Minimal Higgs inflation

We consider a possibility that the Higgs field in the Standard Model (SM) serves as an inflaton when its value is around the Planck scale. We assume that the SM is valid up to an ultraviolet cutoff scale \Lambda, which is slightly below the Planck scale, and that the Higgs potential becomes almost flat above \Lambda. Contrary to the ordinary Higgs inflation scenario, we do not assume the huge non-minimal coupling, of O(10^4), of the Higgs field to the Ricci scalar. We find that \Lambda must be less than 5*10^{17}GeV in order to explain the observed fluctuation of the cosmic microwave background, no matter how we extrapolate the Higgs potential above \Lambda. The scale 10^{17}GeV coincides with the perturbative string scale, which suggests that the SM is directly connected with the string theory. For this to be true, the top quark mass is restricted to around 171GeV, with which \Lambda can exceed 10^{17}GeV. As a concrete example of the potential above \Lambda, we propose a simple log type potential. The predictions of this specific model for the e-foldings N_*=50--60 are consistent with the current observation, namely, the scalar spectral index is n_s=0.977--0.983 and the tensor to scalar ratio 0


Introduction
It is more and more plausible that the particle discovered at the CERN Large Hadron Collider (LHC) [1,2] around 126 GeV is the Standard Model (SM) Higgs boson. Its couplings to the W and Z gauge bosons, to the top and bottom quarks, and to the tau lepton are all consistent to those in the SM within one standard deviation even though their values vary two orders of magnitude, see e.g. Ref. [3]. No hint of new physics beyond the SM has been found so far at the LHC up to 1 TeV. It is important to examine up to what scale the SM can be a valid effective description of nature.
The determination of the Higgs mass finally fixes all the parameters in the SM. We can now obtain the bare parameters at Λ. These parameters are important. If a ultraviolet (UV) theory such as string theory fails to fit them, it is killed.
The parameters in the SM are dimensionless except for the Higgs mass (or equivalently its vacuum expectation value (VEV)). The dimensionless bare coupling constants can be approximated by the running ones at Λ, see e.g. Appendix of Ref. [4]. Once the low energy inputs are given, we can evaluate the running couplings through the renormalization group equations (RGEs) of the SM. The detailed RGE study of the SM tells us that both the Higgs quartic coupling and its beta function become tiny at the same scale ∼ 10 17 GeV for the input value of the Higgs mass around 126 GeV; see e.g. Refs. [5,6,7,8,9,4,10,11] for latest analyses.
After fixing all the dimensionless bare couplings, the last remaining parameter in the SM is the bare Higgs mass. The quadratically divergent bare Higgs mass is found to be suppressed too when the UV cutoff is Λ 10 17 GeV [4]; see also Refs. [10,12,13,14], and also Refs. [15,16,17,18,19]. The absence of the bare mass at Λ, along with the vanishing quartic coupling and its beta function, implies that the Higgs potential is approximately flat there and that its height is suppressed compared to (the fourth power of) the cutoff scale.
Following the evidence of the top quark with mass 174±10 +13 −12 GeV [20] in 1994, Froggatt and Nielsen have predicted [21] that the top and Higgs masses are 173 ± 5 GeV and 135±9 GeV, respectively. This prediction is based on the multiple point principle (MPP) that the SM Higgs potential must have another minimum at the Planck scale and that its height is (order-of-magnitude-wise [22,23]) degenerate to the SM one. This assumption is equivalent to the vanishing Higgs quartic coupling and its beta function at the Planck scale. The success of this prediction indicates that at least the top-Higgs sector of the SM remains unaltered up to a very high UV cutoff scale Λ. 1 In the MPP, it is assumed that there are two vacua that are separated by a potential barrier, as illustrated in the lowermost (green) solid and dashed lines in Fig. 2. In this paper, we consider the case where there is no potential barrier, as illustrated in the middle (blue) and uppermost (red) lines in Fig. 2 so that the Higgs potential can be used for an inflation. In order to have an inflation consistent with the current observational data, we assume that the low energy SM of potential above Λ. In the last section, we summarize our results.

