The form factor program a review and new results

The form factor program a review and new results the nested SU(N)-off-shell Bethe ansatz H. Babujian, A. Foerster, and M. Karowski FU-Berlin Budapest, June 2006 Babujian, Foerster, Karowski (FU-Berlin) The form factor program Budapest, June 2006 1 / 20

Contents 1 The Bootstrap Program Construct a quantum field theory explicitly Integrability 2 SU(N) S-matrix 3 Form factors Form factors equations Locality and field equations SU(N) Form factors 2-particle form factors The general formula Bethe ansatz state 4 Wightman functions Short distance behavior Babujian, Foerster, Karowski (FU-Berlin) The form factor program Budapest, June 2006 2 / 20

The Bootstrap Program Construct a quantum field theory explicitly in 3 steps Babujian, Foerster, Karowski (FU-Berlin) The form factor program Budapest, June 2006 3 / 20

The Bootstrap Program Construct a quantum field theory explicitly in 3 steps 1 S-matrix using 1 general Properties: unitarity, crossing etc. 2 Yang-Baxter Equation 3 bound state bootstrap 4 maximal analyticity 2 Form factors using 1 the S-matrix 2 LSZ-assumptions 3 maximal analyticity 0 O(x) p 1,..., p n in = e ix(p 1+ +p n) F O (θ 1,..., θ n ) Babujian, Foerster, Karowski (FU-Berlin) The form factor program Budapest, June 2006 3 / 20

The Bootstrap Program Construct a quantum field theory explicitly in 3 steps 1 S-matrix using 1 general Properties: unitarity, crossing etc. 2 Yang-Baxter Equation 3 bound state bootstrap 4 maximal analyticity 2 Form factors using 1 the S-matrix 2 LSZ-assumptions 3 maximal analyticity 3 Wightman functions 0 O(x) p 1,..., p n in = e ix(p 1+ +p n) F O (θ 1,..., θ n ) 0 O(x)O(y) 0 = n 0 φ(x) n in in n φ(y) 0 Babujian, Foerster, Karowski (FU-Berlin) The form factor program Budapest, June 2006 3 / 20

The Bootstrap Program Construct a quantum field theory explicitly in 3 steps 1 S-matrix using 1 general Properties: unitarity, crossing etc. 2 Yang-Baxter Equation 3 bound state bootstrap 4 maximal analyticity 2 Form factors using 1 the S-matrix 2 LSZ-assumptions 3 maximal analyticity 3 Wightman functions 0 O(x) p 1,..., p n in = e ix(p 1+ +p n) F O (θ 1,..., θ n ) 0 O(x)O(y) 0 = n 0 φ(x) n in in n φ(y) 0 The bootstrap program classifies integrable quantum field theoretic models Babujian, Foerster, Karowski (FU-Berlin) The form factor program Budapest, June 2006 3 / 20

Integrability Integrability in (quantum) field theories means: there exits -many conservation laws µ J µ L (t, x) = 0 (L Z) Babujian, Foerster, Karowski (FU-Berlin) The form factor program Budapest, June 2006 4 / 20

Integrability Integrability in (quantum) field theories means: there exits -many conservation laws µ J µ L (t, x) = 0 (L Z) Consequence in 1+1 dim.: The n-particle S-matrix is a product of 2-particle S-matrices S (n) (p 1,..., p n ) = i<j S ij (p i, p j ) Babujian, Foerster, Karowski (FU-Berlin) The form factor program Budapest, June 2006 4 / 20

Integrability Integrability in (quantum) field theories means: there exits -many conservation laws µ J µ L (t, x) = 0 (L Z) Consequence in 1+1 dim.: The n-particle S-matrix is a product of 2-particle S-matrices S (n) (p 1,..., p n ) = i<j S ij (p i, p j ) Further consequences Babujian, Foerster, Karowski (FU-Berlin) The form factor program Budapest, June 2006 4 / 20

Integrability Yang-Baxter equation S 12 S 13 S 23 = S 23 S 13 S 12 = 1 2 3 1 2 3 Babujian, Foerster, Karowski (FU-Berlin) The form factor program Budapest, June 2006 5 / 20

Integrability Yang-Baxter equation S 12 S 13 S 23 = S 23 S 13 S 12 = 1 2 3 1 2 3 bound state bootstrap equation S (12)3 Γ (12) 12 = Γ (12) 12 S 13 S 23 (12) = 1 3 2 (12) 1 2 3 Babujian, Foerster, Karowski (FU-Berlin) The form factor program Budapest, June 2006 5 / 20

