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Functional equations of form factors for diagonal scattering theories
~:~
NUCLEAR PHYSICSg [FS]
ELSEVIER
Nuclear Physics B 466 [FS] (1996) 361-382
Functional equations of form factors for diagonal scattering theories Takeshi Oota l Department of Physics, Faculty of Science, Osaka University, Toyonaka, Osaka 560, Japan
Received 29 November 1995; revised 1 February 1996; accepted 15 February 1996
Abstract The form factor bootstrap approach is applied for diagonal scattering theories. We consider the ADE theories and determine the functional equations satisfied by the minimal two-particle form factors. We also determine the parameterization of the singularities in two-particle form factors. For A~I/ affine Toda field theory, which is the simplest non-self-conjugate theory, form factors are derived up to four-body ones and identification of the operator is done. Generalizing this identification to the ac~) affine Toda cases, we find the two-particle form factors. We also "'N determine the additional pole structure of form factors which comes from the double pole of the S-matrices of the A~ ) theory. For AN theories, the existence of the conserved ZN+j charge leads to the division of the set of form factors into N + 1 decoupled sectors. Keywords: Form factor bootstrap; Functionalequations; Parameterizationof the singularities;
Non-self-conjugate
1. Introduction For two-dimensional factorizable scattering theories, the bootstrap framework gives strong constraints on physical quantities. For example, under some assumptions such as unitarity, crossing symmetry and analyticity, the bootstrap determines the S-matrices almost completely and non-perturbatively. The factorizable and diagonal scattering theories are closely related to the notion of integrability which has a deep connection to underlying Lie algebras. The S-matrices for these theories are well known and are determined from the data of the associated Lie algebras. Less known objects for these I This work was supported by JSPS Research Fellow. 0550-3213/96/$15.00 (~ 1996 Elsevier Science B.V. All rights reserved PII SO 5 50- 3 213 ( 96 ) O007 9- X
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T. Oota/Nuclear Physics B 466 [FS] (1996) 361-382
theories are form factors, i.e. matrix elements of operators. If we know all form factors then the correlation function can be calculated in principle. In integrable theories, the form factors can also be determined by the form factor bootstrap approach [ 1,2]. The form factors for the diagonal scattering theories are known for the thermal perturbation of the Ising model [3,4], the scaling Lee-Yang model [5], the sinhGordon model [6,7], the Bullough-Dodd model (A(22)) [8] and the Ol,3-perturbed non-unitary M3,5 model [9], all of which contain only one type of particle. For models whose spectrum have many types of particles, form factors are hardly known, except for minimal a(2) "~2N theories which contain N kinds of particles [ 10] Two-body form factors for the magnetic perturbation of the Ising model can be found in [ 11 ]. More knowledge of form factors for models whose spectrum have many-particle species is desirable to understand the off-critical properties of the models. One way of treating the models with many species of particles is proposed by Koubek. She shows that for minimal Za2N A(2) theories (~bl,3-perturbed M22N+3 minimal conformal field theories), all recursion relations for form factors can be simplified to the recursion relations for the form factors which contain only one kind of particles [ 12]. All particles (or excitations) in these theories are self-conjugate, i.e. a particle and its anti-particle are identical. But for the case of non-self-conjugate theories, the kinematical residue equations cannot be used directly to determine the form factor of one-particle species. In this case, the recursion relations for one particle become rather difficult to solve. In this paper, we take a straightforward approach: trying to solve simultaneously the system of recursion relations. We mainly deal with the A~1) affine Toda field theories, especially the A~1) case. The A~ ) is a theory of N scalar fields with the lagrangian ~_. = 1
tz
_
m2 N Z
ega~.O,
where ai are the simple roots of the AN algebra. Analyzing the additional pole structures, we determine all recursion relations for the form factors. Using the recursion relations, we determine the two-body form factors for ~b<, (a = 1. . . . . N) and for the trace of the energy-momentum tensor O ( x ) . From this knowledge we are able to derive the next leading term of the two-point functions of ~b~ (and those of O ( x ) ),
(4,<,(x) @<~(o))=
~lF~°l'Ko(malxl)+Z
f
d.a(2:).d~' IF:° ('~ - ~')1'
b,c P2(~)>PT(/~')
× exp (-Ixl
T. Oota/Nuclear Physics B 466 [FS] (1996) 361-382
363
form factors. In Section 3 we derive the form factors of A~ 1) affine Toda field theory up to four-body ones, and we identify the fundamental operators. The generalization to AN affine Toda field theories is discussed in Section 4, and we determine the two-particle form factors. Section 5 is devoted to conclusions and discussion.
2. The form factor bootstrap 2.1. Equations f o r the f o r m factors
The matrix elements of a local operator (.9(x) t
/
!
]~alLI 2 •..a m [ I~ [
f~!
alO2...a,, t b-'l,/a2 . . . . . t
/
t
!
/3mill,/32 .....
/3n)
/
= ~, a ~ ° , (/3,,/3~ . . . . .
/3" I O ( 0 ) I / 3 ' , / 3 2 . . . . .
/3.)o,a2
~,,,
(2.1)
are called form factors of general kind. Here/3i is the rapidity of a particle of species ai. Consider the following form of the matrix elements:
rata2...a.
(/31,/32
. . . . .
/3n) = ( 0 1 0 ( 0 ) I / 3 ' , / 3 2
.....
/3n)o,o: ...... .
(2.2)
The general form factors (2.1) are related to the functions (2.2) by analytic continuation t r ¢/1 , . . ~ 3
r;,, ...... : ( / 3 ' , , . .
. .ca'lbt . .
•,/3.,I/31 ..... /3.) ' ,~a' b,, ~. i ot + i~ . . . . . /3m t + i~',/31 . . . . . /3,), ~ °' rb,...b,,a~...a,,~loj
(2.3)
provided that the set of rapidities/3' are separated from the set/3. Here Cab is the inverse of the charge conjugation matrix C~b and for ADE scattering theories Cab = 6~b. Watson's equations for diagonal scattering theories take a simple form, Fa,...a,,,,,,...a,, (/31 . . . . . /3i,/3i+1 = Saiai,l
(/3i -
. . . . .
/3n)
/3i+1) Fa,...ai+,ai...a,, (/31 . . . . . /3i+1, fii . . . . . /3,),
F,,,,,2...a,,(fll + 2~i,/32 . . . . . /3,) = Fa2...~,,a,(/32 . . . . . /3,,/31).
(2.4)
The simple pole structure of the form factors is summarized by the following two types of the recursion relations 2 . The first kind of relations are called kinematical residue equations: - i res/3,=~+/~ Faaa,...a,, (/3',/3,/31 . . . . . /3,) /
\
= Fdl...d,,(/3l . . . . . /3n) ( l -- f ' l S a d j ( / 3 - - / 3 j ) ~x
j=l
~ • /
And the second kind of relations are called bound-state residue equations; 2 If the S-matrices contain double poles an additional simple pole structure appears.
(2.5)
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T. Oota/Nuclear Physics B 466 [FS] (1996) 361-382
--i resfl,=fl+io2b Fabd, ...d,, (/3',/3,/31
. . . . .
/3n )
= FCaaFea,...d,,(/3 + iOgc,/31 . . . . . fin),
(2.6)
where 0 = ~- - 0. The on-shell three-point vertex F~b is given by - i res~=i0,;~, S~b (/3) = (Fcb) 2
2.2. Minimal form factors In the case of n = 2, Watson's equations reduce to
Fab(B) = Sab(/3) F o a ( - f l ) , F,,b (/3 + 27"ri) = Fba (--fl).
(2.7)
The general solution to Watson's equations takes the form [ 1 ]
Fa, .......(/3, . . . . . /3,,) = Ka,...a. (/3, . . . . . fin) I I F(a,~n) (/3i - /3j ),
(2.8)
i
~(min)
where • ab is the solution to Eq. (2.7) which is analytic in the strip 0 ~< Ira/3 ~< 2~" and has no zeros in 0 < Im/3 < 2~r. The building block of the diagonal S-matrices is (x)~-
(x)+(~) (-x)+~)'
where 1 sinh 1 (
(x) +(~) = tTr
7r)
-2 fl + i-£x
.
We write the basic building block of a minimal (two-particle) form factor corresponding to (x)B as the following form: .,
I,,
gx(/3)
fx(/3) = stun 2Pgzh----f~.(/3)
(2.9)
which has no poles and zeros in the strip 0 < Im/3 < 277"for 0 ~< x < 2h. The function gx(/3) is given by
gx(fl) -- I - [ ,,=,
F ( n +/_P__ _ ~ + 1) _ 27r 2h __
/'(n ~
(2.10)
1)
The function g x ( ~ ) has poles at /3 = - i ~ r x / h + 2 ( m + 2)¢ri for m = 0, 1,2 . . . . and zeros a t / 3 = - i q r x / h - 2m~ri for m = 0, l, 2 . . . . . The introduction of the function gx simplifies the calculation. Using the following properties of gx:
T. Oota/Nuclear Physics B 466 [FS] (1996) 361-382 . "B"
gx (fl + t'~y) = gx+y(fl),
365
(2.11) 1
gx (fl + 27ri) = gx+2h(fl) = (--~+gx(fl), 1 gx(iTr - fl) = g3h-x(fl)
(2.12)
(h-x)+ gh-x(fl) '
(2.13)
g~ (0)g4h-x(0) = 1,
(2.14)
the behavior of the minimal form factors under the recursion relation is easily determined. For ADE scattering theories, the basic building blocks of the diagonal S-matrices are (x)# = (x)+(#)/(-x)+(#) with (x - 1 ) + (x + 1 ) + (x)+ =
for perturbed conformal,
(x - l ) + ( x + 1)+
(~ T~~i--
B)+
for affine Toda,
where h is the Coxeter number of the associated Lie algebra. So the building block corresponding to (x) is Gx(fl)
F~min) ( # ) -
(2.15)
G2h-x(fl) ' where
Gx(fl) =
gx-I ( f l ) g x + l ( f l )
for perturbed conformal,
gx-l(fl)gx+l(fl) gx-t+B(fl)gx+l-B(fl)
for affine Toda.
We list some properties of Gx: .77"
(2.16)
Gx (fl + t - ~ y ) =Gx+>,(fl), G x ( # + 2~-i) =
Gx( iTr - fl) =
Gx+2h(fl) ~--~+Gx(fl), =
1 G3h-x(fl)
(2.17) (2.18)
- Gh-x(fl)'
Gx(O)Gah-x(O) = 1.
(2.19)
If the S-matrices have the following form:
s,,b(#) = l-I (x>~,
(2.20)
x C A,,#,
then the minimal solutions of Eq. (2.7) are written as Ft( rain
,h
(#) = ]-I Fx~m~"~(#) xEA~I,
(2.21)
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Here Aab is the minimal set of numbers which gives the correct Sao. The multiplicity of p in A,b is denoted by mp(Aab). The sets Aao are chosen such that m p ( A , b ) = 0 for p~<0orforp~>h. The crossing condition of the S-matrix S~6(irr - / 3 ) = Son(/3) is equivalent to the condition mh-p ( A~b ) = mp ( Aba). The minimal form factor (2.21) has the following property: F(min) :_x ab t-~ , ~ ' "_/._ )% b L-(min)(/3) = l/sCab(/3) •
(2.22)
Eq. (2.22) is an analog of the S-matrix relation Sab(/3 + irr)Sab(/3) = 1. There is a one-to-one correspondence between the constituent of minimal form factors and the Smatrices. Many properties of Fx~min) are similar to those of (x) but the monodromy property is quite different. F~ rain) has a diagonal monodromy which comes from Eq. (2.17) while (x) is 2rri-periodic. Due to these monodromy factors, additional functions (,b(/3) appear in the right-hand side of Eq. (2.22). The factor 1/(ab is given by the product of (x)+ 1/(ab(/3) = H {X)+(/~). xE A,t,
(2.23)
In particular, for affine Toda cases,
(x)+ =
( x - - 1)+(x q- 1)+ (x- I + B)+(x + I - B)+
implies that B dependent parts appear only in the numerator of ~b. The form of sc has been determined for the scaling Lee-Yang model [5], the sinh-Gordon model [7] and for the Bullough-Dodd model [ 8] by explicit calculation. Using Eq. (2.7), we get the following relations: Sab(/3) - (ab(/3 + irr) _ C a b ( - ~ 3 ) (ab(/3)
~ab(/3)
(2.24) "
Under the factorization of Eq. (2.8), the kinematical residue equations reduce to --i res#,=/~+irr Kaad,...d,, (/3',/3,/31 . . . . . /3n ) = Kd,...a,,(/31 . . . . . fin)
(j_I~1( a d j ( / 3 - - / 3 j ) -- H ~ a d j ( / 3 j - - / 3 ) ) / Faa(min)(tzr). . j=l (2.25)
In accordance with the S-matrix bootstrap Sad(~3 q- i ~ b c ) S b d ( / 3 --
iOgc) = Sed(fl),
(2.26)
the minimal form factors have the following properties:
"adP(mi(/n~)-t---~l'lT"abac ~) ~(min) r ( fl -- lO-abc
"~ "~( min ) ( /3 ) / l~Cb;d ( /3 )
(2.27)
T. Oota/Nuclear Physics B 466 IFS] (1996) 361-382
367
The function A is known for the scaling Lee-Yang model [ 5] and the Bullough-Dodd model [8,13]. These models contain only one-particle species. The extra factor 1/aCb;d(/3) comes from the diagonal monodromy and is given by the product of (x)+:
l/<;h:d(/3) =
(U~c -b - x)+(e)
II (xEA.dIX
II
(x - Ubc}+(e ). -a
(2.28)
{xeAbdlX
c = hO~b/rr and Uab -c = h -Here blab From Eq. (2.28), we can see
Uab.
a,'ib;~ (/3) = a;o;a ( - / 3 ) . The associativity of Eq. (2.22) and Eq. (2.28) leads to
Aco,,;d(/3 - iOCab) A~ab;d(/3 -- iObac) = (aa (--/3). The bound-state residue equations reduce to •
!
