exact calculation for ab-phase effective potential via ...seminar/pdf_2013_zenki/akinori.pdf ·...
TRANSCRIPT
![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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/1.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/2.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/3.jpg)
Todayʼs concern ispurely theoretical...
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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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/5.jpg)
![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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/6.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/7.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/8.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/9.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/10.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/11.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/12.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/13.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/14.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/15.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/16.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/17.jpg)
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)
![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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/18.jpg)
Introduction
Hosotani mechanism
How to perform?
(1983)�D(A + θ)e−S(A+θ)
=�
dθ
�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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/19.jpg)
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 + ...
![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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/20.jpg)
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 + ...
![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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/21.jpg)
Introduction
Hosotani mechanism
2. Lattice
(1983)�D(A + θ)e−S(A+θ)
=�
dθ
�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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/22.jpg)
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�
![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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/23.jpg)
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�
![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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/24.jpg)
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�
![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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/25.jpg)
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�
![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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/26.jpg)
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�
![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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/27.jpg)
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�
![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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/28.jpg)
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�
![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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/29.jpg)
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�
![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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/30.jpg)
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�
![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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/31.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/32.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/33.jpg)
Hosotani mechanismSUSY localization
1. Geometry setup
×2. SUSY on ×
3. Localizationd
dtZ(t) = 0
λ
σ
D
λ†
Aµ
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/34.jpg)
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
![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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/35.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/36.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/37.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/38.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/39.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/40.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/41.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/42.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/43.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/44.jpg)
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
![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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/45.jpg)
Hosotani mechanismSUSY localization
2. SUSY on
λ
σ
D
λ†
Aµ
×
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/46.jpg)
2. SUSY on ×
Spin 1
Spin 1/2
Spin 0
λ
σ
D
λ†
Aµ
![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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/47.jpg)
2. SUSY on
λ
σ
D
λ†
Aµ
×
![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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/48.jpg)
2. SUSY on
λ
σ
D
λ†
Aµ
×
∇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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/49.jpg)
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
�
![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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/50.jpg)
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”
![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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/51.jpg)
2. SUSY on
λ
σ
D
λ†
Aµ
×
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/52.jpg)
2. SUSY on
λ
σ
D
λ†
Aµ
×
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/53.jpg)
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
![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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/54.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/55.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/56.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/57.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/58.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/59.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/60.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/61.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/62.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/63.jpg)
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
![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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/64.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/65.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/66.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/67.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/68.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/69.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/70.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/71.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/72.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/73.jpg)
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
![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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/74.jpg)
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
![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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/75.jpg)
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
![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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/76.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/77.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/78.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/79.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/80.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/81.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/82.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/83.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/84.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/85.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/86.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/87.jpg)
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
![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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/88.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/89.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/90.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/91.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/92.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/93.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/94.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/95.jpg)
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
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![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](https://reader036.vdocuments.us/reader036/viewer/2022081617/604182ca9a355346010d0536/html5/thumbnails/97.jpg)
Thank you.