Critical Higgs inflation in a Viable Motivated Model

An extension of the Standard Model with three right-handed neutrinos and a simple invisible axion model can account for all experimentally confirmed signals of new physics (neutrino oscillations, dark matter and baryon asymmetry) in addition to solving the strong CP problem, stabilizing the electroweak vacuum and satisfying all current observational bounds. We show that this model can also implement critical Higgs inflation, which corresponds to the frontier between stability and metastability of the electroweak vacuum. This leads to a value of the non-minimal coupling between the Higgs and the Ricci scalar that is much lower than the one usually quoted in Higgs inflation away from criticality. Then, an advantage is that the scale of perturbative unitarity breaking on flat spacetime can be very close to the Planck mass, where anyhow new physics is required. The higher dimensional operators are under control in this inflationary setup. The dependence of the cutoff on the Higgs background is also taken into account as appropriate when the Higgs is identified with the inflaton. Furthermore, critical Higgs inflation enjoys a robust inflationary attractor that makes it an appealing setup for the early universe. In the proposed model, unlike in the Standard Model, critical Higgs inflation can be realized without any tension with the observed quantities, such as the top mass and the strong coupling.


Introduction
It has been shown that extending the Standard Model (SM) with three right-handed neutrinos (with a generic flavor structure) and with the extra fields required by a simple invisible axion model can solve the observational problems of the SM (neutrino oscillations, dark matter (DM) and baryon asymmetry) and eliminate a number of unsatisfactory aspects of the SM [1,2,3]. These include the strong CP problem and the metastability 1 of the electroweak (EW) vacuum.
The invisible axion model considered in [1] and later further studied in [2,3] is perhaps the simplest model of this sort (originally proposed by Kim, Shifman, Vainshtein and Zakharov (KSVZ) [4]), in which one introduces the following extra fields • An extra Dirac fermion. This Dirac fermion Q consists of a pair of two-component Weyl fermions q 1 and q 2 in the following representation of the SM gauge group G SM ≡ SU(3) c × SU(2) L × U(1) Y q 1 ∼ (3, 1) 0 , q 2 ∼ (3, 1) 0 .
The q i are charged under a spontaneously broken and anomalous axial U(1) symmetry present in any axion model, the Peccei-Quinn (PQ) symmetry [5].
• An extra complex scalar. This scalar A is charged under the PQ symmetry and neutral under G SM . The PQ charge of A is twice as large as that of the q i , such that a Yukawa coupling between A and Q can be non-zero.
Given that a single quark flavor carrying a non-vanishing PQ charge is present, the model avoids the domain wall problem [6], as discussed in Ref. [7].
The above-mentioned extra fields can render the EW vacuum absolutely stable and, therefore, one can identify the inflaton with the Higgs [1]. Indeed, the condition to have successful Higgs inflation (HI) [8] turns out to be very similar to that of vacuum stability [9,10,11]. However, it was pointed out that the original implementation of HI proposed in [8] leads to the breaking of perturbative unitarity well below the Planck scale when a perturbative expansion around the flat spacetime is performed [12]. This is due to the fact that the HI of [8] requires a large non-minimal coupling ξ H between the Higgs and the Ricci scalar and, consequently, a new scalē M Pl /ξ H is generated [12], whereM Pl is the reduced Planck mass. Although this does not necessarily invalidate the HI of [8] as the SM can enter strong coupling when collisions occur at energiesM Pl /ξ H and on flat spacetime 2 , another possible interpretation of the breaking of perturbative unitarity is the onset of new physics, which could change the inflationary predictions. For this reason Refs. [2,3] proposed to identify the inflaton with |A| or a combination of |A| and the Higgs. Furthermore, in [13] it was shown 3 that, unless some parameters are strongly fine-tuned, a large ξ H can generate higher order operators in the quantum effective action, which can change the inflationary predictions.
However, the large value of ξ H used in [8] can be drastically reduced by taking quantum corrections into account [10,11,15]. The minimal value of ξ H is achieved by living very close to the frontier between metastability and stability of the EW vacuum, implementing the so-called critical Higgs inflation (CHI) [16,17,18]. In this case, the scale of perturbative unitarity breaking can be essentially identified with the scale at which Einstein's theory of gravity breaks down. Of course, at those Planckian energies, some new physics is anyhow required to UV-complete gravity. Moreover, in CHI higher dimensional operators do not significantly change the predictions and the inflationary dynamics enjoys a robust attractor [19]. The latter property 2 Indeed, the spacetime is not flat during inflation and, therefore, it is still possible that during this phase of the early universe perturbation theory is reliable. 3 See also [14] for a subsequent discussion.
is very important: if it were not satisfied one would have to fine-tune the initial conditions of the inflaton, and this would make the whole idea of inflation less attractive. The way inflation takes place in CHI is substantially different from the original HI of [8] as the potential in the critical case features a quasi-inflection point 4 . Reheating can also be successfully implemented in HI [21] (both in the critical and non-critical versions) because the Higgs has sizable couplings to other SM particles; this leads to a high reheating temperature, T RH 10 13 GeV. Given these advantages of CHI we here explore whether this version of HI can be implemented in the well-motivated SM extension that includes the KSVZ axion model and three right-handed neutrinos [1]. Moreover, we investigate whether the cutoff of the theory is always bigger than the typical energies taking into account the background Higgs field, as appropriate when the Higgs is identified with the inflaton and, therefore, has a large field value during inflation. We here focus on the original model of [1] because the action of [1] is simpler than that of [2,3] thanks to a different choice of symmetries (see the next section).
The article is organized as follows. In the next section we give further details of the model, which will give us the opportunity to introduce the notation. In Sec. 3 we discuss the current observational bounds updating the analysis of [1] with new experimental and observational results. The renormalization group equations needed to compute the effective potential are presented in Sec. 4 including those of the nonminimal couplings between the scalars and gravity and 2loop extensions. Sec. 5 is instead devoted to the analysis of the stability of the EW vacuum, which is more involved than in the SM due to the presence of an extra scalar, A. The actual analysis of inflation is only performed in Sec. 6 because the new insight provided by the previous sections is necessary for a detailed inflationary analysis. Finally, in Sec. 7 the cutoff of the theory is investigated taking into account the Higgs background in CHI. Our conclusions are presented in Sec. 8.