- - t res/3,=/3+io; b Kabd, ...d,, ( / 3 , / 3 , / 3 1 . . . . .
/3n ) n
C
'--O
C
"
•
= VabKe&...d,,(/3 + tObc,/3, . . . . . fin) IIaaO;dj(/3 + iOgc -- flj)/F(a~un)(tOCab). j=l
(2.29) For ADE scattering theories, the S-matrix can be written as [ 14] h--1
Sab(/3) = 1"I ((2p + 1 + eab)+(,a)) u'"''w Pga~.
(2.30)
p--O
Or equivalently
mx(Aab) = tx (a) • w - P ~ b
for 0 < x = 2p + 1 + Gab < h.
H e r e / z (") is the fundamental weight of the algebra. So the minimal form factors can be given by 3 h-I
Fab(min) (iS)
=
6(cob l)/~(a~.w~b (0 )~._ , II(G2pq_l_}_ffab(/3))~l,(a)'w P4'b.
(2.31)
p=O
Here e,,o = ½( c ( a ) - c(b) ). Depending on the two colors of the Dynkin diagram of the associated algebra, c ( a ) = 1 for white nodes and c ( a ) = - 1 for black nodes [ 15]. Note that we can see c(fi) = ( - 1 ) h c ( a ) . If the Coxeter number h is even, which holds except for A2N theories, Gab = Cab. 3 For the minimal cases, if the S-matrix has a fermionic nature: Sab(O) = -- 1, then one more factor (0)+03) is needed.
T. Oota/Nuclear Physics B 466 [FS] (1996) 361-382
368
The first adjustment factor on the right-hand side of Eq. (2.31) is introduced in order that Fa(fffin) is constructed from Gx (0 ~< x < 2h) for Cab = 1. On the right-hand side of Eqs. (2.25) and (2.29), for the affine Toda cases, coupling dependent parts (x dz B)+ only appear in positive powers, so the singularities of K can be factorized by products of 1/(x)+. 1
K,,,...,,,, (ill . . . . . fin) =Qat...a,,(fll . . . . . fin) 1-I (eft, +eflj'C""J'",aj'°il 'ra i
~P (2.32)
The function Wab contains the factor (UCb)+(--UCb) + for bound-state poles. The factor (--Uab) + is needed to make Wab symmetric: Wba(--fl) = Wab(fl). In general, Wab must contain more factors to factorize the higher-order poles in SCab. The polynomial Q~...a, carries the information about operators. Counting the number of independent solutions, Qat...~,, can be used to classify the operator content of the model [3,6,12,21,16]. We expect that Qal,...,a,, are polynomials in ( x ) + ( - x ) + . Substituting Eq. (2.32) into Eq. (2.25), the function Wab is determined from the requirement that W~b(fl + iTr)Wab(fl)~ab(t~) is a product of the factors (x)+ in positive powers. Using the expression (2.23), we write the singularity of ~ab a s follows: h
1 1-I ( x -
xG A,,j,
1 ) + ( x + 1)+
= l-i(p)+m,_,(Ao~)-m~+,(A~) p=O
Even-order poles do not correspond to the bound state. If (~b contains the factor 1/(x) 2k, the half 1/(x)~+ is canceled by W~b(fl) and the other half is canceled by W~b(fl + i~r). Then W~b(fl) must contain (x)k+ (--X)k+, and Wnb(fl) has ( x - h )k+( h Odd-order poles can be interpreted as the production of a bound state. W~b contains the factor (U~b)+(--U~b)+ for the bound state in the forward channel. If (,~b contains the factor 1 / ( x ) 2k+l then Woo has the factor (x)k++1 and Wab has the factor (x - h)k+ for the forward channel. Dorey's "uphill/downhill" mnemonic [ 14] implies that mp-i - mp+l = + l , 0 , - - 1 and the case m p - i -- mp+l = -k-1 corresponds to the forward channel. nip 1 The order of the pole at Imfl = plr/h is mp-i -}- mp+l. We need the factor (p)+(~) for Wab(~). The above consideration leads to the following form of the parameterization:
where Pab(t~) =
1-I (X)+(fl). (xeAab[x4=h--1}
369
T. O o t a / N u c l e a r Physics B 466 [FS] (1996) 3 6 1 - 3 8 2
a~
~ b k
/
k f
b
a
Fig. 1.
The singularity structure of (ab is similar to that of Sab. The singularities of the Smatrices for ADE theories are explained in terms of multi-scattering processes [ 17,18 ]. So it is natural to expect that the factorization in the above admits such an interpretation. The singularity
1
1
,-,,.,
(x) +(/~) (x) +(_#)
coshfl - c o s ( T r x / h )
is explained by the Coleman-Thun mechanism [ 17]. Recalling i (Pa + P~,) 2
i __
m2
= mamb
1 cosh fl - cos OCb '
this singularity comes from a delta function O ( p o ) ~ ( p 2 _ m 2) according to the Cutkosky rules. For perturbed conformal theories, Delfino and Mussardo [ 11 ] have derived the parameterization of the singularities of two-particle form factors from the Feynman diagrammatic analysis of multi-particle processes. Their factorization agrees with our result. For the Nth order pole in form factors corresponding to the fusing ab --+ e, the parameterization is explained by the diagram of Fig. 1, whose blob represents a connected diagram and contains 2N - 1 three-point vertices. This is essentially the half of the diagram which explains the (2N - 1)th order pole of the S-matrix. The Nth order pole of the form factors, which corresponds to the 2Nth order pole of the S-matrix, may be explained by a diagram ab --+ cd whose blob contains 2N three-point vertices. We determined the parameterization both for the perturbed conformal theories and for the affine Toda field theories. For affine cases, no additional coupling dependent pole appears in the physical strip. So the parameterization is consistent with the mass spectrum of the theory.
T. Oota/Nuclear Physics B 466 [FS] (1996) 361-382
370
3. T h e A~l) affine Toda theory The A2(j) theory is the simplest model based on the Lie algebra which contains nonself-conjugate particles. It contains two kinds of particles, which are denoted by 1 and 2. The particle 2 is the anti-particle of 1 and vice versa. The S-matrices of this theory are given by Sll = $22 = (1) and S12 -- S21 -- (2) [1820]. For definiteness, we consider the A O) affine Toda theories. The A(21) affine Toda theories is a theory of two scalar fields ~bl (x) and ~b2(x). So we concentrate on form factors of scalar operators. We define FI,,,.,,1 (ill,
f12 . . . . .
]~m;/~,/~2 .....
= F l l .. 122 . . 2 (. / ~.1 , ./ ~ 2. , . .
/~'n)
t) • f l m , f l lt , f l 2t . . . . , a,'-n
(3.1)
And we factorize KIm.n] as follows: Kim,nl(fll,.. • ,/3m,/31,. . t
. . , /3'.) Q[m,n] (Xl . . . . . Xm; Yl . . . . . Yn) "1-yj) I-Ii
l-Ii
(D-2yj) (3.2)
with xi = e/~i, Yi = e #' and w = e i'n/3. T h e degree of the polynomial QL,n,nl is given by 1) - mn. For simplicity, we use the vector notation deg(Q[m.,]) = ( m + n ) ( m + n x = (Xl,XZ . . . . ) etc. Then the kinematical residue equation is reduced to -
(3.3)
Qt,.,,+l.n+l j( X, - x ; x, y ) = x D [ m , n 1 (X; x,Y)QIm,,,I ( x ; y ) ,
where Dlm,n I (x; x , y ) = ( - 1 ) n H2 (Dm( x; x; -(D ) Dn( x; y; (D - l ) - O m (X; x; _(D-1 )Dn(x; y; (D)).
The function Dm is defined by m
Din(x; x; Z ) = I I ( x
+ ZXj) ( x - z q x j ) ( x - z q - l x j )
j=l
nl rn-- k
=
Z~-~'(--1)k[k+ljq
1 x3m-2r-k-lz2r+k+l
O./~i X X) S (2",1), k [ X J, "~
l--0 k=0 r--0
(3.4) where [ k ] q = (qk _ q - k ) / ( q
_ q-l)
and the Schur function S(2r,lk/ is
S(2r,lk ) = ( O'r+kO'r_ 2 -- Orr+k_lO'r_l ).
Here o-j are the elementary symmetric polynomials defined by
T. Oota/Nuclear Physics B 466 [FS] (1996) 361-382
371
I1|
111
I-[(x + xj) = j=l
j=0
The B dependent parameter H2 is defined by (-i) H2 - F(~nin)(i77. ) •
The coupling dependent parameter q = e it(B-I)~3 transforms into q - i under the weakstrong transformation B ~ 2 - B. The bound-state residue equations are reduced to Q Im+2,,,I(X, w y , w - l y ; y) = Hy2Dm (y; x; 1)Q[m,n+ll( x; y, y), Q[m,n+21 (x; wx, w - I x , y) = HxZ D,n( X; y; 1)Q[m+l.n]( X, x; y) ,
(3.5)
where H=
v~F
(min) 2__:'~ =
F11
( ~"" J
V~F F2(min)(27ri) " 2
',
The function /" is given by ( r ) 2 = (F{1)2 = (VzZ2)2 = ~
sin ~7rB sin ~ 7 r ( 2 - B) sin 1~-(4 - B) sin ~ ( 2 + B)"
(3.6)
There is the relation between H and H2
H2/H2 = (1 + w) (1 + w[2]q),
(3.7)
which is equivalent to the minimal form factor relation (F/~nin) ( 2 _ : , ~2 / ~:,(min)
T"')J
/"12
(i7"r)= (3)+(0).
(3.8)
We first pay attention only to the indices [m,n]. The bound-state residue equations relate QIm.nl to QI,,,-2,,+I1 or Qp,,+I,,-21 and the kinematical ones relate Qlm,nl to Qlm-J,n-ll. We identify [m,n] with a two-dimensional vector and introduce the following equivalence relations: Ira, n] ~ [re, n] - 1112,-1] - 1 2 [ - 1 , 2 ] - 1311, 1],
(3.9)
where li C •. Because [1,1] is equal to [ 2 , - 1 ] + [ - 1 , 2 ] , the above definition is redundant. But for later convenience, we write here the [ 1,1 ] term. Then there are three equivalence classes
{[m,n]}/~=[[O,O]]+[[1,O]]+[[O,l]].
(3.10)
In other words, if [m, n] and [m ~, n ~] belong to different classes then Q[,,,,nl and QIm',n'l are not linked by residue equations.
T. Oota/NuclearPhysicsB 466 [FS] (1996) 361-382
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So the set o f form factors is divided into three sectors. In each sector, higher polynomials are determined iteratively from lower ones. In the minimal polynomial space, the solutions can be determined uniquely except for the kernel ambiguity. Note that [ 2 , - 1 ] and [ - 1 , 2 ] are equal to the first and second rows of the Cartan matrix of the A2 algebra, respectively. The index [m,n] can be identified with the Dynkin indices of weights and with corresponding weights/z /z
= [mj, m2] =
ml/z Cl) + m2/z ~z),
(3.1 1)
w h e r e / z ~') are the fundamental weights. Then the equivalence relation can be rewritten as tbllows: /z ~ / z - llCel - 12ce2 - 13(cel +
(3.12)
O~2).
