The Bootstrap Program for integrable quantum eld theories...
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The ”Bootstrap Program”for integrable quantum field theories in 1+1 dimensions
H. Babujian, A. Foerster, and M. Karowski
Natal, September 2016
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program Natal, September 2016 1 / 29
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3 Lectures
I. The general Idea:
S-Matrix , Form Factors, Wightman Functions
II. Sine-Gordon Model
III. SU(N) and O(N) Models
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program Natal, September 2016 2 / 29
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Contents
1 The “Bootstrap Program”General idea
2 Examples:Sine-Gordon modelMassive Thirring model
3 Form factorsForm factor definitionExamples: Sine GordonGeneral form factor formula
“Bethe ansatz” state
4 Field equation and Wightman functionsQuantum field equationShort distance behavior
5 References
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program Natal, September 2016 3 / 29
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Contents
1 The “Bootstrap Program”General idea
2 Examples:Sine-Gordon modelMassive Thirring model
3 Form factorsForm factor definitionExamples: Sine GordonGeneral form factor formula
“Bethe ansatz” state
4 Field equation and Wightman functionsQuantum field equationShort distance behavior
5 References
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program Natal, September 2016 3 / 29
![Page 5: The Bootstrap Program for integrable quantum eld theories ...users.physik.fu-berlin.de/~kamecke/t/v162.pdfThe "Bootstrap Program" for integrable quantum eld theories in 1+1 dimensions](https://reader033.fdocuments.in/reader033/viewer/2022042404/5f1be49deb4c9c0cf9663575/html5/thumbnails/5.jpg)
Contents
1 The “Bootstrap Program”General idea
2 Examples:Sine-Gordon modelMassive Thirring model
3 Form factorsForm factor definitionExamples: Sine GordonGeneral form factor formula
“Bethe ansatz” state
4 Field equation and Wightman functionsQuantum field equationShort distance behavior
5 References
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program Natal, September 2016 3 / 29
![Page 6: The Bootstrap Program for integrable quantum eld theories ...users.physik.fu-berlin.de/~kamecke/t/v162.pdfThe "Bootstrap Program" for integrable quantum eld theories in 1+1 dimensions](https://reader033.fdocuments.in/reader033/viewer/2022042404/5f1be49deb4c9c0cf9663575/html5/thumbnails/6.jpg)
Contents
1 The “Bootstrap Program”General idea
2 Examples:Sine-Gordon modelMassive Thirring model
3 Form factorsForm factor definitionExamples: Sine GordonGeneral form factor formula
“Bethe ansatz” state
4 Field equation and Wightman functionsQuantum field equationShort distance behavior
5 References
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program Natal, September 2016 3 / 29
![Page 7: The Bootstrap Program for integrable quantum eld theories ...users.physik.fu-berlin.de/~kamecke/t/v162.pdfThe "Bootstrap Program" for integrable quantum eld theories in 1+1 dimensions](https://reader033.fdocuments.in/reader033/viewer/2022042404/5f1be49deb4c9c0cf9663575/html5/thumbnails/7.jpg)
Contents
1 The “Bootstrap Program”General idea
2 Examples:Sine-Gordon modelMassive Thirring model
3 Form factorsForm factor definitionExamples: Sine GordonGeneral form factor formula
“Bethe ansatz” state
4 Field equation and Wightman functionsQuantum field equationShort distance behavior
5 References
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program Natal, September 2016 3 / 29
![Page 8: The Bootstrap Program for integrable quantum eld theories ...users.physik.fu-berlin.de/~kamecke/t/v162.pdfThe "Bootstrap Program" for integrable quantum eld theories in 1+1 dimensions](https://reader033.fdocuments.in/reader033/viewer/2022042404/5f1be49deb4c9c0cf9663575/html5/thumbnails/8.jpg)
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 Natal, September 2016 4 / 29
