aainstitutetext: Pennsylvania State University-New Kensington, New Kensington,
PA 15068bbinstitutetext: Kavli IPMU (WPI), UTIAS, The University of Tokyo,
Kashiwa, Chiba 277-8583, Japanccinstitutetext: Department of Physics and
Astronomy, UCLA, Los Angeles, CA 90095, USA

# Generating Non-topological Solitons Via Thermal Corrections: Higgs Balls

Lauren Pearce<EMAIL_ADDRESS>b Graham White<EMAIL_ADDRESS>c,b
Alexander Kusenko<EMAIL_ADDRESS>
###### Abstract

Scalar fields which carry charge can generally form non-topoligical solitons
(Q-balls), if the energy in the extended configuration is less than the energy
of an equivalent number of free quanta. For global Q-balls, such solitons
exist whenever the potential grows slower than quadratically. We show that
even in the absence of attractive interactions, finite temperature corrections
can generate Q-ball solutions, as the coefficient of cubic corrections is
generally negative. As an illustration of this, we consider the possibility of
constructing Q-balls using the Higgs field. We first show that the finite
temperature corrections would enable the existence of Higgs balls if the
Standard Model symmetry was ungauged. We then show that Higgs self-
interactions mediated by the Standard Model gauge bosons are sufficient to
prevent the existence of these states in the actual Standard Model, but can be
present in a variety of extensions.

††preprint: IPMU22-0030

##  1 Introduction

###

Non-topological solitons, or Q-balls, are extended configurations of a scalar
field whose stability is typically ensured by a conserved charge. Self-
interactions ensure that the energy of the extended configuration with charge
$Q$ is less than the energy of $Q$ quanta of the scalar field Rosen:1968mfz ;
Friedberg:1976me ; Coleman:1985ki ; Kusenko:1997ad . For models with a global
charge, this occurs when $\sqrt{V(\phi)/\phi^{2}}$ is minimized at nonzero VEV
$\phi$ Coleman:1985ki .

###

In this work, we first observe that in a wide range of models, finite
temperature corrections can allow Q-balls which do not existence at zero
temperature. (This is different than studying thermal corrections in systems
which admit Q-balls at zero temperature, as in e.g. Laine:1998rg .) In
particular, symmetry principles generally forbid a tree level cubic $\phi^{3}$
term in the potential. However, finite temperature corrections to the
potential generally scale as $V_{\rm 1-loop}\propto
T^{4}\int_{0}^{\infty}dx\,x^{2}\ln(1\mp e^{-\sqrt{x^{2}+y}})$, where
$y=m^{2}/T^{2}$. In the high temperature limit, the expansion of the logarithm
generates terms $\propto T|\phi|^{3}$ (assuming the mass is proportional to
the scalar VEV). Furthermore, because the coefficient is generically negative,
the potential grows slower than quadratically when this term dominates.

###

The Standard Model provides an excellent playground to explore these states.
Because the Higgs field is responsible for mass generation, the finite
temperature corrections have the necessary form. Furthermore, because the
running quartic term is small, there is a comparatively large regime in which
the cubic term is relevant. In this paper, we will explore the properties of
these “Higgs balls”. We emphasize that these non-topological solitons are made
of Higgs quanta; this is different from other models in which non-topological
solitons form from other scalar quanta whose interactions are mediated by
Higgs bosons Endo:2022uhe ; Bishara:2017otb .

###

However, one complication is that the Standard Model Higgs carries gauged, not
global, charge. When a field carries gauged (as opposed to global) symmetry,
Q-balls may still exist Rosen:1968mfz ; Lee:1988ag ; Lee:1991bn , although
only for sufficiently small charges Lee:1988ag ; Gulamov:2015fya . (See alose
the review Ref. Nugaev:2019vru .) As we discuss, this has implications for the
existence of Q-balls, both in the Standard Model and in extensions. We also
focus on Higgs balls because they illustrate many non-trivial aspects that can
arise in thermal balls. In addition to being gauged (not global) Q-balls, they
are also not absolutely stable.

###

Like global Q-balls, gauged Q-balls may be long-lived but not absolutely
stable. As mentioned, our Higgs balls will be of this type, because the Higgs
quanta can decay (for example, through the Yukawa interactions). Since the
Higgs field is the only scalar field that carries $\mathrm{SU}(2)$ and
$\mathrm{U}(1)$ charge, we note that we can consider the Higgs balls as
carrying a global “Higgs charge”. When the rate of Higgs-particle-number-
changing processes inside the soliton is smaller than the Hubble parameter,
Higgs charge is effectively conserved. Therefore, we show the Higgs balls are
long lived for sufficiently high temperatures. This can be generalized to
other gauged thermal balls.

###

This paper is organized as follows: First, in section 2 we discuss how thermal
effects can produce Q-balls at finite temperature in models which do not have
Q-balls at zero temperature. We then turn to Higgs balls as a specific
illustration of these “thermal balls”. In section 3, we first study Higgs
balls in a Standard Model-like scenario in which the
$\mathrm{U}(1)\times\mathrm{SU}(2)$ symmetry is a global symmetry. We see that
the finite temperature corrections allow for the existence of (global)
Q-balls.

###

In section 4, we then include gauge interactions by generalizing results for
$\mathrm{U}(1)$ gauged Q-balls to $\mathrm{SU}(2)\times\mathrm{U}(1)$. We see
that even in the static charge limit, the gauge interactions are sufficient to
destabilize Higgs balls in the Standard Model; they have higher energy than an
equivalent number of free Higgs quanta.

###

However, it is not necessarily the case that the gauge interactions always
destabilize the Q-balls, and in section 5, we consider an extension of the
Standard Model in which non-topological solitons remain stable against decay
into separate Higgs quanta.

###

Finally, although the Higgs balls are stable against decay into individual
Higgs quanta, the Standard Model Yukawa couplings enable them to decay into
fermions. This decay rate is suppressed since it can only occur at the
surface, and we make brief remarks on this in section 6.

###

Finally we conclude, emphasizing both the general features that enable the
existence of “thermal balls” in various models, as well as the specific
features of Higgs balls.

##  2 Thermal Balls

###

In this section, we discuss how thermal effects can generally create non-
topological solitons in models which have no Q-balls at zero temperature.

###

We consider a scalar field $\phi$ with vacuum expectation value (VEV) $v$,
which at zero temperature is determined by minimized a potential $V(v)$. The
one-loop finite temperature corrections to this potential can be expressed as
Arnold:1992rz ; DelleRose:2015bpo

$\displaystyle V_{1-\mathrm{loop}}(v,T)$
$\displaystyle=\sum_{\mathrm{bosons}}\dfrac{n_{i}T^{4}}{2\pi^{2}}J_{B}\left(\dfrac{m_{i}^{2}}{T^{2}}\right)+\sum_{\mathrm{fermions}}\dfrac{n_{i}T^{4}}{2\pi^{2}}J_{F}\left(\dfrac{m_{i}^{2}}{T^{2}}\right),$
(1)

where

$\displaystyle J_{B}(y)$
$\displaystyle=\int_{0}^{\infty}dx\,x^{2}\ln\left(1-e^{-\sqrt{x^{2}+y}}\right),$
$\displaystyle J_{F}(y)$
$\displaystyle=\int_{0}^{\infty}dx\,x^{2}\ln\left(1+e^{-\sqrt{x^{2}+y}}\right).$
(2)

###

Although we use the full finite temperature corrections, the high temperature
limit is helpful in understanding this phenomenon. In this limit, the
functions above become

$\displaystyle J_{B}(x)$
$\displaystyle=-\dfrac{\pi^{4}}{45}+\dfrac{\pi^{2}}{12}x-\dfrac{\pi}{6}x^{3/2},$
$\displaystyle J_{F}(x)$
$\displaystyle=\dfrac{7\pi^{4}}{360}-\dfrac{\pi^{2}}{24}x,$ (3)

with the next order corrections of order $x^{2}\ln(x)$. If the bosons have a
mass $\sim gv$, the bosonic loops will produce a term of the form
$-g^{3}T|v|^{3}$, which is effectively an attractive interaction.

###

As has been noted Buchmuller:1993bq , second order loop corrections also
contribute to this cubic term, due to the regularization of infrared
divergences. These can be calculated through the ring (or daisy) diagrams,
which we include in our calculations below. We also note that although the
conceptual explanation above makes use of the high temperature expansion, in
the results presented below we numerically integrate the $J_{B}$ and $J_{F}$
functions in the regime in which the high temperature expansion is not valid.

###

The importance of these cubic terms in spontaneous symmetry beaking has long
been noted; when they are sufficiently large the phase transition is first
order Buchmuller:1993bq . In this work, we demonstrate their importance to the
existence of solitons.

###

The bosonic contributions can come from either scalar or vector fields. Scalar
fields can have a mass term generated by interactions with the $\phi$ field.
We assume that $\phi$ carried a global charge, in which case non-topological
solitons (Q-balls) exist if $V(v,T)/v2$ is minimized at non-zero $v$. We see
immediately that the existence of a $-Ag^{3}T|v|^{3}$ ensures that this is
satisfied. We note that this can occur even in a model in which $\phi$ is the
only field, provided that at sufficiently large VEVs its mass is given by its
self-interaction times its VEV. As we will see below, in the Standard Model
the small Higgs self-coupling suppressed this contribution, but this will be
explored further in Section 5, when we consider BSM (beyond the Standard
Model) scenarios.

###

This situation is more complicated when vector bosons are used. Vector bosons
generally acquire masses of the form $gv$ through spontaneous symmetry
breaking, in which case the scalar field $\phi$ carries gauge charge. The
gauge fields mediate self-interactions, which can increase the energy of the
Q-ball. If the gauge group in question is $\mathrm{U}(1)$, the energy per unit
charge of the gauged Q-ball (in the thin wall limit) is given by Heeck:2021zvk

$\displaystyle\omega(R)$ $\displaystyle=\omega_{0}(R)gv_{0}R\coth(gv_{0}R)$
(4)

where $\omega_{0}$ is the energy per unit charge of a global Q-ball (i.e.,
calculated ignoring gauge self-interactions of the scalar field), and $v_{0}$
is the VEV inside this Q-ball. Provided that $\omega(R)<m_{\phi}$, the mass of
a scalar quanta outside the soliton, non-topological soliton solutions exist.
For sufficiently small $gv_{0}R$, the additional energy from gauge-self
interactions is small and non-topological solitons will exist.

###

However, $\omega_{0}$ (and $v_{0}$) will generally depend on $g$, and
therefore the existence of gauged thermal Q-balls has to be considered in each
individual model. Since the only scalar field confirmed to exist is the Higgs
field, we are particularly interested in it, which requires us to generalize
the above results.

The second condition for a Q-ball is traditionally the presence of a
conservation law for particle number, such that the quanta cannot lose energy
by decaying into lighter particles. In the early Universe, we argue that this
condition is too strict and one merely requires that the decay rate of the
quanta be smaller than the expansion rate of the Universe.

###

We will call the non-topological solitons produced by thermal corrections, and
therefore which do not exist at zero temperature, thermal balls. We now turn
our attention to determining whether such thermal balls exist in the Standard
Model Higgs sector.

##  3 Higgs Balls in the Global Standard Model

###

As the Standard Model of particle physics has a single scalar field, the Higgs
field, it is natural to first ask whether these exist within the Standard
Model. To answer this question, we first study the non-topological solitons
that would exist if the Standard Model $\mathrm{SU}(2)\times\mathrm{U}(1)$
symmetry were not gauged. More specifically, we ignore repulsive interactions
mediated by the gauge bosons, which allows us to use global soliton equations.
(We do keep all Standard Model interactions when calculating corrections to
the Higgs potential, including those involving the gauge bosons.) We do this
for two reasons: first, to show the qualitative features that determine the
existence of thermal balls, and secondly, as we demonstrate in section 4, the
energy of gauged Higgs balls can be found using the energy of this global
model. (This is expected from the $\mathrm{U}(1)$ results in Ref.
Heeck:2021zvk ; see equation (4).)

