Holographic QCD at finite (imaginary) chemical potential

Holographic QCD at finite (imaginary) chemical potential Università di Firenze CRM Montreal, October 19, 2015 Based on work with Francesco Bigazzi (INFN, Pisa), JHEP 1501 (2015) 104

Contents: The Roberge-Weiss phase diagram. Holographic QCD & chemical potential: context. Holographic QCD & chemical potential: results. Conclusions and perspectives.

Contents: The Roberge-Weiss phase diagram. Holographic QCD & chemical potential: context. Holographic QCD & chemical potential: results. Conclusions and perspectives.

The Roberge-Weiss phase diagram QCD with imaginary baryon chemical potential µ B Motivation: sign problem in lattice QCD, real µ B challenging. Analytic continuation from imaginary µ B. Rich phase diagram: Roberge-Weiss [Roberge-Weiss 1986]. θ B µ B /T c (0) is an angle: Z = Tr(e βh+iθ BN B ). Free energy density f at T > T c : Periodic: f (θ B ) = f (θ B + 2πk). Depends on θ B /N c.

The Roberge-Weiss phase diagram QCD with imaginary baryon chemical potential µ B For N f massless Weyl (anti)fundamental fermions: f (θ B ) = N cn f 12 T 4 min k ( θb 2πk Large T : f has first order discontinuities at θ B = (2k + 1)π. Critical temperature for deconfinement: N c T c (θ B ) T c (0) 1 + a θ2 B a > 0 ) 2

The Roberge-Weiss phase diagram Phase diagram: Phase diagram in holography? Top-down model closest to (planar) QCD ( Holographic QCD ): Witten-Sakai-Sugimoto [Witten 1998, Sakai-Sugimoto 2004]. D4 D8/ D8: non supersymmetric, confining theory with only quarks in fundamental.

Contents: The Roberge-Weiss phase diagram. Holographic QCD & chemical potential: context. Holographic QCD & chemical potential: results. Conclusions and perspectives.

Context: Holographic YM D4-branes wrapped on S 1 x 4, at low energy: 4d YM theory + (adjoint) KK modes [Witten 1998] Dual gravity solution: ( u ds 2 = R ) 3/2 [dxµ dx µ + f (u)dx4 2 ] ( u ) [ 3/2 du 2 + R f (u) + u2 dω 2 4 ] f (u) = 1 u3 0 u 3 e φ = g s ( u R ) 3/4 F 4 =... In IR: R 1,3 cigar (u,x4 ) S 4. g 00 (u 0 ) 0 (regular): confinement. Period of x 4 1/M KK. In gravity regime KK modes are NOT decoupled. If x 0 compact: theory at finite T < T c (period of x 0 1/T ).

Context: Holographic YM D4-branes wrapped on S 1 x 4, at low energy: 4d YM theory + (adjoint) KK modes [Witten 1998] Dual gravity solution at T > T c : ( u ds 2 = R ) 3/2 ] ( u ) [ 3/2 du [ f (u)dx0 2 + dx a dx a + dx4 2 2 + R f (u) + u2 dω 2 4 ] f (u) = 1 u3 T u 3 e φ = g s ( u R ) 3/4 F 4 =... In IR: R 3 cigar (u,x0 ) Sx 1 4 S 4. g 00 (u T ) = 0: deconfinement. Period of x 0 1/T. Symmetry between solutions: x 0 x 4, u 0 u T, M KK T.

Context: Holographic YM D4-branes wrapped on S 1 x 4, at low energy: 4d YM theory + (adjoint) KK modes [Witten 1998] Difference of free energy densities (from on-shell actions): ( ) 2N 2 f = c λ 4 [(2πT ) 6 2187π 2 MKK 2 MKK 6 ] Notation: λ 4 = g 2 YM N c = 2πg s l s M KK N c. Critical temperature for deconfinement: T c = M KK 2π.

