Peristaltic modes of single vortex

1
Peristaltic modes of single vortex
in U(1) and SU(3) gauge theories
based on PRD75, 105015 (2007)
Toru Kojo (Kyoto University)
in collaboration with
Hideo Suganuma (Kyoto University)
Kyosuke Tsumura (Fuji film corporation)
This work is supported by the Grand-in-Aid for
the 21st Century COE.
?Exploring QCD ? at Isaac Newton Institute, 2007.
8. 23
2
Contents
I, Dual superconductor (brief review)
I-1, Dual superconducting picture for string
I-2, Dual Ginzburg Landau model
II, Peristaltic modes (Main results)
II-1, Static vortex solution and classification
of vortex
II-2, Fluctuation analysis Peristaltic modes
III, Summary and outlook
3
Dual superconducting picture
DSC picture connects the string picture and QCD.
Abrikosov vortex in U(1) theory
Color flux tube in QCD
A.A.Abrikosov, Soviet Phys.JTEP 5, 1174(1957)
Y.Nambu, PRD.122,4262(1974)
t Hooft , Nucl.Phys.B190.455(1981)
B
Mandelstam, Phys.Rep.C23.245(1976)
electric Cooper-pair condensation
dual
magnetic monopole condensation
B
E
squeeze magnetic field
squeeze color electric flux
periodicity of the phase of Cooper-pair wave
function
quantization of the color electric flux
(periodicity of the phase of monopole )
color confinement (static level)
quantization of the total magnetic flux
linear potential between quarks
(topologically conserved)
4
Dynamics of color flux tube
In most cases, we consider the moduli-dynamics of
strings, i.e.,
rotation
translation
stringy vibration
5
D.O.F for the dual string
t Hooft, Nucl.Phys.B190.455(1981)
fix the gauge of the off diagonal elements
(Abelian projection)
SU(3) gauge theory (QCD)
U(1)3U(1)8 gauge theory
8-gluon
remaining U(1)3U(1)8 sym.
2 -gluon related to t3 ,t8 generators
photon
6 -gluon related to other generators
charged matter
(electric charge)
topological configuration
monopoles with U(1)3U(1)8 magnetic charge
6
Model dual Ginzburg-Landau model
Ezawa-Iwazaki, PRD25(1982)2681 Maedan-Suzuki,
PTP81(1989)229
After the Abelian gauge fixing, we get the D.O.F,
especially magnetic monopole,
necessary to construct the dual strings.
Next question Do monopoles really condense? Do
the effects of off-diagonal gluon
fluctuations make theory untractable?
7
Static solution (n 1 vortex)
G-L parameter
(under rescaled unit)
8
Excitation modes under the static vortex
background
Consider only the axial symmetric fluctuation
around the static vortex solution
neglect 3rd and 4th order terms of fluctuations
because we focus on the case where the quantum
fluctuation is not so strong
9
Peristaltic modes of the vortex
eqs. for fluctuations in the radial direction
radial mass
dispersion relation
10
Vortex-induced potential for fluctuations
Only the radial direction of the potential is
nontrivial.
11
Energy spectrum ( the effect of the diagonal
potential )
12
1st excited state wavefunction in the radial
direction
fluctuation of electric field
fluctuations of f? A?
small
squeezed by monopole
Type-I
( total flux is conserved to 0)
(around)
squeezed by monopole
( total flux is conserved to 0)
BPS
corporative
oscillation eipr / r1/2
Type-II
? resonant scattering
oscillation eipr / r1/2
corporative
large
r
r
13
Summary
For the general vortex case
We consider the vortex vibration
with changing its thickness.
We found the characteristic discrete pole
around BPS value of GL parameter. ?
coherent vibration of Higgs and photon fields.
For the application to QCD
flux-tube in the vacuum
DGL parameters are taken to fit the QQ potential
results.
?2 3 ? Type-II
monopole self-coupling ? 25
resonant scattering type of vibrations appear.
dual-gauge coupling gdual 2.3
value of monopole cond. v 0.126 GeV
excitation energy 0.5 GeV
14
Outlook and speculation
For the application to hot QCD
Then, if the strength of effective
monopole self-interaction ?(T) becomes weak,
becomes weak, and the
property of color-electric flux approach to the
Type-I vortex.
The monopole - dual photon coherent vibration
can appear in non-zero temperture.