Here al and ce2 are the simple roots of the A2 algebra. For the A~ j) theory, the equivalence relation is generated by all positive roots. In the polynomial space we are considering, the degree of a polynomial is equal to the degree of the kernel and the kernel is one dimensional. So at every recursion step, only one parameter enters into the solution space and the parameter Au is attached to the point u in the dominant weight lattice. The most general solution of the residue equations (3.3) and (3.5) has the following form:
Q ~ ( x , y ) = ~ Hlu-~IA~Q~,~(x,y). u<<.l~
(3.13)
Here the ordering of weights/z ~> ~, means that the difference/x - z, lies in the dominant root lattice and I/x[ = ml + m2 f o r / x = [ml,m2]. The polynomial Qu carries the information about the operators. Each Qu,~ satisfies the residue equations, so gives an independent form factor of some operator O~.
3.1. The [O,O]-sector We start from the [0, 0]-sector to solve the recursion relations. Although we call it the [0, 0]-sector, the kinematical recursion relation is not applied to QI l,ll ---+Qlo,ol 4 So the first polynomial is QIx.n I. The most general degree 1 polynomial is QI1AI (x, y) = All,1 ix + A~ 1,1]Y,
(3.14)
where A 11,11 and A~ 1,11 are constants. But the recursion relation Q13,01 ~ Q[ 1,1~ has no solution unless A I 1.11 = A~l,ll. Similar phenomena occur at higher stages of recursion processes. These additional constraints come from Z2 symmetry corresponding to the charge conjugation. These Z2 constraints are imposed on the constants: Aim,n] = A[n,ml. 4 If the recursion process was started from QI0,0I formally,we would only get QIm,nl solution corresponds to the "form factor of the identity operator".
=
Q[0,018m.o8n.o.This
T. Oota/Nuclear Physics B 466 [FS] (1996) 361-382
373
The first few solutions in the [0, 0]-sector are given by QI l,I I (x, y) = All,I] (x + y ) , QI3,01 = AI3,0]BI13,01 + HA[1,1]0-10-2 (0"10"2- (2 + [2]q)0-3) , Q[ 2,2] = A[2,2]B1 [2,21KI2,21B212,2] + HA[3,01Q[2,21,[3,01 + H2AII,I 1QI2,21 ,I Lll. Here Q12,21,1o,31 = ( ~ ( x ) ~ ( y ) ×((~2(X)
- ff2(x)~2(y)) --~2(y)) 2 +~l(X)~l(y)(~2(X)
+~2(x)~(y)
+~2(y))
+ ~(x)~z(y)),
Q[2,21,11,11 = (0-1(x) + 0 - 1 ( y ) ) ( 0 - 2 ( x ) 0 - 1 ( y ) + 0 - 1 ( x ) 0 - 2 ( y ) )
×(0-1(x)0-1(y)(0-2(x) + 0-2(Y) + 0-1(X)0-t(y)) --(1 + [2]q)0-2(X)0-2(y)). The polynomial KIn,, 1 is given by m
n
Ktm'"l(x'Y) = H I I
(xi + YJ) = Z
i=1 j=l
sa(x)s;~(y),
(3.15)
a
which is the kernel of the kinematical residue equations. Here sa is the Schur function and the summation is taken for the partition A = (AI . . . . . Am), i.e. non-increasing sequences of non-negative integers under the condition ,~l = n and ~ = (m - A~,. . . . . m A~l) [22]. The partition ,V is the conjugate of the partition A. The polynomials BaIm,nl (a = 1,2) are given by
BII .... I ( x , y ) = 1--[ (xi -- tO2Xj) (Xi -- ¢O--2Xj) = S2sm(X),
(3.16)
i
B21m,nl ( x , y ) = 1-I (Yi - w2Yj)(Yi - w - 2 y j ) = sz~,.(y),
(3.17)
i
which are the kernel of the bound-state residue equations. Here ~m = (m - l, m 2 . . . . . 1). The Schur function can be expressed as [22]
s a = det (0- a;-i+J ) l <~i,j<~l(a, ) ' where I(A) is the length of the partition A. The polynomials Q[n,,nl,[1A] ( [ m , n ] v~ [1, 1]) have the following forms:
QIm,.1,I IA I( x; Y) = (0-1 ( x ) + 0-1 (y) ) ( O'm-l ( X )O'n(y) + 0-m( X )O'.-l (y) ) X P[m,,,l ( x ; y ) .
(3.18)
374
T. Oota/Nuclear Physics B 466 [FS] (1996) 361-382
From the stress-energy conservation, it is possible to show that the polynomials which enter the form factors of the trace of the stress-energy tensor O are factorized as (3.18). So the operator OiL11 is identified with O. This fixes the constant A[l,ll to be 7rM 2 A[I,11 - 2F~nin)(iT.r) ' where M is the mass of the particles. In this sector, we determined the form factor up to four-body ones. For example, the explicit form of the two-body form factor is given as Fl2(fl) = F~(fl) =
"n'M2 F(2nin) (fl) 2 F(~n)(iTr) "
(3.19)
3.2. The [1,O]-sector and the [O,l]-sector The solutions in the [0, 1]-sector are simply obtained from the [ 1,0]-sector using Z2 symmetry. So we only deal with the [ 1,0]-sector. The first few solutions in the [ 1,0J-sector are given by Q[ l,Ol = All,01, Q[0,21 = A[o,2]B2[0,21 + HAl 1,0]0-(22), Q[2,1t = A[2,I]B212,l]K[2,,l-t- Ha[o,210"~2) KILl I + H2AlI,O]0-12)0-~ 2) 0"I 1), Q[4,01 = A[4,01Bl[4,0] + HAl2,11Q[4,0],[2,1] + H2A[o,2]Q[4,0],[o,2]
+ H3A[1,0lQI4,0],[ 1,01, Q[l.31 = A[ j,3IB2[ I,3IKI I,3I + ttA[2,1IQ[1,3I,[2,1I + H2A[o,2]Q[1,31,[0,21 +H3A[1,0IQ[ 1,31,[1,0l. The explicit forms of the polynomials Ql.,,nl,lm',n'l in the above equations are given by Ql4,01,[2,1] -- (2 + [2]q)0-2(-0-1o-~
- 0-~0-3o-4 -t- 2o'10"2o'3o4 -q- 0"20"4 + 0"~0"2) 2 2 2 - 0"20"4 - 20"10"30"4 -t- 0"2), + (0"92 + ( 2 1 2 ] q + [2]q)0"4)(0-10"3
Q[4,o1,[o,2] = 0"4
((2 +
Q[4,0],[l,0] --- 0"4
((2
[ 2 ] q ) (0"4 - 0-10"3) (0-4 --~ 0"2 _ 0"10-3) 3t- ( [3]q - 5)0"20"4),
+ [ 2 1 q ) 0 " 2 ( - 0 -2 - 020"4 + 0"20"4)
-t-(0"2 + ( [3]q -k- 312]q -k- I)0"4)(0"1o"3 -- o'4)) , ,..r(3) .,r ( 1) /' .~.(3)._,.(3) 3))(3) Q[l.31,[1,ol = v 3 vl ~,v I v 2 - ( 2 + [2]q)0"~ ..0"(2
el, 31,i0,21--0"I
~'-'1 v2
-- (2
+ [21q)0"~3))
(3)_(1)~ +0"1 t~l ) ,
[ " (3).2)0"3(3) -1-'v2/"(3)~2~(1)~, ~,1,0-1 Vl ,]'
,1.(3),1.(3) //.1.(3),1.(3) 3)) QI1,31,12,11 = v l ~2 ~ 1 v2 - ( 2 + [2]q)O-~ K[I,3].
The form factor of the fundamental operator is factorized as follows:
T. Oota/Nuclear Physics B 466 [FS] (1996) 361-382 Q~,~( x , y ) = ¢r(mm)( X )Cr(nn) ( y ) Pu,~( x , y ) .
375 (3.20)
The operators 011,0 ] and 0[0,1 ] correspond to these fundamental operators, and others are composite operators. Note that [ 1, O] and [0, 1 ] are fundamental weights of the Az algebra. The operators labeled by the fundamental weight correspond to the fundamental operators. From the conservation of the Z3 charge, Ou~,, = dpa (a = 1,2). Here ~ba are affine Toda fields. The requirement F . ( f i ) = F ~ ( / 3 ) = (0l~ba(0)l/~)a = ~
1
(a = 1,2)
sets the constant Atl.Ol = A[0,1l to be 1/x/~. In the [1,0]-sector and [0,1]-sector, we also determined the form factors up to four-body ones. For example, the two-body form factor is given by )=_
4. T h e
~.(min) r ,22 (/3) 2cosh/3 + 1 •K'(rain)t" 2__:'~ " 22 t ~'~lt)
V• 3
AN a f f i n e
Toda theories
The AN theory contains N kinds of particles, which are denoted by 1 . . . . . N. The anti-particle of a is ~ = h - a = N + 1 - a. The mass of the particle of type a is given by M,, = 2 M s i n ( a T r / h ) [19]. As for the case of the A2 theory, the AN form factors are divided into N + 1 sectors. The sector specified by the fundamental sector contains the fundamental operator, and the zero sector contains the identity operator and the stress-energy operator. Physically, these sectors simply come from the decomposition of the states into the different ZN+l-charge sectors. For the AN theory, the explicit form o f , lt;,(min) ab can be written as a+b- 1
F(min) ,t, ( / 3 ) =
II
h-]a+b-h l - 1
Fx(min)(/3) =
x=la-bl+l
step 2
1-1
Fx(min)(/3) "
x=la-bl+l
step 2
(min) Using /";x' ( m i n~.r-'.,~ ) f a ~ 2h-x ( m i n )~.PI ¢ ~ = 1, we can show F~b (fl) = Fa(~un)(fl)
The monodromy factors ~ and ,~ are given as follows: h-la+b-hl-I
=
I-[ x=la-bl+l
step 2
T. Oota/Nuclear Physics B 466 [FS] (1996) 361-382
376
b-la-dl-I
II
(x)+(#)
for a + b + c = h,
v(a,b,d)+l step 2
1/ a~b;d(fl) =
(4.1)
b-la-dl-I
II
(x)+(#)
for a + b + c = 2h,
v(a,b,d)+l step 2
where v ( a , b , d ) = [b-all + b - d - l a + b - d [ . The explicit form of on-shell three-point vertex is uCab--3
(FCb) 2 = F(h)(2UCab --
1-[
1}+(0)
(u~ + x}+~o)
(4.2)
x=la_b]+ 1 (Uab -- 2 ) + ( 0 ) ' step 2
where 2
2~" / sin
F(h)=
7/"
for perturbed conformal, "/7"
_ (cos h _ cos ~ ( B _ 1)) / sin_~
for affine Toda.
c = h - [ a + b - hi for a + b + c = 0 mod h. For the AN theories, Uab 77"
q'/"
2
for perturbed conformal,
( os x-cos )/(cos x 77"
77"
77"
_ COS ~
for affine Toda. Especially for the A2 affine Toda theory, ( F ) 2 = F(3)(3}+(0) which agree with the previous result. The on-shell three-point vertex /~cb can be expressed as
{(r~/rorb) 2 (r~b)2= (r~/r~rb) ~
for a + b + c = h, fora+b+c=2h,
(4.3)
where Fa = IId-i 1 Fhd d-I • The S-matrices for the AN theories ( N ~> 3) contain double poles. The singularity of form factors are parameterized by the following functions: %b--2 8 a+b.h
Wab(fl) =
fl
(x)+(--x)+.
(4.4)
x=la-bl+2 step 2
In the above factorization, we allow the case c = 0, i.e. a + b = h. For a + b = h, the c is taken that Uab 0 = h. Corresponding to the double pole of the S-matrices, constant Uab additional simple pole appears at (relative) rapidity
377
T. Oota/Nuclear Physics B 466 [FS] (1996) 361-382
/
'i
I
b " Fig. 2.
fl=ih(u]b-2k)
• . , ~1( u , bc _ l a _ b ] )
(k=l,.
- 1 ).
