Natural inflation and moduli stabilization in heterotic orbifolds
aa r X i v : . [ h e p - t h ] M a y DESY-15-040
Natural inflation and moduli stabilization in heterotic orbifolds
Fabian Ruehle , Clemens Wieck Deutsches Elektronen-Synchrotron DESY, Notkestrasse 85, 22607 Hamburg, Germany
Abstract
We study moduli stabilization in combination with inflation in heterotic orbifold compact-ifications in the light of a large Hubble scale and the favored tensor-to-scalar ratio r ≈ . η function. We present two setups inspiredby the mini-landscape models of the Z − II orbifold which realize aligned inflation and stabi-lization of the relevant moduli. One has a supersymmetric vacuum after inflation, while theother includes a gaugino condensate which breaks supersymmetry at a high scale. [email protected] [email protected] ontents Precision measurements of the CMB radiation increasingly favor the paradigm that the veryearly universe can be described by a phase of single-field slow-roll inflation [1, 2]. In particular,recent observations of polarization fluctuations in the CMB indicate the possibility of substantialtensor modes among the primordial perturbations [3, 4]. This necessitates large-field models ofinflation, i.e., the inflaton field must traverse a trans-Planckian field range during the last 60 e -folds of inflation [5]. Since large-field inflation is potentially susceptible to an infinite series ofPlanck-suppressed operators, this requires an understanding of possible quantum gravity effects.Thus, there has been renewed interest in obtaining inflation models from string theory.In this context natural inflation, first proposed in [6], is among the most promising can-didates. Here the flatness of the inflaton potential is guaranteed by an axionic shift sym-metry which is exact in perturbation theory, but potentially broken by non-perturbative ef-fects [7, 8]. Nevertheless, discrete symmetries may survive which protect the potential even attrans-Planckian field values. However, while axions are abundant in string theory compactifi-cations we still face a problem: trans-Planckian inflaton values require an axion decay constantwhich is larger than the Planck scale. However, in string theory one generically expects thedecay constant to be smaller than the string scale [9, 10].Different paths have been proposed to address this problem. In N-flation [11,12], for example,many axions with sub-Planckian decay constants contribute to the trans-Planckian field range ofthe inflaton, which is a linear combination of axions. However, this typically requires a very largenumber of axions which might be challenging to realize explicitly while maintaining control overthe models. Another option was considered in [13], where the authors obtain trans-Planckian1xions by choosing large gauge groups and by stabilizing the K¨ahler moduli at values much belowthe Planck scale. In that case, in principle one has to worry about perturbative control of thesupergravity approximation, i.e., stringy corrections may be important. Furthermore, there isaxion monodromy inflation [14, 15] which uses a single sub-Planckian axion with a multi-valuedpotential to create an effectively trans-Planckian field range during inflation.Another way of obtaining a large effective axion decay constant from a few number of axionsis by alignment as proposed in [16] and further developed in [17,18], or by kinetic alignment [19]. In the minimal setup of [16] there are two axions which appear as a linear combination in multiplenon-perturbative contributions to the superpotential. If the axion decay constants are almostaligned one obtains an effective axion with a large decay constant, although the individualdecay constants were small. In this paper we focus on the KNP alignment mechanism andits realization in E × E heterotic string theory [22] on orbifolds [23, 24]. Progress in thisdirection has recently been made in [25], where the authors embedded aligned natural inflationin a supergravity model motivated by heterotic string compactifications on smooth Calabi-Yaumanifolds with vector bundles. However, the authors did not specify the mechanism of modulistabilization or an underlying reason for the alignment of the non-perturbative terms. Theauthors of [26] proposed a related model of hierarchical axion inflation and how it could beembedded in type IIB string theory. For other attempts to embed aligned natural inflation intype IIB string theory see [27–32], and [33] for a related analysis.We study whether alignment of heterotic axions may be achieved by considering world-sheetinstantons or a combination of the latter with gaugino condensates. Since the contributionsarise from completely different mechanisms a natural question arises: why should the two effectsbe aligned? We attempt to answer this question, focusing our discussion on heterotic orbifoldswhere the moduli dependence of both effects, the condensing gauge group and the world-sheetinstantons, can be computed using methods of conformal field theory. We argue that an align-ment of the two terms is not as unnatural as one may think, essentially because the modulidependence of both effects is determined by modular weights and Dedekind η functions.Furthermore, we address the issue of consistent moduli stabilization. Whenever inflation isdiscussed in string theory one desires a hierarchy of the form M s , M KK > M moduli > H , (1)where M s denotes the string scale, M KK the Kaluza-Klein scale, and H is the Hubble scaleduring inflation. This hierarchy is essential to ensure that inflation can be described by aneffective four-dimensional supergravity theory where the inflaton is the only dynamical degreeof freedom. In addition, in case of metastable vacua the barriers protecting the minima of themoduli must be larger than H . This is to avoid moduli destabilization during inflation aspointed out in [34, 35]. Using the terms needed for successful inflation and other contributionsto the superpotential we provide such a hierarchy