# Walls, bubbles and doom - the cosmology of HEFT

R. Alonso  J. C. Criado  R. Houtz  and M. West

###### Abstract

As experiment charts new territory at the electroweak scale, the enterprise to
characterise all possible theories becomes all the more necessary. In the
absence of new particles, this ambitious enterprise is attainable and has led
to the Higgs Effective Field Theory (HEFT) as the most general characterising
framework, containing the Standard Model Effective Field Theory (SMEFT) as a
subspace. The characterisation of this theory space led to the dichotomy SMEFT
vs. HEFT$\backslash$SMEFT as the two possible realisations of symmetry
breaking. The criterion to distinguish these two possibilities is non-local in
field space, and phenomena which explore field space beyond the neighbourhood
of the vacuum manifold are in a singular position to tell them apart.
Cosmology allows for such phenomena, and this work focuses on
HEFT$\backslash$SMEFT, the less explored of the two options, to find that
first order phase transitions with detectable gravitational wave remnants,
domain wall formation and vacuum decay in the far, far distant future can take
place and single out HEFT$\backslash$SMEFT. Results in cosmology are put
against LHC constraints, and the potential of future ground- and space-based
experiments to cover parameter space is discussed.

##  1 Introduction

The direct experimental exploration of the electroweak (EW) scale is in full
swing at the LHC, and we are closing in on the answers to questions
historically central to particle physics: the hierarchy problem, the mechanism
for electroweak symmetry breaking, and mass generation. Looking out into the
cosmos, on the other hand, has proven to be a source of invaluable input to
our theories of Nature, the latest potential window into early
cosmology—possibly including electroweak transitions—being gravitational
waves.

At the front line of this exploration is the Higgs scalar and its properties.
Given the absence of particles beyond the Standard Model (SM) spectrum thus
far, the formalism of Effective Field Theories (EFTs) provides a general,
model independent framework to characterise the Higgs particle. This formalism
presents a dichotomy in that the electroweak EFT might or might not admit a
linear representation. These two options are dubbed SMEFT and
HEFT$\backslash$SMEFT where backslash is the mathematical symbol for the
difference of sets. The HEFT is the most general gauge and Lorentz invariant
EFT we can write, and thus it includes both options. However, the literature
often refers to the HEFT$\backslash$SMEFT simply as HEFT. Here, we use
$\textrm{non-linear theory space}\equiv\textrm{HEFT$\backslash$SMEFT}$ to make
it clear that we are referring to EFT’s that do not admit a linear
represenation.

Figure 1: Visual depiction of the space of HEFT theories. There is some
evidence for the boundary beetween non-linear and SMEFT theories belonging to
the former, see e.g. Ref. [1].

The characterisation of these theories has been laid out [2, 3, 4], examples
of UV completions and general results for certain classes have been derived
[5, 6, 7, 8, 9] and new features continue to emerge and be explored [10, 11,
12, 13, 14, 15, 16, 17, 18, 19, 20]. A pertinent remark is that the SMEFT vs
HEFT$\backslash$SMEFT characterisation is an IR one, and there are known cases
in which the UV description that completes a non-linear HEFT$\backslash$SMEFT
theory presents linear scalar representations; what sets apart such linear UV
theories is that they are non-decoupling with an upper bound on the mass of
new states of $\sim 4\pi v$. If such a contrast of UV and IR descriptions
seems counter-intuitive, an example in nature is QCD and its natural low
energy description using the chiral Lagrangian. Fully comprehensive
characterisations are nonetheless elusive, in essence because the question is
non-local in field space. Scattering experiments can only probe our theory
around the vacuum to higher terms in our Taylor expansion of the Lagrangian in
fields, incrementally improving our knowledge of the theory with more
particles involved in the scattering process, yet still inherently local. It
is here that cosmology offers a global view of our theory and the possibility
of testing it non-locally via phenomena such as phase transitions or
topological defect formation. It is the aim of this work to examine the window
that cosmology opens on HEFT$\backslash$SMEFT theories, especially those
hardest to distinguish locally from the SM, and chart its complementarity with
LHC data.

The elementary result for EFT characterisation is the presence or absence of a
point in scalar field space which is invariant, i.e. stays fixed, under the
gauge group action. A useful visualisation in a two-dimensional field space
with rotation as the gauge group has the fixed point at the origin. The
extension to the electroweak theory led to naming it an $O(4)$ fixed point
with $SU(2)_{L}\times U(1)_{Y}\subset O(4)$. If this fixed point is present,
it is possible to cast the EFT as a SMEFT, and if absent the EFT is non-
linear. This absence can arise in two broad forms:

  * •

type A, theories entirely without a fixed point,

  * •

type B, theories with a singularity at the would-be fixed point in scalar
space.

This distinction with SMEFT and within non-linear theories is one about the
possible realisation of the symmetry, and therefore one can expect the
electroweak phase transition epoch of the universe to shed light on the
question—we do not consider here low energy inflation or other scenarios that
bypass a universe at $T_{ew}$. The cosmology of ultraviolet completions giving
rise to theories with a singularity, i.e. type B, has in fact been studied
already in Ref. [21] (see also [22]); this work focuses on type A. For a
discussion of possible UV completions of type A theories see [1].

Non-perturbative dynamics depend generically on extended field configurations
which hence probe the theory globally in field space. As such, these would be
sensitive even to theories that locally resemble the SM. One of the prominent,
phenomenologically relevant examples of such non-pertubative dynamics is B-
and L-number violating sphaleron processes. The sphaleron energy is given by
field configurations which will be modified in non-linear theories, and Sec. 3
explores this modification.

(a)

(b)

Figure 2: In Fig. 2(a), from top to bottom and left to right, 2D
representations of the 4D scalar manifold for the SM case (top left) and type
A theories with the metric of Eqs. (2.1, 2.8) and $\sin^{2}\chi=0.01$, 0.1,
and 0.2. The Goldstone bosons correspond to the angle around the $z$ axis,
while the Higgs $h$ parametrises the surface in the orthogonal direction to
them. Fig. 2(b) shows $M_{W}(h)/M_{W}(0)$ for the SM and the type A theories
in Fig. 2(a) with the same colour coding.

The absence of a fixed point in type A theories requires, in turn, a revision
of the behaviour of the theory at high temperature. Indeed strictly speaking,
electroweak symmetry cannot be restored at high temperature since there is not
a point in the manifold for such restoration. One can still talk about
discrete symmetry restoration nonetheless; a type A theory does not present a
fixed point where the electroweak gauge boson masses would vanish, but it
might contain a point where they reach their minimum. Let us suppose this
point is a field distance $v_{\star}$ away from our current vacuum. One can
then define reflection around this point: if $h$ is the Higgs fluctuation
around the vacuum, let us define $\phi=h+v_{\star}$, and the reflection as
$\phi\to-\phi$. The case of SMEFT has no such possible symmetry: $v_{\star}$
can be taken instead to be the distance to the fixed point, and by definition
$\phi$, a radius, is always positive. The geometric approach helps visualise
such theory. The possibility of extending a ‘radius’ to negative values leads
to a wormhole-like structure as shown in Fig. 2(a).

$h_{\star}$$h_{+}$$h_{0}$4.5{$t$ (a)

$h_{\star}$$h_{+}$$h_{0}$ (b)

$h_{\star}$$h_{+}$$h_{0}$ (c)

Figure 3: The possible extrema histories in SMEFT, minima in blue and maxima
in green.

Even with this discrete parity introduction, one might not have symmetry
restoration at high temperature. In fact, given the variety of possibilities
for minima in these theories, it is convenient to consider the history of
extrema of the finite temperature potential in this extended range for the
Higgs field including ‘negative’ values. We represent such histories as
diagrams where time flows from left to right and the Higgs field value
increases upwards, with lines showing the evolution, emergence and
disappearance of extrema of the effective potential, see for example Fig. 3.
For reference, the characterisation of possibilities for the extrema history
in the case of the Standard Model with variable Higgs mass are well known but
still useful to cast in this diagrammatic approach. One can have a transition
from the symmetry-restored high temperature phase –with a potential of a
single minimum at $h_{\star}$– to the broken low temperature phase – with a
minimum at $h_{0}$ and a maximum $h_{+}$ sitting where $h_{\star}$ used to be–
via either (i) the splitting of the original minimum into a maximum and
minimum (Fig. 3(b)), or (ii) the formation of a maximum-minimum pair at a
finite distance and with a barrier between the new minimum and the symmetric
minimum (Fig. 3(c)). The latter gives rise to a first order phase transition
while the former contains the case of the SM with its measured couplings,
yielding a cross-over phase transition. The extension to SMEFT allows for
changing the EW transition to possibility (ii), see e.g. Ref. [23], but still
not changing the qualitative picture of possibilities as drawn above.

In non-linear theories by contrast, the range for $h$ need not have bounds,
and the symmetric minimum might lose its natural extrema status (see Ref.
[24]) but further, new minima arise from small deviations to SM couplings. The
extension of the Higgs manifold shown in Fig. 2 is symmetric under the
discrete reflection ($\phi\to-\phi$), which introduces a doubling of the
minima, with a new one arising on the other side of the wormhole. This
symmetric limit with Higgs parity restoration at high temperature – i.e. a
single minimum $h_{\star}$– is shown in Fig. 4(b), history P, and without
parity restoration in Fig. 5. For no high temperature symmetry restoration we
can find high and low temperature configurations as in Fig. 6.

$h_{\star}$$h_{-}$$h_{+}$$h_{0}$4.5{ (a)

$h_{\star}$$h_{-}$$h_{+}$$h_{0}$ (b) History P

$h_{\star}$$h_{-}$$h_{+}$$h_{0}$ (c) History Q-

Figure 4: Extrema histories we encounter in non-linear theories with one
extrema at high temperature and three at low temperature.

All the diagrams shown in Figs. 4–6, are selected not by an artistic whim but
rather because they do occur in the theories here considered. Generically one
needs the further input of potential difference between extrema or barrier
height to determine the phenomenology, but certain diagrams do however
necessarily lead to processes markedly distinct from the SM. An example is
Fig. 5(b) with a single minimum for all history and hence no phase transition;
while this in itself is a qualitative difference from the SM case, it is not
one easily testable. History R in Fig. 5(c) by contrast does necessarily lead
to a phase transition before the original minimum meets a sudden end
encountering a maximum. Histories P,Q and R—Fig. 4 shows Q-, while Q0 is the
parity-reflected case—will be the main cases of this study since they leave
potentially observable traces or are partially ruled out.

$h_{\star}$$h_{0}$4.5{ (a)

$h_{\star}$$h_{0}$ (b)

$h_{\star}$$h_{0}$ (c) History R

Figure 5: Possible extrema histories for a single minimum at high and low
temperature

In Figs. 4–6 we have also introduced notation to label extrema; we will use
$h_{0}$ for the minimum of the potential we find ourselves in today, $h_{+}$
and $h_{-}$ for possible maximum and minimum in the $T=0$ potential. Note that
our definition of $h_{0}$ as our vacuum today, –around which we have e.g.
measured the mass of the Higgs to be $125$ GeV– while conventional, does mean
that any evolution of the universe that leads to the vacuum at $h_{-}$ today
is discarded, without loss of generality.

$h_{\star}$$h_{\star}^{+}$$h_{\star}^{-}$$h_{-}$$h_{+}$$h_{0}$ (a)

$h_{\star}$$h_{\star}^{+}$$h_{\star}^{-}$$h_{0}$ (b)

Figure 6: Initial and $T=0$ extrema we find as realisations in non-linear
theories without high-temperature symmetry restoration.

##  2 Classical action

The tree level Lagrangian for the electroweak sector we use here to model non-
linear theories reads

$\displaystyle\mathcal{L}$ $\displaystyle=\frac{1}{2}\partial
h^{2}+\frac{F^{2}(h)}{2}v^{2}\frac{1}{2}\mbox{Tr}\left[D_{\mu}UD^{\mu}U^{\dagger}\right]-V(h)-\left[\frac{F_{\psi}(h)v}{\sqrt{2}}\bar{\psi}_{L}YU\psi_{R}+\text{h.c.}\right]$
(2.1)
$\displaystyle\equiv\frac{1}{2}d_{\mu}\Phi^{i}G_{ij}d^{\mu}\Phi^{j}-V(h)-\left[\frac{F_{\psi}(h)v}{\sqrt{2}}\bar{\psi}_{L}YU\psi_{R}+\text{h.c.}\right],$
(2.2)

where $F(0)=F_{\psi}(0)=1$ and $v=246$ GeV. Here, $U$ is a special unitary
$2\times 2$ matrix parameterised by 3 scalar degrees of freedom $\varphi^{a}$,
the Nambu-Goldstone bosons, which, together with the Higgs degree of freedom
$h$, span a four-dimensional manifold with coordinates
$\Phi^{i}=(\varphi^{a},h)$. We have used the latin letters $i,j,k$ for the
components of $\Phi$, running from 1 to 4, and $a,b,c$ for Nambu-Goldstone
bosons running from 1 to 3. The gauged covariant derivative
$d_{\mu}\Phi=\partial_{\mu}\Phi+t_{C}A_{\mu}^{C}$, with electroweak bosons
$A_{\mu}^{C}=\\{W_{\mu}^{I},B^{\mu}\\}$, where $I=1,2,3$ and $t_{C}$ are the
killing vectors [3] defined as, together with the metric

$\displaystyle\tilde{g}_{ab}=$
$\displaystyle\frac{v^{2}}{2}\mbox{Tr}\left(\frac{\partial
U}{\partial\varphi^{a}}\frac{\partial
U^{\dagger}}{\partial\varphi^{b}}\right)=v^{2}\frac{\partial
u^{T}}{\partial\varphi^{a}}\frac{\partial u}{\partial\varphi^{b}},$
$\displaystyle G_{ij}$
$\displaystyle=\left(\begin{array}[]{cc}F^{2}\tilde{g}_{ab}&\\\
&1\end{array}\right),$ (2.5) $\displaystyle\frac{t_{I}^{a}}{v^{2}}$
$\displaystyle=\frac{ig\tilde{g}^{ab}}{4}\mbox{Tr}\left(U\frac{\partial
U^{\dagger}}{\partial\varphi^{b}}\sigma_{I}\right)$
$\displaystyle\frac{t_{Y}^{a}}{v^{2}}$
$\displaystyle=\frac{ig_{Y}\tilde{g}^{ab}}{4}\mbox{Tr}\left(U^{\dagger}\frac{\partial
U}{\partial\varphi^{b}}\sigma_{3}\right)$ (2.6)
$\displaystyle=\frac{g\tilde{g}^{ab}}{2}u^{T}T_{I}\frac{\partial
u}{\partial\varphi^{b}},$
$\displaystyle=\frac{g_{Y}\tilde{g}^{ab}}{2}u^{T}T_{Y}\frac{\partial
u}{\partial\varphi^{b}},$ (2.7)

where $u$ is a real four vector of norm one, $u^{T}u=1$, and $\sigma$ are the
Pauli matrices. We use $T$ to stand for a subset of the generators of $SO(4)$
such that $T^{T}=-T$, $Tr[TT]=-4$. There are a number of relations these
objects obey due to unitarity such as $(\partial U)U^{\dagger}=-U\partial
U^{\dagger}$. For simplicity we will take a universal $F$, $F_{\psi}=F$ with
no flavour structure, which for concreteness will read, as a function of the
Higgs field,

$\displaystyle F(h)$
$\displaystyle=\sqrt{s_{\chi}^{2}+c_{\chi}^{2}(1+h/v_{\star})^{2}}=\sqrt{s_{\chi}^{2}+c_{\chi}^{2}(1+\gamma_{a}h/v)^{2}}\,,$
(2.8)

where one has $\gamma_{a}=v/v_{\star}$. In the general formulation of the
HEFT, $F^{2}(h)$ can be an arbitrary analytic function
$F^{2}(h)=1+\sum_{n=1}^{\infty}f_{n}h^{n}$. Our choice corresponds to setting

$f_{1}=\frac{2\cos^{2}\chi}{v_{\star}},\quad
f_{2}=\frac{\cos^{2}\chi}{v_{\star}^{2}},\quad f_{n>2}=0\,,$ (2.9)

which just amounts to imposing that $F^{2}(h)$ is quadratic in $h$ with
generic coefficients. This is enough to model the main feature of our non-
linear theory: a scalar manifold with a double sheet structure, connected by a
throat at a distance $v_{\star}$ from our vacuum $h=0$. It is at the narrowest
point of this throat where $F$ reaches its minimum, and so do electroweak
particle masses. For $\chi>0$, Min$[F(h)]>0$ which we maintain throughout, a
limit of decreasing $\chi$ is depicted in Fig. 2 for both $F$ and the shape of
the manifold. The field-dependent mass term for gauge bosons then reads

$\displaystyle m_{W}(h)$ $\displaystyle=\frac{gv}{2}F(h)\,,$ $\displaystyle
m_{Z}(h)$ $\displaystyle=\frac{v}{2}F(h)\sqrt{g^{2}+g_{Y}^{2}}\,,$ (2.10)

with $F\geq\sin\chi$ over the full range of $h$, and the minimum is achieved
for

$\displaystyle\textrm{Min}[m_{W}(h)]\equiv\frac{gv}{2}F(-v_{\star})=\frac{gv\sin\chi}{2}\,.$
(2.11)

We dub the reflection transformation introduced in the previous section Higgs
parity—not to be confused with other acceptations [25]—whose action on $\phi$,
the distance to the throat of the manifold, reads

$\displaystyle\phi=h+v_{\star}$ $\displaystyle\equiv h+v\gamma_{a}^{-1}\,,$
(2.12) $\displaystyle\boxed{\textrm{Higgs Parity}}\quad$
$\displaystyle\phi\to-\phi\,,$ (2.13)

which as remarked in the introduction is not a symmetry applicable to SMEFT.