###

In this section, we consider the Standard Model Higgs, whose running quartic
coupling becomes small (and even negative) at large field values before
increasing again, and therefore there is a regime in which the potential grows
slower than quadratically. If the Higgs field carried only global charge, then
$V(h)/h^{2}$ would be minimized at non-zero $h$, indicating the existence of
thin wall Q-balls Coleman:1985ki . However, such Q-balls would typically have
$V(h)<0$ in their interiors, and they would induce a phase transition to the
true vacuum at large VEVs via the process of solitosynthesis Kusenko:1997hj ;
Postma:2001ea ; Metaxas:2000qf ; Pearce:2012jp ). We do not consider these
states.

###

The measured top quark pole mass is $173.1\pm 0.9\,\mathrm{GeV}$, and for the
central value, the Higgs potential becomes unstable around
$10^{10}\,\mathrm{GeV}$. However, the scale of the instability depends
strongly on the top quark mass. In order to better illuminate our solitons, we
will impose that the Higgs potential is stable up to $10^{17}\;\mathrm{GeV}$,
which corresponds to a pole top quark mass of $170.5\,\mathrm{GeV}$, which is
within $3\sigma$ of current observations ParticleDataGroup:2020ssz .

###

To generate our solitons, we note that in the early universe, finite
temperature corrections modify the Higgs potential Quiros:1994dr . In the work
the follows, we have included one-loop corrections to the Standard Model
potential and one-loop and ring (“daisy”) finite temperature corrections. The
renormalization group equations governing the running coupling constants have
been evaluated to two loop order. For the finite temperature corrections, we
include contributions from the gauge bosons as well as the Higgs and the top
quark, but other fermions are neglected due to their small Yukawa masses.
Details on how we calculated the finite temperature Standard Model potential
and evaluated the running coupling constants can be found in Appendix A, and
details of how we matched the parameters to observables are in Appendix B.

###

The left side of Fig. 1 shows the $V(h)/h^{2}$ at temperature
$T=10^{14}\,\mathrm{GeV}$, and is clearly minimized at nonzero field values
$h\sim 10^{15}\,\mathrm{GeV}$. To demonstrate that this is not a
solitosynthesis scenario, we show the potential $V(h)$ as a function of the
Higgs VEV on the right; as expected, it is positive up to the stabilization
scale of $10^{17}\,\mathrm{GeV}$.

Figure 1: Left: $V(h)/h^{2}$ as a function of Higgs VEV $h$, at a fixed
temperature of $T=10^{14}\,\mathrm{GeV}$. Because this is minimized at nonzero
VEV, thin wall Q-ball solutions exist. Right: $V(h)$ as a function of Higgs
VEV, at a fixed temperature of $T=10^{14}\,\mathrm{GeV}$. The Higgs potential
remains positive throughout the regime of interest, avoiding solitosynthesis.

###

Fig. 2 illustrates concretely how finite temperature contributions lead to the
Q-ball minimum. This figure compares the potential to the temperature-
dependent quadratic and quartic terms. At low VEVs, we see that the
temperature-dependent quadratic term dominates. However, as the VEV increases,
the potential begins to drop below the quadratic terms, indicating the
existence of thin wall Q-balls. This continues until $h\sim
10^{15}\,\mathrm{GeV}$, when the quartic term dominates.

Figure 2: The potential $V(h)$ (black) compared to the quadratic (blue,
dashed) and quartic (red, dotted) terms at $T=10^{14}\,\mathrm{GeV}$. Before
the quartic terms dominate around $h=10^{15}\,\mathrm{GeV}$, the potential is
below the quadratic contribution due to finite temperature corrections.

###

Before discussing gauge effects, it is helpful to study the properties of
these Q-balls, including their stability. Despite being large classical
states, they behave like radiation. The left plot of Fig. 3 shows the energy
per unit charge of the Higgs ball, as a function of temperature, and we see
that it scales linearly. This plot also shows the Higgs VEV in the interior of
the thin wall Q-ball, determined by minimized $V(h,T)/h^{2}$. This is also
approximately linear in temperature, although the linearity begins to break
down at larger scales.

###

Finally, we discuss the stability of the Higgs balls. Fig. 2 shows that the
$V(h,T)/h^{2}$ is only slightly smaller than the quadratic term, which is
likewise dominated by finite temperature corrections. Frequently this would be
sufficient to show that the Higgs ball has lower energy the equivalent number
of free quanta, but we must be careful when evaluating the mass of the free
Higgs quanta. Inside the Higgs ball, we have used the maximum of the
temperature or Higgs VEV as our renormalization scale for our running
couplings, but in the plasma outside the Higgs ball the temperature should be
used. Therefore, the mass of a Higgs quanta outside the Q-ball is not
necessarily the same as the one determined from the quadratic contribution to
the potential inside the Q-ball. We compare the energy per unit charge
$\omega_{0}$ to the mass of a Higgs quanta outside the Q-ball in the left plot
of Fig. 4, and see that the Q-ball is stable against decay into free Higgs
quanta for $T\gtrsim 10^{10}\,\mathrm{GeV}$. (As we will discuss below, Yukawa
couplings still allow these to decay into fermions, although they are long-
lived.)

###

This ensures the stability of thin wall Q-balls, in which the surface tension
contribution is neglected. Given the small difference between $\omega_{0}$ and
$m_{h,{\rm ext}}$, we should be concerned that the small contribution to the
energy from the surface tension may be sufficient to destabilize them. (By
this, we mean that the energy per unit charge is greater than the mass of a
Higgs quanta outside the Q-ball.) For a Q-ball of charge $Q$, the total energy
including the surface contribution is

$\displaystyle E$
$\displaystyle=Q\omega_{0}+\dfrac{4\pi}{3}h_{0}^{2}R\cdot\sqrt{m_{h,{\rm
ext}}^{2}-\omega_{0}^{2}},$ (5)

and we calculate the radius using the thin wall approximation, since we are in
the regime in which the surface tension energy is small compared to
$Q\omega_{0}$. This is plotted for a variety of charges in the right plot of
Fig. 4, showing that for charges $Q\gtrsim 10^{6}$, surface tension is not
sufficient to destabilize the Q-ball.

Figure 3: The energy per unit charge $\omega_{0}$ of the thin-wall Higgs ball
(black) and the Higgs VEV $h_{0}$ inside the Higgs ball (blue, dashed) as a
function of temperature.

Figure 4: Left: The ratio of the energy per unit charge of the Higgs ball to
the mass of a Higgs quanta outside of the Higgs ball, as a function of
temperature. The Higgs ball is stable when this is less than 1. Right: The
total energy of the Higgs ball, including surface effects, as a function of
temperature for a variety of charges. (Color online.)

###

Therefore, in the (unphysical) scenario in which the Standard Model
$\mathrm{SU}(2)\times\mathrm{U}(1)$ is ungauged, we see that thermal effects
produce non-topological solitons with charge greater than $10^{5}$ in the
temperature range $10^{10}\,\mathrm{GeV}$ and $10^{14}\,\mathrm{GeV}$.

##  4 Gauge Effects: Higgs Balls in the Standard Model

###

The Standard Model Higgs carries gauged, not global charge. Q-balls carrying
gauged $\mathrm{U}(1)$ charge have been studied in Ref. Heeck:2021zvk , which
showed that in the thin wall limit the energy per unit charge of the gauged
Q-ball $\omega$ is related to that of a global Q-ball, $\omega_{0}$. In
Appendix C, we generalize these first to a scalar field carrying gauged
$\mathrm{SU}(2)$ charge, and then to the Higgs field, which carries both
$\mathrm{SU}(2)$ and $\mathrm{U}(1)$ gauged charge.

###

We note that as in Ref. Heeck:2021zvk , we make the static charge
approximation. This approximation is perhaps more problematic for gauged
charge, as the assumption that only the zero component of the gauge fields is
nonzero removes $\mathrm{SU}(2)$ self-interactions between the gauge fields in
the Q-ball. The breakdown of the static charge approximation is connected to
the confining aspect of $\mathrm{SU}(N)$ gauge interactions. In this work, we
will ensure that the radii of our Q-balls is less than the $\mathrm{SU}(2)$
confinement scale, as calculated outside the Q-ball where the symmetry is
unbroken. We discuss this further in Appendix C.

###

As shown in the appendix, the energy per unit charge of the Higgs ball is
related to that of the ungauged Higgs ball by

$\displaystyle\omega=\frac{1}{2}Rh_{0}\omega_{0}\sqrt{g_{W}^{2}+g_{Y}^{2}}\coth\left(\frac{1}{2}Rh_{0}\sqrt{g_{W}^{2}+g_{Y}^{2}}\right).$
(6)

A Q-ball made of the Higgs field will carry both $\mathrm{U}(1)$ and
$\mathrm{SU}(2)$ charge; the amount of $\mathrm{SU}(2)$ charge is described by
the weak isospin. A quanta of the Higgs field has hyperchange $1/2$ and weak
isospin $-1/2$, so the charges of the Higgs ball satisfy
$Q_{Y}=-Q_{W}=-\dfrac{i}{2}\int
d^{3}x\,\left(\Phi^{\dagger}D_{0}\Phi-h.c.\right)$, as it must for it to be
electrically neutral. As shown in the appendix, the charge is related to the
large radii behavior of the gauge fields. As shown in the appendix,

$\displaystyle Q_{Y}=-Q_{W}$ $\displaystyle=\frac{8\pi
R\omega_{0}\left(Rh_{0}\sqrt{g_{W}^{2}+g_{Y}^{2}}\coth\left(\frac{1}{2}Rh_{0}\sqrt{g_{W}^{2}+g_{Y}^{2}}\right)-2\right)}{g_{W}^{2}+g_{Y}^{2}}.$
(7)

We note that the Appendix includes explicit expressions for the gauge fields
are derived in the static charge limit, thin wall case.

###

For a chosen charge $Q=|Q_{Y}|=|Q_{W}|$, equation (7) can be numerically
solved to find the radius of the gauged Q-ball, which can then be substituted
into equation (6) to find the energy per unit charge, including the potential
energy from gauge interactions between the Higgses. To determine stability,
this should be compared to the mass of a Higgs quanta in the plasma outside
the Higgs ball. This is shown in Fig. 5, which shows that the ratio is
generally larger than one. This means the Q-ball could lower its energy by
becoming separate free Higgs quanta. In fact, such states would not be
produced, as the repulsion force mediated by the gauge bosons would push the
Higgs quanta apart, to the lower energy state.

We could solve for the radius at which the energy per unit charge for the
gauged Q-ball is equal to the external Higgs mass, and then use equation (7)
to determine the charge of this state. Across the temperature range of
interest it is $\mathcal{O}(0.1)$, well outside the thin wall regime of
validity. Therefore, we conclude that Higgs balls do not exist in the Standard
Model, as the contribution to the energy from gauge interactions is sufficient
to destabilize them.

Figure 5: Ratio of energy per unit charge to Higgs mass outside the Q-ball, as
a function of temperature, for a range of charges. Because this ratio is
larger than one, if such Higgs balls were produced they would immediately
decay into individual quanta.

##  5 Higgs Balls Beyond the Standard Model

###

As the above makes clear, non-topological solitons cannot be constructed with
the Standard Model Higgs because the increased energy from gauge interactions
makes them unstable into decay into individual Higgs quanta. However, there is
no fundamental reason why such states could not exist, and since the behavior
of the Higgs potential at high scales is unknown, the existence of such states
is an open question. In this section, we explore Higgs balls in an extension
of the Standard Model.

###

Higgs balls are unstable due to the increased energy from gauge self-
interactions; these are decreased if the gauge couplings $g_{Y}$ and $g_{W}$
decrease. Although this can be accomplished by changing their running (for
example, through additional fermionic contributions with a negligible coupling
to the Higgs, which would provide a negative contribution to the $beta$
functions of the gauge couplings while leaving the Higgs quartic unaffected),
this is insufficient to stabilize the Higgs balls. The finite temperature
corrections which produce the minimum in $V(h)/h^{2}$ arise from the bosonic
$J_{B}$ functions and scale as powers of the bosonic masses. The contributions
of the gauge bosons disappear as the gauge couplings decrease, leaving only
the much suppressed Higgs and ghost contributions.