explicitly for the complex dilaton field and See [20, 21] for related alignment mechanisms. Z − II orbifold of the mini-landscape models [39, 40]. Section 5 contains our conclusionsand an outlook. In this section we briefly review those properties of heterotic orbifolds relevant for our discussion.A good and detailed review can, for example, be found in [41]. References [42, 43] discuss therelevance of these ingredients for moduli stabilization.In the construction of Abelian heterotic toroidal orbifolds one starts with a six-torus T parameterized by three complex coordinates z , , and mods out a discrete Z N symmetry θ , θ : ( z , z , z ) ( e π i n /N z , e π i n /N z , e π i n /N z ) = ( e π i v z , e π i v z , e π i v z ) , (2)where we have defined the twist vector v = ( v , v , v ). Requiring that the resulting singu-lar space is Calabi-Yau imposes v + v + v ∈ Z . The Z − II orbifold, for example, has v = (1 / , − / , / N twisted sectors θ k , k = 0 , . . . , N −
1. To ensure modular invariance of the one-loop stringpartition functions, these twists have to be accompanied by a shift in the E × E gauge degreesof freedom. This shift is parameterized by the shift-vector V . In addition, depending on thegeometry one can allow for up to six independent Wilson lines W i on the torus.The massless string spectrum is given in terms of the twist v , the shift V , and the Wilsonlines W i . In addition to the usual untwisted strings in the θ sector, which close already onthe torus, it contains new string states in twisted sectors θ k which are called twisted strings.These only close under the orbifold action and are thus forced to localize at orbifold fixed points. A similar discussion applies to the case of Z M × Z N orbifolds. T i , i = 1 , . . . , h , and complex structure moduli U j , j = 1 , . . . , h , may vary. In the Z − II case,for example, one has h , = 3 and h , = 1. The T i parameterize the size of the three T sub-toriwhile U parameterizes the shape of the T on which the orbifold has a Z action. The K¨ahler and complex structure moduli have an SL(2 , Z ) symmetry under which the T i transform as T i → a i T i − i b i i c i T i + d i , (3)and likewise for the moduli U j . Here, a i , b i , c i , d i ∈ Z and a i d i − b i c i = 1.At zeroth order the K¨ahler potential of the moduli reads K moduli = − h , X i =1 ln( T i + T i ) − h , X j =1 ln( U j + U j ) . (4)It is readily checked that under the transformation (3) the K¨ahler potential transforms as K moduli → K moduli + h , X i =1 ln | i c i T i + d i | + h , X j =1 ln | i c j U j + d j | . (5)Hence the shift symmetry of the moduli in the K¨ahler potential is protected by the modularsymmetry. Since G = K moduli + K matter +ln | W | , which appears in the supergravity Lagrangian,has to be invariant we find that the superpotential has to transform with modular weight − W → W h , Y i =1 (i c i T i + d i ) − h , Y j =1 (i c j U j + d j ) − . (6)In addition to the K¨ahler and the superpotential, also the chiral fields have non-trivial modulartransformations, Φ α → Φ α h , Y i =1 (i c i T i + d i ) m iα h , Y j =1 (i c j U j + d j ) ℓ jα . (7)The modular weights m iα and ℓ jα depend on the orbifold twisted sector k and oscillator numbers.Defining w i ( k ) = kv i mod 1, they are given by [44–46] m i = ( , if w i = 0 ,w i − − e N i + e N i ∗ , if w i = 0 .ℓ j = ( , if w j = 0 ,w i − e N j − e N j ∗ , if w j = 0 . (8)4ere, the e N i and e N i ∗ are integer oscillation numbers. In the p th complex plane of the untwistedsector we have m ip = − δ ip , ℓ jp = − δ jp . From this we find for the K¨ahler potential for the matterfields at lowest order K matter = X α h , Y i =1 (cid:0) T i + T i (cid:1) m iα h , Y j =1 (cid:0) U j + U j (cid:1) ℓ jα | Φ α | . (9)Since the matter fields transform non-trivially and the superpotential has to have modularweight −
1, the coupling “constants” y α ...α L of the L -point correlator W ⊃ y α ...α L Φ α . . . Φ α L (10)have to be appropriate modular functions such that the overall modular weight is −
1. Specifi-cally, y α ...α L Φ α . . . Φ α L ∝ h , Y i =1 h , Y j =1 η ( T i ) r i η ( U j ) s j Φ α . . . Φ α L , (11)where η denotes the Dedekind η function defined by η ( T ) = e − πT ∞ Y ρ =1 (cid:0) − e − πρT (cid:1) , (12)and the constant parameters r i and s j are determined by the modular weights, r i = − − X α m iα , s j = − − X α ℓ jα . (13)The Dedekind η function transforms under modular transformations up to a phase, η ( T ) → (i cT + d ) / η ( T ) . (14)For T > η ( T ) = e − πT . (15)As a result the non-perturbative superpotential terms are of the schematic form W WSNP = A (Φ α ) e − π ( P i r i T i + P j s j U j ) . (16)Note that, if the fields Φ α are charged under an anomalous U(1) symmetry, S may appear in theexponent as well. In particular, this is the case when the model-independent axion contained in S cancels the anomalies, as explained in more detail below. In general, other modular functions can appear as well [47–50]. .2 Anomalous U(1) and FI terms In orbifold models with shift embeddings, the primordial E × E gauge symmetry is broken rank-preservingly into Abelian and non-Abelian gauge factors. Generically one U(1) is anomalous,henceforth denoted by U(1) A . This anomaly is canceled via a Green-Schwarz (GS) mechanism[51]. More precisely, the dilaton S transforms under such an anomalous gauge variation as S → S − iΛ δ GS , where Λ is the superfield gauge parameter and δ GS is a real constant. As aconsequence the combination S + S − δ GS V A is gauge-invariant, where V A is the vector multipletassociated with U(1) A .The non-trivial U(1) A transformation of S has two important consequences. First, we observethat GS anomaly cancellation results in a field-dependent Fayet-Iliopoulos (FI) term of the form ξ = δ GS ( S + S ) . (17)In order to preserve D -flatness, this means that some chiral orbifold fields Φ α with appropriatecharge must get a vacuum expectation value (VEV) to cancel ξ . The VEV of these fields can,at the same