# Higgs Inflation via the Metastable Standard Model Potential, Generalised
Renormalisation Frame Prescriptions and Predictions for Primordial
Gravitational Waves

J.McDonald<EMAIL_ADDRESS>Dept. of Physics, Lancaster University,
Lancaster LA1 4YB, UK

###### Abstract

Higgs Inflation via the unmodified metastable Standard Model Higgs Potential
is possible if the effective Planck mass in the Jordan frame increases after
inflation ends. Here we consider the predictions of this model independently
of the dynamics responsible for the Planck mass transition. The classical
predictions are the same as for conventional Higgs Inflation. The quantum
corrections are dependent upon the conformal frame in which the effective
potential is calculated. We generalise beyond the usual Prescription I and II
renormalisation frame choices to include intermediate frames characterised by
a parameter $\alpha$. We find that the model predicts a well-defined
correlation between the values of the scalar spectral index $n_{s}$ and
tensor-to-scalar ratio $r$. For values of $n_{s}$ varying between the
2-$\sigma$ Planck observational limits, we find that $r$ varies between 0.002
and 0.005 as $n_{s}$ increases, compared to the classical prediction of 0.003.
Therefore significantly larger or smaller values of $r$ are possible, which
are correlated with larger or smaller values of $n_{s}$. This can be tested
via the detection of primordial gravitational waves by the next generation of
CMB polarisation experiments.

##  I Introduction

Higgs Inflation bs  seeks to explain inflation using the only known scalar
particle. However, the electroweak vacuum of the Standard Model (SM) is
metastable, due to quantum corrections which cause the Higgs effective
potential to become negative at a scale $\phi=\Lambda\sim 10^{10}{\rm\ GeV}$
frog ; sher ; unst1 ; unst2 ; unst3 . Therefore the Standard Model Higgs
potential cannot serve as a basis for Higgs Inflation in a conventional
cosmological framework.

It is possible to stabilise the SM Higgs potential, for example by adding
scalars with a sufficiently strong coupling to the Higgs boson. However, it is
quite possible that such scalars do not exist. In this case Higgs Inflation
must use the metastable Higgs potential. The only way that this can work is if
the cosmological framework is modified. Specifically, if the effective Planck
mass in the Jordan frame increases after the end of inflation, without
introducing a significant change in the cosmological energy density in the
Einstein frame, then Higgs Inflation via the metastable potential becomes
possible.

In pt1  we presented a specific model to illustrate that such a Planck mass
transition is possible, based on adding a second non-minimally coupled scalar
field. In particular, it was shown that the energy density associated with the
Planck mass transition can be negligible compared to the energy density from
the decay of the inflaton.

In this paper we discuss the general predictions of metastable Higgs
Inflation, which are independent of the dynamics of the Planck mass
transition. In pt1  we showed that the classical predictions of the model are
exactly the same as for conventional Higgs Inflation. However, quantum
corrections can have a strong effect on the predictions of the model. In Higgs
Inflation, the form of the quantum corrections to the potential depends upon
the conformal frame in which the corrections are calculated. The choice of
frame defines the frame in which the renormalisation cut-off is field
independent. Since the cut-off represents the scale of the UV completion, the
correct renormalisation frame can only be determined once the full UV-
completion of the model is known. In the absence of such knowledge, all
renormalisation frames are possible111In rf1  it is shown that different
renormalisation frames correspond to different measures for the path integral
of the effective action. Knowledge of the UV-completion is necessary to
determine the correct measure.

The two commonly considered frame choices are known as Prescription I, where
the effective potential is calculated in the Jordan frame, and Prescription
II, where it is calculated in the Einstein frame. However, in the absence of
the UV-complete theory, there is no reason to exclude other frames which are
intermediate between the initial Jordan and the final Einstein frame. In such
frames there is both a non-minimal coupling of the Higgs field to the Ricci
scalar and non-canonical kinetic terms for the SM fields.

Here we will consider an initial transformation to intermediate
renormalisation frames ("$\alpha$-frames") defined by a conformal factor
$\hat{\Omega}=\Omega^{\alpha}$, where $\Omega$ is the conventional conformal
factor of Higgs Inflation. $\alpha=1$ corresponds to Prescription I and
$\alpha=0$ to Prescription II. We will show that the value of $\alpha$ has a
strong effect on the predictions of the model.

The paper is organised as follows. In Section II we introduce the general
framework for our analysis. In Section III we introduce the $\alpha$-frames
for calculating the effective potential. In Section IV we calculate the
predictions of the model for $n_{s}$ and $r$. In Section V we show how any
renormalisation close to Prescription I can be described by an $\alpha$-frame.
In Section VI we present our conclusions. Some details of the calculations are
discussed in the appendices.

##  II Metastable Higgs Inflation with a Planck mass transition

In this section we explain how an increase in the effective Planck mass at the
end of inflation can allow inflation with values of the SM Higgs field which
are smaller than the instability scale $\Lambda$. We will consider the
classical theory, with the quantum corrections discussed in the following
sections.

The action of the model in the Jordan frame is

$\mbox{$$}S=\int
d^{4}x\sqrt{-g}\left[\left(M_{Pl;\,eff}^{2}+\xi_{\phi}\phi^{2}\right)\frac{R}{2}-\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-V(\phi)+{\cal
L}_{SM}\right]~{}.\vspace{0.1cm}$ (1)

$M_{Pl;\,eff}$ is the Jordan effective Planck mass in the $\phi\rightarrow 0$
limit and ${\cal L}_{SM}$ is the Lagrangian of the SM fields excluding the
Higgs kinetic term and potential.

Metastable Higgs Inflation requires that the following two conditions are
satisfied:

(i) There is an increase in $M_{Pl;\,eff}$ at or after the end of inflation
from $M_{0}\ll M_{Pl}$ to $M_{Pl}$, occurring on a timescale short compared to
$H^{-1}$.

(ii) Any energy released by the Planck mass transition is negligible compared
to the energy from the decay of the inflaton.

If these conditions are satisfied then the model is predictive, independently
of the dynamics responsible for the Planck mass transition.

To analyse inflation and the post-inflation era, we transform the action to
the Einstein frame via a conformal transformation to
$\tilde{g}_{\mu\nu}=\Omega_{0}^{2}g_{\mu\nu}$, where

$\mbox{$$}\Omega_{0}^{2}=\left(\frac{M_{Pl,\,eff}^{2}}{M_{Pl}^{2}}+\frac{\xi_{\phi}\phi^{2}}{M_{Pl}^{2}}\right)~{}.\vspace{0.1cm}$
(2)

The Einstein frame action is then

$\mbox{$$}S=\int
d^{4}x\sqrt{-\tilde{g}}\left[\frac{M_{Pl}^{2}}{2}\tilde{R}-\frac{3M_{Pl}^{2}}{4\Omega_{0}^{4}}{\partial_{\mu}\Omega_{0}^{2}\partial^{\mu}\Omega_{0}^{2}}-\frac{1}{2\Omega_{0}^{2}}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{V(\phi)}{\Omega_{0}^{4}}+\frac{{\cal\tilde{L}}_{sm}}{\Omega_{0}^{4}}\right]~{},\vspace{0.1cm}$
(3)

where $\tilde{{\cal L}}_{SM}$ is the SM Lagrangian in terms of
$\tilde{g}_{\mu\nu}$. Since $M_{Pl,\,eff}/M_{0}$ is constant before and after
the Planck mass transition, the second term can generally be expanded to give

