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The critical O(N) σ-model at dimensions 2 < d < 4: Fusion coefficients and anomalous dimensions
NUCLEAR PHYSICS B [FS]
Nuclear Physics B400 [FS] (1993) 597—623 North-Holland
________________
The critical O( N) u-model at dimensions 2
1
and W. Rühl
Fachbereich Kaiserslautern, D-6750 Kaiserslautern, Germany Received 16 September 1992 (Revised 10 November 1992) Accepted for publication 2 December 1992
Fusion coefficients and anomalous dimensions of the quasi-primary fields are extracted from the 1/N expansion using representation theory of the conformal group whose formulae are analytically extended to 2
1. Introduction Nonlinear a--models can directly be applied to phenomenological solid state physics. But the fact that they are building blocks for other theories such as string theories and gravitational theories is a stronger motivation for us to study them in great detail. At d> 2 a vector nonlinear a--model possesses a critical point at which it can be described by a conformal field theory. This implies that only leading order scaling terms are taken into account. At d = 2 conformal field theories possess an algebraic structure which enables us to apply representation theory, to construct explicit Hubert space realizations, and to classify these field theories. In turn, this representation theory relies heavily on free field theories. To find any kind of algebraic structure beyond the operator product expansion algebra (whose existence is guaranteed) at d> 2, is one of the goals of our investigations. The study of critical nonlinear 0(N) vector a--models involves the following issues:
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(1) to find the type of 0(N) and conformal group representations to which the conformal fields (synonymously “quasi-primary” fields or “qp-fields”) in these models belong; (2) to classify these qp-fields, in particular to determine the spectrum of field dimension 8 in relation to the 0(N) and 0(d) tensor representations; (3) to determine the fusion coefficients YAB for the emergence of the qp-field C from the operator product expansion of two qp-fields A and B. Since conformal field theory allows one to reconstruct any Green function from the information contained in (1), (2), (3) (involving, eventually, infinite sums over qp-fields), it can be expected that these issues can be solved only approximately or partially. In our previous work [1—51we showed that the qp-fields occurring are irreducible 0(N) and 0(d) tensors with a trivial spin-transformation behaviour under special conformal transformations. Thus (1) can be considered to be solved completely. Our contributions to (2) and (3) are approximative in the sense that we limit our investigations to the lowest orders in a 1/N expansion. Moreover, though the number of qp-fields treated is infinite, an infinite number of qp-fields is still out of reach at the end of this work. We suggested in ref. [5]to order the conformal fields of this model into classes (Y, p)
(1.1)
where Y denotes the Young frame of the irreducible 0(N) representation and p is the degree of the normal product of free bosonic massless fields which represents the qp-field in question at N The number of blocks #Y of Y satisfies =
~.
p=#Y+2n,
(1.2)
where n is the number of 0(N) contractions. Within a class (Y, p) the qp-fields are ordered into a two-dimensional pattern according to their normal dimension [6],their (symmetric traceless) tensor rank 1, and their twist t (fig. 1) which obey the relation [6]
—
[6~]
=
1—10 +
t,
(1.3)
Here [6~], 10 are the minimal values belonging to the “primary field” in the class. In certain cases, e.g. if the qp-fields are only accessible by crossing symmetric fusion (see sect. 3) qp-fields with odd 1 or even 1 may be missing for some fixed t. The two fundamental fields Sa(X), the 0(N) vector field, and a(x), the auxiliary field, on which the path integral formalism of the a--model is based, belong to exceptional classes (0, 1) (fig. 2) and (0, 0) (fig. 3) (0 stands for the
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[~5] ~0
10 —
1~+ 3 —
1~+2
[6~1+i
I~
—
10+1
ID 0
2
4
6
1
Fig. 1. The generic class (Y, p) of qp-fields. 10 is the minimal tensor rank, the “primary field” ([&~], la).
[~] the
normal dimension of
empty set). In these classes there are no higher qp-fields in t 0 towers due to the appearance of Sa(X) and a(x) propagators in one-particle reducible graphs. The labels (Y, p; [6], 1) may not discriminate between the qp-fields uniquely but we observe a phenomenon of “degeneracy”. This degeneracy is lifted if we calculate the field dimension of the ‘I completely. =
dim cP=8,1,=p(/.L—1)+q+v~(cP),
(1.4) (1.5)
[8]
1=0
1=2
1=0
1=1 l=0
S 0
2
4
I
6
Fig. 2. The qp-field content of the class (D, 1). The field [,5] = ~.o+ 1, order in 1/N higher than the rest.
t
=
2, 1
=
0 appears probably one
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—
—
1=2
14
—
—
—
—
1=2
12
—
10
—
8
~
—
—
—
1=2
—
1=4
—
1=2
1=0
1=0
—
1=0
1=3 ~-
~-
6
1=0
a I
0
Fig. 3. The class
(0,
2
4
6
I
I
I
8
10
12
1
0) of qp-fields. The degeneracy as obtained in sect. 5 is indicated.
[6~] =p(j.t
—
1)
+
q,
(1.6)
(1.7)
where q is the minimal number of derivatives in the normal product at N and ~(ct) is the anomalous dimension. So we determine different anomalous dimensions for the same position (Y, p, [6], 1). Of course the field algebra of the critical a--model is generated from multiple operator product expansion of the fields Sa(X) and ct(x). This product is associative. Nevertheless we may end up with different resulting qp-fields at the same position (Y, p; [8], 1) if we force the path of expansion to run through different intermediate positions (different “field couplings”). In sect. 5 we will investigate two exemplary cases (k q +p 4) =
=
~,
~‘
(0,0; 2, 0)k®_., (0,0; 2p, 0)
®
(0,0; 2q, 0)
—~(O,0;2k+1,l),
(1.8)
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(o,1;~—1,0)~~(iii ip,p;p(~1),0)®(iii
(~ii
1k,
601
q,q;q(~1),0)
k; k(~—1) +1,1),
(1.9)
and find that 7J(cP) depends besides p + q (trivial) on pq if 1 ~ 4. The results of these degeneracies are plotted in fig. 3 and fig. 8. In this article we calculate anomalous dimensions and fusion coefficients for the primary fields (Y, p; p(~—1) +q, q)
(1.10)
p=#Y
(1.11)
with
and Young frames of two rows of length
11 =p—q,
12=q,
11 >~l2
(1.12)
the anomalous dimensions at 0(1/N) were derived in ref. [2]. Here we compute anomalous dimensions for infinite towers (Y, p, t fixed, 1 running) of several classes. The best studied class is the class (0, 0). Our procedure is as follows. We calculate a four-point function of two qp-fields A and B (A(y1)B(y3)A(y2)B(y4))
(1.13)
in a (1/N)-expansion. In a conformal field theory it depends essentially only on two biharmonic ratios 22
22
Y13Y24 2
2’
Y14Y23 ~=
Y12Y34
2
2’
(1.14)
Yij=Yi~Yj•
Y12Y34
If we let y13—O
or
y24—*0
(1.15)
u—40
and
v-41.
(1.16)
then
An operator product expansion connected with (1.15) is
A( y)B(0)
=
(~2)88A
_1~2~
(n ~
+
...
(1.17)
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(A, B scalar, C rank-I tensor). The four-point function (1.13) appears as a sum over conformal one-particle reducible amplitudes
A
C
(ABAB)=~ B
B
(1.18)
and at the same time as a generalized hypergeometric function [3,5]
(ABAB)=
~
U~(l~V)
~
[...].
(1.19)
n!m!
n,m—0
Comparison of (1.18) with (1.19) gives all information on C: the anomalous dimension and the fusion coefficient (Y~B)’~2. The coefficients Y.~Bdepend on the normalization of the qp-fields A, B, C. For the fundamental fields Sa(X) and a(x) we set ~,
(1.20)
KSa(X)Sb(0))=6ab(X2)a,
(a(x)a(0))
=
(1.21)
(x2)~,
(1.22) (1.23) The normalization (1.20), (1.21) can be obtained by a finite wave function renormalization which puts a coupling constant z 1/2 in front of the constraint term in the action (see (2.10)). The quantities a(S), ‘~j(a)and z are calculable as series expansions in 1/N. The expansion coefficients were listed in ref. [3]. The original calculations at order 1/N2 can be found in ref. [6] and at order 1/N3 in refs. [7,8]. The normalization of all other qp-fields is fixed in sect. 2. The one-particle reducible conformal four-point amplitudes (1.18) are derived in sect. 2 using the group theoretical analysis described in ref. [9]. In sect. 3 we deal with the four-point functions of four equal qp-fields and the implications of crossing symmetry. In sect. 4 we study towers of higher than minimal twist. Instead of using the conformal four-point amplitudes derived in sect. 2 we use the fact that (Y, p; [6], 0) ®a
=
~(Y,
p; [8] + 2+!,!)
Sect. 5 is devoted to the degeneracy problem. *
It gives the anomalous dimensions of A and B, too.
+
o(~).
