LIGHT MESON SPECTROSCOPY AND REGGE TRAJECTORIES IN THE RELATIVISTIC QUARK MODEL Rudolf Faustov Dorodnicyn Computing Centre RAS, Moscow (in collaboration with Dietmar Ebert and Vladimir Galkin) Ebert, Faustov, Galkin — Eur. Phys. J.C 60, 273-278 (2009) Ebert, Faustov, Galkin — Phys. Rev. D 79, 114029 (2009)
OUTLINE 1. Introduction 2. Relativistic quark model 3. Mass spectra of light quark-antiquark mesons 4. Regge trajectories of light mesons 5. Masses of light tetraquarks in the diquark-antidiquark picture 6. Masses of light scalar mesons
INTRODUCTION • Vast amount of experimental data on light meson with masses up to 2500 MeV is available. = ⇒ The classification of these new data requires a better theoretical understanding of light meson mass spectra. • Light exotic states (such as tetraquarks, glueballs, hybrids) predicted by quantum chromodynamics (QCD) are expected to have masses in this range. • It is argued by Glozman et al. that the states of the same spin with different isospins and opposite parities are approximately degenerate in the interval 1700-2400 MeV. An intensive debate is going on now in the literature about whether the chiral symmetry is restored for highly excited states. • Renewed interest to the Regge trajectories both in ( M 2 , J ) and ( M 2 , n r ) planes ( M is the mass, J is the spin and n r is the radial quantum number of the meson state): their linearity, parallelism and equidistance. = ⇒ Assignment of experimentally observed mesons to particular Regge trajectories. • Problem of scalar mesons: • Abundance and peculiar properties of light scalars • Experimental and theoretical evidence for the existence of f 0 (600)( σ ) , K ∗ 0 (800)( κ ) , f 0 (980) and a 0 (980) indicates that lightest scalars form a full SU (3) flavour nonet. • Inversion of the mass ordering of light scalars, which cannot be naturally understood in the q ¯ q picture. = ⇒ Various alternative interpretations: ⋆ four-quark states (tetraquarks) and in particular diquark-antidiquark bound states ⋆ proximity of f 0 /a 0 to the K ¯ K threshold led to the K ¯ K molecular picture.
Comparison of a traditional ideally mixed q ¯ q nonet of light mesons (like vector mesons) with the scalar diquark-antidiquark nonet and experimentally known light scalar mesons. Diquarks are considered in the colour antitriplet state.
RELATIVISTIC QUARK MODEL Relativistic quasipotential equation of Schr¨ odinger type: ! b 2 ( M ) − p 2 d 3 q Z Ψ M ( p ) = (2 π ) 3 V ( p , q ; M )Ψ M ( q ) 2 µ R 2 µ R p - relative momentum of quarks M - bound state mass ( M = E 1 + E 2 ) µ R - relativistic reduced mass: = M 4 − ( m 2 1 − m 2 2 ) 2 E 1 E 2 µ R = 4 M 3 E 1 + E 2 b ( M ) - on-mass-shell relative momentum in cms: b 2 ( M ) = [ M 2 − ( m 1 + m 2 ) 2 ][ M 2 − ( m 1 − m 2 ) 2 ] 4 M 2 E 1 , 2 - center of mass energies: E 1 = M 2 − m 2 E 2 = M 2 − m 2 2 + m 2 1 + m 2 1 2 , 2 M 2 M