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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 Natal, September 2016 4 / 29
![Page 10: The Bootstrap Program for integrable quantum eld theories ...users.physik.fu-berlin.de/~kamecke/t/v162.pdfThe "Bootstrap Program" for integrable quantum eld theories in 1+1 dimensions](https://reader033.fdocuments.in/reader033/viewer/2022042404/5f1be49deb4c9c0cf9663575/html5/thumbnails/10.jpg)
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 Natal, September 2016 4 / 29
![Page 11: The Bootstrap Program for integrable quantum eld theories ...users.physik.fu-berlin.de/~kamecke/t/v162.pdfThe "Bootstrap Program" for integrable quantum eld theories in 1+1 dimensions](https://reader033.fdocuments.in/reader033/viewer/2022042404/5f1be49deb4c9c0cf9663575/html5/thumbnails/11.jpg)
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 Natal, September 2016 4 / 29
![Page 12: The Bootstrap Program for integrable quantum eld theories ...users.physik.fu-berlin.de/~kamecke/t/v162.pdfThe "Bootstrap Program" for integrable quantum eld theories in 1+1 dimensions](https://reader033.fdocuments.in/reader033/viewer/2022042404/5f1be49deb4c9c0cf9663575/html5/thumbnails/12.jpg)
Example: 1 type of particles + a bound state
no backward scattering =⇒ S-matrix = c-number
Assumptions:
unitarity: |S(θ)|2 = S(−θ)S(θ) = 1crossing: S(θ) = S(iπ − θ)‘maximal analyticity’
=⇒ S(θ12) = •
@@
@@
θ1 θ2
=sinh θ12 + i sin πν
sinh θ12 − i sin πν
(θ12 = θ1 − θ2, p± = p0 ± p1 = me±θ
)= S-matrix of sine-Gordon breather b1
The pole belongs to the breather b2 as a breather-breather bound state
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program Natal, September 2016 5 / 29
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Example: 1 type of particles + a bound state
no backward scattering =⇒ S-matrix = c-number
Assumptions:
unitarity: |S(θ)|2 = S(−θ)S(θ) = 1crossing: S(θ) = S(iπ − θ)‘maximal analyticity’
=⇒ S(θ12) = •
@@
@@
θ1 θ2
=sinh θ12 + i sin πν
sinh θ12 − i sin πν
(θ12 = θ1 − θ2, p± = p0 ± p1 = me±θ
)= S-matrix of sine-Gordon breather b1
The pole belongs to the breather b2 as a breather-breather bound state
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program Natal, September 2016 5 / 29
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Example: 1 type of particles + a bound state
no backward scattering =⇒ S-matrix = c-number
Assumptions:
unitarity: |S(θ)|2 = S(−θ)S(θ) = 1crossing: S(θ) = S(iπ − θ)‘maximal analyticity’
=⇒ S(θ12) = •
@@
@@
θ1 θ2
=sinh θ12 + i sin πν
sinh θ12 − i sin πν
(θ12 = θ1 − θ2, p± = p0 ± p1 = me±θ
)= S-matrix of sine-Gordon breather b1
The pole belongs to the breather b2 as a breather-breather bound state
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program Natal, September 2016 5 / 29
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More details
unitarity:
S(θ21)S(θ12) = 1 :@@@@
=
1 2 1 2
crossing:
S(θ1 − θ2) = C−1 S(θ2 + iπ − θ1)C
@@
@@
1 2
=
AAAA
1 2
C = θ θ + iπ , C−1 =
θ θ − iπ
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program Natal, September 2016 6 / 29
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More details
unitarity:
S(θ21)S(θ12) = 1 :@@@@
=
1 2 1 2
crossing:
S(θ1 − θ2) = C−1 S(θ2 + iπ − θ1)C
@@
@@
1 2
=
AAAA
1 2
C = θ θ + iπ , C−1 =
θ θ − iπ
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program Natal, September 2016 6 / 29
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The classical sine-Gordon model
is given by the wave equation
ϕ(t, x) +α
βsin βϕ(t, x) = 0.
Perturbation theory in terms of Feynman graphs agreeswith the expansion of the exact S-matrix
S(θ) =sinh θ + i sin πν
sinh θ − i sin πν
= 1 + 2iπν
sinh θ− 2π2 ν2
sinh2 θ+O
(ν3)
if
ν =β2
8π − β2
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program Natal, September 2016 7 / 29
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The classical sine-Gordon model
is given by the wave equation
ϕ(t, x) +α
βsin βϕ(t, x) = 0.
