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Conformal symmetry and the spectrum of anomalous dimensions in the N-vector model in 4−ϵ dimensions
NUCLEAR PHYSICS B [ES]
Nuclear Physics B402 IFS] (1993) 669—692 North-Holland
________________
Conformal symmetry and the spectrum of anomalous dimensions in the N-vector model in 4 dimensions * —
Stefan K. Kehrein 1 and Franz J. Wegner Institut für Theoretische Physik, Ruprecht-Karls-Universitàt, D-6900 Heidelberg, Germany
Yurej M. Pismak Department of Theoretical Physics, State University of Saint-Petersburg, Ul~yanovskaya1, Stary Petergof, 198904 Saint-Petersburg, Russian Federation Received 16 October 1992 Accepted for publication 26 March 1993
The subject of this paper is to study the critical N-vector model in 4— dimensions in one-loop order. We analyse the spectrum of anomalous dimensions of composite operators with an arbitrary number of fields and gradients. For composite operators with three elementary fields and gradients we work out the complete spectrum of anomalous dimensions, thus extending the old solution of Wilson and Kogut for two fields and gradients. In the general case we prove some properties of the spectrum, in particular a lower limit 0 + O(2). Thus one-loop contributions generally improve the stability of the nontrivial fixed point in contrast to some 2 + expansions. Furthermore we explicitly find conformal invariance at the nontrivial fixed point.
1. Introduction In several recent papers anomalous dimensions of composite operators with a general number of fields and gradients have been classified in one-loop order in 2 + e expansions. The models investigated so far are the nonlinear matrix model discussed by Kravtsov, Lerner and Yudson [3,41, the N-vector model [131, the unitary matrix model [14] and the orthogonal matrix model [7]. All these models share the surprising feature that the dominant scalar operator with 2s gradients has in addition to its naive dimension 2(1 s) a positive one-loop order contribution to the full scaling dimension that is proportional to s2. In the N-vector model —
*
Research supported in part by the Sonderforschungsbereich 123 Stochastic Mathematical Models of the Deutsche Forschungsgemeinschaft. E-mail: ej6ftyvm.urz.uni-heidelberg.de
0550-3213/93/$06.00 © 1993
—
Elsevier Science Publishers B.V. All rights reserved
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for example one finds an operator with 2s gradients and the full scaling dimension [13] s(s—1) Y2(1_S)+E.(1+ N—2 )+O(2). (1) Recently Castilla and Chakravarty have obtained a similar positive contribution of order e2 s3/(N 2)2 in a two-loop calculation [1]. It is an open question whether contributions beyond second order in e are negligible. If this were the case, operators with sufficiently large s would become relevant for a given positive c, thus making the conventional fixed point unstable against perturbations by these operators. This paper considers the related problem of anomalous dimensions in the critical N-vector model in 4 c dimensions in one-loop order. It will become apparent that the behaviour in 4 e dimensions is different from eq. (1). Here one-loop contributions always make canonically irrelevant operators even more irrelevant. Furthermore we explicitly see that the spatial symmetry group of the N-vector model at the critical point is the conformal group in four dimensions 0(5,1) to one-loop order. Thus one finds conformal invariance at the phase transition, which goes beyond the obvious scale invariance. Conformal invariance in the N-vector model is a well-known feature (the most complete treatment is due to Schafer [8]). The spectrum of anomalous dimensions of conformal invariant field theories can be derived in terms of conformal invariant operators [61. Employing our explicit knowledge of the conformal symmetry group, we use these properties to calculate the complete spectrum for composite operators with three elementary fields and an arbitrary number of gradients in one-loop order. This extends the well-known solution for two fields and gradients solved a long time ago by Wilson and Kogut [16] in two-loop order. For composite operators consisting of more than three elementary fields, we have not yet derived a similar complete classification of the spectrum of anomalous dimensions. Still we can establish some general properties like upper and lower limits for the spectrum and the fact that all eigenvalues are real numbers. It is a desirable feature that anomalous dimensions of eigenoperators are real numbers, but nontrivial due to operator mixing [12]. The organization of this paper is as follows. Sect. 2 gives a short definition of the N-vector model. In sect. 3 we calculate the one-loop pole contribution of an operator insertion K 0) 11~for a general composite operator 0. This can be done in terms of an integral operator V. Some properties of V connected with the concept of redundant operators [11] are discussed in sect. 4. In sect. 5 we introduce S0(4) tensor operators cI~1m,m 2) and expand V in this basis. V is hermitian with respect to the metric induced in this manner and has formally the structure of a two-par—
—
—
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671
tide interaction operator. Its spectrum corresponds to the spectrum of anomalous dimensions. Properties of V and its spectrum are discussed in sects. 6 and 7. In sect. 8 the diagonalization problem for V is solved for two and three fields. Sect. 9 closes with a summary of consequences and an outlook on open problems. 2. Definition of the model
S
The N-vector model of the field S0 + S11~with a free contribution
4
=
(4~,. 4~)is . . ,
governed by the action
=
S0
=
f ddx(~(04.)2+
(2)
~m~2)
and the interaction part
Sn~=~fddx(4~)~.
(3)
We are investigating the critical behaviour of this model in one-loop order in d = 4 e dimensions. Thus the renormalized mass vanishes and the critical coupling constant is given by [151 —
3 N+8 +0(e2).
(4)
For a local composite operator 0(x) consisting of elementary fields ~d(x), d = 1,..., N and derivatives thereof at one point x, the divergent part of the one-particle irreducible diagrams in one-loop order will be denoted by K0(x))ii~. We are only interested in this divergent contribution, the index lip will be suppressed everywhere in the sequel. As usual, the pole terms yield the Z-factors and so the anomalous dimensions. Notice that in general composite operators are tensor operators with respect to 0(4) spatial rotations and with respect to internal 0(N) rotations. Anomalous dimensions have been calculated to high orders in e for composite operators without gradients like 5d q~ (compare e.g. ref. [91for traceless tensors Sd ~) or for operators consisting of two elementary fields only (compare ref. [161).Due to the relation (5) these anomalous dimensions are reproduced for operators with gradients. But not all composite operators can be constructed in the manner ~~(4d, 4d). In the next section we will therefore calculate the one-loop pole contribution K 0(x)) for a general composite operator with gradients. . . .
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3. One-loop pole contributions For notation purposes it is useful to introduce the argument contracting operation (ACO). If two arguments of a function of n arguments are denoted with the same letter and one of these letters is barred, this denotes ACO applied to the function. The result is a function of n 2 arguments defined by —
f(...,
z,...,2,...)
def
U
~
=exp
~
(6)
Some basic properties of the ACO are =
.h
f(x+h) =a,~...a~f(x)
(7)
and the shift operation e~f(x+b)=f(x+a).
(8)
The last equation also holds for vectors x, a, b if one employs a euclidean scalar product a b in the exponent. Using the ACO we can write down a scalar generating function for composite operators consisting of n fields ~d(x) with gradients .
~~(x; h, A)
=
flA~~(x+h~).
(9)
By contracting ~I’~(x;h, A) with At’s and h1’s one can construct all composite operators with n fields. Our aim is to calculate KW~(x;h, A)). In one-loop order the only diagrams contributing to pole terms in an c-expansion are of the type in fig. 1, where one
is
—
2 legs
Fig. 1. One-loop pole contributions.
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673
has to sum over all different possibilities to close two legs with a loop. We can express this by writing (~~(x;h,A))=—~g0 ~
fddyG(x+hj_y)G(x+hj_y)
i,j=1
i
x[2(~(y) .A~)(~(y) .A1) +~2(y)(A~.A1)]~
A~(x+h~).
k # i ,j
(10) G(z) denotes the free propagator in configuration space 1 G(z)=
(11)
(2)dfPp2+m2
We have to evaluate the integral Jddy G(x+h~—y)G(x+h~—y)(~(y) .A~)(q~(y) .A.) =
V(h1, h~g~,.~~)(~(x + g~).A1)(~(x + g~).A.),
(12)
where V(h~,h1 g~,g~)=fddyi d”y2 G(y1)G(y2)6(y1 —y2—h1+h1) xexp[(h,—yi) ddp
~
is a standard exercise to
=
=
~ (2~ [(q
—
i~~)
+
+
(13)
~zqa
+
i~)2+ m~1[(q
-
i~)2+ m~] 2+ m~I t
fdt
2+ mg]) (14)
—
1 dt2f
m~I
calculate
d”q
,.
{(~i~3.)2
+2m~]
(21r)d [(p It
~j~+ (h3—y2).~]
d exP(iq
a
—
11[(q
+
i~)
2[(q
—
i~)
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and after performing the integral over q and a change of variables one obtains 1=
1 (
4)d/2E
xexp a
f dt
1
du
tE/21etf
0
0
2 (2u_1)+4(1_u)u/32~_~_a2_~~ t m
0 The integral
over
u
generates a sum E~
(15)
-
m
4t
(t/m~Y’~ck(a, /3), thus
1
=
4n-) d/2m
(
~ F( —k + ~E)ck(a,
/3)m~’+ (finite terms for c
0 k=0
=
0). (16)
The pole contribution I~,in the massless theory comes from k 0, 2f1d exp[a./3(2u 1)J ~ ((1 _u)ua2/32)~ 2_2 (4~)~ (4~) c m 0 0 =
(_~[
—
~
=
(4)2
(~)
2/32
f’du exp[ —au(l
—
+a
u)a
-/3(2u
—
l)]J 0(2i~/~),(17)
with the Bessel function (18) Inserting (17) back into (13) we obtain the following closed expression for the pole contribution of (10). (V’~(x;h, A))
=
V(h, A;
~,
B)V’~(x;g, B)
~,
‘<~
fl
~
k*i,j
exP(hk~k)exP(Ak~k)]
(19) with 1du exp[—+au(1 ~
fr~(h,A;
~,
~)=I0.f
+
(1
—
u)h 1)
.