(1.24)
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2. One-particle reducible four-point functions Application of group theory is possible only if the space dimension d is an integer. This leaves us d 3. Afterwards we may analytically extend d to the interval 2
—
=
[6, 1]
(2.1)
where Iis the rank of the symmetric traceless 0(d) (0(3)) tensor and 8 is the dimension of the field. That these representations are sufficient to describe the qp-fields was pointed out already in ref. [5] and is once again supported in this paper. If A, B are scalar qp-fields with labels [6A,O],
[6B,0],
(2.2)
then qp-fields C coupling to them are elementary representations and belong to [6~, 1]. The three-point function is V(y
6A;
Y2,
8B;
y
866A8~2
3, [6~, 1])A1A2
1,
A~=
C~B~ (y~23)( ~f~
2
.1
2
~Yl2) —
tracesl
(2.3)
with
e=
~
S
2
Y
S ~=
13 —i— Yi3
Y23
2 ~
~
Y13
_______ =
2
2
(2.4)—(2.6)
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The two-point function of a qp-field [6, 1] is the intertwining operator for the representation [6, 1] with its “shadow representation” [d 6, 1] (see ref. [9], (4.19)) —
~ ~
A,(
y1)C~1~2~,( y2) )fl a~.fl b~
(A,,~}
t1
(laIlbi)’ =
(y~2~c(1!)2 ~
Jl
a
b
-1) C~(flr(YI2)~)~
(2.7)
where 6Av
(2.8)
and C~’ (~ ~d) is the Gegenbauer polynomial (see ref. [11], 8.930). The normalization in (2.7) is ad hoc. As long as =
d>6
(2.9)
the shadow qp-fields can play a physical role. In fact, the auxiliary field a(x), introduced as a Lagrange multiplier field in the action by the term [5]
z1/2fdx
a(x)
a=1
(Sa(X))2,
(2.10)
has the same quantum numbers as the shadow field of the field
a~1(Sa(X))2
(2.11)
which it eliminates. This feature shows up explicitly in sect. 3. In order to construct the conformally invariant one-particle reducible four-point function
where both the qp-field C and its shadow field C are exchanged
(~is eliminated
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later), we contract an ABC three-point function (2.3) with an ABC three-point function
EfdY5 V(y1,
2 =f2
6B~ Y
4,
XV(y2, =
y 5, [6~, l])A1A2A,
Y
6A;
Y3~6B;
6A;
5~[d—6~,
(2.12)
!])A,A2A,.
1)
We notice first that (e
E [(eA)’~
—
traces] [fA,~
traces]
—
=
(G~1))~.
CfL~(e.f)
(2.13)
where
C[~(t)
=
~
G~tM
(2.14)
M=0
is again the Gegenbauer polynomial (see eq. (2.7)) which is even (odd) for even (odd) 1. Inserting (2.13), (2.14) into (2.12) and expanding (e .f)M into monomials of y,~, we end up with a finite sum of four-star functions [3] (3.17), (3.18)
S~4~( y 1,
a1
y2, a2 y3,
a3
y4, a4)
=
fdy5~(y~)~’,
(~ai
=
2~ d). =
(2.15) We discard now the second series in ref. [3] (3.18) since it describes the exchange of the shadow field C and, denoting the expansion coefficients in the first series by anm, we obtain
~
2M
x
(n,3} ~ —
1)
2_fl34(
2~2~34~1
)
Vfl23n,~=o
u(lv)
m
anm(ai, a2, a3, a4). (2.16)
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In this formula we use the biharmonic ratios u, v (1.14) and the exponents a, (2.15) 6A
a
1
=
~(6~
—
6B) +
~(n 12
+
—
n23
—
n34
+
n41),
6B) +
6C+6A
a2— ~(2~ 6A + 6B) +
a 3
=
~(—n
~(6~—
12 +n23 +n34 —n41),
(2.17) 3) terms in (2.16) for each fixed n, m. If n is The summation overin{n13} small (we will use factyields only0(M n 0), it is useful to perform a resummation rendering it a sum over 0(n3) terms. The result is =
~
ufl(1_v)mA(M)(6A,6
(2.18)
8,6c)
n.m.
n,m=0
for the last line of (2.16) with ~
~ r,s,f
M!(m +n +s —t)!(A3 + (M t r)!(m + n + s —
—
—
M)!
XF(,u. —A1 —A3—n, A2+A4+M—t—r— 1,
A1
—
~M+m +n, A3— ~M+m +n +s)
x F(A2 +A4
—
1, A1 +A3 + m
+
n
+
1,
s
—
I,
(2.19)
A2+ ~M, A4+ ~M—t—r)
where A 1
=~.t
—A4= ~(6c+6A —6~)’
A3=/i—A2=~(6c—6A+6B).
(2.20)
The summation in (2.18) runs only over 2n +m ~sM.
(2.21)
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In the sequel we will normalize the four-point function (2.12) by replacing C~BC~B2’A~’1
(2.22)
by 1. This is a normalization of the qp-field C as the following argument shows. If the operator product expansion (1.17) is inserted into (1.13), we get (2.7) with a, b replaced by y13, y24, respectively. The term containing no (a, b) contractions in (2.7) is then (2)8C2
1.
{
~-~‘13’34~
‘24’34~
(2.23)
y34
If we take into account that 1
—
v
+
2
=
higher order terms in y13, y24,
(2.24)
y34 we find the same expression in (2.16), (2.18) with (2.21), (2.22) at M=1,
n=0,
m=!.
(2.25)
Thus if we define the C-field by (1.17) then the normalizations (2.7) and (2.22) are identical. This is the normalization entering the fusion constants Y~B~ The master formulae (2.16), (2.18), (2.19) can be exploited to study almost all qp-fields in a class (due to crossing symmetry not all qp-fields may be obtained if A B). But except at n 0, which leads to towers of minimal twist, the algebra is complicated. For towers of non-minimal twist we prefer to apply (1.24). =
=
3. Equal quasi-primary fields and towers of minimal twist We consider first those cases for which 6A6B6’
(3.1)
A=B, as for example in the four-point functions of S-fields (see ref. [5], (3.20)) or of a-fields (see ref. [51,(5.4)). From these four-point functions we draw conclusions on the classes
(m~2),
(9,2),
(0, 0)(O, 2).
(3.2)
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At 0(1) we have from ref. [5] (3.24), (5.5), (5.7) after extraction of the factor (see eq. (2.16)) (3.3)
±1+~-~=
~
m=0
(1—v)tm m.
[±6mo+
(6)m]
(3.4)
(9, 2), + for the classes (ElI, 2) and (0, 0)). The towers of minimal 0 for (ED, 2) and (9, 2) and 2 for (0, 0)) contain qp-fields C
(— for the class
twist
(t
=
t
=
1, 1 ~ 0,
with dimension =
26+!
+
(~(C1) 2?~(A)). —
(3.5)
The fusion coefficients y~follow from m
A~’~
(3.6) with (at 0(1)) 4(1) Om
!\ (m \ [ía L.Y-’ ),n/J ~! (26 + 21)m_i L
=
form (2.19) and a-m= ±6mo+(6)m
(3.8)
from (3.4). The solution of the system (3.6) is
Yi
(26_i+l)1(1 ±(_1)1).
(3.9)
The fact that y~vanishes for I odd (in the cases (ED, 2) and (0, 0))) or for 1 even (for 2)) is the effect of “crossing symmetry”, i.e. the consequence of two fields being identical. It means that qp-fields are missing or not accessible from the operator product considered. The assessment of the first alternative to any order in 1/N is not easy. Next we calculate the anomalous dimensions at 0(1/N). We denote
(9,
=
2~(S)[~(A)
+
~(B) — ~(C,)]
(3.10)
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which from the operator product expansion (1.17) is inspected to multiply
—log u
(1—c)’ 1!
(3.11)
This leads to a system of equations
[(6+I)~,]2
~ E,Y,(7) 1=0
For (ED, 2) and
(3.12)
=
rn—i
(9, 2) we get from ref. [5], (3.25) (~—2)(~t—1+m)[±m!+(~—1)rn].
~rn
(3.13)
Eqs. (3.12), (3.13) do not obviously lead to solutions
e,y,=O
if
y,=0.
(3.14)
This is guaranteed, however, if the ‘I~, satisfy “crossing symmetry constraints”. They are fulfilled if the J~are correctly calculated from the crossing symmetric graphical expansion. Thus these constraints are valuable checks on the correctness of the algebraic forms of I~,(such as (3.13)). In all cases we have made this check. Solving for ~rn with odd labels m we obtain as constraints (2(n —k)
k ~2n+1
=
k=0
—1)
Ck
+
1)2k+1(6 +2(n —k))2k+i (2k + 1)!
with coefficients Ck given in table 1. Inserting finally (3.13) into (3.12) with 6 —2
n(S)
=
—
~~2(n—k)’
1 we get
(I—1)(2~—2+l)
316
(~-1+l)(~-2+l)
-
(3.15)
TABLE
~
I
Coefficients ck from the crossing symmetry constraint equations (3.15)
k
0
1
2
3
4
5
6
7
8
9
ck
I 2
I 4
I 2
17 8
31 2
691 4
5461 2
929569 16
3202291 2
221930581 4
)
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(9~2)). The expression has a limit
limi7(C,) =2i1(S).
(3.17)
The limit 1 -~ ~ for fixed twist seems to exist in all cases considered (we do not always have analytic expressions to prove this). For the class (0, 0) and twist two we have 6 2 and =
m!
