††institutetext: Rudolf Peierls Centre for Theoretical Physics, University of
Oxford, Clarendon Laboratory, Parks Road, Oxford OX1 3PU, United Kingdom

# Q-balls with Multiple Charges and Cores

Olivier Lennon<EMAIL_ADDRESS>
###### Abstract

Q-balls – whether in the single-field or multi-field context – are usually
studied in theories containing only one stabilising symmetry. However, this is
not the most general scenario. In this paper, we study a class of theories
with multiple symmetries. We consider both the traditional thin- and thick-
wall limits of these theories, deriving sufficient conditions for existence in
the latter case. Moreover, we also introduce a new state that could exist in
this class of theory – a cored Q-ball. We show that this new state can be
energetically stable, but leave a detailed phenomenological study to later
work.

††preprint: OUTP-21-31P

##  1 Introduction

The states with the lowest mass-to-charge ratio that exist in many theories of
complex scalar fields are referred to as Q-balls Coleman:1985ki . These
objects are an example of a non-topological soliton Lee:1991ax . These semi-
classical objects are kept stable by a combination of energy and Noether
charge conservation. Usually, this charge is taken to correspond to be some
global $\displaystyle U(1)$ symmetry, but the analysis has been extended to
accommodate more complicated symmetry groups Safian:1987pr , or even gauging
the symmetry Lee:1988ag ; Heeck:2021zvk .

Finding the functional form of these states is a complex process that involves
extremising the energy functional of the theory. This is not analytically
tractable in the most general of theories. However, analytic limits do exist
and can be studied. These are the familiar thin- and thick-wall limits
Coleman:1985ki ; Kusenko:1997ad , with language borrowed from the
mathematically similar study of bounce equations Coleman:1977py ;
Callan:1977pt ; Coleman:1977th . These descriptions are valid in the limits of
large and small charge, respectively. Not every theory that possesses an
energetically stable thin-wall limit has a stable thick-wall limit
PaccettiCorreia:2001wtt ; Postma:2001ea ; Lennon:2021uqu , implying that there
is some gap over which enough charge must accumulate before stability is
achieved. Recently, work has been done in Ref. Heeck:2020bau  to analytically
extend the thin-wall limit to smaller charges.

Q-balls are theoretically interesting objects in their own right. However,
they also are interesting phenomenological objects. Due to their stability,
they are often considered as candidates for the perceived dark matter in our
universe Kusenko:1997si ; Kusenko:2001vu ; Graham:2015apa ; Ponton:2019hux .
They could also appear in supersymmetric extensions to the standard model
Kusenko:1997si ; Kusenko:1997zq , and have been studied in the context of
extra dimensions Demir:2000gj ; Abel:2015tca . Despite not being yet seen in
experiments, it is believed that Q-balls will have striking signals
Gelmini:2002ez ; Kusenko:1997vp ; Croon:2019rqu .

The previous works on single-field theories – with canonical Coleman:1985ki
and non-canonical Lennon:2021uqu  kinetic terms – or multi-field theories –
either coupling a charged field to an arbitrary number of scalars
Friedberg:1976me ; Lennon:2021zzx , or coupling multiple fields charged under
the same symmetry Kusenko:1997zq  – do not represent the most general of
theories. One can generalise these ideas further to incorporate multiple
symmetries into the theory. Under these circumstances, Q-balls emerge with
properties that are different than those from simpler sectors. This work does
not represent an exhaustive list of all structures that can exist in this kind
of theory, but we hope that it can be the beginning of work towards such a
list. One such structure – a “barnacled Q-ball” in the same vein as “bounces
with barnacles” studied in Refs. Balasubramanian:2010kg ; Czech:2011aa ;
Scargill:2017zsz  – will be returned to in future work.

In Section 2.1, we outline the class of theory of study of this paper. The
most straightforward example is a multi-field theory in which each scalar
transforms non-trivially under its own symmetry, and is a singlet under all
others – in some sense, this is similar to multiple copies of a single-field
theory. The key to this analysis is then the requirement that the potential
associated to a single field does not admit Q-ball solutions, and that
stability is only achieved due to the couplings between the fields. We then
perform the minimisation procedure analogously to the previous Q-ball works.
In Section 2.2, we perform the thin-wall analysis. We determine the
differential equation governing the value of the vacuum expectation value
(VEV) of the fields inside the resultant Q-ball. Contrary to previous
findings, we find that this condition is dependent on the total number of each
of the quanta that comprise the Q-ball – previously, these equations have been
charge-independent. In Section 2.3, we perform the thick-wall analysis, once
again making use of the assumption that all fields have the same spatial
profile up to a positive semi-definite normalisation constant. Under this
assumption, we produce sufficient constraints on the theories that may possess
a thick-wall limit. Interestingly, despite the added complexity in the
analysis, we find the same result as for single-field theories – that the
next-to-quadratic term must have have an index $\displaystyle p$ such that
$\displaystyle 2<p<10/3$. This result certainly adds to the idea that this
scenario is similar to multiple copies of a single field theory.

In Section 3, we close out this work with a predominantly heuristic discussion
of a new structure within multi-field theories: a cored Q-ball. We envisage a
thin-wall Q-ball in one field which possesses a core composed of a thick-wall
Q-ball of another field that is stabilised precisely in the homogeneous VEV of
the larger Q-ball.111In principle, these objects are a subclass of “speckled”
Q-balls, whereby the larger Q-ball is populated by many smaller ones
throughout its core – this is beyond this work, but is an interesting avenue
of research for the future. This is an interesting addition to the
consideration of Q-balls that interact with other fields. This line of
thinking has obvious similarities to theories with multiple vacua in the early
universe such that the vacuum bubbles develop “barnacles” of other vacua – see
Refs. Balasubramanian:2010kg ; Czech:2011aa ; Scargill:2017zsz . Each field is
stabilised by its own global $\displaystyle U(1)$ symmetry, and so this
structure is in principle stable. We first consider the scenario whereby one
field exists in the background VEV of another. We seek Q-ball solutions that
are stabilised precisely due to this background VEV. We perform the thick-wall
analysis in this scenario, but we note that the calculation proceeds precisely
as it would for the single-field case.222For completeness, we include the
thin-wall analysis as an appendix to this paper. We include these calculations
here for completeness and for introducing the notation in the rest of the
section. We then consider the case when the background VEV is provided by a
large, thin-wall Q-ball. We then merely seek to see whether a cored Q-ball is
a stable structure. This leads to two inequalities to be satisfied: the first
arises from the requirement that the Q-ball be stable against decay into the
individual quanta of both fields; the second arises from the demand that the
core be smaller than the Q-ball that houses it. These requirements are model-
dependent.

