The D3-D7 Model of AdS/CFT with Flavour (with Special...
Transcript of The D3-D7 Model of AdS/CFT with Flavour (with Special...
The D3-D7 Model of AdS/CFT with Flavour(with Special Feature)
René Meyer
Max-Planck-Institute for PhysicsMunich, Germany
November 13, 2008
Lot’s of known stuff (Review: hep-th/0711.4467 ) +JHEP 0712:091,2007 (with J. Erdmenger & J. P. Shock)
René Meyer (MPI Munich) Flavour AdS/CFT November 13, 2008 1 / 43
Outline
1 Introduction to AdS/CFT
2 The D3-D7 model: Adding Flavour to AdS/CFT
3 Electric and Magnetic Field BackgroundsMagnetic FieldElectric Field
4 Summary
René Meyer (MPI Munich) Flavour AdS/CFT November 13, 2008 2 / 43
Outline
1 Introduction to AdS/CFT
2 The D3-D7 model: Adding Flavour to AdS/CFT
3 Electric and Magnetic Field BackgroundsMagnetic FieldElectric Field
4 Summary
René Meyer (MPI Munich) Flavour AdS/CFT November 13, 2008 2 / 43
Outline
1 Introduction to AdS/CFT
2 The D3-D7 model: Adding Flavour to AdS/CFT
3 Electric and Magnetic Field BackgroundsMagnetic FieldElectric Field
4 Summary
René Meyer (MPI Munich) Flavour AdS/CFT November 13, 2008 2 / 43
Outline
1 Introduction to AdS/CFT
2 The D3-D7 model: Adding Flavour to AdS/CFT
3 Electric and Magnetic Field BackgroundsMagnetic FieldElectric Field
4 Summary
René Meyer (MPI Munich) Flavour AdS/CFT November 13, 2008 2 / 43
Introduction to AdS/CFT
Outline
1 Introduction to AdS/CFT
2 The D3-D7 model: Adding Flavour to AdS/CFT
3 Electric and Magnetic Field BackgroundsMagnetic FieldElectric Field
4 Summary
René Meyer (MPI Munich) Flavour AdS/CFT November 13, 2008 3 / 43
Introduction to AdS/CFT
Motivation
New tools for strongly coupled gauge theories?
QCD in the infrared is strongly coupled (Conformal window?)[Deur, Korsch et. al: hep-ph/0509113]
0.050.060.070.080.090.1
0.2
0.3
0.4
0.50.60.70.80.9
1
2
10-1
1Q (GeV)
α s(Q
)/π
αs,g1/π world data
αs,τ/π OPAL
Burkert-Ioffe
pQCD evol. eq.
αs,g1/π JLab
αs,F3/π
GDH constraint
QGP produced at strong coupling (T >≈ ΛQCD)
René Meyer (MPI Munich) Flavour AdS/CFT November 13, 2008 4 / 43
Introduction to AdS/CFT
Our new tool: AdS/CFT Holography
René Meyer (MPI Munich) Flavour AdS/CFT November 13, 2008 5 / 43
Introduction to AdS/CFT
Our new tool: AdS/CFT Holography
The Original Correspondence (weakest form)
IIB Supergravity on AdS5 × S5 with (R/`s)4 = 2λ 1
ds2AdS5×S5 = R2
(dxµ2 + du2
u2 + dΩ25
)⇔
large Nc limit of N = 4 SU(Nc) Super Yang-Mills with λ = g2YMNc
1 Strong-Weak Coupling Duality: GN ∝ g2s = g4
YM2 Gubser-Klebanov-Polyakov-Witten relation:
〈ei∫
d4xJOO〉 = eiSIIB,onshell[JO]
3 Operator-Field Dictionary:
φm2=∆(∆−4) ' u4−∆JO + u∆〈O〉
René Meyer (MPI Munich) Flavour AdS/CFT November 13, 2008 6 / 43
Introduction to AdS/CFT
A short look at string theory
Basic Objects:Open and Closed Strings, D-Branes
Strings have length: `sStrings have tension: Ts = 1
2πα′ = 12π`2
s
“Minimal Surface” action principle: Nambu-Goto Action
SNG =1
2π`2s
∫dτdσ
