Algebraic renormalization of N = 1 supersymmetric gauge theories

NUCLEAR PHYSICSB ELSEVIER

Nuclear Physics B 458 (1996) 403-429

Algebraic renormalization of N = 1 supersymmetric gauge theories* Nicola Maggiore,

Olivier Piguet, Sylvain Wolf

DOpartement de Physique ThOorique, UniversitO de GenOve, 24, quai E. Ansermet, CH-1211 Geneva 4, Switzerland

Received 10 July 1995; accepted 13 October 1995

Abstract The complete renormalization procedure of a general N = 1 supersymmetric gauge theory in the Wess-Zumino gauge is presented, using the regulator free "algebraic renormalization" procedure. Both gauge invariance and supersymmetry are included into one single BRS invariance. The form of the general non-abelian anomaly is given. Furthermore, it is explained how the gauge BRS and the supersymmetry functional operators can be extracted from the general BRS operator. It is then shown that the supersymmetry operators actually belong to the closed, finite, Wess-Zumino superalgebra when their action is restricted to the space of the "gauge invariant operators", i.e. to the cohomology classes of the gauge BRS operator.

1. Introduction and conclusions The first papers on supersymmetric gauge theories and on their renormalization appeared a long time ago [ 1 - 3 ] . The renormalization of these theories as well as the construction of gauge invariant operators like, e.g., the supercurrent, have been completely performed, for the N = 1 case, using the superspace formalism [4,5]. Theories with supersymmetry breaking remain to be studied systematically. The superspace renormalization schemes, well adapted to exact supersymmetry, may be extended to the case of broken supersymmetry [ 4]. However, in such a situation, the main benefits of superspace renormalization, such as manifest supersymmetry, the non-renormalization theorem of the chiral interactions, etc. [6,7,4], are lost in great part. It seems therefore desirable to be able to master the renormalization procedure in terms of component * Supported in part by the Swiss National Science Foundation. 0550-3213/96/$15.00 (~ 1996 Elsevier Science B.V. All rights reserved SSDI 0550-321 3 (95)00545-5

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fields which, in the Wess-Zumino gauge, presents the advantage of using a much smaller number of field variables - a non-negligible aspect if one intends to perform explicit calculations. In the Wess-Zumino gauge, however, the supersymmetry transformations act nonlinearly on the fields and, what is more critical, their algebra is not closed [8 ]. Renormalization of supersymmetric gauge theories within this framework has yet been performed in the cases of N = 4 [9], N = 2 [ 10,11 ] and N = 1 [ 12] supersymmetry. The approach is based on a niipotent generalized BRS operator which combines the gauge invariance of the theory with its rigid invariances - supersymmetry in the present case. The whole symmetry algebra is translated into the nilpotency of the generalized BRS operator. Generalized BRS invariance is expressed, as usual, by a super Slavnov-Taylor (SST) identity. The renormalizability proof and the search for the possible anomalies are done using the method of algebraic renormalization [13], which has the effect of reducing the analysis to a cohomology problem in the space of the local field polynomials. The present paper deals with N = 1 supersymmetric gauge theories, without breaking, the case of broken supersymmetry being left for subsequent publications [ 14]. Beyond presenting the formalism, giving a simple proof of renormalizability and deriving, as a new result, the explicit form of the gauge anomaly, our aim is to answer the following question: rigid symmetries - supersymmetry in our case - being "hidden" in the generalized BRS operator which also contains gauge invariance, how can the rigid invariance - or covariance - of gauge invariant operators be characterized? Answering this question means finding a way to extract the rigid symmetry generators from the BRS operator. We shall see that this can be done, and that it leads to an equivariant cohomological structure [15]. The nice outcome is that these generators, when restricted to gauge invariant operators, i.e. to cohomology classes of the gauge BRS operator, obey a closed algebra. We conclude with the derivation of the Callan-Symanzik equation.

2. The

model

N = l supersymmetry, involving fields with spin not greater than one, is realized by means of two multiplets [16]: (i) the matter multiplet (tpff,~ba), consisting of a Weyl fermion ~p~ and a complex scalar field ~ba, where ot is a spinorial index and the index a runs over an arbitrary representation of the gauge group, carried by the anti-Hermitian matrices ( T i)a/,; (ii) the Yang-Mills multiplet ( A~, i Ai,~), formed by the gauge field and a Weyl fermion, both belonging to the adjoint representation of the gauge group n In the Wess-Zumino gauge, the supersymmetry transformation laws are realized nonlinearly, the non-linearities being concentrated into the variations of the spinors of the theory. As a consequence, the supersymmetry algebra does not close simply on the I Our conventionsare listed in Appendix C.

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translations, but it exhibits two kinds of obstructions, represented by terms which vanish when equations of motion are used (i.e. on-shell) and, in addition, by field-dependent gauge transformations [ 16 ] : 2 ,/~ I j Ouqb+6gauge( wi)@+field equations,

(2.1)

where @ collectively denotes all the fields of the theory, ~ describes the supersymmetry transformations having e ~ as infinitesimal fermionic parameter, and 6g~uge(oji)q' stands for gauge transformations with field-dependent parameter w', which, for N = 1 SYM is i_

(e~o-~Y~- e~o-~)

a~.

(2.2)

Such an algebraic structure is a common feature of all supersymmetric gauge field theories in the Wess-Zumino gauge, and it has been shown in Ref. [8] that the usual approach consisting in treating separately gauge invariance and supersymmetry leads to a theory which requires infinitely many external sources to be renormalized, because the presence of the field-dependent gauge transformations 8g~uge(W') entails a hopelessly open algebra. The introduction of auxiliary fields in order to put the formalism off-shell, i.e. to eliminate from (2.1) the presence of the equations of motion, even in the cases in which that is possible, does not help to solve this problem. An alternative and more convenient way to proceed is to collect all the symmetries of the theory into a unique operator, and promoting the parameters e '~ to the rank of global ghosts, with Faddeev-Popov (c/,F/) charge and negative dimensions (see Refs. [17,912,18] ). The theory is consequently defined by a SST identity, and the supersymmetry algebra is contained into the simple nilpotency relation of the corresponding linearized Slavnov-Taylor (ST) operator. This same approach allowed to prove the perturbative finiteness of the topological models [19], which exhibit a supersymmetry-like algebraic structure, and to study the renormalizability of super Yang-Mills theories (SYM), also in the presence of extended supersymmetry [ 9-11 ]. The generalized BRS transformations, including ordinary BRS, supersymmetry, translations and R-symmetry, are i + ~,aO'#afi~it3 ' -~

sAi~ = ( D # c ) sAiC~=

fijkcJhk

S~)a

,~,~

i

i

i

i

a -

Aiaor

!~,ycrlx i __ ~I g 2 8 a ( ~-. T a hi~ ) b ) 2 ~ - v ~'alZ, " u~

a

= - T a h c ~Ph - i o ' * ' " [ ~ ' ~ ( D ~ O ) a

s c ' = - ~ f J iikcac s~i= sb i =

bi

k _ 2id, o.~Ai

i(ucgu~i ' 2isao-~3/ff

s( u = -2~

o-~g~

SE a = - - ~ l ~ c~ ,

.s'r/= O,

. ,~/~ -- t•( v O u A ai , ixotflo

i -- i ( I Z c ) ~ b i '

,

4-

i ( # a # / ~ ia -- 7]A ia

2e'~.~,,h,.~qSbq~,. - ts c' ~'3/~g,.'~ + 3r/tb,,'" ,

_ i(,,3/,c i '

(2.3)

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where c, ? and b are respectively the ghost, the antighost and the Lagrange multiplier commonly introduced in order to fix the gauge, and e", sc~' and r/are the global ghosts associated to the supersymmetry, the translations and the R-symmetry2. The covariant derivatives and the field strength are defined as follows:

F'~,

"

=O~A~

i

-

,

i

fijkAJ - - u ~Ak

G,A u + ~

( DI~qg)i= {)izqv'' + f ijk A ~j k , i i ( D~6 ),, = c 9 ~ + T~bAu&b,

(2.4)

'

~oi C adjoint representation ~a E matter field representation.

