Cancellation of quantum corrections on the soft curvature perturbations

We study the cancellation of quantum corrections on the superhorizon curvature perturbations from subhorizon physics beyond the single-clock inflation from the viewpoint of the cosmological soft theorem. As an example, we focus on the transient ultra-slow-roll inflation scenario and compute the one-loop quantum corrections to the power spectrum of curvature perturbations taking into account nontrivial surface terms in the action. We find that Maldacena's consistency relation is satisfied and guarantees the cancellation of contributions from the short-scale modes. As a corollary, primordial black hole production in single-field inflation scenarios is not excluded by perturbativity breakdown even for the sharp transition case in contrast to some recent claims in the literature. We also comment on the relation between the tadpole diagram in the in-in formalism and the shift of the elapsed time in the stochastic-$\delta N$ formalism. We find our argument is not directly generalisable to the tensor perturbations.


Introduction
A hierarchy of scales allows one to integrate out ultraviolet (UV) modes to obtain a low-energy effective field theory (EFT).For example, descriptions of the scattering of the Standard Model particles at the Large Hadron Collider do not require treatment in quantum gravity.Similarly, chemical reactions can be understood without the knowledge of the Standard Model of particle physics.The essential underlying features are the scale separation and the decoupling of the UV modes [1]. 1 Applications of EFTs are quite successful in wide areas of physics.
Analogously, scale separation and decoupling of small-scale modes should be expected in quantum field theory in curved spacetime in the cosmological context.Indeed, one can formulate cosmological EFTs such as the EFT of inflation [3] and the EFT of large-scale structure [4] in the same manner as in the usual EFT in flat spacetime.Somewhat relatedly, it is well known that the primordial curvature perturbations are conserved on superhorizon scales [5][6][7][8] in adiabatic time evolution.They are not affected by subhorizon dynamics.Moreover, it was shown in Refs.[9][10][11] that quantum corrections to the primordial curvature perturbations on superhorizon scales from much smaller scales are absent in single-clock inflation models.
Our focus in this paper is the quantum corrections to the large-scale cosmological perturbations themselves rather than PBHs.While it is important to answer the question of whether PBH formation in single-field inflation is possible without large backreactions to the large scale where the cosmic microwave background (CMB) fluctuations are measured, discussions on this question so far largely depend on the results that the one-loop corrections to the CMB scale can be sizable [12, 13, 24-27, 29, 30, 34, 36, 38].However, the full one-loop calculation including all the relevant vertices has not yet been performed.If the large contributions cancel so that there is no total large quantum correction from small-scale loops (even in the case of sharp SR/USR/SR transitions) as suggested in Refs.[35,37], the threat to PBH formation in single-field inflation disappears completely.In particular, Ref. [37] pointed out the importance of a total derivative term to see such a cancellation.We augment his calculation with another relevant surface term and emphasise the role of Maldacena's consistency relation [5] along the line of Ref. [9] to understand the fact that the quantum corrections vanish as a result of a cosmological soft theorem on the one-loop diagrams.
In addition to the issue of the quantum corrections to the soft (large-scale) curvature perturbations from the small-scale loop effects, quantum corrections to the primordial power spectrum of gravitational waves on large-scales from small-scale loop effects were found in the same SR/USR/SR scenario [39] and in a phenomenological setup where a mode function of a spectator scalar field is enhanced on small scales during SR inflation [40,41].Although these quantum corrections would not violate the perturbativity on CMB scales (at least when we restrict the mode function of the spectator field to be of the Bogoliubov transformation form [41]), it is in serious tension with the notion of decoupling.We will briefly discuss this issue as well.
Another purpose of this paper is to point out the relation between the in-in (Schwinger-Keldysh) formalism [42][43][44] (for review, see Refs.[45,46] and references therein) and the stochastic-δN formalism (see, e.g., Refs.[47][48][49][50][51]) for the calculation of cosmological correlation functions.The stochastic formalism (see Refs. [52][53][54][55][56][57][58][59][60][61] for the first papers on the subject) is the EFT for matter fields (such as the inflaton) coarse-grained on a superhorizon scale, called the infrared (IR) mode.There, the IR mode is interpreted as a (non-quantum) stochastic variable (or a Brownian motion), governed by the effective action improved by the UV loops.Combining it with the δN formalism [6,[62][63][64], one can calculate the statistical properties of the curvature perturbations in the stochastic formalism.Similarly to the ordinary EFT approach, the stochastic computation should be consistent with the direct calculation in the in-in approach.We point out that the shift of the average elapsed time of inflation can be a correction on large-scale perturbations from small-scale ones in the stochastic approach and it is understood as a tadpole contribution in the in-in formalism.
The structure of the paper is as follows.In Sec. 2, we review the consistency relation and discuss why the cancellation of one-loop corrections is generically expected.As a concrete and important example, we consider the transient USR inflation in Sec. 3. We confirm the consistency relation in this setup and use the results in the calculation of the one-loop corrections to the power spectrum of the large-scale curvature perturbations.We find the cancellation of dominant contributions, so there is no significant one-loop correction to the power spectrum of the curvature perturbations.In Sec. 4, we discuss the tadpole diagram and its relation to the counterpart in the stochastic-δN formalism.We argue that the tadpole diagram, which can be absorbed by the redefinition of the scale factor, is the only possible nonzero contribution to the quantum corrections of the power spectrum of the curvature perturbations in the soft limit.We briefly discuss the tensor power spectrum in Sec. 5. Sec.6 is devoted to our conclusions.Throughout the paper, we take the Planck unit c = ℏ = 8πG = 1.

