Advances in High Energy Physics

Volume 2015 (2015), Article ID 397410, 12 pages

http://dx.doi.org/10.1155/2015/397410

## Helical Phase Inflation and Monodromy in Supergravity Theory

^{1}State Key Laboratory of Theoretical Physics and Kavli Institute for Theoretical Physics China (KITPC), Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China^{2}School of Physical Electronics, University of Electronic Science and Technology of China, Chengdu 610054, China^{3}George P. and Cynthia W. Mitchell Institute for Fundamental Physics and Astronomy, Texas A&M University, College Station, TX 77843, USA^{4}Astroparticle Physics Group, Houston Advanced Research Center (HARC), Mitchell Campus, The Woodlands, TX 77381, USA^{5}Division of Natural Sciences, Academy of Athens, 28 Panepistimiou Avenue, 10679 Athens, Greece

Received 3 July 2015; Accepted 17 November 2015

Academic Editor: Ignatios Antoniadis

Copyright © 2015 Tianjun Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

We study helical phase inflation which realizes “monodromy inflation” in supergravity theory. In the model, inflation is driven by the phase component of a complex field whose potential possesses helicoid structure. We construct phase monodromy based on explicitly breaking global symmetry in the superpotential. By integrating out heavy fields, the phase monodromy from single complex scalar field is realized and the model fulfills natural inflation. The phase-axion alignment is achieved from explicitly symmetry breaking and gives super-Planckian phase decay constant. The -term scalar potential provides strong field stabilization for all the scalars except inflaton, which is protected by the approximate global symmetry. Besides, we show that helical phase inflation can be naturally realized in no-scale supergravity with symmetry since the supergravity setup needed for phase monodromy is automatically provided in the no-scale Kähler potential. We also demonstrate that helical phase inflation can be reduced to another well-known supergravity inflation model with shift symmetry. Helical phase inflation is free from the UV-sensitivity problem although there is super-Planckian field excursion, and it suggests that inflation can be effectively studied based on supersymmetric field theory while a UV-completed framework is not prerequisite.

#### 1. Introduction

Inflation plays a crucial role in the early stage of our universe [1–3], and supersymmetry was found to be necessary for inflation soon after its discovery. A simple argument is that the inflation process is triggered close to the unification scale in Grand Unified Theory (GUT) [4, 5]. At this scale physics theory is widely believed to be supersymmetric. To realize the slow-roll inflation, it requires strict flat conditions on the potential of inflaton . The mass of inflaton should be significantly smaller than the inflation energy scale due to the slow-roll parameter where is the reduced Planck mass; otherwise, inflation cannot be triggered or last for a sufficient long period. However, as a scalar field, the inflaton is expected to obtain large quantum loop-corrections on the potential which can break the slow-roll conditions unless there is extremely fine tuning. Supersymmetry is a natural way to eliminate such quantum corrections. By introducing supersymmetry, the flatness problem can be partially relaxed but not completely solved since supersymmetry is broken during inflation. Moreover, gravity plays an important role in inflation, so it is natural to study inflation within supergravity theory.

Once combining the supersymmetry and gravity theory together, the flatness problem reappears known as problem. supergravity in four-dimensional space-time is determined by three functions: Kähler potential , superpotential , and gauge kinetic function. The -term scalar potential contains an exponential factor . In the minimal supergravity with , the exponential factor introduces a term on the inflaton mass at Hubble scale, which breaks the slow-roll condition (1). To realize inflation in supergravity, the large contribution from to scalar mass should be suppressed, which needs a symmetry in . In the minimal supergravity, problem can be solved by introducing shift symmetry in the Kähler potential as proposed by Kawasaki, Yamaguchi, and Yanagida (KYY) [6]: is invariant under the shift . Consequently, is independent of Im, so is the factor in the -term potential. By employing Im as inflaton, its mass is not affected by and then there is no problem anymore. The shift symmetry can be slightly broken; in this case, there is still no problem and the model gives a broad range of tensor-to-scalar ratios [7, 8]. problem is automatically solved in no-scale supergravity because of symmetry in the Kähler potential. Historically, the no-scale supergravity was proposed to get vanishing cosmology constant. At classical level, the potential is strictly flat guaranteed by symmetry of the Kähler potential, which meanwhile protects the no-scale type inflation away from problem. Moreover, symmetry has rich structure that allows different types of inflation. Thus, inflation based on no-scale supergravity has been extensively studied [9–19]. In this work, we will show that, in no-scale supergravity with symmetry, one can pick up subsector, together with the superpotential phase monodromy to realize helical phase inflation.

