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AdS/QCD
D. T. Son
INT, University of Washington
AdS/QCD – p.1/20
The Gauge/Gravity Duality
(Maldacena; Gubser, Klebanov, Polyakov; Witten)
Stack of N D3-branes in type IIB string theory: described in two different pictures:
As a quantum field theory of de- As string theory on a the curved
grees of freedom on the branes: spacetime (induced by the matter
a supersymmetric gauge theory density on the branes)
=
{
N
The limit of infinitely strong coupling in gauge theory is the limit when string theory
becomes Einstein’s general relativity
AdS/QCD – p.2/20
Reminder of AdS/CFT
AdS/CFT correspondence
(J.Maldacena hep-th/9711200, review: hep-th/9905111)
large Nc , d=4, N =4 SYM = IIB strings on AdS5 × S 5
2
R2
λ↔ α string corrections to SUGRA
λ
4πNc ↔ gs string loops
horizon boundary
R
h(x)T (x)
e field = Zstring [g(x, z→0) = h(x)]
AdS 5 R3,1 Tµν (x) ↔ hµν (x, z→0)
Jµ (x) ↔ Aµ (x, z→0)
field theory trF 2 (x) ↔ ϕ(x, z→0)
.
.
.
strings
δ 2 ln Zstring [h] δ2
∴ Tµν Tαβ ∼ δhµν δhαβ ∼ δhµν δhαβ Scl [h]
z=zh z=0
Duality unproven, but many consistency checks performed
(stolen from Pavel Kovtun)
AdS/QCD – p.3/20
Application of AdS/CFT: thermodynamics
AdS/QCD – p.4/20
Application of AdS/CFT: thermodynamics
AdS/QCD – p.4/20
Application of AdS/CFT: thermodynamics
=
Thermal gauge theory = black hole in anti de-Sitter space
AdS/QCD – p.4/20
Application of AdS/CFT: thermodynamics
=
Thermal gauge theory = black hole in anti de-Sitter space
Consequence:
entropy of thermal gauge theory = Bekenstein-Hawking entropy of the black hole
∼ area of the event horizon. Factor 3/4 for strongly coupled N = 4 SYM theory.
AdS/QCD – p.4/20
Dynamics of the horizon
x
r
perturbed horizon
T ∼ r0 = r0 (x)
unperturbed horizon
Generalizing black hole thermodynamics M , Q,... to black brane hydrodynamics
T = TH (x), µ = µ(x)
Event horizon behaves as a viscous fluid with viscosity. Ratio of viscosity and
volume entropy density is universal:
η
=
s 4π
AdS/QCD – p.5/20
AdS/CFT inpired hadronic models
So far we have considered N = 4 SYM theory which is a conformal field theory,
but
AdS/QCD – p.6/20
Confining theories with gravity duals
Many more examples of gauge/gravity duality have ben constructed
including theories with confinement and chiral symmetry breaking
To be able to compute using string theory, the gauge coupling needs to be large at
all scales
g2N
1
Λ QCD E
Consequence: 2 different scales
1
ΛQCD
α
masses of lowest hadrons masses of string excitations
spin ≤ 2 with arbitrary spin
AdS/QCD – p.7/20
Approaches to QCD
top-down: start from N = 4 SYM, adding perturbations to break
supersymmetry, conformal invariance,... to get theories with features similar
to QCD
bottom-up:
Start from known phenomenology
Build a model based on the qualitative feature
We will see that the simplest model works better than one would expect a
priori (i.e„ does not work at all).
Refs: Erlich, Katz, DTS, Stephanov hep-ph/0501128, Da Rold & Pomarol hep-
ph/0501218
AdS/QCD – p.8/20
Hadronic physics in AdS/CFT
4D 5D
hadrons normalizable modes ψ(x)
hadron mass2 eigenvalue of a 5D operator
ψ (z)
decay constant 0|Jµ |ρ lim
z→0 z#
meson coupling overlap integral
dz z # ψA (z)ψB (z)ψC (z)
R
gABC
In QCD the decay constants determine electromagnetic and weak decay rates
(i.e., gρ determines Γ(ρ → e+ e− )
AdS/QCD – p.9/20
Construction of the model
We want a holographic model which exhibits some properties of QCD
For lack of imagination, we take the metric to be AdS5
R2 2
2
ds = 2 (dt − dx2 − dz 2 )
z
Confinement is modeled by cutting off spacetime at some value of the fifth
coordinate
0 < z < zm
and imposing some boundary conditions (e.g., Neuman) at z = zm
AdS/QCD – p.10/20
Implementation of chiral symmetry
Chiral symmetry: two sets of conserved currents in QCD:
µ µ1 − γ5 µ µ1 + γ5
JL ¯
= qγ q, JR ¯
= qγ q
2 2
This means that in 5D we have to introduce two bulk gauge fields:
AL ,
µ AR
µ
This is due to a rule of AdS/CFT correspondence:
operator in 4D ↔ field in 5D
conserved currents ↔ massless gauge fields
AdS/QCD – p.11/20
Chiral symmetry breaking
¯
In QCD chiral symmetry breaking is characterized by the chiral condensate qR qL
¯α β
qR qL ∼ δ αβ , α, β = 1 · · · Nf
⇒ in 5D we introduce a bulk scalar X αβ , bifundamental with respect to
SU (Nf )L × SU (Nf )R
Mass of X: ∆(∆ − 4) = m2 R25D
¯
In QCD dimension of q q is ∆ = 3 (ignorning anomalous dimension)
⇒ m2 = −3/R2
X
χSB ⇒ nonzero vev for X: X = X0 (z)
at small z: X αβ = 1 (mq z + σz 3 )δ αβ
2
explicit symmetry breaking spontaneous χSB by
by quark masses chiral condensate
AdS/QCD – p.12/20
The model
» –
√
Z
3 1
S= d5 x 2 2
g Tr |Dµ X|2 + 2 |X|2 − 2 (FL + FR )
R 4g5
Dµ X = ∂µ X − iAL X + iXAR ,
µ µ FL = dAL + A2
L
FR = dAR + A2
R
on truncated AdS: ds2 = z1 (dt2 − dx2 − dz 2 ), 0 ≤ z ≤ zm
2
The X background should arise dynamically, but we simply take
1
X αβ = (mq z + σz 3 )δ αβ
2
and impose boundary conditions on z = zm (“infrared brane”)
Fzµ = 0, z = zm
Dz X = 0
4 free parameters: mq , σ, zm , g5
cf. 3 free parameters in QCD: mq , ΛQCD , Nc .
