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