Soft theorem on the cosmological loop corrections
In this section, we schematically discuss the possible form of the self-energy of the soft curvature perturbation from the viewpoint of the cosmological soft theorem.We particularly focus on the corrections from UV fluctuations whose wave numbers are much larger than that of the external lines.
Symmetries of the theory restrict the allowed form of Green's functions, one-particle irreducible (1PI) vertices, or their generating functional, via the Ward-Takahashi identity [65,66].In cosmology, the so-called Maldacena's consistency relation [5] (see also Refs.[67][68][69][70] and Refs.[71,72] for its generalisation to N -point functions and to the USR models, respectively) is understood as an example of the Ward-Takahashi identity on correlation functions of cosmological perturbations including at least one soft external/internal momentum [73][74][75].The curvature perturbation2 ζ is regarded as the Nambu-Goldstone (NG) boson associated with the dilatation [76][77][78][79][80][81][82].The squeezed limit of the bispectra in the adiabatic universe for example is then given by the (Fourier-space) dilatation of the power spectrum of the other operator Ô as [5,75,83] Two representative diagrams of the one-loop corrections on the propagator of cosmological perturbations through the cubic interactions.The wave lines represent the external curvature perturbation, while the plain lines are internal particles also for which we assume the curvature perturbation in this paper.Lines without the arrow are the statistical propagators and ones with the arrow indicate the retarded propagators (see Sec. 3 and also Ref. [31]).Left: the so-called cut-in-the-middle diagram representing the induction of long-wavelength modes by short-wavelength modes through the second-order effect such as the induced gravitational waves [93][94][95][96][97]. Right: the cut-in-the-side diagram which causes the effective mass correction.It has the potential ability to alter even the soft propagator.
as the NG soft theorem, where k S := |(k 2 − k 3 )/2| and the power spectrum and bispectrum are defined by It means that the short-long correlation is dominated by the "artefact" due to the modulation of the local spatial metric, which can be renormalised into the redefinition of the local metric [83][84][85][86][87][88][89][90][91][92].Therefore, the satisfaction of Maldacena's consistency relation implies no "physical" correlation between the short-and long-wavelength modes.We numerically confirm the consistency relation beyond the SR inflation in Sec. 3. Does this soft theorem have an implication for self-energy?
In the Schwinger-Keldysh formalism, the (possible) dominant contributions for the self-energy via the cubic interaction3 are schematically classified into the two Feynman diagrams shown in Fig. 1 (see Sec. 3 or Ref. [31] for the details of the Feynman rule).The left one called the cut-in-the-middle diagram [9] represents the induction of long-wavelength modes by short-wavelength modes through the second-order effect such as the induced gravitational waves [93][94][95][96][97] (see Ref. [98] for a review).This diagram is independent of p because the retarded propagator (the line with the arrow) is independent of p at the leading order. 4As the corresponding dimensionless power spectrum includes the factor of p 3 via the Fourier-space volume factor, this diagram is suppressed in the soft limit p → 0, whose feature has been accepted in the literature.On the other hand, the right cut-in-the-side diagram [9] is understood as the effective mass correction, which is proportional to the statistical propagator (the line without the arrow) of p.It is proportional to p −3 and hence this diagram may cause a scale-invariant correction on the soft propagator.In terms of the power spectrum, the diagram formally has a structure of If the short-wavelength perturbations P O (k) are enhanced for PBH formation for example, it could potentially cause a significant correction even on the CMB-scale perturbation, which is the reason why the loop correction has recently attracted much attention.
However, one should note that the coefficient C(k) is not necessarily positive-or negative-definite and there can be cancellation in the integration over the loop momentum k.In fact, the cut-in-the-side diagram includes the bispectrum one shown in Fig. 2  with a constant c.The loop contribution is hence proportional to In principle, the integration should be taken over the whole momentum as k min → 0 and k max → ∞.In order to utilise the squeezed bispectrum, we however set k min ≫ p and neglect the contribution of k < k min , which is enough for our purpose.Nevertheless, k min should be sufficiently smaller than the enhancement scale k enh so that the corresponding power spectrum is much smaller than the enhanced one: P O (k min ) ≪ P O (k enh ).The UV contribution P O (k max → ∞) can be dropped by the iε prescription, i.e., by rotating the time axis as τ → (1 + iε)τ with a small positive parameter ε and replacing the UV mode function as e −ikτ → e −ikτ +εkτ .Indeed, this procedure is necessary to correctly evaluate the correlation functions in the vacuum state of the interacting theory in the infinite past.
As a result, if Maldacena's consistency relation holds for the squeezed bispectrum, the one-loop correction on the soft propagator from the cut-in-the-side diagram is summarised as ∼ P ζ (p)P O (k min ), which is small enough compared with the tree-level result P ζ (p).It should be emphasised that the correction is independent of the enhanced power P O (k enh ) and the enhancement of the small-scale power spectrum does not spoil the perturbativity of the large-scale perturbation as long as Maldacena's consistency relation holds true.In the next section, we confirm this speculation in a specific inflation model: transient USR inflation.

