Higgs Mechanism at Finite Chemical Potential - PowerPoint Presentation

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