Higgs Mechanism at Finite

Chemical Potential with Type-II

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

YITP Kyoto Japan

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 pattern

GrarrH

Broken

generators

( BG)

NG boson NG

boson

dispersion

2-flavor

Massless QCD

SU(2)Ltimes SU(2)R

rarr SU(2)V

3

pion 3 E(k) ～k

Anti-

ferromagnet

O(3) rarr O(2) 2 magnon 2 E(k) ～k

Ferromagnet O(3) rarr O(2) 2 magnon 1 E(k) ～k2

Kaon

condensation

in color

superonductor

U(2) rarrU(1)

3 ldquokaonrdquo 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)

bull analyticity of dispersion of type-II

bull 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

Purpose

Analyze the Higgs mechanism with type-Ⅱ

NG boson at finite chemical potential

m ne 0 type-I amp type-II

NBGneNNG= 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 at

Finite Chemical Potential

minimal model to show type-II NG boson

Lagrangian

SSB Pattern

Field parametrization

2 component

complex scalar

Quadratic Lagrangian mixing by m

U(2) Model at Finite Chemical Potential Miransky and Schafer (2002)

Hamiltonian

Hypercharge

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 pattern

GrarrH

Broken

generators

( BG)

NG boson NG

boson

dispersion

2-flavor

Massless QCD

SU(2)Ltimes SU(2)R

rarr SU(2)V

3

pion 3 E(k) ～k

Anti-

ferromagnet

O(3) rarr O(2) 2 magnon 2 E(k) ～k

Ferromagnet O(3) rarr O(2) 2 magnon 1 E(k) ～k2

Kaon

condensation

in color

superonductor

U(2) rarrU(1)

3 ldquokaonrdquo 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)

bull analyticity of dispersion of type-II

bull 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

Purpose

Analyze the Higgs mechanism with type-Ⅱ

NG boson at finite chemical potential

m ne 0 type-I amp type-II

NBGneNNG= 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 at

Finite Chemical Potential

minimal model to show type-II NG boson

Lagrangian

SSB Pattern

Field parametrization

2 component

complex scalar

Quadratic Lagrangian mixing by m

U(2) Model at Finite Chemical Potential Miransky and Schafer (2002)

Hamiltonian

Hypercharge

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 pattern

GrarrH

Broken

generators

( BG)

NG boson NG

boson

dispersion

2-flavor

Massless QCD

SU(2)Ltimes SU(2)R

rarr SU(2)V

3

pion 3 E(k) ～k

Anti-

ferromagnet

O(3) rarr O(2) 2 magnon 2 E(k) ～k

Ferromagnet O(3) rarr O(2) 2 magnon 1 E(k) ～k2

Kaon

condensation

in color

superonductor

U(2) rarrU(1)

3 ldquokaonrdquo 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)

bull analyticity of dispersion of type-II

bull 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

Purpose

Analyze the Higgs mechanism with type-Ⅱ

NG boson at finite chemical potential

m ne 0 type-I amp type-II

NBGneNNG= 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 at

Finite Chemical Potential

minimal model to show type-II NG boson

Lagrangian

SSB Pattern

Field parametrization

2 component

complex scalar

Quadratic Lagrangian mixing by m

U(2) Model at Finite Chemical Potential Miransky and Schafer (2002)

Hamiltonian

Hypercharge

The number of NG bosons and Broken Generators

system SSB pattern

GrarrH

Broken

generators

( BG)

NG boson NG

boson

dispersion

2-flavor

Massless QCD

SU(2)Ltimes SU(2)R

rarr SU(2)V

3

pion 3 E(k) ～k

Anti-

ferromagnet

O(3) rarr O(2) 2 magnon 2 E(k) ～k

Ferromagnet O(3) rarr O(2) 2 magnon 1 E(k) ～k2

Kaon

condensation

in color

superonductor

U(2) rarrU(1)

3 ldquokaonrdquo 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)

bull analyticity of dispersion of type-II

bull 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

Purpose

Analyze the Higgs mechanism with type-Ⅱ

NG boson at finite chemical potential

m ne 0 type-I amp type-II

NBGneNNG= 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 at

Finite Chemical Potential

minimal model to show type-II NG boson

Lagrangian

SSB Pattern

Field parametrization

2 component

complex scalar

Quadratic Lagrangian mixing by m

U(2) Model at Finite Chemical Potential Miransky and Schafer (2002)

Hamiltonian

Hypercharge

Nielsen-Chadha Theorem Nielsen and Chadha(1976)

bull analyticity of dispersion of type-II

bull 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

Purpose

Analyze the Higgs mechanism with type-Ⅱ

NG boson at finite chemical potential

m ne 0 type-I amp type-II

NBGneNNG= 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 at

Finite Chemical Potential

minimal model to show type-II NG boson

Lagrangian

SSB Pattern

Field parametrization

2 component

complex scalar

Quadratic Lagrangian mixing by m

U(2) Model at Finite Chemical Potential Miransky and Schafer (2002)

