Perturbative unitarity in quasi-single field inflation
PPrepared for submission to JHEP
KOBE-COSMO-21-01
Perturbative unitarity in quasi-single field inflation
Suro Kim, a Toshifumi Noumi, a Keito Takeuchi, a and Siyi Zhou b a Department of Physics, Kobe University, Kobe 657-8501, Japan b The Oskar Klein Centre for Cosmoparticle Physics & Department of Physics, Stockholm Univer-sity, AlbaNova, 106 91 Stockholm, Sweden
E-mail: [email protected] , [email protected] , [email protected] , [email protected] Abstract:
We study implications of perturbative unitarity for quasi-single field inflationwith the inflaton and one massive scalar. Analyzing high energy scattering, we show thatnon-Gaussianities with | f NL | (cid:38) f NL and thescale of new physics that is required for UV completion. In particular we find that for theHubble scale H (cid:38) × GeV, Planck suppressed operators can easily generate too largenon-Gaussanities and so it is hard to realize successful quasi-single field inflation withoutintroducing a mechanism to suppress quantum gravity corrections. Also we generalizethe analysis to the regime where the isocurvature modes are heavy and the inflationarydynamics is captured by the inflaton effective theory. Requiring perturbative unitarity ofthe two-scalar UV models with the inflaton and one heavy scalar, we clarify the parameterspace of the P ( X, φ ) model which is UV completable by a single heavy scalar. a r X i v : . [ h e p - t h ] F e b ontents α = λ = λ = 1) 154.2 General values of α , λ , λ A.1 Four-point amplitude 21A.2 Five-point and six-point amplitudes 22
B Field redefinition for amplitude computation 24
Perturbative unitarity plays an important role in discovering new physics. Historically, theHiggs boson was predicted by studying unitarity of high-energy scattering of weak bosons.In particular, its mass scale is estimated by the unitarity violation scale in a theory withoutthe Higgs sector [1–4]. Indeed, the Higgs boson with the mass 125 GeV was discovered,which completed the Standard Model as a UV complete theory of particle physics [5–7].The purpose of this paper is to apply a similar idea in the study of primordial non-Gaussianities [8] , in light of recent studies of perturbative unitarity in the Standard ModelEffective Field Theory (SMEFT) [17–20] (see Refs. [21, 22] for review articles). In SMEFT,effects of new physics beyond the Standard Model are encoded into low-energy effectiveinteractions. For example, the Higgs potential in the Standard Model has two parameters See, e.g., Refs. [9–16] for application of perturbative unitarity in the inflation context, especially in thecontext of Higgs inflation. Perturbative unitarity was used there to constrain the background dynamics ofinflation, whereas the present paper discusses implications for primordial non-Gaussianities. – 1 –haracterizing the Higgs vacuum expectation value (vev) and the Higgs mass. In particular,the Higgs trilinear coupling is no more free parameter once the Higgs vev and the Higgsmass are specified. Then, a deviation from the Standard Model value implies a new physicsat a higher energy scale, which is specified by studying perturbative unitarity [23]. Anothertypical example is derivative interactions in the scalar sector, i.e., the Higgs boson and theNambu-Goldstone bosons that are eaten by the gauge bosons. If we focus on two-derivativeinteractions, perturbative unitarity requires flatness of the field space in the UV completetheory, which in turn specifies the scale of new physics to be the curvature scale of theSMEFT field space [24]. In this way, recent studies in the SMEFT context have clarifiedimplications of perturbative unitarity for scalar field theories in particular.In this paper, following these developments, we address two issues in the inflationcontext. The first is about primordial non-Gaussianities in quasi-single field inflation [25]:To maintain the flatness of the inflaton potential, it is natural to assume an approximateshift symmetry of the inflaton field. A simplest realization of such a symmetry is toembed the inflaton field as a phase direction of a complex scalar field, analogously tothe Nambu-Goldstone boson in the Higgs potential. If the radial direction has a massnear the Hubble scale, the dynamics of primordial fluctuations is governed by the inflatonfluctuations and the isocurvature modes with the Hubble scale mass. Then, the modelis called quasi-single field inflation. In particular, primordial non-Gaussianities in thismodel accommodate interesting features which can be used to identify the particle spectraduring inflation and so they have been studied intensively in the context of “CosmologicalCollider Program” [25–28] (see also [29–83]). In this paper, we discuss implications ofperturbative unitarity for primordial non-Gaussianities in quasi-single field inflation withthe inflaton and one massive scalar (which we call the two-field quasi-single field inflationin the following) and demonstrate that observable non-Gaussianities with the nonlinearlityparameter | f NL | (cid:38) f NL .On the other hand, if the isocurvature modes are heavier than the Hubble scale, aneffective theory of the inflaton is obtained by integrating out the heavy modes. A usefultemplate for such effective theories of single-field inflation is the so-called P ( X, φ ) model,whose Lagrangian contains an arbitrary function of the inflaton φ and its kinetic operator X = − ( ∂ µ φ ) on top of the Einstein-Hilbert term. In particular, higher derivative inter-actions can source observable non-Gaussianities [84]. Since the function P ( X, φ ) containsinformation of the UV theory such as masses, spins, and number of heavy fields, it is ofgreat interests to clarify which parameter space can be realized by which types of UV com-pletion. By doing so, we can draw a “country map” of the EFT landscape . Based on thismotivation, we use perturbative unitarity to clarify the parameter space scanned by two- Following recent developments in the swampland program [85, 86], our understanding on the consistentEFT parameter space (the EFT landscape) has been increased considerably. The next step useful forphenomenology would be to separate the EFT landscape into several regions (countries) based on detailedproperties of the UV theory beyond the universal consistency conditions such as unitarity and causality. – 2 –calar UV models, i.e., models with the inflaton and one heavy scalar . We demonstratethat cubic and higher order terms in X cannot be generated at IR unless we introduce newphysics beyond the two-scalar UV models. We also provide a relation between the size ofthe X coupling and the scale Λ of new physics beyond the two-scalar UV models.The organization of the paper is as follows. In Sec. 2, we review quasi-single fieldinflation [25] with an emphasis on typical energy scales of the model and their relationsto primordial non-Gaussianities. In Sec. 3, we study the requirement of perturbative uni-tarity on quasi-single field inflation and discuss its implication for non-Gaussianities. Inparticular, we demonstrate that observable non-Gaussianities with | f NL | (cid:38) P ( X, φ ) model scannedby two-scalar UV models that contain the inflaton and one heavy scalar. We conclude inSec. 5 with discussion of our results.
In this section we review quasi-single field inflation [25] with an emphasis on typical energyscales of the model and their relations to primordial non-Gaussianities. After summarizingphenomenological aspects of the model, we demonstrate that in the two-field quasi-singlefield inflation, the nonlinearity parameter f NL is too small to observe if its main sourceis renormalizable interactions. This motivates us to study implications of perturbativeunitarity in the next section. The model consists of two real scalars, φ and φ , with a flat field space metric: S = (cid:90) d x √− g M R − (cid:88) i =1 , ∂ µ φ i ∂ µ φ i − V ( φ i ) . (2.1)The potential, V ( φ i ), has an approximate U (1) symmetry under the phase shift of φ + iφ ,which guarantees the flatness of the potential along the angular direction in the field space.More explicitly, in the polar coordinates, φ + iφ = re iθ , (2.2)the action takes the form, S = (cid:90) d x √− g (cid:20) M R − r ∂ µ θ∂ µ θ − ∂ µ r∂ µ r − V r ( r ) − V soft ( θ ) (cid:21) . (2.3)Here and in what follows we use V for the potential in Cartesian coordinates, ( φ , φ ), and V for that in the polar coordinates, ( r, θ ). V r ( r ) is a U (1) symmetric potential invariant Perturbative unitarity in the EFT of inflation was discussed in Refs. [87–92], where the strong cou-pling scale within the EFT framework was estimated for example. In contrast, the present paper studiesperturbative unitarity constraints on the UV theory and the corresponding EFT of inflation realized at IR. – 3 –nder the constant shift of θ and V soft ( θ ) breaks the shift symmetry softly. We assume that V r ( r ) has a minimum at r = r min (cid:54) = 0 away from the origin and the symmetry breakingpotential, V soft ( θ ), is flat enough for the angular coordinate, θ , to play the role of inflaton. Inflationary backgrounds.