The model
Let us now give a detailed description of the model. Here we consider the SM plus three right-handed neutrinos N i and the extra fields of the first viable invisible axion model (the KSVZ model [4]) [1]. The gauge group of the model is the SM group G SM .
The Lagrangian is given by We define in turn the various terms in L above. L SM corresponds to the SM Lagrangian, while L N represents the part of the Lagrangian that depends on the N i : M i j is the Majorana mass matrix of N i and Y i j is the neutrino Yukawa coupling matrix governing the interaction with the SM Higgs doublet H and the standard lepton doublets L i . Notice that the matrix M can be taken symmetric without loss of generality, but generically it has complex elements. However, we assume it to be diagonal and real without loss of generality thanks to the complex Autonne-Takagi factorization. So where the M i (i = 1, 2, 3) are the Majorana masses of the three right-handed neutrinos. L axion gives the additional terms in the Lagrangian due to the KSVZ model: The full classical potential is where and v and f a are real and positive parameters, which can be interpreted as the EW and PQ breaking scales, respectively. The Yukawa coupling y of Q is chosen real without loss of generality. The quartic couplings λ H , λ A and λ HA have to satisfy some bounds to ensure the stability of the EW vacuum, as we will see in Sec. 5. The PQ symmetry acts on q 1 , q 2 and A as follows where α is an arbitrary real parameter. This symmetry forbids a tree level mass term M q q 1 q 2 + h.c.. The SM fields and the right-handed neutrinos are not charged under U(1) PQ . The model has the accidental symmetry Finally, L gravity are the terms in the Lagrangian that include the pure gravitational part and the possible nonminimal couplings between gravity and the other fields: whereM Pl 2.4 × 10 18 GeV is the reduced Planck mass, R is the Ricci scalar, ξ H and ξ A are the non minimal couplings of the Higgs and the new scalar to gravity and Λ is the cosmological constant, which is introduced to account for dark energy.
The EW symmetry breaking is triggered by the vacuum expectation value (VEV) v 174 GeV of the neutral component H 0 of H (while all the other components of H have a vanishing VEV). After that the neutrinos acquire a Dirac mass matrix m D = vY, which can be parameterized in terms of column vectors m Di (i = 1, 2, 3): Integrating out the heavy neutrinos N i , one then obtains the following light neutrino Majorana mass matrix By means of a unitary (Autonne-Takagi) redefinition of the left-handed neutrinos we can diagonalize m ν obtaining the mass eigenvalues m 1 , m 2 and m 3 (the left-handed neutrino Majorana masses). Calling U ν the unitary matrix that implements such transformation (a.k.a. the Pontecorvo- , with s i j ≡ sin(θ i j ), c i j ≡ cos(θ i j ); θ i j are the neutrino mixing angles and P 12 is a diagonal matrix that contains two extra phases: Even in the most general case of three right-handed neutrinos, it is possible to express Y in terms of low-energy observables, the heavy masses M 1 , M 2 and M 3 and extra parameters [22]: where and R is a generic complex orthogonal matrix (that contains the extra parameters). This is useful for us because the observational constraints are not directly on Y, but they are rather on the low-energy quantities m i , U ν and on M i (see section 3). One can show that the simplest and realistic case of two right-handed neutrinos [23] below M Pl can be recovered by setting m 1 = 0 and where z is a complex parameter and ξ = ±1.
The PQ symmetry is spontaneously broken by f a ≡ A , leading to the following squared mass of Q: Moreover, A contains a (classically) massless particle, the axion, and a massive particle with squared mass Given the lower bound on f a that will be reviewed in Sec. 3, are very small and will be neglected in the following.
When the scalars are set to their VEV, L gravity reduces to the standard pure Einstein-Hilbert action (with a cosmological constant), which is why we added the extra terms proportional to v 2 and f 2 a in Eq. (7).

