Exact form factors in integrable quantum field theories: the scaling Z(N)-Ising model

H. Babujian^{1}, A. Foerster^{2,3}, and M. Karowski^{3}

1 Yerevan Physics Institute, Alikhanian Brothers 2, Yerevan, 375036 Armenia e-mail: babujian@physik.fu-berlin.de

2 Instituto de Física da UFRGS, Av. Bento Gonçalves 9500, Porto Alegre, RS - Brazil e-mail: angela@if.ufrgs.br

3 Institut für Theoretische Physik Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany e-mail: karowski@physik.fu-berlin.de

November 28, 2005

Abstract:

A general form factor formula for the scaling Z(N)-Ising model is constructed. Exact expressions of all matrix elements are obtained for several local operators. In addition, the commutation rules for order, disorder parameters and para-Fermi fields are derived. Because of the unusual statistics of the fields, the quantum field theory seems to be not related to any classical Lagrangian or field equation.

PACS: 11.10.-z; 11.10.Kk; 11.55.Ds

Keywords: Integrable quantum field theory, Form factors

Contents

1 Introduction 2

2 Z(2)-form factors 6 2.1 The general formula for n-particle form factors . . . . . . . . 7 2.2 Examples of fields and their p-functions: . . . . . . . . . . . . 8 3 Z(3)-form factors 10

3.1 The general formula for n-particle form factors . . . . . . . . 10 3.2 Examples of fields and p-functions . . . . . . . . . . . . . . . 14

4 Z(N)-form factors 18 4.1 The general Z(N)-form factor formula . . . . . . . . . . . . . 19 4.2 Examples of fields and p-functions . . . . . . . . . . . . . . . 21

5 Commutation rules 22 5.1 The general formula . . . . . . . . . . . . . . . . . . . . . . . 22 5.2 Application to the Z(N)-model . . . . . . . . . . . . . . . . . 26

A Some useful formulae 27

B Integrals for the Z(2)-model 29

C Proof of the main lemma 29

D Proof of the commutation rules 31

1 Introduction

The ‘form factor program’ is part of the so-called ‘bootstrap program’ for integrable quantum field theories in 1+1-dimensions. This program classifies integrable quantum field theoretic models and in addition it provides their explicit exact solutions in terms of all Wightman functions. This means, in particular, that we do \underline{not} quantize a classical field theory. In fact the quantum field theory considered in this paper is not related (at least to our knowledge) to any classical Lagrangian or field equations of massive particles. The reason for this seems to be the unusual anyonic statistics of the fields, turning this form factor investigations even more fascinating. The bootstrap program consists of three main steps: First the S-matrix is calculated by means of general properties as unitarity and crossing, the Yang-Baxter equations and the additional assumption of ‘maximal analyticity’. Second, matrix elements of local operators

out ⟨ p′_{m}, …, p′_{1} | O(x) | p_{1}, …, p_{n} ⟩^{in}

are calculated using the 2-particle S-matrix as an input. As a third step the Wightman functions can be obtained by inserting a complete set of intermediate states.

The generalized form factors [1] are defined by the vacuum - n-particle matrix elements

⟨ 0 | O(x) | p_{1}, …, p_{n} ⟩_{α_{1}...α_{n}}^{in} = e^{−ix(p_{1}+···+p_{n})} F_{α_{1}...α_{n}}^{O}(θ_{1}, …, θ_{n})

where the α_{i} denote the type (charge) and the θ_{i} are the rapidities of the particles (p_{i} = M_{i}(cosh θ_{i}, sinh θ_{i})). This definition is meant for θ_{1} > ⋯ > θ_{n}, in the other sectors of the variables the function F O ; \underline{α}(\underline{θ}) = F_{α_{1}...α_{n}}^{O}(θ_{1}, …, θ_{n}) is given by analytic continuation with respect to the θ_{i}. General matrix elements are obtained from F O ; \underline{α}(\underline{θ}) by crossing which means in particular the analytic continuation θ_{i} → θ_{i} ± iπ. Using general LSZ assumptions and maximal analyticity in [2] the following properties of form factors have been derived:^{1}

(o) The form factor function F O ; \underline{α}(\underline{θ}) is meromorphic with respect to all variables θ_{1}, …, θ_{n}.

(i) It satisfies Watson’s equations

F_{...α_{i}α_{j}...}^{O}(…, θ_{i}, θ_{j}, …) = F_{...α_{j}α_{i}...}^{O}(…, θ_{j}, θ_{i}, …) S_{α_{i}α_{j}}(θ_{ij}).

(ii) The crossing relation means for the connected part (see e.g. [4]) of the matrix element

ᾱ_{1}⟨ p_{1} | O(0) | p_{2}, …, p_{n} ⟩_{α_{2}...α_{n}}^{in,conn.} = σ^{O}(α_{1}) F_{α_{1}α_{2}...α_{n}}^{O}(θ_{1} + iπ, θ_{2}, …, θ_{n})
= F_{α_{2}...α_{n}α_{1}}^{O}(θ_{2}, …, θ_{n}, θ_{1} − iπ)

where σ^{O}(α) is the statistics factor of the operator O with respect to the particle α.

(iii) The function F O ; \underline{α}(\underline{θ}) has poles determined by one-particle states in each sub-channel. In particular, if α_{1} is the antiparticle of α_{2}, it has the so-called annihilation pole at θ_{12} = iπ which implies the recursion formula

Res_{θ_{12}=iπ} F O ; \underline{α}(θ_{1}, …, θ_{n}) = 2i C_{α_{1}α_{2}}F_{α_{3}...}^{O}(θ_{3}, …, θ_{n})

× (1 − σ^{O}(α_{2})S_{α_{1}α_{n}}(θ_{2n}) … S_{α_{2}α_{3}}(θ_{23})) .

(iv) Bound state form factors yield another recursion formula

Res_{θ_{12}=iη} F_{αβ...}^{O}(θ_{1}, θ_{2}, θ_{3}, …) = \sqrt{2}F_{γ...}^{O}(θ_{(12)}, θ_{3}, …) Γ_{αβ}^{γ}

if iη is the position of the bound state pole. The bound state intertwiner Γ_{αβ}^{γ} (see e.g. [5, 4]) is defined by

i Res_{θ=iη} S_{αβ}(θ) = Γ_{γ}^{βα}Γ_{αβ}^{γ}

1The formulae have been proposed in [3] as a generalization of formulae in [1]. The formulae are written here for the case of no backward scattering, for the general case see [4].

(v) Since we are dealing with relativistic quantum field theories Lorentz covariance in the form

F O ; \underline{α}(θ_{1}, …, θ_{n}) = e^{sμ} F O ; \underline{α}(θ_{1} + μ, …, θ_{n} + μ)

holds if the local operator transforms as O → e^{sμ}O where s is the “spin” of O.

Note that consistency of (ii), (iii) and (v) imply a relation of spin and statistics σ^{O}(α) = e^{−2πis} and also σ^{O}(α) = 1/σ^{O}(ᾱ) where ᾱ is the antiparticle of α, which has the same charge as O. All solutions of the form factor equations (i) - (v) should provide the matrix elements of all fields in an integrable quantum filed theory with a given S-matrix.

Generalized form factors are of the form [1]

F O ; \underline{α}(\underline{θ}) = KO ; \underline{α}(\underline{θ}) ∏_{1≤i<j≤n} F (θ_{ij}) , (θ_{ij} = θ_{i} − θ_{j})  (1)

where F (θ) is the ’minimal’ form factor function. It is the solution of Watsons equation [6] and the crossing relation for n = 2

\begin{matrix}F (θ) = F (−θ)S(θ) \\ F (iπ − θ) = F (iπ + θ) \end{matrix}  (2)

with no poles and zeros in the physical strip 0 < Im θ ≤ π (and a simple zero at θ = 0). In [4] a general integral representation for the K-function KO ; \underline{α}(\underline{θ}) in terms of the off-shell Bethe Ansatz [7, 8] has been presented, which transforms the complicated equations (i)-(v) for the form factors to simple ones for the p-functions (see [4] and (41) below)

KO ; \underline{α}(\underline{θ}) = ∑_{m} c_{nm} ∫_{C\underline{θ}} dz_{1} ⋯ ∫_{C\underline{θ}} dz_{m} h(\underline{θ}, \underline{z}) p^{O}(\underline{θ}, \underline{z}) Ψ\underline{α}(\underline{θ}, \underline{z}) .  (3)

The symbols C\underline{θ} denote specific contours in the complex z_{i}-planes. The function h(\underline{θ}, \underline{z}) is scalar and encodes only data from the scattering matrix. The function p^{O}(\underline{θ}, \underline{z}) on the other hand depends on the explicit nature of the local operator O. We discuss these objects in more detail below. For the case of a diagonal S-matrix, as in this paper, the off-shell Bethe Ansatz vector Ψ\underline{α}(\underline{θ}, \underline{z}) is trivial. The K-function KO ; \underline{α}(\underline{θ}) is an meromorphic function and has the ’physical poles’ in 0 < Im θ_{ij} ≤ π corresponding to the form factor properties (iii) and (iv). It turns out that for the examples we consider in this paper there is only one term in the sum of (3).

In this paper we will focus on the determination of the form factors of the scaling Z(N)-Ising quantum field theory in 1+1 dimensions. An Euclidean field theory is obtained as the scaling limit of a classical statistical lattice model in 2-dimensions given by the partition function

Z = ∑_{\{σ\}} exp (−\frac{1}{kT} ∑_{⟨ij⟩} E(σ_{i}, σ_{j})) ; σ_{i} ∈ \{1, ω, …, ω^{N−1}\} , ω = e^{2πi/N}

as a generalization of the Ising model. It was conjectured by Köberle and Swieca [9] that there exists a Z(N)-invariant interaction E(σ_{i}, σ_{j}) such that the resulting massive quantum field theory is integrable. In particular for N = 2 the scaling Z(2)-Ising model is the well investigated model [10, 11, 12, 13] which is equivalent to a massive free Dirac field theory. In this paper we investigate the general Z(N)-model. It has also been discussed as a deformation [14, 15] of a conformal Z(N) para-fermi field theory [16]. The Z(N)-Ising model in the scaling limit possesses N − 1 types of particles: α = 1, …, N − 1 of charge α, mass M_{α} = M sin π \frac{α}{N} and ᾱ = N − α is the antiparticle of α. The n-particle S-matrix factorizes in terms of two-particle ones since the model is integrable. The two-particle S-matrix for the Z(N)- Ising model has been proposed by Köberle and Swieca [9]. The scattering of two particles of type 1 is

S(θ) = \frac{sinh \frac{1}{2}(θ + \frac{2πi}{N})}{sinh \frac{1}{2}(θ − \frac{2πi}{N})} .  (4)

This S-matrix is consistent with the picture that the bound state of N − 1 particles of type 1 is the anti-particle of 1. This will be essential also for the construction of form factors below. We construct generalized form factors of an operator O(x) and n particles of type 1 and for simplicity we write F_{n}^{O}(\underline{θ}) = F_{1...1}^{O}(\underline{θ}). Note that all further matrix elements with different particle states of the field operator O(x) are obtained by the crossing formula (ii) and the bound state formula (iv). As an application of this form factor approach we compute the commutation relations of fields. In particular, we consider the fields ψ_{QQ̃}(x) , (Q, Q̃ = 0, …, N − 1) with charge Q and ‘dual charge’ Q̃. There are in particular the order parameters σ_{Q}(x) = ψ_{Q0}(x), the disorder parameters μQ̃(x) = ψ_{0Q̃}(x) and the para-fermi fields ψ_{Q}(x) = ψ_{QQ}(x). We show that they satisfy the space like commutation rules:

σ_{Q}(x)σ_{Q′}(y) = σ_{Q′}(y)σ_{Q}(x)
μQ̃(x)μQ̃′(y) = μQ̃′(y)μQ̃(x)
μQ̃(x)σ_{Q}(y) = σ_{Q}(y)μQ̃(x)e^{θ(x^{1}−y^{1})2πiQQ̃/N}  (5)
ψ_{Q}(x)ψ_{Q}(y) = ψ_{Q}(y)ψ_{Q}(x)e^{ϵ(x^{1}−y^{1})2πiQ^{2}/N} .

It turns out that the nontrivial statistics factors σ^{O}(α) in the form factor equations (ii) and (iii) lead the nontrivial order- disorder and the anyonic statistics of the fields.

The form factor bootstrap program has been applied in [13] to the Z(2)- model. Form factors for the Z(3)-model were investigated by one of the present authors in [17]. There the form factors of the order parameter σ_{1} were proposed up to four particles. Kirilov and Smirnov [18] proposed all form factors of the Z(3)-model in terms of determinants. Some related work can be found in [19]. For general N form factors for charge-less states (n particles of type 1 and n particles of type N − 1) were calculated in [20]. In the present paper we present for the scaling Z(N)-Ising model integral representations for all matrix elements of several field operators.

Recently, there has been a renewed interest in the form factors program in connection to condensed matter physics [21, 22, 23] and atomic physics [24]. In particular, applications to Mott insulators and carbon nanotubes as well as in the field of Bose-Einstein condensates of ultra-cold atomic gases have been discussed and in some instances correlation functions have been computed.

The paper is organized as follows: In Section 2 we construct the general form factor formula for the simplest N = 2 case, which corresponds to the well-known scaling Z(2) - Ising model, and show that the known results can be reproduced by our new general approach. In Section 3 we construct the general form factor formula for the Z(3) case, which is more complex due to the presence of bound states, and discuss several explicit examples. We extend these results in Section 4, where the general form factors for Z(N) are constructed and discussed in detail. Section 5 contains the derivation of the commutation rules of the fields and some applications of this formalism are presented. Some results of the present article have been published previously [25] without proves.

