Why dilatation generators do not generate dilatations

ANNALS

OF PHYSICS:

Why

67, 552-598 (1971)

Dilatation

Generators

Do Not Generate

Dilatations*

SIDNEY COLEMAN Lyman Laboratory, Harvard University, Cambridge, Mass 02138 AND ROMAN

JACKIW’

Laboratory for Nuclear Science and Physics Department, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Received December 28, 1970

We show that the Ward identities associated with broken scale invariance contain anomalies in renormalized perturbation theory. In low orders, these anomalies can be absorbed into a redefinition of the scale dimensions of the fields in the theory, but in higher orders this is not possible. Also, these anomalies cannot be removed by studying the Green’s functions for objects other than canonical fields, e.g., currents. These results are established to tirst nontrivial order in perturbation theory by explicit Feynman calculations (which give us information at all momentum transfers), and in higher orders by the method of Callan and Symanzik (which gives information only at zero momentum transfer). The two approaches are consistent within their common domain of validity. Two appendices contain self-contained treatments of the formal canonical theory of scale and conformal transformations and of the derivation of Ward identities. In another appendix, we derive the Callan-Symanzik equations for Green’s functions of currents, and show that no redefinition of scale dimension is necessary for these objects, although the other anomalies remain.

I. INTRODUCTION There is a recurrent idea in particle physics that, in some sense, at high energies, particle masses should be unimportant. In recent years, this idea has led to the concept of scale transformations (dilatations) as approximate symmetries of nature, with possible applications to the study of high-energy scattering [l], low-energy phenomenology [2,3], and short-distance behaviour in field theory [4]. Since in * This work is supported in part through funds provided by the Atomic Energy Commission under Contract At(30-1)2098, and by the Air Force Office of Scientific Research, Contract F44620-70-C-0030.

552

DILATATION

GENERATORS

553

nature, scale invariance is surely not an exact symmetry, it is important for realistic applications to have a clear understanding of the precise mechanism of symmetry violation. The purpose of the present work is to attempt to gain such an understanding by examining, in explicit perturbation-theory calculations, the scaling behaviour of several four-dimensional field theories with only dimensionless coup1ings.l In these theories, it appears that only the mass terms in the Lagrangian are responsible for the breaking of scale invariance, and one would expect that there would remain many traces of exact scale invariance. At first glance, this expectation seems to be fulfilled: Using formal canonical reasoning, one can construct a dilatation current, whose divergence is proportional to the mass terms in the Lagrangian, and the integral of whose fourth component transforms the fields appropriately under equal-time commutation. From these statements about operators one can, by conventional methods, derive Ward identities, and from these Ward identities, one can, in turn, derive theorems about the asymptotic behaviour of Green’s functions. Unfortunately, these theorems are false in perturbation theory. For example, one of them states that the inverse propagator for a spinless meson is, at high energies, proportional to p”, while, in perturbation theory, it is proportional to p3 times a power series in log p2. This means that the equations from which these false theorems are derived must themselves be false; in current parlance, the Ward identities associated with broken scale invariance must contain anomalies. The technical reason for the occurrence of these anomalies is not difficult to understand: The manipulation of canonical commutators required to prove Ward identities is justified only if we introduce a cutoff. However, this does us no good unless the cutoff is chosen in such a way that the cutoff theory still obeys the Ward identities.2 For such familiar cases as quantum electrodynamics or the sigma model, for example, this condition presents no difficulties; it is straightforward to introduce a cutoff in such a way that the relevant equations of the theory (gauge invariance in one case and PCAC and current algebra in the other) remain true. For scale invariance, though, the situation is hopeless; any cutoff procedure necessarily involves a large mass, and a large mass necessarily breaks scale invariance in a large way. This argument does not, of course, show that the occurrence 1 An earlier version of this work appeared in preprint form, but was not submitted for publication [5]. Also the results presented here were reported at the Symposium on DeSitter and Conformal Groups, University of Colorado, Boulder, Colorado (June 1970); and at the Eastern Theoretical Physics Conference, Stevens Institute of Technology, Hoboken, New Jersey, (October 1970). Calculations similar to those of Section III have been done independently by Beg, Bernstein, and Sirlin [6]. 2 The above is an old story, first told about the anomalies in the current-algebra predictions for neutral pion decay [7].

554

COLEMAN AND JACKIW

of anomalies is inevitable, but it does show that there is no reason to believe that it is impossible. In lowest order of perturbation theory, we find that all anomalies can be explained by a change in the scaledimensionsof the fields, induced by the interaction. (The scale dimension is a constant which occurs in the equal-time commutator of the field with the dilatation generator. It should not be confused with the dimension of the field in the senseof dimensional analysis, although the two are equal for free field theories and naive manipulation of canonical commutators would lead one to believe that they are always equal.) If the altered scaledimension is used to derive altered Ward identities, these do predict correct asymptotic forms for Green’s functions. We also find that, in higher orders, the structure of the anomalies is more complicated, and cannot be obtained simply from a change in scale dimension. The first of these results is a verification of Wilson’s argument [4] based on Johnson’s analysis of the Thirring model, that the dimension of a field is changed by interactions. The second tends to confirm his assertion that the divergence of the scalecurrent is not proportional to the massterm in the Lagrangian. The organization of the paper is as follows: In Section II we show how to derive (false) theorems about the asymptotic behaviour of Green’s functions from the equations of broken scale invariance. In Section III we set up the general formalism for calculating anomalies in the equal-time commutators of canonical fields and the generator of dilatations, and we apply this formalism to detailed calculations in lowest nontrivial order of perturbation theory, for several renormalizable field theories. In Section IV we extend our arguments to higher orders, using the Callan-Symanzik equations [8]. Two long appendices contain developments of the formal theory of scale invariance and of the related Ward identities. Two shorter appendixes give the details of some Feynman calculations used in Section III and discussthe Callan-Symanzik equations for Green’s functions of currents.

II. SOME FORMAL

CONSEQUENCES OF BROKEN

SCALE INVARIANCE

1. General Considerations We shall begin by stating some results of a formal investigation of scale transformations. (A derivation of theseresults is given in Appendices I and II.3) We shall restrict ourselves here to renormalizable field theories with only dimensionless 8Theyarealsofound in an earlierpaperwritten by us in collaborationwith CurtisCallan[9]. However,the derivationgivenin the appendices is more complete, and, we believe, clearer.

DILATATION

555

GENERATORS

couplings, although most of the statements we shall make can also be derived for a wider class of theories4 We assemble all the fields in the theory into a vector, ‘p, and define the spin matrix, 6uV, by i[hP) q(x)] = (x@a”- x”au + C”“) cp(x), (2.1) where MuY is the generator of homogeneous Lorentz transformations. Under infinitesimal dilatations, 8DXU = -x”,

(2.2)

the fields can be chosen to transform according to the rule a,(~(-4 = (x3

+ 4 cp(4,

(2.3)

with d some matrix. For the class of theories under discussion, these transformations are invariances, when all the masses are set equal to zero, if d is chosen to multiply every Bose field by one and every Fermi field by 3/2. These numbers are called the scale dimensions of the fields. (These numbers are also the dimensions of the fields as conventionally defined for the purposes of dimensional analysis. However, logically, “dimension” in this sense is completely different from “scale dimension” as we have defined it, and it will be important in the sequel not to confuse the two concepts. To make the difference clear: The transformations of dimensional analysis map the exact solutions of one field theory into the exact solutions of a diferent field theory, with different masses. Scale transformations (dilatations) stay within a given field theory, and do not map solutions into solutions, unless the masses are zero.) For the class of theories under discussion, it is possible to construct a conserved, symmetric, energy-momentum tensor, B,,, , such that D, = xveu, )

(2.4)

is the current of dilatations5 By this we mean that this current is conserved when all masses vanish; more precisely, (2.5)

where Ym is the term in the Lagrangian that breaks scale invariance, i.e., for the theories we are considering, it is the mass term. Also, even if D, is not conserved, if we define D(t) = j d3x D,(x, t), 4 For details, see the appendices. 5 e,, is not, in general, the symmetric energy-momentum Appendix A.

(2.6)

tensor of Belinfante. For details, see

556

COLEMAN

AND

JACKIW

then

Q’(t), cp(x,01 = (-0” + 4 cp(x,0.

(2.7)

2. Ward Identities

By the usual methods, the operator statements of the preceding section can be transformed into statements about Green’s functions-Ward identities. We restrict ourselves here to the Ward identities associated with the renormalized propagator, G(p). (Note that if there are n fields in the theory, this is an n x n matrix.) We define I’&, q) and IQ, q) by d4x d”u eiq.reip.r (0 1 T*O,,(x) cp(y) ~(0) I 0),

G(P) ruv(p, q) G(P + q) =

(2.8)

and G(p) r(p, q) G(p + q) = s d4x d4y e”q’xeip.g x (0 1 T* [se

dv - 4%]

(4 V(Y) do)

I 0). (2.9)

In these equations, cp is the set of renormalized fields, assembled into a vector, and the T* symbol indicates the covariant time-ordered product. (A precise definition of this object in terms of functional derivatives is given in Appendix B.) It follows from the general discussion of Ward identities in Appendix B that quWp,

d

=

W-YP +

; quP”F

+

d

G-YP

i(p

+ +

s>” G-~(P) q) +

; a$-l(p)

X”“,

(2.10)

and g,,Wp,

q) = r(p, q) - idG-l(p)

- G-Q

+ q) d.

(2.11)

The first of these equations is a Ward identity that follows from the Poincare invariance; the second, as might be guessed from its similarity to PCAC Ward identities, is a consequence of the equations of broken scale invariance, Eqs. (2.5) and (2.7). In our subsequent discussion, we will reserve the name “Ward identity” for Eq. (2.10), and refer to Eq. (2.11) as “the trace identity”. 3. A False Theorem

For simplicity, let us restrict ourselves to the case of a single scalar field p; the matrices of the previous section then become ordinary numbers. If we differentiate the Ward identity with respect to q and then set q equal to zero, we obtain P’(p,

0) = ip@G-l(p)

- ig”“G-l(p).

