The form factor program - a review and new results - the nested SU(N)-off-shell Bethe ansatz - 1/N expansion

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1 The form factor program - a review and new results - the nested SU(N-off-shell Bethe ansatz - 1/N expansion H. Babujian, A. Foerster, and M. Karowski Yerevan-Tbilisi, October 2007 Babujian, Foerster, Karowski (FU-Berlin The form factor program Yerevan-Tbilisi, October / 24

2 Contents 1 The Bootstrap Program Construct a quantum field theory explicitly Integrability 2 SU(N S-matrix S-matrix Bound states - crossing 3 Form factors Form factors equations SU(N Form factors Bethe ansatz state p-function Examples Energy momentum tensor T ρσ (x The fields ψ α(x The SU(N-current J µ α(β (x 4 1/N expansion SU(N Gross-Neveu model Babujian, Foerster, Karowski (FU-Berlin The form factor program Yerevan-Tbilisi, October / 24

3 The Bootstrap Program for integrable models Construct a quantum field theory in 1+1 dim. 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 0 O(x p 1,..., p n in = e ix(p 1+ +p n F O (θ 1,..., θ n 1 the S-matrix 2 LSZ-assumptions 3 maximal analyticity 3 Wightman functions 0 O(xO(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 Yerevan-Tbilisi, October / 24

8 The Bootstrap Program for integrable models The convergence of the sum over the intermediate states has been proved up to now only for some trivial cases However, G. Lechner math-ph/ An Existence proof for interacting quantum field theories with a factorizing S-matrix in the framework of algebraic quantum field theory in the sense of Haag and Kastler for nontrivial cases Babujian, Foerster, Karowski (FU-Berlin The form factor program Yerevan-Tbilisi, October / 24

9 The Bootstrap Program for integrable models The convergence of the sum over the intermediate states has been proved up to now only for some trivial cases However, G. Lechner math-ph/ An Existence proof for interacting quantum field theories with a factorizing S-matrix in the framework of algebraic quantum field theory in the sense of Haag and Kastler for nontrivial cases Babujian, Foerster, Karowski (FU-Berlin The form factor program Yerevan-Tbilisi, October / 24

10 Integrability Yang-Baxter equation S 12 S 13 S 23 = S 23 S 13 S 12 = bound state bootstrap equation S (123 Γ (12 12 = Γ (12 12 S 13 S 23 (12 = ( Babujian, Foerster, Karowski (FU-Berlin The form factor program Yerevan-Tbilisi, October / 24

11 Integrability Yang-Baxter equation S 12 S 13 S 23 = S 23 S 13 S 12 = bound state bootstrap equation S (123 Γ (12 12 = Γ (12 12 S 13 S 23 (12 = ( Babujian, Foerster, Karowski (FU-Berlin The form factor program Yerevan-Tbilisi, October / 24

12 SU(N S-matrix Particles α, β, γ, δ = 1,..., N vector representation of SU(N δ γ S δγ αβ (θ = = δ αγ δ βδ b(θ + δ αδ δ βγ c(θ. θ α 1 θ 2 β 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 The S-matrix has a bound state pole at θ = iη = 2πi/N. Bound state of N 1 particles = anti-particle [Swieca et al] β = β 1,..., β N 1 (β N + θ 2πi Γ ( 1 1 N θ 2πi Babujian, Foerster, Karowski (FU-Berlin The form factor program Yerevan-Tbilisi, October / 24

17 Bound states - crossing One derives an unusual crossing relation Γ (β β (S(θ 1... S(θ N 1 δ β αδ = S δ (β (αδ (θγ(α α C (ββ S δ (β (αδ (θγ(α α.. δ δ. =... δ δ α 1 α 2 α N 1 β α 1 α 2 α N 1 with the charge conjugation matrix = ( 1 N 1 C (αγ Γ (α α β S γδ δβ (iπ θ = ( 1 N 1... γ δ δ α 1 α 2 α N 1 β C (α1...α N 1 α N = ɛ α1...α N Babujian, Foerster, Karowski (FU-Berlin The form factor program Yerevan-Tbilisi, October / 24

