Covariant supergraphs

Nuclear Physics B252 (1985) 591-620 (~) North-Holland Publishing C o m p a n y

COVARIANT S U P E R G R A P H S (II). Supergravity M.T. G R I S A R U I

Physics Department, Brandeis University, Waltham, MA 02254, USA D. Z A N O N 2

Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138, USA Received 14 August 1984 We describe covariant D-algebra methods for the evaluation of supergraphs for q u a n t u m matter or supergravity, in the presence of background supergravity fields. These methods lead to improved power counting for the individual supergraphs. We examine two-loop matter contributions to the supergravity two-point function and conclude that there are no corrections to the one-loop (super)trace anomaly.

1. Introduction In a recent work [1] we have explained how the use of covariant D-algebra in conjunction with the background field method leads to significant simplifications in the higher-loop calculation of supergraphs. For SSYM [2] the covariant D-algebra generates Feynman graphs where the only dependence on the background fields is through the vector connection F ~ and the field strength W~. The Yang-Mills prepotential V or the spinor connection F~ never appear explicitly and consequently the resulting graphs are fewer in number and more convergent. In this paper, we explain the method for supergravity. Our main result is that the effective action depends on the background fields only through the vielbein components Ea M and E~ ~"but not E , " , and connection ~ba but not ~b,. We have fewer diagrams to compute, and they are more convergent than conventional background field diagrams. Background field methods in superspace rely heavily on the existence of covariant Feynman rules [3, 4] and are useful when considering processes with external gauge field lines. In the present context, we shall be mainly interested in calculating radiative corrections in pure N --- 1 supergravity, and also corrections due to supersymmetric matter. In both cases significant simplifications occur if the background supergravity fields are on-shell, and we expect that the main applications of our techniques (e.g. a calculation of the three-loop divergence in supergravity) will be L Supported in part by NSF grant 83-13243. 2 On leave of absence from lstituto di Fisica di Milano and INFN, Italy. Supported in part by NSF grant PHY-82-15249. 591

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for such situations. However, we shall also discuss and give some formulae for the case with off-shell background fields. Our p a p e r is organized as follows. In sect. 2 we summarize the essential features of the background field method in supergravity, and the covariant Feynman rules. In sect. 3 we describe covariant D-algebra and in sect. 4 we analyze the two-loop contribution of a self-interacting W e s s - Z u m i n o multiplet to the supergravity selfenergy. Sect. 5 is devoted to a m o m e n t u m space evaluation of the contribution in sect. 6. Sect. 7 considers similar contributions from a supersymmetric Q E D model. In sect. 8 we discuss the general structure of higher-loop divergences in these models. We conclude that there are no higher-loop on-shell divergences and therefore no radiative corrections to the one-loop (super)trace anomalies. The appendices contain a number of ancillary results. We use the notation and conventions of ref. [4].

2. Covariant Feynman rules Covariant Feynman rules for supergravity or matter superfields in the presence of background supergravity were derived in ref. [3] and are described in [4]. We summarize in this section the essential results. The supergravity background is described by a set of covariant derivatives in the vector representation: VA = (Va, V6e, ~7or&)= EAMDM + ~bA.

(2.1)

The quantum system is described by the action

S = f d'xd4OE-'Lg(Th V,H~c~...)+ f d4xd20~b3Lc(71 . . . . ) + h . c . ,

(2.2)

with background covariantly chiral superfields r/, V~? = 0, and real superfields V, H ~ , etc. Any derivatives appearing in Lg, Lc, are background covariant. Here E = sdet EA M is the background vielbein superdeterminant and ¢ the background chiral compensator. In general the quadratic action (after gauge-fixing for gauge fields) leads to covariant propagators such as

(,In)= G.,

(n,7)= G_, (H~eH~j) : G ~ : 3 j ,


(2.3)

etc., where, for massless fields [] O = 1 or, more explicitly,

[]± G±(z, z') ^

=

E88(z

-

z') ,

^

[S]G(z, z') = E 8 8 ( z - z') ,

(2.4)

etc. (G± is defined only when acting on a chiral (antichiral) superfield.) The

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d'Alembertian operators are defined by [-]+r/= (Vz + R)(V2 +/~) r/,

[ ] _ ( / = (V2+ lq)(V2+ R)(/,

[~+ = [] -- 2|['J 1 -..-~a&V,~Votti - RV 2 - ½(V~R)V~ + R R + ( ~ z g ) + W ~ t 3 W ~ M f + l i ( v z"~ G o,~ ) M~fl ,

[-1 = [ ] + * , = {Vz + R, Vz +/~} - ½V"(92 + R)V, - ½Va(V 2 +/~)V ~, =

+~G

[V~,,Va]+(V2R)+(V2g)+2RR+~(V'~R)Va+½(V~'R))V,~

- ½( W~OVVv - V~GO~V ~) M~o + h.c.

(2.5)

(spinor superfields, e.g. ~0~ have propagators proportional to (V~a)-~). One derives the following Feynman rules for the background effective action [4]: at one loop, it is given by the product of the usual determinants of the kinetic operators. At higher loops we draw v a c u u m diagrams, with vertices extracted from the interaction lagrangians and (covariant) propagators G±, t~, with additional factors (V2+R), (V2+/~) for each chiral or anti-chiral line leaving a vertex, etc. These diagrams lead to expressions involving products of propagators, covariant spinor and space-time derivatives acting on them from the vertices, and E -~ d8z integrals for each vertex. In a conventional background field calculation we would expand the propagators G in terms of free propagators and interactions with the background, i.e. G = Go+ GolGo+

GolGolGo+

•••,

(2.6)

and express the interaction terms I, as well as the covariant derivatives at the vertices in terms of ordinary space-time and spinor derivatives 0~a, D~, and background vieibein Ea M - 6a M and connections 6a. We would then do D-algebra, and obtain contributions to the effective action that depend on all components of the vieibein and connections. In particular, from a covariant spinor derivative V~ we would pick up the vielbein component E Z . Instead, just as in the Yang-Mills case [2], we will perform covariant D-algebra that leads to an expression for the effective action that depends only on the vielbein components Ea M, E~" and connection ~b~, as well as the field strengths. We will explain in the next section the details of the procedure. We summarize in the remainder of this section some of the formulae we shall use. For simplicity we give them for on-shell background fields. The off-shell expressions can be found in appendix A and in ref. [5]. On-shell (R = G~a = 0) the d'Alembertians take the form ff] + = VI + W ~ ' o ~ V ~ M : f l , ~2

[]-

2,,

B -~,,,*,

(2.7)

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where the Lorentz operators act on spinor indices according to (2.8)

[M,, m ~b:,] = ½[C:4d,,~ + Cv,~q, t3], []

1 =3V

c,t&

V~a,

v ~ -= vo = E.ma.~ + Eo"D. + &'~D. + ,/,~, 4~ = ~ b ~ ' M r ~ + &~*/~}~ -

{2.9)

Still on-shell we have the c o m m u t a t i o n relations

{v~, % } = o, {V,,, ¢~} = i v a n , [V~, Vb] = i C ~ lg'~f,*lVI, ~ , [V~, Vb] = C ~ t ~ [ W ~ e ' V v + V ~ W t 3 ~ a M a ~ ] + h . c . ,

