Aligned Natural Inflation and Moduli Stabilization from Anomalous $U(1)$ Gauge Symmetries

To obtain natural inflation with large tensor-to-scalar ratio in string framework, we need a special moduli stabilization mechanism which can separate the masses of real and imaginary components of K\"ahler moduli at different scales, and achieve a trans-Planckian axion decay constant from sub-Planckian axion decay constants. In this work, we stabilize the matter fields by F-terms and the real components of K\"ahler moduli by D-terms of two anomalous $U(1)_X\times U(1)_A$ symmetries strongly at high scales, while the corresponding axions remain light due to their independence on the Fayet-Iliopoulos (FI) term in moduli stabilization. The racetrack-type axion superpotential is obtained from gaugino condensations of the hidden gauge symmetries $SU(n)\times SU(m)$ with massive matter fields in the bi-fundamental respresentations. The axion alignment via Kim-Nilles-Pelroso (KNP) mechanism corresponds to an approximate $S_2$ exchange symmetry of two K\"ahler moduli in our model, and a slightly $S_2$ symmetry breaking leads to the natural inflation with super-Planckian decay constant.


I. INTRODUCTION
Natural inflation was proposed to explain the unnatural flatness of inflationary potential which is introduced ad hoc at tree level and remains flat under radiative corrections [1]. The flatness of inflationary potential is protected by continuous shift symmetry of an axionic field φ. But this symmetry is spontaneously broken to a discrete shift symmetry at inflation scale, and the following inflationary potential is generated which is invariant under the discrete shift symmetry φ → φ + 2πf with f as an axion decay constant. Recent observation of the B-mode polarization by the BICEP2 Collaboration suggests a large tensor-to-scalar ratio r = 0.16 +0.06 −0.05 excluding the dust effects [2], which can be obtained from natural inflation with trans-Planckian axion decay constant f ∼ O(10) M Pl , where M Pl = 2.4 × 10 18 GeV is the reduced Planck mass [3]. While the value of r could be much smaller [4,5] if the dust polarization effect plays more important role than estimated in Ref. [2]. Besides, natural inflation with large r also agrees with the Planck observations [6], in which a lower bound of trans-Planckian axion decay constant is needed f 5M Pl .
It is an attractive destination to realize natural inflation in string theory or its efffective no-scale supergravity (SUGRA) theory. Axions arise from anti-symmetric tensor fields in string theory through compactification of the extra space dimensions, and may play important roles in cosmology and particle physics [7]. A more fundamental reason for stringy inflation is that, according to the Lyth bound obeyed by general single field slow-rolling process [8], large tensor-to-scalar ratio requires trans-Planckian excursion during inflation. Therefore, corrections from the Planck-suppressed operators are non-ignorable, and a reliable inflation theory has to be constructed based on the ultra violet (UV) complete theory such as string theory.
Axion (as imaginary component of Kähler modulus T ) inflation in string theory needs a moduli stabilization mechanism, which can separate the masses of real and imaginary components of Kähler modulus T at different scales. Specifically, the real component of modulus should be frozen during inflation, so it obtains a large mass from modulus stabilization: M Re(T) > H, where H is the Hubble constant during inflation, at the order 10 14 GeV from the BICEP2 results, while the "effective mass" of axion M Im(T) = Λ 2 /f is of order 10 13 GeV. In the well-known KKLT mechanism [9], once the real component of modulus is stabilized, the imaginary component obtains a large mass comparable to the real component which destroys axion inflation [10].
Moduli stabilization, which is consistent with axion inflation, has been proposed recently in Refs. [11,12]. In these works, an anomalous U(1) X gauge symmetry has been introduced to split the masses of real and imaginary components of Kähler moduli. The anomalous U(1) X gauge symmetry introduces the moduli-dependent Fayet-Iliopoulos (FI) term as a result of the non-trivial moduli transformation under U(1) X . The D-term scalar potential of U(1) X depends only on the real component of moduli, and actually is close to string scale by taking suitable gauge charges. Consequently, the D-term flatness provides a strong stabilization on real component of moduli once the U(1) X charged matter fields are stabilized by F-terms. The stabilization of matter fields can be directly done by tree-lever superpoential.
