Holographic Walking Technicolor

David Kutasov
EFI, University of Chicago
DK, Jennifer Lin and Andrei Parnachev
arXiv:1107.2324, 1201.4123
1201.41231201.412301.4123

Introduction
Strongly coupled fermions play an important role inmany physical systems:• QCD• Technicolor• Condensed matter

Typically, at weak coupling the dynamics is simple,
while at strong coupling one finds many interesting
phenomena, such as dynamical symmetry breaking,mass generation and confinement.
The focus of this lecture will be the transition
between the two regimes. Examples of systems inwhich such transitions occur are:r ΛQCD and below.

n the other hand, when F " N the theory is believed to confine and break the chiral

QCD
try SU (F ) ×SU (F ) → SU (F ). The ’t Hooft large N analysis of perturbation
L

R

diag

[1] provides a nice picture of what happens in this regime. To leading order in F/N ,SU(N) gauge theory coupled to F Dirac fermions iny fields that run in loops are the adjoint degrees of freedom (the gauge fields). The
the fundamental
representation. Expected phase2
um includes F − 1 massless Nambu-Goldstone bosons (“pions”), and a discrete
structure:
um of massive glueballs and mesons with masses of order Λ
.
QCD

µ

!QCD

F

Fc

11
N
2

Fig. 1: Schematic phase diagram of QCD as a function of the number of flavors
F . ΛQCD is the scale above which the theory becomes free; µ is the meson massscale.

• For F slightly below 11N/2 the theory is

weakly coupled at all scales (Banks-Zaks).

• For small F , the theory looks like realworld QCD: one scale, ΛQCD ; well abovethat scale the theory is free; hadron
masses are at that scale as well.

• Near the transition, F � F , scale
separation, µ � ΛQCD .

c

The physics near the transition is not wellunderstood. It is believed that it is driven by
the fermion bilinear ψ̄ψ .
The UV scaling dimension of this operator
is three, but its IR scaling dimension islower due to gauge interactions. It isbelieved that (at large N) as F → Fc , theIR dimension approaches two and for F < Fc
it becomes complex.

This behavior signals an instability of the Coulomb
phase due to a non-vanishing beta function for the``double trace'' operator (Trψ̄ψ) , which leads tothe appearance of the small mass gap µ .
2

Near the transition one has:�µ ∼ ΛQCD exp(−a/ F − Fc )Miransky scaling

Conformal Phase Transition.

Defect fermions
Another class of systems that exhibits conformal
phase transitions is systems of fermions localized
on a d-1 dimensional defect interacting withgauge fields and scalars propagating in the bulk. In
that case the parameter controlling the transitionis the gauge coupling, which can be treated as afree parameter by taking the bulk theory to havea line of fixed points (e.g. N=4 SYM).

Such systems can be embedded in string theory
as low energy theories on D-branes. The gauge
fields live on D-branes, while the fermions are
localized at intersections. Their dynamics can bestudied using standard string theory techniques.
In some regimes in parameter space this analysis
can be done and one finds Miransky scaling at a
critical value of the `t Hooft coupling λ .

More generally, conformal phase transitions
occur when two fixed points of the RG approacheach other, merge, and move off into thecomplex plane (Kaplan, Lee, Son, Stephanov).
In the examples mentioned above, the two fixed
points that merge differ by the coefficient of adouble trace operator O2 , with O a fermion
bilinear.

In general such transitions are difficult to study
since the fixed points in question are strongly
coupled. It is natural to ask whether holography
can be used to study them. The purpose of thiswork is to develop such a description.

The fact that the transition is continuous makes
it natural to look for a description in terms of
the order parameter O . In the bulk thisoperator correspondence to a scalar field T
whose mass approaches the BreitenlohnerFreedman (BF) bound. Thus, to describe thetransition in d dimensional QFT, we need to
study the dynamics of T in AdSd+1 .

What Lagrangian should we take for T ? Intuition
from Landau-Ginzburg theory of phase transitions
might lead one to expect that we only need toconsider the behavior of the Lagrangian for smalland slowly varying T, but this is not true, essentiallybecause CPT’s are infinite order transitions, as isclear from the Miransky scaling formula.

We will pick a class of Lagrangians and study
their dynamics. This should be understood as abottom-up approach that is useful for exploring
possible universality classes. Which universality
class corresponds to a particular transition is aseparate question, that we will not address.
One advantage of this class is that any CPT that
occurs in a system which has a (probe) D-brane
description is described by a Lagrangian of thissort.

