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Exact calculation for AB-phase effective potential via supersymmetric localization A.T, A. Tomiya, T. Shimotani

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Page 1: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

Exact calculation for AB-phase effective potential via supersymmetric localization

A.T, A. Tomiya, T. Shimotani

Page 2: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

Exact calculation for AB-phase effective potential via supersymmetric localization

A.T, A. Tomiya, T. Shimotani

work in progress

Page 3: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

Todayʼs concern ispurely theoretical...

Page 4: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

Do these plots represent different vacua?

Page 5: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272
Page 6: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

Introduction

Page 7: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

Introduction

Phenomena Tools

Page 8: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

Introduction

...SSB

Higgs mechanism

Hosotani mechanism

Phenomena Tools

Page 9: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

Introduction

...SSB

Higgs mechanism

Hosotani mechanism

Phenomena Tools

Page 10: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

Introduction

...SSB

Higgs mechanism

Hosotani mechanism

...Quantum correctionperturbationLattice

Phenomena Tools

Page 11: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

Introduction

...SSB

Higgs mechanism

Hosotani mechanism

...Quantum correctionperturbationLattice

Phenomena Tools

Page 12: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

Introduction

Hosotani mechanism (1983)

Page 13: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

Introduction

Hosotani mechanism

×M S1(1983)

Page 14: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

Introduction

Hosotani mechanism

×M S1(1983)

Page 15: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

Introduction

Hosotani mechanism

×M S1FMN = 0(1983)

Page 16: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

Introduction

Hosotani mechanism

×M S1FMN = 0

Aµ = 0

AS1 = θ

(1983)

Page 17: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

Introduction

Hosotani mechanism

×M S1

Aµ = 0

AS1 = θ

�D(A + θ)e−S(A+θ)

=�

�DAe−S(A+θ)

(1983)

=�

dθe−Γ(θ)

Page 18: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

Introduction

Hosotani mechanism

How to perform?

(1983)�D(A + θ)e−S(A+θ)

=�

�DAe−S(A+θ)

=�

dθe−Γ(θ)

Page 19: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

Introduction

Hosotani mechanism

1. perturbation

(1983)�D(A + θ)e−S(A+θ)

=�

�DAe−S(A+θ)

=�

dθe−Γ(θ)

Veff = V treeeff + V 1−loop

eff + V 2−loopeff + ...

Page 20: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

Introduction

Hosotani mechanism

1. perturbation

0 Finite !

(1983)

arXiv:hep-ph/0504272

SU(3)

�D(A + θ)e−S(A+θ)

=�

�DAe−S(A+θ)

=�

dθe−Γ(θ)

Veff = V treeeff + V 1−loop

eff + V 2−loopeff + ...

Page 21: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

Introduction

Hosotani mechanism

2. Lattice

(1983)�D(A + θ)e−S(A+θ)

=�

�DAe−S(A+θ)

=�

dθe−Γ(θ)

Page 22: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

Introduction

Hosotani mechanism�D(A + θ)e−S(A+θ)

=�

�DAe−S(A+θ)

2. Lattice

(1983)

PR

PR

PRPolyakov loop via lattice action

=�

dθe−Γ(θ)

∝ �PR�

Page 23: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

Introduction

Hosotani mechanism�D(A + θ)e−S(A+θ)

=�

�DAe−S(A+θ)

2. Lattice

(1983)

PR

PR

Polyakov loop via lattice action

arXiv:0904.1353PR=

�dθe−Γ(θ)

∝ �PR�

Page 24: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

Introduction

Hosotani mechanism�D(A + θ)e−S(A+θ)

=�

�DAe−S(A+θ)

2. Lattice

(1983)

PR

PR

Polyakov loop via lattice action

arXiv:0904.1353PR=

�dθe−Γ(θ)

∝ �PR�

Page 25: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

Introduction

Hosotani mechanism�D(A + θ)e−S(A+θ)

=�

�DAe−S(A+θ)

2. Lattice

(1983)

PR

PR

Polyakov loop via lattice action

arXiv:0904.1353PR=

�dθe−Γ(θ)

∝ �PR�

Page 26: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

Introduction

Hosotani mechanism�D(A + θ)e−S(A+θ)

=�

�DAe−S(A+θ)

2. Lattice

(1983)

