Toru Kojo Bielefeld U with K Fukushima Y Hidaka L McLerran RD Pisarski 1 11 Confined arXiv 11072124 Confined Interweaving Chiral Spirals Toru Kojo ID: 486199
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Slide1
Interweaving Chiral Spirals
Toru Kojo (Bielefeld U.)
with:
K. Fukushima, Y. Hidaka, L.
McLerran, R.D. Pisarski
1/11
(Confined)
(
arXiv: 1107.2124)Slide2
(Confined)
Interweaving Chiral Spirals
Toru Kojo (
Bielefeld U.)
(arXiv:
1107.2124)
with:
K. Fukushima, Y. Hidaka, L. McLerran, R.D.
Pisarski
1
/11
This talk (T=0)Slide3
Chiral restoration
Ch.
freeze out
T
2
/11
μ
B
/Nc
Chiral restoration line
(
Lattice
)
〜
M
N
/
Nc
(〜
300 MeV)Slide4
Chiral restoration
?
NJL, PNJL, PQM, etc.
Ch.
freeze out
T
2
/11
μ
B
/
Nc
Chiral restoration line
(
Models
)
Conf. model
(
Schwinger-Dyson eq.
)
Chiral restoration line
(
Lattice
)
(〜
300 MeV)
quark Fermi sea is formed
(
Glozman
)
〜
M
N
/
NcSlide5
Chiral restoration
?
Ch.
freeze out
T
2
/11
μ
B
/
Nc
Chiral restoration line
(
Lattice
)
(〜
300 MeV)
quark Fermi sea is formed
1, In conventional models,
chiral restoration happens
quickly after
the formation of the
quark Fermi sea
.
2,
Assumption
:
Chiral condensate is
const. everywhere
.
NJL, PNJL, PQM, etc.
Chiral restoration line
(
Models
)
Conf. model
(
Schwinger-Dyson eq.
)
(
Glozman
)
〜
M
N
/
NcSlide6
If we allow non-uniform
condensates...
Ch.
freeze out
T
3
/11
μ
B
/
Nc
Chiral restoration line
(
Lattice
)
M
N
/
Nc
(〜
300 MeV)
quark Fermi sea is formed
Chiral restoration line
(
Models
)
・
GSI-Frankfurt
・
Stony BrookSlide7
If we allow non-uniform
condensates...
Ch.
freeze out
T
3
/11
μ
B
/
Nc
Chiral restoration line
(
Lattice
)
M
N
/
Nc
(〜
300 MeV)
quark Fermi sea is formed
Chiral restoration line
(
Models
)
Deconf
. line
Deconfinment
line would be also shifted because:
・
GSI-Frankfurt
・
Stony BrookSlide8
If we allow non-uniform
condensates...
Ch.
freeze out
T
3
/11
μ
B
/
Nc
Chiral restoration line
(
Lattice
)
M
N
/
Nc
(〜
300 MeV)
quark Fermi sea is formed
Chiral restoration line
(
Models
)
Deconf
. line
Non-uniform
chiral condensate creates the
mass gap
of
quarks
near the Fermi surface.
→
The
pure glue
results are
less
affected by
massive
quarks.
Deconfinment
line would be also shifted because:
・
GSI-Frankfurt
・
Stony BrookSlide9
4/11
Why restoration? (T=0)
E
Pz
Dirac Type
P
Tot
=0
(uniform)
L
R
・
Candidates of chiral pairing Slide10
4/11
E
Pz
Dirac Type
P
Tot
=0
(uniform)
L
R
It costs large energy,
so does not occur
spontaneously
.
・
Candidates of chiral pairing
Why
restoration
?
