Inverse magnetic catalysis in dense matter

Trento, 11/14/12 1 Andreas Schmitt Institut für Theoretische Physik Technische Universität Wien 1040 Vienna, Austria Inverse magnetic catalysis in dense matter Outline: F. Preis, A. Rebhan, A. Schmitt JHEP 1103, 033 (2011); JPG 39, 054006 (2012) (holographic approach) arxiv:1208.0536 [hep-ph] (comparison with field theory) arxiv:1209.4468 [hep-ph] (short summary) Magnetic catalysis Sakai-Sugimoto model Inverse magnetic catalysis Including baryonic matter

Trento, 11/14/12 2 Magnetic catalysis (page 1/2) K. G. Klimenko, Theor. Math. Phys. 89, 1161-1168 (1992) V. P. Gusynin, V. A. Miransky, I. A. Shovkovy, PLB 349, 477-483 (1995) (massless) fermions in Nambu-Jona-Lasinio (NJL) model L NJL = ψ iγ µ µ ψ + G [( ψψ) 2 + ( ψ iγ 5 ψ) 2] mean-field approximation: ψψ = ψψ + fluctuations Zero magnetic field: dynamical fermion mass M ψψ 0 for coupling g > g c = 1 dimensionless coupling g GΛ 2 /π 2

Trento, 11/14/12 2 Magnetic catalysis (page 1/2) K. G. Klimenko, Theor. Math. Phys. 89, 1161-1168 (1992) V. P. Gusynin, V. A. Miransky, I. A. Shovkovy, PLB 349, 477-483 (1995) (massless) fermions in Nambu-Jona-Lasinio (NJL) model L NJL = ψ iγ µ µ ψ + G [( ψψ) 2 + ( ψ iγ 5 ψ) 2] mean-field approximation: ψψ = ψψ + fluctuations Nonzero magnetic field: M 0 for arbitrarily small g, M eb e const./ebg at weak coupling g 1 dimensionless coupling g GΛ 2 /π 2

Trento, 11/14/12 3 Magnetic catalysis (page 2/2) Analogy to BCS Cooper pairing: BCS superconductor Magnetic catalysis Cooper pair condensate ψψ chiral condensate ψψ µ e const./gν F M eb e const./gν 0 (ν F : d.o.s. at E = µ Fermi surface) (ν 0 : d.o.s. at E = 0 surface) pairing dynamics effectively (1+1)-dimensional because of Fermi surface effectively (1+1)-dimensional in lowest Landau level (LLL) because of magn. field = µ2 G 2π 2 gap equation 0 dk (k µ) 2 + 2 M = q BG 2π 2 gap equation (LLL) dk z M k 2 z + M 2

Trento, 11/14/12 4 Magnetic catalysis in the real world and in holography 2 Experiment 0.8 M q 1 0.7 Σ xy e 2 h 0 1 0.6 0.5 2 2 1 Theory T 0.24 0.22 0.5 1 1.5 2 ΧS Restored H Σ xy e 2 h 0 1 2 b 0.04, T 30 mk Γ 18 K, Γ tr 6 K 20 10 0 10 20 30 V g V V.P.Gusynin et al., PRB 74, 195429 (2006) graphene: appearance of additional plateaus in strong magnetic fields [B = 9 T (pink), B = 45 T (black)] 0.20 0.18 0.16 ΧSB 0 2 4 6 8 10 12 H C.V.Johnson, A.Kundu, JHEP 0812, 053 (2008) Sakai-Sugimoto: magnetic field enhances dynamical mass M q and critical temperature T c see also: J. Erdmenger et al., JHEP 0712, 091 (2007) V.G.Filev, R.C.Rashkov, AHEP 473206 (2010)

Trento, 11/14/12 5 Sakai-Sugimoto model: background geometry (p. 1/2) 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 λ 1 λ 1 dual to large-n c QCD x (at energies M KK ) Λ QCD M KK Λ QCD M KK gravity approximation x

Trento, 11/14/12 6 Background geometry (page 2/2): two solutions Confined phase Deconfined phase ds 2 conf = ( u ) 3/2 [dτ 2 + dx 2 + R f(u)dx 2 4] ( ) 3/2 [ ] R du 2 + u f(u) + u2 dω 2 4 ds 2 deconf = ( u ) 3/2 [f(u)dτ 2 + dx 2 + dx 2 R 4] ( ) 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 u 3 KK u 3 Wick rotated regular geometry T = 3 u 1/2 T 4π R 3/2 u= u T f(u) 1 u3 T u 3 Wick rotated black brane

Trento, 11/14/12 7 Chiral transition in the Sakai-Sugimoto model (p. 1/3) T τ x 4 deconfined u L χs restored T c D8 D8 confined χs broken µ not unlike expectation from large-n c QCD in probe brane approximation: chiral transition unaffected by quantities on flavor branes (µ, B,...)