√−det P[G]ab
P[G]ab = Gµν∂Xµ
∂σa∂X ν
∂σb
[Becker, Becker, Schwarz]
René Meyer (MPI Munich) Flavour AdS/CFT November 13, 2008 7 / 43
Introduction to AdS/CFT
A short look at string theory
D-Branes: Dirichlet boundaryconditions for the string (e.g.X 1(σ, τ) = 0 for σ = 0, π)
1st Quantized Strings:Open String: “Gauge Theory”
Aµ ,ΦI ,Fermions
Closed String: “Gravity”
Gµν , Bµν , Φ ,Fermions [hep-ph/0701201]
René Meyer (MPI Munich) Flavour AdS/CFT November 13, 2008 8 / 43
Introduction to AdS/CFT
A short look at string theory
D-Branes: Dirichlet boundaryconditions for the string (e.g.X 1(σ, τ) = 0 for σ = 0, π)
D-Branes Curve Spacetime:
Open-Closed String Duality
René Meyer (MPI Munich) Flavour AdS/CFT November 13, 2008 8 / 43
Introduction to AdS/CFT
A short look at string theory
D-Branes: Dirichlet boundaryconditions for the string (e.g.X 1(σ, τ) = 0 for σ = 0, π)
D-Branes Curve Spacetime:
“Graviton” Absorption/Emission
René Meyer (MPI Munich) Flavour AdS/CFT November 13, 2008 8 / 43
Introduction to AdS/CFT
A short look at string theory
D-Branes: Dirichlet boundaryconditions for the string (e.g.X 1(σ, τ) = 0 for σ = 0, π)
Dp-Branes Have Tension:
Tp =1
(2π)p`(p+1)s gs
gs . . . string coupling
“Graviton” Absorption/Emission
René Meyer (MPI Munich) Flavour AdS/CFT November 13, 2008 8 / 43
Introduction to AdS/CFT
A short look at string theory
D-Branes: Dirichlet boundaryconditions for the string (e.g.X 1(σ, τ) = 0 for σ = 0, π)
Dp-Branes Have Tension:
Tp =1
(2π)p`(p+1)s gs
gs . . . string coupling String Splitting/Merging[“D-Branes”, C. V. Johnson]
René Meyer (MPI Munich) Flavour AdS/CFT November 13, 2008 8 / 43
Introduction to AdS/CFT
A short look at string theory
D-Branes: Dirichlet boundaryconditions for the string (e.g.X 1(σ, τ) = 0 for σ = 0, π)
Dp-Branes are “Minimal Hypersurfaces”: (α′ = `2s)
SDp = SDBI + SCS
SDBI = −Tp
∫d(p+1)ξ
√−det (P[G + B]ab + 2πα′Fab)
SCS = +Tpgs∑
p
∫topform
P[C(p+1) ∧ eB] ∧ e2πα′F
René Meyer (MPI Munich) Flavour AdS/CFT November 13, 2008 8 / 43
Introduction to AdS/CFT
A short look at string theory
D-Branes: Dirichlet boundaryconditions for the string (e.g.X 1(σ, τ) = 0 for σ = 0, π)
Dp-Branes carry Gauge Theories:e.g. flat D3-Brane in flat space in the limit α′ << 1
SDBI,p=3 = − 1(2π)3α′2gs
∫d4ξ
√−det (P[G]ab + 2πα′Fab)
≈ −T3
∫d4ξ
√−det P[η]− 1
8πgs
∫d4ξ
√−det P[η]FabF ab
= −T3Vol(R4)− 18πgs
∫d4ξFabF ab
René Meyer (MPI Munich) Flavour AdS/CFT November 13, 2008 8 / 43
Introduction to AdS/CFT
N = 4 d = 4 Super-Yang-Mills-Theory
More precisely: The gauge theory on a stack of Nc D3-branes is theunique maximally supersymmetric gauge theory in 4D,N = 4 Super-Yang-Mills Theory (SYM)
SUSYs: 16 Poincaré + 16 Conformal = 642 real supercharges = 1
2 BPS
Fields: N = 4 Vector Multiplet in 4D = (Aµ, λαA, φi) = (V ,ΦI)
L = tr
− 1
2g2YM
F 2µν +
ΘI
8π2 Fµν Fµν − i4∑
A=1
λAσµDµλA −6∑
i=1
DµφiDµφi
+g2
YM2
∑i,j
[φi , φj ]2 + gYM
∑A,B,i
(CAB
i λA[φi , λB] + h.c.)