Neutrino masses and oscillations
As far as the neutrino masses m i (i = 1, 2, 3) are concerned, we have several data from oscillation and nonoscillation experiments. For example, Refs. [24,25] presented some of the most recent determinations of ∆m 2 21 , ∆m 2 3l (where ∆m 2 i j ≡ m 2 i − m 2 j and ∆m 2 3l ≡ ∆m 2 31 for normal ordering and ∆m 2 3l ≡ −∆m 2 32 for inverted ordering), the mixing angles θ i j and δ.
Here we take the central values reported in [25] for normal ordering. Indeed, normal ordering is currently favored over inverted ordering. Currently no significant constraints are known for β 1 and β 2 ; thus we will set these parameters to zero for simplicity from now on.

Baryon asymmetry
Successful leptogenesis occurs if neutrinos are lighter than 0.15 eV and the lightest right-handed neutrino mass M l fulfills [26] M l 1.7 × 10 7 GeV.
In order to be conservative we have reported the weakest bound, but depending on assumptions one can have stronger bounds 5 .

Constraints on the axion sector
In order not to overproduce DM through the misalignment mechanism [27] and to elude axion detection one obtains respectively an upper and lower bound (see e.g. [28] and [29], respectively) on the order of magnitude of the scale of PQ symmetry breaking f a : 10 8 GeV f a 10 12 GeV.
The upper bound is obtained by requiring that the axion field takes a value of order f a at early times, which is what we expect but is not necessarily the case; also the precise value of the lower bound is model dependent. Therefore, (14) should not be interpreted as sharp bounds, but it certainly gives a plausible range of f a . The window in (14) also allows us to neglect PQ symmetry breaking effects due to gravity: nonperturbative gravitational effects can violate PQ invariance, but lead to a sizable correction only for f a 10 16 GeV (see Ref. [30] for a recent review).
In addition to contributing to DM, the axion also necessarily leads to dark radiation because it is also thermally produced [31,32,33]. This population of hot axions contributes to the effective number of relativistic species, but the size of this contribution is currently well within the observational bounds although, interestingly enough, within the reach of future observations in some models [33,34] .
In the case of the KSVZ-based model considered here a more precise version of the lower bound in (14) is f a 4 × 10 8 GeV [29]. In any case bounds on f a can only constrain the ratio M A / √ λ A as it is clear from (12). When M q v and M A v the EW constraints are fulfilled. The size of y is also very mildly constrained: we have a lower bound from the lack of observation, which is not more stringent than M q 1 TeV (indeed one has to take into account that the extra quark Q is not charged under the electroweak part of the SM gauge group). Moreover, in this model the bounds on f a , which allows the axion to account for the whole DM, is [2] 2 × 10 10 GeV f a 0.9 × 10 11 GeV.

