Download Time-Dependent Meson Melting in External Magnetic Field

Time-Dependent Meson Melting in
External Magnetic Field
Hajar Ebrahim
University of Tehran and IPM
Based on 1503.04439 in collaboration with:
M. Ali-Akbari, F. Charmchi, A. Davody, L. Shahkarami
Nafplion, July 2015
1
Outline:
Why Time-Dependence?
Holographic Meson
Time-Dependent Set-Up
2
Why Time-Dependence?
system in equilibrium
non-dynamical, stationary
late-time state of a generic system
time-dependent processes
perturbative methods, Linear Response Theory
H ! H + Hpert
coarse-grained view of the system, low energy effective description,
Hydrodynamics, Universal features
lack of adequate techniques for strongly correlated condensed matter
theories, quark-gluon plasma
Focus on field theories with holographic dual. Two dual theories are just
different descriptions of the same physical system.
3
Thermalization in QGP
Collision of two heavy nuclei (Gold or Lead) at the relativistic speed
Production of an Anisotropic Plasma
Hydrodynamics applies after a very short time-scale, 1 fm
Far-from-equilibrium effects at the early stages of QGP production
Strongly Coupled Plasma
Shuryak, 2004; 2005
Presence of a magnetic field at the early stages of QGP production
Kharzeev, McLerran, Warringa, 2008
y
!
x
hadronic gas
described
mixed phase
by hydrodynamics
QGP
pre-equilibrium stage
Figure 2.5: Sketch of the collision of two nuclei, shown in the transverse plane perpendicular to the
beam. The collision region is limited to the interaction almond in the center of the transverse plane.
Spectator nucleons located in the white regions of the nuclei do not participate in the collision. Figure
taken from Ref. [60].
Figure 1: Description of QGP
ion collisions. The kinematic
! formation in1 heavy
x +x
2
2
4
landscape is defined by τ =
short direction of the almond as they are in the long direction. And, as we shall see, it turns
x0 − x1 ; η =
2
log x00 −x11 ; xT = {x2 , x3 } , where the
the initial time where the gauge field is zero and ✏ is the cut-o↵
undary. We have assumed ✏ = 0.0001 and ✓˜ = ⇡2 . The ansatz for
eld is
Z
˜ = h(t, z, ✓)
˜
˜ dt
A✓˜(t, z, ✓)
E(t, z, ✓)
(4.23)
Holography:
Q
(1 + tanh(!t))
(4.24)
2 ˜
R sin ✓
a parameter showing
strength
typethe
IIB
string of the electric field and ! is
N=4
su(N)
of the time interval where
the
electric
field
varies
from
zero
to
a
theory on
Conformal SYM
lue.
5
AdS5 ⇥ S
ts are fixed by
˜ =
E(t, z, ✓)
R4
2
=
4⇡g
N
=
g
s
Y MN =
↵0 2
Strong/Weak (4.25)
>>1
Duality
Strongly Coupled CFT
7
Classical Gravity on Asymptotically
AdS Space-time
The dual gravitational description of a strongly coupled gauge theory provides an
efficient way to study the thermodynamic properties of gauge theories.
Correspondence both in static and dynamical situations: time-dependence on the
boundary manifested by time-dependence in the bulk
5
andstudy
second, the
D7-branes terminate at r = U (0) <
modified: In
First, a horizon
appears
at
r=
r0 > 0,we
1.2
e change
in the
thesince
probe
dynamically.
toinachieve
this
goal
1 brane
from the
bulk
black of
hole
the
final equilibrium
the The
future
cannot
be
corresponds
to shape
thermalization
in the
gluon
sector
of theconfiguration
dual order
gauge theory.
background
isthe
time-dependent
and
L. This ‘termination point’ corresponds to the tip of the branes in Fig. 8.3.
2
2
ane
in the
AdS-Vaidya
background.
the
hole
formation
in within
theThe
bulk
which
4in40.8
8 ItaAdS
2
2 however,
2 black
black
hole
brane
configuration,
to
be AdS
singular
aAdS
finite
1
the
shape
the
brane
such
background
can
deform
dynamically.
x2therefore
+dx
Lembedding.
is theof
AdS
radius.
The
boundary
is2 tends
where
f (z)
=The
1−r
xdescribes
the
radius.
boundary
is
3 and
3 = dx1 +dx2 +dx3 and L is
h z /L , d⃗
3
2
2
ation
in
the
gluon
sector
of
the
dual
gauge
theory.
The
background
is
time-dependent
and
time
as
the
intersection
locus
approaches
the
pole
where
the
S
wrapped
by
the
D7-brane
Let/r
us.start
with
the
AdS-Vaidya
metric
thatcoordinates
2 L
0.8
z =radius.
0,2 while
the
event
horizon
z = L /rh . We use the same world-volume coordinates
0.6 boundary
Weat
use
the
same
world-volume
x=
AdS
The
AdS
isis at
43 and
8 L his
2 the
2
2
in
subsequent
sections,
whether
the
holographic
quarks
described by the D7-branes are the
zbrane
/L
, d⃗
x3such
= dx
and
is the
AdS
radius.orThe
AdS
boundary
0
1 istaken7 into account
shrinks
to
zero
size.
expected
that
ifdynamically.
stringy
quantum
effects
are
in
a
background
can
deform
0 target
1It is3space
7L
1 +dx
2 +dx
2
0.6 The
as
the
coordinates
themselves
such
that
(σ
,
σ
,
·
·
·
,
σ
)
=
(t,
z,
⃗
x
,
Ω
).
3
3
⇥
⇤
lves
that
, σworld-volume
, 2· M
··, N
σ )0.4 =
(t, z, ⃗x3 , Ωanalogues
Theof heavy 2or light 2quarks 2in QCD2 will depend
/rh . such
We use
the2(σ
same
1coordinates
3 ).
on how their mass (or, more precisely,
2
2
event
horizon
is
at
z
=
L
/r
.
We
use
the
same
world-volume
coordinates
there,
themetric
brane
will
reconnect
near
that
locus,
and
then
go
back
the
embedding.
ds
F
(V,
z)dV
d Minkowski
=
G
dx
dx
=
2dVits
dzposition
+ d~x
+ in
costhe d⌦
d 2 ,0.4as a (II.1)
dS-Vaidya
that
is
h
M
N
0 embedding
1
7 of the D7-brane
3to +
3 + sin
is
specified
by
(φ,
ψ)
transverse
space
2
(σ , σ , ·(φ,
· · , ψ)
σ ) in
= the
(t, z,0.2transverse
⃗xz3 , Ω0 3 ). 1 Thespace
the7mass
of
corresponding mesons) compares to the temperature.
dsuch
byWethat
itsshow
position
as2(c),
a thetogether
schematic
illustration
of
this
process
in
Fig.
the
sub-critical
e coordinates
themselves
such
that
(σ
,
σ
,
·
·
·
,
σ
)
=
(t,
z,
⃗
x
, Ω3set
).with
The
function
of z:transverse
φ = Φ(z)space
and ψas=⇤a0. Note
thatconcluded
we 3can
ψ =at
0zero
without
lossand
of generality
We have
that,
temperature,
the 0.2D7-branes lie at U = L and are
itsM
position
(φ,
ψ)
in
the
⇥
Note
that
we
can
set
ψ
=
0
without
loss
of
generality
1
0
where
2
N
2
2
2
2
2
2
super-critical
cases
in
Figs.
2(a)
and
2(b),
respectively.
We
argue
that
this
phenomenon
may
0
0.5
1
1.5
2
D7-brane
is
specified
by
its
position
(φ,
ψ)
in
the
transverse
space
as
a
induced
thanks
towithout
the U(1)-symmetry
generated
F
(V,
z)dV
d by
dx dx
= set
2dVofdzgenerality
+ d~
x3 +parametrised
+ ∂cos
d on, the D7-brane
(II.1)is given
ψ . The
0
, u,sin
⌦metric
the
induced1 metric1.5on the 2
by d⌦
{t,
3xi+
3 }. In terms of these coordinates,
we
can
ψ
=
0
loss
2 induced
0
0.5
.
The
metric
on
the
D7-brane
is
given
dhat
by
∂
z
ψ
be
the
gravity
dual
of
quarks
recombining
into
mesons
in
the
boundary
field
theory.
Φ(z) and ψ =as0. Note that we can set ψ =
0 D7-branes
without
loss
of
4generality
"
!
takes
the
F
(V,
z)
=
1
M
(V
)z
, form 1
(II.2)
induced
metric
on
the
D7-brane
is
given
∂ψ . TheThe
f
(z)
1
remaining
of
this
paper
is
organized
as
follows.
In
Section
will
review
the
static
"
−2∂ψ . The
a
b
2
′
2 2,
2we
2
2
2
induced
metric
on
the
D7-brane
is
given
-symmetry generated by
2
+ Φ (z) 2 dzu2+
L 1hab dσ dσ = − 2 dt + 2
x3 +2 cos Φ(z)dΩ
(2) R2 u2
+ L22d⃗
R
3 ,
1
2
2
"
embeddings
to
fix
notations.
In
Section
3,
we
describe
the
setup
for
our
dynamical
computation.
′
2
2
2
2
2
z
z
f
(z)
z
dsabove
= metric
+ dxi in+Eddington-Finkelstein
du + 2
d⌦23 .
(8.7)
dtwritten
andFigure
we
have
set
the
radius
of
AdS
space-time
to
be
one,
R
=
1.
The
is
5
2
2
2
2
+
Φ
dz
(z)
+
d⃗
x
+
cos
Φ(z)dΩ
,
(2)
"
!
1 2 and
3:
Minkowski
black
hole
embeddings
×S
R
u
+
L
u
+
L
3
3 of the D7-brane in the Schwarzschild-AdS
5
′
2
2
2
2
2
2
Results
of
numerical
computations
are(V
shown
in2 Section
4. VSection
5the
is null
devoted
to discussions
z +
fcoordinate
zcosz)
14(2)
1+
radial
direction
by z and
shows
The
boundary,
the of the D7-brane in the Sch
Φ (z)f (z)
dzwhere
+ 2the
d⃗′x≡
Φ(z)dΩ
Figure
3:direction.
Minkowski
and
black holewhere
embeddings
(V,
=
1
)z
,
(II.2)
3F
3is ,represented
b (z)
2
′horizontal
2 M
2 axes
2ρ, respectively.
2 the
d/dz.
where
spacetime.
The
vertical
and
are
w
and
In
the
plot,
the
horizon
of
quark
mass
a
number
of
solutions
exist.
Among
them
the physi
3
+
Φ
dz
dσ
=
−
dt
+
(z)
+
d⃗
x
+
cos
Φ(z)dΩ
,
(2)
) gauge
z
S axes
. Theare
AdS
factor
suggests
that
ifthe
L =Technical
thisdetails
metric
is
exactly
that
ofhorizontal
AdS5 ⇥function,
3We see
30 then
on the
boundary
theory
interpretations
and
future
directions.
for
numerical
5 and
theory
lives,
is
at
z
=
0
and
V
is
the
time
coordinate
on
boundary.
M
(V
),
which
is
an
arbitrary
2
2
2
D7-branes
spacetime.
The
vertical
and
w
ρ,
respectively.
In
D3-branes
z★
z rembedding
fh (z)
zsymmetry.
The
of
the
D7-brane
is
determined
by
the
Dirac–Born–Infeld
(DBI)
action.
In This
Flavour
added
to
the
gauge
theory
by
introducing
probe
radius
is
fixed
as
=
1
using
the
scaling
For
m
!
0.92
and
m
"
0.92,
we
obtain
lution
is
the
most
energetically
favourable
one.
The
presence
of
the
co
that
the
dual
gauge
theory
should
still
be
conformally
invariant.
is
indeed
the
case
in
represents
the mass
of
the
black
hole
which
changes
asmetric
time passes
by until
itEddington-Finkelstein
reaches
constant
Thescaling
function
computations
are
provided
in R
appendices.
radius
is fixedaas
rh = 1 value.
using the
symmetry. For m ! 0.92 and
s of
AdS
space-time
to
be
one,
=
1.
The
above
is
written
in
the
absence
of
the
world-volume
field
strength,
the
action
is
the
Minkowski
and
blackwith
holeinembeddings,
Forconsideration:
m ≃ the
0.92,Minkowski
there
exist
thehole
the limitIn
under
If Lor
=
0 and
theboth
quarks
are
massless
and the
theory
classically
determined
byD7-branes
the
Dirac–Born–Infeld
(DBI)isrespectively.
action.