,a—2
~
(~t)~
—3)
[(~
1)~+ ~
—
1)~+~ — (m
—
+
1)! 1I (3.18)
(i~2)rn+i(m+1)!}
—(2,u—5) /L
—3
This is taken from ref. [5], eqs. (5.8), (5.10), (A3.1), (A3.5), (A3.7), (A3.9), (A3.11). The solution of system (3.12) with (3.18) and 6 2 can be brought in analytic form. We use the inverse matrix of the system (3.12) given in (4.12) and summation formulae for 2F1 series and saalschützian 3F2 series (for the criterion of saalschützian 3F2 summability see e.g. ref. [12], vol. 1, p. 103): =
—
3)
__________
j,
(1+ 1)(!+2)(~ _2)2 {(2~ The pole at ~
=
—1)— ~ —1—3)~~ (/L), !even (3.19)
2 is of order one. We insert (3.19) into i1(C,) ~
—2~,.
(3.20)
a(S) This gives ~(C0) n(S)
=
4(2k
—
1),
(3.21)
as known from ref. [5], (5.30). In the limit 1 ~ we find —*
—
3)F(JL + 1) 4(2~t— (~—2)2F(4—~t)l22~,
(3.22)
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1.2
1.6
611
18
2
-05
-1.5
-2
o
-2,5
2 4
Fig. 4. Anomalous dimensions for the tower
(0, 0; 4+1, 1), 1 even.
which tends to zero uniformly in any interval l+EI~/L~2.
(3.23)
In fig. 4 the curves for i~(C,)are plotted together with the limit curve at 1 ~: 2~(a). For d 3 (~t ~) all ~(C,) are equal as a consequence of (see eq. (3.18)) —*
=
=
~mI~*~3/20
(3.24)
Therefore we obtain simply
I ~*=3/2
=
2~j(a)I ~i=3/2
=
—
16~(S) I ~*=3/2
(3.25)
(see ref. [5], (5.29)). The tower t 0 of the class (0, 2) fits also into this context. It contains the energy—momentum tensor field at I 2. The leading contribution comes at 0(1/N) from the graphs (see ref. [3], section 4) =
=
1
(3.26)
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The system of equations for the fusion coefficients is (3.6), (3.7) with 6 1
______
6rno+(Ii~1)rn~2 (21.t2)rn
~
=
—
1 and
(3.27)
Comparison with (3.8) shows that only the last term on the r.h.s. of (3.27) which originates from B 2, is essentially new. It leads only to a minor modification of the solution (3.9) 2 Yi
~
—
1)112
(2~—3+l),
N
0,
I
even
1
odd
(3.28)
The sole contribution of the graph B2 is the Kronecker delta which renders (3.29) The graph B2 is the graph for the exchange of the field a and its shadow field B2:SøS—~a~--S®S.
a,
(3.30)
The spin and dimension of c~fits at leading order with the spin and dimension of a possible 1 0 ancestor field of the energy—momentum tensor, which in a free bosonic massless field theory would be =
(3.31)
2. ~Sa(X)
As follows from (3.28) the ~ field cancels this ancestor (up to 0(1/N2)), see ref. [4]) and thus guarantees that the a--model constraint (2.10) is fulfilled. The anomalous dimensions follow from (3.12) with 6 1 and =
—
m
—
___________
(~—2)(~—1+m)
x
(~ 1)rn(/2)rn —
—2(4~t—5)
(2~ 2)~ —
+2(2~t—3)
((,L_1)~)2
(2/2
—
3)m
+m! ~
p—O
(~L—2)~(~L—l)~ . (3.32) P~(2~—
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This expression was calculated from the graphs ~(B1
2(C +B3) +z
21 + C22)
(3.33)
2) radiative vertex function corrections of the graph B and the 0(1/N 2 which were all calculated in ref. [3]. In consistency with (3.29) we have ~o0.
(3.34)
Moreover we obtain =
0,
(3.35)
which was first proved in ref. [3]. This is necessary if C2 is to be identified with the conserved energy—momentum tensor. The general solution of (3.12), (3.28), (3.32) at 6 p. 1 and I ~ 4 is =
____
n(S)
—
—
—
2(p.—1)(2p.+1—1) (2p.
—
1)(p. + 1—2)2
~/—2 2 1 X{2(I1)+~((P+1)!)2(~~1)(/2(2~142P}.
From this expression we can calculate the limit 1
(3.36)
—~ ~
(3.37)
~2+O(!22~).
The curves i7(C,) are depicted in fig. 5. In fig. 6 we draw the curves ~ (C,)
showing that the limit 1
—*
is not uniform at p.
=
1.
4. The class (D, 1) Starting from the four-point function Ka( Yi)Sa( y3)a( Y2)Sb( Y4))
(4.1)
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0.25
8 0.2
6
4
0.15
0.1
0.05
1.2
1.4
1.6
mu 2
1.8
2
Fig. 5. Anomalous dimensions of the tower
(0, 2; 2~.s—2 + 1, 1), 1 ~ 2,
even.
we calculate the t 2 tower of the class (D, 1) applying our master formulae (2.16)—(2.20). However, we consider directly the (2L + 2)-point function =
IL
L
fla(~1) Sa(y3) fla(~1) Sb(Y4)) 1=1 j=1 /
(4.2)
and let ~,—‘y1,
all i,
~—~y2,
allj.
(4.3)
so that in the first step the qp-field a~=(O,0;2L,0)
(4.4)
is produced. In the second step the four-point function (aL®(yi)Sa( y3)a~(y2)S6(y4))
(4.5)
generates the tower of qp-fields C1’-~=(D,1; (p.—1) +2L+I, 1)
(4.6)
/
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0(N) if-model
615
1.5
6
4
I
0.5
1.2
1.4
1.6
1.8
2
2
Fig. 6. The ratio s~(C,)/s~(S)for the tower (0, 2; 2~i—2 + 1, 1), 1 ~ 2, even.
at 0(1). We normalize the fields a~to (a~(y
1)a~(y2))=
At 0(1) we obtain the fusion coefficients a®L 0 S
(4.7)
(y~)_2L~fl(a~).
‘~
—~
for (4.8)
CIL)
from the system of equations L)fm\ ~.4 i=0
i
(2L+I)~,(p.
—
1+!)~_,
i)
=a-rn
(4.9)
(2L+p.—1+2I)~..,
with (m + l)!(p. 1)rn a-~=6~0—L . (p. +1) —
(4.10)
The second term in (4.10) arises by “order mixing” from graph B11 in ref. [5], section 4. Namely, the graph B11 is 0(1/N) but only after partial compensation between the two parts describing the exchange of the S-field and the shadow field S. Extracting the part of the shadow S alone which is relevant here, gives a contribution of order 0(1) (ref. [5], (A2.9)). We have learnt from a more general approach that degeneracy sets in at L ~ 2, 1 ~ 2. Moreover the mechanism of order mixing changes at L ~ 2 invalidating
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K Lang, W. Rühl
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(4.10). Therefore, we limit here L to the values zero and one. The general case will be resumed in a future publication. A system of equations Im
m
(a +!)rn_i(b
1~Yi~1)
(a
l)~, 6~n
(4.11)
~ ~ (a+n), (b+n) 1) i—nil, . \t~/ (a +b +n +1— 1),..~
~4.12
+
b
+
+
21)~,
=
is solved in general by —
‘I
Y,~
Inserting this solution into (4.9), (4.10) gives for L
=
0 (4.13)
For L
=
1 we get (1+ 1)!(p. —1), y(l)
=
(—1)’
(p.
+
1),
—
(4.14)
Thus the shadow field contribution (the Kronecker delta in (4.14)) cancels all other 0(1) terms at 1=0, .y~l) =
0.
(4.15)
This was observed already in ref. [5]. The anomalous dimensions are calculated from ~
E~)~)( 1)
l=0
(2L +l)rn_i(p. —1 +l)rn_i ~ (2L+p.—1 +21)rni
(4.16)
where (from ref. [5], (4.17), (A2.4), (A2.6), (A2.9)) p.(p.—1) (p. —2)(p. —1 +m) m! 4—(4p.—5)(m+1)
p. +2(2p.—3)+(m+1)~. p.+m
(4.17)
We find immediately cI~=0.
(4.18)
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0.5
mu 1.4
1.6
1.6
-0.5
2
-1.5
3
Fig. 7. Anomalous dimensions for the tower (~,1; ,a
+1 +
1, 1), 1 ~ 1.
This is consistent with (4.14), (4.15). The general solution of (4.16), (4.17) at L is obtained from (4.12) as
(1)_~{2(2p._3)/21
+(_1)1~),}.
This expression can be evaluated asymptotically for I
1~ —
E~
1
(4.19)
—*
(—1)’ F(p. + 1) l’~ + 2 p.(p. l)(2p. —3) 1-2, p.—2 p.—2 —
=
(4.20)
so that uniformly 1irne~’~ = 0
(4.21)
l+~p.~2.
(4.22)
in
The anomalous dimensions are plotted in fig. 7.
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5. Different coupling schemes and the degeneracy problem We consider 2n-point functions of the fundamental fields Sa or a. We split n into n=p+q
(5.1)
and couple these fields by p-fold (q-fold) operator product expansion to qp-fields S~®and ~ (a~and 4):
~,,,p;
S~:(iii
p(p. —1), 0),
(5.2)
4: (0, 0; 2p, 0).
