Nambu-Jona-Lasinio Theory and Dynamical Breaking of Supersymmetry

A recently proposed new mechanism of D-term triggered dynamical supersymmetry breaking is reviewed. Supersymmetry is dynamically broken by nonvanishing D-term vacuum expectation value, which is realized as a nontrivial solution of the gap equation in the self-consistent approximation as in the case of Nambu-Jona-Lasinio model and BCS superconductivity.


Introduction
Supersymmetry (SUSY) is one of the attractive solutions to the hierarchy problem, but it has to be broken spontaneously at low energy because of undiscovered superparticles. SUSY should be broken nonperturbatively or dynamically [1,2] according to nonrenormalization theorem [3]. Although the mechanism of dynamical SUSY breaking (DSB) by F-term has been much explored [4][5][6][7][8], models of DSB by D-term were not known. Several years ago, such a simple mechanism of D-term DSB (DDSB) was proposed by H. Itoyama and the present author [9][10][11] in which the nonvanishing vacuum expectation value (VEV) of D-term is dynamically realized as a nontrivial solution of the gap equation in the self-consistent Hartree-Fock approximation as in the case of Nambu-Jona-Lasinio (NJL) model [12,13] and BCS superconductivity [14,15].
In our mechanism, the gauge sector is extended to be N = 2 supersymmetric, and gaugino becomes massive by the D-term VEV through the Dirac mass term with N = 2 partner fermion of gaugino. Thus, our mechanism can be directly applied to Dirac gaugino scenario [16] which is an interesting alternative extension of the minimal SUSY Standard Model (MSSM). Much attentions have been paid to various phenomenological studies based on Dirac gaugino scenario and its extensions .
This paper is organized as follows. In the next section, we start out from exhibiting the component action from that of the superspace, state the set of assumptions we have made.
We review the original reasoning that has led us to the D-term triggered DSB. We set up the background field formalism to be used in the subsequent sections, separating the three kinds of background from the fluctuations. In section three, we elaborate upon our treatment of the effective potential with the three kinds of background fields as well as the point of the Hartree-Fock approximation in Refs. [9][10][11]. Section four is the main thrust of this paper.
We present our variational analyses of the effective potential in full detail. Treating one of the order parameters F-term as an induced perturbation, we demonstrate that the stationary values of D-term and an adjoint scalar field are determined by the intersection of the two real curves, namely, the simultaneous solution to the gap equation and the equation of stationarity for the adjoint scalar field. Numerical analysis is provided that demonstrates the existence of such solution as well as the self-consistency of our analysis. The second variation of the scalar potential is computed and the local stability of the vacuum is shown from the numerical data.
Section five will discuss the lifetime of our SUSY breaking vacuum. Unlike non-SUSY case, our SUSY breaking vacuum is necessarily meta-stable because of positive semi-definiteness of the vacuum energy in the rigid SUSY theory. Namely, the vacuum energy of our SUSY breaking vacuum is higher than the SUSY vacuum which is a trivial solution of the gap equation. We will show that the lifetime of our vacuum can be sufficiently large by adjusting parameters in the theory. In section six, a realization of observed Higgs mass by extra U(1) D-term contributions to Higgs mass will be discussed [55]. Summary is given in the last section.

The action, assumptions and some properties
The action we discuss is the general N = 1 supersymmetric action consisting of chiral superfield Φ a = (φ a , ψ a , F a ) in the adjoint representation and the vector superfield V a = (λ a , V a µ , D a ) with three input functions, the Kähler potential K(Φ a ,Φ a ) with its gauging, the gauge kinetic superfield τ ab (Φ a ) that follow from the second derivatives of a generic holomorphic function F (Φ a ), and the superpotential W (Φ a ), The gauge group is taken to be U(N) and, for simplicity, we assume that the theory is in the unbroken phase of the entire gauge group, which can be accomplished by tuning the superpotential. We also assume that third derivatives of F (Φ a ) at the scalar vacuum expectation values (VEV's) are non-vanishing.
The component Lagrangian of Eq. (1) reads where where and f b ac is the structure constant of SU(N). Note that an equation of motion for F a is F a = −g ab ∂ b W + fermions. We also assume F a tree = − g ab ∂ b W tree = 0 at the tree level. At the lowest order in perturbation theory, there is no source which gives VEV to the auxiliary field D 0 : D 0 tree = 0. The U(N) gaugino is massless at the tree level while the fermionic partner of the scalar gluon receives the tree level mass m a = m 0 = g 00 ∂ 0 ∂ 0 W tree .

