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The critical O(N) σ-model at dimensions 2 < d < 4: a list of quasi-primary fields
NUCLEAR PHYSICS B [FS]
Nuclear Physics B402 (1993) 573—603 North-Holland
________________
The critical O( N) o--model at dimensions 2
Classes of quasi-primary fields are defined. There are infinitely many classes and each class is infinite. We study those classes arising from four-point functions of the basic fields. All calculations are performed by a 1/N expansion.
1. Introduction
The critical nonlinear euclidean 0(N) u-model for space dimension d > 2 is a conformal field theory. Corrections to scaling are neglected. Our aim is to derive a list of all the fields in such a theory. We find it convenient to interpolate the dimension and restrict it to the interval 2
=
JD[S] DEal exP[_ _Jdx(+3~S(x)d~S(x) +~a(x)(S(x)2- 1) + fdx S(x)A(x))j.
(1.1)
Here S(x) is an 0(N)-vector O(d)-scalar field, A is an analogous source, and a(x) is an 0(N)-scalar 0(d)-scalar “auxiliary” or “Lagrange multiplier” field which guarantees that the constraint S(x)2=1
(1.2)
is fulfilled. This model has a critical point at which the theory is conformally covariant. 0550-3213/93/$06.0O © 1993
—
Elsevier Science Publishers B.V. All rights reserved
574
K Lang, W. Rühl
/
0(N) o--model
A saddle point expansion of the path integral (1.1) leads to a perturbative series in 1/N. It allows us to calculate any n-point function to arbitrary order in 1/N. We apply operator product expansions to the fields S(x) and a(x) and afterwards to the fields obtained by these expansions. This defines a free algebra of fields. Our aim is an insight into the structure of this algebra. At d 2 the critical 0(N) cr-models are supposed to be exactly solvable. But only the principal cr-models (with matrix field variables) which belong to the WZWN series [51have been solved so far. Exact solutions of vector models (only these are accessible by a 1/N expansion) are still unknown. Consequently our results cannot be compared with exact d 2 results. Of course it is possible to study critical vector-0(N) u-models around d = 2 or 4 in an expansion or at d 2 in a 1/N expansion. Such program has been followed by Wegner and his group since some time [61.Whenever possible we compare our results with his. But since he does not use conformal field theory nor the notion of a conformal field, a comparison necessitates a reinterpretation of this results in such terms. In a conformal field theory fields are conveniently chosen as conformal fields (which, in agreement with the terminology of Belavin, Polyakov and Zamolodchikov [71, are also called “quasi-primary”) or derivatives thereof. In a conformally covariant operator product expansion a conformal field appears with a well-defined sequence of “derivative fields” (they are not related with the “derivative fields” of Wegner [61). This sequence can be summed resulting in a nonlocal operator product expansion related to conformal three-point functions. After a multiple application of operator product expansions we end up with two-point functions of fields which we have then to identify. We have to decide whether these are conformal fields or derivative fields of conformal fields, and what representations of the conformal group these fields belong to. We have found that a consistent analysis is achieved with conformal fields which under euclidean motions (g, a), g E S0(d), transform as =
=
(T(g,a)~I/)~(X) =
S~(g)ç1s~(g1(x a)), —
(1.3)
where S(g) is an irreducible tensor representation of S0(d). In addition these fields have a dimension 6 and transform under special conformal transformations in a standard way following from (1.3). Though antisymmetric tensors exist, they do not appear in this paper. Symmetric tensor conformal fields possess only the quantum numbers (1, 6) where / is the tensor degree. A computation of n-point functions till order 1/Nk does not entail that anomalous dimensions of identified fields are also determined to that order (the anomalous dimension of the energy—momentum tensor to order 1/N necessitates a calculation of the four-point function of the S fields to order 1/N2, see ref. [3]). So our knowledge of these dimensions is very rudimentary still. According to this present-state knowledge all fields with equal anomalous dimensions belonging to
K Lang, W. Rühl / 0(N) o- -model
575
the same 0(N) representation can be interpreted as derivative fields of one common conformal field. In the limit N = ~ the 0(N) nonlinear cr-model tends to a free-field theory which is essentially obtained by the replacements S0(x)
~4Sa(X),
L%Sa(X)
=0,
a(x)—*0.
(1.4)
Correspondingly the dimensions of fields can be decomposed into a normal part (free-field part) and an anomalous part which is 0(1/N). By a (p 1)-fold operator product expansion of p fields S we obtain fields (among others) of normal dimensions —
p(/.L—1)+integer,
.t=+d.
(1.5)
Such field may have any 0(N) representation contained in the p-fold tensoring of the 0(N) vector: a tensor with a Young frame of p blocks, of p 2 blocks if one contraction is permitted, of p 4 blocks if two contractions are applied, etc.. The normal dimension of the auxiliary field a( x) is two. Now consider the set of fields with a specific 0(N) representation Y and a normal dimension —
—
p(~t—1)+n,
(1.6)
flE~>.
All quasi-primary fields with the same Y and p form a class (Y, p). The field with minimal possible n, n = n~,seems to be unique. It possesses a certain (not necessarily trivial) 0(d)-tensor type. Call it the primary field belonging to (Y, p). The auxiliary field can also be considered as primary with Y = trivial representation (denoted 0) p = 0 and n~= 2. From an operator product expansion producing a primary field of a class (Y, p) usually all fields in this class are accessible. It is therefore convenient to investigate the quasi-primary fields class after class. The content of a class can be represented in a plot such as fig. 1, labelling each quasi-primary field (qp-field) by its normal dimension [6] + n and its twist t. ~ The tensor rank of ~P(’~at the point (t, n) is rank
çp(r) =
n
—
t +
rank
(1.7)
~(O)
As compared with fig. 1 the content of a class can be reduced for two reasons: (a) some places ct~remain empty since symmetry forbids the existence of such fields. Typical are the cases n odd, t = 0 in the classes (1E, 1) and 1) which are forbidden by crossing symmetry (sect. 3);
(9,
K Lang, W. Rühl / 0(N) or-model
576
normal dim.
-
4~(IO)
~(ii)
~(7)
[8]-i-3
[~] ±2
-
____
~(8)
____
~(2)
~(3)
[~] +1 [~] ~(o) I
I
I
0
2
4
t
Fig. 1. The qp-fields contained in a general class (Y, p). (is the “twist”, Z(°)is the primary field.
(b) the levels n > 0, t = 0 in the classes (0, 1), (0, 0) (containing the basic primary fields S and a, respectively) are empty for dynamical reasons. This is one of the results of this paper and is discussed in sect. 6. A multiple occupancy of a level (t, n) in a class is not excluded. The interpolation of space dimensions 2
(5a(Y1)Sb(Y2)~
(1.8)
A(Y~2)~8ab,
2 (a(y1)a(y2)) =B(y12)
, ~
=y~—y1),
(1.9)
K Lang,
W. Rühl /
0(N) cr-model
577
where the “critical exponents” (field dimensions) are a’p—1+?7
p
=
2— 217
(‘q=i~(S)),
(1.10) (1.11)
2K. —
There is moreover a scale-invariant amplitude ratio (if the coupling constant is one) z=A2B. The three quantities
77, K,
z can be expanded asymptotically, 7?k
~
Kk ~
k—i
where
(1.12)
~
k=1
(1.13) k~1
771,2,3’ K
1, z12 can be found in the literature (see the list in ref. [3], ~ was given in refs. [8,9]). Anomalous dimensions of other primary fields ~ are expanded similarly, ~ 77k~ Yp Nk (1.14) k=1
In the cases that Y possesses two rows of lengths 11, 12, respectively, and p = 1~+ 12, ~ was calculated in ref. [2]. The energy—momentum field is probably primary (this question of its “ancestor” field was discussed in ref. [4]), has (Y, p) = (0, 2) and ~ = 0 was verified in ref. [3]. This paper is devoted to the issue of proving the asserted representation properties of the conformal fields, to work out the scheme of classes and to study a finite number of low lying elements in some classes, i.e. calculating anomalous dimensions and fusion constants. These classes are (0, 1) (sect. 4), (0, 0) (sect. 5) and the classes (1D, 2) and 2) (sect. 3). For these classes only connected graphs at 0(1/N) are needed in the graphical expansion. A combinatorial analysis such as developed in ref. [2] is needed for more complicated classes possessing disconnected graphs. After the analytic work of this paper, computer programs for both the combinatorics and the algebra will help us to investigate “all” fields in “all” classes. This paper is closed with sect. 6 which relates the content of a class to the underlying graphs.
(9,
2. Derivative fields Expanding the product of two qp-fields into new qp-fields and their derivatives, these derivatives form a series whose structure is determined by conformal
K Lang, W. Rühl / 0(N) cr-model
578
invariance. The description of such series is a classical topic in conformal field theory (see ref. [10]),but the formulas given there are incomplete and of not much use for us. Assume three scalar qp-fields A, B, C are given with dimensions 6A’ 6B’ 6c so that A(x)B(0)
=
(x2)
+
-(~A±ôB+~c)/2 r~O
Qab(x
ö)C(0)
further qp-fields,
(2.1)
a)
where the operators Q~b(x, are homogeneous in x of degree r and of order r in the derivative These operators follow from the two- and three-point functions,
a.
a,b
k
a~
X~
1 — —
k’(—a~ (—b~
[k/21
~
___________
(—a—b)k
/~\
1k—rn
0m!(k—2m)!(—~t---a—b+1)rn (2.2)
or after summation 1 k=O~
ab
—
(x,
(—a)rn(—b)rn
‘~‘
rn~o(~~+1)rnm!
1 X
—a
—
~
1F1(—b±m;—a—b+2m; xa~)
U12rn
(2.3) with a= —~(6B+8c—6A), b=
—~(6A+6c—6B).
(2.4)
Only the term with m = 0 in (2.3) was given in ref. [10]. This result can be generalized in the case that A, B, C are tensor fields. We are interested here in a particular case, namely
=
(x2)~’
~ 4~(x, r=i
+
where
~Aab
other fields,
r.
d)JAab(O) (2.5)
is the 0(N)-symmetry Noether current which is conserved and has
K Lang, W. Rühl / 0(N) cr-model
dimension 2h
—
579
1. J is normalized ad hoc here and in the sequel. The operators
z1~kx,~)are of degree r in x and of order r
a~)e’~”
Qka~b(X
=
1 in a. Denote r) e’~’~,
—
~~b(u,
u’ix~p,
(2.6)
T—~X2p2.
Then the solution for z1~can be obtained after Fourier transformation as 4~(x,ip)=
x A
4~(~t—
1)
x [(4/h(~ a
1)
—
+
2(3k
—
1)T~—+
~[~(u,
+ 2T2~)r
T)
-
r
—
1)Q~~~(u, r)
+
longitudinal terms,
(2.7)
where “longitudinal terms” (l.t.) are defined to be proportional to p~.In particular we have from (2.7) ~t~(x, ip) ~
XA +
ip) —crxA ~(x,
ip)
=
l.t., +
l.t.,
2(2~ 1) xA[(~ + 1)u2 + T} + l.t.
(2.8)
If A, B are equal scalar fields, (2.4) implies (2.9)
a=b=—~6c,
so that
Qab
does not depend on the anomalous dimension of A
=
B. Analogously
we do not find dependence on -q(S) in (2.7), (2.8). 3. Classes with p
=
2
3.1. THE FREE-FIELD LIMIT
The 0(N) nonlinear u-model degenerates in the limit N fields Sa(X) can be represented by free bosonic fields Sa(X), 4Sa(X)
=0,
m~The
0(N)-vector (3.1)
K Lang, W. Rühl / 0(N) cr-model
580
which form an 0(N) vector as well. By operator product expansions of these free fields Sa(X) we create a free algebra of fields. This algebra is the N = ~ limit of the “deformation sector” of the critical nonlinear 0(N) cr-model. We consider the operator product expansion Sa(X)Sb(X) =6ab(X2)
/L+l
+m~(0)
+ n~1 ~
x~(m~2.~ab(0)+J~)~ab(0))
(3.2) with mY~ab(y) =~(:a,~...a~sa(y)sb(y): + :sa(y)a~a~2...a~sb(y):)
(3.3) J~..i~,ab(Y) =~(:a~...a,~sa(y)sb(y): — :sa(y)a~...3~sb(y):).