time, break unwanted extra gauge groups and lift vector-like exotics and otherextra hidden fields in a Higgs-like mechanism. Generically, the primordial E × E is brokento many U(1) factors under which the orbifold fields are charged simultaneously. Hence, D -flatness of the other U(1) symmetries requires that many fields obtain a non-vanishing VEV.Second, superpotential terms involving the dilaton in the exponent have to be such that thewhole correlator is gauge-invariant.Moreover, S has a non-trivial modular transformation to ensure anomaly cancellation in theunderlying sigma-model [54, 55]: S → S + 18 π h , X i =1 δ i ln(i c i T i + d i ) + h , X j =1 δ j ln(i c j U j + d j ) , (18)where δ i and δ j are real constants of order 1 that can be computed from the sigma-modelanomaly cancellation condition. As a consequence, the modular invariant K¨ahler potential ofthe dilaton reads K dilaton = − ln( Y ) = − ln S + S + 18 π h , X i =1 δ i ln( T i + T i ) + h , X j =1 δ j ln( U j + U j ) . (19)Due to the loop suppression factor 8 π , these corrections are small as long as the T i are notstabilized at substantially larger field values than S . This is not the case in the models we study. Notice that this commonly used terminology is slightly misleading. A field-dependent FI term is usually the D -term of a complex field with a logarithmic K¨ahler potential, which, if integrated out at a high scale, may mimica constant FI term as the one introduced in [52]. We refer to the original discussion in [53] for more details. .3 Gauge kinetic function and gaugino condensation The one-loop gauge kinetic function of a gauge group G a at Kaˇc–Moody level 1 is given by[44, 54, 56] f a ( S, T, U ) = S + 18 π h , X i =1 b ia ( m ) g i N ln( η ( T i )) + 18 π h , X j =1 b ja ( ℓ ) g j N ln( η ( U j )) , (20)where b ia are the β -function coefficients in the i th torus of the gauge group G a . They are non-vanishing in the N = 2 twisted sub-sectors of the theory and depend on the Dynkin indices andon the modular weights of the states charged under G a . Furthermore, the g i are the order ofthe little group of the orbifold action in the i th torus, i.e., the order of the group that leavesthe i th torus fixed. Depending on the lattice and the presence of Wilson lines, the modularsymmetry group SL(2, Z ) might be reduced such that only a subgroup Γ ( N/g i ) or Γ ( N/g i ) isrealized [57–59]. In the example of the factorized Z − II orbifold the N = 2 twisted sectors are θ k with k = 2 , , N = 6, g = 2, g = 3 and the modular group is not reduced.The gauginos of G a may condense at a scale Λ GC a which depends on the low-energy effective N = 1 β -function, given by β a = 113 C ( Ad a ) − C ( Ad a ) + X ψ R a C ( R a ) − X φ R a C ( R a ) , (21)where C ( R a ) is the quadratic Casimir operator of the irreducible representation R a . Λ GC a canthen be written in terms of the gauge kinetic function as [60, 61],Λ GC a = e − π βa f a ( S,T,U ) . (22)As discussed above, all extra fields become massive. If their mass is larger than the condensationscale they can be integrated out. The β -function (21) is then simply 3ˇ c , where ˇ c denotes thedual Coxeter number. The effective superpotential term generated by gaugino condensation is ∝ (Λ GC a ) . In addition, it depends on the fields charged under the condensing gauge group andon the fields that get a VEV and give an effective mass term to those fields. The final expressioninvolves, in addition to the N = 2 beta function of the condensing gauge group, the modularweights of the fields that enter in the condensate. To obtain the final expression, we insert (20)into (22), and include a field-dependent pre-factor from integrating out the heavy fields [62].Using the transformation behavior of the dilaton (18) and requiring that the result has againmodular weight −
1, we find W GCNP = B (Φ ρ ) e − π c S + P i ( − δi ˇ c ) ln η ( T i )+ P j ( − δj ˇ c ) ln η ( U j ) . (23)Hence, we observe that both the non-perturbative world-sheet instanton contributions (16) andthe non-perturbative gaugino condensation terms (23) depend on the modular weights and onthe Dedekind η function. The combined superpotential, using (20) and (15), has the schematicform W ⊃ Y α Φ α e − P α qαδ GS Y − π ( P i r i T i + P j s j U j ) + B (Φ ρ ) e − π c S + π ( P i b i T i + P j b j U j ) , (24)7here q α are the U(1) A charges of the fields Φ α . Note that the modular weights m and ℓ arenegative and such that r i , s j ≥ b i and b j also depend onthe modular weights and in addition on the N = 2 beta function coefficients, b i = 1 − X i δ i ˇ c , b j = 1 − X j δ j ˇ c . (25)As mentioned before, the δ i and δ j are typically of order 1 so that b i , b j ≈
1, especially for largegauge groups. Note that in many couplings at least some of these constants are zero and hencethe corresponding modulus does not appear in those superpotential terms.
Let us now discuss how inflation can be realized in heterotic orbifold compactifications. Webriefly review the alignment mechanism proposed in [16,17] and subsequently put the ingredientsof Section 2 together to build an aligned axion inflation model with all moduli stabilized at ahigh scale.
Remember that alignment means, on the level of the effective potential for two axions τ , , V = κ (1 − cos( β τ + β τ )) + κ (1 − cos( n τ + n τ )) , (26)that there is a flat direction if β n = β n . (27)Notice that the coefficients β i and n i are the inverse of the axion decay constants. To slightlylift this flat direction one can introduce a small misalignment parameterized by [17] k := 1 n − β β n , (28)which vanishes for perfect alignment. After rotating to a convenient field basis, ( τ , τ ) ( ϕ , ϕ )and canonically normalizing the kinetic terms, we obtain for the almost flat direction ϕ an ef-fective decay constant f eff which reads [17, 25] f eff = β q ( β − + β − )( β − + n − ) kn β . (29)It is arbitrarily large for arbitrarily small k and hence closely aligned axions τ i . A sizeabletensor-to-scalar ratio r ≈ .