Example: transient ultra-slow-roll (USR) inflation
We show the concrete one-loop calculation in a specific inflation model: the transient USR inflation considered in Refs.[13,27].There, the second SR parameter η = ε/(ϵH), where ϵ = − Ḣ/H 2 is the first SR parameter, shows sharp transitions at specific conformal time τ s and τ e as so that the inflation dynamics is given by the SR one with η = 0 for τ < τ s or τ > τ e while it is in the USR phase with η = −6 during τ s ≤ τ ≤ τ e .The first SR parameter is solved at the leading-order SR approximation as with a certain initial value ϵ SR .
In each phase, the formal solution of the Mukhanov-Sasaki equation for the mode function of the curvature perturbation is given by The initial condition is given by the Bunch-Davies vacuum A k = 1 and B k = 0 in the first SR phase:  during the transient USR period (τ s < τ ≤ τ e ) and in the second SR period (τ > τ e ).With these expressions, we see explicitly that |ζ k (τ )| 2 decreases exponentially as a function of k upon the contour deformation τ → (1 + iε)τ .This fact is important when we discuss the UV behaviour of the power spectrum P ζ (k).
An example of the power spectrum and its spectral index n s − 1 = d ln P ζ /d ln k at late time τ → 0 is shown in Fig. 3.One sees the amplification of the curvature perturbation due to the transient USR phase.