TDBI description
In the examples mentioned above, the bulk field T
is dual to a bilinear in fields trasforming in thefundamental representation of the gauge group.
Thus, it is an open string tachyon.
The dynamics of such tachyons has been
extensively studied, and is known to be welldescribed by the Tachyon Dirac-Born-Infeld
action.

d (r, θ) here isρ = r ρcos
= θ;
r cos θ;φ = rφsin= θ.
r sin θ.

(2.7)(2action(2.6) hasthe
TDBI
form form
that wasstudiedin thecontext
of openstring
The action
(2.6)
has
the
TDBI
that
was
studiedinthecontextofopen
str
For a real tachyon, it takes the form:
condensation[16-19,22-24],
hyon condensation[16-19,22-24],

# √
$ $
√
√
√
d+1 d+1
d+1
M N ∂M N
d+1
−G
=
−
d
x
−gV
(T
)
1
+
g
T∂, N T ,= −S =d−xV(T)
NT
d xV (T ) −G = − d x −gV (T ) 1 + g M T∂∂M

#

#

#

(2.8)(2ere

GM N G=MgNM=+M∂NM+T ∂M
T ∂N TN g
NT(2.9)(2ucedmetric,
G = detGM, TM N=, Lθ
theis“tachyon”
field,field,
and andheinducedmetric,G=
det
T =is Lθ
the “tachyon”
NGwhere gM N is the%bulk(AdS)
metric, and V(T)%&n−1&n−1TT= cos
V (T )V=(T )cos
L L
is the tachyon potential.

4

We omitted
an overall
multiplicative
mittedan overall
multiplicative
factor.factor.

7

7

(2.
(2.10)

starting point of our discussion is the TDBI action (2.8). This action describes

literature on open string tachyon dynamics, there have been proposals

hyon field T on AdSd+1 , whose dynamics depends on the choice of a potential

generally interpret the open string tachyon geometrically (see e.g. [22-2e will assume that the field has a mass slightly below the BF bound, i.e. theForsmall T it fits
behaveslikeaninterpretation
naturally
with our results, we will not assume it

V (T ) has a local maximum at the origin:

The starting point of our discussion is the TDBI action (2.8). Th

1 2 2(T ) =T1 on+ AdS
m Td+1+, · whose··a real tachyonV field
dynamics depends on(3.1)
the ch2

V (T ). We will assume that the field has a mass slightly below the B< m2BF = −d2 /4.6 We will further assume that V (T ) is a monotonicallypotential
V
(T
)
has
a
local
maximumattheorigin:where the mass m is taken to be close to the BF

g function of T that goes to zero at a particular value of T , T = TIR . As we

1 2 2
bound
V (T ) = 1 + m T + · · ·and below we take the AdS radius to be L = 1.
2

112 /4.6 We will further assume that V (T )where m2 < m2BF = −ddecreasing function of T that goes to zero at a particular value of T
6

Here and below we take the AdS radius to be L = 1.

11

TIR (see figure 4). In the system of [7] TIR was finite, while for the potentials that arise
in open string tachyon condensation it is typically infinite [42,43]. We will consider both

For larger T it has the formcases below.

V

1

(a)

T IR

!TIR

T

V

1

(b)

T

We will see that there is a qualitative difference
The fact that the potential vanishes somewhere in field space is important. In the
betweenthe two cases.
Fig. 4: The tachyon potential, for finite (a) and infinite (b) TIR .

study of open string tachyon condensation this value of the tachyon corresponds to thestate in which the unstable D-brane (or brane-antibrane pair) disappears [16]. In the defectfermion system of [7] it is the state where the probe D-brane is pushed to infinite φ, so thatthe fermions have an infinite current mass and decouple. Such a decoupling limit shouldIn the remainder of the talk I will describe thephysics of the phase transition described by thisclass of Lagrangians (see papers for details).
I will discuss:(1) Vacuum structure.
(2) Small excitations (mesons).
(3) Finite temperature phase transition.

