Some insights into the magnetic QCD phase diagram from the Sakai-Sugimoto model

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1 Some insights into the magnetic QCD phase diagram from the Sakai-Sugimoto model Department of Physics and Astronomy Ghent University, Belgium Twelfth Workshop on Non-Perturbative Quantum Chromodynamics, l Institut d Astrophysique de Paris, June 10-13, 2013 In collaboration with: Nele Callebaut (Ghent/Brussels)

2 Overview 1 Motivation 2 (Magnetic) holographic setup 3 The ρ meson mass in a magnetic field 4 Chiral transition in a magnetic field

3 Overview 1 Motivation 2 (Magnetic) holographic setup 3 The ρ meson mass in a magnetic field 4 Chiral transition in a magnetic field

4 Why study strong magnetic fields? experimental relevance: appearance in QGP after a heavy ion collision (order eb 1 15mπ 2 ) (work of Skokov, Tuchin, Kharzeev, McLerran, Deng, Huang...) lifetime constantb 10 fm McLerran, Skokov, arxiv: ; Tuchin, arxiv: lifetime QGP 1 10 fm Incentive to take eb constant (ignoring spatial decay as well!)) from a holographic viewpoint: interesting for comparison with recent lattice efforts

5 Why study strong magnetic fields? Figure : Tuchin, arxiv: Figure : McLerran, Skokov, arxiv:

6 Studied effects split between T c (eb) and T χ (eb)? ρ meson condensation?

7 ρ meson condensation Studied effect: ρ meson condensation in vacuum (T = 0) (see: Chernodub, Van Doorsselaere, Verschelde; PRD85 (2012) ; Chernodub, PRL106 (2011) , first suggestion made in Schramm, Muller, Schramm, MPLA7 (1992) 973, inspiration from Ambjorn, Olesen on W -condensation (80ies) ) QCD vacuum instable towards forming a superconducting state of condensed charged ρ mesons at critical magnetic field eb c Small note: academic exercise ( hubris ) since by the time ρ enters, B might have already (long) left...

8 ρ meson condensation: Landau levels The energy levels ε of a free relativistic spin-s particle moving in a background of the external magnetic field B = B ez are the Landau levels Landau levels ε 2 n,s z (p z ) = p 2 z + m 2 + (2n 2s z + 1) eb. Appropriate polarization combinations (spin s z = 1 parallel to B) can condense, since in the lowest energy state (n = 0, p z = 0): M 2 ρ(eb) = m 2 ρ eb, tachyonic if the magnetic field is strong enough. Important: gyromagnetic ratio = 2!

9 ρ meson condensation: Landau levels Mρ(eB) 2 = mρ 2 eb, = The fields ρ and ρ should condense at the critical magnetic field eb c = mρ. 2 M 2 Ρ B GeV^ Landau

10 ρ meson condensation phenomenological models: eb c = m 2 ρ = 0.6 GeV 2 (effective DSGS ρ-model,chernodub, PRD82 (2010) ), eb c 1 GeV 2 (NJL) Chernodub, PRL106 (2011) lattice simulation: eb c 0.9 GeV 2 Braguta et al, PLB718 (2012) 667 holographic approach: can the ρ meson condensation be modeled? can this approach deliver new insights? e.g. taking into account strong magnetic effects on constituents ( ρ-substructure), effect on eb c Callebaut, Dudal, Verschelde, JHEP 1303 (2013) 033; work in progress

11 Overview 1 Motivation 2 (Magnetic) holographic setup 3 The ρ meson mass in a magnetic field 4 Chiral transition in a magnetic field

12 The Sakai-Sugimoto model Holographic QCD What is holographic QCD? QCD dual = (super)gravitation in a higher-dimensional background: 4D QCD lives on the boundary of a 5D space where the (super)gravitation theory is defined Origin of the QCD/gravitation duality idea? AdS/CFT-correspondence (Maldacena 1997): supergravitation in AdS 5 space dual = conformal N =4 SYM theory

13 The Sakai-Sugimoto model The D4-brane background ds 2 = ( u ) ( ) 3/2 3/2 ( ) (ηµν dx µ dx ν R du + f(u)dτ 2 2 ) + R u f(u) + u2 dω 2 4, e φ = g s ( u R ) 3/4, F 4 = N c V 4 ε 4, f(u) = 1 u3 K u 3,

14 The Sakai-Sugimoto model The Sakai-Sugimoto model ds 2 = ( u ) ( ) 3/2 3/2 ( ) (ηµν dx µ dx ν R du + f(u)dτ 2 2 ) + R u f(u) + u2 dω 2 4, e φ = g s ( u R ) 3/4, F 4 = N c V 4 ε 4, f(u) = 1 u3 K u 3,

