Higgs as heavy-lifted physics during inflation

Signals of heavy particle production during inflation are encoded as non-analytic momentum scaling in primordial non-Gaussianity. These non-analytic signatures can be sourced by Standard Model particles with a modified Higgs scale uplifted by the slow-roll dynamics of inflation. We show that such a lifting mechanism becomes more efficient with the presence of a strong-coupling, where the Higgs mass scale is further increased by a small speed of sound in the effective theory of inflation. As a primary step towards detecting new particles in the cosmological collider program, non-Gaussianity due to heavy Higgs production in the strong-coupling regime can act as important background signals to be tested by future cosmological surveys.


I. INTRODUCTION
Heavy fields are expected to arise from the UV completion in most of the single-field inflationary scenarios [1]. These heavy states, while stablizing a flat direction for the slowroll inflation, could have typical masses around the size of the Hubble parameter H during inflation, especially when the UV physics is associated with the supersymmetry breaking or is constructed in the framework of supergravity [2][3][4][5]. Although the energy scale of the UV physics seems too high to be probed directly by the ground-based particle accelerators, these heavy fields can leave imprints in the primordial non-Gaussianity once they are spontaneously produced by the high energy quantum fluctuations during inflation [4,[6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]].
Imprints of heavy particle production include characteristic signals from the non-analytic scaling of the momentum restricted by the symmetry of inflation [10,11,21] (see also [4,[6][7][8]). These non-analytic signals are not covered by any local effective operator that is converted from integrating out the heavy degree of freedom.
Recently, attention has been paid to the possible non-analytic signals sourced from those particles with accessible masses to ground-based accelerators, yet their mass scales may be lifted above H during the primordial epoch of cosmic inflation [22,23,29]. For instance, the Standard Model (SM) Higgs boson h, if not the inflaton φ itself, is known to gain a mass m h ∼ H much larger than the electroweak scale in a de Sitter background via a nonminimal coupling to gravity or non-trivial interactions with inflaton. With an unbroken gauge symmetry, the heavy Higgs field (and of course all the SM gauge fields) can enter the ζ-correlation functions through loop interactions, leaving meaningful non-Gaussianity for observations only in limited parameter space [22,23].
The production of heavy SM particles becomes more efficient if the gauge symmetry is (partially) broken where Higgs spontaneously obtains a non-zero vacuum expectation value (VEV) during inflation. The symmetry breaking is manifested by a tachyonic Higgs mass from the wrong-sign non-minimal coupling term or the higher-dimension Higgs-inflaton interactions [29][30][31]. By assuming that higher dimensional Higgs-inflaton interactions are a series of well-controlled low-energy effective operators, perhaps reduced from grand unified theories, Kumar and Sundrum [29] concluded that non-analytic signals in bispectra are always dominated by the heavy Higgs production processes.
However, the cutoff scale for not entering the dynamical region of heavy fields becomes non-trivial if the dispersion relation of the system is non-linearly modified [33,45]. The main focus for the current study is to show that the mass spectrum of SM particles can be non-trivially lifted by a modified dispersion relation due to the presence of non-perturbative interactions with inflaton. In this work, we use the Higgs-inflaton system as an example to show that cubic or higher-order perturbations in the Lagrangian can be well-defined, even if quadratic perturbations are made strongly coupled. One can identify a modified mass scale corresponding to the strong coupling, below which Higgs acts as a heavy degree of freedom.
Integrating out the heavy Higgs field in the strong-coupling regime shall result in an effective speed of sound for inflaton, which captues the leading analytic (local) contribution of the heavy field to the primordial spectra [33,45,47].
Due to the implicit symmetries of the inflationary background, one can expand mode functions into power-laws ∼ η ∆ of the conformal time in the late-time limit |η| → 0, where ∆ characterizes the spatial dimension of an operator residing on the boundary of future infinity (η = 0) [21] (see also [4,7,8]). For the simplest non-Gaussian observable led by the three-point function ζ 3 , the late-time expansion implies that the squeezed limit exhibits the scaling as where k L /k S 1 and k L is a long-wavelength mode that exits the horizon much earlier than the short-wavelength mode k S . w i are coefficients depending on the mass of the heavy Higgs field. If the SM gauge symmetry is unbroken during inflation, one expects the non-analytic scaling takes [10] ∆ non−analytic = 3 ± 2 iL h , since the Higgs field must enters the correlation functions as a pair. For m h /H > 3/2, the non-analytic contribution creates oscillatory featurs in the bispectrum, which can be taken as the signature of heavy particle production during inflation.