##  2 Multi-Charged Q-balls

We begin this work by analysing the most straightforward example of a multi-
field theory with multiple stabilising symmetries – when each of the fields is
charged under its own symmetry.

###  2.1 Minimising the Energy in a Sector of Fixed Charge

We consider a theory of $\displaystyle N$ complex scalars,
$\displaystyle\Phi_{i}(\vec{x},t)$. The Lagrangian density describing the
theory is given by

$\mathcal{L}=\sum^{N}_{i}\partial_{\mu}\Phi_{i}\partial^{\mu}\Phi_{i}^{*}-U(\Phi_{i},\Phi_{i}^{*}),$
(2.1)

where $\displaystyle U(\Phi_{i},\Phi_{i}^{*})$ is some generic potential that
is a function of the fields only, and not their derivatives, and vanishes for
vanishing field. The Euler-Lagrange equations for this theory are

$\partial_{\mu}\partial^{\mu}\Phi_{i}+\frac{\partial
U}{\partial\Phi_{i}^{*}}=0,$ (2.2)

with a similar equation governing the dynamics of $\displaystyle\Phi^{*}_{i}$.

We demand that the theory be invariant under an $\displaystyle N$-fold global
symmetry,

$U(1)^{N}=U(1)_{1}\times U(1)_{2}\times\dots\times U(1)_{N},$ (2.3)

that is, $\displaystyle N$ independent $\displaystyle U(1)$ symmetries. We
consider here the special case that each field $\displaystyle\Phi_{i}$ is
charged under its own independent $\displaystyle U(1)$ symmetry and is
uncharged under all other $\displaystyle U(1)$ symmetries. For each
$\displaystyle U(1)$, its corresponding field transforms as

$\Phi_{i}\to e^{iq_{i}\alpha}\Phi_{i},$ (2.4)

where $\displaystyle\alpha\in\mathbb{R}$ and $\displaystyle q_{i}$ is the
charge of the $\displaystyle i$-th species of complex scalar under the
$\displaystyle i$-th $\displaystyle U(1)$ symmetry. Associated to each
$\displaystyle U(1)$ symmetry is a Noether current density given by

$j^{\mu}_{i}=iq_{i}\left(\Phi_{i}\partial^{\mu}\Phi_{i}^{*}-\Phi_{i}^{*}\partial^{\mu}\Phi_{i}\right).$
(2.5)

This symmetry places the constraint on the potential that it be a function of
the absolute value of the individual fields. We further demand that this
potential does not support Q-balls if all but one of the fields vanish:
otherwise, this system reduces to that of the single-field case as first
discussed in Ref. Coleman:1985ki . We thus assume that the field content is
the minimal set of fields, greater than a single field, required to support a
Q-ball state.

A Q-ball is the state in a theory which minimises the energy per unit charge.
The Hamiltonian for this theory is

$H=\int\mathrm{d}^{3}x\left[\sum^{N}_{i}\left(\dot{\Phi}_{i}\dot{\Phi}_{i}^{*}+\vec{\nabla}\Phi_{i}\cdot\vec{\nabla}\Phi_{i}^{*}\right)+U(\Phi_{i},\Phi_{i}^{*})\right].$
(2.6)

To determine if this system admits Q-ball solutions, we introduce a set of
Lagrange multipliers, $\displaystyle\\{\omega_{k}\\}$, similar to Ref.
Kusenko:1997zq , that each enforce charge conservation upon minimisation with
respect to them:

$\mathcal{E}_{\omega}=H+\sum_{k}^{K}\omega_{k}\left(Q_{k}-\int\mathrm{d}^{3}x\,j^{0}_{k}\right),$
(2.7)

where $\displaystyle j^{0}_{k}$ is the zeroth component of the Noether current
density associated to the $\displaystyle k$-th $\displaystyle U(1)$ symmetry
given in Eq. (2.5). Thus, the functional we wish to analyse is given by

$\begin{split}\mathcal{E}_{\omega}=&\sum_{i}^{N}\omega_{i}Q_{i}\\\
&+\int\mathrm{d}^{3}x\left[\sum^{N}_{i}\left(\dot{\Phi}_{i}\dot{\Phi}_{i}^{*}-i\omega_{i}q_{i}\left(\Phi_{i}\dot{\Phi}_{i}^{*}-\Phi_{i}^{*}\dot{\Phi}_{i}\right)+\vec{\nabla}\Phi_{i}\cdot\vec{\nabla}\Phi_{i}^{*}\right)+U(\Phi_{i},\Phi_{i}^{*})\right].\end{split}$
(2.8)

We may complete the square on the first two terms under the integral to give

$\begin{split}\mathcal{E}_{\omega}=&\sum_{i}^{N}\omega_{i}Q_{i}\\\
&+\int\mathrm{d}^{3}x\left[\sum^{N}_{i}\left(\left|\dot{\Phi}_{i}-i\omega_{i}q_{i}\Phi_{i}\right|^{2}+\vec{\nabla}\Phi_{i}\cdot\vec{\nabla}\Phi_{i}^{*}-\omega_{i}^{2}q_{i}^{2}\Phi_{i}^{*}\Phi_{i}\right)+U(\Phi_{i},\Phi_{i}^{*})\right].\end{split}$
(2.9)

The terms containing derivatives with respect to time are the only terms with
explicit time dependence. These are positive semi-definite and are minimised
if they vanish. Thus, we require, for each species of scalar, that

$\Phi_{i}(\vec{x},t)=e^{i\omega_{i}q_{i}t}\phi_{i}(\vec{x}),$ (2.10)

where $\displaystyle\phi_{i}(\vec{x})$ are functions purely of the spatial
coordinate which we take, without loss of generality, to be real-valued. The
energy functional is then

$\mathcal{E}_{\omega}=\sum_{i}^{N}\omega_{i}Q_{i}+\int\mathrm{d}^{3}x\left[\sum^{N}_{i}\left(\vec{\nabla}\phi_{i}\cdot\vec{\nabla}\phi_{i}-\omega_{i}^{2}q_{i}^{2}\phi_{i}^{2}\right)+U(\phi_{i})\right].$
(2.11)

Notice that spatial profiles for the $\displaystyle\phi_{i}$ that vanish
everywhere lead to a configuration of zero charge. Thus, a configuration of
non-zero charge must have spatial profiles for the charged fields that differs
from zero in some finite domain. The spatial profiles satisfy the equations

$\nabla^{2}\phi_{i}=\frac{1}{2}\frac{\partial}{\partial\phi_{i}}\left(U(\phi_{i})-\omega_{i}^{2}q_{i}^{2}\phi_{i}^{2}\right).$
(2.12)

These differential equations take the form of a bounce equation Coleman:1977py
; Callan:1977pt ; Coleman:1977th , under the potentials given by
$\displaystyle U(\phi_{i},\psi_{i})-\omega^{2}q_{i}^{2}\phi_{i}^{2}$. The
lowest energy configurations are known to be spherically symmetric
Coleman:1977th , under the boundary conditions of the field vanishing at
spatial infinity, and being constant at the coordinate origin, and the first
derivative vanishing at spatial infinity and the coordinate origin.

