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Anomalous dimensions of relevant operators in the O(M)-symmetric ϑ2N-theory
Nuclear Physics B350 (1991) 789-817 North-Holland
ANOMALOUS DIMENSIONS OF RELEVANT OPERATORS IN THE O(M)-SYMMETRIC +2 kTHEORY Jiirgen HOFMANN
Institut für Theoretische Physik, Ruprecht-Karls-Universität, D-~Heidelberg; Gernwny Received 28 May 1990 (Revised 6 August 1990) The calculation of the critical exponent q for the class of OM-symmetric 02N-theories is presented to order O(E ;) in an E-expansion around the upper critical dimension d c - 2N/ (N - 1) using field theoretic methods. For the case of a scalar order parameter 0 a comparison of the exponent q obtained by an extrapolation of the E-expansion to two dimensions with the exact result pertaining to conformal invariance is provided. Basically it is found that the ratio of the O(E 3 )-correction and the O(E 2 ) result is a monotonically increasing function of N diverging exponentially as N tends to infinity thereby suggesting an approach toward the exact value. A further modification affects the behavior of q in the spherical limit M - ac . Additionally the exponent (w as well as the anomalous dimensions of the relevant operators (02)N' with 1 < N' < N - 1 are determined to order O(E 2 ) throughout. In those cases, however, the various contributions to the E-expansion seem to be alternating in sign.
l. Introduction Several years ago Wegner [1] calculated the lowest-order contribution to the ,E-expansion of the critical exponent q for the class of O(M)-symmetric 02 Ntheories which describe the critical behavior of a physical system in the vicinity of its multicritical point [2]. Here E := dc - d denotes the difference between the spacial dimension d of the system and the upper critical dimension d c == 2 N/(N - 1). It is well known that the result for q extrapolated to two dimensions exhibits an exponential decay in the limit N --1- oo at least when restricted to the scalar case M = 1 . On the other hand the two-dimensional situation is known to be special since conformal invariance is powerful enough to allow for a complete determination of the critical indices. The notion of conformal invariance in statistical physics was originally due to Polyakov [3] who suggested that a statistical mechanics system at its critical point not only be invariant under a global change of *This work was supported in part by the Sonderforschungsbereich 123 Stochastic Mathematical Models of the Deutsche Forschungsgemeinschaft . 0550-3213/91/$03 .50 0 1991 - Elsevier Science Publishers B.V. (North-Holland)
7190
J. Hofmann / Anomalous dimensions
scale but also under the more general group of conformal transformations. Unfortunately in dimensions higher than two the conformal group is only finite dimensional thus giving us not much more information than is implied by ordinary scale invariance. On the contrary in two dimensions any analytic function generates a conformal mapping thereby rendering the conformal group infinite dimensional. In a series of papers [4,51 it was shown that the implications are so strong that in some cases the critical indices and the various correlation functions may be calculated exactly. Especially it was shown by Cardy [6] that there exists a unique correspondence between the scalar version of the above-mentioned .02N-theories considered in the present paper and the minimal unitary series of conformal field theories. Additionally the exact expression for the exponent q leads to a very weak decay q - N- 2 in the limit N -+ w which strongly contrasts the behavior found via extrapolating the expansion. In a previous paper [10] Howe and West estimated the renormalization group beta-function to third order in the coupling constant for the scalar (A2N-theory in the limiting case of large values of N by making use of Lipatov's considerations about those diagrams that contribute most at a given order [11]. From their result they suggested that the lowest-order terms in the E-expansion of the critical exponents are not the leading contributions. These authors, however, do not give any results for the critical exponents to higher order. In the present paper the rigorous calculation of the critical exponent q for the O(M)-symmetric 42N-theories up to terms of order ME') is presented using field theoretic methods [7,81. No restriction to the case of a scalar order parameter will be made despite the fact that we do not expect a phase transition for M > 2 as a result of the Mermin-Wagner theorem [9]. Even though the final result turns out to be rather complicated it may easily be recognized that for M = 1 the ME 3)-contribution yields an enormous correction so that it must not be neglected. Basically it is found that in the result q = qi E 2 + 77 2 E3 + ME') the ratio 712/71, diverges exponentially to infinity as N -), x thereby providing concise analytic expressions to the suggestion made by Howe and West [10]. A further modification of current results concerns the behavior of q in the spherical limit M -3. oo . The organization of the current paper is as follows: In sect. 2 the action for the field-theoretic models under consideration is introduced. It also contains an account of the basic results including the situation in two dimensions. In sect. 3 a brief outline of the calculation of the exponent q is given. The result for the exponent w that governs the deviations from scaling is presented to order O(E2). Sect. 4 is devoted to a study of the scalar case M = 1. Sect. 5 extends the calculations of sect . 3 to a determination of the anomalous dimensions of the isotropic relevant operators (42)N' for 1 < N' < N - 1. In sect. 6 some conclusions are drawn . The paper closes with an appendix that contains a complete list of the various tensorial contractions which come along with the required Feynman
J. Hofmann / Anomalous dimensions
1
diagrams . The list is supplied by enumerating the e-expansions of the inter associated with them to the order they are needed.
2. The model
The 4a-theory which is usually considered in calculating the critical properties of physical systems may be generalized to a 0 2 '-theory to incorporate multicritical points of order N [2]. In the present paper the critical exponents are examined that follow for the class of O(M)-symmetric 02N-theories described by the bare euclidean action:
S141 with
=
02 :=
d
i I=1 (a oÎ2 + IIL
ddx
J
N-1
2+
2' (02 )' +
;-2 (21)!
2N (2N)! (102)N
(1)
M
~ 0i .
,j=1
For N = 2 the 0a-theory of the ordinary critical point is recovered which has been extensively investigated and for which calculations have been carried to very high order [7]. The case N = 3 belongs to tricritical phenomena about which a review was given by Lawrie and Sarbach [12] who also presented the lowest-order results for the tricritical exponents. Some of these were extended to the next order by Lewis and Adams [13]. Wegner [1] obtained the exponent q to lowest order for general values of N and M: 77
2 3 __ 4N (N - 1 ) ( N! ) T(±M +N) 2[(2N) ! l 2
T ( 2M + 1 )
1
~(2N)(M)
2
E 2 + O(E3) ,
where ~QN)(M) is one example of a tensorial contraction and may be found in the appendix. For M = 1 the result (2) shows the asymptotic behavior: .1
N--
1
16zr(TrN )312 x 64 -N
when extrapolated to two dimensions. In two dimensions the scalar version of the theory (1) coincides with the unitary minimal model with central charge [6]:
The fusion rules [4,14] then yield the correspondence between the relevant
!. Hofmann / Anomalous dimensions
operators O' with 1
for N
4P2N H
From this identification the scaling dimensions of the relevant operators are obtained as a result of the Kac formula [6,14]:
2N'( N' + 1) (N+ 1)(N+ 2) ' d,~2N .
0
d,2N .
=
2(N'+ 1)(N' + 2) (N+ 1)(N+2) '
=2
N+3 N+ 1 '
for N < 2N'< 2 N - 2, for 2N'= 2N .
(4)
In particular the critical exponent rl, which is connected with the anomalous dimension of the 0 - or 02,2 - field, is given by
rl=2de=
3
(N+1)(N+2)
One notices the weak decay of rl in the limit N -> oc which strongly contrasts (3). In eq. (4) the whole set of operators has to be split into three groups for each of which a separate formula may be given. It is nice that the same feature appears in the calculation of these quantities within the E-expansion presented in sect. 5 where it creeps in as a consequence of the Feynman diagrams .
3. Outline of the calculation . The exponents q and w The calculation of critical exponents proceeds most economically when performed directly at the multicritical point . Hence the bare mass 1,C as well as the coupling constants A21 with 1 < j < N - 1 appearing in (1) are no independent quantities. The index c attached to them indicates that they attain their critical values as functions of J1 2N in order that the corresponding renormalized counterparts vanish . This choice of dealing with the critical theory is rather convenient
!. Hofmann / Anomalous dimensions
793
TABLE 1
Feynman diagrams for ri(,2;(p) to order O(A2N). To every external line an index i is a which is derived from the M-vector structure of the theory whereas the numbers placed near the bubbles indicate the number of lines running through them. The quantities appearing in the last two columns are listed in appendix A. These conventions apply as well to tables 2,4-6. The wiggly lines appearing in the diagrams of tables 4-6 denote insertions of an operator (42)N' with momentum k Symmetry factor
Diagram
Tensorial contraction
1
P 2N-1
Integral P2
1 (2N-1)!
(2N) . D,~," _~(M)S-r1r,
1
eé N )(M)8; J;,
(N- 1)!(N!)2
N~
DZN
B( 2N k
P)
P)
1V-1
since merely diagrams which contain vertices of type 2N exclusively have to be considered . After eliminating all vertices of type different from 2 N one is left with the calculation of the 2-point and the 2N-point vertex. The relevant Feynman diagrams contributing up to order O(Jl2N ) are enlisted in tables 1 and 2. Along with them some terminology is introduced concerning the associated integral expressions and also the tensorial contractions which are due to the M-vector structure of the theory. All of these are explicitly given in the appendix. To remove the divergencies, dimensional regularization is used in conjunction with a set of normalization conditions: rRita ape r(2) R,,,,.
i2
p) P
(0)
for 1
=0,
2-K_
(
rRi ) .i2nr( pi) ISP = UK N
1)eTi,2N~2N'
794
A Hofmann / Anomalous dimenNions
= ~ Q6 * . . . *
~~ ~
. ^
. , . . ~
t!~l Q6
r.
8
k fli
une
LQ
Q
.
.
;e ~ ^=
A~t wd
~~
In
»n
»0)
! ~~ ! ~~ «~~
-
!
!
rv
m
~t
"=
I tdo
ann 1 An
alons dimensions
795
where the tensor T'' ~, , is also defined in appendix A. The quantity dimensional parameter that derives from a symmetry point (SP) given by SP : Pi -pj =
s
K
is a
- ic2(sâ"~ -1), 1
in which s is usually taken to be 2 N but will not be explicitly specified in the following. After rather lengthy calculations the renormalization group beta-function P(v) and the scaling function y,(t) of the O-field are obtained in a double expansion in E and the renormalized dimensionless coupling constant r." : .8(v) = - (N - 1)v[E - (a; +b3 E + 0(4E2)) . L' +2(Nß,+b4 -2a,b3 +O(E)) . 17 2 1 y4(v)
+O(L' 4 ),
= -(N- 1)11,2 [(2a, + 2b,E + O(E2)) + (4a,b 3 + 4b,a 3 - 3b2 + 0('E))-VI + O(c,4) .
The various quantities appearing in these equations are declared through _ _
_ 2(N- 1) 2 (2N) D,N-~(M)' alT (2N) !
a 3 'b2
b
2 N!
y(2N)( Tb
N-1 2N
( N!)