Consistency relation on the squeezed bispectrum
Let us first check if Maldacena's consistency relation is satisfied between the CMB-and PBH-scale perturbations, reviewing the Feynman diagrammatic approach in the Schwinger-Keldysh formalism.See Ref. [31] for the detailed derivation.The propagator is defined by the two-point function time-ordered along the closed-time path (denoted by T C ) as with arrows are retarded (or advanced ) propagators.In Fourier space, they are given by where we have neglected higher-order terms in ϵ and terms that vanish by the equation of motion in the interaction picture as in Ref. [37]. 6This action leads to the vertices listed in Fig. 5. 7 In the squeezed limit, the diagrams including the statistical propagator of the long mode k L dominate because P ζ (k L ) is divergent as ∝ k −3 L in the limit of k L → 0. Therefore, for the tree-level bispectrum calculation and the one-loop power spectrum calculation, the cubic vertices involving three ζ ∆ and/or ζ ′ ∆ (not shown in Fig. 5) give subleading-order (in k −1 L ) contributions, so we neglect these.With use of these diagrams, one can calculate the bispectrum.As noted above, we focus on the leading-order contribution in the squeezed limit.Noting that both η and ζ ′ are suppressed in the SR phase, one finds that the main diagrams for the squeezed bispectrum during the second SR phase are given by Here and hereafter we neglect terms of order (k L /k S ) 2 , (k L τ s ) 2 , and (k L τ e ) 2 .If one fixes k L to the CMB scale such that it is well frozen throughout the USR and second SR phase, one finds G cc (τ, τ s ; k L ) ≃ G cc (τ, τ e ; k L ) ≃ P ζ (τ, k L ).The squeezed bispectrum is then summarised into the following form, Defining the generalised non-linearity parameter f NL (k L , k S ; τ ) by the numerically calculated contribution of each term is presented in the left panel of Fig. 6.In the right panel, the total f NL is compared with 5  12 (1 − n s (k S )).One finds that Maldacena's consistency relation completely holds.The squeezed bispectrum during the USR phase is also calculated similarly.The main diagrams are given by It is expressed as Each contribution is plotted in the left panel of Fig. 7 and the total f NL is compared with  8 The deviation from Maldacena's consistency relation cannot be seen even in the non-squeezed configuration, kL ∼ kS, in Fig. 6.This is because only terms dominant in the squeezed limit are kept in Eq. (3.12).The fNL parameter plotted in the right panel of Fig. 6 should be understood to be valid only in the squeezed limit, though the equilateral fNL is suppressed by slow-roll parameters anyway.
(1 -ns) 0.001 0.010 0.100 (3.16) Once one finds the expression of the squeezed bispectrum, one can also calculate three-point functions including the generalised momentum as where we neglected the time derivative of the k L mode as it is well frozen.The following formula hence holds up to

One-loop corrections to the power spectrum
Armed with all the weapons, one can systematically calculate (cut-in-the-side) one-loop corrections on the soft two-point function.Evaluated at a time well after the USR phase, (would-be) contributions which are not suppressed in the soft limit are summarised as Making use of the soft theorem (3.16), the contribution of the first term, dubbed P The coefficient does not cause any singular behaviour as G c ∆(τ, τ s/e ; k L ) ∼ i/(2a 2 (τ s/e )ϵ(τ s/e )) (see Appendix A).The k max → ∞ contribution can be dropped by the iε prescription as discussed in the previous section.Explicitly, A similar comment applies to P ζ (τ e , k max ): it involves an exponential suppression factor e 2εkmaxτe → 0 though the prefactor becomes more complicated.We take k min as k min ≳ k L to utilise the squeezed bispectrum10 but k min ≪ k enh so that P ζ (k min ) ≪ P ζ (k enh ), neglecting the contribution of q < k min which is expected to be small anyway.After all, the ratio of the correction to the tree propagator, is negligibly small.For example, one can numerically evaluate it as ∼ O(10 −10 ) in our setup for τ → 0 and −k L τ s = −k min τ s = 10 −3 .
The second and third diagrams in (3.19) , are also negligible.The soft theorem (3.18) leads to The coefficients −ia 2 (τ s/e )ϵ(τ s/e )G c∆ (τ, τ s/e ; k L ) are again independent of k L at the leading order and harmless (see Appendix A).The summation of the first and second terms in the square brackets is numerically ∼ O(10 −11 ) for τ → 0 and −k L τ s = −k min τ s = 10 −3 and it becomes even smaller for smaller k L and k min because The third term in the square brackets vanishes in the limit τ → 0.
In the above calculations, it is crucial to cover the whole integration domain [k min , k max ] → [k L , ∞).In contrast, many authors in the recent discussions [13, 24, 26-30, 32-34, 37, 38] restricted the integration domain to or similar ones by hand, where k s and k e are the wavenumbers that cross the Hubble horizon at τ = τ s and τ e , respectively.Although it may apparently make sense to extract the contribution only from the transient USR period, the above calculation shows that this introduces an artificially large contribution to the one-loop correction to the power spectrum of the soft modes of the curvature perturbations.As long as Maldacena's consistency relation holds (and we have confirmed it does hold), the integrand is written as the total derivative, so the large contribution is exactly cancelled by the adjacent integration domains.
In summary, Maldacena's consistency relation as the cosmological soft theorem ensures the cancellation of the one-loop correction from small-scale perturbations.This is our main result.It should be noted that we do not claim the one-loop correction vanishes on an arbitrary scale in the transient USR model.For modes exiting the horizon right before the USR onset τ s , even though they are superhorizon throughout the USR phase, they can grow at the tree level, Maldacena's consistency relation will be violated [104], and the one-loop correction can be sizable for them.Self-loop corrections of the enhanced modes can also be non-negligible effects (see, e.g., Refs.[105][106][107]).

