Starobinsky-like inflation and soft-SUSY breaking

Published for SISSA by Springer Received: March 26, 2019 Revised: May 14, 2019 Accepted: May 23, 2019 Published: May 31, 2019 Stephen F. King and Elena Perdomo School of Physics and Astronomy, University of Southampton, SO17 1BJ Southampton, U.K. E-mail: , Abstract: We study a version of Starobinsky-like inflation in no-scale supergravity (SUGRA) where a Polonyi term in the hidden sector breaks supersymmetry (SUSY) after inflation, providing a link between the gravitino mass and inflation. We extend the theory to the visible sector and calculate the soft-SUSY breaking parameters depending on the modular weights in the superpotential and choice of Kähler potential. We are led to either no-scale SUGRA or pure gravity mediated SUSY breaking patterns, but with inflationary constraints on the Polonyi term setting a strict upper bound on the gravitino mass m3/2 < 103 TeV. Since gaugino masses are significantly lighter than m3/2 , this suggests that SUSY may be discovered at the LHC or FCC. Keywords: Beyond Standard Model, Cosmology of Theories beyond the SM, Supersymmetric Standard Model ArXiv ePrint: 1903.08448 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP05(2019)211 JHEP05(2019)211 Starobinsky-like inflation and soft-SUSY breaking Contents 1 2 The hidden sector 2 3 The visible sector 4 4 Potential and soft-SUSY breaking parameters 4.1 Hidden sector potential and inflation 4.1.1 Stabilizing the modulus field 4.2 Visible sector potential and SUSY breaking 4.2.1 Case I: no-scale SUSY breaking 4.2.2 Case II: pure gravity SUSY breaking 5 6 7 8 9 10 5 Conclusion 11 A Soft-supersymmetry breaking parameters 12 1 Introduction Inflation [1–6] is well known to solve the flatness and horizon problems, diluting cosmological relics and providing an origin of cosmological fluctuations. In slow-roll inflation [7, 8], the inflaton rolls along a quite flat potential and inflation end as it falls into some basin. Inflation is supported and constrained by current observational data [9], which measures a spectral index ns ≈ 0.96 ± 0.007 and low tensor-to-scalar ratio r < 0.08, excluding the simplest chaotic models based on polynomial potentials such as φ2 or φ4 [10]. Surviving models include Starobinsky inflation [3, 11, 12], Higgs inflation [13] and related models [14, 15], and low scale hybrid inflation [16, 17]. Supersymmetry (SUSY) may be naturally combined with inflation since it allows better control over the high energy dynamics of scalars [18–20]. SUSY inflation is also motivated by the Lyth bound [21] on the low tensor-to-scalar ratio, which prefers a scale of inflation below the Planck scale. Since inflation is sensitive to UV scales, it is necessary to consider supergravity (SUGRA) inflation, as in e.g. [22–42]. In no-scale SUGRA [44], the Kähler potential takes a logarithmic form which circumvents the η problem. Alternatives to no-scale SUGRA have also been proposed which also address the η problem based on a Heisenberg symmetry [45–48] or a shift symmetry [49–51] (see also [52–57]). It has been shown by Ellis, Nanopoulos, Olive (ENO) that no-scale SUGRA can behave like a Starobinsky inflationary model [58–60]. However, in this approach, a term with constant modular weight is used to break SUSY, and there is no connection between –1– JHEP05(2019)211 1 Introduction 2 The hidden sector In general supergravity theory, the tree-level supergravity scalar potential can be found using the Kähler function G, which is given in terms of the Kähler potential K and the superpotential W as, K W W ∗ G = 2 + ln + ln . (2.1) MP MP3 MP3 The effective scalar potential is then given by,   ∂G G ∂G V =e Kij ∗ − 3 MP4 , ∂φi ∂φj ∗ (2.2) ∗ where Kij ∗ is the inverse of the Kähler metric K ij ≡ ∂ 2 K/∂φi ∂φ∗j . When, at the minimum of the scalar potential, some of the hidden sector fields acquire VEVs in such a way that at least one of their auxiliary fields, F i , is non-vanishing, then SUSY is spontaneously broken and soft SUSY-breaking terms are generated in the observable sector. The gravitino becomes massive and its mass 2 m23/2 = eG = eK/MP –2– |W |2 MP4 (2.3) JHEP05(2019)211 inflation and SUSY breaking. Recently we considered the above ENO model, but with a linear Polonyi term added to the superpotential [61]. The purpose of adding this term was to provide an explicit mechanism for breaking SUSY in order to provide a link between inflation and SUSY breaking. Indeed we showed that inflation requires a strict upper bound for the gravitino mass m3/2 < 103 TeV [61]. In the present paper we show how the Polonyi-extended ENO model may be generalised to include the fields in the visible sector of the minimal supersymmetric standard model (MSSM). Such a generalisation has been done for the ENO model [58–60] and we perform a similar analysis for the Polonyi-extended ENO model. We calculate the soft-SUSY breaking parameters depending on the modular weights in the superpotential and choice of Kähler potential and we are led to new phenomenological possibilities for supersymmetry (SUSY) breaking, based on generalisations of no-scale SUSY breaking and pure gravity mediated SUSY breaking. The Polonyi-extended ENO model is especially interesting to consider because of the upper bound on the gravitino mass discussed in the previous paragraph which allows the much lighter gauginos to be discovered in future collider experiments. This motivates the present investigation of the soft SUSY breaking parameters, which could form the basis for future phenomenological studies. The layout of the remainder of the paper is as follows. In section 2 we discuss the hidden sector of the supergravity theory where inflation takes place. In section 3 we discuss the visible sector of the supergravity theory and show how the MSSM matter and Higgs fields may be included. In section 4 we discuss the supergravity scalar potential, showing how inflation emerges from the hidden sector and soft-SUSY breaking parameters emerge from the full theory including the visible sector, leading to new examples of no-scale SUSY breaking and pure gravity mediated SUSY breaking. Section 5 concludes the paper. where MP is the reduced Planck scale. It was found in [58] this Kähler potential together with the Wess-Zumino superpotential [62, 63] can lead to the Starobinsky-like inflationary potential. When the modulus field T is fixed with a vacuum expectation value of hRe T i = 1/2 and hIm T i = 0, the no-scale Kähler potential together with the Wess-Zumino superpotential is equivalent of an R + R2 model of gravity, in which Starobinsky inflation emerges at a particular point in parameter space [59]. A simple modification to this superpotential has been done in [61], adding the Polonyi term to provide an explicit and simple mechanism for supersymmetry breaking at the end of inflation. The Wess-Zumino superpotential [62] in the hidden sector, with quadratic and trilinea (...truncated)