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The ”Bootstrap Program”for integrable quantum field theories in 1+1 dimensions
S-matrix - Form Factors - Wightman Functions
H. Babujian, A. Foerster, and M. Karowski
HU-Berlin, August 2016
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program HU-Berlin, August 2016 1 / 27
Contents
1 The “Bootstrap Program”General ideaIntegrability
2 S-matrixExamples: Sine Gordon and SU(N)
3 Form factorsForm factors equations
Examples: Sine Gordon and SU(N)
General form factor formula“Bethe ansatz” state
Examples:The Sine-Gordon ≡ Massive Thirring model
4 Wightman functionsShort distance behavior
5 References
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program HU-Berlin, August 2016 2 / 27
Contents
1 The “Bootstrap Program”General ideaIntegrability
2 S-matrixExamples: Sine Gordon and SU(N)
3 Form factorsForm factors equations
Examples: Sine Gordon and SU(N)
General form factor formula“Bethe ansatz” state
Examples:The Sine-Gordon ≡ Massive Thirring model
4 Wightman functionsShort distance behavior
5 References
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program HU-Berlin, August 2016 2 / 27
Contents
1 The “Bootstrap Program”General ideaIntegrability
2 S-matrixExamples: Sine Gordon and SU(N)
3 Form factorsForm factors equations
Examples: Sine Gordon and SU(N)
General form factor formula“Bethe ansatz” state
Examples:The Sine-Gordon ≡ Massive Thirring model
4 Wightman functionsShort distance behavior
5 References
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program HU-Berlin, August 2016 2 / 27
Contents
1 The “Bootstrap Program”General ideaIntegrability
2 S-matrixExamples: Sine Gordon and SU(N)
3 Form factorsForm factors equations
Examples: Sine Gordon and SU(N)
General form factor formula“Bethe ansatz” state
Examples:The Sine-Gordon ≡ Massive Thirring model
4 Wightman functionsShort distance behavior
5 References
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program HU-Berlin, August 2016 2 / 27
Contents
1 The “Bootstrap Program”General ideaIntegrability
2 S-matrixExamples: Sine Gordon and SU(N)
3 Form factorsForm factors equations
Examples: Sine Gordon and SU(N)
General form factor formula“Bethe ansatz” state
Examples:The Sine-Gordon ≡ Massive Thirring model
4 Wightman functionsShort distance behavior
5 References
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program HU-Berlin, August 2016 2 / 27
The “Bootstrap Program”
Construct a quantum field theory explicitly in 3 steps
1 S-matrixusing 1 general Properties: unitarity, crossing etc
2 ”Yang-Baxter Equation”3 ”bound state bootstrap”4 ‘maximal analyticity’
2 “Form factors”
〈 0 | φ(x) | p1, . . . , pn 〉in = e−ix(p1+···+pn) F φ (θ1, . . . , θn)
using 1 the S-matrix2 LSZ-assumptions3 ‘maximal analyticity’
3 “Wightman functions”
〈 0 | φ(x)φ(y) | 0 〉 = ∑n
∫〈 0 | φ(x) | n 〉in in〈 n | φ(y) | 0 〉
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program HU-Berlin, August 2016 3 / 27
The “Bootstrap Program”
Construct a quantum field theory explicitly in 3 steps
1 S-matrixusing 1 general Properties: unitarity, crossing etc
2 ”Yang-Baxter Equation”3 ”bound state bootstrap”4 ‘maximal analyticity’
2 “Form factors”