15 The Sakai-Sugimoto model The Sakai-Sugimoto model Flavour N f pairs of D8-D8 flavour branes are added to the D4-brane background. Karch, Katz, JHEP 0206 (2002) 043 Probe approximation N f N c : backreaction of flavour branes on background is ignored quenched QCD. Stack of N f coinciding pairs of D8-D8 flavour branes U(N f ) L U(N f ) R theory, to be interpreted as the chiral symmetry in QCD background geometry ( -shape) enforces L = R (joining of flavour branes): U(N f ) L U(N f ) R U(N f ) V. Sakai, Sugimoto, PTP113 (2005) 843; PTP114 (2005) 1083

16 The Sakai-Sugimoto model The flavour gauge field The U(N f ) gauge field A µ (x µ,u) that lives on the flavour branes describes a tower of vector mesons v µ,n (x µ ) in the dual QCD-like theory: U(N f ) gauge field A µ (x µ,u) = v µ,n (x µ )ψ n (u) n 1 with v µ,n (x µ ) a tower of vector mesons with masses m n, and {ψ n (u)} n 1 a complete set of functions of u, satisfying the eigenvalue equation [ ] u 1/2 γ 1/2 B (u) u u 5/2 γ 1/2 B (u) u ψ n (u) = R 3 mnψ 2 n (u),

17 The Sakai-Sugimoto model Why use the Sakai-Sugimoto model the way it works: dynamics of the flavour D8/D8-branes: 5D YM theory S DBI [A µ ] =, A µ (x µ,u) = n 1 v µ,n (x µ )ψ n (u) integrate out the extra radial dimension u effective 4D meson theory for v n µ (x µ ) ideal holographic QCD model to study low-energy QCD confinement and chiral symmetry breaking effective low-energy QCD models drop out: Skyrme (π, also: baryons as skyrmions), HLS (π,ρ coupling), VMD

18 The Sakai-Sugimoto model Approximations of the model Duality is valid in the limit N c and large t Hooft coupling λ = g 2 YM N c 1, and at low energies (where redundant massive d.o.f. decouple). Approximations (inherent to the model): quenched approximation (N f N c ) chiral limit (m π = 0, bare quark masses zero) Choices of parameters: N c = 3 N f = 2 to model charged mesons

19 Introducing the magnetic field How to turn on the magnetic field A non-zero value of the flavour gauge field A m (x µ,z) on the boundary, A m (x µ,u ) = A µ, corresponds to an external gauge field in the boundary field theory that couples to the quarks ψiγ µ D µ ψ with D µ = µ + A µ. To apply an external electromagnetic field A em µ, put A µ (u + ) = iq em A em µ = A µ

20 Introducing the magnetic field How to turn on the magnetic field To apply a magnetic field along the x 3 -axis, A em 2 = x 1 B, in the N f = 2 case, Q em = ( 2/ /3 ) = σ 3, we set A µ = ieq em A em µ = ieq em x 1 Bδ µ2

21 Overview 1 Motivation 2 (Magnetic) holographic setup 3 The ρ meson mass in a magnetic field 4 Chiral transition in a magnetic field

22 Plan DBI action: S DBI = T 8 d 4 x 2 du u 0 ε 4 e φ STr det[g D8 mn + (2πα )if mn ], with STr(F 1 F n ) = 1 n! Tr(F 1 F n + all permutations) the symmetrized trace, g D8 mn = g mn + g ττ (D m τ) 2 the induced metric on the D8-branes (with covariant derivative D m τ = m τ + [A m,τ]) τ = the brane embedding (+ fluctuations)

23 First approximation: Landau levels Simplest embedding to start: u 0 = u K L 5D = d 4 x du { 1 Embedding trivial: u τ = 0 for all values of the magnetic field 2 Determine EOM for ρ µ : STr reduces to regular Tr (because of coincident branes) à m and τ automatically decouple à µ = ρ µ (x)ψ(u) (retain only lowest meson of the tower, most likely to condense) f 1(Fµν) a f 2(Fµu) a f 3 F 3 µνε 3ab à a µãb ν µ,ν=1 }

24 First approximation: Landau levels EOM for ρ for u 0 = u K L 5D = d 4 x du 1 4 f 1 (Fµν) a 2 }{{} (F a µν) 2 ψ f 3 F 3 µνε 3ab à a µãb ν µ,ν=1 }{{} (ρ a µ )2 ( u ψ) 2 ρ a µρ b νψ f 2 (F a µu) 2 }{{} demand du f 1 ψ 2 = 1 and du f 2 ( u ψ) 2 = m 2 ρ, then du f 3 ψ 2 = k L 4D = d 4 x { 1 4 (F a µν) m2 ρ(ρ a µ) k F 3 µνε 3ab ρ a µρ b ν µ,ν=1 (with F a µν = D µ ρ a ν D ν ρ a µ ) standard 4D Lagrangian for a vector field in an external EM field 2 }