A scalar manifold and metric differing from the SM flat case will imply
different couplings and deviations in observables. Scattering matrix elements
are given in terms of tensors in field space projected on the different
directions with vierbeins; in particular, the tensor that dictates scalar
couplings to order $p^{2}$ is the curvature tensor and its covariant
derivatives [2, 4]. One has in our case a Riemann tensor $\mathcal{R}$ that
follows from the metric $G$ as

$\displaystyle\mathcal{R}_{a\,\,c\,\,}^{\,\,b\,\,d}$
$\displaystyle=R_{\varphi}(\tilde{g}_{ac}\tilde{g}^{bd}-\delta_{a}^{d}\delta_{c}^{b})\,,$
$\displaystyle\mathcal{R}_{ah}{}^{bh}$ $\displaystyle=R_{h}\delta_{a}^{b}\,,$
(2.14)

and the only two independent functions

$\displaystyle v^{2}R_{\varphi}(h)$
$\displaystyle\equiv\frac{1}{F^{2}}-v^{2}[(\log
F)^{\prime}]^{2}=\frac{c_{\chi}^{2}(1+h/v_{\star})^{2}(1-c_{\chi}^{2}\gamma_{a}^{2})+s_{\chi}^{2}}{(s_{\chi}^{2}+c_{\chi}^{2}(1+h/v_{\star})^{2})^{2}}\,,$
$\displaystyle v^{2}R_{\varphi}(0)$
$\displaystyle=1-c_{\chi}^{4}\gamma_{a}^{2}\,,$ (2.15) $\displaystyle
v^{2}R_{h}(h)$
$\displaystyle\equiv-\frac{v^{2}F^{\prime\prime}}{F}=-\frac{\gamma_{a}^{2}s_{\chi}^{2}c_{\chi}^{2}}{(s_{\chi}^{2}+c_{\chi}^{2}(1+h/v_{\star})^{2})^{2}}\,,$
$\displaystyle v^{2}R_{h}(0)$
$\displaystyle=-\gamma_{a}^{2}s_{\chi}^{2}c_{\chi}^{2}\,,$ (2.16)

where we also present curvature evaluated at the origin because this is what
LHC is sensitive to and can set bounds on, as in Ref. [9].

Lastly the tree level potential, which might generate new extrema, is a
quartic potential accommodating the Higgs mass but with free cubic and quartic
terms

$\displaystyle V(h)$
$\displaystyle=\frac{m_{h}^{2}}{2}h^{2}+\frac{m_{h}\sqrt{\lambda}}{2}\gamma_{4}(1-\epsilon)h^{3}+\frac{\lambda}{8}\gamma_{4}^{2}h^{4}\,,$
(2.17)

with the SM limit $\gamma_{4}\to 1,\epsilon\to 0$. We have defined
$\sqrt{\lambda}=m_{h}/v\simeq 125/246$ as a way to parametrise our system, but
note this does not imply an SM-like quartic coupling.

(a)

(b)

Figure 7: Schematic diagram of the tree level Higgs potential $V(h)$ in blue
and in red the Higgs-dependent W mass, $v^{2}m_{W}^{2}(h)$, in arbitrary units
to set our conventions. Along the horizontal axis runs the Higgs field $h$.
Fig. 7(a) shows the SM and SMEFT case, with $h\in[\,-v_{+},\infty)\,$, and
Fig. 7(b) shows the non-linear case where $h\in(\,-\infty,\infty)\,$. The
quantities $v_{\pm},v_{*}$ and $\delta$ are defined in Eqs. (2.20), (2.8) and
(2.26) respectively.

The tree-level potential has a minimum by design at $h=0$, where we define the
vacuum we inhabit today. There are two other possible extrema at a distance
$v_{\pm}$ which read, in terms of $\gamma_{4}$, $\epsilon$:

$\displaystyle h_{\mp}(T=0)\equiv-
v_{\mp}\equiv-v\gamma_{4}^{-1}\frac{3(1-\epsilon)\pm\sqrt{1+9(\epsilon^{2}-2\epsilon)}}{2}\equiv\left\\{\begin{array}[]{c}-2v\gamma_{4}^{-1}\gamma_{\epsilon}^{-1}\\\
-v\gamma_{4}^{-1}\gamma_{\epsilon}\end{array}\right.,$ (2.20)

where we have defined

$\gamma_{\epsilon}\equiv\left\\{\begin{array}[]{ll}\frac{3(1-\epsilon)-\sqrt{1+9(\epsilon^{2}-2\epsilon)}}{2}&\qquad\epsilon\leq
1-\sqrt{8/9}\\\ \sqrt{2}&\qquad\epsilon>1-\sqrt{8/9}\end{array}\right..$
(2.21)

Only in the former range of $\epsilon$ are two other extrema present because
for the solutions to be real one needs

$\displaystyle 9(1-\epsilon)^{2}$ $\displaystyle\geq 8\,.$ (2.22)

That is, numerically, $\epsilon\leq 1-\sqrt{8/9}\simeq 0.057$. At the upper
limit of this inequality $\gamma_{\epsilon}=\sqrt{2}$, and the two extrema
meet at $h/v=-\sqrt{2}\gamma_{4}^{-1}$ to form an inflection point, which
informs our definition of $\gamma_{\epsilon}$ for the upper interval. It
should be stressed that this small range for positive $\epsilon$ where two
extrema are present does not guarantee an expansion in small epsilon is
viable, in particular not when close to the upper limit.

The values of the potential at the different extrema are

$\displaystyle\\!V(-v_{\mp})=\frac{\lambda
v^{4}\gamma_{4}^{-2}}{4}\left(\frac{v_{\mp}}{v\gamma_{4}^{-1}}\right)^{\\!2}\\!\left(1+\frac{1-\epsilon}{2}\left(\frac{-v_{\mp}}{v\gamma_{4}^{-1}}\right)\right)=\left\\{\begin{array}[]{c}\gamma_{4}^{-2}\gamma_{\epsilon}^{-2}\lambda
v^{4}(1-\gamma_{\epsilon}^{-1}(1-\epsilon))\\\
\gamma_{4}^{-2}\gamma_{\epsilon}^{2}\lambda
v^{4}(2-\gamma_{\epsilon}(1-\epsilon))/8\end{array}\right..$ (2.25)

Note that only differences in these values are significant, and so we have
chosen $V(0)=0$ in the definition of the potential in Eq. (2.17). It is useful
to define the distance between the local maximum of the potential and the
minimum value of electroweak masses as

$\delta\equiv\left\\{\begin{array}[]{ll}\frac{v_{+}-v_{*}}{v}=\gamma_{4}^{-1}\gamma_{\epsilon}-\gamma_{a}^{-1}&\qquad\epsilon\leq
1-\sqrt{8/9}\\\
\gamma_{4}^{-1}\sqrt{2}-\gamma_{a}^{-1}&\qquad\epsilon>1-\sqrt{8/9}\end{array}\right.,$
(2.26)

and a visual definition of $\delta$ in the presence of two tree-level minima
is given in Fig. 7.

In the limit of small deviation from (local) SM couplings $\epsilon,\delta\ll
1$, we have—noting that this expansion is in the $\epsilon<1-\sqrt{8/9}$
interval,

$\displaystyle\gamma_{\epsilon}$ $\displaystyle\simeq 1+3\epsilon,$
$\displaystyle V(h_{\mp})$
$\displaystyle\simeq\left\\{\begin{array}[]{c}4\gamma_{4}^{-2}\lambda
v^{4}\epsilon+\mathcal{O}(\epsilon^{2})\\\ \gamma_{4}^{-2}\lambda
v^{4}/8+\mathcal{O}(\epsilon)\end{array}\right..$ (2.29)

Figure 8: The effect of the 4 free parameters of our effective Lagrangian on
the tree-level potential $V(h)$ and the $F(h)$ function. Top: variation of the
potential $V(h)$ with $\epsilon$ (left) and $\gamma_{4}$ (right). Bottom:
variation of the $F(h)$ function with $\chi$ (left) and $\delta$ (right).

Now through the definition of $v_{\star}$ in Eq. (2.12), those of $v_{\pm}$ in
Eq. (2.20) and the above, we can write the theory in terms of 4 free
parameters $\delta,\chi,\epsilon$ and $\gamma_{4}$ which control respectively,

  * $\chi$

the width of the throat, defined in Eq. (2.8) and depicted in Figs. 2 and 8.

  * $\epsilon$

the energy difference between the two tree-level minima and the Higgs triple
coupling; this is defined in Eq. (2.17), and its effect depicted in Fig. 8.

  * $\gamma_{4}$

the distance between the two tree level minima and the triple and quartic
Higgs coupling, defined in Eq. (2.17), and depicted in Fig. 8.

  * $\delta$

the distance between the throat and the potential tree level maximum, defined
in Eqs. (2.8) and (2.26) and depicted in Figs. 7 and 8.

As we shall see, this modelling of non-linear theories is versatile enough to
unveil a rich phenomenology of which here we purport to show what we believe
to be a representative sample.

##  3 Modification of sphaleron solutions

The absence of a $O(4)$-symmetric point in the HEFT$\backslash$SMEFT leads to
manifolds topologically different from the SMEFT case. Thus, one may be able
to differentiate both theories, even when they appear to be the same locally
(and hence inseparable by e.g. the LHC), by studying the physics of field
configurations that depend on the global properties of this manifold. In this
section, we focus on the sphaleron configurations which, in principle, rely on
the existence of $O(4)$-symmetric point to exist.

In the SM, a sphaleron is the maximum-energy point in a minimal-energy non-
contractible loop through scalar and gauge field space. This loop can be
approximated by a family of static field configurations depending on a
parameter $\alpha\in[0,\pi]$ as

$\displaystyle h$ $\displaystyle=h(r),$ $\displaystyle U$
$\displaystyle=e^{i\alpha\sigma_{3}/2}\left(\cos(\alpha)+i\sin(\alpha)\,\hat{x}\cdot\vec{\sigma}\right)e^{i\alpha\sigma_{3}/2},$
(3.1) $\displaystyle\ W_{0}$ $\displaystyle=0,$ $\displaystyle W_{j}$
$\displaystyle=-\frac{i}{g}a(r)U\partial_{j}U^{\dagger}$ (3.2)

where $\hat{x}=\vec{x}/|x|$ is the unit vector in space, $U$ and $W$ are the
$SU(2)$ Goldstone and gauge fields111The effects of the $U(1)$ sector are
neglected here which is an approximation valid up to corrections of order
$g^{\prime}/g$, see Ref. [26]., and $h(r)$ and $a(r)$ are radial functions
that minimise the energy and satisfy the boundary conditions

$h(0)=h_{\odot},\qquad
a(0)=0,\qquad\lim_{r\to\infty}h(r)=0,\qquad\lim_{r\to\infty}a(r)=1,$ (3.3)

with $h=h_{\odot}$ being the $O(4)$-invariant point. The fact that
$U|_{\alpha=0}=U|_{\alpha=\pi}=1$ ensures that the curve described in field
space by varying $\alpha$ is indeed a loop. Furthermore, $U$ can be viewed as
a topologically non-trivial mapping from the 3-sphere into itself, and
therefore the loop is not contractible (see Ref. [27] for the original proof).

The sphaleron is the configuration at the midpoint of the loop, with
$\alpha=\pi/2$. In order for it to be well-defined everywhere in space, it is
crucial that $h(0)=h_{\odot}$. This is because $U|_{\alpha\neq 0,\pi}$ is
singular at the origin. Indeed, taking the limit $r\to 0$ from different
spatial directions will yield different $U$ matrices. This makes the field
configuration undefined at $r=0$, unless all values of $U$ can be identified
and collapsed into a single point, making it single-valued. This can only be
done if there is a $O(4)$-invariant point $h_{\odot}$ where $F(h_{\odot})=0$.
Therefore, fully well-defined sphaleron configurations cannot exist in
HEFT$\backslash$SMEFT theories which have $F(h)>0$ for all $h$.

One can nevertheless work with the definition we have given for $U$ away from
the origin, and see if the singularity there has physical consequences. In
particular, in order for it to be physically meaningful, the sphaleron
solution must have finite energy, using an ansatz as that of Eqs. (3.1) with
$\alpha=\pi/2$:

$\displaystyle E$ $\displaystyle=\int
d^{3}x\left[\frac{4}{g^{2}r^{2}}\left(\frac{da}{dr}\right)^{2}+\frac{8a^{2}(1-a)^{2}}{g^{2}r^{4}}+\frac{1}{2}\left(\frac{dh}{dr}\right)^{2}+\frac{v^{2}F(h)^{2}}{r^{2}}(1-a)^{2}+V(h,\Phi)\right].$
(3.4)

While the energy density around $r=0$ would diverge as $1/r^{2}$ if $F(h)\neq
0$, the volume integral cancels this divergence for a finite result. Another
road to reach the same finiteness conclusion is to remember that the sphaleron
energy remains finite when $\lambda\to\infty$ in the SM, i.e. we freeze the
Higgs field value on its vacuum and take the particle out of the spectrum. It
should be noted nonetheless that had we been dealing with a 2-dimensional
theory, the solutions would have indeed disappeared.

The question that follows is then how is the sphaleron energy, which we now
know to be finite, modified. This energy does know about the Higgs manifold
non-locally and is, as such, qualitatively different from, say, scattering
experiments. In particular, the relevant question (even if just theoretical)
as posed in Ref. [9] is whether a non-linear theory locally SM-like around the
vacuum but globally different can be told apart by the sphaleron energy.

A smooth set of functions $F_{n}$ for non-linear theories that look locally
ever more SM like for increasing integer $n$ were proposed in Ref. [9], here
we rewrite them as

$\displaystyle F_{n}$
$\displaystyle=1+\frac{h}{v}+c_{n}\left(\frac{h}{v}\right)^{n},$
$\displaystyle c_{n}$
$\displaystyle=(-1)^{n}\frac{(n-1)^{n-1}}{n^{n}(1-F_{\star})^{n-1}}.$ (3.5)

These functions have a minimum $F_{\star}$ at
$f_{\star}=-v_{\star}/v=-n(1-F_{\star})/(n-1)$, we take $0<F_{\star}<1$ with
the lower limit in particular to keep a non-linear type A theory. We use them
as a probe for approaching the SM while in non-linear theory space. As the
origin boundary condition for the Higgs, we take as $h(0)=-v_{\star}$ where
$F^{\prime}(-v_{\star})=0$.

(a)

(b)

Figure 9: Fig. 9(a): sphaleron energy as a function of $n$ for different values of $F_{\star}$. Fig. 9(b): square root of the difference between the limit value of the energy as $n\to\infty$ (taken from $n=10$) and the SM energy as a function of $F_{\star}$, together with a linear fit. Ansatz |  | $n$
---|---|---
| 1(SM) | 2 | 3 | 4 | 5 | 6
|  | $E_{Sph}$ | 1.948 | 2.079 | 1.991 | 1.971 | 1.966 | 1.966
A) | Eqs. (3.11), (3.14) | $\zeta_{a}$ | 1.4 | 1.2 | 1.4 | 1.4 | 1.4 | 1.4
|  | $\zeta_{h}$ | 1.6 | 2.4 | 2.0 | 1.8 | 1.7 | 1.6
B) | Neural network | $E_{Sph}$ | 1.919 | 2.053 | 1.976 | 1.952 | 1.943 | 1.938
Table 1: Values of the sphaleron energy (and corresponding radii $\zeta_{a}$
and $\zeta_{h}$) at $F_{\star}=0.2$ and varying $n$, as computed with methods
A) and B).

The dimensionless radius $\zeta=gvr$ simplifies the energy to read

$\displaystyle E=$ $\displaystyle\frac{4\pi v}{g}\int
d\zeta\left[4\dot{a}^{2}+\frac{8a^{2}(1-a)^{2}}{\zeta^{2}}+\frac{\zeta^{2}}{2}\dot{f}^{2}+F(vf)^{2}(1-a)^{2}+\zeta^{2}\frac{\lambda}{8g^{2}}(f^{2}+2f)^{2}\right],$
(3.6)

with $f=h/v$. The sphaleron is obtained by minimising this energy as a
function of the profiles $a(\zeta)$ and $f(\zeta)$. We use two different
approaches to minimize it for different values of $n$ in the $F_{n}$ functions
defined above:

A)

Analytical ansatz. While, due to their non-linear character, one cannot solve
the Euler-Lagrange equations that follow from the variational principle for
the energy $E$:

$\displaystyle-8\ddot{a}+16\frac{a(1-1)(1-2a)}{\zeta^{2}}-2(1-a)F(f)^{2}=0,$
(3.7)
$\displaystyle-\zeta^{2}\ddot{f}-2\zeta\dot{f}+2FF^{\prime}(f)(1-a)^{2}+\zeta^{2}\frac{\lambda}{g^{2}}f(1+f/2)(1+f)=0,$
(3.8)

one can solve their asymptotic form in the limits $\zeta\to 0,\infty$.
Following Ref. [26], we devise an ansatz as a piece wise function with the
solutions to the asymptotic equations which we join at a radius $\zeta_{a}$
($\zeta_{h}$ for the Higgs function). Such a function has therefore 4
integration constants, two of which are fixed by the boundary conditions,
while the other two are found by imposing the same value for the function and
its first derivative at $\zeta_{a}$ ($\zeta_{h}$). This leads to

$\displaystyle\hat{a}(\zeta)$
$\displaystyle=\left\\{\begin{array}[]{lcr}\zeta^{2}\left(\frac{\zeta_{a}F_{\star}^{2}+6}{6\zeta_{a}(4+\zeta_{a})}-\frac{F_{\star}^{2}}{12}\log(\zeta/\zeta_{a})\right)&&\zeta<\zeta_{a}\\\
1-\frac{4-F_{\star}^{2}\zeta_{a}^{2}/6}{4+\zeta_{a}}e^{-(\zeta-\zeta_{a})/2}&&\zeta>\zeta_{a}\end{array}\right.,$
(3.11) $\displaystyle\hat{f}(\zeta)$
$\displaystyle=\left\\{\begin{array}[]{lcr}f_{\star}\left(1-\frac{p+\sigma\zeta_{h}}{2p+\sigma\zeta_{h}}\left(\frac{\zeta}{\zeta_{h}}\right)^{p}\right)&&\zeta<\zeta_{h}\\\
f_{\star}\frac{p\zeta_{h}}{2p+\sigma\zeta_{h}}\frac{1}{\zeta}e^{-\sigma(\zeta-\zeta_{h})}&&\zeta>\zeta_{h}\end{array}\right.,$
(3.14)

where $\sigma=m_{h}/(2m_{W})$,
$2p=(\sqrt{1+8F^{\prime\prime}_{\star}F_{\star}}-1)$. Substituting this ansatz
back in the energy expression Eq. (3.7) allows for minimisation in two
variables $\zeta_{a},\zeta_{h}$ which we do numerically. This semi-numerical
method will not yield the true minimum energy since the ansatz are not
solutions to the full equations, but it does have the advantage of treating
boundary conditions analytically. Results are shown in Tab. 1.

