A magnetic instability of the Sakai-Sugimoto model Nele Callebaut - - PowerPoint PPT Presentation

Introduction Holographic set-up The ρ meson mass Tχ

A magnetic instability of the Sakai-Sugimoto model

Nele Callebaut

Ghent University and Vrije Universiteit Brussel

February 25, 2014 Work in collaboration with David Dudal arxiv: 1105.2217, 1309.5042

Holography Seminar, Oxford

Introduction Holographic set-up The ρ meson mass Tχ

Overview

1

Introduction

2

Holographic set-up The Sakai-Sugimoto model Introducing the magnetic field

3

The ρ meson mass Taking into account constituents Full DBI-action Effect of Chern-Simons action and mixing with pions

4

Chiral temperature

Introduction Holographic set-up The ρ meson mass Tχ

Overview

1

Introduction

2

Holographic set-up The Sakai-Sugimoto model Introducing the magnetic field

3

The ρ meson mass Taking into account constituents Full DBI-action Effect of Chern-Simons action and mixing with pions

4

Chiral temperature

Introduction Holographic set-up The ρ meson mass Tχ

Why study strong magnetic fields?

Introduction Holographic set-up The ρ meson mass Tχ

Why study strong magnetic fields?

experimental relevance: appearance in QGP (order B ∼ 15m2

π)

Introduction Holographic set-up The ρ meson mass Tχ

Why study strong magnetic fields?

experimental relevance: appearance in QGP (order B ∼ 15m2

π)

from a holographic viewpoint: interesting for comparison with lattice

Introduction Holographic set-up The ρ meson mass Tχ

Why study strong magnetic fields?

experimental relevance: appearance in QGP (order B ∼ 15m2

π)

from a holographic viewpoint: interesting for comparison with lattice excellent probe for largely unknown QCD phase diagram

Introduction Holographic set-up The ρ meson mass Tχ

Why study strong magnetic fields?

experimental relevance: appearance in QGP (order B ∼ 15m2

π)

from a holographic viewpoint: interesting for comparison with lattice excellent probe for largely unknown QCD phase diagram in other strong interaction systems: interior of dense neutron stars (magnetars), cosmology of early universe

Introduction Holographic set-up The ρ meson mass Tχ

ρ meson condensation

Studied effect: ρ meson condensation (Maxim Chernodub)

QCD vacuum instable towards forming a superconducting state of condensed charged ρ mesons at critical magnetic field Bc

Introduction Holographic set-up The ρ meson mass Tχ

ρ meson condensation

Studied effect: ρ meson condensation (Maxim Chernodub)

QCD vacuum instable towards forming a superconducting state of condensed charged ρ mesons at critical magnetic field Bc

Introduction Holographic set-up The ρ meson mass Tχ

ρ meson condensation: Landau levels

The energy levels ǫ 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 ǫ2

n,sz(pz) = p2 z + m2 + (2n − 2sz + 1)|B|.

Introduction Holographic set-up The ρ meson mass Tχ

ρ meson condensation: Landau levels

The energy levels ǫ 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 ǫ2

n,sz(pz) = p2 z + m2 + (2n − 2sz + 1)|B|.

The combinations ρ = (ρ−

x − iρ− y ) and ρ† = (ρ+ x + iρ+ y ) have spin

sz = 1 parallel to B.

Introduction Holographic set-up The ρ meson mass Tχ

ρ meson condensation: Landau levels

The energy levels ǫ 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 ǫ2

n,sz(pz) = p2 z + m2 + (2n − 2sz + 1)|B|.

The combinations ρ = (ρ−

x − iρ− y ) and ρ† = (ρ+ x + iρ+ y ) have spin

sz = 1 parallel to B. In the lowest energy state (n = 0, pz = 0) their effective mass, m2

ρ,eff (B) = m2 ρ − B,

can thus become zero if the magnetic field is strong enough.

Introduction Holographic set-up The ρ meson mass Tχ

ρ meson condensation: Landau levels

m2

ρ,eff (B) = m2 ρ − B,

0.2 0.4 0.6 0.8

B GeV2

0.2 0.2 0.4 0.6

mΡ,eff

2 GeV2

Introduction Holographic set-up The ρ meson mass Tχ

ρ meson condensation: Landau levels

m2

ρ,eff (B) = m2 ρ − B,

0.2 0.4 0.6 0.8

B GeV2

0.2 0.2 0.4 0.6

mΡ,eff

2 GeV2

= ⇒ The fields ρ and ρ† condense at the critical magnetic field Bc = m2

ρ.

Introduction Holographic set-up The ρ meson mass Tχ

Abrikosov lattice ground state

Figure : Absolute value of the superconducting condensate ρ at B = 1.01Bc in the transversal (x1, x2)- plane.

[Chernodub, Van Doorsselaere and Verschelde, 1111.4401]

Similar result in holographic toy model [Bu, Erdmenger, Shock &

Strydom, 1210.6669]

Introduction Holographic set-up The ρ meson mass Tχ

ρ meson condensation: different approaches

phenomenological models: Bc = m2

ρ = 0.6 GeV2 (bosonic effective

model), Bc ≈ 1 GeV2 (NJL) [1008.1055,1101.0117]

Introduction Holographic set-up The ρ meson mass Tχ

ρ meson condensation: different approaches

phenomenological models: Bc = m2

ρ = 0.6 GeV2 (bosonic effective

model), Bc ≈ 1 GeV2 (NJL) [1008.1055,1101.0117] lattice simulation: Bc ≈ 0.9 GeV2 [1104.3767]

