Inflation from High-Scale Supersymmetry Breaking

Supersymmetry breaking close to the scale of grand unification can explain cosmic inflation. As we demonstrate in this paper, this can be achieved in strongly coupled supersymmetric gauge theories, such that the energy scales of inflation and supersymmetry breaking are generated dynamically. As a consequence, both scales are related to each other and exponentially suppressed compared to the Planck scale. As an example, we consider a dynamical model in which gauging a global flavor symmetry in the supersymmetry-breaking sector gives rise to a Fayet-Iliopoulos D term. This results in successful D-term hybrid inflation in agreement with all theoretical and phenomenological constraints. The gauged flavor symmetry can be identified with U(1)_B-L, where B and L denote baryon and lepton number, respectively. In the end, we arrive at a consistent cosmological scenario that provides a unified picture of high-scale supersymmetry breaking, viable D-term hybrid inflation, spontaneous B-L breaking at the scale of grand unification, baryogenesis via leptogenesis, and standard model neutrino masses due to the type-I seesaw mechanism.

If realized around the electroweak scale, supersymmetry provides a natural solution to the large hierarchy problem in the standard model. For decades, this observation has reinforced the paradigm of low-scale supersymmetry that would be testable in collider experiments. In the present experimental situation, the null results at the LHC, however, bring about the little hierarchy problem. Sparticles with masses of at least O (1) TeV can only be reconciled with the O (100) GeV value of the electroweak scale at the cost of fine-tuning. One is therefore led to adopt one of two possible attitudes. Either one gives up on supersymmetry as a well-motivated extension of the standard model, or one challenges the concept of naturalness and accepts a certain degree of fine-tuning. In this paper, we shall take the latter approach. Our understanding of naturalness may be flawed and, for some reason or another, not apply to the physics of the electroweak scale. Moreover, dismissing supersymmetry altogether would do injustice to supersymmetry's other merits. One must not forget that, irrespective of its relation to the electroweak scale, supersymmetry also (i) provides a natural particle candidate for dark matter, (ii) facilitates the unification of the SM gauge coupling constants at a high energy scale, and (iii) sets the stage for the ultraviolet (UV) completion of the standard model in string theory.
In fact, supersymmetry broken at a high scale [11][12][13] has received significant interest in recent years. Many authors have proposed models of high-scale SUSY breaking and its mediation to the visible sector, including scenarios such as universal high-scale supersymmetry [14], split supersymmetry [15][16][17], mini-split supersymmetry [18], minimal split supersymmetry [19], spread supersymmetry [20,21], and pure gravity mediation [22][23][24] (see also [25]). Another intriguing feature of these models is that a high SUSY-breaking scale implies a very heavy gravitino. The gravitino, thus, decays very fast in the early Universe, which solves the cosmological gravitino problem [26][27][28][29]. In addition, large sfermion masses help reduce the tension with constraints on the SUSY parameter space from flavor-changing neutral currents and CP violation [30]. For these reasons, we consider supersymmetry broken at a high energy scale to be a leading candidate for new physics beyond the standard model.
In this paper, we will take the idea of high-scale SUSY breaking to the extreme and consider SUSY-breaking scales Λ SUSY as large as the scale of gauge coupling unification in typical grand unified theories (GUTs), Λ SUSY ∼ Λ GUT ∼ 10 16 GeV. Electroweak naturalness is then certainly lost. But at the same time, another intriguing possibility emerges which is out of reach in low-scale supersymmetry.
If Λ SUSY is large enough, cosmic inflation in the early Universe [31][32][33][34] can be driven by the vacuum energy density associated with the spontaneous breaking of supersymmetry, V ⊃ Λ 4 SUSY . This represents a remarkable connection between particle physics and cosmology. Inflation is a pillar of the cosmological standard model. Not only does inflation explain the large degree of homogeneity and isotropy of our Universe on cosmological scales, it is also the origin of the primordial fluctuations that eventually seed structure formation on galactic scales (see, e.g., [35,36] for reviews on inflation).
At present, there is, however, no consensus on how to make contact between inflation and particle physics. Against this background, the unification of inflation with the dynamics of spontaneous SUSY breaking provides an elegant and economical embedding of inflation into a microscopic theory.
The interplay between inflation and SUSY breaking has been studied from different angles in the past (see, e.g., [37][38][39]). In the context of supergravity (SUGRA), SUSY breaking in a hidden sector can, in particular, result in severe gravitational corrections to the inflationary dynamics [40].
In this case, one can no longer perform a naive slow-roll analysis that only considers the properties of the inflaton sector and disregards its gravitational coupling to other sectors. Instead -and this is exactly what we will do in this paper -one has to resort to a global and combined analysis that accounts for the presence and interaction of all relevant sectors, including the inflaton sector, SUSYbreaking sector, and visible sector. The first unified model that illustrates how inflation and soft SUSY breaking in the visible sector may originate from the same dynamics has been presented in [41]. In this model, supersymmetry is broken dynamically by the nonperturbative dynamics in a strongly coupled supersymmetric gauge theory. Dynamical SUSY breaking (DSB) first occurs in the hidden sector and is then mediated to the MSSM. More recently, a number of related models have been constructed in [42][43][44]. 1 All these models have in common that the energy scales of inflation and SUSY breaking end up being related to the dynamical scale Λ dyn of the strong dynamics. The scale Λ dyn is generated via dimensional transmutation, analogously to the scale of quantum chromodynamics (QCD) in the standard model. That is, at energies around Λ dyn , the gauge coupling constant in the hidden sector formally diverges. The unified models in [41][42][43][44] therefore do not require any dimensionful input parameters to explain the origin of the energy scales of inflation and SUSY breaking. The generation of Λ dyn is, in particular, a nonperturbative effect in the infrared (IR). This explains the exponential hierarchy between Λ SUSY and Λ inf on the one hand and the Planck scale M Pl on the other hand.
Meanwhile, several perturbative models of inflation and SUSY breaking have recently been discussed in the literature [51][52][53]. These models also draw a unified picture of inflation and SUSY breaking.
But in contrast to strongly coupled models, they depend on dimensionful input parameters which need to be put in by hand. They, thus, fail to provide a dynamical explanation for the separation of scales.
In this paper, we shall revisit our model in [43], which gives rise to a viable scenario of D-term hybrid inflation (DHI) [54,55] in the context of high-scale SUSY breaking. 2 Hybrid inflation [60,61] is an interesting scenario on general grounds, as it establishes another connection between particle physics and cosmology. In hybrid inflation, the inflationary era ends in a rapid second-order phase transition, the so-called waterfall transition, which can be identified with the spontaneous breaking of a local gauge symmetry in models of grand unification. As shown in [43], this symmetry can be chosen to correspond to U (1) B−L , where B and L denote baryon and lepton number, respectively.
Inflation therefore ends in what is referred to as the B−L phase transition [62][63][64][65][66][67][68][69]. That is, the end of inflation coincides with the spontaneous breaking of B−L in the visible sector. This can be used to generate L-violating Majorana masses for a number of sterile right-handed neutrinos. These neutrinos then lead to baryogenesis via leptogenesis [70] and generate the SM neutrino masses via the seesaw mechanism [71][72][73][74][75]. The model in [43], thus, provides a consistent picture of particle physics and early Universe cosmology. It unifies the dynamics of inflation, high-scale SUSY breaking, and spontaneous B−L breaking. Remarkably enough, all of these phenomena occur at energies close to the GUT scale. 1 Besides, there is a more general class of models, sometimes referred to as dynamical inflation, where inflation is a consequence of dynamical SUSY breaking in a hidden sector. In these models, the breaking of supersymmetry during inflation is, however, not responsible for the breaking of supersymmetry in the MSSM at low energies (see, e.g., [45][46][47][48][49][50]). 2 This model has recently been employed in a supersymmetric realization of the relaxion mechanism [56]. For other models of hybrid inflation and high-scale SUSY breaking, see [57,58]. For more recent work on D-term inflation, see [59].
The discussion in [43] mostly focused on model building aspects. We gave a brief summary of our model construction and only touched upon phenomenology. The purpose of the present paper is therefore threefold. We will (i) review the construction of our model in more detail, including many aspects that were left out in [43] (see Sec. 2). We will (ii) perform a more comprehensive scan of parameter space (see Sec. 3). In particular, we will identify new regimes of successful inflation that were overlooked in [43]. And finally, we will (iii) provide a much broader phenomenological discussion, including the implications of our model for the MSSM particle spectrum, the B−L phase transition, dark matter, and cosmic strings (see Sec. 4). In the last section of this paper, we will conclude and give an outlook on how our main observation -the fact that SUSY breaking close to the GUT scale might be the key to a unified picture of particle physics and cosmology -could lead to a new understanding of SUSY's role in nature (see Sec. 5). In the appendix, as a supplement to Secs. 2 and 3, we collect various technical formulas that help to translate between the Einstein-frame and Jordanframe formulations of supergravity (see Appendix A). This also includes a detailed comparison of our Jordan/Einstein-frame expressions for the inflationary slow-roll parameters.
2 Model: Strong dynamics and gauged B−L in the Jordan frame We begin by reviewing the model constructed in [43]. This will also allow us to introduce our notation and conventions. The starting point of our analysis is the idea to build a viable SUGRA model of hybrid inflation that ends in the spontaneous breaking of B−L, i.e., in the B−L phase transition.

Preliminary remarks on hybrid inflation in supergravity
In the absence of supersymmetry, hybrid inflation is incompatible with the statistical properties of the temperature fluctuations in the cosmic microwave background (CMB). Nonsupersymmetric hybrid inflation predicts the primordial scalar CMB power spectrum to be blue-tilted, i.e., it predicts a scalar spectral index n s greater than one [60,61]. This needs to be compared with the recent measurement by the PLANCK collaboration, n obs s = 0.9677 ± 0.0060 [76]. Nonsupersymmetric hybrid inflation is, thus, ruled out with a statistical significance of more than 5 σ. This conclusion serves as an independent motivation to introduce supersymmetry, in addition to supersymmetry's other advantages (see Sec. 1).
In supersymmetry, hybrid inflation is understood to be a consequence of (temporary) spontaneous SUSY breaking, which can be accomplished either by a nonvanishing F term [77,78] or D term [54,55].
In the following, we will outline both scenarios and explain why we will eventually focus on D-term inflation. In both cases, the scalar inflaton σ is contained in a chiral multiplet S that transforms as a singlet under all gauge symmetries. S couples to charged chiral multiplets in the superpotential W inf , where κ is a dimensionless Yukawa coupling. Φ andΦ denote the so-called waterfall fields, which transform in conjugate representations of a gauge group G. We shall identify G with U (1) B−L and assign B−L charges +q and −q to Φ andΦ. This sets the stage for the B−L phase transition at the end of inflation. Any self-interaction of S can be forbidden by invoking R symmetry. If we assign R charges such that S R = 2 and ΦΦ R = 0, terms such as S 2 and S 3 are not allowed. The Yukawa coupling in Eq. (1) does not suffice to break supersymmetry. This is, however, necessary to obtain a nonvanishing vacuum energy density that can drive inflation. In F-term hybrid inflation (FHI), one therefore equips S with an F term, |F S | = µ 2 S . Supersymmetry is then brokenà la O'Raifeartaigh [79] during inflation. In D-term inflation, one assumes instead a nonzero Fayet-Iliopoulos (FI) D term in the D-term scalar potential V D . This breaks supersymmetry during inflation via the FI mechanism [80].
F-term and D-term inflation are then characterized by the following expressions for W inf and V D , DHI: The gauge coupling constant g belongs to the gauge group G. In our case, g consequently denotes the B−L gauge coupling. The gauge charge q 0 is factored out of the FI parameter ξ for later convenience.
In both F-term and D-term inflation, the field S parametrizes a completely flat direction in the scalar potential, at least at tree level and in global supersymmetry. This explains why it is natural to identify S with the chiral inflaton field. At large field values of S, the Yukawa coupling in Eq. (1) induces large S-dependent masses for the waterfall fields. Φ andΦ can therefore be integrated out, which results in a logarithmic effective potential for S. This singles out the origin in field space, S = 0, as the unique vacuum in the quantum theory. Inflation is, thus, characterized by the slow-roll motion of S from large field values towards the origin in field space. The SUSY-breaking parameters µ S (FHI) and ξ (DHI) induce a mass splitting between the scalar components of Φ andΦ. Below a certain critical value of the inflaton field, this results in one (linear combination of) scalar waterfall field(s) becoming unstable. This triggers the B−L phase transition. Inflation ends and the system approaches a ground state in which B−L is spontaneously broken and supersymmetry restored. As we will discuss in Sec. 4.2, this can be used to generate Majorana masses for the right-handed neutrinos N i in the seesaw extension of the MSSM. All we have to do is to choose appropriate gauge charges and introduce Yukawa couplings between Φ and three generations of sterile neutrinos, W ⊃ 1 2 h ij Φ N i N j . Hybrid inflation is sensitive to gravitational corrections in supergravity [81,82]. In the case of F-term inflation, R symmetry breaking leads, in particular, to a SUGRA term in the F-term scalar potential V F that is linear in the complex inflaton field s ⊂ S [39,[83][84][85][86]. To see this, recall that R symmetry breaking can be accounted for by introducing a constant term in the superpotential, w ⊂ W . This term ultimately sets the gravitino mass m 3/2 in the low-energy vacuum, w ∝ m 3/2 M 2 Pl (where M Pl denotes the reduced Planck mass, M Pl 2.44 × 10 18 GeV). In F-term inflation, the inflaton field itself possesses a nonzero F term, |F S | = µ 2 S . This F term couples to w in the SUGRA scalar potential, Here, K s * stands for the derivative of the Kähler potential K w.r.t. s * and K −1 ss * denotes the ss * entry in the inverse of the Kähler metric K. For a canonical Kähler potential, K ⊃ S † S, one obtains This tadpole term is a vivid example for the potential impact of hidden-sector SUSY breaking on the dynamics of inflation (see the discussion in Sec. 1). Any analysis of F-term inflation that ignores Eq. (3) is incomplete. The tadpole term breaks the rotational invariance in the complex inflaton plane and, hence, complicates the analysis of the inflationary dynamics. In fact, it renders F-term inflation a two-field model [86], which needs to be treated with special care. Depending on the size of m 3/2 , the tadpole term also potentially spoils slow-roll inflation. In addition, it generates a false vacuum at large field values, which limits the set of viable initial conditions for inflation in phase space.
For these reasons, we will restrict ourselves to D-term inflation in this paper. A dynamical realization of F-term inflation that avoids the tadpole problem can be found in [42]. The model in [42] also establishes a connection between inflation and dynamical SUSY breaking around the scale of grand unification. In [42], there is, however, no U (1) symmetry that could be identified with U (1) B−L .
Moreover, inflation does not end in a phase transition in the waterfall sector. Together, these features of [42] eliminate the possibility to unify inflation with the dynamics of the B−L phase transition. A dynamical model of F-term inflation that ends in the B−L phase transition will be presented elsewhere.

Ingredients for a unified model of D-term inflation
The absence of the linear tadpole term in D-term inflation tends to make SUGRA corrections more manageable. In particular, the issue of initial conditions appears more favorable compared to F-term inflation (however, see also Sec. 3.4). Nonetheless, D-term inflation still faces a number of challenges.
In this section, we will discuss these challenges one by one and outline how they are respectively met in our model. This serves the purpose to explain the bigger physical picture behind our specific setup.
In Sec. 2.3, we will then become more explicit and present the details of our construction.