So we must determine these additional pole structure of the form factors. From now on, we consider the affine cases for definiteness. For example for a + b < h, the double poles of the AN S-matrix are explained by the diagram of Fig. 2, or the crossed channel version of the above diagram [ 18]. The problem to be solved is which diagram explains the additional simple pole of the form factors. In order to fix the additional simple pole structure, we analyze some low-order form factors. The one-particle form factor F~ is constant, 1
F. =
=
The elementary two-particle form factor is given as sina ~ - a - ' F ~ " ) (/3) via ;~+l cosh/3 - c ° s Olh~-~-~ --laP(rnin)':zlh-a-l~,tVla)
Fk,(/3) = r.h-a-l ~ -'1~
for a 4: N,
F~in) (/3) =
= 2
F(~i") (iTr) "
These two-body form factors play the role of the initial conditions of the recursion equations. Solving the recursion relation for Fjj~, we determined the explicit form of F2u for
a
F2a(/3)=-t2a
f
(min) F~ (13) F~ (i~(a+2))
Va+2 (min)
sin~(a+
2)
~. cosh/3 - cos 7 (a + 2)
sin~(a +2) cosh/3 - cos 7 a
( c o s ~ - cos ~ ( B - 1) x~
Also f o r a = N -
1 =2,
F2~(/3)=2 M2F~z~n)(/3)(min)
~"
1 - 2 sin 2 "rr { cos ~ -
CO ~r
s ~ ( B - 1)
)
T. Oota/Nuclear Physics B 466 [FS] (1996) 361-382
378
x
(
Coshfl
,
-
cos~(h
,
2) + l + c o s - ~ ~"(
-
h
)}
-2)
"
One can show that "it?(min)ab \(irr ~UabC __ Fa(min)
k=l,.
.rr
( i ~ ] a - O]) 2k)) = "I7(min) l(u~b-l)
c
"
. . , 7l ( U acb -- ]a --
.Tr c
(4.5)
bl) - 1,
wherel=kfora+b~h. Using Eq. (4.5), we can see that --ires~=i~,aF2a(fl)=--ll
,-,,-2 ,~h-o-I ~ 11a rl(a+l)
(.Tr t-~(a-2)
)
.
(4.6)
For the case of a = N - 1, the factor --l(N-1)/--ll el / r N - I appears. From the form of F~t,, it holds that (F~(N_I)) 2 = ( F ~ - l ) 2. In showing Eq. (4.6), we take the phase -l
= I.
In general, it is expected that • r,h_l_m,r,h-u~b+l r:," --I resB=i~,(u~,,_2k) Fao(fl) = lira I ~mz(ml_l)rl(u~n_l) k=
1 .....
ml -
(.7"g
)
t-~]a - b I
(4.7)
1,
where l = k for a + b ~< h, l = k for a + b > h, ml = ~1 ( U acb _ l a _ b l ) and m2 = 1 c 7(uab + ]a - b]). And F h j - l = F la( h _ a _ l ) . We checked the above equation for the case of a = 3 . The additional simple pole structure is determined as follows: - i res/~,=#+i~, (a+b-2k) Fabd,...d,, (t ~t, fl, fll . . . . . t~n ) h-a r h - a - b + k F . = r k ( a - k ) (a-k)b k(a+b-k)db..d,
(
x
fl+i
(b-k),fl+t-~(a-k),fll
.....
fin
)
(4.8)
,
f o r a + b <, h a n d a <~ b.
Similar relation holds for other cases. The above pole structure is similar to that of A(2) 2N theories [2] Now we have fixed the additional recursion relations, we are ready to determine the two-body form factors• The ,,rea(1) form factors only have simple poles. The two-body form factors for elementary scalar fields can be written in the following form: g?(min) ( ( ~
F~b(fl)
-
--c -- " ab ~,l"'J - s m" ~c ttablabt' e L - ~ x Pab (tlTab)
l(u~.b--la--bD--1 Z lo--O
Bab;lo cosh fl -
CO
s" c • ~(Uab -- 210) (4.9)
This expression is regarded as an analog of the partial fraction.
T. Oota/NuclearPhysicsB 466 [FS] (1996)361-382
379
'\,
°
\\,
J
\
k ~
a+b-k \
/'
/
a-k
',,
/
',
\
a
a
b
b
Fig. 3. The additional pole structure for a + b ~ h and a ~< b.
Eq. (4.7) is equivalent to the recursion relation for the coefficient
Bah;t,
k-I
Bab;k = Cab;k -~- Z
Dab;k;IBk(a+b-k);l'
l=l
where 71"
[ Fh_ a
,~2
Cal,;k = --Aab sin - ~ ( a + b - 2k) \-~{~-k~/
sin
Aab =
,
~(a + b)FCabFe (i~(a + b))'
-- "/7(min)ab
and =--
Oab;k;l
sin~(a+b-2k) F,h_ a 2 s i n ~ ( a _ l ) sin~(b_t~--k(a-k)~!
Fh-a-b+
b(a-k)
(i~_rb_a)) k v{~a,) " k ( a + b - k ) ~,~h ~ "~-T~-~ \{ i ~~xt a ~ ---b - 2 k ) )
Using the following relations:
/ r k(a+b-k) h-a-b "X Ak(~+b-k) _ |* I F(a~n) ( i ~ ( a + b - 2k)) \ ) F~(a+b_k) (t-~(b- )) D,~b;k;t = - - 2 s i n
the c o e f f i c i e n t
Bab;lo is
sin ~ ( a + b - 2 k ) aab ~ ( a -- l ) s i n - ~ ( b - l) A k ( a + b - k )
( F kh( a- a_ k ) )2 ,
d e t e r m i n e d as lo--1 1t--1
V~(_!). Bab;Io=/_.~ 2 Z~''" .~0
t,,_,-I
ll =1
12=1 n
"rr
I-[j--0 sin Z (U~b --
× ~I. = 1 IIj=lsin~(lj_2
ff,h--lj-1
x2
2lj) \ lj(lj-i-lj) ] lj) sin~(UCb lj_2--1j)
(4.10)
T. Oota/Nuclear Physics B 466 [FS] (1996) 361-382
380
Here I-1 --'-- ~1(UabC_ la - b[) and Bab;O = 1. This gives the solution of the two-body form factors of ~bc(x). Also the two-body form factors of the trace of the energy-momentum tensor can be parameterized in the following way:
,,j - 1
x
1- ~
lo=l
1
Ba;lo ( c o s h f l - cos ~(h - 2•0)
+
'
1 ÷ cos ~ ( h - 2•0)
/)
(4.11) Similarly, Eq. (4.7) is equivalent to the recursion relation
B,i;k=_sin27rk /sin-~ k h ~ \ 2
x
(4.12)
1 + ½~'~" Bk;t s i n - ~ ( a + l ) s i n ~ ( a - l ) 1=1
Solving these relation recursively leads to the solution: --
Ba;lo
/'sin ~lO Fh_l_ I .~ 2 . zcr t sin ~a t0(t-,-to)) l,,_,-1
(santa)212-
× ~t,,:t ( - ½ ) " \ s i ~ l
r
10--1 11--1
"n~>0 l,=l tz--1 [
sin~lj\
l-lj)J,~2
f Fh_lj_l lj(ly
j=l s i n ~ ( l j _ 2 + l j ) s i n ~ ( l j _ 2 - l j )
} "
Here I-i = min(a, ~). Above expression gives the two-body form factors of O(x).
5. Conclusions and discussion
We have derived the minimal two-particle form factors for ADE scattering theories. Using monodromy properties of the building blocks of minimal form factors, we have determined the functional equations satisfied by the minimal form factors. The function A"ab;d will play a key role in constructing the solutions of the form factor bootstrap equations. We have determined the parameterization function Wab for the perturbed conformal theories and for the affine Toda field theories. For the A2 affine Toda theory, form factors are derived up to four-body ones, and the identifications of the fundamental operators have been done. For the AN affine Toda field theories, the form factors are divided into N + 1 sectors characterized by the ZN+I charges. To the chargeless sector, they correspond to the stress-energy operator. To the charge-k sector, they correspond to the elementary operator
T. Oota/Nuclear Physics B 466 [FS] (1996) 361-382
381
with charge N + 1 - k. This is a generalization of the known result for the sinh-Gordon theory (i.e. A1 Toda theory) [6,7] to the AN cases. For the AN affine Toda field theories, we have determined the two-particle form factors. Also, the additional simple pole structure of form factors has been determined. The determination of higher-order form factors remains to be solved. Before concluding this article, we state a relation between roots and equivalence relation of weights. Dorey's fusion rule for the simply laced theories [ 14] is that the fusion process a x b --, ~ occurs if W~(a) [,z(a) + WsC(b)]Lt(b) "Jr-w((C) ].~(c) = 0
(5.1)
for some integer ~:(a), sO(b) and ~:(c). The particle b is the anti-particle of a if Wsc(a)[,Z(a) "-[-w((b) ]£(b) = 0
for some integer We define the Eq. (5.1) or Eq. For the fusion e~ih =/z(a) r =Z
(5.2)
( ( a ) and ~ ( b ) . c = /z(a) +/~(b) //(c) or eab "fusion base" vector eab o = /z (a) + iz (b) if c = 0 otherwise. (5.2) is satisfied and eab a x b ~ ~, if its fusion vector is expanded in simple roots
+/Z
(b) __ /[Z(~)
(5.3)
md°Ld '
d=l then the coefficient md takes integer values. This is equivalent to the statement that for a x b ~ (: the following condition is necessary: (C-1)ad + (c-l)bd
_
_
( C - l ) e d E Z.
Here C is the Cartan matrix of the algebra. For the E8 algebra, all elements of C -1 are integers, so the above condition is always satisfied for any (a, b,c). Only for the AN algebra, the above condition is also sufficient. Using the explicit form of the inverse of the Cartan matrix for the AN theories, (C-t)ab =
1
N + I min(a'b)(N+ 1 - max(a,b)),
one can show that the above condition is equivalent to a + b + c = 0 mod N + 1.
Acknowledgements I am grateful to Professor R. Sasaki for instructive suggestions in the early stage of this work, and to Professor H. Itoyama for useful discussions.
382
T. Oota/Nuclear Physics B 466 [FS] (1996) 361-382
References [ 1 I B. Barg, M. Karowski and P. Weisz, Phys. Rev. D 19 (t979) 2477; M. Karowski and P. Weisz, Nucl. Phys. B 139 (1978) 455; M. Karowski, Phys. Rep. 49 (1979) 229. 121 EA. Smirnov, Form Factors in Completely Integrable Models of Quantum Field Theory, Adv. Series in Math. Phys. 14 (World Scientific, Singapore, 1992) and references therein. [31 J.L. Cardy and G. Mussardo, Nucl. Phys. B 340 (1990) 387. 14] V.P. Yurov and A1.B. Zamolodehikov, Int. J. Mod. Phys. A 6 (1991) 3419. [ 5 ] AI.B. Zamolodchikov, Nucl. Phys. B 348 ( 1991 ) 619. 16] A. Koubek and G. Mussardo, Phys. Lett. B 311 (1993) 193. 171 A. Fring, G. Mussardo and P. Simonetti, Nucl. Phys. B 393 (1993) 413. I81 A. Fring, G. Mussardo and P. Simonetti, Phys. Lett. B 307 (1993) 83. [9f G. Delfino and G. Mussardo, Phys. Lett. B 324 (1994) 40. [101 EA. Smirnov, Nucl. Phys. B 337 (1990) 156; Int. J. Mod. Phys. A 4 (1989) 4213. [ I II G. Delfino and G. Mussardo, Nucl, Phys. B 455 (1995) 724. [ 12] A. Koubek, Nucl. Phys. B 428 (1994) 655. [ 13] A. Fring, Form Factors in Affine Toda Field Theories, USP-IFQSC/TH/93-07, hep-th/9304140. [141 E Dorey, Nucl. Phys. B 358 (1991) 654; B 374 (1992) 741. 115] A. Fring, H.C. Liao and D.I. Olive, Phys. Lett. B 266 (1991) 82. [ 16] EA. Smimov, Counting the local fields in SG theory, hep-th 9501059. [ 171 S. Coleman and H. Thun, Commun. Math. Phys. 61 (1978) 31. [ 18] H.W. Braden, E. Corrigan, P.E. Dorey and R. Sasaki, Phys. Lett. B 227 (1989) 411; Nucl. Phys. B 338 (1990) 689. [ 19] A.E. Arinshtein, V.A. Fateev and A.B. Zamolodchikov, Phys. Lett. B 87 (1979) 253. [201 T.R. Klassen and E. Melzer, Nucl. Phys. B 338 (1990) 485. [21] A. Koubek, Nucl. Phys. B 435 (1995) 703; Phys. Lett. B 346 (1995) 275. 122] I.G. MacDonald, Symmetric Functions and Hall Polynomials (Clarendon, Oxford, 1979).