05 requires a misalignment of k ≈ . .2 Alignment and moduli stabilization on orbifolds A complete treatment of stabilizing all moduli while keeping three MSSM generations of particlesand one pair of Higgs fields with realistic Yukawa couplings, decoupling extra vector-like exotics,and breaking additional U(1) symmetries generically present in these models is beyond the scopeof this paper. Moduli stabilization in similar setups without considering inflation has beeninvestigated in [42, 43]. However, the mechanisms used there typically yield masses below thecurrently favored large Hubble scale and are thus incompatible with single-field inflation.From the discussion in Section 2 it should be clear that the effective potential (26) is sourcedby a superpotential with two non-perturbative terms, both of which contain two K¨ahler moduli T and T . In particular, we mostly focus on the two K¨ahler moduli which correspond to thetori that have an N = 2 sub-sector. In fact, all orbifolds have at least three untwisted K¨ahlermoduli and up to three untwisted complex structure moduli. Concerning their stabilization, notethat those K¨ahler moduli which correspond to tori that have fixed points in all twisted sectors θ k do not enter in the gauge kinetic function and thus can only be stabilized via world-sheetinstantons. Whether they appear in a world-sheet instanton coupling depends on the modularweights as discussed above. For the sake of simplicity we assume that the moduli not involved inthe stabilization or alignment mechanism, as well as other potentially present fields, have beenstabilized at a scale above H and consequently decouple from inflation.The real parts of the T i govern the size of the compactification manifold. The imaginaryparts, albeit not involved in the anomaly cancellation except for the small one-loop contri-bution, enjoy an axionic shift symmetry inherited from the SL(2, Z ) symmetry. They yield acosine-potential as in (26) and can consequently be used as inflaton candidates. The real partof the complex dilaton field determines the gauge coupling strength while its imaginary partis the so-called universal axion which is responsible for Green-Schwarz anomaly cancellation,cf. Section 2.2. For a suitable choice of the superpotential, comprised of the terms genericallyavailable in orbifold compactifications, the effective potential after integrating out all moduliand the universal axion takes the form (26). In the following we discuss which parts of thesuperpotential may achieve this while ensuring consistent stabilization of the aforementionedrelevant moduli. Inflation with world-sheet instantons only
A first option is to employ only world-sheet instanton contributions. For two aligned K¨ahlermoduli this has recently been discussed in [18], based on the mechanism proposed in [63]. Weextend this to include dilaton stabilization by considering the part of the orbifold superpotential In the prime orbifolds Z and Z no torus has an N = 2 sub-sector while in the Z M × Z N orbifolds all threetori do. W = χ (cid:2) A ( φ α , χ β ) e − n T − n T − P ( χ γ ) (cid:3) + χ (cid:2) A ( φ µ , χ ν ) e − n T − n T − P ( χ ρ ) (cid:3) + χ h A ( χ σ ) e − qδ GS S − P ( χ λ ) i , (30)where the χ i and φ i are untwisted and twisted chiral superfields, respectively, and n i = π r i in the notation of (24). Note that we have neglected the loop contributions to S . To obtainthe correct T i dependence in the various terms twisted fields necessarily enter the functions A i ,while the P i are functions of untwisted moduli. Since untwisted fields have modular weight − A and A arises from the twisted fields. The discussion has again been tailoredto the Z − II orbifold. For other orbifolds, especially for Z M × Z N orbifolds, there also existcouplings that involve only twisted states which nevertheless have modular weight −
1, so thatno extra T i occur in these terms.In the above parameterization we assume that the fields entering A i and P i obtain non-vanishing vacuum expectation values via F - and D -terms. In our supergravity analysis we treatthem as numerical constants given by the VEVs of these fields. Those VEVs are genericallyof the order of the string scale, M s . .
