Superpropagators for explicitly broken supersymmetric theories

Volume 135, number 1,2,3

PHYSICS LETTERS

2 February 1984

SUPERPROPAGATORS FOR EXPLICITLY BROKEN SUPERSYMMETRIC THEORIES J.A. HELAYEL-NETO International School for Advanced Studies, Trieste, Italy Received 18 August 1983 By considering cases of explicit softly and hardly broken global supersymmetry,superfield propagators are written down which correctly accommodate the ordinary propagators of a component-field formalism. As an application, tadpole supergraphs are calculated.

The issue of explicit breaking of globally supersymmetric theories has systematically been analysed by Girardello and Grisaru [1], who have classified the possible breakings in two main categories: soft and hard, according to whether or not they induce quadratic divergences which are not present, as is well known [2], in models exhibiting exact supersymmetry. By treating the problem in the framework of a superfield formalism, these authors gave an exhaustive list of the possible soft breakings with the new logarithmic divergences they generate. The study of models with explicit softly broken global supersymmetry is well justified by the fact that they bypass the rigid constraints imposed by the mass formula 2;j (-1)2J(2J + 1)m2 = 0 [3] in the construction of realistic models of fundamental interactions without, however, the risk of spoiling one of the nicest features of exact supersymmetric theories, namely, the absence of ultraviolet quadratic divergences. This means that models based on explicit softly broken supersymmetry can be built up which are of some interest for phenomenological applications [4,5]. More recently, supersymmetric grand unified models coupled to N = 1 supergravity, with interesting low-energy phenomenological consequences, have been proposed and discussed [6-10]. The nonrenormalizability o f N = 1 supergravity however constitutes a serious problem if one wishes to go beyond the tree level and to compute quantum loop corrections. Nevertheless, by using the fact that the grand-unification mass scale, MGUT, and the gravitino mass, m3/2, are both small compared with the Planck mass,Mp, one can formally circumvent this problem by taking the limit Mp ~ ,,o with MGUT and m3/2 fixed. As shown by Barbieri et al. [ 11 ], the supergravity couplings induce, after this limit has been taken, terms which softly break supersymmetry, so that at an intermediate scale physics turns out to be described by an effective theory with explicit softly broken global supersymmetry [12] for which perturbation theory is well-defined and the whole machinery of a perturbative programme can be applied. Loop computations in the framework of supersymmetric theories can much more compactly be performed by means of supergraph methods [ 13-15 ]. Specially in the improved version due to Grisaru et al. [ 15], supergraph techniques became rather simple, as one makes full use of supersymmetry in all stages of the calculations. These methods have remarkably allowed a rather quick derivation of the three-loop ~3-function for the Wess-Zumino model [ 16] and the vanishing three-loop/3-function for the N = 4 super Yang-Mills theory [ 17,18 ]. In dealing with explicitly broken supersymmetric theories totally written in terms of superfields, one usually makes use of the Feynman rules derived for the case in which supersymmetry is exact and treat the breaking terms as spurion vertex insertions into the superpropagators of the exact model. It is the main purpose of this letter to show, through some concrete examples, that superfield propagators can be written down even if supersymmetry is explicitly broken and that they correctly accommodate the ordinary propagators of a component-field formalism. They are however more complicated than those corresponding to the exact case, as they exhibit a rather non-trivial structure in terms of the fermionic coordinates and covariant derivatives. 78

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Volume 135, number 1,2,3

PHYSICS LETTERS

2 February 1984

We shall consider here only those breakings (soft and hard) quadratic in the superfields, which therefore correspond to mass terms for component fields. Propagators shall then be written down which account for the effect of the breaking terms. In other words: by working with these propagators, it will not be necessary to worry about the insertions of spurion superfields into the various propagators appearing in a given supergraph. With them and the Feynman rules derived for the true vertices of the theory, one has all that is required to compute supergraphs % la Grisaru-Ro~ek and Siegel". Even though they lead to more involved algebraic manipulations in the course of a supergraph evaluation, their utilisation is of some advantage if one is concerned with an exact evaluation of finite parts extracted from radiative corrections calculated in the context of a broken model. Another possibility is that our superfield propagators be of some interest for the computation of effective potentials through the tadpole supergraph method recently treated by Miller [19], for which the knowledge of the modified propagators in the case supersymmetry is broken becomes relevant. Before proceeding to the analysis of the different cases we shall be considering, a short remark concerning technical points is worthwhile: we shall here adopt the same conventions as those given in ref. [15] for doing the spinor algebra and for working with the supersymmetry covariant derivatives. We can now start presenting and discussing our results.

