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

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 2007 1 / 24

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 2007 2 / 24

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 2007 3 / 24

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/0601022 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 2007 4 / 24

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 (123 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 Yerevan-Tbilisi, October 2007 5 / 24

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 2007 6 / 24

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 2007 7 / 24

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 2007 8 / 24

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 2007 9 / 24

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(123...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 where s is the spin of O Babujian, Foerster, Karowski (FU-Berlin The form factor program Yerevan-Tbilisi, October 2007 10 / 24

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 2007 11 / 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 2007 12 / 24

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 2007 13 / 24

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 2007 14 / 24

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 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 2007 15 / 24

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 2007 16 / 24

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 2007 17 / 24

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 2007 18 / 24

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 2007 19 / 24

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 2007 20 / 24

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 2007 21 / 24

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 13 13 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 2007 22 / 24

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 R. Köberle, V. Kurak, J.A. Swieca: Scattering theory and 1/N expansion in the chiral Gross-Neveu model, Phys.Rev D20 (1979 897 Babujian, Foerster, Karowski (FU-Berlin The form factor program Yerevan-Tbilisi, October 2007 23 / 24

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; SIGMA 2 (2006, 082; SU(N off-shell Bethe ansatz hep-th/0611012; In preparation:o(n σ- and Gross-Neveu model 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 Yerevan-Tbilisi, October 2007 24 / 24