564±24±86 and 492±54±102 MeV, respectively.In conclusion, theπ11600 was observed decaying into b1π, f1π, and η′π and recently confirmed for ρπ mode by COMPASS collaboration [31], it is considered by the PDG to be an established state [29]. Table 1 shows masses and the corresponding decay widths of the π11600 reported by different experiments.Table 1
Theπ11600 as seen in different experiments. The resonance masses and the corresponding decay widths are reported in GeV.
ρ
π
b
1
π
f
1
1235
π
η
′
π
VES
~
1.6
0.3
∗
0.34
=
0.10
a
1.56
±
0.06
0.34
±
0.06
1.64
±
0.03
0.24
±
0.06
1.56
±
0.06
0.34
±
0.06
E852
1.593
±
0.00
8
−
0.047
+
0.029
0.168
±
0.02
0
−
0.012
+
0.150
b
1.664
±
0.008
±
0.010
0.185
±
0.025
±
0.028
1.709
±
0.024
±
0.041
0.403
±
0.080
±
0.115
1.597
±
0.01
0
−
0.010
+
0.045
0.340
±
0.040
±
0.060
COMPASS
1.60
0
−
0.060
+
0.100
0.58
0
−
0.230
+
0.100
—
—
1.564
±
0.024
±
0.086
0.492
±
0.054
±
0.102
c
CLEO
—
—
—
1.670
±
0.030
±
0.020
0.240
±
0.050
±
0.060
Crystal Barrel
—
~
1.6
seen
—
—
aAn experimental decay width limit. bThis state is excluded by a more refined E852 analysis (Ref. [38]). cFrom JLAC collaboration [45] using COMPASS data [44].


## 3. The QMCG Predictions

The nature of the gluonic field inside the hybrid meson is not yet clear because the gluon plays a double role: it propagates the interaction between color sources and, being itself colored, it undergoes the interaction. In an attempt to achieve a clearer understanding about the hybrid nature, two important hypothesis can be retained from the literature. The first one consider gluonicdegrees of freedom as “excitations” of the “flux tube” between quark and antiquark, which leads to the linear potential, that is familiar from the quark model (flux tube model).In the second one, the framework of the so-calledQuark Model with Constituent Gluon (QMCG) supported by this work, the hybrid meson is considered as a QCD-bound state composed of a quark-antiquark pair and (a massive) constituent gluon which interact through a phenomenological potential. We can adapt this scheme with the idea of confined and confining gluons (in the Landau and Coulomb gauges and in interpolating gauges between them) [46]. Confining gluons establish an area law behavior of the Wilson loop and the linearly rising interquark confinement, while confined gluons do not propagate over long distances, we can accommodate the confined (massive, constituent) gluon in coexistence with an effective quark interaction which is confining (more details can be found in Ref [24, 25]).“It is important to realise that the more complicated picture emerging for QCD in the covariant gauge can certainly accommodate confined (but not confining) gluons in coexistence with an effective quark interaction which is confining, however.” [47].


### 3.1. Ingredients of the QMGC

The QMCG is a natural expansion of the naive quark model where the confined gluon within the hadron matter acquires a (constituent) massmg. As for quarks, this important parameter represents a dynamical mass which is responsible for the infrared finiteness of the gluon propagator and the ghost dressing function observed using continuum methods (the Schwinger-Dyson Equations) and large-volume lattice simulations or combining continuum methods with lattice data (a more complete presentation of the subject is given for example in Refs. [48–51]).From the phenomenological point of view, a nonvanishing gluon mass is welcome by diffractive phenomena [52] and inclusive radiative decays of J/ψ and Y [53]. For the glueball states, color singlet bound states of gluons are considered to be fairly massive, e.g., about 1.5 GeV for the lowest 0++ and about 2 GeV for the lowest 2++, as indicated in lattice QCD calculations [54–56], a simple constituent gluon picture may be approximately obtained as MGB≃2mg for the glueball mass MGB.Using the continuum strong QCD, one infersmg≃0.4−0.6 GeV [51] which is consistent with the lattice results: mg~0.5 GeV [57, 58].In the present work, we fix this parameter as:(2)mg=0.5±0.1GeV.The decay parameterαs (the effective quark-gluon vertex coupling) is the second ingredient of the model. There are many theoretical evidences that the QCD effective charge αs freezes at small momenta. Therefore, the infrared finiteness of the effective charge can be considered as one of the manifestations of the phenomenon of dynamical gluon mass generation. Phenomenology sensitive to infrared properties of QCD gives αs0≃0.7±0.3 [59–61], while the phenomenological evidences for the strong coupling constant freezing in the infrared are much more numerous, as with models where a static potential is used to compute the hadronic spectra that make use of a frozen coupling constant at long distances (for more details, see for example the Ref. [62]).The effective charge obtained within the pinch technique (PT) framework [63, 64], to be denoted by αPT, constitutes the most direct non-Abelian generalization of the familiar concept of the QED effective charge. Since our decay model is obtained in the Feynman gauge [20], it is natural to choose αs≃αPT0 corresponding to the pinch technique gluon propagator, i.e., the background field propagator calculated in the Feynman gauge. αPT0 is correlated to the gluon mass mg [65, 66]:
(3)αPT0~0.6formg~0.5GeV.