〈 0 | φ(x) | p1, . . . , pn 〉in = e−ix(p1+···+pn) F φ (θ1, . . . , θn)
using 1 the S-matrix2 LSZ-assumptions3 ‘maximal analyticity’
3 “Wightman functions”
〈 0 | φ(x)φ(y) | 0 〉 = ∑n
∫〈 0 | φ(x) | n 〉in in〈 n | φ(y) | 0 〉
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program HU-Berlin, August 2016 3 / 27
The “Bootstrap Program”
Construct a quantum field theory explicitly in 3 steps
1 S-matrixusing 1 general Properties: unitarity, crossing etc
2 ”Yang-Baxter Equation”3 ”bound state bootstrap”4 ‘maximal analyticity’
2 “Form factors”
〈 0 | φ(x) | p1, . . . , pn 〉in = e−ix(p1+···+pn) F φ (θ1, . . . , θn)
using 1 the S-matrix2 LSZ-assumptions3 ‘maximal analyticity’
3 “Wightman functions”
〈 0 | φ(x)φ(y) | 0 〉 = ∑n
∫〈 0 | φ(x) | n 〉in in〈 n | φ(y) | 0 〉
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program HU-Berlin, August 2016 3 / 27
The “Bootstrap Program”
Construct a quantum field theory explicitly in 3 steps
1 S-matrixusing 1 general Properties: unitarity, crossing etc
2 ”Yang-Baxter Equation”3 ”bound state bootstrap”4 ‘maximal analyticity’
2 “Form factors”
〈 0 | φ(x) | p1, . . . , pn 〉in = e−ix(p1+···+pn) F φ (θ1, . . . , θn)
using 1 the S-matrix2 LSZ-assumptions3 ‘maximal analyticity’
3 “Wightman functions”
〈 0 | φ(x)φ(y) | 0 〉 = ∑n
∫〈 0 | φ(x) | n 〉in in〈 n | φ(y) | 0 〉
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program HU-Berlin, August 2016 3 / 27
The “Bootstrap Program”
We do not define a quantum field theory by a Lagrangian,
but we solve the S-matrix and form factor equations
The bootstrap program classifiesintegrable quantum field theories
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program HU-Berlin, August 2016 4 / 27
Integrability
“Yang-Baxter equation” S12S13S23 = S23S13S12
@@@@@
=
@@@@@
12 3 1 2
3
“bound state bootstrap equation” S(12)3 Γ(12)12 = Γ
(12)12 S13S23
@@
@@
1 23
(12)
• =
@@
@@@
12 3
(12)•
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program HU-Berlin, August 2016 5 / 27
Integrability
“Yang-Baxter equation” S12S13S23 = S23S13S12
@@@@@
=
@@@@@
12 3 1 2
3
“bound state bootstrap equation” S(12)3 Γ(12)12 = Γ
(12)12 S13S23
@@
@@
1 23
(12)
• =
@@
@@@
12 3
(12)•
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program HU-Berlin, August 2016 5 / 27
Sine-Gordon S-matrix or SUq(2)
Sδγαβ (θ12) =
•
@@@@
α β
γδ
θ1 θ2
α, β, γ, δ = s soliton, s anti-solitonθ = rapidity, p± = p0 ± p1 = me±θ
S ssss (θ) = a(θ), S ss
ss (θ) = b(θ), S ssss (θ) = c(θ)
crossing + unitarity etc [A.B. Zamolodchikov (1977)]
[Karowski Thun Truong Weisz 1977]
Yang-Baxter =⇒ c(θ) = b(θ)sinh iπ/ν
sinh (iπ − θ) /ν, q = −e−iπ/ν
a(θ) = − exp∫ ∞
0
dt
t
sinh 12 (1− ν)t
sinh 12νt cosh 1
2 tsinh t
θ
iπ
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program HU-Berlin, August 2016 6 / 27
Sine-Gordon S-matrix or SUq(2)
Sδγαβ (θ12) =
•
@@@@
α β
γδ
θ1 θ2
α, β, γ, δ = s soliton, s anti-solitonθ = rapidity, p± = p0 ± p1 = me±θ
S ssss (θ) = a(θ), S ss
ss (θ) = b(θ), S ssss (θ) = c(θ)
crossing + unitarity etc [A.B. Zamolodchikov (1977)]
[Karowski Thun Truong Weisz 1977]
Yang-Baxter =⇒ c(θ) = b(θ)sinh iπ/ν
sinh (iπ − θ) /ν, q = −e−iπ/ν
a(θ) = − exp∫ ∞
0
dt
t
sinh 12 (1− ν)t
sinh 12νt cosh 1
2 tsinh t
θ
iπ
