# Vector-like Quark Stabilised Higgs Inflation: Implications for Particle
Phenomenology, Primordial Gravitational Waves and the Hubble Tension

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

###### Abstract

The Standard Model (SM) Higgs potential is very likely to be metastable, in
which case Higgs Inflation is likely to require an extension of the SM to
sufficiently stabilise the Higgs potential. Here we consider stabilisation by
adding $n_{Q}\leq 3$ Vector-Like Quarks (VLQs) of mass $m_{Q}$. We consider
isosinglet $T$ vector quarks transforming under the SM gauge group as $({\bf
3},{\bf 1},2/3)$ and $B$ vector quarks transforming as $({\bf 3},{\bf
1},-1/3)$. Requiring stability of the finite-temperature effective potential
after instant reheating, and assuming that the $t$-quark mass $m_{t}$ equals
the mean value of its experimental range, we find that the upper bounds on
$m_{Q}$ for $T$ quarks are 5.8 TeV (for $n_{Q}=2$) and 55 TeV (for $n_{Q}=3$).
The corresponding absolute stability upper bounds are 4.4 TeV and 29 TeV. For
$n_{Q}=1$ there is stability only for $m_{t}$ at its -2-$\sigma$ value, in
which case $m_{Q}\leq 1.6{\rm\ TeV}$ for one $T$ quark. The upper bounds are
generally smaller for $B$ vector quarks, with finite temperature stability for
$m_{Q}$ less than 2.8 TeV (for $n_{Q}=2$), 18 TeV (for $n_{Q}=3$) and 1.0 TeV
(for $n_{Q}=1$). The upper bounds on $m_{Q}$ are sensitive to the $t$-quark
mass, becoming smaller as $m_{t}$ increases. The inflation predictions depend
upon the conformal frame in which the model is renormalised. For
renormalisation in the Einstein frame (Prescription I) the predictions are
almost indistinguishable from the classical values: $n_{s}=0.966$ and
$r=3.3\times 10^{-3}$. In this case the stability upper bounds on $m_{Q}$
apply. Renormalisation in the Jordan frame (Prescription II) predicts larger
values of $n_{s}$ and $r$, with $n_{s}$ generally in the range 0.980 to 0.990
and $r$ of the order of 0.01. The predicted range of $n_{s}$ is consistent
with the CMB range obtained in Hubble tension solutions which modify the sound
horizon at decoupling, whilst the predicted values of $r$ will be easily
observable by forthcoming CMB experiments. The observational upper bound on
$r$ generally imposes a stronger constraint on $m_{Q}$ in Prescription II than
the requirement of stability, with the $T$ quark upper bound equal to 2.4 TeV
for $n_{Q}=2$ and 13 TeV for $n_{Q}=3$, assuming $m_{t}$ equals its mean
value. $n_{Q}=1$ is generally ruled out by the large value of $r$. The $m_{Q}$
upper bounds rapidly decrease with decreasing $r$. We conclude that VLQ-
stabilised Higgs Inflation with Prescription II renormalisation favours 1-10
TeV vector-like quarks that will be accessible to future colliders, and
predicts a tensor-to-scalar ratio that will be observable in forthcoming CMB
experiments and values of $n_{s}$ that favour an early-time solution to the
Hubble tension.

##  I Introduction

The electroweak vacuum is very likely111In [7] it was found that, using 2023
PDG inputs, there is only a small region of the 4-$\sigma$ ellipse in $m_{t}$
and $\alpha_{s}(M_{z})$ for which stability is possible, and using correlated
CMS inputs, which have larger errors but take into account correlations
between the measurements, there is only a very small region of the 2-$\sigma$
ellipse for which stability is possible. In [6] it was also found that only a
small region of the 2-$\sigma$ ellipse is still compatible with stability. A
factor of 2 improvement in the errors will exclude stability to 5-$\sigma$
[7]. to be metastable due to quantum corrections [1, 2, 3, 4, 5, 6, 7]. In the
case of the SM this does not present any cosmological or phenomenological
problem, as the Universe can naturally evolve into the electroweak vacuum once
finite-temperature evolution is taken into account [8, 9]. However, in the
case of Higgs Inflation [10, 11], it is essential that the Higgs potential is
sufficiently stable to support inflation, which requires that the instability
scale is greater than the Higgs field during inflation, of the order of
$M_{Pl}/\sqrt{\xi}$, where $\xi$ is the non-minimal coupling222In this case it
is possible to have Higgs Inflation in the unmodified SM if the SM instability
scale is larger than the Higgs field during inflation, which requires a low
value for $m_{t}$. In [12] it is proposed that this is possible if the PDG
world average pole value of $m_{t}$ is considered rather than the smaller
error direct measurement value used here.. To achieve this, it is likely that
additional particles must be added to the SM. Here we consider the addition of
vector-like fermions, focusing on isosinglet vector-like quarks (VLQs) in the
$({\bf 3},{\bf 1},Y_{Q})$ representation. Vector-like fermions are anomaly-
free and allow a mass term. We will specifically focus on vector-like quarks
in the same repesentation as SM quarks: $T$ vector-like quarks transforming as
$({\bf 3},{\bf 1},2/3)$ and $B$ vector-like quarks transforming as $({\bf
3},{\bf 1},-1/3)$. These can mix with the SM quarks and so decay, as is
necessary to avoid cosmologically excluded stable coloured particles [13]. It
is known that vector-like fermions can stabilise the Higgs potential through
modification of the renormalisation group (RG) running of the gauge couplings
[14, 15, 16, 17]. TeV-scale fermions have also been motivated by vacuum
selection considerations in [6] and [18].

In this paper we will determine the upper bounds on the VLQ mass $m_{Q}$ from
the requirement of sufficient Higgs stability for inflation and the
predictions of the model for inflation observables. An important distinction
should be made between absolute stability of the zero temperature Higgs
potential and stability of the finite-temperature effective potential. For
successful Higgs Inflation, it is only necessary that the Universe evolve into
the electroweak vacuum following inflation and reheating. We will show that
the upper bounds on $m_{Q}$ from the finite-temperature effective potential
are weaker than those from absolute stability of the Higgs potential.

We will consider $n_{Q}$ VLQs of type $T$ or $B$. We will run the 3-loop SM RG
equations [2], modified to include the Higgs propagator suppression due to the
non-minimal coupling [19, 20, 21], together with the leading order 2-loop
corrections due to the VLQs [14]. We use the 2022 PDG direct measurement value
for the $t$-quark mass, $m_{t}=172.69\pm 0.30{\rm\ GeV}$ [22].