Consider an inflationary background, θ = θ ( t ) , r = r ( t ) , ds = − dt + a ( t ) d x . (2.4)The background equations of motion for θ and r read r ¨ θ + 3 Hr ˙ θ + 2 r ˙ r ˙ θ + V (cid:48) soft ( θ ) = 0 , (2.5)¨ r + 3 H ˙ r + V (cid:48) r ( r ) − r ˙ θ = 0 , (2.6)where H = ˙ a/a is the Hubble parameter. The Friedmann equations are3 M H = 12 r ˙ θ + 12 ˙ r + V r ( r ) + V soft ( θ ) , (2.7) − M ˙ H = r ˙ θ + ˙ r . (2.8)In the following we use the slow-roll approximation under which ˙ θ and r are approximatelyconstant. Then, the equation of motion for r reduces to r ˙ θ = V (cid:48) r ( r ) , (2.9)which specifies how much the background trajectory, r = r , deviates from the minimum, r = r min , of the potential, V r ( r ), due to the “centrifugal force.” Primordial fluctuations.
Next let us introduce fluctuations around the inflationarybackground (2.4) as θ = θ + δθ , r = r + σ . (2.10)Under the slow-roll approximation, the action of matter fluctuations in the spatially flatgauge reads S = (cid:90) dtd xa (cid:20) − r ∂ µ δθ ∂ µ δθ − ∂ µ σ∂ µ σ − V ( σ ) − r σ (cid:104) − θ ˙ δθ + ( ∂ µ δθ ) (cid:105) − σ (cid:104) − θ ˙ δθ + ( ∂ µ δθ ) (cid:105) (cid:21) , (2.11)where we neglected metric fluctuations because their contributions to scalar spectra aresubdominant. Also we introduced an effective potential, V ( σ ) = V r ( r + σ ) − V r ( r ) − (cid:34) ˙ θ r + σ ) − ˙ θ r (cid:35) , (2.12)which has no term linear in σ because of the equation of motion (2.9).– 4 –t is convenient to introduce a canonically normalized field, ϕ = r δθ , in terms ofwhich the action (2.11) may be reformulated as S = (cid:90) dtd xa (cid:20) − ∂ µ ϕ∂ µ ϕ − ∂ µ σ∂ µ σ − V ( σ )+ 2 ˙ θ ˙ ϕσ + ˙ θ r ˙ ϕσ − σr ( ∂ µ ϕ ) − σ r ( ∂ µ ϕ ) (cid:21) , (2.13)where the first two terms in the second line are proportional to ˙ θ and break the Lorentzsymmetry. In particular, the Lorentz symmetry breaking sector accommodates a linearmixing between ϕ and σ . For later use, we parameterize the effective potential V ( σ ), as V ( σ ) = m r (cid:34) (cid:18) σr (cid:19) + λ (cid:18) σr (cid:19) + λ (cid:18) σr (cid:19) + O ( σ ) (cid:35) , (2.14)where we introduced dimensionless parameters λ i . Energy scales.
The model (2.13) contains (at least) three energy scales characterized by m , ˙ θ , and r . In this paper except for Sec. 4, we consider the following typical parameterregime of quasi-single field inflation:1. Isocurvature mass m .First, we assume that the mass of the isocurvature mode, σ , is of the Hubble scaleorder: m ∼ H . This is why the model is called quasi-single field in comparison tomulti-field and effective single-field after integrating out heavy fields.2. Mixing scale ˙ θ .As a consequence of the turning background trajectory, there appears a linear mixingbetween the adiabatic mode ϕ , and the isocurvature mode σ . We assume that themixing is in a perturbative regime during inflation. In other words we assume thatthe mixing scale is well below the Hubble scale ˙ θ (cid:46) H , which means that the turningangle per e -fold is small enough. Our typical benchmark point will be ˙ θ = 0 . × H .3. Decay constant r of the isocurvature σ .The radius, r , of the turning trajectory characterizes the scale of higher derivativeinteractions, σ ( ∂ µ ϕ ) and σ ( ∂ µ ϕ ) . For later convenience, in Eq. (2.14), we used thesame energy scale as the decay constant of σ , even though there are no reasons toassume λ , = O (1) at least at this moment. Primordial spectra.
Finally, we summarize the scalar power spectrum and bispectrumof this model (we refer the readers to the original paper [25] for details). First, the powerspectrum of the curvature perturbation ζ , is given by (cid:104) ζ k ζ k (cid:105) = (2 π ) δ ( k + k ) 2 π k P ζ , P ζ = H (2 π ) r ˙ θ (cid:32) ˙ θ H (cid:33) C ( ν ) , (2.15)– 5 – .0 0.2 0.4 0.6 0.8 1.0 1.20100200300400 ν g ( ν ) ν h ( ν ) ν l ( ν ) Figure 1 : Numerical values of g ( ν ), h ( ν ), and l ( ν ).where ν = (cid:113) − m H and the formulae provided in this subsection are for m < H , i.e., forreal ν . Also C ( ν ) is a mass-dependent numerical coefficient defined by C ( ν ) = π (cid:34)(cid:90) ∞ dx x / (cid:90) ∞ x dx x / (cid:16) H (1) ν ( x ) e ix H (2) ν ( x ) e − ix − H (1) ν ( x ) e − ix H (2) ν ( x ) e − ix (cid:17)(cid:105) , (2.16)where H (1) ( z ) and H (2) ( z ) are Hankel functions of the first and second kinds, respectively.As we mentioned, ˙ θ /H characterizes the mixing of the adiabatic and isocurvature modes.The nonlinearity parameter f NL , characterizing bispectra is defined such that (cid:104) ζ k ζ k ζ k (cid:105) → (2 π ) δ ( k + k + k ) P ζ (cid:18) f NL (cid:19) k k k (2.17)in the equilateral limit k = k = k . Each cubic coupling in Eq. (2.13) sources non-Gaussianities of the magnitude [25], f σ NL = − g ( ν ) λ (cid:32) ˙ θ H (cid:33) , f σ ϕ NL = − h ( ν ) (cid:32) ˙ θ H (cid:33) , f σϕ NL = − l ( ν ) (cid:32) ˙ θ H (cid:33) . (2.18)Here l ( ν ), h ( ν ) and g ( ν ) are mass-dependent numerical coefficients defined by (see alsoFig. 1 for their numerical values) g ( ν ) = 5 π (cid:18) mH (cid:19) Im (cid:90) ∞ dx x I ( x ) I ( x ) I ( x ) , (2.19) h ( ν ) = 5 π
12 Im (cid:90) ∞ dx e − ix x I ( x ) I ( x ) , (2.20) l ( ν ) = − π (cid:90) ∞ dx e − ix I ( x ) + 5 π