Case I £=fd40~,-l(fd2OM,I,2+h.c.)-fd4Om20202~o~,

(1)

i

where • (x; 0, tg) = exp(0o0 • i0)(z + 0 4 + 02h),

(2)

z - (.4 + ~ ) / x / - Y ,

(3)

h -- ( F -

iG)lv~,

A, B and 'II being respectively the scalar, pseudo-scalar and spin-1/2 physical components of the chiral superfield, whereas F and G are its scalar and pseudo-scalar auxiliary components. The superfield propagators stemming from this lagrangian, already written in momentum space, are given by: _7~q~_= 1 ~ 1 1 p 2 p2+M 2 } q~ ~ 1 p 2

m2 !n-2ta2ti2n2 M2m2 1 ~2 2 2 - 2 ~ 2 2'~84(012), 16"'1Vl V l ~ ' l 172D1DlOlO1D1DlO ! p2+M 2+m 2 p 4 ( p 2 + M 2 + m 2)

M 1 2{ p 2 ( p 2 + M 2 ) aD1 1

m2 1 a2t~2~2n2 p2 + M 2 + m 2 a6 "1"11-'1"--'1

m2 ) .,lVP, 1,-,1 84(012 ) . p2 + M 2 + m 2 ~ •n-2a2£2n2

(4)

Upon use of the equalities D 2 [ 0 2 0 2 D 7 8 4 ( 0 1 2 ) ] : 160?02 '

D 2 [0202D~D284(O12)] = - 6 4 0 ~ exp(02aO 2 • p ) , 2 ~ 21 [OlO1D18 2 - 2 2 4 (012)] = _ 6 4 0 2 e x p ( _ 0 1 o b l D1D

• p) ,

D12D~I[0101D1D1 2-2 2 - 2 64 (012)] = 256 exp [-(01001 + 02o02) - p] ,

(5)

we can read from the superpropagators (4) the following component-field propagators:

(T(z*z)) =

1

p2 + M 2 + m 2 '

2a~5 • p ,
(T(zh)) =

(T('I,~'I'O)) -

M

p2 + M 2 + m 2 '

(T(h*h)) =

p2 + m 2 p2 + M 2 + m 2 '

2M p2 + M 2

(6)

in agreement with a direct component-field calculation from the lagrangian (1). 79

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CaseH £ = f d 4 0 g,~ - ~

d20 M ~ 2 +

+ h.c..

(7)

In this case, our superfield propagators read 1

(T(U~qb))=p2 +M------~54(012)

/22

_

(p2 +M2)2 _/24

M2/22 2 -2 p2 +M 2 0101

(M02 +M~2

/22 ! ~ 2 4 2 4 2 n 2 ) 54(012) p2 +M 2 16"1Vl~'l*"l (T(dp~)) =

_

/1 n2t72 M2 ln202 M ~D284(0p ) + /22 p2(p2 +M 2) ~ (p2 +M2)2 _ /24 ~~ u l u l - - 7 4*~1v1

M/22 1a2gn2a2 p2 +M ~ 4~1~*"1~1

+02D2))84(012)

(8)

In trying to make contact with component fields, the relations below may be helpful: D12 [02D2184(012)] = 16022 exp(-01o01 • p ) ,

D2D] [02D284(012)] = -64

exp [-(01 o61 - 02o02) • p] .

(9)

Indeed, by using them, we can check that the following component-field propagators are correctly included in expressions (8):

(T(z*z))-

p2 +M 2 (p2 + M2)2 _/24 '

(T(hh)) = -

(T(zz)) -

M2/22 (p2 +M2)2 _/24 '

/22

(T(h*h))=

M/22

(T(zh)) = M(p2

(p2 +M2)2 _/24 '

(T(z*h)) =

p2(p2 +M2)_/24 (p2 +M2)2 _/24 ' +M2) (p2 +M2)2 _/24

(p2 +M2)2 _/24'

(10)

CaseIII. In this case, we consider chiral superfields without a supersymmetric mass term, but with soft and hard breakings so as all the physical component fields have different masses: £= Sd40 ~cb- f d40m20202~-(l f d20/2202~2 -1Sd40/2'O202(Dadp)(Do/b)+ h.c. ).

(11)

The superfield propagators have been found to be: (T(U~cb)) =

~2 (

1-

m2p2 +(m4 --/24) (p2 + m2)2

/22 (T(q)q~)) = (p2 + m2)2 _/24

1 ~-24242n2 /24 T-6L'lUlUlUl

u '2 ~ p2(p2 +/1,2)

/2' +//,2)

-

I02D284(012) +p2(p2

1

2--&

2-22184(012 ) 1 P~a-2 D1DlcfllO1DI&D1

2-2--

2

4

-g2D1DIOIO1DI&D18 (012)"

(12)

The equalities

£)2Dl0~[0202D1~284(012)] = -640

l~02t~(1 + 01o01 • p) exp(-02o02 • p),

O2/~l& [012-2 ~-2 4 (012)] = 6401&02#(1 01DIt3D18

80

OlaO1 P) exp(-02a02

"P)