SU(N) S-matrix Example S δγ αβ (θ) = δ γ = δ αγ δ βδ b(θ) + δ αδ δ βγ c(θ). θ α 1 θ 2 β particles α, β, γ, δ = 1,..., N vector representation of SU(N) Babujian, Foerster, Karowski (FU-Berlin) The form factor program Budapest, June 2006 6 / 20

SU(N) S-matrix Example S δγ αβ (θ) = δ γ = δ αγ δ βδ b(θ) + δ αδ δ βγ c(θ). θ α 1 θ 2 β particles α, β, γ, δ = 1,..., N vector representation of SU(N) Yang-Baxter = c(θ) = 2πi Nθ b(θ) Babujian, Foerster, Karowski (FU-Berlin) The form factor program Budapest, June 2006 6 / 20

SU(N) S-matrix Example S δγ αβ (θ) = δ γ = δ αγ δ βδ b(θ) + δ αδ δ βγ c(θ). θ α 1 θ 2 β particles α, β, γ, δ = 1,..., N vector representation of SU(N) Yang-Baxter = c(θ) = 2πi Nθ b(θ) = a(θ) = b(θ) + c(θ) = Γ ( 1 θ 2πi Γ ( 1 + θ 2πi ) Γ ( 1 1 N + θ 2πi ) Γ ( 1 1 N θ 2πi ) ) Babujian, Foerster, Karowski (FU-Berlin) The form factor program Budapest, June 2006 6 / 20

SU(N) S-matrix Example S δγ αβ (θ) = δ γ = δ αγ δ βδ b(θ) + δ αδ δ βγ c(θ). θ α 1 θ 2 β particles α, β, γ, δ = 1,..., N vector representation of SU(N) Yang-Baxter = c(θ) = 2πi Nθ b(θ) = a(θ) = b(θ) + c(θ) = Γ ( 1 θ 2πi Γ ( 1 + θ 2πi ) Γ ( 1 1 N + θ 2πi ) Γ ( 1 1 N θ 2πi ) ) The S-matrix has a bound state pole at θ = iη = 2πi/N. Babujian, Foerster, Karowski (FU-Berlin) The form factor program Budapest, June 2006 6 / 20

SU(N) S-matrix Example S δγ αβ (θ) = δ γ = δ αγ δ βδ b(θ) + δ αδ δ βγ c(θ). θ α 1 θ 2 β particles α, β, γ, δ = 1,..., N vector representation of SU(N) Yang-Baxter = c(θ) = 2πi Nθ b(θ) = a(θ) = b(θ) + c(θ) = Γ ( 1 θ 2πi Γ ( 1 + θ 2πi ) Γ ( 1 1 N + θ 2πi ) Γ ( 1 1 N θ 2πi ) ) The S-matrix has a bound state pole at θ = iη = 2πi/N. Bound state of N 1 particles = anti-particle [Swieca] Babujian, Foerster, Karowski (FU-Berlin) The form factor program Budapest, June 2006 6 / 20

Form factors Definition: Let O(x) be a local operator 0 O(0) p 1,..., p n in α 1...α n = Fα O 1...α n (θ 1,..., θ n ) = O... Babujian, Foerster, Karowski (FU-Berlin) The form factor program Budapest, June 2006 7 / 20

Form factors Definition: Let O(x) be a local operator 0 O(0) p 1,..., p n in α 1...α n = F O α 1...α n (θ 1,..., θ n ) LSZ-assumptions + maximal analyticity } = O... = Properties of form factors Babujian, Foerster, Karowski (FU-Berlin) The form factor program Budapest, June 2006 7 / 20

Form factor equations (i) Watson s equation (ii) Crossing F O...ij...(..., θ i, θ j,... ) = F...ji...(. O.., θ j, θ i,... ) S ij (θ i θ j ) O O =............ ᾱ 1 p 1 O(0)..., p n in,conn....α n = σ O α 1 F O α 1...α n (θ 1 + iπ,..., θ n ) = F O...α nα 1 (..., θ n, θ 1 iπ) O... conn. = σα O 1 O... = O... Babujian, Foerster, Karowski (FU-Berlin) The form factor program Budapest, June 2006 8 / 20