Introduction Holographic set-up The ρ meson mass Tχ

ρ meson condensation: different approaches

phenomenological models: Bc = m2

ρ = 0.6 GeV2 (bosonic effective

model), Bc ≈ 1 GeV2 (NJL) [1008.1055,1101.0117] lattice simulation: Bc ≈ 0.9 GeV2 [1104.3767] holographic approach:

can the ρ meson condensation be modeled? can this approach deliver new insights? e.g. taking into account constituents, effect on Bc N.C., Dudal & Verschelde [1105.2217,1309.5042]; Ammon, Erdmenger, Kerner & Strydom [1106.4551], Cai et al [1309.2098]

Introduction Holographic set-up The ρ meson mass Tχ

Overview

1

Introduction

2

Holographic set-up The Sakai-Sugimoto model Introducing the magnetic field

3

The ρ meson mass Taking into account constituents Full DBI-action Effect of Chern-Simons action and mixing with pions

4

Chiral temperature

Introduction Holographic set-up The ρ meson mass Tχ

Holographic QCD

Holographic QCD: QCD dual = supergravitation in a higher-dimensional background

Introduction Holographic set-up The ρ meson mass Tχ

Holographic QCD

Holographic QCD: QCD dual = supergravitation in a higher-dimensional background Anti de Sitter / Conformal Field Theory (AdS/CFT)-correspondence (Maldacena 1997): supergravitation in AdS5 space dual = conformal N =4 SYM theory

Introduction Holographic set-up The ρ meson mass Tχ

Holographic QCD

Holographic QCD: QCD dual = supergravitation in a higher-dimensional background Anti de Sitter / Conformal Field Theory (AdS/CFT)-correspondence (Maldacena 1997): supergravitation in AdS5 space dual = conformal N =4 SYM theory Witten: supergravitation in D4-brane background dual = non-conformal non-susy pure QCD-like theory

Introduction Holographic set-up The ρ meson mass Tχ

The D4-brane background

ds2 = u R 3/2 (ηµνdxµdxν + f (u)dτ2) + R u 3/2 du2 f (u) + u2dΩ2

4

• ,

eφ = gs u R 3/4 , F4 = Nc V4 ǫ4 , f (u) = 1 − u3

K

u3 ,

Introduction Holographic set-up The ρ meson mass Tχ

The Sakai-Sugimoto model

To add flavour degrees of freedom to the theory, add Nf pairs of D8-D8 flavour branes [Sakai and Sugimoto, hep-th/0412141]. Probe approximation Nf ≪ Nc: backreaction of flavour branes on background is ignored ∼ quenched approximation.

Introduction Holographic set-up The ρ meson mass Tχ

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 Ar=0..9(x) which can be decomposed into a U(1) gauge field Aa=0..p(x) living on the brane (“on a D-brane lives a Maxwell field” and (9 − p) scalar fields φm=p+1..9(x) describing the fluctuations of the Dp-bane in its (9 − p) transversal directions.

Introduction Holographic set-up The ρ meson mass Tχ

The flavour D8-branes

• “On a D-brane lives a Maxwell field.”

Introduction Holographic set-up The ρ meson mass Tχ

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.”

Introduction Holographic set-up The ρ meson mass Tχ

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 Nf coinciding pairs of D8-D8 flavour branes lives a U(Nf )L × U(Nf )R theory, to be interpreted as the chiral symmetry in QCD.”

Introduction Holographic set-up The ρ meson mass Tχ

Chiral symmetry

• Massless QCD-Lagrangian

ψiγµDµψ − 1 4F 2

µν

invariant under chiral symmetry transformations (gL, gR) ∈ U(Nf )L × U(Nf )R ψL → gLψL, ψR → gRψR with ψL = 1 2(1 − γ5)ψ, ψR = 1 2(1 + γ5)ψ.

Introduction Holographic set-up The ρ meson mass Tχ

Chiral symmetry

• Massless QCD-Lagrangian

ψiγµDµψ − 1 4F 2

µν

invariant under chiral symmetry transformations (gL, gR) ∈ U(Nf )L × U(Nf )R ψL → gLψL, ψR → gRψR with ψL = 1 2(1 − γ5)ψ, ψR = 1 2(1 + γ5)ψ.

• But chiral symmetry U(Nf )L × U(Nf )R not reflected in mass

spectrum of the mesons...

Introduction Holographic set-up The ρ meson mass Tχ

Chiral symmetry

• Massless QCD-Lagrangian

ψiγµDµψ − 1 4F 2

µν

invariant under chiral symmetry transformations (gL, gR) ∈ U(Nf )L × U(Nf )R ψL → gLψL, ψR → gRψR with ψL = 1 2(1 − γ5)ψ, ψR = 1 2(1 + γ5)ψ.

• But chiral symmetry U(Nf )L × U(Nf )R not reflected in mass

spectrum of the mesons... Explanation: spontaneous chiral symmetry breaking U(Nf )L × U(Nf )R → U(Nf )

Introduction Holographic set-up The ρ meson mass Tχ

Chiral symmetry in the dual picture

“On the stack of Nf coinciding pairs of D8-D8 flavour branes lives a U(Nf )L × U(Nf )R theory, to be interpreted as the chiral symmetry in QCD.”

Introduction Holographic set-up The ρ meson mass Tχ

Chiral symmetry in the dual picture

“On the stack of Nf coinciding pairs of D8-D8 flavour branes lives a U(Nf )L × U(Nf )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(Nf )L × U(Nf )R → U(Nf ).

