QCD at finite temperature and density from holography

Transcription

1 Vienna, 06/17/10 1 Andreas Schmitt Institut für Theoretische Physik Technische Universität Wien 1040 Vienna, Austria QCD at finite temperature and density from holography The Sakai-Sugimoto model T-µ phase diagram (and comparison to large-n c QCD) Effects of magnetic fields: chiral spirals, chiral magnetic effect A. Rebhan, A. Schmitt, S.A. Stricker, JHEP 0905, 084 (2009); JHEP 1001, 026 (2010)

2 Vienna, 06/17/10 2 The gauge/gravity duality (page 1/2): basic idea J. M. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998) For a pedagogical review, see S. S. Gubser, A. Karch, Ann. Rev. Nucl. Part. Sci. 59, 145 (2009) string theory (in higher dimensions) gauge theory (on boundary) original AdS/CFT correspondence : string theory on AdS 5 S 5 N = 4 SU(N c ) SYM theory on R 3,1 l s R supergravity limit (easy!) R 4 l 4 s = g 2 YM N c λ R curvature radius; l s string length λ 1 strong coupling limit (difficult!)

3 Vienna, 06/17/10 3 The gauge/gravity duality (page 2/2): overview QCD compare with N = 4 SYM compute plasma properties ( heavy-ion collisions) viscosity G. Policastro, D. T. Son, A. O. Starinets, PRL 87, (2001) jet quenching H. Liu, K. Rajagopal, U. A. Wiedemann, PRL 97, (2006) expanding plasma R. A. Janik, R. B. Peschanski, PRD 73, (2006) find gravity dual of QCD add flavor to AdS/CFT A. Karch, E. Katz, JHEP 0206, 043 (2002) Sakai-Sugimoto model T. Sakai, S. Sugimoto, Prog. Theor. Phys. 113, 843 (2005) bottom-up approach Erlich, Katz, Son, Stephanov, PRL 95, (2005) Other applications: cold atoms (unitary Fermi gas), D. T. Son, PRD 78, (2008); K. Balasubramanian, J. McGreevy, PRL 101, (2008) (high-t c ) superconductivity, etc... S. A. Hartnoll, C. P. Herzog, G. T. Horowitz, PRL 101, (2008)

4 Vienna, 06/17/10 4 The Sakai-Sugimoto model in two steps 1. Background geometry with D4-branes E. Witten, Adv. Theor. Math. Phys. 2, 505 (1998) M. Kruczenski, D. Mateos, R. C. Myers, D. J. Winters, JHEP 0405, 041 (2004) 2. Add flavor D8-branes T. Sakai, S. Sugimoto, Prog. Theor. Phys. 113, 843 (2005)

5 Vienna, 06/17/10 5 The Sakai-Sugimoto model: background geometry (page 1/3) E. Witten, Adv. Theor. Math. Phys. 2, 505 (1998) M. Kruczenski, D. Mateos, R. C. Myers, D. J. Winters, JHEP 0405, 041 (2004) N c D4-branes compactified on circle x 4 x 4 + 2π/M KK D4 branes Nc 4-4 strings adjoint scalars & fermions, gauge fields periodic x 4 break SUSY by giving mass M KK to scalars & fermions SU(N c ) gauge theory λ = g2 5 N c 2π/M KK dual to large-n c QCD (at energies M KK ) supergravity works λ 1 λ 1 x x

6 Vienna, 06/17/10 6 Background geometry (page 2/3): two solutions Confined phase Deconfined phase ds 2 conf = ( u ) 3/2 [dτ 2 + dx 2 + f(u)dx 2 R 4] ( ) 3/2 [ ] R du 2 + u f(u) + u2 dω 2 4 ds 2 deconf = ( u ) 3/2 [ f(u)dτ 2 + δ ij dx 2 + dx 2 4 R ] ( ) 3/2 [ ] R du 2 + u f(u) + u2 dω 2 4 τ 1/(2 πt) x 4 1/M KK τ 1/(2 πt) x 4 1/M KK u= 8 u u= 8 u M KK = 3 2 u 1/2 KK R 3/2 u= u KK f(u) 1 u3 KK u 3 Wick rotated regular geometry T = 3 u 1/2 T 4π R 3/2 u= u T f(u) 1 u 3 T u 3 Wick rotated black brane