2 Z(2)-form factors

To make our method more transparent and with the hope that our construction will help to calculate form factors for all primary and descendant fields, we start with the simplest case N = 2, which corresponds to the well known Ising model in the scaling limit. This model, already investigated in [10, 11, 13, 20], is equivalent to a massive free Dirac field theory. The model possesses one particle with mass M and the 2-particle S-matrix is S = −1. In [13] the form factor approach has been applied to this case with the result for the order parameter field σ(x)

F_{n}^{σ}(\underline{θ}) = (2i)^{(n−1)/2} ∏_{1≤i<j≤n} tanh \frac{1}{2}θ_{ij}  (6)

for n odd. It is easy to see that this expression satisfies the form factor equations (i) - (iii) with statistics factor σ^{σ} = 1. For the Z(2) case in this section we skip the proof that the functions obtained by our general formula satisfy the form factor equations (i) - (v), the reader may easily reduce the proofs for the Z(3) and the general Z(N) case of the following sections to this simple one. Rather, we present the results for several fields, in particular, we will show that our general formula reproduces the known results.

2.1 The general formula for n-particle form factors

We propose the n-particle form factors of an operator O(x) as given by (1) with the minimal form factor function

F (θ) = sinh \frac{1}{2}θ  (7)

which is the minimal solution of Watson’s equations and crossing (2) for S = −1. The K-function is given by our general formula (3) where the Bethe Ansatz vector is trivial (because there is no backward scattering) and the sum consists only of one term

K_{n}^{O}(\underline{θ}) = N_{n}I_{nm}(\underline{θ}, p^{O}) .  (8)

The fundamental building blocks of form factors are

I_{nm}(\underline{θ}, p^{O}) = \frac{1}{m!} ∫_{C\underline{θ}} \frac{dz_{1}}{R} ⋯ ∫_{C\underline{θ}} \frac{dz_{m}}{R} h(\underline{θ}, \underline{z}) p^{O}(\underline{θ}, \underline{z})  (9)

h(\underline{θ}, \underline{z}) = ∏_{i=1}^{n} ∏_{j=1}^{m} ϕ(z_{j} − θ_{i}) ∏_{1≤i<j≤m} τ(z_{i} − z_{j}) .  (10)

The h-function does not depend on the operator but only on the S-matrix of the model, whereas the p-function depends on the operator. Both are analytic functions of θ_{i} (i = 1, …, n) and z_{j} (j = 1, …, m) and are symmetric under θ_{i} ↔ θ_{j} and z_{i} ↔ z_{j}. For all integration variables z_{j} the integration contours C\underline{θ} = ∑ C_{θ_{i}} enclose clock wise oriented the points z_{j} = θ_{i} (i = 1, …, n). This means we assume that ϕ(z) has a pole at z = 0 such that R = ∫_{C_{θ}} dzϕ(z − θ). The functions ϕ (z) and τ(z) are given in terms of the minimal form factor function as

ϕ(z) = \frac{1}{F (θ)F (θ + iπ)} = \frac{−2i}{sinh z}  (11)
τ(z) = \frac{1}{ϕ(z)ϕ(−z)} = \frac{1}{4} sinh^{2} z .  (12)

The following properties of the p-functions guarantee that the form factors satisfy (i) - (iii)

(i’_{2}) p_{nm}^{O}(\underline{θ}, \underline{z}) is symmetric under θ_{i} ↔ θ_{j}
(ii’_{2}) σ^{O}p_{nm}^{O}(θ_{1} + 2πi, θ_{2}, …, \underline{z}) = p_{nm}^{O}(θ_{1}, θ_{2}, …, \underline{z})
(iii’_{2}) if θ_{12} = iπ : p_{nm}^{O}(\underline{θ}, \underline{z})|_{z_{1}=θ_{1}} = σ^{O}p_{nm}^{O}(\underline{θ}, \underline{z})|_{z_{1}=θ_{2}}
= σ^{O}p_{n−2m−1}^{O}(θ_{3}, …, θ_{n}, z_{2}, …, z_{m}) + p̃

where σ^{O} is the statistics factor of the operator O with respect to the particle. The function p̃ must not contribute after integration, which means in particular that is does not depend on the z_{i} (in most cases p̃ = 0). For convenience we have introduced the indices _{nm} to denote the number of variables. For the recursion relation (iii) in addition the normalization coefficients have to satisfy

N_{n} = iN_{n−2} .  (13)

One may convince oneself that the form factor satisfies (i) and (ii). Not so trivial is the residue relation (iii), however, it follows from a simplified version of the proofs for the Z(3) and the general Z(N) case below.

2.2 Examples of fields and their p-functions:

We present the following correspondence of operators and p-functions which are solutions of (i’_{2}) - (iii’_{2}). For the order parameter σ(x), the disorder parameter μ(x), fermi field ψ(x), and the higher conserved currents J_{L}^{μ}(x), (L ∈ ℤ) we propose the following p-functions and statistics parameters

σ(x) ↔ p_{nm}^{σ}(\underline{θ}, \underline{z}) = 1  for n = 2m + 1 with σ^{σ} = 1
μ(x) ↔ p_{nm}^{μ}(\underline{θ}, \underline{z}) = i^{m}e^{(∑ z_{i}− \frac{1}{2} ∑ θ_{i})} for n = 2m  with σ^{μ} = −1
ψ^{±}(x) ↔ p_{nm}^{ψ^{±}}(\underline{θ}, \underline{z}) = e^{±(∑ z_{i}− \frac{1}{2} ∑ θ_{i})} for n = 2m + 1 with σ^{ψ} = −1
J_{L}^{±}(x) ↔ p_{nm}^{J_{L}^{±}}(\underline{θ}, \underline{z}) = ∑ e^{±θ_{i}} ∑ e^{Lz_{i}} for n = 2m  with σ^{μ} = 1

(14)

Note that p̃ ≠ 0 in (iii’_{2}) only for J_{L}^{±}. The motivation of these choices will be more obvious when we investigate the commutation rules of fields in section 5 and from the properties of the form factors which we now discuss in more detail.

Explicit expressions of the form factors:

Now we have to check that the proposed p-functions really provide the well known form factors for the order, disorder and fermi fields. In order to get simple expressions for these form factors, we have to calculate the integral (9) with (10) for each p-function separately.

For the order parameter: Only for odd numbers of particles the form factors are non-zero. We calculate

I_{nm}(\underline{θ}, 1) = 2^{m} ∏_{1≤i<j≤n} \frac{1}{cosh \frac{1}{2}θ_{ij}} for n = 2m + 1.  (15)

The proof of this formula can be found in Appendix B. The general formulae (1), (3), (7), and (13) with N_{n} = i^{(n−1)/2} then imply for n odd

F_{n}^{σ}(\underline{θ}) = 2^{m} ∏_{1≤i<j≤n} \frac{F (θ_{ij})}{cosh \frac{1}{2}θ_{ij}} = (2i)^{(n−1)/2} ∏_{1≤i<j≤n} tanh \frac{1}{2}θ_{ij}  (16)

which agrees with the known result (6). The proof that the integral I_{nm}(\underline{θ}, 1) vanishes for even n and m > 0 is simple: If m(m−1) < n we may decompose for real θ_{i} the contours C\underline{θ} = C_{0} − C_{0−iπ} where Re C_{0} goes from −∞ to ∞ and Im (θ_{i} + iπ) < Im C_{0} < Im (θ_{i}). The shift z_{i} → z_{i} − iπ implies h(\underline{θ}, \underline{z}) → (−1)^{n}h(\underline{θ}, \underline{z}) such that the integrals along C_{0} and C_{0−iπ} cancel for even n.

For the disorder parameter: Only for even numbers of particles the form factors are non-zero. We calculate with p^{μ} as given in (14)

I_{nm}(\underline{θ}, p^{μ}) = 2^{m} ∏_{1≤i<j≤n} \frac{1}{cosh \frac{1}{2}θ_{ij}} for n = 2m .

The proof of this formula is analog to that in Appendix B, therefore with N_{n} = i^{n/2} the form factors are for n even

F_{n}^{μ}(\underline{θ}) = (2i)^{n/2} ∏_{1≤i<j≤n} tanh \frac{1}{2}θ_{ij} .  (17)

Similar as above for the order parameter one can show that the integral I_{nm}(\underline{θ}, p^{μ}) vanishes for odd n and m > 0. It is also interesting to investigate the asymptotic behavior of the form factors when one of the rapidities goes to infinity. From the integral representation it is easy to check that

F_{n}^{σ}(\underline{θ}) →^{θ_{1}→∞} F_{n−1}^{μ}(\underline{θ}′) →^{θ_{2}→∞} 2iF_{n−2}^{σ}(\underline{θ}′′) .

Of course, this result follows easily from the explicit expressions (16) and (17). This asymptotic behavior is another motivation for our choice (14) of the p-function for σ(x) and μ(x).

For the fermi field: Only for n = 1 the form factors are non-zero. We calculate with p^{ψ^{±}} as given in (14)

I_{nm}(\underline{θ}, p^{ψ^{±}}) = δ_{n1} e^{∓ \frac{1}{2} θ} for n = 2m + 1 .

The proof that I_{nm}(\underline{θ}, p^{ψ^{±}}) = 0 for n = 2m + 1 odd and m > 0 is the same as for the disorder parameter. Therefore with the normalization N_{1} = \sqrt{M} we obtain

F_{1}^{ψ}(θ) = ⟨ 0 | ψ(0) | θ ⟩ = u(θ) = \sqrt{M}\binom{e^{− \frac{1}{2} θ}}{e^{\frac{1}{2} θ}} .  (18)

The property that all form factors of the fermi field vanish except the vacuum one-particle matrix element reflects the fact that ψ(x) is a free field, in particular for the Wightman functions one easily obtains

⟨ 0 | ψ(x_{1}) ⋯ ψ(x_{n}) | 0 ⟩ = ⟨ 0 | ψ(x_{1}) ⋯ ψ(x_{n}) | 0 ⟩_{free} .

For the infinite set of conserved higher currents: Only for n = 2 the form factors are non-zero. We calculate with p^{J_{L}^{±}} as given in (14)

I_{nm}(\underline{θ}, p^{J_{L}^{±}}) = δ_{n2} (e^{±θ_{1}} + e^{±θ_{2}}) 2i (\frac{e^{Lθ_{1}}}{sinh θ_{12}} + \frac{e^{Lθ_{2}}}{sinh θ_{21}} ) for n = 2m

The proof that I_{nm}(\underline{θ}, p^{J_{L}^{±}}) = 0 for n = 2m > 2 is again similar as above. With the normalization c_{21} = ±iM the form factors are

⟨ 0 | J_{L}^{±}(0) | θ_{1}, …, θ_{n}⟩^{in} = ∓δ_{n2}2M (e^{±θ_{1}} + e^{±θ_{2}}) (e^{Lθ_{1}} − e^{Lθ_{2}}) \frac{1}{sinh θ_{12}}

such that as in [26] the charges Q_{L} = ∫ dxJ_{L}^{0}(x) satisfy the eigenvalue equation

(Q_{L} − ∑_{i=1}^{n} e^{Lθ_{i}}) | θ_{1}, …, θ_{n} ⟩^{in} = 0 for L = ±1, ±3, ….

Obviously we get the energy momentum tensor from J_{±1}^{±}(x).

The property F_{n}^{ψ} = 0 for odd n > 1 and F_{n}^{J} = 0 for even n > 2 is related to the fact that in the recursion relation (iii) the factor (1 − σ^{O} ∏ S) is zero in both cases.

3 Z(3)-form factors

Let us now consider N = 3, which corresponds to the scaling Z(3)-Ising model. In this case we have two particles, 1 and 2, and the bound state of two particles of type 1 is the particle 2, which in turn is the antiparticle of particle 1. Conversely, the bound state of two particles of type 2 is the particle of type 1, which in turn is again the antiparticle of 2. So, our construction should take into account this structure of the bound states. We construct the form factors for particles of type 1, the others can then be obtained by the form factor bound state formula (iv).

3.1 The general formula for n-particle form factors

In order to obtain a recursion relation where only form factors for particles of type 1 are involved, we have to apply the bound state relation (iv) to get the anti-particle and then the creation annihilation equation (iii) to obtain

Res_{θ_{23}=iη} Res_{θ_{12}=iη} F_{1111...1}^{O}(θ_{1}, …) = Res_{θ_{(12)3}=iπ} F_{211...1}^{O}(θ_{(12)}, θ_{3}, …)\sqrt{2}Γ

= 2iF_{1...1}^{O}(θ_{4}, …) (1 − σ_{1}^{O} ∏_{i=4}^{n} S(θ_{3i})) \sqrt{2}Γ (19)

where θ_{(12)} = \frac{1}{2}(θ_{1}+θ_{2}) is the bound state rapidity, η = \frac{2}{3}π is the bound state fusion angle and Γ = i |Res_{θ=iη} S_{11}(θ)|^{1/2} is the bound state intertwiner (see [5, 4]). In the following we use again the short notation F_{1...1}^{O}(θ_{1}, …, θ_{n}) = F_{n}^{O}(\underline{θ}) and also write the statistics factor as σ^{O}(1) = σ_{1}^{O}. We write the form factors again in the form (1) where minimal form factor function

F (θ) = c_{3} exp ∫_{0}^{∞} \frac{dt}{t} \frac{2 cosh \frac{1}{3}t sinh \frac{2}{3}t}{sinh^{2} t} (1 − cosh t (1 − \frac{θ}{iπ} ))  (20)

is the solution of Watson’s equations (2) with the S-matrix (4) for N = 3. The constant c_{3} is given by (59) in appendix A. Similar as above we make the Ansatz for the K-functions

K_{n}^{O}(\underline{θ}) = N_{n}I_{nmk}(\underline{θ}, p^{O})  (21)

with the fundamental building blocks of form factors

I_{nmk}(\underline{θ}, p) = \frac{1}{m!k!} ∫_{C\underline{θ}} \frac{dz_{1}}{R} ⋯ ∫_{C\underline{θ}} \frac{dz_{m}}{R} ∫_{C\underline{θ}} \frac{du_{1}}{R} ⋯ ∫_{C\underline{θ}} \frac{du_{k}}{R} h(\underline{θ}, \underline{z}, \underline{u}) p(\underline{θ}, \underline{z}, \underline{u}) (22)

h(\underline{θ}, \underline{z}, \underline{u}) = ∏_{i=1}^{n} (∏_{j=1}^{m} ϕ(z_{j} − θ_{i}) ∏_{j=1}^{k} ϕ(u_{j} − θ_{i}))  (23)

× ∏_{1≤i<j≤m} τ(z_{ij}) ∏_{1≤i<j≤k} τ(u_{ij}) ∏_{1≤i≤m} ∏_{1≤j≤k} κ(z_{i} − u_{j}) .

