diana vaman mctp, university of michigan

Holography with backreacted flavor

Diana VamanMCTP, University of Michigan

Miami 2006 Conference

– p. 1

Based on:– Holographic Duals of Flavored N=1 super Yang-Mills: Beyond the ProbeApproximationJHEP 0502:022,2005, hep-th/0406207, B.Burrington, J.T.Liu,

L.Pando Zayas and D.V.

– Regge Trajectories for Mesons in the Holographic Dual of Large-NcQCDJHEP 0506:046,2005, hep-th/0410035, M. Kruczenski, L.Pando Zayas,

J.Sonnenschein and D.V.

– The D3/D7 Background and Flavor Dependence of Regge TrajectoriesPhys.Rev.D72:026007,2005, hep-th/0505164, I.Kirsch and D.V.

– Holograpic phase transition with backreacted flavorJ. Shao and D.V., hep-th/0612...

– p. 2

Motivation:

The AdS/CFT is most conspicuously a duality without open strings.

– p. 3

Motivation:


As a consequence the dual gauge theory has no fields in the fundamentalrepresentation of the gauge group.

– p. 3

Motivation:



Where are the quarks? Adding flavor dof:

To reintroduce the open strings, one adds probe branes to the AdS/CFTscenario [KK]. But, with probe branes, we are forced to consider onlycases Nf ≪ Nc. We’re missing:-chiral phase transition;-quantum moduli of susy gauge theories with flavor dof...

Can one do better?

– p. 3

Motivation:



Where are the quarks? Adding flavor dof:

To reintroduce the open strings, one adds probe branes to the AdS/CFTscenario [KK]. But, with probe branes, we are forced to consider onlycases Nf ≪ Nc. We’re missing:-chiral phase transition;-quantum moduli of susy gauge theories with flavor dof...

Can one do better?

Yes, include the backreaction of the probe branes.Backreated flavor branes ↔ dynamical (virtual) light quarks.

– p. 3

Outline

The D3-D7 system at T = 0 (beyond the probe approximation):

susy variations

a Monge-Ampere eqn

the warp factor: an analytic solution

Regge trajectories

The D3-D7 system at T 6= 0 (beyond the probe approximation):

the solution

the quark condensate

chiral phase transitions

– p. 4

The D3-D7 system (at T=0)

Idea: modify the AdS background by including the supergravity fields thatare sourced by the probe branes.

– p. 5



Any Dp-brane is charged under a certain supergravity field which is ap+ 1 form.

– p. 5



Any Dp-brane is charged under a certain supergravity field which is ap+ 1 form.

D7 branes source a cplx scalar field: the axion-dilaton τ = χ+ ie−φ.

A purely Nf D7 brane geometry:

ds2 = dx2|| + e−φdzdz

τ(z) = −iNf2π

ln

(

z

ρL

)

, ρL = e2π/(gsNf ), z = ρeiϕ.

Pathology at ρ = ρL. For Nf ≤ 12, ∃ solution with a well-behaved dilatonj(τ) = (z/ρL)−Nf [GSVY: stringy cosmic strings].

– p. 5

The fully localized D3-D7 system:

0 1 2 3 4 5 6 7 8 9

D3 − − − − · · · · · ·D7 − − − − − − − − · ·

The ansatz:

ds2 = h−1/2(xm)dxµdxµ + h1/2(xm)

6∑

m,n=1

gmndxmdxn,

(F5)M1...M5 = −ǫM1...M5M6FM6 + 5ǫ[M1...M4

FM5], τ = τ(xm)

– p. 6

The fully localized D3-D7 system:

0 1 2 3 4 5 6 7 8 9

D3 − − − − · · · · · ·D7 − − − − − − − − · ·

The ansatz:

ds2 = h−1/2(xm)dxµdxµ + h1/2(xm)

6∑

m,n=1

gmndxmdxn,

(F5)M1...M5 = −ǫM1...M5M6FM6 + 5ǫ[M1...M4

FM5], τ = τ(xm)

The program:

solve the Killing spinor equations;

the space transverse to the D3’s is Kähler;

the problem factorizes: first solve for the 6d Kähler potential (MA eqn),then solve for the warp factor.

– p. 6

In detail:

the susy variations:

Pm(1 ⊗ Γm)ǫ∗ = 0,

∂µǫ−(

s

8∂n log(h) +

1

2Fn

)

(Γµ ⊗ Γn)ǫ = 0,

∇(6)m ǫ− s

2Fmǫ+

(

1

8∂n log(h) +

s

2Fn

)

(1 ⊗ Γmn)ǫ− i

2Qmǫ = 0,

where PM = i2∂Mττ2

, QM = −∂Mτ12τ2

and γ5ǫ = sǫ.

