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 / 29

3 Lectures

I. The general Idea:

S-Matrix , Form Factors, Wightman Functions

II. Sine-Gordon Model

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

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

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 〉

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

Assumptions:

=⇒ S(θ12) = •

θ1 θ2

(θ12 = θ1 − θ2, p± = p0 ± p1 = me±θ

Assumptions:

=⇒ S(θ12) = •

θ1 θ2

(θ12 = θ1 − θ2, p± = p0 ± p1 = me±θ

More details

unitarity:

S(θ21)S(θ12) = 1 :@@@@

1 2 1 2

crossing:

S(θ1 − θ2) = C−1 S(θ2 + iπ − θ1)C

C = θ θ + iπ , C−1 =

θ θ − iπ

More details

unitarity:

S(θ21)S(θ12) = 1 :@@@@

1 2 1 2

crossing:

S(θ1 − θ2) = C−1 S(θ2 + iπ − θ1)C

C = θ θ + iπ , C−1 =

θ θ − iπ

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

ν =β2

8π − β2

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

ν =β2

8π − β2

fermion s and anti-fermion s

with backward scattering

Sδγαβ (θ12) =

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

Sδγαβ (θ12) =

θ1 θ2

α, β, γ, δ = s, s

Sδγαβ (θ12) =

θ1 θ2

α, β, γ, δ = s, s

Sδγαβ (θ12) =

θ1 θ2

α, β, γ, δ = s, s

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∫ ∞

sinh 12 (1− ν)t

sinh 12νt cosh 1

2 tsinh t

q = −e−iπ/ν

[M. Karowski, H.J. Thun, T.T. Truong and P. Weisz 1977]

Massive Thirring Lagrangian

LMTM = ψ(iγ∂−M)ψ− 12g(ψγµψ)2

perturbation expansion ←→ the exact SUq(2) S-matrix

ifν =

π + 2g

Coleman:sine-Gordon soliton ←→ massive Thirring fermionsine-Gordon breathers ←→ massive Thirring bound states

ν =β2

8π − β2=

π + 2g

↑Coleman

Massive Thirring Lagrangian

LMTM = ψ(iγ∂−M)ψ− 12g(ψγµψ)2

perturbation expansion ←→ the exact SUq(2) S-matrix

ifν =

π + 2g

Coleman:sine-Gordon soliton ←→ massive Thirring fermionsine-Gordon breathers ←→ massive Thirring bound states

ν =β2

8π − β2=

π + 2g

↑Coleman

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

(12)•

(i) s + s → (ss) = b1 : massiveThirring −→ sine-Gordon S-matrix

(ii) s + b1 → (sb1) = s : sine-Gordon −→ massiveThirring S-matrix

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

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

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

Example: Sine Gordon

sine-Gordon breather-breather form factor

Fbb(θ) = exp∫ ∞

t sinh t

cosh t(12 + ν

)− cosh 1

cosh 12 t

cosh t

(1− θ

sine-Gordon soliton-soliton

Fss(θ) = exp1

∫ ∞

t sinh t

sinh 12 t (1 + ν)

sinh 12νt cosh 1

(1− cosh t

(1− θ

))[Karowski Weisz (1978)]this the highest weight SUq(2) ’minimal’ form factor

Example: Sine Gordon

sine-Gordon breather-breather form factor

Fbb(θ) = exp∫ ∞

t sinh t

cosh t(12 + ν

)− cosh 1

cosh 12 t

cosh t

(1− θ

sine-Gordon soliton-soliton

Fss(θ) = exp1

∫ ∞

t sinh t

sinh 12 t (1 + ν)

sinh 12νt cosh 1

(1− cosh t

(1− θ

))[Karowski Weisz (1978)]this the highest weight SUq(2) ’minimal’ form factor

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 Natal, September 2016 15 / 29

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)

(iii) ←→ φ (z) =1

F (z) F (z + iπ)

Examples: SU(2) :