[v~, D] = "i[(v ~e ff%~) + 2 ~ e ~ v ~ ] ~ ,

~,

[V 2, V~] = i v o r y ~ + I ~ * * M , ~ , [V z, Vb] = - i lg'~* V e M , * ,

For what follows it is i m p o r t a n t to reexpress V~a in (2.9) in terms of O,, and the covariant spinor derivatives V~, fgs. F r o m am = E m A ( V A -- ~')A)

= E,.~(V~-&~)+

Em'*(V,,-qb,~)+ E m a ( ~ 7 , ~ - ~ a ) ,

(2.11)

we can write Va = e~mo,. + ok, + e,"~Em~O, + e ~ ' E , , ~

- eZE,.Wv

=- Da - e~VV, - e~';V ~,

- ea"E~V ~

(2.12)

where

eo'(# EZ)= (E,£)', e J ~ e~"~Em ~ ,

ea ~'=- ea'Em ~ ,

(2.13)

and Da = ea"0,, + q~a

( & = 4,~ + e o ~ + eo~4;~)

(2.14)

is a space-time " c o v a r i a n t " derivative. [At 0 = 0 and in W e s s - Z u m i n o gauge it reduces to the ordinary gravitational covariant derivative (with the i m p r o v e d connection), whereas ea r] = ~/'a~, the gravitino field; see ref. [4], sect. 5.6.b.] In general we write eA M 8A M "~ hA M. =

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Given the above expression for Va, it is clear that the on-shell propagator 6 = [~-l can be written in the form (~ = [[~ - A ] - ' ,

(2.15)

where A

= A~V~+A'~V,~+BVZ+BC72+

C ~"~[V~,¢~],

(2.16)

and [~ contains n o spinor derivatives. The explicit expressions for A S , . . . , C ~ are given in (7.6). We can then use the relation

6= 6 + 6 a 6 ( ~ = [~]-i)

(2.17)

as the starting point of a perturbative expansion of 6 with respect to the terms involving spinor derivatives only. We observe that on-shell, if 6 carries no spinor indices, as a consequence of the fifth relation in (2.10), the spinor derivatives commute with it: IVy, G] = [V~, 6 ] = 0 ,

(2.18)

and can be moved past it in the above relation. We shall also find it convenient to split E into a part that is independent of the spinor components, E~ M, Ea M and the rest: we write E -1 = sdet E ~ A = sdet ( E " a E " r ~ \ E A a EA r ]

= [det ema][det e a r ] =- e - ' e - I ,

(2.19)

where e a r = EA r

--

r] Z , A a ea, , lY.m r r ,

(2.20)

and we are using the notation F = (a, t~), A = (/~, 12). We note that e -I = (det EIA),

(2.21)

where Er a is the spinor, spinor part of the vielbein EA M. I n the calculations, when we reduce supergraphs to ordinary space-time graphs, we shall sometimes leave e - 1 a s part of the integration measure whereas e - l will be treated separately.

3. Covariant D-algebra As in the Yang-Mills case [2], the expression (2.15) for the propagator is the starting point for doing covariant D-algebra. Given any background field supergraph, with propagators 6 and vertices extracted from (2.2), we consider the corresponding

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expressions

f dSz,

''dsz,, U ( z l ) G ( z l , z2) U ( z2 ) a ( z2~ z 3 ) " " ,

(3.1)

where U(z;) consists of products of covariant derivatives acting on the propagators, and factors E - l ( z i ) = e-I(zi)e l(zi). We start the D-algebra by concentrating on a given line, and pushing all the explicit covariant derivatives to its right end, say. (On-shell they commute with (3, off-shell we may pick up R and G ~ dependent terms.) We now use (2.17), writing as a sum of two terms, and for the second term we again push the covariant derivatives to the right vertex, past the G. We repeat the procedure any number of times, generating, for example, terms such as + GA~GAaGV~V~ + . . .

(3.2)

corresponding to strings of propagators G separated by vertices A ~, A~, etc. that depend on the background fields. It is important to note that G is diagonal in 0-space:

G(z~, zj) = G(x~, xj, O~)6a( O,- Oj)e(zj)e(zj) .

(3.3)

For any term in the expansion, we use now the commutation relations {V~, V~} = iV,~ to reduce the number of spinor derivatives at the right end of the line to four or less, as in ordinary D-algebra. All the resulting derivatives may be integrated by parts, freeing the given line of all covariant derivatives (including space-time V,a ; this is important, because V ~ - eaVV~ + • • • still contains spinor derivatives). We now repeat the procedure on another line, going around a loop until, as in ordinary D-algebra, all lines in the loop but one have been freed of covariant derivatives, and on that one line all the derivatives are concentrated at one end. At this point we can do all but two 0-integrations around the loop, using the 84(0~O~)e(zj) in G(zi, z~), and we are left with an expression

84(01- On)(VaVb"

" " VyV6"

" " ~t~'

"

")84(0n Ol) ." --

(3.4)

We use again (2.12) and the commutation relations to generate a sum of terms with a number of external field factors e~Veb~..., or "covariant" derivatives D,, and exactly four spinor derivatives V2V2 (terms with fewer spinor derivatives give zero). Finally we make use of the identity

81,V2V 2 6,; = 8;. det Er a = e -I 8;.,

(3.5)

and do the 0, integration, shrinking the loop to a point in 0-space. Repeating this procedure loop by loop we end up, as in ordinary D-algebra, with contributions to the effective action that are local in O, consisting of a set of terms that correspond to Feynman diagrams with propagators G and vertices containing derivatives D~, and factors e~v, e,, ~, but no spinor derivatives and no E Z components of the background vielbein. In fact at this stage, we are dealing with what looks

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like component supergravity, described by the "vierbein" e Z and "gravitino" e~v except that these objects are still superfields. Although we have described the procedure for on-shell background felds, it can also be used in the off-shell case. However the commutators are more complicated, and we generate many additional terms proportional to R and Ga. Supergraphs computed using covariant D-algebra satisfy improved power counting rules. In the presence of external supergravity fields ordinary supergraphs lead to contributions to the effective action that are functions of the supergravity prepotentials. Since H ~ has dimension -1 individual graphs lead to contributions that can be highly divergent. Some improvement is obtained by working in the background field formalism. The background fields enter only through quantities that are contained in the covariant derivatives, namely vielbein and connections. Since the lowest dimensional object present, the spinor vielbein E Z , has dimension -½, supergraphs computed in the background field method will be more convergent; compared to ordinary methods some cancellations due to gauge invariance have automatically taken place and therefore background field supergraphs must represent sums of ordinary supergraphs. (In the background field method the degree of divergence of the effective action is ultimately determined from the requirement that it be a function of the field strengths and covariant derivatives; but individual supergraphs are more divergent in general.) With covariant D-algebra a further reduction in the degree of divergence is achieved. Since the method completely removes the spinor covariant derivatives from the calculation, the lowest-dimensional objects that we encounter are the vielbein component E~M or E~z, of dimension /> 0. Again some cancellation has automatically taken place and this means that supergraphs computed in this fashion represent sums of ordinary background field supergraphs and are therefore fewer in number and simpler. We will discuss in sect. 6 some consequences of this result.

4. Two-loop self-energy corrections We describe in this section the calculation of the two-loop contribution to the supergravity two-point function from a self-interacting covariantly chiral superfield with action

(4.1) According to the covariant Feynman rules we must calculate an expression corresponding to the diagram of fig. la, to second order in the background fields, with chiral propagators G_(z, z ' ) = [ ] Z 1 E 6 S ( z - z ' ) , and factors V2+/~, 9 2 + R on two of the three lines at each vertex. We have used one factor V2+/~, Vz+ R to convert the chiral vertices to dSz integrals. The factors ~ z + R can be shifted to the right vertex, as shown in fig. lb, by using (~2+ R ) G _ = G+(92+ R).