7
M)
a2VN'
4
2 1 9 NI
8 N -1
R
N -1
b,
2N
T3
a,UN
(
) 2N (M)
'
N- 1
b3 :=2 a3WN [N121 Ni=,
1
) [2WN + EN(l)] N S
ON (NI, N2) S(2N M) Ni , N2( ) NI !N2 !(2N -NI - N2)!
M
(12)
Hofm
/ Anomalous dimensions
with
A := 1l(N- 1),
T== (4ar)-N(r(A))N-1,
( 13) (14)
UN :=(2N-1)(N+ 1) +2N(log47r-C),
N :=3N2 +2N-2+3N(log47r-C),
(15)
~-N N :=N - C+log4r-log N ( S-1),
(16)
~N(NI) -N- 1-C+ jr(A) -«AN,) - «A(N -NI », N( N.'
N2)
:=
(17)
F(A)r(A(N-N,)) r(A(N - N2))I'(A(N, +N2 -N)) . r(AN,)r(AN2 )r(A(2N-N, -N2))
18
Herein C and (z) denote Euler's constant and the digamma function respectively. It should be mentioned that in eq. (17) the constant C always drops out. This is due to a formula of Gauss [151 that applies to "the psi-function in case of rational argument. The expansions of the two renormalization group functions given above together with the nontrivial zero v* of ß(0 then yield the critical exponent 71 to order O(E 3 ): (t,*) = = Y4 iIE 2 + 712E 3 + O(E4) with -2(N - 1)(a,la3) , 772 = -
N -1 4
a3
( 19)
[8Na1 + 8a,b4 - 3a 3 b 2 + 6a23b, - 16a,a 3b3 l .
(20)
Now, substituting (7) and (8) into (19) leads back to eq. (2) thus reproducing Wegner's result. Some more algebra is involved in obtaining the new coefficient 712 through introducing eqs. (7)-(18) into (20): *12
1)2(N!) 4 = (N -( 2N ) ! (N-
(
1 )2 )
2N(N - 1) (2N) i
~ (N),(M)
[C(2N)(M)j 4
~(2N)
M 2+
1-
N
-1 ( )
[N121 Ni-1
~(2N)
D2N-~(
N ~
M)
N
S(2w)
ON( N, 1 ry2) S(2N) _ N ) i NI , N2( M) !N2 !(2N N N1 " " 1 2
M
!. Hofinann / Anomalous °
It should be remarked that in the process of deriving this result various tions occur. Actually these cancellations are necessary in order to obtain finite renormalization group functions. They are due--to certain relations among the tensorial contractions which are given in eqs. (A.3) and (A.4) of appendix A. These
same relations are responsible for the contributions depending on the symmetry point SP to drop out of the final expressions which descries physical ` of the theory. In fact all terms containing the variable s vanished in the result for t exponent 71 . For some low values of 1V the above formula in conjunction with the to contractions listed in the appendix yields the following special results far is
7I(N =2) =
M+2 2 E21 6(3M + 14) + 2 2(M+8) ( M+8)
1
4
E
+O(E 4),
(M+2) (M+4) M3 +34M2 + 620M +2720 E2 1 + ~( N = 3) = îi2 12(3M + 22) 2 8(3M+ 22) 2 141M 2 +2310M+8944 + 3 (M+2)(M +4)(M+6) 2 (3M 2 + 150M + 1072)
3(3M+ 22)2 712
)
E
I
4 + 0(1E )
711
with
112 19,
9 87M4 + 10140M 3 +355452M 2 + 4 146 240M + 15 298 816 8 (3M 2 + 150M + 1072)2 -
18[ vC 3 - 3 log 31
M4 + 64M3 + 2264M2 + 26 936M+ 99 360 2 (3M 2 + 150M + 1072)
3 15M3 + 636M 2 + 8 360M + 33 864 + 64(T(3)) 2 (3M 2 + 150M + 1072) In the first two cases the results coincide with the exponent 71 for the 04-theory and the 06-theory respectively given by Brezin et al. [7] and Lewis and Adams [13] respectively .
Hofmann / Anomalous dimensions
Another modification of the lowest order result for q concerns the spherical limit M -* ac which was noticed by Wegner [1] to be of the form _
~im
0 c,(N) IE2
for N even for N odd
to order O(E2 ). The coefficient q(N) is independent of M. A detailed study of the tensorial contraction SN,N~,(M) shows that it is a polynomial in M of degree N or N - 1 according to whether N, and N, are both even or at least one of them is odd. After some algebra this leads to a modified behavior of the exponent rl in the spherical limit to order O(E 3 ) : _
m M
0 c,(N) E' + c2(N) - ME'
for N even for N odd '
where c2(N) is also independent of M. Another one of the critical indices which may be given with the aid of the renormalization group functions worked out so far is the exponent co that governs the corrections to the leading scaling behavior . It is simply the derivative of the beta-function at the nontrivial zero . A short calculation yields the following order O(E 2 ) result: cù :-
dß(t)
- ( N - 1)E + ui E 2 + O(E3)
(21)
with co, given by
-
(N-
1
1)(N!) 2
2
[~(M)(M)]2
2N (N - 1 ) (2N) .
e(2N) 2N
-
(M)
N- 1
~ N12~ (
(Ni) 2
NI-1
N)
Nl
ON (NI, 1-
1b I
NI +N2 :A N
1
N2 )
2
S (2N)
~N(Nl)
N2) 1
2
-N(M)
SN N(M)
J. Hofmann / Anomalous dïnre
For N = 2 this agrees to order et al. [7].
O(E2)
`
with the same exponent given
4. The scalar case
=1
The descent from the general outline of the preceding section to the special case of a one-component order parameter proceeds via eq. (A .2) of the appendix. Only little effort is required to establish the result: 17
=
711E2[1 + ('%2/171)E + O(E 2 )]
s
with 71, =4(N- 1)2(N!)6[(2N)! ] -3
and the ratio 712 /71 1 composed out of four pieces: 172/711 = YI(N) + Y2(N)
+ S1(N) + S2(N) .
They are respectively defined by Y1 (N)
:= -8N(N- 1)(N!)6[(2N)! ] -3,
SI(N)
:=
4(N - 1)(N!)4 (2N)!
6 _ 4( N! ) S2(N) (2N)!
f
[NI21
4
E N!1 ) 2 [(N -NI) ,-l 2~N N,-1(
(NI),
~'ON(N1,N2) N !)2 2N-N -N !)2( N
)!
2
N,+N,-*N
where the symmetry factor 0 is defined in the appendix. This expression does not coincide with the conjecture of Lewis and Adams [13] for the general case of a scalar theory of type (1) although it yields the same result for the ordinary critical and the tricritical theory. Indeed, for the lowest values of N the corresponding
d. f%fin
/ An
alous dimensions
formulae in the preceding section reduce into 1 54 -1(N = 3) = S ~(
N
109 ) 108
(
1 E2
9
- 4) - 85750
1
+
62[l
+ o(e4),
1(27 7r2 8
+
+
2 279 ) E 75
+
O( E 4 )
(5094189 343000 +
2 35
[32(T( ;))~ - 277r~ + 811og3,
E
+ O(E4),
35 815 924 100 _ 25 _ 1 (r( .1))4 _ 25 2 1 1+ n(N = 5) = (I(4) ) + 32 2 250047 E 1250235 + 63 [ 36 v 2N=
2:
;-'
E + O(E 4 ) , (r(4))2 -47r + 81og2 j)]
71 = 1/54E 2 (1 + 1 .01E + O(E2)),
N= 3:
vl = 1/500E 2 (1 + 12.7E + O(E2)),
N=4 :
v1=9/85750E 2 (1+46 .7E+O(E2 )),
N = 5:
q = 1/250047E 2 (1 + 117.1E + O(E2)),
where the result for the ip'°-theory has been added . Obviously the ratio %/'rl, increases with increasing values of N. This is strongly emphasized by the asymptotic behavior of the four constituents forming the ratio 712/71, YI(N) -
Y2( N
-
87r 3 / 2 N 7 /2 -64 -nr 2N,
S, (N) - 81og 2 - N,
S2(N) -
9
4~
T(3)
r2) (;
y N312 (K)
2N .
The sums S,(N) and S2(N) are plotted in figs. 1 and 2 up to N = 100. It is quite
Hofmann / Anomalous dérnensions
z N
J. Hofmann / Anomalous dimensions TABLE 3
Values for 71 due to the O(E-' ),ME ;)-term and the exact result (eq. (S)) N 3
-D to 0(E-') 0.074 0 0
7
-Q to 04 3 )
Exact value in d = 2
0.224 0.027 0.0015 0.000060
astounding to have S,(N) growing linearly for large N. On the other hand the sum S,(N) clearly denotes the major contribution . To conclude this section the corrections of the exponent n in two dimensions due to the O(E3 Merm are tabulated in table 3 for some values of N together with the exact results according to eq. (5).
5. Anomalous dimensions of composite operators The present section may be regarded a sequel of sect. 3 as it contains an investigation to order O(E 22) of the anomalous dimensions of the relevant isotropic operators (4 -)1' with 1 < N' < N - 1 which may be added to the action (1) as small perturbations . This amounts to examining the stability of the nontrivial fixed int t.* against elongations of the system under consideration in a direction in parameter space that corresponds to such perturbations . The anomalous dimension of the operator (,02)^'' with 1 < N'< N - 1 may be calculated when dealing with the 2N'-point vertex function containing one insertion of such an operator . In these cases all vertex functions endowed with an arbitrary number of insertions renormalize multiplicatively, and for each operator one new renormalization constant Z(& -)N " has to be introduced . This does not apply to insertions of the operator (0 - ) N since for this operator there exists 0)2 - that has the same canonical dimension. another one - which is K(N-l Hence multiplicative renormalization is lost as the two operators mix during the renormalization procedure. In the following the operator (02 )N will be excluded since insertions of this are just equivalent to a modification of the coupling constant therefore resembling corrections to scaling. But this would lead again to the exponent w already given in eq. (21) of sect. 3. A consideration of the renormalization group shows that the full dimension of ' the operator (0 2 ) N is given by Y(O=)N- = d - N'(d - 2) -
y«> > N'( U*)
for 1 < N' < N - 1,
(22)
!. Hofmann
/ Anomalous dimensions
where y(.6,)N.(V)
a log Z(¢~)
:=
.
au
is the scaling function associated with the operator (,0 2 ) N' and c-* as in sect. 3 denotes the nontrivial zero of P(r) . The dimensions y( 42 ) -v , defined in eq. (22) are connected to those given in eq. (4) via d( , b ,)N . = 2 -
y( ,O '') ,v -(
r *) .
