Large Field Inflation Models from Higher-Dimensional Gauge Theories

Motivated by the recent detection of B-mode polarization of CMB by BICEP2 which is possibly of primordial origin, we study large field inflation models which can be obtained from higher-dimensional gauge theories. The constraints from CMB observations on the gauge theory parameters are given, and their naturalness are discussed. Among the models analyzed, Dante's Inferno model turns out to be the most preferred model in this framework.


Introduction
Cosmic inflation [1,2,3,4,5,6] is a leading paradigm in the study of very early universe. Inflation can explain not only the observed homogeneity and isotropy of the universe over the super-horizon scale but also the tiny deviations from them [7,8,9,10,11]. The agreement between the general theoretical predictions of the standard slow-roll inflation and the recent precise CMB measurements [12] is rather impressive.
Recently, another important clue from CMB observations came in. BICEP2 team reported detection of B-mode polarization at degree angular scales [13]. While the important foreground analysis remains to be worked out in the future, if the detected B-mode polarization turns out to be of primordial origin, it will have tremendous impacts on inflationary cosmology and the understanding of our universe at its very beginning: The tensor-to-scalar ratio fixes the energy scale at the time of inflation; another important consequence of the large tensor-to-scalar ratio is that it requires trans-Planckian inflaton field excursion via the Lyth bound [14]. This poses a challenge for constructing viable inflation models, since it is difficult to protect the flatness of the potential from quantum corrections over trans-Planckian field range in effective field theory framework. Thus the large tensor-to-scalar ratio might require the knowledge of physics near the Planck scale. However, this is not the only theoretical possibility: Even if the effective field range of the inflaton is trans-Planckian, field ranges in the defining theory can be sub-Planckian [15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30]. 1 A subclass of this type of models which is specified below will be of our interest.
It has been known that a gauge symmetry in higher dimensions gives rise to an approximate shift symmetry in a four-dimensional scalar potential [31,32], and this mechanism was employed in [33] (see also [34]) to construct a version of natural inflation [35] (extra-natural inflation). The original aim of [33] was to construct a large field inflation model (inflation model in which inflaton makes trans-Planckian field excursion) within the framework of effective field theory. But it was already noticed by the authors of [33] that the embedding of extra-natural inflation to string theory was difficult, and this point was further examined in [36]. Then, it was suggested that the underlying reason for the difficulty was the extremely small gauge coupling which was required to explain the CMB data in extra-natural inflation [37]. The authors of [37] proposed that the tiny gauge coupling causes an obstacle for coupling the effective field theory to gravity. It was motivated by the well-known argument against the existence of global symmetry in quantum gravity based on processes involving black holes (see [38] for recent discussions and references for earlier works). When the gauge coupling is turned to zero, the gauge symmetry is physically indistinguishable from a global symmetry. If the limit to the zero gauge coupling is smooth, something must prevent the occurrence of the global symmetry. The answer suggested in [37] was that when the gauge coupling becomes small, the UV cut-off scale where the effective field theory breaks down must be lowered. More precisely, they proposed that there is an upper bound on the UV cut-off scale Λ: where g is the gauge coupling and M P is the four-dimensional Planck mass. The authors of [37] showed that the bound (1.1) follows from a conjecture that there must be a particle whose mass is smaller than its charge in certain unit (Weak Gravity Conjecture, abbreviated as WGC below). The basis of their arguments which lead to WGC are quite robust, and in this paper we will take WGC seriously. A brief review on WGC is given in appendix B.