$\mbox{$$}S=\int
d^{4}x\sqrt{-\tilde{g}}\left[\frac{M_{Pl}^{2}}{2}\tilde{R}-\frac{1}{2\Omega_{0}^{2}}\left(1+\frac{6\xi_{\phi}^{2}\phi^{2}}{\Omega_{0}^{2}M_{Pl}^{2}}\right)\partial_{\mu}\phi\partial^{\mu}\phi-\frac{V(\phi)}{\Omega_{0}^{4}}+\frac{{\cal\tilde{L}}_{SM}}{\Omega_{0}^{4}}\right]~{}.\vspace{0.1cm}$
(4)

In the Einstein frame the only effect of the Jordan frame Planck mass
transition is to change the definition of the canonically normalised scalar
fields. In particular, cosmological energy densities are not altered by the
Planck transition, since the Planck mass and the metric $\tilde{g}_{\mu\nu}$
in the Einstein remain unchanged throughout.

During inflation $M_{pl,\,eff}=M_{0}$. Therefore

$\mbox{$$}\Omega_{0}^{2}=\left(\frac{M_{0}^{2}}{M_{Pl}^{2}}+\frac{\xi_{\phi}\phi^{2}}{M_{Pl}^{2}}\right)~{}.\vspace{0.1cm}$
(5)

To express $S$ during inflation in terms of the conventional conformal factor
of Higgs inflation, we define a rescaled Higgs field by

$\mbox{$$}\tilde{\phi}=\frac{M_{Pl}}{M_{0}}\phi~{}\vspace{0.1cm}$ (6)

The conformal factor is then

$\mbox{$$}\Omega_{0}^{2}=\frac{M_{0}^{2}}{M_{Pl}^{2}}\left(1+\frac{\xi_{\phi}\phi^{2}}{M_{0}^{2}}\right)=\frac{M_{0}^{2}}{M_{Pl}^{2}}\left(1+\frac{\xi_{\phi}\tilde{\phi}^{2}}{M_{Pl}^{2}}\right)~{}\vspace{0.1cm}$
(7)

and action becomes

$\mbox{$$}S=\int
d^{4}x\sqrt{-\tilde{g}}\left[\frac{M_{Pl}^{2}}{2}\tilde{R}-\frac{1}{2\Omega^{2}}\left(1+\frac{6\xi_{\phi}^{2}\tilde{\phi}^{2}}{\Omega^{2}M_{Pl}^{2}}\right)\partial_{\mu}\tilde{\phi}\partial^{\mu}\tilde{\phi}-\frac{V(\phi)}{\Omega^{4}}\left(\frac{M_{Pl}}{M_{0}}\right)^{4}+\frac{\tilde{{\cal
L}}_{SM}}{\Omega^{4}}\left(\frac{M_{Pl}}{M_{0}}\right)^{4}\right]~{},\vspace{0.1cm}$
(8)

where

$\mbox{$$}\Omega^{2}=\left(1+\frac{\xi_{\phi}\phi^{2}}{M_{0}^{2}}\right)\equiv\left(1+\frac{\xi_{\phi}\tilde{\phi}^{2}}{M_{Pl}^{2}}\right)~{}.\vspace{0.1cm}$
(9)

The classical Higgs potential during inflation is
$V(\phi)=\lambda_{\phi}\phi^{4}/4$, so in terms of $\tilde{\phi}$ the Einstein
frame action purely for the Higgs boson, $S_{\tilde{\phi}}$, becomes

$\mbox{$$}S_{\tilde{\phi}}=\int
d^{4}x\sqrt{-\tilde{g}}\left[\frac{M_{Pl}^{2}}{2}\tilde{R}-\frac{1}{2\Omega^{2}}\left(1+\frac{6\xi_{\phi}^{2}\tilde{\phi}^{2}}{\Omega^{2}M_{Pl}^{2}}\right)\partial_{\mu}\tilde{\phi}\partial^{\mu}\tilde{\phi}-\frac{V(\tilde{\phi})}{\Omega^{4}}\right]~{}.\vspace{0.1cm}$
(10)

This action is identical in form to that of conventional Higgs inflation but
it is expressed in terms of $\tilde{\phi}$ which is related to the SM Higgs
field by $\phi=(M_{0}/M_{Pl})\tilde{\phi}$. So a large value of $\tilde{\phi}$
can correspond to a small value of the SM Higgs field $\phi$. In particular,
it is possible to have $\phi<\Lambda$ if $M_{0}/M_{Pl}$ is small enough. Thus
inflation can be achieved using the metastable SM Higgs potential.

It also follows that the predictions of the model when expressed in terms of
the number of e-foldings of inflation $N$ are the same as in conventional
Higgs Inflation. Therefore, provided that the value of $N$ at the pivot scale
is the same in conventional Higgs inflation, the predictions of the model will
be the same as conventional Higgs Inflation. This will generally be satisfied,
since by condition (ii) the energy density prior to and after the Planck mass
transition is the same. Therefore the temperature of the Universe will be
unchanged by the Planck mass transition; all that changes in the Einstein
frame is the definition of the canonically normalised fields which form the
radiation background222In particular, the massless gauge bosons are
conformally invariant and therefore do not change when $\Omega$ changes due to
the Planck mass transition.. Thus the model will be indistinguishable from
conventional Higgs inflation in both its inflation and post-inflation
evolution, with the same value of $N$ at the pivot scale and the same
predictions for inflation observables. The only difference is in the value of
the canonically normalised Higgs field during inflation relative to that of
the SM Higgs field, which allows inflation with a small value of $\phi$.

##  III Generalised Renormalisation Frames and Quantum Corrections

In the case of the non-minimally coupled Higgs scalar, the calculation of the
quantum-corrected effective potential depends upon the conformal frame in
which the theory is renormalised. The correct frame requires knowledge of the
UV-completion of the theory, therefore from the point of view of the low-
energy effective theory all frames are possible. There are two renormalisation
frames considered in the literature, known as Prescription I and Prescription
II. Prescription I first transforms the classical theory to the Einstein frame
and then computes the effective potential. Prescription II first calculates
the effective potential in the Jordan frame and then transforms the complete
effective potential to the Einstein frame. These are not physically
equivalent. In particular, in Prescription I the $\phi$-dependence of the
logarithms of the 1-loop Coleman-Weinberg (CW) correction becomes suppressed
at $\phi\;^{>}_{\sim}\;M_{Pl}/\sqrt{\xi_{\phi}}$. As a result, the effect of
quantum corrections is milder in Prescription I than in Prescription II.

Prescriptions I and II are special cases of an infinite number of possible
renormalisation frames, any one of which could be the correct frame for
calculating the effective potential. The Jordan frame used in Prescription II
is the special case in which there is a non-minimal coupling of the Higgs to
gravity but the SM fields have canonical kinetic terms. At the other extreme,
the Einstein frame used in Prescription I is the limit where the Higgs is
minimally coupled to gravity but the SM fields have non-canonical kinetic
terms. However, there are an infinite number of alternatives, in which there
is both a non-minimal coupling of the Higgs to gravity and non-canonical
kinetic terms for the SM fields. In the following we will consider the effects
of a generalised renormalisation frame on the predictions of the model.

