The Bootstrap Program for integrable quantum eld theories...

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 / 26

3 Lectures

I. The general Idea:

S-Matrix , Form Factors, Wightman Functions

II. Sine-Gordon Model

III. SU(N) and O(N) Models

3 Lectures

Contents

1 The “Bootstrap Program”General idea

2 The S-matrix and Integrability

3 Examples: Sine-Gordon Breather + SU(N) S-matrix

4 Form factorsForm factors equationsExamples: Sine-Gordon and SU(N)General form factor formula

“Bethe ansatz” state

Examples:The SU(N) Gross-Neveu model

5 Wightman functionsShort distance behavior

Contents

The “Bootstrap Program”

Construct an integrable quantum field theory explicitly in 3 steps

I. step S-matrix

input: S-matrix equations:

1) general Properties: unitarity, crossing2) symmetry3) integrability: ”Yang-Baxter Equation”4) “bound state bootstrap”5) ‘maximal analyticity’

Construct an integrable quantum field theory explicitly in 3 steps

I. step S-matrix

input: S-matrix equations:

1) general Properties: unitarity, crossing2) symmetry3) integrability: ”Yang-Baxter Equation”4) “bound state bootstrap”5) ‘maximal analyticity’

II. step “Form factors”

〈 0 | φ(x) | p1, . . . , pn 〉in = e−ix(p1+···+pn) F φ (θ1, . . . , θn)

rapidity θ defined byp± = p0 ± p1 = me±θ

input 1) the S-matrix2) form factor equations (i) – (v)3) ‘maximal analyticity’

III. step “Wightman functions”

〈 0 | φ(x)φ(y) | 0 〉 = ∑n

∫〈 0 | φ(x) | n 〉in in〈 n | φ(y) | 0 〉

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

afterwards we compare our exact resultswith perturbation theory of Lagrangian field theories etc.

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

afterwards we compare our exact resultswith perturbation theory of Lagrangian field theories etc.

Integrability

An infinite set of conservation laws:

∂µJµk (x) = 0 .

The charges Qk =∫dxJ0k (x) satisfy the eigenvalue equation

Qk |p1, . . . , pn〉in,out =(∑ p±i

)k |p1, . . . , pn〉in,out .

This implies for scattering

out⟨p′1, . . . , p′n′ |p1, . . . , pn

⟩in= S (n) in

⟨p′1, . . . , p′n′ |p1, . . . , pn

⟩inwhere the n-particle S-matrix factorize

S (n) = ∏ S (2)

because of invariance under parallel shifting of the particle trajectories.Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program Natal, September 2016 8 / 26

Integrability

∂µJµk (x) = 0 .

Qk |p1, . . . , pn〉in,out =(∑ p±i

)k |p1, . . . , pn〉in,out .

out⟨p′1, . . . , p′n′ |p1, . . . , pn

⟩in= S (n) in

⟨p′1, . . . , p′n′ |p1, . . . , pn

S (n) = ∏ S (2)

Integrability

∂µJµk (x) = 0 .

Qk |p1, . . . , pn〉in,out =(∑ p±i

)k |p1, . . . , pn〉in,out .

out⟨p′1, . . . , p′n′ |p1, . . . , pn

⟩in= S (n) in

⟨p′1, . . . , p′n′ |p1, . . . , pn

S (n) = ∏ S (2)

Integrability

If there exists backward scattering the order of the 2-particle S-matriceshas to be specified:

E.g. 3-particle S-matrix:

S(3) = S12S13S23 = S23S13S12

12 3 1 2

“Yang-Baxter equation”

Bound states

S(θ) has a pole at θ = iη, (0 < η < π)⇐⇒ there exist a bound state

“bound state bootstrap equation”

S(12)3 Γ(12)12 = Γ

(12)12 S13S23

(12)•

Bound states

S(θ) has a pole at θ = iη, (0 < η < π)⇐⇒ there exist a bound state

“bound state bootstrap equation”

S(12)3 Γ(12)12 = Γ

(12)12 S13S23

(12)•

Example: Sine-Gordon Breather S-matrix

The sine-Gordon equation ϕ +α

βsin βϕ = 0

Assumptions:unitarity, crossing, a bound state, ‘maximal analyticity’ =⇒

S(θ12) = •

θ1 θ2

=sinh θ12 + i sin πν

sinh θ12 − i sin πν

(θ12 = θ1 − θ2, ν = β2

8π−β2

)unitarity: |S(θ)|2 = S(−θ)S(θ) = 1crossing: S(θ) = S(iπ − θ)The pole belongs to the breather b2 as a breather-breather bound state