These coupled differential equations cannot, in general, be solved
analytically. However, under certain assumptions, analytic progress can be
made. Namely, in analogy with the discussion in Ref. Kusenko:1997ad , for
stable Q-ball solutions to be found, we require that
$\displaystyle\omega_{i,0}\leq\omega_{i}<m_{i}/q_{i}$, where $\displaystyle
m_{i}$ is the mass of the $\displaystyle i$-th complex scalar. The thin-wall
limit corresponds to the limit that $\displaystyle\omega_{i}=\omega_{i,0}$ –
in fact, the thin-wall limit defines $\displaystyle\omega_{i,0}$. The thick-
wall limit corresponds to the limit $\displaystyle q_{i}\omega_{i}\to
m_{i}^{-}$.

###  2.2 Thin-Wall Q-balls

Q-balls in the thin-wall limit are characterised by a spherical, homogeneous
core and a thin-shell through which the field interpolates between its core
and vacuum values. The Q-ball properties are then well-approximated by those
of the core. In this regime, we have that

$\mathcal{E}_{\omega}\approx\sum_{i}^{N}\omega_{i}Q_{i}+V\left[U(\phi_{i})-\sum^{N}_{i}\omega_{i}^{2}q_{i}^{2}\phi_{i}^{2}\right].$
(2.13)

Minimisation of this expression with respect to each of the
$\displaystyle\omega_{k}$ yields

$Q_{k}=2\omega_{k}Vq_{k}^{2}\phi_{k}^{2},$ (2.14)

as would be expected from the Noether current density given in Eq. (2.5).
Eliminating the Lagrange multipliers from the energy yields

$E=\frac{1}{4V}\sum_{i}^{N}\frac{Q_{i}^{2}}{q_{i}^{2}\phi_{i}^{2}}+U(\phi_{i})V.$
(2.15)

Minimisation with respect to the volume gives us that

$V^{2}=\frac{1}{4U(\phi_{i})}\sum_{i}^{N}\frac{Q_{i}^{2}}{q_{i}^{2}\phi_{i}^{2}},$
(2.16)

which, upon substitution, finally gives us that

$m_{Q}=\sqrt{U(\phi_{i})\sum_{i}^{N}\frac{Q_{i}^{2}}{q_{i}^{2}\phi_{i}^{2}}}.$
(2.17)

This expression must be minimised with respect to the field content subject to
the condition

$m_{Q}<\sum_{i}^{N}\frac{Q_{i}}{q_{i}}m_{i},$ (2.18)

where $\displaystyle m_{i}$ are the masses of the quanta associated to the
field $\displaystyle\phi_{i}$. This ensures that these Q-balls are classically
stable against evaporation into the field content. The result of this
minimisation is the constraint on the potential:

$\frac{\partial
U(\phi_{i})}{\partial\phi_{k}}\sum_{i}^{N}\frac{Q_{i}^{2}}{q_{i}^{2}\phi_{i}^{2}}=2\frac{Q_{k}^{2}}{q_{k}^{2}\phi_{k}^{3}}U(\phi_{i}).$
(2.19)

Since this must be true for all $\displaystyle k$, we therefore find that

$\frac{\partial
U(\phi_{i})}{\partial\phi_{j}}\frac{q_{j}^{2}\phi_{j}^{3}}{Q_{j}^{2}}=\frac{\partial
U(\phi_{i})}{\partial\phi_{k}}\frac{q_{k}^{2}\phi_{k}^{3}}{Q_{k}^{2}}.$ (2.20)

Note, for the single field case – for a field with unit charge – the rest mass
of a Q-ball is given simply by

$m_{Q}=Q\sqrt{\frac{U(\phi)}{\phi^{2}}},$ (2.21)

where $\displaystyle\phi$ satisfies

$\frac{\mathrm{d}U(\phi)}{\mathrm{d}\phi}=2\frac{U(\phi)}{\phi}.$ (2.22)

In the single-field case, the VEV of the field inside the core is independent
of the charge of the Q-ball. As the field has unit charge, this means that it
is independent of the total number of quanta which make up the Q-ball.333This
is also true in the single-symmetry multi-field case, both when there is only
a single charged scalar Lennon:2021zzx  and when there are multiple fields
charged under the same symmetry Kusenko:1997zq . However, in the multi-
symmetry case above, we note that the VEVs of the fields inside the Q-ball do
depend on the total number of quanta (as $\displaystyle N_{i}=Q_{i}/q_{i}$).
This is in stark contrast to the known Q-ball cases. As the mass of the core
does not simply scale with particle number, we must conclude that it is not
composed of ordinary Q-matter.

###  2.3 Thick-Wall Q-balls

We now consider the thick-wall limit of this theory. This corresponds to the
small-field limit of the field theory, as discussed at length in Ref.
Kusenko:1997ad . Thus, we will perform a small-field expansion on the function
denoting the potential to proceed analytically.

As in the multi-field case with only a single symmetry Postma:2001ea ;
Bishara:2017otb ; Lennon:2021zzx , this theory suffers from the fact that it
is not analytically tractable due to the number of fields. As in earlier
works, this problem must be circumvented by assuming that all the spatial
profiles of the fields are the same up to some semi-positive-definite
normalisation constant – we must then minimise the energy with respect to each
of these normalisation constants. We thus write that

$\phi_{i}=\alpha_{i}\phi,$ (2.23)

where $\displaystyle\phi\in\\{\phi_{i}\\}$ is some reference field – it does
not affect the analysis in this case which field is used. The purpose of this
ansatz is to provide sufficient conditions for the existence of thick-wall
Q-balls in this class of multi-field theories, as well as to provide
approximate values for the Q-ball properties. In reality, the spatial profiles
of the fields might differ, but this extra freedom in the minimisation process
can only further lower the Q-ball energy. To provide necessary conditions for
the thick-wall limit, we must pursue a dedicated numerical analysis of the
true spatial profiles of all individual fields. Since we have effectively
reduced our multi-field system into one of a single-field, we expect to find
results similar to those in Ref. PaccettiCorreia:2001wtt ; Postma:2001ea ;
Lennon:2021uqu . We will thus be unsurprised when this does indeed happen
below.