18 Form factors Definition: Let O(x be a local operator 0 O(x p 1,..., p n in α 1...α n = F O α 1...α n (θ 1,..., θ n e ix p i LSZ-assumptions + maximal analyticity } = O... e ix p i = Properties of form factors Babujian, Foerster, Karowski (FU-Berlin The form factor program Yerevan-Tbilisi, October / 24

19 Form factors Definition: Let O(x be a local operator 0 O(x p 1,..., p n in α 1...α n = F O α 1...α n (θ 1,..., θ n e ix p i LSZ-assumptions + maximal analyticity } = O... e ix p i = Properties of form factors Babujian, Foerster, Karowski (FU-Berlin The form factor program Yerevan-Tbilisi, October / 24

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 Yerevan-Tbilisi, October / 24

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(123...n O (θ (12, θ Γ ( 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 where s is the spin of O Babujian, Foerster, Karowski (FU-Berlin The form factor program Yerevan-Tbilisi, October / 24

22 SU(N 2-particle form factor 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 Yerevan-Tbilisi, October / 24

23 SU(N 2-particle form factor 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 Yerevan-Tbilisi, October / 24

24 General SU(N form factors 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 (zψ β 1...β m α 1...α n (θ, z C θ h(θ, z = n i=1 j=1 m φ(θ i z j τ(z = 1 i<j m 1 φ(zφ( 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 Yerevan-Tbilisi, October / 24

28 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 Yerevan-Tbilisi, October / 24

29 Integration contour C θ θ n + 2πi(1 1 N θ 2 + 2πi(1 1 N θ 1 + 2πi(1 1 N θ n θn 2πi 1 N... θ 2 θ2 2πi 1 N θ 1 θ1 2πi 1 N θ n 2πi θ 2 2πi θ 1 2πi as in the definition of Meijer s G-function Babujian, Foerster, Karowski (FU-Berlin The form factor program Yerevan-Tbilisi, October / 24

30 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 The off-shell Bethe ansatz is to be iterated N 1 times nested off-shell Bethe ansatz Babujian, Foerster, Karowski (FU-Berlin The form factor program Yerevan-Tbilisi, October / 24

31 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 The off-shell Bethe ansatz is to be iterated N 1 times nested off-shell Bethe ansatz Babujian, Foerster, Karowski (FU-Berlin The form factor program Yerevan-Tbilisi, October / 24

32 p-function O(x highest weight SU(N representation with weight vector w O = ( w1 O,..., wn O E.g. energy momentum tensor T ρσ (x w T = (0,..., 0 the fundamental fields ψ α (x w ψ = (1,..., 0 the SU(N-current J µ α(β (x w ϕ = (2, 1,..., 1, 0 the p-function satisfies the periodicity relations (for l = 1,..., N 1 p O (θ,..., z (l,... = σ O p O (..., θ i + 2πi,..., z (l,... σ O statistics factor = ( 1 w l O +wl+1 O p O (θ,..., z (l i + 2πi,... Babujian, Foerster, Karowski (FU-Berlin The form factor program Yerevan-Tbilisi, October / 24

33 p-function O(x highest weight SU(N representation with weight vector w O = ( w1 O,..., wn O E.g. energy momentum tensor T ρσ (x w T = (0,..., 0 the fundamental fields ψ α (x w ψ = (1,..., 0 the SU(N-current J µ α(β (x w ϕ = (2, 1,..., 1, 0 the p-function satisfies the periodicity relations (for l = 1,..., N 1 p O (θ,..., z (l,... = σ O p O (..., θ i + 2πi,..., z (l,... σ O statistics factor = ( 1 w l O +wl+1 O p O (θ,..., z (l i + 2πi,... Babujian, Foerster, Karowski (FU-Berlin The form factor program Yerevan-Tbilisi, October / 24

