Superspace approach to the renormalization of the O'Raifeartaigh model up to the second order in the linear delta expansion parameter

Superspace approach to the renormalization of the O’Raifeartaigh model up

to the second order in the linear delta expansion parameter

M. C. B. Abdalla,1,*J. A. Helaye¨l-Neto,2,†Daniel L. Nedel,3,‡and Carlos R. Senise, Jr.4,§ 1_{Instituto de Fı´sica Teo´rica, UNESP-Universidade Estadual Paulista,}

Rua Dr. Bento Teobaldo Ferraz 271, Bloco II, Barra-Funda, Caixa Postal 70532-2, 01156-970, Sa˜o Paulo, SP, Brazil 2

Centro Brasileiro de Pesquisas Fı´sicas, Rua Dr. Xavier Sigaud 150, Urca, Rio de Janeiro, RJ, 22290-180, Brazil 3_{Universidade Federal do Pampa, Rua Carlos Barbosa S/N, Bairro Getu´lio Vargas, 96412-420, Bage´, RS, Brazil} 4_{Universidade Federal do Pampa, Avenida Pedro Anunciac¸a˜o S/N, Vila Batista, 96570-000, Cac¸apava do Sul, RS, Brazil}

(Received 24 March 2012; published 21 June 2012)

We adopt a superspace/supergraph formalism to pursue the investigation of the structure of one- and two-loop divergences in the frame of the minimal O’Raifeartaigh model that realizes the F-term spontaneous supersymmetry breaking. The linear delta expansion (LDE) procedure is introduced and renormalization is carried out up to the second order in the LDE expansion parameter. In agreement with the nonrenormalization theorem for the (chiral/antichiral) matter potential ofN ¼1,D¼4 supersym-metry, our explicit supergraph calculations confirm that only the Ka¨hler potential is actually renormalized. DOI:10.1103/PhysRevD.85.125032 PACS numbers: 12.60.Jv, 11.10.Gh

I. INTRODUCTION

Supersymmetry (SUSY) and viable procedures to inves-tigate its explicit, spontaneous and dynamical breaking mechanisms [1] are topics of constant and renewed interest in the literature, in view of the structural role SUSY plays in the construction of field-theoretic frameworks for funda-mental interactions and realistic models for elementary particle physics. In the early days of the dawn of SUSY in high-energy particle theories, it was realized that spontane-ous breakdown of this new fermion/boson symmetry was a topic of major relevance in order to make contact between SUSY and observations and communicate high-energy SUSY imprints with the low energy sector of the standard model. Ever since, in different scenarios like the minimal supersymmetric standard model [2], Kaluza-Klein super-gravities [3], string inspired models for elementary particle interactions [4], Seiberg-Witten super-Yang-Mills dualities [5], and brane-world scenarios [6], the problem of SUSY breaking has been suitably reassessed. More recently, SUSY breaking in connection with brane physics in general and M2-brane modeling [7], more specifically, has triggered a great deal of attention to the understanding of a number of issues related to (2þ1)-dimensional supersymmetric field theories [8], like the so-called Aharony-Bergman-Jafferis-Maldacena models [9].

If, on the one hand, SUSY breaking mechanisms and their consequences are relevant for connecting high-energy fundamental physics to the regime of accelerator energies and for establishing the consistency of more formal field-theoretic models, on the other hand, one must develop technical methods to suitably carry out the SUSY breaking

program and to pursue its investigation perturbatively and, hopefully, by means of some nonperturbative scheme. In this case, if it is not possible to derive an exact result, one could somehow come over this problem by devising some sort of semiperturbative scheme based upon the resumma-tion of a certain class of (perturbative) Feynman diagrams and attain, thereby, a result that incorporates all orders in some coupling parameter.

In the frame of perturbative and resummation methods, the calculation and study of the effective potential is a viable path to be followed. In this context, different methods for effective potential calculation have been proposed in the literature soon after the idea of spontaneous symmetry breaking and the Higgs mechanism were adopted to realize the breakdown of gauge symmetries and to introduce the hierarchy of energy scales in unified models for elementary particle interactions. Here, we shall be concentrating our efforts to apply the so-called linear delta expansion (LDE) [10], suitably extended [11,12] to incorporate superspace and superfield techniques, to compute corrections at the one- and two-loop orders to the effective potential of the minimal O’Raifeartaigh model [13] which spontaneously breaksN ¼1SUSY in four space-time dimensions. We stress that, even though SUSY is broken, a superfield ap-proach is still the most appropriate setup to describe the problem. In a previous contribution [14], it was actually shown, in a systematic way, how all order spurion corrected superpropagators and super-Feynman rules can be em-ployed to compute explicit 1- and 2-loop corrections to effective potentials in a scenario with an F-term breaking of N ¼1, D¼4 supersymmetric matter self-coupled model. The main characteristic of the LDE is to use a tradi-tional perturbative approach together with an optimization procedure. So, in order to derive a result in all orders of the coupling constant, it is just necessary to work with a few diagrams and use perturbative renormalization techniques. *mabdalla@ift.unesp.br

†_{helayel@cbpf.br}

‡_{daniel.nedel@unipampa.edu.br} §_{carlossenise@unipampa.edu.br}

Having in mind the ever increasing importance of under-standing and proposing new scenarios for SUSY breaking, our paper sets out to tackle a specific problem—the appli-cation of the LDE at the second order to the minimal O’Raifeartaigh model—mainly motivated by the reasons that follow below:

(i) to exploit superfield techniques and superspace methods in connection with the LDE procedure to compute higher-order corrections to the effective potential of a supersymmetric field model, even if SUSY is spontaneously broken and we are obliged to deal with terms that explicitly break SUSY in super-space. This may show us nontrivial technicalities and features whenever we insist in performing superfield calculations to describe SUSY breaking. Our view-point is that superspace, with its corresponding ten-sor calculus expressed in terms of superfields, is still the most suitable tool to deal with even if SUSY is lost;

(ii) to use the supergraph approach to carry out the superspace renormalization of supersymmetric models if one adopts the LDE procedure to compute the loop corrected effective potential. In the present paper, we shall be concerned with the structure of divergences and the renormalization of the model we pick out to work with. The optimization proce-dure and the attainment of the full (LDE) two-loop corrected effective potential shall be reported on in a forthcoming work [15];

(iii) once the whole treatment is understood for this more traditional case ofF-type breaking, we shall be able to apply it to other interesting situations, such as SUSY breaking by aD-term in supersym-metric gauge theories, R symmetry spontaneous breakdown in connection with the existence of metastable SUSY breaking vacua [16] and the computation of loop quantum contributions to the effective Ka¨hler and chiral potentials [17].

Keeping in mind this whole framework and the motiva-tions mentioned above, we organize the present paper according to the following outline: in Sec. II, we recall the main features of the LDE in superspace, highlighting the superfield techniques in the presence of terms that explicitly break SUSY. Section III is devoted to report and discuss the superspace evaluation of the results at-tained at order one in the parameter; next, in Sec. IV, we go a step further and analyze the divergent structure of the order two contributions, which encompass one- and two-loop supergraphs, to show how to renormalize the model to the first and second order in the parameter of LDE. With our calculations, we shall see that only the Ka¨hler potential is actually renormalized, as expected by the nonrenormalization of the chiral potential inN ¼1, D¼4SUSY. Finally, our concluding remarks are cast in Sec.V. Two Appendices follow: in AppendixA, we work

out the main results in connection with the integration over the Grassmannian sector of superspace; in AppendixB, we present the explicit answers to the momentum-space loop integrals that appear in the course of our calculations.

II. THE LINEAR DELTA EXPANSION IN SUPERSPACE

In this section, we are going to make a brief review of the application of the linear delta expansion to supersym-metric theories. We follow Refs. [11,12]. Starting with a Lagrangian L, let us define the following interpolated LagrangianL:

L_{¼LðÞ þ ð1 ÞL}

0ðÞ; (1)

where is an arbitrary parameter, L0ðÞ is the free

Lagrangian, and is a mass parameter. Note that, when ¼1, the original theory is retrieved. The parameter labels interactions and is used as a perturbative coupling instead of the original one. The mass parameter appears in

L0andL0. Thedependence ofL0is absorbed into the

propagators, whereas L0 is regarded as a quadratic

interaction.

Let us now define the strategy of the method. We apply a usual perturbative expansion in and, at the end of the calculation, we set ¼1. Up to this stage, traditional perturbation theory is applied, working with finite Feynman diagrams, and the results are purely perturbative. However, quantities evaluated at finite order inexplicitly depend on. So it is necessary to fix theparameter. There are two ways to do that. The first one is to use the principle of minimal sensitivity (PMS) [18]. It requires that a physical quantity, such as the effective potential VðkÞ_{ðÞ, calculated}

perturbatively to order k_{, must be evaluated at a point} where it is less sensitive to the parameter. According to the PMS,¼0 is the solution to the equation

@VðkÞ_ðÞ

@

¼0;¼1

¼0: (2)

After this procedure, the optimum value, 0, will be a

function of the original coupling and fields. Then, we re-place0 into the effective potentialVðkÞand obtain a

non-perturbative result, since the propagator depends on. The second way to fixis known as the fastest apparent convergence (FAC) criterion [18]. It requires that, for anyk coefficient of the perturbative expansion

VðkÞ_{ðÞ ¼}Xk

i¼0

c_iðÞi_{; (3)}

the following relation must be fulfilled:

½VðkÞ_{ðÞ V}ðk 1Þ_ðފj

¼1 ¼0: (4)

Again, the0 solution of the above equation will be a

result. Equation (4) is equivalent to taking the kth coefficient of (3) equal to zero (c_k ¼0). If we are inter-ested in an order-k _{result [V}ðkÞ_{ðÞ] using the FAC}

criterion, it is just necessary to find the solution to the equation c_kþ1ðÞj¼0 ¼0 and plug it into V

ðkÞ_ðÞ.

References [19,20] provide an extensive list of successful applications of the method.

Let us now further develop the LDE for superspace applications. Following Ref. [11], for general models with chiral and antichiral superfields, we need to imple-ment two mass parameters,and, instead of just one. In order to fix these parameters, we employ two optimization equations. Also, we need to take care of the vacuum diagrams. In general, when the effective potential is calcu-lated in quantum field theory, we do not worry about vacuum diagrams, since they do not depend on fields. However, the vacuum diagrams depend on and are important to the LDE, since the arbitrary mass parameter will depend on fields after the optimization procedure. So, in the LDE, it is necessary to calculate the vacuum dia-grams order by order. On the other hand, it is well known that, in superspace, vacuum superdiagrams are identically zero, by virtue of Berezin integrals. To avoid this, we have to consider, from the very beginning, the parameters, as superfields and keep the vacuum supergraphs until the optimization procedure is carried out. In order to make the procedure clear, let us write the interpolated Lagrangian,

L_{, for the Wess-Zumino model discussed in [11]:}

L_{¼Lð;Þ þ ð1 ÞL}

0ð;Þ

¼Z d4 _þZ _d2

_M

2

2_þ

3!

