Supergravity corrections to the static quantities of the gauge supermultiplet

Nuclear Physics B316 (1989) 225-237 North-Holland, Amsterdam

SUPERGRAVITY CORRECTIONS TO THE STATIC QUANTITIES OF THE GAUGE SUPERMULTIPLET* A. MI~NDEZ and F.X. ORTEU

Departament de Fisica, UAB, 08193 Bellaterra, Barcelona, Spain Received 25 March 1988

In the context of an unbroken N = 1 locally supersymmetric version of the standard model we compute, at the lowest order in the gravitational coupling constant, the contribution of the graviton and the (massless) gravitino to the electromagnetic moments of the spin-1 charged gauge boson and its spin-~ superpartner. All these corrections are shown to be finite although regularization dependent. We verify the expected equality between the anomalous magnetic moments of the gauge boson and the corresponding gaugino. For the electric quadrupole moment of the gauge boson we do not find the expected vanishing result.

1. Introduction

The problem of the static quantities in supersymmetry has been considered since the early days of these theories. Ferrara and Remiddi [1] showed that the anomalous magnetic moment of a spin-½ matter field must vanish in any unbroken supersymmetric model and they confirmed this result at one-loop level by explicit calculation in the case of global supersymmetry. This result has also been extended to the local case in refs. [2, 3] where it has been shown that the cancellation between the graviton and gravitino contributions takes place if the divergent integrals are regularized by using the dimensional reduction method [4], but not by using the standard dimensional regularization [5]. The ( g - 2) of the electron has also been studied in the framework of a spontaneously broken minimal and non-minimal supergravity (SUGRA) theories [6]. The analysis, in a supersymmetric context, of the static quantities of the particles belonging to a gauge supermultiplet has been carried out by Robinett [7] and by Bilchak et al. [8]. These authors conclude that the term that generates the quadrupole moment (AQ) of a spin-1 boson is forbidden by supersymmetry and consequently ,AQ must be zero if supersymmetry is unbroken. They also show that the anomalous magnetic moment of the spin-~ partner of the W gauge boson need not vanish * Work partially supported by research project CICYT. 0550-3213/89/$03.50@Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

226

A. Mdndez, F.)~ Orteu / Gauge supermultiplet

(contrarily to what happens in the case of spin-½ particles of a chiral supermultiplet), but it must be equal to the anomalous magnetic moment of the gauge boson. The authors of ref. [8] checked these results, at the one-loop level, in a version of the standard model with exact global supersymmetry, using the standard dimensional regularization method that, in this case, gives the same result as that which employs dimensional reduction. In this paper we study the anomalous magnetic moment and the quadrupole m o m e n t of the W gauge boson as well as the (g - 2) of the associated gaugino, 05, in the framework of an unbroken minimal ( N = 1) S U G R A - s t a n d a r d model. To our knowledge, the only previous calculation of static quantities of a spin-1 particle at order x 2 ( ~ 2 = 8~rGN, where G N is the Newton's gravitational constant) was performed in the context of quantum gravity (i.e., only the graviton contribution) by van Proeyen [9] who used the standard dimensional regularization. We have repeated this calculation in dimensional reduction and have also evaluated the contribution of the massless gravitino in both schemes. We obtain, as expected, the same result for the supergravity correction to the anomalous magnetic moment of the W boson and the graviton plus gravitino contributions to 51( g - 2),~ when the dimensional reduction method is used. Surprisingly, we find that the quadrupole m o m e n t of the W does not vanish in either regularization method. The outline of the paper is as follows. In sect. 2 we specify the model and the F e y n m a n rules needed in our calculation. The diagrams that contribute to the static quantities of the W boson are shown in fig. 5 (see below) and the formulae and results are given in sect. 3. In sect. 4 we compute the correction to the anomalous magnetic m o m e n t of the gaugino. The corresponding diagrams are given in fig. 6 (see below). Finally, we devote sect. 5 to the conclusions. We shall work with four-component spinors. Our Dirac matrices satisfy the anticommutation relation {Tu, 7,} = 2,1~ where 7/,~ = ( + - - - ) is the flat spacetime metric. Our convention for 75 is such that ~ ( 1 - 75) is the left-handed projector.

2. The model and Feynman rules The most general lagrangian for N = 1 supergravity coupled to matter is given in ref. [10]. We shall restrict ourselves to the case of minimal coupling defined as ~ , , i = _ ~Sj,

f~B = 6~a.