(2.5)

The most general action of dimension 4, invariant under the transformation s, is S = Ssy M -7 Sgf,

(2.6)

where

SsYM= / d 4 x ( ~

t/ 1 itzv F~l, _ i A i ~ o . ~ ( D ~ ) i )

+~[D~,412 - - i O a o ~ ( D u ~ B ) a

l 2 iq~aTab(bb 12

- 2A(at,,')q~h4c]h(a,le)4d~e

--ida~Tih4bi i~

i -fl-t2t• i a - (baTat,~Oho, + 2a(abc)~ace ~boe~c + 2A(ah,.)~9 ,1~6~, O,'J"~ ,

(2.7)

and

Sgf = s f d4x ~ia~'A~ =

/d4x(bi~)#A~q_(cg#pi)((D#c)iq_8O,

o.tzaB}tilgq_

ai"o-

~a~. ~ j ) .

(2.8)

The action S depends on two kinds of coupling constants: the gauge coupling constant g and the Yukawa couplings A(,h,), which are completely symmetric in their indices. Notice that the fact of dealing with a generalized BRS operator which contains all the symmetries of the theory, allows us to solve easily, by means of the trivial cocycle (2.8), also another drawback affecting the usual approach which keeps separate the BRS transformations and the supersymmetry, i.e. the possibility of defining an invariant gauge fixing term [8] (here we have adopted the Landau gauge). The important property of the operator s is to be nilpotent on-shell. More precisely, s is exactly nilpotent on all the fields but the spinors, on which its square gives the equations of motion s 2 = field equations.

(2.9)

In order to get the on-shell nilpotency of the generalized BRS operator s, it has been crucial that the ghost c transforms into the field-dependent parameter of the gauge 2 The global ghosts r/ and sc are imaginary: ~/= -'q, ~ = -~', and we recall the following conjugationrule:

sBos = sBos, sKer= -sFer, where Bos and bk,r represent some bosonie or fermionic field, respectively.

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transformations present in the algebra (2.1), besides translations and ordinary BRS variation. We have thus successfully managed to transform the algebraic structure (2.1), which is very difficult to handle, into a simple on-shell nilpotency relation. Quantizing a theory defined by a on-shell nilpotent operator is a well-known problem [20]. To write the SST identity corresponding to the generalized BRS transformations (2.3), it is sufficient to add to the action a source term &xt, which contains, besides the usual external sources coupled to the non-linear s-variations of the quantum fields, also nonstandard terms, quadratic in the external sources. This corrects for the fact that the s-operator is on-shell rather than off-shell nilpotent: Se~t=

/(

d4~c A*iu(sA~)i + A,i,,(sA/) + 7~*iSsT~i~) +fb,*,(s05,,) +(b~(s3,,) •

S +6 • or,,(6.,.) + ~*,,~(~,~) + c*i ( sc i )

g2

.

) g:~/;.~

2 ( e ~ a, ~.i _e/~.~*i/~)2+28,~6"

.

(2.10)

We are now able to write, for the total classical action v = &YM + S~f + &×t,

(2.11)

the SST identity S(S) =0,

(2.12)

where

/" ,5(.S) =

(62~" 6Z 6:62 6~" 6v 6Z 6.S 6£ 6S d4~: 6~7-i~ 6A~ + 6A*~ 6A i~ + 6}t*i~ 67,~ + 6¢b,~640~ + 63~ 33,, 6~

6Z

32

+(bi_is~ave~ ) ~

~ 2 ~~ ~

6£

+

6Z 3£

.~

~ -~. -i

"V 3.S _ _ 027 t~13• o2 - rle"Oe,, rle[~ . "l~ 0£*' 3g/3

(2.13)

For renormalization purposes, a relevant object is the linearized SST operator

f

( &S

6

62

32' - - 6+ - - - - +32 ---- 6

6~ti# 6~ *i

6v

6

32

6

6

62"

6

6S

6

+

+ rfi2a m6 + - -3£- - 6+

a~6~,~

6 ( ~ 6~a

6v

6S

6

63,1 6&

m

6

+

6~,~, a(.*,,e

6& 63~

6Y_ 6 ____+

6v 6 m m

ac*, 6c,

6c, ac*,

6~' 6Z

6 6

a~..~&~

408

N. Maggiore et al./ Nuclear Physics B 458 (1996) 403~129

Table I Dimensions d, Grassmann parity GP, ghost number q~ll and R-weights

d GP qSII R

Ap.

h

q~

0

c

(

h

£,u

e

r/

Au

A*

qS*

0*

c*

l 0 0 0

3/2 1 0 --1

1 0 0 --2/3

3/2 1 0 1/3

0 1 1 0

2 1 --1 0

2 0 0 0

-1 I I 0

-I/2 0 1 --1

0 1 1 0

3 1 --1 0

5/2 0 -1 I

3 1 --I 2/3

5/2 0 I --I/3

4 0 -2 0

+(b i

i(iZcg#(i) 6 ~"i ~_ (

/~

(~ ) -2ie~o-~g~O~,? i - i(uO/,b i ) ~-ff

9sc/~ - r/e O~7 - r/e a 0-ga '

(2.14)

which, by virtue of (2.12), is off-shell nilpotent /32 = 0.

(2.15)

The dimensions, Grassmann parities and R-weights of the fields are shown in Table 1. The fields commute or anticommute according to the formula q~j~o2 = (-1)GP"GP2~o2~p~.

