PHYSICAI, REVIELV D VOLUME 2, XUMBER 8 15 OCTOBER 1970

Anomalous Dimensions and the Breakdown of Scale Invariance

in Perturbation Theory*

Sianjord Linear Accelerator Cetzter, Stanford Unicersity, Stanjord, California 94305

and
Laboratoryjor n'z~clearStztdies, Cornell L'niz'ersity, Ithaca, Neal York 1485Ot
(Received 25 May 1970)

Canonical field theory predicts that a zero-mass scalar field theory with a A+4 interaction is scale invariant.
I t is shown here that the renormalized perturbation expansion of the X44theory is not scale invariant in order
XZ. Matrix elements of the divergence of the dilation current D,(x) are computed in order XZ using Ward
identities; it is found that VPL),(x) is proportional to X2+'(x). I t is also shown that the dimension of the field
d4differs from the canonical value in order A, and that this result leads one to expect a X2+*term in V'D,.
I t is also found that matrix elements of the composite field + 4 ( x ) in perturbation theory have troublesome
singularities at short distances which force one to give careful definitions for equal-time commutators and
Fourier transforms of T products in the Ward identities involving this field.

I. INTRODUCTION where K is a reference momentum that is introduced as

part of the Gell-Mann-Low renormalization procedure,
I N a previous paper a new theory of the short-distance
behavior of strong interactions was proposed.' The
theory involved several unfamiliar ideas, in particular,
and e, is a renormalized coupling constant defined rela-
tive to the reference momentum. The reference momen-
tum is necessary, for without i t the renormalization
the idea of an "operator-product expansion" and the procedure would replace ultraviolet divergences by
idea that the dimensions of quantum fields are changed infrared divergences. The form (1.1) is a sum of lead-
by interactions between the fields. The present paper ing logarithms for each order in e,. I n contrast, if the
is one of a series2 designed to make these ideas come renormalized perturbation expansion were scale in-
alive. These papers concern nontrivial problems in variant, the leading logarithms would be required to
perturbation theory or soluble models; they show how sum to a power of k2.
operator-product expansions or dimensions changing A tentative explanation -will be proposed here for this
with the coupling constant are involved in the solution
puzzle. T o simplify matters, the interaction of a
of these problems.
scalar field 4 with zero mass will be discussed instead of
The purpose of this paper is to study a puzzle in zero-mass electrodynamics. At the heart of the explana-
renormalization theory. The puzzle is as follows. Nor- tion is the result (to be derived in Sec. 111) that when a
mally, when the unrenormalized Lagrangian is in-
renormalized Heisenberg composite field is defined start-
variant to a symmetry, the renormalized perturbation
ing from the product 94(x), the resulting field changes its
expansion for the Lagrangian is also invariant to the dimension in the presence of interaction. However, the
symmetry. This is true for internal symmetries such
dimension of the Lagrangian cannot change, so X must
as isotopic spin; it is also true of Lorentz invariance. acquire compensating dimensions. Then h ceases to be
However, there is an exception, the exception being
a dimensionless constant, and there is no longer any
scale i n ~ a r i a n c e .For
~ example, the unrenormalized
reason to expect the theory to be scale invariant. This
Lagrangian for the electrodynamics of zero-mass elec-
is the essence of the explanation given in Sec. 111.What
trons is scale invariant (because the only parameter in
is meant by a change of dimension for +4 will also be
the zero-mass Lagrangian is the bare coupling constant
explained precisely. The idea of the constant X changing
eo, which is dinlensionless). However, the renormalized
dimensions, however, will not be discussed in detail;
perturbation expansion for zero-mass electrodynamics
instead it will be argued that the change of dimension of
is not scale invariant. The renormalized zero-mass per-
+4 leads to a term proportional to appearing in the
turbation expansion was defined by Gell-Nann and
divergence of the dilation current, thereby spoiling
LOW?The photon propagator in the zero-mass theory
scale invariance.
has the approximate form6
In this paper the scaling properties of the theory
D (k) = (k2)-'11 - (e2/12n2)ln(k2/~2)]-1, (1.1) will be inferred from Ward identities involving vacuum
expectation values of the fields +(x), $ J ~ ( Xand) , the
*Work supported by the U, S. Atomic Energy Commission.
t Permanent address. divergence of the dilation current, called S(x). These
K. Wilson, Phys. Rev. 179, 1499 (1969). Ward identities will be used to calculate matrix elements
2 The other paper in the series is K. Wilson, Phys. Rev. D 2,
1438 (1970). of the divergence S(x), given matrix elements involving
For a discussion of scale invariance according to canonical only +(x) and 94(x). I t is possible to calculate matrix
field theory, see G. Mack and A. Salam, Ann. Phys. (N.Y.) 53, elements of S directly without using the Ward identi-
174 (1969). See also Ref. 6.
"1. Gell-Mann and F. E. Lo~v,Phys. Rev. 95, 1300 (1954). Tlzeory of Quantized Fields (Interscience, New York, 1959),
N. N. Bogoliuhov and D. V. Shirkov, Zntroductiolz to the Chap. VIII.
2 478
2 ANOMALOUS DIMENSIONS AND T H E BREAKDOWN OF SCALE. +. 1419

lies; doing so would provide a check on the calculations 11. DEFINITIONS OF T PRODUCTS
of this paper. A start on such calculations has been
made by Callan, Coleman, and J a ~ k i w Direct
.~ calcula- ?'he problem of defining T products will be discussed
tions of the matrix elements of S are not made in this primarily in terms of an example, the example being
paper because there are many problems involved with the T product of two currents.* Consider in particular
such calculations which do not appear in the calculation the propagator
+
of matrix elements of alone. Some of these problems do
appear in the calculation of matrix elements of +4(x)
and will be discussed later. But as far as possible this
paper relies on uncontroversial Feynman-diagram for-
mulas; this is for simplicity and to make clear that the
breakdown of scale invariance is an inevitable conse-
auence of these formulas. where j, is a conserved current in an unspecified field
I n calculating matrix elements of the operator +4(x) theory and f , means fd4x. The problem to be dis-
and in checking Ward identities involving these matrix
elements, problems arise which can be traced to an cussed here is this: How is the integral in Ey. (2.1) to
age-old problem: What does a T product of operators be calculated, assuming the function D,,(x) is lmown?
such as T + ( z ) + ~ ( ~mean
) when x =y? Axiomatic field This is a question which does not arise much in practice
theorists answer that it is arbitrary in the sense that since one is more likely to have an explicit formula for
one is free to add any term proportional to 6"x-y) or D,,(p) (via Feyninan graphs, or whatever) than for
derivatives of a4(x-y) to the T p r ~ d u c t Other
.~ field D,,(x). However, Ward identities are derived in x space
theorists take i t for granted that the T product is and then Fourier-transformed to momentum space; if
uniquely defined, without making clear what that defi- one is deriving a Ward identity for DM,($),then D,,(p)
nition is. I n order to get consistent results in this paper, is defined by Eq. (2.1) and it becomes a legitimate ques-
it will be necessary to specify a definition of the T tion to ask whether ambiguities arise in computing the
~ r o d u c twhich eliminates the arbitrariness. There will integral, and how to avoid them if they do occur.
be a corresponding, precise definition of the equal-time
commutators which occur in Ward identities. I t will The reason that the integral in Eq. (2.1) can cause
be shown that under normal circuinstances the defini- difficulties is that D,, (x) is singular a t x = O ; the singu-
tion of equal-time commutators given in this paper larity a t x=O is such that the integral may be condi-
agrees with the customary one, but in abnormal~c~ses tionally convergent or divergent a t x =O. If the integral
(one of which occurs later in this paper) the two defini- is conditionally convergent, it can be defined by specify-
tions do not agree. There will also be circumstances ing an order of integration for the four integrations (over
where the definition of the T product given here has to the components of x), but the result may depend on
be modified to include subtractions; an example of this which order is chosen. If the integral is divergent then
..
also occurs later in this DaDer. The definition of the T i t can only be defined by subtracting the divergent
product given in this paper may or may not be one that terms.
field theorists can agree upon ; what is essential is that An example of conditional convergence is provided
in all future discussions of Ward identities the definition by a free vector-meson propagator. I n this case i t will
of the T product be stated, so that one can handle more be shown below that the integral in Eq. (2.1) gives
easily the kind of problem that arises later in this paper. different answers depending on whether the x integral
I n Sec. I1 the problem of defining T products is or the xo integral is solved first. I t will also be shown
analyzed, with examples showing the problems that that the usual noncovariant form of D,,(p) is obtained
can arise. I n Sec. 111, which is the heart of this paper, by solving thex integral first. These results will be shown
the Ward identities and explicit formulas for vacuum by using one of the standard derivations of the non-
ex~ectationvalues of d and d4 are written down. These covariant propagator and being careful when the order
fo;mulas are used to show that scale invariance holds of integration is changed. The standard derivation will
in order X and breaks down in order X2, to compute the first be stated without being careful; the careful deriva-
dimension of +4 in order A, and to infer that S(n) in tion will be given afterwards.
order X2 is proportional to +4. I n Sec. IV the operator- The non-time-ordered matrix element
+
product expansion for (x)d4(y) is discussed ; also, the
dimensions of the composite field +,(X)+~(X) in an iso- -.- ppv(x)= (Ql j,(x)jv(0) 19) (2.3)
spin-1 +4 theory are computed and shown to be different 8 The 'Lnoncovariance"of the propagator of a free vector-meson
for the isospin-0 and isospin-2 components. field is discussed in Ref. 5 , pp. 141-142. For more general cur-
-- rents, the problem is discussed in K. Johnson, Nucl. Phys. 25,
T. G. Callan, S. Coleman, and R. Jackiw, Ann. Phys. (N. Y . ) 431 (1961). For more recent discussions of the "noncovariance"
59, 42 (1970). of T products, see R. F. Dashen and S. Y. Lee, Phys. Rev. 187,
' An excellent discussion of the ambiguity in T products is 2017 (1969), and referencescited therein; D. Gross and R. Jackiw,
given in Ref. 5, pp. 144-145 and 168-191. Nucl. Phys. B14, 269 (1969).
(where j, is now the vector-meson field) is correctly before taking any other limits), + 0 second,
8 -+ 0 third, and 7 -+ 0 last. T o do the x integration
last, one takes the limits in the order e + 0 first, then
7 4 0, then a --t 0, then O -+ 0.
The integral with convergence factors present can be
conlputed explicitly when a , e, 7, and 8 are small,
neglecting srnall terms. The result is
where m is the vector-meson mass, f p means
( 2 ~ )f- ~d4p, and O(x0) is the usual 0 f u n ~ t i o nThe
.~ T
product D,,(x) =p,,(x) when xo> 0 ; for mo<O, D,,(x)
=pY,(-x). The propagator D,,(p) is