In this paper, we examine large field inflation models which can be obtained from higher-dimensional gauge theories. We restrict ourselves to one-form gauge fields in higher dimensions, though these can appear from higher-form gauge fields in even higher dimensions with smaller compactification size. While in this paper we restrict ourselves to the simplest Abelian gauge groups, it is straightforward to extend or embed our models to those with non-Abelian gauge groups. Non-Abelian higher-form fields are known to be theoretically quite involved (see e.g. [39]), and our strategy of first concentrating on oneform gauge fields may have an advantage in bypassing these theoretical complications while still covering large portion of theory space. Such one-form gauge fields are also essential ingredients in the Standard Model of particle physics, and it is natural to expect that one-form gauge fields will continue to be an essential part of the new physics beyond the Standard Model. These constitute our basic motivations to consider one-form gauge theories in higher dimensions.
We are particularly interested in the consequences of WGC, and will assume that it is correct. 2 Thus the original extra-natural inflation will be excluded from our study. 3 This naturally lead us to consider models of the type mentioned above: Those in which the field ranges in the defining theory are sub-Planckian but the inflaton effectively travels trans-Planckian field range. As higher-dimensional gauge theories reduce to so-called 2 Another possibility would be that WGC does not always hold, but holds in the dominant majority of string vacua. While this is an interesting theoretical possibility, it is not relevant for the discussion of naturalness below as long as it is extremely likely to be in a vacuum in which WGC holds. 3 There is a possibility that WGC completely excludes natural parameter space for effective field theory. In this case, one may respect the constraints from WGC and accept the unnatural values of the parameters. See [40] for an argument on an example in particle physics model. In this paper we will be interested in natural parameter space allowed by WGC.
axion models, we examined all the major axionic inflation models of the above mentioned type so far known to us, at least in their simplest form. These include: Single-field Axion Monodromy model (AM) [16,17], Dante's Inferno model (DI) [18], Axion Alignment model (AA) [15,21,22,23] and Axion Hierarchy model (AH) [19,20]. We will examine the constraints from CMB data on gauge theory parameters and discuss their naturalness in the effective field theory framework. However, for the tensor-to-scalar ratio, the above mentioned BICEP2 result does not give conclusive value due to the uncertainty in the foreground [41,42]. In this paper, we would like to explore the possibility that the large tensor-to-scalar ratio is real considering its impact if it turns out to be the case. We choose r = 0.16 at the pivot scale as a reference value [43], but this should be taken as an assumption at this moment. Table 1 summarizes the expected parameter ranges in our models. While we will not go into full Bayesian model comparison (see e.g. [44,45]), in principle we can go through it, and in that case our prior can be built based on Table 1. In Table 1, g stands for four-dimensional gauge coupling which is obtained from higher-dimensional gauge theory as 1 where g 5 is the five-dimensional gauge coupling and L is the compactification radius of the fifth dimension. g 2 5 has dimension of length which can be independent from the compactification radius. A priori, we do not have knowledge of their corresponding energy scales besides the upper bound by the Planck scale and lower bound from high energy experiments like LHC. Therefore, the log-flat prior would be appropriate for g and L, if we were to proceed to Bayesian model comparison. The lower bound in g in Table  1 is imposed by WGC, while the upper bound comes from applicability of perturbation theory. The expected value of charges is shown in Table 1 in unit of the minimal charge in the model. It reflects the theoretical belief of the current authors that extraordinary large charge is unlikely or rare in nature. Table 2-4 show the allowed parameter ranges after taking into account CMB data and assuming r = 0.16. Strictly speaking, it is more appropriate to show the allowed parameter range in multi-dimensional parameter space, as the allowed range for one parameter depends on other parameters in general. However, even in the current simplified analysis, one immediately notices that somewhat unusual parameter ranges appear in Table 4: 4 AA and AH have at least one charge which is more than O(100) in unit of minimal charge in the model. Although theories with such a large charge number have been considered, (e.g. see [46] for the so-called milli-charged dark matter, where an issue related to WGC Gauge couplings Compactification radius Charges − log 10 [(LM P ) 2 ] log 10 [g 2 ] 0 log 10 [1/(L GeV)] ∼ 3 − 17 n ∼ O(1) Table 1: Expected parameter ranges from higher-dimensional gauge theory. g is the gauge coupling in four-dimension. L is the compactification radius of the fifth dimension. n represents charge of a matter measured in unit of the minimal charge in the model.

