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The Physics of Glueball The Physics of Glueball Vincent Mathieu Universidad de Valencia February 2011 V. M., N. Kochelev, V. Vento, \The Physics of Glueballs", Int. J. Mod. Phys. E18 (2009) 1 V. M., V. Vento, \Pseudoscalar Glueball and η − η0 Mixing", Phys. Rev. D81, 034004 (2010) C. Degrande, J.-M. G´erardand V. M., in preparation, Vincent Mathieu (Univ. Valencia) Glueballs Spectroscopy February 2011 1 / 38 Introduction QCD = gauge theory with the color group SU(3) 1 νµ X µ LQCD = − Tr Gµν G + q¯(γ Dµ − m)q 4 Gµν = @µAµ − @ν Aµ − ig[Aµ;Aν ] Quark = fundamental representation 3 Gluon = Adjoint representation 8 Observable particles = color singlet 1 Mesons: 3 ⊗ 3¯ = 1 ⊕ 8 Baryons: 3 ⊗ 3 ⊗ 3 = 1 ⊕ 8 ⊕ 8 ⊕ 10 Glueballs: 8 ⊗ 8 = (1 ⊕ 8 ⊕ 27) ⊕ (8 ⊕ 10 ⊕ 10) 8 ⊗ · · · ⊗ 8 = 1 ⊕ 8 ⊕ ::: Three light quarks ! nine0 ± mesons : 3π (I = 1) ⊕ 4K (I = 1=2) ⊕ 2η (I = 0) Glueball can mix with two isoscalars ! glue content in η's wave function and maybe a third isoscalar Vincent Mathieu (Univ. Valencia) Glueballs Spectroscopy February 2011 2 / 38 Lattice QCD Quenched Results Investigation of the glueball spectrum (pure gluonic operators) on a lattice by Morningstar and Peardon, Phys. Rev. D60 (1999) 034509] Identification of 15 glueballs below 4 GeV M(0++) = 1:730 ± 0:130 GeV M(0−+) = 2:590 ± 0:170 GeV M(2++) = 2:400 ± 0:145 GeV Quenched approximation (gluodynamics) ! mixing with quarks is neglected Unquenched Results Lattice studies with nf = 2 exist. The lightest scalar would be sensitive to the inclusion of sea quarks but no definitive conclusion. Theoretical status of glueballs : V. M., N. Kochelev, V. Vento, \The Physics of Glueballs", Int. J. Mod. Phys. E18 (2009) 1 Vincent Mathieu (Univ. Valencia) Glueballs Spectroscopy February 2011 3 / 38 Phenomenology - Strong Coupling Constant Deur, arXiv:0901.2190 2 Saturation of αs(Q ) at large distances Gribov, Eur. Phys. J., C10,71 (1999) Dokshitzer and Webber, Phys. Lett., B352, 451 ('95) Dasgupta and Salam, J. Phys. G 30, R143 (2004) Power correction in event shapes (µI = 2 GeV) Z µI −1 2 2 α0 = µI α(Q )dQ ∼ 0:5 0 Vincent Mathieu (Univ. Valencia) Glueballs Spectroscopy February 2011 4 / 38 Landau Gauge Propagators 7 Gluon Propagator SU(3) Gluon Propagator SU(3) 10 L=64 and =6.0 L = 64 and =5.7 6 L=64 and =6.2 L = 72 and =5.7 L=80 and =6.0 L = 80 and =5.7 Fit 8 Fit Gluon propagator from 5 ] ] 4 -2 6 lattice QCD in Landau -2 3 )[GeV )[GeV 2 gauge 2 (q 4 (q 2 2 1 0 0 1E-3 0,01 0,1 1 10 100 Cucchieri and Mendes, 0,01 0,1 1 10 100 2 2 2 2 q [GeV ] q [GeV ] PoS (Lattice 2007) 297 Gluon Propagator SU(2) 4 L = 128 and = 2.2 Bogolubsky et al., Fit PoS (Lattice 2007) 290 3 ] - 2 2 )[GeV 2 Oliveira and Silva, (q PoS (QCD-TNT 2009) 033 1 0 1E-3 0,01 0,1 1 10 100 2 2 q [GeV ] Predicted by Cornwall Phys. Rev. D26, 1453 (1982) by introducing a dynamically generated gluon mass Vincent Mathieu (Univ. Valencia) Glueballs Spectroscopy February 2011 5 / 38 Dynamical