Poly-instanton axion inflation

TL;DR

This paper explores a string theory-based axion inflation model derived from poly-instanton effects, demonstrating its potential to produce a flat inflationary potential suitable for cosmic inflation and analyzing its cosmological predictions.

Contribution

It introduces a novel axion inflation model based on poly-instanton effects in type II superstring theories, with detailed numerical analysis of inflationary dynamics and observables.

Findings

Potential can support slow-roll inflation due to exponential suppression

Numerical results show compatibility with observed spectral index

Model predicts specific values for cosmological parameters

Abstract

We investigate the axion inflation model derived by poly-instanton effects in type II superstring theories. Poly-instanton effects are instanton effects corrected by another instanton and it can generate the modulus-axion potential with the double exponential function. Although the axion has a period of small value, this potential can have a flat region because its derivatives are exponentially suppressed by non-perturbative effects. From the view point of the cosmic inflation, such potential is interesting. In this paper, we numerically study the possibilities for realizing the cosmic inflation. We also study their spectral index and other cosmological observables, numerically.

Tables

Table 1. Table 1: Examples of parameters and observables.

$\delta$	$\theta$	$A$	$n_s$	$r$	$N_e$	${\alpha }_s$	$m_{\varphi }^2$
8.0	4.13	$3.29\times {10}^{-16}$	$0.953$	$6.02\times {10}^{-7}$	$61.9$	$-3.66\times {10}^{-4}$	$6.70\times {10}^{-11}$
9.0	3.16	$3.46\times {10}^{-16}$	$0.959$	$9.30\times {10}^{-7}$	$58.6$	$-4.66\times {10}^{-4}$	$1.27\times {10}^{-10}$
10.0	2.11	$2.18\times {10}^{-16}$	$0.963$	$7.65\times {10}^{-7}$	$56.8$	$-4.70\times {10}^{-4}$	$1.26\times {10}^{-10}$
11.0	7.25	$1.08\times {10}^{-16}$	$0.966$	$4.24\times {10}^{-7}$	$61.7$	$-3.98\times {10}^{-4}$	$8.31\times {10}^{-11}$
12.0	6.28	$8.89\times {10}^{-17}$	$0.966$	$4.89\times {10}^{-7}$	$55.7$	$-4.61\times {10}^{-4}$	$1.12\times {10}^{-10}$

Equations

$$W=W_p+W_{np}.$$

$$W_{np}=\sum _i{A}_i(U_m)e^{-a_i{\mathcal{S}}_i}.$$

$$T_i={\tau }_i+i{\varphi }_i,$$

$$W_{np}=A_a{e}^{-24{\pi }^2{f}_a/b_a},$$

$$f_a=f_{tree}+f_{1-loop}+f_{np}.$$

$$f_{np}=\sum _j{B}_j(U_m)e^{-b_j{T}_j},$$

$$W_{np}=\sum _a{A}_a\exp [-b_a({\alpha }_{a,i}{T}_i+\sum _j{B}_{a,j}{e}^{-b_{j,k}{T}_k})],$$

$$V_F=e^K[K^{T_i{\bar{T}}_j}{D}_{T_i}W{D}_{\overline{T_j}}\bar{W}-3|W{|}^2],$$

$$W=W(T_i){|}_{i\ne 0}+A_0\exp [-a(T_i+C{e}^{-a_0{T}_0})]{|}_{i\ne 0}.$$

$$W=W_0^{\prime }+A_0^{\prime }\exp [-aC{e}^{-a_0{T}_0}],$$

$$-3e^K|W{|}^2=-3e^K(|W_0^{\prime }{|}^2+{\bar{W}}_0^{\prime }{A}_0^{\prime }\exp [-aC{e}^{-a_0{T}_0}]+h.c.)+\mathcal{O}(A_0^{\prime 2}),$$

$$\begin{array}{rl}{e}^K{K}^{T_0{\bar{T}}_0}|D_{T_0}W{|}^2\approx & e^K{K}^{T_0{\bar{T}}_0}\{K_{T_0}^2(|W_0^{\prime }{|}^2+{\bar{W}}_0^{\prime }{A}_0^{\prime }\exp [-aC{e}^{-a_0{T}_0}]+h.c.)\\ & +K_{T_0}{\bar{W}}_0^{\prime }{A}_0^{\prime }{a}_{0}aC{e}^{-a_0{T}_0}\exp [-aC{e}^{-a_0{T}_0}]+h.c.\}+\mathcal{O}(A_0^{\prime 2}).\end{array}$$