The upper bounds on $m_{Q}$ from stability are independent of the
renormalisation frame of the quantum corrections to the Higgs potential.
However, the conformal frame in which the model is renormalised is important
when calculating the inflation observables; the scalar spectral index $n_{s}$
and tensor-to-scalar ratio $r$. Two cases are commonly considered [23, 24]:
renormalisation in the Einstein frame (Prescription I) and in renormalisation
in the Jordan frame (Prescription II). These frames correspond to different UV
completions of the theory and should therefore be considered as different
Higgs Inflation models [25]. We will show that whereas Prescription I predicts
essentially the same values as classical Higgs Inflation, Prescription II
predicts values of $n_{s}$ and $r$ that are considerably larger than classical
Higgs Inflation. This allows Higgs Inflation to be compatible with the CMB
spectral index of Early Dark Energy (EDE) solutions of the Hubble tension,
which typically predict larger values of $n_{s}$ from CMB than $\Lambda$CDM
[26, 27, 28]. In addition, the predicted primordial gravitational waves are
close to the present observational upper bound and easily observable by
forthcoming CMB experiments [29].

The paper is organised as follows. In Section II we review the Higgs Inflation
model and VLQ stabilisation. In Section III discuss the quantum corrections to
the Higgs potential. In Section IV we discuss the finite-temperature effective
potential. In Section V we present our results and in Section VI we discuss
our conclusions.

##  II Higgs Inflation with Vector-Like Quarks

We first review the essential aspects of Higgs Inflation and its modification
via VLQs. The action of the model in the Jordan frame is

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

where $\hat{{\cal L}}$ is the Lagrangian of the SM and VLQ fields excluding
the Higgs kinetic and potential terms. To analyse inflation and the post-
inflation era, we transform the action to the Einstein frame via a conformal
transformation $\tilde{g}_{\mu\nu}=\Omega^{2}g_{\mu\nu}$, where

$\mbox{$$}\Omega^{2}=\left(1+\frac{\xi\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{1}{2\Omega^{2}}\left(1+\frac{6\xi^{2}\phi^{2}}{\Omega^{2}M_{Pl}^{2}}\right)\partial_{\mu}\phi\partial^{\mu}\phi-
V_{E}(\phi)+\frac{\hat{\cal{L}}}{\Omega^{4}}\right]~{},\vspace{0.1cm}$ (3)

where $V_{E}(\phi)=V(\phi)/\Omega^{4}$ is the Higgs potential in the Einstein
frame.

At field values $\phi>\phi_{c}=M_{Pl}/\sqrt{\xi}\sim 10^{16}{\rm\ GeV}$, the
conformal factor $\Omega$ strongly deviates from 1 and the classical Einstein
frame potential has a plateau suitable for inflation. To obtain the
predictions for the model, we will numerically solve for the end of inflation,
$\sigma_{end}$, defined as when either $|\eta(\sigma)|>1$ or
$\epsilon(\sigma)>1$ first occurs as $\sigma$ decreases. Here $\sigma$ is the
canonically normalised inflaton, which is related to $\phi$ by

$\mbox{$$}\frac{d\sigma}{d\phi}=\frac{1}{\Omega}\left(1+\frac{6\xi^{2}\phi^{2}}{\Omega^{2}M_{Pl}^{2}}\right)^{1/2}~{}.\vspace{0.1cm}$
(4)

We then numerically integrate for the number of e-foldings of inflation as a
function of $\sigma$

$\mbox{$$}N=-\frac{1}{M_{Pl}^{2}}\int_{\sigma}^{\sigma_{end}}\frac{V_{E}(\sigma)}{V^{\prime}_{E}(\sigma)}d\sigma~{}\vspace{0.1cm}$
(5)

to determine $\sigma$ and so $\phi$ at the pivot scale $N_{*}$. The scalar
spectral index, tensor-to-scalar ratio and curvature perturbation power
spectrum are then calculated in the standard way

$\mbox{$$}n_{s}=1+2\eta-6\epsilon~{},\vspace{0.1cm}$ (6)
$\mbox{$$}r=16\epsilon~{},\vspace{0.1cm}$ (7)

and

$\mbox{$$}{\cal P}_{\zeta}=\frac{V_{E}(\sigma)}{24\pi^{2}\epsilon
M_{Pl}^{4}}~{},\vspace{0.1cm}$ (8)

where the slow-roll parameters are
$\eta=M_{Pl}^{2}V_{E}^{\prime\prime}(\sigma)/V_{E}(\sigma)$ and
$\epsilon=(M_{Pl}^{2}/2)(V_{E}^{\prime}(\sigma)/V_{E}(\sigma))^{2}$, where
primes denote derivatives with respect to $\sigma$.

Once quantum corrections are included, the SM potential becomes negative once
$\phi\;^{>}_{\sim}\;\Lambda$ and Higgs Inflation is no longer possible if
$\Lambda<\phi_{c}$. In the next section we will show that this is true over
the whole 2-$\sigma$ range of $m_{t}$. The introduction of VLQs modifies the
RG evolution primarily by modifying the running of the strong gauge coupling
$g_{3}$, which in turn modifies the running of the $t$-quark Yukawa $y_{t}$
and so the running of $\lambda_{h}$. This can be seen by considering the
contribution to the 1-loop RG equations due purely to $y_{t}$ and $g_{3}$ [3]

$\mbox{$$}\mu\frac{\partial\lambda_{h}}{\partial\mu}=-6y_{t}^{4}...~{}\vspace{0.1cm}$
(9) $\mbox{$$}\mu\frac{\partial
y_{t}}{\partial\mu}=-8y_{t}g_{3}^{3}+\frac{9}{2}y_{t}^{3}+...~{}\vspace{0.1cm}$
(10) $\mbox{$$}\mu\frac{\partial
g_{3}}{\partial\mu}=-7g_{3}^{3}+\frac{2}{3}n_{Q}g_{3}^{3}+...~{}\vspace{0.1cm}$
(11)

Increasing $n_{Q}$ reduces the rate of decrease of $g_{3}$ with $\mu$ and so
increases $g_{3}$ at a given $\mu$. This also reduces the rate of increase of
$y_{t}$ and so reduces $y_{t}$ for a given $\mu$. Both of these effects reduce
the rate of decrease of $\lambda_{h}$ with $\mu$, decreasing the instability
of the Higgs potential.

