Cancellation of divergences in the one-loop calculation of g - 2 in broken supergravity

Nuclear Physics B255 (1985) 532-548 e) North-Holland Publishing Company

CANCELLATION OF DIVERGENCES IN THE ONE-LOOP CALCULATION OF g - 2 IN BROKEN SUPERGRAVITY A. GEORGES and P. LE DOUSSAL

lxlboratoire de PtTvsique Thboriquede I'Ecole Normale Supbrieure*

Received 18 January 1985

We show that all divergences encountered in the one-loop calculation of the anomalous magnetic moment of the electron in the N = 1 broken supcrgravity cancel out. This result holds for either a spontaneous or an explicit supersymmetry breaking, provided the photino is kept massless.

1. I n t r o d u c t i o n

A general a r g u m e n t due to F e r r a r a a n d R e m i d d i [1] establishes the absence of an a n o m a l o u s m a g n e t i c m o m e n t in s u p e r s y m m e t r i c Q E D , to all orders of p e r t u r b a t i o n theory. This is a consequence of s u p e r s y m m e t r y which can be checked at o n e - l o o p o r d e r b y an explicit calculation. A n o t h e r result, not s u p p o r t e d by a s y m m e t r y a r g u m e n t a n d hence m o r e surprising, is the discovery b y Berends a n d G a s t m a n s [2] o f the c a n c e l l a t i o n of the o n e - l o o p ultraviolet divergences in the calculation of g - 2, in Q E D c o u p l e d to p u r e gravitation. T h e g e n e r a l a r g u m e n t [1] is still valid for local s u p e r s y m m e t r y . This, together with the result o f ref. [2], implies that the gravitino c o n t r i b u t i o n s at one loop ( o r d e r ~2) in unbroken s u p e r g r a v i t y - Q E D have to c o m p e n s a t e the graviton c o n t r i b u t i o n s exactly. In p a r t i c u l a r , they m u s t also be free of divergences. This has been c o n f i r m e d recently for the N = 1 supergravity, by explicit calculations using various techniques [3-5]. In the p r e s e n t paper, we c o n s i d e r this p r o b l e m for the N = 1 s u p e r g r a v i t y c o u p l e d to Q E D w h e n the local s u p e r s y m m e t r y is broken, with a non-zero gravitino mass. O w i n g to this breaking, one no longer expects g - 2 to vanish. Nevertheless, as a c o n s e q u e n c e of the local c h a r a c t e r of the s u p e r s y m m e t r y , it seems likely that the t o t a l divergent p a r t still vanishes at o n e - l o o p order. This is actually w h a t we find for the g r a v i t o n plus gravitino contributions. M o r e o v e r , this cancellation arises for the graviton a n d gravitino separately, which m a y be v i e w e d as a " m i r a c l e " as long as no s y m m e t r y p r o p e r t y is discovered to * Laboratoire Propre du Centre National de la Recherche Scientifique, associd h l'Ecole Normale Supdrieure e t a l'Universitd de Paris-Sud. Postal address: 24, rue Lhomond, 75231 Paris Cedex 05, France. 532

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2 in broken supergrat~i(v

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explain it. For the graviton this result is already known, the calculation of ref. [2] being still valid here. We show in this paper that the order K 2 contribution of the gravitino (now massive) is divergenceless again, for either spontaneously or explicitly broken supersymmetry. (This result is derived in the case of a massless photino.) Nevertheless, since the theory is not renormalizable, one cannot give a precise meaning to the finite contributions to g - 2. Other cancellations are known to occur when matter is coupled to gravity or supergravity [6-8]. A particularly interesting example is the absence, at one-loop order in the gravity Yang-Mills system, of counterterms which would a priori have been allowed. As shown by Deser [6], this results from an effective chiral invariance of the associated supergravity theory. We also discuss the relevance to our problem of some symmetry properties. We show that some of the observed cancellations can be understood by comparing the exchange of a gravitino with the exchange of a goldstino in spontaneously broken global supersymmetry. This paper is organized as follows: in sect. 2, we briefly recall the construction of the relevant lagrangian density for the two supersymmetry breaking schemes considered. In sect. 3, we perform an explicit calculation showing the cancellation of the divergences for a massive gravitino, at order x 2. In sect. 4, we discuss some symmetry properties and investigate the consequences of the non-renormalizability of the theory. We also discuss the massive photino case.

