Hybrid Meson Interpretation of the Exotic Resonance <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.199599pt" id="M1" height="16.4602pt" version="1.1" viewBox="-0.0657574 -12.2606 61.3997 16.4602" width="61.3997pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-238" d="M574 449L545 460C526 432 516 430 487 430C404 430 311 435 226 435C104 435 56 379 25 341L43 318C81 354 121 372 181 372C161 246 87 53 23 3L30 -12C48 -12 88 -4 113 11C157 75 207 248 232 371L386 367L326 109C321 86 318 66 318 50C318 4 339 -12 366 -12C410 -12 461 21 505 69L492 96C467 79 434 60 418 60C406 60 400 78 411 147C422 217 439 300 457 366C487 366 524 367 536 370C547 385 558 408 574 449Z"/></g><g transform="matrix(.012,0,0,-0.012,9.713,4.134)"><path id="g50-50" d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z"/></g><g transform="matrix(.017,0,0,-0.017,16.241,0)"><path id="g113-41" d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"/></g><g transform="matrix(.017,0,0,-0.017,22.179,0)"><path id="g113-50" d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z"/></g><g transform="matrix(.017,0,0,-0.017,30.415,0)"><path id="g113-55" d="M137 343C167 482 260 545 321 574C357 591 397 603 429 609L423 641C382 634 335 622 295 608C189 570 37 457 37 238C37 84 125 -12 242 -12C362 -12 447 89 447 209C447 311 374 393 267 393C247 393 226 386 204 376L137 343ZM227 337C318 337 361 256 361 173C361 105 336 22 258 22C176 22 126 120 126 240C126 266 127 291 132 310C155 323 189 337 227 337Z"/></g><g transform="matrix(.017,0,0,-0.017,38.654,0)"><path id="g113-49" d="M241 635C89 635 35 457 35 312C35 153 89 -12 240 -12C390 -12 443 166 443 312C443 466 390 635 241 635ZM238 602C329 602 354 454 354 312C354 172 330 22 240 22C152 22 124 173 124 313S148 602 238 602Z"/></g><g transform="matrix(.017,0,0,-0.017,46.89,0)"><use xlink:href="#g113-49"/></g><g transform="matrix(.017,0,0,-0.017,55.127,0)"><path id="g113-42" d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"/></g></svg>

Benhamida, Azzeddine; Semlala, Lahouari

doi:https://doi.org/10.1155/2020/9105240

Advances in High Energy Physics

On this page

Abstract Introduction Results and Discussion Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 9105240 | https://doi.org/10.1155/2020/9105240

Hybrid Meson Interpretation of the Exotic Resonance

Azzeddine Benhamida¹and Lahouari Semlala²

Academic Editor: Roelof Bijker

Received12 Sept 2019

Accepted06 Mar 2020

Published21 Mar 2020

Abstract

The exotic resonance is examined in the framework of the Quark Model with Constituent Gluon (QMCG). We report the possibility of interpreting that resonance as meson, with a masse GeV, and a decay width to GeV.

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 have quantum numbers which are not allowed by the naive quark model, like , , , and , 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 exotic resonances have been claimed to be identified, especially and 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 candidates.

The was observed decaying into , , , and . But although the first three modes have been confirmed for a long time, the 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 and 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 the 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 decay width of the lightest quite compatible with the observed exotic candidate .

This paper is organized as follows. In Section 2, we briefly present the experimental situation of the exotic . We give predictions of the model QMGC in Section 3, and we conclude in Section 4.

2. The Experiemental Status of

We consider here only the status of the resonance (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 , , and 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 final state, they are unable to conclude that the is present, while the following experimental relationship between the branching fractions of the decays is obtained (and therefore a limit on the branching fraction of ):

E852 Collaboration at BNL reported an evidence for the resonance decaying into [33, 34], [35], [36], and [37], regarding the channel, in the earlier E852 analysis [33, 34] of K events showed a possible evidence for a 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 and 3 M events.

The exotic is observed decaying to 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-wave amplitude, which, if interpreted as a resonance, would have parameters consistent with the state with a mass of MeV and a width of MeV [40].

A search for exotic mesons in the system photoproduced by the charge exchange reaction was carried out by the CLAS Collaboration at Jefferson Lab., and no evidence is shown of the exotic decaying to three charged pions [41, 42].

