towards a holographic approach to the QCD phase diagram Pietro Colangelo INFN - Sezione di Bari - Italy in collaboration with F. De Fazio, F. Giannuzzi, F. Jugeau and S. Nicotri Continuous Advances in QCD Minneapolis 12-15 May 2011
outline - hadrons in medium expectations - scalar hadrons at T=0 in an AdS/QCD model - thermal effects on scalar hadrons - the deconfinement critical line in the T- µ plane - perspectives
increasing T: chiral symmetry restoration deconfinement common expectations hadron masses change widths broaden hadrons melt at some temperature experiments at RHIC, LHC th. analyses by lattice QCD, QCD sum rules, effective theories (NJL), models
increasing µ experiments at Fair/GSI th. analyses by effective theories (NJL), models
AdS/QCD at finite temperature/density The holographic approach inspired to the AdS/CFT duality has been applied to several problems shear viscosity/entropy density ratio jet quenching photon emission rate thermodynamic functions Son Starinets Stephanov Policastro Rajagopal Wiedemann Gursoy Kiritzis. what about hadrons increasing T? what about hadrons increasing µ? what about the deconfinement critical line in the T µ plane?
AdS/CFT correspondence conjecture (or Maldacena conjecture): a remarkable connection conjectured between certain string theory in certain curved space-times and certain field theories in flat (3+1) dimensional space-time Maldacena Gubser Klebanov Polyakov Witten AdS d+1 embedded in R d+2 Giannuzzi generalization suggested for QCD-like theories at strong coupling conditions to be implemented: UV -> conformal behaviour -> AdS holographic space IR -> modification (at least) of the AdS geometry of the bulk AdS/QCD models
bottom-up approach: AdS/QCD hard wall model M 4 AdS 5 = M 4 + radial (holographic) coordinate z ds 2 = R2 z (η 2 µν dx µ dx µ + dz 2 ) x λx maps scale transformations into z z λz maximum separation -> maximum value of z: z max Polchinski, Strassler, Erlich, Katz, Son, Stephanov, Pomarol, Da Rold, Radyushkin dictionary 4D gauge invariant operator O hadron mass 2 conformal dimension Δ large coupling UV z=0: UV brane 5D field ψ(x,z) z max z z=z max IR brane eigenvalue of a 5D eq. of motion 5D mass m 5 small coupling small z scale Λ QCD z max
bottom-up approach: soft wall model Karch, Katz, Son, Stephanov, Andreev, Gherghetta, Brodsky, de Teramond confinement implemented by a background dilaton field in the bulk dilaton profile φ(z) µ 1 γ chiral symmetry: - two conserved currents in QCD : J L,R = q γ µ 5 2 - two bulk gauge fields in 5D : A L,R (x,z) chiral symmetry breaking: a massive scalar bulk field in 5D X=X 0 e 2iΠ at small z: X 0 αβ z 0 S = 1 k (m q z + σ z 3 ) δαβ 2 d 5 x g e ϕ(z) DX 2 + m 2 5 X 2 + 1 2 4g Tr F 2 2 L + F R 5 z (σ q q ) ( ) q D M X = M X ia LM X + ixa MR ϕ = c 2 z 2 m 2 (ρ n ) = 4c 2 ( n +1) c=m ρ /2
same dilaton Light glueballs glueball operator <-> bulk field according to the AdS/CFT rules Scalar glueball J PC = 0 ++ 4D O S = Tr (G 2 ) Δ = 4 5D X(x,z) m 5 2 = 0 AdS 5 back. dilaton g MN = e 2A(z) η MN ϕ(z) = c 2 z 2 A(z) = ln R z action S = 1 2k d 5 x g e ϕ(z) g MN M X N X Spectrum m 2 n = 4c 2 ( n + 2) De Fazio Nicotri Jugeau PC, PL563,73
Light glueballs J PC = 0 -+ 4D 5D O = Tr(G G ) Δ = 4 Y(x,z) m 5 2 = 0 spectrum m 2 n = 4c 2 ( n + 2) J PC = 1 -- 4D 5D O = Tr(G(DG)G) Δ = 7 Landau-Yang- Pomeranchuk spectrum A M (x,z) m 5 2 = 24 m 2 n = 4c 2 ( n + 3) 2 m 0 ++ 2 m 0 + 2 m 1 = 2m ρ 2 2 = m 0 ++ 2 = 3m ρ
Two-point correlation functions Φ(x,z) = d 4 y K(x,z;y) Φ 0 (y) M 4 bulk-to-boundary propagator Poles: q 2 n = m 2 n = 4c 2 ( n + 2) residues (decay constants 2 ) F 2 n c 6 (n +1) ( n + 2)
Π QCD vs Π AdS OPE (SVZ) all condensates in terms of the scale parameter c
Π QCD vs Π AdS X D=2 condensate term (absent in QCD)
Scalar mesons S = 1 k d 5 x g e ϕ(z) DX 2 + m 2 5 X 2 + 1 2 4g Tr F 2 2 L + F R 5 ( ) X=(X 0 +S ) e 2iΠ m 2 n = c 2 ( 4n + 6) F n 2 = 3 π 2 c 4 (n +1) mass of a 0 (980) reproduced 2 m a0 m = 3 2 2 ρ (exp =1.59) decay constants compatible with QCD estimates De Fazio, Giannuzzi, Nicotri, Jugeau, PC