(~j
+
+~1)jJo(2iV~)
~
(20)
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and
g ‘0=
—
3(4)2
=
1 N+8 + 0(c)
—
(21)
at the critical point. 4. Properties of V(h, A; j,
.Th
Before we make use of (19) in order to calculate some examples of anomalous dimensions, let us prove some general properties of the operator V(h, A; B). In a straightforward calculation one shows ~,
A;
~,
~)=jldu(gj+j)2uexp[_~au(1_u)(hj_hj)2(gj+~j)2
+(uh~+(1 —u)h1) 2u(l ><
[i
—
—
a + ~a
.
u)(h
2(~~ + )2] J 1
—
h~)
0(2ii/i) (22)
2c) e~cJ since (1 a + a 0(2i~) 0. The harmonicity of J’1(h, A; ~) is connected with the concept of redundant operators [11]. Due to reparametrization invariance of the field 4d(x) under renormalization group transformations, the critical exponents belonging to the class of redundant operators 4-theory bear no physical is given meaning. The first redundant operator that one encounters in 4 by the classical equation of motion —
=
R(x)
~,
2(x). (23) 04~(x)~ Derivatives of R(x) are also redundant operators. The one-loop divergent part of an operator that can be written as the product of a redundant operator R(x) and an arbitrary composite operator 0 1(x), =
—44~d(x)+m~4d(x)+~g
0(x)=R(x)~01(x),
(24)
is due to (22) (0(x))=
—44d(xYKOl(x)), (25) 2(x) yield divergent contributions only in order ~2• since ~g0t~~(x)~ Thus mO~d(x)and operators of the type (24) only reproduce the anomalous dimensions of —(.14~d(xY0l(x))=
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composite operators with lower canonical dimensions. In the sequel we therefore omit operators that can be written in the form (24) by formally setting 1i4d(x) 0. The space of composite operators constructed in this manner will be denoted by W. A suitable basis for W in terms of S0(4) harmonic polynomials is introduced in the next section. By restricting our discussion to we have also excluded all redundant operators from ourselves. Thus we will not encounter problems with non-physical critical exponents belonging to redundant operators. Acting on the space the integral operator ~1(h, A; ~) assumes a simpler form =
~‘
~‘
~,
1du exp[_~au(1 ~
l’~~(h, A;
~,
~) =J~f
(1— u)h
+
1).
~ +~J)IJO(2i%/~) (26)
V(h, A; A) manifestly exhibits the symmetries of the ~ lagrangian. The internal 0(N) symmetry and the 0(4) symmetry of spatial rotations are obvious since (26) is expressed in terms of scalar products. Translation invariance of V(h, A; B) follows immediately from the commutation relation of the translation operator ~,
~,
DA(h, A; with fr~1(g,B;
~,
A)= flexp[(hk+A)~gk] exp(Ak~Ak)
f, ~),
DA(h, A; ~, B)V~(g,B;
f, ~) = J’.~(h,A;
~, A)DA(g, B;
f, ~).
(27)
In the next section we will see that V is a hermitian operator with respect to a suitable metric. Thus the adjoint translation operators also commute with V. These adjoint operators turn out to correspond to special conformal transformations and this yields an extension of the spatial symmetry group to the conformal group in four dimensions 0(5,1). Let us discuss some examples before we go on with this program. For n = 2 fields we can explicitly work out the pole contributions (19) in some simple cases: =~ =
+
[4~2~,
+
10~~[~23,
1r Kt~dô~8~4d~) =
2~d~d~J, 24~d4~d,I,
(~a~
1a i~i~) —
K~9~da~4~d,> = I~(ka,~a~ + ~4)
[~26,
+
2~d~d~],
[~26dd,
+
2~dt~d~I.
(28)
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One easily concludes that in one-loop order, operator mixing only occurs between operators with the same number of fields and the same number of gradients. The critical exponent x0 of an eigenoperator 0 with eigenvalue a, (0>=10a0,
(29)
is due to (21) N±8)+0~),
X0=fl+i+(_~+
(30)
where n denotes the number of fields in 0 and I the number of gradients in 0. The fuH scaling dimension is thus yo=d—xo=4—n—l—c(1—
~
+
N+8)+0(E)
(31)
and the operator 0 is relevant, marginal or irrelevant for y0 > 0, y0 = 0, y0 <0 respectively. As a first 2)P example we calculate well-known of the Application of thethe first equation anomalous in (28) and dimensions induction over p operators (41 proves = jo
~
+N
—
4)(4~2)P
(32)
which yields the full dimensions
Y(~2)P=
4— 2p
+
—
6~2))
+ 0(c2)
(33)
in agreement with earlier calculations [101.The cases p = 1 and p = 2 correspond to the relevant operator ~2 with y = 2 y2(g~)and the marginal operator (~2)2 with y(4~2)2 = W. A nontrivial example of operator mixing is e.g. provided by the 4(a~a~)2, T 2aa~a~0(4) scalar in operators T1 = ~ (N = 1) for n ~ 24=fields. The mixing ~ and 7’3 is= given ~n— scalar 4~ theory matrix by —
3n2
—
3n 2
—
12 T 1+4T2 2—5n+2
3n—9 3n 2)=I0~ 2 T1+
2
T2+T3
2—7n+8
8n—16 3n 3)=I0~ 2 T2+
2
1’3 ,
(34)
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and diagonalization yields anomalous dimensions AE =
3n2—3n 18
‘
)tE =
3n2—5n—2 18
,
=
~
3n2—7n 18
(35)
corresponding to three eigenoperators E 1, E2 and E3. The anomalous dimension is just the anomalous dimension of 4I~in (32), which is consistent with a E1. This reproduction of anomalous dimensions for operators with additional gradients according to (5) simplifies the spectrum of anomalous dimensions. In sect. 7 we will explicitly see that one only needs to calculate the anomalous dimensions of conformal invariant operators. At first sight it is quite surprising that after the diagonalization of (34) one actually finds real anomalous dimensions in spite of the lack of apparent symmetry in the mixing matrix. In the next section we will construct a metric on the space and in fact mixing matrices are hermitian with respect to this metric. ~‘,
5. SO(4) tensor operators
.~ml,m2)
S0(4) tensor operators are defined via ~J~l;mI.m2)
= h~”~aai . . . ô~1t~(0),
d
=
1,..., N,
(36)
with symmetric and traceless S0(4) tensors h~”~(we consider only operators in W). These tensors form a basis for the symmetric irreducible representations of the Lie algebra SO(4) S0(3) ~ S0(3). Symmetric irreducible representations have the same spin in both S0(3) sectors (~I,~I), and the possible magnetic quantum numbers corresponding to the S0(3) sectors are therefore m1 = ~l, -~I+ i,...Ji and in2= ~ Thus there are (1+1)2 basis tensors for each I = 0, 1, 2 The complete space ~ can be constructed by products of these elementary S0(4) tensor operators. Employing the one-to-one correspondence with homogeneous harmonic polynomials of degree 1, —
H1m1~m2)(x)=
—
(37)
h~i~~~2,)xa . ..~
the basis (36) can be constructed explicitly with a generating functional 1/2 = mi,m2=
~(mi
1 ±1/2)(m2+ 1/2)
2sm2+l/2, Hfm1~m2)(x)tm1+l/
(38)
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= (x
1, x2, x3, x4) and u = (i its, it is, t + s,1 + ts) (compare ref. [2]). Important properties of the S0(4) harmonic polynomials H/m1~m2kx) are summarized in the appendix. We will later see that the normalization factor in (38) is not arbitrary, but basically determined by requiring that V is hermitian with respect to the metric induced by (36). The only freedom are additional factors like c’ with a constant real number c in the normalization factor. Let us calculate —
(~.J~l;ml.m2)
—
—
—
1’;m~~m’2))
= H,m1~m2(hl)H,(,m~m’2)(h2)K4~d(hl)~d,(h2))
1(1 —u)”~ ~~(g =10 fdu u
1
~2)~(~)’
x (HfmI~m2)(.~1+ .~2)Hi,m~.m~)(~, + ~2))
x [6~’~(g1)
.
+
çt(g2)
~
(39)
where we have used (26) and (8). The Laplace operator on the right-hand side of (39) is a Laplace operator with respect to ~ + g2. Application of the coupling rule (A.4) from the appendix for the product of harmonic polynomials in (39) yields after a short calculation /~(1;m1,m2) ~11(l’;m~,mi) \
a
‘
(L—2k)!
[L/2]
k=0 (L+1—k)!k!
m1 ~-l’, m~I~L—k, M1)(fl, m2 il’, m~I~L—k, M2) (40)
~
with L~I+l’, M1~m1+m~and M2~m2+m~.(j, m; j’, m’IJ, M) denotes an S0(3) Clebsch—Gordan coefficient. With the decoupling rule (A.5) one finally obtains
J~l,’;mi~m’2)>
[L/2] ‘0
~
1 L—2k+1
k0
~ (i;nL,n2)
m1 ~1’, m~I
—
~L,j+j’
(j’;n,n~)
k, M1)(+l, m2 ~l’, rn’2 I~L k, M2) —
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~j’, n~ ~L —k, M1)(~j,n2 ~j’, n~I~L —k,
M2)
N
x
~
+ ~
~
(41)
.
e= I
The sums and E(j’n~,n~) run over all possible sets of S0(4) quantum numbers (j; n1, n2) and (j’; n~,n~). Now we introduce creation and annihilation operators corresponding to S0(4) tensor operators a~m1,m2)t
._
~J~(l;m~.m~)
(42)
a~jm1m2)
with Bose commutation relations 611’~m [a~ml~m2),
41,’m~,m9tJ
=
(43)
1m~m2m’~dd’
all other commutators vanish. The vacuum state of the Hubert space ~ generated in this way will be denoted by I 0). This explicit realization of composite operators as vectors in a Hilbert space is somehow similar to the construction used in two-dimensional conformal field theories. In terms of the Bose operators (42) one can rewrite (41) as a two-particle interaction operator V,
v=
~
d,d’,e,e’=I
vQa~1;1~ 2)ta(~i.
2) a~
“
2)a~, 1,
2)
(44)
Q
The sum runs over all S0(4) quantum numbers of the creation and annihilation operators. The interaction kernel consists of a product of four SO(3) Clebsch—Gordan coefficients VQ = (6ee’~~~~ + ~ed~e’d’ [(l±1’)/21
~
X
k=0
+ ~ed’~e’d)
-
~i+i’,j±j’
—
k, m1
1 I+1’—2k+1
(il, rn1 ~I’, rn
I ~(l +
1’)
+ m~)
m2 ~-1’,m~I~(l+l’)—k, m2+m~)
n~~j’, n ~(i+j’) ~2;
~j’, n~(j+j’)
—k, m~+m) -k,
m2+m2).
(45)
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An eigenvector I 0) of V with n fields and 1 gradients VI0)=aIO)
(46)
thus corresponds to an eigenoperator in one-loop order with the anomalous dimension AEN8+0(c)
(47)
and the full scaling dimension y=4_n_1_e(1_~+
N+8)+0).
(48)
6. The spectrum-generating operator V V is a hermitian operator according to (44) and (45). Its eigenvalues are real and thus anomalous dimensions and critical exponents of eigenoperators in oneloop order are real numbers in spite of operator mixing. This property is not manifest from the beginning [12]. Notice that the hermiticity of V singles out a natural metric on This gives a well-defined meaning to topological terms like “small perturbation from the nontrivial fixed point”. Let us discuss the symmetries of V. The internal 0(N) symmetry of the N-vector theory is obviously also a symmetry of the operator V. In order to discuss the spatial symmetries of V it is sufficient to notice that due to (5) the momentum operators P~ ~ commute with V, ~‘.
=
—
[P~,v]=o.
