# Encoder complexity control in SVT-AV1 by speed-adaptive preset switching

###### Abstract

Current developments in video encoding technology lead to continuously
improving compression performance but at the expense of increasingly higher
computational demands. Regarding the online video traffic increases during the
last years and the concomitant need for video encoding, encoder complexity
control mechanisms are required to restrict the processing time to a
sufficient extent in order to find a reasonable trade-off between performance
and complexity. We present a complexity control mechanism in SVT-AV1 by using
speed-adaptive preset switching to comply with the remaining time budget. This
method enables encoding with a user-defined time constraint within the
complete preset range with an average precision of 8.9 % without introducing
any additional latencies.

Index Terms— SVT-AV1, encoder, complexity control, time control

##  1 Introduction

Advances in computing power enable solutions to problems that weren’t even
solvable only a few decades ago. But those rapid improvements come with a
cost, as keeping up with the latest technology is quite expensive for end
users as well as companies. Thus, in order to keep on track with state-of-the-
art technology to reach the best performance for compute-demanding
applications, cloud computing centers provide a solution to get the current
technological advancements in terms of processing power for a reasonable price
[13] through which those services gained popularity. To make an economic
decision concerning the trade-off between costs and computational power, it is
beneficial to know the processing time of programs in advance. Also, being
able to control the complexity to a certain degree is crucial to efficiently
using the computing time in data centers. Since video streaming is responsible
for 60% of web traffic [2] and the encoding part of video streaming has a huge
computational demand, one important application field for complexity control
is video encoding.

In 2018, the Alliance of Open Media (AOM) released a new coding standard, the
AOMedia Video 1 (AV1) [3], which achieves higher bitrate savings compared to
its predecessor VP9 [12] but at the expense of a higher encoding time [7].
Intel’s open-source implementation, SVT-AV1 [10], has been released one year
later and reduces some of the complexity overhead. However, the overall
complexity is still high which leads to the necessity of presets. Those
restrict the encoder optimizations in order to save computing time and enable
a rough decision on the trade-off between complexity and compression quality.
However, accurate time control cannot be achieved.

The first approach for complexity control in AV1 is introduced in [14] and
uses multi-objective optimization of preselected configuration parameter sets.
During encoding, a controller decides whether the current configuration is
kept or changed to a faster or slower one. With this method, complexity
reductions ranging from 10% to 40% can be reached. Complexity control methods
in Versatile Video Coding (VVC) restrict the coding unit (CU) partitioning
process by using Support Vector Machines [18]. Intra complexity control is
achieved by [15] et al. by incorporating texture information in the CU-level
partitioning process. Also focusing on intra-coding, assigning a target time
to each coding tree unit (CTU), and adjusting the preset accordingly [8, 9] is
another approach in VVC. In High Efficient Video Coding (HEVC), complexity
control is achieved by adapting the depth of the CUs [5, 6] as well. Further
improvements are achieved by e.g. including spatial and temporal information
[4]. Other methods introduce deep neural network approaches for CU
partitioning [17], or CTU prediction [19] and control encoding complexity by
pruning the weight parameters of those networks [11].

This work proposes a method that enables controlling complexity in SVT-AV1
with a high accuracy within the complete preset range that is in two orders of
magnitude, in contrast to other methods that often remain at a lower range.
Thereby, it bypasses the requirement of the user to set a preset by replacing
this setting with a target time with only a small loss in compression
performance.

In Section 2, first, the encoding speed estimation is introduced. For the
preset-speed relationship, a look-up table is then presented, and it is
explained how it supports the preset switching decision during encoding.
Finally, Section 3 shows the target speed accuracy when encoding sequences
with different resolutions and frame rates.

##  2 Complexity control with quasi-continuous presets

A rough encoder complexity selection is already available in many encoders and
is done via presets, which enable, disable, or restrict specific compression
options. These include e.g., filters that enhance the visual quality, allowing
global motion compensation, or options for block partitioning sizes. They
range from 0 to 12 for SVT-AV1, where preset 0 relates to the highest encoding
quality and uses all encoder optimizations, resulting in also a high
processing time. Vice versa, preset 12 is the option for very fast or real-
time encoding solutions at a reduced compression performance.