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program HU-Berlin, August 2016 6 / 27
Sine-Gordon S-matrix or SUq(2)
Sδγαβ (θ12) =
•
@@@@
α β
γδ
θ1 θ2
α, β, γ, δ = s soliton, s anti-solitonθ = rapidity, p± = p0 ± p1 = me±θ
S ssss (θ) = a(θ), S ss
ss (θ) = b(θ), S ssss (θ) = c(θ)
crossing + unitarity etc [A.B. Zamolodchikov (1977)]
[Karowski Thun Truong Weisz 1977]
Yang-Baxter =⇒ c(θ) = b(θ)sinh iπ/ν
sinh (iπ − θ) /ν, q = −e−iπ/ν
a(θ) = − exp∫ ∞
0
dt
t
sinh 12 (1− ν)t
sinh 12νt cosh 1
2 tsinh t
θ
iπ
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program HU-Berlin, August 2016 6 / 27
Sine-Gordon S-matrix or SUq(2)
Sδγαβ (θ12) =
•
@@@@
α β
γδ
θ1 θ2
α, β, γ, δ = s soliton, s anti-solitonθ = rapidity, p± = p0 ± p1 = me±θ
S ssss (θ) = a(θ), S ss
ss (θ) = b(θ), S ssss (θ) = c(θ)
crossing + unitarity etc [A.B. Zamolodchikov (1977)]
[Karowski Thun Truong Weisz 1977]
Yang-Baxter =⇒ c(θ) = b(θ)sinh iπ/ν
sinh (iπ − θ) /ν, q = −e−iπ/ν
a(θ) = − exp∫ ∞
0
dt
t
sinh 12 (1− ν)t
sinh 12νt cosh 1
2 tsinh t
θ
iπ
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program HU-Berlin, August 2016 6 / 27
Sine-Gordon - Massive Thirring model
exact S-matrix ↔ perturbation theory for the
Lagrangians:
LSG = 12 (∂µφ)2 +
α
β2(cos βφ− 1)
LMTM = ψ(iγ∂−M)ψ− 12g(ψγµψ)2
if
ν =β2
8π − β2=
π
π + 2g
↑Coleman
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program HU-Berlin, August 2016 7 / 27
Example: SU(N) S-matrix
Particles α, β, γ, δ = 1, . . . ,N ↔ vector representation of SU(N)
Sδγαβ (θ12) =
•
@@@@
α β
γδ
θ1 θ2
= δαγδβδ b(θ12) + δαδδβγ c(θ12).
Yang-Baxter =⇒ c(θ) = − 2πiN
1θb(θ) + crossing + unitarity =⇒
a(θ) = b(θ) + c(θ) = −Γ(1− θ
2πi
)Γ(1− 1
N + θ2πi
)Γ(1 + θ
2πi
)Γ(1− 1
N −θ
2πi
)[Berg Karowski Kurak Weisz 1978]
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program HU-Berlin, August 2016 8 / 27
Example: SU(N) S-matrix
Particles α, β, γ, δ = 1, . . . ,N ↔ vector representation of SU(N)
Sδγαβ (θ12) =
•
@@@@
α β
γδ
θ1 θ2
= δαγδβδ b(θ12) + δαδδβγ c(θ12).
Yang-Baxter =⇒ c(θ) = − 2πiN
1θb(θ) + crossing + unitarity =⇒
a(θ) = b(θ) + c(θ) = −Γ(1− θ
2πi
)Γ(1− 1
N + θ2πi
)Γ(1 + θ
2πi
)Γ(1− 1
N −θ
2πi
)[Berg Karowski Kurak Weisz 1978]
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program HU-Berlin, August 2016 8 / 27
Form factors
Definition
Let O(x) be a local operator
〈 0 | O(x) | p1, . . . , pn 〉inα1...αn= FOα1...αn
(θ1, . . . , θn) e−ix ∑ pi
= O
. . .
FOα (θ) = form factor (co-vector valued function)
αi ∈ all types of particles
LSZ-assumptions+ ’maximal analyticity’
=⇒ Properties of form factors
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program HU-Berlin, August 2016 9 / 27
Form factors
Definition
Let O(x) be a local operator
〈 0 | O(x) | p1, . . . , pn 〉inα1...αn= FOα1...αn
(θ1, . . . , θn) e−ix ∑ pi
= O
. . .
FOα (θ) = form factor (co-vector valued function)
αi ∈ all types of particles
LSZ-assumptions+ ’maximal analyticity’
=⇒ Properties of form factors
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program HU-Berlin, August 2016 9 / 27
Form factors equations
[Karowski Weisz (1978)] [Smirnov (World Scientific 1992)]
(i) Watson’s equation
FO...ij ...(. . . , θi , θj , . . . ) = FO...ji ...(. . . , θj , θi , . . . ) Sij (θi − θj ) O... ...