(1)

In the standard SU(2)L X U(1)v model minimally coupled to the S U G R A multiplet we have the vector-supermultiplets corresponding to the gauge sector as well as the chiral superfields that contain the Higgs and matter fields. However, as we are only interested in gravitational corrections to the static quantities of the gauge particles,

A. M~ndez, F.X. Orteu / Gaugesupermultiplet

227

we can ignore the lepton and quark sectors which are irrelevant to our calculation. Thus, the field content of our model will be: (a) gauge sector SU(2)L:

(V;,~.'~),

U(1)y :

(V~',),') ;

(a=1,2,3),

(b) Higgs sector

( Hli' I~1i) Y=

1/2'

( H2i, It2i)Y=l/2,

(i = 1 , 2 ) ;

(c) gravity sector

(h,*", 4,,*), where the second component of each multiplet is the fermionic one. As usual, we have considered a Higgs sector consisting of two doublets (the subscript i labels the components of each doublet). In order not to break supersymmetry the F-terms must not acquire a vacuum expectation value (v.e.v.) and, therefore, we will assume the superpotential and the function accordingly defined. (In fact, as we are not worried about the lepton and quark sectors, we could assume a vanishing superpotential.) On the other hand, as we want to break SU(2)L X U(1)y to U(1)o, at least one of the neutral components of the Higgs doublets must acquire a v.e.v. This would induce a spontaneous supersymmetry breaking through the D-terms unless we impose the constraint that both components have the same v.e.v.

= =

(2)

With these requirements one can easily check that there are an infinite number of minima where the potential vanishes (i.e., zero cosmological constant). Thus, supersymmetry is exact and the gauge symmetry is spontaneously broken. The discussion of the mass eigenstates is given in detail in refs. [8,11] and we will only quote the expressions that are useful for obtaining our Feynman rules. In particular, the charged Higgs scalars are defined as

-

-

H 2 1 + H~2* ,

(3a)

and H = ( H + ) *.

(3b)

228

A. Mdndez, F.X. Orteu / Gauge supermultiplet

It ie[ go,~(p + p ' ) . - (gul~P~ + g.~Pl~) + (g.~qlJ

gl~.q.)]

ieQco,Yu. 1

1

,~'g'g ieQH - (p + P').

H-,,"P P'"• H-

+_~ie(1 -

~1(2)L

"/5)7~,

W.l.t - ~ieQco? (1 -

o~

~'5)

~• H-

Fig. 1. Non-gravitational

Feynman rules.

On the other hand, the higgsinos mix with the gauginos via the non-diagonal terms of the mass matrix (see ref. [10]) ig¢2 ( n * J ) ( Ta) )~aIqLi Jr h.c.

(4)

After diagonalizing one obtains the mass eigenstates

°51 = (/4~)c + (iX-)R,

(5a)

052- = ( / 4 ~ ) c + (i)t+)R ,

(5b)

which are the fermionic partners of the (massive) W ± bosons. With the standard definitions of the W -+ bosons and the photon, 1', in terms of V" and V' we obtain the part of the spectrum of our model that we are interested in* (a) gauge-Higgs sector

(W-,05i-,H-) with m = m w , W 05+ H +) with m = m w (~, "~) with r n = 0 ;

* There

are also the Z ° and the neutral Higgs supermultiplets which will not appear in the

calculation.

229

A. Mdndez, F.X. Orteu / Gauge supermultiplet

(b) gravity sector (G,G)

with

m=0.

The Feynman rules that we will use in the next sections (see figs. 1, 2, 3) can involve the interaction of one graviton, G, one gravitino, G, or be non-gravitational. The latter are the ordinary gauge couplings of the photon to a particle of spin 0, or 1 and their supersymmetric versions. They are shown in fig. 1 (a factor i has been absorbed in the photino field). The graviton interactions can be obtained by expanding at order x the metric tensor g,, = ~/~ + 2xh,,,

(6)

ix2( g~,~q,~q~ + g~,~g#~q-q' - g#~q,~q~ - g/#,Gq~

,f4t~ f l

¢,v

+~g~/~(Gq~

g,~.q'q')}

i~2[ g~,,,q,~q~ + go,t~g#,,q" q' - gfi,,q,~q~ - gfi.q,,q;

WT~t~ a ~ W T v

+~g,~fl(Gq~-gt,~q • q') +±m2 ~ w ~zg ~ g . ~

2gjB~)]

-~ixy,~(p+p')#

w T ~ ' p p" " WTv

+ (g.,oga~q~, - go,~g~,aq~) - g~,.gp/~ (P +P'),~ + (g..g,,#P~, + g~og.BP;) + (g,~fig~,q,,, - g,,.og,'fiq.)]