The calculation of the 2N-point vertex function with one insertion of an operator (,02)N' requires a distinction between three different cases: .
with 2<2N'
(i)
(IA2 ) N
(ii)
2 N' (,0 )
with N<2N'<2N-2,
(iii)
2 N' (10 )
with 2N'= 2N,
the reason being certain Feynman diagrams that do not appear for all operators. This is conveyed quite clearly by tables 4-6 which display the relevant Feynman diagrams to order O(Jl2 N ). Thus, as was mentioned in sect. 2 the calculation of the anomalous dimensions via the E-expansion leads to a grouping of the various operators that is in remarkable accordance with eq. (4) pertaining to conformal field theory. In the process of calculation the finite part of the vertex function h(2 Nv(p- . k; t', K) is fixed by the renormalization condition : 1 (~ 2 )N R( Pi'
k ' ~~'
! _ K)ISP
(2N')
I l . . .i,%"'e
where k denotes the momentum of insertion and the tensorial indices on the left-hand side have been suppressed. The symmetry point SP declared through : Pi -pj SP:
s
1 - 1
K 2 (s3ij
- 1)
for 1 < i, j < 2N',
s-2 N+2N' s-1 K2 pi-k=-(2N-2N')
is an extension of eq. (6) and assures that the integrals enlisted in the appendix may be used without modifications .
J. Hofinann / Anomalous dimensions
804
Z
3
NZ
v z N N v
aN
C
ZZ
b V L
l'!
^ N z
N
Hofmann / Anomalous dimensions
805
z
a
a
z
N
+
a
Uli,
II
aa
ar
V
PJZ
z_ P9
a
' m
_
zPJ
°
Ir
PI
z
PI
3
a
a `-`1
^z z . NZ v
zN
'
:e
:e ..vs
..`/
z
z
zz PJ 1
N_G~
Z ~I
PI z z
VI VI VI
806
J. Hofinann / Anomalous dimensions
z c + zi . rl zl (~ C. C rl~ " 3
ry C r12 '~
rl~
z u
t ri
a+
z
N
O
U U i
C
Û
E
b
"!CA
NC
I y
I r~
I r~
rl '
rl '
rl '
rl '
z rl
z
2 rl
z rl~l
rl
zr_12
2? _
zz
i., U s.. rl
ar L
z
r
u
11 aa
a
c zÏ rl rl
L
^J
A i
-iz
z
Z
v
v
N
N
z
Z
z N rl z z
J. Hofmann / Anomalous dimensions
807
After quite lengthy calculations the following expressions for the anomalous dimensions of the operators appearing in groups G) and (ii) ensue: (i)2<2N'
y(02)N. = 2
with
N-N' - 1 + (N'- 1)E + TE 2 + O(E3 ) N
(N- 1)(N !)2
1
2 N'(N-1)
2
[C(2N)(M)]2
(2N)!
-
(23)
19N (NI, N2) S(2N,2N') M N,,N, NIlN2NIN2 !(2N)! 1
T262 + O(E 3 )
(24)
with (
TI
N'- 1g(2N)(M) -(N- 1)y(2N,2N')(M) y(2N)(M )
1 ( N- 1) 2 T2 ,2 ~(2N)(M) 3 [N/2l N, =1
2( N !) 2
(2N) (M)[N~(2N,2N')(M) - N'y(2N)(M)l (2N)1 bUIN_,
N [y(2N,2N')(M)S(2N~ NI(M) - C(2N)(M)S(2N, NN)(M) NI , N I , NN J EN(NI) 1
+C (2N) (M)
2N'-N N,=1
N NI
! )2 ~N(N)S(2N,2N,)(M - ( N y(2N,2N') M 1 N,, N ) N- 1 ( )
ON (NI, N2) 1-
+
( N! )
2
N-1~
(2N)
N I ! N2!(2N-NI -N2)!
S(2N) M N,, N-'( )
ON( NI 1 N2)
(M) 1 N et N2< 2N'- i NI +Nr * N, N,~N
NIl1V2!(2N--Nl
_ N )1
S(2N,2N') M ( ) Ni .N2
J. Hofmann / Anomalous dimensions
For the special case N' =1 which describes insertions of the operator 02, the critical exponent v is obtained from eq. (22) via the relation v = ;1 (i) N = 2:
v
M+2 (M+2)(M 2 +23 M+60) 1 E+ E2+O(E3), 2 + 4(M+g) 8(M +8 ;
(ü) N > 3: (N!)'
1
(N+ 1)(N- 1)3 v= + 2 2(N-2) (2N) .
D; '_(M)
E2+O(E3) .
(25)
The first result reproduces to order ME 2) the exponent v given by Brezin et al. [7] for the ordinary critical point whereas the second expression restricted to the scalar case in conjunction with the scaling relation y = v(2 - ,q) yields agreement with the exponent y given by Lewis and Adams [131 . Actually eqs. (23) and (24) are due for some more coincidences with exponents that are known for the tricritical point with M arbitrary. Setting N = 3, the variable N' is bound to take the values N' = 1 and N' = 2. In the first case eqs. (23) or (25) leads to a result for the exponent y that equals the expression given by Stephen and McCauley [161. On the other side for N' = 2 the critical index resulting from eq. (24) together y,, with the previous result for yields a crossover exponent 0, := yc,,,r/y, .2 that agrees exactly with the one reported by Lewis and Adams [13]. It should be boldly emphasized that the general expressions for the critical indices obtained in sects. 3-5 of the present paper coincide with all results calculated in special situations. As was anticipated earlier the foregoing discussion may not be extended to insertions of the operator (022 ) N since it gets mixed with another operator when renormalized . Hence ordinary multiplicative renormalization has to be replaced by the scheme : rOh
NR(pi ;k )
=Z Z111(0iN(pi'k) +Z-0Z,2Tr4~2(pj ;k),
r( <0)2 R(pj' k ) -Z4Z?1r~)"(pi ;k) +Z0Z22Tc4~2(pj ;k),
where Zl,, Z12, Z2 , and Z22 denote a set of four new renormalization constants . However, the lowest-order Feynman diagrams contributing to r N~(p . ; k) are of Tcr~r order O"2 N ). Therefore to order O(A 2 N ) the two equations decouple thus restoring multiplicative renormalizability of (02)N to that order. As a consequence
J. Hofmann / Anomalous dimensions
809
the result (24) may be applied to yield the dimension of the operator order O(E): y(,62 )N=
to
-(N--1)E+O(E2) = _w . 6. Conclusions
The calculation of the exponent 9 to order O(E 3 ) obviously shows that the corrections introduced by the O(E 3)-term are overwhelmingly large for sufficiently high values of N at least when restricted to the case of a scalar order parameter. Thus an approach is gained toward the results known in two dimensions from an exploitation of conformal invariance although the absolute numbers are still too small. Actually it is indicated that even higher-order corrections tend to be quite large and therefore may not be neglected. The correction to order O(E 3 ) also affects the behavior of q in the spherical limit M --4 ac leading to a linear increase with M in case of N odd whereas the decay of q for N even is not modified. Another remark concerns the necessary classification of the relevant operators N ' with 1 < N' < N - 1 into three distinct (,02 ) groups when calculating their anomalous dimensions via the E-expansion . These turn out to be just the same as those required by conformal invariance. However, unlike the E-expansion of the exponent q, the various contributions to the E-expansion of the critical indices associated with the operators mentioned before seem to be alternating in sign. Thus no accurate results can be expected unless these expansions are combined with some resummation techniques of the Borel type as they were discussed by Brezin et al. [17] and Le Guillou and Zinn-Justin [18). It is a pleasure to thank Prof. Dr. F.J. Wegner for his great interest in this work and a lot of stimulating discussions. Appendix A TENSORIAL CONTRACTIONS
In an 1Vl-vector theory the usual symmetry factors of Feynman diagrams get enhanced by tensorial contractions which are polynomials in the variable M. To calculate these polynomials a tensor Ti2NizN is introduced for every vertex of type 2N, whose 2N indices are associated with the 2N lines emanating from that vertex. The O(M)-invariance of the theory requires T~ . . i'YN to be of the entirely symmetric form: 1 (ai
J i,si3i4 + si,i3sizi4 + si l i;ai,i;) 2N
Ti, . . . i' .
2N-
1 ~=2 ~ .ii
i~ . . . i ;_,~;+1 . . .12N 1)
for N > 3 .
(A.1)
!. tlofmanit / Anomalous distensions
810
This definition leads after rather lengthy calculations to the following expressions for the tensorial contractions defined through tables l, 2,4-6:
2(2N)! r(IM+ 1) '
(
_ 8N(N!)4 F( ;M +N) M _ l' 1M+ 1 ( ) 2 2N
_N 2~
(2N- 2v) ! 2v v ( N- v)!
T('-,M +N) N 2v F(-1M +N- v '
N
N! ~' (2N- 2v) ! 2v ( N ) F(-2 M +N) N- t (M) -2v T(_-',M+N-v)' (2N)! 0, 0 (N-v)! ( v [NJ
~(2N .2N°)(M)
(2N) 5N1 , k(
M) =
N!N'! 2 (2N'- 2v) ! 2N v =2 v (2N) ! 2N'- N -! 1-o v! (
F'
YON) N)
N1 2
NI +N_-N-q- 1 q- 1
q= 1
p=t)
1-o
N'-
A
2N-N1 -N_+p 2 A=O
T('-,M + N' ) N 2 v ) T( _'-,M+N'v) 1
v!A!p!q!
(2N-2q-2A)! (N-q-A)!(q-v+A)!(N1 +N2-N-p-q+v)! X
(2N-N1 -N2 +p+2q-2v)!(2N-N1 -N,+p)! (N1 - p-2q)!(2N-2N 1 - N2 +p+2q - 2v)!(2N-N1 - N2 +p-2A)!
X
2 (T(-M+N)) T('-,M+N-q-A)T('-,M+2N-N1 -N,+p+q-v)
+Y ( 2 N)
X
Ni 2
I 1
q=0
N i +N,-N-q-1
2N- 2N, -N,+p+2q 2
p=0
v=q
FI
1:
11
2N-N i -N,+p+2q-2v 2
FI
A=0
1 (2N - 2v - 2A) ! v!A!p!q! (N-v-A)!(v-q+A)!(N1+N2-N-p-q+v)!
X(2N-N1 - N2 +p+2q - 2v)!(2N-N 1 -N2+p)!I[(N1 -p-2q)! X(2N-2 ^!, -N2 -;-p+2q-2v)!(2N-N1-N2+p+2q-2v-2A)!]
