Form Factors in Completely Integrable Models of Quantum Field Theory

FORM FACTORS IN COMPLETELY INTEGRABLE MODELS OF QUANTUM FIELD THEORY ADVANCED SERIES IN MATHEMATICAL PHYSICS Editors-i...

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FORM FACTORS IN COMPLETELY INTEGRABLE MODELS OF QUANTUM FIELD THEORY

ADVANCED SERIES IN MATHEMATICAL PHYSICS Editors-in-Charge H Araki (RIMS, Kyoto) V G Kac (M17) D H Phong (Columbia University) S-T Yau (Harvard University)

Associate Editors L Alvarez-Gaume (CERN) J P Bourguignon (Ecole Polytechnique, Palaiseau) T Eguchi (University of Tokyo)

B Julia (CNRS, Paris) F Wilczek (Institute for Advanced Study, Princeton)

Published Vol. 1: Mathematical Aspects of String Theory edited by S-T Yau Vol. 2: Bombay Lectures on Highest Weight Representations of Infinite Dimensional Lie Algebras

by VG Kac and AKRaina Vol. 3: Kac-Moody and Virasoro Algebras: A Reprint Volume for Physicists edited by P Goddard and D Olive

Vol. 4: Harmonic Mappings, Twistors and a-Models edited by P Gauduchon Vol. 5: Geometric Phases in Physics edited by A Shapere and F Wilczek Vol. 7: Infinite Dimensional Lie Algebras and Groups edited by V Kac

Vol. 8: Introduction to String Field Theory by W Siegel Vol. 9: Braid Group, Knot Theory and Statistical Mechanics edited by C N Yang and M L Ge Vol. 10: Yang-Baxter Equations in Integrable Systems edited by M Jimbo Vol. 11 : New Developments in the Theory of Knots edited by T Kohno

Vol. 12: Soliton Equations and Hamiltonian Systems by L A Dickey Vol. 15: Non-Perturbative Quantum Field Theory - Mathematical Aspects and Applications

Selected Papers of Jurg Frohlich Vol. 16: Infinite Analysis - Proceedings of the RIMS Research Project 1991 edited by A Tsuchiya,T Eguchi and M Jimbo

Advanced Series in Mathematical Physics - Vol. 14

FO RM FACTO RS IN CO M PLETELY INTEGRABLE MO D ELS OF QUANTUM FIEL D TH EO RY

F. A. SMIRNOV St. Petersburg Branch of Steklov Mathematical Institute

World Scientific Singapore • NewJersey • London • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P. O. Box 128, Fatter Road, Singapore 9128 U.S.A. office: Suite 1B, 1060 Main Street, River Edge, NJ 07661 U.K. office: 73 Lynton Mead, Totteridge, London N20 8DH

FORM FACTORS IN COMPLETELY INTEGRABLE MODELS OF QUANTUM FIELD THEORY Copyright © 1992 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form orbyany means, electronic ormechanical, including photocopying, recording orany information storage and retrieval system now known or to be invented without written permission from the publisher.

ISBN 981-02-0244-X 981-02-0245- 8 (pbk)

Printed in Singapore by Utopia Press.

CONTENTS

Introduction

vii

0. Completely Integrable Models of Quantum Field Theory

1

1. The Space of Physical States . The Necessary Properties of Form Factors.

7

2. The Local Commutativity Theorem

17

3. Solition Form Factors in SG Model

29

4. The Main Properties of the Soliton Form Factors

47

5. Breathers Form Factors in SG Model

69

6. Properties of the Operators jµ, T.,,, exp (±i!) in SG Model

81

7. Form Factors in SU ( 2)-Invariant Thirring Model

99

8. Form Factors in 0(3 )-Nonlinear o-Model

109

9. Asymptotics of Form Factors

145

10. Current Algebras

163

Appendix A. Form Factors in SU (N)-Invariant Thirring Model (SU(N) Chiral Gross-Neveu Model)

181

Appendix B. Phenomenological Reasonings

203

References

207

This page is intentionally left blank

II

INTRODUCTION

During the last several years the theory of completely integrable models in (1 + 1)-dimensions has attracted much attention of the experts in quantum field theory. The possibility to obtain exact, non-perturbative results for these models is of great importance. Recent activity in the field was initiated by the papers by Faddeev and Korepin [1,2] devoted to the quasiclassical quantization of solitons in sine-Gordon model (SG). The connection of SG with massive Thirring model established by Coleman [3] stimulated further interest to SG model in particular and to the quantization of solitons in general. This interest was stimulated also by the analogy of (1 + 1)-dimensional nonlinear o-model (NLS) and (3 + 1)-dimensional Yang-Mills theory outlined by Polyakov [4]. Korepin and Faddeev [1] realized the most important feature of the completely integrable quantum models, which is the factorizability of scattering. They predicted the exact S-matrix for SG model with the special values of coupling constant corresponding to the absence of the reflection of solitons. This hypothesis was used by Zamolodchikov as the basis for phenomenological calculation of S-matrix for SG model [5]. Further development of the phenomenological (bootstrap) approach based on factorizability, unitarity and crossing-symmetry led to the exact calculation of S-matrices for many completely integrable models (see the review by A.B. and Al. B. Zamolodchikovs [6] and the papers by Karowski and collaborators [7-9]). The quantum inverse scattering method (QISM) formulated by Faddeev, Sklyanin and Takhtajan [10-12] allows us to unify the theories of the completely integrable quantum models. QISM makes it possible to pass from local fields to the physical creation-annihilation operators. Many vii

viii

Introduction

S-matrices predicted phenomenologically have been confirmed dynamically in the framework of QISM (the paper by Korepin [13] was important in this respect). The achievements of QISM are reflected in several reviews [14-16]. In the context of this book, the paper by Thacker and collaborators [17] is worth special attention. In this paper the quantum version of the Gelfand-Levitan-Marchenko (GLM) equations was formulated. These equations allow us to get the inverse transform from the creation-annihilation operators of physical particles to the local fields. Further development of GLM method was given in the papers by Gockeler, Kulish and the author [18-20]. Let us note, however, that all the papers mentioned above deal with models whose physical vacuum coincides with the Fock vacuum of the local fields. These models are rather trivial (in particular relativistic models cannot be of this kind). That is why their consideration is mostly of methodological character. The first model with nontrivial ground state considered in GLM method was XXZ ferromagnet with domain wall [21]. Let us describe the situation in QISM up to 1983. Up to that time the method had been applied to many models and had allowed one to calculate exact S-matrices. The theory of factorizable S-matrices was further developed (the paper by Kulish, Reshetikhin and Sklyanin [22] is of particular importance). However, no remarkable out-of-shell results had been obtained. Thus the calculation of the Green functions and the objects connected with them became the most important problem. In 1984 Korepin proposed a method of presenting the Green functions for Bose gas with nonzero density in the form of special series [23]. We cannot describe this method in more detail (see review [24]) than saying that its greatest advantage is the calculation of the long distance asymptotics of the Green functions for several interesting models. The development of GLM method was also continued. In the paper [25] the author obtained GLM equations for SG model. These equations allowed one to calculate the matrix elements of the local field (form factors) corresponding to the physical states which contain SG bosons (breathers) [26]. The spectrum of SG contains, besides breathers, two-component particulessolitons . According to Ref. [27] the GLM equation when applied to solitons lead to a set of matrix Riemann problems. At first sight there is no hope of solving these Riemann problems. Here, however, two circumstances help. First, for certain values of the coupling constant, the reflection of solitons disappears . In that case the Riemann problems become rather simple and

Form Factors in Completely Integrable Models of Quantum Field Theory ix

soliton form factors for the reflectionless case have been calculated in the papers [28 , 29]. Second, GLM equations provide some additional information. This additional information, known soliton form factors in reflectionless case and known breather form factors, provided the author with the heuristic base necessary for the calculation of soliton form factors [27, 30] (the details are given in [31]). The very possibility of obtaining exact formulae for form factors reflects the remarkable selfconsistency and beauty of completely integrable models. It is necessary to mention the papers by Kirillov and Khamitov [32, 33]. In these papers the following programme proposed by E.K. Sklyanin was realized . It is known that the sinh-Gordon model, possessing phenomenologically very simple spectrum which involves only one boson, cannot be investigated in the usual framework of QISM. However, one can try to write down the hypothetical GLM equations for the model by analogy with SG. The proposed method of verification of the equations is very natural: one has to make sure that the local fields defined by the equations do satisfy local commutativity. Analyzing the results of the papers [32, 33] and having in mind possible applications to more complicated models, the author realizes that the GLM equations are not themselves essential for the proof of the local commutativity and all the necessary requirements can be formulated directly in terms of form factors. It should be mentioned that attempts to formulate the full set of requirements on form factors had been made by Karowski and collaborators [34], but their set was not complete and possessed a number of drawbacks because the factorizability was not taken into account in all its aspects. In the paper [35, 36] Kirillov and the author, considering the particular example of SU(2)-invariant Thirring model (ITM), realized the following programme. The requirements on form factors were formulated which were shown to ensure local commutativity. These requirements had the form of a system of equations, which were solved explicitly. This last step was the most complicated one because we had to solve a matrix Riemann problem. The SG experience was of great importance. In the paper [37] Kirillov and the author realized a similar programme for the 0(3) nonlinear o-model (NLS). The importance of this model was mentioned above. In spite of the fact that the spectrum and the S-matrix for this model were predicted several years ago, for a long time the model could not be treated by QISM because the usual formalism of the method

x

Introduction

was hardly applicable to it. It should be mentioned that the structure of the form factors described in Refs. [30, 35, 37] is very general . It was demonstrated in the paper [38] where the formulae for the form factors in SU(N)-invariant Thirring model (chiral Gross-Neveu model) were presented. In the present work we summarize the results of the papers [27, 30, 31, 35-38]. We follow the axiomatic approach outlined in the papers [35-38]. The main goals of the present work are: first, to describe the general scheme of the calculation of form factors; second, to present complete proofs and reasonings which had been omitted in the original papers because of their limited volume. Let us describe briefly the organization of the work. It consists of eleven sections and appendices. Section 0 contains the necessary general information about completely integrable models of quantum field theory. The spectra and S-matrices of SG, ITM, NLS are presented. In Sec. 1 we formulate a set of requirements on form factors (axioms). The origination of these axioms is clarified by considering the well-known example of two-particle form factors. A sketch of possible phenomenological deduction of the axioms is given in Appendix 2. Section 2 is the ideological base of our programme. It contains the local commutativity theorem which is formulated as follows. If the form factors of some operator satisfy the axioms formulated in Sec. 1 then the operator is local. In Sec. 3 we present the explicit formulae for SG soliton form factors. In Sec. 4 we show that the SG form factors defined in Sec. 3 do satisfy the axioms. In Sec. 5 we discuss the SG form factors corresponding to both solitons and breathers. In Sec. 6 some additional properties of SG form factors are established which allow us to identify the operators defined by the form factors with natural local operators occurring in the model. In Sec. 7 the form factors in ITM are obtained. In Sec. 8 a procedure is described of obtaining NLS form factors from ITM ones. The explicit formulae for NLS form factors are presented and their properties studied. Section 9 deals with the asymptotics of the form factors for large momentum transfer.

Form Factors in Completely Integrable Models of Quantum Field Theory

Xi

In Sec . 10 the singularities of the commutators at the origin of coordinates are studied . The asymptotics obtained in Sec. 9 are essential here. Appendix 1 considers the SU ( N)-invariant Thirring model. The content of Appendix 2 has already been mentioned. I am grateful to L.D. Faddeev for his interest in my work and continued support. I also gratefully acknowledge my thanks for many useful discussions with my colleagues at the Leningrad Branch of Steklov Mathematical Institute: A.G. Izergin , V.E. Korepin , P.P. Kulish , N. Yu. Reshetikin , M.A. SemenovTian- Shansky, E.K. Sklyanin and L . A. Takhtajan . Special thanks are due to A.N . Kirillov for fruitful collaboration.

0 COMPLETELY INTEGRABLE MODELS OF QUANTUM FIELD THEORY

In this section we present briefly the necessary facts concerning the completely integrable models of quantum field theory with massive spectra. We parametrize the massive particles by their rapidities 0 (po = m ch ,Q, pl = m sh /Q, m being the mass of the particle ), and the isotropic index e if the particle possesses internal degree of freedom . The scattering in these models is factorized : the many-particle scattering process is reduced to two-particle processes , momenta of particles are not subjected to any changes. The Smatrix of two particles with rapidities 31, 02 will be denoted by SE1;E2 (/3), where 0 = 1,31 - N21, E1 ,E2 are characteristic of the internal degrees of freedom of "in", ei,e2 of "out" particles . The factorization of scattering is equivalent to the validity of the equation of triangles [6] (Yang-Baxter equation [38, 40]): SE/ EZ(^1 - /32)SE 1/ ,£s(Q1 - /33)SE 1 £ii/(/32 - /33) 1 2

qq 11/ 3 22 3 RR E3(^1 - 03)SE^ EÎ(^1 - N2) = SE2iE3(l^2 C 2 1 C 3 - N3)SE/ 1 3 1 2

Note that summation over repeated indices is always implied . The S-matrix should also satisfy the requirements of unitarity and crossing symmetry: E1,E2 1

[[''E1 b£2

# E1 1E2 2

1

2

1

SE1^£2 (iri - l-) =

1

2

CE1

E// 1

S£E1

; ,L2 // £2 1

(1) (/.^)C£1 E1

2 Form Factors in Completely Integrable Models of Quantum Field Theory

where C is a matrix satisfying two requirements: Ct = C, C2 = I. The value of C will be important. In the basic part of the book three models are considered which are the sine-Gordon model (SG), SU(2)-invariant Thirring model (ITM) and 0(3) nonlinear o-model (NLS). SG model is described by the Lagrangian z CSG = J((Ou)2 + Q2 (cos( 3u) - 1))dx where u is a Bose-field. Great attention has been paid to the quantization of SG model [1,2,5,12,13,41-45]. Let us summarize the necessary results. We use the following quantity as the coupling constant, 2 = Sir )32 , 0 < f < oo .

SG model is equivalent [3] to the neutral sector of massive Thirring model (MTM) whose Lagrangian is

GMTM =

J

(ZW YPaµ -

2 ( yµ02

-

m3V, )

dx ,

where bi = {t)i }i=1 2 is the Fermi field, -yo = ol, yl = io2 , of (i = 1 , 2,3) axe the Pauli matrices , g = Z. (9L) . The correspondence between local operators was established by Coleman [3]: 2a eµvavu= -

.7µ

a Cos flu = _VV'

(2)

where jµ is the fermion current The spectrum of SG model contains solitons possessing an internal degree of freedom (soliton-antisoliton ). The possible values of the corresponding index a are chosen as ± z . Sblitons correspond to fermions in MTM. Two ranges should be distinguished. The range ar < { < oo is the repulsive one, there are no bound states for these values of {. The point t = a corresponds to the free fermion case , interaction disappears and the model is equivalent to the massive free Fermi field. The range 0 < ^ < it is an attractive one ; here bound states (breathers ) exist . These are [] breathers. The

Completely Integrable Models of Quantum Field Theory 3

mass of m-breather (m = 1, ... , [!]) is equal to 2M sin (M is the soliE ton mass). For the sake of uniformity we allow ourselves an inaccuracy by prescribing to m-breather the index e = m. Strictly speaking, m does not correspond to any internal degree of freedom. However, this prescription will be very useful. Soliton-soliton scattering matrix is given by [5]: Si'a] (P) = S_ 2 z i (/3) = So(Q) , ]1 h

Si'- i2 (Q) 2 = S_2 1'i (+l)

sh

S0(13) {A (/3{F^ - iri) h

S' # (Q) = S.L 21--' f (/3) = sh

A/1

Ali

SO

(3)

SEl,es (i3) = 0 el + e2 0 e1 + e2

So(/) = - exp I_i

j - sin(ic/) sh (1 2 fk) dre s'c ch (2) sh ( )

Soliton-breather scattering matrix is given by

S

2(fl) ny

=

,f

Sm,-# ( N) F1,£2

12{n1 (fl) =

S6

b 26' SSr m(/3) (^

Sy

2;F1

= -y,m

z

,

/j (/3) = 6126f111SS,m(N) 1.

-

icosz+sh (/3-,z(rn+1-2j)) (4) SS,m('3)

11=1icosI-sh

(/3- Z(m+1-2j))

Breather-breather scattering matrix is given by

S,`,;, 2(13) = bn bn'cth 2 1 13- Z(m2 n)1; I cth x th x th 2

(13 + i

2

(fl+

(m 2 n )

Im nl

th

1

I /3- i Im2 nlt I min(m,n)-1

a +i f\1 11

(/3- 2 (Im_ nJ+2j) I cth2 2(fl-nI+2j)) (5)

From Eq. (3) it follows that the values of the coupling constant a/v, v = 1, 2, ... , are very specific because at these points S j ' (/3) _ ]' 2


S"1 3 (/l) = 0, i.e., the reflection of solitons is absent. In particular, for s12

= 7r the S-matrix is equal to -1, which corresponds to the free fermion case. Nontrivial matrix C in Eq. (1) exists only for soliton-soliton scattering: C=0.1.

An important circumstance should be mentioned which was a subject of controversy a few years ago. In the papers [42, 43] it is established that there are phase transitions at the points ^ = iric, ic > 2. The existence of the phase transitions results in changes of the spectrum of excitations at these points. In the paper [45] it is established that the spectrum of excitations is the same for all the values of coupling constant in the range 1; > 7r (this is the picture we follow). The contradiction has the following solution. In papers [42, 43] and [45], different regularizations were used for the quantization of SG model which is sensitive to the regularization used for C > 7r. Thus papers [42, 43] and [45] deal with different regularized models. As is mentioned above, we follow the picture presented in [45], which is in our opinion more natural from the phenomenological point of view: there is no need for phase transitions because the S-matrix is a regular function of the coupling constant. This choice does not mean that we reject the regularization given in [42, 43]; we only consider this regularization more useful, rather, for statistical mechanics than for quantum field theory. The second model we deal with is ITM with the Lagrangian GITM = J

(Z^^µaµT^ - g( ° lµ )1 âyµ^))dx ,

is the Fermi field , i = 1, 2 being the spinor index, a = 1, 2 where being the isotopic index, a' are the Pauli matrices acting on the isotopic indices . The model possesses the property of asymptotic freedom. Mass is generated dynamically via the mechanism of dimensional transmutation [46] . Physical excitations are two-component massive particles (kinks) which will be parametrized by their rapidities and isotopic & = 1, 2. Mass of the kink will be denoted by M. The S-matrix is given by

SE

1

ES (N)

(Z-

(m =

Sp( a )

1 ( a-'lrt

3

'l 2

-

(2+) r(- 1

atb61 C2

(6)

Completely Integrable Models of Quantum Field Theory 5

Kinks do not create any bound states. The S-matrix satisfies (1) with C = io2. Also the S-matrix is SU(2)-invariant, which means that Sew 1el2l (Q )9 e I 9E' = 9` l ell S£.I/"2it (I) E1 f, 1 , 1 , 1 ,

(7)

for arbitrary g E SU(2). Notice that the charge conjugation matrix C for ITM does not satisfy the condition C' = C (actually C' = -C). This fact reflects the unusual property of ITM kinks: they are spin-4 particles. However it is possible to modify the base in the space of states and to introduce auxiliary particles which possess the usual statistics. The details are given in Sec. 7. The connection between ITM and SG is also discussed in Sec. 7. NLS model exhibits the greatest difficulty for quantization [47, 49], although phenomenologically the spectrum of excitations has been predicted several years ago [50-54]. The model is described by the Lagrangian £NLS = 1 J(a naona)dx 9

where a = 1, 2, 3, and the constraint (nl)2 + (n2)2 + (n3 )2 = 1 applies. The model possesses asymptotic freedom. Physical excitations are massive (due to dimensional transmutation ) vector particles. We parametrize them by the rapidity /3 and isotopic index E = 1, 2, 3 and denote the mass of the particle by M. The S-matrix is given by

iel 1 E'

x

1 (Q

+

tri )(/3

- 2ai)

{i3(a - ai) 5el bE , - 2

7ri (,3

b`, + 2aib£ 1,e, 5c'1 - iri)b"2 El E

1J

(8)

The S-matrix is 0 ( 3)-invariant , i.e., satisfies (7) with g E 0(3). Matrix C in Eq . ( 1) is equal to the unit matrix . Bound states are absent. Formally, this is all the necessary preliminary information about the completely integrable models we need . If the reader should think this information is too brief we refer him to the original papers cited in this section.

1 THE SPACE OF PHYSICAL STATES. THE NECESSARY PROPERTIES OF

FORM FACTORS.

