Conservation of $\zeta$ with radiative corrections from heavy field

In this paper, we address a possible impact of radiative corrections from a heavy scalar field $\chi$ on the curvature perturbation $\zeta$. Integrating out $\chi$, we derive the effective action for $\zeta$, which includes the loop corrections of the heavy field $\chi$. When the mass of $\chi$ is much larger than the Hubble scale $H$, the loop corrections of $\chi$ only yield a local contribution to the effective action and hence the effective action simply gives an action for $\zeta$ in a single field model, where, as is widely known, $\zeta$ is conserved in time after the Hubble crossing time. Meanwhile, when the mass of $\chi$ is comparable to $H$, the loop corrections of $\chi$ can give a non-local contribution to the effective action. Because of the non-local contribution from $\chi$, in general, $\zeta$ may not be conserved, even if the classical background trajectory is determined only by the evolution of the inflaton. In this paper, we derive the condition that $\zeta$ is conserved in time in the presence of the radiative corrections from $\chi$. Namely, we show that when the scaling symmetry, which is a part of the diffeomorphism invariance, is preserved at the quantum level, the loop corrections of the massive field $\chi$ do not disturb the constant evolution of $\zeta$ at super Hubble scales. In this discussion, we show the Ward-Takahashi identity for the scaling symmetry, which yields a consistency relation for the correlation functions of the massive field $\chi$.


Introduction
Inflation provides us with a natural experimental instrument to explore the high energy physics. Measurements of the temperature anisotropies and polarization of the cosmic microwave background can constrain the Hubble parameter H at the time when the fluctuation was generated. The current data puts an upper bound on H at around 10 14 GeV [1,2], which is much higher than the accessible energy scale in particle accelerators. The precise measurements of the primordial perturbations generated during inflation may place a constraint on the theory of high energy physics independently of the particle experiments.
In string theory, compactification of the extra dimensions typically yields a number of scalar fields, which may have masses bigger than the Hubble parameter during inflation. Investigating a possible imprint of these massive fields might allow us to explore the high energy physics behind. While one field model is consistent with the current data [1], there is still room to include a contamination of such massive fields, which act as isocurvature modes. If such a massive field has a mass much bigger than the Hubble scale, integrating out the -1 -JCAP06(2016)020 massive field only gives local contributions to the effective action for the inflaton (relevant works can be found, e.g., in refs. [3,4]). In such a case, since we are ignorant of the high energy theory, it is impossible to disentangle the radiative corrections of the massive field. However, if one of the isocurvature modes has a mass of order H, the radiative correction may yield a distinctive non-local contribution.
Chen and Wang studied an impact of a massive field on the primordial curvature perturbation ζ in ref. [5] (see also ref. [6]). In their setup, the inflaton has a non-minimal coupling with the massive field, which yields the cross-correlation between them. As emphasized in ref. [7], where a more extensive analysis, including higher spin fields, was done, the massive field leaves more direct information in the squeezed configuration of the correlation functions, which has a soft external leg, than in other configurations. The massive scalar field with 0 < m/H ≤ 3/2 decays as η ∆ − with at large scales. Then, as was computed in refs. [6][7][8] the contribution of the massive field to the squeezed bi-spectrum, when the shorter mode k crosses the Hubble scale, is given by where P ζ (k) is the power spectrum of ζ. Notice that (q/k) ∆ encodes the evolution between the Hubble crossing time for the mode k and the one for q. For m > 3H/2, the massive field oscillates, while decaying, as η∆ ± with which gives the same momentum dependence as in eq. (1.2) except that ∆ − is replaced with∆ ± . When the curvature perturbation stops evolving after the Hubble crossing time of the shorter mode k, eq. (1.2) gives the squeezed bispectrum at the end of inflation. As far as the massive fields do not contribute to the background evolution (an example where a massive field modulates the background evolution was studied, e.g., in refs. [10][11][12][13]) and the tree level contribution is concerned, the curvature perturbation is conserved after all modes cross the Hubble scale [14][15][16][17][18][19]. 1 In this case, eq. (1.2) indeed gives the bi-spectrum at the end of inflation [5,7,9]. The argument in ref. [7] is based on the symmetry of the de Sitter spacetime. Therefore, one may speculate that the loop correction of the massive field may also be still given by eq. (1.2), while the scaling dimension ∆ will no longer be given as in eq. (1.1) nor eq. (1.3). In this generalization, a non-trivial point may be in showing the conservation of ζ after the Hubble crossing. In refs. [24,25], the conservation of ζ was addressed, including the loop correction of ζ in the setup of single field inflation.
In this paper, we address the conservation of the curvature perturbation ζ which is affected by loop corrections of a heavy field χ, assuming that the heavy field does not contribute to the classical background trajectory. The constant non-decaying mode of ζ is called the adiabatic mode. To compute the evolution of the curvature perturbation, we integrate JCAP06(2016)020 out the heavy field and derive the effective action for ζ. If the mass of the heavy field M is much bigger than the Hubble scale, the loop corrections of χ only yields local terms in the effective action and then following the argument in the single field case, we can show the conservation of ζ at large scales. On the other hand, if the mass M is not large enough compared to the Hubble scale, the loop corrections of χ can give non-local contributions to the effective action. The presence of the non-local contribution can yield a qualitative difference from single field models.
In single field models of inflation, the conservation of the curvature perturbation at large scales is implemented by the dilatation invariance x → e s x with a constant parameter s, which changes ζ(t, x) to ζ(t, e −s x) − s. The dilatation invariance is one of the gauge transformations and hence classically it should be preserved for a diffeomorphism invariant theory. However, when we quantize the system, the dilatation invariance is not always preserved, particularly when we allow an arbitrary initial quantum state [26,27]. When the dilatation invariance is preserved, a part of the IR divergences is canceled out [26][27][28][29][30][31][32][33][34][35][36]. (In order to eliminate all the IR divergences, we also need to preserve the invariance under other gauge transformations. For a detailed explanation, see, e.g., ref. [37].) In refs. [36,38], it was shown that when we choose the Euclidean vacuum, a.k.a., the adiabatic vacuum or the Bunch-Davies vacuum in the de Sitter limit, there exists a set of quantities which is free from the IR divergences.
In quantum field theory, a symmetry implies a corresponding identity, the so-called Ward-Takahashi (WT) identity. For one field model of inflation, the dilatation invariance yields the consistency relation, which relates the (n + 1)-point function of ζ with one soft external leg to the n-point function of ζ [39][40][41]. The consistency relation is indeed the WT identity for the dilatation invariance [41]. The consistency relation was first shown for the bi-spectrum in the squeezed limit by Maldacena in ref. [39] and it was extended to more general single field models in ref. [40]. The consistency relation for the arbitrary n-point function was derived in ref. [41]. In a single field inflation with diffeomorphism invariance, when the initial state is the Euclidean vacuum and the background trajectory is on attractor, the consistency relation generally holds. When one of these assumptions is not fulfilled, the consistency relation can be violated [42][43][44].
In this paper, we derive the consistency relation for the heavy field χ from the requirement of the dilatation invariance. When the dilatation invariance is preserved at the quantum level, we obtain the corresponding WT identity. The WT identity for the dilatation invariance yields the consistency relation which relates the (n + 1)-point function of the n χs and one soft curvature perturbation ζ to the n-point function of the χ field. The derivation of the consistency relation also applies in the presence of the loop corrections of the heavy field. Using the effective action for ζ, obtained by integrating out χ, we show that when the consistency relation for χ holds, the curvature perturbation ζ is conserved at the super Hubble scales.