(2.12)

DILATATION

GENERATORS

557

The trace identity then implies that r(p,

0) = zj~Q,G-~(p) - i(4 - 2d) G-l(p).

(2.13)

For the case we are considering, the term in square brackets in the definition of Q, q), Eq. (2.9), is simply a constant times I$. Therefore, the famous analysis of Weinberg [lo] tells us that at high negative (space-like) p2, in any finite order of perturbation theory, r(p, 0) grows, at worst, like a polynomial in log(p2).6 On the other hand, the inverse propagator grows like p2 times a polynomial in log p2. Thus, if we are willing to neglect terms of merely logarithmic growth, we can obtain an asymptotic equation for the behaviour of the asymptotic propagator G,(p) at high negative p2: p”&G;l(p)

= (4 - 2d) G;‘(p).

(2. 14)

This is a first-order differential equation; its solution is G-,l = A(p2)2-d

(2.15)

with A a constant. Since for a scalar field, d is one, we have deduced that, in any finite order or perturbation theory, the inverse propagator grows like p2, without any logarithmic multiplying factors. This is a lie: Explicit calculations show that the logarithmic factors are in fact present. By the same methods, a similar false theorem can be deduced for Fermi fields; it states that the inverse propagator grows like p, again without the logarithmic multiplying factors that we know are in fact present. 4. Post-Mortem

Three ingredients were used to construct the debacle of the last section: (1) The Ward identity, Eq. (2.10). (2) The trace identity, Eq. (2.11). (3) The statement that only the mass terms in the Lagrangian break scale invariance, Eq. (2.5). This was the condition that enabled us to use Weinberg’s analysis to neglect the left-hand side of Eq. (2.13) in the asymptotic region. If, for example, eAAhad involved terms proportional to (3,~)~ or v4, Weinberg’s analysis would have told us that the left-hand side grew as rapidly as G-l, and we would not have been able to compIete the argument. The Ward identity is a consequence of Poincare invariance. Since almost all of the usual cutoff procedures (in particular, all procedures that employ regulator 6 Another way of obtaining the same result is to observe that, on dimensional grounds, IQ, 0) must be of the form m2F(p2/ma), and should vanish as n? goes to zero, since scale invariance then becomes an exact symmetry. Thus, for high pZ, IQ, 0) must grow less rapidly than p2. 595/67/=4

558

COLEMAN AND JACKIW

fields in the Lagrangian) preserve Poincare invariance, it can contain no anomalies, by the arguments sketched in the Introduction. Therefore, either Eq. (2.5) or Eq. (2.11) or both, must be false. Since Eq. (2.11) is a consequence of the equal-time commutator (2.7), which expresses the scale-transformation properties of the fields, we can say that either the dilatation current has an anomalous divergence, or the fields transform anomalously under scale transformations, or both. The preceding analysis requires one small qualification: Its phrasing implies that there is a unique choice of the dilatation generator, D(t). Actually, this is not so; in a theory of a single scalar field, for example, we can always redefine the energy-momentum tensor through the transformation tw -+ 4” + 4v”

- sLl”)

vJ2,

(2.16)

with a: an arbitrary constant. Such a redefinition would not affect the Poincare generators, but it would change D(t), defined by Eq. (2.4) and (2.6). If Eqs. (2.5) and (2.7) were both true, it would be madness to make such a redefinition; however, since they cannot both be true, it might be useful to redefine D(t), if, by doing so, we can save one or the other of these two desirable properties. III.

ANOMALOUS

DIMENSIONS

1. General Considerations

In the preceding section, we argued that the trace identity, Eq. (2.1 l), must be afflicted with anomalies. We wish now to calculate explicitly these anomalies, to lowest nontrivial order in perturbation theory, for some typical renormalizable field theories. The most direct method would be simply to calculate all the terms that occur in the trace identity. This would be a tedious job; fortunately, the calculation can be shortened considerably by an astute use of regulator fields. We argue as follows: Suppose we cut off our field theory by introducing regulator fields directly into the Lagrangian. We emphasize that, if we do this, we must, for consistency, include the contributions of the regulator fields in the definition of the energy-momentum tensor. Once the theory is cut off, canonical arguments are legitimate; in particular, the trace identity is valid: = IQ, q, MJ - idG-l(p, MJ - iG-l(p + q, MS) d,

(3.1)

where we have explicitly indicated the dependence of the Green’s functions on the regular masses, Mi . If we now let the Mi go to infinity

J& G-V, Mi) = G-W,

(3.2)

DILATATION

because the theory is renormalizable.

559

GENERATORS

Also

because this is how the energy-momentum tensor of the renormalized theory is defined [9]. However, we have no a priori information about the value of the limit of I’@, q, MJ. This is because, if the regulator fields are explicitly included in the Lagrangian, the expression

which enters into the definition of l?, contains terms that have as coefficients the masses of the regmator fields. The usual estimates of renormalization theory are inadequate to the task of determining the behaviour of these terms as the regulator masses go to infinity. We can also calculate r(p, q) directly, without the use of regulator fields, defining it by Eq. (2.9), i.e., as the off-mass-shell matrix element of the naive divergence of the dilatation current. Of course, since this operator is logarithmically divergent, this definition is ambiguous. Traditionally, such an ambiguity in the definition of a logarithmically divergent operator is removed by specifying the on-mass-shell one-particle matrix elements of the operator at zero momentum transfer. We will follow this procedure; in particular, we will choose the on-mass-shell zero momentum transfer matrix elements to obey the trace identity. Clearly, any other choice would be folly, since it would introduce anomalies by definition. Putting all this together, we obtain a simple expression for the true equation that replaces the false trace identity, Eq. (2.11). It is g,,rw,

4) = w,

4) - idG-l(p)

- iG-l(p

+ q) d + A@, q).

(3.4)

where A(p, q), the anomaly, is given by A(P, 4) = ;,ym r(p, I

The ambiguity in the definition of IQ, vanishes on the mass shell. 2. A Quartically

4, Mi) -

q) is eliminated

r(p,

(3.5)

d.

by demanding that A(p, 0)

SeEf-Coupled Meson

We begin with the simplest example of a renormalizable field with quartic self-interaction. The Lagrangian is

field theory-a

scalar

(3.6)

560

COLEMAN

AND

JACKIW

Here q+ is the unrenormalized field, p,, is the bare mass, and A,, is the bare coupling constant. We cut off this theory by introducing two regulator fields, q+ and y2 . The Lagrangian now becomes

(3.7) where Q, =

T +

Cl%

+

c2qJ2

(3.8)

3

1 + Cl2 + c22 = 0,

(3.9)

p2 + c12M12 + c,2A422 = 0.

(3.10)

and Here we have written the Lagrangian in renormalized form: y is the renormalized field, p and A are the renormalized mass and coupling constant, and a, b, and c are renormalization counterterms to be determined order by order in perturbation theory in the usual way. For the regularized Lagrangian (3.7),

ehA= p2cp2 + M,2q2,2+ A422tp22 + bSP2.

(3.11)

It is trivial to calculate the matrix elements of this object, to first order in A; the relevant Feynman diagrams are shown in Fig. 1, where the cross indicates eAA.Thus we obtain, (k2

+

c,yp

+

MI")

+

_

c,2(p2

p2,,[,f+

-

q]2

M22)

+ I

by

_

p2)

(3.12)

where the expression (JL”+ M12) indicates the same integrand as in the preceding

FIG.

1. Single-particle

matrix elements of ~9,A, through

first order in the A@ theory.

DILATATION

561

GENERATORS

term, but with p2 replaced by M12. Also, b is to be expanded only to first order in h. (However, as we shall see shortly, we will not need to calculate b for our purposes.) Precisely the same diagrams enter into the calculation of F(p, q): r(p,

q) = 2p2 + &

s

d4k

(k2

_

.,,[;;

q]2

_

p2)

+

const’

(3’13)

where the constant is to be chosen such that (3.12) and (3.13) are equal at q = 0, th e mass shell condition is irrelevant in this case, since these functions depend only on q2.) Thus, if we define the convergent integral

p2 = p2. (Actually,

J- d4k (k2 - /L~)([:;

q12 -

p2)

(3.14)

we find that qp,

q, iui2) - r(p,

q) = h 1;’ dx [c$

(+)

+

c2”f

(y&)-j.

(3.15)

Hence, &P, 4) = $+QYP>

42 a?

-

JYP, 4)

= -Af(O) 42.

(3.16)

However, this anomaly can be trivially removed by a redefinition of the energymomentum tensor, in the manner suggested at the end of Section II. If we make the change I3Gy-

0," + ~f@w,a" - &"02) fD2,

(3.17)

then (3.18) (3.19) and &P, 4) -+ 4P, 4) + fJf(O>q2 = 0.

(3.20)

This observation-that, for a properly defined energy-momentum tensor, the trace identity has no anomalies to first order in h-does not contradict the arguments of Section II, since, to this order, the renormalized propagator is the same as the free propagator. Logarithmic factors, and, therefore, necessary anomalies, only appear in order X2.

562

COLEMAN AND JACKIW

3. Yukawa-Coupled Mesons and Nucleons

We now consider a somewhat more complicated theory, that of a pseudoscalar meson interacting with a spin one-half nucleon through Yukawa coupling. The Lagrangian is 2 = 4b@% + i(add2 - w$,*, - bo2%2 + lQh%~o% * As before, we cut off the theory by introducing nucleon and quafor the meson. The regularized normalized fields, is 2 = ihP#

+ 4ha%

- Mf$,z/,

+ k &4*

(3.21)

two regulator fields, I/& for the Lagrangian, in terms of the re-

+ f C%d2 - 44

- g cp2- ; Mb2y12 + igTPysY@ - a!?T

- 4 rP2+ icPyPWP + k d(aW@)2 + e’Yy,Y@,

(3.22)

where Y = 9 -I- ih ,

(3.23)

@ = p + iy,,

(3.24)

and a, b, c, d, and e, are, as before, renormalization constants. (In (3.22), we have neglected the hv* term required for renormalization; it does not contribute to our calculation in the order to which we work.) For this Lagrangian eAA= p2y2 + Ma2p12+ bQ2 + m$# + M&y&

+ aY?P.

