Holographic study of magnetically induced ρ meson condensation

Transcription

1 Holographic study of magnetically induced ρ meson condensation Nele Callebaut Ghent University November 13, 2012 QCD in strong magnetic fields, Trento

2 Overview 1 Introduction 2 Holographic set-up The Sakai-Sugimoto model Introducing the magnetic field 3 The ρ meson mass First approximation: Landau levels Taking into account constituents

3 Overview 1 Introduction 2 Holographic set-up The Sakai-Sugimoto model Introducing the magnetic field 3 The ρ meson mass First approximation: Landau levels Taking into account constituents

4 Introduction Why study strong magnetic fields?

5 Introduction Why study strong magnetic fields? experimental relevance: appearance in QGP (order B 15mπ) 2

6 Introduction Why study strong magnetic fields? experimental relevance: appearance in QGP (order B 15mπ) 2 excellent probe for largely unknown QCD phase diagram

7 Introduction Studied effect: ρ meson condensation (Maxim Chernodub) QCD vacuum instable towards forming a superconducting state of condensed charged ρ mesons at critical magnetic field B c

8 Introduction Studied effect: ρ meson condensation (Maxim Chernodub) QCD vacuum instable towards forming a superconducting state of condensed charged ρ mesons at critical magnetic field B c

9 Introduction Studied effect: ρ meson condensation (Maxim Chernodub) QCD vacuum instable towards forming a superconducting state of condensed charged ρ mesons at critical magnetic field B c phenomenological models: B c = m 2 ρ = 0.6 GeV 2 (bosonic effective model), B c 1 GeV 2 (NJL) [ , ]

10 Introduction Studied effect: ρ meson condensation (Maxim Chernodub) QCD vacuum instable towards forming a superconducting state of condensed charged ρ mesons at critical magnetic field B c phenomenological models: B c = m 2 ρ = 0.6 GeV 2 (bosonic effective model), B c 1 GeV 2 (NJL) [ , ] lattice simulation: B c 0.9 GeV 2 [ ]

11 Introduction Studied effect: ρ meson condensation (Maxim Chernodub) QCD vacuum instable towards forming a superconducting state of condensed charged ρ mesons at critical magnetic field B c phenomenological models: B c = m 2 ρ = 0.6 GeV 2 (bosonic effective model), B c 1 GeV 2 (NJL) [ , ] lattice simulation: B c 0.9 GeV 2 [ ] holographic approach: can the ρ meson condensation be modeled? can this approach deliver new insights? e.g. taking into account constituents, effect on B c [ ], extended version in preparation; [ ]

12 ρ meson condensation: Landau levels The energy levels ɛ 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 ɛ 2 n,s z (p z ) = p 2 z + m 2 + (2n 2s z + 1) B.

13 ρ meson condensation: Landau levels The energy levels ɛ 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 ɛ 2 n,s z (p z ) = p 2 z + m 2 + (2n 2s z + 1) B. The combinations ρ = (ρ x iρ y ) and ρ = (ρ + x + iρ + y ) have spin s z = 1 parallel to B. In the lowest energy state (n = 0, p z = 0) their effective mass, M 2 ρ (B) = m 2 ρ B, can thus become zero if the magnetic field is strong enough.

14 ρ meson condensation: Landau levels M 2 ρ (B) = m 2 ρ B,

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

16 Overview 1 Introduction 2 Holographic set-up The Sakai-Sugimoto model Introducing the magnetic field 3 The ρ meson mass First approximation: Landau levels Taking into account constituents

17 Holographic QCD What is holographic QCD? QCD dual = supergravitation in a higher-dimensional background: four-dimensional QCD lives on the boundary of a five-dimensional space where the supergravitation theory is defined

18 Holographic QCD What is holographic QCD? QCD dual = supergravitation in a higher-dimensional background: four-dimensional QCD lives on the boundary of a five-dimensional space where the supergravitation theory is defined Origin of the QCD/gravitation duality idea? Anti de Sitter / Conformal Field Theory (AdS/CFT)-correspondence (Maldacena 1997): supergravitation in AdS 5 space dual = conformal N =4 SYM theory

19 Holographic QCD What is holographic QCD? QCD dual = supergravitation in a higher-dimensional background: four-dimensional QCD lives on the boundary of a five-dimensional space where the supergravitation theory is defined Origin of the QCD/gravitation duality idea? Anti de Sitter / Conformal Field Theory (AdS/CFT)-correspondence (Maldacena 1997): supergravitation in AdS 5 space dual = conformal N =4 SYM theory Witten: supergravitation in D4-brane background dual = non-conformal non-susy pure QCD-like theory