$$\begin{array}{rl}V\approx & 2e^K|{\bar{W}}_0^{\prime }{A}_0^{\prime }|\exp [-\delta \cos (a_0{\varphi }_0)]\{(K^{T_0{\bar{T}}_0}{K}_{T_0}^2-3)\cos [\delta \sin (a_0{\varphi }_0)+\theta ]\\ & +K^{T_0{\bar{T}}_0}{K}_{T_0}{a}_0\delta \cos [-a_0{\varphi }_0+\delta \sin (a_0{\varphi }_0)+\theta ]\}+(\mathrm{const}),\end{array}$$

$$V=A(e^{-\delta \cos a\varphi }\cos [\delta \sin a\varphi -a\varphi +\theta ]+V_0).$$

$$V$$

$$\varepsilon\equiv\frac{1}{2}\Bigl{(}\frac{V_{\phi}}{V}\Bigr{)}^{2},\qquad\eta\equiv\frac{V_{\phi\phi}}{V},\qquad\xi\equiv\frac{V_{\phi}\cdot V_{\phi\phi\phi}}{V^{2}}.$$

$$N_e({\varphi }_{\ast })={\int }_{{\varphi }_{end}}^{{\varphi }_{\ast }}d\varphi \frac{V}{V_{\varphi }},$$

$$P_{\xi }=2.20\pm 0.10\times 10^{-9},n_s=0.9655\pm 0.0062,r<\mathrm{0.12.}$$

$$P_{\xi }=\frac{V}{24{\pi }^2\epsilon },n_s=1+2\eta -6\epsilon ,r=16\epsilon .$$

$${\alpha }_s\equiv \frac{d{n}_s}{dlnk}\simeq -24{\epsilon }^2+16\epsilon \eta -2\xi .$$

$$r\sim 10^{-2}(\Delta \varphi {)}^2,$$

$$V_{\varphi }=A\delta a{e}^{-\delta \cos a\varphi }\left[-\sin (\delta \sin (a\varphi )-2a\varphi +\theta )+\frac{1}{\delta }\sin (\delta \sin (a\varphi )-a\varphi +\theta )\right].$$

$$V_{\varphi }\sim A\delta a{e}^{-\delta \cos a\varphi }(-1)\sin (\delta \sin (a\varphi )-2a\varphi +\theta ).$$

$$V_{\varphi \varphi }$$

$$V_{\varphi \varphi \varphi }$$

$$|\eta |\sim \frac{|V_{\varphi \varphi \varphi }|\Delta \varphi }{V}\sim \frac{A{\delta }^3{a}^3{e}^{-\delta \cos a\varphi }|\sin (\delta \sin (a\varphi )-4a\varphi +\theta )|\Delta \varphi }{A{V}_0}\sim {\delta }^2{a}^2\sqrt{\epsilon }\Delta \varphi .$$

$$(\Delta \varphi {)}^2\sim \frac{{\eta }^2}{{\delta }^4{a}^4\epsilon }.$$

$$r\sim 10^{-2}\frac{{\eta }^2}{{\delta }^4{a}^4\epsilon }.$$

$${\epsilon }^2\sim \frac{10^{-7}}{{\delta }^4{a}^4}.$$

$$\epsilon \sim 10^{-7},r=16\epsilon \sim 10^{-6}.$$

$$V=A{V}_0=5.2\epsilon \times 10^{-7}\sim \frac{10^{-10}}{{\delta }^2{a}^2}.$$

Full text

EPHOU-17-008

KUNS-2678

MISC-2017-05

**Poly-instanton axion inflation

**

Tatsuo Kobayashi1,  Shohei Uemura2,  and  Junji Yamamoto3

1**Department of Physics, Hokkaido University, Sapporo, 060-0810 Japan

2**Maskawa Institute for Science and Culture, Kyoto Sangyo University, Kyoto 603-8555, Japan

3**Department of Physics, Kyoto University, Kyoto 606-8502, Japan

( Abstract

We investigate the axion inflation model derived by poly-instanton effects in type II superstring theories. Poly-instanton effects are instanton effects corrected by another instanton and it can generate the modulus-axion potential with the double exponential function. Although the axion has a period of small value, this potential can have a flat region because its derivatives are exponentially suppressed by non-perturbative effects. From the view point of the cosmic inflation, such potential is interesting. In this paper, we numerically study the possibilities for realizing the cosmic inflation. We also study their spectral index and other cosmological observables, numerically.