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This paper is organized as follows. In section 2, we review the conservation of ζ in single field models of inflation, emphasizing the crucial role of the dilatation invariance. In section 3, after we describe our setup of the problem, we introduce the effective action for ζ by integrating out the heavy field in the in-in (or closed time path) formalism. In section 4, we derive the consistency relation for the heavy field from the Ward-Takahashi identity for the dilatation invariance. In section 5, using the consistency relation, derived in section 4, we show the conservation of ζ in the presence of the loop corrections of the heavy field. In section 6, we briefly discuss the renormalization of the heavy field. Finally, in section 7, we conclude.

Conservation of ζ and dilatation invariance in single field inflation
In single field models of inflation, it is known that the curvature perturbation is conserved in the large scale limit. In this section, we show that the conservation of ζ is a direct consequence of the dilatation invariance.

Single field inflation
For illustrative purpose, we start our discussion by considering a single scalar field with the standard kinetic term, whose action is given by In this paper, we set the gravitational constant κ 2 ≡ 8πG to 1. Using the ADM form of the line element: where we introduced the lapse function N , the shift vector N i , and the spatial metric h ij , we can express the action (2.1) as where s R is the three-dimensional Ricci scalar, and κ ij and κ are the extrinsic curvature and its trace, defined by Here, the spatial indices i, j, · · · are raised or lowered by the spatial metric h ij , and D i denotes the covariant differentiation associated with h ij . Taking the variation of the action with respect to N and N i , which are the Lagrange multipliers, we obtain the Hamiltonian and momentum constraint equations as

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We determine the time slicing, employing the uniform field gauge: We express the spatial metric h ij as where the background scale factor is expressed as a ≡ e ρ and δγ ij is set to traceless. As spatial gauge conditions, we impose With this gauge choice, the constraint equations are given by Inserting N and N i , which are expressed in terms of ζ by solving these constraint equations, into the action (2.3), we can derive the action for ζ [39,66].