###

Therefore, in addition to modifying the gauge couplings at large scales, we
additionally include a singlet field $S$. The potential is

$\displaystyle
V(H,S)=-\mu^{2}H^{\dagger}H+\lambda_{H}(H^{\dagger}H)^{2}+\dfrac{m_{S}^{2}}{2}S^{2}+\lambda_{S}S^{4}+\lambda_{HS}H^{\dagger}HS^{2},$
(8)

where we have imposed a $\mathbb{Z}_{2}$ symmetry on the scalar. We assume the
mass of the scalar singlet $m_{S}$ is above the LHC scale, but when the Higgs
acquires a large VEV $h$ the singlet’s mass is given by

$\displaystyle m_{S,\mathrm{eff}}$
$\displaystyle=\sqrt{m_{S}^{2}+\lambda_{HS}h^{2}}\approx\sqrt{\lambda_{HS}}h.$
(9)

This mass is independent of the gauge couplings, which consequently decouples
the depth of the $V(h)/h^{2}$ minimum from the gauge coupling. We note that
the scalar field $S$ does not acquire a vacuum expectation value and will not
mix with the Standard Model Higgs boson. Furthermore, in our benchmark
scenario we will take $\lambda_{HS}=0.9$, forbidding Higgs decays into these
scalars even at large VEVs.

###

Now that we have decoupled the depth of the H-ball minimum from the gauge
couplings, we modify the running gauge couplings. We assume that at some scale
above the LHC scale but below the temperatures of interest here, new states
cause the Standard Model gauge couplings to run to significantly smaller
values. As noted above, this can be implemented through the introduction of
additional fermions; if these carry gauge charge, they contribute to the
$g_{Y}$ and $g_{W}$ gauge functions. We note that at least at the one loop
level, it is impossible for the running couplings to become negative, as each
beta function is proportional to the cube of the coupling. Consequently, it
becomes arbitrarily small.

###

We note that the fermions do not necessarily couple to the Standard Model
Higgs, although in the absence of a new gauge interaction such couplings are
generically expeccted. We assume that either a new gauge interaction forbids
such couplings or the couplings are exceedingly small, so that the effect of
the new fermions on the Higgs potential is negligible.

###

Since the only impact of the fermions is to decrease the running gauge
couplings from their relatively large Standard Model values, we do not present
a complete discussion of the fermionic sector, but rather parameterize our
results by the decrease in the gauge couplings. We will focus primarily on a
benchmark in which the gauge couplings at large VEVs are $1/100$th their
running Standard Model values, although we will also briefly discuss the case
in which they are $1/10$th their running Standard Model values. Finally, to
preserve the running of the Standard Model Higgs quartic coupling to small
values over many orders of magnitude, we also introduced a fermion which at
the same scale as the singlet, with a Yukawa coupling related $\lambda_{HS}$.
Full details of the modifications made in this scenario are in Appendix D.

###

Fig. 6 is the analog of Fig. 2 in our beyond-the-Standard-Model (BSM)
scenario. By comparison, we see that the new scalar field causes the full
potential to be beneath the quartic contribution, indicating the existence of
Q-balls. We also see that the potential is positive throughout this region,
and thus this scenario is protected from solitosynthesis. Although this plot
may appear similar to the Standard Model plot in Fig. 2, we emphasize that the
source of the relevant finite temperature terms is different. In the pure
Standard Model scenario, the gauge bosons were responsible for the dip. Here,
their contribution is negligible because the gauge couplings are significantly
smaller. Instead, the dip is caused by the scalar field. However, the
effective coupling $\lambda_{HS}$ is of order $\mathcal{O}(0.1)$, the same
order as the (original) Standard Model gauge couplings. For this reason, the
dip produced by the singlet in our BSM scenario is comparable to the dip
produced in the pure Standard Model scenario produced by the gauge bosons.
(Additionally, the fact that we adjust the top pole mass to the maximum value
consistent with the potential being stable to scales of
$10^{17}\,\mathrm{GeV}$ also contributes to the similar appearance of the
potentials in the two scenarios.)

Figure 6: Left: The potential $V(h)$ (black) compared to the quadratic (blue,
dashed) and quartic (red, dotted) terms at $T=10^{14}\,\mathrm{GeV}$, for our
beyond-the-Standard-Model scenario. Before the quartic terms dominate around
$h=10^{15}\,\mathrm{GeV}$, the potential is below the quadratic contribution
due to finite temperature corrections.

###

As discussed above, gauged Q-balls are closely related to their global
counterparts, and therefore as above, we first consider global Q-balls in this
scenario. (As above, by global Q-balls we mean neglected gauged self-
interactions.) The VEV and energy per unit charge $\omega_{0}$ are shown in
the left plot of Fig. 7, and the ratio of the energy per unit charge to the
mass of a Higgs quanta is shown on the right. These again are very similar to
the Standard Model values, for the reasons outlined above. Additional, we have
studied the thin wall range of validity like we did for the Standard Model;
the contribution of surface tension is negligible for global Higgs balls with
charges $\gtrsim 10^{5}$.

Figure 7: Left: The energy per unit charge $\omega_{0}$ of the thin-wall
global Higgs ball (black) and the Higgs VEV $h_{0}$ inside the global Higgs
ball (blue, dashed), as a function of temperature, for our BSM scenario.
Right: The ratio of the energy per unit charge of the global Higgs ball to the
mass of a Higgs quanta outside of the Higgs ball, as a function of
temperature. The global Higgs ball is stable when this is less than 1.

###

Thus, the energy of these Higgs balls (and their energy difference compared to
an equivalent number of free Higgs quanta) is not significantly different from
the pure Standard Model scenario. However, because we have decreased the
strength of the hypercharge and weak gauge couplings by a factor of 100, the
destabilizing gauge self-interactions are significantly weaker. We can use
equations (6) and (7) to determine the energy per unit charge of a thin-wall
gauged Higgs ball, and compare this to the mass of a Higgs quanta outside the
Higgs ball. This is shown in the plot of the left side of Fig. 8. We see that
Higgs balls with charges of order $\mathcal{O}(10^{6})$ are stable across our
temperature range. On the right side, we present the analog of Fig. 5 in our
BSM scenario. As expected from the left figure, we see that across the
temperature range of interest, Higgs balls with charges up to $\mathcal{O}(6)$
are stable.

###

Furthermore, while surface effects have not been studied for gauged Q-balls,
we expect the thin wall regime to be valid for these. We note that global
Q-balls with these charges are indeed in the thin wall regime, as mentioned
above, and repulsive gauge self-interactions generally increase the size of
the Q-balls, pushing them towards the thin wall regime.

###

We note that we also studied a model in which the gauge couplings $g_{Y}$ and
$g_{W}$ were modified to their running Standard Model values, divided by a
factor of 10. In this case, the maximum charge of stable Higgs balls was
around $5000$. As we were not confident these would be in the thin wall
regime, we instead will focus on this benchmark scenario, in which the
couplings are modified at high scales to be their running Standard Model
values, divided by a factor of 100.

Figure 8: Left: The maximum charge for which a gauged thin-wall Q-balls is
stable, in our BSM scenario, as a function of temperature. Right: Ratio of
energy per unit charge to Higgs mass outside the Q-ball, as a function of
temperature, for a range of charges.

##  6 Decay Rate of Higgs Balls

###

We have ensured that the Higgs balls discussed exist; that is, that they are
stable against decay into individual Higgs quanta. However, they are not
absolutely stable as inside the Q-ball, where the Higgs VEV is nonzero, the
Yukawa couplings allow Higgs quanta to decay into fermions (and to photons via
fermionic loops). Additionally, the gauge couplings allow the Higgs quanta to
decay into vector bosons. These decays of individual quanta can lower the
energy of the Higgs ball, and ultimately lead the overall soliton to decay.

###

First, let us discuss new decays which may appear in our BSM mode.In the
specific BSM scenario outlined above, the scalar singlet does not carry gauge
charge and so the Higgs ball cannot decay into scalar quanta. We also ensure
the inside the Higgs ball, where $\mathrm{SU}(2)\times\mathrm{U}(1)$ is
broken, decays to the scalar and its associated fermion are kinematically
forbidden. As noted above, we have assumed that the fermions which modify the
running of the gauge couplings do not directly couple to the Higgs boson, and
so any decays to these are loop suppressed (and furthermore, those loops
involve the now-small gauge couplings).

###

Therefore, in this section, we calculate the decay rate of the Q-balls using
Standard Model decays. In alternative BSM scenarios, new decay channels may be
opened and in such cases, the rates considered here should be considered a
lower bound on the decay rates.

###

In Appendix E, we present the rates for a single Higgs quanta inside the
Q-ball to decay into gauge bosons, photons, gluons, and fermions. Bosonic
decays can occur throughout the Q-ball, and therefore the Higgs ball’s rate of
decay into bosons is given by

$\displaystyle\Gamma_{\mathrm{Q-ball}\rightarrow\mathrm{bosons}}=Q\Gamma_{h\rightarrow\mathrm{bosons}}.$
(10)

###

It has been noted that frequently fermionic decays occur only at the surface
of a Q-ball Cohen:1986ct ; Hong:2017uhi , due to one of two factors
White:2021hwi : First, decays can be kinematically forbidden due to the large
VEV inside the Q-ball, which makes the fermions heavy. However, our energy per
unit charge is only about one order of magnitude smaller than the Higgs VEV,
and therefore only the top quark decay is kinematically forbidden.

###

Secondly, inside the Q-ball the Fermi sea can fill up with the produced decay
products, preventing further fermionic decays. This occurs only if the decay
products cannot diffuse out efficiently. The mean free path is $\lambda\sim
1/{\sigma_{\psi\phi}n}$, where number density $n=3Q_{0}/(4\pi R^{3})$ refers
to the density of Higgs quanta inside the Q-ball. To calculate the cross
section, we consider scatterings via the gauge interactions and the Yukawa
couplings, and approximate the momentum as $\omega/2$. Thus the cross section
is taken to be
$\sigma\sim\mathrm{Max}\left(g_{Y}^{4}/(\omega/2)^{2},g_{W}^{4}/(\omega/2)^{2},y_{i}^{4}/(\omega/2)^{2}\right)$.
Comparing the mean free path to the radius of the gauged Higgs ball, we see
that all the fermions are able to efficiently diffuse out; results for the
b-quark are shown in Fig. 9. Therefore, fermionic decays also occur throughout
the volume of the Higgs ball, giving a total decay rate for the overall Higgs
ball of

$\displaystyle\Gamma_{\mathrm{Q-ball}\rightarrow
f\bar{f}}=Q\Gamma_{h\rightarrow f\bar{f}}.$ (11)

Figure 9: The ratio of the mean free path of a b-quark produced via decay to
the radius of the Higgs ball, for a variety of charges. We observe that the
quark can efficiently diffuse out of the Higgs ball.

###

The production of Higgs balls is beyond this paper, as the production of
gauged Q-balls is a complicated and not yet well-understood process. However,
as thermal Q-balls, Higgs balls require a large temperature, and a natural
environment for them would be the early universe. Therefore, we will consider
their lifetime in this environment. Cosmologically, Q-balls decay efficiently
once the decay rate per unit charge is larger than the Hubble parameter, and
therefore we consider

$\displaystyle\dfrac{\Gamma_{\mathrm{Q-ball}}}{QH}=\dfrac{\Gamma_{h}}{H},$
(12)

since all of our decays occur throughout the volume of the Q-ball. The only
dependence on $Q$ is through the frequency of the gauged Q-balls, which varies
weakly with charge as shown in the right plot of Fig. 8. As shown on this
plot, the variation between $Q=10^{6}$ and $Q=100$ (assuming the thin wall
approximation is valid) is only a few percent, and therefore the results
presented below are to high degree of precision independent of charge
(although they are temperature dependent).

This ratio is shown in Fig. 10, along with the net gauge boson and net
fermionic contribution. As explained in the Appendix, the gauge boson decay
rate is dominated by longitudinal modes, while the fermion decay rate is
dominated by b quarks. The photon and gluon decay rates are highly suppressed.
We see that if they are produced in the early universe, Higgs balls are
generally long lived at temperatures above about $2.6\times
10^{13}\,\mathrm{GeV}$.