)

1 Introduction

The cosmic inflation is the important stage in the history of the Universe solving the problems of the standard Big Bang cosmology such as the flatness problem and horizon problem, and deriving the observed fluctuation of the cosmic microwave background. From the theoretical viewpoint, what is the field deriving the inflation, i.e. inflaton, is one of key questions to understand the Universe from underlying theory such as superstring theory. It should have sufficiently flat potential. Indeed, various studies have been done within the framework of superstring theory [1].

Axions as well as moduli are good candidates for the inflaton, because axions have continuous shift symmetries. Non-perturbative effects break continuous shift symmetries to discrete ones, and the potential of the axion can be generated. However, such a potential would be controlled well by discrete shift symmetries. Thus, non-perturbative effects are quite important. Non-trivial backgrounds can also break continuous shift symmetries, generating the axion potential.

The natural inflation [2] and the axion monodromy inflation [3]111 See also [4]. are of famous axion inflation models. The simple natural inflation requires a large decay constant, although the typical decay constant would be smaller than the string and Planck scales. A large decay constant can be realized effectively, e.g. by the alignment mechanism [5] and loop effects [6, 7]. The simple natural inflation may lead to disfavored predictions on observables, but modulation terms would change significantly predictions on observables [7, 8, 9]. Correction terms in monodromy inflation are also important and they change significantly predictions on observables [9, 10].

String world-sheet instaton corrections are stringy-calculable effects. The world-sheet instanton effects can be controlled by modular symmetries, which are characteristic symmetries in string theories. Then, certain modular functions appear as the potential of moduli and axion. The inflation scenario with such modular function terms has been also studied [7, 11, 12]. Moreover, the mixture of polynomial functions and sinusoidal functions was derived by quantum corrections and studied to realize inflation models [13, 14, 15].

Furthermore, one can compute couplings of moduli and axions to gauge bosons and quarks and leptons in four-dimensional (4D) low energy effective field theory. Thus, we can explicitly study the thermal history after the axion inflation. For example, we can analyze explicitly the reheating processes after the axion inflation. The axion couplings to the gauge bosons and quarks and leptons may be rather weak, and the reheating temperature may be rather low [9, 13, 15]. The axions would also appear in phases of couplings among quarks, leptons and the Higgs fields. After the inflation, the axion would oscillate, and then coupling phases would vary. Such an oscillation would have important effects on the history of the Universe. For example, the baryon asymmetry may be generated by such oscillation [16].

As mentioned above, various types of the axion inflation models have been studied and several aspects have been discussed after the inflation. It would be interesting to study various possibilities for the axion inflation, and their several aspects. Recently, a new type of non-perturbative effects was studied, that is, the so-called poly-instanton [17, 18]. That leads to the exponential function of exponential function as the moduli potential term. It would be interesting to study the possibility for the axion inflation by use such a new type of the potential. Indeed, the moduli inflation has been studied by use of such a new potential term [19].222 The moduli stabilization has been also studied [20, 21]. The purpose of this paper is to study the possibilities for the axion inflation by use the new type of the axion potential.

This paper is organized as follows. In section 2, we show which type of axion potential is generated by poly-instanton effects. In section 3, we study numerically the poly-instanton axion inflation. Section 4 is conclusion. In Appendix A, we discuss some examples, that only the axion remains light, while the other moduli are stabilized by heavy masses.

2 Poly-instanton potential

The 4D low energy effective field theory of superstring theory is described by supergravity theory, and it includes the dilaton $S$, the Kähler moduli $T_{i}$ and complex structure moduli $U_{m}$. Here, we focus on type IIB superstring theory, although our scenario would be applicable to other superstring theories. Hereafter, we use the unit that $M_{Pl}=1$, where $M_{Pl}$ denotes the reduced Planck scale, i.e. $M_{Pl}=2.4\times 10^{18}$ GeV.