M (V
) that
we will
work
this
paper,
Kalb-Ramond
equivalently
constant
magnetic
fieldis leads
black the
embeddings,
respectively.
Fortomnonzero
≃ 0.92,
in
the
bulk
A.
Karch
and
E.
Katz,
2002
ialfor
direction
is
represented
by
z
and
V
shows
the
null
direction.
The
boundary,
where
mined
by
the
Dirac–Born–Infeld
(DBI)
action.
In
#
and √
in
the
probeInlimit
Nat
-function,
Minkowski
and
hole embeddings.
f /Nc ! 0 the quantum mechanical
8 conformal,
of
D7-brane
is black
determined
by the
(DBI)
action.
rength,
the
action
is coordinate
condensate
even
zero
quark
mass. In the following
wewhich
will3 discus
Minkowski
and
black
hole
embeddings.
= the
0the
and
V is
on Dirac–Born–Infeld
the boundary.
M
(V
),
which
is
an
arbitrary
function,
8
th,
action
is the time
4
4
8
2
2
2
2
0
V
<
0,
/N
,
vanishes.
If
L
=
6
0
then
the
metric
above
becomes
AdS
⇥
SAdS bou
is
proportional
to
N
<
h
,
(3)
∝
d
σ
S
c , d⃗
5
4 1−r
2x =2dx +dx
2
2
Static
embeddings
where
f1−r
(z) 4=
zx2f /L
+dx
and
L isradius.
the AdS
radius.
The
⇥= D7
⇤ 8 ,results
world-volume
field
strength,
the
action
is
3
1
2
3
h
where
f
(z)
z
/L
d⃗
=
dx
+dx
+dx
and
L
is
the
AdS
The
AdS
boundary
is
in
more
details.
#
⇡V
1
3i.e.
2 reflecting
3 function
e#black hole √
which changes as time
passes
by until
itasymptotically,
reaches
a0 6
constant
The
h
M
(V
)
=
M
(II.3)
only
for 1V,
u value.
L,
the2fact that in the
gauge theory conformal
1
cos(
)
V
6
f
2
V
2
√ 8
5 same world-volume
at
z = 0,the
while
the
event
horizon
at
=
/rhorizon.
.is We
use
the
coo
:
hthe
regularity
at# the horizon.atThe
solution
is
then
obtained
z =asymptotic
0,1while
event
horizon
is
atbyzthe
=asisL
/rofhz.mass
WeL
use
same
world-volume
coordinates
The
background
interest
AdS
⇥
S
geometry
describing
th
8we impose
we
impose
regularity
at
the
The
asymptotic
solution
is
then
ob
rk d∝
with
in
this
paper,
is
5
h
,
(3)
d
σ
V
>
V,
/
L,
but
is
restored
asymptotically
at
invariance
is
explicitly
broken
quark
M
√
q
D7
start
from reviewing
static
embeddings
of
the
D7-brane
to
the
Schwarzschild-AdS
×
h ,★
(3)
σ We
0
1
7
Fluctuations
of
theequation
fields
on
the
brane
produce
the
5
. The
of
motion
is
a
second-order
ordinary
differential
equation
for
where
h = det hab8the
the
target
space
coordinates
themselves
such
that
σ) =
,"
· (t,
· · ,z,σ⃗x3), Ω
=is3 ).(t,
z, ⃗x3 , Ω
as the as
target
space
coordinates
themselves
such
that
(σ 0 ,the
σ 1radius
, · ·(σ
· , of,σ 7the
"
h
,
(3)
∝
d
σ
S
horizon
geometry
N
D3-branes
energies
E
M
.
We
also
note
that,
if
L
=
6
0,
then
three-sphere
notThe
5
5 of
q
D7
c
8
!
2
2
in
order
to
fix
notations.
The
metric
of
the
Schwarzschild-AdS
×
S
background
S spacetime
!
5
2
2
Φ(z),
ρHof
rh −
ρH D7-brane
rin
ρH space
which
is (φ,
constant.
Note
where V ismeson
the
time<
interval
in which
the2mass
the
black
increases
toits
Mfposition
spectrum
%,from
$
embedding
ofhole
the
D7-brane
iszero
by
its
position
(φ,
ψ)
the
transverse
H
2 embedding
0
V
<
0,
of
the
by
ψ)
inρ2at
the
transverse
constant,
as
displayed
in· ·specified
fig.
8.4;
inspecified
particular,
it(7)
shrinks
to
u ρ=
0 (corresponding
toas a spa
h −
2 zero
′ is
(ρ
−
ρ
)
+
·
r
−
ρ
+
W
(ρ)
=
⇥
⇤
(ρ
−
ρ
)+··· ,
r
−
+
W
(ρ)
=
H
1
can
be
given
as
3
z
3
tan
Φ
f
h
H
H
2
2
h
H
is the
aM
second-order
ordinary
differential
equation
for
2 the projection
2
′′
′
′3
′2
′
⇡V
1
sion
athat
second-order
ordinary
differential
equation
for
4
ρ
+
3r
u
R
r
=
L),
at
which
point
the
D7-branes
‘terminate’
from
the
viewpoint
of
on
ρ
+
3r
function
of
z:
φ
=
Φ(z)
and
ψ
=
0.
Note
that
we
can
set
ψ
=
0
without
loss
of
function
of
z:
φ
=
Φ(z)
and
ψ
=
0.
Note
that
we
can
set
ψ
=
0
without
loss
of
generality
(8f
−
zf
−
Φ
=
0
.
(4)
Φ
−
)Φ
+
(3
tan
Φ)Φ
+
+
(V
)
=
M
(II.3)
H
radius
of
the
event
horizon
is
r
=
M
.
1
cos(
)
0
6
V
6
V,
h
2
2
2
2
2
2
2
2
2
2
2 ge
H
h
f
f "
2 a second-order
Vh
2f
The equation of motion
is
ordinary
differential
equation
for
)
)
=
(
(
dt
+
dx
)
+
(
(d⇧
+
⇧
d⇥
+
d⌃
+
⌃
d
),
ds
!
2
f
z
z
:
3
2
2 inVthe
In $
order to
add
the
fundamental
matter
gauge
theory
side
we
have
to
add
the
probe
branes
to
the
bulk.
The
1
>
V,
.
The
induced
metric
on
the
D7-brane
is
given
thanks
to
the
U(1)-symmetry
generated
by
∂
.
The
induced
metric
on
the
D7-brane
%
thanks
to
the
U(1)-symmetry
generated
by
∂
$
L
dz
R
u
%2′
ψ
ψ
2value of ρ at the
2 horizon
2
2
2 the 2(ρ,
2<On
′ ρH ds
where
(0It
<is
ρalso
r
is
the
found
in
w)-plane.
where
ρ2 Hφdψ
(0
≤ 201
rother
is2the
+
d⃗
x
−f
(z)dt
+
L
=
+
(dφ
+
cos
φdΩ
+
sin
)
,
2ρ
−1
−1 value of ρ at the horizon found in the (ρ, w
H the
h ) (1)
H3≤
h ) freedom
3
3
tan
Φ
f
3
tan
Φ
f
%
$
4
dynamics
of
the
degrees
of
living
on
the
brane
is
explained
by
the
DBI
action
as
10
3
3
to
introduce
new
bulk
coordinates
(w,
ρ)
=
(L
z
sin
φ,
L
z
cos φ).
′2
′ z 2 convenient
′2
′
as
′
u
as
"
!
f
(z)
!
− ′3 the
Φ−+
02.intersect
)Φ
+
0 brane
1does not
2 intersect
3 " with ⇤the
4 hole, the brane
4
Φ2does
. (4)
(4)
tan
+
+′2=not
z Φ)Φ
3
tan
Φ
f =30 with
⇥ black
hand,
when
the
hand,
when
brane
the
black
hole,
the
brane
terminates
at
the
point
′
′
g
C
=
dx
⇤
dx
⇤
dx
⇤
dx
,
g
=
e
,
R
=
4⌅g
N
l
, te
Z−2 be
f
(z)
1
1
s
s
s
c
(4)
which
is
constant.
Note
erval
which
the
mass
of
the
black
hole
increases
from
zero
to
M
f )Φ
z
z
f
The
new
embedding
function
can
then
a
function
of
ρ,
w
=
W
(ρ).
In
practical
numerical
f
(z)
1
1
s
(8f in
− zf
−
Φ
=
0
.
(4)
+
(3
tan
Φ)Φ
+
+
f
z
z
f
f
a
b
2
′
2
2
2
2
2
4
p
123
3
5
−2
a
b
2
′
2
2
2
2
2Φ =
5where
Rwrapped
+
Φ
dz
L
h
dσ
=
−
dt
+
(z)
+
d⃗
x
+
cos
Φ(z)dΩ
,
(2)
S
by
the
brane
shrinks
to
zero
size
at
a
pole
of
S
:
82
1by the branefshrinks
abfdσ
to
zero
size
at
a
pole
of
S
:
Φ
=
π/2,
corresponding
+
Φ
dz
L
h
dσ
dσ
=
−
dt
+
(z)
+
d⃗
x
+
cos
Φ(z)dΩ
0
2 where S 3 wrapped
z
z
3
3
ab
2
2
2
3
3 ,
d ⇠W (ρ)gab
+ (2⇡↵ zby
)Fab
, z 2 z Eq.
(II.4) z 2
Sequation
= ⌧7 6 for
2f
computations,
we
solve
the
obtained
rewriting
(4)
in
terms
of
W
(ρ).
4
f
(z)
z
2
−1
2
−1
z
(z)
2 L
ntbulk
horizon
is (w,
rThe
M
. ρ)asymptotic
coordinates
ρ)regular
=
z = sin
zsincos
φ).
to ρ = 0. The regular asymptotic solution at ρ = 0 is given by
h =
to coordinates
ρ = 0.
solution
at2 zρ−1
= cos
02 is
given
by 2boundary
f (L
(w,
(Lφ,
z −1
φ,
L
φ).
−1
−1 C
The
asymptotic
behavior
of
the
field
at
the
AdS
of theand
dualDilaton fields, respectively. In t
andto⌥information
arebulk.
four-form
where
nt
to introduce
new
bulk
coordinates
(w, we
ρ)′ ≡=
(L′to
z add
sinthe
φ, Lprobe
z cos
φ).gives
(4)
damental
matter
in
the
gauge
theory
side
have
branes
the
The
⌅
d/dz.
where
function
of
ρ,
w
=
W
(ρ).
In
practical
numerical
89 and
6
≡ d/dz.
where
bewhere
a function
of
ρ,theory
w=
W (ρ).
practical
numerical
2
2
is the
induced
metric
andIn
FNear
iswthe
the
gauge
strength
on
the asymptotic
brane.
a,⇧2b +
are⌃the
brane
coordinates
gab can
⇥ is the
field
operators.
AdS
boundary
ρ
=
∞,
the
solution
behaves
as
4field
ab
rh4 length
ordinate
u
=
and
l
=
string
scale.
In or
g)f obtained
function
then
be
a
function
of
ρ,
=
W
(ρ).
In
practical
numerical
2
r
s
freedom
living
on
the
brane
is
explained
by
the
DBI
action
as
The
embedding
of
the
D7-brane
is
determined
by
the
Dirac–Born–Infeld
(DBI)
action.
In
2
h
by
rewriting
Eq.
(4)
in
terms
of
W
(ρ).
W
(ρ)
=
W
−
ρ
+
·
·
·
,
P
is obtained
defined as by rewriting Eq. W
embedding
D7-brane is determined
by the Dirac–Born–Infeld
(DBI) ac
(ρ)in
=terms
WP − The
ρ(ρ).
+in
· · terms
· , ofofthe
(8)
Wgab
(ρ)
(4)
ofEq.
W
5 (4)
4WP5a space-filling
olve
the
equation
for
W
(ρ)
obtained
by
rewriting
W
(ρ).
introduce
the
fundamental
matter
we
have
to
add
flavo
c
4W
Z
4567
the
absence
of
the
world-volume
field
strength,
the
action
is
the
AdS
boundary
gives
information
of
the
dual
P= m +
p
+
·
·
·
,
(5)
W
(ρ)
the
absence
of
the
world-volume
field
strength,
the
action
is
dbehavior
at the AdS
boundary
gives
information
of
the
dual
M
N 2 of the dual
8 AdS
the
boundary
information
6ggives
in theWprobe
limit
to(II.4)
this
background.