(5.3)
In a second step we couple these qp-fields to a whole tower of new qp-fields:
~
(I I I
~P,q)~
In,
a~Xa~—~ ~
n; n(p.
1) +1,
—
2n +1,
(0,0;
1)(p,q)
I)(P~a)
(5.4) (5.5)
We are interested in the p-dependence of these fields C1”~for fixed p + q, which we call the “coupling scheme” of forming ~ As usual we calculate the fusion coefficients 4~and the anomalous dimensions ~(C1~°’~).The intermediate fields 4 are normalized as in (4.7). Expanding the 2n-point functions till 0(1/N) gives the anomalous dimensions of the scalar fields S~,,4 first ‘q(S~)
1
n(S)
p.—2
[n(n—2)p.+2n]
(5.6)
(see ref. [2], (1.3)), 4(2p.
_____ _____
n(S)
=
—
—
p.—2
1) [n(n—2)(p.—1)+~n(n—1)(p.—2)(2p.—1)j. (5.7)
For these scalar qp-fields the coupling procedure is irrelevant. For the fusion coefficients of the coupling (5.4), (5.5) we obtain the system of equations 6 +!)m,(q6 +l)rn_i (pq) (58) Yi(p,q)(m\ ~ ~ J (P((p+q)6+2I)~, a-Dfl =
where m
a-~,’~’= ~a~’(6)’ r=0
f—~’i~ I
~
l~
~ /r\
q),.
Jr
(5.9)
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Here and in the sequel such formulae have the following interpretation: First insert q if p ~ q and divide all possible factors, then insert p. The auxiliary coefficients a~(6)are obtained from expanding (6s)~in powers of s, m
(6s)~= ~ (_1)r(_s)ra~(6).
(5.10)
r=0
This equation is solved by ifm=0
~~
a~’(6)
r—1
—
E((r—s)6)rn
—
(_1)s ~
if m>0.
(5.11)
The simplest solutions of the system (5.8), (5.9) follow immediately y(P~0)=
a-~”~ 1,
(5.12)
=
and (p,q) 0. Yi The anomalous dimensions result from the system
(5.13)
=
m
(P,a)( m \ (P6 + l)m,(q6 + l)~_~ Yi \ I) ((p+q)6+21)m, =
~
~P~0)
i=0
~
(5.14)
3(p,q) m
where
~~ m
r
(5.15)
rn—r ~ç(p,q)~ rn
r=0
(~1)r( ~Pr+1
m rn
=pq
—q)~+i
~ a~(6) r=0
(5.16) (—p—q)r+2
~(p,q)= .Lpq ~ a~’(6)(_1)r(_p)(_q) rn 2 r=0
x[(p—1)(p—r)
+(q—1)(q—r)].
For ‘15,,, in (5.15) we insert (3.13) (with plus sign) for the field (3.18) for the field a (6 2). At 1 0 we obtain from (5.14)
(5.17) ~a
(6
=
p.
—
1) and
=
=
=
~pq’I0 =pqf~),
(5.18)
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620
which shows that or
T1(C~P’~)) ~(5p+q)
~j(a~”).
(5.19)
Moreover =
0
(5.20)
(in consistency with (5.13)) follows from the crossing symmetry constraint (5.21) which is known from (3.15). We can therefore interpret (5.13) as a consequence of crossing symmetry. There is another consequence of crossing symmetry visible in the coefficients ~ namely (p —q)2,
~
!~3 odd.
(5.22)
With the shorthands n=p+q, we find e.g. for 6
t=pq,
(5.23)
2
=
E
(5.24)
2—E0(2fl+1)[—0+~I22j,
3—0=(n+
(5.25)
1)[—~o+~~2},
and 24M(n,
C4E0
t)
[Q0(n, t)~0+Q2(n, t)~2+Q4(n, t)~4],
where M, Q0, Q2, Q4 are given by 4 2n3 19n2 + 39n + 72— 12t(13n2 + 4n + 3) + 18t2(n M(n, t) 20n Q 5 192n4 2652n3 6372n2 + 6048n + 13608 0(n, t) —240n + t(240n4 + 3720n3 + 6540n2 + 3060n + 2160) —
—
(5.26)
+
19),
=
—
—
—
=
t2(2304n2
—
Q
+
9432n
+
12744)
4 + 1944n3 + 4164n2
2(n, t)
=
t(1248n3
+
5
53n3
Q t)
=
t3(216n
+
4104),
5076n —9936
—
—96n —
4(n,
+
4n
+
8n4
—
2256n2 —
+
864n
102n2
+
+
432)
153n
+
+
t2(144n2
270.
+
2952n
+
4104),
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621
= 8
4~i+2 6
4~L
=
4~z—1
—
4~—2
—
4/~—4
0
2
1
Fig. 8. The degeneracies of the tower (~, 4; 4j.u —4 + 1, 1).
Thus degeneracy sets in at rank 1 4. The degree of degeneracy found is equal or less the number of values t (5.23), =
n—i
for n odd
for n even, 1 even
forneven,Iodd. This degeneracy is plotted in figs. 3 and 8. In case of degeneracy the pure qp-fields are obtained by a diagonalization of a secular matrix. The values for 1 4 represent some average over the anomalous dimensions. =
6. Discussion We have described a method to obtain qp-fields at any level ([6], 1) in any class (Y, p), when Y possesses at most one row (symmetric traceless 0(N) tensors or 0(N) scalars). If Y possesses two rows of lengths n1, n2,
‘~1
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then the 0(d) tensor rank of the primary field is at least n2 [2]. In such a case we need a master formula for the decomposition 6A’1A)®(6B’0)—~(6c’1) (6.1) (
which offers no problem in principle. The master formula can be used to obtain qp-fields of arbitrary twist and tensor rank by a single operator product expansion. Multiple operator product expansions enable us to reach the same goal with less algebra. Operator product expansions of four or more qp-fields admit different coupling schemes which for sufficiently high 0(d) tensor rank (I ~ 4) leads to the degeneracy phenomenon. In this work we proved only its existence. A full analysis will be attacked in a future publication. Consider two pairs of scalar fields A, B and A’, B’, in e.g. A
=
4,
B
=
4,
A’
=
4’,
B’
=
a~’
(6.2)
with p+q=p’+q’
(6.3)
and the conformal four-point function
In a corresponding non-diagonal master formula replace
~
by (6.4)
and e 1y, by l E fl~ ,
(ff)\
1 A’B’—.C~
)
AB-.C~”
C~”~ are true qp-fields with degeneracy label a-, CAB ~0~) are the fusion coefficients (they are the “linear” coefficients compared with the “quadratic” coefficients y,). The labels (A, B), (A’, B’), which must observe (6.3), are matrix labels defining a “metric matrix” (6.4) and the “secular matrix” (6.5). Finally we want to make a remark about the subalgebra structure of our field algebra. Naively one could expect that the class (0, 0) of qp-fields forms a subalgebra. In fact, the situation is more complicated. Two auxiliary fields contribute to the energy—momentum field (this is proved in ref. [5])and therefore at 0(1/N)
(0,
2) 4—(Ø,
0)0(0, 0).
(6.6)
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-model
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At still higher orders of i/N we expect all classes (0, 2n) to become visible. The primary fields of these classes are rank-2n 0(d)-tensors of dimension 2np., which we call the W-fields. Our suggestion is that
(6.7) is the minimal subalgebra. Whenever possible, the algebra was checked with the aid of MAPLE V installed on an IBM RISC Station of the RHRK Kaiserslautern. This work was partially supported by DFG Grant Ru 227/9-i.