Dynamical generation of the FI term in the strongly coupled SUSY-breaking sector
First of all, the origin of the FI term and its embedding into supergravity are subtle issues that have been the subject of a long debate in the literature. At this point, it is important to distinguish between genuine FI terms and effective FI terms. The former refer to field-independent FI parameters ξ a that parametrize constant shifts in the auxiliary D a components of Abelian vector multiplets V a . The latter denote field-dependent FI parameters ξ a that depend on the vacuum expectation values (VEVs) of dynamical scalar moduli ψ i . Genuine FI terms, ξ a = const, are FI terms in the original sense. That is, they preserve the underlying gauge symmetry and are compatible with massless vector multiplets.
By contrast, in the case of effective FI terms, ξ a = ξ a ( ψ i ), the underlying gauge symmetry is always spontaneously broken by the modulus VEVs ψ i . Effective FI terms are therefore FI terms only in a slightly more general sense. The embedding of genuine FI terms into supergravity always requires the underlying gauge symmetry to be promoted to a gauged U (1) R symmetry as well as the presence of an exact global continuous symmetry [87,88]. While the requirement of a gauged U (1) R symmetry poses a challenge as soon as one wants to make contact with low-energy phenomenology, the requirement of a global continuous symmetry is problematic from the viewpoint of quantum gravity. As can be shown on very general grounds, quantum gravity is likely to violate any global symmetry [89]. Coupling genuinely constant FI terms to gravity, thus, appears to be almost impossible. 3 Effective FI terms promise to offer a possible way out of this problem. Effective FI terms are frequently encountered in string theory [92,93], where they arise in consequence of the Green-Schwarz mechanism of anomaly cancellation [94]. However, such constructions typically suffer from the presence of a shift-symmetric modulus field [95]. This modulus field needs to be adequately stabilized [96]. Otherwise, it will absorb the effective FI term in its VEV or cause a cosmological modulus problem [97,98].
To avoid all of the problems listed above, we will assume that the FI term responsible for inflation is dynamically generated in the strongly coupled SUSY-breaking sector. That is, we will not resort to string theory, but work in the context of field theory. To be precise, we will employ the dynamical mechanism devised in [99]. A dynamical FI term generated via this mechanism is automatically an effective field-dependent FI term that is controlled by the VEVs of moduli in the hidden sector, Here, ψ andψ belong to chiral multiplets Ψ andΨ that carry B−L charges +q 0 and −q 0 , respectively.
This ansatz complies with our philosophy outlined in the introduction (see Sec. 1). Being a dynamically generated quantity, the FI parameter ξ does not need to be added by hand. Instead, it is related to the dynamical scale Λ dyn in the hidden sector which is generated via dimensional transmutation. Let us identify the UV embedding scale of our theory with the Planck scale, Λ UV = M Pl . The renormalization group (RG) running of the hidden-sector gauge coupling g hid then results in the following relation, where b hid is the coefficient of the hidden-sector RG beta function. This relation explains why Λ dyn and, hence, ξ end up being exponentially suppressed compared to the Planck scale (which is the only available mass scale in our setup). The advantage of our approach is that it comes with a builtin mechanism for modulus stabilization. As usual, the generation of ξ results in a shift-symmetric modulus field. In our case, this will be a linear combination of Ψ andΨ (see Sec. 2.3). However, as the FI parameter ξ is generated in the SUSY-breaking sector, the shift-symmetric modulus couples to degrees of freedom (DOFs) involved in the dynamical breaking of supersymmetry. The F term of the SUSY-breaking Polonyi field therefore induces a mass for the modulus field. This stabilizes all dangerous directions in field space and prevents us from running into any modulus problem.
Spontaneous B−L breaking in the hidden sector before the end of inflation The B−L phase transition at the end of D-term inflation is accompanied by the production of topological defects in the form of cosmic strings [100,101] (for reviews on cosmic strings, see [102,103]). Such cosmic strings can leave an imprint in the CMB temperature anisotropies, the spectrum of gravitational 3 Shortly after completion of our work, two proposals appeared in the literature that demonstrate how a novel type of genuine FI terms, based on nonstandard supersymmetric invariants, can be consistently coupled to supergravity [90,91].
These FI terms do not require R symmetry to be gauged and, hence, do not suffer from the presence of a global symmetry.
On the other hand, they result in highly nonlinear terms in the fermionic action. It would be interesting to employ these novel FI terms in phenomenological applications in future work and investigate, e.g., their potential use for inflation.
waves (GWs), and in the diffuse gamma-ray background (DGRB). Cosmic string decays can also affect outcome of big bang nucleosynthesis (BBN). However, no signs of cosmic strings have been detected thus far. Recent limits on the properties of cosmic strings can be found in [104,105] (CMB), [106][107][108] (GWs), and [109] (DGRB and BBN). These bounds allow to put severe constraints on the parameter space of hybrid inflation [110,111]. In fact, the minimal scenario of D-term inflation is already ruled out by the nonobservation of cosmic strings [112]. A possible way out of this problem is to consider scenarios in which B−L is spontaneously broken, in one way or another, already during inflation.
Cosmic strings then form at early times and are sufficiently diluted before the end of inflation.
Fortunately, the dynamical generation of ξ in the hidden sector (see Eq. (5)) provides the ideal starting point for implementing this solution to the cosmic string problem. The modulus VEVs ψ and ψ spontaneously break B−L in the hidden sector already during inflation. Therefore, to prevent the formation of cosmic strings in the waterfall sector at the end of inflation, all we have to do is to communicate the breaking of B−L in the hidden sector to the waterfall fields. This is readily done by adding marginal couplings between the two sectors in the superpotential or Kähler potential that are otherwise irrelevant for the dynamics of inflation [56]. We will come back to this issue in Sec. 4.4.

Sequestered sectors in Jordan-frame supergravity
A notorious problem of any model of D-term inflation is that additional charged scalar fields, other than the waterfall fields Φ andΦ, threaten to destabilize the FI term during inflation. Without any additional physical assumption, there is no reason why Φ andΦ should be the only fields charged under the U (1) symmetry whose FI term drives inflation. In general, one should rather expect a whole set of N charged pairs, Φ i ,Φ i , with the inflaton only coupling to a subset of M such pairs, In this case, there are N − M pairs that are not sufficiently stabilized by an inflaton-induced mass during inflation. At the same time, these fields also enter into the D-term scalar potential, where they threaten to absorb the FI parameter ξ in their VEVs. In our scenario based on U (1) B−L , the role of these dangerous scalar directions is played by the scalar partners of the MSSM quarks and leptons.
The squarksq i and sleptons˜ i are also charged under B−L, but do not couple to the inflation field.
Accounting for the presence of these fields, the D-term scalar potential needs to be rewritten as follows, The squarks and sleptons therefore acquire D-term-induced masses proportional to m D = g |q 0 ξ| 1/2 . As evident from the sign relations in Eq. (8), half of these masses end up being tachyonic (see also [113]).
This renders the corresponding directions in field space tachyonically unstable. The inflationary trajectory, thus, decays into a vacuum in which B−L is broken by nonvanishing sfermion VEVs.
To avoid this problem, one needs to stabilize the MSSM sfermions by means of additional mass contributions during inflation. Here, the simplest solution is to make use of the soft scalar masses induced by the spontaneous breaking of supersymmetry in the hidden sector. Let us assume that these soft masses are all more or less close to a common value m 0 . Then, to stabilize the MSSM sfermions during inflation, we must require that m 0 m D . This is, however, too strong a condition if SUSY breaking is communicated to the visible sector only via ordinary gravity mediation in the Einstein frame (for a review on gravity mediation, see [114]). In gravity mediation, we expect the soft scalar masses to be of the order of the gravitino mass, m 0 ∼ m 3/2 . This soft mass is universal such that also the waterfall fields obtain soft masses of O m 3/2 . As a consequence, the stabilization of the MSSM sfermions also stabilizes the waterfall fields. This is an unwanted but unavoidable side effect. In such a scenario, the waterfall fields would never become unstable and inflation would never end.
A possible solution to this problem is to presume a separation of scales of the following form, In this case, the MSSM sfermions remain stabilized at all times, while the waterfall fields can become unstable at the end of inflation. Parametrically large soft sfermion masses can, e.g., be achieved by adding a direct coupling between the visible and the hidden sector in the Kähler potential, Here, Q i and X stand for a generic MSSM matter field and the SUSY-breaking Polonyi field, respectively. M * denotes the mass scale at which the effective operator in Eq. (10) Now, however, we need a conspiracy among certain parameters. Successful inflation is only possible as long as the parameters g, q 0 , ξ, m 3/2 , and M * conspire in order to satisfy the following relation, Einstein frame: We do not see any compelling argument why this relation should be automatically fulfilled. For this reason, we will go one step further and solve the MSSM sfermion problem in a more elegant way.
Let us assume that the canonical description of hybrid inflation in supergravity corresponds to an embedding into a (specific) Jordan frame rather than an embedding into the Einstein frame [115,116].
At this point, recall that every Jordan-frame formulation of supergravity is characterized by a specific choice for the so-called frame function Ω. The frame function is an arbitrary function of the complex scalar fields in the theory, Ω = Ω (φ i , φ * ı ). For a given Ω, the metric tensor in the Jordan frame, g J µν , is related to the metric tensor in the Einstein tensor, g µν , via the following Weyl rescaling, Here, C denotes what we will refer to as the conformal factor. We emphasize that the Weyl transformation in Eq. (13) does not change the physical predictions of the theory. The physical content of the Jordan frame is equivalent to the physical content of the Einstein frame, even at the quantum level [117,118]. In what follows, we will simply assume that the SUGRA embedding of hybrid inflation is most conveniently described in the Jordan frame. More details on the conversion between the Einstein-frame and Jordan-frame formulations of supergravity can be found in Appendix A.
Given the freedom in defining the frame function Ω, there is, in principle, an infinite number of possible Jordan frames. In the following, we will, however, focus on one particular choice which stands out for several reasons. In this frame, the frame function Ω is determined by the Kähler potential K, This relation is understood to hold in superspace, such that the frame function Ω becomes a function of chiral multiplets, Ω = Ω Φ i , Φ † ı . This choice for Ω is motivated by the curved superspace approach to old minimal supergravity in the Einstein frame [119,120]. In this derivation of the SUGRA action, the function Ω as defined in Eq. (14) is identified as the generalized kinetic energy on curved superspace.
Meanwhile, Ω is also a meaningful quantity in the derivation of the Einstein-frame action based on local superconformal symmetry [121,122]. In this approach to old minimal supergravity, the function Ω is identified as the prefactor of the kinetic term of the chiral compensator superfield. For our purposes, the advantage of the choice in Eq. (14) is that it sets the stage for canonically normalized kinetic terms for the complex scalar fields in the Jordan frame. Indeed, to obtain canonically normalized kinetic terms, the defining relation in Eq. (14) needs to be combined with the following ansatz for Ω [115], Here, we introduced F as the kinetic function of the chiral matter fields. The additional −3M 2 Pl term in Ω accounts for the kinetic term of the gravitational DOFs. J is an arbitrary holomorphic function.
We mention in passing that the relations in Eqs. (14) and (15) also provide the basis for a class of SUGRA models known as canonical superconformal supergravity (CSS) models [116]. In these models, the pure supergravity part of the total action is invariant under local Poincaré transformations as usual. At the same time, the matter and gauge sectors of the theory can be made invariant under a local superconformal symmetry by setting the holomorphic function J to zero. This larger set of symmetry transformations renders CSS models particularly simple. In the Jordan frame, one obtains canonical kinetic terms for all fields. Moreover, one finds that the Jordan-frame scalar potential coincides with the scalar potential in global supersymmetry. In our scenario, we will, however, break the superconformal symmetry via the holomorphic function J to a large degree (see Eq. (27) below).
For this reason, one should not regard our model to be of the CSS type. A model of D-term inflation based on the idea of superconformal symmetry has been constructed in [123]. This model employs a constant FI term that does not depend on the inflaton field value in the Einstein frame. Our model will by contrast involve an effective FI term that does not depend on the inflaton field value in the Jordan frame (see Sec. 2.5). In [123], the dynamics of inflation are moreover described by a two-field model, whereas we will only deal with a single inflaton field. Interestingly enough, the model in [123] reproduces the predictions of Starobinsky inflation [31] in the limit of large inflaton field values [124].
Together with Eq. (14), the ansatz in Eq. (15) results in the following Kähler potential, This is an important result. If we choose the kinetic function F or, equivalently, the holomorphic function J appropriately, this Kähler potential can be readily used to sequester the different sectors of our model. In fact, Eq. (16) turns into a Kähler potential of the sequestering type [125] if the function F can be split into separate (canonical) contributions from the hidden, visible, and inflation sectors, Kähler potentials of this form have been derived in the context of extra dimensions [125] as well as in strongly coupled conformal field theories (CFTs) [126][127][128][129]. They are also similar to the Kähler potential in models of no-scale supergravity [130][131][132] that can be derived from string theory [133]. 4 If the various sectors do not couple to each other in the superpotential, the Kähler potential in Eq. (17) leads to vanishing soft scalar masses at tree level in all sectors except for the hidden sector.
The possibility to sequester different sectors is a crucial property of Eq. (16) which we will use to solve the MSSM sfermion problem. Altogether, we will choose the function F in our model as follows, From now on, we will refer to the total kinetic function as F tot and reserve the symbol F for the kinetic function of the inflaton field S (see further below). We will also assume that the holomorphic function J in Eq. (15) is a function of S only. That is, the kinetic functions F hid , F vis , and F inf are supposed to consist of standard canonical terms for all fields except for S. The kinetic function F tot as defined in Eq. (18) combines two important features. (i) It leads to a sequestering between the hidden sector and the inflaton sector. The waterfall fields consequently obtain no soft masses at tree level. This is necessary to be able to trigger the B−L phase transition at the end of inflation, irrespective of the size of the gravitino mass. (ii) The MSSM sfermions are stabilized thanks to higher-dimensional operators in F tot that couple the visible sector to the SUSY-breaking sector. At this point, we stick to the mechanism that we already discussed in the case of the Einstein frame (see Eq. (10)). Together, these two features allow us to realize successful inflation and solve the MSSM sfermion problem.
Our solution of the MSSM sfermion problem in the Jordan frame is conceptually different from the solution in the Einstein frame discussed around Eq. (12). Now, as the waterfall fields do not acquire a soft mass at tree level, the requirement in Eq. (12) turns into the following two conditions, Jordan frame: The first inequality is a consequence of our decision to work in a Jordan frame with canonical kinetic terms. It is trivially fulfilled. We are, thus, left with only one sensible physical condition, M * m 3/2 M Pl /m D . To satisfy this condition, we no longer have to rely on a conspiracy among different parameter values. Instead, we simply have to deal with an upper bound on the scale M * which derives from the requirement that all dangerous scalar directions in field space must be sufficiently stabilized during inflation. We therefore manage to solve the MSSM sfermion problem in B−L D-term inflation by means of model-building decisions rather than by resorting to a specific part of parameter space.
In the following, we will remain agnostic as to the UV origin of F tot in Eq. (18). We settle for the observation that, apart from additional Planck-suppressed interactions, Eq. (18) can be motivated by demanding canonically normalized kinetic terms in the (standard) Jordan frame. This is the reason why we will formulate parts of our analysis in the language of Jordan-frame supergravity. Beyond that, it might be possible to embed our model into extra dimensions, strongly coupled CFTs, no-scale supergravity and/or string theory. But such a task is beyond the scope of this paper and left for future work. For our purposes, the formalism of Jordan-frame supergravity simply provides a convenient technical framework. We shall not speculate about the underlying physics at higher energies.