B)

Numerical ansatz. Directly minimise $E$ numerically, employing a small neural
network as an ansatz for $f$ and $a$. We use the Elvet package [28] for this
purpose. The neural network is densely connected, with 1 input ($\zeta$), two
outputs ($f$ and $a$) and 2 hidden layers with 5 units each.

Both approaches lead to similar results, shown in Fig. 9, with B) giving
slightly lower energies than A) as expected. From these results, it is clear
that, as $n\to\infty$, the sphaleron energies tend to a value differing from
the SM one, provided $F_{\star}\neq 0$. We find that the difference between
this value and the SM one is approximately proportional to $F_{\star}^{2}$, as
displayed in Fig. 9(b). This non-pertubative phenomenon is hence sensitive to
non-linear theories that would be indistinguishable for a scattering
experiment. In practice, however, finding percent deviations in sphaleron
energy is not a feasible strategy to explore these theories. As we shall show,
phase transitions on the other hand offer a much more promising avenue.

##  4 Finite temperature potential

The key quantity to study phase transitions and defect formation is the
effective potential at finite temperature. The computation of this potential
is not without obstacles, long since identified and discussed in the
literature extensively, see e.g. Ref. [29] for a review. In such context, here
we opt for laying out our derivation with all assumptions and choices explicit
while emphasising the limit of applicability of our results. In addition by
facing old problems from a new theory, we believe this work might bring novel
approaches and perspective.

The effective potential will be computed to one loop at finite temperature.
This suffices to chart possible new phenomena that HEFT brings to the
electroweak phase transition. Before proceeding to the computation, however,
let us first discuss its limitations. The potential estimate here is not valid
in all the field domain since perturbation theory breaks down in thermal field
theory for gauge theories in the limit of vanishing transverse mode mass (i.e.
the infrared problem [30])– for the EW theory this occurs around the $O(4)$
fixed point. While the perturbative expansion can be supplemented with
resummation techniques and dimensional reduction as detailed in Ref. [29,
31],222Another promising direction using nonperturbative methods is presented
in [32]. the deciding say lies in lattice. Rather than attempting to extend
the potential computation to this domain beyond perturbativity, we will mark
its limits and base our conclusions on results outside of it. It is worth
noting already that the IR problem arises as the masses of gauge bosons
approach zero and the presence of a lower bound on such masses in our theory
ameliorates the problem.

A second relevant consideration is that our potential, since it is computed in
an EFT, is not valid to arbitrarily high temperatures. The amplitude for
longitudinal boson scattering scales as $R\,s$, where $s$ is the square of the
centre of mass energy. This points to a cutoff where new states would appear
at $\Lambda\sim 4\sqrt{\pi}|R(0)|^{-1/2}$ [9], see also [6] where unitarity
leads to an estimate of $\Lambda\sim 4\pi v$. Experimental data from LHC
constrains the curvature around the EW vacuum at zero temperature to be small
so that the cut-off is above the TeV. In the evolution of the universe,
however, the electroweak vacuum changes position in field space and scans over
a range of values. Approximating as customary the thermally averaged cross-
section in this environment as the zero temperature result convoluted with the
thermal distributions, one can expect the cut-off estimation outlined above
for $h=0$ to be extended to different manifold values $\Lambda(h)\sim
4\sqrt{\pi}|R(h)|^{-1/2}$. This naive extension of the cut-off estimate to
thermal field theory ignores the extra Boltzmann suppression that the
contribution of heavy states receive (indeed UV divergences in thermal field
theory are much softened with respect to zero temperature), but doing so is
erring on the conservative side. We therefore consider the cut-off for the
whole range that the Higgs field explores, and one has that it would be lowest
at the highest curvature, i.e. the throat, see Fig. 10. At the throat one has
$\Lambda\sim
4\sqrt{\pi}[R_{\varphi}(h=-v_{\star})]^{-1/2}=4\sqrt{\pi}v\sin(\chi)=1.7\sin(\chi)$
TeV, imposing a minimum value for $\chi$ for our EFT to be applicable, and we
impose a similar expression for $R_{h}$. In the following phenomenological
study it is ensured that the cut-off is at least $450$ GeV at the throat for
the allowed parameter space.

Figure 10: Field-dependent cut-off as estimated from perturbative unitarity to
be $4\sqrt{\pi}\left|R_{I}(h)\right|^{-1/2}$ for sectional curvatures
$R_{\varphi}(h)$ (solid) and $R_{h}(h)$ (dashed) as defined in Eqs. (2.15) and
(2.16). The parameter $\chi=\sqrt{0.1}$ and the following
$\gamma_{a}=v/v_{\star}$ values are chosen as those that lead to domain walls,
$\gamma_{a}=1$ (see Sec. 5); the doom scenario, $\gamma_{a}=1.1$ (see Sec. 6.2
and Fig. 27 in Sec. 8); and $\gamma_{a}=1.2$ a conservative value for the
observable bubble region shown in Fig. 26. For comparison, the nucleation
temperature in phase transitions studied here is around $100$ GeV.

###  4.1 Calculation of the one loop finite temperature potential

The geometric description presents relevant differences with the standard
computation of one loop corrections for the effective potential, so let us
sketch them here in some length. Path integral methods allow us to write one
loop corrections in a few lines of algebra, while the notation is abstract
enough to encompass both the finite and zero temperature case which we discuss
simultaneously. In this functional computation one expands around a background
value for the fields $\bar{\Phi}$, which satisfy the EoM at one loop order,
along geodesics if the field space has curvature. This expansion reads, with
$\delta\Phi$ being the field variation we will integrate over,

$\displaystyle\Phi^{i}[\delta\Phi]$
$\displaystyle=\bar{\Phi}^{i}+\delta\Phi^{i}-\frac{1}{2}\Gamma^{i}_{jk}(\bar{\Phi})\delta\Phi^{j}\delta\Phi^{k}+\mathcal{O}(\delta\Phi)^{3},$
$\displaystyle A^{\mu}$ $\displaystyle=\bar{A}^{\mu}+\delta A^{\mu},$ (4.1)

with $\Gamma$ the connection derived from the metric $G$. An expansion for
fermions can be similarly derived to that for gauge bosons.

The path integral for the effective action $\Gamma_{\textrm{eff}}$ to one loop
reads

$\displaystyle
e^{i\Gamma_{\textrm{eff}}}=e^{iS[\bar{\Phi},\bar{A}]}\int\sqrt{G}[\mathcal{D}\delta\Phi][\mathcal{D}\delta
A]e^{i\frac{1}{2}\delta_{\Phi}^{2}S+i\frac{1}{2}\delta^{2}_{A}S+i\delta_{\Phi}\delta_{A}S},$
(4.2)

where $\sqrt{G}$ is the scalar metric inserted for an invariant volume element
and the second variation around the background field for gauge bosons is a
standard result, whereas for the scalars we have, when applied to the action
in Eq. (2.1),

$\displaystyle\delta_{\Phi}^{2}S=$ $\displaystyle\int
d^{4}x\delta\Phi^{i}\left(-G_{ij}D^{2}-\nabla_{i}\nabla_{j}V\right)\delta\Phi^{j},$
$\displaystyle\nabla_{i}\nabla_{j}V$
$\displaystyle=\left(\begin{array}[]{cc}V^{\prime\prime}&0\\\
0&FF^{\prime}V^{\prime}\tilde{g}_{ab}\end{array}\right),$ (4.5)

with $\nabla$ the covariant derivative w.r.t. field coordinates and $D$ the
covariant spacetime derivative acting in the scalar manifold tensor space [3],
which hereafter setting the background field to a constant equals the ordinary
derivative. The extension to higher loop order can be found in Ref. [14]. It
is well worth remarking that the mass term for Goldstones,
$FF^{\prime}V^{\prime}$, (which vanishes at the vacuum) will be altogether
missed in this coordinate system if not doing a covariant treatment. This non-
covariant treatment would lead to inconsistency as one can realise by
resorting to the SM with linear scalar coordinates where all 4 d.o.f. have a
mass term away from the vacuum; in essence the point is that even in flat
space, when in spherical coordinates, one has to use a covariant formulation.
The mixed term in gauge and scalar fields one can borrow from Ref. [33]:

$\displaystyle\delta_{\Phi}\delta_{A}S=\int d^{4}x[-\delta
A_{B}D_{\mu}t_{B,i}\delta\Phi^{j}].$ (4.6)

This term can be absorbed in a redefinition of the path-integral fields
without changing the measure as done in Ref. [33], but here instead as is more
common practice in thermal field theory we cancel it out with the introduction
of a gauge fixing term

$\displaystyle\mathcal{L}_{g.f.}=-\frac{1}{2\xi}\sum_{B}\left(\xi
t^{i}_{B}G_{ij}\delta\varphi^{j}+D_{\mu}\delta A^{\mu}_{B}\right)^{2},$ (4.7)

which puts our gauge bosons in the $R_{\xi}$ gauge while producing an extra
mass term for the scalars as

$\displaystyle\delta^{2}_{\Phi}(S+S_{g.f.})=-\delta\Phi(D^{2}+\nabla^{2}V+\xi
m_{gf}^{2})\delta\Phi,$ (4.8)

with the additional gauge dependent term

$\displaystyle\left[m_{gf}^{2}\right]^{i}_{\,j}=$
$\displaystyle\sum_{B}t_{B}^{i}t_{j,B}=\frac{(Fv)^{2}}{4}\left(\begin{array}[]{cc}g^{2}\delta_{a}^{b}+g_{Y}^{2}\tilde{g}^{ac}v^{2}u^{T}T_{Y}\frac{\partial
u}{\partial\varphi^{c}}\,u^{T}T_{Y}\frac{\partial u}{\partial\varphi^{b}}&0\\\
0&0\end{array}\right),$ (4.11)

with Tr$[m_{gf}^{2}]=2m_{W}^{2}(h)+m_{Z}^{2}(h)$. We note that the last term
does depend on the angular degrees of freedom, the Goldstones $\varphi$; this
does not mean they will feature in the effective potential (which could only
come about via explicit breaking) but that the final result, when all tensors
have had their indices contracted, will make any Goldstone boson dependence
disappear. The second variation of the action for gauge bosons on the other
hand is

$\displaystyle\delta^{2}_{A}(S+S_{g.f.})=\delta
A^{\mu}\left(\eta_{\mu\nu}D^{2}-(1-\xi^{-1})D_{\mu}D_{\nu}+m_{A}^{2}\right)\delta
A^{\nu}.$ (4.12)

When it is only the effective potential one is after, derivatives of the
background field can be neglected, and we obtain loop corrections as, for the
scalar correction

$\displaystyle-i\int d^{4}xV_{\hbar\Phi}$
$\displaystyle=-\frac{1}{2}\mbox{Tr}\left[\log\left(G\partial^{2}+\nabla^{2}V+\xi
m_{gf}^{2}\right)\right]+\frac{1}{2}\mbox{Tr}\log(G)$ (4.13)
$\displaystyle=-\frac{T}{2}\mbox{Tr}\left[\log\left(\delta_{i}^{j}\partial^{2}+\nabla_{i}\nabla^{j}V+\xi[m_{gf}^{2}]_{i}^{j}\right)\right].$
(4.14)

The trace is over internal indexes, position and momentum, and for the full
one loop corrections one adds over fermions and gauge bosons. This trace can
now be made specific to thermal (with periodic time of interval $1/T$) and
$T=0$ corrections as (with momentum in Euclidean space)

$\displaystyle V_{1-\textrm{loop}}$
$\displaystyle=\frac{1}{2}\int\frac{d^{4}\ell}{(2\pi)^{4}}\mbox{tr}\left[\log\left(\ell^{2}+\nabla^{2}V+\xi
m_{gf}^{2}\right)\right],$ (4.15) $\displaystyle V_{\textrm{Th}}$
$\displaystyle=\frac{T}{2}\sum_{n}\int\frac{d^{3}\ell}{(2\pi)^{3}}\mbox{tr}\left[\log\left(E_{n}^{2}+\ell^{2}+\nabla^{2}V+\xi
m_{gf}^{2}\right)\right],$ (4.16)

where the trace is now over scalar indexes, and the 0 components of the
momenta are the Matsubara frequencies [34] in $V_{\textrm{Th}}$. While the
first term requires counterterms, the second only has a field independent
divergence that one can leave aside. Let us renormalise the corrections in
vacuo next by dividing our effective potential into

$\displaystyle V_{\textrm{eff}}(h,T)$ $\displaystyle=V(h)+\Delta
V_{CW}(h)+\Delta V_{\textrm{Th}}(h,T),$ (4.17)

where the one loop correction at $T=0$, $\Delta V_{CW}$ has the form, with cut
off regularisation,

$\displaystyle\Delta V_{CW}$ $\displaystyle=\sum_{i}\left[\Delta
V_{\textrm{1-loop},i}+\Delta V_{\textrm{1-loop}.i}^{c.t}\right],$ (4.18)
$\displaystyle\Delta V_{\textrm{1-loop},i}$
$\displaystyle=\frac{1}{64\pi^{2}}n_{i}\left(m_{i}^{4}\left(\log(m_{i}^{2}/\Lambda^{2})-\frac{1}{2}\right)+2\Lambda^{2}m_{i}^{2}\right),$
(4.19)

where $i$ runs over $\Phi$ scalars, electroweak gauge bosons, ghosts and
fermions, for the latter for practical purposes, as all other fermions
contribute negligibly, only the top. There are three types of masses as far as
the field dependence is concerned which will dictate the number of counterterm
operators needed

$\displaystyle F^{2}(h),$ $\displaystyle V^{\prime\prime}(h),$
$\displaystyle\frac{V^{\prime}(h)F^{\prime}(h)}{F(h)}.$ (4.20)

The counterterms needed are of the form either of the three above or their
square and due to functional dependence similarities they amount to 6 terms; 4
could be taken as the tree level terms present in the Higgs potential and the
other two from $(VF^{\prime}/F)$ are rational rather than polynomial and
require an additional two counterterms. Equivalently here we take

$\displaystyle V^{c.t}_{\textrm{1-loop},i}=$ $\displaystyle
a_{i}m_{i}^{4}(h)+b_{i}m_{i}^{2}(h),$ Counter-terms (4.21)

Which are fixed by the renormalisation conditions here imposed as

$\displaystyle\left.\frac{dV_{CW,h}}{dh}\right|_{h=0}=\left.\frac{d^{2}V_{CW,h}}{dh^{2}}\right|_{h=0}=\left.\frac{dV_{CW,\varphi}}{dh}\right|_{h=0}=\left.\frac{d^{2}V_{CW,\varphi}}{dh^{2}}\right|_{h=0}=0,$
(4.22)
$\displaystyle\sum_{A_{B},t}\left.\frac{dV_{CW,i}}{dh}\right|_{h=0}=\sum_{A_{B},t}\left.\frac{d^{2}V_{CW,i}}{dh^{2}}\right|_{h=0}=0.$
(4.23)

These imply that $h=0$ will stay a minimum of the potential with mass $m_{h}$.
The vanishing of the field dependent mass for $\varphi$ conflicts with these
renormalisation conditions in Landau’s gauge. While one can, as in Ref. [35],
approach this problem by resummation, here we avoid this gauge and rather in
the following select for concreteness Feynman’s gauge, $\xi=1$.