NUCLEAR PHYSICSg [FS]
ELSEVIER
Nuclear Physics B 466 [FS] (1996) 361-382
Functional equations of form factors for diagonal scattering theories Takeshi Oota l Department of Physics, Faculty of Science, Osaka University, Toyonaka, Osaka 560, Japan
Received 29 November 1995; revised 1 February 1996; accepted 15 February 1996
Abstract The form factor bootstrap approach is applied for diagonal scattering theories. We consider the ADE theories and determine the functional equations satisfied by the minimal two-particle form factors. We also determine the parameterization of the singularities in two-particle form factors. For A~I/ affine Toda field theory, which is the simplest non-self-conjugate theory, form factors are derived up to four-body ones and identification of the operator is done. Generalizing this identification to the ac~) affine Toda cases, we find the two-particle form factors. We also "'N determine the additional pole structure of form factors which comes from the double pole of the S-matrices of the A~ ) theory. For AN theories, the existence of the conserved ZN+j charge leads to the division of the set of form factors into N + 1 decoupled sectors. Keywords: Form factor bootstrap; Functionalequations; Parameterizationof the singularities;
Non-self-conjugate
1. Introduction For two-dimensional factorizable scattering theories, the bootstrap framework gives strong constraints on physical quantities. For example, under some assumptions such as unitarity, crossing symmetry and analyticity, the bootstrap determines the S-matrices almost completely and non-perturbatively. The factorizable and diagonal scattering theories are closely related to the notion of integrability which has a deep connection to underlying Lie algebras. The S-matrices for these theories are well known and are determined from the data of the associated Lie algebras. Less known objects for these I This work was supported by JSPS Research Fellow. 0550-3213/96/$15.00 (~ 1996 Elsevier Science B.V. All rights reserved PII SO 5 50- 3 213 ( 96 ) O007 9- X
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T. Oota/Nuclear Physics B 466 [FS] (1996) 361-382
theories are form factors, i.e. matrix elements of operators. If we know all form factors then the correlation function can be calculated in principle. In integrable theories, the form factors can also be determined by the form factor bootstrap approach [ 1,2]. The form factors for the diagonal scattering theories are known for the thermal perturbation of the Ising model [3,4], the scaling Lee-Yang model [5], the sinhGordon model [6,7], the Bullough-Dodd model (A(22)) [8] and the Ol,3-perturbed non-unitary M3,5 model [9], all of which contain only one type of particle. For models whose spectrum have many types of particles, form factors are hardly known, except for minimal a(2) "~2N theories which contain N kinds of particles [ 10] Two-body form factors for the magnetic perturbation of the Ising model can be found in [ 11 ]. More knowledge of form factors for models whose spectrum have many-particle species is desirable to understand the off-critical properties of the models. One way of treating the models with many species of particles is proposed by Koubek. She shows that for minimal Za2N A(2) theories (~bl,3-perturbed M22N+3 minimal conformal field theories), all recursion relations for form factors can be simplified to the recursion relations for the form factors which contain only one kind of particles [ 12]. All particles (or excitations) in these theories are self-conjugate, i.e. a particle and its anti-particle are identical. But for the case of non-self-conjugate theories, the kinematical residue equations cannot be used directly to determine the form factor of one-particle species. In this case, the recursion relations for one particle become rather difficult to solve. In this paper, we take a straightforward approach: trying to solve simultaneously the system of recursion relations. We mainly deal with the A~1) affine Toda field theories, especially the A~1) case. The A~ ) is a theory of N scalar fields with the lagrangian ~_. = 1
tz
_
m2 N Z
ega~.O,
where ai are the simple roots of the AN algebra. Analyzing the additional pole structures, we determine all recursion relations for the form factors. Using the recursion relations, we determine the two-body form factors for ~b<, (a = 1. . . . . N) and for the trace of the energy-momentum tensor O ( x ) . From this knowledge we are able to derive the next leading term of the two-point functions of ~b~ (and those of O ( x ) ),
(4,<,(x) @<~(o))=
~lF~°l'Ko(malxl)+Z
f
d.a(2:).d~' IF:° ('~ - ~')1'
b,c P2(~)>PT(/~')
× exp (-Ixl
T. Oota/Nuclear Physics B 466 [FS] (1996) 361-382
363
form factors. In Section 3 we derive the form factors of A~ 1) affine Toda field theory up to four-body ones, and we identify the fundamental operators. The generalization to AN affine Toda field theories is discussed in Section 4, and we determine the two-particle form factors. Section 5 is devoted to conclusions and discussion.
2. The form factor bootstrap 2.1. Equations f o r the f o r m factors
The matrix elements of a local operator (.9(x) t
/
!
]~alLI 2 •..a m [ I~ [
f~!
alO2...a,, t b-'l,/a2 . . . . . t
/
t
!
/3mill,/32 .....
/3n)
/
= ~, a ~ ° , (/3,,/3~ . . . . .
/3" I O ( 0 ) I / 3 ' , / 3 2 . . . . .
/3.)o,a2
~,,,
(2.1)
are called form factors of general kind. Here/3i is the rapidity of a particle of species ai. Consider the following form of the matrix elements:
rata2...a.
(/31,/32
. . . . .
/3n) = ( 0 1 0 ( 0 ) I / 3 ' , / 3 2
.....
/3n)o,o: ...... .
(2.2)
The general form factors (2.1) are related to the functions (2.2) by analytic continuation t r ¢/1 , . . ~ 3
r;,, ...... : ( / 3 ' , , . .
. .ca'lbt . .
•,/3.,I/31 ..... /3.) ' ,~a' b,, ~. i ot + i~ . . . . . /3m t + i~',/31 . . . . . /3,), ~ °' rb,...b,,a~...a,,~loj
(2.3)
provided that the set of rapidities/3' are separated from the set/3. Here Cab is the inverse of the charge conjugation matrix C~b and for ADE scattering theories Cab = 6~b. Watson's equations for diagonal scattering theories take a simple form, Fa,...a,,,,,,...a,, (/31 . . . . . /3i,/3i+1 = Saiai,l
(/3i -
. . . . .
/3n)
/3i+1) Fa,...ai+,ai...a,, (/31 . . . . . /3i+1, fii . . . . . /3,),
F,,,,,2...a,,(fll + 2~i,/32 . . . . . /3,) = Fa2...~,,a,(/32 . . . . . /3,,/31).
(2.4)
The simple pole structure of the form factors is summarized by the following two types of the recursion relations 2 . The first kind of relations are called kinematical residue equations: - i res/3,=~+/~ Faaa,...a,, (/3',/3,/31 . . . . . /3,) /
\
= Fdl...d,,(/3l . . . . . /3n) ( l -- f ' l S a d j ( / 3 - - / 3 j ) ~x
j=l
~ • /
And the second kind of relations are called bound-state residue equations; 2 If the S-matrices contain double poles an additional simple pole structure appears.
(2.5)
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T. Oota/Nuclear Physics B 466 [FS] (1996) 361-382
--i resfl,=fl+io2b Fabd, ...d,, (/3',/3,/31
. . . . .
/3n )
= FCaaFea,...d,,(/3 + iOgc,/31 . . . . . fin),
(2.6)
where 0 = ~- - 0. The on-shell three-point vertex F~b is given by - i res~=i0,;~, S~b (/3) = (Fcb) 2
2.2. Minimal form factors In the case of n = 2, Watson's equations reduce to
Fab(B) = Sab(/3) F o a ( - f l ) , F,,b (/3 + 27"ri) = Fba (--fl).
(2.7)
The general solution to Watson's equations takes the form [ 1 ]
Fa, .......(/3, . . . . . /3,,) = Ka,...a. (/3, . . . . . fin) I I F(a,~n) (/3i - /3j ),
(2.8)
i
~(min)
where • ab is the solution to Eq. (2.7) which is analytic in the strip 0 ~< Ira/3 ~< 2~" and has no zeros in 0 < Im/3 < 2~r. The building block of the diagonal S-matrices is (x)~-
(x)+(~) (-x)+~)'
where 1 sinh 1 (
(x) +(~) = tTr
7r)
-2 fl + i-£x
.
We write the basic building block of a minimal (two-particle) form factor corresponding to (x)B as the following form: .,
I,,
gx(/3)
fx(/3) = stun 2Pgzh----f~.(/3)
(2.9)
which has no poles and zeros in the strip 0 < Im/3 < 277"for 0 ~< x < 2h. The function gx(/3) is given by
gx(fl) -- I - [ ,,=,
F ( n +/_P__ _ ~ + 1) _ 27r 2h __
/'(n ~
(2.10)
1)
The function g x ( ~ ) has poles at /3 = - i ~ r x / h + 2 ( m + 2)¢ri for m = 0, 1,2 . . . . and zeros a t / 3 = - i q r x / h - 2m~ri for m = 0, l, 2 . . . . . The introduction of the function gx simplifies the calculation. Using the following properties of gx:
T. Oota/Nuclear Physics B 466 [FS] (1996) 361-382 . "B"
gx (fl + t'~y) = gx+y(fl),
365
(2.11) 1
gx (fl + 27ri) = gx+2h(fl) = (--~+gx(fl), 1 gx(iTr - fl) = g3h-x(fl)
(2.12)
(h-x)+ gh-x(fl) '
(2.13)
g~ (0)g4h-x(0) = 1,
(2.14)
the behavior of the minimal form factors under the recursion relation is easily determined. For ADE scattering theories, the basic building blocks of the diagonal S-matrices are (x)# = (x)+(#)/(-x)+(#) with (x - 1 ) + (x + 1 ) + (x)+ =
for perturbed conformal,
(x - l ) + ( x + 1)+
(~ T~~i--
B)+
for affine Toda,
where h is the Coxeter number of the associated Lie algebra. So the building block corresponding to (x) is Gx(fl)
F~min) ( # ) -
(2.15)
G2h-x(fl) ' where
Gx(fl) =
gx-I ( f l ) g x + l ( f l )
for perturbed conformal,
gx-l(fl)gx+l(fl) gx-t+B(fl)gx+l-B(fl)
for affine Toda.
We list some properties of Gx: .77"
(2.16)
Gx (fl + t - ~ y ) =Gx+>,(fl), G x ( # + 2~-i) =
Gx( iTr - fl) =
Gx+2h(fl) ~--~+Gx(fl), =
1 G3h-x(fl)
(2.17) (2.18)
- Gh-x(fl)'
Gx(O)Gah-x(O) = 1.
(2.19)
If the S-matrices have the following form:
s,,b(#) = l-I (x>~,
(2.20)
x C A,,#,
then the minimal solutions of Eq. (2.7) are written as Ft( rain
,h
(#) = ]-I Fx~m~"~(#) xEA~I,
(2.21)
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Here Aab is the minimal set of numbers which gives the correct Sao. The multiplicity of p in A,b is denoted by mp(Aab). The sets Aao are chosen such that m p ( A , b ) = 0 for p~<0orforp~>h. The crossing condition of the S-matrix S~6(irr - / 3 ) = Son(/3) is equivalent to the condition mh-p ( A~b ) = mp ( Aba). The minimal form factor (2.21) has the following property: F(min) :_x ab t-~ , ~ ' "_/._ )% b L-(min)(/3) = l/sCab(/3) •
(2.22)
Eq. (2.22) is an analog of the S-matrix relation Sab(/3 + irr)Sab(/3) = 1. There is a one-to-one correspondence between the constituent of minimal form factors and the Smatrices. Many properties of Fx~min) are similar to those of (x) but the monodromy property is quite different. F~ rain) has a diagonal monodromy which comes from Eq. (2.17) while (x) is 2rri-periodic. Due to these monodromy factors, additional functions (,b(/3) appear in the right-hand side of Eq. (2.22). The factor 1/(ab is given by the product of (x)+ 1/(ab(/3) = H {X)+(/~). xE A,t,
(2.23)
In particular, for affine Toda cases,
(x)+ =
( x - - 1)+(x q- 1)+ (x- I + B)+(x + I - B)+
implies that B dependent parts appear only in the numerator of ~b. The form of sc has been determined for the scaling Lee-Yang model [5], the sinh-Gordon model [7] and for the Bullough-Dodd model [ 8] by explicit calculation. Using Eq. (2.7), we get the following relations: Sab(/3) - (ab(/3 + irr) _ C a b ( - ~ 3 ) (ab(/3)
~ab(/3)
(2.24) "
Under the factorization of Eq. (2.8), the kinematical residue equations reduce to --i res#,=/~+irr Kaad,...d,, (/3',/3,/31 . . . . . /3n ) = Kd,...a,,(/31 . . . . . fin)
(j_I~1( a d j ( / 3 - - / 3 j ) -- H ~ a d j ( / 3 j - - / 3 ) ) / Faa(min)(tzr). . j=l (2.25)
In accordance with the S-matrix bootstrap Sad(~3 q- i ~ b c ) S b d ( / 3 --
iOgc) = Sed(fl),
(2.26)
the minimal form factors have the following properties:
"adP(mi(/n~)-t---~l'lT"abac ~) ~(min) r ( fl -- lO-abc
"~ "~( min ) ( /3 ) / l~Cb;d ( /3 )
(2.27)
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367
The function A is known for the scaling Lee-Yang model [ 5] and the Bullough-Dodd model [8,13]. These models contain only one-particle species. The extra factor 1/aCb;d(/3) comes from the diagonal monodromy and is given by the product of (x)+:
l/<;h:d(/3) =
(U~c -b - x)+(e)
II (xEA.dIX
II
(x - Ubc}+(e ). -a
(2.28)
{xeAbdlX
c = hO~b/rr and Uab -c = h -Here blab From Eq. (2.28), we can see
Uab.
a,'ib;~ (/3) = a;o;a ( - / 3 ) . The associativity of Eq. (2.22) and Eq. (2.28) leads to
Aco,,;d(/3 - iOCab) A~ab;d(/3 -- iObac) = (aa (--/3). The bound-state residue equations reduce to •
!