φSU(2) (z) = Γ( z

2− z

SUq(2) : sine-Gordon solitons

φSUq(2) (z) =∞

∏k=0

Γ(12kν +

)Γ(12kν + 1

2 −z

)Γ(12 (k + 1) ν + 1

)Γ(12 (k + 1) ν + 1− z

)Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program Natal, September 2016 16 / 29

Equation for φ(z)

(iii) ←→ φ (z) =1

F (z) F (z + iπ)

Examples: SU(2) :

φSU(2) (z) = Γ( z

2− z

φSUq(2) (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)

(iii) ←→ φ (z) =1

F (z) F (z + iπ)

Examples: SU(2) :

φSU(2) (z) = Γ( z

2− z

φSUq(2) (z) =∞

∏k=0

Γ(12kν +

)Γ(12kν + 1

2 −z

)Γ(12 (k + 1) ν + 1

)Γ(12 (k + 1) ν + 1− z

Example: SU(2) or SUq(2) ≡ sine-Gordon

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

S-matrix

• •

α1 αn

θ1 θn

...(1 ≤ αi ≤ 2)

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.

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)

Ψ(θ, z) = Bethe state = ∏ S(θi − zj )

h(θ, z) =n

∏i=1

∏j=1

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

FO(θ1, . . . , θn) = KO(θ) ∏1≤i<j≤n

Fbb(θij )

KOnm(θ) =∫Cθ

dz1 · · ·∫Cθ

h(θ, z) =n

∏i=1

∏j=1

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

FO(θ1, . . . , θn) = KO(θ) ∏1≤i<j≤n

Fbb(θij )

KOnm(θ) =∫Cθ

dz1 · · ·∫Cθ

h(θ, z) =n

∏i=1

∏j=1

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

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 )

∑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

(− 1

∫ πν

sin tdt

)is the finite wave function renormalization constant[Karowski Weizs (1978)]

O(x) = : e iγϕ(x) :

∑m=0

where N =√Z ϕ β

Z ϕ = (1 + ν)12πν

sin 12πν

(− 1

∫ πν

sin tdt

O(x) = : e iγϕ(x) :

∑m=0

where N =√Z ϕ β

Z ϕ = (1 + ν)12πν

sin 12πν

(− 1

∫ πν

sin tdt

Example: the field

the form factors of ϕ(x) are obtained from F e iγϕ(θ) by

F ϕ(θ) = −i ∂

∂γF e iγϕ

∣∣∣∣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 =

sin πνm2

Example: the field

F ϕ(θ) = −i ∂

∂γF e iγϕ

∣∣∣∣γ=0

ϕ(t, x) +α

α = m2bare =

sin πνm2

Example: the field

F ϕ(θ) = −i ∂

∂γF e iγϕ

∣∣∣∣γ=0

ϕ(t, x) +α

α = m2bare =

sin πνm2

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

∫dθ1 . . .

∫dθne

−ix ∑ pign(θ) .

gn(θ) =1

(4π)n〈 0 | O(0) |θ1, . . . , θn 〉〈θn, . . . , θ1 | O(0) | 0 〉

Wightman functions

We use a cumulant transformation and write

lnw(x) =∞

∑n=1

∫dθ1 . . .

∫dθne

−ix ∑ pihn(θ)

where the g ’s and h’s are related by

1. . .

∑i=1

+ · · · + h

. . . h

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” ∆

ϕ +α

βsinh βϕ = 0

is obtained from sine-Gordon by β→ iβ =⇒ ν < 0.

w(x) ∼(√−x2

)−4∆for x → 0

“Dimension” ∆

Wightman functions

The two-point function

w(x) = 〈 0 | O(x)O′(0) | 0 〉

ϕ +α

βsinh βϕ = 0

w(x) ∼(√−x2

)−4∆for x → 0

“Dimension” ∆Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program Natal, September 2016 25 / 29

Wightman functions

The two-point function

w(x) = 〈 0 | O(x)O′(0) | 0 〉

ϕ +α

βsinh βϕ = 0

w(x) ∼(√−x2

)−4∆for x → 0

“Dimension” ∆Babujian, Foerster, Karowski (FU-Berlin) The Bootstrap Program Natal, September 2016 25 / 29

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

[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

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. 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

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