##  III Quantum corrections to the Higgs potential

###  III.1 The Metastable Standard Model Higgs potential

To demonstrate the instability of the SM Higgs potential, we first compute the
SM Higgs potential using the RG equations and initial $\overline{{\rm MS}}$
values of the SM couplings that we will later use for VLQ-stabilised Higgs
Inflation. We run the 3-loop SM RG equations given in [2] and use the relation
between the $t$-quark mass and the $\overline{{\rm MS}}$ $t$-quark coupling at
$\mu=m_{t}$ [2],

$\mbox{$$}y_{t}=0.93690+0.00556\left(m_{t}-173.34{\rm\
GeV}\right)~{}.\vspace{0.1cm}$ (12)

For the range of t-quark mass, we use the 2022 PDG release direct measurement
value, $m_{t}=172.69\pm 0.30{\rm\ GeV}$ [22]. Since metastability is less
sensitive to the errors in the other SM inputs, in this analysis we will use
the mean values for those quantities given in [2]: $g_{3}=1.1666$,
$g=0.64779$, $g^{\prime}=0.35830$ and $\lambda_{h}=0.12604$.

In Figure 1 we show the Higgs self-coupling $\lambda_{h}(\mu)$, calculated at
the RG scale $\mu=\phi$, for the mean, -1-$\sigma$ and -2-$\sigma$ values of
$m_{t}$, corresponding to progressively weaker instability. In all cases the
potential runs to negative values.

In Figure 2 we show the Higgs potential calculated using the 1-loop Coleman-
Weinberg (CW) correction at $\mu=\phi$. In the figure we show ${\rm
log_{10}}|V(\phi)|$ multiplied by the sign of $V(\phi)$, which is useful for
visualising the negative gap in the Higgs potential. For all $m_{t}$ within
the 2-$\sigma$ range the potential is metastable, with the instability scale
given by $\Lambda=2.2\times 10^{11}$ GeV, $7.9\times 10^{11}$ GeV and
$3.0\times 10^{12}$ GeV for the mean, -1-$\sigma$ and -2-$\sigma$ values of
$m_{t}$, respectively. The potential eventually becomes positive again, at
$\phi>\phi_{upper}=10^{29}{\rm\ GeV}$ for the mean value of $m_{t}$, where
$\phi_{upper}$ is the upper bound of the negative gap in the potential.
Therefore it is highly likely that the SM Higgs potential is metastable and
unable to support Higgs Inflation.

Figure 1: SM Higgs coupling $\lambda_{h}(\mu)$ at $\mu=\phi$ for the mean
-1-$\sigma$ and -2-$\sigma$ values of $m_{t}$. Figure 2: SM Higgs potential
calculated at $\mu=\phi$ for the mean -1-$\sigma$ and -2-$\sigma$ values of
$m_{t}$.

###  III.2 Quantum corrections in VLQ-stabilised Higgs Inflation

We again use the 3-loop ${\rm\overline{MS}}$ RG equations for the SM given in
[2], now modified to take into account the Higgs propagator suppression due to
the kinetic term mixing of the physical Higgs boson with the graviton in the
presence of a background $\phi$, which suppresses the physical Higgs boson
propagator by a factor $s(\phi)$ [19, 20],

$\mbox{$$}s(\phi)=\frac{1+\frac{\xi\phi^{2}}{M_{P}^{2}}}{1+\left(6\xi+1\right)\frac{\xi\phi^{2}}{M_{P}^{2}}}~{}.\vspace{0.1cm}$
(13)

This results in a suppression of the contribution of the physical Higgs to the
RG equations, but does not affect the Goldstone contribution. We have included
the $s(t)$ factors at 1-loop given in [20, 21] and at 2-loop given in [19]. In
practice, the propagator suppression has a small effect on the RG evolution of
the couplings and Higgs potential. Finally, we supplement the 3-loop SM RG
equations with the 1-loop and leading 2-loop VLQ corrections given in [14],
which we summarise in Appendix A. The $T$ and $B$ vector quarks will also
generally have unknown Yukawa couplings to the SM quarks and Higgs boson. We
will assume that these couplings are small enough to not significantly modify
the Higgs potential.

Figure 3: Upper bound on $m_{Q}$ as a function of $m_{t}$ for $T$ vector
quarks with $n_{Q}=3$, from absolute stability and from stability of the
finite-temperature effective potential. Figure 4: The $T=0$ Prescription II
Higgs potential as a function of $m_{Q}$, for the case of $T$ vector quarks
with $n_{Q}=3$ and mean $m_{t}$. The negative potential gap progressively
closes as $m_{Q}$ is reduced. Figure 5: Quantum corrected $T=0$ potential and
finite-temperature effective potential, for $T$ vector quarks with $n_{Q}=3$,
$m_{Q}=30{\rm\ GeV}$ and mean $m_{t}$. Figure 6: Prescription II values of $r$
and $n_{s}$ as $m_{Q}$ varies from 1 TeV to 55 TeV, for the case of $T$ quarks
witn $n_{Q}=3$ and mean $m_{t}$. The observational upper bound on $r$ imposes
an upper bound on $m_{Q}$ of around 13 TeV. The range of values of $n_{s}$ is
0.984 to 0.990 as $m_{Q}$ increases from 1 TeV to 13 TeV, with the smallest
value of $r$ being 0.011. Figure 7: Prescription II values of $r$ and $n_{s}$
as $m_{Q}$ varies from 1 TeV to 18 TeV, for the case of $B$ quarks with
$n_{Q}=3$ and mean $m_{t}$. The observational upper bound on $r$ imposes an
upper bound on $m_{Q}$ of around 10 TeV. The range of values of $n_{s}$ is
0.979 to 0.989 as $m_{Q}$ increases from 1 TeV to 10 TeV. with the smallest
value of $r$ being 0.0079. Figure 8: Upper bound on $m_{Q}$ as a function of
$m_{t}$ for $T$ vector quarks with $n_{Q}=2$, from absolute stability and from
stability of the finite-temperature effective potential. Figure 9:
Prescription II values of $r$ and $n_{s}$ as $m_{Q}$ varies from 1 TeV to 5
TeV, for the case of $T$ quarks with $n_{Q}=2$ and mean $m_{t}$. The
observational upper bound on $r$ imposes an upper bound on $m_{Q}$ of around
2.4 TeV. The range of values of $n_{s}$ is 0.988 to 0.990 as $m_{Q}$ increases
from 1 TeV to 2.4 TeV, with the smallest value of $r$ being 0.017. Figure 10:
Prescription II values of $r$ and $n_{s}$ as $m_{Q}$ varies from 1 TeV to 2.8
TeV, for the case of $B$ quarks with $n_{Q}=2$ and mean $m_{t}$. The
observational upper bound on $r$ imposes an upper bound on $m_{Q}$ of around
1.75 TeV. The range of values of $n_{s}$ is 0.987 to 0.990 as $m_{Q}$
increases from 1 TeV to 1.75 TeV. with the smallest value of $r$ being 0.015.