Constraints on SM parameters
Finally, in order to have "initial conditions" for the renormalization group equations (RGEs) 6 , we also have to fix the values of the relevant SM couplings at the EW scale, say at the top mass M t 172.5 GeV [35,36]. We take the values computed in [37], which expresses these quantities in terms of M t , the Higgs mass M h 125.09 GeV [38], the strong fine-structure constant renormalized at the Z mass, α s (M Z ) 0.1184 [39] and M W 80.379 GeV [40] (see the quoted literature for the uncertainties on these quantities).

Inflation
In 2018 Planck released new results for inflationary observables [41], which are relevant for our purposes.
For example, for the scalar spectral index n s and the tensor-to-scalar ratio r one has now while for the curvature power spectrum P R (q) (at horizon exit 7 q = aH) These constraints are particularly important for us as the main goal of the article is to study whether CHI is viable.

Renormalization group equations
Given that we want to obtain the predictions of this model at energies much above the EW scale, we need the complete set of RGEs. We adopt the MS renormalization scheme to define the renormalized couplings. Moreover, for a generic coupling g we write the RGEs as where d/dτ ≡μ 2 d/dμ 2 andμ is the MS renormalization energy scale. The β-functions β g can also be expanded in loops: where β (n) g /(4π) 2n is the n-loop contribution. We start from energies much above M A , M q and M i j . In this case, the 1-loop RGEs are (see [42,43,44,45,1] for previous determinations of some terms in some of these RGEs) 7 We use a standard notation: a is the cosmological scale factor, H ≡ȧ/a and a dot denotes the derivative with respect to (cosmic) time, t.
where g 3 , g 2 and g 1 = √ 5/3g Y are the gauge couplings of SU(3) c , SU(2) L and U(1) Y , respectively, and y t is the top Yukawa coupling. In addition to the β-functions presented in [1] we have added here the RGEs for the non-minimal couplings ξ H and ξ A , which, as we will see, play some role in inflation.
Since the SM couplings evolve in the full range from the EW to the Planck scale it is appropriate to use for them the 2-loop RGEs 8 , which we present explicitly here for the first time including the new physics contribution: In the absence of gravity the RGEs for a generic quantum field theory were computed up to 2-loop order in [46].  Tr The RGEs in the MS scheme are gauge invariant as proved in [37]. Next, we consider what happens in crossing the threshold M A : as discussed in [47,44,1] one has to take into account a scalar threshold effect: in the low energy effective theory below M A one has the effective Higgs quartic coupling In practice one should do the following: below M A the RGEs are the ones given above with β λ HA and β λ A removed, λ A and λ HA set to zero and λ H replaced by λ. Above M A one should include λ A , λ HA , β λ A and β λ HA and find λ H using the full RGEs and the boundary condition in (20) atμ = M A . As far as the new fermions are concerned, following [48] we adopt the approximation in which the new Yukawa couplings run only above the corresponding mass thresholds; this can be technically implemented by substituting Y i j → Y i j θ(μ − M j ) and y → yθ(μ − M q ) on the right-hand side of the RGEs.

Stability analysis
Since we use the 1-loop RGEs of the non-SM parameters, we approximate the effective potential V eff of the model with its RG-improved tree-level potential: we substitute to the bare couplings in the classical potential the corresponding running parameters.
Let us find the conditions that ensure the absolute stability of the vacuum H 0 = v and A = f a . We offer a more detailed treatment than the one in [44] although we will agree with their conclusions. For v f a , which is amply fulfilled thanks to (14), the conditions are