Recently, it was shown that problem can be naturally solved in helical phase inflation [20]. This solution employs a global symmetry, which is a trivial fact in the minimal supergravity with . Using the phase of a complex field as an inflaton, problem is solved due to the global symmetry. The norm of needs to be stabilized; otherwise, it will generate notable isocurvature perturbations that contradict the observations. However, it is a nontrivial task to stabilize the norm of while keeping the phase light as the norm and phase couple with each other. In that work, the field stabilization and quadratic inflation are realized via a helicoid type potential. The inflationary trajectory is a helix line, and this is the reason for the name “helical phase inflation.” In addition, the superpotential of helical phase inflation realizes monodromy in supersymmetric field theory. Furthermore, helical phase inflation gives a method to avoid the dangerous quantum gravity effect on inflation.

The single field slow-roll inflation agrees with recent observations [4, 5]. Such kind of inflation admits a relationship between the inflaton field excursion and the tensor-to-scalar ratio, which is known as the Lyth bound [21]. It suggests that, to get large tensor-to-scalar ratio, the field excursion during inflation should be much larger than the Planck mass. The super-Planckian field excursion challenges the validity, in the Wilsonian sense, of inflationary models described by effective field theory. At Planck scale, the quantum gravity effect is likely to introduce extra terms which are suppressed by the Planck mass and then irrelevant in the low energy scale, while for a super-Planckian field, the irrelevant terms become important and may introduce significant corrections or even destroy the inflation process. In this sense, the predictions just based on the effective field theory are not trustable. A more detailed discussion on the ultraviolet (UV) sensitivity of the inflation process is provided in [22].

A lot of works have been proposed to realize inflation based on the UV-completed theory, for example, in [23–30]. However, to realize inflation in string theory, it needs to address several difficult problems such as moduli stabilization, Minkowski or de Sitter vacuum, and - and higher string loop-corrections on the Kähler potential. However, one may doubt whether such difficult UV-completed framework is necessary for inflation. In certain scenario, the super-Planckian field excursion does not necessarily lead to the physical field above the Planck scale. A simple example is the phase of a complex field. The phase factor, like a pseudo-Nambu-Goldstone boson (PNGB), can be shifted to any value without any effect on the energy scale. By employing the phase as an inflaton, the super-Planckian field excursion is not problematic at all as there is no polynomial higher order quantum gravity correction for the phase component. Besides helical phase inflation, inflationary models using PNGB as an inflaton have been studied [31–39]. For natural inflation, it requires super-Planckian axion decay constant, which can be obtained by aligned axions [34] (the axion alignment relates to symmetry among Kähler moduli [40]) or anomalous gauge symmetry with large condensation gauge group [41]. In helical phase inflation, as will be shown later, the phase monodromy in superpotential can be easily modified to generate natural inflation and also realize the super-Planckian phase decay constant, which is from the phase-axion alignment hidden in the process of integrating out heavy fields. Furthermore, all the extra fields are consistently stabilized based on the helicoid potential.

Like helical phase inflation, “monodromy inflation” was proposed to solve the UV sensitivity of large field inflation [42, 43]. In such model, the inflaton is identified as an axion obtained from -form field after string compactifications. The inflaton potential arises from the DBI action of branes or coupling between axion and fluxes. During inflation, the axion rotates along internal cycles and reduces the axion potential slowly, while all the other physical parameters are unaffected by the axion rotation. Interesting realization of monodromy inflation is the axion alignment [34], which was proposed to get super-Planckian axion decay constant for natural inflation, and it was noticed that this mechanism actually provides an axion monodromy in [44–46]. Actually, a similar name “helical inflation” was firstly introduced in [45] for an inflation model with axion monodromy. However, a major difference should be noted; the “helical” structure in [45] is to describe the alignment structure of two axions, while the “helical” structure in our model is from a single complex field with stabilized field norm. The physical picture of axion monodromy is analogical to the superpotential in helical phase inflation. For , there is monodromy around the singularity : The phase monodromy, together with symmetry in the Kähler potential, provides flat direction for inflation. In the following, we will show that this monodromy is corresponding to the global symmetry explicitly broken by the inflation term.

In this work, we will study helical phase inflation from several aspects in detail. Firstly, we will show that the phase monodromy in the superpotential, which leads to the helicoid structure of inflaton potential, can be effectively generated by integrating out heavy fields in supersymmetric field theory. Besides quadratic inflation, the phase monodromy for helical phase inflation can be easily modified to realize natural inflation, in which the process of integrating out heavy fields fulfills the phase-axion alignment indirectly and leads to super-Planckian phase decay constant with consistent field stabilization as well. We also show that helical phase inflation can be reduced to the KYY inflation by field redefinition; however, there is no such field transformation that can map the KYY model back to helical phase inflation. Furthermore, we show that the no-scale supergravity with symmetry provides a natural frame for helical phase inflation, as symmetry of no-scale Kähler potential already combines the symmetry factors needed for phase monodromy. Moreover, we argue that helical phase inflation is free from the UV-sensitivity problem.