Global Symmetries: SO(4,2)× SU(4)R (β(gYM) = 0 !)
René Meyer (MPI Munich) Flavour AdS/CFT November 13, 2008 9 / 43
Introduction to AdS/CFT
D3-Branes in IIB Supergravity
Have seen:1 D3-Branes carry SU(Nc) N = 4 SYM theory2 D3-Branes curve space in a supersymmetric way
D3-Branes are solitonic 12 BPS solutions to IIB Supergravity [Horowitz, Strominger
Nucl.Phys.B360(1991)]
ds2 = H(r)−12 dxµ2 + H(r)
12 (dr2 + r2dΩ2
5)
C4 = H(r)−1dt ∧ dx ∧ dy ∧ dz , H(r) = 1 +R4
r4 ,R4
α′2= 4πgsNc
Asymptotically flatKilling Horizon: gtt = 0 = H(r)−
12 at r = 0
Near horizon geometry: AdS5 × S5 (w. Nc units of 5-form flux on S5)
ds2AdS5×S5 =
r2
R2 dxµ2 +R2
r2 dr2 + R2dΩ25
Q: Connection between these two descriptions of D3-Branes, i.e.between open and closed string physics?
René Meyer (MPI Munich) Flavour AdS/CFT November 13, 2008 10 / 43
Introduction to AdS/CFT
Maldacena’s Decoupling Argument [Juan Maldacena ’98]
1 Field Theory Perspective: D3-Branes in flat space,GN = g2
s `8s = κ2 << 1 (backreaction ∼ GN )
Seff = SIIB︸︷︷︸1
2κ2
∫ √|g|R+...
+ SD3︸︷︷︸∼ 1
κ
+ Sint︸︷︷︸∼κ#>0
→ Low Energy Limit `s → 0, gs, Nc , . . . fixed :
Free IIB SUGRA & N = 4 SYM Theory
2 Soliton Perspective: Observer at Infinity sees (ω << 1/`s)
1 Low-Energy Bulk Modes σD3 ∼ ω3R8 ≈ 0→ Decoupling of Near-Horizon Region→ Free IIB SUGRA
2 Redshifted modes of arbitrary (!) energy from near the horizon→ IIB STRING THEORY on AdS5 × S5
René Meyer (MPI Munich) Flavour AdS/CFT November 13, 2008 11 / 43
Introduction to AdS/CFT
AdS/CFT Forte, Mezzo & Piano
1 SU(Nc) N = 4 SYM ⇔IIB String Theory on AdS5 × S5 (Ncunits of five-form flux,R4
AdS5/S5 = 4πα′2gsNc = 2α′2g2YMNc )
’t Hooft Limit: Nc →∞, λ = g2YMNc fix
2 Planar SU(Nc =∞)N = 4 SYM
⇔Semiclassical Strings on AdS5 × S5
2πgs = g2YM → 0
Strong Coupling Limit: R4
`4s
= 2λ >> 1
3
Planar SU(Nc =∞)N = 4 SYM atstrong ’t Hooft coupling
⇔ IIB Supergravity on AdS5 × S5
René Meyer (MPI Munich) Flavour AdS/CFT November 13, 2008 12 / 43
Introduction to AdS/CFT
The AdS/CFT Correspondence
The Original Correspondence (weakest form)
IIB Supergravity on AdS5 × S5 with (R/`s)4 = 2λ 1
ds2AdS5×S5 = R2
(dxµ2 + du2
u2 + dΩ25
)⇔
large Nc limit of N = 4 SU(Nc) Super Yang-Mills with λ = g2YMNc
1 Strong-Weak Coupling Duality: GN ∝ g2s = g4