Figure 10: The ratio of the decay rate of a Higgs ball to the Hubble parameter
times charge, assuming radiation domination (for $Q=10^{6}$). Q-balls decay
efficiently when this is larger than 1, which is indicated by the horizontal
line.

###

As mentioned in the introduction, this can be usefully thought of in terms of
an approximate conservation of Higgs number. Because the Higgs field is the
only scalar field that carries $\mathrm{SU}(2)$ and $\mathrm{U}(1)$ charge, to
change the charge carried by the Q-ball the number of Higgs quanta must
change. As long as these processes are slow compared to the Hubble rate, there
is approximate conservation of Higgs number charge, and the Higgs balls are
long-lived. Once these decays become efficient, this is no longer a conserved
charge and the Higgs balls decay. (We note that the decay of the Higgs quanta
does not violate $\mathrm{SU}(2)$ and $\mathrm{U}(1)$ conservation, although
the charge of the Q-ball changes; the gauge charge is carried out of the
Q-ball by the decay products.)

###

Since the gauge charge must be carried by scalar fields to produce a soliton,
conserving gauge charge inside the Q-ball will generally be connected to
keeping some sum of scalar field quanta constant. Therefore, an approximate
global symmetry can be usefully defined, which can be used to study stability,
as we did here with Higgs number.

##  7 Discussion and Conclusion

###

In this work, we have demonstrated the possibility of thermal balls, non-
topological solitons that exist due to finite temperature corrections, even in
models which have no attractive interactions at zero temperature and thus no
Q-ball solutions at zero temperature. The generic condition for the existence
of such thermal balls is bosonic fields whose mass is determined by the scalar
VEV.

###

If the bosonic field is a scalar, then the mere existence of a term
proportional to $-AT|\phi|^{3}$ at high temperatures ensures the existence of
these states, unless the scale at which these terms become relevant is
unphysical (e.g., above the Planck scale).

###

If the bosonic field is a gauge boson, then it’s necessary to consider gauge
effects, which can prevent such thermal balls from existing. We have shown
that this occurs in the Standard Model Higgs sector.

###

To show that thermal balls made of Higgs bosons (Higgs balls) can exist in
extensions of the Standard Model, we have considered a model with two
modifications: first, to weaken the gauge self-interactions, we have modified
the running of the gauge couplings so that they are smaller at high scales.
Secondly, we have introduced an additional scalar field, which contributes to
the thermal corrections as explained above. Even with gauge effects included,
Higgs balls exist in this model.

###

We noted that Higgs balls won’t be absolutely stable; even though they cannot
decay into a set of Higgs quanta, they can decay into fermions and gauge
bosons. While the production of gauged Q-balls in the early universe has not
been studied, we have shown that if they are produced, then for sufficiently
high temperatures they will be long-lived. Of course, in other models, the
thermal balls may be completely stable if no decays are permitted.

###

We generally leave potential phenomenological implications for future work, as
we have not discussed production of thermal balls. However, we will note one
intriguing connection. Thermal balls require the existence of large cubic
terms from thermal corrections. If the model undergoes spontaneous symmetry
breaking and these terms are relevant at those scales, then the phase
transition will be first order Buchmuller:1993bq . In the minimal modification
of the Standard Model here, we have been agnostic about the scale of new
physics; the existence of Higgs balls does not necessarily ensure a first
order electroweak phase transition. However, a first order electroweak phase
transition, as envisioned in many models of baryogenesis Turok:1991uc ;
Funakubo:1993jg ; McDonald:1993ey ; Profumo:2006yu ; Profumo:2014opa ;
Kozaczuk:2014kva ; Vaskonen:2016yiu ; Chiang:2017nmu ; Baldes:2018emh ;
Kainulainen:2019kyp ; Brdar:2019fur ; Athron:2019teq ; Huang:2020bbe ;
Niemi:2021qvp  and other well motivated scenarios involving phase transitions
Croon:2018kqn ; Croon:2018erz ; Croon:2019rqu ; Greljo:2019xan ; Dev:2019njv ;
Fornal:2020ngq ; Fornal:2021ovz , does require significant cubic terms and
hence we would expect thermal Higgs balls to exits in such models.

###

We also note that if produced in the early universe, Higgs balls could provide
a novel means of dark matter production. As Fig. 7 demontrates, as the
temperature changes the Higgs VEV also changes, and therefore the effective
mass of the Higgs quanta inside the Higgs ball similar changes. We consider
models in which a new annihilation channel exists, which has a resonance at a
specific Higgs mass (see for instance Davoudiasl:2004be ; Patt:2006fw ;
GAMBIT:2018eea ; Beniwal:2019xop ; Arcadi:2019lka ). As the Higgs mass evolves
with temperature it will can pass through this resonance and thus produce dark
matter Croon:2020ntf . However, since Higgs balls are only a fraction of the
the volume, production will be suppressed. This is desirable in e.g. Wimpzilla
scenarios, in which one must avoid overproducing dark matter and overclosing
the universe. However, specific abundances cannot be calculated without
knowing the Higgs ball production rate which we leave to future work.

### Acknowledgements:

The authors thank Daniel Vagie and Volodymyr Takhistov for helpful
conversations. The works of AK and GW were supported by World Premier
International Research Center Initiative (WPI), MEXT, Japan. A.K. was
supported the U.S. Department of Energy (DOE) grant No. DE-SC0009937 and by
Japan Society for the Promotion of Science (JSPS) KAKENHI grant No.
JP20H05853.

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##  Appendix A Standard Model Potential

###

In this appendix, we discuss how we calculate the Standard Model potential at
zero and finite temperature. Our conventions and resummations follow
Arnold:1992rz . We use the Landau gauge, following Quiros:1994dr ;
DelleRose:2015bpo . The renormalization group improved tree potential is

$V_{\rm tree}(h)=-\dfrac{\mu(h)^{2}}{2}h^{2}+\dfrac{\lambda(h)}{4}h^{4},$ (13)

where, inside the H-ball, we set our renormalization scale to be the Higgs
VEV. We use SARAH Staub:2013tta  to evaluate our running coupling constants,
which evaluates the renormalization group equations to two-loop order.

###

We initialize the Standard Model parameters in SARAH at the Z pole, following
the procedure in Croon:2020cgk , although with all parameters related to the
singlet set to zero. In particular, in matching the $\overline{\mathrm{MS}}$
parameters to observed pole masses, we include one loop corrections for the
Standard Model Higgs parameters $\mu$ and $\lambda$, the top quark Yukawa
$y_{t}$, and the electroweak gauge coupling $g$. All other parameters are
matched at tree-level. This is explained further in Appendix B.

###

In the potential, we also include the one-loop corrections Quiros:1994dr ;
DelleRose:2015bpo

$V_{1-{\rm
loop}}(h)=\sum_{W,Z,t,\chi,h}\dfrac{n_{i}}{64\pi^{2}}m_{i}(h)^{4}\left[\ln\left(\dfrac{m_{i}(h)^{2}}{h^{2}}\right)-C_{i}\right]$
(14)

(where the contributions of all other fields are subdominant). In this
expression, we again take the renormalization scale to be the Higgs VEV; the
coefficients are:

$\displaystyle n_{W}=6,n_{Z}=3,n_{t}=-12,n_{\chi}=3,n_{h}=1,$ $\displaystyle
C_{W}=C_{Z}=5/6,C_{t}=C_{\chi}=C_{h}=3/2.$ (15)

The masses are

$\displaystyle m_{W}(h)^{2}$ $\displaystyle=\dfrac{1}{4}g_{2}(h)^{2}h^{2},$
$\displaystyle m_{Z}(h)^{2}$
$\displaystyle=\dfrac{1}{4}(g_{1}(h)^{2}+g_{2}(h)^{2})h^{2},$ $\displaystyle
m_{t}(h)^{2}$ $\displaystyle=\dfrac{1}{2}y_{t}(h)^{2}h^{2},$ $\displaystyle
m_{\chi}(h)^{2}$ $\displaystyle=-\dfrac{\mu(h)^{2}}{2}+\lambda(h)h^{2},$
$\displaystyle m_{h}(h)^{2}$
$\displaystyle=-\dfrac{\mu(h)^{2}}{2}+3\lambda(h)h^{2},$ (16)

with all running parameters evaluated at the Higgs VEV inside the H-ball. (We
note that at the scales of interest, $-\mu(h)^{2}\ll\lambda(h)h^{2}$ and this
contribution to the Higgs and $\chi$ masses can be neglected.)

###

As noted in Section 3, finite temperature corrections are important to H-ball
existence. We include the one-loop finite temperature correction,

$\displaystyle V_{1-\mathrm{loop}}(h,T)$
$\displaystyle=\sum_{\mathrm{bosons}}\dfrac{n_{i}T^{4}}{2\pi^{2}}J_{B}\left(\dfrac{m_{i}^{2}}{T^{2}}\right)+\sum_{\mathrm{fermions}}\dfrac{n_{i}T^{4}}{2\pi^{2}}J_{F}\left(\dfrac{m_{i}^{2}}{T^{2}}\right),$
(17)

where

$\displaystyle J_{B}(y)$
$\displaystyle=\int_{0}^{\infty}dx\,x^{2}\ln\left(1-e^{-\sqrt{x^{2}+y}}\right),$
$\displaystyle J_{F}(y)$
$\displaystyle=\int_{0}^{\infty}dx\,x^{2}\ln\left(1+e^{-\sqrt{x^{2}+y}}\right),$
(18)

Again we neglect contributions from fermions other than the top, as they are
highly suppressed. (We note that the constant term proportional to $T^{4}$,
which does not depend on the Higgs VEV, has no impact on our solitons and
therefore can be neglected.)

###

We also include the so-called daisy diagrams via

$\displaystyle V_{\rm
ring}(h,T)=\sum_{W_{L},Z_{L},\gamma_{L},\chi,h}\dfrac{n_{i}T^{4}}{12\pi}\left(\left(\dfrac{m_{i}^{2}}{T^{2}}\right)^{3/2}-\left(\dfrac{\mathcal{M}_{i}^{2}}{T^{2}}\right)^{3/2}\right),$
(19)

where the Debye masses for $h$, $\chi$, and the longitudinal mode of the $W$
boson are given by $\mathcal{M}_{i}^{2}(h)=m_{i}^{2}(h)+\Pi_{i}(h,T)$ with

$\displaystyle\Pi_{h}(h,T)$
$\displaystyle=\Pi_{\chi}(h,T)=\left(\dfrac{3g_{2}^{2}+g_{1}^{2}}{16}+\dfrac{\lambda}{2}+\dfrac{y_{t}^{2}}{4}\right)T^{2}$
$\displaystyle\Pi_{W_{L}}(h,T)$ $\displaystyle=\dfrac{11}{6}g_{2}^{2}T^{2}.$
(20)

Due to the mixing between the $Z$ boson and the photon, their Debye masses are

$\displaystyle\mathcal{M}^{2}_{Z_{L}}(h)$
$\displaystyle=\dfrac{1}{2}\left(m_{Z}^{2}+\dfrac{11}{6}\dfrac{g_{2}^{2}T^{2}}{\cos^{2}(\theta_{W})}+\Delta(h,T)\right)$
$\displaystyle\mathcal{M}^{2}_{Z_{L}}(h)$
$\displaystyle=\dfrac{1}{2}\left(m_{Z}^{2}+\dfrac{11}{6}\dfrac{g_{2}^{2}T^{2}}{\cos^{2}(\theta_{W})}-\Delta(h,T)\right),$
(21)

where

$\displaystyle\Delta^{2}(h,T)$
$\displaystyle=m_{Z}^{4}+\dfrac{11}{3}\dfrac{g^{2}\cos^{2}(\theta_{W})}{\cos^{2}(\theta_{W})}\cdot\left(m_{Z}^{2}+\dfrac{11}{12}\dfrac{g_{2}^{2}T^{2}}{\cos^{2}(\theta_{W})}\right)T^{2}.$
(22)

In this expression, all couplings (including the Weinberg angle) are running
couplings, evaluated at a renormalization scale set by the Higgs vacuum
expectation value. As above, we have also subtracted off the $h$-independent
terms proportional to $T^{4}$, which do not impact our solitons.