However, the actual processing time is highly dependent on, e.g., the video
sequence and can vary in a large range. Our work proposes a method that
strives to resolve the existing rigid preset decisions and enables encoding in
a quasi-continuous way within the full preset range by

  1. 1.

making a rough data-driven initial preset decision and

  2. 2.

switching adaptively between presets according to the current encoder speed

in order to reach a given target time $t_{\text{target}}$.

###  2.1 Introducing Encoder Speed Feedback in SVT-AV1

To enable controlling complexity in SVT-AV1 by speed-adaptive preset switching
(SAPS), the current processing time as a measure for complexity has to be
determined first.

####  2.1.1 High-level Encoder Adaptions

In SVT-AV1, each encoding step is swapped out in one isolated process and all
of them work simultaneously on the video data in a FIFO fashion. After the
first frame is fully encoded, the measured time from the last process is
transferred to the first process where an encoding speed estimate is made as
well as a decision for a preset change for the upcoming frames. To keep memory
management intact, the allowed geometries for the splitting decisions are kept
constant.

####  2.1.2 Estimation of the Encoding Speed

The parallel processing within one buffer poses challenges in encoding speed
calculation, i.e. when measuring the processing time for one frame, this value
will also include the computing time of the other frames that are inside the
buffer. As a consequence, without introducing additional latencies, this value
cannot be determined exactly but only estimated.

Let the current encoding speed be given as

$v_{\text{enc}}=\frac{n_{\text{enc}}}{t_{\text{CPU}}},$ (1)

where ${t_{\text{CPU}}}$ is the accumulated time and ${n_{\text{enc}}}$ the
frames that contribute to this compute time. This value is always larger or
equal to the number of fully encoded frames $n_{\text{out}}$ due to the
parallel processing of buffered frames whose time also adds up to the measured
value. Finally, we count the number of incoming frames $n_{\text{in}}$ and add
an estimate of the number of frames that are processed while encoding one
frame, to get an approximation for the number of contributing frames:

$n_{\text{enc}}=\frac{1}{2}\left(n_{\text{out}}+n_{\text{in}}\right).$ (2)

As a side note, the difference between $n_{\text{out}}$ and $n_{\text{in}}$ is
equal to the processing buffer size. Thus the number of frames in the
processing loop contributing to the current time measure $n_{\text{enc}}$ is
chosen as the arithmetic mean between the input frame number $n_{\text{in}}$
and completely encoded frames $n_{\text{out}}$.

Analogously to Eq. (1), one can define a speed budget for the remaining frames
by using the total number of frames $n_{\text{total}}$:

$v_{\text{budget}}=\frac{n_{\text{total}}-n_{\text{enc}}}{t_{\text{target}}-t_{\text{CPU}}}.$
(3)

This value guides the preset decision for encoding the queued frames in order
to fulfill the remaining time constraint.

###  2.2 Preset-Complexity Relationship

Preset | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12
---|---|---|---|---|---|---|---|---|---|---|---|---
Pixel rate (in kpps) | 62.6 | 119.8 | 284.3 | 564.3 | 1048 | 2610 | 4450 | 7907 | 11328 | 13664 | 17838 | 24463
Table 1: Look-up-table for the preset-encoding speed relations

The preset switching decision is guided by a look-up table that is created as
follows: Its initial values are created by measuring the encoding time of 13
sequences from the SJTU 4K dataset [16] and their downsampled versions with
factors 2, 4, and 8, with QPs of 23, 27, 31, 34, presets
$p\in\\{1,2,\dots,12\\}$, and random access (RA) configuration. Averaging each
time result $s$ over all QPs and resolutions and converting to a pixel rate,
which corresponds to a pixel-wise encoding speed with unit kilo pixel per
second, leads to a look-up table consisting of a rough speed estimate for all
measured presets:

$v_{\text{look-
up}}(p)=\frac{1}{n_{\text{SJTU}}}\cdot\sum_{s=1}^{n_{\text{SJTU}}}\frac{W^{(s)}\cdot
H^{(s)}\cdot n_{\text{total}}^{(s)}}{1000\cdot
t_{\text{total}}^{(s)}(p)}[\text{kpps}].$ (4)

$W$ and $H$ denote the width and height of the sequence, respectively, and the
resulting values are displayed in Tab. 1. Since the actual speed also depends
on other parameters instead of only the resolution, so far, the QP dependency
is also included in form of a scaling factor:

$\gamma_{\text{QP}}=\frac{1}{1-0.015\cdot\left(\text{QP}-17\right)}.$ (5)

The corresponding preset to a desired target speed can then be used as an
initial guess for the encoder by first converting the encoding target speed
$v_{\text{target}}$ into pixel rate units and then searching for the preset
that is closest to the converted target speed in the look-up table.
Accordingly, the speed estimates from Eq. (1) and Eq. (3) are transformed to
pixel rates as well.