Given this approximation scheme, the full bounce potential can be written in
the small field limit as

$U_{\Omega}(\phi,\alpha_{i})\approx\left[\sum_{i}^{N}\alpha_{i}^{2}m_{i}^{2}(1-\Omega_{i}^{2})\right]\phi^{2}-g(\alpha_{i})\phi^{p},$
(2.24)

where $\displaystyle p>2$ and we have defined

$\Omega_{i}\equiv\frac{q_{i}\omega_{i}}{m_{i}},$ (2.25)

with $\displaystyle\Omega_{i}\in(0,1)$, such that the coefficient of the
quadratic term is always positive. The function $\displaystyle g(\alpha_{i})$
is model-dependent and we leave it general apart from the fact that it must be
positive definite for the potential to allow Q-ball solutions. We have omitted
the next term in the series, but it is assumed that it is positive definite
and stabilises the potential at high field values. The energy in Eq. (2.11) is
then

$\mathcal{E}_{\omega}=\sum_{i}^{N}m_{i}\Omega_{i}\frac{Q_{i}}{q_{i}}+\int\mathrm{d}^{3}x\left[\sum^{N}_{i}\alpha_{i}^{2}\left(\vec{\nabla}\phi\cdot\vec{\nabla}\phi+m_{i}^{2}(1-\Omega_{i}^{2})\phi^{2}\right)-g(\alpha_{i})\phi^{p}\right].$
(2.26)

We wish to render the integral dimensionless. To do so, we define the
dimensionless variables

$\varphi\equiv\phi\left[\dfrac{g(\alpha_{i})}{\sum_{i}^{N}\alpha_{i}^{2}m_{i}^{2}(1-\Omega_{i}^{2})}\right]^{1/(p-2)}\quad\mathrm{and}\quad\xi_{i}\equiv
x_{i}\left[\dfrac{\sum_{i}^{N}\alpha_{i}^{2}m_{i}^{2}(1-\Omega_{i}^{2})}{\sum_{i}^{N}\alpha_{i}^{2}}\right]^{1/2}.$
(2.27)

The functional of study is then given by

$\mathcal{E}_{\omega}=\sum_{i}^{N}m_{i}\Omega_{i}\frac{Q_{i}}{q_{i}}+\frac{S_{\varphi}}{g(\alpha_{i})^{2/(p-2)}}\left[\sum_{i}^{N}\alpha_{i}^{2}\right]^{3/2}\left[\sum_{i}^{N}\alpha_{i}^{2}m_{i}^{2}(1-\Omega_{i}^{2})\right]^{(6-p)/(2p-4)},$
(2.28)

where $\displaystyle S_{\varphi}$ is a dimensionless integral, defined as

$S_{\varphi}=\int\mathrm{d}^{3}\xi\left[\vec{\nabla}_{\xi}\varphi\cdot\vec{\nabla}_{\xi}\varphi+\varphi^{2}-\varphi^{p}\right].$
(2.29)

For different values of $\displaystyle p$, this has been numerically minimised
in Ref. Linde:1981zj , with the general trend being that $\displaystyle
S_{\varphi}$ increases for increasing $\displaystyle p$. This expression must
be minimised with respect to the $\displaystyle\\{\Omega_{i}\\}$ and
$\displaystyle\\{\alpha_{i}\\}$. Without specifying the model, it is difficult
to make progress. However, as in previous work PaccettiCorreia:2001wtt ;
Postma:2001ea ; Lennon:2021uqu ; Lennon:2021zzx , we can place limits on the
theories that can house Q-ball states.

We minimise the above function in Eq. (2.28) with respect to some
$\displaystyle\Omega_{k}$. This yields the condition

$\frac{Q_{k}}{q_{k}\alpha_{k}^{2}m_{k}\Omega_{k}}=\frac{S_{\varphi}}{g(\alpha_{i})^{2/(p-2)}}\left[\frac{6-p}{p-2}\right]\left[\sum_{i}^{N}\alpha_{i}^{2}\right]^{3/2}\left[\sum_{i}^{N}\alpha_{i}^{2}m_{i}^{2}(1-\Omega_{i}^{2})\right]^{(10-3p)/(2p-4)}.$
(2.30)

Notice, the left hand-side is $\displaystyle k$-dependent, but the right-hand
side is not. Since this condition holds for all $\displaystyle k$, we can
readily write

$\frac{Q_{k}}{q_{k}\alpha_{k}^{2}m_{k}\Omega_{k}}=\frac{Q_{i}}{q_{i}\alpha_{i}^{2}m_{i}\Omega_{i}}$
(2.31)

for all $\displaystyle k,i\in\\{1,\cdots,N\\}$. Given that
$\displaystyle\alpha=1$ for, say, the field $\displaystyle\phi_{1}$, we have
that

$\alpha_{k}^{2}=\frac{q_{1}}{Q_{1}}\frac{Q_{k}}{q_{k}}\frac{m_{1}}{m_{k}}\frac{\Omega_{1}}{\Omega_{k}},$
(2.32)

and so $\displaystyle\alpha$ is now defined for all fields in the theory.

Using our above condition for minimisation, we can rewrite Eq. (2.28) as

$\mathcal{E}_{\omega}=\sum_{i}^{N}m_{i}\Omega_{i}\frac{Q_{i}}{q_{i}}+\frac{Q_{k}}{q_{k}\alpha_{k}^{2}m_{k}\Omega_{k}}\left[\frac{p-2}{6-p}\right]\left[\sum_{i}^{N}\alpha_{i}^{2}\right]^{3/2}\left[\sum_{i}^{N}\alpha_{i}^{2}m_{i}^{2}(1-\Omega_{i}^{2})\right].$
(2.33)

At its minimum, this expression must satisfy

$\mathcal{E}_{\omega}<\sum_{i}^{N}\frac{Q_{i}}{q_{i}}m_{i}$ (2.34)

in order for the resulting Q-balls to be classically stable against decay to
the constituent quanta of the complex scalar fields. From Eq. (2.31), we see
that we can write

$\sum_{i}^{N}m_{i}\Omega_{i}\frac{Q_{i}}{q_{i}}=\frac{Q_{k}}{q_{k}\alpha_{k}^{2}m_{k}\Omega_{k}}\sum_{i}^{N}\alpha_{i}^{2}m_{i}^{2}\Omega_{i}^{2},$
(2.35)

and

$\sum_{i}^{N}\frac{Q_{i}}{q_{i}}m_{i}=\frac{Q_{k}}{q_{k}\alpha_{k}^{2}m_{k}\Omega_{k}}\sum_{i}^{N}\alpha_{i}^{2}m_{i}^{2}\Omega_{i}.$
(2.36)

Thus, our condition on the minimum becomes

$\sum_{i}^{N}\alpha_{i}^{2}m_{i}^{2}\left[\Omega_{i}^{2}+\left(\frac{p-2}{6-p}\right)(1-\Omega_{i}^{2})-\Omega_{i}\right]<0.$
(2.37)

For this condition to hold, we necessarily require that that term in brackets
be negative for some $\displaystyle i$ in the range
$\displaystyle\Omega\in(0,1)$. However, this is nothing more than the same
requirement on the canonical single-field case – see Ref. Lennon:2021uqu . We
thus infer that, for stable thick-wall Q-balls to form in this theory, we must
have that $\displaystyle 2<p<10/3$.