Context: Holographic QCD Probe D8/ D8-branes at antipodal points on S 1 x 4 : massless (anti)quarks in the fundamental [Sakai-Sugimoto 2004] T > T c : branes fall separately into horizon (χsr) u x 0 D8 x4 D8 T>T c T < T c : branes connect at tip of cigar (χsb) u T D8 D8 T<T c U(N f ) U(N f ) SU(N f ) U(1) B u 0 Critical temperature for chiral symmetry restoration coincides with critical temperature for deconfinement T c. [Aharony-Sonnenschein-Yankielowicz 2007]

Context: Smearing technique Dynamical quarks: backreaction of D8/ D8-branes on Witten s background Dynamical flavors in holography: flavor brane backreaction (probe branes quenched approximation). Localized D8/ D8 configuration: too difficult. (PDEs: a limiting case in [Burrington-Kaplunovsky-Sonnenschein 2007]). Difficult technical problem (PDEs) made simpler by smearing technique (ODEs). [Bigazzi-Casero-Cotrone-Kiritsis-Paredes 2005, Casero-Nunez-Paredes 2006]

Context: Smearing technique Dynamical quarks: backreaction of D8/ D8-branes on Witten s background Smearing: homogeneous distribution of flavor branes along transverse dimensions recover symmetry. In dual field theory: U(N f ) U(1) N f. Picture from [Nunez-Paredes-Ramallo 2010]

Context: Smearing technique Is it a good approximation? E.g. Free energy F in flavored ABJM [Conde-Ramallo 2011]: F (S 3 ) = π ( ) 2 3 k1/2 N 3/2 Nf ξ k Comparison of smeared gravity with localized field theory result:

Context: Smearing technique Dynamical quarks: backreaction of D8/ D8-branes on Witten s background Complete solution (D8/ D8-branes uniformly distributed on s.t. U(1) symmetry recovered): not yet. S 1 x 4 (Still complicated second order non linear coupled equations). Smeared configuration at first order in N f N c : analytic solutions!

Contents: The Roberge-Weiss phase diagram. Holographic QCD & chemical potential: context. Holographic QCD & chemical potential: results. Conclusions and perspectives.

Holographic QCD & chemical potential: results Metric: Confined phase (T < T c ) ds 2 = e 2λ ( dt 2 +dx a dx a )+e 2 λ dx 2 4 +l 2 s e 4φ+8λ+2 λ+8ν dρ 2 +l 2 s e 2ν dω 2 4 Action: 2k0 2 S = d 10 x g [e 2φ ( R + 4( φ) 2) 12 ] F 4 2 2k2 0 N f T 8 M KK π d 10 x (g + 2πα F ) g44 e φ smeared DBI Solution for brane gauge field: A t = µ F = 0: zero-density, finite-chemical potential (validity: small µ).

Holographic QCD & chemical potential: results Confined phase (T < T c ) Field expansion in N f /N c : Ψ(ρ) = Ψ Witten (ρ) + ɛ f Ψ 1 (ρ) + O(ɛ 2 f ) where: ɛ f 1 N f 12π 3 λ2 4 N c 1

Holographic QCD & chemical potential: results (Horrible but) analytic solution (in r u3 0 ρ): ls 3 gs 2 λ 1 = 3 8 f + y 1 4 (A 2 A 1 ) 1 4 (B 2 B 1 )r λ 1 = 1 8 f + y 1 4 (A 2 + B 2 r) 3 4 (A 1 + B 1 r) φ 1 = 11 8 f + y 1 4 (A 1 + B 1 r) 3 4 (A 2 + B 2 r) with: ν 1 = 11 24 f + q f = 4 9 e 3r/2 3F 2 ( 1 2, 1 2, 13 6 ; 3 2, 3 2 ; e 3r ) ( ) ( ( )) 3r 3r y = C 2 coth C 1 + C 2 2 2 + 1 + z q = 1 12 (A 1 5A 2 + r(b 1 5B 2 )) + 5 ( ) 3r 3 z coth (M 1 + M 2 (3r + 2)) + 2M 2 ( 2 ) ( e 9r/2 e 3r + 1 (9e ) ( )) 3r 12 3F 2, 2 1, 19 6 ; 3 2, 3 2 ; e 3r + 3 F 32 2, 3 2, 19 6 ; 2 5, 5 ; e 3r 2 z = 162 ( 1 e 3r ) ( ) ( ) ( ) 8e 3r/2 10e 3r + 3 16 2F 1, 2 1 ; 3 2 ; e 3r 15r/2 e 38e 3r + 8e 6r 40 819 ( 1 e 3r ) + 273 ( 1 e 3r ) 13/6