The
configuration of the
0 )F
=
@
x
@
x
(II.5)
ρGMbrane
ab
a
b
N.
#(0)
d
⇠
g
+
(2⇡↵
,
S field
= ⌧at
ary
ρ = ∞, of
thethe
asymptotic
solution
behaves
as
7
ab
ab
6
where
≡
W
>
r
is
the
value
of
W
at
the
pole.
P
h
√
#
6
oundary
ρ
=
∞,
the
asymptotic
solution
behaves
as
wherethe
WPAdS
≡ Wboundary
(0) > rh isρthe
value
of asymptotic
W at the pole.
ors. Near
= ∞,
the
solution behaves
8
√ by using either the boundary conditio
branes as
isSolving
the
equations
of
motion
h
,
(3)
∝
d
σ
S
8
D7
and
condensate
where
the
two
integral
constants
m
and
c
are
related
to
the
quark
mass
M
c Solving the equations of motion by using either the boundary condition (7) or (8) when
the
qD7
h
,
∝
d
σ
S
0
1
2
3
4
5
6
7
8
9
brane
intersects
with theand
black hole or not, respectively, we obtain a seq
+ · ·cintersects
·F,ab⟨Oismthe
(5)
m
+ brane
c 35]
metric
field
strength
on
the
brane.
a, obtain
b are the
brane
coordinates
⟩ ,with
as gauge
[25,the
29–32,
2 and
black
hole
or
not,
respectively,
we
a
sequence
of
embeddings.
+
·
·
·
(5)
) = ρm
+
4
4
8
2
2
2
2
D3
⇥
⇥
⇥ ⇥
· ·3· =, dx1 +dx2 +dx3 and L is theInAdS
(5)radius.
W=
(ρ)1−r
= mz+/L2 ,+d⃗x
Fig.
3, we show
the
solutions
for several
parameter values. For m/r
where
The
AdS
boundary
is differential
2 f (z)
h
ρ
equation
of hmotion
a second-order
ordinary
equation for
h=
det h2ab . The
ρ
In Fig. 3, we show the solutions
forwhere
several
parameter
values.
For m/r
≃ 0.92,isD7
we
can
see
⇥
⇥
⇥Minkowski
⇥ ⇥ embeddings
⇥ ordinary
⇥ ⇥ are
that
both
the
black
hole
and
allowed, and
t
.
The
equation
of
motion
is
a
second-order
differential
equa
where
h
=
det
h
m
N
f
ab
at
z
=
0,
while
the
event
horizon
is
at
z
=
L
/r
.
We
use
the
same
world-volume
coordinates
h
and
condensate
re related
to the
Φ(z),
that both
thequark
blackmass
hole M
and
Minkowski
embeddings
are
allowed,
and
they
are
characterized
q M
,
⟨O ⟩ = − 0 by4 different
c , values
N Mq =
%(6)
$
of the
quark
condensate
c for a given m. We plot c/rh3
2mass M m
13s ℓ6
7
′
Φ(z),
g
=
@
x
@
x
G
.
(II.5)
2πℓ
16π
g
and
condensate
dgral
c are
related
to
the
quark
mass
M
and
condensate
constants
m
and
c
are
related
to
the
quark
ab
a
b
M
N
q
q
as
the
target
space
coordinates
themselves
such
that
(σ
,
σ
,
·
·
·
,
σ
)
=
(t,
z,
⃗
x
,
Ω
).
The
s
s
the
probe
low
energy
action
a<D7-brane
in aim
z WeInplot
31.0802
tan
f 3 33 e⇤ective
% Φ < rfor
$ ′ for
by different values of the quark condensate c for a given
m.
c/r
a limit
function
rh /m
3 the
′′
′ in
′3
′2of
h as 4.
′
Fig.
c/r
becomes
multivalued
/m
1.0913.
This
h
(8f
−
zf
−
Φ
=
0
.
(4)
Φ
−
)Φ
+
(3
tan
Φ)Φ
+
+
h
3
z
3
tan
Φ
f
3
, 35] inembedding
′′ trary
′3
′2 zDirac-Born-Infeld
the
D7-brane
is for
specified
its <
(φ,
ψ)
the
space
aMinkowski
background
istransverse
described
by
(DBI)
and
Fig. 4. c/r
becomes
multivalued
1.0802 <byrh /m
1.0913.
This
implies
that
there
isΦ)Φ
2position
fa between
z 2 f Φ′ + and
h of
−
zfin′ )Φ
=hole
0Chern-S
. embe
Φ
− (8f
+
(3
tanquark
+ as
first-order
phase
transition
the−
black
where
ℓ
coupling
constant.
O
is
the
bilinear
s is the string length and gs is the string
m
2
N
f
2
f
z
z
f
phase
between
Minkowski
and black
hole
embeddings,
and
it can
be
(CS)
actions
function
of
z:
φ=
Φ(z)
and ψItthe
=
0.
that
weto
can
set
ψ =
0 by
without
loss
of free
generality
confirmed
comparing
the
energy
−1
associated
with
its
supersymmetric
counterparts
(See
Refs.
[34,
47]
for
further
details).
Here⟨Om ⟩first-order
=−
cN
,transition
(6)Note
m
N
is
also
convenient
introduce
new
bulk
coordinates
(w,
ρ) = [32,
(L2 z34].
sin φ, L2 z −1 cos φ).
f
4
6
f
16π
g
ℓ
confirmed
by
[32,
s the
−1
M
= after,
, U(1)-symmetry
⟨Othe
⟩ free
= proportional
−energy
c ,34].
(6)c as on
scomparing
mthe
.convenient
The
induced
metric
the
D7-brane
thanks
generated
∂ψ(6)
we
ignore
and
refer
to6to
m
quark
mass
,
⟨O
−
is by
also
introduce
bulk
(w,Inρ)practical
= (L2 znumerical
sin φ, L2 z −
m ⟩q =to
4g ℓ
6 Itconstants
The
new
embedding
function
canand
then bethe
a new
function
ofcoordinates
ρ,and
wisthe
=given
W (ρ).
2πℓ2 4 6 c ,
16π
1.02
1.02
1
Holographic Mesons
0.98
0.96
0
0.2
1
0.98
0.4
0.96
0
0.2
0.4
sition is always of first order, and in confining theories with
re higher than the deconfinement temperature for the glue.
nce is a
of gauge
rgravity
oft coustudy of
portant
nement,
ons, etc.
QCD ittions of
the deoing behis does
of QCD
dictions
niversal
regimes.
calculaegime of
es. The
class of
gravity
generihis ratio
rom ex(RHIC)
any unidence as
ntal repa small
Nc , may
te graviperature
this pasystem
ular, we
ansition
e quark
e the atoutside
the horizon in a ‘Minkowski’ embedding — see fig. 1 and
2
5
0
1.75
"0.05
Minkowski embedding
1.5
A
"0.05
"0.06
"0.1
"0.07
Critical embedding
1.25
1
c
Black hole embedding
c
"0.15
0.75
"0.08
B
0.5
"0.2
"0.09
0.25
"0.25
0.5
1
1.5
2
2.5
3
FIG. 1: Profiles of D7-brane embeddings in a D3-brane background. The thick black circle is the horizon (ρ = 1).
0
1
2
!
T!M
0
3
4
0.762
0.764
0.766
!
T!M
0.768
0.77
0.772
"0.029
"0.0295
"0.1
"0.03
below. In this phase the meson spectrum (i.e., the spec"0.0305
"0.2
trum of quark-antiquark bound states) is discrete and
I!! "0.031
possesses a mass gap. At sufficiently large T /Mq , theI!!
"0.3
"0.0315
gravitational attraction overcomes the brane tension and
the branes fall into the horizon yielding a ‘black hole’ em"0.032
bedding. In this case, the meson spectrum is continuous "0.4
"0.0325
and gapless. In between, a limiting, critical solution exists. We will show that the phase diagram in the vicinity "0.5
0.762 0.764 0.766 0.768 0.77 0.772 0.774
0
1
2
3
4
!
!
T!M
of this solution exhibits a self-similar structure. While
T!M
this structure went unnoticed, the phase transition that
occurs as T /Mq increases from small to large values was
FIG. 3: Quark condensate and free-energy density for a D7 in a D3 background; note that N ∝ T 3 . The blue dashed
observed in two specific models [5, 6]. In fact, as was
(red continuous) curves correspond to the Minkowski (black hole) embeddings. The dotted vertical line indicates the precise
first noted in [5] for a D6-brane in a thermal D4 backtemperature of the phase transition.
ground, the transition is of first order (in the approximations stated above). Rather than dropping continuously
through the critical solution, the probe brane jumps disdiffers from (17) with p = 2, d = 2. This discrepancy
found here. Note that this effect is of order 1/Nc2 , and
continuously from a Minkowski to a black hole embedis not at all a contradiction, and has the same origin as
therefore subleading with respect to the order-Nf/Nc cording at some T = Tfund . This leads to a discontinuity rection
in
the discrepancy found for the meson spectrum [10]. This
to physical quantities from the presence of the
several field theory quantities, such as, for example, the
is the fact that the calculation in [19] applies in the far
brane probes.
quark condensate ⟨ψ̄ψ⟩ or the entropy density. In the Finite ’t Hooft coupling corrections correspond to
infrared of the gauge theory, whereas that presented here
8.7:behavior
Variousand,
D7-brane
configurations
in a black
backgroundapplies
with at
increasing
temper- i.e., at T ≫ g 2 .
following, we will see thatFigure
the critical
ashigher-derivative
a
high temperatures,
corrections
both D3-brane
to the supergravity
YM
ature from
left to right.
At
lowand
temperatures,
probe
branes
o↵ smoothly
above
the
result, the first order transition
are essentially
universal
action
the D-brane the
action.
These
may close
also blur
We will
return
to horizon.
these and other issues in [20].
to all Dp/Dq systems. At high temperatures, the
the
structure
above.
Forhorizon.
example,Inhigherbranes
fall discussed
through the
event
between, a critical solution exists
derivative
corrections
to
the
D-brane
equation
of
motion
Acknowledgments:
We thank
in which the branes just ‘touch’ the horizon at a point. The critical configuration
is never realized:
a V. Frolov for a converare likely to spoil the scaling symmetry of eq. (6), and
sation
and
for
sharing
ref.
first-order phase transition
occurs from a Minkowski to a black hole embedding (or vice versa) before[7] with us prior to publihence the self-similar behaviour. These corrections also
2007
cation. WeMateos,
also thank Myers,
P. Kovtun,Thomson,
A. Starinets and
L.
the critical solution is reached.
become important as the lower part of a Minkowski brane
Yaffe for discussions. Research at the Perimeter Institute
approaches the horizon, since the (intrinsic) curvature of
is supported in part by funds from NSERC of Canada
the brane becomes large there.
and MEDT of Ontario. We also acknowledge support
Yet another type of correction one may consider is due
from NSF grant PHY-0244764 (DM), NSERC Discovery
hole phase.
to the backreaction on the background spacetime of the
grant (RCM) and NSERC Canadian Graduate ScholarDq-branes,
magnitude
is controlled
the ratio
Although we will come
back whose
to this
important
point by
below,
we wish
to emphasize
rightwould also like to thank the
ship (RMT).
DM and RCM
mesonic
phase
melted
phase
7
1
etermined
Dirac–Born–Infeld
action.
Mby
(V )the
=M
(II.3)
1 cos( ⇡VV(DBI)
) 06
V 6 InV,
f
2
eength,
are following
same
introduced in [9] and will not repeat the details. The D7the actionthe
is :
1 notations asV ones
> V,
bedded
the six directions of the bulk metric, ~x and ⌦3 . We choose the other two coordinates on th
# along
√
ischoose
constant.
Note
erval
in which
the mass
of thev.black
hole rest
increases
from
zero
to Mf which
8coordinates
be
null
u
and
For
the
of
the
bulk
coordinates
we
the
ansatz
h
,
(3)
∝
d
σ
1
7
nt horizon is rh = Mf4 .
V =
V (u,
Z(u,
(u, to
v),the bulk.