(2.16)

3. Renormalization According to the approach presented in the previous section, the theory is formally characterized by the same set of constraints defining the ordinary Yang-Mills theories [13]. In fact, N = 1 SYM is defined by the following identities: (i) SST identity 8(2:) =0,

(3.1)

which contains the ordinary ST identity and the Ward identities for the supersymmetry, for the translations and for the R-symmetry; (ii) Gauge condition 6Y

6b--7 = 0UA~ ;

(3.2)

(iii) Ghost equation f / . S = Ai~ ,

(3.3)

which is peculiar to the Landau gauge [21 ], with

.Ui ~

d4x

- fiikg.i=vw,

(3.4)

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and where Agi ~ f d 4 x ( f i j k ( - A * J ~ A ~ + h * J ~ A ~ + , . 3 -~*j ~k~ + C*iC • k'~

(3.5) ,;b. ~t, + 4',,@h

4' <,~'h<~ 0 .S'h

is a classical breaking, i.e. linear in the dynamical fields. (iv) Global ghost equations t = Au '

0(~

--

Or/

= An

,

(3.6)

where A~, j t and AR arc classical breakings given by

i. / (d4x

A~t

A .i..3~A it, + a .i,r-3~a(,i + ~*io#~i[3 . +. C*'c)#C'

- qS,,agOS. + 0 aOu4t.(~ + ~ .Ba~g'. AR

f d4x (_/~,iol i/~c~ + ~*~ifl

•_

+ ~a~ga2 *

_

2-*-

5da,,d)<,

/

,

(3.7) + I,it*,e,lt

_~" . ' . < ~

_ I ~* .,It~

_~'," .he<, ) "

(3.8) We can then write the following algebra, valid for any functional y with zero GP: B ~ S ( ~ , ) = o,

(3.9)

6 ~ S ( Y ) - I3, (~b~

a~,m~ ) = , ~ y ,

(3.10)

~¢~S(r) + B~-¢~, = o,

(3.11)

mis(~l) @ eT (£~'i~1 __ ~ig) = "V~rtig,)/,

(3.12)

"~iigS

(,)/)

i B~W[ij = O,

0

(3.13) (3.14) (3.15)

WRS(y) - B~WRT = O,

(3.17)

which also contains an antighost operator .~'i and Ward operators for the rigid transformations W~g, for the translations ~ and for the R-transformations WR, given by

410

N. Maggiore et al./ Nuclear Physics B 458 (1996) 403-429

6

6

5e ' i - - - +

6

+

(3.18)

b~ --'l~t'ig =~

f d4x(- fJik(Akt*~-~ 4-A*k#6A*j~6 +ak~ ~_.`a _~_6A4- A * ka 6 A*6 j c~ \ - a~ ~

~ 6 7~, j /~ +

-T:ih ,,

+(b h6~+ckh~+~b ~+&,,-~-~+g,*X&l,,, ~

-~ha 6
.

6

2-

+ 6

+5 ~° 6~--7+ 5~°ago _lea

+ c

6 c *~ +

~c J

he6d.oa ] ) ,

i8- ~ - - ~

2

~

(3.19)

*6/~ i~ 2-.

4- ~i/) .

6;re.

a

1 ~

.

.

6

1

5 ~° 6 ~ + 5 0o 6,v

1_, e 6 )

-

~ - -e- ~ a - -

a

6i**a -if°ago 6

5~*xae,*a '

(3.21)

where "all fields" includes all the fields listed in Table 1. Setting y _= X in the relations (3.9)-(3.17) and using the conditions (3.1)-(3.6) satisfied by the classical action X, leads to , f " i x = 0,

}/~[igX= 0,

WuX = 0,

WkX = 0.

(3.22)

Eqs. (3.3) and (3.6) express thus the linearity of the rigid transformations, of the translations and of the R-transformations. The aim of the renormalization is to construct a quantum extension of the theory, described by the I PI generating functional

l" = S+ O(h),

(3.23)

obeying the conditions (3.1)-(3.6). By consequence of the algebra (3.9)-(3.17), it will also obey Eqs. (3.22). Supersymmetric gauge field theories are characterized by the lack of a coherent regularization scheme under which all the symmetries of the theory, i.e. BRS and supersymmetry, are preserved. This implies the adoption of a renormalization procedure not relying on a particular kind of regularization. The algebraic procedure of renormalization, based on the general grounds of power counting and locality, satisfies this requirement [13]. Following this method, the discussion of the quantum extension of the theory is organized according to two independent parts:

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(i) The study of the stability of the classical action under radiative corrections. This amounts to the search of the possible invariant counterterms and to check that they all correspond to a renormalization of the free parameters of the classical theory, i.e. of the coupling constants and of the field amplitudes. (ii) The search for anomalies, i.e. the investigation whether the symmetries of the theory survive at the quantum level.

3.1. Stability In order to check that the classical action is stable under radiative corrections, i.e. that these can be reabsorbed through a redefinition of the fields and the parameters of the theory, we perturb the classical action X by a local functional So, having the same quantum numbers as X, namely canonical dimensions four and vanishing ~ H charge:

X

, X' = X + (X,,

(3.24)

where ( is an infinitesimal parameter. We then require that the perturbed action X' satisfies the constraints (3.1) to (3.6) defining the theory. This implies that the perturbation .S,, ( 1) does not depend on b, (, rl; (2) does depend on the ghost c only if differentiated. Furthermore, it follows from the SST identity ( 3.1 ) and the algebra (3.9) - (3.17) that Xc (3) obeys the condition ~'iX,. = 0, which implies that X,, depend on O and A*~' only through the combination (3.25)

t~ *ilx =_ Otz? i + A ' i # ;

(4) obeys the conditions of invariance i W~ig2c =0,

W ~ c =0,

WRY,: =0.

(3.26)

Finally, at first order in sr, the SST identity (3.1) imposed to the perturbed action X', translates into the following condition on the perturbation: (5) B~,Xc = 0. (3.27) Eq. (3.27) constitutes a cohomology problem, due to the nilpotency of the linearized SST operator/3, (see (2.15)). Its solution can always be written as the sum of a trivial cocycle B_,~,, corresponding to fields renormalizations, which are unphysical, and one or more elements belonging to the cohomology of B~, i.e. which cannot be written as B , variations: (3.28)

2c = Xph ~'- ~ 2 ~ - ~ .

The strategy we applied to construct the explicit form of these terms is explained in Appendix B and we give here the solution: OX

Xph =

0S

ZU8Xag+ Z(ah,.)--OA(~(.) + 2(,h,.) OA(ah,,) '

(3.29)

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2 =

/(

d4x

z

ZA,__(A*iitAi__lt- a*ic~,~ai _ ~ * ~ i f l ) •

2i_ Z~ab(~)a~) b*

-*--~* aBd/~) ),

+ ~ah(fbafbO where

_ i/l*al~tba

(3.30)

Z e, ZA are arbitrary constants and Z(,,h,,), Z(,#,,,), Zc~,h, Zsa h are invariant ten-

sors. The relations between the renormalizations constants, appearing in .S, of fields belonging to a same supermultiplet, are the consequence of the observation, stated in Appendix A, according to which the counterterm cannot depend on terms containing eft,*, e~b* and their complex conjugates. Notice also that the cohomology part Sob depends on parameters which correspond to possible multiplicative renormalizations (and hence non-vanishing beta functions) of the gauge coupling constant and of the Yukawa couplings. This is an algebraic result which just shows that the radiative corrections can be reabsorbed and that no new terms appear at the quantum level, which was our aim.

Remark. This result is in agreement with the generic situation. In certain cases, however, a non-renormalization theorem [6] states that the Yukawa couplings remain unrenormalized. But this has been shown only within the superspace formalism [7]. And even in this case, finite renormalizations may occur if there are massless particles [22]. Note also that a class of N = 1 SYM theories does exist, where no coupling constant renormalization occurs at all [23]. 3.2. Anomalies In this subsection we will deal with the problem of defining a quantum vertex functional which satisfies the SST identity, or, in other words, which preserves the symmetries defining the classical theory. For what concerns super Yang-Mills theories within the superspace approach, it has been demonstrated in [5,4], that the only anomaly affecting N = 1 SYM is the supersymmetric extension of the standard Adler-Bardeen anomaly, and its explicit form has been given in [24]. To the best of our knowledge, the same result for N = 1 SYM in the Wess-Zumino gauge has been obtained in [12] for the abelian case, the expression for the non-abelian supersymmetric gauge anomaly written in the Wess-Zumino gauge being still lacking. In fact, the aim of the renormalization is to show that it is possible to define a quantum vertex functional

I'= X + O ( h )

(3.31)

such that, as for the classical theory,

S(F) =0, i~I" =cgitAi# ~b

(3.32) '

i .~"iF = Ag,

8F t 8S¢u = Ait,

OF --Or/ = ~IR.

