The form factor program a review and new results

Transcription

1 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 / 20

2 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 / 20

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

4 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 Babujian, Foerster, Karowski (FU-Berlin) The form factor program Budapest, June / 20

5 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 / 20

6 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 / 20

7 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 / 20

8 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 / 20

9 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 / 20

10 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 / 20

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

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

13 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 / 20

14 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 / 20

15 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 / 20

16 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 / 20

17 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 / 20

18 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 / 20

19 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 / 20

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 / 20

21 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 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 / 20

22 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 / 20

23 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 Babujian, Foerster, Karowski (FU-Berlin) The form factor program Budapest, June / 20

24 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 / 20

25 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 / 20

26 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 / 20

27 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 / 20

28 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 / 20

29 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 Babujian, Foerster, Karowski (FU-Berlin) The form factor program Budapest, June / 20

30 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 / 20

31 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 / 20

32 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 / 20

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

34 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 / 20

35 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 / 20

36 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 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 / 20

37 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 / 20

38 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 / 20

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

40 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 / 20

41 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) H. Babujian and M. Karowski Phys. Lett. B411 (1999) 53-57, Nucl. Phys. B620 (2002) 407; Journ. Phys. A: Math. Gen. 35 (2002) ; Phys. Lett. B 575 (2003) H. Babujian, A. Foerster and M. Karowski Nucl.Phys. B736 (2006) and in preparation Babujian, Foerster, Karowski (FU-Berlin) The form factor program Budapest, June / 20

42 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) H. Babujian and M. Karowski Phys. Lett. B411 (1999) 53-57, Nucl. Phys. B620 (2002) 407; Journ. Phys. A: Math. Gen. 35 (2002) ; Phys. Lett. B 575 (2003) H. Babujian, A. Foerster and M. Karowski Nucl.Phys. B736 (2006) and in preparation Quantum field theory: G. Lechner math-ph/ An Existence proof for interacting quantum field theories with a factorizing S-matrix. Babujian, Foerster, Karowski (FU-Berlin) The form factor program Budapest, June / 20