# The utilization of spherical camera in simulation for service robotics
††thanks: This work was funded within the INCARE AAL-2017-059 project
,,Integrated Solution for Innovative Elderly Care” by the AAL JP and co-funded
by the AAL JP countries (National Centre for Research and Development, Poland
under Grant AAL2/2/INCARE/2018).

Krystian Chachuła  Warsaw University of Technology
Institute of Control and Computation Engineering
Warsaw, Poland,
Email<EMAIL_ADDRESS>Maciej Stefańczyk  Warsaw University
of Technology
Institute of Control and Computation Engineering
Warsaw, Poland,
Email<EMAIL_ADDRESS>
###### Abstract

Safety is one of the most critical factors in robotics, especially when robots
have to collaborate with people in a shared environment. Testing the physical
systems, however, must focus on much more than just software. One of the
common steps in robotic system development is the utilization of simulators,
which are very good for tasks like navigation or manipulation. Testing vision
systems is more challenging, as the simulated data often is far from the real
camera readings. In this paper, we show the advantages of using the spherical
camera for recording the sequences of test images and a way to integrate those
with existing robotic simulator. The presented system also has the possibility
to be extended with rendered objects to further improve its usability.

###### Index Terms:

spherical camera, service robotics, robot simulation, ROS

##  I Introduction

###  I-A The problem

People use their sight as the most important sense when they move through
their surroundings. It is therefore essential for service robots to use image
processing. When robots coexist with humans, safety is highly important [1].
One of many measures to be taken is software testing, which requires test data
and, in the domain of computer vision, this data consists of images or videos
from a robot’s camera [2]. However, using a real robot for testing can be
dangerous. The danger stems from the fact that, while testing, the robot is
controlled by yet untested software, which, in case of service robots, can be
complicated [3]. This can lead to damaging the environment, the robot itself
or in the worst case scenario can harm the people. Using a real robot for
testing can also be tedious – after each test iteration one has to return the
environment to the initial state and move robot back to the staring pose. To
speed up the process it is possible to run the application on multiple robots
[4], but this is not the cost-effective solution. This creates the need to
simulate the tested robot [5] and, specifically, its one or more cameras. In
this article, we examine the existing robotic simulators as a tool for testing
vision algorithms and propose their extension with the use of spherical
images.

###  I-B State of the art

A practical method of gathering image data for testing is recording a video
while the robot is executing a given path in the testing environment for
future use. Another way is using simulators such as Gazebo [6] or UnrealROX
[7]. Gazebo is meant to be a universal simulator, with physics and 3D
rendering engine. It allows for accurate robots and environment modeling and
robot behavior simulation. Its integration with the Robot Operating System
makes it very popular amongst the research and industry community. It lacks
the most in the vision field. Although the 3D rendering is possible,
environments created in the simulation are often filled with repetitive
objects with very simple textures (Fig. 1). The lighting system is also very
simplified, without any shadows and reflections, which makes the renders
unsuitable for the vision algorithms evaluation.

Figure 1: A real (left) and a simulated (right) camera in Gazebo

On the other hand, simulators like UnrealROX are designed to be used as a tool
for training visual algorithms, especially those based on deep learning,
requiring a lot of training data. Simulator, based on Unreal Engine, is able
to produce photo-realistic scenes in real-time (Fig. 2). To prepare such
scenes, however, an artist is required most of the time. A solution like this
would be perfect, but lack of integration (and no possibility thereof) with
robotics systems makes it a tool only for off-line learning, not interactive
testing.

Figure 2: Images generated by the UnrealROX simulator [7]

From this short review, we can point out some features that are desired in
simulators [8] apart from the photorealism. One is the capability to modify
the world model used for image generation quickly. This allows the tester to
create new test scenarios rapidly. Another essential feature is the ability to
edit the robot’s path dynamically. Of course, this is not possible when
simulating using a prerecorded video, but should be achievable in every
generic simulator, as well as the capability of simulating robot’s
interactions with the environment, such as collisions and manipulation.
Integration with ROS is also important. From the summary in Tab. I we can see
that fulfilling all the requirements at once is hard to achieve.

TABLE I: Comparison of different camera simulation methods Feature | Gazebo | video | UnrealROX
---|---|---|---
environment modification | yes | no | yes
editable robot path | yes | no | yes
interaction with the environment | yes | no | yes
photorealism | no | yes | yes
ROS integration | yes | yes | no

###  I-C The solution

Using pre-recorded video/images has the biggest photorealism (as those are
photos). Using the 3D simulators gives possibility of free robot movement and
interaction with the environment. To combine both worlds, we study the
possibility of using the photos as a source for the camera image in the
simulation. To give the robot the possibility to look around, photos should
cover the full sphere (360° panorama or full spherical photo). To have freedom
of movement images from (almost) all possible robot locations should be
available. The position of the virtual camera can be used to select exactly
one (the closest) spherical image from the set. Finally, to not limit the
solution to only a single pre-recorded environment state, there has to be a
way of adding the simulated/rendered objects to the scenes. This will also
allow for interaction. Schematic diagram of the proposed system is presented
in Fig. 3.

Figure 3: Proposed simulation system structure

The rest of the paper is structured as follows. In section II, we show the
methods of creating a single photo from a spherical image. Section III
introduces a way of dealing with the robot movement. Integration with the 3D
simulator and merging with the rendered overlays are presented in section IV.
A case study for a prepared system is presented in section V, followed by the
conclusions.

There are different camera types used throughout the article. A spherical
camera is a type of camera with a field of view that covers the entire sphere,
and a regular camera is a type of camera with a field of view close to $90$
degrees. The camera that belongs to a robot in simulation is referred to as a
virtual camera. A projection camera is an entity that projects a 3D scene to a
2D image in computer graphics.

##  II Transforming spherical images

In order to transform a spherical image into a regular image, one can utilize
3D computer graphics. Not only is this method simple, but its implementation
can utilize hardware acceleration. A sphere is a 3D closed surface where every
point on the sphere is the same distance from its origin. Since only a finite
number of points can be drawn, a sphere will need to be approximated by
drawing a subset of its points. Those points can be selected by creating a UV
Sphere. A UV Sphere consists of vertices that are created by performing
angular steps in the spherical coordinate system (Fig. 4). Cartesian points
are generated using these equations:

$x=r\cdot\cos\phi\cdot\cos\theta\quad y=r\cdot\cos\phi\cdot\sin\theta\quad
z~{}=r\cdot\sin\phi$

where $r$ is the radius of the sphere, $\phi\in[0,\pi]$ – polar angle and
$\theta\in[0,2\pi)$ – azimuthal angle.