In this work, we consider the heavy Higgs production by virtue of a non-perturbative interaction with inflaton that spontaneously breaks the gauge symmetry and give a non-zero VEV to the Higgs field. Our expectation for the non-analytic scaling in a strong-coupling system is thus modified as where the half-integer from single Higgs production is allowed due to the broken gauge symmetry, and c 2 φ is the effective speed of sound for inflaton which accounts for the local effect via integrating out the Higgs field. Comparing (2) and (3), one can identify a modified mass scale µ h ≡ m h /c φ for the strong-coupling system, which recovers the non-local extension of the effective-field-theory approach for inflation [45]. The modified late-time scaling behavior opens a new parameter space for observing Higgs (or other gauge fields) in primordial non-Gaussianity. To be specific, we show that the oscillatory feature in the bispectrum is This paper is organized as the follows. In Section II, we review the framework for the heavy-lifting mechanism in the effective theory approach. We consider a model in Section III as an example to show that the Higgs mass scale can be heavy-lifted by a strong coupling with inflaton, resulting in a modified dispersion relation without breaking the perturbativity of the loop expansion. Imprints involved with the heavy Higgs production in the non-Gaussian correlation functions of ζ are studied in Section IV. Finally, conclusions are given in Section V.

II. THE EFFECTIVE THEORY FOR HEAVY-LIFTING
In this section we clarify the framework in which the effective theories for the heavy-lifting mechanism is built. We first recall the most general effective action of the Goldstone boson, π, corresponding to the spontaneously broken time-translation symmetry with one extra scalar field, σ, as a massive degree of freedom during inflation [8,27,28]. This formalism (sometimes called the π-σ model [33]) assumes no specific fundamental physics but includes all possible terms with respect to the time-dependent spatial diffeomorphisms. 1 We then specify the class of theory proposed in [29] for realizing a spontaneous symmetry breaking of the Higgs potential during inflation. One can directly write down an effective Lagrangian of the Higgs field h as a low-energy expression of some unified theory with 1 The π-σ model is also known as quasi-single field inflation [6,7] for the mass m σ of the σ field is near or larger than the Hubble parameter H. inflaton φ at very high energy scales. Separating Higgs and inflaton as different particles (namely non-Higgs inflation) generically makes it easier to obtain observable signatures in non-Gaussianity. To make sure such an effective expansion is well-controlled several conditions have been imposed in [29] to keep the Higgs-inflaton interactions weakly coupled.
Although it is in general very difficult to identify the exact couplings in the unified theory, φ could be naturally light if it is associated with the "pion" of the spontaneous symmetry breaking of the full theory and h would obtain a mass of the size of H. This possibility will be discussed in the next section. Note that in the effective theory of the φ-h model we always assume the slow-roll inflation paradigm that exhibits a well-defined homogeneous time evolution, and thus it can be cast into as a special case of the π-σ model.

A. The cosmological Goldstone boson
We follow the standard precedure to construct the most general effective action of inflation [1,27]. The first step is to write down all time-dependent operators that preserve spatial diffeomorphisms in the unitary gauge where there are only metric fluctuations. At leading order the terms with the metric perturbation δg 00 = g 00 + 1 are where operators with the extrinsic curvature perturbation δK µν and with higher derivatives are not shown here since the action (4) is sufficient for our purpose.
Following [8] with the same setup as for S π , the effective action for an extra scalar field σ during inflation is given by where we introduce the interactions between π and σ fields through S πσ . We refer the combination of actions (4), (5) and (6) as the π-σ model.
To simplify the formalism for our later convenience, we want to rewrite the π-σ model in the decoupling limit with gravity. Following the discussion in [1, 33], the decoupling limit with gravitational fluctuations is realized by taking M p → ∞,Ḣ → 0 with M 2 pḢ fixed, which is a trick similar to making the "pion" and the "sigma field" to be decoupled in the well-known linear sigma model.
For this purpose we shall restore the full gauge-invariance to the actions by virtue of the replacements t → t+π, g 00 → g 00 +2∂ 0 π +∂ µ π∂ µ π, and ∂ 0 σ → ∂ 0 σ +∂ µ π∂ µ σ. One can solve gravitational fluctuations in terms of the standard ADM variables (and up to first order in π and σ is enough for our case), as performed in [8]. Putting solutions of the ADM variables back into the actions, such as (4) and (6), and taking H 2 /M 2 p → 0 withḢ/H 2 → 0, we find the simplified actions as where c −2 π = 1 − 2M 4 2 /(M 2 pḢ ), and We can see that (7) and (8) capture nothing but the leading terms of the φ-h model in the following discussion.

B. The Higgs boson during inflation
While the π-σ model contains every operator that is invariant under the spatial trans- , it is also our desire to start with a high energy theory that specifies the explicit Lagrangian for inflaton, Higgs, or all the SM gauge fields. By specifying the Lagrangian of interest we can pin down a large number of free parameters to be tested by observations.
In this work, we will assume that heavy states corresponding to physics much higher than the energy scale of inflation have been integrated out, resulting in an effective low-energy Lagrangian with non-trivial interactions between Higgs and inflaton (putting gauge fields aside for the moment). Masses of SM particles may be lifted up to the size of the Hubble parameter during inflation due to these interactions. For example, the heavy-lifting of Higgs mass considered in [29] has a general Lagrangian of the form where Φ H = (0, h) T / √ 2 is the Higgs doublet in terms of the SM unitary gauge. The inflaton collects all of the viable single-field inflation models that satisfie slow-roll conditions. The Higgs-inflaton interactions are parametrized by a series of dimensionless coefficients c i in which can be taken as an effective expression of the full theory at the energy scale of inflation.