For the description of the space of physical states we use the Zamolodchikov-Faddeev operators. The existence of these operators is an axiom in our approach. Their concrete realizations in the framework of QISM for some models can be found in [25, 55]. Generally speaking, we could avoid using these operators but in that case the methodology would become more complicated and bulky. The Zamolodchikov-Faddeev operators ZE(//3), Z, *(,8) create the following algebra: pp (Nl - N2)ZE^(/322))ZE1( N1 )

ZE1 (#p {l)ZE2 (/02) = SE1 E

//^^

ZE1(Y 1)ZE2(N2) ZEZ(N2)ZE1(^1)SE1,E22(31 ZE1(,

1)ZE2 (Q2)

2

RR 02) ,

- 31)ZE1(N1) +6E^

S,2C

(9)

6(01

- N2)

There is the physical vacuum (ph), and the operator ZE(Q) annihilates the vacuum. Operator Z,*(/3) creates a particle with rapidity ft and internal degree of freedom (for breathers in SG - with the number) C. The space of states is generated by vectors ,8.....

Z%( /lri) ... ZE1(N1)Iph)

7h'1) f,,

and the dual space E1,....

/^

(/^1,... ,/ =

/^ r7 (phIZ'1(/31)...Z(,8.) E° /^

7


Let us order the rapidities /31i ... , /3n in accordance with increasing values , r il < /3i, ... < /3tn. "In" and "out" states are given by the formulae 1 in A

/j out I#,i...

,Pn)£1,...

_ ,En

-

n' ...

i(7j , /îl) fin,...,til ,

Ip(7

1 //7^, /Eil....tin i1,... ,N1

"In" and "out" states form the complete base in the space of state. It is clear from Eq. (9) that "in" states transform to "out" states via the total S-matrix. Consider any local operator O(xo, x1) and construct the matrix element /'mlO(x0, xOP .... ,Y1)£n.... ,t1

exp(iEpµ(/3j')xµ - iEpµ()3j)xµ) Emr • ,Ei /^ / X fE1,.,En Wm,... ,N1IN1,...

The will be called "form factor". e function m , , 1 Q1, ... , fl,••• ,En Usually, in the Quantum Field Theory the term "form factor" is used for the matrix element calculated between "in" or "out" states (frequently this term is used only for two-particle matrix elements). The term "form factor" in our meaning is a special generalization of the usual one for the theory with factorized scattering. If the sets /3 and /3' are ordered we obtain the matrix element taken between "in" or "out" states. For arbitrary sets and /3' we have some "mixed", neither "in" nor "out" states. In the present section the requirements which form factors should satisfy are formulated. To give an intuitive idea about these requirements we would like to consider, following [34], the two-particle form factors whose properties are well-known from the general course of Quantum Field Theory (see for example [56, 57]). Consider for example the two-fermion form factor of current jµ in MTM (two-soliton form factor of e,, 2v&,u in SG): 1

(ph Ijµ(0)I a1, 132) 1,E9 =-F(,31 1

M E 1,C2

2 E(eP' - (-1)1'e-#j) =1

From manuals on Quantum Field Theory it is known that F depends really on the variable S = (p(/31) + p(/32))2 = 2M2(1 + ch (/31 - /32)), being the boundary value of an analytical function of S defined on the complex plane with a cut from 4M2 to 00 (Fig. 1). The possible poles are situated on the

The Space of Physical States. The Necessary Properties of Form Factors. 9

O

F

0

4M2

Fig. 1.

segment [0, 4M2]. These poles correspond to bound states. The value of the analytical function at the point S - iO(S E R) coincides with

(phIj,. (0)I)31, #2)€1 e2 Evidently the plane with the cut transforms into the half-strip 0 < Im /3 < 21r, Re /3 > 0, in terms of the variable /3 = 1,61 - /321 . The line Im /3 = 0 corresponds to the upper, while Im /3 = 2ir to the lower shore of the cut. The segment [0, 4M2] is a pre-image of the segment [0 , 2iri], being covered twice by inverse transformation. Consider for definiteness case N2 > /31 and return to the function f (/31, /32),1,6, . It is clear from the very definition that M1,

Sc1 ez 031 - 02) = f(02, N1) ,2,61 .

(10)

The general properties described above result in the following relation for /32>01 (11) f(01, #2 + 2iri )61,62 = f (N2) h'1 )e3,e, Equation ( 11) applies to all the values of /31i /32 due to continuity. Combining ( 10) and ( 11) one obtains the following equation: f(l31) 132 +2ai)61i62 = f(/31)l32)Ei,2S, : 2'(Q1 -Qz)

(12)

which is the Riemann problem for the functions f (#,, ,32), being considered as functions of /32. Function f (/31,/32) has simple poles on the segment Re /32 = /31, which corresponds to bound states. Two simple poles at the points /32 = 31 + ia , l2 = 27ri + /31 - is correspond to the bound state with the mass 2M cos 2 . These facts taken together provide us with the possibility of explicit calculation of f (Q1 ) /32)e1ie2. The Riemann problem

10


in this case is solvable trivially because the S-matrix can be diagonalized via a transformation which does not depend on Nl, a2. Consider the form factors

/Let us return to the general case .

f (Ql, ... , ^n )£1,... ,En • It will become clear from what follows that other form factors can be reconstructed from these . It follows from the very definition that f (h61, ... ,flu, Pi+l, ... = f (Ni,...

... ,En Jf ;;E }1 Pi i ,+1

//jj /ii+1, fi) ...

04 1) (13)

Equation (13) will be called the symmetry property. How can one generalize Eq. (12) to the case of arbitrary n? In the paper [34] some attempt was made in the framework of the general field theoretical approach. However, the meaning of the equations presented in [34] is not very clear. Form factors which really depend on n - 1 real variables enter these equations as functions of all possible S-variables, i.e. of n 2 1 variables. Using GLM equations, the author came in papers [27, 30] to the simple and fruitful generalization of Eq. (12): f(F^1,...

fn + 27ri)E1,...£n = f(On,011... ,Qn-1)£n,£1.... )£n-1 . (14)

In this equation the specific features of factorized scattering are taken into account . We will not present here any derivation of Eq. (14). Its validity will be proven a posteriori in the next section. Using Eq. (13), Eq. (14) can be rewritten in the form f(131,...

Nn + 27ri)£1,...,En =

f(01,...

n ,fl

)£1.. . . , e.

/^ \ En-y , rl I T_' /j /j X ^'Cn-1,T1 n l e' ( )3n-1 - Nn)S£n-4,1'. (Nn-Z - Qn) . . SE,£n (N1 -fin)

(15) Equation (14) is an axiom in our approach. The equivalent Eq. (15) is a Riemann problem for the tensor-valued function. At first sight there is no hope of solving this problem. However, the miraculous fact is that for all the models considered this problem can be solved explicitly. In our opinion this is a consequence of some hidden symmetry of the problem. What should be said about f (#1, ... fn )E1,... ,en , considered as a function of fn? We require it to be an analytical function in the strip 0 < Im fn < 27r

,,..

The Space of Physical States. The Necessary Properties of Form Factors.

11

whose only singularities are simple poles . In contrast to the two-particle form factor, the n-particle one has annihilation poles at the points Nn = ,0j + 7ri, j = 1, ... , n - 1, with residues 27ri res

(,01, • • •

,fln) E1,...,En

f

X btl ...

aEj -1

SEn -

1

Nj ) sEn-2rs (fln -2 - Qj) ...

,r 1

C1. T J+1, n-)-Z q X Sj+1, ) (N)+1 - Nj) A - N1) . • • r)_8,Ej-2 R r)- 2,E1 /^ (fl1 X STj_ Z,Ej-2 (fl - Ni-2)SI i

tEn (/En-1

E + jl

//^^ A, = Nj + 7ri

(16)

where C is the matrix involved in Eq. (1), and for SG breathers CE1E, = tE1iE2 . These singularities are the only singularities of the form factors for models containing no bound states . Eq. (16 ) is an axiom in our approach. For models with bound states (SG ^ < 7r) there are additional poles. It is sufficient to indicate these poles in the strip 0 < Im f3n < in because, using Eqs . ( 13) and ( 14) one can obtain from them the poles situated in the strip in < Im,3n b Vi, j. Under these

conditions we postulate the formula q ,Qn)fl,...,c

, n CE j-1

Xfxi,...,8

Nri

In .

fi,e1.

E„ ,

(22)


which is equivalent (14) to the following: m /^^/ /R^ /^ N1^Q17... 7f riJE1 ^.. ,fn CE7 E7 f(Ym7 j=1 x f(N17... flri7Qm iri,... 7/3 + iri)El ... C„ E;nI... fi

Why did we require the sets /3 and /3' to be separated? From (22) and (16) it follows that f , ,Ql 01, ... , /3„) as defined by (22) has simple poles at the points f3i = Qj'. The understanding of these singularities should be clarified. That will be done in the next section. Let us formulate explicitly our axioms. Axiom 1 . Form factor f (Nl7 ... , Qn)E1,... , En possesses the symmetry property (13).

Axiom 2 . Form factor f (,81,. . . , i33) E1,,.. satisfies Eq. (14). Axiom 3 . Form factor f (f3 7 ... , I3 ) e 1 r... ,E n has annihilation poles with the residues (16).

Axiom 4. If the spectrum of the model contains different kinds of particles and bound states, the form factors have additional singularities. The positions of these singularities can be found from general physical reasonings . For example, for SG Eqs. (17-20) hold. , #,'I, {Ql , ... , Q„ }, form factor f Q4 dal, ... , 6.)e'- '-" '1 is defined by Eq. (22). A refinement of the last statement suitable for the case of and not being separated will be given in the next section. In the next section , from these axioms the local commutativity theorem will be deduced which is an ideological foundation of the book. In SG model we shall deal with, besides others, the operators For separated sets

exp (f ` 2°) . Axioms 1, 2 are to be slightly changed when dealing with these operators which should be called quasi-local. The fact is that solitons are not very common for the Bose field theory excitations. Specific properties of the operators exp (±`2") may be more easily understood in terms of equivalent MTM, in which the operators are equal to exp(faix f o. jo(x)dx'). It is clear that local operators exist which do not commute with exp(±iri fz, jo( x')dx') on the space-like interval. This specific character of the operators exp (±` 20 stipulates some modifications in Eqs. (14, 16), namely, in the RHS of Eq. (14) the multiplier (-1) 2, 2+1 appears. The

The Space of Physical States. The Necessary Properties of Form Factors.

15

same multiplier appears in the second term of the RHS of Eq . ( 16). Besides these, operators exp (±!) have nonzero vacuum averages, and the corresponding two-soliton form factors have an annihilation pole with residue 27ri res f (#I, 02 )cl,c2 = Crlcz

#2 = 01 + 7ri

We put the vacuum averages of exp (f' 2") to equal unity. Although the programme Axioms 1-4 H local commutativity, which will follow below in this monograph , seems to be most beautiful, the realization of the programme of paper [45], i.e. the obtaining of Axioms 1-4 from general axioms of quantum field theory, is also of interest . In Appendix 2 we outline the latter programme heuristically.

2 THE LOCAL COMMUTATIVITY THEOREM

In the present section, the theorem is proven which states that two local operators commute on a space-like interval if their form factors satisfy the system of axioms formulated in Sec. 1. First, let us introduce a system of brief notations. Consider the set of rapidities /3k,... Let us connect a space h; with every /3;. The spaces h; are isomorphic to C2 for ITM and SG with > ir, to C3 for NLS, to c2® C[A/f] for SG with ^ < ir. The base in the space h; is enumerated by the index e. S-matrix S(/3; - /3j) is considered as an operator in the space h; ®hj, its matrix elements being given by Eqs. (3, 4, 5, 6, 8). The unitarity condition, crossing-symmetry condition and Yang-Baxter equations in these notations look as follows: S((31- 32)S(i32-Pl)/^=I, /^ qq S(fl1 - /N2 + in ) = li(/1)St(P')(/32 - / 1)C(Nl) S(Ql - N2)S(Nl - /33)S(/2 - /33)

= S(12 - /33)S(/31 - #3)S(,31 3)S(/31 - N2) ,

(23)

where C(/3=) is the matrix C acting on h;, t(/3=) is the transposition with respect to h;, in SG model with ^ < it the matrix C is equal to v1 ® Iwel. Matrix C satisfies the relations Ct = C, C2 = I. Besides , we suppose that the following relation holds: Ci(/31)C(/32)S(,81 - /32) = S(Q1 - 82)C(31)C(N2) . 17 1

18


create a (m Form factors f (a m. ... > a 1 N l) • • • f ,l3ri )ee' Ei...e„ ), n) tensor. Let us assume fli to be always connected with hi, which means that when Ni and ,Oj change places the corresponding spaces hi and hj also change places. The meaning of the last, nebulous phrase will become clear after rewriting Eqs. (13, 14, 16) in new notations: f(Q1...Friffli+1f... ,fln)S(fi -Qi+l) = f(Nl) ... fl3i +lffli f... fNn) , f()31i... ,Qn-1,an +27ri) = f(h'n,F3l,... , hen-1) f

res f (01i... fln) = 21 f(^31f... ,^j,... hen-1)e(#jf, n)

xS [

(Nn

- 1 - aj) ... S(,6j +1 - )3j)

S (aj - al) ... S(aj - )3j -l)l qq Nn=i3 +Ira,

where e (,(31, /2) = Cc,E'el,e ® e2 ,,', with ei, forming the base in hi. Equation (22) rewrites as follows: f(am,... ,a1If1,... )Nn) = C( al

f ... , a.)(f ( a,,, - az,... , al - 7raf

t(al,... ,a...)

Nl f ... f Nn)) f

where C(a1 i ... , am) = C(al) ... C(a,..), t(al, ... , am) denotes transposition with respect to the spaces connected with al, ... , a,,,. It has been mentioned above that Eq. (22) gives the correct formula for form factors if the sets al, ... am and )31,... „Qn are separated. From Eqs. (13, 14, 16) it follows that f (ae, ... , al 101, ... , Q.) possesses the properties: 1. f (a., ... al 1,31 =

R R R , ... f. Ni "841' ... , 13.)S//(/3i - A+ 1)

f (aK, ... , al 1,311 ...

, Ni+1Qi, ... ,13m) )

2. S(ai+1 -ai)f(aK,... ,ai +lfail... ) a,1)31i... ,1m) = f(a.,... ) a il ai+li...a11131,... ,13m) 3. f(a. -7ri,aK-1i...

=f

t QK

,alIQI,...

f/ m)

(aK-1i...allplf... fflmfaK)C(aK))

. alI/31i... fQm- 1 Pm +Ira) ft( P" )(Qm, a, ) ... , ail/1f... =C( Nm)

4. f(a.....

(24)

,

,13m -1

) • (25)

The Local Commutativity Theorem 19

5. There are simple poles at the points ai =,8j with residue 21ri res f(a,,,...

,

a lIa l,•••

Nm)

= -S(ai - ai+1) ... S(ai - a,,) x f(a.,... x I(ai , /3j)

,

ai ,...

S(,am

-

,

a lI#,,...

Qj)

,/ j,... ,/3m)

... S(aj+l - Nj )

+ S(ai_1 - at ) ... S(al - at )I(ai, Nj) x f (a,,..

x

a1IQ1,... , ai ... S(/ 1 S(Nj -,81) - Nj-1) ) ... ,

,/j,.• him)

(26)

where I(ai, flj) is the tensor identifying the spaces connected with ai and ,3j, its notation in components being 5 . Let us further shorten our notations. Namely, let us denote the set of the rapidities by capital Latin letters, for example, A = {aj};=1, B = {/3 j }Tl . We are still associating a space h with every rapidity. The number of elements in A will be denoted by n(A). The set of rapidities ordered in accordance with increasing (decreasing) values will be denoted by -!('A T ). Suppose al < a2 < ... < a,., i.e.

A = fall... I a.},

the product Z(al) ... Z( a,) will be denoted by Z(A ) while the product Z* (a,,) ... Z*(a1) by Z*(A). The last definition needs some refinement. We understand Z(a)(Z*(a)) to be a vector ( covector) in the space connected with a . The products of these operators are always understood as tensor ones with respect to this internal structure. Let the set A be divided into two subsets A = Al U A2, and consider the operators S(A I A 1), S(A 1 I A) defined by

Z(A) =

S( AI A 1 )Z(A2)Z(Al) ,

Z *(A) = Z *(A1)Z *(A2)S(AIIA) . It can be easily shown that S(A1I A)S(AI Al)=I. (27) Supposing that a,. > arc-1 ... > a1, Q,,, > ,Qm - 1 > ...,Q1, we define "bfunction" A(AI B) by the formula

A(AI B) =

p

/

qq

6,,,n ll b(ai - )8 ) I (ai, Ni) i=1

20


where I(ai, f3i ) identifies the spaces associated with ai and ,6i. The full base of "in"-states consists of the vectors

1'B --) = Z*(B)Iph) , the dual base being

(11 = (phlZ(A) In what follows we shall deal with several sets of this type. The writings A 1 U A 2 and Al U A2 will be distinguished; the first one means that the ordered set A2 is added to Al while the second means that Al U A2 is ordered. Now we are in a position to refine the definition of the matrix elements of the general type. Let A and B be arbitrary sets of rapidities. We postulate the following form of the matrix element:

S(A IA1)f(A1+i01 B1)

( 7 10 (0, o)IB) = A=A1UAs B=B1UB2

x A(A2, B2)S(B 11

B)(-1)n(Ba) •

(28)

A 1 + iO means that all the rapidities involved in Al are slightly moved onto the upper half-plane. It is clear that we find f (A I B) to be the RHS of (28) for separated sets A and B. The following Lemma is very useful for the proof of local commutativity. Lemma 1. The formula (28) is equivalent to the following:

(Alo(o,O)IB) =

E

S(!IA2)f(A1- i 0jB1)

A=A1 U A2 B=B1UB3

x A(A2, B2) S (B 2 I B)(- 1) n(B2) • (29) Proof. From the formula for the residues (26), it follows that

f(A

A3)S(A1uA3IA3) +i01B)= S(AIA l U A=A 1 UA2UA3 B=BIUB2UB3 X f (A- 1 - iOI B 1 )A(A2,

B2)A(A3, B 3) S(B 318 1

x S(B1 U3B I B)(- 1)n(B3)

B3)

(30)


Substituting Eq. (30) into Eq. (28) one obtains

-T (A IO(01E0) I B) =

S(AIA, U A2 ) A=AjUA2uA3uA4 B=B1uB2UB3UB4

x S(A1 U A2 U IA1 U A3)S(A1 U A31 A 3) x f (A 1 - i0I B i)A (A2, B2 ) A(A3, B3 )A(A4, B4) X S(B 3IB1 U B3)S(B1 U B3IB1 U B2 U B3)

x

B 1 U B U B I B)(- 1)n(B3) +n( B4) S(B1

(31)

Let us transform Eq. (31), replacing A(A2i B2)A(A4i B4) by [S(A2 U U A4I and denote A2 UA4 by A5, A 2)]-1A(A2, B2 )A(A4, B4) [ S(B 21B4)]-1, B2 U B2 U B4 by B5 . Then Eq . ( 31) transforms to

(A 10(0, 0)I B) _ S(A IA1 U A3)S(A1 U A31 A 3) A= A1uA3UA5 B=BIUB3uB5 B2CB5

x f (A 1 - i0I B 1)A( A5, B5 ) A(A3, B3 )S(B 31B1 U B3) X S(B i U B3 I B )(- 1)n(B)+ n ( B1)+n(B2) Notice now that the summation over B2 gives

(-1) n(B2) B2CB5

1 , B5 =

0,

0

B5 96 0

and consequently

AI0(0,0)IB) _

> S(AI A3)f(A1 -i01B1) A=A1uA3 B=B1UB3

x

Q.E.D.