12 Im (cid:90) ∞ dx x e − ix (1 + ix ) I ( x ) , (2.21) See Ref. [93] for its analytic form in terms of special functions. See also Ref. [94] for a related work. – 6 –ith I ( x ) being defined by I ( x ) = x / (cid:34) (cid:34) H (1) ν ( x ) (cid:90) ∞ dx x / H (2) ν ( x ) e − ix (cid:35) (2.22)+ iH (1) ν ( x ) (cid:90) x dx x / H (2) ν ( x ) e − ix − iH (2) ν ( x ) (cid:90) x dx x / H (1) ν ( x ) e − ix (cid:35) . Note that f σϕ NL and f σ ϕ NL are small in the perturbative regime, ˙ θ (cid:46) H , of the linear mixing.On the other hand, the cubic self-coupling σ , of the isocurvature mode may generateobservable non-Gaussianities with | f σ NL | ≥ O (1), if λ is large enough to compensate thesuppression by ˙ θ /H . We have reviewed that in quasi-single field inflation non-Gaussianities with | f NL | > O (1)may be sourced by the cubic self-coupling λ if it is large enough. In the following we discussif such a large non-Gaussianity is realized consistently. To illustrate our motivation, weend this section by providing a simple observation that large non-Gaussianities cannot berealized if the cubic coupling is originated from a renormalizable Mexican hat potential.In the Cartesian coordinates of the field space, the U (1) symmetric renormalizablepotential is parameterized by two parameters, λ and r min , as V U (1) = λ φ + φ − r ) , (2.23)and so the potential of the radial coordinate, r , is V r ( r ) = λ r − r ) . (2.24)Then, the effective potential (2.14) of the isocurvature mode σ reads V ( σ ) = V r ( r + σ ) − V r ( r ) − (cid:34) ˙ θ r + σ ) − ˙ θ r (cid:35) = m r (cid:34)(cid:18) σr (cid:19) + (cid:18) σr (cid:19) + 14 (cid:18) σr (cid:19) (cid:35) , (2.25)where we used the equation of motion (2.9) and defined the isocurvature mass m = 4 λr .Note that λ and λ are no longer free parameters, but rather they are specified as λ = λ = 1. In particular, the nonlinearity parameter f σ NL ∼ ( ˙ θ /H ) , is too small to observeat least in the perturbative regime of the linear mixing.This simple observation suggests that it is hard to obtain observable non-Gaussianitiesin two-field quasi-single field inflation without turning on nonrenormalizable interactions.In the following, we study perturbative unitarity of scattering amplitudes and clarify underwhich conditions observable non-Gaussianities are realized in quasi-single field inflation.– 7 – Implications of perturbative unitarity
Based on the aforementioned motivation, we study high energy scattering of fluctuationsaround inflationary backgrounds and discuss implications of perturbative unitarity. Forlater convenience, we slightly generalize the model (2.13) in the previous section as S = (cid:90) dtd xa (cid:20) −
12 ( ∂ µ ϕ ) −
12 ( ∂ µ σ ) − V ( σ ) − σr (cid:104) − r ˙ θ ˙ ϕ + ( ∂ µ ϕ ) (cid:105) − α σ r (cid:104) − r ˙ θ ˙ ϕ + ( ∂ µ ϕ ) (cid:105) (cid:21) (3.1)= (cid:90) dtd xa (cid:20) −
12 ( ∂ µ ϕ ) −
12 ( ∂ µ σ ) − V ( σ )+ 2 ˙ θ ˙ ϕσ + α ˙ θ r ˙ ϕσ − σr ( ∂ µ ϕ ) − α σ r ( ∂ µ ϕ ) (cid:21) , (3.2)which reproduces Eq. (2.13) for α = 1. Also we use the same parameterization (2.14) of theisocurvature potential. In the rest of the section, we begin by a brief review of the unitaritybound (Sec. 3.1). Then, in Sec. 3.2 we discuss implications of perturbative unitarity for themodel (3.2). There we demonstrate that α = λ = λ = 1 and so small non-Gaussianities, | f NL | (cid:46)
1, are required if we decouple gravity and assume that the model is UV completeat the tree-level. This confirms our simple observation in the previous section. In Sec. 3.3we generalize the argument to include theories with a finite cutoff scale Λ. We provide arelation between the nonlinearity parameter f NL and the scale Λ of new physics beyondquasi-single field inflation which is required for UV completion. The starting point is the S -matrix unitarity: SS † = . (3.3)Using the transition matrix T defined by S = + i T , (3.4)we write Eq. (3.3) as − i ( T − T † ) = T T † . (3.5)Next we rewrite it in term of the scattering amplitude: (cid:104) B | T | A (cid:105) = (2 π ) δ ( p A − p B ) M AB , (3.6) Note that derivatives of ϕ always appear in the combination, − r ˙ θ ˙ ϕ + ( ∂ µ ϕ ) , because of the Lorentzinvariance of the original action for backgrounds. Also this form of action is determined by the symmetryof fluctuations alone, independent of details of the background dynamics. See, e.g., Ref. [27]. – 8 –here | A (cid:105) and | B (cid:105) are the initial and final states, respectively. Also p A is the total mo-mentum of the initial state | A (cid:105) and similarly for p B . In this language, Eq. (3.5) reads − i (2 π ) δ ( p A − p B )( M AB − M ∗ BA )= (cid:88) C N C (cid:89) i =1 (cid:90) d p i (2 π ) E i ( p i ) (2 π ) δ ( p A − p B )(2 π ) δ ( p A − p C ) M CB M ∗ CA , (3.7)where N C is the number of external particles in the intermediate state C and E i ( p i ) isthe on-shell energy of the i -th particle with the spatial momentum p i . In particular, foridentical initial and final states, A = B , we have2 Im M AA = (cid:88) C N C (cid:89) i =1 (cid:90) d p i (2 π ) E i ( p i ) (2 π ) δ ( p A − p C ) | M CA | . (3.8)Since each summand in the right-hand side is non-negative, we have2 | M AA | ≥ N C (cid:89) i =1 (cid:90) d p i (2 π ) E i ( p i ) (2 π ) δ ( p A − p C ) | M CA | , (3.9)which implies a bound on high-energy scattering with a typical energy scale E as | M AA | ≤ E − N A . (3.10)Substituting this bound back into Eq. (3.9), we have N C (cid:89) i =1 (cid:90) d p i (2 π ) E i ( p i ) (2 π ) δ ( p A − p C ) | M CA | ≤ E − N A , (3.11)which implies a bound on more general high-energy scattering as | M CA | ≤ E − ( N A + N C ) . (3.12)For example, bounds on four-, five-, and six-point amplitudes read | M | ≤ E , | M | ≤ E − , | M | ≤ E − . (3.14)If we assume weakly coupled UV completion, these bounds have to be satisfied by tree-levelamplitudes. Note that we have suppressed O (1) coefficients in the inequalities (3.10)-(3.14),but they can be fixed in the standard manner, e.g., using angular momentum eigenstates.We will take care of these O (1) coefficients when necessary. See Appendix A for details. Let us use the bounds (3.14) to constrain the model parameters α, λ , λ in two-fieldquasi-single field inflation (3.2) and discuss their implication to non-Gaussianities. In thissubsection we focus on the case where gravity is decoupled and derive conditions for themodel to be UV complete at the tree-level. Note that the bounds presented here are applicable in four-dimensional spacetime. Bounds in generalspacetime dimensions d are given by | M | ≤ E − d , | M | ≤ E − d , | M | ≤ E − d . (3.13) – 9 – φ σσ φφ σσ Figure 2 : Feynman diagrams which generate E contributions in σσ → ϕϕ scattering. Four-point scattering.
