EPJ Web of Conferences 3, 03014 (2010) DOI:10.1051/epjconf/20100303014 © Owned by the authors, published by EDP Sciences, 2010 Two-particle Bound States: Mesons and Glueballs Gurjav Ganbold1;2;a 1 Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia 2 Institute of Physics and Technology, 210651 Ulaanbaatar, Mongolia Abstract. A relativistic quantum-field model based on analytic confinement is considered to study the two- quark and two-gluon bound states. For the spectra of two-particle bound states we solve the ladder Bethe-Salpeter equation. We provide a new, independent and analytic estimate of the lowest glueball mass and found it at 1660 MeV. The conventional mesons and the weak decay constants are described to extend the consideration. By using a few parameters (the quark masses, the coupling constant and the confinement scale) we obtain numerical results which are in reasonable agreement with experimental evidence in the wide range of energy scale from 140MeV up to 9 GeV. The model can serve a reasonable framework to describe simultaneously different sectors in low-energy particle physics. 1 Introduction reliable result in low-energy hadronization region. The cou- pled Schwinger-Dyson equation is a continuum method The calculations of hadron mass characteristics on the level without IR- and UV-cutoffs and describes successfully the of experimental data precision still remain among the un- QCD vacuum and the long distance properties of strong in- solved problems in QCD due to some technical and con- teractions such as confinement and chiral symmetry break- ceptual difficulties related with the color confinement and ing (e.g., [8]). However, an infinite series of equations re- spontaneous chiral symmetry breaking. We are far from quires to make truncations which are gauge dependent. The understanding how QCD works at longer distances. The Bethe-Salpeter equation (BSE) is an important tool for study- well established conventional perturbation theory cannot ing the relativistic two-particle bound state problem in a be used at low energy, where the most interesting and novel field theory framework [9]. The BS amplitude in Minkowski behavior is expected [1]. space is singular and therefore, it is usually solved in Eu- The confinement and dynamical symmetry breaking are clidean space to find the binding energy. The solution of two crucial features of QCD, although they correspond to the BSE allows to obtain useful information about the under- different energy scales [2,3]. The confinement is an expla- structure of the hadrons and thus serves a powerful test for nation of the physics phenomenon that color charged par- the quark theory of the mesons. Numerical calculations in- ticles are not observed, the quarks are confined with other dicate that the ladder BSE with phenomenological poten- quarks by the strong interaction to form bound states so tial models can give satisfactory results (e.g., see [10]). that the net color is neutral. However, there is no analytic proof that QCD should be color confining and the reasons There exist different suggestions about the origin of for quark confinement may be somewhat complicated. confinement, some dating back to the early eighties (e.g., At longer distances, it is useful to investigate the cor- [11,12]) and some more recent based on the Wilson loop responding low-energy effective theories instead of tack- techniques [13], string theory quantized in higher dimen- ling the fundamental theory itself. Although lattice gauge sions [14] and lattice Monte-Carlo simulations (e.g., [15]) theories are the way to describe effects in the strong cou- etc. It may be supposed that the confinement is not oblig- pling regime, other methods can be applied for some prob- atory connected with the strong-coupling regime, but may lems not yet feasible with lattice techniques. So data in- be induced by the nontrivial background fields. One of the terpretations and calculations of hadron characteristics are earliest suggestion in this direction is the Analytic Con- frequently carried out with the help of phenomenological finement (AC) based on the assumption that the QCD vac- models. Different nonperturbative approaches have been uum is realized by the self-dual vacuum gluon fields which proposed to deal with the long distance properties of QCD, are stable versus local quantum fluctuations and related to such as chiral perturbation theory [4], QCD sum rule [5], the confinement and chiral symmetry breaking [11]. This heavy quark effective theory [6], etc. Along outstanding vacuum gluon field could serve as the true minimum of advantages these approaches have obvious shortcomings. the QCD effective potential [16]. Particularly, it has been Particularly, rigorous lattice QCD simulations [7] suffer shown that the vacuum of the quark-gluon system has the from lattice artifacts and uncertainties and cannot yet give minimum at nonzero self-dual homogenous background field with constant strength and the quark