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Mesons and Glueballs

Mesons and Glueballs

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: and

Gurjav Ganbold1,2,a 1 Bogoliubov Laboratory of Theoretical , 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- and two- 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 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 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 .

1 Introduction reliable result in low-energy region. The cou- pled Schwinger-Dyson equation is a continuum method The calculations of 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 break- ceptual difficulties related with the color confinement and ing (e.g., [8]). However, an infinite series of equations re- spontaneous . 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 . 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 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 are confined with other dicate that the ladder BSE with phenomenological poten- quarks by the strong 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], 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 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 ’’ 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 to the Regge behaviors of 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 , color and flavor explicit quark and gluon 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 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   ipˆ + m [1 ± γ ω(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 ω(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 = {1, ..., 4}); 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 ({A, B, C} = {1, ..., 8}); q f - quark spinor of flavor the quark spin with the background gluon field generates f with color a = {1, 2, 3} 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. m f → 0. This corresponds to the zero-mode solution (the Consider the partition function lowest Landau level) of the massless in the ZZ Z ( Z ) presence of external gluon background field and generates Z(g) = Dq¯ Dq DA exp − dx L[¯q, q, A] , a nontrivial quark condensate Z Z(0) = 1 . (2) D E d4 p h i q¯ (0)q (0) = − Tr S˜ (p ˆ) f f (2π)4 ± We allow that the coupling remains of order 1 (i.e., αs =   3  m2  g2/4π ∼ 1) in the hadronization region. Then, the consid- 6Λ  f  = − exp −  , 0 (5) eration may be restricted within the ladder approximation π2  2Λ2 

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indicating the broken chiral symmetry as m f → 0. A mass and a colorless bi-quark current localized at the center of splitting appears between vector and pseudoscalar mesons masses: (MV > MP) consisting of the same quark content. Z   Our model has a minimal number of parameters, namely, JN (x)  dy q¯ f1 (x) VQJ(x, y) q f2 (x) , the αs, the scale of confinement Λ and † the quark masses {mud, ms, mc, mb}. Hereby, we do not dis- JN (x) = JN (x) , N = {QJ f1 f2} . tinct the masses of lightest quarks, so mu = md = mud. Below we describe the main steps in our approach on Then, (7) can be rewritten as follows the example of the quark-antiquark bound state [21]. Z Allocate the one-gluon exchange between colored bi- g2 X L = dx J (x)J (x) . quark currents 2 2 N N N 2 X ZZ   g A L2 = dx1dx2 q¯ f (x1)iγµt q f (x1) Represent the exponential by using a Gaussian path in- 2 1 1 f1 f2 tegral   AB B Dµν (x1, x2) q¯ f2 (x2)iγνt q f2 (x2) . (6) 2 P g (J2 ) D E 2 N g(B J ) e N = e N N , The color-singlet combination is isolated: B Z Y 1 2 − 2 (BN ) 0 0 4 0 0 1 0 0 h(•)iB  DBN e (•) , h1iB = 1 (tA)i jδAB(tB) j i = δii δ j j − (tA)ii (tA) j j . 9 3 N

Perform a Fierz transformation by introducing auxiliary meson fields BN (x). Then, X *ZZ + (iγ )δµν(iγ ) = C · O O , n o µ ν J J J Z = Dq¯Dq exp −(¯qS −1q) + g(B J ) . J qq¯ N N B where J = {S, P, V, A, T}, C = {1, 1, 1/2, −1/2, 0} and J Now we can take explicit path integration over quark OJ = {I, iγ5, iγµ, γ5γµ, i[γµ, γν]/2}. For systems consisting of quarks with different masses variables and obtain it is important to pass to the relative co-ordinates (x, y) in    Z → Z = exp Tr ln 1 + g(B V )S , the center-of-masses system: qq¯ N N B P i where Tr  TrcTrγ ±, Trc and Trγ are traces taken on xi = x − (−1) ξi y, ξi = m fi /(m f1 + m f2 ) , i = 1, 2 . P color and spinor indices, correspondingly, while ± im- Then, we rewrite (6) plies the sum over self-dual and anti-self-dual modes.