The thermal contributions on the other hand are

$\displaystyle\Delta V_{\textrm{Th}}$
$\displaystyle=\frac{T^{4}}{2\pi^{2}}\sum_{i}n_{i}J_{S_{i}}\left(\frac{m_{i}(h)^{2}}{T^{2}}\right),$
(4.24)

with $n_{i}$ the number of degrees of freedom and $S_{i}$ identifying the
statistics, Bose-Einstein ($b$), or Pauli-Dirac ($f$); explicitly

$\displaystyle
J_{b/f}(x^{2})=\int_{0}^{\infty}y^{2}dy\log\left(1\mp\exp{\left[-\sqrt{x^{2}+y^{2}}\right]}\right).$
(4.25)

All the above leads to the 1-loop finite temperature potential

$\displaystyle
V_{\textrm{eff}}=V(h)+\sum_{i}n_{i}\left\\{\frac{m_{i}^{2}(h)}{64\pi^{2}}\left(m_{i}^{2}(h)\left[\log\left(\frac{m_{i}^{2}(h)}{m_{i}^{2}(0)}\right)-\frac{3}{2}\right]+2m_{i}^{2}(0)\right)+\frac{T^{4}}{2\pi^{2}}J_{S_{i}}\left(\frac{m_{i}^{2}(h)}{T^{2}}\right)\right\\}.$
(4.26)

Each term in this sum, in our Feynman gauge, is

$\displaystyle m_{t}^{2}(h)$ $\displaystyle=m^{2}_{t}(0)F(h)^{2},$
$\displaystyle n_{t}$ $\displaystyle=-12,$ $\displaystyle m^{2}_{h}(h)$
$\displaystyle=V^{\prime\prime}(h),$ $\displaystyle n_{h}$ $\displaystyle=1,$
(4.27) $\displaystyle m^{2}_{W_{T}}(h)$ $\displaystyle=m^{2}_{W}(0)F(h)^{2},$
$\displaystyle n_{W_{T}}$ $\displaystyle=4,$ $\displaystyle m^{2}_{W_{L}}(h)$
$\displaystyle=\frac{F^{\prime}V^{\prime}}{F}+m^{2}_{W_{T}}(h),$
$\displaystyle n_{W_{L}}$ $\displaystyle=2,$ (4.28) $\displaystyle
m^{2}_{Z_{T}}(h)$ $\displaystyle=m^{2}_{Z}(0)F(h)^{2},$ $\displaystyle
n_{Z_{T}}$ $\displaystyle=2,$ $\displaystyle m^{2}_{Z_{L}}(h)$
$\displaystyle=\frac{F^{\prime}V^{\prime}}{F}+m^{2}_{Z_{T}}(h),$
$\displaystyle n_{Z_{L}}$ $\displaystyle=1.$ (4.29)

In the following, to avoid cluttered notation, a mass with no argument
($m_{W}$) is to be understood as a constant, the mass measured at the vacuum;
for a field-dependent mass the dependence will be made explicit, e.g.
$m_{W}(\phi)$. It is useful to extend the extrema defined in Sec. 2 to be the
extrema of the finite temperature effective potential, $h_{\pm}(T)$,
$h_{0}(T)$ so that the end-point of the temperature ‘trajetory’ returns
$h_{0}(0)=0$, $h_{\pm}(0)=-v_{\pm}$. On the other hand, potential differences
and barriers read

$\displaystyle\Delta V(T)$
$\displaystyle=V_{\textrm{eff}}(h_{-}(T),T)-V_{\textrm{eff}}(h_{0}(T),T),$
(4.30) $\displaystyle U_{0}(T)$
$\displaystyle=V_{\textrm{eff}}(h_{+}(T),T)-V_{\textrm{eff}}(h_{0}(T),T),$
(4.31) $\displaystyle U_{-}(T)$
$\displaystyle=V_{\textrm{eff}}(h_{+}(T),T)-V_{\textrm{eff}}(h_{-}(T),T).$
(4.32)

At high temperatures, given the importance of electroweak particle
corrections, the relevant point in field space is to be found around
$h=-v_{\star}$ rather than our vacuum at $h=0$. For this reason, we will write
in the following the potential as a function of $\phi$ as in Eq. (2.12) where
e.g.

$\displaystyle\phi_{\pm}$ $\displaystyle=h_{\pm}+v_{\star}\,,$
$\displaystyle\phi_{0}$ $\displaystyle=h_{0}+v_{\star}\,.$ (4.33)

###  4.2 Limitations of the calculation

Were one to aim at more precision, the present approximation can be improved
by higher zero temperature or thermal loop corrections. These however suffer
from an infrared illness as outlined in the beginning of this section. This
problem is identified when the expansion parameter controlling the IR
divergences

$\displaystyle\varepsilon_{IR}\equiv\frac{g^{2}T}{\pi m_{W}(\phi)},$ (4.34)

ceases to be small.333A similar effective expansion parameter to Eq. (4.34)
exists for both the $Z$ and $h$ bosons. While some amelioration can be
provided by daisy resummation [36], the ‘magnetic modes’ stay massless in
perturbation theory and make these divergences unavoidable [30]. Whenever the
expansion parameter is larger than one, we simply cannot trust perturbation
theory and must resort instead to non-perturbative methods in the form of
lattice. Luckily, we do not need access to this non-perturbative region of the
parameter space to unveil a range of novel phenomenology. We address the IR
problem by identifying the region of the field values $\phi$ that possess a
controlled expansion $\varepsilon_{IR}(\phi)<1$, and only there do we trust
the analytically and perturbatively computed potential. The expression that
determines this region in $\phi$ reads

$\displaystyle\left(\frac{\phi}{T}\right)^{2}$
$\displaystyle>\gamma_{a}^{-2}\left[\left(\frac{2g}{c_{\chi}\pi}\right)^{2}-\tan^{2}\chi\frac{v^{2}}{T^{2}}\right],\qquad\boxed{\textrm{IR
Bound}}$ (4.35)

where we note that for temperatures below $T_{IR}=\pi s_{\chi}v/(2g)$, the RHS
is negative and this constraint disappears for a controlled expansion in the
entire range of $\phi$. Above $T_{IR}$ nonetheless the bound applies, and it
implies that for small values of $\phi$ the computation of the potential is
unreliable.

Let us exemplify how this IR bound is used here in practice by taking the SM
as a case study. The SM potential for high temperature is a 4th degree
polynomial in the fields and the extrema can be solved for analytically, for a
pedagogical exposition see Sec. 4.1 of [34]. One finds that a minimum-maximum
pair appears at a temperature $T_{1}$ as an inflection point at a distance
$\phi_{0}(T_{1})/T_{1}=3(2m_{W}^{3}+m_{Z}^{3})/(4\pi\lambda v^{3})\propto
1/m_{h}^{2}$ from the origin. This minimum becomes degenerate in potential
energy with the origin at the critical temperature $T_{c}$ and sits at a
distance $\phi_{0}(T_{c})/T_{c}=(4/3)\phi_{0}(T_{1})/T_{1}$ from it. After
$T_{c}$, we would naively expect to have a first order phase transition. For
this perturbative argument to be trustworthy, however, one should require the
new minimum to be in the region $\varepsilon_{IR}(\phi_{0})<1$. The region
$\varepsilon_{IR}(\phi)>1$ is an interval around the origin ($\phi=0$) which
shrinks with time and disappears at $T_{IR}$, whereas the distance of
$\phi_{0}$ to $\phi=0$ generally increases monotonically. It suffices then to
specify the earliest time/highest temperature at which this condition is
imposed: if satisfied at a temperature $T$, lower temperatures will continue
to respect the bound. We distinguish two possibilities:

  * •

Weak IR constraint. The new minimum should be at a position where the IR
expansion is under control at the critical temperature, that is

$\displaystyle\varepsilon_{IR}(\phi_{0}(T_{c}))<$ $\displaystyle 1,$ (4.36)

which when applied to the SM returns

$\displaystyle\textrm{SM:}\quad\frac{g}{\pi}\leq\frac{(2m_{W}^{3}+m_{Z}^{3})}{2\pi\lambda
v^{3}},\qquad\qquad
m_{h}\leq\sqrt{\frac{(2m_{W}^{3}+m_{Z}^{3})}{2gv^{3}}}v\simeq 75\text{ GeV},$
(4.37)

which translates into an upper bound on the Higgs mass for a reliable first
order phase transition prediction, a well known qualitative condition here
made quantitative as outlined.

  * •

Strong IR constraint. The new minimum should be at a position where the IR
expansion is under control already at the temperature when it first appears,
that is

$\displaystyle\varepsilon(\phi_{0}(T_{1}))<$ $\displaystyle 1,$ (4.38)

which applied to the SM gives

$\displaystyle\textrm{SM:}\quad\frac{g}{\pi}\leq\frac{3(2m_{W}^{3}+m_{Z}^{3})}{8\pi\lambda
v^{3}},\qquad\qquad
m_{h}\leq\sqrt{\frac{3(2m_{W}^{3}+m_{Z}^{3})}{8gv^{3}}}v\simeq 65\text{ GeV}.$
(4.39)

This a stronger demand and as such demands a smaller allowed ranged of the
Higgs field.

None of these bounds, however, are satisfied in the SM given the measured
Higgs mass, and one cannot affirm there is a first order phase transition; in
fact lattice computations reveal that the SM with the measured couplings
presents instead a crossover transition [37, 38]. The bounds we obtain are not
far from the actual value for the first order phase transition (1OPT) endpoint
as determined by lattice $72\pm 7$ GeV [39] and are also in line with more
explicit estimates of the loop expansion breakdown [40].

In our scenario with an extended range for $\phi$ the application is
analogous, and we abstain from making predictions at temperatures where the
minima fall in the $\varepsilon_{IR}>1$ region. To be explicit and consistent
across cases, we demand that in a 1OPT, the minimum $\phi_{0}$ satisfies
$\varepsilon_{IR}(\phi_{0})<1$ at the critical temperature, $T_{c}$, (weak
constraint) or the temperature when the minima first appears, $T_{1}$ (strong
constraint). The diagrams for the histories of extrema allow for a
visualisation of the strong IR constraint, as shown in Fig. 11.

$h_{\star}$$h_{-}$$h_{+}$$h_{0}$Strong IR constraint compliant

$h_{\star}$$h_{-}$$h_{+}$$h_{0}$Strong IR constraint non-compliant

Figure 11: Schematic representation of thermal histories for the extrema of
the effective potential satisfying (left) and violating (right) the strong IR
constraint.

Lastly, the large field regime of the $T=0$ potential might suffer from the
same instability as the SM [41] but in some instances is aggravated. One has,
in the theory under study

$\displaystyle V_{\text{eff}}(\phi\gg v)$
$\displaystyle\simeq\frac{\lambda\gamma_{4}^{2}}{8}\phi^{4}\left(1+\frac{3L(\phi)}{8\pi^{2}\gamma_{4}^{2}}\left[\frac{2m_{W}^{4}+m_{Z}^{4}-4m_{t}^{4}}{m_{h}^{2}v^{2}}\gamma_{a}^{4}c_{\chi}^{4}+\frac{m_{h}^{2}}{v^{2}}\gamma_{4}^{4}+\frac{2m_{W}^{2}+m_{Z}^{2}}{3v^{2}}\gamma_{a}^{2}c_{\chi}^{2}\gamma_{4}^{2}\right]\right),$
$\displaystyle\equiv\frac{\lambda\gamma_{4}^{2}}{8}\phi^{4}\left(1+a_{\rm
loop}L(\phi)\right)$ (4.40)

with $L(\phi)=\log(\phi^{2}/v^{2})$. In the SM case, top quark contributions
drive the effective quartic coupling towards negative values and the
electroweak minimum is metastable when field values are extended to the Planck
mass. In the present case we will address the same problem with two
prescriptions, perturbativity and boundedness-from-below.

The constraint that we denote perturbativity follows from demanding a loop
correction subleading to the tree level one;444Please note this constraint
differs from unitarity bounds since we demand a convergent loop expansion
rather than an effective coupling value below the unitarity limit, which in
this study would be a subdominant constraint to perturbativity as here
defined. given the dependence in Eq. (4.40) this demand is strongest at the
edges of the field range, and it explicitly reads $a_{\rm loop}L(4\pi v)<1$.
These edges should be chosen to capture a large enough domain of the potential
to resolve the features of both minima, which roughly fall around
$-v\leq\phi\leq v$. We require that this constraint holds instead over the
field range $-4\pi v\leq\phi\leq 4\pi v$, about an order of magnitude larger
in field space, which should safely capture the important features and ensure
the probability of tunneling in some destabilized direction outside this range
is considerably suppressed.

A related but independent constraint arises from demanding the potential be
bounded from below in the range we consider, which again we choose to be field
values within $4\pi v$ of the origin $\phi=0$. To be conservative, we required
that the potential does not start to turn downwards at high field values. We
first checked whether the potential was concave up or concave down at the $\pm
4\pi v$ boundaries. A concave up potential passes the test. If the potential
was concave down at the boundary, however, we required that the potential was
increasing (decreasing) with $\phi$ at the positive (negative) $4\pi v$
boundary. Lastly, we note that the 1-loop contribution to Eq. (4.26) is
complex for some values of the field value $\phi$ –e.g. when
$V^{\prime\prime}$ changes sign for a concave to convex tree-level potential.
However it is the real part of the effective potential that is relevant for
phase transitions [42, 43], given the imaginary component of $\Delta V_{CW}$
is very small relative to the real part, which is found to hold in all cases
studied here. It is therefore the real part of the effective potential which
will be considered in the following for numerical computations.

###  4.3 Symmetry (non)restoration and roads to the SM

The study of phenomenology will make use of numerical methods guided by
analytical estimates; here we present some of the latter. Much in the way of
the well studied electroweak phase transition in the SM for small Higgs mass,
here a high temperature expansion helps draw the features of the problem and
provide an understanding of the underlying dynamics. In addition, it also
sheds some light on the question of limits in non-linear theories that might
lead to the SM.

At high temperatures, the thermal corrections can be approximated by
polynomials in the masses provided we are in the regime [44]:

$\displaystyle\frac{m_{i}(\phi)}{\pi T}$ $\displaystyle<1,$ (4.41)

and hence this high temperature approximation is valid for a neighbourhood
around the point $\phi=0$ which shrinks with decreasing temperature. We note
that (4.35) excludes a neighbourhood around the origin as opposed to the
above; these two exclusion regions fortunately do not overlap.

The field coordinate $\phi=h+v_{\star}$ as outlined above is convenient to
write thermal corrections in terms of which

$\displaystyle V_{\textrm{eff}}^{\textrm{HT}}=$ $\displaystyle G_{o}v^{3}\phi-
D_{o}v^{2}\phi^{2}+E_{o}v\phi^{3}+\frac{\lambda\gamma_{4}^{2}}{8}\phi^{4}$
(4.42)
$\displaystyle+\sum_{B}n_{i}\left(\frac{m^{2}_{i}(\phi)}{8\pi^{2}}\left[\frac{m_{i}^{2}}{4}+\frac{\pi^{2}T^{2}}{3}\right]-\frac{T(m_{i}^{2}(\phi))^{3/2}}{12\pi}-\frac{m_{i}^{4}(\phi)}{64\pi^{2}}\log\left(\frac{m_{i}^{2}}{T^{2}A_{B}}\right)\right)$
$\displaystyle+\sum_{F}n_{i}\left(\frac{m^{2}_{i}(\phi)}{16\pi^{2}}\left[\frac{m_{i}^{2}}{2}-\frac{\pi^{2}T^{2}}{3}\right]-\frac{m_{i}^{4}(\phi)}{64\pi^{2}}\log\left(\frac{m_{i}^{2}}{T^{2}A_{F}}\right)\right),$

where keeping up with our notation masses with no arguments are constants, the
values measured at the vacuum, $n_{i}$ are given in Eqs. (4.27)–(4.29) whereas

$\displaystyle\log(A_{B})$ $\displaystyle=5.4076-3/2,$
$\displaystyle\log(A_{F})$ $\displaystyle=2.6351-3/2,$ (4.43) $\displaystyle
G_{o}$
$\displaystyle=-\frac{\lambda\delta\gamma_{\epsilon}^{2}}{2}(1-\gamma_{\epsilon}^{-1}\gamma_{4}\delta)(1+\gamma_{\epsilon}^{-1}\gamma_{4}\delta+3((1-\epsilon)\gamma_{\epsilon}^{-1}-1))$
(4.44)
$\displaystyle=-\frac{\lambda\gamma_{4}\gamma_{a}^{-1}\gamma_{\epsilon}^{-1}}{2}(\gamma_{4}^{-1}\gamma_{\epsilon}-\gamma_{a}^{-1})(2-\gamma_{\epsilon}\gamma_{a}^{-1}\gamma_{4}),$
(4.45) $\displaystyle D_{o}=$
$\displaystyle\frac{\lambda}{4}\left(6(1-\epsilon)\gamma_{4}\gamma_{a}^{-1}-3(\gamma_{a}^{-1}\gamma_{4})^{2}-2\right),$
(4.46) $\displaystyle E_{o}=$
$\displaystyle-\frac{\lambda}{2}\gamma_{4}\gamma_{a}^{-1}(\gamma_{4}-\gamma_{a}(1-\epsilon)).$
(4.47)

It is useful for phenomenological purposes to highlight the terms that break
Higgs parity. Couplings to the electroweak bosons and top quark are symmetric
so that all asymmetry is sourced by the potential either at tree level, or
through the effective masses of Goldstones and the Higgs. Note that, as in the
SM case, a cubic term would arise from thermal loops of gauge bosons, but this
would be parity symmetric as $(\phi^{2})^{3/2}$. All sources of parity
violation are turned off for $\epsilon$ and $\delta$ vanishing which aligns
the maximum of the tree level potential with the throat of the manifold and
sets the energy difference between minima to zero. This limit of vanishing
$\delta$ and $\epsilon$ simplifies eqs. (4.44)–(4.47) so that the coefficients
of parity violating terms are linear in $\epsilon$ and $\delta$ as
$G_{o}\simeq-\lambda\delta/2$ and
$E_{o}\simeq\lambda\delta\gamma_{4}^{2}/2-2\lambda\gamma_{4}\epsilon$.

(a) $T=246$ GeV

(b) $T=190$ GeV

Figure 12: One-loop effective potential plotted for $\chi=\sqrt{0.1}$, obeying
the high-temperature symmetry restoration bound in Eq. (4.65). The benchmarks
chosen here are $\gamma_{4}=1,\ \epsilon=0.02,\ \delta=-0.1$. Fig. 12(a) and
Fig. 12(b) show the potential evolution with temperature for $T=246$ GeV and
$T=190$ GeV, respectively. Notice, in particular, the large $\chi$ smooths the
potential around $\phi/v=0$ in contrast to Fig. 13.

(a) $T=246$ GeV

(b) $T=190$ GeV

Figure 13: One-loop effective potential plotted for $\chi=0.05$, which unlike
Fig. 12, does not satisfy the high-temperature symmetry restoration bound in
Eq. (4.65). The benchmarks chosen here are $\gamma_{4}=1,\ \epsilon=0.02,$ and
$\delta=-0.1$. Fig. 13(a) and Fig. 13(b) show the potential evolution with
temperature for $T=246$ GeV and $T=190$ GeV respectively. For this choice of
$\chi$, the potential is sharply peaked around $\phi/v=0$ in contrast to Fig.
12.

While the high energy expansion yields a potential as a polynomial in the
fields and is amenable to analysis in SMEFT, in our non-linear theory the
potential is instead a rational function of the fields due to the Goldstones
contribution,
$V^{\prime}F^{\prime}/F\propto(c_{\chi}^{2}\gamma_{a}^{2}\phi^{2}+s_{\chi}^{2})^{-1}$.
Naively recovering a polynomial, as in the SM and SMEFT case, would require
taking $\chi\to 0$ while having the numerator polynomial start at $\phi^{2}$
to avoid poles, attainable by $\delta\to 0$. Such a limit, in the direction of
recovering the SM, is in fact a discontinuous one in what is a distinguishing
feature of this non-linear theory. This feature is hinted at in the small
$\chi$ limit with $\delta\neq 0$, where plots of the potential in Fig. 13 show
a sharp rise around the throat ($\phi=0$).555We have checked that the sharp
features of this potential around the origin come primarily from the shape of
the manifold. The imaginary parts of the potential are still much smaller than
the real parts here.

