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)
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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
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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
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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^ E^I(^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
4 Form Factors in Completely Integrable Models of Quantum Field Theory
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 ^ayµ^))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
8 Form Factors in Completely Integrable Models of Quantum Field Theory
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 , /^il) 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
Form Factors in Completely Integrable Models of Quantum Field Theory
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 re- constructed 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 <, 27r. The SG model possesses a complicated hierarchy of bound states for ^ < 7r : breather is the bound state of two solitons, soliton is the bound, state of soliton and breather , higher breather is the bound state of lower ones . These possibilities correspond to the following singularities of f((31, • • • , 0n_1,fin) E1,...,En _ 1,En (for the sake of simplicity we consider only the poles at the points fin = ,0n _ 1 + ia; the poles at the points Nn = /3j + ia,j < n - 1 can be considered via the symmetry):
1. If En = ±i,En_1 = ±z, there are simple poles at the points fin = Qn-1 + 7ri - ism with residues res f(,61,.. .
s. 1 m
, Qn- l, ^n) E 1 ,••• ,En-1,En q R
=
am (-1)
'
CEn -1,En
7ri ism
X f ( iii... ,Qn-2) fn-1+ 2 , E1,... ,En_g,m
/ 2 am = 4 27rSo(7ri - itm) m sin
(17)
2. If En = m ,En_1 = ±2, there are simple poles at the points fin =
12 Form Factors in Completely Integrable Models of Quantum Field Theory
Nn-1 +
Zi + a (m - 2p), p = 0, ... , m - 1, with the residues res
f (Nl, ... , Nn- 2, Nn-1 , Nn)£l...en-s,f 2
= amf (31 ... , fn-2, fin-1
7ri
+ i^(m - P), fan-1 +
i^p
-
2 E1,•••
2,1 z,P
a,Pn = 21ram-p am-P,P, p > 0 , z
and
(21rires
(18)
, l3= (7r+gym)i
Sm(fl))
where a,,,,,,,,2 will be defined later. We set
-
f(Ni, . ..
f(,'31,
. . . A-
0 ,.,... . -1 .
3. If EK = E, En-1 = m, there are simple poles at the points (m - 2p), p = 0, ... , m - 1, with the residues /ten-1 + +2
On =
res f (Nl, ... , 13n-2,22Nn-1, Nn )E1,... ,En-2,m,} 2 = amf , /in-2, Nn-1
i. +
(m
/^
7ri - iem l
- P), 13n -
2
1
+
2
,
J
(19)
E1 ,..., En-z,P,} a
4. If En ml, En-1 = m2i there are simple poles at the points ,13, = ,fan-1+ 2 (ml+'m2-2p),p= 0 , . . . ,min(m,,m2)-1 and f n = /n-1+7ri2 (ml + m2 - 2p), p = 1,... , min(mi, m2), (for ml = m2 the value p = ml is omitted) with residues P res f (//^^ h^l, ... , n-2, #n-1, ,8n) E1,... ,En - zml,mz = a ml,mz /^ qq x f #I, • • • , Nn-2, fln-1 +
i^ qq 2 (m2 - P), Nn-1
+-i (ml - P) E1,... ,£n-2 ,m 1'fm2-P,P
The Space of Physical States. The Necessary Properties of Form Factors.
13
//^^ q R q reS f Nl ... I'tt-2^ h'n-1^ F'n) el,... ,E„_2mlm2 - a P m1,m2 X f 01, ... , Qn-2, an-1 + Z2p, Qn-l + 7ri
-t( m1+m2-2p) ) 11 .... ,en-2 ,mtt1-P,m2-P P p a m,m2 = 27fdml ,n+2-Pam2 -P,P> > 0
am,m2 P a ml,m2
=
(2^
i^m1+m2 2
i res Sm l, m2(13)
= 27fan+
(20)
1-P,Pa+n2-P,P
This system of relations is axiomized . However one should not be afraid of the complexity of the system: form factors corresponding to states which contains breathers are secondary objects. As it follows from (17) they can be obtained as residue of soliton form factors. So the properties (18), (19), (20) should be indirectly contained in the soliton form factors. Once more about the analytical properties. It follows from (15) that f (Ql • • • ^n ) f 1 i... ,E„ can be continued analytically to all the complex planes of the variable ,Qtt although its structure of singularities in the plane is rather complicated. From the symmetry property (13) it is clear that f()31, • • • ,..) fl,,,.,fn is an analytical function of all its arguments. In what follows we shall be interested in the case where the points are moved from the real axis, for example,
(21)
f(/31,...,/m, /3m+1+o,...,Nn+o) fl,...,e„
What are the singularities of (21) in the strip 0 < Im o < a? It is clear that there are only simple poles on the segments Re o = fli -,3j, the position of which can be found from (17)-(20). In the absence of bound states the function (21) is regular up to the line Im o = a. Let us turn to form factors of the general kind / 3 )f1,...,E„ . Suppose the set fl is separated from the set 3, which means that there exists such a constant b that I,Oi
pj' I > 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)
14 Form Factors in Completely Integrable Models of Quantum Field Theory
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
Form Factors in Completely Integrable Models of Quantum Field Theory
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
Form Factors in Completely Integrable Models of Quantum Field Theory
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)
The Local Commutativity Theorem 21
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)
22 Form Factors in Completely Integrable Models of Quantum Field Theory
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
The Local Commutativity Theorem 25
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
26 Form Factors in Completely Integrable Models of Quantum Field Theory
term one has
J0
(A
I [01(x), 02(0)11 B ) e iw(A)x^O - (x)dx
_ S(AIA1)S(A2UA3^A3) 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).
28 Form Factors in Completely Integrable Models of Quantum Field Theory
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
30 Form Factors in Completely Integrable Models of Quantum Field Theory
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,•••
,F^i +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, ^i, 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
Form Factors in Completely Integrable Models of Quantum Field Theory
Using this relation and the identity 2n sh F(o-Ij) (^I`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
x
(/3i
sh {(Qi-^j+7ri)
Soliton Form Factors in SG Model
33
ose all e j = -12 e1 ...£2n (q1) • • • , ,l32n ) and suppose for j > 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)
34 Form Factors in Completely Integrable Models of Quantum Field Theory
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
Soliton Form Factors in SG Model
(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) (
^I(^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
36 Form Factors in Completely Integrable Models of Quantum Field Theory
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 (^AP
- - 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
Form Factors in Completely Integrable Models of Quantum Field Theory
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
Soliton Form Factors in SG Model
3.
q ";^` pp
^p((^ - `^) _
co(N
(i3 +
ch
)
1
2)
xi
ch 2 (Q - z -k)
()0
sh
{ (,Q --sh 3x2 i) 'p(Q - 7ri + i^k)'o(Q) =
k-1
^^_O sh 2 (i3 + iEj - 2 )
2sh f (Q - a) sh 2 ( 3 - 'K') ^j.o ch 2 (18 + ie j
k>0. Take some polynomial P(x) and consider the integral
J
00 2n
00
fl ^p(a - )P (exp a)) exp (air j =1
-
(n - 2)a) da ,
(50)
where Ik! < n - 1. From the asymptotics (49) it follows that the integral (50) is defined correctly when deg (P) < n - 1. We shall define a certain regularization which will provide us with the possibility of dealing with the integral for deg (P) < 2n - 2. The regularized integral will be denoted by
Definition 2. Let P be a polynomial of the degree < 2n - 2, then , oo 2n
11 cp(a - ,^3j)P
exp
oo j_1
(2 a))
exp
ak -
(n - 2)ada
-
2n
1 3xi exp E Qj ^(a - F'j) x m+i`-io j-1 sh (a - Qj - z)
x Pi
(
exp 2 a exp (ak -
(3n - 2))) da
2n +
11
p(a-(31)P2 Ir4exp(2aa Ilexp(air -T(n-2)a)da
_1
\
\\
IJ
(51)
40
Form Factors in Completely Integrable Models of Quantum Field Theory
where t envelops the points a = ,Qj - a - ilk, k = 0, ... , [7r/6 , T exp ("{ `) , and the polynomials Pi and P2 are defined by the relation
II (xr_3 - exp \ S ti)) P(x) = Pi(x)
+1 (zr2 -7 -ieXp \ S Nj)/ P2(x)
-7-
2(n-2)
JT (xr_3 - exp
(
i3
)
) P2(xr4 )
(52)
S
1
and the requirement deg Pi < 3n-1 (evidently in this case RHS of Eq. (51) is defined correctly).
Let us prove two statements which established the selfconsistency of Definition 1. Lemma 6 . It is always possible to find polymials Pi and P2 satisfying (52) and the requirement deg Pi < 3n - 1. Proof. If one takes in the capacity of P2(x) the monomial xk, the maximal degree of the two latter terms in the RHS of (52) is equal to 2n + k - T4- 8n }4k ) This coand the coefficient before x2n+k is equal to (T-4 efficient does not vanish for k > 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
Soliton Form Factors in SG Model
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 ^o(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
+ ^io
- )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) =
(56) sh (a - 3z ff )
42 42
Form Factors in Completely Integrable Models of Quantum Field Theory
we transform the second integral in the RHS of (55) to 2n
sh { (a - X31 - 2 )
3Ai exp r `P(a - Qi) J^+*^, -io j_i sh E (a - Qj - a
x
ak - (n - 2)a
x 72(n-2) Q2 (exp (2a/ /
Now Eq. (54) obviously follows from Eq. (53). Q.E.D. Lemma 7 clarifies the meaning of the regularization (51). Informally, it can be explained as follows : we move the contour of integration in RHS of (51) to the line Im a = E. - 0 (the singularities of cp(a - ,8j) do not hinder making it). Then using the property (56) of
(P).
Definition 3. Consider an antisymmetric polynomial P (xl,... ,xn_1) whose degree with respect to every xi does not exceed 2n - 2. The transformation -tn,o (v = 0, ±1) associates with this polynomial a function of #I, • • • , /32n : 1 () fl_1 1
^n,,(P) =
(n - 1)!i
n-1 2n 00 dal ... 00 # dan_1
1 H p(ai -)3j) 0o J-00 i=1 j=1
x 11 sh ( ai - aj) exp (- 7 (n - 2)Eai + oEai I i<j
///
X P (exp (w a1) ,...exp (2,r an_1) III. Polynomial An. The third necessary object is the polynomial An (XI , ... , xn_1 jai , ... ) an lbl , ... , bn) which we are going to describe.
Soliton Form Factors in SG Model
43
This polynomial is antisymmetric with respect to 01) ... , xn_1 and symmetric with respect to a1, ... , an and bl, ... , bn separately. Definition 4. Consider an (n - 1) x (n - 1) matrix with the following matrix elements:
A;^) =A1n^( xjlal ... anlb1...bn) where
l
A in) ( x al, ... , an Ibl, ... , bn) = 2-n-i i -i ( X IJ(x - air) (E(1 - T2(i_k))(-1)kx'-k-17kQk(bl, ... , bn) n
1=1 k=0 n
i-1
+,r 2i [J(x -
b 1T-1)
1=1
E(1
- T2(i-k))(- 1)kxi- k-lTkok( al, ... , an)
k=0
(57) Qk is the elementary symmetric polynomial of degree k (Qk(a1i ... , an) = 0 for k > n or k < 0). Polynomial An (x1, ... )xn-1l ai, ... , an lbl, ... , bn) is the determinant of this matrix. Evidently An is symmetric with respect to al, ... , an and bl,... , bn separately and antisymmetric with respect to x1i ... , xn-1 polynomial. Its degree with respect to xi is 2n - 2, its degree with respect to ai(bi) is n - 1. Polynomial An is a homogeneous polynomial of the total degree n(n - 1) + 2(n - 1)(n - 2). The main property of An is described by the following statement. Lemma 8. An(xl, ... , xn -1
lal , ... ) an -1, albl, ... , bn_1 i aT z )
n-1 = 2-3(n-1) ll (xj - ar) j=1
n-1 E(-1)kAn_ 1(x1, ... k=1
-1 xn-l lal , ... ) an-1lbi , ... , bn-1)xk
n X (xk bj r)(xk - a j T) - 7- 4(n-1) [ i'1=1
n-1 fJ(x j=1
k -
bj T -1)( x k
- aj T-1)
(58)
Form Factors in Completely Integrable Models of Quantum Field Theory
44
Proof. Using the identity ok(al,... , an) = ok ( al,... , an-1 ) + anok-1 (al, ... , an_ 1 ) one can show that
A;n)(xl a, ... , an-1, a Ibl,
... , bn-1 , aT2)
/x - aT = I 2 ) { Ajn-1)( xlal, ... , an-11b1, ... , bn-1) 1 2 (n-') - tar Ai_1 ( x lal,
... , a n-l l bl , ...
, bn-1) IT
while (n-1)
2(n -1)
A n -1 (xlal , • • • , a n-1 lbl, • • • , I n-1 X
ii(x -
ajT)(x
j=1
-
bn_1)
=
2-
x-1
n-1 -,r4(n -1) (x - a jT -1 )( x - bjT-1) fJ
bj T )
j=1
Now the proof reduces to the successive adding of the first row with the coefficient ar 2/2 to the second one , of the second row with the coefficient aT2/2 to the third one , etc., and following expansion of determinant along the last row.
Q.E.D. IV. The formulas for the soliton form factors . Consider the functions n (7 ... , An Fn") (xi, ... , xn-1
lal ,
l µl ) ...
µn)
=
exp
E µi 2` ( S i=1
Ai
- 7ft)
21r 21r 27r
X An (Xj ) ... , xn-1l eXp ( S A,
C2^ ))
x exp ^ µn
)
, ... , exp \ S An / I exp \ 'U 1
2n
exp
- -(n - 1) 1: #j j=1
7=f1
(59)
which are antisymmetric polynomials of xl,... , xn_1 and entire symmetric functions of al, ... , An and µl , ... , µn. The form factors of topological current jµ = e,,V -L8„u (f, ), of energy-momentum tensor Tµ„ Up.), of
Soliton Form Factors in SG Model
45
exp(±if p/2) (f') are given by the formulas
Mdn fJC(i3i _
fM(/1r.../3 n) _
,3j) {^n,l((F(1)
i<j + (-
1)µl^n,-1((
F(1)
-
F(
- F(-1)) n)e -in pi
l
1))n)e'DP' Y , 2n
fµv(13 1'... , /12n) = M2dn 1I C(,8i - /3j) r(eA1 + (- 1)µ-1e-Qij) i<j
j=1
.)e2EAI + (-1)°-14Pn -1 X {'tn,1((F, 1) + Fn-1))
Xq((F(1) + F(-1))n)e2Ep'} , q f
f(fi ,...
, F'2n) =
do
[((ai
- flj)'tn,0((F(t 1))n)
(60)
i<j
where
(0)
0o sin2 (z K( /3+ wi)) sh ( z€ic) 0 = csh -exp d^c (1 2 tcsh (Z) sh(aic)ch (z )
0
c - 4
sh (z) sh ("
tc) dtc
1 ex
p 4
S
✓ U tcsh
ch2 (2 ) (2
d=c-1ir 1 (2ir • The components of fµ(01, • • • , /32 n), fµv()31, • • • , 132n), f' ()31, ... ,8 2,) the basis e,,. ... .E2, are denoted by fµ(01, • • • A n ) .,1.... ,E2n , fµv(N1) N2n) E1,...
.c2, ' f: (Pl, ... ,
N 2n ) e,i._C
in
2n •
The function C is regular in the strip 0 > Imf > -2a and satisfies the identities
((#)So(#) = C(/3 -
21ri)
= C(-l3) ,
((/3)((/3 - 7ri) = ^0-1 (/3 + 2
(61)
It behaves as f exp (±4 ({ + 1) 0) for 0 -> foo. It is useful to mention that the operation ( ) n and the integral transformation ^n,a can be rearranged in Eq. (60).
Form Factors in Completely Integrable Models of Quantum Field Theory
46
In the following sections it will be shown that the form factors satisfy all the requirements formulated in Sec . 1. It will be also explained why we associate them with the operators jµ, T,,,,, exp (±i P) . To finish this section let us write down explicitly the formulas for twosoliton form factors (n = 1) because they will be important later and the formulas (60) cannot be applied to (n = 1) directly:
fµv(PI, #z) =
2 (e 0 11 2 ) + (_ 1)µe-;(RI +#2))(e1(01+Ps) 4
+ (-1)°e ('1 +^'))dC(Q1 - /32 qq
fo(Q1,i 2) =
M(e1(#. + Y9)
) ch
1 2 (01 - Q2)
sh 1 (#, -,8z + 7ri
+ (- 1)µe-"(a1+Q2 ))
)(ej,-3 - e_^ 1) x 1 ch 2f (f1-N2+7fi
f }(I31 i32)_ x
(
-d((al -Q2) sh f(P1
Q2 +7ri)
exP(±(13i_/32+ini))e,_ a
+exPe-„,
) (e 1 -
d(((l1
e_ ) ; +
- /32)
4 THE MAIN PROPERTIES OF THE SOLITON FORM FACTORS
In the present section it is shown that the form factor fµ, fh,,, f f do satisfy Axioms 1,2,3. Corresponding statements are formulated as Theorems. Theorem 1 . Form factors fµ, fµ,,, fl satisfy Axiom 1 which means that fµ()lr ... .2n)t,,... , tanq^t^ t^ +l (Ni - Pi+1) =
//^^ , /32n) t,,... ,t: +,,t^.... tsn
fµ(/31, ...
(62)
the same equations are valid for Proof. Form factors are constructed through the functions Fn(7) via the operation ( )n and the integral transformation 4tn. The property of ( )n presented by Eq. (47) implies that q (Fury )n()31i... 032n)t,,..., E]nRst^ ,t^Wi -Ni+1) =
(F(ry))n(/31, ...
Ni+1 q , fli ... 82n) t,,... ,e.+l,c' .... tsn
The application of the transformation 'tn does not disturb this relation. The consequent ulti lication by H(((#3 -,33) changes (61) S(,ai - ,6i +1) to S(Qi -, i+ 1) So(3im - Qpi+1) = S(,81 -, i+1). Q.E.D. 47
48
Form Factors in Completely Integrable Models of Quantum Field Theory
Now we pass to the proof of a considerably more difficult statement that the form factors satisfy Axiom 2 . To this end we establish first some properties of function (Fnry))n. Combining Eqs. (48 ), ( 58) one comes to the conclusion that the only singularities of (Fnry)) n are the simple poles at the points /3 + ai + ilk (j > 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
]Fj
sh(ae /^
p
x.'j-l,i()j-1 - 1i) ... Si+l, i(Qi+1 -
sh - 1
(3y -
Qi - ai)
g>j
,Qi)
(63)
,aj =/3i+ai+ i^k .
Lemma 1 . The equation (Fnry))n(I31, ... ,/32n)€ 1 i... ,£sn !; £ !,£!+l (/i - Nti+1) i ,+1 ('Y)
/^
_ (Fn )n(^i, ... #i+1 Ni ...
(64)
allows one to express all the components of (Fnry))n through one of them. The components of (Fnry))n can be presented as follows:
The Main Properties of the Soliton Form Factors
2n (F(-") n(/31.... „2n)el,...,e2„ =exp y` 9
49
n
E(ei/3i+iei)) j=1 j=1
T X exp
2n
1: Q
j
RCj,... ( Xj) ... ,
xn -1
j =1
xexpi1),...,expa2n
(65)
where REi)••., e2,.(xl , • • • , xn-l 1b1, • • • , b2n) is a rational function of all its arguments . The only singularities of this function are the simple poles i1 + at the points bj = 6,,r2, j > i, 21 E . ell < j - i. Besides is an 1=i -
antisymmetrical function with respect to x1, ... , xn_1- It is homogeneous: R(i)... ,e2n (axl, ... , aXn _i jab, i ... , ab2n) = a ' (n-1)(n-2
(66)
)-nR(Ci)... e2„ (xl.... )xn-1'bl, ... , b2n) .
Proof. The proof of the Lemma is very straightforward; that is why we present only the main formula it needs:
Qi
(F(7))n(1, • • • , ,Ni+l, ... , )32n)c,,...
j, 1 2,ei+2, ... ,e2,.
(Ai - ai +1) =s h-1 X I(F'("))n(O1,••• i+l,/' ,. 3A , xsh`(Ii-ai+l-ai)
s
h
9f 2i
L (F("))n(fl1,...
Q q Ni+1,... ,N2n) e1,...,Ei2,2,...,£2,.
1 (67)
Equation (67) being a consequence of Eq. (64) shows that one can express 11" ei +2,--- n (F(7)) 1 1 a 2 through 1,..., 1r 2 ,- 2, 2, 2,... ,e2n• This fact is sufficient to prove the first statement of the Lemma. The possibility to present (F(ry))n(,a1, ... , /3 n)el,... ,e2„ in the form (65) can be also established using Eq. (67). Singularities of (F (ry))n(,81, • • • ,/32n)el,...,e2„ can be easily found using (63); the homogeneity is quite obvious.
Q.E.D.
50
Form Factors in Completely Integrable Models of Quantum Field Theory
The function R(i)... ,c,n (2:11 ... , xn-ilbl) ... , b2n) is antisymmetrical with respect to x1i... , xn_1 and, consequently, contains as a multiplier the polynomial II (xi - xj) whose degree is equal to z (n - 1)(n - 2). This i<j
fact and Eq. (66) allow to reconstruct this function through its residues which can be taken in number less than the total number of poles by n - 1. The most convenient form can be obtained for R(ry) -1 1 1,
1} 1 , R(ryi R1Y) ^ and R(ti) 1 ;,...,;,;,...,; ;,j,...,;,;,...,z 5,-;,...,-;,, ,...2 Functions R(') ;
i (x1, • • • , xn-l lbl, • . • , b2n) are symmetri-
cal with respect to b1 ... bn and bn+1 ... b2n separately. They have n2 simple poles at the points bj = birz, j > 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<j
1( xl) ... ,xn -1Ibl,... ,b2n) 1 2 1--- IT 12 n-1
2n
2n
= 2 fJ(bl -bir2)-l
n-1
II (bl - bl)-1 IJ (x1 - bir)
1=2 1=n+2 1#j
1=1
n-1 X E(-1)kxk 1 (xk - b1r) - r4(n-1) II (xk - blr-1) k=1
X
lj41,j
(7)
R^^^^( xl,
... ,
1#1,j
xk,
... , zn -1
i
n-I n-1
b2,... ,bj ,... , b2n) , R(ry)-,:F a,...IT
res n+
b i =b:r2 j < n , 2
1-
n+ 1
n
n+1
2n
1J( bi - b,)-1
1_1 1¢i
b , 2n
n-1
n-1
= 8 JJ(b, - bjr2)-1 1=2
(x1 ... ,xn- 1lb 1 ,...
,± . ...,fi , -
n-1
11 (b1 - bir2)-1 fJ(x1 _ bar)
1=n+2
1=1
52
Form Factors in Completely Integrable Models of Quantum Field Theory n-1 X ^( -1)kxk l
II (xk -
k=1
bi r)
l^i,j
- r4(n-1) II (xk -
bi r -1)
Iti,j ... ,xn_11
X R('Y) n-1
n
-2
b1,... ,bi,... ,bj,... ,b2n) . (70) Equations (70) indicate the residues n2 poles and so are sufficient for the recurrent procedure. The initial conditions are the same:
R") 1 (bl, b2) = R(1) 21
2
I(
bl, b2) = R^i
212
21
1)
1 2
(bl, b2)
= R(-i)1(b1, b2) = 2(b2rr-1 - b1T)-1 consequently
R(1) I
1(x1i... ,xn_llbl,... b2n)
1
1 2,...12 2, 2,...,-2, n
=
R( 1 )1
n-1
i -3L _ , (xl, ... xn-llbl, ... , b2n)
1
n-1
1 = R1 1 121 1 ( x1, ... , xn-l I bl) ... , b2n) 21 21...1 2,...,a n
R
n-1
(-1)
_,(Xl,... , Xn_1Ib1 ,... , b2n)
1 1 1 _1 n
n-1
- 2n2+2n _2 0n (x1 i ... , xn_1 Ib1 Ib2 i ... , bn+l lbn+2 , ... , b2n) 2n 2n n+1
II (bjr 1 - blr) ll ll (bjr-1 - bir) j=2 j=n+2 i=1
The last equality defines A. The following Lemma provides us with an explicit expression for On. Lemma 2 . The polynomial &n(xl,... , xn_1lbnl n 1
bn_1) is equal to 2-n+1 ll (bn - bir2) det 1=1
IIB,; )II,
al ,
;n)(xjibni al,...
I bl, ...
,
where B( n) is a (n -
1) x (n - 1) matrix with the following matrix elements: B;j) = B
... ,a n
,dnlbl ,... , bn_1) ,
The Main Properties of the Soliton Form Factors
53
where Bin)(xlbnlal ..... an Ibl , ... )bn_1) = 1 x-bnT3 X
{A ;n)(xla l,
...
