Spectra of Light and Heavy Mesons, Glueball and QCD Effective Coupling Gurjav GANBOLD Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna

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1 Spectra of Light and Heavy Mesons, Glueball and QCD Effective Coupling Gurjav GANBOLD Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna XIV International Conference on Hadron Spectroscopy June 011, München

2 Outline: Take into account the correct symmetry structure of the quark-gluon interaction in the confinement region. Find simple forms of the quark and gluon propagators in the hadroniz. region. Build a relativistic quantum field model of the interacting quarks and gluons with the Analytic Confinement and obtain reasonable description of different processes in hadron physics: spectrum of conventional mesons in a wide range of mass scale; the lowest-state glueball mass, radius, etc.; weak decay constants of light mesons; qualitative description of the QCD effective coupling in the low-energy region by exploiting hadron spectrum; by introducing a minimal set of model parameters. s, mud, ms, mc, mb,

3 QCD at long distances Many interesting and novel behaviors are expected at low energies (IR, infrared region) below 1 GeV. Confinement and dynamical symmetry breaking are crucial features of QCD. Color confinement is the result of strong interaction and in the hadron scale (~ 1fm ~ 00MeV) QCD becomes non-perturbative. Green s functions in QCD are tightly connected to confinement and are ingredients for hadron phenomenology. However, any widely accepted and rigorous analytic solutions to these propagators are still missing. The matrix elements of hadron processes at large distance are integrated characteristics of the vertices and propagators, and the solution should not be too sensitive on the details of propagators. Taking into account the correct global symmetry properties and their breaking (and by introducing additional physical parameters) may be more important than the working out in detail of propagators.

4 Confinement There is no analytic proof that QCD should be color confining. The reason for confinement may be somewhat complicated. E.g.: Analytic Confinement S 1 ( z ) 0 for S 1 0 ( x ) 0 z C 0 ( x) 0 S ( p ) entire analytic function in complex plane The QCD vacuum is realized by nontrivial gluon background field with constant strength (the energy of the quark-gluon system is minimal on this background). The GBF leads to the AC of quarks. (H.Leutwyler [1981], G.V.Efimov et al. [1996]) B ( x) t a n a b x ; n a n a 1; b b / b ; B ( x) B ( x) b const ; b b S ( p ) B S QCD ( p ) B D ( p ) B D QCD ( p ) B entire analytic functions

5 Model Consider a relativistic quantum-field model of quark-gluon interaction. 1 L F A g f 4 ABC B q C A A f a f ab m f g C A q bf F B A B A B C C i t C analytic confinement (AC) takes place. the coupling remains weak (~1) in the hadronization region. Entire Analytic Propagators S ab ( pˆ ) ab p mf 1 ipˆ m f 1 5 (m f / ) exp mf D ( p ) BC [G.Ganbold PRD79 (009)] BC p 1 exp p 4 q f (0) q f (0) 6 3 ( z ) 1 / (1 z / 4) parity-symmetry violation mf exp m 0

6 Quark-Antiquark Bound States Leading-order contributions to quark-antiquark and two-gluon BS Z ( qq ) q q exp q S 1q Lqq g Lqq dx dx f1 f 1 BC J B f1 f ( x1, x ) D ( x1, x ) J Cf1 f ( x, x1 ), J B f1 f ( x1, x ) q f1 ( x1 ) t B q f ( x ). Orthonormalized system U_Q, where Q={n,l, } are quantum numbers ( x y ) U Q ( x) U Q ( y ); QQ ' dyu Q ( y ) U Q ' ( y ) J Jf1 f ( x, y ) D ( y ) q f1 ( x 1 y ) J q f ( x y ) Q J QJf1 f ( x ) U Q ( y ) J Q J f1 f ( x) q f1 ( x) VJ Q ( ) q f ( x) l VJ Q ( ) i dy D ( y) J U Q ( y ) exp ( y / ) A new path integration over auxiliary fields B_N: where N={Q,J,f_1,f_} e Lqq B B N exp ( B N B N ) g [( B N J N ) ( J N B N )] N N N

7 Diagonalization of the quadratic part is equivalent to the solution of the ladder Bethe-Salpeter equation on the orthonormalized system {U_N} Symmetric Bethe-Salpeter kernel: g C J S N ( p ) 9 d 4k V ( k ) ( )4 N Tr J S kˆ 1 pˆ J S kˆ pˆ Renormalization: U REN ( x) S N ( M N ) U N ( x) U N 1 S N ( p ) U N U N 1 S N ( M N ) S N ( M N )( p M N ) U N U REN ( p M N ) U REN Meson mass equation: p M N 1 S N ( M N ) 0

8 Conventional Meson Spectrum Fixing model parameters: s 1.503, MeV, [G.Ganbold PRD79 (009)] mu,d 06.9 MeV, ms 33.6 MeV, mc 1454 MeV, mb 4699 MeV. P-mesons PDG Our estim. V-mesons PDG Our estim. π ρ K ω D K* Ds Φ ηc D* B Ds* Bs J/ψ Bc B* ηb Y uu dd / ss V 740 relat. errors < %