3

2

2

þZ d2M

2

2_þ

3!

3

2

2_{; (5)}

where m is the original mass, M¼mþ and M ¼ mþ. Now, one has a new chiral and antichiral quadratic interaction proportional to and . Also the super-propagator will have a dependence onand. From the generating superfunctional in the presence of the chiral (J) and antichiral (J) sources

~

Z½J;JŠ ¼exp

iSINT

₁

i J;

1

i J

exp _i

2ðJ;JÞG

ðM;_MÞJ

J

; (6)

we can write the supereffective action:

½;Š ¼ i

2ln½sDetðG

ðM;_MÞ_{ފ iln ~Z½J;JŠ}

Z

d6_zJðzÞðzÞ Z _d6_{zJðzÞðzÞ; (7)}

whereGðM;_MÞ_{is the matrix propagator andsDetðG}ðM;_MÞ_Þis

the superdeterminant ofGðM;_MÞ

, which, in general, is equal

to one; but here we keep it, becauseGðM;_MÞ_{depends on}

and. Also, due to the and dependence, the super-generator of the vacuum diagrams, Z~½0;0Š, is not identi-cally equal to one. We can define the normalized functional generator asZ_N¼Z_Z~_~½_½J;JŠ

0;0Š, and write the effective action as

½;Š ¼ i

2ln½sDetðGފ iln ~Z½J0;J0Š þN½;Š; (8)

where the sourcesJ0 andJ0are defined by the equations

W½J;JŠ JðzÞ

J¼J0

¼W½J;JŠ JðzÞ

_J¼_J0¼

Z~½J;JŠ JðzÞ

J¼J0

¼Z~½J;JŠ JðzÞ

_J¼_J0¼0: (9)

In (8), the first two terms represent the vacuum diagrams (which are usually zero) and N½;Š is the usual con-tribution to the effective action.

Let us now derive the interpolated Lagrangian and the new Feynman rules for the O’Raifeartaigh model. The simplest O’Raifeartaigh model is described by the follow-ing Lagrangian:

L¼Zd4

ii

Z

d2_ð

0þm12þg021Þ þH:c:

; (10)

wherei¼0;1;2.

Following Ref. [12], in order to take into account the nonperturbative contributions of all fields of the model, we need to implement the LDE with the matrix mass parame-ters_ijand_ij. Adding and subtracting these mass terms in the Lagrangian of a general O’Raifeartaigh model we obtain

Lð;Þ ¼L0ð;Þ þLintð;Þ; (11)

where

L0ð;Þ ¼

Z d4

ii

Z d2

iiþ 1

2Mijij

þH:c:

; (12)

Lintð;Þ

¼ Z d2

₁

3!gijkijk 1

2ijij

þH:c:

;

(13)

withM_ij¼m_ijþ_ijandi; j; k¼0;1;2are symmetrical indices.

_ij¼_ijk’_k¼_ijkð_kþ2

kÞ ¼ijkkþijkk2

¼ijþbij2; (14)

so that

M_ij¼m_ijþ_ij ¼ ðm_ijþ_ijÞ þb_ij2 _¼a

ijþbij2: (15)

Now, the interpolated Lagrangian (1) becomes

L_¼L

0 þLint; (16)

where the free Lagrangian,L₀, is

L

0 ¼

Z d4

ii Z

d2

_iiþ

1

2aijij

þ1 2bij

2

ij

þH:c:

; (17)

and the interaction Lagrangian reads as follows:

L

int¼

Z d2

3!gijkijk

2ijij

þH:c:

:

(18)

Notice that the interaction Lagrangian has now soft breaking terms proportional to thecomponents. We are going to treat these terms perturbatively in , like all interactions.

Now, in order to get the simplest O’Raifeartaigh model when¼1[10], we make the choices

0¼; M01¼a01¼01¼a; M11¼b112¼b2;

M₁₂¼a₁₂¼m₁₂þ₁₂¼mþ¼M; g₀₁₁¼g; (19)

and all other_iandM_ij set to zero. With that, we obtain

L

0 ¼

Z d4

ii Z

d2

0þM12þa01

þ1 2b

22 1

þH:c:

;

L

int¼

Z d2

g021 12 a01

2b

22 1

þH:c:

: (20)

As is well known, this O’Raifeartaigh model has anR symmetry. TheRcharges of chiral superfields0,1,2

are respectivelyR0 ¼2,R1¼0, andR2¼2. In order to

preserve the R symmetry in the interpolated Lagrangian, the R charges of the parameters a and b are R_a¼0 andRb¼0, which must be preserved after the optimiza-tion procedure.