### 3.2. The Hybrid Bound State

We assume that the hybrid meson is a bound state of quark-antiquark and a constituent gluon which interact through a phenomenological potential, precisely Coulomb plus linear potential supplemented by spin-spin, spin-orbit, and tensor correction terms. The use of relativistic kinetics is appropriate for the study of the light flavor systems [24].For the representation of the hybrid states, the following notations are used:(1)lg: the relative orbital momentum of the gluon in the qq¯ center of mass(2)lqq¯: the relative orbital momentum between q and q¯(3)Sqq¯: the total quarks spinConsidering the gluon moving in the framework of theqq¯ pair, the Parity of the hybrid will be:
(4)P=−lqq¯+1·−1·−lg=−lqq¯+lg,−
1 being the intrinsic parity of the gluon.The Charge Conjugation is given by:(5)C=−lqq¯+Sqq¯+1.S
q
q
¯ can take the values 0 or 1; P and C impose parity restrictions on lqq¯ and lg.For lower values of the orbital excitations (lqq¯ and lg⩽1) and parity P=−1, the hybrid states can be built by two modes: lqq¯=0 and lg=1 which we shall refer as the gluon-excited hybrid (GE hybrid), and lqq¯=1 and lg=0 which we shall refer as the quark-excited hybrid (QE hybrid) (see Table 2 for the case JPC=1−+).Table 2
The lowestJPC=1−+ hybrid meson quantum numbers. Modes and the decay selection rules are shown.
l
q
q
¯
S
q
q
¯
l
g
j
g
L
Hybrid mode
Gluon mode
Preferred decay mode
0
1
1
0
0
GE
Long.
“L + S”
0
1
1
1
1
GE
Long., TM, TE
“L + S”
0
1
1
2
2
GE
Long., TM, TE
“L + S”
1
0
0
1
1
QE
Long., TM
“S + S”In the potential model, the simplest approximation is to factorise theqq¯-wave function with the wave function of the gluon respective to the qq¯ center of mass (the cluster approximation). We shall use the following lowest-lying state qq¯-cluster spin-space wave function:
(6)ΨJMPCρ→,λ→=eμg⊗ψlgmgjgMg⊗ψlqq¯mqq¯Lm⊗χμqq¯Sqq¯JMPC=∑Llgjglqq¯Sqq¯;PCΨJM;Llgjglqq¯Sqq¯ρ→,λ→,where eμg is the gluon polarisation, χμqq¯Sqq¯ is the diquark spin representation, and the sum runs over the values of L,lg,jg,lqq¯andSqq¯ excluding those not consistent with P and C and:
(7)ΨJM;Llgjglqq¯Sqq¯ρ→,λ→=∑mmgμgMgmqq¯μqq¯lgmg1μg∣jgMglqq¯mqq¯jgMg∣LmLmSqq¯μqq¯∣JM×ψlglqq¯mgmqq¯ρ→,λ→eμgχμqq¯Sqq¯;here, the Jacobi coordinates are introduced:
(8)ρ→=r→q¯−r→q,λ→=r→g−Mqr→q+Mq¯r→q¯Mq+Mq¯.The Hamiltonian is constructed, containing a phenomenological potential which reproduces the QCD characteristics; its expression has the mathematical “Coulomb + Linear” form, we take into account also some relativistic effects, i.e., spin-dependent interaction terms and relativistic kinetics; a more detailed description can be found in our previous work [24].In order to make a comparison with the lattice results, we note that our1−+ wave function (Eq. (7)) is related to the so-called TE, TM, and longitudinal gluon states as follows (see Table 2):
(9)Ψ1M−+=Ψjg=0long+Ψjg=1TE+mixΨjg=2TM,Ψjg=2long+mixΨjg=1TM,Ψjg=1long,where mixφ,ψ means a mixture of the states φ and ψ. The “magnetic” TE, “electric” TM, and longitudinal gluons correspond to the following hybrid states [20]:
(10)ΨjgTE≡ΨJM;Llgjglqq¯Sqq¯TE=ΨJM;Llgjglqq¯Sqq¯lg=jg,ΨjgTM≡ΨJM;Llgjglqq¯Sqq¯TM=jg+12jg+1ΨJM;Llgjglqq¯Sqq¯lg=jg−1+jg2jg+1ΨJM;Llgjglqq¯Sqq¯lg=jg+1,Ψjglong≡ΨJM;Llgjglqq¯Sqq¯long=−jg2jg+1ΨJM;Llgjglqq¯Sqq¯lg=jg−1+jg+12jg+1ΨJM;Llgjglqq¯Sqq¯lg=jg+1.We can rewrite Eq. (9) according to the GE and QE hybrid modes as:
(11)Ψ1M−+=ψGE+ψQE,where
(12)ψGE=Ψjg=0long+Ψjg=1TE+mixΨjg=2TM,Ψjg=2long,(13)ψQE=mixΨjg=1TM,Ψjg=1long.Since our gluon is assumed to be massive, the longitudinal component must be present in Eq. (9), mixed with the TM and TE gluon modes jg≠0. This is not true in the lattice hybrid calculations where the low-laying 1−+ states are made with the particular jgPgCg=1+− TE-gluon mode. Indeed, although in principal lattice construction of the hybrid 1−+ states involves the TM and the TE modes, in the light sector, only the last mode results are widely reported since it gives the best and the clearest signal [2–5]. From Table 2, we notice that the TE gluon appears only in the GE-hybrid and is totally absent in the QE-hybrid 1−+ state. We will come back to this later.