2. Breaking supergravity in QED In this section, we consider the N = 1 supergravity coupled to QED. We use two different ways of breaking local supersymmetry: the spontaneous breaking (superHiggs effect) and soft explicit breaking by mass terms. In these two cases, the physical fields of the theory are the supergravity doublet ( G , , +~,)' the Maxwell doublet (A~, ?t) and two charged doublets (Yl, X~C), (Y2, X 2L) describing the electron and its scalar superpartners. These fields transform under the U(1) gauge group as fl

~

e

~eq°)'l ,

Y2 ~ eieq'.}'2,

X1L ~

e

ieq;x1 L ,

X 2 L --~ eie°2X2L ,

(2.1)

where e is the electron charge. The Dirac spinor e describing the electron field is identified as

eL=X2L,

(eC)L=XlL.

Our metric and Dirac matrix conventions are displayed in the appendix.

(2.2)

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A. Georges, P. Le Doussal / g

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2.1. SPONTANEOUS BREAKING

We apply the general results of ref. [9] to the case of QED, with a hidden sector consisting of a single neutral chiral multiplet (Yo, XOL). This is the simplest way to obtain a super-Higgs effect with zero cosmological constant. The model is specified by two functions, f(Ky~) and G(Ky i, K~i). f induces interactions between the scalar fields and the Maxwell doublet, and G is the K~ihler potential for the scalars. In what follows we shall limit ourselves to the case of "minimal coupling", defined as follows: G'"=j -½x2~,

f = 1.

(2.3)

This implies the absence of a mass term for the photino at the classical level. We shall adopt the choice made in ref. [10] for the K~hler potential:

G(~y,, ~fi,) = - ~r2fi, y , - lnlPo(rYo) [2 - l n [ p ( K y x, Ky2)l 2 .

(2.4)

This leads to universal properties for the super-Higgs mechanism. However, it turns out that we only need the lagrangian density up to order J¢ and thus the following expansions of P0 and p:

po(xYo) = Km3/2(1 + axYo) + O ( x 3 ) , K2m

P(~cYl, xY2) = 1 + 2rn3/2),y 2 + O ( x 3 ) .

I2.5)

The conditions of minimization of the scalar potential under the constraints of zero cosmological constant are satisfied for Yi = 0, without breaking Q E D gauge invariance, provided we choose a = ~/~.

(2.6)

This specifies completely P0 up to this order, m and m3/z can be identified as the electron and gravitino masses, respectively. The complete resulting lagrangian at order ~ is not yet expressed in terms of the physical fields after spontaneous breaking of supersymmetry. We still have to eliminate the would-be Goldstone fermion. The eigenstates of the mass matrix for scalar fields are identified as the following: (i) Physical sector: A =~(.pt +Y2),

m4=m+m3/2,

B = ~i(y 2-fil),

m B = In , - m3/21,

(2.7)

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A. Georges, P. Le Doussal / g - 2 in broken supergrat~iO'

A and B are the scalar partners of the electron and have both charge e. One has the mass relation m 2 + m 2 - 2m 2 = 2m~/2.

(2.8)

(ii) Hidden sector: m Ao 2 --- 2(2 - VC3)m2/2,

A0 = ½(y0 +Yo), =

-

m 2Bo = 2~[3m2/z .