We begin by four-point scattering. Perturbative unitarity con-straints on general scalar field theories were nicely discussed in Ref. [24]. A conclusionthere is that in spacetime four dimensions, the model is UV complete at the tree-level onlywhen the field space is flat . In other words, the field space curvature gives a cutoff scaleof the theory. In our model (3.2), the field space curvature tensor at the origin σ = ϕ = 0is given in terms of the parameter α as R σϕσϕ | σ = ϕ =0 = 1 − αr , (3.15)and similarly for other components. Then, a nontrivial constraint is available from σσϕϕ scattering : M σσϕϕ = 2( α − p .p r + O ( E ) , (3.16)where we kept terms which grow up faster than E at high energy E (cid:29) m, | ˙ θ | . SeeFig. 2 for the corresponding Feynman diagrams. Also p i is the four-momentum (in theall-incoming notation) of the i -th external particle φ i in the amplitude M φ ...φ n , and p i .p j is the inner product of p i and p j . Then, the unitarity bound (3.14) implies α = 1 . (3.17) Five-point scattering.
Next we consider five-point scattering. In our model (3.2) thereare two five-point scattering processes relevant for our purpose: σσσϕϕ and σϕϕϕϕ . Theunitarity bound (3.14) requires that they are bounded by O ( E − ). First, we consider theamplitude, M σσσϕϕ . Its high energy behavior reads (see Fig. 3 for the relevant diagrams) M σσσϕϕ = − α − (cid:18) p .p r + O ( E ) (cid:19) + 3 (3 λ − λ − m r + O ( E − ) , (3.18)where the O ( E ) contribution generated by the diagrams (d) and (e) cancel each other outfor α = 1 in particular (see also Appendix B for field redefinition useful in the amplitudecomputation). Now the unitarity bound (3.14) implies α = 1 , λ − λ = 2 . (3.19) Note that in d = 2 a nonzero internal space curvature is allowed since the unitarity bound is milder asshown in Eq. (3.13), which is consistent with the fact that 2d CFT accommodates a curved target space. When we need to specify which particles are in the initial/final states, we call scattering process, e.g.,as σσ → ϕϕ scattering, but otherwise we use the terminology such as σσϕϕ scattering. Correspondingly,we use, e.g., both of M σσϕϕ and M σσ → ϕϕ , to denote scattering amplitudes. – 10 – φ σσσ :(a) φφ σσσ :(b) φφ σσσ :(c) φφ σσσ :(d) φφ σσσ :(e) Figure 3 : The leading order Feynman diagrams of σσσϕϕ scattering. σ φφφφ σ φφ φφ σ φφφφ Figure 4 : The leading order Feynman diagrams of σϕϕϕϕ scattering.Similarly, the high energy behavior of M σϕϕϕϕ reads M σϕϕϕϕ = − ( α − m r (cid:18) p .p p .p + 5 perms. (cid:19) − ( α + 9 λ − m r + O ( E − ) . (3.20)See Fig. 4 for relevant diagrams. Then, the unitarity bound (3.14) implies α = λ = 1 . (3.21)To summarize, perturbative unitarity of five-point scattering implies Eqs. (3.19) and (3.21),so that we conclude that α = λ = λ = 1 (3.22)when we decouple gravity and assume that the model is UV complete at the tree-level.Note that this precisely reproduces the renormalizable potential (2.25) . As we discussedin the previous section, non-Gaussianities are too small to observe for this parameter set. Inother words, quasi-single field inflation cannot generate | f NL | (cid:38) f NL and the scale Λ of such new physics beyond quasi-single field inflation. Six-point scattering.
Before proceeding to the next subsection, let us provide six-pointscattering amplitudes of ϕ for later convenience (see Fig. 5 for relevant diagrams): M ϕϕϕϕϕϕ = − · ( α + 3 λ − m r + O ( E − ) . (3.23) Note that one may easily show that any O ( σ ) term in the potential, V σ , violates the unitarity bound,so that the allowed potential is a quartic one with the conditions (3.22). – 11 – φφφφφ φφφφφφ φφφφφφ Figure 5 : The leading order Feynman diagrams of six-point scattering of ϕ .We find that the O ( E ) term vanishes and so the unitarity bound is satisfied under theconditions (3.22). In the previous subsection, we have shown that observable non-Gaussianities | f NL | (cid:38) m (cid:46) Λ . Inthis subsection, we consider deviations from the UV complete parameter set α = λ = λ =1 and discuss a relation between non-Gaussianities f NL and the scale Λ of new physics.In our model (3.2), there are three types of cubic interactions, σ , σ ˙ ϕ , and σ ( ∂ µ ϕ ) .When the mixing between ϕ and σ is small at the Hubble scale ( | ˙ θ | (cid:46) H ), each cubiccoupling contributes to the nonlinearity parameter f NL as f σ NL = − λ g ( ν ) (cid:32) ˙ θ H (cid:33) , f σ ϕ NL = − αh ( ν ) (cid:32) ˙ θ H (cid:33) , f σϕ NL = − l ( ν ) (cid:32) ˙ θ H (cid:33) , (3.24)where note that an overall factor α is multiplied in the second expression compared toEq. (2.18) since we are allowing a nonzero field space curvature ( α (cid:54) = 1). Our task is nowto write down the cutoff scale Λ derived from the perturbative unitarity in terms of thenonlinearity parameter f NL . Deformations of isocurvature potential.