Again the integration contours C\underline{θ} = ∑ C_{θ_{i}} enclose the points θ_{i} such that R = ∫_{C_{θ}} dzϕ(z − θ). Equations (iii) and (iv), in particular (19) lead to the relations

∏_{k=0}^{1} ϕ(θ + kiη) ∏_{k=0}^{2} F (θ + kiη) = 1  (24)
τ(z)ϕ (z) ϕ (−z) = 1 , κ(z)ϕ(z) = 1  (25)

as an extension of (11) and (12) for the Z(2) case. The solution for ϕ is

ϕ(z) = \frac{1}{sinh \frac{1}{2}z sinh \frac{1}{2}(z + iη)}  (26)

if the constant c_{3} is fixed as in (59). The phi-function satisfies the ‘Jost function’ property

\frac{ϕ(−θ)}{ϕ(θ)} = S(θ) .  (27)

We will now show again that by the Ansatz (21)-(23) we have transformed the form factor equations (i)-(v) in particular equation (19) to simple equations for the p-functions.

Assumptions for the p-functions: The function p_{nmk}^{O}(\underline{θ}, \underline{z}, \underline{u}) is analytic in all variables and satisfies:

(i’_{3}) p_{nmk}^{O}(\underline{θ}, \underline{z}, \underline{u}) is symmetric under θ_{i} ↔ θ_{j}
(ii’_{3}) σ_{1}^{O}p_{nmk}^{O}(θ_{1} + 2πi, θ_{2}, …, \underline{z}, \underline{u}) = p_{nmk}^{O}(θ_{1}, θ_{2}, …, \underline{z}, \underline{u})
(iii’_{3}) if θ_{12} = θ_{23} = iη

\begin{aligned}p_{nmk}^{O}(\underline{θ}, \underline{z}, \underline{u})|\begin{matrix}z_{1}=θ_{1} \\ u_{1}=θ_{2} \end{matrix} & = σ_{1}^{O}p_{nmk}^{O}(\underline{θ}, \underline{z}, \underline{u})|\begin{matrix}z_{1}=θ_{2} \\ u_{1}=θ_{3} \end{matrix} \\ & = σ_{1}^{O}p_{n−3m−1k−1}^{O}(\underline{θ}′, \underline{z}′, \underline{u}′) + p̃ \end{aligned}

(v’_{3}) p_{nmk}^{O}(\underline{θ} + μ, \underline{z} + μ, \underline{u} + μ) = e^{sμ}p_{nmk}^{O}(\underline{θ}, \underline{z}, \underline{u})

(28)

where \underline{θ}′ = (θ_{4}, …, θ_{n}), \underline{z}′ = (z_{2}, …, z_{m}) and \underline{u}′ = (u_{2}, …, u_{k}). In (ii’_{3}) and (iii’_{3}) σ_{1}^{O} is the statistics factor of the operator O with respect to the particle of type 1 and in (v’_{3}) s is the spin of the operator O. Again p̃ must not contribute after integration (in most cases p̃ = 0). Again one may convince oneself that the form factor satisfies (i) and (ii) if h(\underline{θ}, \underline{z}) is symmetric under θ_{i} ↔ θ_{j} and periodic with respect to θ_{i} → θ_{i} + 2πi. Not so trivial is the residue relation (iii) which is proved in the following lemma.

Lemma 1 The form factors F_{n}^{O}(\underline{θ}) defined by (1) and (20)-(23) satisfies (i)–(v), in particular (19), if the p-functions satisfy (i’_{3})–(v’_{5}) of (28), the functions ϕ, τ and κ the relations (24), (25) and the normalization constants in (21) the recursion relation

N_{n} (Res_{θ=iη} ϕ(−θ))^{2} ϕ(iη) F^{2}(iη)F (2iη) = N_{n−3}2i\sqrt{2}Γ .  (29)

Proof. The form factor equations (i), (ii), and (v) follow obviously from the equations for the p-functions (i’_{3}), (ii’_{3}), and (v’_{3}), respectively. As already stated, we will prove properties (iii) and (iv) together in the form of (19). Taking the residues Res_{θ_{23}=iη} Res_{θ_{12}=iη} there will be mk equal terms originating from pinchings for z_{i} and u_{i}. We pick them from z_{1} and u_{1} and rewrite the products that appear in the expression for I_{nmk} in a convenient form, such that the location of the poles turns out to be separated from the general expression. Then, the essential calculation to be performed is

Res_{θ_{23}=iη} Res_{θ_{12}=iη} I_{nmk}(\underline{θ}, p^{O}) = \frac{mk}{m!k!} ∫_{C_{\underline{θ}′}} \frac{dz_{2}}{R} ⋯ ∫_{C_{\underline{θ}′}} \frac{dz_{m}}{R} ∫_{C_{\underline{θ}′}} \frac{du_{2}}{R} ⋯ ∫_{C_{\underline{θ}′}} \frac{du_{k}}{R}

× ∏_{i=4}^{n} (∏_{j=2}^{m} ϕ(z_{j} − θ_{i}) ∏_{j=2}^{k} ϕ(u_{j} − θ_{i}))

× ∏_{2≤i<j≤m} τ(z_{ij}) ∏_{2≤i<j≤k} τ(u_{ij}) ∏_{i=2}^{m} ∏_{j=2}^{k} κ(z_{i} − u_{j})

× ∏_{i=1}^{3} (∏_{j=2}^{m} ϕ(z_{j} − θ_{i}) ∏_{j=2}^{k} ϕ(u_{j} − θ_{i})) r

with

r = Res_{θ_{23}=iη} Res_{θ_{12}=iη} ∫_{C\underline{θ}} \frac{dz_{1}}{R} ∫_{C\underline{θ}} \frac{du_{1}}{R} ∏_{i=1}^{n} (ϕ(z_{1} − θ_{i})ϕ(u_{1} − θ_{i}))

× ∏_{j=2}^{m} τ(z_{1j}) ∏_{j=2}^{k} τ(u_{1j}) κ(z_{1} − u_{1}) ∏_{j=2}^{k} κ(z_{1} − u_{j}) ∏_{j=2}^{m} κ(z_{j} − u_{1}) p_{n}^{O}(\underline{θ}, \underline{z}, \underline{u}).

Replacing C\underline{θ} by C\underline{θ}′ where \underline{θ}′ = (θ_{4}, …, θ_{n}) we have used τ(0) = τ(±iη) = κ(0) = κ(−iη) = 0 and the fact that the z_{1}, u_{1}-integrations give nonvanishing results only for (z_{1}, u_{1}) at (θ_{1}, θ_{2}) and (θ_{2}, θ_{3}). This is because for θ_{12}, θ_{23} → iη pinching appears at z_{1} = θ_{2}, u_{1} = θ_{3} and z_{1} = θ_{1,} u_{1} = θ_{2}. Defining the function

f(z_{1}, u_{1}) = ∏_{j=2}^{m} τ(z_{1j}) ∏_{j=2}^{k} τ(u_{1j})κ(z_{1}−u_{1}) ∏_{j=2}^{k} κ(z_{1}−u_{j}) ∏_{j=2}^{m} κ(z_{j}−u_{1})p_{n}^{O}(\underline{θ}, \underline{z}, \underline{u})

and using the property that f(z, z) = f(z, z − iη) = 0, we calculate

r = Res_{θ_{23}=iη} Res_{θ_{12}=iη} ∫_{C\underline{θ}} \frac{dz_{1}}{R} ∫_{C\underline{θ}} \frac{du_{1}}{R} f(z_{1}, u_{1}) ∏_{i=1}^{n} (ϕ(z_{1} − θ_{i})ϕ(u_{1} − θ_{i}))

= (Res_{θ=iη} ϕ(−θ))^{2} ϕ^{2}(iη) ∏_{i=4}^{n} (ϕ(θ_{2i})ϕ(θ_{3i})) (f(θ_{2}, θ_{3}) − f(θ_{1}, θ_{2}) ∏_{i=4}^{n} S(θ_{3i}))

We have used the symmetries of the phi-function ϕ(θ) = ϕ(θ + 2πi) = ϕ(−θ − iη) the Jost property (27) and θ_{12}, θ_{23} = iη = \frac{2}{3}iπ which implies that

Res_{θ_{12}=iη} ϕ(θ_{21}) Res_{θ_{23}=iη} ϕ(θ_{32}) = Res_{θ_{12}=iη} ϕ(θ_{13}) Res_{θ_{23}=iη} ϕ(θ_{21}) = (Res_{θ=iη} ϕ(−θ))^{2}
ϕ(θ_{23})ϕ(θ_{31}) = ϕ(θ_{12})ϕ(θ_{23}) = ϕ^{2}(iη)
\frac{ϕ(θ_{1i})}{ϕ(θ_{3i})} = \frac{ϕ(θ_{3i} − iη)}{ϕ(θ_{3i})} = \frac{ϕ(−θ_{3i})}{ϕ(θ_{3i})} = S(θ_{3i}) .

With the help of the defining equations (25) for τ and κ which imply

(∏_{i=1}^{3} ϕ(z − θ_{i}))^{−1} = τ(θ_{1} − z)κ(z − θ_{2}) = τ(θ_{2} − z)κ(θ_{1} − z)
= τ(θ_{2} − z)κ(z − θ_{3}) = τ(θ_{3} − z)κ(θ_{1} − z)

we obtain the relations for f(θ_{2}, θ_{3}) and f(θ_{1}, θ_{2})

∏_{i=1}^{3} (∏_{j=2}^{m} ϕ(z_{j} − θ_{i}) ∏_{j=2}^{k} ϕ(u_{j} − θ_{i})) f(θ_{1}, θ_{2}) = κ(θ_{12})p_{n}^{O}(\underline{θ}, θ_{1}, \underline{z}′, θ_{2}, \underline{u}′)

∏_{i=1}^{3} (∏_{j=2}^{m} ϕ(z_{j} − θ_{i}) ∏_{j=2}^{k} ϕ(u_{j} − θ_{i})) f(θ_{2}, θ_{3}) = κ(θ_{23})p_{n}^{O}(\underline{θ}, θ_{2}, \underline{z}′, θ_{3}, \underline{u}′) .

Finally we obtain using the defining relation (24) for the phi-function

Res_{θ_{23}=iη} Res_{θ_{12}=iη} I_{nmk}(\underline{θ}, p^{O}) = (Res_{θ=iη} ϕ(−θ))^{2} ϕ^{2}(iη)κ(iη) ∏_{i=1}^{3} ∏_{j=4}^{n} (F (θ_{ij}))^{−1}

× I_{n−3m−1k−1}(\underline{θ}′, p^{O}) (1 − σ_{1}^{O} ∏_{i=4}^{n} S(θ_{3i}))

if the p-function satisfies (iii’_{3}). Therefore the form factor given by (1) and (20)–(23) satisfies (14) and (iii) if the normalization constants satisfy (29).

3.2 Examples of fields and p-functions

We present solutions of the equations for the p-functions (i’_{3})–(v’_{3}) of (28) and some explicit examples of the resulting form factors. We identify the fields by the properties of their matrix elements. In section 5 we show that the field satisfy the desired commutation rules. This motivates to propose a correspondence of fields ϕ(x) and p-functions p_{nmk}^{ϕ}(\underline{θ}, \underline{z}, \underline{u}). The order parameter σ_{Q}(x): We look for a solution of (i’_{3})-(v’_{3}) with

charge  Q = 1, 2
spin  s = 0
statistics σ_{1}^{σ_{Q}} = 1 .

Since the fields carry the charge Q the only non-vanishing form factors with n particles of type 1 are the ones with n = Q mod 3. We propose the correspondence of the field and the p-function:

σ_{Q} ↔ p_{nmk}^{σ_{Q}} = 1 with n = 3l + Q, \{ \begin{matrix}m = l + 1 & k = l & for Q = 1 \\ m = l + 1 & k = l + 1 & for Q = 2 . \end{matrix}

The normalization constants N_{n} follow from (29).

Examples for Q = 1: The form factors of the order parameter σ_{1}(x) for one and four particles of type 1 are

F_{1}^{σ_{1}} = ⟨ 0 | σ_{1}(0) | p⟩_{1} = N_{1}I_{110} = 1
F_{1111}^{σ_{1}}(\underline{θ}) = ⟨ 0 | σ_{1}(0) | p_{1}, p_{2}, p_{3}, p_{4}⟩_{1111}^{in} = N_{4}I_{421}(\underline{θ}, 1) ∏_{1≤i<j≤4} F (θ_{ij})

where we calculate from our integral representation (22)

I_{421}(\underline{θ}, 1) = const (∑ e^{−θ_{i}} ∑ e^{θ_{i}} − 1) ∏_{i<j} \frac{1}{sinh \frac{1}{2}(θ_{ij} − iη) sinh \frac{1}{2} (θ_{ij} + iη)} .

This result has already been obtained in [17] where also the form factor equation (iv) has been discussed, in particular (up to normalizations)

Res_{θ_{34}=2πi/3}⟨ 0 | σ_{1}(0) | p_{1}, p_{2}, p_{3}, p_{4} ⟩_{1111}^{in} = const⟨ 0 | σ_{1}(0) | p_{1}, p_{2}, p_{3} + p_{4} ⟩_{112}^{in}

with

⟨ 0 | σ_{1}(0) | p_{1}, p_{2}, p_{3} ⟩_{112}^{in} = \frac{const F (θ_{12})}{sinh \frac{1}{2}(θ_{12} − iη) sinh \frac{1}{2} (θ_{12} + iη)} ∏_{i=1}^{2} \frac{F_{12}^{min}(θ_{i3})}{cosh \frac{1}{2}θ_{i3}}

where F_{12}^{min} is the minimal form factor function for the S-matrix S_{12}. Further it has been found in [17] that

Res_{θ_{12}=2πi/3}⟨ 0 | σ_{1}(0) | p_{1}, p_{2}, p_{3} ⟩_{112}^{in} = const⟨ 0 | σ_{1}(0) | p_{1} + p_{2}, p_{3} ⟩_{22}^{in}

with

⟨ 0 | σ_{1}(0) | p_{1}, p_{2} ⟩_{22}^{in} = \frac{const F (θ_{12})}{sinh \frac{1}{2}(θ_{12} − iη) sinh \frac{1}{2} (θ_{12} + iη)}  (30)

and the form factor equation (iii) has been checked

Res_{θ_{23}=iπ}⟨ 0 | σ_{1}(0) | p_{1}, p_{2}, p_{3} ⟩_{112}^{in} = const (S(θ_{12}) − 1) .

