-term Triggered Dynamical Supersymmetry Breaking
H. Itoyama***e-mail: firstname.lastname@example.org and Nobuhito Maru†††e-mail: email@example.com
Department of Mathematics and Physics, Graduate School of Science
Osaka City University and
Osaka City University Advanced Mathematical Institute (OCAMI)
3-3-138, Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, Japan
Department of Physics, and Research and Education Center for Natural Sciences,
Keio University, Hiyoshi, Yokohama, 223-8521 JAPAN
We present the mechanism of the dynamical supersymmetry breaking at the metastable vacuum recently uncovered in the supersymmetric gauge theory that contains adjoint superfields and that is specified by Kähler and non-canonical gauge kinetic functions and a superpotential whose tree vacua preserve supersymmetry. The overall serves as the hidden sector and no messenger superfield is required. The dynamical supersymmetry breaking is triggered by the non-vanishing term coupled to the observable sector, and is realized by the self-consistent Hartree-Fock approximation of the NJL type while it eventually brings us the non-vanishing term as well. It is shown that theoretical analysis is resolved as a variational problem of the effective potential for three kinds of background fields, namely, the complex scalar, and the two order parameters and of supersymmetry, the last one being treated perturbatively. We determine the stationary point and numerically check the consistency of such treatment as well as the local stability of the scalar potential. The coupling to the supergravity is given and the gravitino mass formula is derived.
Spontaneous breaking of rigid supersymmetry occurs much less frequent compared with that of internal symmetry in quantum field theory and has attracted much interest [1, 2] of theorists for over the three decades. Mass hierarchy in elementary particle physics indicates that it is most desirable to break supersymmetry dynamically. In fact, under the non-renormalization theorem , no holomorphic operator is generated in perturbation theory and instanton generated nonperturbative superpotentials have been the major source of dynamical supersymmetry breaking (DSB).
In this paper, we focus our attention on general rigid theory in four spacetime dimensions consisting of vector superfields and chiral superfields in the adjoint representation which permits a non-canonical gauge kinetic function (that may follow from the second derivative of the prepotential) and hence the D term-gaugino-matter fermion (or D term-Dirac gauginos) nonrenormalizable coupling. It has recently been shown in refs. [4, 5] that, in this general situation, supersymmetry is dynamically broken in the metastable vacuum. The mechanism that triggers the DSB is the condensate of the Dirac bilinear above, forcing one of the order parameters of supersymmetry to be non-vanishing. This is very much reminiscent of the Nambu-Jona Lasinio (NJL) theory [6, 7] of broken chiral symmetry and hence the BCS superconductivity [8, 9], being formulated in terms of the effective action of the auxiliary field whose stationary value is the order parameter. The method of approximation employed is the self-consistent Hartree-Fock approximation where the tree and the one-loop contributions are regarded as comparable. Once this mechanism operates, non-vanishing F term is shown to be induced and contributes, for instance, to the mass of the fermions. The mechanism requires massive adjoint scalars, in particular, the scalar gluons and, together with the feature that the D term triggers the breaking, is quite distinct from the previous proposals [10, 11, 12, 13, 14] of DSB both from theoretical and experimental perspectives. The overall where the non-vanishing and the Nambu-Goldstone fermion (NGF) reside serves as the hidden sector and no messenger field is necessary  as non-vanishing third prepotential derivatives connect the sector with the observable sector [15, 16].
While our treatment of the theory bears much resemblance with that of the NJL theory, there is one important complexity which has no counterpart in the NJL and which we did not emphasize in ref. . In NJL, aside from the pseudoscalar auxiliary massless singlet field commonly denoted by , there is only one singlet auxiliary scalar field denoted by in the effective action, which is the order parameter of chiral symmetry. (See appendix A.) In other words, the stationary condition of energy with respect to the scalar is at the same time the stationarity with respect to the order parameter (the gap equation). This is not the case here. After treating the singlet real auxiliary field as perturbation, we have two kinds of background fields in the effective potential: these are the singlet complex scalar and the singlet auxiliary field . The stationarity of energy with respect to the scalar and that with respect to the order parameter are one and the other and both must be imposed simultaneously. In this paper, we will mainly deal with such multi-variable variational problem in depth and present the solution which is the local minimum of the scalar potential. We will also include a few other materials which have phenomenological implications. We work in the unbroken phase of and invoke invariance of the expectation values to suppress indices often.