First, we consider the case when the cubicself-coupling λ of the isocurvature mode σ is the dominant source of non-Gaussianities.For simplicity, we assume λ (cid:54) = 1, but α = 1. As we discussed in the previous subsection, adeviation of λ from unity breaks the perturbative unitarity of the five-point and six-pointscattering amplitudes at a high-energy scale, which gives the cutoff scale of the theory. Inappendix A, we use unitarity of the S -wave amplitudes of M σϕϕϕϕ and M ϕϕϕϕϕϕ to derivethe following cutoff scales taking care of numerical coefficients:Λ = 64 √ π · r m | λ − | (cid:39) · r m | λ − | , (3.25)Λ = 16 √ π / √ · r m | λ − | / (cid:39) · r m | λ − | / . (3.26) Another possibility is that the model becomes strongly coupled at the scale Λ and it is UV completedin a non-perturbative manner, which typically implies a phase transition at the scale Λ just like QCD. – 12 – H r = 0.002 [GeV][GeV] Planck scale V [GeV] Small Field Inflation Large Field Inflation Figure 6 : Cutoff scale Λ vs Hubble scale H : As a benchmark point, we plot Λ =4 × H , which corresponds to the choice H m g ( ν ) = 100 and | f NL | = 1.Having in mind observable non-Gaussianities | f NL | (cid:38)
1, let us assume that λ is largeenough to compensate the suppression by the factor | ˙ θ /H | (cid:46)
1. Then, by using Eq. (3.24),the cutoff scales (3.25)-(3.26) are translated asΛ (cid:39) × H · (cid:18) × − P ζ (cid:19) / · | ˙ θ/H | . · (cid:32) H m g ( ν )100 (cid:33) · (cid:18) | f NL | (cid:19) , (3.27)Λ (cid:39) × H · × − P ζ · (cid:32) H m g ( ν )100 (cid:33) / · (cid:18) | f NL | (cid:19) , (3.28)where note that H m g ( ν ) (cid:39)
100 for m (cid:39) H . We find that in the regime of our interests,unitarity of six-point amplitudes provide a stronger constraint.Now let us take a closer look at Λ . See Fig. 6 for a plot of Λ = 4 × H , taking H m g ( ν ) = 100 and | f NL | = 1 as a benchmark point. First, we find that the cutoff scale isnear the Planck scale M Pl (cid:39) . × GeV when H (cid:39) × GeV, or equivalently forthe vacuum energy V (cid:39) (5 × GeV) . This implies that Planck suppressed operatorscan easily generate large non-Gaussanities when H (cid:38) × GeV ( V / (cid:38) × GeV).In particular, for H (cid:38) × GeV ( V / (cid:38) × GeV), Planck suppressed operatorsgenerate too large non-Gaussianities | f NL | (cid:38)
100 (assuming H m g ( ν ) = 100) and so it is hardto realize successful quasi-single field inflation without introducing a mechanism to suppressquantum gravity corrections. Qualitatively, this is analogous to the Lyth bound [95],beyond which a mechanism to suppress quantum gravity effects is needed to accommodatea super-Planckian inflaton excursion. Notice however that our bound is stronger than theLyth bound quantitatively. On the other hand, if the inflation scale is sufficiently low, H (cid:46) × GeV ( V / (cid:46) × GeV), non-Gaussianities with | f NL | (cid:38) (cid:39) × GeVfor H (cid:39) × GeV ( V / (cid:39) × GeV) and | f NL | = 1.– 13 – H r = 0.002 [GeV][GeV] Planck scale V [GeV] Small Field Inflation Large Field Inflation Figure 7 : Cutoff scale Λ vs Hubble scale H : As a benchmark point, we plot Λ =6 × H , which corresponds to the choice | ˙ θ /H | = 0 . h ( ν ) = 10, and | f NL | = 1. Curved field space.
Let us move on to the case with a curved field space α (cid:54) = 1. Forsimplicity, we assume λ = 1, but the argument is qualitatively the same even when thecubic self-coupling of σ generates non-Gaussianities comparable to the field space curvatureeffects λ (cid:39) α . In the regime of our interests, unitarity of four-point scattering providesthe strongest condition among the constraints discussed in the previous subsection:Λ = 2 √ π · r | α − | / (cid:39) · r | α − | / , (3.29)where the numerical coefficient was fixed by the strong coupling scale of the S -wave scatter-ing. See Appendix A for details. Having in mind observable non-Gaussianities | f NL | (cid:38)
1, letus assume that α is large enough to compensate the suppression by the factor | ˙ θ /H | (cid:46) (cid:39) × H · (cid:18) × − P ζ (cid:19) / · | ˙ θ /H | . · (cid:18) h ( ν )10 (cid:19) / · (cid:18) | f NL | (cid:19) , (3.30)where note that h ( ν ) (cid:39)
10 for m (cid:39) H . See Fig. 7 for a plot of Λ = 6 × H , taking | ˙ θ /H | = 0 . h ( ν ) = 10 and | f NL | = 1 as a benchmark point. We find that the cutoff scaleis near the Planck scale M Pl (cid:39) . × GeV for H (cid:39) × GeV ( V / (cid:39) × GeV).Notice that this scale corresponds to the tensor-to-scalar ratio r (cid:39) . α is a dominantsource of non-Gaussianities, | f NL | (cid:38) (cid:39) × GeV for H (cid:39) × GeV and | f NL | = 1.To summarize, we have shown that observable non-Gaussianities with | f NL | (cid:38) f NL and the cutoff scale Λ is summarized in Eq. (3.28) and Eq. (3.30). In particular, if– 14 –he Hubble scale is H (cid:38) GeV, a Planck suppressed σ coupling can easily generatetoo large non-Gaussianities and so some mechanism to suppress quantum gravity effects isneeded to realize a successful quasi-single field inflation. In the previous section, we focused on the case when the isocurvature mass is comparableto the Hubble scale m ∼ H , which is the mass regime of quasi-single field inflation. Inthis section, on the other hand, we consider the heavy mass regime m (cid:29) H and study theeffective theory after integrating out the isocurvature mode σ . Our primary interest in thissection is in classifying the parameter space of the P ( X, φ ) model, S = (cid:90) d x √− g (cid:34) M R + P ( X, φ ) (cid:35) , (4.1)based on properties of the UV theory behind. Here φ is the inflaton, X = − ( ∂ µ φ ) ,and P ( X, φ ) is a function of X and φ . Note that the model is equivalent to the EFT ofinflation with a unitarity gauge Lagrangian of the form L = L ( δg , t ) [97]. In general, theLagrangian contains higher derivative interactions, which source non-Gaussianities, and soit is interpreted as an effective Lagrangian of the inflaton after integrating out UV degreesof freedom. Then, the function P ( X, φ ) contains information of the UV theory, such asmasses, spins, and the number of heavy fields. Below, we identify the EFT parameter spacescanned by the UV theory with the inflaton and a single heavy isocurvature mode. α = λ = λ = 1 ) First, we study the heavy mass regime of the UV complete parameter set α = λ = λ = 1.As we discussed, this parameter set is equivalent to the quasi-single field inflation model S = (cid:90) d x √− g (cid:20) M R − r ∂ µ θ∂ µ θ − ∂ µ r∂ µ r − V r ( r ) − V soft ( θ ) (cid:21) , (4.2)with the isocurvature potential V r ( r ) = λ r − r ) . (4.3)To derive the inflaton effective action, we neglect the kinetic term of r (since the isocurva-ture mode is frozen) and complete the