=
O
AA... ...
(ii) Crossing
α1〈 p1 | O(0) | . . . , pn 〉in,conn....αn=
Cα1α1σOα1FOα1 ...αn
(θ1 + iπ, . . . , θn) = FO...αnα1(. . . , θn, θ1 − iπ)Cα1α1
O. . .
conn. = σOα1 O
. . .=
O. . .
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program HU-Berlin, August 2016 10 / 27
Form factors equations
[Karowski Weisz (1978)] [Smirnov (World Scientific 1992)]
(i) Watson’s equation
FO...ij ...(. . . , θi , θj , . . . ) = FO...ji ...(. . . , θj , θi , . . . ) Sij (θi − θj ) O... ...
=
O
AA... ...
(ii) Crossing
α1〈 p1 | O(0) | . . . , pn 〉in,conn....αn=
Cα1α1σOα1FOα1 ...αn
(θ1 + iπ, . . . , θn) = FO...αnα1(. . . , θn, θ1 − iπ)Cα1α1
O. . .
conn. = σOα1 O
. . .=
O. . .
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program HU-Berlin, August 2016 10 / 27
Form factors equations
(iii) Annihilation recursion relation
1
2iRes
θ12=iπFO12...n(θ1, . .) = C12 FO3...n(θ3, . .)
(1− σO2 S2n . . . S23
)1
2iRes
θ12=iπ
O...
= O...
− σO2
O...
(iv) Bound state form factors
Resθ12=iη
FO123...n(θ) =√
2FO(12)3...n(θ(12), θ′) Γ(12)12
Resθ12=iη
O...
=√
2
O...(v) Lorentz invariance (with s = “spin” of O)
FO1...n(θ1 + u, . . . , θn + u) = esu FO1...n(θ1, . . . , θn)
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program HU-Berlin, August 2016 11 / 27
Form factors equations
(iii) Annihilation recursion relation
1
2iRes
θ12=iπFO12...n(θ1, . .) = C12 FO3...n(θ3, . .)
(1− σO2 S2n . . . S23
)1
2iRes
θ12=iπ
O...
= O...
− σO2
O...
(iv) Bound state form factors
Resθ12=iη
FO123...n(θ) =√
2FO(12)3...n(θ(12), θ′) Γ(12)12
Resθ12=iη
O...
=√
2
O...(v) Lorentz invariance (with s = “spin” of O)
FO1...n(θ1 + u, . . . , θn + u) = esu FO1...n(θ1, . . . , θn)
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program HU-Berlin, August 2016 11 / 27
Form factors equations
(iii) Annihilation recursion relation
1
2iRes
θ12=iπFO12...n(θ1, . .) = C12 FO3...n(θ3, . .)
(1− σO2 S2n . . . S23
)1
2iRes
θ12=iπ
O...
= O...
− σO2
O...
(iv) Bound state form factors
Resθ12=iη
FO123...n(θ) =√
2FO(12)3...n(θ(12), θ′) Γ(12)12
Resθ12=iη
O...