(49)
In terms of creation and annihilation operators one has P1=D4—D1,
P2=D3+D2,
P3=i(D3—D2),
P4=—i(D4+D1), (50)
with 2m2~/2)ta~m1,m2),
D1
=
d1 (l;m~
~(~i
1
—
rn1)(~l+ 1
—
m2) alm1/
~/(~l + 1
—
m1)(~I+ 1
+
m2)
+
1m2) 2,m2+l/2)ta~m1,m2),
D2
=
d I
~ (1;m
am1/
1m2) 2~m2/2)ta~m1m2),
D3= d~
I (l;m~ 1m2)
~/(~l+ 1 +m1)(~I+1
_m2)aIm1+l/
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/
Conformal symmetry and anomalous dimensions
~/(~I+ 1 +m1)(~l+1
I (l;m
2.m2±I/2)ta~m1~m2). (51) +m2)a~m1±l/
1m2)
One can also check the commutation relations (49) directly in a lengthy calculation. Since V is a hermitian operator the adjoint operators D~commute with V too, and so does the entire algebra generated by the commutators of and D~.It is convenient to consider the anti-adjoint momentum operators K,~ —P~(special conformal transformations) =
K1= —(D~---Dfl,
K2= —(D~+D~),
K3
K4
=
i(Dt3
D~),
—
=
—i(D~+ Dfl
(52)
and the algebra generated in this way is the conformal algebra S0(5,1) in four dimensions: [Pu,
for all ~.t 1,.. defined as =
. ,
K,~I
=
—2iD,
[D,
P~J=iP~,
[D,
K,~I
=
—iK~,
(53)
4 (no summation is implied here) with the dilatation operator D
N
D
=
i ~
~
(1 +
(54)
1)am1~m2)ta~m1.m2).
d1 (l;m1,m2)
The dilatation operator counts the number of gradients plus the number of fields, that is the critical exponent in order e°.The commutators [P,~,K~]for ~ Il yield another six generators corresponding to the S0(4) subalgebra of the 15-dimensional algebra S0(5,1). The additional Z2-symmetry due to the exchange of both S0(3) sectors extends the symmetry group to 0(5,1). Two good quantum numbers for composite operators are obviously the number of fields n and the total number of gradients 1 (eigenvalue of the operator iD n). n and 1 will always have these meanings in the sequel. It is remarkable that conformal symmetry to one-loop order is a consequence of hermiticity of V and translation invariance of the lagrangian alone. We did not have to calculate any commutators with V explicitly. For later purposes it will be j.t
—
—
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683
convenient to have another expression of V in terms of the generators P~and K~. From (A.5) in the appendix one establishes H~~j~l2)(adP) ((fad
(L-f 1 —k)!k! 1L—2k+1~’ I
P)2) k1 a~°~O)t4°~O°)t
I
~ ~L,1+1’ 1,m~)(1’;mj,m~)
(l;m
=
x (fl, rn1 fi’, m~I -
—
k, M1)(fI, m2 fl’, m~I
—
k, M2) (55)
amI,m2)ta~,mi,m2)t
2[ ~ where [P ad P,j X] denotes the commutator [P,~, X] and e.g. (ad P1) 1, [P1, a~OO)ta~O.°)t]]. A similar relation holds for the generators of special conformal transformation K~.Inserting everything in (44) we obtains =
6dd’ + ~ed~e’d’
N
V=
f d,d’,e,e’= ~ 1 (~ee’ [L/2]
+
L+M
—
1±M2(’~ —
~
(L;M
2k)!(L
—
2k
+
2k’2
1)!
~L+1—k~’
1,M2) k~0
,‘.
2) k [~oo0)t~ooo)t] H~1j~2)(ad P) ((fad
x HLI’
P)
~(ad K)((f ad
K)2)[a~0~0)a~0~0)]
(56)
7. General properties of the spectrum of anomalous dimensions It is straightforward to conclude from (56) that V is a positive semi-definite operator. Similarly one proves the upper bound N
V~ f
~
(8ee’~dd’ + ~~d~e’~’ + ~ed’6e’d)
d,d’,e,e”=l
~
X (l;mI,m2)
a(lm1,m2)tam1m2)tam1,m2)a~m1,m2)
(57)
(1’;m~,m~)
in a simple calculation. It is easy to see by explicit construction that for composite operators with n elementary fields, the largest eigenvalue a of the operator on the
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right-hand side of (57) is given by fn(3n+N—4) —
The anomalous dimension
-
1)(3n +N— 1)
forevenn
(58)
for odd n.
A in the full dimension (48)
y=4—n—1—e(1—fn+A)+0(2)
(59)
therefore obeys 1 0~A~
for even n
1 2(N+8)~3~~’1~
(60)
foroddn.
The upper limit is just reached by the operators (42)~? and (~2)’~’4~ or total derivatives thereof (compare eq. (33)). Due to the lower limit in (60) one concludes that one-loop order contributions make all operators “more” irrelevant in contrast to the result (1) for the N-vector model in 2 + dimensions [13]. The stability of the nontrivial fixed point is even improved by one-loop contributions in 4 dimensions. The conformal symmetry group of V yields another simplification of the spectrum of anomalous dimensions. It is well-known that the spectrum of conformal invariant hamiltonians is generated by conformal invariant eigenvectors I i/i) [5,6], —
VI~i)=aj~/i) and
K~I~/j)=0Vj~t.=1,...,4.
(61)
All eigenoperators are either conformal invariant or derivatives of a conformal invariant eigenoperator (primary operator) reproducing the anomalous dimension of this primary operator. Since we have restricted the space of local operators to composite operators in it is actually more precise to speak of conformal invariant operators up to terms containing 1.1k or derivatives thereof. In order to derive the spectrum of anomalous dimensions it is obviously possible to restrict the diagonalization of V to the subspace of conformal invariant operators obeying K,~I~r)=0. ~‘,
8. Solutions for the cases of two and three fields It is manifest that V annihilates all tensors that are completely antisymmetric with respect to the internal 0(N) symmetry. Anomalous dimensions of these
S.K Kehrein et aL
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Conformal symmetry and anomalous dimensions
685
tensors vanish for any number of fields n. We will not discuss this trivial class of operators here. The diagonalization of V can be restricted to the space of conformal invariant operators. Acting on a conformal invariant operator I ~F>, the operator V from (56) assumes a simpler form N
V I il’)
=
~
6dd’ + ‘3ed6e’d’ + ~ed’~e’d)
d,d’,e,e’=l
(~ee’
(L—2k\’
EL/2] x
~
~
(pL±MI±M2
‘
I.
(L+1—kUkt
(L;M 1,M2) k0 X
~
/.
~LJ+J’(~J,
n1
fj’, n~I
—
k, M1)(fj, n2
f.i’,
n~I
—
k, M2)
(j;n1,n2) (j’;n,n’2)
a~00~a~0’0~ I ~).
2)
X
a
(62)
,fl2)ta’~fl~)t . H~~,_M2)(K)((fK)
Notice that two annihilators for fields 4, without gradients act directly on I II’). The spectrum of anomalous dimensions for operators with n 2 fields and gradients is well-known [16]. It is easy to see that these results are consistent with the approach here. There are only two conformal invariant eigenoperators with nonvanishing eigenvalues: (1) The 0(N) scalar operator •2 with eigenvalue N + 2 and a full dimension y = 2— (N + 2)/(N + 8) + 0(c2). (2) The second-rank symmetric and traceless 0(N) tensor with eigenvalue 2 and a full dimension y 2 2/(N + 8) + 0(2). This eigenoperator does not exist in scalar (N 1) 4~theory. Conformal invariant operators with n 2 fields and 1> 0 are annihilated by (62) and thus have vanishing anomalous dimensions in order e. It is easy to see that for an even number of gradients I there exists one 0(N) symmetric traceless and one 0(N) scalar tensor that are conformal invariant. No conformal invariant operators exist for odd I (except completely antisymmetric with respect to the internal symmetry). These conformal invariant eigenoperators correspond to S0(4) symmetric tensors of even rank 1. For I 2 one finds e.g. the stress tensor. It is well-known that except for the stress tensor, these conformal invariant operators acquire anomalous dimensions in order ~2 [16]. We can also work out the complete spectrum of anomalous dimensions for n 3 fields. The two annihilators a~°’°~a~°’°~ in (62) effectively reduce the problem to a one-particle problem that can be solved easily. In order to concentrate on the essential points, let us consider the scalar (N 1) theory first. =
~
=
—
.
=
=
=
=
=
—
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By explicit construction one can show that at least one conformal invariant operator with three fields exists for every 1 ~ 1. We have to distinguish two cases: (1) The conformal invariant operator contains a term ~00,O 0;0,0)n~(l;m,m 2) All other terms are annihilated by V according to (62) and one calculates
V(~(o;o.o)~(o;o.o)ck(l;m1.m2))=
3(1
+
+
(
—
1)~’
)
~p(O;OO)~(O;OO)~(l;mIm2)
other terms.
(63)
The contribution independent of 1 comes from the term (L; M1, M2) (0; 0, 0) in (62) and the factor (~ 1)~2/(l + 1) from (L; M1, M2) (1; m1, m2). Thus we find conformal invariant eigenoperators with anomalous dimensions =
=
VI~l.
(64)
One can check that the spin structure of these operators is (fI, fI). They exist for any number of gradients except for 1 1. (2) There is another class of conformal invariant eigenoperators that becomes especially important for larger values of I (their number increases faster than linear with 1): These conformal invariant operators do not contain a term ~,(0;0,0)~1~(O;0,0)~(/;m1,m2) Therefore the anomalous dimension of these operators vanishes in one-loop order. The first operator of this type that one encounters has 1 4 with the spin structure (0, 2) Notice that there are also tensors with the spin structure (f1, f1) in this class. The first operator with this structure has for example 1 6 gradients ** as one can see by doing a diagonalization of V in this subspace using a computer. This yields the interesting feature that 0(4) scalar operators with three fields and vanishing anomalous dimensions exist (the first with 12 gradients). For N> 1 some additional complications arise, though nothing really modifies the general structure that we have found in the scalar theory. There is again the important class of conformal invariant operators that do not contain the term ~ and therefore yield vanishing anomalous =
=
~.
=
2—1,— l),p(2l.— 1)_ *
+
**
~~(O;°.°)cp(2O—1)~~(2;O,— I)
—
~(L;1/2,— 1/2)~1;1/2,—1/2)~(2;—1.—i)
= ~ ~,(O;O,O)~p(1/2)~J~(I;_ 1/2,—1/2)~(2;O,—I) — ~p(i;—1/2,— I/2)~p(l;—1/2,— I/2)~p(2I.—1) 0 = h~ ~, 6[3~~d,,2cba,~“~*6~ ~3..~ 8* ~ ~ — ~ . . + 34~8~ . . . c 3~ 4. ~ with any completely symmetric and traceless tensor ~
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687
dimensions in one-loop order. For the nonzero eigenvalues one evaluates
=
.