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M. T. Grisaru, D. Zanon / Covariant supergraphs (II)

[]-I

a

_~

b

Fig. 1. Covariant two-loop supergraph for a self-interacting chiral superfield.

We shall discuss first the case with on-shell background fields and describe the off-shell situation later on. For the model described by (4.1) we can drop the Lorentz rotation terms in (2.5) or (2.7) so that the propagators are G = [] ~. We use on any one line the expansion (2.17) for G with the simplification that we need only keep terms linear in the background fields (quadratic terms would give vanishing tadpoles): G = G + G[A~V~ +e{~V~]G = G + GA~GV~

+ GA~GVa,

(4.2)

where A ~ = ~{~ , b, hb ~ } = ½(Obhb ~) + hb~a b ,

(4.3)

and hA M = eA M -- 8A M. At the linearized level we need not distinguish between tangent space and world indices. On the top and bottom lines of the graph we can drop the V~ term (since ~ 2 =0 on-shell) while on the middle line we can drop (after integration by parts) the V~ term. We next use the commutation relations to replace, on the top line for example, We now expand G one more time, either on the same line, or on a different line and again push the spinor derivatives to the right vertex. At this stage, up to second order in the external fields, we obtain the diagrams shown in fig. 2, where diagram (a) corresponds to the zeroth-order expansion in A ~ or A ~ of the G-propagators, (b) and (c) to the first-order terms on either of the three lines, while (d), (e) and (f) show the contributions of the second-order terms on the same line or different lines. All the propagators are now G and at the vertices we still have E -1 factors. The last three diagrams are explicitly second order in the external fields so all other dependence on the background fields may be dropped. We do now the 0-integration. This is trivial for the last three diagrams (figs. 2d, e, f), since we can replace the covariant derivatives by fiat derivatives. For the diagrams with one A s or A s (figs. 2b, c) we use first the expression (2.12) for V~a, which gives us on the top line, at the right vertex, -iha~V2V2: now the diagram is explicitly second order in the background and again the 0-integration becomes trivial. Finally, for the first diagram (fig. 2a) we write E - I = e-~e -~ at the right vertex, and also use (3.3) for the three propagators. Using (3.5), the 0-integration

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M. T. Grisaru, D. Zanon / Covariant supergraphs ( H)

~ ~ , ~ a

b

2 x ~

°"

e

d

f

Fig. 2. Two-loop contributions before covariant D-algebra.

in the two loops will p r o d u c e two more factors o f e -I, so that all such factors cancel except for the one contained in the E -t at the left vertex. We have thus reduced the calculation to one in ordinary space and we have the contributions o f fig. 3, where in the first diagram we must expand n o w ~ - 1 , e-i or E -1. Its perturbative evaluation can be carried out by expanding the "vierbein" ea" = ~a m + h Z in V~-1,

E"~e-'

2x

t=,,i-z a

2x

.,.,,~a b

2x

d

2,

e c

2x

e

Fig. 3. Two-loop contributions after covariant D-algebra.

f

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M. T. Grisaru, D. Zanon / Covariant supergraphs ( H)

and in the determinant factors, and writing [~-1= V301+ [~O 1T[~-l, etc. It is easy to verify that, to linear order in the background fields, T is given by T =½{i3 ~, h~'}i3,. +½i~baiOa + h~Vqo,

(4.4)

where (see appendix A) c~a = - d ) ~ , j ~ - c ~ , ~ ~ = 8bhab -

Oahb b .

(4.5)

The vertex in (4.4) looks exactly like the minimal coupling of a scalar field to gravity. We must evaluate now ordinary two-loop diagrams for a "scatar" field with the following type of interactions with the background: I. Ordinary gravity-type interactions with the minimal coupling of (4.4). To second order in the fields these will give rise to contributions that depend on ha m (fig. 3a). 2. Interactions that involve h~~, h~a from the lowest-order expansion of the determinant e -1 contained in the E -I factor (fig. 3a). 3. Nonminimal interactions with one A s (A a) and one haa(ha s) factor at the cubic vertex (figs. 3b, c). 4. Additional nominal interactions with derivative couplings given by the A s, * a factors from (4.3), and the additional derivatives indicated on the diagrams in figs. 3d, e, f. If the background fields are off-shell, there are two sources of additional contributions: 1. Off-shell terms proportional to R or G ~ , from the explicit form of ~ ± (see eq. (2.5)). Such terms would be picked out in the expansion of G±. 2. Off-shell terms that come from commutators such as [V~, V~], etc. (see eq. (A.1)) again proportional to R or G~a. Their inclusion presents no difficulty.

5. Momentum space considerations

We now analyze the momentum-space contributions corresponding to the diagrams discussed in the previous section, as well as the additional off-shell contribution. The diagram in fig. 3a gives rise to the following "gravity-type" contributions (we do not indicate the 0-dependence): q

x¼h(am)(-p)h(b")(p){kak,, + kap,, - ½3a,,[k2 + ( k + p ) 2 _p2]} X{kbk, + kbp, --½8b,[k2 + ( k + p ) 2 - p 2 ] } , 0

0

x¼h(am)(-p)h(b~)(p){kak,, + kap,, -½8a,,Ek2 + (k +p)2 _p2]} x {qbqn + qbP, -- ½($bn[q2+ (q +p)2 _p2]},

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601

k x¼h('~")(-p)h(b")(p)Sam{qbq,, + qbP,, -½8b,[q 2+ (q +p)2 __p2]},

k

xlh("")(-p)h(b")(p)~b.(k~k,. + k,p,, - ½/~..,[k2 + ( k + p ) 2 _ p2]}, q÷P q

P

~

x l h("")(--p)h(bn)(p)t~,,~b,,

p

(5.1)

k+9 where h ("") = h"" + h and every term is multiplied by scalar propagators associated with the diagrams with a factor I d"k dnq d40/(27r)2"]. The vertices were obtained from (4.4). From the same diagram in fig. 3a, we obtain additional contributions corresponding to the first-order expansion of the factor e-1 at the left chiral vertex: raa

~

- ~ ( hv "y+ hs,5')(-p ) h (am)( p ){ kak m -JrkaPra -½t~a,,[ k2-F ( k +p)2 _ p2]}, -q q

~

"

- ~ ( h ~ + h~)(-p)h(am)(p)8,,,, .

~ k+p

(5.2)

The remaining diagrams (figs. 3b-f) give

½ha~'(-p)h~@(p)(2k + p)a, q

~h~,c,V(-P)haa(P)( 2k + P)a, q q

- l h ~ : ' ( - p ) h b ~ ( p ) ( 2 k + p)~k:,~,(2k + p)b,

- ~ h a V ( - p ) h b ~ ( p ) ( 2 k + p)~(k + p)~,~,(2q+ p)b.