Introduction Holographic set-up The ρ meson mass Tχ

The flavour gauge field

The U(Nf ) gauge field Aµ(xµ, u) that lives on the flavour branes describes a tower of vector mesons vµ,n(xµ) in the dual QCD-like theory: U(Nf ) gauge field Aµ(xµ, u) = ∑

n≥1

vµ,n(xµ)ψn(u) with vµ,n(xµ) a tower of vector mesons with masses mn, and {ψn(u)}n≥1 a complete set of functions of u, satisfying the eigenvalue equation u1/2γ−1/2

B

(u)∂u

• u5/2γ−1/2

B

(u)∂uψn(u)

• = −R3m2

nψn(u),

Introduction Holographic set-up The ρ meson mass Tχ

Flavour gauge field and mesons

• the way it works:

dynamics of the flavour D8/D8-branes: 5D YM theory SDBI[Aµ] = · · · , Aµ(xµ, u) = ∑n≥1 vµ,n(xµ)ψn(u)

↓

integrate out the extra radial dimension u

effective 4D meson theory for vn

µ(xµ)

Introduction Holographic set-up The ρ meson mass Tχ

Flavour gauge field and mesons

• the way it works:

dynamics of the flavour D8/D8-branes: 5D YM theory SDBI[Aµ] = · · · , Aµ(xµ, u) = ∑n≥1 vµ,n(xµ)ψn(u)

↓

integrate out the extra radial dimension u

effective 4D meson theory for vn

µ(xµ)

• 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

Introduction Holographic set-up The ρ meson mass Tχ

Approximations of the model

Duality is valid in the limit Nc → ∞ and large ’t Hooft coupling λ = g2

YMNc ≫ 1, and at low energies (where redundant massive

d.o.f. decouple). Approximations (inherent to the model):

quenched approximation (Nf ≪ Nc) chiral limit (mπ = 0, bare quark masses zero)

Choices of parameters:

Nc = 3 Nf = 2 to model charged mesons

Introduction Holographic set-up The ρ meson mass Tχ

How to turn on the magnetic field

A non-zero value of the flavour gauge field Am(xµ, z) on the boundary, Am(xµ, u → ∞) = Aµ, corresponds to an external gauge field in the boundary field theory that couples to the quarks ψiγµDµψ with Dµ = ∂µ + Aµ.

Introduction Holographic set-up The ρ meson mass Tχ

How to turn on the magnetic field

A non-zero value of the flavour gauge field Am(xµ, z) on the boundary, Am(xµ, u → ∞) = Aµ, corresponds to an external gauge field in the boundary field theory that couples to the quarks ψiγµDµψ with Dµ = ∂µ + Aµ. To apply an external electromagnetic field Aem

µ , put

Aµ(u → +∞) = −iQemAem

µ

= Aµ

[Sakai and Sugimoto hep-th/0507073]

Introduction Holographic set-up The ρ meson mass Tχ

How to turn on the magnetic field

A non-zero value of the flavour gauge field Am(xµ, z) on the boundary, Am(xµ, u → ∞) = Aµ, corresponds to an external gauge field in the boundary field theory that couples to the quarks ψiγµDµψ with Dµ = ∂µ + Aµ. To apply an external electromagnetic field Aem

µ , put

Aµ(u → +∞) = −iQemAem

µ

= Aµ

[Sakai and Sugimoto hep-th/0507073] Aem

2

= x1B Qem = 2/3 −1/3

• = 1

612 + 1 2σ3

Introduction Holographic set-up The ρ meson mass Tχ

Overview

1

Introduction

2

Holographic set-up The Sakai-Sugimoto model Introducing the magnetic field

3

The ρ meson mass Taking into account constituents Full DBI-action Effect of Chern-Simons action and mixing with pions

4

Chiral temperature

Introduction Holographic set-up The ρ meson mass Tχ

Plan

• Action:

SDBI = −T8

• d4x 2

∞

u0

du

• ǫ4 e−φ STr
• − det [gD8

mn + (2πα′)iFmn],

with STr(F1 · · · Fn) = 1 n!Tr(F1 · · · Fn + all permutations) the symmetrized trace, gD8

mn = gmn + gττ(Dmτ)2

the induced metric on the D8-branes (with covariant derivative Dmτ = ∂mτ + [Am, τ]), and Fmn = ∂mAn − ∂nAm + [Am, An] = F a

mnta

the field strength

Introduction Holographic set-up The ρ meson mass Tχ

Plan

• Action:

SDBI = −T8

• d4x 2

∞

u0

du

• ǫ4 e−φ STr
• − det [gD8

mn + (2πα′)iFmn],

• Gauge field ansatz:
• Am = Am + ˜

Am τ = τ + ˜ τ

1

Determine embedding τ(u) as a function of Aµ (put ˜ Am = ˜ τ = 0)

2

Determine EOM for ρµ:

Introduction Holographic set-up The ρ meson mass Tχ

Plan

• Action:

SDBI = −T8

• d4x 2

∞

u0

du

• ǫ4 e−φ STr
• − det [gD8

mn + (2πα′)iFmn],

• Gauge field ansatz:
• Am = Am + ˜

Am τ = τ + ˜ τ

1

Determine embedding τ(u) as a function of Aµ (put ˜ Am = ˜ τ = 0)

2

Determine EOM for ρµ:

Plug total gauge field ansatz into SDBI , expand to 2nd order in the fluctuations and integrate out u-dependence

Introduction Holographic set-up The ρ meson mass Tχ

Plan

• Action:

SDBI = −T8

• d4x 2

∞

u0

du

• ǫ4 e−φ STr
• − det [gD8

mn + (2πα′)iFmn],

• Gauge field ansatz:
• Am = Am + ˜

Am τ = τ + ˜ τ

1

Determine embedding τ(u) as a function of Aµ (put ˜ Am = ˜ τ = 0)

2

Determine EOM for ρµ:

Plug total gauge field ansatz into SDBI , expand to 2nd order in the fluctuations and integrate out u-dependence Expand to order (2πα′)2 ∼

1 λ2

(λ ≫ 1) vs use full DBI-action

Introduction Holographic set-up The ρ meson mass Tχ

General embedding u0 > uK

u0 > uK to model non-zero constituent quark mass which is related to the distance between u0 and uK.

