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Home , Boson, Glueball, Gluon, Lattice QCD

The of

Vincent Mathieu

Universidad de Valencia

February 2011

V. M., N. Kochelev, V. Vento, “The Physics of ”, 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 = with the color group SU(3)

1 νµ µ LQCD = − Tr Gµν G + q¯(γ Dµ − m)q 4 Gµν = ∂µAµ − ∂ν Aµ − ig[Aµ,Aν ]

Quark = fundamental representation 3 = Adjoint representation 8 Observable = color singlet 1

Mesons: 3 ⊗ 3¯ = 1 ⊕ 8 : 3 ⊗ 3 ⊗ 3 = 1 ⊕ 8 ⊕ 8 ⊕ 10 Glueballs: 8 ⊗ 8 = (1 ⊕ 8 ⊕ 27) ⊕ (8 ⊕ 10 ⊕ 10) 8 ⊗ · · · ⊗ 8 = 1 ⊕ 8 ⊕ ...

Three light → nine0 ± : 3π (I = 1) ⊕ 4K (I = 1/2) ⊕ 2η (I = 0) Glueball can mix with two isoscalars → glue content in η’s 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

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

7

Gluon 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

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)| = pole q2

Composite 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) dˆ(q2) = q2 + m2(q2)

  2 2  −12/11 ln q +4m0 Λ2 2 2 2   m (q ) = m0      4m2  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 -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 theories in AdS 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 ∞ ImΠ(s) Π(Q2) = i d4x eiq·xh0|TJ (x)J (0)|0i = 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 h0|αsGµν Ga |0i,... 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 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 xh0|TJG(x, t)JG(0)|0i ∝ e

Sum Rules: Spectral Decomposition

Z 1 Z ∞ ρ(s) i d4x eiq·xh0|TJ (x)J (0)|0i = ds G G 2 π 0 s + Q

2 2 ρ(s) = (2π)hG|JG|0i 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 h0|TJS (x)O(0)|0i = h0|O|0i q2→0 b0

2 32π a µν ΠS (Q = 0) = h0|αsGµν Ga |0i b0 2 2 mumd ΠP (Q = 0) = (8π) h0|qq¯ |0i 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 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 : 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/ψ)

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/ψ → γ(PP ) f0(1500) 1505 ± 6 109 ± 7 ππ,KK,¯ ηη J/ψ → γ(PP ), pp → pp(PP ) pp¯ → PPP f0(1710) 1720 ± 6 135 ± 8 ππ,K K¯, ηη J/ψ → 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. Valencia) Glueballs Spectroscopy February 2011 16 / 38 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, Phys. Rev. D74 (2006) 094005 Close and Kirk, PLB483 (2000) 345

Mnn¯ < Mss¯ < Mgg Mnn¯ < Mgg < Mss¯

Vincent Mathieu (Univ. Valencia) Glueballs Spectroscopy February 2011 17 / 38 Pseudoscalar

Mark III : observation of two pseudoscalar in J/ψ → γP P P γγ fusion : η(1475) seen but not η(1405)

Results from Crystal Barrel (pp¯), OBELIX (pp¯)

Name Masse (MeV) Width (MeV) Decays Production η(1295) 1294 ± 4 55 ± 5 γγ,KKπ,¯ a0π not seen in pp¯ η(1405) 1409.8 ± 2.5 51.1 ± 3.4 KKπ,¯ a0π, ηππ not seen in γγ ∗ η(1475) 1476 ± 4 87 ± 9 γγ,KK¯ ,KKπ,¯ a0π

η(1205) and η(1475) radial excitations of η and η0. η(1405) glueball candidate

Possible glue content in η0

2 3 0 0 ! 2 2 ! Γ(J/ψ → η γ) h0|GG˜|η i MJ/ψ − Mη0 = = 4.81 ± 0.77 ˜ 2 2 Γ(J/ψ → ηγ) h0|GG|ηi MJ/ψ − Mη

Vincent Mathieu (Univ. Valencia) Glueballs Spectroscopy February 2011 18 / 38 Physical Pseudoscalar States