As shown in [29], these higher-dimensional couplings could arise due to the integrating out of some heavy particles that mediate between Higgs and inflaton separately only on very high energy scales.
A spontaneous symmetry breaking of the Higgs vacuum during inflation is the key for the heavy-lifting mechanism. One of the possible origin to have a sizable VEV, namely h ≡ h 0 ∼ H, is the presence of the nonminimal coupling ξRh 2 with ξ < 0 [29]. To gain a Higgs mass m h ∼ H implies ξ ∼ O(1). Alternatively one can consider a spontaneous symmetry breaking caused by an effective tachyonic mass term, for instance, due to the c 2 -term in (11). This second possibility will be the main focus of the current study.
Let ue perform a quick search to see the correspondence in between the φ-h coefficients c i and the parameters β i in the π-σ model. We consider for simplicity a well-defined decomposition h = h 0 + δh with h 0 a constant VEV. In the unitary gauge where δφ = 0, the linear expansion of (11) gives where φ 0 is the inflaton VEV during slow-roll. Since metric components in the decoupling limit with gravity read g 00 → −(1 +π) 2 + (∂ i π) 2 a 2 , and g 0i → the Lagrangian L φh then introduces the π-h interactions up to quadratic order as Replacing h by σ we find from (6) thaṫ These results show how the c 1 and c 2 terms in the φ-h theory can be cast into the π-σ model. One can also check the above correspondence in the flat-slicing gauge, where the

III. ENERGY SCALES OF HEAVY-LIFTING
In this section we consider the heavy-lifting scenario induced by φ-h interactions (11).
We will use an example to demonstrate the spontaneous symmetry breaking of Higgs VEV, and identify a characteristic energy scale µ h below which the Higgs field h represents a heavy degree of freedom. In the simplist case, µ h = m h is characterized by its mass scale and a heavy degree of freedom means that h exhibits a constant dispersion relation ω ≈ m h for modes with physical wavenumbers p = k/a(t) m h . Thus if H 10 2 GeV one would expect that the Higgs field is simply a light degree of freedom during inflation since the SM value m h ≈ 125 GeV and the Higgs self-coupling λ becomes small when running up to high energy scales [24][25][26].
For a strongly-coupled φ-h system, we will examine the energy scale Λ p at which the higher-order perturbation expansion breaks down. In fact, we will show that the cutoff scale Λ p for the perturbative expansion does not rely on the Λ i 's parametrized in (11). To simplify our discussion, we turn off the non-minimal coupling ξ and assume a positive λ 1.

A. The Higgs-inflaton system
We are interested in a system made by two fundamental scalars, which are the inflaton φ and the Higgs field h. To realize a spontaneous symmetry breaking during inflation, we consider as an example the classical Lagrangian of the form where it can be taken as a special case of the φ-h theory (11) with Λ 2 = Λ, c 2 = 1 and otherwise c i = 0. By taking the SM unitary gauge and omitting all SM gauge fields, the kinetic terms of the φ-h system read which represents a two-field limit of the multi-field inflation scenario based on the nonlinear sigma model [36,37]. At a first glance, Λ looks like a cutoff scale of the non-canonical kinetic interaction to make sure that (16) is a well-behaved low-energy effective Lagrangian.
However, we will show that in the presence of a strong-coupling the conclusion is nontrivial. For example, a well-defined perturbative expansion for the φ-h system (16) is in fact independent of the ratio h/Λ. We want to emphasize that the model (16) is used as an simple example for the heavy-lifting mechanism so that there is in fact no primary assumption for its origin. 2 The cutoff scale of the system (16) is non-trivial since the target field space of φ-h can be curved. For convenience, we perform the reparametrization for both fields as so that the kinetic part of the system becomes In this representation, the classical value of R acts as the canonical radius for θ, and the rescaled inflaton θ behaves as the angular mode in the polar coordinate system. In general, the target field space is not flat since the radial mode R is not canonically normalized. There are two interesting limits of this system.
1. For h 2 Λ 2 , the radial mode R → Λ and the non-canonical φ-h interaction is suppressed by the factor h 2 /Λ 2 1. In this limit the field space is nearly flat and it is nothing but the conventional single-field inflation with Higgs as an additional degree of freedom. We refer this regime as the flat-decoupling limit of the φ-h system (to be distinguished from the gravitational decoupling in the π-σ model).

For h 2
Λ 2 , the radial mode R → h coincide with the Higgs field. The field space is flat as the factor R 2 /(R 2 − Λ 2 ) → 1 becomes canonically normalized in the polar coordinate representation. We refer this regime as the curvelinear limit of the φ-h system, since the inflationary trajectory is now curved. Heavy-lifting in this regime was not covered by previous literatures.