βsin βϕ = 0

S(θ12) = •

θ1 θ2

(θ12 = θ1 − θ2, ν = β2

8π−β2

βsin βϕ = 0

S(θ12) = •

θ1 θ2

(θ12 = θ1 − θ2, ν = β2

8π−β2

Example: SU(N) S-matrix

Particles α, β, γ, δ = 1, . . . ,N ↔ vector representation of SU(N)

Sδγαβ (θ12) =

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

)Γ(1− 1

N + θ2πi

)Γ(1 + θ

)Γ(1− 1

N −θ

)[Berg Karowski Kurak Weisz 1978]

Example: SU(N) S-matrix

Particles α, β, γ, δ = 1, . . . ,N ↔ vector representation of SU(N)

Sδγαβ (θ12) =

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

)Γ(1− 1

N + θ2πi

)Γ(1 + θ

)Γ(1− 1

N −θ

)[Berg Karowski Kurak Weisz 1978]

Form factors

Definition

Let O(x) be a local operator

〈 0 | O(x) | p1, . . . , pn 〉inα1...αn= FOα1...αn

(θ1, . . . , θn) e−ix ∑ pi

FOα (θ) = form factor (co-vector valued function)

αi ∈ all types of particles

LSZ-assumptions+ ’maximal analyticity’

=⇒ Properties of form factors

Form factors

Definition

Let O(x) be a local operator

〈 0 | O(x) | p1, . . . , pn 〉inα1...αn= FOα1...αn

(θ1, . . . , θn) e−ix ∑ pi

FOα (θ) = form factor (co-vector valued function)

αi ∈ all types of particles

LSZ-assumptions+ ’maximal analyticity’

=⇒ Properties of form factors

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... ...

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. . .

(σOα1= statistics factor, C = charge conjugation matrix)

[Karowski Weisz (1978)] [Smirnov (World Scientific 1992)]

(i) Watson’s equation

FO...ij ...(. . . , θi , θj , . . . ) = FO...ji ...(. . . , θj , θi , . . . ) Sij (θi − θj ) 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. . .

(σOα1= statistics factor, C = charge conjugation matrix)

(iii) Annihilation recursion relation

θ12=iπFO12...n(θ1, . .) = C12 FO3...n(θ3, . .)

(1− σO2 S2n . . . S23

θ12=iπ

= O...

− σO2

(iv) Bound state form factors

Resθ12=iη

FO123...n(θ) =√

2FO(12)3...n(θ(12), θ′) Γ(12)12

Resθ12=iη

O...(v) Lorentz invariance (with s = “spin” of O)

FO1...n(θ1 + u, . . . , θn + u) = esu FO1...n(θ1, . . . , θn)

θ12=iπFO12...n(θ1, . .) = C12 FO3...n(θ3, . .)

(1− σO2 S2n . . . S23

θ12=iπ

= O...

− σO2

Resθ12=iη

FO123...n(θ) =√

2FO(12)3...n(θ(12), θ′) Γ(12)12

Resθ12=iη

θ12=iπFO12...n(θ1, . .) = C12 FO3...n(θ3, . .)

(1− σO2 S2n . . . S23

θ12=iπ

= O...

− σO2

Resθ12=iη

FO123...n(θ) =√

2FO(12)3...n(θ(12), θ′) Γ(12)12

Resθ12=iη

2-particle form factor

”Watson’s equations”

F (θ) = F (−θ) S (θ)F (iπ − θ) = F (iπ + θ)

⇒ unique minimal solution [Karowski Weisz (1978)]

minimal: F (θ) analytic in 0 ≤ Im θ ≤ π

2-particle form factor

”Watson’s equations”

F (θ) = F (−θ) S (θ)F (iπ − θ) = F (iπ + θ)

⇒ unique minimal solution [Karowski Weisz (1978)]

minimal: F (θ) analytic in 0 ≤ Im θ ≤ π

Examples: Sine-Gordon + SU(N)

The Sine-Gordon (breather) minimal form factor

F SG (θ) = exp∫ ∞

t sinh t

(12 + ν

cosh 12 t

)cosh t

(1− θ

)[ Karowski Weisz (1978)]

The highest weight SU(N) minimal form factor

F SU(N) (θ) = c exp

∞∫0

tN sinh t

(1− 1

)t sinh2 t

(1− cosh t

(1− θ

[Babujian Foerster Karowski (2006)]

Examples: Sine-Gordon + SU(N)

The Sine-Gordon (breather) minimal form factor

F SG (θ) = exp∫ ∞

t sinh t

(12 + ν

cosh 12 t

)cosh t

(1− θ

)[ Karowski Weisz (1978)]