=⇒ s = 1, Fm ∝ ∂mh, ǫ = cov. const.

The F5 ansatz is the same as without the D7 branes. The 6d space ⊥to D3 is Kähler.

=⇒ The problem factorizes.

– p. 7

In detail (cont.):

use the integrability condition of the 6d susy variation

Rmn = PmP∗n ↔ ∂m∂n

(

ln(detg(6))

)

= ∂m∂n lnℑτ

which yields a Monge-Ampere eqn for the Kähler potential

det(∂m∂nK) = f(z)f(z)ℑ(τ)

Info about the D7 branes configuration determines the rhs.

solve for the warp factor from the Einstein equation

(6)h = − Nc√

detg(6)δ6(xm − xm0 )

– p. 8

The method for solving the D3-D7 system:-choose a holomorphic axion-dilaton with appropriate monodromies-solve the Monge-Ampere equation-solve the warp factor.

– p. 9

The method for solving the D3-D7 system:-choose a holomorphic axion-dilaton with appropriate monodromies-solve the Monge-Ampere equation-solve the warp factor.

Example: A single stack of D7 branes-the source

ℑ(τ) =1

gs− Nf

4πln(z3z3)

-the Kähler potential: K = z1z1 + z2z2 + f(z3z3) obeys

∂3∂3f =1

gs− Nf

4πln(z3z3)

-the solution

ds2(6) = dz1dz1 + dz2dz2 +

(

1

gs− Nf

4πln(z3z3)

)

dz3dz3

– p. 9

The warp factor

Solve the transverse Laplacean

(6) = ∂1∂1 + ∂2∂2 + e−Ψ(z3z3)∂3∂3, eΨ =1

gs− Nf

4πln(z3z3)

Have the D3 and D7 placed at the origin: SO(4) × SO(2) symmetry:r2 = |z1|2 + |z2|2, z3 = ρeiϕ.

Solve the Green’s function:(6)G(ρ, ϕ,~r; ρ′, ϕ′, ~r′) = Nc√

g(6)

δ4(~r− ~r′)δ(ρ− ρ′)δ(ϕ−ϕ′) by decomposing

into spherical harmonics

G = 1 +QD3

∑

l

∫

d4qei~q(~r−~r′)eil(ϕ−ϕ

′)yl,~q(ρ, ρ′)

where(

− ∂2

∂ρ2− 1

ρ

∂

∂ρ+ V (ρ) +

l2

ρ2

)

yl,q(ρ; ρ′) = 0 , V (ρ) =

(

1

gs− Nf

2πlog ρ

)

q2 .

– p. 10

With a change of variable x = log(ρ/ρL), the warp factor reduces tosolving the diff eqn [GP]:

∂2xyq(x) = λxe2xyq(x) , λ =

−Nf2π

ρ2Lq

2 ,

where we defined y(x) ≡ yl=0,~q=0(ρ(x); ρ′ = 0).

There are 2 asymptotic soln: for x→ −∞: y → c, y → ax+ b, which canbe found in terms of a unif. convergent series on x ∈ (−∞, 0]:

y(x) = 2π2∞∑

n=0

λne2nxpn(x) ,

where the polynomials pn(x) are defined recursively by(

4n2 + 4nd

dx+

d2

dx2

)

pn(x) = xpn−1(x) , n = 1, 2, 3, ... ,

p0(x) = −x− x0 − γ .

– p. 11

In the near-core region (ρ≪ ρL ≡ x→ −∞) with ρ2 = λxe2x,

y = c1I0(ρ) + c2K0(ρ)

the warp factor is given by

h(r, ρ) = 1 +QD3

∫ ∞

0

dqq2J1(qr)

2rK0(

√λxe2x)

= 1 +QD3

(r2 + ρ2eΨ)2

100 200 300 400 500

1· 10-7

2· 10-7

3· 10-7

4· 10-7

5· 10-7

6· 10-7

ρy( )

ρ

– p. 12

Decoupling limit

The N = 4 fields correspond to 3-3 strings. In addition we have 3-7strings and 7-7 strings. In the limit α′ → 0, only the 3-3 strings and 3-7(N = 2 hypermultiplets) strings localized at the D3D7 intersection remain:

gDp = gslp−3s ⇒ the D7 worldvolume gauge theory decouples. Also

gs << 1, gsNc=fixed.

The SO(6) R-symmetry group is broken to SU(2)hyper × SU(2)R × U(1)R.

