The Physics of Glueballs Vincent Mathieu Universit´ede Mons, Belgique Zakopane, February 2009 Vincent Mathieu (UMons) The Physics of Glueballs Zakopane, February 2009 1 / 21 Outline 1 Introduction 2 Lattice QCD 3 AdS/QCD 4 QCD Spectral Sum Rules 5 Gluon mass 6 Two-gluon glueballs C = + 7 Three-gluon glueballs C = − 8 Helicity formalism for transverse gluons 9 Conclusion 10 References Vincent Mathieu (UMons) The Physics of Glueballs Zakopane, February 2009 2 / 21 References V. Mathieu, N. Kochelev, V. Vento “The Physics of Glueballs” invited review for Int. J. Mod. Phys. E arXiv:0810.4453 [hep-ph] V. Mathieu, F. Buisseret and C. Semay “Gluons in Glueballs: Spin or Helicity ? ” Phys. Rev. D 77, 114022 (2008), [arXiv:0802.0088 [hep-ph]] V. Mathieu, C. Semay and B. Silvestre-Brac “Semirelativistic Potential Model for Three-gluon Glueballs.” Phys. Rev. D77 094009 (2008), [arXiv:0803.0815 [hep-ph]] N. Boulanger, F. Buisseret,V. Mathieu, C. Semay “Constituent Gluon Interpretation of Glueballs and Gluelumps.” Eur. Phys. J. A38 317 (2008) [arXiv:0806.3174 [hep-ph]] Vincent Mathieu (UMons) The Physics of Glueballs Zakopane, February 2009 3 / 21 Introduction - QCD QCD = gauge theory with the color group SU(3) 1 X L = − Tr G Gνµ + q¯(γµD − m)q QCD 4 µν µ 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 ⊕ ... Colored gluons → color singlet with only gluons Vincent Mathieu (UMons) The Physics of Glueballs Zakopane, February 2009 4 / 21 Introduction - Glueballs Prediction of the QCD 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 Candidates: f0(1370) f0(1500) f0(1710) One scalar glueball between those states [Klempt, Phys. Rep. 454]. Vincent Mathieu (UMons) The Physics of Glueballs Zakopane, February 2009 5 / 21 Physical States Pure states: |ggi, |nn¯i, |ss¯i hnn¯|ggi hss¯|ggi |Gi = |ggi + |nn¯i + |ss¯i Mgg − Mnn¯ Mgg − Mss¯ Analysis of Production: J/ψ → γf0, ωf0, φf0 Decay: f0 → ππ, KK,¯ ηη Two mixing schemes Cheng et al [PRD74, 094005 (2006)] Close and Kirk [PLB483, 345 (2000)] Mnn¯ < Mss¯ < Mgg Mnn¯ < Mgg < Mss¯ Vincent Mathieu (UMons) The Physics of Glueballs Zakopane, February 2009 6 / 21 Lattice QCD Investigation of the glueball spectrum (pure gluonic operators) on a lattice by Morningstar and Peardon [Phys. Rev. D60, 034509 (1999)] 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 Lattice studies with nf = 2 exist. The lightest scalar would be sensitive to the inclusion of sea quarks but no definitive conclusion. Vincent Mathieu (UMons) The Physics of Glueballs Zakopane, February 2009 7 / 21 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, 249 (2000)] Vincent Mathieu (UMons) The Physics of Glueballs Zakopane, February 2009 8 / 21 QCD Spectral Sum Rules a a a ea Gluonic currents: JS (x) = αsGµν (x)Gµν (x) JP (x) = αsGµν (x)Gµν (x) Z Z ∞ 2 4 iq·x 1 ImΠ(s) Π(Q ) = i d x e h0|TJG(x)JG(0)|0i = 2 ds 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 h0|αsGµν Ga |0i,... Phenomenological side: X 2 4 2 Cont ImΠ(s) = πfGi mGi δ(s − mGi ) + πθ(s − s0)ImΠ(s) i Vincent Mathieu (UMons) The Physics of Glueballs Zakopane, February 2009 9 / 21 Sum Rules → Constituent models Sum rules calculation by Forkel [Phys. Rev. D71, 054008 (2005)] M(0++) = 1.25 ± 0.20 GeV M(0−+) = 2.20 ± 0.20 GeV (1) Two gluons in interaction Two gluons → C = + Three gluons → C = − Works on constituent models by Bicudo, Llanes-Estrada, Szczepaniak, Simonov,... Vincent Mathieu (UMons) The Physics of Glueballs Zakopane, February 2009 10 / 21 Gluon Mass Gluons massless in the Lagrangian Non-perturbative effects → dynamical mass Cornwall [PRD26 1453 (1982)] 2 2 2 3γ q +ρm0 ln 2 2 2 2 4 Λ 5 m (q ) = m0 2 ρm0 ln Λ2 Gluons ∼ heavy quarks Gluonium models for the low-lying glueballs with Spinless Salpeter Hamiltonian p 2 2 Hgg = 2 p + m + V (r) p bare mass m = 0 and effective mass µ = hΨ| p2|Ψi state dependent (Simonov 1994) Gluons spin-1 particles with the usual rules of spin coupling Vincent Mathieu (UMons) The Physics of Glueballs Zakopane, February 2009 11 / 21 Two-gluon glueballs C = + Brau and Semay [Phys. Rev. D70, 014017 (2004)] Two-gluon glueballs → C = + and P = (−1)L J = L + S with S = 0, 1, 2. p 9 α H0 = 2 p2 + σr − 3 s . 