Perturbation theory in terms of Feynman graphs agreeswith the expansion of the exact S-matrix
S(θ) =sinh θ + i sin πν
sinh θ − i sin πν
= 1 + 2iπν
sinh θ− 2π2 ν2
sinh2 θ+O
(ν3)
if
ν =β2
8π − β2
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program Natal, September 2016 7 / 29
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fermion s and anti-fermion s
with backward scattering
Sδγαβ (θ12) =
•
@@
@@
α β
γδ
θ1 θ2
α, β, γ, δ = s, s
S ssss (θ) = a(θ), S ss
ss (θ) = b(θ), S ssss (θ) = c(θ)
[A.B. Zamolodchikov (1977)]
crossing + unitarity + extra assump. → a, b, c
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program Natal, September 2016 8 / 29
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fermion s and anti-fermion s
with backward scattering
Sδγαβ (θ12) =
•
@@
@@
α β
γδ
θ1 θ2
α, β, γ, δ = s, s
S ssss (θ) = a(θ), S ss
ss (θ) = b(θ), S ssss (θ) = c(θ)
[A.B. Zamolodchikov (1977)]
crossing + unitarity + extra assump. → a, b, c
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program Natal, September 2016 8 / 29
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fermion s and anti-fermion s
with backward scattering
Sδγαβ (θ12) =
•
@@
@@
α β
γδ
θ1 θ2
α, β, γ, δ = s, s
S ssss (θ) = a(θ), S ss
ss (θ) = b(θ), S ssss (θ) = c(θ)
[A.B. Zamolodchikov (1977)]
crossing + unitarity + extra assump. → a, b, c
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program Natal, September 2016 8 / 29
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fermion s and anti-fermion s
with backward scattering
Sδγαβ (θ12) =
•
@@
@@
α β
γδ
θ1 θ2
α, β, γ, δ = s, s
S ssss (θ) = a(θ), S ss
ss (θ) = b(θ), S ssss (θ) = c(θ)
[A.B. Zamolodchikov (1977)]
crossing + unitarity + extra assump. → a, b, c
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program Natal, September 2016 8 / 29
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SUq(2) S-matrix
Yang-Baxter+ crossing + unitarity
=⇒
c(θ) = b(θ)sinh iπ/ν
sinh (iπ − θ) /ν, b(θ) = a(iπ − θ), |a| = |b± c | = 1
a(θ) = − exp∫ ∞
0
dt
t
sinh 12 (1− ν)t
sinh 12νt cosh 1
2 tsinh t
θ
iπ
q = −e−iπ/ν
[M. Karowski, H.J. Thun, T.T. Truong and P. Weisz 1977]
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program Natal, September 2016 9 / 29
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Massive Thirring Lagrangian
LMTM = ψ(iγ∂−M)ψ− 12g(ψγµψ)2
perturbation expansion ←→ the exact SUq(2) S-matrix
ifν =
π
π + 2g
Coleman:sine-Gordon soliton ←→ massive Thirring fermionsine-Gordon breathers ←→ massive Thirring bound states
ν =β2
8π − β2=
π
π + 2g
↑Coleman
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program Natal, September 2016 10 / 29
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Massive Thirring Lagrangian
LMTM = ψ(iγ∂−M)ψ− 12g(ψγµψ)2
perturbation expansion ←→ the exact SUq(2) S-matrix
ifν =
π
π + 2g
Coleman:sine-Gordon soliton ←→ massive Thirring fermionsine-Gordon breathers ←→ massive Thirring bound states
ν =β2
8π − β2=
π
π + 2g
↑Coleman
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program Natal, September 2016 10 / 29
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Sine-Gordon ≡ Massive Thirring
This equivalence is also proved in the Bootstrap program:
Using “bound state bootstrap equation”
S(12)3 Γ(12)12 = Γ
(12)12 S13S23
@@
@@
1 23
(12)
• =
@@
@@@
12 3
(12)•
(i) s + s → (ss) = b1 : massiveThirring −→ sine-Gordon S-matrix
(ii) s + b1 → (sb1) = s : sine-Gordon −→ massiveThirring S-matrix
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program Natal, September 2016 11 / 29
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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
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program Natal, September 2016 12 / 29
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2-particle form factor
〈 0 | O(0) | p1, p2〉in/out = F((p1 + p2)
2 ± iε)= F (±θ12)
where p1p2 = m2 cosh θ12.
”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 Natal, September 2016 13 / 29
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2-particle form factor
〈 0 | O(0) | p1, p2〉in/out = F((p1 + p2)
2 ± iε)= F (±θ12)
where p1p2 = m2 cosh θ12.