COMPASS collaboration observed the spin-exotic in their partial-wave analysis of the final state. They reported the observation of the in the decay mode initially with a mass MeV and width MeV [43], superseded by a mass GeV and width GeV in their recent analysis [31]. COMPASS collaboration has also examined the exclusive production of and and reported that odd partial waves, which carry non- quantum numbers, are suppressed in the system relative to the system. Even though they saw the exotic wave in 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 system measured at COMPASS [44] and reported a single exotic with mass and width determined to be and MeV, respectively.

In conclusion, the was observed decaying into , , and and recently confirmed for 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 reported by different experiments.

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 gluonic rees 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-called Quark 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) mass . 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 and [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 and about 2 GeV for the lowest , as indicated in lattice QCD calculations [54–56], a simple constituent gluon picture may be approximately obtained as for the glueball mass .

Using the continuum strong QCD, one infers GeV [51] which is consistent with the lattice results: GeV [57, 58].

In the present work, we fix this parameter as:

The decay parameter (the effective quark-gluon vertex coupling) is the second ingredient of the model. There are many theoretical evidences that the QCD effective charge 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 [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 , 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 corresponding to the pinch technique gluon propagator, i.e., the background field propagator calculated in the Feynman gauge. is correlated to the gluon mass [65, 66]:

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) : the relative orbital momentum of the gluon in the center of mass

(2) : the relative orbital momentum between and

(3) : the total quarks spin

Considering the gluon moving in the framework of the pair, the Parity of the hybrid will be:

being the intrinsic parity of the gluon.

The Charge Conjugation is given by:

can take the values 0 or 1; P and C impose parity restrictions on and .

For lower values of the orbital excitations ( and ) and parity , the hybrid states can be built by two modes: and which we shall refer as the gluon-excited hybrid (GE hybrid), and and which we shall refer as the quark-excited hybrid (QE hybrid) (see Table 2 for the case ).

In the potential model, the simplest approximation is to factorise the -wave function with the wave function of the gluon respective to the center of mass (the cluster approximation). We shall use the following lowest-lying state -cluster spin-space wave function: where is the gluon polarisation, is the diquark spin representation, and the sum runs over the values of excluding those not consistent with and and: here, the Jacobi coordinates are introduced:

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 our wave function (Eq. (7)) is related to the so-called TE, TM, and longitudinal gluon states as follows (see Table 2): where means a mixture of the states and The “magnetic” TE, “electric” TM, and longitudinal gluons correspond to the following hybrid states [20]:

We can rewrite Eq. (9) according to the GE and QE hybrid modes as: where

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 . This is not true in the lattice hybrid calculations where the low-laying states are made with the particular TE-gluon mode. Indeed, although in principal lattice construction of the hybrid 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 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): where is the color index and , , and are the spin indices. In the nonrelativistic limit, we have: where is the antiquark spinor in the complex conjugate representation.

The standard meson ( or ) and the hybrid () states are written in the nonrelativistic approximation: with where , , and are the color charge of the quark, antiquark, and gluon with and We have also

The matrix element between a hybrid state and two standard mesons and is given by: where is the decay amplitude:

The amplitude involves the flavor , the color , the nonrelativistic spin , and the spatial overlaps defined as follows. label the hadron (quark) isospins, if the gluon goes into strange quarks and if it goes into nonstrange ones. is the number of diagrams contributing to the decay. Indeed, one can check that two diagrams contribute with the same sign and magnitude for , , and -Parity allowed decays while they cancel for forbidden ones. In the case of two identical final particles . The term between brackets in Eq. (20) is the symbol.

Finally, the spatial overlap is represented by the term: where label the orbital momentum between the two final mesons.

The partial width is given by: where 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)The QE-hybrid is allowed to decay into two S-wave mesons only (the so-called “” selection rule)(2)The GE-hybrid is allowed to decay into a channel with one S-wave meson and one P-wave meson only (the so-called “” 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, the 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 . 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 the hybrid masses and decay widths for GeV are summarised in Tables 3 and 4, we add Table 5 for comparison purposes.

It is difficult to get a hybrid masse lower than GeV ( GeV for ).

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 and an S-wave ), we have GeV for GeV.

Our calculated mass is: which is very close to the latest PDG average [29]: and quite far from 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 () 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 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 and produced with the glue interpolation field , it does not mean that they are glueballs [69](ii)In the light sector, lattice authors report only results related to the TE-gluon since it has the best signal with the smallest statistical errors while the explicit masses of the TM-gluon are not yet published(iii)The lattice calculation sill uses an unrealistic mass of the meson ( MeV) which is much greater than the observed one ( MeV).