(3.33)

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Our strategy will thus be the following: we begin by imposing (3.33) and try to solve (3.32). But, before doing the latter, it will be convenient to require the validity of the conditions ~i F

=

O,

i ~l)~igF = 0,

]/V~zF= 0,

W R F = 0,

(3.34)

which anyhow follow from (3.32) and (3.33) through the algebra (3.9)-(3.17). Actually this program will fail, namely the SST (3.32) will turn out to be anomalous: S ( F ) = rASAB,

( 3.35 )

where ASABis the anomaly to be derived in the following (see (3.42) for the result) and r is a well-known function of the parameters of the theory of order h [4,25], which however cannot be determined by the pure algebraic method used here. On the one hand, the extension of the classical conditions (3.2), (3.3) and (3.6) to their quantum counterparts (3.33) is trivial (see Ref. [ 13]), and the extension of the classical rigid invariances (3.22) to their quantum counterparts (3.34) has been proven in [26,13]. The search of the breaking ASAB of the SST identity (3.35), on the other hand, requires some care. The rest of this section will be devoted to this end. According to the quantum action principle [27,13 ], the SST identity gets a quantum breaking $(F)

(3.36)

= _4. F,

which, at the lowest order in h, is a local integrated functional with canonical dimension four and Faddeev-Popov charge one (3.37)

_4. F = _4 + O( h_4).

The algebra (3.9)-(3.17), written for the functional F, at the lowest non-vanishing order in h implies the following consistency conditions on the breaking _4: 8b i8~'1= O,

.Ti A = O,

a~'iA = O,

i W~ig-4 = 0,

/3,A = 0.

OA = O, O("

WuA = 0,

0_4 O---~ = 0, WR-4 = 0,

(3.38) (3.39) (3.40)

Notice that the consistency conditions (3.38)-(3.40) formally coincide with the relations determining the counterterm. The difference is that now the solution must belong to the space of local functionals having Faddeev-Popov charge one instead of zero. Therefore, the first eight conditions tell us that the breaking -4 does not depend on b i, scu and r/, that the ghost c i must always be differentiated, that ?i and A *i~' appear only in the combination j~.iu (3.25) and that _4 is invariant under the rigid transformations, the translations and the R-transformations. The last consistency condition (3.40) is often called the Wess-Zumino consistency condition. Like the corresponding one in the zero-@// sector (3.27), it constitutes a

N. Maggiore et al./ Nuclear Physics B 458 (1996) 403-429

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cohomology problem. Its solution can always be written as the sum of a trivial cocycle /3zJ, which can be absorbed in F as a local counterterm - J , and one or more elements belonging to the cohomology of/3z, i.e. which cannot be written as 13,, variations: zl = / 3 , J + A# .

(3.41)

The method applied to construct the explicit form of the anomaly A# is explained in Appendix B and leads to

A#=-rasA.=rJd4x(e"P"{di'~(,~ci-2ai'o,,'- 2~.'o'/~.,~i~)AJgpA~. q_DUm, (lc~/~ci

I ic~

;. I c~

-ii.'~AJA,,,A,"~

-3dij'(.'/li AJ~''j-~/~'i'/lJ'A:) ),

3.42)

where d ijk is the totally symmetric invariant tensor of rank three given by diJk =_ d ~iJk) = ~ Tr (r i { r J, ~.k} ) ,

3.43)

and D ijmk is an invariant tensor of rank four given by D ijmk ~ d n i j f nmk -Jr-dnikf njm q- dnimf nkj.

3.44)

4. Symmetry generators All the rigid symmetries of the theory (supersymmetry, translations and R-invariance) have been included, together with the original gauge BRS invariance, into a single ST identity, which we assume from now on to be free of anomaly (see (3.35)): $(F)

=0.

(4.1)

From its construction, based on the nilpotency of the generalized BRS operator s (2.3), this identity also incorporates the full algebra formed by the various symmetries. However, since the original algebra was not closed, it not obvious how one can recover the individual symmetry generators and their algebra from (4.1). The purpose of the present section is to show that it is in general possible to obtain functional operators which generate the ordinary gauge BRS transformations and the rigid symmetries. Moreover, the generator of the gauge BRS transformations is nilpotent, it commutes with those of the rigid symmetries, and the latter obey an algebra which closes when their action is restricted to the space of gauge invariant operators, i.e. to the cohomology space of the gauge BRS operator. Such generators are useful in the construction of (super)multiplets of gauge invariant operators, e.g. in the construction of the Ferrara-Zumino supercurrent [ 28 ] in the WessZumino gauge [ 14].

N. Maggiore et al./ Nuclear Physics B 458 (1996) 403-429

415

Let us expand the linearized, nilpotent SST operator/3r, corresponding to the SST identity (4.1), according to the filtration operator =

__

08a +

c9~

+ ~c ~

+

r/

.

(4.2)

This operator counts the degree in the global ghosts. The expansion reads /3r = Z/3,~ ,

with

[.A/, Bn] = n/3n •

(4.3)

n)0

The nilpotency of Br, B 2, = 0 ,

(4.4)

which follows from the SST identity (4.1), implies, at the orders 0, 1 and 2, the identities B02 = 0 ,

{/3o,U, } = o , /312 -1- {/30,/32} = 0 .

(4.5)

respectively, where {-, .} means the anticommutator. We first remark that /3o, i.e. the operator/3r taken at vanishing global ghosts, coincides with the "usual" linearized ST operator, i.e. the one corresponding to the gauge BRS operator. It is this BRS operator which is used to define the physical, i.e. gauge invariant, quantum operators, as follows. Definition. A gauge invariant quantum operator is defined by the cohomology classes of the BRS operator /30. More explicitly, such a gauge invariant operator is defined by ( l ) an insertion Q - F, i.e. the generating functional of the 1PI Green functions with the composite field Q inserted, obeying the invariance condition

/30 (Q" F ) = 0 ,

(4.6)

(2) the equivalence relation Q . F ,-~ Q ' . F

(4.7)

if and only if Q . F - Q' • F = / 3 0 ( 0 - F ) for some insertion 0. The second identity of (4.5), which expresses the commutativity of the gauge transformations with/3~, allows to define the action of/31 in the space of the gauge invariant operators defined above. Indeed, if Q • F is a representative of a/30 cohomology class, then/3~ (Q • F ) is still a representative of such a class. The third identity of (4.5) means that the restriction of the operator/3~ to the space of the gauge invariant operators is nilpotent. Indeed, on any representative Q of a/30 cohomology class, one has /3 2 ( Q . [,) = -/3o/32(Q" I ' ) "-'-'0 .