Constraints on inflation models
We briefly review and summarize our notation on the cosmological constraints from the CMB data observed at the Planck experiment, basically following Ref. [37]. The curvature and tensor power spectra are expanded around a pivot scale k * as We take the slow-roll approximation hereafter. The slow-roll parameters at a given position ϕ of the inflaton potential V are defined as The number of e-folding before the end of inflation t end from a time t * becomes where we have taken V ϕ > 0 in the last step. The end of inflation is defined by the field value ϕ end below which the slow roll condition is violated: For most reasonable inflation models, the scale that we are observing from the CMB data corresponds to the e-folding in the range [37] 50 < N * < 60.
In the following, we evaluate the slow-roll parameters at ϕ * that satisfies Eq. (5).
(We will also consider the range 40 < N * < 50 in Section 4.2 for comparison.) The cosmological parameters are given by 4 where the quantities are evaluated at the field value ϕ * . At the pivot scale k * = 0.05 Mpc −1 , the scalar amplitude A s and the spectral index n s are constrained by the Planck+WMAP data as [37] A s = 2.196 +0.051 −0.060 × 10 −9 , assuming dn s /d ln k = d 2 n s /d ln k 2 = r = 0. If we include the tensor-to-scalar ratio r as an extra parameter, the Planck+WMAP+highdata [37] at the pivot scale k * = 0.002 Mpc −1 give the 1σ range for n s and the 95% CL limit on r as n s = 0.9600 ± 0.0071, r < 0.11.
On the other hand, if we include dn s /d ln k as an extra parameter, we obtain the constraint at the pivot scale k * = 0.05 Mpc −1 [37] n s = 0.9561 ± 0.0080, dn s d ln k = −0.0134 ± 0.0090.
One may vary both r and dn s /d ln k to fit the Planck+WMAP data at the pivot scale k * = 0.05 Mpc −1 , and obtains [37] n s = 0.9583 ± 0.0081, where the constraint on r is given at 95% CL. 5 We note that the upper bound on r gives that of the inflaton energy scale [37]

SM Higgs potential
The SM Higgs potential much above the electroweak scale but below the cutoff scale Λ is governed by the RGE running of the Higgs quartic coupling λ, which highly depends on the top Yukawa coupling y t . Therefore we first review how y t is determined; then we show the numerical results of the RGEs; finally we present the resultant Higgs potential around Λ. 5 In terms of the slow-roll parameters, these conditions become where the constraint on V is given at 95% CL.

Coupling constants at the electroweak scale
The most precise determination of the top quark mass is given by a combination of the Tevatron data for the invariant mass of the top quark decay products [52]: The problem of the Tevatron determination (14) is that the invariant mass, which is reconstructed from the color singlet final states, cannot be the pole mass of the colored top quark [8]. Instead, the authors of Ref. [8] proposed to get the top mass by fitting the tt + X inclusive cross section, and obtained the pole mass: 6 The numerical value of the MS Yukawa coupling at the top mass scale can be read off from Ref. [7] as where we have employed a combined Higgs mass M H = 125.6 ± 0.4 GeV. The electroweak gauge couplings at the Z mass scale are [59] g Y (M Z ) = 0.357418 (35), g 2 (M Z ) = 0.65184 (18).
The MS strong and quartic couplings at the top mass scale are [7] g s (M t ) = 1.1644 + 0.0031

