Spontaneously broken supersymmetry (SUSY) and a vanishingly small cosmological constant imply that symmetry must be spontaneously broken at low energies. Based on this observation, we suppose that, in the sector responsible for low-energy symmetry breaking, a discrete symmetry remains preserved at high energies and only becomes dynamically broken at relatively late times in the cosmological evolution, i.e., after the dynamical breaking of SUSY. Prior to symmetry breaking, the Universe is then bound to be in a quasi-de Sitter phase—which offers a dynamical explanation for the occurrence of cosmic inflation. This scenario yields a new perspective on the interplay between SUSY breaking and inflation, which neatly fits into the paradigm of high-scale SUSY: inflation is driven by the SUSY-breaking vacuum energy density, while the chiral field responsible for SUSY breaking, the Polonyi field, serves as the inflaton. Because symmetry is broken only after inflation, slow-roll inflation is not spoiled by otherwise dangerous gravitational corrections in supergravity. We illustrate our idea by means of a concrete example, in which both SUSY and symmetry are broken by strong gauge dynamics and in which late-time symmetry breaking is triggered by a small inflaton field value. In this model, the scales of inflation and SUSY breaking are unified; the inflationary predictions are similar to those of F-term hybrid inflation in supergravity; reheating proceeds via gravitino decay at temperatures consistent with thermal leptogenesis; and the sparticle mass spectrum follows from pure gravity mediation. Dark matter consists of thermally produced winos with a mass in the TeV range.

April 2016 IPMU 16-0051

Polonyi Inflation

Dynamical Supersymmetry Breaking and Late-Time

Symmetry Breaking as the Origin of Cosmic Inflation

Kai Schmitz ***Corresponding author. E-mail: and Tsutomu T. Yanagida 

[3mm] Max Planck Institute for Nuclear Physics (MPIK), 69117 Heidelberg, Germany

Kavli IPMU (WPI), UTIAS, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan


1 Introduction: Inflation and supersymmetry breaking unified

1.1 High-scale supersymmetry breaking as the origin of inflation

The paradigm of cosmic inflation [1, 2] is one of the main pillars of modern cosmology. Not only does inflation account for the vast size of the observable Universe and its high degree of homogeneity and isotropy on cosmological scales; it also seeds the post-inflationary formation of structure on galactic scales. In this sense, inflation is a key aspect of our cosmic past and part of the reason why our Universe is capable of harboring life. From the perspective of particle physics, the origin of inflation is, however, rather unclear. After decades of model building, there exists a plethora of inflation models in the literature [3]. But a consensus about how to embed inflation into particle physics is out of sight. In this situation, it seems appropriate to take a step back and ask ourselves what avenues have been left unexplored so far. In particular, we should question our dearly cherished prejudices and re-examine whether inflation might not be connected to other high-energy phenomena which, up to now, have been taken to be mostly unrelated to inflation. As we are going to demonstrate in this paper, an important example in this respect might be the interplay between inflation and the spontaneous breaking of supersymmetry (SUSY).111For related earlier work on the relation between inflation and supersymmetry breaking, see, e.g., [4, 5, 6, 7].

In recent years, the picture of supersymmetry as a solution to the hierarchy problem has become increasingly challenged by the experimental data. The null results of SUSY searches at the Large Hadron Collider (LHC) [8] and the rather large standard model (SM) Higgs boson mass of a  [9] indicate that supersymmetry, if it exists in nature, must be broken at a high scale [10]. Based on this observation, one could feel tempted to give up on supersymmetry as an extension of the standard model altogether. But this would not do justice to supersymmetry’s other virtues. Provided that supersymmetry is broken at a high scale [11, 12], such as in the minimal framework of pure gravity mediation (PGM) [13, 14],222For closely related schemes for the mediation of supersymmetry breaking to the visible sector, see [15, 16]. it may no longer be responsible for stabilizing the electroweak scale. But in this case, supersymmetry is still capable of providing a viable candidate for dark matter [14, 16, 17], ensuring the unification of the SM gauge couplings [18] and setting the stage for a UV completion of the standard model in the context of string theory. In addition, high-scale supersymmetry does not suffer from a number of phenomenological problems that low-scale realizations of supersymmetry breaking are plagued with. A high SUSY breaking scale does away with the cosmological gravitino problem [19] and reduces the tension with constraints on flavor-changing neutral currents and violation [20]. Moreover, in PGM, the SUSY-breaking (or “Polonyi”) field is required to be a non-singlet [21], which solves the cosmological Polonyi problem [22].

In this paper, we will now concentrate our attention to yet another intriguing feature of supersymmetry which comes into reach, once we let go of the notion that supersymmetry’s main purpose is to solve the hierarchy problem in the standard model. The spontaneous breaking of supersymmetry at a scale results in a nonzero contribution to the total vacuum energy density, . If we allow to take values as large as, say, the unification scale, , this SUSY-breaking vacuum energy density might, in fact, be the origin of the inflationary phase in the early universe! Such a connection between inflation and supersymmetry breaking not only appears economical, but also very natural.

First of all, supersymmetry tends to render inflation technically more natural, independent of the scale at which it is broken. Thanks to the SUSY nonrenormalization theorem [23], the superpotential in supersymmetric models of inflation does not receive any radiative corrections in perturbation theory. This represents an important advantage in preserving the required flatness of the inflaton potential. Besides, all remaining radiative corrections (which can be collected in an effective Kähler potential to leading order [24]) scale with the soft SUSY-breaking mass scale [25] and are, thus, under theoretical control. Supersymmetry, therefore, has the ability to stabilize the inflaton potential against radiative corrections; and it is, thus, conceivable that supersymmetry’s actual importance may lie in the fact that it is capable of taming the hierarchy among different mass scales in the inflaton sector rather than in the standard model. Second of all, the spontaneous breaking of global supersymmetry via nonvanishing F-terms, i.e., via the O’Raifeartaigh mechanism [26], always results in a pseudoflat direction in the scalar potential [27]. Together with the constant vacuum energy density , such a flat potential for a scalar field is exactly one of the crucial requirements for the successful realization of an inflationary stage in the early universe. In principle, the necessary ingredients for inflation are, therefore, already intrinsic features of every O’Raifeartaigh model. Inflation may be driven by the SUSY-breaking vacuum energy density and the inflaton field may be identified with the pseudoflat direction in the scalar potential.