Since the actual encoding speed highly depends on the sequence to be encoded,
this table is updated during encoding by

$v_{\text{look-up}}(p_{i})=(1-w)\cdot v_{\text{look-up}}(p_{i})\\\
+w\cdot\frac{v_{\text{enc}}}{v_{\text{look-up}}(p_{\text{avg}})}\cdot
v_{\text{look-up}}(p_{i})\quad\forall p_{i}\in(1,12)$ (6)

with an update weight $w$ and the average preset used during encoding
$p_{\text{avg}}=\frac{1}{n_{\text{enc}}}\sum_{i=1}^{n_{\text{enc}}}p^{(i)}$.
Speed estimates for those non-discrete values are obtained by linearly
interpolating the preset-speed table.

###  2.3 Switching Criterion

For each frame, $v_{\text{enc}}$ decides whether the preset needs to be
increased or decreased, by comparing it with the target speed
$v_{\text{target}}$. If changing is necessary, then the required acceleration
for the remaining frames when staying at the current preset $p$ is calculated
as

$a(p)=\frac{v_{\text{budget}}}{v_{\text{table}}(p)}.$ (7)

Whenever this factor exceeds or falls below a given threshold, the preset
adaption $\Delta$ is set to $+1$ or $-1$. For thresholds, we have selected
$a(p)>1$ for the decision to increase the speed and $a(p)<0.9$ for a decrease,
such that the encoder is systematically biased towards faster speeds which is
important, e.g., in real-time encoding.

Then we decide if a higher or lower preset will result in reaching the time
constraint, which is tested by calculating the acceleration factor for the
potential new preset $a(p+\Delta)$. If $a$ is high enough (for speed
increment) or low enough (for speed decrement), $\Delta$ will be either kept
or its absolute value will additionally be increased by one, see Algorithm. 1.
Otherwise, it will be set to $0$. Finally, the preset of frame $i$ is set to
$p_{i}=p_{i-1}+\Delta$.

if  _$a(p) >1$_ then

if  _$a(p+1) >0.5$_ then $\Delta\leftarrow+1$

else if  _$a(p+1) >2$_ then $\Delta\leftarrow+2$

else $\Delta\leftarrow 0$

else if  _$a(p) <0.9$_ then

if  _$a(p-1) <1.8$_ then $\Delta\leftarrow-1$

else if  _$a(p-1) <0.45$_ then $\Delta\leftarrow-2$

else $\Delta\leftarrow 0$

$p\leftarrow p+\Delta$

Algorithm 1 Speed-adaptive preset switching (SAPS)

##  3 Evaluation

The proposed method is tested on 26 sequences from class A2, A3, and A3 of the
AOM Common Test Condiditions [20]. The complexity is measured as the sum of
user and system time with single-core execution on an AMD 7452 processor with
2.35 GHz.

For evaluation, we introduce a desired target speed $v_{\text{target}}$ that
describes the encoder speed in terms of encoded frames per second (fps). Here,
we are interested in reaching those arbitrary speed targets only, in contrast
to real-time encoding methods that strive to achieve encode in recorded time.

###  3.1 Validation of Speed Estimation

The prerequisite for complexity control to work properly is correct speed
feedback. Therefore, we first analyze the speed approximation before we
evaluate the actual speed control.