Example for Q = 2: The form factor of the order parameter σ_{2}(x) for two particles of type 1 is

F_{11}^{σ_{2}}(\underline{θ}) = ⟨ 0 | σ_{2}(0) | p_{1}, p_{2}⟩_{11}^{in} = N_{2}I_{211}(\underline{θ}, 1)F (θ_{12})

where we calculate

I_{211}(\underline{θ}, 1) = const\frac{1}{sinh \frac{1}{2}(θ_{12} − iη) sinh \frac{1}{2}(θ_{12} + iη)}

which agrees with the result (30) of [17]. This is to be expected because of charge conjugation. The disorder parameter μQ̃(x): We look for a solution of (i’_{3})-(v’_{3}) with

charge  Q = 0
spin  s = 0
statistics σμQ̃_{1} = ωQ̃

where ω = e^{iη}, η = 2/3. We call the number Q̃ = 1, 2 the ‘dual charge’ of the field μQ̃(x). Since the fields carry the charge Q = 0 the only non-vanishing form factors with n particles of type 1 are the ones with n = 0 mod 3. We propose the correspondence of the field and the p-function:Q̃

μQ̃ ↔ \{ \begin{matrix}p_{nmk}^{μ_{1}} = ρ exp (∑_{i=1}^{m} z_{i} − \frac{1}{3} ∑_{i=1}^{n} θ_{i}) \\ p_{nmk}^{μ_{2}} = ρ exp (∑_{i=1}^{m} z_{i} + ∑_{i=1}^{k} u_{i} − \frac{2}{3} ∑_{i=1}^{n} θ_{i}) \end{matrix} with \{ \begin{matrix}n = 3m \\ k = m \end{matrix}

where ρ = \sqrt{ω}^{Q̃(Q̃−N+2)m}. Again the normalization constants N_{n} follow from (29).

Examples for Q̃ = 1, 2: The form factors of the disorder parameter μQ̃(x) for 0 and 3 particles of type 1 are

F^{μQ̃} = ⟨ 0 | μQ̃(0) | 0⟩ = 1
F_{111}^{μQ̃}(\underline{θ}) = ⟨ 0 | μQ̃(0) | p_{1}, p_{2}, p_{3}⟩_{111}^{in} = N_{3}I_{311}(\underline{θ}, p^{μQ̃}) ∏_{1≤i<j≤3} F_{11}(θ_{ij}) .

We calculate from our integral representation (22)

I_{311}(\underline{θ}, p^{μQ̃}) = const e^{∓ \frac{1}{3} ∑_{i=1}^{n} θ_{i}} ∑_{i=1}^{3} e^{±θ_{i}} ∏_{i<j} \frac{1}{sinh \frac{1}{2}(θ_{ij} − iη) sinh \frac{1}{2} ((θ_{ij} + iη)}

where the upper sign is for Q̃ = 1 and the lower one for Q̃ = 2. Using the form factor bound state formula (iv) we obtain (up to a constant)

F_{12}^{μQ̃}(\underline{θ}) = e^{∓ \frac{1}{6} θ_{12}} \frac{1}{cosh \frac{1}{2}θ_{12}} F_{12}^{min}(θ_{12}).

It is interesting to note that for Re θ_{1} → ∞ we have the relation of order and disorder parameter form factors (up to constants)

lim_{Re θ_{1}→∞}⟨ 0 | σ_{1}(0) | p_{1}, p_{2}, p_{3}, p_{4}⟩_{1111}^{in} = ⟨ 0 | μ_{2}(0) | p_{2}, p_{3}, p_{4}⟩_{111}^{in}

which follows from the asymptotic behavior

F (θ_{1i}) → e^{\frac{2}{3} θ_{1i}}
I_{421}(\underline{θ}, 1) → e^{θ_{1}} (∑_{i=2}^{4} e^{−θ_{i}}) ∏_{j=2}^{4} e^{−θ_{1j}} ∏_{1<i<j} \frac{1}{sinh \frac{1}{2}(θ_{ij} − a) sinh \frac{1}{2}(θ_{ij} + a)}.

The para-fermi field ψ_{Q}(x): We look for a solution of (i’_{3})-(v’_{3}) with

charge Q = Q
spin  s = Q(3 − Q)/3
statistics σψ_{Q}_{1} = ω^{Q} .

These fields have charge Q = 1, 2 and dual charge Q̃ = Q. The only nonvanishing form factors with n particles of type 1 are the ones with n = Q mod 3. We propose the correspondence of the field and the p-function:

ψ_{Q} ↔ \{ \begin{matrix}p_{nmk}^{ψ_{1}} = ρ exp (∑_{i=1}^{m} z_{i} − \frac{1}{3} ∑_{i=1}^{n} θ_{i}) \\ p_{nmk}^{ψ_{2}} = ρ exp (∑_{i=1}^{m} z_{i} + ∑_{i=1}^{k} u_{i} − \frac{2}{3} ∑_{i=1}^{n} θ_{i}) \end{matrix} with \{ \begin{matrix}n = 3l + Q \\ m = l + 1 \\ k = l + Q − 1 \end{matrix}

where ρ = \sqrt{ω}^{Q̃(Q̃−1)l}. Again the normalization constants N_{n} follow from (29).

Examples for Q = 1, 2: The form factors of the para-fermi field ψ_{1}(x) for 1 and 4 particles of type 1 are

F_{1}^{ψ_{1}}(θ) = ⟨ 0 | ψ_{1}(0) | p⟩_{1} = N_{1}I_{110}(θ, p^{ψ_{1}}) = e^{\frac{2}{3} θ}
F_{1111}^{ψ_{1}}(\underline{θ}) = ⟨ 0 | ψ_{1}(0) | p_{1}, p_{2}, p_{3}, p_{4}⟩_{1111}^{in} = N_{4}I_{421}(\underline{θ}, p^{ψ_{1}}) ∏_{1≤i<j≤4} F (θ_{ij})

= const e^{− \frac{2}{3} ∑_{i=1}^{4} θ_{i}} ∑_{i<j} e^{θ_{i}+θ_{j}} ∏_{1≤i<j≤4} \frac{F (θ_{ij})}{sinh \frac{1}{2}(θ_{ij} − iη) sinh \frac{1}{2}(θ_{ij} + iη)}

and the one of the para-fermi field ψ_{2}(x) for 2 particles of type 1 is

F_{11}^{ψ_{2}}(\underline{θ}) = ⟨ 0 | ψ_{2}(0) | p_{1}, p_{2}⟩_{11} = N_{2}I_{211}(\underline{θ}, p^{ψ_{2}})F (θ_{12})

= const e^{^{\frac{1}{3}} (θ_{1}+θ_{2})} \frac{F (θ_{12})}{sinh \frac{1}{2}(θ_{12} − iη) sinh \frac{1}{2}(θ_{12} + iη)} .

All these examples agree with the results of [18]. The higher currents J_{L}^{±}(x): We look for a solution of (i’_{3})-(v’_{3}) with

charge Q = 0
spin  s = L ± 1
statistics σJ_{L}^{±}_{1} = 1 .

Since the currents are Z(3)-charge-less the only non-vanishing form factors with n particles of type 1 are the ones with n = 0 mod 3. We propose the correspondence of the currents and the p-functions for L ∈ ℤ

J_{L}^{±} ↔ p_{nmk}^{J_{L}^{±}} = ± (∑_{i=1}^{n} e^{±θ_{i}}) (∑_{i=1}^{m} e^{Lz_{i}} + ∑_{i=1}^{m} e^{Lu_{i}}) for \{ \begin{matrix}n = 3m \\ k = m . \end{matrix}

Note that for this case the function p̃ in (28) is non-vanishing, however, it does not contribute because I_{nmm}(\underline{θ}, 1) = 0 for n = 3m. The proof of this fact is similar to that one in appendix B. The higher charges Q_{L} = ∫ dxJ_{L}^{0}(x) satisfy the eigenvalue equations

(Q_{L} − ∑_{i=1}^{n} e^{Lθ_{i}}) | p_{1}, …, p_{n} ⟩^{in} = 0 .

Obviously, from J_{±1}^{±}(x) we obtain the energy momentum tensor.

Examples: The form factors of energy momentum tensor that is J_{L}^{±}(x) for L = ±1 for 0 and 3 particles of type 1 are

F^{J_{L}^{±}} = ⟨ 0 | J_{L}^{±}(0)| 0⟩ = 0
F_{111}^{J_{L}^{±}}(\underline{θ}) = ⟨ 0 | J_{L}^{±}(0) | p_{1}, p_{2}, p_{3}⟩_{111}^{in} = c_{311}I_{311}(\underline{θ}, p^{J_{L}^{±}}) ∏_{1≤i<j≤3} F (θ_{ij})

= ±const (e^{±θ_{1}} + e^{±θ_{2}} + e^{±θ_{3}}) (e^{Lθ_{1}} + e^{Lθ_{2}} + e^{Lθ_{3}})

× ∏_{1≤i<j≤3} \frac{F (θ_{ij})}{sinh \frac{1}{2}(θ_{ij} − iη) sinh \frac{1}{2}(θ_{ij} + iη)} .

By (iv) we obtain the bound state form factor (up to a normalization) for L = ±1

F_{12}^{J_{L}^{±}}(\underline{θ}) = ±const e^{^{\frac{1}{2}} (L±1)(θ_{1}+θ_{2})}F_{12}^{min}(θ_{12}) .

Notice that this last expression agrees with the results of [20] when N = 3.

4 Z(N)-form factors

The scaling Z(N)-Ising model possesses particles of type α = 1, …, N − 1 with Z(N)-charge Q_{α} = α such that the anti particle of α is ᾱ = N −α. The bound state fusion rules are (αβ) = α + β mod N, in particular the bound state of N − 1 particles of type 1 is the antiparticle ¯1. Therefore applying N − 2 times formula (iv) and once (iii) we obtain the recursion relations for form factors where only particles of type 1 are involved

Res_{θ_{N−1N} =iη} … Res_{θ_{12}=iη} F_{n}(θ_{1}, …, θ_{n}) = 2iF_{n−N}(θ_{N+1}, …, θ_{n})

× (1 − σ_{1}^{O} ∏_{i=N+1}^{n} S(θ_{Ni})) ∏_{α=1}^{N−2} \sqrt{2}Γ_{1α}^{1+α} (31)

where η = \frac{2πi}{N} and the Γ_{1α}^{1+α} = i |Res_{θ=iη_{α}} S_{1α}(θ)|^{1/2} are the bound state intertwiners of the fusion (1α) = 1 + α. We will construct the form factors of particles of type 1, all the others are then obtained by the bound state formula (iv).

4.1 The general Z(N)-form factor formula

Following [1] we write the form factors again in the form (1)

F O ; \underline{α}(\underline{θ}) = KO ; \underline{α}(\underline{θ}) ∏_{1≤i<j≤n} F (θ_{ij})  (32)

where minimal form factor function [17]

F (θ) = c_{N} exp ∫_{0}^{∞} \frac{dt}{t} \frac{2 cosh \frac{1}{N} t sinh \frac{N−1}{N}t}{sinh^{2} t} (1 − cosh t (1 − \frac{θ}{iπ} )) (33)

is the solution of Watson’s equations (2) with the S-matrix (4). The constant c_{N} is given by (59) in appendix A. The K-function K_{n}^{O}(θ_{1}, …, θ_{n}) is totally symmetric in the rapidities θ_{i}, 2πi periodic, containing the entire pole structure and determines the asymptotic behavior for large values of the rapidities. Similar as above we make the Ansatz for the K-functions

K_{n}^{O}(\underline{θ}) = N_{n}I_{n\underline{m}}(\underline{θ}, p^{O})  (34)

with the fundamental building blocks of form factors

I_{n\underline{m}}(\underline{θ}, p^{O}) = ( ∏_{k=1}^{N−1} \frac{1}{m_{k}!} ∏_{j=1}^{m_{k}} ∫_{C\underline{θ}} \frac{dz_{kj}}{R} ) h(\underline{θ}, \underline{z})p^{O}(\underline{θ}, \underline{z})  (35)

h(\underline{θ}, \underline{z}) = ∏_{k=1}^{N−1} (∏_{j=1}^{m_{k}} ∏_{i=1}^{n} ϕ(z_{kj} − θ_{i}) ∏_{1≤i<j≤m_{k}} τ(z_{ki} − z_{kj}))  (36)

× ∏_{1≤k<l≤N−1} ∏_{i=1}^{m_{k}} ∏_{j=1}^{m_{l}} κ(z_{ki} − z_{lj})

where \underline{m} = (m_{1}, …, m_{N−1}) and \underline{z} = (z_{ki}), k = 1, …, N − 1, i = 1, …, m_{k}. Again the integration contours C\underline{θ} = ∑ C_{θ_{i}} enclose the points θ_{i} such that R = ∫_{C_{θ}} dzϕ(z − θ). Equations (iii) and (iv), in particular (31) lead to the relations

∏_{k=0}^{N−2} ϕ(z + kiη) ∏_{k=0}^{N−1} F (z + kiη) = 1  (37)
τ(−z)ϕ (z + iπ) ϕ (z) = 1 , κ(z)ϕ(z) = 1  (38)

N_{n} (Res_{θ=iη} ϕ(−θ))^{N−1} ∏_{k=1}^{N−2} ϕ^{k}(kiη) ∏_{k=1}^{N−1} F^{N−k}(kiη) = N_{n−N}2i ∏_{k=1}^{N−2} ∏_{α=1}^{N−2} \sqrt{2}Γ_{1α}^{1+α} (39)

where \underline{m} − 1 = (m_{1} − 1, …, m_{N−1} − 1). The solution of (37) for ϕ is again

ϕ(z) = \frac{1}{sinh \frac{1}{2}z sinh \frac{1}{2}(z + iη)}  (40)

if the constant c_{N} is fixed as in (59). The phi-function satisfies again the ‘Jost function’ property ϕ(−θ)/ϕ(θ) = S(θ). The p-function p_{n\underline{m}}^{O}(\underline{θ}, \underline{z}) is analytic in all variables and satisfies:

(i’) p_{n\underline{m}}^{O}(\underline{θ}, \underline{z}) is symmetric under θ_{i} ↔ θ_{j}
(ii’) σ_{1}^{O}p_{n\underline{m}}^{O}(θ_{1} + 2πi, θ_{2}, …, \underline{z}) = p_{n\underline{m}}^{O}(θ_{1}, θ_{2}, …, \underline{z})
(iii’) if θ_{kk+1} = iη = 2πi/N for k = 1, …, N − 1

\begin{aligned}p_{n\underline{m}}^{O}(\underline{θ}, \underline{z})|_{z_{k1}=θ_{k}} & = σ_{1}^{O}p_{n\underline{m}}^{O}(\underline{θ}, \underline{z})|_{z_{k1}=θ_{k+1}} \\ & = σ_{1}^{O}p_{n−N\underline{m}−1}^{O}(\underline{θ}′, \underline{z}′) + p̃ \end{aligned}

(v’) p_{nmk}^{O}(\underline{θ} + μ, \underline{z} + μ) = e^{sμ}p_{nmk}^{O}(\underline{θ}, \underline{z})

(41)

where \underline{θ}′ = (θ_{N+1}, …, θ_{n}), \underline{z}′ = (z_{ki}), k = 1, …, N − 1, i = 2, …, m_{k}. In (ii’) and (iii’) σ_{1}^{O} is the statistics factor of the operator O with respect to the particle of type 1 and in (v’) s is the spin of the operator O. Again p̃ must not contribute after integration (in most cases p̃ = 0).