Original reasoning of DDSB
In Ref. [9], it was shown that the VEV of an auxiliary field D 0 is non-vanishing in the Hartree-Fock approximation. Therefore, the theory realizes the D-term dynamical supersymmetry breaking.
The part of the Lagrangian providing the fermion mass matrix of size 2N is It was observed that the auxiliary D a field, which is an order parameter of N = 1 supersymmetry, couples to the fermionic (but not bosonic) bilinears through the third prepotential derivatives: the non-vanishing VEV of D 0 immediately gives a Dirac mass of the fermions. The equation of motion for the auxiliary field D 0 implies telling us that the condensation of the Dirac bilinear is responsible for D 0 = 0. This feature reminds us of the electromagnetic U(1) symmetry breaking in BCS theory by Cooper pair condensation or the chiral symmetry breaking in QCD by the quark-antiquark condensation.
We diagonalize the holomorphic part of the mass matrix: Note that the non-vanishing third prepotential derivatives are F 0aa where a refers to the generators of the unbroken gauge group. By an orthogonal transformation, we obtain the two eigenvalues of Eq. (9) for each generator, which are mixed Majorana-Dirac type : Introducing we obtain It was also shown in ref. [9] that the non-vanishing F 0 term is induced by the consistency of our procedure of computation. (See also [56,57]). This is because the stationary value of the scalar fields gets shifted upon the variation (the vacuum condition). The final mass formula for the SU(N) fermions is to be read off from We will write down the explicit form in the next subsection. See Eqs. (15), (16), (17) and (18).
A main remaining point is how to establish the procedure in which the stationary values of the scalar fields, D 0 and F 0 perturbatively induced are determined, which we will resolve in this paper.

Quadratic part of the quantum action
In this subsection, we write down parts of the action with the background fields for the computation of the one-loop determinant in the next section.

Fermionic part
Let us extract the fermion bilinears from Eqs. (3), (4) and (5) Here the fermion fields ψ a ,ψ a , λ a ,λ a are to be integrated to make a part of the effective potential, while the gauge kinetic function F aa , the Kähler metric g aa and their derivatives are functions of the U(N) singlet c-number background scalar field ϕ 0 . The order parameters of supersymmetry F 0 ,F 0 , and D 0 are taken as background fields as well.
From the lagrangian L F , the holomorphic part of the mass matrix is read off as We parametrize this matrix such that, in the case of F 0 =F 0 = 0, its form reduces to that of Ref. [9,10]. The quantities with multiple indices such as F 0aa receive U(N) invariant expectation values: F 0aa = F 000 e.t.c. We suppress the indices as we work with the unbroken U(N) phase in this paper.
The two eigenvalues of the holomorphic mass matrix are written as where These provide the masses for the two species of SU(N) fermions once the stationary values are determined.

Bosonic part
Next, we extract the bosonic quantum bilinears from Eqs. (3), (4), and (5). Let where ϕ 0 are the background field whileφ a ,Ã a µ ,F a andD a are the quantum scalar, vector and auxiliary fields respectively.
We obtain We have also ignored − 1 8 (Re F ) ab ǫ µνρσ F a µν F b ρσ as we eventually set ϕ a to be constant in our analysis and this term becomes a total derivative.