(13)

Volume 135, number 1,2,3

PHYSICS LETTERS

2 February 1984

may be helpful in showing that the propagators (T(z*z)) =

(T(CI'kqt~))

p2 + m 2 (p2 + m2)2 _/24 ' _

2p~ p2 + tt,2 '

( T(zz)> = -

(T(~aqtt~)) _

/.t2 (p2 + m2)2 _/24 '

2ta' p2 + tt,2 eat3 '

(14)

of a component-field analysis can also be extracted from the superfield propagators (12). Case IV. Our last case is of relevance for supersymmetric gauge models where a soft-breaking term is introduced which gives mass to the gauge-fermion field. The lagrangian in such a case is £ = £gauge + £ gauge-fixing + £mass ,

(15)

where _ _ _ 1 tr £gauge - 64g 2

fd20 IV'~IV,~, £gauge-fixing = --16~1 tr f d 4 0

£mass =

fd20 mO2WC~We, + h.c.

1 tr 64g2

(DZV)(D2V)

(16,17)

(18)

with V(X; 0, 0) = p + 0X + 0X + OZM+ 02M * + 0o{J" v + 020~ + 02{j~ + 0202D ,

(19)

W~ =.~2 [ e x p ( _ g V ) D c, exp(gV)] .

(20)

By working out the quadratic pieces in V of £gauge and £mass, one gets £(2) = tr f d 4 0 ( - ½ ) V[-}DC~D2Da + a-1~6 (D2/) 2 +D~2D 2) - (81-mOC~02OZDc~+ b.c.)] V.

(21)

This then leads to the following vector-superfield propagator (T(VV)) =

_1 m 1 a--2 2 p2 84(012) + p2(p2 + m 2) g(DID10lDl°e + h.c.) 84(012 )

m2 1 c~-2 2-2 2 - " -~ 2 2 - 2 - 2 c~ • 4 ~ ( D 1 D t OIO1D1D ~ +D1DIOIO1DtD1 )Po¢ 6 (012) p 4 ( p 2 + m 2)

(22)

and it can be verified that the following component-field propagators (in Feynman gauge, c~ = 1) are nicely accomodated into the expression (22): (T(oD))-

1 p2 '

(T(xaX#)) =

(T(M*M)) = - 1 p2 '

2m p 2 ( p 2 + m 2) e ~ ,

( T ( X c ~ ) ) - p2(p2m + rn 2) P ~ . '

(T(OaOb)) = 2rlab p2 '

(T(~&X~)) =

m2 (T(~'&Xt3)) = pZ(p2 + m 2) 2P~1" '

1 )ea~ (T(x~Xt3)) = ( 7 1 +p2 - -+ m 2 "

2m 2 p4(p2 + m 2 ) p~&'

m 1 (T(~'~XI3)) - p2 + m 2 2ec~ '

(23)

81

Volume 135, number 1,2,3

PHYSICS LETTERS

2 February 1984

Before turning to the applications of our results to loop computations, it is perhaps worthwhile to mention the procedure followed for the evaluation of our superpropagators. In all cases, we could do it in two different ways. The first one consisted in the standard procedure of coupling the superfields to chiral (in cases I, II and III) and vector (in case IV) external sources. Our problem was then reduced to the task of inverting the following operators: MD2 /40

(1 - mQAP)

(1 - m202J2)

MD2/40

(

(M + /_A?)@/40 1’

1

(1

(A4 t /.&2)D2/40

1’

p2$fi2/40 t ip,‘D&e2J2DA (1 - m28282) (1 - m202i2) (iD”02D,

(24)

+ &L~D~Y~WD 01 ’

&Po2/40

t b2D2) - kmDoLt12D2D, - ~mL%2D2fiG),

t a-1&(D2a2

(25)

for the cases I, II, III and IV, respectively. Their explicit answers will not be quoted here. In the second way, we just treated the soft-breaking terms as vertices, inserted them in all possible ways into the propagators of the exact case and summed up the whole series of contributions. Both methods coincided and, moreover, our answers can be supported by a component-field analysis. Finally, as a quick application of our results, let us quote the contribution of the tadpole graph of fig. 1 to the effective action of cases I, II and III, where we adjoin a supersymmetric coupling term -(X/3!) J d20 @3 + h.c. With the help of the Feynman rules of ref. [15] and our superfield propagators, one can readily evaluate the contribution the the effective action of the supergraph of fig. 1: Case I: Mm2 (27~)~ (k2 tM2)(k2

t M2 t m2)

s

d4e e282a(p

= 0; 8, S) ,

(26)

corresponding to a logarithmically divergent linear term in A, in agreement with the results of ref. [I 1. Upon extraction of the pole part, which is done accordingly to the dimensional regularisation scheme of ‘t Hooft and Veltman [20], we obtain that this supergraph contributed the following finite correction: _ (4n)-2h(Mm2)

(In(M2 t m2)

t$

In MT)

ld4e

e2i2@

(27)

Case II: h d4k P2 d40 1-X ( k2 tM2 z s (27r)4 (k2 t M2)2 - p4 J

e2

so that linear terms in F and A are induced

are respectively

all finite correction

8, f-7), logarithmically

(28) divergent

is given by

Fig. 1

82

which

P+(o;

Fig. 2.