Dilatation invariance
The transverse condition imposed on δγ ij is non-local and hence to determine the coordinates, we need to employ boundary conditions. For example, at linear order in perturbation, the tensor perturbation transforms under the spatial coordinate transformation The transverse condition on δγ ij gives which does not determine δx i uniquely without specifying boundary conditions to solve eq. (2.13). For the scalar mode, all spatial coordinate transformations δx i = ∂ i δx which satisfy ∂ 2 ∂ i δx = 0 still keep the transverse condition after the transformations. When we consider a compact support on each time slicing, we find an infinite way to impose the boundary conditions in solving ∂ 2 ∂ i δx = 0. We analyzed these gauge degrees of freedom in detail in refs. [26,27], where we used the italic font for "gauge" to discriminate the gauge transformations defined within the compact support from those in the infinite spatial support.
(See ref. [41] for a more recent work.) So far, we kept the tensor perturbation δγ ij for the illustrative purpose, but in the rest of this paper, we neglect it. Among these transformations, the important one for ζ is the dilatation δx i = sx i , which is extended to x i → x i s = e s x i at non-linear orders. Here, s is a constant parameter. 2 Under 2 In order to evaluate an observable quantity, one may want to compute a quantity which is solely determined by the causal history of the universe. When we solve the evolution of the fluctuation in the causally connected region to us, O, an influence of the causally disconnected region can appear as a boundary condition on the causal boundary. We can show that changing the boundary condition is equivalent to changing the spatial coordinates in O. One of the coordinate transformations is the dilatation with a time dependent s(t) [26,27]. Requesting the diffeomorphism invariance in the causally connected region is crucial to show the absence of the infrared (IR) divergence. While the IR issue is related to the current study, such a time dependent s(t) is not needed in this paper. Therefore, we do not consider the change of the boundary condition and assume that the parameter s is time independent.
-5 -JCAP06(2016)020 this transformation, the spatial line element is recast into and then the curvature perturbation changes to This is a purely geometrical argument, and hence this transformation law also should apply to multi-field models.
Preserving the dilatation invariance is crucial to ensure the infrared (IR) regularity of the curvature perturbation. In refs. [26,27], we introduced the spatial Ricci scalar evaluated in the geodesic normal coordinates as a gauge invariant quantity. Since the contribution from the IR modes, which give rise to the IR divergence, can be eliminated by performing the corresponding gauge transformation, the IR divergence also can be removed from the gauge invariant quantity.
By construction, the spatial Ricci scalar evaluated in the spatial geodesic normal coordinates is gauge invariant and it serves a conceptually clear example of the gauge invariant quantity. Nevertheless, using the geodesic normal coordinates can alter the UV behaviour [67,68]. For a practical use, we may use the smeared geodesic coordinates x g (t) given by [35,36,38] x where gζ is the averaged ζ at a compact support on each time slicing, given by where W t (x) is a window function which vanishes outside the compact support on the time constant slicing. The spatial Ricci scalar evaluated at x g is not invariant under all the gauge transformations, but it is invariant under the dilatation.

Conservation of ζ in single field inflation
In single field inflation, solving the Hamiltonian and momentum constraint equations, we can eliminate N and N i and write down the action only in terms of ζ. Since the action for any diffeomorphism invariant theory remains invariant under the dilatation, the action for ζ should take the following form: where the Lagrangian density L ζ includes ζ only in the form ∂ t ζ or e −(ρ+ζ) ∂ i ζ. (A detailed explanation can be found in ref. [35].) To address the conservation of ζ in the large scale limit, we neglect the terms which include ζ with the spatial derivative. Then, the action for ζ, written in the form (2.18), is given by

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where we schematically wrote the non-linear terms which includeζ. Here, the time dependent function f n (t) is expressed only in terms of the background quantities such asρ and the slowroll parameters. Varying the action with respect to ζ, we obtain the equation of motion in the large scale limit as This equation motion has the anticipated constant solution in time as the non-decaying mode.
I.e., if ζ(x) = F (x) is a solution of eq. (2.20), F (x) + C with a constant shift should also satisfy the equation. Therefore, when the deviation from the constant mode decays as it occurs in the large scale limit of the standard setup, ζ should be conserved at large scales.
The relation between the shift symmetry and the conservation of ζ was pointed out also in Horndeski's theory [69].