where Ds,,(p) is the standard form for D,,(p) given by

Ey. (2.8). If 7 and 8 are both small but of the same
order, the second term is of order 1 ; terms of order 7,
8, etc. have been dropped. If 0 -+ 0 Beeping 7 fixed, the
second term vanishes, leaving the standard form; but
Exchanging the order of integration so that the x inte- if 7 + 0 keeping 0 fixed, one gets
gration is done first, one gets a 6 function [either
63(p-q) or @(p+q)]. Doing the q integration next
DpY +
(p) =D S ~ ~ ( P (imp')
) (-$g,,+ . (2.1 1)
+6,06~0)
eliminates the 6 function; then one does the xo integra- Hence the order of integration in Eq. (2.1) is significant;
tion, leaving to get the standard form for D,,(p), one must write

For any finite 7 the point x=O is excluded from the

Using the explicit form for p,,(p) gives
integral. However, except for this point the function
D,,(x) is covariant [the Fourier transform of the non-
covariant piece of D,,(p) is proportional to a4(x) and
vanishes if x,.f0]. So the noncovariance in D,,(p) is
where 6,0 is the Kronecker 6; thc 6,06,0 term is the non- entirely due to the noncovariant definition of the inte-
covariant piece. gral in Eq. (2.12). This result can be shown directly. If
The integrals in Eq. (2.6) can be evaluated more care- D,,(p) is computed in ;L Lorentz frame moving in the
fully using convergence factors to make all integrals z direction with velocity v, using Eq. (2.12) in the mov-
absolutely convergent. The orders of integration can ing frame, and then Lorentz-transformed back to the
then be exchanged legitimately. \Yith convergence fixed frame, one gets [by transforming Eq. (2.8)] a
factors, one has function D,,(p,v) :
D,,(p,v) =i(p2-m2+ie)-'

where the x integral has been written in terms of polar

coordinates (d0 is the solid angle differential). The con- The function D,,(p,v) must also result if one transforms
stants cr and e (which must be positive) make the in- the integral of Eq. (2.12) from the moving frame to the
tegrals absolutely convergent. By putting lower limits fixed frame. Since D,,(x) is covariant, the only change
9 and 0 on the xo and x integrations one can study differ- is in the boundary of integration; one gets
ent orders of integration for the x integral. ~ h u to
s find
the result of performing the x integration before the
xo integration in Eq. (2.1), one takes the following
limits in Eq. (2.9) : e -+0 first (to get the q integration
--
The metric of this paper is (1, - 1, - 1, -1).
2 ANOlIALOUS DIAIENSIONS AND THE BREAKDOITS OF S C X I , E . 1481

I
i.e., the region xO-ax3 1 <7(1 - . L ' ~ ) is
" ~excluded from The regions R1 and R2 are now
the range of integration. Since the scale of q does not
matter one can also specify the excluded region as R1: jxol < 7 , lxo-z\x3l > 7 , jxoj <to, and 1x1 < Y O ,
IxO-VX~~ <q. The difference between D,,(p,c) and R z : /x01> ? , / X O - C X ~ / < v , jx01 <to, and 1x1 < Y O .
D,,(p) must come from the difference in the excluded
regions. That is, The x integrations can be done explicitly; it is easily
seen that terms depending on Y o will not contribute in
the lirnit q + 0. With such terms dropped, the inte-
grals have the explicit form

where Rl is the region I xoI <v, I xo-z'x3 1 >7, and R 2 is

the region 1 xo/ >7, I xo-vx31 <q.
The regions R1 and R2 both collapse in the limit 7 -+ 0,
so for the limit 7 -+ 0 to be nonzero, D,,(x) has to be
singular within these regions. Both regions are spacelike
relative to the origin except for a region of linear size 7.
The function D,,(x) is singular only on the light cone
and a t x = 0 ; these singularities lie in the region of linear
size 7, and must be strong enough to overcome the
small volume of integration. I t is worth showing ex- where
plicitly how the singularity of D,,(x) a t z = 0 results in Y,= (xo-7)/~', (2.22)
a nonzero limit, for in doing so one can deduce a general
rule for when the integral of a I' product may be
noncovariant .
The explicit form of D,,(x) is known; it is1O [the symmetry for xo -+ -xo of Eq. (2.20) was used to
eliminate integrals with xo<O]. The xo integrations can
also be done explicitly; the result is independent of to
where Do(x)is the free propagator in x space for a scalar when 7 is small and gives
particle :

which agrees with Eq. (2.13).

The reason one can generalize the above calculation
For small 2, one has easilv is that its clualitative features can all be deter-
Do(x) = - (47r2)-'(x2-i€)-I. (2.18) mined by scaling arguments. The terms in 4 which re-
niain finite for 7 + 0 are unaffected by to and YO, and
The most singular term in D,,(x) for small x is in the leading approximation D,,(x) depends only on x
not on m2, except as an over-all factor. Hence 7 becomes
D,, (x)= (27r2m2)-I(- glrvx2+4x,x,)(x2- j c ) - - ~ . (2.19) the only dinlensional parameter in the integrals. Thus
Without affecting the lirnit (2.13, the regions R1 and R2 to get qualitatively the dependence of the integrals on
can be redefined to lie within the region I xo / <to, 1 x / < Y O , 7, one can replace xo and x by the dimensionless vari-
where to and r o are small but held fixed as 7 -+ 0. Within ables y o = x ~ / ~y=x/7,
, and collect factors of 7. When
this region, both DPv(x)and e L p ' z can be approximated yo and y are of order 1, the limits defining R1 and R2 do
by small-x expansions; as will be shown later, only the not depend on 7. So in Eq. (2.20) the substitution gives
leading terms from these expansions contribute to the
limit (2.15). Only the leading terins will be discussed
elplicitly. Also, for sinlplicity only the 00 corrlponent
of D,,(O) will be tiiscussed. Approliinating Doo(0) by
Eq. (2.19) gives

which is independent of 7; the regions R1 and R2

l0Equstions (2 16) and (2 18) can be derived from formulas in are incorrect by a minus sign and there are factors of i relating
.ippendix I of Ref. 5 (the equations a t the top of p. 652 of Ref. 5 the propagators of this paper to those of Ref. 5).
1482 K E N N E T H G. W I L S O N 2

v ~ c u u mpolarization here, so the divergences cannot be

removed b y a renormalization. The calculation here is
of the Fourier transform of the propagator of the cur-
rent; to remove these divergences, the Fourier trans-
Thus from a scaling argument one sees that A will be a form integral must be subtracted. As usual with sub-
constant for 7 --+ 0 (however, only an explicit calculation tractions, there is some arbitrariness in the exact form
can show that the constant does not vanish). One might of the subtracted integral. The calculation will be de-
worry about the effects of the light-cone singularity scribed briefly. The current j,(x) is
(yo= 1 y 1 but yof0) on the scaling analysis, but one can
see by tracing through the detailed calculation that the j p = :$(x)-!P$ (x) : , (2.28)
i c in x2-ie makes the light-cone singularity integrable where $ is a free Dirac field and :. . . : denotes Wick
and does not destroy the scaling arguments (provided ordering. The propagator D,,(x) is nomr
one does not choose t o and ro so that to2-ro2=0).
The importance of the scaling argument is that if one DPY(x)= -TrY,So(x)~ySo(-~), (2.29)
had extra powers of x or xo in the numerator of Eq. where
(2.20), the scaling argument shows that A would vanish.
This can be verified by explicit calculation. This means
that A does not change if one puts ewp'Z in the integral,
since the terms p.x, etc. in the expansion of
ezp."do not contribute in the limit 7 -+ 0. Likewise, less
singular terms in D,,(x) do not contribute to the limit. When x is small, the inost singular term in So(x) is
Hence, the explicit calculation gives the more general
result,
Doo(p,v)-Doo(p)=(-i/m2)v2/(t--v2), (2.26) As a result,

in agreement with Eq. (2.13).

