Higgs as heavy-lifted physics during inflation
Yi-Peng Wu








[Electronic address: ]ypwu@resceu.s.u-tokyo.ac.jp
^aResearch Center for the Early Universe (RESCEU),	Graduate School of Science, The University of Tokyo,	Tokyo 113-0033, Japan

















	
		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 Higgs-inflaton mixing, 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-mixing regime can act as important background signals to be tested by future cosmological surveys.
	

RESCEU-17/18


Higgs as heavy-lifted physics during inflation
    Yi-Peng Wu^a
    December 30, 2023
==============================================


	






§ INTRODUCTION


Heavy fields are expected to arise from the UV completion in most of the single-field inflationary scenarios <cit.>. 

These heavy states, while stablizing a flat direction for the slow-roll 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 <cit.>.
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 <cit.>.
Imprints of heavy particle production include characteristic signals from the non-analytic scaling of the momentum restricted by the symmetry of inflation <cit.> (see also <cit.>). 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 <cit.>. 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 non-minimal 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 <cit.>.

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 <cit.>. 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 <cit.> 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 <cit.>. 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 strong mixing terms 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 strongly mixed. One can identify a modified mass scale corresponding to the strong mixing, below which Higgs acts as a heavy degree of freedom. Integrating out the heavy Higgs field in the strong-mixing 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 <cit.>.

Due to the implicit symmetries of the inflationary background, one can expand mode functions into power-laws ∼η^Δ of the conformal time in the late-time limit |η|→ 0, where Δ characterizes the spatial dimension of an operator residing on the boundary of future infinity (η = 0) <cit.> (see also <cit.>). 
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

    ⟨ζ^3⟩/⟨ζ^2⟩_S ⟨ζ^2⟩_L∼∑_i w_i (k_L/k_S)^Δ_i,

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 <cit.>

    Δ_ non-analytic = 3± 2  i L_h,       L_h = √(m_h^2/H^2-9/4),

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 large mixing 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-mixing system is thus modified as

    Δ_ non-analytic = 3/2±  i L_h,       L_h = √(m_h^2/H^2 c_ϕ^2-9/4),

where the half-integer from single Higgs production is allowed due to the broken gauge symmetry, and c_ϕ^2 is the effective speed of sound for inflaton which accounts for the local effect via integrating out the Higgs field. Comparing (<ref>) and (<ref>), one can identify a modified mass scale μ_h ≡ m_h/c_ϕ for the strong-mixing system, which recovers the non-local extension of the effective-field-theory approach for inflation <cit.>.  
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 generated when m_h/(c_ϕ H) > 3/2, instead of m_h/H > 3/2.




This paper is organized as the follows. In Section <ref>, we review the framework for the heavy-lifting mechanism in the effective theory approach. We consider a model in Section <ref> as an example to show that the Higgs mass scale can be heavy-lifted by a strong mixing 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 <ref>. Finally, conclusions are given in Section <ref>.



§ 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 <cit.>. This formalism (sometimes called the π-σ model <cit.>) assumes no specific fundamental physics but includes all possible terms with respect to the time-dependent spatial diffeomorphisms.
[The π-σ model is also known as quasi-single field inflation <cit.> for the mass m_σ of the σ field is near or larger than the Hubble parameter H. 

]

We then specify the class of theory proposed in <cit.> 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 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 <cit.> 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.


 §.§ The cosmological Goldstone boson

We follow the standard precedure to construct the most general effective action of inflation <cit.>. 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 δ g^00 = g^00+1 are   

    S_π = ∫ d^4x √(-g)[ M_p^2/2 R  + M_p^2Ḣ g^00 - M_p^2 (Ḣ+3H^2) .
    .      + M_2^4(t) /2!(δ g^00)^2 +  M_3^4(t)/3! (δ g^00)^3 + ⋯],

where operators with the extrinsic curvature perturbation δ K_μν and with higher derivatives are not shown here since the action (<ref>) is sufficient for our purpose. 

Following <cit.> with the same setup as for S_π, the effective action for an extra scalar field σ during inflation is given by

    S_σ   = ∫ d^4x √(-g)[ -α_1(t)g^μν∂_μσ∂_νσ + α_2(t) (∂^0 σ)^2 
    - α_3(t) σ^2 + α_4(t)σ∂^0σ + ⋯], 
    
    S_πσ   = ∫ d^4x √(-g)[
    β_1(t) δ g^00σ + β_2(t)δ g^00∂^0σ + β_3(t)∂^0σ. 
                   . -(β̇_3(t) +3Hβ_3(t))σ+ ⋯],

where we introduce the interactions between π and σ fields through S_πσ. We refer the combination of actions (<ref>), (<ref>) and (<ref>) 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 <cit.>, the decoupling limit with gravitational fluctuations is realized by taking M_p →∞, Ḣ→ 0 with M_p^2 Ḣ 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 + ∂_μπ∂^μπ, and
∂^0 σ→∂^0 σ + ∂_μπ∂^μσ. 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 <cit.>. 
Putting solutions of the ADM variables back into the actions, such as (<ref>) and (<ref>), and taking H^2/M_p^2 → 0 with Ḣ/H^2 → 0, we find the simplified actions as

    S_π→∫ d^4x √(-g) M_p^2 Ḣ/c_π^2[- (π̇^2 -c_π^2 (∂_iπ)^2/a^2)-(1-c_π^2) (π̇^3 -π̇(∂_iπ)^2/a^2)] +⋯,

where c_π^-2 = 1- 2M_2^4/(M_p^2 Ḣ), and

    S_πσ→∫ d^4x √(-g) [(-2β_1 +β̇_3)π̇σ + (2β_2 - β_3)π̇σ̇ 
    + β_3 ∂_iπ∂_iσ/a^2
    + ⋯].

We can see that (<ref>) and (<ref>) capture nothing but the leading terms of the ϕ-h model in the following discussion.