square with respective to r as S (cid:39) (cid:90) d x √− g (cid:34) M R − r ∂ µ θ∂ µ θ + 18 λ ( ∂ µ θ∂ µ θ ) − V soft ( θ ) − λ (cid:18) r + 12 λ ( ∂ µ θ∂ µ θ ) − r (cid:19) (cid:35) . (4.4)Integrating out r and defining φ = r min θ gives the following inflaton effective action: S eff = (cid:90) d x √− g (cid:20) M R + X + 12 λr X − V soft ( φ/r min ) (cid:21) , (4.5)– 15 –here X = − ( ∂ µ φ ) as before. We find that for the UV complete parameter set α = λ = λ = 1, there appear no O ( X ) terms in the inflaton effective action after integrating outthe isocurvature mode. In other words, the P ( X, φ ) model with non-vanishing O ( X ) termscannot be UV completed by a single heavy scalar in the weakly coupled regime. Note thatthe only approximation we have used here is that the isocurvature mode is heavy m (cid:29) H .In particular, we have not made any assumption on the size of the mixing between theinflaton fluctuation and the isocurvature mode.It is also useful to rephrase our observation in terms of the EFT of inflation [97]. Underthe slow-roll approximation, the effective action of the Nambu-Goldstone boson π is givenin the decoupling limit as S = (cid:90) dtd xa (cid:34) − M ˙ H (cid:18) ˙ π − ( ∂ i π ) a (cid:19) + ∞ (cid:88) n =2 M n n ! (cid:18) − π − ˙ π + ( ∂ i π ) a (cid:19) n (cid:35) . (4.6)Here we neglected terms which contain second and higher order derivatives of π , which isthe same assumption as the P ( X, φ ) model. In this language, the effective action (4.5) is M ˙ H = − r ˙ θ (cid:32) θ λr (cid:33) , M = ˙ θ λ , M n = 0 ( n ≥ , (4.7)where the Nambu-Goldstone boson π and the fluctuation δθ of the angular variable θ arerelated with each other as δθ = ˙ θ π under the slow-roll approximation. In terms of r and m defined in Eq. (2.9) and Eq. (2.14) (see also Sec. 2.2), we may also write M ˙ H = − r ˙ θ , M = r ˙ θ m , M n = 0 ( n ≥ . (4.8) α , λ , λ We have seen that the P ( X ) model with non-vanishing O ( X ) terms cannot be UV com-pleted by a single heavy scalar in the weakly coupled regime. A natural question to ask isat which scale we need new physics beyond the two-field UV model (the UV model withan inflaton and a heavy scalar) when the P ( X ) model contains O ( X ) terms.To answer this question, we start with the two-field model (3.2) and assume that theisocurvature mode σ is heavy m (cid:29) H . Having non-Gaussianity phenomenology in mind,let us rewrite the action (3.2) in terms of the Nambu-Goldstone boson π = ϕ/ ( r ˙ θ ) as S = (cid:90) dtd xa (cid:20) − r ˙ θ ∂ µ π ) −
12 ( ∂ µ σ ) − V ( σ ) − r ˙ θ σ (cid:20) − π − ˙ π + ( ∂ i π ) a (cid:21) − α ˙ θ σ (cid:20) − π − ˙ π + ( ∂ i π ) a (cid:21) (cid:21) , (4.9)with the isocurvature potential V ( σ ) = m r (cid:34) (cid:18) σr (cid:19) + λ (cid:18) σr (cid:19) + λ (cid:18) σr (cid:19) + O ( σ ) (cid:35) . (4.10)– 16 –ssuming that cubic and higher order interactions are in the perturbative regime (itsvalidity is discussed at the end of the section), we may easily integrate out the isocurvature σ to obtain the effective action (4.6) with the EFT parameters (see, e,g, Ref. [98, 99]), M ˙ H = − r ˙ θ , M = r ˙ θ m , M = 3 ( λ − α ) r ˙ θ m , (4.11)and similarly for M n ( n ≥ M vanishes for the UV complete parameterset α = λ = λ = 1.To discuss implications to non-Gaussianities, let us rewrite the effective action as S eff = (cid:90) dtd xa (cid:34) − M ˙ Hc s (cid:18) ˙ π − c s ( ∂ i π ) a (cid:19) + M ˙ H ( c − s − (cid:18) ˙ π ( ∂ i π ) a − (cid:18) c c s (cid:19) ˙ π (cid:19) + O ( π ) (cid:35) , (4.12)where the speed of sound c s and the dimensionless parameter ˜ c are defined by c − s = 1 − M M ˙ H = 1 + 4 ˙ θ m , (4.13)˜ c ( c − s −
1) = 2 c s M M ˙ H = −
34 ( λ − α ) c s ( c − s − . (4.14)In this language, the power spectrum is given by P ζ = 1 c s H (2 π ) ( − M ˙ H ) = (cid:118)(cid:117)(cid:117)(cid:116) (cid:32) ˙ θ m (cid:33) H (2 π ) r ˙ θ . (4.15)Also, the nonlinearity parameter f NL reads f NL = − (cid:0) c − s − (cid:1) (cid:20) (cid:18) ˜ c + 32 c s (cid:19)(cid:21) . (4.16)See Fig. 8 for the current observational constraints on the parameters ( c s , ˜ c ) [100]. Scale of new physics.
In the previous subsection, we demonstrated that the EFT witha nonzero ˜ c cannot be UV completed by a single heavy scalar. Now let us discuss a relationbetween the value of ˜ c and the scale of new physics beyond the UV two-field model. Asin Eq. (4.14), ˜ c is sourced by a cubic self-coupling λ of the heavy isocurvature mode anda curvature α of the two-field space.First, let us consider the case when λ (cid:54) = 1, but α = 1: | (cid:101) c ( c − s − | = 34 | λ − | ( c − s − c s . (4.17)– 17 – s ˜ c ( c − s − ) Figure 8 : The dark blue, blue, and light blue regions are the observationally allowedregions on the c s -˜ c ( c − s −
1) plane with the statistics of 1 σ , 2 σ , and 3 σ , respectively [100].The orange curve is a benchmark plot of the relation (4.20) for m = 30 H and Λ = 1000 H .The inside of the orange curve corresponds to m = 30 H and Λ > H . The cutoff scaleis determined by Λ for c s < c (cid:63)s (cid:39) .
15, whereas Λ = Λ for c s > c (cid:63)s (cid:39) . | (cid:101) c ( c − s − | (cid:39) × · H ( c − s − / c / s m Λ · (cid:18) × − P ζ (cid:19) / , (4.18) | (cid:101) c ( c − s − | (cid:39) × · H m Λ · (cid:18) × − P ζ (cid:19) . (4.19)We find that six-point scattering provides a stronger constraint for small c s , whereas five-point scattering gives a stronger one for c s (cid:39)
1. Then, the relation between ˜ c and thecutoff scale Λ is given by | (cid:101) c ( c − s − | (cid:39) × · H ( c − s − / c / s m Λ · (cid:18) × − P ζ (cid:19) / for c s > c (cid:63)s , × · H m Λ · (cid:18) × − P ζ (cid:19) for c s < c (cid:63)s , (4.20)– 18 – s ˜ c ( c − s − ) Figure 9 : The dark blue, blue, and light blue regions are the observationally allowedregions on the c s -˜ c ( c − s −
1) plane with the statistics of 1 σ , 2 σ , and 3 σ , respectively [100].The orange curve is a benchmark plot of the relation (4.23) for m = 10 H and Λ = 300 H .The inside of the orange curve corresponds to m = 10 H and Λ > H .where the critical speed of sound c (cid:63)s for a fixed cutoff scale Λ is determined by ( c (cid:63)s ) − − c (cid:63)s (cid:39) × · H m Λ · × − P ζ . (4.22)For illustration, the relation (4.20) for m = 30 H and Λ = 1000 H is provided in Fig. 8.Next we consider the case when α (cid:54) = 1, but λ = 1. Then, the cutoff scale is dictatedby four-point scattering as Eq. (3.29), which gives | (cid:101) c ( c − s − | = 34 | α − | ( c − s − c s (cid:39) × · H ( c − s − c s m Λ · (cid:18) × − P ζ (cid:19) . (4.23)We also find c s -dependence which is analogous to ˙ θ dependence of the relation (3.30). Therelation (4.23) for m = 10 H and Λ = 300 H is provided in Fig. 9. Perturbativity.