Even more generally one deduces the following gen- for small 2. The integral JD,,(x)e""."d3x diverges as
eral rule. Let TOl(x)Oz(O) be a T product of two arbi- xo+ 0 ; froin a scaling argument the divergence should
trary local operators 01(x) and Oa(0). I t does not matter be proportional to xoP3.The divergence can come only
whether these operators are scalars, spinors, tensors, or froin I x j -xo in the integral, so i t is legitimate to use
whatever. Let the approximation (2.32) in doing the calculation of the
divergence. The integration can now be done explicitly
and gives

be the Fourier transform of an arbitrary matrix eleinent

of the T product. If the matrix element itself scales as
x - ~ +as~ x -+ 0, with d>O, then M(p) is covariant and There can also be terms of order I X ~ I - ~jxoj-',
, etc.
independent of the order of integration. The hypothesis Thus computing the integral of Eq. (2.12) gives a diver-
of operator-product expansions1 predicts that no matter gent result. The way to avoid this divergence is to sub-
what matrix element is considered, the leading short- tract the integral so that the scaling argument predicts
distance behavior of the matrix element will be a func- convergence. The simplest subtraction is to subtract a
tion of x only, except for an over-all factor [as was the Taylor's series expansion of e i ~ . Zone
; defines12
case for D,,(x)], so that the scaling analysis applies.
The conventional integral for D,,(p) can be divergent.
The current of a free Dirac field gives a simple example
of this. The divergence is simply the well-known diver-
gence in the lowest-order vacuum polarization diagram The leading singularity of the integrand now scales as
for electrodynamics. However, we are not calculating xP3 instead of N-~.AS a result, the scaling arguments
I T h e original limits I x01 <to, I xl <YO become 1 yo1 <to/?, show that D,,(p) is finite and covariant. The terms
/y1 I n the integrals of Eq. (2.25) these upper limits can subtracted are a quadratic polynomial in p. I n effect,
be replaced by rn without changing A, when a is small. I n scaling one has subtracted infinite constants multiplying p2, p,
analyses of more general problems [discussed after Eq. (2.26)],
replacing to/q, ro/q by m may lead to divergent integrals. Then and 1 from the old form of D,,(p). As usual, one is
one must make a more sophisticated analysis, using the scaling
argument only for values of y-1 and computing explicitly the " T h e subtraction i p . x might seem unnecessary since the inte-
integral for y large, i.e., for yss-l. However, the large-y region gral of zD,, ( x ) should vanish by Lorentz invariance. Unfortunately
will only give terms of order 7 since this region is away from the one often has to use a noncovariant definition of the integral, as
singularity of D,,; hence the scaling analysis will still determine in Eq. (2.12), in which case the integral of xll,,(z) might not
whether A can be nonzero for 7 4 0. vanish.
2 ANOMA1,OTJ.S DIhIENSIONS AND TfIE B1'\EAKDOLf1N O F S C A L E . . . 1453

always free to add finite constants times p2, P, or 1 to I t may help in understanding the problem of the
D,,(p); to keep D,,(p) covariant, the added terms il-~ust ainbiguity in U,,(p) if one can understand why it was
also be covariant. possible for nonaxioinaticists to conclude that D,,(p) is
Even for cases like the free vector-ii~esonpropagator, unique. The reason lies, I believe, in a conscious or un-
where the unsubtracted integral is finite. one is free to
u
conscious assumption that nonaxiomaticists make about
use a subtracted integral to define D,,(p). One can make the nature of field theory. The assumption is this: Any
as many subtractions as one likes, but one subtraction local operator, such as a current, becoines an observable
is sufficient to define a covariant form for D,,(P). when averaged over a region of space, the time being
Axiomatic field theorists have long asserted that the held fixed. By an "observable," I niean an operator
Fourier transforms of T products are ambiguous. There which can be multiplied by itself or by other fields,
is an excellent discussion of the role of these ambiguities without producing singularities. The best way to show
in renormalization theorv in Boeoliubov
" and Shirkov.5 that this assumption is made is to loolr a t the popularly
Nevertheless the popular view is that a Fourier trans- accepted form for an equal-time commutator. The
form such as D,,(p) is a unique and even physical equal-time comillutator of two local fields 01(x) and
quantity, a t least relative to a given Lorentz frame. The Oy(y) is expected to be a sum of 6 functions and deriva-
axiomatic view must in the end replace the popular view, tives of 6 functions in the spatial variables x and y.
since the ambiguity in D,,(p) in examples like the Dirac These G functions can be eliminated by averaging 01(x),
current of a free fernlion field is beyond question. Un- say, over a region of space; if p(x) is an averaging func-
fortunately, much experience has been acquired with tion, then [fp(x)O~(x~,x)d3x,0~(x~,y)] is completely
the unsubtracted form of the definition of D,,(p) and free of singularities. Even more, one assumes that the
more general .transforins like M(p) in Eq. (2.27). One unequal-time commutator [fp(x)Ol(zo,~)td~x,O:!(yo,~)]
inust now distinguish two problems. The first is, given is continuous and differentiable in yo for yo=x@ This
that the standard definition of the Fourier transform assuinption is implicit in the equal-time commutator
exists, to show in practical situations that no physics forniula
is changed by using a subtracted formula instead. This
may not be trivial to demonstrate but is not a very
rewarding subject to pursue. The second question is where H i s the Hainiltonian and the double commutator
what happens to the physics when subtractions are is again expected to be a suin of 6 functions. If the
necessary. There is already one example known where unequal-time coinmutator were not differentiable in yo
the necessity for a subtraction changes a current-a!gebra a t Y O = T O , then the equal-time commutator with 0,
prediction, namely, the Adler-Bell-Jackiw-Schwinger would diverge.
anoinaly which changes the current-algebra prediction Given the assunlption that integration w-ith x niakes
of the no lifetime.I3 One must be prepared to find other operator products be sinooth in time, i t is easy to derive
applications where subtractions have nontrivial effects. the usual form of the Ward identity for D,,(p) from the
I t is certainly worth looking for such effects, especially definition (2.12). One writes
when the use of conventional Ward identities gives un-
satisfactory results, as in ?1 decay.14

'" J. Schwinger, Phys. Rev. 82, 664 (1951); J. S. Bell and R.

Jackiw, Nuovo Cimento 60A, 47 (1969); S. L. adler, Phys. Rev.
177, 2426 (1969). For a discussion explicitly in terms of divergences Integrating by parts, one gets
and subtractions in Fourier transforms of T products, see K. G.
Wilson, ibid. 181, 1909 (1969). For further references see R. il'.
Brown, C-C. Shih, and B. L. Young, ibid. 186, 1491 (1969).
l4 See Ref. 1 for a possible resolution of the ?-decay problem and
further references. The explanation of 7 decay offered in Ref. 1
.fails if all nine pseudoscalar fields are di.vergences of currents, as
in the quark model. The reason is as follows: According to Ref. 1,
the ?-decay amplitude when the r o four-momentum is zero is
given by a matrix element (? I [,fu3(0),Qa3]17rC~-), where f is a
coupling constant, u3 is the third component of the isovector u
field, and QAJ is the third component of the axial charge. Since Since j, is assunled to be conserved, V@j,(?z)is zero, and
/
the TOhas zero four-momentum, the full four-momentum of the since xois never zero in the integral, V,(O T j , (x)j, (0) j O)
7 is carried by the 7r4 and 7r-. Hence the commutator must not
equal a divergence, for any divergence has a zero matrix element I I
= (0 TV.j,(x) j,(O) Q ) =O. So the first term vanishes,
between states of the same four-momentum. But in conventional and one is left with the surface terms. These terms may
SU(3)XSU(3) the commutator is one of the pseudoscalar fields. be written as follows. Let
One can arrange that the commutator is not a divergence by
assuming that there are only eight axial-vector currents instead
of nine (this was done in Ref. I), or by assuming that the field zo
introduced in Ref. 1 does not commute with the ninth axial charge.
See S. Glashow, in H n d ~ o n sand I'izeir Interactions, edited by
A. Zichichi (Academic, New York, 1968); and A4. Gell-Mann,
Hawaii Summer School lecture notes (Caltech report, 1970) out by G. Preparata (private commu~~ication) ; see also K. Ura~ldt
(unpublished). This difficulty in explaining 7 decay was pointed and G. Preparata (unpublished).
1484 K E N N E T H G . IVI1,SON 2

Then Ward identities rrlust be defined as a limit as in Eq.