Model
Gauge coupling(s) is discussed), such theories look somewhat artificial. This view of the current authors had been reflected in the expected charge number in Table 1. On the other hand, the charge of AM is in a natural range, but this model has its own naturalness issue which will be explained in section 2. Charges in DI are in the expected range given in Table 1. From these analysis, one immediately sees that DI is preferred among the models considered. The organization of the rest of the paper is as follows. We start with single-field Axion Monodromy model in section 2. In section 3 we study Dante's Inferno model. In section 4 Axion Alignment model and Axion Hierarchy model are studied. 5 For each model we obtain it from higher-dimensional gauge theory, study the constraints from the CMB observations to the parameters of the gauge theory and discuss naturalness of the 5 In [23] aligned natural inflation from higher-dimensional gauge theory similar to ours was studied, but the four-dimensional WGC was not imposed.

Model
Compactification radius Table 3: Constraints on compactification radius after taking into account CMB data with the assumption r = 0.16.

Model
Charge parameters. We summarize with discussions on future directions in section 5.

Single-Field Axion Monodromy
We begin with single-field axion monodromy inflation [16,17]. The relevant inflaton potential is of the form The potential (2.1) can be obtained from a five-dimensional gauge theory with an action 6 We introduced the Stueckelberg mass term which gives rise to the quadratic potential in (2.1). 7 We take the gauge group to be compact U (1). 8 Then, the Stueckelberg field θ is an angular variable with the identification This allows θ to have a winding mode: 6 We chose the massless charged fermion for an illustrative purpose. We can introduce mass term for the fermion or include charged massive scalars in a similar way. 7 Massive gauge fields can arise via the Higgs mechanism. However, the expectation value of the radial component of the Higgs field, which determines the mass of the gauge field, is affected by the large inflaton expectation value, as the inflaton originates from gauge field in the current model and couples to the Higgs field as such. Then the current analysis does not apply. For a recent review on the use of Stueckelberg fields in axion monodromy inflations in string theory, see [47]. 8 It has been argued that in models which can be consistently coupled to quantum gravity, all the continuous gauge symmetries are compact [38].
Here, x are coordinates in visible large space-time dimensions, and x 5 is the coordinate of the fifth direction compactified on a circle with radius L. The winding number w is an integer. If one takes into account all the winding sectors, the spectrum of the model is invariant under the shift of A by 2πf , while starting from a sector with given winding number the shift leads to the monodromy property [16,17]. At one-loop, the following potential is generated: See appendix A.1 for the outline of the calculation of the one-loop effective potential. For a sector with a given winding number, by redefining A by a constant shift one obtains (2.1). The inflaton field A in the potential (2.1) is the zero-mode of the gauge field: The parameters of the axion monodromy model (2.1) are related to the parameters of the higher-dimensional gauge theory as follows: , where g is the four-dimensional gauge coupling which is related to the five-dimensional gauge coupling g 5 as g = g 5 √ 2πL . (2.8) The constant c in (2.7) depends on the matter contents charged under the gauge group. In (2.7) we have assumed that both the number of the matter fields and their charges are of order one, which we think natural. If one considers all possible winding numbers of θ, the whole theory is invariant under the shift A → A + 2πf . Thus the field A takes values on a circle with radius f . Starting from a given winding number sector, the quadratic potential reveals the phenomenon of monodromy: The potential energy does not return the same under the shift of A by 2πf . Thus one can effectively achieve trans-Planckian field excursion of A even if the original period of A was below the reduced Planck scale M P = 2.4 × 10 18 GeV, by going round the circle several times. This is an important feature of the model, because examples in string theory so far constructed and WGC suggest 2πf M P for an axion decay constant f , which forbids trans-Planckian field excursion of the axion if there were no monodromy (see appendix B for the assertions of WGC we adopt in this paper).
When the slope of the sinusoidal potential is much smaller than that of the mass term in (2.1), the model effectively reduces to chaotic inflation. 9 This condition is written as where A * is the value of A when the pivot scale exited the horizon. Using (2.7), this condition becomes 3g We review the constraints from CMB observations on chaotic inflation in appendix C.1.
Putting the values of m 2 and A * given in (C. 15) and (C.14) for r = 0.16 and N * 50, we obtain 1 Note that the energy scale of the compactification should not be smaller than the Hubble scale during inflation, otherwise the use of the four-dimensional Einstein equation is not justified. From (C.11), this gives If there were no sinusoidal potential, when one takes m 2 to zero the shift symmetry A → A + c (c: constant) recovers. Thus small m 2 is natural in the sense of 't Hooft [50]. In order for the inflaton to achieve trans-Planckian field excursion, this shift symmetry must be a good symmetry at the Planck scale. Whether this is the case or not is a problem beyond the scope of the higher-dimensional gauge theory, which is an effective field theory. One needs to work in a theory of quantum gravity to study this issue. In other words, while the whole theory is invariant under the shift of the field A by 2πf , starting from a given winding number the potential of A is not periodic. And the large A behavior of the non-periodic part of the potential has the usual UV issue of effective field theory.