25 First approximation: Landau levels Landau levels for Sakai-Sugimoto u 0 = u K Standard 4D Lagrangian for a vector field in an external EM field with k = 1 ( f 3 = f 1 ) Landau levels and M 2 Ρ B GeV^ M 2 ρ(eb) = m 2 ρ eb Landau eb GeV^2

26 Taking into account constituents General embedding u 0 > u K u 0 > u K to model non-zero constituent quark mass which is related to the distance between u 0 and u K. Aharony, Sonnenschein, Yankielowicz, Ann.Phys.322 (2007) 1420

27 Taking into account constituents Numerical fixing of holographic parameters There are three unknown free parameters (u K, u 0 and κ( λn c )). In order to get results in physical units, we fix the free parameters by matching to the constituent quark mass m q = GeV, the pion decay constant f π = GeV and the ρ meson mass in absence of magnetic field m ρ = GeV. Results: u K = 1.39 GeV 1, u 0 = 1.92 GeV 1 and κ = cross-check: QCD string tension σ 0.18 GeV 2 (= standard lattice estimate)

28 Taking into account constituents eb-dependent embedding for u 0 > u K Keep L fixed: u 0 (eb) rises with eb. This models magnetic catalysis of chiral symmetry breaking Johnson, Kundu JHEP0812 (2008) 053; in general: Miransky et al. Non-Abelian: u 0,u (eb) > u 0,d (eb)! U(2) U(1) u U(1) d

29 Taking into account constituents eb-dependent embedding for u 0 > u K Change in embedding models: chiral magnetic catalysis m u (eb) and m d (eb) eb explicitly breaks global U(2) U(1) u U(1) d Effect on ρ mass? expect m ρ (eb) as constituents get heavier split between branes generates other mass mechanism: 5D gauge field gains mass through holographic Higgs mechanism

30 Taking into account constituents eb-induced Higgs mechanism The string associated with a charged ρ meson (ud, du) stretches between the now separated up- and down brane because a string has tension it contributes to the mass.

31 Taking into account constituents EOM for ρ for u 0 > u K? Non-trivial embedding ( τu (u)θ(u u τ(u) = 0,u ) 0 0 τ d (u)θ(u u 0,d ) ) 1, describing the splitting of the branes, utterly complicates the analysis (but necessary for a realistic modeling!). { L 5D = STr.. ( [Ã m,τ] + D m τ )2 +..(F µν ) 2 +..(F µu ) 2 +..F µν [Ã µ,ã ν ] +..( ( u τ)f [Ã,τ] + D τ ) } F with all the.. different functions H ( u τ,f;u) of the background fields u τ, F.

32 Taking into account constituents STr-prescription Myers, JHEP 9912 (1999) 022; Denef, Sevrin, Troost, Nucl.Phys. B581 (2000) 135 STr = symmetric average over all orderings of F ab, D a φ i, [φ i,φ j ] and the individual non-abelian scalars φ k appearing in the non-abelian Taylor expansions of the background fields. STr ( H (F) F 2 ) = a=1 F 2 a I(H ) + a=0,3 with I(H ) = 1 0 dαh (F 0 + αf 3 ) dαh (F 0 αf 3 ) 2 The integral functions I(H ) are complicated functions of eb and u, even discontinuous in u (at u = u 0,u ).

33 Taking into account constituents STr-prescription After many pages of computations (analytical + numerical)

34 Taking into account constituents EOM for ρ for u 0 > u K L 5D = d 4 x du f 3(eB) µ,ν=1 1 4 f 1(eB) (Fµν) a 2 }{{} (F a µν) 2 ψ 2 F 3 µνε 3ab à a µãb ν }{{} 1 2 f 2(eB) (F a µu) 2 }{{} (ρ a µ) 2 ( u ψ) 2 1 ρ a µρ b νψ 2 2 f 4(eB) (à a µ) 2 (τ }{{} (ρ a µ) 2 ψ 2 demand du f 1 (eb)ψ 2 = 1 and du f 2 (eb)( u ψ) 2 +f 4 (eb)(τ 3 ) 2 ψ 2 = mρ(eb), 2 then du f3 (eb)ψ 2 = k(eb) 1( f 3 (eb) f 1 (eb)) { } L 4D = d 4 x 1 (F a 4 µν) m2 ρ(eb)(ρ a µ) k(eb) F 3 µνε 3ab ρ a µρ b ν µ,ν=1 modified 4D Lagrangian for a vector field in an external EM field, with eb-dependent gyromagnetic coupling! 3 ) 2

