Jordan Frame Supergravity and Inflation in NMSSM

Dedicated to the memory of Lev Kofman Jordan Frame Supergravity and Inflation in NMSSM Sergio Ferrara1,2,3 , Renata Kallosh4 , Andrei Linde4 , arXiv:1004.0712v2 [hep-th] 8 Jul 2010 Alessio Marrani4 , and Antoine Van Proeyen5 1 Physics 2 INFN Department, Theory Unit, CERN, CH 1211, Geneva 23, Switzerland - Laboratori Nazionali di Frascati, Via Enrico Fermi 40, 00044 Frascati, Italy 3 Department of Physics and Astronomy, University of California, Los Angeles, CA USA 4 Department 5 Instituut of Physics, Stanford University, Stanford, CA 94305 USA voor Theoretische Fysica, Katholieke Universiteit Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium Abstract We present a complete explicit N = 1, d = 4 supergravity action in an arbitrary Jordan frame with non-minimal scalar-curvature coupling of the form Φ(z, z̄) R. The action is derived by suitably gauge-fixing the superconformal action. The theory has a modified Kähler geometry, and it exhibits a significant dependence on the frame function Φ(z, z̄) and its derivatives over scalars, in the bosonic as well as in the fermionic part of the action. Under certain simple conditions, the scalar kinetic terms in the Jordan frame have a canonical form. We consider an embedding of the Next-to-Minimal Supersymmetric Standard Model (NMSSM) gauge theory into supergravity, clarifying the Higgs inflation model recently proposed by Einhorn and Jones. We find that the conditions for canonical kinetic terms are satisfied for the NMSSM scalars in the Jordan frame, which leads to a simple action. However, we find that the gauge singlet field experiences a strong tachyonic instability during inflation in this model. Thus, a modification of the model is required to support the Higgs-type inflation. Contents 1 Introduction 2 2 Complete N = 1 Supergravity Action in a Jordan Frame 4 3 Bosonic Action of N = 1 Supergravity in Einstein and Jordan Frames 6 3.1 The Einstein frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.2 The Jordan frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.3 Canonical kinetic terms for scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4 Supergravity embedding of the NMSSM and Cosmology 4.1 Classical approximation of the Higgs-type inflation with non-minimal ξ-coupling . . . 4.2 Embedding of the NMSSM into Supergravity and the Einhorn-Jones cosmological in- 10 10 flationary model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4.3 Cosmology in the Jordan frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.4 Switching to the Einstein frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 5 Derivation of the Complete N = 1 Supergravity Action in a Jordan Frame 17 5.1 The Superconformal Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 5.2 Partial gauge fixing and modified Kähler geometry . . . . . . . . . . . . . . . . . . . . 21 5.3 The physical fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 6 Conclusions 29 A Stability with respect to the angle β 30 1 1 Introduction Supersymmetry imposes certain restrictions on the non-supersymmetric models of particle physics and cosmology. A well known example of such restrictions is the fact that the supersymmetric version of the Standard Model (SM) of particle physics requires at least two Higgs superfields. Meanwhile, for cosmology Einstein equations have to be solved, therefore the supersymmetry embedding of the Higgs model inflation requires local supersymmetry, i.e. supergravity. Thus, one can try to see how the potential discovery of supersymmetry may affect various models of inflation, derived in the past in the context of general relativity coupled to scalar fields without supersymmetry. It would be interesting to find general restrictions, as well as to study particular models. Here we are motivated by a particular issue in cosmology, the so-called ξφ2 R coupling, which attracted a lot of attention starting from the early days of inflation [1]. Recently, it became also quite important in the context of SM inflation [2]. Until now, the N = 1, d = 4 supergravity action in an arbitrary Jordan frame described by the frame function Φ(z, z̄), with arbitrary Kähler potential K(z, z̄), holomorphic superpotential W (z) and holomorphic function fab (z), was not known. Here we will derive this action, which is the first goal of this paper. This will be achieved by starting with the superconformal theory developed in [3], and by gauge-fixing all extra symmetries in order to get a general supergravity action in Jordan frame. Our results generalize the formulation of N = 1 supergravity in Jordan frame for the particular case in which the Kähler potential K and the frame function are related by K(z, z̄) = −3 log(− 31 Φ(z, z̄)). The corresponding action in Jordan frame was derived in components in [4, 5], and in superspace in [6, 7]. In our treatment, we will also specify the conditions required for the frame function to make the kinetic terms of the scalar fields canonical in the Jordan frame. The non-minimal coupling of scalar fields to curvature is allowed by all known symmetries of the SM and general relativity. If one tries to describe the early universe using the particle physics SM coupled to gravity in the Einstein frame, one finds that: 1) the coupling λ of the Higgs field has to be of the order 10−13 ; 2) the mass of the Higgs field has to be of the order 1013 GeV. These conditions may be satisfied in a general theory of a scalar field, but not in the simplest version of the standard model. However, if the ξφ2 R coupling is included, i.e. if the embedding of the particle physics SM into the Jordan frame gravity is considered, a satisfactory description of cosmology for the Higgs mass in the interval between 126 and 194 GeV can be found [2]. This is possible for very large values of the non-minimal scalar-curvature coupling ξ ∼ 104 . The model predicts the cosmological parameters ns ≈ 0.97, and r ≈ 0.003, which are consistent with cosmological observations. Thus, this model 2 provides very interesting predictions, which will be testable both at LHC and by a Planck satellite. When this work was in progress, a very interesting proposal [8] was made how to generalize the model of Bezrukov-Shaposhnikov [2] in presence of supersymmetry. Under certain assumptions, it was found that slow regime inflation is not possible within the supergravity embedding of the Minimal Supersymmetric Standard Model (MSSM), but rather it is possible for the NMSSM (see e.g. [14] for a recent review of NMSSM). In the present paper we will study the supergravity embedding of the NMSSM and look for a consistent cosmological models of the Higgs-type inflation. Firstly, we will derive the complete N = 1 action in the general Jordan frame, where it is very simple and has interesting features. This will help to clarify the meaning of the large non-minimal ξφ2 R coupling in the context of supergravity. In particular, the origin of the canonical kinetic terms of all scalars of the NMSSM in the Jordan frame is explained, whereas in the Einstein frame scalar kinetic terms are generally very complicated. Secondly, we will study the theory as a function of all three chiral multiplets, namely two Higgs doublets and a singlet, and analyze various directions in the space of scalar fields. In particular, in [8] it was shown that a slow-roll inflationary regime is possible in NMSSM when the Higgs fields move in the D-flat direction of the two Higgs doublets Hu and Hv , assuming that the gauge singlet S is small. However, it was not clear whether this last assumption is justified, i.e. whether S = 0 corresponds to a minimum of the potential with respect to the field S when inflation takes place in the D-flat direction of the two doublet Higgs fields. We will show that, unfortunately, the potential of the field S has a sharp maximum near S = 0 in this regime. This means that the inflationary regime studied in [8] is unstable, and a search for more general models is required to find a supersymmetric version of the Higgs-type inflation. The paper is organized as follows. In Sec. 2 we present the complete explicit N = 1, d = 4 supergravity action in an arbitrary Jordan frame with non-minimal scalar-curvature coupling of the form Φ(z, z̄)R. This includes the bosonic as well as fermionic action. In the special case in which the frame function Φ(z, z̄) is related to the Kähler potential by the relation K(z, z̄) = −3 log(− 13 Φ(z, z̄)), the action reduces to the one derived in [4, 5]. In the case Φ = −3, the action becomes the well known action of N = 1 supergravity in the Einstein frame. Sec. 3 is devoted to a detailed discussion of the bosonic part of the supergravity action, which is especially important for cosmology. In particular, sufficient conditions for the kinetic terms of scalars to be canonical are specified. 