(5.3)

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To obtain the complete two-loop self-energy we must include the off-shell terms that contain G,a or R explicitly. They are given by the following contributions: q

{¼Ga(-p)Gb(p)(k + p)akb + V 2 R ( - p ) V 2 R ( p )

-½(V2R + V2R)(-p)h(~")(p) x{ kok,. + kop,. -½GreEk2+ ( k + p )2-p2]} +~i[V~R(-p)hba(p)+½VaR(-p)hb~(p)]k~(2k +p)b +~[ihO~'(-p)(2k+p)a + VVR(-p)] ×[½V(13av)B(p)k b + V B R ( p ) k ~ -- VvV2R(p)] + l[ih"~(-p)(2k +p)~ + V ~/~(-p)]

x [ - ½¢(~a¢~)(p) k b + V~R (p) ko~ - V~V2R( p)]},

x{~V2R(-p)V2R(p)

+ ~ G ~ ( - p ) G b ( p ) ( k + p)a(q+ p)b - ½ G " ( - p ) V 2 R ( p ) ( k + p)~ +~h(°m)(-p){kakm + kapm -13~m[k2 + ( k + p ) 2 _p2]} × { - ½Gb( p)(q + p)b -- 2¢2/~ (p) -- V2R(p)} -~[ih~V(-p)¢~'R(p)(2k + p)a(k + p)v~, - iVVR(-p)hb~(p)(k+p)v+(2q+p)b + V V R ( - p ) V ~ R ( p ) ( k + p)v~]},

p . ~ ~

{1(h:]' + h ~ ) ( - p ) [ 2 V 2 R ( p ) + V2R(p) -½Ga(p)k.]

q

_ ½h(.,,,)(_p)3a,~(Va R + ~ 2 ~ ) ( p ) - ~Il- V a g ( - p ) h ~

c~

---o; -( p ) - ~1W R(-p)h,,~ (p)}.

(5.4)

The background field effective action is invariant under the background transformations of the covariant derivatives ~V A

=

[iK, VA],

K = KMiDM = K~i3m + K"iD~ + Kailg;,.

(5.5)

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From ¢~(EAMDM + dpA) = --[ K NDN, EAMDM + t~A]

(5.6)

we obtain at the linearized level the gauge transformations t$h,," = 3 aK " , 8h~ ~ = 8h~~ = 0,

(5.7)

and ¢3ham = O,

tSha~ = G K ~ , 3h~/" = D , K ~".

(5.8)

In both eases 8Ga = 3R = 0.

(5.9)

The two-point function must be invariant under these separate gauge transformations. This invariance serves as a useful check of our calculations. We give details in appendix B. The momentum integrals in (5.1)-(5.4) can be performed and the results expressed in terms of a standard set of integrals associated with two-loop scalar diagrams. We list these in appendix C. We discuss here only the divergence structure of the result. We define the projection operators Hab= Gb

P~Pb 2p2 •

(5.10)

The integrals in (5.1) give the following 1/e 2 contribution: 1

2

~4[2Ha,.Hb. - (H~blI,,, + Ha,Flbm)]h~'(-p)hbn(p) E2

(5.11)

Those in (5.2) and (5.3) give respectively 1 p2 e2 6 (hvV+h~)(-P)IIa"h~m(P)'

(5.12)

1

24ei(h"~(-P)Pt,hv;/l~(P) + p[ahv~l~(-p)ha~(p)) .

(5.13)

Finally from (5.4) we obtain the following 1/e 2 contributions: 1

-~{R(-p)R(p)+

I G"(-p)G~(p)-~iV~R(-p)h~(p)-~iV~R(-p)h~(p)

+~ihb~,(_p)VvGb(p) - ~ l1" h b;/( - p ) V--~ G b ( p ) + ~ipahaV(-p)V ff72/~(p) + ~ipah ~ ( - p ) V ~ V E R ( p ) +¼(hy* + h ~ ) ( - p ) ( V 2 R

+ ~72/~)(p)}.

(5.14)

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All the above 1/e 2 terms can be expressed in terms of field strengths by using the linearized Bianchi identities of appendix A, as follows. First,

2p Ha,h 2

am

=c9a cg[mh~lm

=3(V2R+V21~),

(5.15)

where we have used the second and third equations of (A.4). The first term in (5.11) can then be rewritten after integration by parts as 3 8e2R/~.

(5.16)

Next we rewrite

(H~FI~,. + FI~Hb,.) h~mhb~ = ~H~Fl~.,iHt h(~,,,)h(b~), Hm~rta ~ ~tl

cll(am)

-

li_qH,,toa{O[ahdm+Otahm]c+Ot,~hc]a}

(5.17)

which allows us to use again the equations of (A.4). The second and third terms in (5.11) give then 1

48e2(5GaGa

+26R/~).

(5.18)

The contributions from (5.12) and (5.13) can be rewritten as [using (A.4) and (A.5)]

-'j{ ll GaGa - l Rff,-ah:,VV2R-ahS,5'V2R } .

(5.19)

In the same way the various terms in (5.14) give

--132 GaGa+-~e2hff~2R+8~hs,5"V2R.

(5.20)

The sum of the terms in (5.16), (5.18), (5.19) and (5.20) is zero; the two-loop self-energy has no l / e 2 divergence [1]. It is not difficult to isolate the l / e divergence, using the integrals of appendix C and the rules of dimensional reduction. However, in this example, we can obtain the complete contribution to the self-energy as follows. In general the two-point function can be written in terms of quadratic expressions in the field strengths R, Ga, W,~v which in turn are functions of the prepotential Ha and compensator 4~. However, in our case, the result must be independent of the compensator because in the action (4.1) ~b can be removed by a rescaling of the chiral superfield ~7 (note that E - l ~ ~ b ) . [We have verified that this is the case by determining the dependence on 4) of the individual terms in (5.1)-(5.4) and checking that it cancels.] We can therefore drop all dependence on ~b even at intermediate stages. Furthermore, we imagine that we are working in transverse gauge

D"H,~,~ = / ~ H ~

= 0. H = 0,

(5.21)

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which leads to significant simplifications (see the linearized expression of subsect. 7.5 in ref. [4]). In this gauge (and in chiral representation), dropping the dependence on

t~,

ham = L)aD,.H", h~~'= iE)2D ~H~ , Ga = - DO C)2D~Ha , R=O, h~)' = ha a = O , , , h Z = h~,a ~' = O ,

io~hZ = - G " ,

(5.22)

and all the terms in (5.1)-(5.4) reduce to I d40 d"k d"q l (27r) 2- q2k2(q+ p)2(k + p)2(k_q) 2[¼h"mhb"(2k + p)ak"(Eq+ p)bq" 2 -~h

+

I

a3,

I a 1 h b ~ kaqb(k+p),~,+~G G b k~qb-~h

am

G b (2k+p),k,,qb]

4 d"kd"q 1 I a~ b. d 0 (27r)2. q2k4(k+q)2(k+p)2[~h h (2k+p)ak,.(2k+p)bk.

1 a 1 - a'), 1 • a"~ --h":'hb~'kakbk:,~, +~G G b kakb +~zh V(~G:,)~kak b -~zh V(~G~q,)kak b ]

=

I

4 d"k d"q kak,,qbq, ,,, bn d Oq2k2(q+p)2-~---~p)2(k_q)2h (-p)h (p). (2~) 2.