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 in [4, 5] and in this paper and give the Noether current associated with rigid supersymmetry. We review the original reasoning that has led us to the D-term triggered dynamical supersymmetry breaking. We set up the background field formalism to be used in the subsequent sections, separating the three kinds of background from the fluctuations. The action can be coupled to supergravity and we derive the gravitino mass formula via the super-Higgs mechanism associated with the non-vanishing D-term. The action contains a sequence of special cases in which the gauge coupling function and the superpotential are related in a specific form, including the one where the rigid supersymmetry is partially broken to at the tree level [17, 15]. 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. [4, 5]. 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 as an induced perturbation, we demonstrate that the stationary values are determined by the intersection of the two real curves, namely, the simultaneous solution to the gap equation and the equation of stationarity (the energy condition). 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. We finish our paper with summary and brief comments on the issue of regularization and subtraction schemes.
In two of the appendix on rudimentary materials to be referred to in the text, we take a brief look at the NJL effective action and recall a formula of the second variation of a multivariable function. Phenomenological applications of our finding and the estimate of the longevity of our metastable vacuum have been given in [4, 5], which we do not repeat in this paper.
2 The action, assumptions and some properties
The action we work with in this paper is the general supersymmetric action consisting of chiral superfield in the adjoint representation and the vector superfield with three input functions, the Kähler potential with its gauging, the gauge kinetic superfield that follow from the second derivatives of a generic holomorphic function , and the superpotential .
The gauge group is taken to be 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 at the scalar vev’s are non-vanishing.
2.1 action and component expansion
The component Lagrangian of eq. (2.1) reads
and is the structure constant of . Note that an equation of motion for is + fermions. We also assume at the tree level. At the lowest order in perturbation theory, there is no source which gives vev to the auxiliary field : . The gaugino is massless at the tree level while the fermionic partner of the scalar gluon receives the tree level mass .
While we have already stated, it is useful to recapitulate here a set of assumptions made in order to address better the question of dynamical supersymmetry breaking within our framework.
1) a general supersymmetric action of chiral superfield in the adjoint representation and the vector superfield with the three input functions, namely, the Kähler potential with its gauging, the gauge kinetic superfield that follow from the second derivatives of a generic holomorphic function , and the superpotential .
2) third derivatives of at the scalar vev’s are non-vanishing.
3) the superpotential at tree level preserves supersymmetry.
4) the gauge group is and the vacuum is taken to be in the unbroken phase of . It is in principle straightforward to extend this to the (partially) broken cases where is broken into the product groups. The variational analyses we carry out in section four, however, become more complex and we will not address this in this paper, See the comment at eq. (3.22)
We give here an off-shell form of the supercurrent.
Equations of motion for auxiliary fields are
Once the invariant components of the auxiliary fields, and receive non-vanishing vev’s together with invariant scalar vev’s, the second and the fifth terms of the RHS of eq. (2.7) at these vev’s clearly develop a one-body fermionic operator non-vanishing at zero momentum [18, 19, 20]: this particular combination of and creates the one-particle state which is identified with the Nambu-Goldstone fermion .
2.4 original reasoning of DDSB
In ref. , it was shown that the vacuum state of the theory, albeit being metastable, develops a non-vanishing vev of an auxiliary field in the Hartree-Fock approximation. The theory, therefore, realizes the D-term dynamical supersymmetry breaking. The relatively simple estimate has shown that the vacuum can be made long lived. Let us recall a few more key aspects.
The part of the lagrangian which produces the fermion mass matrix of size is
It was observed that the auxiliary field, which is an order parameter of supersymmetry, couples to the fermionic (but not bosonic) bilinears through the third prepotential derivatives: the non-vanishing vev of immediately gives a Dirac mass to the fermions. Eq. (2.8) implies
telling us that the condensation of the Dirac bilinear is responsible for .