Mass Generation Not a violation of gauge invariance Schwinger Mechanism like in (1 + 1)−QED Phys. Rev. 128 (1962) 2425 Self-energy gains a dynamical pole 2 qµqν −i ∆(q )µν = gµν − q2 q2 − q2Π(q2) Mass is the residue m2(q2) Π(q2)j = pole q2 Composite bound state not present in physical processes R. Jackiw and K. Johnson, Phys. Rev. D 8, 2386 (1973) D. Binosi and J. Papavassiliou, Phys. Rept. 479, 1 (2009) A. C. Aguilar, D. Iba~nez,V. M. and J. Papavassiliou, \The Schwinger Mechanism in QCD", in preparation Vincent Mathieu (Univ. Valencia) Glueballs Spectroscopy February 2011 6 / 38 Gluon Mass Cornwall Phys. Rev. D26, 1453 (1982) Gluons massless in the Lagrangian Couplings and nonperturbative effects !Dynamical mass extracted from gauge-invariant propagator g¯2(q2) d^(q2) = q2 + m2(q2) 2 2 2 3−12=11 ln q +4m0 Λ2 2 2 2 6 7 m (q ) = m0 6 7 4 4m2 5 ln 0 Λ2 How degrees of freedom has the gluon ? On-shell decoupling of the pole ! 2 d.o.f vs Gluon mass generation ! 3 d.o.f Gluon massless in the Lagrangien VincentAnswer Mathieu from glueball (Univ. Valencia) spectroscopy Glueballs Spectroscopy February 2011 7 / 38 Two Models p 2 9 αs Hgg = 2 p + 4 σr − 3 r + VOGE Gluons spin-1 usual rules of spin couplings Gluons transverse Gluons helicity-1 particles J = L + S with S = 0; 1; 2 J 6= L + S OGE no needed VOGE = 0 No vector states 3 d.o.f. not compatible with J+− Instanton contribution V.M. , PoS [QCD-TNT09], 024 (2009) Gluon has only 2 d.o.f in bound state w. f. Vincent Mathieu (Univ. Valencia) Glueballs Spectroscopy February 2011 8 / 38 Bag Model R. L. Jaffe and K. Johnson, Phys. Lett. B 60 (1976) 201 J. Kuti, Nucl. Phys. Proc. Suppl. 73 (1999) 72 Free Particles Confined in a Cavity Gluonic Modes in a Cavity P + TE mode J =1 xTE = 2:74; P − TE mode J =2 xTE = 3:96; P − TM mode J =1 xTM = 4:49: Mass Spectrum 3 4πBR X xi α a a E = + ni − λ λ S~1 · S~2 3 R 4R 1 2 i 2 2 2 X xi M = E − ni R i α = 0:5 B = (280 MeV)4 J. F. Donoghue, Phys. Rev. D 29 (1984) 2559 Gluon mass 740 ± 100 MeV in the bag model Vincent Mathieu (Univ. Valencia) Glueballs Spectroscopy February 2011 9 / 38 AdS/QCD AdS/CFT correspondance: Correspondance between conformal theories and string theories in AdS spacetime QCD not conformal ! breaking conformal invariance somehow Introduction of a black hole in AdS to break conformal invariance Parameter adjusted on 2++ Same hierarchy but some states are missing (spin 3,...) R. C. Brower et al., Nucl. Phys. B587 (2000) 249 Vincent Mathieu (Univ. Valencia) Glueballs Spectroscopy February 2011 10 / 38 QCD Spectral Sum Rules µν µν Gluonic currents: JS (x) = αS Tr Gµν G JP (x) = αS Tr Gµν Ge Z 1 Z 1 ImΠ(s) Π(Q2) = i d4x eiq·xh0jTJ (x)J (0)j0i = ds G G 2 π 0 s + Q Theoretical side (OPE): a µν a b b JG(x)JG(0) = C(a)+(b)+(e)1 + C(c)Gµν Ga + C(d)fabcGαβ Gβγ Gγα + ··· a µν Confinement parameterized with condensates h0jαsGµν Ga j0i;::: Phenomenological side: X ImΠ(s) = πf 2 m4 δ(s − m2 ) + πθ(s − s )ImΠ(s)Cont Gi Gi Gi 0 i Vincent Mathieu (Univ. Valencia) Glueballs Spectroscopy February 2011 11 / 38 Correlators Glueball masses extracted from correlators ++ β(g) µν 0 JS (x) = Tr Gµν G g −+ β(g) µν 0 JP (x) = Tr Gµν Ge g ++ α 1 2 JT,µν (x) = Tr GµαG − gµν JS ν 4 Lattice: Euclidean Spacetime Z 3 −MGt d xh0jTJG(x; t)JG(0)j0i / e Sum Rules: Spectral Decomposition Z 1 Z 1 ρ(s) i d4x eiq·xh0jTJ (x)J (0)j0i = ds G G 2 π 0 s + Q 2 2 ρ(s) = (2π)hGjJGj0i δ(s − MG) + Cθ(s − s0) Vincent Mathieu (Univ. Valencia) Glueballs Spectroscopy February 2011 12 / 38 Low Energy Theorems µν µν Gluonic currents: JS (x) = αS Tr Gµν G JP (x) = αS Tr Gµν Ge Z 4 iq·x 8πdO ΠO(0) = lim i d x e h0jTJS (x)O(0)j0i = h0jOj0i q2!0 b0 2 32π a µν ΠS (Q = 0) = h0jαsGµν Ga j0i b0 2 2 mumd ΠP (Q = 0) = (8π) h0jqq¯ j0i mu + md Instantons contribution essential for LETs Forkel, Phys. Rev. D71 (2005) 054008 M(0++) = 1:25 ± 0:20 GeV M(0−+) = 2:20 ± 0:20 GeV Vincent Mathieu (Univ. Valencia) Glueballs Spectroscopy February 2011 13 / 38 Pade´ Approximation and GZ propagators Dudal et al, arXiv:1010.3638 Construction of spin-J glueball propagators Z Σ0 σJ (s) X F (k2) = ds = ν (−1)n(k2)n J 2 n 0 1 + sk n from GZ gluon propagator k2 + M 2 D(k) = k4 + (m2 + M 2)k2 + λ4 Lowest order Pad´eapproximation p after a subtraction analysis: mJ = ν0/ν1 GZ Pad´e m0++ = 1:96 m0−+ = 2:19 m2++ = 2:04 Models m0++ = 1:98 m0−+ = 2:22 m2++ = 2:42 Lattice m0++ = 1:73 m0−+ = 2:59 m2++ = 2:40 Coherent with Instanton contribution ∼ 300 MeV in 0±+ Problem induced by scale anomaly for the 2++ Vincent Mathieu (Univ. Valencia) Glueballs Spectroscopy February 2011 14 / 38 Glueballs in the Real World Crede and Meyer, The Experimental Status of Glueballs, Prog. Part. Nucl. Phys. 63, 74 (2009) Production in gluon rich processes (OZI forbidden,...) Closely linked to the Pomeron: J = 0:25M 2 + 1:08 Mixing between glueball 0++ and light mesons Scalar Candidates: f0(1370) f0(1500) f0(1710) f0(1810) ::: Pseudoscalar Candidates: η(1295) η(1405) η(1475) η(1760) ::: 0++ and 0−+ glueballs shared between those states Three light quarks ! 3 × 3 = 9 (pseudo)scalar mesons A 10th light mesons would be the realization of the glueball Vincent Mathieu (Univ. Valencia) Glueballs Spectroscopy February 2011 15 / 38 Scalar Glueball Chiral Suppression Chanowitz, Phys. Rev. Lett. 95 (1999) 172001 ++ A 0 −! qq¯ / mq Decay to KK¯ favoured over ππ Controversy about mq (current or constituent mass ?) A 0++ −! KK¯ R = > 1 A (0++ −! ππ) f0(1710) good glueball candidate Results from Crystal Barrel (pp¯), OBELIX (pp¯), WA102 (pp), BES (J= ) Name Masse (MeV) Width (MeV) Decays Production f0(1370) 1200 − 1500 200 − 500 ππ;KK;¯ ηη pp¯ ! PPP , pp ! pp(PP ) weak signal in J= ! γ(PP ) f0(1500) 1505 ± 6 109 ± 7 ππ;KK;¯ ηη J= ! γ(PP ), pp ! pp(PP ) pp¯ ! PPP f0(1710) 1720 ± 6 135 ± 8 ππ;K K¯; ηη J= ! γ(PP ), pp ! pp(PP ) not seen in pp¯ Belle and BaBar puzzle : f0(1500) strong coupling to KK¯ and weak to ππ Vincent Mathieu (Univ.
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