By means of the off-shell Bethe Ansatz (3) and (35) we have transformed the complicated form factor equations (i) – (v) to simple equations for the p-functions (i’) – (v’). Again one may convince oneself that the form factor satisfies (i) and (ii) if h(\underline{θ}, \underline{z}) is symmetric under θ_{i} ↔ θ_{j} and periodic with respect to θ_{i} → θ_{i} + 2πi. Not so trivial is again the residue relation (iii) which is proved in the following lemma.

Lemma 2 The form factors given by equations (32) – (36) satisfy the form factor equations (i) – (v) if the functions ϕ, τ, κ satisfy (37) and (38), the normalization constants satisfy (39) and the p-functions satisfy (i’) – (v’) of (41).

The proof of this lemma follows the same strategy of the previous Z(3) case. Here, however, the essential calculation is much more involved, due to the existence more types of particles. Details of this proof can be found in Appendix C.

4.2 Examples of fields and p-functions

We present solutions of the equations for the p-functions (i’)–(v’) of (41) and some explicit examples of the resulting form factors. We identify the fields by the properties of their matrix elements. In section 5 we show that the fields satisfy the desired commutation rules. This motivates to propose a correspondence of fields ϕ(x) and p-functions p^{ϕ}(\underline{θ}, \underline{z}). The fields ψ_{Q,Q̃}(x): These fields have the charge Q = 0, …, N − 1 and the dual charge Q̃ = 0, …, N − 1. We look for a solution of (i’)-(v’) with

\{ \begin{matrix}charge & Q mod N \\ spin & s^{ψ} = min(Q, Q̃) − QQ̃/N \\ statistics & σ_{1}^{ψ} = ωQ̃ \end{matrix}  (42)

with ω = e^{iη} = e^{2πi/N}. The phase factor σ_{1}^{ψ} is the statistics factor of the field ψ_{Q,Q̃}(x) with respect to the particle of type 1. Since the fields carry the charge Q the only non-vanishing form factors with n particles of type 1 are the ones with n = Q mod N. We propose the correspondence of the field and the p-function:

ψ_{Q,Q̃} ↔ p_{n\underline{m}}^{QQ̃} = ρ exp (∑_{k=1}^{Q̃} ∑_{j=1}^{m_{k}} z_{kj} − \frac{Q̃}{N} ∑_{i=1}^{n} θ_{i})  (43)

with n = 3l + Q, l = 0, 1, 2, … and \{ \begin{matrix}m_{k} = l + 1 & for k ≤ Q \\ m_{k} = l & for Q < k \end{matrix}

where ρ = \sqrt{ω}^{Q̃(Q̃−N+2)n/N}. One easily checks that this p-function satisfies the equations (i’)-(v’) and the requirements (42). The normalization constants N_{n} follow from (39). In particular we have for

Q̃ = 0 the order parameters σ_{Q}(x) = ψ_{Q0}(x)
Q = 0 the disorder parameter μQ̃(x) = ψ_{0Q̃}(x)
Q = Q̃ the para-fermi fields  ψ_{Q}(x) = ψ_{QQ}(x) .

They satisfy space like commutation rules (5), derived in the next section. The para-fermi fields ψ_{Q}(x) are the massive analogs of the para-fermi fields in the conformal quantum field theory of [14, 15]. One obtains a second set of fields ψ̃_{Q,Q̃}(x) by changing the sign in the exponent of (43). The higher currents J_{L}^{±}(x): These fields are charge-less, have bosonic statistics and spin L ± 1. The only non-vanishing form factors with n particles of type 1 are the ones with n = 0 mod N. We propose the correspondence of the currents and the p-functions for L ∈ ℤ

J_{L}^{±} ↔ p_{n\underline{m}}^{J_{L}^{±}} = ± ∑_{i=1}^{n} e^{±θ_{i}} ∑_{k=1}^{N−1} ∑_{j=1}^{m} e^{Lz_{kj}} for n = 3m .

The higher charges Q_{L} = ∫ dxJ_{L}^{0}(x) satisfy again the eigenvalue equations

(Q_{L} − ∑_{i=1}^{n} e^{Lθ_{i}}) | p_{1}, …, p_{n} ⟩^{in} = 0 .

Obviously, from J_{±1}^{±}(x) we obtain the energy momentum tensor.

Examples: Up to normalization constants we calculate for the order parameters σ_{1}(x) and σ_{2}(x)

⟨ 0 | σ_{1}(0) | θ⟩_{1} = 1
⟨ 0 | σ_{2}(0) | θ_{1}, θ_{2}⟩_{11}^{in} = \frac{F (θ_{12})}{sinh \frac{1}{2}(θ_{12} − 2πi/N) sinh \frac{1}{2}(θ_{12} + 2πi/N)}

and for the para-fermi fields ψ_{Q}(x) and ψ_{2}(x)

⟨ 0 | ψ_{Q}(0) | θ⟩_{Q} = e^{\frac{Q(N−Q)}{N} θ}  (44)
⟨ 0 | ψ_{2}(0) | θ_{1}, θ_{2}⟩_{11}^{in} = \frac{e^{(1− \frac{2}{N})(θ_{1}+θ_{2})}F (θ_{12})}{sinh \frac{1}{2}(θ_{12} − 2πi/N) sinh \frac{1}{2}(θ_{12} + 2πi/N)}

where | θ⟩_{Q} denotes a one-particle state of charge Q and | θ_{1}, θ_{2}⟩_{11}^{in} a state of two particles of charge 1.

5 Commutation rules

5.1 The general formula

Techniques similar to the ones of this section have been applied for the simpler case of no bound states and bosonic statistics in [3, 27]. A generalization for the case of bound states has been discussed in [28]. Here we generalize these techniques for the case of more general statistics and also discuss the contribution of poles related to the double poles of bound state S-matrices^{2}. In order to discuss commutation rules of two fields ϕ(x) and ψ(y) we have to use a general crossing formula for form factors which was derived in [4] (see also [3]). For quantum field theories with general statistics we introduce assumptions on the statistics factor of a field ψ(x) and a particle α. It is easy to see that for consistency of (ii) and (iii) the condition σ^{ψ}(α)σ^{ψ}(ᾱ) = 1 has to hold if ᾱ is the anti-particle of α. We assume that

2For bound state form factors there are also higher order ’physical poles’ (see e.g.

[29, 30, 31, 32]).

σ^{ψ}(α) = σ^{ψ}(Q_{α})  (45)

depends on the charge of the particle such that σ^{ψ}(Q + Q′) = σ^{ψ}(Q)σ^{ψ}(Q′). A stronger assumption (which holds for the Z(N)-model) is the existence of a ‘dual charge’ Q̃_{ψ} of the fields such that

σ^{ψ}(α) = ω^{Q̃_{ψ}Q_{α}}  (46)

where |ω| = 1.

In order to write the following long formulae we introduce short notation: For a field O(x) and for ordered sets of rapidities θ_{1} > ⋯ > θ_{n} and θ′_{1} < ⋯ < θ′_{m} we write the general matrix element of O(0) as

O\underline{β} ; \underline{α}(\underline{θ}′_{β}, \underline{θ}_{α}) := ^{out}⟨β_{m}(θ′_{m}), …, β_{1}(θp′_{1}) | O | α_{1}(θ_{1}), …, α_{n}(θ_{n}) ⟩^{in} (47)

where \underline{θ}_{α} = (θ_{1}, …, θ_{n}) and \underline{θ}′_{β} = (θ′_{1}, …, θ′_{m}). The array of indices \underline{α} = (α_{1}, …, α_{n}) denote a set of particles (α_{i} ∈ {types of particles}) and correspondingly for \underline{β} (we also write |α| = n etc.). Similar as for form factors this matrix element is given for general order of the rapidities by the symmetry property (i) for both the in- and out-states which takes the general form:

O\underline{β} ; \underline{α}(\underline{θ}′_{β}, \underline{θ}_{α}) = S\underline{β} ; \underline{δ}(\underline{θ}′_{δ})O\underline{δ} ; \underline{γ}(\underline{θ}′_{δ}, \underline{θ}_{γ})S\underline{γ} ; \underline{α}(\underline{θ}_{α})

if \underline{θ}_{γ} is a permutation of \underline{θ}_{α} and \underline{θ}′_{δ} a permutation of \underline{θ}′_{β}. The matrix S\underline{γ} ; \underline{α}(\underline{θ}_{α}) is defined as the representation of the permutation π(\underline{θ}_{α}) = \underline{θ}_{γ} generated by the two-particle S-matrices S_{α_{1}α_{2}}^{γ_{2}γ_{1}}(θ_{12}), for example S_{α_{1}α_{2}α_{3}}^{γ_{3}γ_{2}γ_{1}}(θ_{1}, θ_{2}, θ_{3}) = S_{α_{1}λ_{3}}^{γ_{3}γ_{1}}(θ_{13})S_{α_{2}α_{3}}^{λ_{3}γ_{2}}(θ_{23}) (c.f. [4]).

We consider an arbitrary matrix element of products of fields O = ϕ(x)ψ(y) and O = ψ(y)ϕ(x). Inserting a complete set of intermediate states | \underline{θ̃}_{γ} ⟩in ; \underline{γ} we obtain

(ϕ(x)ψ(y))\underline{β} ; \underline{α} (\underline{θ}′_{β}, \underline{θ}_{α}) = e^{iP′_{β}x−iP_{α}y} \frac{1}{γ!} ∫_{\underline{θ̃}_{γ}} ϕ\underline{β} ; \underline{γ}(\underline{θ}′_{β}, \underline{θ̃}_{γ})ψ\underline{γ} ; \underline{α}(\underline{θ̃}_{γ}, \underline{θ}_{α})e^{−iP̃_{γ}(x−y)}

(48)

where ϕ = ϕ(0), ψ = ψ(0), P_{α} = the total momentum of the state | \underline{θ}_{α} ⟩in ; \underline{α}

etc. and ∫_{\underline{θ̃}_{γ}} = ∏_{k=1}^{|γ|} ∫ \frac{dθ̃_{k}}{4π}. Einstein summation convention over all sets \underline{γ} is assumed. We also define γ! = ∏_{α} n_{α}! where n_{α} is the number of particles of type α in \underline{γ}. We apply the general crossing formula (31) of [4] which is obtained by taking into account the disconnected terms in (ii) and iterating that formula. Strictly speaking, we apply the second version of the crossing formula to the matrix element of ϕ

ϕ\underline{β} ; \underline{γ}(\underline{θ}′_{β}, \underline{θ̃}_{γ}) = ∑_{\begin{matrix}\underline{θ}′_{ρ}∪\underline{θ}′_{τ} =\underline{θ}′_{β} \\ \underline{θ̃}_{ς}∪\underline{θ̃}_{κ}=\underline{θ̃}_{γ} \end{matrix}} S\underline{β} ; \underline{ρ}\underline{τ} (\underline{θ}′_{ρ}, \underline{θ}′_{τ}) ϕ_{\underline{ς}\underline{ρ̄}}(\underline{θ̃}_{ς}, \underline{θ′}ρ̄ − iπ_{−})C^{\underline{ρ}\underline{ρ̄}} 1\underline{τ} ; \underline{σ}(\underline{θ′}_{τ} , \underline{θ̃}_{κ}) S\underline{ς}\underline{σ} ; \underline{γ}(\underline{θ̃}_{γ})

(49)

where \underline{ρ̄} = (ρ̄_{|ρ|}, …, ρ̄_{1}) with ρ̄ = antiparticle of ρ and \underline{θ′}ρ̄ − iπ_{−} means that all rapidities are taken as θ′ − i (π − ϵ). The matrix 1\underline{τ} ; \underline{σ}(\underline{θ′}_{τ} , \underline{θ̃}_{κ}) is defined by (47) with O = 1 the unit operator. The summation is over all decompositions of the sets of rapidities \underline{θ}′_{β} and \underline{θ̃}_{γ}. To the matrix element of ψ we apply the first version of the crossing formula

ψ\underline{γ} ; \underline{α}(\underline{θ̃}_{γ}, \underline{θ}_{α}) = σ_{(\underline{γ̄})}^{ψ} ∑_{\begin{matrix}\underline{θ̃}_{ν}∪\underline{θ̃}_{π}=\underline{θ̃}_{γ} \\ \underline{θ}_{μ}∪\underline{θ}_{λ}=\underline{θ}_{α} \end{matrix}} S\underline{γ} ; \underline{ν}\underline{π}(\underline{θ̃}_{ν}, \underline{θ̃}_{π}) 1\underline{ν} ; \underline{μ}(\underline{θ̃}_{ν}, \underline{θ}_{μ}) C^{\underline{π}\underline{π̄}}ψ_{\underline{π̄}\underline{λ}}(\underline{θ̃}π̄+iπ_{−}, \underline{θ}_{λ}) S\underline{μ}\underline{λ} ; \underline{γ}(\underline{θ}_{γ})

