318. Glueballs and gluelumps as bound states of transverse constituent gluons

Posted by Fabien Buisseret

March 23, 2009

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Fabien Buisseret is a postdoc at the Nuclear Physics department of the University of Mons-Hainaut. His interests include various approaches to QCD. Dmitry.

1. Generalities

Among other exotic hadrons like hybrid mesons and tetraquarks, QCD allows the existence of purely gluonic bound states, called glueballs, whose structure and properties deserve a lot of interest theoretically. A recent review on glueballs can be found in arXiv:0810.4453. Much effort is also devoted to the experimental detection of a clear glueball signal, but no unambiguous candidate has been found yet (arXiv:0812.0600). An important achievement in the field has been the computation of the glueball spectrum in quenched lattice QCD (hep-lat/9901004, hep-lat/0510074), that is the mass spectrum of pure gauge QCD. Gluelump masses have also been obtained on the lattice (hep-lat/9811010, hep-ph/0310130). These are another hypothetical type of gluonic hadrons, where a spinless, static, color-octet source is added to the gauge field. Gluelumps have actually been a first attempt to model gluino-gluon bound states. A plot of some lattice data concerning glueballs and gluelumps is given in Figure 1, where the masses are expressed in units of the lattice energy scale $r_0^{-1}$ in order to avoid larger error bars due to the determination of r_0 .

Glueball and gluelump masses

Figure 1: Some low-lying glueball and gluelump masses obtained in quenched lattice QCD, plotted in lattice units. The boxes separate the spectrum into states with a given number of constituent gluons.

Apart from lattice QCD, the glueball spectrum has also been computed by using effective approaches like Coulomb gauge QCD (hep-ph/0308268) and potential models – see for example the pioneering work of Barnes [Z. Phys C 10, 275 (1981)]. Potential models have been very succesful in the meson and baryon sectors, where those hadrons are described as bound states of valence quarks and antiquarks. The idea that low-energy QCD allows for an effective description of hadrons as bound states of constituent particles is actually not new: The classification of baryons and mesons with the quark hypothesis is a first historical example of the viability of such a picture. Still in potential models, glueballs and gluelumps are assumed to be bound states of constituent gluons. In the present post I give arguments, taken from previous works (arXiv:0802.0088, arXiv:0806.3174, arXiv:0808.2399, arXiv: 0902.1028, arXiv:0902.4836), showing the relevance of such a picture. Notice that the properties of constituent gluons are not unanimously accepted: Should they be massive or massless? With a spin- or helicity-degree of freedom? As shown in the aforementioned works, it is crucial for the constituent gluons to be transverse, that is with helicity-1, in order to understand the lattice data. This will appear more clearly in the following. Concerning the mass, I assume that a constituent gluon has a zero bare mass, as a “usual” gluon does. But, since it is confined inside a glueball, it can gain a dynamical, or constituent, mass generated by the confining interaction. Recall that confinement is due to low-energy interactions between color-charged particles, either quarks or gluons.

2. The constituent gluon picture

Here is our basic assumption: A description of gluelumps and glueballs in terms of states with a given number of transverse constituent gluons is a satisfactory approximation of what these gluonic hadrons exactly are. Large-Nc QCD provides a hint of the relevance of that hypothesis (see the reviews hep-ph/9802419, hep-ph/0701061). In the limit where the number of colours becomes infinite, baryons are pure N_c -quark states and model-independent mass formulae can be obtained within this framework, leading to a very accurate description of light and heavy baryon resonances. The Large- N_c limit thus appears to capture the essential features of QCD, in the baryonic sector in particular. But the number of valence gluons is a good quantum number for glueballs in this limit (arXiv:0710.4185). By analogy, it suggests that the Fock-space expansion of a given glueball at N_c=3 may be dominated by a particular component, characterised by its number of constituent gluons. Let us now identify that component for the different states of Figure 1.