(3.25)

Our task is to calculate the matrix elements of this object up to order g2. The

+* FIG. 2. One-meson matrix elements of 0,$, through second order in the Yukawacoupled meson-nucleon theory.

DILATATION

563

GENERATORS

relevant Feynman diagrams for the one-meson matrix elements are shown in Fig. 2. The cross represents OAA,directed lines are nucleon lines, undirected lines are meson lines, and p’ is p + q. As is evident from the figure, all the relevant integrals can be expressed in terms of

+ (p t--) -p’)

(3.26)

+ const,

where m, and m2 are two arbitrary masses, and the constant is chosen to make Z vanish at p2 = p2, q = 0. By definition, T(P, 4) = 2v2 + g2Wp, 4,m, m>,

(3.27)

while, from Eq. (3.25), T(P, 4, M,) = 2p2

41 - g2Mf[Z@,9, m, MA - Z(P,4, Mf, MJI.

+ g2m[4p, 4, m, 4 - Z(P, 4, Mf,

(3.28)

Just as before, there is no need to compute the mass counterterm, since, to this order, it only contributes a constant. If it were not for the occurrence of M, instead of m as the coefficient of the last two terms, this expression would just be a (somewhat eccentrically) regularized version of (3.27) and there would be no anomaly. It is these two terms which can be traced to the explicit dependence of Eq. (3.25) on M f-that is to say, on the breaking of scale invariance by the cutoffthat lead to the anomaly. Thus,

For these values of the masses, the integral (3.26) has, for any hxed p and q, a convergent power series expansion in p and q for sufficiently large regulator mass. If we perform such an expansion, it is easy to see that only the terms of quadratic order or Iower can survive the limit; the higher terms have too many factors of Mf in the denominator. Thus, 4P, 4) = kT2h(P2+ PIZ - 2P2) +

g2b2q2,

(3.30)

where b, and b, are constants. Here we have (finally) explicitly invoked the condition that determines the constant term in (3.26). The b, term can be removed by a

564

COLEMAN

AND

JACKIW

redefinition of the energy-momentum tensor, just as in our previous discussion of the self-interacting meson field. Thus, 4P,

4) = 8bl(P2

(3.31)

+ P’2 - 2p2).

(The coefficient b, is computed in Appendix C. However, its exact numerical value will not be needed in our discussion.) By Eq. (3.4), the correct version of the trace identity is thus g,vWp,

d = T(P, 4) - iG-YP) - iG-Q’)

+ g2b,(p2 + pf2 - 2~3.

(3.32)

But, to the order in which we are working, this is equivalent to guJ’““(p,

d = T(P, 4) - W - g2b,) G-~(P)

- i(1 - g2b,) G-l(p’).

(3.33)

This is of the same form as the original (false) trace identity, except that the dimension does not have its naive value (one), but is instead given by d’ = I - g2b, .

(3.34)

Thus, to order g2, the only efect of the anomalies is to change the scale dimension of the mesonjield.

We now turn to the calculation of the one-nucleon matrix elements. The calculation is almost identical to that in the meson case. The relevant Feynman diagrams are shown in Fig. 3. We define the two integrals

= i s$$

]%

&

7%

(k

_

p)i

_ (k m22

_

pIi2

_ m22/ + consty (3-35)

-H--b+-2YLP

k

P’

k + P FIG. 3. One-nucleon meson-nucleon theory.

k+p

k+p’

p’

matrix elements of 02, through second order in the Yukawacoupled

DILATATION

565

GENERATORS

and I,(P,

4, ml,

m2)

1

1

F+g-m2k+Pf--2

y5 + const,

(3.36)

where the constants are such that the integrals vanish at p = p’ = m. By the same arguments as in the preceding calculation,

(3.37) Here, only the terms of first order or lower in a power-series expansion of these integrals survive the limit. Thus, (3.38) A@, 4) = s2f(P + P’ - 24, where f is a constant. (We compute f in Appendix C.) The correct form of the trace identity is therefore

g,P‘“(p, 4) = %, d - W-W - @G-Q’) Equivalently,

+ g2f(p + p’ - 2m).

(3.39)

to the order in which we are working, guJ”‘(p,

d = F(P, d - i@ -aLiz”) G-W - i($ -fg2) G-Q’).

(3.40)

This is of the same form as the original (false) trace identity, except that the nucleon field does not have its naive dimension (3/2), but is instead given by d’ = 312 - g”J

(3.41)

Thus to order g2, for the one nucleon matrix elements as well as for the one meson ones, the only effect of the anomalies is to change the scale dimension of the fields. 4. Equal-Time Commutators The calculations we have just performed have shown that, to lowest nontrivial order in perturbation theory, the trace identity remains valid, except that the scale dimension of the field does not have its naive value, but acquires a correction as a

566

COLEMAN

AND

JACKIW

result of the interaction. The most natural interpretation of this phenomenon in terms of operator equations is that the equation for the divergence of the dilatation current, Eq. (2.5), is still true, but that the equal time commutator (2.7) is replaced by

W(t), cp(x,01 = WA + d? cp(x,0,

(3.42)

where d’ is the modified dimension matrix. However, this interpretation is not unique; it might be the divergence equation that is modified, with the correction, for the matrix elements we have studied, coincidentally simulating the effect of a change in the commutator. In this section, we eliminate this possibility by calculating the commutator and showing that it is indeed of the form (3.42). To the order in which we are working, the only matrix elements of the commutator which we need compute are those between vacuum and one-particle states; all others are given by tree graphs and have a trivial structure. Also, by translational invariance, we need only check the commutator for v(O). Finally, for simplicity, we shall restrict ourselves to the case of a meson field; a fermion field can be treated by almost identical arguments. Thus, we wish to establish that (p I[D(O), p(O)]1 0) = -id’(2E)-112

where I p) is a one-meson state of three-momentum define (2~r)~/~ (2E)lj2 F(w, p) = / d4x eiwtxi(p

(3.43)

(24-3’3,

p, and E is its energy. We [ T*B&t,

x) ~(0) 10)

By the standard arguments of Bjorken, Johnson, and Low [ll], established Eq. (3.43) if we can show that lim wF(w, p) = d’.

(3.44)

we will have (3.45)

w4m

We will now prove this equation. In terms of the Green’s functions we have been using,

By virtue of the Ward identity, Eq. (2.10), q”roi(p,

4) = - qir&,

4) + WWP

- i(p + d P(P).

+ 4) (3.47)

DILATATION

567

GENERATORS

Inserting this in Eq. (3.46), we find wf’b, P) = @‘jr&,

(3.48)

q)G(p + q),

where all the functions are to be evaluated at q = (w, 0). In (3.48) we have dropped terms proportional to G-l(p) which vanish on thep-mass shell. From the corrected trace identity we learn that

PrdP, 4)= m 4)- roo(PY 4) - id’G-l(p) - i&G-Q

where d’ is now the modified scale dimension. also know that

+ q),

Using the Ward identity again, we

rock4 4) = w-lqur,,(P, 4) = icd[EG-l(p

(3.49)

(3.50)

+ q) - (E + co) G-Q)].

Thus, apart from terms which vanish on the mass shell, we have wE;(w, P) = ir(p, dG(p + 4)

+ d’(1 + w-lE).

(3.51)

By the standard Weinberg bounds [IO] P(p, q) does not grow fast enough as u -+ ice to compensate for the decrease of G(p + q). Hence, lim oF(w, p) = d’.

w-tic.2

(3.52)

This is the desired result.

IV. FURTHER ANOMALIES 1. A Simple Argument In the preceding section we showed that, in lowest order or perturbation theory, all the anomalies in the trace identity simply served to modify the scale dimension of the fields. However, it is fairly easy to see that this cannot continue to be true if we go to higher orders. For simplicity, let us consider the theory of a single scalar field with quartic self-interaction, discussed in lowest order in Section III. We shall assume that, to all orders in perturbation theory, the trace identities are valid, except that the

568

COLEMAN

AND

JACKIW

scale dimension is anomalous. We shall then show that this assumption leads to false conclusions. Let T(p, q, r) be the off-mass-shell elastic scattering amplitude, with momenta labelled as indicated in Fig. 4. If we study T(pp, pq, pr) with p, q, and Y fixed linearly independent Euclidean four vectors and p a real parameter, it is easy to construct an argument parallel to that given in Section II-3 to show that (4.1)

UPP, pq, pr> = P~-‘~‘,

for large p, We also know that for Euclidean p, G-l(pp) a p4+,

(4.2)

from Eq. (2.15). Thus, WPP)

in all orders of perturbation

(4.3)

T(PP, pq, pr> x P-~,

theory, whatever the value of d’.

p)yjj?(Y?< r

p+q+r

+ +H+X 8

FIG. 4. Off-mass-shell elastic scattering amplitude in the A@ theory.

However, inspection

of the relevant Feynman diagrams

(Fig. 4), shows that

T(pq, pq, pr) = iA + A2 In p + O(h3),

(4.4)

for large p, with c some constant. We also know that G-Ypp)

= p2p2 + 0@3,

(4.5)

for larger p. Thus, G2(pp) T(pp, pq, pr) = ~-~p-~[iX

This is in contradiction

with Eq. (4.3).

+ d2 In p] + 0(X3).