evelops
a non-trivial
vacuumas
expectation value. The rest of the components of T ,from (3.4)
can be written
(L,R)
&
%
SU
(Nf )∂L ×SUr(N)R" gauge
fields structure
AM
which are dual to the global symmet
d+1
fT
Vacuum∂thelnVT=treated√perturbatively.
, In particular, to study(3.5)
s,r 2canbethespectrumofmesons
"2
2
"2
∂r
T
1+r T
ed to Towork
toquadratic
order
in
these
fields,
in
which
(3.2)
is
unambiguous.
find the vacuum we need to minimize the energy

e vacuum of the model (2.8) is described by specifying a field configuration T
"
3 "3
2 ""
ln V +
(1
+
d)rT
+
dr
T
+
r
T = 0.
(3.6)
function,hich minimizes the energy function obtained from (2.8),!
#
in general, that for any potentialform
d−1 of the qualitative2"
2
E = drr
V (T ) 1 + r T (r) .
(3(i.e. T0 (r) = 0 for all r) when the mass (3.1) is above theuationof motion
following
from
(3.4)
can bewrittenas seee evidence
supporting
this
claim
below.
Wewill
also
where r is theradialdirection
in
AdS.
&
%
d−1
d+1 "
∂
r
∂TtolnaVnon-trivialrT
the vacuum corresponds
solution,T0,(r),√√!
=(3
dary condition
φ
(ρ
=
0)
=
0
translates
in
the
(r,
T
)
parametrization
to2 T "2
2 T "2The
boundary
conditions
are:
∂r
1
+
r
1
+
r
depicted in figure 3. The boundary conditions satisfied
that r(T ) is stationary at TIR ,valently,
hose ofUV:
section 2: T (r = Λ) = 0 in the UV, and (2.13) in

2∂r"2
"
3 "3
2 ""
+ rexcised
T )∂T regionln V + and(1 +Td)rT
+
dr
T
+
r
T = 0.
→ ∞, there−(1
is no
(r)
takes
the
form
0
(T = T ) = 0,(2.13)

IR:

IR

(3

∂T the solution in figure 3, obtained by
ught
of
as
a
limit
of
eve, but have not proven in general, that for any potential of the qualitative fo
a
ly,
∂
T
diverges
as
T
→
T
.
r(T,
x
), thought
as r)a field
livingthe (3.1) is above t
e 4, rthe vacuum is trivial= 0 forofall
when
theon
mass
IR (i.e. T0 (r)
Tnd.
≤ TWe
Dirichlet
boundary
conditions
at T =
0 and
Neumannwill mention
someevidence
supporting
this
claim
below. We will also s
IR , satisfies
13

r

µ

The form of the vacuum depends on the sign of22
κ = mBF − m

For µ
the vacuum
trivial,, and
T (r) = 0 constants
κ is< a0 scale,be positive,
and A, φisaredimensionlesswho

is
unbroken.
. As r conformal
decreases, Tsymmetry(r)increases,
until eventually the linear app0> 0totheFor
unstable,own andoneκ hasgo trivial
back tovacuum
the full isequationof themotion. µ canT (r)�= definition
0 andlowestconfiguration
has ofe at whichthisenergy
happens.
The advantage
thisis thasymmetryis broken.
ls with thefiniteconformal
TIR , whereµ here is of
the order of the scale µ defin

nfinite TIR , where there is no excised region. Of course, this defint can be made precise e.g. by setting µ to be the mass of thedefinitions differ from this one by numerical factors. The ambi

he dynamically generated scale is familiar in QFT, and it can b

µThe formof the vacuum solution is different forr

the two classesof potentials mentioned above.
$#"

For potentials with finite TIR , the vacuum isFig. 2: The shape of the brane parametrized in terms of (ρ, φ) and (r, θ).
T
TIR

µ

#

In particular, a region of sizeris excised fromµ in terms of T (r).
Fig. 3: The shape of the brane in figure 2 parametrized

the space. The resulting models are similar to

In terms of the TDBI variables T , r, the shape of the brane is depicted in figure 3.
that:

hard wall models of AdS/QCD.

In the trivial vacuum T (r) = 0 which preserves conformal symmetry, r ranges fromzero to infinity. In contrast, in the vacuum with broken symmetry the radial coordinate

The dynamically generated scale µ is given by√µ � Λ exp(−c/ κ)Thus, as one approaches the transition it becomesparametrically smaller than the UV cutoff. One
can focus on the physics at the scale µ by taking
the double scaling limitΛ → ∞; κ → 0

with µ held fixed.

For infinite TIR the vacuum solution takes the
form
T

!

r

In this case there is no excised region in space.

Fig. 5: The vacuum configuration T0 (r) for infinite TIR .