References [1] [2] [3] [4]
K. Lang and W. Rühl, Z. Phys. C50 (1991) 285 K. Lang and W. Rühl, Z. Phys. C51 (1991) 127 K. Lang and W. Rühl, NucI. Phys. B377 (1992) 371 K. Lang and W. Rühl, Phys. Lett. B275 (1992) 93 [5] K. Lang and W. Rühl, The critical 0(N) if-model at dimension 2 < d < 4: A list of quasiprimary fields, Kaiserslautern University preprint KL-TH-92/7, May 1992 [6] A.N. Vasil’ev, Yu.M. Pismak and Yu.R. Khonkonen, Theor. Mat. Fiz. 46 (1981) 157; 47 (1981) 291 [7] AN. Vasil’ev, Yu.M. Pismak and Yu.R. Khonkonen, Theor. Mat. Fiz. 50 (1982) 195 [8] W. Bernreuther and F. Wegner, Phys. Rev. Lett. 57 (1986) 1383 [9] V.K. Dobrev. G. Mack, V.B. Petkova, S.G. Petrova and IT. Todorov, Harmonic analysis on the n-dimensional Lorentz group and its application to conformal quantum field theory, Lecture Notes in Physics, vol. 63 (Springer, Berlin, 1977) [10] R. Abe and S. Hikami, Frog. Theor. Phys. 49 (1973) 1851 [11] IS. Gradshteyn and I.M. Ryzhik, Table of integrals, series and products, 4th ed. (Academic Press, New York, 1965) [12] Y.L. Luke, The special functions and their approximations, 2 Vols. (Academic Press, New York, 1969)
Nuclear Physics B400 [FS] (1993) 597—623 North-Holland
________________
The critical O( N) u-model at dimensions 2
1
and W. Rühl
Fachbereich Kaiserslautern, D-6750 Kaiserslautern, Germany Received 16 September 1992 (Revised 10 November 1992) Accepted for publication 2 December 1992
Fusion coefficients and anomalous dimensions of the quasi-primary fields are extracted from the 1/N expansion using representation theory of the conformal group whose formulae are analytically extended to 2
1. Introduction Nonlinear a--models can directly be applied to phenomenological solid state physics. But the fact that they are building blocks for other theories such as string theories and gravitational theories is a stronger motivation for us to study them in great detail. At d> 2 a vector nonlinear a--model possesses a critical point at which it can be described by a conformal field theory. This implies that only leading order scaling terms are taken into account. At d = 2 conformal field theories possess an algebraic structure which enables us to apply representation theory, to construct explicit Hubert space realizations, and to classify these field theories. In turn, this representation theory relies heavily on free field theories. To find any kind of algebraic structure beyond the operator product expansion algebra (whose existence is guaranteed) at d> 2, is one of the goals of our investigations. The study of critical nonlinear 0(N) vector a--models involves the following issues:
E-mail address: [email protected]
0550-3213/93/$06.00 © 1993
—
Elsevier Science Publishers By. All rights reserved
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K Lang, W. Rühl
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(1) to find the type of 0(N) and conformal group representations to which the conformal fields (synonymously “quasi-primary” fields or “qp-fields”) in these models belong; (2) to classify these qp-fields, in particular to determine the spectrum of field dimension 8 in relation to the 0(N) and 0(d) tensor representations; (3) to determine the fusion coefficients YAB for the emergence of the qp-field C from the operator product expansion of two qp-fields A and B. Since conformal field theory allows one to reconstruct any Green function from the information contained in (1), (2), (3) (involving, eventually, infinite sums over qp-fields), it can be expected that these issues can be solved only approximately or partially. In our previous work [1—51we showed that the qp-fields occurring are irreducible 0(N) and 0(d) tensors with a trivial spin-transformation behaviour under special conformal transformations. Thus (1) can be considered to be solved completely. Our contributions to (2) and (3) are approximative in the sense that we limit our investigations to the lowest orders in a 1/N expansion. Moreover, though the number of qp-fields treated is infinite, an infinite number of qp-fields is still out of reach at the end of this work. We suggested in ref. [5]to order the conformal fields of this model into classes (Y, p)
(1.1)
where Y denotes the Young frame of the irreducible 0(N) representation and p is the degree of the normal product of free bosonic massless fields which represents the qp-field in question at N The number of blocks #Y of Y satisfies =
~.
p=#Y+2n,
(1.2)
where n is the number of 0(N) contractions. Within a class (Y, p) the qp-fields are ordered into a two-dimensional pattern according to their normal dimension [6],their (symmetric traceless) tensor rank 1, and their twist t (fig. 1) which obey the relation [6]
—
[6~]
=
1—10 +
t,
(1.3)
Here [6~], 10 are the minimal values belonging to the “primary field” in the class. In certain cases, e.g. if the qp-fields are only accessible by crossing symmetric fusion (see sect. 3) qp-fields with odd 1 or even 1 may be missing for some fixed t. The two fundamental fields Sa(X), the 0(N) vector field, and a(x), the auxiliary field, on which the path integral formalism of the a--model is based, belong to exceptional classes (0, 1) (fig. 2) and (0, 0) (fig. 3) (0 stands for the
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[~5] ~0
10 —
1~+ 3 —
1~+2
[6~1+i
I~
—
10+1
ID 0
2
4
6
1
Fig. 1. The generic class (Y, p) of qp-fields. 10 is the minimal tensor rank, the “primary field” ([&~], la).
[~] the
normal dimension of
empty set). In these classes there are no higher qp-fields in t 0 towers due to the appearance of Sa(X) and a(x) propagators in one-particle reducible graphs. The labels (Y, p; [6], 1) may not discriminate between the qp-fields uniquely but we observe a phenomenon of “degeneracy”. This degeneracy is lifted if we calculate the field dimension of the ‘I completely. =
dim cP=8,1,=p(/.L—1)+q+v~(cP),
(1.4) (1.5)
[8]
1=0
1=2
1=0
1=1 l=0
S 0
2
4
I
6
Fig. 2. The qp-field content of the class (D, 1). The field [,5] = ~.o+ 1, order in 1/N higher than the rest.
t
=
2, 1
=
0 appears probably one
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—
—
1=2
14
—
—
—
—
1=2
12
—
10
—
8
~
—
—
—
1=2
—
1=4
—
1=2
1=0
1=0
—
1=0
1=3 ~-
~-
6
1=0
a I
0
Fig. 3. The class
(0,
2
4
6
I
I
I
8
10
12
1
0) of qp-fields. The degeneracy as obtained in sect. 5 is indicated.
[6~] =p(j.t
—
1)
+
q,
(1.6)
(1.7)
where q is the minimal number of derivatives in the normal product at N and ~(ct) is the anomalous dimension. So we determine different anomalous dimensions for the same position (Y, p, [6], 1). Of course the field algebra of the critical a--model is generated from multiple operator product expansion of the fields Sa(X) and ct(x). This product is associative. Nevertheless we may end up with different resulting qp-fields at the same position (Y, p; [8], 1) if we force the path of expansion to run through different intermediate positions (different “field couplings”). In sect. 5 we will investigate two exemplary cases (k q +p 4) =
=
~,
~‘
(0,0; 2, 0)k®_., (0,0; 2p, 0)
®
(0,0; 2q, 0)
—~(O,0;2k+1,l),
(1.8)
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0(N) if-model
(o,1;~—1,0)~~(iii ip,p;p(~1),0)®(iii
(~ii
1k,
601
q,q;q(~1),0)
k; k(~—1) +1,1),
(1.9)
and find that 7J(cP) depends besides p + q (trivial) on pq if 1 ~ 4. The results of these degeneracies are plotted in fig. 3 and fig. 8. In this article we calculate anomalous dimensions and fusion coefficients for the primary fields (Y, p; p(~—1) +q, q)
(1.10)
p=#Y
(1.11)
with
and Young frames of two rows of length
11 =p—q,
12=q,
11 >~l2
(1.12)
the anomalous dimensions at 0(1/N) were derived in ref. [2]. Here we compute anomalous dimensions for infinite towers (Y, p, t fixed, 1 running) of several classes. The best studied class is the class (0, 0). Our procedure is as follows. We calculate a four-point function of two qp-fields A and B (A(y1)B(y3)A(y2)B(y4))
(1.13)
in a (1/N)-expansion. In a conformal field theory it depends essentially only on two biharmonic ratios 22
22
Y13Y24 2
2’
Y14Y23 ~=
Y12Y34
2
2’
(1.14)
Yij=Yi~Yj•
Y12Y34
If we let y13—O
or
y24—*0
(1.15)
u—40
and
v-41.
(1.16)
then
An operator product expansion connected with (1.15) is
A( y)B(0)
=
(~2)88A
_1~2~
(n ~
+
...
(1.17)
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(A, B scalar, C rank-I tensor). The four-point function (1.13) appears as a sum over conformal one-particle reducible amplitudes
A
C
(ABAB)=~ B
B
(1.18)
and at the same time as a generalized hypergeometric function [3,5]
(ABAB)=
~
U~(l~V)
~
[...].
(1.19)
n!m!
n,m—0
Comparison of (1.18) with (1.19) gives all information on C: the anomalous dimension and the fusion coefficient (Y~B)’~2. The coefficients Y.~Bdepend on the normalization of the qp-fields A, B, C. For the fundamental fields Sa(X) and a(x) we set ~,
(1.20)
KSa(X)Sb(0))=6ab(X2)a,
(a(x)a(0))
=
(1.21)
(x2)~,
(1.22) (1.23) The normalization (1.20), (1.21) can be obtained by a finite wave function renormalization which puts a coupling constant z 1/2 in front of the constraint term in the action (see (2.10)). The quantities a(S), ‘~j(a)and z are calculable as series expansions in 1/N. The expansion coefficients were listed in ref. [3]. The original calculations at order 1/N2 can be found in ref. [6] and at order 1/N3 in refs. [7,8]. The normalization of all other qp-fields is fixed in sect. 2. The one-particle reducible conformal four-point amplitudes (1.18) are derived in sect. 2 using the group theoretical analysis described in ref. [9]. In sect. 3 we deal with the four-point functions of four equal qp-fields and the implications of crossing symmetry. In sect. 4 we study towers of higher than minimal twist. Instead of using the conformal four-point amplitudes derived in sect. 2 we use the fact that (Y, p; [6], 0) ®a
=
~(Y,
p; [8] + 2+!,!)
Sect. 5 is devoted to the degeneracy problem. *
It gives the anomalous dimensions of A and B, too.
+
o(~).