Shift symmetry in the direction of the inflaton field
Working in the Jordan frame not only helps to protect the waterfall fields against large soft masses.
In ordinary gravity mediation in the Einstein frame, also the inflaton acquires a soft mass of the order of the gravitino mass. This results in the notorious eta problem in supergravity [77,134]. To see this, recall that the gravitino mass is related to the F term of the SUSY-breaking Polonyi field as follows, At the same time, the Hubble rate during D-term inflation is controlled by the size of the FI term, General arguments in supergravity indicate that D terms are always accompanied by an F term which is at least as large or even larger [135,136]. In our case, we intend to dynamically generate the FI term in conjunction with the Polonyi F term in the SUSY-breaking sector. On general grounds, we, thus, expect that |F X | D . Therefore, if the inflaton indeed obtained a soft mass of O m 3/2 , we would immediately encounter an eta problem, i.e., a slow-roll parameter η much larger than one, where V denotes the second derivative of the scalar potential w.r.t. the inflaton field. This serves as an additional motivation for our specific Jordan frame. There, the soft mass of the inflaton vanishes (at least as long as F = S † S and J = 0), which renders the most dangerous contribution to η zero. This is, however, not the end of the story. To fully solve the eta problem, we need to work a bit harder. In the Jordan frame, the complex scalars are nonminimally coupled to the Ricci scalar R J via the frame function Ω. This follows from the Jordan-frame equivalent of the Einstein-Hilbert action, which contains the nonminimal term −F tot /3 R J . This coupling yields additional mass contributions for the scalar fields. Consider, e.g., the canonical terms in the total kinetic function, which describes the special case of a set of conformally coupled scalars. Each complex scalar with a canonical term in the kinetic function F tot therefore acquires a universal gravity-induced mass m R , where H J denotes the Hubble parameter in the Jordan frame and where we used the relation between Ricci scalar and Hubble parameter in exact de Sitter space, R J = 12H 2 J . This gravitational mass correction spoils slow-roll inflation as long as it is not sufficiently suppressed. That is, an inflaton kinetic function that only consists of a canonical term, F = S † S, results in too large an η parameter, Thus, to fully solve the eta problem, we have to make use of the holomorphic function J in Eq. (15).
The freedom in defining J allows us to realize an approximate shift symmetry in the direction of the inflaton field. Such a shift symmetry is a common tool in SUGRA models of inflation, as it allows to suppress the most dangerous SUGRA contributions to the inflaton potential [137]. In our case, an approximate shift symmetry is realized for the following kinetic function of the inflaton field, Here, χ is a positive shift-symmetry-breaking parameter which we will assume to be small, 0 < χ 1.
To see that Eq. (27) indeed features a shift symmetry, it is convenient to rewrite F as follows, This form of F illustrates that, for χ 1, the kinetic function is approximately invariant under shifts in σ, i.e., the real scalar part of the inflaton field S. Conversely, χ values close to one, 1 − χ 1, lead to an approximate shift symmetry in τ , i.e., the imaginary scalar component of S. In the following, we will focus w.l.o.g. on the first of these two cases. In passing, we also mention that the trivial case χ = χ CSS = 1/2 (which renders the holomorphic function J vanishing) corresponds to an inflaton field that is conformally coupled to the Ricci scalar. This choice for the parameter χ would allow to construct a SUGRA model that is invariant under local superconformal symmetry (see the discussion below Eq. (15)). However, as argued above, we would then fail to solve the eta problem (see Eq. (26)).
For this reason, we need to break the superconformal symmetry. In fact, by choosing χ χ CSS , we break the superconformal symmetry in a maximal sense in favor of an approximate shift symmetry.
Given the kinetic function in Eq. (28), it is straightforward to solve the eta problem. In consequence of the approximate shift symmetry, all contributions to the inflaton mass m σ end up being suppressed by χ. This follows from an explicit computation of m σ in the Einstein frame (see Eq. (116) in Sec. 2.5), As expected, m σ reduces to m R in the limit χ → 1/2. On the other hand, if χ is chosen small enough, m σ becomes suppressed, so that the slow-roll parameter η remains sufficiently small during inflation. 5

Explicit breaking of the shift symmetry
An exact shift symmetry is out of reach in our model, as the Yukawa coupling in the superpotential, W inf = κ S ΦΦ, breaks any inflaton (or waterfall field) shift symmetry explicitly. Therefore, while χ may be zero at tree level, a nonvanishing value of the shift-symmetry-breaking parameter χ is always generated via radiative corrections. To see this, let us consider the one-particle-irreducible (1PI) effective action in global supersymmetry. The superpotential does not receive any quantum corrections in consequence of the SUSY nonrenormalization theorem [138]. The renormalization of the Kähler potential is, however, nontrivial and results in a one-loop effective Kähler potential K 1 [139][140][141][142].
Along the inflationary trajectory, Φ =Φ = 0, a calculation in the MS renormalization scheme yields whereμ denotes the MS renormalization scale. Note that this result for K 1 corresponds to a wavefunction renormalization of the inflaton field S. Next, let us embed the effective Kähler potential in Eq. (30) into supergravity. In the Einstein frame, the relevant quantity is the total Kähler potential K tot , which simply follows from adding K 1 to the tree-level Kähler potential, K tot = K tree + K 1 . In the Jordan frame, we are by contrast interested in the total frame function, Ω tot = Ω tree + Ω 1 . One can show that the one-loop correction to the tree-level frame function is related to K 1 as follows, Here, the higher-order terms correspond to Planck-suppressed radiative corrections which are negligibly small. Together, Eqs. (28), (30), and (31) allow us to determine the effective χ parameter that is induced by the breaking of shift symmetry in the superpotential. Along the direction of the real inflaton component, τ = 0, we obtain the following one-loop kinetic function for the inflaton field, In the absence of any tree-level contribution, the shift-symmetry-breaking parameter χ is therefore expected to be of the order of κ 2 / 16π 2 . This is an important result which was overlooked in [43]. 5 Imposing an approximate shift symmetry in the direction of the inflaton field would also allow to solve the eta problem in the Einstein frame. There, the inflaton mass also vanishes in the limit of an exact shift symmetry. From this perspective, our solution to the eta problem actually does not serve as an additional motivation to work in the Jordan frame. However, our arguments regarding the MSSM sfermion problem remain unchanged. This problem is best solved in the Jordan frame (see the discussion around Eq. (19)). We will therefore continue to work in the Jordan frame.
There, we simply varied χ as a free parameter for fixed κ. Of course, this is a valid procedure, given the fact that χ can very well receive further tree-level contributions (or further radiative corrections from inflaton couplings to extra heavy states). In this case, χ is simply the sum of various contributions, χ = χ tree + χ 1 , which can take any arbitrary value. But the case χ = χ 1 -which we had neglected thus far -is special, as it corresponds to a scenario with minimal field content and number of free parameters. We will study this scenario in more detail in Sec. 3.3. This will represent one of the main results of this paper and a significant step forward beyond our analysis in [43]. In particular, we will find that χ = χ 1 leads to inflation in new parts of parameter space that we had dismissed before.
Finally, we point out that the fact that we are unable to realize an exact shift symmetry is a virtue rather than a shortcoming of our model. A slightly broken shift symmetry allows us to get a handle on the scalar spectral index n s which we would otherwise lack in the case of an exact shift symmetry.
The prediction for n s in standard D-term inflation in global supersymmetry roughly corresponds to where N e denotes the number of e-folds between the end of inflation and the time t CMB when the CMB pivot scale, k CMB = 0.05 Mpc −1 , exits the Hubble horizon during inflation. The prediction in Eq. (33) exceeds the current best-fit value, n obs s = 0.9677 ± 0.0060 [76], by at least 2 σ. This puts some phenomenological pressure on the simplest version of D-term inflation. To improve on the predicted value of n s , various SUGRA models have been proposed in the literature [123,124,[143][144][145][146]. However, in our scenario, no extra effort is needed to enhance the absolute value of the slow-roll parameter η and, thus, reproduce the best-fit value for n s . The inflaton mass in Eq. (29) approximately results in Therefore, to realize n s values around n s 0.96, all we have to do is to choose χ small enough, In this sense, the approximate shift symmetry in the kinetic function of the inflaton field automatically provides a possibility to achieve a scalar spectral index consistent with the observational data.

Three physical assumptions to solve five problems of D-term inflation
So far, we have mainly outlined the ingredients of our construction in physical and less technical terms.
We hope that this part of our discussion will be accessible also to readers without a strong background in SUGRA model building. A more technical description of our model will be given in the next three sections (see Secs. 2.3, 2.4, and 2.5). Readers less interested in the technical aspects of our model and more interested in its phenomenological implications may skip directly to Sec. 3.
Before entering the technical part of our discussion, let us summarize our insights up to this point.
On the one hand, we showed that D-term inflation faces a number of challenges. We discussed the following five problems: (i) The generation of the FI term in the D-term scalar potential and its embedding into supergravity, (ii) the production of dangerous cosmic strings at the end of inflation, (iii) the stabilization of dangerous MSSM sfermion directions in the scalar potential during and after inflation, (iv) the eta problem in supergravity, and (v) the tension between the lower bound on n s in D-term inflation and the current best-fit value. On the other hand, we argued that all five of these problems can be solved if one makes the following three assumptions: (i) The FI term is dynamically generated in the hidden SUSY-breaking sector. (ii) The canonical description of D-term inflation in supergravity corresponds to the embedding into the (standard) Jordan frame with canonically normalized kinetic terms for all scalar fields. (iii) The kinetic function of the inflaton field exhibits a slightly broken shift symmetry. Our model therefore turns out to be a viable SUGRA realization of B−L D-term inflation that is consistent with all theoretical and phenomenological constraints.

SUSY-breaking dynamics in the hidden sector
In the previous section, we summarized our physical ideas about how to realize a viable SUGRA model of D-term inflation that (i) unifies the dynamics of supersymmetry breaking and inflation and that (ii) ends in the B−L phase transition at energies around the GUT scale. In the following, we will show how these ideas can be implemented into a specific model of dynamical SUSY breaking: the Izawa-Yanagida-Intriligator-Thomas (IYIT) model [147,148], which represents the simplest vectorlike model of dynamical SUSY breaking. Despite this choice, we believe that our general ideas extend beyond our specific model. In future work, it would be interesting to study alternative DSB models and assess which other models might give rise to unified dynamics of supersymmetry breaking and inflation.

IYIT sector at high and low energies
We begin by reviewing the IYIT model. In its most general form, the IYIT model corresponds to a strongly coupled supersymmetric Sp(N ) gauge theory. 6 At high energies, its charged matter content consists of N f = N + 1 vector-like pairs of quark flavors, where each quark field Ψ i transforms in the fundamental representation of Sp(N ). The theory becomes confining at energies around the dynamical scale Λ dyn which is generated via dimensional transmutation. Below Λ dyn , the dynamical DOFs in the IYIT sector correspond to a set of where M ji = −M ij and where η denotes a numerical factor of O (4π) that follows from naive dimensional analysis (NDA) [149][150][151][152]. It turns out to be useful to absorb the NDA factor η in the dynamical scale Λ dyn . In the following, we will therefore work with the reduced dynamical scale Λ, At low energies, the scalar meson VEVs parametrize a moduli space of degenerate supersymmetric vacua. This moduli space is subject to a constraint equation, which, in the classical limit, corresponds to the requirement that the Pfaffian of the antisymmetric meson matrix M ij must vanish, Pf (M ij ) = 0.
This constraint, however, becomes deformed in the quantum theory. There, it reads [153] Pf To break supersymmetry in the IYIT sector, one needs to lift the flat directions in moduli space. This is readily achieved by coupling the IYIT quarks Ψ i to a set of At high energies: where λ ij = −λ ji are dimensionless coupling constants. At high energies, these couplings are nothing but ordinary Yukawa couplings which do not affect the vacuum structure of the theory. At low energies, the terms in Eq. (39), however, turn into mass terms for the meson and singlet fields M ij and Z ij , At low energies: These mass terms single out the origin in field space as the true supersymmetric ground state. The quantum-mechanically deformed moduli constraint in Eq. (38), however, prevents the system from reaching the origin in field space. This breaks supersymmetry. The theory is forced to settle into a vacuum away from the origin, M ij = 0, where some of the singlet F-term conditions, F Z ij = 0, cannot be satisfied. Supersymmetry is, hence, brokenà la O'Raifeartaigh by nonvanishing F terms [79].
In the following, we shall focus on the minimal N = 1 realization of the IYIT model, for simplicity.
In this case, the Sp(1) gauge dynamics are equivalent to those of an SU (2) theory, Sp(1) ∼ = SU (2), and we have to deal with four quark fields Ψ i and six singlet fields Z ij at high energies. This translates into six meson fields M ij (and six singlet fields Z ij ) at low energies. As we will see shortly, it turns out to be convenient to label the fields in the low-energy theory in a suggestive manner. To do so, we first note that Eq. (39) exhibits a global U (1) A flavor symmetry that corresponds to an axial quark phase rotation. The U (1) A charges of the two quark flavors at high energies can be chosen as follows, This normalization ensures that the charged meson fields at low energies carry U (1) A charges ±q 0 , and similarly for the Z ij . In the second step, we relabel all fields according to their U (1) A charges, In this notation, the low-energy superpotential in Eq. (40) takes the following form, where we also relabeled the λ ij . Meanwhile, the constraint in Eq. (38) can now be written as follows, Together, Eqs. (44) and (45) allow to explicitly calculate the VEVs in the SUSY-breaking vacuum.
As it turns out, the location of the true ground state in meson field space depends on the hierarchy among three geometric means of Yukawa couplings, λ = (λ We will assume that the first of these three cases is realized, λ < min λ 14 0 , λ 23 0 . In this case, it is the charged meson fields that obtain a nonzero VEV, M + M − Λ 2 . This case is special in the sense that the global U (1) A flavor symmetry becomes spontaneously broken at low energies. In the other two cases, the flavor symmetry remains unbroken even in the SUSY-breaking vacuum.

Properties of the low-energy vacuum
Let us now discuss the properties of the U (1) A -breaking vacuum in a bit more detail. In this vacuum, all neutral fields are stabilized by their supersymmetric masses in Eq. (44). The relevant terms in the superpotential and Pfaffian constraint are therefore only those involving charged fields, The constraint is most easily accounted for by adding a Lagrange multiplier term to the superpotential, where the field T represents the actual Lagrange multiplier. The physical nature of the field T depends on strong-coupling effects in the Kähler potential. If it acquires a nonperturbative kinetic term from the strong gauge dynamics, T becomes physical. On the other hand, if no kinetic term is generated, T is merely an auxiliary field that remains unphysical. Unfortunately, it is unknown which of these cases is realized, as the Kähler potential for T in the strong-coupling regime is incalculable. At any rate, the difference between the two cases is mostly irrelevant for our purposes. All effects in the case of a physical Lagrange multiplier field T are suppressed by powers of λ/ (4π) [154]. Thus, as long as we stay in the perturbative regime, λ 4π, our results will not be affected by the physical status of the field T . In the following, we will therefore assume that T remains unphysical, for simplicity. In practice, this means that we will take the limit λ T → ∞ wherever possible. For discussions of the IYIT model based on the assumption of a physical Lagrange multiplier field T , see, e.g., [42,155].
Given the superpotential in Eq. (48) (and taking the limit λ T → ∞ at the end of the calculation), one can easily show that the vacuum energy density is minimized for the following meson VEVs, These meson VEVs induce SUSY-breaking F terms for the singlet fields Z ± . To determine the total F-term SUSY breaking scale µ, it is useful to transform the fields M ± and Z ± to a new basis, where we introduced f A to denote the total energy scale of spontaneous U (1) A breaking, In the new field basis, the fields M and T share a supersymmetric Dirac mass term, m M T = λ T f A , that formally diverges in the limit λ T → ∞. This allows us to identify M as the meson field that becomes eliminated by the Pfaffian constraint. The remaining meson DOFs are then described by the orthogonal linear combination, i.e., by the field A. Note that this automatically implies that the field A plays the role of the chiral Goldstone multiplet of spontaneous U (1) A breaking. From this perspective, the energy scale f A may also be regarded as the Goldstone decay constant. To obtain the superpotential describing the low-energy dynamics of A, X, and Y , we proceed as follows: (i) We perform the field rotation in Eq. (50), (ii) shift the meson fields A and M by their nonvanishing VEVs, and (iii) integrate out the heavy fields M and T . This results in the following superpotential, Here, µ denotes the total F-term SUSY breaking scale, By construction, the singlet field X is the only field with a nonzero F term, |F X | = µ 2 . It can therefore be identified with the SUSY-breaking Polonyi field. The orthogonal field Y shares a Dirac mass term with the U (1) A Goldstone field A which is given in terms of the mass scales µ and f A , Here, ρ measures the degeneracy between λ + and λ − . For λ + → λ − , the parameter ρ approaches one, ρ → 1. For λ + λ − or λ + λ − , it goes to zero, ρ → 0. In the following, we will assume that both λ + and λ − are sufficiently small, so that we always stay in the perturbative regime. For definiteness, let us require that both couplings are always at least half an order of magnitude smaller than 4π, This translates into a lower bound on the hierarchy parameter ρ in dependence of λ = (λ Moreover, to simplify our analysis, we will replace ρ by its expectation valueρ in the following. We computeρ by averaging ρ over all possible values of λ ± , varying both couplings on a linear scale, Note that this result is independent of the concrete value of λ pert . With ρ fixed at this value, the perturbativity constraint in Eq. (57) turns into an upper bound on the Yukawa coupling λ, This yields in turn an upper bound on the F-term-induced mass in the superpotential, m F 1.94 Λ.
The lesson from this analysis is the following: From now on, we will work with ρ 0.80 and λ 2.41.
The Yukawa couplings λ ± are then guaranteed to assume "typical values" (in the sense of the average in Eq. (58)) which are, at the same time, consistent with the requirement of perturbativity, λ < λ pert .
Similar to the mass m F , also the Yukawa coupling λ X in Eq. (53) is given in terms of µ and f A , This Yukawa coupling between the Polonyi field X and the Goldstone field A is is a direct consequence of the T M + M − term in Eq. (48). Similar couplings also exist between X and the neutral meson fields.
This is shown explicitly in Appendix A of [155] (see also [154]). Together, these Yukawa couplings result in an effective Polonyi mass m 1 at the one-loop level. An explicit calculation yields [155] Here, n eff M denotes the effective number of meson loops contributing to the Polonyi mass. Let us assume that all neutral mesons share the same Yukawa coupling, λ 1,2,3,4 0 ≡ λ 0 . In this case, n eff M can be brought into the following compact form (the full expression is complicated and can be found in [155]), where 0 = (λ/λ 0 ) is a loop function that can be approximated by a simple quadratic power law, In the following, we will set λ 0 = 4π to account for the presence of heavy composite states with masses around the dynamical scale, m heavy = λ 0 Λ ∼ Λ dyn . Just like in QCD, such heavy resonances are expected to appear in the spectrum. Our perturbative language, however, does not suffice to capture their dynamics at low energies. For this reason, we will instead follow an effective approach and mimic the effect of additional heavy states by means of a particular choice for λ 0 . For a Yukawa In global supersymmetry, the Polonyi field X corresponds to a tree-level flat direction (see Eq. (53)).
The loop-induced Polonyi mass m 1 is therefore crucial to stabilize the SUSY-breaking vacuum against gravitational corrections in supergravity [154]. We will discuss this issue in more detail in Sec. 2.4.