###

In our analysis we have integrated $J_{B}$ and $J_{F}$ numerically; however,
we note that in the high temperature limit,

$\displaystyle V(h,T)$ $\displaystyle=V_{1-\mathrm{loop}}(h,T)+V_{\rm
ring}(h,T)$ $\displaystyle\approx\kappa T^{2}h^{2}-ATh^{3},$ (23)

where

$\displaystyle\kappa$
$\displaystyle=\frac{1}{32}\left(g_{1}^{2}+3g_{2}^{2}+8\lambda+4y_{t}^{2}\right)-\frac{\lambda}{6\pi}\sqrt{g_{1}^{2}+3g_{2}^{2}+8\lambda+4y_{t}^{4}}+\frac{1}{32\pi}\sqrt{\dfrac{11}{6}}(g_{1}^{3}+3g_{2}^{3}),$
$\displaystyle A$
$\displaystyle=\frac{8\left(\sqrt{3}+1\right)\lambda^{3/2}+\left(g_{1}^{2}+g_{2}^{2}\right)^{3/2}+2g_{2}^{3}}{32\pi},$
(24)

which illustrates the $-ATh^{3}$ term that allows for Higgs balls to exist.

###

In discussions of the stability of the H-balls, we have compared the energy of
the H-ball to the energy of $Q$ free Higgs quanta. For this, we used the
thermal mass of the Higgs boson in the true vacuum, given by
$m_{h}(T)=\sqrt{2\kappa}$. In the true vacuum (where the Higgs VEV is zero) we
have evaluated the running couplings at a renormalization scale set by the
temperature.

##  Appendix B Matching Standard Model Parameters to Observables

###

As mentioned in section 3, we want to avoid Q-balls that induce
solitosynthesis, which means that we want to ensure the Standard Model vacuum
remains stable. The stability of the Standard Model vacuum is sensitive to the
precise values of measured parameters such as the top mass, and therefore one
must take care when matching observables to the running Standard Model
parameters. We describe our procedure here.

###

Following Ref. Croon:2020cgk , the one loop self energy for the top quark can
be written as

$\frac{1}{2}y_{t}^{2}v^{2}=M_{t}^{2}\left(1+2{\rm
Re}\left(\Sigma_{v}(M_{t}^{2})+\Sigma_{s}(M_{t}^{2})\right)\right),$ (25)

where

$\displaystyle\Sigma_{v}(M^{2}_{t})+\Sigma_{s}(M^{2}_{t})$
$\displaystyle=\frac{3}{16}\frac{g^{2}_{2}}{(4\pi)^{2}}\bigg{(}-2-4\frac{1}{h^{2}}-2h^{2}-\frac{256}{9}s^{2}+2t^{2}+16\frac{t^{4}}{h^{2}}$
$\displaystyle-\frac{2}{27}\Big{(}39-\frac{64}{z^{2}}+25z^{2}+18\frac{z^{4}-1}{h^{2}}\Big{)}$
$\displaystyle+\Big{(}4h^{2}-\frac{8}{3}t^{2}+\frac{4}{3}t^{2}\frac{2t^{2}+h^{2}}{t^{2}-h^{2}}\Big{)}\ln(h)-\frac{8}{9}\Big{(}-9\frac{z^{4}}{h^{2}}+4\frac{(4-5z^{2}+z^{4})}{t^{2}-z^{2}}\Big{)}\ln(z)$
$\displaystyle+\Big{(}\frac{128}{3}s^{2}-32\frac{t^{4}}{h^{2}}-\frac{4}{3}\frac{t^{2}(2t^{2}+h^{2})}{t^{2}-h^{2}}-\frac{32}{9}\frac{(z^{2}-1)(t^{2}-4)}{z^{2}-t^{2}}\Big{)}\ln(t)$
$\displaystyle+\frac{2}{3}\Big{(}4t^{2}-h^{2}\Big{)}F(M_{t},M_{t},M_{h})+\frac{2}{3}\frac{(t^{2}+2)(t^{2}-1)}{t^{2}}F(M_{t},M_{W},0)$
$\displaystyle-\frac{2}{27}\Big{(}\frac{64-80z^{2}+7z^{4}}{z^{2}}+\frac{32-40z^{2}+17z^{4}}{t^{2}}\Big{)}F(M_{t},M_{t},M_{Z})$
$\displaystyle+\bigg{[}2\Big{(}-6\frac{1}{h^{2}}-h^{2}-\frac{32}{3}s^{2}+t^{2}+8\frac{t^{4}}{h^{2}}\Big{)}$
$\displaystyle-\frac{4}{9}\frac{(z^{2}-1)(9+4h^{2}+9z^{2})}{h^{2}}\bigg{]}\ln\Big{(}\frac{Q^{2}}{M^{2}_{W}}\Big{)}\bigg{)}$
$\displaystyle-64c_{6}\frac{M^{2}_{W}}{g^{4}_{0}}\bigg{[}1-\ln\Big{(}\frac{Q^{2}}{M^{2}_{h}}\Big{)}\bigg{]}\bigg{)}\
,$ (26) $\displaystyle\frac{\delta g^{2}}{g^{2}_{0}}$
$\displaystyle=\frac{1}{(4\pi)^{2}}g^{2}_{0}\bigg{(}-\frac{257}{72}-\frac{1}{24}h^{2}+\frac{20}{9}N_{F}+\frac{1}{4}t^{2}-2\ln(t)$
$\displaystyle+\frac{1}{12}(12-4h^{2}+h^{4})F(M_{W},M_{h},M_{W})-\frac{(t^{2}+2)(t^{2}-1)}{2}F(M_{W},M_{t},0)$
$\displaystyle-\frac{33}{4}F(M_{W},M_{W},M_{W})+\Big{(}\frac{4}{3}N_{F}-\frac{43}{6}\Big{)}\ln\Big{(}\frac{Q^{2}}{M^{2}_{W}}\Big{)}\bigg{)},$
(27)

and where the mass ratios are
$\\{h,t,z\\}=\\{M_{h}/M_{W},M_{t}/M_{W},M_{Z}/M_{W}\\}$. (In this section, we
use $v$ for the Higgs VEV.) The $\overline{\mathrm{MS}}$ RG scale is denoted
by $Q$, and $F(k,m_{1},m_{2})$ is a loop function given in Ref.
Kajantie:1995dw . We adjusted the top mass pole mass away from its central
value of $172.4\pm 0.7$ GeV ParticleDataGroup:2020ssz  as explained in section
3 to stabilize the vacuum up to the desired scale.

For the Higgs sector parameters we again use the self energies of the W and
Higgs bosons,

$\displaystyle\mu_{h}^{2}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}M_{h}^{2}\left(1+\frac{{\rm
Re}\Pi_{h}(M_{h}^{2})}{M_{h}^{2}}\right)$ (28) $\displaystyle\lambda$
$\displaystyle=$
$\displaystyle\frac{1}{8}g_{2}^{2}\frac{M_{h}^{2}}{M_{W}^{2}}\left(1+\frac{{\rm
Re}\Pi_{h}(M_{h}^{2})}{M_{h}^{2}}-\frac{{\rm
Re}\Pi_{W}(M_{W}^{2})}{M_{W}^{2}}\right)$ (29)

where to one loop, the self energies have the form

$\displaystyle\Pi_{h}(M^{2}_{h})$
$\displaystyle=\frac{3}{8}\frac{g^{2}_{0}M^{2}_{h}}{(4\pi)^{2}}\bigg{(}-\frac{4}{3}-8\frac{1}{h^{2}}-2h^{2}+16\frac{t^{4}}{h^{2}}-\frac{2}{3}z^{2}-4\frac{z^{4}}{h^{2}}$
$\displaystyle+3h^{2}F(M_{h},M_{h},M_{h})+4t^{2}\Big{(}1-4\frac{t^{2}}{h^{2}}\Big{)}F(M_{h},M_{t},M_{t})$
$\displaystyle+\frac{2}{3}\frac{h^{4}-4h^{2}+12}{h^{2}}F(M_{h},M_{W},M_{W})$
$\displaystyle+\Big{(}\frac{1}{3}\frac{1}{h^{2}}-\frac{4}{3}z^{2}+4\frac{z^{4}}{h^{2}}\Big{)}F(M_{h},M_{Z},M_{Z})$
$\displaystyle-2h^{2}\ln(h)-8t^{2}\ln(t)+\Big{(}-\frac{2}{3}h^{2}+4z^{2}\Big{)}\ln(z)$
$\displaystyle+\Big{(}-4+2h^{2}+4t^{2}-2z^{2}\Big{)}\ln\Big{(}\frac{Q^{2}}{M^{2}_{W}}\Big{)}$
$\displaystyle+64c_{6}\frac{M^{2}_{W}}{g^{4}_{0}h^{4}}\bigg{[}-2+12t^{4}-z^{4}+3h^{4}F(M_{h},M_{h},M_{h})-6h^{4}\ln(h)$
$\displaystyle-24t^{4}\ln(t)+6z^{4}\ln(z)+\Big{(}-2+h^{4}+4t^{4}-z^{4}\Big{)}\ln\Big{(}\frac{Q^{2}}{M^{2}_{W}}\Big{)}\bigg{]}$
$\displaystyle+3072c_{6}^{2}\frac{M^{4}_{W}}{g^{4}_{0}h^{2}}\bigg{[}-1+F(M_{h},M_{h},M_{h})\bigg{]}\bigg{)}\;,$
(30)

and

$\displaystyle\Pi_{W}(M^{2}_{W})$
$\displaystyle=\frac{3}{8}\frac{g^{2}_{0}M^{2}_{W}}{(4\pi)^{2}}\bigg{(}-\frac{212}{9}-\frac{8}{3}\frac{1}{h^{2}}-\frac{22}{9}h^{2}+\frac{4}{27}(40N_{F}-17)-\frac{4}{3}t^{2}+16\frac{t^{4}}{h^{2}}+\frac{14}{9}z^{2}-\frac{4}{3}\frac{z^{4}}{h^{2}}$
$\displaystyle+\frac{4h^{2}(h^{2}-2)}{h^{2}-1}\ln(h)-8\Big{(}\frac{2}{3}-t^{2}+4\frac{t^{4}}{h^{2}}\Big{)}\ln(t)+4\Big{(}2\frac{z^{4}}{h^{2}}-\frac{z^{4}-4z^{2}-8}{z^{2}-1}\Big{)}\ln(z)$
$\displaystyle+\frac{2}{9}\Big{(}12-4h^{2}+h^{4}\Big{)}F(M_{W},M_{h},M_{W})-\frac{4}{3}(t^{2}+2)(t^{2}-1)F(M_{W},M_{t},0)$
$\displaystyle-\frac{32}{3}\frac{z^{2}-1}{z^{2}}F(M_{W},M_{W},0)+\frac{2}{9}\frac{(z^{4}+20z^{2}+12)(z^{2}-4)}{z^{2}}F(M_{W},M_{W},M_{Z})$
$\displaystyle+2\bigg{[}-1+\frac{2}{h^{2}}+\Big{(}-\frac{59}{9}-6\frac{1}{h^{2}}-h^{2}+\frac{16}{9}N_{F}-2t^{2}+8\frac{t^{4}}{h^{2}}\Big{)}+z^{2}-2\frac{z^{4}}{h^{2}}\bigg{]}\ln\Big{(}\frac{Q^{2}}{M^{2}_{W}}\Big{)}$
$\displaystyle-\frac{8}{9}\pi
i(4N_{F}-3)-64c_{6}\frac{M^{2}_{W}}{g^{4}_{0}}\bigg{[}1-\ln\Big{(}\frac{Q^{2}}{M^{2}_{h}}\Big{)}\bigg{]}\bigg{)}\;.$
(31)

##  Appendix C Gauged Q-balls in the Static Charge Approximation

###

In this appendix, we consider gauged Q-balls in the static charge
approximation following Ref. Heeck:2021zvk , which studies gauged
$\mathrm{U}(1)$ Q-balls. We first consider a gauged $\mathrm{SU}(N)$ scenario
before discussing the Standard Model Higgs, which carries
$\mathrm{SU}(2)\times\mathrm{U}(1)$ charge.