, an lbl , ... , bn _l, bn T4 )( x
n-1 - bnx-lr6
(n -1)
JJ(bn 1-1
X
[
fI ( x
- bnr)
n
b IT-2)-1
JJ( b
n - a IT-2) - 1
1=1
- a1T)(x - bIT) - T4( n-1)(x - bnr3)
1=1 n
n-1
11(x - aI7-1) II (x - b1T-1) 1=1 11=1
X
Ain)(bnT3lal, ... an
Ibl , ...
, bn_1i bnr4)} . (71)
Proof. It is evident that det IIB"')II is a function antisymmetric with respect to xi,... , xn_l and symmetric with respect to al, ... , an and b1, ... , bn_1. This function is a homogeneous function of the proper degree (66). It is also evident that the poles at the points bn = aIT-2 which occur in (71) are fictitious because A(n)(bnr3ja1i ... , aklbl, ... ) bn-1i bn r4) contains the multiplier lI(bn - aIT-2). The pole at the point x = bnT3 is also fictitious. At first sight it is not quite clear whether the poles of det IIBn)II at the points bn = b1-T-2(1 < n - 1) which appear in all the matrix elements are actually simple ones. Let us clarify the fact. Let b, for some 1 < 1 < n - l be equal to b1 = bnr2 + A, A « 1, then B^^) can be presented as follows: B Vin) = A-lfigi + 0(1) • (72) The explicit expressions for f; and gj can be obtained from Eq. (71). It is easy to show that the determinant of a matrix with matrix elements of this kind is O(0-1), which means that the pole at the points b, = bnT2 is simple. Consequently the function 2-n+'II(bn - bIT-2) det IIBtt)II is a 13 polynomial. Thus 2-n+1ll(bn - bIT-2) det IIBn)II satisfies all the structural requirements on On . We have now to show that it satisfies the recurrent relations following Eq. (70).
Form Factors in Completely Integrable Models of Quantum Field Theory
54
The calculation of det llBs7)II at the point an = bnT2 (ai = bnr2 (i < n - 1) can be treated due to the symmetry) is based on the use of the identities B ,n)(xlbn +
lal, ... , a n-1, bnr2l bl, ... , bn-1)
1bnT4B(n) 2 2-1 (xlbnlal
,...
,an_1i bnr2lb1,... ,bn -1)
1 = 2(x - bnT){Ain-1)(xlal, ... , an-l lbl, ... , bn-1) n-1 I n-1 bir) - T4(n-1) IJ(x - al7-1)(x - biT-1) - x-1 IJ(x - alr)(x 1-1
1-1
a
X -')Ci ( bn a
Ib
b n-1 )) i < n - 2
Bn-1(xIbnlal,.... ,an-l,bnr2lbli... ,bn-1)
+ 2br4Bnn)2(xlbn lal , ...
, an_l,
bnr2lbi, ... b n -1)
n-1
= 2(x - bnT) fJ (bn - b1T-4)(bn - b1T-2)- 1 2- 2(n-1)x-1 l-1 rn - i
n-1
X IJ (x - a,T)(x - bir) - 4(n-1) H (x - al r-1)(x - bl r-1)
1-1
1_1
where Ci is some function. Adding successively the i-th row of the determinant with coefficient z bnT4 to the (i + 1)-th row, and then subtracting from the 1, 2, ... , n - 2 rows the (n - 1)-th one with the coefficients Cill(bn - b1T-2)(bn - bl r-4)-1 and expanding the determinant along the last row, one obtains the required relation. The calculation of det IIB^1)II at the point bl = anr2 (at b3 = air 2, j < n-1, i < n-1, due to the symmetry) can be performed in a similar manner, being based on use of identities
B(n)(xlbnlal,... ,an
l a nr2,b2) ...
,bn -1)
1 + 1an74B(n ) (,IbnJaI... ,anlanr2 ,b2) ... ,bn-1)
1
2 (x - anT ){B(n-1) ( x
l bn j ai,... , an-1Ib1, ...
, bn-1 )
- 2-
2(n -1)x-1
The Main Properties of the Soliton Form Factors n
J(x
X
1
1 =1 X lii( b njai ,...
n Bn- 1 (xlbn lal ,
+
n-1 - r4(n-1)
, a n _1jb 2i... , bn-1 )} ,
... , a n l a nr 2 )
- anr
)(bn
...
- ar-2
n-1
X
[
- alr-1)(x -
b l+lr -1)
1=1
a r42Bnn>2(xl bn l al ,
= 1( x
fJ( x
- atr)(x - bl +lr)
55
ll(x - alr)( x -
b2,... ,
i
bn-1 )
l
an anr2 ,
b2) ...
,
)(b n - an )-12- 2(n
bn-1 ) _ l) X
-1
n-1
bl +lr )
1-1
- 74(n
-1) [J (x - air -')( x - bl+lr-1)
l-1
Q.E.D. Now we are in a position to show that the form factors satisfy Axiom 2. Theorem 2. Form factors f,, f,,l,, fl satisfy Axiom 2, which means fp(/31, ... , /#2n + 21ri)£1,... 12. = fµ(Q2n, /31, ... , Q2n-1 )£2..,e1,... £2n-1 fµv (01, ... , N2n + 27ri)£1)... ,£2„ = fµv(/32n, /31,, • ,32n + 2irz)£1,... e2,. = -f±(N2n,, 1, .
f±(,31, ...
, #2n- 1)'2. , '1-. ,12.-I , /2n- 1)£2n,E1, ... , E2n-I -
(73) These identities are to be understood in the sense of analytic continuation with respect to N2n . Proof. From the definition of f1„ fµ,,, f t and the identity ((Q - 27ri) = ((-a) , it follows that Eqs. (73) are equivalent to 0n,o,((Fnry))n)(,Q1, • • • , /32n-1, 132n + 21ri) ,,,••• e2n-I,E2n =
fory=±1,a
0n,o,((Fnry))n)(a2n) Nl,
... , #2n- 1) E2n, e 1 .... ,e2,.-I (74)
= 0, ±1.
As was explained in the proof of Lemma 1 using the relation ^++,a((Fnry))n)(^1, ,Ni,Nti+1,. ,N2n )£i....,Ei,ei+l ..... E2n
X s;/'E;+I E i,E
i+I
(Ni
-tn a((FE'rl)n)(Ql) ... ,fli+1, 3i, ... , 92n)£ 1i...,£^.+l , f ^ .,... ,f2n
Form Factors in Completely Integrable Models of Quantum Field Theory
56
one can change the order of El ... e2n-1 in Eq. (74) and thus reduce the proof of Eq. (73) for arbitrary el ... E2n-1 to the proof of the two identities: -tn,o,((F17))n)(131,... ,/2nn+27ri)- a,...,-2i1a1 ...1a 1 _ 1 _1 1 1 ((Fn7))n)(82n) l q'1, ... Q2n-1) 2 2 2 5 2
-tn,a((Fn7))n)(/31, .. • ,,32n + 21ri) Z,... , z,-;,... ,- z 2 2 ,..., 2,...-'
_ ^n,o((Fn7))n)(^2n^^1,...
(75)
Equations (42), (69), (71) and the definition of 'n,,, imply that the proof of Eqs. (75) reduces to the proof of the following statement. Continuing analytically with respect to 32n :,82n -+,32n + 27ri the integral
I1(,81,...
,02n)
00
1 =
( n-11 )
x fl sh(ai - aj) exp
(
n-1 2n r^ da l ...* da n_1 H H c2(ai - I3j )
00 00
ci -
i<j
x An
(
exp
i-1 j-1
- (n - 2)Eai
27r 21r 2v 2v al, ... , exp an_l I exp exp Nn
-
27r 27r
x exp exp /32n
one obtains the integral 1
I2(
Nlr
... ,a2 n)
00
= (n - 1)!
x fl sh (ai - aj ) exp
C
0o
n-1
dal... $ dan_1 f
QEai -
i<j
o°
2n 1111 W( ai -,8j) i=1 j=1
7 (n - 2)Eai)
27r 27r 27r27r 27r x on exp a1) ... , exp an-1 I exp /32n exp ,d1, ... , exp Nn 27r 27r 1 x exp C /3n+1 i ... , exp
2n1
x (exp j-n+ 1
27r
n_1
N2n-1) 2
27r
1
/3. - 7 exp ^32n^
The Main Properties of the Soliton Form Factors 57
Let us prove it. During the proof we shall manipulate f as with the usual f. The necessary rigour can be achieved by using Definition 2 of Sec. 3, but in that case the proof becomes very bulky. Notice that the antisymmetry of An(xl,... , xn_1I ...) and On(xl, ... , xn_1I ...) with respect to x1i ... ) xn_1 allows one to replace in 11, I2 the product n_1 1 ^lxsh (a; - a-^) by 2- (n-1)2 n-2) exp(Eai(n - 2i)). Now noting that every column of matrices det IIA(n)II, det II B(n )II depends only on one ak we can integrate the columns separately and rewrite I7i 12 in the form
Il = 2_ (, - 1)2(n - 2)
72 = 2-
d et II Qtii II
2)
2 det II`Bi.i II
where 2n
2(12
f da H cp(a - ,Qp) exp oo
x AI
p=1
(
CexP aIeXP Q1 ...,exp £ On
2a 21r x exp )3n+,,... , exp hqq'2n oo
=
J
a(n - 2j) + va - (n - 2)a)
J
2n
darj cp(a-,(3p)exp(a(n-2j)-}-oa- ( n-2)a) °O
pct
x B^nl (exp aI exp
)2n
I
exp fl1.... exp
f
h'n
I eXP
a
n +l
...
21r X exp c f32n_1
We shall show that 2t, being continued analytically over 82n : Q2n -' Q2n + tai gives (Bid . Let us first deform the contours of integration for %-j and (Bij to the line Im a - a - 0 (
Form Factors in Completely Integrable Models of Quantum Field Theory
58
im p
im p 3
7tt
52
x x 3TC
TC T
x
0
I
x
0 2,
01 TC 2
x + Re 2n 3TC
a
b
Fig. 2.
from (56) an integral of 2n
l cp(a - #I) exp ((n - 2j)a } tra - (n - 2)a) i-1
x A;n)
(
exp a exp
,a1, ... , exp
)3.
1 exp ^n
+
1
•••,
x r4 exP 21r 7r iri N2n f sh (a - /2n - 2 I sh-1 (a - f32n - 32 z
taken over the joint of contours represented on Fig. 2b. Let us transform the integral over I'1. It can be evaluated through its residues . Notice that poles of the integrand are spaced by it. This fact implies the possibility of replacing of the integrand by
The Main Properties of the Soliton Form Factors
59
2n 2n-1 exp f a-Texp f 3j
rj ^(a-#j) 11 j=1 (exP Lt"-,32n ) T3 - (exp L.-#j T j=1 sh { (a - 82n X
sh { (a - 82n - 3V 2
xexp(a(n-2j)+cra- (n-2)a+ f (02n-a+321))
x A1n) 1 T3 exp fan I exP 31, ... , exp fan I exp ^3n+1 i .. . x exp
T4
2Jr N2n
(76)
It is clear that the function (76) has no other singularities in the strip z - 0 < Im a < 2 - 0 but those enveloped by r1; hence rl can be deformed to (JR + 52 `` - i0) U (- (JR + 2` - iO)). Using the identity sh(a cp(a +21ri) _ sp(a)
+2' )
sh { ( a+
32i
)
we transform the integral over R+ 2- iO to the integral over JR+ a - i0. As a result the integral over r1 transforms to 2n
11
- m+ -;o " j=1 ^(a 3i x
A(n)
C
73
exp
-,(3j)exp (a( n - 2j) + ca - £( n-2)a )
2pi 132n ( exp fl,.... , exp f
Nn
exp
Fln
+
1
.... )T 4
2a /^ X exp T 62n zn=
x
1 (exp
27r
02n -
7
-2
exp
1 (exp 2 a
2a ^3j)
2a a - T 3 exp N/^2n) - 1
j-1
I exp 2c a - T exp (3j) - T4(n-1) (exp 2C a - T3 exp 2C ^32n) S S 1\ S S
2n-1
x 11 (exp
\
a - T-1 exp 2=,Qj I
j_1
S
/^
exP - (N2n - a)T-sln-ll da .
/
(77)
60
Form Factors in Completely Integrable Models of Quantum Field Theory
Adding to (77)
2n
(
m+" -io11cp(a-/3,)exp j=1 F x ASnl
C
exp al eXP
a(n - 2j)+va- ( n - 2)aI
f31, ... exp
Nn
2ir 4 2ir
x exp f fan+1, ... , T exp T f32n exp
a - r exp f #2n
X d exp f a - T-3 exp ',32n a'
one obtains
--io x B^nl
fJ cp(a - f32) exp ( a( n - 2j) + ca -
(
expc exP__ /32n IexP
27r X eXp C
q
f32n-1
(n - 2)a I
T f31, ... , exp -
Nn
I
exp
-
/3n +1,
... ,
da
Q.E.D. Remark . The theorem remains valid if exp(a(n - 2j) + oa) in the integral over aj is replaced by exp(kj a), where k1,... ,kn_1 are arbitrary integers which do not exceed n - 1 for `di. Let us now pass on to the calculation of the residue of the form factors at the point f3, = 3i+ri, j > i. From the symmetry property (62) it follows that we can restrict ourselves to considering the case j = 2n, i = 2n - 1. Theorem 3 . Form factors fµ(,01, ... „Q2n),• • • , f32n), f + (#1, ... , f32n) have no other singularities in the strip 0 < Im ,32n < it but the simple poles at the points 92n = f3, + iri - ilk, j < 2n -1, 0 < k < [ *, The residues at the point )32n = N2n-1 + ri are given by the formulae
The Main Properties of the Soliton Form Factors 27ri res fµ(#I, ... , /^2n)E1i... ,c
X
1
61
= fl1(N1, ... , )32n -2)E'1,... ,E2n_2be2n-ESn-1
o: .. bESn-1 - S ErlZ,nEl-1'E1 ('32n-1 C2-1
- 01) ...
111 E1 '
S.7"22- 13:,62-2 q /^ q fl2n = /92n-1 + 7ri E 2n_1E2n-2 (i32n-1 - N2n-2)
27ri res f+(/3l, ... , 02n)E3,... 12.
x
/
I
I
= f+( F'1, ...
,
g ) 2n-2) e/3^"' ^ E/2n-2 5E2n,2n-1 -E/
I
I611 fe'n- 1 E21>E1 n-
1 2n-1 + Sr1,E 1
R //^^ (N2n -1 - 01) . .
'r2n-3,E2n-2
(78)
SE2n -1,E2n -2(^2ri-1 - /32n-2)} , N2n = N2n-1 + 7fi
The formula for the residues of fµ„ is quite similar to the formula for the residue of fµ. Proof. Function ((/3) is regular in the strip 0 < Im ,Q < 21r. This is why singularities of form factors are caused by singularities of the function 'tn,a((Fn7))n)• Singularities of the latter are of double origin. First, the poles at the points f32n = X33 + 7ri + iek are implied in the definition of the operation ( )n. The residue of (F(ry))n at the point Q2n = N2n-1 + iri + il;k is given by Eq. (63): res(Fn (ti) )n
exP 27r al,
27r a ... exP
n-1 IQl, ... , 132n
l
)
= ik in (e 2n-1,- z ® e 2n, 2 + (_1 )ke2 2n-1, ® e2n,- i ) n-1
® ^(-1) ^ (F(7)1)n -1 11=1
x exp
27r an-1I
8
1, ...
C
exp
2 7r
^
2
al'... , exp
at, ... ,
q N 2n-2
2n2
7r
7ri
2n-2
in
7r21
x sh (ai_/3p_ -r2(-1) sh i-/+ II 11 - (al p=1
) x exp f (2n - 3)ai + in ai - in (n - 1),(32n-1 i#l n-1
x J sh 'r (ai - /32-1 I=1
k
)
I
2n-2 / [I sh + 1 'r (/2-1 p=1
J
132n = 82n-1 + iri + il;k .
62 Form Factors in Completely Integrable Models of Quantum Field Theory
Second, the singularities can appear during the analytic continuation with respect to 132n of the integrals over al, ... , an-1. The fact is that for Re 1 2n --+ /33 and certain imaginary part of ,32n the contour of integration appears to be clutched between two singularities of the integrand. The points nearest to the real axis where this situation is realized are fl2n = /33 + iri (see Fig. 3). Now it is clear that the poles at the point 32n = are caused only by the poles of (FA"l)n.
/3j + ai + ilk, k
Residues at these poles are connected with breather form factors and will be considered in the next section. The poles at the points /32n = /3j + 7ri have to be considered individually. Let us consider the pole at the point /32n = /32n_1 + in. The situation of the poles of the integrand on the plane of the variable aj, Im /32n = in for j = 1 ... n - 1, is indicated in Fig. 3a.
Im p
1
3TL 2
rn x
x
x
x x
x
x x
TL 2
^--^ --aa
^`
iE pX
I2X ^R? +Zn
0 02n-1 Re32n 02n-1 Re 02, P2n-1 X
x
3Tt 2
x
x x x x x x x x x x E u
a
b
Fig. 3.
C
The Main Properties of the Soliton Form Factors
63
It is clear that when 82n tends to 82n-1 + 7fi the contour appears to be clutched between two simple poles of the integrand. Hence the integral over aj, considered as a function of Q2n, has a simple pole at the point Q2n = Q2n-1 + ai. At first glance the pole of 'tn,o((Fnry))n) at the point #2n-1+7ri is of the n-th degree ((n-1) integrations plus the pole of (Fnry))n). However this is not the case. The pole is simple. Let us explain this fact. n-1 First, it is clear that 11 , iH sh (ai - aj) in the definition of tn,o can be n-2 n-2
replaced by n i2 , iI: sh (ai - ai) iH?ch (ai +,32n) due to the antisymmetry of An with respect to xl,... , xn-1. After the replacement, the poles at the point 132n = 82n-1 + iri of the integrals over al,... , an-2 disappear. The poles remain which are caused by the integration over an_1 and by the pole of ( )n. These poles do not interfere because the RHS of Eq. (63) contains a factor sh f (a -,Q2n-1 - 2) (see Fig. 3a). Thus 'tn,o((Fn( ))) has a simple pole at the point ,Q2n = Q2n-1+xi and we have only to calculate the residue. To this end the following trick is proposed. Consider the function
G(IQl,...
,2n)
() fl_i00 dai...
=
00 ^^ n-1 2n
x dan _1 H H
n-1
x 2(1-exp ( an_1-o ))exp --(n-2)ai+0, Cei i=1 i=1 n-2
n-2
x (n 12)!` [Jsh(ai-aj)[Jch(ai-,Q2n)
x (F, ('Y))n
exp ` 2x eel,... , exp 2w C an-1IN1, ...
l /^ ,32n
)
Evidently, '1^n,o ((Fnry))n) = Gory) (sfl - 2Z ()31, ... , N2n)
C -ti
+ Gory) 32n +
2
Ql,... , Q2n
)
Notice now that the pole of GV) (fl2n - a 1 ,81, ... An) at the point ,Q2n = ,82n-1 + -Xi is caused only by the pole of ( ) n, because 1 - exp (an_l - Q2n
64
Form Factors in Completely Integrable Models of Quantum Field Theory
-{ z) cancels the poles at the points an_1 = 02n - 2 , an-1 = /32n - 52s (on Fig. 3 these poles are encircled). Let us calculate the residue of G(a' (,132n at the p oint = /32n-1 + 7ri: *i /^1) • • • , ,62n ) ^2n 2 IF' / 7ri I res G( -Y) I /32n - 2 Nl, • • • , N2n n\-1
1
n-1
00
n-12n-2
fm
_ E( -1)k (2ai) dal... dan-1 fl II 00
k =1
^(ai
-Qi)
00 i=1 j=1
n-2
1 -11 i p 7r 1 H sh ( ai - aj) 2sh (an-1 - N2n-1 + (n - 2)! i<j n-1
n-1
- E(n-2ai+oai+ -(n- 1)ak
X exp
i =1
i-1
- f (n - 1)/32n -1 +
2(
an
iri i - #2n -1 -
n_1 2 )) (F 2 )
-1
2a 27r 27r
exp al, ... , exp ak ,... , exp can -1IQ1, ... , /32n-2 2n-2
)
®e2n-1, ^ ) sh-17r (/32n-1 - ^i
0 (e2n, ®e2n -1,- s + e2n,-
i=1 S 2n-2
1= 1
x sh
a
( ak
-
13l
2n-2
- 2
)
n^
- 2(n- 1) sh
a
(ak - A
+ 2)
N2n = F32n-1 + 7ri
In deriving this relation we have used the identity 1
V(a),*, - 7ri) = x xti sh£(a- cha
Transform the integral over ak in k-th term using (56): 1 Hl roo-
r 2n-2 2,i
27ri + /- oo-2,xi )
n p(
ak - /3j )sh
(ak -
j= 1
a
j
+
2
n-2
xflsh(ak - aj)exp ( 1#k
(n -1)ak + oak dak
,
k
The Main Properties of the Soliton Form Factors 1
oo
oo - 2ai
65
1 2n-2
_ r l 7r ?rt -J_ + / ij ^( 'P(an-1 - )3j)3-'h (an-1 - /j 2 2ai 00 oo-2ai j-1
a 1 x exp ( n - 1)an - 1 + Zan-1 + cran-1 f x 1 dan_1 2 sh2 (an-1- f2n -1+ )
,
k =n-1.
The integrals from the first group are equal to zero because the integrands have no singularities in the strip -27r < Im ak < 0. The last integral is equal to
2n-2
(
R
/32n -1
-
a
j -
Iri 2
)
exp
a x
n
(/32n -1
1
1
- + 2 + Q
J
x s h - ( # 2 n - 1 - /j) •
Hence res G
( fl2fi -
2
1a
2n 1 ...
/^
N2n
ll
)
=
2
ePs -i j_1
C
N2n -1
^j -
( 7) X ^n_l,v (Fn -1)tt - 1 ) ®(e2n, 2 ®e2n_1,- z +e2n ,- z ®e2n-1, 2) .
Let us consider now G(ry) (82n + 2 1/31, ... ,32n) rem 2 implies G o7)+ (/3 = G
The remark to Theo-
2 81, • • • N2n
1 (7) ir1 //^^ N2n + N2n, - 27rt , Nl, • • • An - 1 )
2
Continuing RHS with respect to 82n : N2n 82n + ?ri one obtains the picture of poles presented on Fig. 3b. The poles which are cancelled by 1 exp (an-1 - f32n - Zr) are encircled. It is clear that the pole of G(47') (82n + - 21ri , ,01,•• • ,02n-1) is caused by the pole of (F('))n(^32n - 27ri,,01i 2 1#2n, „32n_l). The calculation of the residue is similar to the calculation for
Form Factors in Completely Integrable Models of Quantum Field Theory
66
res G(,132n - a ...) presented above and leads to the following result:
res G;-Y) (/32
ai ,Q
2n-2 (in
_i
l - j +
/ qq exp( (N2n1
+
WO)
pp -tn-1,a ((F('Y)1)n-1)(Q11/, ... , N2n-2)( e2n,; ® e2n-1 ,- z + e2n,-j ® e2n-1, J) /32n-1 - f3) ... Stn-1,2n-2(fl2n-1 - 132n-2) , XS2n-1,1(//^^ 132n = )32n-I + at .
Hence 2n-2 //^^ res 'Dn,a (( Fiiy))n)( h'1 , ... ,
0
2n) = 11 c (/2n_1 - Nj j=1
F'2n-2) 'Y)1)n-1)(Nl,... ,/^ X -tn-1,o((Fn(® (e2n, J 0 e2n-1 ,- l + e2n-1 , -l 0 e2) x (I - (-1)°S0(Q2n-1 - /31)S2n-1,1(N2n-R1 - fl1) .. . XSO(j32n-1 - I32n-2)S2n-1,2n-2(N2n-1 - N2n-2)) /32n = N2n-1 + at
(79)
In deriving Eq. (79) we have used the identity
cP(a- ai) 2 So(a) - cc(a+
Now, using the identitites
C(Q)C((j- in) ='p-1 (, Q+ 2E f ,
d = C-1(ai)2ai -
one obtains formulae (78). Q.E.D. We have shown that soliton form factors ' as defined in the previous section do satisfy Axioms 1, 2, 3. For e < in it should be shown that the
1 11
The Main Properties of the Soliton Form Factors 67
breather form factors defined by the soliton ones satisfy Axiom 4. We shall make it in Sec. 5. Remark . The reasonings presented after the proof of Lemma 1 provides us with the possibility of the calculation of form factor norms which will be important in Secs . 9, 10. In the previous section the base uwE, ,... ,E,n /91, .. . 132n) has been defined which is dual to , /32n). This base possesses the triangularity opposite to that of the base wfl ,,,. 12n ('31, ... , ,32n). This fact and the formula (69) imply the existence of another representation for (F(7))n:
(F(7 ))n
F(,...