The superpotential $W$ can be written by a combination of perturbative and non-perturbative terms,

[TABLE]

The gaugino condensations and D-brane instanton effects are well-known computable non-perturbative effects in type II superstring theory. The superpotential terms induced by D-brane instantons can be written as [22, 23],

[TABLE]

Here, ${\cal S}_{i}$ denotes the action of the D-brane instantons, and it can be obtained as ${\cal S}_{i}=\int_{\Gamma_{i}}dV=MS+m_{i,j}T_{j}$, where $\Gamma_{i}$ represents a p-cycle wrapped by the D-brane instanton and $m_{i,j}T_{j}$ represent the Kähler moduli corresponding to the volume of $\Gamma_{i}$. Moduli $T_{i}$ are written as,

[TABLE]

and the imaginary parts $\phi_{i}$ represent axions. In addition, $a_{i}$ is a constant parameter and its value is typically 2$\pi$ for D-brane instantons. In general, $A_{i}(U_{m})$ is a function of the complex structure moduli $U_{m}$. Here, we assume that all of the complex structure moduli $U_{m}$ and the dilaton $S$ are stabilized by three-form flux background [25, 26].

On the other hand, the gaugino condensation in the hidden sector $G_{a}$ would generate the following term:

[TABLE]

where $f_{a}$ and $b_{a}$ are gauge kinetic function and beta-function coefficient of $G_{a}$, respectively. The real part of the gauge kinetic function provides with the gauge coupling $g_{a}$, i.e. $1/g_{a}^{2}=Re(f_{a})$. Including quantum corrections, the gauge kinetic function can be written by

[TABLE]

Here, the tree-level gauge kinetic function $f_{tree}$ is a linear combination of $S$ and $T_{i}$ corresponding to the volume of the cycle, where the hidden sector $G_{a}$ wraps. The 1-loop correction $f_{1-loop}$ is a function of complex structure moduli $U_{m}$ and independent of Kähler moduli $T_{i}$. The non-perturbative correction, $f_{np}$ is given by

[TABLE]

and such a correction can be induced by D-brane instantons having one complex position moduli on the compact space. Thus, in general, we can write non-perturbative terms in the superpotential,

[TABLE]

where we have redefined coefficients including VEVs of $S$ and $U_{m}$, because we have assumed that the dilaton and complex structure moduli are stabilized. A similar non-perturbative superpotential term would appear through the D-brane instanton terms (2.2) with corrections. Since it is an instanton correction corrected by other instanton effects, it is called poly-instanton [17, 18]. It can generate potential for Kähler moduli and stabilize them [19, 20, 21].

The $F$-term scalar potential is given by,

[TABLE]

where $K$ denotes the Kähler potential and $D_{T_{i}}W=(\partial_{T_{i}}K)W+\partial_{T_{i}}W$. The Kähler potential is given by $K=-2\ln\mathcal{V}$ where $\mathcal{V}$ is the volume of the compact space, and the Kähler potential is a function of $T_{i}+\bar{T}_{i}$, i.e. $K(T_{i}+\bar{T}_{i})$. In addition, $K^{T_{i}\bar{T_{j}}}$ represents the inverse of $K_{T_{i}\bar{T_{j}}}=\partial_{T_{i}}\partial_{\bar{T_{j}}}K$.

In this paper, we focus on the axion potential generated by the poly-instanton. Suppose that there is a modulus field $T_{0}$, which appears only in poly-instanton superpotential, but dose not appear at the tree level and single instanton potential, i.e.,

[TABLE]

When $A_{0}=0$, $\phi_{0}$ remains massless, since $\phi_{0}$ does not appear in the $F$-term scalar potential. However, all the other moduli including $\tau_{0}$ appear in the potential. Here, we assume that these moduli including $\tau_{0}$ are stabilized with heavy masses. We show some models realizing this assumption in Appendix A. Their masses would be estimated by $W(\braket{T_{i}})|_{i\neq 0}$. Thus, when $A_{0}\ll W(\braket{T_{i}})|_{i\neq 0}$, the mass of $\phi_{0}$ is much smaller than those of the other moduli fields. Therefore, we integrate all the other fields out, and obtain the superpotential,

[TABLE]

where $W_{0}^{\prime}=W(\braket{T_{i}})|_{i\neq 0}$ and $A_{0}^{\prime}=A_{0}\exp[-a\braket{T_{i}}]$.