Dante's Inferno
Next we study Dante's Inferno model [18], which is a two-axion model with the following potential: See [48,49] for the case in which the sinusoidal potential is not totally negligible. From appendix A of [49] one can show that the effect of the sinusoidal potential is proportional to L −3 and thus quickly suppressed as one moves away from the bound in (2.11).
The potential (3.1) can be obtained from a gauge theory in higher dimensions with the action We consider the case where both of the gauge groups are compact U (1), which we refer to as U A (1) and U B (1). Here, as an illustration, we consider fermionic matter, but the case with bosonic matters can be studied in essentially the same way. The one-loop effective potential of this model produces the second term in (3.1) with and Here, g A and g B are four-dimensional gauge couplings which are related to the fivedimensional gauge couplings g A5 and g B5 as It is convenient to rotate the fields as where Then the potential (3.1) takes the form where In this model, the regime of interest is 10
where A in is the initial condition set at the beginning of the observable inflation and we require it to be in the range f A in < M P . Notice that the condition (3.10) implies in the leading order in We require that the excitation inÃ direction is much heavier than the Hubble scale during inflation so that they can be safely integrated out: From (3.11) and f A in this reads (3.14) After integrating outÃ, we obtain the following effective potential forB which we rewrite as φ ≡B [18]: to leading order in f A /f B . Thus Dante's Inferno model effectively reduces to chaotic inflation, with φ being the inflaton. The constraints from CMB observations on chaotic inflation are summarized in appendix C.1. Using these inputs, now we examine the CMB constraints on the parameters of the higher-dimensional gauge theory. We will take the number of e-fold N * 50 and the tensor-to-scalar ratio r = 0.16 (see appendix C for the detail and the notations used below). From (3.3), the condition (3.10) reads in terms of gauge theory parameters as Chaotic inflation is a large field inflation model in which the inflaton travels trans-Planckian field distance ∆φ ≡ φ * − φ e 14M P , see (C.14). However, the original fields in the current model, A and B (which were the zero-modes of the higher-dimensional gauge theory), do not need to make trans-Planckian field excursion.
Regarding the field A, its initial value A in is restricted as This condition should be compared with (3.16). On the other hand, field B is periodic and its field range 2πf B is bounded from above by M P , as noted in (3.10). There is also a lower bound on the inverse compactification radius. Using (3.15) and (3.18), the condition (3.11) can be rewritten as See Fig. 1 for the values of g B in between. We observe that the allowed values of the gauge couplings and the compactification radius of the gauge theory are rather restricted, which will be advantageous for the model to be predictive. Note that the above compactification scales are high enough so that the use of the four-dimensional Einstein gravity is justified, 1/L H ∼ 10 14 GeV (see (C.11)). For completeness, we check that (3.13) is satisfied. It gives Putting the value from appendix C (C.11) we obtain g A 2πL 2 × 10 14 GeV. (3.28) This is readily satisfied for the above values of g A and L. Now we turn to another feature of the model which could be potentially constrained by CMB data. The shift symmetry allows the following axionic coupling to gauge fields: where σ i is an axion, f i is its decay constant and α i is a constant parameter. i labels axions when there are more than one, in the current case i labels the field A and the field B (we just label them as i = A and i = B, respectively). How the coupling (3.29) arises from higher-dimensional gauge theory is explained in appendix A.2. Contributions to CMB power spectrum, non-Gaussianity and primordial gravitational waves through this coupling have been studied in [51,52,53,54,55]. These effects are mainly controlled by the following parameter: The current observational bound is given as [53,55] ξ i 3.