 §.§ The Higgs boson during inflation

While the π-σ model contains every operator that is invariant under the spatial transformation x^i ↦ x^i +ξ^i(t, x), 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 <cit.> has a general Lagrangian of the form 

    ℒ= ℒ_ sr(ϕ) -ξ R Φ_H^†Φ_H  -| D_μΦ_H|^2  - λ(Φ_H^†Φ_H)^2 + ℒ_ϕ h,

where Φ_H  = (0, h)^T   /√(2) is the Higgs doublet in terms of the SM unitary gauge. The inflaton Lagrangian

    ℒ_ sr(ϕ) = -1/2(∂_μϕ)^2 - V(ϕ),

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

    ℒ_ϕ h = c_1/Λ_1∂_μϕ (Φ_H^† D^μΦ_H) 
    + c_2/Λ_2^2 (∂_μϕ)^2 Φ_H^†Φ_H 
    + c_3/Λ_3^4(∂_μϕ)^2 | D_μΦ_H|^2 + ⋯,

which can be taken as an effective expression of the full theory at the energy scale of inflation. As shown in <cit.>, 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 non-minimal coupling ξ R h^2 with ξ < 0 <cit.>. 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 (<ref>). 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 +δ h with h_0 a constant VEV. In the unitary gauge where δϕ = 0, the linear expansion of (<ref>) gives

    ℒ_ϕ h = c_1/2Λ_1ϕ̇_0 δ h δḣ + c_1/2Λ_1δ g^00ϕ̇_0 h_0 δḣ
    +c_1/2Λ_1δ g^0iϕ̇_0 h_0 ∂_i δ h 
    
    + c_2/2Λ_2^2 g^00ϕ̇_0^2 δ h^2 +c_2/Λ_2^2δ g^00ϕ̇_0^2 h_0 δ h + ⋯,

where ϕ_0 is the inflaton VEV during slow-roll. Since metric components in the decoupling limit with gravity read

    g^00→ -(1+π̇)^2 +(∂_iπ)^2/a^2,   and   g^0i→δ^ij∂_j π/a^2,
 
the Lagrangian ℒ_ϕ h then introduces the π-h interactions up to quadratic order as

    ℒ_ϕ h⊃ -c_1/Λ_1ϕ̇_0 h_0 π̇δḣ + c_1/2Λ_1ϕ̇_0 h_0 ∂_i π∂_i δ h/a^2
    -2 c_2/Λ_2^2ϕ̇_0^2 h_0 π̇δ h.

Replacing h by σ we find from (<ref>) that

    β̇_3 -2β_1 = -2 c_2/Λ_2^2ϕ̇_0^2 h_0,    
    2β_2 - β_3 = -c_1/Λ_1ϕ̇_0 h_0,   and  β_3 = c_1/2Λ_1ϕ̇_0 h_0 .

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 gauge-invariant variable ζ = - Hπ = - H δϕ/ϕ̇_0 implies π = δϕ/ϕ̇_0.



§ ENERGY SCALES OF HEAVY-LIFTING


In this section we consider the heavy-lifting scenario induced by ϕ-h interactions (<ref>). 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 <cit.>.

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 (<ref>). To simplify our discussion, we turn off the non-minimal coupling ξ and assume a positive λ≪ 1.





 §.§ 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

    ℒ= ℒ_ sr(ϕ) -Φ_H^†Φ_H (∂_μϕ)^2/Λ^2 -| D_μΦ_H|^2  - λ(Φ_H^†Φ_H)^2,

where it can be taken as a special case of the ϕ-h theory (<ref>) 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

    ℒ⊃ -1/2(1 + h^2/Λ^2) (∂_μϕ)^2 -1/2(∂_μ h)^2,

which represents a two-field limit of the multi-field inflation scenario based on the non-linear sigma model <cit.>.
To justify (<ref>) from the effective theory formulation, we shall consider an approximated shift symmetry in the inflaton sector so that non-derivative ϕ-h couplings are not appear (or highly suppressed) in the system. To our purpose, we should consider a fine cancellation of higher-order terms in the series of h^2/Λ^2, which allows us to explore the parametric region with h > Λ. On the other hand, higher-order terms in the series of ∂ϕ/Λ^2 can still be controlled by the Naturalness condition introduced in Section <ref>.
At a first glance, Λ looks like a cutoff scale of the non-canonical kinetic interaction to make sure that (<ref>) is a well-behaved low-energy effective Lagrangian. 
However, we will show that in the presence of a strong-mixing the conclusion is non-trivial. For example, a well-defined perturbative expansion for the ϕ-h system (<ref>) is in fact independent of the ratio h/Λ.







[
It is also desirable to specify the origin of the system (<ref>) from explicit theories at high energy. For example,
the non-canonical kinetic interaction in (<ref>) can be found in the presence of an approximated U(1) symmetry in the extended Higgs sector beyond the SM <cit.>.
This type of non-canonical kinetic interaction can also be found in inflation models based on supergravity with a soft-breaking of the shift-symmetry <cit.>.
We want to emphasize that the model (<ref>) is used as an simple example for the heavy-lifting mechanism so that there is in fact no primary assumption for its origin.]



The cutoff scale of the system (<ref>) is non-trivial since the target field space of ϕ-h can be curved.
For convenience, we perform the reparametrization for both fields as

    R = (Λ^2 +h^2)^1/2,         θ = ϕ/Λ,

so that the kinetic part of the system becomes

    ℒ⊃ -1/2 R^2 (∂_μθ)^2 -1/2R^2/R^2 - Λ^2(∂_μ R)^2.

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.  



	
  * 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 (<ref>).
If the Higgs field develops a well-defined background value h_0 during inflation, the homogeneous field equations of the system are given by

    3M_p^2H^2    = 1/2 R_0^2 θ̇_0^2 +1/2ḣ_0^2 +V(Λ θ_0) + λ/4 h_0^4, 
    
    -2M_p^2 Ḣ   = R_0^2 θ̇_0^2 + ḣ_0^2, 
    θ̈_0 +3 Hθ̇_0 + 2h_0ḣ_0/R_0^2θ̇_0+   V_θ/R_0^2   = 0, 
    ḧ_0 + 3H ḣ_0 + λ h_0^3    =  h_0 θ̇_0^2,

where R_0 = (Λ^2 +h_0^2)^1/2 and V_θ≡∂ V/∂θ.
For λ > 0, there exists a static solution h_0^2 = θ̇_0^2/λ such that the Higgs field 

can develops a non-zero vacuum expectation value (VEV) h_0 = ±θ̇_0/√(λ). 
This non-trivial VEV is sourced by the essential dynamics θ̇^2 = ϕ̇^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

    V_ eff = λ/4 h^4 -1/2θ̇_0^2 h^2,

around h_0 one finds the effective mass m_h^2 = 2 θ̇_0^2. In the gravitational decoupling limit the mass m_h^2 becomes a constant. A stable h_0 asked by the stochastic condition m_h^2 ≳ H^2 <cit.> can be satisfied if θ̇_0^2 ≳ H^2/2.
Note that the first slow-roll parameter is related to the non-zero Higgs VEV as

    ϵ = - Ḣ/H^2≃R_0^2 θ̇^2/2 M_p^2 H^2 = λ R_0^2 h_0^2/2 M_p^2 H^2.