Numerical Results of SM RGEs
We use the two-loop RGEs in the SM which are summarized in Ref. [4]. We show our result of running MS couplings in Fig. 1. The gauge couplings g Y , g 2 , g 3 are drawn by thick lines. The thickness of the curves for y t , λ and β λ comes from the 6 As clarified in Refs. [53,54], currently there are two ways to define the MS running top mass for a given MS running Yukawa yt(µ). The MS mass used in QCD [55,56,8], which we call m QCD t (µ), can be approximately written as m QCD t (µ) yt(µ)V / √ 2 with V = 246. 22 GeV, up to electroweak corrections less than 1%. In Refs. [57,58], MS mass is defined as mt(µ) := yt(µ)v(µ)/ √ 2, where v(µ) is given by the relation −m 2 (µ) = λ(µ) v 2 (µ), with m 2 (µ) being the running mass parameter in the tree potential in the MS scheme: There are ∼ 7% difference between m QCD t (Mt) and mt(Mt) [58], which is mainly due to the tadpole contribution from the top quark. Though the bound on the pole mass (15) has been derived from that on m QCD t (Mt) = 163.3 ± 2.7 GeV [8], it is consistent to use Eq. (15) in obtaining the Yukawa coupling (16) since the pole mass Mt should be the same in both schemes. B /I 1 , y t (µ), λ(µ), and 10β λ (µ); see text for more details. The intervals for the gauge couplings g Y , g 2 , and g 3 are too small to be seen. Right: Enlarged view around the horizontal axis of Left. Darker bands are the 95% confidence intervals under the (theoretically unjustified) assumption that the Tevatron mass (14) can be identified with the top pole mass M t . Similarly, we plot the bare Higgs mass-squared m 2 B , divided by the quadratically divergent integral I 1 = Λ 2 /16π 2 , as a function of Λ [4,53]. Note that the bare mass m 2 B is not the running mass. We see that the Higgs quartic coupling λ has a minimum around 10 17 GeV. This is due to the fact that the beta function of λ receives less negative contribution from the top loop since y t becomes smaller at high scales.
We can fit the parameters at the reduced Planck scale 7 M P := 1/ √ 8πG = 2.4 × 10 18 GeV as where the dependence of β λ on α s (M Z ) is of O(10 −7 ) and is not shown.  (23). Beyond the UV cutoff Λ, which we have taken in this figure to be 4.5 × 10 17 GeV as an illustration, we assume that the potential becomes flat as depicted by the dot-dashed lines.

Higgs inflation?
We have seen in Fig. 1 that both the Higgs self coupling λ and its beta function β λ become very small at high scales µ 10 17 GeV. This fact suggests that the SM Higgs field could be identified as an inflaton. In the following, we show that the SM Higgs potential becomes flat around 10 17 GeV if we tune the top quark mass. However, we will see that it is difficult to reconcile this potential with the cosmological observation [60,23]. For a field value V = 246.22 GeV ϕ < Λ, the Higgs potential becomes, with RGE improvement, Around the scale 10 17 GeV, this potential strongly depends on the top quark mass, which we show by the solid and dashed lines in Fig. 2. The dot-dashed lines are irrelevant to the argument of this subsection. If we fine tune the top mass, we can have a saddle point in the Higgs potential: V ϕ (ϕ) = V ϕϕ (ϕ) = 0, as indicated by the middle (blue) line in Fig. 2. When we slightly lower the top mass, the saddle point disappears and the potential becomes monotonically increasing, as the upper (red) line. On the contrary, when we slightly raise the top mass, there appears another minimum at a high scale, as the lower (green) line [61,62].
at the saddle point ϕ c = 4.2 × 10 17 GeV. One might think of using this saddle point for a Higgs inflation, but it is impossible due to the following reasons: With Eq. (8), this height of potential necessitates V ∼ 10 −3 . However, the point of N * 50 becomes too close to the saddle point and gives V 10 −3 . In order to avoid this problem, one might try lowering the top mass slightly to reproduce the value of V ∼ 10 −3 at the inflection point V ϕϕ = 0. This still does not work because we cannot have enough e-foldings, N * 50, at the inflection point.
As the third trial, one might choose V freely at the inflection point so that one can have enough e-folding in passing the inflection point. In this case, one tries to reproduce V ∼ 10 −3 at the higher point with N * ∼ 50. However, η V at this point turns out to be too large to satisfy the slow-roll condition.
We present more detailed discussion in Appendix A.
Note that the precise value of M t to give the saddle point in Fig Therefore the value (24) should be taken as an indication of the order of magnitude.

Minimal Higgs inflation
We pursue the possibility that the Higgs potential above Λ becomes sufficiently flat to realize a viable inflation, as the dot-dashed lines in Fig. 2. The inflaton potential is bounded from above as in Eq. (8). In order to avoid the graceful exit problem, the Higgs potential must be monotonically increasing in all the range below and above Λ. Therefore, even if we allow an arbitrary modification above Λ, we still can get a bound: Λ < 5 × 10 17 GeV. As a concrete example of the modification above Λ, we propose a log type potential and study its cosmological implications.

