Light glueball spectrum from the Light glueball spectrum from the - PowerPoint PPT Presentation

Light glueball spectrum from the Light glueball spectrum from the AdS/CFT correspondence AdS/CFT correspondence Frederic Jugeau INFN BARI Orsay 14/11/2007 In collaboration with P. Colangelo, F. De Fazio, S. Nicotri Euroflavour ‘07

AdS/CFT correspondence provides a new way to address Physics at strong coupling • vector r meson (Erlich et al. ‘05) • Hadronic spectrum: • glueballs (Pietro Colangelo, Fulvia De Fazio, F.J., Stefano Nicotri ‘07) the subject of this talk • Hadronic ( r , p ) form factors (Brodsky, de Teramond, Radyushkin ‘07) _ • QQ potential (Andreev, Zakharov ‘07) • Gluon condensate (Andreev, Zhakarov ‘07) : • U(1) sector of QCD/ η ’ mass (Katz, Schwartz, Schäfer ‘07) A • Low Energy Constants, c SB (Da Rold, Pomarol ‘05) • Deep Inelastic Scattering (Braga ’07) - strongly coupled plasma features • Heavy ion collisions/QGP : - confinement/deconfinement transition (Rajagopal, Shuryak, Iancu, Mueller ‘07)

Maldacena’ ’s AdS/CFT duality conjecture ( s AdS/CFT duality conjecture (‘ ‘98) 98) Maldacena IIB (oriented closed) superstring theory N = 4 Superconformal YM theory SU(N ) c in in the boundary (z → 0) â compact space Anti de Sitter space Holographic spacetime / bulk : (no physical extra dimensions) R : AdS typical size ‘t Hooft coupling : l : string typical size s - classical limit : g g → → 0 0 - large N ( λ fixed) : g → 0 c YM string string - supergravity limit : l << R - large ‘t Hooft limit : λ λ >> 1 >> 1 s Weakly - coupled effective theory Weakly Strongly - coupled gauge theory Strongly in a warped higher dim. space Classical bulk field Boundary value : Source for Z → 0

Scale invariance and its breaking (or what is z ?) Scale invariance and its breaking (or what is z ?) AdS/CFT provides 2 languages for deriving correlator functions ! inv. / scale transf. mapped into the 5 th holographic coord. z • • AdS modes in z : extension of the hadron wave functions into the 5 th holo. coord. z • different values of z : different scales at which the hadron is examined : - (z → 0) i.e. q → • : UV regime 2 - max. separation of quarks (~ x ) → max. value of z at IR boundary : hard f (z) Hard wall approx. (Polchinski, Strassler ‘01) : quadratic Regge trajectories: Soft wall approx. (Karch et al ‘06) : background dilaton Soft wall approx. soft linear Regge trajectories for vector mesons : z z m 0 (a, z ) break break conformal inv. of CFT : introduction of QCD scale QCD scale Λ Λ m QCD QCD

- AdS/CFT : String-like theories → QCD-like gauge theories (up-down approach) - AdS/QCD : QCD properties → weakly-coupled effective theory ( bottom bottom- -up up approach) in a warped higher dim. space AdS/QCD Model of light glueballs (scalar, vector) AdS/QCD Model of light glueballs (scalar, vector) Glueballs : Bound-states of gluons well defined in large N limit c QCD in Dual theory in Dilaton : Boundary operators : bulk fields : R=1 Scalar, vector glueball Scalar, vector fields : ( AdS radius : ) operators O ,O , s v 5d mass m 5 dim. O z IR UV

• Scalar bulk field : • Vector bulk field : 5-dim. bulk Dilaton Bulk field mass • Broken AdS isometries/conformal sym. ( energy scale [a]=1 ) • Regge behaviour of the mass spectrum - (Classical) eq. of motion : M n ² - Bulk field decomposition (mode) : holo. wave V(z) function IR UV Ψ n (z) dilaton metric Effective QCD theory z

- Schrödinger eq. : with c = 1 : A M (x,z) metric dilaton c = 3 : X(x,z) ( IR : z ¶ ) ( UV : z 0) • Mass spectrum : z ¶ • Holo. wave function : 0 z 0 (regular) polynomial • Spectrum given by a simple 1d. QM Shrödinger-like equation to resolve ! • AdS/CFT provides another language with tractable computations for non-perturbative Physics!

• Scalar glueball : • Vector glueball : Regge behaviour (n >>1) : (dilaton ) • up-down approach ≠ (KK spectrum) • Hard wall ( z : IR brane ) m (a=m /2 (Karch et al.’06)) Ground states : r AdS/QCD Lattice QCD AdS/QCD QCDSR QCDSR Lattice QCD Narison Hang, Zhang Morningstar Meyer Dominguez, PLB 652:73-78,2007 Paver (‘86) (hep-ph/9801214) (hep-lat/9901004) (hep-lat/0508002) (hep-ph/9612457) 1.096 1.5 (0.2) 1.580(150) 1.475(30)(65) 1.730(50)(80) < 1 1.342 3.850(50)(190) 3.240(330)(150) too light (?) too light (?) Let us see if this picture can be improved with further deviations of conformality Background corrections (but still close to AdS ) 5