###  III.1 $\alpha$-frames and the Calculation of the Effective Potential

We consider a family of renormalisation frames defined by a conformal
transformation from the Jordan frame with conformal factor $\hat{\Omega}$,
where

$\mbox{$$}\hat{\Omega}^{2}=\Omega^{2\alpha}~{},\vspace{0.1cm}$ (11)

where $\Omega$ is given by Eq. (8). The transformation from the Jordan frame
to the final Einstein frame therefore occurs in two stages:

$\mbox{$$}g_{\mu\,\nu}\rightarrow\hat{g}_{\mu\,\nu}=\Omega^{2\alpha}g_{\mu\,\nu}~{},\vspace{0.1cm}$
(12)

followed by

$\mbox{$$}\hat{g}_{\mu\,\nu}\rightarrow\overline{g}_{\mu\,\nu}=\Omega^{2-2\alpha}\hat{g}_{\mu\,\nu}\equiv\Omega^{2}g_{\mu\,\nu}~{},\vspace{0.1cm}$
(13)

with the quantum corrections being calculated in the intermediate frame. This
gives us a simple one parameter family of frames which contains the
Prescription I and Prescription II frames as special cases ($\alpha=1$ and
$\alpha=0$, respectively). We will refer to these intermediate renormalisation
frames as $\alpha$-frames.

Since $\Omega\rightarrow 1$ when $\phi<M_{0}/\sqrt{\xi_{\phi}}$, we can safely
run the SM renormalisation group (RG) equations up to a scale $\mu_{c}$ close
to $M_{0}/\sqrt{\xi_{\phi}}$, since up to this scale the SM fields have
canonical kinetic terms and the metric is the Minkowski metric. We can then
use the 1-loop CW potential in terms of the masses squared of the canonically
normalised SM fields in the intermediate frame to compute the effective
potential at values of $\phi$ where $\Omega>1$, as long as the logarithms in
the potential are not very large. We finally transform the complete effective
potential to the Einstein frame in which inflation is analysed.

As we are calculating the effective potential during inflation, we can set
$M_{Pl,\,eff}=M_{0}$. We start from the Jordan frame action

$\mbox{$$}S=\int
d^{4}x\sqrt{-g}\left[\left(M_{0}^{2}+\xi_{\phi}\phi^{2}\right)\frac{R}{2}-\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-V(\phi)+{\cal
L}_{SM}\right]~{}.\vspace{0.1cm}$ (14)

We then transform to the $\alpha$-frame via the conformal factor

$\mbox{$$}\hat{\Omega}^{2}=\Omega^{2\alpha}=\left(1+\frac{\xi_{\phi}\phi^{2}}{M_{0}^{2}}\right)^{\alpha}~{}.\vspace{0.1cm}$
(15)

The action in the $\alpha$-frame is then

$\mbox{$$}S=\int
d^{4}x\sqrt{-\hat{g}}\left[\frac{M_{0}^{2}}{2}\Omega^{2-2\alpha}\hat{R}-\frac{1}{2\hat{\Omega}^{2\alpha}}\left(1+\frac{6\xi_{\phi}^{2}\phi^{2}}{\hat{\Omega}^{2\alpha}M_{0}^{2}}\right)\partial_{\mu}\phi\partial^{\mu}\phi-\frac{V(\phi)}{\hat{\Omega}^{4\alpha}}+\frac{{\cal\hat{L}}_{SM}}{\hat{\Omega}^{4\alpha}}\right]~{},\vspace{0.1cm}$
(16)

where $\hat{g}_{\mu\nu}=\Omega^{2\alpha}g_{\mu\nu}$. To obtain the canonically
normalised SM fields, with the exception of the Higgs boson, the scalars and
fermions are rescaled according to $\hat{\phi}=\phi/\Omega^{\alpha}$ and
$\hat{\psi}=\psi/\Omega^{3\alpha/2}$. The masses of the canonically normalised
SM fields, $\hat{M}$, are then related to the Jordan frame mass terms,
$M_{J}(\phi)$, by

$\mbox{$$}\hat{M}(\phi)=\frac{M_{J}(\phi)}{\Omega^{\alpha}}~{}.\vspace{0.1cm}$
(17)

The ${\rm\overline{MS}}$ scheme 1-loop CW potential in the $\alpha$-frame is
given by

$\mbox{$$}\Delta\hat{V}_{1-loop}=\sum_{i}\frac{C_{i}\hat{M}_{i}^{4}}{64\pi^{2}}\left[\ln\left(\frac{\hat{M}_{i}^{2}}{\mu^{2}}\right)-K_{i}\right]~{},\vspace{0.1cm}$
(18)

where $(C_{i},K_{i})=(3,3/2)$ for the Goldstone bosons, (6, 5/6) for the $W$
bosons, (3, 5/6) for the $Z$ boson, and (-12, 3/2) for the t-quark. In these
we have summed over the 3 Goldstone bosons, 2 $W$ bosons and all t-quark
colours. We do not include the physical SM Higgs boson as its contribution is
suppressed by the non-minimal coupling wilczek ; corr1 ; corr2 ; barv . The
terms in the 1-loop CW potential in the $\alpha$-frame therefore have the form

$\mbox{$$}\Delta\hat{V}_{1-loop}\sim\frac{C\,M_{J}^{4}}{16\pi^{2}\Omega^{4\alpha}}\left(\ln\left(\frac{M_{J}^{2}(\phi)}{\mu_{c}^{2}\Omega^{2\alpha}}\right)-K\right)~{}.\vspace{0.1cm}$
(19)

We then transform to an Einstein frame in which the Einstein-Hilbert term is
expressed in terms of $M_{0}$, via a conformal factor $\Omega^{2-2\alpha}$.
This gives for the action of the Higgs scalar

$\mbox{$$}S_{\phi}=\int
d^{4}x\sqrt{-\overline{g}}\left[\frac{M_{0}^{2}}{2}\overline{R}-\frac{1}{2\Omega^{2}}\left(1+\frac{6\xi_{\phi}^{2}\phi^{2}}{\Omega^{2}M_{0}^{2}}\right)\partial_{\mu}\phi\partial^{\mu}\phi-\overline{V}_{E}(\phi)\right]~{},\vspace{0.1cm}$
(20)

where
$\overline{g}_{\mu\nu}=\Omega^{2-2\alpha}\hat{g}_{\mu\nu}=\Omega^{2}g_{\mu\nu}$
and

$\mbox{$$}\overline{V}_{E}(\phi)=\frac{V(\phi)}{\Omega^{4}}+\frac{C\;M_{J}^{4}}{16\pi^{2}\Omega^{4}}\left(\ln\left(\frac{M_{J}^{2}(\phi)}{\mu_{c}^{2}\Omega^{2\alpha}}\right)-K\right)~{}.\vspace{0.1cm}$
(21)