We now study the symmetry breaking of the Higgs potential in the system (16). If the Higgs field develops a well-defined background value h 0 during inflation, the homogeneous field equations of the system are given by where R 0 = (Λ 2 + h 2 0 ) 1/2 and V θ ≡ ∂V /∂θ. For λ > 0, there exists a static solution h 2 0 =θ 2 0 /λ such that the Higgs field can develops a non-zero vacuum expectation value (VEV) This non-trivial VEV is sourced by the essential dynamicsθ 2 =φ 2 /Λ 2 of slow-roll inflation and is invariant under a constant shift of the inflaton value θ 0 → θ 0 + c.
Expanding the effective potential around h 0 one finds the effective mass m 2 h = 2θ 2 0 . In the gravitational decoupling limit the mass m 2 h becomes a constant. A stable h 0 asked by the stochastic condition m 2 h H 2 [34,35] can be satisfied ifθ 2 0 H 2 /2. Note that the first slow-roll parameter is related to the non-zero Higgs VEV as In the flat-decoupling limit (h 2 0 Λ 2 ) where R 0 ≈ Λ, the slow-roll condition implies Λ 2M p H/θ 0 . In the curvelinear limit (h 2 0 Λ 2 ) where R 0 ≈ h 0 , the smallness of instead guarantees the subdominance of the Higgs vacuum energy to the background energy density.

B. Scales of heavy Higgs
The non-zero Higgs VEV h 0 is led by the slow-roll inflation dynamics −2M 2 pḢ = R 2 0θ 2 0 and 3Hθ 0 = −V θ /R 2 0 at the first-order of . We can treat h 0 as a stable constant during the slow-rolling of θ given thatḣ 0 is at least second-order in the slow-roll parameters. Performing (16), we obtain the quadratic Lagrangian as where O( ) means quadratic perturbations that are suppressed by the slow-roll parameters (which includes the mass term of inflaton). The terms shown in (26) can also be derived from the general perturbation theory, and one can check that metric perturbations only To see the dynamics of the system, it is convenient to use the canonically normalized field θ c = R 0 θ with respect to the canonical commutation relation for canonical quantization. The quadratic Lagrangian (26) is rewritten as where the coupling plays a key role in the dynamics of our system. Note that once λ and Λ are fixed, both µ and m 2 h are controlled by the same parameterθ so that they are not independent from each other. This is a fundamental difference from the constant-turn quasi-single-field inflation [6,7,43] (or the strongly coupled π-σ model [33]).
For H > µ, the interaction 2µδhδθ c does not play an important role during inflation and the system (27) simply describes two weakly interacted scalar fields. In this case we expect ± the parameters Λ = 2H, λ = 0.01 andθ 0 = 0.1H are used, which gives ± the parameters Λ = 2H, λ = 0.01 andθ 0 = 2H are used, which gives the standard picture of a massless Higgs field. The condition µ > H for the presence of a strong-coupling impliesθ To understand how the two degrees of freedom decoupled with the energy scales, let us write down the equations of motion of the perturbations In the long-wavelength regime with k/a → 0, we expect the usual solution δθ c → constant and δh → 0 of the single-field inflation. For p = k/a H we are allowed to neglect the cosmic expansion so that the equations of motion are reduced to The solutions in the subhorizon regime thus take the form of δθ c ∼ δθ ± e iω ± t and δh ∼ δh ± e iω ± t [36,37], where the two frequencies are found as Here the speed of sound c 2 h is simply defined in the limit of p (m 2 h + µ 2 )/µ such that the low-energy frequency can be expanded as and thus it indicates The result (37) appears to be the same as using the effective field approach for a curvelinear trajectory after neglecting the slow-roll parameter suppressed effective mass [36,37] With ± are in the case with µ < H and ω (2) ± are in the case with µ > H. For µ < H, ω (1) ± become almost degenerate at the Hubble scale during inflation (p ∼ H) and they recover the usual linear dispersion relation ω ≈ p.
On the other hand, in the limit of p (m 2 h + µ 2 )/µ the high-energy mode ω where µ h → 0 asθ 0 → 0. We therefore identify the heavy-Higgs condition as: µ h > H, which is namely m 2 h > c 2 h ≥ H 2 /9. This implies the lower limit for a heavy Higgs aṡ which is to be compared with (29). The corresponding values of the examples µ h /H ≈ 0.17 and µ (2) h /H ≈ 4.88 are given as the vertical lines in Fig. 1.

C. Perturbativity and Naturalness
We now check if the perturbative expansion for higher-order terms are well-defined in the presence with a strong quadratic coupling µ > H. We shall identify a scale Λ p as the cutoff for the higher-order expansion. If the condition Λ p > µ h can be satisfied, the Higgs field recovers the usual dispersion relation ω = k as a relativistic degree of freedom before the break down of the perturbative expansion of the theory. This case is illustrated by the left panel of Fig. 2.