Class Target in fps | 16 | 8 | 4 | 2 | 1 | 0.5 | 0.25 | 0.125 | Average
---|---|---|---|---|---|---|---|---|---
A2 (1920$\times$1080) | $(^{\star})$ | 13.7 % | 11.5 % | 6.8 % | 5.0 % | 6.3 % | 7.6 % | 8.8 % | 8.5 %
A3 (1280$\times$720) | 5.6 % | 6.4 % | 3.7 % | 10.0 % | 8.9 % | 10.1 % | 14.1 % | $(^{\star})$ | 8.4 %
A4 (640$\times$360) | 6.3 % | 5.6 % | 10.4 % | 12.4 % | 10.5 % | 15.8 % | $(^{\star})$ | $(^{\star})$ | 10.2 %
All |  |  |  |  |  |  |  |  | 8.9 %
Table 2: Complexity control errors over all test classes and targets measured
as the relative deviation of the target from the actual speed. Particular
target speeds are in general not reachable for some classes and thus excluded
from evaluation. Those are indicated with (⋆). Fig. 1: Validation of the speed
estimation by comparing the current approximated speed with the actual average
speed. The grid lines indicate the buffer sizes.

We calculate the actual encoding speed $v_{\text{real}}$ as the number of
encoded frames divided by the measured encoding time for 160 frames and
compare this value with the estimated speed. In order to use the complexity
control mechanism, encoding speed should be approximately constant. Fig. 1
shows that the method can only work properly when enough frames are processed
since the time approximation is only possible after processing the first
buffer and is sufficiently correct after the second. Buffer sizes are
indicated in the figure by vertical grid lines.

###  3.2 Speed Control Accuracy

In order to ensure obtaining correct speed estimates, 300 frames are encoded
for every sequence for speed control evaluation. The encoding is performed
with four constant rate factors $\text{QP}\in\left\\{23,27,33,37\right\\}$
with random access configuration, and a keyframe every 10 seconds, i.e. each
test consists of one or two GOPs. Target times are set such that the encoder
has a predefined encoding speed range of
$v_{\text{target}}\in\left\\{16,8,4,2,1,0.5,0.25,0.125\right\\}$ fps. The
average speed errors are calculated for each class containing $n_{\text{AOM}}$
sequences for each target speed:

$\epsilon_{\text{v}}=\frac{1}{4\cdot n_{\text{AOM}}}\sum_{1}^{4\cdot
n_{\text{AOM}}}\frac{\left|v_{\text{real}}-v_{\text{target}}\right|}{v_{\text{target}}}$
(8)

The performance for each target and class is shown in Tab. 2 and we obtain an
average deviation over all of $8.9\%$. Average errors of the AV1 complexity
controller from [14] range from 3.4% to 11.3% in a target complexity range
from 10% to 30%. In comparison, the proposed method reaches relative
complexities from below 1% to 100%.

###  3.3 Complexity-Compression Performance

To measure the overhead in terms of compression performance loss, the
Bjøntegaard Delta [1] bitrate (BDBR) increase compared to preset 1 is
calculated for the original encoder implementation as well as the time control
method. The dependency between BDBR and encoding time over all sequences is
shown in Fig. 2. Small resolutions (from classes A3 and A4) show a good
performance regarding their trade-off since the curve of the time control
method lies close to the reference curve.

Fig. 2: Trade-off between complexity and compression performance. Dashed lines
denote the reference implementation over the preset range, solid lines show
the proposed time control. The compression performance is calculated as the
BD-rate increase compared to the anchor of preset 1, which corresponds to the
slowest preset as well as the theoretical minimum reachable encoding speed in
terms of speed control.

##  4 Conclusion

This work proposes a complexity control approach in SVT-AV1 using the existing
presets. Over a range of two orders of magnitude we can reach target
processing times with a mean error of 8.9 %. In future works, we will improve
the time estimation for the first two buffers, which currently is the limiting
factor for short video sequences, by identifying which encoding steps mostly
contribute to the processing time and making speed estimations before fully
encoding one frame. Since the initial preset guess helps to reach the accurate
target and to keep the rate increase low, we will also focus on introducing a
more complex speed prediction model.

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# A nontopological soliton in an $\mathcal{N}=1$ supersymmetric gauge Abelian
model

A.Yu. Loginov<EMAIL_ADDRESS>Laboratory of Applied Mathematics and
Theoretical Physics, Tomsk State University of Control Systems and
Radioelectronics, 634050 Tomsk, Russia

###### Abstract

A version of $\mathcal{N}=1$ supersymmetric scalar electrodynamics is
considered here, and it is shown that an electrically charged nontopological
soliton exists in this model. In addition to the long-range electric field,
the soliton also possesses a long-range scalar field, which leads to a
modification of the intersoliton interaction potential at large distances. The
supersymmetry of the model makes it possible to express fermionic zero modes
of the soliton in terms of bosonic fields. The properties of the
nontopological soliton are investigated using analytical and numerical
methods.