The dynamically generated scale can be thought

As is clear from figures 3 and 5, at large r the solution of (3.6) is small, and we can

eplace the full equation of motion by the linearized one,of as the value of r for which T takes somer 2 T !! + (1 + d)rT ! − m2 T = 0.
prescribed value, say T=1.

The general solution of this equation is(3.7)

• The resulting dynamics is very similar tothat of soft wall AdS/QCD.

• The ``open string’’ metric G

isapproximately AdS; the soft wall is provided
by a non-trivial effective dilaton.
MN• The dynamically generated scale again
satisfies Miransky scaling.

Mesons
One can use the bulk description to study the spectrumof small excitations, which give rise to mesons. We will
only discuss scalar mesons. One can study vector
mesons by including gauge fields in the TDBI action, andpseudoscalars and axial vectors by studying thecomplex TDBI.

section, we The
willpresence
derivetheequationsdescribing
fluctuations
of
the
TDB
of a non-trivial tachyon condensate T (r) leads to a deformation of the0
he closed string (AdS) metric g and the dilaton approaches
a constant, but for sm

AdS
d+1 metric gM N tovacuumT(r).
0One
expands
he open and closed
string backgrounds are in general quite different. In the rest
GM N = gM N + ∂M T0 ∂N T0

M(3.12)

Wewillstart
withfluctuationsof the field
T (x ),of which
giveaction
rise toσection, we
derivetheequationsdescribing
fluctuations
the
TDBI
arou
and to a non-trivial dilaton Φ(r). Excitations described by the TDBI action live in thisMM

the fluctuation
viastring,
T (xbackground.
) = T0 (r)), plug(2.8), and e
modified, oryopen
As r +y(x→ ∞, the open
string thismetric into
G approaches
acuum
T0 (r).
the closed string (AdS) metric g and the dilatonMapproaches a constant, but for small r,

We startwithfluctuations
ofaction
the field
T the
(x ),
which give riseto σ-mesons. We
orderin they.
Theresultingforfluctuationsisopen and closed string backgrounds are in general quite different. In the rest of thisMM!
he fluctuation
y
via
T
(x
)
=
T
(r)
+y(x
),
plug
this
into
(2.8),
andexpandsection, we will derive the equations
describing
fluctuationsofthe TDBI
actionaroundthe to qua
0√" yM Nandwritesthequadratic
actionfor
1d−Φ
2
vacuumT(r).
0
S2action= − for the−Ge is G
d xdr∂M y∂N y + m (r)yrder in y. The resulting
fluctuations
2
We start with fluctuationsof the field T (xM ), which give rise to σ-mesons. We define
!

where

M
" M
#
the fluctuation y via(r) +y(x−Φ), plugthis
1 T (xMd) = T0√
N into (2.8), and expand
2 to quadratic
2
Sy. =
−
d xdr −GeG
∂ y∂N y + m (r)yorder in 2
The resulting action for the fluctuations is M

21
S2 = −
2!

√d xdr −Ge
(3.13)V (T0 ),
e where
=
"2
2T )
(1
+
r
where
0
V
(T
)
0
%
$
,
e−Φ =
V 2(T0 )"2
2
"
"2
,
e2−Φ
=
(1
+
r
T
)
1
+
r
T
∂
V
1
∂V
rT
∂V
0 (1 + r 2 T0"2 )
0
2
% 0 %
&
$
$
m (r)
−
(d
−
1)
$=
&
'( −
$
2
2
" ∂V ∂
2"2
2 T "2∂V∂V rT " rT∂T
2 T "2
2
2 2
"
"2
2T
V
∂T
∂T
1
+
r
∂
V
1
r
T
1
+
r
∂
V
1
∂V
∂
r
T
1
+
r
1
+
r
T
0
0
2
0
0
2
0
0
0
$
$
$

where −Φm (r) =

m (r) =

V

d

−Φ

" MN
#
2
2
G
∂M y∂N y + m (r)y− (d(d −−1)1)
−
,
$
$
$
−
−
2
"2
"2
"2
2
2
2
∂T
∂T
∂T
∂r
1 + r∂r
T0∂T 2 1 + 1r 2+Tr0"2T0∂T1 + r 1T0+ r 2 T0"2∂T
1+
(3.14)

VG
is the open string metric (3.12). It is easy to check using (3.1), (3.10), thatand GM andN isM Nthe open string metric (3.12). It is easy to check usinglargeopen
r the dilaton
approaches(3.12).
a constant,Itand
masstofunction(r) goes (3.1),
to the (3.10
nd GM N is atthe
stringΦ metricis theeasy
checkm2using

at large tachyonr themassdilatonΦ approaches a constant, and the mass function
m2 .