Let us consider the cubic interactions introduced by the Higgs-inflaton coupling from (17) With a linear dispersion relation ω = k, the temporal derivative and spatial derivative has the same dimension so that we can easily identify Λ p = R 2 0 /h 0 from the first two cubic interactions in (41). Therefore Λ p > µ h implies that One can find that even in the curvelinear limit whereθ 2 0 λΛ 2 the condition holds if λ < 1/6. The coefficient of the third interaction in (41) is dimensionless so that perturbativity simply requiresθ c /R 2 0 < 1, which asks λ < 1 in the curvelinear limit. Note that in the case with µ h > Λ p , the system may enter to the non-perturbative region with a non-linear dispersion given by (36), as illustrated by the right panel of Fig. 2 where time-derivatives have dropped out. with the redefinition of fields as δh r = µ −1/4 δh and δθ r = µ −1/4 δθ c , the momentum conjugate is normalized as p δθr ≡ ∂L 2 /∂δθ r = δh r . The cubic interactions are thus parametrized as where∂ i means the spatial derivative with respect to x r , and Here the curvelinear condition R 0 → h 0 is used. Since each loop integral of the cubic interactions asks at least two insertions of the vortex, the cutoff is raised by a factor of � �� Λ/� 16π 2 [33]. As a result, one can check that in order to realize µ h > Λ p , the cutoff scales Λ p1 , Λ p2 and Λ p3 all ask λ 1. These results are inconsistent with the parameter space of our consideration. In summary, for λ 1 the two fields in the system always become weakly coupled before they reach the non-perturbative region.
As shown in Fig. 3, a heavy Higgs field can be realized in the curvelinear limit with respect to the perturbativity and Naturalness in Region I with a weak quadratic coupling, or in Region II with a strong quadratic coupling.

IV. OBSERVATIONAL IMPACT OF HEAVY HIGGS
A. power spectrum Observational constraints. We study in this section the corrections to the power spectrum led by the Higgs-inflaton interactions at linear order. Given that the quadratic interaction of the system (27) is a derivative coupling, these corrections are suppressed on superhorizon scales so that they do not change the scale dependence of the power spectrum.
However, in the strong-coupling regime these corrections may become comparable to the leading term [43] and thus one has to rescale the amplitude to meet the observational constraint. Treating the background valueθ 0 as a free parameter, we shall rescale the power spectrum with respect to the observational constraint where P * θ = H 2 * /(4π 2 ) is a reference field spectrum for a given reference angular velocityθ * , which is more convenient to choose in the flat-decoupling limit (namelyθ * √ λΛ) such that R * = Λ. H * is the reference Hubble parameter as viewed in the flat-decoupling limit.
To keep a constant ∆ 2 ζ while varyingθ * , we shall adjust parameters in P * θ . In general, we can rescale H * or Λ.
• H-rescaling. The first choice is to rescale H * for a givenθ 0 . For numerical convenience, the change of the field spectrum evaluated under a constant Hubble parameter H * is parametrized as P θ = f P * θ . We define a rescaled Hubble parameter H such that where the rescaled field spectrum is P new θ ≡ P θ H 2 /H 2 * = H 2 f /(4π 2 ). The rescaled Hubble parameter is then given by The numerical results for f with Λ = 2H and λ = 0.01 are given in Fig. 4, where the threshold value for the heavy Higgs condition isθ 0 /H * = 0.24 andθ 0 /H * > 0.27 becomes strong-coupling. We refer this method as the H-rescaling scheme.
• Λ-rescaling. Another choice to fit the observational constraint is to rescale Λ with the change ofθ 0 . For the flat-decoupling case, we can define that where the possible change of the power spectrum is absorbed in H 2 = H 2 * f −1 for convenience. In this scheme Λ is determined by a givenθ 0 as Λ = H 2 * /(2π∆ ζθ0 ). We refer this method as the Λ-rescaling scheme, and the numerical results for f with  usually applied to define as the vacuum of "free fields" for the in-in formalism [38][39][40] in the interaction picture. However, according to the first-principles of the in-in formalism, these initial states in general need not to be fully decoupled from each other, and therefore the EoM approach is also useful to deal with mixed initial states arised from a strongly-coupled system. Initial mode functions for the θ-h system (27) are found to be where the derivation is given in Appendix A. The dimensionless spectrum is given by P θ = In-in formalism (in-in). As a crosscheck of our numerical results, we provide the analytical computation for the power spectrum based on the in-in formalism. The formulae for the φ-h system (27) computation can be found in [7,19,42], and for inflaton with a generalized speed of sound c 2 θ ≤ 1 is considered in [15,47]. As given by the dotted line in Fig. 4, the correction from the quadratic interaction δL 2 = µδhδθ c up to first order in the perturbative expansion is led by the form where ψ (1) (x) ≡ d 2 ln Γ(x)/dx 2 is the polygamma function, and ∆P ζ = P ζ − P * ζ is the deviation of the power spectrum from the expectation value of the standard single-field inflation P * ζ = ∆ 2 ζ . Therefore we have f = ∆P ζ /P * ζ + 1. One can check that ∆P ζ /P * ζ → 1 in the limit ofθ 0 /H * → ∞ where m 2 h → ∞. The deviation from the EoM results in the strong-coupling regime is due to the fact that δθ aquires an effective speed of sound c 2 θ < 1 (see the discussion in EFT method). Note that the expression (51) does not diverge in the limit ofθ 0 /H * → 0 since µ also goes to zero. This is one of the essential difference from the quasi-single-field inflation results [6,7,43].