[Aharony et.al. hep-th/0604161]

Introduction Holographic set-up The ρ meson mass Tχ

Numerical fixing of holographic parameters

There are three unknown free parameters (uK, u0 and κ(∼ λNc)). In order to get results in physical units, we fix the free parameters by matching to

the constituent quark mass mq = 0.310 GeV, the pion decay constant fπ = 0.093 GeV and the rho meson mass in absence of magnetic field mρ = 0.776 GeV.

Results: uK = 1.39 GeV−1, u0 = 1.92 GeV−1 and κ = 0.00678

Introduction Holographic set-up The ρ meson mass Tχ

B-dependent embedding for u0 > uK

Keep L fixed: u0(B) rises with B. This models magnetic catalysis of chiral symmetry breaking

[Bergman 0802.3720; Johnson and Kundu 0803.0038].

Introduction Holographic set-up The ρ meson mass Tχ

B-dependent embedding for u0 > uK

Keep L fixed: u0(B) rises with B. This models magnetic catalysis of chiral symmetry breaking

[Bergman 0802.3720; Johnson and Kundu 0803.0038].

Non-Abelian: u0,u(B) > u0,d(B)! U(2) → U(1)u × U(1)d

Introduction Holographic set-up The ρ meson mass Tχ

B-dependent embedding for u0 > uK

Change in embedding models:

chiral magnetic catalysis ⇒ mu(B) and md(B) ր

• B explicitly breaks global U(2) → U(1)u × U(1)d

Introduction Holographic set-up The ρ meson mass Tχ

B-dependent embedding for u0 > uK

Change in embedding models:

chiral magnetic catalysis ⇒ mu(B) and md(B) ր

• B explicitly breaks global U(2) → U(1)u × U(1)d

Effect on ρ mass?

Introduction Holographic set-up The ρ meson mass Tχ

B-dependent embedding for u0 > uK

Change in embedding models:

chiral magnetic catalysis ⇒ mu(B) and md(B) ր

• B explicitly breaks global U(2) → U(1)u × U(1)d

Effect on ρ mass?

expect mρ(B) ր as constituents get heavier

Introduction Holographic set-up The ρ meson mass Tχ

B-dependent embedding for u0 > uK

Change in embedding models:

chiral magnetic catalysis ⇒ mu(B) and md(B) ր

• B explicitly breaks global U(2) → U(1)u × U(1)d

Effect on ρ mass?

expect mρ(B) ր as constituents get heavier split between branes generates other mass mechanism: 5D gauge field gains mass through holographic Higgs mechanism

Introduction Holographic set-up The ρ meson mass Tχ

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.

Introduction Holographic set-up The ρ meson mass Tχ

EOM for ρ for u0 > uK?

Non-trivial embedding τ(u) = τu(u)θ(u − u0,u) τd(u)θ(u − u0,d)

• ∼ 1,

describing the splitting of the branes, severely complicates the analysis.

Introduction Holographic set-up The ρ meson mass Tχ

EOM for ρ for u0 > uK?

Non-trivial embedding τ(u) = τu(u)θ(u − u0,u) τd(u)θ(u − u0,d)

• ∼ 1,

describing the splitting of the branes, severely complicates the analysis. L5D = STr

• ..

[ ˜ Am, τ] + Dm ˜ τ 2 + ..(Fµν)2 + ..(Fµu)2 + ..F µν[ ˜ Aµ, ˜ Aν] +..(∂uτ)F [ ˜ A, τ] + D ˜ τ

• F
• with all the .. different functions H(∂uτ, F; u) of the background

fields ∂uτ, F.

Introduction Holographic set-up The ρ meson mass Tχ

Fixing the gauge to disentangle ˜ A and ˜ τ

Faddeev-Popov gauge fixing: The functional integral Z =

• DADτ ei L[A,τ]

= C ′

• DADτ ei (L[A,τ]− 1

2 G2) det

δG[Aα, τα] δα

• is restricted to physically inequivalent field configurations, by

imposing the gauge-fixing condition G[fields] = 0.

Introduction Holographic set-up The ρ meson mass Tχ

Fixing the gauge to disentangle ˜ A and ˜ τ

We choose the gauge condition on the fields Ga[ ˜ A, ˜ τ] = 1 √ξ Hm(∂uτ, F; u)Dm ˜ Aa

m +

• ξǫabc ˜

τbτc (a = 1, 2) such that the gauge fixed Lagrangian L[ ˜ A, ˜ τ] − 1 2G2 no longer contains ˜ A ˜ τ mixing terms. Then we choose ξ → ∞ (”unitary gauge”): ˜ τ1,2 decouple. Remaining gauge freedom in Abelian direction fixed by Aa

u = 0

(a = 0, 3).

Introduction Holographic set-up The ρ meson mass Tχ

Fixing the gauge to disentangle ˜ A and ˜ τ

In the chosen gauge the Higgs-mechanism is more visible:

˜ τ1,2 are ’eaten’ = Goldstone bosons ˜ A1,2

µ

eating the ˜ τ1,2 = massive gauge bosons (mass ∼ τ2) ˜ τ0,3 in the direction of the vev τ = Higgs bosons

Introduction Holographic set-up The ρ meson mass Tχ

Fixing the gauge to disentangle ˜ A and ˜ τ

In the chosen gauge the Higgs-mechanism is more visible:

˜ τ1,2 are ’eaten’ = Goldstone bosons ˜ A1,2

µ

eating the ˜ τ1,2 = massive gauge bosons (mass ∼ τ2) ˜ τ0,3 in the direction of the vev τ = Higgs bosons

We are left with L5D = L[ ˜ τ] + L[ ˜ A]

Introduction Holographic set-up The ρ meson mass Tχ

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 ˜ τ0,3 must fulfill E =

∞

u0,d

H > 0 We checked that this is the case.