~(0;0,O)~(1;m~,m~)
+
(
—
~
.
+ 2~~oO,o)t~9;o,o)~D(l;m1,m2)
+
(
+
other terms.
—
1)’—_~~— ~
+~
.
(65)
Now we have to distinguish three classes of conformal invariant operators corresponding to the 0(N) transformation properties with respect to the internal indices. We write 0 5d1d2d3~QcQ~pdl12)~I2i2)tp~3I2) where the sum EQ =
runs over S0(4) quantum numbers, and there are three possibilities for the tensors Sd
d2d2:
(1) Completely symmetric and traceless tensors S~1~2~3: The respective anoma-
lous dimensions are VI~1.
(66)
(2) Antisymmetric with respect to two indices and traceless: The anomalous dimensions are VI~1.
(67)
(3) Tensors S~1~ ~, with a nonvanishing trace: This case leads to the diagonalization of the matrix N+2 (N+2)(b1—1) with b1
=
1
+ (~1)1.
2/(I
+
b1 2b1
1). Thus the anomalous dimensions are
A±=2(N+8)[N+2+2bl±~/(N+2)2+4b~+4(N+2)bl(bl_2)1
+0(E2).
(68)
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Notice that for 1 0, 1 (or for N 1) only one of the solutions A corresponds to a nonzero eigenoperator. We were not yet able to derive a similar complete classification for the spectrum of anomalous dimensions for n > 3 fields. E.g. even for four fields in the scalar theory, one finds conformal invariant operators with nonrational anomalous dimensions. The solution for n 3 relies heavily on the conformal symmetry which allows a complete classification. This sounds familiar from three-point functions in conformal field theories, but cannot directly be carried further than three fields. Work on the spectrum for n> 3 is in progress. =
=
±
=
9. Conclusions and outlook We will briefly sum up the main results of this paper. We have discussed the N-vector model in 4 c dimensions in one-loop order. It was shown that the pole contribution of an operator insertion K 0) for a general composite operator 0 with n elementary fields and 1 gradients can be written as the action of a hermitian operator V on 0. The operator 0 is represented as a vector in a Hilbert space of composite operators and V has formally the structure of the two-particle interaction potential (44) acting on this space. Thus the eigenvectors of V correspond to scaling eigenoperators and the eigenvalues correspond to the respective anomalous dimensions. The diagonalization of V in an n-particle space with a given total number of gradients yields the spectrum of anomalous dimensions in this class of mixing composite operators. Obviously anomalous dimensions of eigenoperators are therefore real numbers in spite of operator mixing. By establishing upper and lower bounds for the spectrum-generating operator V, we found upper and lower limits (60) for the spectrum of anomalous dimensions. Since all anomalous dimensions are positive due to (60), one-loop order contributions make all canonically irrelevant operators even more irrelevant. One does not encounter the unusual behaviour of some 2 + expansions where the stability of the nontrivial fixed point is endangered by high-gradient operators in one-loop order [3,4,13,14,7]. The spatial symmetry group of V was found to be the conformal group in four dimensions 0(5,1). This conformal invariance should be expected from general grounds (compare e.g. ref. [8]). The restriction of the diagonalization of V to the subspace of conformal invariant operators allows a complete classification of the spectrum of anomalous dimensions for composite operators consisting of three elementary fields with an arbitrary number of gradients. This extends the wellknown solution by Wilson and Kogut [16] for two fields and gradients. It is quite remarkable that for a sufficiently large number of gradients, the spectrum for three elementary fields is dominated by the eigenvalue 0. —
11~
~‘
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Conformal symmetry and anomalous dimensions
689
The complete spectrum of V for more than three fields has not yet been worked out. Already the case of n 4 fields seems to possess the complexity of the general problem. Apart from this diagonalization problem, it should be interesting to extend the present work to two-loop order. Work on these problems is in progress. =
One of the authors (S.K.K.) would like to thank H.D. Conradi and K.J. Wiese for valuable discussions.
Appendix A. Properties of SO(4) harmonic polynomials S0(4) harmonic polynomials
Hfml,m2)(x),
m12
—1/2,—
=
1/2 + 1,.. .,1/2 of
degree 1 can be constructed using a generating functional 1/2 =
2Sm2+t/2
~(m1
ml,m =—i/2
±I/2)(m2 +1/2)
H/ml.m2)(X)tml÷t/
(A.l) with four-vectors x
=
(x
1, x2, x3, x4) and u (1 its, it is, t + s, 1 + ts) (compare ref. [2]). The polynomials Hf m l.m2)(.~) provide a complete set of functions on the unit sphere S3. Orthogonality and normalization conditions can be deduced from ref. [2], =
fdf~ ~
—
—
—
= 2~2~ii~5m m’~rnm’
4i(l
where Is3 dQ is the surface integral over S3 and conjugate of Hf m~,m~)(X): Hf,mc~~~~)( x) =
(—
~
—
J~If,m~~mi)(x) is
(A.2) the complex
(A.3)
m~m~)(x).
We need the following two theorems for the polynomials H/rnI,m2kx): Coupling rule: Hfml.m2)( x)Hfm~.m~( x) [L/2]
= k0
(L—2k)’ 1!1’!
(fi,
m1 fi’, rn’1 fL
XH/,~~2)(x). (~x2)”,
I
—
k, M1)(fI, m2 fi’, rn’2 I fL
—
k,
-~~~2)
(A.4)
690
/
S.K Kehrein et al.
where L
1’, M12 m12 + m~2and the brackets on the right-hand side denote S0(3) Clebsch—Gordan coefficients (j, m; j’, rn’ I J, M). Decoupling rule: =
I
Conformal symmetry and anomalous dimensions
+
=
H~~2)(x1+x2)
.
(Xi.X2)k
(L +1 —k)!k! =
(L
—
2k +
n1
5L,j±j’
/ - (j;n 1~’ 1,n2)
ff’,
~ (j’;n~,n~)
n~IfL —k, M1)(fj, n2 +x~ (
XHJ~1’~2)(x1)I-IJi’i”12)(x2)
...)
fj’,
n~IfL—k, M2)
+x~ (
(A.5)
...).
We will prove (A.4) first. Consider the surface integral over the unit sphere S3,
f dQ(u1 .x)1(u2 Ox)’ (v
(A.6)
.x)L_2k,
S3
with u1, u2 and v defined similar to (A.1), =
(i
—
it1s1,~it1
u2
=
(i
—
it252,
v
=
(i
—
iTT~_
—
it2
iT
—
—
is1,— ~1+ s~,1 ~
io,— T
t2 +52,
1
+ o,
+
1
t1s1),
+
t2s2), (A.7)
+ TcT).
The integral (A.6) can only depend on euclidean invariants made up of u1, u2 and v. The only possibility is
f
1 (v
1(u
dfl(u1 ~x) S3
.x)L2k
=
C(l, 1’; k)(u 1 u2)k(u1
2-x)
.
~)l_k(u2
.
(A.8) with a factor C(l, 1’; k) and L I + 1’. Expanding the scalar products on the right-hand side of (A.8), employing the definition (A.1) of the polynomials Hfml.m2)(x) on the left-hand side and comparing coefficients of t1, s~,12, ~2’ T, U =
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Conformal symmetry and anomalous dimensions
691
yields after a straightforward calculation
f dQ
HfmIm2)(x)Hicm~.m~)(x)H~~2)(x)
S
3 I.
—
‘.
I
‘.
‘
~‘
‘
2
Ic.
2’—~~l!I’!
(L—k+1\/(L—2k k k )/~i—k
x(fi, rn1 fI’, m~IfL —k, M1)(fl, m2 fi’, m~IfL —k, M2). (A.9) By choosing the specifically simple set of magnetic quantum numbers rn1 m2 —1/2, m~ m~ —l’/2 one can explicitly work out the integral (A.9) and obtain the unknown factor C(l, 1’; k), =
=
=
=
2 C(i, I’; k)
=
2~
l!l’!(L —2k)!
2L
k(l
—
k)!(l’
—
k) !(L
—
k
+
1) !kL
(A.10)
Inserting this back into (A.9) and comparing with the orthogonality relation (A.2) proves the theorem. In order to derive the decoupling rule (A.5) we will first consider the special case k 0 in (A.5), =
~ L
~,X1
— X21
—
V ~.s (j;n1,n2) (j’;n,n~)
(fi, ~
~ L,j±j’
fj’, n~I fL, M1)(fj,
~2;
X H~~”2)(x1)HJi~”~2)( x2).
fj’, n’~IfL,
M2) (A.11)
Now (u (x1 +x2))L .
=
~
(~
)(u .x)~(u~
(A.12)
with u from (A.1). Application of the definition (A.1) for the polynomials identification of the coefficients proves (A.11). The general theorem (A.5) follows from (A.11) by induction over k. It is a somehow tedious but straightforward exercise where one makes use of 2”2~( x 2”/2~(x x1 x2 = H~l/2._l/2)( x1) H~’~’ 2) + H1”/ 1) H~1/2,—1/2)( x2) 2’ 1/2)( x 2’’/2~(x Hf 1/2.1/2)( x1) H~” 2) H/ 1) H~1/2.1/2)( x2) (A.13) H/?~~l.m2)(x) and
-
—
—
and frequently employs the coupling rule (A.4) in the special case that one of the coupling spins equals 1.
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References [1] G.E. Castilla and S. Chakravarty, Preprint University of California, Los Angeles (1992) [2] A. Erdélyl, W. Magnus, F. Oberhettinger and F.G. Tricomi, Higher transcendental functions, vol. 2 (McGraw-Hill, New York, 1953) [3] yE. Kravtsov, IV. Lerner and VI. Yudson, Zh. Eksp. Teor. Fiz. 94 (1988) 255; Soy. Phys. JETP
67 (1988) 1441 [4] yE. Kravtsov, LV. Lerner and V.!. Yudson, Phys. Lett. A134 (1989) 245 [5] M. Lüscher and G. Mack, Commun. Math. Phys. 41(1975) 203
G. Mack, Commun. Math. Phys. 53 (1977) 155 [7] H. Mall and F.J. Wegner, to be published in NucI. Phys. B [8] L. Schafer, J. Phys. A9 (1976) 377 [9] D.J. Wallace and R.K.P. Zia, J. Phys. C8 (1975) 839 [10] F.J. Wegner, Phys. Rev. B6 (1972) 1891 [11] F.J. Wegner, J. Phys. C7 (1974) 2098 1121 F.J. Wegner, in Phase transition and critical phenomena, vol. 6, ed. C. Domb and MS. Green [6]
(Academic Press, New York,
1976)
[131 F.J. Wegner, Z. Phys. B78 (1990) 33 [14] F.J. Wegner, NucI. Phys. B354 (1991) 441 [15] KG. Wilson, Phys. Rev. Lett. 28 (1972) 548 [16] K.G. Wilson and J. Kogut, Phys. Rep. 12 (1974) 75
Nuclear Physics B402 IFS] (1993) 669—692 North-Holland
________________
Conformal symmetry and the spectrum of anomalous dimensions in the N-vector model in 4 dimensions * —
Stefan K. Kehrein 1 and Franz J. Wegner Institut für Theoretische Physik, Ruprecht-Karls-Universitàt, D-6900 Heidelberg, Germany
Yurej M. Pismak Department of Theoretical Physics, State University of Saint-Petersburg, Ul~yanovskaya1, Stary Petergof, 198904 Saint-Petersburg, Russian Federation Received 16 October 1992 Accepted for publication 26 March 1993
The subject of this paper is to study the critical N-vector model in 4— dimensions in one-loop order. We analyse the spectrum of anomalous dimensions of composite operators with an arbitrary number of fields and gradients. For composite operators with three elementary fields and gradients we work out the complete spectrum of anomalous dimensions, thus extending the old solution of Wilson and Kogut for two fields and gradients. In the general case we prove some properties of the spectrum, in particular a lower limit 0 + O(2). Thus one-loop contributions generally improve the stability of the nontrivial fixed point in contrast to some 2 + expansions. Furthermore we explicitly find conformal invariance at the nontrivial fixed point.