= 0. The
damental matter in the gauge theory
side
we v),
have zto=add
thev),
probe=
branes
on is a second-order ordinary differential equation for
Erdmenger, Meyers, Shock, 2007
of freedom living on the brane is explained by the DBI action as
# Johnson,
nce we are$interested
in
studying
the
e↵ect
of
the
external
magnetic
field Rashkov,Viswanathan,
on the dynamical2007
meson mel
%
Filev,
Z
′
B
$0
pΦ
3 field′ to38be
tan
f
′2 magnetic
oose
the
0
an Φ)Φ +
,
(II.4)
S−= ⌧Φ7 + d ⇠2 g=
ab 0+. (2⇡↵ )Fab (4)
f
z
z f
L!Ρ"
L!Ρ"
#
#
Fx1 x2 = B,
2 B−1
2 −1
$0
B$5
bulk
coordinates
(w,
ρ)
=
(L
z
sin
φ,
L
z
cos
φ).
metric and Fab is the gauge field strength on the brane. 2a, b are the brane coordinates
and
2
be
of ρ,L!Ρ"
w = W (ρ). In practical
L!Ρ"
at ais function
constant.
# numerical 1
#
1
W (ρ) obtained
rewriting
Eq.
(4) inN terms
of Wto(ρ).
B
$0
B
$5
Since
we wouldbylike
the
u
and
stay
null,
we
have
to
impose
the
following
constraint
equa
M v coordinates
g
=
@
x
@
x
G
.
(II.5)
2
2
ab
a
b
M
N
the AdS ofboundary
givesthe
information
of the
dual obtained# from the DBI action
Ρ
e at
equations
motion for
fields on the
brane,
L!Ρ" solution behaves as6
undary ρ = ∞, the1 asymptotic
4L!Ρ"
6
8
B1$02
"1
"1
2
2
2
F
V
+
2Z
V
Z
(Z
)
=
0,
,v
,v
,v
,v
c
Ρ
Ρ
# 2
2
L!Ρ"
L!Ρ"
= m+ 2 +··· ,
2 $10
"2
2
4
6 V 2 +(5)
82Z "2
2= 0, 4
6
8
F
V
Z 2 (Z )B
ρ
Non-Zero Magnetic Field:
"1
1
,u
,u ,u
"1
,u
1#
#
2
2
L!Ρ"
L!Ρ"
c are related to"2
the
quark mass
Mq and condensate
(u,v)
B
$10
B
$17
"2
brane. With the above assumptions the DB
ere (u, v) ⌘ Z(u,v) which, in fact, gives the shape of the Ρ
2
1
1 62
L!Ρ"
comes
2
4
6
8 L!Ρ"
2
4
8
"1
"1
r
1
1
N
Ρ
Z
f
3
2
2
#
#
⟨Om ⟩ = −
c,
cos
1 (6)
2
4 g ℓ6
0 )2 B 2 F4V V "2
2
6
8
2
16π
B
$10
B
$17
dudv
V
+
(2⇡↵
+
V
Z
+
Z
V
Z
S = ⌧7sV"2
Ρ
⌦
,u
,v
,u
,v
,u
,v
,u ,v .
x
s 3 ~
3
4
1
1
Z "1Z
"1
2
4
6
8
#
L!Ρ" Om is the quark bilinear "1
L!Ρ"
he string coupling constant.
Ρ
Ρ "1
"2 the
"2magnetic fie
e di↵erence
action
and
one
in [9]Bis$10
that we 2have considered
non-zero
terparts
(See between
Refs. [34,the
47] 2above
for further
Here4 details).
6
8
4
6
8
"2motion
"2
pears
inrefer
the square
root.
of
"1 read
2c asTherefore,
2
nts and
to "1
m and
the quark the
massequations
and
the
L!Ρ"
L!Ρ"
✓
◆
"2
1
11
ZB1+"2
3
F B2
oundary
terminates.
V,uv conditions
(Zat )where
F,Z
V,u V,v 2,
=
),v D7-brane
+ tan(Z
2)[(Z ),u V,v + (Z ),v V,u ] +
,u (Z the
2 embedding, that is,2 whether
Figure
4: IncreasingΡvalues of2 B̃ for fixedZ T show the repulsive n
topology of the
the D78
+
˜
1
1
the mesonic (stable mesons) and melted (unstable mesons) phases, respectively [8]. The process of dynamical meson
of the4:probe
brane condensate
in the asymptotically
AdS /m.
background.
In fact,isina the
AdS-blackfunction
hole background, the shape of
Figure
The quark
against
condensate
multi-valued
melting
is described
by the changec/r
inh3the
shaperhof
theThe
probe
brane dynamically.
In order to achieve this goal we study
the brane can
into Minkowski
(ME) and
Black Hole Embedding (BE) which correspond to
II.be categorized
PROBE D7-BRANE
INEmbedding
THE VAIDYA
BACKGROUND
in shape
1.0802of<the
rh /m
< brane
1.0913,inand
a finitebackground.
jump between
the points
and hole
B byformation
the
the
probe
the makes
AdS-Vaidya
It describes
theAblack
in the bulk which
the mesonic (stable mesons) and melted (unstable mesons) phases, respectively [8]. The process of dynamical meson
corresponds
to
thermalization
in the gluon sector of the dual gauge theory. The background is time-dependent and
first-order
transition.
melting isphase
described
by the change in the shape of the probe brane dynamically. In order to achieve this goal we study
In this section
wethe
will
study
thebrane
set-up
ofsuch
the
meson
melting
in the
presence of a constant external
therefore
shape
of the
brane
a dynamical
background
can deform
dynamically.
the shape
of
the probe
ininthe
AdS-Vaidya
background.
It describes
the black hole formation in the bulk which
Let
us start with
the AdS/CFT
AdS-Vaidya
metric
magnetic field.
According
tothermalization
the
theis mesons
in the
gauge
theory
fluctuationsand
corresponds
to
in dictionary
the
gluonthat
sector
of the dual
gauge
theory.
Thecorrespond
backgroundtois the
time-dependent
3
Time-Dependent Set-Up
Dynamical embeddings
of the probe brane
in the
AdS
background.
In fact,
in the dynamically.
AdS-black
hole background, the shape of
therefore
the asymptotically
shape of the brane
in 1
such
can deform
⇥ a background
⇤
2
2
M
N
2
2
2
2
2
2
the brane can beLet
categorized
into
(ME)
and
Black
Hole
Embedding
(BE)
which
correspond
to(II.1)
ds =
F
(V,
z)dV
+
d
GMthe
dxAdS-Vaidya
dx =Embedding
2dV
dz
+
d~
x
+
cos
d⌦
+
sin
d
,
us start
with
metric
that
is
NMinkowski
3
3
2
3.1 Vaidya-AdS5 spacetime
z
the mesonic (stable mesons) and melted (unstable ⇥mesons) phases, respectively
[8]. The process of dynamical meson
⇤
1
2 goal
2 we consider
Mthe
Nvery-far-from-equilibrium
2
2of the
2 Murata,
2 achieve
2 inthis
2 we study
zero
magnetic
field:
Ishii,
Kinoshita,
Tanahashi;
2014
dynamics
probe
D7-brane
melting is described
bysection,
thedschange
the
shape
the
probe
brane
dynamically.
In
order
to
whereIn this
F
(V,
z)dV
+
d
= GMin
dx
= offor
2dV
dz
+
d~
x
+
cos
d⌦
+
sin
d
,
(II.1)
N dx
3
3
z2
dynamical
spacetime
focusing
on the thermalization
Thermalization
in theformation
boundary in the bulk which
the shape of athe
probe brane
in the
AdS-Vaidya
background. process.
It describes
the
black hole
4
F (V,
z) gauge
=
1theM
(Vdue
)z ,to
(II.2)
Webyare
following
same
notations
as collapse
ones
introduced inand
[9]
an
is realized in in
thethe
gravity
the
hole
formation
gravitational
corresponds theory
towhere
thermalization
gluondual
sector
of black
the
dual
theory.
The
background
is time-dependent
in the as
field
theory
ismetric,
dual ~
inshape
the bulk
AdS
space.
In this
paper,
we use
theThermalization
Vaidya-AdS
spacetime
the
dynamical
therefore the
of
the
brane
in
such
a
background
can
deform
dynamically.
embedded
along
the
six
directions
of
the
bulk
x and ⌦3 . W
4
and we have set the radius of AdS space-time to F
be(V,
one,
Rblack
= 1.MThe
above
metric isinwritten
in Eddington-Finkelstein
z)
=
1
(V
)z
,
(II.2)
to
hole
formation
gravity.
background
with
the
black
hole
formation.
Hereafter,
we
set
units
where
the
AdS
radius
is
Let us start
with
the
AdS-Vaidya
metric
that
is
to
be
null
coordinates
u
and
v.
For
the
rest
of
the
bulk
coordinates
coordinate where the radial direction is represented by z and V shows the null direction. The boundary, where the
AdS-Vaidya Background
unity,
L =have
1.lives,
The
Vaidya-AdS
metric
bycoordinate
5AdS
andtheory
we
set the
to
be one, R =on1.
Theboundary.
above metric
is ),
written
gauge
is
atradius
z =10of⇥and
Vspace-time
is is
thegiven
time
M (V
whichinisEddington-Finkelstein
an arbitrary function,
⇤ the
2
2
M
N
2
2
2
2
2
2
thethe
radial
direction
is represented
by
zd~
and
V
shows
the
null
direction.
The
boundary,
where
the
represents
massdx
of
hole
which
changes
as
time
passes
by
until
it
reaches
a
constant
value.
function
V
=
V
(u,
v),
z
=
v),
=
dscoordinate
F
(V,
z)dV
+
d
= Gthe
dx
= black
2dV
dz
+
x
+
cos
d⌦
+
sin
d
, Z(u,The
(II.1)
M Nwhere
3
3
1
2
2
(5) z 2 µ
ν
lives,
= dx
0 and
V 2 is
the(V,
time
M (V ), which
[−F
z)dV
− 2dV dzon+the
d⃗x3boundary.
],
(9)is an arbitrary function,
gatµνzdx
for gauge
M (V )theory
that we
williswork
with
in=this
paper,
is coordinate
represents the mass of the black holez which changes
as time passes by until it reaches a constant value. The function
8
4
Since
we
are
interested
in studying the e↵ect
of the external mag
where
,
(10)
F (V,with
z) =in1this
− M(V
)z
for M (V ) that we will work
paper,
is
0
V
<
0,
< ⇥
⇤ to be
choose
the
magnetic
field
⇡V
1
8
M (VF) (V,
=M
cos(
) 0 6 V 6 V,
2 1M (V
z)f =
)z 4V,the
(II.2)(II.3)
where the mass
function M(V ) is a free
function
Bondi
mass
(density)
of
this
:<1representing
0
V
<
0,
Period
of
the
Boundary
Time
⇤ V > V,
11⇥
⇡V
spacetime. This metric is an exact M
solution
five-dimensional
equation
(V ) = of
Mfthe following
(II.3)
1
cos(
)
0 6 V 6 Einstein
V,
2
V
EnergyFx1 x2 = B,
:
and
we
have
set
the
radius
of
AdS
space-time
to
be
one,
R
=
1.
The
above
metric
is
written
in
Eddington-Finkelstein
tric γab are evolution
the in
equations
V > V, from zero to Mf which is constant. Note
with null
dust,
where
Vequations,
is the timeand
interval
which the (16)
mass of31 the black hole increases
coordinate
where
the radial
direction is
represented
by1 z3zand
boundary, where the
dMV shows the null direction. TheInjection
duced
metric are
constraint
equations.
In
the
following,
(5)
4
that
is
constant.
kν boundary.
, hole in
(11)
that
the
radius
the
event
horizon
rh6g
=µν
M=
. ofon
µνis−
µblack
which isthe
constant.
Note T
where
V is
the
interval
inGtime
which
the
mass
thekthe
increases
from
zero not
to
f
gauge theory
lives,
is
at
z of
=
0time
and
V is
the
coordinate
M (V
), which
is M
an
arbitrary
function,
f
We
are
following
the
same
notations
as
ones
introduced
[9]
and
will
repeat
details.
2
dV
1
gs on the Vaidya-AdS
background
spacetimes
solving
Inmass
orderoftothe
addblack
the fundamental
matter
inaswe
the
theory
side
we
have
add
the probevalue.
branes
tonull,
the
bulk.
The
Since
would
like
the
uitand
vto coordinates
to
stay
we have
4gauge
represents
the
hole
which
changes
time
passes
by
until
reaches
a
constant
The
function
µ
that kthe
radius
of
the
event
horizon
is
r
=
M
.
embedded
along
the
six
directions
of
the
bulk
metric,
~
x
and
⌦
.
We
choose
the
other
two
coordinate
h
f
where
denotes
the
ingoing
null
vector
defined
by
k
dx
=
−dV
.