[l] and finite.

The over

Volume 135, number 1,2,3

_

_

PHYSICS LETTERS

+ i/2 +//2 ln(M4 (471)-2 ~ (M 2 lnM22 __//~

2 February 1984

fd40 62,

//4))

fd40 0262,.

(29)

021 62~(0; O' 6)

(30)

- (47r)-2-~(M//21nM~I2+//~2 +M31n M 4M4__//4~] Case III: ?t ( d4k -2 3(271)4

f d40I

//2

//'

(k 2 + m2)2 _//4

k 2 +//'2

as already expected from the analysis of ref. [1]. Its finite correction to the effective action reads -

m2+//2 (47r)-2 4~___( m 2 In m 2 _//~ +//2 ln(m 4 _//4) ) f d 4 0 ~2~i,

_ (47r)-2 ~ # ' 3 In//'2

fd40 0202qb .

(31)

Finally, just to illustrate a calculation involving the superpropagator worked out in case IV, let us couple massive chiral superfields to the vector supermultiplets of the lagrangian (15) and then compute the contribution of the supergraph of fig. 2, for example, to the effective action. Its evaluation yields the expression

_)tg2(Mm)(

d4k d41 1 1 a(271)4 (27r)4 (k 2 +M2)2 12(12 + m 2)

fd40(02+O2 -mO202)rb(O;O,O),

(32)

where we can read logarithmically divergent terms in A and F [ 1] and the following finite correction

- (471)-4~72(Mm)(lnm 2 - 1) lnM 2 f d40(0 2 + 62 - m0202) cI' •

(33)

In summary: we have seen that, for lagrangians containing terms which explicitly break supersymmetry, superfield propagators can be written down which correctly accommodate the'ordinary propagator~ of a qomponent-field formulation. By using them, one can quickly calculate exact expressions for tadpole supergraphs which, according to the results of ref. [19], are fundamental for the obtention of effective potentials in the framework of broken supersymmetric theories. Thanks are due to Dr. M.A. Namazie, Professor J. Strathdee and Professor L. Girardello for many patient discussions and several crucial remarks in the course of these calculations. I would also like to express my appreciation to the National Council for Scientific and Technological Development of Brazil (CNPq) for my Graduate fellowship.

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

L. GirardeUoand M.T. Grisaru, Nucl. Phys. B194 (1982) 65. J. Iliopoulos and B. Zumino, Nucl. Phys. B76 (1974) 310. S. Ferrara, L. GirardeUoand F. Palumbo, Phys. Rev. D20 (1979) 403. S. Dimopoulosand H. Georgi, Nucl. Phys. B193 (1981) 150. N. Sakai, Z. Phys. Cll (1981) 153. H.P. Nilles, M. Srednickiand D. Wyler, Phys. Lett. 120B (1983) 346. H.P. Nilles, M. Srednicki and D. Wyler, Phys. Lett. 124B (1983) 337. R. Arnowitt, A.H. Chamseddine and P. Nath, Phys. Rev. Lett. 49 (1982) 970; Phys. Lett. 120B (1983) 145. 83

Volume 135, number 1,2,3 [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

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S. Ferrara, D. Nanopoulos and C.A. Savoy, Phys. Lett. 123B (1983) 214. J.A. Helay~l-Neto, I.G. Koh and H. Nishino, SISSA preprint 6/83/E.P. R. Barbieri, S. Ferrara and C.A. Savoy, Phys. Lett. 119B (1982) 343. J. Polchinsky and L. Susskind, Phys. Rev. D26 (1982) 3661. A. Salam and J. Strathdee, Nucl. Phys. B86 (1975) 142. K. Fujikawa and W. Lang, Nucl. Phys. B88 (1975) 77. M.T. Grisaru, M. Ro~ek and W. Siegel, Nucl. Phys. B159 (1979) 429. L.F. Abbott and M.T. Grisaru, Nucl. Phys. B169 (1980) 415. M.T. Grisaru, M. Ro~ek and W. Siegel, Nucl. Phys. B183 (1981) 141. W.E. Caswetl and D. Zanon, Nucl. Phys. B182 (1981) 125. R.D.C. Miller, Phys. Lett. 124B (1983) 59; CERN preprint TH. 3533 (1983). G. 't Hooft and M. Veltman, Nucl. Phys. B44 (1972) 189.

2 February 1984