2 X ZZ 2g † L2 = CJ dxdyJJ f f (x, y)D(y)J (x, y), (7) 9 1 2 J f1 f2 J f1 f2 3 Mesons

where In particle accelerators, scientists see ’jets’ of many color-   J (x, y) = q¯ (x + ξ y) O q (x − ξ y) . neutral particles in detectors instead of seeing the individ- J f1 f2 f1 1 J f2 2 ual quarks. This process is commonly called hadroniza- tion and is one of the least understood processes in particle Introduce a system of orthonormalized functions {UQ(x)}: physics. Z X QQ0 Introduce a hadronization Ansatz and identify BN (x) dx U (x) U 0 (x) = δ , U (z) U (y) = δ(z − y) . Q Q Q Q fields with mesons carrying quantum numbers N. Isolate Q 2 all quadratic field configurations (∼ BN ) in the ’kinetic’ Expand the bi-quark nonlocal current on the basis term and rewrite the partition function for mesons [23]:  † Z Y  X D(y) J (x, y)   h 0 i  J f1 f2  1 NN Z Z = DBN exp − BN δ + ΠNN0 BN0 p p  2 = D(y) dz δ(z − y) D(z) J† (x, z) N NN0 J f1 f2 −Wres[BN ]} , (8) X Z p p † = dz D(y)UQ(y) · D(z)UQ(z) J (x, z) . J f1 f2 where the interaction between mesons is described by the Q 3 residual part Wres[BN ] ∼ 0(BN ). Define a vertice function VQJ(x, y) The leading-order term of the polarization operator is

q¯ (x) V (x, y) q (x) Π 0 (z − z ) f1 QJ f2 NNZZ 1 2 2 p p  C D(y)U (y)q¯ (x + ξ y) O q (x − ξ y)  dxdy UN (x) αsλ(z1 − z2, x, y) UN0 (y) , (9) 3 J Q f1 1 J f2 2

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where the Fourier transform of the kernel reads momentum ` the is P = (−1)`+s and the total spin is Z |` − s| < J < |` + s|. Below we consider the meson ground ipz PC −+ αsλJJ0 (p, x, y) = αs dz e λJJ0 (z, x, y) states (` = 0, nr = 0), the pseudoscalar (P : J = 0 ) and vector (V : JPC = 1−−) mesons, the most established 2 √ Z 4 4g C C 0 p d k sectors of hadron spectroscopy. = J J D(x)D(y) e−ik(x−y) 9 (2π)4 We should derive the meson masses from equation (11). h    i The polarization kernel λ (−p2) is real and symmetric that ˜ ˆ ˜ ˆ N ·Tr OJS k + ξ1 pˆ OJ0 S k − ξ2 pˆ . (10) allows us to find a simple variational solution to this prob- lem. For the ground-state we choose a trial function [21, Diagonalize the polarization kernel on the orthonormal 23]: basis {UN }: ( ) Z p aΛ2 x2 2 U(x, a) ∼ D(x) · exp − , dyλ 0 (p, x, y) U 0 (y) = λ (−p ) U 0 (x) 4 JJ N N N Z 2 or, dx |U(x, a)| = 1 , a > 0 . (14) ZZ NN0 2 Substituting (14) into (11) the following variational equa- dxdyUN (x)λJJ0 (p, x, y)UN0 (y) = δ λN (−p ) tion defining the masses of P and V mesons as follows:

that is equivalent to the solution of the corresponding lad- 1 = −αs · λJ(Λ, MJ, m1, m2) 2  2 2 2 2 2  der BSE. αsCJΛ  MJ (ξ1 + ξ2) − m1 − m2  In relativistic quantum field theory a stable bound state = exp  2  3πm1m2 2Λ of n massive particles shows up as a pole in the S-matrix  " #2  2 2  with a center of mass energy. Accordingly, the meson mass  (6a − 1)(1 − 2a)  aMJ (ξ1 − ξ2)  · max  · exp −  may be derived from equation: 1/4

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where JA is the axial vector part of the weak current and

2980.95799 UR(p) is the normalized vector of state.

We estimate 1096.63316 √ Z Z p 2 g dk 403.42879 f · p = dx e−ikx U (x) D(x) P µ 3 (2π)4 R

h    i 148.41316 Tr iγ5S˜ kˆ + ξ1 pˆ γ5γµS˜ kˆ − ξ2 pˆ

q 54.59815 u 2 (MeV)massQuark ˙ s 32 Λ αs 2λ(MP) (1 − 2a )(6a − 1) P P 20.08554 c = pµ · 3/2 2 b 3 π (m1 + m2) (1 + 2aP)

 7.38906 " 2 # 2 2 2 aP (m1 − m2)  M (ξ + ξ )  J 1 2 380 400 420 440 460 480 1 + exp  1 + 2aP m1m2 2 (MeV)  m2 + m2  Fig. 1. Solutions for constituent quark masses versa the confine- 1 2 aP 2 2 − − MP(ξ1 − ξ2)  , (16) ment scale value Λ. 2 1 + 2aP

6000 where aP is the value of parameter a calculated for given meson with mass MP.

