Unified Model of Chaotic Inflation and Dynamical Supersymmetry Breaking

The large hierarchy between the Planck scale and the weak scale can be explained by the dynamical breaking of supersymmetry in strongly coupled gauge theories. Similarly, the hierarchy between the Planck scale and the energy scale of inflation may also originate from strong dynamics, which dynamically generate the inflaton potential. We present a model of the hidden sector which unifies these two ideas, i.e., in which the scales of inflation and supersymmetry breaking are provided by the dynamics of the same gauge group. The resultant inflation model is chaotic inflation with a fractional power-law potential in accord with the upper bound on the tensor-to-scalar ratio. The supersymmetry breaking scale can be much smaller than the inflation scale, so that the solution to the large hierarchy problem of the weak scale remains intact. As an intrinsic feature of our model, we find that the sgoldstino, which might disturb the inflationary dynamics, is automatically stabilized during inflation by dynamically generated corrections in the strongly coupled sector. This renders our model a field-theoretical realization of what is sometimes referred to as sgoldstino-less inflation.


I. INTRODUCTION
Cosmic inflation not only solves the flatness and horizon problems of big bang cosmology [1][2][3][4], but also explains the origin of the primordial density fluctuations that seed the large-scale structure of the universe [5][6][7][8][9]. To satisfy the upper bound on the tensor-to-scalar ratio in the power spectrum of the cosmic microwave background (CMB) [10], the potential energy during inflation must be much smaller than the scale of gravity, Λ inf = V 1/ 4 10 −2 M Pl . The smallness of the energy scale of inflation, Λ inf , is nicely explained if the inflaton potential V is generated by means of dimensional transmutation in a strongly coupled gauge theory. Refs. [11][12][13][14] and [15][16][17][18] proposed models of small-field and largefield inflation along this idea, respectively.
The electroweak scale also suffers from a hierarchy problem, v ew M Pl , which can be solved by supersymmetry and its breaking at a low energy scale [19][20][21][22]. Again, a plausible explanation for the smallness of the supersymmetry breaking scale, Λ SUSY M Pl , would be to presume that supersymmetry is broken dynamically by strong dynamics [21]. So far, no evidence for superpartners of the standard model particles has been found at the LHC, which has brought about the little hierarchy problem, v ew m SUSY (where m SUSY denotes a typical soft superparticle mass). But supersymmetry nonetheless solves the large hierarchy problem, predicts the unification of the standard model gauge couplings and provides a particle candidate for dark matter. For these reasons, we take up the attitude that supersymmetry as well as its dynamical breaking are some of the leading candidates for new physics beyond the standard model.
In this letter, we propose a model of the hidden sector which unifies these two ideas of dynamically generated energy scales. The model resembles that of Refs. [15][16][17] during inflation; but the potential energy is non-zero even after the end of inflation, which breaks supersym-metry. The inflationary dynamics are those of chaotic inflation [23] with a fractional power-law potential. The model is thus free from an initial conditions problem; and it is consistent with the recent PLANCK data [10]. See Refs. [24,25] for other models of chaotic inflation with fractional power-law potentials. We also refer to Ref. [13] for an earlier proposal for the unified and dynamical generation of the energy scales of inflation and supersymmetry breaking, which results in a scenario of hybrid inflation [26,27]. This work has been followed up more recently in Refs. [28,29], where it is demonstrated how the dynamical breaking of supersymmetry at a very high energy scale may result in scenarios of Fterm and D-term inflation, respectively. Finally, we refer to Ref. [30], which considers a perturbative model (as opposed to our strongly coupled models) in which the inflaton potential as well as the breaking of supersymmetry are both provided by the F term of a single chiral field.