4.2. The Decay Results

As shown in Eq. (23), the decay width is proportional to the parameter which is in turn correlated to the mass of the gluon , as mentioned above. In Table 3, we represent the results for 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 observed resonance seen by several collaborations as shown in Table 5. This is especially true for the controversial channel which is forbidden by the gluonic excitation models (the selection rule [10]).

In the constituent glue model, the nonvanishing width comes from the QE-hybrid mode ( whith -wave , Eq. (13)) decaying preferably into two S-wave mesons, i.e.,

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 resonance , 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: 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.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

We are grateful to Professor F. Iddir for her help and valuable advice. This work was supported by the PRFU research program (under No. B00L02EP310220190001).

References

C. A. Meyer and E. S. Swanson, “Hybrid mesons,” Progress in Particle and Nuclear Physics, vol. 82, pp. 21–58, 2015.
View at: Publisher Site | Google Scholar
C. Bernard, J. E. Hetrick, T. A. DeGrand et al., “Exotic mesons in quenched lattice QCD,” Physical Review D, vol. 56, no. 11, pp. 7039–7051, 1997.
View at: Publisher Site | Google Scholar
P. Lacock, K. Schilling, and SESAM Collaboration, “Hybrid and orbitally excited mesons in full QCD,” Nuclear Physics B, vol. 73, p. 261, 1999.
View at: Google Scholar
J. N. Hedditch, W. Kamleh, B. G. Lasscock, D. B. Leinweber, A. G. Williams, and J. M. Zanotti, “1−+exotic meson at light quark masses,” Physical Review D, vol. 72, no. 11, 2005.
View at: Publisher Site | Google Scholar
J. J. Dudek, “The lightest hybrid meson supermultiplet in QCD,” Physical Review D, vol. 84, no. 7, 2011.
View at: Publisher Site | Google Scholar
C. Bernard, T. Burch, E. B. Gregory et al., “Lattice calculation of 1−+ hybrid mesons with improved Kogut-Susskind fermions,” Physical Review D, vol. 68, no. 7, 2003.
View at: Publisher Site | Google Scholar
C. McNeile and C. Michael, “Decay width of light quark hybrid meson from the lattice,” Physical Review D, vol. 73, no. 7, 2006.
View at: Publisher Site | Google Scholar
N. Isgur, R. Kokoski, and J. Paton, “Gluonic excitations of mesons: why they are missing and where to find them,” Physical Review Letters, vol. 54, no. 9, pp. 869–872, 1985.
View at: Publisher Site | Google Scholar
F. E. Close and P. R. Page, “The production and decay of hybrid mesons by flux-tube breaking,” Nuclear Physics B, vol. 443, no. 1-2, pp. 233–254, 1995.
View at: Publisher Site | Google Scholar
P. R. Page, “Why hybrid meson coupling to two S-wave mesons is suppressed,” Physics Letters B, vol. 402, no. 1-2, pp. 183–188, 1997.
View at: Publisher Site | Google Scholar
P. R. Page, E. S. Swanson, and A. P. Szczepaniak, “Hybrid meson decay phenomenology,” Physical Review D, vol. 59, no. 3, 1999.
View at: Publisher Site | Google Scholar
T. Barnes, F. E. Close, and F. de Viron, “QG hybrid mesons in the MIT bag model,” Nuclear Physics B, vol. 224, no. 2, pp. 241–264, 1983.
View at: Publisher Site | Google Scholar
M. Flensburg, C. Peterson, and L. Sköld, “Applications of an improved bag model,” Zeitschrift für Physik C Particles and Fields, vol. 22, no. 3, pp. 293–300, 1984.
View at: Publisher Site | Google Scholar
J. Govaerts, F. de Viron, D. Gusbin, and J. Weyers, “QCD sum rules and hybrid mesons,” Nuclear Physics B, vol. 248, no. 1, pp. 1–18, 1984.
View at: Publisher Site | Google Scholar
F. de Viron and J. Govaerts, “Some decay modes of1−+hybrid mesons,” Physical Review Letters, vol. 53, no. 23, pp. 2207–2210, 1984.
View at: Publisher Site | Google Scholar