(3.4)
Applying the differential operators (2.2), (2.8) we can first show that mW is a derivative field of m~°~ and ~(2) a derivative field of ~(l) We denote the qp-fields M~°~=m~°~, J~=j~).
(3.5)
New qp-fields arise at the next levels,
~
X~iX~2M~2ab(Y)
=
x,~x~2m~2 ab(
Y)
=
~
~
+ 1,1L±‘(x,
—
y)
—
4~(x,
a~)m~( y), (3.6)
a)j~’)(y).
(3.7)
This procedure can be continued. With A~=n(n+1) (3.8) 2’0, we find new qp-fields ~ j(A~) in m’ 1(2n + 1) respectively. These qp-fields are symmetric traceless 0(d) tensors which are moreover conserved. This unexpected (except for j(O)) conservation follows from conformal covariance (appendix A). The picture emerging for the spectra of qp-fields at N = is described by the t = 0 towers in figs. 2, 3.
K Lang, W. Rühl / 0(N) cr-model
581
normal dim.
2~c+2
6~
M~8~
M~
M~5~ 2j~
-
_______
2~—2
-
_____
______
I
I
I
0
2
4
t
Fig. 2. The spectrum of qp-fields in the class (EEl, 2).
normal dim.
_______
j(6)
_______
J(7)
_______
j(8)
2iz+2
j(5) 2~+1 j(2)
2~i
j(3)
-
2~z—1 -
_____
j(o)
I
I
0
2
Fig. 3. The spectrum of qp-fields in the class
I
4
(9 2).
t
K Lang, W. Rühl / 0(N) cr-model
582 3.2. THE cr-MODEL AT 011/N]
The operator product expansion ansatz for two 0(N)-vector fields is Sa(Yi)Sc(Y3)
=
(y~3) ~‘8ac1
x~
[(
+ ~
2)~?(M(~)7?(S) 1~4’,.~~..p,,ac(Y3)
+ other fields].
+(y~)~(J(A))_fl(S)J,~)~ac(y3)
(3.9)
A labels the qp-fields as in fig. 1 so that the anomalous dimension depends only on A. The tensor rank n may be trivially enlarged by multiplication with 4t Kronecker deltas so that the proper tensor rank is n t. t is denoted “twist”. The qp-field itself may have a proper tensor rank n(A) (see (1.7)) so that e.g. —
jq(A)
~
Y3 appears as the derivative field of M~ of order n dimensions of the fields in (3.9) are i~IP~,’~’
2/L
dim
Mi~~,ac
dim J
=
..p.,,ac2/2
—
2
+n
—
t
—
n(A). The total field
+ 77(M(A)),
—2+n+~(J~).
(3.10)
The 1/N correction introduces new qp-fields with twist nonzero so that the scheme of fig. 1 is completed. The gaps in the towers t = 0 at odd levels remain. Moreover all qp-fields obtain an anomalous dimension (except J~) and are nonconserved (except J~). The 0(d) tensors remain traceless symmetric by definition. The final list of qp-fields is described in figs. 2, 3. The calculations start from the four-point function KSa(Yl)Sb(
Y2)S~(Y3)Sa(Y4P~
(3.11)
which is expanded in powers of 1/N (see fig. 4). At each order we develop further in powers of Y13YlY3’
Y24Y2Y4-
(3.12)
Then we compare with (3.9) inserted into (3.11). We apply a finite wave function
K Lang, W. Rlihl / 0(N) cr-model
583
10
02
lo
03
lo
04
30
04
2°
04
2o
03
A1
30
A2
A3
o2
10
p3
04
2o
04
B1
2o
03
B3
B2
Fig. 4. Graphs A123 of order one and B123 of order 1/N to the four-point function (3.11).
2
renormalization to S andoperators a so that A introduce the projection
=
to the vertices. We
B = 1 and ascribe ~‘~
+
Hab,cd(M)
=
~(6ad6bc
6ac6bd)
11abcd(J)
=
~(6ad6bc~6ac6bd),
—
~6ab6cd’
(3.13) (3.14)
the biharmonic ratios 22
22
Y13Y24 2
2’
2
Y12Y34
(3.15)
2’
Y 12Y34
and use the technical apparatus of conformal functions developed elsewhere [3]. After extraction of 11(M) (3.13) (upper sign) or 11(J) (3.14) (lower sign) we get the contributions A1 +A3 = ±(y~y~)a±(y~y~)~,
B1±B3+
~77(5)(y~y~ya
(3.16) ~
~
U ~
n,m~O
x
±(~L
—
\m
~
n.m.
l)~(j.L 1)n+rn(n+m)! —
(P~)2n+rn
K Lang, W. Rühl / 0(N) cr-model
584
x[—log u+~í(n+1)—~(n+m+1)+2~p(p.+2n±m) 2 —
—~(~—1+n)—~(M—1+n+m)I +
1)n+rnl n!
(/.L)2n+rn
x [—log
u
+
2~(~+ 2n
+
m)
—
2~(~—1
+
n
+
m)}).
(3.17)
Expressions (3.16), (3.17) must be expanded in powers of y13, y42. It is of advantage to use the tensor bases described in appendix A (with x1 = y13, x2 = y42, say) and the help of a computer. The comparison with the operator product expansion (3.9) is done in the order of increasing polynomial degree of y13, y42. After identification of one field one subtracts all its derivative fields. This is done by applying the differential operators of sect. 2. The anomalous dimensions of the fields result from the log u coefficients in (3.17). From (3.9) and (3.16), (3.17) we deduce (M(°)(y3)M’~°~( y4)) =
(2
+
—
1)(~~
±o(l))(y2)_2~_2_~Mo~H(M)
(3.18)
The contribution of the field M~°~ to (3.11) is then 101)
[2
+
~
=
(2 +
?l(S)+~?(M(o))( 2) —2~+2—~i(M
o(~4){i + (
77(S)
-
-log u
~77(M(0)))(
-
log~2)}(y~4y2a
(3.19) Comparison with (3.17) gives =
4 2—p. a(S)
+
ul\
01~NJ I. —~
(3.20)
At the rank-one level no new qp-field arises and at the rank-two level we try the ansatz with mutually orthogonal qp-fields ~ ~ ~ Q~~M~)
M~”(x,
a)M~°)(y)+x~x~(M~(y) + (3.21)
K Lang, W. Rühl / 0(N) cr-model
585
The field M,~)is symmetric, traceless (and conserved in the leading order of 1/N). We find (appendix A) Xl~XlVX
2AX2~(MILP(X)MA~(0))=(X2)2~_2~~M~(air2s2 +...),
(3.22)
with a1 ... a5 satisfying (A.4) and (A.6). Here 2
8~i(~—1) a1=
/1\
+01—I.
(3.23)
The anomalous dimension is
2~)= 4
77(M~
177(s)+0(~).
(3.24)
The leading terms in (3.31) are of course the free-field expressions. More interest-3~ ing than verifying (A.6) also at order 1/N, is the appearance of the new field M~ at this order. After some algebra we find (M~3~(X)M~3~(0)) = 8(j~— 1)
2~+ 77(S)(X2)
o(~).
(3.25)
It is remarkable that (3.26) is necessary for positivity (it is in fact fulfilled). The lowest level qp-field in the J spectrum is the vector field J(O)• It can in fact be proved to any order of 1/N that a scalar field does not arise. The vector field is the Noether current of 0(N) symmetry. We obtain from (3.9), (3.16), (3.17) (J(o)(X)J(o)(O))
(2(~-1)
2~ 1 77(S)
+o(~))
26~~)(X2) _2(~fl(J),
x (2x~x~ _X
(3.27)
with 0, consistent with conservation of the Noether current.
(3.28)
586
K. Lang, W. Rühl
/
0(N) cr-model
At the rank-two level we find no new qp-field. At the rank-three level we obtain two new qp-fields j(2) and J(3) (see (2.15)),
~)J~~)( y) + ~
x,
y)
+ x2x~J~3~( y)
1\
~j,
+0
(3.29)
where j(2) is symmetric traceless (and conserved at leading order only) and appears first at order 1/N. We obtain
j(3)
1 =
4(~ + 1)(~+ 2) 17(S)
o(~)
+
(3.30)
and (see appendix A) ~
J,~2~3( X) J,~2~3(0)) 2~(b =
~
(X2)_2~_1
1r~s~ + ...).
(3.31)
At leading order we have in addition to (A.9) b5=b6=b1+8b4=O
(3.32)
and 16 j.L(~+ 1)(~
—
b1=-~--
2~+1
1
1)2
+0(N),
(3.33)
which corresponds to the free-field case. According to (A.13), (A.11) we expect (3.32) and (A.9) to hold at all orders 1/N. After some algebra we find also
26A~)(X2) 2~2H(J)
(p.±l) (J~~(X)J~(0) = ~
—
1
+0 —
N2
77(S)(2XAX~ X
~
—
.
(3.34)
The main obstacle for extending this analysis to higher-rank tensor fields is the subtraction of the derivative fields.
K Lang, W. Rühl
/
0(N) cr-model
587
4. The 0(N)-vector field The class of qp-fields labelled by
(Y, p)
=
(0,1)
(4.1)
can be obtained by expanding the operator product a(yt)Sa(y3).
(4.2)
In this case the spectrum of qp-fields turns out to be quite different from fig. 1.
We analyse the four-point function (a(yl)Sa(y3)a(y2)Sb(y4)).
(4.3)
It can be expanded graphically as in fig. 5. The graph B11 is 1P-reducible in the channel (1,3) (2,4). Moreover, the S-field propagator satisfies —,
2) a
=
4a77(S)(x2)
_~
4(x
=
o(~).
This leads to the phenomenon of “order mixing” in the 1/N expansion. o2
l~
o4 A
10
~2
10
B 11
B12
o2 30
B2 Fig. 5. Graph A is of order one and B11, B12, B2 are of order 1/N.
(4.4)
588
K. Lang, W. Rühl
/
0(N) cr-model
We renormalize the propagators (1.8), (1.9) so that A = B = 1 and z1”2 is ascribed to each vertex. We extract powers of N for each S-field loop and obtain for (4.3) 2NB
+z(B 11
+B12)
+z
2+
o(~))~
(4.5)
with A ~1 2 ~~i—~Yl2)
B11
=
2 \—a ~Y34j
(Y~1~(;~)_a(u;1c2Fi(1,~
—
2; ~
—
1; 1
—
v)
(4.7)
(_a~lo~u+b~2))~
+ rn,n=O
n.m. ~
flf1_
B12= (~~)_P(~~4)_a ~
~
rn,n=O
\rn
~ n.m.
(—a~,’,~ log u +b,~),
(4.8)
(—a~logu+b~).
(4.9)
~
B2=(Y~)P(Y~4)a ~
~
U ~
n.m.
rn,n’=O
The explicit forms of the coefficients anrn, bnrn, c are given in appendix B. The 1P-reducible feature of the B11 graph is visible only in the appearance of a gaussian hypergeometric function in addition to the usual two-variable series. The phenomenon of 1/N order mixing has also been obscured in (4.7) where each term is 0(1). In order to work this out we calculate the complete (to any order in 1/N) 1P-reducible graph B11~which contains B11 and all radiative vertex corrections in
a conformal-invariant way. The procedure was explained in ref. [3]. The result is ~ =
Both ~
(y~)(y~)~
1—
2c~ + u~’c~). (4.10)
n~m~ (u~
m,n~O ~ U
are 0(1). But expanding U’~ in powers of 1/N leads to a cancellation
and, as leading 0(1/N) term, we obtain zB 11 (4.7). The coefficients c~1’,’,~ are given in appendix B.