-Y0),

½iCyo

(2.9)

The Goldstone fermion to be eliminated is, as usual, identified as the combination of fields which couples to the "gauge condition" f.yu: 'I~L= I e - G / Z G t i X L i ~

g

1 ..'-.i~,i yj ~kL, ~I(~jL¢

(2.10)

where Q is the diagonal matrix: d i a g ( 0 , - e , e ) . In order to obtain the final lagrangian in terms of physical fields, one has to perform the local supersymmetry transformation which amounts to setting tic = 0. This reads

O=TJL=

3

IX 2

--~¢m3/2XoL--5

m3/2

[(_ m ) ( m ) ] Yl +m~3/2Y2 XL1 q- f 2 + ~ 3 / 2 Y l XL2

- ¼iex2(y2Ya--f, yl)XL + .., ,

(2.11)

where we have not written the terms that do not contribute to the lagrangian at order K. Using this constraint, we obtain the following interaction terms relevant to our calculation, at first order in x and e: f~¢e = ½ix~O.( A + 75B)y~y"~,~ + ½iK~y°y~Op( A ++ ysB +) e.

= - ½exY(A + ysB)y~A~bu + ½ e x ~ . A T " ( A ++ 7~B+)e,

[~gee = e X ( A + + 7 5 B + ) e - e Y ( A + 3,sB)X, = -ieA.(Aa~A

+-

A + OVA) + ( A <--)B ) ,

~yee ~eey{/= --iK~/~e(

m A 0 -- I~[P'y5O oBo ) e .

(2.12)

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A. Georges, P. Le Doussal / g

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The last coupling involves both the hidden sector and the physical one. In the following, the sum of all these interaction terms will be denoted as EsH ("superHiggs"). 2.2. EXPLICIT SOFT BREAKING Another way of breaking supergravity is to start from the usual lagrangian of unbroken supergravity [9, 11] and to add an explicit mass term for the gravitino and arbitrary masses for the electron and its scalar partners, lifting the degeneracy of the supersymmetric theory. The interactions of the gravitino up to order • in the unbroken theory are given by - ½ix~-/~J",

(2.13)

where J " is the total current of global supersymmetry [11], the electron and its scalar partners having the same masses m. The breaking of supersymmetry is then achieved by giving arbitrary masses to the scalar partners of the electron, and by adding a mass term for the gravitino. The resulting interaction lagrangian, relevant to our calculation, differs from ~SH obtained in subsect. 2.1 since: (i) it does not involve the term ~ y 0 since no hidden sector is introduced here. (ii) there is an additional contribution, which reads

A~gee=½iKm~.y(A+-ysB+)e-liKmO(A -ysB)'~.+.

(2.14)

The resulting interaction lagrangian will be called fie×p1 in the following. The fact that the gravitino interacts differently from the case of spontaneous breaking is only a consequence of the choice made in subsect. 2.1 for the Goldstino field BL" Indeed, let us come back to the case of a spontaneous breaking and show that, had we made a different choice for ~/L in subsect. 2.1, we would have found the same gravitino interactions as in ~expl" AS for other spontaneous symmetry breaking mechanisms, only the linear part of the Goldstone field is uniquely defined, and one has the freedom to add higher-order combinations of the fields: L = -- ~-2 Km 3/2 X 0L + higher-order terms.

(2.15)

The use of the constraint ~L = 0 leads to a different lagrangian, but should not change physical amplitudes. Instead of the choice (2.11) made in subsect. 2.1, we can set the following combination equal to zero: ~L = -- ~-~3-~rn3/2XoL -- ½KZrn3/Z(YlX1L +Y2X2L) -

(2.16)

A. Georges, P. Le Doussal / g - 2 in broken supergravity

537

This adds precisely the term AE~ee to the interaction E~e of formula (2.12): we recover the same couplings of the gravitino as in E~xplThis freedom which exists in the choice of ~L amounts to different definitions of the gravitino field, related by supersymmetry transformations. ~SH and Esn + z~Eg~e are thus equivalent forms of the lagrangian density of spontaneously broken supergravity at this order. This will be illustrated explicitly in our calculation.