(2.39).

111. SCALE INVARIANCE AND

PERTURBATION T H E O R Y
According to the assunlption stated above, the products
Q(p,4)jp(0) and jv(0)Q(p, -7) should be free of any T o begin this section, the commutators of the gen-
singularity for 7 7 - 0, in which case the limit gives erator of scale transformations will be derived. Ward
identities for the dilation current will then be written
for matrix elements involving the fields (b and (b4 of the
X(b4 theory. I t will be assumed, to start with, that all
which is the usual IYard identity relating p@D,,(p) to an
integrals of T products are conventionally defined and
equal-time commutator. If the assumption that Q(p,?)
all \Yard identities have their customary form. The
is an observable breaks down,15 the linlit (2.39) inay
exceptions will be discussed later.
not behave like a commutator, since the expression for
If the field theory is scale invariant,17then there exists
finite 4 is not a commutator. An example of this occurs a set of unitary transformations U(s) with the property
in Sec. 111.
The assumption that integrating an operator over
space only gives an observable is a basic tenet of
canonical field theory, since one builds the Hamiltonian The constant d is called the dimension of (b. The unitary
of a canonical theory out of space-averaged operators, transformations C(s) can be written in terms of an
and the Hainiltonian has to be an observable. The as- infinitesimal generator D :
sumption has been rejected by axiomatic field theory
fro& the beginning since the currents and other local
products in free-field theories violate this assunlption
(as is shown by the example of a divergent propagator The logarithm of s appears in the exponent so that U(s)
discussed earlier). I n axiomatic field theory one assumes will satisfy the composition law
only that operators averaged over space anti time give
observables; this hypothesis was formally stated by c- (s) U (s1) = L-(ss1) . (3.3)
Wightman, but the idea dates back to the discussion of Let s be l f t with e small. Then from Eq. (3.1) one
the measurability of fields by Bohr and Rosenfeld.16 derives
Unfortunately the assumption that space-time averages i[D,(b(x)] = ( d f ~ @ V , ) ( b (.~ ) (3.4)
give observables is not very helpful in dealing with the
specific problems posed by the singularities of T For each conlposite field in the theory there will be a
products. corresponding commutator. I n particular,
Some general conclusions of this section are as
follows.
(1) The precise definition for the Fourier transforin where d l is the dimension of (b4(x). The generator D is
of a T procluct in coininon usage is exemplified by Eq. expected to be the integral of a local "dilation current"
(2.12). D, (") :
(2) T products in x space are covariant; any non-
covariance in their Fourier transforms are entirely due
to the noncovariant 4 limit chosen to define the Fourier
integral.
D=
J DO(x)rl"~.
(3.6)

(3) The definition (2.12) is capable of giving diver- The current D, nlust be conserved if scale invariance
gent results, in which case a subtracted definition, as in holds, in which case D is time independent.
Eq. (2.34), will have to be used instead. Now consider the \Yard identities. T o allow for the
(4) If the integral of a T product is defined as in I3q. brealrdown of scale invariance, let L), have a diver-
(2.12), then the equal-time comnlutators appearing in gence S ,
-- VND, (x) =S (x) , (3.7)
l6 The operator Q(0,xo) is independent of xo because j, is
conserved; therefore it automatically satisfies the smoothness l7 For more detailed discussions of scale invariance, see, for free-
assumption. But Q(p,xo) need not be smooth in xo for nonzero field theories, J. Weiss, Nuovo Cimento 18, 1086 (1960); for
p. The problem of defining equal-time commutators within the interacting-field theories (including Ward identities), G. Mack,
framework of axiomatic field theory is discussed in R. Schroer Nucl. Phys. B5, 499 (1968). See also Refs. 1, 3, and 6, and D.
and P. Stichel, Commun. Math. Phys. 3,258 (1966) ; A. H. V8lke1, Gross and J. LVess, Phys. Rev. D 2, 753 (1970). Some recent
Phys. Rev. D 1, 3377 (1970); Free University of Berlin report papers are S. P. deAlwis and P. J. O'Donnell, Phys. Rev. D 2,
(unpublished):. 1023 (1970); L. N. Chang and P. G. 0. Freund, Caltech report,
"See A. Rightman and 1,. GBrding, Arkiv Fysik 28, 129 (1970) (unpublished); P. de Mottoni and H. Genz, Nuovo Ci-
(1965), especially pp. 131-133 and 153-154, and references cited mento 67B, 1 (1970); K. G. L\'ilson, STAC Report No. S1>,4C-
therein. PUB-i37 (unpublished); M. Gell-Mann (Ref. 14).
 -
2 ANOlIALOUS DIiLIENSIONS A N D THE BREAI<DO\YN O F S C A L E . . . 1485

and consider the matrix element

I?IG.1. ( a ) Feynman graph (a i

for self-energy function 2 ; (b)
Feynman graph for p.

where lfij is the vacuum state. Substituting VPD, for S

and integrating by parts, the conventional calculation
gives's
interaction representation, one defines (before re-
nl(x1. x.) =I] V,(QI7-+(11) . .+(X,)DV(Y)I a ) i?orrnalization)

The integral of the gradient vanishes and one is left

with the commutators. I t is convenient to bring the where + l ( h ) is the scalar field in the interaction repre-
derivatives 81, etc., outside the T product, which results sentation It', is the connected part of TI'. The matrix
in further equal-time coinlnutator terms. However, elements K , will be quoted to order A" the inatrix ele-
these further commutators cancel in pairs.lg Consider ments TI', to order A only. The vacuum expectation
the case n = 2 , for example. Then the result of moving value LTTC(y)wlll not be conlputed since it can be re-
the gradients is normalized to zero by subtracting a c number from the
Heisenberg field :+4:. Alatr~xelements involving pro-
ducts of two or more Heisenberg fields :c$~: will not be
discussed; hopefully the analysis of the ITr, functions is
sufficient to determine the properties of .+k. T h e non-
zero, unrenormalized graphs for K c and ITr, (to order
The two cornnlutator terms cancel. This is true for all X2 and X, respect~vely)are
lz ; thus
M ( x ~ . . . ~ , ) = i ( ~ t d + x l . ~. .l +. +x,.V,)K(xl...x,),
(3.11)
where
R(xl...r,)=(QiT+(xl)...+(x,)iC2) . (3.12)
The Ward identity (3.11) is the starting point of the
analysis of this section. If scale invariance is exact, M
must vanish. Therefore, we shall try to ~naliethe func-
where I)()) is the interacting-meson propagator, DO@)
tions M(z1.. .x,) vanish ill perturbation theory. The
is the free meson propagator with Lero mass, and
climensio~ld will be treated as a fudge factor chosen to
Zl (p2,A2)is the Fey11111an graph shown in Fig. I (a) com-
make M vanish if possible. This will be possible in order
puted with a cutoff A. Formul:is are
X but not in order X2. Having found that the functions M
cannot vanish in order X2, they will be calculated ex-
plicitly and used to infer the form of VfiD,.
Next some explicit perturbation forrnulas will
be written out for vacuum expectation values in-
volving +(x) and +4(x). Only connected graphs will
Z(P',A~)= 1 , ~(~-B)
P(~~~~\~)D (3.17)

be considered (disconnected graphs will be discussed

later). Let K,(xl...x,) be the connected part of
(QT+(xl). . .+(x,) la) and let Tlr,(xl. . .x,,y) be the
connected part of the matrix element
p(q2,A2)= i
I: DO(h)Do(q-k)D~(lz,li)D~(q--7~,

Do(lz,A)=A2(ii"--k2-ie)-l.
A), (3.18)

(3.19)
p(q2,A2)is the Feynman graph shown in Fig. 1 (b), also
with a cutoff. Calculation of p and 2 in the limit of
By :+4(y): is meant a Heisenberg field that reduces to
large cutoff gives (see the Appendis)
the Nick product :+4(x): in the free-field limit. I n the
l8 Surface terms at t i m e y o f 0 are neglected. In a zero-mass
theory this can be a mistake; i t is assumed here that t h e neglect
is legitimate.
l9 D. Gross anti J. Wess ( R e f . 17).
u h ~ r ec and cl are i~unleric:~l
c o t ~ s t m ~ tterins
s; of order formulas are relatively silnyle because tile mais of 4
q2/A' or sillaller for large A have been dropped. These i q zero F i ~ r t h formulas
~r are

