Hadrons in a holographic approach to finite temperature (and density) QCD Pietro Colangelo INFN - Sezione di Bari - Italy in collaboration with F. De Fazio, F. Giannuzzi, F. Jugeau, S. Nicotri EMMI Workshop: Quarkonia in Deconfined Matter 2011 Acitrezza 28-30 September 2011
Holographic methods inspired to the AdS/CFT duality have been applied to several cases of interest: η/s jet quenching photon emission rate thermodynamic functions, AdS/QCD at finite temperature/density what about the deconfinement line in the T µ plane? what about hadrons increasing T and µ? Son Starinets Stephanov Policastro Rajagopal Wiedemann Gursoy Kiritzis Mateos Peschanski Yaffe Iancu Casalderrey-Solana. Outline: - about the deconfinement line in the T- µ plane - hadrons at T=0 in a holographic approach to QCD - medium effects: hadronic spectral functions
N=4 SYM with gauge group SU(N C ) at large N C in 4d The AdS/CFT correspondence d=10 Type IIB string theory in AdS 5 x S 5 g YM, N c g s, R/ s Maldacena Gubser Klebanov Polyakov Witten more general: equivalence (duality) between a gravity theory defined in AdS d+1 x C (C a compact manifold) and a conformal field theory (CFT) defined on the boundary of AdS d+1 (M d ) ds 2 = R2 z 2 ( dt 2 + d x 2 + dz 2 ) + R 2 dω 5 AdS 5 x S 5
Peculiar role of the d+1 dimensional Anti de Sitter (AdS d+1 ) space: - solution of Einstein s eqs. in vacuum with negative cosmological constant (Λ =-6/R 2 ) - negative curvature - if embedded in R d+2 (with coordinates (X 0, X d+1 ) ) it is defined by ( X 0 ) 2 d X i i=1 ( ) 2 + ( X d +1 ) 2 = R 2 AdS d+1 embedded in R d+2 - group of isometries SO(2,d) ( (d+2)(d+1)/2 generators) - it has a boundary M d - on the boundary M d the coordinate transformation belonging to SO(2,d) are conformal transformations ( (d+2)(d+1)/2 generators)
The AdS/CFT correspondence matching the parameters: 2 g YM 2 = g s λ = g YM N c = R s 4 g YM 0 N c λ g s 0 R s large t Hooft coupling limit in gauge theory supergravity limit in the string theory dynamics of a 4d CFT encoded by classical gravity in 5d
The AdS/CFT correspondence Φ 0 (y) boundary value M d Φ (x,z) bulk field in AdS d+1 z At Φ 0 (y) an operator O(y) of CFT on M d is associated Φ 0 (y) coupled to O via CFT generating functional with the source Φ 0 M d d d y Φ 0 (y) O(y) AdS/CFT duality provided by the generating functionals { } Z CFT [Φ 0 ] = da exp S CFT + d d x Φ 0 O gravity partition function in AdS d+1 x C with boundary value Φ 0 Z grav [ Φ 0 ] AdS/CFT correspondence conjecture Z CFT [ Φ 0 ] = Z grav [Φ 0 ] Gubser Klebanov Polyakov Witten
AdS/CFT and QCD AdS/CFT correspondence conjectured for conformal and supersymmetric field theories in M 4 QCD is not conformal (neither supersymmetric): it has a mass scale Λ QCD however, in the UV, neglecting quark mass and radiative effects, QCD is a nearly conformal theory QCD a candidate for a description inspired by AdS/CFT conditions to be implemented: UV -> conformal behaviour -> AdS holographic space IR -> modification (at least) of the AdS geometry of the bulk AdS/QCD: extradimensional models, motivated by the AdS/CFT correspondence conjecture, developed to compute QCD observables
bottom-up AdS/QCD hard-wall 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 z=0: UV brane z max z z=z max IR brane soft-wall Karch, Katz, Son, Stephanov, Andreev, Gherghetta, Brodsky, de Teramond, Lebed scale provided by a background dilaton field in the bulk z example of dilaton profile φ(z)
finite temperature: black hole AdS vs thermal-ads (N=4 at T 0 on S 3 xs 1 and N c -> ) two solutions E. Witten, Adv. Theor. Math. Phys. 2, 505 black hole - AdS ds 2 = R2 z 2 f (z)dt 2 + dx 2 + dz2 f (z) holographic space horizon f (z) =1 z4 z h 4 T = 1 π z h 0 z z h 0 τ β' Hawking temperature increase the temperature z h z thermal-ads ds 2 = R2 z 2 (dt 2 + d x 2 + dz 2 ) periodic Euclidean time τ extended to β T=1/ β
finite temperature and quark density: AdS-RN finite quark density: term added to the QCD lagrangian J D (x) = µ ψ + (x) ψ(x) µ : boundary value of the time component of a U(1) gauge field: A 0 (z) Reissner/Nordstrom black hole metric (charged spherically symmetric) ds 2 = R2 e c 2 z 2 z 2 f (z) =1 1 4 z + q2 2 z h h f (z)dt 2 + dx 2 + dz2 f (z) z 4 + q 2 z 6 Euclidean q charge of the black hole T = 1 df 4π dz z= zh temperature A 0 (0) = µ A 0 (z h ) = 0 µ = k q z h 2 k parameter
finite temperature and quark density: AdS-RN Static (infinitely heavy) quark-antiquark pair P x ( 1 )P + ( ) = e x 2 P = e 1 2T F (T ) 1 F(r,T )+γ (T ) T correlation function of Polyakov loops Polyakov loop AdS/CFT prescription: F(r,T) given by the classical Nambu-Goto action on the string worldsheet (Maldacena) S NG = 1 2πα' d 2 ξ [ ( )( b X N )] det g MN a X M Nambu-Goto action ξ 0 = τ ξ 1 = x F(r,T) = T S NG
The problem reduces to extremizing the string configurations in the background AdS-RN space configurations of minimal surface horizon horizon one brane connecting the two static quark lines -> confinement Maldacena, Sonnenshein, Rey, Andreev, Zakharov, Albacete.