In the flat-decoupling limit (h_0^2 ≪Λ^2) where R_0 ≈Λ, the slow-roll condition implies Λ≪ 2 M_p H/θ̇_0.
In the curvelinear limit (h_0^2 ≫Λ^2) where R_0 ≈ h_0, the smallness of ϵ≃λ h_0^4/(2M_p^2 H^2) instead guarantees the subdominance of the Higgs vacuum energy to the background energy density.




 §.§ Scales of heavy Higgs


The non-zero Higgs VEV h_0 is led by the slow-roll inflation dynamics -2M_p^2 Ḣ = R_0^2 θ̇_0^2 and 3H θ̇_0 = -V_θ/R_0^2 at the first-order of ϵ. We can treat h_0 as a stable constant during the slow-rolling of θ given that ḣ_0 is at least second-order in the slow-roll parameters.


Performing the scalar perturbations θ(t,  x) = θ_0(t) + δθ(t,  x) and h(t,  x) = h_0(t) + δ h(t,  x) to (<ref>), we obtain the quadratic Lagrangian as

    ℒ_2 = 1/2[R_0^2 δθ̇^2- R_0^2/a^2(∂_iδθ)^2 + δḣ^2 -1/a^2(∂_i δ h)^2 - m_h^2 δ h^2 + 4h_0 θ̇_0 δ h δθ̇] + 𝒪(ϵ)⋯,

where 𝒪(ϵ) means quadratic perturbations that are suppressed by the slow-roll parameters (which includes the mass term of inflaton). 
The terms shown in (<ref>) can also be derived from the general perturbation theory, and one can check that metric perturbations only contribute to 𝒪(ϵ).


To see the dynamics of the system, it is convenient to use the canonically normalized field θ_c = R_0θ with respect to the canonical commutation relation for canonical quantization.
The quadratic Lagrangian (<ref>) is rewritten as

    ℒ_2 ⊃1/2[ δθ̇_c^2- 1/a^2(∂_iδθ_c)^2 + δḣ^2 -1/a^2(∂_i δ h)^2 - m_h^2 δ h^2 + 2μδ h δθ̇_c],

where the mixing parameter 

    μ≡2h_0 θ̇_c/R^2 = 2θ̇_0^2/√(θ̇_0^2 + λΛ^2) ,

plays a key role in the dynamics of our system. Note that once λ and Λ are fixed, both μ and m_h^2 are controlled by the same parameter θ̇ so that they are not independent from each other. This is a fundamental difference from the constant-turn quasi-single-field inflation <cit.> (or the strongly mixed π-σ model <cit.>).   

We classify the ϕ-h system with respect to the mixing parameter μ as

    
  * weak-mixing: μ / H < 1, and
    
  * strong-mixing: μ / H > 1.

For H > μ, the interaction 2μδ h δθ̇_c does not play an important role during inflation and the system (<ref>) simply describes two weakly interacted scalar fields. In this case we expect the standard picture of a massless Higgs field.
The condition μ > H for the presence of a strong-mixing implies 

    θ̇_0^2 > H^2/8 + H^2/8√(1+16λΛ^2/H^2).
 






To understand how the two degrees of freedom decoupled with the energy scales, let us write down the equations of motion of the perturbations

    δθ̈_c + 3H δθ̇_c+ k^2/a^2δθ_c    = -μ(δḣ + 3H δ h ), 
    δḧ + 3H δḣ+ (k^2/a^2 + m_h^2 ) δ h    = μδθ̇_c.

In the long-wavelength regime with k/a → 0, we expect the usual solution δθ_c → constant and δ h → 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

    δθ̈_c + p^2 δθ_c    = -μδḣ , 
    δḧ +  (p^2 + m_h^2 ) δ h    = μδθ̇_c .

The solutions in the subhorizon regime thus take the form of δθ_c ∼δθ_± e^i ω_± t and δ h ∼δ h_± e^i ω_± t <cit.>, where the two frequencies are found as 

    ω_±^2    = p^2 + m_h^2+μ^2/2±√(p^2μ^2 + (m_h^2 + μ^2)^2/4), 
       = p^2 + m_h^2/2 c_h^2±√(p^2 μ^2 + m_h^4/4 c_h^4).

Here the speed of sound c_h^2 is simply defined in the limit of p ≪ (m_h^2 + μ^2)/μ such that the low-energy frequency can be expanded as

    ω_-^2 → c_h^2 p^2 + μ^4 c_h^6/m_h^6 p^4 = c_h^2 p^2 + (1- c_h^2)^2c_h^2/m_h^2 p^4,

and thus it indicates

    c_h^2 = m_h^2/m_h^2 + μ^2 = θ̇_0^2 + λΛ^2/3θ̇_0^2 + λΛ^2.

The result (<ref>) appears to be the same as using the effective field approach for a curvelinear trajectory after neglecting the slow-roll parameter suppressed effective mass <cit.>
With the definition (<ref>) the low-energy mode has a linear dispersion relation ω_- ≈ c_h p for p^2 ≪ m_h^6 μ^-4 c_h^-4 and a nonlinear dispersion relation ω_- ∝  p^2 for  p^2 ≫ m_h^6 μ^-4 c_h^-4.

The dispersion relation of the two frequency modes ω_± given by (<ref>) is depicted in Fig. <ref>, yet keeping in mind that these solutions are only valid for subhorizon scales. The modes ω_±^(1) 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_h^2 + μ^2)/μ the high-energy mode

    ω_+^(2)→ m_h/c_h,

describes a heavy degree of freedom during inflation as long as m_h ≫ c_h H. Thus, with a mixing μ > H, the existence of Higgs as a heavy field during inflation does not necessarily requires m_h^2 ≳ H^2. In fact, in the π-σ model one can make σ a heavy mode merely due to a strong-mixing μ/H ≫ 1 with a mass m_σ≪ H, provided that c_h^2 ≪ 1 <cit.>.
However, in our scenario the two parameters μ and m_h are not independent, and one can check that c_h^2 → 1 in the flat-decoupling limit of Higgs and inflaton where θ̇_0^2 ≪λΛ^2 and c_h^2 → 1/3 in the curvelinear limit where θ̇_0^2 ≫λΛ^2. Based on these findings one can identify the energy scale to have a heavy Higgs field during inflation as

    μ_h ≡ (m_h^2 + μ^2 )^1/2 = m_h/c_h,

where μ_h → 0 as θ̇_0 → 0.
We therefore identify the heavy-Higgs condition as: μ_h > H, which is namely m_h^2 > c_h^2 ≥ H^2/9. This implies the lower limit for a heavy Higgs as

    θ̇_0 > H /√(18),

which is to be compared with (<ref>). 