=- ~ ~ ~ ~ D ~ ( ~ I ) D o ( P ~ ) ~ o ( P ~ ) D o ( - P I - P ~ - P ~ )
JCc(p~l)2,p3)
x {1- 1 2 ~ ~ [ ( p l + p a ) ~ , h ~12hp[(p~+~3)',-k2]-
]- 1 2 ~ p [ ( p z + p 3 ) ~ ~ ~ " } (3.22)
.
I t is a nuisance to write out terms which differ only by a permutation of the momenta, so in the following for-
nlulas only the number of such terms will be given:

K,(pl.. a*:) = -576X"~o(pl). . .Do(p,)Do(--$l--. . . - p ~ ) { D o ( p l + p 2 + ~ ~ ) + ( 9permutations)), (3.23)

I V c ( ~ *. .*xW,Y) = lL.l,,L
.. ~-~,,I.(.*I-u). .. TVc(pl. . .pu),
e-i~n. (~vL-u) (3.24)

.JVc(Pl,pz)= - ~ ~ X D O ( P ~ ) D O ( P Z ) [ ~ ( ~ I ~ , ~ ~ ~ ) + L : ( P ~ ~ , ~ ~ ) ] , (3.23)
I ) o ( ~ . ~ ) ( ~ d2]+(5 permutations of the X term)) ,
[Vc($i,p2,p3,p4)= ~ ~ I ) o ( P ~ ) L ) o ( P ~ ) D o ( P ~-)12Xp[($l+$~)', (3.26)
Ml,(pl. . .PC)= -S'i6iXDo(pl). . .IJ)o(ps){Uo(p1-tp,i-p3)-t(19permutations)} . (3.27)
---

The renormalized formulas for Kc and 1;1', are ob- When L: is a first order contribution to l~t7,(pl,p2),
tained by modifying 2: and p and redefining the coupling the nlodifications have a different interpretation. If the
constant but otherwise using the fornlulas given above. unrenormalized forinula for vPc(pI,p2)is Fourier-trans-
The renorinalized Z is obtained by dropping the con- formed to x space, one obtains (see the Appendis)
stants c and cl and replacing A2 by an arbitrarily chosen
but fixed "reference momentum" K ~Likewise,
. the re- ~r',(rl,xz,y)= ( & T - ~ ) X { D o ( x 2 - Y ) [ ( x l - y ) 2 - i e ] - 2
normalized p is obtained by replacing 1i2by K ~The . re- + - D o ( x l - y ) [ ( x z - y ) 2 - i e ] - ~ -192X(c112)
normalized functions ZB and pn are
XDo(zl-y)Do(x~--y) -96hicl[Do(x2- y)P(xl-y)
ZR(q2)=-(512~4)-1q21n[(-q2-it)j~2] (3.28) S D O ( ~ L - ~ ) ~ ~ ( ~ ~(3.31) -Y)],
and
~~((1~1 (3.29) where Do(%)is the Fourier transform of Do@), and the
The rationalization of these modifications is as f o l l o ~ ~ ~first
s , term is correct only for xl-y and x2-y nonzero.
~h~ function occurs in tTvo different forlnulas; the The term proportional to c can be rewritten -96hc1i2
modifications have a different significance in the two X (Ql T@r(x1)4r(x2):$r2(y) : 1 :. Replacing c by 0 is
cases, hi^ is also true of the function p, 1\'hen 2 is a epiva.lent to subtracting -96cXii2:4'(x) : fronl the un-
correction to the propagator, the nlodifications amount renori~lalizedoperator :+4(x:):. This subtraction is one
to a lnass and wave-function renormalization, T~ par. of two needed to define a finite reilornlalized form of the
titular, replacing bJ- zero ensures that the renor.nalized Heisenberg field :qj4(x):. The other subtraction needed
l-r,ass is zero through order 12; replacing cl by 0 ~"0 define the renorrnalized form of :d4(x): is a subtrac-
by Kz are both wave.function renormalizatio,ls. ~t is tion proportional OX,:$^:. Thissubtractionis generated
necessary to introduce the arbitrary parameter K (which when one A b!- K ill the function P, P being con-
has the dimensions of a mass) into the theory because sidered as a correction to the function wC(p1,p2,p3,p4).
there is no naturally occurring parameter with the Replacing cl by 0 in TV,(pl,p2) is sinlply a redefinition
dilnensions of a mass to replace the cutoff inside the of the Fourier transforln of TI.'c(xl,xz,Y). When
logarithm. The value of K is unilnportant since challging LV.(~I,.~~,Y) Fourier transformed, the cl term in
only changes the normalization of the field 4, which is 1VC(x1,"2,y)will not contribute because by definition
arbitrary. Similarly, whenp is a correction to K,(p,pl,p,) the points x l = y and xz = y are excluded froin the region
the modification of is a coupling.constaIlt renormaliza- of integration (see Sec. 11). However, the unsubtracted
tion; when is replaced by one lnust also replace Fourier transform of FP,(xl,xz,y) diverges because of
by a renorlnalized coupling constant 1,. The renor- the singularities [(XI- y)2-ie]-2 and [(x2- yj2-i
lnalized coupling constant depends on in the sense that in the first term of Eq. (3.31).21This means that the
if ti is changed to one must A, to A,,, with Fourier transfornl must be subtracted. The unsub-
tracted Fourier transforin would be
(3.30)

of K."
X,p=A,+ (9h,2/4~9ln(~"/'K~)+[order (X:)],
in order that K,(pl,pz,p3) be independent of the choice
-
-
il',(pl,p2) = llL2 c"71Llei~"Ti7 ,(~l,xa,O).
. (3.32)

20 For further discussion of the dependence of the coupling con- 21 These singularities cause a logarithmic divergence; this can
stant on the parameter K, see Ref. 4. be shown using the methods of Sec. 11.
2 ANOiVTALOUS D I M E N S I O N S A N D TI-IE B R E A K D O i T ' N OF SCALE... 1487

The singular term for s1-i 0 in the integrand has the part of K by the same ecluation. One can also define
Form
~"""'""&T-~)XD~(xz) (x12- i ~ ) + .
The singular term in xl is present for any xz so onc
cannot approximate the x2 dependence of the singular
term. One cannot subtract this term unchanged because
it does not go to zero fast enough when x l + = . To avoid and obtain
an infrared divergence, one subtracts

where K ,is
, any four-vector with magnitude K ~ K @ = -K?.
Putting in the factor e f K . " l does not change the depen-
dence of the subtraction on pl and $2, and so i t is a
legitimate modification. The renorn~alized,subtracted
formula for TY,(pl,pz) is
I t is straightforward to obtain explicit formulas for the
connected parts of M to second order in A, and the
connected parts of V to first order in A,. The dimensions
ii and dl will be left as unknowns for the moment. For
example,
Mc(p) =i(2d-4-p.vp)D(p)
=i(2d-4-p.~,)Xi(p~+i~)-'
~ { 1 (3X,2/167i4)
+ ln[(-p2--i~)/~2]}. (3.40)
with X replaced by A, in W,(xl,xz,O) (and the c and CI
terms dropped). This formula reproduces the renor- Separating the tern1 where V, acts on (p2+ie)-l from
nlalized form of FV, (p1,pS) [given by Eq. (3.25) with A, the term where V, acts on the logarithm, this becomes
replacing X and ZR replacing 81. M L ( p )=i(2il--2)D(p) -i(6X,Z/16a4)Do(p). (3.41)
The subtractions in Eq. (3.33) depend on pl and $2
in the form [Do(pz)+Do(pl)]; hence one is always free Hut to order XK2, one can replace Do@) by D(p) in the
to change the formula for W,(pl,pz) by adding a finite second term. The resulting formula for M,(p) and
constant times [Do(pl)+Do(pz)]. Changing Z n back ~~nalogousformulas for other M , and V , functions are
towards 2 by replacing K by A and adding the cl term
M , (pi =i[2tl-2 -3XK2/8rr4]D(p) , (3.42)
is exactly a change in I.V,(pl,pz) of this type. Hence cl
is a subtraction constant which one is free to set equal
to zero.
iYow study the matrix elements of the divergence of . .ps) =i(6d--6)RC(pl..
AWzir,(p~. .pd, (3.44)
the dilat.ion current, using the Ward identity (3.11). V,(pl,pz) =i(2d+dr-6)pV,(pi,pz)
First note that +3X,(8~9-~[Do(pi)+Do(pa)], (3.45)
V,(pl. . .p4) =i(4d+dr-8-9h,/7r2)
x rr ,(p, . .p4) , (3.46)
iT,(p1. . .p6) =i(htdfd~--10)TV,(pl.. ' ~ 6 ) . (3.47)
Using Eqs. (3.11) and (3.14) and an integration by Equation (3.45) for V, (pl,pz) is incorrect because its
parts, one gets derivation assumes that 11',(p~,p2)is unsuhtracted. Thc
~ o r r e c fornlula
t will be derived later.
The first application of Eqs. (3.42)- (3.47) is to show
that scale invariance breaks down in order A?. TOdeter-
mine the validity of scale invariance the equations for
M e will be discussed order by order (the equations for
V, will be discussed later). I n the free-field limit, the
only nonzero M , is M , (p) and i t too is zero if d = 1. This
agrees with the known result that the free-field theory
is scale invariant and 4 has dimension 1. T o first order
in A,, M,(p) and Mc(pl,p2,p3)do not trivially vanish,
The connected part of iCf is related to the connected but by setting d = l , both are zero. So we infer that
1488 K E N N E T H C . IYILSOl\r

scale invariance holds to order A, and d is 1 to this order. grate by parts, getting
I n order XX2 the situation is as follows. M c ( p l . ..p6)
vanishes because Kc(pl. . .p5) is already of order X,2 and V,(pl,p2) =lim i(2d+dr--8--ps. V,, -92%V p 2 )
?,-*O
6d-6 is zero to order 1. 'The function iMc(pl,pz,p3)
cannot vanish: Kc(pl,p2,p3)is of order A, and d is al-
ready determined to be l through order A,, so