The new propagators can be derived from the free Lagrangian, which also has an explicit dependence on

andfrom theand components. Using the techniques developed in [21], the propagators can be written as

h00i ¼ ðk2þ jMj2ÞAðkÞ412þ jaj2jbj2BðkÞ2121412;

h01i ¼abC ðkÞ

1 16D

2

1D2121412;

h02i ¼ MaA ðkÞ412þMajbj2BðkÞ2121412;

h11i ¼EðkÞ412þ jbj2BðkÞ

1 16D

2

12121D21412;

h12i ¼ MbF ðkÞ21412;

h22i ¼ ðk2þ jaj2ÞAðkÞ412þ jMj2jbj2BðkÞ2121412;

h00i ¼ jaj2bC ðkÞ

1 4D

2 121412;

h01i ¼aA ðkÞ

1 4D

2

1412 ajbj2BðkÞ

1 4

2

121D21412;

h02i ¼ M ab C ðkÞ

1 4D

2 121412;

h11i ¼bF ðkÞ

1 4

2 1D21412;

h12i ¼MA ðkÞ

1 4D

2

1412 Mjbj2BðkÞ

1 4D

2

12121412;

h22i ¼ jMj2bC ðkÞ

1 4D

2

121412; (21)

with

AðkÞ ¼ 1

k2_ðk2_{þ jMj}2_{þ jaj}2_Þ;

BðkÞ ¼ 1

ðk2_{þ jMj}2_{þ jaj}2_Þ½ðk2_{þ jMj}2_{þ jaj}2_Þ2 _jbj2_Š;

CðkÞ ¼ 1

k2_½ðk2_{þ jMj}2_{þ jaj}2_Þ2 _jbj2_Š;

EðkÞ ¼ 1

k2_{þ jMj}2_{þ jaj}2;

FðkÞ ¼ 1

ðk2_{þ jMj}2_{þ jaj}2_Þ2 _jbj2:

In usual quantum field theories, the optimized parame-ters appear at the poles of the propagators, as mass terms. Here, it should be emphasized the nontrivial dependence on the parametersa,b, and, which appear not only at the poles, but also in the numerators.

We can also write the new Feynman rules for the vertices:

021vertex:2g

Z

d4_;

12vertex:

Z d4_;

01vertex: a

Z d4_;

11vertex: b

2 Z

d42_:

Now we have all the necessary ingredients to calculate the effective potential using perturbation theory in. In the next section we show the order one results.

III. ORDER ONE RESULTS

The perturbative effective potential can now be calcu-lated in powers of using the one particle irreducible functions, defined in the expansion of the effective action, taking into account vacuum diagrams. In Ref. [12], it was shown that, after the optimization procedure at order one, the optimized effective potential provides the sum of all one-loop diagrams. In that case, analytical solutions were obtained for the optimization procedure before calculating the superspace and momentum integrals. However, in order to go beyond the one-loop approximation, it is necessary to go beyond the order1_{. In this case, it is not possible to}

find solutions before evaluating the superspace and mo-mentum integrals and the optimization procedure must be carried out after the renormalization of the theory. Owing to the nontrivial dependence of the propagators on the optimized parameters, it is not clear that the method does not alter the divergences structure of the model. We inves-tigate this fact here.

In Fig. 1, one can see the diagrammatic sum of the effective potential up to the order1_(Vð1Þ

eff).

Note that, by virtue of the-dependent propagators, the tadpole diagrams are not identically zero, as usual in super-space. The first diagram is of order0 _{and corresponds to}

the first term of the effective action expansion, defined in (8) .

Using the Feynman rules and the results of [12], the expressions for the diagrams of the figure above, in the order they appear are:

(i) Order0 _{(vacuum diagram):}

Gð1Þ0

¼1 2

Z d4

12421Tr ln½PTKŠ421; (23)

whered4

12¼d41d42andTris the trace over the

chiral multiplets defined in the real basis by ðT_;ÞT_{. Details of this calculation can be seen in} Refs. [12,22]. The matrix Pis defined by the chiral projectorsPþ¼

D2_D2

16h andP ¼

D2_D2 16h as

P¼ 0 P

Pþ 0

!

; (24)

and

K¼

AP þB 1 h1=2

D2

4h 13

13

APþþB_h11=2þ

D2 4h

0

B B B @

1

C C C A;

(25)

with

A¼

0 a 0

a 0 M

0 M 0 0

B B @

1

C C

A; B¼

0 0 0

0 b 0 0 0 0 0

B B @

1

C C A;

¼h1=2P 2_{P ;}

þ¼h1=2Pþ2Pþ;

(26)

is the quadratic part of the free Lagrangian (there are also quadratic terms in the interaction Lagrangian, which depend on the optimization parameters). Using the results of [12], Eq. (23) can be written as

Gð1Þ0

¼1 2

Z d4_k

ð2 Þ4 ln

1 jbj

2

ðk2_{þ jMj}2_{þ jaj}2_Þ2

:

(27)

(ii) Order1_:

Gð1Þ1

1 ¼ 2gb

Z d4k

ð2 Þ4FðkÞ

Z

d42

0þH:c:;

Gð1Þ1 2 ¼

1 2jbj

2Z d 4_k

ð2 Þ4FðkÞ

Z

d422_þH:c:;

Gð1Þ1

3 ¼4gajbj2

Z d4k

ð2 Þ4BðkÞ

Z

d422 1þH:c:;

Gð1Þ1

4 ¼ jaj2jbj2

Z d4k

ð2 Þ4BðkÞ

Z

d422_þH:c:;

Gð1Þ1

5 ¼ Mjbj2

Z d4k

ð2 Þ4BðkÞ

Z

d422_þH:c::

(28)

Note that all the expressions for the superdiagrams were written as an integral over d4_{, according to the}

nonrenormalization theorem. In Appendix Awe give the final results for the superspace integrals. Using the results