###  III.3 Renormalisation prescription

The form of the quantum correction to the potential during inflation depends
upon the conformal frame in which the model is renormalised. Two frames are
commonly considered, known as Prescription I, where the model is renormalised
in the Einstein frame, and Prescription II, where the model is renormalised in
the Jordan frame and the complete quantum corrected Higgs potential is
transformed to the Einstein frame [23, 24]. These are in effect two completely
different versions of Higgs Inflation, which correspond to different UV
completions of the model [25]. In the case of Prescription II, we will run the
SM RG equations (including the Higgs propagator suppression) up to $\mu=\phi$.
In the case of Prescription I, the SM RG equations become invalid once the
conformal factor significantly deviates from 1 and
$\mu=\phi\;^{>}_{\sim}\;\phi_{c}$. We will therefore run to RG equations up to
$\mu_{c}=0.1\phi_{c}$ and then use the 1-loop CW correction to compute the
potential at $\phi>\phi_{c}$. Since the quantum corrections become rapidly
suppressed at $\phi>\phi_{c}$ in Prescription I, this choice of
renormalisation scale is large enough to prevent large logarithms in the
1-loop CW potential.

###  III.4 Perturbative unitarity violation

Higgs scalar scattering in the cosmological background during inflation is
expected to violate unitarity at energies $E\approx\phi$, corresponding to the
mass of the $W$ and $Z$ bosons during inflation, where $\phi$ is the
background Higgs field during inflation [30, 31]. If perturbative unitarity
indicates true unitarity violation, then a change in the theory is necessary,
characterised by the energy scale $\phi$. In this case it is possible that new
physics could modify the effective potential. However, perturbative unitarity
violation in Higgs scattering may instead indicate a breakdown of perturbation
theory, with unitarity conserved non-perturbatively [32, 33, 34, 35]. In this
case no new physics is necessary to conserve unitarity, and since the effect
of the non-renormalisable coupling on the RG equations and the effective
potential is taken into account by the Higgs propagator suppression, there is
no reason for the effective potential to be modified. At present it is not
known whether perturbative unitarity violation will significantly modify the
Higgs potential during inflation. In the following we will assume that
corrections to the potential due to unitarity conservation are negligible.

###  III.5 The 1-loop Coleman-Weinberg correction

To calculate the quantum corrected Higgs potential, we run the couplings up to
the renormalisation scale $\mu$ and then add the 1-loop CW correction. In
general, the ${\rm\overline{MS}}$ scheme SM 1-loop CW potential is given by
[4]

$\mbox{$$}\Delta
V_{1-loop}=\sum_{i}\;\frac{C_{i}M_{i}^{4}}{64\pi^{2}}\left[\ln\left(\frac{M_{i}^{2}}{\mu^{2}}\right)-K_{i}\right]~{},\vspace{0.1cm}$
(14)

where $M_{i}$ the mass in the renormalisation frame and
$(C_{i},K_{i})=(3,3/2)$ 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 propagator suppression. The particle masses are
$M_{i}=M_{J,\,i}$ in the Jordan frame and $M_{J,\,i}/\Omega$ in the Einstein
frame. The terms in the 1-loop CW potential in the chosen renormalisation
frame therefore have the form

$\mbox{$$}\Delta
V_{1-loop}=\frac{C\,M_{J}^{4}}{16\pi^{2}\Omega^{4\alpha}}\left(\ln\left(\frac{M_{J}^{2}(\phi)}{\mu^{2}\Omega^{2\alpha}}\right)-K\right)~{},\vspace{0.1cm}$
(15)

where $\alpha=1$ for Prescription I and $\alpha=0$ for Prescription II. Once
transformed the Einstein frame, the final 1-loop potential is of the form

$\mbox{$$}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}$
(16)

##  IV The finite temperature effective potential

The condition for successful Higgs Inflation is that the Higgs expectation
value can evolve into the electroweak vacuum. This does not require absolute
stability but only that the minimum of the finite-temperature effective
potential (FTEP) after inflation, which gives the thermal equilibrium
expectation value of the Higgs field, is at $\phi=0$. We refer to this as
finite temperature stability. Due to the large couplings of the Higgs boson to
the SM fields, after inflation the Higgs field oscillations rapidly preheat
and decay to SM fields [36, 37, 38]. We will therefore assume that reheating
is instantaneous and compute the FTEP after inflation to see if it is stable.
We calculate the FTEP in the Einstein frame, which is the frame in which
inflation is analysed, so the mass terms entering the FTEP are calculated in
the Einstein frame. These are generally related to the conventional (i.e.
Jordan frame) SM masses $M_{J,\,i}$ by $M_{i}=M_{J,\,i}/\Omega$. In practice,
the Einstein frame is essentially the same as the Jordan frame when
calculating the FTEP, since $\Omega\approx 1$ after inflation and reheating.

The FTEP is given by $V_{E}(\phi,T)=V_{E}(\phi)+\Delta V(\phi,T)$, where

$\mbox{$$}\Delta
V(\phi,T)=\frac{T^{4}}{2\pi^{2}}\sum_{i}n_{B,\,i}\,I_{B}\left[\frac{M_{i}^{2}(\phi)}{T^{2}}\right]-\frac{T^{4}}{2\pi^{2}}\sum_{i}n_{F,\,i}\,I_{F}\left[\frac{M_{i}^{2}(\phi)}{T^{2}}\right]~{}\vspace{0.1cm}$
(17)

where the integrals $I_{B}$ and $I_{F}$ are given by

$\mbox{$$}I_{B}\left[\frac{M_{i}^{2}(\phi)}{T^{2}}\right]=\int_{0}^{\infty}dx\,x^{2}\ln\left[1-\exp\left(-\sqrt{x^{2}+\frac{M_{i}^{2}(\phi)}{T^{2}}}\right)\right]~{}\vspace{0.1cm}$
(18)

and

$\mbox{$$}I_{F}\left[\frac{M_{i}^{2}(\phi)}{T^{2}}\right]=\int_{0}^{\infty}dx\,x^{2}\ln\left[1+\exp\left(-\sqrt{x^{2}+\frac{M_{i}^{2}(\phi)}{T^{2}}}\right)\right]~{},\vspace{0.1cm}$
(19)

where $n_{B,\,i}$ and $n_{F,\,i}$ are the number of degrees of freedom of the
bosons and fermions. In the figures we show $V_{E}(\phi,T)-\Delta V(0,T)$ so
that the finite temperature contribution equals zero at $\phi=0$.