A(A3i B3)S(B 31 B)(- 1)n(B3)


Remark . Everything is slightly different when dealing with quasi-local operators exp (f' 2°) . Recalling that the space h for SG model is isomorphic to C2® C[i], let us. introduce an operator d which acts in h as (-12) ®(I[ i ] ), where 12, I[ i ] are unit operators in C2 and C[T'] respectively, i.e. the operator d reflects the soliton space and leaves unchanged the breather one. For the quasi-local operators the matrix d(ai) should be put in before the second term in the RHS of (26). Eq. (29) remains valid without any modifications while in Eq. (28) the multiplier (-1)"(B2) is to be replaced by (-1)"(Bl)d(B1). From Lemma 1 follows the existence of two presentations for the matrix element of the commutator of two local operators. These presentations are given by the following. Lemma 2. Consider two operators 01, 02 which are constructed through the form factors fl(/fl, ... f2(Ql...... ") via the procedure described above. There are two possible presentations of an arbitrary matrix element of the commutator [Oi(x), O2(y)] (hereafter x, y refer to space coordinates). The first is (A 1[01(X), 02(Y)II B) exp(isc(A)x - iic(B)y)

E S(AI A1)S(A2 A3IA3) A=A,UA2UA3 B=B1UB2UB3

x E

J dCG_( x_ICAi

Bi

A3)

n(C) 00 =o

B 3 )S(B B3 U2 U2 x 0(A3, B3)S(B 2I B ) J +B) ,

(32)

and the second is (A

I[01(x), 02(y)]I B) exp(itc (A)x - ix(B)y) S(A I A 1)S(A U2 AIA3) A=A,UA2UA3

B=B,uB2uB3

x > JdCG+(x_ICAi 1 BiA3) n(C) 00 =O

x 0(A3, B3) S( B 3I B U2 B3 )S(B 1 J 'B ' ) (33)

The Local Commutativity Theorem where tc(A) =

23

E rc (a), a(a) = M sh a , M is the mass (in SG model with aEA

is < a, M is to be understood as the matrix diag (M, M, 2M sin 2) ... 2M sin (2 [ f ])) acting in the space h connected with a ). Integration over C means integration over all rapidities involved in C. The function G_ is given by

G-(x -

yIC, A 1, B1, A3) = (- 1)n( Bl) +n(B2) +n(C)

x fl(A1+iOIC U B2)S(CIA3U C)f2(C U A2-iOIB1) x exp (i(x - y)(ic(C) + k(B2) + ic(A3) + i(A2))) - f2(A2- i0IB1 U C )S(C U A3IC)fl(A1 U C + i0jB2) x exp ( i(x - y)(-tc (C) + tc(B2 ) + c(A3) + tc ( A2))) • (34) The function G+(x - yjC, Al, Bl, A3) can be obtained from -G-(x yIC, Al, B1i A3) by the replacement x y, fl H f2. Proof. Consider the matrix element (A IO1(x) O2(y)IB). ' Let us put a full set of states between O1 ( x) and O2(y):

(A

IO 1 (x)O2(y) I B) 00 n(D)=0

J

B)

Let us use the formula (28) for (A IOl (x) I D) and the formula (29) for (F 102(y)IB): '

(A IO1(x) O2(y)I B ) E

00 1: 1:

A=AIUA4 B=B1uB 4 n(D)=0

JdDdF S(-AI A1) D=DIuD2 F=F,uF2

x f1(A1 +i0ID 1)A(A4, D2)S(D I I D)A(D,F) x S(FI F2)f2(F1I B1+ i0)A(F2 , B4)S(B4IB) x exp (-i tc(A)x + itc(D)x - itc ( F)y + ir.(B)y)

24 Form Factors in Completely Integrable Models of Quantum Field Theory 00

1 JdCS(AIA1) A-A1UA,UA3 B=B1UB2UB3 n(C) =O

X f1(A1+i0ICUB2)S(CUB21CUB2UA3UA2) x i(A3i B3)S(C U B2 U B3 U A21B2 U B3) x f2(A2 UCIBl + iO)S(B2 UB31B) x exp (-itc(Aj) x + iK( B2)x + iic ( C)(x - y) - iK ( Bl)y - iK (A2)y) Let us use two identities:

Z* (C U B2)Z* (A3 U A2)S(C U B2 IC U B2 U A2 U A3) x S(C U B2 U A2 U A31 B2 U A3) = Z*(B U A3)Z*(CUB2) Z*(C U B2)Z*(A3 A2)S(A2 U A3^ A3)S(C B UIB2) A2 2 S(B2IB2UA3) X S(CIA3U C)S(CICUA) = Z*(B2U A3)Z*(CUA2) .

(35)

When deriving these identities Eq. (27) should be used. From Eqs. (35) it follows that S(C U BflC U B2 U A2 U A3)0(A3i B3)S(C U B2 U A2 U B3IB2 B3)

= S(A2 UA31 A3)S(CUB21 B2)0(A3iB3)

A2)S(B21B X S(ClB3U C)S(CICU `^ 2U' , 21B hence

(A101(x)02(y)1B ) _ > J dCS(AI A1) 00 A=A1UA,UA3 B=BIUB2UB3 n(C)=O

X S(A2UA3IA3)fl(A1+i0+C U B2)S(CJA3U C)0(A3iB3) x f2(C U A2IB1+iO)S(B2IB2UB3) x S(B U B 1 B )(-1)n(B')+n (B3)+n(C) exp(iic(C)(x - y) - itc(Al)x + iic( B2)x + itc(Bi)y - itc (A2)y) The matrix element (A 1O2(y)Ol(x)lB) can be considered in a similar way using again the formulae ( 28), (29 ) for 01, 02 respectively. In this way we


obtain in Eq. (32). Equation (33) can be obtained using the formulae (29), (28) for 01, 02 respectively. Q.E.D. Remark. When dealing with quasi-local operators exp (t' 2u) some modifications have to be made. If 02 is local and 01 is quasi-local, the multipliers d(C U B2) and d(B2) appear in the formula for 0 before the first and second terms respectively. If both 01 and 02 are quasi-local the common multiplier (-1)n(Bl)+n(B2)+"(C) disappears while the multipliers d(CU B2) and d(B2) appear before the first and second terms respectively. Now we are in a position to prove the main theorem. Theorem 1. Suppose the form factors of two operators satisfy Axioms 1-5 and the additional requirement

fs(l^ 1, ... , a., /3K+1 + T, ...

, On

+ a) = O (es101) , Jul - 00

where S is some number common for all rc, n. Then every matrix element of the commutator [01(x), 02(y)] is a distribution whose support consists of one point x = y. Proof. Let us consider for simplicity a model without bound states (SG 1; > ir, ITM, NLS). The matrix element of the commutator (A ^[Ol(x), 02(y)] B ) depends essentially only on x - y ; that is why we can put y = 0. Consider any finite function from C°°-cp(x). This function can be presented in the form ,p(x) ='P+( x) + V- (x), V± ( x) = cp(x ) O(±x), 9 being Heaviside 's function. Consider the convolution

J ((A l[01(x), 02(O)]I B )e'r-,A)x))So(x)dx = f 0 ((AI[O1( x), 02(6 )] I B ao 00

+f ((A I [O1( x), 02(6)] I 0

)e:K(A)x ) ,P -(x)dx

B)eir(A ) x)w+(x)dx .

(36)

A trick first used in Ref. [32] concerns the operation of substituting (32) for the first term in (36) and (33) for the second term in (36). For the first


term one has

J0

(A

I [01(x), 02(0)11 B ) e iw(A)xÔ - (x)dx

_ S(AIA1)S(A2UA3Â3) A=A1UA2uA3 B=B 1UB2UB3 r

x E { 00

° dxcp_(x) j 00

J dCG_(xICA,B1iA3

A(A3i B3)

X S(B 2I'32 `' '3)S(BB 2 I 3) B .

Consider the expression in the brackets above: 0

1-00 =

r dxcp_ (x)

J dCG_ (x I C, A,, B1, A3)

J dCf1(A1+i0IC U B2)S(CI + A C)f2(C U A2IB1+i0)

x 0- (,(C) +

-

ic(B2)

+ tc ( A3) + Ic( A4))

J dCf2(A2-iOIB1U C)S(C U A3IC)fl(AiU CIB2-i0)

x 0- (-ic(C) + Kc(B 2 ) + K(A3) + tc(A4)) , (37) where cp_ is the Fourier transform of the function cp_ . Function cp_ (tc) is analytical for Im Kc < 0 because cp_(x) = .0 for x > 0. Suppose that the support of cp(x) does not contain the point x = 0, then cp_ (tc) would decrease faster than any power of K-1 for rc -+ oo. Let C = {y1, ... , 7n(c)} and consider new variables Ai = yi -yi+l, i = 1,... , n(C)-1, o = n C Eyi For fixed A1, • .. , A(c)_1, the function 0_(ic(C)+Kc(B2)+tc(A3)+r.(A4)) decreases faster than any power of a-"°1 for o -* ±oo while the function f1(A I I C U B 2) S(C I A 3 U C) f 2 (C U '1 ' 1) increases no faster than a fixed power of el°I as it follows from the estimation presented in the statement of the theorem. From the reasoning presented at the end of the previous section it is clear that fl and f2 are regular in the strip -a + 0 < Im v < 0 (the absence of bound states is essential here). The same can be said about the matrix S(C I A 3 U V C). All these facts together with an understanding of the singularities prescribed in (37) allow us to move the contour of integration of the first integral in Eq. (37) from the real axis to

The Local Commatativity Theorem 27

the line Im a = -a + 0. Making this transformation we can rewrite the first integral in Eq. (37) in the form

J dCfi( Ai +iOI ( C - 7ri +iO)UB2) x S( /C - aiIA3 U C - 7ri ) f2((C /- 7ri + iO ) U A2 I B 1 + iO) x c-(-ic ( C) + K(B2 ) + ,c(A3 ) + ic(A4)) Notice now that it is possible to change the ordering of C in fl, f2 and S simultaneously. Using ( 23), (25 ), ( 26) we can write the above as

J dCfi(C)(A, U CI B2 -i0)St(C)(C UA.1' I VC x ff(C)(A 2 - i0I B i U C )cP-(-ic(C) + ic(B2 ) + ic(A3) + ic(A4)) (38) where t (C) is the transposition with respect to all the spaces associated with the rapidities involved in C. It is clear now that ( 38) is cancelled by the second integral in Eq. (37) ( evidently fi(C>St(c) f2(C) = f2Sf1) The integral

J ((A

1 [01( x), C2(0)1I B)eix (A )x)cp+(x)dx

(39)

can be treated in a similar way. Using the presentation (33) for the matrix element we make sure that the integral (39) is equal to zero if the support of cp does not contain the point x = 0. For the case with bound states (SG, ^ < a), the proof encounters serious complications when deforming the contour of integration across the poles of the integrand. As a result, we obtain a discrepancy between the first and the second integral in Eq. (37). The system of relations (17), (18), (19), (20) stipulates the vanishing of the discrepancy. The proof of this fact is very tangled and we would not present it here. Q.E.D. Remark 1. The theorem can be generalized to take into consideration the operators exp (f' 2°) . The additional signs which were considered in the Remark to Lemma 2 are compensated by signs appearing during the analytical continuation a -* o - iri + iO due to the changes in (14).


Remark 2. The Lorentz transformation corresponds to simultaneous shifts of the rapidities in form factors. That is why the generalization of Theorem 1 to the space-like interval is trivial. Remark 3 . In proving Theorem 1 we have ignored completely the problem of convergence. We shall return to this question in Sec. 10.

3 SOLITON FORM FACTORS IN SG MODEL

In this section formulae are presented for the soliton form factors in SG model. All the necessary properties of these form factors are established in the following sections. We shall consider the coupling constant ^ of generic position under e # ; the reflectionless case _ will be considered in Sec. 6. The section consists of four subsections; three of them are devoted to the description of some auxiliary objects and the fourth contains the formulae for the form factors. I. The definition of the operation "( )". Consider a set of rapidities ,61 i ... , 02n (we deal only with operators preserving the topological charge, which is why only the even-number soliton form factors differ from zero). Associate a space hi Cz with every In the space h; we consider a base ei,e (e = ±2, ei (0, 1), e; (1, 0)). The natural base of the tensor product Hn = II 0 hj consists of the vectors eE1...FZr = e1,.1 0 ... (9 ezn ,,,,,. The uncharged sub-space stretched on with EEC = 0 will be denoted by H,, o. Consider the operator Sid (,r3) acting nontrivially only in hi 0 hj : Sii (,Q) =

2

sh-1

(

(/3 -

7n

))

{I ( i 0 II

sh

(/- ai) - sh

• ll +03®0^ (sh(/3_inz)+shfl)_sh1(1 ® + i(0 o?) where o; o?, o , Ii are the Pauli matrices and the unit matrix acting in JJJhi. 29


It is evident that Si j differs from the soliton S-matrix only by the absence of the multiplier SO(/3). Our first purpose is to construct certain special base in Hn 0. To this end, let us introduce a new object. Consider an auxiliary space ho _ C2 and construct a matrix

C(uI131,••• C A(o1)31,...

, 132n) , N2n)

B ( OINl,...

D(Olal,•••

,

13

2n ) ,fl2n)

,132n)

(40)

= 50,1(9 - 131)50,2(0-132)...S0,2n(O-12n) .

The product and division to blocks are due to the space ho. This definition is standard in the framework of QISM [14-16]. The matrix (40) is a monodromy matrix of "high level Bethe Ansatz" [58]. The following involutions hold when o E R:

A

(olal,... ,/32n)= D(o +iilal,...

/^ 2n ,/32n) 11 -1

sh { (o - lj ) s

h f ( O-aj+7ri)

2n sh £(O-j33/-iri) B (o + iril/31,... , 132n) = C(OI/3i...... 82n) ]] Sh ^ (O - i9 )

i-1 f (41) The operators Sij satisfy the Yang-Baxter equation: S 0,i(0 -

,ai) 50,i+1(0 - /i + 1)Si,i+1( Qi

- 0i +1)

Si,i+1(Ni - Ni+1)S0,i+1(0- - Qi+1)So,i(o - A)

hence

( A(alal,••• C

Si,41( i - Ni+1) X

,13i,8i+1,••• ,12n)

B(o lal) ...

,Ni,/3i+1)._. ,62n)

C(0'I01,... ,f3i,Pi+1i... An)

D(oh31,.. •

/'i,)i +1i... ,/2n)

B (OI/31,..• q

)3i+l,

X &i+1 (/3i - , i +1)

A(OI11,•••

,Fî +1,

C(oh31,•••

, /3i +1,8i,••• ,12n)

Qi,•••

An)

,3i,•••

,i12) 1

D(OI)1 ,..• ,/q3i+l,#i,••• ,/32n)J

(42) As in Sec. 2, we imply hi to be always associated with 1i, which means that if (3i and 1i}1 change places their spaces hi and hi+1 also change places. In what follows we shall often denote, when it does not cause confusion, the operators A(o1,61, • • • ,12n), etc. by A(o), etc. From the Yang-Baxter

Soliton Form Factors in SG Model

31

equations it follows that operators A, B, C, D satisfy a great number of commutation relations which can be written in the form [14] R(ol - o2 )T(ol) 0 T( o2) = T(92 ) 0 T(ol)R(ol - 02) ,

(43)

where the tensor product is taken only with respect to the space ho, and R(o) is a (4 x 4)-matrix acting in C2® (C2: sh £(o-7ri)

0

0

0

0

-sh *e i

sh f o,

0

0

sh T a

-sh *{i

0

0

0

0

sh f (o - vi)

R(o)

In particular, one finds from Eq. (43) that the operators A(o1) and A(o2) commute; the same can be said about the operators B(o1) and B(o2). Consider a set of covectors we 1,...,e2,.(/I1) ... A.) _

(01

II B(NpI/3i, ... ,82n) , .t

where ei = f 2, Eci = 0, (01 = e_ _ _ . From what follows it will be clear that Wet,--- '2,. (Nl,... )32n) create a base in H'n,0• Let us list the properties of the base, formulating them as Lemmas. Lemma 1. well ...

C2. (#,) ...qq, î, Ni+l, ...qq, /92n)'si ,i +l(F3i - /i+1)

i,q = 'wet,... ,t:+t , ti.... 2. (F'1 , ... , A+1' Ni, ... , N2n) .

(44)

Proof. The proof follows immediately from (42). Lemma 2. The covectors wtt,,.. ,c2, (Ql) ... „ 62 n) are eigencovectors for q

A(crlfl,... ,,62n).

Proof. From (43) one gets in particular

B(ol)A(o2) (02

sh

f

- of

sh f (02 - ol )

-

7ri) sh ( A o2)B( o1) + - Ol )A(^1)B(192) sh '(0-2

32


Using this relation and the identity 2n sh F(o-Ij) (Î`4(^) = (0^ 11 sh "(o- - /3j - 7ri) J=1 f one can derive [14, 59] an identity k (011

2n

sh E (a - 35) k sh £ (O - Tj - 7ri) 11

1 i B(Tj)`4(^) = 11 i sh { (Q - ,Qj F' - Sri) ; i sh { (Q - Tj ) j= 1

j= 1

k

x

sh

fi

(0111B(7•j) +u(0I11B(Tj)B(tr)sh (a j=1 1=1 j$l

sh {(^-Tj)

X 2n sh {(T7-pj) " 11=11 sh (7y -

,131

- 7ri) 11-11 sh {

(Ti - Tj -

7ri)

It is quite obvious that if r1i ... , Tk is a subset of 61, ... , 82n all the terms except the first one disappear. Hence wElr ... .E2n (P1) ... ,#2n)A(o,)

sh £(01 -#p)

sh

(o

- ,13p -

7r1)

(^ ^ /^ { we1i ... rE2n (N l,... ,82n) .

(45)

P:Ep=-.

Q.E.D. Lemma 3 . The bases e,,,._ C2. and wE1r ... rc2n (/31 i ... „132tt ) are connected via a triangular transformation , which means that in the decomposition u/Elr ...,E2n (F'1, ... ,E2n) _ Ccl,...:ec (Nl, ... , N2n)eel ,.. ,ern , Er

only those CEi,... ;;: for which (cc, ... , £Zn) < (E1, ... , E2n) differ from zero . The multi-indices are ordered as binaries, for example, (1, 1, - 1, - 1) 2

1 _1 1 _1 1 1 1 _1 1 _1 _1 1 1 2 1 21 2 1 < (2, 2, 2, 2) < (-2^ 2, 2, 2) < (2 2 2 2) < ( 2 ) < 1 1 1 1 i...- 2 2•••i

(- 2 2 2 2) The coefficient C_ 2 is equal to 2 2 2 2

_ 1 1

I

C_32... 22...2

sh

(131i...

0 2n) ,N

x

i k (k is arbitrary). Present matrix (40) as a product of two matrices: Proof. Consider some w

C

Al(u) Ci(o,)

B1(a) 50,10 D1(a)

C2 (a) A2(o)

D2(v)

B2(0.)) = 5o,k(

91)

Q -,Qk) ... So,2n (O - 32n)

Evidently the matrix elements of these matrices commute. The operator B(o) can be presented as follows: B(o) = Al(0)B2(0') + B1(Q)D2(0-) .

To obtain w£1,... E2n (,O1i... ,,Q2n) we shall successively apply the operators B(#3), p : ep = 2 to (01. Taking into account that the covectors (0IB1(,Qj1) ... B1(Qjl) are the eigencovectors for A1(a) with the eigenvalues (45) and (0ID2(u) = ( 0I, one gets WE1,...,£2n (fl1) ... ,N2n) _

(01

11

B 1( f3p)

EP=1

( i j' ®ej,.. 2 j=1

II Bl(ip) ® (

fl ej,

_.

2. EP=1

Consequently the decomposition of wE1,...,E2n(131, ... , f32n) contains only those eE1 .„ ,E2n for which Ek = ... = -2n The calculation of C ?'"' 2:3,. .2, is rather trivial. Q.E.D. Lemma 4. The covector

wE1,...,£2n(.1, • • •

,82n) has simple poles only

at the points ,Qj = 13; = xi + ilk, k E Z for j > i, ej = 2, ej = -1, the corresponding residues being

res

z,E:

z E2. 01) ... , 02n)

7r 2 = sin

X P1,... )A-1)A+1,... ,flj-1,Nj+1,... ,,82n)


1

®(e

,8p

®ej ,

sh X sh k>j

+iri)

sh £(Nik (- 1) e- 1 ®ej,1) x 2 sh (fli - /'p ) Per=-1 p#i

(Qk ^ /3i) / - //^^ h i) (Qk - - 7ri) Sj-l,i(I3j-1 / /^ - Ni

Proof. Consider first the case j = 2n, i = 2n - 1. Present matrix (40) as a product of two matrices:

A,

B1(o) = 50 ,10 - t3 ) ... 5o ,2n-2(O - N2n-2) C1 (o) Dl(v) = a(tr) b ( o) - So,2n-1 (o- - 82n-l)50,2n(O' - /32n) c(tr) d(v)) -

At first glance it is clear that the pole at the point /32n = N2n-1 + 7ri + il;k may exist only for E2n = 2. Let us act on (01 by B(j32n) = B1(#2n)d(f32n)+ Al (Q2n)b(f32n ):

(OIB(Q2n)

=

(0I { B l( a2n) +

2'2 j=1

#j) shsh F^^(#zn /// ,13j - 7ri) (/^2n -

sh " q _ sh W 2i sh F (/^2n - l2n - 1 - ini) N2n + sh 1(82n - 82n-1 -

i

hence

res (0IB (i32n) = 2il2 shsh(Q(flzn-lr'Nj) t) ` z x S sin in

_( 01(0, -

q ( ( + l-1)kcT;n-1 ) , (f2n = /32n-1 +7ri - i£k)

Now let us apply B(o) to (0I(o + (-1) kO'2n-1 ), (0' 0 N2n,,82n = #2n-l+ 7ri + il;k). Notice that


(OI(0'2n

_ sh + (-1)k,72n

sh f (C - 132n-1 - 7ri)

(01

+ (- 1)ke•2n- 1)d(c) =

35

0'2n sh { (^ -N2n-1 -

27ri)

(O - /Q2n-1)

-1 sh (O' - Q2n-1 - 7ri

sh *f' 0'2n sh { (

qq

0 - 82n-1 - 7ri)sh { (O - N2n - 1 - 27ri) }

sh { (0 82n-1) / k

sh f (Q - N2n-1 - 7ri ) (

(OI(o- 2n + (- 1)ko;_1

)b(r) sh

(OI

_

DI(^2n + (-1) Q2n-1)

2na2n-1 sh

(-1) k

- Q2n- 1 - 7ri

)sh

F l(O' - Q2n-1 - 27ri)sh L (O' - 82n-1 - 7ri)

sh W2i (0- - 62n - 1 - 21r0

sh

" (0'

0,

2n

2n

hence (0I(,T2n + (- 1)ko,2n -1)B(o) sh { (? -,62n-1) sh

f

k _

(Q - 82n -1 - 7ri) (

Î(^2n + ( - 1) Q2n -1)Bl(o) .