(4.8)

416

N. Maggiore et al./ Nuclear Physics B 458 (1996) 403-429

One may call this property "equivariant nilpotency" [ 15]. Eq. (4.8) is the desired result. In order to see this, let us define the rigid symmetry generators W by the expansion 13~ = ~ w ~ . + ~ v ~

+ ¢'~w,~ + . w R

- 2~ o - o ~ 7 ( ;

- n~ ~

+ ~e ~ ~ae

.

(4.9) where we have separated the terms expressing the transformation laws of the global ghosts from the rest which is linear in these ghosts. Then it is obvious that the equivariant nilpotency of 13j implies the "equivariant algebra"

[WIC, W~] ~ W ( , ,

[WR, W,~] ~ -'1~,~ ,

(4.10)

(other (anti)commutators ~ 0) .

Remark. This result reproduces at the quantum level the classical supersymmetry algebra which obviously closes on the translations, when restricted to the space of the classical gauge invariant operators. Since the effective calculations are always done on representatives of the cohomology classes defining the gauge invariant operators, it is worthwhile to write down explicitly the algebra of the symmetry generators including the elements pertaining to /32. The latter are defined by

132 = e"eaX, a + ½e%/~Y./~ + ½ga#~Ya~ .

(4.11)

(There is no term in 7/ and ( due to the third and fourth of the conditions (3.33) expressing the linearity of the R-transformations and of the translations.) The second and third of Eqs. (4.5) then yield {130,w~} = [130,wR] = o ,

{W,, WE} = - {/30, G~}

and conjugate equation ,

This algebra does not close, each order in the filtration bringing new symmetry generators.

5. Non-renormalization theorem for the anomaly We still have to show that the anomaly coefficient r in the anomalous SST identity (3.35) is not renormalized or, more precisely, that it vanishes to all orders if its one-loop order is equal to zero.

N. Maggiore et al./ Nuclear Physics B 458 (1996) 403-429

417

This theorem follows from the observation that the present theory reduces to an ordinary gauge theory if one sets the global ghosts (u, e~ and 7/to zero. Then it suffices to refer to the known proofs [29].

6. C a l l a n - S y m a n z i k e q u a t i o n

Although the theory we are considering is massless, a mass parameter /z must be introduced in order to define the quantum theory. This parameter fixes the scale where the normalization conditions fixing the free parameters of the theory are taken. Being the only dimensionful parameter in the present case, it also determines the overall scale of the theory. This scale is controlled by the Callan-Symanzik equation, which follows from the renormalizability of the model [13]. In order to derive the Callan-Symanzik equation, we first observe that the scale operator tz 0/0/z "commutes'" with the ST operator: #-~$(F)

=/31. k, ~-~j

(6.1)

The assumed validity of the SST identity without anomaly and the quantum action principle thus imply that the local, dimension-4, insertion # OF/Olz is /3F-invariant. Let us expand the latter into a basis of insertions with the same properties. An appropriate basis is given by the following insertions, which are the quantum extensions of the classical counterterms (3.29), (3.30): OF Og

OF

OF

CL~ab,. '

~ ~ah ;

HAr=t3,,f a4x (~*a -

a*a - ~*~)

= (NA + N a + N;~ - N A . -- Na* - N~. - N5 - Ne) F,

.A/'abF = I3t" f d4x ( q~aqb*h -- OaO; ) = ( N4~ + U4, - N¢~* - Ue,* ) ab F, J ~ i ~ s F = B 1 " f d4x ((ba(b,*,-Oi, O ; ) = (N 6 + N 6 - N6. - N ~ . ) a s l .

(6.2)

where we have introduced the counting operators N~ = { d 4 x q ~ -8- ,

J

q~ = A, A, A, A * ,C, -*A , b, c- ,

&P~ ~oa 6q~t, '

'

(6.3) '

~

"

Expanding now the scale operator in this basis we obtain the Callan-Symanzik equation (

0

O

0

O

)

?)~asc

(6.4)

418

N. Maggiore et al. / Nuclear Physics B 458 (1996) 4 0 3 - 4 2 9

The/?-functions fig, /3~bc, as well as the anomalous dimensions TA, Tab are of order h and are calculable in terms of vertex functions using the normalization conditions.

Acknowledgements We would like to thank Carlo Becchi and Claudio Lucchesi for useful discussions.

Appendix A. Non-renormalization of the linear supersymmetry transformations The linearity of the supersymmetric transformations of the bosonic fields A., q~ is expressed, at the classical level, by the fact that the right-hand sides of the equations O SX Oea 6A *i 3 f~

O.tX ~i~

'~'~- '

Oe'~ 64~,]

3 f£ Oe'~ 67t~ ,9 f 5 'gea fO,~

2.i

ix&

"% 0-a '

(a.1)

2~a fa

are at most linear in the dynamical fields. In the simplifying notation {B,M} =

{Au,

(b a, } ,

{ g A } = {~i,~, Oao }

(and similarly for the associated fields BM, F~), this reads O fix Oea fB,M

A

%MFA,

O 62 Oea fFa

A n*M CaMP "

(A.2)

We want to show that, at all orders of perturbation theory, 3 fF 0 _6..,,~

557

aS a fFa

cAMFA + Aa • AM . F A B*M -~- A a~ • CaM

= CA aMFA +

A~A • F

O(F*),

a n*M+O(F*F) = CaMIJ

(a.3) +O(F*F*),

i.e. that the classical results hold up to terms vanishing with F*. This implies in particular that the vertices a,,,M IJ CA a M P,-A , i.e. ~ . gear i i xa s i,

2a.ui .a i o - tz e, -

• 2q~,eO~,

-. - 2~afOa,

(A.4)

are not renormalized. Remark. All the graphs contributing to the radiative corrections to (A.1) or (A.2) are superficially convergent.

The proof of (A.3) goes by induction. Let us assume it to hold at order h ' - J : 0

6F fB,M OeO f F Oe a f F A

A

%MFA + O( F*) + O( h"), A ~*M CaMIJ

Av O ( F * F ) + O(F*F*) + O(hn)

(A.5)

N. Maggiore et al./ Nuclear Physics B 458 (1996) 403~¢29

419

The quantum action principle together with the induction hypothesis imply (for n/> 1 ) 0

~F

A

Oes 6B.M 3 6F 3s s

csMFA + As • AM • F + hnAaM + O(hn+l), A +B*M CaM

6FA

t AtA hnAtsA O(hn+l ~- Act " " F ~@ ).

(A.6)

Let us begin with the first equation. The double insertion A s . AM. F corresponds to Feynman graphs where the derivatives with respect to s and B* act on two different vertices, whereas the single, local, insertion AsM corresponds to both derivatives acting on the same vertex. For the latter we have made explicit that only the tree graphs contribute at its lowest non-vanishing order - order h n by the induction hypothesis. The double insertion graphs containing at least one loop, As and AM are produced by terms of the action of order tz - 1 at most. Furthermore, since one of them must contain a factor e and since, by hypothesis, s couples to B* only in a trivial manner (i.e. linearly in the dynamical fields) up to this order, it is its coupling with F* which is involved. Thus (A.7)

A s • A M . F = O( F*).

Similarly, for the second of Eqs. (A.6), one obtains A2 • A 'a • F = O ( F * F ) + O ( F * F * ) .