This feature can be understood analytically if we inspect the very high
temperature potential while still keeping in mind $T<4\pi v$:

$\displaystyle V_{\textrm{eff}}^{\textrm{HT}}(\phi)$
$\displaystyle=\sum_{B}n_{i}\frac{T^{2}m_{i}^{2}(\phi)}{24}-\sum_{F}n_{i}\frac{T^{2}m_{i}^{2}(\phi)}{48}+\mathcal{O}(T^{0})$
(4.48)
$\displaystyle=\frac{T^{2}}{24}\left((6m_{W}^{2}+3m_{Z}^{2}+6m_{t}^{2})F^{2}(\phi)+V^{\prime\prime}(\phi)+3V^{\prime}(\phi)\frac{F^{\prime}(\phi)}{F(\phi)}\right)+\mathcal{O}(T^{0})$
(4.49)
$\displaystyle=V_{\textrm{eff}}^{\textrm{HT}\,\prime}(0)\phi+\frac{V_{\textrm{eff}}^{\textrm{HT}\,\prime\prime}(0)}{2}\phi^{2}+\mathcal{O}(\phi^{3},T^{0}),$
(4.50)

with

$\displaystyle V_{\textrm{eff}}^{\textrm{HT}\,\prime}(0)$
$\displaystyle=T^{2}v\left(\frac{E_{o}}{4}+\frac{G_{o}c_{\chi}^{2}\gamma_{a}^{2}}{8s_{\chi}^{2}}\right),$
(4.51)
$\displaystyle\frac{V_{\textrm{eff}}^{\textrm{HT}\,\prime\prime}(0)}{2}$
$\displaystyle=T^{2}\left(\frac{2m_{W}^{2}+m_{Z}^{2}+2m_{t}^{2}}{8v^{2}}c_{\chi}^{2}\gamma_{a}^{2}+\frac{\lambda\gamma_{4}^{2}}{16}-\frac{D_{o}c_{\chi}^{2}\gamma_{a}^{2}}{4s_{\chi}^{2}}\right).$
(4.52)

This expansion around the Higgs parity symmetric field value $\phi=0$ allows
one to discuss symmetry restoration in analytic but approximate terms as
follows.

  * •

$V_{\textrm{eff}}^{\textrm{HT}\,\prime}(0)\neq 0$. The linear term in $\phi$
breaks Higgs parity and will prevent $\phi=0$ from being an extremum, its sign
for small $\delta,\epsilon$ determining if this extremum has shifted to
positive or negative values. Assuming $V^{\prime\prime}_{\rm eff}>0$ and that
the minimum is in a neighbourhood of the throat ($\phi=0$), one has a minimum
shifted to negative $\phi$ values for positive slope, a condition expressed in
terms of the model parameters as when subsituting in Eq. (4.51) as

$\displaystyle-\frac{\lambda}{8}\gamma_{4}\gamma_{a}^{-1}(\gamma_{4}-\gamma_{a}(1-\epsilon))-\frac{\lambda
c_{\chi}^{2}\gamma_{4}\gamma_{a}\gamma_{\epsilon}^{-1}}{16s_{\chi}^{2}}(\gamma_{4}^{-1}\gamma_{\epsilon}-\gamma_{a}^{-1})(2-\gamma_{\epsilon}\gamma_{a}^{-1}\gamma_{4})>0,$
(4.53)

and the opposite inequality for a minimum in the positive $\phi$ axis. In the
limit of small $\epsilon$ and $\delta$ we can simplify the expression to
derive

$\displaystyle\delta\frac{c_{\chi}^{2}-2s_{\chi}^{2}}{16s_{\chi}^{2}}$
$\displaystyle<-\frac{\epsilon}{2\gamma_{4}}$
$\displaystyle\begin{array}[]{c}\textrm{high T min at}\,\phi<0\\\
\textrm{History Q}{}_{-}\end{array}$ (4.56)
$\displaystyle\delta\frac{c_{\chi}^{2}-2s_{\chi}^{2}}{16s_{\chi}^{2}}$
$\displaystyle>-\frac{\epsilon}{2\gamma_{4}}$
$\displaystyle\begin{array}[]{c}\textrm{high T min at}\,\phi>0\\\
\textrm{History Q}{}_{0}\end{array}$ (4.59)
$\displaystyle\delta\frac{c_{\chi}^{2}-2s_{\chi}^{2}}{16s_{\chi}^{2}}$
$\displaystyle\simeq-\frac{\epsilon}{2\gamma_{4}}$
$\displaystyle\begin{array}[]{c}\textrm{high T min at}\,\phi\simeq 0\\\
\textrm{History P}\end{array}\,.$ (4.62)

The equivalence with histories is not exact given that this analytic
approximation does only hold for small $\epsilon$ and $\delta$, but it is
useful in sketching the possibilities. This result divides the
$(\epsilon,\delta)$ plane in half with a negative slope line that goes through
the origin as in Fig. 16: above this line one would find history Q0, below it
Q- .

  * •

$V_{\textrm{eff}}^{\textrm{HT}\,\prime}(0)=0$. If the linear term vanishes,
which is the case for $\epsilon,\delta\to 0$, $\phi=0$ is an extremum: a
minimum for $V_{\textrm{eff}}^{\textrm{HT}\,\prime\prime}(0)>0$ and maximum
for $V_{\textrm{eff}}^{\textrm{HT}\,\prime\prime}(0)<0$. The sign of the
second term, $s_{0}$, will hence determine if Higgs parity is restored or not
so

$\displaystyle s_{0}\equiv\textrm{Sign}\big{[}$ $\displaystyle
V_{\textrm{eff}}^{\textrm{HT}\,\prime\prime}(0)\big{]}$ (4.63)
$\displaystyle=\textrm{Sign}\Big{[}$
$\displaystyle\frac{2m_{W}^{2}+m_{Z}^{2}+2m_{t}^{2}}{8}c_{\chi}^{2}\gamma_{a}^{2}$
$\displaystyle-\frac{m_{h}^{2}\gamma_{4}^{2}}{16}\left(\gamma_{a}^{2}\gamma_{4}^{-2}t_{\chi}^{-2}\left(6(1-\epsilon)\gamma_{4}\gamma_{a}^{-1}-3(\gamma_{a}^{-1}\gamma_{4})^{2}-2\right)-1\right)\Big{]}\,,$
(4.64)

which to first order in an $\epsilon,\delta$ expansion returns

$\displaystyle s_{0}$ $\displaystyle\simeq\textrm{Sign}\left(\frac{(\sin
2\chi)^{2}}{\cos(2\chi)}-\frac{m_{h}^{2}}{m_{W}^{2}+m_{t}^{2}+m_{Z}^{2}/2}\right),$
(4.65)
$\displaystyle\left\\{\begin{array}[]{cl}s_{0}>0\Rightarrow\chi>0.3\,,&\textrm{Symmetry
restoration at high }T\\\ s_{0}<0\Rightarrow\chi<0.3\,,&\textrm{Symmetry stays
broken at high }T\,.\end{array}\right.$

The small $\chi$ limit leading to no symmetry restoration leads to an apparent
contradiction: one expects this limit to yield SM-like couplings and with it
the symmetry should naively be restored at high temperature. This finding
sheds light on the question posed in Ref. [9] of whether non-linear theories
have a limit in which the SM is recovered. Approaching SM-like couplings from
our non-linear theory, we obtain a locally identical theory around $h=0$ which
nonetheless does not have symmetry restoration at high temperature for any
infinitesimally small value of $\chi$ since the theory is different at
$\phi=0$. This discontinuous limit can be spotted opening up the term
$V^{\prime}F^{\prime}/F$ in Eq. (4.50)

$\displaystyle V_{\textrm{eff}}^{\textrm{HT}}$ $\displaystyle\supset V_{\rm
peak}=-\frac{T^{2}D_{o}}{4}\frac{\phi^{2}}{\phi^{2}/v^{2}+s_{\chi}^{2}/(c_{\chi}^{2}\gamma_{a}^{2})}$
(4.67)

Figure 14: Contribution $V_{\rm peak}$ of the $V^{\prime}F^{\prime}/F$ term to
the effective potential.

This same limit of $\chi\to 0$ would give an ever smaller cut-off at the
throat as the curvature increases as outlined in the discussion at the
beginning of Sec. 4. New light states would be accessible in the thermal
history which would change our EFT to a theory with extra degrees of freedom.
This suggests that for a consistent theory, the functions in our manifold
should be sufficiently smooth. The exploration of this interplay in the
$\chi\to 0$ limit and its phenomenology, however, we leave for future work on
the connection of UV models and HEFT$\backslash$SMEFT.

Fig. 14 plots the contribution for different values of $\chi$ and the SM, and
shows that at no finite stage is the SM limit (a constant for all field range,
in black) reached. Sign of an anomalous limit is also present in the linear
term of Eq. (4.50) since it reads in this expansion
$G_{o}/s_{\chi}^{2}\sim\delta/\chi^{2}$, giving a different result depending
on the order of limits that one takes to the SM. Unlike the behaviour in Eq.
(4.67) however, the SM can be approached by first taking $\delta\to 0$.

##  5 Domain walls

The finite temperature effective potential is symmetric under Higgs parity in
the limit $\epsilon,\delta\to 0$. This limit contains couplings that are
locally those of the SM yet the doubling of the Higgs domain implies two
degenerate minima at low temperature. One has different possible phases for
the discrete symmetry in the universe history and, although this theory might
locally resemble the SM, the cosmological dynamics change drastically. As the
most salient new possible feature, one has spontaneous breaking of Higgs
parity and the creation of domain walls. The formation of domain walls in the
Higgs field is, to the best of our knowledge, a unique feature of non-linear
theories. In this section we describe and identify the conditions for this
approximate symmetry to lead to walls with the assumption, validated below,
that $\delta$, $\epsilon$ are small enough to expand on.

Sec. 4.3 showed that for $\chi>0.3$ and small $\epsilon$ and $\chi$ there will
be symmetry restoration in the form of a single minimum at high energies,
whereas the zero temperature potential presents two parity reflected minima.
This well studied case of cosmological spontaneous breaking of discrete
symmetry leads in general to the formation of walls and will be the focus of
our study here. The late time phenomenology of walls is quite insensitive to
whether the transition is first or second order, but let us sketch how it
occurs in both cases for illustration. For a first order PT, different patches
in the universe will bubble to one of the two vacua, and after the bubbles
collide and lose energy to the bath a network of walls will form, the
characteristic size of this network dependent on the transition properties.

For a second order PT, depicted in Fig. 4(b) (history P), shortly after the
critical temperature the small separation and barrier between minima implies
that thermal fluctuations for correlation-volume ($l^{3}$ with
$l^{-2}=V^{\prime\prime}(h_{0})\simeq V^{\prime\prime}(h_{-})$) patches allow
for transitions from one vacuum to another. As temperature decreases, the
probability of these jumps will decrease with the Boltzmann factor, $\sim
e^{-l^{3}U/T}$ where $U$ is the potential barrier given our approximate
degeneracy assumption $U=U_{0}=U_{-}$ with $U_{0,-}$ in Eq. (4.31), until this
suppression is effective enough to in practice forbid the transitions. At this
time patches of the correlation volume will be stuck in whichever vacuum they
happened to be at, and the network of walls will form. Formation is marked by
Ginzburg’s temperature defined as the temperature when the the negative of the
exponent in Boltzmann’s factor equals one,

$\displaystyle
T_{G}\equiv\left[V^{\prime\prime}_{\textrm{eff}}(h_{0}(T_{G}),T_{G})\right]^{-3/2}U(T_{G})\,.$
(5.1)

This description allows for a qualitative picture of the symmetry non-
restoration case for $\chi<0.3$. In this instance two minima survive until
high temperature, but noting that the barrier scales like $v^{2}T^{2}$ and
naively correlation the length is $T^{-1}$, one can expect that above $T\sim
v$ the Boltzmann factor is large enough to allow for these jumps. If this is
so, even without symmetry restoration, the Ginzburg temperature is well
defined and one can expect formation of walls. This must be the case at least
in a neighbourhood below $\chi=0.3$, but we leave its detailed study for
future work.

The definition of the Ginzburg temperature is implicit; in practice however it
is well approximated by the zero temperature potential, in our case

$\displaystyle T_{G}\simeq\frac{\lambda v^{4}}{8\gamma_{4}^{2}m_{h}^{3}}\sim
60\gamma_{4}^{-2}\textrm{GeV},$ (5.2)

where we have expanded on $\epsilon,\delta$ and kept only the first term. The
probabilities $P_{0}$ and $P_{-}$ to find a patch of each vacua $h_{0}$ and
$h_{-}$ at the Ginzburg temperature can then be estimated as

$\displaystyle\frac{P_{0}}{P_{-}}\sim\exp\left(\frac{l^{3}V_{\textrm{eff}}(h_{-},T)}{T_{G}}-\frac{l^{3}V_{\textrm{eff}}(h_{0},T)}{T_{G}}\right)=\exp\left(\frac{\Delta
V(T_{G})}{U(T_{G})}\right),$ (5.3)

where $\Delta V$ is as defined in Eq. (4.30). A large $\Delta V/U$ ratio would
imply one vacuum is selected predominantly and very few walls form. We instead
assume this ratio, which depends only on $\epsilon$, is small, which implies
small $\epsilon$ according to Eq. (2.29). This translates into $\Delta
V/U=32\epsilon\ll 1$.

We note that regardless of how the network formed and the typical scale of
structure, both types of phase transition have a large wall stretching out to
the horizon. One can realise this in a 2D case by rolling a dice to fill each
correlation patch inside a causal box with either of the two vacua. Once the
filling is done, a zoom out to see the global structure will reveal the
presence of this large wall.

Let us then continue taking $l$ as the typical scale of the network regardless
of how it was formed. The small scale structure dynamics is ruled by the
balance of pressure $p=\Delta V(T_{G})$ and tension $\mu$. The tension is well
approximated by the zero temperature potential and to first non-vanishing
order in small $\epsilon,\delta$

$\displaystyle\mu=\int_{-\infty}^{\infty}d\ell\,T^{0}_{\,\,0}(h_{w}(\ell))=\int
d\ell\left(\frac{1}{2}(\nabla
h_{w})^{2}+V(h_{w})\right)=\frac{2\sqrt{\lambda}v^{3}}{3\gamma_{4}^{2}},$
(5.4)

where the profile function is found as a solution to the static field
equations and reads
$\gamma_{4}\sqrt{\lambda}h_{w}(\ell)/m_{h}=\tanh(m_{h}\ell)-1$.

The potential energy difference, Eq. (4.30), vanishes in the parity symmetric
limit and here it suffices to estimate it to linear order in $\delta$ and
$\epsilon$. On top of this expansion, we also perform a loop expansion, with
minima at $h_{-}=h_{-}^{(0)}+\kappa h_{-}^{(1)}$, $h_{0}=h_{0}^{(0)}+\kappa
h_{0}^{(1)}$ and $\kappa=(4\pi)^{-2}$ to find the energy difference
($V_{\textrm{eff}}=V+V_{\kappa}$ , $V_{\kappa}=V_{CW}+V_{\textrm{Th}}$)

$\displaystyle\Delta V=$ $\displaystyle
V(h_{-}^{(0)})+V_{\kappa}(h_{-}^{(0)})+\frac{dV(h_{-}^{(0)})}{dh}\kappa
h_{-}^{(1)}+\mathcal{O}(\kappa^{2})$ (5.5)
$\displaystyle-\left[V(h_{0}^{(0)})+V_{\kappa}(h_{0}^{(0)})+\frac{dV(h_{0}^{(0)})}{dh}\kappa
h_{0}^{(1)}+\mathcal{O}(\kappa^{2})\right]$ (5.6) $\displaystyle=$
$\displaystyle
V(h_{-}^{(0)})+V_{\kappa}(h_{-}^{(0)})-[V(h_{0}^{(0)})+V_{\kappa}(h_{0}^{(0)})]+\mathcal{O}(\kappa^{2}),$
(5.7)

where the derivative term cancels since it is evaluated at a minimum to the
given approximation. One can now use that $h_{i}^{(0)}$, the minima of the
tree level potential, do not depend on $\delta$ and are the Higgs parity
conjugate of one another to write

$\displaystyle h_{-}^{(0)}$ $\displaystyle=-2v\gamma_{4}^{-1}+\epsilon
h_{a,-}^{(1)}+\mathcal{O}(\epsilon^{2}),$ (5.8)

whereas we recall by definition and for all values of the tree level potential
$h^{(0)}_{0}=0$. The expanded energy difference is then

$\displaystyle\Delta V=$ $\displaystyle\epsilon\left.\left(\frac{\partial
V(-2v\gamma_{4}^{-1})}{\partial\epsilon}-\frac{\partial
V(0)}{\partial\epsilon}\right)\right|_{\epsilon=0}+\epsilon
h_{a,-}^{(1)}\left.(\frac{dV(-2v\gamma_{4}^{-1})}{dh}\right|_{\epsilon=0}$
(5.9) $\displaystyle+\delta\left.\left(\frac{\partial
V_{\kappa}(-2v\gamma_{4}^{-1})}{\partial\delta}-\frac{\partial
V_{\kappa}(0)}{\partial\delta}\right)\right|_{\epsilon,\delta=0},$ (5.10)

where the potential derivative w.r.t. the field will cancel once more given it
is evaluated at a minimum to the given approximation. The leading $\epsilon$
contribution comes at tree level and is straightforward to obtain. For
$\delta$, a number of intermediate steps leads to