!. Hofmann / Anomalous dimensions
x
811
(l'( ; M+N))2 T('-,M+N-v-A)T('-M+2N-NI-N2+p+q-tr N, [ 2
+ YON) E
1
N, -2q
q-1
1:
E
[Nl +IV,-p-2q] 2
F.
q= l p=N, +N,-N-q v=0
A=0
1 veA!P!gr
(2N-2q-2A)! (N-q-A)!(q-v+A)!(N-NI-N2+p+q+v)! (NI +N2-p-2v)!(NI+N,-p-2q)! (NI -p-2q)!(N2-p-2v)!(NI +N,-p-2q-2A)! x
(T(,,,M+N))2 T( ;M+N-q-A)T( ;M+NI+N2-p-q-v) N, 2
+ YON) F,
I
1
Ni -2q
E
q=0 p=Nj +N,-N-q
X
x
x
N
2 J ,-p
NI+N:-p-2v~ 22
1:
E
v=q
A=0
1 vtA!P!gt
(2N - 2v - 2A)! (N-v-A)!(v-q+A)!(N-NI-N2+p+q+v)! (NI + N2 -p- 2v)!(NI +N2 - p - 2g)!
(NI -p-2q)!(N2-p-2v)!(NI+N2-p-2v-?A)! (T('- M+N))2 T(M+N-v-A)T('-,M+NI +N2-p-q-v)
with YON) given by
Y(2N) : = 0 . 4N(
N ~)3
[(2N)11
2
and
:=
for N I < N2 < 2 N - NI - N2,
1 1 2! 1
for N I =N2 < 2N-NI -N2
ii2 for N, =N2 = 2N-N, -N
or
N I
J. Hofmann / Anomalous dimensions
For N, + 11% - N - q < 0 the first two quadruple sums are to be neglected and in the remaining two sums the lower bound of the p-summation is to be replaced by P = 0. (2N.2N')(M) SN,.N_
= y(2N, 2N')
[
N,
NI +N,-N-q-1 N-N'+q-1
q=0
p=0
2N'-N, -N, +p 2
a'=0
v!A!p!q!
A=0
2q - 2A)! (N'-q-A)!(N-N'+q-v+A)!(NI +N,-N-p-q+v)! (2N'-
X
X
+
(2N-Nj-N,+p + 2q-2v)!(2 N'-Nl-N2+p)! (NI -p-2q)!(2N-2N, -N2 +p+2q-2v)!(2N'-Nl -N2 +p-2A)! T( M+N'-q-A)T( ;M+N+N'-NI -N2+p+q-v)
Y(2N,2N")
Na l 1 2 1 N, +N__N_q_1 q=0
2N -2N, -1!,':+p+ 2g 2
2N-N I -N,+p+ 2q -2v 2
v=N-N'+q
11=0
p=0
v!A!P!q!
X
(2N - 2v - 2A)! (N-v-A)!(N'-N+v-q+A)!(NI +N2-N-p-q+v)!
X
(2N-Nl -N,+p+2q- 2v)!(2N'-N I -N2 +p)! (N i -p-2q)!(2N-2NI -N2 +p+2q-2v)!(2N-NI -N2 +p+2q-2v-2A)!
X
I'( 1 M+N-v-A)T(2M+N+N'-N I -N2 +p+q-v)
+ y(2N,2N')
[
Ni 1 2 1
NI -2q
q-1
q=1 p=N, +N,-N-q v=0
[ Ni +N2 -P - 24 2 1 11=0
v!A rP!q!
zq - 2A)! (N'-q-A)!(q-v+II)!(N-N i -N2+p+q+v)! (2N'-
J. Hofmann / Anomalous dimensions
X
X
813
(NI +N2 -p - 2v)!( NI +N2 -p-2q)!
(NI -P-2q)!(N2-P-2v)!(N,+N,-p-2q-2A)! T(M+N')T( '-, M +N) T(2M+N'-q-A)l'(2M+NI +N2 -p-q-v) NI
+ Y(2N,2N')
N,-p 2
N, +N,-p-2v 2
q=0 p=N, +N,-N-q
v=q
A=0
(2N'- 2v
- 2A) !
E 2
1
NI -2q
1 v!A!P!q!
(N' -v-A)!(v-q+A)!(N-Nl-N2+p+q+v)!
X
X
(Ni + N2 -p-2v)!(NI +N2 - p - 2q)!
(Ni -p-2q)!(N2 -p-2v)!(N1 +N2 -p-2v-2A)t T(2M +N')T(2M+N) T('-,M+N'-v-A)T('-,M+N I +N2-p-q-v)
with Y(2N,2N') given by
or :=
1 1 2!
for N, * N2, for N, = N2,
For NI + N2 - N - q < 0 the first two quadruple sums are to be neglected and in the remaining two sums the lower bound of the p-summation is to be replaced by p = 0. In calculating the tensorial contraction SN~NÏV,N ")(M) one will occasionally encounter the situation that the expression 2N - 2N, - N2 + p + 2q - 2v occurring in the first two quadruple sums becomes negative for some values of q, p and v. These contributions are to be skipped.
814
J. Hofinar:r: / Anomalous dimensions
In the cases N, = 0 and N, = 1 the last two quadruple sums reduce to simple summations : 2N( N!)` aN) (M) 1 _ S(Z(a.I N (2N)!
N
1 N 2N-2v T(;M+N +v) 92 ( )( N-2v )F 2
1-0
N,
4N(2N-N,)!N,!(N!)2
2N !" 2
1-(IM +N) 2 1 T _'-,M + N,_) v-0 v!(N - N2 + v) !
(N) 2N- 2.) ,r( -, M +N+ v) v N,-2P T('-,M + N - v) ' Sn-''N-2N')(M) =
F
(2N)! I T('2 M+N) N 1
X
N) Si,_N_(
M) _ ~ .
x
2N'-2v T( 1M+N+v) N-2v ) F(-2 M +N'-v ) '
N' 2 v
4N (2N - N2) !N2!(N!)2 112 2 " [(2N) T('-, M+N) T('-,M+N2 )
2 v-0
( N2 -2v ) M + 2N2 (N-v)-2(N2 -2v) v!(N-N,+ v)! _
x (N) 2N-2v T( '-, M +N+v-1) + N2 +1
v
N2 - 2v
N, + 1 2
X
x
N (v
T( _2 M+N _ v)
T( '-, M +N) 2N-N2 T( _2 M+N2 +1)
(N2 -2v+ 1)M+2(N2 + 1) ( N- v) -2(N2 -2v+ 1)
T('-,M + N + v - 1) 2N-2v N2 -2v+ 1 ' T( 2~ M+N- v)
!. Hofmann / Anomalous dimensions
815
The definition in eq. (A.1) is introduced in such a way that in the scalar case M = 1 the tensorial contractions reduce to the multiplicities of the corresponding diagrams: ) CpN 1(1) (2N)
= 1,
fBN)(1)
1 2N (1) = 2 N
S(2N) (2N) (1) N, -
S(2N,2N,)(1) N1 , N,
_ d,
C(2N,2N')(1)
,
2N
= 1,
(2N- N, N2
Nl
2N')
Ni
2N'-
Nz
)
= 2N' N ,
,
Ni .
(A .2)
)
As a consequence of the finiteness of the beta-function and the various scaling functions there exist a number of relations among the tensorial contractions enlisted above. They read
Sr~`N_ NI(M) = 0,
_
[S(2N)(M)l2-S( N)(M)
2~DN)'(M)C(2N)(M) -
2N ( N
~(2N)(M) + ~(2N .2N')(M )J y(2N,2N')(M) N 2
N =1
N N~
_ SN NN NN1( M )
2 N'
N
FNI = 1
e(,Nr )(
M) = 0,
(A .3) (A.4)
7S ~2N.2N ,)(M )
N ( NI
SN1;NN')(M)
=0
and may serve as a check of the calculations . The second equation can easily be established using the explicit expressions while for the first and third relations no proof could be given in view of the complexity of the tensorial contractions involved . Nevertheless they may be verified for any definite value of N. For N'== N the third equation reduces to the first as might be seen from the following relations : C(2N,2N)(M)
SN N,N)( M)
=
-
2~(2N)(M)
'
r SN Nmin N,,2N-N -N,)(M)
1
d between the ten rial contractions with one and two upper indices th th ten ri to the order nee
contractions the E-e
4_2,v1(
t2
l
1)UNE + O(E2)] ,
2=1z2
?qx? -1k
nsion of the various i te
1 73
P)
E
,
Ne + O(e2)], I
[2 N + ( N-
1)WN e+O(E2 )], "(Pl + . . . +P .V) Iqzu = -[2+(NE _I),-
_N N - 1
(NI , N2)+O(1), for N,+N,*N,
42N1
EN., ,\,_(p,
a 7-
sp
` [2 + (N - 1 -
2W + .;
N
E +O E
)],
for N, +N2 = N, where the quantities -r, Uv , VN , WN, EN(N,), and ON(N,, N2) appearing on the right-hand side are declared through eqs. (13)-(18) .
[1] [21 [31 [41 [51 [61 [71
References F.J. Wegner, in Phase transitions and critical phenomena, vol. 6, ed. C. Domb and M.S. Green (Academic Press, New York, 1976) p. 7 C. Itzykson and J.M. Drouffe, Statistical field theory, vol . 1 (Cambridge Univ. Press, Cambridge, 1989)321 A.M. Polyakov, Pis'ma Zh. Eksp. Teor. Fiz. 12 (1970) 538 [JETP Lett. 12 (1970) 3811; 66 (1974) 23 [39 (1974) 101 A .A. Belavin, A.M . Polyakov and A.B. Zamolodchikov, Nucl. Phys. B241 (1984) 333 D. Friedan, Z. Qiu and S. Shenker, Phys. Rev. Lett. 52 (1984) 1575 J.L. Cardy, Conformal invariance and statistical mechanics, in Les Houches, Session XLIX, 1988, Fields, Strings and Critical Phenomena, ed. E. Brézin and J. Zinn-Justin (North-Holland, Amsterdam) to be published E. Brézin, J.C. Le Guillou and J. Zinn-Justin, in Phase transitions and critical phenomena, vol. 6, ed. C. Domb and M.S. Green (Academic Press, New York, 1976) p. 125
matin:1s
bp
~~cb% `:
ate= D.J. Anit, F ""_ c 1984) [9] D. e in and Ha Wagner, . LetL 17 (1966) 1133 [1®] P.S. Howe and P.C. West Phn Leit. 09») 371 [ID] LN. J h t , . ) (1 [12] I.D. Lawrîe and S. in Phase transitierts and J.L. itz ( Y 19SA) . 1 [13] A.L ° and . 18 F.W. (1 [14] P. Gi tormal field theory. in and Critical na, ed. E. J. Zia J [8]
published us, F. [151 W.