YM2 Gubser-Klebanov-Polyakov-Witten relation:
〈ei∫
d4xJOO〉 = eiSIIB,onshell[JO]
3 Operator-Field Dictionary:
φm2=∆(∆−4) ' u4−∆JO + u∆〈O〉
René Meyer (MPI Munich) Flavour AdS/CFT November 13, 2008 13 / 43
Introduction to AdS/CFT
The AdS/CFT Correspondence
closed string sectorAdS x S geometry
open string sectorlarge N D3stack at bottomof throat
55
René Meyer (MPI Munich) Flavour AdS/CFT November 13, 2008 14 / 43
The D3-D7 model: Adding Flavour to AdS/CFT
Outline
1 Introduction to AdS/CFT
2 The D3-D7 model: Adding Flavour to AdS/CFT
3 Electric and Magnetic Field BackgroundsMagnetic FieldElectric Field
4 Summary
René Meyer (MPI Munich) Flavour AdS/CFT November 13, 2008 15 / 43
The D3-D7 model: Adding Flavour to AdS/CFT
Adding Flavour to AdS/CFT [Karch & Katz ’02]
N = 4 Vector Multiplet (V ,ΦI), I = 1,2,3: All Adjoint
How to introduce “Quarks” = fundamental degrees of freedom ?→ Open String Sector in Gravity Dual → Probe Branes
Nc D3-Nf D7 Intersection:
L
N D3s
D7s
7−7
3−3
3−7
7−3
D3:(x0, x1, x2, x3)D7:(x0, x1, x2, x3, y4, y5, y6, y7)
N = 2 Supersymmetric
Field Content:3-3: N = 4 vector multiplet7-3: Nc-Nf chiral multiplet QI
3-7: Nc-Nf chiral multiplet QI7-7: U(Nf ) DBI theory
“Quark” masses: mq = LTs = L2πα′
René Meyer (MPI Munich) Flavour AdS/CFT November 13, 2008 16 / 43
The D3-D7 model: Adding Flavour to AdS/CFT
Adding Flavour to AdS/CFT [Karch & Katz ’02]
How to A) Not spoil Maldacenas argument?B) Decouple 7-7 U(Nf ) theory from the field theory?
’t Hooft Limit = Probe Limit Nc →∞, Nf = fix
1 Field TheoryDecoupling limit still holds: T7 ∼ 1
(gs`8s)
= gsκ2
U(Nf ) becomes global : λ8 = g2YM,8Nf ∼ gs`
4sNc
NfNc∼ λ`4
sNfNc→ 0
→ Quenched Approximation!2 Gravity Side:
VNewton = GNNf T7 ∼ Nf gs ∼ λNfNc→ 0
→ Neglect D7 backreaction on D3 soliton background
René Meyer (MPI Munich) Flavour AdS/CFT November 13, 2008 17 / 43
The D3-D7 model: Adding Flavour to AdS/CFT
Adding Flavour to AdS/CFT [Karch & Katz ’02]
Extended Correspondence: Nf D7 Branes in AdS5 × S5
4d N = 4 SU(N) SuperYang-Mills theory
coupled to4d Nf N = 2 hypermultiplets
in theQuenched Approximation
λ>>1↔
type IIB SUGRA on AdS5 × S5
+Dirac-Born-Infeld & Chern-Simons
theory on D7with
Neglected Backreaction
89
0123
4567
D3N
4R
AdS5