2
on
mass
m
. 1 ! of mesons,tudy the spectrum
it is" convenient to change coordinatesfrom r to√#
dd+1 x −he−Φ hαβ ∂α y∂β y + m2 (r(z))y 2
S2 = −
To
spectrumof mesons,it stringis convenient
coordinatesfromnedstudyby thetherequirementthat
the open
metric Gto
takes theform
2
M Nchange(3.17)
After1 a! change of coordinates"#
defined by=the
requirement
that
d+1
d−1
−Φ the open
α string
2 metric
2 2 GM N takes the form
d Nxr
e
−
M
α∂α y∂β y+
2 m (r)rα yβ
GM N2 dx dx = hαβ dx dx = r (z)dx dx ηαβ .
(3.15)
M
N
α
β
2
α
β
G
dx
dx
=
h
dx
dx
=
r
(z)dx
dx
ηαβ . and lowered (MviewedNαβthesecond
line
r
is
as
a
functionofz,andindicesareraised) = (z, xa ). Comparing (3.12) and (3.15) we see that the coordinates z and r are
−B/2
flat
metricη.
Wenext
definethewavefunction
ψ
=
ye
where
αβ
α
a
s(x
follows:) = (z, x ). Comparing (3.12)$and (3.15)
we
see
that
the
coordinateszanddz
1 + r 2 T0"2=−.
(3.16)d as follows:$2
B(z) =(3.18)drΦ(z) − (d −r 1) ln r(z),"22dz
1 + r T0
=−
.
(
2
r
α
a dr 15
it as ψ(x) = gets
ψ(z)exp(ikequation of motionproblem
for ψ(z) that followsa x ). The SchroedingerOne
an effective
157) takes the Schrödinger form−ψ "" + Veff (z)ψ = M2 ψ

(3.19)

prime denotes differentiation w.r.t. z, M2 = −ka k a is the meson mass, and theotential is

1 " 2 1 ""Veff (z) = (B ) − B + r(z)2 m2 (r(z)).
42(3.20)

from (3.17) takes the
form"
√
#
1 Schrödinger
d+1
−Φ
αβ
2a
2
2
d x −heS2 = − differentiation+m
prime denotes
w.r.t.hz, ∂M
is the meson mass, and the
α y∂=
β y−k
a k (r(z))y2""2!
(3.17)(3.19)
−ψ
+
V
(z)ψ
=
M
ψ
eff
potential is"#1
dd+1 xr d−1 e−Φ ∂α y∂ α y + m2 (r)r 2 y 2=−
2
1 " 2 1 w.r.t.
""
22 =2 −ka k a is the meson mass, and thewhere the prime Vdenotes
differentiationz,M(3.20)eff (z) = (B ) − B + r(z) m (r(z)).
4 as a function2
neffective
the secondline risis viewedof z, and indices are raised and lowered
potential

whereflat metric ηαβ . We next define the wavefunction ψ = ye−B/2 wherearge z (IR) behavior of the potential
on the details of the tachyon
1 " 2Veff1(z)"" depends
2 2(3.20)
Veff (z) = (B ) − B + r(z) m (r(z)).
4the potential
2
V (T ). For small z, B(z)however,
is universal: T0 (r) goes to(3.18)
zero as= Φ(z) − (d − 1) ln r(z),he dilaton
Φ(r)andbehavior
mass function
m(r) approach
constants,
r =of1/z.
Thus,
The largez (IR)of the potential
Veff (z) depends
on theanddetails
the tachyon
α
te
it as
)).=
ψ(z)exp(ikxaln). zTheequationof ismotionfor ψ(z)potential
Vfinds(TFor
smallz, a1)
however,thepotential
universal:
T0 (r)that
goesfollows
to zero asgimeoneψ(xB(z)= (d −
andIf one takes T(r)=0(the symmetric solution), one finds
2

theSchrödinger
form
r17)
→ takes∞, thedilaton
Φ(r)$andmass function% m(r) approach$%constants, and r = 1/z. Thus,

d −1

1

1

1

2ln z andin this regimeVone
finds
B(z)
=
(d
−
1)
+
m
=−
+κ
.
(3.21)
(z)
=
eff
2
2
""
2
4
z
−ψ4 $
+ Veff (z)ψ z= Mψ
(3.19)%$%
1
1
1
d2 − 12+m=−+κ
.
(3.21)eff (z) =22nterested in κ > 0 V(themassive4 phase),inthepresenceofaUV
cutoff,which
4 the mesonz
he prime denotes differentiation w.r.t. z, M2 =z −k k a ismass, and theaa potential
lower boundis z ≥ zUV = 1/Λ on z. The boundary condition there is ψ(zUV ) = 0.