Figure 4: Sphere coordinates generation

Then the sphere has to be textured with the spherical image (Fig. 5a). Most
spherical cameras create images in the form of equirectangular projections,
and, without the loss of the generality of the solution, we assumed images in
that format are available. The process of texturing a UV sphere with an image
in the form of this projection is fairly straightforward [9]. Every point on
the sphere can be mapped to a point on the texture by simply assigning angles
to texture coordinates $x_{t}=\theta$ and $y_{t}=\phi$.

a)  b)

Figure 5: Sphere textured with equirectangular photo

Having textured the sphere, now a projection camera can be placed in its
center (Fig. 5b). Its parameters (field of view and resolution) are set to
match the robot’s virtual camera. The image from this camera is a regular
image as seen by the robot looking in a given direction.

##  III Interpolation

The fact that always the nearest spherical image is used for rendering poses a
significant issue. If two virtual camera positions are close to each other it
is possible, that both of them will use the same spherical image to create the
projected camera view. Moreover, if those cameras have the same orientation
relative to the global reference frame then exactly the same regular image
would be produced for both cameras.

Figure 6: Tthe method of interpolating camera views inside the single
spherical image

A simple solution to this problem would be to mimic the position of the
virtual camera relative to a region with the position of the projection
camera, thus creating the illusion of moving closer to certain objects as
visualized in Fig. 6. This solution is not perfect, as there will be no
parallax visible, but with the photos sampled sufficiently dense the effect is
satisfactory.

##  IV 3D objects overlay

So far, the robot can look around and move through the prepared environment.
Generated images can be further processed in order to make this solution more
versatile. As the position and orientation of virtual camera is known, and its
parameters are the same as the projection camera, objects generated using 3D
rendering software can be overlaid onto the photo. Fig. 7 depicts an example
of such a process.

In our solution, we decided to stick with the Gazebo simulator for this
purpose. In order to remove unwanted objects from the image, additional plugin
to the Gazebo simulator was prepared to hide the objects from rendering in a
virtual camera, keeping them visible for other sensors (like laser scanner or
sonars), thus keeping the obstacle avoidance still working. This provides
additional simulation features mentioned in the introduction. It enables the
tester to modify the testing environment and provides the means of interaction
of the simulated robot with the environment utilizing the Gazebo simulator.

Figure 7: Rendered object (can) overelaid on the photo

##  V Evaluation

The simulation system presented above was tested in our robotics laboratory.
At first, individual modules and system parameters were evaluated, followed by
the partial scan of the room. Prepared environment is meant to be used with
the TIAGo robot applied to service tasks related to Elderly care and
assistance.

Figure 8: Example images generated with grid resolutions of 20, 100 and 200
cm.

###  V-A Camera selection

In order to acquire a full spherical photo, a dedicated camera setup is
required. Nowadays, there are different possibilities to achieve that, without
the need for building multiple camera rig from scratch. If the price doesn’t
matter, Ladybug5+ from Flir is the option. Its precalibrated six cameras
provide 360° panorama with 8K resolution. Custom solutions use single camera
with specially shaped mirror [10] to achieve the omnidirectional field of
view. On the consumer side of the market, there are a few different devices to
choose from, most of them built from two fish-eye cameras facing in opposite
directions. We chose the Theta V camera from Ricoh. This camera is capable of
on-board image stitching to the equirectangular projection with final
resolution $5376\times 2688$ pixels (created from two 4K photos). Another
useful feature is the possibility to capture and transfer photos using WiFi.
The average angular resolution of the selected camera is around 15 px/°, which
is more than the 10 px/° of a robot’s camera.

###  V-B Rendering performance

All the advantages of even the most advanced and photorealistic simulation are
useless if its performance doesn’t allow for (near)real-time work. There are
two independent factors in photo rendering that can limit the simulation
performance. When the robot moves inside a single cell (in range of a single
spherical photo), only the rendering time is important. For our system, it
takes on average 0.365 ms to render a single VGA view. This allows us to
simulate the camera with 2740 FPS.

The other latency is a result of loading the spherical photo from disk,
decoding it, and loading it into the memory of the video card. A single 5376
$\times$ 2688 pixels JPEG image takes on average $t=198.672\,ms$ to load and
decode on our system. This further limits the FPS to 5 if the images are
loaded synchronously. However, if the images can be loaded in advance, this
would mitigate the negative impact of the load latency on the robot’s maximum
speed. In this case simulated robot’s maximum speed $v_{max}$ depends on the
amount of primary memory.

To achieve scaling $v_{max}$ with memory, one can utilize a pool of threads or
processes. Now a significant problem is choosing which spherical images should
be cached in RAM. The most straightforward strategy is to always store four
images that were taken closest to the current position of the robot. Such a
solution often loads many images that will not be used for simulation. A more
sophisticated method would be to predict the movement of the robot. For
example, the robot’s velocity vector could be projected forward until it
intersects the border of a cell. Then, this image would be decoded and loaded
in advance.

With just one thread or process for decoding images, the grid resolution of
$20\,cm$ and image load time of $200\,ms$, the theoretical aggregated speed
limit of the simulated robot is equal to $1\,\frac{m}{s}$. According to
TIAGo’s datasheet, this value coincides with real robot’s maximum speed.

###  V-C Grid resolution and interpolation parameters

Proper distance between consecutive photos is not easy to decide and depends
on the actual environment. Photos captured to sparse will result in big jumps
when the camera moves between different grid points. Too dense and image
loading and decoding times will limit the speed of the robot in simulation to
unacceptable values.

In order to make such a decision, we created a relatively small but dense
spherical photo set. After that, example images were generated for a given
position and orientation of the virtual camera using that set. Next, we made
the grid more sparse by selecting every second, third, and so forth photo from
the set and generated images for each step. Some of the generated images can
be seen in Fig. 8 (notice the vertical line distortions for too large step).
The grid resolution of 20 cm provided a good balance between the quality of
the simulation and the time needed for the creation of the photo set.

###  V-D Collecting the photos of the target environment

Finally, when all the parameters are selected, a set of spherical images can
be created. To facilitate the whole process, we’ve created helper hardware and
software. For accurate photo capture and straight camera movement, the device
in Fig. 9 was used. It consists of a rail and a cart to which a monopod has
been fixed. At the other end of the monopod, there is a mounting point for the
camera. A grid marks were put on the rail in order to allow precise
positioning of the cart. Similar marks were put on the floor in order to guide
the rail.

Figure 9: Hardware and software used for the creation of the image set

To simplify the process and keep track of photos that are already taken and
still waiting to be captured, an application was created (Fig. 9). It allows
for selecting points and sending requests to the camera for taking pictures
using the Open Spherical Camera API. It also automatically tags the images
with the correct position. In the selected room, 404 pictures were taken in
$20\,cm$ intervals, covering the living room part, the kitchen, and part of
the corridor. The whole process took 2 hours and 17 minutes, which yields
approximately 3 pictures taken per minute.

