Holographic study of magnetically induced QCD effects:

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1 Holographic study of magnetically induced QCD effects: split between deconfinement and chiral transition, and evidence for rho meson condensation. Nele Callebaut, David Dudal, Henri Verschelde Ghent University February 23, 2011 Excited QCD 2011, Les Houches

2 Overview 1 Introduction 2 Holographic set-up The Sakai-Sugimoto model Introducing the magnetic field 3 The rho meson mass First approximation: Landau levels Also taking into account the chiral magnetic catalysis effect 4 The chiral symmetry restoration temperature The Sakai-Sugimoto model at finite T The Sakai-Sugimoto model at T and B

3 Overview 1 Introduction 2 Holographic set-up The Sakai-Sugimoto model Introducing the magnetic field 3 The rho meson mass First approximation: Landau levels Also taking into account the chiral magnetic catalysis effect 4 The chiral symmetry restoration temperature The Sakai-Sugimoto model at finite T The Sakai-Sugimoto model at T and B

4 Introduction Growing interest in magnetic-field induced QCD effects, mainly due to possible creation of strong magnetic fields in heavy ion collisions in LHC. Prediction of M. Chernodub [PRD 82 (2010) , [hep-ph]; [hep-ph]]: the QCD vacuum is unstable towards condensation of charged rho mesons in the presence of a very strong magnetic field eb m 2 ρ, turning into a superconducting vacuum. Goal of our work: study the QCD vacuum in strong magnetic field, in search of a possible rho meson condensation, using a holographic approach. Paper to appear, main results in proceeding [hep-ph]

5 Overview 1 Introduction 2 Holographic set-up The Sakai-Sugimoto model Introducing the magnetic field 3 The rho meson mass First approximation: Landau levels Also taking into account the chiral magnetic catalysis effect 4 The chiral symmetry restoration temperature The Sakai-Sugimoto model at finite T The Sakai-Sugimoto model at T and B

6 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

7 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): conformal N =4 SYM theory dual = supergravitation in AdS 5 space

8 ds 2 = ( u R with R N 1/3 c The D4-brane background ) 3/2 (η µν dx µ dx ν + f (u)dτ 2 ) + ( R u and f (u) = 1 u3 K u 3 ) 3/2 ( du 2 f (u) + u2 dω 2 4 ),

9 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. Sakai and Sugimoto, Prog. Theor. Phys. 113 (2005) 843, hep-th/

10 D-branes Dp-brane = (p + 1)-dimensional hypersurface in spacetime in which an endpoint of a string (with Dirichlet boundary conditions) is restricted to move. D1-brane D2-brane D3-brane Figure from Peeters and Zamaklar, hep-ph/ The spectrum of vibrational modes of an open string with endpoints on the Dp-brane contains p massless photon states: On a D-brane lives a Maxwell field.

11 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 ).

12 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),

13 Approximations and choices of parameters Approximations: quenched approximation chiral limit (m π = 0) We choose N c = 3 N f = 2 to model charged mesons u 0 > u K to model non-zero constituent quark mass Aharony et.al., Annals Phys. 322 (2007) 1420, hep-th/

14 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 =

15 How to turn on the magnetic field 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. Duality (g L, g R ) (g +, g ) = (lim z + g, lim z g) with g U(N f ) a residual gauge symmetry transformation for the gauge field on the D8-branes, in the gauge A µ (x µ, ± ) = 0. Sakai-Sugimoto bulk theory: Under a gauge symmetry transformation g U(N f ) the flavour gauge field A µ (x µ, z) in the bulk transforms as A µ (x µ, z) ga µ (x µ, z)g 1 + g µ g 1.

16 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 )): ψiγ µ µ ψ ψiγ µ ( µ + g 1 µ g )ψ = ψiγ µ D µ ψ, }{{} A µ (z ± ) so corresponds to applying an external gauge field in the field theory that couples to the quarks via a covariant derivative D µ = µ A µ (z ± ).