3. Cancellation of divergences In this section we show, by an explicit calculation, that the divergent contributions to the anomalous magnetic moment of the electron cancel out, for the two models L~sn and Eexpl- Five diagrams are common to these two models: they are represented in fig. 1. In the case of a spontaneous breaking, one has two more diagrams arising from the hidden sector of the theory, represented in fig. 2. The corresponding contributions to the vertex function F,(r, p) are all displayed in the appendix. There, we also make some technical remarks about their computation. They are all superficially divergent, at most quartically in the case of F2 la) and /-(lb)+(lc). However, as the divergent part of g - 2 is obtained from F. on-shell by

½(g-

2)div = F2div(0) ,

(3.1)

where

--

i u div 2 ] u(r)r~divu(p)=gt(r) I y,F~div(q 2 )+~-~m[V~,y,]qF~ (q ) u(p),

(3.2)

we expect, on dimensional grounds, that only quadratic and logarithmic divergences will occur in g - 2. We postpone the questions of the finite parts until sect. 4. We proceed as follows: we first perform the algebra of Dirac matrices in four dimensions for each integrand and then develop it in powers of 1/k up to order 1/k 4. The resulting integrals are understood to be regularized according to a given scheme that we do not need to specify explicitly. The only requirement is Lorentz invariance which allows us to reduce all encountered tensorial structures to the two following integrals:

I2

=[ d4k 1 )reg(27r)4 k z '

f d4k 1 IO=Jreg(21r)4 ( k 2 + ~ 2 ) 2 '

(3.3)

where ~ is an arbitrary mass designed to avoid spurious infrared divergences. This method has the advantage of being easily performed using a symbolic computer program such as " S C H O O N S C H I P " . Our results for ½(g - 2) are displayed in table 1 for the explicitly broken theory, and in table 2 for the spontaneously broken one. (In the latter case, we have introduced a parameter 3 = 0 or 1 corresponding to EsH and Esn + A E, respectively.)

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A. Georges, P. Le Doussal / g

2 in broken super grat,i(v

q

/

/

I \

\

A,B / /

\ \ A,B

/

\

-,J+~ /

\

q

(a)

k

q

% /

/

(b)

4

(c)

r

,

\

q

?_ ....

L~

,e,

Fig. 1. Contributions to I~ of order ~2 c o m m o n to spontaneous and explicit breaking,

A. Georges, P. Le Doussal / g - 2 in broken supergraoi(v q

539

q

(a)

(b)

Fig. 2. Contributions to F~,of order t¢ 2 due to the hidden sector, in the case of spontaneous breaking.

B o t h q u a d r a t i c a n d l o g a r i t h m i c divergences cancel out for the two cases considered, l e a v i n g o n l y finite c o n t r i b u t i o n s to g - 2 . F u r t h e r m o r e , our m e t h o d clearly e x h i b i t s t h a t this cancellation is i n d e p e n d e n t of the regularization scheme, as is e x p e c t e d for d i v e r g e n t parts. L e t us n o t i c e that the l o g a r i t h m i c divergence which is a priori e x p e c t e d for the d i a g r a m i n v o l v i n g the scalar B 0 of the h i d d e n sector, vanishes b y itself. (The d i a g r a m i n v o l v i n g A 0 is less divergent b y two orders.) This is a r e m a r k a b l e fact, since a n o n - v a n i s h i n g result for the divergence of the h i d d e n sector d i a g r a m could have led to d i f f e r e n t results for the two s u p e r s y m m e t r y b r e a k i n g schemes considered. W e also n o t i c e that & d e p e n d e n t terms cancel b y themselves, which c o n f i r m s for o u r calculation the e q u i v a l e n c e between fish a n d ~sn + A~, as c l a i m e d in subsect. 2.2 (let us

TABLE l

Divergent contributions to 12(g- 2) of the diagrams of fig. 1 in the case of an explicit breaking of supersymmetry (STR denotes m 2 + m 2 - 2rn 2) Order In A (coefficient of I 0)

O r d e r A2

Diagram

(coefficient of 12)

la

0

lb + lc

0

ld+le

0

m2K 2 l

m~/2 18(STR

8m~/2)

m2~2 1 (2STR + 8m2/2) m~/2 18 m2g 2 1 ----STR+ m23/2 6

/ ( m 2 - m 2) 2m~2

2

- 3m3/~ (bIB -

rng 2 m3/2

( m ~ - m 2)

nG) ]