Connection with the previous work
We here stop shortly to address the connection of ref. [9] with the previous work. Models of dynamical supersymmetry breaking with non-vanishing F-and D-terms have been previously proposed: they are, for instance, the 3-2 model [6] and the 4-1 model in [56]. In these models, supersymmetry is unbroken at the tree level and is broken by the non-vanishing VEV of the F-term through instanton generated superpotentials. Non-vanishing VEV of the D-term is also induced, but is much smaller than that of the F-term.
In our mechanism, supersymmetry is unbroken at the tree level, and is broken in a selfconsistent Hartree-Fock approximation of the NJL type that produces a non-vanishing VEV for the D-term. A non-vanishing VEV for the F-term is induced in our Hartree-Fock vacuum that shifts the tree vacuum and we explore the region of the parameter space in which F-term VEV is treated perturbatively.
We should mention that the way in which the two kinds of gauginos (or the gaugino and the adjoint matter fermion) receive masses is an extension of that proposed in [16]: the pure Dirac-type gaugino mass is generated in [16] while the mixed Majorana-Dirac type gaugino masse is generated in our case, the Majorana part being given by the second derivative of the superpotential. In [16], the dynamical origin of non-vanishing D-term VEV was not addressed.
As for the application to dynamical chiral symmetry breaking, a supersymmetric NJL type model has been considered [58][59][60][61]. Chiral symmetry is not spontaneously broken in a supersymmetric case. Even in softly broken supersymmetric theories, the chiral symmetry broken phases are degenerate with the chirally symmetric ones. Thus, in supersymmetric theories, the phase with broken chiral symmetry is no longer the energetically preferred ground state.
3 The effective potential in the Hartree-Fock approximation The goal of this section is to determine the effective potential to the leading order in the Hartree-Fock approximation. We will regulate one-loop integral by the dimensional reduction [62]. We prepare a supersymmetric counterterm, and set the normalization condition. We make brief comments on regularization and subtraction schemes in the end of section 4. We also change the notation for expectation values in general from ... to ... * as our main thrust of this paper is the determination of the stationary values from the variational analysis.
In the Hartree-Fock approximation, one begins with considering the situation where oneloop corrections in the original expansion in become large and are comparable to the tree level contribution. The optimal configuration of the effective potential to this order is found by matching the tree against one-loop, varying with respect to the auxiliary fields. We start the analysis of this kind for our effective potential. There are three constant background fields as arguments of the effective potential: ϕ ≡ ϕ 0 (complex), U(N) invariant background scalar, D ≡ D 0 (real) and F ≡ F 0 (complex). The latter two are the order parameters of N = 1 supersymmetry.
We vary our effective potential with respect to all these constant fields and examine the stationary conditions. We also examine a second derivative at the stationary point along the constraints of the auxiliary fields to understand better the Hartree-Fock corrected mass of the scalar gluons. Let us denote our effective potential by V . It consists of three parts: The first term is the tree contributions, the second one is the counterterm and the last one is the one-loop contributions. After the elimination of the auxiliary fields, the effective potential is referred to as the scalar potential so as to be distinguished from the original V .

The tree part
To begin with, let us write down the tree part and find a parametrization by two complex and one real parameters. We also introduce simplifying notation g 00 (ϕ,φ) ≡ g(ϕ,φ), (Im F (ϕ)) 00 ≡ As a warm up, let us determine the vacuum configuration by a set of stationary conditions at the tree level: Eq. (26) determines the stationary value of D: while from Eq. (27), we obtain Eq. (28) together with these two gives as well as The negative coefficients of the RHS of Eq. (25) imply that both D and F profiles of the potential have a maximum for a given ϕ. These signs are, of course, the right signs for the stability of the scalar potential as is clear by completing the square. This is a trivial comment to make here but will become less trivial later. The mass of the scalar gluons at tree level |m s * | 2 is read off from the second derivative at the stationary point: As we have already introduced in Eq. (16), ∆ and r are defined by Recall that we have suppressed the indices, invoking the U(N) invariance of the expectation values. Also where We obtain We also see that the mass scales of the problem are set by m s * , the scalar gluon mass and ′′′ * , the third prepotential derivative, (and g −1 ∂g), once the stationary value of the scalar is determined.