S. Narison, “Gluonia, scalar and hybrid mesons in QCD,” Nuclear Physics A, vol. 675, no. 1-2, pp. 54–63, 2000.
View at: Publisher Site | Google Scholar
S. Narison, “1−+ light exotic mesons in QCD,” Physics Letters B, vol. 675, no. 3-4, pp. 319–325, 2009.
View at: Publisher Site | Google Scholar
Z. R. Huang, H. Y. Jin, and Z. F. Zhang, “New predictions on the mass of the 1−+ light hybrid meson from QCD sum rules,” Journal of High Energy Physics, vol. 2015, no. 4, 2015.
View at: Publisher Site | Google Scholar
D. Horn and J. Mandula, “Model of mesons with constituent gluons,” Physical Review D, vol. 17, no. 3, pp. 898–908, 1978.
View at: Publisher Site | Google Scholar
A. Le Yaouanc, L. Oliver, O. Pène, J. -C. Raynal, and S. Ono, “qq̄g Hybrid mesons in ψ → γ+hadrons,” Zeitschrift für Physik C Particles and Fields, vol. 28, pp. 309–315, 1985.
View at: Publisher Site | Google Scholar
F. Iddir, A. Le Yaouanc, L. Oliver, O. Pène, J.-C. Raynal, and S. Ono, “qq̄g hybrid and qqq̄q̄ diquonium interpretation of the GAMS 1−+ resonance,” Physics Letters B, vol. 205, no. 4, pp. 564–568, 1988.
View at: Publisher Site | Google Scholar
S. Ishida, M. Oda, H. Sawazaki, and K. Yamada, “Is the f1(1420) our first hybrid meson?” Progress of Theoretical Physics, vol. 82, no. 1, pp. 119–126, 1989.
View at: Publisher Site | Google Scholar
S. Ishida, H. Sawazaki, M. Oda, and K. Yamada, “Decay properties of hybrid mesons with a massive constituent gluon and search for their candidates,” Physical Review D, vol. 47, no. 1, pp. 179–198, 1993.
View at: Publisher Site | Google Scholar
F. Iddir and L. Semlala, “Hybrid states from constituent glue model,” International Journal of Modern Physics A: Particles and Fields; Gravitation; Cosmology; Nuclear Physics, vol. 23, no. 32, pp. 5229–5250, 2008.
View at: Publisher Site | Google Scholar
L. Semlala and F. Iddir, “The hybrid meson: new results from the updated mg and αs parameters,” International Journal of Modern Physics A, vol. 26, no. 23, pp. 4101–4110, 2011.
View at: Publisher Site | Google Scholar
Y. S. Kalashnikova and A. V. Nefediev, “Spectra and decays of hybrid charmonia,” Physical Review D, vol. 77, no. 5, 2008.
View at: Publisher Site | Google Scholar
F. J. Llanes-Estrada, S. R. Cotanch, A. P. Szczepaniak, and E. S. Swanson, “Hyperfine meson splittings: chiral symmetry versus transverse gluon exchange,” Physical Review C, vol. 70, no. 3, 2004.
View at: Publisher Site | Google Scholar
I. J. General, F. J. Llanes-Estrada, and S. R. Cotanch, “Coulomb gauge approach to qq̄g hybrid mesons,” European Physical Journal C: Particles and Fields, vol. 51, no. 2, pp. 347–358, 2007.
View at: Publisher Site | Google Scholar
M. Tanabashi, K. Hagiwara, K. Hikasa et al., “Review of particle physics,” Physical Review D, vol. 98, article 030001, 2018.
View at: Publisher Site | Google Scholar
F. E. Close and J. J. Dudek, ““Forbidden” decays of hybrid mesons to πρ can be large,” Physical Review D, vol. 70, no. 9, 2004.
View at: Publisher Site | Google Scholar
M. Aghasyan, M. G. Alexeev, G. D. Alexeev et al., “Light isovector resonances in π−p→π−π−π+p at 190 GeV/c,” Physical Review D, vol. 98, no. 9, 2018.
View at: Publisher Site | Google Scholar
D. V. Amelin, Y. G. Gavrilov, Y. P. Gouz et al., “Investigation of hybrid states in the VES experiment at the Institute for High Energy Physics (Protvino),” Physics of Atomic Nuclei, vol. 68, no. 3, pp. 359–371, 2005.
View at: Publisher Site | Google Scholar
G. S. Adams, T. Adams, Z. Bar-Yam et al., “Observation of a new JPC=1−+ exotic state in the reaction π−p→π+π−π−pat18GeV/c,” Physical Review Letters, vol. 81, no. 26, pp. 5760–5763, 1998.
View at: Publisher Site | Google Scholar