20 The D4-brane background

21 The Sakai-Sugimoto model

22 The Sakai-Sugimoto model To add flavour degrees of freedom to the theory, N f pairs of D8-D8 flavour branes are added to the D4-brane background. Probe approximation N f N c : backreaction of flavour branes on background is ignored quenched approximation. [Sakai and Sugimoto, hep-th/ ]

23 D-branes Dp-brane = (p + 1)-dimensional hypersurface in (10-dim) spacetime in which an endpoint of a string is restricted to move. The spectrum of vibrational modes of an open string with endpoints on the Dp-brane contains a massless photon field A r=0..9 (x) which can be decomposed into a U(1) gauge field A a=0..p (x) living on the brane ( on a D-brane lives a Maxwell field and (9 p) scalar fields φ m=p+1..9 (x) describing the fluctuations of the Dp-bane in its (9 p) transversal directions.

24 The flavour D8-branes On a D-brane lives a Maxwell field.

25 The flavour D8-branes On a D-brane lives a Maxwell field. On a stack of N coinciding D-branes lives a U(N) YM theorie.

26 The flavour D8-branes On a D-brane lives a Maxwell field. On a stack of N coinciding D-branes lives a U(N) YM theorie. = On the stack of N f coinciding pairs of D8-D8 flavour branes lives a U(N f ) L U(N f ) R theory, to be interpreted as the chiral symmetry in QCD. The U-shaped embedding of the flavour branes models spontaneous chiral symmetry breaking U(N f ) L U(N f ) R U(N f ).

27 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),

28 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 )

29 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

30 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

31 How to turn on the magnetic field Sakai-Sugimoto bulk theory: Under a gauge symmetry transformation g U(N f ) the flavour gauge field A m (x µ, z) in the bulk transforms as A m (x µ, z) ga m (x µ, z)g 1 + g m g 1.

32 How to turn on the magnetic field Sakai-Sugimoto bulk theory: Under a gauge symmetry transformation g U(N f ) the flavour gauge field A m (x µ, z) in the bulk transforms as A m (x µ, z) ga m (x µ, z)g 1 + g m g 1. In the gauge A m (x µ, ± ) = 0: residual gauge symmetry {g U(N f ) m g(x µ, ± ) = 0}.

33 How to turn on the magnetic field Sakai-Sugimoto bulk theory: Under a gauge symmetry transformation g U(N f ) the flavour gauge field A m (x µ, z) in the bulk transforms as A m (x µ, z) ga m (x µ, z)g 1 + g m g 1. In the gauge A m (x µ, ± ) = 0: residual gauge symmetry {g U(N f ) m g(x µ, ± ) = 0}. Duality (g L, g R ) (g +, g ) = (lim z + g, lim z g)

34 How to turn on the magnetic field Sakai-Sugimoto bulk theory: Under a gauge symmetry transformation g U(N f ) the flavour gauge field A m (x µ, z) in the bulk transforms as A m (x µ, z) ga m (x µ, z)g 1 + g m g 1. In the gauge A m (x µ, ± ) = 0: residual gauge symmetry {g U(N f ) m g(x µ, ± ) = 0}. Duality (g L, g R ) (g +, g ) = (lim z + g, lim z g) Dual QCD theory: Under a global chiral symmetry transformation (g L, g R ) U(N f ) L U(N f ) R a quark ψ transforms as ψ = ψ L + ψ R g L ψ L + g R ψ R.

35 How to turn on the magnetic field Lightly gauging the global symmetry U(N f ) L U(N f ) R, the Dirac action for the quarks transforms to (g L = g R = g(z = ±, x µ ) = g(x µ )): with ψiγ µ µ ψ ψiγ µ (D µ + g 1 µ g )ψ = ψiγ µ D µ ψ, }{{} A µ (z ± ) µ D µ = µ + A µ (z ± ) so this corresponds to applying an external gauge field in the field theory that couples to the quarks via a covariant derivative D µ.

36 How to turn on the magnetic field Lightly gauging the global symmetry U(N f ) L U(N f ) R, the Dirac action for the quarks transforms to (g L = g R = g(z = ±, x µ ) = g(x µ )): with ψiγ µ µ ψ ψiγ µ (D µ + g 1 µ g )ψ = ψiγ µ D µ ψ, }{{} A µ (z ± ) µ D µ = µ + A µ (z ± ) so this corresponds to applying an external gauge field in the field theory that couples to the quarks via a covariant derivative D µ. To apply an external electromagnetic field A em µ, put [Sakai and Sugimoto hep-th/ ] A µ (u + ) = iq em A em µ = A µ

37 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

38 Overview 1 Introduction 2 Holographic set-up The Sakai-Sugimoto model Introducing the magnetic field 3 The ρ meson mass First approximation: Landau levels Taking into account constituents