We are interested in κ > 0 (the massive phase), in the presence of a UV cutoff, which

tentialwasboundgivenz byfor onallz.zThe≥ zboundary
spectrumthereof (3.19)
UV , the condition
providesVaefflower≥ z(3.21)= 1/Λ
is ψ(z would) = 0.

UV
1 UV " 2 1 ""222 ) − B + r(z) m (r(z)).
(B(3.20)
Veff (z)=Mormalizable
states
with
<
0,
i.e.
tachyons.
The
mass
squaredoftheIf the potential Veff was givenby (3.21)
for all z ≥ zUV , the spectrum of (3.19)lowest
would
4
2
√
2
2
κ). mass
It is easy
to understandhese
tachyons
is of order, whereΛexp(−π/containnormalizable
states−µ
with
M < µ0,∼i.e.
tachyons. The
squaredof the lowest
e large z (IR) behavior of the potential Veff (z) depends on thedetails of the tachyon√
It is easy
to understand
lying
theseoftachyonsof order−µ2system.
, where µ The∼ Λexp(−π/the ofpoint
view ofis the
TDBIpotentialκ).(3.21)
describes
small

1
d −1 $ 2 1%+ md2 − 12 = − 1+
Veff (z) =
4
z
2 4
+m
Veff (z) =4
z2
We are interested in κ > 0 (the massive phase), in the preκ>0This is a well known
potential
in
QM.
For
We are interested in κ > 0 (the massive phase),provides a lower bound z ≥ zUV = 1/Λ on z. The boundary cprovidesaUVlowerbound z ≥ zUV = 1/Λ on z. The b
and in theIf presenceofacutoffthe potential Veff was given by (3.21) for all z ≥ zUV , t
If the potential Veff was givenby (3.21) for all z
2normalizable
stateswith M < 0, i.e.
there arecontainnormalizable
stateswithi.e.2tachyons. Th
contain normalizable states2 with M < 0, i.e. tach√lying of these tachyons is of order −µ , where µ ∼ Λexp(−π/tachyons.
lying of these tachyons is of order −µ2 , where µ ∼ Λthis from the point of view of the TDBI system. The pot
this fromthe point interpretation:of view of the TDBI system.
These tachyons havean obvious
16
they signal the instability of the unbroken (T=0) 16
2−µvacuum. Their mass squared is of order.

Expanding around the stable vacua above
modifies the potential at z ∼ 1/µ and lifts thesetachyons. Interestingly, the lowest lying of themesons is relatively light, since the tachyoniccontribution from small z is approximately
canceled by that of larger z. This light scalar canbe thought of as a pseudo dilaton; it is ofinterest in the context of technicolor.

]. We will refer to them as soft wall potentials.

potential (5.6) goes to zero as z → ∞. Hence the spectru

The highly
excitedspectrumis ingeneralgeneralhe previous
section,we willmeson
not analyze
the most
potential wi

scattering”
statesliving
at
large
z.
For
β
>
d,
the
potentialginteresting (for soft wall potentials). E.g. if we

orm of figure 4(b). As a first step, we restrict to the class of potential∞,takeandtheonepotential
expects V(T)a discrete
spectrum.
The
to behaveat large T
likeasymptotic

rge T liketermined by the behavior of thepotentialat large z. The pr2− 21 βT
V (T ) # e

analyzed in [29], and one can read off the asymptotic spectru

find thespectrumitive weconstant.
Potentialsof this form (with a fixed value of β) naturally

ven byopen string tachyon condensation 1[27,28].
There is no particular read1− β )(mn ∼ µn.
2

to play a role in AdS/QCD, but we will see that they give rise to intere,ussionphysics.was for scalar mesons, but it is easy to see that the bing
in section
4, weThe
substitute(5.2)between
into (3.6),
to twofind cases
the form
for asvectormesons.
difference
the
is in