This paper is organized as follows. In Section 2, we review the minimal supergravity construction of helical phase inflation. In Section 3, we present the realization of phase monodromy based on supersymmetric field theory. In Section 4, natural inflation as a special type of helical phase inflation is studied. In Section 5, the relationship between helical phase inflation and the KYY model is discussed. In Section 6, we study helical phase inflation in no-scale supergravity with symmetry. In Section 7, we discuss how helical phase inflation dodges the UV-sensitivity problem of large field inflation. Conclusion is given in Section 8.

#### 2. Helical Phase Inflation

In four dimensions, supergravity is determined by the Kähler potential , superpotential , and gauge kinetic function. The -term scalar potential is given by To realize inflation in supergravity, the factor in the above formula is an obstacle as it makes the potential too steep for a sufficient long slow-roll process. This is the well-known problem. To solve problem, usually one needs a symmetry in the Kähler potential. In the minimal supergravity, there is a global symmetry in the Kähler potential . This global symmetry is employed in helical phase inflation. As the Kähler potential is independent of the phase , the potential of phase is not affected by the exponential factor . Consequently, there is no problem for phase inflation. However, the field stabilization becomes more subtle. All the extra fields except inflaton have to be stabilized for single field inflation, but normally the phase and norm of a complex field couple with each other and then it is very difficult to stabilize norm while keeping phase light.

The physical picture of helical phase inflation is that the phase evolves along a flat circular path with constant, or almost constant, radius—the field magnitude, and the potential decreases slowly. So even before writing down the explicit supergravity formula, one can deduce that phase inflation, if realizable, should be particular realization of complex phase monodromy, and there exists a singularity in the superpotential that generates the phase monodromy. Such singularity further indicates that the model is described by an effective theory.

Helical phase inflation is realized in the minimal supergravity with the Kähler potentialand superpotentialThe global symmetry in is broken by the superpotential with a small factor ; when , symmetry is restored. Therefore, the superpotential with small coefficient is technically natural [47], which makes the model technically stable against radiative corrections. As discussed before, the superpotential is singular at and exhibits a phase monodromyThe theory is well defined only for away from the singularity.

During inflation, the field is stabilized at , and the scalar potential is simplified aswhere , and the kinetic term is . Interestingly, in the potential (7), both the norm-dependent factor and reach the minimum at . The physical mass of norm is therefore, the norm is strongly stabilized at during inflation and the Lagrangian for the inflaton is which gives quadratic inflation driven by the phase of complex field .

In the above simple example given by (4) and (5), the field stabilization is obtained from the combination of supergravity correction and the pole in , besides an accidental agreement that both the factor and the term obtain their minima at , while for more general helical phase inflation, such accidental agreement is not guaranteed. For example, one may get the following inflaton potential: in which the coefficient admits a minimum at but . In this case, the coefficient still gives a mass above the Hubble scale for , while is slightly shifted away from in the early stage of inflation, and after inflation evolves to rapidly. Also, the term gives a small correction to the potential and inflationary observables, so this correction is ignorable comparing with the contributions from the super-Planckian valued phase unless it is unexpectedly large.

*Potential Deformations*. In the Kähler potential, there are corrections from the quantum loop effect, while the superpotential is nonrenormalized. Besides, when coupled with heavy fields, the Kähler potential of receives corrections through integrating out the heavy fields. Nevertheless, because of the global symmetry in the Kähler potential, these corrections can only affect the field stabilization, while phase inflation is not sensitive to these corrections.

Given a higher order correction on the Kähler potential one may introduce an extra parameter in the superpotential

Based on the same argument, it is easy to see the scalar potential reduces to The factor reaches its minimum at , below for . To get the “accidental agreement” it needs the parameter , and then inflation is still driven by the phase with exact quadratic potential.

Without , the superpotential comes back to (5) and the scalar potential is shown in Figure 1 with . During inflation for small , the term contribution to the potential, at the lowest order, is proportional to . After canonical field normalization, the inflaton potential takes the form in which the higher order terms proportional to are ignored. With regard to the inflationary observations, take the tensor-to-scalar ratio** r**, for example, as where is the phase when inflation starts and is the -folding number. So the correction from higher order term is insignificant for .