Effective action for ζ with loop corrections of heavy field
Next, we consider a two-field model with one inflaton and one heavy field. The latter field does not contribute to the classical background evolution. Following the Feynman and Vernon's method [70], in this section, we compute the effective action for the curvature perturbation with loop corrections of the massive field.

Two field model
In this paper, we consider a light scalar field φ and a massive scalar field χ whose action is given by where V (φ, χ) is a potential for the scalar fields: We decomposed the potential V (φ, χ) into the χ independent part V ph and the rest V ch . When the mass of χ field, M , depends on the inflaton φ, this model includes the direct interaction between φ and χ, e.g., φ 2 χ 2 , which was addressed in refs. [71,72]. We assume that the mass of the inflaton m is much smaller than the Hubble parameter H as m ≪ H, while the mass of the field χ is bigger than H as M > H. Then, the classical background evolution is determined solely by the inflaton φ and (the linear perturbation of) χ becomes the pure isocurvature perturbation. In this paper, we only allow the renormalizable interactions for χ, while V ph (φ) may include non-renormalizable interactions. To perform the dimensional regularization, we consider a general (d + 1) dimensional spacetime. An extension to include non-renormalizable interactions for χ is straightforward as long as a finite order of loop corrections is concerned.

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As in the single field case, we determine the time slicing, imposing For our later use, we discriminate the part which explicitly depends on χ from the rest as where S ad agrees with the action in single field models, given in eq. (2.3), and S χ is given by Even if there is no direct interaction between φ and χ, gravity yields the non-linear interaction between ζ and the heavy field χ.

Effective action
Since the mass of the field χ is much bigger than the Hubble scale H, it is natural to set the background value of χ to 0. Then, the field χ does not contribute to the classical background evolution. Meanwhile, because of the interaction between ζ and χ, the quantum fluctuation of χ can affect the evolution of the field ζ. In order to compute the evolution of ζ under the influence of χ, we compute the Feynman and Vernon's influence functional [70,73,74], which can be obtained by integrating out the field χ, in the closed time path (or the in-in) formalism.
As in the single field case, the action S also includes the lapse function and the shift vector, which can be removed by solving the Hamiltonian and momentum constraint equations. These constraint equations are also modified due to the quantum fluctuation of the heavy field χ.
Using the closed time path, the n-point function of the curvature perturbation is given by where we used an abbreviation δg = (δN, N i , ζ). In the closed time path, we double the fields: δg + and χ + denote the fields integrated from the past infinity to the time t and δg − and χ − denote the fields integrated from the time t to the past infinity. Here, T denotes the time ordering. Inserting ζ − (x) into the path integral in the numerator, we can compute the n-point function ordered in the anti-time ordering. Since N and N i are not independent variables, we perform the path integral only regarding ζ and χ.
Introducing the effective action S eff as we rewrite the n-point function for ζ as (3.9) -8 -

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By inserting the action S into eq. (3.8), the effective action is recast into where S ′ eff is the so-called influence functional, given by where we factorized S ad [δg ± ] which commute with the path integral over χ ± . The effective action S eff [δg + , δg − ] describes the evolution of the curvature perturbation affected by the quantum fluctuation of the heavy field χ.

Computing the effective action
Performing the path integrals about χ + and χ − , we can compute the effective action where S ′ eff(n) denotes the terms which include n δg α s, given by . (3.14) In eq. (3.13), each δg αm with m = 1, · · · , n should add up δN αm , N i,αm , and ζ αm . Here and hereafter, for notational brevity, we omit the summation symbol over δg unless necessary. Using eq. (3.11), we can express the variation of S ′ eff with respect to δg ± by using the propagators for χ. Notice that the shift symmetry is not manifest in this series expansion.
The linear term in the effective action is given by where W (1) δgα is given by the expectation value as Next, we compute the quadratic terms in S ′ eff . Taking the second variation of S ′ eff with respect to δg + , we obtain where δg and δg are either δN , N i , or ζ, and they can be different metric perturbations.
Here, we introduced the expectation value: Since the action S χ [δg + , χ + ] includes only local terms, the variation of S χ [δg + , χ + ] with respect to δg + (x 1 ) and δg + (x 2 ) yields the delta function δ(x 1 − x 2 ) in eq. (3.17). Similarly, the second variation of S ′ eff with respect to δg − is given by Taking the derivative with respect to both δg + and δg − , we obtain and These functions W (2) δgα 1 δgα 2 (x 1 , x 2 ) can be expanded by the propagators of χ for λ = 0, i.e., the time-ordered (Feynman) propagator: the anti-time ordered (Dyson) propagator: 23) and the Wightman functions: Here, S χ,0 [χ] denotes the action given by Recall that these propagators are mutually related as where θ is the Heaviside function.