Treatment of UV infinity
In the NJL theory [12,13], there is only one coupling constant carrying dimension −1 and the dimensionless quantity is naturally formed by combining it with the relativistic cutoff, which is interpreted as the onset of UV physics. In the theory under our concern, UV physics is specified by the three input functions, K, F , W and the UV scales and infinities reside in some of the coefficients. Our supersymmetric counterterm [9,10] is It is a counterterm associated with ImF ′′ . We set up a renormalization condition and relate (or transmute) the original infinity of the dimensional reduction scheme with that of ImF ′′ . We have indicated that this condition is set up at D = 0 and the stationary point of the scalar which we will determine. We stress again that the entire scheme is supersymmetric.

The one-loop part
The entire contribution of all particles in the theory to i· (the 1PI to one-loop) ≡ iΓ 1−loop is easy to compute, knowing (17), (18) and (2). It is given by In the unbroken U(N) phase, it is legitimate to replace a by N 2 and drop the index a as we have said before. We obtain Note that |m s | 2 , whose stationary value give the tree mass squared of the scalar gluon, differ from |trM| 2 : To evaluate the integral in d-dimensions, we just quote where We obtain This again depends upon ∆, f and ϕ.

Stationary conditions and gap equation 4.1 Variational analyses
Now we turn to our variational problem. It is stated as in the tree case as We will regard the solution to be obtained by considering Eqs. (49) and (51) first and solving D and ϕ for F andF : Eq. (50) is then and its complex conjugate. These will determine F = F * ,F =F * .
In this paper, we are going to work in the region where the magnitude |F * | is small and can be treated perturbatively. This means that, in the leading order, the problem posed by Eq.
(49) and Eq. (51) becomes Eq. (54) is nothing but the gap equation given in [9,10], while Eq. (55) is the stationary conditions for the scalar. This is the variational problem which we should solve. A set of stationary values (D * , ϕ * ,φ * ) is determined as the solution.

The analysis in the region F * ≈ 0
Let us first determine V (D, ϕ,φ, F = 0,F = 0) explicitly. We need to solve the normalization condition.
where J has been introduced in Eq. (43). At F,F → 0, where essentially reducing the situation to that of Refs. [9,10].
Note, however, that r and ∆ (or r 0 , ∆ 0 ) are complex in general except those special cases which include the case of the rigid N = 2 supersymmetry partially broken to N = 1 at the tree vacua. For |∆ 0 | ≪ 1, We solve the normalization condition for the number A to obtain We obtain After some calculation, this is found to be expressible as δ(ϕ,φ) = 1 2 Note thatδ * ≡δ(ϕ * ,φ * ) = 0, and If r 0 (and ∆ 0 ) is real, this is rewritten as where c ′ ≡ c r 2 0 * |ms * | 4 is the rescaled number, and Clearly, there are two scales in our current problem |r 0 * | −1/2 and |m s * |, which are controlled by the second superpotential derivative and the third prepotential derivative at the stationary value ϕ * .
On the other hand, for Eq. (70) with ∆ 0 being real, N 2 |m s | 4 is scaled out and it is simply given by the ∆ 0 derivative: which is our original gap equation. 1 In both cases, the solutions are given by the extremum of the potential V 0 (D, ϕ,φ) in its D profile. We stress again that the D profile is not a direct stability criterion of the vacua, which is to be discussed with regard to the scalar potential V 0 (D * (ϕ,φ), ϕ,φ).
We next examine ∂V 0 ∂ϕ D,φ = 0 and its complex conjugate. For Eq. (64), we obtain and its complex conjugate wherê The second term of the RHS of Eq. (76) is proportional to the gap equation Eq. (73). As for the third term, after some calculation, we obtain In the RHS of Eqs. (76) and (78), we have regarded ∆ 0 ,∆ 0 , ϕ andφ as independent variables.
For Eq. (70), with ∆ 0 real, we obtain and its complex conjugate. Here in the last term of the RHS, we have regarded ∆ 0 , ϕ,φ as independent variables.  As for the former case, as in the latter case, we can safely replace Eq. (76) by The values (∆ 0 * ,∆ 0 * , ϕ * ,φ * ) can be determined by the intersection of Eq. (73) and Eq. (81).
We will not carry out the (numerical) analysis for this case further in this paper.