3 Sec. 4 starts with a short description of the Higgs-type inflation with non-minimal scalar-curvature coupling. Then, we proceed with an attempt to generalize this model to the supersymmetric case. For this purpose, we study the embedding of the NMSSM into supergravity, focussing on the EinhornJones cosmological model [8]. We study this model in the Jordan as well as in the Einstein frame. The dependence of the potential on the singlet gauge field S, as well as at large values of the Higgs fields in a D-flat direction of the two Higgs doublets, is explicitly computed. We find that this potential has a maximum for small values of S near the inflationary trajectory. The resulting instability disallows the inflationary regime in the model of [8], unless some way of stabilizing the field S is found. Sec. 5 provides a detailed derivation of the Jordan frame supergravity action presented in Sec. 2, by gaugefixing the extra symmetries of the superconformal action. Finally, the Appendix contains a discussion of the cosmological behavior of the angle β between the two components of the Higgs field. Complete N = 1 Supergravity Action in a Jordan Frame 2 The N = 1, d = 4 supergravity action in a Jordan frame with arbitrary scalar-curvature coupling is uniquely defined by the frame function Φ(z, z̄), Kähler potential K(z, z̄), holomorphic superpotential √ W (z), holomorphic kinetic gauge matrix fAB (z) and momentum map1 PA . It is given by2 (e ≡ −g)   e−1 L = − 16 Φ R(e) − ψ̄µ Rµ − 16 (∂µ Φ)(ψ̄ · γψ µ ) + +L0 + L1/2 + L1 − V + Lm + Lmix + Ld + L4f , (2.1) where the curvature R(e) uses the torsionless connection ωµ ab (e), and the gravitino kinetic term is defined using   Rµ ≡ γ µρσ ∂ρ + 41 ωρ ab (e)γab − 23 iAρ γ∗ ψσ . (2.2) Here Aµ is the part of the auxiliary vector field containing only bosons, namely:
Aµ = 61 i ∂µ z α ∂α K − ∂µ z̄ ᾱ ∂ᾱ K − 13 Aµ A PA , (2.3) where Aµ A is the Yang-Mills gauge field. 1 This is also equivalently named Killing potential, and it encodes the Yang-Mills transformations of the scalars (it may include Fayet-Iliopoulos terms, as well). 2 A derivation of this action, as well as a detailed notation, is given in Sec. 5. 4 The kinetic terms of spin 0, 21 , 1 fields in (2.1) are respectively given by: 1 1 (∂µ Φ)(∂ µ Φ) + gαβ̄ Φ(∂ˆµ z α ) (∂ˆµ z̄ β̄ ) , 4Φ 3   β̄ / α 1 1 = − 2 g̃αβ̄ χ̄ Dχ + 2 Φχ̄α γ µ χβ̄ ∂ˆµ z γ − 13 gγ β̄ Lα + 41 Lαγ Lβ̄ − 14 Lα Lγ β̄ + h.c. ,   A µν B / B F − 21 λ̄A Dλ = (Re fAB ) − 14 Fµν h i A µν B + 41 i (Im fAB ) Fµν F̃ + (∂ˆµ Im fAB ) λ̄A γ∗ γ µ λB . L0 = − L1/2 L1 (2.4) (2.5) (2.6) The covariant derivatives of scalars and fermions are defined as follows: α ∂ˆµ z α ≡ ∂µ z α − AA µ kA ,   ∂kA α (z) β χ + Γαβγ χγ ∂ˆµ z β , D µ χα ≡ ∂µ + 41 ωµ ab (e)γab + 23 iAµ χα − AA µ β ∂z   B A D µ λA ≡ ∂µ + 14 ωµ ab (e)γab − 23 iAµ γ∗ λA − AC µ λ fBC . (2.7) (2.8) The theory has a modified Kähler geometry. In particular, as given by (2.5), the kinetic term of fermions depends on the metric g̃αβ̄ ≡ − 13 Φgαβ̄ + 41 ΦLα Lβ̄ , where gαβ̄ is the Kähler metric and Lα ≡ ∂α ln (−Φ), Lᾱ ≡ ∂ᾱ ln (−Φ) = Lα (see (5.57) and (5.43) further below). Concerning the kinetic terms of scalars, see Sec. 3.2 . The potential reads h  i  V = 91 Φ2 eK −3W W + ∇α W g αβ̄ ∇β̄ W + 21 (Re f )−1 AB PA PB . (2.9) The fermion mass terms are given by Lm = m3/2 = mαβ = µν 1 2 m3/2 ψ̄µ PR γ ψν − 31 Φ
3/2 − 21 mαβ χ̄α χβ − mαA χ̄α λA − 12 mAB λ̄A PL λB + h.c. , eK/2 W ,
3/2 K/2   − 13 Φ e ∇β ∇α W + 2L(α ∇β) W , mAB = − 21 (− 31 Φ)1/2 eK/2 fAB α g αβ̄ ∇β̄ W , √  
mαA = − 31 i 2Φ ∂α + 21 Lα PA − 14 fAB α (Re f )−1 BC PC . (2.10) The remaining terms read   1 1 Lmix = ψ̄ · γPL − i Φ PA λA + √ χα eK/2 (∂α + (∂α K)) (− 13 Φ)3/2 W + h.c. , 6 2   ab A A 1 b Fab + Fab γ µ λB Ld = 8 (Re fAB )ψ̄µ γ i h 1 n + √ ψ̄µ γ ν γ µ χα (− 31 Φ)gαβ̄ ∂ˆν z̄ β̄ + 14 Lα ∂ν Φ 2 o 1 − fAB α χ̄α γ ab Fb −A λB − 1 ΦLα χ̄α γ µν Dµ ψν + h.c. , 4 ab 3 (2.11)   A AB AC + ψ̄ γ λA (see (5.6)). The explicit expression for the where Fbab A ≡ ea µ eb ν 2∂[µ AA + gf BC [µ ν] µ ν ν] 4-fermion terms L4f will be presented in Sec. 5. Remarkably, also L4f contains a significant dependence on the frame function Φ and its derivatives. 5 Bosonic Action of N = 1 Supergravity in Einstein and Jordan 3 Frames 3.1 The Einstein frame By setting Φ = −3 in (2.1), the general N = 1 action in a Jordan frame reduces to the well known action of N = 1 supergravity in the Einstein frame [4, 5]. It is here worth recalling some basic facts about the structure of the bosonic sector of N = 1, d = 4 supergravity. In MP = 1 units, the action of N = 1 supergravity coupled to chiral and vector matter multiplets is usually given in the Einstein frame, where the curvature R appears in the action only √ µν is the Einstein frame space-time metric. through the Einstein-Hilbert term 21 −gE R(gE ), where gE The theory is defined by a real Kähler function K(z, z̄), by an holomorphic superpotential W (z) and by an holomorphic matrix fAB (z) defining the action of the vector multiplets [5]. A particular feature of the theory is the Kähler geometry of the complex scalar fields. The purely bosonic Lagrangian density reads grav Lbos + Lscalar + Lvec E = LE E E , (3.1) where h i √ scalar 1 ˆµ z α ∂ˆν z̄ β̄ g µν − VE , Lgrav = + L −g R(g ) − g ∂ E E E αβ̄ E E 2 i h √ A µν B A µν B F + 41 i (Im fAB ) Fµν F̃ . −gE − 14 (Re fAB ) Fµν Lvec E = (3.2) (3.3) Note that the contractions of spacetime indices and the definition of the dual field strength are perµν formed using the Einstein frame metric gE . The strictly positive-definite metric gαβ̄ (z, z̄) of the non-linear sigma model of scalars z α , z̄ β̄ is given by the second derivative of the real Kähler potential gαβ̄ (z, z̄) ≡ ∂ ∂ K(z, z̄) > 0 , ∂z α ∂ z̄ β̄ (3.4) and ∂ˆµ z α is the Yang-Mills gauge covariant derivative of a scalar field, defined by (2.7). Concerning the potential VE , the F -term potential VEF depends on K and W . On the other hand, the D-term potential VED depends on the values of the auxiliary D-fields, obtained by solving the corresponding equations of motion:   VE = VEF + VED = eK −3W W + ∇α W g αβ̄ ∇β̄ W + 21 (Re f )−1 AB PA PB . (3.5) Notice that (3.5) yields that VJ = Φ2 VE , 9 6 (3.6) where the potential in Jordan frame VJ ≡ V is given by (2.9). ∇α W denotes the Kähler-covariant derivative of the superpotential. The D-term potential can be presented also in the form VED =
1 AB D A D B , where D A is the value of the auxiliary field of the vector multiplets. This is the 2 Re f standard form of the purely bosonic part of the N = 1, d = 4 supergravity action in the Einstein frame. Of course, such a bosonic action can be made supersymmetric by adding suitable fermionic terms (see e.g. [4, 5]). 3.2 The Jordan frame In order to find the action in an arbitrary Jordan frame, one can perform a change of variables from the Einstein to the Jordan frame. Only the metric and the fermions have to be rescaled, the scalars and the vector fields do not change. The metric in a Jordan frame is related to the metric in the Einstein frame as follows (subscripts “E” and ”J” respectively stand for Einstein and Jordan frames throughout) µν gJµν = Ω2 gE , Ω2 = − 31 Φ(z, z̄) > 0 . (3.7) Within our treatment, we will consider the scale factor Ω2 as an arbitrary real function of the complex scalar fields (z, z̄). Its positivity, through (3.7), correspondingly constrains Φ(z, z̄). Since the new action in a Jordan frame is related to the standard one in the Einstein frame by a change of variables, it is supersymmetric, as the original one. Instead of performing the above change of variables by “brute force”, in Sec. 5 we use as a starting point an N = 1, d = 4 superconformal theory [3] with local SU(2, 2|1) symmetry. Such a superconformal theory has a set of local symmetries which includes all N = 1 supergravity symmetries and, in addition, a set of extra local symmetries: local dilatation, U(1) symmetry and special supersymmetry. The superconformal theory has no dimensionful parameters. In [3] the local dilatation, U(1) symmetry and special supersymmetry were gauge fixed in a way that allowed to reproduce the standard N = 1, d = 4 supergravity action in the Einstein frame. In fact, the purely bosonic action of N = 1 supergravity in a Jordan frame is already suggested by Eq. (C.5) of [3]. The complete N = 1 supergravity action in d = 4 in a generic Jordan frame has been presented in Sec. 2, and it is thoroughly derived in Sec. 5 through a suitable gauge-fixing of superconformal supergravity theory [3]. This is a symmetry-inspired approach, alternative to the “brute force” computation based on the change of variables (3.7). Here we will just present the results for the purely bosonic part of the supergravity action in a Jordan frame, which is the most relevant one for cosmology. 7 As mentioned above, the locally supersymmetric action is defined by the choice of four independent functions: a real Kähler potential K(z, z̄), an holomorphic superpotential W (z) and an holomorphic matrix fAB (z), determining the kinetic vector matrix. This suffices to define the N = 1, d = 4 supergravity in the Einstein frame. When dealing with a Jordan frame, an additional fourth function, namely the real frame function Φ(z, z̄), has to be specified. Thus, the purely bosonic part of the N = 1, d = 4 supergravity in a generic Jordan frame reads    2   β̄ α−Φ ∂ ˆ ˆ Φ ∂z z̄ 2 α Φα Φβ̄ β̄ √ Φ 1   1 Φgαβ̄ − − VE + Lbos Lbos = ∂ˆµ z α ∂ˆµ z̄ β̄ − −gJ − ΦR(gJ ) + J 1  . 