Axial Anomaly → mixing with η1

µ αS µν µ 1 ∂ (¯qγµγ5q) = 2imqγ¯ 5q + Gµν G˜ → gg|∂ J |0 6= 0 4π µ5 Mixing Scheme with pseudoscalar glueball         |ηi cos ϕ − sin ϕ 0 1 0 0 |η8i 0  |η i  = sin ϕ cos ϕ 0 0 cos φG − sin φG |η1i 00 |η i 0 0 1 0 sin φG cos φG |ggi

2 0 2 Measurement of ZG = hη |ggi V → P γ and P → V γ → 0.04 ± 0.09 Escribano and Nadal (2007) Br(φ(1020) → η0γ) → 0.12 ± 0.04 KLOE collaboration (2008) Br(φ(1020) → ηγ) J/ψ → VP → 0.28 ± 0.21 Escribano (2008)

Where is the physical pseudoscalar glueball |η00i ?

η(1295),η (1405), η(1475),... Cheng et al., Phys. Rev D 79 (2008) 014024

Vincent Mathieu (Univ. Valencia) Glueballs Spectroscopy February 2011 19 / 38 Chiral

QCD = gauge theory with the color group SU(3)

1 µν X µ LQCD = − Tr Gµν G + q¯(γ Dµ − m)q 4 Gµν = ∂µAµ − ∂ν Aµ − ig[Aµ,Aν ]

Degrees of freedom at High Energies: Quarks q and Gluons Aµ Degrees of freedom at Low Energies: , ,... Goldstone of (Global) Chiral Symmetry: U(3)V ⊗ U(3)A

a U(3)V : q → exp(iθaλ )q a U(3)A : q → exp(iγ5θ5aλ )q

U(3)A broken spontaneously by condensate h0|qq¯ |0i 6= 0 U(3)A broken explicitly by quark masses m → 9 Goldstone pseudoscalar bosons with a small mass ∝ mh0|qq¯ |0i

3π, 4K and 2η

Vincent Mathieu (Univ. Valencia) Glueballs Spectroscopy February 2011 20 / 38 Chiral Anomaly - U(1) Problem

Observation of only8 Goldstone bosons Anomaly: Classical symmetry broken by quantum effects

Z Z[J] = DqDq¯DA exp(LQCD + J · A +ηq ¯ + ηq¯)

U(1)A αβµν DqDq¯ −→ DqDq¯exp(θ Gµν Gαβ )

Variation is a total derivative αβµν µ  Gµν Gαβ = ∂µK But non trivial gauge configuration (Instanton) with different winding number θ (topological charge)

U(1)A is not a symmetry of QCD 0 → η is NOT a Goldstone Mη0 ∼ 958 MeV > Mη ∼ 548 MeV But Anomaly vanishes for large N, for a gauge group SU(N → ∞)

Vincent Mathieu (Univ. Valencia) Glueballs Spectroscopy February 2011 21 / 38 Chiral Lagrangian

Lagrangian with9 Golstone Boson πa in the large N limit Nonlinear parametrization

0  √π + √η8 + √η0 π+ K+  2 6 3 √ √ 0 U = exp(i 2π/f) π = 2  π− √η8 − √π + √η0 K0   6 2 3   q  − 0 2 η0 K K¯ − η8 + √ 3 3

† † U ∈ U(3) UU = 1 U → LUR L ∈ U(3)L,R ∈ U(3)R Effective Lagrangian based on Symmetry with a Momentum Expansion p2

2 2 2 f D † µ E f D † ED µ †E Kinetic term at O(p ) ∂µU ∂ U but not 1 U ∂µU U∂ U 8 8 Symmetry breaking terms

f 2 D E Explicit breaking B mU † + Um† 8    2 α0 f det U 3 2 U(1) Anomaly ln ∼ − α0η 2 4 det U † 2 0

Vincent Mathieu (Univ. Valencia) Glueballs Spectroscopy February 2011 22 / 38 Mass Matrix at Leading Order

Chiral Lagrangian at leading order (in p2 and 1/N)

2 (p2) f D † µ † † E 1 2 L = ∂µU ∂ U + B(mU + Um ) − α0η 8 2 0 √ a a 2 2 Expand U = 1 + 2πaλ /f − (πaλ ) /f + ···

