Holographic QCD at finite (imaginary) chemical potential

Transcription

1 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

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

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

4 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.

5 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

6 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.

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

8 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 ).

9 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 + dx 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.

10 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 ) π 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.

11 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]

12 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]

13 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]

14 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:

15 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!

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

17 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).

18 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

19 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 = 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 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 F 1, 2 1 ; 3 2 ; e 3r 15r/2 e 38e 3r + 8e 6r ( 1 e 3r ) ( 1 e 3r ) 13/6

20 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 ɛ f 2 5/6 (1 x) 1/6 Physics reliable at x x LP 1. ] x LP = (3/7) 6 ɛ 6 f

21 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 [ ɛ f ] From wrapped D4: baryon (vertex) mass m B = 1 27π λ 4N c M KK [ ɛ 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 Without flavors : mρ Experiment : mρ ma 2 1 (1260) ma 2 mρ Without flavors : 1 (1260) ma 2 mρ Experiment : 1 (1260) mρ

22 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!

23 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 n φ ) ] Features: covariant (good!), non local (bad?).

24 Holographic QCD & chemical potential: results Free energy density From renormalized on-shell action: free energy density [ ] f = p = 2N2 c λ 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).

25 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 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.

26 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 n φ ) ] (In backreacted case numeric coeffs. are different).

27 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 [ 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 MKK 2 T ɛ f T ( )] q2 )]

28 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 MKK 2 T ɛ f T ( )] q2 From ε = Ts + f Helmholtz free energy density (canonical ensemble) f = 1 ( 256N 2 c π 4 ) [ λ MKK 2 T 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 ɛ f T (1 13 )] q2

29 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 ɛ f T ( )] q2 From s c V,µ speed of sound (squared) c 2 s = 1 5 [ ɛ f T (1 12 )] q2 From ε 3p interaction energy IE = 1 ( 256N 2 c π 4 ) [ λ MKK 2 T ɛ f T ( )] q2

30 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.

31 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 π 3/2 ) ( ) Γ 16 ( Γ π 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) = π 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.

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

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

34 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.

35 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.