Introduction Holographic set-up The ρ meson mass Tχ

L[ ˜

A]: back to the ρ meson EOM

L5D = STr

• ..[ ˜

Am, τ]2 + ..(Fµν)2 + ..(Fµu)2 + ..F µν[ ˜ Aµ, ˜ Aν]

• with all the .. different functions H(∂uτ, F; u) of the background

fields ∂uτ, F.

Introduction Holographic set-up The ρ meson mass Tχ

L[ ˜

A]: back to the ρ meson EOM

L5D = STr

• ..[ ˜

Am, τ]2 + ..(Fµν)2 + ..(Fµu)2 + ..F µν[ ˜ Aµ, ˜ Aν]

• with all the .. different functions H(∂uτ, F; u) of the background

fields ∂uτ, F.

STr-prescription [Myers, Hashimoto and Taylor, Denef et.al.] STr

• H(F)F 2

= −1 2

2

∑

a=1

F 2

a

I(H) + ∑

a=0,3

· · · with I(H) = 1

0 dαH(F 0 + αF 3) + 1 0 dαH(F 0 − αF 3)

2

Introduction Holographic set-up The ρ meson mass Tχ

L[ ˜

A]: back to the ρ meson EOM

L5D = STr

• ..[ ˜

Am, τ]2 + ..(Fµν)2 + ..(Fµu)2 + ..F µν[ ˜ Aµ, ˜ Aν]

• with all the .. different functions H(∂uτ, F; u) of the background

fields ∂uτ, F.

STr-prescription [Myers, Hashimoto and Taylor, Denef et.al.] STr

• H(F)F 2

= −1 2

2

∑

a=1

F 2

a

I(H) + ∑

a=0,3

· · · with I(H) = 1

0 dαH(F 0 + αF 3) + 1 0 dαH(F 0 − αF 3)

2

L5D = −1 4f1(B)(F a

µν)2 − 1

2f2(B)(F a

µu)2 − 1

2f3(B)F 3

ijǫ3ab ˜

Aa

i ˜

Ab

j

−1 2f4(B)( ˜ Aa

µ)2(τ3)2−1

2f5(B)( ˜ Aa

i )2(τ3)2

Introduction Holographic set-up The ρ meson mass Tχ

EOM for ρ for u0 > uK

S5D =

• d4x
• du

         −1 4f1(B) (F a

µν)2 (F a

µν)2ψ2

−1 2f2(B) (F a

µu)2 (ρa

µ)2(∂uψ)2

−1 2f3(B)F 3

ijǫ3ab ˜

Aa

i ˜

Ab

j ρa

i ρb j ψ2

−1 2f4(B) ( ˜ Aa

µ)2 (ρa

µ)2ψ2

(τ3)2−1 2f5(B) ( ˜ Aa

i )2 (ρa

i )2ψ2

(τ3)2          with ˜ Aµ = ρµ(x)ψ(u)

Introduction Holographic set-up The ρ meson mass Tχ

EOM for ρ for u0 > uK

S5D =

• d4x
• du

         −1 4f1(B) (F a

µν)2 (F a

µν)2ψ2

−1 2f2(B) (F a

µu)2 (ρa

µ)2(∂uψ)2

−1 2f3(B)F 3

ijǫ3ab ˜

Aa

i ˜

Ab

j ρa

i ρb j ψ2

−1 2f4(B) ( ˜ Aa

µ)2 (ρa

µ)2ψ2

(τ3)2−1 2f5(B) ( ˜ Aa

i )2 (ρa

i )2ψ2

(τ3)2          with ˜ Aµ = ρµ(x)ψ(u) demand du f1(B)ψ2 = 1 and du f2(B)(∂uψ)2+f4(B)(τ3)2ψ2 = m2

ρ(B),

then du f3(B)ψ2 = k(B)= 1 and du f5(B)(τ3)2ψ2 = m2

+(B)

S4D =

• d4x
• −1

4(Fa

µν)2 − 1

2m2

ρ(B)(ρa µ)2 − 1

2k(B)F 3

ijǫ3abρa i ρb j − 1

2m2

+(B)(ρa i )2

• (with Fa

µν = Dµρa ν − Dνρa µ)

Introduction Holographic set-up The ρ meson mass Tχ

EOM for ρ for u0 > uK

S5D =

• d4x
• du

         −1 4f1(B) (F a

µν)2 (F a

µν)2ψ2

−1 2f2(B) (F a

µu)2 (ρa

µ)2(∂uψ)2

−1 2f3(B)F 3

ijǫ3ab ˜

Aa

i ˜

Ab

j ρa

i ρb j ψ2

−1 2f4(B) ( ˜ Aa

µ)2 (ρa

µ)2ψ2

(τ3)2−1 2f5(B) ( ˜ Aa

i )2 (ρa

i )2ψ2

(τ3)2          with ˜ Aµ = ρµ(x)ψ(u) demand du f1(B)ψ2 = 1 and du f2(B)(∂uψ)2+f4(B)(τ3)2ψ2 = m2

ρ(B),

then du f3(B)ψ2 = k(B)= 1 and du f5(B)(τ3)2ψ2 = m2

+(B)

S4D =

• d4x
• −1

4(Fa

µν)2 − 1

2m2

ρ(B)(ρa µ)2 − 1

2k(B)F 3

ijǫ3abρa i ρb j − 1

2m2

+(B)(ρa i )2

• (with Fa

µν = Dµρa ν − Dνρa µ)

modified 4D Lagrangian for a vector field in an external EM field

Introduction Holographic set-up The ρ meson mass Tχ

Solve the eigenvalue problem

The normalization condition and mass condition on the ψ combine to the eigenvalue equation f −1

1

∂u(f2∂uψ) − f −1

1

f4(τ3)2ψ = −m2

ρψ with b.c. ψ(x = ±π/2) = 0, ψ′(x = 0) = 0

which we solve with a numerical shooting method to obtain m2

ρ(B).