Dynamical generation of an effective FI term
The IYIT model can also be used to dynamically generate an effective FI term. This was pointed out for the first time in [99]. In this paper, we will make use of this mechanism to generate the effective FI term required for B−L D-term inflation. All we have to do now is to promote the global U (1) A flavor symmetry in the IYIT superpotential (see Eq. (48)) to a local U (1) B−L gauge symmetry. The B−L gauge interactions then result in the following D-term scalar potential in the IYIT sector, Here and in the following, lowercase symbols (m ± , z ± , etc.) denote the complex scalar components of the corresponding chiral multiplets (M ± , Z ± , etc.). The charged mesons M ± acquire nonzero VEVs as a result of the dynamical breaking of supersymmetry (see Eq. (49)). These VEVs spontaneously break B−L which leads to the following effective FI term, This FI parameter is exactly of the form that we anticipated in Eq. (5) in Sec. 2.2. In particular, we now see that, in our dynamical model, the roles of the moduli Ψ andΨ are played by the mesons M ± .
To evaluate the expression in Eq. (65), it is, in principle, necessary to account for the backreaction of the D-term scalar potential in Eq. (64) on the meson VEVs in Eq. (49). In the following, we will, however, restrict ourselves to the weakly gauged limit, |gq 0 | λ, where this backreaction is negligible.
In this case, we can simply continue to work with our results in Eq. (49), such that Eq. (65) turns into Here, we assumed w.l.o.g. that λ + > λ − such that the FI parameter is always positive, ξ > 0. Once we set the parameter ρ to its expectation value,ρ 0.80, we obtain the following simple relation, In Sec. 2.5, we will use this result as a key ingredient in our construction of the inflaton potential.
However, before we are able to do so, we need to make sure that the FI parameter ξ is not "eaten up" by the charged singlet fields Z ± in the IYIT sector. To this end, let us rewrite Eq. (64) as follows, where we integrated out the meson fields and replaced the fields Z ± by the Polonyi field X and the stabilizer field Y . From Eq. (68), it is evident that ξ results in a mass mixing between X and Y , At the same time, X obtains an effective mass m 1 at the one-loop level, while Y acquires a tree-level mass m F from the superpotential. Taken as a whole, this results in the following two mass eigenvalues, In the limit of vanishing mass mixing, ∆m xy → 0, these mass eigenvalues reduce to m 1 and m F , respectively. For too large mass mixing, the eigenvalue m − xy can, however, become tachyonic. In this  (72)). Similarly, the lower bound on λ guarantees that the VEV of the Polonyi field is located in the quadratic part of the effective potential (see Eq. (102)).
Both bounds need to be satisfied to sufficiently stabilize the SUSY-breaking vacuum in the IYIT sector. In both plots, the solid red line indicates where in parameter space the perturbativity constraint in Eq. (57) becomes violated.
case, one singlet mass eigenstate becomes unstable and absorbs ξ in its VEV. To prevent this from happening, the absolute value of ∆m xy needs to be smaller than the geometric mean of m 1 and m F , This requirement translates into an upper bound on the gauge coupling constant g, For ρ =ρ and λ λ pert , we hence obtain |g max q 0 | 7.8 × 10 −2 λ 3/2 . In Fig. 1, we plot the upper bound on |gq 0 | as a function of λ and ρ. This figure illustrates that the stability condition in Eq. (72) is always stronger than the mere requirement of a weak gauge coupling, |gq 0 | λ. The constraint in Eq. (72) is therefore a sufficient condition to justify our analysis in the weakly gauged limit.

Modulus stabilization and mass spectrum at low energies
The dynamically generated FI term in Eq. (65) is an effective FI term that depends on the VEVs of the meson fields M ± . As shown in [95], such FI terms are typically accompanied by a shift-symmetric modulus field (see Sec. 2.2). Our dynamical model is no exception to this statement. In our case, the role of the modulus field is played by the B−L Goldstone multiplet A which contains all meson DOFs after imposing the Pfaffian constraint in Eq. (47). The field A is a chiral multiplet. It, hence, consists of a real scalar c, a real pseudoscalar ϕ, and a Weyl fermionã. In analogy to supersymmetric models of the QCD axion, these particles may also be referred to as the saxion c, axion ϕ, and axinoã (see, e.g., [155]). In our model, the pseudoscalar ϕ corresponds to the Goldstone boson of spontaneous B−L breaking. It remains massless and exhibits a derivative coupling to the B−L vector boson A µ .
To see this, we have to apply the field transformation in Eq. (50) to the kinetic terms of the scalar meson fields. The kinetic part of the Lagrangian then ends up containing the following terms, Note that this is nothing but the Stückelberg Lagrangian of an Abelian gauge field with mass m V , In view of Eq. problems. At the same time, the real scalar c may become problematic, as it threatens to destabilize the FI parameter ξ. This is the notorious modulus problem in the presence of an effective FI term [95]. In our model, this problem is, however, absent. The scalar c and the fermionã are automatically stabilized by the F-term-induced mass m F in the superpotential (see Eq. (53)). To make this statement more precise, let us now review the mass spectrum in the IYIT sector at low energies (see also [99,155]).
The relevant superfields at low energies are the Polonyi field X = (x,x), the stabilizer field Y = (y,ỹ), the Goldstone field A = (a,ã), and the vector field V = (λ, A µ ). Here, x, y, and a are complex scalars. The complex scalar a contains the Goldstone boson and its scalar partner, a = 2 −1/2 (c + iϕ).
The fieldsx,ỹ,ã, and λ are Weyl fermions, where λ denotes the B−L gaugino. A µ is the B−L vector boson. To determine the mass spectrum for these fields, we work in the weakly gauged limit where ∆m xy in Eq. (69) is negligible. A standard calculation in global supersymmetry then yields Here, the indices refer to the respective mass eigenstates which are not necessarily identical to the corresponding "flavor" eigenstates (see, e.g.,ỹ,ã, and λ). As expected, we find that the Goldstone boson ϕ remains massless, m ϕ = 0. Similarly, also the fermionic component of the Polonyi field X remains massless, mx = 0. This is becausex corresponds to the goldstino field of spontaneous supersymmetry breaking. It is absorbed by the gravitino field in supergravity. Finally, also the B−L gaugino λ remains massless (however, see also Eq. (77)). This is due to the following reason. The original Weyl fermion λ shares a Dirac mass with the axino, L ⊃ i m V λã. One would, thus, think that λ ends up forming a Dirac fermion with the axino, i.e., the fermionic component of the B−L Goldstone multiplet. The axino, however, also shares a Dirac mass with the fermionic component of the stabilizer field, L ⊃ m Fãỹ , that is parametrically larger, m F m V , in the weakly gauged limit.
ỹ therefore "steals" the axino from the gaugino, so that the gaugino no longer has a mass partner to form a Dirac fermion. In more technical terms, this goes back to the fact that the diagonalization of two Dirac masses for three Weyl fermions necessarily results in one massive Dirac fermion and one massless Weyl fermion. For our purposes, the most important lesson from Eq. (75) is that the superpartners of the Goldstone boson, the saxion c and the axinoã, are indeed stabilized. Both fields obtain masses from the gauge sector 7 as well as from the F-term-induced mass in the superpotential.
This solves the modulus problem and assures us that ξ is a viable input for the construction of our inflation model.
Up to now, our discussion only dealt with the properties of the IYIT model in global supersymmetry. The next important step is to embed the IYIT model into supergravity. As discussed in Sec. 2.2, we especially intend to work in Jordan-frame supergravity with canonically normalized kinetic terms.
However, before going into any details, let us briefly discuss the implications of supergravity for the low-energy mass spectrum. On general grounds, we essentially expect two effects: (i) In supergravity, R symmetry is broken to ensure the vanishing of the cosmological constant (CC) in the true vacuum.
The order parameter of R symmetry breaking is the gravitino mass m 3/2 . We therefore expect that supergravity leads to corrections to the various mass eigenvalues of O m 3/2 . This should, in particular, also hold true if all other sectors are sequestered from the hidden SUSY-breaking sector. (ii) As a consequence of R symmetry breaking, the Polonyi field X acquires a nonzero VEV that is typically parametrically larger than the gravitino mass, X ∝ m 3/2 / c 2 1 λ 3 (see Sec. 2.4). This induces an effective Majorana mass for the Goldstone multiplet A in the superpotential (see Eq. (53)), The Majorana mass m A breaks some of the mass degeneracies in Eq. (75) and helps to make sure that the B−L gaugino λ acquires a nonzero mass after all. To illustrate the effect of nonzero m A , we compute the fermion masses in the limit of vanishing gravitino mass, m 3/2 → 0, and small Majorana mass, m A m F . This exercise leads to the following mass eigenvalues (in the Jordan frame), and again mx = 0. Now, the vector boson mass m V also receives contributions from the Polonyi VEV, where we parametrize the two contributions from f A and X to v A in terms of a mixing angle θ, Eq. (77) demonstrates that, for X = 0, the gaugino λ indeed obtains a nonvanishing mass. Moreover, we recognize that the Dirac fermion (ỹ,ã) splits into two nondegenerate Majorana fermions. Making 7 This follows from the super-Higgs mechanism. In consequence of spontaneous B−L breaking, the massless vector multiplet V (one Weyl fermion, one massless vector boson) absorbs an entire chiral multiplet (one complex scalar, one Weyl fermion), so that it eventually contains the DOFs of a massive vector multiplet (one real scalar, one Dirac fermion, one massive vector boson). In the absence of supersymmetry breaking, all these DOFs would have a common mass mV .
the same assumptions as before, we also compute the scalar masses for a nonvanishing Polonyi VEV, where and m x = m 1 and m ϕ = 0. For a dominant F-term-induced mass, m A , m V m F , this simplifies to We, thus, find that the Majorana mass m A breaks the degeneracy among the two real scalar components of Y and shifts the saxion mass to even larger values. This concludes our analysis of the IYIT mass spectrum. We will now discuss the embedding into Jordan-frame supergravity in more detail.

Effective Polonyi model and embedding into supergravity
In global supersymmetry, the Polonyi field X is stabilized at the origin in field space, X = 0, by its one-loop effective mass m 1 (see Eq. (61)). Gravitational corrections in supergravity, however, threaten to destabilize this vacuum solution. We shall now explain why this is a serious problem. At field values larger than some critical value, |x| x c , the one-loop effective potential for the complex Polonyi field x changes from a quadratic to a logarithmic behavior (see, e.g., [42] for an explicit calculation), Here, V 0 1 denotes the height of the logarithmic plateau at large field values. The critical field value x c is reached once the Polonyi field induces effective masses for the quarks in the IYIT sector of O (Λ dyn ).
The Yukawa interactions of the Polonyi field with the IYIT quarks follow from Eqs. (39) and (50), This allows us to estimate x c . For definiteness, we will work with x c = √ 2 Λ dyn /λ in the following.
At |x| x c , the IYIT quarks decouple perturbatively, which gives rise to the logarithmic corrections in Eq. (83). Dangerous SUGRA corrections can shift the Polonyi VEV from the origin in field space towards the logarithmic plateau at large field values. Once this happens, the Polonyi field is no longer stabilized and the system settles into a completely different vacuum at field values of O (M Pl ). In the following, we will illustrate where in parameter space this unwanted conclusion can be avoided. This will, in particular, provide us with a useful lower bound on the Yukawa coupling λ (see Eq. (102)).
For the purposes of this section, it will suffice if we exclusively focus on the Polonyi field X and integrate out the heavier fields A and Y . The low-energy superpotential in Eq. (53) then turns into Here, w denotes a constant contribution to the superpotential that we added by hand. w is meaningless in global supersymmetry. In supergravity, it accounts for the fact that R symmetry must be broken to ensure a vanishing CC in the true vacuum. In the presence of the constant w, Eq. (85) is nothing but the superpotential of the standard Polonyi model of spontaneous supersymmetry breaking [156].
In this sense, the IYIT model can be regarded as a UV completion of the Polonyi model that offers a dynamical explanation for the origin of the SUSY-breaking scale µ. This is expected as the IYIT model is, after all, just a dynamical realization of spontaneous supersymmetry breakingà la O'Raifeartaigh.
The IYIT model does not explain the UV origin of the constant w. In this paper, we will not speculate about this issue any further. That is, we do not have anything new to say about the CC problem.
Eq. (85) needs to be supplemented by the following Kähler potential (see Eqs. (16) and (28)), where we set all terms to zero that are irrelevant for the present discussion. The functions in Eq. (85) and (86) allow us to calculate the total SUGRA potential for the Polonyi field in the Jordan frame, Here, the second term on the right-hand side, i.e., the Jordan-frame one-loop effective potential V J 1 , is equivalent to the global-SUSY expression V 1 in Eq. (83). The third term on the right-hand side corresponds to a tree-level SUGRA correction in the Jordan frame, while the last term represents the gravity-induced mass discussed around Eq. (25). The function f in Eq. (87) is defined as follows, which may be regarded as a dimensionless measure for the amount of superconformal symmetry breaking in the kinetic function of the inflaton field. In the following, we will refer to f as the reduced kinetic function of the inflaton field. Given our choice for F in Eq. (28), the function f evaluates to As expected, this expression vanishes for χ → χ CSS = 1/2. The imaginary component of the complex inflaton field will be stabilized during inflation, τ = 0. During inflation, we therefore have to deal with The constant w controls the value of the CC. To ensure that inflation ends in a Minkowski vacuum with vanishing CC, we need to impose the following two conditions after inflation, i.e., for f = 0, These two conditions can be solved for the two unknowns X and w 0 . Let us focus on w 0 for now, 8 Here, k is a convenient measure for the relative importance of the SUGRA corrections in Eq. (87), The definition of k is chosen such that it mimics the effect of a higher-dimensional operator in F hid , that induces a mass correction ∆m 2 k = m 2 1 + m 2 R in the SUGRA potential. Large values of k, thus, indicate that the Polonyi field is strongly stabilized by its one-loop effective mass, ∆m 2 k m 2 3/2 . Given the result for the constant w 0 in Eq. (92), we can go one step back and determine the time-dependent Polonyi VEV during inflation. To do so, we just need to solve one equation, Assuming that X is located in the quadratic part of the potential, Eq. (95) has the following solution, Indeed, in the limit of a large one-loop effective mass, k 1, the Polonyi field remains stabilized close to the origin, X M Pl . For f = 0, Eq. (96) turns into the solution of Eq. (91). Both the constant w 0 and the Polonyi VEV X break R symmetry. This is because both the superpotential W as well as the IYIT singlets Z ± carry R charge 2. We can therefore use our results in Eqs. (92) and (96) to determine the order parameter of R symmetry breaking, i.e., the gravitino mass in the Jordan frame, This result allows us to write the total mass of the Polonyi field in the Jordan frame, m J x , as follows, In the following, we will restrict ourselves to the large-k limit. This is justified because k is typically at least of O (10) in the part of parameter space that we are interested in (see Eqs. (54) and (61)), In the large-k limit, the Polonyi VEV in Eq. (96) can be approximately written as follows, confirming that X is parametrically enhanced compared to the gravitino mass, X ∝ m 3/2 / c 2 1 λ 3 . For large values of k, also the relations between w 0 , µ, and m 3/2 become significantly simpler, In the remainder of this paper, we will restrict ourselves to working with these approximate expressions.
We derived the result in Eq. (100) under the assumption that the leading term in the one-loop effective potential is an effective mass term, V J 1 = m 2 1 |x| 2 + O x 4 . The derivation of Eq. (100) is therefore only self-consistent and valid as long as X x c (see Eq. (83)). This implies a lower bound on the Yukawa coupling λ that depends on the energy scale Λ as well as on the hierarchy parameter ρ. We plot this lower bound λ min as a function of Λ and ρ in the right panel of Fig. 1. The exact numerical result shown in this figure is well approximated by the following analytical expression, λ min is sensitive to ρ at large values of ρ where n eff M in Eq. (62) is dominated by ρ rather than the neutral meson contribution 4 0 . At small ρ values, where n eff M ≈ 4 0 , the ρ dependence disappears. In addition to the lower bound λ min , we also require that λ must remain perturbative, λ < λ pert = 4.
In the following, we will eliminate λ from our analysis and set it to the following value, for simplicity, That is, we fix λ just at the central value of the allowed range of values, λ min < λ < λ pert . Together with our choice for the hierarchy parameter, ρ =ρ 0.80, this removes all dimensionless parameters from the IYIT sector, so that the only remaining free parameter is the scale Λ. We then obtain forλ, At the same time, λ must not be larger than λ 2.41, since otherwise λ + or λ − will exceed λ pert (see