= (01
IAnAl
...^3jn
{1...2n} -{il •••in}U{ji.... in} n
n
X Sh
7r 1 (N iy - N jv + 7fZ)tUe1_ . rzn(Nl) ... A n)
p=1 q=1
1 1 sip = 2 , cjp = -2
Fn(-') ( 8l, • • • , Qn I Nn + 1 ,• • •, I32n ) = F( n'Y)(pn -^- 1, • •• , N 2n 101, ... , an) •
Using these facts and Eq. (46), one obtains
((F(7))n, (F(7'))n) F(")(3i ,,...
i3 1 ,8j
q .,l1 )
{1,... ,2n }={il ... i, }U {jl. ••• .j, } //^^
X F(-7')(A1,... An j jl,... ,Njn) n
n
sh 1(/3i p Xp=1 II q=1 f
1 -
Qj q ) sh" (/3,
(80) - 13ja
+ 7ri)
This formula allows us to calculate the norms of form factors.
5 BREATHERS FORM FACTORS IN SG MODEL
Formula ( 17) implies that the form factors corresponding to both solitons and breathers can be obtained as the residues of soliton form factors. In the present section we consider for the sake of brevity only the form factors f+. Other form factors can be treated in a similar way. According to Eq. ( 17) for the calculation of form factors corresponding to 2n - 2 solitons and a breather, one has to calculate the residues of f+(f31, ... , f32n) at the point I32n = f32n _ 1+iri-iem , the rapidity of breather being equal to /32n_ 1 + i(Z - m2). Let us calculate the residue. As it has been established in the previous section the pole at the point f32n = #n-1+iri-iem , m > 1 is caused only by the pole of ( ) n. Applying Eq. (63) one obtains
rest+(f31,..., ,82n)=
ldm (n -1)I
2n-2 ^
/^ ( i3 - /32n-1 -
x
1 2n-2
a m dn_
i.1
H ((Qi-3j)
i<j
-iem 2
X e2n l ® e2n-1 -1 + (-1)me2n,-; ® e2n-1,z) r 00 0o n^-71 2n-2
® (n - 1) $ dal ... $ dan-1 11 II V(ai - /3j) JJ oo JJ JJ 00 i=1 j=1 n-1
7r2 X ^(Pm ( ai - #2n-1 - 2 + 2m flsh( ai - aj) i-1
i<j
69
70 Form Factors in Completely Integrable Models of Quantum Field Theory (1) 21r 2 /^ /^ X (Fn_1)n_1 exp 1 a1i... ,exp t an-2I /31 ,••• ,Q2n-2 2n - 2
x
7 2
[ [I 1=1
n-z
x 2H 1=1
(exp a an-1 -- 7 exp
(
exp
2ir { an-1 - T
_1
2x
13i
2,
exp
l 1
)
- T4( n
- 1)
/31
n-1 2n-2
x exp
- - (n - 3) > ai i =1
sh- 1 = ( # 2 n -
- 01) ,
1= 1 132n = 02n-1
+ in - i(m
where 2n-2
II shz (a- i
+i^k)
1=1 / ^m (a) - m H chi 2 (a - i + i(k k=0 (s,m (Q) = ( \~ -
2 + t( ) ( 2
= C ex P 2
(13+ -__-_ ) 7ri
t m
' sin 2 2 (/3 + tri)ch{ i sh
0 k sh akch a sh a
2 A
M-1 xi _ s m-2 1 11 chi ( Q + 2 2 2 J j=0
C = exp
00
msh
2 ch 2
- ch 2 L-k sh
f k 2
J0 k sh Z sh irk ch 2 dm = ((in - iem)daml
dk)
A more explicit formula for dm will be given later. Now let us apply the familiar trick with the transformation of the contour of integration over an_1 and obtain the result
27n--2 reef+(Q,...fl2n) = amd"-ldm (n 12)1
II ((Q
- 1j)
i<j
aiX (s,m (fli - /32n-1 X (e2n 1 0 e 2n-1 - .21 + (- 1)me2n,-.2 ® e2n-1,1 2)
Breather, Form Factors in SG Model 71
I 2n-2 n-1
O
x $ dal ... $ dan-2 Ain-1 11 II `a(ai -,8j) Jll Jll oo ; ,, i=1 f -1 Too x
ca m ( ai
am i is
)
- 32.-l -
n-1
2
+ 2
n-2
x II sh(ai - ai) exp (n - 3) ai^ i<j
S
i=1
1 27r 27r x (Fn )1)n_1 exp C a1, ... , exp C an-21 0 1, ..
#2n-2J
/2n=/2n-1+in - im , where F 'M is the contour enveloping the point 82n-1 - 2 - ism + i{ j, j = 0,... ,m.
Continuing along the same line one can obtain a formula for the factors corresponding to arbitrary numbers of solitons and breathers. We present only the final result denoting, to avoid a confusion of terms, the rapidities of breathers by 0.
To this end it is useful to introduce an integral
transformation On m1,... ,mk
Definition 1. Transformation On,m1.., mk;a prescribes for every antisymmetric polynomial P(x1i... xn_1) a function of 81.... „82n, el, ... , 01 by the following rule: da
( n 1 1 )! ( 2^ri
oo 1
n+k-1 2n n+k-1 k rk
dan +k -i
11 1 1 cP ( ai i =1
n+k-1
x exp
/3j )
j= 1
II
ll ^Pmk (ai - 0i )
i=1 j =1
n - 1 n+k-1
o ai - ^ (n - 2)
E a; I
sh( ai - ai )
i=1 S i=1 I i<j
2ir 2ir l x P exp al , ... exP an -1
J
where Fk are the contours enveloping the points Bk - ai - i z mk + i{ j, j = 0,... MkLet us introduce the functions
sh 1, ^m,s(^) = Ss,m(fI) f ,
shE(/3+ 2 -'2 )
72 Form Factors in Completely Integrable Models of Quantum Field Theory
and for ml>m2 \ ( ml-m2) I13+ 2(m1-m2)
^ml,+n2(13)=(-
(
x(
fl+ ir i _4(mi
+
m 2)
) c(/3_
7r i +1 (mi
+m2) )
x sh-1 (P + 2 (ml + m2) + in I
=
°O sin 2 2(P +iri)k ch(2 - Zml)k sh
CM, ,nl2 eXp
z2k
ch
2
(4 dk o ksh 2ch shirk
m1-m2
II
2 x
sh z (/3 + 2 (ml - m2 - 2 j))
j =0
m1+m2-1 II ch 2 (a + z ('ml + m2 - 21)) 1=1
Cm1,m2
= ex p
C
°O m2 ch
2
z
sh 2 - ch (2 - fml )k sh Z2k ch
o ksh7rksh 2ch z
k dk
(m2,m1(13 ) = Cml,m2(1) •
Form factors corresponding to arbitrary numbers of solitons and breathers are given by + J
q
(Q1,... r h'j1B1Yj1 +1...,(3j2B2 k
...Bk#jk +1 ...)32n)m
k
l
.... mk
h
= do lldmj HC(13i -aj) HCmI,mk (B1 -Bk) HHCm,,s (B1 -Ni) j=1 i<j l
do = ll sin 1= 1
21
-21
(sm^mcmlm-
(81)
where the operation ( )n is applied of course only to solitons. The component of (81) in the decomposition with respect to the base eC1 ...E2i is denoted by f
+(al, ... , /3jAAAflj l +1 ... 13j 2 82 ... B k)3j k +l ...1
Breathers Form Factors in SG Model 73
Our next goal is to show that these form factors possess the necessary properties. Theorem 1. Breather form factors satisfy Axiom 1, which means f+(... ,Pi,el,... .... Ss,mt('j -el) = f+(... , el, f'j.... )... M" ,j .... , p+(... , Bl, e1+1, ...) ... J )M1mt+1 , ... Smt,nlt+1 (Bl - B1+1) - f ( . . . , e1+1, el, ...) ... ,mt}1,mt 7****
Proof. The function cn m1 ,.. ,mk;O ((F,il))n) is invariant with respect to permutations of breathers with breathers and breathers with solitons. That is why the statement of the theorem is equivalent to Sm s(1 )Sm,s(N) _ ^s,m(-N) ,, ,sq ,^ bml,m2(N)Sml,m2(/3) = Cms,m1(-N)
These identities can be checked directly. Q.E.D. Theorem 2 . Breather form factors satisfy Axiom 2, which means f +(... Bk + 27ri)... ,mk = f+(Bk, ... )mk,...
Proof. The simplest way to prove the theorem is to cite the corresponding formula for soliton form factors and to use the identity res f+(... 7ri - iem = ambeon -1,-E4nf 1 . • . , N2n-1 +
2 132n = N2n-1 + Ti - i m .
Actually, everything is much simpler. It can be shown that ¢n, m1,... ,mk;o ((Fn 1))n) fl 9-1(,8j - 0k + 7ri - i ) is a periodic function of breather rapidities with the period equal to 27ri, and that functions C3,m, Cm1,m2 satisfy the identities t /^ 1 (s,m(N - 27ri)cO (13 - Ti - iT2
CM, (- ,8 )9 (-,Q + 7ri - Z m J ,
(mm.(N - 21ri) = Cm.,ml(-)3) •
Q.E.D.
74 Form Factors in Completely Integrable Models of Quantum Field Theory
Let us pass onto the analytic properties of breather form factors. Consider f (... , 9k) ... ,,n,; as a function of Bk. The function (,,,n((3) is regular in the strip -a < Imp < 0 and has simple zeros at the points Q = - Z` + i i (m + 2j); j = 0, .... The function has simple poles at the points Q = -ai + i z (ml + m2) - i{ j, j = 1, ... , and simple zeros at the points 8 = - 2 Jm1 - m21 + ir; j, j = 0, ... . Now we have to understand what singularities appear during the analytical continuation of /n,ml,,.. ,mk;0- The singularities of the integrand with respect to al, ... , an-1 are presented on Fig. 4a.
Imaj j Lc n-1 A
Imanit-1
Iman,k-1 31t
31t '2
2
x
x x
It
x
x x it 2
0
0
0
Pj e, 0ek
it 2
n x x x
-n
x
x x x
31t 2
2t
a
b
C
Fig. 4.
It is clear that during the analytical continuation into the strip 0 < ImOk < a, the contour of integration over al.... an-1 cannot be clutched
Breathers Form Factors in SG Model 75
between singularities of the integrand. Hence the integrals over 01, ... , an_ 1 give no singularities in the strip 0 < Im 0k < 7r. Consider now the integral over an+1-1, 1 < k. It is clear that the singularities of this integral considered as a function of Bk can be situated on the line Re Bk = Re 0I. The contour of integration, poles and zeros of the integral are presented on Fig. 4b. Due to the antisymmetry of the integrand with respect to permutation of an+1-1 and an+k_1 the contour of integration can be replaced by r1 U rk. After this replacement the integral has no singularities for Re 0k = Re O1 because both the poles on the line Re an+1_1 = Re e1 and those on the line Re an+1-1 = Re Bk are situated inside the contour. And, what is more, the integral over an+1_1 vanishes for Mk > m1 + 1 and Ok = Cl +7ri - 2 (m1 + mk - 2 j), j = m1 + 1... , because all the singularities inside r1 u rk for these values of Bk are cancelled by zeros. These zeros of the integral over an+1_1 cancel a part of the poles of (m,,mk( B1 - Bk). Finally, let us consider the integral over an+k_1. The locations of the poles and zeros of the integrand are shown on Fig. 4c. It is clear that for Bk = A + z +z (Mk - 2p), j = 1,... , 2n, p = 0,--- , Mk -1, and Ok = 01+ 2 (mk+mi -2p) 1 = 1, ... , k-1, p = 0, ... , mk+m1, the contour of integration appears to be clutched between two poles of the integrand, hence there are simple poles of the integral at these points. A part of these poles is cancelled after the multiplication by Cs,mk(/3j - 0k),(m,,mk(01 - Bk). For m1 > mk+l, the integrand vanishes when Bk = O1+7ri-2(mk+mi-2j), j = Mk + 1, ... . Summarizing all the above reasonings we obtain the following picture of poles of the form factor: form factor f , B1, ... , Bk) has simple poles at the points Bk = /35+ r -i z(mk-2p), p = 0, ... , Mk-1; Bk = 01+ 2(mk+m1)-i&,p=0,...,min(m1,mk)-1;Bk=Of +Ti -2(mk+ ml - 2p), P = 1.... , min(ml, Mk). This picture is in agreement with that required in Sec. 1. We have now to calculate the residues at these poles.
Theorem 3 . Form factor f+(... , Bk),,, ,nk satisfies Axioms 3, 4, which means than
21ri res f+(... , Ok -1, Ok)...
, mk-1,mk
2n k-2 ff vmk,mk-1f+(...) [ 1_llSmks(Ok_l_fl)HSmkmP(0k_l_0P) ] 1=1
p=1
Bk = Ok-1 + in,
76 Form Factors in Completely Integrable Models of Quantum Field Theory res f+(... ,N2n,Ok)..., e^,,,mk
aIn
(...
,
+
82n
t( ai - i^p k(Mk-P),#2n+ 2 ...EZ... o
ai i^ Ok=N2n+ 2 + 2(mk-2P ),P= O...mk-1, res f+(... , O k-1, Ok)... ,mk-l,mk
i = ami,m2f+ ( ...Ok_1+(mk _P)Ok_1 i^ + 2 (Mk-1 - P) . Ok =
0k -1 +
i 2
( m k + Mk-1 - 2P) ,P
resf+(... ,O k-1, ami,m2
Ok )
= 0, ... , min(mk_1i Mk) - 1 ,
... mk-1,mk Z^
f+
, B k-1
Z
- 2 (Mk + Mk-1 - 2P) ...mk-1-P,mk-P
Ok = Ok-1 + ai - 2 (mk + Mk-1 - 2p), P = 1, ... , min(mk, mk_1) .
(82) Proof. The proofs of the above identities are quite similar ; that is why we shall prove only the first one which seems to be the most difficult. We leave the proofs of other identities to the reader. The pole of f+(... , Ok_1i Ok)...mk_l ,,^k at the point Ok = Ok _1+ii exist only for Mk = mk_1i being caused by the pole of the function Cmk_l,mk (Ok-1 -Ok). Let us evaluate the residues at 0k = Ok_1 + ai: tat res f
(h'1,
...
Bk -1, Bk )el,... ,m k_ l,mk 2n
=
sin fmkbmk -^, mk
k-2
lS(F'i - /3j) fl Cmv,mv(0p - B9)
i<j P<4
k-2 j, k-2 2n X llHCs,m,(13i-Bo)H
II Cm,,s(01-Nti)
1-1 i-1 1-1 i=jt+1 2n
- 1 (o_
-
-
i m 1 k
J
im -1
Ok-1 - + -
Breathers Form Factors in SG Model 77
(
k-2
x H ^0-' p=1 f oo
B
-
Ok-1
a'2
p
k
l I SUMP /
1 Bk _ 1 - Bp + Z^2 k - ^i
oo
x $ dal ... $ dan-1 f dan ... „ll oo J oo k-2
J
dan+k-1
n-1
x 11pm, (ai-Op ) exp(- f(n-2 p=1 n+k-1
x sh-1
i=1
(
ai
11 ^P ( ai - /j )
_Ok _ i_
i=1
ai)11 sh( ai - aj)
i<j
k) 1 (
k
sh _
Z(
(
2 27r 27r 1 X (Fn )n (exP1 ai... exP ` an-11#1 . ... f2n ,
(83)
where we have used the identitites S 7ri) = cp-1 (-ii - 2 m I so-' (-,3 - 7ri + Z 2m^ (ml,msW)Sml, m2(18 - 7rii)
=
^ml
tl;m2
(- ,
)
6 - 2 So-
i
I -/3-7ri Zl m2 \
Spm(a)cpm( a - vi ) = sh-1 (a -
z 2m
\ 2 sh-1 (a +
27rid t2,^(m,m(-7ri) = sin m(. Consider the integrals over an+k_1 and an+k_2, 2n
k-2
1
27ri x
Ir ', - 1
dan+k-2 fl c (an+k-2 - Nj) 11 9mv(an+ k-2 - Bp) j=1 p=1
H
sh(a n +k_2 -
ai)
i
1
x
sh(an+k_2 - 9k_1 - f z )sh ( an+k_2 - Bk-1 + f z ) 2n
H
k-2
H
27r rkc(an j=i p dan+ k _1
X
H
sh
+k _1 - 13j)
pmn( an+k_1 -
Bp)
=1
(an +k-1 - ai)
i
1 x imk imk sh(an+k_1 - Bk_1 - 2 ) sh(an+k-1 - 9k_1 + 2 )
78 Form Factors in Completely Integrable Models of Quantum Field Theory
where the contours rk_1 i rk envelop the points an+k-2 = Bk-1 ± ' 2 7ri and an +k-1 = Ok_1 ± ' 2 k respectively (other poles being cancelled). Calculate the integral over an+k-1: 2n
sinlym
tl;mk
1 1=1
x
k _2
2^mk ) \Bk_1 - Bp + 2 l/ 2 / 11 gym'
( H `P \Bk_1 - (3j +
p=1
i{mk
sh Bk_1 - a; +
2 27ri
i
2n
1
'P(an+k -2 - Qj)
dan+k-2 ir
k-,
i i
k-2 x IJ'Pm,(an+k_2 - Op) 11
sh(an+k_2 - ai)
p=1 i
1 x
sh (an +k_2 - ek-1 +' 2k)
O
2n 2n -
k-2 k) k-1-NjIT SPmp (ok_1
=^2
-
OP
= kj=1 l
X sh ek-1 ai -
mkMk 1 k
i
2n
dan_k_2 11 'P(an+k-2 - /j) k _1 j_1
k-2 X II lomp( an+k_2 - Bp)
II sh( an+k_2 - ai)
p=1 i
1
x
(84)
sh(an+k_2 - Bk-1 - ` 2 ) }
It is notable that one singularity under the integral over an+k-2 has disappeared in each term in Eq. (84). Take the integral over an+k_2:
sin m k
11 'P j= 1
(
' 2 l2k f
k-2 X fl 40mp (9k-1 p
1
(
Z^mk
6k-1 - Q 2 J'P I\Bk_1 - Pj - 7ri+ 2
8p
C k - 7ri + as2 ) lpmp
(
c
1
Ok- 1 - Op - ZS2 k
Breathers Form Factors in SG Model 79
2
x II shCcrf -Ok -1i
i
2mk l
)
k-2
Sm,s(Ok-1-/31)II Smk,,(0k-1-BP)
(85)
P=1
i.l
where we used the identities 'P(f3-iri iri
)'P(a +f2 ) +`Z)c(a -i2 ) , -i2
'Smi,ms(/') _ mi(P ^ - 7r1 - i 2 )cPmi(/3+ f 2 ) `P / Vmi( F' -7ri+i2 )corm(F'-f2 )
Substituting ( 85) to ( 86) we obtain the required identity. Q.E.D. Finally, in order to check the identity of Axiom 4 completely one has to show that form factor f (131, ... , #2n)...,, . possesses proper analytical properties in the presence of breathers. We formulate the corresponding theorem but do not prove it as the proof is quite similar to that of Theorem 3. Theorem 4 . Besides the poles at the points / 2n = a3 + 7ri - ilk, k = 0,... , [1], the form factor f +(131, ... , 01, .. • , Nzn )sl m, ez„ has simple poles at the points /32n = Bl + a + 2 (ml - 2p), p = 0,1, ... , ml - 1, the residues being res f (. . . , Bk,,8zn)... ,mk,ssn
amff+ =
(.
, ..
,02n=ek+
2
, Bk +
7r 2 -
2
.-P,62.
7ri k 2(mk -2p), p=1,...,mk-1. 2+
6 PROPERTIES OF THE OPERATORS jµ, T,,,,, exp(±i P;) IN SG MODEL
In Sec . 3 formulae have been presented for the soliton form factors fµ, fµv, ft. Their properties, which entail local commutativity, have been established in Secs. 4, 5. In the present section we established some additional properties of the form factors which demonstrates that it is natural to identify the operators defined by these form factors with j,,, T,,,, exp(±i ^ ). We consider only soliton form factors ; a generalization to form factors corresponding to both solitons and breathers is straightforward. Theorem 1. Soliton form factors fµ(fl1,... ,. 32n),fµvA,••• , 32n),
f}(/3i, ... ,82n ) possess the properties
f'`(91, +A,... f,.,, (#1 + f
0, ...
N2n
+A )
, 82n +
0)
= Lµv(A)fv (131,... ,,82n), = Lµµ/,
( A) L .., (A)fµ^v^ ( ft,. ... N2n)
:' (81 + 0, ... , N2 n + 0 ) = f + (Q 1, ...
A.)
(86)
Equations (86) means that with respect to Lorentz transformations jµ is a vector , Tµ„ is an ( 1, 1)-tensor, and exp ( ±i^) are scalars. Proof. Eqs. (86) follow evidently from Eqs. (60) and the identities ((F.01)).)
(81
+ 0, ... , ,Q2n + O)
= exp((n - 1)Dv)Qin,o((F("))n)(fl1, ... A n ) ) .
81
82 Form Factors in Completely Integrable Models of Quantum Field Theory
A Lorentz transformation corresponds to a simultaneous shift of rapidities; that is why Eqs. (86) imply that j,,TN„ and exp(±i-) are vector, (1, 1)-tensor and scalar respectively. Q.E.D. It is possible to introduce an operator of charge conjugation postulating its action in the space of states:
Clph) = Iph ), CZ: (a)C CZZ (Q)C
-1
1=
Z' E(#)),
e
= ±1
= (- 1) mZm(a)
The following theorem establishes the properties of operators defined through fN , fN,,, f^ with respect to charge conjugation. Theorem 2 . Soliton form factors satisfy the identities fp(/317 ...
, /3 2n)E1
... ,e2 n = -fµ(,6 1, ... ,,62n)- E1 ,...
fNvll^l ,... ,/^2n)£1 ,...,£2n = fpi(/3lf... f
+(61, ...
,/2n) E1i ... ,E2n
,- E2n
7
£2n)-£ 1,...,-£2,.
f - 01, ... ,Q2n)-£ 1,...,-e2n
which mean that
CjNC -1
= -jp,
CTNvC - 1
= TNv 7
Cexp(±iLu C-1=exp(Fi 2u)
Proof. From the reasonings presented in Sec. 4 it follows that (F,^,*'))n(xl, ... , xn-1I/ /'17 ... ,, /32n)- j... ,- ,... _
(F, (- 7)) n( x1 , ... `
, xn-11,617... #2n) 1 ... 121
21
1
1 ,..., -;
.
(87)
Successively used (64) one can obtain from (87) a relation (F, 1r))n(x1, ... , xn-1I Nl, ...
E,... , E2n
(-7) _ (Fn )n (x17... ,xn-ll ,66lf... ,fl2n)- e1,...,-e2n ,
(88)
where E1, ... , E2n is an arbitrary set of indices. The statement of the theorem follows immediately from this relation. Q.E.D.
Properties of the Operators...
83
Theorem 3. Soliton form factors satisfy the identities 2n
2n
Echaj
f0(N1,... ,N2n) =
j -1
Es
h,8j fl(fl1,... ,62n)
j =1
(89)
f0l(/'1, • • • An) = f10(/31, • • • An)
which mean that
8µjµ = 0 , T,,,, = T,,,, . Proof. From the definition of fµ, fµ„ it follows that Eqs. ( 89) are equivalent to the following relation: exp I -2 > J3j i (>ePi) On, 1((F(ti))n) = eXp
(2
j) (:e-p') On,-
Ea
1(( Fn"))n)
,
which, in turn , is equivalent to
exp (-
>/3)
= exp (
(>el3')
On,i(An)
>'i) (Ee-A1) On,-l(An) . (90)
Let us prove Eq. (90). Present, as in the proof of Theorem 4 of Sec. 4, the function On,o(On) in the form
O n,, (A n) = det 112(i(j )II , where '11^^) is an (n -1) x (n -1) matrix with the following matrix elements.
C
^)= $^daIIp(a-flj )exp -(n- 2) Ta+ (n-2j+c)a) JJJ 00 2r 2w 27r 27r 2v x Ai (exp al exp Q1, ... , exp On Iexp 8n+1, ... , exp T #2n
Obviously the first n-2 columns of 21(-) coincide with the last n-2 columns of Qt(+); hence the difference (90) can be presented as a determinant of the
84
Form Factors in Completely Integrable Models of Quantum Field Theory
matrix whose first (n - 2) columns coincide with those of 21(-), the last column being co
dal
2(i,n_1 = $ JJ 00
2a 2a 2v x Ai ( exp a exp Q1 i ... , exp
x [exP(_
2a '0n
Ir
Qn +1,..