39 Action: S DBI = T 8 d 4 x 2 du u 0 Plan ɛ 4 e φ STr det [gmn D8 + (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]), and F mn = m A n n A m + [A m, A n ] = F a mnt a the field strength

40 Action: S DBI = T 8 d 4 x 2 du u 0 Plan ɛ 4 e φ STr det [gmn D8 + (2πα )if mn ], Gauge field ansatz: { Am = A m + à m τ = τ + τ 1 Determine embedding τ(u) as a function of A µ (put à m = τ = 0) 2 Determine EOM for ρ µ :

41 Action: S DBI = T 8 d 4 x 2 du u 0 Plan ɛ 4 e φ STr det [gmn D8 + (2πα )if mn ], Gauge field ansatz: { Am = A m + à m τ = τ + τ 1 Determine embedding τ(u) as a function of A µ (put à m = τ = 0) 2 Determine EOM for ρ µ : Plug total gauge field ansatz into S DBI, then expand to order (2πα ) 2 1 (λ 1) and to second λ order in the fluctuations, 2 then integrate out u-dependence

42 Simplest embedding to start: u 0 = u K 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)

43 Simplest embedding to start: u 0 = u K 1 Embedding trivial: u τ = 0 for all values of the magnetic field 2 Determine EOM for ρ µ : L 5D = d 4 x 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) { } du 1 4 f 1(Fµν) a f 2(Fµu) a f 3 F 3 µνɛ 3ab à a µãb ν µ,ν=1

44 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 2 (Fµu) a 2 }{{} (ρ a µ) 2 ( u ψ) f 3 F 3 µνɛ 3ab à a µãb ν µ,ν=1 }{{} ρ a µρ b ν ψ 2

45 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 2 (Fµu) a 2 }{{} (ρ a µ) 2 ( u ψ) f 3 F 3 µνɛ 3ab à a µãb ν µ,ν=1 }{{} ρ a µρ b ν ψ 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 } 2 F 3 µνɛ 3ab ρ a µρ b ν µ,ν=1 (with F a µν = D µ ρν a D ν ρ a µ) standard 4D Lagrangian for a vector field in an external EM field

46 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) = m 2 ρ B m 2 Ρ GeV Landau eb GeV 2

47 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 et.al. hep-th/ ]

48 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 Results: the constituent quark mass m q = GeV, the pion decay constant f π = GeV and the rho meson mass in absence of magnetic field m ρ = GeV. u K = 1.39 GeV 1, u 0 = 1.92 GeV 1 and κ =

49 B-dependent embedding for u 0 > u K Keep L fixed: u 0 (B) rises with B. This models magnetic catalysis of chiral symmetry breaking [Johnson and Kundu, hep-th/ ]. Non-Abelian: u 0,u (B) > u 0,d (B)! U(2) U(1) u U(1) d

50 B-dependent embedding for u 0 > u K Change in embedding models: chiral magnetic catalysis m u (B) and m d (B) B explicitly breaks global U(2) U(1)u U(1) d

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

52 B-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 gets a mass.

53 EOM for ρ for u 0 > u K? Non-trivial embedding ( ) τu (u)θ(u u τ(u) = 0,u ) 0 1, 0 τ d (u)θ(u u 0,d ) describing the splitting of the branes, severely complicates the analysis.

54 EOM for ρ for u 0 > u K? Non-trivial embedding ( ) τu (u)θ(u u τ(u) = 0,u ) 0 1, 0 τ d (u)θ(u u 0,d ) describing the splitting of the branes, severely complicates the analysis. { L 5D = STr.. ( [Ã m, τ] + D m τ ) } 2 +..(Fµν ) 2 +..(F µu ) 2 +..F µν [Ã µ, Ã ν ] with all the.. different functions H( u τ, F ; u) of the background fields u τ, F.

55 Fixing the gauge to disentangle à and τ Faddeev-Popov gauge fixing: The functional integral Z = DADτ e i L[A,τ] = C DADτ e i ( (L[A,τ] 1 2 G2 ) δg [A α, τ α ) ] det δα is restricted to physically inequivalent field configurations, by imposing the gauge-fixing condition G[fields] = 0.

56 Fixing the gauge to disentangle à and τ We choose the gauge condition on the fields G[Ã, τ] = 1 ξ H m ( u τ, F ; u)d m à a m + ξɛ abc τ b τ c (a = 1, 2) such that the gauge fixed Lagrangian L[Ã, τ] 1 2 G2 no longer contains à τ mixing terms. Then we choose ξ ( unitary gauge ): τ 1,2 decouple.