The main obstacle in implementing this idea in realistic models is gravity. Here, the crucial point is that the vanishingly small value of the cosmological constant (CC) tells us that we live in a near-Minkowski vacuum with an almost zero total vacuum energy density, . Note that, as pointed out by Weinberg, this not a mere observation, but a necessary condition for a sufficient amount of structure formation in our Universe, so that it can support life [28]. In the context of supergravity (SUGRA) [29], the fact that means that the SUSY-breaking vacuum energy density must be balanced, with very high precision, by a nonvanishing vacuum expectation value (VEV) of the superpotential, ,


where denotes the reduced Planck mass. If the SUSY breaking scale is indeed of , the requirement of a zero CC results in a huge VEV of the superpotential, which, in turn, generates dangerously large SUGRA corrections to the scalar potential. These corrections then easily spoil the flatness of the potential and render inflation impossible [30].

The most attractive and, in fact, only way out of this problem is symmetry. Under symmetry, the superpotential carries charge , so that , as long as symmetry is preserved [31]. In other words, by imposing symmetry, we promote to the order parameter of spontaneous symmetry breaking, in analogy to which acts as the order parameter of spontaneous supersymmetry breaking. In the true vacuum, symmetry must be broken, so that , in order to satisfy the condition in Eq. (1). But this does not necessarily mean that must be nonzero during the entire cosmological evolution. It is conceivable that, at early times, symmetry is, in fact, a good symmetry, so that , thereby “switching off” the most dangerous SUGRA corrections to the Polonyi potential. The main intention of our paper now is to present a minimal dynamical model, in which this is indeed the case, so that inflation driven by the SUSY-breaking vacuum energy density, i.e., Polonyi inflation, becomes a viable option.

1.2 Two avenues towards a vanishing cosmological constant

Let us now outline our general philosophy in, to some extent, simplified terms. We suppose that, from the perspective of the low-energy effective theory, the breaking of supersymmetry and symmetry are two distinct dynamical processes, taking place in two different hidden sectors at two different times, and . In particular, we assume that the parameters and may be sampled in the UV theory, so that in some patches of the early universe (or in the landscape of string vacua [32] for that purpose) and in other patches . Note that this differentiation equally includes the case of a maximally symmetric initial state, , as well as the possibility that either supersymmetry or symmetry is already broken from the very beginning, . Our crucial observation is that regions in space where symmetry is broken before supersymmetry correspond to bubbles of anti-de Sitter (AdS) space with an AdS radius equal to the inverse of the gravitino mass , while regions in space where supersymmetry is broken before symmetry correspond to bubbles of de Sitter (dS) space with a dS Radius equal to the inverse of the Hubble parameter ,


As long as supersymmetry is unbroken, the AdS bubbles represent open Friedmann-Lemaître-Robertson-Walker (FLRW) universes with a negative CC and an oscillatory scale factor. The dS bubbles, on the other hand, turn, as long as symmetry is unbroken, into (asymptotically) flat FLRW universes with a positive CC and an exponentially growing scale factor [33]. This is to say that the dS bubbles experience inflation, while the AdS bubbles remain limited in their spatial extent. Furthermore, if we suppose that there is no other source of inflation present in the theory, this means that the AdS bubbles will never develop into habitable universes. Intelligent observers and humans can, therefore, only life in regions where, initially, symmetry remains unbroken up to a certain high energy scale. This applies, in particular, to our own observable Universe. Under the above assumptions, the inflationary period in the early history of our Universe must have been a consequence of spontaneous supersymmetry breaking and symmetry must have been broken only at late times, i.e., after a sufficient amount of inflation. Put differently, we can say that supersymmetry breaking and Weinberg’s argument regarding the size of the CC postdict a period of inflation and late-time symmetry breaking in our cosmic past.

This conclusion also sheds new light on the role of the CC itself. The fine-tuning of the CC is now a dynamical process that takes place only after inflation. In order to obtain zero CC, we have to require that the gravitino mass generated during symmetry breaking matches the inflationary Hubble rate,333Note that the relation between and in our scenario is conceptually quite different from other inflation models, such as, e.g., the one in [4], where on purely phenomenological grounds. In our case, is a property of the low-energy effective Lagrangian of our Universe that ensures that we live in a vacuum with an almost zero CC.


If symmetry breaking results in a gravitino mass smaller than , inflation never ends; if it “overshoots” and the gravitino mass is eventually larger than , our Universe becomes AdS. We, thus, recognize the requirement that inflation must terminate at one point or another as part of the reason why the CC in our Universe is fine-tuned. A certain amount of fine-tuning during late-time symmetry breaking is inevitable, as inflation would otherwise not exit into a near-Minkowski vacuum. This situation needs to be contrasted with standard scenarios of inflation, such as chaotic [34] or hybrid [35] inflation, where the vacuum energy density driving inflation is neither related to symmetry breaking nor to low-energy supersymmetry breaking. These scenarios require an independent reason for the vanishing of the inflationary vacuum energy density (e.g., a tuning of the inflaton potential or some kind of waterfall transition), whereas in our case this reason is already inherent to the fine-tuning of the CC in the course of spontaneous symmetry breaking at the end of inflation.

1.3 Ingredients for a realistic model of Polonyi inflation

The above reasoning is just a rough sketch. To construct a realistic model of Polonyi inflation, we need to be more specific. This pertains, first of all, to the kind of symmetry that we have in mind. Naively, our first attempt might be to protect by means of a global symmetry. On general grounds, quantum gravity is, however, expected to explicitly break all global symmetries (see [36] and references therein), so that a global does not appear to be a viable possibility. Meanwhile, gauging a continuous symmetry is a subtle issue that easily results in conflicts with anomaly constraints at low energies (see [37] for a recent discussion). This leaves us with a discrete gauged (i.e., anomaly-free [38]) symmetry, , as a unique choice to ensure the vanishing of the superpotential at early times.