##  3 Cored Q-balls

We now turn our attention to another type of Q-ball that can plausibly exist
in multi-field theories with multiple global symmetries. We refer to these as
“cored Q-balls”. In this section, we merely show that these objects can be
classically stable against decay into constituent quanta.

###  3.1 Q-balls in the Background of Another Field

A key component of this section is the idea of a Q-ball stabilised in the
background of another field. We therefore formulate this idea for a background
VEV extending over all space.

####  3.1.1 The Minimisation Procedure

Consider the $\displaystyle N=2$ case of the Lagrangian given in Eq. (2.1). We
write the potential of this theory as

$U(\Phi_{1},\Phi_{2})=f(\Phi_{1})+g(\Phi_{2})+h(\Phi_{1},\Phi_{2}),$ (3.1)

where it is understood that $\displaystyle h$ contains all terms that couple
the two fields together. We assume that the field $\displaystyle\Phi_{1}$
acquires some constant VEV – in this subsection, we leave the method of
acquisition of this VEV open, but in the next subsection, we will assume that
it comes after the formation of a thin-wall Q-ball. We thus seek Q-ball
solutions for the field $\displaystyle\Phi_{2}$ in this constant background.
Furthermore, this constant background will merely change the coefficients of
the field $\displaystyle\Phi_{2}$ by constant amounts, and so this analysis
will proceed exactly as for a single-field system. For definiteness, we
perform the analysis below.

The energy of a configuration of $\displaystyle\Phi_{2}$ in the background of
$\displaystyle\bar{\phi}_{1}$ is

$H=\int\mathrm{d}^{3}x\left[\dot{\Phi}_{2}\dot{\Phi}_{2}^{*}+\vec{\nabla}\Phi_{2}\cdot\vec{\nabla}\Phi_{2}^{*}+g(\Phi_{2})+h(\bar{\phi}_{1},\Phi_{2})\right].$
(3.2)

A Q-ball is a state that minimises the energy for a given charge and so we
introduce a Lagrange multiplier, $\displaystyle\omega_{2}$, that ensures
charge conservation upon minimisation with respect to it. The functional we
wish to analyse is then

$\mathcal{E}_{\omega}=H+\omega_{2}\left(Q_{2}-\int\mathrm{d}^{3}x\,j_{2}^{0}\right),$
(3.3)

where $\displaystyle j_{2}^{0}$ is the zeroth component of the Noether current
density associated to the $\displaystyle U(1)_{2}$ symmetry,

$j_{2}^{0}=iq_{2}\left(\dot{\Phi}_{2}^{*}\Phi_{2}-\Phi_{2}^{*}\dot{\Phi}_{2}\right).$
(3.4)

We may thus rewrite our functional of study as

$\begin{split}\mathcal{E}_{\omega}=&\int\mathrm{d}^{3}x\left[\left|\dot{\Phi}_{2}-i\omega_{2}q_{2}\Phi_{2}\right|^{2}+\vec{\nabla}\Phi_{2}\cdot\vec{\nabla}\Phi_{2}^{*}+g(\Phi_{2})+h(\bar{\phi}_{1},\Phi_{2})-\omega_{2}^{2}q_{2}^{2}\Phi_{2}^{*}\Phi_{2}\right]\\\
&+\omega_{2}Q_{2}.\end{split}$ (3.5)

Note, the term containing explicit time-dependence is positive semi-definite.
This is therefore minimised when it vanishes, i.e., if

$\Phi_{2}(\vec{x},t)=e^{i\omega_{2}q_{2}t}\phi_{2}(\vec{x}),$ (3.6)

where we take the spatial profile to be real-valued, without loss of
generality. Reinsertion of this into the functional of study leads to

$\mathcal{E}_{\omega}=\omega_{2}Q_{2}+\int\mathrm{d}^{3}x\left[\vec{\nabla}\phi_{2}\cdot\vec{\nabla}\phi_{2}+g(\phi_{2})+h(\bar{\phi}_{1},\phi_{2})-\omega_{2}^{2}q_{2}^{2}\phi_{2}^{2}\right].$
(3.7)

Minimisation with respect to the spatial profile yields a differential
equation governing a “bounce” solution associated to the formation of vacuum
bubbles during phase transitions Coleman:1977py ; Callan:1977pt ;
Coleman:1977th . These equations are well-studied and, for a set of boundary
conditions appropriate for Q-balls, are known to yield spherically symmetric
solutions for the lowest energy configurations.

####  3.1.2 The Thick-Wall Limit

We now pursue the thick-wall limit of this analysis, which is also known as
the small field limit.444We note that “small-field” here only relates to
$\displaystyle\phi_{2}$, as it is from this that this Q-ball is being formed.
In principle, we could also form thin-wall Q-balls in the background of
another field – this will not be relevant to the discussion of cored Q-balls,
but we include the analysis in Appendix A for completeness.