##  V Results

The potential during inflation depends upon the value of the non-minimal
coupling. This is determined by requiring that the curvature perturbation
power spectrum at the pivot scale is equal to its observed value. The number
of e-foldings at the pivot scale is discussed in Appendix B. We find that in
general the potential at the end of inflation is $V_{end}\approx 10^{64}{\rm\
GeV}^{4}$, where $V_{end}=V_{E}(\sigma_{end})$ is the Einstein frame potential
at the end of inflation, and so $N_{*}\approx 57$ for the case of instant
reheating.

The present experimental lower bound on the mass of vector quarks is around 1
TeV, with the exact lower bound depending on the assumed decay mode of the
vector quark [39, 40, 41, 42]. We will therefore impose a lower bound of 1 TeV
on $m_{Q}$ in our results.

$m_{t}({\rm GeV})$ |  $m_{Q}$ (${\rm\,Abs}$) |  $m_{Q}$ (${\rm\,Finite\;T}$)
---|---|---
173.29 ($+$2-$\sigma$) | $6.1{\rm\ TeV}$ | $11{\rm\ TeV}$
172.99 ($+$1-$\sigma$) | $12{\rm\ TeV}$ | $25{\rm\ TeV}$
172.69 (mean) | $29{\rm\ TeV}$ | $55{\rm\ TeV}$
172.39 ($-$1-$\sigma$) | $88{\rm\ TeV}$ | $175{\rm\ TeV}$
172.09 ($-$2-$\sigma$) | $380{\rm\ TeV}$ | $680{\rm\ TeV}$
Table 1: $m_{Q}$ upper bound as a function of $m_{t}$ for $T$ vector quarks with $n_{Q}$ = 3, for absolute and finite temperature stability.  $m_{Q}$ | $n_{s}$ | $r$ | $\xi(m_{t})/10^{3}$ | $V_{end}\,({\rm\ GeV}^{4})$
---|---|---|---|---
$55{\rm\,TeV}$ | $0.9744$ | $0.101$ | $0.45$ | $1.31\times 10^{64}$
$29{\rm\,TeV}$ | $0.9833$ | $6.99\times 10^{-2}$ | $0.58$ | $1.89\times 10^{64}$
$20{\rm\,TeV}$ | $0.9872$ | $5.25\times 10^{-2}$ | $0.70$ | $1.86\times 10^{64}$
$15{\rm\,TeV}$ | $0.9890$ | $4.21\times 10^{-2}$ | $0.84$ | $1.63\times 10^{64}$
$13{\rm\,TeV}$ | $0.9895$ | $3.80\times 10^{-2}$ | $0.91$ | $1.53\times 10^{64}$
$12{\rm\,TeV}$ | $0.9897$ | $3.59\times 10^{-2}$ | $0.93$ | $1.55\times 10^{64}$
$10{\rm\,TeV}$ | $0.9898$ | $3.17\times 10^{-2}$ | $1.05$ | $1.37\times 10^{63}$
$5{\rm\,TeV}$ | $0.9889$ | $2.11\times 10^{-2}$ | $1.45$ | $1.05\times 10^{63}$
$3{\rm\,TeV}$ | $0.9874$ | $1.65\times 10^{-2}$ | $1.84$ | $8.31\times 10^{63}$
$1{\rm\,TeV}$ | $0.9836$ | $1.12\times 10^{-2}$ | $2.70$ | $6.07\times 10^{63}$
Table 2: Prescription II inflation observables and parameters as a function of $m_{Q}$, for $T$ vector quarks with $n_{Q}=3$ and mean $m_{t}$. $m_{Q}$ | $n_{s}$ | $r$ | $\xi(m_{t})/10^{3}$ | $V_{end}\,({\rm\ GeV}^{4})$
---|---|---|---|---
$18{\rm\,TeV}$ | $0.9739$ | $9.20\times 10^{-2}$ | $0.35$ | $1.15\times 10^{64}$
$14{\rm\,TeV}$ | $0.9814$ | $6.94\times 10^{-2}$ | $0.42$ | $1.70\times 10^{64}$
$10{\rm\,TeV}$ | $0.9882$ | $4.06\times 10^{-2}$ | $0.60$ | $1.60\times 10^{64}$
$9{\rm\,TeV}$ | $0.9890$ | $3.48\times 10^{-2}$ | $0.68$ | $1.44\times 10^{64}$
$8{\rm\,TeV}$ | $0.9892$ | $2.97\times 10^{-2}$ | $0.75$ | $1.36\times 10^{64}$
$6{\rm\,TeV}$ | $0.9885$ | $2.12\times 10^{-2}$ | $1.02$ | $1.00\times 10^{64}$
$3{\rm\,TeV}$ | $0.9844$ | $1.23\times 10^{-2}$ | $1.65$ | $6.61\times 10^{63}$
$1{\rm\,TeV}$ | $0.9792$ | $7.87\times 10^{-3}$ | $2.70$ | $4.56\times 10^{63}$
Table 3: Prescription II inflation observables and parameters as a function of $m_{Q}$, for $B$ vector quarks with $n_{Q}=3$ and mean $m_{t}$. $m_{t}({\rm GeV})$ | $n_{s}$ | $r$ | $\xi(m_{t})/10^{3}$ | $V_{end}\,({\rm\ GeV}^{4})$
---|---|---|---|---
173.29 ($+$2-$\sigma$) | 0.9896 | $3.24\times 10^{-2}$ | 1.07 |  $1.43\times 10^{64}$ GeV
172.99 ($+$1-$\sigma$) | 0.9890 | $2.21\times 10^{-2}$ | 1.46 |  $1.04\times 10^{64}$ GeV
172.69 (Mean) | 0.9873 | $1.65\times 10^{-2}$ | 1.84 |  $8.31\times 10^{63}$ GeV
172.39 ($-$1-$\sigma$) | 0.9854 | $1.34\times 10^{-2}$ | 2.21 |  $6.98\times 10^{63}$ GeV
172.09 ($-$2-$\sigma$) | 0.9839 | $1.12\times 10^{-2}$ | 2.57 |  $6.08\times 10^{63}$ GeV
Table 4: Prescription II inflation observables and parameters as a function of $m_{t}$, for $T$ vector quarks with $n_{Q}=3$ and $m_{Q}=$ 3 TeV.  $m_{t}({\rm GeV})$ |  $m_{Q}$ (${\rm\,Abs}$) |  $m_{Q}$ (${\rm\,Finite\;T}$)
---|---|---
173.29 ($+$2-$\sigma$) | $1.0{\rm\ TeV}$ | $1.3{\rm\ TeV}$
172.99 ($+$1-$\sigma$) | $2.0{\rm\ TeV}$ | $2.3{\rm\ TeV}$
172.69 (mean) | $4.4{\rm\ TeV}$ | $5.8{\rm\ TeV}$
172.39 ($-$1-$\sigma$) | $12{\rm\ TeV}$ | $15{\rm\ TeV}$
172.09 ($-$2-$\sigma$) | $45{\rm\ TeV}$ | $60{\rm\ TeV}$
Table 5: $m_{Q}$ upper bound as a function of $m_{t}$, for $T$ vector quarks with $n_{Q}=2$.  $m_{Q}$ | $n_{s}$ | $r$ | $\xi(m_{t})/10^{3}$ | $V_{end}\,({\rm\ GeV}^{4})$
---|---|---|---|---
$5.0{\rm\,TeV}$ | $0.9763$ | $9.14\times 10^{-2}$ | $0.40$ | $1.46\times 10^{64}$
$4.0{\rm\,TeV}$ | $0.9821$ | $7.25\times 10^{-2}$ | $0.46$ | $1.84\times 10^{64}$
$3.0{\rm\,TeV}$ | $0.9876$ | $5.04\times 10^{-2}$ | $0.59$ | $1.76\times 10^{64}$
$2.4{\rm\,TeV}$ | $0.9895$ | $3.84\times 10^{-2}$ | $0.72$ | $1.55\times 10^{64}$
$2.0{\rm\,TeV}$ | $0.9900$ | $3.10\times 10^{-2}$ | $0.85$ | $1.33\times 10^{64}$
$1.5{\rm\,TeV}$ | $0.9895$ | $2.34\times 10^{-2}$ | $1.06$ | $1.10\times 10^{64}$
$1.0{\rm\,TeV}$ | $0.9877$ | $1.68\times 10^{-2}$ | $1.40$ | $8.54\times 10^{63}$
Table 6: Prescription II inflation observables and parameters as a function of $m_{Q}$, for $T$ vector quarks with $n_{Q}=2$ and mean $m_{t}$.  $m_{Q}$ | $n_{s}$ | $r$ | $\xi(m_{t})/10^{3}$ | $V_{end}\,({\rm\ GeV}^{4})$
---|---|---|---|---
$2.8{\rm\,TeV}$ | $0.9734$ | $9.76\times 10^{-2}$ | $0.29$ | $1.28\times 10^{64}$
$2.6{\rm\,TeV}$ | $0.9765$ | $8.93\times 10^{-2}$ | $0.31$ | $1.61\times 10^{64}$
$2.0{\rm\,TeV}$ | $0.9877$ | $5.06\times 10^{-2}$ | $0.45$ | $1.84\times 10^{64}$
$1.75{\rm\,TeV}$ | $0.9899$ | $3.78\times 10^{-2}$ | $0.56$ | $1.55\times 10^{64}$
$1.5{\rm\,TeV}$ | $0.9903$ | $2.77\times 10^{-2}$ | $0.85$ | $1.29\times 10^{64}$
$1.25{\rm\,TeV}$ | $0.9893$ | $2.05\times 10^{-2}$ | $0.71$ | $1.26\times 10^{64}$
$1.0{\rm\,TeV}$ | $0.9874$ | $1.55\times 10^{-2}$ | $1.15$ | $7.91\times 10^{63}$
Table 7: Prescription II inflation observables and parameters as a function of
$m_{Q}$, for $B$ vector quarks with $n_{Q}=2$ and mean $m_{t}$.