The
null
dust
injected
from
3 DBI action as
µ of the degrees of freedom living on the brane
µ is explained by the
dynamics
the
equations
of
motion
for
the
fields
the
brane,
obtained
In
order
to
add
the
fundamental
matter
in
the
gauge
theory
side
we
have
towe
addon
the
probe
branes
to the bulk.from
The th
for M
(V
)
that
we
will
work
with
in
this
paper,
is
the
AdScoordinates
boundary infalls
into v.
the For
bulk,the
andrest
thenZ
thethe
event
horizon
is formed.
When
the
mass
null
u
and
of
bulk
coordinates
choose
the
ansatz
a to be
inates σ (a = 0, 1,
· · · , 7)ofonthe
the
brane.
For six8
of them,
dynamics
degrees
of freedom
living
on the branepis explained by the DBI action as
function
M(V
)
stops
depending
on
V
,
namely,
when the energy injection
ceases, the spacetime
2
2 (II.4)
gab<+0,(2⇡↵0 )Fab ,
S = ⌧7 Zd8 ⇠
0
V
F
V
+
2Z
V
Z
(Z )
<
·...,
· ,⇠σ77)isometric
) = (⃗x3 , to
Ω3the
) because
we
assume
emselves as (σ(⇠2is,2·,locally
,v
,v
,v 0.
Schwarzschild-AdS
horizon.
5 with
V =1 ⇥V (u,
v), a⇡Vplanar
z8 ⇤p
= Z(u,
v), 0 = (u, v),
=
+ (2⇡↵
)Fab ,
S = cos(
⌧7 dV ⇠) 0 g6
(V those
) = Mfdirections.
(II.3)
6 V,
ab V
as the background0 bulk
spacetimeMin
2
2(II.4)
2 1
1
:
F
V
+
2Z
V
Z
(Zand)
,u coordinates
,u
,u
(⇠
,
⇠
)
=
(u,
v)
is
the
induced
metric
on
the
brane.
a,
b
are
the
brane
where
g
0
1 and Fab 1is the gauge field strength
V
>
V,
ab
oduce the double
nullare
coordinates
(σin, σstudying
) = (u, v),
and
Since
we
interested
the
e↵ect of the external magnetic field on the dynamical mes
is defined
as
gab where
gab is the induced metric and Fab is the gauge field strength on the brane. a, b are the brane coordinates and
re parametrized
as
(u,v)
gives
the
shape
the
magnetic
field
to the
be mass
is shape
constant.
Note
where choose
V is the
time
interval
which
of the(u,
black
hole
increases
from
zero
to M
is defined
as in
gab
f which
which,
in
fact,
gives
the
of
the bran
where
v)
⌘
M
N
Z(u,v)
Null Coordinate on the
1
g
=
@
x
@
x
G
.
(II.5)
ab
a
b
M
N
of
the
brane
4
M
N
that
of
the
Mf . (17) gab = @a x @b x GM N .
= Z(u,
v)the
, radius
φ = Φ(u,
v) event
, Brane
ψ horizon
= 0 , is rh = becomes
(II.5)
Probe
F
=
B,
x
x
10
1
2
In order to add the fundamental matter in the gauge theory side we have to add the probe branes to the bulk. The
Freedom to choose eight world-volume coordinates
Z
as3
dynamics
of the
degrees
of freedom living
on the by
brane
is explained by the DBI actioncos
.
erality
because
of
the
U(1)-symmetry
generated
∂
ψ
that is constant.
S = ⌧7 V⌦3 V~x dudv
Z
9
Z3
hat the ∂u and ∂v are null generators. Then, the branep
r
1
0 )2 B 2 F V V
+
(2⇡↵
,u
Z4
embedded along the six directions of the bulk metric, ~x and
the
thebrane,
braneobtained from the
F⌦x31.the
=choose
B,= B,
(II.7)
(u,v)
of other
motiontwo
forcoordinates
the fields ononthe
DBI
action
xWe
2 equations
F
(II.7)
which,
in
fact,
gives
the
shape
of
the
brane.
With
the
above
assumptions
the
where
(u,
v)
⌘
x
x
1
2
to be null coordinates u and v. For the rest of the bulk coordinates
we choose the ansatz
Z(u,v)
becomes
F V,v2 + 2Z,v V,v Z 2 (Z )2,v = 0,
that is
constant.
V = V (u, v), z = Z(u, v),
= Z
(u, v),
= 0. r
(II.6)
that is constant.
2
F
V
2Z,u V,u equations
Z 2 (Z )2,u =
3to impose the following
SinceSince
we would
like
the
u
and
v
coordinates
to
stay
null,
we
have
constraint
on0,on
,u + constraint
we would like the u and v coordinates to stay null, wecoshave to1impose the
following
equations
0
2
2
dudv
F V,u V,v we
Vx from
+ (2⇡↵meson
) B melting,
+ V,u Z,v + Z,u V,v Z 2 ,u ,v .
S=
⌧obtained
we are interested
in for
studying
the e↵ect
of the
external
fieldthe
on3 the
dynamical
7 V⌦magnetic
3 ~
theSince
equations
of
motion
the
fields
on
the
brane,
DBI
action
4
the equations of motion for the fields on the brane, where
obtained
from
action
Z the
ZDBI
(u,v)
choose the magnetic field to be
which, in fact, gives the shape of the brane. With the above
(u, v) ⌘ Z(u,v)
2
becomes
2 ,v V
2 )2 and
F V,v2Fbetween
2Z
Z
(Z
The di↵erence
the
action
we
have considered(II.8a)
non-zero
,v,vabove
,v =
Fx+
V
+
2Z
V
Z
(Z
)2,v0,the
= 0,one in [9] is that
(II.8a)magnetic
=
B,
(II.7)
,v
1 x2,v
r
Z of motion
appears in the2 square
root. Therefore,
the equations
2
cos3 read1
2 ,u V,u
2 )2 =20,
F V,uF+V 2Z
Z
(Z
0 )2 B 2 F V(II.8b)
Z (Z
V~x dudv
+ (2⇡↵
+ V,u Z,v + Z,u V
S,u= ),u
⌧7 V=
⌦30,
,u V
,u + 2Z,u V,u
that is constant.
✓
◆,v(II.8b)
3
4
Z
Z
+
ZBnull,
3 the following constraint equations on1
F B2
Since we would like the u and v coordinates to stay
we have to impose
1
(u,v) (u,v)
V,uv =
(Z ),u (Z ),v + tan(Z )[(Z ),u V,v + (Z ),v V,u ] +
F,Z
V,u V,v ,
the where
equations
of⌘motion
forwhich,
the fields
the
brane,
obtained
DBI
action
inonfact,
givesgives
the
ofthethe
brane.
WithWith
above
assumptions
DBI
where
(u, v)(u,
The
di↵erence
between
thethe
above
action
and
the one2 inthe
[9] the
is
that
we action
have considered n
2 shape
2 brane.
Z action
which,
in fact,
thefrom
shape
of
the
the
above
assumptions
DBI
v)Z(u,v)
⌘ Z(u,v)
in the square root. Therefore, the equations of motion read
+ 2 appears
becomes
2
becomes
3
B2
F,V
FZV,v2 +=2Z,vFVZB
Z
(Z
)
=
0,
1
,v
(Z ),u,v(Z ),v + +tan(Z )[(Z ),u Z,v + (Z (II.8a)
Z,u Z,v
V✓,u V,v
),v Z,u ] +
,uv r
r2 2
Z Z F V32 + 2Z
2
2
Z
2
ZB
3
1
F B2
(II.8b)
cos ,ucos3 1,u1✓V,u 1 Z F(ZB0 2)◆,u 2V=,uv0, = 1 (Z ),u (Z ),v + tan(Z )[(Z
)
F
V
+
(Z
)
V
]
+
,u
,v
,v
,u
,Z
2
0 )2 B
2VZ,v,u V,vZ Z,u2 ,v
2
Z
F 2VV,uFV2+
(II.9)
+ (2⇡↵
)2 B(F
+
V,u
ZV,vV,u+
S = S ⌧=
dudv3
V,v,u
(II.9)
+ (2⇡↵
V,v
++
ZZ,vZ
+
7 V⌦3⌧V
,u
~
x ⌦3 Vdudv
7V
,u .,v .
~
x
4
F
V
V
Z
)
,
3
4
,Z
,u
,v
,u
,v
,v
,u
Z
Z
(u,v)
Z
Z
+ assumptions the DBI action
2
Z
of the brane.
With theF ZB
above
where (u, v) ⌘ Z(u,v) which, in fact, gives the shape
3
B2
1
✓
(Z ),u (Z ),v + tan(Z )[(Z ),u Z,v + (Z ),v Z,u ] +
Z,u Z,v
Z,uv+=◆
becomes
2
2
Z
F
B
B
+
3Z
tan(Z
)
3
tan(Z
)
1is that
1
The di↵erence
between
the above
action
and the
that
we
considered
non-zero
magnetic
✓ have
◆considered
The di↵erence
between
the above
and one
the in
one[9]
have
non-zero
magnetic
+inis[9]
(Z
)we
[(Z ),ufield
Z,vfield
+which
(Zwhich
),v Z,u ]
raction
,uv =
,u (ZF B)2,v
Z
1
2
3
2Z
2
2Z
appears
in theinsquare
root.root.
Therefore,
the
of motion
readread
appears
the square
Therefore,
the equations
of motion
F
cos
1 equations
2(F V,u V,v + V,u Z,v + V,v Z,u ) ,
0 )2 B 2 F V V + V Z2 + ,Z
✓
◆
(II.9)
+
(2⇡↵
Z
V
Z
S = ⌧7 V⌦3 V~x dudv
Z
,u
,v
,u
,v
,u
,v
,u ,v .
+
Z3
Z4
3
tan(Z
)
F
B
B
F,V
◆
✓
2
✓
◆
✓
◆
1
+
+
+
F
+
(F
V
V
+
V
Z
+
V
Z
)
Z
Z
+
+
F
B
B
+
3Z
tan(Z
) V,u V,v ,
3
tan(Z
)
,Z
,u
,v
,u
,v
,v
,u
,u
,v
B2),u (Z ),v
1 2
ZB1 ZB1
3 3
1 1+
F1 B2F
2
(Z
=
[(Z ),u Z,v +
2Z
Z
Z
Z
2Z
,uv
V,uv between
),u)(Z
)[(Z
),u+V
= (Z the
+ thetan(Z
+ (Z
),v] V
+ F,Z F2,Z
(II.10a)
2
)(Z
tan(Z
)[(Z
),u
+),vand
(Zhave
),vconsidered
V,u
+2Z
(II.10a)
The Vdi↵erence
above
action
one
in [9]
is V
that
we
magneticVfield
which
,v
,u ] non-zero
,u,V,v ,
,uv =
,u (Z
,v
,v
,u VV,v
2Z
2root. Therefore, the
2
2
2
2 equations
Z Z ◆
✓ 2
appears in the square
of motion read
3
tan(Z
)
F
B
B1+
where
2
+
+
F,Z ◆ B2 +
Z,u Z,v
+
ZB1
B2 2 F,V (F
F,VV,u V,v + V,u Z,v + V,v Z,u )
✓
F+ZBF
3 3
2
1
2Z
Z
Z
Z
ZB=
3
1
F
B
(Z
)
tan(Z
)[(Z
)
Z
V
(Z
)
+
Z
+
(Z
)
Z
]
+
Z
V
Z,uv
2
(Z
)
tan(Z
)[(Z
)
Z
V
=
(Z
)
+
Z
+
(Z
)
Z
]
+
Z
V
Z,uv
,u ,v
,u ,v
1
,v,u ,v F ,v ,u,v ±,uV V ,u
,u ,v
V,uv =
(Z
),u2(Z ),u
tan(Z ,v
)[(Z 2),u V,v + (Z ),u
, ,v,u ,v22 (II.10a)
,v +
,v V,u ] +
,Z
,u1 ±
,vZ
2
2
2
Z
=
,
B
2
2
Z 1
✓
✓
◆ 2◆
1 + (2⇡↵0 )2 Z 4 B 2
where
1
F
B
2
1 F ZB+ F F B2
3 V,u
(F
Z,v
(II.10b)
,v)[(Z
2 ±
F2,Z1 (Z ,Z),u (Z Z(F
ZV
V +ZV,u,v) Z
,),v,uZ),u, ] + B2 Z,u Z,v F,V V,u V,v
,v +VV
,u+
,v,u
2 (II.10b)
)+
),v V
+,u Vtan(Z
Z,uv =
,u Z,v,v+ (Z
=
3
+
.