The corresponding values of the examples μ_h^(1)/H ≈ 0.17 and μ_h^(2)/H ≈ 4.88 are given as the vertical lines in Fig. <ref>.







 §.§ Perturbativity and Naturalness









We now check if the perturbative expansion for higher-order terms are well-defined in the presence with a strong quadratic mixing 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. <ref>.

Let us consider the cubic interactions introduced by the Higgs-inflaton coupling from (<ref>)

    ℒ_3 ⊃h_0/R_0^2[δθ̇_c^2 -1/a^2(∂_i δθ_c)^2] δ h
    + θ̇_c/R_0^2δ h^2 δθ̇_c.

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_0^2/h_0 from the first two cubic interactions in (<ref>). Therefore Λ_p > μ_h implies that

    (θ̇_0^2 + λΛ^2)^2 >  2 λθ̇_0^4/c_h^2.

One can find that even in the curvelinear limit where θ̇_0^2 ≫λΛ^2 the condition holds if λ < 1/6. 

The coefficient of the third interaction in (<ref>) is dimensionless so that perturbativity simply requires θ̇_c/R_0^2 < 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 (<ref>), as illustrated by the right panel of Fig. <ref>. In this case the space and time coordinate can have different dimensions, making the discussion much complicated. To determine the cutoff scale Λ_p, we write down the non-relativistic version of the Lagrangian (<ref>) as

    ℒ_2 ≈μδ h δθ̇_c - 1/2 a^2 (∂_i δθ_c)^2 -1/2a^2 (∂_i δ h)^2 -1/2m_h^2 δ h^2,

where time-derivatives have dropped out. 

For p ≫ m_h/(1-c_h^2), the low-energy mode ω_- ≈ (1-c_h^2)(c_h/m_h) p^2 ∝ p^2/m_h so that the energy density has the dimension [T^00] = [ω_-][k]^3 = [ω_-]^5/2[m_h]^3/2. Identifying each term in (<ref>), we find that [m_h] = [δ h]= [δθ_c] =[ω_-]. Rescaling the spatial coordinate as x_r = μ^1/2 x with the redefinition of fields as δ h_r = μ^-1/4δ h and δθ_r = μ^-1/4δθ_c, the momentum conjugate is normalized as 
p_δθ_r≡∂ℒ_2/ ∂δθ̇_r = δ h_r.
The cubic interactions are thus parametrized as

    ℒ_3 ⊃1/(Λ_p1)^7/4δθ̇_r ^2 δ h_r 
    - 1/(Λ_p2)^3/4(∂̃_i δθ_r)^2 δ h_r 
    +1/(Λ_p3)^3/4δθ̇_r δ h_r^2,

where ∂̃_̃ĩ means the spatial derivative with respect to x_r, and 

    Λ_p1 = h_0^4/7μ^3/7,   Λ_p2 =  h_0^4/3μ^-1/3  and  Λ_p3 = μλ^-2/3.

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 16π^2 <cit.>. 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. 



One can impose the condition ϕ̇/Λ^2 < 1 to suppress the higher-order corrections from (∂_μϕ)^n/Λ^n to the system (<ref>). In terms of θ = ϕ/Λ, the Naturalness condition, namely θ̇ < Λ, includes all the flat-decoupling region θ̇ < √(λ)Λ since the perturbativity asks λ < 1. As shown in Fig. <ref>, 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 mixing, or in Region II with a strong quadratic mixing. 




§ OBSERVATIONAL IMPACT OF HEAVY HIGGS



 §.§ 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 (<ref>) 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-mixing regime these corrections may become comparable to the leading term <cit.> and thus one has to rescale the amplitude to meet the observational constraint. Treating the background value θ̇_0 as a free parameter, we shall rescale the power spectrum with respect to the observational constraint 

    Δ_ζ^2 ≡H_∗^2/ R_∗^2 θ̇_∗^2P^∗_θ≈ 2.2 × 10^-9,

where P^∗_θ = H_∗^2/(4π^2) is a reference field spectrum for a given reference angular velocity θ̇_∗, which is more convenient to choose in the flat-decoupling limit (namely θ̇_∗≪√(λ)Λ) such that R_∗ = Λ. H_∗ is the reference Hubble parameter as viewed in the flat-decoupling limit. To keep a constant Δ_ζ^2 while varying θ̇_∗, 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 given θ̇_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

    Δ_ζ^2 = H^2/R^2_0 θ̇_0^2P_θ^  new 
        = H^2/R_0^2 θ̇_0^2H^2/H_∗^2 f  P_θ^∗,

where the rescaled field spectrum is P_θ^ new≡ P_θ H^2/H^2_∗ = H^2f/(4π^2). The rescaled Hubble parameter is then given by

    H/H_∗ = 
        (θ̇_0/θ̇_∗)^1/2(R_0/R_∗)^1/2 f^-1/4,


The numerical results for f with Λ = 2H and λ = 0.01 are given in Fig. <ref>, where the threshold value for the heavy Higgs condition is θ̇_0/H_∗ = 0.24 and θ̇_0/H_∗ > 0.27 becomes strong-mixing.

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 of θ̇_0. For the flat-decoupling case, we can define that

    Δ_ζ^2 = H_∗^2/Λ^2  θ̇_0^2 P_θ^  new
        = H^2_∗/Λ^2  θ̇_0^2H^2/H_∗^2P_θ
        = H^2_∗/Λ^2  θ̇_0^2H^2/H_∗^2 f  P_θ^∗,

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 given θ̇_0 as Λ = H_∗^2/(2πΔ_ζθ̇_0). We refer this method as the Λ-rescaling scheme, and the numerical results for f with λ = 0.01 are given in Fig. <ref>. For λ = 0.01, the threshold value for the heavy Higgs condition μ_h/H_∗ = 1 gives θ̇_0/H_∗ = 0.7, the Naturalness condition asks θ̇_0/H_∗ < 58.3 and the strong-mixing μ/H_∗ > 1 requires θ̇_0/H_∗ > 5.5. 