(4.6)

DILATATION

2. The Callan-Symanzik

569

GENERATORS

Equations

Further insight into the nature of the anomalies in the trace identity can be obtained by studying the equations derived recently by C. Callan and K. Symanzik.’ In a sense, the Callan-Symanzik equations give us information about the anomalies in the trace identities complementary to the information obtained by the explicit calculations of the last section. The Callan-Symanzik equations, as we shall see, give us information about the form of the anomalies to all orders in perturbation theory, but only at zero momentum transfer. Our explicit calculations, on the other hand, give us information at all momentum transfers but only in lowest order of perturbation theory.* We begin by giving a short derivation of the Callan-Symanzik equations. For simplicity, we restrict ourselves to the case of a single scalar field with quartic self-interaction. We denote the n-point renormalized Green’s function in position space by G(“)(x, ... xn) and its Fourier transform (with the energy-momentum conserving delta-function factored out) by Gfn)(p, ... p,). We add to the Lagrangian a coupling to an external c-number function of space and time, J(X),

and we define

iry

y;

x1

...

x,)

=

z---

6 ~J(Y)

G’“‘(x, ... x,)

, I=0

where Z is a cutoff dependent constant, chosen to make P) cutoff independent. (We know from the usual considerations of renormalization theory that it is always possible to choose Z to do this. Note that this condition leaves Z undetermined up to a cutoff independent multiplicative factor; we will choose this factor shortly.) We denote the Fourier transform of P)(y, x1 a** x,) by Fn)(k, PI ... P,). From these definitions, it follows that

iryo, p1 *-*pm)= z &

GYPI ..* PTJ.

(4.9)

However, in the limit of high cutoff, G(“) is determined completely, to every order ’ Our treatment of these equations closely follows that of the original papers [8]. We emphasize that this subsection contains no material that is not in these papers, and is included here only for completeness. * A third method of analysis would be to use the renormalization group of Gell-Mann and Low [12] to study the asymptotic forms of Green’s functions. This was the method we followed in the first version of this work [5]. However, since it gives us information at zero momentum transfer, but only in the asymptotic region, it is definitely less useful, for our purposes, than the CallanSymanzik approach.

570

COLEMAN

AND

JACKIW

in perturbation theory, by the renormalized mass p, the renormalized coupling constant h, and by our convention for normalizing the field. Thus, simply by the chain rule for differentiation iP’(0,

p1 ** * pn) = (z$k)

5

+ (Z&)

F

+ ; (ZT)

G(“),

(4.10)

where the partial derivatives are to be taken with fixed cutoff. In this equation, the first two terms arise from the change in X and p caused by a change in the bare mass. The third term arises because the vacuum-to-one-particle matrix elements of the fields will, in general, be changed by a change in the bare mass, and therefore, it is necessary to rescale the field in order to preserve the usual definition of the renormalized Green’s functions. This change can be expressed in terms of the change in Z, , the wave function renormalization constant, because the scale of the unrenormalized fields is fixed by the canonical commutation relations, which are not altered by a change in the bare mass. From this equation, it is evident that the three coefficients in parentheses must be cutoff independent. We fix Z by demanding that (4.11) Also, we define

and _ Zalnz2 = 2y acL2

(4.13)

Thus we obtain aG’“’

aG’“’

nyGtn).

(4.14)

These are the Callan-Symanzik equations.g We can apply identical reasoning to the one-particle irreducible tions, which we denote by an overbar. In this way we obtain

Green’s func-

iP)(O;

p1 -*- pn) = O1ah

@cn)

iP)(O;

p1 ... p,)

= 01ah

+ 2P2 -

ap2

a@0

+ 2/..L2w

-

+ nyw.

(4.15)

B If we solve these equations in the asymptotic region, using the same methods as we used in Section II, we obtain precisely the predictions of the renormalization group of Gell-Mann and Low [121.

DILATATION

571

GENERATORS

The last term has changed sign because when we rescale the definition of the field, the G’s transform like the inverses of the G’s. Of course, we could also have obtained Eq. (4.15) by algebraic manipulation of Eq. (4.14)-it is just that it is quicker to derive it afresh. To establish a connection with the formulae of Section II we observe that, in the notation of that Section, ~(2)(o; P, P) = np,

O),

(4.16)

and @‘(p,p) Also, from dimensional

= G-l(p).

(4.17)

analysis, p’ $

G-l + 2$ a G-1 = 2G-1. aP2

(4.18)

Thus, we obtain F(p, 0) = -ia

T

aG-1 + &Y F - i(2 - y) G-l.

If 01were to vanish, this would be identical with Eq. (2.13), with d-+d’

= 1 + y/2.

(4.20)

In this case, the only effect of the anomalies, at least at zero momentum transfer, would be to change the scale dimension of the field, with y a measure of the change. Unfortunately 01 does not vanish, at least in perturbation theory. We have already shown this for the Xy4 theory in the preceding subsection. Now, we can see it even more directly, from the definition of 01,and the observation that h = A, + ch,2 In(ll%/~$) + O(hz),

(4.21)

with c some constant and II the cutoff. Likewise, we can see that similar anomalies will arise in order g3 in meson-nucleon theory, from g = go + cl go3 WV~o>

+ c2 go3 lWWpO2) + O( go51

(4.22)

Moreover, it is clear that these difficulties cannot be removed by studying the Green’s functions of other dynamical variables (e.g., currents) instead of those of the renormalized canonical fields. Such a change of variables would effect the y term (and also introduce new terms of similar type if the new dynamical variables require more than just a multiplicative renormalization to make their Green’s functions cut-off independent) but would not change the (Yterm at all.10 lo These remarks are explained in more detail in Appendix D.

572

COLEMAN AND JACKIW

V. CONCLUSIONS We summarize our conclusions: The Ward identities associated with broken scale invariance contain anomalies in renormalized perturbation theory. To low orders, these anomalies can be absorbed into a redefinition of the scale dimensions of the fields of the theory, but to higher orders this is not possible. Also, these anomalies cannot be removed by studying the Green’s functions for objects other than canonical fields, e.g., currents. Although we have used the asymptotic behaviour of Green’s functions as a guide throughout this investigation, we emphasize that the anomalies we have found exist at all momenta; their derivation does not depend on asymptotic estimates derived from perturbation theory. This is important. If all one knew was that the asymptotic behaviour derived from perturbation theory was inconsistent with that derived from broken scale invariance, one might well choose to disbelieve perturbation theory in the asymptotic region. It is a much more radical step to disbelieve perturbation theory at small momenta. Finally, there remains the possibility that the a-term in the Callan-Symanzik equations vanishes for the real world. That is to say, that as a function of h, 01has zeroes, and that the real value of A is one of these zeroes. (This is a conjecture of Wilsonlo&, among others; in the case of quantum electrodynamics it corresponds to the Gel&Mann-Low eigenvalue equation.) In this case, we would obtain naive scaling behaviour for Green’s functions (probably with anomalous dimensions). Obviously, nothing can be said either for or against this possibility from perturbation theory alone.

APPENDIX

A. CANONICALTHEORYOFSCALEANDCONFORMALTRANSFORMATIONS

In this Appendix we present a self-contained discussion of the canonical development of scale and conformal transformations, in the context of Lagrangian theory. The present treatment duplicates some of the brief discussion of this topic which was given in an earlier paper [9]: A less general treatment and references to older literature can be found in [2]. Dilatations are discussed in Section A I and conformal transformations in Section A II. A I. Dilatations in Field Theory A dilatation

is a change of the space time coordinates;

thus it is akin to

loa To be more precise, Wilson [4] merely conjectures that OLhas zeroes, not that the real value of X is one of these zeroes. This (weaker) condition is sufficient for asymptotic scaling (Wilson’s primary interest in the papers cited) but not for the validity of the naive scaling Ward identities at all momenta (our concern in this paper).

DILATATION

a Poincart

transformation

573

GENERATORS

rather than to an internal

symmetry

transformation.

x” ---f x’” = e-ppu;

(A.la)

s*xL” = -X@.

(A. lb)

(It is our convention to omit the infinitesimal parameter which characterizes the infinitesimal transormation.) By investigating the combined effects of dilatations and Poincare transformations, one can abstract a Lie algebra of commutators. Upon defining D to be the generator of infinitesimal dilatations, it follows that D commutes with all the Lorentz generators M uv, but does not commute with the translation generator Pp. i[D, P”] = P”.

64.2)

We wish to determine the effect of this transformation on particle fields q, which are assumed to transform in the usual fashion under Poincare transformations

v?(x) = W”, CPWI= @cpW, SE8cp(x) = i[Mao, v(x)]

(A.3a)

= (xa2B - xsau + W)

q(x).

(A.3b)

For determining the dilatation transformation of y, it is sufficient to specify the variation of cp at the origin of X, since the translation operator can then be used, together with (A.2), to determine the transformation for nonvanishing X. Because the origin is left invariant by (A.l), it is plausible to require that the effect of transforming the field at the origin be expressible in terms of the field at the origin. Thus, we set

SDCP@) = i[D, v(O)1= d4% with d some matrix.

For finite x the transformation

(A.4a)

is therefore determined.

6,d.x) = i[D, cp(x)l = (x . a + 4 cp(x)

(A.4b)

We now inquire what conditions must be satisfied by a Lagrangian, 9, which depends on a set of fields cp and their derivatives Scp in order that a dilatation be a symmetry operation. Recall that one may show a transformation to be a symmetry operation by demonstrating without the use of the Euler-Lagrange equations of motion, that the infinitesimal transformation changes the Lagrangian only by a total divergence, so that the action, j”d4x 9(x), is invariant; i.e., one wishes to show that (A.5a) 595/6712-15

574

COLEMAN

AND

JACKIW

where

From this equation the conserved current is immediately deduced with the help of an alternate expression for 69 which does use Euler-Lagrange equation 69 = a,p

* 6cp].

(A.5b)

Eqs. (A.5a) and (A.5b) imply that 0 = a,p

* &p - A”).

(A.5c)

We now consider the change of a Lagrangian under an infinitesimal dilatation. Poincare invariance, which is always assumed, impose the following constraints on S? (A.6a) adi”

=

X,

(A.6b)

( nI, . ava*cp + $

. au,) + n, . (d + 1) a”cp + $$

. dq. (A.7a)

Translation form

invariance, S&?(x)

as expressed by (A.6a) allows

= a,[x9]

(A.7a) to be written

- 49 + l-II, * (d + 1) auv + g

This shows that a theory possesses dilatation I$, . (d + 1) a“(p + g

* dep.

in the (A.7b)

invariance when . dv - 49 = 0.

(A-8)

From this equation, it is easy to see that massless free field theories possess dilatation symmetry, if we choose d such that it multiples every Bose field by 1, and every Fermi field by 3/2. This “scale dimensionality” coincides with the physical dimensions of such fields, in units of inverse length (fi = c = 1).