Interestingly enough, such a discrete symmetry readily comes with a number of other advantages in the context of SUSY phenomenology: (i) A discrete symmetry prevents too rapid proton decay via perilous dimension-5 operators [39]; (ii) it may give rise to an accidental approximate global Peccei-Quinn symmetry and, thus, help in solving the strong problem [40, 41]; (iii) and it may account for the approximate global continuous symmetry which is required to realize stable [42] or meta-stable [43] SUSY-breaking vacua in a large class of models of dynamical supersymmetry breaking (DSB). Moreover, if we restrict ourselves to the special case of a symmetry, symmetry can also help us in solving the problem [44] in the minimal supersymmetric standard model (MSSM). Any discrete symmetry suppresses the bilinear Higgs mass term (i.e., the term) in the superpotential. But only in the case of a symmetry, we are, in addition, allowed to include a Higgs bilinear term in the Kähler potential, (see [41] and references therein for an extended discussion of this point). We are then able to generate the term in the course of symmetry breaking [45], which directly relates the parameter to the gravitino mass.444This solution to the problem is not to be confused with the Giudice-Masiero mechanism [46], which relates the generation of the term to the spontaneous breaking of supersymmetry rather than to symmetry breaking. It is for this reason that we will assume a discrete symmetry in the following.

For our purposes, it will not be necessary to specify the origin of this symmetry. But it is interesting to note that orbifold compactifications of the heterotic string have the ability to yield discrete symmetries in the low-energy effective theory. In this case, the discrete symmetry at low energies is nothing but a remnant of the higher-dimensional Lorentz symmetry that survives the compactification of the internal space (for early as well as more recent work on this topic, see [47] and [48], respectively). We also mention that the special case of a discrete symmetry has received particular attention in the context of orbifold compactifications in recent years [49]. The assumption of a discrete symmetry in the low-energy effective theory is, therefore, well motivated and stands theoretically on a sound footing.

Second of all, inflation is more than just a pure dS phase. It represents a stage of quasi-dS expansion, in the course of which the Hubble parameter slowly varies. We, thus, need to specify the dynamics of supersymmetry breaking more precisely and check whether the corresponding Polonyi potential is, in fact, suitable for slow-roll inflation. Here, we shall work within the framework of dynamical supersymmetry breaking [50], in which supersymmetry is assumed to be broken by the dynamics of a strongly coupled SUSY gauge theory. This gives us the advantage that the SUSY breaking scale is generated dynamically via dimensional transmutation. As far as supersymmetry breaking is concerned, we will, therefore, not have to rely on any dimensionful input parameters. Instead, and, thus, the energy scale of inflation, , will be controlled by the dynamical scale of the SUSY-breaking hidden sector,


In this sense, our inflation model should be regarded as a variant of dynamical inflation [51, 52, 53], as we assume the energy scale of inflation to be generated by strong dynamics. Similarly, our model is closely related to natural inflation [54], which treats the inflaton as an axion-like field that is likewise subject to a scalar potential generated by nonperturbative dynamics (for recent dynamical implementations of natural inflation in field theory, see [55]), as well as to modulus inflation [56], where one identifies the inflaton as a (composite) modulus in the effective low-energy regime of strongly coupled gauge theories.

In our case, the role of the inflaton is played by the scalar component of the chiral Polonyi field , which breaks supersymmetry via its nonzero -term. In global supersymmetry and at the classical level, the scalar Polonyi potential is exactly flat and, thus, an ideal starting point for the realization of inflation. At the quantum level and in supergravity, the Polonyi potential, however, receives corrections, which may or may not spoil the flatness of the potential. Here, the SUGRA corrections lead, in particular, to the notorious problem [57], which typically requires a parameter fine-tuning at the level of or so. The quantum and gravity corrections to the effective scalar potential scale with the coupling strengths of the Yukawa interactions between the Polonyi field and matter fields, , as well as with the coefficients of the higher-dimensional Polonyi terms in the effective Kähler potential, , respectively. In order to assess the prospects of successful Polonyi inflation, one, therefore, has to study the viability of inflation as a function of the parameters and and identify those parameter ranges that lead to consistency with the observational data on the cosmic microwave background (CMB) [58].

1.4 Our setup: minimal model based on two strongly coupled gauge theories

To this end, we will present in this paper a minimal realization of the idea of Polonyi inflation. Our model is based on two strongly coupled hidden gauge sectors (featuring two quark/antiquark pairs each), which we take to be responsible for the dynamical breaking of supersymmetry and symmetry, respectively. We supplement both sectors with an appropriate number of gauge singlet fields, so that the SUSY-breaking sector becomes identical to the simplest version of the IYIT DSB model [59], while the -symmetry breaking sector turns into a strongly coupled SQCD theory with a quantum mechanically deformed moduli space and field-dependent quark masses [60]. More precisely, we assume the quark masses in the symmetry-breaking sector to be controlled by the VEV of a singlet, which we call .

Inflation is then driven by the SUSY-breaking vacuum energy density in the IYIT sector, which results in an inflaton potential equivalent to that of supersymmetric F-term hybrid inflation (FHI) [61], including corrections from supergravity [62] as well as from higher-dimensional terms in the tree-level Kähler potential [63, 64]. This is reminiscent of the inflation models presented in [6] and [51]. The model in [51], however, corresponds to a reduced version of the IYIT model with less singlets, which leads to the vanishing of the inflationary vacuum energy at the end of inflation. Contrary to our approach, it, thus, does not establish a connection between inflation and supersymmetry breaking. Meanwhile, the model in [6] identifies a different singlet field, other than the Polonyi field, as the inflaton. This allows the author of [6] to separate the scales of inflation and supersymmetry breaking by imposing a hierarchy among the Yukawa couplings of the inflaton and the SUSY-breaking field. We, on the other hand, will show how to implement inflation into the full IYIT model, sticking to the notion that the best motivated inflaton candidate in the IYIT model is still the Polonyi field itself. An important consequence of this approach is that, in our case, F-term hybrid inflation does not end in a waterfall transition in the inflationary sector. Instead, we simply retain the inflationary vacuum energy density at low energies, which then continues to act as the vacuum energy density associated with the spontaneous breaking of supersymmetry. This also means that our “waterfall transition-free” scenario of F-term hybrid inflation does not suffer from the usual production of topological defects, such as cosmic strings in the case of a waterfall transition, which would otherwise exert some serious phenomenological pressure on our model (see, e.g., [65]).