7 Vienna, 06/17/10 7 Background geometry (page 3/3): deconfinement phase transition temperature τ x 4 τ x 4 τ x 4 confined deconfined T c = M KK 2π

8 Vienna, 06/17/10 8 Add flavor (page 1/2) T. Sakai, S. Sugimoto, Prog. Theor. Phys. 113, 843 (2005) add N f D8- and D8-branes, separated in x D4 x x x x x D8/D8 x x x x x x x x x D4 D8 D8 x 5 9 x , 4-8 strings fundamental, massless chiral fermions under U(N f ) L U(N f ) R L x 4 quarks & gluons

9 Vienna, 06/17/10 9 Add flavor (page 2/2): Chiral symmetry breaking background geometry unchanged if N f N c ( probe branes ) quenched approximation gauge symmetry on the branes global symmetry at u = x 4 x 4 L L u SU(N f) L D8 D8 SU(N ) f R SU(N ) f L+R chiral symmetry breaking SU(N f ) L SU(N f ) R SU(N f ) L+R

10 Vienna, 06/17/10 10 Conjectured QCD phase diagram in T-µ plane real QCD N. Cabibbo, G. Parisi, PLB 59, 67 (1975) large-n c QCD L. McLerran, R. D. Pisarski, NPA 796, 83 (2007) L. McLerran, K. Redlich, C. Sasaki, NPA 824, 86 (2009)

11 Vienna, 06/17/10 11 More (conjectured) details T heavy ion collider hadronic gas nuclear superfluid liq QGP non CFL neutron star CFL µ Physical systems: heavy-ion collisions (intermediate T, small µ) neutron stars (small T, intermediate µ) M. G. Alford, A. Schmitt, K. Rajagopal, T. Schäfer, RMP 80, 1455 (2008) Challenging (strongly coupled!) regions at intermediate T, µ Try large-n c (Sakai-Sugimoto) approach

12 Vienna, 06/17/10 12 T-µ phase diagram (page 1/4) chemical potential introduced through gauge fields on D8 branes confinement chiral symmetry breaking T deconfined χs restored T c confined χ S broken µ M KK = 949 MeV (fit to ρ mass) T c 150 MeV

13 Vienna, 06/17/10 13 T-µ phase diagram (page 2/4) deconfined, chirally broken phase for L < 0.3π/M KK O. Aharony, J. Sonnenschein, S. Yankielowicz, Annals Phys. 322, 1420 (2007) N. Horigome, Y. Tanii, JHEP 0701, 072 (2007) T deconfined χs broken L deconfined χs restored T c confined χ S broken µ

14 Vienna, 06/17/10 14 T-µ phase diagram (page 3/4) NJL limit L π/m KK T deconfined χs broken deconfined χs restored confined χ S broken T c µ differences to QCD limit : crystalline phases vs. chiral spiral? D. Nickel, PRD 80, (2009) T. Kojo, Y. Hidaka, L. McLerran, R. D. Pisarski, arxiv: [hep-ph] F. Preis, A. Rebhan, A. Schmitt, work in progress

15 Vienna, 06/17/10 15 T-µ phase diagram (page 4/4) T deconfined χs restored Holographic nuclear matter: T c 4 brane source n = 0 B n = 0 B baryons introduced as instantons H. Hata, T. Sakai, S. Sugimoto, S. Yamato, Prog. Theor. Phys. 117, 1157 (2007) homogeneous baryon number from 4-branes wrapped on S 4 O. Bergman, G. Lifschytz, M. Lippert, JHEP 0711, 056 (2007) same diagram from large-n c QCD, quarkyonic matter L. McLerran, R. D. Pisarski, Nucl. Phys. A 796, 83 (2007) µ

16 Vienna, 06/17/10 16 Supercurrents in chirally broken phase (page 1/2) A. Rebhan, A. Schmitt, S.A. Stricker, JHEP 0905, 084 (2009) Stress on ψ R ψ L through chemical potential µ anisotropic (or even crystalline) pairing Sakai-Sugimoto: magnetic field B needed to create σ π 0 π 0 σ = µf(b) µ (for large B) Μ solution to the EOMs on D8-branes with A 1 (z = ± ) = x 2 B: gauge fields Π 0 B z u KK