###  V.1 ${\bf n_{Q}=3}$

For the mean t-quark mass and three $T$ vector quarks, we find that the
absolute stability bound is $m_{Q}\leq 29{\rm\ TeV}$, and the finite
temperature stability bound is $m_{Q}\leq 55{\rm\ TeV}$. For $B$ vector
quarks, the corresponding bounds are $m_{Q}\leq 14{\rm\ TeV}$ and $m_{Q}\leq
18{\rm\ TeV}$ respectively.

In Table 1 and Figure 3 we show how the stability bounds vary with $m_{t}$
over the 2-$\sigma$ observed range for three $T$ quarks. The upper bounds on
$m_{Q}$ are sensitive to $m_{t}$, becoming smaller as $m_{t}$ increases and
the SM instability strengthens.

In Figure 4 we show the effect of decreasing $m_{Q}$ on the $T=0$ potential
and its stability for Prescription II with three $T$ vector quarks. For the
case without VLQs, the upper bound on the gap is around $10^{29}$ GeV and the
lower bound is around $10^{11}{\rm\ GeV}$. Once the vector quarks are
introduced, the lower bound of the gap increases and the upper bound (more
rapidly) decreases as $m_{Q}$ decreases, until the gap disappears and the
potential becomes absolutely stable at $m_{Q}=29{\rm\ GeV}$.

In Figure 5 we show the $T=0$ and FTEP potential for $m_{Q}=30$ TeV and three
$T$ vector quarks, showing that the $T=0$ potential is metastable, with a deep
minimum at $\phi\neq 0$, whereas the FTEP has a minimum only at $\phi=0$. In
this case $\phi$ will cool into the electroweak vacuum after inflation, even
though the SM potential is metastable.

The stability upper bounds apply to both Prescription I and II, as
$\phi_{upper}<\phi_{c}$ when stability breaks down and so the Jordan and
Einstein frames are equivalent as far as stability of the potential is
concerned.

We next consider how the inflation predictions vary as a function of $m_{Q}$.
For Prescription I, we find that the inflation predictions are almost exactly
equal to the classical Higgs Inflation predictions, $n_{s}=0.966$ and
$r=3.2\times 10^{-3}$, for all $m_{Q}$. This is due to the suppression of the
Einstein frame mass terms by $\Omega$, which means that at $\phi>\phi_{c}$ the
1-loop CW corrections quickly becomes independent of $\phi$.