Scalar potential in the inflaton sector
We now have all ingredients at our disposal to construct our inflationary model. In a first step, we add the usual superpotential of D-term inflation (see Eq. (2)) to the Polonyi superpotential in Eq. (85), The inflaton S as well as the waterfall fields Φ andΦ belong to a separate sector that is sequestered from the IYIT sector. The relevant Kähler potential is of the following form (see Eqs. (16) and (28)), The waterfall fields Φ andΦ carry B−L charges +q and −q, respectively. They, thus, appear in the D-term scalar potential, together with the dynamically generated FI parameter ξ (see Eq. (65)), V J D denotes the D-term scalar potential in the Jordan frame which is identical to the D-term scalar potential in global supersymmetry, V 0 D . In addition, the total tree-level scalar potential V J tree also receives an F-term contribution V J F which can be computed by making use of Eqs. (106) and (107), Here, the first contribution, V 0 F , denotes the usual F-term scalar potential in global supersymmetry, while the second term, ∆V J F , corresponds to the tree-level SUGRA correction in the Jordan frame, For more details on the computation of ∆V J F , see Appendix A. In Eqs. (110) and (111), we introduced the masses squared m 2 eff and δm 2 eff as well as the quartic coupling δκ 2 . These are defined as follows, Note that all three parameters are field-dependent. The real mass parameter m 2 eff denotes the effective inflaton-dependent mass that stabilizes the waterfall fields during inflation. The complex mass parameter δm 2 eff is a bilinear mass that originates from the interference between the supersymmetric mass of the waterfall fields in the superpotential, κ S , and the gravitino mass m 3/2 . This so-called B term is, hence, a consequence of R symmetry breaking in the superpotential. It is only generated for the scalar waterfall fields and not for the corresponding fermions, which is why it breaks supersymmetry.
Just like the usual A terms in models of broken supersymmetry, the B term results in a soft breaking of supersymmetry. The coupling δκ 2 will be irrelevant for our purposes as it constitutes just a small correction to κ 2 , i.e., the quartic coupling of the waterfall fields in global supersymmetry.
During inflation, the waterfall fields are stabilized at their origin, φ = φ = 0. Along the inflationary trajectory, the tree-level scalar potential in the Jordan frame therefore reads as follows, Here, D 2 0 /2 denotes the contribution to the vacuum energy density from the D-term scalar potential. The large F-term contribution, V 0 F ⊃ +F 2 0 , is canceled by the contribution from R symmetry breaking that is contained in the SUGRA correction to the scalar potential, ∆V J F ⊃ −F 2 0 . This explains why the vacuum energy density during inflation is dominated by the D-term contribution in our model. During the B−L phase transition at the end of inflation, the D term is absorbed by the VEV of one of the waterfall fields. In the true vacuum after inflation, all contributions to the CC therefore approximately cancel. In the next section, we will perform a standard slow-roll analysis of our inflationary model. This is most easily done in terms of the usual slow-roll parameters in the Einstein frame. To compute these parameters, we need to convert the potential in Eq. (113) from the Jordan frame to the Einstein frame. This is achieved by rescaling V J tree by the fourth power of the conformal factor C (see Eq. (13)), On the inflationary trajectory, this can be rewritten as a function of the parameter z (see Eq. (90)), The second derivative of V tree w.r.t. the field σ provides us with the mass parameter m 2 σ (see Eq. (29)), As anticipated, m σ is suppressed by the shift-symmetry-breaking parameter χ. By appropriately choosing χ, we will therefore be able to adjust the scalar spectral index n s so that it matches the observed best-fit value. We also note that the mass parameter m σ is not a physical mass eigenvalue in the actual sense. This is because the scalar inflaton field σ is not properly normalized. The mass of the canonically normalized inflaton fieldσ is instead given in terms of the slow-roll parameter η in the Einstein frame, m 2 σ = 3 ηH 2 ∼ m 2 σ (see Sec. 3.2). Moreover, m σ is just a tree-level parameter, whereas the scalar potential also receives important contributions at the one-loop level.
The radiative corrections are encoded in the one-loop effective Coleman-Weinberg potential [157], Here, STr [·] stands for the supertrace over a (matrix-valued) function of the total tree-level mass matrix squared M 2 J in the Jordan frame. We evaluate V J 1 in the MS renormalization scheme and only consider contributions from scalars and fermions. This fixes the numerical constant C in Eq. (117) to C = 3/2 (see, e.g., [158]). The energy scale Q J denotes the MS renormalization scale. To determine the radiative corrections to the inflaton potential, it is sufficient to focus on the inflaton-dependent masses in the waterfall sector. From Eqs. (24), (108), (110), and (111), we find for the scalar fields, where the mass parameters m 2 D , m 2 R , m 2 eff , and δm 2 eff are respectively defined below Eq. (8) as well in Eqs. (25) and (112). In addition to these tree-level masses, the scalar waterfall fields also obtain gaugemediated masses at the loop level. This is because supersymmetry breaking in the IYIT sector results in a mass splitting among the components of the massive B−L vector multiplet (see Eq. (75)) [159].
Including these one-loop masses in Eq. (117) would result in a two-loop contribution to the effective potential. For this reason, we will ignore the effect of gauge-mediated supersymmetry breaking (GMSB) for now. We will come back to this issue in Sec. 4.1. In the next step, we diagonalize the mass matrix in Eq. (118). This results in two complex mass eigenstates φ ± with inflaton-dependent mass eigenvalues, Here, δ is related to the rotation angle β φφ that diagonalizes the scalar mass matrix, δ 2 = tan 2 β φφ .
We note that the parameter δ depends on the inflaton field value, which makes it a time-dependent quantity. For this reason, the mass eigenstates φ ± do not coincide with the charge eigenstates φ,φ We are now ready to evaluate V J 1 in Eq. (117). Our final result can be written as follows, where L is a one-loop function that takes the same form in the Jordan frame as in the Einstein frame, and where the variable x, the parameter α, and the renormalization scale Q J are introduced such that All of these quantities depend on the inflaton field value by virtue of the parameters m 2 eff , m 2 R , and δ. In the following, we will, however, neglect the field dependence of m 2 R and approximate it instead by the constant expression in Eq. (116). The fact that Q J is field-dependent does not pose any problem for our model. Recall that one usually encounters a field-independent renormalization scale Q J in the Jordan frame and a field-dependent renormalization scale Q = C Q J in the Einstein frame -or vice versa. In our model, the renormalization scale is, by contrast, field-dependent in both frames. This is a priori a perfectly valid choice. Independent of whether Q J is field-dependent or not, we merely have to make sure that our final results are not overly sensitive to our particular choice for Q J . This is a requirement that we will have to check a posteriori as part of our slow-roll analysis (see Sec. 3).
The main field dependence of Eq. (120) is encoded in x which may also be written as follows, x = 1 therefore corresponds to the critical point along the inflationary trajectory at which m − vanishes, At this point in field space, the mass eigenstate φ − becomes tachyonically unstable which triggers the B−L waterfall transition. We also note that Eq. To convert Eq. (120) into the one-loop effective Coleman-Weinberg potential in the Einstein frame, we need to multiply again by C 4 , just like in the case of the tree-level scalar potential (see Eq. (114)), The Weyl transformation from the Jordan frame to the Einstein frame therefore corresponds to nothing but a rescaling of the Q 4 J factor in Eq. (120). The loop function L remains unchanged. This is consistent with the fact that the Weyl transformation in Eq. (13) only affects dimensionful parameters.
A mass scale m J in the Jordan frame is, e.g., mapped onto m = C m J in the Einstein frame. Dimensionless ratios of mass parameters, thus, remain invariant under the Weyl transformation [117,118]. In passing, we also mention that the effective scalar potential V 1 in Eq. (125) cannot be derived from the effective Kähler potential K 1 in Eq. (30). The reason for this is that, in D-term inflation, the effective Kähler potential K 1 can enter into the total scalar potential only via the D-term scalar potential, This, however, only constitutes a contribution to the one-loop effective potential for the waterfall fields which we are not interested in. The one-loop effective potential for the inflaton field in Eq. (125) has a different origin. This can be explicitly seen in the superspace formulation of global supersymmetry.
There, Eq. (125) does not follow from the effective potential for the chiral multiplets S, Φ, andΦ, i.e., from the effective Kähler potential K 1 , but from the effective potential for the auxiliary D component of the vector field V [139]. This quantity is discussed less often in the literature. Alternatively, Eq. (125) can also be derived in a superspace language that applies to models of softly broken global supersymmetry [142]. In this approach, one first integrates out the heavy vector multiplet V such that supersymmetry is softly (and explicitly) broken in the effective theory at low energies. Then, one calculates the radiative corrections to the soft SUSY-breaking terms in the Lagrangian. This allows one to recover Eq. (125) as the one-loop renormalization of the so-called soft Kähler potential K. In an explicit calculation, we convince ourselves that Eq. (125) indeed satisfies the relation V 1 = − K 1 . It is interesting to note that this result differs from the situation in F-term inflation. There, the effective Kähler potential directly contributes to the effective inflaton potential via the F-term scalar potential.
The total inflaton potential follows from the combination of our results in Eqs. (114) and (125), The individual parameters and functions appearing in this potential can be found in the following Compared to this potential, our total inflaton potential V receives four different SUGRA corrections: (i) The total potential is rescaled by C 4 to account for the transition from the Jordan frame to the Einstein frame. (ii) The approximate shift symmetry in the inflaton kinetic function in combination with F-term SUSY breaking in the IYIT sector results in a small contribution from the F-term scalar potential. (iii) The soft B term mass δm 2 eff modifies the prefactor of the one-loop effective potential as well as the definition of the field variable x. (iv) The gravity-induced mass m R gives rise to the parameter α. All of these effects vanish in the global-SUSY limit, such that V → V 0 . One of our main claims is that the SUGRA corrections in Eq. (127) are instrumental in realizing a viable scenario of D-term inflation that is in agreement with all theoretical and phenomenological constraints.

Phenomenology: A viable scenario of D-term hybrid inflation
In the previous section, we introduced a supergravity embedding of the IYIT model of dynamical In this section we turn to the phenomenology of the resulting DHI model, outlined in Sec. 2.1.
This will essentially fix the only remaining free parameter of our DSB sector, the dynamical scale Λ. After briefly reviewing the standard picture in global supersymmetry, we proceed to an analytical study of the parameter space, highlighting the most important effects of the different contributions to the scalar potential calculated in the previous section. We then present a full numerical study of the relevant parameter space, supplemented by a discussion on the initial conditions in different parts of the parameter space.

D-term inflation in global supersymmetry
The key ingredients of globally supersymmetric DHI are the superpotential and D-term potential given in Eq. (2). The waterfall fields Φ,Φ obtain masses which depend on the scalar component s of the chiral multiplet S, which stabilize them for values of the inflaton field above the critical value |s glob c | 2 = g 2 qq 0 ξ/κ 2 . These field-dependent masses result in a Coleman-Weinberg one-loop contribution to the effective potential of the inflaton, so that the scalar potential for the inflaton above the critical field value is given by Eq. (128). At the critical field value (corresponding to x 0 = 1) one of the waterfall fields acquires a nonvanishing vacuum expectation value, absorbing the FI term ξ.
Identifying the inflaton field as the radial component of s, ρ = √ 2 |s|, its classical evolution during inflation is described by the slow-roll equation, where N = − Hdt counts the number of remaining e-folds until the end of inflation (with N = 0 at the end of inflation). At field values much larger than the critical field value ρ c , the scalar potential (128) can be approximated as Eq. (129) is an accurate description of the inflationary dynamics as long as the slow-roll parameters, are much smaller than one. The CMB observables, describing the statistical properties of quantum vacuum fluctuations during inflation, can be expressed in terms of these variables as evaluated at N * 55 e-folds before the end of inflation.
DHI ends at the critical field value ρ c or even earlier, when the second slow-roll parameter η becomes large, ρ η = gqM Pl /(2π), depending on the size of ρ η /ρ c . For ρ η /ρ c 1, i.e., if the slow-roll condition is violated before the critical point, the value of ρ at N * e-folds before the end of inflation is given by ρ 2 * (g 2 q 2 N * M 2 Pl )/(2π 2 ). The amplitude of the scalar spectrum is mainly controlled by the FI parameter, and its spectral index, governed by the second slow-roll parameter η, is obtained as n s 1 − 1/N * 0.98 in the limit of gq 4π. For values of ξ around the GUT scale, this yields the correct scalar amplitude, albeit with a somewhat too large spectral index, disfavored by about 2σ by the current data [76].
On the other hand, if the slow-roll conditions are satisfied all the way down to the critical field value, we find ρ * ρ c . The small value of the inflaton coupling κ in this region of parameter space implies that the field excursion during N * 55 e-folds of inflation is typically small compared to the field value at the end point of inflation ρ c . The observed value of the scalar amplitude fixes enabling lower values of ξ for smaller values of κ. The spectral index in this region is found to be n s 1, excluded at more than 5σ by the PLANCK data [76].
Despite its simplicity and obvious connection to particle physics, this model has several major shortcomings, as discussed in Sec. 2.2. These are connected to the origin of the FI mass scale in supergravity, the stability of scalar fields during inflation, gravitational corrections to the inflaton trajectory in supergravity, and phenomenological constraints from CMB observations. In the following, we demonstrate how all these shortcomings can simultaneously be overcome in our setup.

Inflationary dynamics in SUGRA
In the following we implement DHI with the dynamically generated FI term of Sec. 2.3, supplemented by the assumption of an approximate shift symmetry in the direction of the inflaton field. As discussed in Section 2.2, this shift symmetry is broken by one-loop effects in the scalar potential and Kähler potential. The interplay of these two small contributions will enable us to identify regions in parameter space which comply with all experimental constraints. 9 The dynamics of inflation is determined by the scalar potential (127), which contains all relevant supergravity and one-loop contributions. Our choice of kinetic function F (see Eq. (28)) with χ 1 ensures that σ, the real part of the complex scalar s, plays the role of the inflaton. The supergravity version of Eq. (129) in the Einstein frame reads where V = C 4 V J is the Einstein-frame scalar potential and K ss * = ∂ 2 K/(∂s∂s * ) is the prefactor of the kinetic term for the inflaton. The initial condition (i.e., the end of inflation) is given by σ(N = 0) = max(σ c , σ η ). This enables us to evaluate the (Einstein-frame) slow-roll parameters ε(σ) and η(σ) and hence the CMB observables at N * = 55 e-folds before the end of inflation. Evaluating the slow-roll parameters requires derivatives of the scalar potential w.r.t. the canonically normalized fieldσ, which we perform by exploiting ∂σ/∂σ = √ K ss * , as follows from the canonical normalization of the kinetic terms in the Einstein frame, 9 This includes the nonobservation of cosmic strings, as will be demonstrated in Sec. 4.4. For convenience, we recall here a few key quantities (evaluated along the inflationary trajectory) introduced earlier (see Eqs. (13), (15), (28), and (90)) For more details on translating between the Einstein and Jordan frames, see App. A.
The results of the numerical analysis are shown in Fig. 2. Before discussing them in detail, we will give an analytical analysis of the parameter space in the vicinity of the globally supersymmetric limit.
This will prove instructive for interpreting the numerical results.