6 3 Φ 4Φ 9 (3.8) Here VE is the Einstein frame potential defined in (3.5) Φ2 9 VE = VJ ≡ V is the Jordan frame potential given by (2.9), and Φα ≡ ∂ Φ(z, z̄) , ∂z α Φβ̄ ≡ ∂ Φ(z, z̄) = Φβ . ∂ z̄ β̄ (3.9) Notice that (3.8) is implied by (2.1), (2.4), and (2.7), observing that ∂µ Φ = ∂ˆµ Φ because in general bos bos is conformal invariant (and therefore frame Φ is gauge-invariant. Furthermore, Lbos 1,J = L1,E = L1 independent), and it is given by the purely bosonic part of (2.6), or equivalently by ((−gE )−1/2 times) (3.3): bos bos A µν B A µν B 1 Lbos F̃ . + 14 i (Im fAB ) Fµν 1,J = L1,E = L1 = − 4 (Re fAB ) Fµν F (3.10) In the Jordan frame, the contractions of space-time indices and the definition of the dual field strength are performed using the Jordan frame metric gJµν given by (3.7). It should be remarked that (3.8) yields that the geometry of the non-linear sigma model of scalars is  2 ˆ α − Φ ∂ˆz̄ β̄ , the metric is of a modified Kähler type: indeed, due to the term proportional to Φα ∂z β̄ not Hermitian, i.e. there are terms of the form dzdz and complex conjugate; furthermore, the metric term of dzdz̄ is not of the Kähler type. As a consequence of the previous treatment and computations, by setting Φ = −3 in (3.8) the purely bosonic part of the N = 1, d = 4 supergravity action in the Einstein frame [4, 5] is recovered. 1 With the choice Φ = −3e− 3 K(z,z̄) , (3.8) yields to the purely bosonic action of N = 1 supergravity in the particular Jordan frame considered in [4, 5]. 8 3.3 Canonical kinetic terms for scalars In the Einstein frame, the kinetic term of scalar fields is given by gαβ̄ ∂ˆµ z α ∂ˆµ z̄ β̄ , where gαβ̄ (z, z̄) is given by (3.4). Thus, canonical kinetic terms are possible for the following choice of a Kähler potential: K(z, z̄) = δαβ̄ z α z̄ β̄ + f (z) + f¯(z̄) , (3.11) where f (z) is a holomorphic function (associated to the considered Kähler gauge). A 1-modulus example of the canonical Kähler potential (3.11) is provided by the shift-symmetric function K(z, z̄) = − 12 (z − z̄)2 , often used in cosmology. As pointed out above, an early version of N = 1, d = 4 supergravity theory in a (particular) Jordan, as well as in the Einstein, frame was derived in [4, 5] on the basis of the superconformal calculus, with Φ and K related as follows: K(Φ(z, z̄)) = −3 log(− 13 Φ(z, z̄)) , (3.12) and Ω2 given by (3.7). Within such a framework, the following simpler form of Lbos J given by (3.8) is obtained:   √ Φ2 bos,K(Φ) bos,K(Φ) α ˆµ β̄ 2 1 ˆ LJ = −gJ − 6 ΦR(gJ ) − Φαβ̄ ∂µ z ∂ z̄ + ΦAµ − . VE + L 1 9 (3.13) The kinetic term for the scalar action is partly determined by the value Aµ of the bosonic part of µ the auxiliary field of supergravity, entering in the action Lbos J (3.8) as ΦAµ A . In the case of gauge- invariant K, Aµ reads3   1  i ˆ α Aµ = i ∂ˆµ z α ∂α K − ∂ˆµ z̄ ᾱ ∂ᾱ K = − ∂µ z ∂α Φ − ∂ˆµ z̄ ᾱ ∂ᾱ Φ . 6 2Φ (3.14) The purely bosonic action (3.13) yields to the following statement: within the relation (3.12) between K and Φ, in order to have canonical kinetic terms in the Jordan frame it is sufficient a) to choose the frame function Φ as follows: ¯ , Φ(z, z̄) = −3 + δαβ̄ z α z̄ β̄ + J(z) + J(z̄) (3.15) where J(z) is holomorphic. Note that, through (3.12), (3.15) implies K to read: 3   ¯ K(z, z̄) = −3 log 1 − 31 δαᾱ z α z̄ ᾱ − 31 J(z) − 13 J(z̄) ; (3.16) When the Kähler potential is not gauge-invariant in direction A, the auxiliary pseudovector has an additional contribution depending on a gauge field, + 61 iAA µ (rA − r̄A ) , where rA is the holomorphic part of the transformation of the Kähler potential under gauge symmetry, δK(z, z̄) = θA [rA (z) + r̄A (z̄)], see [3]. 9 b) to consider only (scalar) configurations for which the contribution from the bosonic part of the auxiliary vector field vanishes: Aµ = 0 . (3.17) The embedding of the NMSSM into supergravity along the lines suggested in [8] requires only the knowledge of the simple case in which the relation (3.12) between K and Φ holds. Moreover, concerning the canonicity of the kinetic terms of scalars, in the treatment below we will see that condition a) is always satisfied, and condition b) given by (3.17) is satisfied during the cosmological evolution, when the system under consideration depends on three real fields: h1 , h2 , s. Thus, apart from the frame function Φ given by (3.15), the action of the NMSSM embedded in supergravity in Jordan frame (3.12) along the lines of [8] has canonical kinetic scalar terms and a potential Φ2 9 VE (see Secs. 4.2, 4.3 for details). In particular, when only the Higgs field h is non-vanishing in the D-flat direction, the Jordan frame supergravity potential is extremely simple and is given by 4 λ2 4 4 h , see Eq. (4.29). Supergravity embedding of the NMSSM and Cosmology 4.1 Classical approximation of the Higgs-type inflation with non-minimal ξ-coupling The essential reason for the new version of the SM inflation [2] to work successfully is the following. The SM potential with canonical kinetic term for the Higgs field h is coupled to a gravitational field in a suitable Jordan frame. In other words, the Lagrangian density to start with reads:  2  √ λ M + ξh2 (4.1) R (gJ ) − 12 ∂µ h∂ν hgJµν − (h2 − v 2 )2 . LJ = −gJ 2 4 √ At present, h = v ∼ 10−16 MP , and MP2 = M 2 + ξv 2 . Therefore M ≈ MP for ξ < 1016 . In the subsequent investigation we will consider ξ < 106 . In this case M = MP with a very good accuracy. In our paper we will use the system of units where M = MP = 1. In general, the cosmological predictions have to be compared with the observations in the Einstein frame, related to the Jordan one through the conformal rescaling (3.7), with Φ = −3(1 + ξh2 ) , Ω2 = 1 + ξh2 . (4.2)
µν − 12 ∂µ ψ∂ν ψ gE − U (ψ) , (4.3) By switching to the Einstein frame, (4.1) yields to LE = √ −gE 1 2 R(gE ) where ψ is a canonically normalized scalar in the Einstein frame, defined by r Ω2 + 6ξ 2 h2 dψ ≡ dh . Ω4 10 (4.4) where λ U (ψ) = 4  h2 (ψ) − v 2 1 + ξh(ψ)2 2 . (4.5) The relation between the field h and the canonically normalized field ψ looks very different in three one has ψ ≈ h. In the interval 1ξ ≪ h ≪ √1ξ , the relation between q h and ψ is more complicated: ψ ≈ 32 ξh2 . Finally, for h ≫ √1ξ (or, equivalently, ψ ≫ 1) one has different ranges of h. At h ≪ h∼ √1 ξ e ψ √ 6 1 ξ . In this regime, the potential in the Einstein frame is very flat, which leads to inflation: U (ψ)ψ→∞ λ ⇒ 4ξ 2  1+e 2ψ −√ 6 −2 . As one can see from (4.6), the constant (ψ-independent) term in the potential U (ψ) is (4.6) λ , 4ξ 2 so nothing would work without the non-minimal scalar curvature coupling proportional to ξ. q 1 . For the non-supersymmetric The Hubble constant during inflation in this model is H ≈ λ3 2ξ standard model, λ = O(1), so one could worry that this energy scale is dangerously close to the possible unitarity bound Λ ∼ 1/ξ discussed in [9–11]. One should note, however, that most of the arguments suggesting the existence of this bound are based on the investigation of the theory in the small field approximation ψ ≈ h, where one can use an expansion ψ = h(1 + ξ 2 h2 + ...). This approximation is valid only for h ≪ 1ξ , which is parametrically far from the inflationary regime at h ≫ √1 . ξ We are going to return to this issue in a forthcoming publication; see also a discussion in [12], and especially in [13], where it was noticed that in NMSSM one may consider the regime with λ ≪ 1, where the concerns about the unitarity bound do not seem to appear. It is worth noting that potentials exponentially rapidly approaching a constant positive value have been proposed in one of the first models of chaotic inflation in supergravity [15], but at that time models of this type were lacking a compelling motivation. Therefore, it is very tempting to use the intuitively appealing and simple model discussed above as a starting point, in order to analyze the Einhorn-Jones approach [8] to embed NMSSM into N = 1, d = 4 supergravity, and its relevance for the issue of inflation. 4.2 Embedding of the NMSSM into Supergravity and the Einhorn-Jones cosmological inflationary model The Higgs field sector of NMSSM has one gauge singlet and two gauge doublet chiral superfields, namely [14]: z α = {S, H1 , H2 } , 11 (4.7) with S = seiα ,   Hu+ , Hu =  0 Hu  Hd0 Hd =  Hd−  . (4.8) As in [8], the frame function is chosen as follows: Φ(z, z̄) = −3 + (S S̄ + Hu Hu† + Hd Hd† ) + 32 χ(Hu · Hd + h.c.) , (4.9) Hu · Hd ≡ −Hu0 Hd0 + Hu+ Hd− . (4.10) where Note that (4.9) is of the form (3.15), with J = 32 χHu · Hd . In this framework, the Kähler potential is related to Φ through (3.12), and the superpotential is chosen to be ρ W = −λSHu · Hd + S 3 . 3 (4.11) Thus, the action of such an implementation of NMSSM depends on five chiral superfields. Through explicit computations, we checked that such an action admits a consistent truncation in which the charged superfields, namely Hu+ and Hd− , are absent. Therefore, below we deal with a simplified action of NMSSM, containing only three superfields: S, Hu0 and Hd0 , such that:  H1 =  0 Hu0   , H2 =  Hd0 0  . (4.12) Within this truncation, the frame function and the superpotential respectively read:
Φ(z, z̄) = −3 + |S|2 + |Hu0 |2 + |Hd0 |2 − 23 χ(Hu0 Hd0 + Hu0 Hd0 ) ; ρ W = λSHu0 Hd0 + S 3 . 3 bos,K(Φ) Thus, by recalling Eqs. (3.6), (3.13) and (3.15 ), and by disregarding L1 (4.13) (4.14) in (3.13), one obtains the following Jordan frame supergravity scalar-gravity action for this implementation of NMSSM: i h M SSM LN J √ = 21 R(gJ ) + 16 δαβ̄ z α z̄ β̄ + 23 χ(Hu0 Hd0 + Hu0 Hd0 ) R − δαβ̄ ∂ˆµ z α ∂ˆµ z̄ β − ΦA2µ − VJ , −gJ (4.15) where Aµ is given by (3.14). Remarkably, the scalar-curvature coupling exhibited by (4.15) breaks the discrete Z3 symmetry of the theory due to the chosen cubic superpotential (4.14) of NMSSM. Such a symmetry may generate domain walls after the spontaneous breaking of a symmetric phase in the early universe. In such a 12 case, unacceptably large anisotropies of CMB may be generated. This is a well known domain wall problem of NMSSM (see e.g. [14]). The scalar-curvature coupling in (4.13 ) and in (4.15) breaks the discrete Z3 symmetry. This may help to remove the eventual domain wall problem. Thus, it is challenging and interesting to formulate a consistent cosmology within this framework. As usual, VJ = VJF + VJD . In the present framework, VJF has a zero, second and fourth power of the S field: VJF = λ2 |Hu0 |2 |Hd0 |2 + λρ(S̄ 2 Hu0 Hd0 + c.c.) − On the other hand, VJD reads VJD 2λ2 |S|2 |HA0 |2 (χ(Hu0 Hd0 + c.c.) − 2) + ρ2 |S|4 . 4 + 3χ2 |HA0 |2 − 2χ(Hu0 Hd0 + h.c.) (4.16) ′ g2 g2 (|Hu0 |2 − |Hd0 |2 )2 + ((Hu )†~τ Hu + (Hd )†~τ Hd )2 , = 8 8 (4.17) where ~τ is the 3-vector of Pauli σ-matrices. In [8] this model was described at the vanishing value of the gauge singlet field S. In order to analyze the theory consistently, in the present treatment we keep the full dependence on S. 4.3 Cosmology in the Jordan frame We start by checking that the CP -invariant solution found in [8], in which S, Hu0 and Hd0 are real, corresponds to a(n at least local) minimum of VJ itself. In order to do so, a priori we assume that these three fields are complex, namely ( s, h1 , h2 ∈ R+ , α, α1 , α2 ∈ [0, 2π)): S = seiα , Hu0 = h1 eiα1 , Hd0 = h2 eiα2 . (4.18) By computing (4.15), it follows that the scalar-gravity action depends only on the combination angles γ ≡ α1 + α2 and δ ≡ 2α − α1 − α2 . More precisely, the dependence on δ enters via λρ cos δ and the dependence on γ is via χ cos γ. In order to study CP -invariant solution(s) with α = α1 = α2 = 0, one has to analyze the minima of the potential VJ , also taking into account the R -dependent terms in (4.15) (notice that VJD does not depend on any phase). Firstly, we notice that Eqs. (4.16), (4.17) and the definition of δ yield that the dependence on δ enters only in one term in the potential, namely: VJ (δ) = 2λρ|S|2 |Hu0 ||Hd0 | cos δ . (4.19) This potential has a minimum at δ = 0, under the condition that λρ is negative: λρ = −|λρ|. Secondly, in order to deal correctly with the dependence on γ, one can look at the expected minimum of the potential at S = 0 [8]. (4.16) implies that the Jordan frame potential at S = 0 is very simple: VJF S=0 = λ2 |Hu0 |2 |Hd0 |2 . 13 (4.20) At S = 0 the dependence on γ enters only through the frame function Φ given by Eq. (4.13). By switching to the Einstein frame, and recalling the relation (3.6), one obtains: (VE (γ))S=0 = 9λ2 |Hu0 |2 |Hd0 |2 = Φ2 1− 1 3 λ2 |Hu0 |2 |Hd0 |2
2 . |Hu0 |2 + |Hd0 |2 + χ|Hu0 ||Hd0 | cos γ (4.21) Since during inflation 1 − 31 (|HA0 |2 ) > 0 [8], it can be checked that during inflation γ = 0 is a minimum of VJ , under condition that χ > 0. Thus, the CP -invariant solution with three real fields is confirmed to be a minimum in the directions of angles δ and γ during inflation. Therefore we can take S = s, Hu0 = h1 , Hd0 = h2 , (4.22) provided that the coupling constants of the model under consideration satisfy λρ < 0 , χ > 0. (4.23) Notice that (4.13) and (4.22) yield that the kinetic scalar terms in the Jordan frame are canonical, since both sufficient conditions ( 3.15) and (3.17) are satisfied (in particular, Aµ = 0 on scalar configurations (4.22)):   √ M SSM (LN )kinetic = − −gJ (∂µ s)2 + (∂µ h1 )2 + (∂µ h2 )2 . J (4.24) It is now convenient to switch to the standard mixing of the Higgs fields, defined as: h1 ≡ h cos β , h2 ≡ h sin β , (4.25) which leaves us with two real fields, h and β, instead of h1 and h2 . Through Eq. (4.17), the D-flat direction, defined by VJD = 0, (4.26) requires that sin(2β) = 1; h21 = h22 = h2 /2. Thus, along the D-flat direction, the curvature term of (4.15) simplifies to: √ 
−gJ  N M SSM (LJ 1 − 13 s2 + h2 + 12 χh2 R(gJ ) . )curv = 2 (4.27) (4.28) On the other hand, along the D-flat direction (4.26)-(4.27) the F -term potential reads VJF = 2λ2 s2 h2 (χh2 − 2) λ2 4 h − |λρ|s2 h2 − + ρ2 s 4 . 4 4 + 3χ2 h2 − 2χh2 14 (4.29) In [8] the inflationary regime driven by the Higgs within NMSSM was shown to take place for χh2 ≫ 1 ≫ h2 , s ≈ 0, β= π , 4 in Planck units MP2 = 1. For small s, (4.29) can be simplified as follows:   2λ2 λ2 4 F h − |λρ| + s 2 h2 . VJ ∼ 4 3χ (4.30) (4.31) The effective mass of the s field is negative, but one actually has to take into account an effective contribution from the curvature-scalar coupling. This latter provides a positive contribution, however, it does not remove the tachyonic instability of the system in the s direction. Indeed, for small s, the complete expression of the effective potential is   λ2 4 λ2 F ṼJ ∼ h − |λρ| + s 2 h2 . 4 3χ (4.32) As we will see in the next Sec., the instability in the s direction is very strong, corresponding to a large tachyonic mass and a slow-roll parameter |η| ≥ 2/3. As a result, a rapidly developing tachyonic instability does not allow inflation to occur in the regime studied in [8]. Note that in general instead of λρ < 0 one could take λρ > 0. Correspondingly, such a choice of coupling constants would stabilize the real part of the field S, but it would lead to an equally strong instability in the direction of its imaginary part. In other words, independently of the sign of λρ, the potential with respect to the complex field S has a saddle point at S = 0, which results in the tachyonic instability in one of the two directions. 4.4 Switching to the Einstein frame In the Einstein frame, (3.6) and (4.16) yield that the F -term potential is VEF 9 = 2 VJ = Φ λ2 4 4 h 2 2 2 2 s h (χh −2) − |λρ|s2 h2 − 2λ + ρ2 s4 4+3χ2 h2 −2χh2 . 2  1 − 31 (s2 + h2 ) + 12 χh2 (4.33) Let us compute the effective mass of the s field also in the Einstein frame, where by definition there is no contribution from the curvature coupling. During the inflationary regime (4.30) [8], the leading behavior of the potential is VEF   λ2 λ2 4s2 + O(s4 ) . ∼ 2 − |λρ| + χ 3χ χ2 h2 (4.34) The shape of the potential is shown in Fig. 1. The trajectory with s = 0 at large h, which was expected to be an inflationary trajectory in [8], is unstable. It corresponds to the top of the ridge for the potential VEF , see Fig. 1. 15 Figure 1: The F -term potential V F in the Einstein frame. The inflationary trajectory s = 0 is unstable. In order to find whether this instability is dangerous, one should calculate the tachyonic mass of the s field and compare it to the Hubble constant. This will allow us to check whether the tachyonic instability develops rapidly, or whether it occurs on a time scale much smaller than the cosmological time scale H −1 . An alternative way to approach this issue is to find the related value of the relevant slow-roll parameter η. To find the effective mass of the s field, attention must be paid to the non-minimal normalization of the field S = seiα . At constant α, the kinetic term of field S is given by gSS ∂S∂ S̄ = 2 2 ∂S∂ S̄ = (∂s)2 . 2 χh χh2 (4.35) Thus, in the vicinity of the inflationary trajectory s ≈ 0 (4.30 ), the Lagrangian density of the field s is LE,s   2 λ2 λ2 4s2 2 = − 2 (∂s) − 2 + |λρ| + + O(s4 ) . χh χ 3χ χ2 h2 In terms of the canonical scalar field s̃ = LE,es = − 12 (∂s̃)2 √2s , χh (4.36) such a Lagrangian at small fields s̃ is λ2 − 2+ χ  λ2 |λρ| + 2 χ 3χ  s̃2 + O(s̃4 ) , resulting in the mass squared of the s̃ field to be tachyonic:  2  λ |λρ| 2 ms̃ ∼ −2 + < 0. 3χ2 χ 16 (4.37) (4.38) Taking into account that during inflation H 2 = V /3 ≈ m2s̃ ≤ − λ2 , 3χ2 it thus follows that 2V 2λ2 =− = −2H 2 = R/6 . 3χ2 3 (4.39) Interestingly, m2s̃ resembles the conformal mass m2 = −R/6, but has an opposite sign. Since |m2s̃ | > H 2 , the trajectory s̃ = 0 is exponentially unstable and unsuitable for inflation. One can also reach the same conclusion by computing the relevant slow-roll parameter η in the s̃ direction: ηse ≡ m2s̃ 2 2|λρ|χ 2 =− − <− . 2 V 3 λ 3 (4.40) We did not find any way to solve this problem of the Einhorn-Jones model [8]. It should also be clearly stated that there are many other scalar fields in this model, and the field s is not the only one which may experience a tachyonic instability. This is supported by the results obtained in the Appendix, where the dependence of the potential on the angular variable β is studied. Therein, we find that in certain cases the post-inflationary cosmological trajectory may experience an additional tachyonic instability, and deviate from the value β = π 4 characterizing the D-flat direction (4.30). We should emphasize, however, that these results are model-dependent. We believe that the cosmological models based on N = 1, d = 4 supergravity in Jordan frame can be very interesting, and they certainly deserve further investigation. In the past, a systematic study of such models was precluded by the absence of the corresponding formalism, which we presented in a complete form in Sec. 2. In the next Section we will give a detailed derivation of the complete N = 1, d = 4 supergravity action in a generic Jordan frame. 