We diagonalize the holomorphic part of the mass matrix:
Note that the non-vanishing third prepotential derivatives are where refers to the generators of the unbroken gauge group. By an orthogonal transformation, we obtain the two eigenvalues of eq. (2.11) for each generator, which are mixed Majorana-Dirac type :
It was also shown in ref.  that the non-vanishing term is induced by the consistency of our procedure of computation. (See also [21, 27]). This is because the stationary value of the scalar fields gets shifted upon the variation (the vacuum condition). The final mass formula for the fermions is to be read off from
We will write down the explicit form in the next subsection. See eqs. (2.17), (2.18), (2.19) and (2.20). A main remaining point is how to establish the procedure in which the stationary values of the scalar fields, and perturbatively induced are determined, which we will resolve in this paper.
2.5 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. The linear terms that arise upon separation into quantum fields and background fields are dropped as they always cancel with source terms in .
2.5.1 the fermionic part
Let us extract the fermion bilinears from eqs. (2.3), (2.4) and (2.5) which are needed for our analysis in what follows. Rescaling the fermion fields so that their kinetic terms become canonical, we obtain
Here the fermion fields , , , are to be integrated to make a part of the effective potential, while the gauge kinetic function , the Kähler metric and their derivatives are functions of the singlet -number background scalar field . The order parameters of supersymmetry , , and are taken as background fields as well.
From the lagrangian , the holomorphic part of the mass matrix is read off as
We parametrize this matrix such that, in the case of , its form reduces to that of ref. [4, 5]. The quantities having multiple indices such as receive invariant expectation values: e.t.c. See, for instance, . We suppress the indices as we work with the unbroken phase in this paper.
The two eigenvalues of the holomorphic mass matrix are written as
These provide the masses for the two species of fermions once the stationary values are determined.
2.5.2 the bosonic part
where are the background -number field while , , and are the quantum scalar, vector and auxiliary fields respectively.
We have also ignored as we eventually set to be constant in our analysis and this term becomes a total derivative.
2.6 coupling to supergravity and super-Higgs mechanism
The fermionic part of the lagrangian relevant to the super-Higgs mechanism is given by
where is the determinant of the vierbein and the covariant derivatives of several kinds are defined as follows,
Now, we focus on the gravitino mass terms to discuss super-Higgs mechanism associated with eq. (2.26).
The field redefinition of the gravitino
eliminates the mixing terms of the gravitino with the gaugino and the adjoint fermion :
where the NG fermion absorbed in the massive gravitino is read
The eq. (2.34) tells us that the gravitino mass is given by
Requiring the cosmological constant to be almost vanishing
the gravitino mass can be expressed in terms of the vev’s of the auxiliary fields
2.7 special cases
As is mentioned in the introduction, the theory permits a sequence of interesting limiting cases. If we demand the Kähler function to be special Kähler, are expressible in terms of as
and . If we further demand such that the action possesses the rigid supersymmetry with one input function by choosing the superpotential to be a particular form, the tree vacua are shown to break supersymmetry to spontaneously [17, 15, 16, 23] 111This superpotential consists of the terms referred to as the electric and magnetic Fayet-Iliopoulos terms. This FI term is very special in the sense that, by the rigid rotation, it can be represented as a part of the superpotential. In this way, it avoids the difficulty (see, for instance,  for a recent discussion) of coupling the system to supergravity [25, 26]. We list the transformation laws for the doublet of fermions in this special case
2.8 connection with the previous work
We here stop shortly to address the connection of ref.  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  and the 4-1 model in .222Application of these models to the mediation mechanism, see for example [27, 28, 29]. 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 self-consistent 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 : the pure Dirac-type gaugino mass is generated in  333Attention has been paid to Dirac gaugino in many papers [28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44]., 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 , 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 [45, 46, 47, 48]. 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 . We prepare a supersymmetric counterterm, setting 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.
3.1 the point of the approximation
In the Hartree-Fock approximation, one begins with considering the situation where one-loop corrections in the original expansion in become large and are comparable to the tree 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. In this section, we start the analysis of this kind for our effective potential. There are three constant background fields as arguments of the effective potential: , invariant background scalar, and . The latter two are the order parameters of 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 . 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 .
3.2 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 etc.
As a warm up, let us determine the vacuum configuration by a set of stationary conditions at the tree level:
Eq. (3.3) determines the stationary value of :
while from eq. (3.4), we obtain
Eq. (3.5) together with these two gives
as well as
The negative coefficients of the RHS of eq. (3.2) imply that both and profiles of the potential have a maximum for a given . 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 is read off from the second derivative at the stationary point:
As we have already introduced in eq. (2.18), and are defined by