”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 Natal, September 2016 13 / 29
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Example: Sine Gordon
sine-Gordon breather-breather form factor
Fbb(θ) = exp∫ ∞
0
dt
t sinh t
cosh t(12 + ν
)− cosh 1
2 t
cosh 12 t
cosh t
(1− θ
iπ
)
sine-Gordon soliton-soliton
Fss(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)]this the highest weight SUq(2) ’minimal’ form factor
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program Natal, September 2016 14 / 29
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Example: Sine Gordon
sine-Gordon breather-breather form factor
Fbb(θ) = exp∫ ∞
0
dt
t sinh t
cosh t(12 + ν
)− cosh 1
2 t
cosh 12 t
cosh t
(1− θ
iπ
)
sine-Gordon soliton-soliton
Fss(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)]this the highest weight SUq(2) ’minimal’ form factor
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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 Natal, September 2016 15 / 29
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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 Natal, September 2016 15 / 29
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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 Natal, September 2016 15 / 29
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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 Natal, September 2016 15 / 29
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Equation for φ(z)
(iii) ←→ φ (z) =1
F (z) F (z + iπ)
Examples: SU(2) :
φSU(2) (z) = Γ( z
2πi
)Γ(
1
2− z
2πi
)
SUq(2) : sine-Gordon solitons
φSUq(2) (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 Natal, September 2016 16 / 29
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Equation for φ(z)
(iii) ←→ φ (z) =1
F (z) F (z + iπ)
Examples: SU(2) :
φSU(2) (z) = Γ( z
2πi
)Γ(
1
2− z
2πi
)
SUq(2) : sine-Gordon solitons
φSUq(2) (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 Natal, September 2016 16 / 29
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Equation for φ(z)
(iii) ←→ φ (z) =1
F (z) F (z + iπ)
Examples: SU(2) :
φSU(2) (z) = Γ( z
2πi
)Γ(
1
2− z
2πi
)
SUq(2) : sine-Gordon solitons
φSUq(2) (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 Natal, September 2016 16 / 29
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“Bethe ansatz” state
Example: SU(2) or SUq(2) ≡ 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)
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Integration contour for SU(N)
• θn − 2πi
•θn − 2πi 1N
• θn
• θn + 2πi(1− 1N )
. . .
• θ2 − 2πi
•θ2 − 2πi 1N
• θ2
• θ2 + 2πi(1− 1N )
• θ1 − 2πi
•θ1 − 2πi 1N
• θ1
• θ1 + 2πi(1− 1N )
-
-
Figure: The integration contour Cθ. The bullets refer to poles of the integrand.
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General form factor formula
for sine Gordon breathers
FO(θ1, . . . , θn) = KO(θ) ∏1≤i<j≤n
Fbb(θij )
KOnm(θ) =∫Cθ
dz1 · · ·∫Cθ
dzm h(θ, z) pO(θ, z)Ψ(θ, z)
(1)
Ψ(θ, z) = Bethe state = ∏ S(θi − zj )
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)
φb(z) =S(z)
Fbb(z)Fbb(z + 1)= 1 +
i sin πν
sinh z
and∫Cθ
dz · · · = ∑ Resz=θi
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General form factor formula
for sine Gordon breathers
FO(θ1, . . . , θn) = KO(θ) ∏1≤i<j≤n
Fbb(θij )
KOnm(θ) =∫Cθ
dz1 · · ·∫Cθ
dzm h(θ, z) pO(θ, z)Ψ(θ, z)
(1)
Ψ(θ, z) = Bethe state = ∏ S(θi − zj )
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)
φb(z) =S(z)
Fbb(z)Fbb(z + 1)= 1 +
i sin πν
sinh z
and∫Cθ
dz · · · = ∑ Resz=θi
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General form factor formula
for sine Gordon breathers
FO(θ1, . . . , θn) = KO(θ) ∏1≤i<j≤n
Fbb(θij )
KOnm(θ) =∫Cθ
dz1 · · ·∫Cθ
dzm h(θ, z) pO(θ, z)Ψ(θ, z)
(1)
Ψ(θ, z) = Bethe state = ∏ S(θi − zj )
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)
φb(z) =S(z)
Fbb(z)Fbb(z + 1)= 1 +
i sin πν
sinh z
and∫Cθ
dz · · · = ∑ Resz=θi
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Example: Exponential field
O(x) = : e iγϕ(x) :
with : · · · := normal ordering.
We derived all form factors [Babujian Karowski (2002)]
F e iγϕ(θ) = Nn ∏
1≤i<j≤nFbb(θij )
n
∑m=0
qn−2m(−1)mKnm(θ)
where N =√Z ϕ β
2πν and q = exp(iπνγ/β)and Knm(θ) is given by (1) for p = 1
Z ϕ = (1 + ν)12πν
sin 12πν
exp
(− 1
π
∫ πν
0
t
sin tdt
)is the finite wave function renormalization constant[Karowski Weizs (1978)]
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Example: Exponential field
O(x) = : e iγϕ(x) :
with : · · · := normal ordering.
We derived all form factors [Babujian Karowski (2002)]
F e iγϕ(θ) = Nn ∏
1≤i<j≤nFbb(θij )
n
∑m=0
qn−2m(−1)mKnm(θ)
where N =√Z ϕ β
2πν and q = exp(iπνγ/β)and Knm(θ) is given by (1) for p = 1
Z ϕ = (1 + ν)12πν
sin 12πν
exp
(− 1
π
∫ πν
0
t
sin tdt
)is the finite wave function renormalization constant[Karowski Weizs (1978)]
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Example: Exponential field
O(x) = : e iγϕ(x) :
with : · · · := normal ordering.