DILATATION

575

GENERATORS

The canonical current DC““, associated with scale transformation, from (A.4b), (A.~c), and (A.7b).

(A.9a)

D,” = x,8? +- W . dv

? a,D,” = II, . (d + 1) Pep + g The canonical energy-momentum (y

is determined

. d

(A.9b)

tensor ~97 has been introduced. zz

nu

f pep

-

(A.lO)

g”“Z.

Obviously the current is conserved in the presence of the symmetry, though it may, of course, be delined even in the absence of scale invariance. It is useful to express DC” in terms of the symmetric Belinfante tensor @ : (A.1 la) where puu” = &[l[IB . ptYq - J-p . puqJ - l-p . puqJ].

(A.llb)

The Belinfante tensor may be used instead of er for purposes of generating translations, while the current associated with the Lorentz transformations, Mns“, is easily expressed in terms of it. ~@u. = -ffjfp - x*gy

(A.12)

In terms of @, the canonical dilation current has the form D,” = x,&$= + II’” . dq + Xp -

=

x,ep

ag(xBuax,)

(A.13)

+ VU - ~,(x*~~~J,

where Vu E n, - [ g”“d - P:““] ‘p.

(A. 14)

The object Vu is called the$eld-uirial. The reason for the nomenclature is that Vu, ignoring the spin term, consists of the product between field momentum II” and field ‘p, a structure analogous to the virial in nonrelativistic Lagrangian theory. The last term in (A.13) is a total divergence of an antisymmetric tensor. Therefore it makes no contribution to the space integral of DCo, nor does it contribute to auD,‘. Without loss of physical content, we may drop it, and introduce a modified dilatation current. DB” = x,0;;” + V”

(A.15)

576

COLEMAN

AND

JACKIW

From (A. 15) we see that, in the general case, the vanishing of the trace of &” is not a necessary condition for scale invariance, since && may be cancelled by aUVu. However, if the field virial is itself a total divergence, V” = aauua,

(A.16)

further simplifications may be effected. We shall see shortly that (A.16) is aftvays true for a scale invariant theory, as a consequence of the equations of motion. However, for the derivation of Ward identities, it is important that the equations of motion never be used to simplify the conserved current. (This point is explained in Appendix B.) Since our main interest in this paper is the study of Ward identities, we will be concerned mainly with the case where (A.16) is an identity (i.e., true for all functions of space and time with the stated transformation properties, not just for solutions of the equations of motion). This is what happens in all renormalizable, massive field theories. It is also the case for all theories involving fields carrying spin 0, l/2, and 1 if the kinetic energy is of the standard form and the interaction involves no derivative couplings. In the next section we shall demonstrate the connection of (A. 16) to conformal transformations. To simplify (A. 15) further, we define 5y = &(u”” + 5”U), (A.17) 0”” = &(rJ’” - cJ”U), and

where -

6~~ may be used instead of &” in constructions of the Poincare generators, since the added term is symmetric and conserved, and does not contribute to the relevant space integrals. In terms of B@“,D# is Dsu = xj””

- ~a,a,(XAow”xv) + a,o!?

(A.20)

The last two terms in (A.20) again may be dropped; our final expression for the dilatation current is D” = x,0?

(A.21)

DILATATION

577

GENERATORS

The divergence of (A.21), which is also given by the left-hand side of (A.8) is

a,w = e,u.

(A.22)

Thus, it is seen that the vanishing of the trace of the energy momentum tensor becomes related to scale invariance, only when the additional condition (A.16) is satisfied. Furthermore, it is only the trace of 0uy, rather than that of @” or t9: that is relevant in the general case. When the current is not conserved, one may still define a time-dependent dilatation charge by

D(t) = S d3xx,eyx) = Tao+ j

xieyx).

d3x

(A.23)

Unlike the situation for broken internal symmetries, the Lie algebra is not satisfied by this charge in the absence of dilatation symmetry. The commutators of D(t) with the Poincare generators may be computed. To to this, one uses the formula (A.23), the known commutators of PO and eoi with P” and Ma%,and the conservation of 0,“. The result is i[D(t), i[D(t),

P”] = Pa - gEo s d3x 8,D”(x), MaB] = 1 d3x(gao9 - gB”xa)

a,o+).

(A.24b)

Thus the nonconserved dilatation charge D(t) has improper commutators with those PoincarC generators which make reference to time. In spite of this defect, D(t) generates at equal times the correct dilatation transformation on the fields, (A.4b). To see this, we return to the canonical formula for Dcu(x), (A.9a). (Of course D(t) is the same when it is constructed from D,O(x), D,O(x) or Do(x).)

dW>, 441 = i [top + 1 d3y no@,Y) . biai + dl cpk Y), cp(t,XI] = (d + x * a) q(x) = 8,9(x).

(A.25)

This coincides with (A.4b).11 I1 In deriving (A.29, we have made use of the canonical equal time commutators i[lr”(t, x), qp(t, y)] = 6(x - y) and i[&, x), cp(r, y)] = 0. This requires the assumption that all field variables v are dynamically independent, which is not always true on theories involving fields carrying spin > 1. However, our main results, the Ward and trace identities derived in Appendix B, do not require these equal time commutators, and are quite general.

578

COLEMAN

AND

JACKIW

The time derivative of a field is not properly transformed by D(t) in the general case.

= i[P=, (d + x - a) p(x)]

- i [v(x),

= (d + 1 + x . 4 @cp(x) + W”

P@ - gao J d3y @,“(t, y)]

[v(x),

1 d3y 8x”k Y)]

= ~iaP(x) + 4Tuo[dx), j d”Y@Y%Yl].

(A.26)

Therefore although ai cpis properly transformed by D(r), a”‘p possessesthis property only if q(x) commutes with J d3yOaff(t,y), In particular a Boson mass contribution to the trace, p2$, commutes with the fields; however a Fermion mass contribution, m$t,h, does not commute with $. We conclude this investigation of scale transformations by discussing the most general scale invariant theory involving onZy one Boson field. The Lagrange function may be written in the form (A.27a) Applying the condition equation for f

for scale invariance, (A.8) with d = 1, yields a differential

;f(KY)

= 0

(A.27b)

Evidently the most general scale invariant 9 is of the form (A.27~) We may further restrict f by imposing the condition that the field virial be identically a total divergence, (A. 16). In the present case this requires (A.28a) The quantity auoris a function of v and a,v. However it cannot depend on ay, for

DILATATION

579

GENERATORS

then the indicated differentiation on the right hand side of (A.28a) would produce second derivatives of y, which are not present on the left-hand side. Thus, (A.28b) and (A.28a) becomes

It is now seen that f’ must be constant, since a’(~~) cannot depend on a”~. We conclude therefore that the most general scale invariant single Boson Lagrangian, which also satisfies (A. 16) is (A.29a) with (A.29b) and

In the next section it will be shown that this is a conformally since (A.16) is satisfied as an identity. A II. Conformal Transformations in Field Theory A conformal transformation is a space-time transformation xu +

x’”

=

X”

-

c”x2

1 - 2c . x + c2x2 ’

8CUXL”= 2X@X” - guex2.

invariant

theory,

of the form (A.30a) (A.30b)

Since the sign of x2 is not invariant under global conformal transformations, the relevance (if any) of these operations to relativistic quantum physics is restricted to the infinitesimal transformations (A.30b). The lo-parameter Poincare group, the one-parameter dilatation group, and the conformal transformations make up the 15-parameter conformal group. By considering the combined effects of conformal, dilatation and Poincare transformations, the Lie algebra may be deduced. In addition to the usual Poincare commutators and the previously discussed

COLEMAN AND JACKIW

580

commutators with D, the following relations are satisfied by Ku, the generator of conformal transformations. i[P”, K”] = -2gu”D + 2iWa,

(A.31a)

i[Muv, K] = guaKv - gv”K@,

(A.31b) (A.31~)

i[D, K*] = -K*,

(A.31d)

i[K”, KB] = 0.

We now study the transformation of fields which are assumed to transform conventionally under Poincare transformations, (A.3), and dilatations, (A.4). Just as for dilatations it is sufficient to consider only the origin of the coordinates. Since this point is left invariant by (A.30), we demand that the effect of a conformal transformation of the field at the origin be expressible again in terms of the field at the origin. Sca(p(0) = i[K”, (p(O)] = x”cp(0). Here H~ is a matrix in the component

(A.32)

space of cp. From (A.31), it now follows that

[w, d] = 0, [Km, C”“] = guax”- gum@,

(A.33a) (A.33b) (A.33~)

[IP, d] = --x~.

Equations (A.33b) or (A.33~) imply that X” is nilpotent. to the case in which x vanishes.12

We will restrict ourselves

Scmcp(0)= i[K”, v(O)] = 0

(A.34a)

It now follows from the Lie algebra of commutators

that

&“‘P(x) = iW, cpW1

(A.34b)

= (2x%x” - g”“x2) Qp + 2x,(g”“d - X”,) q(x). I2 For Fermions an acceptable solution to these equations is d=

i-$6,

x= = cw

f

iYsY1,

with c an arbitrary constant. Note that X= is, as promised, nilpotent. The resemblance of these structures to those occurring in the V - A theory of weak interactions has been exploited by W. Hepner [13], and also by Mack and Salam [2]. Of course, these solutions must be rejected if one demands that dilations commute with space reflections.

DILATATION

581

GENERATORS

To establish the conditions under which the transformation (A.34) is a symmetry of a Lagrangian, we compute SCa,Ep,without the use of Euler-Lagrange equations.

= au[(2x9u

- ga92) 21

ap + 2x” [atp

* dq + rII, * (d + 1) a“cp - 491 + 2V*.

(A.35)

In deriving (A.35) we have made use of Poincart invariance (A.6), and of the definition (A.14) for the field virial V”. Equation (A.35) exhibits the fact that two conditions must be satisfied by a Poincart invariant theory in order that it be conformally invariant. Firstly it must be true that % ’ Cd + 1) a”cp + g

. dq - 49 = 0.