We make the ansatz for the operator product, 2)~1+(S)+~(S(~)+2K) a(
yi)Sa( ~
=
n~O~
Y13)®~1~
(y
5((
Y3)’ (4.11)
K Lang, W. Riihl / 0(N) cr-model
589
where S~’yiSa.
(4.12)
We conclude from (4.11) that the S-field propagator is contained in the u -1312 series of (4.10). By definition this series contains the S field together with all its derivative fields. From (B.8) we know that =
(4.13)
=~.
So ~ plays the role of the square of the fundamental aS S structure constant in the critical cr-model. Now we turn our attention to other quasi-primary fields. The series UKC~, in (4.10) contains terms 0(1) resulting from the subtraction of derivative fields of S. In fact, operating twice with the derivative operators 4~(x, for aS S on (X2)_a, e.g. with —*
a)
a)
4~2~(x,
f3(f3+2)
=
—‘
________
4a(a+ 1) ((x.a)2_ 2)x4).
(4.14)
leads only to the cancellation of one power of 77(5) in the denominator of (4.14) due to (4.4). Thus we have to add the series UKC~ to the contribution (4.6) of the A graph (this is what we call “order mixing”),
(Y~2)13(Y~4ia(1+ ~
m,n~O
Ufl(1_v)UKc(~)
n.m.
(4.15)
At n = m = 0 there is no 0(1) contribution in (4.15), but at n = 0, m = 1 there is one. This gives rise to an 0(d)-vector 0(N)-vector qp-field S~(y) (with t, r, s defined in (A.2)),
~ 2
—
2rs)(1 +
~
(4.16)
il (t The log u terms have coefficients (from (4.8), (4.9) and (4.15)) =
z(a~’,~+zNa~) —KC~.
(4.17)
At n
= m = 0 there is no 0(1/N) contribution (this is a nontrivial consistency check for our approach) but at n = 0, m = 1 there is one,
—2
(~t.—1)(2~i-—1) 17(5), JL+1
(4.18)
590
K. Lang, W. Rühl
normal dim.
/
0(N) cr-model
_______
_______
s~7~
s~8~
~(5)
/2+1
/2—1
I
I
I
0
2
4
t
Fig. 6. The spectrum of qp-fields in the class (~,1). The field S131 appears one order in 1/N higher than the rest.
and thereby 4~L2—j.~—2 —2
=
1
77(5)
+
o(_~).
(4.19)
Whether a scalar qp-field S~3~ exists can be decided upon only after has been calculated. Its two-point function is proportional to
‘2
(B.10)
F(a+K)2F(~+K)2F(a—~)F(a) —
—
a
—
K)F(2a —/L + K)F(2/L
—
a)F(~
—
a)
same expression at N = —
22/L + 17)
+
2
77(S).
(4.20)
(~L.—1)(~—2)
7~ appears at 0(1) with (see appendix A) The qp-field S~ 4xI,~xlpx2~x2,,(S~va(X)SA,~b(0)) 2s2 + =
(a1r
...
)(x2)
_~_3_fl(S(7))ôab,
(4.21)
K Lang, W. Rühl / 0(N) cr-model
591
with a, satisfying (A.6), (A.7) and 48~.t(p—1) (~t+2)(js.+3)
(1
+01— \N
.
(4.22)
To obtain (4.21), (4.22) we have to subtract the derivative field of S~ with derivative operator 2~(x,d)M= 1
4~
3[6x~(x~a) +x28~].
(4.23)
Of course the existence of the qp-fields ~(1), 5(2),2)~(4)~ - has not been excluded at least. rigorously, but their two-point functions are 0(1/N 5. The auxiliary field The class of qp-fields labelled by (Y,p)=(O,0)
(5.1)
contains the auxiliary field a( y) and is accessible from the operator products 5a(Y1)Sb(Y2)
(5.2)
and
a(y 1)a(y2).
(5.3)
Fields of (5.1) appear in the product (5.2) at order 1/N but in (5.3) already at order one. Hoping for a reduced computational effort we prefer to start from (5.3). Again we conjecture a spectrum of qp-fields in the class (5.1) similar to fig. 1 with ~ replaced by a(T) and [6] by two (see fig. 7). The four-point function of four auxiliary fields is then (a(y1)a(y2)a(y3)a(y4)) =A1 +A2+A3+ ~[z?(B11
+B12+B13) +z~(B21+B22+B23)1
(5.4)
592
K Lang, W. Rühl
/
0(N) cr-model
10-
02
19
02
lo
02
30
04
3O
04
3p.•
04
A1
A2
A3
lo--
-02
1?
92
lo.
30--
-04
36
64
3~/
B11
lo----
B12
~
3~....
~4
~
B13
lo---
p2
lo----
~°_
o4
30:-..
B21
--02
.
B22
B23
Fig. 7. Graphs for the four-point function of four fields a(y). The graphs A~are of order one, B,1 are of order 1/N.
Evaluation of these graphs gives
(5.5) A2
=
(y~y~)~,
(5.6)
13,
(5.7)
A3
=
(y~y~)
B
13~ 11=(y~2y~4)
B
m,n=O
13 ~ 12= (y~3y~4)~m,n=O
Ufl(lV)(_a~IogU+b~))
(5.8)
n.m.
U~(1—v)m (UaC~)+UM_adnrn),
n.m.
(5.9)
K Lang, W. Rühl
/
0(N) cr-model
593
rn
13 ~
B13 = (y~4y~3)~ m,n=O
u(1 v) n!m!
B
U~(1—v)
(—a”3) log u
—
+ b~’3~ nrn
(5.10)
nrn m
13~ 21=(y~y~) m,n=O x(—a~)log ~
n!m!
(5.lfl nrnJ~ rn
B
13~ 22=(y~y~)
x (—a
U~(1—V)
n!m!
m,n=O
log u
b~22~ + U”13c~22~
+
nm
(5.12)
nrn)~ rn
13~
B 23
=
U~(i
—
v)
‘—a~
log u + b~23~
(y~2y~4)
(5.13)
nrn)~ m,n=O
n!m!
nrn
‘
The coefficients a~j~b~1~b~13~C~’~d am are presented in appendix C. In the operator product expansion (5.3) we take into account the unit operator, the qp-fields (0, 0) and the qp-fields (0, 2). The energy—momentum tensor field is a twist-zero field from (0, 2), a(y 1)a(y3)
=
(y~)~1
~ + ~
i n ~(Yi~)®~~(
0))
a~
r~_1~_~~
(~)
n=O
+
~
( y132
)_3+JL+~?1(T(~))_t1(a(O))T(S) ILl
(~)
s=O
+further fields).
(5.14)
Further fields in (5.14) are expected from (0,4), (0,6), etc. Such fields exist because the free-field theory of sect. 3 contains corresponding examples (“W currents”). Moreover, a~°~ is proportional to the auxiliary field and therefore, by (1.13), 17(a~°~) = —2(77(5)
+K).
(5.15)
594
K Lang, W. Rühl
/
0(N) cr-model
The contribution of the identity operator in (5.14) appears in A2 (5.6). The fields ~ give (neglecting anomalous dimensions) a factor
x polynomials of y13,
~
y24.
(5.16)
Their contributions can be recognized in B12, B21, B22 as having the factor 13U~~
=
(y~
(5.17)
13Ua.
3y~4)
(y~y~)
From (C.3), (C.8), (C.10) we obtain z 1c~+c~,+c~~=0 if(n, m) =(0, 0) or (0, 1).
(5.18)
It follows then that the lowest tensor rank of the fields T~ of twist zero is two. We denote this field as and obtain (see appendix A) XlILXlVX2AX2U(1~V(X)T)~O)(0))
(5.19) where (A.6), (A.7) are fulfilled and (using ref. [3], eq. (2.20)) 128j.~ 1 1 \ 2) —+0 —I. (5.20) 2/1—iN N is proportional to the energy—momentum tensor of the cr-model. Comparing (5.19) with the corresponding expression for the free-field model ([1], eq. (4.10)), x1ILx1px2Ax2UKt~(X)tAU(0))= (a~r2s2+
.. .
)(X~)2~’,
(5.21)
where (A.6), (A.7) are also fulfilled and N
a 1
=
—i—
21(/L)F(/.L+1) (5.22)
2/1—1
fixes the relative normalization. t0~field produces a factor (neglecting anomalous Inserting (5.14) into (5.4) the a dimensions) (y~y~)’(y~)2xpolynomial in y 13, y24. Such an expression appears only in B12 (5.9) in the form 13uIL~ = (y~y~) _P+IL_a(22yIL+a (y~y~)
(5.23)
(5.24)
K Lang, W. Ru/cl
/ 0(N) cr-model
595
We proved that for polynomial degrees of y13, y24 less or equal to two this term (namely the dnm series in (59)) can be attributed solely to the field a~°~ and its derivatives 4Wa(°),~ We obtain a~°~(y) =y2a(y),
8 (2~
—
~N
3)21(2
—
(5.25)
2)
1(3-~)1(~-1)~
1
+O(~)~
(5.26)
t3~appears in fact at order one, namely in A The scalar field a
1 (5.5) and A3
(5.7),
3~(X)a~3~(0)> = (8 + 0(~))(X2)_4~3~.
(5.27)
(a~ Its anomalous dimension is as usual extracted from the log u terms, —
+~(a~3~) + 77(a~°~) =
~[z~(a~j1)
a~)I±o(-~).
43k) + z~(a~±a~2~ +
+
(5.28)
We use (5.15) and ref. [3], eq. (2.22), (2/.L—l)(~—l)
1
(5.29)
and obtain ~(a~3~)
=
—4(2k
—
1)277(5)
+o(~).
(5.30)
The fields ~ at8~and probably all higher qp-fields of nonvanishing twist appear also at order one. The n = 4 term of (5.14) is replaced by +
~
3)(
n(a10~) 4(2)(~13
+ ~ {(y~) 2)
+
(y
~
+ I + ~n(abB>)_71(ab°>)a(8)(
y 3)
d)a(
y3)
p 3)]
,
(5.31)
K Lang, W. Rühl
596
/
0(N) cr-model
is assumed to be traceless symmetric, ~ where a~7~
at7~,a~8~ to be pairwise
orthogonal, and (5.13) is fitted into the four-point function (5.4). We find xlILxlVx2Ax 2U(a,~V(X)a~(0)
)
2s2 + =
.
)( X2) —6—~(a~~~)
..
(5.32)
(a’r
where the a~’satisfy (A.6), (A.7) and 1
21333
a~’= —~--—
+0(k).
(5.33)
Moreover we obtain
(at8~(X)a~8~(0)) = (21032 (5)
+
o( ~))(x2)
-6-~(a~~~)
(5.34)
The qp-fields a~”kA~’° as in (3.8)) at twist zero have so far not been identified. it is straightforward to show that they are neither contained in the operator product expansion of two S fields including the order i/N. In fact the relevant graph is B 2 in fig. 4, and applying the integration techniques and conformal functions of ref. [3], we obtain for the relevant series
_______________
2
IL
(~—2)F(~t—1)
2
2
Y13Y24
—~f~-.
md
x
uFl(1_v) n,m~0
n.m.
IL+13
2
2
Yi2Y34)
(5.35)
00
Here dnm is the same as in (5.9). Therefore the same holds true what we found above (after eq. (4.24)): this series describes only the field a~°~ and its derivative fields. These derivative fields are in fact identical in the aa and in the SS channel. We conclude that the only qp-field of twist zero is ~ The whole class (0, 0) is presented in fig. 8.
6. Conclusions Though the content of each class may sometimes be surprising, the whole program of fitting operator products of conformal fields carrying elementary representations of the conformal group into the graphically expanded n-point functions have proved to be consistent. The connection between graphs and the content of classes can be derived from a more general scheme. We express the
K Lang, W. Ru/cl / 0(N) cr-model
597
normal
dim.