Constraint on top mass from minimal Higgs inflation
We have seen that the scale 10 17 GeV gives the vanishing beta function β λ . This scale is close to the string scale in the conventional perturbative superstring theory. Above the string scale, a conventional local field theory is altered. We have shown that the bare Higgs mass becomes very small around this scale [4]. This fact strongly suggests that the Higgs boson is a zero mass state of string theory. If it is the case, after integrating out all the massive stringy states, we get the effective potential, which is meaningful for field values beyond the string scale. The resultant potential beyond the string scale would be greatly modified from that in the SM. 8 As we have discussed in Introduction, it is plausible that the effective potential for the field value beyond Λ becomes almost flat. This opens up a possibility that this flat potential can be used for the inflation. Firstly in this subsection, we consider a necessary condition for the SM potential at ϕ < Λ to allow such a modification in the region ϕ > Λ.
To avoid the graceful exit problem [39,40,41], the Higgs potential must be a monotonically increasing function of ϕ in all the range below and above Λ. 9 Therefore, we have for all the scales below the cutoff: ϕ < Λ, and the upper bound (13) leads to We show excluded regions on the Λ-M t plane from the above two constraints in the left panel of Fig. 3. The left (red) region is excluded by the condition (27) within 95% CL and the right (blue) region is forbidden by Eq. (26). The right panel is an enlarged view. The dashed and dot-dashed lines correspond to the to be integrated out. 9 If there is a potential barrier, then the phase transition becomes first order. This is problematic: The false vacuum decays only through the tunneling. The bubbles of true vacuum expand with speed of light in the exponentially expanding medium of the false vacuum. They can hardly collide each other. See also Ref. [9] for a possible false vacuum inflation assuming a lowered Planck scale, which we do not employ in this paper. exclusion limits at the 95% CL, r < 10 −2 and 10 −3 , that are expected from the future experiments EPIC [63] and COrE [64], respectively.
We see that the top quark mass needs to be M t 171 GeV if we want to have the cutoff scale to be at the string scale Λ ∼ 10 17 GeV. If the top quark mass turns out to be heavier, say M t 173 GeV, then this minimal scenario breaks down. However, it is possible that there exists an extra gauge-singlet scalar X that couples to the SM Higgs boson e.g. as where ρ and κ are coupling constants. Then X contributes to the running of λ positively, and the vacuum stability condition becomes milder. Such a scalar naturally arises in the Higgs portal dark matter scenario; see e.g. Refs. [65,66,67,68,69]. 10