It is clear now that //^^ res wE1 ...., e2nO31 ,... , 82n)

2 sin 7r

2n-2

q

sh

been,;

£ (N2n-1 - Pp + 7ri)

sh

C(Q2n-1 - 13p)

P:ep-- j x (e2n - 1,-; ® e2n ,; +//(- 1)k e2n-1„ ® e2n,-; )

(9 wel,••• ,e2n- 2 (P1, • • • , N2n -2) ,

82n = N2n- 1

+ 7ri

+ i1 k

The proof for arbitrary i, j can be obtained using (44) and the formula (ei,-; 0 ei,; + (-1)Pei,; 0 e1 ,-;)Ski(u)Ski ( u - 7ri)

sh {o sh f (o -

Q.E.D.

(-1)Pei , ® e1,-; ) 0 ej 7ri) (ei,_ 2 i+ 2


In Sec. 9 we shall need a formula for the scalar products of vectors decomposed via WEI.... E2n {,Ql,... ,Q2n). It is evident that the base {we 1,..., e2n A, ... ,132n)} is not orthogonal. However, a base dual to {wE1-• e2n (i31i ... , 162n)} can be easily constructed. Consider the covectors W-1,... ,-2.

01

(01

, ... , #2n) =

11 `'(/p + in )

_ 1

Per -2

where / qq

B/ (, p + 7ri ) =

lm

7ri)(a

B(c +

^p

2

- f3P) C (sh =^

1 -1

The base { GJe1 ... ,esn(Q1i ... „62n )} possesses properties similar to those of the base /32n)}, the most important property being the triangularity: in the decomposition w£1,...,f2n (31,... ,/3 n) _ Q1^...EEn 1 ,..., 2n (Nl,... P2.) e4..., ESn

only those ' '...... differ from zero for which (Ei, • • • , Ein) ^ (El, ... , E211). This triangularity is opposite to the triangularity of { wE1,•••,£2n (31) • • • , f32n)}. The following statement holds.

Lemma 5 . The bases (31, ... , /32 n)} are dual:

{w£1, , E2n (/3 i, • • . #2 n )} and

{w£1,•-• E2n

2n bed (46)

WEI,... E2. (P1) ... , E2n)wi,.. 1£Zn (31, ... , /2n) = j=1

Proof. From Eq. (41) it follows that 2n s

h

f (ÂP

- - j 7r1)

E2n) _ If If sh

P Ey=2

(^P -,6j)

j#P

C(`3p )I )

p:Ey=2

Consider the scalar product (46). Let (El, ... , E2n) differ from \E1, ... , E2n), then necessarily a number j exists for which Ej = - 2e j = z Then the scalar product (46) can be presented as follows,

(01 11 B(/9P)C(Q3) P'EP-]

11 C(w)10)

q : cq-i,q#j

Soliton Form Factors in SG Model 37

Let us move C(131) to the left using the commutation relation which follows from Eq. (43) (the second term in RHS can be omitted because Lemma 2 implies B(,Q,)A(,a,n) = 0):

(01d 1

/^

sh "'

B(/3)C(pj) = C(aj)B(a3) - sh f (Qp{ aj)(A(Qp)D(flj) - ACS2)D(/p)) Hence

(01

rj

B(/P)C(/9j)

P:ep=3

= (OIC(aj) jl B(ap) = 0 . p:cp=

I

Thus the scalar product differs from zero only for (El, ... , E2n) coinciding with (Ei, ... , EZn). The calculation of the scalar product in this case is a technical problem we will not go into. Q.E.D. Now we are in a position to define the operation ( )n. Definition 1. Consider a function F(Al, ... ) an 1µl1... , µn) which is invariant under independent permutations of elements in the sets al, ... , An and µl, ... , µn. With every function of this kind is associated the following vectors from Hn 0: E F(Qil,... ,An l/ji,... ,h3jn)

(F)n(Ql,... ,P2n) _

{1,... ,2n} ={il,... ,in}U{jI,... Jn}

n

n

1 wcl,...,E2, (/31, ... , 82n) X H II sh r F (h'ip - Njq) P=l q=1

where e1, =

Z,

Ei p = - Z , p = 1 , ... , n.

Lemmas 1 , 2, 4 imply the following properties of the operation ( )n1. (F)n ( F( ), 1, 9( 19i 1 , . ,/31,/3i+1,... , N2n)Si,i+1(A - Ni+l) = (F)n(/31,... ,/3i+1,Qi,... ,N2n) (47)

2. Denoting the components of (F)(/1,... , N2n) in the base e£1 „. . by (F)n(,31,... ,,32n) el,...,c2„ one has (F)n(N1, ... , N2n)_ i _ i i i 2,..., 2121... 12 n n 1

fl

)F(Pi,... a ($n+p - ,(3q - 7ri P=1 9=1 sh f

n In+1,... ,L32n) . Nn

38


3. If F(A1, ... An 1/U1,• • • , pn) is an entire function if all its arguments then the only singularities of (F)n(al, • • • , fl2n) are the simple poles of the points /3j = Ni + 7ri + ilk with residues

res (F) n(f31, • , N2n ) _

(ei,.i 0 ej,- i + (-1)kei ,- ®ej, i

(//^^

x fJsh

-1 'r(/3l

//^ ,,32n)

7< Sh-1

)

s

/^ //^^ - Qi - 7ri)Sj- l,i(Qj-1 - 13i) ... i+l,i(Ni+1 - Ni). ,

Nj =

Pi

+ 7ri + i^ k , (48)

where Qr(A1, ... , )n-1I/p1, ... , /tn-1) =

F(A1i...

,An-

1,TIµ1,... ,µn_l,r+7ri+i^k)

At this point we finish the discussion of the properties of ( )n and pass on the second important object which is the integral transformation -bn,o. II. The integral transformation

7(3

)

(

,(p). Consider the function

)2

=

x ex oo 2 (sin2 (!SA) sh ("z t) + sh2 ( 4) sh ( j^)) d7c P o lcsh (2) shWK Having in mind the importance of this function for what follows let us list its main properties.

1. W(Q) ~ 2 exp (T-1 ({ + 1) /3) for

8

±oo

.

(49)

2. 0, simple zeros at the point /3 = 32 i + ilk, k > 1.

39


3. n (recall that we consider ^ in generic position). That is why one can successively reduce the degree of the polynomial II (x - exp (f ,Ql)) P(x) by subtracting II ( x r _2 - T- i

exp

k \ S #'

)) I I xr4k

- II (xr3 - eXp \ w i3

with the proper coefficients. It is clear that the/ / process terminates when the degree of the remainder is equal to 3n - 1. It should be noticed that the polynomials Pi and P2 are defined ambiguously. Q.E.D. Lemma 7. The ambiguity of the definition of Pi and P2 does not affect the value of the RHS of Eq. (51). Proof. Consider two different pairs of polynomials Pi, P2 and Pl, PZ satisfying (52) for the same P. Polynomials Qi = Pi - Pl and Q2 = P2 - P2


41

satisfy the equation Q1(x) = I

( (

xr

3 - exp C

xr 2 -

- fi

T -1

i3)) Q2(xr4

exp

C

) T2

2)

aj)) Q2(x) ,

(53)

with deg Q < 3n - 1. Eq. (53) implies (see the proof of Lemma 6) that deg Q2 < n -1. To prove the uniqueness of this definition one has to show that 2n

J + ^- io^sh

^p(a - ,(3j) E( a_

aj _2i ) exp

C-

E,8i)

Q l Cexp

2n

x exp Cak -

(3n - 2)a) } r p(a -,3j) rj=1

x exp (ak - (n - 2)a) Q2 (r4exp (ja)) da = 0 . (54) The inequality deg Q2 < n-1 and the fact that ô(a-fj) has no singularities in the strip -I*- < Im,Q < 2 except those enveloped by r imply the possibility of deformation of the contour r in the second integral in Eq. (54) to

( (

2aa r4 exp ))exp - x (3n - 2)a) do Qj )Q2 (ak 2n

+ îo

- )Qz ^'p( a Pj

(

T4

ex P

2

a) )

C

m(ak x exp - 1(3n - 2)a) de 2n

+

2`p(a -Qj)Qz

18- 92' -i0 j=1

C 4 C2a)) T

exp

T

x exp (ak - (3n - 2)a)I .

(55)

Taking into account sh { (a - zi ) *a - 2^ri) = i, k E Z ) with residues res(F,,ry)) n(xl,••• , xn-1I01,••• , 02n)

= a(ei,^ 0 ej,-; + (-1)kei,-; 0 ej„) n-1 r 1 ('Y) ® 1) (Fn- 1)n-l(x1i... ) 2l,... ,xn-1I /^ 1=1 /^ X flj,...

x 2-3(n-1)

... ,3 N2n)

fi (x1 - e2sPT) - 7 4(n-1) p#i,j O PT

x fl

p #i ,j n-1

x

(xp - e Z of r) p=1

II

9 n, i < n with residues (63): b' rbsr^ R

('Y)

xl, ... , xn-llbl , ... , b2n) n

=

n

1

2'(n -z) k

11 bk - b

k#j

1

iT2 kll

n--1 bk - bi 11(xl - bir)

kqi

n-1 X E(

II

- 1 )kxk 1

k=1

X R (-Y)

(xk - T

bl)

l0ij

... :F , ± 121... ,f ; (xl, ...

- 7 4( n-1)

II

b

( xk - 7'1 l)

loi,j

, xk, ... , xn -1 l

x b1i..,bj,... , bj,... ,b2n)

(68)

E uations (68) provide us with the possibility to construct R^y^ :F ^ t 21 t ^ recurrently. It can be shown that ) 1(bl, bz) = Ri R}l) (bl, bz) = R(11 ]' 2 ass ]'

(bl, bz)

= 2(b2T -1 - R( ii(bl,bz) Z'S

b1T) -

consequently

R(1)I

1(x1, ... , xn

_1 1

-ll bl ) ... , b2n)

n

R1), .1 .1 1 (xl, ...

^r

^r

xn-1

lbl, ... An)

The Main Properties of the Soliton Form Factors 51

R( -11)

I 1 1 ( 1 "51 21... 12

n

=

xl i...

, xn-l Ibl, ... , b2n)

n

1(xl,... R(13,...,^-2,..., 2 1) 1 3 -

xn -l l bl,

= 2n2 On(xl, ...

,xn-

1j bl ,...

... bn

, b2n)

l bn +l, ... , b2n)

(69)

2n n

II II (b-r-1 - bs•r) j=n+li=1 9

The latter equality follows from Eq. (59) and Lemma 3 of Sec. 3. Functions

Rfryi+ z,...,Tz,f^,...,±1(xl,...)xn-

ll bl,...,b 2n) are sym-

metric with respect to b2, ... , bn+1 and bn+2, ... , b2n separately. They have n2 + n - 1 simple poles at the points bj = bir2, j > n + 2,1 < i < n + 1, bj = b1r2, 2 < j < n + 1. For the recurrent reconstruction of these functions one has to indicate the residues at n2 poles, the thing we are doing now (see (63)):

res R(1 bj=bir2 f3, 2<j2(xl bn l al ,

= 1( x

fJ( x

- atr)(x - bl +lr)

55

ll(x - alr)( x -

b2,... ,

i 1 are given by the usual formulae (57), Ao(xlai, .... , ani bl) ... , bn) is an arbitrary polynomial of x whose degree is equal to n - 1 and the higher coefficient is equal to f Proof. Multiply G(/31, ... , Nn hn+1, ... , 82n) by E exp(-Q3 ). The result can be presented as a determinant whose first column is multiplied by E exp(-,3j), while others remain unchanged. Now add to the first column the second one with coefficient o3(exp(-/31), ... , exp(-/32n)), the third one with coefficient Q5(exp(-/31),... ,exp(-/32n)) etc., (recall that we denote by ok the elementary symmetrical polynomials). Consequently the first column changes to 6i

o = 27ri fo0

da jj 11'P(a - ^3) exp

(

-(n - 2) a

a exp /31, ... exp

x Ai ( exp

x22"-1 [ii ch

(a-a3+

On

exp 2r 21r )3n+1 , ... , exp

-(-1)fJ ch

1

/32

n

(a-i33-

2 ai)

x exp ( H

As it follows from the proof of Theorem 3, 6io = 0, i > 1. Consider 60o. The integral in the formula for 600 is to be understood as a usual one because the degree of A0 is equal to n - 1. Using Eq. ( 92) one obtains rn a) da fJ cp(a - ,Q3) exp 600 = lim 1 n-.o/o 27ri \

J

/

\

x A0 I exp 2 aI exp Ql.... I exp 2 /3n+1, ... I exp I 2 x 2 2n-1 l rj

1 ( c 2 a - Q3 + h (J A

nim

- _ 00-if

7ri1

2

- (-1)

n

1 (

^ ch2 a -/33 -

^ ^p(a - /3j) exp \-(n - 2)

7rt

2

l C a/

)l

90


(

X Ao

exp

T alA ... )

2 2"

-

1

11 ch 2

-if

(

a - ,Qj +

2 // I exp (

2 r pj)

7

11 `^(a - Qi )

A-moo 2ai JA

xexp

a(n-2)IAolexp { al...)22n-1 llch2(a-i3i+ 2z

x exp (-

2 ^Qi) =

2n- 1 exp (-

2 Ea')

where we \\\\ also used the fact that the higher coefficient is equal to Now we have only to expand the determinant along the first column, obtaining (Ee-A,)

(Gnry)) n = exp

(_

/3)

gn,l ((Fn7))n)

.

(101)

The equation (êf') (G

Y)) n'

n = exp 2

E,8)

On,- 1(( F, ))n)

(102)

can be proven in a similar way. Eqs. (101), (102) are equivalent to Eqs. (99), (100). Q.E.D. Remark . As functions G(') do not depend on the particular form of A but only on its higher coefficient, it is not necessary to choose A0 as symmetric with respect to al , ... , an, bl , ... , bn. What are the analytical properties of g± (,81 i ... , ,82n), considered as functions of Q2n? For n > 2, one can choose the polynomial Ao(exp f aI in the form (exp f a - exp f O32n + 2))P(exp f a) where P is a polynomial of degree n - 2. The multiplier (exp - exp f (/ 2n + Zi )) Ea cancels the pole of cp(a - 32n) at the point a = ,62n + 2 (see Fig. 3). Moreover, for this choice of A0 the following relation is satisfied for the matrix Aij = A i ( x, Ial , ... ) an l bl,... , bn), i,1 = 0.... : n -1 det IIAiillnxnlb„ = anrs = 2-3( n-1) ][I (xi - r- t an) i=o n-1

X

E(-1)k det IIAii II(n- 1)x(n-1 )(x0) ... k=0

X xnlal , ...

an -1

1bl , ... bn-1)

X I JJ(xk - ajr)(xk - bar) - T4(n 1 ) IJ(xk - aj7-1)(xk - bjT-1) l 1

91

Properties of the Operators...

These facts taken together allow us to conclude that the function g± (/31, ... , ,lzn) has for n > 2 the same singularities as fµ(j3,... , f2n), fµv(01, • • • ) 02n), i.e., simple poles at the points 02n = /3j +7ri - i.k, where the residues are given by the formulae similar to (78) and (17). The case n = 1 needs a special consideration. For n = 1, one has

E1-E2 9-(01,/2)E1,E2=d((/il-/32)2ch1 ( chx 2(Ql - 02) 2f(01 - Q2 + 7ri)b /^ q qq 9+101, ^2)e1,ES = d( (01 - 02)

,r 4£

1 a - #2) sh Ch 1(81 2 2f(Pi

-/^ #2 + 7r i)

bE1 -C2

,

which means that g_ has a simple pole at the point 32 = Q1 + 7ri and g+ has a double pole at the same point. The following theorem establishes the most important properties of the operator j., T,,,,, exp(±ii). Theorem 5 . Operators Q = f jo(x )dx, Pµ = f

Tµo(x ) dx, and

W- lim exp(±i 2u) represent the charge, energy momentum and exp ( 7riQ), X 00 which means that n QZE1 (/31)...ZEn ( )3n)IPh) =2 ( t ej)7Ei(^1)...Zn(pn)IPh) /q

PNZe1 (Q1) ... ZE^. (/n)I Ph ) = x

1

( E(e P' + (

Z1 M E

/

-1)µe-P;))

ZEn (/3n)IPh)

exp( 7riQ ) ZE 1(/31 ) ... ZE„ (/n ) I Ph) = (- 1)n7' E1\Ql )

/

1

... ZE,.(/n)IPh)

Proof. Construct all the matrix elements of jµ,T,4 ,exp(fi^) through fµ, fµ,,, fx using Eq. (28). It is clear that the matrix elements of f00 jo(x)dx, f z. Tµo(x)dx are constructed using Eq. (28) through g_, (E exp(/3j) - (-1) µ exp(-/3j ))g+ respectively. As was to be expected, /jU w jo(x)dx-0, Tµo(x)dxZ0, exp fi 2 ->1 T. f0, 00

x-->

-00

because the way of understanding the poles in Eq. (28) is in agreement with the sign of the exponent exp(i(p1(A) - p1(B))x). We want to consider

92


the limit of the above operators for x --> oo. To this end it is natural to use Eq . ( 29) in which the poles are understood in the opposite way. The equivalence Lemma (Lemma of Sec. 1 ) gives Eq. ( 29) for exp (±i^) with (-1)n(62 ) changed to (- 1)n(B) (see the Remark to this Lemma). Taking the limit x -+ oo we obtain

l im(A I exp

(

2

)I

B) = (-1)"^8)A(A, B) .

The Equivalence Lemma is almost applicable to the operators f x, jo(x)dx and f x TMo(x)dx, the only discrepancy between Eq. (28) and Eq. (29) being caused by the existence of the annihilation poles of the functions g_(aI/3), (e'-e,-(-1)µ(e-o -e-Q))g+(al)3). For the matrix element of the operators f. jo(x)dx and f. Tµo(x)dx this discrepancy is tr3 (A)A(AI B) and Pµ(A)A(AI B) respectively. In Eq. (29) one can take the limit x -+ oo and obtain zero . That is why

f C-0

(A 1

00 Tµo(x)dxl B) = A(AI

B)Pµ(A) ,

(A I f jo(x)dxI B) = A(AjB)o-3(A) . 00 Q.E.D. Now we want to show that the formulae for fµ, fµ,,, fl give the correct result for ^ = it where the model is equivalent to the free fern-don one. But first we would like to show what simplification appears in the formulae at the points 1; = I for an arbitrary positive integer v . At these points the reflection of solitons disappears and the soliton S-matrix becomes diagonal:

S,

4

(/3)

So(p) ve16E7 (- 1)(d1_

e2)(v_l^

So(d) _ chz((3+ ^fk) k-o ch 2 (a - Vt k) Straightforward application of the formulae of Sec. 3 to this case is difficult. For example the proof of Lemma 5 of Sec. 3 becomes senseless . We shall obtain the necessary formulae from those for + 8, taking the limit b -+ 0.

93


Consider for example the form factor

f

n

+()31i...

)32n qq)_T,...,-^,

X exp ( ^^^``

;...,1 = d 2

-(n-1)(n-Z)

2

(i3+p - Qp - 7ri)

`C p=1 n

2n

det11Qtijll, (103)

1

x i<j p=1 4 = n+1 sh t

(,8p - R9 + 7rt)

where

27ri J-00

(

(n - 2)a + ( n - 2j)a

da 119(a - /3i) exp 2ir

27r

) 21r

x Ai e x p aIexp / 3 1 i ... , exp

/3n

I

27r

27r q 11

)3n+1 , ... , exp

0

2n

)

.

From the very definition of the functions A2(xl a1 i ... , an Ib1, ... , bn) it follows that Ai( xja1 .... ) an lb1, ... , bn) = 00) for 6 --p 0. Consider the relation [J(xT-3 - e t /3i)Ai(xl ...) = X

PZ `)(x)

- 7-

p(2)(x)

2( n-2) II( xT- 3 - e c

+ II( x T-2 - 7--1e p3 )

Q')P2

)(xT4)

which occurs in the definition of the regularization f. Let us choose for n+i-1

27r 27r R xn-k+i-1T3kOy k (exP C 31, ... , exp C /32n

P(f^(x) = 2-n-= 2 k=O

Evidently P1`) = 0(S) for 6 -+ 0; this is why 00 21ri

11 'P(a - Qj) exp (_(n_2)a+(n_2)a ) 27r

I

x Ai (exp a ...

27ri

j fl(a_/3i)ex

0

daa +

(- ( n - 2)a + (n - 2 )a^ P2 (exp 2a^ d.

(104)


Notice now that for l; =

v

W(a) = i/i(a)sh-1v (a - 2, "-1 ,O(a)=2' 1llsh a-irj) , v>2 ;Vi(a)=1, v=1. 2( i=1 The integrand in (104) becomes a periodic function of a with the period 27ri. Hence the contour of integration (see Fig . 5a) can be deformed to the joint of two segments (Al, Al - 27ri) and (A2i A2 - 2ai ), with Al < ,8j, A2 > Q9 for Vj (see Fig. 5b).