=√
2
O...(v) Lorentz invariance (with s = “spin” of O)
FO1...n(θ1 + u, . . . , θn + u) = esu FO1...n(θ1, . . . , θn)
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program HU-Berlin, August 2016 11 / 27
2-particle form factor
”Watson’s equation””crossing equation”
F (θ) = F (−θ) S (θ)F (iπ − θ) = F (iπ + θ)
“maximal analyticity” ⇒ unique solution [Karowski Weisz (1978)]
”maximal analyticity” ↔F (θ) meromorphic and all poles have a physical interpretation
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program HU-Berlin, August 2016 12 / 27
2-particle form factor
”Watson’s equation””crossing equation”
F (θ) = F (−θ) S (θ)F (iπ − θ) = F (iπ + θ)
“maximal analyticity” ⇒ unique solution [Karowski Weisz (1978)]
”maximal analyticity” ↔F (θ) meromorphic and all poles have a physical interpretation
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program HU-Berlin, August 2016 12 / 27
Example: Sine Gordon
The highest weight SUq(2) ’minimal’ soliton-soliton form factor
F (θ) = exp1
2
∫ ∞
0
dt
t sinh t
sinh 12 t (1 + ν)
sinh 12νt cosh 1
2 t
(1− cosh t
(1− θ
iπ
))[Karowski Weisz (1978)]
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program HU-Berlin, August 2016 13 / 27
Example: SU(N)
The highest weight SU(N) minimal 2-particle form factor
F (θ) = exp
∞∫0
dte
tN sinh t
(1− 1
N
)t sinh2 t
(1− cosh t
(1− θ
iπ
))
=∞
∏k=1
Γ(k − 1
N + 1− 12
θiπ
)Γ(k − 1
N + 12
θiπ
)Γ(k − 1
2θiπ
)Γ(k − 1 + 1
2θiπ
) Γ2(k − 1
2
)Γ2(k − 1
N + 12
)[Babujian Foerster Karowski (2006)]
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program HU-Berlin, August 2016 14 / 27
General form factor formula
FOα1...αn(θ1, . . . , θn) = KOα1 ...αn
(θ) ∏1≤i<j≤n
F (θij )
”Off-shell Bethe Ansatz”
KOα1...αn(θ) =
∫Cθ
dz1 · · ·∫Cθ
dzm h(θ, z) pO(θ, z)Ψα1 ...αn(θ, z)
Ψα(θ, z) = Bethe state
h(θ, z) =n
∏i=1
m
∏j=1
φ(θi − zj ) ∏1≤i<j≤m
τ(zi − zj ) , τ(z) =1
φ(z)φ(−z)
depend only on the S-matrix (see below),
pO(θ, z) = depends on the operator OBabujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program HU-Berlin, August 2016 15 / 27
General form factor formula
FOα1...αn(θ1, . . . , θn) = KOα1 ...αn
(θ) ∏1≤i<j≤n
F (θij )
”Off-shell Bethe Ansatz”
KOα1...αn(θ) =
∫Cθ
dz1 · · ·∫Cθ
dzm h(θ, z) pO(θ, z)Ψα1 ...αn(θ, z)
Ψα(θ, z) = Bethe state
h(θ, z) =n
∏i=1
m
∏j=1
φ(θi − zj ) ∏1≤i<j≤m
τ(zi − zj ) , τ(z) =1
φ(z)φ(−z)
depend only on the S-matrix (see below),
pO(θ, z) = depends on the operator OBabujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program HU-Berlin, August 2016 15 / 27
General form factor formula
FOα1...αn(θ1, . . . , θn) = KOα1 ...αn
(θ) ∏1≤i<j≤n
F (θij )
”Off-shell Bethe Ansatz”
KOα1...αn(θ) =
∫Cθ
dz1 · · ·∫Cθ
dzm h(θ, z) pO(θ, z)Ψα1 ...αn(θ, z)
Ψα(θ, z) = Bethe state
h(θ, z) =n
∏i=1
m
∏j=1
φ(θi − zj ) ∏1≤i<j≤m
τ(zi − zj ) , τ(z) =1
φ(z)φ(−z)
depend only on the S-matrix (see below),
pO(θ, z) = depends on the operator OBabujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program HU-Berlin, August 2016 15 / 27
General form factor formula
FOα1...αn(θ1, . . . , θn) = KOα1 ...αn
(θ) ∏1≤i<j≤n
F (θij )
”Off-shell Bethe Ansatz”
KOα1...αn(θ) =
∫Cθ
dz1 · · ·∫Cθ
dzm h(θ, z) pO(θ, z)Ψα1 ...αn(θ, z)
Ψα(θ, z) = Bethe state
h(θ, z) =n
∏i=1
m
∏j=1
φ(θi − zj ) ∏1≤i<j≤m
τ(zi − zj ) , τ(z) =1
φ(z)φ(−z)
depend only on the S-matrix (see below),
pO(θ, z) = depends on the operator OBabujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program HU-Berlin, August 2016 15 / 27
Equation for φ(z)
Example: SU(2)
(iii)←→ φ (z) =1
F (z) F (z + iπ)= Γ
( z
2πi
)Γ(
1
2− z
2πi
)
Sine-Gordon
φ (z) =∞
∏k=0
Γ(12kν +
z
2πi
)Γ(12kν + 1
2 −z
2πi
)Γ(12 (k + 1) ν + 1
2 +z
2πi
)Γ(12 (k + 1) ν + 1− z
2πi
)
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program HU-Berlin, August 2016 16 / 27
Equation for φ(z)
Example: SU(2)
(iii)←→ φ (z) =1
F (z) F (z + iπ)= Γ
( z
2πi
)Γ(
1
2− z
2πi
)
Sine-Gordon
φ (z) =∞
∏k=0
Γ(12kν +
z
2πi
)Γ(12kν + 1
2 −z
2πi
)Γ(12 (k + 1) ν + 1
2 +z
2πi
)Γ(12 (k + 1) ν + 1− z
2πi
)
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program HU-Berlin, August 2016 16 / 27
“Bethe ansatz” state
Example: SU(2) or sine-Gordon
Ψα(θ, z) = (ΩC (θ, zm) . . .C (θ, z1))α1...αn
=
S-matrix
• •
• •
α1 αn
2
2
1 1
1
1
θ1 θn
z1
zm
. . .