Z

d42 0¼

Z d2

0;

Z

d422_{¼1 e}

Z

d422 1¼

Z

d22

1; (29)

the effective potential up to the order1 _{is given by}

Vð1Þ eff ¼G

ð1Þ0

þX

5

i¼1

Gð1Þ1

i ¼ 1 2

Z d4

12412Tr ln½PTKŠ412

þZ d 4_k

ð2 Þ4FðkÞ

2gbZ d2 0þ

1 2jbj

2_þH:c:

þZ d 4_k

ð2 Þ4BðkÞ

4gajbj2Z _d22

1 jaj2jbj2 Mjbj2þH:c:

: (30)

Now we are going to regularize the integrals that appear in the effective potential up to the order1_{. The diagrams of}

order1 _{are shown in Fig.2.}

We are going to use the notation defined in (B9). Using the results of Appendix B, these one-loop diagrams are written as

Gð1Þ1 1 ¼2g

Z d4_kd4 12

ð2 Þ4 0ð1Þ 4 12

₁

4D

2

1ðkÞh11i

þH:c:

¼ 2gbZ d 4_k

ð2 Þ4FðkÞ

Z

d42

0þH:c:

¼ 4gbhF0i

1

þ

2ghF0i

ðþ_lnþ _{ln 2bÞ; (31)}

Gð1Þ1 2 ¼

b

2

Z d4_kd4 12

ð2 Þ4 2 1412

₁

4D

2

1ðkÞh11i

þH:c:

¼jbj

2

2

Z d4_k

ð2 Þ4FðkÞ

Z

d422_þH:c:

¼b

2

1

b

2ð

þ_lnþ _{ln 2bÞ; (32)}

Gð1Þ1 3 ¼4g

Z d4_kd4 12

ð2 Þ4 1ð1Þ412

₁

4D

2

1ðkÞh01i

þH:c:

¼4gajbj2Z d4k

ð2 Þ4BðkÞ

Z

d422

1þH:c:

¼ 4gah’1i

ð2

2_ln2 þ_lnþ _{ln Þ;}

(33)

Gð1Þ1

4 ¼ a

Z d4_kd4 12

ð2 Þ4 4 12

₁

4D

2

1ðkÞh01i

þH:c:

¼ jaj2_jbj2Z d 4_k

ð2 Þ4BðkÞ

Z

d422_þH:c:

¼a

2

ð2

2_ln2 þ_lnþ _{ln Þ;}

(34)

Gð1Þ1

5 ¼

Z d4_kd4 12

ð2 Þ4 4 12

₁

4D

2

1ðkÞh12i

þH:c:

¼ Mjbj2Z d4k

ð2 Þ4BðkÞ

Z

d422_þH:c:

¼M

ð2

2_ln2 þ_lnþ _{ln Þ; (35)}

with0ð1Þ ¼0ðp¼0; 1;1Þ.

Let us deal with the divergent ones, Gð1Þ1 1 andG

ð1Þ1 2 ,

and apply the renormalization procedure. We are going to use the MS scheme. The diagram Gð₂1Þ1 is a vacuum diagram, so its renormalization is trivial. We just need to cancel the divergence with a constant counterterm, which implies a redefinition of the vacuum energy. Since the tadpole diagrams are not identically zero, the interpolated theory has a new divergence, which apparently is not canceled by a counterterm of the Kahleria€n-type potential. To renormalize the divergence in Gð₁1Þ1, we need the counterterm

2gb

Z

d42

0RþH:c:

¼2gb

Z d2

0RþH:c:; (36)

where 0R denotes the renormalized superfield. It seems that the chiral potential is renormalized, which is strange because the renormalization structure of the theory should not be modified by the soft terms introduced by the method. However, this counterterm depends on b, which must be a solution of the optimization procedure.

The renormalized effective potential up to the order1_is

Vð1Þ eff¼

1 ð4 Þ2

₁

4ðM

2_þa2_Þ2_ln

1 b

2

ðM2_þa2_Þ2

þb

2ðM

2_þa2_Þln

_M2_þa2_þb

M2_þa2 _b

þb

2

4 ln

_ðM2_þa2_Þ2 _b2

4

_3b2

4

þ

ð4 Þ2

bðb 4ghF0iÞ þ2½aða 4gh’1iÞ þMŠðM2þa2Þln½M2þa2Š þ

að4gh’1i aÞ þ

1

2ð4ghF0i bÞ M

ðM2_þa2_þbÞln½M2_þa2_{þbŠ þ}

að4gh’₁i aÞ 1

2ð4ghF0i bÞ M

ðM2_þa2 _bÞln½M2_þa2 _bŠ

: (37)

At this stage, we have a perturbative result for the effective potential. In order to get a nonperturbative result, we apply the optimization procedure. Since we split the parameters M_ij into a -independent (a_ij) and a -dependent (bij) part, and recalling (19), the optimized parameters will bea01¼a,b11¼b, and12¼. Using

the PMS criterion to find the optimized parameters a, b, and, we have to solve the three coupled equations

@V2 eff @a

a¼a0

¼@V 2 eff @b

b¼b0

¼@V 2 eff @

¼0

¼0; (38)

at¼1, and plug the optimized valuesa0,b0, and0into

(37). We find the following analytical solutions:

a₀¼4gh’₁i ¼a₀

b0¼4ghF0i ¼b0;