6
171
a
4
-
___
a~3~
2
I
I
0
2
I
4
Fig. 8. The spectrum of qp-fields in the class (0, 0). In the
t = 2 tower
t odd tensor fields are missing.
conformal function of the general box graph fig. 9 as a series expansion in u and
1
—
(4)
R (y1, y2, y3, y4)
a
2 =
1 (y34) 2 a1a3 2 (y24)
(y13) ~
x n,m=O ~
U~(1~V) n.m.,
~2
133). (6.1)
(c~u’~+ C~UU2+ C~U
Fig. 9. A ring graph for scalar fields with arbitrary dimensions a, f3. satisfying only uniqueness at each vertex: the sum of dimensions equals 2/2.
K Lang, W. Ru/cl
598
~1
/
0(N) cr-model
~3
Fig. 10. Three vertical cuts define sums of exponents which according to (6.3) are the dimensions of product fields (6.2).
The coefficients ~ were discussed in ref. [3], in the appendix. Let ~a denote a scalar field of dimension a and assume that an operator product expansion ~aj(
Yl )
~a3(
~)
=
(y~3)-(as
+a3-ô)/2
~
y3)
+
...
(6.2)
exists. Then each of the three series in (6.1) yields one solution for 6, ual:
61=a1+a3,
u’~2:
62=a2+a4,
13~: 83
=
f~1+ /33,
(6.3)
u
It is easy to see that log u terms of higher-order graphs may add a correction 0(1/N) to these dimension formulas (6.3). Let 3 be decomposed into a normal dimension [6] and an anomalous part 37(8), (6.4) Then the graphical rule of fig. 10 and its possible generalizations where the cuts meet more than two lines, determine [6] and moreover the class (Y, p) to which the qp-field cI~ belongs. Within this class (Y, p) it is not only one qp-field which follows from a single hypergeometric series in (6.1) but it is a whole subtree of qp-fields, i.e. all qp-fields fitting into fig. 1 with, say, t ~ t 0.
We consider now a single-particle reducible graph such as B12 in fig. 7. In this case the darn series in (5.9) corresponds to the propagation of the single-particle field and all its derivative fields. Consequently this series produces only one qp-field namely the S or the a field. The other qp-fields of twist zero whose existence was only conjectured, can therefore not be created at all according to these graphical rules.
K Lang, W. Rühl / 0(N) cr-model
599
Appendix A TENSOR BASES
Let I~(y) be a symmetric tensor of dimension y. Its two-point function xlILxlPx2Ax2UKrI3ILP( X)’IA~(O)) can be expressed in terms of the invariants 2, s = (x 2, r = (x1X)(X2)~/’ 2X)(X2)~
(A.1)
t
=
(x 1x2)
(A.2)
as the polynomial 2+a
2+ x~r2)+ a 2) 4(x~s 5x~xfl( X If ‘I~~(y) is traceless then the following relations hold:
~‘.
(A.3)
[air2s2 + a2rst + a3t
a 1+a2+2j.~a4=0, If
a3+a4+2~a5=0.
(A.4)
is moreover conserved,
a~I~IL~( y) 0,
(A.5)
=
and has dimension y
=
2j.~,then in addition to (A.4) we have a1
=
4a3,
a4
=
0,
(A.6)
or together a1:a2:a3:a5=1: —1:-i:
—i——.
(A.7)
In an analogous fashion we treat a symmetric tensor field of rank three D(y). The two-point function gives 2s2t ±b 2 + b 3+b 2+x~r2)rs = [b1r3s3 ±b2r 3rst 4t 5(x~s +b 2+x~r2)t+ b 6(x~s 7x~x~rs + b8x~x~t}(X2)Y. (A.8) Requiring I~ to be traceless entails 3b1 +2b2+2(/L b2
+
2b3
+
2(/L
+
1)b5=0,
+
1)b6 = 0,
b3+3b5+2b6+2(~+1)b7=0, 3b4 + b6
+
2(/L
+
1)b8
=
0.
(A.9)
600
K Lang, W. Rühl
/
0(N) cr-model
Consider a generating function for Gegenbauer’s polynomial,
[1
—
2p(2rs
—
t)
+x~x~p2]IL+1
n=O
=
x2, X),
(A.10)
where 7, is homogeneous in both x1, x2 of degree n. This gives [n/2l
1,(x1, x2, X)
=
k=0
1k 1 k!(n— 2k)
(x?x~[2(2rs
—
1)]n2k
(A.11)
It is easy to show that the generating function is harmonic in x1 and x2, so that 417~(x1,x2, X) =421~(x1,x2, T) =0.
(A.12)
Let ..1JX) be a traceless symmetric qp-field of dimension y. The conformal invariance of the two-point function can be shown to imply ~ILlIL2
XIIL1X1IL2...
xlIL x2~1x2~2...
=N,Dl~(xI,x2, X)(X2)~’.
(A.13)
For n = 2 we find (A.6), (A.7) and for n = 3 we obtain (A.9) and (3.32). Thus we can formulate another result: If the dimension of the conformal field is moreover canonical, ~ILIL2
y=2~—2+n
(A.14)
and n ~ 1, then IL2
IL(X)
=
0.
(A.15)
This explains all conservation laws found by us.
Appendix B COEFFICIENTS OF CONFORMAL FUNCTIONS B11, B12, B2, SECT. 4
B 1P.
aW= nm
—
~.2M F(/1—l)
(~i —
1)~(/.L 1)n+m(n +m F(,u+2n+m+1) —
+
1)!
‘
B 1
)
K Lang, W. Rühl / 0(N) cr-model j.(11)
=
a~[~/i(n + 1)
—
601
c/I(n + m + 2) + 2qi(j~+ 2n + m + 1)
Unm
—~fr(~t—1+n)—~/i(j~—1+n+m)},
(B.2)
2 ~
B
(
(~_2)F(~_1))’
(B.3)
~-IL
(l.L2)n(Ml)n+m(n+m+1)!
a~’2~
(B.4)
=
.
(~—2)F(p.—l)
am
12~
j,(12)
=
a~,’~[cfr(n+ 1)
+
F(ji+2n+m)
2i~(j.~ + 2n + m)
1~nm
—qi(j~—2+n)—q.~(p.—1+n±m)---~)j(n+m+2)],(B.5)
B 2:
a~
1(3 p,) 2)~F(2,L—3)
(~ —
—
2~
nm
=
(/1
—
1)~(~
1(.t
—
1)n+m(fl +
+
2n + m)
m)!
(B.6)
b~2~ =a~2’[c/1(n+ 1) —c~(n+m+1) +2c/i(~+2n +m) am
am
—&fi(~—1+n)—~/i(~t—1+n+m)].
(B.7)
of B
The coefficients ~
1 1,c are given by am
=
am
=
(~ — a
—
K)a+m(,LL /3 K)~(/.L (a)2a+m(a—i+1)a
K)~(,LL — a
—
—
—
21(a)F(a ~F(~
x
—
/3
—
K)n+m
(B.8)
—
F(a + )2p(p—/3+—K)2F(~ K) —a—K)21(~ —a)F(2~—a
(a + K)~(a + K)n+m(13 + K)~(/3 + K)a+m
(2.t
—
(B.9)
a) 2n±m(
—
a + 1)~
and ~can be expanded as E ~k37’
(B.10)
k= 1
,~2~
‘1
(p. (see ref. [3], eqs. (2.38), (2.39), (2.22)).
‘2
—
is still unknown.
(B.11)
602
K Lang, W. Ru/cl
/
0(N) cr-model
Appendix C COEFFICIENTS OF THE CONFORMAL FUNCTIONS
F(p. )2 6~ 2)1(2 a~~,~=ir(p.—2)21’(2p.—3)2 —
B11:
(p.
—
b~’1~ 1) nm = a~11~[c/i(n+ nm
—
B,~,i E (1,2), j ~ (1,2,3),
i)~(p.
—
SECT.
1)n+m(n +
—
m)!
1(p.+2n+m)
c/i(n + m + 1) +
2n
21/J(p. +
(12)
n.6IL 1(p.
=
—
2)~1(2 p.)3 (p. 2)1(2p. 3)2 —
12.
Cam
1(2p.
(2p.
2
—
a”3~[2c/i(n + 1) nm —2i/i(p.
1(2
B
(n!)2[(n+m)!12
—
p.)
1
—
—
(2n
3)2
—
m + 1) !(3
+
i)a]2[(n
p.)2 [(p.—
—
—
am =
(C.3)
(C.4) 2)41(2p.
—
1(p. 2)1(2 6~ a~=~ (p. 2)21(2p.
13:
i)n+m12
—
2)2n+m
~.6IL
=
(p.
B
—
—
1(2—p.) dam
(C.2)
i)~[(p.
—
—
(C.1)
m)
+
—~i(p.—1+n)—~(p.—1+n+m)],
B
5
n!1(p.
3)2
—
2~/i(n+ m
+
1)
+
+
—
+
m) !]2
2c/i(p.
+
2n
+ m)
n)],
+
(C.5) 2n +m)
(C.6)
[(n + m + 1) !}2
4IL
21:
(p.
a~=ir
—
2)21(p.)1(2p. —4)
(~ 1)r(~ + s)!
r!s!
(p.
(p.)
—2—
r)a(p.
—
1)n+m+s
(C.7)
2~±~~r(n+m+r+s+2)!’
r,s
r,s~0,n~r+s
2 F(p. F(2p. 2)21(23)2 p.)
am =
—
a
X
~
—
—
n! [(p._1)a~mJ2(2p._4)a±m+s(p._2)~ s!
(2p.
(C.8) —
3)
2p. 2a+m(
—
3)a+m+s
K Lang, W. RuhI
22
B22:
4
a~~=~rIL
/
0(N) cr-model
603
1(2—p.)(n+m+1)!(n+1)! 2) 1(p.)F(2p.—4) (p.— 2
n
(p+r)!(n+m—r)!
n+rn—r
X r~0 ~-o r!p!(n—r)!(n+r+p+2)!
x
(3~p.)r(p.
~2)n—r(p.
—
l)n+p
(C.9)
,
(p.)~,~ +m —r
22 4 [‘(p. — 2)21(2 — p.)2 ~am )=~r12 F(2p. — 3) 2
n+m
(n + m)! (p.
x ~~-0 B 23:
p!n!
—
1)~(p.
2)~(2p.
—
1)n±m(p.—
—
3)~+~(2p. 3)2a+m
(2p.
—
,
(C.10)
—
1(4—p.) (p.—2)41(p.)1(2p.—4)
= nm23~
a~
~n~rn+~—r(r+p)!(r+q)!(n±rn_r)!(n+m+1)! r=0 p=O
(4— p.)~(p. 2)ar(p. (2r+p+q+ 1)!(p.) —
X
r.p.q.
q=0 —
2)r+p+q
2n_r+rn 1,b~, ~
.
(C.11)
are known so far only as integrals of Barnes type.
The coefficients b~ References [11K. Lang
and W. Rühl, Z. Phys. C50 (1991) 285 [2] K. Lang and W. Rühl, Z. Phys. C51 (1991) 127 [3] K. Lang and W. RühI, Nuci. Phys. B377 (1992) 371 [4] K. Lang and W. Rühl, Phys. Lett. B275 (1992) 93 [5] see e.g. M. Kaku, Strings, conformal fields, and topology, an introduction (Springer, New York, 1991) ch. 3 [6] F. Wegner, Z. Phys. B78 (1990) 33, and earlier papers quoted there [7]A. Belavin, A. Polyakov and A. Zamolodchikov, Nuci. Phys. B241 (1984) 333 [81 A.N. Vasil’ev, Yu.M. Pismak and Yu.R. Khonkonen, Teor. Mat. Fiz. 50 (1982) 195 [91W. Bernreuther and F. Wegner, Phys. Rev. Lett. 57 (1986) 1383 [101 S. Ferrara, R. Gatto and A.F. Grub, Conformal algebra in space—time and operator product expansion, Springer Tracts in Modern Physics, Vol. 67 (Springer, Heidelberg, 1973)
Nuclear Physics B402 (1993) 573—603 North-Holland
________________
The critical O( N) o--model at dimensions 2
Classes of quasi-primary fields are defined. There are infinitely many classes and each class is infinite. We study those classes arising from four-point functions of the basic fields. All calculations are performed by a 1/N expansion.