We assume that the SUSY-breaking sector and the -symmetry breaking sector only communicate with each other via interactions in the Kähler potential (i.e, not via interactions in the superpotential). This is sufficient to stabilize the scalar field during inflation at by means of its Hubble-induced mass. During inflation, the (fermionic) quarks in the -symmetry breaking sector are, therefore, massless and the discrete symmetry remains unbroken in this sector. As anticipated, the superpotential then lacks a constant term during inflation, which relieves the inflationary dynamics from the most dangerous gravitational corrections in supergravity. In particular, the inflaton potential is free of the notorious “tadpole term” linear in the inflaton field [7, 65, 66]. Towards the end of inflation, the Hubble-induced mass of the scalar field decreases. Adding an appropriately chosen superpotential for the field , we can use this fact to trigger a waterfall transition in the symmetry-breaking sector at small inflaton field values. The field then acquires a large VEV and the quarks in the symmetry-breaking sector become very massive. Consequently, the symmetry-breaking sector turns into a pure super-Yang-Mills (SYM) theory and symmetry becomes spontaneously broken via gaugino condensation [67]. This external waterfall transition is associated with the breaking of a parity, which is, however, only an approximate symmetry. For this reason, we do not have to fear the production of topological defects (i.e., domain walls) during the waterfall transition in the symmetry-breaking sector.

After these introductory remarks, we are now in the position to present our analysis. The remainder of this paper is organized as follows. In the next section, we will show how the IYIT DSB model may give rise to dynamical F-term hybrid inflation. Here, we will first argue why the original Polonyi model [68] of supersymmetry breaking is not sufficient for a successful realization of Polonyi inflation, even if we assume zero constant in the superpotential during inflation. We, therefore, conclude that we only have a chance of successfully realizing Polonyi inflation in the presence of radiative corrections—which leads us to consider the IYIT model as a possible UV completion of the original Polonyi model. We then derive the scalar potential of F-term hybrid inflation in the IYIT model and discuss its embedding into supergravity. As an interesting aside, we demonstrate that Polonyi inflation is incompatible with the concept of an approximate shift symmetry in the inflaton direction [69]. Instead, it turns out that Polonyi inflation requires a near-canonical Kähler potential. In Sec. 3, we expand on our mechanism of late-time symmetry breaking, showing how a small inflaton field value may trigger gaugino condensation in a separate hidden sector. Related to that, we comment on the backreaction of the symmetry-breaking sector on the inflationary dynamics and discuss how the two sectors of supersymmetry and symmetry breaking have to conspire to yield a zero CC in the true vacuum after inflation. In Sec. 4, we turn to the phenomenological implications of our scenario. Here, we identify the viable region in parameter space that leads to agreement with the latest PLANCK data on the inflationary CMB observables [58]. As we are able to show, a scalar spectral index of can be easily achieved for an Yukawa coupling, , and a slightly suppressed coefficient in the noncanonical Kähler potential, . The amplitude of the scalar power spectrum, , fixes the dynamical scale of the SUSY-breaking sector to a value close to the unification scale, , suggesting that our setup may eventually be part of a grand unified theory (GUT). As a characteristic feature of Polonyi inflation, we highlight the fact that the relation between the gravitino mass and the inflationary Hubble rate in Eq. (3) directly translates into a one-to-one correspondence between the gravitino mass and the tensor-to-scalar ratio ,


At this point, it is interesting to note that the observed value of the scalar spectral amplitude, , might be the result of anthropic selection [70]. Together with the paradigm of slow-roll inflation (which implies ), the anthropic value of could, therefore, explain why the soft SUSY mass scale is so much higher than the electroweak or TeV scale. Moreover, we point out in Sec. 4 that, after inflation, the Polonyi field mostly decays into gravitinos. Polonyi inflation is, thus, followed by a phase of gravitino domination [71], which leads to reheating around temperatures of . This paves the way for thermal wino dark matter (DM) as well as thermal leptogenesis [72] enhanced by resonance effects [73].

Appendix A contains some technical details regarding the derivation of the effective inflaton potential. In particular, we show how the one-loop corrections to the effective potential may be obtained, to leading order, from an effective Kähler potential. In Appendix B, we explain in more detail why Polonyi inflation in the IYIT model does not work, if the superpotential already contains a constant term from the very beginning. This completes our argument that, in the context of our minimal model, successful Polonyi inflation requires (i) radiative corrections, (ii) a near-canonical Kähler potential, (iii) as well as late-time symmetry breaking. Among all possible choices regarding (i) the type of interactions that the Polonyi field participates in, (ii) the shape of the Kähler potential, and (iii) the chronology of supersymmetry and symmetry breaking, this leave one unique possibility for how to realize Polonyi inflation.

2 Dynamical inflation in the IYIT supersymmetry breaking model

2.1 Inflation in the original Polonyi model and the need for radiative corrections

The Polonyi model [68] is the simplest O’Raifeartaigh model of supersymmetry breaking via a nonvanishing F-term. Its superpotential consists of a tadpole term and a constant,


Here, denotes the chiral Polonyi field, is the scale of supersymmetry breaking and is a constant that breaks symmetry and which determines the potential energy density in the ground state. For a canonical Kähler potential and fine-tuning , so that it takes the particular value , this model has a Minkowski vacuum at , in which supersymmetry is broken by the Polonyi F-term, . It has been known for a long time, that the scalar potential for the Polonyi field around this vacuum is unfortunately too steep to support slow-roll inflation [30]. In Appendix B, we review and extend this argument, showing for various choices of the Kähler potential, that, with from the very beginning, the Polonyi model does not give rise to inflation. We consider, in particular, a canonical Kähler potential supplemented by higher-dimensional corrections as well as Kähler potentials featuring an approximate shift symmetry either along the real or the imaginary axis in the complex plane. In none of the cases under consideration inflation is viable—either because we fail to satisfy the slow-roll conditions or because the scalar potential does not exhibit a global Minkowski vacuum in the first place.

This immediately raises the question whether inflation might perhaps become possible in the Polonyi model, if we impose a discrete symmetry at high energies, so that initially. Let us address this question for a canonical Kähler potential supplemented by a higher-dimensional correction,555In Sec. 2.4, we will discuss the same question for an approximately shift-symmetric Kähler potential. In this case, inflation turns out be unfeasible because, with the superpotential being given as , the SUGRA term in the scalar potential induces a tachyonic mass for the Polonyi field. This results in a global AdS minimum.