Propagators for χ
In this subsection, solving the mode function for the heavy field χ, we compute the propagators introduced in the previous subsection. At the linear order of χ, the equation of motion is given byχ where χ k is the Fourier mode of χ. Changing the variable from χ k to X k (t) = e d−1 2 ρ(t) χ k (t) and using the conformal time η, the mode equation (3.29) is recast into where the dash denotes the derivative with respect to the conformal time η and the time dependent frequency Ω k is given by Using W k which satisfies Using the mode function χ k , we quantize the non self-interacting heavy field as follows where a k denotes the annihilation operator. With this expansion, the Wightman function G + (x 1 , x 2 ) is given by

Ward-Takahashi identity from the dilatation invariance
As was discussed in section 2.2, the action for the single field model preserves the invariance under the dilatation, which changes ζ(t, x) to ζ(t, e −s x) − s. This invariance is preserved classically also for multi-field models of inflation, since it is a part of the spatial diffeomorphism. In a quantum field theory, it is known that a symmetry leads to a corresponding Ward-Takahashi (WT) identity. When the dilatation invariance is also preserved at the quantum level, we obtain the WT identity. 3

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In section 4.1, we discuss the WT identity from the dilatation invariance of the correlators of χ. In single field models of inflation, it was shown that the WT identity of the dilatation invariance yields the consistency relation which relates the (n+1)-point function of ζ with one soft external leg to n-point function of ζ [39][40][41]. Likewise, in section 4.2, we find that the WT identity derived in section 4.1 gives the consistency relation which relates the (n + 1)-point cross-correlation with n χs and one soft ζ to the n-point auto-correlation of χ.

Ward-Takahashi identity
In single field models, an invariant quantity regarding the dilatation was constructed in refs. [26,27] by using the smeared geodesic normal coordinates, defined in eq. (2.16). Using which is invariant under the dilatation with the constant parameter s. Here, gζ ≡ gζ (t f ). When the dilatation invariance is also preserved for the quantum system, the correlation functions of g χ(t, x g ) should be invariant under the dilatation as χ α 1 (t 1 , e − gζ x g1 ) · · · χ αn (t n , e − gζ x gn ) ± δg = χ α 1 (t 1 , e − gζ +s x g1 ) · · · χ αn (t n , e − gζ +s x gn ) ± δgs (4.2) with α i = ±. Here, δg s denotes the metric perturbations after the dilatation. Under the dilatation, δN and N i change as and ζ changes as in eq. (2.15), and then gζ changes to gζ s = gζ − s. Equation (4.2) holds only when the quantum state also preserves the dilatation invariance. This is the WT identity for the dilatation invariance. At O(s), setting δg = 0, the WT identity yields Since the changes of δN and N i under the dilatation are linear in N and N i and their derivatives, they vanish after setting δg to 0. Using the WT identity (4.5) with x 1 = · · · = x p ≡ x and x p+1 = · · · = x n ≡ x ′ , we obtain where α = ±. In the next section, using these identities, we show the conservation of ζ, including the loop corrections of the heavy field.

Consistency relation (soft theorem)
In single field models of inflation, it is known that the WT identity for the dilatation invariance gives the consistency relation. The consistency relation for ζ is an example of the soft theorem, which was first shown for the soft graviton scattering by Weinberg [75]. Recently, Weinberg's soft theorem was recaptured by Strominger et al. and was shown to be equivalent to a Ward-Takahashi identity in an asymptotically flat spacetime [76][77][78].
Here, we show that the WT identity (4.5) also gives a consistency relation in multi-field models. Performing the Fourier transformation of the WT identity (4.5) evaluated at an equal time t with all α i s chosen as +, we obtain where we abbreviated t in the argument of χs. The correlator in the first line is simply the in-in n-point function of χ(t, k). The correlator in the second line is given by the product of the Wightman function, the Feynman propagator, and the Dyson propagator, which appear by contracting χ ± with χ ∓ , χ + with χ + , and χ − with χ − , respectively. The correlator in the third line is given by the product of the Feynman propagator. The interaction vertices inserted at any time after t are canceled between the terms in the second and third lines. This ensures the causality in the closed time path formalism. Taking into account this cancellation, we can rewrite eq. (4.7) as where the correlation function with dash denotes the correlation function from which (2π) d and the delta function are removed, e.g., In deriving eq. (4.8), we used (4.10) -13 -JCAP06(2016)020 Figure 1. As an example, we consider a three point interaction vertex in S χ with two χs and one ζ. Since ζ is removed from the interaction vertex by operating the functional derivative, only two χs remain in the vertex. Contracting these two χs with χ included in the Heisenberg operator χ(k i ) where i = 1, · · · n, we obtain the diagram in the right of the arrow. The red dotted line represents the amputated ζ.
The correlation function in the second line of eq.
where P ζ (k) is the power spectrum of the free ζ. 4 This is the consistency relation for the heavy field χ. The correlation function in the second line contains only one gravitational interaction vertex with ζ, but it can contain more than one self interaction vertexes for the heavy field χ. This is one example of the soft theorem. Using the WT identity at O(s), we derived the consistency relation for the correlation functions with one soft ζ. Using the WT identity at O(s p ), we can derive the consistency relations with p soft ζs.