9 Decay Constants of Light Mesons these are important value in particle physics: i f P p 0 J (0) U renorm ( p ) Ds l g d 4k ikx kˆ pˆ i S kˆ pˆ i f p p dxe U ( x ) D ( x ) Tr i S R ( )4 s 1.503, MeV, f 18.8 MeV mu, d 06.9 MeV, m s 33.6 MeV. f K MeV [G.Ganbold PRD79 (009)] PDG: f exp MeV f Kexp MeV

10 Glueball Lowest State Theoretical status: The existence of glueballs is predicted by QCD because of the self-interaction of gluons. The lightest glueball is a scalar. J PC ±50±58 MeV 1750±50±80 MeV MeV 1475 MeV Experimental status: Signatures for glueballs: -- no place in (q-qbar) nonets, -- enhanced production in gluon-rich (short distance) channels of rad.decays, -- decay branching fractions incompatible with (q-qbar) states LEP and LHC: see talk by W.Ochs PDG: Particle Data Group C.J.Morningstar, M.Peardon (000). C.Amsler, N.A.Tornqvist (004); S.Narison(000);D.V.Bugg (004); H.B.Meyer, M.J.Teper (005)

11 Two-Gluon Bound States Z ( AA) exp LAA g BB CC BB CC LAA dx1 dx J ' ( x1, x ) J ' ( x1, x ) J ' ( x1, x ) J ' ( x1, x ) 1 'W ' ( x1, x ) 'W ' ( x1, x ) 'W ' ( x1, x ) 'W ' ( x1, x ), BC B C J ( x1, x ) A ( x1 ) A ( x ), W ( x1, x ) D ( x1 x ) x1 x The glueball mass is derived from: 8g izp 1 dz e ( z ), p M G 3 t s t s ( z ) dt dsu (t ) W (t ) D z D z W ( s )U ( s ) W (t ) exp( t ) / ( )

12 Lowest-State (Scalar) Glueball MG crit crit ln s [G.Ganbold PRD79 (009)] s M ev d x x D( x) d x D( x) 4 rg 4 M G 1661 MeV fm 95MeV rg M G ln crit 5.64 s ~ 4.6 in quenched lattice, Y.Chen (006)

13 QCD Effective Coupling Due to the polarization of QCD vacuum, the color charge g is a) screened by the virtual quark-antiquark pairs b) antiscreened by the polarization of virtual gluons. a variation of the physical coupling under changes of distance 1/Q, or energy scale Q. Many quantities in hadron physics are affected by the IR behavior of alpha. However, the latter is not well defined yet. Determination of QCD coupling remains at forefront of experimental studies. Process Q (GeV) decays 1.78 QQ states 4.1 decays QQ states decays e e jets s Reference S. Bethke (009) S. Bethke (009) S. Bethke (009) P. A. M. Fernandez (00) S. Davies (003) A. Penin (1998)

14 Measurements of s as a function of energy scale Q versa QCD predictions. A self-consistent and physically meaningful prediction of the QCD effective charge in the IR regime remains one of the actual problems in particle physics.

15 Long-distance behavior Recent theoretical results predict an IR behavior of the gluon propagator. We consider a gluon propagator: BC D ( p ) BC 1 exp p / p BC 1/ ds e s p 0 The quark propagator remains the same. Meson mass is defined from equation: m m M 1 s J, 1,, J, J P,V, S, A, T Fixing model parameters: s (9460) , s (3097) 0.619, s (11) , s (010) MeV mud 19.56, ms 93.56, mc , mb

16 Solving Inverse Problem: Derive effective coupling in region below 1 GeV: s (138) P 1 (,138, mud, mud ) , [G.Ganbold PRD81 (010)] s (89) V 1 (,89, mud, ms ) s (495) P 1 (, 495, mud, ms ) , s (770) V 1 (, 770, mud, mud ) ,

17 Meson Masses Estimated with Running Coupling relative errors < 3% our estimate -- PDG [G.Ganbold PRD81 (010)]

18 IR-finite Behavior of Effective Coupling The possibility that the QCD coupling constant features an IR-finite behavior has been extensively studied in recent years. Consider the origin point: M=0, m1= m=m_{ud} Particularly, for m=19.56 MeV and =345 MeV s0 s (0) m / m / s , or s0 / 0.41 s0 / [ S.Godfrey 1985], s0 / 0.65 [T.Zhang 1991], s0 / 0.6 [ F.Halzen 1993], s0 / 1GeV 0. [ M.Baldicchi 008]

19 Conclusion and Outlook Our guess about the symmetry structure of the quark-gluon interaction in the confinement region has been tested and the use of simple forms of propagators has resulted in quantitatively reasonable estimates in different sectors of the low-energy particle physics. We provide a new, independent and analytic estimate of the lowest (scalar) glueball mass. Despite its pure model origin, the approximations used, our model gives a new glance at the long-distance behavior of QCD coupling. Particularly, we found a specific IR behaviour of QCD coupling below 1 GeV. The consideration can be extended to actual problems in hadron physics: - Other mesons (scalar, axial : c-cbar, b-bbar, X, Yb, Z ) - Exotic, mixed, many-body states (qqg, GG+qq, qqqq, ) - Other glueball states (pseudoscalar, tensor), pomeron exchange - Hadronic decay processes - Further analyses for the behaviour of QCD coupling below 1 GeV.