. exp 2a 822
I3i+(n _1)c) i (> e'0' 1
exp ( 2 E #j - (n - 1)a) (E a-0j) 1 where (-1)n appears because of the permutation of the columns in 2t(+). Notice now that the column 2(i,n_1 can be changed to ft
cc
da 11
2ti,n-1 = * 00
x [llch(&_iii+1)_(_1llch(a_fli_)J 21r 27r x Ai (exp t a I t )31, ... , exp t ,l3n lexp ^ ,^3n+1 i ... , exp T /.32n
(91) f because the difference 2(i,n_1 - 2(i,n-1 is a linear combination of the first = 0. At first we shall present n - 2 columns . We want to show that 2(i,n-1 Ps a "wrong" proof to clarify the idea. Then we shall give the rigorous proof. The idea of the proof is to use the identity ch2(a+ 2) 'p(a - k) = -'p(a) 1 vi
ch2(a-2-il;)
(92)
and to transform the integral (91) in the following way: 2(i,n-1 = (. - J _f f r H co(a - Qi) exp (_(n_2)a) x II ch 2 (a -,(3i + 2
2r 2v P1, ) Ai (eXP
ai exP
(93)
85
Properties of the Operators...
Notice now that the integrand in (93) has no singularities in the strip 0 > Im a > -^; hence if we deal with a usual integral the identity i,n-1 = 0 would be evident. But the fact is that we are dealing with a regularized integral f and it is essential here; as it will be clear later, only specific properties of the function Ai allow us to transform this "wrong" proof to a rigorous one. Recall the definition of f : 00 a) ek«da 11 ,P( a - Q.i) exp _(n - 2) a) P exp
(
(
J 00
j^^(a-
-Z exp ,8') sh "(a-a' 1 H
x exp (ak - (3n - 2 ) a) PI I exp +
j ll
J
3j) exp (ak (a - ,
:/i
a) da
( n - 2)a) p2 (r4exp T a) da (94)
where IkI < n - 1, deg P < 2n - 2 , Pl and P2 are polynomials defined by II(xr-3 - e < P3)p(x ) = Pi(x) + H( xr-2 - r-1e p')P2(x) - 7- 2(n-2 ) fJ(xT-3 - e i'' ) P2(xr4), deg Pl < 3n - 1 . (95) The condition IkI < n - 1 is essential because for degP1 = 3n - 1 the integrand of the integral over R + z` i0 behaves as exp ( k - n)a when a -+ oo. That is why, generally speaking it is impossible to divide the integral (91) into the sum of two integrals containing II ch i ( a - i3 + 2 ) and II ch 2 (a - /35 - i ). The only case that it is possible is when it appears possible to find such a decomposition as (95 ) for which deg Pl = 3n - 2. This requirement is a restriction on the polynomial P. We are going to show that the polynomial A1(xI ...) does satisfy this requirement.
Recall the definition n
A i( xla l ,... an Ibl ,... bn) = 2- n-i [J( x
i-1 ( - air) ( E(1 - r2(i-k))
1=1 l k-0 l n X (-1)kxi-k-lrkc.k(bl, ... , b,) T +,r 2i JJ(x - bl r-1) l=1 X
i-1 E(1
k _0
- T2(`- k))(- 1)kx'- k-lTkQk(al,... a++) I
86
Form Factors in Completely Integrable Models of Quantum Field Theory
Notice that for x - ^ oo, i-1 ^(rk - r2i- k)(-l) kxi - k-lo.k(bl, ... , bn) k=0 = x-n+i-l I IJ(x - rbj) -,r 2i IJ( x
+ O(x-2)
Hence n A.(xlal ,... , an
Ib1,
... , bn ) = 2-n-i
[f (x - ra ^ )(x - r ,)
b
1=1 n 1 1 n+i-1 - [J(x r- a1)(x - r b1) x- +n-2 O(x ) . 1=1
(96)
Take as P2 ( x) the following polynomial:
P2( x )
2-n -i
> r3ko
(ex
p
exp f32n)
n-k+i-l . (97)
k=0
Evidently
P2 (x) = 2-n-i [J
C
x
- r3 exp (
f3 )) x' + 0(.T-1)
for x --^ oo. Substituting Eqs. (97) and (96) in Eq. ( 95) one obtains 2n H(xr3 -
e U13)O(xn -2) = Pl(x)
j=1 2n
2n
+ t l (xr -2 - r-le 'i)O(x-1) - 7.2(n-2) [f (xr 3 j=1
ez#j
)O( x -l)
j=1
and consequently P1(x) = O(x3"-2) , which means that it is possible to give meaning to f for Ai taking for P2 in Eq. (94) the polynomial (97).
Properties of the Operators... 87
Let us return to Eq. (91): _
oo
21i,n_1 = $ 00
da 11 ,'(a -,6j) exp ( _(n - 2)
2w 2w x Ai exp a I exp N1 , ...
x
ch
2
C a - Qj
/ B+*,-`-io^^p(c, r
- ✓
C- -
exp /3n
+ .... )1 1
1
sh{(a_#j _ 32i)exp
(3n - 2)cv) Pl (exp -0) 11, ch 2 Ca + 2 I
-(-1)n ch 2 (a _i31 + fl'P(a - 6j) exp x
a
2 f (-1)n ch 2 Ca - 2 / J
_
x exp
I
T
da
2/
(-
J
( n - 2)a/ PZ CeXP C a/ T4/
7r
27r
[rj ch2 ( Ce -,Oj + 2) - (- 1)n fj ch Ca - /3j - T)] 7 ri
(98)
where P2 is given by (97), Pl is defined by (95). Now all the problems concerning convergence have been overcome. Hence it is possible to rewrite Eq. (98), using Eq. (56), as follows: _ ^cp(a-i3j)ch2(a-,aj+ 2) ^i,n-1
-io
x exp
(
+Cf (
_
E/3i) exP
=
(_ ,
r r -if
X A
1+
z4
h
3ai
(3n_ 2)a) Pl
(
exp
a) - 7r
H(P(a- 8j) ch2Ca-13j+ (
exp a 2x )
The integrand of the first integral has 2n singularities in the strip z - 0 > Ima > z - 0 - ^, which are simple poles at the points a = ^3j + 32i
88
Form Factors in Completely Integrable Models of Quantum Field Theory
-i([f ] + 1)^, the corresponding residues being ^7io (/j -Qi+32` -i([£]+ 1)S)
j
it#l
sh
exp
(3 - 2 ),8j
(lr
n
L - Qi
C 27r )
x P1 7.3exp TQj
The integrand of the second integral has 2n singularities between the contours F, F - ic, which are simple poles at the points a = 0j - a - i ([ f ]+ 1), the corresponding residues being
tn^ (Pi
7ri
-A -
2-
t \L^J +1)^)
(n - 2),Qj ) p2 (r3exp i3j )
x exp
As it follows from these formulae for residues and Eq. (95), the first and second integrals cancel each other; hence %i,n-1 = 0. Q.E.D. Theorem 4 . From factors f,,, f,,, can be rewritten as follows: fµ(31, ... An ) =
E(el6i
- (- 1)µe-Qi)g-(#,, ... , /32n)
(99)
i //^^ elji fµv(fl1,... ,/92n) = E(
i,j
(100)
x 9+ (01.... /32n)
where 9± (81 ,
... , 132n) =
((G(' )n
±
(G( 1)) n) f
Coi - i9 )
i<j
qq G(f1)(01, ... , fln INn+1, ... , /32n) -..(n-1) = 2 2 exp
n
(
±j
(, +- 1i - 7f)/lI 7r
x n+1, • • • ,,32n) exp - (n - 1)
89
Properties of the Operators...
Gn(191, ... , 3nl)n+l, i2n) is the determinant of a n x n matrix 6i3(i = 0, ... , n - 1, j = 0, ... , n - 1) with the following matrix elements:
6i3 = 271ri, X Ai
(
cp(a -
,a3) exp ( _(n_2)a
+ (-n + 1 + 2j)a
00
-
2a 27r 21r 27r 21r an+l, ... , exp,- #2.) ,3 1, ... , exp n Iexp
exp aIeXp
-
-a
where A1, 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
Form Factors in Completely Integrable Models of Quantum Field Theory
(
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 (^ef') (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
Form Factors in Completely Integrable Models of Quantum Field Theory
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
Properties of the Operators...
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)
94 Form Factors in Completely Integrable Models of Quantum Field Theory
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
Properties of the Operators...
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
Form Factors in Completely Integrable Models of Quantum Field Theory
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<1
fµ(131,
f,..
-13,) II
f[ sh- 1(13p
132)-1„ = -
( (( exp
(131, ,62 ) - i i = - exp P 87ri ( "' x
(
exp
(
-
P=1 9=n+1
(/3 + 132)
)
/3 1+/32 ) )
13
(107)
9) ,
exP (2 (/31 +,82))
) (108)
62) I - (-1)µ exP ( -1(131 + #2)/ l (2 (131 +l 2
- (- 1)v exP ( -(/3 + 132)) / sh - 132)
(109)
Let us check the agreement of these formulae with those corresponding to the massive free Fermi-field. The SG model for 1s = 7r is identified with the massive free Fermi-field in accordance with the rule that Z*( Ze ((3) are the creation-annihilation operators of fermions. Hence
2 ^j(x) = r (eh1' (lj)sv (e L )z(fl) + e-Pµ (16)-,- (: f 1\ e^ -2
+
2
eipv(Q)zv
Ce
! ) Z^ (13) + e-P,•(Q)=,.
Ce
)
Z - 21 (13)
2
/ Z* 1 13)^ d13 .
\ e 2 J _ r
The operators jµ, Tµv are constructed for the massive free fermion field in the usual way:
2(YyµavY - avY yµW) 'Y0=131 , -Y1=i02
Properties of the Operators... 97
Evidently (phijµ(0) = (Phi
f
(P,+P 2) _ (_1)µe-',-(P1+P d/3id/32 (e12
(PhITµ,, (0) = (PhI
f
7))
Z-I(Qi)Z*(Qi)
d/3ld)32 (e2(#1+0 2) _ (_1) µe- 12 (161 +32) )
x (e#(P^+Ps) _ (_1)µe-1VP,+162)) sh 1()3 -/3z)Z '(/31)Z'(Q2) which coincide with (108), (109). /l The operators exp(±i^) are identified with exp(fai fs. jo(x)dx). The operators f:. jo(x)dx are presented through the creation-annihilation operators as follows: xo
j
io(xoxi)dxo = Jdthd/32( ch
'(/3i
x exp(-ixµ(Pµ(/31) + Pµ(/32))) + Z*
- /32)(Z '( /31) Z'(Qz)
; (Q1 )Z; (/32)
x exp(-ixµ(Pµ(/31) +pµ(/32 )))) + sh -1(/31 - /32 - i0)(Z* (/31)Z' (/32) x exp(ixµ(PI()3i) - Pµ(/32))) + Z* 1 (/31)Z- '(/32) x exp(ixµ(Pµ(Qi) - Pµ(/32)))) 1 • / To calculate the matrix elements of the operators exp(i^) and exp(-i^) we have to rewrite these operators in the normal form. To this end one can use the well-known procedure of normal order ring of an exponent of a bilinear form. Using the corresponding formulae from the book [62] one obtains exp (firi : J
1o(xl)dxl
1 JJ (x) = Jd/31d/32 exP {± 2 /3i - Q2)
r 1 t ch 1(Q, - /32)
x (Z ' (#,) Z 12 (/32) exp(-ixµ(Pµ (/31) + Pµ(/3z))) + Z'1(/31)Z*(/3 2)eXp ( ixµ(Pµ (01) + Pµ(/32)))) + ah 2(Q1 1 02 - i0) (Z^
(01)Z' (Q2) eXp(ixµ (Pµ(Q1) - Pµ(/32))
+ Z` ,(/31)Z 3(,32)eXp(ix,(pI (/31) -p"(,32)))) } 2
98
Form Factors in Completely Integrable Models of Quantum Field Theory
Hence (phi eXP
(
±
0)) =
(ph i E J
d,3 1 . .. dp ..Z -12 ( 0 1 ) ... Z-
(Qn)
n=0 n X Z
I
//^^ (Pn+1) ... Z s(/32n)
exP
f2
^()3 n +p-
/3P )
p=1
xII sh 1 (Qi-Q.i)11 1 pn
which coincides with Eq. (107). This section contains several arguments in favour of identifying the operators defined by fN, f,,,,, ff with j,4,T,,,,exp(±i^) respectively. For the currents jr, one more argument will be obtained in Sec. 10 which is the calculation of the singularities of the commutators [jµ(x), j„(y)] at the point x=y.
7 FORM FACTORS IN SU(2)-INVARIANT THIRRING MODEL
As it has been said in the Introduction, ITM particles (kinks) possess unusual statistics: they are spin-4 particles. That is why the charge conjugation matrix C = io2 is skew-symmetric and the reasonings of Sec. 2 cannot be applied to ITM directly. However we can pass from spin-4 physical particles to auxiliary particles with the usual statistics. Consider the base of ITM space of states which consists of the vectors
Z,%
(,8.) ...
ZE
1 (131) I
ph)
Let us introduce formally the operators ZE , Z` which satisfy the Zamolodchikov- Faddeev algebra with the following S-matrix:
S
fl E7(3)
= (-
1)E1 +`'
1 c^ (fl)
where S is the ITM S-matrix (e = 1, 2). The matrix S satisfies the YangBaxter equation , unitarity and crossing-symmetry; the matrix of charge conjugation c = vl, i. e., the associated particles satisfy the usual statistics. Let us identify the spaces of states generated by Z* and Z* via n
ZEn(an)
...
Zf1(N 1)Iph)
1)i`iZ,%(/3n)
_
...
Zf1(,3 1)Iph)
j=1
Evidently this prescription is in agreement with the commutation relations. Now one can construct local operators whose auxiliary-particle form factors
99
Form Factors in Completely Integrable Models of Quantum Field Theory
100
satisfy the usual equations. Locality is guaranteed by the theorem of Sec. 2. In terms of the physical particles the main equations for the form factors are written as follows: f(/3l,••• ,fl,/3+1,•.. ,fn)Sii^,i+l(h#i -Ni+1) q= f(/3,.... ,Ni//+1rh'i,... fl n) , /^
f(,61, • • •
, Nn1, Nn
+ 27ri)
= i
nf(#n, #l, • • •
, Bn-1)
27ri res f (/31, ... , Qn )Fl,... ,Er n-1 = Sn,n_1(I - i
/^ Sn-1,1(Nn-1 - N1) ... Sn-1,n-2(in-1- Nn-2)) On = N3n-1 + 7ri
(110) where Sn,n-1 = en,l ® en-1, 2 - en,2 0 en-1,1 ,
ek,e being the base in the isotopic space associated with the k-th particles. Notice now that the matrix S coincides with the SG S-matrix in the limit ^ -+ oo. This is why one can construct the form factors of the operators T,,,,, jµ taking the corresponding limit of SG form factors. Then one has to show that Tµ„ is SU(2)-invariant while jN is the component of the SU(2) triplet, and to present the form factos of jµ, jµ. This programme can be realized but we would not like going into the details of the limiting process l; -+ oo. We present only the final results directly in the terms of ITM. Recall the S-matrix of ITM
'Je1,e3 (p) -
So(d)((3bEl
C2 - 7r -
vEZ)
so(F ) EZZ (/^) f
7ri) E 1
SO(N) = r(2 + 2ri Q)
r(2
21)r(E 2vi
Now we present several definitions similar to those given in Sec. 3. Consider a set of rapidities 81,. .. „ 62n connected with isotopic spaces hi " C2. Introduce an auxiliary space ho and a matrix
C
A(oif33l,... , 02n) C(o1/1,... , N2n)
B(oj/l, ... , N2n) 1 = So,1(0_ - / 3 1 ) ... So,2n( Q - 92n) fi, ... , 132n) J
D(oi
(111)
Form Factors in SU(2)-Invariant Thirring Model
101
where Sil(o)=
(
1 ai ai 1 o-- o;ai o-7ri 2 2
o; (a = 1, 2, 3) are the Pauli matrices acting in the spacing hi. The matrix S differs from S by the absence of the multiplier So. In every space h; we consider a base e; E(E = 1, 2), the covector II 0 e1,1 being denoted by (Of. Besides the natural base of the tensor product e£1r...E2n = II 0 eirE;, we introduce a base consisting of the vectors
WE1.... re2, (/ 1) ... ,/32n) = (OI
11 B(/pI)31, • • • , /32n) p Ep=2
In contrast with the SG model we do not impose any restriction on E,-j. The base wE1r••• rE2n is connected with e£1 ,,, rE,n by a triangular transformation and satisfies several requirements similar to those presented in Sec. 3 (the correspondence is established via the rule: all sh 1,8 are to be changed to ,Q). A dual base 17V,l, --- ,C2n 031, • • • , /2n) is constructed as in the SG model. Lemma 5 of Sec. 3 remains valid. The base w1 r ••• re2n(/31 , . . • An) possesses some specific properties associated with the SU(2)-invariance of the model. The algebra SU(2) acts in the space h10 ... (9 h2n whose generators are Ea = Eioa. The following Lemma describes the properties of wE 1. ... ,£2n 0i3i, • • • ,82n) with respect to this action. Lemma 1 . The covectors w£1 .... E2n (131, ... , fl2n) are transformed under the action of the algebra SU ( 2) in the following way:
wEI,- r£]n
(N1) ... , N2n)E3
( - 2E3) = 1
W1....
A2n( al,
•••
(112) wEl,... rE,n (f31)
...
,
#2ri)E
+
+ 7ri R / ^ wE1r... rEP-1r1,£P+lr ...r E 2n(h' 1 ,. . ., N 2n ) J /3P _)3'
p:Ep=2
q:£q=2
flp _ Nl
qp
(113)
Form Factors in Completely Integrable Models of Quantum Field Theory
102
wC1,...,C2n (fl11 ... ,#2.)E-
/p- Q9-xi //1 , ...., f2n) E wel,..., er- 1, 2 , Cr+1,...e2n ( p:cr=1 grey=l )6P Qy 4$p (114) Proof. Eq. (112 ) is evident because B(o) is a lowering operator:
B(Q)E3 = (E3 + 1) B(Q) . Let us prove Eq. (113). The covector wet .., e2n (31, • .. , /3 )E+ has simple poles at the points /3j = /3i + lri, j > i, Ej = 2, ei = 1, with residues
res wel ,... , e2, (/31 , ... , EZn)E+ _ ( ei,l (9 ej,2 - ei ,22 0 ej,l) /
®We1 ,... C;,... ,E'j,...
N2.) E ak ki6ij
Irk - Qi sj-1 /^ p /^ q i) . . . Si+1 7fi,i(flj-1 -/3,i(F' i+1 - i) ii (115
X II 8k -,8i k>j
)
where we have rearranged E+ and Sk,i and made use of the SU(2)-invariance of the S-matrix. The covector Wei,--- 'V2. (,81.... , /32n)E+ can be reconstructed by means of Eqs. (115). The RHS of (113) has the same singularities (the poles at the points /3p = /3y are obviously fictious). The residue of the RHS of (113) at the point /3j = /3i + lri, ej = 2, Ei = 1, is given by
(
//^^
( /3p -
/3y
+ 7ri
we1,•••,er - 1,1,er+1, ...,e2n(Nl) ... ,N2n) 11
res p:ep=2
6p 0
rj
-
P9 g:ey=2 9$P
/3p-Ny+7ri
- (ei,l ® ej , 2 - ej l 0 ei,z)
pier=Z q :ey=2 QQ 6q p$j v#j X well....f':,...
....e2n 01 , ... , /^i ... /3j, ... , N2n)
X II A
k>jj
- Fri §j-1,i(/3j-1 - pi) ... Si+l,i(Qi+l - Ni)
/3i Ti
(116)
Form Factors in SU(2)-Invariant Thirring Model
103
The RHS of (113) decreases for (3j -> oo, hence it can be reconstructed by means of Eqs . (116). Comparing Eqs. (115) and Eqs. (116), we can see that they provide us with the possibility of an inductive proof of Eq. (113). The verification of Eq. (133) for n = 1 is an easy exercise. The identity (114) can be proven in a similar way. Q.E.D. Definition 1. Let F(A1, ... An+cjµl, • • • µn-c) be a function symmetric with respect to Al, ... , An+c and pi,... , lin-1 separately, c being an integer -n < c < n. The operation associates with the function F a vector from h' ® ... R® hen in the following way: (F )n(NA, • • • , 132n) E F(Qf1,...
,3i.+. 1Qj. ..... f3 -)
{1,... ,2n} ,i,.tc} U{j1,... ,.In_c} x
^j w£1,... EZn (N1, ...
^
,
#2n)
,
(
117 )
p,9 ^sr - '8jq
where eip = 1, ejq = 2)p = 1, ... , n + c, q = 1, ... , n - c. This definition is quite similar to Definition 1 of Sec. 3, the only difference being that here we do not restrict ourselves to the consideration of the Q3 neutral sector. Definition 2. Consider an antisymmetric polynomial P(al,... ) an-1)• The integral transformation On,a(0' = 0, ±1) associates with this polynomial a function of /32n in the following way: qq q (2xi)-n o0 fo On,o, (P)(hll, ... , N2n
(n 1).
J
oo
(dal ... dan_1 1I t0(ai - 9j) ao i,j
x IJ sh(ai - aj) exp (u ai ) P(al, ... , an-1) , i<j
where a _ a _^r 1 a l 'P( ) 2 7r 2xi r 4 + 2xi In contrast with Definition 2 of Sec. 3, there is no necessity of regularization of integrals here because they are well-defined for an arbitrary degree of P. Definition 3. Polynomials A c , an_11a1, ...''n+clµl, ... , µn-c), c = 0, ±1 are the determinants of (n - 1) x (n - 1) matrices Ac with the
Form Factors in Completely Integrable Models of Quantum Field Theory
104
following matrix elements:
Ate = A i (aj Ial, ... , An+cl/t1, ... , µ n -c) , A; (ala1, ... , )ln+cl µl, ... , µ n-c) n-c 7rZ
_ (a -)1j - 2
[Qi -c(a
7r ilpi
-
-
ir2,
... , µn -c -
7r2)
j= 1 n-c
.fin +c)l
7r2
2
+ fl
[Qi+c(al'\1 i ... , .\n+c)
j=1 (118)
72)]
where Qi(aIAi , ... , \k) 7r2^1
=
LL^^
a + 2
(
7r2 )1
Ak)
a -
1=0
(119)
This definition is similar to Definition 3 of Sec. 3. Let us formulate some Lemmas which describe the properties of On. Lemma 2 . The polynomials On satisfy the recurrent relation c
On (al, ... , an-1
(ai
n-1
j=11
lal,
7r2
-
µ n-c
µ ,.-c=a^}c+xi
n-1
kk=JO
p=1 q=1 n+c-1 n-c-1
11 p-1 g=1
... ,
+c - 2) [^(_1)k
x
H
µ l)
.An
n+c-In-c-1
-
. , An +cl
(
(
ak -
-
ri f (ai - - 2
7r i ak _ AP + .
)
(
7r2
ak _ Ii9 + 2 )}
X An (al, ... , &k, ... ) an-1IA1.... , An+c-1Iµ, ... , µn-c-1)
(120) Proof. The proof follows immediately from two simple identities (see the proof of similar lemma of Sec. 4):
Form Factors in SU(2)-Invariant Thirring Model Qi(alAI, ... , Ak) =
105
QW(alkl, ... , Ak-1) - AkQi-1(Al , ... , Ak_1) , (121)
k k/ 7rt \ 7ri \ Qk(ajal,... , ak) = jj a - a - 2 I - y I a - Aj + 2 I . j=1 / \ / ( 122)
Q.E.D. Remark. The polynomial An(a1, • • • An+elP1, ... , µn_,) has a total degree n(n - 1) + 2 (n - 1)(n - 2). Being antisymmetric with respect to a1,... , an_1 it should contain IIi<j (ai - a1) as multiplier. Lemma 2 gives the values of this polynomial at n2 - c2 points, defining it uniquely. Lemma 3. Consider the functions (An)n, (An)n, (An 1)n, (On)n whose range of values is hi ®... ® hen (0° is defined as On = E(µj - Aj - 7ri)An). This statement means that the first of these functions belongs to a singlet subspace of hi ®... ® hen while the others belong to triplet subspaces; more precisely (An)n E'3 = (A0)nE+ = (An)nE = 0 ; (An)n F'3 = C(An)n , C = ±1 ; (An)E'3 = 0 (An ) n Et = 0 ; (An )nE:F = (An)n (An)n Fit = (An )n .