We derived all form factors [Babujian Karowski (2002)]
F e iγϕ(θ) = Nn ∏
1≤i<j≤nFbb(θij )
n
∑m=0
qn−2m(−1)mKnm(θ)
where N =√Z ϕ β
2πν and q = exp(iπνγ/β)and Knm(θ) is given by (1) for p = 1
Z ϕ = (1 + ν)12πν
sin 12πν
exp
(− 1
π
∫ πν
0
t
sin tdt
)is the finite wave function renormalization constant[Karowski Weizs (1978)]
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Example: the field
the form factors of ϕ(x) are obtained from F e iγϕ(θ) by
F ϕ(θ) = −i ∂
∂γF e iγϕ
(θ)
∣∣∣∣γ=0
We proved the quantum field equation
ϕ(t, x) +α
β: sin βϕ(t, x) := 0
for all matrix elements. [Babujian Karowski (2002)]
The bare and the renormilized masses are related by
α = m2bare =
πν
sin πνm2
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Example: the field
the form factors of ϕ(x) are obtained from F e iγϕ(θ) by
F ϕ(θ) = −i ∂
∂γF e iγϕ
(θ)
∣∣∣∣γ=0
We proved the quantum field equation
ϕ(t, x) +α
β: sin βϕ(t, x) := 0
for all matrix elements. [Babujian Karowski (2002)]
The bare and the renormilized masses are related by
α = m2bare =
πν
sin πνm2
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Example: the field
the form factors of ϕ(x) are obtained from F e iγϕ(θ) by
F ϕ(θ) = −i ∂
∂γF e iγϕ
(θ)
∣∣∣∣γ=0
We proved the quantum field equation
ϕ(t, x) +α
β: sin βϕ(t, x) := 0
for all matrix elements. [Babujian Karowski (2002)]
The bare and the renormilized masses are related by
α = m2bare =
πν
sin πνm2
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Wightman functions
Example: the two-point function
w(x) = 〈 0 | O(x)O(0) | 0 〉 .
Inserting a complete set of states
w(x) = 1 +∞
∑n=1
1
n!
∫dθ1 . . .
∫dθne
−ix ∑ pign(θ) .
where
gn(θ) =1
(4π)n〈 0 | O(0) |θ1, . . . , θn 〉 〈θn, . . . , θ1 | O(0) | 0 〉
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Wightman functions
We use a cumulant transformation and write
lnw(x) =∞
∑n=1
1
n!
∫dθ1 . . .
∫dθne
−ix ∑ pihn(θ)
where the g ’s and h’s are related by
g
1. . .
n
= h
1. . .
n
+n
∑i=1
h
. .h
i
+ · · · + h
1
. . . h
n
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Example: The sinh-Gordon model
ϕ +α
βsinh βϕ = 0
is obtained from sine-Gordon by β→ iβ =⇒ ν < 0.
Short distances behavior for O(x) = exp βϕ(x)
w(x) ∼(√−x2
)−4∆for x → 0
“Dimension” ∆
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Example: The sinh-Gordon model
ϕ +α
βsinh βϕ = 0
is obtained from sine-Gordon by β→ iβ =⇒ ν < 0.
Short distances behavior for O(x) = exp βϕ(x)
w(x) ∼(√−x2
)−4∆for x → 0
“Dimension” ∆
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Wightman functions
The two-point function
w(x) = 〈 0 | O(x)O′(0) | 0 〉
Example: The sinh-Gordon model
ϕ +α
βsinh βϕ = 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 Natal, September 2016 25 / 29
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Wightman functions
The two-point function
w(x) = 〈 0 | O(x)O′(0) | 0 〉
Example: The sinh-Gordon model
ϕ +α
βsinh βϕ = 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 Natal, September 2016 25 / 29
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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
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[H. Babujian and M. Karowski (2004)]
∆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
)
B = 2β2
8π+β2 = −2ν
Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program Natal, September 2016 27 / 29
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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 Natal, September 2016 28 / 29
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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. ZapletalNucl. 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, Journal of High Energy Physics 2013:89 ;O(N) Gross-Neveu model, in preparation
H. Babujian and M. Karowski, . . . Constructions of Wightman Functions. . . ,
International Journal of Modern Physics A, 19 (2004) 34-49Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program Natal, September 2016 29 / 29