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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

Inflation plays a crucial role in the early stage of our universe [

Once combining the supersymmetry and gravity theory together, the flatness problem reappears known as

Recently, it was shown that

The single field slow-roll inflation agrees with recent observations [

A lot of works have been proposed to realize inflation based on the UV-completed theory, for example, in [

Like helical phase inflation, “monodromy inflation” was proposed to solve the UV sensitivity of large field inflation [

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

This paper is organized as follows. In Section

In four dimensions,

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 potential

During inflation, the field

In the above simple example given by (

Given a higher order correction on the Kähler potential

Based on the same argument, it is easy to see the scalar potential reduces to

Without

The helicoid structure of potential (

As discussed before, phase inflation naturally leads to the phase monodromy (in mathematical sense) in the superpotential. The phase monodromy requires singularity, which means the superpotential proposed for phase inflation should be an effective theory. It is preferred to show how such phase monodromy appears from a more “fundamental” theory at higher scale. In [

Historically, the monodromy inflation as an attractive method to realize super-Planckian field excursion was first proposed, in a more physical sense, for axions arising from string compactifications [

The more “fundamental” field theory for the superpotential in (

As shown in [

To integrate out heavy fields, we need to consider the

Based on the above construction, it is clear that the phase monodromy in

As to the inflation term, a question appears: as global

When integrating out the heavy fields, they should be replaced both in superpotential and in Kähler potential by the solutions from vanishing

In the superpotential

To realize natural inflation, the superpotential

The supersymmetric field theory given by (

Near vacuum, fields

Helical phase inflation is described by the effective superpotential

The

After integrating out the heavy fields, the Kähler potential is

The helicoid structure of potential (

It is known that, to realize aligned axion mechanism in supergravity [

Here, we will show that although the physical pictures are much different in helical phase inflation and KYY type model, just considering the lower order terms in the Kähler potential of redefined complex field, helical phase inflation can reduce to the KYY model.

Because the phase of

Nevertheless, helical phase inflation is not equivalent to the KYY model. There are higher order corrections to the Kähler potential in the map from helical phase inflation to KYY model, which have no effect on inflation after field stabilization but indicate different physics in two models. By dropping these terms, certain information is lost so the map is irreversible. Specifically, the inverse function

The no-scale supergravity is an attractive frame for GUT scale phenomenology; it is interesting to realize helical phase inflation in no-scale supergravity. Generally, the Kähler manifold of the no-scale supergravity is equipped with

The Kähler potential with

The helicoid structure of potential (

The physical mass of

Instead of stabilizing

The Kähler potentials of superfields

For the slow-roll inflation, the Lyth bound [

In consideration of the UV sensitivity of large field inflation, a possible choice is to realize inflation in UV-completed theory, like string theory (for a review, see [

The UV-completion problem is dodged in helical phase inflation. Since the super-Planckian field excursion is the phase of a complex field and the phase component is not directly involved in the gravity interaction, there are no dangerous high-order corrections like in (

Helical phase inflation is free from the UV-sensitivity problem, and it is just a typical physical process at the GUT scale with special superpotential that admits phase monodromy. So it provides an inflationary model that can be reliably studied just in supersymmetric field theory.

In this work, we have studied the details of helical phase inflation from several aspects. Helical phase inflation is realized in supergravity with global

An amazing fact of helical phase inflation is that it deeply relates to several interesting points of inflation and naturally combines them in a rather simple potential with helicoid structure. The features of helical phase inflation can be summarized as follows:

The global

The phase excursion requires phase monodromy in the superpotential. So helical phase inflation provides, in the mathematical sense, a new type of monodromy in supersymmetric field theory.

The singularity in the superpotential, together with the supergravity scalar potential, provides strong field stabilization which is consistent with phase inflation.

The super-Planckian field excursion is realized by the phase of a complex field instead of any other “physical” fields that directly couple with gravity. So there are no polynomial higher order corrections for the phase and thus inflation is not sensitive to the quantum gravity corrections.

To summarize, helical phase inflation introduces a new type of inflation that can be effectively described by supersymmetric field theory at the GUT scale. Generically, the super-Planckian field excursion makes the inflationary predictions based on effective field theory questionable, since the higher order corrections from quantum gravity are likely to affect the inflation process significantly. One of the solutions is to realize inflation in a UV-completed theory, like string theory; nevertheless, there are many difficult issues in string theory to resolve before realizing inflation completely. Helical phase inflation is another simple solution to the UV-sensitivity problem. It is based on the supersymmetric field theory and the physics are clear and much easier to control. Furthermore, helical phase inflation makes the unification of inflation theory with GUT more natural, since both of them are triggered at the scale of

Besides, we have shown that helical phase inflation also relates to several interesting developments in inflation theory. It can be easily modified for natural inflation and realize the phase-axion alignment indirectly, which is similar to the axion-axion alignment mechanism for super-Planckian axion decay constant [

Our inflation models are constructed within the supergravity theory with global

The authors declare that there is no conflict of interests regarding the publication of this paper.

The work of DVN was supported in part by the DOE Grant DE-FG03-95-ER-40917. The work of Tianjun Li is supported in part by the Natural Science Foundation of China under Grants nos. 10821504, 11075194, 11135003, 11275246, and 11475238 and by the National Basic Research Program of China (973 Program) under Grant no. 2010CB833000.

^{2}supergravity with non-minimal superpotentials

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