The whole solution was evaluated by comparing generated images with images
from a real robot. For this purpose, the TIAGo robot [11] was used. It was
booted up and manually controlled in order to take pictures. Then the proposed
solution was used to generate images corresponding to pictures taken by the
TIAGo. Finally, images from both sources were compared, as seen in Fig. 10.
The images differ mainly in color appearance and are otherwise nearly
identical. The mentioned color issue can be resolved by the means of color
correction.

Figure 10: Images generated in simulation (first row) and taken by the robot
(second row). Last two rows presents sample source spherical photos.

##  VI Conclusions

The article presented the idea and application for the spherical camera-based
simulator for service robots. The system was tested by comparing images
generated in simulation with those gathered by the real robot. Those results
prove that using the proposed solution a set of spherical images can be used
to replace a real robot in software testing. One important feature of
presented system is that it doesn’t replace existing simulators, but rather
coexists with them and provides additional features. Presented system in its
current form can be utilized to test the robotic vision algorithms involving
visual inspection [12] or even objects grasping [13] when the 3D overlays are
used.

There are some issues in the system that can be improved. The process of photo
acquisition at the moment is highly manual labor. For each photo, one has to
move the camera, hide from the camera view, and capture a photo. In its
current form, it takes around 8 minutes per square meter. The natural
extension would be to utilize a robotic platform to move the camera around and
take photos automatically.

Manual photo capturing results also in slight misalignments between
consecutive frames. Photo stabilization techniques could be implemented to
calculate angular offsets for each image.

Created photo collection could be extended with depth information calculated
from multiview structure estimation. This would make the interpolation process
work better and allow for, possibly, grid with bigger steps.

## References

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

# Multi-Field Q-balls with Real Scalars

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

Multi-field Q-balls, in which some, but not all, of the constituent fields are
real scalars, are studied. Uncharged fields may classically contribute to
Q-balls provided that their effect is to not destabilise the resulting object.
The thin-wall limit is reviewed based on existing literature. Sufficient
conditions for the theories that can plausibly contain thick-wall solutions
are derived. Necessary conditions require a detailed numerical analysis,
outside the scope of this work. These conditions show that when the additional
real scalars are massive, a minimum charge must accumulate in order to render
the Q-ball energetically stable.

††preprint: OUTP-21-30P

##  1 Introduction

Q-balls Coleman:1985ki  are fascinating objects that can be found in theories
of complex scalar fields. They are an example of a non-topological soliton
(see Ref. Lee:1991ax  and references therein) that are kept stable by a
combination of conservation of energy and a Noether charge. Thus, Q-balls
represent the state of minimum energy for a given charge. Though the theory is
usually taken as that of a single complex scalar field symmetric under a group
of $\displaystyle U(1)$ transformations, the principles have been extended to
include multiple fields Kusenko:1997zq , more complicated symmetry groups
Safian:1987pr , as well as gauging the stabilising symmetry Lee:1988ag ;
Heeck:2021zvk .

Finding the extremum of the energy functional is not analytically tractable in
the most general case. However, two analytic limits exist, and represent the
most studied examples of Q-balls. Specifically, these are the thin- and thick-
wall limits Coleman:1985ki ; Kusenko:1997ad . These are appropriate to
describe Q-balls in the large and small charge limits, respectively. At least
in the single-field case, not every theory that possesses a thin-wall limit
has a corresponding thick-wall limit PaccettiCorreia:2001wtt ; Postma:2001ea ;
Lennon:2021uqu . Recently, it has been shown that the thin-wall limit can be
extended to describe Q-balls with smaller charges Heeck:2020bau .

Q-balls are not just interesting theoretical objects. Owing to their rich
phenomenology, they have been studied in the context of real-world scenarios.
In the absence of couplings to other light fields, Q-balls are absolutely
stable, and so have been considered as candidates for dark matter
Kusenko:1997si ; Kusenko:2001vu ; Graham:2015apa ; Ponton:2019hux . Moreover,
they have been shown to arise in supersymmetric theories Kusenko:1997si ;
Kusenko:1997zq , or in theories of extra dimensions Demir:2000gj ;
Abel:2015tca . At the time of writing, Q-balls have not been seen in any
experiment, but it is believed that their signature will be striking if
evidence for them is ever found Gelmini:2002ez ; Kusenko:1997vp ;
Croon:2019rqu ; White:2021hwi .

As mentioned above, the canonical Q-ball analysis has been extended to include
multiple fields. However, the single-field scenario is still the most studied
example. Given that extensions to the Standard Model often come with a
plethora of scalar fields with non-trivial potentials, the multi-field
scenario is relatively under-explored. In this paper, we consider the simplest
multi-field example: a theory of a single complex scalar field and an
arbitrary number of real scalar fields. The case of a non-topological soliton
composed of a single complex scalar field with one additional real scalar
field was first explored in Ref. Friedberg:1976me . It was noted that the real
scalar, despite not carrying charge, would figure in the Q-ball if it allowed
the resulting Q-ball to lower its energy either to one of greater stability,
or even to achieve stability in the first place.

In this work, we review this original analysis according to the more modern
approach as given in Ref. Kusenko:1997zq . Moreover, we also explore the
thick-wall limit of the theory. As discussed in Refs. PaccettiCorreia:2001wtt
; Postma:2001ea ; Lennon:2021uqu , not every theory with a thin-wall limit
possesses a thick-wall limit, which has ramifications for early universe
phenomenology in solitosynthesis scenarios Frieman:1989bx ; Griest:1989bq ;
Kusenko:1997hj . The added complexity introduced by considering multiple
fields makes it impossible to derive an exact constraint in the same way.
However, sufficient conditions can be found to restrict the theories to those
with an analytic limit. Necessary conditions require a numerical study, which
we leave to future work.

Specifically, in Section 2, we perform the minimisation procedure on the
energy functional appropriate for the multi-field case, as first outlined in
Ref. Kusenko:1997zq . In Section 3, we review the work of Ref.
Friedberg:1976me , which considered thin-wall Q-balls comprising one charged
scalar field with a single real scalar field, using the more modern approach
of Ref. Kusenko:1997zq . In this section, we determine the differential
equations that govern the vacuum expectation values (VEVs) of the complex and
real scalars within the resulting thin-wall Q-balls, together with their
physical properties of mass and volume. In Section 4, we analyse the thick-
wall limit in the approximation scheme that the fields all have the same
spatial profile up to a positive normalisation – this is a necessity to make
analytic progress. 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. Thus, any constraints found under this approximation
scheme are sufficient to show the existence of thick-wall Q-balls, but are not
strictly necessary. With this in mind, we show the existence of thick-wall
Q-balls in a class of theories by demanding that the minimum be energetically
stable against classical decay into the quanta of the charged scalar fields.
We fully solve a special case – one complex scalar with an arbitrary number of
massless (or very light) scalars. Work with this assumption on the spatial
profiles has been done in an ad-hoc manner in Refs. Postma:2001ea ;
Bishara:2017otb , but we seek to generalise these works here.