1. Introduction In several recent papers anomalous dimensions of composite operators with a general number of fields and gradients have been classified in one-loop order in 2 + e expansions. The models investigated so far are the nonlinear matrix model discussed by Kravtsov, Lerner and Yudson [3,41, the N-vector model [131, the unitary matrix model [14] and the orthogonal matrix model [7]. All these models share the surprising feature that the dominant scalar operator with 2s gradients has in addition to its naive dimension 2(1 s) a positive one-loop order contribution to the full scaling dimension that is proportional to s2. In the N-vector model —
*
Research supported in part by the Sonderforschungsbereich 123 Stochastic Mathematical Models of the Deutsche Forschungsgemeinschaft. E-mail: ej6ftyvm.urz.uni-heidelberg.de
0550-3213/93/$06.00 © 1993
—
Elsevier Science Publishers B.V. All rights reserved
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Conformal 3ymmetiy and anomalous dimensions
for example one finds an operator with 2s gradients and the full scaling dimension [13] s(s—1) Y2(1_S)+E.(1+ N—2 )+O(2). (1) Recently Castilla and Chakravarty have obtained a similar positive contribution of order e2 s3/(N 2)2 in a two-loop calculation [1]. It is an open question whether contributions beyond second order in e are negligible. If this were the case, operators with sufficiently large s would become relevant for a given positive c, thus making the conventional fixed point unstable against perturbations by these operators. This paper considers the related problem of anomalous dimensions in the critical N-vector model in 4 c dimensions in one-loop order. It will become apparent that the behaviour in 4 e dimensions is different from eq. (1). Here one-loop contributions always make canonically irrelevant operators even more irrelevant. Furthermore we explicitly see that the spatial symmetry group of the N-vector model at the critical point is the conformal group in four dimensions 0(5,1) to one-loop order. Thus one finds conformal invariance at the phase transition, which goes beyond the obvious scale invariance. Conformal invariance in the N-vector model is a well-known feature (the most complete treatment is due to Schafer [8]). The spectrum of anomalous dimensions of conformal invariant field theories can be derived in terms of conformal invariant operators [61. Employing our explicit knowledge of the conformal symmetry group, we use these properties to calculate the complete spectrum for composite operators with three elementary fields and an arbitrary number of gradients in one-loop order. This extends the well-known solution for two fields and gradients solved a long time ago by Wilson and Kogut [16] in two-loop order. For composite operators consisting of more than three elementary fields, we have not yet derived a similar complete classification of the spectrum of anomalous dimensions. Still we can establish some general properties like upper and lower limits for the spectrum and the fact that all eigenvalues are real numbers. It is a desirable feature that anomalous dimensions of eigenoperators are real numbers, but nontrivial due to operator mixing [12]. The organization of this paper is as follows. Sect. 2 gives a short definition of the N-vector model. In sect. 3 we calculate the one-loop pole contribution of an operator insertion K 0) 11~for a general composite operator 0. This can be done in terms of an integral operator V. Some properties of V connected with the concept of redundant operators [11] are discussed in sect. 4. In sect. 5 we introduce S0(4) tensor operators cI~1m,m 2) and expand V in this basis. V is hermitian with respect to the metric induced in this manner and has formally the structure of a two-par—
—
—
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Conformal symmetry and anomalous dimensions
671
tide interaction operator. Its spectrum corresponds to the spectrum of anomalous dimensions. Properties of V and its spectrum are discussed in sects. 6 and 7. In sect. 8 the diagonalization problem for V is solved for two and three fields. Sect. 9 closes with a summary of consequences and an outlook on open problems. 2. Definition of the model
S
The N-vector model of the field S0 + S11~with a free contribution
4
=
(4~,. 4~)is . . ,
governed by the action
=
S0
=
f ddx(~(04.)2+
(2)
~m~2)
and the interaction part
Sn~=~fddx(4~)~.
(3)
We are investigating the critical behaviour of this model in one-loop order in d = 4 e dimensions. Thus the renormalized mass vanishes and the critical coupling constant is given by [151 —
3 N+8 +0(e2).
(4)
For a local composite operator 0(x) consisting of elementary fields ~d(x), d = 1,..., N and derivatives thereof at one point x, the divergent part of the one-particle irreducible diagrams in one-loop order will be denoted by K0(x))ii~. We are only interested in this divergent contribution, the index lip will be suppressed everywhere in the sequel. As usual, the pole terms yield the Z-factors and so the anomalous dimensions. Notice that in general composite operators are tensor operators with respect to 0(4) spatial rotations and with respect to internal 0(N) rotations. Anomalous dimensions have been calculated to high orders in e for composite operators without gradients like 5d q~ (compare e.g. ref. [91for traceless tensors Sd ~) or for operators consisting of two elementary fields only (compare ref. [161).Due to the relation (5) these anomalous dimensions are reproduced for operators with gradients. But not all composite operators can be constructed in the manner ~~(4d, 4d). In the next section we will therefore calculate the one-loop pole contribution K 0(x)) for a general composite operator with gradients. . . .
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3. One-loop pole contributions For notation purposes it is useful to introduce the argument contracting operation (ACO). If two arguments of a function of n arguments are denoted with the same letter and one of these letters is barred, this denotes ACO applied to the function. The result is a function of n 2 arguments defined by —
f(...,
z,...,2,...)
def
U
~
=exp
~
(6)
Some basic properties of the ACO are =
.h
f(x+h) =a,~...a~f(x)
(7)
and the shift operation e~f(x+b)=f(x+a).
(8)
The last equation also holds for vectors x, a, b if one employs a euclidean scalar product a b in the exponent. Using the ACO we can write down a scalar generating function for composite operators consisting of n fields ~d(x) with gradients .
~~(x; h, A)
=
flA~~(x+h~).
(9)
By contracting ~I’~(x;h, A) with At’s and h1’s one can construct all composite operators with n fields. Our aim is to calculate KW~(x;h, A)). In one-loop order the only diagrams contributing to pole terms in an c-expansion are of the type in fig. 1, where one
is
—
2 legs
Fig. 1. One-loop pole contributions.
/
S.K. Kehrein et al.
Conformal symmetry and anomalous dimensions
673
has to sum over all different possibilities to close two legs with a loop. We can express this by writing (~~(x;h,A))=—~g0 ~
fddyG(x+hj_y)G(x+hj_y)
i,j=1
i
x[2(~(y) .A~)(~(y) .A1) +~2(y)(A~.A1)]~
A~(x+h~).
k # i ,j
(10) G(z) denotes the free propagator in configuration space 1 G(z)=
(11)
(2)dfPp2+m2
We have to evaluate the integral Jddy G(x+h~—y)G(x+h~—y)(~(y) .A~)(q~(y) .A.) =
V(h1, h~g~,.~~)(~(x + g~).A1)(~(x + g~).A.),
(12)
where V(h~,h1 g~,g~)=fddyi d”y2 G(y1)G(y2)6(y1 —y2—h1+h1) xexp[(h,—yi) ddp
~
is a standard exercise to
=
=
~ (2~ [(q
—
i~~)
+
+
(13)
~zqa
+
i~)2+ m~1[(q
-
i~)2+ m~] 2+ m~I t
fdt
2+ mg]) (14)
—
1 dt2f
m~I
calculate
d”q
,.
{(~i~3.)2
+2m~]
(21r)d [(p It
~j~+ (h3—y2).~]
d exP(iq
a
—
11[(q
+
i~)
2[(q
—
i~)
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Conformal symmetry and anomalous dimensions
and after performing the integral over q and a change of variables one obtains 1=
1 (
4)d/2E
xexp a
f dt
1
du
tE/21etf
0
0
2 (2u_1)+4(1_u)u/32~_~_a2_~~ t m
0 The integral
over
u
generates a sum E~
(15)
-
m
4t
(t/m~Y’~ck(a, /3), thus
1
=
4n-) d/2m
(
~ F( —k + ~E)ck(a,
/3)m~’+ (finite terms for c
0 k=0
=
0). (16)
The pole contribution I~,in the massless theory comes from k 0, 2f1d exp[a./3(2u 1)J ~ ((1 _u)ua2/32)~ 2_2 (4~)~ (4~) c m 0 0 =
(_~[
—
~
=
(4)2
(~)
2/32
f’du exp[ —au(l
—
+a
u)a
-/3(2u
—
l)]J 0(2i~/~),(17)
with the Bessel function (18) Inserting (17) back into (13) we obtain the following closed expression for the pole contribution of (10). (V’~(x;h, A))
=
V(h, A;
~,
B)V’~(x;g, B)
~,
‘<~
fl
~
k*i,j
exP(hk~k)exP(Ak~k)]
(19) with 1du exp[—+au(1 ~
fr~(h,A;
~,
~)=I0.f
+
(1
—
u)h 1)
.
