Large Field Inflation Models from Higher-Dimensional Gauge Theories
LLarge Field Inflation Models FromHigher-Dimensional Gauge Theories
Kazuyuki Furuuchi a and Yoji Koyama ba Manipal Centre for Natural Sciences, Manipal UniversityManipal, Karnataka 576104, India b Department of Physics, National Tsing-Hua UniversityHsinchu 30013, Taiwan R.O.C.
Abstract
Motivated by the recent detection of B-mode polarization of CMB by BICEP2which is possibly of primordial origin, we study large field inflation models whichcan be obtained from higher-dimensional gauge theories. The constraints from CMBobservations on the gauge theory parameters are given, and their naturalness arediscussed. Among the models analyzed, Dante’s Inferno model turns out to be themost preferred model in this framework. a r X i v : . [ h e p - t h ] M a r 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 overthe super-horizon scale but also the tiny deviations from them [7, 8, 9, 10, 11]. Theagreement between the general theoretical predictions of the standard slow-roll inflationand the recent precise CMB measurements [12] is rather impressive.Recently, another important clue from CMB observations came in. BICEP2 teamreported detection of B-mode polarization at degree angular scales [13]. While the impor-tant foreground analysis remains to be worked out in the future, if the detected B-modepolarization turns out to be of primordial origin, it will have tremendous impacts on in-flationary cosmology and the understanding of our universe at its very beginning: Thetensor-to-scalar ratio fixes the energy scale at the time of inflation; another importantconsequence of the large tensor-to-scalar ratio is that it requires trans-Planckian inflatonfield excursion via the Lyth bound [14]. This poses a challenge for constructing viableinflation models, since it is difficult to protect the flatness of the potential from quantumcorrections over trans-Planckian field range in effective field theory framework. Thus thelarge 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 ofthe 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]. A subclass of this type ofmodels which is specified below will be of our interest.It has been known that a gauge symmetry in higher dimensions gives rise to an ap-proximate shift symmetry in a four-dimensional scalar potential [31, 32], and this mech-anism 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 inflationmodel (inflation model in which inflaton makes trans-Planckian field excursion) withinthe 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 thispoint was further examined in [36]. Then, it was suggested that the underlying reasonfor the difficulty was the extremely small gauge coupling which was required to explainthe CMB data in extra-natural inflation [37]. The authors of [37] proposed that the tinygauge coupling causes an obstacle for coupling the effective field theory to gravity. Itwas motivated by the well-known argument against the existence of global symmetry inquantum gravity based on processes involving black holes (see [38] for recent discussionsand references for earlier works). When the gauge coupling is turned to zero, the gauge This is a partial list of references on such models, we picked up papers whose interests are relativelyclose to that of the current paper. (cid:46) gM P , (1.1)where g is the gauge coupling and M P is the four-dimensional Planck mass. The au-thors of [37] showed that the bound (1.1) follows from a conjecture that there must be aparticle 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 quiterobust, and in this paper we will take WGC seriously. A brief review on WGC is givenin appendix B.In this paper, we examine large field inflation models which can be obtained fromhigher-dimensional gauge theories. We restrict ourselves to one-form gauge fields in higherdimensions, though these can appear from higher-form gauge fields in even higher dimen-sions with smaller compactification size. While in this paper we restrict ourselves to thesimplest Abelian gauge groups, it is straightforward to extend or embed our models tothose with non-Abelian gauge groups. Non-Abelian higher-form fields are known to betheoretically quite involved (see e.g. [39]), and our strategy of first concentrating on one-form gauge fields may have an advantage in bypassing these theoretical complicationswhile still covering large portion of theory space. Such one-form gauge fields are alsoessential ingredients in the Standard Model of particle physics, and it is natural to expectthat one-form gauge fields will continue to be an essential part of the new physics beyondthe Standard Model. These constitute our basic motivations to consider one-form gaugetheories in higher dimensions.We are particularly interested in the consequences of WGC, and will assume that it iscorrect. Thus the original extra-natural inflation will be excluded from our study. Thisnaturally lead us to consider models of the type mentioned above: Those in which thefield ranges in the defining theory are sub-Planckian but the inflaton effectively travelstrans-Planckian field range. As higher-dimensional gauge theories reduce to so-called Another possibility would be that WGC does not always hold, but holds in the dominant majorityof string vacua. While this is an interesting theoretical possibility, it is not relevant for the discussion ofnaturalness below as long as it is extremely likely to be in a vacuum in which WGC holds. There is a possibility that WGC completely excludes natural parameter space for effective fieldtheory. In this case, one may respect the constraints from WGC and accept the unnatural values of theparameters. See [40] for an argument on an example in particle physics model. In this paper we will beinterested in natural parameter space allowed by WGC. r = 0 .