17 Vienna, 06/17/10 16 Supercurrents in chirally broken phase (page 1/2) A. Rebhan, A. Schmitt, S.A. Stricker, JHEP 0905, 084 (2009) Stress on ψ R ψ L through chemical potential µ anisotropic (or even crystalline) pairing Sakai-Sugimoto: magnetic field B needed to create σ π 0 π 0 σ = µf(b) µ (for large B) Μ solution to the EOMs on D8-branes with A 1 (z = ± ) = x 2 B: gauge fields Π 0 B z u KK

18 Vienna, 06/17/10 16 Supercurrents in chirally broken phase (page 1/2) A. Rebhan, A. Schmitt, S.A. Stricker, JHEP 0905, 084 (2009) Stress on ψ R ψ L through chemical potential µ anisotropic (or even crystalline) pairing Sakai-Sugimoto: magnetic field B needed to create σ π 0 π 0 σ = µf(b) µ (for large B) Μ solution to the EOMs on D8-branes with A 1 (z = ± ) = x 2 B: gauge fields Π 0 B z u KK

19 Vienna, 06/17/10 16 Supercurrents in chirally broken phase (page 1/2) A. Rebhan, A. Schmitt, S.A. Stricker, JHEP 0905, 084 (2009) Stress on ψ R ψ L through chemical potential µ anisotropic (or even crystalline) pairing Sakai-Sugimoto: magnetic field B needed to create σ π 0 π 0 σ = µf(b) µ (for large B) Μ solution to the EOMs on D8-branes with A 1 (z = ± ) = x 2 B: gauge fields Π 0 B z u KK

20 Vienna, 06/17/10 17 Supercurrents in chirally broken phase (page 2/2) usual U(1) superfluid (e.g. 4 He): superflow via φ(x) = φ 0 e iqz superflow in z direction with velocity q (rotation in U(1) space) Sakai-Sugimoto: σ π 0 π 0 σ µ (axial) supercurrent (rotation in chiral space) chiral spiral σ iπ 0 e 2iµz (rotation in chiral space) G. Basar, G. V. Dunne, M. Thies, PRD 79, (2009) quarkyonic chiral spiral (rotation in spin space) T. Kojo, Y. Hidaka, L. McLerran, R. D. Pisarski, arxiv: [hep-ph]

21 Vienna, 06/17/10 18 The Chiral Magnetic Effect (CME) spin momentum B q R q R q L electric current parallel to B J = e2 N c 2π 2 µ 5B A.Y. Alekseev, V.V. Cheianov, J. Fröhlich, PRL 81, 3503 (1998) K.Fukushima, D.E.Kharzeev, H.J.Warringa, PRD 78, (2008) analogously for axial current: M.A. Metlitski, A.R. Zhitnitsky, PRD 72, (2005) J 5 = e2 N c 2π 2 µb

22 Vienna, 06/17/10 19 CME in heavy-ion collisions + B simplified description with µ 5 : take snapshot with conserved N 5 event-by-event charge separation B. I. Abelev et al. [STAR Collaboration], PRL 103, (2009) non-central collisions: eb G V. Skokov et al., Int. J. Mod. Phys. A24, 5925 (2009) N 5 from QCD axial anomaly dn 5 = g2 dt 16π 2 d 3 xfµν a (large T: sphalerons) F µν a

23 Vienna, 06/17/10 20 CME in the Sakai-Sugimoto model A. Rebhan, A. Schmitt, S.A. Stricker, JHEP 1001, 026 (2010) maximal separation L = π/m KK one flavor N f = 1 holographic coordinate u z [, ] (broken) z [0, ] (symmetric) gauge choice A z = 0 S YM = κm 2 KK S CS = N c 24π 2 d 4 x d 4 x dz [ k(z)f zµ F zµ + h(z) ] 2MKK 2 F µν F µν dz A µ F zν F ρσ ǫ µνρσ S = S YM + S CS k(z) 1 + z 2 κ λn c 216π 3 equations of motion δ(l YM + L CS ) δa µ = 0 boundary conditions A 0 (z = ± ) = µ L/R A 1 (x 2,z = ± ) = x 2 B