(A.36)

This is just the requirement of scale invariance-every conformally invariant theory is scale invariant, as is already seen from the commutator (A.31a). The second condition, which is the intrinsic conformal invariance requirement, is

i.e., the field virial must be expressible, without the use of equations of motion, as a total derivative, of some local quantity CJaa. This is precisely the condition which we encountered previously, (A.16), when we wished to express the dilatation current simply in terms of an energy-momentum tensor.13 According to (A.5b), A.~c), (A.35) and (A.37) the canonical current associated with conformal transformations K,“‘” is K,“” = (2x*x,

- g/x’)

0:” + 2x,lT‘

. (g”“d -

a,Ky = 2x”a,D”.

Z”“) cp’- 2cr”“, (A.38a) (A.38b)

I3 Note that in a scale invariant theory, (A.35) can be written, wirh the help of the equations of motion, as 2P = a,[rP . S& - 2x”x” 2 + g*” x2 Pi”]. This verifies our previous assertion, that the field virial is always a total divergence for a scale invariant theory, for solutions of the equations of motion. This means that for such a theory, we can always construct an energy-momentum tensor with vanishing trace. However, the Green’s functions associated with such a tensor will, in general, not obey simple Ward identities, for reasons which will be explained in Appendix B.

582

COLEMAN

AND

JACKIW

In deriving (A.38) we have assumed that (A.37) holds, an assumption which we shall make throughout the subsequent discussion. Eq. (A.38b) expresses the fact that when (A.37) holds, the conservation of the conformal current is equivalent to the conservation of the dilatation current, i.e., conformal invariance is broken by the same mechanism which violates scale invariance. Just as for the dilatation current, the expression for the conformal current can be considerably simplified by expressing it in terms of the various energy-momentum tensors &” and 0~“. In terms of the Belinfante tensor, (A.ll), K;” has the following foim: K,“” = [29X” - g,Y]

e;;u + 2fa”dy

+ a,(xflp”12xk”

- 2uy

(A.39a)

- gy”x”] + XW~).

The total divergence does not contribute to space integrals of K;‘, nor to a K,““. It may be dropped without loss of physical content, leaving the Belinfante coiformal current. Kjp = [2X”X” - g,“x2] e

(A.39b)

+ 2x”aa”dy - 2ay.

Finally the current may be written in terms of the tensor (A.18). We find K;“ = [2x”x, - gy”x”] 8”” -

~aAap[P~“(2x~x”

+ 3a,[g~~x”oft” - f@xyoA+y + A:”

- g”V)]

- Ay

+ fg%dq”

- gg~%b;“)].

(A.4Oa) The second two terms on the right hand side of (A.4Oa) are again physically uninteresting due to their antisymmetry properties. Dropping them, we arrive at the completely simplified formula for the conformal current. Kmw = [2.19x” - g/x2] e””

(A.4Ob)

The final expression for Km“ may also be written in terms of Du and MaBU, where the latter is the current associated with Lorentz transformations, (A.12), expressed in terms of 0~” rather than 0;. K”” = x,[ g”“D” - M”mu] The charges associated with the conformal

(A.41)

current

Kw(t) = 1 d3x K”O(x)

(A.42)

do not satisfy the Lie algebra when the current is not conserved. The commutators

DILATATION

583

GENERATORS

with the Poincare generators may be calculated analogously to the computation of commutators satisfied by D(t), (A.24). One uses the expression (A.40b) for Keo and the known commutators of P” and MuY with &. The result is i[P”, Km(t)] = -2g”“D(t) i[M”“,

K”(r)]

= g”%“(t)

+ 2Mua + 2guo I d3x x”+DB(x),

(A.43a)

- g”“K”(t)

- 2 s d3x x”(gu”x”

- guoxu) a&@(x).

(A.43b)

It is seen that commutators with those Poincare generators which make reference to time contain symmetry breaking effects. The commutators between the nonconserved charges D(t) and Km(t) can be studied only with the help of canonical expressions for these objects which follow from (A.9) and (A.38). In general, no explicit results can be given since u Uyis not known a priori. We do not discuss these commutators here. The fields are transformed properly by the time dependent charges. To establish this we consider the canonical formula for K”(t) which follows from (A.38).11 KO(t) = 2t D(t) - t2Po - 1 d3y yjyje%, -

Y)

2 / d3y no@, y) yj . ZjOq(t, y) - 2 1 d3y oOO(~, y);

Ki(t) = 2tMio + t2Pi +

s

(A.44a)

d”y I-IO@,y)

. [2y”y,aj - y’y$Y + 2yid - 2yyjZji] cp(t, y) - 2 1 d3y uio(t, y).

(A.44b)

In order to evaluate the commutator of the above with q(x), we use the fact that @’ commutes at equal times with cp. The reason for this is that uUycannot depend on first derivatives of the fields, for if it did then VU = 3,~ would depend on second derivatives. This, according to (A.37), would indicate that the Lagrangian depends on second derivatives-a possibility which we reject, The remaining commutators are evaluated with the help of canonical relations satisfied by no and ‘p, as well as14

j[e:“(t, Y), cpk XII = a”vw 6(x -

Y>.

(A.45)

Thus we find i[K”(t), v(x)]

= (2x0x” - g”“x”) a. v(x)

+ 2x,( g”“d - Z”“) C&K) = Q+(x). I4 This formula follows from the definition of e0,“; see ref. (14).

(A.46)

584

COLEMAN

AND

JACKIW

Of course the time derivatives of the fields are not properly transformed by K”(t) since [K”(t), PO] does not satisfy the algebra. The significance of the field virial and of the condition (A.37) may be understood as follows. Consider the response of the Lagrange function to a conformal transformation at the origin of the coordinate system. From (A.35) we have S,“LqO) = 2V”(O). When conformal

(A.47a)

symmetry is present, S,aLqO) = 2a,@(o>.

(A.47b)

This shows that in the general case, when u aBis nonvanishing the Lagrange function of a conformally invariant theory is not manifestly conformally covariant. That is, it does not transform like a field under the conformal transformation, since S,9yO) # 0. The object aorwsatisfies the following constraints in a conformally invariant theory. From its definition (A.38) it is seen that 0~” possesses the well-defined scale dimension 2. i[D, V(O)] = 2aU”(0) (A.48a) Therefore, since au” does not depend on derivatives of fields, (A.48b) Since CY depends only on fields, we have i[K*, uuv(0)] = 0.

(A.49)

Finally, i[K”, V(O)] = i[K*, i[P, , aBu(O

= [[P, >K”l, ~YO>l = 2g”%,“(o) - 2uaB(O) - 2&(O).

(A.50)

In deriving (A.50) use has been of (A.31a), (A.48), and (A.49) as well as of the property of uUVthat it is as a second-rank tensor under Poincart transformations. Evidently the virial does not transform covariantly under conformal transformations, since (A.50) is nonzero. The symmetry of the right hand side of (A.50) is a consequence of the commutativity of conformal transformations. i[K”, P(O)] = i [K”, i[P,

U(O)]] (A.51)

= i [P, i[K”, L?(O)]].

DILATATION

GENERATORS

585

Explicit computation for conformally invariant spin 0, l/2, 1 theories without derivative coupling, shows that I’” is zero for spin l/2. For spin 0, unu is given by *gtic”$; see (A.29). Thus we assert that a conventional spin-zero Lagrangian is not manifestly conformally covariant, even in a conformally invariant theory. For spin 1, the Lagrangian is manifestly conformally covariant and no&is zero if the kinetic term is of the usual form 2 = -$F”“F LL.> (A.52a) where Fu” x= +A” - a”Au (A.52b) However, if the kinetic term is of the form 2

= - ; F”“F,”

+ ; @,A” a”A” - a”A” a,A”),

(A.53a)

then we findl” aau = --a(A”lAu

+ $g,=‘A,AA).

(A.53b)

For spin 2, a conformally invariant theory does not seem to exist. We have not understood the special role of spin zero. We remark that the usual energy momentum tensor of a spin-zero conformallyinvariant theory is not manifestly conformally covariant, while the tensor which we have introduced in (A.18) possesses this property. Explicitly for the theory given in (A.29)16,

&v:“(o) = &Vg”(O) = 2do)p+(o) g"=+ q~(o) gue - aaT gq;

(A.54a)

&Y+“(O) = 0.

(A.54b)

Finally we inquire whether there are any other conserved currents of the form J“(x) = f”(x) &‘, where BUYis symmetric and traceless. It is easy to show that the Poincart, dilatation, and conformal currents are the only possible ones. For if a, Ju = 0, then

0 = a,fy(x) eqx)

(A.55)

zzz I5 A simple exercise shows that the Belinfante tensors for 2 and 9’ are different, though the modified tensors (A. 18) coincide. Note that 2 and 2” differ by a total divergence, hence they lead to the same dynamics. In this connection see also [15]. I6 This point was developed in conversations with Prof. K. Johnson.

586

COLEMAN

AND

JACKIW

In deriving (A.%), we have used the symmetry and tracelessness of @‘. The only way (A.55) can be zero is if the term in brackets is zero. (A.56a) But it is well known [16] that in four dimensions, the only solution to this equation is

fy(x> = a, + bx, + [c,, - cvul.P + m%J” - &“XZl, where a, , b, c,, and 4‘ are x-independent. which we have already considered.

APPENDIX

B.

(AS6b)

This solution reproduces the currents

WARD

IDENTITIES

1. General Considerations

We begin by an infinitesimal we assemble all the infinitesimal ing to

recapitulating the general formula for the current associated with transformation of the fields in a Lagrangian field theory. As before, of the fields of the theory into a vector ‘p, and assume that under transformation cp-+ ‘p + 6 cp, the Lagrangian 2 changes accord-

= a,& + A. (When d is zero, the transformation If we define Ju=

(B-1)

is a symmetry of the theory.) l-P.8cp-h‘,

(B.2)

then (B.3a) = A,

(B.3b)

where we have used the equations of motion to eliminate the last two terms. (We emphasize that no assumptions have been made about the Lorentz-transformation properties of 6~; therefore, the reader should not be misled by our notation into thinking that J, is necessarily a four-vector.)