Log type potential
So far we have not specified anything about the potential shape above Λ. In the following, let us examine the log-type potential: We note that the Coleman-Weinberg potential with an explicit momentum cutoff Λ leads to a log type potential; see Appendix B. 11 The potential (29) leads to the slow roll parameters The end point of the inflation is determined from Eq. (4) for a given constant C: where W is the Lambert function defined by z = W (z)e W (z) . Equivalently, the constant C is fixed as a function of ϕ end : 10 No matter X is included or not, our scenario is not altered by the right-handed neutrinos if their Dirac Yukawa couplings are 0.1. 11 In terms of the parameters given there, The e-folding becomes To summarize: For a given ϕ end , we fix the constant C by Eq. (32). Then we can obtain the slow roll parameters from Eq. (30) at any field value ϕ. The field value ϕ * corresponding to a relevant e-folding N * is determined from Eq. (33).
This way the slow roll parameters (30) at a given N * is completely fixed. Note that they are independent of V 1 , the overall normalization of the potential. In Fig. 4, we plot the slow roll parameters V , η V , ξ 2 V , and 3 V at the field value ϕ * as functions of ϕ end /M P . The dotted, dashed and solid lines correspond to the values N * = 40, 50 and 60, respectively. 12 Once the slow roll parameters are given, the spectral indices, their running, and their running of running are completely fixed. In Fig. 5, we plot them as functions of ϕ end . The solid (dotted) lines represent the values for N * from 50 to 60 (40 to 49). The values for N * below 50 are just for reference; e.g. the late time thermal inflation [70] can reduce the corresponding N * to the observed value 12 We have chosen the highest end point of the horizontal axis of Fig. 4 to be ϕ end /MP = e W (1/ √ 2) = 1.57 at which C = 0 and V(Λ) = 0. In this case, we cannot connect the potential V to VSM, even if the latter were zero at Λ.
where the order of the inequality corresponds to that of 0 < ϕ end < 1.57M P ; the range of numbers denoted by the en-dash "-" corresponds to the range N * = 50-60. So far we have not considered V 1 , since it is sufficient to fix C to determine the slow-roll parameters. Now we determine V 1 by the magnitude of the density perturbation (6): Then the potential and its derivatives at an e-folding N * are completely fixed. In Fig. 6, we plot ϕ * , V end , and V * as functions of ϕ end . We indicate N * the same as in Fig. 5. For a given ϕ end we have obtained the constants V 1 and C. If we demand that the high scale potential (58), fixed by these values, is directly connected with the SM potential at Λ, then we can fix Λ by In Fig. 7 we present left and right hand sides of Eq. (36) for N * = 50 and M t = 170.5 GeV to illustrate the situation. We see that the low energy SM potential can be directly connected to the high energy one when and only when ϕ end 0.5M P . This critical value of ϕ end is not sensitive to the choice of N * and M t . Then we plot r vs n s , r vs dns d ln k , dns d ln k vs n s , and d 2 ns d ln k 2 vs dns d ln k for 0 < ϕ end < 0.5M P in Fig. 8, which can be compared with Figs. 1-5 in Ref. [37]. When we where the order of the inequality corresponds to that of 0 < ϕ end < 0.5M P ; see Eq. (34). From Fig. 5, we see that we need rather small N * ∼ 40 for 0 < ϕ end < 0.5M P in order to account for the observed value of n s in Eq. (10). We note that a large field inflation with scale 10 17 GeV tends to require relatively high value of N * , barring the late-time thermal inflation mentioned above. When we approximate that the Higgs field decays into the SM modes instantaneously after the inflation [42], the reheating temperature is given by 13 where g * 106.75 is the effective number of degrees of freedom in the SM; the resultant reheating temperature is T reh 4 × 10 15 GeV for ϕ end = 0.5M P . Then the e-folding number, corresponding to the pivot scale k * = 0.002 Mpc −1 and the Hubble parameter H 0 = 67.3 km s −1 Mpc −1 , is given by [37]: Using the value of V * and V end as depicted in Fig. 6 , we get the e-folding number as a function of ϕ end , which is plotted in Fig. 9. This indicates that the large field inflation requires N * 60.
On the other hand, as we have seen around Eq. (36) and Eq. (37), we need a small value of N * ∼ 40, if we want to directly connect the log potential with the SM one. To do so, ϕ end needs relatively small, ϕ end < 0.5M P , and N * should be around 40 in order to obtain a realistic value of n s .
There are two ways to solve this apparent inconsistency. One is to note that indeed we do not have to connect the log potential to V SM so strictly, as it is unclear what happens around ϕ ∼ Λ. 14 All we need is that the end point value of the inflaton potential is larger than the SM potential at its UV cutoff scale: V end > V SM (Λ) for ϕ end > Λ. Then we can take larger ϕ end to obtain smaller value of n s .
The other is adding a small correction to the log potential: For example, if we choose V 1 = 4 × 10 −11 M 4 P , C = 5, c 1 = 0.1, c 2 = −0.01, and c n = 0 for n ≥ 3, then we get ϕ end = 0.48 M P , ϕ * = 4.7M P , N * = 64, r = 0.008 and n s = 0.978. The resultant potential is illustrated in Fig. 10.  (40) as a function of ϕ is drawn in the loglog plot as the solid line in the right, which is the modification of the line for ϕ end = 0.5M P in Fig. 7; ϕ * and ϕ end are also indicated. The SM lines are the same as in Fig. 2 but the SM cutoff is taken to be slightly smaller Λ 10 17 GeV.