15
Vortex vortex potential per unit length
( for DGL, per 1 fm )
16
String picture of hadrons
String picture of hadrons gives natural
explanation for
The string picture may share important part of
QCD.
17
Static solution
G-L parameter
topological quantization (topological charge n)
18
Importance of mixing
around the core, Higgs photon are mixed
static vortex Higgs bound state
Type-II
BPS
Type-I
around BPS
Same threshold leads the corporative behavior of
Higgs photon
then, lowest excitation energy is considerably
decreased.
19
The property of the potential in the radial
direction
energy threshold for continuum states
V(r) potential induced by static
vortex
2 Md-photon2
(independent of ?2)
asymptotic region ( r ? 8 )
central region ( r ? 0 )
state below threshold
state above threshold
for all states
a(r) ? rm const.
a(r) ? 0
a(r) ? eipr
ß(r) ? rm const.
ß(r) ? 0
a(r) ? eipr
(m ? 2 )
20
Static profile for Type-I II
?d/?(?1/2/ e) G-L parameter
condensed matter
DGL (QCD effective theory)
penetration depth d 500 A,
0.3 - 0.4 fm
coherence length ? 25 104 A,
0.16 fm
G-L parameter ? 0.05-20,
1.6 - 2.0
ex)
pure metal
ex)
high Tc SC, metal with inpurity
BPS
1/2
?
d
(usually not considered)
finite thickness
string like
21
We will consider color flux tube linking specific
charges.
To discuss the color flux linking specific
charges, we have only to consider this part.
dual photon field
monopole field
same form as the Ginzburg-Landau type action
Ginzburg-Landau action
Higgs (Cooper-pair) field
photon field
for Abrikosov vortex
We have only to consider the GL-type action.
22
1st excited state wavefunction in the radial
direction
fluctuation of electric field Ez
fluctuations of f? a?
small
squeezed by monopoles
Type-I
( total flux is conserved to 0)
Ez
Ez
(around)
squeezed by monopoles
( total flux is conserved to 0)
BPS
corporative
Ez
oscillation eipr / r1/2
Type-II
? long tail
oscillation eipr / r1/2
corporative
large
r
r
23
Thermodynamical Stability
( Mnvortex mass with topological charge n )
vortex-vortex interaction
vortex-vortex interaction
exact solution de Vega-Schaposnik,
PRD14,1100(1976)
attractive
repulsive
no interaction between vortices
vortex lattice with topological charge n1 (
thermodynamically stable)
Type-I vortices system is thermodynamically
unstable
(at least in tree level)
Usually, Type-I vortex is not considered,
B
not uniform
but we consider the external magnetic
field squeezed enough to generate only one vortex
We study not only Type-II vortex but also Type-I
vortex
24
n gt 1 vortex, classical profiles potentials
(?2 1/2 case )
n1
n2
n3
profile
increasing n
total magnetic flux ( 2pn ) increases
Cooper-pair around core is suppressed
potential
Cooper-pair around core is suppressed
fluc. of Hz is enhanced
large potential around core for f
surface between Cooper-pair and magnetic flux
shifts outward
mixing potential shifts outward
25
n 2, 3 energy spectrum
giant vortex
the lowest excitation becomes softer one
The topological defect of Cooper-pair
condensation is enlarged, then photon can
easily excite around the core.
The property as static vortex photon excitation
becomes strong.
The threshold is unchanged, then continuum states
behave
like n 1 continuum states.
26
Summary for single Abelian vortex
We have discussed the peristaltic modes of
single vortex.
We found, new discrete pole around ?2 1/2.
This discrete pole is characterized by the
corporative behavior of the Higgs and
photon fields.
As ?2 is increased, the low excitation modes
change from the Higgs dominant modes to
the photon dominant modes.
As n is increased, photon can excite more
easily, and lowest excitation
becomes softer one.
27
Summary for single color electric flux
We can directly apply the previous arguments
to the color flux linking specific color charge.
R
profile in radial direction
DGL gives ?2 3 - 4. ? Type-II
(mass of color electric flux per 1fm 1.0
GeV/fm.)
Only resonant scattering type of
excitations appear.
r
excitation energy 0.5 GeV.
electric flux vibrate with long tail.
28
4, Calculation in 2 - D (Preliminary)
Motivation We would like to discuss
1, excitation modes around 1-vortex without
cylindrical symmetry.
