Exact calculation for AB-phase effective potential via supersymmetric localization
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1 Exact calculation for AB-phase effective potential via supersymmetric localization
2 Exact calculation for AB-phase effective potential via supersymmetric localization
3 Todayʼs concern is purely theoretical...
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13 M S 1
14 M S 1
15 F M MN =0 S 1
16 F M MN =0 S 1 A µ =0 A S 1 = θ
17 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)
18 D(A + θ)e S(A+θ) = dθ DAe S(A+θ) = dθe Γ(θ)
19 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 +...
20 SU(3) D(A + θ)e S(A+θ) = dθ DAe S(A+θ) arxiv:hep-ph/ = dθe Γ(θ) V eff = V tree eff + V 1 loop eff + V 2 loop eff +...
21 D(A + θ)e S(A+θ) = dθ DAe S(A+θ) = dθe Γ(θ)
22 D(A + θ)e S(A+θ) P R = dθ DAe S(A+θ) P R = dθe Γ(θ) P R P R
23 D(A + θ)e S(A+θ) P R = dθ DAe S(A+θ) P R = dθe Γ(θ) P R arxiv: P R
24 D(A + θ)e S(A+θ) P R = dθ DAe S(A+θ) P R = dθe Γ(θ) P R arxiv: P R
25 D(A + θ)e S(A+θ) P R = dθ DAe S(A+θ) P R = dθe Γ(θ) P R arxiv: P R
26 D(A + θ)e S(A+θ) P R = dθ DAe S(A+θ) P R = dθe Γ(θ) P R arxiv: P R
27 D(A + θ)e S(A+θ) P R = dθ DAe S(A+θ) P R = dθe Γ(θ) P R arxiv: P R
28 D(A + θ)e S(A+θ) P R = dθ DAe S(A+θ) P R = dθe Γ(θ) P R arxiv: P R
29 D(A + θ)e S(A+θ) P R = dθ DAe S(A+θ) P R = dθe Γ(θ) P R arxiv: P R
30 D(A + θ)e S(A+θ) P R = dθ DAe S(A+θ) P R = dθe Γ(θ) P R arxiv: P R
31
32
33 A µ λ λ σ D Mass:k d Z(t) =0 dt
34 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
35
36
37
38 β
39 β lim β 0 β =
40 lim β 0 β =
41 lim β 0 β =
42 lim β 0 β =
43 lim β 0 β =
44 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
45 A µ λ λ σ D Mass:k
46 A µ λ λ σ D
47 A µ λ λ σ D
48 A µ λ σ D λ µ = 1 2 γ µγ 3
49 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
50 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
51 A µ λ σ D λ δ S SY M =0 δ η S SY M =0
52 A µ λ σ D λ δ S SY M =0 δ η S SY M =0 S SY M = δ V
53 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
54 d Z(t) =0 dt
55 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
56 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( )
57 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( )
58 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
59 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
60 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
61 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 )
62 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 )
63 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
64 V eff D(fields)e S = dθe V eff (θ)
65 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 =
66 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 =
67 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 =
68 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 =
69 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
70 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 )
71 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 )
72 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=
73 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
74 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
75 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
76 P 3
77 P 3
78 P 3
79 P 3
80 P 3
81 P 3
82 P 3
83 P 3 δ P 3 =0
84 P 3 δ P 3 =0
85 P 3 δ P 3 =0
86 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)
87 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
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89 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
90 P 3
91 P 3 arxiv:
92 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
93 P 3
94 P 3
95 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
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