Trento, 11/14/12 8 Chiral transition in the Sakai-Sugimoto model (p. 2/3) less rigid behavior for smaller L deconfined, chirally broken phase for L < 0.3 π/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

Trento, 11/14/12 9 Chiral transition in the Sakai-Sugimoto model (p. 3/3) L π/m KK corresponds to (non-local) NJL model E. Antonyan, J. A. Harvey, S. Jensen, D. Kutasov, hep-th/0604017 J. L. Davis, M. Gutperle, P. Kraus, I. Sachs, JHEP 0710, 049 (2007) T deconfined χs broken deconfined χs restored T NJL chiral transition (no confinement) µ confined χ S broken T c decompactified limit gluon dynamics decouple this limit is considered in the following calculation... µ

Trento, 11/14/12 10 Setup of our calculation D8-brane action in deconfined geometry for N f = 1 O. Bergman, G. Lifschytz, M. Lippert, PRD 79, 105024 (2009) S = S DBI + S CS = N du u 5 + b 2 u 2 1 + fa 2 3 a 2 0 + u3 fx 2 4 + 3N u T 2 b du (a 3 a 0 a 0 a 3) u T a 0 accounts for µ magnetic field b = F 12 b, µ induce a 3 ( anisotropic condensate π 0 ) E.G.Thompson, D.T.Son, PRD 78, 066007 (2008) A. Rebhan et al., JHEP 0905, 084 (2009) x 4 (u) = { const. χs nontrivial χsb equations of motion: ( ) a 0 u5 + b 2 u 2 u 1 + fa 2 3 a 2 0 + u3 fx 2 4 ( ) f a 3 u5 + b 2 u 2 u 1 + fa 2 3 a 2 0 + u3 fx 2 4 ( ) u 3 f x 4 u5 + b 2 u 2 u 1 + fa 2 3 a 2 0 + u3 fx 2 4 = 3ba 3 = 3ba 0 = 0 solve for a 0 (u), a 3 (u), x 4 (u)

Trento, 11/14/12 11 T = 0 phase diagram 0.4 0.3 b 0.2 "LLL" mesonic 0.1 "hll" 0.0 0.0 0.2 0.4 0.6 0.8 Apparent Landau level transition G. Lifschytz, M. Lippert, PRD 80, 066007 (2009): Μ 0.06 n 2ΠΑ R 0.15 0.10 0.05 t 0.1 t 0.05 t 0 Μ 0.5 0.00 0.00 0.02 0.04 0.06 0.08 0.10 b nfree NcΜq 3 0.05 0.04 0.03 T 0.03Μ q 0.02 0.01 free fermions T 0.2Μ q T 0 T 0.1Μ q 0.00 0.0 0.5 1.0 1.5 2.0 2B Μ q 2

Trento, 11/14/12 12 Inverse magnetic catalysis Why does B restore chiral symmetry for certain µ? free energy gain from ψ ψ pairing increases with B (magnetic catalysis) µ induces free energy cost for pairing; this cost depends on B! NJL (weak coupling): E. V. Gorbar et al., PRC 80, 032801 (2009) Ω B[µ 2 M(B) 2 /2] Sakai-Sugimoto: large B: Ω B[µ 2 0.12 M(B) 2 ] no inverse catalysis just like Clogston limit δµ = 2 in superconductivity A. Clogston, PRL 9, 266 (1962) B. Chandrasekhar, APL 1, 7 (1962) no inverse catalysis small B: Ω µ 2 B const M(B) 7/2 inverse catalysis possible

Trento, 11/14/12 13 Comparison with NJL calculation (T = 0) F. Preis, A. Rebhan and A. Schmitt, arxiv:1208.0536 [hep-ph] 0.4 Sakai-Sugimoto NJL (strong coupling) T 0, g 4.1061 0.3 0.8 b 0.2 mesonic "LLL" qb ² 0.6 0.4 ΧSb ΧS LLL 0.1 "hll" 0.0 0.0 0.2 0.4 0.6 0.8 Μ 0.2 higher LL 0.0 0.4 0.5 0.6 0.7 0.8 NJL at large g: inverse magnetic catalysis like in Sakai-Sugimoto! inverse magnetic catalysis in NJL and related models: D. Ebert, K. G. Klimenko, M. A. Vdovichenko and A. S. Vshivtsev, PRD 61, 025005 (2000) T. Inagaki, D. Kimura and T. Murata, Prog. Theor. Phys. 111, 371 (2004) B. Chatterjee, H. Mishra and A. Mishra, PRD 84, 014016 (2011) S. S. Avancini, D. P. Menezes, M. B. Pinto and C. Providencia, PRD 85, 091901 (2012) J. O. Andersen and A. Tranberg, JHEP 1208, 002 (2012)

Trento, 11/14/12 14 Physical units original version of Sakai-Sugimoto model (L = π M KK ): choose M KK 949 MeV and κ λn c 216π 3 0.007 to fit m ρ and f π T. Sakai and S. Sugimoto, Prog. Theor. Phys. 114, 1083 (2005) ( deconfinement temperature T c 150 MeV) here (non-asymptotic separation L π M KK ): µ q = R3 µl 2 2πα L, T = tl 2 L, B = R3 bl 3 (l = L 2πα L 3 R ) ( B 5.1 10 19 µq,c ) ( ) T c G bl 3 400 MeV 150 MeV inverse magnetic catalysis reduces critical µ q from 400 MeV (B = 0) to 230 MeV (B 1.0 10 19 G)