(~j
+
+~1)jJo(2iV~)
~
(20)
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675
and
g ‘0=
—
3(4)2
=
1 N+8 + 0(c)
—
(21)
at the critical point. 4. Properties of V(h, A; j,
.Th
Before we make use of (19) in order to calculate some examples of anomalous dimensions, let us prove some general properties of the operator V(h, A; B). In a straightforward calculation one shows ~,
A;
~,
~)=jldu(gj+j)2uexp[_~au(1_u)(hj_hj)2(gj+~j)2
+(uh~+(1 —u)h1) 2u(l ><
[i
—
—
a + ~a
.
u)(h
2(~~ + )2] J 1
—
h~)
0(2ii/i) (22)
2c) e~cJ since (1 a + a 0(2i~) 0. The harmonicity of J’1(h, A; ~) is connected with the concept of redundant operators [11]. Due to reparametrization invariance of the field 4d(x) under renormalization group transformations, the critical exponents belonging to the class of redundant operators 4-theory bear no physical is given meaning. The first redundant operator that one encounters in 4 by the classical equation of motion —
=
R(x)
~,
2(x). (23) 04~(x)~ Derivatives of R(x) are also redundant operators. The one-loop divergent part of an operator that can be written as the product of a redundant operator R(x) and an arbitrary composite operator 0 1(x), =
—44~d(x)+m~4d(x)+~g
0(x)=R(x)~01(x),
(24)
is due to (22) (0(x))=
—44d(xYKOl(x)), (25) 2(x) yield divergent contributions only in order ~2• since ~g0t~~(x)~ Thus mO~d(x)and operators of the type (24) only reproduce the anomalous dimensions of —(.14~d(xY0l(x))=
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composite operators with lower canonical dimensions. In the sequel we therefore omit operators that can be written in the form (24) by formally setting 1i4d(x) 0. The space of composite operators constructed in this manner will be denoted by W. A suitable basis for W in terms of S0(4) harmonic polynomials is introduced in the next section. By restricting our discussion to we have also excluded all redundant operators from ourselves. Thus we will not encounter problems with non-physical critical exponents belonging to redundant operators. Acting on the space the integral operator ~1(h, A; ~) assumes a simpler form =
~‘
~‘
~,
1du exp[_~au(1 ~
l’~~(h, A;
~,
~) =J~f
(1— u)h
+
1).
~ +~J)IJO(2i%/~) (26)
V(h, A; A) manifestly exhibits the symmetries of the ~ lagrangian. The internal 0(N) symmetry and the 0(4) symmetry of spatial rotations are obvious since (26) is expressed in terms of scalar products. Translation invariance of V(h, A; B) follows immediately from the commutation relation of the translation operator ~,
~,
DA(h, A; with fr~1(g,B;
~,
A)= flexp[(hk+A)~gk] exp(Ak~Ak)
f, ~),
DA(h, A; ~, B)V~(g,B;
f, ~) = J’.~(h,A;
~, A)DA(g, B;
f, ~).
(27)
In the next section we will see that V is a hermitian operator with respect to a suitable metric. Thus the adjoint translation operators also commute with V. These adjoint operators turn out to correspond to special conformal transformations and this yields an extension of the spatial symmetry group to the conformal group in four dimensions 0(5,1). Let us discuss some examples before we go on with this program. For n = 2 fields we can explicitly work out the pole contributions (19) in some simple cases: =~ =
+
[4~2~,
+
10~~[~23,
1r Kt~dô~8~4d~) =
2~d~d~J, 24~d4~d,I,
(~a~
1a i~i~) —
K~9~da~4~d,> = I~(ka,~a~ + ~4)
[~26,
+
2~d~d~],
[~26dd,
+
2~dt~d~I.
(28)
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One easily concludes that in one-loop order, operator mixing only occurs between operators with the same number of fields and the same number of gradients. The critical exponent x0 of an eigenoperator 0 with eigenvalue a, (0>=10a0,
(29)
is due to (21) N±8)+0~),
X0=fl+i+(_~+
(30)
where n denotes the number of fields in 0 and I the number of gradients in 0. The fuH scaling dimension is thus yo=d—xo=4—n—l—c(1—
~
+
N+8)+0(E)
(31)
and the operator 0 is relevant, marginal or irrelevant for y0 > 0, y0 = 0, y0 <0 respectively. As a first 2)P example we calculate well-known of the Application of thethe first equation anomalous in (28) and dimensions induction over p operators (41 proves = jo
~
+N
—
4)(4~2)P
(32)
which yields the full dimensions
Y(~2)P=
4— 2p
+
—
6~2))
+ 0(c2)
(33)
in agreement with earlier calculations [101.The cases p = 1 and p = 2 correspond to the relevant operator ~2 with y = 2 y2(g~)and the marginal operator (~2)2 with y(4~2)2 = W. A nontrivial example of operator mixing is e.g. provided by the 4(a~a~)2, T 2aa~a~0(4) scalar in operators T1 = ~ (N = 1) for n ~ 24=fields. The mixing ~ and 7’3 is= given ~n— scalar 4~ theory matrix by —
3n2
—
3n 2
—
12 T 1+4T2 2—5n+2
3n—9 3n 2)=I0~ 2 T1+
2
T2+T3
2—7n+8
8n—16 3n 3)=I0~ 2 T2+
2
1’3 ,
(34)
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and diagonalization yields anomalous dimensions AE =
3n2—3n 18
‘
)tE =
3n2—5n—2 18
,
=
~
3n2—7n 18
(35)
corresponding to three eigenoperators E 1, E2 and E3. The anomalous dimension is just the anomalous dimension of 4I~in (32), which is consistent with a E1. This reproduction of anomalous dimensions for operators with additional gradients according to (5) simplifies the spectrum of anomalous dimensions. In sect. 7 we will explicitly see that one only needs to calculate the anomalous dimensions of conformal invariant operators. At first sight it is quite surprising that after the diagonalization of (34) one actually finds real anomalous dimensions in spite of the lack of apparent symmetry in the mixing matrix. In the next section we will construct a metric on the space and in fact mixing matrices are hermitian with respect to this metric. ~‘,
5. SO(4) tensor operators
.~ml,m2)
S0(4) tensor operators are defined via ~J~l;mI.m2)
= h~”~aai . . . ô~1t~(0),
d
=
1,..., N,
(36)
with symmetric and traceless S0(4) tensors h~”~(we consider only operators in W). These tensors form a basis for the symmetric irreducible representations of the Lie algebra SO(4) S0(3) ~ S0(3). Symmetric irreducible representations have the same spin in both S0(3) sectors (~I,~I), and the possible magnetic quantum numbers corresponding to the S0(3) sectors are therefore m1 = ~l, -~I+ i,...Ji and in2= ~ Thus there are (1+1)2 basis tensors for each I = 0, 1, 2 The complete space ~ can be constructed by products of these elementary S0(4) tensor operators. Employing the one-to-one correspondence with homogeneous harmonic polynomials of degree 1, —
H1m1~m2)(x)=
—
(37)
h~i~~~2,)xa . ..~
the basis (36) can be constructed explicitly with a generating functional 1/2 = mi,m2=
~(mi
1 ±1/2)(m2+ 1/2)
2sm2+l/2, Hfm1~m2)(x)tm1+l/
(38)
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= (x
1, x2, x3, x4) and u = (i its, it is, t + s,1 + ts) (compare ref. [2]). Important properties of the S0(4) harmonic polynomials H/m1~m2kx) are summarized in the appendix. We will later see that the normalization factor in (38) is not arbitrary, but basically determined by requiring that V is hermitian with respect to the metric induced by (36). The only freedom are additional factors like c’ with a constant real number c in the normalization factor. Let us calculate —
(~.J~l;ml.m2)
—
—
—
1’;m~~m’2))
= H,m1~m2(hl)H,(,m~m’2)(h2)K4~d(hl)~d,(h2))
1(1 —u)”~ ~~(g =10 fdu u
1
~2)~(~)’
x (HfmI~m2)(.~1+ .~2)Hi,m~.m~)(~, + ~2))
x [6~’~(g1)
.
+
çt(g2)
~
(39)
where we have used (26) and (8). The Laplace operator on the right-hand side of (39) is a Laplace operator with respect to ~ + g2. Application of the coupling rule (A.4) from the appendix for the product of harmonic polynomials in (39) yields after a short calculation /~(1;m1,m2) ~11(l’;m~,mi) \
a
‘
(L—2k)!
[L/2]
k=0 (L+1—k)!k!
m1 ~-l’, m~I~L—k, M1)(fl, m2 il’, m~I~L—k, M2) (40)
~
with L~I+l’, M1~m1+m~and M2~m2+m~.(j, m; j’, m’IJ, M) denotes an S0(3) Clebsch—Gordan coefficient. With the decoupling rule (A.5) one finally obtains
J~l,’;mi~m’2)>
[L/2] ‘0
~
1 L—2k+1
k0
~ (i;nL,n2)
m1 ~1’, m~I
—
~L,j+j’
(j’;n,n~)
k, M1)(+l, m2 ~l’, rn’2 I~L k, M2) —
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/
Conformal symmetry and anomalous dimensions
~j’, n~ ~L —k, M1)(~j,n2 ~j’, n~I~L —k,
M2)
N
x
~
+ ~
~
(41)
.
e= I
The sums and E(j’n~,n~) run over all possible sets of S0(4) quantum numbers (j; n1, n2) and (j’; n~,n~). Now we introduce creation and annihilation operators corresponding to S0(4) tensor operators a~m1,m2)t
._
~J~(l;m~.m~)
(42)
a~jm1m2)
with Bose commutation relations 611’~m [a~ml~m2),
41,’m~,m9tJ
=
(43)
1m~m2m’~dd’
all other commutators vanish. The vacuum state of the Hubert space ~ generated in this way will be denoted by I 0). This explicit realization of composite operators as vectors in a Hilbert space is somehow similar to the construction used in two-dimensional conformal field theories. In terms of the Bose operators (42) one can rewrite (41) as a two-particle interaction operator V,
v=
~
d,d’,e,e’=I
vQa~1;1~ 2)ta(~i.
2) a~
“
2)a~, 1,
2)
(44)
Q
The sum runs over all S0(4) quantum numbers of the creation and annihilation operators. The interaction kernel consists of a product of four SO(3) Clebsch—Gordan coefficients VQ = (6ee’~~~~ + ~ed~e’d’ [(l±1’)/21
~
X
k=0
+ ~ed’~e’d)
-
~i+i’,j±j’
—
k, m1
1 I+1’—2k+1
(il, rn1 ~I’, rn
I ~(l +
1’)
+ m~)
m2 ~-1’,m~I~(l+l’)—k, m2+m~)
n~~j’, n ~(i+j’) ~2;
~j’, n~(j+j’)
—k, m~+m) -k,
m2+m2).
(45)
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681
An eigenvector I 0) of V with n fields and 1 gradients VI0)=aIO)
(46)
thus corresponds to an eigenoperator in one-loop order with the anomalous dimension AEN8+0(c)
(47)
and the full scaling dimension y=4_n_1_e(1_~+
N+8)+0).
(48)
6. The spectrum-generating operator V V is a hermitian operator according to (44) and (45). Its eigenvalues are real and thus anomalous dimensions and critical exponents of eigenoperators in oneloop order are real numbers in spite of operator mixing. This property is not manifest from the beginning [12]. Notice that the hermiticity of V singles out a natural metric on This gives a well-defined meaning to topological terms like “small perturbation from the nontrivial fixed point”. Let us discuss the symmetries of V. The internal 0(N) symmetry of the N-vector theory is obviously also a symmetry of the operator V. In order to discuss the spatial symmetries of V it is sufficient to notice that due to (5) the momentum operators P~ ~ commute with V, ~‘.