16 at the pivot scale as a reference value [43], but this should be taken asan assumption at this moment.Table 1 summarizes the expected parameter ranges in our models. While we will notgo into full Bayesian model comparison (see e.g. [44, 45]), in principle we can go throughit, and in that case our prior can be built based on Table 1. In Table 1, g stands forfour-dimensional gauge coupling which is obtained from higher-dimensional gauge theoryas 1 g = 2 πLg , (1.2)where g is the five-dimensional gauge coupling and L is the compactification radiusof the fifth dimension. g has dimension of length which can be independent from thecompactification radius. A priori, we do not have knowledge of their corresponding energyscales besides the upper bound by the Planck scale and lower bound from high energyexperiments 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 Table1 is imposed by WGC, while the upper bound comes from applicability of perturbationtheory. The expected value of charges is shown in Table 1 in unit of the minimal chargein the model. It reflects the theoretical belief of the current authors that extraordinarylarge charge is unlikely or rare in nature.Table 2-4 show the allowed parameter ranges after taking into account CMB data andassuming r = 0 .
16. Strictly speaking, it is more appropriate to show the allowed param-eter range in multi-dimensional parameter space, as the allowed range for one parameterdepends on other parameters in general. However, even in the current simplified analysis,one immediately notices that somewhat unusual parameter ranges appear in Table 4: AAand AH have at least one charge which is more than O (100) in unit of minimal chargein 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 As can be seen from the derivation of Table 4 in the main body, this conclusion does not depend onother parameters. − log [( LM P ) ] (cid:46) log [ g ] (cid:46) [1 / ( L GeV)] ∼ − n ∼ O (1)Table 1: Expected parameter ranges from higher-dimensional gauge theory. g is thegauge 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)AM − (cid:46) log [ g ] (cid:46) − (cid:46) log [ g A ] (cid:46) − (cid:46) log [ g B ] (cid:46) − − (cid:46) log [ g A ] , log [ g B ] (cid:46) − − (cid:46) log [ g A ] (cid:46) − − (cid:46) log [ g B ] (cid:46) r = 0 . For eachmodel we obtain it from higher-dimensional gauge theory, study the constraints from theCMB observations to the parameters of the gauge theory and discuss naturalness of the 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 radiusAM log [1 / ( L GeV)] ∼ − [1 / ( L GeV)] ∼ [1 / ( L GeV)] ∼ − [1 / ( L GeV)] ∼ − r = 0 .
16. 4odel Charge(s)AM O (1)DI O (1)AA max( | m , m | ) (cid:38) O (100)AH m (cid:38) O (100)Table 4: Constraints on charges after taking into account CMB data with the assumption r = 0 . We begin with single-field axion monodromy inflation [16, 17]. The relevant inflatonpotential is of the form V ( A ) = 12 m A + Λ (cid:18) − cos (cid:18) Af (cid:19)(cid:19) . (2.1)The potential (2.1) can be obtained from a five-dimensional gauge theory with an action S = (cid:90) d x (cid:104) − F MN F MN − m ( A M − g ∂ M θ ) + (matters) (cid:105) . (2.2)We introduced the Stueckelberg mass term which gives rise to the quadratic potential in(2.1). We take the gauge group to be compact U (1). Then, the Stueckelberg field θ isan angular variable with the identification θ ∼ θ + 2 πg . (2.3)This allows θ to have a winding mode: θ ( x, x ) = x g L w + (cid:88) n θ n ( x ) e i nL x (2.4) We chose the massless charged fermion for an illustrative purpose. We can introduce mass term forthe fermion or include charged massive scalars in a similar way. Massive gauge fields can arise via the Higgs mechanism. However, the expectation value of the radialcomponent of the Higgs field, which determines the mass of the gauge field, is affected by the largeinflaton expectation value, as the inflaton originates from gauge field in the