To proceed in the thick-wall limit, we expand $\displaystyle\phi_{2}$ to its
lowest order terms. As stated above, this analysis is essentially identical to
a single field case and so is bound by the requirement that the next-to-
quadratic term must be cubic in the bounce potential – see Refs.
PaccettiCorreia:2001wtt ; Postma:2001ea ; Lennon:2021uqu . We thus assume that
this term exists. We therefore have, in the small-field limit, that

$g(\phi_{2})+h(\bar{\phi}_{1},\phi_{2})\approx\left(\mu^{2}+h_{2}(\bar{\phi}_{1})\right)\phi_{2}^{2}-\left(h_{3}(\bar{\phi}_{1})-A\right)\phi_{2}^{3},$
(3.8)

where $\displaystyle h_{2}(\bar{\phi}_{1})$ and $\displaystyle
h_{3}(\bar{\phi}_{1})$ are the components, functionally dependent on
$\displaystyle\bar{\phi}_{2}$, of the terms quadratic and cubic in
$\displaystyle\phi_{2}$ contained within $\displaystyle
h(\bar{\phi}_{1},\phi_{2})$, respectively, and similarly for
$\displaystyle\mu^{2}$ and $\displaystyle A$ in $\displaystyle g(\phi_{2})$.
We assume that some positive definite term of a higher order exists to
stabilise the potential at large field. In this scenario, a Q-ball may only
form if the coefficients of the quadratic and cubic terms are positive and
negative, respectively:

$\mu^{2}+h_{2}(\bar{\phi}_{1})>0\quad\mathrm{and}\quad
h_{3}(\bar{\phi}_{1})-A>0.$ (3.9)

However, recall that we demanded that the pure $\displaystyle\Phi_{2}$
potential could not support Q-ball solutions. This sets $\displaystyle A\geq
0$, hence the sign assignment on the cubic term.

For notational purposes, we will write the effective quadratic and cubic
couplings in the background as

$\mu^{2}_{\mathrm{eff}}(\bar{\phi}_{1})\equiv\mu^{2}+h_{2}(\bar{\phi}_{1})\quad\mathrm{and}\quad
A_{\mathrm{eff}}(\bar{\phi}_{1})\equiv h_{3}(\bar{\phi}_{1})-A,$ (3.10)

noting that both of these new parameters are positive definite. The bounce
potential can now be written in the familiar form,

$U_{\omega}(\bar{\phi}_{1},\phi_{2})\approx(\mu_{\mathrm{eff}}^{2}-\omega_{2}^{2}q_{2}^{2})\phi_{2}^{2}-A_{\mathrm{eff}}\phi_{2}^{3}.$
(3.11)

For a barrier to form in the bounce potential, and thus for a bounce solution
to exist – see Ref. Kusenko:1997ad  – we see that

$\omega_{2}^{2}<\frac{\mu_{\mathrm{eff}}^{2}}{q_{2}^{2}},$ (3.12)

which is the familiar requirement of Q-ball solutions. The thick-wall limit is
thus equivalent to the limit
$\displaystyle\omega_{2}q_{2}\to\mu_{\mathrm{eff}}^{-}$.

We redefine the spatial coordinate and field by choosing

$\begin{split}\xi_{i}=&\left(\mu_{\mathrm{eff}}^{2}-\omega_{2}^{2}q_{2}^{2}\right)^{1/2}x_{i}\\\
\psi=&\left(\mu_{\mathrm{eff}}^{2}-\omega_{2}^{2}q_{2}^{2}\right)^{-1}A_{\mathrm{eff}}\phi_{2}.\end{split}$
(3.13)

Our functional of study thus reduces to

$\mathcal{E}_{\omega}=\mu_{\mathrm{eff}}^{3}\dfrac{\left(1-\Omega^{2}\right)^{3/2}}{A_{\mathrm{eff}}^{2}}S_{\psi}+\omega_{2}Q_{2},$
(3.14)

where we have defined
$\displaystyle\Omega\equiv\omega_{2}q_{2}/\mu_{\mathrm{eff}}$ and

$S_{\psi}=\int\mathrm{d}^{3}\xi\left[\vec{\nabla}_{\xi}\psi\cdot\vec{\nabla}_{\xi}\psi+\psi^{2}-\psi^{3}\right]$
(3.15)

is a dimensionless integral that may be minimised numerically. This has been
numerically found to be approximately 38.8 Linde:1981zj .

A stable Q-ball in the background of $\displaystyle\bar{\phi}_{1}$ is found
when minimising this expression with respect to $\displaystyle\Omega$ under
the constraint that $\displaystyle\Omega^{2}<1$. If this constraint is not
satisfied, it is more energetically favourable for the quanta of
$\displaystyle\Phi_{2}$ to remain seperate within the background of
$\displaystyle\bar{\phi}_{1}$. This requirement leads us to the condition

$\epsilon=\Omega(1-\Omega^{2})^{1/2},$ (3.16)

where we define

$\epsilon\equiv\frac{1}{3S_{\varphi}}\frac{Q_{2}}{q_{2}}\frac{A^{2}_{\mathrm{eff}}}{\mu^{2}_{\mathrm{eff}}}.$
(3.17)

The solution to this condition is that

$\Omega^{2}=\frac{1+\sqrt{1-4\epsilon^{2}}}{2},$ (3.18)

where $\displaystyle 0<\epsilon<1/2$. The mass of the resulting Q-ball is then
given by

$m_{Q}=\mu_{\mathrm{eff}}\frac{Q_{2}}{q_{2}}\left(1-\frac{1}{6}\epsilon^{2}+\mathcal{O}(\epsilon^{4})\right).$
(3.19)

We note that

$m_{Q}<\mu_{\mathrm{eff}}\frac{Q_{2}}{q_{2}},$ (3.20)

which is precisely the requirement for classical stability of this Q-ball
within the background of $\displaystyle\bar{\phi}_{1}$. The radius of the
resulting Q-ball is given by $\displaystyle\xi\sim 1$, and so

$R^{-1}\sim\epsilon\mu_{\mathrm{eff}}\left(1+\mathcal{O}\left(\epsilon^{2}\right)\right).$
(3.21)

This solution is also subject to the constraint that
$\displaystyle\epsilon<1/2$, which translates to the requirement on the charge
that

$\frac{Q_{2}}{q_{2}}<58.2\frac{\mu_{\mathrm{eff}}}{A^{2}_{\mathrm{eff}}}.$
(3.22)

The solution is also subject to the constraint that the higher order term that
stabilises the potential is indeed small enough inside the Q-ball to ignore in
our analysis – see Ref. Kusenko:1997ad  for details.

