The Physics of Glueballs

Home , Glueball, Gluon

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

X3 −µr 9 αS 1 d e − 2 Lij · Sij U(rij ) with U(r) = 4µ rij drij r i

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 . − 5 2 5 3

Vincent Mathieu (UMons) The Physics of Glueballs Zakopane, February 2009 17 / 21 Application

Same hierarchy as the lattice QCD Application with a simple Cornell potential p 9 α H0 = 2 p2 + σr − 3 s . 4 r Parameters:

2 σ = 0.185 GeV αs = 0.45

Addition of an instanton induced interaction to split the degeneracy between the 0++ and 0−+ ∆HI = −P I δJ,0 with I = 450 MeV. Instanton attractive in the scalar channel and repulsive in the pseudoscalar and equal in magnitude Very good agreement without spin-dependent potential Extension for three-body systems ?

Vincent Mathieu (UMons) The Physics of Glueballs Zakopane, February 2009 18 / 21 Three-gluon glueballs with transverse gluons

SU(2) × SU(3) decomposition for two gluons → to the lowest J: a b (ab) [ab] ⊗ = ⊕ •(ab) ⊕ Lowest J allowed for 3 gluons with helicity: a b c (abc) (abc) ⊗ ⊗ = ⊕ ⊕ · · · ⊕ •[abc] Low-lying states are J = 1 and J = 3 with symmetric colour function and J = 0 with an antisymmetric colour function

Low-lying states are the 1±−, 3±− and the 0±+

J = 0P − are not allowed for three-gluon glueballs → 0+− four transverse gluons

The result should be conﬁrmed by a detailed analysis of the three-body helicity formalism These developments are under construction

Vincent Mathieu (UMons) The Physics of Glueballs Zakopane, February 2009 19 / 21 Conclusion Model with spin-1 gluons reproduces the pure gauge spectrum but spin-dependent potentials should be added and the spectrum is plagued with unwanted states Three-gluon glueballs with massive gluon cannot reproduce the lattice data With two transverse gluons, a simple linear+Coulomb potential reproduce the lattice QCD results. Implementation of the helicity formalism for three transverse gluons should solve the hierarchy problem

Vincent Mathieu (UMons) The Physics of Glueballs Zakopane, February 2009 20 / 21 References

Vincent Mathieu (UMons) The Physics of Glueballs Zakopane, February 2009 21 / 21