5 Derivation of the Complete N = 1 Supergravity Action in a Jordan Frame Here we use the superconformal action [3] and gauge fix it to get a complete N = 1 supergravity action, including fermions, in an arbitrary Jordan frame. Superconformal invariance means that the action is invariant under the local symmetries of the superconformal algebra. This involves, apart from the super-Poincaré transformations, local dilatations, a local U (1) R-symmetry, local special conformal transformations, and an extra special supersymmetry, denoted as S-supersymmetry. One first constructs a “superconformal action”, i.e. an action that is invariant under all symmetries of the superconformal algebra. Then one gets rid of the extra symmetries by imposing gauge conditions. 17 The vierbein eaµ and gravitino ψµ are the gauge fields of the translations and Q-supersymmetry, which belong to the super-Poincaré algebra. The gauge field of local Lorentz rotations is the spin connection ωµ ab which is a constrained field, i.e. it has as usual a value that depends on eµ a and ψµ . We will write here the expressions in terms of ωµ ab (e) which is the usual torsionless spin connection of gravity. Also the gauge fields of special conformal transformations and of S-supersymmetry are such composite fields. In the expressions below, they have been substituted by their values. On the other hand, the gauge field of the U (1) R-symmetry, Aµ , is an auxiliary field. It is value will be given below. Finally, the gauge field of dilatations is a field bµ , which will later be set to zero by a gauge condition for the special conformal symmetry. The superconformal transformations of the vierbein and gravitino are (apart from general coordinate transformations) δeµ a = −λa b eµ b − λD eµ a + 21 ǭγ a ψµ ,     δψµ = − 41 λab γab − 21 λD + 32 iλT γ∗ ψµ + ∂µ + 12 bµ + 41 ωµ ab γab − 23 iAµ γ∗ ǫ − γµ η , (5.1) where λab are the parameters of local Lorentz transformations, λD are those of dilatations, λT are those of the U (1) R-symmetry. ǫ and η are the spinor parameters of Q and S-supersymmetry, respectively. 5.1 The Superconformal Action We first repeat the result for the full superconformal action using the notation that we will use in this paper. The action contains 3 superconformal-invariant terms   L = [N ]D + [W]F + fAB λ̄A PL λB F . (5.2) The first one is defined by a Kähler potential N (X, X̄) for the superconformal fields, the second uses a superpotential W(X), and the third involves the chiral kinetic matrix fAB (X) (where A are the gauge indices), and gauginos λA . The matrix PL = 12 (1 + γ∗ ) projects on the left-handed fermions. The dilatation symmetry implies that N should be homogeneous of first order in both X and X̄, W should be homogeneous of third degree and fAB (X) is of zeroth order, i.e. XI ∂ ¯ ∂ N =N, N = X̄ I I ∂X ∂ X̄ I¯ XI ∂ W = 3W , ∂X I XI ∂ fAB = 0 . ∂X I (5.3) The superconformal chiral multiplets contain the bosonic fields X I and fermions ΩI = PL ΩI . We assume that they transform under the gauge symmetries depending on Killing vectors kA I (X) δX I = θA kA I , δΩI = θA ∂J kA I ΩJ . 18 (5.4) These Killing vectors should satisfy homogeneity equations due to the conformal symmetry, and leave N and W invariant4 . These statements can be encoded in the following equations ∂J¯kA I = 0 , X J ∂ J kA I = kA I , ¯ NI kA I + NI¯kA I = 0 ,   ¯ ¯ PA = 12 i NI kA I − NI¯kA I = iNI kA I = −iNI¯kA I , ∂I¯PA = iNJ I¯kA J , WI kA I = 0 . (5.5) We use here the notation that derivatives on N and W are denoted by adding indices, similar to (3.4). The physical fields of the chiral and gauge multiplets transform as follows under the superconformal transformations: δX I δΩI δAA µ δλA Dµ X I Fbab A 1 = (λD + iλT ) X I + √ ǭΩI , 2   √
1 / I + F I ǫ + 2X I PL η , = − 41 λab γab + 23 λD − 12 iλT ΩI + √ PL DX 2 A 1 = − 2 ǭγµ λ , h  i    = − 41 λab γab + 23 λD + 12 iλT γ∗ λA + 23 λD + 23 iγ∗ λT λA + 41 γ ab Fbab A + 12 iγ∗ DA ǫ , 1 I = (∂µ − bµ − iAµ ) X I − √ ψ̄µ ΩI − AA µ kA , 2   A B C A = ea µ eb ν 2∂[µ AA . ν] + gfBC Aµ Aν + ψ̄[µ γν] λ (5.6) After elimination of the auxiliary fields, the terms in (5.2) mix. The scalars form a Kähler manifold with metric, connection and curvature given by GI J¯ = NI J¯ , ΓIJK = GI L̄ NJK L̄ , RI K̄J L̄ = NIJ K̄ L̄ − NIJ M̄ GM M̄ NM K̄ L̄ . (5.7) The superconformal action5 can be split in several parts e−1 L = 1 6N  −R(e, b) + ψ̄µ Rµ + e −1 ∂µ (e ψ̄ · γψ µ )  L0 + L1/2 + L1 − V + Lm + Lmix + Ld + L4f . (5.8) The leading kinetic terms of the matter multiplets are ¯ L0 = −GI J¯Dµ X I Dµ X̄ J , h i ¯ ¯ /̂ J + 21 Ω̄J DΩ /̂ I , L1/2 = − 21 GI J¯ Ω̄I DΩ   A µν B / B F − 12 λ̄A Dλ L1 = (Re fAB ) − 14 Fµν h i A µν B + 41 i (Im fAB ) Fµν F̃ + (Dµ Im fAB ) λ̄A γ∗ γ µ λB . 4 (5.9) Note that this does not imply that the Kähler potential or superpotential of the Einstein theory should be invariant under the gauge transformations, as we will see below. 5 There is a possible generalization including a Chern-Simons term, see [17], which we neglect here. 19 The potential in the conformal form is ¯ V = VF + VD = GI J WI W J¯ + 12 (Re f )−1 AB PA PB . (5.10) Bilinear fermion terms can be divided in those that give rise to physical masses, terms relevant for the super-BEH mechanism and terms with derivative couplings to bosonic fields: Lmix Ld ¯ − 21 ∇I WJ Ω̄I ΩJ + 14 GI J W J¯fABI λ̄A PL λB √   + 2 i −∂I PA + 14 fABI (Re f )−1 BC PC λ̄A ΩI + h.c. ,   1 1 I A = ψ̄ · γPL iPA λ + √ WI Ω + h.c. , 2 2   1 1 n ¯ A A / J γ µ ΩI = + (Re fAB )ψ̄µ γ ab Fab γ µ λB + √ GI J¯ψ̄µ DX + Fbab 8 2 o − 1 fAB I Ω̄I γ ab Fb A λB − 2 NI Ω̄I γ µν Dµ ψν + h.c. . Lm = µν 1 2 W ψ̄µ PR γ ψν 4 ab 3 (5.11) Finally, the 4-fermion terms are L4f =  1  ρ µ ν N (ψ̄ γ ψ )(ψ̄ρ γµ ψν + 2ψ̄ρ γν ψµ ) − 4(ψ̄µ γ · ψ)(ψ̄ µ γ · ψ) 96  1 1 + − √ fAB I ψ̄ · γΩI λ̄A PL λB + ∇I fAB J Ω̄I ΩJ λ̄A PL λB + h.c. 8 4 2   ¯ ¯ 1 −1 µνρσ + 16 e ε ψ̄µ γν ψρ Ω̄J γσ ΩI + 12 i Re fAB λ̄A γ∗ γσ λB − 21 GI J¯ψ̄µ ΩJ ψ̄µ ΩI 1 I J¯ G fAB I λ̄A PL λB f¯CD J¯λ̄C PR λD + 41 RI K̄J L̄ Ω̄I ΩJ Ω̄K̄ ΩL̄ − 16     ¯ ¯ 2 1 + 16 (Re f )−1 AB fAC I Ω̄I − f¯AC I¯Ω̄I λC fBD J Ω̄J − f¯BD J¯Ω̄J λD + N (AF µ ) . (5.12) This superconformal action contains the bosonic and fermionic parts of the auxiliary field Aµ , which are 1 2N 1 = i 2N 1 ≡ i 4N Aµ = i AF µ h  
i ¯ I¯ I NI¯ ∂µ X̄ I − AA − N I ∂ µ X I − AA µ kA µ kA h i 1 ¯ PA , NI¯ ∂µ X̄ I − NI ∂µ X I + AA N µ     √ 3 I¯ I I J¯ A B 2ψ̄µ NI Ω − NI¯Ω + GI J¯Ω̄ γµ Ω + (Re fAB )λ̄ γµ γ∗ λ . 2 (5.13) R(e, b) is defined with the spin connection ωµ ab (e, b), which is intermediate between the full connection ωµ ab (e, b, ψ) and the torsionless one ωµ ab (e): ωµ ab (e, b, ψ) = ωµ ab (e, b) + 21 ψµ γ [a ψ b] + 41 ψ̄ a γµ ψ b , ωµ ab (e, b) = ωµ ab (e) + 2eµ [a eb]ν bν , ωµ ab (e) = 2eν[a ∂[µ eν] b] − eν[a eb]σ eµc ∂ν eσ c . (5.14) Fermion terms are extracted from covariant derivatives Dµ , whose superconformal U (1) connection 20 involves only the bosonic part Aµ . Thus, explicitly, Dµ X I D̂µ ΩI D µ λA I I = ∂ µ X I − b µ X I − AA µ kA − iAµ X ,   I J I K J = ∂µ − 23 bµ + 14 ωµ ab (e, b)γab + 21 iAµ ΩI − AA µ ∂J kA Ω + ΓJK Ω Dµ X ,   B A = ∂µ − 23 bµ + 14 ωµ ab (e, b)γab − 23 iAµ γ∗ λA − AC µ λ fBC . (5.15) We also defined in a similar way   Rµ ≡ γ µρσ ∂ρ + 21 bµ + 41 ωρ ab (e, b)γab − 23 iAρ γ∗ ψσ , (5.16) while Dµ ψν contains also ψ-torsion in the derivative. The action (5.8) is invariant under the superconformal transformations. We now will break those symmetries that are not required for super-Poincaré supergravity: special conformal transformations, dilatations and S-supersymmetry. 5.2 Partial gauge fixing and modified Kähler geometry First, we eliminate the special conformal transformations, by imposing the special conformal transformations gauge choice: bµ = 0 . (5.17) Next, we discuss the gauge choice for dilatations. The dilatational gauge, D-gauge, that has been chosen in the past is [16] N =− 3 . κ2 (5.18) This brings the Einstein-Hilbert term in its canonical form. We further put κ = 1. To solve such a gauge condition, an appropriate way [3] is to change variables from the basis {X I } to a basis {y, z α }, where α = 1, . . . , n using X I = y Z I (z) . (5.19) We do not specify the (n + 1) functions Z I of the base space coordinates z α , so that we keep the freedom of arbitrary coordinates on the base. The Z I must be non-degenerate in the sense that the (n + 1) × (n + 1) matrix  ZI    ∂α Z I (5.20) should have rank n + 1. There are many ways to choose the Z I . One simple choice, labeling the I index from 0 to n, can be Z0 = 1 , Z α = zα . 