5000

Particularly, for an ’asymmetric’ meson containing an *

K (495)

infinitely heavy quark (m1  m2 ∼ 1) we obtain the cor- D(1870)

B(5279) rect asymptotic behavior 4000

(2980) √ C fP ∼ 1/ m1

3000 due to the localization of meson field at the center of two

quark masses. 2000 Meson mass (MeV)

1000 3.3 Numerical Results

380 400 420 440 460 480 To calculate the meson masses we need to fix the model (MeV) parameters. We determine the quark mass mud and the cou- pling constant α from equations: Fig. 2. Solutions for some meson masses in dependence on the s confinement scale value Λ.

1 + αsλP(Λ, 138 MeV, mud, mud) = 0 , √ 1 + αsλV (Λ, 770 MeV, mud, mud) = 0 (17) and the solution for quark mass behaves mud ∼ αs → 0. This picture is observed in Fig.1. by fitting the well established mesons π(138) and ρ(770) By using these quark masses and coupling constant we at different values of Λ. The remaining constituent quark can estimate other meson masses in dependence on Λ and masses ms, mc and mb are determined by fitting the known the some results are shown in Fig. 2. mesons K(495), J/Ψ(3097) and Υ(9460) as follows: To fix the value of parameter Λ we calculate the weak 1 + α λ (Λ, 495, m , m ) = 0 , decay constants fπ and fK to compare with experimental s P ud s data. Note, these constants considerably depends on Λ (see 1 + αsλV (Λ, 3097, mc, mc) = 0 , Figure 3) that allows us to fix it unambiguously at Λ = 1 + αsλV (Λ, 9460, mb, mb) = 0 . 416.4 MeV. The final set of model parameters are fixed as follows: The dependencies of the estimated constituent quark masses

on Λ are plotted in Fig. 1. αs = 1.5023 ,Λ = 416.4 MeV , m = 206.9 MeV , m = 323.6 MeV The sharp drop of all quark mass curves in Fig.1 may ud s be shortly explained as follows. Note, two equations in mc = 1453.8 MeV , mb = 4698.9 MeV . (18) Eqs. (17) mostly differ by meson masses in exponentials With these parameters we have estimated the pseudoscalar along different numerical factors CJ, ρJ and χJ. They have and vector meson masses shown in Table 1 and compared general solutions {mud , αs} not for any Λ. Suppose, at fixed with experimental data [24]. The relative error of our es- Λ = Λ0 they are solvable. Then, for finite coupling αs timate does not exceed 3.5 per cent in the whole range of the solution mud is obviously finite to obey both equations. mass (from 140 MeV up to 9.5 GeV). However, for vanishing αs → 0 the equations take form There are mainly two schemes describing ω − Φ and α C η−η0 mixings [24]. The octet-singlet scheme uses the mix- 1 ≈ s J · const(Λ , M , ρ ) √ 2 0 J J ing angle θ between states (uu¯ + dd¯ − 2ss¯)/ 6 and (uu¯ + mud

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300 this expectation because it is estimated to fit the π meson f mass and so, the corresponding energy scale is ∼ 140 MeV.

f

250 We keep this value for further calculations. The weak decay constants of light mesons are well es- tablished data and many groups (MILC [29], NPLQCD 200 [30], HPQCD [31], etc.) have these with accuracy at 2 per- cent level. Therefore, these values are often used to test any model in QCD. By substituting optimal values of 150 {mud, ms, αs,Λ} (18) into (16) we calculate

100 fπ = 128.8 MeV , Weak DecayWeak Constants(MeV) fK = 157.7 MeV . (19)