(p2) 1 a µ b 1 a b 3 2 L = ∂µπ ∂ π δ − Bπ π hλaλ mi− α0η 2 ab 2 b 2 0

2 2 Symmetry m = diag (m, ˜ m,˜ m˜ s) → mπ = Bm˜ and mK = B(m ˜ + ms)/2

Mass matrix in the octet-singlet (η8 − η0) √ 1  2 2 2 2  2 √4mK − mπ −2 2(mK − mπ) M80 = 2 2 2 2 3 −2 2(mK − mπ) 2mK + mπ + 9α0

Mass matrix in the flavor basis (ηq − ηs) √  2  2 mπ√+ 2α0 2α0 Mqs = 2 2 2α0 2mK − mπ + α0 Anomaly only source of mixing No anomaly for vector 1−− −→ ω and φ ideal mixed Vincent Mathieu (Univ. Valencia) Glueballs Spectroscopy February 2011 23 / 38 Mass Matrix at Leading Order

Effective Lagrangian for vector (without anomaly)

2 (p2) f 2 0 µ L = − (∂µVν − ∂µVµ) − 2B mVµV 8

Mass matrix in the flavor basis (ηq − ηs)

 2  2 mρ 0 Mqs = 2 2 0 2mK∗ − mρ Theory at LO: Physical states are qq¯ and ss¯ Experiments: Decay Properties ω → πππ, φ → K+K−

ω(782) ∼ qq¯ φ(1020) ∼ ss¯

2 mass predictions at leading order:

2 2 mω = mρ Satisfied at 3% Gell-Mann−Okubo mass formula

2 2 2 mφ = 2mK∗ − mρ Satisfied at 8%

Vincent Mathieu (Univ. Valencia) Glueballs Spectroscopy February 2011 24 / 38 Mass Matrix at Leading Order

Mass matrix in the flavor basis (ηq − ηs) √  2  2 mπ√+ 2α0 2α0 Mqs = 2 2 2α0 2mK − mπ + α0 Rotation to Physical States

 2  † 2 mη 0 R (φ)MqsR(φ) = 2 0 mη0 φ determines Decay Properties       η cos φ − sin φ ηq 0 = . η sin φ cos φ ηs Conservation of trace and determinant

2 2 2 mη + mη0 = 2mK + 3α0 2 2 2 2 2 2 2 mηmη0 = (4mK − mπ)α0 + mπ(2mK − mπ)

Vincent Mathieu (Univ. Valencia) Glueballs Spectroscopy February 2011 25 / 38 η − η0 Mixing at Leading Order

Only 1 parameter α0 but 2 states (or 2 invariants)

2 2 2 Trace α0 = (mη + mη0 − 2mK )/3 2 2 2 2 2 mηm 0 − mπ(2mK − mπ) Determinant α = η 0 2 2 4mK − mπ ! Not Equal ! 2 decays : η → γγ and η0 → γγ

Degrande and G´erard, JHEP 0905 (2009) 043

◦ θ ∼ −(20 − 15) in the U(3) basis (η8 − η0) ◦ φ ∼ (40 − 45) in the flavor basis (ηq − ηs)

θ = φ − θi with the ideal√ mixing angle ◦ θi = arccos(1/ 3) ∼ 54.7

Vincent Mathieu (Univ. Valencia) Glueballs Spectroscopy February 2011 26 / 38 Decays with respect to θ

Γ(η0 → ωγ) Γ(η0 → ργ) Γ(φ → η0γ) Γ(ω → ηγ) Γ(ρ → ηγ) Γ(φ → ηγ)

Γ(J/Ψ → η0γ) Γ(J/Ψ → ρη0) Γ(J/Ψ → φη0) Γ(J/Ψ → ηγ) Γ(J/Ψ → ρη) Γ(J/Ψ → φη)

Vincent Mathieu (Univ. Valencia) Glueballs Spectroscopy February 2011 27 / 38 Improvements