While axion potential for natural inflation is obtained from non-perturbative effects, so it is generally much weaker than the perturbative terms [12]. In the model, besides moduli stabilization consistent with axion inflation, U(1) X symmetry also provides an elegant solution to the problem of super-Planckian axion decay constant. A general review on the anomalous U(1) gauge symmetry in SUGRA and its applications on cosmology is provided in Ref. [13].
However, the super-Planckian axion decay constant in natural inflation required by the BICEP2 results is problematic in string theory. String theory predicts the axion decay constants cannot surpass the string scale [7,14,15], while as a controllable theory, the scale of weakly coupled string theory should be lower than the Planck scale. The Kim-Nilles-Pelroso (KNP) mechanism was proposed to obtain the effective super-Planckian axion decay constant [16]. In this proposal, two axions with sub-Planckian decay constants are aligned to have a flat direction along which the effective decay constant can be large enough for natural inflation. Another solution to this problem was provided in Ref. [12] based on an anomalous U(1) gauge symmetry. In this work the axion decay constant is directly determined by the flatness of anomalous U(1) D-term, and can be super-Planckian by taking a reasonably large condensation gauge group.
Recently, a lot of works on the KNP mechanism have been done after the BICEP2 results [17][18][19][20][21]. Specifically, in Ref. [20] a stringy geometrical realization of aligned axions was proposed based on the assumption that the moduli are well stabilized. While a complete realization of the KNP mechanism based on string framework, which is consistent with moduli stabilization, is still absent.
In this paper, we will realize both the KNP mechanism and moduli stabilization in string inspired no-scale supergravity with anomalous gauge symmetries U(1) X × U(1) A . Similar to our previous study [11,12], we stabilize the matter fields by F-term potential and the moduli by the D-terms of anomalous U(1) symmetries. Two axions are imaginary components of two Kähelr moduli, which transform non-trivially under two anomalous gauge symmetries The axion superpotential is of racetrack-type, and is from gaugino condensations of two gauge groups SU(n) × SU(m) with massive quark representations. The alignment of axions as in the KNP proposal, actually corresponds to an approximate S 2 exchange symmetry between two Kähler moduli. This S 2 symmetry is explicitly broken slightly so that one linear combination of the axions gives us the natural inflation with super-Planckian effective decay constant. This paper is organized as follows. In Section 2 we present string inspired no-scale SUGRA with anomalous gauge symmetries U(1) X × U(1) A . In Section 3 we provide stabilizations of matter fields and Kähler moduli based on F-term and D-term potentials, respectively.
In Section 4 we show that the aligned natural inflation can be realized by superpotential obtained from gaugino condensations of hidden gauge groups. Conclusions are given in Section 5.

II. ALIGNED AXIONS IN NO-SCALE SUGRA
We start from the no-scale type SUGRA with Kähler potential where i = 1, 2, and T 1 and T 2 are Kähler moduli. No-scale SUGRA [22] was realized naturally in the compactifications of weakly coupled heterotic string theory [23] or M-theory on S 1 /Z 2 [24]. For the third Kähler modulus T 3 , we assume that it is neutral under anomalous gauge symmetries U(1) X ×U(1) A and then is ignored in this work. Two axions θ i are the imaginary parts of the Kähler moduli, i.e., θ i ≡ Im(T i ), i = 1, 2.
As we know, n stacks of D7-branes, which wrap a 4-cycles of the Calabi-Yau space, gives U(n) gauge group. As in the KKLT scenario, the condensation gauge group is SU(n), and typically there is another anomalous U(1) gauge symmetry. For two Kähler moduli, there can be two copies of such gauge sectors, saying SU(n) × SU(m) × U(1) X × U(1) A . In particular, two anomalous U(1) gauge symmetries can play special role in moduli stabilization and inflation.