Finally, we comment on the validity of the dictionary (4.11). In theheavy mass regime m (cid:29) H , the equation of motion of the isocurvature σ is approximated The corresponding value ˜ c (cid:63) of ˜ c is given by | (cid:101) c (cid:63) ( c (cid:63)s − − | = 4 × · H ( c (cid:63)s − − c (cid:63)s m · (cid:18) × − P ζ (cid:19) . (4.21) – 19 –round the Hubble scale as σr = − ˙ θ m (cid:20) − π − ˙ π + ( ∂ i π ) a (cid:21) − λ (cid:18) σr (cid:19) − α ˙ θ m σr (cid:20) − π − ˙ π + ( ∂ i π ) a (cid:21) + . . . , (4.24)where the dots stand for cubic and higher orders in perturbations. The dictionary (4.11)was derived by assuming that the second line is smaller than the first line, which is justifiedif the following conditions are satisfied: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ ˙ θ m (cid:20) − π − ˙ π + ( ∂ i π ) a (cid:21)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:28) , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α ˙ θ m (cid:20) − π − ˙ π + ( ∂ i π ) a (cid:21)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:28) . (4.25)By noticing the linear order relation ζ (cid:39) − Hπ and the smallness of the scalar powerspectrum P ζ (cid:39) × − , these conditions imply (cid:12)(cid:12) ˜ c ( c − s − (cid:12)(cid:12) = 3(1 − c s ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( λ − α ) ˙ θ m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:28) − c s | ˙ ζ/H | . (4.26)If we estimate the size of ζ as ζ (cid:39) P / ζ , the right hand side is O (10 ) in the regime of ourinterests. Therefore, the dictionary (4.11) provides a good approximation at least withinthe regime allowed by the present observations. In this paper we studied implications of perturbative unitarity for two-scalar inflation mod-els with the inflaton and one massive scalar. When the massive scalar has a Hubble scalemass m ∼ H , the model is classified into quasi-single field inflation, which is of interestsespecially in the context of Cosmological Collider Program. While it has been expectedthat observable non-Gaussianities with | f NL | (cid:38) | f NL | (cid:38) H (cid:38) × GeV ( V / (cid:38) × GeV), Planck suppressedoperators can easily generate too large non-Gaussanities and so it is hard to realize suc-cessful quasi-single field inflation without introducing a mechanism to suppress quantumgravity corrections. It would be interesting to explore such a UV mechanism explicitly.We also performed a similar analysis for non-Gaussianities sourced by the cubic couplingassociated with the field space curvature, whose results are summarized in Eq. (3.30).When the massive scalar is heavy m (cid:29) H , the inflationary dynamics is captured bythe effective field theory of the inflaton. Using perturbative unitarity, we identified theparameter space of the P ( X, φ ) model scanned by two-scalar UV models. In particular,– 20 –he cubic and higher order terms in X cannot be generated at IR unless we introducednew physics beyond the two-scalar UV model. The scale of such new physics beyondthe two-scalar model can again be identified by perturbative unitarity and the results aresummarized in Eqs. (4.19)-(4.23). It would be interesting to generalize our analysis to UVmodels with multiple scalars and clarify which parameter space of the P ( X, φ ) model canbe UV completed by scalar field theories. It would also be interesting to include heavyspinning fields in the discussion. By pushing this direction further, we would be able toenlarge the scope of the cosmological collider to higher energy.
Acknowledgements
We would like to thank Xingang Chen for useful discussion and also for pointing out anerror in the earlier draft. S.K. is supported in part by the Senshu Scholarship Foundation.T.N. is supported in part by JSPS KAKENHI Grant Numbers JP17H02894 and 20H01902.S.Z. is supported in part by the Swedish Research Council under grants number 2015-05333and 2018-03803.
A Evaluation of cutoff scales
In this appendix, we derive the cutoff scales from unitarity of partial wave amplitudestaking care of numerical coefficients. For this purpose, it is convenient to use a canonicallynormalized basis of initial/final states which satisfy (cid:104)A|B(cid:105) = (2 π ) δ ( p A − p B ) δ AB , (A.1)where A , B are discrete labels of the states. In such a basis, the unitarity bound for elasticscattering reads | M AA | ≤ , ≤ Im M AA ≤ , − ≤ Re M AA ≤ . (A.2)Similarly, the bound for A (cid:54) = B reads | M AB | ≤ . (A.3) A.1 Four-point amplitude
We first derive the cutoff scale from unitarity of four-point scattering amplitudes, whosepartial wave expansion reads M ( s, t ) = 16 π ∞ (cid:88) n =0 a n (cid:0) E (cid:1) (2 n + 1) P n (cos θ ) . (A.4)Here E cm is the total energy in the center-of-mass frame, θ is the scattering angle, and P n ( x ) is the Legendre polynomial. By employing the angular momentum eigenstates as acanonically normalized basis, the bound (A.3) for non-elastic scattering process is trans-lated in terms of the partial wave amplitudes a n as (cid:12)(cid:12) a n ( E cm ) (cid:12)(cid:12) ≤ , (A.5)– 21 –here we assumed m (cid:28) E cm . For elastic scattering, we may use (A.2) to derive thefollowing bounds: (cid:12)(cid:12) a n ( E cm ) (cid:12)(cid:12) ≤ , ≤ Im a n ( E cm ) ≤ , − ≤ Re a n ( E cm ) ≤ . (A.6)Now we apply these bounds to 2 → σσ → ϕϕ scattering, the amplitude is expanded as M σσ → ϕϕ = − ( α − E r + O ( E ) , (A.7)which gives the partial wave amplitudes, a (cid:0) E (cid:1) = − ( α − E πr , a n (cid:0) E (cid:1) = 0 ( n ≥ . (A.8)Then, Eq. (A.5) implies the following cutoff scale:Λ σσ → ϕϕ = 2 √ π · r | α − | / . (A.9)Similarly, for ϕσ → ϕσ scattering, the amplitude reads M ϕσ → ϕσ = ( α − E r − ( α − E r P (cos θ ) + O ( E ) , (A.10)which gives a (cid:0) E (cid:1) = ( α − E πr , a (cid:0) E (cid:1) = − ( α − E πr , a n (cid:0) E (cid:1) = 0 ( n ≥ . (A.11)Note that for this process the S -wave amplitude a provides a stronger bound than a .Then, Eq. (A.6) implies the following cutoff scale:Λ ϕσ → ϕσ = 4 √ π · r | α − | / . (A.12)We conclude that the cutoff scale dictated by four-point scattering isΛ = 2 √ π · r | α − | / . (A.13) A.2 Five-point and six-point amplitudes
Let us perform a similar argument for five-point and six-point amplitudes. For simplicity,we focus on the S -wave amplitudes and introduce the following zero angular momentumstates (see also Ref. [23]): | P ; n ϕ , n σ (cid:105) = C n ϕ ,n σ (cid:90) d xe iP.x (cid:104) ϕ ( − ) ( x ) (cid:105) n ϕ (cid:104) σ ( − ) ( x ) (cid:105) n σ | (cid:105) (A.14)= C n ϕ ,n σ n ϕ + n σ (cid:89) i (cid:20) (cid:90) d p i E ( p i )(2 π ) (cid:21) (2 π ) δ (cid:16) P − (cid:88) i p i (cid:17) | p , · · · , p n ϕ + n σ (cid:105) , – 22 –here the ( n ϕ + n σ )-particle state | p , · · · , p n ϕ + n σ (cid:105) is defined in the standard manner. Also ϕ ( − ) is the creation operator part of ϕ , ϕ ( − ) ( x ) = (cid:90) d p (2 π ) (cid:112) E ( p ) e − ip.x a † ϕ,p , (A.15)and similarly for σ ( − ) . C n ϕ ,n σ is the normalization factor defined as1 | C n ϕ ,n σ | = n ϕ ! n σ ! n ϕ + n σ (cid:89) i (cid:20) (cid:90) d p i E ( p i )(2 π ) (cid:21) (2 π ) δ ( P − (cid:88) i p i ) (A.16)such that the states are canonically normalized: (cid:104) P (cid:48) ; n (cid:48) ϕ , n (cid:48) σ | P ; n ϕ , n σ (cid:105) = (2 π ) δ ( P (cid:48) − P ) δ n ϕ ,n (cid:48) ϕ δ n σ ,n (cid:48) σ . (A.17)More explicitly, C n ϕ ,n σ for n ϕ + n σ = 2 and n ϕ + n σ = 3 are given by | C n ϕ ,n σ | = 8 πn ϕ ! n σ ! for n ϕ + n σ = 2 , (A.18) | C n ϕ ,n σ | = 16 πn ϕ ! n σ ! (cid:18) πE cm (cid:19) for n ϕ + n σ = 3 . (A.19)In this language, the S -wave amplitudes are read off by projecting the initial/final statesonto the zero angular momentum states (A.14). Five-point amplitudes.
