Spurion operators in the generalized ORaifeartaigh model

Carlos R. Senise Jr.1

1Instituto de Fsica Teorica, UNESP, Rua Pamplona 145,

Bela Vista, Sao Paulo, SP, 01405-900, Brazil

Abstract

The generalized ORaifeartaigh model, described by a renormalizable superpotential with an

arbitrary number of chiral superfields, is a model which describes a mechanism for supersymmetry

(soft) breaking. We can consider this breaking to be owned by spurion insertions in the action.

This approach allows us to deal with quantum corrections, which are represented by functional

determinants in superspace. These spurion operators generate a Clifford algebra, and possess

interesting algebraic properties. We can use these properties to calculate the quantum corrections

directly in superspace, generally not very easy to solve.

senise@ift.unesp.br

1

I. INTRODUCTION AND SUMMARY

The study of models with explictly broken supersymmetry is of great interest for many

reasons, either theoretically and phenomenologically. One of the most important of these

reasons is the fact that supersymmetric theories possess important ultraviolet properties, like

the absence of quadratic divergences. This property still holds in theories with spontaneously

broken supersymmetry, and so one could work with the construction of realistic models. It is

this property that led to the development of the Minimal Supersymmetric Standard Model

(MSSM).

When dealing with supersymmetric theories, one may work in the component-field

approach or directly in superspace. In this work we use the superfield formalism, and make

the computations directly in superspace. In this approach, the soft parameters are described

by spurion insertions, 2 and 2.

For the case of spontaneously broken supersymmetric theories, which are of great phe-

nomenological interest, the soft explicitly breaking terms have been carefully classified and

studied by Girardello and Grisaru in [1]. Also, the propagators and some quantum correc-

tions have been achieved [24].

In this work, we use the techniques introduced in [5], where the use of spurion ope-

rators is made, to compute quantum corrections for the effective action of a generalized

ORaifeartaigh-like model in N = 1, d = 4 superspace [4, 6]. These quantum corrections are

represented by functional determinants, and can be splitted in various contributions.

Dealing with such functional determinants is very important when one needs to work with

vacuum supergraphs - which are generally zero in supersymmetric theories - as is the case

when one intends to apply the resummation method of the Linear Delta Expansion (LDE),

where the vacuum diagrams play an important role. The application of this method in

superspace may present interesting features, and the computation of the vacuum supergraphs

is essential for this propose. The application of the LDE to the Wess-Zumino model [7] have

been done in [8], where the Coleman-Weinberg potential [9] plus a two loop contribution

was obtained.

In view of that, the main goal of the present work is the computation of functional

determinants using the techniques introduced in [5], which are related with vacuum diagrams

in the LDE, for the case of a generalized ORaifeartaigh-like model with soft spontaneously

2

broken of supersymmetry.

The work is organized as follows: in Sec. II, we present the model and introduce the

spurion insertions to the action. In Sec. III, we define the spurion operators and their

properties, and rewrite the action in terms of these operators. In Sec. IV, we use these

operators and their properties to compute the one loop effective action, demonstrating that

the total result can be splitted in various contributions. The concluding remarks are finally

cast in Sec. V.

II. THE GENERALIZED ORAIFEARTAIGH MODEL WITH SPURION INSER-

TIONS

The action of the generalized ORaifeartaigh model is given by [4]

S =

d8zii

d6zf()

d6zf() , (1)

where i = 1, ..., n and

f() = i +1

2mijij +

1

3!gijkijk , (2)

with , m and g real constants, is the superpotential.

Adding the terms 12ijij and

12ijij to the action, where is a new mass parameter,

we have (disregarding the linear terms in the s)

S =

d8zii

[d6z

(1

2Mijij +

1

3!gijkijk

)+ h.c.

], (3)

where d8z = d4xd4 = d4xd2d2, d6z = d4xd2 and

Mij = mij + ij . (4)

We may now consider the mass parameter () to be a (anti)chiral superfield, so that

Mij = mij + ijkk

= mij + ijk(Ak + 2Fk)

= mij + ijkAk + ijkFk2

= aij + bij2 (5)

3

and

S =

d8zii 1

2

d6zi(aij + bij

2)j 12

d6zi(aij + bij

2)j

13!gijk

(d6zijk +

d6zijk

)= S0 + Sint , (6)

with

S0 =

d8zii 1

2

d6zi(aij + bij

2)j 12

d6zi(aij + bij

2)j ,

Sint = 13!gijk

(d6zijk +

d6zijk

). (7)

This action is essentiallly identical to the supersymmetric action (1), except for the

explicit appearance of the spurions 2 and 2, which is the signal of supersymmetry breaking

[1]. The use of these spurions to parametrize supersymmetry breaking terms proves to be

very useful when one intends to work in superspace, which is the main goal of the present

work.