...(1 ≤ αi ≤ 2)
If rank > 1⇒ nested Bethe Ansatz
⇒ Bethe Ansatz of level 1, 2, . . . ,
rank(SU(N)) = N − 1rank(O(N)) = [N/2]
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program HU-Berlin, August 2016 17 / 27
“Bethe ansatz” state
Example: SU(2) or sine-Gordon
Ψα(θ, z) = (ΩC (θ, zm) . . .C (θ, z1))α1...αn
=
S-matrix
• •
• •
α1 αn
2
2
1 1
1
1
θ1 θn
z1
zm
. . .
...(1 ≤ αi ≤ 2)
If rank > 1⇒ nested Bethe Ansatz
⇒ Bethe Ansatz of level 1, 2, . . . ,
rank(SU(N)) = N − 1rank(O(N)) = [N/2]
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program HU-Berlin, August 2016 17 / 27
“Bethe ansatz” state
Example: SU(2) or sine-Gordon
Ψα(θ, z) = (ΩC (θ, zm) . . .C (θ, z1))α1...αn
=
S-matrix
• •
• •
α1 αn
2
2
1 1
1
1
θ1 θn
z1
zm
. . .
...(1 ≤ αi ≤ 2)
If rank > 1⇒ nested Bethe Ansatz
⇒ Bethe Ansatz of level 1, 2, . . . ,
rank(SU(N)) = N − 1rank(O(N)) = [N/2]
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program HU-Berlin, August 2016 17 / 27
Integration contour for SU(N)
(ii) ←→(∫Cθ
−∫Cθ′
)dz h(θ, z)a(θ − z) . . . = 0 , θ′ = θ + 2πi
• θn − 2πi
bθn − 2πi 1N
• θn
• θn + 2πi(1− 1N )
. . .
• θ2 − 2πi
bθ2 − 2πi 1N
• θ2
• θ2 + 2πi(1− 1N )
• θ1 − 2πi
bθ1 − 2πi 1N
• θ1
• θ1 + 2πi(1− 1N )
-
-
Figure: The integration contour Cθ.The bullets refer to poles of the integrand due to the amplitude aThe circles refer to poles of the integrand due to the amplitudes b and c
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program HU-Berlin, August 2016 18 / 27
General form factor formula
The Ansatz
KOα1...αn(θ) =
∫Cθ
dz1 · · ·∫Cθ
dzm h(θ, z) pO(θ, z)Ψα1 ...αn(θ, z)
transforms the complicated matrix equations intosimple equations for the scalar functions pO(θ, z)
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program HU-Berlin, August 2016 19 / 27
p-Equations
(i) Watson’s equation:pO(θ, z) is symmetric with respect to the θ’s and the z ’s
(ii) Crossing
pO(θ, z) = pO(θ1 + 2πi , . . . , z) = pO(θ, z1 + 2πi)
(iii) Annihilation recursion relation - Residue equation
pO(θ1 = θ2 + iπ, . . . , z1 = θ1, . . . ) = pO(θ3, . . . , z2, . . . )
(v) Lorentz invariance
pO(θ + µ, z + µ) = esµpO(θ, z)
where s is the ‘spin’ of the operator O(x).