50

380 400 420 440 460 480 The obtained estimates are in agreement with the ex- perimental data [32,24]: (MeV) Fig. 3. Weak decay constants depending on the confinement scale PDG fπ− = 130.4 ± 0.04 ± 0.2 MeV , value Λ. PDG fK− = 155.5 ± 0.2 ± 0.8 ± 0.2 MeV . (20) Table 1. Estimated spectrum of conventional mesons (in MeV). Our model represents a reasonable framework to de- PC −+ PC −+ scribe the conventional mesons and the parameters are fixed. J = 0 MP J = 0 MP Below we can consider two-gluon bound states. π(138) 138 ηc(2979) 3012 K(495) 495 B(5279) 5437 η(547) 547 Bs(5370) 5551 4 Glueball Lowest State D(1870) 1840 Bc(6286) 6522 D (1970) 1970 η (9300) 9434 s b Because of the confinement, gluons are not observed, they PC −− PC −− J = 1 MV J = 1 MV may only come in bound states called glueballs. Glueballs ∗ are the most unusual particles predicted by the QCD but ρ(770) 770 Ds(2112) 2078 ω(782) 785 J/Ψ(3097) 3097 not found experimentally yet [33]. There are predictions K∗(892) 909 B∗(5325) 5464 expecting non-qq¯ scalar objects, like glueballs and multi- Φ(1019) 1022 Υ(9460) 9460 quark states in the mass range ∼ 1500÷1800 MeV [34,35]. D∗(2010) 1942 Experimentally the closest scalar resonances to this energy range are the f0(1500) and f0(1710) [36]. Some references favor the f0(1500) as the lightest scalar glueball [37], while √ ¯ others do so for the f0(1710) [38,39]. Recent scalar hadron dd+ss¯)/ 3. We use the quark-fla√ vor based mixing scheme ¯ f0(1810) reported by the BES collaboration may also be a between states (uu¯ + dd)/ 2 and ss¯ with mixing angle ϕ. glueball candidate [40]. These two schemes√ are equivalent to each other by θ = ϕ − The study of glueballs currently deserves much interest π/2 + arctan(1/ 2) when the SU(3) symmetry is perfect. from a theoretical point of view, either within the frame- id ◦ Particularly, for ’ideal’ vector mixing the angle is ϕV = 90 work of effective models or lattice QCD. The glueball spec- id ◦ or, θV = 35.3 . trum has been studied by using effective approaches like With fixed parameters (18) we calculate a relatively the QCD sum rules [41], Coulomb gauge QCD [42] and heavy mass MV (ss¯) = 1064 MeV of vector ss¯ state. To potential models (e.g., [43,44]), etc. The potential mod- obtain correct masses of ω(782) and Φ(1019) one needs els consider glueballs as bound states of two or more con- a considerable mixing to light quark-antiquark state with stituent gluons interacting via a phenomenological poten- ◦ mixing angle ϕV ' 73.2 which differs significantly from tial [43,45,46]. It should be noted that potential models the ’ideal’ value. By using the same parameters (18) we have difficulties in reproducing all known lattice QCD data. obtain a pseudoscalar ss¯ state with mass MP(ss¯) = 705 Different string models are used for describing glueballs MeV. We cannot describe the physical mass of η0(958) by [47,48], including combinations of string and potential ap- any mixing to light-quark pair and can only fit the correct proaches [44]. It has been shown that a proper inclusion of ◦ mass MP(η) = 547 MeV at angle ϕP ' 58.5 . Our model the helicity degrees of freedom can improve the compati- fails to describe simultaneously η − η0 mixing. This prob- bility between lattice QCD and potential models [49]. lem obviously deserves a separate consideration. An important theoretical achievement in this field has Note, the infrared behavior of effective QCD coupling been the prediction and computation of the glueball spec- αs is not well defined and needs to be more specified [25– trum in lattice QCD simulations [50,51]. Recent lattice 27]. In the region below the τ- mass (Mτ = 1.777 calculations, QCD sum rules, ”tube” and constituent glue GeV) the strong coupling value is expected between models predict that the lightest glueball has the quantum PC ++ ++ αs(Mτ) ≈ 0.34 [24] and the infrared fix point αs(0) = numbers of scalar (J = 0 ) and tensor (2 ) states [52]. 2.972 [28]. Our parameter αs = 1.5023 does not contradict Gluodynamics has been extensively investigated within