open/closed string duality
7−7
AdS5brane
flavour open/open string duality
conventional
3−7quarks
3−3
SYM
N probe D7f
René Meyer (MPI Munich) Flavour AdS/CFT November 13, 2008 18 / 43
The D3-D7 model: Adding Flavour to AdS/CFT
D3/D7-Model: Field TheoryN = 4 Glue & Nf N = 2 Quarks
L = =[τ
∫d2θd2θ
(tr(ΦieV Φie−V ) + Q†
I eV QI + QI†e−V QI
)+
+τ
∫d2θ
(tr(WαWα) + tr(εijkΦiΦjΦk ) + QI(mq + Φ3)QI
) ],
Φ1 = φ1 + iφ2 + . . . ,Φ2 = φ3 + iφ4 + . . . ,Φ3 = φ5 + iφ6 + . . . ;
τ =θ
2π+ i
4πg2
ym
1 Field Content:N = 4 “Glue” Vector Multiplet: V , Φi , i = 1,2,3N = 2 “Quark” Hypermultiplet: QI , QI
2 Conformal in the Nc →∞ limit: β(λ) = Ncβ(g2ym) ∝ λ2 Nf
Nc→ 0
3 Global Symmetries (mq = 0, VEVs=0):
SO(4,2)× SU(2)Φ × SU(2)R × U(1)R × U(Nf )
René Meyer (MPI Munich) Flavour AdS/CFT November 13, 2008 19 / 43
The D3-D7 model: Adding Flavour to AdS/CFT
D3/D7-Model: Field TheoryN = 4 Glue & Nf N = 2 Quarks
L = =[τ
∫d2θd2θ
(tr(ΦieV Φie−V ) + Q†
I eV QI + QI†e−V QI
)+
+τ
∫d2θ
(tr(WαWα) + tr(εijkΦiΦjΦk ) + QI(mq + Φ3)QI
) ],
Φ1 = φ1 + iφ2 + . . . ,Φ2 = φ3 + iφ4 + . . . ,Φ3 = φ5 + iφ6 + . . . ;
τ =θ
2π+ i
4πg2
ym
components spin SU(2)Φ × SU(2)R U(1)R ∆ U(Nf ) U(1)fΦ1,Φ2 Φ1,Φ2,Φ3,Φ4 0 ( 1
2 ,12 ) 0 1 1 0
λ1, λ212 ( 1
2 , 0) −1 32 1 0
Φ3, Wα Φ5,Φ6 0 (0, 0) +2 1 1 0λ3, λ4
12 (0, 1
2 ) +1 32 1 0
vµ 1 (0, 0) 0 1 1 0Q, Q qm = (q, ¯q) 0 (0, 1
2 ) 0 1 Nf +1ψi = (ψ, ψ†) 1
2 (0, 0) ∓1 32 Nf +1
→ U(1)R acts like a chiral symmetry: . . .mq(iψψ + h.c.) . . .
René Meyer (MPI Munich) Flavour AdS/CFT November 13, 2008 20 / 43
The D3-D7 model: Adding Flavour to AdS/CFT
Supersymmetric Embeddings (Nf = 1)Rewrite AdS5 × S5: r2 = (y4)2 + (y5)2 + (y6)2 + (y7)2 + (z8)2 + (z9)2 = ρ2 + L2
ds2AdS5×S5 =
r2
R2dxµ2 +
R2
r2dr2 + R2dΩ2
5
=ρ2 + L2
R2dxµ2 +
R2
ρ2 + L2
dρ2 + ρ2dΩ23︸ ︷︷ ︸
d~y2
+ dL2 + L2dΦ2︸ ︷︷ ︸d~z2
Match Geometric Symmetries :
SO(4,2) SU(2)Φ SU(2)R U(1)R U(Nf )Iso(AdS5) SU(2)L,~y SU(2)R,~y U(1)~z U(Nf )
Embedding: ξα = (xµ, ρ,S3), L = L(ρ), Φ = 0
Constant L: LDBI ∝ ρ3√
1 + L′2 ⇒(
ρ3L′√1+L′2
)′
= 0 ⇒ L = 2πα′mq
N = 2 SUSY?: ∂Sonshell∂mq
= 0 (or check κ-symmetry)
René Meyer (MPI Munich) Flavour AdS/CFT November 13, 2008 21 / 43
The D3-D7 model: Adding Flavour to AdS/CFT
Supersymmetric Embeddings (Nf = 1)
Dual Operator: Φ3 = φ+ . . .