24 Vienna, 06/17/10 21 Chiral currents YM and CS contributions to chiral currents H. Hata, M. Murata, S. Yamato, PRD 78, (2008) K. Hashimoto, T. Sakai, S. Sugimoto, Prog. Theor. Phys. 120, 1093 (2008) S = S YM + S CS J µ L/R δs δa µ (x,z = ± ) = J µ L/R,YM + J µ L/R,CS sometimes CS contribution is ignored H. U. Yee, JHEP 0911, 085 (2009) D. T. Son, P. Surowka, PRL 103, (2009) only YM part in asymptotics of A µ : A µ (x,z) = A µ (x,z = ± ) ± J µ L/R,YM 2κMKK 2 1 z + O ( ) 1 z 2.

25 Vienna, 06/17/10 22 Anomalies axial and vector anomalies [F L/R µν (x) F µν (x, z = ± )] µ J µ 5 = N c 24π 2 ( F V µν F µν V + FA µν ) µν F A µ J µ = N c 12π 2 FV µν F µν A consistent anomaly W.A. Bardeen, Phys. Rev. 184, 1848 (1969); C.T. Hill, PRD 73, (2006) need Bardeen s counterterm S = c d 4 x(a L µa R ν Fρσ L + A L µa R ν Fρσ)ǫ R µνρσ (here interpreted as holographic renormalization) determine c to get QED ( covariant ) anomaly

26 Vienna, 06/17/10 23 Correct anomaly with Bardeen s counterterm Bardeen s counterterm S new chiral currents J µ L/R J µ L/R,YM + J µ L/R,CS + J µ L/R µ J µ 5 = N c 8π 2 FV µν µ J µ = 0 F µν V + N c 24π 2 FA µν F µν A covariant anomaly J.S. Bell, R.Jackiw, Nuovo Cim. A60, 47 (1969); S.L. Adler, Phys. Rev. 177, 2426 (1969) conservation of vector current & correct decay rate π 0 2γ

27 Vienna, 06/17/10 24 Use YM currents? notice: YM part alone gives µ J µ YM,5 = N ( c 8π 2 Fµν V F µν V + FA µν ) µν F A µ J µ YM = N c 4π 2FV µν F µν A seems OK for F A = 0 J µ YM J µ even if F A = 0 B +++ µ 5 = 0 J µ YM not strictly conserved (need µ 5 for charge separation at RHIC!) µ 5 µ 5 = 0 = 0

28 Vienna, 06/17/10 25 Results for currents A. Rebhan, A. Schmitt, S.A. Stricker, JHEP 1001, 026 (2010) absence of CME axial current as expected

29 Vienna, 06/17/10 26 Possible problems in current approach Need to consider conserved N 5? V. A. Rubakov, arxiv: [hep-ph] A.Y. Alekseev, V.V. Cheianov, J. Fröhlich, PRL 81, 3503 (1998) Careful distinction of chemical potential and source for currents A. Gynther, K. Landsteiner, F. Pena-Benitez, A. Rebhan, arxiv: [hep-th] Need to work in canonical ensemble? H.U. Yee, Talk at BNL Workshop, April 26-30, 2010 A. Rebhan, A. Schmitt, S.A. Stricker, work in progress Or take result seriously... strong-coupling vector current differs from weak-coupling current (unlike axial current) similar to NJL-like models E. V. Gorbar, V. A. Miransky, I. A. Shovkovy, PRC 80, (2009) K. Fukushima, M. Ruggieri, arxiv: [hep-ph]

30 Vienna, 06/17/10 27 Summary the Sakai-Sugimoto model provides a top-down gauge/gravity duality for a non-supersymmetric, strongly-coupled SU(N c ) gauge theory with confinement and chiral symmetry breaking ( large-n c QCD) for finite magnetic field and chemical potential, the model exhibits supercurrents ( chiral spirals?) and finite baryon number in the chirally broken phase the chiral magnetic effect can be computed via introducing an axial chemical potential