1. Introduction
The critical nonlinear euclidean 0(N) u-model for space dimension d > 2 is a conformal field theory. Corrections to scaling are neglected. Our aim is to derive a list of all the fields in such a theory. We find it convenient to interpolate the dimension and restrict it to the interval 2
=
JD[S] DEal exP[_ _Jdx(+3~S(x)d~S(x) +~a(x)(S(x)2- 1) + fdx S(x)A(x))j.
(1.1)
Here S(x) is an 0(N)-vector O(d)-scalar field, A is an analogous source, and a(x) is an 0(N)-scalar 0(d)-scalar “auxiliary” or “Lagrange multiplier” field which guarantees that the constraint S(x)2=1
(1.2)
is fulfilled. This model has a critical point at which the theory is conformally covariant. 0550-3213/93/$06.0O © 1993
—
Elsevier Science Publishers B.V. All rights reserved
574
K Lang, W. Rühl
/
0(N) o--model
A saddle point expansion of the path integral (1.1) leads to a perturbative series in 1/N. It allows us to calculate any n-point function to arbitrary order in 1/N. We apply operator product expansions to the fields S(x) and a(x) and afterwards to the fields obtained by these expansions. This defines a free algebra of fields. Our aim is an insight into the structure of this algebra. At d 2 the critical 0(N) cr-models are supposed to be exactly solvable. But only the principal cr-models (with matrix field variables) which belong to the WZWN series [51have been solved so far. Exact solutions of vector models (only these are accessible by a 1/N expansion) are still unknown. Consequently our results cannot be compared with exact d 2 results. Of course it is possible to study critical vector-0(N) u-models around d = 2 or 4 in an expansion or at d 2 in a 1/N expansion. Such program has been followed by Wegner and his group since some time [61.Whenever possible we compare our results with his. But since he does not use conformal field theory nor the notion of a conformal field, a comparison necessitates a reinterpretation of this results in such terms. In a conformal field theory fields are conveniently chosen as conformal fields (which, in agreement with the terminology of Belavin, Polyakov and Zamolodchikov [71, are also called “quasi-primary”) or derivatives thereof. In a conformally covariant operator product expansion a conformal field appears with a well-defined sequence of “derivative fields” (they are not related with the “derivative fields” of Wegner [61). This sequence can be summed resulting in a nonlocal operator product expansion related to conformal three-point functions. After a multiple application of operator product expansions we end up with two-point functions of fields which we have then to identify. We have to decide whether these are conformal fields or derivative fields of conformal fields, and what representations of the conformal group these fields belong to. We have found that a consistent analysis is achieved with conformal fields which under euclidean motions (g, a), g E S0(d), transform as =
=
(T(g,a)~I/)~(X) =
S~(g)ç1s~(g1(x a)), —
(1.3)
where S(g) is an irreducible tensor representation of S0(d). In addition these fields have a dimension 6 and transform under special conformal transformations in a standard way following from (1.3). Though antisymmetric tensors exist, they do not appear in this paper. Symmetric tensor conformal fields possess only the quantum numbers (1, 6) where / is the tensor degree. A computation of n-point functions till order 1/Nk does not entail that anomalous dimensions of identified fields are also determined to that order (the anomalous dimension of the energy—momentum tensor to order 1/N necessitates a calculation of the four-point function of the S fields to order 1/N2, see ref. [3]). So our knowledge of these dimensions is very rudimentary still. According to this present-state knowledge all fields with equal anomalous dimensions belonging to
K Lang, W. Rühl / 0(N) o- -model
575
the same 0(N) representation can be interpreted as derivative fields of one common conformal field. In the limit N = ~ the 0(N) nonlinear cr-model tends to a free-field theory which is essentially obtained by the replacements S0(x)
~4Sa(X),
L%Sa(X)
=0,
a(x)—*0.
(1.4)
Correspondingly the dimensions of fields can be decomposed into a normal part (free-field part) and an anomalous part which is 0(1/N). By a (p 1)-fold operator product expansion of p fields S we obtain fields (among others) of normal dimensions —
p(/.L—1)+integer,
.t=+d.
(1.5)
Such field may have any 0(N) representation contained in the p-fold tensoring of the 0(N) vector: a tensor with a Young frame of p blocks, of p 2 blocks if one contraction is permitted, of p 4 blocks if two contractions are applied, etc.. The normal dimension of the auxiliary field a( x) is two. Now consider the set of fields with a specific 0(N) representation Y and a normal dimension —
—
p(~t—1)+n,
(1.6)
flE~>.
All quasi-primary fields with the same Y and p form a class (Y, p). The field with minimal possible n, n = n~,seems to be unique. It possesses a certain (not necessarily trivial) 0(d)-tensor type. Call it the primary field belonging to (Y, p). The auxiliary field can also be considered as primary with Y = trivial representation (denoted 0) p = 0 and n~= 2. From an operator product expansion producing a primary field of a class (Y, p) usually all fields in this class are accessible. It is therefore convenient to investigate the quasi-primary fields class after class. The content of a class can be represented in a plot such as fig. 1, labelling each quasi-primary field (qp-field) by its normal dimension [6] + n and its twist t. ~ The tensor rank of ~P(’~at the point (t, n) is rank
çp(r) =
n
—
t +
rank
(1.7)
~(O)
As compared with fig. 1 the content of a class can be reduced for two reasons: (a) some places ct~remain empty since symmetry forbids the existence of such fields. Typical are the cases n odd, t = 0 in the classes (1E, 1) and 1) which are forbidden by crossing symmetry (sect. 3);
(9,
K Lang, W. Rühl / 0(N) or-model
576
normal dim.
-
4~(IO)
~(ii)
~(7)
[8]-i-3
[~] ±2
-
____
~(8)
____
~(2)
~(3)
[~] +1 [~] ~(o) I
I
I
0
2
4
t
Fig. 1. The qp-fields contained in a general class (Y, p). (is the “twist”, Z(°)is the primary field.
(b) the levels n > 0, t = 0 in the classes (0, 1), (0, 0) (containing the basic primary fields S and a, respectively) are empty for dynamical reasons. This is one of the results of this paper and is discussed in sect. 6. A multiple occupancy of a level (t, n) in a class is not excluded. The interpolation of space dimensions 2
(5a(Y1)Sb(Y2)~
(1.8)
A(Y~2)~8ab,
2 (a(y1)a(y2)) =B(y12)
, ~
=y~—y1),
(1.9)
K Lang,
W. Rühl /
0(N) cr-model
577
where the “critical exponents” (field dimensions) are a’p—1+?7
p
=
2— 217
(‘q=i~(S)),
(1.10) (1.11)
2K. —
There is moreover a scale-invariant amplitude ratio (if the coupling constant is one) z=A2B. The three quantities
77, K,
z can be expanded asymptotically, 7?k
~
Kk ~
k—i
where
(1.12)
~
k=1
(1.13) k~1
771,2,3’ K
1, z12 can be found in the literature (see the list in ref. [3], ~ was given in refs. [8,9]). Anomalous dimensions of other primary fields ~ are expanded similarly, ~ 77k~ Yp Nk (1.14) k=1
In the cases that Y possesses two rows of lengths 11, 12, respectively, and p = 1~+ 12, ~ was calculated in ref. [2]. The energy—momentum field is probably primary (this question of its “ancestor” field was discussed in ref. [4]), has (Y, p) = (0, 2) and ~ = 0 was verified in ref. [3]. This paper is devoted to the issue of proving the asserted representation properties of the conformal fields, to work out the scheme of classes and to study a finite number of low lying elements in some classes, i.e. calculating anomalous dimensions and fusion constants. These classes are (0, 1) (sect. 4), (0, 0) (sect. 5) and the classes (1D, 2) and 2) (sect. 3). For these classes only connected graphs at 0(1/N) are needed in the graphical expansion. A combinatorial analysis such as developed in ref. [2] is needed for more complicated classes possessing disconnected graphs. After the analytic work of this paper, computer programs for both the combinatorics and the algebra will help us to investigate “all” fields in “all” classes. This paper is closed with sect. 6 which relates the content of a class to the underlying graphs.
(9,
2. Derivative fields Expanding the product of two qp-fields into new qp-fields and their derivatives, these derivatives form a series whose structure is determined by conformal
K Lang, W. Rühl / 0(N) cr-model
578
invariance. The description of such series is a classical topic in conformal field theory (see ref. [10]),but the formulas given there are incomplete and of not much use for us. Assume three scalar qp-fields A, B, C are given with dimensions 6A’ 6B’ 6c so that A(x)B(0)
=
(x2)
+
-(~A±ôB+~c)/2 r~O
Qab(x
ö)C(0)
further qp-fields,
(2.1)
a)
where the operators Q~b(x, are homogeneous in x of degree r and of order r in the derivative These operators follow from the two- and three-point functions,
a.
a,b
k
a~
X~
1 — —
k’(—a~ (—b~
[k/21
~
___________
(—a—b)k
/~\
1k—rn
0m!(k—2m)!(—~t---a—b+1)rn (2.2)
or after summation 1 k=O~
ab
—
(x,
(—a)rn(—b)rn
‘~‘
rn~o(~~+1)rnm!
1 X
—a
—
~
1F1(—b±m;—a—b+2m; xa~)
U12rn
(2.3) with a= —~(6B+8c—6A), b=
—~(6A+6c—6B).
(2.4)
Only the term with m = 0 in (2.3) was given in ref. [10]. This result can be generalized in the case that A, B, C are tensor fields. We are interested here in a particular case, namely
=
(x2)~’
~ 4~(x, r=i
+
where
~Aab
other fields,
r.
d)JAab(O) (2.5)
is the 0(N)-symmetry Noether current which is conserved and has
K Lang, W. Rühl / 0(N) cr-model
dimension 2h
—
579
1. J is normalized ad hoc here and in the sequel. The operators
z1~kx,~)are of degree r in x and of order r
a~)e’~”
Qka~b(X
=
1 in a. Denote r) e’~’~,
—
~~b(u,
u’ix~p,
(2.6)
T—~X2p2.
Then the solution for z1~can be obtained after Fourier transformation as 4~(x,ip)=
x A
4~(~t—
1)
x [(4/h(~ a
1)
—
+
2(3k
—
1)T~—+
~[~(u,
+ 2T2~)r
T)
-
r
—
1)Q~~~(u, r)
+
longitudinal terms,
(2.7)
where “longitudinal terms” (l.t.) are defined to be proportional to p~.In particular we have from (2.7) ~t~(x, ip) ~
XA +
ip) —crxA ~(x,
ip)
=
l.t., +
l.t.,
2(2~ 1) xA[(~ + 1)u2 + T} + l.t.
(2.8)
If A, B are equal scalar fields, (2.4) implies (2.9)
a=b=—~6c,
so that
Qab
does not depend on the anomalous dimension of A
=
B. Analogously
we do not find dependence on -q(S) in (2.7), (2.8). 3. Classes with p
=
2
3.1. THE FREE-FIELD LIMIT
The 0(N) nonlinear u-model degenerates in the limit N fields Sa(X) can be represented by free bosonic fields Sa(X), 4Sa(X)
=0,
m~The
0(N)-vector (3.1)
K Lang, W. Rühl / 0(N) cr-model
580
which form an 0(N) vector as well. By operator product expansions of these free fields Sa(X) we create a free algebra of fields. This algebra is the N = ~ limit of the “deformation sector” of the critical nonlinear 0(N) cr-model. We consider the operator product expansion Sa(X)Sb(X) =6ab(X2)
/L+l
+m~(0)
+ n~1 ~
x~(m~2.~ab(0)+J~)~ab(0))
(3.2) with mY~ab(y) =~(:a,~...a~sa(y)sb(y): + :sa(y)a~a~2...a~sb(y):)
(3.3) J~..i~,ab(Y) =~(:a~...a,~sa(y)sb(y): — :sa(y)a~...3~sb(y):).