In the subsequent sections, we will be particularly interested in the free part of the action,

so we rewrite this in the matrix form

S0 =

d8z 1

2

d6zT (a+ b2) 1

2

d6z(a+ b2)T , (8)

which will be very useful in our latter calculations.

III. SPURION OPERATORS

To work directly in superspace, using the spurions 2 and 2, it is now convenient to

present some operators [5] that will be very useful in our calculations. We will see latter that

we can write the action in such a way that the spurions appearing in it will be surrounded

by the chiral projections

P+ =D2D2

16 , P =D2D2

16 , (9)

where the Ds are the covariant derivatives in superspace. Next, we define the chiral spurion

operators

= 1/2P2P , = 1/2P2P . (10)

4

It is important to note that the products ++ and ++, and similarly and ,

are not equal, as can be seen by

++ =1

16D222D2 , ++ = P+22P+ ;

= P22P , =1

16D222D2 . (11)

The spurion operators + and + have interesting algebraic properties. They generate a

Clifford algebra: {+, +} = 1+, where the combination 1+ = +++ ++ plays the role ofthe identity, since we have +1+ = 1++ = + and +1+ = 1++ = +. Therefore, we can

interpret the combination

A+ = A11++ + A12+ + A21+ + A22++ (12)

as a 2 2 matrix

A+ =

A11 A12A21 A22

+

. (13)

We can also define the projection operator

+ = P+ 1+ = P+ ++ ++ , (14)

which is perpendicular to any A+, i.e., +A+ = A++ = 0.Finally, we define the trace of A+ as

trA+ =

trA11 trA12trA21 trA22

+

= trA11++ + trA12+ + trA21+ + trA22++ . (15)

This definition will be very useful in what follows in the next section.

We note that all of these properties can be identically obtained for and , so we do

not describe them here explicitly.

Now that we have defined those operators and their properties, we are able to write the

action in a convenient manner. Using the operators defined in eqs. (9) and (10), we see that

the free part of the action (8) can be writen as

S0 =

d8z

{ +

1

2T

(aP + b

1

1/2)D2

4 +1

2

(aP+ + b

1

1/2 +)D2

4T

}, (16)

which will be our starting point for the computation of quantum corrections.

5

IV. THE ONE LOOP EFFECTIVE ACTION

In this section we illustrate how we can use the spurion operators to compute quantum

corrections. We will determine the one loop effective action for the model, and see that

these contribution is splitted into terms that are proportional to 2, terms proportional to

2, and terms proportional to 22.

The effective action at one loop is given by [5]

i =

|D|2 eiS0 , (17)

where S0 is the quadratic part of the action, given in (16).

After a somewhat lenghty manipulations, we can show that [5]

i = 12

d8z1221Tr ln[P

TK]21 , (18)

where d8z12 = d8z1d

8z2, 21 = 8(z2 z1) = 4(2 1)4(x2 x1) =

2(2 1)2(2 1)4(x2 x1), and the notation Tr refers to the trace over the chiralmultiplets in the real basis defined by the vector (T , )T . In this expression, K is the

quadratic operator of the free part of the action, and P is the matrix

P =

0 PP+ 0

. (19)So, in order to determine the effective action at one loop, we write the action S0 as

S0 =1

2

d8z

(T

)K

T

= 12

d8z

(T

) K KK K

T

, (20)and comparing these equation with (16) we see that

K =

K KK K

= (aP + b 11/2) D24 1

1(aP+ + b

11/2 +

)D2

4

. (21)From the eqs. (19) and (21) we write

P TK =

P+ KK P

= P+ C

C P

, (22)with C and C given in (21).

6

Now, writing P TK as P+ 00 P

1 CC 1

,the one loop effective action (18) reads

i = 12

d8z1221{tr lnP+ + tr lnP + Tr ln[1 + Z]}21 , (23)

where

Z =

0 CC 0

. (24)From eq. (23) we see that we can separate the computation of the one loop effective action

in three parts: a contribution from P+, a contribution from P and a contribution from the

last term, which is proportional to C and C. We first compute the contributions from P+

and P.