###  3.2 Cored Q-balls

We assume that the potential $\displaystyle f(\Phi_{1})$ admits Q-balls
composed of the field $\displaystyle\Phi_{1}$. In the thin-wall limit, these
Q-balls are spherically-symmetric, extended objects comprised of a large core
of homogeneous “Q-matter”, and a thin shell which interpolates between the
core and the vacuum of the theory. The VEV in the core is therefore the origin
of the VEV of $\displaystyle\Phi_{1}$ in the previous subsection. The physical
properties of the Q-ball – it’s mass and volume – are well-approximated by the
properties of the core in this limit. Inside the Q-ball, where the field is
homogeneous, $\displaystyle\Phi_{1}$ takes on the well-known functional form

$\Phi_{1}(t)=e^{iq_{1}\omega t}\bar{\phi}_{1},$ (3.23)

where $\displaystyle\bar{\phi}_{1}\in\mathbb{R}$. The rest mass and volume of
the Q-ball are then given by

$m_{Q}=Q_{1}\sqrt{\frac{f(\bar{\phi}_{1})}{q_{1}^{2}\phi_{1}^{2}}}\quad\mathrm{and}\quad
V=\frac{Q_{1}}{2\sqrt{q_{1}^{2}\bar{\phi}_{1}^{2}f(\bar{\phi}_{1})}},$ (3.24)

where $\displaystyle\bar{\phi}_{1}$ must satisfy

$\frac{\partial
f(\bar{\phi}_{1})}{\partial\bar{\phi}_{1}}=\frac{2f(\bar{\phi}_{1})}{\bar{\phi}_{1}},$
(3.25)

such that $\displaystyle E/Q_{1}$ must be less than the mass per unit charge
of individual quanta of the $\displaystyle\Phi_{1}$ field. It is in this way
that these Q-balls are said to be classically stable.

We assume that, if anything, $\displaystyle h(\Phi_{1},\Phi_{2})$ provides, at
most, a small perturbation to this Q-ball solution. For definiteness, we
assume that $\displaystyle g(\Phi_{2})$ does not admit Q-ball solutions in
isolation from $\displaystyle\Phi_{1}$, and we further assume, for simplicity,
that $\displaystyle U(\Phi_{1},\Phi_{2})$ does not admit Q-balls charged under
both symmetries when all terms are considered.

Now, consider a configuration where a thin-wall Q-ball composed of
$\displaystyle\Phi_{1}$ is surrounded by quanta of the field
$\displaystyle\Phi_{2}$. The potential $\displaystyle h(\Phi_{1},\Phi_{2})$
defines how these entities interact. Let us consider an attractive potential
such that the quanta of $\displaystyle\Phi_{2}$ become bound within the
Q-ball. We will only consider a small amount entering the Q-ball such that the
value of $\displaystyle\bar{\phi}_{1}$ is still consistent with that defined
above. Though the core of the Q-ball is, in principle, homogeneous, the
presence of the new quanta slightly breaks this homogeneity, and we expect
these to settle in the core of the Q-ball as this is the symmetric point of
the Q-ball.555We could also consider the case that many thick-wall Q-balls
form within the homogeneous background of the thin-wall Q-ball. However, we do
not consider this “speckled” Q-ball here.

If enough quanta are collected, it is reasonable to ask whether a Q-ball
composed of $\displaystyle\Phi_{2}$, in the background of
$\displaystyle\bar{\phi}_{1}$, can form. Moreover, if this Q-ball does not get
too large, such that the terms in $\displaystyle h(\Phi_{1},\Phi_{2})$ greatly
perturb the Q-ball background, it should not greatly affect the value of
$\displaystyle\bar{\phi}_{1}$. We will assume that the core is a thick-wall
Q-ball, as this is more consistent with our assumption that the new Q-ball
does not overly affect the value of $\displaystyle\bar{\phi}_{1}$, since the
VEV is smaller. We can take the results directly from the previous section.

We require that the whole system be energetically stable against classical
decay to quanta of both fields. The total mass of the cored Q-ball is

$E_{\mathrm{tot}}=\frac{Q_{1}}{q_{1}}\sqrt{\frac{f(\bar{\phi}_{1})}{\bar{\phi}_{1}^{2}}}+\frac{Q_{2}}{q_{2}}\mu_{\mathrm{eff}}\left(1-\frac{1}{6}\epsilon^{2}\right).$
(3.26)

For this to be classically stable against decay to the quanta of both of the
fields, we require that

$E_{\mathrm{tot}}<\frac{Q_{1}}{q_{1}}m+\frac{Q_{2}}{q_{2}}\mu,$ (3.27)

where $\displaystyle m$ is the mass of quanta of the field
$\displaystyle\Phi_{1}$ and, as before, $\displaystyle\mu$ is the mass of the
quanta of the field $\displaystyle\Phi_{2}$ in the vacuum of the theory,
$\displaystyle\Phi_{1}=0$. Note that, the first term of the total energy is
less than the first term of this constraint. If
$\displaystyle\mu_{\mathrm{eff}}<\mu$, then this constraint is satisfied. If
this is not true, then a sufficient condition is for666The necessary condition
is that the the binding energy of the thin-wall Q-ball is greater than any
energy cost that the thick-wall Q-ball expanded in the background of
$\displaystyle\bar{\phi}_{1}$ brings over the vacuum of the theory,
$\frac{Q_{1}}{q_{1}}\left(m-\sqrt{\frac{f(\bar{\phi}_{1})}{\bar{\phi}_{1}^{2}}}\right)>\frac{Q_{2}}{q_{2}}\left[\mu_{\mathrm{eff}}\left(1-\frac{1}{6}\epsilon^{2}\right)-\mu\right].$
(3.28) This is a model-dependent condition, so we say nothing more of this
here.

$\mu_{\mathrm{eff}}\left(1-\frac{1}{6}\epsilon^{2}\right)<\mu.$ (3.29)

This constraint, in terms of the charge of the thick-wall Q-ball, evaluates to

$\frac{Q_{2}}{q_{2}}>201.6\sqrt{h_{2}(\bar{\phi}_{1})}\frac{\mu_{\mathrm{eff}}}{A_{\mathrm{eff}}^{2}}.$
(3.30)

This implies that, for this structure to be classically stable, there is a
minimum charge that must accumulate in the thick-wall core. For this to be
compatible with Eq. (3.22), we require that

$11h_{2}<\mu.$ (3.31)

We thus see that we require that the coupling between the two fields not be
too large. If this is satisfied, then it is sufficient to prove that this
structure is classically stable over some range of charge. The total system is
therefore adequately described as a thin-wall Q-ball, stabilised by the
conservation of some charge, with a core composed of a thick-wall Q-ball,
stabilised by the conservation of some other charge, over the background field
of the thin-wall Q-ball. This is because the field $\displaystyle\phi_{2}$
will remain “small” and thus will not overly affect the background field,
$\displaystyle\bar{\phi}_{1}$.

By definition, a Q-ball is the state that minimises the energy per unit charge
of a sector of a theory. Thus, the combination of energy and Noether charge
conservation implies that this configuration is more stable than emitting the
individual quanta of $\displaystyle\Phi_{2}$ outside the Q-ball formed from
$\displaystyle\Phi_{1}$. One could also question whether a Q-ball composed of
$\displaystyle\Phi_{1}$ and $\displaystyle\Phi_{2}$ could be emitted as a
decay channel. However, the ability for the potential defined through
$\displaystyle g(\Phi_{2})$ and $\displaystyle h(\Phi_{1},\Phi_{2})$ in the
background of $\displaystyle\bar{\phi}_{1}$ to house stable Q-ball solutions
does not imply that the full potential $\displaystyle U(\Phi_{1},\Phi_{2})$
will allow for stable “doubly-charged” Q-balls to exist in isolation. In the
absence of this decay mode, the cored Q-ball is indeed stable.