(3.4)
Applying the differential operators (2.2), (2.8) we can first show that mW is a derivative field of m~°~ and ~(2) a derivative field of ~(l) We denote the qp-fields M~°~=m~°~, J~=j~).
(3.5)
New qp-fields arise at the next levels,
~
X~iX~2M~2ab(Y)
=
x,~x~2m~2 ab(
Y)
=
~
~
+ 1,1L±‘(x,
—
y)
—
4~(x,
a~)m~( y), (3.6)
a)j~’)(y).
(3.7)
This procedure can be continued. With A~=n(n+1) (3.8) 2’0, we find new qp-fields ~ j(A~) in m’ 1(2n + 1) respectively. These qp-fields are symmetric traceless 0(d) tensors which are moreover conserved. This unexpected (except for j(O)) conservation follows from conformal covariance (appendix A). The picture emerging for the spectra of qp-fields at N = is described by the t = 0 towers in figs. 2, 3.
K Lang, W. Rühl / 0(N) cr-model
581
normal dim.
2~c+2
6~
M~8~
M~
M~5~ 2j~
-
_______
2~—2
-
_____
______
I
I
I
0
2
4
t
Fig. 2. The spectrum of qp-fields in the class (EEl, 2).
normal dim.
_______
j(6)
_______
J(7)
_______
j(8)
2iz+2
j(5) 2~+1 j(2)
2~i
j(3)
-
2~z—1 -
_____
j(o)
I
I
0
2
Fig. 3. The spectrum of qp-fields in the class
I
4
(9 2).
t
K Lang, W. Rühl / 0(N) cr-model
582 3.2. THE cr-MODEL AT 011/N]
The operator product expansion ansatz for two 0(N)-vector fields is Sa(Yi)Sc(Y3)
=
(y~3) ~‘8ac1
x~
[(
+ ~
2)~?(M(~)7?(S) 1~4’,.~~..p,,ac(Y3)
+ other fields].
+(y~)~(J(A))_fl(S)J,~)~ac(y3)
(3.9)
A labels the qp-fields as in fig. 1 so that the anomalous dimension depends only on A. The tensor rank n may be trivially enlarged by multiplication with 4t Kronecker deltas so that the proper tensor rank is n t. t is denoted “twist”. The qp-field itself may have a proper tensor rank n(A) (see (1.7)) so that e.g. —
jq(A)
~
Y3 appears as the derivative field of M~ of order n dimensions of the fields in (3.9) are i~IP~,’~’
2/L
dim
Mi~~,ac
dim J
=
..p.,,ac2/2
—
2
+n
—
t
—
n(A). The total field
+ 77(M(A)),
—2+n+~(J~).
(3.10)
The 1/N correction introduces new qp-fields with twist nonzero so that the scheme of fig. 1 is completed. The gaps in the towers t = 0 at odd levels remain. Moreover all qp-fields obtain an anomalous dimension (except J~) and are nonconserved (except J~). The 0(d) tensors remain traceless symmetric by definition. The final list of qp-fields is described in figs. 2, 3. The calculations start from the four-point function KSa(Yl)Sb(
Y2)S~(Y3)Sa(Y4P~
(3.11)
which is expanded in powers of 1/N (see fig. 4). At each order we develop further in powers of Y13YlY3’
Y24Y2Y4-
(3.12)
Then we compare with (3.9) inserted into (3.11). We apply a finite wave function
K Lang, W. Rlihl / 0(N) cr-model
583
10
02
lo
03
lo
04
30
04
2°
04
2o
03
A1
30
A2
A3
o2
10
p3
04
2o
04
B1
2o
03
B3
B2
Fig. 4. Graphs A123 of order one and B123 of order 1/N to the four-point function (3.11).
2
renormalization to S andoperators a so that A introduce the projection
=
to the vertices. We
B = 1 and ascribe ~‘~
+
Hab,cd(M)
=
~(6ad6bc
6ac6bd)
11abcd(J)
=
~(6ad6bc~6ac6bd),
—
~6ab6cd’
(3.13) (3.14)
the biharmonic ratios 22
22
Y13Y24 2
2’
2
Y12Y34
(3.15)
2’
Y 12Y34
and use the technical apparatus of conformal functions developed elsewhere [3]. After extraction of 11(M) (3.13) (upper sign) or 11(J) (3.14) (lower sign) we get the contributions A1 +A3 = ±(y~y~)a±(y~y~)~,
B1±B3+
~77(5)(y~y~ya
(3.16) ~
~
U ~
n,m~O
x
±(~L
—
\m
~
n.m.
l)~(j.L 1)n+rn(n+m)! —
(P~)2n+rn
K Lang, W. Rühl / 0(N) cr-model
584
x[—log u+~í(n+1)—~(n+m+1)+2~p(p.+2n±m) 2 —
—~(~—1+n)—~(M—1+n+m)I +
1)n+rnl n!
(/.L)2n+rn
x [—log
u
+
2~(~+ 2n
+
m)
—
2~(~—1
+
n
+
m)}).
(3.17)
Expressions (3.16), (3.17) must be expanded in powers of y13, y42. It is of advantage to use the tensor bases described in appendix A (with x1 = y13, x2 = y42, say) and the help of a computer. The comparison with the operator product expansion (3.9) is done in the order of increasing polynomial degree of y13, y42. After identification of one field one subtracts all its derivative fields. This is done by applying the differential operators of sect. 2. The anomalous dimensions of the fields result from the log u coefficients in (3.17). From (3.9) and (3.16), (3.17) we deduce (M(°)(y3)M’~°~( y4)) =
(2
+
—
1)(~~
±o(l))(y2)_2~_2_~Mo~H(M)
(3.18)
The contribution of the field M~°~ to (3.11) is then 101)
[2
+
~
=
(2 +
?l(S)+~?(M(o))( 2) —2~+2—~i(M
o(~4){i + (
77(S)
-
-log u
~77(M(0)))(
-
log~2)}(y~4y2a
(3.19) Comparison with (3.17) gives =
4 2—p. a(S)
+
ul\
01~NJ I. —~
(3.20)
At the rank-one level no new qp-field arises and at the rank-two level we try the ansatz with mutually orthogonal qp-fields ~ ~ ~ Q~~M~)
M~”(x,
a)M~°)(y)+x~x~(M~(y) + (3.21)
K Lang, W. Rühl / 0(N) cr-model
585
The field M,~)is symmetric, traceless (and conserved in the leading order of 1/N). We find (appendix A) Xl~XlVX
2AX2~(MILP(X)MA~(0))=(X2)2~_2~~M~(air2s2 +...),
(3.22)
with a1 ... a5 satisfying (A.4) and (A.6). Here 2
8~i(~—1) a1=
/1\
+01—I.
(3.23)
The anomalous dimension is
2~)= 4
77(M~
177(s)+0(~).
(3.24)
The leading terms in (3.31) are of course the free-field expressions. More interest-3~ ing than verifying (A.6) also at order 1/N, is the appearance of the new field M~ at this order. After some algebra we find (M~3~(X)M~3~(0)) = 8(j~— 1)
2~+ 77(S)(X2)
o(~).
(3.25)
It is remarkable that (3.26) is necessary for positivity (it is in fact fulfilled). The lowest level qp-field in the J spectrum is the vector field J(O)• It can in fact be proved to any order of 1/N that a scalar field does not arise. The vector field is the Noether current of 0(N) symmetry. We obtain from (3.9), (3.16), (3.17) (J(o)(X)J(o)(O))
(2(~-1)
2~ 1 77(S)
+o(~))
26~~)(X2) _2(~fl(J),
x (2x~x~ _X
(3.27)
with 0, consistent with conservation of the Noether current.
(3.28)
586
K. Lang, W. Rühl
/
0(N) cr-model
At the rank-two level we find no new qp-field. At the rank-three level we obtain two new qp-fields j(2) and J(3) (see (2.15)),
~)J~~)( y) + ~
x,
y)
+ x2x~J~3~( y)
1\
~j,
+0
(3.29)
where j(2) is symmetric traceless (and conserved at leading order only) and appears first at order 1/N. We obtain
j(3)
1 =
4(~ + 1)(~+ 2) 17(S)
o(~)
+
(3.30)
and (see appendix A) ~
J,~2~3( X) J,~2~3(0)) 2~(b =
~
(X2)_2~_1
1r~s~ + ...).
(3.31)
At leading order we have in addition to (A.9) b5=b6=b1+8b4=O
(3.32)
and 16 j.L(~+ 1)(~
—
b1=-~--
2~+1
1
1)2
+0(N),
(3.33)
which corresponds to the free-field case. According to (A.13), (A.11) we expect (3.32) and (A.9) to hold at all orders 1/N. After some algebra we find also
26A~)(X2) 2~2H(J)
(p.±l) (J~~(X)J~(0) = ~
—
1
+0 —
N2
77(S)(2XAX~ X
~
—
.
(3.34)
The main obstacle for extending this analysis to higher-rank tensor fields is the subtraction of the derivative fields.
K Lang, W. Rühl
/
0(N) cr-model
587
4. The 0(N)-vector field The class of qp-fields labelled by
(Y, p)
=
(0,1)
(4.1)
can be obtained by expanding the operator product a(yt)Sa(y3).
(4.2)
In this case the spectrum of qp-fields turns out to be quite different from fig. 1.
We analyse the four-point function (a(yl)Sa(y3)a(y2)Sb(y4)).
(4.3)
It can be expanded graphically as in fig. 5. The graph B11 is 1P-reducible in the channel (1,3) (2,4). Moreover, the S-field propagator satisfies —,
2) a
=
4a77(S)(x2)
_~
4(x
=
o(~).
This leads to the phenomenon of “order mixing” in the 1/N expansion. o2
l~
o4 A
10
~2
10
B 11
B12
o2 30
B2 Fig. 5. Graph A is of order one and B11, B12, B2 are of order 1/N.
(4.4)
588
K. Lang, W. Rühl
/
0(N) cr-model
We renormalize the propagators (1.8), (1.9) so that A = B = 1 and z1”2 is ascribed to each vertex. We extract powers of N for each S-field loop and obtain for (4.3) 2NB
+z(B 11
+B12)
+z
2+
o(~))~
(4.5)
with A ~1 2 ~~i—~Yl2)
B11
=
2 \—a ~Y34j
(Y~1~(;~)_a(u;1c2Fi(1,~
—
2; ~
—
1; 1
—
v)
(4.7)
(_a~lo~u+b~2))~
+ rn,n=O
n.m. ~
flf1_
B12= (~~)_P(~~4)_a ~
~
rn,n=O
\rn
~ n.m.
(—a~,’,~ log u +b,~),
(4.8)
(—a~logu+b~).
(4.9)
~
B2=(Y~)P(Y~4)a ~
~
U ~
n.m.
rn,n’=O
The explicit forms of the coefficients anrn, bnrn, c are given in appendix B. The 1P-reducible feature of the B11 graph is visible only in the appearance of a gaussian hypergeometric function in addition to the usual two-variable series. The phenomenon of 1/N order mixing has also been obscured in (4.7) where each term is 0(1). In order to work this out we calculate the complete (to any order in 1/N) 1P-reducible graph B11~which contains B11 and all radiative vertex corrections in
a conformal-invariant way. The procedure was explained in ref. [3]. The result is ~ =
Both ~
(y~)(y~)~
1—
2c~ + u~’c~). (4.10)
n~m~ (u~
m,n~O ~ U
are 0(1). But expanding U’~ in powers of 1/N leads to a cancellation
and, as leading 0(1/N) term, we obtain zB 11 (4.7). The coefficients c~1’,’,~ are given in appendix B.