Axion Alignment and Axion Hierarchy
In this section we study aligned axion inflation [15,21,22] and hierarchical axion inflation [19,20] from higher-dimensional gauge theory perspective. Both models can be described by the potential of the form the potential (4.1) takes the form The potential (4.1) can be obtained from a higher-dimensional gauge theory with following action: The parameters in the potential (4.1) and the higher-dimensional gauge theory are related as , (4.8) where g A and g B are four-dimensional gauge couplings Anticipating UV completions such as string theory, it is natural that charges are quantized with respect to the unit charge. Thus we assume m 1 , m 2 , n 1 , n 2 are all integers. 11 Aligned axion inflation is obtained in the regime In this regime one obtains |f l | f A , f B from (4.6). Notice that |f l | is at largest the order of max(|m 1 |f B , |n 1 |f A ). On the other hand, as explained in appendix C.2, r 0.16 requires |f l | 20M P . Since from WGC we have 2πf A , 2πf B M P , this requires 11 As we have assumed that the gauge groups are compact U (1), charges are quantized. Here we made a stronger assumption that charges are all integer multiples of the minimal charge in the theory. This can be regarded as for simplicity, the result does not change qualitatively unless one assumes highly exotic charge spectrum. max(|m 1 |, |n 1 |) 20 × 2π. A matter with such a large charge seems to us quite unnatural, considering that the energy scale under consideration is rather high (H ∼ 10 14 GeV).
Next we turn to the hierarchical axion inflation in higher-dimensional gauge theory. This model corresponds to taking n 2 = 0 in (4.1). Then (4.6) reduces to One further requires a hierarchy Then (4.11) can be approximated as From WGC we have 2πf B M P , thus |f l | 20M P requires |m 1 | 20|n 1 m 2 | × 2π. Such a large hierarchy between the charges in the same gauge group seems quite unnatural. 12

Summary and Discussions
In this paper we studied large field inflation models which can be obtained from higherdimensional gauge theories. We accept WGC as our working hypothesis, and studied the constraints from CMB data on the gauge theory parameters. We consider the case with large tensor-to-scalar ratio, and used r = 0.16 as a reference value. We found that the allowed range of gauge theory parameters are quite constrained. Among the models studied in this paper, Dante's Inferno model appears as the most preferred model. The allowed values of the gauge couplings and the compactification radius turned out to be quite restricted but fell within a natural range, making the model attractive for being predictive. Single-field axion monodromy model leaves the problem that whether the shift symmetry is a good symmetry or not to its UV completion. Axion alignment model and axion hierarchy model require large hierarchy among charges in the same gauge group, which makes the models rather unnatural.
The allowed values of gauge couplings in Dante's Inferno model are in the range 0.04−O(1). This is in contrast to the extremely small gauge coupling O(10 −3 ) required for extra-natural inflation [33,56]. The above values of gauge couplings for Dante's Inferno model would be large enough to have interesting consequences in cosmological history or particle physics experiments in model dependent ways, which will be interesting to investigate. In particular, since gauge symmetry is a basic ingredient of the Standard Model of particle physics, it is natural to expect that the higher-dimensional gauge theories responsible for inflation are also relevant for the new physics beyond the Standard Model. If this is the case, particle physics experiments would provide complimentary data for such models. See [57,56] for earlier investigations along this line in the case of extra-natural inflation. In this appendix we outline the calculation of the one-loop effective potential in higherdimensional gauge theories compactified on a circle. We start with the five-dimensional action