We assume that the vector-like massive quarks are in the fundamental representations of Yang-Mills gauge groups SU(n)×SU(m). After integrating out the heavy chiral superfields, the gaugino condensations of gauge groups SU(n) × SU(m) generate the following effective where φ 1 and φ 2 are matter fields charged on both SU(n) and SU(m) [25][26][27]. Here, we have assumed that the gauge kinetic functions of SU(n) and SU(m), which relate to the The superpotential in Eq. (3) and moduli dependent D-term are employed to stabilize the matter fields and real parts of the moduli, respectively. Once the matter fields φ i obtain vacuum expectation values (VEVs), the real part of the Kähler moduli are fixed by the Dterm flatnesses. The effects of non-perturbative terms on moduli stabilization and inflation based on anomalous U(1) have been studied in Refs. [28][29][30][31][32][33]. In these works, normally the D-terms are non-cancellable so that they can uplift the AdS vacua to dS vacua from moduli stabilization. The non-cancellability of anomalous U(1) D-term arises from the massless fundamental representation of condensation gauge group. While here the cancellable anomalous U(1) D-terms are preferred, the condensation gauge groups are equipped with massive fundamental representations. Recently, under the stimulation from the BICEP2 results, the potential roles of gaugino condensation on inflation have been studied in [34,35].
In our strategy, fields φ i are stabilized by their superpotential terms, which break anomalous U(1) X (also the continous shift symmetry of Kähler modulus) spontaneously. This procedure was first proposed in Ref. [12] for string-inspired no-scale SUGRA, and later it was applied to the case of the minimal SUGRA [36].
The U(1) a charges of the Kähler moduli and matter fields are provided in Table I, so the overall superpotential, which is invariant under gauge transformations of U(1) a with a ∈ {A, X}, is given by in which the constant term w ⋆ is arising after integrating out all complex-structure moduli.
Similar to Ref. [12], we may realize the above superpotential or its equivalent.
The U(1) a gauge invariance of matter couplings in (6) is clearly based on the charges provided in Table I. While for the non-perturbative terms, gauge invariance requires the where α ∈ {A, X} and q i α means the U(1) α charge of φ i . Without c the equation is degenerate and the moduli charges cannot be uniquely determined based on charges of the matter fields.
In the superpotential (6), the parameter c is very small. There is an exact S 2 exchange symmetry in Kähler potential K and also in superpotential W without term cT 2 . The exact S 2 symmetry corresponds to exact aligned axions, then the potential is independent with bT 1 − aT 2 , which becomes an exact flat direction in the scalar potential. The S 2 symmetry is broken explicitly by cT 2 while reserves approximately as long as c ≪ b. This approximate symmetry is crucial that it provides sufficient flat potential for inflation.
In this work, we will take m = n + 1 for simplicity, besides, the approximate S 2 symmetry is useful to simplify calculations.
Given the charges of φ i in Table I, the U(1) a charges of moduli T i are uniquely fixed In this work the degree of gauge group n is taken as O (10), while the ratio b/c is close to O(10 2 ), so we have 1 − b cn < 0, i.e., the U(1) a charges of modulus T 1 are negative in unit q orq. Negative charges of T 1 are directly determined by the smallness of ratio c/b, i.e., the approximate S 2 symmetry between two Kähler moduli. This property is greatly appreciated in quantum anomaly cancellation, as will be shown later.
The gravitational anomalies are cancelled by higher derivative terms R 2 . The mixed anomalies such as U(1) 2 a × U(1) b and SU(n) × U(1) a can be cancelled as well.
For cubic U(1) 3 a anomalies, the fermionic contributions are Gauge kinetic functions of U(1) a are in which k 1 a are positive parameters. Consequently, the gauge kinetic terms are where W a is the U(1) a gauge field strength. There are two parts in the gauge kinetic term: Re(f )F 2 and Im(f )FF . The first part is U(1) a invariant, while the second part shifts under U(1) a due to non-trivial U(1) a gauge transformations of Kähler moduli T i , actually the U(1) a variation of gauge kinetic term cancels the cubic gauge anomaly from fermionic contributions. Vanishing of cubic U(1) 3 a anomaly requires where an extra coefficient 1/3 is introduced as a symmetry factor of U(1) 3 a anomaly graphs. Specifically, we have Here all the coefficients k i a are positive. The above conditions cannot be fulfilled unless for each U(1) a , at least one of the Kähler moduli has negative charge (with unit of q orq).
Fortunately, as shown in (9), we do have one Kähler modulus T 1 whose U(1) a charges are negative resulting from the approximate S 2 symmetry. Then for any values of q/q, it is easy to adjust the parameters k i a so that the cubic U(1) 3 a anomalies vanish.