Slow-roll parameters
The slow-roll parameters in the Einstein frame can be expressed in terms of derivatives of the scalar potential and of the kinetic function in the Jordan frame as (see Appendix A.4), These expressions are equivalent to those found in Appendix A of Ref. [160]. In the following, we will use Eqs. (138) and (139) to analyze the inflationary predictions, since this format enables us to nicely disentangle the different contributions in various parts of the parameter space.
With the definitions above, simplified expressions for the Jordan-frame slow-roll parameters ε J and η J can be obtained by approximating the Coleman-Weinberg one-loop potential for x 1 and α 1 as with x, α and Q J given in Eq. (122). As in the globally supersymmetric case, x = 1 denotes the critical point. Eq. (141) is a good approximation as long as 1 x 1/α, which will hold in most of the parameter regime of interest. We then find with f given in Eq. (90) and where δ was introduced in Eq. (119) and F 0 and D 0 denote the F-and D-term contributions from global supersymmetry, respectively. For both ε J and η J , the term proportional to D 2 0 stems from the Coleman-Weinberg one-loop potential whereas the term proportional to F 2 0 is a supergravity effect, induced by the noncanonical terms in the Kähler potential. Moreover, we note that the Coleman-Weinberg term splits into the expression familiar from global supersymmetry (indicated by the "1" in the parentheses) and the supergravity contribution to the waterfall field sector, parametrized by δ 4 ε,η . Note that, as in global supersymmetry, ε J is suppressed compared to η J .
Eqs. (142) and (143) illustrate the main effect of the F-term SUGRA corrections in our model.
indicating that the supergravity contributions from the tree-level F-term scalar potential can induce the Turning to the effects of a small, positive value for χ on the first slow-roll parameter, see Eq. (142), we note that a positive χ will lead to a decrease in ε. The one-loop and SUGRA contributions may even cancel each other, indicating the presence of a hilltop or a saddle point in the scalar potential.
We will come back to the consequences of such a scenario below.
The explicit expressions for the remaining auxiliary Jordan-frame slow-roll parameters are Here, ζ J captures the gravity-induced mass of the inflaton and emphasizes once more the need for an approximate shift symmetry (χ 1) for the inflaton field: for χ 1/2, the inflaton picks up a gravity-induced mass just as all the other scalars do, with m 2 R D 2 0 /(3M 2 Pl ). This implies a contribution to the slow-roll parameter η of ∆η ζ 2/3 (see Eq. (26)) and, hence, leads to an η problem. A purely canonical term in the inflaton kinetic function, F S † S, is therefore not viable in our model. The approximate shift symmetry resolves the problem, suppressing this contribution as ∆η ζ 4χ/3. Of course, Eq. (145) and (147) directly correspond to the second and first terms of Eq. (116), respectively. Due to F 0 > D 0 , the contribution from Eq. (145) will always dominate for χ 1.

Viable parameter space
Starting from Eqs. (148) and (149) for σ = σ * . This implies a lower bound on δ (governing the sizes of both δ ε and δ η ). At the same time, δ η yields a positive contribution to η and a too large value will lead to an enhancement of the spectral index n s . Together this implies δ ∼ O(0.1 · · · 1), where to leading order δ 4 κ 2 σ 2 m 2 3/2 /(2q 2 0 q 2 g 4 ξ 2 ). We can estimate δ by exploiting the analytical results for σ * in globally supersymmetric D-term hybrid inflation, see Sec. 3.1. In addition, we note that the requirement that the tree-level supergravity term contributes ∆η ∼ −0.01 implies − ∆η 2χ 3 where we have inserted the relations (67) and (105).
In the regime of large κ (and taking for simplicity |q| ∼ |q 0 | ∼ 1), the constraints on δ thus roughly fix the parameter combination κ 2 √ ξ/g 2 . Using Eq. (151) we immediately see that the correct spectral index can be obtained for The amplitude of the scalar power spectrum A s is mainly dependent on ξ, see Sec. 3.1. This essentially fixes ξ 10 −5 M 2 Pl , and determines the preferred range of the gauge coupling, e.g. g ∼ 0.1 for κ = 0.1. Note that this constraint can be circumvented if one allows inflation to begin very close to the hilltop of the scalar potential, ε * 0, which can be obtained by tuning the contributions in Eq. (148). From Eq. (132) we see that in this case, we can in principle arbitrarily lower ξ. However, this corresponds to a very tuned situation and we will not focus on this regime of the parameter space.
In the regime of small κ we note from the expressions of the globally supersymmetric limit that σ * σ c and n s 1. This indicates that (i) the leading-order term in the expansion of V CW in 1/x becomes a poor approximation and (ii) to obtain the correct spectral index, we must rely nearly exclusively on the supergravity terms in η. As a result of the first point, the lower bound on δ in fact becomes irrelevant when using the full expression for the one-loop potential. We are thus left with ξ 3/2 /g 2 8.5 × 10 −3 M 3 Pl /2 δ 4 . Imposing the observed value for A s and approximating σ * by the corresponding expression in globally supersymmetric DHI, this yields Inserting this into Eq. (151), we find

Scan of parameter space and numerical results
In this section we present our results for a numerical scan of parameter space, focusing on the regions identified analytically in the previous section. Starting from the full scalar potential (127)  Our results are depicted in Fig. 2, for κ = 0.1 (left panel) and κ = 10 −3 (right panel), as well as in Fig. 3 where we have imposed the additional relation χ = χ 1 = κ 2 /(16π 2 ), see Eq. (30). In all figures, the green band indicates the region of parameter space in accordance with all constraints.
In the parameter space of interest, we find values for the tensor-to-scalar ratio r of O(10 −6 · · · 10 −4 ), Eq. (149), as well as with our estimates for the shift-symmetry-breaking parameter χ in Eqs. (152) and (154). This underlines that although our numerical analysis takes into account all contributions to the scalar potential, the most relevant contribution to lower the spectral index is the shift-symmetrysuppressed soft mass for the inflaton, leading to Eq. (145).
Our choices for the coupling κ are designed to cover the relevant aspects of the parameter space, while focusing on particularly interesting benchmark points. As it is responsible for explicit shift symmetry breaking in the superpotential, we expect κ 1. In the left panel of Fig. 2, we consider κ = 0.1.
This enables us to reproduce the observed CMB observables (in particular n s ) with loop and SUGRA contributions of comparable size, leading to χ ∼ 10 −4 . In global-SUSY DHI the parameter space splits into two regimes, characterized by the size of |σ η /σ c | and consequently by different parameter dependencies of s * , see Sec. 3.1. The value of κ = 0.1 falls into the regime of σ η σ c . In the right panel of Fig. 2 we turn to the opposite regime, σ η σ c . To reproduce the observed spectral index, we here need to require the SUGRA contributions to clearly dominate over the one-loop contributions.
Note that for even smaller values of κ, the critical value σ c can take super-Planckian values, enabling a phase of "subcritical hybrid inflation" after the inflation field has passed σ c [161,162].

Initial conditions
In the viable parameter space, inflation occurs either near a hilltop (i.e., a local maximum in the scalar potential) or near an inflection point, depending on the exact values of χ and g. In Fig. 4 we depict these two possibilities (for κ = 0.1), together with the decomposition of the total scalar potential into its dominant components. The solid line shows the full scalar potential, while the labeled dashed lines indicate the following components: (1) leading-order term in the Coleman-Weinberg potential in the global-SUSY limit, (2) supplemented with the leading supergravity terms to the inflaton F-term potential and to the waterfall mass spectrum and (3) in addition supplemented with the next-to-leading-order term in the expansion of the Coleman-Weinberg potential (see Eq. (141)), . The latter term becomes relevant as x(σ) increases to x ∼ 1/α. The remaining discrepancy compared to the full scalar potential (in the left panel) is mainly due to the σ dependence of the D-term potential in the Einstein frame, induced by the conformal factor. Implementing inflation in the left panel of Fig. 4 requires some fine-tuning in the initial conditions, to ensure the correct vacuum is reached. However, we point out two further observations: (i) in the entire parameter space of interest, we find H * /(2π) ≪ (σ max − σ * ), i.e., if (by accepting some tuning), the initial conditions are in the desired regime, they are at least stable against quantum fluctuations.
(ii) lowering the B−L gauge coupling g, the energy level of the false minimum is raised compared to V (σ * ). An interesting (albeit fine-tuned) situation arises if the vacuum energy density of the false minimum lies just a tiny bit above V (σ * ), allowing for a phase of eternal inflation, followed by N * e-folds of inflation arising once the inflaton field tunnels through the potential barrier.
On the other hand, in the right panel of Fig. 4 inflation can start at large field values, avoiding an initial conditions problem. There is however some degree of tuning required in the model parameters to ensure this shape of the potential. For κ = 0.1, this becomes particularly relevant for small values for the structure of the scalar potential are (i) inflection point inflation (black "+" symbol, same situation as in the right panel of Fig. 4), (ii) hilltop inflation with a local minimum at σ σ * (white "x" symbol, same situation as in the left panel of Fig. 4) and (iii) hilltop inflation without a local minimum (red "o" symbol). This last situation arises due to the effect described below Eq. (155), and requires tuning the initial conditions. As in Figs. 2 and 3, the green band indicates that the scalar spectral index lies within the 2σ band.
of g, when large SUGRA contributions to the slow-roll parameters need to be carefully balanced. This results in the 'fine-tuning' constraint on the parameter space in Fig. 2. An overview of these different regions in parameter space is given in Fig. 5 for the cases of κ = 0.1 and χ = κ 2 /(16π 2 ), in both cases focusing on the region of parameter space which reproduces the correct CMB observables.
Note that for very large field values, σ 2 ∼ 3M 2 Pl /χ ∼ 10 4 M 2 Pl , both the conformal factor C and the F-term potential exhibit a pole (see Eqs. (152) and (114)): After canonical normalization of the inflaton field, the pole in the conformal factor will be pushed to infinity. The pole in the F-Term potential is always at larger field values and is hence never reached. 11 However, for χ 1, σ ∞ F approaches σ ∞ C and the F-term potential begins to dominate the tree-level potential already for σ < σ ∞ C . This can generate a false, negative-valued vacuum at large field values. In the regions of parameter space where the inflaton field reaches values of order M Pl , this can impact the vacuum structure. However at these large field values, σ M Pl , higher-order operators may significantly modify the scalar potential.
For κ = 10 −3 , we find that inflation typically occurs in a small field region in the vicinity of a hilltop, accompanied by a false, often negative-valued vacuum at large field values. The amplitude of this vacuum is lifted as g and χ are decreased, until at values of g 5 × 10 −4 inflation occurs close 11 The pole in the conformal factor implies that in the field space of the canonically normalized fieldσ, the scalar potential asymptotes to lim σ→∞ V (σ) = V (σ ∞ C ) and is thus always bounded from below.
to an inflection point. The shape of the potential is well described by the D-term potential, C 4 D 2 0 /2, supplemented by the full globally supersymmetric one-loop potential (not truncated at O(1/x)) and the leading-order SUGRA contribution from the F-term potential.

Discussion: Particle spectrum and cosmology after inflation
During inflation, supersymmetry is broken through F-and D-term contributions. After inflation, when the D-term is absorbed into the VEV of the waterfall field, only F-term supersymmetry breaking remains. This is communicated to the particles of the MSSM through (i) higher-dimensional terms in the Kähler potential (Planck-scale-mediated supersymmetry breaking, PMSB) (ii) anomalymediated supersymmetry breaking (AMSB) and (iii) couplings to the B−L multiplet which receives a supersymmetry-breaking mass splitting at tree level (gauge-mediated supersymmetry breaking, GMSB), see e.g. [164] for an overview. While PMSB will play a crucial role for the SM squarks and sleptons and for the mass parameters of the Higgs sector, the standard model gauginos will only receive a loop-suppressed AMSB contribution. At low energies, the particle spectrum thus resembles the results obtained in pure gravity mediation (PGM) [19,[22][23][24], 12 with the overall scale of the spectrum determined by the inflationary observables (which determine the value of the FI parameter ξ), see Eqs. (67) and (105). In this section, we discuss the mass spectrum of our model during and after inflation, and discuss consequences for early-Universe cosmology, including reheating, leptogenesis, dark matter and the production of topological defects.

Stabilization of squarks and sleptons during and after inflation
The total scalar mass of an MSSM matter field Q i with gauge charge q i is given by where the first term denotes the D-term-induced mass present only during inflation, the second term is a tree-level supergravity contribution induced by higher-dimensional operators in the Kähler potential, the third term is the gauge-mediated SUSY-breaking contribution and the fourth term is the gravity-induced mass (present only during inflation). Assuming that the MSSM sector couples to the supersymmetry-breaking sector via higher-dimensional operators in the Jordan frame (see Eq. (18)), yields and hence ∆m 2 0 = 3 m 2 3/2 M 2 Pl /M 2 * . The quantities m D and m R have been introduced in Eqs. (8) and Eq. (25), respectively. m gm denotes the mass contribution obtained through gauge mediation, with the dominant (one-loop) effect arising due to the mass splitting within the B−L gauge multiplet [159] with m V , m c and mã given in section 2.3. This contribution is clearly subdominant compared to the tree-level D-term contribution during inflation. The stabilization of the MSSM scalars during inflation (m 2 0,i H 2 ) requires a ii < 0 and which for |q i | |q 0 | |a ii | 1 implies a value relatively close to the dynamical scale Λ.
After the end of inflation, only the second and third terms in Eq. (156) remain, leading to very heavy MSSM squarks and sleptons,

MSSM gauginos
Similar to the squarks and sleptons of the previous subsection, the µ and B parameters of the MSSM receive tree-level supergravity contributions. Consequently, the heavy Higgs scalars and the Higgsinos obtain masses of O(m 3/2 ). For the MSSM gauginos on the other hand tree-level supergravity contributions are strongly suppressed (since our supersymmetry-breaking field X is not a total SM singlet) [19,[22][23][24]. The dominant contributions are thus obtained at one loop trough anomaly mediation and (in the case of binos and winos) through Higgsino threshold effects. As detailed in Ref. [24] in the context of PGM, these are generically both of the same order (∼ g 2 a m 3/2 /(16π 2 )), where g a is the respective SM gauge coupling. Depending on the size of the Higgsino threshold effects, either the wino or the bino can take the place of the lightest neutral MSSM particle 13 -and hence of the dark matter candidate. In the following we will focus on the case where the Higgsino threshold contributions do not dominate over the AMSB contribution, rendering the wino the lightest supersymmetric particle (LSP). The motivation for this is twofold. First, this is the more likely scenario in PGM [24]. Second, due to its smaller annihilation cross section, a thermal bino LSP population with mb 300 GeV leads to the overproduction of dark matter [165,166]. In combination with the bounds set by ATLAS [1] and CMS [2], this excludes a thermal bino as a viable dark matter candidate.
A particularly interesting situation arises if the two contributions to the wino mass are tuned to very similar values, leading to a cancellation of these two terms and hence to a wino mass which can be arranged to be much lighter than its generic mass scale m 3/2 /(16π 2 ). For example, in the notation 13 Due to their larger gauge coupling at low energy scales, the gluinos are typically significantly heavier. of Ref. [24] this is achieved for −µ H −2B m 3/2 (for tan β = v u /v d 1). Such a fine-tuning might be justified from the anthropological requirement of dark matter, see Sec. 4.3. Additional contributions to the gaugino masses may arise from threshold and anomaly-mediated corrections from heavy vector matter multiplets charged under SU (2) L and/or threshold corrections from the F terms of flat directions in KSVZ-type axion models [167]. In this case, these contributions would also play a role in tuning the wino mass.