Finally, to analyse inflation, we rescale the metric by a constant factor to
$\tilde{g}_{\mu\nu}=(M_{0}^{2}/M_{Pl}^{2})\overline{g}_{\mu\nu}$ and rescale
the Higgs field to $\tilde{\phi}=(M_{Pl}/M_{0})\phi$, which gives the action
for the Higgs scalar $\tilde{\phi}$ in terms of a conventional Planck mass
Einstein-Hilbert term

$\mbox{$$}S_{\tilde{\phi}}=\int
d^{4}x\sqrt{-\tilde{g}}\left[\frac{M_{Pl}^{2}}{2}\tilde{R}-\frac{1}{2\Omega^{2}}\left(1+\frac{6\xi_{\phi}^{2}\tilde{\phi}^{2}}{\Omega^{2}M_{Pl}^{2}}\right)\partial_{\mu}\tilde{\phi}\partial^{\mu}\tilde{\phi}-V_{E}(\tilde{\phi})\right]~{},\vspace{0.1cm}$
(22)

where

$\mbox{$$}V_{E}(\tilde{\phi})=\left(\frac{M_{0}}{M_{Pl}}\right)^{4}\overline{V}_{E}(\phi)~{}.\vspace{0.1cm}$
(23)

Therefore

$\mbox{$$}V_{E}(\tilde{\phi})=\frac{V(\tilde{\phi})}{\Omega^{4}}+\frac{CM_{J}^{4}(\tilde{\phi})}{16\pi^{2}\Omega^{4}}\left(\ln\left(\frac{M_{J}^{2}(\tilde{\phi})}{\tilde{\mu}_{c}^{2}\Omega^{2\alpha}}\right)-K\right)~{},\vspace{0.1cm}$
(24)

where $M_{J}(\tilde{\phi})$ has $\tilde{\phi}$ in place of $\phi$ and
$\tilde{\mu}_{c}=(M_{Pl}/M_{0})\mu_{c}$. All couplings are calculated at the
renormalisation scale $\mu_{c}$. In writing Eq. (24) we have assumed that
$M_{J}(\phi)\propto\phi$, which is true for all SM particles at large $\phi$.

##  IV Predictions for the scalar spectral index and primordial gravitational
waves

###  IV.1 Method

We next discuss the predictions of the metastable Higgs Inflation model for
$n_{s}$ and $r$ as a function of $\alpha$. To do this we first run the 2-loop
SM RG equations, extended to include the non-minimal coupling $\xi_{\phi}$,
from $\mu_{0}=m_{t}$ to $\mu_{c}=0.1M_{0}/\sqrt{\xi_{\phi}}$. Since we are
calculating the effective potential, the renormalisation scale is equal to the
Higgs field $\phi$. We run the RG equations up to $\mu_{c}$, since for
$\phi\leq\mu_{c}$ we have $\Omega\approx 1$ and so the RG equations for the SM
couplings are the same in all frames, whilst the value of $\phi$ during
inflation, $\phi\sim\sqrt{N}M_{0}/\xi_{\phi}$, is small enough compared to
$\mu_{c}$ to not cause large logarithms in the CW potential. The RG equations
for the non-minimally coupled SM are same as the conventional SM RG equations,
up to a modification due to the suppression of the contribution of the
physical Higgs scalar at $\mu\equiv\phi>M_{0}/\xi_{\phi}$ due to kinetic term
mixing of the physical Higgs scalar with the graviton. This causes a
suppression of the physical Higgs propagator by a factor $s(\phi)$, where
wilczek ; corr1 ; corr2 ; barv

$\mbox{$$}s(\phi)=\frac{1+\frac{\xi_{\phi}\phi^{2}}{M_{0}^{2}}}{1+(1+6\xi_{\phi})\frac{\xi_{\phi}\phi^{2}}{M_{0}^{2}}}~{}.\vspace{0.1cm}$
(25)

Therefore we can apply the SM RG equations, with the physical Higgs
contribution to the RG equations suppressed by the factor $s(\phi)$. We use
the 1-loop RG equations for the non-minimally coupled SM including the
$s(\phi)$ factors from corr1 ; corr2  (given in Appendix B) and the 2-loop
corrections from wilczek . The RG evolution is dominated by the 1-loop terms.

Once we have determined the SM couplings and $\xi_{\phi}$ at the
renormalisation scale $\mu_{c}$, we compute the 1-loop CW potential in the
$\alpha$-frame, using $\overline{{\rm MS}}$ scheme SM inputs for the RG
equations at $\mu=m_{t}$. The 1-loop correction to the potential in the
$\alpha$-frame is

$\Delta\hat{V}_{1-loop}=\frac{1}{64\pi^{2}}\left[3\left(\frac{\lambda_{\phi}\phi^{2}}{\Omega^{2\alpha}}\right)^{2}\left(\ln\left(\frac{\lambda_{\phi}\phi^{2}}{\mu^{2}\Omega^{2\alpha}}\right)-\frac{3}{2}\right)-12\left(\frac{y_{t}^{2}\phi^{2}}{2\Omega^{2\alpha}}\right)^{2}\left(\ln\left(\frac{y_{t}^{2}\phi^{2}}{2\mu^{2}\Omega^{2\alpha}}\right)-\frac{3}{2}\right)\right.$
$\mbox{$$}\left.+6\left(\frac{g_{2}^{2}\phi^{2}}{4\Omega^{2\alpha}}\right)^{2}\left(\ln\left(\frac{g_{2}^{2}\phi^{2}}{4\mu^{2}\Omega^{2\alpha}}\right)-\frac{5}{6}\right)+3\left(\frac{(g_{2}^{2}+g_{1}^{2})\phi^{2}}{4\Omega^{2\alpha}}\right)^{2}\left(\ln\left(\frac{(g_{2}^{2}+g_{1}^{2})\phi^{2}}{4\mu^{2}\Omega^{2\alpha}}\right)-\frac{5}{6}\right)\right]~{},\vspace{0.1cm}$
(26)

where the contributions are from the Goldstone bosons, the t-quark and the W
and Z bosons, respectively, with the physical Higgs contribution being
suppressed by the non-minimal coupling. $g_{i}$ are the gauge couplings and
$y_{t}$ is the top quark Yukawa coupling. After the change of frame to the
final Einstein frame, the Einstein potential in terms of $\phi$ becomes

$\mbox{$$}\overline{V}_{E}(\phi)=\frac{V(\phi)}{\Omega^{4}}+\frac{\Delta\hat{V}_{1-loop}}{\Omega^{4-4\alpha}}~{},\vspace{0.1cm}$
(27)

where all couplings are calculated at the renormalisation scale
$\mu_{c}=0.1M_{0}/\sqrt{\xi_{\phi}}$. The potential $V_{E}(\tilde{\phi})$ is
finally obtained by rescaling $\overline{V}_{E}(\phi)$ as in Eq. (23).

For the $\overline{{\rm MS}}$ inputs at the renormalisation scale
$\mu=m_{t}=173.1$ GeV, we use the values listed in martin : $v=246.60109$ GeV,
$\lambda_{h}=0.12604$, $g_{3}=1.16362$, $g_{2}=0.64766$, $g_{1}=0.35854$,
$y_{t}=0.93480$. $\xi_{\phi}(m_{t})$ is a free parameter of the model, which
we choose by adjusting the amplitude of the power spectrum at the pivot scale
to be equal to its observed value, $A_{s}=2.1\times 10^{-9}$. In our analysis
we assume that the pivot scale corresponds to $N=57$ e-foldings, which is a
good estimate for instant reheating after slow-roll inflation (Appendix A).

