Conformal Ward identities in massive quantum electrodynamics

Nuclear Physics B65 (1973) 413-426. North-Holland Publishing Company

CONFORMAL MASSIVE

WARD IDENTITIES

QUANTUM

IN

ELECTRODYNAMICS

N.K. NIELSEN Institute of Physics, Unirersity o f Aarhus, DK-8000 Aarhus C, Denmark

Received 3 August 1973

Abstract: Integrated conformal Ward identities of the Callan-Gross-Parisi type are derived in massive quantum electrodynamics, including external axial-vector currents, by means of the normal product formalism. It is found that conformal symmetry in the Gell-Mann Low limit is violated for gauge-variant vertex functions through a mechanism which can be anticipated from classical field theory and from free field theory.

1. Introduction The concept of conformal invariance has recently aroused considerable interest because it leads to useful constraints on the form of vertex functions (the literature on this subject can be traced from refs. [1,2]). The real world cannot be described by a conformal invariant field theory since this would prevent the elementary particles from having a discrete mass spectrum. However, the hypothetical Gell-Mann Low [3] (GML) limiting theory, which is supposed to be the short-distance asymptote of the ordinary massive theory, could be conformal invariant. This is indeed the case in the ~b4 theory [4,5]. On the other hand, the GML limit of a gauge theory such as quantum electrodynamics cannot be expected to be conformal-invariant for two reasons: (a) It is known that no conformal-covariant propagator exists for a free vector field [1 ]. (b) In classical field theory scale invariance will only imply conformal symmetry if the so-called field-virial is the divergence of a rank-two tensor [6]. This condition is violated by the gauge-dependent terms of the Lagrangian density. This situation necessitates a precise criterion, allowing one to determine which quantities are conformal-covariant in the GML limit, and which ones are not (cf. [ 7 - 9 ] ) . Such a criterion will be obtained in the present paper through the derivation of conformal Ward identities (CWls) like those found within 44 theory by Callan and Gross [10] and Parisi [1 1]. The derivation is carried out by means o f the normal product algorithm [12,13], which guarantees that all anomalies are taken properly into account.

N.K. Nielsen, Conformal Wardidentities

414

It is found that only those of the vertex functions of the GML limit are conformal-covariant for which the corresponding functions of the ordinary theory are annihilated by the differential vertex insertion in the sense of Lowenstein [13,14]:

f d4xN3 [Av3.Au](x), where Au is the photon field. The condition can only be fulfilled for vertex functions in the charge-zero sector, and these functions will, anyhow, vanish in the GML limit due to a general theorem [ 15]. Also vertex functions containing an external axial-vector current are considered and found to be subject to this condition. Thus the axial-vector symmetry current [ 16], which is not gauge-invariant, will have no conformal-covariant vertex functions in the GML limit, whereas those of the gauge-invariant axial-vector current in the charge-zero sector will be conformal covariant in the same trivial way as above. The treatment of quantum electrodynamics without any foreign operators is contained in sect. 2, while the properties of axial vector currents are investigated in sect. 3. The Pauli metric is used throughout.

2. CWIs for ordinary vertex functions Massive quantum electrodynamics is, within the BPHZ formalism, described by the effective Lagrangian density [17] •~ : -½(1 + d ) ~ ' 1 , ' 3 ¢ - ( m

_l(la2+a)AuA u

c ) @ ~ - - ~ ( 1 - b ) ( ~ v A v O v A v - - 3uA**OvAv)

x /~2+a 2 ~ 2 buAu3vA v + ie (1 + d ) ~ 7 . SAv

(1)

where m and 12 are the electron and photon masses, e is the electric charge, and a, b, c and d are functions of e to be determined in perturbation theory from the normalization conditions imposed upon the propagators. ~ is a real parameter characterizing the gauge. It follows from the work of Coleman and Jackiw [6] on classical field theory that the following expressions is a reasonable ansatz for the conformal current Kxv corresponding to the Lagrangian 22 of eq. (1):

Kxv=

~

c3_~_~ [2xv(dr+xu31~)__x23v+2xuEvu]~) r

r =A,¢,~ 3elkqSr

-- (2xxx v - x26Kv) d2÷ (1 - b) (6xvA 2 + 2AxAv) , where d r = 1 for r = A and d r = ~ for r = ~, ~. We have also introduced the spin matrices

(2)

N.K. Nielsen, Conformal WardMentities ~vu =~ [7v,7 u ]

415

for spinor fields,

~vu,xo = 8vxfuo -

6vp6ux

for the vector field.