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quenched lattice QCD simulations and the lightest glue- auxiliary field B(x) as follows ball is found a scalar object with a mass of ' 1.66 ± 0.05 Z ( ) GeV [53]. A use of much finer isotropic lattices resulted in 1   Z → Z = DB exp − B G−1B + L [B] , a value 1.475 GeV [51]. Recently, an improved quenched AA G 2 I lattice calculation of the glueball spectrum at the infinite 3 volume and continuum limits based on much larger and where LI[B] ∼ O(B ) and the BS kernel is finer lattices have been carried out and the scalar glueball mass is calculated to be 1710 ± 50 ± 80 MeV [54]. 8 g2 G−1(x − y) = δ(x − y) − Π(x − y) , Two-gluon bound states are the most studied purely ZZ 3 gluonic systems in the literature, because when the spin- p t + s  Π(z)  dtds U (t) W(t) D + z orbital interaction is ignored (` = 0), only scalar and ten- n 2 sor states are allowed. Particularly, the lightest glueballs t + s  p with positive charge parity can be successfully modeled ·D − z W(s) Un(s) . 2 by a two-gluon system in which the constituent gluons are massless helicity-one particles [55]. The hadronization Ansatz allows us to identify B with Below we consider a two-gluon scalar bound state. We scalar glueball field. To find the glueball mass we should isolate the color-singlet term in the bi-gluon current in ZAA diagonalize the Bethe-Salpeter kernel Π(z). The glueball (4) by using the known relations : mass MG is defined from equation [21]:

2 2 Z Nc − 1 1 8 g tC tC = δilδ jk − tCtC , izp 2 2 ik jl 2 il jk 1 − dz e Π(z) = 0 , p = −MG . (22) 2Nc Nc 3 0 0 2  0 0 0 0  f ABE f A B E = δAA δBB − δAB δBA For the lightest ground-state scalar glueball choose a 3 AA0 E BB0 E AB0 E BA0 E Gaussian wave function: + d d − d d . Z 2c 2 U(x) = e−cx , dx |U(x)|2 = 1 , c > 0 . The second-order matrix element containing color-singlet π two-gluon current reads [20] ZZ Then, we derive (22) as follows g2  AA BB √ LAA = dxdy Jµµ0 (x, y)Jνν0 (x, y)  2  2 12 αs  M  3π(3 + 2 2)  h 1 = exp G , α  . AA BB νν0 µν0  2  crit −Jµν0 (x, y)Jνµ0 (x, y) · δ Wµµ0 (x, y) − δ Wνµ0 (x, y) αcrit 4Λ 4 i νµ0 µµ0 −δ Wµν0 (x, y) + δ Wνν0 (x, y) , The final analytic result for the lowest-state glueball mass reads where " !#1/2 BC B C αcrit Jµν (x, y)  Aµ (x)Aν (y) , MG = 2Λ ln . (23) αs ∂ ∂ µν Wµν(x, y)  D(x − y) = δ W(x − y) + . .., 2 ∂xµ ∂yν The solution MG ≥ 0 exists for any αs < αcrit ≈ 80.041. Note, the scalar glueball mass depends linearly on the 1 2 W(z) = e−z . confinement scale Λ and the scaled mass M /Λ depends (2π)2 G only on coupling αs (see Fig. 4). Particularly, if take values This part consists of spin-zero (scalar) and spin-two Λ ∼ ΛQCD ≈ 360 MeV and αs ' αs(Mτ) = 0.343 then we (tensor) components. Below we consider the scalar com- estimate MG ≈ 1710 MeV. ponent. However, our purpose is to describe simultaneously dif- ZZ ferent sectors of low-energy particle physics. Accordingly, g2 with values α = 1.5023 and Λ = 416.4 MeV determined LS = dx dx J(x , x )W(x − x )J(x , x ) , s AA 3 1 2 1 2 1 2 1 2 by fitting the meson masses and weak decay constants we BB calculate the scalar glueball mass as follows J(x1, x2)  Jµµ (x1, x2) . M = 1661 MeV . (24) By introducing the relative co-ordinates (x1  x + y/2 G and x  x − y/2) we rewrite 2 Our estimate (24) is in reasonable agreement with other ZZ g2 predictions expecting the lightest glueball located in the LS = dxdy J(x, y)W(y)J(x, y) . (21) AA 3 scalar channel in the mass range ∼ 1500 ÷ 1800 MeV [34, 41,51,56]. The often referred quenched QCD calculations One can see that matrix element (21) is similar to (7) predict 1750 ± 50 ± 80 MeV for the mass of the lightest by the very construction. By omitting details of intermedi- glueball [50]. The recent quenched lattice estimate with ate calculations (similar to those represented in the previ- improved lattice spacing favors a scalar glueball mass MG = ous section) we rewrite the partition function in terms of 1710 ± 50 ± 58 MeV [54].

03014-p.7 EPJ Web of Conferences

5.5 4 4 = 6παsΛ ≈ 0.8 GeV (26) which is the same order of magnitude with the reference

5.0 value [58] D E 2 µν 4 g Tr GµνG ≈ 0.5 GeV .