1. We assume that the other fields we have not madeexplicit obtain a mass in a similar way from couplings to fields that get a VEV.The effective theory defined by (30) and the K¨ahler potential discussed in Section 2 has asupersymmetric Minkowski vacuum at h χ i = h χ i = h χ i = 0. The auxiliary fields of the χ i stabilize the complex scalars S , T , and T at mass scales determined by the A i and P i . In theheterotic mini-landscape models of [39, 40] there are many examples in which the coefficients n i are such that sufficient alignment is possible. There is then a light linear combination of T and T whose imaginary part is the inflaton field. All other degrees of freedom can besufficiently stabilized in many examples. More details and an explicit example which realizesthe hierarchy (1) are given in Section 4.After inflation has ended supersymmetry must be broken to avoid phenomenological prob-lems. As pointed out in [18] the above scheme can accommodate low-energy supersymmetrybreaking, for example, via the F -term of a Polonyi field. A more generic situation on orbifoldsis supersymmetry breaking via gaugino condensates. As is well-known, these can also lead toa suppression of the supersymmetry-breaking scale compared to the Hubble scale. From theperspective of aligned inflation this is desirable in the above setup, since the gaugino condensatemust not interfere with the alignment of T and T . No matter how supersymmetry is broken,the resulting vacuum will have a positive cosmological constant which is determined by the scaleof supersymmetry breaking. This must be cancelled by a fine-tuned constant contribution tothe superpotential to high accuracy.Since many non-Abelian gauge groups arise from the breaking of E × E the appearanceof gaugino condensates is quite generic in orbifolds. In the following we discuss an example inwhich a gaugino condensate participates in the alignment mechanism.10 nflation with world-sheet instantons and gaugino condensates A second option to achieve alignment and moduli stabilization is to use a combination of gauginocondensates and world-sheet instantons. In this case supersymmetry is necessarily broken ata high scale in order to stabilize all fields above the Hubble scale. In many models we findsuperpotentials of the form (24), or more specifically W = χ (cid:20) B e − π c S + β T + β T − P (cid:21) + χ (cid:2) A e − n T − n T − P (cid:3) + χ h A e − qδ GS S − P i , (31)with β i = π b i . The notation and the field dependence of the A i , P i , n i is as in the previousexample, and again we assume them to be constants arising from other fields that obtain a VEV.As explained in Section 2.3 the β i depend on the particle content of the N = 2 sub-sector andthe modular weights of χ and the fields entering B . In the effective theory of inflation B isassumed to be constant as well. It is in general a non-analytic function of mesonic degrees offreedom which are integrated out above the scale of gaugino condensation. As explained in moredetail in [42], B h χ i determines the meson mass in the vacuum, which must be larger than H and the condensation scale to ensure decoupling in the effective theory and during inflation.This means that h χ i 6 = 0 in such setups. This is typically guaranteed by D -terms associatedwith U(1) A or, as in the above case, other U(1) symmetries. This means that the superpotentialin (31) yields a type of racetrack potential for the moduli, which are stabilized by their own F -terms and those of the χ i .Moduli stabilization via the superpotential in (31) generically yields dS vacua since su-persymmetry is broken by the gaugino condensate. The scale of supersymmetry breaking isproportional to B h χ i and necessarily lies, as explained above, close to the inflationary Hubblescale. However, to avoid a potentially destructive back-reaction of the auxiliary fields respon-sible for supersymmetry breaking, cf. the discussion in [64], one must find examples in whichthe gravitino mass is not substantially larger than H . We demonstrate that this is possible in asecond benchmark model in Section 4. Let us now turn to two examples. We chose to discuss inflation in the context of the Z − II mini-landscape models because these are the most-discussed models in the literature. However,the mechanisms discussed here apply to most orbifolds in a similar vein. Let us start with the situation described in Section 3.2, where we stabilize the moduli via world-sheet instantons only. We assume that some untwisted and twisted fields Φ α have obtained astring-scale VEV from D -terms which we do not include explicitly here. As explained above, wetake the χ i to be untwisted and the φ i to be twisted matter fields. Furthermore, we consider χ to carry U(1) A charge q = 1. 11 A A P P P . · − . · − . · − . · − . · − . · − Table 1: Input parameters for the constants used in Example 1. The A i and P i arise from 3- and 4-point couplings. h T i = h T i h T i = h T i f eff n s r .
06 1 .
24 5 . .
96 0 . Table 2: CMB observables and other relevant parameters for 60 e -folds of inflation in Example 1. The K¨ahler potential in this case reads K = − ln (cid:0) S + S (cid:1) − ln (cid:0) T + T − | χ | (cid:1) − ln (cid:0) T + T − | χ | (cid:1) + | χ | , (32)where we have neglected the loop contributions to S . The contributions (9) of the twisted fieldsdo not affect the results of our discussion as long as all VEVs of the matter fields are of theorder of the string scale or below. We consider the part of the full superpotential given in (30).The possible values for the modular weights n i are taken from the orbifolder [65], in thiscase n = π/ n = π/ n = π/
3, and n = π/
2. In typical models δ GS ∼ O (0 .
1) and theU(1) A charges of the fields entering A are O (1), such that we obtain an overall prefactor of S of order 1. The VEVs of the fields entering A cancel the D -term induced by δ GS . Theremaining input parameters for this example are summarized in Table 1. The resulting theoryhas a supersymmetric vacuum at h χ i = h χ i = h χ i = h Im S i = 0 and h Re S i ≈ .
8. Thelightest eigenvalue in the mass matrix corresponds to the aligned linear combination of T and T . A convenient field basis is therefore T → ˜ T = aT + bT , T → ˜ T = − bT + aT , (33)with a ≈ − .
64 and b ≈ − .
77 in this case. ˜ T is the lightest direction and Im ˜ T is the inflaton.In the vacuum its real part is as heavy as the inflaton because supersymmetry is unbroken. Thus,one may worry that it contributes quantum fluctuations to the system, yielding a multi-fieldinflation model. However, during inflation Re ˜ T receives a soft mass term of the same order asthe Hubble scale. Indeed, a numerical analysis of the coupled equations of motion, similar tothe one carried out in [18], reveals that all fields except the inflaton are sufficiently stabilizedduring inflation. For 60 e -folds of slow-roll inflation we summarize the predictions for the CMBobservables and other relevant parameters in Table 2.Apparently, successful inflation in line with recent observations is possible in this setup. How-ever, since we have chosen to only employ a portion of the total superpotential of such orbifoldmodels, one may worry about additional terms which can interfere with moduli stabilization orthe alignment mechanism. In particular, there may be terms of the form W ⊃ C (Φ α ) e − f ( T ,T ) , (34)12 A A P P P . . · − . · − . · − . · − . · − Table 3: Input parameters for the constants used in Example 2. The A i and P i arise from 3- and 4-point couplings. h S i = h S i h T i = h T i h T i = h T i h χ i h χ i h χ i . .
97 1 .