(
T=0
)Slide11
4/11
E
Pz
Dirac Type
P
Tot
=0
(uniform)
L
R
E
Pz
E
Pz
Exciton
Type
Density wave
P
Tot
=0
(uniform)
P
Tot
=
2μ
(non-uniform)
L
R
R
L
Why non-uniform? (
T=0
)
・
Candidates of chiral pairing Slide12
4/11
E
Pz
Dirac Type
P
Tot
=0
(uniform)
L
R
E
Pz
E
Pz
Exciton
Type
Density wave
P
Tot
=0
(uniform)
P
Tot
=
2μ
(non-uniform)
L
R
R
L
・
Kinetic
energy:
comparable
・
Potential
energy:
Big difference
Why non-uniform? (
T=0
)
・
Candidates of chiral pairing Slide13
5/11
Single Chiral Spiral・ Choose one particular direction :
p
zSlide14
5/11
Single Chiral Spiral・ Choose one particular direction :
・
Two kinds of condensates appear :
linear
comb.
p
z
space-dep.
P-oddSlide15
5
/11 Single Chiral Spiral
・
Choose one particular direction :
・
Two kinds of condensates appear :
linear
comb.
・
Chiral rotation with fixed radius :
p
z
V
period of rotation
Δ
Z 〜
1/2p
F
radius
(for 1-pair)
〜
Λ
QCD
3
space-dep.
P-oddSlide16
6/11
So far we have considered only the Chiral Spiral in one direction.
YES!
Is it possible to have CSs in multiple directions?
Interweaving Chiral Spiral
p
z
Then
, the free energy
becomes
comparable to
the
S-wave color super conductor.
Pairs around the
entire Fermi surface
can
condense.Slide17
7/11
p
F
U
(1)
Z
2Np
(
N
p
: Num. of patches )
Rotational Sym. :
(2+1) D Example
Θ
Q
Variational
parameter :
angle
Θ
~
1/
N
p
We use canonical ensemble :
Q
→
Q
(
Θ
,
p
F
)
・
We will optimize the angle
Θ
SSBSlide18
8/11
Energetic gain v.s. cost
・
Cost
: Deformation
Θ
Q
p
F
equal vol. (
particle num.
)
(dominant for
large
Θ
)Slide19
8/11
Energetic gain v.s. cost
・
Cost
: DeformationCondensation effects
・ Gain : Mass gap origin
Θ
Q
p
F
equal vol. (
particle num.
)
M
E
Q
p
(dominant for
large
Θ
)Slide20
8/11
Energetic gain v.s. cost
・
Cost
: DeformationCondensation effects
・ Gain : Mass gap origin
・
Cost : Interferences among CSs
Θ
Q
p
F
equal vol. (
particle num.
)
M
E
Q
p
Condensate – Condensate int.
destroy one another, reducing gap
(dominant for
large
Θ
)
(dominant for
small
Θ
)
(
Model dep.
!!
)Slide21
A schematic model
9/11 Strength of interactions is determined by
Momentum transfer
,
NOT by quark momenta.
→ Even at high density,
int. is strong
for some processes.
Q
gluon
exchangeSlide22
A schematic model
Q
IR
enhancement
UV
suppression
strength
9
/11
Strength of interactions is determined by
Momentum transfer
,
NOT
by
quark momenta.
Therefore we use the int. with the following properties:
Q
gluon
exchange
→
Even at high density
,
int.
is strong
for
some processes
.
?Slide23
A schematic model
Q
IR
enhancement
UV
suppression
strength
9
/11
Strength of interactions is determined by
Momentum transfer
,
NOT
by
quark momenta.
Therefore we use the int. with the following properties:
・
The
detailed form in the IR region
does not matter
.
Q
gluon
exchange
→
Even at high density
,
int.
is strong
for
some processes
.
?Slide24
A schematic model
Q
IR
enhancement
UV
suppression
strength
9
/11
Strength of interactions is determined by
Momentum transfer
,
NOT
by
quark momenta.
Therefore we use the int. with the following properties:
Q
gluon
exchange
Λ
f
→
Even at high density
,
int.
is strong
for
some processes
.