Dilatation invariance and WKB solution
In this subsection, in order to provide an intuitive understanding of the consistency relation (4.8), we consider the case with λ = 0. We will find that in this case, the condition (4.8) restricts the mode function for the massive field χ and the condition can be satisfied, e.g., for the WKB solution.

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Inserting eq. (3.34) into eq. (4.8), we obtain Integrating by parts and using the mode equation, we obtain where d|χ k (t)| 2 in the left hand side was canceled with the term which appears by operating the time derivative on the Heaviside function θ(t − t ′ ). We can show that the WKB solution satisfies the WT identity (4.13). In order to show this statement, we rewrite eq. (4.13) as The last two terms in L k (η) are canceled between the two terms in eq. (4.14). The time integral of the second term converges by rotating the time path as η → −∞(1 + iǫ) where ǫ is a positive constant. For L * k (η), the time integral of the second term should be rotated as η → −∞(1 − iǫ). We chooseη at a time in the distant past when the mode function can be well approximated by the leading order WKB solution with W k = k. (To be precise,η differs for a different wavenumber k.) Since L k (η) satisfies the mode equation for χ k (η), i.e., 16) and the initial conditions L k (η) = L ′ k (η) = 0, L k (η) vanishes all the time. Thus, we find that the WKB solution satisfies eq. (4.14).
For the exact de Sitter space, in the limit ke −ρ ≪ M and H ≪ M , we can easily check that the WKB solution, given by satisfies eq. (4.13) as (4.18)

Conservation of ζ with loop corrections of heavy field
In this section, we show that when the dilatation invariance is preserved, the curvature perturbation ζ is conserved in time at super Hubble scales, including the loop correction of the heavy field χ. For this purpose, first we rewrite the effective action, using the WT identities derived in the previous section. Then, using the obtained effective action, we show the conservation of ζ.

Effective action with dilatation invariance
As discussed for the single field inflation in section 2.1, the presence of the constant solution is implied by the dilatation invariance. In this subsection, using the WT identity, we rewrite the effective action in such a way that the dilatation invariance becomes manifest. Taking the variation of S χ with respect to δg, we can compute W (n) δgα 1 ···δgα n and the effective action. For instance, taking the n-th derivative of S χ with respect to ζ and setting δg to 0, we obtain In this subsection, we show that when the WT identity for χ, given in eq. (4.6), is fulfilled, the effective action for ζ preserves the dilatation invariance.
To show this, we further rewrite the WT identity (4.6). Operating and performing the integration by parts, we obtain Here, after rewriting δ(x − y)(x · ∂ x + y · ∂ y ) as δ(x − y)(y · ∂ x + x · ∂ y ), we performed the integration by parts and then we used Similarly, operating on eq. (3.14) with n = 2 and p = 1, where µ, ν = 0, i, we obtain As is clear from the derivation, eq. (5.6) also holds, even if we replace δg ± (x) included in each term with an arbitrary function. Therefore, replacing δg ± (x) with a constant nonzero number, we obtain By adding the left hand side of eq. (5.6) multiplied by a constant parameter −s and eq. (5.7) with δg ± = ζ ± multiplied by −s 2 /2, the linear and the quadratic terms in the effective action can be given by δgα (x) where δg s are related to δg as given in eqs. (2.15), (4.3), and (4.4). Here, each δg i,α (i = 1, 2) sums over δN α,s , N i,α,s , and ζ α,s . We can drop the term with one shift vector, because W N i,α , which is proportional to χ∂ i χ , vanishes. In deriving eq. (5.8), we used and gζ + = gζ − , which holds since ζ + (t f , x) = ζ − (t f , x). The first term in eq. (5.6) changes the argument of the metric perturbations in the linear term of eq. (5.8). We also changed the arguments of the quadratic terms, since the modification appears only in higher orders of δg. Equation (5.8) shows that with the use of the WT identity, δg α (x) in S ′ eff can be replaced with δg α, s (x). Since the rest of the effective action, S ad , is simply the classical action for the single field model, it also should be invariant under this replacement. Therefore, when the WT identity (4.2) holds, the total effective action S eff preserves the invariance under the change of δg α to δg α,s .
The effective action (5.8) includes the lapse function and the shift vector. By solving the Hamiltonian and momentum constraint equations, we can express δN s and N i,s in terms of ζ s . Using these expressions, we can eliminate δN s and N i,s in the effective action as in the single field model [39]. Since the constraint equations for δg s are given by replacing δg with δg s in the constraint equations for δg, the effective action for ζ s obtained after eliminating δN s and N i,s should be given simply by replacing ζ with ζ s in the effective action expressed only in terms of ζ.