35 Taking into account constituents Solve the eigenvalue problem The normalization condition and mass condition on the ψ combine to the eigenvalue equation f 1 1 u (f 2 u ψ) f 1 1 f 4 (τ 3 ) 2 ψ = m 2 ρψ with b.c. ψ(x = ±π/2) = 0,ψ (x = 0) = 0 which we solve with a numerical shooting method to obtain m 2 ρ(eb). m Ρ 2 B GeV^

36 Taking into account constituents Landau vs Sakai-Sugimoto u 0 > u K Modified 4D Lagrangian for a vector field in an external EM field with k(eb) 1( f 3 (eb) f 1 (eb)) modified Landau levels and, with ξ = al, Theor.Math.Phys. 55 (1983) 536 M 2 ρ(eb) = m 2 ρ(eb) eb +(1 k(eb))m 2 ρ(eb) M 2 Ρ B GeV^ eb, mρ(eb) 2 Tsai, Yildiz PRD4 (1971) 3643; Obukhov et ( ξ ξ 1 ξ + ξ2 4 ) 0.2 Sakai Sugimoto Landau eb GeV^2

37 Taking into account constituents Summary ρ meson Studied effect: possibility for ρ meson condensation phenomenological models: eb c = mρ 2 = 0.6 GeV2 lattice simulation: slightly higher value of eb c 0.9 GeV 2 holographic approach: can the ρ meson condensation be modeled? yes can this approach deliver new insights? e.g. taking into account constituents, effect on eb c Up and down quark constituents of the ρ meson can be modeled as separate branes, each responding to the magnetic field by changing their embedding. This is a modeling of the chiral magnetic catalysis effect. We take this into account and find also a string effect on the mass, leading to a eb c 0.78 GeV 2

38 Overview 1 Motivation 2 (Magnetic) holographic setup 3 The ρ meson mass in a magnetic field 4 Chiral transition in a magnetic field

39 Chiral temperature T χ = temperature at which chiral symmetry is restored (chiral limit is understood) Studied effect: possible split between T c (eb) and T χ (eb)

40 Split between T c and T χ Expected behaviour (Fig from 08): T χ (eb) : chiral magnetic catalysis seen in chirally driven models (e.g. NJL) Miransky,Shovkovy, PRD66 (2002) T c (eb) : quark gas thermodynamically favoured over pion gas (e.g. MIT bag)agasian, Fedorov, PLB663 (2008) 445; Fraga, Palhares, PRD86 (2012) ; Fukushima, Hidaka, PRL110 (2013)

41 Some results in different models PLSM q model Mizher, Chernodub, Fraga, PRD82 (2010) Lattice D Elia, Mukherjee, Sanfilippo, PRD82 (2010) Different PNJL models Gatto, Ruggieri, PRD82 (2010) ; PRD83 (2011)

42 Motivation (Magnetic) holographic setup The ρ meson mass in a magnetic field Chiral transition in a magnetic field Sakai-Sugimoto at finite temperature Black D4-brane background ds2 = u 3/2 R (f (u )dt 2 + δij dx i dx j + d τ2 ) + f (u ) = 1 ut3 u3, ut T 3/2 R du 2 u f (u ) + u 2 d Ω24 2

43 Numerical fixing of holographic parameters Input parameters at eb = 0 f π = GeV and m ρ = GeV fix all holographic parameters except brane separation L. Choice of L a priori free, determines a kind of choice of holographic theory: L small NJL-type boundary field theory Antonyan, Harvey, Jensen, Kutasov, hep-th/ L = δτ/2 maximal maximal probing of the gluon background (original antipodal Sakai-Sugimoto)

44 Sakai-Sugimoto at finite T and eb no backreaction T c independent of eb eb-dependent embedding of flavour branes T χ (eb): S merged S separated = 0 T χ

45 Conclusion on T χ (eb) The appearance of a split between T χ (GeV) (blue) and T c (GeV) (purple) depends on the choice of L! B B B Plots for fixed L (from small to large) respectively corresponding to m q (eb = 0) = 0.357,0.310 and GeV and T c = 0.103,0.115 and GeV Callebaut, Dudal, PRD87 (2013) Left: split for L small enough NJL results Middle and right: split only at large eb or no split at all for parameter values that match best to QCD SU(2),N f = 2 (chirally extrapolated) lattice data of Ilgenfritz et al, PRD85 (2012) (no split)

46 (Locally) Inverse magnetic catalysis BIG BUT Latest lattice data disagree with most previous results: T χ (eb) quenched vs. true QCD Bali, Bruckmann, Endrodi, Fodor, Katz, Krieg, Schafer, Szabo, JHEP 1202 (2012) 044; Phys.Rev. D86 (2012) Important task for future holographs: construct a magnetized bulk geometry that is sufficiently QCD-like

47 Fin Merci!