(50)

where we assume that the statistics factor σ_{(\underline{γ̄})}^{ψ} of the field ψ with respect to all particles in \underline{γ} is the same for all \underline{γ} which contribute to (48) (see below). Inserting (49) and (50) in (48) we use the product formula S\underline{ς}\underline{σ} ; \underline{γ}(\underline{θ̃}_{γ})S\underline{γ} ; \underline{ν}\underline{π}(\underline{θ̃}_{ν}, \underline{θ̃}_{π}) = S\begin{matrix}\underline{ς} & \underline{σ} \\ \underline{ν} & \underline{π} \end{matrix}(\underline{θ̃}_{ν}, \underline{θ̃}_{π}). Let us first assume that the sets rapidities in the initial state \underline{θ}_{α} and the ones of the final state \underline{θ}′_{β} have no common elements which implies that also \underline{θ̃}_{ν} ∩ \underline{θ̃}_{σ} = ∅. Then we may use (ii) to get S\begin{matrix}\underline{ς} & \underline{σ} \\ \underline{ν} & \underline{π} \end{matrix}(\underline{θ̃}_{ν}, \underline{θ̃}_{π}) = 1 and we can perform the \underline{θ̃}_{ν}- and \underline{θ̃}_{κ}-integrations. The remaining θ̃-integration variables are \underline{θ̃}_{ω} = \underline{θ̃}_{ς} ∩ \underline{θ̃}_{π}, then we may write for the sets of particles \underline{ς} = \underline{μ}\underline{ω}, \underline{π} = \underline{ω}\underline{τ} and \underline{γ} = \underline{μ}\underline{ω}\underline{τ} and similar for rapidities and momenta. Equation (48) simplifies as

(ϕ(x)ψ(y))\underline{β} ; \underline{α} (\underline{θ}′_{β}, \underline{θ}_{α}) = ∑_{\begin{matrix}\underline{θ}′_{ρ}∪\underline{θ}′_{τ} =\underline{θ}′_{β} \\ \underline{θ}_{μ}∪\underline{θ}_{λ}=\underline{θ}_{α} \end{matrix}} \frac{μ!τ!}{μωτ!}S\underline{β} ; \underline{ρ}\underline{τ} (\underline{θ}′_{ρ}, \underline{θ}′_{τ}) ∫_{\underline{θ̃}_{ω}} X\begin{matrix}\underline{ρ} & \underline{τ} \\ \underline{μ} & \underline{λ} \end{matrix}

× S\underline{μ}\underline{λ} ; \underline{α}(\underline{θ}_{α})e^{i(P′_{ρ}−P_{μ})x−i(P_{λ}−P′_{τ})y} (51)

where

X\begin{matrix}\underline{ρ} & \underline{τ} \\ \underline{μ} & \underline{λ} \end{matrix} = σ_{(\underline{γ̄})}^{ψ}ϕ_{\underline{μ}\underline{ω}\underline{ρ̄}}(\underline{θ}_{μ}, \underline{θ̃}_{ω}, \underline{θ′}ρ̄ − iπ_{−})C^{\underline{ρ̄}\underline{ρ}}C^{\underline{τ}\underline{τ̄}}C^{\underline{ω}\underline{ω̄}}

× ψ_{\underline{τ̄}\underline{ω̄}\underline{λ}}(\underline{θ}′ ; τ̄ + iπ_{−}, \underline{θ̃}ω̄ + iπ_{−}, \underline{θ}_{λ})e^{−iP̃_{ω}(x−y)} . (52)

The integrand X\begin{matrix}\underline{ρ} & \underline{τ} \\ \underline{μ} & \underline{λ} \end{matrix} may be depicted as

X\begin{matrix}\underline{ρ} & \underline{τ} \\ \underline{μ} & \underline{λ} \end{matrix} = σ_{(\underline{γ̄})}^{ψ} [figure: ϕ; \underline{μ}; \underline{ρ}; \underline{ω}; \underline{ω̄}; ψ; \underline{τ}; \underline{λ}]

Similarly, if we apply for the operator product ψ(y)ϕ(x) again the second crossing formula to the matrix element of ϕ and the first one the matrix element of ψ we obtain equation (51) where X\begin{matrix}\underline{ρ} & \underline{τ} \\ \underline{μ} & \underline{λ} \end{matrix} is replaced by

Y\begin{matrix}\underline{ρ} & \underline{τ} \\ \underline{μ} & \underline{λ} \end{matrix} = σ_{(\underline{β̄})}^{ψ}ϕ_{\underline{μ}\underline{ω}\underline{ρ̄}}(\underline{θ}_{μ}, \underline{θ̃}_{ω} − iπ_{−}, \underline{θ′}ρ̄ − iπ_{−})C^{\underline{ρ̄}\underline{ρ}}C^{\underline{τ}\underline{τ̄}}C^{\underline{ω}\underline{ω̄}}

× ψ_{\underline{τ̄}\underline{ω̄}\underline{λ}}(\underline{θ}′ ; τ̄ + iπ_{−}, \underline{θ̃}ω̄, \underline{θ}_{λ})e^{iP̃_{ω}(x−y)} (53)

which means that only σ_{(\underline{γ̄})}^{ψ} is replaced by σ_{(\underline{β̄})}^{ψ}, P̃_{ω} by −P̃_{ω} and the integra-

tion variables \underline{θ̃}_{ω} by \underline{θ̃}ω̄ − iπ_{−}.

No bound states: In this case there are no singularities in the physical strip and we may shift in the matrix element of ψ(y)ϕ(x) (51) with (53) for equal times and x^{1} < y^{1} the integration variables by θ̃_{i} → θ̃_{i} + iπ_{−}. Note that the factor e^{iP̃_{ω}(x−y)} decreases for 0 < Re θ̃_{i} < π if x^{1} < y^{1}. Because P̃_{ω} → −P̃_{ω} we get the matrix element of ϕ(x)ψ(y) (51) with (52) up to the statistics factors. Therefore we conclude

ϕ(x)ψ(y) = ψ(y)ϕ(x)σ^{ψϕ} for x^{1} < y^{1}  (54)

where σ^{ψϕ} = σ_{(\underline{γ̄})}^{ψ}/σ_{(\underline{β̄})}^{ψ}. Using the assumption (45) we have with Q\underline{γ̄} = ∑_{γ∈\underline{γ}} Qγ̄

σ_{(\underline{γ̄})}^{ψ} = ∏_{γ∈\underline{γ}} σ^{ψ}(γ̄) = σ^{ψ}(Q\underline{γ̄}) = σ^{ψ}(Q\underline{β̄} − Q_{ϕ})

which is the same for all \underline{γ}, as assumed above. The last equation follows from Qγ̄ = −Q_{γ} and charge conservation which means that the the matrix elements ϕ\underline{β} ; \underline{γ} in (48) are non-vanishing if Q\underline{β} + Q_{ϕ} = Q\underline{γ}. Therefore the statistics factor of the fields ψ with respect to ϕ is

σ^{ψϕ} = \frac{σ^{ψ}(Q\underline{γ̄})}{σ^{ψ}(Q\underline{β̄})} = σ^{ψ}(−Q_{ϕ}) = 1/σ^{ψ}(Q_{ϕ})  (55)

which is in general not symmetric under the exchange of ψ and ϕ. Finally, we obtain the space like commutation rules

ϕ(x)ψ(y) = ψ(y)ϕ(x) \{ \begin{matrix}1/σ^{ψ}(Q_{ϕ}) & for & x^{1} < y^{1} \\ σ^{ϕ}(Q_{ψ}) & for & x^{1} > y^{1} \end{matrix}  (56)

where the second relation is obtained from (54) by exchanging ϕ ↔ ψ and x ↔ y . The same result appears when there are bound states. This is proved in appendix D where also the existence of double poles in bound state S-matrices is taken into account.

5.2 Application to the Z(N)-model

The statistics factors in this model are of the form (46) σ^{ψ}(α) = ω^{Q̃_{ψ}Q_{α}} where Q̃_{ψ} is the dual charge of the field ψ and Q_{α} is the charge of the particle α, therefore σ^{ψϕ} = ω^{−Q̃_{ψ}Q_{ϕ}}. The general equal time commutation rule (56) for fields ψ_{QQ̃}(x) defined by (43) in section 4 reads as

ψ_{QQ̃}(x)ψ_{RR̃}(y) = ψ_{RR̃}(y)ψ_{QQ̃}(x) \{ \begin{matrix}ω^{−R̃Q} = e^{−2πiR̃Q/N} & for & x^{1} < y^{1} \\ ω^{Q̃R} = e^{2πiQ̃R/N} & for & x^{1} > y^{1} . \end{matrix} (57)

Notice that in this model we have a more general anyonic statistics. Examples:

1. The order parameters have bosonic commutation rules with respect to

each other

σ_{Q}(x)σ_{Q′}(y) = σ_{Q′}(y)σ_{Q}(x) .

2. The disorder parameters have again bosonic commutation rules with respect to each other.

3. For the order-disorder parameters we obtain the typical commutation rule

μQ̃(x)σ_{Q}(y) = σ_{Q}(y)μQ̃(x) \{ \begin{matrix}1 & for & x^{1} < y^{1} \\ ω^{Q̃Q} = e^{2πiQ̃Q/N} & for & x^{1} > y^{1} . \end{matrix}

4. The para-fermi fields have anyonic commutation rules

ψ_{Q}(x)ψ_{R}(y) = ψ_{R}(y)ψ_{Q}(x)e^{ϵ(x^{1}−y^{1})2πiQR/N} .  (58)

These results prove the commutation rules (5) in the Introduction. The 2-point Wightman function: In order to compare these commutation rules with the explicit results of the previous section we calculate the 2-point Wightman function for the para-fermi fields ψ_{Q} and ψ_{N−Q} (with spin s = Q(N − Q)/N) in 1-particle (charge Q) intermediate state approximation. Using the result (44) we obtain

⟨ 0 | ψ_{Q}(x) ψ_{N−Q}(0)| 0⟩ = ∫ \frac{dθ}{4π} ⟨ 0 | ψ_{Q}(x)| θ⟩_{Q} _{Q}⟨ θ|ψ_{N−Q}(0)| 0⟩ + …

= \frac{1}{2π} (\frac{x^{−} − iϵ}{x^{+} − iϵ})^{ν/2} K_{ν} (M\sqrt{i (x^{+} − iϵ)}\sqrt{i (x^{−} − iϵ)}) + …

where ν = 2Q(N −Q)/N and x^{±} = t∓x. This agrees with the commutation rule (58), because for t = 0 and x > 0 using the symmetry Q ↔ N − Q, x → −x and translation invariance we obtain

⟨ 0 | ψ_{Q}(x) ψ_{N−Q}(0)| 0⟩ = ⟨ 0 | ψ_{N−Q}(x) ψ_{Q}(0)| 0⟩
= e^{iπν}⟨ 0 | ψ_{N−Q}(−x) ψ_{Q}(0)| 0⟩
= e^{iπν}⟨ 0 | ψ_{N−Q}(0) ψ_{Q}(x)| 0⟩

where ((x − iϵ) / (−x − iϵ))^{ν/2} = e^{iπνϵ(x)/2} has bee used. The asymptotic behavior is obtained from

2K_{ν} (z) → \{ \begin{matrix}Γ(ν) (\frac{z}{2})^{−ν} + Γ(−ν) (\frac{z}{2})^{ν} & for & z → 0 \\ \sqrt{\frac{2π}{z}} e^{−z} & for & z → ∞ \end{matrix}

for ν ≠ 0. Therefore the leading short distance behavior is up to constants

⟨ 0 | ψ_{Q}(x) ψ_{N−Q}(0)| 0⟩ ∼ (x^{+} − iϵ)^{−ν}
⟨ 0 | ψ̃_{Q}(x) ψ̃_{N−Q}(0)| 0⟩ ∼ (x^{−} − iϵ)^{−ν}

where the fields ψ̃_{Q}(x) are obtained by changing the sign in the exponent of (43).

Acknowledgments

We thank V.A. Fateev, R. Flume, A. Fring, A. Nersesyan, R. Schrader, B. Schroer, J. Teschner, A. Tsvelik, Al.B. Zamolodchikov and A.B. Zamolodchikov for discussions. In particular we thank V.A. Fateev for bringing the preprint [18] to our attention and F.A. Smirnov for sending a copy. H.B. was supported by DFG, Sonderforschungsbereich 288 ‘Differentialgeometrie und Quantenphysik’, partially by the grants INTAS 99-01459 and INTAS 00-561 and in part by Volkswagenstiftung within in the project ”Nonperturbative aspects of quantum field theory in various space-time dimensions”. A. F. acknowledges support from PRONEX under contract CNPq 66.2002/1998-99 and CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico). This work is also supported by the EU network EUCLID, ’Integrable models and applications: from strings to condensed matter’, HPRN-CT-2002-00325.

Appendix

A Some useful formulae

In this appendix we provide some explicit formulae for the scattering matrices and the minimal form factors which we frequently employ in the explicit computations. The S-matrix of two ’fundamental’ particles (i.e. of type 1) is [9]

S(θ) = \frac{sinh \frac{1}{2}(θ + \frac{2πi}{N})}{sinh \frac{1}{2}(θ − \frac{2πi}{N})} = − exp ∫_{0}^{∞} \frac{dt}{t} 2\frac{sinh t(1 − 2/N)}{sinh t} sinh tx

where θ is the rapidity difference defined by

p_{1}p_{2} = m^{2} cosh θ .

A particle of type α (0 < α < N) is a bound state α = (α_{1} ⋯ α_{l}) of particles of type α_{i} where α = α_{1} + ⋯ + α_{l}, in particular α = (\underbrace{1 ⋯ 1}_{α}) for all α_{i} = 1.

For the scattering of the bound state α and β we have [17]

S_{αβ}(θ) = exp 2 ∫_{0}^{∞} \frac{dx}{x} \frac{cosh x(1 − \frac{|β−α|}{N}) − cosh x(1 − \frac{β+α}{N})}{sinh x tanh(x/N)} sinh x\frac{θ}{iπ} .