0.2 0.4 0.6 0.8 B GeV2 0.65 0.70 0.75 0.80 mΡ

2 GeV2

0.2 0.4 0.6 0.8 B GeV2 1.005 1.010 1.015 k

Introduction Holographic set-up The ρ meson mass Tχ

Solve the eigenvalue problem

0.2 0.4 0.6 0.8 B GeV2 0.0012 0.0010 0.0008 0.0006 0.0004 0.0002 m

2 GeV2

0.2 0.4 0.6 0.8 B GeV2 0.65 0.70 0.75 0.80 mΡ

2 GeV2

0.2 0.4 0.6 0.8 B GeV2 1.005 1.010 1.015 k

Introduction Holographic set-up The ρ meson mass Tχ

Landau vs Sakai-Sugimoto u0 > uK

Modified 4D Lagrangian for a vector field in an external EM field with k(B)= 1 modified Landau levels and m2

ρ,eff (B) = m2 ρ(B)+m2 +(B) − k(B) B

Introduction Holographic set-up The ρ meson mass Tχ

Landau vs Sakai-Sugimoto u0 > uK

Modified 4D Lagrangian for a vector field in an external EM field with k(B)= 1 modified Landau levels and m2

ρ,eff (B) = m2 ρ(B)+m2 +(B) − k(B) B

0.2 0.4 0.6 0.8

B GeV2

0.2 0.2 0.4 0.6

mΡ,eff

2 GeV2

Landau SakaiSugimoto

Introduction Holographic set-up The ρ meson mass Tχ

ρ meson condensation in Sakai-Sugimoto

Antipodal embedding (u0 = uK ) ⇒ Landau levels

0.2 0.4 0.6 0.8 B GeV2 0.2 0.2 0.4 0.6 mΡ,eff

2 GeV2

Non-antipodal embedding (u0 > uK ) ⇒ modified Landau levels

0.2 0.4 0.6 0.8 B GeV2 0.2 0.2 0.4 0.6 mΡ,eff

2 GeV2 Landau SakaiSugimoto

Introduction Holographic set-up The ρ meson mass Tχ

Full DBI-action

Reasons for considering full DBI-action:

Introduction Holographic set-up The ρ meson mass Tχ

Full DBI-action

Reasons for considering full DBI-action:

Expansion parameter in action det(g + iF) = det g × det(1 + g−1iF) is g−1iF ⇒ most strict condition eB ≪ 3

2

u0,d(B=0)

R

3/2 (2πα′)−1 ≡ 0.45 GeV2

Introduction Holographic set-up The ρ meson mass Tχ

Full DBI-action

Reasons for considering full DBI-action:

Expansion parameter in action det(g + iF) = det g × det(1 + g−1iF) is g−1iF ⇒ most strict condition eB ≪ 3

2

u0,d(B=0)

R

3/2 (2πα′)−1 ≡ 0.45 GeV2 α′-corrections can cause magnetically induced tachyonic instabilities

• f W -boson strings, stretching between separated D3-branes, to

disappear; the Landau level spectrum for the W -boson receives large α′-corrections in general [Bolognesi 1210.4170; Ferrara hep-th/9306048].

Introduction Holographic set-up The ρ meson mass Tχ

Full DBI-action

Reasons for considering full DBI-action:

Expansion parameter in action det(g + iF) = det g × det(1 + g−1iF) is g−1iF ⇒ most strict condition eB ≪ 3

2

u0,d(B=0)

R

3/2 (2πα′)−1 ≡ 0.45 GeV2 α′-corrections can cause magnetically induced tachyonic instabilities

• f W -boson strings, stretching between separated D3-branes, to

disappear; the Landau level spectrum for the W -boson receives large α′-corrections in general [Bolognesi 1210.4170; Ferrara hep-th/9306048]. S4D =

• d4x
• −1

4(Fa

µν)2 − 1

2m2

ρ(B)(ρa µ)2−1

2b(B)(Fa

12)2

−1 2k(B)F 3

ijǫ3abρa i ρb j − 1

2m2

+(B)(ρa i )2−1

2a(B)((Fa

i3)2 + (Fa i0)2)

Introduction Holographic set-up The ρ meson mass Tχ

Full DBI-action

Reasons for considering full DBI-action:

Expansion parameter in action det(g + iF) = det g × det(1 + g−1iF) is g−1iF ⇒ most strict condition eB ≪ 3

2

u0,d(B=0)

R

3/2 (2πα′)−1 ≡ 0.45 GeV2 α′-corrections can cause magnetically induced tachyonic instabilities

• f W -boson strings, stretching between separated D3-branes, to

disappear; the Landau level spectrum for the W -boson receives large α′-corrections in general [Bolognesi 1210.4170; Ferrara hep-th/9306048]. S4D =