Holographic QCD & chemical potential: results Features IR regular (Ricci and Kretschmann scalars, with B 1 = 6C 2, B 2 = 0, M 2 = C 2 6 ). UV divergent (for whatever choice of integration constants): Landau pole. Compact radial variable: x = e 3r/2 IR: x 0, UV: x 1. In UV: 1 g 2 YM,x 1 [ g YM 2 1 3 7 ɛ f 2 5/6 (1 x) 1/6 Physics reliable at x x LP 1. ] x LP = 1 2 5 (3/7) 6 ɛ 6 f

Holographic QCD & chemical potential: results Examples of observables From fundamental string: string tension T s = g 00 g 11 x=0 = 2 27π λ 4M 2 KK [1 + 1.13ɛ f ] From wrapped D4: baryon (vertex) mass m B = 1 27π λ 4N c M KK [1 + 0.95ɛ f ] From fluctuation of D8 gauge fields: vector meson masses (ɛ f = 0.02, naive comparison ) mρ(1450) 2 mρ(1450) 2 mρ(1450) 2 mρ 2 3.7 Without flavors : mρ 2 4.3 Experiment : mρ 2 3.5 ma 2 1 (1260) ma 2 mρ 2 2.37 Without flavors : 1 (1260) ma 2 mρ 2 2.39 Experiment : 1 (1260) mρ 2 2.51

Holographic QCD & chemical potential: results Holographic renormalization Divergent on-shell action S ren E = (S E + S GH ) + S D4 c.t. + S D8 c.t. Standard counter-term for D4-branes [Mateos-Myers-Thomson 2007]: S D4 c.t. = g s 1/3 k0 2R d 9 x h 5 2 e 7 3 φ For D8-branes no existing covariant counter-terms!

Holographic QCD & chemical potential: results Holographic renormalization Strategy: Go to dual frame: d s 2 e 2 3 φ ds 2 [Kanitscheider-Skenderis-Taylor 2008]. Reduce on S 4 metric is asymptotically AdS. Use standard AdS counter-terms: volume + GH (written in 8d) S D8 c.t. d 8 x h 8 e 2φ [ 1 2α K ] 9 Bring back to original frame (probe case): S D8 c.t. = 2N f T 8 d 8 x h 8 [ 16 7 R g 1/3 s (In backreacted case numeric coeffs. are different). e 2φ/3 R2 g 2/3 s e φ/3 ( K 9 8 3 n φ ) ] Features: covariant (good!), non local (bad?).

Holographic QCD & chemical potential: results Free energy density From renormalized on-shell action: free energy density [ ] f = p = 2N2 c λ 4 3 7 π 2 M4 KK 1 4 λ 2 4 N f π 3/2 7 π 3 N c Γ ( ( 3) 2 Γ 1 ) 6 T -independent zero entropy (O(1): confining phase).

Holographic QCD & chemical potential: results Metric: Deconfined phase (T > T c ) ds 2 = e 2 λdt 2 +e 2λ dx a dx a +e 2λs dx 2 4 +l 2 s e 4φ+6λ+2 λ+2λ s +8ν dρ 2 +... Solution for brane gauge field complicated small charge ( q ) expansion simple brane gauge field A t q(1 1 e 3r ). Fields expanded in ɛ f T = λ2 4 12π 3 2πT M KK N f N c 1 and q 2 1. Another (horrible but) analytic solution (let s skip it). Features: Horizon in IR. Landau pole in UV.

Holographic QCD & chemical potential: results Holographic renormalization Divergent on-shell action S ren E = (S E + S GH ) + S D4 c.t. + S D8 c.t. Sub-leading divergence from D8-branes does not match with T < T c one cannot use background subtraction! Have to add the counter-term (probe case): S D8 c.t. = 2N f T 8 d 8 x h 8 [ 2 7 R g 1/3 s e 2φ/3 2 7 R 2 g 2/3 s e φ/3 ( K 9 8 3 n φ ) ] (In backreacted case numeric coeffs. are different).