Summary and discussions
The Higgs potential in the Standard Model (SM) can have a saddle point around 10 17 GeV, and its height is suppressed because the Higgs quartic coupling becomes small. These facts suggest that the SM Higgs field may serve as an inflaton, without assuming the very large coupling to the Ricci scalar of order 10 4 , which is necessary in the ordinary Higgs inflation scenario. In this paper, we have pursued the possibility that the Higgs potential becomes almost flat above the UV cutoff Λ. Since a first order phase transition at the end of the inflation leads to the graceful exit problem, the Higgs potential must be monotonically increasing in all the range below and above Λ. From this condition, we get an upper bound on Λ to be of the order of 10 17 GeV.
In this paper, we have pursued the bottom-up approach from the latest Higgs data, without assuming any other structure than the SM below Λ. We have shown in Fig. 3 that even if we allow arbitrary potential above Λ, still the restriction is rather severe to achieve this minimal Higgs inflation scenario.
It is curious that the upper bound on Λ from the minimal Higgs inflation coincides with the scale where the quartic coupling λ and its beta function (and possibly the bare Higgs mass) vanish. This coincident scale ∼ 10 17 GeV is close to the string scale in the conventional perturbative superstring scenario. 15 This fact may suggest that the physics of the SM, string theory, and the universe are all directly connected.
There are possibilities that realizes the flatness from the gauge symmetry as in the gauge-Higgs unification scenario [31,32,33,34,35]. See also Refs. [29,30] for other stringy attempts. Furthermore, Ref. [72,73] derives the log potential of the type (58). It would be interesting to construct a realistic string model that breaks the supersymmetry at string scale 16 and realizes the flat Higgs potential above Λ consistent to the cosmological observations. One can even go beyond the symmetry argument of the ordinary quantum field theory/string theory to realize the flat potential, such as with the MPP [21,22,23], the classical conformality around Λ [77,24,25,26,27], the multiverse [78], the anthropic principle [79], etc.
Let us expand the potential around the point ϕ 0 (∼ 10 17 GeV) of vanishing beta function β λ = 0: where b i are given by with λ i representing [4] the Yukawa coupling squared, y 2 t etc., the gauge coupling squared, g 2 Y , g 2 2 , g 2 3 , and the quartic coupling λ. Note that each β i = dλ i /d ln µ has a loop suppression factor 1/16π 2 . We see that the SM Higgs potential can always have a saddle point by choosing a particular value of λ 0 by adjusting the top quark mass. For example, when we approximate b 3 = 0, the saddle point is realized at ϕ = e −1/4 ϕ 0 by choosing λ 0 = b 2 /16. In the SM, b 2 takes values b 2 (1.9-4.6)×10 −5 for the 95% confidence interval from the top quark mass (15), see Fig. 11.
In order to show the difficulty of the inflection point inflation with the SM Higgs potential, it suffices to expand the potential around its inflection point ϕ c that satisfies V c := V ϕϕ (ϕ c ) = 0: where V c := V ϕϕϕ (ϕ c ) and we tune the top quark mass in order to make V c := V ϕ (ϕ c ) very small. The e-folding from ϕ c + δϕ * to ϕ c − δϕ end becomes In the following, we discuss in detail the three cases that are sketched in the text: • First possibility is to put V c = 0 and earn the e-folding near the saddle point. The e-folding (44) for δϕ end > 0 and δϕ * < 0 becomes Close to the saddle point, we have Putting Eq. (45) into Eq. (46), the slow roll parameter reads and hence the scalar perturbation becomes From Eq. (41), we can compute the values in the SM: which results in A s 1, far larger than the allowed value (8).
This does not work. • Finally one may take very tiny V c at ϕ c to earn enough e-folding, in order to realize the inflection point inflation scenario [80,81,82,83,84,85,86,87,88,89], while obtaining necessary amount of V at a point above the inflection point: ϕ c + δϕ * with δϕ * > 0. In passing through the inflection point ϕ c from ϕ * (> ϕ c ) to ϕ end (< ϕ c ), we earn the e-folding and hence we can have as large an e-folding as we want by tuning V c small. More concretely, we need to get N * ∼ 50 with Eq. (49). However, to keep the slow roll parameter sufficiently small, we need to be close to the inflection point: Within this range, we get -When δϕ 2 * 2V c /V c , we have the same expression as Eq. (46). From Eq. (55), we get Putting the SM values, this results in A s 10 −5 which is far larger than the observation (8).
-On the contrary when δϕ 2 * 2V c /V c , we get V = M 2 P (V c ) 2 /2V 2 c and hence where we have put Eq. (52). We see that this is too large again.