$\displaystyle\delta\left.\left(\frac{\partial}{\partial\delta}\left[V_{\kappa}(-2v\gamma_{4}^{-1})-V_{\kappa}(0)\right]\right)\right|_{\delta,\epsilon=0}=\delta\sum_{i}\left[\frac{\partial
m_{i}^{2}(-2v\gamma_{4}^{-1})}{\partial\delta}-\frac{\partial
m_{i}^{2}(0)}{\partial\delta}\right]\frac{\partial V_{\kappa}}{\partial
m_{i}^{2}}$ (5.11) $\displaystyle=$
$\displaystyle\frac{T^{2}}{2\pi^{2}}\frac{\partial
F^{2}(-2v\gamma_{4}^{-1})}{\partial\delta}\sum
n_{i}m_{i}^{2}(0)J^{\prime}_{i}(m_{i}^{2}(0)/T^{2})$ (5.12) $\displaystyle=$
$\displaystyle\frac{6\gamma_{4}\delta
c_{\chi}^{2}T^{2}}{\pi^{2}}\left(2m_{W}^{2}(0)J^{\prime}_{b}\left(\frac{m_{W}^{2}(0)}{T^{2}}\right)+m_{Z}^{2}(0)J^{\prime}_{b}\left(\frac{m_{Z}^{2}(0)}{T^{2}}\right)-4m_{t}^{2}(0)J^{\prime}_{f}\left(\frac{m_{t}^{2}(0)}{T^{2}}\right)\right)$
(5.13) $\displaystyle\equiv$ $\displaystyle\lambda
v^{4}\gamma_{4}^{-2}\delta\left(b(x)x^{-2}\right),$ (5.14)

where $x=v/T$, and we used that our renormalisation conditions of Eq. (4.23)
imply the variation w.r.t. the mass vanishes so that the linear term in
$\delta$ comes from the thermal $J$ functions, which are $F$-dependent (and
$F$ is $\delta$-dependent). The result is the potential energy difference to
leading order in $\epsilon,\delta$,

$\displaystyle p_{V}(T)$ $\displaystyle=\Delta V(T)=4\lambda
v^{4}\gamma_{4}^{-2}(\epsilon+b(x)x^{-2}\delta/4)+\mathcal{O}(\kappa\epsilon,\delta^{2})\,.$
(5.15)

Wall dynamics will be determined by the balance of the volume pressure
$p_{V}=\Delta V$ and an equivalent tension pressure $p_{T}=\mu/l$. Volume
pressure $\Delta V$ domination at $T_{G}$ means the lowest energy vacuum
patches will quickly expand against the other vacuum patches and make the $l$
size structure of walls disappear shortly after formation. The opposite case,
approximately given by $(\epsilon+b(x)x^{-2}\delta/4)<1/6$ [45], has that
tension would drive dynamics together with the friction of the thermal bath.
Indeed in vacuo, the tension would make the walls oscillate converting
potential energy stored in tension into kinetic energy and back. The thermal
bath however introduces friction with a pressure $T^{4}v_{w}$ with $v_{w}$ the
wall velocity which dissipates the energy of the walls efficiently back into
the bath, some of it in the form of gravitational radiation. Both instances
then lead to the disappearance of the $l$ size network shortly after
formation.

The dynamics of the large wall are dictated by the horizon rather than $l$ and
one can translate the above discussion by $l^{-1}\to H$. As long as $\mu H\geq
p_{V}(T)$, structure of size $H^{-1}$ will enter the horizon, oscillate and
radiate as controlled by tension and friction leaving a potentially detectable
trace. The opposite case wipes even this large wall away leaving virtually no
trace. Focusing on detectable phenomenology, we restrict to the condition for
long lived large walls, which is $p_{T}>p_{V}$ as outlined above, and reads
explicitly

$\displaystyle p_{T}(H(T_{G}))=\mu H(T_{G})$ $\displaystyle>p_{V}=\Delta
V(T_{G}),$ (5.16)
$\displaystyle\left|\epsilon+b(x_{G})x_{G}^{-2}\delta/4\right|$
$\displaystyle\lesssim\frac{x_{G}^{-2}v^{2}}{6m_{h}M_{\textrm{pl}}}\sqrt{\frac{4\pi^{3}g_{\star}}{45}},$
(5.17)

where $x_{G}^{2}=v^{2}/T_{G}^{2}$. This condition can be met by a cancellation
of $\epsilon$ and $\delta$, however, if they themselves are individually much
larger than the RHS of the inequality above, the temperature dependence would
mean that the condition will, shortly after $T_{G}$, not be satisfied. For
this reason we also impose a condition on the logarithmic derivative w.r.t.
temperature of the LHS; explicitly the change after an e-fold variation in
temperature should also be smaller than the RHS of Eq. (5.17),

$\displaystyle\left.\frac{d}{d\log
x}\left(b(x)x^{-2}\delta/4\right)\right|_{x_{G}}\lesssim\frac{x_{G}^{-2}v^{2}}{6m_{h}M_{\textrm{pl}}}\sqrt{\frac{4\pi^{3}g_{\star}}{45}}.$
(5.18)

All the above conditions are met for $\epsilon=\delta=0$, a case in which the
perfect degeneracy would make walls endure the history of the universe to be
around today. If so, their effect must have been negligible in the
cosmological evolution and in particular any anisotropic impact on the CMB
less than one part in $10^{5}$. Given the energy density stored in walls is
$\mu H$, the bound can be translated into the tension [45]

$\displaystyle\mu H_{0}<M_{\rm
pl}^{2}H_{0}^{2}\quad\Rightarrow\quad\mu=\frac{2\sqrt{\lambda}vv_{+}^{2}}{3}=\frac{2\sqrt{\lambda}v^{3}}{3\gamma_{4}^{2}}\leq(0.1\textrm{GeV})^{3}.$
(5.19)

Avoiding this bound would require smaller tension, attainable with larger
$\gamma_{4}$, of order $\gamma_{4}\sim 10^{4}$, yet perturbativity from e.g.
$h+h\to h+h$ scattering demands $\lambda\gamma_{4}^{2}<8\pi/3$, and this
possibility is ruled out. The avenue of reducing the tension in our degenerate
potential means bringing the vacua closer together, which is a modification at
small field values only. As a result this conclusion is robust against higher
dimensional operator insertions in the potential.

(a)

(b)

Figure 15: Fig. 15(a) shows the $(\epsilon,\gamma_{4})$ plane with excluded
regions in darker green (from the upper bound on walls energy density, Eq.
(5.22)), light gray horizontal (perturbativity, Eq. (4.40)), dark gray (wall
annihilation before BBN, (5.21)). The region in dark purple is discarded if
one is to have walls that survive long past the original phase transition (Eq.
(5.16)). Also shown is the gravitational wave spectrum value at peak frequency
$\Omega_{\rm GW}(f_{\rm peak})$ in the yellow gradient, see Eq. (7.5) and Sec.
7. Dashed lines for the peak frequencies run from $f_{\rm peak}=10^{-6}$ Hz on
the far right to $10^{-10}$ Hz on the far left in intervals of $10$, and in
blue the region SKA [46] would be sensitive to (this region is however
excluded by LHC bounds see Sec. 8). Fig. 15(b) shows the $(\epsilon,\delta)$
excluded by wall annihilation before BBN in dark gray vertical Eq. (5.21), and
the regions discarded for long lived walls following from Eq. (5.17) (purple)
and Eq. (5.18) (gray horizontal).

Having ruled out the $\delta=\epsilon=0$ limit, we now turn to finite but
small values of the Higgs parity breaking parameters. We note that the
condition that ensures wall survival at the Ginzburg time, Eq. (5.16), is more
demanding if translated to later time and larger $x$; for finite $\Delta V$ it
will cease to be satisfied at $T_{w}$, i.e.

$\displaystyle\mu H_{w}\equiv p_{V}(T_{w})=\Delta
V(T_{w})\quad\Rightarrow\quad
T_{w}^{2}=\sqrt{\frac{45}{4\pi^{3}g_{\star}}}\frac{M_{\textrm{pl}}\Delta
V}{\mu}=\sqrt{\frac{45}{4\pi^{3}g_{\star}}}6\epsilon m_{h}M_{\rm pl}.$ (5.20)

Below this temperature, tension pressure gives way to vacuum pressure and we
have that walls are swept away towards the horizon at a velocity close to the
speed of light.

A conservative bound [47] is for this temperature to be higher than our
earliest direct evidence of universe history, big bang nucleo-synthesis. This
constraint can be translated into a lower bound on $\epsilon$:

$\displaystyle
T_{w}>T_{BBN}\quad\Rightarrow\quad\epsilon\geq\frac{T_{BBN}^{2}}{6M_{\textrm{pl}}m_{h}}\sqrt{\frac{4\pi^{3}g_{\star}}{45}}.$
(5.21)

In addition, the energy density of the walls should always be subdominant if
one is to avoid entering an inflationary phase. Given its scaling, this
constraint is strongest at the latest time, i.e. $T_{w}$

$\displaystyle\mu H(T_{w})\leq
g_{\star}T_{w}^{4}\quad\Rightarrow\quad\epsilon>\frac{m_{h}^{2}}{9\lambda\gamma_{4}^{2}M_{\textrm{pl}}^{2}}.$
(5.22)

These constraints are put together in Fig. 15 for an illustration of the
parameter space region compatible with experimental data. We note that the
lower bound in $\epsilon$ from Eq. (5.22) means a locally SM-like potential
(which has $\epsilon=0$) is excluded, whereas an upper bound follows from the
LHC (non)measurement of the triple Higgs coupling, see Eq. (8.3). Nevertheless
the large disparity of scales involved leaves the possibility to close this
window out of reach.

##  6 Past and future first order phase transitions

The presence of more than one minimum at a given temperature allows for
tunnelling phenomena with potentially detectable imprints in our universe.
This requires an evolution of the universe whose dynamics initially puts it in
the false vacuum and a tunneling rate large enough to trigger a universe-wide
transition, in the past or in the future.

A well studied case that satisfies these conditions is the SM with small Higgs
mass. The history for this case is depicted in Fig. 3; the high temperature
symmetric minimum $\phi=0$ stops being the only extremum after the appearance
of a new minimum (and maximum) at a finite (and large enough, see Sec. 4.3)
field distance. The potential energy difference between minima changes sign at
a critical temperature $T_{c}$, after which tunneling is energetically viable
but suppressed by the negative exponential of the bounce solution action.
Given that the $\phi=0$ symmetric point turns eventually into a maximum, we
have that the potential barrier between minima decreases and the energy
difference increases, so sooner or later the transition will occur at what is
dubbed the nucleation temperature $T_{N}$, where $T_{N}<T_{c}$.

Non-linear theories allow for first order phase transitions as that in the
case of the light SM Higgs but also a set of qualitatively different ones. One
could indeed attempt at the equivalent of the light Higgs SM in our non-linear
theory; leaving every other parameter SM-like, decreasing $\gamma_{4}$
effectively increases field distances (i.e. decorrelates the quartic coupling
and $m_{h}^{2}$, see Eq. (2.17)) and could yield a first order phase
transition. It is the case now, however, that there exist two degenerate true
minima, $\phi_{0},\phi_{-}$, given this limit is Higgs parity preserving, and
uncorrelated patches of the universe will tunnel into one or the other minimum
with equal likelihood. This distribution implies that a network of domain
walls would form in the boundary between different-vacua patches, the
subsequent phenomenology of domain walls having been studied in Sec. 5.

This illustrates the variety of phenomena in non-linear theories. While the
picture just described is more convoluted that typical case studies, one can
break it down into staged transitions, each of which with a known theory
formalism. For this reason we will study in this section scenarios that lead
solely to a first order phase transition. Results derived here could be put
together with those of Sec. 5 for more intricate cases as the one described
above with two transitions, however we leave this for future work. Explicitly
the cases studied will be histories Q of Fig. 4(c) and R of Fig. 5(c), where
we note that both are far from symmetric under Higgs parity and require
sufficiently large $\epsilon$ and $\delta$ as shown below. For history Q we
have two possibilities by flipping the diagram vertically, these are Q0 and Q-
for a continuous connection of the high temperature with $h_{0}$ or $h_{-}$
respectively. History R leads necessarily to a phase transition given the
original minimum disappears, while history Q might not lead to a phase
transition if the barrier and separation are never small enough. Following the
discussion above we define the nucleation temperature as the temperature when
the tunneling rate equals the Hubble 4-volume, which will signal the
transition in the early universe

$\displaystyle v^{4}e^{-B(T_{N})}\equiv H^{4}(T_{N}).$ (6.1)

History Q does not specify which is the true minimum at low energies; this is
dictated by the sign of $\Delta V(0)$ and in our parametrisation, $\epsilon$.
The two options for the sign of $\epsilon$ then have to be put against the two
histories Q0,- their selection being in turned mapped to $\delta$. Small
$\chi$ leads to more than one minimum at high temperature so we choose
$\chi>0.3$ to ensure we remain over the limit of Eq. (4.65).

For exposition purposes, here we use the small $\epsilon$ and $\delta$ limit
to give a connection of history and parameter space through Eq. (4.56);
although this expansion does not hold for the whole regime we explore (where
we use the full expressions), it does identify possibilities and gives a rough
outline of the regions. One has a contrast of high and low energy extrema
dependent on the parameters as shown on Tab. 2. Note that out of the four, one
option is discarded since it never had $h_{0}$, where we are today, as a true
vacuum.

|  | $\epsilon<0$ |  | $\epsilon>0$
---|---|---|---|---
|  | High $T$ | Low $T$ |  | High $T$ | Low $T$
$(\mbox{ctg}^{2}_{\chi}-2)\delta>-8\epsilon/\gamma_{4}$ |  | $h_{0}(T)$ | $h_{-}(T)$ |  | $h_{0}(T)$ | $h_{0}(T)$
$(\mbox{ctg}^{2}_{\chi}-2)\delta<-8\epsilon/\gamma_{4}$ |  | $h_{-}(T)$ | $h_{-}(T)$ |  | $h_{-}(T)$ | $h_{0}(T)$
Table 2: True vacua at high and low temperatures for small $\epsilon$ and
$\delta$.

It is for a mismatch of high and low temperature minima that transitions are
likely and that is the case for two out of remaining three options. These
conditions are visualised in Fig. 16, which schematically displays the
different phenomena that may be found in the $(\epsilon,\delta)$ plane:

  * •

In the first quadrant, both the tree-level potential and the finite-
temperature corrections favor the $h_{0}$ SMEFT-like vacuum, and so we expect
its cosmological phenomenology to be similar to that of the SMEFT. Note that a
phase transition in this quadrant would require higher powers of $h/v$ (such
as dim-6 or higher operators) in the tree-level potential.

  * •

We mark as _unphysical_ the region where the thermal history selects the
vacuum $h_{-}$ at present times, because, by definition, $h_{-}$ does not have
the known measured masses and couplings of the SM particles. This is the case
for the third quadrant, in which both the tree-level potential and the thermal
corrections favor $h_{-}$; but also for part of the second quadrant, where
$h_{-}$ is the tree-level vacuum; and part of the fourth quadrant, where the
system may become trapped at $h_{-}$ and high temperatures, and not decay
until after today.

  * •

In the rest of the second quadrant, we find ourselves currently in the false
vacuum $h_{0}$, a situation that we refer to as the _doomsday_ scenario, since
a phase transition could be triggered at any time. However, as we will show in
Sec. 6.2, the lifetime of the false vacuum we are in is much larger than the
age of the universe. Notice that this possibility of the long-term survival of
a false vacuum is in sharp contrast with the SM case.

  * •

In the fourth quadrant, the true vacuum is $h_{0}$, but the finite-temperature
corrections favor $h_{-}$. There will thus be a region where $h_{-}$ is
selected at high temperatures, and a first-order phase transition happens
before today. We have labelled this region _bubbles_ , since the phase
transition will happen through bubble nucleation, as described in Sec. 6.1.

  * •

Finally, around the origin, when both $\epsilon$ and $\delta$ are small, long-
lived domain walls are generated, as described in Sec. 5.

These considerations have to be put up against the IR problem; Sec. 4.3
outlines our treatment of this issue. In summary we will discuss a phase
transition if the new minimum appears in the perturbative regime,
$\varepsilon_{IR}(\phi_{0}(T_{1}))<1$ (strong IR constraint) or if the new
minimum is in the perturbative region by the critical temperature
$\varepsilon_{IR}(\phi_{0}(T_{c}))<1$ (weak IR constraint)

Figure 16: Schematic representation of the of the different cosmological
phenomena arising in the $(\epsilon,\delta)$ plane. The small diagrams at the
top and the bottom borders represent the tree-level potential in black, and
the finite-temperature corrections in gray.

More complicated histories arise for small $\chi$ which we shall comment on
but not study in detail. In this case two minima coexist at high temperature
and we might have one or more minima at low energy. If the minima coexist
above a temperature where thermal fluctuations of correlation-distance-size
patches overcome the barrier height (above a Ginzburg temperature, see Sec. 5)
one has a mixed spatial coexistence of the vacua. Above this temperature
either the two minima are degenerate to a good approximation and the evolution
is described in Sec. 5, or if they have a sizeable energy difference, thermal
jumps from the true to the false vacuum will cease first to leave
predominantly only the true vacuum. This vacuum might still not be the same as
the zero temperature one, in which case a phase transition would be possible.
We leave such cases for future study.

###  6.1 First order electroweak phase transition

The possibility of nucleation in the past history of the universe is, given
the considerations above, realised in the quadrant $\epsilon>0$, $\delta<0$ of
the $(\epsilon,\delta)$ plane so that the true vacuum is today $\phi_{0}$ and
at high temperatures $\phi_{-}$. In this quadrant we can find histories Q- and
R, the latter for $\epsilon>1-\sqrt{8/9}$ where there is a single minimum at
tree level. For $\phi_{-}$ to be the true vacuum at high temperature, however,
not only negative but sufficiently sizeable $\delta$ is required; we found in
a small $\epsilon,\delta$ expansion that this condition is
$\delta<-8s^{2}_{\chi}/[\gamma_{4}(c_{\chi}^{2}-2s_{\chi}^{2})]\epsilon$. This
analytic result is validated around the origin but also extended to large
values of the parameters (where one finds history R) by our numerical results
in this section.