III. THE KÄHLER MODULI AND MATTER FIELD STABILIZATION
The matter fields are stabilized by F-term potential. Once the matter fields obtain VEVs, the real components of the moduli are fixed by vanishing of U(1) a D-terms. Normally field stabilization happens at scale much higher than inflation scale. The couplings between matter fields and inflation potential can only slightly modify the VEVs of matter fields, and more detailed analysis is presented in Ref. [11,12]. Here, we just ignore the scalar potential relating to inflation at this stage.
The F-term scalar potential is determined by the Kähler potential K and superpotential in which K ij is the inverse of the Kähler metric K ij = ∂ i ∂jK and D i W = W i + K i W . The D-term scalar potential is given by where X i a are the components of Killing vectors X a = X i a (φ)∂/∂φ i which generate the isometries of the Kähler manifold that are gauged to form U(1) a . If the superpotential W is gauge invariant instead of gauge covariant, the D a components reduce to For the U(1) a charged matter fields z n , they transform linearly under U(1) a , and the Killing vectors linearly depend on the matter field z n X zn a = iq a zn z n .
For the Kähler moduli T i , they shift under U(1) a gauge transformations, and the Killing vectors are which are purely imaginary constants.

A. Matter Field Stabilization
Considering the renormalizable matter couplings in (6), it is clear that the neutral matter fields X i and Y i have global minimum at the origin, while the charged matter fields φ i , χ i and ψ i obtain non-vanishing VEVs. During inflation, these matter fields will evolve to the global minimum very fast in consequence of the F-term exponential factor e znzn and the large masses obtained from the matter couplings in (6).
Part of the F-term potential is where we have ignored the terms proportional to X i , Y i or containing φ i , χ i , ψ i while several orders smaller. The small term 2c 0 (ψ i χ iW +ψ iχi W ), although ignorable for field stabilization, has considerable contribution to inflation potential.
As shown in [12], for λ 2 ≫ c 2 0 , above potential admits a global minimum at The U(1) a gauge symmetries are broken spontaneously by non-zero VEVs. Through matter field stabilization, potential V m obtains VEVs as well and uplifts the vacuum energy Normally the non-perturbative superpotential associated with no-scale Kähler potential leads to the AdS vacua. To obtain the Minkowski or dS vacua, an uplifting mechanism is needed. Here, the positive vacuum energy obtained from matter field stabilization provides a natural solution to this problem.
Up to now we have ignored the effects of the lower order terms on matter field stabilization. In [12] these effects have been studied, it was shown that the small couplings can only lead to tiny corrections on these VEVs, and reduce the vacuum energy slightly, which are totally ignorable in a general estimation.
According to the U(1) a charges provided in Table I, the D-term potentials associated with U(1) a are where z n ∈ {φ i , χ i , ψ i } and q a zn is U(1) a charge of field z n . Gauge kinetic function f U (1)a is provided in (13). The matter fields z n obtain VEVs through F-term stabilization, then the U(1) a D-terms become which are vanished at vacuum. Together with the charges of moduli in (9), the real components of Kähler moduli T i ≡ T Ri + iθ i can be uniquely determined at vacuum Even though the U(1) a gauge symmetries are broken after field stabilization, the approximate discrete symmetry S 2 is sustained. The symmetry breaking factor is very small, c ≪ b, therefore the VEVs of T i satisfy a T R1 ≃ b T R2 ≡ r/2 which is guaranteed by the approximate S 2 symmetry.
The imaginary components θ i remain free in the perturbative potential, so actually they only appear in the potential through non-perturbative effects.

IV. INFLATION POTENTIAL
Field stabilization is happened at scale much higher than the inflation scale. After stabilization, we get the following effective superpotential where w 0 = w ⋆ + 2c 0 λ 2 , and there is a positive cosmology constant term 4c 2 0 λ 2 e K , which is necessary to uplift the AdS vacua to Minkowski or dS vacua.
For the matter fields φ i , we have where the terms proportional to X i are ignored. The F-term potential contains and Terms like φ iφi WW will be dropped in the following discussions as they are several orders smaller than others.