I. λ H > 0 and λ
The origin of Condition I is obvious. Notice, however, that once λ H > 0 and λ 2 HA < 4λ H λ A are fulfilled then λ A > 0 is fulfilled too. The origin of Condition II is provided in Appendix A.
An important remark is in order now. Suppose that, taking into account the dependence of the couplings onμ, one finds that Conditions I and II are violated at some energyμ = µ * . Can we really conclude that there is an instability? The answer to this question is "yes" only if µ * is close enough to the field configurations at which the potential is lower than its value at the EW vacuum (henceforth the instability configurations); indeed, if this is not the case this instability would be outside the range of validity of the RG-improved treelevel potential. For this reason it is interesting to find the instability configurations. This is done in Appendix B.
In Fig. 1 an example of the running of the SM parameters close to criticality (and compatible with absolute stability) is provided (the example is specified in the caption). In that plot the threshold effect in (20) has been taken into account, but the jump of λ H cannot be appreciated in the plot because a λ HA λ A has been chosen there. In Fig. 2 the corresponding running of the couplings in the axion sector is shown. No pathologies (such as Landau poles) appear below the Planck scale and Condition I for stability is sat- isfied. In Fig. 3 the corresponding instability configurations for Condition II (the configuration space defined in (B.3) and (B.4)) is shown: this space opens up only atμ |H ± | meaning that we do not encounter any instability of the EW vacuum (see the discussion in the previous paragraph). This is not in contradiction with Figs. 1 and 2 of [1] because the region marked as "λ 2 HA (μ) < 4λ H (μ)λ A (μ)" there corresponds to having that condition satisfied for allμ up to the Planck scale; as explained in the paragraph above this is only a sufficient condition for absolute stability (not a necessary one). Note that for those parameter values neutrino data are reproduced, the axion accounts for the full DM abundance (see (15)) and leptogenesis can provide the observed matterantimatter asymmetry.
In the next section it will be shown that a successful inflation can also be achieved with the Higgs close to criticality.