We also require that the Q-ball in the core be smaller than the thin-wall
Q-ball it resides in, as otherwise we cannot assume the results of the
previous section. From the sizes of the respective Q-balls given in Eqs.
(3.21) and (3.24), together with the definition for $\displaystyle\epsilon$
given in Eq. (3.17), we find that

$36\pi(S_{\varphi})^{3}\frac{q_{2}^{3}}{Q_{2}^{3}}\frac{\mu_{\mathrm{eff}}^{3}}{A_{\mathrm{eff}}^{6}}\ll\frac{Q_{1}}{2\sqrt{q_{1}^{2}\bar{\phi}_{1}^{2}f(\bar{\phi}_{1})}},$
(3.32)

which is a model-dependent condition which relates the charge accumulation of
both Q-balls.

##  4 Summary

In this work, we studied a subclass of multi-field and multi-symmetry theories
for Q-ball solutions. Specifically, we considered a theory of $\displaystyle
N$ complex scalar fields, each charged under their own $\displaystyle U(1)$
symmetry. Thus, the stabilising symmetry is an $\displaystyle N$-fold
$\displaystyle U(1)$ symmetry. In this scheme, we studied both the thin- and
thick-wall limits, using the standard multi-field approximation of similar
spatial profiles in order to progress analytically.

In the thin-wall case, we determined the physical properties of the Q-ball,
together with the differential equation that governs the VEV of the fields
inside the Q-ball. In sharp contrast to all previous studies, we have found
that the VEVs depend upon the charge of the Q-ball itself. The differential
equation must then be solved for each charge configuration, independently. It
would be interesting to solve this in future work in, say, the “doubly-
charged” case.

In the thick-wall case, we demanded that the minimum of energy be classically
stable against decay into constituent quanta of the fields. We thus gave
sufficient constraints on the theories that possess a thick-wall limit. The
general set up of the theory was such that we had, in some sense,
$\displaystyle N$ copies of the single-field case, and so it comes to no
surprise that the restriction of theories is the same as in Refs.
PaccettiCorreia:2001wtt ; Postma:2001ea ; Lennon:2021uqu , namely, that
$\displaystyle 2<p<10/3$, where $\displaystyle p$ denotes the index of the
next-to-quadratic term in the bounce potential.

In the latter half of this work, we heuristically studied a new type of object
that we called a “cored Q-ball”. This is a thin-wall Q-ball whose homogeneous
interior acts as a stabilising background VEV for a thick-wall Q-ball composed
of another field. In this work, we merely concerned ourselves with the
question of stability of these objects. Realistic theories that lead to the
formation of these objects is a direction for future work. Specifically, a
phenomenological analysis is required in which all timescales are taken into
account, together with a mechanism for the release of the additional energy.
Moreover, one could also extend the ideas here to the case where a stable,
“doubly-charged” Q-ball can exist. This would open up the decay channel
whereby a mixed Q-ball is emitted, leaving behind a pure thin-wall Q-ball.

Intriguingly, these objects could be important in DM experiments. Consider the
case that the Q-ball at the core is charged under the SM, whereas the larger
thin-wall Q-ball is not directly coupled to the SM. If this object were to
pass through a direct detection experiment, we would only see the core, and
not the Q-ball at large. This would cause a misidentification of the DM until
experiments are sensitive enough to determine the other component, which could
indirectly couple to the SM via the cored field.

## Acknowledgments

OL would like to thank John March-Russell for useful comments on this work,
which was completed while OL was supported by the Colleges of St John and St
Catherine, Oxford. Conversations with Fady Bishara on Q-ball physics over the
years have also been particularly fruitful.

##  Appendix A Thin-Wall Q-balls in the Background of Another Field

For completeness, we now include the thin-wall analysis. A thin-wall Q-ball’s
physical properties are well-described by those of its homogeneous core. We
thus write

$\mathcal{E}_{\omega}\approx\bar{\omega}_{2}Q_{2}+V\left[g(\bar{\phi}_{2})+h(\bar{\phi}_{1},\bar{\phi}_{2})-\bar{\omega}_{2}^{2}q_{2}^{2}\bar{\phi}_{2}^{2}\right],$
(A.1)

where $\displaystyle V$ is the volume of the core, and we use bars to
represent values of variables within the core. Minimisation of this with
respect to the Lagrange multiplier yields, as expected, the expression for the
charge of this configuration,

$Q_{2}=2\bar{\omega}_{2}q_{2}^{2}\bar{\phi}_{2}^{2}V.$ (A.2)

Eliminating $\displaystyle\bar{\omega}_{2}$ and minimising with respect to the
volume leads to

$V^{2}=\frac{Q_{2}^{2}}{4q_{2}^{2}\bar{\phi}_{2}^{2}}\frac{1}{\left[g(\bar{\phi}_{2})+h(\bar{\phi}_{1},\bar{\phi}_{2})\right]}.$
(A.3)

Finally, upon elimination of the volume, we obtain the rest mass of the
configuration

$m_{Q}=Q_{2}\sqrt{\frac{g(\bar{\phi}_{2})+h(\bar{\phi}_{1},\bar{\phi}_{2})}{q_{2}^{2}\bar{\phi}_{2}^{2}}},$
(A.4)

which is completely expected from the single-field theory, since
$\displaystyle
U(\bar{\phi}_{2})=g(\bar{\phi}_{2})+h(\bar{\phi}_{1},\bar{\phi}_{2})$. For the
resultant Q-ball to be stable in the background of the VEV of
$\displaystyle\Phi_{1}$, we must have that

$m_{Q}<\mu_{\mathrm{eff}}\frac{Q_{2}}{q_{2}},$ (A.5)

where $\displaystyle\mu_{\mathrm{eff}}(\bar{\phi}_{1})$ is defined as in the
previous section. The VEV of $\displaystyle\bar{\phi}_{2}$ must obey the
differential equation given by

$\frac{\partial g}{\partial\bar{\phi_{2}}}+\frac{\partial
h}{\partial\bar{\phi_{2}}}=2\left(\frac{g}{\bar{\phi_{2}}}+\frac{h}{\bar{\phi_{2}}}\right).$
(A.6)

This concludes the thin-wall analysis of this theory.

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