PR

PR

Polyakov loop via lattice action

arXiv:0904.1353PR=

�dθe−Γ(θ)

∝ �PR�

Page 27: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

Introduction

Hosotani mechanism�D(A + θ)e−S(A+θ)

=�

�DAe−S(A+θ)

2. Lattice

(1983)

PR

PR

Polyakov loop via lattice action

arXiv:0904.1353PR=

�dθe−Γ(θ)

∝ �PR�

Page 28: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

Introduction

Hosotani mechanism�D(A + θ)e−S(A+θ)

=�

�DAe−S(A+θ)

2. Lattice

(1983)

PR

PR

Polyakov loop via lattice action

arXiv:0904.1353PR=

�dθe−Γ(θ)

∝ �PR�

Page 29: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

Introduction

Hosotani mechanism�D(A + θ)e−S(A+θ)

=�

�DAe−S(A+θ)

2. Lattice

(1983)

PR

PR

Polyakov loop via lattice action

arXiv:0904.1353PR=

�dθe−Γ(θ)

∝ �PR�

Page 30: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

Introduction

Hosotani mechanism�D(A + θ)e−S(A+θ)

=�

�DAe−S(A+θ)

2. Lattice

(1983)

PR

PR

Polyakov loop via lattice action

arXiv:0904.1353PR=

�dθe−Γ(θ)

∝ �PR�

Page 31: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

Introduction

...SSB

Higgs mechanism

Hosotani mechanism

...Quantum correctionperturbationLattice

Phenomena Tools

Page 32: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

Introduction

...SSB

Higgs mechanism

Hosotani mechanism

...Quantum correctionperturbationLattice SUSY localization

Phenomena Tools

Page 33: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

Hosotani mechanismSUSY localization

1. Geometry setup

×2. SUSY on ×

3. Localizationd

dtZ(t) = 0

λ

σ

D

λ†

Mass:k

Page 34: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

Hosotani mechanismSUSY localization

1. Geometry setup

×2. SUSY on ×

3. Localizationd

dtZ(t) = 0

4. Results on Veff�D(fields)e−S =

�dθe−Veff (θ)

?

5. Results on �P3�

6. Preliminary results

λ

σ

D

λ†

Mass:k

Page 35: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

Hosotani mechanismSUSY localization

1. Geometry setup

×

Page 36: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

1. Geometry setup

×

Page 37: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

1. Geometry setup

×

×

Page 38: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

1. Geometry setup

×

×

β

Page 39: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

1. Geometry setup

×ClaimIf CFT,

×β

limβ→0

= ×

β

Page 40: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

1. Geometry setup

×ClaimIf CFT,

×β

limβ→0

= ×

Page 41: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

1. Geometry setup

×ClaimIf CFT,

×β

limβ→0

= ×

Page 42: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

1. Geometry setup

ClaimIf CFT,

βlimβ→0

=

××

Page 43: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

1. Geometry setup

ClaimIf CFT,

βlimβ→0

=

××

Page 44: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

Hosotani mechanismSUSY localization

1. Geometry setup

2. SUSY on

3. Localizationd

dtZ(t) = 0

4. Results on Veff�D(fields)e−S =

�dθe−Veff (θ)

?

5. Results on �P3�

6. Preliminary results

λ

σ

D

λ†

×

Mass:k

Page 45: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

Hosotani mechanismSUSY localization

2. SUSY on

λ

σ

D

λ†

×

Mass:k

Page 46: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

2. SUSY on ×

Spin 1

Spin 1/2

Spin 0

λ

σ

D

λ†

Page 47: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

2. SUSY on

λ

σ

D

λ†

×

Page 48: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

2. SUSY on

λ

σ

D

λ†

×

∇µ� =12γµγ3�

curved effect

Page 49: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

2. SUSY on

λ

σ

D

λ†

×

δ�SSCS = 0 δη†SSCS = 0

SUSY invariance

SSCS =14π

�d3x

√gTr

� 1√

g�µνλ(Aµ∂νAλ +

2i

3AµAνAλ)− λ†λ + 2Dσ

Page 50: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

2. SUSY on

λ

σ

D

λ†

×

δ�SSCS = 0 δη†SSCS = 0

SUSY invariance

SSCS =14π

�d3x

√gTr

� 1√

g�µνλ(Aµ∂νAλ +

2i

3AµAνAλ)− λ†λ + 2Dσ

“Mass”