B
2✓ 2
Z◆
2 2
=21 ±
,
2
Z
0 )2 Z 4 B 2
1 + (2⇡↵0 )2BZ14 B
◆
✓
1
+
(2⇡↵
◆
✓
+
1
F B2
+F B
B13Z+ 3Z
tan(Z
3 tan(Z
B
tan(Z
) ) [(Z ) Z + (Z
3=
) (F)VF
Ftan(Z
V
+ V,u1 Z,v (Z
+ V,v)Z,u(Z
) , ) B1 +
(II.10b)
2
,Z
,u
,v
1
+
),u,v3]Z+,u ]
,v recover the results
,u+,v
(Z ),uB(Z
[(Z
)
Z
(Z
)
Z
2
Z2Z+
=
.
B
2
,uv =,uv
,v can
,u
,v
,v
By2setting
=,u0)we
of
[9].
2
2 2Z
◆
✓
1 + (2⇡↵0 )2 Z 4 B 2
2Z ✓
2Z
+2
◆
The above
equations
motion) are solvable, in spite of
F B1
B1 + 3Z of tan(Z
3 tan(Z
)
+ being non-linear and time-dependent, if one use
✓
◆
+
+
(Z
)
=
(Z
)
[(Z
)
Z
+
(Z
)
Z
]
3
tan(Z
)
F
B
B
F,V[9]. The solution to these equatio
,uv
,u
,v
,u
,v
,v
,u
2
1the
2finite
3
tan(Z
)
F
B
B
F
methods.
Here
we
apply
di↵erence
method
as
has
been
used
in
2
,V
By
setting
B
=
0
we
can
recover
results
of
[9].
2Z
2Z
1
F
+
(F
V
V,u,V,v , (II.10c)
V
+
V
Z
+
V
Z
)
Z
Z
+
(II.10c)
+
,u+ ,v
,u+ ,v
,v) ,u
,u+,v
F,Z ,Z
+Z us how
(F
V
V
V
V
Z
V
Z
Z
V
+
2
2 Z,v
,u
,v
,u
,v
,v
,u
,u
,u
,v
✓ 2Z
◆Z
Z
2Z
2
2
+due to of
the
background
the presence
ofand
the time-dep
non-zer
The changes
above Bequations
are solvable,
of being
non-linear
2Z
Zmotion
2Z in spiteand
F B2Z 3 tan(ZZ ) the shape of the brane
Fdynamical
,V
1
F,Z
+
(F V,u V,v +to
Vdi↵erence
V,uthe
Z,v +
V,v Z,u ) Here
Z Z,vthe
+ finite
(II.10c)
+
methods.
method
as has
used inof[9].
solutio
,u V,v ,
AdS/CFT
dictionary
near
boundary
expansion
of been
the shape
the The
brane
give
2field. According
2 we,u apply
2Z
Z
Z
Z
2Z
us how the shape
of thespecifically
brane changes due to the dynamical background and the prese
of quark (m) and the condensation
(c). More
wherewhere
field. According to the AdS/CFT dictionary the near boundary expansion of the shap
where
◆
2and the condensation✓(c). Morem(V
3
±
of
quark
(m)
specifically
2
)
,
(II.11a)
± B1 = 21 ±
2
0Z)|
2Z
4
2= m(V ) + c(V ) +
Z
+
....
(V,
=
1
±
,
(II.11a)
B
Z!0
±
1
+
(2⇡↵
)
B
1
◆
✓
, 0 2 4 2
(II.11a)
B1 = 1 ±
6
m(V )3
1 + (2⇡↵10 )2+
Z 4(2⇡↵
B2 ) Z B
2
Z 2 + ....
(V, Z)|Z!0 = m(V ) + c(V ) +
6 instance B is
.
B
2 = 23 +of the2magnetic
Note that
the
presence
field
parameters with (II.11b)
m, (II.11b)
for
0 )2 Z 4
2 we rescale all the(II.11b)
.
B2 =in3B+
=
3
+
.
1
+
(2⇡↵
B
2
4B2 0 2 4 2
B
1 + (2⇡↵10 )2+
)inZ
Bresults.
=Z (2⇡↵
1 that
in all
our
solving
the equations
of motion
will be able
the
presence By
of the
magnetic
field we rescale
all thewe
parameters
withto
m,s
m2 . This means that m Note
B
condensation
behaves
in time.
The boundary and initial conditions are summarized in the following subs
setting
0 we
can the
recover
results
of [9].
By By
setting
B = 0Bwe=can
recover
resultsthe
of [9].
m2 . This means that m = 1 in all our results. By solving the equations of motion w
10
By setting
B equations
= 0 we can
recover
the
results
of [9].
condensation
in time.
boundary
and initial
conditions
are summarized in
TheThe
aboveabove
of motion
solvable,
spite of being
non-linear
andbehaves
time-dependent,
iftime-dependent,
one
uses numerical
equations
of are
motion
areinsolvable,
in spite
of being
non-linear
andThe
if one
uses numerical
Constraint equations
to keep u,v null
equations of motion:
begin we have to distinguish between two configurations mentioned before; Minkowski embedding and Black Hole
begin
wedoes
havenot
to cross
distinguish
betweenwe
two
configurations
mentioned
before;
Minkowski
embedding. Since the
ME
the horizon,
have
to impose two
boundary
conditions;
one embedding
at the AdSand Bl
embedding.
theto
ME
doesthe
notappropriate
cross the horizon,
we and
haveinitial
to impose
two boundary
one at
In order toboundary
solve theZ|
equations
of motion Since
we need
define
boundary
conditions.
To ⇡ conditions;
|u=v+
= ⇡2 , where ⇡
u=v = 0, where we have used the same definition as [9], and the other at the pole
2
we have used theMinkowski
same definition as [9], and
and Black
the other
pole |
=
boundary
Z|
u=v = 0, where
begin we have
distinguish
between
configurations
Holeat the
theto3-sphere
on which
thetwo
D7-brane
wraps, mentioned
shrinks to before;
zero. For the BE,embedding
the 3-sphere never
shrinks
to zero as u=v+
the 2
the
3-sphere
on
which
the have
D7-brane
wraps, shrinks
zero. data
For theone
BE,atthe
never shrinks to ze
embedding. D7-brane
Since the ME
doesthe
nothorizon
cross the
horizon,
we
impose
boundary
the3-sphere
AdS
A.
Boundary
conditions
and to
initial
crosses
and
therefore
we dotonot
need two
to impose
anyconditions;
boundary condition
at
the pole. In order
⇡
D7-brane
crosses
thedefinition
horizon and
therefore
do not
to impose
condition at the pole.
⇡ = boundary
= 0, where
we have
used
the same
as [9],
and in
thewe
other
at need
the of
pole
|u=v+any
boundary Z|to
u=vderive
2 , where
2
the appropriate
boundary
and initial conditions
the
presence
the magnetic
field,
we follow the same
to derive
theshrinks
appropriate
boundary
conditions
in the
presence
of the
magnetic field, we follow
the 3-sphere on which the D7-brane
wraps,
to zero.
For theand
BE,initial
the 3-sphere
never
shrinks
to zero
as the
path
as
[9]
where
we
refer
the
reader
to
it
for
more
details.
In order
theand
equations
of
wethe
need
totodefine
the appropriate
4as [9] where
path
reader
it for
more
details.
D7-brane
crossestothesolve
horizon
therefore
we motion
do we
notrefer
need
to
impose
any
boundary
condition at boundary
the pole. In and
orderinitial conditions.
we appropriate
have
to distinguish
between
two
configurations
mentioned
before;
embedding
and Black Ho
to begin
derive the
boundary
and
initial
conditions
in the
presence
ofconditions
the magnetic
field,Minkowski
we follow
the
same
• Boundary
conditions
at
the
AdS
boundary
A.
Boundary
and
initial
data
• to
Boundary
conditions
at thewe
AdS
boundary
as [9] where Since
we referthe
the ME
reader
itnot
for more
details.
embedding.
does
cross
the horizon,
have
to impose two4 boundary conditions; one at the A
4
data path
The physical observables
are obtained
on the
AdS
boundary,
u = v and
therefore
have
to include
Thehave
physical
are
obtained
on
AdS
u =pole
vwe
and
we⇡have
t
boundary Z|u=v
= 0, where we
usedobservables
the same
definition
as the
[9],where
andboundary,
the
otherwhere
at
the
|therefore
=
, whe
u=v+ ⇡
2
2
the
appropriate
boundary
conditions
there.
The
boundary
on appropriate
Z and
areon
simply
Z|u=v
=simply
0initial
andZ|u=v
• Boundary
at the
AdS
boundary
A. Boundary
conditions
and
initial
data
the
appropriate
boundary
conditions
there.
TheBE,
boundary
condition
Z and
are
Inconditions
order
to solve
the
equations
ofshrinks
motion
wezero.
need
to condition
define
the
boundary
and
con
the
3-sphere
on
which
the
D7-brane
wraps,
to
For
the
the
3-sphere
never
shrinks
to
zero
as
t
|
=
m.
In
order
to
obtain
the
rest
of
the
boundary
conditions
we
expand
the
fields
near
the
boundary
and
iate boundary
and
initial
conditions.
To
oundary
conditions
andobservables
initial data
u=v
|
=
m.
In
order
to
obtain
the
rest
of
the
boundary
conditions
we
expand
the
fields
near
the
boun
The
physical
are
obtained
on
the
AdS
boundary,
where
u
=
v
and
therefore
we
have
to
include
u=v
begin
we
have
to distinguish
between
two
configurations
mentioned
before;
Minkowski
embedding
D7-brane
crosses
the
horizon
and therefore
we
not
need to
impose
anyconsistency
boundary
condition
at the
pole. In and
ord
byand
imposing
the
regularity
condition
ondo
the
equations
of
motion
and
with
the
constraint
equations
e;the
Minkowski
embedding
Black
Hole
by
imposing
the
regularity
condition
on
the
equations
of
motion
and
consistency
with
the
constraint
e
the
appropriate
boundary
conditions
there.
The
boundary
condition
on
Z
and
are
simply
Z|
=
0
and
u=v
equations
of motion
we
need
to
define
the
appropriate
boundary
and
initial
conditions.
To
embedding.
Since
the
ME
does
not
cross
the
horizon,
we
have
to
impose
two
boundary
conditions;
one
to
derive
the
appropriate
boundary
and
initial
conditions
in
the
presence
of
the
magnetic
field,
we
follow
the
sam
we
(II.8)
|u=v = m. (II.8)
Inone
order
obtain
the
restwe
of get
the boundary conditions we expand the fields near the boundary and
conditions;
attoget
the
AdS
nwo
weboundary
need between
to
define
the
appropriate
boundary
and
initial
conditions.
Todefinition
istinguish
two
configurations
mentioned
before;
Minkowski
embedding
and Black
Hole
⇡refer
boundary
Z|
=
0,
where
we
have
used
the
same
as
[9],
and the equations
other at the pole |u=v+ ⇡2
path
as [9]
where
we
the
reader
to
it
for
more
details.
u=v
by pole
imposing
the
regularity
condition
on
the
equations
of
motion
and
consistency
with
the constraint
⇡
he
other
at
the
|
=
,
where
u=v+
2 we
2
he ME does not
cross
the
horizon,
have tothe
impose
two boundary
conditions;
one
at the
AdS
d wraps,
configurations
mentioned
before;
Minkowski
embedding
and Black
Hole
d to
(II.8)
we
get
the
3-sphere
onas
which
D7-brane
zero.
For
BE,
the=3-sphere
never (II.13)
shrinks to
V0 (v) = shrinks
2Z,u |u=v
,V0 (v)
Z⇡,uv
|
=
0the
, Z
the
3-sphere
never
shrinks
to
zero
the
⇡,u |u=v
u=v
=
2Z
,
|
0
,
,uv
u=v
0,
where
we
have
used
the
same
definition
as
[9],
and
the
other
at
the
pole
|
=
,
where
he horizon, we• have
to impose
two boundary
conditions;
u=v+ 2
dv one at the AdS
dv
Boundary
conditions
at
the
AdS
boundary
D7-brane
crosses
the
horizon
and
therefore
we
do
not
need
to2 impose
any boundary condition at the po
boundary
condition
at
the
pole.
In
order
⇡
ich
the
D7-brane
wraps,
shrinks
to
zero.