The last term in eq. (2) takes into account the part of the field virial [6] that can be written as the divergence of a rank-two tensor. More explicitly, Kay becomes 2 +a ) Kxv = - I I - b ) ~ x A ¢ - (1-b _ ' u ~U 2

c.)oAoax- ]

> {[2xv(1 +xo3o)-xZOv]A¢+2XuZvu,~ojA~} }(l

+d)(2xvxu(~Tx'~ )

x2(@'yx~v~)

+ ~ {77,, 2x.E~u ) qJ) - (2xxx~-x2ax~) .8 +(1

b)(ax~,A2+2AxAv).

(3)

Of course, the manipulations so far only make sense on the heuristic level. We can, however, with eq. (3) as a motivation, define Kay by

(TKxv(x) X) pr°p = (2XvXu - x26ug) X (TN4 C-(1-b)OxA oOuA~o+ ( 1 - b - / ' t 2,~u + a ~ OoAoOuAx 2] -- ½(I +d)~TX~'.~ - 6~..22] (x)X) pr°p

- 2Xv(TN3 [((1-b)OxAu- (1-b-"2+a) ~U2 OoAofxv) Au] (x)X) pr°p + (1 - b)(

TN 2 [6XvA2 +2AxA v] (x)X)Pr°P ~u2 !

---~(1 +d)~(')'X,]~ug } qJ] (x)X)Pr°P ~

(4)

with B

F

i=1

/=1

and where the superscript prop means that only amputated, one-particle irreducible diagrams are taken into account. The symbol N 6 , 6 = 2, 3, 4, indicates the subtraction prescription [ 1 2 - 1 4 ] . Eq. (4) does not define Kxv as an operator [18], but

N.K. Nielsen, Conformal Wardidentities

416

this point is unimportant for our purposes since we are only interested in integrated CWIs from which Kay has disappeared completely. Equations of motion are obtained by inspection of all relevant Feynman diagrams [12], [ 13], [17]. After a rather tedious calculation these equations allow the following CWIs to be obtained from eq. (4): B

~x x(TKxv(x)X>pr°p=ii=l ~

v

l+xp~

~

6(x-ui)

×
+ 2ix. ~ 6(X ui)~,uu.~ui(TX(Aui(ui) --~Ao(ui))) pr°p i=1

F j=l

F

- 2ix u ~

[6(x- vj) (~vu)~j"r (TX(~e~/(o/) -+ ~'r(uj))

/=1

>propJ

- 6(x wj)(~v#),,/~j(TX(~o(wj) -+ ~,y(wj))} pr°p] + 4(U 2 + a)

~t12


+ 2Xv(TN 4 [(02 +a)A 2 + ( m - c ) ~ ]

(x)X) pr°p .

(5)

This equation has the naively expected form, but the last term on the right-hand side is oversubtracted. This circumstance will give rise to anomalies (cf. [17]). The oversubtracted term can be redu'ced by means of Callan-Symanzik equations [ 19,20] and Zimrnermann identities [ 12]. Introducing the differential vertex insertions

A 0 = -½ i f d 4 x N 3 [A 2](x), A0=i

'

f d4xN3[fa~](x),

(6a) (6b)

we have the following Callan-Symanzik equation in the present theory [20-23]"

N.K. Nielsen, Conformal Ward identities

t

= {areA 0 + (2-- p) (/32 + a)A 0 ) ( T X )prop - A ( T X )prop ,

417

(7)

where a, 13, 7 and p are functions o f e to be determined in perturbation theory. Restricting ourselves to two-point functions, we obtain from (7) ~-~ +/3 ~ + 5~3 1 (, ~/ 3 - 2 ) ~/3 rn~-m+/3 ~