Holographic QCD & chemical potential: results From asymptotics of A t Thermodynamics T 2 chemical potential µ = 8π 27 qλ 4 Note : q T 2 (fixed µ) M KK From electric displacement δl δf t ˆρ n q = 32π 729 qn cn f λ 2 4 T 5 M 2 KK From area of horizon entropy density s = 256N2 c π 4 λ 4 729MKK 2 T [1 5 + 2 3 ɛ f T quark number density Note : q T 5 (fixed n q ) (1 + q2 2 From ADM energy energy density ε = 5 ( 256N 2 c π 4 ) [ λ 4 6 729MKK 2 T 6 1 + 24 35 ɛ f T (1 + 79 )] q2 )]

Holographic QCD & chemical potential: results Thermodynamics From on-shell action Gibbs free energy density (grand canonical ensemble) ω = p = 1 ( 256N 2 c π 4 ) [ λ 4 6 729MKK 2 T 6 1 + 4 7 ɛ f T (1 + 76 )] q2 From ε = Ts + f Helmholtz free energy density (canonical ensemble) f = 1 ( 256N 2 c π 4 ) [ λ 4 6 729MKK 2 T 6 1 + 4 7 ɛ f T (1 76 )] q2 From ( ) ε T heat capacity at fixed quark density V,n ( 256N 2 c V,n = 5 c π 4 ) [ λ 4 729MKK 2 T 5 1 + 4 5 ɛ f T (1 13 )] q2

Holographic QCD & chemical potential: results From ( ) ε T V,µ Thermodynamics heat capacity at fixed chemical potential ( 256N 2 c V,µ = 5 c π 4 ) [ λ 4 729MKK 2 T 5 1 + 4 5 ɛ f T (1 + 13 )] q2 From s c V,µ speed of sound (squared) c 2 s = 1 5 [ 1 2 15 ɛ f T (1 12 )] q2 From ε 3p interaction energy IE = 1 ( 256N 2 c π 4 ) [ λ 4 3 729MKK 2 T 6 1 + 6 7 ɛ f T (1 + 718 )] q2

Holographic QCD & chemical potential: results Thermodynamics Consistency checks: s = ( f / T ) (considering T -dependence of ɛ f T and q in canonical ensemble) s = ( ω/ T ) (considering T -dependence of ɛ f T and q in grand canonical ensemble) ω = f µn q Ok with probe approximation.

Holographic QCD & chemical potential: results The critical temperature From p conf (T c ) = p deconf (T c ): 2πT c = 1 1 N f M KK 126π 3 λ2 4 N c 1 + 12π 3/2 ) ( ) Γ 16 ( Γ 2 3 27 16π 2 N f µ N c M KK 2 Note: N f -behavior is comparison scheme dependent. At imaginary baryon chemical potential µ B (θ B = µ B /T c (0)): T c T c (θ B = 0) = 1 + 27 64π 3 N f N c θ 2 B N 2 c Also in [Rafferty 2011], where it is shown that free energy has first order discontinuities at θ B = (2k + 1)π.

Holographic QCD & chemical potential: results Phase diagram: Qualitatively the same as lattice QCD

Contents: The Roberge-Weiss phase diagram. Holographic QCD & chemical potential: context. Holographic QCD & chemical potential: results. Conclusions and perspectives.

Conclusions and perspectives Done: Derived gravity solutions dual to Witten-Sakai-Sugimoto model with dynamical flavors, at first order in N f /N c and q 2, in confined and deconfined phases. Studied a few observables (e.g. string tension, hadron masses, etc.). Analyzed phase diagram at finite imaginary chemical potential ([Rafferty 2011]). Phase diagram of holographic model observed to be in qualitative agreement with lattice QCD. Introduced covariant (non-local) counterterms for D8-branes.

Conclusions and perspectives To do (among others): Fields/operators dictionary. Holographic renormalization. Study of other observables (e.g. probe energy loss, entanglement entropy). Go beyond leading orders in ɛ f, q 2.