(1.24)
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2. One-particle reducible four-point functions Application of group theory is possible only if the space dimension d is an integer. This leaves us d 3. Afterwards we may analytically extend d to the interval 2
—
=
[6, 1]
(2.1)
where Iis the rank of the symmetric traceless 0(d) (0(3)) tensor and 8 is the dimension of the field. That these representations are sufficient to describe the qp-fields was pointed out already in ref. [5] and is once again supported in this paper. If A, B are scalar qp-fields with labels [6A,O],
[6B,0],
(2.2)
then qp-fields C coupling to them are elementary representations and belong to [6~, 1]. The three-point function is V(y
6A;
Y2,
8B;
y
866A8~2
3, [6~, 1])A1A2
1,
A~=
C~B~ (y~23)( ~f~
2
.1
2
~Yl2) —
tracesl
(2.3)
with
e=
~
S
2
Y
S ~=
13 —i— Yi3
Y23
2 ~
~
Y13
_______ =
2
2
(2.4)—(2.6)
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The two-point function of a qp-field [6, 1] is the intertwining operator for the representation [6, 1] with its “shadow representation” [d 6, 1] (see ref. [9], (4.19)) —
~ ~
A,(
y1)C~1~2~,( y2) )fl a~.fl b~
(A,,~}
t1
(laIlbi)’ =
(y~2~c(1!)2 ~
Jl
a
b
-1) C~(flr(YI2)~)~
(2.7)
where 6Av
(2.8)
and C~’ (~ ~d) is the Gegenbauer polynomial (see ref. [11], 8.930). The normalization in (2.7) is ad hoc. As long as =
d>6
(2.9)
the shadow qp-fields can play a physical role. In fact, the auxiliary field a(x), introduced as a Lagrange multiplier field in the action by the term [5]
z1/2fdx
a(x)
a=1
(Sa(X))2,
(2.10)
has the same quantum numbers as the shadow field of the field
a~1(Sa(X))2
(2.11)
which it eliminates. This feature shows up explicitly in sect. 3. In order to construct the conformally invariant one-particle reducible four-point function
where both the qp-field C and its shadow field C are exchanged
(~is eliminated
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later), we contract an ABC three-point function (2.3) with an ABC three-point function
EfdY5 V(y1,
2 =f2
6B~ Y
4,
XV(y2, =
y 5, [6~, l])A1A2A,
Y
6A;
Y3~6B;
6A;
5~[d—6~,
(2.12)
!])A,A2A,.
1)
We notice first that (e
E [(eA)’~
—
traces] [fA,~
traces]
—
=
(G~1))~.
CfL~(e.f)
(2.13)
where
C[~(t)
=
~
G~tM
(2.14)
M=0
is again the Gegenbauer polynomial (see eq. (2.7)) which is even (odd) for even (odd) 1. Inserting (2.13), (2.14) into (2.12) and expanding (e .f)M into monomials of y,~, we end up with a finite sum of four-star functions [3] (3.17), (3.18)
S~4~( y 1,
a1
y2, a2 y3,
a3
y4, a4)
=
fdy5~(y~)~’,
(~ai
=
2~ d). =
(2.15) We discard now the second series in ref. [3] (3.18) since it describes the exchange of the shadow field C and, denoting the expansion coefficients in the first series by anm, we obtain
~
2M
x
(n,3} ~ —
1)
2_fl34(
2~2~34~1
)
Vfl23n,~=o
u(lv)
m
anm(ai, a2, a3, a4). (2.16)
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In this formula we use the biharmonic ratios u, v (1.14) and the exponents a, (2.15) 6A
a
1
=
~(6~
—
6B) +
~(n 12
+
—
n23
—
n34
+
n41),
6B) +
6C+6A
a2— ~(2~ 6A + 6B) +
a 3
=
~(—n
~(6~—
12 +n23 +n34 —n41),
(2.17) 3) terms in (2.16) for each fixed n, m. If n is The summation overin{n13} small (we will use factyields only0(M n 0), it is useful to perform a resummation rendering it a sum over 0(n3) terms. The result is =
~
ufl(1_v)mA(M)(6A,6
(2.18)
8,6c)
n.m.
n,m=0
for the last line of (2.16) with ~
~ r,s,f
M!(m +n +s —t)!(A3 + (M t r)!(m + n + s —
—
—
M)!
XF(,u. —A1 —A3—n, A2+A4+M—t—r— 1,
A1
—
~M+m +n, A3— ~M+m +n +s)
x F(A2 +A4
—
1, A1 +A3 + m
+
n
+
1,
s
—
I,
(2.19)
A2+ ~M, A4+ ~M—t—r)
where A 1
=~.t
—A4= ~(6c+6A —6~)’
A3=/i—A2=~(6c—6A+6B).
(2.20)
The summation in (2.18) runs only over 2n +m ~sM.
(2.21)
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In the sequel we will normalize the four-point function (2.12) by replacing C~BC~B2’A~’1
(2.22)
by 1. This is a normalization of the qp-field C as the following argument shows. If the operator product expansion (1.17) is inserted into (1.13), we get (2.7) with a, b replaced by y13, y24, respectively. The term containing no (a, b) contractions in (2.7) is then (2)8C2
1.
{
~-~‘13’34~
‘24’34~
(2.23)
y34
If we take into account that 1
—
v
+
2
=
higher order terms in y13, y24,
(2.24)
y34 we find the same expression in (2.16), (2.18) with (2.21), (2.22) at M=1,
n=0,
m=!.
(2.25)
Thus if we define the C-field by (1.17) then the normalizations (2.7) and (2.22) are identical. This is the normalization entering the fusion constants Y~B~ The master formulae (2.16), (2.18), (2.19) can be exploited to study almost all qp-fields in a class (due to crossing symmetry not all qp-fields may be obtained if A B). But except at n 0, which leads to towers of minimal twist, the algebra is complicated. For towers of non-minimal twist we prefer to apply (1.24). =
=
3. Equal quasi-primary fields and towers of minimal twist We consider first those cases for which 6A6B6’
(3.1)
A=B, as for example in the four-point functions of S-fields (see ref. [5], (3.20)) or of a-fields (see ref. [51,(5.4)). From these four-point functions we draw conclusions on the classes
(m~2),
(9,2),
(0, 0)(O, 2).
(3.2)
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At 0(1) we have from ref. [5] (3.24), (5.5), (5.7) after extraction of the factor (see eq. (2.16)) (3.3)
±1+~-~=
~
m=0
(1—v)tm m.
[±6mo+
(6)m]
(3.4)
(9, 2), + for the classes (ElI, 2) and (0, 0)). The towers of minimal 0 for (ED, 2) and (9, 2) and 2 for (0, 0)) contain qp-fields C
(— for the class
twist
(t
=
t
=
1, 1 ~ 0,
with dimension =
26+!
+
(~(C1) 2?~(A)). —
(3.5)
The fusion coefficients y~follow from m
A~’~
(3.6) with (at 0(1)) 4(1) Om
!\ (m \ [ía L.Y-’ ),n/J ~! (26 + 21)m_i L
=
form (2.19) and a-m= ±6mo+(6)m
(3.8)
from (3.4). The solution of the system (3.6) is
Yi
(26_i+l)1(1 ±(_1)1).
(3.9)
The fact that y~vanishes for I odd (in the cases (ED, 2) and (0, 0))) or for 1 even (for 2)) is the effect of “crossing symmetry”, i.e. the consequence of two fields being identical. It means that qp-fields are missing or not accessible from the operator product considered. The assessment of the first alternative to any order in 1/N is not easy. Next we calculate the anomalous dimensions at 0(1/N). We denote
(9,
=
2~(S)[~(A)
+
~(B) — ~(C,)]
(3.10)
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which from the operator product expansion (1.17) is inspected to multiply
—log u
(1—c)’ 1!
(3.11)
This leads to a system of equations
[(6+I)~,]2
~ E,Y,(7) 1=0
For (ED, 2) and
(3.12)
=
rn—i
(9, 2) we get from ref. [5], (3.25) (~—2)(~t—1+m)[±m!+(~—1)rn].
~rn
(3.13)
Eqs. (3.12), (3.13) do not obviously lead to solutions
e,y,=O
if
y,=0.
(3.14)
This is guaranteed, however, if the ‘I~, satisfy “crossing symmetry constraints”. They are fulfilled if the J~are correctly calculated from the crossing symmetric graphical expansion. Thus these constraints are valuable checks on the correctness of the algebraic forms of I~,(such as (3.13)). In all cases we have made this check. Solving for ~rn with odd labels m we obtain as constraints (2(n —k)
k ~2n+1
=
k=0
—1)
Ck
+
1)2k+1(6 +2(n —k))2k+i (2k + 1)!
with coefficients Ck given in table 1. Inserting finally (3.13) into (3.12) with 6 —2
n(S)
=
—
~~2(n—k)’
1 we get
(I—1)(2~—2+l)
316
(~-1+l)(~-2+l)
-
(3.15)
TABLE
~
I
Coefficients ck from the crossing symmetry constraint equations (3.15)
k
0
1
2
3
4
5
6
7
8
9
ck
I 2
I 4
I 2
17 8
31 2
691 4
5461 2
929569 16
3202291 2
221930581 4
)
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/
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(9~2)). The expression has a limit
limi7(C,) =2i1(S).
(3.17)
The limit 1 -~ ~ for fixed twist seems to exist in all cases considered (we do not always have analytic expressions to prove this). For the class (0, 0) and twist two we have 6 2 and =
m!