(5.23)

To obtain the last form, we have used some integration by parts and identities between the integrals in appendix C, as a consequence of which most terms cancel trivially. Using the integrals in (C.4) only the A~ and B2"terms contribute in transverse gauge (with n-dimensional 3 functions). We write

f h°"hb"=f f)~D~H"OaD~Hn=-IH"D~D2D~H"C

~ ,

(5.24)

and using the rules of dimensional reduction [6, 4] ~d~3~ ~ = (2 - e)3~ ~ ,

(5.25)

we obtain the complete two-point function

f d40 Ha(-p)E)~D2L)~Ha(p)[(-3 + e ) A l - pEB2] - ~ [ = Y d4OHa(-p)D~D2D~Ha(P)

31 124--nS n o + & + 2 - -- ~ X ] '

(5.26)

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which has only a 1/e divergence. Note that in transverse gauge, at the linearized level,

Id4Op2Haff)BD2ff)BHa=Id4OGaaa=-Id20 W,~t3vW~ ' ,

(5.27)

so that the result cannot be expressed unambiguously in terms of the field strengths unless we study higher-order contributions. Such ambiguities are generally encountered when trying to determine the detailed divergence structure from momentum-space considerations. On dimensional grounds (the constant 3. is dimensionless), we expect the two-loop divergences of the effective action to have the same form as the one-loop divergence [4]:

Fo~= f d4xd40E '[c2RR+c3G2]+ f

d4xd2Oqb3[ClW2]+h.c. ,

(5.28)

with c~ - A2/e. (Recall dim (/~R) = dim G 2 = 2, dim W 2 = 3.) Further, since in the action (4.1) all dependence on the compensator 4) can be removed by a rescaling of 7, it follows that c2 = 2 c 3 [the combination E-J(G2+2RR) is independent of 4), as is ~b3 W2]. However, the relative values of c~ and c3 cannot be determined from a direct m o m e n t u m - s p a c e calculation in four dimensions at the linearized level of the two-point function. Such a calculation assumes trivial topology and integration by parts so that the result can always be written in terms of any two of the three field strengths, an example of the G a u s s - B o n n e t theorem at work. The same problem occurs at the one-loop level but is easily circumvented by the use of elegant techniques that do not rely on perturbation theory, both in ordinary space [7] and in superspace [8]. Some of these techniques have been extended to the higher-loop level [9-11 ] but not in superspace as yet. In sect. 7 we shall attempt to get around this difficulty. In closing this section we should mention another possible problem: we have assumed in the calculations that at this order it is sufficient to use the dimensional regularization method of global supersymmetry. It is known that this violates local gauge invariance if ordinary field theory is used [6]. The background field method presumably avoids this. However, if locally supersymmetric dimensional regularization must be used instead [12, 4] the scheme should be consistently implemented and one must solve the Bianchi identities in n dimensions. This may introduce some additional e-dependent terms on the right-hand side of commutation relations such as (2.10) and those of ref. [5]. Such terms, proportional to e, would combine with the 1/e 2 and 1/e dependence of integrals to give additional contributions.

6. Supersymmetrie QED It is not difficult to carry through the analysis for the case where one of the chiral lines in fig. 1 is replaced by a real scalar line, corresponding to the coupling of a vector multiplet to a charged scalar multiplet. We note the following differences:

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in fig. l the factors V2+/~, V 2 + R are interchanged on one of the lines, and the propagator for the middle line is G. In transverse gauge the result is

I

a

-/~ 2 -

p2

0

2--3n

(23

(6.1) 7. The higher-loop divergence structure In the preceding sections we have concentrated on a m o m e n t u m space analysis of the two-point function. Our main purpose was to develop some tools and expertise for more complicated calculations. In this section we consider the general form of the results and in particular the structure of the divergences. We begin by looking in more detail at the objects Da = earn(x, O)O,,+~b~(x, 0), ear(X, 0), ea+(x, 0), that we introduced in sect. 2. We shall refer to them as the "covariant derivative", "vierbein", "connection", and "gravitino fields", respectively. Since Da contains only a space-time derivative and Lorentz operators, we can define a "torsion" and "curvature" by IDa, Db] = tabCDc + tab.

(7.1)

We can express these objects in terms of the superspace field strengths. We use (2.12) Da = V a + e~VV v + e ~ V ~ =- eaC V c ,

(7.2)

with e j -- 6 j , and substituting into (7.1) obtain IDa, Db]

=

( D[ aeb ]F ) • F -- e~CebO TocP V F -- eaCeb° R o c ,

(7.3)

from which we deduce tabc = T J

- i e J eb )~/,

D[aeb] v = tabCecv - Tabv - e[aST~b]v - e[a~Tbb]v r~b = Rab + eEaVRvbl + e[~R+b]- et~Veb]~Rv~ , et~Yeb]~R~ •

(7.4)

Here TAn c, RAB are the superspace torsion and curvature, expressible in terms of

R, Q, W~,. In a similar manner we find the explicit form of the expressions in (2.16). We start with

= DAD,, - { D ", ea*}Vv - { D ", eaS"}¢q+ e"V[V., Vv]+ e"~[V., ~7~/]

-- e~Ve~vV2 - e~'te~v92 + e"Vea4"[9~, Vv].

(7.5)

As long as the d'alembertian is acting on scalars we can drop curvature terms from

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M. T. Grisaru, D. Zanon / Covariant supergraphs (1I)

[Vo, Vv] and we obtain for the quantities in (2.15), (2.16) :½D"D., A • = ea~Da - ½ ( e ~ T ~

~ + ea~To~ ~ - ( D a e ~ ) ) ,

,,Yi~ = e a~/Da - ½( e ~ Tat3~"+ e "~ Ta~ ~ - ( D " ea f ) ) ,

B = ! 2e a~ear,

/~ =2e _l ~-~ea~,

C~-~ = - ~ et ~ e~-; .

(7.6)

We note that on-shell T ~ v = T ~ ~ = Rv~ = R ~ = 0. Finally, we work out the transformation laws that follow from the'superspace transformation laws of the covariant derivatives: ¢~ A =

[ i g , VA].

(7.7)

In particular from (2.12) we have 8Va = [iK, V~] = 8D~ - (Se~V)V~ - ((Se~)V ~ - e~r[iK, V~] - e j [ i K , ¢~].

(7.8)

We choose first i K = evVv and, comparing the two sides of (7.8) we find 8Da = ie~/eVD~ + eVRv~ + e ~ e V R v ~ , Be, v = D~e v + iea~e~eb r -- e¢T~a v ' 8e, ~ = - e¢ T o j + iej3e~ eb~ .

(7.9)

Similarly, choosing i K = K"V~ we find 8Da = - ( D ~ K ~ ) D ~ + KbTba~D~ + K b ( R~a + e, VRvb + ejR~b), (Seav

=

8e~ ~ =

-- (

D ~ K ~) e J + K b ( TbaCe V _ Tb V + e a T~ v + e * T~ r),

(~eaV)

* .

(7. I 0)

These transformation laws, especially on-shell where many of the terms on the right-hand side of (7.9), (7.10) vanish, are replicas of the corresponding gravitongravitino c o m p o n e n t general coordinate and local supersymmetry transformations. This is not surprising since, at 0 = 0 and in W e s s - Z u m i n o gauge these objects equal the corresponding component quantities [4, 13]. As discussed in sects. 2, 3, we perform the covariant D-algebra by expanding E] -~ with respect to the spinor derivatives, one at a time, in the form [] t = V~ l+ ~-~AK]-I, pushing the spinor derivatives to the right, and then repeating the procedure. Given the expression above, a f t e r D - a l g e b r a we obtain results that depend only on the "gravitino" fields and their "covariant" derivatives, and on the superspace field strengths, all evaluated at the same 0. These results are associated with diagrams such as those of fig. 3 and they formally look like those for component scalar and spinor fields in a graviton-gravitino background, with "scalar" propagators ~ - ~ , and "spinor" propagators [~-~D, sandwiched between "gravitino" vertices ear, eb~, etc. This is particularly the case when we impose the on-shell conditions on the background fields.