0¼0¼0:

(39)

This result shows that the optimized parameters are functions of the original coupling and fields, as we ex-pected. It is easy to see that this is also a solution of the FAC criterion. However, as we have three optimized pa-rameters, it is not clear how we may write three optimiza-tion equaoptimiza-tions using this criterion. Replacing these values in (30), all the1_{terms vanish and the optimized potential}

is written as

Vð1Þ eff ¼G

ð1Þ0

¼ 1

ð8 Þ2

ðm2_þ16g2_h’ 1i2Þ2ln

1 16g

2_hF 0i2

ðm2_þ16g2_h’ 1i2Þ2

þ8ghF0iðm2þ16g2h’1i2Þln

_m2_þ16g2_h’

1i2þ4ghF0i m2_þ16g2_h’

1i2 4ghF0i

þ16g2_hF 0i2ln

_ðm2_þ16g2_h’

1i2Þ2 16g2hF0i2 4

48g2_hF 0i2

: (40)

This is the Coleman-Weinberg potential for the O’Raifeartaigh model [21] and represents the sum of all one-loop diagrams, i.e., a nonperturbative result, because it takes into account infinite orders of the original coupling constant.

Let us come back to the counterterm defined in (36). Since0 and 1 are classical superfields, the optimized

parameters can be written as

a0 ¼4g

Z d2

1 ¼4g

Z d2

1 ¼4gh’1i;

b0 ¼4g

Z d2

0 ¼4g

Z d2

0 ¼4ghF0i: (41)

This is in agreement with theRcharges of the parametersa andb. Thus, putting the expression above forb0into (36)

we see that the counterterm needed to renormalize the divergence inGð₁1Þ1 is of the form

8g2

Z d4

0R0RþH:c:¼ 16g2

Z

d4 0R0R;

(42)

and we note that in fact, after the optimization procedure, only the Ka¨hler potential is renormalized, in agreement with the nonrenormalization theorem. In the next section we discuss the divergences at order2_.

IV. ORDER TWO RESULTS

Gð1Þ2

1 ¼2ð2gÞð2gÞ

Z d4_kd4 12

ð2 Þ4 0ð1Þ0ð2Þ

₁

4D

2

1ðkÞh11i

₁

4D

2

2ð kÞh11i

¼8

2_g2

16

Z d4_k

ð2 Þ4

EðkÞEðkÞJ1ð;Þ þ2

jbj2

16 EðkÞBðkÞJ2ð;Þ þ jbj4

ð16Þ2BðkÞBðkÞJ3ð;Þ

¼8

2_g2_hF 0i2

₁

ln

2_þ22g2hF0i2 b ½4bln

2 _ð2_þ2bÞlnþ_{þ ð}2 _{2bÞln þ2bŠ: (43)}

Gð1Þ2

2 ¼2ð2gÞ

_b

2

Z d4_kd4 12

ð2 Þ4 0ð1Þ 2 2

₁

4D

2

1ðkÞh11i

₁

4D

2

2ð kÞh11i

þH:c:

¼ 2

2_g

16 b Z d4_k

ð2 Þ4

EðkÞEðkÞJ4ð;Þ þ2

jbj2

16 EðkÞBðkÞJ5ð;Þ þ jbj4

ð16Þ2BðkÞBðkÞJ6ð;Þ

þH:c:

¼ 4

2_gbhF 0i

₁

ln

2 2gbhF0i b ½4bln

2 _ð2_þ2bÞlnþ_{þ ð}2 _{2bÞln þ2bŠ: (44)}

Gð1Þ2 3 ¼2

_b

2 _b

2

Z d4_kd4 12

ð2 Þ4 2 122

₁

4D

2

1ðkÞh11i

₁

4D

2

2ð kÞh11i

¼

2

2ð16Þjbj

2Z d 4_k

ð2 Þ4

EðkÞEðkÞJ7ð;Þ þ2

jbj2

16 EðkÞBðkÞJ8ð;Þ þ jbj4

ð16Þ2BðkÞBðkÞJ9ð;Þ

¼

2_b2

2 ₁

ln

2

þ

2_b2

8b½4bln

2 _ð2_þ2bÞlnþ_{þ ð}2 _{2bÞln þ2bŠ: (45)}

There are four two-loop diagrams (plus the Hermitian conjugates). They are all vacuum diagrams and differ from each other by the propagators appearing in the loops. Out of these four diagrams, only one gives a divergent contri-bution, and we show it in Fig.4.

As the one-loop diagramGð1Þ2

3 , this is a vacuum

dia-gram, and its renormalization is trivial. Again, we just need to cancel the divergence with a constant counterterm, which implies a redefinition of the vacuum energy.