・
The
detailed form in the IR region
does not matter
.Slide25
Energy Landscape (for fixed pF
)
Λ
QCD
/p
F
(
Λ
QCD
/p
F
)
1/2
(
Λ
QCD
/p
F
)
3
/5
×
δE
tot
.
Θ
deformation energy
too big
gap too small
Θ
10/11
−
M
×
Λ
QCD
QSlide26
Energy Landscape (for fixed pF
)
Λ
QCD
/p
F
(
Λ
QCD
/p
F
)
1/2
(
Λ
QCD
/p
F
)
3
/5
×
δE
tot
.
Θ
deformation energy
too big
gap too small
−
M
×
Λ
QCD
Q
N
p
~
1/
Θ
~
(
p
F
/
Λ
fQCD
)
3/
5
Θ
・
Patch num. depends upon density.
10/11
Slide27
Λ
QCD
N
c
1/2
Λ
QCD
μq
Nuclear physics
CSC
11/11
(2+1) dim. ICSSlide28
Λ
QCD
N
c
1/2
Λ
QCD
μq
Nuclear physics
CSC
Very likely
Chiral
sym. restored
・
Deeply inside:
(
perturbative
quarks)
11/11
(2+1) dim. ICSSlide29
Λ
QCD
N
c
1/2
Λ
QCD
μq
Nuclear physics
CSC
Very likely
Chiral
sym. restored
Local
violation of
P
&
Chiral
sym.
Quarkyonic
Chiral
Spirals
・
Near the Fermi surface:
・
Deeply inside:
(
perturbative
quarks)
Quarks acquire the
mass gap
,
delaying the
deconf
. transition at finite density.
(2+1) dim. ICS
11/11
Slide30
Summary & Outlook
The ICS has large impact for chiral restoration &
deconfinement
.
・ 1, The low energy effective Lagrangian
→ coming soon.
・ 2
, Temperature effects & Transport properties
( → hopefully next CPOD )Slide31
Summary & Outlook
The
ICS
has large impact for
chiral restoration & deconfinement.
・ 1
, The low energy e
ffective Lagrangian → coming soon.
・ 2, Temperature effects
& Transport properties
(
→ hopefully next CPOD )
My guess :
V
T=0
(
inhomogeneous
)
V
0 <<
T <
Tc
(
homogeneous
)
T
〜
Tc
(linear realization)
V
Slide32
Appendix Slide33
( →
McLerran’s talk)
This talk (T=0)
(Confined)
Large
Nc phase diagram (2-flavor)Slide34
Consequences of
convolutional effectsNonpert. gluon dynamics
Fermi surface
effects
emphasized by
Large
Nc
emphasized by
Large μ×
We will discuss
small
fraction
flatter
for larger μSlide35
How useful is such regime ?
Large μ :
small
fraction
Vacuum:
large
fraction
So gluon sector will be eventually modified.
・
Two approximations compete :
・
When modified? :
gluon
d.o.f
:
Nc
2
quark
d.o.f
:
Nc
Nc
2
Nc
×
(μ/Λ
QCD
)
d
-1
larger
phase space
For (
3
+1)D,
μ 〜 Nc
1/2
Λ
QCD
.
(Large
Nc
picture is no longer valid.)
〜
Λ
QCD Slide36
Strategy
Large
Nc
Large μ
μ
Good
Good
Bad
Bad
ΛQCD
Nc
1/2
Λ
QCD
Nuclear
Quark matter
with
pert.
gluons
This work
We will
1, Solve large
Nc
& μ, theoretically clean situation.
2, Construct the pert. theory of Λ
QCD
/μ expansion.
3
,
Infer
what will happen in the
low
density region.
VacSlide37
Θ
~ QΘ
~ Λ
QCD
~ Λ
QCD
condensation region
Interference
effects
Gap distribution will be
small gap
12/29
Slide38
15/29
A crude model with asymptotic freedom Color S
inglet
p - k
IR
enhancement
UV
suppression
・
ex)
Scalar
-
Scalar
channel
strengthSlide39
15/29
A crude model with asymptotic freedom
Color
Singlet
・
ex)
Scalar
-
Scalar
channel
must be
close
must be
close
p - k
G
Λ
f
IR
enhancement
UV
suppression
strengthSlide40
16/29
Comparison with other form factor models Typical model
quark mom.