57 Fixing the gauge to disentangle à and τ In the chosen gauge the Higgs-mechanism is described: τ 1,2 are eaten = Goldstone bosons à 1,2 µ eating the τ 1,2 = massive gauge bosons (mass τ 2 ) τ 0,3 in the direction of the vev τ = Higgs bosons

58 Fixing the gauge to disentangle à and τ In the chosen gauge the Higgs-mechanism is described: τ 1,2 are eaten = Goldstone bosons à 1,2 µ eating the τ 1,2 = massive gauge bosons (mass τ 2 ) τ 0,3 in the direction of the vev τ = Higgs bosons We are left with L 5D = L[ τ] + L[Ã]

59 L[ τ]: Stability of the embedding L[ τ] stability of the embedding: energy density H = δl δ 0 τ 0 τ L associated with fluctuations τ 0,3 must fulfill E = We checked that this is the case. u 0,d H > 0

60 L[Ã]: back to the ρ meson EOM L 5D = STr {..[Ã m, τ] 2 +..(F µν ) 2 +..(F µu ) 2 +..F µν [Ã µ, Ã ν ] } with all the.. different functions H( u τ, F ; u) of the background fields u τ, F.

61 L[Ã]: back to the ρ meson EOM L 5D = STr {..[Ã m, τ] 2 +..(F µν ) 2 +..(F µu ) 2 +..F µν [Ã µ, Ã ν ] } with all the.. different functions H( u τ, F ; u) of the background fields u τ, F. STr-prescription[Myers hep-th/ , Denef et.al. hep-th/ ] The 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. with STr ( H(F ) F 2) = 1 2 I (H) = 2 F a 2 a=1 I (H) + a=0,3 1 0 dαh(f 0 + αf 3 ) dαh(f 0 αf 3 ) 2

62 STr evaluation The integral functions I (H) are complicated functions of B and u, and discontinuous in u (at u = u 0,u ). Here plotted as functions of x related to u as u = u 0,d cos 2/3 x you see f 1 (blue), f 2 (red) and f 3 (yellow), and f 4 (τ 3 ) 2, all for eb = 0.8 GeV

63 L 5D = d 4 x EOM for ρ for u 0 > u K du 1 4 f 1(B) (Fµν) a 2 1 }{{} 2 f 2(B) (Fµu) a 2 }{{} (Fµν) a 2 ψ 2 (ρ a µ) 2 ( u ψ) f 3(B) F 3 µνɛ 3ab à a µãb ν µ,ν=1 }{{} 1 ρ a µρ b ν ψ 2 2 f 4(B) (à a µ) 2 (τ }{{} (ρ a µ) 2 ψ 2 3 ) 2

64 L 5D = d 4 x EOM for ρ for u 0 > u K du 1 4 f 1(B) (Fµν) a 2 1 }{{} 2 f 2(B) (Fµu) a 2 }{{} (Fµν) a 2 ψ 2 (ρ a µ) 2 ( u ψ) f 3(B) F 3 µνɛ 3ab à a µãb ν µ,ν=1 }{{} 1 ρ a µρ b ν ψ 2 2 f 4(B) (à a µ) 2 (τ }{{} (ρ a µ) 2 ψ 2 3 ) 2 demand du f 1 (B)ψ 2 = 1 and du f 2 (B)( u ψ) 2 +f 4 (B)(τ 3 ) 2 ψ 2 = mρ(b), 2 then du f 3 (B)ψ 2 = k(b) = 1( f 3 (B) = f 1 (B)) { } L 4D = d 4 x 1 4 (F µν) a m2 ρ(b)(ρ a µ) k(b) F 3 µνɛ 3ab ρ a µρ b ν µ,ν=1 (with F a µν = D µ ρν a D ν ρ a µ) modified 4D Lagrangian for a vector field in an external EM field

65 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(b). m Ρ 2 B GeV^

66 Landau vs Sakai-Sugimoto u 0 > u K Modified 4D Lagrangian for a vector field in an external EM field with k = 1 ( f 3 = f 1 ) modified Landau levels and, with ξ = B, [Tsai & Yildiz Phys. mρ(b) 2 Rev. D 4 (1971) 3643] M 2 ρ (B) = m 2 ρ(b) B + (1 k(b))m 2 ρ(b) M 2 Ρ B GeV^ ( ξ ξ 1 ξ + ξ2 4 ) 0.2 Sakai Sugimoto Landau eb GeV^2

67 Conclusion: back to objectives Studied effect: ρ meson condensation phenomenological models: B c = m 2 ρ = 0.6 GeV 2 lattice simulation: slightly higher value of B 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 B 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 B crit 0.78 GeV 2

68 Thank you for your attention! Questions?