Here, we assume that the Kähler potential is always dominated by the canonical term, , also at field values above the Planck scale. An exhaustive study of arbitrary choices for the Kähler potential is beyond the scope of this paper. Under this assumption, the scalar potential always picks up a SUGRA correction, , which spoils the flatness of the potential at super-Planckian field values. For this reason, we only have a chance of realizing slow-roll inflation at field values below the Planck scale. For the Kähler potential in Eq. (7), the scalar potential in supergravity then takes the following form,


where the real scalar field denotes the canonically normalized radial component of the complex Polonyi scalar contained in (see Sec. 2.4) and where we have introduced as the SUSY-breaking vacuum energy density at . From the form of the scalar potential in Eq. (8), it is evident that, even with being set to zero, the Polonyi model fails to yield successful inflation. For instance, if we choose to be negative, the field is driven towards the origin by a positive mass squared, similarly as in chaotic inflation [34]. Inflation may then take place at small field values close to the origin—but not in accord with the observational data. To see this, consider the slow-roll parameters and ,


Independent of the sign of , we have . For negative , the slow-roll parameter is, therefore, bound to be positive, while the slow-roll parameter turns out to be negligibly small during inflation, . According to the slow-roll formula for the scalar spectral index, , we will then always obtain a blue-tilted scalar spectrum (). In view of the latest best-fit value for reported by the PLANCK collaboration,  [58], such a spectrum is clearly ruled out by the observational data. On the other hand, if we choose to be positive, the Polonyi field acquires a tachyonic mass around the origin and inflation proceeds from small to large field values, similarly as in new inflation [2]. It is then hard to imagine how the generation of the constant in the superpotential should be triggered, after a sufficient amount of inflation, at field values close or even above the Planck scale. But more than that, even if we assume that this problem could somehow be solved, the scalar potential in Eq. (8) still does not lead to an acceptable phenomenology. In new inflation, the scalar spectral index turns out to be bounded from above,  [74, 75], which deviates from the observed value by at least . Therefore, also for , we fail to reach consistency with the observational data.

In summary, we conclude that the bare Polonyi model based on the superpotential in Eq. (6)—and for reasonable ansätze regarding the shape of the Kähler potential—does not allow for a successful realization of slow-roll inflation. In respect of this null result, two comments are in order: (i) We emphasize that our analysis of the Polonyi Kähler potential in this paper does not mount up to a general no-go theorem. In the most general case, the Kähler potential for the Polonyi field is given by an arbitrary function of the field and its conjugate, . In absence of any other scale, it is, in particular, clear that can be the only relevant scale in the Kähler potential. It may then well be that certain fine-tuned functions do allow for successful inflation in the Polonyi model, after all (see, e.g., [76] for a discussion of fine-tuned Kähler potentials in the context of SUGRA models of inflation). In the following, we will, however, ignore the possibility of such a biasedly chosen Kähler potential and focus on the usual suspects: Kähler potentials that are either near-canonical or approximately shift-symmetric. (ii) The fact that we are unable to realize successful inflation in the bare Polonyi model is not a serious problem, as the Polonyi model is not expected to be a fundamental description of spontaneous supersymmetry breaking, anyway. It should rather be seen as the effective theory resulting from some UV dynamics that provide a dynamical explanation for the origin of the parameters and in Eq. (6). From this perspective, it is then more likely than not that the Polonyi field is not only subject to its gravitational self-interaction, but that it also participates in Yukawa interactions with heavy matter fields in the UV theory. In the corresponding effective Polonyi model at low energies, these matter fields are integrated out, so that they no longer appear in the superpotential. But their couplings to the Polonyi field still yield radiative corrections to the scalar potential in Eq. (8), which affect the inflationary dynamics. In such a modified setup, i.e., in the original Polonyi model supplemented by radiative corrections, successful Polonyi inflation may, therefore, very well be an option. In the following, we will construct a minimal extension of the Polonyi model where this is indeed the case. We shall consider a minimal UV completion of the Polonyi model—consisting of two strongly coupled sectors that account for the dynamical origin of the parameters and , respectively—and demonstrate that, in the presence of radiative corrections, successful Polonyi inflation is indeed feasible for a natural choice of parameters values.

2.2 Dynamical supersymmetry breaking in the low-energy regime of the IYIT model

One of the simplest ways to generate the SUSY breaking scale in Eq. (6) is to identify the Polonyi field as part of the IYIT model—the simplest vector-like model of dynamical supersymmetry breaking [59]. In its most general formulation, the IYIT model is based on a strongly coupled gauge theory featuring pairs of “quark fields” that transform in the fundamental representation of . The gauge dynamics of this model are associated with a dynamical scale , which denotes the energy scale at which the gauge coupling formally diverges. The low-energy effective theory below the dynamical scale exhibits a quantum moduli space of degenerate supersymmetric vacua, which is spanned by gauge-invariant composite flat directions (or “meson fields”) ,


Here, the parameter is a dimensionless numerical factor, which ensures the canonical normalization of the meson fields at low energies. Naive dimensionful analysis [77] leads us to expect that should be of and, for definiteness, we will, therefore, simply set in the following. The quantum moduli space of the IYIT model is subject to the following constraint pertaining to the meson VEVs,


which represents the quantum mechanically deformed version of the classical constraint  [60]. In order to break supersymmetry in the IYIT model, one has to lift the flat directions , so that Eq. (11) is longer compatible with a vanishing vacuum energy density. This is readily done by coupling the quark pairs to a corresponding number of singlet fields, , in the tree-level superpotential,


where denotes a matrix of Yukawa couplings with at most independent eigenvalues. These Yukawa couplings induce Dirac mass terms for the meson and singlet fields at low energies,


so that the singlet F-term conditions, , are incompatible with the deformed moduli constraint, . Supersymmetry is then broken à la O’Raifeartaigh via nonvanishing F-terms.

In global supersymmetry and for all Yukawa couplings in Eq. (12) being equal, , the anomaly-free global flavor symmetry of the IYIT model is given as follows,


Here, the discrete symmetry is the anomaly-free subgroup of the anomalous that is contained in the full flavor symmetry at the classical level, . Under the symmetry, all quarks carry charge , while the singlet fields carry charge . Meanwhile, the presence of the global continuous symmetry is characteristic for a large class of DSB models [42]. Under , the quark and singlet fields carry charges and , respectively. In the SUSY-breaking vacuum of the IYIT model, where and , symmetry, therefore, remains unbroken. As a global symmetry, the continuous symmetry of the IYIT model is, of course, only an approximate symmetry, which we expect to be broken by quantum gravitational effects [36]. On the other hand, recall that we assume the discrete subgroup to be gauged (see our discussion in Sec. 1.3). This protects the quality of the symmetry; and it is reasonable to assume that all gravity-induced -breaking effects in the IYIT sector are suppressed. In the following, we will, therefore, stick to the effective superpotential in Eq. (13) and neglect the possibility of small symmetry-breaking corrections.