### 3.3. The Hybrid Decay Model

To the lowest order, the decay of a hybrid state A into two ordinary mesons B and C is described by the matrix element of the Hamiltonian annihilating a gluon and creating a quark pair (QPC model):(14)H=g∫dx→ψ¯x→γμλa2ψx→Aaμx→=g∑ss′λ∫dp→dk→dp′→2ω2π92π3δ3p→−k→−p′→×u¯p→sγμλa2u−p′→s′εk→λμφabp→s†d−p′→s′†ak→λ,where a is the color index and s, s′, and λ are the spin indices. In the nonrelativistic limit, we have:
(15)u¯p→sγμλa2u−p′→sεcμ≃χs†σ→χ¯s′ε→k→λ,where χs′ is the antiquark spinor in the complex conjugate representation.The standard meson (B or C) and the hybrid (A) states are written in the nonrelativistic approximation:
(16)qq¯;JM=∑ss′aa¯mμ∫dp→qdp→q¯2π62π3δ3p→−p→q−p→q¯lmSμ∣JM×13χss′μψlmmq¯p→q−mqp→q¯mq¯+mqbp→qsa†dp→q¯s′a¯†0with ∫dp→/2π3ψlmp→2=1;(17)qq¯g;JM=∑ss′aa¯mμ∫dp→qdp→q¯dk→2π32π3δ3p→−p→q−p→q¯−k→×λcqcq¯cg4χss′μψlglqq¯mgmqq¯mq¯p→q−mqp→q¯mq¯+mq,mq¯+mqk→−mgp→q+p→q¯mq¯+mq+mg×lgmg1μg∣jgMglqq¯mqq¯jgMg∣LmLmSqq¯μqq¯∣JMbp→qsa†dp→q¯s′a¯†ak→λ†0,where cq, cq¯, and cg are the color charge of the quark, antiquark, and gluon with cq,cq¯=1,2,3 and cg=1,⋯,8. We have also ∫dp→dk→/2π6ψlglqq¯mgmqq¯p→,k2=1.The matrix element between a hybrid stateA and two standard mesons B and C is given by:
(18)BC∣H∣A=gfA,B,C2π3δ3p→A−p→B−p→C,where fA,B,C is the decay amplitude:
(19)fA,B,C=∑m,μΦΩXμqq¯,μg;μB,μcImqq¯,mg;mB,mc,m×lgmg1μg∣JgMglqq¯mqq¯JgMg∣Lm′Lm′Sqq¯μqq¯∣JM×lBmBSBμB∣JBMBlCmCSCμC∣JCMC.The amplitudef involves the flavor Φ, the color Ω, the nonrelativistic spin X, and the spatial I overlaps defined as follows.
(20)Φ=2IB+12IC+112IA+1i1i3IBi2i4ICIA0IAηε,I’si’s label the hadron (quark) isospins, η=1 if the gluon goes into strange quarks and η=2 if it goes into nonstrange ones. ε2 is the number of diagrams contributing to the decay. Indeed, one can check that two diagrams contribute with the same sign and magnitude for C, P, and G-Parity allowed decays while they cancel for forbidden ones. In the case of two identical final particles ε=2. The term between brackets in Eq. (20) is the 9j symbol.
(21)Ω=124∑aTrλa2=23.Xμqq¯,μg;μB,μc=∑s22SB+12SC+132Sqq¯+11212SB1212SCSqq¯1S×SBμBSCμC∣SμB+μCSqq¯μqq¯1μg∣Sμqq¯+μg.Finally, the spatial overlap is represented by the term:(22)Imqq¯,mg;mB,mc,m=∫dp→dk→2π6ψlglqq¯mgmqq¯p→B−p→,k→ψlBmB∗mq¯ip→Bmq¯i+mq−p→−12k→×ψlCmC∗−mqip→Bmqi+mq+p→−12k→dΩBYlm∗ΩB,where l,m label the orbital momentum between the two final mesons.The partial width is given by:(23)ΓA→BC=4αsfA,B,C2PBEBECMA,where αs represents the infrared quark-gluon vertex coupling. For more details on the decay model, see Refs. [20, 24]; here, we focus on the main (nonrelativistic) results:
(1)
The1−+ QE-hybrid is allowed to decay into two S-wave mesons only (the so-called “S+S” selection rule)(2)
The1−+ GE-hybrid is allowed to decay into a channel with one S-wave meson and one P-wave meson only (the so-called “L+S” selection rule)The last selection rule is also reported in the gluonic excitation models of hybrid where the decay to two S-wave mesons is strongly suppressed (see [67] and references therein).In the decay model which we use, theη′π decay modes are suppressed by the nonrelativistic spin conservation law although the spatial overlap is not vanishing for the QE-hybrid mode, a full relativistic studies shall give nonvanishing answer for both QE and GE modes (this will be the subject of future work). In the other side, it seems that the flux tube model [8] and the QCD sum rules [15] predict a suppression of 1−+hybrid⟶ηπ. This is confirmed using a quite independent model way without any further hypothesis than the quenched approximation [68]. However, this approximate selection rule is related only to the “magnetic” or TE-gluon mode.