(A.8)

The additional dependence on F or F* is implied by the conservation of the total number of these spinor fields, which can easily be checked by inspection of the action. Let us come now to the single insertions AsM and A'~a. They are field polynomials of dimensions bounded by the dimensions of the left-hand sides of (A.6), i.e. by 3/2 and 3, respectively. Moreover, their ghost numbers and R-weights are those of the lefthand sides, too. A detailed analysis then shows that they have exactly the same form as the right-hand sides of (A.I). In our notation, /IsM = E2MFA,

AtaA = r tas M lnJ * M .

(A.9)

The coefficients r and r z are evaluated by computing the three-point vertex Fe"B*MFA

3

¢~

6 F all

Oe s ~ B *M 6 ~ A

fields ¢=0

(A.10)

at the order h" (n ~> I). But, since the coupling of e with B* is linear in F up to order n - 1 by the induction hypothesis, there is no one-particle-irreducible loop graph contributing to (A.IO). Thus the coefficients r and r' vanish. This ends the proof of (A.3).

Appendix B. Computation of the cohomology o f / ~ In Section 3,we met two cohomology problems: the first is (3.27) which leads to the possible counterterms of the theory, and the second is (3.40) which gives the anomaly

N. Maggiore et al./Nuclear Physics B 458 (1996) 403-429

420

of the theory. Both problems are to be solved in the space of local integrated functional with canonical dimension four and Faddeev-Popov charge q (with q = 0 or 1 according to which problem we are dealing with), subject to the constraints (3.38) and (3.39). We will denote these constrained spaces by 5 ~q). The first method, which we applied to find 25,, in (3.27), consists in the following steps: (1) Write a basis {~i}i>>.J of 7 ~ j) (2) A p p l y / 3 , on it to obtain the set of functionals {/3,_~i}i>J, (3) Construct a basis {Oi}i>/l o f the space spanned by this set of functions. Then for each i we can find a ~7i such that

/3,_Z = Oi.

(B.I)

The/3z.~i are thus trivial solutions of the problem. (4) Complete the basis of step 3 by the set {.~//}/~>j such that this whole set of functionals constitutes a basis of the invariant functionals of 5c~°). Since by construction there does not exist a ~ such that /3,_='~ = 2:,a, the functions ~ , are non-trivial solutions of the problem. The general solution of (3.27) can thus be written

i/>l

i/>l

where the eri,/3i are some constants. This leads to the result (3.28)-(3.30). The advantage of this method is that it gives both the trivial and the non-trivial part of the solution. The inconvenience is that it requires very long calculations which can be partially avoided by the more sophisticated method using filtration to which we now turn. The second method, which we used to find /t in (3.40), was developed in Ref. [30] and applied in Refs. [9-12,18], and consists first into passing from functionals to functions, next to use a filtration to get a simpler problem of local cohomology for the lowest order/3~,°} of the operator/3,, and finally to recover the cohomology o f / 3 , . This corresponds in practice into translating the functional operator /3,, which acts on the space of local functionals, into an ordinary differential operator, also denoted by Be, which acts on the space of functions A(x). Thus, (3.40) becomes a problem of local cohomology modulo d,

/3,a(x) + d A ( x ) = 0,

(B.3)

where d is the exterior derivative: d 2 = 0, d ( x ) is a four-form defined by A = f A(x) and zl(x) is some three-form with dimension three and 011 charge two.

Definition.

A

A is/3,-modulo-d equivalent to A' if and only if

A' =/3,/] + dA',

(B.4)

for some /i and /]'. Moreover, d is called /3_,-modulo-d trivial if and only if it is equivalent to zero.

N. Maggiore et al./ Nuclear Physics B 458 (1996) 403-429

Definition.

421

d is/3,-modulo-d invariant if and only if there exists N such that

/3,A + dA' = 0.

(B.5)

Definition.

The cohomology of/3,-modulo-d is then defined as the space of equivalence classes of non-trivial invariant functions.

Remark. Where no confusion may arise, we will speak indifferently of a function A or of the class to which it belongs. The general solution of (B.3) can formally be written

A(x) =/3,~l(x) + d~l'(x) + A#(x),

(B.6)

where /i(x) and /]'(x) are some forms with appropriate degree and cb//charge, and A# belongs to the cohomology of/3-`-modulo-d.

Definition. Let .M be a "filtration" operator mapping the spaces .Tff of p-forms with q~H charge q into itself, the eigenvalues n of A/" being the non-negative integers 3. The expansion of a function Aqp ~ 5r]q and of the operator/3, according to their eigenvalues reads N

/3,-

= ~X--'n(~) .~, ,

Aft, =

n--O

g-,a(,,)u ~ P,

(B.7)

n/>M

with =

n/3(.2>,

(B.8)

=.a">7,,

where M stands for the order of the first non-vanishing term of the expansion of A~. By means of these definitions, we can filter (B.3) into the following set of equations: .4A(M)2 / 32 ( ~0 ) A4 (>TM t ~ m) I 3 / 3 2 ( I ) A ( M ) I + /3t'0)A(M+I)I4 + dA(M+I)23 /3(2)A(M)IE4 + /3~I)AtM~ I)~ + /3('O)AtM+2)I_4 + d A ( M - 2 ) 2 3

=

0,

= O, =

0,

(B.9)

etc., which can be written at any order 4 P (with P ~> M) min{N,P--M} Z N(k)A ( P - k ) j d A (P)2 O. 4+ 3= k--0

The nilpotency of/3_, (2.15) can be filtered into 3 We restrict ourselves to the case of a filtration that commutes with d. 4 In this context, the order of an equation is the eigenvalue obtained by application of .N" to it.

(B.)0)

N. Maggiore et al./ Nuclear Physics B 458 (1996) 403-429

422

/3(`o )

o ) - o,

+/3(,o)/3,_,,) : 0, /3(,27/3(,0) • + ~-~ r~(') ~r~(J> + /3(,0)/3~,2) . . =0,

(B.I 1)

Notice that B~0) is nilpotent, as expressed by the first equation in (B.12), and thus we can as well define the notions of B(,0)-modulo-d equivalence, invariance and cohomology. The procedure we will apply to find the explicit form o f / l # ( x ) in (B.6) consists in the following steps: Step 1. It has been shown in [ 30] that for each element of the cohomology o f / 3 , modulo-d , we can pick up a particular representative which has the property that its lowest order according to the filtration AT is /3(~°)-modulo-d non-trivial. Thus, the first step of our procedure is to find the cohomology of/3(,°)-modulo-d in the sector of 4forms with qSH charge one, which is denoted by D~o. To this end, we apply the method described at the beginning of this appendix for the search of the counterterms but now in the sector of ghost one and for the simpler operator/3(,o). That means that we begin by writing a basis {Ai°}i>>.~ of the space of 4-forms with qs//charge zero and apply/3(,o) 0~ This set of functions to these elements, thus getting a set of functions tr~(0)/~ t - . , ~i4Ji~l. generate the space of /3(,°)-modulo-d trivial invariant 4-forms with qSH charge one. We can then choose a basis of this space and complete it in a basis of the space of the/3(,°)-modulo-d invariant 4-forms with q~H charge one. The functions we just have defined to complete the basis are non-trivial invariant 4-forms with qSH charge one and therefore define a basis of 5~0.