,a—2
~
(~t)~
—3)
[(~
1)~+ ~
—
1)~+~ — (m
—
+
1)! 1I (3.18)
(i~2)rn+i(m+1)!}
—(2,u—5) /L
—3
This is taken from ref. [5], eqs. (5.8), (5.10), (A3.1), (A3.5), (A3.7), (A3.9), (A3.11). The solution of system (3.12) with (3.18) and 6 2 can be brought in analytic form. We use the inverse matrix of the system (3.12) given in (4.12) and summation formulae for 2F1 series and saalschützian 3F2 series (for the criterion of saalschützian 3F2 summability see e.g. ref. [12], vol. 1, p. 103): =
—
3)
__________
j,
(1+ 1)(!+2)(~ _2)2 {(2~ The pole at ~
=
—1)— ~ —1—3)~~ (/L), !even (3.19)
2 is of order one. We insert (3.19) into i1(C,) ~
—2~,.
(3.20)
a(S) This gives ~(C0) n(S)
=
4(2k
—
1),
(3.21)
as known from ref. [5], (5.30). In the limit 1 ~ we find —*
—
3)F(JL + 1) 4(2~t— (~—2)2F(4—~t)l22~,
(3.22)
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1.4
1.2
1.6
611
18
2
-05
-1.5
-2
o
-2,5
2 4
Fig. 4. Anomalous dimensions for the tower
(0, 0; 4+1, 1), 1 even.
which tends to zero uniformly in any interval l+EI~/L~2.
(3.23)
In fig. 4 the curves for i~(C,)are plotted together with the limit curve at 1 ~: 2~(a). For d 3 (~t ~) all ~(C,) are equal as a consequence of (see eq. (3.18)) —*
=
=
~mI~*~3/20
(3.24)
Therefore we obtain simply
I ~*=3/2
=
2~j(a)I ~i=3/2
=
—
16~(S) I ~*=3/2
(3.25)
(see ref. [5], (5.29)). The tower t 0 of the class (0, 2) fits also into this context. It contains the energy—momentum tensor field at I 2. The leading contribution comes at 0(1/N) from the graphs (see ref. [3], section 4) =
=
1
(3.26)
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The system of equations for the fusion coefficients is (3.6), (3.7) with 6 1
______
6rno+(Ii~1)rn~2 (21.t2)rn
~
=
—
1 and
(3.27)
Comparison with (3.8) shows that only the last term on the r.h.s. of (3.27) which originates from B 2, is essentially new. It leads only to a minor modification of the solution (3.9) 2 Yi
~
—
1)112
(2~—3+l),
N
0,
I
even
1
odd
(3.28)
The sole contribution of the graph B2 is the Kronecker delta which renders (3.29) The graph B2 is the graph for the exchange of the field a and its shadow field B2:SøS—~a~--S®S.
a,
(3.30)
The spin and dimension of c~fits at leading order with the spin and dimension of a possible 1 0 ancestor field of the energy—momentum tensor, which in a free bosonic massless field theory would be =
(3.31)
2. ~Sa(X)
As follows from (3.28) the ~ field cancels this ancestor (up to 0(1/N2)), see ref. [4]) and thus guarantees that the a--model constraint (2.10) is fulfilled. The anomalous dimensions follow from (3.12) with 6 1 and =
—
m
—
___________
(~—2)(~—1+m)
x
(~ 1)rn(/2)rn —
—2(4~t—5)
(2~ 2)~ —
+2(2~t—3)
((,L_1)~)2
(2/2
—
3)m
+m! ~
p—O
(~L—2)~(~L—l)~ . (3.32) P~(2~—
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This expression was calculated from the graphs ~(B1
2(C +B3) +z
21 + C22)
(3.33)
2) radiative vertex function corrections of the graph B and the 0(1/N 2 which were all calculated in ref. [3]. In consistency with (3.29) we have ~o0.
(3.34)
Moreover we obtain =
0,
(3.35)
which was first proved in ref. [3]. This is necessary if C2 is to be identified with the conserved energy—momentum tensor. The general solution of (3.12), (3.28), (3.32) at 6 p. 1 and I ~ 4 is =
____
n(S)
—
—
—
2(p.—1)(2p.+1—1) (2p.
—
1)(p. + 1—2)2
~/—2 2 1 X{2(I1)+~((P+1)!)2(~~1)(/2(2~142P}.
From this expression we can calculate the limit 1
(3.36)
—~ ~
(3.37)
~2+O(!22~).
The curves i7(C,) are depicted in fig. 5. In fig. 6 we draw the curves ~ (C,)
showing that the limit 1
—*
is not uniform at p.
=
1.
4. The class (D, 1) Starting from the four-point function Ka( Yi)Sa( y3)a( Y2)Sb( Y4))
(4.1)
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0.25
8 0.2
6
4
0.15
0.1
0.05
1.2
1.4
1.6
mu 2
1.8
2
Fig. 5. Anomalous dimensions of the tower
(0, 2; 2~.s—2 + 1, 1), 1 ~ 2,
even.
we calculate the t 2 tower of the class (D, 1) applying our master formulae (2.16)—(2.20). However, we consider directly the (2L + 2)-point function =
IL
L
fla(~1) Sa(y3) fla(~1) Sb(Y4)) 1=1 j=1 /
(4.2)
and let ~,—‘y1,
all i,
~—~y2,
allj.
(4.3)
so that in the first step the qp-field a~=(O,0;2L,0)
(4.4)
is produced. In the second step the four-point function (aL®(yi)Sa( y3)a~(y2)S6(y4))
(4.5)
generates the tower of qp-fields C1’-~=(D,1; (p.—1) +2L+I, 1)
(4.6)
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1.5
6
4
I
0.5
1.2
1.4
1.6
1.8
2
2
Fig. 6. The ratio s~(C,)/s~(S)for the tower (0, 2; 2~i—2 + 1, 1), 1 ~ 2, even.
at 0(1). We normalize the fields a~to (a~(y
1)a~(y2))=
At 0(1) we obtain the fusion coefficients a®L 0 S
(4.7)
(y~)_2L~fl(a~).
‘~
—~
for (4.8)
CIL)
from the system of equations L)fm\ ~.4 i=0
i
(2L+I)~,(p.
—
1+!)~_,
i)
=a-rn
(4.9)
(2L+p.—1+2I)~..,
with (m + l)!(p. 1)rn a-~=6~0—L . (p. +1) —
(4.10)
The second term in (4.10) arises by “order mixing” from graph B11 in ref. [5], section 4. Namely, the graph B11 is 0(1/N) but only after partial compensation between the two parts describing the exchange of the S-field and the shadow field S. Extracting the part of the shadow S alone which is relevant here, gives a contribution of order 0(1) (ref. [5], (A2.9)). We have learnt from a more general approach that degeneracy sets in at L ~ 2, 1 ~ 2. Moreover the mechanism of order mixing changes at L ~ 2 invalidating
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K Lang, W. Rühl
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(4.10). Therefore, we limit here L to the values zero and one. The general case will be resumed in a future publication. A system of equations Im
m
(a +!)rn_i(b
1~Yi~1)
(a
l)~, 6~n
(4.11)
~ ~ (a+n), (b+n) 1) i—nil, . \t~/ (a +b +n +1— 1),..~
~4.12
+
b
+
+
21)~,
=
is solved in general by —
‘I
Y,~
Inserting this solution into (4.9), (4.10) gives for L
=
0 (4.13)
For L
=
1 we get (1+ 1)!(p. —1), y(l)
=
(—1)’
(p.
+
1),
—
(4.14)
Thus the shadow field contribution (the Kronecker delta in (4.14)) cancels all other 0(1) terms at 1=0, .y~l) =
0.
(4.15)
This was observed already in ref. [5]. The anomalous dimensions are calculated from ~
E~)~)( 1)
l=0
(2L +l)rn_i(p. —1 +l)rn_i ~ (2L+p.—1 +21)rni
(4.16)
where (from ref. [5], (4.17), (A2.4), (A2.6), (A2.9)) p.(p.—1) (p. —2)(p. —1 +m) m! 4—(4p.—5)(m+1)
p. +2(2p.—3)+(m+1)~. p.+m
(4.17)
We find immediately cI~=0.
(4.18)
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0.5
mu 1.4
1.6
1.6
-0.5
2
-1.5
3
Fig. 7. Anomalous dimensions for the tower (~,1; ,a
+1 +
1, 1), 1 ~ 1.
This is consistent with (4.14), (4.15). The general solution of (4.16), (4.17) at L is obtained from (4.12) as
(1)_~{2(2p._3)/21
+(_1)1~),}.
This expression can be evaluated asymptotically for I
1~ —
E~
1
(4.19)
—*
(—1)’ F(p. + 1) l’~ + 2 p.(p. l)(2p. —3) 1-2, p.—2 p.—2 —
=
(4.20)
so that uniformly 1irne~’~ = 0
(4.21)
l+~p.~2.
(4.22)
in
The anomalous dimensions are plotted in fig. 7.
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5. Different coupling schemes and the degeneracy problem We consider 2n-point functions of the fundamental fields Sa or a. We split n into n=p+q
(5.1)
and couple these fields by p-fold (q-fold) operator product expansion to qp-fields S~®and ~ (a~and 4):
~,,,p;
S~:(iii
p(p. —1), 0),
(5.2)
4: (0, 0; 2p, 0).