4.5 / G M In conclusion, the suggested model in its simple form

4.0 is far from real QCD. However, our aim is to demonstrate that global properties of the lowest glueball state and con- ventional mesons may be explained in a simple way in the framework of a simple relativistic quantum-field model

3.5

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 of quark-gluon interaction based on analytic confinement. Our guess about the symmetry structure of the quark-gluon s interaction in the confinement region has been tested and Fig. 4. Evolution of the lowest-state glueball mass scaled to Λ the use of simple forms of propagators has resulted in quan- with the coupling αs. titatively reasonable estimates in different sectors of the low-energy particle physics. The consideration can be ex- Another important property of the scalar glueball is its tended to other problems in hadron physics. size, the ’radius’ which should depend somehow on the glueball mass. We estimate the glueball size by using the ’effective potential’ W(y) (21) connecting two scalar gluon References currents. The glueball radius may be roughly estimated as follows 1. M. Baldicchi and G.M. Prospri, arXiv:hep- ph/0310213 (2003). tvR √ d4 x x2 W(x) 2 2. V. A. Miransky, Dynamical Symmetry Breaking in rG ∼ R = Quantum Theories (Word Scientific, Singa- 4 Λ d x W(x) pore, 1993) 1. 1 3. C. S. Fischer and R. Alkofer, hep-ph/0301094. ≈ ≈ 0.67 fm . (25) 295 MeV 4. J. Gasser and H. Leutwyler, Ann. Phys. 158, (1984) 142; Nucl. Phys. B250, (1985) 465. This means that the dominant responsible for bind- 5. M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, ing gluons must be provided by medium-sized vacuum fluc- Nucl. Phys. B147, (1979) 385. tuations of correlation length ∼ 0.7 fm. Consequently, typi- 6. M. Neubert, Phys. Rep. 245, (1994) 259. cal energy-momentum transfers inside a scalar glueball oc- 7. R. Gupta, hep-lat/9807028. cur at the QCD scale ∼ 360 MeV, rather than at the chiral 8. C. D. Roberts, arXiv:0802.0217 [nucl-th]; C. D. symmetry breaking scale Λχ ∼ 1 GeV (or, ∼ 5 fm). Roberts and S. M. Schmidt, Prog. Part. Nucl. Phys. From (23) and (25) we deduce that 45, (2000) s1. " !#1/2 9. H.A. Bethe and E.E. Salpeter, Phys. Rev. 82, (1951) √ αcrit rG · MG = 2 2 ln ≈ 5.64 . 309. αs 10. C. D. Roberts and A. G. Williams, Prog. Part. Nucl. Phys. 33, (1994) 477. This value may be compared with the prediction (rG ·MG = 4.16 ± 0.15) of quenched QCD calculations [50,54]. A 11. H. Leutwyler, Phys. Lett., 96B, (1980) 154; Nucl. study of the glueball properties at finite temperature using Phys. B179, (1981) 129. SU(3) lattice QCD at the quenched level with the anisotropic 12. M. Sting, Phys. Rev. D29, (1984) 2105. lattice imposes restrictions on the glueball parameters at 13. T. C. Kraan and P. van Baal, Nucl. Phys. B533, (1998) 627. zero temperature: 0.37 fm < rG < 0.57 fm and MG ' 1.49 GeV [57]. The nonprincipal differences of quenched 14. R.Alkofer and J. Greensite, J. Phys. G34, (2007) s3. lattice QCD data from our estimates may be explained by 15. F. Lenz, J. W. Negele and M. Thies, Phys. Rev. D69, the presence of quarks (our parameters have been fixed by (2004) 074009. fitting two-quark bound states) in our model. 16. E. Elisade and J.Soto, Nucl. Phys. B260, (1985) 136. A method of analysis of correlation functions in QCD 17. G.V. Efimov and S.N. Nedelko, Phys. Rev. D51, is to calculate the corresponding condensates. The value of (1995) 176; Phys. Rev. D54, (1996) 4483; Ya.V. Bur- the correlation function dictates the values of the conden- danov and G.V.Efimov, Phys. Rev. D54, (2001) sates. We calculate the lowest non-vanishing gluon con- 014001. densate in the leading-order (ladder) approximation: 18. G.V. Efimov and G. Ganbold, Phys. Rev. D65, (2002) 054012. D E Z 2 A µν 4 4 19. G. Ganbold, AIP Conf. Proc. 717, (2004) 285; ibid g Tr FµνFA = 8NcπαsΛ d z W(z) 796, (2005) 127.

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