Comment on the tadpole contributions and the stochastic-δN formalism
We comment on the tadpole contribution in this section.Cubic interactions correct the two-point function not only by the ordinary self-energy diagrams shown in Fig. 1  Such a tadpole effect is also observed in the so-called stochastic-δN approach [47][48][49][50][51].The stochastic formalism is known as an effective theory of the superhorizon-coarse-grained matter fields such as the inflaton.There, the effective action for the coarse-grained fields is obtained by integrating out the subhorizon perturbations.It causes random Gaussian noise at the leading order, which can be understood as the horizon exit of perturbations.Assuming that the coarse-grained fields are well-classicalised and follow the stationary point of the effective action, they behave as independent Brownian motions.The inflatons' fluctuations are converted into the curvature perturbations by the δN formalism [6,[62][63][64].
The elapsed e-folding number N can fluctuate due to the stochastic noise, even from the same initial condition to the same end-of-inflation condition, and such a fluctuation is nothing but the curvature perturbation according to the δN formalism.The (physical) scale k ph of the power spectrum is then related to the field-space (or phase-space) points N (k ph ) = ln(k f /k ph ) e-folds before the end of inflation, where k f ∼ H f is the coarse-graining scale at the end of inflation.Here, even if the scale of interest corresponds to an attractor phase, if there is a very phase after that, the mean elapsed time ⟨N ⟩ from a point can be shifted from N cl estimated without noise.This difference is understood as the tadpole ⟨ζ⟩ = ⟨N ⟩ − N cl in the stochastic formalism and it causes the correction in the (dimensionless) power spectrum given by at the leading order in the attractor case.This is an interpretation of the tadpole correction in the stochastic approach.We do not check the consistency in the tadpole value ⟨ζ⟩ itself between the in-in and stochastic approach and leave it for future work as the stochastic calculation in the transient USR model is a rich topic in itself.
Other than such a tadpole contribution, the stochastic approach would only predict the volume-suppressed ((k L /k S ) 3 -suppressed) correction, which is typical for the cut-inthe-middle ones.Spatial coarse-graining of the curvature perturbations on a certain scale k ph is implemented as a sampling average over paths branching at the time N (k ph ) e-folds before the end of inflation.The large-scale perturbation generated by small-scale physics is hence understood as a deviation of the sampling average from the true average.However, the central limit theorem guarantees that such a deviation is suppressed by the inverse sampling number, i.e., the volume ratio (k L /k S ) 3 .