16 9 . · − − . · − − . · − Table 4: Vacuum expectation values of all relevant fields in the dS minimum with h V i ≈ · − . In addition,the imaginary parts of the χ i obtain VEVs much smaller than 1. where the function f contains some linear combination of the two moduli. On the one hand,this term clearly breaks supersymmetry if C (Φ α ) = 0. The effects on inflation, however, are notsignificant as long as the resulting gravitino mass is not much larger than H , which is genericallyfulfilled. On the other hand, the additional dependence on the moduli may interfere with thealignment of the effective inflaton field. We have verified that this is negligible as long as C < A i .This means that additional terms of this type must be suppressed up to slightly higher orderthan the ones in the part of the superpotential we consider. The setup which includes a gaugino condensate is slightly more generic, but also more compli-cated. Similar to the previous example the K¨ahler potential reads K = − ln (cid:0) S + S (cid:1) − ln (cid:0) T + T − | χ | (cid:1) − ln (cid:0) T + T − | χ | (cid:1) + | χ | , (35)where we have once more neglected the loop-suppressed correction to the dilaton K¨ahler potentialand the contribution of the twisted matter fields. The superpotential is this time given by (31)with n = π/ n = π/ β = π/ β = π/ q/δ GS = 1. As in the previous example, theFI term of U(1) A is canceled by the VEVs of the fields entering A . Note that χ cannot cancelthis FI term since we assume in our example that its charge has the wrong sign. Nevertheless,on orbifolds fields are typically charged under many U(1) factors simultaneously. To account forthis, we include a D -term ζ originating from another U(1) under which χ has charge − V D = 1 S + S ( χ K χ − ζ ) , (36)with ζ = 10 − . This D -term is canceled by h χ i 6 = 0, which results in a non-vanishing VEVof the other fields, h χ , i 6 = 0. All other input parameters are summarized in Table 3. Theresulting scalar potential has a dS vacuum specified in Table 4. The positive vacuum energycan be cancelled by a fine-tuned constant contribution to W , and the gravitino mass in thenear-Minkowski vacuum is m / ≈ . · − . There is again a lightest direction in the massmatrix which is ˜ T with a ≈ − .
82 and b ≈ − .
56, and its imaginary part is the inflaton. Once13 eff n s r . .
96 0 . Table 5: CMB observables for 60 e -folds of inflation and the effective axion decay constant in Example 2. more we solve the coupled equations of motion to ensure that all other degrees of freedom aresufficiently stable during inflation. The CMB predictions for this second case are summarizedin Table 5. Further contributions to the superpotential must satisfy the same constraints as inExample 1 to not interfere with inflation. We have analyzed the feasibility of natural inflation with consistent moduli stabilization inheterotic orbifold compactifications. To allow for the trans-Planckian axion field range favoredby recent observations of the CMB polarization, we implement aligned natural inflation withtwo axions. Generic properties of orbifolds naturally permit sufficient alignment for 60 e -folds ofslow-roll inflation with a detectable tensor-to-scalar ratio and a scalar spectral index of n s ≈ . × E . The axions which mix are the imag-inary parts of two complex untwisted K¨ahler moduli, governing the size of two tori. A crucialobservation is that both possible non-perturbative effects are determined by the modular weightsof the fields involved in the coupling and the Dedekind η function. This leads to many instan-tonic couplings with similar coefficients in the exponential, corresponding to the individual axiondecay constants, which in turn allows for aligned inflation. Since any embedding of inflation instring theory must address moduli stabilization, we demonstrate how both K¨ahler moduli andthe dilaton can be stabilized at a high scale. This can happen through the terms needed forinflation and additional terms involving the VEVs of twisted and untwisted matter fields.In the case of two world-sheet instantons all axion coefficients are determined by sumsof modular weights and the Dedekind η function. Thus, the more fields are involved in thecorrelator, the larger the coefficients of the moduli in the instantonic terms. This way, couplingsgenerated at fourth or higher order generically have coefficients which allow for just the rightamount of alignment. The case in which inflation is driven by a world-sheet instanton anda gaugino condensate is more constrained, and thus more predictive. The coefficients in thegaugino condensate are fixed by symmetry arguments and the Dedekind η function. Alignmentcan occur when the world-sheet instanton coupling is introduced at sufficiently high order. Inboth cases, additional terms in the superpotential do not interfere with inflation or modulistabilization, as long as their magnitude is below the inflationary Hubble scale.14e provide benchmark models for both cases to illustrate our findings. In the first case wefind a supersymmetric Minkowski vacuum in which the flattest direction is a linear combinationof the two K¨ahler moduli, the imaginary part of which is the aligned inflaton. All other degreesof freedom are stabilized at a higher scale and decouple from inflation. During inflation thereal part of the aligned modulus receives a Hubble-scale soft mass and is sufficiently stable aswell. This situation is similar in the second case, although in the vacuum supersymmetry isspontaneously broken by the gaugino condensate with m / . H . Acknowledgments
We thank Wilfried Buchm¨uller, Emilian Dudas, Rolf Kappl, Hans Peter Nilles, Martin Winkler,and Alexander Westphal for useful discussions. This work was supported by the German ScienceFoundation (DFG) within the Collaborative Research Center (SFB) 676 “Particles, Strings andthe Early Universe”. The work of C.W. has been supported by a scholarship of the JoachimHerz Foundation.