01 ... 3 2n

AIR,... R 2,A2

a l... 32nAi

0

X X

3TC

K-K

M K

-21L

a

b

c

Fig. 5.

Notice that the integrand decreases for a -+ -oo; consequently the integral over (A1, Al- 2iri) can be omitted. Present the integrand as follows:

1 2n-'21ri

n,

f ,- 2^:

^/,(a-,Qj)exp (( n-2j)a + v(n-2i)) R;(e"^)da , (105)

95


where Ri(x) is a rational function of x possessing the following expansion for x -• oo: (106)

R.(x) = 1 + d(1)x-1 + d(2)x-2 + ... .

The function 0(a) behaves as exp (a a - i for a -> oo, hence one can retain only a part of the series (106) in the integral (105):

R ,(x) = 1 + d(1)x-1 +... + d('-1) x-'+1 . Now substitute these formulae into the determinant (103). Adding the first row with coefficient -d(21) to the second, the second with coefficient -d31) and the first with coefficient -d32) to the third, etc., one obtains n

f +(81i...

2E( Qn

, =dnexp #2.)-2 ,,...,-1,1,...,

-9fZ) +P-QP- Qp

p=1 n

2v

x II ((Qi - Qj) II rj sh-1v(Qp -,8q) det

II ^;)II(n- 1)x(n -1)

i<j p=1q=n+1

where

2( (o) = 1

2aa J A- 2xi

II o(a - Qj) exp ((n - 2j) a + v(n - 2i)a) da2n -`

Notice that the value of A is of no importance now. Evidently 2(;°) is a symmetric function of ,Q1 ... Q2n. The S-matrix is diagonal, hence 2n

H(2Ej)jdn

II C(Q $j) det 1121i;)II 1 -

j=1 i<j x fl fl sh-1v(Qp - Nq)eXp

(V EEj/j)

P'eP= 7 q:eq=- 2

Similarly one can consider fµ, fµ,,, expressing them in terms of det

2t^^)

IIQt(})II:

0(a - ,Qj) exp(n - 2j + 1)a + (n - 2i) va)da A = 2lrii J -2ai II

A

For v=1one has 0(a)-1,((Q)=shc) j)=bi , n- i,26

)_2t())

=01 n>2.

96


Hence ff (131, ... 13 ,) 2n _ 21 ...1 -2,2,...,2

(

_ ) '12_2exp

±2

=

E( 13n

6

+p -, p)

p=1

n 2n

x llsh 2(13j i 0. Consider the function

Form Factors in 0(3)- Nonlinear a-Model

111

F(/31,f32 ,. .. , Qn)wl ,...,wn

7rt ri 72 = lim 6-obn-l Q1f- 2 + ib "81 + 2 - ib, Q2 - 2 + ib, Q2 + 2 7ri ai + ib,Qn +2

,6n

ib e Le2, e S,e4r ••• ,e7n-Le2n

PS

x j7e1,e2 $W Peare' ...;z Pe9n-1,e2n wl wZ wn

(130)

Equations (127), (128) and p2 = P imply wi,wi}3 /

)... ,wi,wi+l ....SwI:w:+1 \Qi - Qi+l

=

lira + ^...

6-0

bn-l 72 71 - ib, Qi + 1 + ib, Q i + 2

,6i -

2 + ib ,

2

e2i-1, e2 i e1,a2 ... - ib .. P 10i+1 + IP / 2

wl e2i-l,e2i ••• ,e2i-l,e2i re7i}l , e2i}Z,••• .v eZi+1C2i + Z C2n-1 , e2n :Zi-1re2i+7 eZi+1 " 2i+2 wn t2i - 1"2i}2

,,, ()39. 7ri)

x P / I P S fit

/ //I III I III III , e Zi+2 (Se Zie2i-1,e/2i}1 eZi, e 2i+1 x L7 It /// E /// (Qf - #,+,)SLII. II (Qf - 8f+1)Se11 (Qi - Qf+l + tea) a e 2i,e 2i+2 2i-1, 2i+1 2„ 2i+1 ^"

II

II

^ '{

II

II

X Pe2i,e2i-1 Pe2i} 2,e2i+1 W wi+1

= lim bn- 1 f

6-0

1

Q4

... '16i -

7ri 7ri

2 + ib, Qi +

7ri 7ri +ib ,Qi + 1 +2 2

ib,.. ••• re2i-1 , e2i,e2i+1,e2i+21...

CE 2i-l,e (ê2i,e 2i+1 f x 1J // 7ri) a7 II + (Qi - Q+1 Zi+1\Qi L

Zi,e

II

X Ses.,esl+Z

Z

i+1

II

e2i-"

2i+1

II

l)aSe2i-lre2 :+2

(Qf - Qi+ (Qt ZireZi+2 e2i-1,e2i}2 (Pi

7ri)

ft ] i+lre4 . PC Pel,e7 ... Pe2i-1, e2i7e i+2 2.-1,e2n wn wl w1 w1 +1

= F(

... ,Qi+1,Qi,...) wl,..., w^,+l,w!.,...wn

It is evident from the very definition that F(,81,... , ,6n + 21r2)wl,... ,wn = F(Qn, Ql , ... )wn,wl, ... .

Thus F satisfies the two principal requirements imposed on the form factors.


112

Our further programme is to calculate explicitly the functions F and prove that they satisfy the third necessary requirement which is Eq. (16). We are going to calculate F explicitly. Let us consider as an example the ITM form factor f+ = z (fo + f l ). According to Eqs . ( 124) this form factor can be written as

f+W I ... , PI 2n)

= CIn

i3

\î - fl ) exp 1 2 i<j

n))++

J

To avoid confusion we indicate the ITM kinks' rapidities by primes. Our first purpose is to consider the limit of the function On,-l(An) when /31, ... , Nzn are partitioned into pairs 02i = Qi + z - ib, ,Q'2i-1 2 + ib and b --> 0. Having in mind the following application of the operation ()n we are interested in all possible partitions of Pi', ... , 162n in the function 1&n(al, ... , an-1I/'il, ... flij,8j, ...

, I3)

.

The function ¢n, _ 1 (O) can be presented as follows: On,-1(,&n( al, ... , an -1IQi1...., /3jj3j',, , ...

n // //^^ /^ _ ^ c8jy - Qiy - 7ri) det IIQtij II p=1

where 1 00 2tii

27ri (a

,r3p

P

x exp((n - 2j - 1)a)A;(cel,l3i1....

.. i„

Ia;l,...

,8j.)dce

that is why The function W(a) has simple poles at the points a = f a ; when Qii -' ,Q; + 2 , Q2i-1 -^ Qi - 2 the singularities of the integrand tend to the contour of integration. Extract these singularities explicitly: b-n+1

O(B IB°IB+) B=B-uB°uB+

x

{E(,8+ -,3 - ) - ai(n(B+) + n(B °))}

X 11 (f3+-,Q-// -iri)) R0 - /^ (^+ 7r i 7r -,3- +

i)(, + - /3 0)(/- - 30)(N0 -

)(/3+ -

/3 -)

x ((011 11 B(a°)B (a+)B(Q+ + Ii) ¢^ ( x `>2 e

1

11 ( /3 q sh(/ - i3 )(F'i - Pj ) ) exp `- /7) i<j

the terms with B" # 0 disappearing in agreement with Lemma 2. Now that we have some idea about the degeneration of ITM form factors to the NLS ones , let us pass on to formal definitions. First , let us define a base in H1 0 ... 0 H;, consisting of covectors wwi,...,w,. (31

)

... A) =

((OII

11 j:wi=0

B(i3)

11

B(/31)B((3j

+

in)

j:wj = 1

wj = 0, ±1. This base is an eigenbasis for the operator .A(r) with the eigenvalues

11 j:w^=-1

o, -,6j o -)3j - 27ri

(138)

x ? ^3j - 27ri wwl,... ,w„ (Nl, ... , n j:w^=0

The base ww1.... wn is connected with the natural base of tensor product ew1.... Mn via a triangular transformation which means that in the expansion Cwl,...,w„t^l,...

wwl,...,w,.(Nl,... ,/3 Nn) = Ew,=Ewi

Nn ) ew ,1,•

w rn

Form Factors in O(3)-Nonlinear ,-Model

121

only those C, 1.... ,wn differ from zero for which (w1, ... , w,) < (W1.... wn), the multiindices being ordered as ternary numbers (for example, (0,-l) < (0,0) < (0,1) < (1 - 1) < (10) < (11)). The covector WW1 1... w,. (o1, ... , , n) can possess simple poles at the points ,C3j = f3 i + 2ii, /3j = f3i + ai; the poles at the points f3j = /3i+ai exist only for j > i, wj = 1, wi = -1. All above statements can be proven similarly to the Lemmas of Sec. 3. A base dual to w,,,l,,,//^^,wn (,Q1i ... (3n) can be defined: wwl,...,wn (N1,... ,fln)

_

((0il

11

B'( ,8j

+ 2aa) 11 B((3

+ 7ri)B' (Qj + 2ai) ,

j:w1=O j:w1-1

where B'(i3p + tai) =

lim B(o + 21ri )(o - ,6p). The following equation a^pp

similar to (80) holds (ww1....,wn P1,... ,Nn),,13m)) = bwl,wi ...bw„„w;,,

Definition

1. Consider a set of functions F (1 . ) 1 , . . . al+cIp 1, • • • ) /fin-21+clv1, ... , v,), c = 0,±1,1 = min(0, -c),... , [!5 ]. Operation (( ))n associates with these functions the following vector from Hi ® ... ® Hn:

Fn(B +)(B I B °IB+)

(({F}))n= E B=B- uBouB+ X 11

(#+

-,3- - ai)

(,o+ -,a- - ai)()3+ - P°)(,8 - - a°)(,Q+ - 3-)()31 - a ° - a i)

X wwl,...,wn(B) , (139) where wj = -1,0, 1 for fij E B-, B°, B+ respectively.

Definition 2. Polynomials 4e(B-IB°IB+) (n(B+) - n(B-) = c) are defined as follows:

0c(B jB°IB+)= II 11(Q+-P-)fk(B-jB°IB+)detMk i<j i,jq k fk(B IB°IB+)

1,,8kEB°,

/^

(1I(/3k -,3-)(#k - /3 - ai)(13k - j30 - ai))-1 , Pk E B+ (II(/3k - N+)(/3k - )3+ + ai)(/3k - N° + ai))-1 A E B-

122


where Mk is a ( n -1) x (n -1 ) matrix obtained by omitting the k-th column from the n x (n - 1 ) matrix M` with the following matrix elements:

M, =Qi +c

r

Ti _

MMIB

7ri

7fi

+ 2'B 2,B° 2

( B° - 321 - Sc,1o; I B+ - 2 ,B+ - 321 ,

(140)

for i = 1,... ,n-1„Q3 EB-;

M; = II(a, - a -)(/3 -,8- - 7ri)(,a; - - 7ri) Q°

X {

i_c

- bc, -1Q; x (/33 -

flj - iIB(

(B

#+

- -

B- B° 2

2

2

+ 2 B° + 21 + jj(î - a+)

1B

+7ri)( Qi - (3° + 7ri ) { Q i+c (Qi IB- +

_ 7ri 7ri

B - 2B °-

-

bc1 Q;

l

2

,

7ri 37ri 37ri 1

B+ 2,+-

2

B°

2J

for i=1 ,..., n-1„Qj EB°;

M1 =

,

° (iii - iniIB - 2 , B+-321 B 21) iri bc, -1Oi (B - - 21,B + 21)

,

for13.

The reasonings presented at the beginning of the present section implies the validity of the following statement.

Form Factors in 0(3)-Nonlinear o-Model

125

Theorem 1. Form factors fa , f a, f,," f satisfy the requirements /^ //^^ //^^ w„ ww;+1(f /î /î+1) -)3 f (Qi, ... , Ni,+1.,. Ni+)wi . .. w.,w.+i ,... ,w,.Swi t}1 , f qq1i,. f(^1,...,F'

,.q Ni,•n.. fln) wl,...,w'+,,w', ...,w,.,

(141)

f(fi,... , fin-1, Qn + 27ri) wl,••• ,w+.-l,wn

(142)

= f(i3 , i1, ... , fn-1)w,.,w,,... ,w,._1

The connection with ITM form factors implies that fµ, fa generate triplets and fm,,, f generate singlets with respect to the action of SU(2) whose generators are Ea = ES;. This fact clarifies the equality of the minimal number of arguments of f (Q1...... On) to 3: one-particle space has no singlet subspace. Now we proceed to the most difficult problem which is the calculation of residues of form factors. Unfortunately, in contrast with Eqs. (141), (142) it seems to be impossible to obtain formulae for residues in connection with ITM. First, let us elucidate what singularities form factors as functions of fin have in the strip 0 < Im fn < 21r. The only singularities are simple poles at the points ,Qn = ,ij + in. Let us explain this fact. The operation (( ))n is regular in the strip besides simple poles at the points ,On = ,Qj +27ri. The covector fln) has also simple poles at the points fin = /3j +7ri for wn = 1, wj = -1, but they are cancelled by the multiplier II(,3+,Q--7ri). The multiplier Hi<jth2 2(fi-,Q3)(fi-,13j+7ri)(fi-Qj)-1 cancels the poles at the points Nn = ,13j +21ri and produces simple poles at the points fn = ,3j + in. As usual, the multipliers (E(exp(fj) - (-1)0 exp(-,3j)) in fµ and (E(exp(,3j) - (-1)µ exp(-fj))) x (E(exp(,ij) - (-1)" exp(-,Qj))) in f,," cancel the poles in minimal (two-particles) form factors; (Ech pj )2 (EshPi)2-1 in f serves the same purpose for the three-particle form factors since 3

(

2

ch/3i ' j=1

3

2

3

- Esh,ij -1=2+2Ech(Pi j=1

i<j

3 =8flch2(fi-fi) i<j

For the calculation of residues of the form factors we need the following statement.


126

Lemma 3 . The following equations hold:

O(B U Q I B°IB+U/3)=O(B IB°U QI B+), U fIB°I B+ U (a +i i)) _ iri o (B-IB°I +) c(B x

[ll( ^3

-

/°

+iri)(N-

- ll(,3 -(3° -

(143)

/^ B /^ ar °)(N-N +7ri)(F3-/3+)

-/3°+27ri)(/3 -,3- - 7ri)(,3- /3 ++27ri)]

7r i)(/

B+UB-UBO #0. (144) To avoid piling up of indices we have denoted ¢° by 0. Proof. Consider, at first, Eq. (143). Recall that

1 4( B- U a1 B °i B + u a) = III i< (fl - fl)

H

P+EB+„p

i,i#k A-EB-up

x det Mk(B- UQIB°IB+ U,9 )

fl

1

B

U PIB °IB+ UP)

,

igEk

(145) where Mk is an (n - 1 ) x (n - 1) matrix, n = n(B+) + n (B-) + n(B°) = 2n(B+) + n(B°), with matrix elements given by (140), /3k E B- U B° U B+. The two columns which correspond to ,3 E B- UP and ,3 E B+ U ,3 are placed last . Use Eqs. (112) which imply + - 7ri + - 3ri ° 3ri 7ri 3ri Qi ( iB 2 ,B 2 B - 2 ,#2 ,P2 Q

B+ - 3ri B° - 37ri - 3ri (aiB+ - 7ri ' 2'

2'

2'

2

7ri + - 7ri + - 3ri o 37ri 3ri ,3- 2 Q:-i (ajB 2 B 2 B - 2 ,Q- 2

Qi aIB

_

ai

- 2 B

=Q ( a

B-

_

iri ° Sri

ai ai l

+ 2 ,B + 2 ,N - 2 ,Q+ 2 /

7f°

- 2,B

7ri

+

7ri

B + 2,a+

Ti

( /3 - 7r Qi-1 ( lB

-

-

2

_ -

B +

7fi

2)

7ri ° 7ri Ti

+ 2 ,8 2 ,/9+ 2 /

(146)

Form Factors in 0(9)-Nonlinear o,-Model

127

Equation (146) allows us to rewrite (145) as follows:

O(B U l3I B° I B+ U Q) =

11 (,3+ -# -) Q II (/3i1- /3,i) (i +EB+ i#k lI ( - /3i)

i<j

P-EB-

x det A1k fk(B

IB° U)3 IB +)

where Mk is a (n + 1) x (n + 1) matrix whose left-upper corner coincides with the matrix Mk (B- J B' U /3IB+). Two last columns are equal to + 7ri B - Ti

aIB ' 2'

Q'

B°

- 7ri

+ Ti

Q 2' 2' 2) B+ - 7ri B° - 37ri )3- 37ri Qi (/3- 7riIB+ - 37ri 2 2, 2 2) Elements of the last row and the first (n - 1) columns are equal to

Qn +1 Qn+l

/3i

-7r iIB

+ 2 2) ' QiEB ,

+ - 37ri + - 7ri ° - 37ri 37ri

2 'B

2'B

_ iri , _ 7ri ° ai i , 7r 3 + 8 + 'B +

2

2

2

2)

' /3iEB

- /3- - 7ri )(,Qi - /3° - 7ri )(Ni - /3 - 7ri)

79 x Qn+1 (/3i - 7ril B+ - 32 ZZ' B+ - 2 ' B° - 32 ^'

+ H(ai -

/3+)(, i -,6+ + 7ri)(Yi - 3 77°

0

2t

+ 7ri)(/3i - + 7ri)

7

7°

q xQ+ (/3dB - 2,B - 2,B + 2,,6+ 2) ,

/3iEB .

(147) Notice that the number of arguments in Qn+1 in (147) is equal to n + 1; hence one can use Eq . (129):

Qn(B)(aIB) = II (a_13_

)_ II (-,(3+ 2)

(148)

PEB PEB

Substituting (148) into (147) one proves that all the functions in (147) are equal to zero . For the last two elements of the last row one has 7riIl(/3-,3+)(P -P++iri )(a-N°+7ri) , -7rill(N-F' )(N-l3--7ri)(Q-/3°-iri) .