34 Example: The energy momentum tensor T ρσ (x weight vector w T = (0,..., 0, σ T = 1 : the p-function is n p T ρσ (θ, z = i=1 e ρθ i m e σz i. i=1 E.g.: form factor of the particle α and the bound state (β = (β 1,..., β N 1 F T ρσ α(β (θ 1, θ 2 = K T ρσ K T ρσ α(β (θ 1, θ 2 = α(β (θ 1, θ 2 G(θ 12 ( e ρθ 1 + e ρθ 2 C θ dz φ(θ 1 zl(θ 2 ze σz ɛ δγ S δ1 αɛ(θ 1 zs ɛ(γ (β1 (θ 2 z Babujian, Foerster, Karowski (FU-Berlin The form factor program Yerevan-Tbilisi, October / 24

35 Example: The energy momentum tensor T ρσ (x weight vector w T = (0,..., 0, σ T = 1 : the p-function is n p T ρσ (θ, z = i=1 e ρθ i m e σz i. i=1 E.g.: form factor of the particle α and the bound state (β = (β 1,..., β N 1 F T ρσ α(β (θ 1, θ 2 = K T ρσ K T ρσ α(β (θ 1, θ 2 = α(β (θ 1, θ 2 G(θ 12 ( e ρθ 1 + e ρθ 2 C θ dz φ(θ 1 zl(θ 2 ze σz ɛ δγ S δ1 αɛ(θ 1 zs ɛ(γ (β1 (θ 2 z Babujian, Foerster, Karowski (FU-Berlin The form factor program Yerevan-Tbilisi, October / 24

36 Example: The energy momentum tensor The last integral may be performed 0 T ρσ (0 θ 1, θ 2 in α(β = 4M2 ɛ αβ e 1 2 (ρ+σ(θ 1+θ 2 +iπ sinh 1 2 (θ 12 iπ G(θ 12 θ 12 iπ One can prove the eigenvalue equation ( n dxt ±,0 (x i=1 p ± i θ 1,..., θ n in α = 0 Babujian, Foerster, Karowski (FU-Berlin The form factor program Yerevan-Tbilisi, October / 24

37 Example: The energy momentum tensor The last integral may be performed 0 T ρσ (0 θ 1, θ 2 in α(β = 4M2 ɛ αβ e 1 2 (ρ+σ(θ 1+θ 2 +iπ sinh 1 2 (θ 12 iπ G(θ 12 θ 12 iπ One can prove the eigenvalue equation ( n dxt ±,0 (x i=1 p ± i θ 1,..., θ n in α = 0 Babujian, Foerster, Karowski (FU-Berlin The form factor program Yerevan-Tbilisi, October / 24

38 Example: The fields ψ α (x weight vector w ψ = (1,..., 0, σ ψ = exp iπ ( 1 1 N [Swieca et al] : the p-function is ( p ψ (θ, z = exp 1 m ( z i 1 1 n θ i. 2 N i=1 i=1 E.g.: 1-particle form factor 0 ψ(0 θ α = δ α1 e 1 2(1 1 N θ. Babujian, Foerster, Karowski (FU-Berlin The form factor program Yerevan-Tbilisi, October / 24

39 Example: The fields ψ α (x weight vector w ψ = (1,..., 0, σ ψ = exp iπ ( 1 1 N [Swieca et al] : the p-function is ( p ψ (θ, z = exp 1 m ( z i 1 1 n θ i. 2 N i=1 i=1 E.g.: 1-particle form factor 0 ψ(0 θ α = δ α1 e 1 2(1 1 N θ. Babujian, Foerster, Karowski (FU-Berlin The form factor program Yerevan-Tbilisi, October / 24

40 Example: The fields ψ α (x the 2-particle and 1-bound state form factor F ψ αβ(γ (θ 1, θ 2, θ 3 = K ψ αβ(γ (θ 1, θ 2, θ 3 F (θ 12 G(θ 13 G(θ 23 K ψ αβ(γ = Nψ e 1 2(1 1 N θ i C θ dz R φ(θ 1 zφ(θ 2 zl(θ 3 ze 1 2 z ɛ ζδ S ζ1 αξ (θ 1 zs ξ1 βɛ (θ 2 zs ɛ(δ (γ1 (θ 3 z We were not able to perform this integration but 1/N expansion Babujian, Foerster, Karowski (FU-Berlin The form factor program Yerevan-Tbilisi, October / 24