Particles beyond the MSSM
The masses of all remaining degrees of freedom are set by the dynamical scale Λ, effectively decoupling these particles from low-energy physics.
In the supersymmetry-breaking sector, the only degree of freedom which is present in the lowenergy effective theory of the IYIT model is the pseudomodulus X, which acts as the Polonyi field of SUSY breaking, see Sec. 2.4. The dominant mass contribution after the end of inflation arises from its one-loop effective potential, The masses of the scalar fields of the inflation sector can be obtained from the scalar potential in Eq. (127). For q/(q 0 ξ) > 0, the field φ obtains a VEV of |φ| 2 = q 0 /q ξ after the end of inflation. 14 In this vacuum, the masses of the scalar degrees of freedom are given as m |φ| = 10 14 GeV (q 0 q) 1/2 g 0.1 symmetry massive. In Sec. 4.4 we will achieve this by introducing higher-dimensional operators in the Kähler potential coupling Φ to Z − and/or M − . These operators will come with small coefficients (respecting the level of sequestering necessary to prevent the waterfall fields from obtaining too large masses) and will explicitly break the global U (1) symmetry (in agreement with general arguments that no exact global symmetries should exist in any theory of quantum gravity [89]). As we will see below, this leads to a mass for the Goldstone boson of the order of the Hubble scale during inflation.
Note that this situation is crucially different than in standard DHI, where the VEV of the waterfall field φ spontaneously breaks a local U (1) symmetry at the end of inflation, triggering the super-Higgs mechanism: there, the complex phase is "eaten" by the U (1) gauge boson, providing the longitudinal degree of freedom for the massive vector field. In our case, however, the U (1) B−L vector boson is already massive, having absorbed the corresponding degree of freedom from the meson multiplets.
In fact, when gauging B−L, the introduction of three right-handed neutrinos is the simplest way to ensure anomaly cancellation. Once the waterfall field φ obtains a VEV, this generates the Majorana mass matrix for the right-handed neutrinos, M ij = h ij φ .
A very similar cosmological phase transition was studied in the case of F-term hybrid inflation in Refs. [62][63][64]68]. It was shown that this phase transition can set the initial conditions for the hot early Universe: With the energy initially stored in oscillations of the waterfall field (as well as in degrees of freedom created in tachyonic preheating), this energy is transferred to the thermal bath through the decay into right-handed neutrinos (which obtain their mass from the coupling to the B−L breaking waterfall field). In the course of this process, both thermal and nonthermal processes generate a lepton asymmetry, which, after conversion into a baryon asymmetry through sphaleron processes, can explain the baryon asymmetry observed today. Using a coupled set of Boltzmann equations, Refs. [62][63][64]68] provide a time-resolved picture of the entire reheating and leptogenesis process.
We expect this overall picture to also hold in our model. There are, however, a few differences in the details of the phase transition. Contrary to [62][63][64]68], in the DHI model presented here (i) B−L is broken already during inflation, (ii) there is a tree-level mass splitting in the B−L multiplet, (iii) supersymmetry is broken in the true vacuum with m 3/2 ∼ 10 12 · · · 10 13 GeV (iv) there is an additional pseudoscalar degree of freedom in the waterfall sector, 15 which in the analysis of [64] plays the role 15 This relatively light degree of freedom in the waterfall sector has a decay rate into right-handed neutrinos of Γ of the B−L Goldstone boson and (v) we allow here for smaller values of the B−L gauge coupling.
Consequently, no (local) cosmic strings are formed at the end of inflation, the B−L multiplet is not produced in tachyonic preheating and the gravitino is too heavy to be a dark matter candidate as in [64]. Hence, while we expect the same sequence of events as in [62][63][64]68], leading to successful leptogenesis for a mass of the lightest right-handed neutrino above about 10 10 GeV and a reheating temperature of about T RH 10 8 GeV, the above-mentioned differences require a detailed study to verify these expectations. This is beyond the scope of the current paper.

Particle candidates for dark matter
The high reheating temperature expected in our model (see above) implies an abundant production of gravitinos [168]. Since these gravitinos are very heavy, their decay temperature is much larger than the temperature of BBN (∼ MeV). The gravitinos will thus decay into MSSM gauginos (the lightest particles in our MSSM spectrum) before the onset of BBN, a well-known solution to the classical gravitino problem [27,169,170]. If one of the gauginos (in our setup the wino, see Sec. 4.1) is sufficiently light, so that its freeze-out temperature, T f ∼ mw/28, is lower than the gravitino decay temperature T 3/2 , then its relic abundance will be set by the usual thermal freezeout contribution. 16 This occurs for mw 4 × 10 9 GeV (m 3/2 /(10 12 GeV)) 3/2 , with the correct relic density obtained for mw 2.7 TeV [173,174]. This value is much smaller than the generic gaugino mass scale m 3/2 /(16π 2 ), but may be achieved by fine-tuning the anomaly mediation and Higgsino threshold corrections (see Sec. 4.1). Without such fine-tuning, the strongly enhanced LSP abundance would lead to an overclosure of the Universe. Such a fine-tuning may therefore be justified by anthropological arguments. Alternatively, one can take the gauginos to be at their natural scale m 3/2 /(16π 2 ) and invoke R-parity breaking to ensure a sufficiently fast decay of the LSP into the SM degrees of freedom [175].
In this case, the question of the nature of dark matter remains open and may, e.g., be addressed by the QCD axion.

Topological defects
In standard DHI, the angular degree of freedom of the waterfall field φ is massless, protected by the U (1) gauge symmetry of DHI. Consequently, cosmic strings are formed at the end of standard DHI. This is known as the cosmic string problem of DHI, since the nonobservation of cosmic strings in the CMB [104], together with constraints on the spectral index, essentially exclude the entire for typical values of the Yukawa coupling of h 10 −5 . It thus decays into the SM thermal bath before the onset of BBN and does not create any cosmological problems. 16 On the other hand, if T f > T 3/2 the LSP abundance will be dominated by the nonthermal contribution from gravitino decay. The gravitino abundance in turn receives contributions from thermal production, production from the decay of the Polonyi field and production through oscillations of a field in the inflaton sector [168]. In particular the thermal production [171] and the decay of φ to two gravitinos through a supergravity coupling present when K φ φ = 0 [172] yield large gravitino abundances and thus overclose the Universe for the large LSP masses consistent with T f > T 3/2 . parameter space. The setup we propose here is crucially different. The U (1) B−L symmetry is already broken during inflation by the meson VEVs M ± = 0, and no local cosmic strings are formed at the end of inflation. There is instead an accidental global symmetry which is not expected to be exact, see Sec. 4.1. We can express this by adding higher-dimensional operators 17 in the Kähler all supplemented by their complex conjugate. Here the parameters K IJ are expected to be exponentially small, respecting the sequestering between the inflation and the IYIT sector. By means of a Kähler transformation these holomorphic terms can be equivalently considered as terms in the superpotential, W ⊃ W 0 K ZZ Z 2 − Φ/M 3 Pl , etc. Taking into account the vacuum expectation values for the scalar and auxiliary (F-term) components in Z ± , M ± ,Φ, Φ and S, this leads to linear terms in the scalar potential for the waterfall fields. Schematically, whereλ = 2g 2 q 2 denotes the self-coupling of the waterfall field,m 2 is its (inflaton-dependent) mass and c ∼ K IJ Λ 3 m 3/2 /M Pl is determined by the higher-dimensional operators mentioned above. To study cosmic string formation, 18 we consider the system close to the end of inflation, just when Eq. (167) develops a local maximum. At this point, the local minimum of the potential is given by |φ| = 2(c/λ) 1/3 . The phase of the local minimum is set by the phase of c and we will take it to be zero in the following. The mass of the canonically normalized radial degree of freedom α in this local minimum is given by m 2 α = 1 2 (c 2λ ) 1/3 . To avoid the production of cosmic strings, we require that quantum fluctuations of the angular degree of freedom (see e.g. [176]) cannot overcome the barrier at α/( √ 2 φ ) = π, i.e., This leads to with H inf denoting the Hubble scale at the end of inflation. In this paper, we will consider safely satisfying Eq. (169) but also ensuring that the fluctuations in the radial direction are small compared to the position of the local minimum and that the decay rate of the angular component α in the true vacuum is not significantly smaller than the decay rate of the radial component. For the couplings in the Kähler and/or superpotential, this implies in good agreement with our sequestering ansatz. 17 Here, we assumed q0 = −1. Similar terms (also involving the inflaton field) can be written down for q0 = −2. 18 We thank the authors of Ref. [56] for very helpful discussions on this point.
However, scenarios with a richer phenomenology are possible. For example, imagine that a term such as K ⊃ ΦZ − Z − /M Pl is forbidden by an additional discrete symmetry, which in turn is explicitly broken by Planck-suppressed operators of even higher dimension. If this explicit breaking is of a suitable size, unstable domain walls will form [177][178][179]. A similar situation has been discussed for the QCD axion [180,181], see also [182][183][184]. The decaying domain walls will emit energy in the form of gravitational waves. The resulting gravitational-wave spectrum depends mainly on two parameters, the tension σ of the domain walls and their annihilation temperature T ann . For the high energy scales present in our model, the resulting stochastic gravitational-wave background might be within the sensitivity reach of upcoming advanced LIGO runs [185,186], depending on the details of the discrete symmetry (breaking).

Conclusions: A unified model of the early Universe
In this paper, we constructed a phenomenologically viable SUGRA model of hybrid inflation in which reheating proceeds via the B−L phase transition. We focused on the case of D-term inflation to avoid the notorious complications associated with the inflaton tadpole term in F-term inflation (see Eq. (3)).
This tadpole term turns F-term inflation into a two-field model, potentially spoils the slow-roll motion of the inflaton field, and creates a false vacuum at large field values. D-term inflation does not, by contrast, involve any inflaton tadpole term, which prevents one from running into these problems.
The first part of our paper contains the details of our model-building effort (see Sec. 2). To meet all theoretical and phenomenological constraints, our model combines the following three features: (i) The vacuum energy driving D-term inflation is provided by a Fayet-Iliopoulos (FI) D term.
We assume that this D term is dynamically generated in the hidden SUSY-breaking sector [99].
Our construction involves two steps. First, we suppose that SUSY breaking in the hidden sector is accomplished by the dynamics of a strongly coupled supersymmetric gauge theory. To be specific, we employ the Izawa-Yanagida-Intriligator-Thomas (IYIT) model [147,148] Our dynamically generated FI term has a number of interesting properties. (1) Being an effective field-dependent FI parameter, it can be consistently coupled to supergravity. In this sense, it differs from genuinely constant FI parameters whose coupling to supergravity always requires an exact global continuous symmetry. (2) The generation of field-dependent FI parameters typically results in dangerous flat directions in the scalar potential. In our case, all moduli are, however, automatically stabilized by a large mass term in the superpotential that is induced by the SUSY-breaking F term. For the purposes of this paper, we do not specify the high-energy origin of the additional coupling between the visible MSSM sector and the IYIT sector in the Kähler potential. However, it would be interesting to study different scenarios for the possible origin of these operators in future work.
We caution that one should not attribute too much meaning to our choice to work in Jordan-frame supergravity. The formulation of our model in the language of Jordan-frame supergravity should rather be regarded as a placeholder for a hypothetical completion of our model at high energies. Possible candidates for an ultraviolet completion of our model that feature an appropriate Kähler geometry include models of extra dimensions, strongly coupled conformal field theories, no-scale supergravity, and string theory. Again, any further speculations into this direction are left for future work.
(iii) Our third and final assumption consists in an approximate shift symmetry in the direction of the inflaton field in the Kähler potential. Such an approximate shift symmetry is a popular tool in many SUGRA models of inflation. As usual, it helps us to suppress dangerously large SUGRA corrections to the inflaton mass and, hence, solve the SUGRA eta problem. In our case, the most dangerous such correction, m 2 R = R J /6, stems from the nonminimal coupling between the inflaton field to the Ricci scalar in the Jordan frame, R J . This effect can be completely suppressed by an exact shift symmetry -which is, however, not feasible in our model, since the superpotential of D-term inflation inherently breaks any shift symmetry. But this is not a problem. As we are able to show, also an approximate shift symmetry manages to adequately suppress all dangerous SUGRA corrections. On top of that, we can use the fact that the inflaton shift symmetry must be slightly broken to adjust our prediction for the scalar spectral index n s . The amount of shift symmetry breaking in the Kähler potential is quantified by a parameter χ. By choosing this additional parameter appropriately, we can reach agreement between our prediction for n s and the current best-fit value reported by PLANCK.
Here, an interesting special case arises if χ is zero at tree level and only radiatively generated because of the shift-symmetry-breaking Yukawa coupling in the superpotential, χ = χ 1 = κ 2 / 16π 2 . We are able to demonstrate that even this minimal scenario allows to successfully reproduce the CMB data.
In this case, the constraints A s A obs s and n s n obs s fix all free parameters of our model (see Fig. 3), In summary, we conclude that the above three assumptions allow us to solve five problems of B−L D-term inflation: (i) Our FI term can be consistently coupled to supergravity; (ii) we avoid the formation of cosmic strings at the end of inflation; (iii) all MSSM sfermions are sufficiently stabilized during and after inflation; (iv) we do not encounter any SUGRA eta problem; and (v) our prediction for n s is in agreement with the PLANCK data. This is a highly nontrivial success of our model.
A further outcome of our model is a unified picture of the early Universe (see Secs. 3 and 4).
Provided that we include the right couplings in the superpotential, the B−L phase transition at the end of inflation generates large Majorana masses for a number of right-handed neutrinos. This sets the stage for baryogenesis via leptogenesis as well as for the generation of small standard model neutrino masses via the seesaw mechanism. In the end, our model therefore unifies the scales of dynamical SUSY breaking, inflation, and spontaneous B−L breaking. All of these scales derive from the dynamical scale Λ dyn = 4πΛ in the IYIT sector. We also find that, in order to reproduce the amplitude of the scalar power spectrum, the reduced dynamical scale Λ must take a value close to the GUT scale, This is another highly nontrivial result of our analysis. Before confronting our model with the experimental CMB data, we did not need to make any assumption about the numerical value of Λ. A central prediction of our model is that supersymmetry is broken at a high energy scale. The naturalness of the electroweak scale is therefore lost. This sacrifice is, however, compensated for by the unification of the dynamics of SUSY breaking and inflation. One of our key messages therefore is that pushing the SUSY breaking scale to very high values is not necessarily just a loss. A high SUSY breaking scale also represents an opportunity for novel ideas such as those presented in this paper. In the end, supersymmetry might play a different role in nature than previously expected. Following the arguments presented this paper, it is conceivable that supersymmetry's actual purpose is not to ensure the stability of the electroweak scale, but to provide the right conditions for successful inflation!
In this paper, we only touched upon the implications of a high SUSY breaking scale for the particle spectrum of the MSSM and more work in this direction is certainly needed. In particular, one should reevaluate in more detail how the running of the standard model coupling constants can be matched with the coupling constants in the MSSM provided that supersymmetry is broken at energies close to the GUT scale. This matching of the low-energy parameters with their counterparts at high energies is sensitive to important experimental input data, such as the top quark mass m t and the strong coupling constant α s . Given the current experimental uncertainty in these observables, we expect that it should actually not pose any problem to successfully match the standard model to our high-scale scenario.
On top of that, large threshold corrections due to nonuniversal soft masses at high energies may help us to achieve a successful matching (see [187] for a recent analysis). In fact, given our treatment of the MSSM soft masses (see Eq. (156)), large nondegeneracies in the sparticle mass spectrum at high energies are quite likely. Moreover, one should reevaluate in more detail under which conditions our high-scale scenario is compatible with the idea of gauge coupling unification. Again, such an analysis would be sensitive to the experimental input data at low energies. In addition, it would also depend on the details of the anticipated unification scenario. We are, however, confident on general grounds that it should be feasible to realize gauge coupling unification in our model. After all, unification is also possible in entirely nonsupersymmetric scenarios. We therefore expect that supersymmetry, despite the large value of its breaking scale, will only help in achieving gauge coupling unification [188].
In conclusion, we find that our model provides a consistent cosmological scenario that unifies five different phenomena: (i) dynamical supersymmetry breaking at a high energy scale, (ii) viable D-term hybrid inflation in supergravity, (iii) spontaneous B−L breaking at the GUT scale, (iv) baryogenesis via leptogenesis, and (v) standard model neutrino masses due to the type-I seesaw mechanism. Our model is built around a strongly coupled hidden sector, which puts it on a sound theoretical footing.
We do not need to make any ad hoc assumptions about the dimensionful parameters in our model.
Instead, all important mass scales are related to the dynamical scale of the strong interactions in the hidden sector. Thanks to its precise parameter relations, our model is therefore well suited to be used as a basis for further explicit calculations. It would, e.g., be worthwhile to study the reheating process after inflation in greater detail and determine the corresponding implications for the spectrum of gravitational waves. Similarly, a more comprehensive study of the MSSM particle spectrum and its consequences for dark matter would be desirable. The analysis in the present paper should only be regarded as a first step. It served the purpose to illustrate our main point: the fact that SUSY breaking close to the GUT scale might be the key to a unified picture of particle physics and cosmology.
This is a fascinating observation and we are excited to see where it will lead us in the future. One possibility is that it will eventually cause a paradigm shift in our understanding of SUSY's role in the physics of the early Universe. High-scale SUSY breaking might be the driving force behind inflation! A Technicalities: Supergravity in the Einstein/Jordan frame Our model is based on a particular embedding into supergravity. We assume that the coupling to gravity is most naturally described in a Jordan frame where all scalar kinetic terms are canonically normalized (see Sec. 2). At the same time, we wish to perform a standard slow-roll analysis of the inflationary dynamics (see Sec. 3), which requires a reformulation of our model in the Einstein frame.
To facilitate the transition between these two different frames, this appendix provides a dictionary that allows one to translate back and forth between the two different formulations of our model.