The problem reduces to extremizing the string configurations in the background AdS-RN space configurations of minimal surface horizon horizon one brane connecting the two static quark lines -> confinement horizon horizon two branes attached to the horizon -> deconfinement Maldacena, Sonnenshein, Rey, Andreev, Zakharov, Albacete.
z 0 = z(x = 0) f 0 = f (z 0 ) λ = cz 0 v = z /z 0 g = R 2 /α' unstable branch
T > 0 µ > 0 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 T > 0 µ = 0 lattice QCD Digal, Kaczmarek, Karsch, Satz, 2005
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 meson mass k=1/2 from NJL analyses Giannuzzi, Nicotri, PC, PRD 2011
medium effects on hadrons: spectral functions bottom-up approach: constructing a holographic model of QCD AdS 5 = M 4 + radial (holographic) coordinate z dictionary 4d - CFT gauge invariant operator O(x) hadron mass 2 conformal dimension Δ 5d - gravity field ψ(x,z) eigenvalue of a 5d eq. of motion 5d mass m 5 2 m d +1 R 2 = (Δ p)(δ + p d) UV scale Λ QCD Δ dimension of the p-form operator small z dilaton parameter wall position
light hadrons Karch, Katz, Son, Stephanov, Andreev, Gherghetta, Brodsky, de Teramond scale provided by a background dilaton field in the bulk dilaton profile ϕ = c 2 z 2 z µ 1 γ SU(2) L xsu(2) R 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 +S) e 2iΠ q at small z: X 0 αβ z 0 (m qz + σ z 3 ) δαβ 2 S = 1 k D M X = M X ia LM X + ixa MR (σ q q ) d 5 x g e ϕ(z) DX 2 + m 2 5 X 2 + 1 2 4g Tr F 2 L + F R 5 vector meson spectrum m 2 (ρ n ) = 4c 2 ( n +1) scalar mesons m 2 n = c 2 ( 4n + 6) ( ) c=m ρ /2 scalar glueballs S = 1 k d 5 x g e ϕ(z) g MN M X N X m 2 n = c 2 ( 4n + 4) De Fazio, Giannuzzi, Jugeau, Nicotri, PC, PL657,73
finite temperature: AdS-black hole vs thermal-ads (N=4 at T 0 on S 3 xs 1 and N c -> ) two solutions E. Witten, Adv. Theor. Math. Phys. 2, 505 black hole - AdS ds 2 = R2 z 2 f (z)dt 2 + dx 2 + dz2 f (z) holographic space horizon f (z) =1 z4 z h 4 T = 1 π z h 0 z z h 0 τ β' Hawking temperature increase the temperature z h z thermal-ads ds 2 = R2 z 2 (dt 2 + d x 2 + dz 2 ) periodic Euclidean time τ extended to β T=1/ β
AdS-black hole for all T scalar meson light hadron spectral functions scalar glueball ω = q 0 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
spectral function non zero three-momentum
AdS-black hole for all T Mass vs T Width vs T scalar glueball melting temperature similar results for vector mesons (Fukushima et al) and for scalar mesons (Giannuzzi et al) same qualitative behavior different temperature lattice (N c =3) (Nishii et al)
finite µ soft-wall model AdS-RN scalar glueball Giannuzzi Nicotri PC
finite µ soft-wall model AdS-RN scalar glueball preliminary m 2, T, µ dimensionless Giannuzzi Nicotri PC
deconfinement Hawking-Page transition between the two geometries black hole - AdS vs phase transition between two geometries in the dual space thermal-ads E. Witten, Adv. Theor. Math. Phys. 2, 505
deconfinement Hawking-Page transition between the two geometries black hole - AdS vs phase transition between two geometries in the dual space thermal-ads E. Witten, Adv. Theor. Math. Phys. 2, 505 in AdS/QCD models C.P Herzog, PRL98, 091610 on-shell gravitational action (hard-wall) 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
deconfinement Hawking-Page transition between the two geometries black hole - AdS vs phase transition between two geometries in the dual space thermal-ads E. Witten, Adv. Theor. Math. Phys. 2, 505 in AdS/QCD models AdS-black hole C.P Herzog, PRL98, 091610 thermal AdS 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 unaffected in the thermalized medium in the confined region and melt at the deconfinement transition What about charmonium? proposal: U(1) symmetry connected to the charm quantum number c J /ψ c ρ fixed from J/ψ mass TJ dissociation /ψ T HP Giannuzzi Nicotri PC two-scale soft wall (Fukushima et al.)
Conclusions Deconfinement line in the T-µ plane by AdS/RN At T=0 some hadronic features (mass, decay constants, strong couplings of glueballs, scalar and vector mesons) reproduced At T>0 in the AdS/BH phase: thermal effects at low temperatures Similar effects for spectral functions at finite density and low T If the Hawking-Page transition occurs: the deconfinement HP temperature is close to the generally accepted value; hadrons melt at the deconfinement point
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