Ph.D. Thesis Inflation in Supergravity with a Single Superfield Takahiro Terada Advanced Leading Graduate Course for Photon Science, and Department of Physics, Graduate School of Science, The University of Tokyo Research Fellow of Japan Society for the Promotion of Science arXiv:1508.05335v1 [hep-th] 21 Aug 2015 Submitted: December 16, 2014 Defended: January 20, 2015 Revised for Library: February 17, 2015 Revised for arXiv: August 20, 2015 Thesis Advisor: Prof. Koichi Hamaguchi, Chief Examiner: Prof. Hitoshi Murayama, Co-Examiner: Prof. Masaki Ando, Co-Examiner: Prof. Shigeki Matsumoto, Co-Examiner: Prof. Yuji Tachikawa, Co-Examiner: Prof. Taizan Watari. Abstract Supergravity is a well-motivated theory beyond the standard model of particle physics, and a suitable arena to study high-energy physics at the early universe including inflation, whose observational evidences are growing more and more. Inflation in supergravity, how- ever, can not be trivially described because of restrictions from supersymmetry. The scalar potential has an exponential factor and a large negative term whereas a flat and positive potential is needed to realize inflation. The standard method to obtain a suitable infla- tionary scalar potential requires an additional superfield to the one containing inflaton. In this thesis, we propose and develop an alternative method which does not require the ad- ditional superfield and thus reduces the necessary degrees of freedom by half. That is, we study inflation in supergravity with only a single chiral superfield which contains inflaton. We accomplish it by introducing a higher dimensional term in the inflaton K¨ahlerpotential, which plays an important dual role: fixing the value of the scalar superpartner of the inflaton resulting in effective single field models, and ensuring the positivity of the inflaton potential at the large field region. Our proposal is not just particular models but rather a new frame- work to realize various inflationary models in supergravity. In particular, large field inflation in supergravity using one superfield without tuning has become possible for the first time. In our generic models, supersymmetry breaking at the inflationary scale by inflaton is not completely restored after inflation, so null results for supersymmetry search at the LHC are predicted for the simplest cases. Remarkably, however, it is possible with tuning to embed arbitrary positive semidefinite scalar potentials into supergravity preserving supersymmetry at the vacuum. Our discovery opens up an entirely new branch of model building of inflation in supergravity. Preface for the arXiv version This is the arXiv version of the author's Ph.D. thesis. Only a few minor changes were made for this version. The original version of the thesis will be available at UTokyo Repository. Also, the contents of Section 3.6 were summarized in a proceeding of the HPNP 2015 workshop [270] though some parts have been updated in this version of the thesis. The achievement of the works [44,45] that this thesis is based on is a breakthrough in the field of inflation in supergravity, and it triggered subsequent interesting developments [63, 271{277]. In particular, another approach to inflation in supergravity with a single chiral superfield was proposed by Roest and Scalisi in Ref. [273] based on generalization of no-scale supergravity, which was further developed in Ref. [277]. A similar model was proposed by Linde [274] as a generalization of Goncharov-Linde model [34,62{64]. These works appeared after the completion of this thesis, so these are not covered in the review part of the thesis. Instead, we here briefly comment on some relations between their and our approaches. The models in Refs. [273,274] are unstable for some parameter choice, and they requires stabilization [275] (see also Ref. [277] for the generalization which does not require stabi- lization in the K¨ahler potential). When we take c2 = c4 = 0 in eq. (23) in Ref. [275], it reduces to the model in Ref. [273]. When we take c4 = 0 and c2 = (α−1)=2 in the equation, it reduces to the model in Ref. [274]. Now, when we take c2 = 0 and α = 1 in the same equation, it is essentially the same structure as eq. (39) in Ref. [44] (or eq. (3.34)). (Note also the similarities of superpotentials between eqs. (14) and (17) in Ref. [273] and eq. (32) in Ref. [44] or its extension, eq. (B.2) in Ref. [45], or equivalently eq. (3.46).) Thus, the apparently ad-hoc choice of K¨ahlerpotential in the equation can be understood in terms of the dilatation symmetry of the real part [275]. In other examples of K¨ahlerpotential in Refs. [44, 45], we exploit the shift symmetry of the imaginary part. The two kinds of symmetry transformations form M¨obiusgroup [275], so these models are related in the perspective of K¨ahergeometry. For this