The analysis of inflation is conducted in the Einstein frame with canonically
normalised inflaton $\sigma$, where

$\mbox{$$}\left(\frac{d\sigma}{d\tilde{\phi}}\right)^{2}=\frac{1}{\Omega^{2}}\left(1+\frac{6\xi_{\phi}^{2}\tilde{\phi}^{2}}{\Omega^{2}M_{Pl}^{2}}\right)~{}.\vspace{0.1cm}$
(28)

The number of e-foldings of slow-roll inflation at a given $\sigma$ is
obtained numerically from

$\mbox{$$}N(\sigma)=-\frac{1}{M_{Pl}^{2}}\int_{\sigma}^{\sigma_{end}}\frac{V_{E}}{V_{E}^{\prime}}d\sigma~{},\vspace{0.1cm}$
(29)

where $\sigma_{end}$ is defined to be the value at which either $|\eta|$ or
$\epsilon$ first become greater than 1, with the slow-roll parameters defined
with respect to $\sigma$. The scalar spectral index and the tensor-to-scalar
ratio are computed in the usual way, $n_{s}=1+2\eta-6\epsilon$ and
$r=16\epsilon$, with
$\eta=M_{Pl}^{2}(V^{{}^{\prime\prime}}_{E}(\sigma)/V_{E}(\sigma))$ and
$\epsilon=(M_{Pl}^{2}/2)(V^{{}^{\prime}}_{E}(\sigma)/V_{E}(\sigma))^{2}$,
where primes denote derivatives with respect to $\sigma$, and the amplitude of
the power spectrum is
$A_{s}=V_{E}^{3}/(12\pi^{2}M_{Pl}^{6}V_{E}^{{}^{\prime}\,2})$. We numerically
compute the values of $n_{s}$ and $r$ by using the potential Eq. (24) in terms
of $\tilde{\phi}$ and relating the potential and its derivatives with respect
to $\tilde{\phi}$ to the corresponding quantities in terms of $\sigma$ via Eq.
(28), with the values of $n_{s}$ and $r$ being calculated at
$\tilde{\phi}(N=57)$.

###  IV.2 Results

For reference, we show in Figure 1 the conventional SM Higgs potential using
our SM inputs. The SM potential has an instability at $\phi=\Lambda=4.2\times
10^{10}{\rm\ GeV}$.

Figure 1: The unmodified SM Higgs potential. Figure 2: Einstein frame
potential for different $\alpha$ in the range 0.6 to 1.3 for
$M_{0}=10^{10}{\rm\ GeV}$ and fixed $\xi_{\phi}(m_{t})$.  Figure 3: SM
couplngs for $\mu$ running from $m_{t}$ to $\mu_{c}$, for metastable Higgs
Inflation with $M_{0}=10^{10}{\rm\ GeV}$.

In Figure 2 we show how the Einstein frame potential varies as a function of
$\alpha$ for the case with $M_{0}=10^{10}{\rm\ GeV}$. In this figure we have
set the value of $\xi_{\phi}(m_{t})$ to be equal in all models, in order to
compare their behaviours. For realistic potentials the value of
$\xi_{\phi}(m_{t})$ should be adjusted for each $\alpha$ to give the correct
value of $A_{s}$. For $\alpha<1$ the potential develops a maximum and then
decreases as $\phi$ increases. Nevertheless, inflation with $\alpha<1$ is
still possible if $\phi(N=57)$ is to the left of the maximum of the potential.
In contrast, for $\alpha\geq 1$ the potential is monotonically increasing with
$\phi$. Therefore there is no need to carefully choose the initial value of
$\phi$ in order to achieve inflation in this case.

In Figure 3 we show the RG evolution of the SM couplings from $\mu=m_{t}$ to
$\mu=\mu_{c}$ for the non-minimally coupled model with $M_{0}=10^{10}{\rm\
GeV}$.

In Table 1 we show our results for $n_{s}$ and $r$ as a function of $\alpha$
for $M_{0}=10^{10}{\rm\ GeV}$. We also show the values of $\phi(N=57)$,
$\xi_{\phi}(m_{t})$, $\xi_{\phi}(\mu_{c})$ and $\lambda_{\phi}(\mu_{c})$. In
Figure 4 we show $n_{s}$ as a function of $\alpha$ for $M_{0}=10^{10}{\rm\
GeV}$. We also show the Planck 2-$\sigma$ bounds on $n_{s}$. In Figure 5 we
show $r$ as a function of $\alpha$, with the bounds on $\alpha$ corresponding
to the 2-$\sigma$ Planck bounds on $n_{s}$ indicated. In Figure 6 we show $r$
as a function of $n_{s}$ for $M_{0}=10^{10}{\rm\ GeV}$. The model predicts a
specific correlation of the values of $n_{s}$ and $r$, with $r$ rapidly
increasing as $n_{s}$ increases.

Figure 4: $n_{s}$ as a function of $\alpha$ for $M_{0}=10^{10}{\rm\ GeV}$. The Planck best-fit value and 2-$\sigma$ upper and lower bounds on $n_{s}$ are indicated. Figure 5: $r$ as a function of $\alpha$ for $M_{0}=10^{10}{\rm\ GeV}$ . The upper and lower bounds on $\alpha$ from the Planck 2-$\sigma$ upper and lower bounds on $n_{s}$ are indicated. Figure 6: $r$ as a function of $n_{s}$ for $M_{0}=10^{10}{\rm\ GeV}$, showing how the model predicts a correlation of $n_{s}$ and $r$. $\alpha$ | $n_{s}$ | $r$ | $\phi(57)$ | $\xi_{\phi}(m_{t})$ | $\xi_{\phi}(\mu_{c})$ | $\lambda_{\phi}(\mu_{c})$
---|---|---|---|---|---|---
$1.7$ | $0.9857$ | $0.0239$ | $3.80$ | $2.43$ | $2.86$ | $0.0181$
$1.5$ | $0.9852$ | $0.0177$ | $2.99$ | $2.66$ | $3.10$ | $0.0183$
$1.4$ | $0.9843$ | $0.0143$ | $2.58$ | $2.86$ | $3.33$ | $0.0185$
$1.3$ | $0.9825$ | $0.0110$ | $2.17$ | $3.16$ | $3.68$ | $0.0187$
$1.2$ | $0.9794$ | $0.0080$ | $1.78$ | $3.64$ | $4.24$ | $0.0189$
$1.1$ | $0.9744$ | $0.0053$ | $1.42$ | $4.35$ | $5.06$ | $0.0193$
$1.0$ | $0.9668$ | $0.0031$ | $1.09$ | $5.51$ | $6.41$ | $0.0198$
$0.9$ | $0.9561$ | $0.0017$ | $0.82$ | $7.48$ | $8.69$ | $0.0204$
$0.8$ | $0.9417$ | $0.00077$ | $0.59$ | $10.85$ | $12.60$ | $0.0211$
$0.7$ | $0.9235$ | $0.00031$ | $0.42$ | $17.10$ | $19.81$ | $0.0221$
$0.6$ | $0.9017$ | $0.00011$ | $0.29$ | $28.70$ | $33.18$ | $0.0231$
$0.5$ | $0.8767$ | $0.000034$ | $0.20$ | $51.80$ | $58.80$ | $0.0244$
$0.4$ | $0.8488$ | $0.0000085$ | $0.13$ | $104.99$ | $120.68$ | $0.0259$
Table 1: The scalar spectral index, $n_{s}$, the tensor-to-scalar ratio, $r$,
the SM Higgs field $\phi$ at $N=57$ (in units of $10^{9}{\rm\ GeV}$), the non-
minimal coupling $\xi_{\phi}$ at $\mu=m_{t}$ (in units of $10^{3}$),
$\xi_{\phi}$ at $\mu_{c}$, and the SM Higgs coupling $\lambda_{\phi}$ at
$\mu_{c}$, as a function of $\alpha$ for $M_{0}=10^{10}{\rm\ GeV}$.