For Prescription II, since the quantum corrections are calculated in the
Jordan frame, the quantum corrections to the potential are not cut off at
$\phi>\phi_{c}$. In Figure 6 we show $r$ versus $n_{s}$ for three $T$ vector
quarks for $m_{Q}$ in the range 1 TeV to 55 TeV and in Table 2 we show the
predictions of the model for $n_{s}$, $r$, $\xi(m_{t})$ and $V_{end}$, with
$m_{t}$ equal to its mean value.

We find from Figure 6 that the upper bound on $m_{Q}$ from the observational
2-$\sigma$ upper bound on $r$ is stronger than the bound from stability of the
potential. Imposing the 2-$\sigma$ upper bound $r<0.037$ [43] gives an upper
bound on $m_{Q}$ of 13 TeV for three $T$ quarks, compared to 55 TeV from
finite-temperature stability.

For $1{\rm\ TeV}\leq m_{Q}\leq 13$ TeV the range of $r$ is about 0.01 to 0.04.
Therefore the predicted range of values for $r$ are not far below the present
observational limits and will be easily observable by the next generation of
CMB polarisation experiments, which will be able to observe $r$ down to
$O(10^{-3})$ [29].

The values of $n_{s}$ for $1{\rm\ TeV}\leq m_{Q}\leq 13$ TeV are in the range
0.984 to 0.990. These values are large compared to $\Lambda$CDM CMB range from
Planck, $n_{s}=0.965\pm 0.004$ [44]. However, $\Lambda$CDM is now challenged
by the $H_{0}$ tension between the CMB value of $H_{0}$ and the direct late-
time measurements of $H_{0}$. In the case of early-time solutions of the
Hubble tension, where the tension is resolved by modifying the sound horizon
at decoupling via an additional energy density component, in particular Early
Dark Energy (EDE), the value of $n_{s}$ from CMB is larger than the
$\Lambda$CDM value [26, 27]. This can be understood as due to the additional
scale dependent suppression of perturbation growth at smaller length scales
due to the additional energy density component, which requires a larger
$n_{s}$ to compensate. For the case of a typical EDE model based on an axion-
like potential, the best-fit values are in the range 0.981-0.996, depending on
the data set [28]. Therefore Prescription II Higgs Inflation with VLQ
stabilisation can provide a minimal model for inflation based on TeV scale
physics that is also naturally compatible with early-time solutions to the
$H_{0}$ tension.

In Figure 7 and Table 3 we show the corresponding predictions for three $B$
quarks. In this case the observational bound on $r$ imposes an upper bound on
$m_{Q}$ of 10 TeV, as compared to 18 TeV for finite temperature stability. As
$m_{Q}$ varies from 1 TeV to 10 TeV the value of $r$ varies from about 0.008
to 0.04, whilst $n_{s}$ varies from 0.979 to 0.989.

In Table 4 we show how the Prescription II inflation predictions vary with
$m_{t}$ for three $T$ vector quarks with fixed $m_{Q}=3{\rm\ TeV}$. We see
that the value of $n_{s}$ is greater than 0.984 and $r$ is greater than 0.01
over the whole 2-$\sigma$ experimental range of $m_{t}$.

###  V.2 ${\bf n_{Q}=2}$

We next consider the effect of fewer VLQs. In this case the upper bounds on
$m_{Q}$ are smaller, increasing the likelihood that they will be observed in
future colliders. For two $T$ vector quarks and the mean value
$m_{t}=172.69{\rm\ GeV}$, the absolute stability upper bound is 4.4 TeV,
whilst the finite temperature stability upper bound is 5.8 TeV. The
corresponding upper bounds for the case of two $B$ vector quarks are 2.6 TeV
and 2.8 TeV respectively. In Table 5 and Figure 9 we show how the stability
bounds vary with $m_{t}$ for the case of two $T$ vector quarks.

In Figure 10 and Table 6 we show $r$ and $n_{s}$ for $m_{Q}$ varying from 1
TeV to 5 TeV for the case of Prescription II with two $T$ vector quarks and
$m_{t}$ equal to its mean value. As in the $n_{Q}=3$ case, the observational
upper bound on $r$ imposes a stronger constraint on $m_{Q}$ that stability of
the potential, with $m_{Q}\;^{<}{}_{\sim}\;2.4$ TeV compared with 5.8 TeV from
finite temperature stability. The values of $r$ are in the range 0.017 to 0.04
and $n_{s}$ is in the range 0.988 to 0.990 for $m_{Q}$ in the range 1 TeV to
2.4 TeV.

In Figure 11 and Table 7 we show the corresponding results for the case of two
$B$ vector quarks. In this case the observational upper bound on $r$ imposes
an upper bound on $m_{Q}$ of 1.75 TeV, compared to 2.8 TeV from finite
temperature stability. The values of $r$ are in the range 0.016 to 0.04 and
$n_{s}$ is in the range 0.987 to 0.990 for $m_{Q}$ in the range 1 TeV to 1.75
TeV.

###  V.3 ${\bf n_{Q}=1}$

For the case of a single vector quark, we find stability is only possible if
$m_{t}$ is at its -2-$\sigma$ value or less. For the case of a single $T$
vector quark the absolute stability and the finite temperature stability
bounds are both $1.6{\rm\ TeV}$. For the case of a single $B$ vector quark
there is stability only at $m_{Q}\leq 1.0{\rm\ TeV}$. For Prescription II we
find that there is no value of $m_{Q}\geq 1{\rm\ TeV}$ for which $r$ is less
than the observational upper bound. Therefore $n_{Q}=1$ is possible for
Prescription I but it is ruled out for Prescription II.

##  VI Conclusions

Higgs Inflation is a minimal approach to inflation, using only the SM fields
and TeV-scale extensions to achieve inflation. Here we have considered a
purely fermionic extension with the addition of isosinglet vector-like quarks.
This extension can be thought of as a continuation of the structure of the SM,
where there is a single scalar multiplet plus fermions.

It is likely that the VLQ mass is in the range 1-10 TeV. This will be true if
the $t$-quark mass is at its mean value or in the upper half of its 2-$\sigma$
range. The $m_{Q}$ upper bounds are also lower for smaller numbers of vector
quarks. Thus there is good reason to hope that proposed future particle
colliders such as the HL-LHC and FCC-hh/SppC will be able to detect VLQs if
VLQ stabilised Higgs Inflation is correct [41, 42].

