Spectral
functions
for holographic mesons
with Rowan Thomson, Andrei
Starinets
[
arXiv:0706.0162
]
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with
Aninda
Sinha
[
arXiv:0801.nnnn
]
Motivation:
Exploring
AdS
/CFT as a tool to study
the strongly coupled quark

gluon plasma
See Steve
Gubser’s
talk!
Field
theory story
:
N
=2 SU(
N
c
)
super

Yang

Mills with (N
f
+1)
hypermultiplets
N
f
massive
hyper’s
“quarks”
2 complex scalars
:
2 Weyl fermions:
N
=4 SYM
content
fund. in U(N
c
)
& global U(N
f
)
(Reader’s Digest version)
fundamental
adjoint
adjoint
fields:
vector:
1 hyper:
fundamental fields:
•
work in limit of large
N
c
and large
λ
but
N
f
fixed
“quenched approximation”:
•
low temperatures:
free quarks
mesons ( bound states)
Finite Temperature:
•
phase transition
:
•
high temperatures:
NO
quark
or meson quasi

particles
“quarks dissolved in strongly coupled plasma”
(strong coupling!!)
•
note
not
a confining theory:
free quarks
“mesons”
( bound states)
unusual dispersion relation:
add
N
f
probe D7

branes
horizon
AdS
5
boundary
pole
equator
S
5
S
3
D7
Free quarks appear with mass:
Karch
&
Katz (
hep

th
/0205236 )
Adding
flavour
to
AdS
/CFT
Aharony
,
Fayyazuddin
&
Maldacena
(
hep

th
/9806159
)
add
N
f
probe D7

branes
horizon
AdS
5
boundary
pole
equator
S
5
S
3
D7
Karch
&
Katz (
hep

th
/0205236 )
Adding
flavour
to
AdS
/CFT
Mesons ( bound states)
dual
to open string
states supported by D7

brane
Aharony
,
Fayyazuddin
&
Maldacena
(
hep

th
/9806159
)
Mesons
:
lowest lying open string states are excitations of the
massless
modes on D7

brane: vector, scalars (&
spinors
)
(free)
spectrum
:
•
expand
worldvolume
action
to second order in fluctuations
•
solve
linearized
eq’s
of motion by separation of variables
V
eff
r
Discrete spectrum:
Kruczenski
,
Mateos
,
RCM & Winters [
hep

th
/0304032]
= radial
AdS
#
=
angular
#
on
S
3
Gauge theory thermodynamics = Black hole thermodynamics
Gauge/Gravity thermodynamics:
Witten (
hep

th
/9803131);
…..
•
Replace SUSY D3

throat with throat of black D3

brane
•
Wick rotate and use euclidean path integral techniqes
•
. . . . .
Extend
these ideas to include
contributions of probe
branes
/fundamental matter
Gauge/Gravity thermodynamics with probe
branes
:
put D7

probe in throat geometry of black D3

brane
SUSY embedding
Minkowski
embedding
Black hole embedding
T=0: “
brane
flat”
Low T: tension supports
brane
;
D7 remains outside BH horizon
raise T: horizon expands and increased gravity
pulls
brane
towards BH horizon
High T: gravity overcomes tension;
D7 falls through BH horizon
D7
D3
Phase transition
†
(
†
This
new
phase transition is
not
a
deconfinement
transition.)
Mateos
, RCM &Thomson [
hep

th
/0605046]; . . . . .
Babington,
Erdmenger
, Evans,
Guralnik
& Kirsch [
hep

th
/0306018]
Brane entropy:
1
st
order phase transition
Transition temperature:
Mateos
, RCM &Thomson [
hep

th
/0605046 &
hep

th
/0701132]
Mesons in Motion:
pseudoscalar
scalar
Mateos
, RCM &Thomson [
hep

th
/0701132]
Ejaz
, Faulkner, Liu,
Rajagopal
&
Wiedemann
[arXiv:0712.0590]
Radial profile
k increasing
•
holographic model shows bound states persist above
T
c
and have interesting dispersion relation
•
lattice QCD indicates heavy quark bound states persist above
T
c
Asakawa
&
Hatsuda
[
hep

lat/0308034]
Datta
,
Karsch
,
Petreczky
&
Wetzorke
[
hep

lat/0312037]
Does “speed limit” apply to heavy quark states in QCD?
In experiments (
eg
, RHIC or LHC), these bound states
are created with finite (possibly large)
momenta
.
•
holographic model shows bound states persist above
T
c
and have interesting dispersion relation
•
lattice QCD indicates heavy quark bound states persist above
T
c
Asakawa
&
Hatsuda
[
hep

lat/0308034]
Datta
,
Karsch
,
Petreczky
&
Wetzorke
[
hep

lat/0312037]
Satz
[
hep

ph/0512217]
’s have finite width!
but in Mink. phase, holographic mesons
are absolutely stable (for large
N
c
)
can we do better in
AdS
/CFT?
Spectral functions:
diagnostic for “meson dissociation”
•
simple poles in retarded
correlator
:
yield peaks:
“quasi