The $F$-term scalar potential derived from (2.10) has a very specific form. The second term, $-3e^{K}|W|^{2}$, is calculated as,

[TABLE]

where we have used the approximation, $|W_{0}^{\prime}|\gg|A_{0}^{\prime}|$. Similarly, we approximate

[TABLE]

Then, we obtain the $F$-term scalar potential for the axion generated by a poly-instanton,

[TABLE]

where $\delta=aC\exp[-a_{0}\braket{\tau_{0}}]$, and $\theta$ is the phase of the prefactors. We omitted the terms of ${\cal O}(A^{\prime 2}_{0})$.

In the next section, we study the cosmological inflation using the potential (2.13), which is rather complicated. We estimate that $K^{T_{0}\bar{T}_{0}},K_{T_{0}}^{2}={\cal O}(1)$. Here, we treat $\delta$ and $a_{0}$ as free parameters from the phenomenological viewpoint. When $a_{0}\delta\gg 1$, the second term in the curly bracket is dominant. Then, the potential for the canonically normalized axion $\phi=\sqrt{2K_{T_{0}\bar{T}_{0}}}\phi_{0}$ is approximated to

[TABLE]

Here, we redefined coefficients as, $a=a_{0}/\sqrt{2K_{T_{0}\bar{T_{0}}}}$, and $A=2e^{K}\bar{W}_{0}^{\prime}A^{\prime}_{0}K^{T_{0}\bar{T}_{0}}a_{0}\delta$. We also added the uplifting term $V_{0}$ to set the potential (local) minimum to be almost zero. 333 Such a uplifting term can be realized by supersymmetry breaking D-brane configuration [26] or spontaneous supersymmetry breaking, i.e., the so-called F-term uplifting [27]. To cancel the former term, $V_{0}$ is of the order of $e^{-\delta_{min}}$, where $\delta_{min}=\delta\cos a\phi_{min}$ with the value $\phi_{min}$ at the potential (local) minimum. In the next section, we will concentrate on the axion inflation with the above potential.

When $a_{0}\delta={\cal O}(1)$, both terms in the curly bracket of Eq.(2.13) are comparable with each other. In this case, it is difficult to control the potential to be flat. On the other hand, a large $\delta$ in the factor $e^{-\delta}$ is important to make the potential flat. Thus, we will not discuss the case with $\delta={\cal O}(1)$, but concentrate on the case with $\delta\gg 1$.

When $a_{0}\delta\ll 1$, the potential is approximated to

[TABLE]

where $A_{1}$ and $A_{2}$ represent the prefactors including Kähler potential and its derivatives, This potential may realize the potential for the natural inflation with modulation terms.

3 Numerical analysis

In this section, we investigate the poly-instanton potential (2.14) numerically. First, we define the slow-roll parameters,

[TABLE]

These parameters should be much smaller than 1 during the inflation. We evaluate the e-folding number:

[TABLE]

where $\phi_{end}$ is the end-point of inflation. The e-folding number is required to be more than 50 in order to solve the problems of the standard cosmology. Furthermore, the Planck results constrain the power spectrum of curvature perturbation $P_{\xi}$, its spectral index $n_{s}$, the tensor-to-scalar ratio [28, 29]

[TABLE]

These can be written by use of the slow-roll parameters as,

[TABLE]

Table 1 shows some examples of parameters and observables, which almost satisfy the above constrains. The overall coefficient $A$ is decided to impose that power spectrum of curvature perturbation $P_{\xi}$ is $2.2\times 10^{-9}$. We set $a=2\pi$ in all of the examples in the table. The table also shows the inflation mass $m_{\phi}$ and the running of the spectral index $\alpha_{s}$, which is obtained as

[TABLE]

Figures 1 and 2 show the potential forms with $\delta=8.0$ and $9.0$, which are shown in Table 1. The left panels in the figures show the form of the potential, while the right panels show the slow-roll parameters, $\varepsilon$ and $\eta$. The figures show that there is a flat region in the potential. In order to realize this flatness, the factor $e^{-\delta}$ is important as will be explained later. The shaded regions correspond the field excursions during the inflation, i.e., $\phi_{\ast}$ and $\phi_{end}$. The inflaton moves from left to right toward the potential minimum.444 Because of the potential form, there are many local minima. the inflaton reaches one of the minima, which may be the global minimum or one of local ones. At any rate, those minima are away the Planck scale.