- - t res/3,=/3+io; b Kabd, ...d,, ( / 3 , / 3 , / 3 1 . . . . .
/3n ) n
C
'--O
C
"
•
= VabKe&...d,,(/3 + tObc,/3, . . . . . fin) IIaaO;dj(/3 + iOgc -- flj)/F(a~un)(tOCab). j=l
(2.29) For ADE scattering theories, the S-matrix can be written as [ 14] h--1
Sab(/3) = 1"I ((2p + 1 + eab)+(,a)) u'"''w Pga~.
(2.30)
p--O
Or equivalently
mx(Aab) = tx (a) • w - P ~ b
for 0 < x = 2p + 1 + Gab < h.
H e r e / z (") is the fundamental weight of the algebra. So the minimal form factors can be given by 3 h-I
Fab(min) (iS)
=
6(cob l)/~(a~.w~b (0 )~._ , II(G2pq_l_}_ffab(/3))~l,(a)'w P4'b.
(2.31)
p=O
Here e,,o = ½( c ( a ) - c(b) ). Depending on the two colors of the Dynkin diagram of the associated algebra, c ( a ) = 1 for white nodes and c ( a ) = - 1 for black nodes [ 15]. Note that we can see c(fi) = ( - 1 ) h c ( a ) . If the Coxeter number h is even, which holds except for A2N theories, Gab = Cab. 3 For the minimal cases, if the S-matrix has a fermionic nature: Sab(O) = -- 1, then one more factor (0)+03) is needed.
T. Oota/Nuclear Physics B 466 [FS] (1996) 361-382
368
The first adjustment factor on the right-hand side of Eq. (2.31) is introduced in order that Fa(fffin) is constructed from Gx (0 ~< x < 2h) for Cab = 1. On the right-hand side of Eqs. (2.25) and (2.29), for the affine Toda cases, coupling dependent parts (x dz B)+ only appear in positive powers, so the singularities of K can be factorized by products of 1/(x)+. 1
K,,,...,,,, (ill . . . . . fin) =Qat...a,,(fll . . . . . fin) 1-I (eft, +eflj'C""J'",aj'°il 'ra i
~P (2.32)
The function Wab contains the factor (UCb)+(--UCb) + for bound-state poles. The factor (--Uab) + is needed to make Wab symmetric: Wba(--fl) = Wab(fl). In general, Wab must contain more factors to factorize the higher-order poles in SCab. The polynomial Q~...a, carries the information about operators. Counting the number of independent solutions, Qat...~,, can be used to classify the operator content of the model [3,6,12,21,16]. We expect that Qal,...,a,, are polynomials in ( x ) + ( - x ) + . Substituting Eq. (2.32) into Eq. (2.25), the function Wab is determined from the requirement that W~b(fl + iTr)Wab(fl)~ab(t~) is a product of the factors (x)+ in positive powers. Using the expression (2.23), we write the singularity of ~ab a s follows: h
1 1-I ( x -
xG A,,j,
1 ) + ( x + 1)+
= l-i(p)+m,_,(Ao~)-m~+,(A~) p=O
Even-order poles do not correspond to the bound state. If (~b contains the factor 1/(x) 2k, the half 1/(x)~+ is canceled by W~b(fl) and the other half is canceled by W~b(fl + i~r). Then W~b(fl) must contain (x)k+ (--X)k+, and Wnb(fl) has ( x - h )k+( h Odd-order poles can be interpreted as the production of a bound state. W~b contains the factor (U~b)+(--U~b)+ for the bound state in the forward channel. If (,~b contains the factor 1 / ( x ) 2k+l then Woo has the factor (x)k++1 and Wab has the factor (x - h)k+ for the forward channel. Dorey's "uphill/downhill" mnemonic [ 14] implies that mp-i - mp+l = + l , 0 , - - 1 and the case m p - i -- mp+l = -k-1 corresponds to the forward channel. nip 1 The order of the pole at Imfl = plr/h is mp-i -}- mp+l. We need the factor (p)+(~) for Wab(~). The above consideration leads to the following form of the parameterization:
where Pab(t~) =
1-I (X)+(fl). (xeAab[x4=h--1}
369
T. O o t a / N u c l e a r Physics B 466 [FS] (1996) 3 6 1 - 3 8 2
a~
~ b k
/
k f
b
a
Fig. 1.
The singularity structure of (ab is similar to that of Sab. The singularities of the Smatrices for ADE theories are explained in terms of multi-scattering processes [ 17,18 ]. So it is natural to expect that the factorization in the above admits such an interpretation. The singularity
1
1
,-,,.,
(x) +(/~) (x) +(_#)
coshfl - c o s ( T r x / h )
is explained by the Coleman-Thun mechanism [ 17]. Recalling i (Pa + P~,) 2
i __
m2
= mamb
1 cosh fl - cos OCb '
this singularity comes from a delta function O ( p o ) ~ ( p 2 _ m 2) according to the Cutkosky rules. For perturbed conformal theories, Delfino and Mussardo [ 11 ] have derived the parameterization of the singularities of two-particle form factors from the Feynman diagrammatic analysis of multi-particle processes. Their factorization agrees with our result. For the Nth order pole in form factors corresponding to the fusing ab --+ e, the parameterization is explained by the diagram of Fig. 1, whose blob represents a connected diagram and contains 2N - 1 three-point vertices. This is essentially the half of the diagram which explains the (2N - 1)th order pole of the S-matrix. The Nth order pole of the form factors, which corresponds to the 2Nth order pole of the S-matrix, may be explained by a diagram ab --+ cd whose blob contains 2N three-point vertices. We determined the parameterization both for the perturbed conformal theories and for the affine Toda field theories. For affine cases, no additional coupling dependent pole appears in the physical strip. So the parameterization is consistent with the mass spectrum of the theory.
T. Oota/Nuclear Physics B 466 [FS] (1996) 361-382
370
3. T h e A~l) affine Toda theory The A2(j) theory is the simplest model based on the Lie algebra which contains nonself-conjugate particles. It contains two kinds of particles, which are denoted by 1 and 2. The particle 2 is the anti-particle of 1 and vice versa. The S-matrices of this theory are given by Sll = $22 = (1) and S12 -- S21 -- (2) [1820]. For definiteness, we consider the A O) affine Toda theories. The A(21) affine Toda theories is a theory of two scalar fields ~bl (x) and ~b2(x). So we concentrate on form factors of scalar operators. We define FI,,,.,,1 (ill,
f12 . . . . .
]~m;/~,/~2 .....
= F l l .. 122 . . 2 (. / ~.1 , ./ ~ 2. , . .
/~'n)
t) • f l m , f l lt , f l 2t . . . . , a,'-n
(3.1)
And we factorize KIm.n] as follows: Kim,nl(fll,.. • ,/3m,/31,. . t
. . , /3'.) Q[m,n] (Xl . . . . . Xm; Yl . . . . . Yn) "1-yj) I-Ii
l-Ii
(D-2yj) (3.2)
with xi = e/~i, Yi = e #' and w = e i'n/3. T h e degree of the polynomial QL,n,nl is given by 1) - mn. For simplicity, we use the vector notation deg(Q[m.,]) = ( m + n ) ( m + n x = (Xl,XZ . . . . ) etc. Then the kinematical residue equation is reduced to -
(3.3)
Qt,.,,+l.n+l j( X, - x ; x, y ) = x D [ m , n 1 (X; x,Y)QIm,,,I ( x ; y ) ,
where Dlm,n I (x; x , y ) = ( - 1 ) n H2 (Dm( x; x; -(D ) Dn( x; y; (D - l ) - O m (X; x; _(D-1 )Dn(x; y; (D)).
The function Dm is defined by m
Din(x; x; Z ) = I I ( x
+ ZXj) ( x - z q x j ) ( x - z q - l x j )
j=l
nl rn-- k
=
Z~-~'(--1)k[k+ljq
1 x3m-2r-k-lz2r+k+l
O./~i X X) S (2",1), k [ X J, "~
l--0 k=0 r--0
(3.4) where [ k ] q = (qk _ q - k ) / ( q
_ q-l)
and the Schur function S(2r,lk/ is
S(2r,lk ) = ( O'r+kO'r_ 2 -- Orr+k_lO'r_l ).
Here o-j are the elementary symmetric polynomials defined by
T. Oota/Nuclear Physics B 466 [FS] (1996) 361-382
371
I1|
111
I-[(x + xj) = j=l
j=0
The B dependent parameter H2 is defined by (-i) H2 - F(~nin)(i77. ) •
The coupling dependent parameter q = e it(B-I)~3 transforms into q - i under the weakstrong transformation B ~ 2 - B. The bound-state residue equations are reduced to Q Im+2,,,I(X, w y , w - l y ; y) = Hy2Dm (y; x; 1)Q[m,n+ll( x; y, y), Q[m,n+21 (x; wx, w - I x , y) = HxZ D,n( X; y; 1)Q[m+l.n]( X, x; y) ,
(3.5)
where H=
v~F
(min) 2__:'~ =
F11
( ~"" J
V~F F2(min)(27ri) " 2
',
The function /" is given by ( r ) 2 = (F{1)2 = (VzZ2)2 = ~
sin ~7rB sin ~ 7 r ( 2 - B) sin 1~-(4 - B) sin ~ ( 2 + B)"
(3.6)
There is the relation between H and H2
H2/H2 = (1 + w) (1 + w[2]q),
(3.7)
which is equivalent to the minimal form factor relation (F/~nin) ( 2 _ : , ~2 / ~:,(min)
T"')J
/"12
(i7"r)= (3)+(0).
(3.8)
We first pay attention only to the indices [m,n]. The bound-state residue equations relate QIm.nl to QI,,,-2,,+I1 or Qp,,+I,,-21 and the kinematical ones relate Qlm,nl to Qlm-J,n-ll. We identify [m,n] with a two-dimensional vector and introduce the following equivalence relations: Ira, n] ~ [re, n] - 1112,-1] - 1 2 [ - 1 , 2 ] - 1311, 1],
(3.9)
where li C •. Because [1,1] is equal to [ 2 , - 1 ] + [ - 1 , 2 ] , the above definition is redundant. But for later convenience, we write here the [ 1,1 ] term. Then there are three equivalence classes
{[m,n]}/~=[[O,O]]+[[1,O]]+[[O,l]].
(3.10)
In other words, if [m, n] and [m ~, n ~] belong to different classes then Q[,,,,nl and QIm',n'l are not linked by residue equations.
T. Oota/NuclearPhysicsB 466 [FS] (1996) 361-382
372
So the set o f form factors is divided into three sectors. In each sector, higher polynomials are determined iteratively from lower ones. In the minimal polynomial space, the solutions can be determined uniquely except for the kernel ambiguity. Note that [ 2 , - 1 ] and [ - 1 , 2 ] are equal to the first and second rows of the Cartan matrix of the A2 algebra, respectively. The index [m,n] can be identified with the Dynkin indices of weights and with corresponding weights/z /z
= [mj, m2] =
ml/z Cl) + m2/z ~z),
(3.1 1)
w h e r e / z ~') are the fundamental weights. Then the equivalence relation can be rewritten as tbllows: /z ~ / z - llCel - 12ce2 - 13(cel +
(3.12)
O~2).