current model and couplesto the Higgs field as such. Then the current analysis does not apply. For a recent review on the use ofStueckelberg fields in axion monodromy inflations in string theory, see [47]. It has been argued that in models which can be consistently coupled to quantum gravity, all thecontinuous gauge symmetries are compact [38]. x are coordinates in visible large space-time dimensions, and x is the coordinateof the fifth direction compactified on a circle with radius L . The winding number w isan integer. If one takes into account all the winding sectors, the spectrum of the modelis invariant under the shift of A by 2 πf , while starting from a sector with given windingnumber the shift leads to the monodromy property [16, 17]. At one-loop, the followingpotential is generated: V ( A ) = 12 m ( A − πf w ) + Λ (cid:18) − cos (cid:18) Af (cid:19)(cid:19) . (2.5)See appendix A.1 for the outline of the calculation of the one-loop effective potential. Fora 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: A ≡ A . (2.6)The parameters of the axion monodromy model (2.1) are related to the parameters of thehigher-dimensional gauge theory as follows: f = 1 g (2 πL ) , Λ = cπ (2 πL ) , c ∼ O (1) , (2.7)where g is the four-dimensional gauge coupling which is related to the five-dimensionalgauge coupling g as g = g √ πL . (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 areof order one, which we think natural.If one considers all possible winding numbers of θ , the whole theory is invariant underthe shift A → A + 2 πf . Thus the field A takes values on a circle with radius f . Startingfrom a given winding number sector, the quadratic potential reveals the phenomenon ofmonodromy: The potential energy does not return the same under the shift of A by 2 πf .Thus one can effectively achieve trans-Planckian field excursion of A even if the originalperiod of A was below the reduced Planck scale M P = 2 . × GeV, by going roundthe circle several times. This is an important feature of the model, because examples instring theory so far constructed and WGC suggest 2 πf (cid:46) 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 term6n (2.1), the model effectively reduces to chaotic inflation. This condition is written asΛ f (cid:28) m A ∗ , (2.9)where A ∗ is the value of A when the pivot scale exited the horizon. Using (2.7), thiscondition becomes 3 gπ πL ) (cid:28) m A ∗ , (2.10)or 1 L < π (cid:18) π g m A ∗ (cid:19) / . (2.11)We review the constraints from CMB observations on chaotic inflation in appendix C.1.Putting the values of m and A ∗ given in (C.15) and (C.14) for r = 0 .
16 and N ∗ (cid:39)
50, weobtain 1
L < g − / × . × GeV . (2.12)Note that the energy scale of the compactification should not be smaller than the Hubblescale during inflation, otherwise the use of the four-dimensional Einstein equation is notjustified. From (C.11), this gives 1 L > . × GeV . (2.13)If there were no sinusoidal potential, when one takes m to zero the shift symmetry A → A + c ( c : constant) recovers. Thus small m is natural in the sense of ’t Hooft [50].In order for the inflaton to achieve trans-Planckian field excursion, this shift symmetrymust be a good symmetry at the Planck scale. Whether this is the case or not is aproblem beyond the scope of the higher-dimensional gauge theory, which is an effectivefield theory. One needs to work in a theory of quantum gravity to study this issue. Inother words, while the whole theory is invariant under the shift of the field A by 2 πf ,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 effectivefield theory. Next we study Dante’s Inferno model [18], which is a two-axion model with the followingpotential: V ( A, B ) = 12 m A A + Λ (cid:18) − cos (cid:18) Af A − Bf B (cid:19)(cid:19) . (3.1) See [48, 49] for the case in which the sinusoidal potential is not totally negligible. From appendix Aof [49] one can show that the effect of the sinusoidal potential