Higgs inflation and criticality
The possibility that we study in this article is that inflation is triggered by the Higgs and in particular when one is very close to criticality. In HI the field A is set to its VEV, |A| 2 = f 2 a . In this case, the term in the action that depends on the metric and the Higgs field only (the scalar-tensor part) is where V H = λ H |H| 4 is the classical Higgs potential and we have ignored the EW scale v, which is completely negligible compared to the inflationary scales (that will be discussed in this section and the next one). We assume a sizable nonminimal coupling, ξ H > 1, because this is what inflation leads to as we will see. We start by using the classical approximation, later we will also include quantum corrections. The ξ H |H| 2 R term can be removed through a conformal transformation (a.k.a. Weyl transformation): which, as we will see below, redefines the kinetic term and the potential of the Higgs field. The original frame, where the Lagrangian has the form in Eq. (21), is known as the Jordan frame, while the one where gravity is canonically normalized (after the transformation above) is called the Einstein frame. In the unitary gauge, where the only scalar field is φ ≡ 2|H| 2 , we have (after having performed the conformal transformation) and The non-canonical Higgs kinetic term can be made canon-ical through the field redefinition φ = φ(φ ) given by with the conventional condition φ(φ = 0) = 0. Note that φ(φ ) is invertible because Eq. (25) tells us dφ /dφ > 0. Thus, one can extract the function φ(φ ) by inverting the function φ (φ) defined above. We will refer to φ as the canonically normalized Higgs field. Note that φ feels a potential Let us now recall how inflation emerges in this context in the slow-roll approximation. From Eqs. (25) and (26) (27) are guaranteed to be small. Therefore, the region in field configurations where φ M Pl (or equivalently [8] φ M Pl / √ ξ H ) corresponds to inflation.
The parameter ξ H can be fixed by requiring that the predicted curvature power spectrum equals the observed value, Eq. (17), for a field value φ = φ b corresponding to an appropriate number of e-folds [21]: where φ e is the field value at the end of inflation, computed by requiring In the slow-roll approximation (used here) such constraint can be imposed by using the standard formula For N ∼ 60, this procedure leads to a very large ξ H at the classical level. However, the need of a very large ξ H can be avoided when quantum corrections are included [16,17,18], as we will see below.
We can also compute the scalar spectral index n s and the tensor-to-scalar ratio r: in the slow-roll approximation the formulae are r = 16 H and n s = 1 − 6 H + 2η H . These parameters are important as they are constrained by observations (as we have seen in Sec. 3).
We now discuss the quantum corrections to the Higgs potential. We want to include both the large-ξ H inflationary scenario of [8] and the CHI proposed in [16,17,18]. The latter case permits a drastic decrease of the value of ξ H with respect to the classical result. This indicates that we cannot rely on large-ξ H approximations to analyze this case. Thus, we do not use such approximations here. However, we do assume in the following that ξ H > 1 as this is present both in the original formulation of HI and in CHI.
Note that Eqs. (21), (22), (24) and (25) also hold if ξ H is field-dependent, as dictated by quantum corrections [49]. A second step we should do now is the computation of the effective potential. In defining the quantum theory there are well-known ambiguities [9,10,17,50,51]. We follow here Ref. [17] and choose to compute the loop corrections to the effective potential -a.k.a. Coleman-Weinberg potential -in the Einstein frame (after having performed the conformal transformation (22)). This choice is convenient because we can then use the standard formulae to compute the primordial quantum fluctuations, which assume minimal couplings to gravity. The effective potential is also RG-improved by using the RGEs.
Such prescription to compute the quantum effects is known as Prescription I and it leads to the following renormalization group scalē where κ is an order one factor. Furthermore, we will use the RG-improved potential neglecting the loop corrections: this means that we will take as effective potential the one in Eq. (26) with the constants λ H and ξ H replaced by the corresponding running parameters. There are good reasons to use this approximation. Indeed, taking into account the loop corrections to the potential would only be more precise if supplemented by the loop corrections to the kinetic term of the inflaton; such corrections have not been included in HI and are expected to be comparable to the loop corrections to the potential for moderate values of ξ H , unlike what happens for large ξ H [10]: the large value of ξ H allowed [10] to show that the corrections to the kinetic term are negligible, but the smaller value of ξ H of critical HI does not permit to trust this approximation anymore. Another reason to employ the RG-improved potential is its gauge independence, which is not shared by the Coleman-Weinberg effective potential. Therefore, the use of the RG-improved potential allows us to obtain a more transparent physical interpretation.
Given that we use this approximation we can also compute the RGEs in the Jordan frame. Let us see why.
In an exact computation we should also compute the RGEs in the Einstein frame (just like the Coleman-Weinberg potential) but the approximation in which the RGEs are computed in the Jordan frame is a good approximation because the error Figure 4: RG-improved potential and its approximation (logapproximation) based on the expansions in Eq. (32) for the parameters set in Fig. 1 and for κ 1.8.
one is doing is of order of the Weyl anomaly, which is suppressed by 1/(4π) 2 [52] and we are not including anyway the Coleman-Weinberg corrections to the potential which are of the same order.
Moreover, in computing the inflationary potential a further approximation can be done. One can approximate the running couplings λ H and ξ H by expanding them around the minimum of λ H (henceforth λ 0 ), which typically occurs around the Planck scale, as follows where µ 0 is the value ofμ where this minimum occurs, and ξ 0 ≡ ξ H (µ 0 ). The parameters b λ and b ξ are related to the β-functions as follows and can be computed once the RGEs are solved. Then, one can approximate the potential by inserting these expansions inside (26). Such approximation (which we will call the "log approximation") works very well (see Fig. 4) and we will therefore use it from now on. Now, Eqs. (27), (28) and (30) are still valid as long as one is in the slow-roll regime, but one should now interpret U as the effective potential, not just as the classical potential.
The inflationary observables predicted by the model analyzed here are in agreement with the most recent observational bounds [41] (see for instance Eqs (16) and (17)): e.g.
for the parameters used in Fig. 1 we have n s 0.965, r = 0.0472, P R = 2.12 × 10 −9 (34) and a number of e-folds equal to about 55. Note that for the values of the parameters used in Fig. 1 the quantities satisfy {κ H > 0, κ A < 0} (see also Fig. 2) and, therefore [3], in that setup the inflation along the |A|-direction and the multifield inflation (in which both |A| and |H| are active) can be neglected. Furthermore, in Ref. [13] it was shown that CHI features a robust inflationary attractor in the SM. The same conclusion holds here because the results of the analysis in Ref. [13] were based only on the qualitative features of the inflationary potential, which are the same in the model studied here. Moreover, for {κ H > 0, κ A < 0}, which has been realized in this paper, the other directions in the scalar field space are not inflationary attractors [3].