Comment on the tensor perturbations
The loop correction on the tensor perturbations is also a hot topic.Ota et al. [40,41] claimed that the superhorizon tensor power spectrum can be enhanced by the loop correction of a resonantly enhanced spectator scalar field.Firouzjahi [39] computed the tensor loop correction from the inflaton in the transient USR model and found that it is not significant but non-vanishing.
The soft tensor h also satisfies Maldacena's consistency relation.There, the squeezed bispectrum between the soft tensor and two hard operators O is given by the shear transformation of O's power spectrum [5,83]: where λ labels the tensor polarisation and e λ ij is the corresponding polarisation tensor, normalised by e λ ij ( k)e λ ′ ij ( k) = δ λλ ′ .However, this soft theorem does not ensure vanishing one-loop correction contrary to the scalar case.Assuming that the other vertex brings another polarisation tensor e λ ′ * ij ( k1 )k i 2 k j 2 and making use of the solid angle integration d k2 e λ ij ( k1 )e λ ′ * lm ( k1 ) ki 15 δ λλ ′ , the one-loop correction from the coupling hOO is expected to be proportional to the first term cancels the contribution of the enhanced power P O (k enh ) similarly to the scalar case, the second one is affected by the enhanced power.It can be significant if the enhanced power is large enough.It would be interesting to evaluate the one-loop correction from the coupling hhOO because it can be comparable to Eq. (5.2).
Note that the self-energy of gauge fields, i.e., vacuum polarisation, is renormalised into the gauge coupling in quantum field theory [108].The tensor loop correction would also be absorbed by the renormalisation of the gravitational coupling G.We leave this problem for future work, too.

Conclusions
In this paper, we have studied the quantum corrections of the large-scale curvature perturbations from much smaller-scale modes beyond the standard single-clock inflation paradigm.We have taken into account surface terms in the action neglected in the literature and emphasised the role of Maldacena's consistency relation.We have numerically confirmed the consistency conditions with or without a time derivative of the curvature perturbations ζ ′ k in the SR/USR/SR scenario [Eqs.(3.16) and (3.18)] both in the USR regime and in the second SR regime [Figs.7 and 6].Once the consistency relation has been established in the SR/USR/SR scenario, we have substituted the relation in the subdiagram of the one-loop correction to the power spectrum of the soft curvature perturbations.As in the single-clock case [9], we have found that the one-loop corrections from the enhanced scale k enh cancel.We emphasise that this generally holds true beyond the SR/USR/SR scenario we focused on as an example as long as Maldacena's consistency relation holds.
Yet, our one-loop calculations are not complete since we have not included quartic interactions of ζ.The quartic action of a related quantity δϕ or ζ n [5] can be found in Refs.[109,110].However, our calculation is based on the physical/original curvature perturbation ζ, so we cannot use these results directly.Derivation of the quartic action of ζ including the surface terms is an important subject left for future work toward the complete proof of the absence of one-loop corrections from the small-scale physics.Despite the fact that it is not a proof, we believe our result strongly suggests that the small-scale loop effects do not significantly affect the curvature perturbations on much larger scales.
Finally, let us discuss the implications of our results on the recent claim that primordial black hole production is excluded in single-field inflation scenarios.As we have seen, the quantum correction on a large scale such as the CMB scale is insensitive to the USR phase as long as these scales are hierarchically separated.In our precision of calculation, we have been left with finite residual contributions but they are negligible compared to the tree-level contribution.Thus, the one-loop correction is much smaller than the tree-level result, so there is no concern about the breakdown of perturbative loop expansion.This is true even without the assumption of a smooth transition between either the SR phase and the USR phase.As we emphasised above, the same discussion applies to any PBH production scenarios utilising the small-scale enhanced curvature perturbations that satisfy Maldacena's consistency relation.Therefore, PBH production in single-field inflation scenarios is not excluded.

Figure 2 :
Figure 2: Schematic representation of the Feynman diagram for the squeezed bispectrum B ζOO (p, k, k ′ ).
as a subdiagram.Therefore, if the squeezed bispectrum satisfies Maldacena's consistency relation (2.1), the diagram contribution should be summarised as cP ζ (p)P O (k) d ln P O (k) d ln k

Figure 6 :
Figure 6: Left: each contribution of τ s (blue) and τ e (orange) to the generalised nonlinearity parameter f NL (k L , k S ; τ ) (3.13) as functions of k S for k L = −0.001/τs and τ → 0 with the same model parameters as Fig. 3. Right: a comparison between the total f NL (blue) and Maldacena's consistency relation 5 12 (1 − n s (k S )) (orange-dashed) at τ → 0.

5 12 ( 1 −
n s ) in the right panel.Maldacena's consistency relation is again satisfied.These facts the caption of Fig. 5.