~1 ~

~

~11

- ie~Q,~? g~yfi

Fig. 2. Feynman rules corresponding to the graviton (G) interactions.

230

A. Mdndez, F.X. Orteu / Gauge supermultiplet 7,~t

7

g,~ g,~

-¼i~(1 Ys)[(~ row)%- q.lY. ~iZ

W~

J?

Orl,;~

~(2)#

~,v"

¼i~O - v s ) ( ~ + ,,,w) v.

p~H -

++_}ie~(1 -

Y5)[7.,%]Y,~

¼iegQ11

(1 - ys)'f~,y~,

c W.v

", H-

Fig. 3. Feynmanrulescorrespondingto the gravitino(G) interactions. and its determinant in the kinetic terms of the photon, the W boson and the gaugino 05. After a straightforward calculation we arrive at the Feynman rules given in fig. 2. The couplings involving the gravitino must be found by direct inspection of the lagrangian of ref. [10]. When the kinetic terms of the lagrangian are canonically normalized, the interaction terms that are relevant to our calculation are

~e,~v~,o= ~/~x°v"[Y ', v'] F;;¢, + h.c.,

(7a)

o,~tpI~iH(V ) = ~/~-~(--K)¢tz~Hi*~l~I~iL

Jr" h.c.,

(7b)

T )i Hfl~L + h.c.,

(7c)

=

where V is any of the vector bosons defined at the beginning of this section, D is the gauge-covariant derivative involving the gauge vector fields and ~ is the coupling constant of the SU(2) group. Bearing in mind eq. (5), we obtain from the eqs. (7) the Feynman rules shown in fig. 3 (a factor i has also been absorbed in the ~p~ field).

A. Mdndez, F.X. Orteu / Gaugesupermultiplet

231

After a suitable choice of the gauge-fixing terms for the graviton, gravitino (see, e.g. ref. [12]) and gauge fields, the propagators are given by _gl,. D r~ -- i k 2 + i c '

D~ = i

D? = i k 2 ~+iE ,

-g"~ + kUk"/rn2w k 2 - m 2 + iE ,

(8a, b)

Dco = i k 2

g + mw m2w+i e,

1

(8c, d) (Se)

D n =ik2_m2+i¢, 1 { D~,,B=i2(k2+ie) g,~g,B+g,Bg~

2 2 g~,~g.¢ }

(8f)

n-

1

D ~ f = i ( n _ 2)(k 2 + ie) {Y~Y~+ ( 4 - n)[g"~g- 2 k " k ~ / k 2 ] ) .

(8g)

As shown by the form of the W propagator (8c), the calculations [13] have been performed in the SU(2)L × U(1)y unitary gauge. The divergent loop integrals have been regularized, as explained before, with the standard dimensional regularization and dimensional reduction methods. In the latter, since Lorentz indices of fields are kept four-dimensional, we must set n = 4 in eqs. (Sf) and (8g). Our calculations have been done with the programme REDUCE.

3. Static quantities of the W boson

As it is well known, the W boson may only have, besides the charge, an anomalous magnetic moment, K, and an electric quadrupole moment, AQ. In order to study the one loop SUGRA correction to these quantities, one must calculate the value, at Q 2 = 0, of the order ff2 contribution to the coefficients of the following expression

+2(K-

1)(Qdlt~ . - Q ~ , ~ ) + 4

P,Q~QB '

(9)

which is the most general electromagnetic vertex of the W boson that is CP-

232

A. M~ndez, F.X. Orteu / Gauge supermultiplet

A~Nc~ Fig. 4. Notations for momenta and polarization indices of eq. (9).

w-s'

'-'

'--tw~

(a)

(b)

(c)

(d)

(e)

d2f

w-F.~v.~hwF (f)

'z w-F w- "kw-¢ w -~. w-

~w~

.~zwF wi ~w F ~,-Sw-? ~, hw(g)

(h)

(i)

(j)

Fig. 5. Diagrams contributing to AQ and K at order j¢2.