Bound states of two gluons are the most studied purely gluonic systems in the literature. The color wave function of a two-gluon system is the symmetric configuration [8,8]^1 , leading to a positive charge conjugation and to a symmetric spin-space wave function in virtue of the Pauli principle. Jacob and Wick’s helicity formalism [Ann. Phys.7, 404 (1959)] can be applied to build the spin-space wave function of a two-transverse gluon system. It appears that no $J^{PC}=1^{P+}$ state exists, in agreement with Yang’s theorem stating that a vector meson cannot decay into two photons. The average value of the squared orbital angular momentum can give an idea of the mass hierarchy of the different states. One obtains the following ordering for the lightest two-gluon glueballs: $0^{P+}$ , $2^{++}$ , $2^{-+}$ , $3^{++}$ . Two is the minimal number of constituent gluons to build a color-singlet glueball. Looking at Figure 1, it is rather clear that the lightest glueballs are located in the positive-C sector. Moreover, the mass hierarchy corresponds to the one expected for a two-gluon bound state, and the absence of low-lying $1^{P+}$ glueballs is confirmed.

The lightest C=- glueballs are heavier than those with positive-C. That point can be intuitively understood by remarking that at least three gluons are needed to make a negative-C state, the color wave function being in that case the totally symmetric one $[[8,8]^{8s},8]^{1s}$ . Another color singlet exists; it is totally antisymmetric and leads to C=+ glueballs. Three-gluon helicity states can also be built in principle with the helicity formalism, and it can be deduced that the lightest ones have J=1 , followed by J=3 . If gluons were not transverse, a light $0^{-{}-}$ state would be allowed, in disagreement with lattice QCD. The three-gluon sector corresponds to the green box in Figure 1. It is now worth looking at the $0^{+-}$ glueball. It cannot be made of three gluons since J=0 is not allowed in this case. But a four gluon state can generate those quantum numbers: The $0^{+-}$ glueball could be located in a four-gluon sector, the purple box in Figure 1.

Finally, let us make some comments about gluelumps. They are lighter that the lightest glueballs. Such a low mass can be understood as follows: The presence of the static color-octet source makes possible for a single constituent gluon to be bound in a color singlet. The charge conjugation of such a state is always negative, and the helicity formalism demands that J>0 , in agreement with the lattice data.

Roughly speaking, the mass of a gluonic state should grow like $\mu{}N_g$ , with N_g the number of gluons and $\mu$ the constituent gluon mass. With $r_0\mu=3.1$ , one obtains the squares in Figure 1. They are all located near the average mass of a given sector, as expected. Since $r0=0.41 {\rm MeV}$ typically, $\mu$ is around $1 {\rm GeV}$ .

3. Interactions between gluons

Now we go beyond the above qualitative description by showing how to build an explicit potential model for glueballs. Consider the case of two-gluon glueballs. The simplest Hamiltonian describing a system of two massless particles is a spinless Salpeter one, given by

$H=2\sqrt{\vec{p}^2}+V(\vec{r})$ (1)

It is well-known from lattice QCD that the potential energy between a static quark-antiquark pair is accurately fitted by a funnel form, i.e.

$V(r)=\sigma{}r-\kappa{}/r+D$ .

The linear term stands for the confining interaction generated by a flux tube of tension $\sigma$ , while the Coulomb term encodes one-gluon-exchange effects. The funnel potential has been widely used in quark models to reproduce accurately the meson and baryon experimental mass spectra. Can it be used as a good gluon-gluon potential? The potential energy between two static color-octet sources has been computed in lattice QCD. It is still compatible with a funnel shape for different values of the parameters (hep-lat/0006022). In particular, the string tension has to be scaled by a factor (9/4) following the Casimir scaling hypothesis, and the color factor of the Coulomb term has also to be modified. However, massless constituent gluons are far from being static sources, and it would be interesting to find an alternative way of computing the gluon-gluon potential.

The radial wave function of the scalar glueball has been computed in lattice QCD (hep-lat/0603030), with a mass that is in agreement with previous calculations. The idea is now to find the local radial potential needed in (1) to reproduce the lattice mass and wave function. Such a task can be performed numerically and the result is shown in Figure 2. The computed gluon-gluon potential has a long-range confining part and a short-range attractive singular part. Actually, it is compatible with a funnel form, validating the use of such a potential in effective approaches.

Effective potential between two transverse gluons

Figure 2: Effective potential between two transverse gluons in the scalar channel (dashed line and gray area), computed from the lattice wave function R(r) (dots and dashed-dotted line) and the spinless Salpeter Hamiltonian (1).