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

We consider a theory of a single complex scalar,
$\displaystyle\Phi(\vec{x},t)$, and $\displaystyle M$ real scalars,
$\displaystyle\Psi_{j}(\vec{x},t)$. The Lagrangian density we consider is

$\mathcal{L}=\partial_{\mu}\Phi\partial^{\mu}\Phi^{*}+\frac{1}{2}\sum^{M}_{j}\partial_{\mu}\Psi_{j}\partial^{\mu}\Psi_{j}-U(\Phi,\Phi^{*},\Psi_{j}),$
(2.1)

where $\displaystyle U(\Phi,\Phi^{*},\Psi_{j})$ is some potential that we
leave generic for now, stipulating only that it be a function of the fields,
and not their derivatives, and that it vanish for vanishing field. The Euler-
Lagrange equations for this theory are

$\partial_{\mu}\partial^{\mu}\Phi+\frac{\partial
U}{\partial\Phi^{*}}=0\quad\mathrm{and}\quad\partial_{\mu}\partial^{\mu}\Psi_{j}+\frac{\partial
U}{\partial\Psi_{j}}=0,$ (2.2)

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

We demand that the theory be invariant with respect to a single global
$\displaystyle U(1)$ group of transformations defined through

$\Phi\to e^{i\alpha}\Phi\quad\mathrm{and}\quad\Psi_{j}\to\Psi_{j},$ (2.3)

where $\displaystyle\alpha\in\mathbb{R}$ and the charge of the complex scalar
is set to unity, with the real scalars uncharged. Associated to this symmetry
is a Noether current density given by

$j^{\mu}=i\left(\Phi\partial^{\mu}\Phi^{*}-\Phi^{*}\partial^{\mu}\Phi\right).$
(2.4)

This symmetry places a functional constraint on the potential, namely, that it
be a function of the absolute value of the complex field.

A Q-ball is the state in a theory which minimises the energy for a fixed, non-
zero Noether charge. The Hamiltonian for this theory is

$H=\int\mathrm{d}^{3}x\left[\dot{\Phi}\dot{\Phi}^{*}+\vec{\nabla}\Phi\cdot\vec{\nabla}\Phi^{*}+\frac{1}{2}\sum^{M}_{j}(\dot{\Psi}_{j}\dot{\Psi}_{j}+\vec{\nabla}\Psi_{j}\cdot\vec{\nabla}\Psi_{j})+U(\Phi,\Phi^{*},\Psi_{j})\right].$
(2.5)

To determine if this system admits Q-ball solutions, we introduce a Lagrange
multiplier $\displaystyle\omega$, as in Ref. Kusenko:1997zq , that enforces
charge conservation upon minimisation with respect to it:

$\mathcal{E}_{\omega}=H+\omega\left(Q-\int\mathrm{d}^{3}x\,j^{0}\right),$
(2.6)

where $\displaystyle j^{0}$ is the zeroth component of the Noether current
density given in Eq. (2.4). Thus, the functional we wish to analyse is given
by

$\begin{split}\mathcal{E}_{\omega}=\omega
Q+\int\mathrm{d}^{3}x&\left[\dot{\Phi}\dot{\Phi}^{*}-i\omega\dot{\Phi}^{*}\Phi+i\omega\Phi^{*}\dot{\Phi}+\vec{\nabla}\Phi\cdot\vec{\nabla}\Phi^{*}\right.\\\
&\left.+\frac{1}{2}\sum^{M}_{j}(\dot{\Psi}_{j}\dot{\Psi}_{j}+\vec{\nabla}\Psi_{j}\cdot\vec{\nabla}\Psi_{j})+U(\Phi,\Phi^{*},\Psi_{j})\right].\end{split}$
(2.7)

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

$\begin{split}\mathcal{E}_{\omega}=\omega
Q+\int\mathrm{d}^{3}x&\left[\left|\dot{\Phi}-i\omega\Phi\right|^{2}-\omega^{2}\Phi^{*}\Phi+\vec{\nabla}\Phi\cdot\vec{\nabla}\Phi^{*}\right.\\\
&\left.+\frac{1}{2}\sum^{M}_{j}(\dot{\Psi}_{j}\dot{\Psi}_{j}+\vec{\nabla}\Psi_{j}\cdot\vec{\nabla}\Psi_{j})+U(\Phi,\Phi^{*},\Psi_{j})\right].\end{split}$
(2.8)

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

$\Phi(\vec{x},t)=e^{i\omega
t}\phi(\vec{x})\quad\mathrm{and}\quad\Psi_{j}(\vec{x},t)=\psi_{j}(\vec{x}),$
(2.9)

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

$\mathcal{E}_{\omega}=\omega
Q+\int\mathrm{d}^{3}x\left[\vec{\nabla}\phi\cdot\vec{\nabla}\phi-\omega^{2}\phi^{2}+\frac{1}{2}\sum^{M}_{j}(\vec{\nabla}\psi_{j}\cdot\vec{\nabla}\psi_{j})+U(\phi,\psi_{j})\right].$
(2.10)

Notice that a spatial profile for the field $\displaystyle\phi$ that vanishes
everywhere leads to a configuration of zero charge. Thus, a configuration of
non-zero charge must have a spatial profile for the charged field that differs
from zero in some finite domain – this is not the case for the uncharged
fields, which can be vanishing inside a multi-field Q-ball. The spatial
profiles satisfy the equations

$\nabla^{2}\phi=\frac{1}{2}\frac{\partial}{\partial\phi}\left(U(\phi,\psi_{j})-\omega^{2}\phi^{2}\right),\quad\nabla^{2}\psi_{j}=\frac{\partial
U(\phi,\psi_{j})}{\partial\psi_{j}}.$ (2.11)

These differential equations take the form of a bounce equation Coleman:1977py
; Callan:1977pt ; Coleman:1977th , with bounce potentials defined by
$\displaystyle U(\phi,\psi_{j})-\omega^{2}\phi^{2}$ and $\displaystyle
U(\phi,\psi_{j})$, respectively. 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 highly-coupled differential equations cannot in general be solved
analytically. However, under certain assumptions, analytic progress can be
made. Namely, for stable Q-ball solutions to be found, we must have that
$\displaystyle\omega_{0}\leq\omega<m_{\phi}$, where $\displaystyle m_{\phi}$
is the mass of the complex scalar Kusenko:1997ad . The thin-wall limit
corresponds to the limit that $\displaystyle\omega=\omega_{0}$ – in fact, the
thin-wall limit defines $\displaystyle\omega_{0}$. The thick-wall limit
corresponds to $\displaystyle\omega\to m_{\phi}^{-}$.