DILATATION

587

GENERATORS

We now wish to use Eq. (B.3) to establish Ward identities. Ward identities are relations among Green’s functions; the most common way to derive them is to define the Green’s functions asvacuum expectation values of time-ordered products and then utilize the equations of motion and the equal-time commutation relati0ns.l’ We will useinstead a method basedon Schwinger’s definition of the Green’s functions as variational derivatives. There are two advantages to doing things this way: First, in any but the simplestfield theories, the vacuum expectation values of time ordered products are not the Green’s functions, and, indeed are frequently not covariant; extra terms (“seagulls”) have to be added by hand. Schwinger’s method, on the other hand, always gives a covariant object. Second, Schwinger’s method enables us to work directly with the Lagrangian, and circumvent the computation of equal-time commutators.ls We shall have to deal not only with Green’s functions for many fields, but also with those for many fields and a current, many fields and the divergence of a current, etc. Therefore, we introduce s(cp, %vP, A, B, Cu ,D> = ~(cP, 8,~) + A * ‘p + Be 8qa

+ J”C, + DA,

(B.4)

where, A, B, C and D are arbitrary c-number functions of space and time. We will refer to these collectively as “the external fields”. We are now in a position to define the (connected) Green’s functions: 6 inG(n)(xl ... x,) = W-d

... ___6 Wxn)

ln(0 / S 10;).

(Since we are treating cp as a vector “in the space of field components”, this object is a tensor of rank YEin the samespace.That is to say, if there are r fields in our theory, G(“) has IZindices, each of which runs from 1 to r.). Likewise, we define iJ““‘“‘(y; x1 ... x,) = &

II

G’“‘(x,

... xn),

and 6 iA’“‘( y.2 x1 *** x,) z WY)

G(“)(xl ... x,).

I7 This was the method utilized in the first version of this work 151. I8 The methods of this appendix are so close in spirit to those used by J. Schwinger during the 1950’s that we are convinced that our argument must be a duplication of one of his. Schwinger (private communication) shares this conviction; unfortunately, neither he nor we can find the appropriate reference.

588

COLEMAN

AND

JACKIW

Finally, we define j#l’G’“‘(x, j#2)(-$“)(x1

6 . . . x,) zz WXI) 6 . . . x,) = __ Wd

G-)(x2

0.. x,J,

G’“-l’(xl

, x2 *-* x,),

etc. Our main interest is the equations obeyed by these objects when the external fields are set equal to zero; however, it will be convenient to avoid doing this until the last stage of the argument. (An alternative notation is sometimes used for these objects: G’“‘(x,

a-* x,> = (0 I T*cp(x,)

.** (~(-4

JP(@(y, x1 .a* x,) = (0 I T*J“(y) dcn)( y, x1 -.- x3 = (0 I T*A(y) 6(1’G’“‘(xl 8’2’G’“‘(xl

O>c ,

+I) *** +,)I (0, , cp(xl) 0.0 (~(~31 O>, ,

-.. c,) = (0 1 T* 8cp(xl) cp(x2) ... +,)I

o), ,

a..x,> = (0 I T*cp(x,) Scp(x,>.*- (Al

O>o

etc. This notation-that of “connected T*-ordered products”-stresses the identity of the Green’s functions with the corresponding connected vacuum expectation values of time-ordered products when no two space-time arguments coincide. We emphasize, however, that their primary definition for us is as variational derivatives, not as time-ordered products with some extra terms added to insure covariance.) To derive the Ward identities, let us consider _Ep((P,a,(p,A,

-AD,

ap, D) = 2q(p,aw’p)

+

A ecp - sq .m + Jpap + AD. (B.8)

If we make the change of variables, cp-cp+scpD, then

+ 2.~6~

+ D [A = ~(cP, an)

+

A. up+ ~ua,D

a,Ju -

+ AD + o(D~)

89 - a,nfi + g

+ A * ‘P + O(D2).

* Bcp] + O(D2)

(B.lO)

Here we have dropped the first term in brackets because it is a total divergence, and

DILATATION

589

GENERATORS

the second because of Eq. (B.3a). (Note that this equation is just a consequence of the definition of J, , and does not depend on the equations of motion. Thus, it is completely legitimate to use it to simplify the Lagrangian.) Since, with no loss of generality, D can be chosen to vanish at infinity, the two Lagrangians, (B.8) and (B.lO), must make the same predictions for In (0 / S j 0), and therefore must lead to the same Green’s functions. In particular, if we take IZ variational derivatives with respect to A, one with respect to D, and then set the external fields equal to zero, we find that P‘Jp)(y;

x1 ... x,) - d(y;

x1 1.1 x,) + i i

S14)(y - x,) 8(p)G(n)(xl *.. x,) = 0.

r=1

(B.11) This is because the left-hand side of this equation is obtained by performing the derivatives on the Lagrangian (B.8). while the right-hand side is obtained by performing the derivatives on the equivalent Lagrangian (B.lO). These are the Ward identities. We can also write the Ward identities in momentum space. We define (27~)~ Sc4) [C pi] G’“‘(pl

**- p,J = J d4xl *.. d4x, ei’l’“l

and define the other momentum-space then find

ct‘J?(q; PI . ..Pn)

=

w;p1

. ..p.J

.-- eipn’WP(xl

*.* x$,12)

Green’s functions in a like manner. We

+

f T=l

+)G(“)(pl

.-. pr + q ... pn).

(B.13)

Of course, the passage from Eq. (B. 11) to Eq. (B. 13) can only be made if 6 cp has no explicit dependence on space-time coordinates. Otherwise, derivatives of deltafunctions would appear in the Fourier transform (B.12). Some remarks should be made about these equations: (1) They are precisely the equations we would have obtained if the Green’s functions had been simply the vacuum expectation values of time-ordered products and the equal-time commutators of the time component of the current with the fields had been free of Schwinger terms. However, the derivation we have given is completely general and does not depend on either of these assumptions. The extra terms in the definition of the Green’s functions (seagulls) always cancel the extra terms in the equal-time commutators (Schwinger terms). (2) It is evident from our derivation that a redefinition of the current of the form Ju + Ju + ~,,.iP (B. 14) with ZUv antisymmetric,

does not affect the form of the Ward identities.

590

COLEMAN

AND

JACKIW

(3) On the other hand, it is important that the current be exactly as given by Eq. (B.2). If the equations of motion are used to simplify the expression for the current, this will in general change the form of the Ward identities. (4) Although our analysis has focussed on Green’s functions associated with n fields and one current, it can trivially be extended to multicurrent Green’s functions. (5) Finally, all of these results are only on the level of formal canonical reasoning. Although the method of variational derivatives is more efficient than the manipulation of canonical commutators, it is no more rigorous. 2. Special Cases

We begin by deriving the Ward identities associated with the canonical energymomentum tensor, ey

=

w

. avq

-

(B.15)

gw*2.

This is the conserved current associated, via Eq. (B.2), with the infinitesimal translations, 6 =uv = a*cp. (B.16) If we denote the Green’s functions associated with this object by I’r, then Eq. (B. 11) tells us that a,ry(n)(y;

x1 ... x,) = -i

f

s(‘)(y

- x,) arVGtn)(xl **. &A

(B.17)

7=1

where a,* indicates differention with respect to x, . Next, we consider the canonical angular momentum

current,

This is the conserved current associated with (B. 19) Thus, its Green’s functions obey the Ward identities a

,r

pvA(n) c

_

-

-i

where the notation n indices.

fl

8’4’(y -

x,)(x,ya,A

-

X,‘a,”

f C;;))

@)(x1

..-

X,),

(B.20)

C$, indicates that the matrix CA”acts only on the r-th of Gfm)‘s

DILATATION

The symmetric energy-momentum

591

GENERATORS

tensor of Belinfante may be written as

We use this as our primary definition, rather than Eq. (A.ll), because it defines a symmetric tensor without the use of the equations of motion, and is therefore guaranteed to yield symmetric Green’s functions, I”,“. Without using the equations of motion, we can rewrite this as

The term in brackets is the divergence of an antisymmetric tensor, and cannot contribute to the Ward identities. The contribution of the remainder is readily calculated with the aid of Eq. (B.17) and (B.20); the result is a&(n)(y;

x1

m-e xn)

=

-i

i T=l

S’4’(y

-

XT) atGcn)(Xl

***

Xn)

Since our preferred symmetric energy-momentum tensor, BfiV, differs from the Belinfante tensor only by the divergence of an antisymmetric object, it obeys Ward identities of the same form. In Fourier space,

These are the identities referred to simply as “Ward identities” in the body of the text. We now derive the equations we referred to as “trace identities” in the body of the text. The dilatation current, D, , is a nonconserved current, auo,

=

s,9

-

av[xcq,

(B.25)

associated with the transformation,

8,(P = w

+ d) v.

(B.26)

COLEMAN AND JACKIW

592

In Appendix A, we showed that the dilatation

current could be written in the form

D, = x,&~.

(B.27a)

The transformation of the dilatation current to the form (B.27a) never used the equations of motion. However, it did use the formula (A.ll) for the Belinfante tensor. Since this is not the expression we used to define our Green’s functions, we must add to (B.27a) extra terms, corresponding to the extra terms in (B.22). We thus obtain D, =

X,[eoy

+ +a&fp

We may now apply our general formalism. a,~y,r~~(")(y,

x1

= ryy,

-

;X"aAep

+

(B.27b)

&X'a&].

The result is

*emx,) x1 *** x3 + ; ay, YY i SC4)(Y- x,) W+% r=1

x?J

- i fl Sc4’(y - x,)(x,AaAT + 4,)) G(‘)(x, *-* 4,

(B.28)

where I? denotes the Green’s function associated with the right-hand Eq. (B.26). Using Eq. (B.23), we can simplify this to

side of

&"r""YY,

Xl

-.. x,) = ryy,

x1 *.- x,) - i f S4( y - xT) d(,.)Gcn)(xl *** x,). r=1

(B.29)

In Fourier space, this becomes grrvr~~(wy p1 ... 14 = r94,

p1 *.* PJ - C d&‘%l

..-pl- + q a-p,).