Modification of the background (Karch et al. ‘06) modification of the dilaton modification of the metric UV subleading IR subleading (z 0) (z ¶ ) • dilaton : Mass splitting • metric : Mass splitting increases λ < 0 Maximun effect : metric

Conclusion Conclusion AdS/CFT provides a new way to address Physics at strong coupling Scalar and vector glueball mass spectra Ground states : Perturbation • dilaton modification : of the background (metric/dilaton) • metric modification : max. mass splitting ( λ < 0 ) AdS/QCD at its very beginning : • fashionable at present (bad reason to investigate it) up-down approach (quantitative predictions difficult) predictions • there is the strong hope to identify the Dual Theory of QCD Dual Theory of QCD (at low energy !) bottom- -up up approach bottom

Backup Slides

More about the operator/field correspondence More about the operator/field correspondence • Bulk field A(x,z) : p-form (totally antisymmetric tensor with p indexes) (scalar) 0-form : (vector) 1-form : Bulk (strength field ) 2-form : p , m AdS • eq. of motion mass term m AdS • Superconformal gauge theory : conformal group invariant m AdS m AdS Scale transf. : Field Boundary scaling dim. = canonical dim. : (without anomalous dim.) Operator 4, Δ or

CFT AdS Holographic space : Bulk Our spacetime • strongly coupled λ • weakly coupled String theory • SUSY SU(N) • classical • conformal scaling • p-form Bulk field Operator dimension • massive scaling Source dimension z

AdS/QCD Correspondence (Witten ‘98) AdS/QCD Correspondence • squarks, gluinos SUSY ? QCD SU(3) c Holographic dual • massive quarks theory of QCD CFT • renormalisation scale μ How modifying AdS/CFT towards AdS/QCD ? QCD could be nearly conformal ( UV ) (Brodsky ’02; Alkofer et al. ‘04) QCD could have IR fixed point Dimensionless renormalized Green function : • effective coupling with Renormalization : • effective mass with

Homogeneous RGE : Scale transf. : λ 0 0 Chiral limit m=0 : λ (t) breaks scale invariance scale invariant theory Classical theory or fixed point : β =0 and λ (t) = λ = const. QCD nearly Chiral QCD : m=0 conformal invariant IR fixed point λ * : β ( λ *)=0 AdS/CFT AdS/QCD Effective bulk field action Deformation of the geometry

AdS/QCD spectrum of ρ ρ meson meson (Son et al. ’05) AdS/QCD spectrum of Dual theory in QCD in Chiral symmetry Gauge symmetry Dilaton : Condensate : Scalar field : Left/right currents : Left/right gauge fields : z bulk fields operators massless tachyonic

(Classical) eq. of motion : ρ meson vector field : plane wave holo. wave Schrödinger eq. : function connection Regge behaviour : dilaton/geometry V(z) M n ² = 4n+4 QCD in Dual theory in Ψ (z) IR UV QCD Better than 20% ! ADS geometry dilaton z

Holographic Models of mesons Holographic Models of mesons I) Top-to-bottom approach : M-theory/Supertring Theory QCD-like Gauge theory Bulk : Boundary : II) Bottom-up approach ( AdS/QCD AdS/QCD ) : Properties of QCD SU(3) c : (String-inspired) dual theory : • squarks, gluinos SUSY • massive quarks Holographic spacetime : CFT • renormalisation scale μ distorted distorted Minkowski Confinement, Chiral symmetry breaking, masses, decay constants, form factors, etc... - - ++ • Scalar 0 and vector 1 mass spectrum - + Glueball Spectroscopy (pseudoscalar 0 , hybrid mesons) (Colangelo, de Fazio, Nicotri, F.J. ‘07) • Dual theory of QCD (if exists…)

AdS/QCD Model of light glueballs (scalar, vector) AdS/QCD Model of light glueballs (scalar, vector) Glueballs : Bound-states of gluons (gg...) QCD in Dual theory in Dilaton : Scalar glueball R=1 Scalar field : ( AdS radius : ) Vector glueball Vector field : bulk fields z boundary operators Operators / fields of the model 2 4D : 5D : p Δ m AdS massless 0 4 0 massive 1 7 24

boundary bulk (p=0) ( Δ =4) Scalar glueball Vector glueball (p=1) ( Δ =7) ? AdS/QCD AdS/CFT • Scalar bulk field : • Vector bulk field : 5-dim. bulk Dilaton Bulk field mass • Broken AdS isometries/conformal sym. (energy scale [a]=1) • Regge behaviour of the mass spectrum

- (Classical) eq. of motion : - Bulk field decomposition (mode) : plane wave holo. wave function V(z) M n ² QCD in Dual theory in Ψ n (z) IR UV metric dilaton z - Schrödinger eq. : with c = 1 : A M (x,z) metric dilaton c = 3 : X(x,z) ( IR : z ¶ ) ( UV : z 0)

• Mass spectrum : z ¶ • Holo. wave function : 0 z 0 Kummer confluent hypergeometric function (-n < 0 : polynomial) Scalar glueball Vector glueball Vector ρ meson (Son et al. ‘05) : Boundary ( Δ =4) ( Δ =7) ( Δ =3) Bulk (p=0) (p=1) (p=0) Spectra