Ours
mom. transfer
Strength at large μ :
function of :
weaken
unchange
(at large Nc)
・
As far as we estimate
overall
size of free energy
,
two pictures would not differ so much, because:
Hard quarks
Typical
int. : Hard
(dominant in free energy)
・
However,
i
f we compare
energy
difference
b.t.w
. phases
,
typical
part
largely
cancel out
,
so we
must
distinguish these two pictures.
Slide41
A key consequence of our form factor. 1
Quark Mass Self-energy (vacuum case)
At Large
Nc
, largely comes from Quark -
Condensate int.
(large amplitude 〜
Nc)
Mom. space
Decouple
if p & k
are
very
different
(
Composite
objects
with
internal
momenta)Slide42
Relevant domain of Non-pert. effects
Σ
m
(p)
|p|
Λ
c
(
Λ
f
)
restored
Σ
m
(p)
|p|
p
F
restored
restored
broken
(Fermi sea)
made of
low energy
quark - antiquark
made of
low energy
quark - quark hole
Vac.
Finite
DensitySlide43
3 Messages in this section
・1, Condensates exist
only near the Fermi surface.
・
2, Quark-Condensate int. & Condensate-Condensate int. are local in mom. space
.・3
, Interferences
among differently oriented CSs happens only
at the patch-patch boundaries.
Decouple
C
ouple
( Range 〜
Λ
f
)
~ Q
Θ
Q
Θ
>>
Λ
f
If
Boundary int. is
rare process
, and can be treated as Pert.Slide44
20/29
One Patch : Bases for Pert. Theory
Θ
Particle-hole combinations
for one patch chiral spiralsSlide45
21/29
Picking out one patch Lagrangian
・
Kin. terms:
trivial to decompose
・
Int. terms: Different patches can couple
:
momentum belonging to i
-
th
patch
i
i
j
j
k
k
Patch - Patch int.
All
fermions belong to the
i
-
th
patchSlide46
22/29
i
＋
i
−
eigenvalue: Moving direction
“(
1
+1) D” “
chirality
” in
i
-
th
patch
Dominant terms in One Patch, 1Slide47
22/29
i
＋
i
−
eigenvalue:
Moving direction
“(
1
+1) D” “
chirality
” in
i
-
th
patch
・
Fact :
“Chiral”
Non
- sym. terms
suppressed by
1/
Q
Dominant terms in One Patch, 1Slide48
22/29
i
＋
i
−
eigenvalue:
Moving direction
“(
1
+1) D” “
chirality
” in
i
-
th
patch
・
Longitudinal
Kin. (
Sym
.)
・
Transverse
Kin. (
Non
-
Sym
.)
・
Fact :
“Chiral”
Non
- sym. terms
suppressed by
1/
Q
ex) free theory
Dominant terms in One Patch, 1Slide49
22/29
i
＋
i
−
eigenvalue:
Moving direction
“(
1
+1) D” “
chirality
” in
i
-
th
patch
・
Longitudinal
Kin. (
Sym
.)
・
Transverse
Kin. (
Non
-
Sym
.)
excitation
energy
momentum
measured from Fermi surface
・
Fact :
“Chiral”
Non
- sym. terms
suppressed by
1/
Q
ex) free theory
Dominant terms in One Patch, 1Slide50
23/29
Dominant terms in One Patch, 2
“
Chiral
”
sym. part
Non
- sym. part
1/Q suppressed
( can be treated in Pert
. )
IR
dominant
( must be
resummed
→
MF
) Slide51
23/29
Dominant terms in One Patch, 2
“
Chiral
”
sym. part
Non
- sym. part
1/Q suppressed
( can be treated in
Pert
. )
IR
dominant
( must be
resummed
→
MF
)
・
IR
dominant :
Unperturbed
Lagrangian
Longitudinal
Kin. +
“
Chiral
”
sym. 4-Fermi int.