is proportional to L − and thus quicklysuppressed as one moves away from the bound in (2.11). S = (cid:90) d x (cid:104) − F AMN F AMN − m A ( A M − g A ∂ M θ ) − F BMN F BMN − i ¯ ψγ M ( ∂ M + ig A A M − ig B B M ) ψ (cid:105) . (3.2)We consider the case where both of the gauge groups are compact U (1), which we refer toas U A (1) and U B (1). Here, as an illustration, we consider fermionic matter, but the casewith bosonic matters can be studied in essentially the same way. The one-loop effectivepotential of this model produces the second term in (3.1) with f A = 1 g A (2 πL ) , f B = 1 g B (2 πL ) , (3.3)and Λ (cid:39) π πL ) . (3.4)Here, g A and g B are four-dimensional gauge couplings which are related to the five-dimensional gauge couplings g A and g B as g A = g A √ πL , g B = g B √ πL . (3.5)It is convenient to rotate the fields as (cid:32) ˜ B ˜ A (cid:33) = (cid:32) cos γ sin γ − sin γ cos γ (cid:33) (cid:32) BA (cid:33) , (3.6)where sin γ = f A (cid:112) f A + f B , cos γ = f B (cid:112) f A + f B . (3.7)Then the potential (3.1) takes the form V ( ˜ A, ˜ B ) = m A (cid:16) ˜ A cos γ + ˜ B sin γ (cid:17) + Λ (cid:32) − cos ˜ Af (cid:33) , (3.8)where f ≡ f A f B (cid:112) f A + f B . (3.9)In this model, the regime of interest is πf A (cid:28) πf B (cid:46) M P , (3.10)Λ f (cid:29) m A A in , (3.11) Be aware of the difference between (2.9) and (3.11). A in is the initial condition set at the beginning of the observable inflation and werequire it to be in the range f (cid:28) A in < M P . Notice that the condition (3.10) implies inthe leading order in f A /f B cos γ (cid:39) , sin γ (cid:39) f A f B , f (cid:39) f A . (3.12)We require that the excitation in ˜ A direction is much heavier than the Hubble scale duringinflation so that they can be safely integrated out: ∂ ∂ ˜ A V ( ˜ A, ˜ B ) > H . (3.13)From (3.11) and f (cid:28) A in this reads 3 g A π L > H . (3.14)After integrating out ˜ A , we obtain the following effective potential for ˜ B which we rewriteas φ ≡ ˜ B [18]: V eff ( φ ) = m φ , m ≡ f A f B m A , (3.15)to leading order in f A /f B . Thus Dante’s Inferno model effectively reduces to chaoticinflation, with φ being the inflaton. The constraints from CMB observations on chaoticinflation are summarized in appendix C.1. Using these inputs, now we examine the CMBconstraints on the parameters of the higher-dimensional gauge theory. We will take thenumber of e-fold N ∗ (cid:39)
50 and the tensor-to-scalar ratio r = 0 .
16 (see appendix C for thedetail and the notations used below). From (3.3), the condition (3.10) reads in terms ofgauge theory parameters as g A (cid:29) g B , (3.16)and 1 g B (2 πL ) (cid:46) M P . (3.17)Chaotic inflation is a large field inflation model in which the inflaton travels trans-Planckian field distance ∆ φ ≡ φ ∗ − φ e (cid:39) M P , see (C.14). However, the original fields inthe current model, A and B (which were the zero-modes of the higher-dimensional gaugetheory), do not need to make trans-Planckian field excursion.Regarding the field A , its initial value A in is restricted as A in (cid:39) f A (cid:112) f A + f B φ ∗ (cid:39) f A f B × M P . (3.18)Thus A in is sub-Planckian if f A < f B . (3.19)9rom (3.3), in terms of gauge couplings (3.19) amounts to g A > g B . (3.20)This condition should be compared with (3.16). On the other hand, field B is periodicand its field range 2 πf 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 as3 π (2 πL ) (cid:29) f A (cid:18) m f B f A (cid:19) f A f B × M P . (3.21)Using (3.3) and putting the value of m in (C.15), we obtain g / B L > . × GeV . (3.22)Together with (3.17) we have g − / B × . × GeV < L (cid:46) g B × . × GeV . (3.23)(3.23) immediately implies g B (cid:38) .
04. On the other hand, in order for our one-loop effec-tive potential to be valid, the gauge coupling should not be large, g A (cid:46) O (1). Togetherwith (3.20), we have 0 . (cid:46) g B (cid:46) O (0 . . (3.24)For g B = 0 .