References [1]
Planck
Collaboration P. Ade et al. “Planck 2015 results. XIII. Cosmological parameters,” [ ].[2]
Planck
Collaboration P. Ade et al. “Planck 2015. XX. Constraints on inflation,” [ ].[3]
BICEP2 Collaboration
Collaboration P. Ade et al. “Detection of B -Mode Polarization at Degree AngularScales by BICEP2,” Phys.Rev.Lett. (2014) no. 24, 241101 [ ].[4]
BICEP2 Collaboration, Planck Collaboration
Collaboration P. Ade et al. “A Joint Analysis of BI-CEP2/Keck Array and Planck Data,”
Phys.Rev.Lett. (2015) [ ].[5] D. H. Lyth “What would we learn by detecting a gravitational wave signal in the cosmic microwave back-ground anisotropy?,”
Phys.Rev.Lett. (1997) 1861–1863 [ hep-ph/9606387 ].[6] K. Freese, J. A. Frieman, and A. V. Olinto “Natural inflation with pseudo - Nambu-Goldstone bosons,” Phys.Rev.Lett. (1990) 3233–3236.[7] X. Wen and E. Witten “World Sheet Instantons and the Peccei-Quinn Symmetry,” Phys.Lett.
B166 (1986)397.[8] M. Dine and N. Seiberg “Nonrenormalization Theorems in Superstring Theory,”
Phys.Rev.Lett. (1986)2625.[9] T. Banks, M. Dine, P. J. Fox, and E. Gorbatov “On the possibility of large axion decay constants,” JCAP (2003) 001 [ hep-th/0303252 ].[10] P. Svrcek and E. Witten “Axions In String Theory,”
JHEP (2006) 051 [ hep-th/0605206 ].[11] S. Dimopoulos, S. Kachru, J. McGreevy, and J. G. Wacker “N-flation,”
JCAP (2008) 003[ hep-th/0507205 ].[12] S. A. Kim and A. R. Liddle “Nflation: multi-field inflationary dynamics and perturbations,”
Phys.Rev.
D74 (2006) 023513 [ astro-ph/0605604 ].[13] H. Abe, T. Kobayashi, and H. Otsuka “Towards natural inflation from weakly coupled heterotic stringtheory,” [ ].[14] E. Silverstein and A. Westphal “Monodromy in the CMB: Gravity Waves and String Inflation,”
Phys.Rev.
D78 (2008) 106003 [ ].
15] L. McAllister, E. Silverstein, and A. Westphal “Gravity Waves and Linear Inflation from Axion Monodromy,”
Phys.Rev.
D82 (2010) 046003 [ ].[16] J. E. Kim, H. P. Nilles, and M. Peloso “Completing natural inflation,”
JCAP (2005) 005[ hep-ph/0409138 ].[17] R. Kappl, S. Krippendorf, and H. P. Nilles “Aligned Natural Inflation: Monodromies of two Axions,”
Phys.Lett.
B737 (2014) 124–128 [ ].[18] R. Kappl, H. P. Nilles, and M. W. Winkler “Natural Inflation and Low Energy Supersymmetry,”[ ].[19] T. C. Bachlechner, M. Dias, J. Frazer, and L. McAllister “Chaotic inflation with kinetic alignment of axionfields,”
Phys.Rev.
D91 (2015) no. 2, 023520 [ ].[20] S. H. H. Tye and S. S. C. Wong “Helical Inflation and Cosmic Strings,” [ ].[21] K. Choi, H. Kim, and S. Yun “Natural inflation with multiple sub-Planckian axions,”
Phys.Rev.
D90 (2014)no. 2, 023545 [ ].[22] D. J. Gross, J. A. Harvey, E. J. Martinec, and R. Rohm “The Heterotic String,”
Phys. Rev. Lett. (1985)502–505.[23] L. J. Dixon, J. A. Harvey, C. Vafa, and E. Witten “Strings on Orbifolds,” Nucl. Phys.
B261 (1985) 678–686.[24] L. J. Dixon, J. A. Harvey, C. Vafa, and E. Witten “Strings on Orbifolds. 2,”
Nucl. Phys.
B274 (1986)285–314.[25] T. Ali, S. S. Haque, and V. Jejjala “Natural Inflation from Near Alignment in Heterotic String Theory,”[ ].[26] I. Ben-Dayan, F. G. Pedro, and A. Westphal “Hierarchical Axion Inflation,”
Phys.Rev.Lett. (2014)no. 26, 261301 [ ].[27] C. Long, L. McAllister, and P. McGuirk “Aligned Natural Inflation in String Theory,”
Phys.Rev.
D90 (2014)no. 2, 023501 [ ].[28] X. Gao, T. Li, and P. Shukla “Combining Universal and Odd RR Axions for Aligned Natural Inflation,”
JCAP (2014) no. 10, 048 [ ].[29] H. Abe, T. Kobayashi, and H. Otsuka “Natural inflation with and without modulations in type IIB stringtheory,” [ ].[30] T. C. Bachlechner, C. Long, and L. McAllister “Planckian Axions in String Theory,” [ ].[31] G. Shiu, W. Staessens, and F. Ye “Widening the Axion Window via Kinetic and St¨uckelberg Mixings,”[ ].[32] G. Shiu, W. Staessens, and F. Ye “Large Field Inflation from Axion Mixing,” [ ].[33] T. W. Grimm “Axion inflation in type II string theory,”
Phys.Rev.
D77 (2008) 126007 [ ].[34] W. Buchmuller, K. Hamaguchi, O. Lebedev, and M. Ratz “Dilaton destabilization at high temperature,”
Nucl.Phys.
B699 (2004) 292–308 [ hep-th/0404168 ].[35] R. Kallosh and A. D. Linde “Landscape, the scale of SUSY breaking, and inflation,”
JHEP (2004) 004[ hep-th/0411011 ].[36] T. Rudelius “Constraints on Axion Inflation from the Weak Gravity Conjecture,” [ ].[37] M. Montero, A. M. Uranga, and I. Valenzuela “Transplanckian axions !?,” [ ].[38] J. Brown, W. Cottrell, G. Shiu, and P. Soler “Fencing in the Swampland: Quantum Gravity Constraints onLarge Field Inflation,” [ ].[39] O. Lebedev, H. P. Nilles, S. Raby, S. Ramos-Sanchez, M. Ratz, et al. “The Heterotic Road to the MSSMwith R parity,”
Phys.Rev.