The beta-function (λ = g2YMNc) is non-vanishing:

βλN=2 ≡ NcβN=2 =1

2π

(

λ

4π

)2NfNc

.

For Nf/Nc fixed, the field theory is neither conformal nor asymptoticallyfree.

⇒ the existence of a UV Landau pole (ΛL):

λ(Λ) =1

Nf ln(ΛL/Λ)– p. 13

Matching pathologies

Compare with the wv action for a probe D3 placed in the D3/D7supergravity background:

SD3 = −TD3

∫

d4σe−Φ√

− det(gab + Fab) + TD3

∫

C4 + C0F ∧ F

≈ TD3

∫

d4σ(2πα′)2[

−1

4e−ΦFabF

ab + χFabFab

]

+ ... ,

where χ =Nf

2π ϕ , e−Φ =Nf

4π logρ2Lρ2 .

The D3-brane action relates

g2YM = 4πeΦ, θYM = 2πχ

Upon identifying also Λ2 = ρ2/(2πα′2), ΛL = ρ2L/(2πα

′2), the running ofthe gauge coupling follows from the logarithmic behavior of the dilaton.

The U(1)R chiral anomaly is reflected by the non-trivial axion profile.

– p. 14

Mesons

Mesons of low spin: fluctuations of probe D7.

Mesons of large spin: open spinning strings ending on probe branes.

A phenomenological model of the meson: a spinning open string withmassive end-points.

ω

For a string derivation of this model, introduce probe branes in theconfining supergravity background of choice.

r0

ω

D−brane

– p. 15

Regge trajectories

Solve the classical string eom and find a U-shaped spinning stringconfiguration. The corresponding Regge trajectories J(E2) get acorrection due to the “quark masses” (“vertical arms”)

E =2Tgω

(

arcsinx+1

x

√

1 − x2

)

J =2Tgω2

(

arcsinx+3

2x√

1 − x2

)

, x = speed of the endpoints

⇒: the mass-loaded Chew-Frautschi formula.

Regge regimex→ 1 : J =1

πTgE2

(

1 +

√2

π

(

mQ

E

)3/2

+π − 1

π

mQ

E+ . . .

)

where mQ = (1 − x2)Tg/(ωx).

– p. 16

Flavor dependence of Regge trajectories

Consider an open string rotating in the near-horizon limit of the D3/D7background, ending on a D7 probe at ρR from the stack of D7.

the four-dimensional spacetime: dxµdxµ = −dt2 + dR2 +R2dϕ2 + dz2 ;

the field theory the set-up: Nf massless flavors plus an additionalmassive flavor whose mass is proportional to ρR;

assume a large spin for the meson (semiclassical approx validity);

ansatz for a string rotating with constant angular velocity ω is

t = τ, ϕ = ωτ, R = R(σ), r = r(σ), ρ = ρ(σ) ,

with world sheet coordinates σ and τ

classical Nambu-Goto action:

L = −Ts√

(det ∗ g) = −Ts√

(1 − ω2R2)(h−1R′2 + r′2 + (1 − Nf2π

log ρ)ρ′2) .

– p. 17

the energy and the angular momentum of the spinning string

E =

∫

dσ

(

ω∂L∂ω

− L)

=

∫

dσω

E

√

h−1R′(σ)2 + r′(σ)2 + eψρ′(σ)2 ,

J =

∫

dσ∂L∂ω

=

∫

dσR2

E

√

h−1R′(σ)2 + r′(σ)2 + eψ ρ′(σ)2 ,

with E =√

1 − R2, Ts = 1 and h(r, ρ) = QD3

(r2+ρ2eΨ(ρ))2.

eom in the gauge R = σ:

d

dR

(E2

L ∂Rr

)

=E2

2L∂rh−1,

d

dR

(E2

L eΨ∂Rρ

)

=E2

2L(

∂ρh−1 + (∂Rρ)

2∂ρeΨ

)

.

=⇒ r ≡ 0 is a solution of the equation of motion.

– p. 18

the remaining eom coordinates z = 1/ρ, we find

d

dR

(E2

L eΨ∂Rz−1

)

= −E2

2L z2(

∂zh−1 + (∂Rz

−1)2∂zeΨ

)

.