-p~

2

(/32 +~) = ( 2 - p ) (/32 + a ) ,

(1--b)=(2-p)(/32+a)s2 +amt 2 Of

+-~e~-~

+~/3e~-27

(m--c)=

(8a) (Sb)

am,

23' ( l + d ) = ( 2 - p ) ( / 3 2 + a ) s 4 + a m t 4 ,

(8c)

(8d)

where the fact t h a t a and b do not depend on ~ [171 has been used. s 2, s4, t 2 and t 4 are functions of e which also occur in the Zimmermann identities:

½( T N 3 [A 2 ] (x)X)P r°p = (TN 4 [12A 2 + ½s2(auAv 3uAv - 3uA**3vAv) + s4(½ g/73" $ - ie ~7 u $Au) ] (x)X)Pr°P ,

(9a)

- ( T N 3 [¢/~] (x)X) pr°p = ( TN 4 [-fJ ~ +ltz(OuAvOuAv - 3vAu3vAv) + t4( 1 ~73" ~ - ie fay u ~Au)] (x)X) pr°p .

(9b)

From eq. (8a) we see that if we set a = 0, then we must have O = ft. This leads to considerable simplification of the formalism (cf. [22] sec. IIA), but the transverse photon propagator will only have the conventional normalization for/3 = 0. However,/3 = 0 implies p = 0 no matter how a is fixed. By means of eqs. (8) and (9) we obtain

( TN 4 [(/./2 +a)A 2 + ( m - c)~ ~] (x)X)prop = ( T N 3 [g (2 - p) (/32 + a)A 2 _ am ~ ~ ] ( x ) X )prop 3 .C , c'J.~. . . . . prop + ( TN4 [lfl e --37 - P~ ~ l (x)a~

+ 23' ( T N 4 [(m - c)t~ ~ + (1 +d) (~ t~7~ t~ - ie ~Tu~Au) ] (x)X)P r°p +{fl(TN 4 [(1-b)OuAvauAv-

( l - b - / 3 2 + a ) 3uAu3vA v

\

+ ( / 3 2 + a ) A 2 - i e ~ y u ~ A u ] (x)X) pr°p .

~/32

(10)

418

N.K. Nielsen. Conformal Wardidentities

We are now finally able to express the integrated CWI in a suitable form. Introducing the differential operator ~ v: B

C p ( T X ) p r ° p =i=1 ~ I2uiv(l+1/3)

2 ~-~iv O UipUi,+ -~iuUi 0 2-] j (TX)prop

B

+ 2 ~ uiu Y-WU,~oui(TX(Aui(ui) -+A¢(ui)) )prop i=1

F

+j~=l [2vi~(~+7)- 2 ~uiuvi~viu+°Viv + 2wiu(-~+"/) 2 ~ WiuWi + w (TX)Pr°P Owit~ ;z F 2 ~ [v]u(~vu)ajy (TX($a](o]) -+ ~y(o])) )prop /=i

-w/u(~vu).r#](TX(~/(w/) -+ ~7(w])))pr°p ] ,

(11)

we obtain from eqs. (5) and (11), combined with the equations of motion

~u(TX)prop= i (,46u2+a)+/3 ( l _ b _ / ~ 2 + a ] ) f d4x(TN3[Av~uAu](x)X)Pr°P - ~ -- fi~ 3_~] (x)X) pr°p +ifd4x2xv(TN4~,eOd2