## 3.1. Ingredients of the QMGC

The QMCG is a natural expansion of the naive quark model where the confined gluon within the hadron matter acquires a (constituent) massmg. As for quarks, this important parameter represents a dynamical mass which is responsible for the infrared finiteness of the gluon propagator and the ghost dressing function observed using continuum methods (the Schwinger-Dyson Equations) and large-volume lattice simulations or combining continuum methods with lattice data (a more complete presentation of the subject is given for example in Refs. [48–51]).From the phenomenological point of view, a nonvanishing gluon mass is welcome by diffractive phenomena [52] and inclusive radiative decays of J/ψ and Y [53]. For the glueball states, color singlet bound states of gluons are considered to be fairly massive, e.g., about 1.5 GeV for the lowest 0++ and about 2 GeV for the lowest 2++, as indicated in lattice QCD calculations [54–56], a simple constituent gluon picture may be approximately obtained as MGB≃2mg for the glueball mass MGB.Using the continuum strong QCD, one infersmg≃0.4−0.6 GeV [51] which is consistent with the lattice results: mg~0.5 GeV [57, 58].In the present work, we fix this parameter as:(2)mg=0.5±0.1GeV.The decay parameterαs (the effective quark-gluon vertex coupling) is the second ingredient of the model. There are many theoretical evidences that the QCD effective charge αs freezes at small momenta. Therefore, the infrared finiteness of the effective charge can be considered as one of the manifestations of the phenomenon of dynamical gluon mass generation. Phenomenology sensitive to infrared properties of QCD gives αs0≃0.7±0.3 [59–61], while the phenomenological evidences for the strong coupling constant freezing in the infrared are much more numerous, as with models where a static potential is used to compute the hadronic spectra that make use of a frozen coupling constant at long distances (for more details, see for example the Ref. [62]).The effective charge obtained within the pinch technique (PT) framework [63, 64], to be denoted by αPT, constitutes the most direct non-Abelian generalization of the familiar concept of the QED effective charge. Since our decay model is obtained in the Feynman gauge [20], it is natural to choose αs≃αPT0 corresponding to the pinch technique gluon propagator, i.e., the background field propagator calculated in the Feynman gauge. αPT0 is correlated to the gluon mass mg [65, 66]:
(3)αPT0~0.6formg~0.5GeV.