Omq = i(ψψ + ψ†ψ†) + mq(qq† + q†q) +√
2(qφq† + q†φq + h.c.)︸ ︷︷ ︸∂ 1
2 (|Fq |2+|Fq |2)
∂mq
Why no VEV in AdS5 × S5? A Priori:(ρ3L′√1 + L′2
)′= 0 ⇒ L(ρ)
ρ→∞' 2πα′mq +
(2πα′)3〈Omq 〉ρ2
Reason 1: Om is an F-Term∫
d2θQ(mq + Φ3)QReason 2: Embeddings not well-behaved
ρ3L′√1+L′2
= c1 ⇒ L′ = c1√ρ6−c2
1
ρ→c1+→ ∞ , unless c1 = 0 !
René Meyer (MPI Munich) Flavour AdS/CFT November 13, 2008 22 / 43
The D3-D7 model: Adding Flavour to AdS/CFT
Flavour at Finite Temperature
Flavour Physics at Finite Temperature
AdS-Schwarzschild× S5 (Black brane) , T = THawking ∝ rs
Picture: [hep-th/0611099]
1 Embedding: L(ρ)ρ→∞∼ 2πα′mq + (2πα′)3
ρ2 〈Omq 〉2 No “Spontaneous CSB”: 〈Omq 〉(mq = 0) = 03 Fluctuations: Mesons → Meson Melting Transition
René Meyer (MPI Munich) Flavour AdS/CFT November 13, 2008 23 / 43
The D3-D7 model: Adding Flavour to AdS/CFT
D7 in AdS-Schwarzschild
-1 1 2 3 4 5 6Ρ
-2
-1
1
2L
René Meyer (MPI Munich) Flavour AdS/CFT November 13, 2008 24 / 43
The D3-D7 model: Adding Flavour to AdS/CFT
1st order Phase Transition in AdS-Schwarzschild
0.5 1 1.5 2 2.5 3Quark mass
-0.4
-0.3
-0.2
-0.1
-Quark condensate
1.32 1.33 1.34 1.35 1.36Quark mass
-0.24
-0.23
-0.22
-0.21
-Quark condensate
F (mq,T ) = −SD3,onshell(mq,T )⇒ −〈Omq 〉 =∂F (mq,T )
∂mq
René Meyer (MPI Munich) Flavour AdS/CFT November 13, 2008 25 / 43
Electric and Magnetic Field Backgrounds
Outline
1 Introduction to AdS/CFT
2 The D3-D7 model: Adding Flavour to AdS/CFT
3 Electric and Magnetic Field BackgroundsMagnetic FieldElectric Field
4 Summary
René Meyer (MPI Munich) Flavour AdS/CFT November 13, 2008 26 / 43
Electric and Magnetic Field Backgrounds Magnetic Field
Electric/Magnetic B-Field [see also 0709.1547, 0709.1554 (hep-th)]
Ansatz for the Kalb-Ramond Field
Bel = Bdt ∧ dx , Bmag = Bdy ∧ dz
dB = 0 ⇒ No deformation of AdS-Schwarzschild
−T7
g2
√−det (P[G + B] + 2πα′F ) +
∑p
P[Cp ∧ eB] ∧ e2πα′F
Affects Brane (Flavour) physics: D7 embeddings ,thermodynamics, phase transitions , meson spectra ...Effect: Background for U(1)F gauge field, mimics constantU(1) ∈ U(Nc) field strength
René Meyer (MPI Munich) Flavour AdS/CFT November 13, 2008 27 / 43
Electric and Magnetic Field Backgrounds Magnetic Field
Magnetic Bµν
Magnetic Field
Bmag = Bdy ∧ dz
Zero Temperature: [Filev etal.hep-th/0701001]