41 Example: The fields ψ α (x the 2-particle and 1-bound state form factor F ψ αβ(γ (θ 1, θ 2, θ 3 = K ψ αβ(γ (θ 1, θ 2, θ 3 F (θ 12 G(θ 13 G(θ 23 K ψ αβ(γ = Nψ e 1 2(1 1 N θ i C θ dz R φ(θ 1 zφ(θ 2 zl(θ 3 ze 1 2 z ɛ ζδ S ζ1 αξ (θ 1 zs ξ1 βɛ (θ 2 zs ɛ(δ (γ1 (θ 3 z We were not able to perform this integration but 1/N expansion Babujian, Foerster, Karowski (FU-Berlin The form factor program Yerevan-Tbilisi, October / 24

42 Example: The SU(N-current J µ α(β (x J µ α(β = δ α1ɛ (βn J µ, J µ = ɛ µν ν ϕ weight vector w ϕ = (2, 1,..., 1, 0, σ ϕ = 1 : the p-function is p ϕ (θ, z, u = N ϕ ( n i=1 exp θ i 1 exp 1 2 ( n0 n 1 θ i i=1 i=1 z (1 i n N 1 i=1 z (N 1 i Babujian, Foerster, Karowski (FU-Berlin The form factor program Yerevan-Tbilisi, October / 24

43 1/N expansion SU(N Gross-Neveu model Lagrangian L = ψ iγ ψ + g 2 ( ( ψψ 2 ( ψγ 5 ψ 2 2 = ψ(iγ σ iγ 5 πψ 1 2g 2 (σ2 + π 2 There are infrared divergencies due to a would-be-goldstone boson. Swieca et al. e.g. γ θ 3 i(iγ mψ δ (0 θ 1, θ 2 in αβ = F ψ δ, γ αβ(θ 12, θ 13, θ 23 ( = iπ N 2m δγ αδβ δ u(p 2 cosh 1 2 θ γ5 u(p 2 sinh θ sinh 1 2 θ O 13 θ 13 The same result follows from our exact form factor!! ( 1 N 2 Babujian, Foerster, Karowski (FU-Berlin The form factor program Yerevan-Tbilisi, October / 24

44 Some References S-matrix: A.B. Zamolodchikov, JEPT Lett. 25 ( M. Karowski, H.J. Thun, T.T. Truong and P. Weisz Phys. Lett. B67 ( M. Karowski and H.J. Thun, Nucl. Phys. B130 ( A.B. Zamolodchikov and Al. B. Zamolodchikov Ann. Phys. 120 ( M. Karowski, Nucl. Phys. B153 ( R. Köberle, V. Kurak, J.A. Swieca: Scattering theory and 1/N expansion in the chiral Gross-Neveu model, Phys.Rev D20 ( Babujian, Foerster, Karowski (FU-Berlin The form factor program Yerevan-Tbilisi, October / 24

45 Some References Form factors: M. Karowski and P. Weisz Nucl. Phys. B139 ( B. Berg, M. Karowski and P. Weisz Phys. Rev. D19 ( F.A. Smirnov World Scientific 1992 H. Babujian, A. Fring, M. Karowski and A. Zapletal Nucl. Phys. B538 [FS] ( H. Babujian and M. Karowski Phys. Lett. B411 ( , Nucl. Phys. B620 ( ; Journ. Phys. A: Math. Gen. 35 ( ; Phys. Lett. B 575 ( H. Babujian, A. Foerster and M. Karowski Nucl.Phys. B736 ( ; SIGMA 2 (2006, 082; SU(N off-shell Bethe ansatz hep-th/ ; In preparation:o(n σ- and Gross-Neveu model 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 Yerevan-Tbilisi, October / 24