The minimal form factor functions, which satisfies Watson’s equations, are obtained from the S-matrix formulae [1] and are given as (β > α)

F_{αβ}^{min}(θ) = exp ∫_{0}^{∞} \frac{dt}{t} 2\frac{sinh t (1 − \frac{β}{N} ) sinh t (\frac{α}{N} )}{sinh^{2} t tanh t/N} (1 − cosh t (1 − \frac{θ}{iπ} ))

in particular[17]

F_{11}^{min}(θ) = exp ∫_{0}^{∞} \frac{dt}{t} 2\frac{sinh t (1 − \frac{1}{N} ) cosh t\frac{1}{N}}{sinh^{2} t} (1 − cosh t (1 − \frac{θ}{iπ} ))

= −i sinh \frac{1}{2}θ exp ∫_{0}^{∞} \frac{dt}{t} \frac{sinh t (1 − \frac{2}{N} )}{sinh^{2} t} (1 − cosh t (1 − \frac{θ}{iπ} ))

= −i sinh \frac{1}{2}θ ∏_{k=0}^{∞} \frac{Γ(k + 1 − \frac{1}{N} + \frac{x}{2})}{Γ(k + \frac{1}{N} + \frac{x}{2})} \frac{Γ(k + 2 − \frac{1}{N} − \frac{x}{2})}{Γ(k + 1 + \frac{1}{N} − \frac{x}{2})} (\frac{Γ(k + \frac{1}{2} + \frac{1}{N})}{Γ(k + \frac{3}{2} − \frac{1}{N})})^{2}

and with ¯1 = (N − 1)

F_{1¯1}^{min}(θ) = exp ∫_{0}^{∞} \frac{dt}{t} \frac{sinh t\frac{2}{N}}{sinh^{2} t} (1 − cosh t (1 − \frac{θ}{iπ} ))

= ∏_{k=0}^{∞} \frac{Γ(k + \frac{1}{2} + \frac{1}{N} + \frac{θ}{2πi})}{Γ(k + \frac{1}{2} − \frac{1}{N} + \frac{θ}{2πi})} \frac{Γ(k + \frac{3}{2} + \frac{1}{N} − \frac{θ}{2πi})}{Γ(k + \frac{3}{2} − \frac{1}{N} − \frac{θ}{2πi})} (\frac{Γ(k + 1 − \frac{1}{N})}{Γ(k + 1 + \frac{1}{N})})^{2} .

There are simple relations between the minimal form factors which we essentially use in our construction which are up to constants

F_{11}^{min} (θ + \frac{iπ}{N} ) F_{11}^{min} (θ − \frac{iπ}{N} ) ∝ sinh \frac{1}{2} (θ + \frac{iπ}{N} ) sinh \frac{1}{2} (θ − \frac{iπ}{N} ) F_{12}^{min}(θ)

∏_{k=0}^{N−1} F_{11}^{min}(θ + \frac{k}{N} 2πi) ∝ ∏_{k=0}^{N−2} sinh \frac{1}{2} (θ + \frac{k}{N} 2πi) sinh \frac{1}{2} (θ + \frac{k + 1}{N}2πi)

F_{11}^{min}(θ)F_{1¯1}^{min}(θ + iπ) ∝ sinh \frac{1}{2}θ sinh \frac{1}{2} (θ + 2iπ/N) .

In equations (20) and (33) we used the function F (θ) = c_{N}F_{11}^{min}(θ) with

c_{N} = e^{iπ \frac{N−1}{N}} exp (∫_{0}^{∞} \frac{dt}{t sinh t} ((1 − \frac{2}{N} ) − \frac{sinh t (1 − \frac{2}{N} )}{sinh t})) (59)

such that the normalizations in (37) and (40) hold.

B Integrals for the Z(2)-model

The claim (15) follows from the following lemma

Lemma 3 For n = 2m + 1 odd and x_{i} = e^{θ_{i}}

f_{n}(\underline{x}) := I_{nm}(\underline{θ}, 1) − (2i)^{(n−1)/2} ∏_{1≤i<j≤n} \frac{tanh \frac{1}{2}θ_{ij}}{F (θ_{ij})} = 0 .

Proof. Again as in the proof of lemma 2 in [33] we apply induction and Liouville’s theorem. One easily verifies f_{1}(\underline{x}) = f_{3}(\underline{x}) = 0. As induction assumptions we take f_{n−2} = 0. The functions f_{n}(\underline{x}) are a meromorphic functions in terms of the x_{i} with at most simple poles at x_{i} = −x_{j} since pinchings appear for z_{k} = θ_{i} = θ_{j} ± iπ. The residues of the poles are proportional to f_{n−2} as follows from the recursion relations (iii) for both terms. Furthermore f_{n}(\underline{x}) → 0 for x_{i} → ∞. Therefore f_{n}(\underline{x}) vanishes identically by Liouville’s theorem.

Note that the integrations in the definition (9) of I_{nm} can easily be performed and with N = {1, …, n} and |K| = m

I_{nm}(\underline{θ}, 1) = ∑_{K⊂N} ∏_{k∈K} ∏_{i∈N \K} \frac{2i}{sinh θ_{ki}} .

C Proof of the main lemma

In this appendix we prove the main lemma 2 which provides the general Z(N)-form factor formula.

Proof. Similar as in the proof of lemma 1 we calculate

Res_{θ_{N−1N} =iη} ⋯ Res_{θ_{12}=iη} I_{n\underline{m}}(\underline{θ}, p_{n\underline{m}}^{O}) = \frac{m_{1} … m_{N−1}}{m_{1}! … m_{N−1}!} ( ∏_{k=1}^{N−1} ∏_{j=2}^{m_{k}} ∫_{C_{\underline{θ}′}} \frac{dz_{kj}}{R} )

× ∏_{k=1}^{N−1} ( ∏_{i=N+1}^{n} ∏_{j=2}^{m_{k}} ϕ(z_{kj} − θ_{i}) ∏_{2≤i<j≤m_{k}} τ(z_{ki} − z_{kj}))

× ∏_{1≤k<l≤N−1} ∏_{i=2}^{m_{k}} ∏_{j=2}^{m_{l}} κ(z_{ki} − z_{lj}) ( ∏_{k=1}^{N−1} ∏_{i=1}^{N} ∏_{j=2}^{m} ϕ(z_{kj} − θ_{i})) r

with r =

Res_{θ_{N−1N} =iη} ⋯ Res_{θ_{12}=iη} ( ∏_{k=1}^{N−1} ∫_{C\underline{θ}} dz_{k1}) ∏_{k=1}^{N−1} (∏_{i=1}^{n} ϕ(z_{k1} − θ_{i}) ∏_{2≤j≤m} τ(z_{k1} − z_{kj}))

× ∏_{1≤k<l≤N−1} (κ(z_{k1} − z_{l1}) ∏_{i=2}^{m_{k}} κ(z_{ki} − z_{l1}) ∏_{j=2}^{m_{l}} κ(z_{k1} − z_{lj})) p_{n\underline{m}}^{O}(\underline{θ}, \underline{z}) .

Replacing C\underline{θ} by C\underline{θ}′ where \underline{θ}′ = (θ_{N+1}, …, θ_{n}) we have used τ(0) = τ(±iη) = κ(0) = κ(−iη) = 0 and the fact that the z_{k1}-integrations give non-vanishing results only for z_{k1} = θ_{k} and θ_{k+1}, k = 1, …, N − 1. This is because for θ_{12}, …, θ_{N−1N} → iη pinching appears at (z_{11} …, , z_{N−11}) = (θ_{2}, …, θ_{N}) and (θ_{1}, …, θ_{N−1}). Defining the function

f(z_{11}, …, z_{N−11}) = ∏_{k=1}^{N−1} ∏_{2≤j≤m_{k}} τ(z_{k1} − z_{kj}) ∏_{1≤k<l≤N−1} κ(z_{k1} − z_{l1})

× ∏_{1≤k<l≤N−1} (∏_{j=2}^{m_{l}} κ(z_{k1} − z_{li}) ∏_{i=2}^{m_{k}} κ(z_{ki} − z_{l1})) p_{n\underline{m}}^{O}(\underline{θ}, \underline{z})

one obtains by means of (38) after some lengthy but straight forward calculation r =

Res_{θ_{N−1N} =iη} ⋯ Res_{θ_{12}=iη} ( ∏_{k=1}^{N−1} ∫_{C\underline{θ}} \frac{dz_{k1}}{R} ) ∏_{k=1}^{N−1} (∏_{i=1}^{n} ϕ(z_{k1} − θ_{i})) f(z_{11}, …, z_{N−11})

= (Res_{θ=iη} ϕ(−θ) ∏_{k=1}^{N−2} ϕ(kiη))^{N−1} ( ∏_{k=1}^{N−1} ∏_{i=N+1}^{n} ϕ(θ_{k+1i}))

× (f(θ_{2}, …, θ_{N}) − ( ∏_{i=N+1}^{n} \frac{ϕ(θ_{1i})}{ϕ(θ_{Ni})}) f(θ_{1}, …, θ_{N−1})) .

It has been used that f(…, z, …, z …) = f(…, z, …, z − iη …) = 0 because of κ(0) = κ(−iη) = 0. Using further the defining relation of ϕ in terms of F (37), the Jost property (27) of the ϕ-function and the properties (iii’) of (41) for the p-function we get

Res_{θ_{N−1N} =iη} ⋯ Res_{θ_{12}=iη} I_{n\underline{m}}(\underline{θ}, p_{n\underline{m}}^{O}) = (Res_{θ=iη} ϕ(−θ))^{N−1} ∏_{k=1}^{N−2} ϕ^{k}(kiη)

× I_{n−N\underline{m}−1}(\underline{θ′}, p_{n−N\underline{m}−1}^{O}) (∏_{k=1}^{N} ∏_{i=N+1}^{n} F (θ_{ki}))^{−1} (1 − σ_{1}^{O} ∏_{i=N+1}^{n} S(θ_{Ni}))

which together with the relation for the normalization constants (39) proves the claim.

D Proof of the commutation rules

In this Appendix we prove that we find the same commutation rules for two fields ϕ(x) and ψ(y) when there are bound states poles or even when the S-matrix has double poles^{3}.

Bound states: We now show that the same result (54) appears when there are bound states^{4} which means that there are poles in the physical strip. Let γ = (αβ) be a bound state of α and β with fusion angle η_{αβ}^{γ} which means that at θ_{αβ} = iη_{αβ}^{γ} the S-matrix S_{αβ}(θ) has a pole. The momentum and the rapidity of the bound state are

p_{γ} = p_{α} + p_{β}
θ_{γ} = θ_{α} − i(π − ηβ̄ ; γ̄α) = θ_{β} + i(π − ηᾱ ; βγ̄)

where ηβ̄ ; γ̄α and ηᾱ ; βγ̄ are the fusion angles of the bound states β̄ = (γ̄α) and ᾱ = (βγ̄), respectively.

We start matrix element of ψ(y)ϕ(x) (given by (51) with (53)). First we consider the contribution in the sum over the intermediate states where α ∈ \underline{ω̄} and β ∈ \underline{λ}. All the particles which are not essential for this discussion will be suppressed. Then the function ψ_{αβ}(θ̃_{α}, θ_{β}) has a pole at θ̃_{α} − θ_{β} = iη_{αβ}^{γ} such that by shifting the integration θ̃_{α} → θ̃_{α} + iπ_{−} there will be the additional contribution

\frac{i}{2} Res_{θ̃_{α}=θ_{α}} ϕᾱ(θ̃_{α} − iπ_{−})C^{ᾱα}ψ_{αβ}(θ̃_{α}, θ_{β})e^{iP̃_{α}(x−y)}e^{−iyP_{β}}

= \frac{i}{2}ϕᾱ(θ_{α} − iπ_{−})C^{ᾱα}ψ_{γ}(θ_{γ})\sqrt{2}Γ_{αβ}^{γ}e^{ixP_{α}−iyP_{γ}}

3These poles appear typically for bound state - bound state scattering. The case of higher order poles may be discussed similarly and will be published elsewhere.

4Here we follow the arguments of Quella [28].

with θ_{α} = θ_{β} + iη_{αβ}^{γ}, θ_{γ} = θ_{β} + i(π − ηᾱ ; βγ) and the fusion intertwiner Γ_{αβ}^{γ} (see e.g. [33]). Next we consider the contribution to the sum over the intermediate states where γ ∈ \underline{ω̄} and β = \underline{μ}. Then the function ϕ_{βγ̄}(θ_{β}, θ̃_{γ} − iπ) has a pole at θ_{β} − θ̃_{γ} + iπ = iηᾱ ; βγ̄ such that by shifting the integration θ̃_{γ} → θ̃_{γ} + iπ_{−} there will be the additional contribution

\frac{i}{2} Res_{θ̃_{γ}=θ_{γ}} ϕ_{βγ̄}(θ_{β}, θ̃_{γ} − iπ_{−})C^{γ̄γ}ψ_{γ}(θ̃_{γ})e^{iP̃_{γ}(x−y)}e^{−ixP_{β}}

= −\frac{i}{2}ϕᾱ(θ_{α} − iπ_{−})\sqrt{2}Γᾱ ; βγ̄C^{γ̄γ}ψ_{γ}(θ_{γ})e^{ixP_{α}−iyP_{γ}}

with θ_{α}, θ_{γ} as above and the fusion intertwiner Γᾱ ; βγ. From the crossing relation of the fusion intertwiners

C^{ᾱα}Γ_{αβ}^{γ} = Γᾱ ; βγ̄C^{γ̄γ}

we conclude that these residue terms form bound state poles cancel. The steps of the argument may be depicted as

[figure: ϕ; ψ; ᾱ; α; β] ⟶^{Res_{θ̃_{β} =θ_{β}}} [figure: ϕ; ψ; γ; ᾱ; α; β]= [figure: ϕ; ψ; γ; ᾱ; γ̄; β] ⟵^{Res_{θ̃γ =θγ}} [figure: ϕ; ψ; β; γ̄; γ]

Double poles:^{5} Form factors have more poles which are not related to bound states, they belong to higher poles of the S-matrix. First we consider the contribution to the sum over the intermediate states where the particles ¯1, 1 ∈ \underline{ω̄}, 2 ∈ \underline{μ}. Again we suppress all particles which are not essential for our discussion. Then the function ϕ_{2¯11}(θ, θ̃ − iπ, θ̃′ − iπ) has a pole at θ̃ = θ_{1} = θ + iπ/N which correspond to the bound state (2¯1) = 1 with the fusion angle η_{2¯1}^{1} = π(1 − 1/N). We shift the integrations θ̃ → θ̃ + iπ_{−} and θ̃′ → θ̃′ + iπ_{−} such that during the shift 0 < Im(θ̃ − θ̃′) < ϵ. From the θ̃-integration there will be the additional contribution

\frac{i}{2} Res_{θ̃=θ_{1}} ϕ_{2¯11}(θ, θ̃ − iπ, θ̃′ − iπ)C^{¯111¯1}ψ_{1¯1}(θ̃′, θ̃)e^{iP̃(x−y)}e^{−ixP}

= −\frac{i}{2}ϕ_{11}(θ − iπ/N, θ̃′ − iπ)\sqrt{2}Γ_{2}^{11}C^{¯111¯1}ψ_{1¯1}(θ̃′, θ_{1})e^{iP̃(x−y)}e^{−ixP} .