,i -w
r
lL
Làatr4. oi
c LLl4a
h-1161L,~U
ttinger and R . . P SonL Formulas and theorem
mathematical physics (Springer, Berlin, 1966) p. 15
[16] MJ. Stephen and J.L. Lett. A (1 89 [17] F. Brizin, J.C. Le Guil a J. Zinn-Justin. . D15 (1977) 1 [18] J.C. Le Guillou and J. Zinn-Justin, Rev. Lett. 39 (1977) 95
ta-
ANOMALOUS DIMENSIONS OF RELEVANT OPERATORS IN THE O(M)-SYMMETRIC +2 kTHEORY Jiirgen HOFMANN
Institut für Theoretische Physik, Ruprecht-Karls-Universität, D-~Heidelberg; Gernwny Received 28 May 1990 (Revised 6 August 1990) The calculation of the critical exponent q for the class of OM-symmetric 02N-theories is presented to order O(E ;) in an E-expansion around the upper critical dimension d c - 2N/ (N - 1) using field theoretic methods. For the case of a scalar order parameter 0 a comparison of the exponent q obtained by an extrapolation of the E-expansion to two dimensions with the exact result pertaining to conformal invariance is provided. Basically it is found that the ratio of the O(E 3 )-correction and the O(E 2 ) result is a monotonically increasing function of N diverging exponentially as N tends to infinity thereby suggesting an approach toward the exact value. A further modification affects the behavior of q in the spherical limit M - ac . Additionally the exponent (w as well as the anomalous dimensions of the relevant operators (02)N' with 1 < N' < N - 1 are determined to order O(E 2 ) throughout. In those cases, however, the various contributions to the E-expansion seem to be alternating in sign.
l. Introduction Several years ago Wegner [1] calculated the lowest-order contribution to the ,E-expansion of the critical exponent q for the class of O(M)-symmetric 02 Ntheories which describe the critical behavior of a physical system in the vicinity of its multicritical point [2]. Here E := dc - d denotes the difference between the spacial dimension d of the system and the upper critical dimension d c == 2 N/(N - 1). It is well known that the result for q extrapolated to two dimensions exhibits an exponential decay in the limit N --1- oo at least when restricted to the scalar case M = 1 . On the other hand the two-dimensional situation is known to be special since conformal invariance is powerful enough to allow for a complete determination of the critical indices. The notion of conformal invariance in statistical physics was originally due to Polyakov [3] who suggested that a statistical mechanics system at its critical point not only be invariant under a global change of *This work was supported in part by the Sonderforschungsbereich 123 Stochastic Mathematical Models of the Deutsche Forschungsgemeinschaft . 0550-3213/91/$03 .50 0 1991 - Elsevier Science Publishers B.V. (North-Holland)
7190
J. Hofmann / Anomalous dimensions
scale but also under the more general group of conformal transformations. Unfortunately in dimensions higher than two the conformal group is only finite dimensional thus giving us not much more information than is implied by ordinary scale invariance. On the contrary in two dimensions any analytic function generates a conformal mapping thereby rendering the conformal group infinite dimensional. In a series of papers [4,51 it was shown that the implications are so strong that in some cases the critical indices and the various correlation functions may be calculated exactly. Especially it was shown by Cardy [6] that there exists a unique correspondence between the scalar version of the above-mentioned .02N-theories considered in the present paper and the minimal unitary series of conformal field theories. Additionally the exact expression for the exponent q leads to a very weak decay q - N- 2 in the limit N -+ w which strongly contrasts the behavior found via extrapolating the expansion. In a previous paper [10] Howe and West estimated the renormalization group beta-function to third order in the coupling constant for the scalar (A2N-theory in the limiting case of large values of N by making use of Lipatov's considerations about those diagrams that contribute most at a given order [11]. From their result they suggested that the lowest-order terms in the E-expansion of the critical exponents are not the leading contributions. These authors, however, do not give any results for the critical exponents to higher order. In the present paper the rigorous calculation of the critical exponent q for the O(M)-symmetric 42N-theories up to terms of order ME') is presented using field theoretic methods [7,81. No restriction to the case of a scalar order parameter will be made despite the fact that we do not expect a phase transition for M > 2 as a result of the Mermin-Wagner theorem [9]. Even though the final result turns out to be rather complicated it may easily be recognized that for M = 1 the ME 3)-contribution yields an enormous correction so that it must not be neglected. Basically it is found that in the result q = qi E 2 + 77 2 E3 + ME') the ratio 712/71, diverges exponentially to infinity as N -), x thereby providing concise analytic expressions to the suggestion made by Howe and West [10]. A further modification of current results concerns the behavior of q in the spherical limit M -3. oo . The organization of the current paper is as follows: In sect. 2 the action for the field-theoretic models under consideration is introduced. It also contains an account of the basic results including the situation in two dimensions. In sect. 3 a brief outline of the calculation of the exponent q is given. The result for the exponent w that governs the deviations from scaling is presented to order O(E2). Sect. 4 is devoted to a study of the scalar case M = 1. Sect. 5 extends the calculations of sect . 3 to a determination of the anomalous dimensions of the isotropic relevant operators (42)N' for 1 < N' < N - 1. In sect. 6 some conclusions are drawn . The paper closes with an appendix that contains a complete list of the various tensorial contractions which come along with the required Feynman
J. Hofmann / Anomalous dimensions
1
diagrams . The list is supplied by enumerating the e-expansions of the inter associated with them to the order they are needed.
2. The model
The 4a-theory which is usually considered in calculating the critical properties of physical systems may be generalized to a 0 2 '-theory to incorporate multicritical points of order N [2]. In the present paper the critical exponents are examined that follow for the class of O(M)-symmetric 02N-theories described by the bare euclidean action:
S141 with
=
02 :=
d
i I=1 (a oÎ2 + IIL
ddx
J
N-1
2+
2' (02 )' +
;-2 (21)!
2N (2N)! (102)N
(1)
M
~ 0i .
,j=1
For N = 2 the 0a-theory of the ordinary critical point is recovered which has been extensively investigated and for which calculations have been carried to very high order [7]. The case N = 3 belongs to tricritical phenomena about which a review was given by Lawrie and Sarbach [12] who also presented the lowest-order results for the tricritical exponents. Some of these were extended to the next order by Lewis and Adams [13]. Wegner [1] obtained the exponent q to lowest order for general values of N and M: 77
2 3 __ 4N (N - 1 ) ( N! ) T(±M +N) 2[(2N) ! l 2
T ( 2M + 1 )
1
~(2N)(M)
2
E 2 + O(E3) ,
where ~QN)(M) is one example of a tensorial contraction and may be found in the appendix. For M = 1 the result (2) shows the asymptotic behavior: .1
N--
1
16zr(TrN )312 x 64 -N
when extrapolated to two dimensions. In two dimensions the scalar version of the theory (1) coincides with the unitary minimal model with central charge [6]:
The fusion rules [4,14] then yield the correspondence between the relevant
!. Hofmann / Anomalous dimensions
operators O' with 1
for N
4P2N H
From this identification the scaling dimensions of the relevant operators are obtained as a result of the Kac formula [6,14]:
2N'( N' + 1) (N+ 1)(N+ 2) ' d,~2N .
0
d,2N .
=
2(N'+ 1)(N' + 2) (N+ 1)(N+2) '
=2
N+3 N+ 1 '
for N < 2N'< 2 N - 2, for 2N'= 2N .
(4)
In particular the critical exponent rl, which is connected with the anomalous dimension of the 0 - or 02,2 - field, is given by
rl=2de=
3
(N+1)(N+2)
One notices the weak decay of rl in the limit N -> oc which strongly contrasts (3). In eq. (4) the whole set of operators has to be split into three groups for each of which a separate formula may be given. It is nice that the same feature appears in the calculation of these quantities within the E-expansion presented in sect. 5 where it creeps in as a consequence of the Feynman diagrams .
3. Outline of the calculation . The exponents q and w The calculation of critical exponents proceeds most economically when performed directly at the multicritical point . Hence the bare mass 1,C as well as the coupling constants A21 with 1 < j < N - 1 appearing in (1) are no independent quantities. The index c attached to them indicates that they attain their critical values as functions of J1 2N in order that the corresponding renormalized counterparts vanish . This choice of dealing with the critical theory is rather convenient
!. Hofmann / Anomalous dimensions
793
TABLE 1
Feynman diagrams for ri(,2;(p) to order O(A2N). To every external line an index i is a which is derived from the M-vector structure of the theory whereas the numbers placed near the bubbles indicate the number of lines running through them. The quantities appearing in the last two columns are listed in appendix A. These conventions apply as well to tables 2,4-6. The wiggly lines appearing in the diagrams of tables 4-6 denote insertions of an operator (42)N' with momentum k Symmetry factor
Diagram
Tensorial contraction
1
P 2N-1
Integral P2
1 (2N-1)!
(2N) . D,~," _~(M)S-r1r,
1
eé N )(M)8; J;,
(N- 1)!(N!)2
N~
DZN
B( 2N k
P)
P)
1V-1
since merely diagrams which contain vertices of type 2N exclusively have to be considered . After eliminating all vertices of type different from 2 N one is left with the calculation of the 2-point and the 2N-point vertex. The relevant Feynman diagrams contributing up to order O(Jl2N ) are enlisted in tables 1 and 2. Along with them some terminology is introduced concerning the associated integral expressions and also the tensorial contractions which are due to the M-vector structure of the theory. All of these are explicitly given in the appendix. To remove the divergencies, dimensional regularization is used in conjunction with a set of normalization conditions: rRita ape r(2) R,,,,.
i2
p) P
(0)
for 1
=0,
2-K_
(
rRi ) .i2nr( pi) ISP = UK N
1)eTi,2N~2N'
794
A Hofmann / Anomalous dimenNions
= ~ Q6 * . . . *
~~ ~
. ^
. , . . ~
t!~l Q6
r.
8
k fli
une
LQ
Q
.
.
;e ~ ^=
A~t wd
~~
In
»n
»0)
! ~~ ! ~~ «~~
-
!
!
rv
m
~t
"=
I tdo
ann 1 An
alons dimensions
795
where the tensor T'' ~, , is also defined in appendix A. The quantity dimensional parameter that derives from a symmetry point (SP) given by SP : Pi -pj =
s
K
is a
- ic2(sâ"~ -1), 1
in which s is usually taken to be 2 N but will not be explicitly specified in the following. After rather lengthy calculations the renormalization group beta-function P(v) and the scaling function y,(t) of the O-field are obtained in a double expansion in E and the renormalized dimensionless coupling constant r." : .8(v) = - (N - 1)v[E - (a; +b3 E + 0(4E2)) . L' +2(Nß,+b4 -2a,b3 +O(E)) . 17 2 1 y4(v)
+O(L' 4 ),
= -(N- 1)11,2 [(2a, + 2b,E + O(E2)) + (4a,b 3 + 4b,a 3 - 3b2 + 0('E))-VI + O(c,4) .
The various quantities appearing in these equations are declared through _ _
_ 2(N- 1) 2 (2N) D,N-~(M)' alT (2N) !
a 3 'b2
b
2 N!
y(2N)( Tb
N-1 2N
( N!)
