Volume 71B, number 1 PHYSICS LETTI'RS 7 November 1977

C O N F O R M A L S P I N O R C U R R E N T A N O M A L I E S *

Thomas CURTRIGHT Department of Physics, University of California, Irvine, California 9271 7, USA

Received 24 June 1977

An anomalous divergence of the conformal spinor current is obtained for some massless supersymmetric theories. The corresponding modified Ward identities are discussed for a simple model without gauge fields.

Formal manipulations in field theory often lead to incorrect results which are fundamentally modified by quantum effects. These modifications, or "anomalies", typically arise in a renormalized model as remnants of the regularization needed to define the theory. A well- known example is the anomalous divergence of the singlet axial current [1, 2]. Another anomaly which has recently received at tention appears as a nonzero trace for the energy-momentum tensor of a "massless" gauge theory [3-7].

The trace anomaly is trivial to obtain in lowest or- der, O(h). Consider a nonabelian gauge theory quan- tized in an axial gauge with n" V = O, and containing no other fields. The axial gauge is ghost free and sub- ject to very simple Ward identities [8]. One conse- quence of these identities is that

Zv = 1 / z 2 , ( l )

where Z v is the field renormalization, V = Z 1/2 Vren ' - - u n r - - v

and Z e is the charge renormalization, eun r = Z e e. Be- cause of (1) the field strength is multiplicatively renor- malized

FuU~r =~ Vunr-~VV~um+ieunr[ unr' v ren" ta v Vta V~m~]=z1/ZFUv

(2) Since the axial gauge is ghost free, there is no question of ghost operator mixing [9].

Using dimensional regularization we can now write the energy-momentum tensor as (contracted group in- dices are suppressed)

0 v = F u n r . F ~ r + ~g~'FUm.F ~t3 ,u p . " - c t ~ U l U " t., ~,

(3) = Zv(Fruen.I, ren + ¼g~F~n'F?eOn).

* Work supported by the National Science Foundation Technical Report No. 77-31.

Upon taking the trace with g~u = N (the spacetime di- mension) we have

0 u = ¼(N - 4)ZvFren'F~fn (4) /~ /3 •

Taking the limit N ~ 4 we extract the residue of the pole in the field renormalization which, to O(fi), is well-known to be (again using (1))

lim (N - 4)Z v = _2 fie + O(//2) • (S) N-,4 e

Therefore,

OU = 1 ~ Fren.Fat3 + O(/i2)" (6) u 2e e atJ ren

This is the correct anomaly to lowest order with/3 e

- ~-Cve3.~/167r 2 andfacdfbc d = Cv6ab. We will not discuss here the interpretation of the

operator product on the RHS of (6) which is needed to accommodate the higher order corrections. We pri- marily wished to sketch how residual effects can occur in the renormalized theory as the regulator (N - 4 here) vanishes.

The occurrence of the anomaly in 0uu is no surprise. All regularization methods inevitably break the con- formal invariance o f a naively massless field theory and, as is well-known, the divergence of the conformal current is proport ional to 0uu. The survival of a non- zero trace as the regulator is removed is possible be- cause a unit of mass must be introduced to define the coupling constant in the renormalized, four-dimension- al theory.

Consider now a "massless" globally supersymmetric field theory [10]. In such a theory there are two vec- tor-spinor currents which are formally conserved: 3~a, whose charge is the "square root" of the momentum Pu; and cK~, whose charge is the "square root" of the

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generators of special conformal transformations K u [ 11 ]. We refer to 9 and c'K as the "translational and conformal spinor currents", respectively. For the same basic reasons that the conformal current is not con- served to O(h), leading to the anomaly (6), the confor- mal spinor current is also not conserved.

3 .oK = O(fi). (7)

Any regulator will inevitably destroy the conservation of q( and possibly leave a residual divergence in the re- normalized theory.

The purpose of this letter is to exhibit and discuss the form of the anomalous divergence in "X.