• Parameters of the model fixed from heavy meson sector • q ¯ q quasipotential q 1 q 1 q 1 q 1 q 1 q 1 Conf V QCD = g + (string) q 2 ¯ q 2 ¯ q 2 ¯ q 2 ¯ q 2 ¯ q 2 ¯ ( ) 4 3 α S D µν ( k ) γ µ 1 γ ν 2 + V V conf ( k )Γ µ 1 Γ 2; µ + V S V ( p , q ; M ) = ¯ u 1 ( p )¯ u 2 ( − p ) conf ( k ) u 1 ( q ) u 2 ( − q ) k = p − q D µν ( k ) - (perturbative) gluon propagator Γ µ ( k ) - effective long-range vertex with Pauli term: Γ µ ( k ) = γ µ + iκ 2 mσ µν k ν , κ - anomalous chromomagnetic moment of quark,
0 1 1 s ǫ ( p ) + m u λ ( p ) = A χ λ , σ p @ 2 ǫ ( p ) ǫ ( p ) + m p 2 + m 2 . p with ǫ ( p ) = • Lorentz structure of V conf = V V conf + V S conf In nonrelativistic limit V V ff = (1 − ε )( Ar + B ) conf Sum : ( Ar + B ) V S = ε ( Ar + B ) conf ε - mixing parameter V Coul ( r ) = − 4 α s 3 r V Cornell ( r ) = − 4 α s r + Ar + B 3
Parameters A , B , κ , ε and quark masses fixed from analysis of meson masses and radiative decays: ε = − 1 from heavy quarkonium radiative decays ( J/ψ → η c + γ ) and HQET from fine splitting of heavy quarkonium 3 P J states and HQET κ = − 1 (1 + κ ) = 0 = ⇒ vanishing long-range chromomagnetic interaction (flux tube model) Freezing of α s for light quarks 4 π β 0 = 11 − 2 2 m 1 m 2 α s ( µ ) = , 3 n f , µ = , µ 2+ M 2 m 1 + m 2 0 β 0 ln Λ2 √ M 0 = 2 . 24 A = 0 . 95 GeV Quasipotential parameters: A = 0 . 18 GeV 2 , B = − 0 . 30 GeV, Λ = 0 . 413 GeV (from M ρ ) Quark masses: m b = 4 . 88 GeV m s = 0 . 50 GeV m c = 1 . 55 GeV m u,d = 0 . 33 GeV
• Light tetraquarks in diquark-antidiquark picture V qq ′ = 1 ( qq ′ ) -interaction: 2 V q ¯ q ′ V ( p , q ; M ) = ¯ u 1 ( p )¯ u 2 ( − p ) V ( p , q ; M ) u 1 ( q ) u 2 ( − q ) , where V ( p , q ; M ) = 2 2 + 1 1 Γ 2; µ + 1 3 α s D µν ( k ) γ µ 1 γ ν 2 V V conf ( k )Γ µ 2 V S conf ( k ) ( d 1 ¯ d = ( qq ′ ) d 2 )-interaction: 3 α s D µν ( k ) � d 2 ( P ′ ) | J ν | d 2 ( Q ′ ) � V ( p , q ; M ) = � d 1 ( P ) | J µ | d 1 ( Q ) � 4 2 p E d 1 E d 1 2 p E d 2 E d 2 h i + ψ ∗ d 1 ( P ) ψ ∗ d 2 ( P ′ ) J d 1; µ J µ d 2 V V conf ( k ) + V S ψ d 1 ( Q ) ψ d 2 ( Q ′ ) , conf ( k ) d 1 d 1 d 1 d 1 d 1 d 1 Conf V QCD = g + (string) ¯ ¯ ¯ ¯ ¯ ¯ d 2 d 2 d 2 d 2 d 2 d 2
J d,µ – effective long-range vector vertex of diquark: ( P + Q ) µ 8 for scalar diquark > q > > 2 E d ( p ) E d ( q ) > < J d ; µ = for axial vector ( P + Q ) µ + iµ d 2 M d Σ ν − µ k ν > > q diquark ( µ d = 0 ) > > 2 E d ( p ) E d ( q ) : µ d - total chromomagnetic moment of axial vector diquark (Σ ρσ ) ν µ = − i ( g µρ δ ν σ − g µσ δ ν diquark spin matrix: ρ ) S d - axial vector diquark spin: ( S d ; k ) il = − iε kil ψ d ( P ) – diquark wave function: 1 for scalar diquark ψ d ( p ) = ε d ( p ) for axial vector diquark ε d ( p ) – polarization vector of axial vector diquark � d ( P ) | J µ | d ( Q ) � – vertex of diquark-gluon interaction: Z d 3 p d 3 q (2 π ) 6 ¯ Ψ d P ( p )Γ µ ( p , q )Ψ d Q ( q ) ⇒ F ( k 2 ) � d ( P ) | J µ (0) | d ( Q ) � = Γ µ – two-particle vertex function of the diquark-gluon interaction