Our inflation model corresponds to the so-called small-field inflation scenario, where the field excursion $\Delta\phi$ during the inflation is small, i.e. $\Delta\phi<1$. By using the Lyth relation [30],

[TABLE]

that implies that $r$ and $\varepsilon$ are smaller than ${\cal O}(10^{-2})$. In order to derive $n_{s}=0.9655$, we have to realize $\eta\sim-0.01$. The right panels in Figures 1 and 2 show that $\eta\sim-0.01$ and $\varepsilon$ are very much suppressed.

Now, let us understand the above numerical behavior of our inflation model by using some approximation. The first derivative of the potential is given by,

[TABLE]

Since we assume $\delta\gg 1$, it is approximated by

[TABLE]

Similar to that, the second and third derivatives can be approximated as follows,

[TABLE]

During the inflation, the potential is dominated by $AV_{0}$. On the other hand, the first and second derivatives in a specific region can be much smaller because of the factor $e^{-\delta\cos a\phi}$ when $\delta$ is sufficiently large. Thus, we can realize the flat potential without severe fine-tuning of parameters. In addition, the ratio, $(V_{\phi})^{2}/V_{\phi\phi}$ is suppressed by $e^{-\delta\cos a\phi}$. Then, it is expected that $\varepsilon$ is much smaller that $\eta$ in our model.

The inflation would occur at the very flat region of the potential, i.e. $V_{\phi\phi}\approx 0$, and we approximate $V_{\phi\phi}$ as $V_{\phi\phi}\sim V_{\phi\phi\phi}\Delta\phi$ during inflation. Then, we estimate the slow-roll parameter $\eta$,

[TABLE]

Here, the phases of sine functions in $V_{\phi}$ and $V_{\phi\phi\phi}$ are different from each other, but we estimate values of sine functions are similar. That is, we can write

[TABLE]

By using the Lyth relation (3.6), we obtain

[TABLE]

We have to realize $\eta\sim-0.01$, in order to derive $n_{s}=0.9655$, as mentioned above. Then, by use of $\eta\sim-0.01$ and $r=16\varepsilon$, the relation (3.12) leads to

[TABLE]

For example, for $\delta=10$, $a=2\pi$, we can estimate

[TABLE]

From the relation among $P_{\xi}$, $\varepsilon$ and $V\approx AV_{0}$, we find

[TABLE]

Then, the inflaton mass can be estimated as

[TABLE]

where we used $V_{0}\sim e^{-\delta_{min}}$. Note that the above inflaton mass does not depend explicitly on $\delta$ and $a$. We can etimate $m_{\phi}^{2}\sim 10^{-10}$, i.e., $m_{\phi}\sim 10^{13}$ GeV. Also, we estimate the running,

[TABLE]

which does not depend explicitly on $\delta$ and $a$. The running is negative and rather large. A similar value has been obtained in other axion inflation models [9, 14].

Finally, let us estimate the reheating temperature. The modulus field $T_{0}$ would appear in the gauge kinetic function of the visible sector through the instanton effect,

[TABLE]

Through this coupling, the inflaton-axion $\phi$ can decay into gauge bosons, and its decay rate can be estimated by

[TABLE]

where $\gamma=B_{vis}b_{vis}e^{-b_{vis}\tau_{0}}$. When this process is dominant in the decay of $\phi$, the reheating temperature could be estimated as

[TABLE]

where the effecitve degrees of freedom $g_{\ast}=106.75$. Depending on $\gamma$, the reheating temperature becomes lower.

4 Conclusion

We have investigated the axion potential generated by poly-instanton effects. In superstring theory, there are $h_{1,1}$ Kähler moduli and they remain massless at tree level. The $\alpha^{\prime}$ correction and 1-loop correction in the Kähler potential can generate masses for the Kähler moduli, but the axions would keep flat directions. The axions can be stabilized by non-perturbative effects and their masses could be lighter than other modulus. Thus, the natural inflation with and without modulations would be a good candidate for string derived inflation models. Recently, another type of non-perturbative effects, i.e. the so-called poly-instanton effects were studied. Thus, it is interesting to study the axion potential due to the poly-instanton effects and the possibility toward the cosmic inflation.