##  3 Thin-Wall Q-balls

The thin-wall limit for multi-field Q-balls was first considered in Ref.
Friedberg:1976me  for the case of one complex scalar and one real scalar.111It
should be noted that in Ref. Kusenko:1997zq , the case of multiple charged
scalars – and no real scalars – was analysed for thin-wall Q-balls. For
completeness, we include the analysis for multiple real scalars here. We note,
however, that the physics is largely unchanged in going to this case.

In the thin-wall limit, the properties of the Q-ball are well-approximated by
the properties of a spherical core of homogeneous Q-matter. We have that

$\mathcal{E}_{\omega}\approx\omega
Q+V\left[U(\phi,\psi_{j})-\omega^{2}\phi^{2}\right],$ (3.1)

where $\displaystyle V$ is the volume of the spherical core of the Q-ball.
Minimisation with respect to the Lagrange multiplier yields

$Q=2\omega V\phi^{2},$ (3.2)

as we would expect from the Noether current density given in Eq. (2.4). This
expressions defines $\displaystyle\omega_{0}$, as stated above. Eliminating
$\displaystyle\omega$ from $\displaystyle\mathcal{E}_{\omega}$ yields

$E=\frac{Q^{2}}{4V\phi^{2}}+U(\phi,\psi_{j})V.$ (3.3)

Minimising with respect to the volume gives us

$V^{2}=\frac{Q^{2}}{4U(\phi,\psi_{j})\phi^{2}},$ (3.4)

which, when eliminated, finally gives us an expression for the rest mass of a
multi-field Q-ball:

$m_{Q}=Q\sqrt{\frac{U(\phi,\psi_{j})}{\phi^{2}}}.$ (3.5)

This expression must be minimised such that the resulting Q-ball is
classically stable against decay to the quanta of any of the fields
$\displaystyle\phi$, i.e., that

$m_{Q}<Qm_{\phi}$ (3.6)

where $\displaystyle m_{\phi}$ is the mass of the quanta of the field
$\displaystyle\phi$. The minimisation procedure with respect to the field
content leads to the conditions on the potentials that

$\frac{\partial
U(\phi,\psi_{j})}{\partial\phi}=2\frac{U(\phi,\psi_{j})}{\phi}\quad\mathrm{and}\quad\frac{\partial
U(\phi,\psi_{j})}{\partial\psi_{k}}=0.$ (3.7)

A straightforward example of a potential that allows for multi-field Q-balls
is the following:

$U(\Phi^{*},\Phi,\Psi)=m_{\phi}^{2}\Phi^{*}\Phi-A\Psi\Phi^{*}\Phi+\lambda(\Phi^{*}\Phi)^{2}+\lambda^{\prime}\Psi^{4},$
(3.8)

where $\displaystyle\Phi$ is a massive, charged field (with charge set to
unity) and $\displaystyle\Psi$ is a massless, uncharged field. The
coefficients, $\displaystyle m_{\phi}^{2}$, $\displaystyle A$,
$\displaystyle\lambda$, $\displaystyle\lambda^{\prime}$, are all taken
positive, such that the sign assignment of each term is explicit. The Q-ball
ansatz, given in Eq. (2.9), means that we can rewrite this as

$U(\phi,\psi)=m_{\phi}^{2}\phi^{2}-A\psi\phi^{2}+\lambda\phi^{4}+\lambda^{\prime}\psi^{4}.$
(3.9)

The thin-wall conditions above yield the field values

$\phi=\frac{1}{4}\frac{A}{(\lambda^{3}\lambda^{\prime})^{1/4}}\quad\mathrm{and}\quad\psi=\frac{1}{4}\frac{A}{(\lambda\lambda^{\prime})^{1/2}}.$
(3.10)

The resulting Q-ball has a mass given by

$m_{Q}=Q\sqrt{m_{\phi}^{2}-\frac{1}{8}\frac{A^{2}}{(\lambda\lambda^{\prime})^{1/2}}},$
(3.11)

which is clearly smaller than $\displaystyle Q$ amounts of the mass of the
charged scalar, $\displaystyle m_{\phi}$. Thus, this theory allows for stable,
multi-field Q-balls. Notice, there is special point where

$\frac{1}{8}\frac{A^{2}}{(\lambda\lambda^{\prime})^{1/2}}=m_{\phi}^{2}.$
(3.12)

In this case, the core of the thin-wall Q-ball does not contribute to the
overall mass of the object – it will all come from the wall. This was studied
in the single-field case in Ref. Spector:1987ag

##  4 Thick-Wall Q-balls

We now analyse multi-field Q-balls in the thick-wall limit. In order to
analytically progress further, we must make an assumption about the spatial
profile of the fields. This removal of a degree of freedom of the system leads
to an upper bound on the mass of any stable Q-balls found, as additional
freedom can only reduce the rest energy further. Thus, armed with this, we can
derive sufficient conditions on the existence of thick-wall Q-balls on the
class of theory of study. A full analysis for the necessary conditions would
require a numerical analysis, which is outside the scope of this paper, and so
we leave this to future work.

###  4.1 The Spatial Profile Problem

As noted in Ref. Kusenko:1997ad , the thick-wall limit is equivalent to the
limit of small field values inside the Q-ball. Though this limit offers up the
simplification that only the next-to-quadratic order in the potential is
relevant, this is not enough to make analytic progress. The complication is
purely due to the number of fields.

In Refs. Postma:2001ea ; Bishara:2017otb , this problem was circumvented by
assuming that all spatial profiles of the fields were the same up to a
positive semi-definite normalisation constant for each field, with the
resulting expressions minimised with respect to these normalisation
constants.222In some sense, this is borrowing from the thin-wall analysis,
whereby the field profiles are ignored apart from their homogeneous core
values which differ only by a positive normalisation constant. As stated
above, the purpose of this ansatz is to provide sufficient conditions for the
existence of thick-wall Q-balls in the class of multi-field theories of study,
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 a thick-wall limit, we must pursue a
dedicated numerical analysis of the true spatial profiles of all individual
fields, and this is the topic of future work. This would also allow us to see
how accurate the analytic Q-ball properties are. Moreover, given that we have
effectively reduced our multi-field system into one of a single-field, we
expect to find results similar to those in Refs. PaccettiCorreia:2001wtt ;
Postma:2001ea ; Lennon:2021uqu .

To be explicit, let the reference field be the charged scalar,
$\displaystyle\phi$. As such, we define a set of positive semi-definite
constants, $\displaystyle\\{\beta_{j}\\}$, such that

$\psi_{j}=\beta_{j}\phi.$ (4.1)

The equations governing the spatial profiles of the scalars are now given by

$\nabla^{2}\phi=\frac{1}{2}\frac{\partial
U}{\partial\phi}-\omega^{2}\phi\quad\mathrm{and}\quad\beta_{j}\nabla^{2}\phi=\frac{\partial
U}{\partial\psi_{j}}.$ (4.2)

Whether or not these equations are satisfied in general, and what it means for
the properties of the Q-ball calculated, is the subject of future work.
Nevertheless, we progress.