Here al and ce2 are the simple roots of the A2 algebra. For the A~ j) theory, the equivalence relation is generated by all positive roots. In the polynomial space we are considering, the degree of a polynomial is equal to the degree of the kernel and the kernel is one dimensional. So at every recursion step, only one parameter enters into the solution space and the parameter Au is attached to the point u in the dominant weight lattice. The most general solution of the residue equations (3.3) and (3.5) has the following form:
Q ~ ( x , y ) = ~ Hlu-~IA~Q~,~(x,y). u<<.l~
(3.13)
Here the ordering of weights/z ~> ~, means that the difference/x - z, lies in the dominant root lattice and I/x[ = ml + m2 f o r / x = [ml,m2]. The polynomial Qu carries the information about the operators. Each Qu,~ satisfies the residue equations, so gives an independent form factor of some operator O~.
3.1. The [O,O]-sector We start from the [0, 0]-sector to solve the recursion relations. Although we call it the [0, 0]-sector, the kinematical recursion relation is not applied to QI l,ll ---+Qlo,ol 4 So the first polynomial is QIx.n I. The most general degree 1 polynomial is QI1AI (x, y) = All,1 ix + A~ 1,1]Y,
(3.14)
where A 11,11 and A~ 1,11 are constants. But the recursion relation Q13,01 ~ Q[ 1,1~ has no solution unless A I 1.11 = A~l,ll. Similar phenomena occur at higher stages of recursion processes. These additional constraints come from Z2 symmetry corresponding to the charge conjugation. These Z2 constraints are imposed on the constants: Aim,n] = A[n,ml. 4 If the recursion process was started from QI0,0I formally,we would only get QIm,nl solution corresponds to the "form factor of the identity operator".
=
Q[0,018m.o8n.o.This
T. Oota/Nuclear Physics B 466 [FS] (1996) 361-382
373
The first few solutions in the [0, 0]-sector are given by QI l,I I (x, y) = All,I] (x + y ) , QI3,01 = AI3,0]BI13,01 + HA[1,1]0-10-2 (0"10"2- (2 + [2]q)0-3) , Q[ 2,2] = A[2,2]B1 [2,21KI2,21B212,2] + HA[3,01Q[2,21,[3,01 + H2AII,I 1QI2,21 ,I Lll. Here Q12,21,1o,31 = ( ~ ( x ) ~ ( y ) ×((~2(X)
- ff2(x)~2(y)) --~2(y)) 2 +~l(X)~l(y)(~2(X)
+~2(x)~(y)
+~2(y))
+ ~(x)~z(y)),
Q[2,21,11,11 = (0-1(x) + 0 - 1 ( y ) ) ( 0 - 2 ( x ) 0 - 1 ( y ) + 0 - 1 ( x ) 0 - 2 ( y ) )
×(0-1(x)0-1(y)(0-2(x) + 0-2(Y) + 0-1(X)0-t(y)) --(1 + [2]q)0-2(X)0-2(y)). The polynomial KIn,, 1 is given by m
n
Ktm'"l(x'Y) = H I I
(xi + YJ) = Z
i=1 j=l
sa(x)s;~(y),
(3.15)
a
which is the kernel of the kinematical residue equations. Here sa is the Schur function and the summation is taken for the partition A = (AI . . . . . Am), i.e. non-increasing sequences of non-negative integers under the condition ,~l = n and ~ = (m - A~,. . . . . m A~l) [22]. The partition ,V is the conjugate of the partition A. The polynomials BaIm,nl (a = 1,2) are given by
BII .... I ( x , y ) = 1--[ (xi -- tO2Xj) (Xi -- ¢O--2Xj) = S2sm(X),
(3.16)
i
B21m,nl ( x , y ) = 1-I (Yi - w2Yj)(Yi - w - 2 y j ) = sz~,.(y),
(3.17)
i
which are the kernel of the bound-state residue equations. Here ~m = (m - l, m 2 . . . . . 1). The Schur function can be expressed as [22]
s a = det (0- a;-i+J ) l <~i,j<~l(a, ) ' where I(A) is the length of the partition A. The polynomials Q[n,,nl,[1A] ( [ m , n ] v~ [1, 1]) have the following forms:
QIm,.1,I IA I( x; Y) = (0-1 ( x ) + 0-1 (y) ) ( O'm-l ( X )O'n(y) + 0-m( X )O'.-l (y) ) X P[m,,,l ( x ; y ) .
(3.18)
374
T. Oota/Nuclear Physics B 466 [FS] (1996) 361-382
From the stress-energy conservation, it is possible to show that the polynomials which enter the form factors of the trace of the stress-energy tensor O are factorized as (3.18). So the operator OiL11 is identified with O. This fixes the constant A[l,ll to be 7rM 2 A[I,11 - 2F~nin)(iT.r) ' where M is the mass of the particles. In this sector, we determined the form factor up to four-body ones. For example, the explicit form of the two-body form factor is given as Fl2(fl) = F~(fl) =
"n'M2 F(2nin) (fl) 2 F(~n)(iTr) "
(3.19)
3.2. The [1,O]-sector and the [O,l]-sector The solutions in the [0, 1]-sector are simply obtained from the [ 1,0]-sector using Z2 symmetry. So we only deal with the [ 1,0]-sector. The first few solutions in the [ 1,0J-sector are given by Q[ l,Ol = All,01, Q[0,21 = A[o,2]B2[0,21 + HAl 1,0]0-(22), Q[2,1t = A[2,I]B212,l]K[2,,l-t- Ha[o,210"~2) KILl I + H2AlI,O]0-12)0-~ 2) 0"I 1), Q[4,01 = A[4,01Bl[4,0] + HAl2,11Q[4,0],[2,1] + H2A[o,2]Q[4,0],[o,2]
+ H3A[1,0lQI4,0],[ 1,01, Q[l.31 = A[ j,3IB2[ I,3IKI I,3I + ttA[2,1IQ[1,3I,[2,1I + H2A[o,2]Q[1,31,[0,21 +H3A[1,0IQ[ 1,31,[1,0l. The explicit forms of the polynomials Ql.,,nl,lm',n'l in the above equations are given by Ql4,01,[2,1] -- (2 + [2]q)0-2(-0-1o-~
- 0-~0-3o-4 -t- 2o'10"2o'3o4 -q- 0"20"4 + 0"~0"2) 2 2 2 - 0"20"4 - 20"10"30"4 -t- 0"2), + (0"92 + ( 2 1 2 ] q + [2]q)0"4)(0-10"3
Q[4,o1,[o,2] = 0"4
((2 +
Q[4,0],[l,0] --- 0"4
((2
[ 2 ] q ) (0"4 - 0-10"3) (0-4 --~ 0"2 _ 0"10-3) 3t- ( [3]q - 5)0"20"4),
+ [ 2 1 q ) 0 " 2 ( - 0 -2 - 020"4 + 0"20"4)
-t-(0"2 + ( [3]q -k- 312]q -k- I)0"4)(0"1o"3 -- o'4)) , ,..r(3) .,r ( 1) /' .~.(3)._,.(3) 3))(3) Q[l.31,[1,ol = v 3 vl ~,v I v 2 - ( 2 + [2]q)0"~ ..0"(2
el, 31,i0,21--0"I
~'-'1 v2
-- (2
+ [21q)0"~3))
(3)_(1)~ +0"1 t~l ) ,
[ " (3).2)0"3(3) -1-'v2/"(3)~2~(1)~, ~,1,0-1 Vl ,]'
,1.(3),1.(3) //.1.(3),1.(3) 3)) QI1,31,12,11 = v l ~2 ~ 1 v2 - ( 2 + [2]q)O-~ K[I,3].
The form factor of the fundamental operator is factorized as follows:
T. Oota/Nuclear Physics B 466 [FS] (1996) 361-382 Q~,~( x , y ) = ¢r(mm)( X )Cr(nn) ( y ) Pu,~( x , y ) .
375 (3.20)
The operators 011,0 ] and 0[0,1 ] correspond to these fundamental operators, and others are composite operators. Note that [ 1, O] and [0, 1 ] are fundamental weights of the Az algebra. The operators labeled by the fundamental weight correspond to the fundamental operators. From the conservation of the Z3 charge, Ou~,, = dpa (a = 1,2). Here ~ba are affine Toda fields. The requirement F . ( f i ) = F ~ ( / 3 ) = (0l~ba(0)l/~)a = ~
1
(a = 1,2)
sets the constant Atl.Ol = A[0,1l to be 1/x/~. In the [1,0]-sector and [0,1]-sector, we also determined the form factors up to four-body ones. For example, the two-body form factor is given by )=_
4. T h e
~.(min) r ,22 (/3) 2cosh/3 + 1 •K'(rain)t" 2__:'~ " 22 t ~'~lt)
V• 3
AN a f f i n e
Toda theories
The AN theory contains N kinds of particles, which are denoted by 1 . . . . . N. The anti-particle of a is ~ = h - a = N + 1 - a. The mass of the particle of type a is given by M,, = 2 M s i n ( a T r / h ) [19]. As for the case of the A2 theory, the AN form factors are divided into N + 1 sectors. The sector specified by the fundamental sector contains the fundamental operator, and the zero sector contains the identity operator and the stress-energy operator. Physically, these sectors simply come from the decomposition of the states into the different ZN+l-charge sectors. For the AN theory, the explicit form o f , lt;,(min) ab can be written as a+b- 1
F(min) ,t, ( / 3 ) =
II
h-]a+b-h l - 1
Fx(min)(/3) =
x=la-bl+l
step 2
1-1
Fx(min)(/3) "
x=la-bl+l
step 2
(min) Using /";x' ( m i n~.r-'.,~ ) f a ~ 2h-x ( m i n )~.PI ¢ ~ = 1, we can show F~b (fl) = Fa(~un)(fl)
The monodromy factors ~ and ,~ are given as follows: h-la+b-hl-I
=
I-[ x=la-bl+l
step 2
T. Oota/Nuclear Physics B 466 [FS] (1996) 361-382
376
b-la-dl-I
II
(x)+(#)
for a + b + c = h,
v(a,b,d)+l step 2
1/ a~b;d(fl) =
(4.1)
b-la-dl-I
II
(x)+(#)
for a + b + c = 2h,
v(a,b,d)+l step 2
where v ( a , b , d ) = [b-all + b - d - l a + b - d [ . The explicit form of on-shell three-point vertex is uCab--3
(FCb) 2 = F(h)(2UCab --
1-[
1}+(0)
(u~ + x}+~o)
(4.2)
x=la_b]+ 1 (Uab -- 2 ) + ( 0 ) ' step 2
where 2
2~" / sin
F(h)=
7/"
for perturbed conformal, "/7"
_ (cos h _ cos ~ ( B _ 1)) / sin_~
for affine Toda.
c = h - [ a + b - hi for a + b + c = 0 mod h. For the AN theories, Uab 77"
q'/"
2
for perturbed conformal,
( os x-cos )/(cos x 77"
77"
77"
_ COS ~
for affine Toda. Especially for the A2 affine Toda theory, ( F ) 2 = F(3)(3}+(0) which agree with the previous result. The on-shell three-point vertex /~cb can be expressed as
{(r~/rorb) 2 (r~b)2= (r~/r~rb) ~
for a + b + c = h, fora+b+c=2h,
(4.3)
where Fa = IId-i 1 Fhd d-I • The S-matrices for the AN theories ( N ~> 3) contain double poles. The singularity of form factors are parameterized by the following functions: %b--2 8 a+b.h
Wab(fl) =
fl
(x)+(--x)+.
(4.4)
x=la-bl+2 step 2
In the above factorization, we allow the case c = 0, i.e. a + b = h. For a + b = h, the c is taken that Uab 0 = h. Corresponding to the double pole of the S-matrices, constant Uab additional simple pole appears at (relative) rapidity
377
T. Oota/Nuclear Physics B 466 [FS] (1996) 361-382
/
'i
I
b " Fig. 2.
fl=ih(u]b-2k)
• . , ~1( u , bc _ l a _ b ] )
(k=l,.
- 1 ).