## 3.2. The Hybrid Bound State

We assume that the hybrid meson is a bound state of quark-antiquark and a constituent gluon which interact through a phenomenological potential, precisely Coulomb plus linear potential supplemented by spin-spin, spin-orbit, and tensor correction terms. The use of relativistic kinetics is appropriate for the study of the light flavor systems [24].For the representation of the hybrid states, the following notations are used:(1)lg: the relative orbital momentum of the gluon in the qq¯ center of mass(2)lqq¯: the relative orbital momentum between q and q¯(3)Sqq¯: the total quarks spinConsidering the gluon moving in the framework of theqq¯ pair, the Parity of the hybrid will be:
(4)P=−lqq¯+1·−1·−lg=−lqq¯+lg,−
1 being the intrinsic parity of the gluon.The Charge Conjugation is given by:(5)C=−lqq¯+Sqq¯+1.S
q
q
¯ can take the values 0 or 1; P and C impose parity restrictions on lqq¯ and lg.For lower values of the orbital excitations (lqq¯ and lg⩽1) and parity P=−1, the hybrid states can be built by two modes: lqq¯=0 and lg=1 which we shall refer as the gluon-excited hybrid (GE hybrid), and lqq¯=1 and lg=0 which we shall refer as the quark-excited hybrid (QE hybrid) (see Table 2 for the case JPC=1−+).Table 2
The lowestJPC=1−+ hybrid meson quantum numbers. Modes and the decay selection rules are shown.
l
q
q
¯
S
q
q
¯
l
g
j
g
L
Hybrid mode
Gluon mode
Preferred decay mode
0
1
1
0
0
GE
Long.
“L + S”
0
1
1
1
1
GE
Long., TM, TE
“L + S”
0
1
1
2
2
GE
Long., TM, TE
“L + S”
1
0
0
1
1
QE
Long., TM
“S + S”In the potential model, the simplest approximation is to factorise theqq¯-wave function with the wave function of the gluon respective to the qq¯ center of mass (the cluster approximation). We shall use the following lowest-lying state qq¯-cluster spin-space wave function:
(6)ΨJMPCρ→,λ→=eμg⊗ψlgmgjgMg⊗ψlqq¯mqq¯Lm⊗χμqq¯Sqq¯JMPC=∑Llgjglqq¯Sqq¯;PCΨJM;Llgjglqq¯Sqq¯ρ→,λ→,where eμg is the gluon polarisation, χμqq¯Sqq¯ is the diquark spin representation, and the sum runs over the values of L,lg,jg,lqq¯andSqq¯ excluding those not consistent with P and C and:
(7)ΨJM;Llgjglqq¯Sqq¯ρ→,λ→=∑mmgμgMgmqq¯μqq¯lgmg1μg∣jgMglqq¯mqq¯jgMg∣LmLmSqq¯μqq¯∣JM×ψlglqq¯mgmqq¯ρ→,λ→eμgχμqq¯Sqq¯;here, the Jacobi coordinates are introduced:
(8)ρ→=r→q¯−r→q,λ→=r→g−Mqr→q+Mq¯r→q¯Mq+Mq¯.The Hamiltonian is constructed, containing a phenomenological potential which reproduces the QCD characteristics; its expression has the mathematical “Coulomb + Linear” form, we take into account also some relativistic effects, i.e., spin-dependent interaction terms and relativistic kinetics; a more detailed description can be found in our previous work [24].In order to make a comparison with the lattice results, we note that our1−+ wave function (Eq. (7)) is related to the so-called TE, TM, and longitudinal gluon states as follows (see Table 2):
(9)Ψ1M−+=Ψjg=0long+Ψjg=1TE+mixΨjg=2TM,Ψjg=2long+mixΨjg=1TM,Ψjg=1long,where mixφ,ψ means a mixture of the states φ and ψ. The “magnetic” TE, “electric” TM, and longitudinal gluons correspond to the following hybrid states [20]:
(10)ΨjgTE≡ΨJM;Llgjglqq¯Sqq¯TE=ΨJM;Llgjglqq¯Sqq¯lg=jg,ΨjgTM≡ΨJM;Llgjglqq¯Sqq¯TM=jg+12jg+1ΨJM;Llgjglqq¯Sqq¯lg=jg−1+jg2jg+1ΨJM;Llgjglqq¯Sqq¯lg=jg+1,Ψjglong≡ΨJM;Llgjglqq¯Sqq¯long=−jg2jg+1ΨJM;Llgjglqq¯Sqq¯lg=jg−1+jg+12jg+1ΨJM;Llgjglqq¯Sqq¯lg=jg+1.We can rewrite Eq. (9) according to the GE and QE hybrid modes as:
(11)Ψ1M−+=ψGE+ψQE,where
(12)ψGE=Ψjg=0long+Ψjg=1TE+mixΨjg=2TM,Ψjg=2long,(13)ψQE=mixΨjg=1TM,Ψjg=1long.Since our gluon is assumed to be massive, the longitudinal component must be present in Eq. (9), mixed with the TM and TE gluon modes jg≠0. This is not true in the lattice hybrid calculations where the low-laying 1−+ states are made with the particular jgPgCg=1+− TE-gluon mode. Indeed, although in principal lattice construction of the hybrid 1−+ states involves the TM and the TE modes, in the light sector, only the last mode results are widely reported since it gives the best and the clearest signal [2–5]. From Table 2, we notice that the TE gluon appears only in the GE-hybrid and is totally absent in the QE-hybrid 1−+ state. We will come back to this later.