Proof. The equation (An )E3 = 0 is obvious . (An)nE+ = 0. Equations (113), (117) imply (An)n F+ = (X(l))n
Let us prove that
,
where n+1
Xnl)(al.... an_11A,... )An
+11/1 ,...
i
,Pn-1) = E11 k=1 Ilk Ak - Al
n-1
x fJ ( ak - µ 1 +
1ri)An( a,,
... , Cin
-1IA 1,
1=1 ak, ... , An+1I /11) ... , pn-1 , Ak) •
(123)
106 Form Factors in Completely Integrable Models of Quantum Field Theory
The poles at the points At = al in RHS of Eq. (123) are obviously fictitious, hence X(l) is a polynomial of degree n(n - 1) + n-12n-2 -1. It can be shown that X,(,1) satisfy the same recurrent relation as 0,1„ moreover X(1) = 0. Consequently X,(1) = 0 for every n. Equation (On)nE- = 0 can be proven in a similar way. Equations (An)nE3 = c(On)n are obvious. Let us prove any one of the remaining equations , for example, (,&n)nE+ = (On )n. From Eqs. (113), (117) it follows that (On)nE+ _ (Xn2))n ,
where
Xn2)(al, ... , an-1 IA) ... 3 An+1 Iµ1 ) ... , pn-1) n+1 1
_ IT 11(Ak - µl + ii) k=1 10k 1 n+1 n-1 l x [(Eai-Eµ.i+7rin) -2Ak]l j=1 j=1 x On(Cr1i ... , CYn- 21k , ...
,k,.•.
, An
+ 11p 1, ...
, j.Ln_1, Ak)
It can be shown that Xn2) satisfies the same recurrent relation as On and its degree is equal to (n-1Z(n-2) + n(n - 1). These observations together with Xi 2) = 1 imply X,(,2) = Ott for every n. Proofs of the other equations are quite similar. Q.E.D. Let us present explicit formulae for the form factors:
2n fµv (Q 1 , ... /32n ) = M2dn 11((Qi - Qi)
(eli - (-1)va-1; )
) ii<j 9 =1 x [n,1(( 1)n)exp (-
xexp (2^/3i1
1
E/i
- l-1)µ n,-1\(^n)n)
Form Factors in SU(2)-Invariant Thirring Model
fµ(/31, ...
107
,82n ) = Mdn II S( Pi - #j )
i<j x [ n1(()n)exP (_./3, )
(-1)µ0n,-1((On)n)
x exp G 1:0j) ] fN(F'1,•••, ,2n)=Mdnfl((/i-/j) a=1,2 i <j
x{
[ n1(()n)exP ( -2 1:,6j ) - (-1)µ0n,-1((& )++)
x exp
1 _1 +(-1) a [ 2_aj) \ 2
l
n,1 ((;) n)exP
-(-1)µ n,- 1((on1)n ) x exp (
,8j
(124)
i3)]
where 7ri)kexp(- 121 ) (()3) = cs0exp ^-10o sin2 z(f + 2 ksh Ir k ch 2 } 0 Zak ak) 1 1 00 c = 204 r-1 4 exp sh 4 exp (- z dk ( ) k sh^ irk ch Z
d = (2 7rc) -1,
, ((3)((/3 - iri) _
- 1 (R +
T)
.
Let us list the main properties of these form factors. The form factors fµ,,, fµ satisfy Eqs. (13), (14), (16). This fact is a corollary of the connection with SG model for ^ -+ oo. Form factors fµ, fµ also satisfy Eqs. (13), (14), (16) since f° = fµE°,a = 1,2 and the S-matrix is SU(2)invariant. Lemma 3 implies that fµ„ (Nl, ... 082p) creates a singlet while f;, ()1, ... , Q2n) creates a triplet. From the connection with SG for oo it follows that
fµi, = fi',i , fo
(E c h /3j)
= fi ( >
sh Qj)
.
Operators P. = f Tµodx, Q° = f jo dx1 are the operators of energy-
108
Form Factors in Completely Integrable Models of Quantum Field Theory
momentum and charges: PpZ.,.(Nn) ...
ZF1()31)Iph)
n
=
ML(eOj
+ (_1)"e- "i)Z*,. ( an ) ... Z,*1(al)Iph) ,
j=1
QaZf ,.( fln) n -
... ZZ1 ( 01)Iph)
D0, ) file,.(Nn)...z:1 (/3)...Z1 ( N1)Iph) j=1
Formulae for norms similar to those given in Sec . 4 can also be obtained, for example ((An)n, (An)n) E (On(a1,...
,'j+.))2
11 ... 2n}
x p,q
(Nip _
R
^9
) )(Nip
^^9 + 7rZ
(125)
8 FORM FACTORS IN 0(3)-NONLINEAR o-MODEL
In this section we present the formulae for form factors in NLS. It will be convenient to realize the group 0(3) such that it preserves the bilinear form -i')i +6 71- i - or/o the corresponding indices being denoted as w = 0, ±1. The connection with the usual realization of 0(3) is established by similitude with the matrix
1 0 i U= 72 ( 0 -' 0 1 0 -i Operators Zw, Z'° are defined by Zw=
Uw ZE
, Z'L=
UE Z f.
The S-matrix in this realization is given by Swis
(N) = (^ + 7fi)(fl - tai)
- 27ri (/3 -
7ri)
{(# - 7ri)Nbwl bw2
6wi bws' + 27ri,13(-
1 )w l
+ wa6w, - w
2b w l-w2 S 1 (126)
It is well-known that the NLS S-matrix can be obtained from the ITM one via the fusion procedure [50, 51]. To avoid confusion of terms, we denote 1 09
110
Form Factors in Completely Integrable Models of Quantum Field Theory
in this section the ITM S-matrix by cursive letters SEi E2. The equation for fusion procedure reads S. -Iw' 1,W2
E C CZ e„
w w^
(/3) =
P11L
- ai )SL„ Ls (Q) 2P£,'£,S„'L',(/3 1 4 2 ' 4
) SE^,c,g IN + 7fi )Ptii`'Pw;`4 , (127) X SL,1E,(F' g '21C3
where P = EP, P being the projection operator onto the symmetric subtt space of C2 ® C2, and E identifying this space with C3, P = Pt. The following consequence of the Yang-Baxter equation is of great importance for the fusion procedure:
L1 , 3 E1 , C2 _
f2,£g
SL^ ,es ( li + ^^ )Sf i ,Eg (^) PCI,t,
E1 , E2 PC
El e E f (N)SC1,Lg
i ,f^ ^e1,£g
(/3 + 7ri) . (128)
Consider the form factor of any operator in ITM, f (/31 It satisfies the equations q
f(Ql
,
...
el,ei+1 , N2n) L1,... ,£ 2.SC !,C#.
fs +1)
Yi{•1,, i) • • • , N2n)£1,... , C^+l,C^ ,... C2,.
/^= f (F'1,
f(al, ..
q
(Nt -
...' 2n ) E1 ... C2.•
, /32n + 21r ) ,,,... ,E2,. = (-1)
n f(Q2n1 111 ... )E2,, , E1 i...
Equation (127) means that the NLS particle can be interpreted as a triple part of a "bound state" of two ITM kinks whose rapidities are /3- a , a+ ti .
Clearly the term "bound state" must be understood here very carefully because the difference in the kinks' rapidities corresponds to their annihilation . Consider the following object: 7ri
7ri
7ri
7ri
f a1- 2 + ib,/3i+ 2ib,,62- 2 +ib,/32+ 2-i6,... /^ 7ri lln-
2
7ri
+ib,&
+2
-ib
(129) 41,e2 ,e3,e4 ,••• e2w-2,C2.
At this point the hope appears that the limit of (129) for b --+ 0 is related to the NLS form factor. This limit is singular because ITM form factors have simple poles at the point /32j = 132j-1 + 'r'. As will be clear from what follows, the expression ( 129) increases as 6-n +1 for 6 -> 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 (^e2i,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.
Form Factors in Completely Integrable Models of Quantum Field Theory
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
\^i - 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:
C
a -
2t + ib I
b-.0
Form Factors in 0 (3)-Nonlinear a-Model
113
where po(x) is the function
a a l a l 1 a _ a 1 (00 (a) = T7_ 2xil r( \2xil r 2 + 2ai sha ' 0 2 2af ), which has no singularities on real axis. Hence 2n
Hp(
a -/3)-f
P=1
=
n
j=1
a1 -L A0 ( Q•)2
6(Ce
11 Ph(a -Qj) E (a-Qk)2+b2 11 (Qk -Qj)(2a n
j=1
n
/^
j$k
k=1
-Qj -Qk)
Thus
_
n
1
°°
n
1
da ri
1 x H )`4i(aIQ=1,... Q:nIQjI,... ,Q^,.) j#k (Qk - Qj)(2a - flk -,8j
(131)
The poles at the points a = z (Qk + Qj) are made artificially; as usual in such cases it makes no difference how to understand them. We can suppose, for example , that all Pk are slightly moved to the upper half-plane . Integrals in (131 ) behave as 6'1 for b -> 0. Using the formula lim b = irb(a) 6-o a2 + b2
one obtains nn
%ij pb -1 E
k=lp^k
sll (Qk - Qp)(Qk - )3p )
exp(n - 2j - 1)13k
x Ai(Qkl/il,... ,Q,LIQ;3,... ,Qjn) . Hence
det II j II = 6-n+ 1 det 11(2t'1'2((2))ij 11, (132) where 21 ( 2) is a n x (n - 1) matrix with the following matrix elements:
1= A i( Q;IQ;1,... ,I3 J13,,... ,Q;.) .
114
Form Factors in Completely Integrable Models of Quantum Field Theory
2Q) is a (n - 1) x n matrix with the following matrix elements: 20) = exp((n -2j - 1)9j)fl poi
1 -i3 )sh(Qi - Qp)
Recall that the set {/311, /2, ... , /3in } coincides with the set {,(31 - T" 61 + 2 , ... , Qn + 2 }. Due to a well-known theorem a determinant of the kind (132) can be expanded into the sum n
det 112((l)%(2) 11 = det II2(kl)II det II2((2)II k=1
where 2(k1) is a minor of 20) with the k-th column omitted and 2((k2) is a minor of 2((2) with the k-th row omitted. The determinant of 2(k1) can be evaluated explicitly:
det IIQt(1)II-(2) = 1 (1 IIsh(/3: /3)exP ,) 11 ( /3i-fli) > fls i<j i
(let 1 II2(k2 )IIee
= ;n (ai - ai )
k 1
jok
Notice now that if we consider the form factor f3 = i (fo - f1) the transformation On , -1 would change to On j and for On 1 (On) we would undoubtedly obtain Onj(An)
8 -n+1
/3:t
- 7ri
))
n sll(Qi i<j
exp
n
x II (Q1- ^3j) II (/ - Qj) i<j k=1 i<j
det II21k2)Ile-Pk
i,j$k
According to Theorem 3 of Sec. 6 exp (_Eii) eIj) ¢n,1(,&n) _ = exp ( 2
r , fl; ) (^e-flj') On,-1(1&n) ,.
(/3)
Form Factors in 0(3)- Nonlinear or-Model
115
hence for 6 0
I
n /q Wi - 16i)
k-1
e-13i)
detII`^lk2)Ileflk
i,i #k
E
(Qi1 _ ^^)
k=1 "<j.
det II21k2)IIe-fl. (^ ePi ) . ( 133 )
Notice now that det 1121(2) 11 are polynomials , and the exponent exp(,Q,) are linearly independent over the field of rational functions. Hence the following statement is valid. Lemma 1 . Consider a set ,Q1, ... ,,On . Let {Q! ,Q! } U {,Q! Q3. } be an arbitrary partition of the set {a1 - 2, 01 + #2 - 2,,132 + 2, ... , Nn + 2}. Consider now (n - 1 ) x (n - 1) matrices II ( k2)IIi.i with the following matrix elements:
II` k2)Ilii = Ai(QiIYj,... a,nla,,,... ,,8;,,) , j 0 k where Ai are given by (118). The following identity holds for every k, 1: / 1 det II2(2)II 1 = X det II2k2)II i #k (Qi -,(3i) - ,3j)
i^#r(,(3i
i
k
The proof of Lemma 1 follows immediately from (133). It is also possible to give an algebraic proof but it is very bulky. Let us denote the set,131 i ... ,6n by B and the sets ,Q! Q! ;,l3! ,l3'• by B', B2,B'UB'=B'=(B-2)U(B+ 2),(B+a {Qi +a}).Fixa partition of B' into Bi and B2; it induces the partition of B into four subsets B = B+UB-UB°UB according to the rule,0i E B+ if,Qi+ z „Qi - 2 E B2; Qi EB- ifQi+ 2,6i- ME BI ; )3j EB°if/3i+ jEB',Qi - a EB'; ,13i E B° if Qi + z` E B,/31 - a E Bi B. The function X introduced in Lemma 1 is a function of B-, B°, B+, B : X(B-IB°IB IB+). Evidently n(B+) = n(B-). The following statement holds. Lemma 2 . The function X(B-IB°IB IB+) vanishes if B° # 0. The function X (B- I BO IOI B+) contains the multiplier (except for the case B° _
116
Form Factors in Completely Integrable Models of Quantum Field Theory
0,n(B+) = n (B-) = 1):
II
II (Ni - /3j)2(ai -
aj
11 (Ni - aj - iri)
#(EB+ P, EB°
#iEB + #jEB x
II
- iri)2
II II
(Ni -,8j
+ ri)
.
l3;EB- #,,EB°
Proof. According to (118)
Ai ( a I B 1 I B2)
= 1a-N/- 2
P'EBi \
+
II Q'EB,
(
J
Ti a_13 ' +T )
WGi( a-7riIB2-7ri)
Q i( a IB)
.
(134)
According to Lemma 1, one of n realizations can be chosen for X(B-IB°IB IB+) : 1
X(B-IB°IB IB+) = detlI2t 2>II i
II
(13
-Qj )
i,j#k
Suppose B # 0, i.e .
B contains at least one element 6j.
Take for
X (B- I B- I WO I B+) a realization with k j then 2(k21 contains the column A(aj I Bi I B2). As it follows from (134) this column vanishes. Now let ai E B-,8j E B+. Consider the realization of X with k # i, j. For the column corresponding to ,8i one has
A t( Qi I B' IB2)
=
ll(li - a -)($i - a-
xQl +
( /3i
- 7ri)(f3i - a° - ri)
+ - ri + - 3ri ° 3ri
-riIB 2,B 2 ,B - 2
jj( ,ai - Q+)(ai - a+ + ri)(ai - /3° + ri) ,
( ( ri ri ° /^ T2) i xQl1NiIB-+ 2,B-- 2,B + -
where II(ai - a-) II \(ai - ak), etc. Clearly, the first term disappears PrEB-
(there is ,8i among a-), and hence II(ai -a +)(ai- a ++ri)(ai - a °+ri) can
Form Factors in 0(8)-Nonlinear a-Model
117
be extracted from Al(/3iIBiIBZ). Similarly ll(,3j -/3-)(#j -/3- -7ri)(/3j ,Q° - 7ri ) can be extracted from the columns corresponding to /3g. Thus X contains as a multiplier (/3i -Qj)2 ( #i -/3, -7ri ) 2. The cases /3i E B /3j E B° and /3i E B+, /31 E B° can be treated in a similar ways. Q.E.D. Lemma 2 implies the possibility of introducing the polynomial ( except for the case n(B°) = 0, n ( B+) = n(B-) = 1)
O(B
I B° I B+) = X (B I B ° IOIB +) (/3+ -
/3 - - 7ri ) 2(/3+ - /3-)2(/3+ - /30 - 7ri )(/-
-
/30
- Sri) .
For n ( B+) = n(B-) = 1, n(B°) = 0, the function 0 is equal to
For the constructive definition of q see Definition 2 to be given later. Let us return to formulae (130). We have to understand what the operation ( ) n turns into after the partition of B' into B - T", B + z and the projection onto spaces symmetric with respect to pairs of particles. The operator B(v) defined by A(a)
D(o-)/ Solo-Qi)... Sozn(O-'zn)
(135)
is of fundamental importance for the definition of the operation ( )n. We have to calculate B(aI/31 - a , Nl + 2i , • • • Nn - 2 , fln + 2 x P1,2 ... P2n-1, 2n, where P2i-1,2i is the operator P acting from Hi ^_ C3 to h2i_1® h2i ^, (C2 ® V. Multiply ( 135) from the left by IIP2i-1,2i and use the usual trick : present P2i - 1,2i as Pzi , zi-1P2i,zi-1 and move all P2i _ 1,2i to the left using the equation (see (128))
So,zi-1
C
7ri1
= P2i,2i-1So,zi
(
So,zi
Q - Qi +
(
0, -
Qi
7ri ^ Q - Qi - 2 Pzi-l,zi
- 2
(
)
P zi,zi - 1 P2i , 2i- 1So,2i ^ - /^ i
S0,2i-1
(
17 - /8i
+ 2
7ri1 ( -
2
7ri
0, Ni + 2 J So,zi -1 -
118 Form Factors in Completely Integrable Models of Quantum Field Theory
Hence 50,1(-fi1+
)&
2f
(c -'8 1-
J
...
7ri S0,2n-1 1 -Nn+2/50,2,, (^-Nn-2) rs a 7ri x JJ P2i-1,2i = fJ P2i-1,2iS0,1 - Nl - 2 ...
So,n
C
7ri l - 2
O - Nn
where So,i is an operator acting in ho ® Hi ^-- C2 ® C3 in the following way: asp,i( tT) = 1 (o - 7ri + Trio°Si) , tr-27ri
where tr° are the Pauli matrices acting in ho, Si are the spin-1 generators of SU(2) acting in Hi:
53 ...
1
10 0 00 0 00-1
Sl-
v
010 101 010
1 0 i0 S2^
7
-t Oi 0-i0
Define the operators
C(o)
D(o) /
'(a,) =
S 0,1(Q - ,Q1)
... SO,n(0,
The operators satisfy the same commutation relations as A(o'), B(cr), C(tr), D(tr), which follows from Sa,b(Q1 - D2)Sa,i(Q1)Sb,i(Q2) = Sb,i(O2)Sa,i(Q1)Sa,b(Q1 - 172)
The following relation holds: Si,1(Qi-fij)asa,1( 0-/ii)asa , i(Q-Ni ) = ass,ilk-Ni ) asa,1 (^-N1)Si,7lNi-Qj) ,
(136) where S is the NLS S-matrix ( 126). It can be concluded from (136) that B(ojf1, ... 'A , N41, • • • ,,8.)Si,i+l(Ni - A+ 1) = Si,i+l (fii - Qi+l)B(i7I )3l, ... , )3i+1, N{, ... , Qn)
Form Factors in 0(3)-Nonlinear a-Model
119
As before, we suppose Hi and Hi+1 to change places if /3i and Oi+l change places. Consider some function F(B' IBZ) which is invariant under independent permutations of the rapidities comprising Bi and B. The operation ()n was defined as follows: (F)n(j1, ...
, Q2n)
=
F(B1 IB2)
1
II
B'=BiuB^ 01EB i Pii fl9i T fl EBz
(137)
x (01 lI B(a;) . A^ EB2'
Partition rapidities into pairs ,0'' = 13 + j, Q2j-1 = f3 - zi . This partition induces, as was explained above, the ppartition of B into B-, B°, WO, B+, so we can define a function F(B_ IB°IB IB+) through F(B' IB'2). After B' is partitioned into pairs and projected onto symmetric subspaces , Eq. (137) transforms to
F(B IBOI- IB+) B=B-uBouB uB+
1
X
7ri )(a+ -
X (Q+ X X
- /3° -
7r i)(Q/^ +
+7r i)(a+
f/
- 3
-
/3-)2
1
°)(/l +
- N
°)(a+
- N ° + 7ri)
1 (/3-
-
/3°
- 7ri )(/ - -
,8 0 ) 1
11
(N- - i3°)(h3- - /.' ° - 7ri)( h'° - h'° - 7ri) 1
X (Q ° - Q ° - 7ri) (/3° - R 0)2
x (roll II B (a °)B(a+) B(/3 + + 7ri )B(Q ° + 7ri) , where ((Oil = 110 ei_i, and ei,i„(w = 0,±1) generate the base in Hi. The fact may give rise to misunderstanding that no singularities appear. As a matter of fact, the operation ( ))n for ITM does have singularities at the points /32J = /32j-1 + in. What has become of them? This contradiction
Form Factors in Completely Integrable Models of Quantum Field Theory
120
has a simple resolution: the residues of ( )„ at its poles belong to skewsymmetric subspaces, but we have projected it onto the tensor product of symmetric subspaces. Now we have to substitute the polynomial O(B- I B° I B+) for F:
On-1((An)) n6->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
Form Factors in Completely Integrable Models of Quantum Field Theory
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(^i - 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)
,
for1
0 0 (a-I 0 I)3+)
7r1
- ,Q--7ri
Let us discuss an important point . ITM form factors depend on even numbers of rapidities . NLS form factors obtained from the ITM ones depend on arbitrary numbers of rapidities since an NLS particle is created
123
Form Factors in 0(3)-Nonlinear or-Model
by two ITM kinks. In the space of states of NLS one can introduce the operator of charge conjugation, postulating its properties as follows:
CIph) = I ph) , g
CZw (Q)C - 1
= - Z,,
( )3)
Section 2 implies that even and odd parts of an operator are mutually independent which means that both parts are local operators . That is why the form factors of one local operator in ITM create the form factors of two operators in NLS , one of which is provided with even particle form factors, while the other with odd ones. Hence the form factors fµ, fµ., create four sets of form factors in NLS: i
a liM b-n+1 a i
fµ(^31i... ,h'n) = h_n
f(P1l...n) I
2 " ` +i6 P ei-1=Q;- z
x P1,2 ... P2n-1,2n IJ th2(a, - ,Qj) , n =- O(mod 2) i<j
f a (^1^ ...
p Nn)
= liM 6-n+l a /^, l (NI
f
6^0
.. .r
X P1,2 ... P2n-1,2n (^ sh,Oj) /^ qq fµv(F31,... ,
,(3n)
i
i JI
l n th22(ai
'+i6
- i3) ,
n = 1(mod 2) ,
<j
l
i iM -n+1 fµ= 6^0 6 ... p1"
"'201
T' +ib
X P1,2 ... P2n-1,2n ll th2 2(,Qi q q f(al, ...
- /3j )
,
n O(mod 2) ,
i<j
+1f
00 p1, • • • , 02n)) .Nn ) = liM 6-n ^c_n P2; =P;+ ^ - : 6
'i +ib
.,, 2 1 E(ch)Qj )2 - (Esh pj )2 - 1 XP1,2 ... P2n-1,2n 11th 2 (Ni - aj (Esh /3j)2 i<J
n - 1(mod 2) . We hope that the use of the same notations for the NLS form factors in the RHS's and for the ITM form factors in the LHS's will not cause any misunderstanding. Additional multipliers Hth2(,(3i - ,3j) and (Ech,Qj )2 (Esh /3j )2 -1 are introduced in order that form factors possess proper structure of poles; obviously they do not influence the validity of Eqs. (141), (142).
124
Form Factors in Completely Integrable Models of Quantum Field Theory
The constructive definitions of f;,, fa, fµv, f are
(Ni-Qj +7ri J3(t1,...,n)=2- Mllth22(Qi-13j) i<j ^i-Qj )
x
E(e,6j - (- 1)µe- 3i )(({ ^° }))n
, n
O( mod 2) ,
^°(B- I B°I B+) _ O°(B- I B°I B+) (E(a+ -,3-) - ini( n(B +) + n(B °))) f/t(,8 ,...
fn) = 2-
th21 (Qi
VMII i
x {((¢1))n + (_1 )
-,13j)
<1
+7fi
Ni
13
aj
a ((10-'}))n} , n = 0(mod 2 ), a = 1, 2,
f3(F^1,...^n)=2-*'II th2(Qi-^1) ^'^t^ (({^°}))n i<j
)
n=1(mod2),
fa(N1 ,...
On)
= 2
^
i
a
^s-
th22(Pi-
\ `
j+^i
13i - 13j J
x {(({01}))n + (- 1)a(({0-1} )) n} , n = 1(mod 2 ) ,
fµv(31,. . .
x
pp F' n) =
(E(efii
2-
^
ai - flj +ai
M 2 ^ 21 th 2(Nqqi
fli - Qj
i
-
a = 1, 2 ,
(r(eQi - (We L^
J
(( -Qi)) {0
l
° }))n
n =O(mod 2) ,
f (fl,. ... ,.3n) = 2-g'-M2 th2 2(1i -,3j) CQi R' Nj fl' ^tl (({0°}))n i<j 11
x I(Ech,Qj)2-(Eshfj)
2
-1
]
,
n= 1(mod2),n>3.
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 /^i /^i+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.
Form Factors in Completely Integrable Models of Quantum Field Theory
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.
