exact calculation for ab-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf ·...

Exact calculation for AB-phase effective potential via supersymmetric localization

A.T, A. Tomiya, T. Shimotani

Exact calculation for AB-phase effective potential via supersymmetric localization

A.T, A. Tomiya, T. Shimotani

work in progress

Todayʼs concern ispurely theoretical...

Do these plots represent different vacua?

Introduction

Introduction

Phenomena Tools

Introduction

...SSB

Higgs mechanism

Hosotani mechanism

Phenomena Tools

Introduction

...SSB

Higgs mechanism

Hosotani mechanism

Phenomena Tools

Introduction

...SSB

Higgs mechanism

Hosotani mechanism

...Quantum correctionperturbationLattice

Phenomena Tools

Introduction

...SSB

Higgs mechanism

Hosotani mechanism

...Quantum correctionperturbationLattice

Phenomena Tools

Introduction

Hosotani mechanism (1983)

Introduction

Hosotani mechanism

×M S1(1983)

Introduction

Hosotani mechanism

×M S1(1983)

Introduction

Hosotani mechanism

×M S1FMN = 0(1983)

Introduction

Hosotani mechanism

×M S1FMN = 0

Aµ = 0

AS1 = θ

(1983)

Introduction

Hosotani mechanism

×M S1

Aµ = 0

AS1 = θ

�D(A + θ)e−S(A+θ)

=�

dθ

�DAe−S(A+θ)

(1983)

=�

dθe−Γ(θ)

Introduction

Hosotani mechanism

How to perform?

(1983)�D(A + θ)e−S(A+θ)

=�

dθ

�DAe−S(A+θ)

=�

dθe−Γ(θ)

Introduction

Hosotani mechanism

1. perturbation

(1983)�D(A + θ)e−S(A+θ)

=�

dθ

�DAe−S(A+θ)

=�

dθe−Γ(θ)

Veff = V treeeff + V 1−loop

eff + V 2−loopeff + ...

Introduction

Hosotani mechanism

1. perturbation

0 Finite !

(1983)

arXiv:hep-ph/0504272

SU(3)

�D(A + θ)e−S(A+θ)

=�

dθ

�DAe−S(A+θ)

=�

dθe−Γ(θ)

Veff = V treeeff + V 1−loop

eff + V 2−loopeff + ...

Introduction

Hosotani mechanism

2. Lattice

(1983)�D(A + θ)e−S(A+θ)

=�

dθ

�DAe−S(A+θ)

=�

dθe−Γ(θ)

Introduction

Hosotani mechanism�D(A + θ)e−S(A+θ)

=�

dθ

�DAe−S(A+θ)

2. Lattice

(1983)

PR

PR

PRPolyakov loop via lattice action

=�

dθe−Γ(θ)

∝ �PR�

Introduction

Hosotani mechanism�D(A + θ)e−S(A+θ)

=�

dθ

�DAe−S(A+θ)

2. Lattice

(1983)

PR

PR

Polyakov loop via lattice action

arXiv:0904.1353PR=

�dθe−Γ(θ)

∝ �PR�

Introduction

Hosotani mechanism�D(A + θ)e−S(A+θ)

=�

dθ

�DAe−S(A+θ)

2. Lattice

(1983)

PR

PR

Polyakov loop via lattice action

arXiv:0904.1353PR=

�dθe−Γ(θ)

∝ �PR�

Introduction

Hosotani mechanism�D(A + θ)e−S(A+θ)

=�

dθ

�DAe−S(A+θ)

2. Lattice

(1983)

PR

PR

Polyakov loop via lattice action

arXiv:0904.1353PR=

�dθe−Γ(θ)

∝ �PR�

Introduction

Hosotani mechanism�D(A + θ)e−S(A+θ)

=�

dθ

�DAe−S(A+θ)

2. Lattice

(1983)

PR

PR

Polyakov loop via lattice action

arXiv:0904.1353PR=

�dθe−Γ(θ)

∝ �PR�

Introduction

Hosotani mechanism�D(A + θ)e−S(A+θ)

=�

dθ

�DAe−S(A+θ)

2. Lattice

(1983)

PR

PR

Polyakov loop via lattice action

arXiv:0904.1353PR=

�dθe−Γ(θ)

∝ �PR�

Introduction

Hosotani mechanism�D(A + θ)e−S(A+θ)

=�

dθ

�DAe−S(A+θ)

2. Lattice

(1983)

PR

PR

Polyakov loop via lattice action

arXiv:0904.1353PR=

�dθe−Γ(θ)