## 3.3. The Hybrid Decay Model

To the lowest order, the decay of a hybrid state A into two ordinary mesons B and C is described by the matrix element of the Hamiltonian annihilating a gluon and creating a quark pair (QPC model):(14)H=g∫dx→ψ¯x→γμλa2ψx→Aaμx→=g∑ss′λ∫dp→dk→dp′→2ω2π92π3δ3p→−k→−p′→×u¯p→sγμλa2u−p′→s′εk→λμφabp→s†d−p′→s′†ak→λ,where a is the color index and s, s′, and λ are the spin indices. In the nonrelativistic limit, we have:
(15)u¯p→sγμλa2u−p′→sεcμ≃χs†σ→χ¯s′ε→k→λ,where χs′ is the antiquark spinor in the complex conjugate representation.The standard meson (B or C) and the hybrid (A) states are written in the nonrelativistic approximation:
(16)qq¯;JM=∑ss′aa¯mμ∫dp→qdp→q¯2π62π3δ3p→−p→q−p→q¯lmSμ∣JM×13χss′μψlmmq¯p→q−mqp→q¯mq¯+mqbp→qsa†dp→q¯s′a¯†0with ∫dp→/2π3ψlmp→2=1;(17)qq¯g;JM=∑ss′aa¯mμ∫dp→qdp→q¯dk→2π32π3δ3p→−p→q−p→q¯−k→×λcqcq¯cg4χss′μψlglqq¯mgmqq¯mq¯p→q−mqp→q¯mq¯+mq,mq¯+mqk→−mgp→q+p→q¯mq¯+mq+mg×lgmg1μg∣jgMglqq¯mqq¯jgMg∣LmLmSqq¯μqq¯∣JMbp→qsa†dp→q¯s′a¯†ak→λ†0,where cq, cq¯, and cg are the color charge of the quark, antiquark, and gluon with cq,cq¯=1,2,3 and cg=1,⋯,8. We have also ∫dp→dk→/2π6ψlglqq¯mgmqq¯p→,k2=1.The matrix element between a hybrid stateA and two standard mesons B and C is given by:
(18)BC∣H∣A=gfA,B,C2π3δ3p→A−p→B−p→C,where fA,B,C is the decay amplitude:
(19)fA,B,C=∑m,μΦΩXμqq¯,μg;μB,μcImqq¯,mg;mB,mc,m×lgmg1μg∣JgMglqq¯mqq¯JgMg∣Lm′Lm′Sqq¯μqq¯∣JM×lBmBSBμB∣JBMBlCmCSCμC∣JCMC.The amplitudef involves the flavor Φ, the color Ω, the nonrelativistic spin X, and the spatial I overlaps defined as follows.
(20)Φ=2IB+12IC+112IA+1i1i3IBi2i4ICIA0IAηε,I’si’s label the hadron (quark) isospins, η=1 if the gluon goes into strange quarks and η=2 if it goes into nonstrange ones. ε2 is the number of diagrams contributing to the decay. Indeed, one can check that two diagrams contribute with the same sign and magnitude for C, P, and G-Parity allowed decays while they cancel for forbidden ones. In the case of two identical final particles ε=2. The term between brackets in Eq. (20) is the 9j symbol.
(21)Ω=124∑aTrλa2=23.Xμqq¯,μg;μB,μc=∑s22SB+12SC+132Sqq¯+11212SB1212SCSqq¯1S×SBμBSCμC∣SμB+μCSqq¯μqq¯1μg∣Sμqq¯+μg.Finally, the spatial overlap is represented by the term:(22)Imqq¯,mg;mB,mc,m=∫dp→dk→2π6ψlglqq¯mgmqq¯p→B−p→,k→ψlBmB∗mq¯ip→Bmq¯i+mq−p→−12k→×ψlCmC∗−mqip→Bmqi+mq+p→−12k→dΩBYlm∗ΩB,where l,m label the orbital momentum between the two final mesons.The partial width is given by:(23)ΓA→BC=4αsfA,B,C2PBEBECMA,where αs represents the infrared quark-gluon vertex coupling. For more details on the decay model, see Refs. [20, 24]; here, we focus on the main (nonrelativistic) results:
(1)
The1−+ QE-hybrid is allowed to decay into two S-wave mesons only (the so-called “S+S” selection rule)(2)
The1−+ GE-hybrid is allowed to decay into a channel with one S-wave meson and one P-wave meson only (the so-called “L+S” selection rule)The last selection rule is also reported in the gluonic excitation models of hybrid where the decay to two S-wave mesons is strongly suppressed (see [67] and references therein).In the decay model which we use, theη′π decay modes are suppressed by the nonrelativistic spin conservation law although the spatial overlap is not vanishing for the QE-hybrid mode, a full relativistic studies shall give nonvanishing answer for both QE and GE modes (this will be the subject of future work). In the other side, it seems that the flux tube model [8] and the QCD sum rules [15] predict a suppression of 1−+hybrid⟶ηπ. This is confirmed using a quite independent model way without any further hypothesis than the quenched approximation [68]. However, this approximate selection rule is related only to the “magnetic” or TE-gluon mode.