To renormalize the divergent term inGð₁1Þ2we introduce the counterterm

82_g2

Z d4

0R0R; (46)

and, for the divergent term in Gð₂1Þ2, we introduce the counterterm

22_g

Z

d42_b

0RþH:c: (47)

As in the previous section, when we renormalized the effective potential up to the order 1_{, the counterterm}

introduced for the divergent term inGð1Þ2

2 is proportional

tob, and, at one-loop, the optimized parameterb0 is given

by Eq. (41). It can be shown that this solution is valid for all orders in at one-loop level. Thus, the counterterm is proportional to

2Z _d4

0R0R; (48)

showing that in fact, at one-loop, only the wave function is renormalized. However, owing to the two-loop vacuum diagrams, for the effective potential up to the order 2_,

the optimized parameters defined in (39) are no longer solutions to the PMS equations. In fact, when diagrams like the one in Fig.4are taken into account, we can derive two-loop nonperturbative corrections to the Coleman-Weinberg potential. To this end it is necessary to evaluate all the order2 _{diagrams and solve numerically a}

compli-cated set of equations. This is a work in progress [15]. Here, we are interested in the renormalization structure of the theory. Since the counterterms depend only onb, let us concentrate on this parameter. Although it is not possible to find an analytical solution for thebparameter at order2_,

we can write the general form for it. Based on R charge

FIG. 3. Divergent diagramsGð1Þ1 2,G ð1Þ2

2 , andG ð1Þ2

3 . FIG. 4. Divergent diagramsG ð1Þ2

1 ,G ð1Þ2

2 , andG ð1Þ2

considerations, on the fact thatb₀must be a function of the original component fields and on the Lagrangian depen-dence onb, we can argue that the optimized parameter up to the order2 _{must be}

b₀¼Z d22_½4g

0þA2-loopŠ; (49)

whereA2-loopis a two-loop correction withRcharge equal

to 2. Since all the counterterms are proportional to R

d4

0R0R orRd2b0R, it can be seen that only the Ka¨hler potential is renormalized.

V. CONCLUDING REMARKS

Our efforts in the present work have been focused on the application of superfield techniques and supergraph calculations to study the renormalization of the minimal O’Raifeartaigh model in the LDE scheme to the second order in the expansion parameter. Our calculations show that only the Ka¨hler potential gets renormalized, according to what should be expected from the N ¼1, D¼4 SUSY nonrenormalization theorem for the chiral potential. We point out that, in this paper, we are actually interested in understanding and mastering the superfield approach if we adopt the LDE procedure to compute higher-order corrections to the effective potential in the case of sponta-neously broken SUSY.

Here, we have not yet concentrated on the task of effective potential calculation. As already stated in the Introduction, this is the matter of a forthcoming work. Our main purpose in the present paper was to check the consistency and the efficacy of superfield and supergraph methods to deal with SUSY explicitly breaking terms in a higher-order loop computation in superspace. We have checked, with our explicit supergraph computations, that the extended super-Feynman rules are perfectly consistent even though these explicitly breaking terms show up. The structure of divergences has suitably been treated and the final result of the Ka¨hler potential renormalization is a good check of our manipulations. Once the renormaliza-tion task is accomplished, it remains to be done—and we believe this requires a forthcoming work—the complete two-loop calculation to allow us to proceed to the next step, namely, the optimization in the LDE parameter, to finally write down the full two-loop corrected effective potential. This demands a nontrivial work in terms of Feynman supergraph computation and numerical computation and we shall be soon reporting on our results [15].

Once this whole program of higher-order corrected ef-fective potentials has been accomplished for the F-term SUSY breaking, we believe it would be worthwhile to concentrate efforts on the LDE effective potential calcu-lation in the case ofD-term SUSY breakings in the gauge sector and to pay attention to the problem of metastable SUSY breaking vacua and its connection withRsymmetry spontaneous breaking, which has direct consequences to

the physics of the so-called lightest supersymmetric particle.

ACKNOWLEDGMENTS

J. A. Helaye¨l-Neto expresses his gratitude to FAPERJ-RJ for the financial support. Daniel L. Nedel would like to thank CNPq, Grant No. 501317/2009-0, for financial sup-port. Carlos R. Senise Jr. thanks CAPES-Brazil and Programa Rece´m-Doutor-UNESP for financial support.

APPENDIX A: SUPERSPACE INTEGRALS

This Appendix is devoted to the evaluation of some of the superspace one-loop integrals which arise in order2_.

The integrals that appear in the expressions for the diver-gent one-loop diagrams of order2 _are

J1ð;Þ ¼

Z d4

120ð1Þ0ð2Þ½D21ðkÞ412Š

½D2

2ð kÞ412Š ¼16hF0i2; (A1)

J2ð;Þ ¼

Z d4

120ð1Þ0ð2Þ½D21ðkÞ412Š

½D2

2ð kÞD22ð kÞ2222D22ð kÞ412Š

¼ ð16Þ2_hF

0i2; (A2)

J3ð;Þ ¼

Z d4

120ð1Þ0ð2Þ

½D2

1ðkÞD21ðkÞ2121D21ðkÞ412Š

½D2

2ð kÞD22ð kÞ2222D22ð kÞ412Š

¼ ð16Þ3_hF

0i2; (A3)

J4ð;Þ ¼

Z d4

120ð1Þ½D21ðkÞ124 Š½22D22ð kÞ412Š

¼16hF0i; (A4)

J5ð;Þ ¼

Z d4

120ð1Þ½D21ðkÞ21D21ðkÞ412Š

½2

2D22ð kÞ22D22ð kÞ412Š

¼ ð16Þ2_hF

0i; (A5)

J6ð;Þ ¼

Z d4

120ð1Þ½D12ðkÞD21ðkÞ2121D21ðkÞ412Š

½2

2D22ð kÞD22ð kÞ2222D22ð kÞ412Š

¼ ð16Þ3_hF

0i; (A6)

J7ð;Þ ¼

Z d4

12½21D21ðkÞ412Š½22D22ð kÞ412Š ¼16;

J8ð;Þ ¼

Z d4

12½21D21ðkÞ21D21ðkÞ412Š

½2

2D22ð kÞ22D22ð kÞ412Š

¼ ð16Þ2_{; (A8)}

J9ð;Þ ¼

Z d4

12½21D21ðkÞD21ðkÞ2121D21ðkÞ412Š

½2

2D22ð kÞD22ð kÞ2222D22ð kÞ412Š

¼ ð16Þ3_{: (A9)}

APPENDIX B: MOMENTUM-SPACE INTEGRALS

In this Appendix we collect the expressions for the one-loop integrals, which are not so easy to obtain directly. We use dimensional regularization.