F-term potential is separated into two parts: these independent of axions V 1 and these depending on axions V 2 . For the axion-independent part, it is while the axion-dependent part is Terms proportional to 4c 0 λ 2 in V 1 and V 2 are obtained from the matter field stabilization.
Employing the formula of w 1 and w 2 in the above equations and ignoring the lower order terms, we rewrite scalar potentials as follows and where we have used e znzn ≃ 1.
A and B are parameters that depend on the details of non-perturbative effects. Here we may simply assume they are close to each other, and Ar At the global minimum, the scalar potential V 1 + V 2 decreases to Without uplifting term from matter field stabilization, the non-perturbative superpotential with no-scale-type Kähler potential admits an AdS vacuum, as expected. The constant term 4c 2 0 λ 2 elevates the AdS vacuum to Minkowski vacuum under a constraint and the scalar potential can be simplified as V = 2Λ 4 1 + Λ 4 2 + Λ 4 1 cos(aθ 1 + bθ 2 ) + Λ 4 1 cos(aθ 1 + bθ 2 + cθ 2 ) + Λ 4 2 cos(cθ 2 ).
Because the kinetic terms of Kähler moduli are non-canonical, the field transformations are needed to determine the physical axion decay constants. The kinetic terms are Taking field re-scale θ i → √ 2T Ri θ i , and using aT R1 = bT R2 = r/2, we get where the axions θ i now have canonical kinetic terms. Redefining the axions ϕ 1,2 = (θ 1 ± The effective mass of axion ϕ 1 is where H is the Hubble constant during inflation, and its value is about 10 −4 in Planck units based on the BICEP2 results. Therefore, axion ϕ 1 is frozen out during inflation, and another axion ϕ 2 drives the observed inflationary process if its decay constant f = 2b/cr is of order O(10). Although r ∼ O(10), we have c ≪ b, so we can easily get a large effective decay constant f ∼ 10 by adopting a small S 2 symmetry breaking factor c/b ∼ 10 −2 .

V. CONCLUSIONS
We have proposed a concrete model to realize aligned axion inflation [16] for natural inflation with moduli stabilization based on two anomalous U(1) gauge symmetries. String theory provides abundant axion landscapes, and the natural inflation driven by aligned axions are expected to be true of certain choices in the axion landscapes [17][18][19][20][21]. For the axions as imaginary components of Kähler moduli, generally they appear in the potential through non-perturbative effects. Inflation driven by these axions needs subtle moduli stabilization since it requires to fix real components of moduli while keep axions sufficient light.
The condensation hidden sectors SU(n) × SU(m) are assumed to have massive fundamental representations, from which the gaugino condensations introduce race-track type superpotential and cancellable D-terms. Since the D-terms depend on real components of Kähler moduli only, their cancellations at vacuum state lead to strong stabilizations on the real components of Kähler moduli. The axions, which are imaginary components of Káhler moduli, remain light.
We introduced renormalizable matter couplings for matter field stabilization. Prior to D-term moduli stabilization, the matter fields have to be stabilized and obtain non-zero VEVs. This is done by Higgs-like matter couplings. Gauge symmetries U(1) X × U(1) A are spontaneously broken by VEVs of charged matter fields, besides, the continuous shift symmetries of Kähler moduli are spontaneously broken into discrete shift symmetries. Field stabilization in this model also provides a natural mechanism for uplifting the AdS vaccum to Minkowski or dS vaccum, i.e., it introduces large positive vaccum energy, which is suitable for elevating the AdS vaccum arising from non-perturbative superpotential.
We showed that the alignment of axions in the KNP mechanism corresponds to an approximate S 2 symmetry between two Kähler moduli. The S 2 symmetry is approximate as it is explicitly broken by a small factor c. Different from the U(1) sectors, the discrete S 2 symmetry is sustained after spontaneously gauge symmetry breaking and field stabilization.
After field stabilization, the potential is determined by two axions through non-perturbative effects. The decay constants of the two axions, which are determined by the moduli stabilization and canonical field transformation, are close to f i = 1/r, just about the string scale.
In consequence of S 2 symmetry, the potential forms a steep direction along θ 1 + θ 2 , while its orthogonal direction θ 1 − θ 2 is flat, and is suitable for inflation by taking small S 2 symmetry broken factor c.