=
—
[P~,v]=o.
(49)
In terms of creation and annihilation operators one has P1=D4—D1,
P2=D3+D2,
P3=i(D3—D2),
P4=—i(D4+D1), (50)
with 2m2~/2)ta~m1,m2),
D1
=
d1 (l;m~
~(~i
1
—
rn1)(~l+ 1
—
m2) alm1/
~/(~l + 1
—
m1)(~I+ 1
+
m2)
+
1m2) 2,m2+l/2)ta~m1,m2),
D2
=
d I
~ (1;m
am1/
1m2) 2~m2/2)ta~m1m2),
D3= d~
I (l;m~ 1m2)
~/(~l+ 1 +m1)(~I+1
_m2)aIm1+l/
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S.K Kehrein et al.
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~
/
Conformal symmetry and anomalous dimensions
~/(~I+ 1 +m1)(~l+1
I (l;m
2.m2±I/2)ta~m1~m2). (51) +m2)a~m1±l/
1m2)
One can also check the commutation relations (49) directly in a lengthy calculation. Since V is a hermitian operator the adjoint operators D~commute with V too, and so does the entire algebra generated by the commutators of and D~.It is convenient to consider the anti-adjoint momentum operators K,~ —P~(special conformal transformations) =
K1= —(D~---Dfl,
K2= —(D~+D~),
K3
K4
=
i(Dt3
D~),
—
=
—i(D~+ Dfl
(52)
and the algebra generated in this way is the conformal algebra S0(5,1) in four dimensions: [Pu,
for all ~.t 1,.. defined as =
. ,
K,~I
=
—2iD,
[D,
P~J=iP~,
[D,
K,~I
=
—iK~,
(53)
4 (no summation is implied here) with the dilatation operator D
N
D
=
i ~
~
(1 +
(54)
1)am1~m2)ta~m1.m2).
d1 (l;m1,m2)
The dilatation operator counts the number of gradients plus the number of fields, that is the critical exponent in order e°.The commutators [P,~,K~]for ~ Il yield another six generators corresponding to the S0(4) subalgebra of the 15-dimensional algebra S0(5,1). The additional Z2-symmetry due to the exchange of both S0(3) sectors extends the symmetry group to 0(5,1). Two good quantum numbers for composite operators are obviously the number of fields n and the total number of gradients 1 (eigenvalue of the operator iD n). n and 1 will always have these meanings in the sequel. It is remarkable that conformal symmetry to one-loop order is a consequence of hermiticity of V and translation invariance of the lagrangian alone. We did not have to calculate any commutators with V explicitly. For later purposes it will be j.t
—
—
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Conformal symmetry and anomalous dimensions
683
convenient to have another expression of V in terms of the generators P~and K~. From (A.5) in the appendix one establishes H~~j~l2)(adP) ((fad
(L-f 1 —k)!k! 1L—2k+1~’ I
P)2) k1 a~°~O)t4°~O°)t
I
~ ~L,1+1’ 1,m~)(1’;mj,m~)
(l;m
=
x (fl, rn1 fi’, m~I -
—
k, M1)(fI, m2 fl’, m~I
—
k, M2) (55)
amI,m2)ta~,mi,m2)t
2[ ~ where [P ad P,j X] denotes the commutator [P,~, X] and e.g. (ad P1) 1, [P1, a~OO)ta~O.°)t]]. A similar relation holds for the generators of special conformal transformation K~.Inserting everything in (44) we obtains =
6dd’ + ~ed~e’d’
N
V=
f d,d’,e,e’= ~ 1 (~ee’ [L/2]
+
L+M
—
1±M2(’~ —
~
(L;M
2k)!(L
—
2k
+
2k’2
1)!
~L+1—k~’
1,M2) k~0
,‘.
2) k [~oo0)t~ooo)t] H~1j~2)(ad P) ((fad
x HLI’
P)
~(ad K)((f ad
K)2)[a~0~0)a~0~0)]
(56)
7. General properties of the spectrum of anomalous dimensions It is straightforward to conclude from (56) that V is a positive semi-definite operator. Similarly one proves the upper bound N
V~ f
~
(8ee’~dd’ + ~~d~e’~’ + ~ed’6e’d)
d,d’,e,e”=l
~
X (l;mI,m2)
a(lm1,m2)tam1m2)tam1,m2)a~m1,m2)
(57)
(1’;m~,m~)
in a simple calculation. It is easy to see by explicit construction that for composite operators with n elementary fields, the largest eigenvalue a of the operator on the
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Conformal symmetry and anomalous dimensions
right-hand side of (57) is given by fn(3n+N—4) —
The anomalous dimension
-
1)(3n +N— 1)
forevenn
(58)
for odd n.
A in the full dimension (48)
y=4—n—1—e(1—fn+A)+0(2)
(59)
therefore obeys 1 0~A~
for even n
1 2(N+8)~3~~’1~
(60)
foroddn.
The upper limit is just reached by the operators (42)~? and (~2)’~’4~ or total derivatives thereof (compare eq. (33)). Due to the lower limit in (60) one concludes that one-loop order contributions make all operators “more” irrelevant in contrast to the result (1) for the N-vector model in 2 + dimensions [13]. The stability of the nontrivial fixed point is even improved by one-loop contributions in 4 dimensions. The conformal symmetry group of V yields another simplification of the spectrum of anomalous dimensions. It is well-known that the spectrum of conformal invariant hamiltonians is generated by conformal invariant eigenvectors I i/i) [5,6], —
VI~i)=aj~/i) and
K~I~/j)=0Vj~t.=1,...,4.
(61)
All eigenoperators are either conformal invariant or derivatives of a conformal invariant eigenoperator (primary operator) reproducing the anomalous dimension of this primary operator. Since we have restricted the space of local operators to composite operators in it is actually more precise to speak of conformal invariant operators up to terms containing 1.1k or derivatives thereof. In order to derive the spectrum of anomalous dimensions it is obviously possible to restrict the diagonalization of V to the subspace of conformal invariant operators obeying K,~I~r)=0. ~‘,
8. Solutions for the cases of two and three fields It is manifest that V annihilates all tensors that are completely antisymmetric with respect to the internal 0(N) symmetry. Anomalous dimensions of these
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685
tensors vanish for any number of fields n. We will not discuss this trivial class of operators here. The diagonalization of V can be restricted to the space of conformal invariant operators. Acting on a conformal invariant operator I ~F>, the operator V from (56) assumes a simpler form N
V I il’)
=
~
6dd’ + ‘3ed6e’d’ + ~ed’~e’d)
d,d’,e,e’=l
(~ee’
(L—2k\’
EL/2] x
~
~
(pL±MI±M2
‘
I.
(L+1—kUkt
(L;M 1,M2) k0 X
~
/.
~LJ+J’(~J,
n1
fj’, n~I
—
k, M1)(fj, n2
f.i’,
n~I
—
k, M2)
(j;n1,n2) (j’;n,n’2)
a~00~a~0’0~ I ~).
2)
X
a
(62)
,fl2)ta’~fl~)t . H~~,_M2)(K)((fK)
Notice that two annihilators for fields 4, without gradients act directly on I II’). The spectrum of anomalous dimensions for operators with n 2 fields and gradients is well-known [16]. It is easy to see that these results are consistent with the approach here. There are only two conformal invariant eigenoperators with nonvanishing eigenvalues: (1) The 0(N) scalar operator •2 with eigenvalue N + 2 and a full dimension y = 2— (N + 2)/(N + 8) + 0(c2). (2) The second-rank symmetric and traceless 0(N) tensor with eigenvalue 2 and a full dimension y 2 2/(N + 8) + 0(2). This eigenoperator does not exist in scalar (N 1) 4~theory. Conformal invariant operators with n 2 fields and 1> 0 are annihilated by (62) and thus have vanishing anomalous dimensions in order e. It is easy to see that for an even number of gradients I there exists one 0(N) symmetric traceless and one 0(N) scalar tensor that are conformal invariant. No conformal invariant operators exist for odd I (except completely antisymmetric with respect to the internal symmetry). These conformal invariant eigenoperators correspond to S0(4) symmetric tensors of even rank 1. For I 2 one finds e.g. the stress tensor. It is well-known that except for the stress tensor, these conformal invariant operators acquire anomalous dimensions in order ~2 [16]. We can also work out the complete spectrum of anomalous dimensions for n 3 fields. The two annihilators a~°’°~a~°’°~ in (62) effectively reduce the problem to a one-particle problem that can be solved easily. In order to concentrate on the essential points, let us consider the scalar (N 1) theory first. =
~
=
—
.
=
=
=
=
=
—
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By explicit construction one can show that at least one conformal invariant operator with three fields exists for every 1 ~ 1. We have to distinguish two cases: (1) The conformal invariant operator contains a term ~00,O 0;0,0)n~(l;m,m 2) All other terms are annihilated by V according to (62) and one calculates
V(~(o;o.o)~(o;o.o)ck(l;m1.m2))=
3(1
+
+
(
—
1)~’
)
~p(O;OO)~(O;OO)~(l;mIm2)
other terms.
(63)
The contribution independent of 1 comes from the term (L; M1, M2) (0; 0, 0) in (62) and the factor (~ 1)~2/(l + 1) from (L; M1, M2) (1; m1, m2). Thus we find conformal invariant eigenoperators with anomalous dimensions =
=
VI~l.
(64)
One can check that the spin structure of these operators is (fI, fI). They exist for any number of gradients except for 1 1. (2) There is another class of conformal invariant eigenoperators that becomes especially important for larger values of I (their number increases faster than linear with 1): These conformal invariant operators do not contain a term ~,(0;0,0)~1~(O;0,0)~(/;m1,m2) Therefore the anomalous dimension of these operators vanishes in one-loop order. The first operator of this type that one encounters has 1 4 with the spin structure (0, 2) Notice that there are also tensors with the spin structure (f1, f1) in this class. The first operator with this structure has for example 1 6 gradients ** as one can see by doing a diagonalization of V in this subspace using a computer. This yields the interesting feature that 0(4) scalar operators with three fields and vanishing anomalous dimensions exist (the first with 12 gradients). For N> 1 some additional complications arise, though nothing really modifies the general structure that we have found in the scalar theory. There is again the important class of conformal invariant operators that do not contain the term ~ and therefore yield vanishing anomalous =
=
~.
=
2—1,— l),p(2l.— 1)_ *
+
**
~~(O;°.°)cp(2O—1)~~(2;O,— I)
—
~(L;1/2,— 1/2)~1;1/2,—1/2)~(2;—1.—i)
= ~ ~,(O;O,O)~p(1/2)~J~(I;_ 1/2,—1/2)~(2;O,—I) — ~p(i;—1/2,— I/2)~p(l;—1/2,— I/2)~p(2I.—1) 0 = h~ ~, 6[3~~d,,2cba,~“~*6~ ~3..~ 8* ~ ~ — ~ . . + 34~8~ . . . c 3~ 4. ~ with any completely symmetric and traceless tensor ~
S.K. Kehrein et al.