Trento, 11/14/12 15 Phase structure at nonzero temperature blue: chiral phase transition green: LLL transition b 0.10 0.08 0.06 0.04 0.02 ΧSb t 0.13 t 0.1 t 0.05 t 0 ΧS 0.00 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Μ t 0.20 0.15 0.10 0.05 ΧS Μ 0.2 Μ 0.25 Μ 0.3 Μ 0 ΧSb 0.00 0.0 0.2 0.4 0.6 0.8 1.0 1.2 b t 0.20 0.15 0.10 0.05 b 5 ΧSb b 0.12 b 0 ΧS 0.00 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Μ

Trento, 11/14/12 16 Comparison with NJL calculation (T 0) Sakai-Sugimoto: F. Preis, A. Rebhan and A. Schmitt, JHEP 1103, 033 (2011) b 0.10 0.08 0.06 0.04 0.02 ΧSb t 0.13 t 0.1 t 0.05 t 0 ΧS 0.00 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Μ t 0.20 0.15 0.10 0.05 ΧS Μ 0.2 Μ 0.25 Μ 0.3 Μ 0 ΧSb 0.00 0.0 0.2 0.4 0.6 0.8 1.0 1.2 NJL: b t 0.20 0.15 0.10 0.05 b 5 ΧSb b 0.12 b 0 ΧS 0.00 0.0 0.1 0.2 0.3 0.4 0.5 0.6 T. Inagaki, D. Kimura, T. Murata, Prog. Theor. Phys. 111, 371-386 (2004) Μ m [GeV] 0.35 0.3 0.25 0.2 T = 0.01 T = 0.05 T = 0.03 Symmetric phase T = 0.10 Broken phase H [GeV 2 ] T = 0.15GeV 0 0.2 0.4 0.6 0.8 1 2 H [GeV ] 1 0.5 B. P. B. P. m = 0.34 m = 0.28 m = 0.31 m = 0.31 m = 0.34 m = 0.27 m = 0.24GeV Symmetric phase 0 0 0.05 0.1 0.15 0.2 T [GeV] T [GeV] 0.2 0.1 Broken phase Symmetric phase H = 0.60GeV H = 0.30 H = 0.21 0 0 0.1 0.2 0.3 0.4 m [GeV] 2 H = 0.0 H = 0.18

Trento, 11/14/12 17 Homogeneous baryonic matter in Sakai-Sugimoto baryons in AdS/CFT: wrapped D-branes with N c strings E. Witten, JHEP 9807, 006 (1998); D. J. Gross, H. Ooguri, PRD 58, 106002 (1998) baryons in Sakai-Sugimoto: D4-branes wrapped on S 4 equivalently: instantons on D8-branes ( skyrmions) T. Sakai, S. Sugimoto, Prog. Theor. Phys. 113, 843-882 (2005) H. Hata, T. Sakai, S. Sugimoto, S. Yamato, Prog. Theor. Phys. 117, 1157 (2007) pointlike approximation for N f = 1: O. Bergman, G. Lifschytz, M. Lippert, JHEP 0711, 056 (2007) S = S from above + N 4 T 4 dω 4 dτ e Φ det g + N c A }{{} 8π 2 0 Tr F 2 } R 4 U {{} n 4 N c M q n 4 A 0 (u)δ(u u c ) (n 4 baryon density, M q constituent quark mass, u c location of D4-branes)

Trento, 11/14/12 18 Effect of baryonic matter on T = 0 phase diagram ignoring baryonic matter including baryonic matter 0.4 0.4 b 0.3 0.2 "LLL" b 0.3 0.2 mesonic "LLL" mesonic 0.1 "hll" 0.0 0.0 0.2 0.4 0.6 0.8 Μ small b: baryonic matter prevents the system from restoring chiral symmetry 0.1 baryonic 0.0 0.0 0.2 0.4 0.6 0.8 baryon onset line intersects chiral phase transition line large b: mesonic matter superseded by quark matter with baryonic matter, IMC plays an even more prominent role in the phase diagram Μ

Trento, 11/14/12 19 Summary Dense matter in the Sakai-Sugimoto model...... shows a transition reminiscent of LLL... shows a chiral phase transition with MC at large B and IMC at small B... behaves qualitatively like in the NJL model IMC: if magnetic fields in magnetars sufficiently large, then transition to quark matter at lower densities Baryonic matter changes the phase diagram dramatically for small B

Trento, 11/14/12 20 Outlook Extensions/questions within the holographic model: Saturation of T c for large B at µ = 0 (Why?) Extend to N f = 2 More general: problem of large N c vs. N c = 3! How is IMC affected by the chiral shift? E. V. Gorbar, V. A. Miransky and I. A. Shovkovy, PRC 80, 032801 (2009) Are QCD lattice results related to IMC? G. S. Bali et al., JHEP 1202, 044 (2012) see also: K. Fukushima, Y. Hidaka, arxiv:1209.1319