1 Next to leading order O(p4) Different decay constants fπ 6= fK J. Schechter, A. Subbaraman and H. Weigel, Phys. Rev. D 48, 339 (1993) J. M. Gerard and E. Kou, Phys. Lett. B 616, 85 (2005) V. Mathieu, V. Vento, Phys.Lett.B688:314-318,2010

2 Including a third gluonic state √  2  2 mπ√+ 2α0 2α0 2 Mqs = 2 2 ⇒ 3 × 3 Matrix Mqsg 2α0 2mK − mπ + α0 Description of the “glue” content in η0

2 3 0 0 ! 2 2 ! Γ(J/ψ → η γ) h0|GG˜|η i MJ/ψ − Mη0 = = 4.81 ± 0.77 ˜ 2 2 Γ(J/ψ → ηγ) h0|GG|ηi MJ/ψ − Mη

V. Mathieu, V. Vento, Phys. Rev. D81, 034004 (2010) Degrande, V.M., G´erard, in preparation

Vincent Mathieu (Univ. Valencia) Glueballs Spectroscopy February 2011 28 / 38 Third Gluonic State

Possible small “glue” content in η0

2 3 0 0 ! 2 2 ! Γ(J/ψ → η γ) h0|GG˜|η i MJ/ψ − Mη0 = ˜ 2 2 Γ(J/ψ → ηγ) h0|GG|ηi MJ/ψ − Mη = 4.81 ± 0.77

µν θGˆ µν G˜ →Glueball ∼ massive in the chiral Lagrangian

2 (p2) f D † µ † † E α 2 1 2 2 1 µ L = ∂µU ∂ U + B(mU + Um ) − (η0 + θ) − m θ + ∂µθ∂ θ 8 2 2 θ 2

Inclusion of a gluonic state via the θ−term and the anomaly √ √  2  mπ√+ 2α 2α 2β M2 = 2 2 qsg  √2α 2mK − mπ + α β  2β β γ

Rosenzweig, Salomone, and Schechter, Phys. Rev. D24 (1981) 2545 Degrande, V.M., G´erard, in preparation

Vincent Mathieu (Univ. Valencia) Glueballs Spectroscopy February 2011 29 / 38 Third Gluonic State

What are the theoretical constraints on R ?     |ηi |ηqi 0 |η i = R |ηsi |Gi |ggi

Inclusion of a gluonic state via the θ−term and the anomaly √ √  2  mπ√+ 2α 2α 2β M2 = 2 2 qsg  √2α 2mK − mπ + α β  2β β γ Diagonalization † 2 R MqsgR = D 2 2 2 2 D = diag (Mη ,Mη0 ,MG) with MG unknown

Three unknowns α, β, γ but three rotation invariants

2 2 2 2 2 2 2 α, β, γ ≡ F (M ,M 0 ,M ) ⇒ R(M ,M 0 ,M ) ⇒ Decays(M ) η η ηθ η η G G

1 8 (M 2 − M 2)2 β2 ≥ 0 −→ M 2 ≥ (4M 2 − M 2) + K π ∼ (1.5 GeV)2 G K π 2 2 2 3 3 4MK − Mπ − 3Mη

Vincent Mathieu (Univ. Valencia) Glueballs Spectroscopy February 2011 30 / 38 Decay with Glueball for Mη = 530 MeV and Mη0 = 1030 MeV

Γ(η0 → ωγ) Γ(η0 → ργ) Γ(η(0) → γγ) Γ(ω → ηγ) Γ(ρ → ηγ) Γ(π → γγ)

Γ(J/Ψ → η0γ) Γ(J/Ψ → ρη0) Γ(J/Ψ → φη0) Γ(J/Ψ → ηγ) Γ(J/Ψ → ρη) Γ(J/Ψ → φη)

Vincent Mathieu (Univ. Valencia) Glueballs Spectroscopy February 2011 31 / 38 00 η = η(1405)? decays for Mη = 530 MeV and Mη0 = 1030 MeV

Γ(η00 → ωγ) Γ(η00 → ργ) Γ(η00 → φγ) Γ(ω → ηγ) Γ(ρ → ηγ) Γ(φ → ηγ)