and gluon prop- a e-mail: [email protected] agators in the background gluon field represent entire an- This is an Open Access article distributed under the terms of the Creative Commons Attribution-Noncommercial License 3.0, which permits unrestricted use, distribution, and reproduction in any noncommercial medium, provided the original work is properly cited. Article available at http://www.epj-conferences.org or http://dx.doi.org/10.1051/epjconf/20100303014 EPJ Web of Conferences alytic functions on the complex momentum plan p2 [17]. sufficient to estimate the spectra of two-quark and two- However, direct use of these propagators for low-energy gluon bound states with reasonable accuracy [20,23]. The particle physics problems encounters complex formulae and path integrals defining the leading-order contributions to cumbersome calculations. the two-quark and two-gluon bound states read: It represents a certain interest to combine the AC con- ( ) ZZ 2 D E ception and the BSE method within a phenomenological −1 g 2 Zqq¯ = Dq¯Dq exp −(¯qS q) + (q¯ΓAq) ; model and to investigate some low-energy physics prob- 2 D lems by using the path-integral approach. Particularly, it g is shown that a ’toy’ model of interacting scalar ’quarks’ ZAA = exp − ( f AAF) ; (3) and ’gluons’ with AC could result in qualitatively reason- Z 2 D − 1 (AD−1A) able description of the two- and three-particle bound states h(•)iD DA e 2 (•) : [18] and obtained analytic solutions to the ladder BSE lead to the Regge behaviors of meson spectra [19]. This model was further modified in [20], applied to leptonic decay con- The Green’s functions in QCD are tightly connected to stants in [21] and used to simultaneously compute meson confinement and are ingredients for hadron phenomenol- masses and estimate the mass of the lowest-lying glueball ogy. The structure of the QCD vacuum is not well estab- in [23]. Below we consider an extended and more realis- lished and one may encounter difficulties by defining the tic model by taking into account the spin, color and flavor explicit quark and gluon propagator at the confinement scale. degrees of constituents. Here the aim is to collect all neces- Obviously, the conventional Dirac and Klein-Gordon forms sary formulae, explain the method in detail and show that of the propagators cannot adequately describe confined the correct symmetry structure of the quark-gluon interac- quarks and gluons in the hadronization region. Any widely tion in the confinement region reflected in simple forms accepted and rigorous analytic solutions to these propaga- of the quark and gluon propagators can result in quanti- tors are still missing. Besides, the currents and vertices tatively reasonable estimates of physical characteristics in used to describe the connection of quarks (and gluons) low-energy particle physics. In doing so, we build a model within hadrons cannot be purely local. And, the matrix describing hadrons as relativistic bound states of quarks elements of hadron processes are integrated characteris- and gluons and to calculate with reasonable accuracy the tics of the propagators and vertices. Therefore, taking into hadron important characteristics such as the lowest glue- account the correct global symmetry properties and their ball mass, mass spectra of conventional mesons and the breaking, also by introducing additional physical parame- decay constants of light mesons. ters, may be more important than the working out in detail (e.g., [22]). Due to the complexity of explicit Green functions de- 2 The Model rived in [17], we examine simpler propagators exhibiting similar characteristics. Consider the following quark and gluon (in Feynman gauge) propagators: Because of the complexity of QCD, it is often prudent to examine simpler systems exhibiting similar characteristics 8 9 ipˆ + m [1 ± γ !(m )] > p2 + m2 > first. Consider a simple relativistic quantum-field model of ˜ ab ab f 5 f < f = S ± (p ˆ) = δ exp >− 2 > ; quark-gluon interaction assuming that the AC takes place. Λm f : 2Λ ; The model Lagrangian reads [20]: δ ˜ AB AB µν 2 2 Dµν (p) = δ exp −p =4Λ ; (4) 1 2 p2 L = − FA − g f ABCABAC 4 µν µ ν X h i 2 2 a α α C ab b wherep ˆ = pµγµ and !(z) = 1=(1+z =4Λ ). The sign ’±’ in + q¯ f γα@ − m f + gΓCAα q f ; (1) the quark propagator corresponds to the self- and antiself- f dual modes of the background gluon fields. These propa- C gators are entire analytic functions in Euclidean space and where Aα - gluon adjoint representation (α = f1; :::; 4g); A µ A ν A ABC may serve simple and reasonable approximations to the ex- Fµν = @ Aν − @ Aµ ; f - the SUc(3) group structure a plicit propagators obtained in [17]. Note, the interaction of constant (fA; B; Cg = f1; :::; 8g); q f - quark spinor of flavor the quark spin with the background gluon field generates f with color a = f1; 2; 3g and mass m f ; g - the coupling a singular behavior S˜ ±(p ˆ) ∼ 1=m f in the massless limit α C C strength, ΓC = iγαt and t - the Gell-Mann matrices.