Remark.

Notice that as/3(,o) and d both commute with AT, the elements of f o have

a definite order and we will therefore denote them by A#~).

Step 2.

We then proceed to the "extension" of 5~0, and to this aim we introduce the notion of the extension of an element A#~m E ~0, which consists in constructing, if possible, the /l("') for m > n such that ZI =~ A #(n) +

~--~zl ¢'')

(B.12)

IIl~ll

is B,-modulo-d invariant. ~ is then defined as the space generated by all the B , modulo-d invariant functions with a B(`0)-modulo-d non-trivial lowest order, i.e. by the extension of the elements of 9~0. The explicit procedure of extension consists in solving (B.10) order by order in such a way that we get explicitly the elements of ~ . We proceed by induction: suppose we know the general solutions a(P') ~ ( P - - l ) p,s satisfying (B.10) up to order P - 1"

423

N. Maggiore et al./ Nuclear Physics B 458 (1996) 403-429

min{N,P '-M}

A(P') 1 ~(P') 1

Z

~(P--I)4 = ~(P--I)4 4-

13(k)/](e'-k)o ~_: 4 + d ~ l ( P ')t3,

k=O

for M <~ P ' <~ P - 1 ,

min{N,P '-M} Z B(k)/](P'-k) 3I + d/i(P ')22, A(p-I)3 4k--O

2 ~(P--I)3 A(P')

-(P')

2

(B.13) where the ,~(P'--k)qp are some functions for which we do not add a lower index since, as we will see below, they get neither constraints nor corrections at the next order. The '~I PP--l)p '~ u constitute thus the beginning of the extension we are looking for. Inserting these solutions in (B.10) at order P and using the nilpotency of 13, expressed by (B.12) gives 13',°)AIP)~ + d d le)23 4-

min{N,P--M} Z g.~(k)~(P-k)l k=l

(minIM})/min{N, 13,,o) .

.

__ \

.

k=l

13(k)/i(P-k)o .

d

|\

Zk=i

:

0, (B.14)

-

which entails the following interpretation: first it imposes constraints on the free coefficients of the general solution (B.13) since the third term of (B.14) has to be 13(,0)-modulo-d trivial, and secondly we see that the terms involving z] do not suffer any constraint to be extended at order P since they appear in (B.14) in a trivial form. Let us denote by J(p)i(P')14the constrained lower order solutions and by ,~(P)~, ~(P)23 a special solution of

min{N,P-M} Z i..~.,:r~(k)'i(e--k)],(p) 4 = --13(0) 1 2Z1"(P) 4 --

(B.I 5)

d'~' P) 23-

k=l The general solution of (B.I 4) then reads A(P)I

z~(P)/

min{N,P-M} 13(k)~(P-k)O

A#(P~I

N(0) /~(p)o

d~(P)l

k=l

min{N,P--M} A(P)2

= -

~(P)23 + At(P)23

+

(B.16) +

+

k=l

where za"#(P)I4 belongs to 9~o, and A z(P)23 satisfies 3(o) ,_ A#(P) ~ 4i + dA~(P)~ = 0.

(B.17)

Defining ~fi(P)'l ( p ) p =-- -A(P)qp + A #( P) ~ leads us to the general solution of (B.10) at order P in the same form as in (B.13), thus ending the induction step. In principle this system of equations is endless but in practice it becomes rapidly trivial due to dimensional constraints (all the functions under consideration have Remark.

424

N. Maggiore et al. / Nuclear Physics B 458 (1996) 403-429

dimension four), the spaces ~ ( ' ) of functions of order n being empty for n greater than some constant (in our case it will be five). Then we sum the solutions A(P) ~ ( o c ) given by (B.13) to get the general solution A = Z--(~)'A(e) which we can write in the form P)M

P~M

which shows that the extensions of the/3(0)-modulo-d trivial terms/3(,0 ~ ) A(P)° + d/i(e)l 3 are the/3.,-modulo-d trivial t e r m s / 3 , / i ( e ) ° + d/~(e)~ and that the general element of ..~ we are looking for is Li. We then define the subspaces -~n) of ~ as formed by the elements whosc lowest order is n. Step 3. For each element ,J(,) E - ~ m , which is invariant by construction, we havc to separate the /3~-modulo-d non-trivial part from the trivial one. To this aim, let us introduce the notion of triviality until order M:

Definition.

A (') is/3_,-modulo-d trivial until order n if and only if

3 / ] , /1~

such that

A(')=/3,~il+dZlr+O(n+l),

(B.19)

where O(n + 1) contains terms of order greater than n. It leads to the following decomposition: =

+

(B.20)

where A~I~), contrarily to 3-~o,), has a lowest non-vanishing order /i~) (') which is /3_,modulo-d trivial up to order n. It follows that zl~) is/3_,-modulo-d trivial and that zi~,) constitutes the anomaly. Finally, we can write the explicit form of A# in (B.6) as (B.21)

A# n>/M

where the fl(,, are some arbitrary constants. We have now finished with the description of the general method and turn to the specific case of interest in this paper. We choose a filtration operator N" which assigns the weight I to all the fields and their derivatives as well as to the global ghosts e", g~, sc" and 7/:

x=

/

Z

all fields ~5

{0

o

+

+

0 0,9//

.o

o T]~-~.

(B.22) The B~° ) transformations of the fields are (the transformations of their derivatives being trivially inferred due to this simple choice of the weights)

N. Maggiore et al. /Nuclear Physics B 458 (1996) 403-429 B~0)~*iu = ~cg~,(OVAi#-cg,~Aiv)

I3~O)A#* = O~*c ;,

B(O) c i = O,

425

,

8(0)C *i = Otz,.A*i#, ~

=

-- 7

~#'"

°a/3'

" = o, e~0,~, o,

= 2 <,o,a: = -"

Z~0)4a = 0,

z(o)~b; = _ 1~ a , , a /~-G , _

Bg,°lg, 2 = O, . t 3 ( , 0•~ ¢ , ~ = o ,

13(o),f,. . = _iO#~p2o. u. , ,r aB aft' re~°~,~, " # -~ ~._ v" * .,~ = tcr,.aOu~P.,

B~,O)b ~ = o,

13~,O~e~ = O,

~'.~' ~ = o,

t~'._~'~. = o,

LoS, "

(B.23)

Now, beginning with Step 1, we find the general elements of the cohomology of/3~,°)modulo-d at each order: A#(3J~ = a t - Aijk .c#vp~r , v2t~ ,.,~i . vAJ,a. v p , , ,Ak r,

(B.24)

A#(4,1 = oQdijk (13ot/~i) ( ~ j f~k[3) ~_ ol3dijk (~3fl~i~) (/~,ja~k)

q- OlSab g"or~[3"~t~8 4)aOl~l)b Jr Ol9ab \ i

-&-I xv -~

i

"

oq~ I eoe i.zv 13 dpaOl~A.