First consider a massless, "pure gauge" supersym- metric theory [12, 13] quantized in the axial gauge. The usual vector field is accompanied by a Majorana spinor also in the adjoint representation of the group. The translational spinor current is

i ~ = -~ %,~Fu~nr "Yu ffunr, (8)

and the conformal spinor current is (¢ = xU'),v)

cKu = ¢ 5~.. (9)

Using dimensional regularization, we compute the di- vergence of cK to be

i 3 "eK = ~+ ¢3" J~ = ~ Vu oaf'[. Fu~nr "~ unr + ¢3"

= 2 ( N - 4) oa~Fu~nr'~unr+ ~3"~ (10)

i - ~ (N - 4) Z1/2zlc/2ooa3F~dn • Lkre n + ~3 "~ o

In the ghost free axial gauge we can immediately ex- tract the 0(fi) residue of the divergence a sN ~ 4, using

6Cv e2 /i - - - - + O ( / ~ 2 ) , ( 1 1 )

Z v=Z~, = l + 4 _ N 1 6 , r 2

to obtain

3 "oK = 13~Oc,,,Fa3 "ffren + O(~2)' (12) e ~ ~, ten

Note that/3 e = -3Cve3~i/16zr2 + O(h 2) in this model. The RHS of (12) may be discussed in higher orders

of Pt in a manner similar to the RHS o f (6 ) by using the normal product method [14]. We will postpone

such a discussion in order to later include scalar, pseudo- scalar, and spinor "mat ter" fields interacting with the gauge supermultiplet. The presence of 75 couplings among such matter supermultiplets raises certain ques- tions [ 15] when using dimensional regularization which we have not investigated to our complete satis- faction..It is not obvious what the most sensible defini- tion of 75 is in N dimensions. However, we have con- sidered a self-interacting matter supernmltiplet without gauge fields and avoided all 75 ambiguities by using kinetic instead of dimensional regularization. We will now investigate the conformal and supersymmetric Ward identities in such a theory,.

The theory is the massless version of the Wess- Zumino model [16] defined by the Lagrangian

.(9= ~o 0 + .~g + .~Oreg ' (13)

with (all quantities are unrenormalized)

I [(3A)2 + (3B) 2 + f i~ t~ + F 2 + G2] , (14) "~0 =

_ , [F(A 2 ~g - 5 g B2) + 2GAB ~(A --i')'5B)ff ] ,(15)

Z?reg =-}C[(3 DA) 2 + (3DB) 2 + D~ci~Dff (16)

+ (D]F) 2 + (E]G)2].

The last term is the kinetic regularizing device [17] which introduces enough higher derivatives in ~ to make all diagrams finite• The parameter C has mass di- mension --4 and is to be taken to zero in obtaining the renormalized theory. The infinitesimal supersymmetry transformations are

6A = ~-~9, 8B = ~ i75~, (17a, b)

6 ~ O = - i 3 u ( A - i75B)TUe

-~i(A - i75B)~e + F e + Gi75 e, (17c)

6F = -i'~qJ, 6G = gTs~ff. (17d, e)

The parameter e is a Majorana spinor which can de- pend linearly on x.

e(x) = q + i.~r. (18)

Both q and r are spacetime independent Majorana spi- nors.

I f q 4:0 and r = 0 the Lagrangian (13) changes b3~ a total divergence under the transformation (17). These

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are ordinary supersymmetry transformations and the corresponding spinor current is 9. The regulator (16) respects these transformations. On the other hand, if q = 0 and r d: 0 both (14) and (15) change by total di- vergences but the variation in the regulator is not a spacetime divergence. The corresponding spinor cur- rent is now cK and so the regulator breaks the conser- vation of the conformal spinor current. We may identi- fy the variation in .~°reg as the divergence of oK. For the rest of this paper we will only deal with the zero momentum component of this divergence and will con- sider the effects of the insertion of this component on the associated Ward identities. Explicitly, under (17) we have

/d 4x6.~ = fd4xS.~reg

= 4 C~ f d4x [(A - i 75 B) i@ - (F - i 3, 5 G)] [] 2

= - F ~ " ~ (0). (19)

The tilde indicates the operator is in momentum space and the zero in parenthesis indicates zero momentum.

Naively one would say that b ' ~ ~ 0 as C ~ 0 in eq. (19). Because of the large number of derivatives in the operator 5.~°reg , however, quantum corrections pro- duce compensating factors of 1/C to give residual ef- fects in renormalized quantities in a fashion analogous to the RttS in eq. (12). The form of these residual ef- fects may be obtained to all orders in perturbation theory by combining the Ward identities arising from the transformation (17) with the Ward identities for broken scale invariance, i.e. the renormalization group equations. Let Z[J] be the generating functional

ZlJ] : z o. f @A ~BC-OFC-B G~

(2o)

X exp i fd4x {.~ +JA A +JB B +JFF+JGG +Ja/~O}.