AdS/QCD – p.13/20
OPE matching
Using rules of AdS/CFT correspondence one can compute the current-current
correlators, in particular in the regime Q2 m2 :
ρ
1
Vµ Vν = (qµ qν − q 2 gµν )ΠV (Q2 ), ΠV (Q2 ) = 2
ln Q2
2g5
(Q2 ≡ −q 2 ).
Comparing with the lowest order QCD result ⇒:
12π 2
g5 =
Nc
Therefore there are only two free parameters: confimenent scale zm , and the χSB
scale σ (three if counting the quark mass mq )
For any set of parameters, one compute ρ, π, a1 masses from the lowest
eigenvalues of some 5D differential operators; the decay constants from the
behaviors of renormalized 5D wavefunctions near z = 0, and meson coupling
from overlap integrals
Fit the parameters to reproduce experimental results
AdS/QCD – p.14/20
Fitting
Exp Model A Model B
mπ 140 140∗ 140
mρ 776 776∗ 800
ma1 1230 1363 1223
fπ 92.4 92.4∗ 85
1/2
Fρ 345 329 340
1/2
Fa1 433 452 437
gρππ 6.0 4.5 5.1
Input parameters for model B: zm = (333 MeV)−1 , σ = 301MeV
Note: Gell-Mann–Oakes–Renner relationship
m2 = 2mq σ
π
can be proven for small mq , as one should expect.
AdS/QCD – p.15/20
Good features of holographic models
Automatic implementation of quark-hadron duality
At large Euclidean Q2 : parton-model logs, power corrections can also be
modeled
At the same time, correlations functions = sum over poles: from spectral
representation of 5D Green’s functions
2
X fn
2 1
Π(Q ) =
n
m2 Q2 + m2
n n
With enough terms in 5D Lagrangian, many OPE relations can be
satisifed
Phenomelogically good for low-masses resonances
There exist a very similar model, obtained from string theory “top-down”
(Sakai and Sugimoto)
AdS/QCD – p.16/20
What is bad
Only a model, no direct relationship (except OPE matching) to QCD
Not worse than many models
Analogy with linear sigma model
Not unique: higher-order terms in 5D Lagrangian, background...: need more
work to understand constraints
Resonsance mass in a tower grows two fast: m2 ∼ n2 , while semiclassical
n
strings implies m2 ∼ n
n
An attempt to address the last point is made in A. Karch et al., hep-ph/0602229
AdS/QCD – p.17/20
1/2
Why mn ∼ n is preferred?
Semiclassical quantization of a radially oscillating flux tube: maximal length l,
p ∼ l, x ∼ l, px ∼ l2 ∼ n
The behavior observed in ’t Hooft model
Phenomelogically works much better than mn ∼ n:
6
Experiment
m2 , GeV2 0.93*n
n
5
ρ(2150)
4
ρ(1900)
3
ρ(1700)
2
ρ(1450)
1
ρ(770) n
0
0 1 2 3 4 5 6
The semiclassical argument ignores transverse excitations of the (fat) flux
tube, thus one should expect much more large-mass states then m2 ∼ n
n
would predict. We cannot see that from present data.
. AdS/QCD – p.18/20
A smoother IR cutoff
We connsider the following Lagrangian
√ −Φ 1 2
Z
S= d5 x ge Fµν
4
where Φ ∼ z 2
IR sharp cutoff is now replaced by a smoother one
Higher resonances become extended into larger and larger z.
The result is a slower rise of masses: m2 ∼ n
n
Most direct introduction of high-spin fields leads to:
m2 = const(n + S),
n S = spin
i.e., linear Regge trajectories too
However, the connection of this new background to string theory is more tenuous
than before.
AdS/QCD – p.19/20
Conclusion
If string theory did not lead to a solution of QCD, it has lead to
AdS/CFT-inspired modeling of hadrons
We need to understand connections to QCD: constraints from matching to
QCD OPE? Power corrections and higher order terms in 5D.
DIS: importance of incorporating high-spin fields in 5D (infinite number of
twist-2 operators).
Chiral anomaly
Baryons (Teramond-Brodsky)
Strange mesons (Schvellinger)
AdS/QCD – p.20/20