###

As we show, the static charge approximation does not include the effects of
gauge boson self-interactions, which generally occur in non-Abelian gauge
theories. Therefore it is valid when the size of the Q-ball is less than the
confinement scale. All unbroken $\mathrm{SU}(N)$ interactions other than
$\mathrm{U}(1)$ confine on some scale, and therefore it is expected that the
static charge approximation breaks down. We note that a Q-ball which carries
$\mathrm{SU}(N)$ charge will necessarily be screened by charges exterior to
the Q-ball at the confinement scale. To estimate the $\mathrm{SU}(2)$
confinement scale we run $g_{2}$ using the one loop RGE

$\displaystyle\mu\frac{d\alpha_{W}}{d\mu}=-\frac{b_{W}}{2\pi}\alpha_{W}$ (32)

with

$b_{W}=\frac{22}{3}-\frac{n_{F}}{3}-\frac{n_{s}}{6}\ .$ (33)

As a result we find that the $\mathrm{SU}(2)$ confinement scale is at
$\mu=O(10^{-10})({\rm GeV})$. This gives a confinement scale $R_{\rm conf}\sim
1/\mu$ which is significantly larger than the radii of our Q-balls throughout
the temperature ranges considered in this work. We can therefore ignore
macroscopic confinement effects.

###

We start with a complex scalar field $\phi$ which transforms under a gauged
$\mathrm{SU}(N)$, with generators $t^{a}$ and structure constant $f^{abc}$.
The relevant Lagrangian density is

$\displaystyle\mathcal{L}$
$\displaystyle=|D_{\mu}\phi|^{2}-U(|\phi|)-\dfrac{1}{4}F^{a\dagger}_{\mu\nu}F^{a\mu\nu},$
(34)

where the field strength tensor is

$\displaystyle F^{a}_{\mu\nu}$
$\displaystyle=\partial_{\mu}A_{\nu}^{a}-\partial_{\nu}A_{\mu}^{a}+gf^{abc}A_{\mu}^{b}A_{\nu}^{c}$
(35)

and the gauged covariant derivative is

$\displaystyle D_{\mu}$ $\displaystyle=\partial_{\mu}-igA_{\mu}^{a}t^{a}.$
(36)

###

We introduce a generalized static charge ansatz

$\displaystyle\phi(t,\vec{x})$
$\displaystyle=\dfrac{1}{\sqrt{2}}\phi_{0}F(r)e^{i\omega t},$ (37)

where $F(r)$ is real, although $\phi_{0}$ can be complex. In this ansatz, the
gauge fields are

$\displaystyle A_{0}^{a}(t,\vec{x})$ $\displaystyle=A_{0}^{a}(r),\qquad
A_{a}^{i}(t,\vec{x})=0,$ (38)

where $A_{0}^{a}(r)$ will be taken to be real.

###

After simplifying, we have the Lagrangian

$\displaystyle L$ $\displaystyle=(4\pi)\int
dr\,r^{2}\left[\dfrac{1}{2}\phi_{0}^{\dagger}\phi_{0}\left(\omega^{2}F(r)^{2}-(F^{\prime}(r))^{2}\right)-\omega
gA_{0}^{a}(r)\phi_{0}^{\dagger}t^{a}\phi_{0}F(r)^{2}\right.$
$\displaystyle\qquad\left.+\dfrac{g^{2}}{4}A^{a}_{0}A_{0}^{b}F(r)^{2}\phi_{0}^{\dagger}\left\\{t_{a},t_{b}\right\\}\phi_{0}-U\left(\dfrac{1}{2}\phi_{0}^{\dagger}\phi_{0}F(r)^{2}\right)+\dfrac{1}{2}(A_{0}^{a\prime}(r))^{2}\right].$
(39)

###

We now specialize to $\mathrm{SU}(2)$ and assume that $\phi_{0}$ transforms in
the fundamental representation. Without a loss of generality we take the VEV
to be

$\displaystyle\phi_{0}$ $\displaystyle=\begin{pmatrix}0\\\ v\end{pmatrix},$
(40)

giving

$\displaystyle L$ $\displaystyle=(4\pi)\int
dr\,r^{2}\left[\dfrac{v^{2}}{2}\left(\omega^{2}F(r)^{2}-(F^{\prime}(r))^{2}\right)+\dfrac{\omega
g}{2}A_{0}^{3}(r)v^{2}F(r)^{2}+\dfrac{g^{2}v^{2}}{8}A^{a}_{0}A_{0}^{a}F(r)^{2}\right.$
$\displaystyle\qquad\left.-U\left(\dfrac{v^{2}}{2}F(r)^{2}\right)+\dfrac{1}{2}(A_{0}^{a\prime}(r))^{2}\right],$
(41)

where our choice of the VEV direction selects the $A^{3}$ gauge field.

###

Next, we introduce rescalings like those in Ref. Heeck:2021zvk . In
particular, we define:

$\displaystyle\Phi_{0}$
$\displaystyle=\dfrac{v_{0}}{\sqrt{m_{\phi}^{2}-\omega_{0}^{2}}},\qquad$
$\displaystyle\rho$ $\displaystyle=r\sqrt{m_{\phi}^{2}-\omega_{0}^{2}},\qquad$
$\displaystyle\Omega$
$\displaystyle=\dfrac{\omega}{\sqrt{m_{\phi}^{2}-\omega_{0}^{2}}},$
$\displaystyle A^{a}(\rho)$ $\displaystyle=\dfrac{A_{0}^{a}(\rho)}{v},\qquad$
$\displaystyle\alpha$ $\displaystyle=g\Phi_{0},\qquad$
$\displaystyle\kappa^{2}$ $\displaystyle=\Omega^{2}-\Omega_{0}^{2},\qquad$
$\displaystyle\Omega_{0}$
$\displaystyle=\dfrac{\omega_{0}}{\sqrt{m_{\phi}^{2}-\omega_{0}^{2}}},$ (42)

and now we will use a prime to indicate a derivative with respect to $\rho$.
The Lagrangian is then

$\displaystyle L$
$\displaystyle=4\pi\Phi_{0}^{2}\sqrt{m_{\phi}^{2}-\omega_{0}^{2}}\int
d\rho\,\rho^{2}\left[-\dfrac{1}{2}(F^{\prime}(\rho))^{2}+\dfrac{1}{2}(A^{a\prime}(\rho))^{2}+\dfrac{1}{2}F^{2}\left(\Omega^{2}+\Omega\alpha
A^{3}(\rho)+\dfrac{\alpha^{2}}{4}(A^{a})^{2}\right)\right.$
$\displaystyle\qquad\left.-\dfrac{U\left(F(\rho)^{2}\right)}{\Phi_{0}^{2}(m_{\phi}^{2}-\omega_{0}^{2})^{2}}\right]$
(43)

from which we find the equations of motion

$\displaystyle 0$
$\displaystyle=F^{\prime\prime}+\dfrac{2}{\rho}F^{\prime}+F\left(\Omega^{2}+\Omega\alpha
A^{3}(\rho)+\dfrac{\alpha^{2}}{4}(A^{a})^{2}\right)-\dfrac{1}{\Phi_{0}^{2}(m_{\phi}^{2}-\omega_{0}^{2})^{2}}\dfrac{dU}{df}$
$\displaystyle 0$
$\displaystyle=(A^{3})^{\prime\prime}+\dfrac{2}{\rho}(A^{3})^{\prime}-\dfrac{\alpha}{2}F^{2}\left(\Omega+\dfrac{\alpha}{2}A^{3}\right)$
$\displaystyle 0$
$\displaystyle=(A^{1})^{\prime\prime}+\dfrac{2}{\rho}(A^{1})^{\prime}-\dfrac{\alpha}{2}F^{2}\left(\dfrac{\alpha}{2}A^{1}\right)$
$\displaystyle 0$
$\displaystyle=(A^{2})^{\prime\prime}+\dfrac{2}{\rho}(A^{2})^{\prime}-\dfrac{\alpha}{2}F^{2}\left(\dfrac{\alpha}{2}A^{2}\right).$
(44)

###

Assuming the scalar field carries unit charge, the charge of the Q-ball,
$Q=-i\int d^{3}x\,(\Phi^{\dagger}D_{0}\Phi-h.c.)$, is given by

$\displaystyle Q$ $\displaystyle=4\pi\int
d\rho\,\rho^{2}F(\rho)^{2}\Phi_{0}^{2}\left(\Omega+\dfrac{\alpha}{2}A^{3}(\rho)\right),$
(45)

which after using the equations of motion and an integration by parts, is

$\displaystyle Q$
$\displaystyle=\dfrac{4\pi\Phi_{0}^{3}}{\alpha/2}\lim_{\rho\rightarrow\infty}\rho^{2}A_{3}^{\prime},$
(46)

similar to $\mathrm{U}(1)$ gauged Q-balls. In particular, we see that

$\displaystyle\lim_{\rho\rightarrow\infty}A_{3}=-\dfrac{\alpha
Q/2}{4\pi\Phi_{0}^{2}\rho}.$ (47)

###

We observe that regardless of the function $F(\rho)$, $A^{1}=A^{2}=0$ is a
solution to the equations of motion, and so therefore we set these fields to
zero inside the Q-ball. The resulting equations for $F$ and $A_{0}^{3}$ are
identical to those in Ref. Heeck:2021zvk  with $\alpha\rightarrow-\alpha/2$.
Following this reference, we similarly impose the boundary conditions

$\displaystyle\lim_{\rho\rightarrow
0}F^{\prime}=\lim_{\rho\rightarrow\infty}F=\lim_{\rho\rightarrow
0}A_{3}^{\prime}=\lim_{\rho\rightarrow\infty}A_{3}=0$ (48)

Therefore, if we approximate the scalar field profile with a step function
$F(\rho)=1-\Theta(\rho-R^{*})$, and impose $A^{3}$ and $(A^{3})^{\prime}$ are
continuous at the boundary, we find

$\displaystyle A^{3}(\rho)$
$\displaystyle=-\dfrac{2\Omega}{\alpha}\begin{cases}1-\dfrac{\sinh(\alpha\rho/2)}{\cosh(\alpha
R^{*}/2)(\alpha\rho/2)},\qquad&\rho<R^{*}\\\ \dfrac{\alpha
R^{*}/2-\tanh(\alpha R^{*}/2)}{\alpha\rho/2},\qquad&\rho\geq
R^{*}.\end{cases}$ (49)

###

The derivative of $\alpha A^{3}$ is small if the radius is large, and
therefore in the equation of motion for $F$ we can approximate $A^{3}$ with
its value at the radius. Then we recognize that the equation for $F$ is that
of a global Q-ball, but with the frequency given by

$\displaystyle\Omega_{G}$
$\displaystyle=\Omega+\dfrac{\alpha}{2}A^{3}(R^{*}),$ (50)

giving a relation between the energy per unit charge of the gauged Q-ball and
the energy per unit charge of the ungauged Q-ball,

$\displaystyle\Omega$ $\displaystyle=\Omega_{G}\cdot\dfrac{\alpha
R^{*}}{2}\coth\left(\dfrac{\alpha R^{*}}{2}\right).$ (51)

In the thin wall limit, this becomes

$\displaystyle\omega$ $\displaystyle=\omega_{0}\cdot\dfrac{\alpha
R^{*}}{2}\coth\left(\dfrac{\alpha R^{*}}{2}\right).$ (52)

###

Now we turn our attention to the Standard Model Higgs, which carries both
$\mathrm{SU}(2)$ and $\mathrm{U}(1)$ charge. We start with the Lagrangian
density