Determination of F *
Let us now turn to the analysis of the remaining equation of our variational problem, Eq. (50).
In our current treatment, As the stationary values (D * , ϕ * ,φ * ) are already determined, this equation and its complex conjugate determine F * andF * : Note that, knowing V 1−loop explicitly in Eq. (48), the RHS can be evaluated. We can check the consistency of our treatment through f 3 in Eq. (36) by |f 3 | ≪ 1.

Numerical study of the gap equation
In this subsection, we study some numerical solutions to the gap equation Eq. (75) and the stationary condition for ϕ Eq. (80) in the real ∆ 0 case. The equations we should study are where we note that δ(ϕ,φ) in the gap equation (75) vanishes at the stationary point in the real ∆ 0 case. By using Eqs. (70) and (71), the second condition can be rewritten after dividing by The nontrivial solution ∆ 0 * = 0 to the gap equation (84) is found by some region of the parameters c ′ andÃ, which was already done in [9]. This solution fixes the LHS of Eq. (86) and ϕ * is determined by solving Eq. (86) in principle. In order to find ϕ * explicitly, the form of the prepotential F and that of the superpotential W must be specified. Here, we take a simple prepotential and a superpotential of the following type where c, d are dimensionless constants while m, M are dimensionful parameters. In particular, M is a cutoff scale of the theory. This prepotential is minimal for DDSB. As for the superpotential, at least two terms are required to be supersymmetric at tree level. We can take a quadratic term ϕ 2 instead of the cubic one, but in that case, RHS of Eq. (86) becomes singular because of ∂ ln |m s | 2 = 0.
Substituting these F and W into Eq. (86), we obtain where we utilized the fact that 1/M, d, ϕ * are real and c is pure imaginary, which are necessary for ∆ 0 =∆ 0 . Taking the coefficients c = i, d = 1 for further simplification, we can easily obtain a solution by tuning N and ImΛ. We note 0 ≤ ϕ * /M ≤ 1 for our effective theory to be valid.
In our analysis carried out in this paper, we consider the region where the magnitude of the F-term is smaller compared to that of the D-term. Therefore, we need to check whether our solutions satisfy this property consistently. Let us consider the ratio of the auxiliary fields: where the form of the prepotential and that of the superpotential in Eq.  Table 1.
In these examples, we have taken some values of − N 2 Im(i+Λ) and m just for an illustration and the ratio |F * /D * | and |f 3 * | are evaluated. We can find that the F -term is smaller than the D-term in some of these examples.

Mass of the scalar gluons
We now turn to the question of the second variations of the scalar ϕ. The ratio |F * /D * | and |f 3 * | are also evaluated for consistency check.
For V 0 , y L = D, y R = (ϕ,φ), We know that the D profile of V 0 (D, ϕ,φ) near the stationary point is convex to the top and we fit this by Here α is a positive real function of ϕ,φ and V h (ϕ,φ) = V 0 (D * (ϕ,φ), ϕ,φ). One can check while and The entire contribution of the second variation δ 2 V * = δ 2 V * + δ 2 V 0 * to the leading order in the Hartree-Fock approximation is given by Eqs. (99), (109). The mass of the scalar gluons squared is obtained by multiplying the combined mass matrix by g −1 * : generalizing the tree formula. In practice, we just need a well-approximated formula valid in the region we work with and one can invoke the U(1) invariance to ensure that the two eigenvalues of the complex scalar gluons are degenerate. Let us, therefore, use the expression to check the local stability of the potential and the mass. The above expression is obtained for our simple example of F and W   Table   2, the scalar gluon masses squared are found to be positive for any N, which implies that our stationary points are locally stable. Even in the last case, the stability is ensured for small N.
In these data, we have checked that the inequalities |(∂ F ∂F V 1−loop )| * , |(∂ 2 F V 1−loop )| * ≪ g * are in fact satisfied. As a summary of our understanding, a schematic figure is drawn in Fig. 2, which illustrates the local stability of the scalar potential at the vacuum of dynamically broken supersymmetry in comparison with the well-known NJL potential.