###  4.2 The Thick-Wall Analysis

The bounce potential in the thick-wall limit can now be written as

$U_{\omega}(\phi,\beta_{j})\approx\left[\left(m_{\phi}^{2}-\omega^{2}\right)+\frac{1}{2}\sum_{j}^{M}m_{j}^{2}\beta_{j}^{2}\right]\phi^{2}-g(\beta_{j})\phi^{p},$
(4.3)

where $\displaystyle m_{j}$ is the mass of the $\displaystyle j$-th species of
real scalar, and we take the next to quadratic term to be of order
$\displaystyle p>2$. The coefficient of the quadratic term is clearly positive
for $\displaystyle\omega<m_{\phi}$. The function $\displaystyle g(\beta_{j})$
is model-dependent and so is left general for now, but it is assumed that it
is positive definite, as otherwise there would be no Q-ball solution. The
energy in Eq. (2.10) is then

$\begin{split}\mathcal{E}_{\omega}=\omega
Q+\int\mathrm{d}^{3}x&\left[\left(1+\frac{1}{2}\sum^{M}_{j}\beta_{j}^{2}\right)\vec{\nabla}\phi\cdot\vec{\nabla}\phi\right.\\\
&\left.+\left[\left(m_{\phi}^{2}-\omega^{2}\right)+\frac{1}{2}\sum_{j}^{M}m_{j}^{2}\beta_{j}^{2}\right]\phi^{2}-g(\beta_{j})\phi^{p}\right].\end{split}$
(4.4)

Consider the substitutions

$\begin{split}\xi_{i}&=\left(\dfrac{\left(m_{\phi}^{2}-\omega^{2}\right)+\dfrac{1}{2}\sum_{j}^{M}m_{j}^{2}\beta_{j}^{2}}{1+\dfrac{1}{2}\sum^{M}_{j}\beta_{j}^{2}}\right)^{1/2}x_{i},\\\
&\\\
\varphi&=\left(\frac{g(\beta_{j})}{\left(m_{\phi}^{2}-\omega^{2}\right)+\dfrac{1}{2}\sum_{j}^{M}m_{j}^{2}\beta_{j}^{2}}\right)^{1/(p-2)}\,\phi.\end{split}$
(4.5)

Then,

$\mathcal{E}_{\omega}=\omega
Q+\dfrac{\left(1+\dfrac{1}{2}\sum^{M}_{j}\beta_{j}^{2}\right)^{3/2}\left[\left(m_{\phi}^{2}-\omega^{2}\right)+\dfrac{1}{2}\sum_{j}^{M}m_{j}^{2}\beta_{j}^{2}\right]^{(6-p)/(2p-4)}}{g(\beta_{j})^{2/(p-2)}}S_{\varphi},$
(4.6)

where $\displaystyle S_{\varphi}$ is a dimensionless integral given by

$S_{\varphi}=\int\mathrm{d}^{3}\xi\left[\vec{\nabla}_{\xi}\varphi\cdot\vec{\nabla}_{\xi}\varphi+\varphi^{2}-\varphi^{p}\right].$
(4.7)

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 $\displaystyle p$. Our expression for
the energy must be minimised with respect to $\displaystyle\beta_{j}$ and
$\displaystyle\omega$. To proceed further, we must specify the theory and
therefore $\displaystyle g(\beta_{j})$. However, we can restrict the types of
theories that could plausibly lead to stable thick-wall Q-balls, and thus
derive sufficient conditions for stable Q-balls to exist in the wider context
of our assumption of the spatial profiles of the fields.

Consider the first derivative with respect to $\displaystyle\omega$. Requiring
that this vanishes means that $\displaystyle p<6$ and leads to the condition

$\epsilon=\Omega\left[\left(1-\Omega^{2}\right)+\frac{1}{2}\sum_{j}^{M}\beta_{j}^{2}\mu_{j}^{2}\right]^{(10-3p)/(2p-4)},$
(4.8)

where

$\epsilon\equiv\frac{Q}{S_{\varphi}}\left[\frac{p-2}{6-p}\right]\dfrac{g(\beta_{j})^{2/(p-2)}m_{\phi}^{(2p-8)/(p-2)}}{\left[1+\dfrac{1}{2}\sum_{j}^{M}\beta_{j}^{2}\right]^{3/2}},$
(4.9)

and we define the dimensionless parameters,

$\mu_{j}\equiv\frac{m_{j}}{m_{\phi}}\quad\mathrm{and}\quad\Omega\equiv\frac{\omega}{m_{\phi}}.$
(4.10)

Note, $\displaystyle 0<\Omega<1$, as $\displaystyle\omega<m_{\phi}$, by
definition. From this condition, we can rewrite
$\displaystyle\mathcal{E}_{\omega}$ as

$\mathcal{E}_{\omega}=Qm_{\phi}\left[\Omega+\frac{1}{\Omega}\left(\frac{p-2}{6-p}\right)\left(\kappa-\Omega^{2}\right)\right],$
(4.11)

where

$\kappa=1+\frac{1}{2}\sum_{j}^{M}\beta_{j}^{2}\mu_{j}^{2}.$ (4.12)

Note, $\displaystyle\kappa\geq 1$ and is related to the mass of the
configuration.333Technically, it is related to the mass to charge ratio of the
configuration, but we have taken the charge of the complex scalar to be unity,
and so this factor is not explicit. Notice, $\displaystyle\kappa=1$ if and
only if $\displaystyle\beta_{j}=0$ or $\displaystyle\mu_{j}=0$ for each
$\displaystyle j$. A subset of these requirements is the single-field case,
and so it is entirely reasonable to expect that $\displaystyle\kappa=1$ to
have the same condition as for the single-field case. We show that this is
indeed the case below. The regime in which $\displaystyle\kappa>1$ is distinct
from the case that $\displaystyle\kappa=1$, as we see below.