21 (5.21) Then the gauge condition can be solved for the modulus of y. Its phase is determined by a gauge condition for the R-symmetry. The homogeneity properties then determine that ¯ N = y ȳ Z I (z)GI J¯(z, z̄)Z̄ J (z̄) , GI J¯(z, z̄) = ∂I ∂J¯N (X, X̄) = NI J¯ . (5.22) The function that acts as Kähler potential for this gauge is i h ¯ K(z, z̄) = −3 ln − 31 Z I (z)GI J¯(z, z̄)Z̄ J (z̄) . (5.23) This defines the Kähler metric gαβ̄ = ∂α ∂β̄ K(z, z̄) . (5.24) Note that there is an arbitrariness in the definition (5.19). We may consider redefinitions y ′ = y ef (z)/3 , Z ′ I = Z I e−f (z)/3 . (5.25) These redefinitions lead to a different Kähler potential: K′ (z, z̄) = K(z, z̄) + f (z) + f¯(z̄) . (5.26) Hence these can be identified with Kähler transformations for the Kähler potentials defined by (5.23). In view of these Kähler transformations, it is often useful to define Kähler-covariant derivatives. The gauge field for the parameter f (z) is then ∂α K, while for f¯(z̄) it is ∂ᾱ K. In both cases these are the gauge fields because they transform with a derivative on the parameters. We thus define ∇α Z I = ∂α Z I + 13 (∂α K) Z I , ¯ = ∂ᾱ Z̄ I + 31 (∂ᾱ K) Z I , ∇ᾱ Z̄ I ¯ ∇ᾱ Z I = ∂ᾱ Z I = 0 , ¯ ¯ ∇α Z̄ I = ∂α Z̄ I = 0 . (5.27) We now define weights of functions under Kähler transformations. Any object that transforms like Z I in (5.25) is defined to have weights (w+ , w− ) = (1, 0). Hence, y has weight (−1, 0). The objects that appear in the superconformal formulation do not transform under Kähler transformations. For any quantity that in the superconformal variables is of the form V(X, X̄) = y w+ ȳ w− V (z, z̄) . (5.28) we define that V has weights (w+ , w− ), and the Kähler-covariant derivatives are ∇α V ∇ᾱ V  ∂α + 31 w+ (∂α K) V ,   = ∂ᾱ + 31 w− (∂ᾱ K) V . =  (5.29) Remark that Φ does not transform under Kähler transformations, and thus has weights (0,0). On the other hand eK/3 has weights (−1, −1), and thus ∇α eK/3 = 0. 22 The gauge and U(1) transformations on X I split in those for y and z α as follows: 1 A 3 θ rA (z) δy = y
+ iλT , α (z) , δz α = θA kA (5.30) where 31 rA (z) can be considered as the component of the Killing vectors in the direction of y. Our new setup will assume the following gauge conditions: N = Φ(z, z̄) , D-gauge U (1)-gauge y = ȳ , (5.31) with an arbitrary function Φ(z, z̄). We keep the definition of K as in (5.23), with the associated Kähler transformations and covariant derivatives as in (5.27), and all the above equations remain valid. Furthermore, all the results below will then reduce to those of [3, 5] when Φ = −3.6 The value for y for the new gauge choice is y = ȳ = r − K Φ exp . 3 6 (5.32) However, in many equations we will keep the phase of y arbitrary. The U(1) gauge choice can be taken at any time. The vanishing of the derivative of the D-gauge condition w.r.t. z α leads to ¯ N I ∇α Z I = 0 , NI = ȳ GI J¯Z̄ J . (5.33) Note that this equation does not feel the presence of the function Φ. With these equations one can write the matrix identity   −3 0 0 gαβ̄    = eK/3  ZI ∇α Z I     GI J¯ Z̄ J¯ ∇ Z̄ J¯ . β̄ (5.34) Every matrix here is (n + 1) × (n + 1), and should be invertible. This matrix identity is useful to translate quantities in the X I basis to quantities in the {y, z α } basis. E.g. it implies that the inverse of GI J¯ is   ¯ ¯ ¯ GI J = eK/3 − 31 Z I Z̄ J + g αβ̄ ∇α Z I ∇β̄ Z̄ J . (5.35) We assume that Φ is a (Yang-Mills) gauge-invariant function. Hence the dilatation gauge condition is invariant. However, the U(1)-gauge is not, see the transformations (5.30), and it is not invariant 6 Or Φ = −3κ−2 , where κ is the gravitational coupling constant which has often been set to 1. To restore κ one also replaces exp K with κ6 exp κ2 K, and thus also gαβ̄ with κ2 gαβ̄ , and ψµ with κψµ . 23 under Kähler transformations (5.25) either. It is not invariant under supersymmetry either, but we postpone this for when we have discussed a new basis of the fermions. This implies that we cannot forget the transformations with parameter λT , but should relate it to the gauge transformations and Kähler transformations (and later also to supersymmetry)   λT = 16 iθA [rA − r̄A ] + 61 i f (z) − f¯(z̄) . (5.36) Taking this into account, and also the gauge invariance of Φ, we find that the Kähler potential K transforms under gauge transformations as δK = θA [rA (z) + r̄A (z̄)] . (5.37) The moment map PA defined in (5.5) depends on this quantity rA (z) as
PA = i (kA α ∂α K − rA ) = −i kA ᾱ ∂ᾱ K − r̄A . PA = (− 31 Φ)PA , (5.38) Another convenient way to state this, is to write the Killing vectors in in the X I basis as  α  I kA = y kA ∇α Z I + 31 iPA Z I . (5.39) The bosonic part of the value of the auxiliary field Aµ , see (5.13) is Aµ = =
∂µ z α ∂α K − ∂µ z̄ ᾱ ∂ᾱ K − 13 Aµ A PA   α ᾱ A 1 1 ˆ ˆ 6 i ∂µ z ∂α K − ∂µ z̄ ∂ᾱ K + 6 iAµ (rA − r̄A ) . 1 6i (5.40) Independent of the gauge conditions, one proves that the kinetic terms of the scalars, L0 in (5.9) is h i ∂ ∂ 1 ¯ (∂µ N )(∂ µ N ) − N (∂ˆµ z α ) (∂ˆµ z̄ β̄ ) α ln Z I (z)GI J¯Z̄ J (z̄) , 4N ∂z ∂ z̄ β̄ α ≡ ∂ µ z α − AA µ kA . L0 = − ∂ˆµ z α (5.41) After the gauge choice, this is thus 1 (∂µ Φ)(∂ µ Φ) + 31 gαβ̄ Φ(∂ˆµ z α ) (∂ˆµ z̄ β̄ ) , 4Φ h i
= Φ 31 gαβ̄ − 12 Lα Lβ̄ (∂ˆµ z α ) (∂ˆµ z̄ β̄ ) − 41 Φ Lα Lβ (∂ˆµ z α ) (∂ˆµ z β ) + h.c. , L0 = − (5.42) which is the same as (2.4), and where we introduced L for ln Φ and Lα = ∂α ln(−Φ) , Lᾱ = ∂ᾱ ln(−Φ) . (5.43) For the superpotential, we define W(X) = y 3 W (z). Hence W (z) has Kähler weights (3,0). This leads to WI Z I = 3y 2 W (z) , y −2 WI ∇α Z I = ∇α W ≡ ∂α W + (∂α K)W . 24 (5.44) The F -term in the superpotential therefore reduces to (taking only bosonic terms from the field equation of F I )   VF = 91 Φ2 eK −3W W + ∇α W g αβ̄ ∇β̄ W . (5.45) This agrees with what we already expected in (3.6). Due to (5.38), also the D-term has the same overall Φ-dependence VD = −1 AB 1 2 PA PB 18 Φ (Re f ) , (5.46) This agrees with what we found in (3.6). We introduce now modified Kähler-covariant derivatives, which take the presence of Φ into account. For an object V that has weights (w+ , w− ), we define e αV ∇ e V ∇ ᾱ   ∂α + 31 w+ (∂α K) + 12 (w+ + w− )Lα V ,   = ∂ᾱ + 13 w− (∂ᾱ K) + 12 (w+ + w− )Lᾱ V . = (5.47) We can also define the covariant derivatives in spacetime, using e . e µ = (∂ˆµ z α )∇ e α + (∂ˆµ z̄ ᾱ )∇ ∇ ᾱ (5.48) One can evaluate these before or after gauge fixing of the U(1) symmetry. Before the latter is gaugefixed we have to add the covariantization of the latter. y has then weight (−1, 0), so that we define   e µ y = ∂µ − iAµ − 1 ∂ˆµ z α ∂α K + 1 ∂µ L − 1 AA r y = 0. (5.49) ∇ A µ 3 2 3 The calculation is modified after the U(1) gauge fixing, but the result is still the same. The U(1) transformation is gone, but due to (5.36) and (5.30), the Kähler transformation of y is (in agreement with its value in (5.32))   y ′ = y exp f (z) + f¯(z̄) /6 . Thus y has now Kähler weights (w+ , w− ) = (− 21 , − 12 ), leading again to   e µ y = ∂µ − 1 ∂ˆµ z α ∂α K − 1 ∂ˆµ z̄ ᾱ ∂ᾱ K + 1 ∂µ L − 1 AA (rA + r̄A ) y = 0 . ∇ 6 6 2 6 µ (5.50) (5.51) We thus find that y is invariant under the new covariant derivatives This will facilitate many calculations. e y = 0. e αy = ∇ ∇ ᾱ (5.52) There are some differences between these modified covariant derivatives and the ordinary covariant derivatives (5.27). Most important is that the anti-chiral modified covariant derivative does not vanish on Z I : e ZI = 1 ZI L . ∇ ᾱ ᾱ 2 25 (5.53) The commutator of the covariant derivatives on scalar functions still satisfies the rule i h e V (z, z̄) = 1 (w − w )g V (z, z̄) . e α, ∇ ∇ − + αβ̄ β̄ 3 (5.54) This leads also to an expression that we will need below: e ∇ I e I ∇ β̄ α Z = Z 1 3 gαβ̄
e αZ I . + 12 Lαβ̄ + 21 Lβ̄ ∇ (5.55) The matrix equation (5.34) gets modified:     1 I   ΦL Z Φ β̄  2 e Z̄ J¯ .  GI J¯ Z̄ J¯ ∇  = y ȳ  β̄ 1 e αZ I ∇ g̃αβ̄ 2 ΦLα (5.56) where g̃αβ̄ ≡ − 31 Φgαβ̄ + 14 ΦLα Lβ̄ . (5.57) To obtain the second holomorphic derivative of Z I , one can take a covariant derivative on the second line of (5.56) to obtain where e αZ J ∇ e β Z K + L(α ∇ e β) Z I + Z I e α Z I = −y ΓIJK ∇ e β∇ ∇ 1 2 Lαβ
− 14 Lα Lβ , Lαβ = ∇α Lβ = ∂α Lβ − Γγαβ Lγ . (5.58) (5.59) This can be used further to calculate the curvature of the projective manifold. Indeed, acting with e Z̄ J¯G ∇ e y ȳ ∇ I J¯ ᾱ on this equation, and using that on a vector quantity β̄ i h e e γZI , e α Z I + Rᾱβα γ ∇ e e α Z I = 1 gβ ᾱ ∇ (5.60) ∇ᾱ , ∇β ∇ 3 we obtain after many cancellations of L-dependent terms i h e Z̄ J¯ . e Z̄ I¯∇ e βZJ ∇ e αZ I ∇ (− 13 Φ) Rαᾱβ β̄ − 23 gᾱ(α gβ)β̄ = (y ȳ)2 RI IJ ¯ J¯∇ ᾱ β̄ (5.61) Observe that the cancellations can be explained due to the fact that the dilatational symmetry of the embedding manifold implies that Z I RI IJ ¯ J¯ = 0. 5.3 The physical fermions In order to define the physical bosons in the previous section, we changed from the conformal basis {X I } to the basis {y, z α }. We now make a similar change of basis from the conformal fermions {ΩI } to a new basis7 {χ0 , χα }, using ΩI 7   e αZ I . = y χ0 Z I + χα ∇ We again use the implicit chiral notation, i.e. PL χα = χα and PR χᾱ = χᾱ . 26 (5.62) Our aim is to have χ0 = 0 as gauge condition for the S-gauge transformations. We therefore choose S-gauge NI ΩI = 21 Φα χα , (5.63) e αZ I . which is equivalent to χ0 = 0. Hence the gauge fixed fermions are ΩI = yχα ∇ The covariant derivative of the physical fermion is   ∂kA α (z) β χ + Γαβγ χγ ∂ˆµ z β . Dµ χα = ∂µ + 14 ωµ ab (e)γab + 23 iAµ χα − AA µ ∂z β (5.64) The covariant derivative on the conformal fermions (5.15) can then be rewritten as   e α Z I + y 2 ΓI χα ∇ e αZ K ∇ e β Z J ∂ˆµ z β D̂µ ΩI = Dµ yχα ∇ JK   e ∇ e α Z I + y 2 ΓIJK χα ∇ e αZ K ∇ e β Z J ∂ˆµ z β e β + ∂ˆµ z̄ β̄ ∇ e α Z I + yχα ∂ˆµ z β ∇ = y (Dµ χα ) ∇ β̄ i h