Hamiltonian

Hypercharge

Higgs Mechanism

Purpose

Analyze the Higgs mechanism with type-Ⅱ

NG boson at finite chemical potential

m ne 0 type-I amp type-II

NBGneNNG= 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 at

Finite Chemical Potential

minimal model to show type-II NG boson

Lagrangian

SSB Pattern

Field parametrization

2 component

complex scalar

Quadratic Lagrangian mixing by m

U(2) Model at Finite Chemical Potential Miransky and Schafer (2002)

Hamiltonian

Hypercharge

Type-II Nambu-

Goldstone Spectrum at

Finite Chemical Potential

minimal model to show type-II NG boson

Lagrangian

SSB Pattern

Field parametrization

2 component

complex scalar

Quadratic Lagrangian mixing by m

U(2) Model at Finite Chemical Potential Miransky and Schafer (2002)

Hamiltonian

Hypercharge

minimal model to show type-II NG boson

Lagrangian

SSB Pattern

Field parametrization

2 component

complex scalar

Quadratic Lagrangian mixing by m

U(2) Model at Finite Chemical Potential Miransky and Schafer (2002)

Hamiltonian

Hypercharge

Type-II NG boson spectrum

Equations of

motion

(m=0) (m ne 0)

crsquo1 massive

crsquo2 type-II

crsquo3 type-I

yrsquo 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 NG Boson at Finite Chemical Potential

Gauged SU(2) Model

U(2) Lagrangian

field parametrization

gauged SU(2) Lagrangian

covariant derivative

gauge boson mass

background charge density

to ensure the ldquochargerdquo neutrality

Kapusta (1981)

Gauged SU(2) Model

U(2) Lagrangian

field parametrization

gauged SU(2) Lagrangian

covariant derivative

gauge boson mass

background charge density

to ensure the ldquochargerdquo neutrality

Kapusta (1981)

Rx Gauge Clear separation between unphysical spectra

(A3 m=0 ghost ldquoNG bosonsrdquo) and physical spectra

(A3 m=i Higgs) and by taking the ararrinfin

masses of unphysical particles decouple from

physical particles

Fujikawa Lee and Sanda (1972)

Gauge-fixing

function

a gauge parameter

Landau gauge

Feynman gauge

Unitary gauge

Quadratic Lagrangian

coupling

new

mixing between c12 y and unphysical modes (Aa m=0 )

What remain as physical modes

Dispersion Relation (prarr0 αgtgt1)

diagonal off-diagonal

Dispersion Relation (prarr0 αgtgt1)

diagonal off-diagonal

Field Mass Spectrum and Result

total physical degrees of freedom are correctly conserved

Yrsquo(Higgs) Yrsquo(Higgs) crsquo3 (type-I)

crsquo2 (type-II)

crsquo1 (massive)

A123 T

A123 T L

Fields g=0 μne0

gne0 μne0

massive 2 1

NG boson 1 (Type I) 1(Type II) 0

Gauge boson ３times2T ３times3T L

Total 10 10

Summary We 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 amp 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 conjugate

belong to the same dynamical

degree of freedom NBGneNNG

O(3) algebra anti-ferromagnet

ferromagnet NBG=NNG

NBGneNNG

Nambu (2002) Qa broken generators

independent broken generators NBG=NNG

SU(2) algebra NBGneNNG U(2) model

Examples

Back Up Slides

Counting NG bosons with Algebraic Method

behave canonical conjugate

belong to the same dynamical

degree of freedom NBGneNNG

O(3) algebra anti-ferromagnet

ferromagnet NBG=NNG

NBGneNNG

Nambu (2002) Qa broken generators

independent broken generators NBG=NNG

SU(2) algebra NBGneNNG U(2) model

Examples

Counting NG bosons with Algebraic Method

behave canonical conjugate

belong to the same dynamical

degree of freedom NBGneNNG

O(3) algebra anti-ferromagnet

ferromagnet NBG=NNG

NBGneNNG

Nambu (2002) Qa broken generators

independent broken generators NBG=NNG

SU(2) algebra NBGneNNG U(2) model

Examples

The Spectrum of NG Bosons

V v

vv

Future Work

Glashow-Weinberg-Salam Model

Fields g=0

mne0

gne0

mne0

Gauge 2times4 3times3+2

NGB Type Itimes1

Type IItimes1

0

Massive 2 １

Gauged SU(3) Model

Fields g=0

mne0

gne0

mne0

Gauge 2times5 3times5

NGB 1 (Type I)

2 (Type II)

0

Massive 3 １