The highest weight SU(N) minimal form factor

F SU(N) (θ) = c exp

∞∫0

tN sinh t

(1− 1

)t sinh2 t

(1− cosh t

(1− θ

[Babujian Foerster Karowski (2006)]

General form factor formula

FOα1...αn(θ1, . . . , θn) = KOα1 ...αn

(θ) ∏1≤i<j≤n

F (θij )

”Nested 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

∏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 18 / 26

FOα1...αn(θ1, . . . , θn) = KOα1 ...αn

(θ) ∏1≤i<j≤n

F (θij )

KOα1...αn(θ) =

∫Cθ

dz1 · · ·∫Cθ

h(θ, z) =n

∏i=1

∏j=1

τ(zi − zj ) , τ(z) =1

φ(z)φ(−z)

FOα1...αn(θ1, . . . , θn) = KOα1 ...αn

(θ) ∏1≤i<j≤n

F (θij )

KOα1...αn(θ) =

∫Cθ

dz1 · · ·∫Cθ

h(θ, z) =n

∏i=1

∏j=1

τ(zi − zj ) , τ(z) =1

φ(z)φ(−z)

FOα1...αn(θ1, . . . , θn) = KOα1 ...αn

(θ) ∏1≤i<j≤n

F (θij )

KOα1...αn(θ) =

∫Cθ

dz1 · · ·∫Cθ

h(θ, z) =n

∏i=1

∏j=1

τ(zi − zj ) , τ(z) =1

φ(z)φ(−z)

Equation for φ(z)

Example: SU(2)

(iii)←→ φ (z) =1

F (z) F (z + iπ)= Γ

2− z

Example: SU(2) or sine-Gordon

Ψα(θ, z) = (ΩC (θ, zm) . . .C (θ, z1))α1...αn

S-matrix

• •

α1 αn

θ1 θn

...(1 ≤ αi ≤ 2)

If rank > 1⇒ nested Bethe Ansatz

⇒ Bethe Ansatz of level 1, 2, . . . ,

rank(SU(N)) = N − 1rank(O(N)) = [N/2]

S-matrix

• •

α1 αn

θ1 θn

...(1 ≤ αi ≤ 2)

S-matrix

• •

α1 αn

θ1 θn

...(1 ≤ αi ≤ 2)

Integration contour for SU(N)

(ii) ←→

• 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 ansatz

KOα1...αn(θ) =

∫Cθ

dz1 · · ·∫Cθ

transforms the complicated matrix equations intosimple equations for the scalar functions pO(θ, z)

Example: The SU(N) Gross-Neveu model

Lagrangian

∑α=1

ψα iγ∂ ψα +g2

∑α=1

ψαψα

∑α=1

ψαγ5ψα

p-function for the field ψ(x)

pψ(θ, z) = exp1

∑i=1

zi −(

1− 1

∑i=1

)[Babujian Foerster Karowski (2006,2008,2009)]

n = 1〈0|ψα(0)|θ〉β = δαβe

12 (1−

1N )θ

→ the particles are anyons with statistics σ = e iπ(1−1N )θ

n = 3 has been checked in 1/N expansion.Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program Natal, September 2016 23 / 26

Lagrangian

∑α=1

ψαψα

∑α=1

ψαγ5ψα

pψ(θ, z) = exp1

∑i=1

zi −(

1− 1

∑i=1

n = 1〈0|ψα(0)|θ〉β = δαβe

12 (1−

1N )θ

Lagrangian

∑α=1

ψαψα

∑α=1

ψαγ5ψα

pψ(θ, z) = exp1

∑i=1

zi −(

1− 1

∑i=1

n = 1〈0|ψα(0)|θ〉β = δαβe

12 (1−

1N )θ

Lagrangian

∑α=1

ψαψα

∑α=1

ψαγ5ψα

pψ(θ, z) = exp1

∑i=1

zi −(

1− 1

∑i=1

n = 1〈0|ψα(0)|θ〉β = δαβe

12 (1−

1N )θ

Wightman functions

Example: The sinh-Gordon model

ϕ +α

βsinh βϕ = 0

S-matrix

S(θ) =sinh θ + i sin πν

sinh θ − i sin πν

ν = − β2

8π+β2

Wightman functions

Example: The sinh-Gordon model

ϕ +α

βsinh βϕ = 0

S-matrix

S(θ) =sinh θ + i sin πν

sinh θ − i sin πν

ν = − β2

8π+β2

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” ∆

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” ∆

Short distance behavior

“Dimension” ∆ for sinh-Gordon1- and 1+2-particle intermediate state contributions

0 1 21-particle

1+2-particlewhere B = 2β2

8π+β2

The Bootstrap Program for integrable quantum eld theories...

Documents

Transcript of The Bootstrap Program for integrable quantum eld theories...