II. DYNAMICAL CHAOTIC INFLATION
We first review the idea of dynamical chaotic inflation (DCI) proposed in Refs. [15][16][17]. We start from a strongly coupled gauge theory which generates a potential energy proportional to some power of the dynamical scale Λ, To this theory, we add a pair of particles, q andq, that obtain their mass from a coupling to the inflaton field φ, For a large field value of the inflaton, such that λφ Λ, the fields q andq decouple; and around the dynamical scale the potential energy in Eq. (1) is generated. Since the energy scale at which qq decouples depends on the arXiv:1707.03646v1 [hep-ph] 12 Jul 2017 inflaton field value, the dynamical scale also depends on it, through the running of the gauge coupling constant, d dln µ where µ is the renormalization scale. We shall denote the beta function coefficient b in the high/low-energy theory with/without qq as b HE and b LE , respectively. Then the effective dynamical scale Λ(λφ) follows from where g formally diverges, g (Λ) → ∞, at the dynamical scale. Matching the running of the gauge coupling constant at the qq mass threshold, we obtain the dependence Together with Eq. (1), this results in a power-law potential for the inflaton, φ p , with the power p given as This potential is suitable for inflation at large values of the inflaton field, φ M Pl , which is nothing but a (dynamical) realization of the idea of chaotic inflation.
The implementation of the above scheme into supersymmetric theories is straightforward. We start from a model of dynamical supersymmetry breaking, add chiral multiplets q andq, and couple these chiral multiplets to the inflaton multiplet Φ. To avoid the eta problem in supergravity [31][32][33][34] for a large field value of the inflaton, we introduce an approximate shift symmetry Φ → Φ+iC in the Kähler potential [35,36]. The negative contribution to the potential energy is suppressed as long as the supersymmetry-breaking (Polonyi) field has a field value much smaller than the Planck scale during inflation.

III. DYNAMICAL CHAOTIC INFLATION AND SUPERSYMMETRY BREAKING UNIFIED
In this section, we propose a model of dynamical chaotic inflation in which the gauge dynamics also break supersymmetry in the true vacuum after the end of inflation. The basic idea is the following: We start from a dynamical supersymmetry breaking model with a product group G 1 ×G 2 , such that supersymmetry is broken by the strong dynamics of G 2 , while the gauge interactions of G 1 merely lift flat directions by a classical D-term potential. To this model, we add G 2 -charged matter fields and couple them to an inflaton multiplet, W = λΦ 2 . Supersymmetry is broken by the gauge dynamics of G 2 for large inflaton field values, where the new matter multiplets decouple. But for small field values, the gauge  dynamics flow into a different phase; and the potential energy proportional to the dynamical scale of G 2 and hence the inflaton potential vanish. By a suitable choice of matter fields and couplings, supersymmetry is instead now broken by the strong dynamics of G 1 (or a subgroup of G 1 , if the strong dynamics of G 2 partially break G 1 ). The supersymmetry breaking scale in the vacuum can be naturally much smaller than the scale of inflation, provided there is a hierarchy between the dynamical scales of G 1 and G 2 and/or the breaking of supersymmetry by the strong dynamics of G 1 involves particularly small couplings (realized, e.g., in the form of higher-dimensional operators). In this paper, we shall present a simple realization of this idea based on the groups G 1 = SU (5) and (2). Other examples will be given elsewhere.
Let us apply the idea described in Sec. II to the SU (5) × Sp(2) model of supersymmetry breaking [37], which is a generalization of the so-called 3-2 model [38]. The model is based on SU (5) × Sp(2) gauge dynamics and features chiral multiplets Q,Ū ,D L,q 1,2 in representations of the gauge group as listed in Tab. I. Our convention for Sp(N ) groups is such that Sp(1) ∼ = SU (2). The theory contains the following flat directions, whereQ ∈ D ,Ū ,q i . The flat directions are lifted by introducing the following tree level superpotential, In this paper, we concentrate on the case where the dynamical scale of SU (5) is much smaller than that of Sp(2), Λ SU Λ Sp . Supersymmetry is then broken by the deformed moduli constraint [39] of the Sp(2) dynamics, which results in non-zero F terms forD and the flat direction QQq 1q2 . The potential energy is given by [40] To turn this supersymmetry breaking model into a model of dynamical chaotic inflation, we add Sp(2)-charged chiral multiplets and couple them to the inflaton field Φ, For λΦ Λ Sp the extra multiplets decouple from the gauge dynamics. The theory then exhibits supersymmetry breaking and generates a non-zero potential energy.
The supersymmetry-breaking field is contained inD and QQq 1q2 . Its scalar component, the sgoldstino, is a flat direction at tree level, which could potentially disturb the inflationary dynamics. It, however, obtains a mass from strong-coupling corrections in the Kähler potential, as is the case in generic models of dynamical supersymmetry breaking. Unless y is small, m is much larger than the Hubble scale, which provides a field-theoretical realization of the so-called sgoldstino-less inflation [41]. This is a generic feature in models of dynamical chaotic inflation. We note that the stabilization by a Hubble-induced mass would already be enough to ignore the sgoldstino dynamics [35,36]; but the stabilization via IR quantum corrections is advantageous in the sense that it is independent of the unknown UV physics which determine the sign and the magnitude of the Hubble-induced mass.