finite temperature (N=4 at T 0 on S 3 xs 1 and N c -> ) two solutions E. Witten, Adv. Theor. Math. Phys. 2, 505 + + periodic Euclidean time τ extended to β T=1/ β holographic space horizon + + increasing the temperature z h z
Hawking-Page transition between the two geometries E. Witten, Adv. Theor. Math. Phys. 2, 505 deconfinement transition phase transition between two geometries in the dual space
Hawking-Page transition between the two geometries E. Witten, Adv. Theor. Math. Phys. 2, 505 deconfinement transition in AdS/QCD models on-shell gravitational action (hard wall) phase transition between two geometries in the dual space C.P Herzog, PRL98, 091610 thermal AdS regularized action density AdS-black hole z h horizon on-shell gravitational action (soft wall) thermal AdS regularized action density AdS-black hole z h horizon
Hawking-Page transition between the two geometries E. Witten, Adv. Theor. Math. Phys. 2, 505 deconfinement transition in AdS/QCD models phase transition between two geometries in the dual space AdS-black hole thermal AdS AdS-black hole Soft wall T HP Thermal AdS Hard wall T HP
Soft wall model - AdS-black hole for all T (metastable phase at small T) scalar glueball action with AdS/BH metric bulk-to-boundary propagator q=0 ω=q 0 u=z/z h in-falling out-going (discarded) retarded Green function
Soft wall model - AdS-black hole for all T (metastable phase at small T) scalar glueball spectral function q=0 ω=q 0 temperature from the position of the horizon; scale c fixed from the ρ meson mass increasing the temperature: masses decrease widths increase -> melting at particular T Giannuzzi Nicotri PC, PRD80.094019
Soft wall model - AdS-black hole for all T (metastable phase at small T) scalar glueball Mass vs temperature melting temperature Width vs temperature
spectral function q not zero
Soft wall model - AdS-black hole for all T (metastable phase at small T) scalar glueball spectral function with background subtracted: lightest state
lattice QCD scalar glueball N c =3 Mass vs temperature Width vs temperature Ishii-Suganuma-Matsufuru
The AdS-black hole vs lattice : scalar glueball same qualitative behavior different temperature lattice similar results for vector mesons and for scalar mesons (Giannuzzi et al) (Fukushima et al)
implementing the Hawing-Page transition: thermal AdS / AdS-BH deconfinement in QCD phase transition between two geometries in the dual space AdS-black hole Soft wall T HP Thermal AdS Hard wall T HP
implementing the Hawing-Page transition: thermal AdS / AdS-BH scalar glueball scalar meson hadrons survive in the thermalized medium in the confined phase and melt at the deconfinement transition Giannuzzi Nicotri PC
finite temperature and density: AdS-RN Finite density: term added to the QCD lagrangian J D = µψ + ψ µ : boundary value of the time component of a U(1) gauge field: A 0 (z) Reissner/Nordstrom black hole metric Euclidean q charge of the black hole temperature from A 0 (z h )=0 k parameter
finite temperature and density: AdS-RN Static (infinitely heavy) quark-antiquark pair P x ( 1 )P + ( ) = e x 2 1 F(r,T )+γ (T ) T Correlation of Polyakov loops P = e 1 2T F (T ) Polyakov loop S NG = 1 2πα' d 2 ξ det g MN a X M ( )( b X N ) [ ] Nambu-Goto action F(r,T) = T S NG
finite temperature and density: AdS-RN configurations of minimal surfaces: one brane connecting the two quark lines -> confinement two branes attached to the horizon -> deconfinement
ˆ µ = 0.5 T ˆ = 0.82T * ˆ µ = 0.5 T ˆ =1.23T * ˆ µ = 0.5 T ˆ =1.65T * T * = 0.122
ˆ µ = 0.7 ˆ µ = 0.2 Polyakov loop
deconfinement line T c =134 MeV µ c = 248 MeV scale c from the ρ mass using k=1/2 from NJL analyses Giannuzzi Nicotri PC, PRD83,035015
Conclusions and perspectives At T=0 the AdS/QCD model reproduces some features of glueballs and light mesons; a problem in obtaining the OPE result At T>0 in the AdS/BH (metastable) phase hadron masses decrease and widths broaden, but at low temperatures Considering the Hawking-Page transition, the deconfinement temperature is close to the value obtained from other approaches; hadrons melt at the deconfinement point Confinement-deconfinement line in the T-µ plane Hadron properties at finite density and low T to be investigated