From now on, let us restrict ourselves to the simplest version of the IYIT model: a strongly coupled gauge theory featuring four matter fields and six singlet fields . The non-Abelian flavor symmetry then corresponds to a global , under which and transform as six-dimensional antisymmetric rank-2 tensor representations. Here, note that is the double cover of . This allows us to rewrite the meson and singlet fields as vector representations of ,


As we will see shortly, it will turn out to be convenient to study the SUSY-breaking dynamics of the IYIT sector in terms of this “ language” rather than in terms of the original “ language”.

Trading the mesons for the new fields , the Pfaffian constraint in Eq. (11) can be written as,


which defines a sphere in the six-dimensional space spanned by the six meson coordinates . An elegant way to enforce this constraint is to directly incorporate it into the effective superpotential [60, 59],


Here, the field denotes a Lagrange multiplier, the corresponding F-term condition of which, , is nothing but a reformulation of the moduli constraint in Eq. (16). The physical status of the field is unfortunately rather unclear and depends on whether (uncalculable) strong-coupling effects induce a kinetic term for or not. If the field should be physical, it might represent a dynamical glueball field, , and the dimensionless coupling constant in Eq. (17) would be expected to take some value of . In that case, the Pfaffian constraint in Eq. (16) would be satisfied only approximately, depending on the competition between the different F-term conditions that enter into the determination of the true ground state. If, on the other hand, should be unphysical, we would have to treat it as a mere auxiliary field. In that case, the Pfaffian constraint should be satisfied exactly, which would require us to eventually take the limit in our analysis. In the following, we will suppose that the Lagrange multiplier field is, indeed, a physical (glueball) field and set for definiteness.666For an extended discussion of this point, see also [41, 78].

In terms of the fields and , the effective tree-level superpotential takes the following form,


where we assume, w.l.o.g., the Yukawa couplings to be ordered by size, for all . Note that we will refer to the smallest Yukawa coupling, , simply as in the following, . For generic values of the six Yukawa couplings , the non-Abelian flavor symmetry is completely broken,


The total effective superpotential is given by the sum of in Eq. (17) and in Eq. (18),


We mention once more that, as a consequence of the nonrenormalization theorem, the effective superpotential does not receive radiative corrections in perturbation theory [23]. Given this form of the total superpotential and assuming a quadratic Kähler potential for all meson and singlet fields, the global minimum of the resulting F-term scalar potential is located at


where is undetermined at tree level. The sign ambiguity is a consequence of the flavor symmetry (see Eq. (19)). Moreover, the parameter measures how well the deformed moduli constraint is satisfied,


For perturbative values of the Yukawa coupling , i.e., for or smaller, we have , which tells us that the deformed moduli constraint is fulfilled almost exactly. On the other hand, for nonperturbative values of , i.e., for as large as , the parameter becomes of , indicating that the constraint function significantly deviates from zero. To put this result into perspective, we must remember that, for nonperturbative values of the Yukawa coupling, uncalculable corrections to the effective Kähler potential due to strong-coupling effects become important. In fact, as pointed out by Chacko et al., these nonperturbative corrections are only negligible as long as  [79]. We can, therefore, trust our above analysis, based on a canonical Kähler potential, only as long as remains in the perturbative regime. For this reason, we will, from now on, only consider values at most as large as , so that we may always maintain a hierarchy among and (i.e., so that ). For the parameter , this then means that it can take values at most as large as . This translates into the statement that the moduli constraint is always satisfied in our analysis—up to a deviation of at most , .

In passing, we also mention that the scalar potential exhibits a saddle point at the origin in field space as well as two saddle points along each direction in moduli space. Here, the loci of the saddle points away from the origin have the same functional form as and in Eq. (21), the only difference being that in Eq. (22) needs to be exchanged with the respective Yukawa coupling . The low-energy vacuum along the axis, therefore, not only marks the global (and only local) minimum of the scalar potential, it is also the stationary point at which the deformed moduli constraint is fulfilled best.

In the true vacuum, supersymmetry is broken by the nonvanishing F-terms of the fields and ,


The F-term of the meson field , on the other hand, (which seems to be nonzero at first sight) cancels,


In order to identify the mass eigenstates around the true vacuum, we now shift the field by its VEV,


and rotate the SUSY-breaking fields and by their mixing angle ,


After these field transformations, the effective superpotential for , , , , and reads as follows,


which is (apart from the missing constant term ) exactly of the form anticipated at the end of Sec. 2.1.

First of all, note that the first term on the right-hand side of Eq. (27), , is nothing but the dynamical realization of the SUSY-breaking tadpole term in Eq. (6) within the IYIT model. The field is, thus, to be identified as the chiral Polonyi field which breaks supersymmetry via its nonzero F-term,


where the parameter denotes again the SUSY breaking scale. In this sense, Eqs. (27) and (28) show that the IYIT model serves, indeed, as a viable UV completion for at least half the Polonyi model: The SUSY-breaking dynamics of the IYIT model manage to provide a dynamical explanation for the SUSY breaking scale . However, as the IYIT model preserves symmetry in its ground state, it is not capable of accounting for the origin of the symmetry-breaking constant in the Polonyi superpotential.

Second of all, the effective superpotential in Eq. (27) contains (also as envisaged at the end of Sec. 2.1) Yukawa couplings between the Polonyi field and a number of massive matter fields, and . Here, the meson masses, and , follow from Dirac mass terms together with the singlet fields and ,


where we have introduced the flavor-independent mass scale as well as the respective ratios and between this scale and the masses and . From the fact that (see Eq. (31) below), it immediately follows that the scale represents the amount of SUSY-breaking mass splitting within the respective meson and singlet multiplets that is induced by the tadpole term in Eq. (27), see Appendix A.1 for details. Given the definition of in Eq. (22) and recalling that we assume to be the smallest among all Yukawa couplings, , we also find that the ratios and are bounded from above,


so that the SUSY-breaking mass splitting never exceeds the supersymmetric Dirac masses . Moreover, for the parameter range of interest, and , the ratio always turns out to be the smallest, , which leads to the interesting (and to some extent counter-intuitive) result that the zeroth flavor, i.e., the flavor with the smallest Yukawa coupling, ends up being stabilized the most. In addition to that, the flavor is also singled out by the fact that its Dirac mass partner is none of the original singlet fields , but the linear combination that we introduced in Eq. (26) and which, for small values of , mostly consists of the Lagrange multiplier field . This also explains why, for large (and, hence, small ), the mass diverges. In this limit, the Pfaffian constraint is fulfilled exactly, which results in the decoupling of and removes one meson multiplet (i.e., ) from the spectrum.