(B.30)

This is the desired result. APPENDIX

C.

SOME FEYNMAN INTEGRALS

In this appendix we complete the computation of the Feynman integrals necessary for the calculations of the anomalous dimensions of Section 111.3. We begin with the calculation of the anomalous dimension for the meson field. According to Eq. (3.29),

(C-1)

DILATATION

593

GENERATORS

By the arguments given in the text, it is sufficient to compute Z(P, 0, ml , m3

= -2iTrS~~y5X+d-mlY5~~l

fconst.

(C.2)

Furthermore, by the arguments given in the text, we need only compute the terms of order p2 in the power-series expansion of this integral. Expanding the denominator and performing the traces, we find, m2Z(p, 0, ml , m3 = 8ipzm,/&

1

m12[mlm22 + (m, - 2m2) k2] + (k2 - ml”) k2m2 [k2 - ml”]” [k2 - mz2]” I

+ const + O(p4).

(C.3)

On dimensional grounds, the coefficient of p2 in this expression is a function of (ml/m2) only; thus it is trivial to take the limit: 4P,

0) = g2m4p,

0, M, Ml - Z(P, o,o,

WI 1

=---

(k2 -’ M2)3

= - $$

k2(k2 -

A42)2

(C.4)

+ const.

Following Eqs. (3.30) and (3.34) of the text, we find that the modified dimension of the boson field is given by (C.5)

The calculation

for the nucleon field is similar. By Eq. (3.37),

+

g2Mf[z2(P,

4,

Mb

? Mf)

-

z2z(P,

4,

h

VW

Mf)l).

By the arguments given in the text, it suffices to compute zl(py

O? mlF

m2)

=

is

$$

[Yja

+

j-

m,

%i

(k2

/m22)2/

+

consty

(c-7)

594

COLEMAN

AND

JACKIW

Furthermore, we need only calculate the terms of order p in the power-series expansions of these integrals. Expanding the denominators, we find UP,

0, ml7 m2>

= 4 / A.&

lck2 _’ m22)2(k2 _’ m12)2(k2 - 2m12)/ + const + O(p2), (C.9)

and I,(P,

0, ml,

m2)

= ipm2 J +$

j(k2 -*m22)e (k2 1m12)2/ + const + WP3.

(C.10)

By the same reasoning as before, A(P,O)

= 2tT2Mb2[UP, + g2MrV2(P,

0, Mf 9 &I

-

0, Mb 9 Mt> -

UP,

o,o,

WI

I2(P9 03 0,

MAI

s

d4k Mf2Mb2 = @g2 m 7

x 1(k2- M,2&2- Mfz)+

(k2 - Mf2):(k2 - Mb2) 1

(C.11) Hence, from Eqs. (3.38) and (3.41), we find that the modified nucleon field is given by

d’=;+g.

dimension

of the (C. 12)

It is easy to verify that, to this order of perturbation theory, both the nucleon and meson propagators have the high energy asymptotic form, correctly given by the anomalous dimension (C.5) and (C.12).

DILATATION

APPENDIX

D.

595

GENERATORS

CALLAN-SYMANZIK

EQUATIONS

FOR CURRENTS

In this appendix we derive the Callan-Symanzik equations for Green’s functions associated with currents, rather than canonical fields. As we shall see, in this case the anomalies associated with a change in scale dimension are not present, but anomalies of a new type take their place. For definiteness, we will consider the theory of Yukawa-coupled nucleon and meson fields discussed in Section III.3 ; however, ~lith the graphic meson self interaction which is necessary for renormalizability of the theory. We begin by considering the Green’s functions for the canonical fields, which we denote by G(n.n,r)( p1 . . . Pn ; p1’ . . . Pn’ ; 41 . . . qr), where n is the number of nucleon fields (and also, therefore, the number of antinucleon fields) and m is the number of meson fields. We add to the Lagrangian a coupling to two external c-number functions of space and time. $I and fz , of the form and define

where Z and Z’ are cut-off dependent constants, chosen to make the r’s cutoff independent. (We know from standard power-counting arguments that there are two linearly independent ways to achieve this.) Just as in Section IV.2, it follows from these definitions that

tD.3)

596

COLEMAN

AND

JACKIW

where g is the renormalized Yukawa coupling constant, h the renormalized quartic meson self-coupling constant, p and m the renormalized masses, and where 2, and Z, are the nucleon and meson wave-function renormalization constants. It is trivial to show that the quantities in parentheses are cut-off independent. We now fix Z and Z’ by demanding that

+Z’gr = z2!c acLo2

(D.4a)

q&2,

0

and

am

(D.4b)

and define Zag

ag

+zL&g=

a1

(D.5a)

>

aPO2

Z*

+

z/g

=

a/Jo2

(D.5b)

Q2,

0

zy+z+ 0

-7-n

(D.5c)

3

and z a In Z, apo2 +’ We thus obtain the Callan-Symanzik

, -= a In z, am0

-2

(D.5d)

Y2'

equations for this theory: aG(n,n.r) + 2P2

aG(7Ln.r) +m

i3m

ap2

- (2ny, + ry2) G(n*n*r).

u3.6)

These have the same interpretation as the equations derived in Section IV.2. The only difference is that there are now two y terms, one for the meson field and one for the nucleon field. This theory contains a conserved current .i, =

~JY&.

(D-7)

The usual power counting arguments show that we can define cut-off independent connected Green’s functions for n currents, which we denote by Jrl$&l

... PA

DILATATION

597

GENERATORS

by taking n variational derivatives of ln(0 1S 10) with respect to yP , a c-number function of space and time, coupling according to Z’ -

2 + AJ”

+ BA(%A

- U%)2.

(D-8)

Two features of this formula require comment: (1) In the first added term, there is no cutoff-dependent constant multiplying the current. This is because the Ward identities tell us that the Green’s functions for one current and an arbitrary number of renormalized canonical fields are finite without any renormalization of hle current. (2) The second added term involves the cutoff-dependent constant A. This is necessary to cancel the logarithmic divergence in the two-current connected Green’s function. (If we had coupled the current to a vector meson, a term of precisely this structure would have arisen as the vector-meson wave-function renormalization counterterm.) Just as in Eq. (D.2) we define (D-9) where Z and Z’ are the same as in the previous discussion, i.e., fixed by Eqs. (D.4). We thus obtain 012 2aX + m &

+ 2p2 “) J$!..LLn aP2

(D.lO)

for n # 2. For n = 2, we have I'z'(O;

p, p) =

(al $

+ a2 $ + m &

+ P(PUP”- &“P2),

+ 2p2 &)

J,%,

p)

(D.ll)

where (D.12) These are the desired equations. Some comments should be made: (1) The or-terms, which give us anomalies that cannot be absorbed in a redefinition of the scale dimension, remain when we change from a description of the theory in terms of canonical fields to one in terms of currents. This is not surprising, because Eqs. (D.5a) and (D.5b), which define the 01’s, refer only to the fundamental constants of the theory, not to which local fields we choose to use to describe it. (2) On the other hand, the y-terms, which do describe anomalous dimensions, if the 0~‘svanish, are gone. This also is not sur-

598

COLEMAN AND JACKIW

prising, for if it is possible to give a consistent definition of scale dimension at all, the dimension of the current is fixed by its equal-time commutator with the fields, independent of the dimension of the fields, (3) Finally, a new type of anomaly, the p-term, has appeared in Eq. (D. 11). Its occurrence is essential; it prevents us, even in the case when the LX’Svanish (such as, for example, when g and h are zero and the theory is free) from assigning a scale dimension to the Schwinger term in the current-current commutator. It is easy to see that if such a dimension could be defined, it would lead to immediate paradoxes [17].

ACKNOWLEDGMENTS We are very much in debt to Curtis Callan, Kenneth Johnson, Kurt Symanzik, and Kenneth Wilson for many private communications, both oral and written.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.

9. 10. 11. 12.

13. 14. 15.

16. 17.

H. A. KASTRUP, Phys. Rev. 142 (1966), 1060; see also earlier references cited in this paper. G. MACK, Nucl. Phys. B 5 (1968), 499; G. MACK ANLI A. SALAM, Ann. Phys. 53 (1969), 174. L. S. BROWN AND M. GELL-MANN (unpublished); P. CARRUTHERS,Phys. Reo. D2 (1970), 2265. K. WILSON, Phys. Rev. 179 (1969), 1499; Phys. Rev. D2 (1970), 1473, 1478; Phys. Rev. D3 (1971), 1818. S. COLEMAN AND R. JACKIW, “Canonical and non-Canonical Scale Symmetry Breaking,” Massachusetts Institute of Technology, preprint, Cambridge, Mass., May 1970. M. A. B. BBG, J. BERNSTEIN, AND A. SIRLIN (unpublished). J. S. BELL AND R. JACKIW, Nuovo Cimento 60 (1969), 47; S. L. ADLER, Phys. Rev. 177 (1969), 2426. C. G. CALLAN, JR., Phys. Reu. D2 (1970), 1541; K. SYMANZIK, Comm. Math. Phys. 18 (1970), 227. C. G. CALLAN, JR., S. COLEMAN, AND R. JACK~V, Ann. Phys. 59 (1970), 42. S. WEINBERG, Phys. Rev. 118 (1960), 838. J. D. BJORKEN, Phys. Rev. 148 (1966), 1467; K. JOHNSON AND F. E. Low, Progr. Theoret. Phys. Suppl. 37-38 (1966), 74. M. GELL-MANN AND F. E. Low, Phys. Rev. 95 (1954), 1300. W. HEPNER, Nuovo Cimento 26 (1962), 352. R. JACKIW, Phys. Rev. 175 (1968), 2058. E. R. HUGGINS, Ph.D. thesis, California Institute of Technology, Pasadena, Calif., 1962. L. BIANCHI, Lezioni di Geometria Differenziale, p. 375, Pisa, 1902. M. A. B. Bfi~, J. BERNSTEIN, D. J. GROSS, R. JACKIW AND A. S~RLIN, Phys. Rev. Left. 25 (1970), 1231.