We make the ansatz for the operator product, 2)~1+(S)+~(S(~)+2K) a(
yi)Sa( ~
=
n~O~
Y13)®~1~
(y
5((
Y3)’ (4.11)
K Lang, W. Riihl / 0(N) cr-model
589
where S~’yiSa.
(4.12)
We conclude from (4.11) that the S-field propagator is contained in the u -1312 series of (4.10). By definition this series contains the S field together with all its derivative fields. From (B.8) we know that =
(4.13)
=~.
So ~ plays the role of the square of the fundamental aS S structure constant in the critical cr-model. Now we turn our attention to other quasi-primary fields. The series UKC~, in (4.10) contains terms 0(1) resulting from the subtraction of derivative fields of S. In fact, operating twice with the derivative operators 4~(x, for aS S on (X2)_a, e.g. with —*
a)
a)
4~2~(x,
f3(f3+2)
=
—‘
________
4a(a+ 1) ((x.a)2_ 2)x4).
(4.14)
leads only to the cancellation of one power of 77(5) in the denominator of (4.14) due to (4.4). Thus we have to add the series UKC~ to the contribution (4.6) of the A graph (this is what we call “order mixing”),
(Y~2)13(Y~4ia(1+ ~
m,n~O
Ufl(1_v)UKc(~)
n.m.
(4.15)
At n = m = 0 there is no 0(1) contribution in (4.15), but at n = 0, m = 1 there is one. This gives rise to an 0(d)-vector 0(N)-vector qp-field S~(y) (with t, r, s defined in (A.2)),
~ 2
—
2rs)(1 +
~
(4.16)
il (t The log u terms have coefficients (from (4.8), (4.9) and (4.15)) =
z(a~’,~+zNa~) —KC~.
(4.17)
At n
= m = 0 there is no 0(1/N) contribution (this is a nontrivial consistency check for our approach) but at n = 0, m = 1 there is one,
—2
(~t.—1)(2~i-—1) 17(5), JL+1
(4.18)
590
K. Lang, W. Rühl
normal dim.
/
0(N) cr-model
_______
_______
s~7~
s~8~
~(5)
/2+1
/2—1
I
I
I
0
2
4
t
Fig. 6. The spectrum of qp-fields in the class (~,1). The field S131 appears one order in 1/N higher than the rest.
and thereby 4~L2—j.~—2 —2
=
1
77(5)
+
o(_~).
(4.19)
Whether a scalar qp-field S~3~ exists can be decided upon only after has been calculated. Its two-point function is proportional to
‘2
(B.10)
F(a+K)2F(~+K)2F(a—~)F(a) —
—
a
—
K)F(2a —/L + K)F(2/L
—
a)F(~
—
a)
same expression at N = —
22/L + 17)
+
2
77(S).
(4.20)
(~L.—1)(~—2)
7~ appears at 0(1) with (see appendix A) The qp-field S~ 4xI,~xlpx2~x2,,(S~va(X)SA,~b(0)) 2s2 + =
(a1r
...
)(x2)
_~_3_fl(S(7))ôab,
(4.21)
K Lang, W. Rühl / 0(N) cr-model
591
with a, satisfying (A.6), (A.7) and 48~.t(p—1) (~t+2)(js.+3)
(1
+01— \N
.
(4.22)
To obtain (4.21), (4.22) we have to subtract the derivative field of S~ with derivative operator 2~(x,d)M= 1
4~
3[6x~(x~a) +x28~].
(4.23)
Of course the existence of the qp-fields ~(1), 5(2),2)~(4)~ - has not been excluded at least. rigorously, but their two-point functions are 0(1/N 5. The auxiliary field The class of qp-fields labelled by (Y,p)=(O,0)
(5.1)
contains the auxiliary field a( y) and is accessible from the operator products 5a(Y1)Sb(Y2)
(5.2)
and
a(y 1)a(y2).
(5.3)
Fields of (5.1) appear in the product (5.2) at order 1/N but in (5.3) already at order one. Hoping for a reduced computational effort we prefer to start from (5.3). Again we conjecture a spectrum of qp-fields in the class (5.1) similar to fig. 1 with ~ replaced by a(T) and [6] by two (see fig. 7). The four-point function of four auxiliary fields is then (a(y1)a(y2)a(y3)a(y4)) =A1 +A2+A3+ ~[z?(B11
+B12+B13) +z~(B21+B22+B23)1
(5.4)
592
K Lang, W. Rühl
/
0(N) cr-model
10-
02
19
02
lo
02
30
04
3O
04
3p.•
04
A1
A2
A3
lo--
-02
1?
92
lo.
30--
-04
36
64
3~/
B11
lo----
B12
~
3~....
~4
~
B13
lo---
p2
lo----
~°_
o4
30:-..
B21
--02
.
B22
B23
Fig. 7. Graphs for the four-point function of four fields a(y). The graphs A~are of order one, B,1 are of order 1/N.
Evaluation of these graphs gives
(5.5) A2
=
(y~y~)~,
(5.6)
13,
(5.7)
A3
=
(y~y~)
B
13~ 11=(y~2y~4)
B
m,n=O
13 ~ 12= (y~3y~4)~m,n=O
Ufl(lV)(_a~IogU+b~))
(5.8)
n.m.
U~(1—v)m (UaC~)+UM_adnrn),
n.m.
(5.9)
K Lang, W. Rühl
/
0(N) cr-model
593
rn
13 ~
B13 = (y~4y~3)~ m,n=O
u(1 v) n!m!
B
U~(1—v)
(—a”3) log u
—
+ b~’3~ nrn
(5.10)
nrn m
13~ 21=(y~y~) m,n=O x(—a~)log ~
n!m!
(5.lfl nrnJ~ rn
B
13~ 22=(y~y~)
x (—a
U~(1—V)
n!m!
m,n=O
log u
b~22~ + U”13c~22~
+
nm
(5.12)
nrn)~ rn
13~
B 23
=
U~(i
—
v)
‘—a~
log u + b~23~
(y~2y~4)
(5.13)
nrn)~ m,n=O
n!m!
nrn
‘
The coefficients a~j~b~1~b~13~C~’~d am are presented in appendix C. In the operator product expansion (5.3) we take into account the unit operator, the qp-fields (0, 0) and the qp-fields (0, 2). The energy—momentum tensor field is a twist-zero field from (0, 2), a(y 1)a(y3)
=
(y~)~1
~ + ~
i n ~(Yi~)®~~(
0))
a~
r~_1~_~~
(~)
n=O
+
~
( y132
)_3+JL+~?1(T(~))_t1(a(O))T(S) ILl
(~)
s=O
+further fields).
(5.14)
Further fields in (5.14) are expected from (0,4), (0,6), etc. Such fields exist because the free-field theory of sect. 3 contains corresponding examples (“W currents”). Moreover, a~°~ is proportional to the auxiliary field and therefore, by (1.13), 17(a~°~) = —2(77(5)
+K).
(5.15)
594
K Lang, W. Rühl
/
0(N) cr-model
The contribution of the identity operator in (5.14) appears in A2 (5.6). The fields ~ give (neglecting anomalous dimensions) a factor
x polynomials of y13,
~
y24.
(5.16)
Their contributions can be recognized in B12, B21, B22 as having the factor 13U~~
=
(y~
(5.17)
13Ua.
3y~4)
(y~y~)
From (C.3), (C.8), (C.10) we obtain z 1c~+c~,+c~~=0 if(n, m) =(0, 0) or (0, 1).
(5.18)
It follows then that the lowest tensor rank of the fields T~ of twist zero is two. We denote this field as and obtain (see appendix A) XlILXlVX2AX2U(1~V(X)T)~O)(0))
(5.19) where (A.6), (A.7) are fulfilled and (using ref. [3], eq. (2.20)) 128j.~ 1 1 \ 2) —+0 —I. (5.20) 2/1—iN N is proportional to the energy—momentum tensor of the cr-model. Comparing (5.19) with the corresponding expression for the free-field model ([1], eq. (4.10)), x1ILx1px2Ax2UKt~(X)tAU(0))= (a~r2s2+
.. .
)(X~)2~’,
(5.21)
where (A.6), (A.7) are also fulfilled and N
a 1
=
—i—
21(/L)F(/.L+1) (5.22)
2/1—1
fixes the relative normalization. t0~field produces a factor (neglecting anomalous Inserting (5.14) into (5.4) the a dimensions) (y~y~)’(y~)2xpolynomial in y 13, y24. Such an expression appears only in B12 (5.9) in the form 13uIL~ = (y~y~) _P+IL_a(22yIL+a (y~y~)
(5.23)
(5.24)
K Lang, W. Ru/cl
/ 0(N) cr-model
595
We proved that for polynomial degrees of y13, y24 less or equal to two this term (namely the dnm series in (59)) can be attributed solely to the field a~°~ and its derivatives 4Wa(°),~ We obtain a~°~(y) =y2a(y),
8 (2~
—
~N
3)21(2
—
(5.25)
2)
1(3-~)1(~-1)~
1
+O(~)~
(5.26)
t3~appears in fact at order one, namely in A The scalar field a
1 (5.5) and A3
(5.7),
3~(X)a~3~(0)> = (8 + 0(~))(X2)_4~3~.
(5.27)
(a~ Its anomalous dimension is as usual extracted from the log u terms, —
+~(a~3~) + 77(a~°~) =
~[z~(a~j1)
a~)I±o(-~).
43k) + z~(a~±a~2~ +
+
(5.28)
We use (5.15) and ref. [3], eq. (2.22), (2/.L—l)(~—l)
1
(5.29)
and obtain ~(a~3~)
=
—4(2k
—
1)277(5)
+o(~).
(5.30)
The fields ~ at8~and probably all higher qp-fields of nonvanishing twist appear also at order one. The n = 4 term of (5.14) is replaced by +
~
3)(
n(a10~) 4(2)(~13
+ ~ {(y~) 2)
+
(y
~
+ I + ~n(abB>)_71(ab°>)a(8)(
y 3)
d)a(
y3)
p 3)]
,
(5.31)
K Lang, W. Rühl
596
/
0(N) cr-model
is assumed to be traceless symmetric, ~ where a~7~
at7~,a~8~ to be pairwise
orthogonal, and (5.13) is fitted into the four-point function (5.4). We find xlILxlVx2Ax 2U(a,~V(X)a~(0)
)
2s2 + =
.
)( X2) —6—~(a~~~)
..
(5.32)
(a’r
where the a~’satisfy (A.6), (A.7) and 1
21333
a~’= —~--—
+0(k).
(5.33)
Moreover we obtain
(at8~(X)a~8~(0)) = (21032 (5)
+
o( ~))(x2)
-6-~(a~~~)
(5.34)
The qp-fields a~”kA~’° as in (3.8)) at twist zero have so far not been identified. it is straightforward to show that they are neither contained in the operator product expansion of two S fields including the order i/N. In fact the relevant graph is B 2 in fig. 4, and applying the integration techniques and conformal functions of ref. [3], we obtain for the relevant series
_______________
2
IL
(~—2)F(~t—1)
2
2
Y13Y24
—~f~-.
md
x
uFl(1_v) n,m~0
n.m.
IL+13
2
2
Yi2Y34)
(5.35)
00
Here dnm is the same as in (5.9). Therefore the same holds true what we found above (after eq. (4.24)): this series describes only the field a~°~ and its derivative fields. These derivative fields are in fact identical in the aa and in the SS channel. We conclude that the only qp-field of twist zero is ~ The whole class (0, 0) is presented in fig. 8.
6. Conclusions Though the content of each class may sometimes be surprising, the whole program of fitting operator products of conformal fields carrying elementary representations of the conformal group into the graphically expanded n-point functions have proved to be consistent. The connection between graphs and the content of classes can be derived from a more general scheme. We express the
K Lang, W. Ru/cl / 0(N) cr-model
597
normal
dim.