128

Form F&ctort in Completely Integr&ble Model* of Quantum Field Theory

Now expanding the determinant along the last row one obtains a sum of two n x n determinants with identical first n — 1 columns. This sum can be rewritten as an n x n determinant with the same first n — 1 columns and the last column being

+(-*«) n ^ - /no? - v - 'O^ - /?° - »o Evidently we have obtained det Mk(B~ \B° U P\B+). Let us pass on to the proof of Eqs. (144). Applying (121) twice, one obtains Qi(at\B,p-j,p+jy)=Qi{a\B)-2PQi_1{a\B) + (/J-y)(/J+y)Q«-i(«|B). Further using the identities

QnW(a\B)=n ( « - * - ? ) - n ( « - * + T ) <J.w 2 X s 1

//^^ qq i wz (Nl - Ni + 7ri) W,2

flj + 27ri

w2 (,3l - #j + 27ri)Pwi'£1

Hence /^ n-1 _T7 w' (/31 -,w^ j) P 1£/TE /i(^1 +7ri) Vw 101) 1 1

11 ( fli_/3j+21rj) 1

j=2

r

£1 i£1

(155)

x TE/(l31 +27ri)P.1

Here and later on f(fl) = r(/3 /32, ... , /3n - 1). For example, A(/31 + 7ri)A(/31 + 27ri) ll (/1

a1 Qj ++27ri )

The following identity also holds [22]: (P

)£1 £1/Ta1 (ill + 7ri)7ai'(Nl + 27ri)(P-) o vl//

^ ^

al - Q;- + 27ri )l

a;

°18`1 °1 6£i

(156)

134


which implies for example

.A(/31 + ai) B(/31 + 27ri ) - B(/31 + 7ri).A(Q1 + 27ri) = 0 . Let us prove Eq. (151) for (w1, wn) = (-1,1). Eq. (151) can be written in the form

((oll E

jI B(/3°)B(3+) B(/3+ + 7ri)^(B

IB° IB+)

B=B-uBouB+

X

X

11

/(/3+ /- Q- + iri)

(,13+

- Q- + 7ri

)(/3+

,

- 3 0)(,8- -

00)( Q+

-

l3 - )(/3°

- fl° - 7ri) }

P_ 1,11pn =p 1 +*i

11 B(/30)B(#+)B(#+ + iri)O(B - I B° J B+)

_ ((°II (B\31,fln)= B-UB°UB+ 1

X 2 (Sn )2 X

(/3+

-

9-

-

ri)

(a+ -,Q- + 7ri)(/3+ - l3°)(l3- - /3°)(/3+ -13-)(13° - /30 - 7r) x (A(/31 + 7ri).A(/3i + 27ri )1 Gi /3 1 - +i ) - 1 I (157) -/39+27ri /) where P_1 , 1 is a projection operator onto el ,_ 1 ® en,1 0 Hz 0 . • • , ®Hn. What terms in LHS of Eq. (157) contain ((0II(Sn )z, i .e., are not annihilated by the operator P_11? The triangularity of the base w,,.,1 ,wn(/31 , ... , /3n) implies , at first , that these are the terms with ,Qn E B+. Present B(o) in the form

B(o) =

a1(o) A (o)bn(o)

+

al(o)B

(o)dn(o)

+ b1(o)C(o)bn(o) + bl(o)D(o)dn(o)

(158)

where

C

Cl(o)

dl(o) /

SO,1(o

- /31), \ C

n n(o)

dn (o)

SO,n(o - Nn)

the partition into the blocks being made with respect to ho. Note that

((Ollai(tr) = G -/3i-27ri o -,Qi ) ((°II, ((olldi (o) = ((oll

Form Factors in 0(S)-Nonlinear o-Model

((OIIbi(o) =

xi ) ((Ou1Sf , G - Qi - 2î

135

((O fl ci( o ) = 0

Consider a term with Ql E B°, B+ in LHS of Eq. (154 ). Move it to the left and apply it to the vacuum vector

((OII B(al ) = 2 ((O ISj . (159) The expansion ( 158) shows that B(o) does not contain the raising operator Si and consequently further application of the operators cannot annihilate Si in (159). Hence those terms in LHS of Eq. (157) for which #1 E B° or B+ are annihilated by P-1,1. Thus only those terms in LHS of Eq. (157) are essential for which 61 E B-, On E B+. To calculate ( (An - Pl - x i)

°II (B\P1,Pa )= B-UB°uB+

An - Al + ai )( An - 01)

) B(An ) 13(An + ri)

X r j B (A°)B(P +)B(A+ + ,ri)4(B U P1 IB° I B+ U An ) (Al - P+)(Al - A+ + xi) (A1 - P- + 2wi)(Pl - A+ - * i)(A1 - P° + ai )(P1 - P°)(Ai - P- +,ri)(A1 -,6+)

(P+ -,6- -,ri) P11 X H (P+ - P- +Wi)(P+ - 000- - P°)(P+ - P-00 7P - *i) IPw=Pi+*t-

let us use the following trick: put before P-11 the product

S( Yn-1 -,3n) ... S( 02

- Nn) S(pn

- Q2) ... S(Qn -

Qn -1) = I

obtaining

W

II (B \P1,P2 )= 1

U B_

UROUB+

( (On - P l - * i) B(Pn ) B(An +,r i \(Pn - Pl + ,ri)(An - P l)1 )

X 11 g( P °)g (P +)g (A+ + ,ri)O(B - U PuIB°IB+ U (01 + *i)) X X

(P+- P --si) (P+ - P- +,ri)(A+ - A °)(A- - A°)( A+ - P-)(A° - A° PI - P-)(Pu - P+ +,r i) (P1

(

- P- + 2,ri)(Al - P+ - ,ri )(P1 - P° +,ri )(P1- P°)(Pl - P- + -i)(Pl - P+)

Sn,2 (Pn -

02 ) ... Sn,n -1 (An

- Pn-1)

(160)


136

where ,B(o) is constructed via 7"(o) = T(oI,Q1, Nn, R2, ... , Nn-1), i.e.,

B(o) = (a1(o)an(o) + b1(o)cn(o)B(0)

+ (a1(o)bn(t) + b1(u)dn(o))D(o) Direct calculation gives

((OIIB(Qn)B(/n + 7ri)(, n - 01 - 7ri ) Ip,. =A1 +xi _ (-ai)((01I((S1 ) 2 + (Sn )2

- S1 S.-),

(161) o -1a Q2

((0II((S1)2 + (Sn )2 - ST S.-)e(r) _ (o, (T1 X ((011((S1)2 + (S )2 - Sj S)a(o) .

()(o13

k - in 37ri) (162)

Substituting these equations in Eq. (160) and using Eq. (155) one obtains

II B(a0)B(a +)B()3+ + iri)m(B- I B°I B+)

(roll

(B\#1,f3,.)= B-uBouB+

x (,d+ - Q- - Ti) (l3+ -13- + 7ri)(a+ -,80)(,8- -,80)(,8+ - fl-)(l3° - /3° - 7ri) -N+ 27ri)1 X I-1 -rj (a1 - ^° + â)()/q'1 -^3)(/ l'1Z)a1 +-N+)1a1 ` p (N1 )(N1 00 ) x

A(# , + 7ri )A(N1

+ 21ri)

(Sn 2

2 I

(

(Pi'

+ini)

Recall now that the covectors ( (OlllL6()3°)a(f3 +),6(,a++7ri) are eigenvectors for A(o) with the eigenvalues

(roll II g(a°)e(a +)g(a + + (o =

-

,(3

7ni)

A( o)

)(o - P° 7ri )

(o - ,3- - 27ri )( o -,8 0 - 27ri)

((Oil

+ 7ri) .

(163) This observation finishes the proof of (157) for (wlwn) = (-11). Consider now w1 = - 1, wn = 0 . Eq. (157) reduces to

Form Factors in 0(3)- Nonlinear or-Model

137

((Oil L, 11 B(o°)8(0+)g(#+ + *i)^(B-IB°IB+) t B=B-uB°UB+ (#+ -

x

(P+

P-

-

xi)

- P- + xi)(P+ - P°)(P- - P °)(P+ - P-)(P° - a° - xi) } P-''°

Pw=P1 }xi

E 11 1j (P°)g(P+)AJ(P++xi ) ^(B-IB°IB+)

=-7((°11

(B\P1,Pw )= B-UB°uB+

x

1

f

+

xi)(P+

x sn A(P1 + xi)8(P1 +

- P°)(P- - P°)(P+ - P-)(P° - P° fl ( P1 - P

2xi )

fEB\91 , P,.

xi)

1

P1 - P+2xi

(164)

where P_1,o is the projection operator onto e1,_1 0 en,o ® Hz ... Hn-1. It can be shown that only those terms in LHS of Eq. (164) are not annihilated by P_1,o, for which Qn E B+,B°,N1 E B-. For Qn E B,61 E B- the LHS of Eq. (164) can be calculated similarly to LHS of Eq. (157):

F, fl B(#') B(#+) 13(#+ + xi )6(Pn)Ii(Pn + TO (rB\p1,Pn )= B-UB0UB+ l

^(P1,B - IB°IB+, P1+xi)

x\(Pn (P1+xi )(Pn) 01) x

11

(P+-P--xi)

(P+ - P- + xi)(P+ - P°)(P- - P°)(P+ - P-)(P° - P° -xi)

-)(

x

(Q1 - P P1 - P+ + xi) P -1,0 (P1 - P- + 2xi)(P1 - P+ - xi )( P1 - P° + xi )(P1 - 0 0)(P1 - P- + xi)(P1 - P+)

= ((Oil

E rJ g(P°)d(P +)g(P+ + xi ) ^(B-IB°IB+) (B \91,P2) = B-uBOUB+

x

H (P+ - P- +

(P+-P-x

xi)

i)(P+ - P- )(P- - P °)(P+ - P_)(007 00 - x i) Sn

n-1 PI - P1 1

x i=2 (P1 - P; + 2xi)) (A

(P1

+ 2xi)t33( P1 + 2xi) - A(P1 + xi)d(P1 + 2 xi)) .


138

Substitute this in Eq. (164). As a result, Eq. (164) is reduced to

((Oil

BOO

E

(B \Q1,pw)= B-UB°UB+

(- 7ri)(Qn - i3) II

x m(B , Qi I B °, #1 + iI B +) 11

+

(Q+

(Q+

)

1

- N -)(#° - Q° - 7ri) 11 (Yl - /3+ - 7ri)(N1 - 0°)(/1 -

1 x (i3 -

Q

+

7ri)(,Ql

-

+ in )

- a- (+ 7ri)(,O+ - Q°)(Q- - Q °)

1 x I

B(Q °)B( i3 )B(a+

0 °)( ,31 - a°

a+)

+ 2 7ri ) P_ 1'° I3„=p1 +xi

_ -((Oil jj B(l60)B(a +)B(,6+ + 7ri )O(B I B° I B+) (B\p1,p„ )= B-UB°UB+

x (a+ - /3- - 7ri)

+ 7ri )(/3+ - a°)(a+ - a-)(a° - a° - 7ri )(a- - a °)

- f0 + 7ri) )(a1 -,8 (Q1 - Q 1 + 2-01

Sn B(,Q1 + 27ri)

,

( 165)

where we have used the identity (156). Calculate

((Oil

H B()3°)B(f+)B(Q+ + 7r i) B(,13n)P- 1,o

for some partition of (B\,31 i Qn) into B-, B°, B+. Substitute B(a) in the form (158):

B(O) = a1(cr )A(a)bn (r) + a1(a)B(o,)dn(o,) + b1(a)C( o)bn(i) + b1(o,)15(o, ) dn(or) .

(166)

The last two terms can be omitted because they create ST. We state that

((Oil II B(a°)B(Q+)B(/+ + et) =

11 (a° -,61 2eni)( Q+ -

?1 -

x ((011 II B(Q°)B(a+)B(a+ - 7ri) + (terms containing Si) .

/3 - 7ri)

27ri )(a +±

(167)

Suppose we have proven (167) for some subsets of {,3°} = B1, {/3+} = Bi . Let us apply one more operator B(,(33 ), /33 E B°. The last two terms in (166)

Form Factors in 0(3)- Nonlinear o-Model

139

are omitted, the first term gives nothing because ((O^^HB(/3°)B(/3+)B(/3++ iri) is the eigenvector for A(/31) with the eigenvalue (163) and B1 UB1 does not contain /3i . The case 8i E B+ is treated in a similar way. Now apply the operator B(3n)P-1 0 to

(coil II BcQ°)Bc Q+)g (/3 + in) . The second term can be omitted because it does not contain Sri. As a result one has ((oil J B(P° ) B(P+)B(P+ + ri ) B(Pn)P-1,o

__ - 11

(P1 - P°)(P1 -,6+)(,61 - P+ - -001 -P - + ri )(Pj - P°) (P1 - Po + 2-i )(Pl - P+ + 2ari)(P1 - P+ + ,ri )(P1 - P- - ri )(P1 - P° - *i)

x 1((°II II 6(Po)g( P+)9(P+ + ai)sn

(168)

The following identities hold:

((011

B(Q°)BcQ+)BcQ+ + iri)! (Q1 + 2ai)

_ ( -ai) 11

(Q1

-,0- + 2iri)(Qi - Q° + 7ri)

c/1- Q-)(Q1- Q

x B(Q+)BcQ+ + ai ) P +EB+

I B-\P

( 1

fl_ fl

0)

PEB- PII

2 i

B(Q°)

uB°

1 (/3 - /3 - 21ri Q1

-

Q

+ 2 7ri I J E B o \ )3 - Q° - Ti )

-2lni\ + E BII P EB°

°\P

^

((Oil

8 -6 - 7ri 0

x Q1-Q +ir iBO \# ( Q - Q °

B(Q°) II B(Q + )e()3+ + ii) B+uP

Q - Q - - ir i Q -+7ri ) ,

)B-^ Q -

(169)

whose validity follows from the observations. The LHS and RHS are both rational functions of /31 which decrease for ,Q1 - oo and have n(B-)+n(B°) simple poles and whose values at the points /31 = /3- - 2iri, /31 = /3o - 7ri coincide. Transform LHS of Eq. (165) using (168) and RHS of (165) using (169) and use the identity ^(B ,Qu1B°, Q1 + 7rilB+) = - 2(7ri)2 11('61 -,3- - 7ri)(Q1 - /30 - 7ri) x (fl, -)3+ + 2iri)(Ql - Q° + 2ai)^(B1, Q1 + ai l B° I B+)


140

following from Lemma 3. As a result the proof of Eqs. (165) reduces to the proof of the identity: O(B ,,61 + 7ri I B° I B+) =

jj(,a1 - Q+ + 7ri)(/31 - Q° + 2wi)

x { -2(7ri)2 E ^(B , /3IB°\l3IB+) /31 PEB °

(Q - f3° - 27ri

x T7

-

1 8 +29ri

)(0 - 0° + iri )(f -,6- - 2iri )(j3 - Q+ + in ) (/3 -

B +,B-,B °\P

iri)

1 + E ^(B IB° U,QIB+\Q) /^ PEB+ M1 - l3 + vi

x 11

1 (/-go +'i)(/-Q+)

(170)

From the very definition of 4 it follows that , considered as a function of /31i 5 ( B- U /31 IB°IB+) is a polynomial of degree n(B+) + n(B°) - 1. This is why it is sufficient to prove that RHS and LHS of ( 170) coincide at the points /31 = /3° - 2ai , 33I = /3+ - ai . This fact follows immediately from Lemma 3. Consider finally w1 = 0, w ,i = 1. Eq. (157) reduces to

((Oil E H 8(P°)8(P+)B(i+ + xi)m(B-IB°IB+) l B=B-UBOUB+ x

(P+

-

P-

+ Wi)(

P+

-

(P+- P- - *i) P ° )(P - P °)( P+

P - )( P O

-

PO

-

a i)

P0^1

Pn=^1

+ai

ljg(P °)g(P+)d(P++,.i)m(B- IB°IB+)

E

=((Oil

-

(B\9j„eh )= B- U B° UB+

(P+ - P- - ai)

x 11 (P+ -

n-1

P

+

xi )(P +

- P°)(P -

P°)(P+

2xix(Pl + ,ri) " II Pl - P; + zxi A( ,Ol + Pl - Pj

S-(S -)2 -

P -)( P° =7 - ,ri)

1 n

(171)

It can be shown that in LHS only those terms are essential in which On E B+, Yl E B- or B°. The accounting of terms with /3n E B+, /31 E Bleads to the replacement in RHS of A(l31 + 2iri)C(/31 + ai) by .A-1(/31+

Form Factors in 0(3)-Nonlinear o-Model

141

iri)C(/31 + 7ri). To transform the LHS for /3n E B+, #1 E B° one should use the identity

lim (0II8(P1)II 8(P°)g(P+)g(P+ +xi)13(Pn)13(Pn +ri)Po,1 ((°IIS (Sn )2 fl 8(P°)( (P+)g(P+ + xi)

(Pi - P° + ri)2(Pi - P- + *i)(Pi - P+ + 2ori)(P, - P+) (Pi - P° + 2 ri)(P1 - P° - *i)(01 - P- - ri)(01 - P-)(P3 - P+ + 2xi)

(172) whose proof is similar to that of Eq. (168): operators B should be applied to ((011 in the order they are written in LHS of Eq. (172). Note that it is impossible to use Eqs. (161), (162) here since an indeterminacy appears: in Eq. (172) the multiplier (fan -,81 - in) is absent, in Eq. (162) one has for a = #1 zero in the RHS. Formula (172) is the result of uncovering this indeterminacy. To transform RHS of (171) one should use the identity

(01l rj 8((3°)8(0 +)B (Q + 7ri )C(/31 + in )

X ((011 2

E

11

8(a°)

PEB+ B °UP

xB( p

+ri )/31

H

8(/3+)

B+\P

1 (#- /3° + 27ri ) (13- ,Q+ + 2iri) -/3-ai )(a-!l +)

1

+

II°\P B(/3°) 11 ,R(,8+ + iri) al - (3

PEB° B

B+

x TT (Q - + + iri) (P - )13° + ai) # /x (N- /3 +-ini)(N-0°)

(173)

Equation ( 173) is not quite obvious ; let us outline its proof. Both LHS and RHS are rational functions of fll, decreasing for ,l1 -+ oo and having n(B°)+ n(B-)+n(B+) simple poles at the points f1 =,Q+7ri , 3 E B- UB+U B°. That is why it is sufficient to prove Eq. ( 173) at n ( B-) + n(B°) + n(B+) points . RHS has zeros at the point #1 = /3- - in. Among the commutation

142 Form Factors in Completely Integrable Models of Quantum Field Theory relations on elements off one can find the following:

B(^1)^(^2) - ^(r2)e(o1)

COr1 1

O2

(A(^1)D (o2) - A(o2)D(ri))

(174)

Let /3k = 61 + in, /3k E B-, and move C(/3k) to the left, successively using (174). We obtain zero because

((611

e(Q°)B(^+) g(Q+ + vi).A(Q) = 6 , Q E B

((011 II e(a°)e(a +)g(a+ + 7ri)A(p + ai) = 6 , a E B° .

(175)

Consider now /3k = /31i /3k E B°. In RHS one has

(176)

((611 11 g(a°)e(a+)e(a+ + jr,) II 131 - Ii'i + 7ri B°\Pk j#k In LHS one has

((611 II e(a°)g(a+)B(a+ + in)! (13k )C(ak + ii) . (177) B°\Pr

From Eq. (156) it follows that

B(,6k)C(ak +

7ri)

= •A(/3k)D(Qk + 7ri)

'I

(

j /3k - 6

+

mil

Substitute this identity into Eq. (177). The first term disappears by virtue of Eq. (175) and the remaining term is equal to (176). The case of 13k = P1 - 7ri, Qk E B+ is treated in a similar way. This completes the proof of Eq. (173). Let us return to Eq. (171). Transform LHS using Eq. (172) and RHS using Eq. (173) and exploit

^(B- I B° U ,61 IB+

U (al + 7ri)) = 2(7ri)2¢(B- I B° I B+ UP,)

x fl(,3 - a+ + 27ri)(,ol - 0° + 2r i)(,31 - 0- mni)(al - /3° - vi) .

Form Factors in 0(9)- Nonlinear a-Model

143

As a result the proof of Eq. (171) reduces to the proof of the identity

O(B

I B° J B+ U )31) = jj( /31 - Q° - iri)( /3

x{ ^(B PEB-

+ 2,r2 E

1/31B° U /3 I B +) a1 1

-

Q -) 1

(l3 - '80 - 7ri)(0 - ,8 -)

a

^(B- IB°\/3IB+Ul3)

1

(/31 -'0 - z•i)

PEBo

X II (a -,60 + 27ri)(/3 - /3+ + 2iri)(/3 - a° - ri)(8 - ,3- - ii) } /3- +ai) (178) which follows from Lemma 3 and the fact that ¢(B-IB°IB+ U /31) is a polynomial of degree of n(B°) + n(B+) - 1 with respect to /31.

Q.E.D. We have shown that form factors fµ, f°, fo,,, f satisfy Axioms 1, 2, 3 and thus defined some local operators. What are the properties of these operators? With respect to the Lorentz group the operators defined by f,.' and fµ„ are a vector and an (1, 1) tensor respectively, the operators defined by f °, f are scalars. With respect to the isotopic group the operators defined by fµ, f ° are vectors, the operators defined by f,,,,, f are scalars. The possibility of introducing the nontrivial operator of charge conjugation C in the space of states is connected, of course, with the possibility of introducing it in terms of initial Lagrangian formalism: Cn°C-1 = -n°. Evidently the Lagrangian is invariant under this transformation. The operators defined by f,,,,, fµ are even with respect to the charge conjugation operators f°; f is odd. The above facts show that it is plausible to identify the operators defined by fµ and f;,,, with jµ = ea6cOµnbne and T,,,,. This identification is confirmed by the following theorem. Theorem 3 . If operators j", T,,, are defined by fµ, ff,,, then

O.j a µ -

aµ 'µ. = 0 , Tµ. = Tmµ

(179)

and the operators P. = f°° Tµ0 (x1)dx1iQ° = f jo(x1 ) dx1 are the

144 Form Factors in Completely Integrable Models of Quantum Field Theory operators of energy-momentum and charge, which means PPZwl (91) ... Zw,. (/3n) l ph) n = M>2(ePi + (-1)µe-'i )Zwi (t1) ... Zw,.(Qn)Iph) j=1 QGZ.*,(/3) ... Zwn(fln)IPh) n

E(.S'a)wiZw* (^1)...Zw(Qj)...Zu,,,(Qn)IPh)

(180)

j=1

Proof. Eqs. (179) follow from the very definition of form factors. The proof of Eq . ( 180) is quite similar to those of Theorem 5 of Sec . 6; it is based on the explicit formulae for the two-particle form factors presented above. Q.E.D. The operator defined by fa is C-odd Lorentz scalar transformed under vector representation of the isotopic group . It is natural to identify it with the field na itself. Additional arguments in favour of this identification will be presented in Sec . 10 by calculating the singularities of commutators at the origin of coordinates . At last , the operator defined by f is C-odd Lorentz and isotopic . It is natural to identify it with the operator of the euclidean topological charge q = eabcc,,8,,na8„nbn ° which is the simplest operator of the kind. The norms of the form factors are calculated by means of formulae similar to (80 ), ( 125). For example, ((({0°}))n, (({0o}))n) = (O°(B IB°IB+))2 B=B-uBouB+

(a+

N

- - 7ri)

x ($+ - /3-)(N+ - )3°)(N° - /3 .)(Q° - N° - 7ri)(13+ -)3- + 7ri)

1 x (/3° -)30 - 27ri)(/3° /3- - 27ri)(/3+ - /3- - 27ri)(/.3+ - /3° - 27ri) . (181)

9 ASYMPTOTICS OF FORM FACTORS

The local commutativity theorem (Theorem 1 of Sec. 2) requires besides the validity of Axioms 1, 2, 3, 4 special asymptotic behaviors of f(,61,... ,8k,,6k+l + a,... Q„ + a) for o -> oo, which will be studied in the present section. The asymptotic for ITM will be considered in detail; for SG model only statements will be given; the asymptotic for NLS will be obtained from the ITM ones. We consider in detail ITM asymptotics, but not the SG ones, because of two reasons: first, proofs are more descriptive for ITM; second, for ITM more refined formulae can be obtained which will be used in the next section. Before we calculate the asymptotic of form factors in ITM let us prove some subsidiary statements. Theorem 4 of Sec. 6 implies for the limit t --> oo the following Lemma. Lemma 1 . Form factors fµ, fµ„ in ITM can be presented in the form fj,33

... , 82n) = [ (eP; _

(_1)µe -P;)^

LLL (( /^ fµ( /31,... ,/32n) = I^( e"i

[J (3 _ F'j)(Fn).

i<j

_ (_1)µe-P; )]

[J (,3i

- /j)

i<j x ((F.+). + 1)a(F. )n) , a = 1, 2 fpv(QI,.