The contribution from P+ is given by

iP+ = 1

2

d8z1221tr lnP+21 . (25)

Since 4(21)P+4(21) = 4(21) 1p2 , we obtain iP+ = 0 and, identically, iP = 0,so there is no contribution provenient from P+ neither from P.

The contribution from the last term in (23) is given by

iC = 12

d8z1221Tr ln(1 + Z)21 . (26)

To compute this, we introduce a continuous parameter 0 1 [5] in front of the off-diagonal terms (which are proportional to C and C), so that

iC = 1

2

d8z1221Tr ln(1 + Z)21 , (27)

and

iC =

10

dd

diC . (28)

Differentiating (27) with respect to we obtain

d

diC =

1

2

d8z1221Tr[Z Z2 + 2Z3 3Z4 + ...]21 . (29)

7

Now we note that, since

Z2 =

CC 00 CC

(30)is block diagonal, it follows that odd powers of Z are necessarily off-diagonal in the real

basis of the chiral multiplets. But the trace Tr in this basis is the sum of the traces in the

complex basis of the block diagonal parts, hence the odd powers of Z do not contribute, and

eq. (29) reads

d

diC =

1

2

d8z1221Tr[Z2 3Z4 5Z6 ...]21 . (31)

Integrating with respect to we obtain

iC = 14

d8z1221tr[ln(P+ CC) + ln(P CC)]21 . (32)

To compute this, we first note that

CC =aa

P+ +ab

3/2+ +ba

3/2 + +bb

2 ++ ,

CC =aa

P +ba

3/2 +ab

3/2 +bb

2 , (33)

and next we use the matrix notation introduced in (13) to write

P CC = (1 aa

)+ 1 E ,

P+ CC = +(1 aa

)+ 1+ E+ , (34)

where

E =

aa ab3/2ba3/2

1

(aa+ bb

)

=

,

E+ =

1 (aa+ bb) ba3/2ab3/2

aa

+

=

T TT T

+

. (35)

With these relations, eq. (32) reads

iC = 14

d8z1221tr

{ln[+

(1 aa

)+ 1+ E+

]+

+ ln[

(1 aa

)+ 1 E

]}21 . (36)

8

We again split the computation of eq. (36) in two parts: a contribution from the terms

proportional to and a contribution from the terms proportional to (1 E).First we compute the 1 E contribution, which is given by

i = 14

d8z1221tr[ln(1 E)]221 . (37)

To deal with this contribution, we again make use of a continuous parameter 0 1,introducing it in front of the -matrix part of (37). This gives

i =

1

4

d8z1221tr[ln(1 E)]221 . (38)

Differentiating with respect to ,

d

di

=

1

4

d8z1221trF [E]221 , F [E] = E(1 E)1 . (39)

Explicitly, this gives

d

di

=

1

4

d8z1221tr {F11[E] + F12[E]+

+F21[E] + F22[E]}2 21 . (40)

Now we use the following important relations, which ared8z1221[F]221 =

d8z1221[F ]221

=

d4x1d

4x2d4224(x2 x1)F24(x2 x1) , (41)

with both spurion operators of a given type inserted, andd8z1221[F]221 =

d4x1d

4x2d424(x2 x1)

[F

1

1/2]2

4(x2 x1) ;d8z1221[F ]221 =

d4x1d

4x2d424(x2 x1)

[F

1

1/2]2

4(x2 x1) , (42)

with only a single spurion operator of a given type inserted, to rewrite (40) as

d

di

=

1

4

d8ztr

{(F11[E] + F22[E])22 +

1

1/2F21[E]2 +

1

1/2F12[E]2

}. (43)

This same result is achieved for the 1+ E+ contribution, so we obtaind

di =

d

di

+

d

di

+

=1

2

d8ztr

{(F11[E] + F22[E])22 +

1

1/2F21[E]2 +

1

1/2F12[E]2

}.(44)

9

This expression shows that we can distinguish between contributions that go proportional

to 22 and those that only involve either 2 or 2.