»n(-*+7) we transform det M t ( B _ U /?|B°|S+ U (/? + Tt')) to the following form: its left upper (n - 1) x (n — 1) corner coincides with Mk(B~\B°\B+), its two last rows of the first n — 1 columns are equal to zero, its right-lower corner is a 2 x 2 matrix n(/S - p~)(p -P~ - wi)(0 -p" - xi) -n(/9 - p~ + *i)(/3 -p-)(p-p°)
n(/S - 0+ + iri)(/9 - /3+ + 2«ri)(0 - 0° + 2iri) _n(/9 - /S+X/5 - (5+ + »iX/9 - 4° + wi)
hS-
^ . J n ( / 5 - / 3 - ) ( / 3 - / 3 - -iri)
( /3 + y J 11(0 - 0+ + 2*t)(0 _ /3° + 2iri)
X(/J -
/»" - iri) - (p + y j n(*j - p°)
x(p - p+ + «•) - f/j - y ) n(/j - /s+)
x(P-P~)(P-p-
+ iri)
x(/S - 0+ + irO(/J - /S° + »i)
Form Factors in 0(3)-Nonlinear o-Model
129
The determinant of the last matrix is equal to
ll(a-a-)(8-,Q++in){11(#-a°+ini)(,8-a°)(,a-a-+in ) (,a - a +) - ll(a - ao - iri)(a - a° + 2ai)(Q - /3- - 7ri)(/3 - a+ + 2ai)}
ri
Q.E.D. Remark . It should be noted that the recurrent relation ( 144) has appeared in another branch of the theory of completely integrable models, which is Korepin 's approach to Green 's functions (see, for example review [24]). We hope that the explicit solution of these relations as provided by function 0 will lead to progress in the framework of this method. We approach the calculation of residues of the form factors at the points Qn = /3j + in . The only proof we know of is rather cumbersome, so the reader who is interested in the result rather than the techniques may omit it. Theorem 2 . Form factors fµ, fa, fµ,,, f considered as functions of ,Qn have in the strip 0 < Im/3n < 21r only simple poles at the points ,Qn = /3j + in, j < n - 1, the corresponding residues being
2 7ri res f X
X
N1
...
h' n wl, .. ,wn = (-1 )
f(/ i,... ,/3j,... bwi
qq iNn-
bw .,, -wi
q
b" i-1
Swn-i,wj { w1 wn-1 wn-1, r1 (Nn-1 S,w q -.. Sr'7 wi r1 ,w (^9 S11 h^l) wi,wj -1
S
+l, rn - j -s
wj+l,wj
(F'j+1 - flj)
wj+1 wn-1 (Nj - Nj-1)bwj+1 . . bwn-1 }
Nn=/3j+ 7ri.
(149)
In this formula f can be replaced by fµ, fa, fµ,,. Proof. Consider any form factor, for example, f3(31, ... , /3n). Evidently it is enough to prove (149) for j = 1; other j can be considered using the symmetry (141). Let us introduce the notation a 4 1 F(/31,... ,/3n) = () Hcth(fli_/3i)f3(/3i,... ,/3) . (150) i<j The function F is regular at the point Nn 3= /31+iri. Eq. (150) in conformity to F is equivalent to
F(,81,... ^Qn) ^Pn=P1+ai = F(/2,.... ,,3n-, )singn,l qq
X {U(Nl 132, /33, ... r fln-1) - 1}
Form Factors in Completely Integrable Models of Quantum Field Theory
130
where singn , l = en,-1 ® ee1,1 + en ,l ® el,_1/- en,o ® el,o RR /q //^^ U(/11, 2, ... , fan-1) = Sn-1,1(/n-1 - 01) ... S2,1(N2 - N1)
Later on we shall denote U(/31I,Q2i ... , /3n-1) by U(/31) if the context does not lead to misunderstanding. We shall extract explicitly the matrix structure of U(/31) with respect to H1, writing as Uwi (13l). The proof is divided into two parts. Part I. It is sufficient to prove the identity
F(/31i ...
,i3n) wi,w ,.IPn =^1}xi = (- 1)wnbw^i - w
;F(32 .... ,i3n-1) (151)
61) - bwi } X {Uyw, (i3
for (wl, wn) = (0, 1), (-1, 0), (-1, 1). Proof of Part I. First, the formula
F (81,...
,6n )wl,...
,w,
=
F (81,
(152)
...
implies that if we prove Eq. (151 ) for (wl, wn) = (0,1), (-1, 0), it will also be proven automatically for (w1i wn) = (0, -1 ), ( 1, 0). So only three components remain : ( w1) wn ) = (-1, -1), (1, 1), ( 0, 0). Using once more Eq. (152) one makes sure that it is sufficient to consider (w1, wn) = (1, 1), (0, 0). Thus our purpose is to prove Eq. (151 ) for (w1i wn) = (1, 1), (0, 0) assuming its validity for
(w1, wn ) = (0, 1), (-1, 0), (-1,1), (0, -1), (1, 0), (1, -1) . Consider the case (w1i wn) = (1, 1). Rewrite the identity in question extracting explicitly the indices which correspond to the particle ,Q2:
F(/31, #2. ...
, Nn)1,w2i...
,1I/3*= P1+,r i
= F (#2. ... ) w^
W/ w' (#
X Uwi
(/31)S1,w2 2
2
- Nl) ,
where U(Q1) = U(j31I(33i... „Qn-1), the second term in brackets has disappeared since 61 1 = 0. Three possibilities exist : w2 = 0, -1 , 1. Let us consider these possibilities in turn.
131
Form Factors in 0(3)-Nonlinear o-Model = 0. The identity in question is rewritten as follows: 1. Let w2 #2
F (#2 . ... )0S1,O (Q2 - N1) F(/31 , ,. . , /3n )1,o ,... x Ui 1(Q1)+F(132,...)1Si' (f32 -/31)Uo 1(#l) Using (141), (142) write a sequence of equalities:
F(/31,/32,...
, /3n), ,0,...,1
[SO1(f1 -
Q2)]- 1{F(N2,#1,.../^ Nn)1,0,...,1 /32)1 -1
=?p - S0,1 (61 - 132)F(j31, N2, ... ) Nn)0,1,... ,1} = [S0,1(h'1 -
x {F(13,, • • • ,,Qn, Q2
^21ri )o,... ,,1,1 +/
,^ - So'l(f31 -, 2)F(Q1,Q2,...
_ [So' ( , -
132)]
Nn)0,1,...,1}
{F(/3,,... , 02 + 2ri, Nn)o,... ,1,1Si'
( N2
-
'n
+ 27ri )
13 - Soo: (#1 - 62)F(Pi,N2) ... , #n)0,1,..., 1} .
We suppose the value of F(#,,.. . /n)o,...,1 at the point is known and is given by Eq. (151). Hence L So; 1
F(/31, N2,
P1- /32)1 -1
x {51;1(132 - /31 +7ri)F(... 12 +27ri)w,S,yi 1 2(fl2 - /31 + 27ri)Ul i(N1) q
w^
-50,1 ( fl, - #2)F( ,82,.
F(#2, ,63,. .. x
w'2
N2 - /31 )} w^Uw1 50,1 (/^
,/3n_1)w2,[S0,1( /3, - $2)
s- 1;1 ( /32 - 3, + 2
7ri)Ul
1- 1 {S1,1 (32 - /31 +irt)
l(a1)bi 2 - S;^(al -
/ 2)
So;°(
32
- al)
x Ui 1()3l)bi' - S;i(f1 - l2)S' (f2 - Nl)Uo 1(/tl)bo'} Using the unitarity and crossing symmetry properties, one obtains the required result. 2. Let w2 = -1. Using Eqs. (141), (142 ) one obtains
O'O -1 x {-F(/31, /32, ... , /3n)o ,o,...,1 S-1,l(31 - /32) - S-11(/31 - /32) 1,1(/32 - /3n + 27ri) x F(/31, /32, ... , in)-1,1,... ,1 + S1,1
x F(/3,,... ,/32+27ri,/3n)-1,...,1,1} . (153) Using ( 153) and supposing that F(,31, ... , /3n)O,... ,, and F(,31,... /3n)_1,,.. ,1 for On = /31 + 7ri are known , one obtains Eq. (151).
132
Form Factors in Completely Integrable Models of Quantum Field Theory 3. Let w2 = 1. The following identity holds: / q //^^F031,#2, ... , Qn)1,1,... ,1 = Sl,l1,32 - 3j)Sj,l (N2 '61 + 27ri) x F(,Q1, N3, • • • , 92
+ 27ri, Qn )1,w3,... ,1,1 .
(154)
If w3 = 0, -1 we calculate (154) at the point ,3n = ,131 +7ri using the results of the previous subsections. If w3 = 1 we use (154) one more time, etc., until we get w = 0 or -1 for the first time (we shall reach w = 0 or -1 without fail because Ewe = 0). Let us outline the proof for (w1, wn) = (0, 0). We assume Eq. (151) to be proven for (w1, wn) = (0, 1), (1, 0), (-1, 0), (0, -1), (1, -1), (-1,1) and consequently, according to the proof given above, (wl, wn) = (1, 1), (-1, -1). We consider again the three possibilities w2 = 0, 1, -1 in turn. /1. Let w2 =//^^0. Using Eqs. (141), (142) lone obtains qq F(fl1, f32, ... ,N n)o,o,... ,o = [SO'/,1(N1 - R2)]-1{-5-1,1(131 - N2) R
x F(h31, Q2, ... to
1,-1/
,8n)-1,1,...,O - S-1,1 \/31 - j32)F(31, #2,... , fln)1,-1,... ,o
+ S1'0(/32 - )3n +
27r i)F(,l1,
... , #2 + 27ri, i3n)-1,... ,1,0
of
q + S110(/ 2 - Nn + 27ri)F(,31, ... ,,82 + 27ri, 8n)-1,... ,o,l} .
Substituting F(,Q1, ... , Nn)wl,.•. ,w,. for ,fan = /3 + 7ri, (wl, wn) (1, 0), (-1,1), one obtains Eq. (151) for w1 = w2 = wn = 0.
= (-1, 0),
2. Let W2 = 1. Using Eqs. (141), (142) one obtains //^^ F(#,,,62,.. . , /3n)o,1,... ,o = [S/;o (/31 - /32)] ' {-So1// (131 -N2) x F031,,32,. .. , Nn)1,o,... ,o + S00:00(#2 + Nn + 27ri)
x F(,Q1, ... ,132 +27ri,,fan)1,..., 0,0 + F(931, ...
02 + 27ri, ,(3n) 1,... ,-1,1 So,o' 1(32 - Qn + 27ri)
+ F(,81,... , #2 + 27ri, /3n)1,... ,1,- 1So,o 1(/32 -
,8n
+ 27ri)}
Substituting F(,Q1, ... „ l3n)wl ... ,wn for /3n /31 + 7ri, (w1, wn) _ (1, 0), (1,1), (1, -1), one obtains Eq. (151) for w2 = 1, w1 = wn = 03. The case w2 = -1 is quite similar to the case w2 = 1. The proof of Part I is finished. Part 2. Function F(i31i ...
_ (-1,1), (0,1), (-1, 0).
, Nn)u 1
_,,
v,n
satisfies Eq. (151) for (w1, wn)
133
Form Factors in 0(3)- Nonlinear a-Model
Proof of Part 2 . At first, notice the following circumstance. Since S(f3 -
Ql)
= cistlA - a; + 7ri)c l
t
singl ,nA1 = singl,ncnA „ Cn where cl is the crossing matrix c acting in H1, tl is the transposition with respect to H1, A is an arbitrary matrix, one can change in (151): U(,811,62, ... A -1) -+ C1 Vt1 (Q1 I/32, ... , Qn-1)Cl
where V(1311/32 ,../^. ,Nn) q
= Sn,2(i31 - 02 + 7ri ) ... Sn,n-1 (31 - /3n-l + 7ri) .
The matrix S(/31 - /3j + vi) can be presented as follows: /^ q
N£1
i (- /3j + ) •Swi:
(/31 - F'j +7rt) ( W
// _ .r El>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
Form Factors in Completely Integrable Models of Quantum Field Theory
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 - 2^i
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)
Form Factors in Completely Integrable Models of Quantum Field Theory
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 - ^° + ^a)()/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)) .
Form Factors in Completely Integrable Models of Quantum Field Theory
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+)
Form Factors in Completely Integrable Models of Quantum Field Theory
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)
Form Factors in Completely Integrable Models of Quantum Field Theory
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
( ) ()
Form Factors in Completely Integrable Models of Quantum Field Theory
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( ^i)
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.
Form Factors in Completely Integrable Models of Quantum Field Theory
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<j
Let us now turn to (Fn)n. From the very definition it follows that E (OI fJB(,8jp)F(I3il,... ,fli.Ij..... . IQjn)
(Fn)n =
{1...2n}={il ...i,.}U{jl...jn}
X
11
1
(190)
(j
The factors IIB(,3j,)II(,Qi, - 8jq)-1 give only power contribution; that is why we have gonly to estimate the exponential behaviour of
F3(,li1) ... , /i„II'j1,...
,)3;n)
//^ = (E(/3 , - A, - 7ri)) F(Ni1, ... , ,3i,.
IQj1 ... ,,8jn) ,
where F is the determinant of an n x n matrix with matrix elements ' zn 2- 4Jdft ( a_/3i)Ai_i(aIfli1,... fli. INj1 7 ... , Pjn) j-1
xexp(n+1-2j)a, i>j 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
Form Factors in Completely Integrable Models of Quantum Field Theory
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 ^o(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
Asymptotics of Form Factors
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
Form Factors in Completely Integrable Models of Quantum Field Theory
154
The matrix elements of Dl and D4 have the following asymptotics for it -+ 00: D = ^zk
k+p vi k-p ( ^i 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 ,
Asymptotics of Form Factors
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
Form Factors in Completely Integrable Models of Quantum Field Theory
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
The determinant of Dl for example, is transformed to the determinant of a k x k matrix with the following elements: rj
(
aj
- A9 - 2 Qi( aj - 7ril{ll - 7ri, ...
µk
- Sri)
4 k ai +r (ai q+ Qi(ajI i,...,Ak),ij =1,...,k. 9=1
Recall that
Qk(alvl ,... ,vk
a - vy - 2z I ^ I a-v9+ 2Z I q
158
Form Factors in Completely Integrable Models of Quantum Field Theory
hence in the last row of the determinant one has (a3-Aq- 2t) q
C
ad-µy- 7ri 2)
(aj -Ag+2) (aj _pq+) q
Usual arguments which make use of the identity
2
cp(a - 2ai) _ sp(a) a 3
ai
2
show that the last row vanishes after the integrations in (191) over eel.... , ak are fulfilled. The same happens to the last row of det D2 after the integrations over ak+l, ... , a2k are fulfilled. Hence
detF = O( e-k2v01 k2 - 2 ) , p = 0 . (195) Now we have to collect all the above facts together. Consider Eq. (190). The operator B(r) can be written as follows: B(o) = Bl (rr)D2(r) + Al(r)B2(r) ,
where A1i B1 and D2, B2 are constructed via Eq. (111) through the first and the last 2k particles respectively. Let r - 0, then D2(r) - 1, B2(r) _ O(o-1), hence
B(r) = Bl(r) + O(,7-1) . (196) Let r - a, then
B(r) = B2(r) + O(v-1) ,
(197)
where r = r - O, B2(T) = B2(T I/I2k+1, ... , L) -
Consider a fixed term in Eq. (190). Let ,13i1, ... , Qik +p, )3il,
)3jk-p - 0
Nik+p +l, ... f I3i2k f Pik-p+1 f ... , '832k - v, then
2k
k+p k - p 1 2 _-2k2p 2 1I 1I
s,t.1 ^i• - Qls
2k
2k
1
a=1 t=1 07s 9=k^+^p+l ack^-^p+1 fl'9 - I3.
(198)
Asymptotics of Form Factors
159
Combining/^Eqs. (192, 195, 193, 194, 198, 197 ) one gets (F3)2k(Ql, • • • ,Q4k) =
2k'o.-le-ksa0.
p -kslrf(F+)k(Q1, • • • Q2k)
x (F )k(Q2k+1, ... A k) - (F )k(Ql, • • • ,Q2k) x (F+)k(Q2k+1, ... , Q4k)}(l + O(0'-1)) .
Substituting this in (188) and using the asymptotics ( 189) one obtains finally q1, 93(Q...
2a 3bc e a
,••• rNn)
x 9c(Qn+1,... ,+O(a-1)) & )(l The power contribution for n = 2k + 1 can be obtained in a similar way.
Consideration of the functions g1, g2, g involves no additional difficulties. Q.E.D. Let us present without proof a similar theorem for SG model. Theorem 2 . The functions g+, g- in SG model have the following asymptotics for Ql, ... , Qk ^' 0, Qk+1, • • • , Q2n - 0, o -' oo: 9+ (Ql , ... , Q2n) = O
9-(Qi,•••,Q2n)=0
exp
a 1 4^ Q - 40)
) a 1 exp -4fv-- / , 40
9+(Q1, ... , Q2n) = O (exp ( -2 min 9-(Q1, • • • , 02n) = 0
(
k- 1(mod2),
1,
/ a exp (_2min 11, ) v I I , k - O(mod 2) .
(199) The asymptotics in NLS can be obtained as a corollary of Theorem 1. Theorem 3 (Corollary of Theorem 1). Consider form factors f, fµ", fµ, fa in NLS . Introduce the functions 1 9a(#1,... ,Q2n) =(E chOj)
fl(#I)... ,Q2n)
2 9(Ql,... An) _ ChQj)
9(91,
... , #2n+ 1) = ((chj x
f (Ql, ...
f11(Ql,... ,Q2n) ,
)2
, #2n+ 1)
\ shflj 1 2 - 1)
-1
160
Form Factors in Completely Integrable Models of Quantum Field Theory
Let ,Ql,... , Qk ^' O,,Dk+l, ... , N2n ^' o. The following asymptotics hold: /^ 27r /p-^ 7 Eabc9b (N1, • • • , /3k)9c(Qk+1) ... , N2n) 9°(Q1, ... , f2n) 01
k=0(mod2) 9°(P1, ... ,N2n ) -
011
Eabcfb (P1, ... , fln)fc($k+1) ... , hl2n)
k - 1(mod 2) /^ 27r //^^1,... ,Q2n+1) ^' f°(/'
q1 Eabcfb(/31i... ,,6k) 9c( h^k+1 ,... ) ^2n+1)
k-1(mod2), ,,62n+1) ... 21r Q
Eabcgb qq q (Ni, ... , f k)fc(&+1, ... , ^2n+1)
k = 0(mod 2) ; 9(fl,
... , N2n) =
O(0' -2)
,
(200)
where ,13j =/3i - a. The form factors in NLS are expressed in terms of elementary functions; that is why Eqs. (200) can be nicely illustrated by explicit consideration of the few-particle form factors. One can get from the general formulae of Sec. 9 the following results: '$`N12 +7ri) el-1 - e-1,1) PA) = 1, 93(Pl,F'2) = 1th21312 R 4 8 2 )312(012 + 27ri)
f3()1, #2, Q3) =
1 11 th21 fli (/3ij + 7ri) 7r2{-(/313 + 27ri) 2 Qij (/3ij + 27ri)
16 i<j
x (eo,o,o - el,o,1, -el,o-1) +,Q12(eo,o,o - el,-l,o - e-1,1,o) +,Q23(eo, o,o - eo,l,-l - eo,-l,l)} , where lij = Ni -,3j. In the next section we shall need one more asymptotic formula. Consider the form factor fµ (/31, ... , N2n). Let /3 < ... < ,Q2n and denote ,6i+l-/3iby L. We have shown that fµ(01i ... ,82n) =E ( e'3i _(-1 ) Me-L;) 9°(Q1, ... ,Q2n ), where g° is invariant under simultaneous shift of rapidities, i.e., g° depends only on Al, 02, ... , 02n_1 . We are interested in the behaviour of Ilg°(/31i... , /32n)112 when all Ai's are large.
Asymptotics of Form Factors
161
Theorem 4. The function Ilga(Nl, ... „Q2n)112 in ITM is characterized by the following behaviour for n p
Ilga(h'1,... ,/32n
112
(
n-1
2p2 l exp
..' II
p=1
_x2^i )
II
O 2p
(201)
p=1
Sketch of the Proof. According to the remark made the end of Sec. 7 one can present IIg3(/31, ... , )32.) ll2 as follows:
IIg3(al , ... , /32n)112 -
x (F3(ai"... X /^
laj],... ,ij.))2
1 ,
/^
p,q (NJp - Niy )(^Jp - ^iq - 7rt)
where the function F3 is presented in Lemma 1 . The main difficulty in obtaining on estimation for (201 ) lies in the estimation of F3 (/311,... IQjI,... ). The method of estimating F3 is similar to that used in Theorem 1. One should separate , at first , the exponential contributions . Having done that , one should make sure that the integral over aj (j = 1, ... , n) is mainly concentrated on the segment ,62j_1 < aj < /32j (we supported the /3 's to be sufficiently separated ). Then the power contribution should be estimated . Corresponding calculations are very cumbersome and do not differ radically from those made in Theorem 1. We present without proof a similar Theorem for SG model. Theorem 5. The function Ilg±(,01, • • • 032n )112 in SG model behaves for Ai - ai+1 >> 1 as follows:
Ilg±(al, ...
, /32n
)11 2
fJ exp (_
2_1
p=1
-
L 2-1)
11 exp
p=1
Finally, for NLS the following estimations of
( (
- min 2p/ 2,
11fal1 2
and IIgaI12 hold.
(202)
162
Form Factors in Completely Integrable Models of Quantum Field Theory
Theorem 6 . The functions > 1 as follows: i+1
,6 - ai
I If ° II 2
and
I Ig° I I2 in NLS behave for
Di
2n-1
IIgd(p1, ... , #2n)112 = [
A;2
1 P=1
q 12n Ilf a (f1, ... , N2n = 2 IJ p=1
+l II2
O; 2
(203)
Sketch of the Proof. According to Eq. (181), IIf3(31,... be presented as follows:
, )11 2 13n
can
2 r2
IIf3(P1..... Pn )112 =
rj th41 (Pi i<j 2
(Pi - dj)2 (Pi - Pj)
+ P° - ,ri) ° - ° - , E (m°(B IB°IB+))2 I (P+o (P - P-)(P P)(P P-)(P P ri) X a_a-ua o ua+ T7 1 x 11(P+- P-+,ri)(P° - P° - 2,ri)(P° - P-- 2wi)(P+ - P-- 2ai)(P+- P° - 2,ri)
The main difficulty lies in the estimation of ^°(B-IB°IB+).
For these
determinants one can write formulae similar to (183), multiplying and dividing them by the Vandermond determinant II (,Qi - ,0j). When this is i<j
done the estimation is reduced to complicated but rather straightforward calculations.
10 CURRENT ALGEBRAS
The present section is devoted to the calculation of the singularities of commutators at the origin of coordinates. We consider the following operators: jµ in SG, jµ in ITM, jµ in NLS. Let us recall several formulae from Sec. 2. Suppose the operators O(x) O(y) are defined by the form factors fl, f2. Then there are two equivalent forms of matrix element of )x-ix(B)y (A IO(x)O(y )I B)+eix(A
S(AI A1) A=A1uA2UA3 B=B1uB2UB3
xS(A2UA3IA3)
00 .(C)=O
J dCf1(A1+i0ICUB2)S(CIA3UC)
x f2((C U A 2) - i01-91 ) exp(i(x - y)(,c(C) + i (B2 ) + ic(A2) + ic(A3))) (204) 1I B) , x O(A3i B3)S(B 31 1i2 U (A I0(x)O( y)I B)- eiK( A )x- iI(B)y = S(A IA 2) A=AIUA,UA3 B=BIuB2uB3 00
X S(A1UA 3IA3 ) .I(C)=O
J dCfl(A1-iOIC U B2)S(C UA' 316)
x f2((C U A 2) + i0i B 1 ) eXP(i(x - y)(K(C) + i(B1 ) + ic(A3) + ic(A))) O(A3i B3)S(B 3IB1 B3)S(B 2I B) . 163
(205)
164
Form Factors in Completely Integrable Models of Quantum Field Theory Lemma 2 of Sec. 2 states that
(AIO1(x)O2(y)I B)t = (A101(x)02(y)I B)To prove the local commutativity theorem we have written the convolution f [O(x), O(0)]dx in the form (A
l
J
[O i (x), 02(0)] exp(ix (A)x)'o(x) dx1 B )
= J (1A1O1(x)O2(0)I B>- - (A102(0)01(x)I B> +)e`(A)x,P -( x )dx + J ((1 101(X)02(0)1 B )+ - (A 1°2(0)O1(x)1 B) -) e"`(A)x
J fj(X1+i0Ic'UB2)S(CIA3U C)f2((CUA2)-a0IB1) x So- (ic(C) + c(B2) + K(A3) + ?(A4))do where o E ;j C E yEC-Y, 0_ is the Fourier transform of tp_ . Function tp_ (,c (C) + ...) decreases faster than an arbitrary power of a-° for o -• 00, if cp(x) vanishes with all its derivaties at the point x = 0. This is why it is sufficient to require that fl, f2 do not increase too fast for o -+ oo. Evidently the asymptotics (187), (199), (200) satisfy this requirement. But as we intend to study the singularity of the commutator at the origin of coordinates we have to use the functions tp with a different behaviour at the point x = 0. Here some new possibilities arise. Let us discuss one important circumstance which has been ignored until now. Dealing with the expressions (204), (205) one has to make sure that the integrals and the series in Eqs. (204, 205) are convergent. Actually, very rough estimation of the form factors guarantees the convergence. Consider, for example, the operators T,,,,, jµ in any model discussed in this book. The form factors of these operators can be presented as follows: f(B) = (po(B))K(pl(B))'g(B) ,
(207)
where ic,1 are certain integers (for example, rc = 1 = 1 for To,), the function g(B) satisfies the condition
g(B + o) = g(B) •
(208)
165
Current Algebras
Let us suppose that the function g allows the following estimation:
Ilg(A - in, C - 2 , B)II
< Cn(A)+n(B)+n(C) J cth 1 ( a (209) aEA l3EB
where C is some construct. Consider now the expression (205) for x < y. It is possible to move the contour of integration with respect to o, _ >7EC "Y to the line Imo, = I. . Then , using the estimation (209), one can easily get the following estimation for the matrix element:
t(A3, B3)
I (A IO(x)O(y) I B) I ? A=A,UA2UA3 B=B1uBsuB3
') (a -3)Cn( A1)+n(A2 )+ n(B,)+n(B s) X [J cth aEAluA2 QEB1uB2
)K1 d x (Ipo(Ai) - po(B2 )I + (1P1 A1) - pi (Bl) I + dy x
(I
d
o( A 2 ) _o(Bi)I+ dy/l
"'
(1P1A2_P1B11+
d ) 13
x exp (C1CZKo( m(x - y))) , (210) where KO is the following function: exp(-x ch j3)d(3 .