The determining factor to characterise the transition, once one has arranged
for the high temperature vacuum turning into a false one, is the bounce
action. In particular it will determine whether the transition took place or
whether the probability is too small for it to have occurred yet. At high
temperatures this bounce action can be approximated by a 3-space-rotation
symmetric action times the small time interval $1/T$

$\displaystyle B=\frac{S_{3}}{T}.$ (6.2)

There are no closed formulas for this action but rather a series of
approximations of various accuracies and software for numerical solutions of
the problem. Here we use both for a better understanding of the process. The
software used is CosmoTransitions [48] and Anybubble [49] whereas the analytic
approximations are:

Figure 17: Comparison of the four methods to compute the tunnelling bounce
action, as described in detail in the text of Sec. 6.1. The displayed lines
are evaluated for the parameters $\chi=\sqrt{0.1},\epsilon=0.04,\delta=-0.08$,
with the results from CosmoTransitions in orange, Anybubble in blue, the
quartic potential approximation in red and the thin wall approximation in
green.

  * •

Quartic potential. The formula for the bounce action of a polynomial potential
of degree four is known to a good approximation [50]. The formula gives, for a
quartic potential with a local minimum at $\phi=0$, a true minimum at
$\phi=\phi_{m}$, and maximum at $\phi_{M}$ as:

$\displaystyle\bar{V}=\frac{\bar{\lambda}}{2}\left[\frac{\phi^{4}}{4}-\frac{\phi_{m}+\phi_{M}}{3}\phi^{3}+\frac{\phi_{m}\phi_{M}}{2}\phi^{2}\right],$
(6.3)

and a bounce action as:

$\displaystyle\frac{S_{3}^{L}}{T}=$ $\displaystyle
4.85\sqrt{\frac{8\phi_{m}\phi_{M}}{\bar{\lambda}T^{2}}}\left(\alpha+\frac{\alpha^{2}}{4}\left[1+\frac{2.4}{1-\alpha}+\frac{0.26}{(1-\alpha)^{2}}\right]\right),$
(6.4) $\displaystyle\alpha=$
$\displaystyle\frac{9}{2}\frac{\phi_{m}\phi_{M}}{(\phi_{m}+\phi_{M})^{2}}.$
(6.5)

Our effective potential $V_{\textrm{eff}}$ is not a polynomial, but as an
approximation it can be modelled as such. We do so by identifying the minimum
$\phi_{m}=\phi_{-}-\phi_{0}$ (if tunneling out of $\phi_{-}$) which leaves
still two parameters in the potential, $\phi_{M}$ and $\bar{\lambda}$, which
are not fixed to the maximum and quartic coupling in Eq. (4.26), but rather
are implicitly defined by two conditions of equal energy difference between
minima and barrier height, i.e.

$\displaystyle\bar{V}(\phi_{m};\bar{\lambda},\phi_{M})$
$\displaystyle\equiv\Delta V(T),$
$\displaystyle\bar{V}(\phi_{M};\bar{\lambda},\phi_{M})$ $\displaystyle\equiv
U_{-}(T).$ (6.6)

These implicit definitions for $\bar{\lambda}$ and $\phi_{M}$ that ensure the
model potential has the same energy difference and barrier height.

Figure 18:  A slice of parameter space in $\epsilon,\delta$ for which the
region in colour following history Q- meets the condition of Eq. (6.10) for
some $T=T_{N}$, the nucleation temperature. For this plot, $\chi=\sqrt{0.1}$
and $\gamma_{4}=1.6$. The blank area in the upper region follows history Q0,
as the thermal corrections favour the $h_{0}$ vacuum, leading to a SMEFT-like
cosmological history. In the lower blank region while still following history
Q-, the condition for nucleation, Eq. (6.10), is not met for electroweak scale
temperatures and as such we do not predict a first order phase transition in
the early universe here. This and Fig. 19 occupy roughly quadrant IV of Fig.
16.

  * •

Thin wall and triangular approximation. Shortly after the critical temperature
$T_{c}$, one has that the thin wall approximation (which assumes a small
energy difference between minima) holds and returns [44]

$\displaystyle\frac{S_{3}^{tw}}{T}=$
$\displaystyle\frac{16\pi}{3}\frac{\mu^{3}}{T(\Delta V)^{2}},$
$\displaystyle\mu$
$\displaystyle=\int_{\phi_{-}}^{\phi_{0}}d\phi\sqrt{2(V_{\textrm{eff}}(\phi)-V_{\textrm{eff}}(\phi_{-}))}.$
(6.7)

Fitting the potential between minima to a triangle yields

$\displaystyle\mu_{\triangle}=2\sqrt{2}\int_{\phi_{-}}^{\phi_{+}}\sqrt{(U_{-})\phi/(\phi_{+}-\phi_{-})}=\frac{4\sqrt{2}}{3}(\phi_{+}-\phi_{-})\sqrt{U_{-}},$
(6.8)

and in this approximation

$\displaystyle\frac{S_{3}^{tw}}{T}=\left(\frac{4\sqrt{2}}{3}\right)^{3}\frac{16\pi}{3}\frac{v}{T}\frac{U_{-}^{3/2}(\phi_{+}(T)-\phi_{-}(T))^{3}}{\Delta
V^{2}v}.$ (6.9)

This expression is easy to evaluate, but is only valid for a short time after
$T_{c}$ whereas Eq. (6.5) continues to be valid at lower temperatures.

One can see a comparative plot for the two analytic approximations and the two
numerical estimates of the bounce action for the 1-loop thermal effective
potential defined in Eq. (4.26) as a function of temperature in Fig. 17. The
numerical estimates are performed with Anybubble [49] and CosmoTransitions
[48] respectively. For Anybubble, the potential has been first fitted to a 9th
degree polynomial for technical reasons and then passed into the Anybubble
code, for CosmoTransitions we use the tunneling1D module to compute the bounce
action.

The curves all start with a very large value of the bounce action as right
after the critical temperature, the energy difference between true and false
minima is small and the thin wall approximation applies, giving a bounce
action inversely proportional to the minima potential energy difference. In
the case of history Q-, as temperature decreases the bounce action decreases
and reaches a minimum value after which it grows again. For the case of
history R however, the action decreases until it vanishes since the original
minimum disappears and so does the barrier between minima. The condition for
the transition to occur reads

$\displaystyle\frac{S_{3}(T_{N})}{T_{N}}\simeq-\log\frac{H^{4}(T_{N})}{v^{4}}\simeq
140.$ (6.10)

This condition will be met in history R but not necessarily in history Q-,
where it could be that the bounce action never decreases below 140 and one is
stuck in the wrong vacuum. In Fig. 18 we show a slice of parameter space where
the condition is met for history Q-, where the gradient marks the nucleation
temperature. Additionally Fig. 19 extends this slice to encompass history R.
As discussed in the next section, if one is stuck in $\phi_{-}$ below
electroweak temperatures, one stays there until today. Since this conflicts
with our definition of $\phi_{0}$ as our vacuum today these instances are
excluded; visually that is the lower left white region in Fig. 18. On the edge
of this region we have a sizeable barrier, but just low enough so that the
transition occurs, and it does so strongly. Further away from this lower edge
of the wedge in Fig. 18, we have smaller barriers and weaker first order phase
transitions until at the upper edge one meets the numerical extension of the
condition $(\mbox{ctg}_{\chi}^{2}-2)\delta<-8\epsilon/\gamma_{4}$ where the
history changes to Q0 and the universe is always in the true vacuum which is
connected continuously at low and high temperatures. For history R with
$\epsilon>1-\sqrt{8/9}$, this condition changes qualitatively and the edge of
the first order phase transition region is found numerically.

Figure 19: Extension of Fig. 18 to larger epsilon values and different
histories, with $\gamma_{4}=1.6,\chi=\sqrt{0.1}$ as in the previous plot. The
plot is discontinuous at $\epsilon=1-\sqrt{8/9}$ as a result of Eq. (2.26).
For $\epsilon<1-\sqrt{8/9}$, the region in colour follows history Q- and the
blank, upper region above this follows history Q0 as in Fig. 18. However when
$\epsilon>1-\sqrt{8/9}$, the $T=0$ potential has a single minimum only and
history R is followed for the region in colour. The blank region above only
has a single minimum throughout its evolution. We note that the upper boundary
from history R to a SMEFT-like history is slightly ambiguous as the potential
becomes very flat and difficult to treat numerically.

###  6.2 Doom: future vacuum decay

The previous section showed that the tunneling rate out of a false vacuum
might not be large enough to undergo a transition at temperatures near but
below the critical temperature. At later times, the causal 4-volume increases
as $H^{-4}$, and the bounce action is given by the 4d bounce action, well
approximated by the potential for $T=0$. It could be that the growing Hubble
volume exceeds both the inverse decay per time and volume at this later time,
and vacuum decay would ensue; if so a conservative bound would be to require
this to happen before BBN,

$\displaystyle B\leq 4\log\frac{v}{H(T_{BBN})}\simeq
4\log\frac{vM_{\textrm{pl}}}{T_{BBN}^{2}}\simeq 250\,.$ (6.11)

It is the case however, that for all studied cases of history Q-, if the phase
transition does not occur at $T_{ew}$, it will also not occur before BBN. This
means that any point on Fig. 18 of Sec. 6.1 on the lower left white region,
which marked instances where the universe remained in the false minimum
$\phi_{-}$ past the electroweak epoch, the universe will remain in $\phi_{-}$
until today. For this reason these points are deemed unphysical.

The situation of being stuck in a false minimum is realised in history Q0 in
the opposite quadrant, $\epsilon<0$, $\delta>0$, with the further approximate
constraint of $(\mbox{ctg}_{\chi}^{2}-2)\delta\gtrsim-8\epsilon/\gamma_{4}$ as
sketched in Tab. 2. In practice, this approximate constraint has to be
extended to a numerical search for history Q0. In Fig. 20, we display in blue
the region where the thermal history is that of Fig. 6, with two minima at
high temperature. Above this region, we have history Q0. The infrared problem
as sketched in Sec. 4.2 casts doubt on histories where extrema are to be found
in regions with $\varepsilon_{IR}(\phi_{-})>1$; for such cases we do not trust
our perturbative computation, and so we do not discuss the region below the
red solid and dashed lines in Fig. 20. We also show in gray colour in Fig. 20
the regions excluded by the boundedness-from-below and perturbativity criteria
described above.

Even when history Q0 can be perturbatively treated, the fact that at high
temperature the only minimum is $h_{0}(T)$, which turns into a false vacuum at
$T_{c}$, has to be reconciled with us being at $h_{0}(T)$ today; this demands
that the decay rate be small enough so that the transition has not occurred
yet

$\displaystyle B\geq 4\log\frac{v}{H_{0}}\simeq 410.$ (6.12)

This case with negative $\epsilon$ is however more restricted from the
perturbativity and boundedness from below constraints. This can be understood
since for $\epsilon<0$, $\gamma_{\epsilon}<1$ and given
$\delta\gamma_{4}=\gamma_{\epsilon}-\gamma_{4}/\gamma_{a}$ for positive
$\delta$ we require small $\gamma_{4}/\gamma_{a}$, yet one of the terms in the
perturbative bound of Eq. (4.40) scales with $(\gamma_{4}/\gamma_{a})^{-2}$.
This perturbative bound then translates into a lower bound on negative
$\epsilon$ to obtain a bounded from below potential in a range of $\phi\sim
4\pi v$. On the other hand the limit of $\epsilon\to 0^{-}$ gives an infinite
lifetime and hence is compatible with observation, the question then is if
within this allowed window for $\epsilon$ decay lifetimes around the age of
the universe can be found.

Figure 20: Bounds on the $(\epsilon,\delta)$ parameter space for the doom
scenario at $\gamma_{4}=0.95$ and $\chi^{2}=0.1$. The upper-left gray region
is excluded by the boundedness-from-below (dark) and perturbativity (light)
criteria. The hatched region below the solid red line is excluded by the
strong IR constraint, Eq. (4.38), while the dashed red line shows the weak
version, Eq. (4.36). There is some noise associated with our method of
numerically tracking the temperature evolution of the minima, and this noise
is visible in the strong IR constraint curve. The blue region corresponds to
the history displayed in Fig. 6, in which two minima exist in the high-
temperature potential. The blue line separates the regions where the sign of
the magnitude defined in Eq. (4.64) is positive or negative and is an estimate
for the regions for histories Q-,0 respectively

To answer this question we turn to the zero-temperature bounce action; a
(euclidean) space-time symmetric solution with the $T=0$ potential. A quick
estimate of the thin wall approximation for our zero temperature potential,
which holds if

$\displaystyle\frac{\Delta V}{3\sigma
m_{h}}=\frac{1}{2\gamma_{\epsilon}^{2}}(1-\gamma_{\epsilon}^{-1}(1-\epsilon))\ll
1,$ (6.13)

gives the tension as in the bubble case expanding in $\epsilon$

$\displaystyle\sigma=\int_{h_{+}}^{0}dh\sqrt{2V(h)}=\mu+\mathcal{O}(\epsilon)=\frac{2\sqrt{\lambda}v^{3}\gamma_{4}^{-2}}{3},$
(6.14)

and an expression for the bounce action as [51]

$\displaystyle B=\frac{27\pi^{2}\sigma^{4}}{2\Delta
V^{3}}=\frac{3^{3}\pi^{2}}{2}\frac{\lambda^{2}2^{4}\gamma_{4}^{-8}}{3^{4}}\frac{1}{2^{6}\lambda^{3}\epsilon^{3}\gamma_{4}^{-6}}=\frac{\pi^{2}}{24\lambda\gamma_{4}^{2}\epsilon^{3}}.$
(6.15)

This analytic approximation returns, for $\gamma_{4}=1$, $\epsilon>-0.15$
decay lifetimes longer than today. This is not a particularly accurate result:
the use of CosmoTransitions and $\gamma_{4}=1$ returns $\epsilon>-0.475$ for a
decay after today. All these values are however ruled out by the
perturbativity and boundedness bound. These bounds are relaxed for smaller
$\gamma_{4}$ allowing for larger $|\epsilon|$ values, so one might expect this
limit to allow for shorter universe lifetimes. In this same limit however, the
bounce action, while not well approximated by the thin wall, does factor out
in the same way for $\gamma_{4}$, i.e.
$B=\lambda^{-1}\gamma_{4}^{-2}f(\epsilon)$ and so reducing $\gamma_{4}$
actually increases the lifetime exponentially. One concludes then that the
lifetime of the universe is in all instances well beyond the current age.

Figure 21: Top: excluded values of the lifetime of the universe as a function
of $\gamma_{4}$. Bottom: example points with the minimal lifetime allowed by
boundedness from below in the second quadrant of the $(\epsilon,\delta)$
plane, used to generate the blue line in the top plot. In these three plots,
the shaded region is ruled out from various constraints: boundedness from
below (darker, solid), perturbativity (lighter, solid), and LHC constraints
(dashed). See Sec. 8 and Fig. 27 for more details on all the bounds shown in
these plots.

The question that follows is then what are the allowed values for the
lifetime. This turns our gaze to the future which changes the estimation
qualitatively. Indeed the estimation of the Hubble 4-volume as the relevant
spacetime factor is valid in a decelerated expansion with scale factor $a\sim
t^{p}$ but not in an accelerated expansion. In this case revisiting the
estimation for the relevant 4-volume, i.e. the past light cone of an observer
at time $t_{f}$, one obtains for $a\sim e^{H_{0}t}$

$\displaystyle\int_{\textrm{light cone}}$ $\displaystyle
d^{4}x\sqrt{-\det(g_{\mu\nu})}=\frac{4\pi H_{0}^{-3}t_{f}}{3}.$ (6.16) light
cone eq. $\displaystyle\quad-a(t)d\chi=dt,\qquad\chi(t_{F})=0,$ (6.17)

i.e. an space-time volume which is a sphere of radius $H_{0}^{-1}$ times time,
and one concludes that the accelerated expansion from this perspective is
‘living in a box’, the longtime physicist dream. For a given value of
potential parameters we find the lifetime $t_{f}$ by solving

$\displaystyle\frac{4\pi H_{0}^{-3}t_{f}}{3}v^{4}e^{-B}=1.$ (6.18)

We have used this estimate to generate Fig. 21 which further illustrates the
point of the lifetime of the universe being much larger the its age. To
generate it, we look for the point in the second quadrant of the
$(\epsilon,\delta)$ space that gives the minimal lifetime allowed by the
boundedness-from-below criterion, for different values of $\gamma_{4}$. Such
points are shown in red in the 3 plots at the bottom of Fig. 21, corresponding
to 3 values of $\gamma_{4}$. We then compute the corresponding lifetime and
display it as the blue line in the top plot. The gray region on the left of
the top plot is excluded by a combination of all bounds, as described in Sec.
8. Thus, the only allowed lifetimes are those in the white region, and are all
larger than the age of the universe by a factor $>10^{2000}$.

##  7 Gravitational Waves

The spectrum of gravitational waves is customarily given as

$\displaystyle\Omega_{\rm GW}=\frac{d\rho_{\rm GW}}{d\log k}=\frac{d\rho_{\rm
GW}}{d\log f}\,,$ (7.1)

with $\rho_{\rm GW}$ the energy density in gravitational waves, $k$ the wave
number and $f$ frequency, in natural units $2\pi f=k$. In our study two
sources of gravitational waves have been identified: walls and first order
phase transitions. Let us discuss each in turn.