Form factor equations (iii) Annihilation recursion relation 1 2i 1 2i Res F 1...n(θ O 1,..) = C 12 F3...n(θ O 3,..) ( 1 σ O ) 2 S 2n... S 23 θ 12 =iπ = O σ2 O O... Res θ 12 =iπ O... (iv) Bound state form factors... 1 Res F 123...n(θ) O = F(12)3...n O (θ (12), θ ) Γ (12) 12 2 θ 12 =iη 1 2 Res θ 12 =iη (v) Lorentz invariance O... = O... F O 1...n(θ 1 + u,..., θ n + u) = e su F O 1...n(θ 1,..., θ n ) Babujian, Foerster, wherekarowski s is the (FU-Berlin) spin of OThe form factor program Budapest, June 2006 9 / 20

Locality and field equations (i) (iv) and a general crossing formula in φ [O(x), O(y)] ψ in = 0 for all matrix elements, if x y is space like. Babujian, Foerster, Karowski (FU-Berlin) The form factor program Budapest, June 2006 10 / 20

Locality and field equations (i) (iv) and a general crossing formula in φ [O(x), O(y)] ψ in = 0 for all matrix elements, if x y is space like. Exact form factors = Quantum field equations E.g. sine-gordon equation: in φ ϕ(t, x) + α β : sin βϕ : (t, x) ψ in = 0 Babujian, Foerster, Karowski (FU-Berlin) The form factor program Budapest, June 2006 10 / 20

Locality and field equations (i) (iv) and a general crossing formula in φ [O(x), O(y)] ψ in = 0 for all matrix elements, if x y is space like. Exact form factors = Quantum field equations E.g. sine-gordon equation: in φ ϕ(t, x) + α β : sin βϕ : (t, x) ψ in = 0 the bare mass α is related to the renormalized mass m by α = m 2 πν sin πν where ν = β2 8π β 2 Babujian, Foerster, Karowski (FU-Berlin) The form factor program Budapest, June 2006 10 / 20

2-particle form factors 0 O(0) p 1, p 2 in/out = F ( (p 1 + p 2 ) 2 ± iε ) = F (±θ 12 ) where p 1 p 2 = m 2 cosh θ 12. Watson s equations F (θ) = F ( θ) S (θ) F (iπ θ) = F (iπ + θ) maximal analyticity unique solution Babujian, Foerster, Karowski (FU-Berlin) The form factor program Budapest, June 2006 11 / 20

2-particle form factors 0 O(0) p 1, p 2 in/out = F ( (p 1 + p 2 ) 2 ± iε ) = F (±θ 12 ) where p 1 p 2 = m 2 cosh θ 12. Watson s equations F (θ) = F ( θ) S (θ) F (iπ θ) = F (iπ + θ) maximal analyticity unique solution Example: The highest weight SU(N) 2-particle form factor F (θ) = exp 0 dt t sinh 2 t e t N sinh t ( 1 1 ) ( ( 1 cosh t 1 θ )) N iπ Babujian, Foerster, Karowski (FU-Berlin) The form factor program Budapest, June 2006 11 / 20

The general formula Fα O 1...α n (θ 1,..., θ n ) = Kα O 1...α n (θ) F (θ ij ) 1 i<j n Babujian, Foerster, Karowski (FU-Berlin) The form factor program Budapest, June 2006 12 / 20

The general formula F O α 1...α n (θ 1,..., θ n ) = K O α 1...α n (θ) 1 i<j n F (θ ij ) Nested off-shell Bethe Ansatz Kα O 1...α n (θ) = dz 1 C θ dz m h(θ, z) p O (θ, z) L β1...β m (z)ψ β 1...β m α 1...α n (θ, z) C θ h(θ, z) = n i=1 j=1 m φ(z j θ i ) τ(z) = 1 i<j m 1 φ(z)φ( z) depend only on F (θ) i.e. on the S-matrix (see below), τ(z i z j ), Babujian, Foerster, Karowski (FU-Berlin) The form factor program Budapest, June 2006 12 / 20

The general formula F O α 1...α n (θ 1,..., θ n ) = K O α 1...α n (θ) 1 i<j n F (θ ij ) Nested off-shell Bethe Ansatz Kα O 1...α n (θ) = dz 1 C θ dz m h(θ, z) p O (θ, z) L β1...β m (z)ψ β 1...β m α 1...α n (θ, z) C θ h(θ, z) = n i=1 j=1 m φ(z j θ i ) τ(z) = 1 i<j m 1 φ(z)φ( z) depend only on F (θ) i.e. on the S-matrix (see below), p O (θ, z) = polynomial of e ±z i τ(z i z j ), depends on the operator. Babujian, Foerster, Karowski (FU-Berlin) The form factor program Budapest, June 2006 12 / 20