Remark: there are additional statistics phase factors
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program HU-Berlin, August 2016 20 / 27
p-Equations
(i) Watson’s equation:pO(θ, z) is symmetric with respect to the θ’s and the z ’s
(ii) Crossing
pO(θ, z) = pO(θ1 + 2πi , . . . , z) = pO(θ, z1 + 2πi)
(iii) Annihilation recursion relation - Residue equation
pO(θ1 = θ2 + iπ, . . . , z1 = θ1, . . . ) = pO(θ3, . . . , z2, . . . )
(v) Lorentz invariance
pO(θ + µ, z + µ) = esµpO(θ, z)
where s is the ‘spin’ of the operator O(x).
Remark: there are additional statistics phase factors
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program HU-Berlin, August 2016 20 / 27
p-Equations
(i) Watson’s equation:pO(θ, z) is symmetric with respect to the θ’s and the z ’s
(ii) Crossing
pO(θ, z) = pO(θ1 + 2πi , . . . , z) = pO(θ, z1 + 2πi)
(iii) Annihilation recursion relation - Residue equation
pO(θ1 = θ2 + iπ, . . . , z1 = θ1, . . . ) = pO(θ3, . . . , z2, . . . )
(v) Lorentz invariance
pO(θ + µ, z + µ) = esµpO(θ, z)
where s is the ‘spin’ of the operator O(x).
Remark: there are additional statistics phase factors
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program HU-Berlin, August 2016 20 / 27
Example: Sine-Gordon ≡ Massive Thirring model
[Babujian Fring Karowski Zapletal (1999)]
p-function for the soliton-field (fermi-field) ψ(x)
pψ(θ, z) = exp
(m
∑i=1
zi −1
2
n
∑i=1
θi
)
Massive Thirring model perturbation expansion
〈 0 |ψ(0) | p1, p, p3 〉insss =6
6
@@I
••
p1 p2 p3
+O(g2)
= −ig sinh 12θ23
u(p2) cosh 12θ12 + u(p3) cosh 1
2θ13
cosh 12θ12 cosh 1
2θ13 cosh 12θ23
+O(g2)
aggrees with the expansion of the exact result.Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program HU-Berlin, August 2016 21 / 27
Example: Sine-Gordon ≡ Massive Thirring model
[Babujian Fring Karowski Zapletal (1999)]
p-function for the soliton-field (fermi-field) ψ(x)
pψ(θ, z) = exp
(m
∑i=1
zi −1
2
n
∑i=1
θi
)
Massive Thirring model perturbation expansion
〈 0 |ψ(0) | p1, p, p3 〉insss =6
6
@@I
••
p1 p2 p3
+O(g2)
= −ig sinh 12θ23
u(p2) cosh 12θ12 + u(p3) cosh 1
2θ13
cosh 12θ12 cosh 1
2θ13 cosh 12θ23
+O(g2)
aggrees with the expansion of the exact result.Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program HU-Berlin, August 2016 21 / 27
Wightman functions
Example: The sinh-Gordon model
ϕ +α
βsinh βϕ = 0
Finite wave function and mass renormalizations:
Z ϕ = (1 + ν)12πν
sin 12πν
exp
(− 1
π
∫ πν
0
t
sin tdt
)[Karowski Weisz (1978)]
α = m2 πν
sin πν[Babujian Karowski (2002)]
has been checked in perturbation theory ν = − β2
8π+β2
S-matrix
S(θ) =sinh θ + i sin πν
sinh θ − i sin πν
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program HU-Berlin, August 2016 22 / 27
Wightman functions
Example: The sinh-Gordon model
ϕ +α
βsinh βϕ = 0
Finite wave function and mass renormalizations:
Z ϕ = (1 + ν)12πν
sin 12πν
exp
(− 1
π
∫ πν
0
t
sin tdt
)[Karowski Weisz (1978)]
α = m2 πν
sin πν[Babujian Karowski (2002)]
has been checked in perturbation theory ν = − β2
8π+β2
S-matrix
S(θ) =sinh θ + i sin πν
sinh θ − i sin πν
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program HU-Berlin, August 2016 22 / 27
Wightman functions