• d4x
• −1

4(Fa

µν)2 − 1

2m2

ρ(B)(ρa µ)2−1

2b(B)(Fa

12)2

−1 2k(B)F 3

ijǫ3abρa i ρb j − 1

2m2

+(B)(ρa i )2−1

2a(B)((Fa

i3)2 + (Fa i0)2)

• Further modified 4D Lagrangian for a vector field in an external EM field

Introduction Holographic set-up The ρ meson mass Tχ

4-dimensional EOM

Standard Proca EOM for charged rho meson ρµ = (ρ1

µ + iρ2 µ)/

√ 2 D2

µρν − 2iF 3 µνρµ − DνDµρµ − m2 ρρν = 0,

Dνρν = 0

with Dµ = ∂µ + iA3

µ and Fµν = Dµρν − Dνρµ

Introduction Holographic set-up The ρ meson mass Tχ

4-dimensional EOM

Standard Proca EOM for charged rho meson ρµ = (ρ1

µ + iρ2 µ)/

√ 2 D2

µρν − 2iF 3 µνρµ − DνDµρµ − m2 ρρν = 0,

Dνρν = 0

with Dµ = ∂µ + iA3

µ and Fµν = Dµρν − Dνρµ

replaced by (1 + a)D2

µρν − i(1 + b + k)F 3 µνρµ − (1 + a)DνDµρµ

− (m2

ρ + m2 +)ρν + (b − a)(D2 j ρν − DνDjρj) = 0,

Dνρν = i m2

ρ

(1 + b − k)F

3 µνDνρµ − m2 +

m2

ρ

Diρi

Introduction Holographic set-up The ρ meson mass Tχ

Generalized Landau levels

Landau levels ǫ2

n,sz(pz) = p2 z + m2 ρ + (2n − 2sz + 1)B

Introduction Holographic set-up The ρ meson mass Tχ

Generalized Landau levels

Landau levels ǫ2

n,sz(pz) = p2 z + m2 ρ + (2n − 2sz + 1)B

replaced by

ǫ2

n(pz) = Bp2 z +

m2

ρ + m2 +

1 + a + (2n + 1)B(B − M 2 ) + (1 + b − k) 2 B2 m2

ρ

± B

• M

(2n + 1)2 4 + K − 2B

• + (K − 2B)2

−(1 + b − k)(2n + 1)ξ(K − 2B + M 2 ) + (1 + b − k)2 4 ξ2 1/2

with B = 1 + b 1 + a , K = 1 + b + k 1 + a , M = b − a 1 + a − m2

+

m2

ρ

and ξ = B m2

ρ

Introduction Holographic set-up The ρ meson mass Tχ

Effective ρ meson mass from full DBI-action

Condensing solution n = 0, pz = 0 for transverse charged ρ mesons ρ = (ρ−

x − iρ− y ) and ρ† = (ρ+ x + iρ+ y )

m2

ρ,eff (B) = m2 ρ − B

Introduction Holographic set-up The ρ meson mass Tχ

Effective ρ meson mass from full DBI-action

Condensing solution n = 0, pz = 0 for transverse charged ρ mesons ρ = (ρ−

x − iρ− y ) and ρ† = (ρ+ x + iρ+ y )

m2

ρ,eff (B) = m2 ρ − B

becomes m2

ρ,eff (B) =

m2

ρ(B) + m2 +(B)

1 + a(B) − k(B) 1 + a(B)B

Introduction Holographic set-up The ρ meson mass Tχ

ρ meson condensation in Sakai-Sugimoto

Antipodal embedding (u0 = uK ) ⇒ Landau levels

0.2 0.4 0.6 0.8 1.0 1.2 1.4B GeV2 0.8 0.6 0.4 0.2 0.2 0.4 0.6 mΡ,eff

2 GeV2 Landau full DBI

Non-antipodal embedding (u0 > uK ) ⇒ modified Landau levels

0.2 0.4 0.6 0.8 B GeV2 0.2 0.2 0.4 0.6 mΡ,eff

2 GeV2 Landau F2 approximation full DBI

Introduction Holographic set-up The ρ meson mass Tχ

Effect of Chern-Simons action and mixing with pions

S = SDBI + SCS with SCS ∼

• Tr
• ǫmnpqrAmFnpFqr + O( ˜

A3)

• ρπB mixing terms in the Chern-Simons action:

SCS ∼ B ∂[0π0ρ0

3] + 1

2

• ∂[0π+ρ−

3] + ∂[0π−ρ+ 3]

• + · · · ,

but only between pions and longitudinal ρ meson components so no influence of pions on condensation of transversal ρ meson components (in order ˜ A2 analysis)

Introduction Holographic set-up The ρ meson mass Tχ

Conclusion: back to objectives

Studied effect: ρ meson condensation

phenomenological models: Bc = m2

ρ = 0.6 GeV2

lattice simulation: slightly higher value of Bc ≈ 0.9 GeV2 holographic approach:

can the ρ meson condensation be modeled? yes can this approach deliver new insights? e.g. taking into account constituents, effect on Bc 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 Bc ≈ 0.8 GeV2. Effect of full DBI is further increase of Bc.

Introduction Holographic set-up The ρ meson mass Tχ

Overview

1

Introduction

2

Holographic set-up The Sakai-Sugimoto model Introducing the magnetic field

3

The ρ meson mass Taking into account constituents Full DBI-action Effect of Chern-Simons action and mixing with pions

4

Chiral temperature

Introduction Holographic set-up The ρ meson mass Tχ

Chiral symmetry

• Massless QCD-Lagrangian

ψiγµDµψ − 1 4F 2

µν

invariant under chiral symmetry transformations (gL, gR) ∈ U(Nf )L × U(Nf )R ψL → gLψL, ψR → gRψR with ψL = 1 2(1 − γ5)ψ, ψR = 1 2(1 + γ5)ψ.