opportunity, it is useful to answer the question the author is sometimes asked: whether our models are the low-energy effective descriptions of the double superfield mod- els [36,38,39,108] after integrating out the stabilizer superfield. We think the answer is no for the following reason. In the latter case, inflation is driven by the supersymmetry (SUSY) breaking of the stabilizer superfield and inflaton does not break SUSY during inflation. On the other hand, in our case, it is driven by the SUSY breaking of the inflaton superfield. Thus, SUSY breaking property during inflation is qualitatively different. Because of this argument, the single superfield models [44,45] are distinguished from the double superfield models. Since the mechanism of single-superfield inflation has just been discovered, there is room to be explored. For example, we will show that potentials with very rich structures and SUSY preserving vacuum can be made for a simple polynomial K¨ahlerpotential in an upcoming publication. It is our pleasure that the readers of the thesis further develop this exciting field. Some parts of the preface and revision are motivated or benefited from useful discussions with I. Ben-Dayan, E. Dudas, S. Shirai, F. Takahashi, and A. Westphal. Acknowledgement is attached to the end of the thesis. iii Preface This is a Ph.D. thesis on cosmological inflation in supergravity theory submitted to the University of Tokyo. The thesis is based on the author's works [e, f] in collaboration with S. V. Ketov. It is newly written, and presentations and explanations have been reorganized and integrated. There are some new contents beyond Refs. [e, f] such as Sections 3.4 and 3.6. List of the author's publication in Ph.D. course [a] Motoi Endo, Koichi Hamaguchi, and Takahiro Terada, \Scalar decay into gravitinos in the presence of D-term supersymmetry breaking", Phys. Rev. D 86, 083543 (2012), arXiv:1208.4432 [hep-ph]. [b] Sergei V. Ketov and Takahiro Terada, \New Actions for Modified Gravity and Super- gravity", JHEP 1307 (2013) 127, arXiv:1304.4319 [hep-th]. [c] Sergei V. Ketov and Takahiro Terada, \Old-minimal supergravity models of inflation", JHEP 1312 (2013) 040, arXiv:1309.7494 [hep-th]. [d] Koichi Hamaguchi, Takeo Moroi, and Takahiro Terada, “Complexified Starobinsky In- flation in Supergravity in the Light of Recent BICEP2 Result", Phys. Lett. B 733 (2014), 305, arXiv:1403.7521 [hep-ph]. [e] Sergei V. Ketov and Takahiro Terada, “Inflation in Supergravity with a Single Chiral Superfield”, Phys. Lett. B 736 (2014) 272, arXiv:1406.0252 [hep-th]. [f] Sergei V. Ketov and Takahiro Terada, \Generic Scalar Potentials for Inflation in Su- pergravity with a Single Chiral Superfield”, JHEP 1412 (2014) 062, arXiv:1408.6524 [hep-th]. [g] Takahiro Terada, Yuki Watanabe, Yusuke Yamada, and Jun'ichi Yokoyama, \Reheating processes after Starobinsky inflation in old-minimal supergravity", JHEP 1502 (2015) 105, arXiv:1411.6746 [hep-ph]. Contents 1 Introduction1 2 Review of Inflation in Supergravity6 2.1 Difficulties of inflation in supergravity . .6 2.1.1 Approaches to evade the problem . .7 2.2 Small field inflation with a single chiral superfield . 10 2.2.1 SGoldstino inflation . 10 2.2.2 Model with discrete R-symmetry . 12 2.2.3 Inflection point inflation . 14 2.3 Shift symmetry and a stabilizer superfield . 15 2.3.1 Shift symmetry . 15 2.3.2 Stabilizer superfield . 17 2.3.3 Nilpotent stabilizer superfield . 21 2.3.4 Example: Higgs inflation in (SM, MSSM, and) NMSSM . 25 2.4 Inflation with a linear or real superfield . 27 2.4.1 Model with a massive vector multiplet . 28 2.4.2 Web of dualities . 30 2.5 Purely supergravitational theories . 33 2.5.1 Starobinsky model and its SUSY extension . 33 2.5.2 Inflation driven by gravitino condensation . 38 3 Inflation in Supergravity with a Single Chiral Superfield 41 3.1 Basic strategy and implementations . 41 3.1.1 Minimal K¨ahlerpotential . 42 3.1.2 Logarithmic K¨ahlerpotential . 45 3.2 Embedding arbitrary scalar potentials . 51 3.2.1 Examples of the (super)potential . 55 3.3 Effects of the finite stabilization term . 56 3.3.1 Robustness of SUSY preservation . 66 3.3.2 Comments on small field inflation . 67 3.3.3 Comments on the unitarity bound . 68 3.4 Effects of other terms in the K¨ahlerpotential . 69 3.4.1 Minimal K¨ahlerpotential . 70 3.4.2 Logarithmic K¨ahlerpotential . 71 3.5 Matter coupling and inflaton decay . 74 3.6 Generalization to charged superfields: MSSM Higgs inflation . 80 4 Conclusion 92 A Bird's-eye Review of Supergravity 96 B Short Review of Inflation 100 iv Chapter 1 Introduction \What are the fundamental laws of Nature?" \What happened at the beginning of the universe?" These deep and fundamental questions have been fascinating people since the ancient time. Scientifically, these questions are addressed in particle physics and cosmol- ogy.