invariant and has all particles on the mass shell [14]. The definitions of the m o m e n t a and polarization indices of eq. (9) are given in fig. 4. I n fig. 5 we show the F e y n m a n graphs that contribute to AQ and K. The first five diagrams, 5 a - 5 e , correspond to the graviton contribution and they were already c o m p u t e d in ref. [9] by using dimensional regularization. The other five diagrams, 5f-5j, involve the exchange of a gravitino. It is easy to convince oneself that there are no ghost contributions to this calculation (the graviton and gravitino gaugefixing terms are SU(2)L × U ( 1 ) v invariant). T h e numerical results for the correction to AQ are given in table 1, where the contribution of each diagram in both regularization methods is explicitly shown. We see that the graviton and gravitino contributions are separately finite* although the numerical value is regularization dependent in the case of the graviton. * The divergences written in square brackets arise from infrared divergent integrals. As we can see, there is a cancellation between infrared and ultraviolet divergences. These cancellations occur in dimensional regularization when loops of massless particles are involved [9].

A. Mdndez, F.X. Orteu / Gauge supermultiplet TABLE 1 Contribution to AQ in units of

Diagram

a

d+e

Dimensional regularization

14

14

1

26

f

1

3

3 n-4 28 1

+!~

3 n-4

3 n-4

graviton (a+b+c+d+e)

(GNmZw/~r)

Dimensional reduction

3 n-4 28 1 3 n-4

b+c

233

-~

3 n-4

-

_ as 9

82 9

]o

g+h i+j gravitino (f+g+h+i+j) Total (graviton + gravitino)

+u~

9

q

0 2 z8 9

0 2

- 3

- 6

TABLE 2

Contribution to AK in units of

Diagram

Dimensional reduction 5

a b+c d+e

1

gravitino (f+g+h+i+j) Total (graviton + gravitino)

18 ~3

1

1

3 n-4 35 1 3 n-4 1 -10--+~ n-4

10

3 n-4 2 1

i+j

is

3 10

g+b

Dimensional regularization 5

3 n-4 35 1 3 n-4 1 10--+ n-4

graviton (a+b+c+d+e)

(GNrn~v/Tr)

3 n-4 8 1 3 n-4 1

1

9

3 n-4 2 1

+~

3 n-4 8 1 -----+~ 3 n-4

~-~

__1

-T 0

9

3 3

+~

9

A. M~ndez, F.X. Orteu / Gauge supermultiplet

234

According to the conclusions of ref. [8], we would expect the total contributions to AQ to be zero in dimensional reduction since supersymmetry is unbroken and we use a regularization method that "in principle" respects this symmetry. However, the finite contributions coming from the graviton and gravitino diagrams do not cancel each other in either method. As a check we have calculated the graviton contribution in the 't H o o f t - F e y n m a n gauge obtaining the same result. We also reproduce the results of ref. [9]. We now turn to the anomalous magnetic moment of the W boson. In table 2 we show the numerical values of the contributions to A K -= K - 1. In this case there are no infrared divergences and the results are once more separately finite. The graviton contribution is, again, regularization dependent. Using dimensional reduction we find the result AK = 0. This is not a prediction of supersymmetry. Rather, we expect A K to be equal to the anomalous magnetic m o m e n t of the 03 which will be calculated in sect. 4.

4. The anomalous magnetic moment of the As explained in sect. 2, the supersymmetric partners of the W ÷ bosons are the winos ~1 and o~-. Since they are spin-½ fermions, they can only have (apart from

(a)

(b)

(c)

(d)

(e)

P,j

°


(g)

~T

=-lt

, T

~"N = - f (k)

N=-lt \=-~-=--N~i-~:N < (h)

..

,T

X =~t (I)

(i)

(j)

T

Y

x<°,P~=-,/~ (m)

(n)

Fig. 6. Diagrams contributing to (g - 2); r at order K2.

(o)

A. Mdndez, F.X. Orteu / Gaugesupermultiplet the charge) an anomalous

magnetic moment

235

w h i c h , a c c o r d i n g t o refs. [7, 8], m u s t

verify

= aKw, where d=

(10)

!(2g_ 2). I n t h i s s e c t i o n w e c o m p u t e .~¢~;

to compare

its n u m e r i c a l

v a l u e w i t h t h e r e s u l t o f A K o b t a i n e d i n sect. 3. A s it is w e l l k n o w n , i n o r d e r to e v a l u a t e z ¢ o n e m u s t f i n d t h e v a l u e at q2 = 0 o f the coefficient F 2 of

A~(q 2)

iF2(q 2)

=Fa(q2)7"+

-

2mw

o ~ q~,

(11)

w h i c h is t h e m o s t g e n e r a l e l e c t r o m a g n e t i c v e r t e x of a s p i n - ~ p a r t i c l e . T h e F e y n m a n

TABLE 3 Contribution to ( g - 2),~t. in units of (GNmw/'rr)

Diagram

a b+c d+e graviton (a+b+c+d+e) f

Dimensional reduction

Dimensional regularization

1 1 3n 4 11 1

1 1 3n-4 11 1

1~

~ 3 n-4 1 4 +6 n 4 ± 2 2 1 37 72 3 n--4 1

g+h

29

1

_ _ _ _

!