Introduction Holographic set-up The ρ meson mass Tχ

Chiral symmetry

• Massless QCD-Lagrangian

ψiγµDµψ − 1 4F 2

µν

invariant under chiral symmetry transformations (gL, gR) ∈ U(Nf )L × U(Nf )R ψL → gLψL, ψR → gRψR with ψL = 1 2(1 − γ5)ψ, ψR = 1 2(1 + γ5)ψ.

• But chiral symmetry U(Nf )L × U(Nf )R not reflected in mass

spectrum of the mesons...

Introduction Holographic set-up The ρ meson mass Tχ

Chiral symmetry

• Massless QCD-Lagrangian

ψiγµDµψ − 1 4F 2

µν

invariant under chiral symmetry transformations (gL, gR) ∈ U(Nf )L × U(Nf )R ψL → gLψL, ψR → gRψR with ψL = 1 2(1 − γ5)ψ, ψR = 1 2(1 + γ5)ψ.

• But chiral symmetry U(Nf )L × U(Nf )R not reflected in mass

spectrum of the mesons... Explanation: spontaneous chiral symmetry breaking U(Nf )L × U(Nf )R → U(Nf )

Introduction Holographic set-up The ρ meson mass Tχ

Chiral temperature

Tχ = temperature at which chiral symmetry is restored U(Nf )

Tχ

→ U(Nf )L × U(Nf )R

Introduction Holographic set-up The ρ meson mass Tχ

Chiral temperature

Tχ = temperature at which chiral symmetry is restored U(Nf )

Tχ

→ U(Nf )L × U(Nf )R Studied effect: possible split between Tc and Tχ(B)

Introduction Holographic set-up The ρ meson mass Tχ

Split between Tc and Tχ

Expected behaviour (Fig from ’08):

Tχ(B) ր: “chiral magnetic catalysis” seen in chirally driven models (e.g. NJL) [hep-ph/0205348] Tc(B) ց: paramagnetic gas of quarks thermodynamically favoured

[0803.3156] (e.g. bag model [1201.5881])

Introduction Holographic set-up The ρ meson mass Tχ

Some results in different models

PLSMq model [Mizher et.al., 1004.2712] Lattice [D’Elia et.al., 1005.5365] Different PNJL models [Gatto and Ruggieri, 1012.1291]

Introduction Holographic set-up The ρ meson mass Tχ

Sakai-Sugimoto at finite temperature

“Black D4-brane background”

ds2 = u R 3/2 (ˆ f (u)dt2 + δijdxidxj + dτ2) + R u 3/2 du2 ˆ f (u) + u2dΩ2

4

• ˆ

f (u) = 1 − u3

T

u3 , uT ∼ T 2

Introduction Holographic set-up The ρ meson mass Tχ

Numerical fixing of holographic parameters

Input parameters fπ = 0.093 GeV and mρ = 0.776 GeV fix all holographic parameters except L. Choice of L left free, determines the choice of holographic theory:

L very small ∼ NJL-type boundary field theory L = δτ/2 maximal ∼ maximal probing of the gluon background (original antipodal Sakai-Sugimoto)

Introduction Holographic set-up The ρ meson mass Tχ

Sakai-Sugimoto at finite T and B

no backreaction ⇒ Tc independent of B B-dependent embedding of flavour branes ⇒ Tχ(B):

Smerged − Sseparated = 0 ⇒ Tχ

Introduction Holographic set-up The ρ meson mass Tχ

Conclusion on Tχ(B)

The appearance of a split between Tχ (GeV) (blue) and Tc (GeV)

(purple) depends on the choice of L:

0.0 0.5 1.0 1.5 2.0 2.5 3.0 eB 0.11 0.12 0.13 0.14 0.15 TΧ 0.0 0.5 1.0 1.5 2.0 2.5 3.0 eB 0.105 0.110 0.115 0.120 0.125 TΧ 0.5 1.0 1.5 2.0 2.5 3.0 eB 0.05 0.10 0.15 0.20 0.25 TΧ

Plots for fixed L (from small to large) respectively corresponding to mq(eB = 0) = 0.357, 0.310 and 0.272 GeV and Tc = 0.103, 0.115 and 0.123 GeV

[N.C. and Dudal, 1303.5674]

Left: split for L small enough ∼ NJL results [1012.1291] Middle and right: split only at large B or no split at all for parameter values that match best to QCD ∼ lattice data of

[Ilgenfritz et.al.,1203.3360] (no split, also quenched)

Introduction Holographic set-up The ρ meson mass Tχ

Chiral transition in Sakai-Sugimoto

Antipodal embedding (u0 = uK ) ⇒ no split

0.5 1.0 1.5 2.0 2.5 3.0 eB 0.05 0.10 0.15 0.20 0.25 TΧ

Non-antipodal embedding (u0 > uK ) ⇒ appearance split depending on L

0.0 0.5 1.0 1.5 2.0 2.5 3.0 eB 0.11 0.12 0.13 0.14 0.15 TΧ 0.0 0.5 1.0 1.5 2.0 2.5 3.0 eB 0.105 0.110 0.115 0.120 0.125 TΧ 0.5 1.0 1.5 2.0 2.5 3.0 eB 0.05 0.10 0.15 0.20 0.25 TΧ

Introduction Holographic set-up The ρ meson mass Tχ

Inverse magnetic catalysis

Latest lattice data disagree with all previous results: Tχ(B) ց Unquenched!

[Bali et.al. 1111.4956 and 1206.4205, Bruckmann et.al. 1303.3972]

Introduction Holographic set-up The ρ meson mass Tχ

Thank you for your attention! Questions?