17 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 )): ψiγ µ µ ψ ψiγ µ ( µ + g 1 µ g )ψ = ψiγ µ D µ ψ, }{{} A µ (z ± ) so corresponds to applying an external gauge field in the field theory that couples to the quarks via a covariant derivative D µ = µ A µ (z ± ). To apply an external electromagnetic field A em µ, put A µ (z + ) = A µ (z ) = eq em A em µ = A µ Sakai and Sugimoto, Prog. Theor. Phys. 114 (2005) 1083, hep-th/

18 How to turn on the magnetic field To apply a magnetic field along the x 3 -axis, A em 2 = x 1 eb, in the N f = 2 case, Q em = ( 2/ /3 ) = σ 3, we set A 3 2 = x 1 eb

19 Effect of B on the embedding L = 1.57 GeV 1 (corresponding to u 0 (B = 0) = 1.92 GeV 1 ) kept fixed: u 0 rises with B.

20 Effect of B on the embedding L = 1.57 GeV 1 (corresponding to u 0 (B = 0) = 1.92 GeV 1 ) kept fixed: u 0 rises with B. This models magnetic catalysis of chiral symmetry breaking. Johnson and Kundu, JHEP 0812 (2008) 053, [hep-th]

21 Effect of B on the embedding of the flavour branes: chiral magnetic catalysis u0 GeV 1 mq GeV Figure: u 0 as a function of the magnetic field. Figure: The constituent quark mass as a function of the magnetic field.

22 Overview 1 Introduction 2 Holographic set-up The Sakai-Sugimoto model Introducing the magnetic field 3 The rho meson mass First approximation: Landau levels Also taking into account the chiral magnetic catalysis effect 4 The chiral symmetry restoration temperature The Sakai-Sugimoto model at finite T The Sakai-Sugimoto model at T and B

23 Our gauge field ansatz is Landau levels A µ (x µ, u) = A µ + V µ,n (x µ )ψ n (u). n 1 We plug it it into the non-abelian DBI-action for the flavour gauge field S DBI = T 8 d 4 xdτ ɛ 4 e φ STr det [gmn D8 + (2πα )F mn ] (with T 8 the D8-brane tension, STr the symmetrized trace, g D8 mn the induced metric on the D8-branes, α the string tension, and F mn the field strength),

24 Our gauge field ansatz is Landau levels A µ (x µ, u) = A µ + V µ,n (x µ )ψ n (u). n 1 We plug it it into the non-abelian DBI-action for the flavour gauge field S DBI = T 8 d 4 xdτ ɛ 4 e φ STr det [gmn D8 + (2πα )F mn ] (with T 8 the D8-brane tension, STr the symmetrized trace, g D8 mn the induced metric on the D8-branes, α the string tension, and F mn the field strength), expand the action to order (V µ,n ) 2,

25 Our gauge field ansatz is Landau levels A µ (x µ, u) = A µ + V µ,n (x µ )ψ n (u). n 1 We plug it it into the non-abelian DBI-action for the flavour gauge field S DBI = T 8 d 4 xdτ ɛ 4 e φ STr det [gmn D8 + (2πα )F mn ] (with T 8 the D8-brane tension, STr the symmetrized trace, g D8 mn the induced metric on the D8-branes, α the string tension, and F mn the field strength), expand the action to order (V µ,n ) 2, use the normalization condition for the ψ n (u) s to integrate out the u-dependence,

26 Our gauge field ansatz is Landau levels A µ (x µ, u) = A µ + V µ,n (x µ )ψ n (u). n 1 We plug it it into the non-abelian DBI-action for the flavour gauge field S DBI = T 8 d 4 xdτ ɛ 4 e φ STr det [gmn D8 + (2πα )F mn ] (with T 8 the D8-brane tension, STr the symmetrized trace, g D8 mn the induced metric on the D8-branes, α the string tension, and F mn the field strength), expand the action to order (V µ,n ) 2, use the normalization condition for the ψ n (u) s to integrate out the u-dependence, and derive the mass equation for the rho meson ρ µ = V µ,1.

27 Landau levels We find mρρ 2 µ Dν 2 ρ µ + 2i( µ A em ν ν A em µ )ρ ν = 0, with D µ = µ iea em µ for the negatively charged rho meson ρ µ = ρ 1 µ iρ 2 µ.

