The Bootstrap Program for integrable quantum eld theories...

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

Contents

5 References

Contents

5 References

Contents

5 References

Contents

5 References

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 〉

〈 0 | φ(x)φ(y) | 0 〉 = ∑n

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

〈 0 | φ(x)φ(y) | 0 〉 = ∑n

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

〈 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

Integrability

“Yang-Baxter equation” S12S13S23 = S23S13S12

12 3 1 2

“bound state bootstrap equation” S(12)3 Γ(12)12 = Γ

(12)12 S13S23

(12)•

Integrability

“Yang-Baxter equation” S12S13S23 = S23S13S12

12 3 1 2

“bound state bootstrap equation” S(12)3 Γ(12)12 = Γ

(12)12 S13S23

(12)•

Sine-Gordon S-matrix or SUq(2)

Sδγαβ (θ12) =

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∫ ∞

sinh 12 (1− ν)t

sinh 12νt cosh 1

2 tsinh t

Sδγαβ (θ12) =

θ1 θ2

a(θ) = − exp∫ ∞

sinh 12 (1− ν)t

sinh 12νt cosh 1

2 tsinh t

Sδγαβ (θ12) =

θ1 θ2

a(θ) = − exp∫ ∞

sinh 12 (1− ν)t

sinh 12νt cosh 1

2 tsinh t

Sδγαβ (θ12) =

θ1 θ2

a(θ) = − exp∫ ∞

sinh 12 (1− ν)t

sinh 12νt cosh 1

2 tsinh t

Sine-Gordon - Massive Thirring model

exact S-matrix ↔ perturbation theory for the

Lagrangians:

LSG = 12 (∂µφ)2 +

β2(cos βφ− 1)

LMTM = ψ(iγ∂−M)ψ− 12g(ψγµψ)2

ν =β2

8π − β2=

π + 2g

↑Coleman

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. . .

[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. . .

(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 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

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

Example: Sine Gordon

The highest weight SUq(2) ’minimal’ soliton-soliton form factor

F (θ) = exp1

∫ ∞

t sinh t

sinh 12 t (1 + ν)

sinh 12νt cosh 1

(1− cosh t

(1− θ

))[Karowski Weisz (1978)]

Example: SU(N)

The highest weight SU(N) minimal 2-particle form factor

F (θ) = exp

∞∫0

tN sinh t

(1− 1

)t sinh2 t

(1− cosh t

(1− θ

∏k=1

Γ(k − 1

N + 1− 12

)Γ(k − 1

N + 12

)Γ(k − 1

2θiπ

)Γ(k − 1 + 1

2θiπ

) Γ2(k − 1

)Γ2(k − 1

N + 12

)[Babujian Foerster Karowski (2006)]

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

∏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

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

Sine-Gordon

φ (z) =∞

∏k=0

Γ(12kν +

)Γ(12kν + 1

2 −z

)Γ(12 (k + 1) ν + 1

)Γ(12 (k + 1) ν + 1− z

Equation for φ(z)

Example: SU(2)

(iii)←→ φ (z) =1

F (z) F (z + iπ)= Γ

2− z

Sine-Gordon

φ (z) =∞

∏k=0

Γ(12kν +

)Γ(12kν + 1

2 −z

)Γ(12 (k + 1) ν + 1

)Γ(12 (k + 1) ν + 1− z

“Bethe ansatz” state

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) ←→(∫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

The Ansatz

KOα1...αn(θ) =

∫Cθ

dz1 · · ·∫Cθ

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

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

p-Equations

(ii) Crossing

pO(θ1 = θ2 + iπ, . . . , z1 = θ1, . . . ) = pO(θ3, . . . , z2, . . . )

p-Equations

(ii) Crossing

pO(θ1 = θ2 + iπ, . . . , z1 = θ1, . . . ) = pO(θ3, . . . , z2, . . . )

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

∑i=1

zi −1

∑i=1

Massive Thirring model perturbation expansion

〈 0 |ψ(0) | p1, p, p3 〉insss =6

••

p1 p2 p3

+O(g2)

= −ig sinh 12θ23

u(p2) cosh 12θ12 + u(p3) cosh 1

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

∑i=1

zi −1

∑i=1

Massive Thirring model perturbation expansion

〈 0 |ψ(0) | p1, p, p3 〉insss =6

••

p1 p2 p3

+O(g2)

= −ig sinh 12θ23

u(p2) cosh 12θ12 + u(p3) cosh 1

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

(− 1

∫ πν

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

Wightman functions

ϕ +α

βsinh βϕ = 0

Z ϕ = (1 + ν)12πν

sin 12πν

(− 1

∫ πν

sin tdt

α = m2 πν

8π+β2

S-matrix

Wightman functions

ϕ +α

βsinh βϕ = 0

Z ϕ = (1 + ν)12πν

sin 12πν

(− 1

∫ πν

sin tdt

α = m2 πν

8π+β2

S-matrix

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 = −2ν

∆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

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).

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

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