## 4. Results and Discussion

### 4.1. The Mass Results

Our results related to the1−+ hybrid masses and decay widths for mg=0.4−0.6 GeV are summarised in Tables 3 and 4, we add Table 5 for comparison purposes.Table 3
The1−+ hybrid masses and the decay widths (in GeV) from the QMGC model.
Constituent glue model (this work)
m
g
0.4
0.5
0.6
M
1
−
+
1.61
1.65
1.70
Γ
1
−
+
⟶
ρ
π
0.19
0.28
0.42
Γ
1
−
+
⟶
b
1
π
0.19
0.29
0.46
Γ
1
−
+
⟶
f
1
1235
π
0.06
0.10
0.17Table 4
Our1−+ hybrid masses and the decay widths (in GeV) compared to the lattice QCD and flux tube results.
Constituent glue model (this work)
Flux tube model
Lattice QCD
m
g=0.5 GeV
QE-GE mix.
QE
GE
Ref. [9]
Ref. [11]
Ref. [7]
Γ
1
−
+
hybrid
⟶
ρ
π
0.28
0.28
Forbiddena
0.005-0.020
0.009
—
Γ
1
−
+
hybrid
⟶
b
1
π
0.29
Forbiddenb
0.29
0.170
0.024
0.40
±
0.12
Γ
1
−
+
hybrid
⟶
f
1
1235
π
0.10
Forbiddenc
0.10
0.060
0.005
0.09
±
0.06
Γ
1
−
+
hybrid
⟶
η
π
,
η
′
π
—
Forbiddend
<0.010
0.000-0.010
0.000
—
aSelection rule. bSelection rule. cSelection rule. dAllowed beyond the nonrelativistic approximation. Calculations of this contribution will be the subject of future work.Table 5
The comparison of our work with data and the recent gluonic excitation models.
This work
The experience
Gluonic excitation models
m
g
0.5±0.1
VES
E852
CLEO
COMPASS
Lattice QCD
Flux tubea
M
1
−
+
1.6
5
−
0.04
+
0.05
1.64
±
0.03
1.66
±
0.01
1.67
±
0.04
1.6
0
−
0.06
+
0.10
1.79
±
0.14
b, 1.96±0.04c
~
200
.
ρ
π
0.2
8
−
0.09
+
0.14
≲
0.10
d
0.17
±
0.0
2
−
0.01
+
0.15
—
0.5
8
−
0.23
+
0.10
—
0.005
−
0.020
b
1
π
0.2
9
−
0.11
+
0.16
0.34
±
0.06
0.19
±
0.04
—
—
0.40
±
0.12
e
0.170
f
1
1235
π
0.1
0
−
0.04
+
0.07
0.24
±
0.06
0.40
±
0.14
—
—
0.09
±
0.06
f
0.060
aRef. [9]. bRef. [6]. cFrom Figure 5 of Ref. [5] at mπ=396 MeV (⋙139 MeV, the observed value). dAn experimental limit. eRef. [7]. fRef. [7].It is difficult to get a hybrid masse lower than1.5 GeV (≤1.52 GeV for mg≥0.).We observe a large mixing between the two QE and GE-hybrid modes where all the TE, TM, and longitudinal gluon modes are included in the hybrid wave function (Eq. (9)); for a pure GE-mode (with excited glue lg=1 and an S-wave qq¯), we have M1−+GE≃1.76±0.05 GeV for mg=0.5±0.1 GeV.Our calculated mass is:(24)M1−+≃1.65−0.04+0.05GeV,which is very close to the latest PDG average [29]:
(25)1.660−0.011+0.015GeV,and quite far from ~2. GeV emerged from the lattice QCD [6] and the flux tube [9] studies that systematically discard the QE-mode where the gluon is not excited, i.e., ignore states electric TM (jg=1) and longitudinal (Eq. (13)). In addition, there is some difficulties that taint the lattice masse calculations:
(i)
How to identify interpolation fields used as a hybrid and distinguish them from ordinary mesons? From the criteria for hybrids proposed in [5] (and adopted implicitly by earlier lattice works [2–4]) the hybrid-like character is directly related to the overlap with the appropriate JPC interpolating fields. This is not always true, we cannot understand the nature of a state by the appearance of its interpolation field. This is sufficiently illustrated by the strong projection on η and η′ produced with the glue interpolation field GμνG~μν, it does not mean that they are glueballs [69](ii)
In the light sector, lattice authors report only results related to thejgPgCg=1+− TE-gluon since it has the best signal with the smallest statistical errors while the explicit masses of the jgPgCg=1−− TM-gluon are not yet published(iii)
The lattice calculation sill uses an unrealistic mass of theπ meson (~396 MeV) which is much greater than the observed one (~139 MeV).


### 4.2. The Decay Results

As shown in Eq. (23), the decay width is proportional to the parameter αs which is in turn correlated to the mass of the gluon mg, as mentioned above. In Table 3, we represent the results for mg=0.5±0.1GeV and the final theoretical uncertainty is taken as the deviation from the nominal value and the upper and lower tolerance.Despite the imperfections of the model, our predictions are mostly in reasonable accord with the observed1−+ resonance π11600 seen by several collaborations as shown in Table 5. This is especially true for the controversial ρπ channel which is forbidden by the gluonic excitation models (the "L+S" selection rule [10]).In the constituent glue model, the nonvanishing width comes from the QE-hybrid mode (lg=0 whith P-wave qq¯, Eq. (13)) decaying preferably into two S-wave mesons, i.e.,
(26)Γ1−+→ρπ≃0.28−0.09−0.14GeV.