7
M)
a2VN'
4
2 1 9 NI
8 N -1
R
N -1
b,
2N
T3
a,UN
(
) 2N (M)
'
N- 1
b3 :=2 a3WN [N121 Ni=,
1
) [2WN + EN(l)] N S
ON (NI, N2) S(2N M) Ni , N2( ) NI !N2 !(2N -NI - N2)!
M
(12)
Hofm
/ Anomalous dimensions
with
A := 1l(N- 1),
T== (4ar)-N(r(A))N-1,
( 13) (14)
UN :=(2N-1)(N+ 1) +2N(log47r-C),
N :=3N2 +2N-2+3N(log47r-C),
(15)
~-N N :=N - C+log4r-log N ( S-1),
(16)
~N(NI) -N- 1-C+ jr(A) -«AN,) - «A(N -NI », N( N.'
N2)
:=
(17)
F(A)r(A(N-N,)) r(A(N - N2))I'(A(N, +N2 -N)) . r(AN,)r(AN2 )r(A(2N-N, -N2))
18
Herein C and (z) denote Euler's constant and the digamma function respectively. It should be mentioned that in eq. (17) the constant C always drops out. This is due to a formula of Gauss [151 that applies to "the psi-function in case of rational argument. The expansions of the two renormalization group functions given above together with the nontrivial zero v* of ß(0 then yield the critical exponent 71 to order O(E 3 ): (t,*) = = Y4 iIE 2 + 712E 3 + O(E4) with -2(N - 1)(a,la3) , 772 = -
N -1 4
a3
( 19)
[8Na1 + 8a,b4 - 3a 3 b 2 + 6a23b, - 16a,a 3b3 l .
(20)
Now, substituting (7) and (8) into (19) leads back to eq. (2) thus reproducing Wegner's result. Some more algebra is involved in obtaining the new coefficient 712 through introducing eqs. (7)-(18) into (20): *12
1)2(N!) 4 = (N -( 2N ) ! (N-
(
1 )2 )
2N(N - 1) (2N) i
~ (N),(M)
[C(2N)(M)j 4
~(2N)
M 2+
1-
N
-1 ( )
[N121 Ni-1
~(2N)
D2N-~(
N ~
M)
N
S(2w)
ON( N, 1 ry2) S(2N) _ N ) i NI , N2( M) !N2 !(2N N N1 " " 1 2
M
!. Hofinann / Anomalous °
It should be remarked that in the process of deriving this result various tions occur. Actually these cancellations are necessary in order to obtain finite renormalization group functions. They are due--to certain relations among the tensorial contractions which are given in eqs. (A.3) and (A.4) of appendix A. These
same relations are responsible for the contributions depending on the symmetry point SP to drop out of the final expressions which descries physical ` of the theory. In fact all terms containing the variable s vanished in the result for t exponent 71 . For some low values of 1V the above formula in conjunction with the to contractions listed in the appendix yields the following special results far is
7I(N =2) =
M+2 2 E21 6(3M + 14) + 2 2(M+8) ( M+8)
1
4
E
+O(E 4),
(M+2) (M+4) M3 +34M2 + 620M +2720 E2 1 + ~( N = 3) = îi2 12(3M + 22) 2 8(3M+ 22) 2 141M 2 +2310M+8944 + 3 (M+2)(M +4)(M+6) 2 (3M 2 + 150M + 1072)
3(3M+ 22)2 712
)
E
I
4 + 0(1E )
711
with
112 19,
9 87M4 + 10140M 3 +355452M 2 + 4 146 240M + 15 298 816 8 (3M 2 + 150M + 1072)2 -
18[ vC 3 - 3 log 31
M4 + 64M3 + 2264M2 + 26 936M+ 99 360 2 (3M 2 + 150M + 1072)
3 15M3 + 636M 2 + 8 360M + 33 864 + 64(T(3)) 2 (3M 2 + 150M + 1072) In the first two cases the results coincide with the exponent 71 for the 04-theory and the 06-theory respectively given by Brezin et al. [7] and Lewis and Adams [13] respectively .
Hofmann / Anomalous dimensions
Another modification of the lowest order result for q concerns the spherical limit M -* ac which was noticed by Wegner [1] to be of the form _
~im
0 c,(N) IE2
for N even for N odd
to order O(E2 ). The coefficient q(N) is independent of M. A detailed study of the tensorial contraction SN,N~,(M) shows that it is a polynomial in M of degree N or N - 1 according to whether N, and N, are both even or at least one of them is odd. After some algebra this leads to a modified behavior of the exponent rl in the spherical limit to order O(E 3 ) : _
m M
0 c,(N) E' + c2(N) - ME'
for N even for N odd '
where c2(N) is also independent of M. Another one of the critical indices which may be given with the aid of the renormalization group functions worked out so far is the exponent co that governs the corrections to the leading scaling behavior . It is simply the derivative of the beta-function at the nontrivial zero . A short calculation yields the following order O(E 2 ) result: cù :-
dß(t)
- ( N - 1)E + ui E 2 + O(E3)
(21)
with co, given by
-
(N-
1
1)(N!) 2
2
[~(M)(M)]2
2N (N - 1 ) (2N) .
e(2N) 2N
-
(M)
N- 1
~ N12~ (
(Ni) 2
NI-1
N)
Nl
ON (NI, 1-
1b I
NI +N2 :A N
1
N2 )
2
S (2N)
~N(Nl)
N2) 1
2
-N(M)
SN N(M)
J. Hofmann / Anomalous dïnre
For N = 2 this agrees to order et al. [7].
O(E2)
`
with the same exponent given
4. The scalar case
=1
The descent from the general outline of the preceding section to the special case of a one-component order parameter proceeds via eq. (A .2) of the appendix. Only little effort is required to establish the result: 17
=
711E2[1 + ('%2/171)E + O(E 2 )]
s
with 71, =4(N- 1)2(N!)6[(2N)! ] -3
and the ratio 712 /71 1 composed out of four pieces: 172/711 = YI(N) + Y2(N)
+ S1(N) + S2(N) .
They are respectively defined by Y1 (N)
:= -8N(N- 1)(N!)6[(2N)! ] -3,
SI(N)
:=
4(N - 1)(N!)4 (2N)!
6 _ 4( N! ) S2(N) (2N)!
f
[NI21
4
E N!1 ) 2 [(N -NI) ,-l 2~N N,-1(
(NI),
~'ON(N1,N2) N !)2 2N-N -N !)2( N
)!
2
N,+N,-*N
where the symmetry factor 0 is defined in the appendix. This expression does not coincide with the conjecture of Lewis and Adams [13] for the general case of a scalar theory of type (1) although it yields the same result for the ordinary critical and the tricritical theory. Indeed, for the lowest values of N the corresponding
d. f%fin
/ An
alous dimensions
formulae in the preceding section reduce into 1 54 -1(N = 3) = S ~(
N
109 ) 108
(
1 E2
9
- 4) - 85750
1
+
62[l
+ o(e4),
1(27 7r2 8
+
+
2 279 ) E 75
+
O( E 4 )
(5094189 343000 +
2 35
[32(T( ;))~ - 277r~ + 811og3,
E
+ O(E4),
35 815 924 100 _ 25 _ 1 (r( .1))4 _ 25 2 1 1+ n(N = 5) = (I(4) ) + 32 2 250047 E 1250235 + 63 [ 36 v 2N=
2:
;-'
E + O(E 4 ) , (r(4))2 -47r + 81og2 j)]
71 = 1/54E 2 (1 + 1 .01E + O(E2)),
N= 3:
vl = 1/500E 2 (1 + 12.7E + O(E2)),
N=4 :
v1=9/85750E 2 (1+46 .7E+O(E2 )),
N = 5:
q = 1/250047E 2 (1 + 117.1E + O(E2)),
where the result for the ip'°-theory has been added . Obviously the ratio %/'rl, increases with increasing values of N. This is strongly emphasized by the asymptotic behavior of the four constituents forming the ratio 712/71, YI(N) -
Y2( N
-
87r 3 / 2 N 7 /2 -64 -nr 2N,
S, (N) - 81og 2 - N,
S2(N) -
9
4~
T(3)
r2) (;
y N312 (K)
2N .
The sums S,(N) and S2(N) are plotted in figs. 1 and 2 up to N = 100. It is quite
Hofmann / Anomalous dérnensions
z N
J. Hofmann / Anomalous dimensions TABLE 3
Values for 71 due to the O(E-' ),ME ;)-term and the exact result (eq. (S)) N 3
-D to 0(E-') 0.074 0 0
7
-Q to 04 3 )
Exact value in d = 2
0.224 0.027 0.0015 0.000060
astounding to have S,(N) growing linearly for large N. On the other hand the sum S,(N) clearly denotes the major contribution . To conclude this section the corrections of the exponent n in two dimensions due to the O(E3 Merm are tabulated in table 3 for some values of N together with the exact results according to eq. (5).
5. Anomalous dimensions of composite operators The present section may be regarded a sequel of sect. 3 as it contains an investigation to order O(E 22) of the anomalous dimensions of the relevant isotropic operators (4 -)1' with 1 < N' < N - 1 which may be added to the action (1) as small perturbations . This amounts to examining the stability of the nontrivial fixed int t.* against elongations of the system under consideration in a direction in parameter space that corresponds to such perturbations . The anomalous dimension of the operator (,02)^'' with 1 < N'< N - 1 may be calculated when dealing with the 2N'-point vertex function containing one insertion of such an operator . In these cases all vertex functions endowed with an arbitrary number of insertions renormalize multiplicatively, and for each operator one new renormalization constant Z(& -)N " has to be introduced . This does not apply to insertions of the operator (0 - ) N since for this operator there exists 0)2 - that has the same canonical dimension. another one - which is K(N-l Hence multiplicative renormalization is lost as the two operators mix during the renormalization procedure. In the following the operator (02 )N will be excluded since insertions of this are just equivalent to a modification of the coupling constant therefore resembling corrections to scaling. But this would lead again to the exponent w already given in eq. (21) of sect. 3. A consideration of the renormalization group shows that the full dimension of ' the operator (0 2 ) N is given by Y(O=)N- = d - N'(d - 2) -
y«> > N'( U*)
for 1 < N' < N - 1,
(22)
!. Hofmann
/ Anomalous dimensions
where y(.6,)N.(V)
a log Z(¢~)
:=
.
au
is the scaling function associated with the operator (,0 2 ) N' and c-* as in sect. 3 denotes the nontrivial zero of P(r) . The dimensions y( 42 ) -v , defined in eq. (22) are connected to those given in eq. (4) via d( , b ,)N . = 2 -
y( ,O '') ,v -(
r *) .