3 n-4

9

3 n-4 1 -4 n-4 v 4 2 1

gravitino - H (f+g+h+i+j) k l+m n+o gravitino-W (k+l+m+n+o) Total (graviton + gravitino)

1

+7

55

3 n--4 1

1

13

3 n-4

1

i+j

61

~6

36

1

n

4

+~

-1

n- 4

+1

1 -~ 3 1 2 n-4 1 1 6 n-4 4 1 3 n-4

~ 4

3

-~ 0

e~4

_1 3 1 2 n 4 +~ 1 1 6 n-4 4 1 _ - 3 n 4 1

8

4

9

236

A. M6ndez, F.X. Orteu / Gauge superrnultiplet

graphs that contribute, at order K2, to this expression are shown in fig. 6. We see that the diagrams of the first and third rows (6a-6e and 6k-6o) are analogous to those that contribute to the anomalous magnetic moment of the fermionic component of a chiral supermultiplet (see ref. [2]). However, we have now five extra diagrams (6f-6j) involving the gravitino and the third member of the (massive) gauge boson supermultiplet, i.e., the charged scalar Higgs particle. In table 3 we show the numerical results for Jack1, using both regularization methods. As we can see, the divergences cancel in each group of five diagrams and the finite results depend again on the regularization method*. On the other hand, once all the partial contributions are added we obtain zero in the dimensional reduction column. Thus, we have verified the relation (10) in the context of unbroken supergravity when dimensional reduction is used.

5. Conclusions In sects. 2, 3 and 4 we have obtained the supergravity corrections (at order K2) to the static quantities of the particles of a gauge supermultiplet, in the context of an unbroken locally supersymmetric version of the standard model. We have shown that the one-loop contribution of the graviton and the massless gravitino to the electromagnetic moments of the W boson and its supersymmetric partner ~ are separately finite although regularization dependent. In particular, we have checked the equality between the anomalous magnetic moments of the W and the o5 as predicted by supersymmetry [7, 8]. This equality holds only if the divergent integrals are regularized using dimensional reduction, as happens with the vanishing of the anomalous magnetic moment of a spin-½ member of a chiral supermultiplet [2]. Besides being equal, AK and ( g - 2)~ are both found to be zero. This zero results appear to be accidental since they are not predicted by supersymmetry (for instance, they are not zero in the globally supersymmetric standard model [8]). On the other hand, we do not obtain the expected vanishing result for the electric quadrupole moment of the W in neither of both regularization methods. We have checked this result by repeating the calculation in the 't H o o f t - F e y n m a n gauge. This made us confident about the most divergent part. There are other reasons to trust our results for zlQ. First, the fact that we obtain a finite result. Concerning the graviton correction, our algebraic manipulation programs can be easily adapted to each regularization method and we reproduce the result of ref. [9]. On the other hand, we get the same gravitino contribution in both regularization methods, as happens in the cases of a K and the electron anomalous magnetic m o m e n t [2, 3]. Moreover, the value of AK, which is obtained from the * More specifically,the contribution of the graviton is exactly the same as in the case of the (g - 2) of the spin- 21member of a chiral supermultiplet, and the contribution of the second group of diagrams (6f-6j) is one fourth of the gravitino correction to the same quantity [2].

A. M~ndez, F.X. Orteu / Gauge supermultiplet

237

s a m e set o f d i a g r a m s , is f o u n d to v e r i f y the e q u a l i t y A K = ( g - 2),~. H o w e v e r , an i n d e p e n d e n t c h e c k of o u r results w o u l d b e v e r y i l l u m i n a t i n g . W e t h a n k F. del A g u i l a , L.E. Ibafiez, M. Quir6s, R. T a r r a c h a n d A. T r i a s for d i s c u s s i o n s o n d i f f e r e n t aspects o f this work.

References [1] [2] [3] [4] [5]

[6]

[7] [8] [9] [10] [11] [12] [13] [14]

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