A. Georges, P. Le Doussal / g 2 in broken supergravity

540

TABLE 2 1 Divergent contributions to ~(g - 2) of the diagrams of figs. 1 and 2 in the case of a spontaneous breaking of supersymmetry

Diagram

Order A 2 (coefficient of I 2)

m2l~2 1 (rn2A + m 2 _ 2m 2 _ 8 m ~ / 2 ) - - ~ nlK2 m2/2 18

~2m2

3m2/2 (3 -

la

2 K2m 2 3 -m2/2 (a

l d + le 2a

1)

K2m 2 3m32/2(3-1)

lb+lc

Order In A (coefficient of I 0 )

m2~;2 1

m2/2 l~[(8--63)(m~+rn2B)--4rn2+8m~/2] m2K 2 1

1)

2 (rnB m2A}

2rn~ 2 ~(m~--mA)

mg 2

----[(23--3)(m~+m2)+2rn2]+rn3/z(mB m2/2 6

0 (by power counting)

2

m 2)

0

3 = 0 or 1 corresponds to ~sn and ~sn + ~ff, respectively.

emphasize that this only holds for the sum of diagrams). Of course, when 3 = 1 one recovers the results of the explicitly broken theory for diagrams (la), ( l b ) + (lc), (ld) + (le). In order to make the connection with previous calculations which established the cancellation of the divergent parts for the unbroken theory [3, 4], we also made the same calculation as for ~expl, but with a massless gravitino propagator and with m 3/2 = O, m A = m B = m . Our results for ½(g 2) div are -

(la)"

- - 34 x 2 m 2 I 0 ,

(lb)+(lc)"

-

- ~ x. 2 m 2 1 o ,

(ld)+(le):

2x2m2Io •

(3.4) If we decide to use dimensional regularization we have to set

1 Io -

8q72(4-- d)

(3.5)

(3.4) then confirms the results of ref. [3]. Let us notice that no simple procedure seems to exist in order to recover the unbroken results from the broken ones diagram

by diagram. We now make some remarks and give elements of explanation concerning these cancellations.

A. Georges, P. L e D o u s s a l /

g

541

2 in b r o k e n supergrat, i O'

4. Discussion

Let us first emphasize that, unlike the unbroken case [1], the Pauli counterterm

~o. Fe is not excluded a priori by supersymmetry here. This is obvious for the explicitly broken case. In the case of a spontaneous breaking, one can build a supersymmetric invariant which leads to the desired counterterm after the superHiggs mechanism, namely

f d2Od2OSIL),~S2W'~S~; + h.c.,

(4.1)

where S0,1,2 are the chiral superfields associated with Y0,1,2, and W~ = DDD~V. We thus turn to the study of other symmetry properties which may be useful in understanding these cancellations.

4.1. S Y M M E T R I E S

Some symmetry properties common to the lagrangians previously considered are useful in understanding the form of the divergent parts obtained in our calculation. The following transformation:

(S1)

~p.---+Y5~/~,

B-+ - A ,

A ~ B,

~ ~ -~,s~,

leaves these lagrangians invariant provided one also formally makes the following changes for the masses: m3/2

---+ - - m 3 / 2

,

m A ~

mB,

mR

---+ m A ,

mx

--+ - - r n x

(for the sake of completeness we consider here a non-zero photino mass). This transformation can be considered as the "square root" of the R-parity transformation. In the same way, it has been noticed [4] that the following invariance holds:

($2)

e--+yse,

B~-A,

A~B,

provided m ~

-m,

m . 4 --> m B ,

m B ---+ m A .