Next to the Polonyi field , also the “stabilizer field” couples to the meson fields and . Here, the strengths of the respective Yukawa couplings, and , are given by and the mixing angle ,


Just like the mass , the Yukawa coupling diverges in the limit . This is a trivial consequence of its proportionality to . As for the Polonyi coupling , this divergence is, however, canceled out by the factor. In contrast to , the coupling , therefore, always remains finite. In the limit , it reproduces, in particular, the Yukawa coupling of the singlet at energies above the dynamical scale,


Finally, we mention that the four new parameters , , , and introduced in Eq. (27) are not linearly independent. In fact, they must be dependent, as they can all be expressed in terms of the three old parameters , , and . By making use of Eqs. (28), (29), and (31), one easily convinces oneself that


In order to see how the superpotential in Eq. (27) may give rise to Polonyi inflation, it is instructive to forget about the Yukawa couplings of the field for a moment and to rewrite Eq. (27) as follows,


where the ellipsis stands for the Yukawa couplings involving the stabilizer field and where we have introduced the mass scale ,


Remarkably enough, the first part of the superpotential in Eq. (34) has the same form as the superpotential of supersymmetric F-term hybrid inflation [61] based on . In the context of this interpretation, the Polonyi field plays the role of the chiral inflaton singlet, while the meson fields and act as a multiplet of FHI waterfall fields that transform in the vector representation of . The mass scale is then to be identified as the energy scale of the waterfall transition at the end of inflation, while the mass splitting should be understood as the tachyonic mass of the FHI waterfall fields at . We note that it is this picture that the authors of [51] arrive at. In their model, no other singlets except for the Polonyi field are introduced. The meson fields and , therefore, lack their Dirac mass partners, so that the resulting effective superpotential is exactly identical to the one of F-term hybrid inflation. This allows the authors of [51] to realize F-term hybrid inflation at large inflaton field values, , where the flatness of the inflaton potential is lifted by logarithmic loop corrections.

In our case, the situation at large field values is quite similar, as we will discuss shortly; but at small field values, it is drastically different. The second part of the superpotential in Eq. (34) introduces explicit Dirac mass terms for the “would-be waterfall fields” and that are absent in standard F-term hybrid inflation (as well as in the model in [51]). Accounting for the presence of these Dirac masses in the low-energy effective theory, the tachyonic waterfall mass is always compensated (see Eq. (30)), so that none of the meson fields ever becomes destabilized. Because of that, the total superpotential in Eq. (34) fails to give rise to a waterfall transition and, even in the low-energy vacuum, we retain the vacuum energy density resulting from the nonzero Polonyi F-term, . As anticipated in Sec. 1.4, this is a characteristic feature of our construction, in which we intend to use one and the same vacuum energy density for driving inflation and breaking supersymmetry. Moreover, independent of the symmetry group under which the meson fields transform, the absence of the waterfall transition automatically implies that the end of inflation is not accompanied by the production of topological defects. This may be regarded as a significant phenomenological advantage of our scenario over standard F-term hybrid inflation.

2.3 Pseudomodulus potential in global supersymmetry

To the best of our knowledge, the superpotential in Eq. (34) has not been considered as the dynamical origin of inflation, so far. Here, part of the reason certainly is that successful inflation based on Eq. (34) is bound to require a rather high SUSY breaking scale . As explained in the introduction, in supergravity, this necessitates a large constant in the superpotential to cancel the CC in the true vacuum, which then spoils slow-roll inflation (see Sec. 1.1). In the rest of this paper, we will, however, show that the superpotential in Eq. (34) can yield successful Polonyi inflation, after all, if we generate the constant in the superpotential only towards the end of inflation. To this end, we shall now examine the effective one-loop potential for the complex Polonyi scalar in global supersymmetry more closely. In the next sections, we will then turn to the embedding of the IYIT model into supergravity (see Sec. 2.4) as well as to the generation of the constant term in a separate hidden sector (see Sec. 3).

The Yukawa interactions between the Polonyi field and the meson fields and in Eq. (27) lead to radiative corrections to the Polonyi potential, , that may be calculated according to the Coleman-Weinberg (CW) formula for the effective one-loop potential [80]. The details of our calculation may be found in Appendix A; in the following, we will merely summarize our results. Generally speaking, the effective potential may be divided into two regimes: (i) At large field values, all of the meson fields acquire a large inflaton-dependent Majorana mass . For , the supersymmetric Dirac masses in Eq. (27) are, hence, negligible and the effective potential takes the usual logarithmic form as in standard F-term hybrid inflation, . (ii) On the other hand, at small Polonyi field values, such that , the Dirac masses become more relevant. Integrating out the “heavy fields” then leads to a quadratic Polonyi potential around the origin, . We note that, as has been shown for the first time in [79], the effective potential around the origin has positive curvature. The low-energy vacuum at is, therefore, indeed stable. Moreover, we find that, at large as well as at small field values, the effective potential scales with the soft SUSY-breaking mass scale , i.e., the smallest mass scale in our model (see Eq. (30)). This illustrates how supersymmetry succeeds in protecting the inflaton potential from picking up too large radiative corrections [25].

To quantify the above statements, it is convenient to introduce the following mass ratios,


For large values of , we then obtain a logarithmic one-loop potential, while for small values of , the one-loop corrections take a quadratic form. This behavior can be captured by studying the effective potential as a function of a single order parameter , the geometric mean of all ratios ,


Here, counts the number of meson fields in the IYIT sector, while stands for the geometric mean of all explicit mass parameters in Eq. (27). In this sense, denotes the “supersymmetric mass scale” of the IYIT sector, i.e., a characteristic value for the nonperturbatively generated Dirac masses in the low-energy effective theory. In the following, we will set to the dynamical scale , for definiteness,


This mainly serves the purpose to account, in an effective way, for heavy composite states with masses around the dynamical scale that we expect to be present, but which we can unfortunately not explicitly describe in terms of our perturbative language at low energies. Formally, we can always set to in our calculation by choosing an appropriate value for the effective heavy-flavor Yukawa coupling ,


By fixing at a nonperturbative value, we are, therefore, able to enforce our designated value for ,


The value of indicates whether the Dirac masses are negligible or not and, thus, decides whether we are in the logarithmic or the quadratic part of the effective potential. The transition between both regimes takes place at field values close to what we shall refer to as the critical field value ,


This implies that the order parameter can also be regarded as the ratio of the actual and the critical field value, . Far away from the critical field value, i.e., at and , we now find the following expressions for the effective potential (see Appendix A for details),


with denoting the effective one-loop mass of the Polonyi field around the origin,777A similar expression has been derived for the first time in [79]. Our result differs from the one in [79] to the extent that we allow for nonzero (and, hence, nonzero ), which means that we do not necessarily enforce the moduli constraint exactly. The calculation in [79], on the other hand, is based on the assumption that the moduli constraint is fulfilled exactly, so that . For a recent derivation and discussion of the effective Polonyi mass in language, see [78, 41].