From the 2-$\sigma$ range of $n_{s}$, we find that for $M_{0}=10^{10}{\rm\
GeV}$ the values of $\alpha$ are constrained to be in the range

$\mbox{$$}0.91<\alpha<1.09~{}.\vspace{0.1cm}$ (30)

The corresponding range of $r$ is

$\mbox{$$}0.0018<r<0.0050~{},\vspace{0.1cm}$ (31)

with $r$ increasing as $n_{s}$ increases across the range allowed by Planck.
Thus observation requires that $\alpha$ is close to 1 and so the
renormalisation frame must be close to Prescription I. In particular, in the
case of Prescription I with $\alpha=1$, we find that the values of $n_{s}$ and
$r$ are very close to the values predicted by the classical Higgs inflation,
$n_{s}=0.9667$ and $r=0.0032$. In Appendix C we show that the reason
$\alpha=1$ produces results very close to the classical results for $n_{s}$
and $r$ is not because the quantum corrections to the potential are small, but
because the 1-loop potential has the same functional form to leading order as
the classical potential when $\alpha=1$.

We also find that Prescription II ($\alpha=0$) is completely excluded. We find
no consistent solution which can produce inflation with the observed power
spectrum when $\alpha\;^{<}_{\sim}\;0.1$.

We find that the results for $n_{s}$ and $r$ are not very sensitive to
$M_{0}$. In Table 2 we show the upper and lower bounds on $\alpha$ and $r$ for
$M_{0}$ in the range $10^{4}{\rm\ GeV}$ to $5\times 10^{11}{\rm\ GeV}$. The
range starts to narrow as $M_{0}$ becomes larger than $10^{11}{\rm\ GeV}$. For
$M_{0}\;^{>}{}_{\sim}\;10^{12}{\rm\ GeV}$ there is no solution for inflation
on the metastable part of the potential.

$M_{0}({\rm GeV})$ | $\alpha_{lower}$ | $\alpha_{upper}$ | $r_{lower}$ | $r_{upper}$
---|---|---|---|---
$10^{4}$ | $0.91$ | $1.08$ | $0.0017$ | $0.0047$
$10^{6}$ | $0.90$ | $1.10$ | $0.0017$ | $0.0050$
$10^{10}$ | $0.91$ | $1.09$ | $0.0018$ | $0.0051$
$10^{11}$ | $0.94$ | $1.07$ | $0.0019$ | $0.0055$
$5\times 10^{11}$ | $0.97$ | $1.03$ | $0.0023$ | $0.0049$
Table 2: The upper and lower bounds on $\alpha$ and $r$ corresponding to the
2-$\sigma$ Planck bounds on $n_{s}$, as a function of $M_{0}$.

We also note that inflation can be driven by a SM Higgs field value of the
order of a TeV. For example, in Table 3 we show values of $n_{s}$ and $r$ for
the case $M_{0}=10^{4}{\rm\ GeV}$. In this case $\phi(N=57)$ is in the TeV
range.

We conclude that metastable Higgs Inflation can have a significantly larger or
smaller value of $r$ than the classical Higgs Inflation prediction, and that
these values are correlated with correspondingly larger or smaller values of
$n_{s}$. Renormalisation frames close to $\alpha=1$, corresponding to
Prescription I, are favoured by the observed range of $n_{s}$ from Planck.

$\alpha$ | $n_{s}$ | $r$ | $\phi(57)$ | $\xi_{\phi}(m_{t})$
---|---|---|---|---
$1.2$ | $0.9793$ | $0.0074$ | $3.36$ | $1.1\times 10^{4}$
$1.1$ | $0.9745$ | $0.0051$ | $3.03$ | $1.3\times 10^{4}$
$1.0$ | $0.9669$ | $0.0031$ | $2.71$ | $1.6\times 10^{4}$
$0.9$ | $0.9552$ | $0.0015$ | $2.40$ | $2.1\times 10^{4}$
$0.8$ | $0.9365$ | $0.0006$ | $2.13$ | $3.1\times 10^{4}$
Table 3: Values of $n_{s}$, $r$, $\xi_{\phi}(m_{t})$ and $\phi$ at $N=57$ (in
TeV units) as a function of $\alpha$ for the case $M_{0}=10^{4}{\rm\ GeV}$. In
this case inflation is possible with a TeV scale Higgs expectation value.

The LiteBIRD CMB polarisation satellite is predicted be able to measure $r$ to
an accuracy $\delta r<0.001$ litebird . Therefore if a value of $r$
significantly larger or smaller than the classical prediction $r\approx 0.003$
is observed, and if $n_{s}$ is also found to be significantly larger or
smaller than the Planck best-fit value, then metastable potential Higgs
Inflation would be favoured.

We also note that Early Dark Energy (EDE) models which seek to resolve the
$H_{0}$ tension typically require larger values of $n_{s}$ than $\Lambda$CDM
models ede . Metastable potential Higgs Inflation can be compatible with
models with large $n_{s}$, in which case $r$ could be significantly larger
than the upper limit of 0.005 coming from the $\Lambda$CDM 2-$\sigma$ upper
bound on $n_{s}$.

One issue that we have not yet discussed is perturbative unitarity violation.
This will be the same as in conventional Higgs inflation, with unitarity
violation in the inflationary background occurring at energies of the order of
the Higgs field during inflation. If this is interpreted as a strong coupling
scale rather than a breakdown of the effective theory, then there are no
direct consequences for the Higgs potential in the form of non-renormalisable
terms associated with new unitarity-conserving physics. The only question is
whether the strong coupling may modify the quantum corrections to the
potential. We are therefore assuming that such corrections are small compared
to the perturbative SM corrections.

##  V The significance of $\alpha$-frames with $\alpha$ close to 1

$\alpha$-frames define just one set of possible renormalisation frames. One
may therefore question the significance of these frames and whether there
could be equally valid renormalisation frames that are not described by an
$\alpha$-frame. Here we will argue that if the conformation transformation to
the general renormalisation frame is a purely function of $\Omega$ and tends
to 1 as $\Omega\rightarrow 1$, then $\alpha$-frames with $\alpha$ close to 1
(as preferred by the observed spectral index) will be a good approximation to
any renormalisation frame that is a small deviation from Prescription I. That
the conformal transformation is purely a function of $\Omega$ is a natural
possibility, since $\Omega$ defines the non-minimal coupling in the Jordan
frame. It is also likely that the conformal factor will tend to 1 as $\Omega$
tends to 1, since in this limit the non-minimal coupling in the Jordan frame
becomes negligible.