04 we have 9 . × GeV < L (cid:46) . × GeV , (3.25)while for g B = 0 . . × GeV < L (cid:46) . × GeV . (3.26)See Fig. 1 for the values of g B in between. We observe that the allowed values of the gaugecouplings and the compactification radius of the gauge theory are rather restricted, whichwill be advantageous for the model to be predictive. Note that the above compactificationscales are high enough so that the use of the four-dimensional Einstein gravity is justified,1 /L (cid:29) H ∼ GeV (see (C.11)).For completeness, we check that (3.13) is satisfied. It gives g A πL (cid:38) π √ H. (3.27)10 .4 (cid:180) g B (cid:180) g B (cid:45) (cid:144) (cid:180) (cid:180) (cid:180) g B L (cid:64) G e V (cid:68) Figure 1: Allowed range of L as a function of g B .Putting the value from appendix C (C.11) we obtain g A πL (cid:38) × 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 constrainedby CMB data. The shift symmetry allows the following axionic coupling to gauge fields: S AC = (cid:90) d x α i σ i f i F µν ˜ F µν , (3.29)where σ i is an axion, f i is its decay constant and α i is a constant parameter. i labelsaxions when there are more than one, in the current case i labels the field A and thefield 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. Contributionsto CMB power spectrum, non-Gaussianity and primordial gravitational waves throughthis coupling have been studied in [51, 52, 53, 54, 55]. These effects are mainly controlledby the following parameter: ξ i ≡ α i ˙ σ i f i H . (3.30)The current observational bound is given as [53, 55] ξ i (cid:46) . (3.31)To obtain ξ i ( i = A, B ) in (3.30), we first need to know the time derivatives of fields A and B . In ˜ A direction, we had ˙˜ A = 0 . (3.32)11n the other hand, ˜ B is the inflaton which slowly rolls down the potential. From (C.9)we estimate ˙˜ B H M P ∼ M P (cid:32) V (cid:48) ( ˜ B ) V ( ˜ B ) (cid:33) ∼ . . (3.33)From (3.32) and (3.33) we can estimate ˙ A and ˙ B through˙ A = sin γ ˙˜ B + cos γ ˙˜ A ∼ f A f B ˙˜ B, (3.34)˙ B = cos γ ˙˜ B − sin γ ˙˜ A ∼ ˙˜ B. (3.35)On the other hand, α i ( i = A, B ) can be obtained as in (A.17): α A = g A k A π , α B = g B k B π (3.36)Putting (3.33), (3.34), (3.35) and (3.36) into the definition (3.30), we arrive at ξ A (cid:46) g A k A π LM P × . , (3.37) ξ B ∼ g B k B π LM P × . , (3.38)In deriving (3.37) we have used (3.19). As we have assumed g A (cid:46) O (1), by putting (3.19)and L ∼ O (10 ) GeV, we obtain ξ A (cid:46) k A × O (10 − ) . (3.39)On the other hand, from g B (cid:46) O (0 .
1) in (3.24), we obtain ξ B (cid:46) k B × O (10 − ) . (3.40)As argued in appendix A.2, we expect k A , k B ∼ O (1 − ξ i (cid:46) i = A, B . 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 describedby the potential of the form V ( A, B ) = Λ (cid:18) − cos (cid:18) m f A A + n f B B (cid:19)(cid:19) + Λ (cid:18) − cos (cid:18) m f A A + n f B B (cid:19)(cid:19) . (4.1)Upon field rotation (cid:32) φ s φ l (cid:33) = (cid:32) cos ζ sin ζ − sin ζ cos ζ (cid:33) (cid:32) AB (cid:33) , (4.2)12ith cos ζ = f s f A m , sin ζ = f s f B n , (4.3) f s = 1 (cid:113) m f A + n f B , (4.4)the potential (4.1) takes the form V ( φ s , φ l ) = Λ (cid:18) − cos (cid:18) φ s f s (cid:19)(cid:19) + Λ (cid:18) − cos (cid:18) φ s f (cid:48) s + φ l f l (cid:19)(cid:19) , (4.5)where f l = (cid:112) m f B + n f A m n − m n , f (cid:48) s = 1 f s (cid:16) m m f A + n n f B (cid:17) . (4.6)The potential (4.1) can be obtained from a higher-dimensional gauge theory withfollowing action: S = (cid:90) d x (cid:104) − F AMN F AMN − F BMN F BMN (4.7) − i ¯ ψγ M ( ∂ M + ig A m A M + ig B n B M ) ψ − i ¯ χγ M ( ∂ M + ig A m A M − ig B n B M ) χ (cid:105) . The parameters in the potential (4.1) and the higher-dimensional gauge theory are relatedas f A = 1 g A (2 πL ) , f B = 1 g B (2 πL ) , (4.8)where g A and g B are four-dimensional gauge couplings g A ≡ g A √ πL , g B ≡ g B √ πL . (4.9)Anticipating UV completions such as string theory, it is natural that charges are quantizedwith respect to the unit charge. Thus we assume m , m , n , n are all integers. Aligned axion inflation is obtained in the regime | m n − m n | (cid:28) | m | , | n | . (4.10)In this regime one obtains | f l | (cid:29) f A , f B from (4.6). Notice that | f l | is at largest theorder of max( | m | f B , | n | f A ). On the other hand, as explained in appendix C.2, r (cid:39) .