S. U. Chung, K. Danyo, R. W. Hackenburg et al., “Exotic andqq¯resonances in the π+π−π−system produced in π−pcollisions at18 GeV/c,” Physical Review D, vol. 65, no. 7, 2002.
View at: Publisher Site | Google Scholar
E. I. Ivanov, D. L. Stienike, D. I. Ryabchikov et al., “Observation of exotic meson production in the reaction π−p→ηπ−pat18GeV/c,” Physical Review Letters, vol. 86, no. 18, pp. 3977–3980, 2001.
View at: Publisher Site | Google Scholar
J. Kuhn, G. S. Adams, T. Adams et al., “Exotic meson production in the f1(1285)π− system observed in the reaction π−p→ηπ+π−π−p at 18 GeV/c,” Phys. Lett. B, vol. 595, no. 1-4, pp. 109–117, 2004.
View at: Publisher Site | Google Scholar
M. Lu, G. S. Adams, T. Adams et al., “Exotic meson decay to ωπ0π−,” Physical Review Letters, vol. 94, no. 3, article 032002, 2005.
View at: Publisher Site | Google Scholar
A. R. Dzierba, R. Mitchell, E. Scott et al., “Partial wave analysis of the π−π−π+ and π−π0π0 systems and the search for aJPC=1−+meson,” Physical Review D, vol. 73, no. 7, article 072001, 2006.
View at: Publisher Site | Google Scholar
C. A. Baker, C. J. Batty, K. Braune et al., “Confirmation of a0(1450) and π1(1600) in p̄p→ωπ+π−π0 at rest,” Physics Letters B, vol. 563, no. 3-4, pp. 140–149, 2003.
View at: Publisher Site | Google Scholar
G. S. Adams, J. Napolitano, K. M. Ecklund et al., “Amplitude analyses of the decaysχc1→ηπ+π−and χc1→ηπ+π−,” Physical Review D, vol. 84, no. 11, article 112009, 2011.
View at: Publisher Site | Google Scholar
M. Nozar, C. Salgado, D. P. Weygand et al., “Search for the photoexcitation of exotic mesons in the π+π+π−system,” Physical Review Letters, vol. 102, no. 10, article 102002, 2009.
View at: Publisher Site | Google Scholar
P. Eugenio and B. Stokes, “Photo-production of proton antiproton resonances,” in AIP Conference Proceedings, Florida State University, Tallahassee, FL, 2004.
View at: Publisher Site | Google Scholar
M. G. Alekseev, V. Y. Alexakhin, Y. Alexandrov et al., “Observation of aJPC=1−+exotic resonance in diffractive dissociation of 190 GeV/cπ− into π−π−π+,” Physical Review Letters, vol. 104, no. 24, article 241803, 2010.
View at: Publisher Site | Google Scholar
C. Adolph, R. Akhunzyanov, M. G. Alexeev et al., “Odd and even partial waves of ηπ−and ηπ−in π−p→η()π−p at 191GeV/c,” Physics Letters B, vol. 740, pp. 303–311, 2015.
View at: Publisher Site | Google Scholar
A. Rodas, A. Pilloni, M. Albaladejo et al., “Determination of the pole position of the lightest hybrid meson candidate,” Physical Review Letters, vol. 122, no. 4, article 042002, 2019.
View at: Publisher Site | Google Scholar
K. Büttner and M. R. Pennington, “Infrared behavior of the gluon propagator: confining or confined?” Physical Review D, vol. 52, no. 9, pp. 5220–5228, 1995.
View at: Publisher Site | Google Scholar
R. Alkofer and L. von Smekal, “The infrared behaviour of QCD Green's functions confinement, dynamical symmetry breaking, and hadrons as relativistic bound states,” Physics Reports, vol. 353, no. 5-6, pp. 281–465, 2001.
View at: Publisher Site | Google Scholar
C. S. Fischer, “QCD at finite temperature and chemical potential from Dyson–Schwinger equations,” Progress in Particle and Nuclear Physics, vol. 105, pp. 1–60, 2019.
View at: Publisher Site | Google Scholar
A. Maas, “Gauge bosons at zero and finite temperature,” Physics Reports, vol. 524, no. 4, pp. 203–300, 2013.
View at: Publisher Site | Google Scholar
A. C. Aguilar, D. Binosi, and J. Papavassiliou, “The gluon mass generation mechanism: a concise primer,” Frontiers of Physics, vol. 11, no. 2, 2016.
View at: Publisher Site | Google Scholar
I. C. Cloët and C. D. Roberts, “Explanation and prediction of observables using continuum strong QCD,” Progress in Particle and Nuclear Physics, vol. 77, pp. 1–69, 2014.
View at: Publisher Site | Google Scholar