28 Landau levels We find mρρ 2 µ Dν 2 ρ µ + 2i( µ A em ν ν A em µ )ρ ν = 0, with D µ = µ iea em µ for the negatively charged rho meson ρ µ = ρ 1 µ iρ 2 µ. Inserting the background gauge field ansatz A em 3 2 = x 1 eb, and Fourier transforming ρ µ, we obtain the Landau energy levels ) E 2 (ρ 1 iρ 2 (eb(2n ) = + 1) + p3 2 + mρ 2 2eB (ρ 1 iρ 2 ) with p 3 the momentum of the meson in the direction of the magnetic field.

29 Landau levels Landau levels E 2 ρ = ( ) eb(2n + 1) + p3 2 + mρ 2 2eB ρ The combinations ρ = (ρ 1 iρ 2 ) and ρ = (ρ iρ+ 2 ) have spin s 3 = 1 parallel to B = B e3. In the lowest energy state (N = 0, p 3 = 0) their effective mass, m 2 eff = m 2 ρ eb, can become zero if the magnetic field is strong enough.

30 Landau levels Landau levels E 2 ρ = ( ) eb(2n + 1) + p3 2 + mρ 2 2eB ρ The combinations ρ = (ρ 1 iρ 2 ) and ρ = (ρ iρ+ 2 ) have spin s 3 = 1 parallel to B = B e3. In the lowest energy state (N = 0, p 3 = 0) their effective mass, m 2 eff = m 2 ρ eb, can become zero if the magnetic field is strong enough. = The fields ρ and ρ condense at the critical magnetic field eb c = m 2 ρ.

31 Taking into account chiral magnetic catalysis In the above we have neglected any dependence on the magnetic field of the mass eigenvalue m ρ itself, but actually m 2 eff = m 2 ρ(b) eb. m 2 ρ(b) is the lowest eigenvalue of 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 mn(b)ψ 2 n (u), which depends on B both explicitly and implicitly through the changed embedding of the probe branes, represented by the value of u 0 (B).

32 Taking into account chiral magnetic catalysis m 2 Ρ GeV

33 Taking into account chiral magnetic catalysis m 2 Ρ GeV

34 Taking into account chiral magnetic catalysis m 2 Ρ GeV = The fields ρ and ρ condense at the critical magnetic field eb c = m 2 ρ(b c ) = 1.08m 2 ρ.

35 Overview 1 Introduction 2 Holographic set-up The Sakai-Sugimoto model Introducing the magnetic field 3 The rho meson mass First approximation: Landau levels Also taking into account the chiral magnetic catalysis effect 4 The chiral symmetry restoration temperature The Sakai-Sugimoto model at finite T The Sakai-Sugimoto model at T and B

36 The Sakai-Sugimoto model at finite T At low temperature T < T c, the D4-brane background of the Sakai-Sugimoto model is wick-rotated and the Euclidean time is periodic with period β = 1/T. At the deconfinement temperature T c = 115 MeV (for our numerical values of the holographic parameters) this background changes to the high-t background (T T c ), which only differs from the low-t one in that the time and τ dimensions are interchanged.

37 The Sakai-Sugimoto model at finite T The chiral symmetry restoration temperature T χ is determined as the value of the temperature for which the difference in actions S = S[U-shaped embedding] S[straight embedding] becomes zero. Aharony et.al., Annals Phys. 322 (2007) 1420, hep-th/

38 The Sakai-Sugimoto model at T and B Keeping L(u 0, T, B) fixed to the value for B = 0 ( u 0 (B, T )), we determine T χ from S(B, u 0, T ) = 0 and the assumption that T χ (B = 0) = T c

39 The Sakai-Sugimoto model at T and B We find a split T χ (B) T c of 1.2% at eb = 15m 2 π 0.28 GeV 2 and 3.2% at eb = 30m 2 π 0.57 GeV 2. The percent estimation of the magnitude of the split is in quantitative agreement with Gatto and Ruggieri, [hep-ph];D Elia et.al.,prd 82 (2010) , [hep-lat] in qualitative agreement with Mizher,Chernodub and Fraga,PRD 82 (2010) , [hep-ph]; Gatto and Ruggieri, Phys. Rev. D82 (2010) , [hep-ph].

40 The Sakai-Sugimoto model at T and B Results in the chiral limit can be extracted from the work of Fukushima et.al. wherein T c seems to depend only marginally on B, at least up to eb = 20m 2 π, the maximum value considered there. Fukushima et.al., Phys. Rev. D81 (2010) , [hep-ph]

41 Thank you for your attention! Questions?