Equation for φ(z) bound state ( 1,..., 1 }{{} ) N 1 (N 1) = 1 = anti-particle of 1 and the form factor recursion relations (iii) + (iv) = N 2 k=0 N 1 φ (θ + kiη) k=0 F (θ + kiη) = 1, η = 2π N Solution: ( ) ( θ φ(θ) = Γ Γ 1 1 2πi N θ ) 2πi Babujian, Foerster, Karowski (FU-Berlin) The form factor program Budapest, June 2006 13 / 20

Bethe ansatz state Ψ β α(θ, z) = β 1 β m 1 1... z m... θ 1 α 1 z 1 θ n α n 1. 1 2 β i N 1 α i N Babujian, Foerster, Karowski (FU-Berlin) The form factor program Budapest, June 2006 14 / 20

Wightman functions The two-point function w(x) = 0 O(x) O (0) 0 Summation over all intermediate states w(x) = n=0 1 n! dθ 1... dθ n e ix(p 1+ +p n) g n (θ) where g n (θ) = 1 (4π) n 0 O(0) p 1,..., p n in in p n,..., p 1 O (0) 0 Babujian, Foerster, Karowski (FU-Berlin) The form factor program Budapest, June 2006 15 / 20

Wightman functions The two-point function w(x) = 0 O(x) O (0) 0 Summation over all intermediate states w(x) = n=0 1 n! dθ 1... dθ n e ix(p 1+ +p n) g n (θ) where g n (θ) = 1 (4π) n 0 O(0) p 1,..., p n in in p n,..., p 1 O (0) 0 The Log of the two-point function ln w(x) = n=1 1 n! dθ 1... dθ n e ix(p 1+ +p n) h n (θ) Babujian, Foerster, Karowski (FU-Berlin) The form factor program Budapest, June 2006 15 / 20

Wightman functions The functions g n and h n are obviously related by the cummulant formula g I = h I1... h Ik I 1 I k =I For example g 1 = h 1 g 12 = h 12 + h 1 h 2 g 123 = h 123 + h 12 h 3 + h 13 h 2 + h 23 h 1 + h 1 h 2 h 3... Babujian, Foerster, Karowski (FU-Berlin) The form factor program Budapest, June 2006 16 / 20

Short distance behavior The two-point Wightman function for short distances 0 O(x) O(0) 0 ( x 2) 4 for x 0 with the dimension = 1 8π n=1 1 n! dθ 1... dθ n 1 h n (θ 1,..., θ n 1, 0) Babujian, Foerster, Karowski (FU-Berlin) The form factor program Budapest, June 2006 17 / 20

Short distance behavior The two-point Wightman function for short distances 0 O(x) O(0) 0 ( x 2) 4 for x 0 with the dimension = 1 8π n=1 1 n! dθ 1... dθ n 1 h n (θ 1,..., θ n 1, 0) Example: The sinh-gordon model ϕ + α β sinh βϕ = 0 Operator: O(x) = :exp βϕ:(x) Babujian, Foerster, Karowski (FU-Berlin) The form factor program Budapest, June 2006 17 / 20

Short distance behavior sinh-gordon 1- and 1+2-particle intermediate state contributions where B = 0.4 0.3 0.2 0.1 0 0 1 2 2β2 4π+β 2 B 1-particle 1+2-particle Babujian, Foerster, Karowski (FU-Berlin) The form factor program Budapest, June 2006 18 / 20

Some References S-matrix: A.B. Zamolodchikov, JEPT Lett. 25 (1977) 468 M. Karowski, H.J. Thun, T.T. Truong and P. Weisz Phys. Lett. B67 (1977) 321 M. Karowski and H.J. Thun, Nucl. Phys. B130 (1977) 295 A.B. Zamolodchikov and Al. B. Zamolodchikov Ann. Phys. 120 (1979) 253 M. Karowski, Nucl. Phys. B153 (1979) 244 Babujian, Foerster, Karowski (FU-Berlin) The form factor program Budapest, June 2006 19 / 20

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 Nucl. 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 Nucl.Phys. B736 (2006) 169-198 and in preparation Babujian, Foerster, Karowski (FU-Berlin) The form factor program Budapest, June 2006 20 / 20

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 Nucl. 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 Nucl.Phys. B736 (2006) 169-198 and in preparation Quantum field theory: G. Lechner math-ph/0601022 An Existence proof for interacting quantum field theories with a factorizing S-matrix. Babujian, Foerster, Karowski (FU-Berlin) The form factor program Budapest, June 2006 20 / 20