J. R. Forshaw, J. Papavassiliou, and C. Parrinello, “Massive Yang-Mills model and diffractive scattering,” Physical Review D, vol. 59, no. 7, article 074008, 1999.
View at: Publisher Site | Google Scholar
J. H. Field, “Phenomenological analysis of gluon mass effects in inclusive radiative decays of theJ/ψ and Υ,” Physical Review D, vol. 66, no. 1, article 013013, 2002.
View at: Publisher Site | Google Scholar
H. J. Rothe, Lattice Gauge Theories: An Introduction, World Scientific, 3rd ed edition, 2005.
N. Ishii, H. Suganuma, and H. Matsufuru, “Glueball properties at finite temperature in SU(3) anisotropic lattice QCD,” Physical Review D, vol. 66, no. 9, article 094506, 2002.
View at: Publisher Site | Google Scholar
N. Ishii, H. Suganuma, and H. Matsufuru, “Scalar glueball mass reduction at finite temperature in SU(3) anisotropic lattice QCD,” Physical Review D, vol. 66, no. 1, article 014507, 2002.
View at: Publisher Site | Google Scholar
P. Boucaud, M. E. Gómez, J. P. Leroy et al., “Low-momentum ghost dressing function and the gluon mass,” Physical Review D, vol. 82, no. 5, article 054007, 2010.
View at: Publisher Site | Google Scholar
P. Boucaud, J. P. Leroy, A. Le Yaouanc, J. Micheli, O. Pène, and J. Rodríguez-Quintero, “The infrared behaviour of the pure yang–mills green functions,” Few-Body Systems, vol. 53, no. 3-4, pp. 387–436, 2012.
View at: Publisher Site | Google Scholar
E. G. S. Luna, “Diffraction and an infrared finite gluon propagator,” Brazilian Journal of Physics, vol. 37, no. 1a, 2007.
View at: Publisher Site | Google Scholar
A. A. Natale, “Phenomenology of infrared finite gluon propagator and coupling constant,” Brazilian Journal of Physics, vol. 37, no. 1b, pp. 306–312, 2007.
View at: Publisher Site | Google Scholar
A. C. Aguilar, A. Mihara, and A. A. Natale, “Freezing of the QCD coupling constant and solutions of Schwinger-Dyson equations,” Physical Review D, vol. 65, no. 5, article 054011, 2002.
View at: Publisher Site | Google Scholar
S. Godfrey and N. Isgur, “Mesons in a relativized quark model with chromodynamics,” Physical Review D, vol. 32, no. 1, pp. 189–231, 1985.
View at: Publisher Site | Google Scholar
J. M. Cornwall, “Dynamical mass generation in continuum quantum chromodynamics,” Physical Review D, vol. 26, no. 6, pp. 1453–1478, 1982.
View at: Publisher Site | Google Scholar
J. M. Cornwall and J. Papavassiliou, “Gauge-invariant three-gluon vertex in QCD,” Physical Review D, vol. 40, no. 10, pp. 3474–3485, 1989.
View at: Publisher Site | Google Scholar
A. C. Aguilar, D. Binosi, and J. Papavassiliou, “QCD effective charges from lattice data,” Journal of High Energy Physics, vol. 2010, no. 7, article 731, 2010.
View at: Publisher Site | Google Scholar
A. C. Aguilar, D. Binosi, J. Papavassiliou, and J. Rodríguez-Quintero, “Nonperturbative comparison of QCD effective charges,” Physical Review D, vol. 80, no. 8, article 085018, 2009.
View at: Publisher Site | Google Scholar
N. J. Poplawski, A. P. Szczepaniak, and J. T. Londergan, “Towards a relativistic description of exotic meson decays,” Physical Review D, vol. 71, no. 1, article 016004, 2005.
View at: Publisher Site | Google Scholar
F. Iddir, A. Le Yaouanc, L. Oliver, O. Pène, and J. C. Raynal, “Selection rule for 1−+ hybrid decay into ηπ from QCD,” Physics Letters B, vol. 207, no. 3, pp. 325–328, 1988.
View at: Publisher Site | Google Scholar
Y. B. Yang, Y. Chen, G. Li, and K. F. Liu, “Is the1−+meson a hybrid?” Physical Review D, vol. 86, no. 9, article 094511, 2012.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2020 Azzeddine Benhamida and Lahouari Semlala. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP³.

PDF Download Citation

Download other formats

Order printed copies

Views

571

Downloads

722

Citations