@ OQ2a[bc] (/3aO"~//]ff)t0p.q~bq~,"-~ ~13a[bc] (,t~aaO'~ '~/~) O#~b~gc

= ~18(abc) (g~r/~i) ~a~bdt)c -r- OZ,9(abc )

@ OQOab(cd)(g°'lpao, ) ~9b4c4 d q- OQl(ab)cd~)a~) b (e&~t(~) 6~,,,

(B.26)

where d ijk is defined by (3.43). Doing the extension of these functions 5 according to Step 2 gives rise to the following constraints on the coefficients o~,,: eel

I 1 = --5Cr2 = _~3 = r,

(B.27)

5To do this e x t e n s i o n w e n e e d of course the explicit form of /3 ~m, for n ~> I, which can be read off from (2.3), (2.11) and (2.14) ( in our case B,_ =B(,°J+/3 ILl, +/5 d't, ).

426

N. Maggiore et al./ Nuclear Physics B 458 (1996)403-429

(B.28)

0{4a(bc) = O~4(abc),

(B.29)

O~5a(bc) = Ol5(abc) , !

i

I

i

•

i

i

i

(B.30)

~0"6ab = 10{dab = O~lOab ~ 0{lab, •

i

i

i

(B.31)

--20{7ab = --10{9ab = 0{I lab z_ 0{llab, O{12 = 0{13 = 0{18 = 0{19 = 0 ,

(B.32)

0 { 1 4 = 0 { 1 5 = 0{16 = 0{17 ~

(B.33)

0{111,

(B.34)

0{20ab(cd) =--0{21(ab)cd z 0{IV(ab)(cd),

where 0{lab, i 0{llab / and Oqv¢~,b~(cd) are invariant tensors and 0{1ii an arbitrary constant. (From (B.28) and (B.29) follows that the terms in a4 and a5 vanish.) This shows that the space 5~ defined in Step 2 is of dimension 5. We choose for 5~' the basis elements Asm, ,Jj . . . . . /trv characterized by their lowest a# (3) , 31(4) '" " ' ' --IV a# ~5) given by order terms "SAB A# (3) SAB

_Aijk~r#vp,rA riaj.~ Ak -- ~ ~ ~/z~ " ~ v v p ~ "¢F~

a#(4) i (2e~ga,~-iAB¢-Bh ~I = OLlab

(B.35)

• . - + e-~-~,~ te"o'~A'zfb~Oudpb o'~Ob-~ fbacgua,i ) ,

(B.36)

x

1I1 #(4) =OlIliab

dl#11t4)=--i0{lll

(-- 2 -eB*aA +e"°-:~P~dPaO~Ai~) , (B.37) -D ia ~Pb. +tA'"tr~BeBdpaOu(bh • • -" (eao'a~Bg~) ( - Z ^*i~,cg~ait, + A*ia.O~zAai +

~*i/3t91z~ifl

+

C*tO#Ct

(B.38) ~,v/t#,5, =a,V(,h~(,.d)

( (e"~Pa,~) fbbqSc~bd -- (ba4)b (eaO,. ) ~d ) .

(B.39)

Next, according to Step 3, we test the triviality of these elements. We find that ASAB iS 13z-modulo-d non-trivial, and that i - i - ~ ) + d (tcelab(fbaA~Ob - i -i-¢ ) +0(5), A#I'4)=]32 (Oqab(baAB~b

(B.40)

A#1,(4)=]32(0{lliab(baAia~yba) +d(iOllliab(~aAia~l,a )

(B.41)

A# (4) 111 =

Z~llI = B ,- (0{IUL;) + d (i0{m(L;)

A#wv,5) = z ~ , v = / 3 ; ( l ~0{w~oo~.,l~4'~4'b4',4'. - - ) +( id

+0(5),

(B.42) ~ 0 { w . , 0 ~ . , l ~ 4 ' ~ 4 ' h 4 ' , -4 ' e - ) , (B.43)

where 12 is defined up to a total derivative by

/d4x£=4

(X-/d4xbig4~zZOZ-rlAR-,~ZAt~).

(B.44)

This entails the B.,-modulo-d triviality of the four elements zi~. . . . . A~v, and therefore ASAB constitutes a basis of the cohomology o f / 3 5 - m o d u l o - d .

427

N. Maggioreet al./ Nuclear Physics B 458 (1996) 403-429 We finally write the explicit form of the anomaly: ASAB= (e u~p" {d ijk (Ouc i - 2Ai~o'u~Bg:B - 28'~o'~,,~i)AiB) A,JOpA,k~

+ DiJn,, ( ~ ) i z c i _

1 a iao-,~.a8~ _ -~

1 a% . ~ a-ifl\AiAmAk'~ -~8 )_.,,_~_,.j

-3diJt(8"ai~]tkB-gB~iaAi~aX)), where dUk and D ijmk are defined by (3.43) and (3.44), respectively.

Appendix C. Notations and conventions

Units:

h = c = 1.

Space-time metric:

(guy) = diag(1, - l , - 1 , - 1 )

(#,v .... =0,1,2,3).

Fourier transform: f(x) = ~

dk eik"f(k)

f(k) =

e-i~Xf(k) .

Weyl spinor."

(0,~ , a = 1,2) C repr. (½,0) of the Lorentz group. The spinor components are Grassmann variables: 0,~P~ = -~P~P,~

Complex conjugate spinor: (de, , & = 1,2) C repr. (0, ½). Raising and lowering of spinor indices: 4, 4 = 8"~,e,

~,. = 8.eJ

,

with 8 4/3=

--8 fie

812= 1

,

,

(the same for dotted indices).

Derivative with respect to a spinor component:

o = 8~~

a : ~ = a~ a¢,~

'

ag,.

ag,~

(the same for dotted indices).

Pauli matrices:

(o-,'~)J

1

= ~ [o-,~0 -~

o-"o-,qf

_

,

•

/

_p,

( ° ~ ' ) " B = 2 [o" er

P

gr"crt']~B ,

(B.45)

428

N. Maggiore et al./ Nuclear Physics B 458 (1996) 403-429

with 0-0= (10

O)

0.o = 0_0 ,

,

0-,= (~

~).i =

10) '

0_2= ( 0

0_i = 0_i , 0_oi = _o.oi = _i0_i ,

oi)

,

0_3= (10

?1)

0-u = o-iJ = eijk0-k ,

i,j,k=l,2,3. c o n v e n t i o n s a n d c o m p l e x conjugation." Let ~b and X be two Weyl spinors.

Summation

~Px = g ' " x ~ = - x ~ ¢ ,~ = x ' ~ P ~ = x ¢ , , ¢,2 = ¢,~2 ~ = ~

0-#

2~&~ = 2~& ~ = 26 -a

,

,

= ~- 0 --~ & a X,~

( g ' X ) = YC& = & 2 , 0p0-~ 2 ) = X 0 - ~ & = _~O-~' X , ( ~

x)

= 2a~"g ' •

I n f i n i t e s i m a l L o r e n t z t r a n s f o r m a t i o n s o f the W e y l spinors." If t~Lx t~ = tO#uX u ,

with WU ~ =

W~/~

(WU ~ = g"PWUp)

,

then 6L~p~(X) = ~O~ SL(~(X)

1

= ~0~

Izu

(X~a~ -- X . ~ ) g , ~ f X )

(

-

•

(XF, a . - - X u a ~ ) ~ a ( X )

-- ~ ( O - ~ , ~ ) J g , ~

"

I

_

+ ~(0-~,~)

&

-~ /3~

,

)

.

References [ 1] [2] [3] [4] [5] [6] [7] [8]

[9]

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