,A2n) _

X

[>(e#i

_ (_1)µe-R 1) r(e^1

HC(Ni -

Qj)( Fn)n

i<j 145

,

L^J

_ (_1)ve-Q;

11

11,, (182)


146

where Fn( /31,...

(21ri) -n ,/'nlln +1,... ,12n ) =

nI

h( a i - aj ) lI 0 ( al, ... , a n la l, • • •

J dai...dan

flct(ai_Pi)

q qS , Nn 1)n +1, • • • 032n)

i<j Fn()31,... ,Nn l/n+l.... N2n) n

(an

=

+j - a j - 7r i)

F(,31 ,...

,, n

l,6n+l, ... , i2n)

,

j=1

F} n (81, ... , Qnfl lqq h'n+1:F1 , ... , Y2n) - (27ri)-n

n!

J doi... , dan I ca(ai - #j)

x II sh( ai -

aj ) Il }(al, ... ) an 191, • • • , , n+l INn +1:F1 , • • • , /32n)

i<j

Polynomials IIc(al , ... )an l.l, • • • An+c lµ1, • • • µn-c) are the determinants of the n x n matrices B j (c = 0 , ±) with matrix elements Bj =Ai-1(aj IAl,... ,An+cl/11 )... ,/ln-c),(2 < i < n),B, = Oc(aj) where the functions A; are given in Sec. 7, Oe(a) are arbitrary polynomials of degree (n - 1) whose senior coefficient is equal to 1. Later on we shall take for 4 (a1), c = ±, the following polynomials:

vi

(a)=

n

Ia

7ri

-,Uq+

q=1

2 I , \

n-1

/

q5 (a)= I a

/

-^1q-

q=1

7ri \ 2)

\

/

In this case the terms containing 8c -loi(al, ... , An-l), 8c,i°i(µl - 7ri, ... , Pn-1 - 7ri) can be omitted in the formulae for As (118), since their contributions to the 2-nd, ... , n-th rows are proportional to the first row. For further estimation we shall need the following representation of the polynomials W. Lemma 2 . Polynomials IIc(al,... , an 1a1, ... , An+c lµl, ... , µn-c) can be presented as follows:

det De ll '(al, ... , and.\l, ... , An+elpl, ... , Pn-c =

7ri II (ai - aj) i<j

(183)

Asymptotic* of Form Factors

147

where D` are n x n matrices with matrix elements n+c

DSj 11

n-c -1

7ri ai-Aq --

n-c

1

11 (ai q=1 m=1 q=m+2 ^

-{4 q)1: s=0 (

2s

+

1) !.

2s+1 m

= d

( 2 da ) n+c-1 n+c

, 1 f7ri d \ 2s+1

X IL 11 m=1 q=m+2

(2s + 1)! ` 2 dal

m

x IT(aj - A,) } - 7ribc oO°(a1)O(ai)

(184)

q=1

where q(aj) is an arbitrary polynomial of degree (n - 1) whose senior coefficient is equal to 1. Proof. Consider, for example, II+(al,... , an Jal, ... An+l lµ1, • • • , 14n-l). According to Lemma 1 it is the determinant of an n x n matrix with matrix elements n+1

_^( a, - ay - 2 Qi-z(ai - 7ril{41 - 7ri, ... , /4n_1 - 7ri) 9 =1 n-1 ^

( ai

-

l4 9 + xq

Q i(a1

dal, ...

I

An +1)

,

q

where the polynomials Qi are defined in Sec. 7, Eq. (119) Qi(al.Al, ... , Ak)

[ (a+!)'_ (a_!)

_

'J

(-1)iO-i-r(^1, ... , ak) .

1=o Let us multiply and divide II+ by the Vandermond determinant ( a l - 7ri) n-1 , (al - 7ri) n-2

det

(an - 7ri) n-1 ,

(an - 7ri

,. . . . ..

1

185

( ) ()


148

We obtain

II+( a l, ... , CY n111)

1 l ... , n+1 /1 1 , ... , µn-1 )

det D+

=

7rtn(a{-aj) i<j

where

=n( î)

Q„+(aj - ai lp1 - ai, ... ,

V t aj - ay - 2 g=1

mcl n-1

x µn-1 - ai)(ai - ai)

n-m-2

+ 11 (aj - Pg + 2 4=1

n X

Q.(aj

lal. ..... n+1)(ai - iri) n-m

.

m=1

Thus we have to show that -1 kr

k-1

k

lvl, ... , Vk)a2- 1-"'

= L^ 11 (a2 - v9) m=1 m=1 9=m4-2 Qm(al

2s+1 m

x

11(al - v9)

(2s + 1)! (2 dai)

S=O

(186)

g=1

Let us prove this identity . Notice that Qm can be presented as follows: m 1 7ri -1 [mom [[ Qm(al lvl , ... , Vk) = LL (2) I1 - (- 1)l]C,l,ai 1 = 0 u= t

X (-1)m-uOm-u(vl, ... , vk ri 2s

= 2) s=0 C

+

1 1

(2s+ 1)!

d 28}-1 m

(dal )

Qm-u(v1,... , vk) , u =0

hence k-1 k-m-1 ai 2s+1 1 E Qm(all v1 , , .. , Vk)a2 = (2 (2s + 1)! m=1 a=0 p-1 d 2s+1 kk- k--p x (dal )

E E CIP, 2 -1) p=1 q=O

k-P-9-1Qk

- P-9-1(Vl, ... , vk) .

Asymptotica of Form Factors

149

Now we have only to use the identity k-lk-pp-1 aga2(-1 )k-p-q-lOrk-p-q-1(v1) ... , vk) p=1 q=0 n-1

m

k

- vq) ft (a2 - vq) .

E Mal M=O q=1 q=m+2

The proof of the last identity is not very complicated but a beautiful combinatorial problem which we leave to the reader. Thus the representation (183) for 11+ is proven. H- can be considered in a similar way; to prove Eq. (183) for II0 one should replace the first column of the Vandermond determinant used by 0(a;), i = 1,... , n. + Remark 1. The series of the type sEo 2s+1 , ( 1 da)28 1 g11(a - vq) encountered in Eq. (184) terminates. Evidently 00

7ri d

2s + 1

=1

ir i

7ri d

d

(2s ± 1)! ( 2 da) 2 Cexp ( 2 da) - exp C

2 da))

and consequently 00 1

ai d

2s+1 m

-) H(a - vq)

E (

2 da )! \Cs=0 2s + 1. 2

HC

q=1

-n(

7ri

a-vq+- a-vg-21) 2)

q=1

q=1

Remark 2. From the proof of Lemma 2 it follows that ordering of the sets {a} and {p} is of no importance. For example, in the expression n--c 7ri 2s+ 1 1 q=,+2(ai

d 2s+1 m

Cdaj) ^(ai - µq) - µq ) > (2) (2s + 1)!

all µq can be replaced by µ.wlgl, where in is an arbitrary permutation. The asymptotic behaviour of f (fl1, ... ,13k, Qk+l + o,,... , ,62n + o) is described by the following theorem.


150

Theorem 1 . Consider the form factors fµ, fµ„ in ITM. Denote by g° and g the following functions: 9 3(/31,...

,N2n) =

]I C(13i -

6j)(

F3)n

i <j

9°(Q1,... fl2n) = IJ((Ni -Qj)((Fn )n+ (-1)a(Fn )n) , a = 1,2 , i

9

(0

1, - .. r,62n) =

[JC(Qi - ij)( Fn ) n i<j

which are connected with fµ, fµ,, by Eq. (182). The following asymptotic holds : let /31 i ... , Qk be finite while Qj for j > k + 1 are equal to $ + a, / being finite quantities and o -> oo, then

9a(01, ...

,62n) = 0

0

k-1(mod2), exp ( 4IcT)

9(/11, ... 062n) = O a-' exp I 4 jojI 1 , k =- 1(mod 2) , 9(/31, 9°(Q1,

... , fl2n) =

O(0,- 2 )

... , Q2n) = 27r0'-

k-0(mod2),

,

leabcgb

(01, ... ,

Pk)

X 9e(&+1, • • • , Q2n)(1 + O(0'-1)) , k = 0(mod 2) . (187) We have presented in explicit expression the principal term of the asymptotics in the last formula as it will be important later. Proof. Notice first that we can limit ourselves to the consideration of the case k = n. In fact, suppose we have proven Eqs. (187) for k = n. Then using Eq . ( 110), we get res g (,Ql, Q2, ... Q2n) = 9(/33, ... , Q2n ) 0 (e1,1 ® e2,2 - e 2,1 (9 el,2) X (S(f2n - Q1) ... S(f3 -,61) /- 1) = /^^ 9 (/33, ... An) 0(el,l

0

e2,2 - el,2

0 e2,1)(S(fn - 81) ... S(/33

-

01) - 1 + O(t _l)) in. N2 =Nl+7ri

Here we have used the asymptotics S(v) = 1 + 0(o,-'). The asymptotics of the LHS have been assumed known, so that g(/33i ... , Q2n) also has the

Asymptotics of Form Factors

151

required asymptotics (the operator 1-S(,Qn-,81) ... S(,Q3-,Q1 ) has no kernel independent of f31 i the latter being arbitrary). The case 12n = , 02n_1 + 7ri can be treated in a similar way. Thus we can obtain from Eq. (187) for k=nEgs.(187)fork>nandk 0 and consider the function 93(#,,.. . q g3: /^ (188) ,82n) _

JT C(pi - /3j)(F3)n •

i<j

For the form factors in ITM the combination of exponential and power behaviour is characteristic. Consider, at first, the exponents. The function C(/3) has the following asymptotics:

y^2-* exp 14^ Q^ (1 + O(/-1)) , hence 2n

n

2

11C(Qi-

(3j)^' (2 l

)

i

2n

, e -11(( Qi -)3)

i<j

11

C(1ii-/j) •

(189)

i<jj 2-

2-P-

2n

.Fl; = 7ri Jdaft(a_

/3i) (a) exP(n+

1_2i)a,

j-1

where q( a) is an arbitrary polynomial of degree n - 1. Ai and 0 are polynomials, which is why the exponential behaviour of the integrands is

152


determined by ^p(a - i3) and exp ( n + 1 - 2j ) a. The function ^p(a - /3p) behaves for a -^ ±oo as follows: ^p(a - ap) = 21 exp (_ia - QpI) a-#(1 + O(a-1)) We can estimate this roughly by the principal term of its asymptotics. Consider three regions on the real axis : a < /3 j , j = 1.... n; ,Qi < a < /3i, i = 1,... , n, j = n+1, ... 2n; a > /3i, j = n+1, ... , 2n. The exponential behaviour of the integrands in these regions is described by exp((n + 1 2n 2n n 2j)a + 2na - E /i ), exp((n + 1- 2j)a - E /3j + .E Pi ), exp((n + 1 j=1 j=n}1 7=1 2n

2j)a - 2na + j E 1 6j) respectively. The first exponent is always an increasing one, i .e. it decreases for a -oo; the third exponent is always a decreasing one, i .e. it decreases for a -> oo. The second exponent is an increasing one for n+ 1- 2j > 0 and a decreasing one for n+ 1- 2i < 0; for n + 1- 2j = 0 the exponential behaviour in the intermediate region disappears. These estimations show that for n + 1 - 2j > 0 only the vicinity of a = tr gives n significant contribution to the integral; H ô(a - lap) can then be replaced p=1 by its asymptotics r.j 2' a-Il eXP - 2I f 1) o) 1 (G2 tai

I

0000

di

2n

x ^P(& - Qp)eXP n +1-2j iY ((2 p=n+1

x Ai-1(i••r+-1fli..... . Pi. 1Qi..... . Pi,.)(1+O(o-1)),

n+1 -2j > 0.

We have denoted O(a) by Ao(a). For n + 1 - 2j < 0 only the vicinity of zero gives an essential contribution to the integral fii = 2 2 o- 2 xexp(-

l

1

oo

n

\, 2 1+1-2j)a) 2 / tai,/ °°dafJ`p(a-/p)eXp p=1

x Ai_1(ajfi1,... ,Pi.I)lil,... ,/i,.)(1+O(a-1)) , n+1 -2j < 0 . Finally, for odd n the case n + 1 - 2j = 0 is possible. For this j we have for .Fi,i a plateau between the points 0 and a, which is estimated as


o

1 - (-na\

_

7'j

2'2aio

exP

2

153

A i- l(a I 0i17...

,Nijfl11,...F3i,.)da

0

x (1+O(Q-1)) , n+1-2j =0 Hence, for n = 2k the full exponential contribution to the asymptotics is k

exp((k2+jE1(1-2j))a) = 1, for n = 2k+1 the full exponential contribution k

is exp((14-(2k + 1)(2k + 1) - Z (2k + 1) + jE1(1 - 2 j))a) = exp(- 4 ). These results are in agreement with Eq. (187). Let us now estimate the power contribution. We consider only the case n = 2k which is the most interesting. For n = 2k + 1 the form factor decreases exponentially. The above reasoning implies 2k

det Jr - e-k

° 01

-20

1

J da i... dakd& i ...d&k

^7ri

4k

k 2k /

111

(ai - 3)

k

II alai-

gyp)eXP

i 1 =1 p=1 p=2k+1 X

H

0

(al) ...

,a k, 51 1 + a,... , &

1=

+ a l,ail....

))3i^.IN7i^... ,/3k)

(191) where II° is given by Eq. (183). Our main purpose is the estimation Of 110 (al, ... , a2k IA,.... ) A2k Ip1, ... ,µ2k) for the case al, ... , ak ^- 0, ak+l, ... , a2k ^ ' a; the set X11, ... , A2k, /L1, ... µ2k is divided into two subsets both containing 2k elements, elements of the first subset being - 0, elements of the second subset being - a. More concretely, let al, ... , Ak+p 0, Ak+P+1, ... , A2k - a, 111) ... , Pk-P '" 0,µk-P+1, ... , µ2k -. a, p being an integer such that I pI < k. For the sake of definiteness we consider p > 0 also. 2k-1

Let us use for H° the presentation (183), taking - 911 (aj - /19 + 2) for 2k

0(aj) and H (ai - a9 - z') for 0(ai). The matrix appears to be divided 9=2

into four blocks:

D° =1 D1 D2 V3 V4


154

The matrix elements of Dl and D4 have the following asymptotics for it -+ 00: D = ^zk

k+p vi k-p ( î l 2a+1 TT (a^ - aq - 2) 1 II (ai - µq) 2 qq=11 8=0 m=1 q=m+2 )

k -p

2a+1 m

d

iri

Mai - µ9) + I (aj - µq + 2 )

x (2s + 1)! (da •)

d 2s+1

k+p Ai 2s+1 1

x

E J1

(ai - J^

_m m=1q m x JI(aj -

q) a_

(2s + 1)! . (day )

(2 )

i < k, j < k ,

Ag - iri)(1 + O(tr-1)) ,

q=1

2k

Do

2k

2s+1

or2k (aj - aq - iri) > H - . U , ) m=1 q=m+2 8=0 q=k+p+1 d

23+1

2k

(aj - µq)+

x (2s + 1)! ()

q=k-p+l

11 (aj -

µq

+

2)

q=k-p+l

2s+1

2a+1

x (ai - Aq) 2 ) m=1 2k q=m+2Cri a=0

(2s + 1)! (daj )

m

(aj-Aq-7fi) (1+0(ir-1)),

i>k,j>k

q=k+p+1

From the matrix elements of the blocks V2 and D3 we extract only those terms which increase faster than a2k-2. They originate from the terms 2p-2 2k

2s+1

7ri

D - E J (a' q + ) ( "i) 2 m=1 q=1 a=0 2a+1

m

2k

(2s + 1)! 2k-1

d •) fl(aj -)1q) 11 ( ai - Aq - 7ni ) - t (aj_uq+) X (da q=1 q=m+2 q=1 2k

x

[J(ai - Aq - li) + O(o 2k -2 ) , q=2

7 < k, i > k ,


2p-2 2k 7ri 2s+1 1 r l (cu•-A q

'j - III

D°--

155

2)

m=1 q=1 8=0

\21 (2s + 1)!

28+1 jm 2k 2k-1 d ) X (da

^ (ai - µq) - I (aj - Pq +

11(aj - p.)

q=1 q=m+2 q=1

21

2k

x

jj(ai - .\q -

1r i)

+ O(tr 2k -2) ,

j > k, i < k

q=2 2p-1

Notice that D° is in this case of the form E 1 f^ g;'+O(Q2k-2) where f^ trk+p. One can transform the matrix D°, taking the linear combinations of columns in order to obtain in the changed blocks D2 and D3: 2p-1 frngm

(D2)ij = m=j-k 2p-1 (D3)ij =

fj g{' . m=j

In the course of this transformation the estimation O(0.2k) for the matrix elements of b, and D4 is preserved. Now the dependence on ai can be extracted precisely in D2i D3: (D2)ij = a'nP(

j)(Q)

m=0

amPm)(a)

(D3)ij =

m=0

where P,(nj), P,(„j ) are polynomials which depend , besides a, on aj, A, µ, though this dep endence is not essential for our purposes . The polynomials P,(,{) (a) and P,(j )(a) have the degrees 2k + 2p-(j -k )- 1 and 2k + 2p- j -1 respectively. Let us add to the second column of the determinant the first column multiplied by -Po' -1 to the third column - the second column multiplied by P(3)( tr)[P12 ) - Pi0(Po1))]- 1 and the first column multiplied by Po3 )(o)[P0 ( a)]-1, etc ., up to the k-th column. In the course of this process the asymptotic behaviour of D1 remains unchanged while the first (p - 1) columns of V3 change their asymptotic behaviour to

156


01 2(1+P-i), 7 = 1, ... , p - 1; other columns of D3 still behave as O(o2k). A similar operation can be carried out with the task k columns of the determinant, whose result is: the first p - 1 columns of D2 are O( o2(2k+P-i)), other columns of D2 and all the matrix elements of D4 are O(o2k). Hence detDo = 0( 0,4k2+2P (P-1)) .

Consider in more detail the cases p = O, p = 1. For p = 0 the block D3 is O(o2k-2), hence D° = o4k2

det

[(det D1 + 0(o,-1))(det D4 + O(o-1)) + O(o-2)]

where D1 is a k x k matrix with matrix elements k

(Dl)ij =

ft

( aj -Aq-

)

2

- µq q=1 m=1 q=m+2 s=° d

x (da)

2s+1

O

m

k

1 (2s + 1)!

7ri

fi(aj -pq)+jj (aj_pq+--) q=1

ire 2s+1

x

2s+1

)

2 Y_ 11 (a;

q=1

1

d

2 (2s+1)!

2s+1 m

( da; )

TI(aj -µq) • q=1

The matrix D4 differs from Dl by the replacement al -4 ak+l, Aq

. Ak+q, µq ' µk+q

For p = 1 the block D3 is O(o2k-1); the block D2 is O(o2k). Consequently a

detDo = .4 k detD1(al,... ,ak I A1, ... ,Ak + 1 Iµ1) ...

µk -1)

x det D-1(ak+l, ... , a2k IAk+2, ... , azk l At, ... , µ2k)(1 + O(o-1))

Matrices De are given by Eq. (184). The case p < -1 can be treated in a similar way. One has detDo = O(Q4k2+2P(P+1))

For p = -1 this formula can be more precise:

kl Al, • • • , 14-1I µ1, • • • , /6k+ 1) x det Dl (ak+1, ... , a2k Iak) ... , A2k I µ 1+2, ... , µ2k)(1 + O(0' -1))

detDo = - Q4k^ det D1 (al, ...

)Cr

157

Aaymptotica of Form Factors

Substituting these results into Eq. (191) one obtains k2°trk2+ 21)2

det.F = O(e2 k2 e_k2°Ok2 det.F =

X det.T

(192)

-2 ) , (IpI > 1) .

detF+(f11 ,...

(Nik +2) ... ,fit2k

Pi

(j (^ ,iik+l A l)... Pik -1)

k ,... ,

fi72k)( 1+ 0(O -1))

,

p = 1 (193) where /ail det

....

) Ak

+1 , /3

F-2k2e -

j1) ...

k2°Ok2d et

x det F+(aik)...

) /"

j k -1 '"

0) Y ik}21

.F- (fi1....

,/" i2kl/,k +2

)...

.. ik

... ) i2k ) #jk) ... 03j2k

-llajl,... ,f3j

-

or)

k +l)

, Nj2 k )( 1+0(O -1)) ,

p

- -1 .