(5.3)
In a second step we couple these qp-fields to a whole tower of new qp-fields:
~
(I I I
~P,q)~
In,
a~Xa~—~ ~
n; n(p.
1) +1,
—
2n +1,
(0,0;
1)(p,q)
I)(P~a)
(5.4) (5.5)
We are interested in the p-dependence of these fields C1”~for fixed p + q, which we call the “coupling scheme” of forming ~ As usual we calculate the fusion coefficients 4~and the anomalous dimensions ~(C1~°’~).The intermediate fields 4 are normalized as in (4.7). Expanding the 2n-point functions till 0(1/N) gives the anomalous dimensions of the scalar fields S~,,4 first ‘q(S~)
1
n(S)
p.—2
[n(n—2)p.+2n]
(5.6)
(see ref. [2], (1.3)), 4(2p.
_____ _____
n(S)
=
—
—
p.—2
1) [n(n—2)(p.—1)+~n(n—1)(p.—2)(2p.—1)j. (5.7)
For these scalar qp-fields the coupling procedure is irrelevant. For the fusion coefficients of the coupling (5.4), (5.5) we obtain the system of equations 6 +!)m,(q6 +l)rn_i (pq) (58) Yi(p,q)(m\ ~ ~ J (P((p+q)6+2I)~, a-Dfl =
where m
a-~,’~’= ~a~’(6)’ r=0
f—~’i~ I
~
l~
~ /r\
q),.
Jr
(5.9)
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Here and in the sequel such formulae have the following interpretation: First insert q if p ~ q and divide all possible factors, then insert p. The auxiliary coefficients a~(6)are obtained from expanding (6s)~in powers of s, m
(6s)~= ~ (_1)r(_s)ra~(6).
(5.10)
r=0
This equation is solved by ifm=0
~~
a~’(6)
r—1
—
E((r—s)6)rn
—
(_1)s ~
if m>0.
(5.11)
The simplest solutions of the system (5.8), (5.9) follow immediately y(P~0)=
a-~”~ 1,
(5.12)
=
and (p,q) 0. Yi The anomalous dimensions result from the system
(5.13)
=
m
(P,a)( m \ (P6 + l)m,(q6 + l)~_~ Yi \ I) ((p+q)6+21)m, =
~
~P~0)
i=0
~
(5.14)
3(p,q) m
where
~~ m
r
(5.15)
rn—r ~ç(p,q)~ rn
r=0
(~1)r( ~Pr+1
m rn
=pq
—q)~+i
~ a~(6) r=0
(5.16) (—p—q)r+2
~(p,q)= .Lpq ~ a~’(6)(_1)r(_p)(_q) rn 2 r=0
x[(p—1)(p—r)
+(q—1)(q—r)].
For ‘15,,, in (5.15) we insert (3.13) (with plus sign) for the field (3.18) for the field a (6 2). At 1 0 we obtain from (5.14)
(5.17) ~a
(6
=
p.
—
1) and
=
=
=
~pq’I0 =pqf~),
(5.18)
K Lang, W. ROhl / 0(N) if-model
620
which shows that or
T1(C~P’~)) ~(5p+q)
~j(a~”).
(5.19)
Moreover =
0
(5.20)
(in consistency with (5.13)) follows from the crossing symmetry constraint (5.21) which is known from (3.15). We can therefore interpret (5.13) as a consequence of crossing symmetry. There is another consequence of crossing symmetry visible in the coefficients ~ namely (p —q)2,
~
!~3 odd.
(5.22)
With the shorthands n=p+q, we find e.g. for 6
t=pq,
(5.23)
2
=
E
(5.24)
2—E0(2fl+1)[—0+~I22j,
3—0=(n+
(5.25)
1)[—~o+~~2},
and 24M(n,
C4E0
t)
[Q0(n, t)~0+Q2(n, t)~2+Q4(n, t)~4],
where M, Q0, Q2, Q4 are given by 4 2n3 19n2 + 39n + 72— 12t(13n2 + 4n + 3) + 18t2(n M(n, t) 20n Q 5 192n4 2652n3 6372n2 + 6048n + 13608 0(n, t) —240n + t(240n4 + 3720n3 + 6540n2 + 3060n + 2160) —
—
(5.26)
+
19),
=
—
—
—
=
t2(2304n2
—
Q
+
9432n
+
12744)
4 + 1944n3 + 4164n2
2(n, t)
=
t(1248n3
+
5
53n3
Q t)
=
t3(216n
+
4104),
5076n —9936
—
—96n —
4(n,
+
4n
+
8n4
—
2256n2 —
+
864n
102n2
+
+
432)
153n
+
+
t2(144n2
270.
+
2952n
+
4104),
K Lang, W. Rühl
[6]
/
0(N) if-model
621
= 8
4~i+2 6
4~L
=
4~z—1
—
4~—2
—
4/~—4
0
2
1
Fig. 8. The degeneracies of the tower (~, 4; 4j.u —4 + 1, 1).
Thus degeneracy sets in at rank 1 4. The degree of degeneracy found is equal or less the number of values t (5.23), =
n—i
for n odd
for n even, 1 even
forneven,Iodd. This degeneracy is plotted in figs. 3 and 8. In case of degeneracy the pure qp-fields are obtained by a diagonalization of a secular matrix. The values for 1 4 represent some average over the anomalous dimensions. =
6. Discussion We have described a method to obtain qp-fields at any level ([6], 1) in any class (Y, p), when Y possesses at most one row (symmetric traceless 0(N) tensors or 0(N) scalars). If Y possesses two rows of lengths n1, n2,
‘~1
622
K Lang, W. Rühl
/ 0(N) if-model
then the 0(d) tensor rank of the primary field is at least n2 [2]. In such a case we need a master formula for the decomposition 6A’1A)®(6B’0)—~(6c’1) (6.1) (
which offers no problem in principle. The master formula can be used to obtain qp-fields of arbitrary twist and tensor rank by a single operator product expansion. Multiple operator product expansions enable us to reach the same goal with less algebra. Operator product expansions of four or more qp-fields admit different coupling schemes which for sufficiently high 0(d) tensor rank (I ~ 4) leads to the degeneracy phenomenon. In this work we proved only its existence. A full analysis will be attacked in a future publication. Consider two pairs of scalar fields A, B and A’, B’, in e.g. A
=
4,
B
=
4,
A’
=
4’,
B’
=
a~’
(6.2)
with p+q=p’+q’
(6.3)
and the conformal four-point function
In a corresponding non-diagonal master formula replace
~
by (6.4)
and e 1y, by l E fl~ ,
(ff)\
1 A’B’—.C~
)
AB-.C~”
C~”~ are true qp-fields with degeneracy label a-, CAB ~0~) are the fusion coefficients (they are the “linear” coefficients compared with the “quadratic” coefficients y,). The labels (A, B), (A’, B’), which must observe (6.3), are matrix labels defining a “metric matrix” (6.4) and the “secular matrix” (6.5). Finally we want to make a remark about the subalgebra structure of our field algebra. Naively one could expect that the class (0, 0) of qp-fields forms a subalgebra. In fact, the situation is more complicated. Two auxiliary fields contribute to the energy—momentum field (this is proved in ref. [5])and therefore at 0(1/N)
(0,
2) 4—(Ø,
0)0(0, 0).
(6.6)
K Lang, W. Rühl
/
0(N)
if
-model
623
At still higher orders of i/N we expect all classes (0, 2n) to become visible. The primary fields of these classes are rank-2n 0(d)-tensors of dimension 2np., which we call the W-fields. Our suggestion is that
(6.7) is the minimal subalgebra. Whenever possible, the algebra was checked with the aid of MAPLE V installed on an IBM RISC Station of the RHRK Kaiserslautern. This work was partially supported by DFG Grant Ru 227/9-i.
References [1] [2] [3] [4]
K. Lang and W. Rühl, Z. Phys. C50 (1991) 285 K. Lang and W. Rühl, Z. Phys. C51 (1991) 127 K. Lang and W. Rühl, NucI. Phys. B377 (1992) 371 K. Lang and W. Rühl, Phys. Lett. B275 (1992) 93 [5] K. Lang and W. Rühl, The critical 0(N) if-model at dimension 2 < d < 4: A list of quasiprimary fields, Kaiserslautern University preprint KL-TH-92/7, May 1992 [6] A.N. Vasil’ev, Yu.M. Pismak and Yu.R. Khonkonen, Theor. Mat. Fiz. 46 (1981) 157; 47 (1981) 291 [7] AN. Vasil’ev, Yu.M. Pismak and Yu.R. Khonkonen, Theor. Mat. Fiz. 50 (1982) 195 [8] W. Bernreuther and F. Wegner, Phys. Rev. Lett. 57 (1986) 1383 [9] V.K. Dobrev. G. Mack, V.B. Petkova, S.G. Petrova and IT. Todorov, Harmonic analysis on the n-dimensional Lorentz group and its application to conformal quantum field theory, Lecture Notes in Physics, vol. 63 (Springer, Berlin, 1977) [10] R. Abe and S. Hikami, Frog. Theor. Phys. 49 (1973) 1851 [11] IS. Gradshteyn and I.M. Ryzhik, Table of integrals, series and products, 4th ed. (Academic Press, New York, 1965) [12] Y.L. Luke, The special functions and their approximations, 2 Vols. (Academic Press, New York, 1969)