We have derived a new axion inflation model generated by poly-instanton effects. The form of the poly-instanton potential is specific. Since the derivatives of the potential are exponentially suppressed by non-perturbative effects, the potential can have a flat region without severe fine-tuning of parameters. We have shown that the axion inflation is possible with our new potential, and that corresponds to the small-field inflation scenario. We numerically analyze its potential and find some parameter sets realizing the inflation. The magnitude of the poly-instanton effects is represented by the parameter $\delta$ and the large value of the $\delta$ is favored to realize the inflation. Most of the observables depend on the parameters, and we have shown their dependencies on $\delta a$. For example, the tensor-to-scalar ratio is proportional to $\delta^{-4}a^{-4}$, and it becomes smaller as $\delta a$ becomes larger. On the other hand, the inflaton mass is independent of $\delta$ and $a$ and is of ${\cal O}(10^{13})$ GeV. Also, the running $\alpha_{s}$ is independent of $\delta$ and $a$. Its value is negative and rather large, $\alpha_{s}\sim-10^{-3}$. A similar value has been obtained in other axion inflation models [9, 14]. This behavior might be a generic feature in a certain class of the axion inflation models.

In this paper, we have assumed the existence of poly-instanton, but more precisely, it depends on the geometry of the compact space. For more concrete discussion, one should define the geometry giving rise to poly-instantons. In addition, the value of $\delta$ is significant for the poly-axion inflation, although we treat it as a free parameter. Moreover, the poly-instanton effects in the gauge kinetic function of the visible sector would be important, because it affects the decay of the axion and the reheating processes. In principle, the prefactor can be calculated as 1-loop partition function. It may be interesting to investigate it explicitly in concrete models. That is beyond our scope, and we would study it elsewhere.

Acknowledgments

T. K. is supported in part by the Grant-in-Aid for Scientific Research No. 26247042 and No. 17H05395 from the Ministry of Education, Culture, Sports, Science and Technology in Japan.

Appendix A Moduli stabilization

We give an example stabilizing the real part of a single Kähler modulus as well as the other moduli, but remaining its axionic part massless. Suppose that there are three Kähler moduli fields, $T_{1,2,3}$ as well as complex structure moduli and the dilaton. We assume that all of the complex structure moduli and the dilation are stabilized by three-form flux. We introduce the Kähler potential and the superpotential as follows,

[TABLE]

The $F$-term conditions are given as

[TABLE]

[TABLE]

[TABLE]

Eqs. (A.3) and (A.4) determine the ratio of $\tau_{2}$ and $\tau_{3}$,

[TABLE]

Eliminating $e^{a_{1}T_{1}}$ in (A.2) and (A.3), we obtain

[TABLE]

Similarly, eliminating $e^{-a^{\prime}T^{\prime}}$, we obtain the condition for $T_{1}$,

[TABLE]

When $a^{\prime},a_{1}\gg 1$ and $K_{T_{i}}$ are of the order of 1, the approximate solution is given by,

[TABLE]

[TABLE]

Therefore, the vacuum expectation values of $T_{1}$ and $T^{\prime}$ including their axionic modes are determined. Their values can be large enough when $W_{0}$ is enough small. Since the ratio of the real parts of $T_{2}$ and $T_{3}$ is determined by (A.5), there is only one massless direction, which is the imaginary part of the Kähler moduli which is perpendicular to $T^{\prime}$ and $T_{1}$. The masses of heavy modes are estimated by $W_{0}$. When $a^{\prime}$ and $a_{1}$ is not so large, we can also tune $W_{0}$ such that $T_{1}$ and $T^{\prime}$ have legitimate values. Instead of the above superpotnetial, we could use the following superpotential:

[TABLE]

for our purpose.

Another possibility is suggested in [31]. In this model, the real part of Kähler moduli is stabilized by 1-loop and $\alpha^{\prime}$ corrections, but its imaginary part remains massless. Similar to the above model, non-perturbative effects can stabilize its imaginary part.

More or less, we realize the models having one massless axion. Adding to single instanton, the axion potential can be corrected by poly-instanton effects. We can introduce relatively suppressed poly-instanton superpotential, $\Delta W=A\exp[b_{j}T_{j}+Ce^{\sum a_{i}T_{i}}]$. Then, the massless axion obtain a relatively light mass and the poly-instanton potential (2.13) would appear.