16 requires | f l | (cid:38) M P . Since from WGC we have 2 πf A , πf B (cid:46) M P , this requires As we have assumed that the gauge groups are compact U (1), charges are quantized. Here we madea stronger assumption that charges are all integer multiples of the minimal charge in the theory. This canbe regarded as for simplicity, the result does not change qualitatively unless one assumes highly exoticcharge spectrum. | m | , | n | ) (cid:38) × π . A matter with such a large charge seems to us quite unnatural,considering that the energy scale under consideration is rather high ( H ∼ GeV).Next we turn to the hierarchical axion inflation in higher-dimensional gauge theory.This model corresponds to taking n = 0 in (4.1). Then (4.6) reduces to | f l | = (cid:112) m f B + n f A | n m | . (4.11)One further requires a hierarchy (cid:12)(cid:12)(cid:12)(cid:12) f A m (cid:12)(cid:12)(cid:12)(cid:12) (cid:28) f A | m | , f B | n | . (4.12)Then (4.11) can be approximated as | f l | (cid:39) (cid:12)(cid:12)(cid:12)(cid:12) m n m (cid:12)(cid:12)(cid:12)(cid:12) f B . (4.13)From WGC we have 2 πf B (cid:46) M P , thus | f l | (cid:38) M P requires | m | (cid:38) | n m | × π . Sucha large hierarchy between the charges in the same gauge group seems quite unnatural. In this paper we studied large field inflation models which can be obtained from higher-dimensional gauge theories. We accept WGC as our working hypothesis, and studiedthe constraints from CMB data on the gauge theory parameters. We consider the casewith large tensor-to-scalar ratio, and used r = 0 .
16 as a reference value. We found thatthe allowed range of gauge theory parameters are quite constrained. Among the modelsstudied in this paper, Dante’s Inferno model appears as the most preferred model. Theallowed values of the gauge couplings and the compactification radius turned out to bequite restricted but fell within a natural range, making the model attractive for beingpredictive. Single-field axion monodromy model leaves the problem that whether theshift symmetry is a good symmetry or not to its UV completion. Axion alignment modeland 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 range0 . − O (1). This is in contrast to the extremely small gauge coupling (cid:46) O (10 − ) required The upper bound of the gauge coupling in Table 2 for AA and AH were obtained by requiringapplicability of perturbation theory with these large charge number: In order for the perturbation theoryto be appropriate, we need gn ∼ O (1), where g is the gauge coupling and n is the maximal charge in themodel. Acknowledgments
KF benefited from the discussions on the BICEP2 results, naturalness and WGC at thePhysical Research Laboratory (PRL) and Indian Institute of Astrophysics (IIA) duringhis visits including the workshop “Aspects of Cosmology” at IIA held in April 9-11, 2014.In particular, he would like to express special thanks to Namit Mahajan at PRL andPravabati Chingangbam at IIA for the hospitality. YK’s work is supported in part bythe National Science Council of Taiwan under Grant No. NSC-101-2112-M-007-021 andTaiwan String Theory Focus Group of NCTS under Grant No. NSC-103-2119-M-002-001. The authors would also like to express their gratitude to the anonymous referee forvarious suggestions for improvements, in particular for pointing out possible relevance ofthe effects of the axionic couplings in the current analysis.