e α Z I + yχα ∂ˆµ z β L(α ∇ e β) Z I + Z I 1 Lαβ − 1 Lα Lβ = y (Dµ χα ) ∇ 2 4 h i
e αZ I , +yχα ∂ˆµ z̄ β̄ Z I 13 gαβ̄ + 21 Lαβ̄ + 21 Lβ̄ ∇ (5.65) using (5.58) and (5.55). This can be inserted in the kinetic fermion terms, L1/2 in (5.9). The contribution of the last line of (5.65) can be complex conjugated such that this leads to   / α + 21 Φχ̄α γ µ χβ̄ ∂ˆµ z γ − 31 gγ β̄ Lα + 41 Lαγ Lβ̄ − 14 Lα Lγ β̄ + h.c. . L1/2 = − 12 g̃αβ̄ χ̄β̄ Dχ (5.66) Now we consider the fermion mass terms, Lm in (5.11). We rewrite them as Lm = 21 m3/2 ψ̄µ PR γ µν ψν − 21 mαβ χ̄α χβ − mαA χ̄α λA − 21 mAB λ̄A PL λB + h.c. , (5.67) where the (complex) gravitino mass parameter can be easily recognized as m3/2 = W = y 3 W = − 31 Φ
3/2 eK/2 W . (5.68) For the mass terms of the chiral fermions, we have mαβ χα χβ = ∇I WJ Ω̄I ΩJ . (5.69) e due to the We first observe that in (5.44) we can insert as well the modified covariant derivatives ∇ homogeneity conditions. Then we take a further covariant derivative using again (5.58), gives   e α Z I + y −2 WI L(α ∇ ˜ α W = y −1 ∇J WI ∇ e βZJ ∇ ˜ β∇ e β) Z I + 1 Lαβ Z I − 1 Lα Lβ Z I . ∇ (5.70) 2 4 Therefore, e αZ I e βZJ ∇ mαβ = y 2 ∇I WJ ∇   e β∇ e α W − yWI L(α ∇ e β) Z I + 1 Lαβ Z I − 1 Lα Lβ Z I = y3∇ 2 4 h
i 3 e e 1 1 e β) W − 3W Lαβ − Lα Lβ = y ∇β ∇α W − L(α ∇ 2 4
  3/2 K/2 = − 13 Φ e ∇β ∇α W + 2L(α ∇β) W . 27 (5.71) For the mass terms involving λ, we first need an equation for the derivative of fAB : e α Z I = y fAB I ∇α Z I . fAB α = ∇α fAB = y fAB I ∇ (5.72) A further derivative on this equation is relevant for the 4-fermion terms. Using (5.58) and the homogeneity of degree zero of fAB so that fAB I Z I = 0, we obtain e β fAB α = y 2 ∇J fAB I ∇ e αZ I ∇ e β Z J + L(α ∂β) fAB . ∇β fAB α = ∇ (5.73) ¯ For the λλ mass term we use the expression for GI J in (5.35), the same homogeneity equation of fAB , (5.44) and (5.72) to translate ¯ mAB = − 12 GI J W J¯fABI = − 12 (− 31 Φ)1/2 eK/2 fAB α g αβ̄ ∇β̄ W . (5.74) The conformal expression of the λχ mass term, gives √   e αZ I . mαA = i 2 ∂I PA − 14 fABI (Re f )−1 BC PC y ∇ (5.75) We first calculate ¯ e α Z I ∂I PA + ȳ ∇ e α Z̄ I ∂ ¯PA ∂ α PA = y ∇ I e α Z I ∂ I PA + 1 Lα PA , = y∇ 2 (5.76) ¯ due to (5.53) and the homogeneity equation X I ∂I¯PA = PA . Using (5.38) this gives e α Z I ∂I PA = − 1 Φ∂α PA − 1 PA ∂α Φ . y∇ 3 6 (5.77) Using also again (5.72), we obtain √  
mαA = − 31 i 2Φ ∂α + 12 Lα PA − 41 fAB α (Re f )−1 BC PC . (5.78) For Ld in (5.11) we need only one new calculation: ¯ e αZ I GI J¯Dµ X̄ J y ∇   e Z̄ J¯∇ e αZ I e β + ∂ˆµ z̄ β̄ ∇ = y ȳGI J¯ ∂ˆµ z β ∇ β̄ = 1 ˆ β 4 ΦLα Lβ ∂µ z + g̃αβ̄ ∂ˆµ z̄ β̄ . (5.79) To calculate the 4-fermion terms, we need the fermionic part of the auxiliary field Aµ . Its conformal expression was given in (5.13), which can be evaluated as
i 3i i α ᾱ + g̃αβ̄ χ̄α γµ χβ̄ + (Re fAB )λ̄A γµ γ∗ λB . AF µ = √ ψ̄µ Lα χ − Lᾱ χ 4Φ 8Φ 4 2 28 (5.80) One term in the square of this expression is the χ4 term, which combines (after a Fierz transformation) with the curvature term in L4f , where (5.61) is now convenient. The result is given in the beginning of the paper, in Sec. 2. Here we still give the 4-fermion term: L4f =  1  ρ µ ν Φ (ψ̄ γ ψ )(ψ̄ρ γµ ψν + 2ψ̄ρ γν ψµ ) − 4(ψ̄µ γ · ψ)(ψ̄ µ γ · ψ) 96  1 1 α β A α A B B + − √ fAB α ψ̄ · γχ λ̄ PL λ + χ̄ χ λ̄ PL λ [∇β fAB α − Lα fAB β ] + h.c. 8 4 2   A 1 1 −1 µνρσ ψ̄µ γν ψρ 2 i Re fAB λ̄ γ∗ γσ λB + g̃αβ̄ χ̄β̄ γσ χα + 16 e ε  
1 + 16 Φgαβ̄ ψ̄µ χβ̄ ψ̄ µ χα − 32 Φψ̄µ χα Lα + χᾱ Lᾱ ψ̄ µ χβ Lβ + χβ̄ Lβ̄ 2 9  3 (Re fAB )λ̄A γµ γ∗ λB + + g αβ̄ Φ−1 fAB α λ̄A PL λB f¯CD β̄ λ̄C PR λD 64Φ 16  
1 (Re f )−1 AB fAC α χ̄α − f¯AC ᾱ χ̄ᾱ λC fBD β χ̄β − f¯BD β̄ χ̄β̄ λD + 16
3 − 41 gαβ̄ + 16 Lα Lβ̄ (Re fAB )χ̄α λA χ̄β̄ λB  
3 1 α µ β̄ A B gαβ̄ χ̄ γ χ + (Re fAB )λ̄ γµ γ∗ λ − √ ψ̄µ Lγ χγ − Lγ̄ χγ̄ 2 8 2
1 2 3 1 Lα Lβ Lγ̄ Lδ̄ χ̄α χβ χ̄γ̄ χδ̄ . − 12 Φ Rαγ̄β δ̄ − 2 κ gαγ̄ gβ δ̄ − 41 Lα Lγ̄ gβ δ̄ + 32 − 6 (5.81) Conclusions The main goal of our paper was to derive a complete formulation of N = 1, d = 4 supergravity in a generic Jordan frame. We found that, in general, this formulation is very non-trivial. It involves modified Kähler geometry (in the sense specified in our treatment), and it gives rise to many new complicated terms in the supergravity Lagrangian. However, we identified a subclass of theories where the resulting formulation is remarkably simple. This subclass includes the recently proposed model of Einhorn and Jones [8], which was introduced as an N = 1 supergravity realization of the Higgs field inflation [2]. We found that the inflationary regime in this model is unstable. Hopefully, however, the general formalism developed in our paper may allow one to find new realistic inflationary models in supergravity. As a starting approach, one can simply study in the Jordan frame several classes of inflationary models in supergravity, which were found long time ago in the Einstein frame. As shown by the example of the Higgs inflation, sometimes it is helpful to identify and study various physical features of the cosmological models by switching from one frame to another. 29 Acknowledgments We are grateful to S. Dimopoulos, M. Einhorn, D. Freedman, P. Graham, R. Harnik, T. Jones, L. Kofman, S. Mukohyama, S. Shenker, L. Susskind, A. Westphal for the useful discussions. The work of RK and AL is supported by the NSF grant 0756174. The work of SF is supported by ERC Advanced Grant n.226455, Supersymmetry, Quantum Gravity and Gauge Fields (Superfields), in part by PRIN 2007-0240045 of Torino Politecnico, in part by DOE Grant DE-FG03-91ER40662 and in part by INFN, sez. L.N.F. The work of AM is supported by an INFN visiting Theoretical Fellowship at SITP, Stanford University, Stanford, CA, USA. The work of AVP is supported in part by the FWO Vlaanderen, project G.0235.05, and in part by the Federal Office for Scientific, Technical and Cultural Affairs through the ‘Interuniversity Attraction Poles Programme – Belgian Science Policy’ P6/11-P. A Stability with respect to the angle β As we found in Subsecs. 4.3 and 4.4, the inflationary trajectory with s = 0 (4.30) of the Einhorn-Jones model [8] is unstable with respect to a rapid generation of the s field. Other scalar fields may also have nontrivial dynamical properties. If after a modification of this model one can find a way to stabilize the s field, one would then need to study the cosmological behavior of all other fields. As an example, in this Appendix we will analyze the behavior of the angle β, ignoring the issue of the s field instability. For h2 ≪ 1 (consistent with (4.30)), the Einstein frame potential VE of the fields h and β reads VE (h, β) = 2h4 λ2 sin2 2β + (g 2 + g ′2 )h4 cos2 2β . 2(2 + h2 χ sin 2β)2 (A.1) The first term in the numerator originates from the F -term, the second term from the D-term. During inflation, in the slow roll regime at h2 χ ≫ 1 (see (4.30)), the potential with respect to β is minimized by the condition of D-flatness, corresponding to β = π/4. In this regime, VE (h, β = π/4) = λ2 h4 λ 2 ≈ . (2 + h2 χ)2 χ2 (A.2) One could contemplate the possibility of an additional slow-roll regime with respect to the slow variation of β [8]. However, the stabilization of β during inflation is very firm. Indeed, when we take into account that the non-canonical kinetic terms in the angular direction near the minimum are proportional to 1 χ, we find that the effective mass squared of the fluctuations of the field β is given by m2β   h2 −4λ2 + (g 2 + g ′2 )(2 + h2 χ) ∼ . (2 + h2 χ)2 30 (A.3) During inflation, in the limit χh2 ≫ 1 m2β = g 2 + g ′2 , χ (A.4) and the slow-roll parameter η with respect to the field β thus reads ηβ ≡ m2β V ≈χ g 2 + g ′2 . λ2 (A.5) This means that for χ(g 2 + g ′2 ) ≫ λ2 , one has ηβ ≫ 1. Thus, there is no slow-roll regime with respect to the change of β during inflation, because the mass squared of perturbations of the angle β is much greater than H 2 . Therefore, during inflation the field β rapidly approaches π/4 and stays there. However, the angle β may play an interesting dynamical role at the end of inflation. Our calculations show that the potential vanishes at h = 0 for all β, see Fig. 2. However, in our investigation we did not take into account spontaneous symmetry breaking in the SM, as well as soft terms leading to supersymmetry breaking, which are important at an energy scale much smaller than the energy scale relevant for inflation. Clearly, the low-energy scale dynamics of the field β will depend on the above mentioned effects that we ignored, but also on the value of the field β at the end of inflation. Figure 2: During inflation at large h the angular variable β is stabilized at β = π/4, corresponding to h1 = h2 . For g 2 , g ′2 ≫ λ2 , this stabilization is preserved even after the end of inflation. One could expect that until the low-energy effects become important, the field β remains equal to π/4. However, this is not always the