Choice of regularization and subtraction scheme
In this paper, we have considered the theory specified by the general N = 1 supersymmetric lagrangian Eq. (1), have regularized the theory by the supersymmetric dimensional regularization (dimensional reduction) and have subtracted the part of the 1/ǫ poles of the regularized one-loop effective action in Eq. (48) by the supersymmetric subtraction scheme defined by the condition Eq. (40). The original infinity is transmuted into the infinite constant Λ which is the coefficient of the counterterm and the effective potential has been recast to describe the behavior of the theory well below the UV cutoff residing in the prepotential function.
We now make brief comments on other regularizations and subtraction schemes which we did not employ in this paper. The relativistic momentum cutoff is a natural choice of the NJL theory as we mentioned earlier but regularizing the integral Eq. (41) by the momentum cutoff leads us to a rather unwieldy expression. See Ref. [10]. Unlike supersymmetric dimensional reduction [62], the momentum cutoff perse, while preserving the equality between the Bose and Fermi degrees of freedom, does not have a firm basis on the regularized action which the supersymmetry algebra acts on. Moreover, as is clear from (A.1) of Ref. [10], the result violates the positivity of the effective potential in the vicinity of the origin in the ∆ profile. This violation is a necessity in the broken chiral symmetry of the NJL theory but here it contradicts with the positive semi-definiteness of energy that the rigid supersymmetric theory possesses.
Turning to the choice of the subtraction scheme, one might also like to apply the "(modified) minimal subtraction scheme" in our one-loop integral Eq. (48). While we do not know how to justify this prescription here, the subsequent analyses proceed almost in the same way and the main features of the equations obtained from our variational analyses and the conclusions are unchanged.

Lifetime of metastable SUSY breaking vacuum
Combining the two facts that the trivial solution ∆ 0 = 0 of the gap equation is also a trivial solution and the energy in rigid SUSY theories is positive semi-definite leads us that our SUSY breaking vacuum is a local minimum. For our mechanism to be viable, we have to show that our SUSY breaking vacuum is sufficiently long-lived during the decay into the true vacuum with ∆ 0 = 0; in other words, the lifetime of our vacuum must be much longer than the age of universe. Taking into account the nonvanishing F-term VEV induced by D-term VEV as well, we carry out an order estimate of the lifetime of our SUSY breaking vacuum. Neglecting O(1) quantities, we have where Λ is a cutoff scale. Plugging these VEVs into Eq. (4.1) in [10] | F 0 | 2 + m s g 00 ∂ 0 g 00 leads to provided m s ≪ Λ.
The decay rate of our vacuum to the true one is controlled by the factor exp[−| ∆φ | 4 / ∆V ] as seen in Ref. [64], where ∆φ , ∆V are the scalar field distance and the potential height between two vacua. These two quantities are estimated as follows.
Using these results, the requirement of the longevity of our metastable vacuum is given by the condition which is always satisfied as long as m s ≪ Λ.