B. Flow into SU (5) model in the vacuum
After inflation, at λΦ Λ Sp , the extra multiplets no longer decouple, but participate in the gauge interactions just like the other Sp(2) flavors. In Refs. [15][16][17], the fields as well as their couplings were chosen so that the theory reaches a phase of s-confinement at low energies, where all flat directions are lifted and supersymmetry is restored. In this paper, we are instead going to chose the matter content and couplings such that supersymmetry remains broken even in the true vacuum after inflation.
We add a pair of Sp(2) fundamentals, 1 and 2 , and introduce a coupling to the inflaton multiplet Φ, The beta function coefficient of the Sp(2) gauge coupling at high and low energies is then given as b HE = 5 and b LE = 6, respectively. The potential energy during inflation scales like Λ Sp to the power n = 9/2, see Eq. (9), so that the exponent of the inflaton potential is given by p = 3/4, see Eq. (6). The dynamical scale around the vacuum,Λ Sp , and the dynamical scale during inflation Λ Sp are related to each other as follows, see Eq. (5), Around Φ = 0, the Sp(2) gauge theory reaches a phase of s-confinement; and the low-energy theory is described in terms of 28 gauge-invariant, composite meson fields, The fields (M QL ,D) and (M 1 2 , Φ) obtain their masses from the superpotential in Eqs. (8) and (12), respectively. The inflaton mass around the origin is thus given by After those fields decouple, the theory still contains the following chiral multiplets M QQ (10), M Q 1,2 (5), M L 1,2 (1),Ū (5),q 1,2 (5), (16) where the numbers in bold refer to representations of SU (5). The superpotential in the s-confined phase reads Here, the second line is generated by the Sp (2) dynamics.
The theory now contains one 10, two 5's, and threē 5's of SU (5). By giving masses to two pairs of 5 +5, the theory becomes nothing but the chiral supersymmetry breaking model based on SU (5), featuring one 10 and one5 of SU (5) [42]. The vacuum energy is then given by whereΛ SU is the dynamical scale of SU (5) in the lowenergy effective theory containing only 10 +5. We may obtain a hierarchy between the inflation scale and the supersymmetry breaking scale by choosingΛ SU Λ Sp . The SU (5) singlets M L 1 and M L 2 remain massless. We can stabilize these fields by introducing Sp(2) singlets and coupling them to L 1 and L 2 in the quark picture at high energies. Another possibility would be to simply introduce a higher-dimensional operator, W = L i L j .
Depending on how we give masses to the two pairs of 5 +5, the inflaton potential could be affected. We may, e.g., remove the fields M Q 1,2 andq 1,2 by adding the following superpotential in the quark picture, such that the matter content of the SU (5) supersymmetry breaking model is provided by the chiral fields M QQ andŪ . After s-confinement of Sp(2), those terms give masses to the (M Q 1 ,q 1 ) and (M Q 2 ,q 2 ) pairs. At the same time, during inflation and after integrating out 1 2 , this superpotential also generates the second term in Eq. (8) with M * ∝ Φ. When this inflaton-dependent term dominates over the Φ-independent one, the inflaton potential becomes the one with p = 3/4 − 1/2 = 1/4. If they are comparable to each other, we have p = 1/4 for small field values and p = 3/4 for large field values.