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The next step in the evaluation of the effective action can be carried out perturbatively, as we did in the previous sections or, under special conditions, by any means that work in component calculations. Here we shall discuss the divergence structure of the results. We are considering matter systems governed by a dimensionless coupling constant such as the massless Wess-Zumino model of the previous sections, supersymmetric QED, or Yang-Mills, in the presence of background supergravity. On dimensional grounds the local leading divergences at any loop order are a combination al F + a2G of the invariants F=J

d 4 x d 2 0 W 2,

c=

d'x d'o (o2+2R

)+ f d'x

w2

(7.11)

The second, Gauss-Bonnet, combination is a topological invariant and its coefficient cannot be determined by perturbation theory in four dimensions. Therefore, in attempting to determine al and a2 we cannot assume trivial topology or a specific number of external lines. [At the two-loop level, aside from the uncertainties discussed at the end of sect. 5, our perturbative calculation does determine at but not a2.] We are interested in ultraviolet divergences and shall use power counting to isolate UV divergent parts. We assume we are in the on-shell situation and will set G~ = R = 0 whenever convenient. Therefore, in this section, we are attempting to determine the coefficient al + a2 of ~ d4x d 2° W2: We assume the existence of a suitable regularization scheme that respects the symmetries under (7.9), (7.10), which is tantamount to respecting local supersymmetry. We shall formulate our remarks with specific reference to the two-loop examples of the previous sections, but our conclusions apply to any loop order. As in the previous sections we begin with an expansion with respect to the spinor terms in (2.15), (2.16) or (7.5), (7.6). This expansion leaves us with propagators ~ - t and vertices containing "gravitino" factors e,L By power counting (dim e, ~ = ½), terms with more than four such vertices are finite and need not be considered further. We do not expand I~ -~ with respect to h," so that we will not lose contributions proportional to the Gauss-Bonnet invariant. After covariant D-algebra we are led to " c o m p o n e n t " structures corresponding to diagrams as in fig. 3, with an overall d40 integration (but up to four external "gravitino" lines). The diagram without any "gravitino" lines corresponds to that for an ordinary scalar field with lagrangian -½DadPDaqb+g~b 3, in a "gravity" background. Power counting (g has dimension 1) tells us that the divergent part must have dimension 2, and therefore must be a combination of the " c o m p o n e n t " field strengths

F~o~fd4Ofd4x[ol(tabe)2+~r],

(7.12)

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where r is the scalar "curvature" r-

- _ ~ 6 t P y ~~ 8 ~~r ~ .

tab

~

,

(7.13)

and dim tabC= 1, dim r = 2. Other divergent contributions come from terms with "gravitino" factors. On dimensional grounds we may expect them to have the form (other index contractions are possible) I'~ ~ d40

f

d4x[e"~D[be,~]+ ea:,eb~t,b c + • • • + e"~'eb~ea~'eb~,+ • • -].

•

(7.14)

The expressions in (7.12), (7.14) are not separately invariant under the local supersymmetry transformations (7.10). The sum must be and it is fairly obvious that the "gravitino" action in (7.14) must complete the "gravity" action in (7.12) to a "supergravity" invariant. In principle we can use (7.10) to determine the relative coefficients. However it is simpler to make use instead of (7.4) and argue that the terms in (7.12), (7.14) must enter in such combinations that in the end we can express their sum in terms of the superspace quantities that appear in (7.4). For example the first term in (7.12) and some of the terms in (7.14) must combine to give, according to the first equation in (7.4), (TabC)2. More generally, by using (7.4), we should be able to rewrite (7.12), (7.14) as an expression containing terms that consist of products of two factors, both of them superspace field strengths, or one noncovariant factor (e.g., ea r) multiplied by a field strength satisfying a Bianchi identity (so that the variation, after integration by parts, vanishes). It is a simple matter to examine (7.4) on-shell and determine if anything can survive in the divergent expressions (7.12), (7.14). First of all, examining the on-shell commutation relations (2.10) we see that Tabc= T~b~= Tsh~ =0. In (7.4), tab does not vanish on-shell, since Rab ~ = C ~ V ~ W~ ~ and R~b~ = i C ~ W~ ~. However only the contracted form r of (7.13) appears in the divergent terms and this does vanish since Wvv~= 0. Finally, from the second equation in (7.4), Tab~ = C ~ W ~ ~ could be present but it can only occur multiplied by e, ~ or e~~ and any index contraction between the two factors necessarily leads to its vanishing. Let us summarize. We have argued that the two-loop divergence is a local expression given by the terms in (7.12), (7.14), and that these terms must combine so as to be expressible as invariants constructed out of the superspace field strengths in (7.4). We have concluded that any such invariants vanish on-shell and this means that there is no on-shell two-loop divergence for the models considered. [In contrast, at the one-loop level in the background field method (see ref. [4], subsect. 7.8) an on-shell divergence is generated by local contributions proportional to

dnx d40 ObE ~ O[bEv~]. It is a crucial consequence o f the covariant D-algebra that terms proportional to the vielbein component Eva do not get generated beyond one loop.] Beyond two loops the situation does not change significantly. The degree of divergence of the " c o m p o n e n t " graphs stays the same. [Although the "coupling constant" g has dimension l, we also produce additional momenta in the numerators

M.T. Grisaru, D. Zanon / Covariant supergraphs (II)

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because we have more spinor derivatives at the vertices than needed for the 0integration.] Again, we expect no on-shell divergences. It is conceivable that even off-shell divergences are absent, but this cannot be determined from our general discussion. [At the two-loop level the actual coefficient of the r and (tabc) 2 terms in (7.12) could be computed, for example by the methods of ref. [9] suitably modified to take into account the nonzero torsion and the absence of a one-loop mass renormalization counterterm. Note that, unlike the situation discussed there, the nonlocal divergence is cancelled by the "gravitino" contributions rather than the counterterm contributions.] The degree of divergence does increase if the internal lines include supergravity fields. Our " c o m p o n e n t " counterterms are of higher dimension and could lead to on-shell divergences (although we know in fact that this does not happen at the two-loop level). Furthermore, if we are dealing with quantum superfields that carry Lorentz indices such as H ~ , explicit factors of W~v are present irr the propagators and commutators and affect our arguments. Therefore we cannot draw any conclusion at the present time about the structure of higher (/>3) loop divergences in pure supergravity. In the models discussed here a higher loop on-shell divergence would give rise to higher-order corrections to the familiar one-loop superanomaly - W 2. Similar corrections to the gravitational trace anomaly -(Wabcd)2 (W is the Weyl tensor) have been computed in nonsupersymmetric models [9-11] and found to be nonzero in general. However, in supersymmetric theories the one-loop trace anomalies are topological [7] and it might be expected that they receive no (coupling constant dependent) corrections. Our results support this conclusion. Thus, for self-interacting supersymmetric matter systems coupled to background supergravity the supercurrent satisfies an Adler-Bardeen theorem: there are no higher-loop corrections to the supertrace.