To show that $\alpha$-frames with $\alpha$ close to 1 are a good approximation
to any renormalisation frame close to Prescription I, we start by writing the
transformation to a general renormalisation frame as

$\mbox{$$}\hat{\Omega}^{2}=\Omega^{2}F(\Omega)~{}\vspace{0.1cm}$ (32)

where $F(\Omega)$ is a function of $\Omega$ which tends to 1 as
$\Omega\rightarrow 1$. $F$ represents the deviation of the frame from
Prescription I; for small deviations, $F$ will be close to 1. Without loss of
generality, we can write $F$ in the form $F(\ln(\Omega))$. Since for
renormalisation frames close to Prescription I we expect that $F\approx 1$,
and since $F\rightarrow 1$ as $\Omega\rightarrow 1$, we expect that
$F(\ln(\Omega))$ can be expanded in the form

$\mbox{$$}F(\ln(\Omega))=1+k_{1}\ln(\Omega)+k_{2}\ln^{2}(\Omega)+...\approx
1+k_{1}\ln(\Omega)~{},\vspace{0.1cm}$ (33)

where the leading-order term is small compared to 1. Finally, defining
$k_{1}=2(\alpha-1)$, where $\alpha$ is close to 1, we obtain

$\mbox{$$}F(\ln(\Omega))\approx
1+(\alpha-1)\ln(\Omega^{2})\approx\exp\left((\alpha-1)\ln(\Omega^{2})\right)=\Omega^{2(\alpha-1)}~{}.\vspace{0.1cm}$
(34)

Therefore

$\mbox{$$}\hat{\Omega}^{2}=\Omega^{2}F(\Omega)\approx\Omega^{2}\Omega^{2(\alpha-1)}\approx\Omega^{2\alpha}~{}.\vspace{0.1cm}$
(35)

Thus the $\alpha$-frames are likely to be good approximations to any
renormalisation frame which is defined by a conformal transformation that is
purely a function of $\Omega$ and which is a small deviation from the
Prescription I frame.

##  VI Conclusions

Inflation via the unmodified metastable SM Higgs potential is an interesting
and potentially significant possibility. It requires a change in conventional
assumptions of post-inflation cosmology, with a large increase in the Jordan
frame effective Planck mass after the end of inflation without significant
release of energy. The dynamics responsible for the Planck mass transition may
come from an extension of the non-minimally coupled effective theory, as
considered in pt1 , or may arise at a deeper level, as part of the UV
completion of the theory. In any case, since the Planck mass transition is a
necessity of inflation via the metastable Higgs potential, we can investigate
the consequences of the model independently of the dynamics of the transition.
We believe that the development of specific models for the Jordan frame Planck
mass transition is well-motivated by the possibility of metastable SM Higgs
Inflation.

We find that quantum corrections can have a significant impact on the
predictions of the model. Since the quantum corrections are purely SM
corrections, they are well-defined, allowing the model to make clear
predictions. The predictions depend on the conformal frame in which the
quantum corrections are calculated. We have generalised beyond the usual
Prescription I and II renormalisation frames, since the correct
renormalisation frame can only be determined by the UV completion of the
theory. For the $\alpha$-frames we consider, we find that the observed range
of $n_{s}$ restricts $\alpha$ to be close to the Prescription I frame, which
corresponds to $\alpha=1$. The model predicts a correlation between the values
of $n_{s}$ and $r$, such that the 2-$\sigma$ Planck bounds on $n_{s}$ gives a
possible range of $r$ between 0.0018 and 0.0050. Thus significantly larger or
smaller values of $r$ than the classical value of 0.0032 are possible, which
are correlated with larger or smaller $n_{s}$ and which can be tested by the
detection of primordial gravitational waves by the next generation of CMB
satellites. It is interesting to consider that the observation of a large
value for $r$ by LiteBIRD, combined with a large value for $n_{s}$, could be
the signature of an alternative post-inflation cosmology.

## Appendix A: $N_{*}$ for Instant Reheating

In the Einstein frame, the energy density at the end of inflation is

$\mbox{$$}\rho_{end}=V_{E}\approx\frac{\lambda_{\phi}M_{Pl}^{4}}{4\xi_{\phi}^{2}}~{}.\vspace{0.1cm}$
(A-1)

Therefore, assuming instant reheating, the reheating temperature is

$\mbox{$$}T_{R}=\left(\frac{30\rho_{end}}{\pi^{2}g(T_{R})}\right)^{1/4}~{}.\vspace{0.1cm}$
(A-2)

The number of e-foldings at the pivot scale, $k_{*}$, is obtained from

$\mbox{$$}\frac{2\pi}{k_{*}}\left(\frac{a_{N}}{a_{0}}\right)\equiv\frac{2\pi}{k_{*}}\left(\frac{g(T_{0})}{g(T_{R})}\right)^{1/3}\left(\frac{T_{0}}{T_{R}}\right)e^{-N}=H^{-1}~{},\vspace{0.1cm}$
(A-3)

where $T_{0}$ is the present CMB temperature, $g(T_{i})$ are the effective
number of relativistic degrees of freedom, and $k_{*}=0.05\,{\rm Mpc}^{-1}$ is
the Planck pivot scale. During inflation

$\mbox{$$}H\approx\left(\frac{\rho_{end}}{3M_{Pl}^{2}}\right)^{1/2}~{}.\vspace{0.1cm}$
(A-4)

Therefore we obtain

$\mbox{$$}N_{*}=\ln\left(\frac{2\pi
T_{0}}{k_{*}}\left(\frac{g(T_{0})}{g(T_{R})}\right)^{1/3}\left(\frac{\lambda_{\phi}}{\xi_{\phi}^{2}}\right)^{1/4}\left(\frac{\pi^{2}g(T_{R})}{1080}\right)^{1/4}\right)~{}\vspace{0.1cm}$
(A-5)

Numerically we find

$\mbox{$$}N_{*}=62.6+\frac{1}{4}\ln\left(\frac{\lambda_{\phi}}{\xi_{\phi}^{2}}\right)~{},\vspace{0.1cm}$
(A-6)

where $\lambda_{\phi}$ and $\xi_{\phi}$ are calculated at $\mu=\mu_{c}$.

For $M_{0}=10^{10}{\rm\ GeV}$, using $T_{0}=2.4\times 10^{-13}{\rm\ GeV}$,
$g(T_{0})=3.91$ and $g(T_{R})=106.75$, we find that $N_{*}$ varies from 55.8
to 57.6 as $\alpha$ increases from 0.4 to 1.7. For $M_{0}=10^{4}{\rm\ GeV}$ we
find that $N_{*}$ varies from 56.9 to 57.4 as $\alpha$ varies from 0.8 to 1.2.
Thus $N_{*}=57$ is a good approximation to the number of e-foldings at the
pivot scale.