Page 51: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

2. SUSY on

λ

σ

D

λ†

×

SUSY invarianceδ�SSY M = 0 δη†SSY M = 0

Page 52: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

2. SUSY on

λ

σ

D

λ†

×

SUSY invarianceδ�SSY M = 0 δη†SSY M = 0

SUSY exactnessSSY M = δ�V

Page 53: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

2. SUSY on ×

Hosotani mechanismSUSY localization

1. Geometry setup

×

3. Localizationd

dtZ(t) = 0

4. Results on Veff�D(fields)e−S =

�dθe−Veff (θ)

?

5. Results on �P3�

6. Preliminary results

λ

σ

D

λ†

Mass:k

Page 54: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

Hosotani mechanismSUSY localization

3. Localizationd

dtZ(t) = 0

Page 55: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

3. Localization

d

dtZ(t) = 0

Why?

Z(t) :=�D(fields)eikSSCS−tSSY M

fields : (Aµ,λ†,λ,σ, D)

δ�V

=

d

dtZ(t) =

�D(fields)

d

dteikSSCS−tδ�V

=�D(fields)(−δ�V )eikSSCS−tδ�V

=�D(fields)δ�

�− V eikSSCS−tδ�V

=�D(fields)(total derivative)

=0

Page 56: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

3. Localization

d

dtZ(t) = 0

Z(1) =Z(∞) ←Steepest decent method is exact

Z(t) :=�D(fields)eikSSCS−tSSY M

fields : (Aµ,λ†,λ,σ, D)

δ�V

=

Page 57: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

3. Localization

d

dtZ(t) = 0

Z(1) =Z(∞) ←Steepest decent method is exact

Z(t) :=�D(fields)eikSSCS−tSSY M

fields : (Aµ,λ†,λ,σ, D)

δ�V

=

Page 58: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

3. Localization

d

dtZ(t) = 0

FMN = 0

Aµ = 0

AS1 = θ

Z(1) =Z(∞) ←Steepest decent method is exact

Z(t) :=�D(fields)eikSSCS−tSSY M

fields : (Aµ,λ†,λ,σ, D)

δ�V

=

Page 59: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

3. Localization

d

dtZ(t) = 0

AS1 = θ

Aµ = a(m)

Z(1) =Z(∞) ←Steepest decent method is exact

Z(t) :=�D(fields)eikSSCS−tSSY M

fields : (Aµ,λ†,λ,σ, D)

δ�V

=

Page 60: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

3. Localization

d

dtZ(t) = 0

AS1 = θ

Aµ = a(m)

Z(1) =Z(∞) ←Steepest decent method is exact

Integers

Z(t) :=�D(fields)eikSSCS−tSSY M

fields : (Aµ,λ†,λ,σ, D)

δ�V

=

Page 61: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

3. Localization

d

dtZ(t) = 0

Z(1) =Z(∞) ←Steepest decent method is exact

Integers

= ∞�

m1,m2,...=−∞ψ(m)

� π/β

0dθ1dθ2...e

2kiP

miθiβ�

i<j

�coshβ(mi −mj)− cos 2β(θi − θj)

Z(t) :=�D(fields)eikSSCS−tSSY M

fields : (Aµ,λ†,λ,σ, D)

δ�V

=

Page 62: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

3. Localization

d

dtZ(t) = 0

Z(1) =Z(∞) ←Steepest decent method is exact

Integers

=cannot determine phases

∞�

m1,m2,...=−∞ψ(m)

� π/β

0dθ1dθ2...e

2kiP

miθiβ�

i<j

�coshβ(mi −mj)− cos 2β(θi − θj)

Z(t) :=�D(fields)eikSSCS−tSSY M

fields : (Aµ,λ†,λ,σ, D)

δ�V

=

Page 63: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

Hosotani mechanismSUSY localization

1. Geometry setup

×2. SUSY on ×

3. Localizationd

dtZ(t) = 0

4. Results on Veff�D(fields)e−S =

�dθe−Veff (θ)

?

5. Results on �P3�

6. Preliminary results

λ

σ

D

λ†

Mass:k

Page 64: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

Hosotani mechanismSUSY localization

4. Results on Veff�D(fields)e−S =

�dθe−Veff (θ)

?