For both rescaling schemes the results from three kinds of approaches are summarized in Figures <ref> and <ref>.
We discuss the implication from each approach as the follows.

Equation of Motion (EoM).
The equation of motion (EoM) approach <cit.> solves quantum field fluctuations from a complete set of initial states that satisfy the canonical commutation relation. The Bunch-Davies vacuum states are special examples of these initial states and are usually applied to define as the vacuum of “free fields” for the in-in formalism <cit.> 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 (<ref>) are found to be

    δθ^±_k = H_∗/√(4k^3)e^-i k η(-k η)^1± i μ/(2H_∗),   and  δ h^±_k = ± i δθ^±_k,

where the derivation is given in Appendix <ref>. The dimensionless spectrum is given by P_θ = k^3(|δθ_k^+|^2+|δθ_k^-|^2)/(4π^2).

For both rescaling scheme, in the limit θ̇_0/H_∗≪ 1 Higgs behaves as a light isocurvature mode with negligible corrections to the power spectrum. For θ̇_0/H_∗≫ 1 the EoM results agree with the prediction from the effective field theory (EFT) method by integrating out the heavy Higgs field. In the intermediate regime θ̇_0/H_∗∼𝒪(1), the numerical result from EoM method in the H-rescaling scheme cannot be explained by the analytical approach (in-in formalism) nor the EFT method.
We remark that the numerical results shown in Figs. <ref> and <ref> do not diverge in the limit of m_h → 0 where θ̇_0/H_∗→ 0, since the parameter μ vanishes identically in this limit. 




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 (<ref>) computation can be found in <cit.>, and for inflaton with a generalized speed of sound c_θ^2 ≤ 1 is considered in <cit.>. As given by the dotted line in Fig. <ref>, the correction from the quadratic interaction δℒ_2 = μδ h δθ̇_c up to first order in the perturbative expansion is led by the form

    Δ P_ζ/P_ζ^∗ = 2(μ/H_∗)^2 
    {π^2/4 coshπ L 
    + e^π L/16sinhπ L [ψ^(1)(3/4+i L/2)-ψ^(1)(1/4+i L/2)] .
    . - e^-π L/16sinhπ L [ψ^(1)(3/4- i L/2)-ψ^(1)(1/4- i L/2)]
    },

where ψ^(1)(x) ≡ d^2 lnΓ(x)/dx^2 is the polygamma function, and

    L = √(m_h^2/H_∗^2-9/4).

Δ 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^∗ = Δ_ζ^2.

Therefore we have f = Δ P_ζ/P_ζ^∗ +1.
One can check that Δ P_ζ/ P_ζ^∗→ 1 in the limit of θ̇_0/H_∗→∞ where m_h^2→∞. The deviation from the EoM results in the strong-mixing regime is due to the fact that δθ aquires an effective speed of sound c_θ^2 < 1 (see the discussion in EFT method).
Note that the expression (<ref>) does not diverge in the limit of θ̇_0/H_∗→ 0 since μ also goes to zero. This is one of the essential difference from the quasi-single-field inflation results <cit.>. 

Effective field theory (EFT).
Given that Higgs behaves as a heavy degree of freedom in the strong-mixing 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 <cit.>, which is to be distinguished from the effective theory discussed in Section <ref>. 

The EFT approach for the two-field system (<ref>) has been studied in the large mass regime (m_h ≫μ with m_h ≫ H) <cit.> and in the large mixing regime (m_h ≪μ with μ≫ H) <cit.>. In our case, both m_h and μ are controlled by the parameter θ̇_0 so that the generalized method <cit.> applied for (m_h^2 + μ^2)^1/2 > H is required. Note that the effective mass for the heavy mode ω_+ is nothing but (m_h^2 + μ^2)^1/2 when taking k → 0 into (<ref>). 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) <cit.>. The solution is

    δ h = -a μ R_0/∂_i^2 - a^2 (m_h^2 - 2H^2)δθ^' + ⋯,

where a prime is the derivative with respect to the conformal time η. 

The term shown in (<ref>) only gives the local effect, while non-local effects must come from higher-order corrections. Using (<ref>) we find the effective action

    S_ eff = ∫ d^4x  a^2/2[R_0^2 /c_θ^2δθ^' 2 -R_0^2 (∂_i δθ)^2],

where the effective speed of sound after integrating out the heavy Higgs reads

    c_θ^-2(kη) = 1+ μ^2/H^2k^2η_∗^2 +m_h^2 -2H^2,

where η_∗ = - β/(k c_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) <cit.>.
In the strong-mixing limit with θ̇_0/H ≫ 1, the speed of sound reduces to the prediction (<ref>) as
c_θ^-2→ c_h^-2 = 1 + μ^2/m_h^2 ≈ 3. In the weak-mixing limit with θ̇_0/H ≪ 1, the speed of sound approaches to a constant value c_θ^-2→ 2/ β^2. The power spectrum is enhanced by P_ζ = P_ζ^∗/c_θ where c_θ^-1≈ 6.19. 




 §.§ 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 intermediate region θ̇_0/H∼𝒪(1) 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-mixing as labeled by the Region II in Fig. <ref>. 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 (<ref>).
In general, one can classify the cubic interactions of (<ref>) into

    ℒ^(1)_3    = -a^3 h_0/R_0^2[δθ̇_c^2 -1/a^2(∂_i δθ_c)^2] δ h, 
    ℒ^(2)_3    = -a^3 θ̇_0/R_0δθ̇_c δ h^2, 
    ℒ^(3)_3    = -a^3 λ h_0 δ h^3,

where ℒ^(1)_3,  ℒ^(2)_3 and ℒ^(3)_3 correspond to the vertices of the single, double and triple exchange diagrams, as depicted in Fig. <ref>.

Equilateral limit.
For μ < H the quadratic interaction δℒ_2 = μδ h δθ̇_c is treated perturbatively. We can estimate the size of the non-Gaussianity in the equilateral limit by using the usual in-in formalism <cit.>. For the triple exchange diagram (the middle-right panel of Fig. <ref>), the non-Gaussianity is estimated by

    f_NL∼1/Δ_ζ(μ/H)^3 λ h_0/H.

In the curvelinear region μ≈ 2θ̇_0 and Δ_ζ≃ H^2/(2π h_0θ̇_0) as R_0 ≈ h_0, we find that

    f_NL∼ 16π(θ̇_0/H)^6.