Here, counts the effective number of mesons that contribute to the effective Polonyi mass. The full functional form of is a sum of complicated loop factors . To good approximation, these loop functions, however, happen to coincide with the mass ratios squared, . We can write the result in Eq. (42) more compactly, if we make use of the following two potential energy scales,


The effective potential far away from the critical field value then takes the following form (see Fig. 1),

Figure 1: Effective CW potential for the Polonyi field, , as a function of the order parameter for , see Eq. (45). The potential energy scales and are given in Eq. (44). For values , we can only have limited trust in our perturbative result because of potentially important strong-coupling effects at energies close to the dynamical scale .

The crucial question which we need to answer in the following is: Can we use either the low-energy or the high-energy part of this effective potential to realize successful Polonyi inflation? Let us first investigate whether inflation might occur in the quadratic part close to the origin. As we know from standard chaotic inflation [34], the effective inflaton mass then needs to take a value of to ensure the correct normalization of the scalar power spectrum. This requires the coupling to take a value at least as large , since otherwise the dynamical scale would have to be super-Planckian,


At the same time, we know that chaotic inflation requires a large super-Planckian field excursion to yield a sufficient number of -folds. The scalar perturbations probed in CMB observations, e.g., cross outside the Hubble horizon at a field value . However, in the context of our SUSY breaking model, this large field range does not “fit” into the low-energy part of the effective potential. This follows from the fact that, for and , the critical field value only becomes as large as ,


Therefore, to raise , so as to make the field range required for chaotic inflation fit into the quadratic part of the effective potential, , we would have to go to smaller values of . But then, we are either forced to push beyond the Planck scale or we fail to reproduce the correct scalar spectral amplitude. This eliminates the possibility of Polonyi inflation in the low-energy part of the effective potential, which is why we will focus on inflation in the logarithmic part of the effective potential from now on.

Before continuing, we, however, point out that inflation close to the origin might become possible, after all, if we relax our assumptions. That is, if we allowed for values of the dynamical scale as large as, say, , we would, in fact, be able to raise above . If we then trusted the full effective potential also at field values close to (see Fig. 1), inflation in the transitioning regime between the quadratic and the logarithmic part of the effective potential might become feasible. Such a scenario would promise to interpolate between the predictions of chaotic inflation and F-term hybrid inflation, so that we would expect it to result in interesting predictions for the tensor-to-scalar ratio, . Because of the uncertainties involved in such a scenario, we, however, do not pursue this idea any further in this paper and leave a more detailed study for future work. In closing, we remark that a similar model of subcritical hybrid inflation, based on a dynamically generated D-term [78], may be found in [81]. This model illustrates how to realize chaotic inflation after the waterfall transition of D-term hybrid inflation.

Let us now turn to the possibility of inflation in the logarithmic part of the effective potential. As we will show, in this part of the potential, successful Polonyi inflation is indeed feasible. To be on the safe side, we will limit our analysis in the following to field values that are larger than the critical field value by at least half an order of magnitude, . We do so because the effective Polonyi potential may receive nonperturbative corrections around that we do not have under control. In fact, around the critical field value, the inflaton-dependent mass drops below the dynamical scale (see Eq. (41)). This triggers the IYIT sector to transition from the high-energy quark-gluon regime into the low-energy meson regime. During this mesonic phase transition, the IYIT quarks become confined in the composite mesons and the strongly coupled gauge group becomes completely broken by the nonzero squark VEVs and that contribute to the meson VEV (see Eqs. (15) and (21)),888Here, follows from the fact that the linear combination vanishes in the vacuum, .


Thanks to the fact that the symmetry is spontaneously broken down to “nothing”, no topological defects are formed during the confining phase transition [82]. We emphasize that this is an important phenomenological feature of the IYIT model based on , as it allows for the particular breaking pattern . In summary, our scenario of Polonyi inflation, therefore, crucially differs from ordinary F-term hybrid inflation in the following respect: While ordinary F-term hybrid inflation ends in a waterfall transition—in the course of which the inflationary vacuum energy density is “eaten up” by the FHI waterfall fields and which potentially leads to the production of troublesome topological defects—our scenario of Polonyi inflation undergoes a confining quark-meson phase transition that conserves the inflationary vacuum energy density and that does not lead to the production of topological defects.

2.4 Embedding into supergravity and choice of the Kähler potential

In supergravity, the flatness of the tree-level Polonyi potential in global supersymmetry, , is not only lifted by the radiative corrections in Eq. (45), but also by gravitational corrections (see our discussion at the end of Sec. 1.3). In our case, these SUGRA corrections turn out to be rather mild for basically two reasons: (i) Since the superpotential in Eq. (27) only contains terms linear in , the tree-level SUGRA mass of the Polonyi field accidentally cancels, as long as we assume a canonical Kähler potential. As far as the embedding into supergravity is concerned, this represents an important advantage of F-term hybrid inflation (and of our model) over alternative models that do feature higher powers of the inflaton field in the superpotential. (ii) Since we intend to realize Polonyi inflation in the logarithmic part of the effective potential, we will consistently work with sub-Planckian field values. Our scenario of Polonyi inflation will, hence, turn out to be a small-field model of inflation. Accordingly, the SUGRA corrections in our model are bound to be less significant than in alternative large-field models of inflation.

The total scalar potential for the complex Polonyi field in supergravity now reads,999The field is not necessarily canonically normalized, which we indicate by placing a tilde on top of the symbol .


with being given in Eq. (45) and where denotes the tree-level F-term potential in supergravity,


Here, denotes the norm of the generalized F-term vector induced by the Kähler metric ,


The individual components of the the F-term vector