10
20
Since the vacuum!10with f (ρ)0 != 0 spontaneously
breaks tm2

o O(5 − n), when n < 5 one expects to find massless Golds
(σ)
(σ)
To
study
spectrumneeds to specify
2
Fig.
3: C2 the/C1low
as alying
functionof L4 m2 /µone.
α(T ) describe= (cos T ) themhe cosetO(6 − potentialn)/O(5 −n). For
We Vwilland cthe tachyon
V(T).
, whichn fig. 3 we plot the results of the numerical integration. Note that eq. (is relevant
forthedefectfermionsystem,one
finds√
ymmetryisbrokenexplicitly.
atisfied for negative
m2 , whichis consistent with the stability of the vacuum.
(µ̄ ∼ µ/ λ)w states have

4.1. σ-mesons- mesons:

m2 /µ2 ≈ 0.44, 9.65, 26.63, 51.35, 84, · · ·m This/µ̄ ≈3.08, 15.12,
34.87,
62.32, 97.46,· · general·
here vectorµ is definied
in (4.1).
spectrumis well
described
by thefmesons:2

2

To study radial excitations of the probe brane, we exp
The anomalously
meson
is
the
techni2
2 light scalar2
mn /µ of≈ 3.89n+ 5.32n + 0.44;
n = 0, 1, 2, · · ·around the solution(3.17),
dilaton.

exhibits the characteristic behavior mn ∼ n at large excitation numbeaaf (ρ, x ) = f (ρ) + y(ρ, x ).

milar to other holographic models, although here it does not have an obv

In technicolor there is a long-standing debate about
the fate of the dilaton near the CPT in QCD. There
are two schools of thought:(1) mtd /mmeson → 0(2) mtd /mmeson → const
as µ/Λ → 0.
We find that (2) is correct.

Intriguingly, in QCD it was argued (by M.
Hashimoto and K.Yamawaki) that the massof the techni-dilaton is smaller than that of
the lightest vector meson by a factor of
about 2.8. In our system this ratio is about
2.6...
It would be interesting to explore the range
which this ratio takes as one changes thepotential V(T).

Finite temperature
To introduce finite temperature we replace the
AdS background by an AdS black hole. Even if thezero temperature theory is in the broken phase,the symmetry is restored at a finite temperatureof order µ . Interestingly, the nature of the phasetransition is only sensitive to the behavior of thepotential V(T) near the origin.

= 0 and T (Λ) = 0.

Thus, the leading contribution to F is quartic in Th . To calculate it we need to include

Parametrizing

uartic contribution to the potential V (T ),1 2 2 a 4
V (T ) = 1 + m T + T + · · ·24(6.27) q

ing (6.27)forintoa(6.18),
we findbelow
a critical value the transition is first$%1
a value
1 2it is second
1 2order.
order,
whiled−1" above2
2 2this4" 2=
drr
F (r)(rT ) + m T + T + m F (rT T ) − F (rT " )4 . (6.28) d
2 rh
2
2
4
! Λ

olution to the full e.o.m. (6.3) can be expanded as T (r) = T1 (r) + T3 (r) + . . . where

is of order Thi . In particular, T1 is given by (6.13), and T3 (r) is obtained by varying), and keeping only terms cubic in Th .

Using the e.o.m. of (6.28) and plugging back into F4 one finds (in the BKT limit)
1
F4 =
8

! ∞

2
3d
drr d−1 (−6aT 4 +r 2 F T 2 T "2 + 3r 4 F 2 T "4 )2
rh

(6.29) dSummary• Holography provides a useful tool for studying
conformal phase transitions. In this approach,

one needs to analyze the dynamics of a scalar
field in AdS, with mass close to the BF bound.

• The universal physics near the transition issensitive to the full non-linear Lagrangian for T.

• We took the Lagrangian to have the TDBI form

and found that the dynamics is similar to that ofhard and soft wall AdS/QCD (depending on thepotential V(T)), with a dynamically generated
wall.

• We analyzed the spectrum and thermodynamics
of these models and found some properties
that are different from standard holographic
models, such as a second order symmetry
restoration transition.

• It would be interesting to analyze in moredetail the properties of the lightest scalarmeson, which plays the role of the technidilaton in technicolor theories.

• It would also be interesting to further study
the finite temperature dynamics and look
for applications to condensed matter andother systems.

Thank You!