We expect this f_NL≪ 1 since it is the estimation for μ/H <1, and thus θ̇_0 < H/2. One can perform a similar estimation to obtain that f_NL∼ 4π (θ̇_0/H)^4 for the double exchange diagrams and f_NL∼π (θ̇_0/H)^2 for the single exchange diagrams.



For the strong-mixing case (μ > H) the quadratic interaction δℒ_2 also plays an important role in the equation of motion so that δθ and δ h are not treated independently.
We have shown in the previous section that the perturbativity of all cubic ϕ-h interactions in (<ref>) 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 <ref>. 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 (<ref>).




The three-point diagrams given by Fig. <ref> are computed as

    ⟨δθ^3(t)⟩
    
    
    = i ∫^t_t_0dt_1 ⟨ 0|[H̃_I,3(t_1), δθ^3_I(t)] | 0⟩ ,

where H̃_I,3 collects all cubic interactions (<ref>) with δθ_I and δ h_I resolved from the EoM approach.
We adopt the conventional definition for the bispectrum B_θ as

    ⟨δθ_ k_1(t)δθ_ k_2(t)δθ_ k_3(t) ⟩   ≡ (2π )^3 δ^3( k_1+ k_2 + k_3) B_θ( k_1, k_2 ,k_3), 
       =  (2π )^7 δ^3( k_1+ k_2 + k_3) P_θ^2/(k_1k_2k_3)^2 S_θ( k_1, k_2 ,k_3),

where S_θ is the dimensionless shape function and P_θ = P_ζ×(θ̇_0^2/H^2).




A numerical estimation of the total non-Gaussianity in the equilateral limit (k_1=k_2=k_3) from all cubic interactions (<ref>) is given in Fig. <ref> with λ = 0.01 and Λ = 2. This result is evaluated by the H-rescaling scheme with the definition of f_NL based on <cit.> as

    B^eq_ζ = (H/R_0θ̇_0)^3 B^eq_θ = Δ_ζ^4   f_NL18/5.

Note that the parameter H in (<ref>) is rescaled according to (<ref>) since we evaluate P_θ and B_θ≡ (H/H_∗)^3 B_θ^∗ from the mode functions (<ref>) with a reference parameter H_∗. As a result, the bispectrum amplitude is

    f_NL   = 5/18Δ_ζ^-4(Δ_ζ^2/P_θ^∗)^3/2 f^-3/2(H_∗/H)^3 B_θ^eq, 
       = 5/18Δ_ζ^-1f^-3/2 I(t),

where f and I(t) = (2π)^3 B_θ^∗/H_∗^3 are computed by the EoM approach. The result (<ref>) is independent of the choice of θ̇_∗ or H_∗.
[For the numerical estimation in Fig. <ref>, we have applied the Wick rotation technique <cit.> to mode functions to avoid the slow convergence in the UV limit.]  
The asymptotic value f_NL∼𝒪(10^-2) in the limit θ̇_0/H ≫ 1 is consistent with the estimation by integrating out the heavy Higgs field with an effective speed of sound <cit.>. 




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.[However, the leading non-analytic contribution can be captured if heavy-field operators are only partially integrated out <cit.>.]
For a weakly coupled system, the non-analytic components in the correlation functions can be computed explicitly by the in-in formalism <cit.>. In the weak-mixing 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^-π L (with a canonical speed of sound c_π^2 = 1), where the scaling factor L is given by (<ref>). 



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 <cit.> as (see Appendix <ref> for details)

    δ h_I →H/√(4k^3)(-kη)^3/2[  B_+(-kη)^i L_h + B_-(-kη)^-i L_h +B_∗(-kη)^1/2 + ⋯].

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

    L_h ≡√(μ_h^2/H^2-9/4) = √(m_h^2/H^2c_h^2-9/4).

The imaginary powers ± i L_h in the late-time expansion (<ref>) are the sources of the non-analytic scaling in correlation functions. In order to justify the late-time expansion (<ref>), we rewrite the mixed two-point function (<ref>) as

    ⟨δθ_ k (η) δ h_ q (η) ⟩ 
      =  (2π)^3 δ^3( k + q) [G_θ h^+(η) + G_θ h^-(η)]

with G_θ h^± =  δθ^±_kδ h^±∗_k = G^±∗_hθ, where the two sets of independent mode functions are defined in (<ref>) and (<ref>). 
By using the vanish of the equal time commutator

    [δθ_k(η),δ h_k(η)] =
        G_θ h^+ + G_θ h^- - G_hθ^+ - G_hθ^- = 0,

we finds Im[ G_θ h^+ + G_θ h^- ] =0. Given that the late-time expansion of δθ is led by a constant, where following Appendix <ref> we parametrize as δθ→ H A/ √(4k^3) with a phase | A | = 1, we may express in the limit η→ 0 that

    G_θ h^±→H^2/4k^2 (-kη)^3/2Re[A B_+^∗ (-kη)^-i L_h + AB_-^∗ (-kη)^i L_h].

We use this expression for G^+_θ h to compare with that numerically solved by the EoM method in Fig. <ref>. 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)^Δ_± with Δ = 3/2 ± i L_h. Enhancement of the non-Gaussian spectra due to Δ_+ < 3/2 was investigated in <cit.>.







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 = c k 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  <cit.>, 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 <cit.>. 
As an example, we plot in Fig. <ref> the contribution from the interaction

    ℋ̃_I,3 = a^3 λ h_0 δ h_I^3,

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 (<ref>) as

    B_θ∼λ h_0/H c^-3/2Im[c^-iL_hℐ_- + c^iL_hℐ_+ ],

where

    ℐ_± = ∫dη/η^4 AB_∓^∗(-kη)^3/2± i L_h[G_θ h^+(kη)+G_θ h^-(kη)]^2.


The left panel of Fig. <ref> shows a case with μ_h^2/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 <cit.>). The bispectra in the right panel of Fig. <ref> have μ_h^2/H^2 > 9/4, or S_θ∼ c^1/2sin(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-mixing system the oscillatory signatures must be generated with m_h^2/H^2 > 9/4.