The magnetic moment counterterm reads 2(g - 2)(1/2m)~o. Fe, and we notice that ( 1 / m ) 8 o - Fe is invariant under (S1) and ($2). This clearly implies (,g -- 2 ) ( m 3 / 2

, m , m x, m A , m R )

= (g-

2)(

m3/2,

=(g-

2)(m3/2, - m , m x , rn~,m4).

m,

- - m x, r o B , m A )

(4.2)

One can check that the combinations of masses that appear in our results do indeed

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A. Georges, P. Le Doussal / g - 2 in broken supergravi O'

respect these symmetries. Hence, we think, contrary to the claim of ref. [4], that the use of the two symmetries ($1) and ( & ) is not sufficient to explain the observed cancellations of divergences, even in the case of a massless gravitino (unbroken supersymmetry). We now turn to considerations which explain the cancellations of the quadratic divergences and of some of the logarithmic ones. 4.2. T H E R O L E OF GLOBAL S U P E R S Y M M E T R Y

In this subsection, we show that the cancellation of divergent contributions proportional to K2/m]/2 originates from the absence of divergences for g - 2 in spontaneously broken global supersymmetric QED. These contributions arise from the following term in the numerator of the gravitino propagator:

2 k,k~k

(4.3)

m3/2

This term is responsible for all the quadratic divergences (by power counting), and for the logarithmic divergences of the form K2m 2

2

STR,

(4.4)

m3/2

where the supertrace STR denotes mA2 + m~ -- 2m z. We consider here the lagrangians ~cxp~ and £sn + A£, for which the interactions of the gravitino are

- ½i~;f,,JI",

(4.5)

where J~' is the total current of global supersymmetry. It has been remarked [12] thai if one takes the limit m3/2 ~ O, ~ ---,O, keeping K/m3/2 fixed, the gravitino behaves like the goldstino of a spontaneously broken global supersymmetric theory. In this limit, only the helicities_+ ½ of the gravitino survive. They contribute to the propagator precisely by the term (4.3), which is dominant. As far as this term is concerned, the calculations can be made by replacing the field ~, in the interaction lagrangian by

~

O~X

(4.6)

m 3/2 where X is a massless Majorana spinor describing the goldstino. Making this replacement, one gets after an integration by parts f~

~i~/~

I£

m3/2

2a, J'.

(4.7)

A. Georges, P. Le Doussal / g - 2 in broken supergraviO"

543

This is equivalent to the interaction lagrangian of the goldstino in the spontaneously broken global theory. As it is known that no divergence occurs for g - 2 in such a theory, we conclude that the divergent contributions associated with the term (4.3) add up to zero when the sum of all diagrams is performed. Moreover, one can gain information on each diagram by using the above effective form of the interaction lagrangian. Consider for example the part of the total lagrangian, ~ e ~ + A ~ + ~ve~ leading to diagram la only. One gets an effective interaction (the electron being on-shell):

f~eff

K

Y(e(E]-m2)(A*+ ysB~) +h'c.

(4.8)

m3/2

Simple power counting now shows that no quadratic divergence can occur for this diagram. Furthermore the operator [] - m2 is responsible for the appearance of the supertrace in the results of table 1. A similar argument can be given for diagrams lb and ld. 4.3 THE QUESTION OF THE FINITE PARTS In this subsection we make some remarks concerning questions raised in the literature about possible ambiguities which could arise in the calculation of g - 2 in gravity and supergravity theories [3, 4]. As is well known, neither pure gravitation, nor supergravity theories are renormalizable [13,14]. Even at one-loop order, one is not able to define properly the renormalized charge of the electron e R = fZ3e o, since the one-loop counterterms necessary to renormalize the photon propagator do not only involve the term Fu,F ~'~,as shown in refs. [5,13]. As a consequence, in all these theories, no procedure exists to give a physically meaningful value to g - 2, unless a symmetry property can be used to fix it. In pure gravitation and also here (broken supergravity) this is not the case, and we cannot give a meaning to any calculation of finite parts for g - 2. In unbroken supergravity however, the situation is different since supersymmetry compels g - 2 to be zero. This means that, independently of the regularization scheme used, the divergent contributions due to the graviton plus gravitino have to vanish. This is in fact the case in explicit calculations encountered in the literature [3]. If one were able to use renormalization conditions to fix unambiguously the finite parts, one should also find a vanishing result for them. As it is not the case, all one can calculate is the finite part of the regularized vertex function, and of course one will obtain zero only by using a regularization scheme preserving supersymmetry. This is illustrated by the results of ref. [3]. However, there exists a limit where one can give a meaning to the finite parts: when i¢ ~ O, m 3 / 2 ~ O, with •/m3/2 fixed, the exchange of a gravitino is equivalent to the exchange of a goldstino in the global supersymmetric theory, which is renormalizable.