Tadpole contribution
Before we discuss the conservation, we show that the linear terms in the effective action S eff , which is the tadpole terms, vanish all together. Taking the variation of the effective action with respect to N and N i , we obtain the constraint equations. The Hamiltonian constraint for the FRW background gives (5.10) and at the liner order where ∂ i ≡ δ ij ∂ j . By changing the spatial distance, the curvature perturbation ζ can affect on the evolution of the heavy field χ. In χ 2 and V (φ, χ) , the dependence on ζ disappears after taking the coincidence limit, while (∂ x χ(t, x)) 2 still depends on ζ. Using the proper distance x phys (t) = ae ζ x, where ζ is absorbed, we introduce the expectation value which does not depend on ζ. Here, k is the Fourier mode of x phys . The scalar part of the momentum constraint gives where we used ∂ i χ∂ j χ ∝ δ ij . The momentum constraint equation can be solved as (5.14) We can add a homogeneous solution of the Laplace equation on the right hand side. The action S ad which is accurate at the linear order of δg is given by where, for our purpose, we partially kept the higher order terms in the exponential form. The n-th order effective action S ′ eff(n) includes the local terms given by with α = ±. Adding up these local terms for all n, we obtain where the terms which do not depend on δg are canceled between the two terms on the right hand side. Adding these local terms to S ad and using the Hamiltonian constraint, we obtain a concise expression as Using the Hamiltonian constraint (5.10) and (5.11), 5 we can rewrite the action which is valid up to the linear order as (5.18) The first term vanishes as a total derivative. The second term is proportional to which can be verified by using the time derivative of the Friedman equation and the field equations for φ and χ, given byφ 20) The tadpole terms contained in the last line cancel with each other and the term which does not depend on ζ is canceled between the action for + and the one for −. In this way, using the background equations, we can show that the tadpole contributions all disappear. As we discussed in the previous subsection, the effective action S eff stays invariant under the replacement of ζ α (x) with ζ α,s (x). Therefore, the tadpole contribution for ζ s should be given simply by replacing ζ(x) with ζ s (x) in eq. (5.17). When the background equations are satisfied, the terms in the second line of eq. (5.8), which are linear in the metric perturbations, all vanish. 5 Here, we implicitly assumed that the boundary term of the spatial infinity is chosen such that d d x ∂ i Ni = 0. As far asζ k vanishes in the large scale limit, this boundary condition can be consistently imposed. Notice the dilatation transformation with the constant parameter s does not alter this boundary condition.

Existence of constant solution
Removing the tadpole contribution which vanishes with the use of the background equations, we only consider the quadratic terms about ζ. At the linear level, ζ α,s simply gets the constant shift as Therefore, the symmetry under the change of ζ α into ζ α, s immediately implies the existence of the constant solution also in the presence of the loop corrections of the heavy field. In single filed cases, it is well known that only the constant solution survives while the other independent solution simply decays in the late time limit, as far as the background evolution is on an attractor (see, e.g., ref. [80]). Then, the curvature perturbation becomes time independent at super Hubble scales. When we add a quantum correction from a heavy field, in principle, the "decaying" mode can turn into a growing mode. Such a drastic change of the behaviour of perturbation can occur, in case the trajectory sizably deviates from the attractor solution, for instance, owing to an effect of an additional field. In the present context, the classical background evolution is determined only by the inflaton and we assume that the quantum effects of the heavy field always remain to be perturbative. In such cases, the effect of the heavy field does not drive the "decaying" mode to grow in time. Then, the presence of the constant mode implies the conservation of the curvature perturbation in time as well as in the presence of the loop corrections of the heavy field.