∝ �PR�

Introduction

Hosotani mechanism�D(A + θ)e−S(A+θ)

=�

dθ

�DAe−S(A+θ)

2. Lattice

(1983)

PR

PR

Polyakov loop via lattice action

arXiv:0904.1353PR=

�dθe−Γ(θ)

∝ �PR�

Introduction

Hosotani mechanism�D(A + θ)e−S(A+θ)

=�

dθ

�DAe−S(A+θ)

2. Lattice

(1983)

PR

PR

Polyakov loop via lattice action

arXiv:0904.1353PR=

�dθe−Γ(θ)

∝ �PR�

Introduction

...SSB

Higgs mechanism

Hosotani mechanism

...Quantum correctionperturbationLattice

Phenomena Tools

Introduction

...SSB

Higgs mechanism

Hosotani mechanism

...Quantum correctionperturbationLattice SUSY localization

Phenomena Tools

Hosotani mechanismSUSY localization

1. Geometry setup

×2. SUSY on ×

3. Localizationd

dtZ(t) = 0

λ

σ

D

λ†

Aµ

Mass:k

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†

Aµ

Mass:k

Hosotani mechanismSUSY localization

1. Geometry setup

×

1. Geometry setup

×

1. Geometry setup

×

×

1. Geometry setup

×

×

β

1. Geometry setup

×ClaimIf CFT,

×β

limβ→0

= ×

β

1. Geometry setup

×ClaimIf CFT,

×β

limβ→0

= ×

1. Geometry setup

×ClaimIf CFT,

×β

limβ→0

= ×

1. Geometry setup

ClaimIf CFT,

βlimβ→0

=

××

1. Geometry setup

ClaimIf CFT,

βlimβ→0

=

××

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†

Aµ

×

Mass:k

Hosotani mechanismSUSY localization

2. SUSY on

λ

σ

D

λ†

Aµ

×

Mass:k

2. SUSY on ×

Spin 1

Spin 1/2

Spin 0

λ

σ

D

λ†

Aµ

2. SUSY on

λ

σ

D

λ†

Aµ

×

2. SUSY on

λ

σ

D

λ†

Aµ

×

∇µ� =12γµγ3�

curved effect

2. SUSY on

λ

σ

D

λ†

Aµ

×

δ�SSCS = 0 δη†SSCS = 0

SUSY invariance

SSCS =14π

�d3x

√gTr

� 1√

g�µνλ(Aµ∂νAλ +

2i

3AµAνAλ)− λ†λ + 2Dσ

�

2. SUSY on

λ

σ

D

λ†

Aµ

×

δ�SSCS = 0 δη†SSCS = 0

SUSY invariance

SSCS =14π

�d3x

√gTr

� 1√

g�µνλ(Aµ∂νAλ +

2i

3AµAνAλ)− λ†λ + 2Dσ

�

“Mass”

2. SUSY on

λ

σ

D

λ†

Aµ

×

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

2. SUSY on

λ

σ

D

λ†

Aµ

×

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

SUSY exactnessSSY M = δ�V

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†

Aµ

Mass:k

Hosotani mechanismSUSY localization

3. Localizationd

dtZ(t) = 0

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

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

=

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

=

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

=

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

=

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

=

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

=

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

=

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†

Aµ

Mass:k

Hosotani mechanismSUSY localization

4. Results on Veff�D(fields)e−S =

�dθe−Veff (θ)

?

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)

�

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)

�

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)

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)

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)

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)

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)

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)

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 =

2π

kq, p, q ∈ Z

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 =

2π

kq, p, q ∈ Z

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†

Aµ

Mass:k

Hosotani mechanismSUSY localization

5. Results on �P3�

5. Results on �P3�

×

5. Results on �P3�

×

5. Results on �P3�

×

5. Results on �P3�

×

5. Results on �P3�

5. Results on �P3�

5. Results on �P3�

Calculable if δ�P3 = 0

2 possibilities

5. Results on �P3�

Calculable if δ�P3 = 0

2 possibilities

5. Results on �P3�

Calculable if δ�P3 = 0

2 possibilities

We consider this with SU(3).

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)

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†

Aµ

Mass:k

Hosotani mechanismSUSY localization

6. Preliminary results

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

6. Preliminary results �P3� via various vacua?

6. Preliminary results �P3� via various vacua?

arXiv:0904.1353

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

6. Preliminary results �P3� via various vacua?

6. Preliminary results �P3� via various vacua?

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

Thank you.