We begin with the simple integrals [22]:

Jðm2_{Þ ¼}ð 2_Þ

ð2 ÞD

Z dDk

k2_þm2¼

ð2_Þ

ð4 Þ2 ð 1þÞðm2Þ1

Jnðm2Þ ¼

Z dD_k

ð2 ÞDD 4

1

k2

1 ðk2_þm2_Þ

¼ðm

2_Þ1 n

16 2

4

2

m2

_{2 ðD=2Þ}

ðD=2Þsin ðD=2 nÞ

L_nðm2_{Þ ¼}Z d

D_k

ð2 ÞDD 4

1

k2nln

1þm

2

k2

¼ m

2

D=2 nJnðm 2_Þ;

(B1)

where D¼4 2 and is the renormalization scale. Expanding these expressions, we obtain the useful equalities:

Jðm2_{Þ ¼} 1

16 2

_m2

þm

2_lnm2

2 1

; (B2)

J₀ðm2_{Þ ¼} m 2

16 2

₁

þ1 ln

m2 2

; (B3)

J1ðm2Þ ¼

1 16 2

₁

þ1 ln

m2 2

; (B4)

L₀ðm2_{Þ ¼} 1

2

m4

16 2

₁

þ

3 2 ln

m2 2

; (B5)

L1ðm2Þ ¼ m2

16 2

₁

þ2 ln

m2 2

: (B6)

In the calculation of the order0 _{diagram, we have to}

solve the integral

Kðm2_{; M}2_{Þ ¼}Z dDk

ð2 ÞDD 4 tr ln

1 M

2_M2

ðk2_þm2_m2_Þ2

;

(B7)

which can be written as a sum of integrals of the type (B5)

Kðm2_;M2_{Þ ¼tr½L}

0ðm2þm~2Þ þL0ðm2 m~2Þ 2L0ðm2ފ;

(B8)

withm~2 _{¼ ðM}2_M2_Þ1=2_.

In the following, we define

2_¼M2_þa2_{; ¼M}2_þa2_{b; (B9)}

and we adopt the same notation of [23–25].

Using the definition of the propagators we now show the results for the integrals appearing in the one-loop diagrams of orders1 _and2_{. To solve them we use the method of}

partial fraction, splitting each integral as a sum of other integrals with just one propagator in the integrand. Doing this, we obtain

I1ðkÞ ¼

Z d4_k

ð2 Þ4FðkÞ ¼

Z d4_k

ð2 Þ4

1

ðk2_þM2_þa2_Þ2 _b2

¼ 1

2bJð

þ_{Þ þ} 1

2bJð Þ

¼ 1

1 2bð

þ_lnþ _{ln 2bÞ;}

(B10)

I2ðkÞ ¼

Z d4_k

ð2 Þ4BðkÞ

¼Z d

4_k

ð2 Þ4

1

ðk2_þM2_þa2_Þ½ðk2_þM2_þa2_Þ2 _b2_Š

¼ 1

b2Jð 2_{Þ þ} 1

2b2Jð þ_{Þ þ} 1

2b2Jð Þ

¼ 1

2b2ð2

2_ln2 þ_lnþ _{ln Þ;}

(B11)

I3ðkÞ ¼

Z d4_k

ð2 Þ4EðkÞEðkÞ

¼Z d

4_k

ð2 Þ4

1

ðk2_þM2_þa2_Þðk2_þM2_þa2¼_Þ J~ð 2_Þ

¼ 1

1

ln

2_{; (B12)}

I₄ðkÞ ¼Z d

4_k

ð2 Þ4BðkÞEðkÞ ¼

Z d4_k

ð2 Þ4

1

ðk2_þM2_þa2_Þ½ðk2_þM2_þa2_Þ2 _b2_Šðk2_þM2_þa2_Þ

¼ 1 b4J~ð

2_{Þ þ} 1

4b4J~ð þ_{Þ þ} 1

4b4J~ð Þ þ

3 4b5Jð

þ_Þ 3

4b5Jð Þ ¼

1

2b3ð2bln

2 þ_lnþ_{þ ln þ2bÞ;}

(B13)

I5ðkÞ ¼

Z d4_k

ð2 Þ4BðkÞBðkÞ ¼

Z d4_k

ð2 Þ4

1

ðk2_þM2_þa2_Þ½ðk2_þM2_þa2_Þ2 _b2_Šðk2_þM2_þa2_Þ½ðk2_þM2_þa2_Þ2 _b2_Š

¼ 1 b4J~ð

2_{Þ þ} 1

4b4J~ð

þ_{Þ þ} 1

4b4J~ð Þ þ

3 4b5Jð

þ_Þ 3

4b5Jð Þ

¼ 1

4b5½4bln

2 _ð3þ _bÞlnþ_{þ ð3 þbÞln þ6bŠ:}

(B14)

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