6
171
a
4
-
___
a~3~
2
I
I
0
2
I
4
Fig. 8. The spectrum of qp-fields in the class (0, 0). In the
t = 2 tower
t odd tensor fields are missing.
conformal function of the general box graph fig. 9 as a series expansion in u and
1
—
(4)
R (y1, y2, y3, y4)
a
2 =
1 (y34) 2 a1a3 2 (y24)
(y13) ~
x n,m=O ~
U~(1~V) n.m.,
~2
133). (6.1)
(c~u’~+ C~UU2+ C~U
Fig. 9. A ring graph for scalar fields with arbitrary dimensions a, f3. satisfying only uniqueness at each vertex: the sum of dimensions equals 2/2.
K Lang, W. Ru/cl
598
~1
/
0(N) cr-model
~3
Fig. 10. Three vertical cuts define sums of exponents which according to (6.3) are the dimensions of product fields (6.2).
The coefficients ~ were discussed in ref. [3], in the appendix. Let ~a denote a scalar field of dimension a and assume that an operator product expansion ~aj(
Yl )
~a3(
~)
=
(y~3)-(as
+a3-ô)/2
~
y3)
+
...
(6.2)
exists. Then each of the three series in (6.1) yields one solution for 6, ual:
61=a1+a3,
u’~2:
62=a2+a4,
13~: 83
=
f~1+ /33,
(6.3)
u
It is easy to see that log u terms of higher-order graphs may add a correction 0(1/N) to these dimension formulas (6.3). Let 3 be decomposed into a normal dimension [6] and an anomalous part 37(8), (6.4) Then the graphical rule of fig. 10 and its possible generalizations where the cuts meet more than two lines, determine [6] and moreover the class (Y, p) to which the qp-field cI~ belongs. Within this class (Y, p) it is not only one qp-field which follows from a single hypergeometric series in (6.1) but it is a whole subtree of qp-fields, i.e. all qp-fields fitting into fig. 1 with, say, t ~ t 0.
We consider now a single-particle reducible graph such as B12 in fig. 7. In this case the darn series in (5.9) corresponds to the propagation of the single-particle field and all its derivative fields. Consequently this series produces only one qp-field namely the S or the a field. The other qp-fields of twist zero whose existence was only conjectured, can therefore not be created at all according to these graphical rules.
K Lang, W. Rühl / 0(N) cr-model
599
Appendix A TENSOR BASES
Let I~(y) be a symmetric tensor of dimension y. Its two-point function xlILxlPx2Ax2UKrI3ILP( X)’IA~(O)) can be expressed in terms of the invariants 2, s = (x 2, r = (x1X)(X2)~/’ 2X)(X2)~
(A.1)
t
=
(x 1x2)
(A.2)
as the polynomial 2+a
2+ x~r2)+ a 2) 4(x~s 5x~xfl( X If ‘I~~(y) is traceless then the following relations hold:
~‘.
(A.3)
[air2s2 + a2rst + a3t
a 1+a2+2j.~a4=0, If
a3+a4+2~a5=0.
(A.4)
is moreover conserved,
a~I~IL~( y) 0,
(A.5)
=
and has dimension y
=
2j.~,then in addition to (A.4) we have a1
=
4a3,
a4
=
0,
(A.6)
or together a1:a2:a3:a5=1: —1:-i:
—i——.
(A.7)
In an analogous fashion we treat a symmetric tensor field of rank three D(y). The two-point function gives 2s2t ±b 2 + b 3+b 2+x~r2)rs = [b1r3s3 ±b2r 3rst 4t 5(x~s +b 2+x~r2)t+ b 6(x~s 7x~x~rs + b8x~x~t}(X2)Y. (A.8) Requiring I~ to be traceless entails 3b1 +2b2+2(/L b2
+
2b3
+
2(/L
+
1)b5=0,
+
1)b6 = 0,
b3+3b5+2b6+2(~+1)b7=0, 3b4 + b6
+
2(/L
+
1)b8
=
0.
(A.9)
600
K Lang, W. Rühl
/
0(N) cr-model
Consider a generating function for Gegenbauer’s polynomial,
[1
—
2p(2rs
—
t)
+x~x~p2]IL+1
n=O
=
x2, X),
(A.10)
where 7, is homogeneous in both x1, x2 of degree n. This gives [n/2l
1,(x1, x2, X)
=
k=0
1k 1 k!(n— 2k)
(x?x~[2(2rs
—
1)]n2k
(A.11)
It is easy to show that the generating function is harmonic in x1 and x2, so that 417~(x1,x2, X) =421~(x1,x2, T) =0.
(A.12)
Let ..1JX) be a traceless symmetric qp-field of dimension y. The conformal invariance of the two-point function can be shown to imply ~ILlIL2
XIIL1X1IL2...
xlIL x2~1x2~2...
=N,Dl~(xI,x2, X)(X2)~’.
(A.13)
For n = 2 we find (A.6), (A.7) and for n = 3 we obtain (A.9) and (3.32). Thus we can formulate another result: If the dimension of the conformal field is moreover canonical, ~ILIL2
y=2~—2+n
(A.14)
and n ~ 1, then IL2
IL(X)
=
0.
(A.15)
This explains all conservation laws found by us.
Appendix B COEFFICIENTS OF CONFORMAL FUNCTIONS B11, B12, B2, SECT. 4
B 1P.
aW= nm
—
~.2M F(/1—l)
(~i —
1)~(/.L 1)n+m(n +m F(,u+2n+m+1) —
+
1)!
‘
B 1
)
K Lang, W. Rühl / 0(N) cr-model j.(11)
=
a~[~/i(n + 1)
—
601
c/I(n + m + 2) + 2qi(j~+ 2n + m + 1)
Unm
—~fr(~t—1+n)—~/i(j~—1+n+m)},
(B.2)
2 ~
B
(
(~_2)F(~_1))’
(B.3)
~-IL
(l.L2)n(Ml)n+m(n+m+1)!
a~’2~
(B.4)
=
.
(~—2)F(p.—l)
am
12~
j,(12)
=
a~,’~[cfr(n+ 1)
+
F(ji+2n+m)
2i~(j.~ + 2n + m)
1~nm
—qi(j~—2+n)—q.~(p.—1+n±m)---~)j(n+m+2)],(B.5)
B 2:
a~
1(3 p,) 2)~F(2,L—3)
(~ —
—
2~
nm
=
(/1
—
1)~(~
1(.t
—
1)n+m(fl +
+
2n + m)
m)!
(B.6)
b~2~ =a~2’[c/1(n+ 1) —c~(n+m+1) +2c/i(~+2n +m) am
am
—&fi(~—1+n)—~/i(~t—1+n+m)].
(B.7)
of B
The coefficients ~
1 1,c are given by am
=
am
=
(~ — a
—
K)a+m(,LL /3 K)~(/.L (a)2a+m(a—i+1)a
K)~(,LL — a
—
—
—
21(a)F(a ~F(~
x
—
/3
—
K)n+m
(B.8)
—
F(a + )2p(p—/3+—K)2F(~ K) —a—K)21(~ —a)F(2~—a
(a + K)~(a + K)n+m(13 + K)~(/3 + K)a+m
(2.t
—
(B.9)
a) 2n±m(
—
a + 1)~
and ~can be expanded as E ~k37’
(B.10)
k= 1
,~2~
‘1
(p. (see ref. [3], eqs. (2.38), (2.39), (2.22)).
‘2
—
is still unknown.
(B.11)
602
K Lang, W. Ru/cl
/
0(N) cr-model
Appendix C COEFFICIENTS OF THE CONFORMAL FUNCTIONS
F(p. )2 6~ 2)1(2 a~~,~=ir(p.—2)21’(2p.—3)2 —
B11:
(p.
—
b~’1~ 1) nm = a~11~[c/i(n+ nm
—
B,~,i E (1,2), j ~ (1,2,3),
i)~(p.
—
SECT.
1)n+m(n +
—
m)!
1(p.+2n+m)
c/i(n + m + 1) +
2n
21/J(p. +
(12)
n.6IL 1(p.
=
—
2)~1(2 p.)3 (p. 2)1(2p. 3)2 —
12.
Cam
1(2p.
(2p.
2
—
a”3~[2c/i(n + 1) nm —2i/i(p.
1(2
B
(n!)2[(n+m)!12
—
p.)
1
—
—
(2n
3)2
—
m + 1) !(3
+
i)a]2[(n
p.)2 [(p.—
—
—
am =
(C.3)
(C.4) 2)41(2p.
—
1(p. 2)1(2 6~ a~=~ (p. 2)21(2p.
13:
i)n+m12
—
2)2n+m
~.6IL
=
(p.
B
—
—
1(2—p.) dam
(C.2)
i)~[(p.
—
—
(C.1)
m)
+
—~i(p.—1+n)—~(p.—1+n+m)],
B
5
n!1(p.
3)2
—
2~/i(n+ m
+
1)
+
+
—
+
m) !]2
2c/i(p.
+
2n
+ m)
n)],
+
(C.5) 2n +m)
(C.6)
[(n + m + 1) !}2
4IL
21:
(p.
a~=ir
—
2)21(p.)1(2p. —4)
(~ 1)r(~ + s)!
r!s!
(p.
(p.)
—2—
r)a(p.
—
1)n+m+s
(C.7)
2~±~~r(n+m+r+s+2)!’
r,s
r,s~0,n~r+s
2 F(p. F(2p. 2)21(23)2 p.)
am =
—
a
X
~
—
—
n! [(p._1)a~mJ2(2p._4)a±m+s(p._2)~ s!
(2p.
(C.8) —
3)
2p. 2a+m(
—
3)a+m+s
K Lang, W. RuhI
22
B22:
4
a~~=~rIL
/
0(N) cr-model
603
1(2—p.)(n+m+1)!(n+1)! 2) 1(p.)F(2p.—4) (p.— 2
n
(p+r)!(n+m—r)!
n+rn—r
X r~0 ~-o r!p!(n—r)!(n+r+p+2)!
x
(3~p.)r(p.
~2)n—r(p.
—
l)n+p
(C.9)
,
(p.)~,~ +m —r
22 4 [‘(p. — 2)21(2 — p.)2 ~am )=~r12 F(2p. — 3) 2
n+m
(n + m)! (p.
x ~~-0 B 23:
p!n!
—
1)~(p.
2)~(2p.
—
1)n±m(p.—
—
3)~+~(2p. 3)2a+m
(2p.
—
,
(C.10)
—
1(4—p.) (p.—2)41(p.)1(2p.—4)
= nm23~
a~
~n~rn+~—r(r+p)!(r+q)!(n±rn_r)!(n+m+1)! r=0 p=O
(4— p.)~(p. 2)ar(p. (2r+p+q+ 1)!(p.) —
X
r.p.q.
q=0 —
2)r+p+q
2n_r+rn 1,b~, ~
.
(C.11)
are known so far only as integrals of Barnes type.
The coefficients b~ References [11K. Lang
and W. Rühl, Z. Phys. C50 (1991) 285 [2] K. Lang and W. Rühl, Z. Phys. C51 (1991) 127 [3] K. Lang and W. RühI, Nuci. Phys. B377 (1992) 371 [4] K. Lang and W. Rühl, Phys. Lett. B275 (1992) 93 [5] see e.g. M. Kaku, Strings, conformal fields, and topology, an introduction (Springer, New York, 1991) ch. 3 [6] F. Wegner, Z. Phys. B78 (1990) 33, and earlier papers quoted there [7]A. Belavin, A. Polyakov and A. Zamolodchikov, Nuci. Phys. B241 (1984) 333 [81 A.N. Vasil’ev, Yu.M. Pismak and Yu.R. Khonkonen, Teor. Mat. Fiz. 50 (1982) 195 [91W. Bernreuther and F. Wegner, Phys. Rev. Lett. 57 (1986) 1383 [101 S. Ferrara, R. Gatto and A.F. Grub, Conformal algebra in space—time and operator product expansion, Springer Tracts in Modern Physics, Vol. 67 (Springer, Heidelberg, 1973)