§ SUMMARY AND DISCUSSION


The heavy-lifting mechanism <cit.> is closely related to the cosmological collider research in the sense that one of its goals is to clarify the observability of SM signals in primordial non-Gaussianity. In general, the SM observability is improved with a broken gauge symmetry so that gauge fields (or at least the neutral bosons) can enter the late-time correlation functions via tree-level exchange with inflaton. It is nevertheless also interesting to consider observable SM signals from other possibilities. In any case, what can be 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 velocity θ̇, where both the Higgs mass m_h and the quadratic mixing μ are controlled by the background value θ̇_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_h^2 + 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-mixing 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-mixing regime, we confirmed that the non-Gaussianity in the equilateral limit in general can reach f_NL≳𝒪(1). This value is compatible with current constraints from Cosmic Microwave Background observations <cit.>, and could be tested by the future Large Scale Structure or the 21-cm surveys <cit.>. 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-mixing can still result in an effective speed of sound c_θ^2 < 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_θ^2 <cit.>. Although the specific system (<ref>) considered in this work has a lower limit at c_θ^2 =1/3, it is also straightforward to seek for the realization of c_θ^2 ≪ 1 from extended scenario in the full expression of the effective theory (<ref>).

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 <cit.>. These results might reveal some hints connected to the Naturalness of SM <cit.>, or the issue of Higgs instability <cit.>.




§ ACKNOWLEDGMENTS

We thank Alexander Kusenko, Hayden Lee, Shao-Jiang Wang, Yi Wang, Zhong-Zhi Xianyu, Masahide Yamaguchi, Jun'ichi Yokoyama and Siyi Zhou for many helpful comments and inspiring discussions.
The author is grateful to Xingang Chen for his careful reading of the manuscript and Louis Yang for his contribution at the early stage of this work.
YPW is supported by JSPS International Research Fellows and JSPS KAKENHI Grant-in-Aid for Scientific Research No. 17F17322.









§ THE EQUATION OF MOTION METHOD

In this appendix we describe the formulation of the EoM method used in Section <ref>.
According to the first-principles of the in-in formalism <cit.>, the evolution of field perturbations are governed by the part of Hamiltonian H̃ = H̃[δθ, δ h, p_θ, p_h; t] consisted by quadratic and higher-orders in perturbations, where p_θ and p_h are conjugate momenta for θ and h. In general, one can perform the decomposition

    H̃[δθ, δ h, p_θ, p_h; t] = H_0[δθ, δ h, p_θ, p_h; t] + H_I[δθ, δ h, p_θ, p_h; t],

where H_0 includes only quadratic terms without ϕ-h interactions. H_0 is usually used to define the evolution of free fields δθ_I and δ h_I  in the interaction picture. Since ϕ-h interactions are cast into H_I as perturbations, mode functions of δθ_I and δ h_I must have vanished correlations, namely ⟨δθ^I_ kδ h^I_ q⟩ = 0.

In a strongly coupled system, the quadratic interactions in H_I can also play an important role from the initial time. Therefore, instead of defining mode functions with respect to δθ_I and δ h_I, we impose the definition <cit.>:

    δθ_ k   = 
    δθ^+_ kâ_ k + δθ^+∗_ kâ^†_ -k + δθ^-_ kb̂_ k + δθ^-∗_ kb̂^†_ -k, 
    δ h_ k    =
    δ h^+_ kâ_ k + δ h^+∗_ kâ^†_ -k + δ h^-_ kb̂_ k + δ h^-∗_ kb̂^†_ -k, 
    
    p_θ,  k   = 
    p_θ,  k^+â_ k + p_θ,  k ^+∗â^†_ -k + p_θ,  k^-b̂_ k + p_θ,  k ^-∗b̂^†_ -k, 
    
    
    p_h,  k    =
    p_h,  k^+â_ k + p_h,  k ^+∗â^†_ -k + p_h,  k^-b̂_ k + p_h,  k ^-∗b̂^†_ -k,

which satisfies the canonical commutation relation at initial time t_0 as

    [δθ(x,t_0), p_θ(y,t_0)] = [δ h(x,t_0), p_h(y,t_0)]    = i δ^3(x-y), 
    
    [δθ(x,t_0), δθ(y,t_0)] = [δ h(x,t_0), δ h(y,t_0)]  
       = [δθ(x,t_0), δ h(y,t_0)] = 0, 
    
    [p_θ(x,t_0), p_θ(y,t_0)] = [p_h(x,t_0), p_h(y,t_0)]    =[p_θ(x,t_0), p_h(y,t_0)] =0.

This also implies the mutual operators shall follow

    [â_ k, â^†_ -q]    = [b̂_ k, b̂^†_ -q] 
    = (2π)^3 δ^3(  k + q), 
    
    [â_ k, â_ -q] = [b̂_ k, b̂_ -q] 
       = [â_ k, b̂_ -q] = [â^†_ k, b̂^†_ -q]  
    = [â_ k, b̂^†_ -q] = [b̂_ k, â^†_ -q] = 0.

The two-point correlation function is led by contributions from all vacuum as

    ⟨δθ_ k (z) δθ_ q (z) ⟩ =
    (2π)^3 δ^3( k + q) [δθ^+_  k(z) δθ^+ ∗_  q (z) + δθ^-_  k(z) δθ^-∗_  q(z) ], 
    ⟨δ h_ k (z) δ h_ q (z) ⟩ =
    (2π)^3 δ^3( k + q) [δ h^+_  k(z) δ h^+ ∗_  q (z) + δ h^-_  k(z) δ h^-∗_  q(z) ].
  
It is important to note that the definition (<ref>) and (<ref>) admits the non-vanished correlation

    ⟨δθ_ k (z) δ h_ q (z) ⟩ 
      =  (2π)^3 δ^3( k + q) [δθ^+_  k(z) δ h^+ ∗_  q (z) + δθ^-_  k(z) δ h^-∗_  q(z)].






The non-diagonal mode functions defined in (<ref>) can be taken as modified “free fields” for computing higher-order perturbations in the interaction picture (although they are indeed strongly-coupled). In other words, we are in fact performing the decomposition of H̃ as

    H̃[δθ, δ h, p_θ, p_h; t] = H̃_0[δθ, δ h, p_θ, p_h; t] + H̃_I[δθ, δ h, p_θ, p_h; t],

where H̃_0 includes all quadratic terms that give the coupled EoM (<ref>) and H̃_I contains only cubic or higher-order in perturbations.