case. Indeed, (A.3) yields that when χh2 becomes smaller than O(1) and inflation ends, the mass squared of the field β becomes   g 2 + g ′2 2 2 2 , mβ ≈ h −λ + 2 31 (A.6) thus affecting the slow-roll parameter ηβ as follows (recall Eq. (A.5)):  ηβ ≈ χ χh 2  g 2 + g ′2 −1 + 2λ2  . (A.7) Note that typically |ηβ | ∼ χ ≫ 1, and that when χh2 becomes smaller than O(1) and inflation ends. Therefore, for g 2 , g ′2 > 2λ2 , the D-term continues to dominate the dynamics of the field β even at the end of inflation, ηβ remains large and positive, and β continues to be captured at its original value β = π/4, see Fig. 3. Oscillations of the inflaton field h near the minimum of its potential may lead to perturbative [18, 19], as well as non-perturbative [20, 21] decay of this field, which can be very efficient because the coupling constants of the corresponding interactions are rather large. A detailed discussion of reheating in the original (non-supersymmetric) version of this scenario can be found in the second and third Refs. of [2]. Figure 3: During inflation at large h the angular variable β is stabilized at β = π/4, corresponding to h1 = h2 . For g 2 , g ′2 ≪ λ2 , at the end of inflation the curvature of the potential in β-direction becomes large and negative, much greater than the curvature in the inflaton direction. This leads to tachyonic instability, generation of large fluctuations of the field β, and spontaneous symmetry breaking. The situation is more complicated in the opposite case g 2 , g ′2 < 2λ2 , in which the field moving along the trajectory β = π/4 experiences strong tachyonic instability at the end of inflation, which leads to spontaneous symmetry breaking, see Fig. 3 . This effect, which is called “tachyonic preheating” [22,23], is similar to the waterfall regime in the hybrid inflation scenario [24]. The physical meaning of “tachyonic preheating” within the framework under consideration can be understood as follows. As mentioned, inflationary regime ends when χh2 becomes O(1). At that 32 0.007 0 3π/4 π/2 π/4 0 - π/4 Figure 4: “Tachyonic preheating” effect at the end of inflation (for g 2 , g ′2 ≪ λ2 ). time, the parameter ηh describing the slow-roll in the h direction becomes O(1), which means that the effective mass squared of the field h becomes O(H −1 ). Therefore the field h reaches the minimum of the potential at h = 0 within the time ∆t = O(H −1 ) from the end of inflation. This last, post-inflationary, part of the field evolution is shown in Fig. 4. During that time, quantum fluctuations of the field β start growing, δβ ∼ emβ t , they rapidly reach the minima of the potential in the β direction, which correspond to the two valleys in Fig. 4, at β ≈ 0 and at β ≈ π/2. Spontaneous symmetry breaking occurs within the time m−1 β , which is shorter than H −1 by the factor O(η −1/2 ) ∼ 10−2 . In other words, this process occurs almost instantly, on the cosmological time scale. When this happens, the universe becomes divided into domains with the field β taking values in one of the two valleys in Fig. 4. These domains, of initial size m−1 β , will be separated from each other by domain walls corresponding to the ridge of the potential at β = π/4. Then the field h will continue rolling down to smaller values of h, following the two valleys of the potential. A detailed evolution of the field distribution can be studied by the methods developed in [22, 23]. In order to find out which of the two regimes (g 2 , g ′2 > 2λ2 versus g 2 , g ′2 < 2λ2 ) occurs in the realistic versions of this scenario one should perform an investigation of the running of the coupling constants from their present day values to the end of inflation, similar to the investigation performed in [2]. However, prior to such an investigation, one should find a solution to the main problem of this scenario, which is the tachyonic instability with respect to the field s found in Section 4. 33 References [1] T. Futamase and K. I. Maeda, Chaotic inflationary scenario in models having nonminimal coupling with curvature, Phys. Rev. D39, 399 (1989). D. S. Salopek, J. R. Bond and J. M. Bardeen, Designing density fluctuation spectra in inflation, Phys. Rev. D40, 1753 (1989). N. Makino and M. Sasaki, The Density perturbation in the chaotic inflation with nonminimal coupling, Prog. Theor. Phys. 86, 103 (1991). R. Fakir, S. Habib and W. Unruh, Cosmological density perturbations with modified gravity, Astrophys. J. 394, 396 (1992). E. Komatsu and T. Futamase, Complete constraints on a nonminimally coupled chaotic inflationary scenario from the cosmic microwave background, Phys. Rev. D59, 064029 (1999) [arXiv:astro-ph/9901127]. L. Kofman and S. Mukohyama, Rapid roll inflation with conformal coupling, Phys. Rev. D77, 043519 (2008) [arXiv:0709.1952 [hep-th]]. [2] F.L. Bezrukov and M. Shaposhnikov, The Standard Model Higgs boson as the inflaton, Phys. Lett. B659, 703 (2008), arXiv:0710.3755. F. Bezrukov, D. Gorbunov and M. Shaposhnikov, On initial conditions for the Hot Big Bang, JCAP 0906, 029 (2009), [arXiv:0812.3622 [hep-ph]]. J. Garcia-Bellido, D. G. Figueroa and J. Rubio, Preheating in the Standard Model with the HiggsInflaton coupled to gravity, Phys. Rev. D79, 063531 (2009), [arXiv:0812.4624 [hep-ph]]. A. De Simone, M. P. Hertzberg and F. Wilczek, Running inflation in the Standard Model, Phys. Lett. B678, 1 (2009), [arXiv:0812.4946 [hep-ph]]. F. Bezrukov and M. Shaposhnikov, Standard Model Higgs boson mass from inflation: two loop analysis, JHEP 0907, 089 (2009), [arXiv:0904.1537 [hep-ph]]. A. O. Barvinsky, A. Y. Kamenshchik, C. Kiefer, A. A. Starobinsky and C. Steinwachs, Asymptotic freedom in inflationary cosmology with a non-minimally coupled Higgs field, JCAP 0912, 003 (2009), [arXiv:0904.1698 [hep-ph]]. [3] R. Kallosh, L. Kofman, A. D. Linde and A. Van Proeyen, Superconformal symmetry, supergravity and cosmology, Class. Quant. Grav. 17, 4269 (2000) [Erratum-ibid. 21, 5017 (2004)] [arXiv:hep-th/0006179]. [4] E. Cremmer, B. Julia, J. Scherk, S. Ferrara, L. Girardello and P. van Nieuwenhuizen, Spontaneous symmetry breaking and Higgs effect in supergravity without cosmological constant, Nucl. Phys. B147, 105 (1979). [5] E. Cremmer, S. Ferrara, L. Girardello and A. Van Proeyen, Yang-Mills theories with local supersymmetry: Lagrangian, transformation laws and superHiggs Effect, Nucl. Phys. B212, 413 (1983). 34 [6] G. Girardi, R. Grimm, M. Muller and J. Wess, Superspace geometry and the minimal, nonminimal, and new minimal supergravity multiplets, Z. Phys. C26, 123 (1984). [7] J. Wess and J. Bagger, Supersymmetry and supergravity, Princeton University Press (Princeton), 1992. [8] M. B. Einhorn and D. R. T. Jones, Inflation with non-minimal gravitational couplings and supergravity, JHEP 1003, 026 (2010), [arXiv:0912.2718 [hep-ph]]. [9] C. P. Burgess, H. M. Lee and M. Trott, “Power-counting and the Validity of the Classical Approximation During Inflation,” JHEP 0909, 103 (2009) [arXiv:0902.4465 [hep-ph]]. C. P. Burgess, H. M. Lee and M. Trott, “Comment on Higgs Inflation and Naturalness,” arXiv:1002.2730 [hepph]. [10] J. L. F. Barbon and J. R. Espinosa, “On the Naturalness of Higgs Inflation,” Phys. Rev. D 79, 081302 (2009) [arXiv:0903.0355 [hep-ph]]. [11] M. P. Hertzberg, “On Inflation with Non-minimal Coupling,” arXiv:1002.2995 [hep-ph]. [12] R. N. Lerner and J. McDonald, “Higgs Inflation and Naturalness,” JCAP 1004, 015 (2010) [arXiv:0912.5463 [hep-ph]]. [13] H. M. Lee, “Chaotic inflation in Jordan frame supergravity,” arXiv:1005.2735 [hep-ph]. [14] U. Ellwanger, C. Hugonie and A. M. Teixeira, The next-to-minimal supersymmetric Standard Model, [arXiv:0910.1785 [hep-ph]]. [15] A. B. Goncharov and A. D. Linde, Chaotic Inflation In Supergravity, Phys. Lett. B139, 27 (1984). [16] T. Kugo and S. Uehara, Improved superconformal gauge conditions in the N = 1 supergravity Yang-Mills matter system, Nucl. Phys. B222, 125 (1983). [17] J. De Rydt, J. Rosseel, T. T. Schmidt, A. Van Proeyen and M. Zagermann, Symplectic structure of N = 1 supergravity with anomalies and Chern-Simons terms, Class. Quant. Grav. 24 (2007) , [arXiv:0705.4216 [hep-th]]. [18] A. D. Dolgov and A. D. Linde, Baryon Asymmetry In Inflationary Universe, Phys. Lett. B116, 329 (1982). [19] L. F. Abbott, E. Farhi and M. B. Wise, Particle Production In The New Inflationary Cosmology, Phys. Lett. B117, 29 (1982). 35 [20] L. Kofman, A. D. Linde and A. A. Starobinsky, Reheating after inflation, Phys. Rev. Lett. 73, 3195 (1994) [arXiv:hep-th/9405187]. [21] L. Kofman, A. D. Linde and A. A. Starobinsky, Towards the theory of reheating after inflation, Phys. Rev. D56, 3258 (1997) [arXiv:hep-ph/9704452]. [22] G. N. Felder, J. Garcia-Bellido, P. B. Greene, L. Kofman, A. D. Linde and I. Tkachev, Dynamics of symmetry breaking and tachyonic preheating, Phys. Rev. Lett. 87, 011601 (2001), [arXiv:hep-ph/0012142]. [23] G. N. Felder, L. Kofman and A. D. Linde, Tachyonic instability and dynamics of spontaneous symmetry breaking, Phys. Rev. D64, 123517 (2001), [arXiv:hep-th/0106179]. [24] A. D. Linde, Axions in inflationary cosmology, Phys. Lett. B259, 38 (1991). A. D. Linde, Hybrid inflation, Phys. Rev. D49, 748 (1994), [arXiv:astro-ph/9307002]. 36