## Appendix B: One-loop Renormalisation Group Equations for the Non-Minimally
Coupled Standard Model

The RG equations for the non-minimally coupled Standard Model are333There is a
slight difference between the equations for $\xi_{\phi}$ between corr1  and
corr2 , where corr1  has a factor $s^{2}(t)$ and corr2  has $s(t)$. This has
only a very minor impact on the running. We follow corr2  here. corr1 ; corr2

$\mbox{$$}\left(4\pi\right)^{2}\frac{dg_{1}}{dt}=g_{1}^{3}\frac{\left(81+s(t)\right)}{12}~{}.\vspace{0.1cm}$
(B-1)
$\mbox{$$}\left(4\pi\right)^{2}\frac{dg_{2}}{dt}=-g_{2}^{3}\frac{\left(39-s(t)\right)}{12}~{}.\vspace{0.1cm}$
(B-2)
$\mbox{$$}\left(4\pi\right)^{2}\frac{dg_{3}}{dt}=-7g_{3}^{3}~{}.\vspace{0.1cm}$
(B-3)
$\mbox{$$}\left(4\pi\right)^{2}\frac{dy_{t}}{dt}=y_{t}\left(\left(\frac{23}{6}+\frac{2}{3}s(t)\right)y_{t}^{2}-8g_{3}^{2}-\frac{17}{12}g_{1}^{2}-\frac{9}{4}g_{2}^{2}\right)~{}.\vspace{0.1cm}$
(B-4)
$\mbox{$$}\left(4\pi\right)^{2}\frac{d\lambda_{\phi}}{dt}=\left(\left(6+18s^{2}(t)\right)\lambda_{\phi}^{2}-6y_{t}^{4}+\frac{3}{8}\left(2g_{2}^{4}+\left(g_{1}^{2}+g_{2}^{2}\right)^{2}\right)+12y_{t}^{2}\lambda_{\phi}-3g_{1}^{2}\lambda_{\phi}-9g_{2}^{2}\lambda_{\phi}\right)~{}.\vspace{0.1cm}$
(B-5)
$\mbox{$$}\left(4\pi\right)^{2}\frac{d\xi_{\phi}}{dt}=\left(\xi_{\phi}+\frac{1}{6}\right)\left(6\left(1+s(t)\right)\lambda_{\phi}+6y_{t}^{2}-\frac{3}{2}g_{1}^{2}-\frac{9}{2}g_{2}^{2}\right)~{},\vspace{0.1cm}$
(B-6)

where $t=ln(\mu/\mu_{t})$ and $s(t)$ is given by Eq. (25) with
$\phi=m_{t}e^{t}$.

## Appendix C: Why the $\alpha=1$ predictions for $n_{s}$ and $r$ are almost
the same as the classical predictions.

At first sight it is surprising that Prescription I ($\alpha=1$) produces
essentially identical predictions for $n_{s}$ and $r$ as the classical
potential. Even though there is a suppression of $\phi$ dependence of the
logarithms in the 1-loop CW potential, we would still expect to see some
effect of the 1-loop potential. The reason that the deviation from the
classical predictions for $n_{s}$ and $r$ are strongly suppressed at
$\alpha=1$ is not that the correction is very small, but that it has the same
functional form as the tree level potential to leading order. As a result,
even though there is a significant contribution to the effective potential
from the 1-loop CW potential and the values of $\lambda_{\phi}$ and
$\xi_{\phi}$ are significantly different from the classical values (for
example, for $M_{0}=10^{10}{\rm\ GeV}$ we find $\lambda_{\phi}(\mu_{c})=0.02$
and $\xi_{\phi}(m_{c})=6.4\times 10^{3}$, compared to the classical values
$\lambda_{\phi}=0.126$ and $\xi_{\phi}=1.8\times 10^{4}$), the predictions for
$n_{s}$ and $r$ are hardly modified when expressed as a function of $N$.

To illustrate this, we will consider only the t-quark contribution to the CW
potential, which is the largest contribution during inflation. After
renormalising in the $\alpha$-frame and then transforming to the Einstein
frame (with effective Planck mass $M_{0}$), the potential is given by

$\mbox{$$}\overline{V}_{E}(\phi)=\frac{\lambda_{\phi}\phi^{4}}{4\left(1+\frac{\xi_{\phi}\phi^{2}}{M_{0}^{2}}\right)^{2}}-\frac{3y_{t}^{4}\phi^{4}}{4\left(1+\frac{\xi_{\phi}\phi^{2}}{M_{0}^{2}}\right)^{2}}\left[\ln\left(\frac{y_{t}^{2}\phi^{2}/2}{\mu_{c}^{2}\left(1+\frac{\xi_{\phi}\phi^{2}}{M_{0}^{2}}\right)^{\alpha}}\right)-\frac{3}{2}\right]~{}.\vspace{0.1cm}$
(C-1)

During inflation, we have $\xi_{\phi}\phi^{2}\gg M_{0}^{2}$. Expanding
$\overline{V}_{E}(\phi)$ in $M_{0}^{2}/(\xi_{\phi}\phi^{2})$, we obtain

$\mbox{$$}\overline{V}_{E}(\phi)=A\left(1-\frac{BK}{A}\right)\left(1-\frac{2\left(1-B/A\right)}{\left(1-BK/A\right)}\left(\frac{M_{0}^{2}}{\xi\phi^{2}}\right)+\frac{B(1-\alpha)}{A(1-BK/A)}\ln\left(\frac{M_{0}^{2}}{\xi\phi^{2}}\right)+O\left(\frac{M_{0}^{2}}{\xi\phi^{2}}\right)^{2}\right)~{},\vspace{0.1cm}$
(C-2)

where $A=\lambda_{\phi}^{2}M_{0}^{4}/4\xi_{\phi}^{2}$,
$B=3y_{t}^{4}M_{0}^{4}/4\xi_{\phi}^{2}$ and
$K=\ln[y_{t}^{2}M_{0}^{2}/2\xi\mu_{c}^{2}]-3/2$. Therefore to leading order in
$M_{0}^{2}/\xi_{\phi}\phi^{2}$ the potential has the form

$\mbox{$$}\overline{V}_{E}=V_{0}\left(1-c\frac{M_{0}^{2}}{\xi_{\phi}\phi^{2}}+d(1-\alpha)\ln\left[\frac{M_{0}^{2}}{\xi_{\phi}\phi^{2}}\right]\right)\vspace{0.1cm}$
(C-3)

where $V_{0}$, $c$ and $d$ are constants. The classical potential to leading
order has the form

$\mbox{$$}\overline{V}_{E,\,cl}=V_{0,\,cl}\left(1-2\frac{M_{0}^{2}}{\xi\phi^{2}}\right)~{}.\vspace{0.1cm}$
(C-4)

If $\alpha\neq 1$ then the leading-order effective potential has a different
functional form from the classical potential. But if $\alpha=1$ then the form
of the potential is the same, only the values of the constants $V_{0}$ and $c$
are different. However, the predictions for $n_{s}$ and $r$ are independent of
the constants $V_{0}$ and $c$ when they are expressed as a function of $N$.
Therefore if $\alpha=1$ then the predictions of the classical and quantum
potentials are the same to leading order. This is confirmed numerically, where
there is very little difference between the predictions for $n_{s}$ and $r$.
This is not because the quantum corrections to the potential in Prescription I
are small, but because they do not modify the functional form of the potential
to leading order in $M_{0}^{2}/\xi_{\phi}\phi^{2}$.

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