with TypeII Nambu Goldstone Boson Based on arXiv11024145v2 hep ph Yusuke Hama Univ Tokyo Tetsuo Hatsuda Univ Tokyo Shun Uchino Kyoto Univ 420 2011 Dense Strange Nuclei and Compressed Baryonic Matter ID: 293657
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Slide1
Higgs Mechanism at Finite Chemical Potential
with Type-IINambu-Goldstone Boson
Based on arXiv:1102.4145v2 [
hep
-ph]
Yusuke Hama
(Univ. Tokyo)
Tetsuo
Hatsuda
(Univ. Tokyo)
Shun Uchino (Kyoto Univ.)
4/20 (2011)
Dense Strange Nuclei and Compressed Baryonic Matter
@ YITP, Kyoto, JapanSlide2
Contents
1. Introduction 2. Spontaneous Symmetry Breaking and Nambu
-Goldstone Theorem
3. Type-II Nambu-Goldstone Spectrum at Finite Chemical Potential 4. Higgs Mechanism with Type-II Nambu-Goldstone Boson 5. Summary and Conclusion
*
*
Our original workSlide3
Introduction
Condensed Matter Physics
Elementary Particle Physics
Spontaneous Symmetry Breaking
Background: Spontaneous Symmetry Breaking (SSB) Nambu (1960)
Cutting Edge Research of SSB
Ultracold
Atoms
Color Superconductivity
Extremely similar phenomena
Origin of MassSlide4
The number of NG bosons and Broken Generators
systemSSB patternG
→
H
Broken generators ( BG)NG boson#NG bosondispersion2-flavorMassless
QCD
SU(2)
L
×
SU(2)
R
→
SU(2)
V
3
pion
3
E(k)
~
k
Anti-
ferromagnet
O(3)
→
O(2)
2
magnon 2E(k) ~kFerromagnetO(3) → O(2)2 magnon 1E(k) ~k2Kaon condensation in color superonductorU(2) →U(1)3 “kaon” 2E(k) ~k E(k) ~k2
Chemical potential plays an important role for
the number and dispersion of NG bosons
One of the most important aspects of SSB
The appearance of
massless
Nambu-Goldstone (NG) bosons
Motivation: How many numbers of Nambu-Goldstone (NG) bosons appear?
Relations between the dispersions and
the number of NG bosons?Slide5
Nielsen-Chadha Theorem Nielsen
and Chadha(1976)analyticity of dispersion of type-IIspectral decomposition Classification
of NG bosons by dispersions
E
~p2n+1 : type-I, E~p2n : type-IINielsen-Chadha inequality
NI
+ 2
N
II
≧
N
BG
All previous examples satisfy Nielsen-
Chadha
inequalitySlide6
Higgs Mechanism
PurposeAnalyze the Higgs mechanism with type-Ⅱ NG boson at finite chemical potential .
m
≠ 0: type-I & type-II NBG≠NNG= NI +NII
m
=0: type-I
N
BG
=
N
NG
=
N
I
without gauge bosons
?
N
NG
=(
N
massive
gauge
)/3
with gauge bosons
N
NG
=(
Nmassive gauge)/3Slide7
Type-II
Nambu-Goldstone Spectrum atFinite Chemical PotentialSlide8
minimal model to show type-II NG boson
Lagrangian
SSB Pattern
Field
parametrization
2 component
complex scalar
Quadratic
Lagrangian
m
ixing by
m
U(2) Model at Finite Chemical Potential
Miransky
and Schafer (2002)
Hamiltonian
HyperchargeSlide9
Type-II NG boson spectrum
Equations of motion
(
m
=0)(
m ≠
0)
c
’
1
massive
c
’
2
type-II
c
’
3
type-I
y
’ massive
c
3
type-I
y
massive
Nielsen-
Chadha
inequality:
NI =1, NII =1, NI + 2NII = NBGc1 type-Ic2 type-Idispersionsmixing effectSlide10
Higgs Mechanism with Type-II NG Boson at Finite Chemical Potential
Slide11
Gauged SU(2) Model
U(2) Lagrangian
f
ield
parametrizationgauged SU(2) Lagrangiancovariant derivativegauge boson mass
b
ackground charge density
to ensure the “charge” neutrality
Kapusta
(1981) Slide12
Rx Gauge
Clear separation between unphysical spectra (A3 m=0, ghost, “NG bosons”) and physical spectra (A
3
m=
i ,Higgs) and by taking the a→∞ masses of unphysical particles decouple from physical particles Fujikawa, Lee, and Sanda (1972)
Gauge-fixing function
a
: gauge parameter
Landau gauge
Feynman gauge
Unitary gaugeSlide13
Quadratic
Lagrangian
coupling
new
mixing between c1,2 , y, and unphysical modes (Aa
m=0 )
What remain as physical modes?Slide14
Dispersion Relation (p→0, α>>1)
diagonal
off-diagonalSlide15
Field Mass Spectrum and Result
total physical degrees of freedom are correctly conserved
Y
’
(Higgs)Y’(Higgs)c
’3
(type-I)
c
’
2
(type-II)
c
’
1
(massive)
A
1,2,3
T
A
1,2,3
T, L
Fields
g=0, μ
≠
0
g
≠0, μ≠0massive21NG boson1 (Type I), 1(Type II)0Gauge boson3×2T3×3T, LTotal1010Slide16
Summary
We analyzed Higgs Mechanism at finite chemical potential with type-II NG boson
with R
x
gauge Result: ・Total physical degrees of freedom correctly conserved -- Not only the massless NG bosons (type I & II) but also the massive mode induced by the chemical potential became unphysical
・Models: gauged SU(2) model, Glashow-Weinberg-Salam type
gauged U(2) model, gauged SU(3) model
Future Directions:
・
Higgs Mechanism with type-II NG bosons in
nonrelativistic
systems
(
ultracold
atoms in optical lattice)?
-- What is the relation between the Algebraic method (
Nambu
2002) and the Nielsen
Chadha
theorem
?
・
Algebraic
method: counting NG bosons without deriving dispersions
・Nielsen-Chadha theorem: counting NG bosons from dispersions Slide17
Back Up SlidesSlide18
Counting NG bosons with
Algebraic Method behave canonical conjugatebelong to the same dynamical degree of freedom NBG
≠
N
NGO(3) algebraanti-ferromagnetferromagnet
NBG
=
N
NG
N
BG
≠
N
NG
Nambu
(2002)
Q
a
: broken generators
independent broken generators
N
BG
=
N
NG
SU(2) algebra
N
BG
≠
N
NGU(2) modelExamplesSlide19
The Spectrum of NG Bosons
V
v
vv
Future WorkSlide20
Glashow-Weinberg-Salam Model
Fields
g=0
m
≠0g≠0m≠0Gauge2×4
3×3+2
NGB
Type I
×1
Type II
×1
0
Massive
2
1Slide21
Gauged SU(3) Model
Fieldsg=0m≠0
g
≠
0m≠0Gauge2×53×5NGB1 (Type I)
2 (Type II)0
Massive
3
1