Effective field theory (EFT). Given that Higgs behaves as a heavy degree of freedom in the strong-coupling limit (µ > H), one may consider to integrate out the heavy Higgs to obtain an effective single-field theory. We denote the integrate-out process of the heavy Higgs as the effective field theory (EFT) approach [9,36,37,43,[45][46][47], which is to be distinguished from the effective theory discussed in Section II.
The EFT approach for the two-field system (27) has been studied in the large mass regime (m h µ with m h H) [36,37] and in the large coupling regime (m h µ with µ H) [33,43]. In our case, both m h and µ are controlled by the parameterθ 0 so that the generalized method [45][46][47] applied for (m h + µ) 1/2 > H is required. Note that the effective mass for the heavy mode ω + is nothing but (m h + µ) 1/2 when taking k → 0 into (34). To derive the effective action with a modified speed of sound, one solves the heavy Higgs field in the conformally flat space with a = −1/(Hη) [46]. The solution is where a prime is the derivative with respect to the conformal time η.
The term shown in (53) only gives the local effect, while non-local effects must come from higher-order corrections. Using (53) we find the effective action where the effective speed of sound after integrating out the heavy Higgs reads where ⌘ ⇤ = /(kc ✓ ) is taken to be the value at the sound horizon crossing. The value of can be determined by matching the analytical solutions obtained in the m h ! 0 limit, which gives 1/2 = ( 1/4) 2 /(16⇡) [43,45,46]. In the strong-coupling limit with✓ 0 /H 1, the speed of sound reduces to the prediction (37) as c 2 ✓ ! c 2 h = 1 + µ 2 /m 2 h ⇡ 3. In the weak-coupling limit with✓ 0 /H ⌧ 1, the speed of sound approaches to a constant value The power spectrum is enhanced by P ⇣ = P ⇤ ⇣ /c ✓ where c 1 ✓ ⇡ 6.19.

B. bispectrum
We now investigate the non-Gaussian correlation functions sourced by the Higgs field.
Based on the results of power spectrum, we find that the numerical computation from the ⇤-rescaling scheme is highly consistent with the prediction of in-in formalism for the where the EFT approach is not yet ready to apply. On the other hand, the numerical results from the H-rescaling scheme for the intermediate region can have distinctive behavior from the analytic approaches. The heavy Higgs in this range mainly corresponding to the curvelinear limit with a strong-coupling as labeled by the Region II in Fig. 3. We therefore focus on the study in this region by using the numerical method.
As the simplest example, we compute the three-point correlation functions involved with the -h exchange processes led by (41). In general, one can classify the cubic interactions where η * = −β/(kc θ ) is taken to be the value at the sound horizon crossing. The value of β can be determined by matching the analytical solutions obtained in the m h → 0 limit, which gives β 1/2 = Γ(−1/4) 2 /(16π) [43,45,46]. In the strong-coupling limit withθ 0 /H 1, the speed of sound reduces to the prediction (37) as c −2 θ → c −2 h = 1 + µ 2 /m 2 h ≈ 3. In the weak-coupling limit withθ 0 /H 1, the speed of sound approaches to a constant value c −2 θ → 2/β 2 . The power spectrum is enhanced by P ζ = P * ζ /c θ where c −1 θ ≈ 6.19.

B. bispectrum
We now investigate the non-Gaussian correlation functions sourced by the Higgs field.
Based on the results of power spectrum, we find that the numerical computation from the Λ-rescaling scheme is highly consistent with the prediction of in-in formalism for the where the EFT approach is not yet ready to apply. On the other hand, the numerical results from the H-rescaling scheme for the intermediate region can have distinctive behavior from the analytic approaches. The heavy Higgs in this range mainly corresponding to the curvelinear limit with a strong-coupling as labeled by the Region II in Fig. 3. We therefore focus on the study in this region by using the numerical method.
As the simplest example, we compute the three-point correlation functions involved with the φ-h exchange processes led by (41). In general, one can classify the cubic interactions of (41) into where L (1) 3 and L 3 correspond to the vertices of the single, double and triple exchange diagrams, as depicted in Fig. 6.
Equilateral limit. For µ < H the quadratic interaction δL 2 = µδhδθ c is treated perturbatively. We can estimate the size of the non-Gaussianity in the equilateral limit by using the usual in-in formalism [7]. For the triple exchange diagram (the middle-right panel of Fig.   6), the non-Gaussianity is estimated by In the curvelinear region µ ≈ 2θ 0 and ∆ ζ H 2 /(2πh 0θ0 ) as R 0 ≈ h 0 , we find that We expect this f N L 1 since it is the estimation for µ/H < 1, and thusθ 0 < H/2. One can perform a similar estimation to obtain that f N L ∼ 4π(θ 0 /H) 4 for the double exchange diagrams and f N L ∼ π(θ 0 /H) 2 for the single exchange diagrams.