Gravitational emission occurs for domain walls as smaller curvature (larger
radius) wall structure enters the causal horizon and starts a damped
oscillation in the plasma. While most of the energy goes into the plasma, part
of it is emitted as gravitational waves, with the contribution at time $t$
after the big bang for an interval $\Delta t$ estimated as $\Delta\rho_{\rm
GW}/\Delta t\simeq G\mu^{2}/t$. Today the spectrum is redshifted and more so
the earlier the emission, so that the peak frequency corresponds to the
annihilation time $t_{w}$, and one has (see e.g. Ref. [52])

$\displaystyle f_{\textrm{peak}}$
$\displaystyle=a_{w}H_{w}=\frac{g_{0}^{1/3}T_{0}}{g_{\bar{w}}^{1/3}T_{w}}\sqrt{\frac{4\pi^{3}g_{w}}{45}}\frac{T_{w}^{2}}{M_{\rm
pl}}=\frac{g_{0}^{1/3}}{g_{\bar{w}}^{1/3}}\left(\frac{4\pi^{3}g_{w}}{45}\right)^{1/4}T_{0}\sqrt{\frac{6\epsilon
m_{h}}{M_{\rm pl}}},$ (7.2)
$\displaystyle=\frac{g_{0}^{1/3}}{g_{\bar{w}}^{1/3}}\sqrt{\frac{4\pi^{3}g_{w}}{45}}\frac{T_{0}T_{\rm
BBN}}{M_{\rm pl}}\frac{T_{w}}{T_{\rm BBN}}=1.1\times
10^{-10}\frac{T_{w}}{T_{\rm BBN}}\textrm{ Hz},$ (7.3)

where $g_{\bar{w}}=g_{w}-g_{\nu}=g_{w}-21/4$ and in the last equality we
assumed that $T_{w}$ occurred while $g_{w}=10.75$ which is the latest allowed.
Correspondingly, the power spectrum value at the peak is given by:

$\displaystyle\Omega_{\rm GW}(f_{\textrm{peak}})$
$\displaystyle=\frac{G_{N}\mu^{2}}{\rho_{cr}}\left(\frac{T_{0}}{T_{w}}\right)^{4}=\Omega_{\gamma}^{0}\frac{2^{3}\pi
g_{w}}{3^{5}g_{0}}\frac{v^{4}}{M_{\rm
pl}^{4}}\frac{1}{\epsilon^{2}\gamma_{4}^{4}}$ (7.4)
$\displaystyle=\Omega_{\gamma}^{0}\frac{40}{3\pi^{2}g_{0}}\frac{m_{h}^{2}v^{4}}{M_{\textrm{pl}}^{2}T_{BBN}}\frac{T_{BBN}^{4}}{T_{w}^{4}}\gamma_{4}^{-4}=1.4\times
10^{-17}\frac{T_{BBN}^{4}}{T_{w}^{4}}\gamma_{4}^{-4}.$ (7.5)

One can note, when expressed in terms of the ratio of temperatures for the BBN
and wall annihilation, that the frequency is bounded to be greater than $0.1$
nHz, and the peak in the spectrum increases with decreasing $\gamma_{4}$ and
decreasing frequency. These features are shown in Fig. 15; given that the
strength of the gravitational wave signal is greater for lower frequencies,
one finds that the low frequency Square Kilometer Array (SKA) experiment [46]
would probe part of the parameter space. As we shall see in Sec. 8, when put
against LHC bounds for our model, however, the parameter region that would
yield a signal at SKA is already excluded. The shape of the full spectrum can
be approximated by $k^{-1}$, and $k^{3}$ right and left of the peak
respectively [52], in qualitative agreement with simulations [53].

For a first order phase transition undergoing bubble nucleation, the
gravitational wave signal can be estimated using the thermal parameters
derived from the tunnelling action: the speed of the transition $\beta/H$ and
the strength of the phase transition $\alpha$ related to the latent heat, both
evaluated at the nucleation temperature $T_{N}$ given in Eq. (6.10).666More
precisely, the relevant temperature is the transition temperature $T_{*}$.
Here we assume $T_{*}\approx T_{N}$ and do not distinguish between them, a
safe assumption for fast phase transitions without significant reheating and
in the absence of large supercooling [54, 55] The thermal parameters are
explicitly:

$\displaystyle\alpha$ $\displaystyle=\frac{1}{\rho_{rad}(T_{N})}\left(\Delta
V(T_{N})-T_{N}\frac{d\Delta V}{dT}\big{|}_{T=T_{N}}\right)\,,$ (7.6)
$\displaystyle\frac{\beta}{H_{*}}$
$\displaystyle=T_{N}\frac{d(S_{3}/T)}{dT}\big{|}_{T=T_{N}}\,,$ (7.7)

where $H_{*}$ is the Hubble constant at the time of nucleation, $\Delta V(T)$
is defined in Eq. (4.30), and $\rho_{rad}(T_{N})=g_{*}\pi^{2}T_{N}^{4}/30$ is
the radiation energy density. The quantity $g_{*}$ is the effective number of
relativistic degrees of freedom in the plasma at $T_{N}$, which was calculated
by tracking particle decoupling following [56]. The GW signal also depends on
the velocity of the bubble wall in the rest-frame of the plasma, $v_{w}$,
estimated in Ref. [57]777A more recent, model-independent prescription to
determine the bubble wall velocity is outlined in Ref. [58] which we leave for
future work. to be

$\displaystyle v_{w}$
$\displaystyle=\left\\{\begin{array}[]{cl}\sqrt{\frac{\Delta
v_{\textrm{min}}}{\alpha\rho_{\textrm{rad}}}}\,,&\textrm{for }\frac{\Delta
v_{\textrm{min}}}{\alpha\rho_{\textrm{rad}}}<v_{\textrm{J}}\\\ 1,&\textrm{for
}\frac{\Delta v_{\textrm{min}}}{\alpha\rho_{\textrm{rad}}}\geq
v_{\textrm{J}}\end{array}\right.\,,$ (7.10)

where $v_{\textrm{J}}$ is the Jouguet velocity [59, 60, 61].

As demonstrated in Sec. 6.1, our non-linear theory model is capable of FOPTs
that are qualitatively different than those accessible in SMEFT theories. In
particular, we are interested in bubble nucleation that occurs in the
parameter space identified in Fig. 18. We now present in Fig. 22 the behaviour
of the associated thermal parameters $\alpha,\ \beta/H_{*}$ along with $T_{N}$
and $v_{w}$ as a function of $\epsilon,\delta$, using the same slice of
parameter space as Fig. 18. With these parameters associated with bubble
nucleation, we can proceed to estimate the GW signal observable today.

Figure 22: The thermal parameters $T_{N},\alpha,\beta/H_{*},v_{w}$ for a slice
of $\epsilon,\delta$ parameter space with $\gamma_{4}=1.6$ and
$\chi=\sqrt{0.1}$. The thermal parameters are defined respectively in Eqs.
(6.10), (7.6), (7.7), and (7.10). Note Fig. 18 is included again on the top
left plot.

Next, we discuss our calculation of the gravitational wave signal from the
thermal parameters. The following discussion is largely pedagogical, and we
note that our analysis follows standard techniques. There are three sources of
gravitational wave production during bubble nucleation: the bubble collisions,
plasma sound waves driven by the expanding bubble wall, and turbulence [55].
The total gravitational wave energy density, $\Omega_{\text{GW}}$, is the sum
of the signal from each of these sources is

$\displaystyle\Omega_{\text{GW}}$
$\displaystyle=\Omega_{\text{col}}+\Omega_{\text{sw}}+\Omega_{\text{tu}}\,.$
(7.11)

The initial source for GW radiation occurs when the bubbles first collide,
breaking spherical symmetry. The contribution from bubble collisions, however,
is thought to be small (order percent) so long as the walls do not enter the
runaway regime [54], so here we do not include them in our estimation of the
GW signal.

As the bubbles expand, wave fronts emerge preceding the the bubble walls,
forming acoustic sound waves propagating in the plasma. This is typically the
dominant contribution to the GW signal near the peak frequency of the
gravitational wave power spectrum. Schematically, this power spectrum as
measured today can be estimated by redshifting the GW signal at the source,

$\displaystyle\Omega_{\text{sw}}$
$\displaystyle=F_{\text{redshift}}\times\Omega_{\text{amp}}\times
S_{\text{sw}}(f)\,,$ (7.12)

where $S_{\text{sw}}(f)$ is the spectral shape of the GW signal. Accounting
for the redshift of GW radiation from the time of production introduces the
factor (as outlined in, for example, Refs. [62, 54, 63])

$\displaystyle F_{\text{redshift}}$ $\displaystyle=3.57\times
10^{-5}\left(\frac{100}{g_{*}}\right)^{1/3}\,.$ (7.13)

The GW energy density from the sound waves in the plasma is [64]:

$\displaystyle\Omega_{\text{sw},0}$
$\displaystyle=3\Gamma^{2}\bar{U}_{f}^{4}(H_{*}R_{*})\tilde{\Omega}_{\text{sw}}\,,$
(7.14)

where $\Gamma\sim 4/3$ is the adiabatic index, $\bar{U}_{f}$ is the root mean
square (RMS) fluid four-velocity, and $R_{*}$ is the average separation
between bubbles $R_{*}=(8\pi)^{1/3}v_{w}/\beta$ [54]. The efficiency factor
$\tilde{\Omega}_{\text{sw}}$ stems from converting motion in the fluid to
metric perturbations, estimated from simulation to be
$\tilde{\Omega}_{\text{gw}}=0.012$ [65, 64]. The RMS velocity $\bar{U}_{f}$ is
set by the strength of the phase transition $\alpha$ through

$\displaystyle\bar{U}_{f}^{2}$
$\displaystyle=\frac{3}{4}\kappa_{\text{sw}}\frac{\alpha}{1+\alpha}\,.$ (7.15)

The quantity $\kappa_{\text{sw}}$ controls how much of the vacuum energy is
transferred into kinetic energy of the plasma. This has been studied in detail
in Ref. [61], resulting in numerical fits of
$\kappa_{\text{sw}}(v_{w},\alpha)$ to approximate full solutions to the
relativistic fluid equations of the plasma. The results of these fits are used
here and checked against PTPlot [54].

During the phase transition, shock waves will develop, at which point the
motion of the plasma is better described by turbulence than acoustic sound
waves. If the time scale of shock formation is small compared one Hubble time
$\tau_{\text{sh}}<1/H_{*}$, the resulting sound wave signal is reduced by a
factor $H_{*}\tau_{\text{sh}}=H_{*}R_{*}/\bar{U}_{fl}.$ The total
$\Omega_{\text{amp}}$ is then888The numerical prefactor ensures the total GW
power resulting from integrating the power spectrum $\Omega_{\text{sw}}$
reproduces the total power estimate $\Omega_{\text{sw},0}$ [64]

$\displaystyle\Omega_{\text{amp}}$
$\displaystyle=2.061\,\Omega_{\text{sw},0}\text{Max}[H_{*}\tau_{\text{sh}},1]\,.$
(7.16)

Putting all of this together gives

$\displaystyle h^{2}\Omega_{\text{sw}}(f)$ $\displaystyle=2.59\times
10^{-6}h^{2}\left(\frac{100}{g_{*}}\right)^{1/3}\Gamma^{2}\bar{U}^{4}_{fl}\left(\frac{H_{*}}{\beta}\right)v_{w}\text{Max}[H_{*}\tau_{\text{sh}},1]S_{\text{sw}}(f)\,.$
(7.17)

The spectral shape of the sound wave power spectrum is [55]

$\displaystyle S_{\text{sw}}$
$\displaystyle=\left(\frac{f}{f_{\text{sw}}}\right)^{2}\left(\frac{7}{4+3\left(f/f_{\text{sw}}\right)^{2}}\right)^{7/2}\,,$
(7.18)

written in terms of the peak frequency as observed today,

$\displaystyle f_{\text{sw}}$ $\displaystyle=\left(8.876\ \mu\text{
Hz}\right)\left(\frac{g_{\text{eff}}}{100}\right)^{1/6}\left(\frac{T_{N}}{100\text{
GeV
}}\right)\left(\frac{1}{v_{w}}\right)\left(\frac{\beta}{H}\right)\left(\frac{z_{p}}{10}\right)\,.$
(7.19)

This results from red shifting the frequency profile obtained from numerical
simulations, which predict $z_{p}\sim 10$ [64].

(a)

(b)

Figure 23: The gravitational wave signal for $\gamma_{4}=1.6$, $\chi^{2}=0.1$.
The GW signal curve is the maximal envelope of the power spectra obtained from
varying $\epsilon$ and $\delta$. To give a sense of the $\epsilon$\- and
$\delta$-dependence, the parameter space is broken up into subsets, and a
resulting maximal envelope power spectrum is drawn for each subset. In Fig.
23(a), the $\epsilon$-dependence is emphasised by breaking the parameter space
up into subsets of $\epsilon$ ranges as specified in the legend, while varying
over all $\delta$ for each curve. In Fig. 23(b), the $\delta$-dependence is
correspondingly emphasised, this time varying over all $\epsilon$ for each
curve. The GW sensitivity curves are drawn for SKA [46], LISA [66], the Big
Bang Observer (BBO) [67], and the Einstein telescope (ET) [68].

The next important contribution to the overall signal is the turbulence term.
In this case, the power of the gravitational wave signal at the source is now
generated by turbulence in the fluid [63, 69, 70]. While our understanding of
turbulence during cosmological phase transitions is still evolving [71, 72,
73, 74, 63], the result used here is:

$\displaystyle h^{2}\Omega_{\text{tu}}(f)$ $\displaystyle=3.35\times
10^{-4}\left(\frac{H_{*}}{\beta}\right)\left(\frac{\kappa_{\text{tu}}\alpha}{1+\alpha}\right)^{3/2}\left(\frac{100}{g_{*}}\right)^{1/3}v_{\text{w}}S_{\text{tu}}(f)\,,$
(7.20)

where the efficiency factor $\kappa_{\text{tu}}$ parameterises how much of the
kinetic energy is converted into turbulent motion. Here we use
$\kappa_{\text{tu}}=0.05\,\kappa_{\text{sw}}$, as chosen in Ref. [55]. This
approximation is based on simulated fluid motion [65], though this simulation
does not last long enough to realistically capture turbulence effects. We
emphasise that, as noted in the literature, this efficiency factor is not yet
well-understood [54, 63, 75, 76]. Finally, the spectral shape of the
turbulence signal is [77]:

$\displaystyle S_{\text{tu}}(f)$
$\displaystyle=\frac{\left(f/f_{\text{tu}}\right)^{3}}{\left[1+f/f_{\text{tu}}\right]^{11/3}\left(1+8\pi
f/h_{N}\right)}\,,$ (7.21)

where

$\displaystyle f_{\text{tu}}$ $\displaystyle=27\mu\text{
Hz}\frac{1}{v_{\text{w}}}\left(\frac{\beta}{H_{*}}\right)\left(\frac{T_{N}}{100\text{
GeV }}\right)\left(\frac{g_{*}}{100}\right)^{1/6}\,,$ (7.22) $\displaystyle
h_{N}$ $\displaystyle=16.6\mu\text{ Hz}\left(\frac{T_{N}}{100\text{ GeV
}}\right)\left(\frac{g_{*}}{100}\right)^{1/6}\,.$ (7.23)

The total gravitational wave signal that results from both the sound wave and
turbulence contributions is plotted in Fig. 23 for a benchmark choice of
$\gamma_{4}$, $\chi$ in the nonlinear theory model. We also show the
sensitivity curves for planned gravitational wave detectors. Given
$\gamma_{4}$ and $\chi$, there is still a range of $\epsilon$ and $\delta$
that give bubble nucleation and generate a gravitational wave signal. This
figure summarizes this parameter space by plotting the maximal envelope of the
many power spectra that result from varying $\epsilon$ and $\delta$. While no
single choice of $(\epsilon,\delta)$ will reproduce the whole curve, the
maximal envelope overlapping with the GW detector sensitivity curves indicates
at least one point in $(\epsilon,\delta)$ parameter space with an observable
GW signal.

##  8 Complementarity with LHC

While measurements at the LHC have opened the door to the electroweak symmetry
breaking mechanism and made formidable advances in its exploration, these
measurements are also limited to only probe couplings of the electroweak
sector around the vacuum. As this work has tried to underline, cosmology has
the potential to reach where such scattering experiments cannot, to non-local
effects in field space. Here we establish this complementarity quantitatively
on a concrete case study by confronting our previous cosmological analysis
with LHC bounds.

Local observables correspond to coefficients in a covariant expansion of
fields of the action around the vacuum. These can be given in the scalar
sector in terms of covariant derivatives of the cuvature tensor and scalar
potential, reproducing here for simplicity the curvatures of Sec. 2:

$\displaystyle v^{2}R_{\varphi}(0)$
$\displaystyle=1-c_{\chi}^{4}\gamma_{a}^{2},$ $\displaystyle v^{2}R_{h}(0)$
$\displaystyle=-\gamma_{a}^{2}s_{\chi}^{2}c_{\chi}^{2}\,,$ (8.1)

which are bounded by ATLAS to a $95\%$ confidence level to be

$\displaystyle v^{2}R_{\varphi}(0)$ $\displaystyle=-0.080^{+0.12}_{-0.13},$
$\displaystyle v^{2}R_{h}(0)$ $\displaystyle=+0.080^{+1.0}_{-1.1}\,,$ (8.2)

where these bounds are derived from [78, 79]. Also relevant will be the
element of the third covariant derivative of the potential
$(\mathcal{\nabla}^{3}V)_{hhh}=V^{\prime\prime\prime}(0)=3\lambda\gamma_{4}(1-\epsilon)v$,
probed by measurements of triple Higgs coupling. Bounds from ATLAS [80] give
the following constraints at $95\%$ confidence on the triple-Higgs coupling in
the kappa framework (i.e. ratio of the triple-Higgs coupling to the SM
expectation)

$\displaystyle-1.0<\kappa_{\lambda}=\frac{V^{\prime\prime\prime}(0)}{V^{\prime\prime\prime}_{\text{SM}}(0)}=\gamma_{4}(1-\epsilon)<6.6\,.$
(8.3)

Here, we combine experimental input from the LHC with the cosmological
analyses of Secs. 5 and 6 in order to present a final comprehensive
phenomenological picture. We categorise these results into different different
cosmological processes as follows:

  * •

Symmetry restoration. The small $\epsilon,\delta$ limit allows for