Transverse
Kin. +
Non
- sym. 4-Fermi int.
・
IR
suppressed :
Perturbation
Gap eq. can be reduced to (1+1) D
(
P
T
- factorization
) Slide52
24/29
Quick Summary of 1-Patch results・
Integral eqs
. such as Schwinger-Dyson, Bethe-Salpeter,
can be reduced from (2+1) D to (1+1) D.
cf) kT factorization
・
Chiral Spirals emerge, generating large quark mass gap.
(even larger than vac. mass gap)・
Quark num. is spatially uniform.
(in contrast to chiral density)
Pert.
corrections
At leading order
of
Λ
QCD
/μ
・
Q
uark num
.
oscillation
.
・
CSs
:
Plane wave
→
Solitonic
approach to
Baryonic Crystals
Slide53
2
5/29 Multi
-patches & Optimizing
Θ
Θ
Q
(
Θ
)Slide54
26/29
Multi-Patches: Boundary Effects
・ Interferences among
differently oriented CSs
destroy one another, reducing the mass gap.・ S
uch effects arise only around patch boundaries.
(Remark: Such deconstruction effects are
bigger if
CS’swave vectors take closer value.)Phase space :
〜
N
p
×
Λ
f
2
(
N
p
〜
1/
Θ
)
reduction of gap :
〜
Λ
f
Energetic Cost
:
〜
N
p
×
Λ
f
3
(Checked by Pert. Numerical study by
Rapp.et
al 2000)Slide55
27/29
Energy Landscape
Λ
f
/p
F
(
Λ
f
/p
F
)
1/2
(
Λ
f
/p
F
)
3
/5
×
δE
tot
.
Θ
deformation energy
too big
gap too small
−
M
×
Λ
c
Q
N
p
~
1/
Θ
~
(
p
F
/
Λ
f
)
3/
5Slide56
Chiral Spirals (CSs)
・ One can find (1+1) D solution for the gap equation.
(
except boundaries
of patches)・ The size of mass gap is ~
Λ
f
, if we choose
G ~ 1 / Λf .
・ The subdominant terms can be computed
systematically
as
1 /
Q
or
Θ
expansion
.
・
The form of chiral condensates:
Spirals
&
& Slide57
Energetic cost of deformed Fermi sea
Θ
Q
p
F
・
Constraint
:
Canonical
ensemble
→
Fermi vol. fixed
V.S.
・
Energetic difference : (deformation energy)
δE
deform
.
~
N
p
×
p
F
3
×
Θ
5
(1
+
O
(
Θ
2
) )
(This expression holds even if condensations occur.)Slide58
Condensation effects 1.
・ Gain : Less single particle contributions
(due to
mass gap generated by condensates
)
M
E
E
Q
Q
Q
p
p
Fermions occupy energy levels only up to
Q
−
M
.
δE
1particle
~
−
M
×
Λ
c
×
Q
(phase space)
Λ
c
(after adding
condensation energy)Slide59
Condensation effects 2.
・ Cost : Induced interactions
b.t.w. CSs
i
i
Θ
i-
1
i-
1
QΘ
Patch-Patch boundary
1, Int. between CSs happen
only
within phase space,
〜
Λ
f
2
2
, The strength becomes
smaller
with
smaller
size of
M
B
(mass gap near the
boundary
)
3, The sign is
positive
.
δE
int
.
~
+
N
p
×
f
int
.