The calculation of the 2N-point vertex function with one insertion of an operator (,02)N' requires a distinction between three different cases: .
with 2<2N'
(i)
(IA2 ) N
(ii)
2 N' (,0 )
with N<2N'<2N-2,
(iii)
2 N' (10 )
with 2N'= 2N,
the reason being certain Feynman diagrams that do not appear for all operators. This is conveyed quite clearly by tables 4-6 which display the relevant Feynman diagrams to order O(Jl2 N ). Thus, as was mentioned in sect. 2 the calculation of the anomalous dimensions via the E-expansion leads to a grouping of the various operators that is in remarkable accordance with eq. (4) pertaining to conformal field theory. In the process of calculation the finite part of the vertex function h(2 Nv(p- . k; t', K) is fixed by the renormalization condition : 1 (~ 2 )N R( Pi'
k ' ~~'
! _ K)ISP
(2N')
I l . . .i,%"'e
where k denotes the momentum of insertion and the tensorial indices on the left-hand side have been suppressed. The symmetry point SP declared through : Pi -pj SP:
s
1 - 1
K 2 (s3ij
- 1)
for 1 < i, j < 2N',
s-2 N+2N' s-1 K2 pi-k=-(2N-2N')
is an extension of eq. (6) and assures that the integrals enlisted in the appendix may be used without modifications .
J. Hofinann / Anomalous dimensions
804
Z
3
NZ
v z N N v
aN
C
ZZ
b V L
l'!
^ N z
N
Hofmann / Anomalous dimensions
805
z
a
a
z
N
+
a
Uli,
II
aa
ar
V
PJZ
z_ P9
a
' m
_
zPJ
°
Ir
PI
z
PI
3
a
a `-`1
^z z . NZ v
zN
'
:e
:e ..vs
..`/
z
z
zz PJ 1
N_G~
Z ~I
PI z z
VI VI VI
806
J. Hofinann / Anomalous dimensions
z c + zi . rl zl (~ C. C rl~ " 3
ry C r12 '~
rl~
z u
t ri
a+
z
N
O
U U i
C
Û
E
b
"!CA
NC
I y
I r~
I r~
rl '
rl '
rl '
rl '
z rl
z
2 rl
z rl~l
rl
zr_12
2? _
zz
i., U s.. rl
ar L
z
r
u
11 aa
a
c zÏ rl rl
L
^J
A i
-iz
z
Z
v
v
N
N
z
Z
z N rl z z
J. Hofmann / Anomalous dimensions
807
After quite lengthy calculations the following expressions for the anomalous dimensions of the operators appearing in groups G) and (ii) ensue: (i)2<2N'
y(02)N. = 2
with
N-N' - 1 + (N'- 1)E + TE 2 + O(E3 ) N
(N- 1)(N !)2
1
2 N'(N-1)
2
[C(2N)(M)]2
(2N)!
-
(23)
19N (NI, N2) S(2N,2N') M N,,N, NIlN2NIN2 !(2N)! 1
T262 + O(E 3 )
(24)
with (
TI
N'- 1g(2N)(M) -(N- 1)y(2N,2N')(M) y(2N)(M )
1 ( N- 1) 2 T2 ,2 ~(2N)(M) 3 [N/2l N, =1
2( N !) 2
(2N) (M)[N~(2N,2N')(M) - N'y(2N)(M)l (2N)1 bUIN_,
N [y(2N,2N')(M)S(2N~ NI(M) - C(2N)(M)S(2N, NN)(M) NI , N I , NN J EN(NI) 1
+C (2N) (M)
2N'-N N,=1
N NI
! )2 ~N(N)S(2N,2N,)(M - ( N y(2N,2N') M 1 N,, N ) N- 1 ( )
ON (NI, N2) 1-
+
( N! )
2
N-1~
(2N)
N I ! N2!(2N-NI -N2)!
S(2N) M N,, N-'( )
ON( NI 1 N2)
(M) 1 N et N2< 2N'- i NI +Nr * N, N,~N
NIl1V2!(2N--Nl
_ N )1
S(2N,2N') M ( ) Ni .N2
J. Hofmann / Anomalous dimensions
For the special case N' =1 which describes insertions of the operator 02, the critical exponent v is obtained from eq. (22) via the relation v = ;1 (i) N = 2:
v
M+2 (M+2)(M 2 +23 M+60) 1 E+ E2+O(E3), 2 + 4(M+g) 8(M +8 ;
(ü) N > 3: (N!)'
1
(N+ 1)(N- 1)3 v= + 2 2(N-2) (2N) .
D; '_(M)
E2+O(E3) .
(25)
The first result reproduces to order ME 2) the exponent v given by Brezin et al. [7] for the ordinary critical point whereas the second expression restricted to the scalar case in conjunction with the scaling relation y = v(2 - ,q) yields agreement with the exponent y given by Lewis and Adams [131 . Actually eqs. (23) and (24) are due for some more coincidences with exponents that are known for the tricritical point with M arbitrary. Setting N = 3, the variable N' is bound to take the values N' = 1 and N' = 2. In the first case eqs. (23) or (25) leads to a result for the exponent y that equals the expression given by Stephen and McCauley [161. On the other side for N' = 2 the critical index resulting from eq. (24) together y,, with the previous result for yields a crossover exponent 0, := yc,,,r/y, .2 that agrees exactly with the one reported by Lewis and Adams [13]. It should be boldly emphasized that the general expressions for the critical indices obtained in sects. 3-5 of the present paper coincide with all results calculated in special situations. As was anticipated earlier the foregoing discussion may not be extended to insertions of the operator (022 ) N since it gets mixed with another operator when renormalized . Hence ordinary multiplicative renormalization has to be replaced by the scheme : rOh
NR(pi ;k )
=Z Z111(0iN(pi'k) +Z-0Z,2Tr4~2(pj ;k),
r( <0)2 R(pj' k ) -Z4Z?1r~)"(pi ;k) +Z0Z22Tc4~2(pj ;k),
where Zl,, Z12, Z2 , and Z22 denote a set of four new renormalization constants . However, the lowest-order Feynman diagrams contributing to r N~(p . ; k) are of Tcr~r order O"2 N ). Therefore to order O(A 2 N ) the two equations decouple thus restoring multiplicative renormalizability of (02)N to that order. As a consequence
J. Hofmann / Anomalous dimensions
809
the result (24) may be applied to yield the dimension of the operator order O(E): y(,62 )N=
to
-(N--1)E+O(E2) = _w . 6. Conclusions
The calculation of the exponent 9 to order O(E 3 ) obviously shows that the corrections introduced by the O(E 3)-term are overwhelmingly large for sufficiently high values of N at least when restricted to the case of a scalar order parameter. Thus an approach is gained toward the results known in two dimensions from an exploitation of conformal invariance although the absolute numbers are still too small. Actually it is indicated that even higher-order corrections tend to be quite large and therefore may not be neglected. The correction to order O(E 3 ) also affects the behavior of q in the spherical limit M --4 ac leading to a linear increase with M in case of N odd whereas the decay of q for N even is not modified. Another remark concerns the necessary classification of the relevant operators N ' with 1 < N' < N - 1 into three distinct (,02 ) groups when calculating their anomalous dimensions via the E-expansion . These turn out to be just the same as those required by conformal invariance. However, unlike the E-expansion of the exponent q, the various contributions to the E-expansion of the critical indices associated with the operators mentioned before seem to be alternating in sign. Thus no accurate results can be expected unless these expansions are combined with some resummation techniques of the Borel type as they were discussed by Brezin et al. [17] and Le Guillou and Zinn-Justin [18). It is a pleasure to thank Prof. Dr. F.J. Wegner for his great interest in this work and a lot of stimulating discussions. Appendix A TENSORIAL CONTRACTIONS
In an 1Vl-vector theory the usual symmetry factors of Feynman diagrams get enhanced by tensorial contractions which are polynomials in the variable M. To calculate these polynomials a tensor Ti2NizN is introduced for every vertex of type 2N, whose 2N indices are associated with the 2N lines emanating from that vertex. The O(M)-invariance of the theory requires T~ . . i'YN to be of the entirely symmetric form: 1 (ai
J i,si3i4 + si,i3sizi4 + si l i;ai,i;) 2N
Ti, . . . i' .
2N-
1 ~=2 ~ .ii
i~ . . . i ;_,~;+1 . . .12N 1)
for N > 3 .
(A.1)
!. tlofmanit / Anomalous distensions
810
This definition leads after rather lengthy calculations to the following expressions for the tensorial contractions defined through tables l, 2,4-6:
2(2N)! r(IM+ 1) '
(
_ 8N(N!)4 F( ;M +N) M _ l' 1M+ 1 ( ) 2 2N
_N 2~
(2N- 2v) ! 2v v ( N- v)!
T('-,M +N) N 2v F(-1M +N- v '
N
N! ~' (2N- 2v) ! 2v ( N ) F(-2 M +N) N- t (M) -2v T(_-',M+N-v)' (2N)! 0, 0 (N-v)! ( v [NJ
~(2N .2N°)(M)
(2N) 5N1 , k(
M) =
N!N'! 2 (2N'- 2v) ! 2N v =2 v (2N) ! 2N'- N -! 1-o v! (
F'
YON) N)
N1 2
NI +N_-N-q- 1 q- 1
q= 1
p=t)
1-o
N'-
A
2N-N1 -N_+p 2 A=O
T('-,M + N' ) N 2 v ) T( _'-,M+N'v) 1
v!A!p!q!
(2N-2q-2A)! (N-q-A)!(q-v+A)!(N1 +N2-N-p-q+v)! X
(2N-N1 -N2 +p+2q-2v)!(2N-N1 -N,+p)! (N1 - p-2q)!(2N-2N 1 - N2 +p+2q - 2v)!(2N-N1 - N2 +p-2A)!
X
2 (T(-M+N)) T('-,M+N-q-A)T('-,M+2N-N1 -N,+p+q-v)
+Y ( 2 N)
X
Ni 2
I 1
q=0
N i +N,-N-q-1
2N- 2N, -N,+p+2q 2
p=0
v=q
FI
1:
11
2N-N i -N,+p+2q-2v 2
FI
A=0
1 (2N - 2v - 2A) ! v!A!p!q! (N-v-A)!(v-q+A)!(N1+N2-N-p-q+v)!
X(2N-N1 - N2 +p+2q - 2v)!(2N-N 1 -N2+p)!I[(N1 -p-2q)! X(2N-2 ^!, -N2 -;-p+2q-2v)!(2N-N1-N2+p+2q-2v-2A)!]