Let us derive H̃_0 and H̃_I for the ϕ-h system given by (<ref>) and (<ref>). The conjugate momenta are

    p_θ   = ∂ℒ/∂δθ̇_c = 
         δθ̇_c + μδ h + 2 h_0/R_0^2δθ̇_c δ h + θ̇_c/R_0^2δ h^2, 
    
    p_h    =  ∂ℒ/∂δḣ = δḣ,
 
where ℒ = ℒ_2 +ℒ_3. Replacing δθ̇_c and δḣ by p_θ and p_h while keeping linear in perturbations, we can obtain the modified interaction-picture fields via δθ̇_I = ∂ℋ̃_0/∂ p_θ and δḣ_I = ∂ℋ̃_0/∂ p_h. This results in

    ℋ̃_0 = 1/2[δθ̇_I^2 + 1/a^2(∂_i δθ_I)^2 +δḣ_I^2 + 1/a^2(∂_i δ h_I)^2 + (m_h^2 + μ^2) δ h_I^2 - 2μδθ̇_I δ h_I ],
 
as well as the cubic interactions ℋ̃_I,3 of the form

    ℋ̃_I,3^(1)   = -h_0/R_0^2δθ̇_I^2 δ h_I + h_0/a^2R_0^2 (∂_i δθ_I)^2 δ h_I,
    ℋ̃_I,3^(2)   = (2h_0/R_0^2μ - θ̇_c/R_0^2) δθ̇_I δ h_I^2,
    ℋ̃_I,3^(3)   = (λ h_0 -h_0/R_0^2μ^2 + θ̇_c/R_0^2μ)δ h_I^3.

They are classified by the single, double and triple exchange vertices in Fig. <ref>, respectively. 

To determine the mode functions at the initial time, let us rewrite the equations of motion (<ref>) with respect to the dimensionless time variable z = k η, which read

    ∂^2/∂ z^2δθ -2/z∂/∂ zδθ +δθ   = μ/H(∂/z ∂ zδ h - 3δ h/z^2), 
    ∂^2/∂ z^2δ h-2/z∂/∂ zδ h +(1+ m_h^2/H^2z^2)δ h
       = - μ/H∂/z ∂ zδθ.

In the limit of -z ≫ 1, one can drop terms in proportion to z^-2 and solve the perturbations by the factorization <cit.> of the form

    δθ = H/√(4k^3)A e^-i z,   δ h = H/√(4k^3)B e^-i z,

to obtain A = (-z)^1± i μ/(2H) and B = ± i A. These results give the initial states (<ref>) for the numerical computations.



§ THE LATE-TIME EXPANSION

In this appendix we perform power series expansion of the mode functions to match their coefficients in the late-time limit z→ 0.
A general power series expansion of the solutions for the equations of motion (<ref>) is given by <cit.> as

    δθ_k^±   = H/√(4k^3)∑_n,ν A_nν^± (-z)^n+ν,
    δ h_k^±   = H/√(4k^3)∑_n,ν B_nν^± (-z)^n+ν,

where n is summed from 0 to ∞. The branch value ν can be found from the n = 0 equations in terms of the power series for the vanish of (-z)^ν - 2 as

    (ν A_0ν- μ/HB_0ν) (ν - 3)    = 0, 
    [B_0ν(ν - 3) + μ/HA_0ν]ν + m_h^2/H^2B_0ν   = 0.

There are four possible branch values given by

    ν = 0,   3,  3/2± i L_h.

The n = 1 equations for the vanish of (-z)^ν - 1 are

    [(ν+1) A_1ν- μ/HB_1ν] (ν - 2)    = 0, 
    [B_1ν(ν - 2) + μ/HA_1ν](ν + 1) + m_h^2/H^2B_1ν   = 0.

The n ≥ 2 equations for the vanish of (-z)^ν - 2 + n are

    [(ν+n) A_nν- μ/HB_nν] (ν - 3+n) +A_nν   = 0, 
    [B_nν(ν - 3 + n) + μ/HA_nν](ν + n) + m_h^2/H^2B_nν + B_nν   = 0.

Note that odd-n coefficients can be set to zero to remove redundant solutions. 
In the late-time limit z→ 0, the mode functions (for both ± states of (<ref>) and (<ref>)) are led by

    δθ_k    = H/√(4k^3)(-z)^3/2[A_00(-z)^-3/2+A_0+(-z)^iL_h + A_0-(-z)^-iL_h+⋯],
    δ h_k    = H/√(4k^3)(-z)^3/2[B_0+(-z)^iL_h + B_0-(-z)^-iL_h+ B_20(-z)^1/2⋯],

where B_00 = 0 due to the constraint of (<ref>). One can find the relation between coefficients from (<ref>), where (3/2± i L_h)A_0± =μ B_0±/H.

The two non-integer branch values v = 3/2± i L_h are in fact solved from the part of equation ν A_0ν = μ B_0ν/H in (<ref>). These solutions imply a possible late-time approximation of the theory (<ref>) as

    ℒ_2 ≈1/2[ δθ̇^2_c + δḣ^2 - 1/2a^2(∂_i δ h)^2 - m_h^2δ h^2 + 2μδ h δθ̇_c],

which results in the effective relation δ h = -δθ̇_c/μ. 
Putting this effective relation back into (<ref>), one obtain the Lagrangian only for δ h, which gives the familiar equation of motion as

    ∂^2/∂ z^2δ h -2/z∂/∂ zδ h +( m_h^2 + μ^2/H^2 z^2)δ h =0.

Assuming the usual Bunch-Davies vacuum, the mode function of the equation of motion is well-known:

    δ h = H √(π/4k^3)(-z)^3/2e^i(iL_h+1/2)π/2 H^(1)_i L_h(-z),

where H^(1)_ν is Hankel function of the first kind.
Expanding the Hankel function at z → 0, we find that

    δ h → -i/π e^iπ/4    H √(π/4k^3) (-z)^3/2
    ×   [e^π L_h/2Γ(-i L_h)(-z/2)^i L_h+ e^-π L_h/2Γ(i L_h)(-z/2)^-i L_h].

Therefore it is easy to match the leading coefficients in the late-time expansion up to a phase difference as

    | B_0±| = | e^±π L_h/2Γ(∓ i L_h)/√(π)|.

A comparison of the late-time expansion given by (<ref>) with the numerical result of the EoM approach is shown in Fig. <ref>. 
We remark that although the power series expansion was first derived in Ref. <cit.> for studying the cases with m_h^2+μ^2 < 9H^2/4, the late-time effective theory (<ref>) still holds in cases with m_h^2+μ^2 > 9H^2/4 since the approximation does not rely on the value of the parameter m_h nor μ.




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