So we must determine these additional pole structure of the form factors. From now on, we consider the affine cases for definiteness. For example for a + b < h, the double poles of the AN S-matrix are explained by the diagram of Fig. 2, or the crossed channel version of the above diagram [ 18]. The problem to be solved is which diagram explains the additional simple pole of the form factors. In order to fix the additional simple pole structure, we analyze some low-order form factors. The one-particle form factor F~ is constant, 1
F. =
=
The elementary two-particle form factor is given as sina ~ - a - ' F ~ " ) (/3) via ;~+l cosh/3 - c ° s Olh~-~-~ --laP(rnin)':zlh-a-l~,tVla)
Fk,(/3) = r.h-a-l ~ -'1~
for a 4: N,
F~in) (/3) =
= 2
F(~i") (iTr) "
These two-body form factors play the role of the initial conditions of the recursion equations. Solving the recursion relation for Fjj~, we determined the explicit form of F2u for
a
F2a(/3)=-t2a
f
(min) F~ (13) F~ (i~(a+2))
Va+2 (min)
sin~(a+
2)
~. cosh/3 - cos 7 (a + 2)
sin~(a +2) cosh/3 - cos 7 a
( c o s ~ - cos ~ ( B - 1) x~
Also f o r a = N -
1 =2,
F2~(/3)=2 M2F~z~n)(/3)(min)
~"
1 - 2 sin 2 "rr { cos ~ -
CO ~r
s ~ ( B - 1)
)
T. Oota/Nuclear Physics B 466 [FS] (1996) 361-382
378
x
(
Coshfl
,
-
cos~(h
,
2) + l + c o s - ~ ~"(
-
h
)}
-2)
"
One can show that "it?(min)ab \(irr ~UabC __ Fa(min)
k=l,.
.rr
( i ~ ] a - O]) 2k)) = "I7(min) l(u~b-l)
c
"
. . , 7l ( U acb -- ]a --
.Tr c
(4.5)
bl) - 1,
wherel=kfora+b~
,-,,-2 ,~h-o-I ~ 11a rl(a+l)
(.Tr t-~(a-2)
)
.
(4.6)
For the case of a = N - 1, the factor --l(N-1)/--ll el / r N - I appears. From the form of F~t,, it holds that (F~(N_I)) 2 = ( F ~ - l ) 2. In showing Eq. (4.6), we take the phase -l
= I.
In general, it is expected that • r,h_l_m,r,h-u~b+l r:," --I resB=i~,(u~,,_2k) Fao(fl) = lira I ~mz(ml_l)rl(u~n_l) k=
1 .....
ml -
(.7"g
)
t-~]a - b I
(4.7)
1,
where l = k for a + b ~< h, l = k for a + b > h, ml = ~1 ( U acb _ l a _ b l ) and m2 = 1 c 7(uab + ]a - b]). And F h j - l = F la( h _ a _ l ) . We checked the above equation for the case of a = 3 . The additional simple pole structure is determined as follows: - i res/~,=#+i~, (a+b-2k) Fabd,...d,, (t ~t, fl, fll . . . . . t~n ) h-a r h - a - b + k F . = r k ( a - k ) (a-k)b k(a+b-k)db..d,
(
x
fl+i
(b-k),fl+t-~(a-k),fll
.....
fin
)
(4.8)
,
f o r a + b <, h a n d a <~ b.
Similar relation holds for other cases. The above pole structure is similar to that of A(2) 2N theories [2] Now we have fixed the additional recursion relations, we are ready to determine the two-body form factors• The ,,rea(1) form factors only have simple poles. The two-body form factors for elementary scalar fields can be written in the following form: g?(min) ( ( ~
F~b(fl)
-
--c -- " ab ~,l"'J - s m" ~c ttablabt' e L - ~ x Pab (tlTab)
l(u~.b--la--bD--1 Z lo--O
Bab;lo cosh fl -
CO
s" c • ~(Uab -- 210) (4.9)
This expression is regarded as an analog of the partial fraction.
T. Oota/NuclearPhysicsB 466 [FS] (1996)361-382
379
'\,
°
\\,
J
\
k ~
a+b-k \
/'
/
a-k
',,
/
',
\
a
a
b
b
Fig. 3. The additional pole structure for a + b ~ h and a ~< b.
Eq. (4.7) is equivalent to the recursion relation for the coefficient
Bah;t,
k-I
Bab;k = Cab;k -~- Z
Dab;k;IBk(a+b-k);l'
l=l
where 71"
[ Fh_ a
,~2
Cal,;k = --Aab sin - ~ ( a + b - 2k) \-~{~-k~/
sin
Aab =
,
~(a + b)FCabFe (i~(a + b))'
-- "/7(min)ab
and =--
Oab;k;l
sin~(a+b-2k) F,h_ a 2 s i n ~ ( a _ l ) sin~(b_t~--k(a-k)~!
Fh-a-b+
b(a-k)
(i~_rb_a)) k v{~a,) " k ( a + b - k ) ~,~h ~ "~-T~-~ \{ i ~~xt a ~ ---b - 2 k ) )
Using the following relations:
/ r k(a+b-k) h-a-b "X Ak(~+b-k) _ |* I F(a~n) ( i ~ ( a + b - 2k)) \ ) F~(a+b_k) (t-~(b- )) D,~b;k;t = - - 2 s i n
the c o e f f i c i e n t
Bab;lo is
sin ~ ( a + b - 2 k ) aab ~ ( a -- l ) s i n - ~ ( b - l) A k ( a + b - k )
( F kh( a- a_ k ) )2 ,
d e t e r m i n e d as lo--1 1t--1
V~(_!). Bab;Io=/_.~ 2 Z~''" .~0
t,,_,-I
ll =1
12=1 n
"rr
I-[j--0 sin Z (U~b --
× ~I. = 1 IIj=lsin~(lj_2
ff,h--lj-1
x2
2lj) \ lj(lj-i-lj) ] lj) sin~(UCb lj_2--1j)
(4.10)
T. Oota/Nuclear Physics B 466 [FS] (1996) 361-382
380
Here I-1 --'-- ~1(UabC_ la - b[) and Bab;O = 1. This gives the solution of the two-body form factors of ~bc(x). Also the two-body form factors of the trace of the energy-momentum tensor can be parameterized in the following way:
,,j - 1
x
1- ~
lo=l
1
Ba;lo ( c o s h f l - cos ~(h - 2•0)
+
'
1 ÷ cos ~ ( h - 2•0)
/)
(4.11) Similarly, Eq. (4.7) is equivalent to the recursion relation
B,i;k=_sin27rk /sin-~ k h ~ \ 2
x
(4.12)
1 + ½~'~" Bk;t s i n - ~ ( a + l ) s i n ~ ( a - l ) 1=1
Solving these relation recursively leads to the solution: --
Ba;lo
/'sin ~lO Fh_l_ I .~ 2 . zcr t sin ~a t0(t-,-to)) l,,_,-1
(santa)212-
× ~t,,:t ( - ½ ) " \ s i ~ l
r
10--1 11--1
"n~>0 l,=l tz--1 [
sin~lj\
l-lj)J,~2
f Fh_lj_l lj(ly
j=l s i n ~ ( l j _ 2 + l j ) s i n ~ ( l j _ 2 - l j )
} "
Here I-i = min(a, ~). Above expression gives the two-body form factors of O(x).
5. Conclusions and discussion
We have derived the minimal two-particle form factors for ADE scattering theories. Using monodromy properties of the building blocks of minimal form factors, we have determined the functional equations satisfied by the minimal form factors. The function A"ab;d will play a key role in constructing the solutions of the form factor bootstrap equations. We have determined the parameterization function Wab for the perturbed conformal theories and for the affine Toda field theories. For the A2 affine Toda theory, form factors are derived up to four-body ones, and the identifications of the fundamental operators have been done. For the AN affine Toda field theories, the form factors are divided into N + 1 sectors characterized by the ZN+I charges. To the chargeless sector, they correspond to the stress-energy operator. To the charge-k sector, they correspond to the elementary operator
T. Oota/Nuclear Physics B 466 [FS] (1996) 361-382
381
with charge N + 1 - k. This is a generalization of the known result for the sinh-Gordon theory (i.e. A1 Toda theory) [6,7] to the AN cases. For the AN affine Toda field theories, we have determined the two-particle form factors. Also, the additional simple pole structure of form factors has been determined. The determination of higher-order form factors remains to be solved. Before concluding this article, we state a relation between roots and equivalence relation of weights. Dorey's fusion rule for the simply laced theories [ 14] is that the fusion process a x b --, ~ occurs if W~(a) [,z(a) + WsC(b)]Lt(b) "Jr-w((C) ].~(c) = 0
(5.1)
for some integer ~:(a), sO(b) and ~:(c). The particle b is the anti-particle of a if Wsc(a)[,Z(a) "-[-w((b) ]£(b) = 0
for some integer We define the Eq. (5.1) or Eq. For the fusion e~ih =/z(a) r =Z
(5.2)
( ( a ) and ~ ( b ) . c = /z(a) +/~(b) //(c) or eab "fusion base" vector eab o = /z (a) + iz (b) if c = 0 otherwise. (5.2) is satisfied and eab a x b ~ ~, if its fusion vector is expanded in simple roots
+/Z
(b) __ /[Z(~)
(5.3)
md°Ld '
d=l then the coefficient md takes integer values. This is equivalent to the statement that for a x b ~ (: the following condition is necessary: (C-1)ad + (c-l)bd
_
_
( C - l ) e d E Z.
Here C is the Cartan matrix of the algebra. For the E8 algebra, all elements of C -1 are integers, so the above condition is always satisfied for any (a, b,c). Only for the AN algebra, the above condition is also sufficient. Using the explicit form of the inverse of the Cartan matrix for the AN theories, (C-t)ab =
1
N + I min(a'b)(N+ 1 - max(a,b)),
one can show that the above condition is equivalent to a + b + c = 0 mod N + 1.
Acknowledgements I am grateful to Professor R. Sasaki for instructive suggestions in the early stage of this work, and to Professor H. Itoyama for useful discussions.
382
T. Oota/Nuclear Physics B 466 [FS] (1996) 361-382
References [ 1 I B. Barg, M. Karowski and P. Weisz, Phys. Rev. D 19 (t979) 2477; M. Karowski and P. Weisz, Nucl. Phys. B 139 (1978) 455; M. Karowski, Phys. Rep. 49 (1979) 229. 121 EA. Smirnov, Form Factors in Completely Integrable Models of Quantum Field Theory, Adv. Series in Math. Phys. 14 (World Scientific, Singapore, 1992) and references therein. [31 J.L. Cardy and G. Mussardo, Nucl. Phys. B 340 (1990) 387. 14] V.P. Yurov and A1.B. Zamolodehikov, Int. J. Mod. Phys. A 6 (1991) 3419. [ 5 ] AI.B. Zamolodchikov, Nucl. Phys. B 348 ( 1991 ) 619. 16] A. Koubek and G. Mussardo, Phys. Lett. B 311 (1993) 193. 171 A. Fring, G. Mussardo and P. Simonetti, Nucl. Phys. B 393 (1993) 413. I81 A. Fring, G. Mussardo and P. Simonetti, Phys. Lett. B 307 (1993) 83. [9f G. Delfino and G. Mussardo, Phys. Lett. B 324 (1994) 40. [101 EA. Smirnov, Nucl. Phys. B 337 (1990) 156; Int. J. Mod. Phys. A 4 (1989) 4213. [ I II G. Delfino and G. Mussardo, Nucl, Phys. B 455 (1995) 724. [ 12] A. Koubek, Nucl. Phys. B 428 (1994) 655. [ 13] A. Fring, Form Factors in Affine Toda Field Theories, USP-IFQSC/TH/93-07, hep-th/9304140. [141 E Dorey, Nucl. Phys. B 358 (1991) 654; B 374 (1992) 741. 115] A. Fring, H.C. Liao and D.I. Olive, Phys. Lett. B 266 (1991) 82. [ 16] EA. Smimov, Counting the local fields in SG theory, hep-th 9501059. [ 171 S. Coleman and H. Thun, Commun. Math. Phys. 61 (1978) 31. [ 18] H.W. Braden, E. Corrigan, P.E. Dorey and R. Sasaki, Phys. Lett. B 227 (1989) 411; Nucl. Phys. B 338 (1990) 689. [ 19] A.E. Arinshtein, V.A. Fateev and A.B. Zamolodchikov, Phys. Lett. B 87 (1979) 253. [201 T.R. Klassen and E. Melzer, Nucl. Phys. B 338 (1990) 485. [21] A. Koubek, Nucl. Phys. B 435 (1995) 703; Phys. Lett. B 346 (1995) 275. 122] I.G. MacDonald, Symmetric Functions and Hall Polynomials (Clarendon, Oxford, 1979).