B Motivating log type toy model
We present a motivation for the toy model with the log type potential at ϕ > Λ: In the bare perturbation theory, see e.g. Ref. [4], the one-loop effective potential for the Higgs field ϕ is given by where the integration is performed over Euclidean four momentum. 17 Since we are interested in the behavior of the potential at the field value ϕ very much larger than the electroweak scale ϕ V = 246.22 GeV, we work in the symmetric phase by setting the Higgs VEV to be zero: V = 0. The number of degrees of freedom, N i , and the coupling to the Higgs, c i , are summarized in Table 1 for species i that have non-negligible coupling to the Higgs. h and χ are the physical and Nambu-Goldstone modes of the Higgs around the field value ϕ, respectively. 18 Assuming existence of an underlying gauge invariant regularization, such as string theory, let us cutoff the integral by |p| < Λ: The bare Higgs mass m 2 B is tuned to yield the desired value of the low energy mass-squared parameter, to be zero: m 2 R = 0, see e.g. Ref. [4]. 19 Then we get 17 In Eq. (59), we have tuned the cosmological constant so that we get V eff → 0 as ϕ → 0. 18 Though we show our results in the Landau gauge, we can explicitly show that in the R ξ gauge, depending on the external field ϕ, the one-loop result (59) is independent of the gauge parameter ξ if we expand it by ciϕ 2 Λ 2 . 19 Recall that we are working in the symmetric phase.
To summarize, we generally have We see that the bare mass drops out of the effective potential, as it should be. The form (63) corresponds to the one loop correction to λ B . As a side remark, we comment that the condition m 2 B = 0 at this one-loop order, namely i N i c i = 0, is the celebrated Veltman condition [90,4].
Rigorously speaking, the effective potential (59) or (63) can be trusted only when the field dependent mass in the loop integral is sufficiently small: c i ϕ 2 Λ 2 . Nonetheless, let us venture to assume that the expression (59) or (63) is still valid even for the field values much larger than the cutoff Λ. 20 When we can take c j ϕ 2 Λ 2 for some species j, say j = W, Z, t, and c i ϕ 2 Λ 2 for others i, then We note that the bare mass re-appears in this limit. As one can see from Fig. 1, both the bare coupling λ B , approximated by the MS one λ(Λ), and the bare mass m 2 B are very close to zero for Λ 10 17 GeV. If the UV theory somehow chooses the bare mass to be zero, as is proposed by Veltman, and also λ B = 0, then the effective potential becomes at c j ϕ 2 Λ 2 , where V 0 is an integration constant. We see that the potential at very high scales takes the form of Eq. (58).
We can read off the coefficient V 1 in Eq. (58) from Eq. (65): when we put j = W, Z, t. We see that we need to add extra scalar fields coupling to the Higgs to make V 1 positive at high scales, as in the Higgs portal dark matter scenario. As an illustration, we assume hereafter that the Higgs potential is not modified up to the cutoff Λ and is connected to Eq. (58) directly at Λ with arbitrary constants V 0 and V 1 , though generally the RGE itself can be changed by the inclusion of the extra scalar fields.

C Limiting behavior
We show the limiting behavior of the high energy potential Eq. (58).
• In the limit V 1 V 0 , we get V 0 → V 0 , and hence For an observed value of A s , we get Or if we remove V 1 = V 0 ϕ 2 * /2N * M 2 P , Or else, we can rewrite V 1 = ϕ end M P 2 V 0 to yield • In the opposite limit V 0 V 1 , we get We define the end point of the inflation by the condition: max { V , |η V |} = V = 1, to get: Then the e-folding number becomes which gives ϕ * = 8.54M P (7.96M P ) for N * = 60 (50), and hence n s → 0.994 (0.993), r → 6.2 × 10 −3 (7.2 × 10 −3 ).