+ i f d4x2x~ (TN 3 [½(2-p)(~t 2 + a)A 2 -o~rn~ if] (x)X)Pr°P,

(12)

which are the relations within quantum electrodynamics corresponding to the equations derived by Callan and Gross [ 10] and Parisi [ 11 ] in 4)4 theory. The CallanSymanzik equations can be obtained in a similar way from the Ward identities of the dilatation current (cf. [20] appendix A). There is, however, a crucial difference between eq. (12) and the Callan-Gross [10] -Parisi [11] equations in the 4)4 theory. In the latter case, the right-hand side of the equations are negligible compared with the left-hand sides in the deep euclidean region if the coupling constant is chosen so as to be equal to the hypothetical GML eigenvalue, which makes the function 13of 4)4 theory vanish; from this fact one concludes [11 ] that in this case the GML limiting theory, besides being scale-invariant, is also conformal-invariant. This reasoning cannot be carried through in quantum electrodynamics because the right-hand side of eq. (12) contains the term

N.K. Nielsen, Con]brmalWardidentities i 4(/12 +a)

~i,12

fd4x
419 (13)

which is never negligible compared with the left-hand side for vertex functions with external fermion legs (notice that the operator G v softens the vertex function upon which it acts with one power). Eq. (13) vanishes for vertex functions with only external boson legs because of current conservation [17], but such functions will, anyhow, vanish in the GML limit, due to a theorem of general field theory [15], so conformal symmetry is not useful here either. One can appreciate the role played by the vertex insertion encountered here by the observation that the free massless photon propagator corresponding to the parameter

--i4f ~ d4k (TA(uO)(x)A(f)(Y))=(~

(6uv-(1

kuk~ ~ e ik(x y) -~) k~-~_ie)

(14)

fulfils the relation a +1) +

0° +1) (Yoay----

x2~x

y2

-2x. ~'xo, o. _ 2y. £x#,a~ (TA(uO)(x)A(O)(y)) +

d4z (TA(O)(x)A(O)(y)'AIO)(z)a#A(O)(z):):

O,

(15)

(the superscript (0) denotes a free field). It should also be noted that the expression in eq. (13) is well-defined for ~-+0 (Landau gauge) since the factor OuAu in the vertex insertion has to hook onto a longitudinal photon propagator, which explicitly contains a factor ~.

3. External axial-vector currents

The recent interest in the short-distance properties of vertex functions involving external axial-vector currents [24-26] makes it natural to study also such functions along the lines indicated above. It is known from the work of Adler [16] that within quantum electrodynamics one can construct two different axial-vector currents, since the two requirements of partial conservation and gauge invariance are mutually exclusive. The partially conserved axial-vector symmetry current is denoted by j5s and the gauge-invariant axial-vector current by j5; the latter may have anomalous scaling dimension, because it is not partially conserved [26]. The axial-vector symmetry current is treated first; it is defined by

420

N.K. Nielsen, Conformal Ward identities

/5S(x) = N 3 [i(1 + d - s)~/u~/5 ~ + 2reuv:coA vF: vo] ,

(16)

and fulfils the Ward identity (cf. [17])

Oxu

( T/5S(x)X )Pr°p = 2(m-c)i ( TN 3 [~Y5 if] (x) X)pr°p

+ ~ (8(X -- 0j) (T5)c~j3,(TX(t~cej(v]) -+ t~7(vj)) )prop /

+ (~(X--W/-)('Y5)y~](TX(f/,~.,i(w]) "+ f)"l(w.i)))prop ).

(17)

Here F:,.o = QoAvo - OvoA: is the field strength, the constant r is given by its second order value [27,28,23]: e2 r = -167r2 '

(18)

while s, as seen from eq. (17), is

s = - ~ (im - c )

~

Tr {TttT5 (TN3[~T5~](O)f(O)~(p))pr°P}lp=O,

(19)

where ~(p) = f d4x e-ipx ~(x). Eq. (17) expresses the 75-invariance of the theory in the deep euclidean region. We now define ( TKxv(y)f5S(x)X)P r°p in analogy with eq. (4) and may then derive Ward identities which are similar to eq. (5) with the exception of the expected extra contact terms on the right-hand sides. However, the reduction of the oversubtracted terms here presents additional complications. The Callan-Symanzik equation [23]