For stable Q-balls to form, we require that
$\displaystyle\mathcal{E}_{\omega}<Qm_{\phi}$. Thus,

$\Omega+\frac{1}{\Omega}\left(\frac{p-2}{6-p}\right)\left(\kappa-\Omega^{2}\right)<1,$
(4.13)

which can be rewritten as

$\frac{2(4-p)}{6-p}\Omega^{2}-\Omega+\frac{p-2}{6-p}\kappa<0.$ (4.14)

Notice, if $\displaystyle\Omega\to 0$, this condition is violated for
$\displaystyle 2<p<6$ and $\displaystyle\kappa>0$. However, since
$\displaystyle\omega_{0}\leq\omega<m_{\phi}$, this is acceptable. Notice, if
$\displaystyle\kappa=1$ this reduces to

$(\Omega-1)\left(\frac{2(4-p)}{6-p}\Omega-\frac{p-2}{6-p}\right)<0,$ (4.15)

which is precisely the condition for single-field Q-balls as found in Ref.
Lennon:2021uqu , i.e., that $\displaystyle 2<p<10/3$. As stated above, the
single-field case is an example of a theory for which $\displaystyle\kappa=1$,
and so this was expected. We must therefore now consider theories for which
$\displaystyle\kappa>1$.

For $\displaystyle\kappa>1$ and $\displaystyle\Omega\to 1^{-}$, the condition
given in Eq. (4.14) must be satisfied. Considering $\displaystyle\Omega=1$,
this leads to

$\frac{p-2}{6-p}(\kappa-1)<0.$ (4.16)

We thus see that for any $\displaystyle\kappa>1$, this condition is not
satisfied. Thus, our requirement is broken at both $\displaystyle\Omega\to 0$
and $\displaystyle\Omega\to 1$. As such, if any stable Q-balls were to form,
there would be a minimum charge required to reach stability. In order to
determine if this can be the case, we require that both the roots of the
quadratic given in Eq. (4.14) lie in the range $\displaystyle 0<\Omega<1$. The
roots of this polynomial are

$\Omega_{\pm}=\frac{6-p}{4(4-p)}\pm\frac{6-p}{4(4-p)}\sqrt{1-8\frac{(4-p)(p-2)}{(6-p)^{2}}\kappa}.$
(4.17)

For $\displaystyle\Omega\in(0,1)$, it is a necessary requirement that the
first term on the right-hand side be in the range $\displaystyle(0,1)$, which
leads to the condition

$2<p<10/3.$ (4.18)

Requiring that the roots exist at all – the term under the square root be
positive – leads to a $\displaystyle\kappa$-dependent condition for
$\displaystyle p$,

$1-8\frac{(4-p)(p-2)}{(6-p)^{2}}\kappa>0.$ (4.19)

The function on the left-hand side is a monotonically decreasing function of
$\displaystyle\kappa$ if $\displaystyle 2<p<10/3$. Taking, therefore, the
limit $\displaystyle\kappa\to\infty$ shows us that both roots always exist.
Furthermore, the larger $\displaystyle\kappa$ is, the smaller the difference
between the roots, and so the smaller the range of values which allow for a
stable Q-ball.

To understand these requirements, we return to our definition of
$\displaystyle\kappa$ as given in Eq. (4.12). We note that we cannot form a
stable Q-ball for all $\displaystyle Q>0$ unless the $\displaystyle j$-th
species is either massless (or very light) or $\displaystyle\beta_{j}=0$ for
the species. A theory of this type ($\displaystyle\kappa=1$) was the subject
of Ref. Bishara:2017otb , whereby one real scalar was taken to be effectively
massless, and the other had $\displaystyle\beta=0$.

In the case that $\displaystyle\kappa>1$, we find that we have a lower bound
on the charge of these Q-balls for them to be stable. This makes sense here –
the real scalars increase the mass of the Q-ball without increasing the charge
of the Q-ball. Classical stability requires that the mass to charge ratio of
the Q-balls be less than that of the constituent scalars. Thus, in the
presence of real scalars, the Q-ball must accumulate enough charge to offset
this increase in mass. This is therefore the origin of the requirements given
above: as $\displaystyle\kappa$ gets larger, more charge must be accumulated,
and the range of charges over which we have a stable, thick-wall Q-ball
decreases.

###  4.3 A Worked Example: Arbitrary Number of Massless Real Scalars

As stated above, the minimisation with respect to the $\displaystyle\beta_{j}$
is model-dependent and cannot be completed in general. As an example that can
be fully worked out analytically, consider a simple theory of one complex
scalar $\displaystyle\Phi$ with unit charge and $\displaystyle M$ massless
scalars $\displaystyle\Psi_{j}$ defined through the Lagrangian

$\begin{split}\mathcal{L}=\,\,&\partial_{\mu}\Phi\partial^{\mu}\Phi^{*}+\frac{1}{2}\sum^{M}_{j}\partial_{\mu}\Psi_{j}\partial^{\mu}\Psi_{j}\\\
&-\left(m_{\phi}^{2}-\sum_{j}^{M}A_{j}\Psi_{j}\right)\Phi^{*}\Phi-\mathcal{O}\left(\Psi^{4},(\Phi^{*}\Phi)^{2},\Phi^{*}\Phi\Psi^{2}\right).\end{split}$
(4.20)

Notice, we have taken the leading order interaction between the real and
complex scalars to be cubic444Note, we could have also written down a term
cubic in only the real scalars as it is not forbidden by the $\displaystyle
U(1)$ symmetry. This class of term will only render the Q-ball more unstable
as it will increase the energy (rest mass) of the configuration without
increasing the charge. We omit this for simplicity here, but it is important
to note. – this is because we are in the regime where $\displaystyle\kappa=1$,
and so $\displaystyle 2<p<10/3$. We seek stable thick-wall Q-balls, and assume
that all spatial profiles are, up to some positive semi-definite
normalisation, the same. The functional in Eq. (4.6) is then given by

$\mathcal{E}_{\omega}=\Omega
mQ+m_{\phi}^{3}\dfrac{\left(1+\frac{1}{2}\sum^{M}_{j}\beta_{j}^{2}\right)^{3/2}\left(1-\Omega^{2}\right)^{3/2}}{\left(\sum_{j}^{M}A_{j}\beta_{j}\right)^{2}}S_{\varphi},$
(4.21)

where $\displaystyle S_{\varphi}\approx 38.8$ after numerical minimisation
Linde:1981zj . This expression must be minimised with respect to all
$\displaystyle\beta_{j}$ and $\displaystyle\Omega$. First, we minimise with
respect to all $\displaystyle\beta_{j}$. This process leads to the condition

$\frac{3}{4}\beta_{k}=A_{k}\frac{\left(1+\frac{1}{2}\sum^{M}_{j}\beta_{j}^{2}\right)}{\left(\sum_{j}^{M}A_{j}\beta_{j}\right)}.$
(4.22)

If the $\displaystyle k$-th species of real scalar does not have a coupling to
the complex scalar that is cubic in the fields, it does not contribute to the
Q-ball.555If a real scalar has a coupling to the complex scalar that is
greater than cubic in the fields, it should contribute. However, this
contribution to the properties of the Q-ball would be subleading to those
calculated from only considering from the cubic coupling. Taking
$\displaystyle A_{k}$ to therefore be non-vanishing, we find