For the strong-coupling case (µ > H) the quadratic interaction δL 2 also plays an important role in the equation of motion so that δθ and δh are not treated independently.
We have shown in the previous section that the perturbativity of all cubic φ-h interactions in (41) are well-defined even if the quadratic perturbations appear to be strongly coupled.
Therefore our computation for the bispectrum can be taken as a modified in-in formalism based on the EoM approach given by Appedix A. To be more precisely, we take solutions of the coupled EoM from quadratic perturbations to be the mode functions in the interaction picture and we treat cubic interactions as perturbations, following the scheme of (A8).
The three-point diagrams given by Fig. 6 are computed as δθ 3 (t) = i whereH I,3 collects all cubic interactions (A11) with δθ I and δh I resolved from the EoM approach. We adopt the conventional definition for the bispectrum B θ as where S θ is the dimensionless shape function and P θ = P ζ × (θ 2 0 /H 2 ). A numerical estimation of the total non-Gaussianity in the equilateral limit (k 1 = k 2 = k 3 ) from all cubic interactions (A11) is given in Fig. 7 with λ = 0.01 and Λ = 2. This result is evaluated by the H-rescaling scheme with the definition of f N L based on [5] as Note that the parameter H in (63) is rescaled according to (47) since we evaluate P θ and B θ ≡ (H/H * ) 3 B * θ from the mode functions (50) with a reference parameter H * . As a result, the bispectrum amplitude is independent of the choice ofθ * or H * . 3 The asymptotic value f N L ∼ O(10 −2 ) in the limiṫ θ 0 /H 1 is consistent with the estimation by integrating out the heavy Higgs field with an effective speed of sound [9].
Non-analytic scaling. Away from the equilateral limit we can test the momentum scaling in bispectrum. In general, the particle exchange with a heavy field exhibits both local and non-local processes. These processes result in the analytic and non-analytic momentum scaling in bispectrum, respectively. The non-local process comes from components in the heavy-field mode functions which are oscillating in the late-time limit. In other words, non-analytic signals do not appear if we integrate out the heavy field through the EFT approach from the beginning. 4 For a weakly coupled system, the non-analytic components in the correlation functions can be computed explicitly by the in-in formalism [15,42]. In the weak-coupling limit, the oscillatory feature in the bispectrum are generated by a heavy field with a mass m/H > 3/2, and they have the generic suppression factor ∼ e −πL (with a canonical speed of sound c 2 π = 1), where the scaling factor L is given by (52). To see the non-analytic effect in the strongly coupled φ-h system, it is convenient to expand the mode function in the late-time limit |η| → 0 where the approximated expression is found in [43] as (see Appendix B for details) Here B * and B ± are coefficients that in general can only be fixed by fitting with numerical results, and the parameter L h is defined as The imaginary powers ±iL h in the late-time expansion (65) are the sources of the nonanalytic scaling in correlation functions. In order to justify the late-time expansion (65), we rewrite the mixed two-point function (A7) as with G ± θh = δθ ± k δh ± * k = G ± * hθ , where the two sets of independent mode functions are defined in (A1) and (A2). By using the vanish of the equal time commutator we finds Im[G + θh + G − θh ] = 0. Given that the late-time expansion of δθ is led by a constant, where following Appendix B we parametrize as δθ → HA/ √ 4k 3 with a phase |A| = 1, we may express in the limit η → 0 that We use this expression for G + θh to compare with that numerically solved by the EoM method in Fig. 8. The results confirms that L h features the non-analytic scaling of the late-time mode functions, which is associated with the heavy Higgs scale µ h .
We remark that the oscillatory signature for the heavy particle production vanishes as µ h /H < 3/2 where L h becomes an imaginary number. In this case the non-analytic component has a non-integer scaling (−kη) ∆ ± with ∆ = 3/2 ± iL h . Enhancement of the non-Gaussian spectra due to ∆ + < 3/2 was investigated in [48].
Shape functions. A common classification of the shape function is based on the scaling behavior in the squeezed limit, for example, by taking k 1 = k 2 = k and k 3 = ck with c → 0. For c 1, a typical equilateral bispectrum scales as S θ ∼ c and a typical local bispectrum scales as S θ ∼ c −1 [39], where the former peaks at the equilateral limit c = 1 and the later peaks at the squeezed limit c = 0. For a bispectrum scales as S θ ∼ c ν with −1 < ν < 1 is referred to the intermediate shapes [6,7]. As an example, we plot in Fig. 9 the contribution from the interactionH with respect to different values of m h in Hubble unit and we have used the normalization k/H = 1. We can estimate the scaling of the triple exchange bispectrum in the squeezed limit by using the late-time expansion (69) as where The left panel of Fig. 9 shows a case with µ 2 h /H 2 < 9/4 so that L h is imaginary and that c 2 B θ ∼ S θ ∼ c ν with 0 < ν < 1/2. The bispectrum in these cases peak in between the equilateral and the squeezed limits (they can be referred as the quasi-equilateral shapes [7,39]). The bispectra in the right panel of Fig. 9 have µ 2 h /H 2 > 9/4, or S θ ∼ c 1/2 sin(L h ln c+ϕ) with a phase ϕ. They are basically of the equilateral shapes but they exhibit oscillatory signatures in the squeezed limit sourced by the imaginary part of ν. We remark that for a weak-coupling system the oscillatory signatures must be generated with m 2 h /H 2 > 9/4.