The function Me($) vanishes to order A:, if d is where the integral over xl and xz still excludes xlo;
<7, / xzo / <7, and I xlo-x~, I <s. The term E (q,pl,pg) is
the sum of surface ternls. I t turns out that the surface
The nonvanishing of Mc(pl,p2,p3) in order X,2 ineans terms a t Ixlo-x~ol=s are negligible but the surface
that S(x) is nonzero in order X,2, SO scale invariance terms a t xlo=+7 or x z o = f q have to be computed
breaks down in this order. I t does not help to change d giving
in order to make Mc(pl,pz,p3)vanish in order X,2; this
wonld require a change in d of order A, which would
rnalie M,(p) nonzero in order A,, which would be even
~(q,p1,92) =ill [-~so*(xIo-~)+~Io~(xIo+~)

worse. I t will be assumed in what follows that d is

given by Eq. (3.49).22
I t appears that scale invariance is exact through order
A,. If so, the quantities V, must vanish to order A,. Con-
with the regions jxloI <?I, etc., still excluded. Because
sider first Vc(pl,pz,p3,p4). Since Wc(pl,p2,p3,p4) is of
order 1, V , vanishes only if of the 6 functions, the factors xlo and xzoare of order
7, so only the singular part of W,(xl,xz,O) is impor-
tant in the integral; for example, the integrals with
xlo= f q conle predominately from small xl. Hence, E
Since d is already known, this gives is approximately

Therefore, the dimension of :+4(x): changes in order

A,. T o order A,, Vkie(pl,.. .pe) vanishes [note that
RrC(pl.. .fie) is itself of order A,].
Before examining V,(pl,pz), the correct Ward iden-
tity for V,(pl,p2) must be obtained. To do so requires
careful attention to the definition of Fourier trans-
f o r m ~For
. ~ ~Vc(pl,pz) we shall use the standard defini-
tion [Vc(xl,xz,y) will turn out to be zero, so the standard
definition exists]. Thus
These integrals can be performed explicitly, giving

vC(pl,pz)=lirn
q-0 'i4x1iz20,
iZlO,
>? >v
d4X2ei~~.r~
T o complete the construction of the Ward identity one
must replace the unsubtracted Fourier transform of TIT',
in Eq. (3.54) by its subtracted form. The result is
The region ~ x l o - x z o<q~ is also excluded from the
integral. By analogy with Eq. (3.16).

with
When this is substituted in Eq. (3.52), one can inte-
22 I t was suggested by S. Coleman (private communication via
I<. Jackiw) that 4 has a dimension in second order despite the
breakdown of scale invariance. See the end of Sec. 111 for further
discussion.
23 There are many aspects of the derivation of the Ward identity
, ) should be examined carefully. In practice, only
for V , ( p ~ ~ pthat
one problem seems to cause difficulties, namely, the singularity
in the product T+(x) :+4(y) : for x + y, and only this problem will with i xlo/< q , etc., omitted from the integral. The inte-
be discussed. grals give 7-dependent constants multiplying the func-
2 A N O M A L O U S DIi?IENSIOhTS A N D THE BREAKDOTVN OF S C A L E 1489

tions Do(p2) and Do(pl). Using the values d = l and The operator D must contain an explicit time depen-
d1=4 to lowest order, one finds that F(~,pl,pz)=O. dence proportional to xJZ, where H is the Hamil-
Using these values for d and d l in Eq. (3.58), one has tonian17; this is necessary to give the xoVo+(x)term in
(correct through order A,) the commutator of D with 4. Therefore, let

The formula for dD/dxo is

This Ward identity has an extra term which does not
appear in the conventional form [Eq. (3.39)]. I t
is not caused by the subtractions in W,(pl,pz). I t came
from the surface terms E(~,pl,pz), arising when
x1-VWc(xI,xZ,O) and x~.VW,(x~,x~,O) were integrated The Hamiltonian contains the interaction term
by parts in the integral of Eq. (3.52). According to the
conventional analysis given earlier [cf. Eq. (3.10)],
these surface terms should have canceled. They would
have vanished had the assumption underlying the
conventional analysis been correct. Namely, if The contribution of H I to cdD/dxo is E~I-~[DA,HI]. The
,fd3xlT.V,(xl,a~,0) were a smooth function of xlo a t commutator of DA with :44(x) : is
xlo= O [and likewise for fd3xzWc(xl,x~,0)a t xzo=O],
then the integral (3.55) for E(?,pl,p2) would have been
of order 7. I n practice, the integral Jd3xlW,(x~,x2,0) is Integrating over x, and using an integration by parts on
of order 1 xlo I-' for xlo+ O and cancels the explicit the gradient term, one obtains
factor xlo in Eq. (3.55); hence, E(?l,pl,p2) has a finite,
nonzero limit for 17 -+ 0.
Using the explicit renormalized formula for W,(pl,p2)
Thus the contribution of the interaction to dD/dxo is
to order A,, one finds that Eq. (3.60) gives V,(pl,pz) =O.
- (dr-4)Hr, which is -A,(dl-4)fd3x :~$~(x) : . Using
Thus all the functions V , vanish to order A,, as expected,
Eq. (3.51), this is (-9X,2/7r2) f d3x :+4(x): . According to
ztnd the field :+4(x): has a dinlension dr given by
Eq. (3.62), the total dD/dxo is half of this, so there must
Eq. (3.51).
also be a contribution to dD/dxo from the unperturbed
Since Mc(pl,p2,p3) does not vanish, the operator S(x)
part of the Hamiltonian. This analysis shows that a
[the divergence of D,(x)] is nonzero. Can it be identi-
term of order A,2:44(x): is to be expected in V,D,(x),
fied? I t has been shown that all connected matrix ele-
given that :c$~(x): changes its dimension in order A,.
ments of S(x) vanish in order A,2 except for M,(pl,pz,pa),
T o conclude this section, the various assumptions and
and M,(pl,p2,p3) is proportional to Ko(pl,p2,p3), or, to
undiscussed problen~s will be listed. The above dis-
be precise, M , in order Xe2 is proportional to K c in order
cussion concerned only connected graphs but i t can
A,. Transforming to x space, and using the perturbation
be shown that the conclusions are unchanged by the dis-
formula which defines K c in order A,, Eq. (3.48)
connected graphs (such as the products of two propa-
becomes
~?fc(~1,~2,%3,%4)
= - ($Ar') 11z
(0 T41(~1)41(~2)
gators in the four-point function). The matrix elements
of two or more :c$~(x): fields were not computed (thus
avoiding the problems associated with the product
T :+4(x) :: c # J ~:(when
~ ) x =y). I n deriving V;ard identities,
the surface terms a t time f were assumed to vanish;
A conlparison of this formula with Eq. (3.8) suggests this should be checked by explicit calculation of the
that matrix elements of D,(x), since one is dealing with a
S(x) = - (9A,2/2) :44(x) :. (3.62) zero-mass theory. I n second order in A,, for which D, is
not conserved, it was assumed that the equal-time com-
This hypothesis gives back Eq. (3.61) and also makes +
mutator of D(xo) with could still be conlputed from
all other connected matrix elements M, vanish to the matrix element M,(p) as if D were conserved; this
order X,2. will have to be checked by explicit c a l ~ u l a t i o n How-
.~~
Can one understand how a term proportional to ever, even if this assumption were incorrect it will not
:+"x) : appears in the divergence of D ? I t will be shown change the calculation of M,(p1,p2,p3) to order X,2, since
that this is to be expected, given that the operator this calculation involves the commutator of D with 4
:+4(x): changes its dimension in order A,. To simplify only to order A,. Thus whatever the commutator of D
matters, consider not S(x) but rather the integral with @ is in order A,: there will still be a A,2 :44(x) : term
in S(x) ; there may be other terms also. The presence of
the X,2:+4(x) : term in S(x) makes it likely that the
equal-time commutator of D(xo) with +(x) will diverge
1490 K E N N E T H G. II'ILSON 2

