Exact calculation for AB-phase effective potential via supersymmetric localization

Exact calculation for AB-phase effective potential via supersymmetric localization

Exact calculation for AB-phase effective potential via supersymmetric localization

Todayʼs concern is purely theoretical...

M S 1

M S 1

F M MN =0 S 1

F M MN =0 S 1 A µ =0 A S 1 = θ

A µ =0 M S 1 A S 1 = θ D(A + θ)e S(A+θ) = dθ DAe S(A+θ) = dθe Γ(θ)

D(A + θ)e S(A+θ) = dθ DAe S(A+θ) = dθe Γ(θ)

D(A + θ)e S(A+θ) = dθ DAe S(A+θ) = dθe Γ(θ) V eff = V tree eff + V 1 loop eff + V 2 loop eff +...

SU(3) D(A + θ)e S(A+θ) = dθ DAe S(A+θ) arxiv:hep-ph/0504272 = dθe Γ(θ) V eff = V tree eff + V 1 loop eff + V 2 loop eff +...

D(A + θ)e S(A+θ) = dθ DAe S(A+θ) = dθe Γ(θ)

D(A + θ)e S(A+θ) P R = dθ DAe S(A+θ) P R = dθe Γ(θ) P R P R

D(A + θ)e S(A+θ) P R = dθ DAe S(A+θ) P R = dθe Γ(θ) P R arxiv:0904.1353 P R

D(A + θ)e S(A+θ) P R = dθ DAe S(A+θ) P R = dθe Γ(θ) P R arxiv:0904.1353 P R

D(A + θ)e S(A+θ) P R = dθ DAe S(A+θ) P R = dθe Γ(θ) P R arxiv:0904.1353 P R

D(A + θ)e S(A+θ) P R = dθ DAe S(A+θ) P R = dθe Γ(θ) P R arxiv:0904.1353 P R

D(A + θ)e S(A+θ) P R = dθ DAe S(A+θ) P R = dθe Γ(θ) P R arxiv:0904.1353 P R

D(A + θ)e S(A+θ) P R = dθ DAe S(A+θ) P R = dθe Γ(θ) P R arxiv:0904.1353 P R

D(A + θ)e S(A+θ) P R = dθ DAe S(A+θ) P R = dθe Γ(θ) P R arxiv:0904.1353 P R

D(A + θ)e S(A+θ) P R = dθ DAe S(A+θ) P R = dθe Γ(θ) P R arxiv:0904.1353 P R

A µ λ λ σ D Mass:k d Z(t) =0 dt

V eff D(fields)e S = dθe V eff (θ) P 3 A µ λ λ σ D d Z(t) =0 dt Mass:k

β

β lim β 0 β =

lim β 0 β =

lim β 0 β =

lim β 0 β =

lim β 0 β =

V eff D(fields)e S = dθe V eff (θ) P 3 A µ λ λ σ D d Z(t) =0 dt Mass:k

A µ λ λ σ D Mass:k

A µ λ λ σ D

A µ λ λ σ D

A µ λ σ D λ µ = 1 2 γ µγ 3

A µ λ σ λ δ S SCS =0 δ η S SCS =0 S SCS = 1 4π D d 3 x 1 gtr g µνλ (A µ ν A λ + 2i 3 A µa ν A λ ) λ λ +2Dσ

A µ λ σ λ δ S SCS =0 δ η S SCS =0 S SCS = 1 4π D d 3 x 1 gtr g µνλ (A µ ν A λ + 2i 3 A µa ν A λ ) λ λ +2Dσ

A µ λ σ D λ δ S SY M =0 δ η S SY M =0

A µ λ σ D λ δ S SY M =0 δ η S SY M =0 S SY M = δ V

V eff D(fields)e S = dθe V eff (θ) P 3 A µ λ λ σ D d Z(t) =0 dt Mass:k

d Z(t) =0 dt

d dt Z(t) =0 Z(t) := fields : (A µ, λ, λ, σ,d) D(fields)e iks SCS ts SY M = δ V d dt Z(t) = = = = D(fields) d dt eiks SCS tδ V D(fields)( δ V )e iks SCS tδ V D(fields)δ Ve iks SCS tδ V D(fields)(total derivative) =0

d dt Z(t) =0 Z(t) := fields : (A µ, λ, λ, σ,d) D(fields)e iks SCS ts SY M = δ V Z(1) =Z( )

d dt Z(t) =0 Z(t) := fields : (A µ, λ, λ, σ,d) D(fields)e iks SCS ts SY M = δ V Z(1) =Z( )

d dt Z(t) =0 Z(t) := fields : (A µ, λ, λ, σ,d) D(fields)e iks SCS ts SY M = δ V Z(1) =Z( ) F MN =0 A µ =0 A S 1 = θ

d dt Z(t) =0 Z(t) := fields : (A µ, λ, λ, σ,d) D(fields)e iks SCS ts SY M = δ V Z(1) =Z( ) A µ = a(m) A S 1 = θ

d dt Z(t) =0 Z(t) := fields : (A µ, λ, λ, σ,d) D(fields)e iks SCS ts SY M = δ V Z(1) =Z( ) A µ = a(m) A S 1 = θ

d dt Z(t) =0 Z(t) := fields : (A µ, λ, λ, σ,d) D(fields)e iks SCS ts SY M = δ V Z(1) =Z( ) = π/β ψ(m) m 1,m 2,...= 0 dθ 1 dθ 2...e 2ki P m i θ i β i<j cosh β(m i m j ) cos 2β(θ i θ j )

d dt Z(t) =0 Z(t) := fields : (A µ, λ, λ, σ,d) D(fields)e iks SCS ts SY M = δ V Z(1) =Z( ) = m 1,m 2,...= ψ(m) π/β 0 dθ 1 dθ 2...e 2ki P m i θ i β i<j cosh β(m i m j ) cos 2β(θ i θ j )