V. SUMMARY AND DISCUSSION
The heavy-lifting mechanism [29] is closely related to the cosmological collider research in the sense that its goal is to make SM particles "observable" in primordial non-Gaussianity.
To achieve this goal, it is crucial to break the gauge symmetry so that gauge fields (or at least the scalar bosons) can enter the late-time correlation functions via tree-level exchange with inflaton. More desirable is that the masses of SM fields can be lifted above the inflationary Hubble scale so that they can create non-analytic signals with non-integer or imaginary momentum scaling distinctive from that of the single-field inflation.
In this work, we investigated a heavy-lifting scenario with broken gauge symmetry introduced by the slow-roll dynamics of inflation, characterized by the normalized inflaton velocityθ, where both the Higgs mass m h and the quadratic coupling µ are controlled by the background valueθ 0 . The coupled Higgs-inflaton system can be cast into a special type of quasi-single field inflation with a modified mass scale µ h ≡ (m 2 h + µ 2 ) 1/2 which governs the non-analytic scaling behavior for the late-time mode functions. We have shown that Higgs indeed behaves as a heavy degree of freedom when µ h > H and a strong-coupling µ > H can be realized without violating the loop expansion.
After a proper rescaling of the power spectrum with respect to the observational constraint, especially when the system enters the strong-coupling regime, we confirmed that the non-Gaussianity in the equilateral limit in general can reach f N L O(1). This value is compatible with current constraints from Cosmic Microwave Background observations [49], and could be tested by the future Large Scale Structure or the 21-cm surveys [14,18].
As the signature of heavy Higgs production, we demonstrated shape functions that exhibit non-analytic scaling in the squeezed limit. We have confirmed that the non-analytic scaling of the late-time correlation function is governed by the parameter L h with respect to the modified mass scale µ h .
It is remarkable that, even if the size of the Higgs signal is too weak to be detected in the large mass regime (µ h H), the heavy Higgs field lifted by a strong-coupling can still result in an effective speed of sound c 2 θ < 1 for inflaton. Non-analytic signals sourced by other gauge fields are therefore relatively enhanced (or less suppressed) by the modified sound horizon with a small c 2 θ [15]. Although the specific system (16) considered in this work has a lower limit at c 2 θ = 1/3, it is also straightforward to seek for the realization of c 2 θ 1 from extended scenario in the full expression of the effective theory (9).
A rearrangement of the SM mass spectrum with the modified Higgs scale invoked by heavy-lifting is an important subject to be clarified, which has been postponed as a future study. In a very optimistic scenario where signals of more than one heavy-lifted gauge fields can be detected, it might be possible to test the renormalization group running of SM couplings up to the energy scale of inflation [23]. These results might reveal some hints connected to the Naturalness of SM [29], or the issue of Higgs instability [24][25][26].
whereH 0 includes all quadratic terms that give the coupled EoM (A12) andH I contains only cubic or higher-order in perturbations.
Let us deriveH 0 andH I for the φ-h system given by (27) and (41). The conjugate momenta are δθ c δh +θ c where B 00 = 0 due to the constraint of (B3). One can find the relation between coefficients from (B3), where ( 3 2 ± iL h )A 0± = µB 0± /H. The two non-integer branch values v = 3 2 ±iL h are in fact solved from the part of equation νA 0ν = µB 0ν /H in (B3). These solutions imply a possible late-time approximation of the theory (27) as which results in the effective relation δh = −δθ c /µ. Putting this effective relation back into (B9), one obtain the Lagrangian only for δh, which gives the familiar equation of motion as Assuming the usual Bunch-Davies vacuum, the mode function of the equation of motion is well-known: where H (1) ν is Hankel function of the first kind. Expanding the Hankel function at z → 0, we find that δh → − i π e iπ/4 H π 4k 3 (−z) 3/2 (B12) Therefore it is easy to match the leading coefficients in the late-time expansion up to a phase difference as A comparison of the late-time expansion given by (B13) with the numerical result of the EoM approach is shown in Fig. 8. We remark that although the power series expansion was first derived in Ref. [48] for studying the cases with m 2 h + µ 2 < 9H 2 /4, the late-time effective theory (B9) still holds in cases with m 2 h + µ 2 > 9H 2 /4 since the approximation does not rely on the value of the parameter m h nor µ.