We first compute the 22 part. Using the notation for the E matrix given in (35), we

can show that

F11[E] + F22[E] = dd

[ln(1 ) + ln

(1

2

1 )]

, (45)

and we write the 22 contribution as

d

di

22=

1

2

d8ztr

{(F11[E] + F22[E])22

}= 1

2

d8z

d

dtr

[ln(1 ) + ln

(1

2

1 )]

22 . (46)

Integrating with respect to we obtain

i22

= 12

d8ztr

[ln(1 ) + ln

(1

1 )]

22 . (47)

Next, by noting that

1 1 =

1

{ aa bb aa

}, (48)

it follows that the 22 contribution takes the form

i22

=

d8z22{trL0(aa) 1

2K(aa, b)

}, (49)

where

L0(aa) =

d4p

(2pi)4ln

(1 +

aa

p2

), K(aa, b) =

d4p

(2pi)4tr ln

(1 bb

(p2 + aa)2

). (50)

The contribution proportional to 2 is given by

d

di2 =

1

2

d8z2

1

1/2 trF21[E]

=1

2

d8z2

1

1/2 tr[(

1 2

1 )1

1

]. (51)

By noting that

1 2

1 =1

{ aa bb aa

}, (52)

and

1 =ba

1/2( aa) , (53)

10

it follows that the 2 contribution takes the form

i2 =1

2

d8z2R(b, a) , (54)

where

R(b, a) =

10

d

d4p

(2pi)4tr

[(1 bb

(p2 + aa)2

)1ba

(p2 + aa)2

]. (55)

The contribution proportional to 2 is given by

d

di

2=

1

2

d8z2

1

1/2 trF12[E]

=1

2

d8z2

1

1/2 tr[(

1 2

1 )1

1

]. (56)

By noting that

1 =ab

1/2( aa) , (57)

and using (52), it follows that the 2 contribution takes the form

i2=

1

2

d8z2R(b, a) , (58)

where

R(b, a) =

10

d

d4p

(2pi)4tr

[(1 bb

(p2 + aa)2

)1ab

(p2 + aa)2

]. (59)

The last contribution comes from the terms proportional to in (36). The contri-bution is given by

i = 1

4

d8z1221tr ln

[

(1 aa

)]21

= 14

d8z1221tr ln

[(P )

(1 +

aa

p2

)]21 . (60)

Using that 4(2 1)P+4(2 1) = 4(2 1) 1p2 and the relation (41), we obtain

i =

d8ztr

[1

4L1(aa) +

1

2L0(aa)

22], (61)

where L0(aa) is given in (50) and

L1(aa) =

d4p

(2pi)41

p2ln

(1 +

aa

p2

). (62)

The + contribution will be exactly the same and we have

i = i + i+ =

d8ztr

[1

2L1(aa) + L0(aa)

22], (63)

which is the total contribution of the projection operators .

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V. CONCLUDING REMARKS

In this work, we have applied the techniques introduced in [5] to calculate the one loop

effective action for a generalized ORaifeartaigh-like model. The introduction of spurion

operators allows us to deal with the somewhat complicated structure of the functional de-

terminant representing this quantum correction, owned by the spurion insertions of 2, 2

in the action. We were able to split the total contribution in various ones, each of these

proportional to some operator acting on the superspace coordinates.

The techniques we made use proves to be very useful to calculate quantum corrections in

theories which present spontaneously soft breaking of supersymmetry, due to the fact that

we can deal with the spurion contributions separately.

With these techniques in mind, we can try to apply the same procedures to calculate the

vacuum supergraphs appearing in the LDE in the broken case, which may bring us very

interesting results.

VI. ACKNOWLEDGEMENTS

I would like to thank J. A. Helayel-Neto and Daniel L. Nedel for valuable and clarifying

discussions and CAPES-Brazil for financial support.

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[4] J. A. Helayel-Neto, F. A. B. Rabelo de Carvalho and A. William Smith, Nucl. Phys. B 271

(1986) 175-287.

[5] Stefan Groot Nibbelink and Tino S. Nyawelo, Phys. Rev. D 75 (2007) 045002.

[6] L. ORaifeartaigh, Nucl. Phys. B 96 (1975) 331.

[7] J. Wess and B. Zumino, Phys. Lett. B 49 (1974) 52-54.

[8] M. C. B. Abdalla, J. A. Helayel-Neto, Daniel L. Nedel and Carlos R. Senise Jr. , Phys. Rev. D

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[9] S. Coleman and E. Weyberg, Phys. Rev. D7 (1973) 1888.

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