Validity of the effective theory
We have already commented that CHI leads to an increasing of the cutoff of the effective theory on flat spacetime compared to the ordinary HI case. Let us generalize the discussion now to include the non-trivial background fields characteristics of inflation. Ref. [53] showed that the cutoff of the theory can be studied by dividing the range of the background Higgs fieldφ into three regimes 9 . We use the results of Ref. [53] in the following and further extend them.
•φ M Pl /ξ H . In this small field regime the cutoff of the theory is identified as the coefficients of the dimensionn operators δφ n (for n > 4), where δφ is the fluctuation of φ around its background valueφ . The cutoff obtained in this way reads where n acquires even values. This is the flat spacetime result. Given that in CHI λ H is very small the smallest value of the cutoff is reached in the limit n → ∞. However, for moderate values of n the cutoff Λ (n) is much bigger in CHI than in the ordinary HI case thanks to the smallness of λ H . In Fig. 5   into account the running of the couplings. One finds that the inflationary scale is always much smaller than the cutoff.
and again in CHI the smallest value of the cutoff is obtained in the limit n → ∞ and for moderate n > 4 the CHI features a much larger cutoff than the large-ξ H HI. In this field range the tower of cutoffs is shown in Fig. 6, taking into account the running of the couplings, and compared again with the inflationary scale. That plot shows that the inflationary scale is always much smaller than the cutoff.
• Finally, in the inflationary regime,φ M Pl / √ ξ H , the cutoff is simply Λ ∼M Pl , which coincides with the scale at which sizable quantum gravity effects are expected to emerge. Fig. 4 shows that the inflationary scale is much smaller than the cutoff in this last regime too.

Conclusions
In this article it was found that CHI can be implemented in a well-motivated extension of the SM, which explains with few extra degrees of freedom neutrino oscillations (through three right-handed neutrinos), DM (identified with the axion), baryon asymmetry (through leptogenesis) and the strong CP problem (thanks to the PQ symmetry). Furthermore, all the above-mentioned features can be there together with a stable EW vacuum. The fact that CHI can be implemented in this context is non-trivial: indeed, to establish this result one needs to carefully take into account the stability conditions in the presence of RG-improved parameters. CHI inflation has the advantage of 1) being free from a large non-minimal coupling ξ H and from a consequent scale of violation of perturbative unitarity much below the Planck scale, 2) enjoying a robust inflationary attractor and 3) allowing for an efficient reheating thanks to the sizable couplings between the Higgs and other SM particles.
Moreover, in the proposed model, unlike in the SM [17,54], CHI can be realized without any tension with the observed quantities, such as the top mass or the strong finestructure constant.
This condition can be derived as follows.
Assume λ H > 0 and λ A > 0, which is anyhow required by the stability, and then define to write the potential in a more compact form: Next use polar coordinates, u = r cos θ, w = r sin θ, which give The stability condition is that V should not become negative for any field value, thus When sin θ cos θ acquires its minimum (sin θ cos θ = −1/2), inequality (A.5) becomes λ HA < +2 √ λ H λ A (A. 6) and ensures that V is non-negative for positive λ HA . When sin θ cos θ acquires instead its maximum (sin θ cos θ = +1/2) this inequality becomes and ensures that V is non-negative for negative λ HA . Putting together (A.7) and (A.6) gives

Instability configurations
In this appendix we find the field configurations at which the potential is lower than its value at the EW vacuum if Condition II (see Sec. 5) is violated. We called these configurations the "instability configurations".
Let us focus on the case λ HA > 0, since this case is the one that imposes the weaker constraint from stability [44]. From the discussion provided in Appendix A the most unstable direction is θ = θ 0 ≡ −π/4 (at which sin θ cos θ = −1/2) and the most unstable value of r (henceforth r 0 ) is the maximal one compatible with the ranges in (A.3). To compute r 0 note that the most unstable configuration has w = w 0 ≡ − √ λ A f 2 a (we assume λ H > 0 and λ A > 0 as required by Condition I), which corresponds to From r 0 sin θ 0 = w 0 one then determines r 0 = √ 2λ A f 2 a . Therefore, the most unstable configuration has u = u 0 ≡ r 0 cos θ 0 = √ λ A f 2 a , or, in terms of |H 0 |, defined by √ λ H (|H 0 | 2 − v 2 ) = u 0 , Moreover, the full instability configurations with A = 0 are given by which confirms the result in Ref. [44].