A. CMB observables
Taken all together, the model constructed in Sec. III results in an inflaton potential of the following form, Here, we choose a convention in which both φ and M * are real and positive; and the phase difference between these two complex parameters is accounted for by the phase α. The parameter c is a numerical constant, which we will set to c = 1 in the following. The scalar potential is only monotonically increasing for positive φ as long as |α/π| ≤ 5/6. For values of |α/π| closer to unity, the potential exhibits a false vacuum at small field values. From the potential in Eq. (20), we derive the predictions for the CMB observables, i.e., for the scalar spectral index n s as well as for the tensor-to-scalar ratio r. The result of our analysis is shown in Fig. 1. The predictions for both parameters only depend on M * and α. If there is a clear hierarchy between M * and φ for all times during inflation, we simply recover the predictions for chaotic inflation based on a standard power-law potential, V ∝ φ p , where N e is the number of e-folds at the CMB pivot scale. Pl , it can be neglected throughout inflation and we can effectively work with p = 1/4. For intermediate values of M * , the predictions for n s and r are more complicated, as they become sensitive to the phase α. This is evident from Fig. 1, where we show the variation of n s and r for different values of α. In particular, we observe how, for fixed α, the variation of M * results in orbits in the n s -r plane that connect the predictions for p = 3/4 and p = 1/4.
The parametric freedom of our model makes it easy to achieve consistency with the recent PLANCK data [10]. Our model predicts values of r in the r ∼ 0.01 · · · 0.1 range and is, therefore, in accord with the current upper bound, r 0.1. In particular, close-to-maximal values of the phase, α 5/6π, allow to achieve rather large values of r, which are going to be tested in future CMB experiments. Our model moreover prefers values of n s in the n s ∼ 0.97 · · · 0.99 range, which is slightly above the current best-fit value, n s 0.965. It is however interesting to note that the data still admits such relatively large values of n s , if it is fit by a ΛCDM + r + N eff model, which also accounts for the possibility of dark radiation.

Z2
− + For given values of M * and α, the observed amplitude of the scalar power spectrum, A s 2×10 −9 , fixes the parameter combination λ 1/5 Λ Sp in Eq. (20). We find that, in the entire parameter space of interest, this product is required to take a value of around λ 1/5 Λ Sp ∼ 10 16 GeV. At the same time, λ should not be too small, since otherwise the matter fields 1 and 2 do not decouple for the entire duration of inflation. We demand that λφ Λ Sp at all times during inflation, which roughly translates into λ 10 −2 , see Ref. [17] for details. Given this lower bound on λ, we then find that the required value of Λ Sp is always remarkably close to the scale of grand unification.

B. Reheating
After inflation, the energy density stored in the inflaton field must be transferred into standard model particles. In our model, the inflaton resides in the supersymmetry breaking sector, such that it may dominantly decay into particles in this sector. Those particles eventually decay into gravitinos, which easily leads to an overproduction of gravitinos. We can forbid the decay mode into the supersymmetry breaking sector by symmetry arguments. For example, we can impose the Z 2 symmetry shown in Table II, under which the inflaton is odd. The particles in the SU (5) model, M QQ andŪ , are Z 2 -even and, hence, the inflaton does not decay into these states. The other Z 2 -odd particles obtain masses proportional toΛ Sp . If λ is sufficiently small, the inflaton ends up being the lightest particle in the supersymmetry breaking sector, so that it does not decay into any particles in this sector.
The Z 2 symmetry also forbids the operator QQq 1q2 in Eq. (8). Therefore, if the Z 2 is an exact symmetry, the M −1 * term in Eq. (20) is actually no longer present. In our analysis, this corresponds to taking the limit M * → ∞, such that the scalar potential reduces to an exact powerlaw with p = 1/4. On the other hand, if the Z 2 is only an approximate symmetry, it only suppresses the M −1 * term to some degree. In this case, we have to work with the full scalar potential in Eq. (20) and the predictions for the CMB observables depend on the exact hierarchy between M −1 * and φ −1 , as discussed in the previous section. The inflaton can decay, e.g., via a coupling to the Higgs multiplets H u,d in the minimal supersymmetric standard model, W = ΦH u H d . In this case, the µ term is generated via Z 2 symmetry breaking. We may also identify the Z 2 with R parity and introduce W = i ΦL i H u , where the L i denote the standard model lepton doublets [43].  [10]. The blue contours correspond to the standard ΛCDM + r fit, whereas the red contours also take into account the possibility of a non-standard number of relativistic degrees of freedom, N eff , at the time of photon decoupling. The color scale indicates the value of the phase α, which we vary on a linear scale. For each value of α, we vary the mass scale M * in the interval 10 −3 , 10 3 M Pl on a logarithmic scale. This results in orbits in the ns-r plane that smoothly connect the predictions of the pure power-law potentials φ 3/4 and φ 1/4 . The local density of points in the above plot can be regarded as a measure for how "generic" or "typical" a certain prediction is. A low density of points indicates a rather special parameter choice, while a high density of points indicates that a prediction is stable under small variations of the input parameters M * and α.