A Four-Dimensional Effective Action fromHigher-Dimensional Gauge Theory
A.1 One-loop Effective Potential
In this appendix we outline the calculation of the one-loop effective potential in higher-dimensional gauge theories compactified on a circle. We start with the five-dimensionalaction S = (cid:90) d x (cid:20) − F AMN F A MN − F BMN F B MN + 12 m A ( A M − g A ∂ M θ ) + ¯ ψi Γ M D M ψ (cid:21) + S g.f. , (A.1)where space-time indices M and N run 0 , · · · , F AMN = ∂ M A N − ∂ N A M , F BMN = ∂ M B N − ∂ N B M , (A.2)15nd D M ψ = ∂ M ψ − ig A pA M ψ − ig B qB M ψ. (A.3)We choose the gauge fixing term as S g.f. = (cid:90) d x (cid:20) −
12 ( ∂ M A M ) −
12 ( ∂ M B M ) (cid:21) . (A.4)Then the total action becomes S = (cid:90) d x (cid:104) A N ∂ M ∂ M A N + 12 B N ∂ M ∂ M B N + m A A M − g A ∂ M θ ) + ¯ ψi Γ M D M ψ (cid:105) . (A.5)We compactify the fifth dimension on a circle with radius L . The Fourier expansions ofthe fields in the fifth dimension are A M ( x, x ) = 1 √ πL ∞ (cid:88) n = −∞ A M ( n ) ( x ) e i nL x , similar for B M , ψ, (A.6) θ ( x, x ) = x g L w + 1 √ πL ∞ (cid:88) n = −∞ θ ( n ) ( x ) e i nL x . (A.7)We will be interested in the effective potential for the zero-modes of the gauge fields, A ≡ A and B ≡ B . At one-loop level, only the quadratic part of the matter actionis relevant: S (2) ψ = (cid:90) d x ∞ (cid:88) n = −∞ ¯ ψ ( n ) (cid:16) i Γ µ ∂ µ + g A p Γ A + g B q Γ B + Γ nL (cid:17) ψ ( n ) . (A.8)Here, µ and ν run four-dimensional space-time indices 0 , · · · ,
3. Then, the one-loopeffective potential is expressed as V ( A, B ) − loop = Tr ln (cid:16) − i Γ µE ∂ µE − g A p Γ E A + g B q Γ E B + Γ E nL (cid:17) = 12 Tr ln1l × (cid:20) − ∂ µE + (cid:16) nL − ( g A pA + g B qB ) (cid:17) (cid:21) , (A.9)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 g A = g A √ πL , g B = g B √ πL . (A.10)Employing the ζ function regularization, the effective potential becomes V ( A, B ) − loop = 3 π (2 πL ) ∞ (cid:88) n =1 n cos (cid:20) n (cid:18) pAf A + qBf B (cid:19)(cid:21) , (A.11)16here f A = 1 g A (2 πL ) , f B = 1 g A (2 πL ) . (A.12)In (A.11) we have dropped the constant part, the fine tuning of which is the cosmologicalconstant problem which we will not address in this paper. Taking the leading term n = 1in (A.11) together with the tree-level potential coming from the Stueckelberg mass term,we arrive at the potential V ( A, B ) (cid:39) m A A − πf w ) + 3 π (2 πL ) (cid:20) − cos (cid:18) pAf A + qBf B (cid:19)(cid:21) , (A.13)where we have redefined the field B by an appropriate constant shift. A.2 Axionic Couplings
The shift symmetry allows the following axionic coupling S AC = (cid:90) d x ασ f F µν ˜ F µν , (A.14)where σ is an axion and α is some constant. In higher-dimensional gauge theory, theaxionic coupling (A.14) follows from the Chern-Simons term in five-dimensional gaugetheory [56]: S CS = k π (cid:90) AF , (A.15)where A = A M dx M , F = d A = F MN dx M dx N and k is an integer. Quantum correctionsto k due to parity-violating charged matters are one-loop exact and proportional to thecubic powers of charges [58]. As we assume charges to be O (1), we may expect k ∼O (1 − A M dx M is related to the canonically normalized gauge field A M in five dimensions as A M = 1 g A M , (A.16)where g is the five-dimensional gauge coupling. After integrating KK modes of the fifthdirection we obtain the axionic coupling (A.14) with α = g k π , (A.17)and σ = A g . (A.18)17 Weak Gravity Conjecture
Weak Gravity Conjecture (WGC) [37] asserts the existence of a state with charge andmass ( q, m ) which satisfy gq √ π ≥ (cid:112) G N m = m √ πM P . (B.1)(B.1) is estimated from requiring that the Coulomb repulsive force is greater than theNewtonian attractive force so that extremal black holes can loose their charge by emittingsuch particles. In this paper we assume the existence of a particle with the smallest unitcharge, with respect to which all charges are integers. Generalization is straightforwardand dose not change the result qualitatively, unless one assumes highly exotic chargespectrum. Then, the Dirac monopole with unit magnetic charge has charge and mass q m = 4 πg , m m (cid:39) π Λ UV g , (B.2)where Λ UV is a UV scale which regularizes the mass of the Dirac monopole. Here, weused