8. Conclusions

We believe that the covariant methods described in this paper will open the way for higher-loop calculations in quantum supergravity. The example studied in this work, the two-loop contribution of supersymmetric matter to the supergravity self-energy, is a relatively simple o n e - t h e full power of the method really shows up in cases with more spinor derivatives- but it is sufficient to illustrate the procedure. The explicit calculations in sects. 4, 5 may look complicated but in fact are straightforward and much easier than if done by any other method. The main feature of the covariant D-algebra, the absence of the vielbein component E~m from contributions to the effective action and the corresponding improvement in power counting has two consequences: (a) a practical advantage, reduction in the number of graphs and simplifications in their evaluation; (b) a step forward concerning the nature of higher-loop divergences. In higher loops local terms such

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as ~ d4x d40 ObE ~ o[bEral, responsible for the one-loop 5 d4x d20 W 2 divergence, do not get generated in the examples studied. We concluded in sect. 7 that selfinteracting supersymmetric matter does not give rise to radiative corrections to the one-loop multiplet of anomalies. This result, somewhat reminiscent of the nonrenormalization theorems of N = 1 global supersymmetry or, beyond one loop, of N = 2 supersymmetry, is not obvious at the c o m p o n e n t level nor at the level of ordinary supergraphs. It rekindles the hope that divergences in supergravity may be less severe than one may think. Eventually we hope to apply the methods described in this p a p e r to a computation of the three-loop divergence in supergravity. However three obstacles have to be overcome. (1) We must know the supergravity interactions to sufficiently high order in the quantum field H ~ , in the presence of background supergravity. A start in this direction has been made by Grundberg who has computed the ( H ) 3 term [14]. They are rather complicated and it is to be hoped that some simplifications are possible. (2) We must have a better understanding of the regularization procedure. Can we use locally supersymmetric regularization and, if so, must we know the solution of the Bianchi identities and the whole supergravity formalism in arbitrary n < 4 dimensions? (3) We must have a good algebraic manipulation program. With our covariant methods it may be possible to do the D-algebra by hand, but it would be preferable to let the computer do the work.

Appendix A We collect here the linearized relations that follow from the graded commutators of the covariant derivatives. We write VA = EAMDM + daA, with (at the linearized level) E A M = eA ~ = 6AM+ hAM, and substitute into the solution of the Bianchi identities {V~, Vt3} = -2/~M,~t3, {V~, Vt~} = i v a n , [V~, Vb] = -iC,,t3[RV~ - G~t~V~] + iC~e[ ff"~,*f4* ~ - (V'Ga~) M , 8] - i(V~R)M,#3,

[v., vb] = { [ c ~ w~t,:' + C~r,(~,~G~¢,) - C ~ (V~R)a~,~]V~ + iC~r,G:'f,V~ + [ C ~ (V ~ W,,~~ + (~=~ + 2RR) C~f,a~~) - C~, ( ~ V ~C#)]M~'} + h.c. (A.I) From the anticommutator of two spinor covariant derivatives we obtain at the linearized level D(,,h~ ) ~'+ cb~t3) = O, D~h~) 5' = O,

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613

ih(ij'z3,~) '~ + D(~h,~)" = O, D ( , d ~ ) P " = Ra~"a~ ") , D(,~qba )P°" = O, D,~h,~ ~' + £)ah,, ~' + cba~',~ = i h J , h , i ~ , ~ + h~f3~, ~ - iDa, ha m - zD~,h,~ •,,, = ha m , D.4,a

ptr

--

po"

+ Da6~

= i4a "= •

(A.2)

The expression of the commutator of a spinor covariant derivative with a vectorial one gives D~hb v - tgbh~ v - (~b'rc~ = i C ~ G V f3 , D ~ h b ~' - 3 b h j '

= - iC~SB

~'R ,

- ihb'ZS~ '~ + O ~ h b m - 3bh,~ m + c ~ c ~ t ~ / / ~ d- c~,~/~t~19 ~ = O ,

--D~dPb p¢ + ObC~ °¢ = I i C ~ V ( P G ¢ ) ~ - I iV f3RS~(PSff') , -tC~

Wf3

(A.3)

.

We also find further relations from the commutator of two vectorial covariant derivatives: 3[ahb]V =

. Ca~W~

v

I_ v. +2C~;3V(aG a ) - ~ CI a f 3 V ( ~ R S ~ )

3[ahb]r" + [ t ~ a l j ~ t ~ j S ~ + ~a#~t~13/~ __ I

3[aqbb]~'°'=~C,~[V(,~Wm

po"

a ~ b ] --E- 1~"C ~ G

v

,

~ (~8~) . /~__1" . /~ 3~) 2~Ca~G(~

+(V~R + 2RR)3P(~8,~)"]-¼C,~V(aV(aG")

,

~).

(A.4)

We used also some other identities that easily follow from the previous ones: ~ ~/ha~' ---- 3 ,,h~j' - iG~ , V~h ~ = ~ , V h~

= 2iR-

iVEh~ "~.

(A.5)

Appendix B We discuss in this appendix the gauge invariance of the self-energy graphs computed in sect. 5. We consider first the gauge transformations (5.7) 3ha" =OAK", with hA", h A # invariant, and the contributions The K " gauge transformations of ha m are transformations" of the "vierbein" ea'. We contributions in (5. l ) are exactly like those from

(a.l)

in (5.1). just linearized "general coordinate have already pointed out that the a scalar field in a gravity background.

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Their gauge invariance under (B.I) can be easily checked. If we replace h , ' ( - p ) by i p ~ K ' ( - p ) the vertex T ( - k , k + p ) in (4.4) transforms by

3T = ½iK m ( - p )pa{Zk,,ka + k,,pm + k,,,pa

-

~am[k 2 + ( k + p )2 _p2]}

= iK ~ ( - p ) [ ( k +p)2k m - k2(k +P)m] •

(B.2)

The gauge variation of the various terms in (5.1) is given by the corresponding contributions (we have dropped tadpoles):

½iK "(-p)h~b")(p){kbk,, + kbp,, -- ½~b,[k2 + (k +p)2 _p2]}

½iK m(--p)h(b"'(p){qbq. + qbP. --½8b.[q 2 + (q + p)2 _p21} ._

k+p

½iKm(-p)h(b")(p){%q. + qbP. -½~5b.[q 2+ (q + p ) 2 _ p2]}

xIpm -}

~tK (--p)h~b")(p)Sb, km

~-

½iKm(--p)h~b")(p)Sb,{~pm-" (

~

~

,

} ,

(B.3)

where every term is multiplied by the scalar propagators associated to the diagrams (the dashed line represents the gauge parameter K "). It is easy to see, after some m o m e n t u m relabeling, that the first three terms and the last two add up to zero separately. It is also easy to verify that the gauge variation [under (B.I)] of the terms in (5.2) and (5.4) separately vanishes. It follows that these contributions give a result that depends only on C)[bha]m. We examine now the invariance of the contributions under the gauge variation

(5.8) ~ha" = OaK ~', 8h~, ~ =

DvK ~ ,

8ha ~ = 0.

(B.4)

M.T. Grisaru,D. Zanon/ Covariantsupergraphs(II)

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The variation of the two terms in (5.2) gives

-½K~'(-p)Dvh""(p){(2k + p),,k,, + (kap,. - 6a,.k.

p)}

The sum of the corresponding variation of the contributions in (5.3) is

½iK:'(-p)ha~'(p){(2k+p)ak:,~,-*'k@} -2iK'/(-p)h,ef(p){-"

~ ' - " ~ "

}.