$\frac{\beta_{k}}{A_{k}}=\frac{4}{3}\frac{\left(1+\frac{1}{2}\sum^{M}_{j}\beta_{j}^{2}\right)}{\left(\sum_{j}^{M}A_{j}\beta_{j}\right)}.$
(4.23)

This expression holds for each species of real scalar, but the right-hand side
is a constant for each species, and so we can we can then relate all species
through

$\frac{\beta_{k}}{A_{k}}=\frac{\beta_{l}}{A_{l}}.$ (4.24)

We can use this to eliminate all but a single $\displaystyle\beta$ from the
above condition since

$\sum_{j}^{M}\beta_{j}^{2}=\left(\frac{\beta_{l}}{A_{l}}\right)^{2}\sum_{j}^{M}A_{j}^{2}\quad\mathrm{and}\quad\sum_{j}^{M}A_{j}\beta_{j}=\frac{\beta_{l}}{A_{l}}\sum_{j}^{M}A_{j}^{2}.$
(4.25)

We then find that

$\beta_{l}=\dfrac{2A_{l}}{\left(\sum_{j}^{M}A_{j}^{2}\right)^{1/2}}.$ (4.26)

Minimisation with respect to $\displaystyle\Omega$ yields

$\epsilon=\Omega(1-\Omega^{2})^{1/2},$ (4.27)

where

$\epsilon=\frac{4Q}{3^{5/2}S_{\varphi}m_{\phi}^{2}}\sum_{j}^{M}A_{j}^{2},$
(4.28)

where we have eliminated all the $\displaystyle\beta_{j}$ by invoking the
above condition. Notice, this has the same functional form as the single-field
case given in Ref. Kusenko:1997ad . Thus, we may simply read off the solution
that

$\Omega^{2}=\frac{1+\sqrt{1-4\epsilon^{2}}}{2},$ (4.29)

where $\displaystyle\epsilon<1/2$. Given this, we may now expand
$\displaystyle\mathcal{E}_{\omega}$ in powers of $\displaystyle\epsilon$ to
give the mass of the Q-ball,

$m_{Q}=m_{\phi}Q\left(1-\frac{1}{6}\epsilon^{2}-\mathcal{O}\left(\epsilon^{4}\right)\right).$
(4.30)

We see that $\displaystyle m_{Q}<m_{\phi}Q$, and so the resulting Q-ball is
indeed classically stable to decay to quanta of the field $\displaystyle\Phi$.
The characteristic radius, $\displaystyle R$, and VEV of the field,
$\displaystyle\langle\phi\rangle$, are given when $\displaystyle\xi\sim 1$ and
$\displaystyle\varphi\sim 1$, and so

$R^{-1}\sim\frac{\epsilon
m_{\phi}}{\sqrt{3}}\left(1+\mathcal{O}\left(\epsilon^{2}\right)\right)\quad\mathrm{and}\quad\langle\phi\rangle\sim\frac{\epsilon^{2}m_{\phi}^{2}}{2\left(\sum^{M}_{j}A_{j}^{2}\right)^{1/2}}\left(1+\mathcal{O}\left(\epsilon^{2}\right)\right).$
(4.31)

We thus see that these Q-balls are large compared to the Compton wavelength of
the constituent complex scalars, and that the field inside the Q-ball is
indeed small in some sense.

This analysis rested on two assumptions, namely, that the higher order term
that stabilises the potential be small in comparison to the cubic and
quadratic terms inside the Q-ball, and that
$\displaystyle\mathcal{E}_{\omega}$ has a minimum in the range $\displaystyle
0<\omega<m_{\phi}$. These conditions lead to constraints on the allowed charge
of the thick-wall Q-ball solution. Assuming that this stabilising term comes
in at order $\displaystyle q$ (as opposed to the minimal 4 given in the
Lagrangian for this theory) and using the characteristic VEV given above, we
demand that

$\left(\sum_{j}^{M}A_{j}\beta_{j}\right)\langle\phi\rangle^{3}\gg\frac{c_{q}}{\Lambda^{q-4}}\langle\phi\rangle^{q},$
(4.32)

which leads to the constraint on $\displaystyle Q$,

$Q\ll
604.8m_{\phi}\left[\frac{\Lambda^{q-4}}{c_{q}}\right]^{1/(2q-6)}\left[2\left(\sum^{M}_{j}A_{j}^{2}\right)^{1/2}\right]^{(10-3q)/(2q-6)}.$
(4.33)

Demanding this for the quadratic term leads to the same constraint. The second
condition on $\displaystyle Q$ arises from rearranging the condition
$\displaystyle\epsilon<1/2$,

$Q<151.2\frac{m_{\phi}^{2}}{\sum^{M}_{j}A_{j}^{2}}.$ (4.34)

Again, as in the single-field case, the upper bounds on the charge of these
objects imply that these are small Q-balls.

##  5 Summary

The canonical Q-ball analysis is that of a single complex scalar field
invariant under a $\displaystyle U(1)$ group of transformations. In this
paper, we have reviewed the simplest multi-field generalisation, namely, that
of a single complex scalar field together with an arbitrary number of real
scalars. We found the physical parameters associated with the thin-wall
Q-balls in this class of theory. This section was largely a review of the
existing work found in Ref. Friedberg:1976me , albeit presented in a style
similar to Ref. Kusenko:1997zq .

For the thick-wall limit, we presented a sufficient constraint for the
existence of thick-wall Q-balls by assuming that all spatial profiles of the
fields are the same up to positive semi-definite normalisation constant. To
reiterate, these theories were classified by a parameter
$\displaystyle\kappa\geq 1$, defined in Eq. (4.12). In summary, the case
$\displaystyle\kappa=1$, for which the single-field case is a subset, can
possess energetically stable thick-wall solutions of any charge $\displaystyle
Q>0$ when the index, $\displaystyle p$, of the next-to-quadratic term is in
the range $\displaystyle 2<p<10/3$. Similarly, for the case
$\displaystyle\kappa>1$, the index of the next-to-quadratic term is
constrained to be in the range $\displaystyle 2<p<10/3$, but is only
energetically stable for a range of charges, where $\displaystyle
Q_{\mathrm{min}}>0$. This latter case arises as the real scalars add mass to
the Q-balls, but do not reduce the charge, and so enough charge must
accumulate in order for stability to be achieved.

The key part of the thick-wall analysis was the assumption that all fields
have the same spatial shape. However, as we noted in the main text, this isn’t
necessarily valid from the point of view of the field equations. It is
expected that the additional freedom afforded by the multitude of spatial
profiles would allow for a lower energy solution to exist, and this is the
fundamental reason for our work to represent sufficient, but not necessary,
conditions. This assumption must be checked and verified numerically, with the
necessary conditions for a stable thick-wall limit derived – this will be the
subject of future work.

## 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.

## References

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