For
the
BE,
the
3-sphere
never
shrinks
to
zero
as
the
d
⇡ =
he same definition as [9],
and
the
other
at
the
pole
|
,
where
u=v+
Vobtained
(v)that
=
2Z
|u=v
,initial
Z2,uv
|conditions
= boundary
0 , arein
(II.13)
2Note
0(v)
u=v
(v)
= Vto|u=v
. Note
the
boundary
not
a↵ected
the
presence
thehave
external
where
Vobservables
derive
the
appropriate
boundary
and
conditions
the
presence
of the
magnetic
field,
weofinclu
follo
of horizon
the magnetic
field,
we
follow
same
=
V
|,u
. the
that
the
conditions
not
a↵ected
byofwe
the
presence
the
where
V
Thetophysical
are
on
AdS
boundary,
where
u = are
vbyand
therefore
to
0the
0
u=v
he
and
therefore
we
do
not
need
impose
any
boundary
condition
at
the
pole.
In
order
dv
, shrinks to zero. For the
BE,
the
3-sphere
never
shrinks
to
zero
as
the
magnetic
field.
magnetic
field.
the
appropriate
boundary
conditions
there.
boundary
condition
on Z and
are simply Z|u=v = 0 a
path
as
[9]
where
refer
the
reader
to
itThe
forIn
more
details.
priate
boundary
and
initial
conditions
inwethe
presence
ofatthe
magnetic
field,
we
follow
the same
we do not need
to
impose
any
boundary
condition
the
pole.
order
V |u=v
.order
Noteto
that
the boundary
conditions
are not a↵ected
by the
the fields
external
where | V0 (v)==m.
Indetails.
obtain
the
rest
of theatboundary
conditions
wepresence
expandofthe
near the boundary a
etial
refer
the
reader
to
it
for
more
u=v
•
Boundary
conditions
at
the
Pole
•
Boundary
conditions
the
Pole
conditions
in thefield.
presence of the magnetic field, we follow the same
magnetic
by imposing
the regularity
condition
on the
equations
of motion and consistency with the constraint equatio
•Similar
Boundary
conditions
atprevious
the
AdS
boundary
Similar
to
the
set
of
boundary
conditions
themagnetic
presence field
of the
magnetic
fieldthe
does
not alter the b
to
the
previous
set
of
boundary
conditions
the
presence
of the
does
not alter
boundary
or
more
details.
onditions
the(II.8)
AdSwe
boundary
• therefore
Boundary
conditions
at the Pole
we
get
u = v andat
have
to
include
⇡
⇡
⇡ =
conditions
|u=v+on
vanishing
magneticwhere
field.
Therefore
we have
conditions
at the pole,
|u=v+at⇡2 the
= 2pole,
for vanishing
magnetic
field.
Therefore
we
have
The physical
observables
are
obtained
the
boundary,
u = v and
therefore we hav
2 forAdS
2
nobservables
on
Z
and
are
simply
Z|
=
0
and
are
obtained
on
the
AdS
boundary,
where
u
=
v
and
therefore
we
have
to
include
Similar
to
the
previous
set
of
boundary
conditions
the
presence
of
the
magnetic
field
does
not
alter
the
boundary
u=v
boundary
the
appropriate
boundary
conditions
there.
The Zboundary
condition
on Z and are simply Z|u
⇡
d
⇡ =
=
Z
,
V
=
V
,
conditions
theboundary
pole,
for
vanishing
magnetic
field.
Therefore
we
have
expand
the fields
nearatthe
and
=
Z
,
V
=
V
,
(II.14) (II.1
Z
te
boundary
conditions
there.
The |boundary
condition
on
Z
and
are
simply
Z|
=
0
and
,u
,v
,u
,v
u=v+
,u
,v
,u
,v
u=v
2
2
V (v) = 2Z |
, Z |
=0,
Second order, nonlinear partial differential equations: We use
finite difference method to solve them.
Rescaling the parameters to have m=1.
Boundary and Initial Conditions:
0have
,uthe
u=v
,uv u=v
|u=vboundary
=um.
order
to we
obtain
rest
boundary
conditions
we expand the fields near the bo
d
on the
= vInconditions
and
therefore
wethe
toofinclude
consistency
withboundary,
the rest
constraint
equations
n order
to AdS
obtain
the
of where
the
expand
the
fields
near
the
boundary
and
dv
=
Z,v ,Z|u=v
V,u
Vthe
(II.14) with the constrai
Z⇡2 ,u
at
u = vare
+
.simply
,v
atby
u =imposing
v + ⇡2on
. Zof
the
regularity
condition
on==the
equations
of motion and consistency
there.
The condition
boundaryoncondition
and
0 ,and
he
regularity
the
equations
motion
and
consistency
with
constraint equations
(v) =
Vexpand
|get
Note
that the boundary conditions are not a↵ected by the presence of the extern
where
V0(II.8)
t of the boundary
conditions
we
the fields
we
u=v•. Initial
datanear the boundary and
at u = v +•⇡2 Initial
.
data
on
the equationsmagnetic
of motionfield.
and consistency
with the constraint equations
=0,
(II.13) The background at which the probe brane is embedded is time-dependent where before the energy
d The background at which the probe brane isdembedded is time-dependent where before the energy injection,
• Initial data
<Zisthe
0,
the
space-time
is pureV
AdS.
Therefore,
our ,(II.13)
initial
data
includes
theto
static
(v)0,
=the
2Zspace-time
, at
|u=v
= 0 ,Therefore,
,u |u=v V
,uv
VV0<
pure
AdS.
our
initial
data
includes
the
static
solution
the solution
equationstoofthe equ
(v)
=
2Z
|
Z
|
=
0
,
•
Boundary
conditions
Pole
0
,u
u=v
,uv
u=v
dv
The background at which the probe
brane pure
is embedded
time-dependent
where
before
motion
withisnon-zero
field [3,
4]. the energy injection,
dv
motion
forexternal
pure AdS
with for
non-zeroAdS
magnetic
field [3, magnetic
4].
a↵ected
by
the
presence
of
the
d
V <Similar
0, the space-time
is pure AdS.
ourconditions
initial data the
includes
the static
solution
to thefield
equations
to
the previous
set ofTherefore,
boundary
presence
of the
magnetic
does ofnot alter the bounda
= VV0 (v)
|u=v=
. 2Z
Note
that
the
boundary
conditions
are
not
a↵ected
by
the
presence
of
the
external
, pure
Z,uvAdS
| with
= 0non-zero
,
(II.13)
,u |u=v for
⇡
motion
field
4]. the boundary
(v) = |V magnetic
|u=v⇡. =Note
that
are not
by the presence of
where
conditions
atu=v
theV0pole,
for[3,
vanishing
magneticconditions
field. Therefore
wea↵ected
have
v
.
magnetic field.
u=v+ 2
2
III.
III. RESULTS
NUMERICAL RESULTS
NUMERICAL
e boundary
the presence ofZ,u
the= external
Z,v ,
V,u = V,v ,
(II.1
onditions
at conditions
the Pole are not a↵ected byIII.
NUMERICAL RESULTS
magnetic field does not alter
the
boundary
In
this
section
we
study
the
response
of
the
system
to
the
time-dependent
change
in
the
temperature
in
the
•
Boundary
conditions
at
the
Pole
In
this
section
we
study
the
response
of
the
system
to
the
time-dependent
change
in
the
temperature
in
the
presence
previouswesethave
of boundary conditions
the
of magnetic
the magnetic
does notofalter
the boundary
⇡
of presence
an external
field. field
The response
the system
is described by the behaviour of condensation, c, in
erefore
at
u
=
v
+
.
⇡
of
an
external
magnetic
field.
The
response
of
the
system
is
described
by
the
behaviour in
of the
condensation,
c, in terms of
⇡
2
In
this
section
we
study
the
response
of
the
system
to
the
time-dependent
change
in
the
temperature
presence
the pole, |u=v+ 2 = 2 for Similar
vanishing
field.
Therefore
wemagnetic
haveconditions
tomagnetic
theboundary
previous
set
of boundary
the this
presence
of the
field
doescategories:
not alter M
t
the
time.
When
the
field is zero,
behaviour
can magnetic
be classified
into three
the
boundary
time.
When
the
magnetic
field
is
zero,
this
behaviour
can
be
classified
into
three
categories:
Minkowski
of an external magnetic field. The
response ofBlack
the system
is described
byOvereager
the behaviour
c, in terms of
⇡
embedding,
Hole
embedding
and
case.of condensation,
⇡ = 11
(II.14)
conditions
at
the
pole,
|
for
vanishing
magnetic
field.
Therefore
we have
•
Initial
data
u=v+
embedding,
Black
Hole
and
Overeager
2 case.
2
y conditions
the presence
ofZthe
does
alter
the
boundary
the boundary
time. When
the
field
is zero,
this
behaviour
can be classified into three
categories: Minkowski
=magnetic
Zmagnetic
, embedding
Vfield
=V
, not
(II.14)
5
5
Minkowski Embedding:
Black Hole Embedding:
If the corresponding mass of the
If the corresponding mass of the
initial ME configuration is much
initial ME configuration is less
FIG.
1: This
figure
(right)
shows(right)
that
how
an final
overeager
crosses
the horizon
an
FIG.
1: This
figure
shows
that temperature
howconfiguration
an overeager
configuration
cross
larger than the final
temperature
than
the
of
the
1
1
sin
reperesents
shape
of the
the brane
Accordingly,
graph on thethe
lef
W
(⇢)
=
z
sin the
reperesents
shape[9].
of the
brane [9].theAccordingly,
W
(⇢)
=
z
of the system:
system:
m
m
at the final1atstages,
as itstages,
is expected
ME. This
figure
is<plotted
for B is
= plotted
1.87140, forVB=
1This figure
the final
as it from
is expected
from
ME.
T
T
(a) Sub-critical
FIG.
1: This
figure
shows
that how
an overeager
configuration
crosses
the horizon
and
jumps
out of
it and becomes ME.
FIG. 1: This figure
(right)
shows
that(right)
how an
overeager
configuration
crosses
the horizon
and jumps
out of it
and
becomes
ME.
1
reperesents
the shape
the brane [9].
Accordingly,
the
left
showsoscillating
that c(V ) starts oscillating
(⇢) = z sin
the shape
of the brane
[9]. of
Accordingly,
the graph
on the the
left graph
shows on
that
c(V
) starts
W (⇢) = z 1 sin Wreperesents
at as
theit final
stages, from
as it is
expected
from isME.
Thisfor
figure
plotted forV B==
= 0.5 and rh = 1.25.
at the final stages,
is expected
ME.
This figure
plotted
B =is1.87140,
0.51.87140,
and rh =V1.25.
(a) Sub-critical
0.015
0
B=0.56446
- 0.02
B=0.0
0.010
B=0.56446
B=0.0
- 0.04
- 0.06
c(V)
0.005
c(V)
(b) Super-critical
0
(b) Super-critical
- 0.08
-0.10
- 0.12
- 0.005
(c) Overeager
- 0.14
- 0.16
-0.010
Figure 2: Schematic illustration of the brane dynamics.
0
5
10
15
20
0
1
2
3
V
4
5
6
7
8
V
FIG. 2: c(V ) for ME (left) and BE (right). In both figures blue solid curves show c(V
= 0.0747132.
c2:0 isblue
theboth
initial
configuration
condensation,
before
the
black
hole
formation
cOvereager
0(left)
FIG. 2: c(V ) forFIG.
ME 2:
(left)
and
BE
In both
figures
solid
curves
show
c(V and
)curves
c0 ,BE
with
B=0.56446,
in both
which
FIG.
c(V
)
for
ME
(left)
(right).
figures
blue
solid
(c) (right).
c(V
)
for
ME
and
BE
(right).
In
figures
blue
solid
show
c(V
)
c
with
B=0.56446,
in
which
0 ,In
No
Dissipation
and
thus
initial configuration
condensation,
the black
hole formation
in In
the all
bulk.
The r
red
dashed
c0 = 0.0747132. cc00 is
=
1.25
and
=
1.0.
show
c(Vconfiguration
)cin
the before
absence
of cmagnetic
field.
cases
=the
0.0747132.
c0 is the
initial
condensation,
black
hole
formation
in
bulk.curves
The redVdashed
curves
h the
is thethe
initial
configuration
condensation,
before
the blac
0 = 0.0747132.
0 before
Oscillatory Behavior
= 1.0.
show c(V ) in the absence of Vmagnetic
field.