Page 65: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

4. Results on Veff

�D(fields)e−S =

�dθe−Veff (θ)

?

ClaimIf CFT,

×β

limβ→0

= ×

∞�

m1,m2,...=−∞ψ(m)

� π/β

0dθ1dθ2...e

2kiP

miθiβ�

i<j

�coshβ(mi −mj)− cos 2β(θi − θj)

Page 66: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

4. Results on Veff

�D(fields)e−S =

�dθe−Veff (θ)

?

ClaimIf CFT,

×β

limβ→0

= ×

θ̃ := βθ

∞�

m1,m2,...=−∞ψ(m)

� π/β

0dθ1dθ2...e

2kiP

miθiβ�

i<j

�coshβ(mi −mj)− cos 2β(θi − θj)

Page 67: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

4. Results on Veff

�D(fields)e−S =

�dθe−Veff (θ)

?

ClaimIf CFT,

×β

limβ→0

= ×

βN

� π

0dθ̃1dθ̃2...

θ̃ := βθ

e2kiP

miθ̃i

∞�

m1,m2,...=−∞ψ(m)

� π/β

0dθ1dθ2...e

2kiP

miθiβ�

i<j

�coshβ(mi −mj)− cos 2β(θi − θj)

cos 2(θ̃i − θ̃j)

Page 68: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

4. Results on Veff

�D(fields)e−S =

�dθe−Veff (θ)

?

ClaimIf CFT,

×β

limβ→0

= ×

θ̃ := βθ

cosh 0 = 1βN

� π

0dθ̃1dθ̃2...e

2kiP

miθ̃i

∞�

m1,m2,...=−∞ψ(m)

� π/β

0dθ1dθ2...e

2kiP

miθiβ�

i<j

�coshβ(mi −mj)− cos 2β(θi − θj)

cos 2(θ̃i − θ̃j)

Page 69: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

4. Results on Veff

�D(fields)e−S =

�dθe−Veff (θ)

?

∞�

m1,m2,...=−∞ψ̃(m)

� π

0dθ̃1dθ̃2...e

2kiP

miθ̃i�

i<j

sin2(θ̃i − θ̃j)

Page 70: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

4. Results on Veff

�D(fields)e−S =

�dθe−Veff (θ)

?

Choice of “wave function” ψ̃(m)

∞�

m1,m2,...=−∞ψ̃(m)

� π

0dθ̃1dθ̃2...e

2kiP

miθ̃i�

i<j

sin2(θ̃i − θ̃j)

Example with SU(3) 1:

Veff (θ̃1, θ̃2)

ψ̃(m,n,−m− n) = (δm,0 + δm,0)(δn,0 + δn,0)

Page 71: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

4. Results on Veff

�D(fields)e−S =

�dθe−Veff (θ)

?

Choice of “wave function” ψ̃(m)

∞�

m1,m2,...=−∞ψ̃(m)

� π

0dθ̃1dθ̃2...e

2kiP

miθ̃i�

i<j

sin2(θ̃i − θ̃j)

Example with SU(3) 2: ψ̃(m,n,−m− n) = (δm,1 + δm,−1)(δn,1 + δn,−1)

Veff (θ̃1, θ̃2)

Page 72: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

4. Results on Veff

�D(fields)e−S =

�dθe−Veff (θ)

?

Choice of “wave function” ψ̃(m)

∞�

m1,m2,...=−∞ψ̃(m)

� π

0dθ̃1dθ̃2...e

2kiP

miθ̃i�

i<j

sin2(θ̃i − θ̃j)

Example with SU(3) 3:

Veff (θ̃1, θ̃2)

ψ̃(m,n,−m− n) = 1

=?, but a little bit interesting.

Poisson resummation:∞�

n=−∞e2πixn =

∞�

p=−∞δ(x− p)

Page 73: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

4. Results on Veff

�D(fields)e−S =

�dθe−Veff (θ)

?

Choice of “wave function” ψ̃(m)

∞�

m1,m2,...=−∞ψ̃(m)

� π

0dθ̃1dθ̃2...e

2kiP

miθ̃i�

i<j

sin2(θ̃i − θ̃j)

Example with SU(3) 3:

Veff (θ̃1, θ̃2)

ψ̃(m,n,−m− n) = 1

=?, but a little bit interesting.