!. Hofmann / Anomalous dimensions
x
811
(l'( ; M+N))2 T('-,M+N-v-A)T('-M+2N-NI-N2+p+q-tr N, [ 2
+ YON) E
1
N, -2q
q-1
1:
E
[Nl +IV,-p-2q] 2
F.
q= l p=N, +N,-N-q v=0
A=0
1 veA!P!gr
(2N-2q-2A)! (N-q-A)!(q-v+A)!(N-NI-N2+p+q+v)! (NI +N2-p-2v)!(NI+N,-p-2q)! (NI -p-2q)!(N2-p-2v)!(NI +N,-p-2q-2A)! x
(T(,,,M+N))2 T( ;M+N-q-A)T( ;M+NI+N2-p-q-v) N, 2
+ YON) F,
I
1
Ni -2q
E
q=0 p=Nj +N,-N-q
X
x
x
N
2 J ,-p
NI+N:-p-2v~ 22
1:
E
v=q
A=0
1 vtA!P!gt
(2N - 2v - 2A)! (N-v-A)!(v-q+A)!(N-NI-N2+p+q+v)! (NI + N2 -p- 2v)!(NI +N2 - p - 2g)!
(NI -p-2q)!(N2-p-2v)!(NI+N2-p-2v-?A)! (T('- M+N))2 T(M+N-v-A)T('-,M+NI +N2-p-q-v)
with YON) given by
Y(2N) : = 0 . 4N(
N ~)3
[(2N)11
2
and
:=
for N I < N2 < 2 N - NI - N2,
1 1 2! 1
for N I =N2 < 2N-NI -N2
ii2 for N, =N2 = 2N-N, -N
or
N I
J. Hofmann / Anomalous dimensions
For N, + 11% - N - q < 0 the first two quadruple sums are to be neglected and in the remaining two sums the lower bound of the p-summation is to be replaced by P = 0. (2N.2N')(M) SN,.N_
= y(2N, 2N')
[
N,
NI +N,-N-q-1 N-N'+q-1
q=0
p=0
2N'-N, -N, +p 2
a'=0
v!A!p!q!
A=0
2q - 2A)! (N'-q-A)!(N-N'+q-v+A)!(NI +N,-N-p-q+v)! (2N'-
X
X
+
(2N-Nj-N,+p + 2q-2v)!(2 N'-Nl-N2+p)! (NI -p-2q)!(2N-2N, -N2 +p+2q-2v)!(2N'-Nl -N2 +p-2A)! T( M+N'-q-A)T( ;M+N+N'-NI -N2+p+q-v)
Y(2N,2N")
Na l 1 2 1 N, +N__N_q_1 q=0
2N -2N, -1!,':+p+ 2g 2
2N-N I -N,+p+ 2q -2v 2
v=N-N'+q
11=0
p=0
v!A!P!q!
X
(2N - 2v - 2A)! (N-v-A)!(N'-N+v-q+A)!(NI +N2-N-p-q+v)!
X
(2N-Nl -N,+p+2q- 2v)!(2N'-N I -N2 +p)! (N i -p-2q)!(2N-2NI -N2 +p+2q-2v)!(2N-NI -N2 +p+2q-2v-2A)!
X
I'( 1 M+N-v-A)T(2M+N+N'-N I -N2 +p+q-v)
+ y(2N,2N')
[
Ni 1 2 1
NI -2q
q-1
q=1 p=N, +N,-N-q v=0
[ Ni +N2 -P - 24 2 1 11=0
v!A rP!q!
zq - 2A)! (N'-q-A)!(q-v+II)!(N-N i -N2+p+q+v)! (2N'-
J. Hofmann / Anomalous dimensions
X
X
813
(NI +N2 -p - 2v)!( NI +N2 -p-2q)!
(NI -P-2q)!(N2-P-2v)!(N,+N,-p-2q-2A)! T(M+N')T( '-, M +N) T(2M+N'-q-A)l'(2M+NI +N2 -p-q-v) NI
+ Y(2N,2N')
N,-p 2
N, +N,-p-2v 2
q=0 p=N, +N,-N-q
v=q
A=0
(2N'- 2v
- 2A) !
E 2
1
NI -2q
1 v!A!P!q!
(N' -v-A)!(v-q+A)!(N-Nl-N2+p+q+v)!
X
X
(Ni + N2 -p-2v)!(NI +N2 - p - 2q)!
(Ni -p-2q)!(N2 -p-2v)!(N1 +N2 -p-2v-2A)t T(2M +N')T(2M+N) T('-,M+N'-v-A)T('-,M+N I +N2-p-q-v)
with Y(2N,2N') given by
or :=
1 1 2!
for N, * N2, for N, = N2,
For NI + N2 - N - q < 0 the first two quadruple sums are to be neglected and in the remaining two sums the lower bound of the p-summation is to be replaced by p = 0. In calculating the tensorial contraction SN~NÏV,N ")(M) one will occasionally encounter the situation that the expression 2N - 2N, - N2 + p + 2q - 2v occurring in the first two quadruple sums becomes negative for some values of q, p and v. These contributions are to be skipped.
814
J. Hofinar:r: / Anomalous dimensions
In the cases N, = 0 and N, = 1 the last two quadruple sums reduce to simple summations : 2N( N!)` aN) (M) 1 _ S(Z(a.I N (2N)!
N
1 N 2N-2v T(;M+N +v) 92 ( )( N-2v )F 2
1-0
N,
4N(2N-N,)!N,!(N!)2
2N !" 2
1-(IM +N) 2 1 T _'-,M + N,_) v-0 v!(N - N2 + v) !
(N) 2N- 2.) ,r( -, M +N+ v) v N,-2P T('-,M + N - v) ' Sn-''N-2N')(M) =
F
(2N)! I T('2 M+N) N 1
X
N) Si,_N_(
M) _ ~ .
x
2N'-2v T( 1M+N+v) N-2v ) F(-2 M +N'-v ) '
N' 2 v
4N (2N - N2) !N2!(N!)2 112 2 " [(2N) T('-, M+N) T('-,M+N2 )
2 v-0
( N2 -2v ) M + 2N2 (N-v)-2(N2 -2v) v!(N-N,+ v)! _
x (N) 2N-2v T( '-, M +N+v-1) + N2 +1
v
N2 - 2v
N, + 1 2
X
x
N (v
T( _2 M+N _ v)
T( '-, M +N) 2N-N2 T( _2 M+N2 +1)
(N2 -2v+ 1)M+2(N2 + 1) ( N- v) -2(N2 -2v+ 1)
T('-,M + N + v - 1) 2N-2v N2 -2v+ 1 ' T( 2~ M+N- v)
!. Hofmann / Anomalous dimensions
815
The definition in eq. (A.1) is introduced in such a way that in the scalar case M = 1 the tensorial contractions reduce to the multiplicities of the corresponding diagrams: ) CpN 1(1) (2N)
= 1,
fBN)(1)
1 2N (1) = 2 N
S(2N) (2N) (1) N, -
S(2N,2N,)(1) N1 , N,
_ d,
C(2N,2N')(1)
,
2N
= 1,
(2N- N, N2
Nl
2N')
Ni
2N'-
Nz
)
= 2N' N ,
,
Ni .
(A .2)
)
As a consequence of the finiteness of the beta-function and the various scaling functions there exist a number of relations among the tensorial contractions enlisted above. They read
Sr~`N_ NI(M) = 0,
_
[S(2N)(M)l2-S( N)(M)
2~DN)'(M)C(2N)(M) -
2N ( N
~(2N)(M) + ~(2N .2N')(M )J y(2N,2N')(M) N 2
N =1
N N~
_ SN NN NN1( M )
2 N'
N
FNI = 1
e(,Nr )(
M) = 0,
(A .3) (A.4)
7S ~2N.2N ,)(M )
N ( NI
SN1;NN')(M)
=0
and may serve as a check of the calculations . The second equation can easily be established using the explicit expressions while for the first and third relations no proof could be given in view of the complexity of the tensorial contractions involved . Nevertheless they may be verified for any definite value of N. For N'== N the third equation reduces to the first as might be seen from the following relations : C(2N,2N)(M)
SN N,N)( M)
=
-
2~(2N)(M)
'
r SN Nmin N,,2N-N -N,)(M)
1
d between the ten rial contractions with one and two upper indices th th ten ri to the order nee
contractions the E-e
4_2,v1(
t2
l
1)UNE + O(E2)] ,
2=1z2
?qx? -1k
nsion of the various i te
1 73
P)
E
,
Ne + O(e2)], I
[2 N + ( N-
1)WN e+O(E2 )], "(Pl + . . . +P .V) Iqzu = -[2+(NE _I),-
_N N - 1
(NI , N2)+O(1), for N,+N,*N,
42N1
EN., ,\,_(p,
a 7-
sp
` [2 + (N - 1 -
2W + .;
N
E +O E
)],
for N, +N2 = N, where the quantities -r, Uv , VN , WN, EN(N,), and ON(N,, N2) appearing on the right-hand side are declared through eqs. (13)-(18) .
[1] [21 [31 [41 [51 [61 [71
References F.J. Wegner, in Phase transitions and critical phenomena, vol. 6, ed. C. Domb and M.S. Green (Academic Press, New York, 1976) p. 7 C. Itzykson and J.M. Drouffe, Statistical field theory, vol . 1 (Cambridge Univ. Press, Cambridge, 1989)321 A.M. Polyakov, Pis'ma Zh. Eksp. Teor. Fiz. 12 (1970) 538 [JETP Lett. 12 (1970) 3811; 66 (1974) 23 [39 (1974) 101 A .A. Belavin, A.M . Polyakov and A.B. Zamolodchikov, Nucl. Phys. B241 (1984) 333 D. Friedan, Z. Qiu and S. Shenker, Phys. Rev. Lett. 52 (1984) 1575 J.L. Cardy, Conformal invariance and statistical mechanics, in Les Houches, Session XLIX, 1988, Fields, Strings and Critical Phenomena, ed. E. Brézin and J. Zinn-Justin (North-Holland, Amsterdam) to be published E. Brézin, J.C. Le Guillou and J. Zinn-Justin, in Phase transitions and critical phenomena, vol. 6, ed. C. Domb and M.S. Green (Academic Press, New York, 1976) p. 125
matin:1s
bp
~~cb% `:
ate= D.J. Anit, F ""_ c 1984) [9] D. e in and Ha Wagner, . LetL 17 (1966) 1133 [1®] P.S. Howe and P.C. West Phn Leit. 09») 371 [ID] LN. J h t , . ) (1 [12] I.D. Lawrîe and S. in Phase transitierts and J.L. itz ( Y 19SA) . 1 [13] A.L ° and . 18 F.W. (1 [14] P. Gi tormal field theory. in and Critical na, ed. E. J. Zia J [8]
published us, F. [151 W.
,i -w
r
lL
Làatr4. oi
c LLl4a
h-1161L,~U
ttinger and R . . P SonL Formulas and theorem
mathematical physics (Springer, Berlin, 1966) p. 15
[16] MJ. Stephen and J.L. Lett. A (1 89 [17] F. Brizin, J.C. Le Guil a J. Zinn-Justin. . D15 (1977) 1 [18] J.C. Le Guillou and J. Zinn-Justin, Rev. Lett. 39 (1977) 95
ta-