Example: The sinh-Gordon model
ϕ +α
βsinh βϕ = 0
Finite wave function and mass renormalizations:
Z ϕ = (1 + ν)12πν
sin 12πν
exp
(− 1
π
∫ πν
0
t
sin tdt
)[Karowski Weisz (1978)]
α = m2 πν
sin πν[Babujian Karowski (2002)]
has been checked in perturbation theory ν = − β2
8π+β2
S-matrix
S(θ) =sinh θ + i sin πν
sinh θ − i sin πν
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program HU-Berlin, August 2016 22 / 27
Wightman functions
The two-point function
w(x) = 〈 0 | O(x)O(0) | 0 〉
Intermediate states expansion
〈 0 | O(x)O(y) | 0 〉 = ∑n
∫〈 0 | O(x) | n 〉in in〈 n | O(y) | 0 〉
Short distances behavior for O(x) = exp βϕ(x)
w(x) ∼(√−x2
)−4∆for x → 0
“Dimension” ∆
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program HU-Berlin, August 2016 23 / 27
Wightman functions
The two-point function
w(x) = 〈 0 | O(x)O(0) | 0 〉
Intermediate states expansion
〈 0 | O(x)O(y) | 0 〉 = ∑n
∫〈 0 | O(x) | n 〉in in〈 n | O(y) | 0 〉
Short distances behavior for O(x) = exp βϕ(x)
w(x) ∼(√−x2
)−4∆for x → 0
“Dimension” ∆
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program HU-Berlin, August 2016 23 / 27
Short distance behavior
“Dimension” ∆ for sinh-Gordon1- and 1+2-particle intermediate state contributions
0
0.1
0.2
0.3
0.4
0 1 21-particle
∆
B
1+2-particlewhere B = 2β2
8π+β2 = −2ν
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program HU-Berlin, August 2016 24 / 27
∆1+2 = −sin πν
πF (iπ)+
(sin πν
πF (iπ)
)2 ∫ ∞
−∞dθ (F (θ)F (−θ)− 1)
= − sin πν
πF (iπ)− π
2sin πνF 2(iπ)− π
cos πν− 1
sin πν+ 2
(1− πν cos πν
sin πν
)
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program HU-Berlin, August 2016 25 / 27
Some References
S-matrix:A.B. Zamolodchikov, JEPT Lett. 25 (1977) 468
M. Karowski, H.J. Thun, T.T. Truong and P. WeiszPhys. Lett. B67 (1977) 321
M. Karowski and H.J. Thun, Nucl. Phys. B130 (1977) 295
A.B. Zamolodchikov and Al. B. ZamolodchikovAnn. Phys. 120 (1979) 253
M. Karowski, Nucl. Phys. B153 (1979) 244
V. Kurak and J. A. Swieca, Phys. Lett. B82, 289–291 (1979).
R. Koberle, V. Kurak, and J. A. Swieca, Nucl. Phys. B157, 387–391 (1979).
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program HU-Berlin, August 2016 26 / 27
Some References
Form factors:M. Karowski and P. Weisz Nucl. Phys. B139 (1978) 445
B. Berg, M. Karowski and P. Weisz Phys. Rev. D19 (1979) 2477
F.A. Smirnov World Scientific 1992
H. Babujian, A. Fring, M. Karowski and A. Zapletal, sine-GordonNucl. Phys. B538 [FS] (1999) 535-586
H. Babujian and M. Karowski Phys. Lett. B411 (1999) 53-57,
Nucl. Phys. B620 (2002) 407; Journ. Phys. A: Math. Gen. 35 (2002)
9081-9104; Phys. Lett. B 575 (2003) 144-150.
H. Babujian, A. Foerster and M. Karowski, SU(N) off-shell Bethe ansatz
hep-th/0611012; Nucl.Phys. B736 (2006) 169-198; SIGMA 2 (2006), 082; J.
Phys. A41 (2008) 275202, Nucl. Phys. B 825 [FS] (2010) 396–425;
O(N) σ- model, arXiv:1308.1459, JHEP 2013:89;
O(N) Gross-Neveu model, arXiv:1510.08784, JHEP 2016:42
H. Babujian and M. Karowski, . . . Constructions of Wightman Functions. . . ,
International Journal of Modern Physics A, 19 (2004) 34-49
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program HU-Berlin, August 2016 27 / 27