$\displaystyle\mathcal{L}$
$\displaystyle=|D_{\mu}\phi|^{2}-U(|\phi|)-\dfrac{1}{4}G^{a\dagger}_{\mu\nu}G^{a\mu\nu}-\dfrac{1}{4}F^{\dagger}_{\mu\nu}F^{\mu\nu},$
(53)

where the field strength tensors are

$\displaystyle G^{a}_{\mu\nu}$
$\displaystyle=\partial_{\mu}A_{\nu}^{a}-\partial_{\nu}A_{\mu}^{a}+g_{W}f^{abc}A_{\mu}^{b}A_{\nu}^{c}$
$\displaystyle F_{\mu\nu}$
$\displaystyle=\partial_{\mu}B_{\nu}-\partial_{\nu}B_{\mu}$ (54)

and the gauged covariant derivative is

$\displaystyle D_{\mu}$
$\displaystyle=\partial_{\mu}-ig_{W}A_{\mu}^{a}t^{a}-\dfrac{i}{2}g_{Y}B_{\mu}.$
(55)

We use the convention that the Higgs has hypercharge $1/2$. Proceeding as
above, and making the static charge ansatz for both the $\mathrm{SU}(2)$ gauge
field $A\mu^{a}$ and the $\mathrm{U}(1)$ gauge field $B_{\mu}$, we find the
Lagrangian

$\displaystyle L$
$\displaystyle=4\pi\Phi_{0}^{2}\sqrt{m_{\phi}^{2}-\omega_{0}^{2}}\int
d\rho\,\rho^{2}\left[-\dfrac{1}{2}(F^{\prime}(\rho))^{2}+\dfrac{1}{2}(A^{a\prime}(\rho))^{2}-\dfrac{U\left(F(\rho)^{2}\right)}{\Phi_{0}^{2}(m_{\phi}^{2}-\omega_{0}^{2})^{2}}\right.$
$\displaystyle\left.+\dfrac{1}{2}F^{2}\left(\Omega^{2}+\Omega\alpha_{W}A^{3}(\rho)+\dfrac{\alpha_{W}^{2}}{4}(A^{a}(\rho))^{2}-\Omega\alpha_{Y}B(\rho)+\dfrac{\alpha_{Y}^{2}}{4}(B(\rho))^{2}-\dfrac{\alpha_{W}\alpha_{Y}}{2}B(\rho)A^{3}(\rho)\right)\right].$
(56)

In this expression, $\alpha_{W}=g_{W}\Phi_{0}$, $\alpha_{Y}=g_{Y}\Phi_{0}$,
$A^{a}(\rho)=A_{0}^{a}(\rho)/v$, and $B(\rho)=B_{0}(\rho)v$.

###

We note that again the equation of motion for the radial $F(\rho)$ matches
that of a global Q-ball, with a shifted frequency,

$\displaystyle 0$
$\displaystyle=F^{\prime\prime}+\dfrac{2}{\rho}F^{\prime}+F\left(\Omega+\dfrac{\alpha_{W}}{2}A^{3}-\dfrac{\alpha_{Y}}{2}B\right)^{2}-\dfrac{1}{\Phi_{0}^{2}(m_{\phi}^{2}-\omega_{0}^{2})^{2}}\dfrac{dU}{df}$
$\displaystyle 0$
$\displaystyle=(A^{3})^{\prime\prime}+\dfrac{2}{\rho}(A^{3})^{\prime}-\dfrac{\alpha_{W}}{2}F^{2}\left(\Omega+\dfrac{\alpha_{W}}{2}A^{3}-\dfrac{\alpha_{Y}}{2}B\right),$
$\displaystyle 0$
$\displaystyle=B^{\prime\prime}+\dfrac{2}{\rho}B^{\prime}+\dfrac{\alpha_{Y}}{2}F^{2}\left(\Omega-\dfrac{\alpha_{Y}}{2}B^{3}+\dfrac{\alpha_{W}}{2}A_{3}\right),$
(57)

and the other fields $A_{1}$ and $A_{2}$ may be set to zero. We see that the
equations of motion mix $A^{3}$ and $B$, as is expected for the Higgs sector.

A Q-ball made of the Higgs field will carry both $\mathrm{U}(1)$ and
$\mathrm{SU}(2)$ charge. We note that since we have chosen the VEV in the form
above, the amount of $\mathrm{SU}(2)$ charge is described by the weak isospin.
A quanta of the Higgs field has hyperchange $1/2$ and weak isospin $-1/2$, so
the charges of the Higgs ball satisfy $Q_{Y}=-Q_{W}=-\dfrac{i}{2}\int
d^{3}x\,\left(\Phi^{\dagger}D_{0}\Phi-h.c.\right)$, as it must for it to be
electrically neutral. Using our equation of motion above, we see that this can
be expressed using either gauge field

$\displaystyle Q_{Y}=-Q_{W}$
$\displaystyle=\dfrac{4\pi\Phi_{0}^{2}}{\alpha_{W}/2}\lim_{\rho\rightarrow\infty}\rho^{2}A_{3}^{\prime}=-\dfrac{4\pi\Phi_{0}^{2}}{\alpha_{B}/2}\lim_{\rho\rightarrow\infty}\rho^{2}B^{\prime},$
(58)

which again determines their behavior at large radii.

###

Although the equations are coupled, they can be solved in the thin wall
approximation. Taking $F(\rho)=1-\Theta(\rho-R^{*})$ and imposing the boundary
conditions

$\displaystyle\lim_{\rho\rightarrow
0}F^{\prime}=\lim_{\rho\rightarrow\infty}F=\lim_{\rho\rightarrow
0}A_{3}^{\prime}=\lim_{\rho\rightarrow\infty}A_{3}=0=\lim_{\rho\rightarrow
0}B\prime=\lim_{\rho\rightarrow\infty}B=0,$ (59)

as well as imposing the continuity of $A^{3}$ and $B$ at $R^{*}$ (as well as
their derivatives), gives

$\displaystyle A^{3}(\rho)$
$\displaystyle=\begin{cases}2\alpha_{W}\Omega\left(\frac{2\sinh\left(\frac{1}{2}\rho\sqrt{\alpha_{W}^{2}+\alpha_{Y}^{2}}\right)\text{sech}\left(\frac{1}{2}R^{*}\sqrt{\alpha_{W}^{2}+\alpha_{Y}^{2}}\right)}{\rho\left(\alpha_{W}^{2}+\alpha_{Y}^{2}\right)^{3/2}}-\frac{1}{\alpha_{W}^{2}+\alpha_{Y}^{2}}\right)&\rho<R^{*}\\\
-\frac{2\alpha_{W}\Omega\left(R^{*}\sqrt{\alpha_{W}^{2}+\alpha_{Y}^{2}}-2\tanh\left(\frac{1}{2}R^{*}\sqrt{\alpha_{W}^{2}+\alpha_{Y}^{2}}\right)\right)}{\rho\left(\alpha_{W}^{2}+\alpha_{Y}^{2}\right)^{3/2}}&\rho\geq
R^{*}\end{cases}$

and:

$\displaystyle
B(\rho)=\begin{cases}2\alpha_{Y}\Omega\left(\frac{1}{\alpha_{W}^{2}+\alpha_{Y}^{2}}-\frac{2\sinh\left(\frac{1}{2}\rho\sqrt{\alpha_{W}^{2}+\alpha_{Y}^{2}}\right)\text{sech}\left(\frac{1}{2}R^{*}\sqrt{\alpha_{W}^{2}+\alpha_{Y}^{2}}\right)}{\rho\left(\alpha_{W}^{2}+\alpha_{Y}^{2}\right)^{3/2}}\right)&\rho<R^{*}\\\
\frac{2\alpha_{Y}\Omega\left(R^{*}\sqrt{\alpha_{W}^{2}+\alpha_{Y}^{2}}-2\tanh\left(\frac{1}{2}R^{*}\sqrt{\alpha_{W}^{2}+\alpha_{Y}^{2}}\right)\right)}{\rho\left(\alpha_{W}^{2}+\alpha_{Y}^{2}\right)^{3/2}}&\rho\geq
R^{*}\end{cases}$

We note that in the large $\rho$ regime,
$B(\rho)=-\dfrac{\alpha_{Y}}{\alpha_{W}}A^{3}(\rho)$, as required by
$Q_{Y}=-Q_{W}$. We remind the reader that in this expression, $\alpha_{W}$ and
$\alpha_{Y}$ are the quantities defined under equation (56), and not the
couplings squared over $4\pi$.

From the equation for $F$ above, we observe that the energy per unit charge is
related to that of a global Q-ball by

$\displaystyle\Omega_{G}$
$\displaystyle=\Omega+\dfrac{\alpha_{W}}{2}A^{3}(R^{*})-\dfrac{\alpha_{Y}}{2}B(R^{*}),$
(60)

which gives us

$\displaystyle\omega=\frac{1}{2}Rv_{0}\omega_{0}\sqrt{g_{W}^{2}+g_{Y}^{2}}\coth\left(\frac{1}{2}Rv_{0}\sqrt{g_{W}^{2}+g_{Y}^{2}}\right).$
(61)

###

Substituting the solutions into (58), we find

$\displaystyle Q_{Y}=-Q_{W}$ $\displaystyle=\frac{8\pi
R\omega_{0}\left(Rv_{0}\sqrt{g_{W}^{2}+g_{Y}^{2}}\coth\left(\frac{1}{2}Rv_{0}\sqrt{g_{W}^{2}+g_{Y}^{2}}\right)-2\right)}{g_{W}^{2}+g_{Y}^{2}},$
(62)

which scales as $R^{3}\omega_{0}v_{0}^{2}$ in the appropriate limit. Replacing
$v_{0}$ with $h_{0}$ for the Higgs VEV gives the expression in section 4. We
also note that since we have used the step function ansatz, $F=1$ inside the
Higgs ball, and therefore the VEV is $h_{0}$, the same value as for the global
Higgs ball.

##  Appendix D Modifications for our BSM Scenario

###

In this appendix, we will summarize the changes we made in our BSM (Beyond-
the-Standard-Model) scenario. We assume that somewhere between the scales
probed by current experiments and the lowest scales relevant (temperature
and/or Higgs VEV of order $\mathcal{O}(10^{8})$) new fermions affect the
running of the Standard Model gauge couplings, suppressing them. Since we are
interested in the general phenomenon of Higgs balls, we we implemented this
simply by dividing the running Standard Model couplings by a factor of 100.

###

The primary modification to the Higgs potential comes from the additional
scalar $S$. This scalar field generally helps to stabilize the Higgs potential
through a positive contribution to the running quartic coupling. The quartic
term in the potential will typically dominate at scales slightly above the
mass of the singlet. Consequently, Q-balls would exist only in a narrow
temperature regime around the mass of the singlet, and a broad study of
thermal balls is difficult. (This would also likely decrease any
phenomenological impact they may have.)

###

Therefore, we also introduce a fermion $\psi$ at the same scale as the scalar,
with a coupling constant $y_{f}=\sqrt{2\lambda_{HS}}$. With this choice, the
one-loop corrections to the Standard Model Higgs sector quartic coupling
$\lambda$ from the scalar and fermion cancel with each other. This could
perhaps be justified via an embedding in a supersymmetric model, although we
leave the for future study. Therefore, we use the Standard Model running
couplings (with $g_{Y}$ and $g_{W}$ modified as explained above).

###

As mentioned above, we assume the fermions either do not couple to the
Standard Model Higgs directly, or such couplings are suppressed so that their
contribution to the Higgs potential is negligible.

###

As in Appendix A, we follow the approach in Ref. Arnold:1992rz . As the scalar
does not acquire a vacuum expectation value, $V_{\rm tree}(h)$ is unchanged.
The scalar and fermion contributions to $\Delta V_{1-\mathrm{loop}}$ exactly
cancel, due to the relation between $\lambda_{HS}$ and the Yukawa coupling
discussed above.

###

The contribution to the finite temperature potential is

$\displaystyle\Delta V_{1-\rm{loop}}(h,T)$
$\displaystyle=\dfrac{T^{4}}{2\pi^{2}}J_{B}\left(\dfrac{m_{S,\rm{eff}}^{2}}{T^{2}}\right)-\dfrac{T^{4}}{2\pi^{2}}J_{F}\left(\dfrac{m_{f}^{2}}{T^{2}}\right),$
(63)

where $m_{S,\rm{eff}}=\sqrt{\lambda_{HS}h}$ and the mass of the fermion is
$\sqrt{\lambda_{HS}}h$. We note that this is independent of the gauge
couplings, as desired.