V eff D(fields)e S = dθe V eff (θ) P 3 A µ λ λ σ D d Z(t) =0 dt Mass:k

V eff D(fields)e S = dθe V eff (θ)

V eff D(fields)e S = dθe V eff (θ) m 1,m 2,...= ψ(m) π/β 0 dθ 1 dθ 2...e 2ki P m i θ i β i<j cosh β(m i m j ) cos 2β(θ i θ j ) lim β 0 β =

V eff D(fields)e S = dθe V eff (θ) m 1,m 2,...= ψ(m) π/β 0 dθ 1 dθ 2...e 2ki P m i θ i β i<j cosh β(m i m j ) cos 2β(θ i θ j ) θ := βθ lim β 0 β =

V eff D(fields)e S = dθe V eff (θ) m 1,m 2,...= ψ(m) π/β 0 dθ 1 dθ 2...e 2ki P m i θ i β i<j cosh β(m i m j ) cos 2β(θ i θ j ) β N π 0 d θ 1 d θ 2... e 2ki P m i θi cos 2( θ i θ j ) θ := βθ lim β 0 β =

V eff D(fields)e S = dθe V eff (θ) m 1,m 2,...= ψ(m) π/β 0 dθ 1 dθ 2...e 2ki P m i θ i β i<j cosh β(m i m j ) cos 2β(θ i θ j ) β N π 0 d θ 1 d θ 2...e 2ki P m i θi cosh 0 = 1 cos 2( θ i θ j ) θ := βθ lim β 0 β =

V eff D(fields)e S = dθe V eff (θ) m 1,m 2,...= ψ(m) π 0 d θ 1 d θ 2...e 2ki P m i θi sin 2 ( θ i θ j ) i<j

V eff D(fields)e S = dθe V eff (θ) m 1,m 2,...= ψ(m) π 0 d θ 1 d θ 2...e 2ki P m i θi i<j sin 2 ( θ i θ j ) ψ(m) ψ(m, n, m n) =(δ m,0 + δ m,0 )(δ n,0 + δ n,0 ) V eff ( θ 1, θ 2 )

V eff D(fields)e S = dθe V eff (θ) m 1,m 2,...= ψ(m) π 0 d θ 1 d θ 2...e 2ki P m i θi i<j sin 2 ( θ i θ j ) ψ(m) ψ(m, n, m n) =(δm,1 + δ m, 1 )(δ n,1 + δ n, 1 ) V eff ( θ 1, θ 2 )

V eff D(fields)e S = dθe V eff (θ) m 1,m 2,...= ψ(m) π 0 d θ 1 d θ 2...e 2ki P m i θi i<j sin 2 ( θ i θ j ) ψ(m) ψ(m, n, m n) =1 V eff ( θ 1, θ 2 ) e 2πixn = δ(x p) n= p=

V eff D(fields)e S = dθe V eff (θ) m 1,m 2,...= ψ(m) π 0 d θ 1 d θ 2...e 2ki P m i θi i<j sin 2 ( θ i θ j ) ψ(m) ψ(m, n, m n) =1 V eff ( θ 1, θ 2 ) 2 θ 1 + θ 2 = 2π k p, θ1 +2 θ 2 = 2π k q, p, q Z

V eff D(fields)e S = dθe V eff (θ) m 1,m 2,...= ψ(m) π 0 d θ 1 d θ 2...e 2ki P m i θi i<j sin 2 ( θ i θ j ) ψ(m) ψ(m, n, m n) =1 V eff ( θ 1, θ 2 ) 2 θ 1 + θ 2 = 2π k p, θ1 +2 θ 2 = 2π k q, p, q Z

V eff D(fields)e S = dθe V eff (θ) P 3 A µ λ λ σ D d Z(t) =0 dt Mass:k

P 3

P 3

P 3

P 3

P 3

P 3

P 3

P 3 δ P 3 =0

P 3 δ P 3 =0

P 3 δ P 3 =0

P 3 P 3 = m 1,m 2,...= ψ(m) π/β 0 dθ 1 dθ 2...e 2ki P m i θ i β i<j cosh β(m i m j ) cos 2β(θ i θ j ) Tr Rn (2βiθ + βm)

V eff D(fields)e S = dθe V eff (θ) P 3 A µ λ λ σ D d Z(t) =0 dt Mass:k

P 3 ψ(m, n, m n) =(δ m,1 + δ m, 1 )(δ n,1 + δ n, 1 ) k =4 2 θ 1 + θ 2 = 2π k p, θ 1 +2 θ 2 = 2π k q, p, q Z ψ(m, n, m n) =1

P 3

P 3 arxiv:0904.1353

P 3 ψ(m, n, m n) =(δ m,1 + δ m, 1 )(δ n,1 + δ n, 1 ) k =4 2 θ 1 + θ 2 = 2π k p, θ 1 +2 θ 2 = 2π k q, p, q Z ψ(m, n, m n) =1

P 3

P 3

P 3 ψ(m, n, m n) =(δ m,1 + δ m, 1 )(δ n,1 + δ n, 1 ) k =4 2 θ 1 + θ 2 = 2π k p, θ 1 +2 θ 2 = 2π k q, p, q Z ψ(m, n, m n) =1