Renormalization and dilatation invariance
As is common in a non-linear quantum field theory, the effective action for ζ potentially diverges due to UV corrections. In our case, the bare coefficients of the effective action W (n) δgα 1 ···δgα n , which are expressed in terms of the correlators for χ, can diverge. The UV divergence should be renormalized by introducing counter terms. Depending on a way to introduce the counter terms, the dilatation invariance might be broken. If it were the case, the WT identity would not hold any more and the renormalized effective action does not preserve the dilatation invariance.
When the counter terms are introduced in such a way that the dilatation invariance is preserved, the WT identity holds also after the UV renormalization. Then, inserting the WT identity into the effective action, which can be renormalized following the standard procedure since the theory (before the gauge fixing) is a local theory, and repeating the same argument as we did for the non-renormalized effective action, we can replace ζ α (x) into ζ α, s (x) in the renormalized effective action.
Since only the heavy field is quantized in computing the effective action, the curvature perturbation ζ should be dealt with as a classical external field. We may set the arbitrary constant parameter s to a c-number variable gζ in order to express the effective action in terms of the fluctuation in the local region. Then, the effective action includes the nonlocal contribution gζ . Nonetheless, the renormalization should proceed in the standard way, because the inserted non-local contribution, which is schematically in the following form: 0 = (WT identity, which identically vanishes and is local) × gζ n (n = 1, 2) is fictitious and does not introduce any non-local interactions.

JCAP06(2016)020
This aspect may be instructive to speculate on the UV renormalization of an IR regular quantity. Preserving the dilatation invariance is crucial to cancel out the potentially IR divergent contribution. In refs. [26,27], a quantity which preserves the dilatation invariance was proposed and it contains non-local contributions. Because of that, in refs. [67,68], it was suggested that the quantity which preserves the dilatation invariance may not be able to be renormalized in the standard way by introducing local counter terms.
In this paper, we presented a handy example where the UV renormalization of the heavy field can be performed simply by introducing local counter terms as well as for a quantity which looks to include a non-local contribution. Here, we only considered the UV renormalization of the heavy field χ. It will be important to see if the UV renormalization of the curvature perturbation also can proceed by introducing local counter terms or not. We leave this issue for a future study.

Concluding remarks
String theory predicts the presence of a bunch of massive excitations after reduction to four dimensional spacetime, which may encode, for instance, the information on the structure of the compactification of the extra dimensions. It is important to explore a possible imprint of such massive modes on the curvature perturbation. In this paper, we considered an influence of a heavy scalar field on the curvature perturbation ζ at the super Hubble scales. When the mass of the heavy field χ is of O(H), it can give non-local radiative corrections to the effective action of ζ, which may provide a distinctive imprint of the heavy field. We showed that the time evolution of ζ at the super Hubble scales is not affected by the loop corrections of the heavy field as far as the dilatation invariance, which is entailed in a covariant theory at the classical level, is preserved. The implies that the constant adiabatic mode exists as well as in the presence of the loop corrections of the heavy field.
For simplicity, we considered one heavy field with the standard canonical kinetic term. However, our argument can be extended in a straightforward manner to a more general model which contains more than one heavy fields with a non-canonical kinetic term.
Our result indicates that in order to leave an imprint of massive fields well after the Hubble crossing, we need to break either of the following conditions: 6 • The massive fields do not alter the background evolution at the classical level.
• The quantum system preserves the dilatation invariance, which yields the Ward-Takahashi identity.
• The radiative corrections of the massive fields on the curvature perturbation ζ are perturbatively suppressed.
If the last condition does not hold, we need to perform a non-perturbative analysis to compute the radiative corrections of the massive fields. In this paper, using the WT identity (4.2) for the dilatation invariance at O(s), we showed that the metric perturbation δg(x) in the effective action can be replaced with δg s (x), given in eqs. (2.15), (4.3), and (4.4), keeping up to the quadratic terms. This argument can JCAP06(2016)020 be extended to higher orders in δg. Using the WT identity (4.2), we can derive the WT identity which relates W (n) to W (n ′ ) s with n ′ < n. Adding the WT identity for W (m) with m ≤ n (multiplied by some particular constant factors) to the effective action S ′ eff , we can replace all δg(x) with δg s (x) in S ′ eff up to the n-th order of perturbation. After removing the lapse function and the shift vector, we find that the effective action for the curvature perturbation is invariant under the replacement of ζ(x) with This implies that the curvature perturbation includes a solution which is given by the sdependent terms in eq. (7.1), whose first term is the constant adiabatic mode. In order to keep the terms which explicitly depend on x perturbatively small, we need to confine the perturbation within a finite spatial region on each time slicing. For that, we will need to use other residual gauge degrees of freedom, which are addressed in refs. [26,27].
In this paper, we studied a spin 0 scalar field as the heavy field. It will be interesting to extend the discussion to include a field with a more general spin [7]. Our discussion does not rely on the explicit form of the interaction vertices nor the propagator. Therefore, we expect that this extension will be feasible. We leave this study for a future project [83].