(m~+is~_p~

1_[ 8 - B ) - 2~{(F- 1)) (TN3[~'gS~](x)X)Pr°P ~8 + 7¢4~e~7

= ( A - u) ( TN 3 [f75 if] (x) X)pr°p ,

(20)

(u is a function of e) can be combined with the Ward identity

a~u ~ ( T/~S(x)X)pr°p = [2(m- c) (A-- u)-- 2am] i ( TN 3 [~75 qJ] (x)X)prop

+ ~ (~ (X- Uj) (T5 )oe..iyA (TX(~oe.i(vj) "-+ ~.,/(oj)))prop /

+ ~(X-- Wj) (T5)3,/3].A(TX(~f3/(wj) ~ ~y(w/)))prop ),

(21)

N.K. Nielsen, Conformal Ward Mentities

421

and with eq, (8c) to yield the relation

X ( T N 3 [2(m

c)if~75~] (x)y)prop = ~ A ( T/5S(x)y)prop ox~

- ~ . ( ~ ( x - v/) (~/'5)a/,yA (TX(~aj(oj) -+ ~Jx(vj)) )prop J + ~ ( x - wj) (")/5)'y~jA (TX(~)@(wj) -+ fJ~,(wj)))prop).

(22)

If X consists of two photon field operators, then eq. (22) can be used [28,23] to prove eq. (18), the celebrated Adler-Bardeen low-energy theorem [27], Here we are more concerned with the case where X is a product of two fermion-field operators; in this case eq. (22) implies (_p~+

1 ~ \ + , ~13e~-2"),) s = ( 2 - p ) ( u 2 + a ) ( s 4 + K ) am(t4+K )

(23)

with = - & i Tr (7•75A0 ( TjSus(o)~(O)~(O ) )prop},

(24a)

K'= -~6 iTr (7~75 A~ ( TjSus(o)~(O)~(O ) )prop),

(24b)

The Zimmermann identities are now I ( T N 3 [A 2 ] (y)j~s(x)x)Pr°P 1 I = ( T N 4 [~A a2 +~Sz(OaAvOaAv - OxAaOvAv)

+ s4(~ ~ 7 ~ -

ie~y& ~Aa)] (y)jSS(x)X)pr°p

+ K6 (y - x ) ( T N 3 [~Tu3,5 ~] (x)X)Pr°P , - ( rN 3 [~¢]

(25a)

(y)]Ss(x)X)Pr°P

= ( TN 4 [-~

+½t2(O?~AvB?~A v - B?~Aa~vAv)

+ t4(~ ~ 7 ~ - i e ~ T ? ~ A a ) ] (y)j5S(x)X)pr°p

+ K'6 (y - x) ( T N 3 [ ~3'.~/5 ~ ] (x)X)pr°p , and so we finally obtain

(25b)

N.K. Nielsen, Conformal Ward identities

422

( TN 4 [(/22 + a)A 2 + (m - c)t~ ~] (y)f5S(x)X)prop = ( TN 3 [~(2-p) (/22 +a)A 2 _ +

amPUl()')]5s(x)x)pr°p

-p~-~+~i3e.~

+(TN4 F1

Oi2

OZ-1 (y)]5S(x)X) pr°p

+ 2"7( TN 4 [(m- c)~ff +(1 +d) (~-~3'~'~b-ie~Tx~Ax)] (y)f5S(x)X)pr°p +-~ (3(TN4 [(1-b)OxAvOxAv-

( 1 - b - /22+a ) OxAxOvA,, ~/22

+ (/22+a)A 2 _ ie(1 +d)~Tx~Ax] (y)j5S(x)X)prop ,

(26)

corresponding to eq. (10). Eq. (26) first implies that the axial-vector symmetry current has canonical scaling dimension, as it should have, being partially conserved. This follows from the observation that eq. (26) can be used to transform the integrated dilatation Ward identities for the vertex functions of the axial-vector symmetry current into CallanSymanzik equations. Next eq. (26) allows manipulations similar to those leading to eq. (12) from eq. (5). Since

a (TKxu(y)]5s(x)x)pro Oy~

p

apart from terms analogous to those on the right-hand side of eq. (5), has an extra contact term

