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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:
The AdS/CFT is most conspicuously a duality without open strings.
As a consequence the dual gauge theory has no fields in the fundamentalrepresentation of the gauge group.
– p. 3
Motivation:
The AdS/CFT is most conspicuously a duality without open strings.
As a consequence the dual gauge theory has no fields in the fundamentalrepresentation of the gauge group.
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:
The AdS/CFT is most conspicuously a duality without open strings.
As a consequence the dual gauge theory has no fields in the fundamentalrepresentation of the gauge group.
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
The D3-D7 system (at T=0)
Idea: modify the AdS background by including the supergravity fields thatare sourced by the probe branes.
Any Dp-brane is charged under a certain supergravity field which is ap+ 1 form.
– p. 5
The D3-D7 system (at T=0)
Idea: modify the AdS background by including the supergravity fields thatare sourced by the probe branes.
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
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