CSB, Goldstone Boson with M ∝ √mq for small mq, ZeemansplittingFinite Temperature:
Small B: Meson Melting TransitionNo molten phase & CSB above a critical magnetic field strength(GMOR)Phase diagramSpectrum of Pseudoscalar Mesons
René Meyer (MPI Munich) Flavour AdS/CFT November 13, 2008 28 / 43
Electric and Magnetic Field Backgrounds Magnetic Field
Magnetic Finite Temperature Embeddings
2 4 6 8Ρ
-2
-1
1
2
L@ΡD
B=0
2 4 6 8Ρ
-2
-1
1
2
L@ΡD
B=5
2 4 6 8Ρ
-2
-1
1
2
L@ΡD
B=10
2 4 6 8Ρ
-2
-1
1
2
L@ΡD
B=17
χSB : Bcrit ≈ 16 , B ∝ BT 2
René Meyer (MPI Munich) Flavour AdS/CFT November 13, 2008 29 / 43
Electric and Magnetic Field Backgrounds Magnetic Field
Phase Diagram m = 2mq/(√λT ) , T = m−1
Melted phase
Mesonic phase
0 5 10 15B
0.0
0.2
0.4
0.6
0.8
1.0
1.2
m
Melted phase
Mesonic phase
0 20 40 60 80 100T
0
5
10
15
B
Meson Melting Transition below Bcrit
No molten phase and spontaneous CSB above Bcrit
Magnetic KR-Field acts repells the D7s from the origin
René Meyer (MPI Munich) Flavour AdS/CFT November 13, 2008 30 / 43
Electric and Magnetic Field Backgrounds Magnetic Field
Φ Meson Spectrum (Upper Branch, M = M√
λ√πmq
)
0 1 2 3 4 5m
10
20
30
40
M
B
=0
0 2 4 6 8m
10
20
30
40
50
60
M
B
=5
0 2 4 6m
10
20
30
40
50
60
M
B
=10
0 2 4 6 8m
10
20
30
40
50
60
M
B
=17
René Meyer (MPI Munich) Flavour AdS/CFT November 13, 2008 31 / 43
Electric and Magnetic Field Backgrounds Magnetic Field
Goldstone Boson (B = 16)
0.01 0.02 0.03 0.04 0.05m
0.2
0.4
0.6
0.8
1.0
M
M = 5√
m
René Meyer (MPI Munich) Flavour AdS/CFT November 13, 2008 32 / 43
Electric and Magnetic Field Backgrounds Electric Field
Electric Bµν
Bel = Bdt ∧ dx
Problem: Zero Locus of DBI action√−det P[G + B] = 0 for ρ2
IR + L(ρIR)2 = BR2
2 + 12
√4r4
s + B2R4
Zero Locus
L
rho
René Meyer (MPI Munich) Flavour AdS/CFT November 13, 2008 33 / 43
Electric and Magnetic Field Backgrounds Electric Field
Electric Bµν
Bel = Bdt ∧ dx
Solution: U(1)f gauge field Ax(ρ), At(ρ) [hep-th:0705.3890]
∂a
(δLD7δ∂aAb
)= 0 ⇒ Two Conserved Quantities
At(ρ) ' µ− Dρ2 ↔ finite baryon number density 〈Jt〉 = D
Ax(ρ) ' Bρ2 ↔ baryon number current in x-direction 〈Jx〉 = B
⇒ A′x(ρ) = f ′(D,B, ρ) , A′t(ρ) = g′(D,B, ρ) ⇒ Legendre transform ⇒
Require SD7[L(ρ),D,B] to be well-defined :
B = B(ρIR,T ,B,D)
⇒ Well-defined EOMs for L(ρ)!