non-Abelian gauge-Higgs system as the UV completion to estimate the mass of theDirac monopole. An important constraint for our study is obtained by applying WGCthe Dirac monopole: 4 πg (cid:38) π Λ UV g √ M P . (B.3)It follows that Λ UV (cid:46) √ gM P . (B.4)This condition also follows by requiring that the Dirac monopole with unit magneticcharge is not a black hole [37]. Strictly speaking, one should take into account the runningof the couplings. We assume that those runnings are not significant so that they do notalter our order of magnitude estimate. In order for the higher-dimensional gauge theoryto be applicable, the compactification scale should be sufficiently below the UV cut-offscale: L (cid:28) √ gM P . (B.5)In terms of the axion decay constant f = 1 / ( g πL ),2 πf (cid:28) √ M P . (B.6)Since the above argument is an order estimate, in the main body we adopted slightlymilder bound 2 πf (cid:46) M P . More precisely we consider WGC in five dimensions [37]. In this case electro-magnetic dual to theone-form gauge potential is two-form gauge potential which couples to magnetic strings. Then the analysisof the forces in three spacial dimensions transverse to the string is the same. Relevant Inflation Models in Light of BICEP2
In this appendix we review the constraints from CMB observations, in particular thepossible detection of primordial tensor perturbation by BICEP2 [13], on inflation modelswhich are relevant in this paper. The detection of the B-mode polarization by BICEP2indicates 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 − as a reference value, considering the uncertaintyin the foreground [41] and the constraint from Planck 2013 [12, 43]. C.1 Chaotic Inflation with Quadratic Potential
Consider quadratic potential for the inflaton V ( φ ) = m φ . (C.1)We assume canonical kinetic term for the inflaton φ . The slow-roll parameters are givenby (cid:15) ( φ ) = M P (cid:18) V (cid:48) V (cid:19) = 2 M P φ , (C.2) η ( φ ) = M P V (cid:48)(cid:48) V = 2 M P φ . (C.3)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 n s = 1 − (cid:15) ∗ + 2 η ∗ . (C.4)Using (C.2) and (C.3) we obtain n s = 1 − . × r . . (C.5)The scalar power spectrum and the tensor power spectrum are given as P s = V ( φ ∗ )24 π M P (cid:15) ∗ = 2 . × − , (C.6) P t = 2 V ( φ ∗ )3 π M P , (C.7)where the last value in (C.6) is the COBE normalization. The tensor-to-scalar ratio r isgiven by r ≡ P t P s = 16 (cid:15) ∗ , (C.8)19r equivalently (cid:15) ∗ = 0 . × (cid:16) r . (cid:17) . (C.9)From (C.6) this requires V ( φ ∗ ) (cid:39) (2 . × GeV) × (cid:16) r . (cid:17) . (C.10)Via the Friedmann equation V (cid:39) H M P , (C.10) corresponds to the Hubble scale H ∗ (cid:39) . × × (cid:16) r . (cid:17) / GeV . (C.11)The slow-roll inflation ends when (cid:15) ( φ e ) ∼
1. This gives φ e ∼ √ M P . (C.12)The number of e-folds is given as N ∗ = (cid:12)(cid:12)(cid:12)(cid:12) M P (cid:90) φ ∗ φ e dφ VV (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) = 14 M P (cid:2) φ ∗ − φ e (cid:3) . (C.13)Thus φ ∗ = 2 M P (cid:114) N ∗ − (cid:39) M P × (cid:18) N ∗ − (cid:19) / . (C.14)Putting this value to (C.1) and comparing it with (C.10), we obtain m = (cid:115) V ∗ φ ∗ = 3 . × GeV × (cid:18) N ∗ − (cid:19) / × (cid:16) r . (cid:17) / . (C.15) C.2 Natural Inflation
The typical form of the potential for natural inflation is given by V ( φ ) = V (cid:20) − cos (cid:18) φf (cid:19)(cid:21) . (C.16)From (C.16) the slow-roll parameters are given as (cid:15) ( φ ) ≡ M P (cid:18) V (cid:48) V (cid:19) = M P f (cid:16) φf (cid:17) − cos (cid:16) φf (cid:17) , (C.17) η ( φ ) ≡ M P V (cid:48)(cid:48) V = M P f cos (cid:16) φf (cid:17) − cos (cid:16) φf (cid:17) . (C.18)20he number of e-folds as a function of φ is given by N ( φ ) (cid:39) (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) φ e φ dφ M P VV (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) φφ e dφ fM P − cos φf sin φf (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:18) fM P (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) log (cid:20) (cid:18) φf (cid:19)(cid:21) φφ e (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (C.19)The slow-roll inflation ends when (cid:15) ( φ e ) (cid:39)
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