(194) The case p = 0 is to be considered in more detail . We have obtained the formula

II°(al)...

,a2kl)il)...

1

2

,A2kIP1,... ,

µ 2k)

03k i<j

(ai

^ (detDi

- aj)

{ O(tr-1)) al 1 a') (detD2 + O(O-1)) + O(O-2)1 k A=A1uA2uA3 n(C)-0 B=B1uB2uB3 X

0(A3, B3)S(B 21B2

U B3)S(B2

U B31 B)

f

dCG_(xIC,A1,B1,A3)

(215)

,

where G- (C, A1, B1, A3) = (-1)n

( Bl)+n(B2)+n(C)[ f1

(A1 + i01 U B 2)

S(CIA3U C)f2((C U A 2)-i01B1 ) c _(ic ( C)+ic(B )+ ic(A3)

+ c(A2))-f2(A2-i0IB1U C)S(C U A31C) X f1((A 1 U C) + i0I B 2)co_ (-,(C) + c(B2) + ic (A3) + tc(A2)) (216) The principal statement of Sec. 2 is that the matrix element ( 214) vanishes if e,6j)ga(/31,,... ,)32.) , f b (Nl, ... ,N2n) = M(> e-1i) ga(Nl, ... , 02n)


Let
oo after the cut-off is introduced. Consequently in the integrals

-,/ dcrfl(A1+i0IC U B2)S(CIA3U C)f2(C UA2IB1) ✓r x 'P_ (K (C) + ic(B2 ) + ic(A3 ) + sc(A4))do , (228)

CJII

we are now interested not only in those terms which behave as 0(1) for A -+ oo but also in those which behave as O(A-1) for A -+ oo. But the terms which decrease as o(A - 1) disappear after the summation over n(C) and the limit A -> oo are taken.

Current Algebras

175

Let us check the commutation relations . We start with the commutator [na(x),nb (0)]. The function f a(A1 + t01Z1 U B2 ) S(CIA3U C)fb(C U

A21 B1)

behaves as 0(1) or decreases for A -> 0 while the function 0_ decreases as exp(-I A 1). Hence the integral (228) decreases (at least as O(e-IÎ)). Therefore [na(x),nb (0)] = 0. Now consider the commutator [Q° (x), Ob(0)], where Ob is nb or jµ. The following formula similar to (223), holds:

J (A I [Qa-(x)

O"(0)]I B )e'r(A)xcp(x)dx

= J (AI[Q. (x) , Ob(0)] I B )e",(A) - J(7 I [Q.(x ), 06(0)] 1 B )et + J(

l[Qa,0b(0)}l

z,P - (x)dx

(A)s^

+(x)dx

)e(A)rc^(x)dx .

(229)

Evidently the last term in RHS is responsible for the appearance of iCabc9 (x)Oc(0) in the commutator. Consider the first term. Let Ob = nb, then in the integral (228) the functions f1i f2 are taken as ga , f b, respectively. Obviously the integral goes to zero as exp(-I A I) for A -> oo. The second term in RHS of (228) is treated in a similar way. Hence [Qax), ( nb( 0)] = ie3bcO ( x)nc(0). Now let 06 = jib, and consider the function

(_1)n(C)+n(B, ) +n(B2)ga( A 11 G^ U 8 2 )S(C I A3 U c) x

fib, C U A 21 B 1)c- (ic(C) + i(B2) + ic(A3) + c(A4)) (230)

The form factor fµ is equal to fµ (O U A2)B1)= Epp' (P"' (C)+1 ,(A2)-pµ" ( B1))g" (C U A21B1) Hence, in the vicinity of or„,, the function (230) can /behave both as O(1) and as O ( A-1). It is easy to understand that the function (23) behaves as O(1) only if Al = A2 = B1 = B2 = 0. In that case it is equal to ,P

(0)IIg°(6) 1126ab(1 + O(A -1))

. (231)

176


When does the function (230) behave as O(A-1)? For Al = B2 = 0 one has

(-1)n(B.)Ilg°(C)II2,°d aga( A2-iOIBi ) (1+O(A-1))

(232)

For A2=B1=0 one has

(- 1)n(B2)II9

°(

C)II2E adbgd (A2+iOIB1) A (1+O(A- 1))

(233)

Finally, for Al = A2 = B1 = B2 = 0, the next term of asymptotics after (231) is Eabdllg

° (C)IIZ

(234)

A F'As ,

where Ed is the sum E.C. SH., Ha is the isotopic space connected with the rapidity a. Having derived Eq. (234) we have to use the formula S12(a) = 1+27ria-1S1S2 a + O(a-2) a -> oo . For all A1i A2, B1, B2 which do not satisfy the requirement listed above the function ( 230) behaves as O(A-2) for A - oo. Fix C and compute the sum (218) of the functions ( 230) for o - oma.. The term of order A -1 is equal, as it can be understood from (232 ), ( 233), (234), to

c (0)(A IQ s ( x) + Q+ (x) - QdI B )Eabdn-1 = Hence, the contribution O(A-1) to the sum disappears. The point omin also does not give an O(A-1) contribution. Thus we have only to calculate the 0(1) contributions. Such a contribution from the point Amax is defined by Eq. (231) while the contribution from the point on,in differs by the multiplier (-1)'. Hence

J(Z[Q(x), j, ( O) ]I B )^-(x ) dx

iso(0)(1- (-1)µ)Sab 21

r

Similarly,

-

J (7

B ) ^+(x)dx

I[Q.(x) , 7µ(0)1 '--

iso(0)(1- (-1)µ)b

ab-

Current Algebras 177

Hence [Qa (x),

iµ(0)]

=

is abce (x )jµ(0) +

ib abb( x)(1

- (-1)A)oo

(235)

where oo means a constant diverging as A for A --i oo. Let us now calculate the commutator [na, jµ]. Notice that this commutator has been calculated above, but here we give another method. The coincidence of the results provides us with a strong argument in favour of Hypothesis 2. Consider the function (_l)n(C)+n(Bl)+n(B2 )fa(A1ICU B2)S(CI A3U C)

x

f; (C U A2I B 1)0- (ic (C) + ic(B2 ) + i( A3) + c(A4))

and estimate it for o - an,ax. Everything is quite similar to the case considered above, the only difference being that now n(Ai)+n(C)+n(B2) 1(mod 2), n(C) + n(A2) + n(Bi) - 0(mod 2). This fact implies that the terms with Al = A2 = B1 = B2 = 0 are absent, that is why the 0(1) contribution is absent. The O(A-1) contributions are produced by the terms with Al=B2=0 or A2=B1=0: (_l)n(Bs )Ebda fd(A2

loiBl)II9b(C)II2 A- 1 2ircp(0) - (-1)n(Bl)Eadb fd(A1 + jOl B 2)Il.fa(C/)112 A-1 2ircp(0) -

I

where n(C) - 0(mod 2), n(C') - 1(mod 2), which fact stipulates the presence of the minus sign in the second formula. Summing over A1, B1, A2, B2, A3 one obtains the contribution from o - omax:

'P(0)

27r

a

": (A Inc(0)I B)

(Ilf d (CI)112 + Ilgd(C)I12)

Now summing over C one obtains

(O)E ab, ( A 1 , The contribution from o -

,,P(0)

om;n

I n `(0)I B )

can be calculated in a similar way:

Eabe (A

In`(o)1 B)( -1)" - 1

178

Form Factors in Completely fategrablc Models of Quantum Field Theory

Hence

J(2\[n'{z),

jl(0))\*B) The same formula can be obtained for the convolution with i , kj > ki.

Let us introduce the usual compact notations for the sets of rapidities.

We are going to define the polynomials

A(AIB(1 ) I ... IB(N)), oi ( AIB(1)I

I B(N)), (n(B (`)) = m, n(A) _ (N - 1) m - 1), oij (AIB(1)I ... IB(N)), ( n (B(k)) = m - Ski + bkj, n (A) _ (N - 1)m - 1). The role of these polynomials is similar to that of the polynomials A, A3, A' from Sec. 7. The polynomials A, A', A'j are invariant under the permutation of rapidities within the sets B (j). They are antisymmetric with respect to the rapidities composing A. Polynomial A is the determinant of an ((N - 1)m - 1)((N - 1)m - 1) matrix with the following matrix elements: Aij =

Ai ( aj IB (' ) I

...

I B(N))

,

N

Ai (aIB(1) I ... I B(N))

(a -,3 - N (N + 1- 2s)) s=1 /EB(•)

\\ N x QS N) a N (N - 2s + 1 I I I U (B(P) + Nsgn( s - P)) ( (_ // P=1 \ P#J


where

Q(N)(aI B ) _

k

(

( a

(

Sri iri

)

+

1-

1=0

-

)'

1 1( B) ,

(-1)

\

and o-i is an elementary symmetrical polynomial of degree i. The polynomial A' is 21nirn

A' A B(1) ... I B PEB('+1) PEB(')

x

A(AIB(1)I ... IB(N))

.

The polynomial 0'j are the determinants of ((N -1)m-1) x ((N-1)m-1) matrices A'1 with the following matrix elements:

Ak1

=

Ak ( a 1IB(1)I

... IB(N)) , N

Ak(aIB(1)I...IB(N))=^ T7

II

(a- ,3

+ N(N+1 -2s))

3=1 PEB(

x Qk+b ..- b.i a - N (N - 2s + 1) I U (B(P) + N sgn(s - p) P#s - 631 k U (B(P) + N 7ri sgn(s - 7)) P #i

Let us define also the antisymmetric functions, which are analogs of exp(±Eai) exp( z E,61 )ill sh(ai - aj) involved in the integral transformation On,, from Sec. 7. The functions wf (A) are defined by ((N-1)m-2)(m-1)

w+(A) = 2-

2

exp

(TN 2 E a± 2 E Q aEA PEB

b} (A) ,

where B = U B(P). b} are the determinants of the ((N - 1)m - 1) x ((N p=1

1)m - 1) matrices BI with the following matrix elements: Bk =exp

(±(k+[N-1] )al) .

Form Factors in SU(N)-Invariant ...

187

Otherwise, b± are the Vandermond determinants of the variables exp(±at) with all the rows whose numbers are integer multiples of N omitted. Evidently, for n = 2 we deal with the Vandermond determinants of exp(±2a,). Now we are in a position to define the functions

Fµ,(B(1 )I ... I B(N)) _

PEB

(eP

- (- 1)µe-0 )

(1 2

)n(A) fdA

PEB aEA n(A)

FN (B(1)I ... I B(N)) = f dA

( 27ri )

IT col (a - Q) 11 aEA,PEB

X Aa(AIB(1)I ... IB(N))(w+(A) - (-1)µw-(A)) N_i a 'P(N) (a) = 4- N 7r- ^N^ N - 1 - a r( 2N 27ri)r( 2N +2ri)

(A7)

where f dA = f 00 dal ... f dan(A). Evidently, we are dealing with a generalization of the integral transformation cn,o. It is quite clear that the operators T,,,,, j,; can only transform the vacuum into a state containing Nm kinks (m E 7L+), which means that the corresponding form factors depend on Nm rapidities /#N,n} = B. The form factors of T,,,, are given by the formula fpv()31 .... ,fNm) _ rj (1,1(/3i -)j) E F,tv(B(1)I...IB(N)) _< i B=UB(P) n(B(P))=m,Vp

x fJ P 0? Let fn = r + 27rk, 0 < Imr < 27r, then, as it follows from (A13), h(81,... „ln-i, r + 27rik) = h(#,,... ,

S(f1

(an x S(/3n -1 - 7 - 27ri (k - 1))S(f1 -

X ... x

- r)S

F' n -1, r/^)'s(Nn-1, -r)

-1 - r - 27ri) ... 7

S(f1

- r - 27ri)

- 27ri(k - 1)) . (A14)

The only singularity of S(/3) is the simple pole at the point /3 = 2ia N Consequently, the singularities of (A14) are caused by the singularities of h(81i ... , /3n-1i r), i.e., they are simple poles at the points /3n = r + 27ri + 27rik ± (7r - N )i whose residues can be expressed through h2(/31i ... , /3n). What singularities does h(#,,... , /3n) have in the strip -27r(k + 1) < Im/3n < -27rk, k > 0? Let fn = r - 21ri(k + 1). Then h(,31,... , fn-1, r - 21ri(k + 1)) = h(r - 27ri(k + 1),/31, • • • , /3n-1) x S(rq- 21ri(k + 1) - /31) ... S(r - 27ri(k + 1) - fln-1) = h( 1,... , fln-1, r - 27rik)S(r - 27ri(k + 1) - N1) 6 x ... S(7 - 27ri(k + 1) - /3n-1) = h(fl, ... „an-1, r) x S(r - /31 - 27i). -S(r - /3n-1 - 27ri)S(r - /31 - 47ri) x ... S(7 - fn-1 - 41ri)S(-r - f1 - 27r(k + 1))

x ... S(7 - fn-1 - 27ri(k + 1)) . Since S(/3) has only one simple pole at the point /3 = N , the singularities of h(,81,... , /3n) in the strip are caused by the singularities of


198

h(#,,... , Nn_1, r), i.e., they are simple poles with residues expressed through h01, ... , Nn-2, T)•

Suppose we know the function h2(/31i ... , fn-2, r). Then if we construct two functions h(/31i ... , /.3n) which satisfy all the above requirements, the difference between them Oh(/31i ... , /3n) is an analytical function of Nn which has no singularities in the finite part of the complex plane and decreases exponentially for /3n --+ ±o0, Qn E R. Such a function is, evidently, equal to zero.

Similarly, it can be shown that h2(/31i ... , Nn-2, r) can be reconstructed g 0 V(3))*: ... , Nn-3 , r) E (01) ®... ®Vnl

uniquely through h3 (/31,

h3(h^l , ... , Fin-3 , r) - f (F^1, ...

C

fli

i<j n-3 n-3 X II H (1, 3 Pi - T) ,

where f (/31i ... , f3n_3, r) is form factor corresponding to n - 3 kinks and one rank-3 particle. Further application of this procedure shows that 1(E1) ... 1 (e ,) = 1, is defined uniquely through f (el. ... , fan-N+1)

1 , l(En_N+2) = N - 1. The latter form the vectors f (f31 i ... , Nn-N +2) belonging to (V(1) ®.. . Nn-N+2) C1....

,Cn -N+1,Cn-N+Z,

l(Ej) = 1 ,

j N ( 3 1, ,--and, hence , is defined uniquely through h(/3l, ... , /3n_N). Thus the recurrent procedure is obtained for proving the uniqueness of h(/31,... ,hen),1(El) _ ... = 1(e,) = 1.

Q.E.D. Theorem 1 . The form factors fµ,,, fa satisfy the requirements of Cinvariance:

fµv(^1,... ,Nn) Ei.... ,E,,

= (-

1)'^îlE')f,Y(131) ...

fµ(31,... Nn)Ei, ...,en = -(-1)NEl (Ej)fµ(/31,... ,h'n) Ei,...,En

(A15)

Proof. Functions (- 1)*ErÊ;) fµv(al, • • • ,/3n)Ei,...,E• and -(- 1)lv£t(Es) f,° (Ql1 • • • , Nn ) Ei,... ,En possess properties similar to those of functions fµv(/^1, • • • , Nn)e1,... ,En and fµ (01, • • • , /3 )e1..•• ,En i consequently they can be reconstructed uniquely through fµv(Nl „Qz)E1 E ; and fµ(N1,/32) El,EZ in the spirit of Lemma 1. Thus the proof of the theorems reduces to the checking of (A15) for n = 2, l(el) = 1,1(e2) = N - 1, which is trivial. Q.E.D. C-invariance leads to interesting identities. Consider, for example, form factors f++ = foo + fl, + 2fio which form the vector f++(/31, ... , /3n). Construct the operator C1 = C1,1 0 ... 0 Cn,1, where C1,1 is an operator acting in V^1) via e^,kC^,1 = ej,N_k. It can be shown that

f++(01,••• ,fn)C 1 = f++(a1,••• ,an) •

(A16)

We construct also another operator C2 = C1,2 0 ... 0 C,,2, where C1,2 acts from the space of the first fundamental representation 1, 1) to the space of the (N - 1)-th fundamental representation V(N-1) : ei,kC,,,z = ej,{1,... ,k,...N} • C-invariance together with (A16) imply that f++(/31, • • •, /3n) C1C2 coincides with the form factor corresponding to rank-(N-1) particles with rapidities 61 ,... , /3n. Using the identities


200

'wkl,... ,k„ (#17 • • • Nn)ClC2 = w{kl}',... ,{k„}• (Nl^ • • • , Nn)

N

tai

(

k,-k,-1

X

i^n /3i-#j -j?N f^ kikj

(N_ 1N-1 //^^ (1,1 (/3)F N 2ai r N N-3

N-2

1 I (N-2j+1));_1 (P - L'-'-j =i shz(/3+ rr

H

the definition of form factors (A8) and the linear independence of wkl ()31i ... , /3„), one obtains a nice identity

rj

f dA

,kn

(p ( N ) (a - 3)A(AIB(1)I ... IB(N))w+(A)

aEA,13EB

rir(N-1r(N-1+ l N tai J N 2ai N-3

1 N (N - 2p - 1) 2 (/3i ,Qj + pf=1 sh IT

x

J

dA'

II V^N)1(a - /3)0(A'IB(1)I ... IB(N)) aEA',(3EB

w+(A' U U 2 U (a - iv (2j - 1)) X

7=UPEB

N-2

'

II II II shz(a - /3+ N(2p- 1))

p=OaEA/EB

(A17)

where B = UB(j), n(BCj)) = m for di , n(A) = (N - 1)m - 1, n(A') = m - 1 and ,& is the determinant of an (m - 1) x (in - 1) matrix with the following matrix elements:

201

Form Factors in SU(N)-Invariant ...

...IB(N)) , AiA = Ai( a9IB(1) I N

Ai(aIB(1)I

... I B(N)) = I

(a -,a - N(.7 - s))

II I

s=1.7 #s PEB(i)

X

2s)

Even in the simplest case N = 3, n = 3, the identity (A17) leads to nontrivial result: 3 1 a- #i

l_ da j=1 II r (3

1

2î ) r (3 +

a-,Qi

= TJ r 2 _ fl, - fli r (2 + fi - ' 11 i

3

a

2^ri ) (3a -,Ql -,Q2 -,Q3)e (e#- + e02 + e,63)

27ri 3 27ri

We believe in the convergence of the series 0r0 r q II9- 031 ... Nn)li,... 77= L^ L^ I n >...>P1 n=211,...

Il2dI, ... dan

where a = {i}, i = 1, ... , N - 1, the sum being taken over all possible values of the ranks of particles with rapidities an. Assuming the convergence one can calculate the singularities of commutators of currents at the origin of coordinates: fU (x),IO(O)]7 = ifabcjc(0)b(x) L/1 (x),I1(0)J = 2 fabcj0(0)b(x)

[jo (x),7i(o)] = i fabcjc(0)8(x) + 41rir)babb'(X) At this point we finish the consideration of the SU(N)-invariant Thirring model. The sketchiness of the exposition is caused by the fact that full proofs, while not very instructive, would occupy to much place.

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APPENDIX B PHENOMENOLOGICAL REASONINGS

This Appendix contains some heuristic speculations on the derivation of Axioms 2, 3 in the framework of the axiomatic field theory. Consider for simplicity the form factor of a scalar operator calculated between the vacuum and a state which contains n one-component particles with rapidities ,81, ... Denote this form factor by

f(C31,...

Nn) =

(phl0(0)Z*(1n)

...Z'(N1)Iph)

It is natural to suppose that f (,31,... „ Qn ) allows analytical continuation with respect to all its arguments . Let us concentrate on analytical properties of f (,131, ... , ,3n) considered as a function of Nn. Suppose 131 < ... < an_1 0 is to be understood as meaning that Re/3n is greater than all the remaining rapidities (evidently the initial definition of f (/31, ... , /3n) allow us to shift simultaneously all the rapidities ). More refined consideration needs all the particles to be convoluted . When this is done one makes sure that the "true" value of Re/3n is determined by its value taken relative to /31i ... So we have presented some speculations in favour of the possibility to continue analytically the integral ( B1) into the range 0 < Im/3n < ir. Similarly for f(h'n, ^n-1, • • . , Ql ) =

( Ph IO(O) IR1, ... ) Nn)out

,

one gets f (Nn, Nn-1, • • • , N1) =

J

dxe-' P '('6')xvO(xo )(Phl

[O(0), tl(x)]I /jl,

... ,

,6n -1)out • (B2)

It seems likely that the integral (B2) allows analytical continuation into the range -1r < Im/3n < 0. Consider also two matrix elements f (Nn 1,81, • • • Nn -1) = (fl. l° (0)) l#l, ... , fln-1)in f(Nn 1,8n -1,••• ,,31) _ ((ten

IO(0)I/31, •••

Nn -l).ut

205

Phenomenological Reasonings

One can obtain for them presentations similar to (B1), (B2). There is, however, one difference. When convoluting the particle ,an in the form factor f (Nn I i3 , • • - , Nn-1), one obtains the expression

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