- i 6 ( y - x ) ( 2 X u 3+x o ~x °

~

(

+ 2 i 6 ( y - x ) x w Y~u~ ,,Ou ( Tf5S(x)X)pr°p ' we obtain the following result

(27)

N.K. Nielsen, Conformal Wardidentities

,x>pr°P

(3+x

x2

423

,xprop

+ 2xco ~v~o,sola( T]5S(x)X)Pr°P - / (.4(/-t2 _+a)+l3 (l_b \ ~/d2

(1~ aJ~s

/-t2+al) ~/l2 1 !

aJ~sx

+ 2Xv ( T ~l..~e--~- - p~ ~ - - )

+i fd4y2yv
fdayprop

(x)X) pr°p

~ 1 (Y)f5S(x)X)pr°p

+i fd4yZyu
(28)

Again we see that we cannot have conformai symmetry in the GML limit; the symmetry is destroyed by the term 4(~a)

f d4y
(29)

We have, however, the possibility here thatjuSs through the last term of eq. (16) can hook directly onto the vertex insertion, and hence not even the charge-zero-sector vertex functions of/58 will be conformal-covariant in the GML limit. Specifically, the Schreier [29] ansatz cannot be applied for the two-photon vertex function of /58 (this circumstance does not invalidate the argument of ref. [26] concerning the inapplicability of the Schreier ansatz for the two-photon vertex function of/5, the gauge-invariant axial-vector current, since the main ingredient of this argument is the fact that the difference between the two vertex functions is no longer a seagull when two-photon reducible graphs are taken into account). The gauge-invariant axial-vector current/us is in the present formalism defined by [171

/uS(x) = (1 +d-s-fir)iN 3 [~7u75 4] (x),

(30)

where ~ is a function of e chosen to vary in a suitable manner under gauge transformations. Corresponding to eqs. (10) and (26) we have here

N.K. Nielsen, Conformal WardMentities

424

(TNg[(/j2+a)A2 +(m c)~¢] (y)/5(x)X)Pr°P = ( TN 3 [l(2-p)(uZ+a)A2 -c~m~] ()')]5(x)X)pr°p + {[

3

l^

3

~)(e+d-s

~r) 6(y-x)

X ( TN 3 [ ~7u7 5 ~ ] (x)X)prop (y)f5(x)X)pr°p + 27 ( z g 4 [(/77 ¢)t~ + (1 +d) (½~"[~ - ie~yxCAx)] (y)/Su(x)X)P~°P +½/3(TN4 [(1

b)3xAv3xAv-(l-b--ta2+a-]3xA~,3vAv ~u 2 ,]

+ (/.12+a)A 2 - ie(1 +d)~T~~Ax] (y)j5(x)X)pr°p ,

(31)

with (_ p~, ~~ + ~1~e ~~3 - 2 T - ~ ) ( l + d

s-~r)

- ~ i Tr {7u7 s A (TjS(O)~(O)~(O) )prop]..

(32)

Equation (31) can be used to derive Callan-Symanzik equations for vertex functions ofjSu, and since F need not vanish at the GML eigenvalue, one concludes that j~ may have anomalous scaling dimension [26]. Also eq. (3 l) leads to the following integrated CWIs:

@v(Tj5(x)X)Pr°P- (2xv (3-rT+Xo ~ )

-x2 ~u)(T/5(x)X}pr°p

+ 2xco £vw,~o~ (Tj5(x) X)pr°p =i (4(la2+a)+~ (l_b_t~2+al)fd4y(TN3[Av~xAx](y)f5(x)X)prop \ ~Id2 ~t.t2 ]

aJ5 3j~ +2xv(T(1 ~e~b--p~ ~1

(X)x)pr°p

+ i f d4y2yv ( TN4 [1¢3e O'C~-~- P~ ~;'3£°1(y)f5(x)X)prop

+if d4y2yv
(33)

Eq. (33) allows the same conclusions as eq. (12), viz. that the GML limit vertex functions are only conformal-covariant in the charge-zero sector where they vanish due to the theorem of Strocchi [15].