D77 (2008) 046013 [ ].
40] O. Lebedev, H. P. Nilles, S. Ramos-Sanchez, M. Ratz, and P. K. S. Vaudrevange “Heterotic mini-landscape(II): completing the search for MSSM vacua in a Z orbifold,” Phys. Lett.
B668 (2008) 331–335 [ ].[41] D. Bailin and A. Love “Orbifold compactifications of string theory,”
Phys.Rept. (1999) 285–408.[42] B. Dundee, S. Raby, and A. Westphal “Moduli stabilization and SUSY breaking in heterotic orbifold stringmodels,”
Phys.Rev.
D82 (2010) 126002 [ ].[43] S. L. Parameswaran, S. Ramos-Sanchez, and I. Zavala “On Moduli Stabilisation and de Sitter Vacua inMSSM Heterotic Orbifolds,”
JHEP (2011) 071 [ ].[44] L. J. Dixon, V. Kaplunovsky, and J. Louis “On Effective Field Theories Describing (2,2) Vacua of theHeterotic String,”
Nucl.Phys.
B329 (1990) 27–82.[45] J. Louis “Nonharmonic gauge coupling constants in supersymmetry and superstring theory,”.[46] L. E. Ibanez and D. Lust “Duality anomaly cancellation, minimal string unification and the effective low-energy Lagrangian of 4-D strings,”
Nucl.Phys.
B382 (1992) 305–364 [ hep-th/9202046 ].[47] J. Lauer, J. Mas, and H. P. Nilles “Duality and the Role of Nonperturbative Effects on the World Sheet,”
Phys.Lett.
B226 (1989) 251.[48] S. Ferrara, D. Lust, and S. Theisen “Target Space Modular Invariance and Low-Energy Couplings in OrbifoldCompactifications,”
Phys.Lett.
B233 (1989) 147.[49] J. Lauer, J. Mas, and H. P. Nilles “Twisted sector representations of discrete background symmetries fortwo-dimensional orbifolds,”
Nucl.Phys.
B351 (1991) 353–424.[50] S. Stieberger, D. Jungnickel, J. Lauer, and M. Spalinski “Yukawa couplings for bosonic Z(N) orbifolds: Theirmoduli and twisted sector dependence,”
Mod.Phys.Lett. A7 (1992) 3059–3070 [ hep-th/9204037 ].[51] M. B. Green and J. H. Schwarz “Anomaly Cancellation in Supersymmetric D=10 Gauge Theory and Super-string Theory,” Phys. Lett.
B149 (1984) 117–122.[52] P. Fayet and J. Iliopoulos “Spontaneously Broken Supergauge Symmetries and Goldstone Spinors,”
Phys.Lett.
B51 (1974) 461–464.[53] M. Dine, N. Seiberg, and E. Witten “Fayet-Iliopoulos Terms in String Theory,”
Nucl.Phys.
B289 (1987)589.[54] L. J. Dixon, V. Kaplunovsky, and J. Louis “Moduli dependence of string loop corrections to gauge couplingconstants,”
Nucl.Phys.
B355 (1991) 649–688.[55] J. Derendinger, S. Ferrara, C. Kounnas, and F. Zwirner “On loop corrections to string effective field theories:Field dependent gauge couplings and sigma model anomalies,”
Nucl.Phys.
B372 (1992) 145–188.[56] D. Lust and C. Munoz “Duality invariant gaugino condensation and one loop corrected Kahler potentials instring theory,”
Phys.Lett.
B279 (1992) 272–280 [ hep-th/9201047 ].[57] D. Bailin, A. Love, W. Sabra, and S. Thomas “String loop threshold corrections for Z(N) Coxeter orbifolds,”
Mod.Phys.Lett. A9 (1994) 67–80 [ hep-th/9310008 ].[58] D. Bailin, A. Love, W. Sabra, and S. Thomas “Modular symmetries in Z(N) orbifold compactified stringtheories with Wilson lines,” Mod.Phys.Lett. A9 (1994) 1229–1238 [ hep-th/9312122 ].[59] D. Bailin and A. Love “Reduced modular symmetries of threshold corrections and gauge coupling unification,”[ ].[60] T. Taylor, G. Veneziano, and S. Yankielowicz “Supersymmetric QCD and Its Massless Limit: An EffectiveLagrangian Analysis,” Nucl.Phys.
B218 (1983) 493.[61] I. Affleck, M. Dine, and N. Seiberg “Dynamical Supersymmetry Breaking in Supersymmetric QCD,”
Nucl.Phys.
B241 (1984) 493–534.[62] P. Binetruy and E. Dudas “Gaugino condensation and the anomalous U(1),”
Phys.Lett.
B389 (1996) 503–509[ hep-th/9607172 ].
63] C. Wieck and M. W. Winkler “Inflation with Fayet-Iliopoulos Terms,”
Phys.Rev.
D90 (2014) no. 10, 103507[ ].[64] W. Buchmuller, E. Dudas, L. Heurtier, and C. Wieck “Large-Field Inflation and Supersymmetry Breaking,”
JHEP (2014) 053 [ ].[65] H. P. Nilles, S. Ramos-Sanchez, P. K. Vaudrevange, and A. Wingerter “The Orbifolder: A Tool to studythe Low Energy Effective Theory of Heterotic Orbifolds,”
Comput.Phys.Commun. (2012) 1363–1380[ ].].