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Higgs Mechanism with Type-IINambu-Goldstone Boson at Finite Chemical Potential
Yusuke Hama (Univ. Tokyo)
Collaborators
Tetsuo Hatsuda (Univ. Tokyo)Shun Uchino (Kyoto Univ.)
Based on arXiv:1102.4145v2 [hep-ph]Phys. Rev D in press
5/18 (2011) Seminar @ Komaba
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 work
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 Mass
The number of NG bosons and Broken Generators
system SSB patternG→H
Broken generators ( BG)
NG boson
#NG boson
dispersion
2-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 2 E(k) ~ k
Ferromagnet O(3) → O(2) 2 magnon 1 E(k) ~ k2
Kaon condensation in color superonductor
U(2) →U(1) 3 “ kaon” 2 E(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?
Nielsen-Chadha Theorem Nielsen and Chadha (1976)
• analyticity of dispersion of type-II• spectral decomposition
Classification of NG bosons by dispersions
E~p2n+1 : type-I, E~p2n : type-II
Nielsen-Chadha inequality
NI + 2 NII ≧ NBG
All previous examples satisfy Nielsen-Chadha inequality
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 NBG=NNG= NI
without gauge bosons
?NNG =(Nmassive
gauge)/3
with gauge bosons
NNG=(Nmassive gauge)/3
Type-II Nambu-Goldstone Spectrum
atFinite Chemical
Potential
minimal model to show type-II NG boson
LagrangianSSB Pattern
Field parametriza
tion
2 component complex scalar
Quadratic Lagrangian mixing by
m
U(2) Model at Finite Chemical Potential
Hamiltonian
Hypercharge
Miransky , Schafer and Nambu (2002)
Type-II NG boson spectrum
Equations of motion
(m=0) (m ≠ 0)
c’1
massivec’2 type-II
c’3 type-I
y’ massive
c3 type-I
y massive
Nielsen-Chadha inequality: NI =1, NII =1, NI + 2NII = NBG
c1 type-I
c2 type-I
dispersions
mixing effect
Higgs Mechanism with Type-II Nambu
Goldstone Boson at Finite Chemical Potential
Gauged SU(2) ModelU(2) Lagrangian
field parametrization
gauged SU(2) Lagrangian
covariant derivative
gauge boson mass
background charge densityto ensure the “charge” neutrality Kapusta (1981)
Charge Neutrality in Gauged U(1) Model 1
1. Lagrangian
Covariant Derivative
Equations of motion in mean field approximation
Disappearance of chemical potential
Absense of background charge density
3. Quadratic Lagrangian
2. Ground Expectation Value
finite value
Charge Neutrality in Gauged U(1) Model 2
Equations of motion in mean field approximation
1. Lagrangian
Background charge density
2. Ground Expectation Value
zero value
3. Quadratic Lagrangian
4. Charge Neutrality
Charge Neutrality in Gauged SU(2) Model
Equations of motion in mean field approximation
4. Charge Neutrality
3. Quadratic Lagrangian
2. Ground Expectation Value
zero value
1. Lagrangian
Background charge density
Rx GaugeClear separation between unphysical spectra (A3
=0m , ghost, “NG bosons”) and physical spectra
(A3 =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 gaugeFeynman gauge
Unitary gauge
Quadratic Lagrangian
coupling
new
mixing between c1,2 , ,y and unphysical modes (Aa =0 m )
What remain as physical modes?
Dispersion Relation (p→0, α>>1)
diagonal off-diagonal
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)
A1,2,3 T
A1,2,3 T, L
Fields g=0, μ≠0 g≠0, μ≠0
massive 2 1
NG boson 1 (Type I), 1(Type II)
0
Gauge boson 3 ×2T 3 ×3T, L
Total 10 10
SummaryWe analyzed Higgs Mechanism at finite chemical potential with type-II NG boson with Rx 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
Back Up Slides
Counting NG bosons with Algebraic Method
behave canonical conjugatebelong to the same dynamical degree of freedom NBG≠NNG
O(3) algebra anti-ferromagnet
ferromagnet
NBG=NNG
NBG≠NNG
Nambu (2002)Qa: broken generators
independent broken generatorsNBG=NNG
SU(2) algebra
NBG≠NNGU(2) model
Examples
The Spectrum of NG Bosons
V
vv
Future Work
Gauged SU(2) Model without Background Charge DensityGusynin, Miransky, & Shovkovy (2004)
1. Lagrangian
2. Ground Expectation Value
3. Quadratic Lagrangian
4. Charge Neutrality
finite value
Comparison of Results in Gauged SU(2) Model
Our Work Gusynin, Miransky, & Shovkovy
Ground Expectation
Value
Scalar Field mass
A3 =1,2,3m
mass
A± =1,2,3m
mass
Difference: Reproduction of U(2) model mass spectrum in the limit g→0
Glashow-Weinberg-Salam Model
Fields g=0m≠0
g≠0m≠0
Gauge 2×4 3×3+2
NGB Type I×1
Type II×1
0
Massive
2 1
Gauged SU(3) Model
Fields g=0m≠0
g≠0m≠0
Gauge 2×5 3×5
NGB 1 (Type I)
2 (Type II)
0
Massive
3 1