René Meyer (MPI Munich) Flavour AdS/CFT November 13, 2008 34 / 43
Electric and Magnetic Field Backgrounds Electric Field
Electric Embeddings at T = 0: No CSB, Phase Transition
-1 1 2 3 4 5 6Ρ
-1.0
-0.5
0.5
1.0
1.5
2.0
L
→ Dissociation of Mesons?→ Conical singularities?
René Meyer (MPI Munich) Flavour AdS/CFT November 13, 2008 35 / 43
Electric and Magnetic Field Backgrounds Electric Field
Condensate vs. Mass
1 2 3 4m
-0.30
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
c
René Meyer (MPI Munich) Flavour AdS/CFT November 13, 2008 36 / 43
Electric and Magnetic Field Backgrounds Electric Field
Condensate vs. Mass: Phase Transition
1.3160 1.3165 1.3170 1.3175 1.3180 1.3185 1.3190m
-0.225
-0.224
-0.223
-0.222
-0.221
-0.220
c
Area∝ F
René Meyer (MPI Munich) Flavour AdS/CFT November 13, 2008 37 / 43
Electric and Magnetic Field Backgrounds Electric Field
Φ (l=0) Meson Spectrum at T = 0: ∆M < 0
1 2 3 4 5m
10
20
30
40
50
M
Incoming Wave Boundary Conditions → Dissociation of Mesons!
René Meyer (MPI Munich) Flavour AdS/CFT November 13, 2008 38 / 43
Electric and Magnetic Field Backgrounds Electric Field
Electric Bµν: Finite Temperature
What to expect at Finite Temperature?Meson melting enhanced by dissociationAny nonzero electric field will decrease the melting temperatureNo SCSBFinite T: One or two transitions?
Open QuestionsFate of the conically singular solutions? They are either
physical → What creates the singularity?unphysical → What happens in that mass range?
Energy of the system is time-dependent →Is this still equilibrium physics?Thermodynamics in the presence of external currents?
René Meyer (MPI Munich) Flavour AdS/CFT November 13, 2008 39 / 43
Summary
Outline
1 Introduction to AdS/CFT
2 The D3-D7 model: Adding Flavour to AdS/CFT
3 Electric and Magnetic Field BackgroundsMagnetic FieldElectric Field
4 Summary
René Meyer (MPI Munich) Flavour AdS/CFT November 13, 2008 40 / 43
Summary
Summary: Electric/Magnetic Background Fields
Magnetic BackgroundInduced SCSBMagnetic Field Stabilizes MesonsMesons don’t melt for large enough B (SCSB)Zeeman splitting
Electric BackgroundNo SCSBDissociation (& Meson Melting)Stark shift
René Meyer (MPI Munich) Flavour AdS/CFT November 13, 2008 41 / 43
Summary
Backup Slides
René Meyer (MPI Munich) Flavour AdS/CFT November 13, 2008 42 / 43
Summary
Condensate vs. Mass
1 2 3 4 5 6
-0.30-0.25-0.20-0.15-0.10-0.05
B
=0
1 2 3 4 5 6
-0.30-0.25-0.20-0.15-0.10-0.05
B
=0.5
1 2 3 4 5 6
-0.3
-0.2
-0.1
0.1
0.2
B
=1
1 2 3 4 5 6
12345
B
=5
1 2 3 4 5 6
468
10
B
=10
1 2 3 4 5 602468
1012
B
=12
1 2 3 4 5 6
5
10
15
B
=15
1 2 3 4 5 60
10
15
20
B
=17
-2 2 4 60
1015202530
B
=20
-4 -2 2 4 60
20
30
40
50
B
=30
-4 -2 2 4 6
406080
100120140
B
=60
-10 10 20 30
2000400060008000
10 000
B
=1000
René Meyer (MPI Munich) Flavour AdS/CFT November 13, 2008 43 / 43