V. DISCUSSION
In this letter, we presented a strongly coupled model of the hidden sector based on SU (5) × Sp(2) gauge dynamics. Our model combines the ideas of dynamical supersymmetry breaking and dynamical chaotic inflation and, hence, explains the hierarchy between the scales of supersymmetry breaking, inflation, and gravity, Λ SUSY Λ inf M Pl . During inflation, supersymmetry is broken because of the Sp(2) deformed moduli constraint. This results in an inflaton potential that interpolates between the power-law potentials φ 3/4 and φ 1/4 , see Fig. 1. The pseudoflat sgoldstino direction is automatically stabilized during inflation by dynamically generated corrections in the Kähler potential. After inflation, the Sp(2) sector reaches a phase of s-confinement and supersymmetry is broken by the SU (5) gauge interactions. In fact, at low energies, our model reduces to the chiral SU (5) model of dynamical supersymmetry breaking.
In the SU (5) model, some approximate global sym-metries are believed to be spontaneously broken, which results in the presence of (pseudo-) Nambu-Goldstone bosons. These bosons obtain non-zero field values in the early universe and may affect the cosmological history. Among them, the R axion is potentially dangerous, since it has a mass squared of O(m 3/2ΛSU ) through the explicit breaking of R symmetry [44] and because it dominantly decays into gravitinos. The gravitino eventually decays into the lightest supersymmetric particle (LSP), which may lead to its overproduction. Assuming that the initial amplitude of the R axion is as large asΛ SU , the LSP abundance is estimated as where s is the entropy density and T RH the reheating temperature. Here, we imposed the condition that the universe must reach a flat Minkowski vacuum after inflation, m 3/2 M Pl ∼Λ 2 SU . Requiring that ρ LSP /s < 4 × 10 −10 GeV, we obtain an upper bound on T RH , T RH 10 9 GeV 100 TeV In the SU (5) model, the gaugino masses of the minimal supersymmetric standard model are generated only via anomaly mediation [45][46][47][48][49][50], meaning that they are loop-suppressed compared to the gravitino mass. The scalar masses, on the other hand, follow from the treelevel Kähler potential and are as large as (or larger than) the gravitino mass. For m 3/2 ∼ O (100 · · · 1000) TeV, our model is thus compatible with the scenario of high-scale supersymmetry breaking [46,51,52], which has gained considerable interest after the discovery of the standard model Higgs boson with a mass of 126 GeV [53,54].
By choosing a different gauge group, we may also obtain a model of gauge mediation. For example, we can modify our model by gauging only the SU (3) × SU (2) × U (1) subgroup of SU (5). By adding an appropriate superpotential term, supersymmetry is broken via the 3-2 model in the vacuum. The U (1) symmetry may be used as the messenger hypercharge [55]. We leave a detailed discussion of modifications of our model for future work.