2θ̃1 + θ̃2 =2π

kp, θ̃1 + 2θ̃2 =

kq, p, q ∈ Z

Page 74: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

4. Results on Veff

�D(fields)e−S =

�dθe−Veff (θ)

?

Choice of “wave function” ψ̃(m)

∞�

m1,m2,...=−∞ψ̃(m)

� π

0dθ̃1dθ̃2...e

2kiP

miθ̃i�

i<j

sin2(θ̃i − θ̃j)

Example with SU(3) 3:

Veff (θ̃1, θ̃2)

ψ̃(m,n,−m− n) = 1

=?, but a little bit interesting.

2θ̃1 + θ̃2 =2π

kp, θ̃1 + 2θ̃2 =

kq, p, q ∈ Z

Page 75: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

Hosotani mechanismSUSY localization

1. Geometry setup

×2. SUSY on ×

3. Localizationd

dtZ(t) = 0

4. Results on Veff�D(fields)e−S =

�dθe−Veff (θ)

?

5. Results on �P3�

6. Preliminary results

λ

σ

D

λ†

Mass:k

Page 76: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

Hosotani mechanismSUSY localization

5. Results on �P3�

Page 77: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

5. Results on �P3�

×

Page 78: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

5. Results on �P3�

×

Page 79: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

5. Results on �P3�

×

Page 80: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

5. Results on �P3�

×

Page 81: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

5. Results on �P3�

Page 82: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

5. Results on �P3�

Page 83: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

5. Results on �P3�

Calculable if δ�P3 = 0

2 possibilities

Page 84: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

5. Results on �P3�

Calculable if δ�P3 = 0

2 possibilities

Page 85: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

5. Results on �P3�

Calculable if δ�P3 = 0

2 possibilities

We consider this with SU(3).

Page 86: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

5. Results on �P3�

�P3�=

∞�

m1,m2,...=−∞ψ(m)

� π/β

0dθ1dθ2...e

2kiP

miθiβ�

i<j

�coshβ(mi −mj)− cos 2β(θi − θj)

× TrRn(2βiθ + βm)

Page 87: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

Hosotani mechanismSUSY localization

1. Geometry setup

×2. SUSY on ×

3. Localizationd

dtZ(t) = 0

4. Results on Veff�D(fields)e−S =

�dθe−Veff (θ)

?

5. Results on �P3�

6. Preliminary results

λ

σ

D

λ†

Mass:k

Page 88: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

Hosotani mechanismSUSY localization

6. Preliminary results

Page 89: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

6. Preliminary results

2θ̃1 + θ̃2 =2π

kp,

θ̃1 + 2θ̃2 =2π

kq,

p, q ∈ Z

�P3� via various vacua?

k = 4

ψ̃(m,n,−m− n) = (δm,1 + δm,−1)(δn,1 + δn,−1)

ψ̃(m,n,−m− n) = 1

Running p,q

Importance sampling

Page 90: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

6. Preliminary results �P3� via various vacua?

Page 91: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

6. Preliminary results �P3� via various vacua?

arXiv:0904.1353

Page 92: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

6. Preliminary results

2θ̃1 + θ̃2 =2π

kp,

θ̃1 + 2θ̃2 =2π

kq,

p, q ∈ Z

�P3� via various vacua?

k = 4

ψ̃(m,n,−m− n) = (δm,1 + δm,−1)(δn,1 + δn,−1)

ψ̃(m,n,−m− n) = 1

Running p,q

Importance sampling

Page 93: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

6. Preliminary results �P3� via various vacua?

Page 94: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

6. Preliminary results �P3� via various vacua?

Page 95: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

6. Preliminary results

2θ̃1 + θ̃2 =2π

kp,

θ̃1 + 2θ̃2 =2π

kq,

p, q ∈ Z

�P3� via various vacua?

k = 4

ψ̃(m,n,−m− n) = (δm,1 + δm,−1)(δn,1 + δn,−1)

ψ̃(m,n,−m− n) = 1

Running p,q

Importance sampling

Page 96: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272
Page 97: Exact calculation for AB-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf · Introduction Hosotani mechanism 1. perturbation 0 Finite ! (1983) arXiv:hep-ph/0504272

Thank you.