/
Conformal symmetry and anomalous dimensions
687
dimensions in one-loop order. For the nonzero eigenvalues one evaluates
=
.
~(0;0,O)~(1;m~,m~)
+
(
—
~
.
+ 2~~oO,o)t~9;o,o)~D(l;m1,m2)
+
(
+
other terms.
—
1)’—_~~— ~
+~
.
(65)
Now we have to distinguish three classes of conformal invariant operators corresponding to the 0(N) transformation properties with respect to the internal indices. We write 0 5d1d2d3~QcQ~pdl12)~I2i2)tp~3I2) where the sum EQ =
runs over S0(4) quantum numbers, and there are three possibilities for the tensors Sd
d2d2:
(1) Completely symmetric and traceless tensors S~1~2~3: The respective anoma-
lous dimensions are VI~1.
(66)
(2) Antisymmetric with respect to two indices and traceless: The anomalous dimensions are VI~1.
(67)
(3) Tensors S~1~ ~, with a nonvanishing trace: This case leads to the diagonalization of the matrix N+2 (N+2)(b1—1) with b1
=
1
+ (~1)1.
2/(I
+
b1 2b1
1). Thus the anomalous dimensions are
A±=2(N+8)[N+2+2bl±~/(N+2)2+4b~+4(N+2)bl(bl_2)1
+0(E2).
(68)
688
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Conformal symmetry and anomalous dimensions
Notice that for 1 0, 1 (or for N 1) only one of the solutions A corresponds to a nonzero eigenoperator. We were not yet able to derive a similar complete classification for the spectrum of anomalous dimensions for n > 3 fields. E.g. even for four fields in the scalar theory, one finds conformal invariant operators with nonrational anomalous dimensions. The solution for n 3 relies heavily on the conformal symmetry which allows a complete classification. This sounds familiar from three-point functions in conformal field theories, but cannot directly be carried further than three fields. Work on the spectrum for n> 3 is in progress. =
=
±
=
9. Conclusions and outlook We will briefly sum up the main results of this paper. We have discussed the N-vector model in 4 c dimensions in one-loop order. It was shown that the pole contribution of an operator insertion K 0) for a general composite operator 0 with n elementary fields and 1 gradients can be written as the action of a hermitian operator V on 0. The operator 0 is represented as a vector in a Hilbert space of composite operators and V has formally the structure of the two-particle interaction potential (44) acting on this space. Thus the eigenvectors of V correspond to scaling eigenoperators and the eigenvalues correspond to the respective anomalous dimensions. The diagonalization of V in an n-particle space with a given total number of gradients yields the spectrum of anomalous dimensions in this class of mixing composite operators. Obviously anomalous dimensions of eigenoperators are therefore real numbers in spite of operator mixing. By establishing upper and lower bounds for the spectrum-generating operator V, we found upper and lower limits (60) for the spectrum of anomalous dimensions. Since all anomalous dimensions are positive due to (60), one-loop order contributions make all canonically irrelevant operators even more irrelevant. One does not encounter the unusual behaviour of some 2 + expansions where the stability of the nontrivial fixed point is endangered by high-gradient operators in one-loop order [3,4,13,14,7]. The spatial symmetry group of V was found to be the conformal group in four dimensions 0(5,1). This conformal invariance should be expected from general grounds (compare e.g. ref. [8]). The restriction of the diagonalization of V to the subspace of conformal invariant operators allows a complete classification of the spectrum of anomalous dimensions for composite operators consisting of three elementary fields with an arbitrary number of gradients. This extends the wellknown solution by Wilson and Kogut [16] for two fields and gradients. It is quite remarkable that for a sufficiently large number of gradients, the spectrum for three elementary fields is dominated by the eigenvalue 0. —
11~
~‘
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Conformal symmetry and anomalous dimensions
689
The complete spectrum of V for more than three fields has not yet been worked out. Already the case of n 4 fields seems to possess the complexity of the general problem. Apart from this diagonalization problem, it should be interesting to extend the present work to two-loop order. Work on these problems is in progress. =
One of the authors (S.K.K.) would like to thank H.D. Conradi and K.J. Wiese for valuable discussions.
Appendix A. Properties of SO(4) harmonic polynomials S0(4) harmonic polynomials
Hfml,m2)(x),
m12
—1/2,—
=
1/2 + 1,.. .,1/2 of
degree 1 can be constructed using a generating functional 1/2 =
2Sm2+t/2
~(m1
ml,m =—i/2
±I/2)(m2 +1/2)
H/ml.m2)(X)tml÷t/
(A.l) with four-vectors x
=
(x
1, x2, x3, x4) and u (1 its, it is, t + s, 1 + ts) (compare ref. [2]). The polynomials Hf m l.m2)(.~) provide a complete set of functions on the unit sphere S3. Orthogonality and normalization conditions can be deduced from ref. [2], =
fdf~ ~
—
—
—
= 2~2~ii~5m m’~rnm’
4i(l
where Is3 dQ is the surface integral over S3 and conjugate of Hf m~,m~)(X): Hf,mc~~~~)( x) =
(—
~
—
J~If,m~~mi)(x) is
(A.2) the complex
(A.3)
m~m~)(x).
We need the following two theorems for the polynomials H/rnI,m2kx): Coupling rule: Hfml.m2)( x)Hfm~.m~( x) [L/2]
= k0
(L—2k)’ 1!1’!
(fi,
m1 fi’, rn’1 fL
XH/,~~2)(x). (~x2)”,
I
—
k, M1)(fI, m2 fi’, rn’2 I fL
—
k,
-~~~2)
(A.4)
690
/
S.K Kehrein et al.
where L
1’, M12 m12 + m~2and the brackets on the right-hand side denote S0(3) Clebsch—Gordan coefficients (j, m; j’, rn’ I J, M). Decoupling rule: =
I
Conformal symmetry and anomalous dimensions
+
=
H~~2)(x1+x2)
.
(Xi.X2)k
(L +1 —k)!k! =
(L
—
2k +
n1
5L,j±j’
/ - (j;n 1~’ 1,n2)
ff’,
~ (j’;n~,n~)
n~IfL —k, M1)(fj, n2 +x~ (
XHJ~1’~2)(x1)I-IJi’i”12)(x2)
...)
fj’,
n~IfL—k, M2)
+x~ (
(A.5)
...).
We will prove (A.4) first. Consider the surface integral over the unit sphere S3,
f dQ(u1 .x)1(u2 Ox)’ (v
(A.6)
.x)L_2k,
S3
with u1, u2 and v defined similar to (A.1), =
(i
—
it1s1,~it1
u2
=
(i
—
it252,
v
=
(i
—
iTT~_
—
it2
iT
—
—
is1,— ~1+ s~,1 ~
io,— T
t2 +52,
1
+ o,
+
1
t1s1),
+
t2s2), (A.7)
+ TcT).
The integral (A.6) can only depend on euclidean invariants made up of u1, u2 and v. The only possibility is
f
1 (v
1(u
dfl(u1 ~x) S3
.x)L2k
=
C(l, 1’; k)(u 1 u2)k(u1
2-x)
.
~)l_k(u2
.
(A.8) with a factor C(l, 1’; k) and L I + 1’. Expanding the scalar products on the right-hand side of (A.8), employing the definition (A.1) of the polynomials Hfml.m2)(x) on the left-hand side and comparing coefficients of t1, s~,12, ~2’ T, U =
S.K Kehrein et al.
/
Conformal symmetry and anomalous dimensions
691
yields after a straightforward calculation
f dQ
HfmIm2)(x)Hicm~.m~)(x)H~~2)(x)
S
3 I.
—
‘.
I
‘.
‘
~‘
‘
2
Ic.
2’—~~l!I’!
(L—k+1\/(L—2k k k )/~i—k
x(fi, rn1 fI’, m~IfL —k, M1)(fl, m2 fi’, m~IfL —k, M2). (A.9) By choosing the specifically simple set of magnetic quantum numbers rn1 m2 —1/2, m~ m~ —l’/2 one can explicitly work out the integral (A.9) and obtain the unknown factor C(l, 1’; k), =
=
=
=
2 C(i, I’; k)
=
2~
l!l’!(L —2k)!
2L
k(l
—
k)!(l’
—
k) !(L
—
k
+
1) !kL
(A.10)
Inserting this back into (A.9) and comparing with the orthogonality relation (A.2) proves the theorem. In order to derive the decoupling rule (A.5) we will first consider the special case k 0 in (A.5), =
~ L
~,X1
— X21
—
V ~.s (j;n1,n2) (j’;n,n~)
(fi, ~
~ L,j±j’
fj’, n~I fL, M1)(fj,
~2;
X H~~”2)(x1)HJi~”~2)( x2).
fj’, n’~IfL,
M2) (A.11)
Now (u (x1 +x2))L .
=
~
(~
)(u .x)~(u~
(A.12)
with u from (A.1). Application of the definition (A.1) for the polynomials identification of the coefficients proves (A.11). The general theorem (A.5) follows from (A.11) by induction over k. It is a somehow tedious but straightforward exercise where one makes use of 2”2~( x 2”/2~(x x1 x2 = H~l/2._l/2)( x1) H~’~’ 2) + H1”/ 1) H~1/2,—1/2)( x2) 2’ 1/2)( x 2’’/2~(x Hf 1/2.1/2)( x1) H~” 2) H/ 1) H~1/2.1/2)( x2) (A.13) H/?~~l.m2)(x) and
-
—
—
and frequently employs the coupling rule (A.4) in the special case that one of the coupling spins equals 1.
692
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Conformal symmetry and anomalous dimensions
References [1] G.E. Castilla and S. Chakravarty, Preprint University of California, Los Angeles (1992) [2] A. Erdélyl, W. Magnus, F. Oberhettinger and F.G. Tricomi, Higher transcendental functions, vol. 2 (McGraw-Hill, New York, 1953) [3] yE. Kravtsov, IV. Lerner and VI. Yudson, Zh. Eksp. Teor. Fiz. 94 (1988) 255; Soy. Phys. JETP
67 (1988) 1441 [4] yE. Kravtsov, LV. Lerner and V.!. Yudson, Phys. Lett. A134 (1989) 245 [5] M. Lüscher and G. Mack, Commun. Math. Phys. 41(1975) 203
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(Academic Press, New York,
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[131 F.J. Wegner, Z. Phys. B78 (1990) 33 [14] F.J. Wegner, NucI. Phys. B354 (1991) 441 [15] KG. Wilson, Phys. Rev. Lett. 28 (1972) 548 [16] K.G. Wilson and J. Kogut, Phys. Rep. 12 (1974) 75