Neumann bc for the directions || to the probe D7, Dirichlet bc for the ⊥directions =⇒ the string end ⊥ to D7.

solve the string profile z(R)

substitute into the energy and angular momentum =⇒ obtainE = E(

√J)

fix the quark mass: m =∫ ρR

ε

√g00gρρdρ =

∫ ρR

ε( 1gs

− Nf

2π log ρ)1/2dρ

– p. 19

0.25 0.5 0.75 1 1.25 1.5 1.75

0.5

1

1.5

2

2.5

5

3

2

1

0

10

Q

J/1/4

λ

E/m

Figure 1: Chew-Frautschi plot for Nf = 0, 1, 2, 3, 5, 10 additional masslessflavors. The straight line represents the Nf = 0 trajectory for small spinvalues. All graphs approach the horizontal line E = 2m.

– p. 20

2 4 6 8 10Nf

1.2

1.4

1.6

1.8

Tension

Figure 2: String tension in dependence of the number of flavors Nf .

– p. 21

Finite temperature localized D3D7 system

The non-extremal D3-D7 solution in the Einstein frame is given by

ds210 = h(R)−1/2(−f(R) dt2 + d~xd~x) + h(R)1/2

»

f(R)−1 dR2 + R2 sin2 β dΩ23 + R2 dβ2

+ R2 cos2 β

„

1 − 2α + (5 − 4 ln(R cos β))α2

«

dφ2)

–

+ O(α3)

h(R) = 1 +L4

R4+ α2 2L4 ln(R)

R4+ O(α3)

f(R) = 1 −R4

0

R4− α2 2R4

0 ln(R)

R4+ O(α3)

e−Φ = 1 − 2α ln(R cos β) + 2α2(1 − ln(R cos β)) + O(α3)

χ = 2αφ + O(α3)

C(4) = Q

„

1 +L4

R4− α2 2L4 ln(R)

R4

«

−1

dt ∧ dx1 ∧ dx2 ∧ dx3 + O(α3), Q2 =L4 + R4

0

L4

α =gsNf

2π= λ

Nf

Nc

– p. 22

The probe brane embedding

The probe brane Lagrangian

S = −TD7

Z

dt d3x dσ ω3 eΦq

det ∗gαβ + T7

Z

C(8)

= −TD7

Z

dt d3x dR ω3 R3 sin3 β

„

eΦp

1 + R2fβ′2 − (eΦ− 1 − 2α2)(sinβ + R cos ββ′)

«

0.96

5 15

R cos(beta)

1

0.92

0

0.94

R sin(beta)

0.9

10

0.98

20

0th order in alpha

1st order

2nd order

– p. 23

The quark condensate

Far away, the profile of the probe D7 is given by

z ≃ m+c

r2, r → ∞

The composite mass quark is equal to the energy of a string stretchedbetween the D3 stack and the D7 probe, far away from the black holehorizon (∼ m).

The quark condensate is equal to the vev of the hyperquark bilinear ψψ

〈ψψ〉 =δEδmq

Parametrizing the probe brane embedding as z = z(r), we have

L[z(r)] = −TD7ω3r3eΦ(z)

(

1√r2 + z2

√

(r + zdz

dr)2) + f · (z − r

dz

dr)2 − 1

)

– p. 24

δE = δz TD7r3eΦ(z)

z(r + zdz/dr) − fr(z − rdz/dr)√r2 + z2

√

(r + zdz/dr)2 + f(z − rdz/dr)2ω3

∣

∣

∣

∣

r=∞

r=0

= −δmq (2πl2s)2cTD7eΦ/2(m)ω3

where we have used that δz = δm = 2πl2s e−Φ/2δmq. Then, the quark

condensate is given by 〈ψψ〉 = −(2πl2s) 2cTD7eΦ/2(m)ω3

The effect of the backreacted flavor branes on the quark condensate:

Consider a Minkowski brane lying far outside the black hole horizon:

z(r) = z(r = ∞) + δz(r) = m+ δz(R), δz(r) ≪ 1, m≫ R0

z(r) ≃ m+c0 + c1α+ c2α

2

r2, r → ∞

c0 = −m3

96ǫ2, c1 =

m3ǫ

576(72 − 3ǫ), c2 =

m3ǫ

576

„

144(1 + ln(m)) − 10ǫ(1 + 3 ln(m))

«

The quark condensate value decreases at each order in α.

– p. 25

Chiral phase transitions at finite temperature

0.5 1 1.5 2 2.5 31m

-0.1

-0.08

-0.06

-0.04

-0.02

c

1.05 1.06 1.07 1.08 1.09 1.11 1.121m

-0.04

-0.03

-0.02

-0.01

c

Blue curve: Minkowski branes (have a vanishing S3)Red curve: black branes (have a vanishing S1)

The phase transition remains of first order: the quark condensate jumpsdiscontinuously accros the transition.

– p. 26

diana vaman mctp, university of michigan

Documents