Ko(x) = j
Thus, if the estimation (209) holds the expression (205) for the matrix element is finite for x > y (similarly (204) is finite for x < y). When x approaches y, the matrix element can be singular but the estimation (210) proves that the singularity is of power character because the function Ko(x) behaves for x -> 0 as follows: Ko(x) = -2lnx +... , x --+ 0 . As to the estimation (209) it looks quite natural. The author proved it for SG in the reflectionless case. The investigation of the estimation (209) in the other cases encounters many technical problems but the author believes that they can be overcome.
166
Form Factors in Completely Integrable Models of Quantum Field Theory
For a detailed study of the singularities of the commutators at the origin of coordinates one needs some refined information. Some experiments and general reasonings have convinced us of the validity of the following statements which are formulated as hypotheses , since their rigorous proof is still lacking. Hypothesis 1. In SG model and ITM the series 00
IIg(V)II2dVo
, > J
II9a(e)112 dcVo
(211)
n(C)=0 n(C)=O
converge (the convergence of the integrals f IIg(C )II2d C o follows from Eqs. (201 , 202)). For what follows Hypothesis 1 will be sufficient in the form it is formulated . We have however more precise conjecture about the character of the convergence of the series ( 211). We expect the existence of the estimation
f IIg(•)II2dCo .r
q n(C)( )
for large n(C). The function q(l;) < 1 for 0 < 1; l< oo. This estimation implies the existence of a similar estimation for ITM with q = q(oo). What is the sum of series (211)? This question is very important . It is natural to suppose that according to Coleman 's equivalence Eq. (2)
n(C)
(
J IIg(-e)II2dCo = ir -)
2a2
and consequently for { = oo,
J Ilga (t)II2dcto = 2a 2 n(C)
To prove these formulae directly is an extremely difficult but tempting problem. Let us pass on to NLS. For this model we suppose the following hypothesis to be valid Hypothesis 2. In NLS model the series
J llga(V)II2dCo , J Ilfa(c ) I2dcVo n(C)
n(C)
167
Current Algebras
diverge (the convergence of the integrals f II9aII2dCo, f II faIl2 dCo follows from Eq. (203)). Asymptotically for n -+ oo the n-th term of these series approach as some constant . When the ultraviolet cutoff is introduced, Ai < A, and the series become convergent:
I J
n
r An ( -6) 112 < oo , d01 ... . . f0 0 dAn-1 II9a E J A dA1...
J
A dOn-
n
1IIfa(C)II2
(212)
Assuming this hypothesis to be valid one can prove the following statement. Lemma 2. The series (212) are characterized by the following behaviour for A --> oo:
E J ^ d01...
IA
0
dAn- 1 II9a(
V
)112
..'
(41r)2 '
Proof. Denote the sums of the series f
C)II z
f
Ad01 ...
_
^d01 ...
AdAn-1 119* (
n
n
fo
o ^dzn-1IIfa(V)II2
f
by j1(A), q2(A) respectively. According to Hypothesis 2, ij1,2 (A) < oo. Differentiate q1(A) with respect to A: d oo n+1
A
= f do1
n
, n
J
dn 1191(71 , ...7K-1, 7,c + A ,
,7n +A)II2 . For large A the following estimations (200) hold: 9a (B1, B2 + A) .., A 6abcgb(Bl)9e(B2) ,
n(B1) = 0(mod 2) ,
9a (B1, B2 + A) ... 2^ 6abc fb(B1)fe(B2) , n(B1) = 1(mod 2) .
Form Factors in Completely Integrable Models of Quantum Field Theory
168
Hence for large A the formula is valid:
dd nl ( n) =
2(2ir)2 2 (71 (A)
+ riz (n))
Similarly, dn772 ( A) = 2(21r)2 1 211i (A)7l2(A) Hence , for the function ra(n) = nl(A) + 712(A) one has the equation do i7(A ) = 2(2x)2 n2 271 (A) . The solution of this differential equation is A
It c A +2(2x)2 According to Hypothesis 2, q (A) --> oo for A -+ oo; hence c = 0, which means that for large A the following asymptotic holds : j7(A) ^ 2 21 T 2A. Hence , n;(A) = cin for A - oo. Substituting these rlt into the above equations one gets ", .v 772 (A) 771(A) (47r)2 Q.E.D. The possibility of indicating an explicit asymptotic behaviour of 711,2(A) in the framework of Hypothesis 2 is very important and, as it will be clear later, is a strong indirect argument in favour of this hypothesis. Now, assuming the validity of Hypothesis 1, 2, let us pass on to calculation of the singularities of commutators at the origin of coordinates. Theorem 1. The following commutation relations between currents on the space interval for ITM hold: [70a(x), 70(0)) = i6abcjo(0)b(x) W1(x), 71(0) = Zeabcj o(0)a(x)
Uo(x), 7i(0)] = ieabcji(x)S(x) + 4babij6'(x)ai, where
00 C dCo r (-)112 7 J II9a n(C) =O
(213)
Current Algebras
169
Proof. Look at the formula (206),
(A I J[Oi(x), 02(0))e
p(x) dxjd B )
= J {7Io1 x)o2(o )I B)- - (A 1O2 (0)O1(x)I B)+} e',c(A ) s^o -(x)dx +
f
{ 7 IO 1
x)O2(o I+ - ( A 102(0)01(x) 1 B)- } e'ic(A):p +(x)dx
(214)
and consider the first term in RHS,
J {7IO1( xO2oI_ - (1102 (0)01(x)I B )+} e',(A)xso-(x)dx S(AIA1)S(A3IA3) > 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
(217)
because the first and second terms in (216) are the boundary values of a regular function in the strip -a < Imo- < 0 on the lines Imo = 0 and Imv = - a + 0, and this regular function decreases for a -^ ±oo.
170
Form Factors in Completely Integrable Models of Quantum Field Theory
The asymptotics (187) implies that fµ and f,6, do not increase faster than e"°i for o, - ^ ±oo. Hence the condition that 0_ (ic) decreases faster than an arbitrary power of ic-1 can be replaced by the following: 0_(ic) decreases as is-3 for ic -+ oo. Under this condition on Eq. (217) remains valid. The condition iO_(it) - k-3 for i -+ oo is equivalent to W(0) = (p'(0) = 0. There are no restrictions on other derivatives, hence the commutator (I U,'.-(x), j'(0)] I B) does not contain other singularities but those proportional to b and Y. We want to extract the b and 6' singularities from the commutators. To this end we need to consider a convolution of commutators with the functions which do not vanish at the point x = 0. One should be careful at this point because in the cases we shall consider later the integral
J dofl(A1+i0ICU Bz)S(CIA3U C).fz(C U A2IB1+i0) (218)
x 0- (tc(C) + ic(B2) + ic (A2) + ic(Aa)) diverges , which means that the convolution
J Ol(x)Oz(0)
00 [01
02 (0 )]l B) = ^ m
J ((Ioi(x)i)
00 n(o) -o
X (DIO2(0) IB)_ - (AIO2 (0)I D)(DIO1 (x)I B)) n(O)
x
f f 9(A - bj+l + bj)dD ,
j= 1
where D = {b1, ... , bn(D)}, 6n(D) > ... > bz > bl . In other words, we demand that the difference between two nearest rapidities does not exceed A and then take the limit A -> oo . Lemma 2 of Sec . 1 implies that D = C U A2 U B2 U A3. Let us denote the minimal and maximal rapidities in Az U Bz UA3 by fln"' and #,'n respectively . Let C = {yl) ... , yn(C)}, Di =
Current Algebras
min
171
Y, Cmax Yn (C)
max
A 40-
Y,
Cmin
Yn ( C)
Pmin Pmax Fig. 6.
7i-1 - yi, Q = n j EryEC y and fix 01 i ... , An(C)_1, then o, may vary in the range from Amin to trmax as indicated in Fig. 6.
The integral
r
J dad_ (C, AI, B1i A3)
(219)
changes under the limit sign to
om dvG_ (C,A,,B1,A3) = (J -J) A (7+ioiu2)
J
Il )9
omin
X
.S(li
I A3 U
C)f2(C'
U A3 )
B 1)0-(4C) + ic(B2) + ic(A3) + ic(A2)) , (220)'
where I1, I2 are segments parallel to the imaginary axes II = ( °min) 0min ii), I2 = ( t7max, Amax - 7ni ). Evidently the limit of the integral (220) can exist when the integral (219) diverges. It is desirable to work with such operators for which only the leading term of asymptotics contributes to (220) for a -a ±oo.
The first commutation relation we intend to obtain is
[j+a (x) , jb (0)] = 0 ,
(221)
where jf = 12 (jo ± ji ). Form factors f. , f f are written through ga as follows: f+(fll,... #2.) = M(> e,6j)ga(/31,,... ,)32.) , f b (Nl, ... ,N2n) = M(> e-1i) ga(Nl, ... , 02n)
172 Form Factors in Completely Integrable Models of Quantum Field Theory
Let oo. Fix A,, ... , An(C )_1, then from the asymptotics ( 187) and S(/3) N 1,,3 -+ oo it follows that
f+(A1IC U B2)S(CIA3U C)fb(C U A21B2) x V_(a(C) + c(B2) + c (A3) + ic(A2))
(P(0) n-2+6*(A2)+n(B2 ). o+6.(A2 )+ '(B1).o x exp(- A bn(A2) +n(Bi),O + a ((-1)n(C) - 1)) , if o (0)^_ 2+6.( Al)+n(B2 ). 0+6^(A2)+n(B1)1a t
N
max
T
x exp(- A an(A1)+ n(B2),O + z ((-1)n(C) - 1)) , if o N Om,,, (222) Thus the function (222) decrease when A --* oo in the vicinities of omin and om Hence , the first term in RHS of (214) disappears. The second term can be considered in a similar way. Eq. (222) is now proven. Consider now the operators oo x Q° (x) = f .10 (x)dx, Q + = f .70 (x)dx . co
x
Matrix elements of Q° , Q+ are constructed according to Eq. (28) (Eq. (29)) through the functions g°, -g° According to the results of Sec. 7 the operator Q° = Q_ + Q. is an operator of change. We want to calculate the commutator [Q° (x), jµ(0)]. For its matrix element one has
(A ^ J[Q (a), jb ( 0)]eix (A)xc(x)dxl B ) _ (A J [Q° (x), 7µ(O)]eix(A)xco-(x)dxI B ) - (A J[Q(x), 7µ(0)]eiK(A)x
J (A j[Q°, jµ(0)]eiK(A)xW+(x)dx .
(223)
The last term in RHS can be easily evaluated using the formula [Qa, jb
(0)]
= iEabc
jc (0)
The first term in RHS is considered in the usual manner , and is calculated as the limit (220) for A -+ oo, when g°, f b are taken as fl, f2. The asymptotics
173
Current Algebras
(187) imply ya(AJCU B2)S(CIA3U C)f^,(C U AdBI) x c'(i(C) +,c(B2) + ic(A3) + K(A2)) +6,. ( A2)+, (Hi),o ,P(0) exp(2 ((-1)"(C) - 1)) A- 2+b°(Al) +n( B2),0 if o - Amax 6 ^p(0) exp(z ((-1)" (C) - 1)) A-2+ °(A^ ) +n(B2),0 +6n(A2)+ - (BO,if 0 Omin
(224) Eq. (224) means that the integrand in (220) does not decrease when A oo in the vicinities of Amax and omin in the one case, n(Al) = n(A2) _ n(Bi) = n(B2) = 0. In that case the limit of the integral (220) is equal to 7ri(1 - (-1)0). The condition Al = A2 = B1 = B2 = 0 is equivalent to A3 = A, B3 = B. Hence, f (AI[Q°(x),1µ(0)]IB)etx(A)x^-(x)dx = c'(0)0(A,B)7ri(1 - 1)µ)7 In a similar way, f (A I [Q+(x), j1(0)] l B)e:w(A)X p+(x)dx = -cp(0)A(A, B)7ri(1 - (-1)1)r Hence [Qa (x),.11(0)] = B(x)eabejµ (0) + 2rri ( 1 - (-1)µ)rjb(x) or, differentiating with respect to x U (x), j1(0)] = ieabc j'(0)b(x) + 47rir7(1 - (-1)1)6'(x) .
(225)
Algebra (221), (225) is equivalent to (213). Q.E.D. A similar Theorem holds for SG model. We present it without proof. Theorem 2. The currents in SG model satisfy the commutation relations (jo(x),jo(0)] = (j1(x),j1(0)] = 0 (jo(x), ji(0)] = 47rib'(x)i](^)
174 Form Factors in Completely Integrable Models of Quantum Field Theory
where
17(4) _ J IIg(C)II'dCo. n(C)
Let us pass on to NLS. Theorem 3. The following commutation relations hold in NLS model: [na(x), nb(0)] = 0, X70 (x), nb(0)] = isabc
nc(o)b(x)
UO(x),j0 ( 0)] = ica bcjo(0)b(x) rr rr Wl (x
), nb (0)] = 0, Wl (x), 71(0)] = 0
Uo(x),ji(0)] = iEabcj1c(0)b(x) + oobabb'(x) . (226) The last relation means that the convolution of Ua(x), ji(0)] with an arbitrary function wp(x) from Co whose first derivative at x = 0 vanishes is finite, while the convolution with a function ^p(x) with nonvanishing first derivative at x = 0 is infinite. Proof. The main idea of the proof is similar to that of Theorem 1. We introduce the cut-off and consider the limit (220) instead of (219). At this point, however, an essential difference arises . The fact is that the series
J Ilg(6)II2d
Co
(227)
is divergent. That is why even if the integrals (220) tend to zero for A -+ 00 the sum of the terms with fixed A1, A2, B1, B2 can give a non-zero contribution. So we are dealing with an indeterminacy of the kind 0 • oo and the cut-off should help us to uncover it. As it follows from Lemma 2 the series (227) behaves as A for A --> 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-I^I)). 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
Form Factors in Completely Integrable Models of Quantum Field Theory
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)
^
x ? ( 0 ) ( l + ( - i n ( i y (0)1*8). The same formula can be proven for the convolution with Ji(ty] = 0Evidently, the commutators [io> ii]> Uo >io] having been calculated, it is sufficient to calculate [j+,.7'1], which is what we are going to do now. We have to elucidate the behaviour of the function (_l)»(CH"(»i)+»(*a)/« ( 7 x + f'0|~(5 U ~B2)S(C\*AZ
U *C)
x / ! ( C U *A2\Bi) / ° ( ^ U *A2\Bi)
= (p-(A3)
+ p-(C) - p - ( B O )
where
p^A)
= M*£e±a
.
It is easy to understand that there is no 0(1) contribution from a ~ .
Current Algebras
179
The contribution from a - omin can be calculated similarly and it appears to be equal to
E abc'P (0)
2. (A Ij+(0) I
B)
Hence
f (A l U+(x), jb (O)] I B )so_ (x)dx = 2Eab`s^(o)(A ljo (o) I B > The same formula can be obtained for the convolution with
Combining this with Eq. (235) one gets [ji (x), jb(0)] = 0 . Q.E.D. The commutation relations ( 226) coincide in general with corresponding Poisson brackets in a classical model, the only difference being in the coefficient before the 6' function ( Schwinger term) in the commutator [jo ( x), ji(0)]. In a classical model this coefficient is (bab - nanb). The infiniteness of the Schwinger term in the quantum case is quite logical: usually the Schwinger term is the result of quantum anomaly ; it disappears in the classic case , being proportional to h; in NLS the Schwinger term is finite in the classic case, which is why it is natural to suppose that tin the quantum case it is infinite, being proportional to h-1. This argument is in good agreement with the fact that the coefficient before 6' in (226) can be estimated as A (A is the ultraviolet cut-off of the rapidities which is connected with the ultraviolet cut-off of the moments MO via A = logMM). We have the formula M=Moe-9 , which follows from asymptotic freedom ( M is the mass of the physical particle, g is the coupling constant , MO is the ultraviolet cut-off). This formula is very general : the origin of MO is of no importance. Putting these facts together one realizes that the coefficient before 6' is proportional to g-1. The coupling constant in NLS can be identified with hC . The absence of the nondiagonal term nanb in the quantum Schwinger term needs further understanding.
APPENDIX A FORM FACTORS IN SU(N)-INVARIANT
THIRRING MODEL (SU(N) CHIRAL GROSS-NEVEU MODEL)
In Sec. 7 formulae have been presented for currents and the energymomentum tensor form factors in SU(2)-invariant Thirring model. This Appendix extends these results to the SU(N)-invariant model (chiral GrossNeveu model) with Lagrangian = f (i r ,4O i,J -
gK.b ( Tta-'v ')(Ttb7µ0))dx
,
where {o"} = ii , i = 1, 2, a = 1, 2,... , N, are Fermi fields , i, a being spinor and isotopic indices , to are the generators of SU (N), Kab is the Killing form. It can be shown that the model leads to the same physical results as the chiral SU(N) Gross- Neveu model. The spectrum of physical excitations involves N - 1 massive particles which transform with respect to N - 1 fundamental representations of the group SU ( N) [53, 54 , 55]. The particle which transforms with respect to the 1-fundamental representation will be referred to as rank-1 particle. All the particles are the bound states of rank1 particles (kinks). The rank- 1 particle is the antiparticle of the rank-(N-1) particle. The mass of rank -I particle is M, = M1 sice sin N
where M1 is the mass of the kink. 181
182
Form Factors in Completely Integrable Models of Quantum Field Theory
The rank- 1 particle will be parametrized by the rapidity /3 and the isotopic multi-index e = {k(1), ... , k(l>}( k(l) < ... < k('), O) = I,_ , N). The number of k's in a is denoted by 1(e). The S-matrix of two rank- 1 particles can be presented as an operator in Vi1) 0 Val) (V(1) is the space of the vector representation of SU(N)):
S 1,2(
3) 5°(3)51,2(3) =
,
where
So (P)
2Ai P N 1,2
02vi )r( . --)
r( N1 + = N-1
r(N - -L)r(j--)
N
P1,2 is the permutation operator of spaces V(1) and V2^1)
The S-matrix of rank- 11 and rank- 12 can be presented as an operator in V(11) ®V(12) (V' ) is the space of 1-th fundamental representation of SU(N)). The spaces 01) and V( 12) can be realized as completely skew11) symmetric subspaces in Val) ®... (& Va; j and V ®• • • ®V 1 respectively. The S-matrix can be found in accordance with fusion procedure [50, 51]:
51,2(3) = Pal,... ,a,1
Pb1,...b12
13
( +
Sal,bl
N
(1 1 + 12 - 2)) .. .
iri
b l2 1 3 - N (11 - 12) ) ... Sa,lbl I ,Q + N (11 - 12)
X
Sa
X
Sa l l,bl2
1,
(3 -
7ri
/
2)) N (11 + 12-
)
...
\
Pal,... ,all
Pb 1 ,... ,b,2
where Pal,._. ,all , P^, b,2 are the projection operators onto skew-symmetric subspaces. To make the notations more close to those used in the basic part of this book we write the matrix element of all S-matrices as Sfi;c;. Evidently, only those 5'11"2 can differ from zero for which 1(e1) = l(e1(e2) = l(E' ). The Zamolodchikov-Faddeev operators satisfy the usual algebra of commutation relations. "In" and "out" states are constructed in the usual way. The matrix C involved in the crossing-symmetry relation is
-1) A ,
CE 1,E2
El
U E2 = 7r 1, ... , N)
0, El UE2#7r(1,...,N)
We denote by e* such multi-index that e U e* = 7r(1, ... , N), it being some permutation.
Form Factors in SU(N)-Invariant ...
183
The general matrix elements of the local operator O(x) can be expressed through
f(/31i...
,/3ri)£1,...,En = (phIO(0)
IZEn(fin),
...
,ZE1(N 1)I ph)
in the usual way. We consider form factors of currents ja = T_,,,tao, where to are the generators of SU(N) in vector representation, and the energy-momentum tensor T,,,,. We use the following choice of generators: Hi = diag(0,... ) li,
, 0) and Ei1 with one nonzero element which is unity in the position ij. So a can be made equal to one of N2 - 1 symbols: {i},i = 1,... ,n - 1;{i, j},i 0 j. The fundamental requirements on form factors remain the following: /^ p 1. f(f1, ... , Ni,,6j+l, ...
f(31, ...
Eir
i +1 1
,£ nSE^, :,
(ii
- /3t +1) (Al
,,8i+1, Ni, ...
2. f(Ni,... ,Nn-l,fn + 21ri) ,,.... ,E n-l,En
(A2)
q pEI(Ei) _ (-1) f(/'n, )3l, ... ,^n-1) enrElr ... , f n-1 ,
3. res f (/3 , ...
E. , Nn-2)ci,.. , Ea ,f[n-1,i3)E[1,... , ,En_1,£n = f (Q1, ..E_
. 1) ) y,", 1,£ l,f (Nn-1 q p Tn_gEn-2 X 'SEn- 1,E, 2 (fn-1 - Nn-2)} , Nn = fn-1 + 7ri ' /i
CEO - 1
X
)
)
En
{UCI ,...
,
SEn -3
- (-
(A3)
Besides these identities there is, as in SG model, a number of formulae for residues corresponding to bound states. We would not list all these formulae, noting only one of the most fundamental. Suppose En-1 and En do not contain coinciding k. Then En-1 U En can be reordered into a new multi-index c by some permutation. There we require f ()131, ... , En_1, En)f1,...,En_1,£n to have a simple pole at the point Nn = Nn-1 + y(l(En) + 1(En_1)) with the following residue: res f (01.... , Nn-1, /3n)£i,... , En-1,En _ (
-1)
a ai (En),(fn_1) f
7rt , E n- 2, fn-1 +
,131 i...
N l(E n_1 ) f1 , ...
n_,
(A4) where E = 7r(E1 U E2), and a^( En)i(£n_1 ) is certain constant connected with EnrEn -1 SEn,Cn-1'
184
Form Factors in Completely Integrable Models of Quantum Field Theory
It is convenient to present, first, the formulae for the form factors in the case 1 (el) = ... = 1(en) = 1 (in the sector containing only kinks) and then to consider the general situation. Form factors f (,31, ... , /3 ) ,... ,n, 1(el) _ • • • = l(en) = 1, can be comLet ( bined into a vector f ()31,... , fan) in the space (V) ® ... ®Vl))*. us introduce some orthonormal bases ej,k in the spaces Val). The natural basis of the tensor product consists of the vectors ekl,...,k„ = 11 0 ej,k,. The form factor )3n)E1.... .En is the component of f031,. .. , 8n) in the base eE1i_.,,E, (ej = {kk}).
As usual, it is convenient to consider also
another base wk,.,...,k„ (I31, ... , Nn), which we are now going to describe. Unfortunately, we have not succeeded in the generalization of the definition given in Sec. 7 for the SU(N) case which is why we shall give only an indirect definition of the base wk1.... ,k„ (N1, ... , fn). To this end, let us first divide the space (V^1) 0 ... ,®V„ 1))* into a direct sum of special subspaces. The algebra sl(N) acts on Val) with generators e`k , hi , which are represented as matrices EtI, Hi in the base ek,1i... , ek,N. This extends the action of sl(N) to the space (V1l) ®... ® V„l)) with the generators eij = Eek , xi = Ehk . Obviously this is a restriction of the action of N2 - 1 charges to the n-kinks sector. The space (V^1) ® ... V(1))* appears to be divided into a direct sum of subspaces Vi which are the eigensubspaces for all the V with the eigenvalues 1i - li+1(Elj = n). The basis 1„ (k1, ... , kn is some permutation wk1 i... ,k„ (,Q1, ... , Fin) in the space V i of the set 1,... ,1, 2, ... ) 2, ... , N, ... , N) is defined by two relations:
w(/31, 1n
12
11
= el,N 0 ... 0 el,,N 0 ... 0 ell+...+1N-1+1,1 0 ... ® en,l
(A5)
w(f31 i ... , Pi, #i+1.... Qi+0 = w(/31, ... ,,3i+l, A, ... , )3n)k1,... , kj+jk;,... ,k. .