The inflation observables predicted by the model strongly depend upon the
conformal frame in which the theory is renormalised Prescription I produces
predictions for $n_{s}$ and $r$ that are essentially identical to the
classical predictions of Higgs Inflation, $n_{s}=0.966$ and $r=3.3\times
10^{-3}$, with the spectral index in good agreement with the conventional
$\Lambda$CDM CMB value. However, in light of the Hubble tension, it is not
clear that the $\Lambda$CDM value of $n_{s}$ is correct. Solutions of the
tension that modify the sound horizon, in particular Early Dark Energy,
typically favour values in the range 0.980-0.995. It may therefore be
significant that the Prescription II prediction of $n_{s}$ is also in the
range 0.980-0.995. Moreover, this is accompanied by a values of the tensor-to-
scalar ratio in the range 0.01-0.04, with the upper bound on $m_{Q}$ coming
from the present observational upper bound on $r$ rather than from Higgs
potential stability. Therefore Prescription II Higgs Inflation with VLQs
typically predicts 1-10 TeV VLQ masses, a value of $n_{s}$ compatible with
early-time solutions to the Hubble tension, and primordial gravitational waves
that will be easily detectable by the next generation of CMB observations.

## Appendix A: VLQ modifications of the SM RG equations

The 1-loop and leading 2-loop modifications to the SM RG equations due to
$n_{Q}$ VLQs in the $({\bf 3},{\bf 1},Y_{Q})$ representation are

$\mbox{$$}\Delta\beta_{g_{3},\,1-loop}=\frac{g_{3}^{2}}{16\pi^{2}}\left(\frac{2}{3}n_{Q}\right)~{},\vspace{0.1cm}$
(A-1)
$\mbox{$$}\Delta\beta_{g_{3},\,2-loop}=\frac{g_{3}^{5}}{\left(16\pi^{2}\right)^{2}}\left(10n_{Q}\right)~{},\vspace{0.1cm}$
(A-2)
$\mbox{$$}\Delta\beta_{g^{\prime},\,1-loop}=\frac{g^{\prime\,3}}{\left(16\pi^{2}\right)^{2}}\left(4n_{Q}Y_{Q}^{2}\right)~{},\vspace{0.1cm}$
(A-3)

and

$\mbox{$$}\Delta\beta_{y_{t},\,2-loop}=\frac{y_{t}g_{3}^{4}}{\left(16\pi^{2}\right)^{2}}\left(\frac{40}{9}n_{Q}\right)~{}.\vspace{0.1cm}$
(A-4)

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

The reheating temperature, assuming instant reheating and an Einstein frame
energy density at the end of inflation given by
$V_{end}=V_{E}(\sigma_{end})\approx\lambda_{h}M_{Pl}^{4}/(4\xi^{2})$, where
$\lambda_{h}$ and $\xi$ are calculated at $\mu=\phi_{end}\approx
M_{Pl}/\sqrt{\xi}$, is given by

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

where $g(T_{R})\approx 106.75+10.5n_{Q}$ for the SM plus $n_{Q}$ isosinglet
VLQs.

The number of e-foldings $N_{*}$ 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}$
(B-2)

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{V_{end}}{3M_{Pl}^{2}}\right)^{1/2}~{}.\vspace{0.1cm}$
(B-3)

Therefore

$\mbox{$$}N_{*}=\ln\left(\frac{2\pi
T_{0}}{k_{*}}\left(\frac{g(T_{0})}{g(T_{R})}\right)^{1/3}\left(\frac{\pi^{2}g(T_{R})}{270}\right)^{1/4}\left(\frac{V_{end}^{1/4}}{M_{Pl}}\right)\right)~{}.\vspace{0.1cm}$
(B-4)

For $n_{Q}=3$ we obtain

$\mbox{$$}N_{*}=57.4+\frac{1}{4}\ln\left(\frac{V_{end}}{10^{64}{\rm\
GeV}^{4}}\right)~{}.\vspace{0.1cm}$ (B-5)

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Introductionsec:intro

Unpaired image-to-image translation maps images between two or more domains,
where image instances are not matched. Gathering datasets with exact pixel-to-
pixel mapping is difficult and often impossible as most of scientific
experiments cannot be exactly reproduced by simulations. Differing from
regular generative models
goodfellow2014generative,brock2018large,karras2020analyzing that create images
from random vector inputs, image-to-image translation derives from an input
image, which hypothetically should be an easier task. Along with its potential
applications in art and design, we believe image-to-image translation will
have a profound impact in scientific simulations cranmer_frontier_2020.
Instead of replacing an entire scientific simulation with a generative model,
simulation results could be defined as one domain with experimental data as
the other. Thus, an image-to-image translation model would fill the
simulation-reality gap.

The CycleGAN zhu_unpaired_2017,kim_learning_2017,yi_dualgan_2017 model
connects two generative adversarial network (GAN) models, one for each
translation direction, and introduces a cycle-consistent loss that reinforces
an image should look like itself after a cycle translation, i.e., translated
to the other domain and back. Other methods
huang_multimodal_2018,lee_diverse_2018,nizan_breaking_2019,zhao2020unpaired,kim_u-
gat-it_2020 promote the translation diversity or one-to-many mapping between
two domains. More recently, ACL-GAN zhao2020unpaired relaxes the self-
consistent constraint by introducing a so-called “adversarial-consistency
loss,” a self-consistency loss at the distribution level. CouncilGAN
nizan_breaking_2019 completely discards the idea of self-consistency, employs
an ensemble of generators, and only works in one translation direction. To
boost diversity, both models have injected random noise into the feature
space. Although these models have shown promising improvement on benchmark
data, features like stochasticity and diversity are not desirable for many
scientific applications. The question is: can newer deep learning
architectures and techniques improve the one-to-one deterministic translation
with cycle-consistent loss?

Convolutional neural network (CNN) architecture has been a popular choice for
computer vision tasks. In the natural language processing (NLP) field, the
attention mechanism and transformer-style architecture have surpassed previous
models, such as hidden Markov models and recurrent neural networks in open
benchmark tasks. Compared to CNNs, transformers can capture long-distance
patterns, which are common in nature. Applications of transformers in computer
vision have debuted in dosovitskiy_image_2020, while other recent work has
shown that a CNN-transformer hybrid can achieve better performance
xiao_early_2021,guo_cmt_2021. In this work, we will examine how much
improvement transformer and GAN training techniques can bring to the classic .

gen ! [inner sep=0] (N) at (0, 0) ;

Figure :  Schematic diagrams of : a) CycleGAN, b) generator, and c) pixel-wise
ViT.