in order AZY. This is because the integral (3.8) which given by the Eeynman rules is a term which in a space
defines M(x1,x2)diverges in order h,3 if S(x) is h,2 : + 4 ( ~: ;) involves 6 functions of xl or xz, which one is always
this in turn is a consequence of the nonintegrable sin- allowed to add to a T product, even if one of the opera-
gularity of IV(xl,x2,y) for y --t X I or 2 2 in order h,. tors in the T product vanishes. While there is nothing
Given that the interaction :+4(x): changes its dimen- wrong with adding S functions to the T product, it is
sion in order A,, why does not the free part of the not a sensible thing to do. I n any case, cl is a subtraction
Hamiltonian also change its dimension in order A,? If constant in a Fourier integral. I t does not matter
this were to happen, then scale invariance would break whether it is recognized as such or sneaked in by the
down in order A, instead of A.: This is another question device of subtracting :+V,C"+: from :+4(x): and using
that will not be discussed here. the Feynman rules to introduce a subtraction in the
The analysis of this section has been carried through definition of integrals of T products involving :+V,Vp+ :.
for the zero-mass theory. One may ask: Why not
work with the finite-mass theory instead? The reason IV. MISCELLANY
for not using the nonzero-mass theory is that when the
mass is nonzero the divergence S(x) contai~lsa term I n Sec. 111,it was necessary to know the behavior of
proportional to : # ~ ~ (:,x )which is nonzero in the free- the matrix elenlent (b2 I T+(xl)+(xz) : 4 ~ ~ (:yQ)
) for XI --t y
field limit. This means that the matrix elements or x z + y. This behavior was detein~inedby explicit
il.l(xl . .s,) will be nonzero in the free-field limit. T o calculation. This is a problenl which can be understood
show that S ( a ) contains a term proportional to A,L :+I~(x) :
in general in ternls of operator-product e ~ p a n s i o n s . ~ ~
in addition, one must cdlculate matrix elerrients of I n this section, the operator-product expansion for
:@(x): to order A?; one must also arglle that terms T+(x) :+4(y): will be discussed through order A, using
proportional to : + 4 ( ~ )are: not permitted to occur as the matrix element TV(x1,x,y) of :+4(y):. At the end of
part of the renortllalizatiorl of : + 2 ( ~:.) The argument this section, the dinlension of the field :@(x) : will be
cannot be rigorous, for if one is flexible enough about calculated through order X, for the case of an isospin-1
how one renormalizes, there is no argument that foibids field +: it will be shown that the isospin-2 component
the use of finite : + 4 ( ~: )counteitelms in renormalizing of :@: has a different dinlension (in order A,) than the
: + 2 ( ~:.) Furthennore, the zero-mass case is discussed isospin-0 component of :@(x) :. A similar isospin split-
because it is only for the zero-mass case that the can- ting was postulated in a previous paper1 to explain the
onical Lagrangian fornlulation of the theory predicts A I = g rule in weak interactions.
scale invariance, and, therefore, it is only for the zero- I n the free-field theory the operator-product expan-
nlass case that there is a contradiction between the pre- sion for the product T+(x) :~$~(y) : is derived from the
diction and perturbation-theory calculations. IVicli expansion of this product:
I n the renormalization of TV,(pl,pz), the constant cl T+(x) :+4(y): =4Do(z-y) :@(y) :+:+(x)+~(Y) :
was interpreted as a subtraction constant. I t is possible
to give the constant c1 a different interpretation. If one
:+
=4Do(x- y) :@(y) :+"y) :$(XP- yfi)
defines the renonnalized form of : + 4 ( ~ )to : include a x : m 4 ( y ) v n ( y ) : + . . . . (4.1)
subtraction proportional to cl:+VpV,C$:, this will also I n the final foi-nl of this formula, functions of (x-y)
eliminate the cl teim from TV,(pl,pz). This is because multiply local operators a t the point y ; any such for-
the matrix elenlent mula is called an operator-product expansion. The ex-
pansion is an expansion in terms of x-y and makes
sense when x-y is small. I n perturbation theory, one
looks for a generalization of Eq. (4.1) in the form

computed b y Feynman rules, is ( - - p ~ ~ - -Do(p1) p~~)

XDo(p,). This is proportional to [Do(pl)+Do(pz)], where the C,(x-y) are functions of x-y and O,(y) are
which is exactly the form of the cl terrn in Eq. (3.25) local fields a t y. The functions C,(x-y) may be sin-
[using Eq. (3.21) for Z]. This procedure for eliininating gular as x + y. The operators O,(y) are Heisenberg
the c1 term is more conventional than to interpret cl as a operators whose matrix elements will be functions of
subtraction in a Fourier integral. Unfortunate1~-,the A,; the functions C,(x-y) can also change with A,. One
procedure is nonsensical. The field :+1V,Vfi+1 : vanishes can separate the two dependencies because only
because +I (x) satisfies the free-field equation V,VP+I (a) C,(x- y) can depend on x and because the same func-
=O. This means that :+V,V'+: also vanishes in lowest
tions C,,(x--y) must occur no matter which matrix
order, so subtracting it from :C$4(x): does not change
: + 4 ( ~ ) : in order A,. Eurthenllore, the integral in Eq. 24 For background, see Refs. 1 and 2, and references cited

(3.69) should vanish since the integrand vanishes. How- therein. Ideas completely analogous to operator-product espan-
sions and scale invariance have bcen developed independently
ever, the FeJmman rules give a nonzero result for this for classical statistical mechanics by L. Kadanoff, Phys. T2ev.
integral. There is nothing wrong with this; the term Letters 23, 1430 (1969), and references cited therein.
 2 ANOR/IAI,OUS DIMENSIONS AND T H E BIZEAKDOIVN OF SCALE... 1491

element of T+(x) :+4(y): one studies. T o first order in The first matrix element is in lo~vestorder the free
A,, perturbation theory is scale invariant, which restricts propagator; comparing with Eq. (4.9) gives
the behavior of the functions C,(x--y). As shown in a
previous paper,l Cn(x) must scale as
The nlatrix element (Q 1 T+(XI):+3(y) :1 Q) vanishes in
order 1 and has not been computed here to order A,;
where d, is the dimension of the operator On(%).If the function Cz(z) is linown in order 1 from Eq. (4.1)
to be 4D0(z). Comparison of Eqs. (4.8) and (4.1 1) gives
d, =d,o+A,d,,1 (4.4)
and
cn (x) = C ~ O ( Z ) + L C ~ I ( X )
7 (4.5)
then the expansion of Eq. (4.2) to order A, gives The most singular term in the operator-product ex-
pansion of T + ( z ):d4(y): is the term Cl(x-y)4(y) be-
cause +(y) is the field of lowest dimension in the expan-
sion. I t is this t e ~ mthat has caused all the troubles with
[The dimensio~~s ( I and (11 are tnlten f~ 0111 Eqs. (3.49) subtractions and breakdown of conventional Ward
and (3.51).] identities in Sec. 111. To order A,, this term does not
To learn s o m e t h i ~ ~about
g the functions C', (.L- y) anci affect the other connected functions W,(n-l,xz,x,,x,y),
the operators O,(y) in order A,, we study the matrix etc., because Cl(z) is of order A, and the connected parts
element W(xl,r,y) for n- near y. The function IV(xl,x,y) of (D I T+(?cl)+(n2)+(xd)+(y)Is), etc., vanish in order 1.
has no disconnected diagrams [given that the vacuunl The analysis of the other connected matrix elements
:
expectation value (D / :+*(y): 0) is renormalized to l l ~ , ( ~ ~ , x 2 , ~ ~etc.,
xvill not be given.
, x , yfor
) , small x-y is complicated and
~ero], so W(xl,x,y) =W,(xl,x,y) which is given by the
renormalized form of Eq. (3.31) : I n a previous paper1 it was postulated that there
xvould be specific local fields of isospin $ and $ involved
in nonleptonic weak interactions, and that these fields
have different dimensions, the isospin-3 field being of
lower dimension than the isospin-2 field. If this is true,
I n terms of z=x-y, this is'"
it was shown that the A I = $ rule is universal, with all
AI=$ decays being suppressed by a power of (mlmw),
where m is a strong interaction mass (-1 BeV) and
m v is the weak boson mass or the equivalent. The
There are only two terins when TV(xl,n-,y) is expanded assumption is not true of the free-quark model. I n the
in z. Comparing with the operator-product expansion, free-quark model, the relevant local fields are the iso-
one should h u e spin-4 and # parts of the Wick product :j,, (x) jPst(x) :
~ v i t hjpn(x) being the chiral SU(3) currents of the
nlodel; both A I = + and AI =# components of the Wick
product have dimension 6. So it is worthwhile to con-
From the scaling law (4.6) the tern1 proportional to sider how perturbation theory changes the dimensions
( 9 - i ~ ) - ~n u s t involve an operator 0, of dimension of such a Q'ick product. T o simplify the calculation, a
d,,"= 1, while the term proportional to (z2-ie)-' nlust simple Wick product :+, (x)+, (x) : is discussed, where
involve an operator O,, of dinlension 3. There is only +$(x) (i= 1, 2, or 3) are the components of an isospin-1
one operator of dimension 1, namely, 4 itself. The co- scalar field. The interaction Lagrangian density will be
efficient (z2-ie)-l is a Lorentz scalar so it must in- -A[C,+,2(~)]2. Consider the matrix element
+
volve a scalar field 0,. 0, must be odd in since $ : + 4 :
is odd. The only possibilities are V,Vp+(x) and :@(z) :.
These are not linearly independent because they are 'I'o order A, this matrix elenlent (before renormalization)
related by the field equation of the d4 theory; it is con- is given by
venient to regard V,V"+ as the dependent field, so the
only field left is $3:. Therefore, the evpansion for
TB(xl,x,y) should heJb
rv(x1,x,y) =c,( 2 ) (Q I w ( x l ) + ( y ) I Q )
~ -
- +C2(z)(QI T + ( z ~ ):+"y) :1$1). (4.11)
" The zero-mass propagator Do(z) behaves as (zZ)-I for all z.
I t seems a bit strange that other local fields such as V,Vr:@(y) :
'b
do not occur in this expansion; presuinably they will i ~ involveti
e where p is defined by Eq. (3.18). The field :+i(x)+,(x) :
in higher orders i11 A., has isospin-0 and isospin-2 components. The isospin-0
1492 K E N N E T H G. W I L S O N