Now, we are ready to evaluate the cutoff scales from S -waveamplitudes. First, let us consider five-point scattering amplitudes. In the main text, weconsidered two amplitudes M σσσϕϕ and M σϕϕϕϕ . While both of them break the unitaritybound at high energy for generic EFT parameters, high energy behavior of M σσσϕϕ dependsnot only on α and λ , but also on λ , which does not source bispectra. Therefore, it is notuseful for exploring implications for the bispectrum f NL12 . On the other hand, M σϕϕϕϕ has no λ -dependence and so it is useful for the study of the bispectrum. Hence, we focuson M σϕϕϕϕ for the estimation of the cutoff scale in this paper. Also, in the regime of ourinterests, it is easy to see that the strongest unitarity constraint on α − α = 1and focus on the effect of λ − S -wave amplitude associated with M σϕϕϕϕ . First, let usconsider σϕ → ϕϕϕ scattering. By using Eq. (3.20) and Eq. (A.14), the S -wave component Besides, high energy behavior of M σσσϕϕ is dominated by the nonrenormalizable operator σ ( ∂ µ ϕ ) associated with the field space curvature, which we did not include in the main text. Note that one mightwonder that M σϕϕϕϕ is also affected by a similar operator σϕ ( ∂ µ ϕ ) , but it is prohibited by the shiftsymmetry of ϕ . Moreover, high energy behavior of this operator is as mild as the non-derivative interaction σϕ , essentially because these two are related to each other by partial integrals and field redefinition. – 23 –f the scattering amplitude M σϕ → ϕϕϕ reads M ( S ) σϕ → ϕϕϕ = C , C , (cid:89) i =1 (cid:20) (cid:90) d p i E ( p i )(2 π ) (cid:21) (2 π ) δ ( p + p − P )(2 π ) δ ( p + p + p + P ) M σϕ → ϕϕϕ = 964 √ π m (1 − λ ) r E cm + O ( E ) . (A.20)Similarly, the S -wave component of M σϕϕ → ϕϕ reads M ( S ) σϕϕ → ϕϕ = − √ π m ( λ − r E cm + O ( E ) . (A.21)Then, the unitarity bounds | M ( S ) σϕ → ϕϕϕ | ≤ | M ( S ) σϕϕ → ϕϕ | ≤ = 64 √ π · r m | λ − | . (A.22) Six-point scattering.
Finally, let us look at six-point amplitudes, among which therelevant one for our purpose is M ϕϕϕϕϕϕ given in Eq. (3.23). The S -wave component for ϕϕϕ → ϕϕϕ scattering is M ( S ) ϕϕϕ → ϕϕϕ = − π m ( λ − r E + O ( E ) . (A.23)Then, the unitarity bound | Re M ( S ) ϕϕϕ → ϕϕϕϕ | ≤ = 16 √ π / √ · r m | λ − | / . (A.24) B Field redefinition for amplitude computation
In this appendix we summarize field redefinition useful for the computation of scatteringamplitudes. Essentially because we are interested in high-energy scattering E (cid:29) m, | ˙ θ | ,one may explicitly show that terms which contain ˙ θ are irrelevant for evaluation of thecutoff scale. Then, it is enough to analyze high-energy scattering in the following setup: S = (cid:90) dtd xa (cid:34) − (cid:18) σr (cid:19) ∂ µ ϕ∂ µ ϕ − ∂ µ σ∂ µ σ − α − σ r ( ∂ µ ϕ ) − V ( σ ) (cid:35) . (B.1)To simplify the computation of scattering amplitudes, we perform the field redefinition,( r + σ ) e iϕ/r = r + ϕ + iϕ , (B.2)or in other words, σ = (cid:113) ( r + ϕ ) + ϕ − r , ϕ = r arctan ϕ r + ϕ . (B.3)Note that at the linear level we have σ (cid:39) ϕ and ϕ (cid:39) ϕ , so that scattering amplitudes of σ and ϕ are identical to those of ϕ and ϕ .– 24 – erivative interactions. We begin by the first two terms in Eq. (B.1), which can bereformulated by the field redefinition (B.2) as − (cid:18) σr (cid:19) ∂ µ ϕ∂ µ ϕ − ∂ µ σ∂ µ σ = −
12 ( ∂ϕ ) −
12 ( ∂ϕ ) . (B.4)Note that it is simply a canonical kinetic term without any interaction. On the other hand,the third term in Eq. (B.1) reads − α − σ r ( ∂ µ ϕ ) = − α −
12 ( (cid:112) ( r + ϕ ) + ϕ − r ) r (cid:16) ∂ µ r ϕ r + ϕ (cid:17) (cid:16) ϕ ( r + ϕ ) (cid:17) = − α − (cid:20) ϕ r ( ∂ µ ϕ ) + − ϕ + ϕ ϕ r ( ∂ µ ϕ ) − ϕ ϕ r ( ∂ µ ϕ ∂ µ ϕ )+ 3 ϕ − ϕ ϕ + ϕ r ( ∂ µ ϕ ) + 6 ϕ ϕ − ϕ ϕ r ( ∂ µ ϕ ∂ µ ϕ ) + ϕ ϕ r ( ∂ µ ϕ ) (cid:35) + O ( ϕ i ) . (B.5)Therefore, derivative interactions appear only when the field space curvature is nonzero. Inthis expression, e.g., the O ( E ) contribution to the four-point amplitude (3.16) is sourcedonly by the four-point contact vertex presented in the first term of the third line. Similarsimplification occurs when we evaluate more general amplitudes too. Isocurvature potential.
Next, let us take a look at the isocurvature potential, V ( σ ) = V (cid:18)(cid:113) ( r + ϕ ) + ϕ − r (cid:19) (B.6)with a parameterization V ( σ ) = 12 m r (cid:20) σ r + λ σ r + λ σ r (cid:21) . (B.7)More explicitly, we find V ( σ ) = 12 m r (cid:20) ϕ r + ϕ + ϕ ϕ r + ϕ + 2 ϕ ϕ + ϕ r (cid:21) + λ − m r (cid:20) ϕ r + 32 ϕ ϕ r + 34 − ϕ ϕ + ϕ ϕ r + 18 12 ϕ ϕ − ϕ ϕ + ϕ r (cid:21) + λ − m r (cid:20) ϕ r + 2 ϕ ϕ r + 12 − ϕ ϕ + 3 ϕ ϕ r (cid:21) + O ( ϕ i ) , (B.8)from which it is obvious in particular that non-renormalizable interactions do not appearwhen α = λ = λ = 1 is satisfied.In the main text, we provided Feynman diagrams before the field redefinition forillustration. However, the expressions (B.4), (B.5), and (B.8) simplify the computation ofscattering amplitudes a lot (the results are of course invariant under field redefinition).– 25 – eferences [1] B. W. Lee, C. Quigg and H. B. Thacker, Weak Interactions at Very High-Energies: TheRole of the Higgs Boson Mass , Phys. Rev.
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