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A. Georges, P. Le Doussal / g

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4.4. THE MASSIVE PHOTINO CASE

In this subsection we consider two different ways of introducing a mass term for the photino. The first one is to add explicitly such a mass term in Eexpl: (4.9)

- ½imjt~t.

This only modifies the results found for diagrams (ld) and (le), and leads to a logarithmic divergence for g - 2 of the form ~2 m x

(4.10)

m2io"

m 3/2 W e thus conclude that the previous mass is i n t r o d u c e d by hand. It is also possible to generate a s p o n t a n e o u s l y broken, as shown in different f r o m (2.3) for the function

f=l+

V~

cancellation does not survive when a photino mass for the p h o t i n o when supersymmetry is refs. [9,10]. This can be achieved by a choice f. U p to order x, we m a y choose

-mx - - Ky0 q- O(K2) . m 3/2

(4.11)

This introduces new couplings relevant to our calculation:

~v~,Ao- •

mh m 3/2

2

F~,~Ao,

~vvso- ~

mx m 3/2

-,,

F.~F B o.

(4.12)

\\

Ao,Bo //

~_~

'//1 e k~r

-Sk °

\ \ \ \ t °'g°

Fig. 3. Additional contributions to Up,of order x2 in the case of a massive photino and of a spontaneous breaking.

A. Georges, P. Le Doussal / g - 2 in broken supergrat,itv

545

As a consequence, two new diagrams involving the hidden sector have to be considered; they are represented in fig. 3. They introduce additional logarithmic divergences. These divergences do have the required form (4.10) which would be needed to compensate those arising from diagrams (ld) and (le). However, according to a preliminary calculation, such a cancellation does not occur. Therefore we find that, although the supersymmetry breaking has been achieved in a spontaneous way, a logarithmic divergence does remain.

5. Conclusion We have studied the one-loop contributions to the anomalous magnetic moment of the electron in spontaneously broken N = 1 supergravity. We have shown that the divergences encountered in this calculation cancel out, as was already the case for an unbroken supersymmetry. Remarkably enough, this occurs separately for the graviton and for the gravitino contributions. What is also remarkable is that the cancellation of divergences survives the breaking of supersymmetry. This reminds us of cancellations occurring in gauge theories and which also survive a spontaneous breaking. (In addition our result happens to hold even in the case of an explicit breaking by gravitino and scalar electron mass terms.) The calculation was performed with a massless photino. When a mass term is introduced by hand for the photino, the cancellation no longer holds. As long as no symmetry property is discovered to explain the observed cancellations, they can appear as "miraculous". We think that one still lacks such a general argument, even when supersymmetry is unbroken. However, we have shown that some of these cancellations can be understood in the broken case, by using known properties of global supersymmetry. We thank E. Cremmer, P. Fayet and J. Iliopoulos for suggesting the subject of this work to us, and for enlightening discussions. We are also indebted to L. Baulieu, M. Bellon and C. Kounnas for useful remarks.

Note added

After the completion of this work, we received a preprint (UAB FT-116) by A. Mendez and F.X. Orteu, who have also studied the case of a massive gravitino (with a massless photino). As no "hidden sector" appears in their work, their calculation corresponds to the one made here in the case of an explicit breaking of supersymmetry.