Further the function ϕ_{11}(θ − iπ/N, θ̃′ − iπ) has a pole at θ̃′ = θ_{2} = θ + iπ(1 − 3/N) which correspond to the bound state (11) = 2 with the fusion angle η_{11}^{2} = π2/N. From the θ̃′-integration there will be the additional contribution

5This discussion is new.

(−\frac{i}{2})^{2} \sqrt{2} Res_{θ̃′=θ_{2}} ϕ_{11}(θ − iπ/N, θ̃′ − iπ)\sqrt{2}Γ_{2}^{11}C^{¯111¯1}ψ_{1¯1}(θ̃′, θ_{1})e^{iP̃(x−y)}e^{−ixP}

= −\frac{1}{2}ϕ_{2}(θ − iπ2/N)Γ_{11}^{2}Γ_{2}^{11}C^{¯111¯1}ψ_{1¯1}(θ_{2}, θ_{1})e^{i(P_{1}+P_{2})(x−y)}e^{−ixP}

This procedure may be depicted as

[figure: ϕ; θ; 2; 1; ¯1; θ̃′; θ̃; ψ; ⟶^{Res_{θ̃=θ_{1}}}; ϕ; θ; 2; 1; 1; ¯1; θ̃′; θ_{1}; ψ; ⟶^{Res_{θ̃′=θ_{2}}}; ϕ; θ; 2; 2; 1; 1; ¯1; θ_{2}; θ_{1}; ψ]

We show this the additional term is cancelled by a contribution to the sum over the intermediate states where ¯2 ∈ \underline{ω}, 2 ∈ \underline{λ}. Then the function ψ_{¯22}(θ̃, θ) has a pole at θ̃ = θ_{3} = θ + iπ(1 − 2/N) which correspond to the double pole of the S-matrix

S_{¯22}(θ) = (\begin{matrix}sin & \frac{π}{2} & (\frac{θ}{iπ} + \frac{N−2}{N}) \\ sin & \frac{π}{2} & (\frac{θ}{iπ} − \frac{N−2}{N}) \end{matrix})^{2} \begin{matrix}sin & \frac{π}{2} & (\frac{θ}{iπ} + \frac{N−4}{N}) \\ sin & \frac{π}{2} & (\frac{θ}{iπ} − \frac{N−4}{N}) \end{matrix}

at θ = iπ(1 − 2/N). From the θ̃-integration there will be the additional contribution

\frac{i}{2} Res_{θ̃=θ_{3}} ϕ_{2}(θ̃ − iπ)C^{2¯2}ψ_{¯22}(θ̃, θ)e^{iP̃(x−y)}e^{−iyP}

= \frac{i}{2}iϕ_{2}(θ − iπ2/N)C^{2¯2}(−i) (ψ_{¯11}(θ_{2}, θ_{1})Γ¯1¯1 ¯2C_{¯11}Γ_{2}^{11}) e^{iP_{3}(x−y)}e^{−iyP} .

This procedure may be depicted as

[figure: ϕ; 2; ¯2; 2; θ̃; θ; ψ; ⟶^{Res_{θ̃=θ_{3}}}; ϕ; 1; 1; ¯1; ¯1; 2; 2; θ_{3}; θ; ψ]

The crossing relation of the fusion intertwiners

Γ_{11}^{2}Γ_{2}^{11}C^{1¯1} = C^{2¯2}Γ¯1¯1 ¯2C_{¯11}Γ_{2}^{11}

implies that this contribution again cancelled the one above. It has been used that the form factor of bound states ¯22 has a simple pole where the S-matrix S_{¯22} has a double pole and the residue is

Res_{θ̃=θ_{3}} ψ_{¯22}(θ̃, θ) = −i (ψ_{¯11}(θ_{2}, θ_{3})Γ¯1¯1 ¯2C_{¯11}Γ_{2}^{11}) .

This may be calculated as follows. By the form factor equation (iv) we have

Res_{θ_{12}=iη_{11}^{2}} Res_{θ_{34}=iη_{11}^{2}} ψ_{¯1¯111}(θ_{1}, θ_{2}, θ_{3}, θ_{4}) = 2ψ_{¯22}(θ_{(12)}, θ_{(34)})Γ¯2 ¯1¯1Γ_{11}^{2} .

Therefore using the form factor equation (iii) and the definition of the fusion intertwiners Res iS = ΓΓ we obtain

Res_{θ_{(12)(34)}=iπ(1−2/N)} ψ_{¯22}(θ_{(12)}, θ_{(34)})Γ¯2 ¯1¯1Γ_{11}^{2}

= \frac{1}{2} Res_{θ_{14}=iπ} Res_{θ_{12}=iη_{11}^{2}} Res_{θ_{34}=iη_{11}^{2}} ψ_{¯1¯111}(θ_{2}, θ_{1}, θ_{4}, θ_{3})S_{¯1¯1}(θ_{12})S_{11}(θ_{34})

= \frac{1}{2} Res_{θ_{14}=iπ} Res_{θ_{12}=iη_{11}^{2}} 2iC_{¯11} (ψ_{¯11}(θ_{2}, θ_{3})S_{¯1¯1}(θ_{12})S_{11}(θ_{34}) − ψ_{¯11}(θ_{2}, θ_{3}))

= iC_{¯11} (ψ_{¯11}(θ_{2}, θ_{3})(−i)Γ¯1¯1 ¯2Γ¯2 ¯1¯1(−i)Γ_{2}^{11}Γ_{11}^{2})

which implies the residue formula used above. This procedure may be depicted as

[figure: ψ; θ_{1}; θ_{2}; θ_{3}; θ_{4}; ¯1; ¯1; 1; 1] ⟶^{Res_{θ_{14}=iπ}}[figure: ψ; θ_{1}; θ_{2}; θ_{3}; θ_{4}; ¯1; ¯1; 1; 1]− [figure: ψ; θ_{1}; θ_{2}; θ_{3}; θ_{4}; ¯1; ¯1; 1; 1] ⟶^{Res_{θ_{12}=iη{2 \atop 11}} Res_{θ_{34}=iη{2 \atop 11}}}[figure: ψ; ¯1; ¯1; 1; 1; ¯2; 2]

Note that the last graph, as an on-shell graph, resembles (half of) the ‘box’ Feynman diagram which was used to investigate the double poles of bound state S-matrices (see e.g. [29, 30])

The general case: Finally we consider the general case that the sets rapidities in the initial state \underline{θ}_{α} and the ones final state \underline{θ}′_{β} have also common elements. Then after inserting (49) and (50) in (48) there will be S-matrices S\begin{matrix}\underline{ς} & \underline{σ} \\ \underline{ν} & \underline{π} \end{matrix}(\underline{θ̃}_{ν}, \underline{θ̃}_{π}) which produce additional poles in the physical strip which would produce additional residue contributions while shifting the integration contours. However, we can remove these S-matrices by using again the crossing relation (ii) and move all the lines of common rapidities to the left or right as depicted as follows

[figure: ϕ; ψ]⟶ [figure: ϕ; ψ]

Then we can apply the procedure as above.

References

[1] M. Karowski and P. Weisz, Exact form factors in (1+1)-dimensional field theoretic models with soliton behavior, Nucl. Phys. B139, 455–476 (1978).

[2] H. M. Babujian, A. Fring, M. Karowski, and A. Zapletal, Exact form factors in integrable quantum field theories: The sine-Gordon model, Nucl. Phys. B538, 535–586 (1999).

[3] F. Smirnov, Form Factors in Completely Integrable Models of Quantum Field Theory, Adv. Series in Math. Phys. 14, World Scientific, 1992.

[4] H. Babujian and M. Karowski, Exact form factors in integrable quantum field theories: The sine-Gordon model. II, Nucl. Phys. B620, 407–455 (2002).

[5] M. Karowski, On the bound state problem in (1+1) dimensional field theories, Nucl. Phys. B153, 244 (1979).

[6] K. M. Watson, Some general relations between the photoproduction and scattering of pi mesons, Phys. Rev. 95, 228–236 (1954).

[7] H. M. Babujian, Correlation function in WZNW model as a Bethe wave function for the Gaudin magnetics, In ‘Gosen 1990, Proceedings, Theory of elementary particles’ 12-23. (see high energy physics index 29 (1991) No. 12257).

[8] H. Babujian, Off-shell Bethe Ansatz equation and N point correlators in SU(2) WZNW theory, J. Phys. A26, 6981–6990 (1993).

[9] R. Köberle and J. A. Swieca, Factorizable Z(N) models, Phys. Lett. B86, 209–210 (1979).

[10] R. Z. Bariev, On The Rotational Symmetry Of The Spin Spin Correlation Function Of The Two-Dimensional Ising Model, Phys. Lett. A55, 456–458 (1976).

[11] B. M. McCoy, C. A. Tracy, and T. T. Wu, Two-Dimensional Ising Model As An Exactly Solvable Relativistic Quantum Field Theory: Explicit Formulas For N Point Functions, Phys. Rev. Lett. 38, 793–796 (1977).

[12] M. Sato, T. Miwa, and M. Jimbo, Studies on holonomic quantum fields: I, Proc. Japan Acad. 53, 6–146 (1977).

[13] B. Berg, M. Karowski, and P. Weisz, Construction of Green functions from an exact S matrix, Phys. Rev. D19, 2477–2479 (1979).

[14] A. B. Zamolodchikov, Integrals Of Motion In Scaling Three State Potts Model Field Theory, Int. J. Mod. Phys. A3, 743–750 (1988).

[15] V. A. Fateev, Integrable deformations in Z(N) symmetrical models of conformal quantum field theory, Int. J. Mod. Phys. A6, 2109–2132 (1991).

[16] V. A. Fateev and A. B. Zamolodchikov, Parafermionic Currents In The Two-Dimensional Conformal Quantum Field Theory And Selfdual Critical Points In Z(N) Invariant Statistical Systems, Sov. Phys. JETP 62, 215–225 (1985).

[17] M. Karowski, Field Theories in 1+1 Dimensions with Soliton Behaviour: Form Factors and Green’s Functions, Lecture Notes in Physics (Springer) 126, 344 (1979).

[18] A. N. Kirillov and F. A. Smirnov, ITF preprint 88-73P, Kiev (1988).

[19] G. Delfino and J. L. Cardy, Universal amplitude ratios in the two-dimensional q-state Potts model and percolation from quantum field theory, Nucl. Phys. B519, 551–578 (1998).

[20] M. Jimbo, H. Konno, S. Odake, Y. Pugai, and J. Shiraishi, Free field construction for the ABF models in Regime II, J. Statist. Phys. 102, 883–921 (2001).

[21] F. H. L. Essler and R. M. Konik, Applications of massive integrable quantum field theories to problems in condensed matter physics, (2004), In ‘Shifman, M. (ed.) et al.: From fields to strings, vol. 1’ 684-830.

[22] A. M. Tsvelik, Quantum field theory in condensed matter physics, Cambridge, UK: Univ. Pr. (1995) 332 p.

[23] A. O. Gogolin, A. A. Nersesyan, and A. M. Tsvelik, Bosonization in Strongly Correlated Systems, Cambridge, UK: Univ. Pr. (1999).

[24] J. Links, H. Zhou, R. McKenzie, and M. Gould, Algebraic Bethe ansatz method for the exact calculation of energy spectra and form factors: applications to models of Bose-Einstein condensates and metallic nanograins, J. Phys. A36, R63–R104 (2003).

[25] H. Babujian and M. Karowski, Exact form factors for the scaling Z(N)-Ising and the affine A(N-1) Toda quantum field theories, Phys. Lett. B575, 144–150 (2003).

[26] H. Babujian and M. Karowski, Sine-Gordon form factors and quantum field equations, J. Phys. A35, 9081–9104 (2002).

[27] M. Y. Lashkevich, Sectors of mutually local fields in integrable models of quantum field theory, (1994), LANDAU-94-TMP-4, unpublished.

[28] T. Quella, Formfaktoren und Lokalitaet in integrablen Modellen der Quantenfeldtheorie in 1+1 Dimensionen, (1999), Diploma thesis FU-Berlin (1999), unpublished.

[29] S. R. Coleman and H. J. Thun, On the prosaic origin of the double poles in the sine-Gordon S-matrix, Commun. Math. Phys. 61, 31–51 (1978).

[30] H. W. Braden, E. Corrigan, P. E. Dorey, and R. Sasaki, Affine Toda field theory and exact S matrices, Nucl. Phys. B338, 689–746 (1990).

[31] G. Delfino and G. Mussardo, The Spin-spin correlation function in the two-dimensional Ising model in a magnetic field at T = T(c), Nucl. Phys. B455, 724–758 (1995).

[32] C. Acerbi, G. Mussardo, and A. Valleriani, On the form factors of relevant operators and their cluster property, J. Phys. A30, 2895–2914 (1997).

[33] H. Babujian and M. Karowski, The exact quantum sine-Gordon field equation and other non- perturbative results, Phys. Lett. B471, 53–57 (1999).