###

Finally, the contribution to the daisy diagrams is

$\displaystyle\Delta V_{\rm{ring}}(h,T)$
$\displaystyle=\dfrac{T^{4}}{12\pi}\left(\left(\dfrac{m_{S,\rm{eff}}^{2}}{T^{2}}\right)^{3/2}-\left(\dfrac{\mathcal{M}_{S}^{2}}{T^{2}}\right)^{3/2}\right),$
(64)

where the Debye mass is $\mathcal{M}_{S}^{2}=m_{S,\rm{eff}}^{2}+\Pi_{S}(h,T)$
and

$\displaystyle\Pi_{S}(h,T)=\dfrac{\lambda_{HS}}{12}T^{2}.$ (65)

As before, we have taken care to subtract off terms proportional $T^{4}$ which
are independent of the VEV $h$ are therefore do not affect our solitons.
Finally, we note that the Debye masses of the Higgs and Goldstone bosons
$\chi$ are also modified by the scalar and fermion. We add to these

$\displaystyle\Delta\Pi^{2}_{h,\chi}=\left(\dfrac{\lambda_{HS}}{12}+\dfrac{y_{f}^{2}}{4}\right)T^{2}.$
(66)

##  Appendix E Decay Rates of Higgs Quanta Inside the Higgs Ball

###

In this appendix, we describe our calculations for the decay rate of Higgs
quanta inside the Higgs ball into Standard Model particles, which is used in
section 6 to calculate the decay rate of the Higgs ball. At tree-level, the
Higgs quanta inside the Higgs ball can decay directly to gauge bosons and
fermions. In addition to considering these, we note that at loop level it can
decay to photons and gluons. We also consider these decays, and show that they
are suppressed.

###

Before proceeding, we note the following: In appendix C we showed that the
Higgs VEV inside the gauged Q-balls is the same as in the global Q-ball, and
so we evaluate the decay rates at $h=h_{0}$. We also made sure to evaluate
these at the energy per unit of the gauged, not global, Higgs balls.

### Gauge Boson Decay:

As noted in Appendix A, the longitudinal and transverse modes of the gauge
bosons receive different corrections to their masses at finite temperature.
Consequently, we consider their decay rates separately. The decay rates for a
single Higgs quanta inside the Higgs ball to decay to gauge bosons are given
by

$\displaystyle\Gamma_{h\rightarrow WW,\mathrm{trans}}$
$\displaystyle=\dfrac{g_{W}^{4}h^{2}}{32\pi}\cdot\dfrac{\sqrt{1-4x_{W,\mathrm{trans}}}}{\omega},$
$\displaystyle\Gamma_{h\rightarrow ZZ,\mathrm{trans}}$
$\displaystyle=\dfrac{1}{2}\dfrac{(g_{W}^{2}+g_{Y}^{2})^{2}h^{2}}{32\pi}\cdot\dfrac{\sqrt{1-4x_{Z,\mathrm{trans}}}}{\omega},$
$\displaystyle\Gamma_{h\rightarrow WW,\mathrm{long}}$
$\displaystyle=\dfrac{1}{256\pi}\cdot\dfrac{g_{W}^{4}h^{2}\omega^{3}}{m_{W,\mathrm{long}}^{4}}\cdot\sqrt{1-4x_{W,\mathrm{long}}}\left(1-2x_{W,\mathrm{long}}\right)^{2},$
$\displaystyle\Gamma_{h\rightarrow ZZ,\mathrm{long}}$
$\displaystyle=\dfrac{1}{2}\cdot\dfrac{1}{256\pi}\cdot\dfrac{(g_{W}^{2}+g_{Y}^{2})^{2}h^{2}\omega^{3}}{m_{Z,\mathrm{long}}^{4}}\cdot\sqrt{1-4x_{Z,\mathrm{long}}}\left(1-2x_{Z,\mathrm{long}}\right)^{2},$
(67)

where $x_{i}=M_{i}^{2}/\omega^{2}$, where inside the Higgs ball the effective
mass of a single Higgs quanta is $\omega$, the energy per unit charge. We note
that at zero temperature, in which $m_{W,\mathrm{long}}=g_{W}h/2$ and
$m_{Z,\mathrm{long}}=\sqrt{g_{W}^{2}+g_{Y}^{2}}h/2$, the longitudinal decay
rates have a piece independent of the gauge coupling, as required by the
Golstone Boson Equivalence Theorem. Using these relations and summing over the
modes reproduces the standard result (e.g., Spira:2016ztx ), provided that one
also replaces $\omega$ with the Higgs mass.

###

Because $g_{W}$ and $g_{Y}$ are suppressed in our BSM model, the decays to the
transverse modes are also highly suppressed. However, the decays to the
longitudinal modes are not, and it turns our these will be the dominant decay
modes of the quanta inside the Higgs ball.

### Photon Decay:

The decay of the Higgs quanta to photons (and gluons) is suppressed by loop
effects. At zero temperature, the photon decay rate is given by Spira:2016ztx

$\Gamma(h\rightarrow\gamma\gamma)=\dfrac{G_{F}\alpha^{2}m_{h}^{3}}{128\sqrt{2}\pi^{3}}\bigg{|}\sum_{f}N_{cf}e_{f}^{2}A_{f}^{H}(\tau_{f})+A_{W}^{H}(\tau_{W})\bigg{|}^{2},$
(68)

where

$\displaystyle A_{F}^{H}(\tau)$
$\displaystyle=2\tau\left(1+(1-\tau)f(\tau)\right),$ $\displaystyle
A_{W}^{H}(\tau)$ $\displaystyle=-\left(2+3\tau+3\tau(2-\tau)f(\tau)\right),$
(69)

and $\tau_{i}=4M_{i}^{2}/m_{h}^{2}$. $N_{cf}$ is one for leptons and three for
quarks, and $e_{f}$ is the charge of each species in units of the proton
charge. In these relations, the function $f$ is given by

$\displaystyle f(\tau)=\begin{cases}\arcsin(1/\sqrt{\tau})^{2}&\tau\geq 1,\\\
-\dfrac{1}{4}\left[\log\left(\dfrac{1+\sqrt{1-\tau}}{1-\sqrt{1-\tau}}\right)-i\pi\right]^{2}&\tau<1.\end{cases}$

###

The first term, from fermion loops, can be easily generalized to decays of
Higgs quanta inside Q-balls at finite temperature, by replacing $m_{H}$ with
$\omega$ and using $G_{F}=1/(\sqrt{2}h^{2})$. However, the second term, from
$W$ boson loops, is more complicated. At finite temperature, it should be
evaluated in the Landau gauge, and since the transverse and longitudinal parts
of the gauge boson receive different Debye masses, these should be separately
identified; for example, by using a gauge boson propagator which projects out
the longitudinal and transverse components Buchmuller:1993bq .

###

As we expect the photon contribution to be negligible due to loop suppression,
and the smallness of $\alpha$, we will not carefully calculate it but estimate
it as follows. First, we write $G_{F}=g_{W}^{4}h^{2}/(\sqrt{2}m_{W}^{4})$,
motivated by the fact that the $hWW$ vertex introduces a factor of $g_{W}^{2}$
into the matrix element. At zero temperature, when $m_{W}=gh/2$, this reduces
to the familiar $1/(\sqrt{2}h^{2})$. Secondly, we note that the first term in
$A_{W}^{H}$, which is independent of the $W$ boson mass, must come from the
longitudinal mode by the Goldstone Boson Equivalence Theorem, as this gives
the contribution which is independent of the gauge coupling. The remaining
pieces of $A_{W}^{H}$ are a sum of the longitudinal and transverse
contributions.

###

We will thus overestimate the decay rate if we calculate:

$\Gamma(h\rightarrow\gamma\gamma)\leq\dfrac{\alpha^{2}\omega^{3}}{256\pi^{3}}\bigg{|}\dfrac{1}{h}\sum_{f}N_{cf}e_{f}^{2}A_{f}^{H}(\tau_{f})+\dfrac{g_{W}^{2}h}{m_{W,\mathrm{long}}^{2}}A_{W}^{H}(\tau_{W,\mathrm{long}})+\dfrac{g_{W}^{2}h}{m_{W,\mathrm{trans}}^{2}}\tilde{A}_{W}^{H}(\tau_{W,\mathrm{trans}})\bigg{|}^{2},$
(70)

where $\tilde{A}_{W}^{H}$ simply leaves off the first term of $A_{W}^{H}$, and
we also replace $m_{H}$ with $\omega$ when calculating $\tau$ (in addition to
including thermal masses). Using this, we have verified that
$\Gamma_{h\rightarrow\gamma\gamma}$ is many orders of magnitude smaller than
the gauge boson decay rate throughout the temperature range of interest.

### Gluon Decay:

In general, the Higgs quanta may also decay to gluons via fermion loop
effects. The leading order decay rate is Spira:2016ztx

$\displaystyle\Gamma(h\rightarrow gg)$
$\displaystyle=\dfrac{\alpha_{s}^{2}\omega^{3}}{72v^{2}\pi^{3}}\bigg{|}\sum_{Q}A_{Q}^{H}(\tau_{Q})\bigg{|}^{2},$
(71)

where

$\displaystyle A_{Q}^{H}(\tau)$
$\displaystyle=\dfrac{3}{2}\tau\left(1+(1-\tau)f(\tau)\right)$ (72)

and the function $f(\tau)$ was given above. The sum is over all the quarks,
using thermally corrected mass. However, the gluon also has an induced thermal
mass Comelli:1997ru , which are

$\displaystyle m_{G,\mathrm{long}}$ $\displaystyle=g_{S}^{2}T$ $\displaystyle
m_{G,\mathrm{trans}}$
$\displaystyle=\sqrt{2g_{S}^{2}T^{2}}+\dfrac{3}{4\pi}g_{S}^{2}T\ln\left(\dfrac{\sqrt{3+n_{Q}/2}}{g_{S}}\right),$
(73)

where $n_{Q}$ is the number of quarks in the thermal plasma. These thermal
masses are sufficient to make the decay kinematically forbidden throughout the
region of interest, due to the smallness of the energy per unit charge
$\omega$.

### Fermionic Decays:

Next we turn our attention to fermionic decays. As noted in Section 6,
fermionic decays occur throughout the Higgs ball as the decay products are
able to efficiently diffuse out. The rate for a single Higgs quanta to decay
into fermions is

$\displaystyle\Gamma_{h\rightarrow f\bar{f}}$ $\displaystyle=\dfrac{3\omega
y_{f}^{2}}{8\pi},$ (74)

where we use the running Yukawa couplings. The Higgs can decay to all fermions
except the top quark, although we neglect neutrino decays as they are highly
suppressed. As expected, decays to bottom quarks dominate due to the larger
Yukawa coupling.

Figure 11: The ratio of $2m_{S,\rm{eff}}=2m_{\psi}=2\sqrt{\lambda_{HS}}h$ to
the energy per unit charge inside the Higgs ball. Since this is greater than
one, decays into the scalar $S$ and fermion $\psi$ are kinematically
forbidden. This is evaluated at $Q=10^{6}$, but it depends on the charge very
weakly for reasons explained in Section 6.

### BSM Decays:

Our BSM model introduced several new states: a set of fermions which alter the
running of the $g_{Y}$ and $g_{W}$ couplings, a scalar singlet $S$, and a
fermion $\Psi$ whose role is to cancel the singlet’s contribution to the
running Higgs quartic coupling. All of these could introduce new decay
channels for the Higgs.

###

Since we are primarily interested in long-lived Higgs balls, we want to
suppress these decays. Therefore, we do not couple the fermions which alter
the gauge running to the Higgs boson, or restrict their Yukawa couplings to be
highly suppressed so that these decays are negligible.

###

The large value $\lambda_{HS}=0.9$ induces a large mass for the singlet, and
since $y_{f}=\sqrt{2\lambda_{HS}}$, for the fermion also. In particular,
$m_{S,\rm{eff}}=m_{\psi}=\sqrt{\lambda_{HS}}h$. In Fig. 11, we show the ratio
$2\sqrt{\lambda_{HS}}h/\omega$, which shows that it is kinematically forbidden
in the range of interest.