component is Ci:+i2 :; the isospia-2 conlpoiients can be tailled euactly in closed fornl, the result being
written as the traceless tensor :4i4j--$6ijCk:+k2:.
There is a corresponding decomposition of Nijkl(p,q) :

where Sois the isospin-0 ~oillpoilentof lATz,~l,

alld;l'z the
isospin-2 component. Using Eq. (4.16) and using the
form of p [Eq. (3.29)], one gets
rello~~nalized

with qLbeing replaced by q2+ie if necessary. For q2<<A2,

this reduces to

,> giving Eq. (3.20). For q2>>A" p is proportional t o

l he renornlalizntion is n n 'ive-f unction :eno~r~lnliaatitan .I4 1n ($/A?). The formula for 2 (p2,A?)is
(with different renormalization constants for the iso
spin-0 and isospin-2 con~ponentsof :+,+, :). Let do and
dl be the dimensions of the isospin-0 and isospin-2 con^-
ponents, respectively, of :+,+, :. The $Yard identities
which scale invariance imposes on iVo and N %'ire
The function p drops off rapid!y enough at l u g e q2 so
that the integral for Z converges (for finite A). The
function X will be calculated first for spacelike p, and
then determined for timelike p through analytic con-
tinuation. For spacelike p, one can choose a Lorentz
in the case of the neutral field theorj- ol Sec. li1, lrarrle in which po is 0. I n this frame the integral over q0
tl is 1 through order A,. Explicit calculation using Eqs. can be rotated from the real axis to the imaginary axis
(4.18) and (4.19) gives (counterclockwise). The result can be written in tern~s
of Euclidean four-vectors :

so in order A, the dimensions d o and d2 indeed differ.

Note added i i z $roc$. Other discussions of the break- where q is the four-vector (q1,92,~3,~4) (and siin:larlj- for
d.:)wn of scale invariance in perturbation theory have
heen given by CallenJ27 Coleman and J a c l i i ~ , ~ % n d
p ) 4 ~ similarly for q . p and
and q2 is ~ ; ? + q Z ~ + q 3 ~ + q(and
P". The integral over q can be performed in hyper-
%rnan~ik.~~
spherical coordinates :
ACKNOWLEDGMENTS (j1==qcoso, ('45)
The author thanks Professor R. Jacliiw and Dr. 'l'.
Appelquist for discussions. The author is grateful to
Professor S. Drel! and members of the SLAC iheorj
group for their hospitality during the author's stay at
SLAC.

I n t11is appendix the calculntion of p(p2,Li2)and

Z(p2.A2) [Eqs. (5.17) and (3.18)] will be desciibed L'erlorr~ling the angular integrations gives
1,rieiiy. Then the calculation of the Fou~iertransfori 1
of LV,(p,,p,) [Eqs. (3.25) and (3.21)] will be discussed
The calculation of p(p2.112)is a standard Feynma:~
diagram calculation. The answer for Gnite R can be ob
'7 C. Callen, Phys. Rev. D 2, 1541 (l970j.
j8 S.Coleman and R. Jackiw, MIT report (unpu'olish~.il)
7y I<. Symanzik, DESY Report 70/20 (unpublished)
2 AIiOILIAL,OUS DIhIENSIONS AND 7tlr BREr\KDO\\N OF SCALE... 1493

When p2 is small compared to A2, the integrals can be o>O, Po, . . . , p3 being the components of p. Then one
computed using the approximate form for p [Eq. (A2)] writes
except in a constant term (the second integral with p
replaced b y 0). The reslllt is Eq. (3.21) with

After s:!bs.iit:!ticg this forr~iulain Ey. (rZ12), the p

i;itc-;;,.:!tion c:ln i ) e dolie explicitly, ieaving
and c l = 3 ( 1 0 2 4 ~ ~ ) -The
~ . constant c is independent of
because p depends only on the ratio (q2/1i2).
I n Fourier-transforming W , ( P ~ , P ~the ) , only integ'tl
which is not already known is an integral of the form
[If the convergence factor is inserted in Eq. (A12), the
result is to cutoff the integral (A14) for w<02.] One can
change variables to v =w-I and then conlpute the inte-
gral, obtaining
For x = 0 this is highly divergent, but for x#O the ex-
u(x)= ( l / i ~ ~ ) ( ~ ~ - - i e ) - ~ . (Al5)
ponent serves as a convergence factor. If one wishes to
be careful one can insert an explicit convergence factor, The i e is present because x2 needs an irnaginarg. part
for example, exp(- i p ~ i o - I p ~ l o - Ipzlq- Ip,ld, with -ie to ensure that the integra! (A14) converges.

I'HYSICAL REVIEmT U VOLUME 2 , NUhlBER 8 15 OCTOBER 1970

High-Energy Behavior of Total Cross Sections

P. Yo~zrs*AND R. L. INGRAHA~I
Ilepartmelzt of Phsyics, New Mexico State Uakevsity, Las G u c e s , New Mexico 88001
(Received 26 June 1969; revised manuscript received 21 May 1970)

A method is presented for obtaining an asymptotic series, for large values of the energy, of a four-dimen-
sional Fourier translorm, using only one analyticity assumption. I t is show11 that this method iinplies (1)
asymptotic constancy of hadron total cross sections, as an "upper bound," and ( 2 ) the Pomeranchuk
theorem. h consisteilcy check, which lends some plausibility lo our assumption, is made. The calculations
are done within the contevt of frame-dependent cutoff rluantum field theory.

I. INTRODUCTION The method requires only one assumption, mhich is,

however, rather strong1: I t is that certain "light-plane
B using the Lehmann-Syinanzili-Ziil~mermann
integrals" f*(() admit power-series representntionk
(LSZ) reduction formalism, one can express a about (= 0 which are valid in the interval (= 10, = ).
great many phq~sicallyinteresting qunntities in terms of
At present, we cannot either prove or disprove this
a Fourier transform,
assunmtion on theoretical " nrounds. although some in-
dications of its plausibility are available (see belov ).
-
(1) Its implications are, however, in good agreement v i t h
experiment, a t least for the processes that we ha\c
treated thus far.
~ ~ h c Fr eis typically a matri\ element (;f (possiblj Assuming that the leading term in our asvrn~tclic "
retarded) commutator or anticommutator, and the four- expansion is nonzero, we obtain, in a model-independer~ i
&

alomentum q is on some mass shell. \Ye shall describe fashion, asymptotic constancq of totdl cross section>
---
herein a very simple method for obtaining an as>mptoiic
expansion of such a quantity, for large values (this theI Tconsiderably
h e same asymptotic expansion can be obtained also fiom
weaker assurllption that J,([) admit power series
will be made inore precise below) of the energy q4, and in some interval 4=[O,a), for some a>O, and independent of hon
shall apply this method to the prohlenl of hadron total small a may be, by the use of Watson's lemma [E. T. Copson,
Tlzeory of Fzlnctions of a Conzplex Variable (Oxford U . P., Oxford,
cross sections. 1935), p. 2181. Ko~vever,if one uses this method, the physical
amplitudes must be defined by a diffelent limit than the one uied
*Present address: School of Theoretical Physics, Dublin in the present paper [see Eq (5) J. The limit defined by Eq (5,
Institute for Advanced Studies, Dublin 2, Ireland. reduces to the conventional one for local field theory.