## 4.1. The Mass Results

Our results related to the1−+ hybrid masses and decay widths for mg=0.4−0.6 GeV are summarised in Tables 3 and 4, we add Table 5 for comparison purposes.Table 3
The1−+ hybrid masses and the decay widths (in GeV) from the QMGC model.
Constituent glue model (this work)
m
g
0.4
0.5
0.6
M
1
−
+
1.61
1.65
1.70
Γ
1
−
+
⟶
ρ
π
0.19
0.28
0.42
Γ
1
−
+
⟶
b
1
π
0.19
0.29
0.46
Γ
1
−
+
⟶
f
1
1235
π
0.06
0.10
0.17Table 4
Our1−+ hybrid masses and the decay widths (in GeV) compared to the lattice QCD and flux tube results.
Constituent glue model (this work)
Flux tube model
Lattice QCD
m
g=0.5 GeV
QE-GE mix.
QE
GE
Ref. [9]
Ref. [11]
Ref. [7]
Γ
1
−
+
hybrid
⟶
ρ
π
0.28
0.28
Forbiddena
0.005-0.020
0.009
—
Γ
1
−
+
hybrid
⟶
b
1
π
0.29
Forbiddenb
0.29
0.170
0.024
0.40
±
0.12
Γ
1
−
+
hybrid
⟶
f
1
1235
π
0.10
Forbiddenc
0.10
0.060
0.005
0.09
±
0.06
Γ
1
−
+
hybrid
⟶
η
π
,
η
′
π
—
Forbiddend
<0.010
0.000-0.010
0.000
—
aSelection rule. bSelection rule. cSelection rule. dAllowed beyond the nonrelativistic approximation. Calculations of this contribution will be the subject of future work.Table 5
The comparison of our work with data and the recent gluonic excitation models.
This work
The experience
Gluonic excitation models
m
g
0.5±0.1
VES
E852
CLEO
COMPASS
Lattice QCD
Flux tubea
M
1
−
+
1.6
5
−
0.04
+
0.05
1.64
±
0.03
1.66
±
0.01
1.67
±
0.04
1.6
0
−
0.06
+
0.10
1.79
±
0.14
b, 1.96±0.04c
~
200
.
ρ
π
0.2
8
−
0.09
+
0.14
≲
0.10
d
0.17
±
0.0
2
−
0.01
+
0.15
—
0.5
8
−
0.23
+
0.10
—
0.005
−
0.020
b
1
π
0.2
9
−
0.11
+
0.16
0.34
±
0.06
0.19
±
0.04
—
—
0.40
±
0.12
e
0.170
f
1
1235
π
0.1
0
−
0.04
+
0.07
0.24
±
0.06
0.40
±
0.14
—
—
0.09
±
0.06
f
0.060
aRef. [9]. bRef. [6]. cFrom Figure 5 of Ref. [5] at mπ=396 MeV (⋙139 MeV, the observed value). dAn experimental limit. eRef. [7]. fRef. [7].It is difficult to get a hybrid masse lower than1.5 GeV (≤1.52 GeV for mg≥0.).We observe a large mixing between the two QE and GE-hybrid modes where all the TE, TM, and longitudinal gluon modes are included in the hybrid wave function (Eq. (9)); for a pure GE-mode (with excited glue lg=1 and an S-wave qq¯), we have M1−+GE≃1.76±0.05 GeV for mg=0.5±0.1 GeV.Our calculated mass is:(24)M1−+≃1.65−0.04+0.05GeV,which is very close to the latest PDG average [29]:
(25)1.660−0.011+0.015GeV,and quite far from ~2. GeV emerged from the lattice QCD [6] and the flux tube [9] studies that systematically discard the QE-mode where the gluon is not excited, i.e., ignore states electric TM (jg=1) and longitudinal (Eq. (13)). In addition, there is some difficulties that taint the lattice masse calculations:
(i)
How to identify interpolation fields used as a hybrid and distinguish them from ordinary mesons? From the criteria for hybrids proposed in [5] (and adopted implicitly by earlier lattice works [2–4]) the hybrid-like character is directly related to the overlap with the appropriate JPC interpolating fields. This is not always true, we cannot understand the nature of a state by the appearance of its interpolation field. This is sufficiently illustrated by the strong projection on η and η′ produced with the glue interpolation field GμνG~μν, it does not mean that they are glueballs [69](ii)
In the light sector, lattice authors report only results related to thejgPgCg=1+− TE-gluon since it has the best signal with the smallest statistical errors while the explicit masses of the jgPgCg=1−− TM-gluon are not yet published(iii)
The lattice calculation sill uses an unrealistic mass of theπ meson (~396 MeV) which is much greater than the observed one (~139 MeV).


## 4.2. The Decay Results

As shown in Eq. (23), the decay width is proportional to the parameter αs which is in turn correlated to the mass of the gluon mg, as mentioned above. In Table 3, we represent the results for mg=0.5±0.1GeV and the final theoretical uncertainty is taken as the deviation from the nominal value and the upper and lower tolerance.Despite the imperfections of the model, our predictions are mostly in reasonable accord with the observed1−+ resonance π11600 seen by several collaborations as shown in Table 5. This is especially true for the controversial ρπ channel which is forbidden by the gluonic excitation models (the "L+S" selection rule [10]).In the constituent glue model, the nonvanishing width comes from the QE-hybrid mode (lg=0 whith P-wave qq¯, Eq. (13)) decaying preferably into two S-wave mesons, i.e.,
(26)Γ1−+→ρπ≃0.28−0.09−0.14GeV.


## 5. Conclusion

To conclude, we note that despite the imperfections of the model, the results obtained are encouraging and describe quite well the observed properties of the resonanceπ11600, supporting the fact that this resonance is a hybrid meson with the internal structure suggested by the generalized Quark Model with Constituent Gluon, i.e., a pair of quark-antiquark with a massive constituent gluon: mg≃0.5±0.1 GeV. However, this approximate model needs to be improved by considering more relativistic effects especially for the decay model. On the other hand, it would be advisable to seriously review the hypothesis that hybrids are exclusively build by excited gluonic fields.



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*Source: 9105240-2020-03-21.xml*