Higgs Mass
In order to realize the observed Higgs mass 126 GeV in the MSSM, SUSY breaking scale would be higher since the Higgs boson requires large radiative corrections from the top squarks. Also, we have no signals for SUSY particles from the experiments at Large Hadron Collider (LHC).
Thus, the naturalness of MSSM becomes worse and worse. It is well known fact that Higgs mass in the MSSM at tree level is smaller than the Z-boson mass, but it can be avoided if we consider extensions of the MSSM.
In this section, we investigate implications of the mechanism of DDSB uncovered in [9][10][11], coupling the system to the MSSM Higgs sector which includes the µ and Bµ terms [55]. The pair of Higgs doublet superfields H u , H d is taken to be charged under the overall U(1): We have adopted notation X · Y ≡ ǫ AB X A Y B = X A Y A = −Y · X, ǫ 12 = −ǫ 21 = ǫ 21 = −ǫ 12 = 1.
where c is a pure imaginary number as discussed above, and m, M are mass parameters. Here N = 5 and M (real number) sets the scale in the prepotential, which is the cutoff scale.
We embed the generators of the gauge group into the bases which expand ϕ: We have represented the overall U(1) and U(1) Y generators to be proportional to the unit matrix and the traceless diagonal generator respectively. We analyze the case in which only S receives its VEV, namely, the unbroken U(5) vacuum of the superpotential. We will make a comment for those cases in which these do not hold, which lead to the kinetic mixing. We drop octet T 8 as it is irrelevant to the analysis below.
After a simple calculation, we obtain the non-vanishing prepotential derivatives their VEV's and the derivatives of the superpotential We choose c = 10i but S is complex, not necessarily real.
In this paper, we add Eq. (119) to Eq. (1) and consider a part relevant to 126 GeV Higgs The third prepotential derivatives, which are now real numbers, can be read off from Eq. (123).
In our analysis, we take that the value of D 0 VEV is determined essentially by our Hartree-Fock approximation in [11]. This source of supersymmetry breaking is then fed to the Higgs sector and its effects are given by a tree level analysis. We will argue the validity of this procedure below.

Higgs potential and variations
Let us extract the part relevant to the Higgs potential in (125).
where ϕ C = (T a , Y, S). The one-loop part of the effective potential in [9,11] is denoted by Let us vary L pot with respect to the auxiliary fields, replacing ϕ C by their VEV ϕ C = (0, 0, S ).

Estimate of the Higgs mass
We are now ready to calculate Higgs mass. As in the MSSM, the minimization of the scalar potential ∂V Higgs /∂v 2 = ∂V Higgs /∂β = 0 allows us to express µ and Bµ in terms of other parameters.
It is straightforward to obtain the mass matrix for CP-even Higgs from the second derivative of the potential, where each component is given by In order for the µ-term to be allowed in the superpotential, we must have a condition e u +e d = 0 which is also required from an anomaly cancellation condition for the overall U(1). Then, the Higgs mass can be expressed as It is interesting to see the correspondence between our expression of Higgs mass (143) and that in the MSSM,

Summary
In this paper, we reviewed our recently proposed mechanism of D-term triggered dynamical SUSY breaking [9][10][11]55]. The nonvanishing D-term VEV is dynamically realized as a nontrivial solution of the gap equation in the Hartree-Fock approximation as in the case of NJL model and BCS superconductivity. In our mechanism, since the gauge sector is extended to be N = 2 supersymmetric, gaugino becomes massive by the D-term VEV through the Dirac mass term with N = 2 partner fermion of gaugino. Our mechanism can be directly applied to Dirac gaugino scenario which much attention has been paid to.
A systematic analysis of the scalar potential was performed by treating the order parameters of SUSY breaking D and F , and the adjoint scalar field as the background fields. It was shown numerically that SUSY is indeed broken dynamically and our meta-stable vacuum is locally stable. The lifetime of our meta-stable vacuum was also shown to be sufficiently long-lived.
As a phenomenological application, we have discussed how an observed Higgs mass can be realized in the context of DDSB and have shown that it is naturally realized by an additional overall U(1) D-term contribution to Higgs mass.