By using parameters in agreement with the potential of Figure 2, one can build a glueball model, based on the helicity formalism and on the spinless Salpeter Hamiltonian (1), that reproduces accurately the glueball spectrum in the positive-C sector. The gluelump spectrum is also nicely reproduced, while three- and four-gluon generalizations of that approach remain to be done.

4. Gluon plasma

The constituent gluon picture eventually finds an application in finite-temperature QCD, in relation with the celebrated quark-gluon-plasma. Remember that the “confined world” of the three previous sections is a zero-temperature one. Theoretical as well as experimental results (mainly obtained at RHIC) support the idea that there exists a critical temperature in QCD, beyond which the hadronic matter is deconfined; that is the quark-gluon-plasma. For temperature slightly above the critical one Tc, lattice computations of the free and internal energies between two static color sources in various representations show that the color interactions are still nonnegligible, although no longer confining (see for example hep-lat/0309121 and arXiv:0711.2251). There exists consequently a strongly coupled phase near the critical temperature, where quarks and gluons are deconfined but not free as it could have been expected.

Let us turn our attention to a pure gluon plasma, whose equation of state is known from quenched lattice QCD (hep-lat/9506025 and Figure 3). These last data can be understood thanks to a constituent gluon approach: The gluon plasma is then modeled as an ideal Bose gas of transverse gluons with an appropriate temperature-dependent mass (hep-ph/9710463). The thermal gluon mass has to be fitted on lattice QCD, but from perturbative QCD it can be expected that it grows linearly at very large temperature. Fits are in agreement with that point. Notice that a linearly rising gluon mass is a necessary condition for the thermodynamical quantities to saturate below the Stefan-Boltzmann limit, as it can be observed in Figure 3.

Energy density, entropy density and pressure vs. temperature

Figure 3: Energy density (red line), entropy density (pink line), and pressure (blue line), in reduced form, versus temperature computed in quenched lattice QCD at zero chemical potential. The horizontal black line shows the Stefan-Boltzmann limit for a gas of transverse gluons. Gray curves are the results of the glueball-gluon gas model.

In the paragraphs above, two apparently opposite statements have been made. First, color interactions between gluons are strong near T_c . Second, the gluon plasma can be described as an ideal gas of transverse gluons in the same temperature range. The point is that the thermal gluon mass, fitted on lattice data, takes into account effective color interactions. It even becomes singular near T_c , while the linear behavior only appears at temperature larger than 2.5 T_c . Using a spinless Salpeter Hamiltonian such as (1) in which the gluon-gluon potential is modified according to finite-temperature lattice results, it can be computed that the color interactions are strong enough to bind two gluons in the scalar channel up to 1.6 T_c . Then, it seems relevant to see the gluon plasma as an ideal Bose gas containing a mixture of free gluons and glueballs, with a gueball abundance depending on the temperature. Fitting that quantity on lattice QCD, one reproduces with an excellent accuracy the gluon plasma equation of state, see Figure 3.

5. Conclusions

In the present post were summarized various arguments showing that the glueballs and gluelumps currently observed in lattice QCD can be understood in terms of bound states of a few transverse constituent gluons. In this scheme, the lowest-lying glueballs can be identified with two-gluon states, while the lightest negative-C glueballs are compatible with three-gluon states. Gluelumps should moreover be seen as one-gluon states, allowed because of the static colour octet source.

Not only the glueball masses can be computed on the lattice, but also glueball wave functions. In the scalar channel, it appears that the lattice mass and wave function correspond to those of a spinless Salpeter Hamiltonian with a funnel potential, partly justifying the use – and the success – of such potential models in the description of glueballs.

Finally, modelling the gluon plasma as an ideal mixture of gluons and glueballs allows to understand its equation of state computed in quenched lattice QCD. Although it is no standard way of concluding, let me say that in view of the results known so far and those still to be obtained, the gluon below has good reasons to smile.

A constituent gluon :-)

A constituent gluon, as people from http://www.particlezoo.net imagine it.

Some good QCD reading

1. W. Greiner and A. Schafer, Quantum Chromodynamics, Springer, 1995 – pedestrian introduction to QCD.
2. F. J. Yndurain, The Theory of Quark and Gluon Interactions, Springer, 1998 – a higher-level course.

Journal club, Master Mason, Quantum field theory, String theory

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