(
M
B
,
Θ
)
(Num. of boundary points)
(
f →
0
as
M
B
→ 0
)Slide60
6/21
Consequences of form factor. 1 For
quarks –
condensates int. to happen,
their momentum domains must be close each other.
p
k
k
(+ q)
p
(+ q)
loop
(condensate)
・
Schwinger-Dyson eq. for mass gap: (
q=0
for vacuum)Slide61
6/21
Consequences of form factor. 1 their momentum domains
must be close
each other.
p
k
k
(+ q)
p
(+ q)
loop
must be
close
(condensate)
・
Schwinger-Dyson eq. for mass gap: (
q=0
for vacuum)
For
quarks
–
condensates int.
to happen
,Slide62
6/21
Consequences of form factor. 1 their momentum domains
must be close
each other.
p
k
k
(+ q)
p
(+ q)
loop
must be
close
(condensate)
・
Schwinger-Dyson eq. for mass gap: (
q=0
for vacuum)
・
UV cutoff for
k
is measured from
p
,
NOT from
0
.
・
Condensate created by fermions around momenta
k
can couple
only
to fermions with momenta
p
~
k
.
For
quarks
–
condensates int.
to happen
,Slide63
7/21
Consequences of form factor. 2 Dominant contributions to condensates : Low
energy
modes
When p →
∞ :
・
k must also go to
∞, so
ε(k
)
→
∞
.
(for vacuum)
・
Phase space is
finite
:
Nothing compensates denominator
.Slide64
7/21
Consequences of form factor. 2 Dominant contributions to condensates : Low
energy
modes
When p →
∞ :
・ Phase space is
finite : Nothing compensates denominator
.・
k must also go to
∞
,
so
ε
(
k
)
→
∞
.
Σ
m
(p)
p
p
Λ
c
(
Λ
f
)
Λ
c
(
Λ
f
)
ー
＜
ψ
(
p
)
ψ
(
p
)
＞
ー
・
finite density: Low
energy
modes appear
near the Fermi surface
.
Remark)
(for vacuum)
tr
S
(p)Slide65
3/21
Our goal
(MF
treatments)
・ Large Nc
(high density expansion, T=0)・ Large density
(simple
shape of the Fermi surface)
・ (2+1) D
Simple
analytic insights
Θ
・
To express the energy density
as a function of theta,
・
4-Fermi int. with a strong form factor
and to determine the best shape.
Approximations to be used
2DSlide66
8/21
At very high density: PF >> Λ
f
~
Λ
c
p
F
・
Low
energy
modes:
particle - holes
(
near
the Fermi surface)
・
Domain of condensations:
limited
to Fermi surface region
・
Decoupling
:
For
Δ
P
>>
Λ
f
Q
uarks
do not
couple to condensates
in
very
different momentum domain.
Δ
P
Quark-condensate
int.
is
local
in
momentum space
.Slide67
10/21
Do we need to treat many CSs simultaneously ?
~
Q
Θ~
Λf
Θ
Domain of condensation
~
Λ
c
Λ
f
~
Λ
c
Q
Θ
phase space
( for
1-boundary
)
( for
1- patch
)
QSlide68
10/21
~ Q Θ
~
Λ
f
Θ
Domain of condensation
~
Λ
c
Λ
f
~
Λ
c
Q
Θ
phase space
( for
1-boundary
)
( for
1- patch
)
We consider
Λ
f
/p
F
<<
Θ
<
<
1
where
Phase space:
1- patch
>>
1- boundary
B
oundary
effects
Small Perturbations
to the
1-patch problem
( Patch-Patch interactions
)
Q
Do we need to treat many CSs
simultaneously
?Slide69
14/21
Chiral Spirals (CSs)・ One can find (
1+1) D
solution for the gap equation.
(except boundaries of patches)
・ The size of mass gap is ~
Λf
, if we choose G ~ 1 / Λ
f .
・
The form of chiral condensates:
Spirals
&
& Slide70
1/1
What is the best shape ?・ Gain
: Less
single particle contributions
(due to mass gap generated by condensates)
M
E
E
p
F
p
F
p
F
p
p
Fermions occupy levels only up to
p
F
−
M
.
δE
1-paticle
〜
−
M
×
Λ
×
Q