2, the dynamics of the multi-vortices system, for
example,
vortex-vortex fusion into the giant vortex,
the giant vortex fission to the small vortices,
vortex - anti vortex annihilation and production,
bearing in mind the future application to the
hadron physics
ex) meson-meson reaction
scattering
production of the resonance, especially
exotic hadrons etc.
In this talk, we show only the static profile,
vortex- vortex potential, and
vortex- anti vortex potential.
29
Vortex antivortex potential per unit length
( for DGL, per 1 fm )
30
Energy spectrum New-type discrete pole
V(r)
Type-II
monopoles
dual photon field
Type-I
dual photon field
monopoles
Around BPS saturation, characteristic discrete
pole appear as a result of monopoles d-photon
corporative behavior.
31
Sudden annihilation of fluxes (in DGL unit)
1.2 fm
1.0 fm
Around d 1.0 - 1.2 fm, the fluxes suddenly
annihilate.
This critical distance dcr is related with the
penetration depth d
dcr (1.5 - 2.0) 2d
This value seems to be considerably large.
32
Abelian projection and monopoles
Usually magnetic monopole does not appear in
U(1) gauge theory, but if theory includes SU(N) (
Ngt1) gauge fields, their specific topological
configuration constructs U(1) point like
singularity as a topological object.
mapping
R3 in physical space
SU(2) variables in internal space
33
Two vortices system ( static case )
field degrees of freedom Re?, Im?, Ax, Ay
(without cylindrical sym.)
As the previous 1-vortex system, we first search
for the static profiles which
minimize the static energy.
Step 1)
Starting from the case where the distance between
two vortices is large,
adopt the product ansatz
for initial B.C
?12?1 ?2 /const.
A12A1 A2
(We need this B.C. only at the beginning of the
calculation)
Step 2)
Fix the ?2 0 at the vortices cores, and
minimize the static energy with checking
that the total magnetic flux is quantized
appropriately.
Step 3)
After convergence, change the distance of the
vortex core.
Step 4)
Adopt the previous profile as I.C. and return to
the step 2.
Then we acquire the static profile
and the potential between two vortices .
34
Summary
We have discussed the peristaltic modes of
single vortex.
We found, new discrete pole around ?2 1/2.
This discrete pole is characterized by the
corporative behavior
of the Cooper-pair and photon fields.
To discuss the dynamics of color flux, we need
more careful
treatments to retain the confinement
property.
We have also discussed the potential between
vortices as a preparation for the dynamics
of multi-vortices system.
Future work
Non axial symmetric excitation of single vortex.
Dynamics of two vortices.
Careful treatment of color flux with projection.
35
Introduction
Abrikosov vortex in U(1) theory
Color confinement in QCD
Y.Nambu, PRD.122,4262(1974)
A.A.Abrikosov, Soviet Phys.JTEP 5, 1174(1957)
t Hooft , Nucl.Phys.B190.455(1981)
Mandelstam, Phys.Rep.C23.245(1976)
B
Cooper-pair condensation
magnetic monopole condensation
dual
Meissner effect
dual Meissner effect
squeezed color electric flux
squeezed magnetic field
B
E
36
Static U(1) vortex solution (n1 case)
magnetic field penetrates with small kinetic
energy
magnetic field is strongly squeezed by Higgs field
finite thickness
string
37
gauge fixing
Regge trajectories of hadrons
(hadron mass)2
angular momentum
constant string tension
38
Model dual Ginzburg-Landau model
Ezawa-Iwazaki, PRD25(1982)2681 Maedan-Suzuki,
PTP81(1989)229
1) Abelian gauge fixing ? U(1)2 monopole DOF
naturally appear as topological objects.
2) include the auxiliary field U(1)mxU(1)m dual
photon B which couples with the monopole
current.

(Zwanziger, PRD3(1970)880 )
3) sum up the monopole trajectory ? leading
order kinetic term of monopole field
correction monopole self interaction
( Bardakci, Samuel, PRD18(1978)2849 )
(this is included phenomenologically)
4) integrate out the U(1)e x U(1)e photon field
A
39
Static solution
G-L parameter
Ansatz for topological charge n
at vortex core, f0
minimize static energy with B.C for finite
energy vortex solution
in asymptotic region, sym. is restored