(A6)
As usual, it is implied in the last formula that the rapidity flj is always connected with the space Val), i.e., when rapidities change places the spaces also change places.
Form Factors in SU(N)-Invariant ...
185
Let us explain how to construct Wkl __. ,kn (,Q1, ... , Nn ) using Eqs. (A5, A6). It is quite evident that using Eq . (A6) one can present wk1 .., kn ()31,... , Nn) in the form wN,... , N,... ,1, ... ,1(fix(1), ... , fir(n)) (some
product of operator S), where a is some permutation. From Eq. (A5) it follows that wN,... , N,N - 1,... )N - 1,... ,1, ... 1N
iN-1
1(3r(l ), ... , Nr(n))
11
= er(1) N (9 ... ® er ( IN) ® ... 0 er(n),1 .
This definition cannot lead to any contradiction because of the validity of the Yang-Baxter equation for S and offs the equation ei,k ® ej , kSi9 (Ni - /3j) = ei,k 0 ej,k
It is clear that the logic of Sec. 7 is borne out: Eqs. (A5, A6) which are considered as properties of wk1, ..., k„(,Q1, ... , Qn ) in Sec. 7 are now considered as definitions. It can be shown that , as well as in the SU(2) case, the transformation {ek1... kn } -i {w1 , ,,. , kn } is triangular . Vector wk1,... ,kn (al, ... , Nn) has simple poles at the points /3j = ,Qi + N for j > 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
186 Form Factors in Completely Integrable Models of Quantum Field Theory
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<9
n 0EB(P)
(/ 1 f,)
w kl,...
qq ,k-N (/31, ... ,,3Nm)
P'EB (4)
(A8)
where kj = k, 3, E.B(k) 16
(1,1(8) = c sh- exp 2
{_
2 r
sine ' (3
ai)k exp(- N 1 ak)sh N
ksh2irk
dk }
N+I 1 \ 1 ) sh2 exp (- M irk) sh N ) 2a 2N r2N exp ( _2 f°° dk 1
k(sh rk)2
JJJ
188
Form Factors in Completely Integrable Models of Quantum Field Theory
Similar formulae can be written for fµa). The only difference is that in the formula for fµ=j) one has to take the sum over all the partitions of B into B(P) which satisfy the condition n(B(P)) = m - bpi + bpj. The validity of (Al) for the form factors is contained, as usual, in the very defiition and follows from Eq. (A6) and the formulae (1,1(fl)So(Q) = (1,1(-a)
((Q - 2ai) = ((-/3)
We would not discuss the properties (A2), (A3) at this stage, but pass on to the problem of bound states. It is easy to understand that the problem of calculations of form factors corresponding to bound states (particles of higher ranks) can be re,Qik') _ duced essentially to the calculation of F(B(1)I ... IB(")) in the case r - N (1 - 2j + 1),1 < N - 1, j = 1,... 1,)31 k3) E B(ki). Let us restrict our consideration to one form factor, namely to the form factor f++(,31. • • • . i3 ). The polynomial Q(N) satisfies the relations QiN )(aI
B)
= Q(N)(aI B\/3) i
)3Qi N)(aI B \Q)
n(B)
+ N)
l
Q(N) (aI B) = \a
T7 ( a 11 /EB
i - n(B)
- (a
a11
- N)
(
-# + N7ri ll/ )3
a-
- N)
i > n(B) .
Using these relations one can show that A(AIB(1) I ... I
B(N))I Alk ^^= r-R(1-29+1) j =1 ... I
_
Ak1 ,...,k,(A \( aP1,... ) app- 1)IB(1)I...IB(N)Ir)
Pi <...
X M( aP1 , ... , a PI- 1)(-1) P 1 +...+P1-1 ,]
(A9)
\alki),
is the M(al, ... determinant of an (1 - 1) x (1 - 1) matrix M with the following matrix elements:
where f30) = B(9) if q # k„ Bkj = B(ki)
Mii = Mi( ai) , \ l a - ,^3 + N(N - 1)) (N -1) I (a + ai)i -1 fi 1 / M(a) = C a - r +
N
\J /EB /\ // - l a-r- N(N-1) I (a-ai)i-1 fl I a-,Q- N(N-1)/ /// AEB
Form Factors in SU(N)-Invariant ...
189
where b = UB(P). The polynomial Akl,... k, (AI B(1) I ... IB(N) I r), n(B(P)) = m - Ebp ,k,, n(A) = (N - 1)m - 1, is the determinant of an ((N - 1)m 1) x ((N - 1)m - 1) matrix with the following elements: k,) = Aiki,...,k,)(a,IB(1)I
I B (N)Ir) N
A( kl >... )(aIB(1) I ... IB(N) IT)
s =1 1EB(•)
N (N + 1 - 2s) 2s ))
s#kl,...,ki
N
N -2s+1 N
) x Q, CN
U
(B(1)
+
7ri
sgn(s _k)) U ^ r+
7r
p1 pis
1
C x (-I + q(s, ki, ... , k1))) + L, x
(
II
7ri (a_i_ (N + 1 - 2k1 ))
=1 AE B
a_r_
N(N
\
-1-2kj +2j) I
x QS N) a - N (N - 2kj + 1) I Up$k1 (BP ) + Nsgn(s - p)))
where q(s, k1i ... , k1) is an integer defined by the relation kq < s < kq+1. Relation (A9) is valid also for the case 1 = N. In this particular case, A,....,N(AIB(1)I ... IB(N)Ir) = II(a-7-)A(AIB(1)j ... IB (N)), and we obtain a direct analog Eq. (120). Consider F++(B(1)I... IB(N)). The determinant of matrix M can be transformed to the determinant of matrix .M whose elements are
Mi.I = Mi(ai ) Mi(a) = (a_r+(N_l))'^ (a_(N_l+2P_ 2)+2ini) P=O
7ri xI(a - Q+N (N-1))-(a-r-N(N- 1))
X U ( a_r_ N(N-1+2p-2)) H (a_t3_N(N-1)) P
Form Factors in Completely Integrable Models of Quantum Field Theory
190
Notice now that
(
Sol j-1
a-
-
(I + 1 - 2j)) _ 'p,N)(a)
1-1
x [Jsh -12 (a=4 p
N
N
(N-1+2j-2))
N-1 2N 27ri 2N
( N_i _ r r
Substituting these into Eqs. (A7) and taking into account the antisymmetry with respect to variables aj one obtains
F++(B(1) I ... IB(N)) IPlk^^-r-77(
1-27+1)
l_i
1 n(A)
= ()
JdA J dai... J dazi (
-)
`P1N)(a - Q)
I aEAPEB
1-1
x fl Wl(a-r)flsh-1 (a-r- N(N-1+2j-2)) aEA j=1 1-1
x
Ak..... .
k. (AIB(1) I ... I B(N) I .T) (ai - r + N(N -1)) i-1
i-i
X 11 (ai-rP
N(N-
1+2p-2)+ 27ri
(ai-
) 11
i +N(N -1))
PEB
(ai_r_(N _l))
j
(ai_r_(N_l+2p_2))
P_
x II (a - /3 PEB
N(N
- 1))
J
w+(A)
where A = A U j ai,.. - , al_i }. The integrals over a1 , ... , al_1 can be calculated using the usual trick . The fact is that the integrals over a1, ... ,
191
Form Factors in SU(N)-Invariant ...
a7_1 can be rewritten in the form 1 2ai
(
L +2,i -1
'P(N)(ai
t-1
x11
j-osh2 (ai-
-
13)w, N )(ai
- r)
pEB
r-
1
(
Ti
N( N-1+2j-2)
ai_r _
(N_1))
i-1 x(ai_fl_(N_1) N )
n (ai_
r_(N_l+2P_2))
I3EB
X w+(A)... , where the dots denote a function not depending on ai. Thus these integrals are equal to the sum of residues of the integrand in the strip 0 < Imai < 2iri. Consider first the integral over a;_1. The integrand has only one simple pole at the point al-1 = r+ N(N+ 1- 4). Now consider the integral over a1_2. The integrand has two simple poles at the points al_ 2 = r + N (N + 1- 4) and al-2 = r + N (N + 1 - 6). However, the first pole is cancelled by the zero of w+, whose arguments after calculating the integral over al-1 become A, a1i ... , a;_2 , r + N (N + 1 - 4). Continuing these calculations one makes sure that the integrals over a; (i = 1, ... , 1 - 1) are equal to the residues at the points a; = r + N; (N - 1 + 2i - 2). Finally, one obtains for F'++ F'++(B(1) I . I B(N)) I l-1
Sri
(N) r-,Q+ N(N-1+2j-2) 1=1/3EB n(A)
x (2ai)
J dA JJ
'(a-r)Hco(a_fl)
aEA /3EB
x
Akl,...
x
,k, (AIB (1) I ... I B(N))w +(A U U^-i i_1
1
(
r + N (N - 1+ 2j - 2)))
(A10)
Hsh2( a-r-T,(N-l+2j-2))) j=1 Using the above procedure one can obtain the form factors which correspond to an arbitrary set of particles of all possible ranks. We present
Form Factors in Completely Integrable Models of Quantum Field Theory
192
the final result, restricting our consideration to f++. Consider the set of rapidities B = { Ql, ... „ on). Let the rapidity aj correspond to the rank-lj
N-1 particle . The set B appears to be divided into the sum U Bl, the subsets 1=1 B1 containing rapidities which correspond to particles of fixed ranks. Form factors corresponding to this situation create the vector f++ ()31,... A) belonging to the space (V(11) ®... ®Vri1n))'. This vector is given by the formula f+ +(91i...
an) = fl(11,1i(13 3 )/ 6 j ' lai -+
i<j
r
N(1i
-ii))
x { II II II (a-a, B1=uB(l )
c 1?6c2,F1 , 2={k(j)j
(pp,q' (a) PEB(^1) k1 k2 P'EB(`2)
- N (12 - 11 - 2p - 2q))}F'++(B(1) I ... IB(N) IB( 12) I ... IB(N-1,N) I
IB(1,... ,N-1) I
I B(2,... ,N))
x w91 ,...,cn(131, ... , an) , 1(e) = 1, En(B(`))e = m{1... N} where ej = e if aj E B(`), and w{k(1),...,k1^1 )},{ k21),...,k2^2 )}...{ k(1),...,k("
)}(a1, ... , Nn)
2e E(1;-1)
e1 .0 (N)
x
(
I3 ._
7rN
W k(l),..., k111),kz^1 )I..., k^t2)
ei(Ii - 1 ), al - -N a i(11 - 3) ...
a1 + -Nei(11 - 1), N2 - -Nei( 12 - 1),... a+e a2+ Nei(12-1),...,an- _Nei(ln-1),...,an+ N i(ln-
X Pl,... ,11P11+1,... ,11+12
Form Factors in SU(N)-Invariant ...
cr;,t;(Q) = c1.d;sh2 1 79=
193
ai ) - j2) ^ sh ( /3+ N(j1 2 2=1
l`-1 ( li-Ii-2p 2N-li-Ij+2p Q x 11 r 2N 2^ri) T ( 2N + 21ri ) p= = 11 Ii-1
2N-li-Ij-2p Q Ij -li -2p 2N 2^ri) T ( 2N + 27ri )
x T p=1
X exp
°° sine 1 (/3 + ai)k exp(- N 11 7rk )sh(N k)sh(N k) A 111-Jo ksh2irksh N j
The function F++ is given by the following integral: F++ (B(1) I ...
IB(N) IB(1'2) I ... I B (2,3,... ,N))
= (217ri )fl(A)j dA H 11 `P(N)(a- 3) aEA/3EB1 U
x
w+(A
1 >19 U DQ EBi (N
jY (N - I+ 2j - 2)))
aII P B`sh2 (a-/3+;(N-1+2j-2)) A
x A(AIB(1) I ... IB(N) IB(1'2) I ... I B(N-1'N) I ... IB(2,3,... ,N))
where n (A) = E(N - 1)n(Bi),A is the determinant of the n(A) x n(A) matrix A with the following elements: Aij = A i(aj IBM I
Ai(aIB(1)I ...
... I B(N) IB(1'2)
IB(N) IB(1'2) I ... IB(N _1, N) I ... IB(2,3,... ,N))
n N-1
77-f^
s=1 (=1 £={k ( 1),... ,k(')}: I3EB(') 3k(')=s
x Q(N) N-1
(
I ... I B(N-1'N) I ... IB(2,3,... ,N))
(a_
fl_(N_l_2s+2i))
a - N (N + 1 - 2s) I
x U U
U
1 =1 £={k(1).... k ( 2)}: /EB(•) Vk(i)96s
iri
a + N (-I + 2q(s, k(1), ... , k(i)))
194
Form Factors in Completely Integrable Models of Quantum Field Theory
Relations (A9) imply an enormous number of recurrent relations for A(B(1)I IB(2,3,...,N)) Using these relations one can prove the validity of Eqs. (A2,A3) for f,,,, f. a. The method of proof differs only in detail from that proposed in Sec. 4. We would not like to repeat once more this cumbersome proof. We believe that having read the book a reader interested in techniques should have obtained enough experience to reproduce this proof himself. Let us list several important properties of fµv, fµ. The corresponding proofs are similar to those presented in Secs. 4, 7. 1. The operators T., and jµ defined by fµ„ and fµ transform as a (1, 1) tensor and a vector respectively with respect to Lorentz transformations. 2. The operators Tµ1, are isotopic scalar while jµ form the base of adjoint representation of SU(N). 3. The following equations hold: 1 n
EDe -P, +(_1)µeP^)Mi(r;)fµ(/31...
,
Nn ) El,..., rn
=0
µ -oj_1 1
n
E(e-P, + (_1)µeP')M1(r;)fpv\/31 ... , f3 )rl,... ,P,. = 0 µ-0j=1
} pp
// ?
fµv()311... ,/n) r1,...,En which imply that
a µjµ =0, µTµ.= 01 µv= vµ .
4. The functions 1 /^ 9a( Q 1.../3n) E1,...
g µ(P1...
,En
,Qn)r 1,...,En
=
=
fo(,01i...
,fl
) E 1,... ,En (EMl(E1)shflj)
fµo (,(31... ,P
(EMl( Ej)shaj)-
have no additional singularities in comparison with the functions f,., fµv for n > 2. For n = 2 there are additional simple poles at the point /32 = /j1 + in with the residues /
/) p 1,P2) E1,E 2 resg' W
= Cc1,Ei
resgµ ( /31#2 )E 1,E 2
=
CE1,E2
E'
a) E 2 ((t _ (_1)µe -PI) (eP1
Form Factors in SU(N)-In-Variant...
195
where (ta)E2 is the matrix element of to in the 1(el)-th fundamental representation for 1(el) = l(e2),(ta)El = 0 for 1(el) # 1(e2). This property implies that the operators Qa = f jo (xl)dxl and P. = f T4o(xl)dxl are the charges and energy-momentum respectively. 5. The following asymptotics hold: 9a(/31 ,
... , PL-,,3k +1
+
A,_ ,
an + A)
Ll,...
,Lr
m(N = 0 (A_m)exp - m) A I / \ N q 9µG1)... )Qk, Pk+1 +A,... ,Qn
= O
(
w(m)-1 exp
A-
+
A) ,l,...,Ln
m(N - m)
\ N
AI
) A ->oo (All)
where m = El(e3)(mod N), 0 < m < N - 1, and rp( m) is some function , whose value for m = 0 when the exponentionally decreasing factor disappears is important : cp(0) = 1. All the properties of form factors mentioned above have their analogs in the SU( 2) case . That is why we have not dwelt on their proofs. There is, however, one interesting property, trivial in the SU (2) case , which is to be discussed individually. This property is concerned with C-invariance which requires the following equations to hold: fliv(N1f. 1n) Ll,...,Ln = (-/ 1)N^,(f ')fl^^(/N 1,... ,^n)Li ,..,Ln qq fµ(fl1,... ^Nn) el,...,e„ (-l)71EI (L^) fµ(Nl,... ^^n) L1,..,Ln
(A12) It is not evident from Eqs. (A8) that these equations take place. To prove Eqs. (A12) let us, first, prove the following lemma, which is also of interest by itself. Lemma 1 . The form factors f (,Ql, • • • , an) L,,,,, ,LA are defined uniquely by Eqs. (Al, A2, A3), the analytical properties described at the beginning of the Appendix, the asymptotics (All), the properties with respect to the Lorentz transformations and the two-particle form factors f(/3l, N2)L1re2 for I(ej) = 1,1(e2) = N - 1.
Proof. We do not concretize any particular form factor we deal with because all following reasonings are applicable to both f,,,, and f,,.
Form Factors in Completely Integrable Models of Quantum Field Theory
196
Evidently, it is sufficient to prove the uniqueness of f (,Q,, ... , fn)el,... ,e„ for 1(E1 ) = ... = 1(En) = 1, because all the form factors can be obtained as residues using (A4). Form factors f (131i ... , /3n)E1,.„ ,En for 1(E1) = ... = 1(En) = 1 form the vector f (/31i... on ) belonging to (V(l) ®... (D V(1))*. Consider the vector h(/31, ... , /3n) defined by
h(Ql,
... , fln) _ H 51,1 (Ni - Qj )
f (^1, ... , Nn)
.
i<j
h(/31, ... , /3n ) possesses the properties 1. h(131, ... , fln). i,i +1(Ni - fli +0
= h(131,. .. , i3 +l, A, ... , e3n)/^ /^ 2.
h(/3n, /31, ... , /3n-1) _1 - fln) ... s1,n(3l -,8n) , = h(/31, ... , /3n)Sn-1,n(fln
h(/31 ,
... , Nn -1, / n +
27r i )
=
(A13)
3. h(/31i ... , /3n) is an analytical function of /3n whose only singularities in the strip 0 < Im/3n < 7r are the simple poles at the points /3n = /j + N
with residues N) 7ri /^1, //^^ resh (/3 ... , Nn) = h2 (13i... fln-1, fli + ^T X sn-l,j (in-1 - /3j) ... Sj+1,0j+l - Nj ) n-1
x i =1
N_i fli A "'_ 1j -Ni 2N + 27ri 27ri ) ( t 2N
(
where h2(/31,... ,/3n-2,T) n-2
n-2
rj (1,11/'i -)3j) ll (1,2 (F'i -
i<j
T)f A, ... /3n-2, T)
i=1
1 ®... ®with v, 22 ® V(2))where f (N1, / 03n- 2, T) is a vector from ((1) _ ... = 1(En_2) = 1, components fl/31,... , ,On-2, 1(E1 ) 1(En-1) = 2.
4. h(#,,... , /3n) decreases exponentially for a -* ±oo. What singularities does h(/31i ... , /3n) have as functions of /31 in the strip - 7r < Im/31 < 0? From the properties with respect to Lorentz transformations it follows
197
Form Factors in SU(N)-Invariant...
that it makes no difference to move /31 into lower half plane or to move N2, ... , /9, into upper half plane. That is why this function has only simple poles at the points with Ql 2*' = /^j - N residues expressed in terms of h2(f32,...
,/3j,...
, Nn) 3 7) -
What singularities does h(#,,... , /3n) have as functions of /3n in the strip 7r < Imfn < 27r? Eq. (A13) implies that h(,61,... , fn) = h(/3n 27ri, )31i ... , /3n-1), hence the singularities at the strip +7r < Im/3n < 27r are connected with the singularities with respect to the first argument in the strip -7r < Im/31 < 0. Namely, there are simple poles at the points fn = /3j + 27ri - N the residues of which are expressed in terms of h2(f1,...
fln-1, r)
What singularities does h(/31i ... , /3n) have as functions of /3 in the strip 27rk < Im/3n < 27r(k + 1), k > 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
Form Factors in Completely Integrable Models of Quantum Field Theory
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 N+ )`. Define the function
Vnl N+1 0
hN-1(/31,...
On-N+1, 7) = f(/31i... ,13n-N+1,T) n-N+1 //RR n-N+1 X
H
II
(1,1 (i3 - f3j) i<j i=1
C1,N-1 (ai
- T) .
It can be shown that the only singularities of the function in the strip 0 < Im < 2a are simple poles at the points r = ) 3j + 1ri with residues reshN_1 (/31, X
x
h(/31 ,
... ,
fan-N+1, T)
_ J1 r
a' - aj 1 'Q' -'3i (N 21ri ) r ( N + 2ri
f3j, ... , /3n-N+1 ) singj ,n-N+ 2( S(Nn -N+1
S(f3j+l -'3 - S (13 - 1 3 1 ) . . . S ( / 3 1 -'3 -1)) . j)
- Qj )
j
It is easy to show that all the singularities of hN_1(... , r) in the r plane are simple poles with residues expressed through h(/31i ... , Nn-N). For r --+ 00
Form Factors in SU(N)-Invariant ...
199
the function ... , /3n-N+ 1, •r) decreases exponentially for n > 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)'^^ilE')f,Y(131) ...
fµ(31,... Nn)Ei, ...,en = -(-1)NEl (Ej)fµ(/31,... ,h'n) Ei,...,En
(A15)
Proof. Functions (- 1)*Er^E;) 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
Form Factors in Completely Integrable Models of Quantum Field Theory
200
'wkl,... ,k„ (#17 • • • Nn)ClC2 = w{kl}',... ,{k„}• (Nl^ • • • , Nn)
N
tai
(
k,-k,-1
X
i^n /3i-#j -j?N f^ ki
)3i - )
/33 -
;-/^ 2ai k;-kt-1 x (N Qj)N-ki-k. fl (Ni - /3 )N-1 i <j i<j ki=kj k;>kj
(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
2^i ) 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 <,in, then f(,81,... , Nn) =
( ph l0(0) I/31, • • •
Pn)i. •
Let us use the usual trick (see for example [56, 57]) convoluting the particle Nn:
f (fi , • • • , /3, ) = (phIO(0 ) a n(fn ) Ii1, • • • , Nn-i)in H
SO
lim
/ e-P°(P °)x" 8 q • • • rh^ x )INl, dx1 0(p h I O \ 0 qn-1)in )din(
oo
lim xo-.-00
J
eiPµ(Rn)xjOjo(phj[O(0)r^oin(x)]IN1,... ,Pn-1)in dx l
+ (ph Iain(fn)0(0)I /31, • • •
, Nn-1)in
203
204
Form Factors in Completely Integrable Models of Quantum Field Theory
Evidently the last term is equal to zero. Proceeding in the usual calculation [57] one gets f(,61, ... ,Nn)
=
dxe-`pµ(^" )xµ 9(- xo )(phi [O(0 ), n(x)]101, ...
J
, Nn -l)in
(B1)
where rj (x) is defined by (❑ + M2) cp(x) = n(x ), 0 . The condition Re/3n > 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
- )3n-
1)(phIO(0)l/jl,
• • • , /3n -1)in •
Keeping in mind that ,6n can coincide only with ,Qn-1 one makes sure that this discrepancy is equal to (phI0(O)ain(Qn) Ji3i, - • • , /9n-1)in •
Taking into account this remark one gets f(/3nh1,•-- , Nn-1) =
+
Nn-1)f(N1,... ,Nn-2)
J dxeP"( an)x"O(-xo )(phl[0(o )
f(NnIfln-1, ... , i3) = +
6(fln -
J dxe1
(
6()3n
)zM 0(xo
qq ,i7(x)]I/31,... , Nn-1)in ,
- /3n-1)f(Nn-2, ... , #0
)( phl
[O(0), rl(x)] I/31, ...
, On- 1)out
(B3) Comparing (B1), (B2), (B3) and using the symmetry property f(N1,... ,/3i, Ni+1,... ,F'n )S([3 -/3i+1) = f(31,...
h'i+1,Ni,...
fn) ,
(B4) one gets f(N1...... n+ 7ri-iO ) =f(in -7ri+i0,F'1,... ,Qn-1)
+
b(/n
//^^ - /n -1)f(f1, • • • , /3n-2)
( 1_
2
s(/3fl
l - 0j)
j=1 On >fln-1>Nn-2> •••>^1
This equation means that the function fn) can be continued analytically up to Im,Q, = 7r and has on this line a simple pole at the point fn + iri with the residue n-2 /^ 27rif (fl1, • • • ,3n-2)
1 - 11 S(Nn-1 j=1
206
Form Factors in Completely Integrable Models of Quantum Field Theory
and it coincides with the analytical continuation of f (/3n, N1, ... , /3n_1) to the line Im/3n = -a. The symmetry (B4) implies that if no restriction of thy... , Nn is imposed fln) has simple poles at the points Nn = /33 + 1ri with the proper residues. Although the above speculations cannot be considered as a rigorous proof of Axioms 2, 3 it seems that they can be refined in order to get a proof on the level of rigour acceptable in the axiomatic field theory.
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