I fq 4:0 = r in eq. (18), the invariance of the function- al measure in eq. (20) under the transformation eq. (17) implies

fd4x{(JA+i3"SJB)~-~-(JF+i3"SJG)i~ts~ (21)

- ( 6 j - ~ + 6~-~) i@JqJ + (6~-~---~ + i3'5 6 ],/

l fq = 0 4: r in eq. (18), we similarly obtain

6 6 j /d4xI~c('"}--2i(~-AA--i3"5~-~B) VJ Z[ J]

4 6 . 6 =-4C fdxC(~-j-~-ri5-~-~B)i@ (22)

The term {...} represents the same operator that ap- pears in braces in eq. (21). Finally, if we make a scale transformation on the fields we deduce that

6

5J~oJ

The RHS of this last equation corresponds to a zero momentum insertion Of~reg in every Green function. Taking functional derivatives of (21), (22), and (23) we obtain the appropriate Ward identities for the vari- ous Green functions which may be rewritten in terms of renormalized fields with the limit C ~ 0 then taken.

The RHS of (22) corresponds to an insertion of (0) in any Green function obtained by functional

derivation. The LHS of this equation may thus be used to take the limit C ~ 0 of any matrix element contain- ing one insertion of 0 "cK (0). Eqs. (21)and (23)may be used to simplify the expressions obtained by using eq. (22). In particular, eq. (23) just produces the renor- malization group equations for the various Green func- tions, as C ~ 0, and eq. (21) gives the ordinary super- symmetric Ward identities for this model [ 17]. We now illustrate these remarks by writing the simplest nontrivial insertions of 0 "~ (0) in the four possible two-point functions. Converting to renormalized fields and taking the limit C ~ 0, we obtain

lim C~ren(P)~'~ (O)~ren(O)> C--0 (24a)

= i [fl(g) 0g - 23'(g)] ("~r en (P)Ar en (0)),

lim (Fren(P)~ " ~ (O)~ren(O)) C--, 0

=(_~_) /3(g)()g-23'(g)](Fren(P)Fren(0))' (24b)

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Volume 71B, number 1 PIIYSICS LETTERS 7 Novembe~ 1977

lira (Bren(P)~'°K (0)~ren(0)} C~O (24c)

= 3'5 [ ~ ( g ) 0 g - 23,0g)1 (B'ren(P)Bren(0)),

lira (Gren(P) ~ ''~OK (0) t~ren(0 )) C~0 (1)

= ~ 7 5 [/3(g)ag - 23,@)] (G'ren(P)Gren(0)>. (24d)

Operators with/without tildes are in momentum/coor- dinate space. The brackets (..> on both sides of eq. (24) denote time ordered vacuum expectations or function- al averages using eq. (20) with J,4 . . . . f ¢ = 0.

The two-point Ward identities from all three eqs. (21), (22) and (23) were used in obtaining eqs. (24). We may replace 13Og - 23' on the RHS of eq. (24a) with p'Op - 2 by making use of the renormalization group equations. Similarly_, we can alter eqs. ( 2 4 b ) - (24d). The insertion of ~ ' cg then does not vanish in these theories because of the noncanonical scaling of the renormalized Green functions. In this particular model, all the anomalous field dimensions are equal [17] and also [18] /3(g) = -3go'0g) = 3g 3/z/167 r2 + O(fi2), due to there being only one independent re- normalization constant. These last two facts are well- known consequences of eq. (21) and chiral invariance. Note that the/30g{") terms in eq. (24) contribute to 0@2). To O(h) it is a simple exercise to see that the -23 '<") terms agree with a quick dimensional regular- ization computat ion of the LHS of eq. (24), such as led to eq. (12).

The anomaly in the trace of the energy-momentum tensor provides a logical basis for the generation of masses by radiative corrections in formally massless

gauge theories [19]. Perhaps the anomaly in the con- formal spinor current can help clarify the status of purely radiatively induced masses in supersymmetric models. This possibility is under investigation.

I would like to thank Professor M. Bander for use- ful discussions.

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