Acknowledgments
where space-time indices M and N run 0, · · · , 3 and 5, and We choose the gauge fixing term as Then the total action becomes We compactify the fifth dimension on a circle with radius L. The Fourier expansions of the fields in the fifth dimension are We will be interested in the effective potential for the zero-modes of the gauge fields, A 5(0) ≡ A and B 5(0) ≡ B. At one-loop level, only the quadratic part of the matter action is relevant: Here, µ and ν run four-dimensional space-time indices 0, · · · , 3. Then, the one-loop effective potential is expressed as where we have made Wick rotation and the subscript E indicates the Euclidean space. The four-dimensional gauge couplings are related to the five-dimensional ones as Employing the ζ function regularization, the effective potential becomes .
(A. 12) In (A.11) we have dropped the constant part, the fine tuning of which is the cosmological constant problem which we will not address in this paper. Taking the leading term n = 1 in (A.11) together with the tree-level potential coming from the Stueckelberg mass term, we arrive at the potential where we have redefined the field B by an appropriate constant shift.

A.2 Axionic Couplings
The shift symmetry allows the following axionic coupling where σ is an axion and α is some constant. In higher-dimensional gauge theory, the axionic coupling (A.14) follows from the Chern-Simons term in five-dimensional gauge theory [56]: and k is an integer. Quantum corrections to k due to parity-violating charged matters are one-loop exact and proportional to the cubic powers of charges [58]. As we assume charges to be O(1), we may expect k ∼ O(1 − 10). The 1-form A M dx M is related to the canonically normalized gauge field A M in five dimensions as where g 5 is the five-dimensional gauge coupling. After integrating KK modes of the fifth direction we obtain the axionic coupling (A.14) with

B Weak Gravity Conjecture
Weak Gravity Conjecture (WGC) [37] asserts the existence of a state with charge and mass (q, m) which satisfy is estimated from requiring that the Coulomb repulsive force is greater than the Newtonian attractive force so that extremal black holes can loose their charge by emitting such particles. In this paper we assume the existence of a particle with the smallest unit charge, with respect to which all charges are integers. Generalization is straightforward and dose not change the result qualitatively, unless one assumes highly exotic charge spectrum. Then, the Dirac monopole with unit magnetic charge has charge and mass where Λ U V is a UV scale which regularizes the mass of the Dirac monopole. Here, we used non-Abelian gauge-Higgs system as the UV completion to estimate the mass of the Dirac monopole. An important constraint for our study is obtained by applying WGC the Dirac monopole: 4π g This condition also follows by requiring that the Dirac monopole with unit magnetic charge is not a black hole [37]. Strictly speaking, one should take into account the running of the couplings. We assume that those runnings are not significant so that they do not alter our order of magnitude estimate. In order for the higher-dimensional gauge theory to be applicable, the compactification scale should be sufficiently below the UV cut-off scale: 13 In terms of the axion decay constant f = 1/(g2πL), Since the above argument is an order estimate, in the main body we adopted slightly milder bound 2πf M P .

C Relevant Inflation Models in Light of BICEP2
In this appendix we review the constraints from CMB observations, in particular the possible detection of primordial tensor perturbation by BICEP2 [13], on inflation models which are relevant in this paper. The detection of the B-mode polarization by BICEP2 indicates large tensor-to-scalar ratio r. In this paper we adopt a conservative value r = 0.16 at the pivot scale k = 0.05 Mpc −1 as a reference value, considering the uncertainty in the foreground [41] and the constraint from Planck 2013 [12,43].

C.1 Chaotic Inflation with Quadratic Potential
Consider quadratic potential for the inflaton We assume canonical kinetic term for the inflaton φ. The slow-roll parameters are given by We will use suffix * to indicate that it is the value when the pivot scale exited the horizon. The scalar spectral index is given by The scalar power spectrum and the tensor power spectrum are given as where the last value in (C.6) is the COBE normalization. The tensor-to-scalar ratio r is given by GeV. (C.11) The slow-roll inflation ends when (φ e ) ∼ 1. This gives φ e ∼ √ 2M P . (C.12) The number of e-folds is given as

C.2 Natural Inflation
The typical form of the potential for natural inflation is given by From (C.17) and (C.21), for a given N * , r is determined as a function of f . This is plotted in Fig. 2. Notice that to obtain the tensor-to-scalar ratio as large as r 0.16, we need f 20M P and N * 50. These values were adopted in the main body.