4 r Cornell potential does not lift the degeneracy between states with different S Corrections of order O(1/µ2), with p µ = hΨ| p2|Ψi Structures coming from the OGE " 1 1 π 5 V =λ + S2 U(r) − δ(r) S2 − 4 oge 4 3 µ2 2 # 3 U 0(r) 1 U 0(r) L · S − − U 00(r) T 2µ2 r 6µ2 r with λ = −3αS and U(r) = exp(−µr)/r. Vincent Mathieu (UMons) The Physics of Glueballs Zakopane, February 2009 12 / 21 Two-gluon glueballs C = + J PC (L, S) Smearing of the attractive δ3(r) by a gluon size 0++ (0,0) (2,2) (0,0)∗ 0−+ (1,1) (1,1)∗ ρ(r, γ) = exp(−r/γ)/r 2++ (0,2) (2,0) (2,2) 2−+ (1,1) (1,1)∗ Replacement of potentials by convolutions with 3++ (2,2) the size function Z 4++ (2,2) ++ V (r) → Ve(r, γ) = V (r + r∗)ρ(r∗, γ)dr∗ 1 (2,2) 1−+ (1,1) (1,1)∗ Parameters 2 −1 σ = 0.21 GeV αs = 0.50 γ = 0.5 GeV Good agreement with lattice QCD Gluons = spin 1 → spurious J = 1 states and indetermination of states For instance, 0++ can be (L, S) = (0, 0) or (2, 2) Vincent Mathieu (UMons) The Physics of Glueballs Zakopane, February 2009 13 / 21 Three-gluon glueballs C = − The Model Extension of the model to three-gluon systems 3 q 3 X 9 X H = p2 + σ |r − R | + V i 4 i cm OGE i=1 i=1 Same parameters (σ, αS , γ) as for two-gluon glueballs 3 3 X 1 1 π 5 V = − α + S2 U(r ) − δ(r ) β + S2 OGE 2 S 4 3 ij ij µ2 ij 6 ij i<j=1 X3 −µr 9 αS 1 d e − 2 Lij · Sij U(rij ) with U(r) = 4µ rij drij r i<j=1 We find the eigenvalue of this operator thanks to a Gaussian basis Vincent Mathieu (UMons) The Physics of Glueballs Zakopane, February 2009 14 / 21 Three-gluon glueballs C = − a b c 1 The results dabcAµAν Aρ [(88)8s 8] C = − Gluons with spin → J = L + S a b c 1 fabcAµAν Aρ [(88)8a 8] C = + Good results for 1−− and 3−− but higher 2−− ← symmetry SS Symmetry J PC Disagreement with lattice QCD for int 0 0 1 A 0−+ PC = +− 1 0, 1, 2 1 S, 2 MS 1−− This model cannot explain the splitting 2 1, 2 2 MS 2−− ∼ 2 GeV between 1+− and 0+− 3 2 1 S 3−− 0+−, 1+−, 2+−, 3+− are L = 1 and degenerate in a model with spin J PC (L, S) J PC (L, S) 1−− (0,1) 0+− (1,1) 2−− (0,2) 1+− (1,1) 3−− (0,3) 2+− (1,1) 0−+ (1,1) 3+− (1,2) Solution: Implementation of the helicity formalism for three transverse gluons. Vincent Mathieu (UMons) The Physics of Glueballs Zakopane, February 2009 15 / 21 Helicity Formalism for two-gluon glueballs Yang’s theorem: ρ 9 γγ γγ ∼ gg → J 6= 1 J = L + S cannot hold for relativistic system. J is the only relevant quantum number. Solution: Helicity formalism for transverse gluons with helicity si = 1. Only two projection, λi = ±1. Formalism to handle with massless particles [Jacob and Wick (1959)] PC State J in term of usual (L, S) states (with Λ = λ1 − λ2): E X 2L + 1 1/2 |J, M; λ , λ i = hL0SΛ| JΛ i hs λ s − λ | SΛ i 2S+1L . 1 2 2J + 1 1 1 2 2 J L,S Not eigenstates of the parity and of the permutation operator. J P |J, M; λ1, λ2i = η1η2(−1) |J, M; −λ1, −λ2i , J−2si P12 |J, M; λ1, λ2i = (−1) |J, M; λ2, λ1i , Vincent Mathieu (UMons) The Physics of Glueballs Zakopane, February 2009 16 / 21 Helicity Formalism for two-gluon glueballs Construction of states symmetric (bosons) and with a good parity → selection rules. S ; (2k)+ ⇒ 0++, 2++, 4++,... + S ; (2k)− ⇒ 0−+, 2−+, 4−+,... − D ; (2k + 2)+ ⇒ 2++, 4++,... + + ++ ++ D−; (2k + 3) ⇒ 3 , 5 ,... States expressed in term of the usual basis (useful to compute matrix elements!). Low-lying states: r r + 2 1 1 5 S+; 0 = S0 + D0 , 3 3 S ; 0− = 3P , − r 0 r r + 2 5 4 5 1 5 D+; 2 = S2 + D2 + G2 , r 5 r7 7 + 5 5 2 5 D−; 3 = D3 + G3 , r 7 r 7 2 3 S ; 2− = 3P + 3F .