Figure
2:
Schematic
ofmagnetic
the cases
branefield.
show c(V ) in the
absence
of
magnetic
field.ofIn
all
rdynamics.
h = 1.25
1.25 and
show
c(V
) in
the illustration
absence
In alland
casesVrh==1.0.
• Overeager Case
• Overeager Case
• Overeager Case 12
In all cases rh = 1.25 and
V
Overeager:
If the corresponding mass of the initial ME configuration is
(a) Sub-critical
larger but close to the final temperature
of the system,
provided that the time-scale of the change in the
temperature is small:
Ishii, Kinoshita, Murata, Tanahashi; 2014
m
⇠1
T
V <1
5
(b) Super-critical
Brane
Reconnection
5
(c) Overeager
0.65
Figure 2: Schematic illustration of1.4the brane dynamics.
1.2
0.60
1.0
w(ρ )
c(V)
0.55
0.50
0.8
0.6
V=5.5
FIG. 1: This figure (right) shows that how an overeager
configuration
crosses the horizon and jumps out of it and becom
1 0.45
V=1.5
0.4
[9]. Accordingly, the graph on the left shows that c(V ) starts os
W (⇢) = z sin reperesents the shape of the brane
FIG. 1: This figure (right) shows0.2that how an overeager configuration crosses the horizon and jumps ou
0.40
at the final stages,
as it is1expected from ME. This figure is plotted for B = 1.87140, V = 0.5 and rh = 1.25.
reperesents
the 0shape of
the brane
[9]. Accordingly,
the
graph
on becomes
the left shows
tha
W
(⇢)
= zhowsin
This figure (right) shows
that
an
overeager
configuration
crosses
the
horizon
and
jumps
out
of
it
and
ME.
0
0.5
1.0
1.5
2.0
2.5
0
1
2
3
4
5
6
7
at
the
final
stages,
as
it
is
expected
from ME.
This figure
plotted
B =c(V
1.87140,
V = 0.5 and r
1
brane [9]. Accordingly,
the graph
on theisleft
showsforthat
) starts oscillating
= z sin reperesents the shape of the
ρ
V
final stages, as it is expected from ME. This figure is plotted for B = 1.87140, V = 0.5 and rh = 1.25.
FIG. 1: This figure (right) shows that how an overeager configuration crosses the horizon and jumps out of it and becomes ME.
W (⇢) = z 1 sin reperesents the shape of the brane [9]. Accordingly, the graph on the left shows that c(V ) starts oscillating
at the final stages, as it is expected from ME. This figure is plotted for B = 1.87140, V = 0.5 and rh = 1.25.
7
13
!1
!2
!3
2.82052
4.83518
6.84984
0
3.04731
5.18042
7.21553
0.05
B=0.43264
B=0.0
0
c(V)
c(V)
- 0.05
-0.10
- 0.05
BE
changes to
ME by
increasing
the
magnetic
filed.
-0.10
- 0.15
0
1
2
3
4
5
6
7
0
1
2
3
4
V
5
6
7
V
0.65
3.4
c(V)
0.55
c(V)
B=5.22375
3.6
B=1.87140
0.60
0.50
3.2
0.45
3.0
0.40
2.8
0
1
2
3
V
4
5
6
0
1
2
3
4
5
6
7
8
9
10
V
. 3: This figure shows how the BE changes to overeager and consequently to ME after one raises the magnetic field. We
e set rh = 1.25 and V = 0.5.
FIG. 3: This figure shows
how the BE chang
and
V = 0.5.
have set rh = 1.25
FIG.
3: This
figure shows
have
setthat
rh = 1.25 and V
sidering the linear perturbation around the equilibrium shape of the brane. Figure 2 (right) confirms
the fact
) decreases by applying an external magnetic field. We will elaborate more on this in figure 4.
A remarkable phenomenon has been observed in the presence of the external magnetic field. Let us assume we start
m a ME which results in BE after the injection of energy when the magnetic field
is zero, figure
3 up-left.
we
considering
the
linearIfperturbation
around
e the magnetic field, it can still give BE as can be seen in figure 3 up-right. Interestingly,
by further
increasing
c(V
)
decreases
by
applying an external m
14
value of the magnetic field, with the same initial configuration and time-scale of the energy injection, the final
Power Spectrum of c(V)-ceq for the ME:
FIG.
1: This
shows
that howconfiguration
an overeager
configuration
cros
FIG. 1: This
figure
(right)figure
shows(right)
that how
an overeager
crosses
the horizon
an
1
1 (⇢) = z
sin the
reperesents
the brane
shape[9].
of the
brane [9].
sin reperesents
shape of the
Accordingly,
theAccordingly,
graph on the the
lef
W (⇢) = z W
the final
as itfrom
is expected
ME.
This figure
at the finalatstages,
as itstages,
is expected
ME. Thisfrom
figure
is plotted
for B is
= plotted
1.87140, forV B=
10 -5
P (!)
10 -6
The Fourier transform of the
oscillations of c(V) has
discrete spectrum.
10 -7
10 -8
10 -9
10 -10
0
5
10
TABLE
I: Bound-States
or Stable Mesons obtained from the ME Power Spe
15
20
!
Stable Modes
Bound-States obtained from
ME c(V) Power Spectrum
!1
!2
!3
B=0
2.82052
4.83518
6.84984
B = 0.56446
3.04731
5.18042
7.21553
FIG. 2: c(V ) for ME (left) and BE (right). In both figures blue solid curves show c(V
FIG.
forfield,
ME
(left) andcondensation,
BE (right).before
In both
figures
solid
c2:
the)initial
configuration
the black
holeblue
formation
c0 = 0.0747132.
0 isc(V
presence
of magnetic
configuration
1.25 and V =before
1.0. the bla
show c(V ) cin
the0.0747132.
absence of cmagnetic
field. In
all cases rh =condensation,
0 =
0 is the initial
In the
mesons are more show
resistant
c(V ) into
themelting.
absence of magnetic field.
• Overeager Case 15
In all cases rh = 1.25 and
V
tion can be explained by the fact that due to the non-zero magnetic field the probe
e in its shape. In the field theory side this means that the binding energy of the
FIG.
4: The
plot
on the
showsisc(V
) for BEs
valuessatisfies
of B. The
the right
where
the to
equilibration
time
defined
as for
thedi↵erent
time which
✏(Vgraph
0.003
and shows
✏(V ) ✏(V
sta
re, more energy
is needed
melt
the left
mesons.
eq ) <on
boundary
time,
obtained
from
(III.1),
for
the corresponding
plotsc(V
in )the leftat
figure.
The equilibrationmagn
tim
afterwards.
Note
that
ceq
be
obtained
static
solution
the corresponding
ch, due to the the
energy
injection,
ends
up in
a can
BE,
there
exists
afrom
time the
at which
1.25
andbeen
=
1.0.
intemperature.
table II. We
have
rh =
This
function
has
plotted
in figure state.
4 for di↵erent
values of the magnetic field
We call this equilibration
time
Veqsetthat
the
system
isV at
its equilibrium
We
r
from figure 4 (right) that the presence of the magnetic field delays the evolution of the system to it
The equilibration times corresponding
to the
figure
4which
are given in table II which confirm, quantita
is
defined
as
the
time
c(V ) ceq
statement.
✏(V
)
=
,
(III.1) ✏(Veq ) < 0.003 and ✏(V ) stays
where the equilibration
time
is
defined
as
the
time
which
satisfies
satisfies:
c(V )
afterwards. NoteFIG.
that1:ceq
can
be
from
the
static
solution
at
corresponding
magneti
This
figure
(right)
shows
that
how
an overeager
crosses
the horizon
a
FIG.
1: obtained
This
figure
(right)
shows
that
howconfiguration
an the
overeager
configuration
cro
and
for
the
time
afterwards.
7
1
1
temperature. ThisWfunction
has
in shape
figure ofwith
4 the
for
di↵erent
of the
magnetic
field.
II:
Equilibration
Times
to
the
Magnetic
sin
reperesents
brane
Accordingly,
the
graph Field
on
the th
leW
(⇢) =TABLE
z W
sin the
reperesents
therespect
shape[9].
of values
theExternal
brane
[9].
Accordingly,
(⇢)been
=
z plotted
from
figure 4 (right)
thatfinal
theatstages,
presence
magnetic
field
the ME.
evolution
to its
at the
as itofstages,
isthe
expected
ME.delays
Thisfrom
figure
is plotted
forthe
B system
=
the final
as itfrom
is expected
This of
figure
is 1.87140,
plotted
forVeB
0
The equilibration timesB=0.0
corresponding to the figure 4 are given in table II which confirm, quantitativ
B=0.3137
B
Veq
statement.
B=0.5645
- 0.05
Equilibration Time:
B=0.7129
B=0.7946
ϵ(V)
c(V)
0.0
3.27834
-0.10
0.3137with respect to the
3.33708
TABLE II: Equilibration Times
External Magnetic Field
0.5645
3.49866
- 0.15
0.7129
4.8057
B
Veq
0.7946
5.15314
-0.20
7
0
1
2
3
4
5
6
0.0
3.27834
V
0.3137
3.33708
0.08
Acknowledgment
0.003 c(V ) for BEs for di↵erent values of B. 0.5645
FIG. 4: The plot on the left shows
The graph on the right shows3.49866
✏(V ) with respect to
We
would
like
to
thank
S.corresponding
Sharifi, T.
Ishii,
K. Murata
and
N. Tanahashi for fruitfu
the boundary
time,
obtained
from
(III.1),
for the
plots
in the S.
left Kinoshita,
figure. The equilibration
times are
represented
0.06
0.002
0.7129
4.8057
in table II. We have set rh = 1.25 and V = 1.0.
0.001
0.7946
5.15314
0.04
0
3.5
4.0
4.5
5.0
where the equilibration time is defined as the time which satisfies ✏(Veq ) < 0.003 and ✏(V ) stays below this limit
FIG. 2: c(V ) for ME (left) and BE (right). In both figures blue solid curves show c(
0.02
afterwards.
Note that ceq can be obtained from the static solution at the corresponding magnetic field and final
Acknowledgment
0.0747132.
c2:
the
configuration
condensation,
before
the black
holeProg.
formation
cbeen
0 = plotted
0 4isfor
FIG.
c(V
)initial
for
ME
(left)
and
BE
InTanahashi
both
figures
blue
soli
E. function
Shuryak,
does
the
quark
gluon
plasma
atthe
RHIC
behave
as
nearly
ideal
fluid?,”
Part
temperature.[1]This
has“Why
in
figure
di↵erent
values
of
magnetic
field.(right).
Weacan
conclude
We
would
like
to
thank
S.
Sharifi,
T.
Ishii,
S.
Kinoshita,
K.
Murata
and
N.
for
fruitful
co
=
1.25
and
V
=
1.0.
show
c(V
)
in
the
absence
of
magnetic
field.
In
all
cases
r
from figure
4
(right)
that
the
presence
of
the
magnetic
field
delays
the
evolution
of
the
system
to
its
equilibrium
state.
h
(2004)
[arXiv:hep-ph/0312227].
c0 = 0.0747132. c0 is the initial configuration condensation, before the bl
0
2
3 corresponding
4
5 figure 4 are
6
The equilibration
times
to the
given in table II which confirm, quantitatively, the previous
E. V. Shuryak, “What RHIC
experiments
theory
us about
properties
of quark-gluon
Ali-Akbari,
Ebrahim,
and
show
c(V
) in the and
absence
of tell
magnetic
field.
In 2013
all cases
rh = 1.25plasma?,
V
statement.
64 (2005) [arXiv:hep-ph/0405066].
E.TABLE
Kharzeev,
L.✏(V
D.) with
McLerran
for di↵erent values of B.[2]
TheD.
graph
on
the right shows
respect to and H. J. Warringa, “The E↵ects of topological charge change in
II: Equilibration Times with respect to the External Magnetic Field
corresponding plots in the left’Event
figure. The
times are
byequilibration
event P
and
CPrepresented
violation’,”
Nucl. Phys. A 803, 227 (2008) [arXiv:0711.0950 [hep-ph]].
• Overeager
Case 16
[1] E. Shuryak, “Why does the quark gluon plasma at RHIC behave as a nearly ideal fluid?,” Prog. Part. N
Summary and Future Direction:
Mesons melt at higher temperatures due to the presence of magnetic
field.
The time-dependent system equilibrates later at non-zero magnetic
field.
studying the effect of anisotropy by having a time-dependent
anisotropic background
quark-antiquark bound-state (potential) on the time-dependent
background
looking for universal behaviour at very small energy injection timescale
17
“Thank you”