A. Georges,P. Le Doussal / g

546

2 in broken supergrat'lO'

Appendix

A.1. CONVENTIONS W e work in the metric ( - , + , + , + ) = G~, and use the conventions of ref. [15] for D i r a c algebra. With this choice the propagators are

spin-O field:

-i k2+m 2 ' -i

spin-½ field:

1~ -

spin -3 field:

massless

:

im

P~,, = ~~y ~ k y u / k -+,

in the gauge:

7~'~ = 3"~b, = O,

massive

+ ~ y~-

ik + m

Y"-

,,--7-

"

A.2. DIAGRAMS W e list here the complete expressions for diagrams (la), (lb), (ld), (2a), (2b) adding contributions from A and B for each one (the p r o p a g a t o r of the gravitino is d e n o t e d by P~o): t ~ f d4k eFt,(la)=geK~J~ Y (~-k+iam)

(

x

[

)

(

1

(P~" + Y s P " ° Y s ) ( ( p _ k ) 2 + m 2 A ) ( ( r _ k ) 2 + m ~ )

+ ((v-,)

,

2 + m ~ ) ( ~ r - , ) 2 +m;,)

)

1 +(P~-~,+vo~,+) ((p_k)2+,,~)((r_ k)2 +m~)

547

A. Georges, P. Le Doussal / g - 2 in broken supergral~ity

1 ((p-k)2 4, m2)((r-k)2 4.m 2)

X(p-

eC(lb)= _

1~+ i3m)y°(p + r - 2k )ff

~e~2f d4k

v"

[(11)

X (P~o+ysP~oyS) (r_k)2+m2A

(

1

+ (P~" - YsP~'~Ys) ( r _ k ) 2 + m 2A

eF,(ld) = _

)1

,e~=f

(r_k)2+m ~

1

(r-k)2+m 2

)] Yl'7°'

d4k ~ 4- J~ (2~) 4 ( q + k) 2Y'(~Yu- Y,~')

[(,+1) (,,)]

x (P"°+YsP"°75) ( p _ k ) 2 + m ]

+ (P~°- 7sP~°75)(p_k)2+m2 A

(p-k)2+m~

(p-k)Z+m 2

eF,2~+~2b,=3iex2f( -d4k [_[¢ys(~_[¢4.im)y~_4"im[cT5 ~ k2+m 2 Bo

8m 2 4-

x

1

k2+m 2 Ao

1

(r-k)2+m 2 (p-k)2+rn 2

In these expressions, one has to set 3 = 1 for ~SH + A~ and for ~expl, and 3 = 0 for ~SH"

548

A. Georges, P. Le Doussal / g

2 in broken supergravity

F o r the first three a m p l i t u d e s ( p h y s i c a l sector) we h a v e c o m b i n e d the c o n t r i b u t i o n s of A a n d B i n o r d e r to s e p a r a t e terms even or o d d u n d e r the e x c h a n g e of A a n d B. T h i s i n t r o d u c e s the c o m b i n a t i o n s ½(P,o +_ ysP~o~,5) w h i c h are the parts of the p r o p a g a t o r o f the g r a v i t i n o even or o d d u n d e r a c h a n g e of sign of m 3/2, respectively. T h i s c l e a r l y e x h i b i t s the s y m m e t r y of the l a g r a n g i a n u n d e r ~p" ~ ~,5~p~, A + ~,sB - y s ( A + y s B ) , ?t ~ - y s ) t , as m e n t i o n e d i n subsect. 4.1. T h i s also shows that o d d t e r m s i n rn 3/2 give o n l y l o g a r i t h m i c c o n t r i b u t i o n s to g - 2. A s i m p l e r e m a r k allows u s to d e d u c e the c o n t r i b u t i o n s of d i a g r a m s ( l c ) a n d ( l e ) f r o m those of ( l b ) a n d ( l d ) . U s i n g the p r o p e r t y P ~ = 2+oPo~7,o, one obtains F~(i)(p, r ) = y 0 F y ) + ( r , P ) Y 0 ,

(i, j) = (lc, lb); (le, ld),

w h i c h i m p l i e s t h a t the c o n t r i b u t i o n s of ( i ) a n d ( j ) to g - 2 are equal.

References [1] [2[ [3] [4] [5] [6] [7] [8] [9] [lO] [11] [12l [131 [14] [151

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