particle”
if
•
characteristic high “frequency” tail:
discrete spectrum;
low
temperature Mink. phase
continuous spectrum;
high
temperature BH phase
mesons stable (at large
N
c
)
no quasi

particles
hi

freq tail
Spectral functions:
diagnostic for “meson dissociation”
•
approaching phase transition, structure builds
quasinormal
frequencies approach real axis
Thermal spectral function:
subract
off asymptotic tail:
phase transition
see also:
Hoyos
, Landsteiner & Montero
[
hep

th
/0612169]
RCM,
Rowan
Thomson &
Andrei
Starinets
[arXiv:0706.0162]
Kobayashi,
Mateos
, Matsuura, RCM & Thomson [
hep

th
/0611099]
Mateos
, Matsuura, RCM & Thomson [arXiv:0709.1225]; . . . . .
Need an extra dial: “
Quark” density
D7

brane gauge field:
asymptotically (
ρ→∞
):
Kobayashi,
Mateos
, Matsuura, RCM & Thomson [
hep

th
/0611099]
Mateos
, Matsuura, RCM & Thomson [arXiv:0709.1225]; . . . . .
Need an extra dial: “
Quark” density
electric field lines can’t end in empty
space;
n
q
produces neck
D7

brane gauge field:
asymptotically (
ρ→∞
):
BH embedding with tunable horizon
See also:
Erdmenger
, Kaminski & Rust [arXiv:0710.033]
Increasing
n
q
, increases width of meson states
Spectral functions:
n
q
= 0
= 0.001
= 0.05
= 0.25
at rest: q=0
See also:
Erdmenger
, Kaminski & Rust [arXiv:0710.033]
Increasing
n
q
, increases width of meson states
Spectral functions:
n
q
= 0
= 0.001
= 0.05
= 0.25
at rest: q=0
Spectral functions:
introduce
nonvanishing
momentum
(
n
q
= 0.25)
Spectral functions:
follow positions of peaks
real part of
quasiparticle
frequency,
Ω
(q)
(
n
q
= 0.25)
Spectral functions:
follow positions of peaks
real part of
quasiparticle
frequency,
Ω
(q)
(
n
q
= 0.25)
v
max
= 0.9975
(calculated for
n
q
=0)
Quasiparticles
obey same speed limit!
follow widths of peaks
imaginary part of
quasiparticle
frequency,
Γ
(q)
Γ(q) d
iverges
at finite
q
max
examine Schrodinger potential for
quasinormal
modes
Quasiparticles
limited to maximum momentum
q
max
Conclusions/Outlook
:
•
first order phase transition appears as universal feature of
holographic theories with fundamental matter (T
f
> T
c
)
•
how robust is this transition?
should survive finite 1/
N
c
, 1/
λ
,
N
f
/
N
c
corrections
interesting question for lattice investigations
•
D3/D7 system: interesting framework to study quark/meson
contributions to strongly

coupled
nonAbelian
plasma
•
“speed limit” universal for
quasiparticles
in plasma
•
quasiparticle
widths increase dramatically with momentum
find
in present holographic model
universal
behaviour
? real world effect?
(
INVESTIGATING)
[extra slides]
Meson spectrum:
Minkowski:
discrete stable states
black hole:
continuous gapless excitations
•
feature of QCD ??
•
in a confining theory, will have two phase transitions
for sufficiently heavy quarks
•
simple physical picture:
Matsui & Satz
(Hong, Yoon & Strassler)
structure functions reveal:
(Rey, Theisen & Yee)
Wilson lines reveal:
mesons dissociate:
•
one of most
striking
features
of
transition
is “
meson melting
”:
even
with
m
q
=0,
hypermultiplets
introduce non

vanishing

function; however,
running of
`t
Hooft
coupling
vanishes
with large

N
c
limit
More legal
details:
w
ith large but finite
N
c
to avoid Landau pole need to
introduce additional matter content at some large UV scale
Probe approximation
:
N
f
/
N
c
→ 0
recall above
construction does not take into account the
“gravitational” back

reaction of the D7

branes!
→
at finite
N
f
/
N
c
back

reaction would cause singularity;
introduce
orientifold
at large radius
(see, however:
Burrington
et al; Kirsch &
Vaman
;
Casero
, Nunez &
Paredes
, . . . . )
entropy density:
Reminder about large N counting:
counts # of d.o.f.
entropy density:
counts # of d.o.f.
in our limit, thermodynamics dominated by adjoint fields;
we are calculating small corrections due to fundamental matter
these dominate over quantum effects, eg, Hawking radiation,
p
hase
transition
physical properties of thermal
system are multi

valued
minimizing free
energy
(
euclidean
brane
action)
fixes physical configuration
c
ritical
embedding
Minkowski
embeddings
BH embeddings
See also:
Babington et al (
hep

th
/0306018
)
Kirsch
(
hep

th
/0406274)
Brane entropy:
1
st
order phase transition
Transition temperature:
H
Gauge theory entropy:
λ
enhanced over naïve
large

N counting
phase transition
“small glitch in
extensive quantities”
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