N.K. Nielsen, Con formal Ward identities

425

4. Summary In the present work the following results have been obtained: Integrated CWls of tile Callan-Gross-Parisi type have been derived within quantum electrodynamics by a general technique, which exhibits their connection with the Callan-Symanzik equations. Thus it is found that no new anomalies arise; conformal symmetry in the GML limit is broken, however, by an effect which is active already on the classical and free-field levels and which is due to the gauge-dependent part of the Lagrangian density. Only gauge-invariant vertex functions will be conformal-covariant in the GML limit, but unfortunately in a trivial manner, namely by vanishing. No connection has been established to the approach that uses a non-local gauge [ 7 - 9 ] , althougt this in principle should be possible. The author is deeply indebted to Professor K. Symanzik for suggesting this problem, for helpful discussions, and for reading the manuscript. Also useful comments from Dr. R. Koberle are appreciated, and the kind hospitality of Professor R. Haag and Professor H. Lehmann at II Institut fiJr Theoretische Physik, University of Hamburg, where this work was carried out, is gratefully acknowledged.

References [1] I.T. Todorov, Lecture Notes in Physics 17 (1972) 270; Lecture Notes, 12th Internationale Universit~itswochenfiJr Kernphysik, Schladming (1973). [2] R. Nobili, Nuovo Cimento 13A (1973) 129. [3] M. Gell-Mann and F.E. Low, Phys. Rev. 95 (1954) 1300. [4] K. Symanzik, Comm. Math. Phys. 23 (1971) 49. [5] B. Schroer, Nuovo Cimento Letters 2 (1971) 627. [6] S. Coleman and R. Jackiw, Ann. of Phys. 67 (1971) 552. [7] S.L. Adler, Phys. Rev. D6 (1972) 3445. [8] H.J. Schnitzer, Phys. Rev. D8 (1973) 385. [9] F. Englert, Nuovo Cimento 16A (1973) 557. [10] C.G. Callan and D.J. Gross, Broken conformal invariance, Brinceton report (1972). [11] G. Parisi, Phys. Letters 39B (1972) 643. [12] W. Zimmermann, Lectures on elementary particles and quantum field theory, Brandeis University Summer Institute in theoretical physics, 1970, ed. S. Deser et al. (MIT Press, Cambridge, Mass., 1971). [13] J.H. Lowenstein, Seminars on renormalizatfon theory, vol. II, Maryland University Technical Report No. 73-068 (1972). [14] J.H. Lowenstein, Comm. Math. Phys. 24 (1971) 1. [15] F. Strocchi, Phys. Rev. D6 (1972) 1193. [16] S.L. Adler, Phys. Rev. 177 (1969) 2426. [17] J.H. Lowenstein and B. Schroer, Phys. Rev. D6 (1972) 1553. [18] J.H. Lowenstein, Phys. Rev. D4 (1971) 2281. [19] C.G. Callan, Phys. Rev. D2 (1970) 54l. [20] K. Symanzik, Comm. Math. Phys. 18 (1970) 227.

426 [21] [22] [23] [24] [25] [26] [27] [28] [29]

N.K. Nielsen, Conformal Ward identities

A. Sirlin, Phys. Rev. D5 (1972) 2132. S.L. Adler, Phys. Rev. D5 (1972) 3021. J.H. Lowenstein and B. Schroer, Phys. Rev. D7 (1973) 1929. S.L. Adler, C.G. Callan, D.J. Gross and R. Jackiw, Phys. Rev. D6 (1972) 2982 B. Schroer, Lecture Notes in Physics, 17 (1972) 364. R. Koberle and N.K. Nielsen, Phys. Rev. D8 (1973) 660. S.L. Adler and W.A. Bardeen, Phys. Rev. 182 (1969) 1517. A. Zee, Phys. Rev. Letters 29 (1972) 1198. E.J. Schreier, Phys. Rev. D3 (1971) 982.