Γ(J/Ψ → η00γ) Γ(J/Ψ → ρη00) Γ(J/Ψ → φη00) Γ(J/Ψ → ηγ) Γ(J/Ψ → ρη) Γ(J/Ψ → φη)

Vincent Mathieu (Univ. Valencia) Glueballs Spectroscopy February 2011 32 / 38 Preliminary Results

Degrande, V.M., G´erard, in preparation

Glueball ∼ massive axion in the chiral Lagrangian

2 (p2) f D † µ † † E α 2 1 2 2 1 µ L = ∂µU ∂ U + B(mU + Um ) − (η0 + θ) − m θ + ∂µθ∂ θ 8 2 2 θ 2

Decays in better agreement with data but very sensitive to Mη

◦ ◦ ◦ θ = −11.4 ϕG = (4.7 ± 0.3) ϕ = (46.8 ± 1.8)

Mη = 530 MeV Mη0 = 1030 MeV MG = 1400 − 1500 MeV

η(1405) would be the third state, mainly gluonic ! BUT only leading order

G´erardand Kou, Phys.Rev.Lett.97 (2006) 261804 Other anomalous decays

Br(B0 → K0η0) = 65 10−6, Br(B0 → K0η) < 2 10−6, Br(B0 → K0π0) = 10 10−6.

η0 : η : π = 3 : 0 : 1 via Penguin diagram

Vincent Mathieu (Univ. Valencia) Glueballs Spectroscopy February 2011 33 / 38 Summary

Pure Yang-Mills Various models reproduce lattice spectrum Dynamical gluon mass generation BUT only 2 d.o.f. in wave function Instanton effects → mass difference between 0++ and 0−+

Scalar Mesons ++  Chiral suppression A 0 −→ qq¯ ∝ mq → f0(1710) realization of the glueball ? Not without controversy

A confirmation of three f0 is required Accurate data about their productions and decays would improve the understanding of their structure.

Pseudoscalar Mesons Anomaly in pseudoscalar → no ideal mixing for η and η0 Chiral Lagrangian at LO not enough to describe η and η0 → Inclusion of glueball in chiral Lagrangian Preliminary results favor η(1405) realisation of the glueball Prediction for η00 decays → need experimental confirmation !

Vincent Mathieu (Univ. Valencia) Glueballs Spectroscopy February 2011 34 / 38 Backup Slides

Backup Slides

Vincent Mathieu (Univ. Valencia) Glueballs Spectroscopy February 2011 35 / 38 Decays with respect to θ

Γ(η0 → ωγ) Γ(η0 → ργ) Γ(φ → η0γ) Γ(ω → ηγ) Γ(ρ → ηγ) Γ(φ → ηγ)

Γ(J/Ψ → η0γ) Γ(J/Ψ → ρη0) Γ(J/Ψ → φη0) Γ(J/Ψ → ηγ) Γ(J/Ψ → ρη) Γ(J/Ψ → φη)

Vincent Mathieu (Univ. Valencia) Glueballs Spectroscopy February 2011 36 / 38 Decay with Glueball

0 Γ(η0 → ωγ) Γ(η0 → ργ) Γ(J/Ψ → ωη ) Γ(ω → ηγ) Γ(ρ → ηγ) Γ(J/Ψ → ωη)

0 Γ(J/Ψ → η0γ) Γ(J/Ψ → ρη ) Γ(J/Ψ → φη0) Γ(J/Ψ → ηγ) Γ(J/Ψ → ρη) Γ(J/Ψ → φη) Vincent Mathieu (Univ. Valencia) Glueballs Spectroscopy February 2011 37 / 38 η00 Decays

Γ(η00 → ργ) Γ(η00 → ωγ) Γ(η00 → φγ) Γ(ρ → ηγ) Γ(ω → ηγ) Γ(φ → ηγ)

Γ(J/Ψ → η00γ) Γ(J/Ψ → ωη00) Γ(J/Ψ → φη00) Γ(J/Ψ → ηγ) Γ(J/Ψ → ωη) Γ(J/Ψ → φη)

Vincent Mathieu (Univ. Valencia) Glueballs Spectroscopy February 2011 38 / 38