MASSES OF LIGHT QUARK-ANTIQUARK MESONS The quasipotential of q ¯ q interaction is extremely nonlocal in configuration space for arbitrary quark masses. To make it local ⋆ heavy quarks: nonrelativistic v/c or heavy quark 1 /m Q expansion ⋆ light quarks: highly relativistic, substitution M 2 − m 2 q ′ + m 2 q q q + p 2 → E q = m 2 ǫ q ( p ) ≡ 2 M q ¯ q potential V q ¯ q ( r ) = V SI ( r ) + V SD ( r ) spin-dependent potential − S 1 S 2 + 3 » – V SD ( r ) = a 1 LS 1 + a 2 LS 2 + b r 2 ( S 1 r )( S 2 r ) + c S 1 S 2 + d ( LS 1 )( LS 2 ) where e.g. " 2 „ E 1 − m 1 − (1 + κ ) E 1 + m 1 « ∆ ¯ c = V Coul ( r ) + 3 E 1 E 2 2 m 1 2 m 1 # „ E 2 − m 2 − (1 + κ ) E 2 + m 2 « ∆ V V × conf ( r ) 2 m 2 2 m 2
spin-independent potential ( V SI ( r ) = V Coul ( r ) + V conf ( r ) + ( E 2 1 − m 2 1 + E 2 2 − m 2 2 ) 2 1 V Coul ( r ) 4( E 1 + m 1 )( E 2 + m 2 ) E 1 E 2 " 1 (1 + κ )( E 1 + m 1 )( E 2 + m 2 ) + 1 + (1 + κ ) m 1 m 2 E 1 E 2 «#! ) „ E 1 + m 1 + E 1 + m 2 1 V V V S − conf ( r ) + conf ( r ) E 1 E 2 m 1 m 2 +1 „ 1 1 « E 1 ( E 1 + m 1 )∆ ˜ V (1) E 2 ( E 2 + m 2 )∆ ˜ V (2) Coul ( r ) + Coul ( r ) 4 − 1 1 1 1 1 » „ «– ∆ V V m 1 ( E 1 + m 1 ) + m 2 ( E 2 + m 2 ) − (1 + κ ) + conf ( r ) 4 E 1 m 1 E 2 m 2 ( E 2 1 − m 2 1 + E 2 2 − m 2 L 2 2 ) 1 8 m 1 m 2 ( E 1 + m 1 )( E 2 + m 2 )∆ V S V ′ ¯ + conf ( r ) + Coul ( r ) , E 1 E 2 2 r
Table 1: Masses of excited light ( q = u, d ) unflavored mesons (in MeV). Theory Experiment Theory Experiment n 2 S +1 L J J PC q ¯ q I = 1 mass I = 0 mass s ¯ s I = 0 mass 1 1 S 0 0 − + 154 π 139.57 743 1 3 S 1 1 −− 776 ρ 775.49(34) ω 782.65(12) 1038 ϕ 1019.455(20) 1 3 P 0 0 ++ 1176 a 0 1474(19) f 0 1200-1500 1420 f 0 1505(6) 1 3 P 1 1 ++ 1254 a 1 1230(40) f 1 1281.8(6) 1464 f 1 1426.4(9) 1 3 P 2 2 ++ f ′ 1317 a 2 1318.3(6) f 2 1275.1(12) 1529 1525(5) 2 1 1 P 1 1 + − 1258 b 1 1229.5(32) h 1 1170(20) 1485 h 1 1386(19) 2 1 S 0 0 − + 1292 π 1300(100) η 1294(4) 1536 η 1476(4) 2 3 S 1 1 −− 1486 ρ 1465(25) ω 1400-1450 1698 ϕ 1680(20) 1 3 D 1 1 −− 1557 ρ 1570(70) ω 1670(30) 1845 1 3 D 2 2 −− 1661 1908 1 3 D 3 3 −− ρ 3 ω 3 ϕ 3 1714 1688.8(21) 1667(4) 1950 1854(7) 1 1 D 2 2 − + 1643 π 2 1672.4(32) η 2 1617(5) 1909 η 2 1842(8) 2 3 P 0 0 ++ 1679 f 0 1724(7) 1969 2 3 P 1 1 ++ 1742 a 1 1647(22) 2016 f 1 1971(15) 2 3 P 2 2 ++ 1779 a 2 1732(16) f 2 1755(10) 2030 f 2 2010(70) 2 1 P 1 1 + − 1721 2024 3 1 S 0 0 − + 1788 π 1816(14) η 1756(9) 2085 η 2103(50) 3 3 S 1 1 −− 1921 ρ 1909(31) ω 1960(25) 2119 ϕ 2175(15) 1 3 F 2 2 ++ 1797 f 2 1815(12) 2143 f 2 2156(11) 1 3 F 3 3 ++ 1910 a 3 1874(105) 2215 f 3 2334(25) 1 3 F 4 4 ++ 2018 a 4 2001(10) f 4 2018(11) 2286
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