A.1 Bosonic action
In the usual Einstein frame, the purely bosonic action of our model takes the following form, 20 Here, M Pl denotes the reduced Planck Mass, M Pl 2.44 × 10 18 GeV; g is the determinant of the Einstein-frame spacetime metric g µν ; R is the Ricci scalar constructed from g µν ; g µν stands for the inverse of the metric g µν ; the fields φ i represent the complex scalar fields in our model; D µ denotes the usual gauge-covariant derivative; F µν is the field strength tensor of the Abelian B−L vector field; and V represents the total scalar potential in the Einstein frame. As evident from Eq. (174), the scalar fields φ i couple to gravity only via the inverse spacetime metric g µν . This corresponds to the case of minimal coupling. At the same time, the scalar fields exhibit a nontrivial (Kähler) geometry in field space. This is accounted for by the Kähler metric K which multiplies the scalar kinetic terms in Eq. (174). The Kähler metric K is defined as the Hessian of the real-valued Kähler potential K, 20 Some authors in the literature distinguish between Einstein-frame and Jordan-frame quantities by labeling them with indices E and J, respectively. We will, by contrast, not use any particular label for quantities in the Einstein frame and merely label quantities in the Jordan frame with an index J. This will slightly simplify our notation.
In our model, the Kähler potential K is not canonical, such that Kī j = δī j . The scalar kinetic terms (and, hence, the scalar fields themselves) are, thus, not canonically normalized in the Einstein frame.
To obtain the equivalent of Eq. (174) in the Jordan frame, we need to perform a Weyl rescaling, where C is known as the conformal factor. The frame function Ω is an arbitrary real negative function of the complex scalars, Ω = Ω (φ i , φ * ı ) < 0. Each choice for Ω defines a separate Jordan frame. In Sec. 2, we make a particular choice for Ω, demanding the following relation to the Kähler potential, This results in what may be regarded as the standard Jordan frame. In light of the relation in Eq. (177), the frame function Ω has two possible interpretations. In the curved superspace approach to old minimal supergravity [119,120], Ω can be identified as the generalized kinetic energy on curved superspace, while in the superconformal approach to old minimal supergravity [121,122], Ω can be identified as the prefactor of the kinetic term of the chiral compensator superfield. Eq. (177) allows to relate the partial derivatives of the Kähler potential to the partial derivatives of the frame function, Similarly, we are able to express the Kähler metric K in terms of derivatives of the frame function, Here, we introduced ω as a rescaled field-space metric that is determined by the derivatives of the frame function and that is equivalent to the Kähler metric up to the rescaling factor C 2 . Eq. (179) automatically implies a similar relation between the respective inverse metrics, K −1 and ω −1 , We now apply the Weyl transformation in Eq. (176) to the Einstein-frame action in Eq. (174).
This yields the purely bosonic action of our model in the Jordan frame (see [115] for more details), In view of this action, several comments are in order: (i) The frame function Ω depends on the scalar fields of our model. The Einstein-Hilbert term (i.e., the kinetic term for the metric that is proportional to the Ricci scalar R J ) therefore becomes field-dependent. Or in other words, the scalar fields are now nonminimally coupled to gravity.
(ii) The Planck mass squared in Eq. (174) is now replaced by −Ω/3. This indicates that the square root of −Ω/3 should be interpreted as the effective field-dependent Planck mass in the Jordan frame, which is consistent with the fact that all Jordan-frame mass scales m J pick a factor C when transforming from the Jordan frame to the Einstein frame, m = C m J . From this perspective, the conformal factor C turns out to be nothing but the ratio of the two respective Planck masses, C = M Pl /M J Pl . (iii) The Kähler metric K in Eq. (174) is now replaced by the Hessian of Ω. In our model, we choose Ω such that it only contains canonical as well as purely holomorphic/antiholomorphic terms, where J is an arbitrary holomorphic function. The canonical terms, Ω ⊃ δī j φ * ı φ j , lead to nonminimal couplings between the complex scalars and the Ricci scalar R J that are invariant under a classical conformal symmetry. These conformal couplings can be disturbed by a nonzero function J which explicitly breaks the conformal symmetry. Irrespective of whether J = 0 or J = 0, Eq. (183) leads to In this case, we obtain the following expressions for the rescaled field-space metric ω and its inverse, Here, the dimensionless parameter functions as a measure for the amount of conformal symmetry breaking in the frame function Ω. In the conformal limit, J → 0, it simply reduces to the conformal factor squared, ρ → C 2 . In this sense, plays a similar role as the reduced kinetic function of the inflation field, f , defined in Eq. (89). In fact, in our concrete model, one can show that = C 2 (1 − f ).
(iv) In the Jordan frame, the scalar kinetic terms receive additional contributions from the bosonic part of the auxiliary SUGRA gauge field A µ . This is accounted for by the third term on the right-hand side of Eq. (181). The auxiliary field A µ can be eliminated after solving its equation of motion, This solution illustrates that the A 2 term in Eq. (181) is only relevant as long as we are interested in the dynamics of angular degrees of freedom, i.e., the complex phases of the complex scalars φ i . This is, however, not the case. In our model, inflation occurs along the real direction of the complex inflation field s. The auxiliary field A µ therefore vanishes and the A 2 term in Eq. (181) can be ignored.
Together with Eq. (184), A µ = 0 leads to canonically normalized kinetic terms for all complex scalar fields in our model. This is an important result and the main motivation for our ansatz in Eq. (183).
Combining our above results, the action in Eq. (181) can be simplified to the following expression, This is the starting point for the analysis of our model in the Jordan frame. Thus far, we have not commented on the relation between the potentials V and V J . We will do this now in the next section.

A.2 Scalar potential
The total scalar potential in the Einstein frame, V , has mass dimension four. On general grounds, this implies the following universal relation to the total scalar potential in the Jordan frame, V J , which holds at tree level as well as at the loop level. Eq. (188) implies the following useful relations, Together, Eqs. (188) and (189) illustrate that a Minkowski vacuum in the Einstein frame (V = V i = 0) also corresponds to a Minkowski vacuum in the Jordan frame (V J = V J i = 0), and vice versa. In the next step, we shall become more specific and discuss the individual contributions to V and V J , respectively. The Einstein-frame potential consists of the usual F-term and D-term contributions, To begin with, let us focus on the F-term scalar potential (see [189] for more details), Here, the F i and F * ı stand for the generalized F terms in supergravity and their complex conjugates, where DW denotes the Kähler-covariant derivative of the superpotential on the Kähler manifold, The F-term potential in the Jordan frame is given as V J F = C −4 V F . This can be rewritten as follows, which underlines the similarity between the role of the inverse metric ω −1 in the Jordan frame and the role of the inverse Kähler metric K −1 in the Einstein frame. With our ansatz for the frame function Ω in Eq. (183), V J F can be further simplified to the following compact expression (see, e.g., [123]), Eq. (195) illustrates a remarkable effect. Provided that the scalar kinetic terms in the Jordan frame are canonically normalized, V J F splits into two separate contributions -where the first contribution, V 0 F , is nothing but the ordinary F-term scalar potential in global supersymmetry and the second contribution, ∆V J F , represents an additive SUGRA correction. Eq. (195) provides the basis for the calculation of the F-term scalar potential in our model (see Eqs. (110) and (111)). We first calculate the F-term scalar potential in the Jordan frame according to Eq. (195). Then, we convert our result from the Jordan frame to the Einstein frame making use of the general relation in Eq. (188), Computing V F via this detour is considerably easier than a direct calculation starting with Eq. (191).
The SUGRA correction ∆V J F in Eq. (195) is directly proportional to the mass scales that appear in the superpotential. In our model, these mass scales correspond to the F-term SUSY breaking scale µ, the effective inflaton-dependent mass of the waterfall fields, κ S , and the R-symmetry-breaking constant w 0 (see Eq. (106)). All of these mass scales are responsible for the explicit breaking of superconformal symmetry. Conversely, this means that, if the superpotential does not exhibit any explicit mass scales, the SUGRA correction ∆V J F must vanish. This is exactly what happens in the class of canonical superconformal supergravity (CSS) models studied in [116]. These models are based on the frame function in Eq. (183) with the holomorphic function J set to zero. Moreover, they exhibit a purely cubic superpotential, such that ∆V J F = 0. One generic feature of CSS models therefore is that their Jordan-frame scalar potential coincides with the scalar potential in global supersymmetry, CSS models: Note that this statement applies to the total scalar potential, including the D-term scalar potential.
In our model, the D-term scalar potential in the Einstein frame, V D , is given by [189] where D denotes the auxiliary component of the B−L vector multiplet V . In writing down Eq. (198), we assumed a canonical gauge-kinetic function for the B−L vector field, f V = 1, and absorbed the gauge coupling constant g into the definition of D. On-shell, the auxiliary D field can be replaced by In the language of Kähler geometry, this is equivalent to the Killing potential of the U (1) B−L isometry of our Kähler manifold. Together, Eqs. (198) and (199) result in the following expression for V D , This result can be easily translated into the Jordan frame by making use of Eqs. (178) and (188), In our model, all complex scalars with nonzero gauge charge q i appear with a canonical term in Ω.
Just like in the class of CSS models, V J D therefore obtains the same form as in global supersymmetry, This means in turn that the D-term scalar potential in the Einstein frame can be written as Combining all of our above results, we conclude that V and V J are given as follows in our model, This result is the starting point for our calculation of the inflaton potential in Sec. 2.5.

A.3 Scalar mass parameters
Eq. (204) allows us to derive useful expressions for the scalar mass parameters m J ab in the Jordan frame. 21 The scalar mass matrix is given by the Hessian of the scalar potential, m J ab 2 = V J ab . As a consequence of the simple structure of V J , the scalar masses, thus, split into two contributions: the ordinary masses in global supersymmetry, m 0 ab , as well as additive corrections in supergravity, ∆m ab , In the case of scalar fields that only appear with a canonical term in the frame function, Ω ⊃ |φ i | 2 , the corrections ∆m 2 ab take a particularly simple form. Based on our result in Eq. (195), we find and similarly for the respective conjugate parameters, ∆m 2 i = ∆m 2 ıj * and ∆m 2 ı = ∆m 2 ij * . The diagonal entries of the scalar mass matrix therefore obtain the following compact form, In addition to the masses encoded in the scalar potential, the complex scalar fields acquire further, effective masses from their nonminimal coupling to R J in the Jordan-frame action (see Eq. (181)), All scalar fields with a canonical kinetic function, thus, receive a universal gravity-induced mass m R , In summary, the entries of the total effective mass matrix in the Jordan frame, M 2 J , read as follows, In our model, all scalar fields are canonically normalized by construction. The eigenvalues of the matrix M 2 J therefore directly correspond to the physical scalar mass eigenvalues in the Jordan frame.
The situation is more complicated in the Einstein frame, where the scalar fields parametrize the target space of a nonlinear sigma model (see Eq. (174)). There, the scalar fields are a priori not canonically normalized which makes it more difficult to find the physical mass eigenvalues. Without reference to the Jordan frame, the computation of the scalar mass spectrum in the Einstein frame requires two steps. First, one has to perform a field transformation that renders all fields canonically normalized.
Then, one has to calculate the mass eigenvalues of these canonically normalized fields as usual. Our result in Eq. (210), however, allows us to bypass this complicated procedure. Instead, we can simply make use of the universal scaling behavior of physical mass scales when transforming back and forth between the Jordan frame and the Einstein frame. According to this scaling behavior, we know that the total effective scalar mass matrix in the Einstein frame, M 2 , must obtain the following form, It would be interesting to check the validity of this result by means of an explicit calculation in the Einstein frame. Such a task is, however, beyond the scope of this paper and left for future work.
More details on the mass parameters for all fields with nonzero spin (i.e., the fermions, vector boson, and gravitino in our model) can be found in the literature. The relevant expressions in the Jordan frame are spelled out in [115], while the standard Einstein-frame results are listed, e.g., in [189].

A.4 Slow-roll parameters
Finally, let us discuss the relation between the inflationary slow-roll parameters in the Einstein frame, ε and η, and their counterparts in the Jordan frame, ε J and η J . The results derived in this section will enable us to use our results for the scalar potential in the Jordan frame (see Sec. 2) as input for a standard slow-roll analysis of the inflationary dynamics in the Einstein frame (see Sec. 3).
Let us consider the action of the complex inflaton field s in the Einstein frame (see Eq. (174)), As can be seen from this action, the inflaton field is not canonically normalized in the Einstein frame. This is made explicit by the noncanonical normalization factor of the inflaton kinetic term, N = 1.
However, in our slow-roll analysis, we will have to work with the canonically normalized fieldŝ. The fieldŝ can be constructed as a function of the field s by solving the following differential equations, In terms of the canonically normalized fieldŝ, the action in Eq. (212) obtains its standard form, This action is the starting point of our standard slow-roll analysis. The slow-roll parameters in the Einstein frame, ε and η, are defined in terms of the usual partial derivatives of the scalar potential, Here, we assumed that inflaton occurs along the real componentσ of the complex inflaton fieldŝ. In the next step, we rewrite these expressions making use of the chain rule and the relations in Eq. (213), where V σ = ∂V /∂σ, V σσ = ∂ 2 V /∂σ 2 , and N σ = ∂N /∂σ. With these definitions and relations, we find where we assumed that V σ > 0 and N σ > 0. The parametersε andη represent what can be referred to as the naive slow-roll parameters in the Einstein frame, i.e., the slow-roll parameters that one would obtain if one ignored the noncanonical normalization factor N in Eq. (212). Meanwhile, the factor ν is an auxiliary slow-roll parameter that accounts for the field dependence of the factor N , The expressions in Eq. (217) are now well suited to establish a connection to the Jordan frame.
The naive slow-roll parametersε andη can be readily related to the Jordan-frame slow-roll parameters ε J and η J by employing the relations for the partial derivatives of the scalar potential in Eq. (189), where we assumed again a positive potential gradient, V J σ > 0. The Jordan-frame slow-roll parameters ε J and η J are defined in terms of the usual partial derivatives of the Jordan-frame scalar potential, In Eq. (219), we also introduced the auxiliary slow-roll parameters ξ J and ζ J which account for the field dependence of the frame function Ω. These slow-roll parameters are defined as follows, Unlike in the Einstein frame, we do not have to distinguish between naive and actual slow-roll parameters in the Jordan frame. This is because, in our model, all scalar fields are canonically normalized in the Jordan frame by construction. In the language of Eq. (217), this can be rephrased by saying that the normalization factor of the inflaton kinetic term in the Jordan frame is simply trivial, N J = 1.
Combining our results in Eqs. (217) and (219), we finally obtain the following relations, This is an important result that holds in any model with an Einstein-frame action as in Eq. (212).
Eq. (222) is the starting point for our computation of the Einstein-frame slow-roll parameters in Sec. 3.
We emphasize that computing ε and η according to Eq. (222) is considerably easier than a brute-force calculation in the Einstein frame. In the Einstein frame, we would have to deal with a complicated Kähler potential, a more complicated scalar potential, and a noncanonically normalized inflaton field.
Eq. (222) allows us to circumvent these complications and determine the parameters ε and η simply based on the derivatives of the Jordan-frame scalar potential V J and the frame function Ω.