(B.6)

Using the integrals of appendix C, it is easy to verify that these variations are proportional to the transverse parts of ham and ha~. Using the linearized relations of appendix A, we find (2pEB,fl -

p,.p")ha m = 3(~2/~ + V2R),

(2p2~,," -pmp")h~ ~ = -3iV~C72/~.

(B.7)

The total variation in (B.5) and (B.6) can then be written as

-3Kr(-p)V~f72R(p)[A +O].

(B.8)

Although it is not zero, it vanishes on-shell. To get a complete cancellation we must take into account the off-shell terms in (5.4). Their variation under (B.4) is given by

-~K'(-P)¢~'R(p)kv~- ~ k , p ~ +2KV(-p)[½V(oGv)~(p)kb+¢BR(p)kv~

- Vv~2/~(p)] - ~.

+½KZ'(-p)V~,[2~72~(p)-½Ga(p)ka] - ~. k +½K:'(-p)~'"(p)(k+p)~,~,[q~

-~K "(-p)V~'R( p)p:,~, - ~.

k+p

(B.9)

616

M.T. Grisaru,D. Zanon / Covariantsupergraphs(1I)

After using the integrals in appendix C, the variation reduces to - ~ K V(_p)[V~oGv)~pO~ + 3~p~/~](p)[z~ + O ] .

(B. 10)

Using the Bianchi identity we can rewrite the expression in the bracket as -6VvV2/~ so that, adding (B.10) to (B.8), we find that the gauge variation of the complete contribution is zero.

Appendix C In our work we have encountered a number of two-loop self-energy diagrams which can be expressed in terms of a standard set of scalar Feynman integrals. We list them in this appendix. To begin with we define A-=

-CZ>'

~2e2,

air--=

2e2,

1 4e '

O11-=

-finite,

(C.1)

where the diagrams are massless scalar Feynman diagrams, scaled by factors of p2 to make them dimensionless in four dimensions. We have indicated the leading divergent behavior. We also define

f dnk d"q A=-

(27r) 2"

k4

1

(k+p)2q2(q+k) 2

24e"

With these definitions, we then have

f d"k d"q (27r) 2~

f d"k d"q

k2 (k +p)2q2(q + k)2 - ~P40 ,

q2(p+ k)2 = ~p6[~O - A ] (2rr)2. k 2 ( ~ ( ~ _ - q)2

(c.2)

617

M. T. Grisaru, D. Zanon / Covariant supergraphs (II)

d”k d”q (2~)‘”

(q-k)* q2k2(k+p)2(q+p)2= k2

d”k d”q &?-’

= $*[A - O] .

q2(q +p12(k - d2(k+p)2

We list now a set of Feynman The Kronecker

-“*”

integrals

with momentum

factors

(C.3)

in the numerator.

deltas are in n dimensions.

(C.4a) where 2 $a-$A_ Al=(nt1)

A*= ~~~~-4A+tr+~~-~*]-nA,, B, =

~l-A+~@l--$4+~nA21,

B2= -&[-;~+2A-X+@+;A]-$A,,

C = -$[inA,

+$n2A2+ np2B, +2p2Bz].

qakbkc= APahx + B( w

Pbsec

+

@(lb)

+

(C.4b)

CpaPbpc,

(CSa)

q

where A= &W3A-@l, B= &[“-4A+2X-201,

c=

4(n~l)p2[-~(n+2)fi+f(12-n)A-2E+$@].

%kb = A&b + BPaPb,

(CSb)

(C.6a)

M.T. Grisaru,D. Zanon / Covariantsupergraphs(II)

618

where A=

1

[-½fA+2a-]£+O],

n--1 1

n

(C.6b)

B = 2-~p2[2A- £]-2~p2 A. (

~

~

k

k,,kb = AS,,b + Bpapb

(C.7a)

q

where 1

A 2(n- 1) [ - 1 1 + 3 ~ + O ] ' _lz~

B- 7

n

(C.7b)

-~p2 A-

k. = --~p~p~a,

(c.8)

q

~

k~k,nkbk, = A( ~b6m, + ~am6bn + 6anSmb)

+ B(6~mpbp, + 6~bpmpn + 6anPmPb -4- 8,,bp~p, + 8,,,p~pb + 8b.p~p~) + Cp~pbp,.p.,

(C.9a)

where 2

A=~

[¼~I~- &,+ 70 + ¼A], 1

1

B =4(n - 1) [-½air + O - ½ A ] - ~ p 2A' C

I

22

3

= l~pSp2[ ~ + 4a + TO + A ] - ~ B

q

p ~ ~ ' -

_3

A

4p 4

k,,kbkc = A( 6,,bpc + 6acPb+ 6bop,,) + Bp,,pbpc,

(C.9b) (C.10a)

where 1

A = 4(n _ 1 ) [ ~ - • - 2 0 ] ' 1 n+2 B = --~Sp2[A+ O]---~pZ a .

(C.10b)

M.T. Grisaru, D. Zanon / Covariantsupergraphs (II)

619

q p ~ ~ ~ -

k~kb = A6ab + Bp~pb,

(C.lla)

where 1

A=

[-½qt +a-½0],

n-I

1

n

B = -~ A - ~p2 A .

(C.llb)

q p ~ ~ ~

r

k~ = Ap. ,

(C.12a)

where A= -+[xp+

A].

(C.12b)

~ k~kb = A6.b + Bp.pb,

(C.13a)

q

p ~ ~ ) k

where

p2 A= /I--

1 [ - ~I + g O 5 ],

1 B - 2-------~n ( ) 1[½hA + ~(7 n - 1 2 ) O ] .

p••.-

k,,= Ap,~,

(C. 13b)

C.14a)

k where A = - ½[z~+ O ] .

C.14b)

References

[1] [2] [3] [4]

M.T. Grisaru and D. Zanon, Phys. Lett. 142B (1984) 35 M.T. Grisaru and D. Zanon, Nucl. Phys. B252 (1985) 578 M.T. Grisaru and W. Siegel, Nucl. Phys. B201 (1982) 292 S.J. Gates, M.T. Grisaru, M. Ro~ek and W. Siegel, Superspace (Benjamin-Cummings, Reading, MA, 1983) [5] M,T. Grisaru and D. Zanon, Nucl. Phys. B237 (1984) 32 [6] W, Siegel, Phys. Lett. 84B (1979) 193 [7] S.M. Christensen and M.J. Duff, Nucl. Phys. B154 (1979) 301

620

M.T. Grisaru, D. Zanon / Covariant supergraphs ( H)

[8] I.N. McArthur, Phys. Lett. 128B (1983) 194; Class. Quantum Grav. 1 (1984) 245 [9] I. Jack and H. Osborn, Nucl. Phys. B234 (1984) 331; I. Jack, Nucl. Phys. B234 (1984) 365 [10] S. J. Hathrell, Ann. of Phys. 139 (1982) 136; 142 (1982) 34 [11] M. D. Freeman, Ann. of Phys. 153 (1984) 339 and references therein [12] W. Siegel, Phys. Lett. 94B (1979) 37 [13] S. Ferrara and P. van Nieuwenhuizen, in Supergravity: Proc. Supergravity Workshop at Stony Brook, ed. P. van Nieuwenhuizen and D. Z. Freedman (North-Holland, Amsterdam, 1979) [14] J. Grundberg, Expansion of the N = 1 superspace supergravity action, 1TP Univ. of Stockholm preprint