###### keywords:

nontopological soliton , electric charge , supersymmetry , fermionic zero
modes

††journal: Physics Letters B

##  1 Introduction

Many models of field theory have solutions that describe spatially localised
and nonspreading field configurations with a finite energy [1, 2].
Nontopological solitons [3] represent one of these field configurations. A
necessary condition for the existence of a nontopological soliton is the
symmetry of the corresponding field model, which may be both global and local.
In addition, the interaction potentials of the model must meet a certain
condition [4, 5]. The symmetry of the model results in the existence of a
conserved Noether charge. The field configuration of a nontopological soliton
is an extremum (minimum or saddle point) of the energy functional at a fixed
value of the Noether charge, and this basic property largely determines the
other properties of a nontopological soliton; in particular, it leads to the
characteristic time dependence $\exp\left(-i\omega t\right)$ of a soliton
field.

Nontopological solitons may be formed during a primordial phase transition,
thus making a contribution to various scenarios of the evolution of the early
Universe [6]. Furthermore, they may play an essential role in baryogenesis via
the Affleck-Dine mechanism [7], and are considered to be places where dark
matter may be concentrated [8].

Some field models with local Abelian symmetry admit the existence of
electrically charged nontopological solitons. First described in Refs. [9,
10], they have since been investigated in many other works (see, e.g., Refs.
[11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]). The properties of
electrically charged solitons differ significantly from those of solitons
without an electric charge; in particular, the electric charge and the energy
of a nontopological soliton cannot be arbitrarily large in the general case
[10, 20]. In addition, an electrically charged nontopological soliton can
exist only if the gauge coupling constant does not exceed some maximum value
[10].

The main goal of this work is to study a non-topological soliton in a version
of $\mathcal{N}=1$ supersymmetric scalar electrodynamics. The interaction
potential of this model is expressed in terms of a superpotential, which leads
to relations between the nonlinear interaction constants. In addition, the
superpotential largely determines the form of the scalar-fermion interaction.
The requirements of renormalisability and gauge invariance impose severe
restrictions on the form of the superpotential, all of which significantly
reduces the number of model parameters compared to the nonsupersymmetric case.

Throughout this paper, we use the natural units $c=1$, $\hbar=1$. The metric
tensor and the Dirac matrices are defined according to Ref. [24].

##  2 Lagrangian and field equations of the model

The $\mathcal{N}=1$ supersymmetric gauge model under consideration includes
three left-chiral matter superfields $\Phi_{-1}$, $\Phi_{0}$, and $\Phi_{+1}$,
and one Abelian gauge superfield $V$. The left-chiral superfield $\Phi_{n}$
contains two components: the complex scalar field $\phi_{n}$ and the left-hand
Dirac spinor field $\psi_{nL}$. Written in the Wess-Zumino gauge, the gauge
superfield $V$ also contains two components: the Abelian gauge field $A_{\mu}$
and the Majorana spinor field $\lambda$. The superfields $\Phi_{n}$ and $V$
also contain auxiliary fields, but these can be expressed in terms of the
above mentioned physical fields.

The Lagrangian of the model takes the form

$\displaystyle\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\sum\limits_{n}\left(D_{\mu}\phi_{n}\right)^{\ast}D^{\mu}\phi_{n}-V\left(\phi\right)$
$\displaystyle-\frac{1}{2}\bar{\lambda}\gamma^{\mu}\partial_{\mu}\lambda-\sum\limits_{n}\overline{\psi_{nL}}\gamma^{\mu}D_{\mu}\psi_{nL}$
$\displaystyle-\frac{1}{2}\sum\limits_{nm}\left\\{f_{nm}\left(\psi_{nL}^{\text{T}}\epsilon\psi_{mL}\right)+f_{nm}^{\ast}\left(\psi_{nL}^{\text{T}}\epsilon\psi_{mL}\right)^{\ast}\right\\}$
$\displaystyle+i\sqrt{2}\sum\limits_{n}q_{n}\left\\{\phi_{n}\left(\overline{\psi_{nL}}\lambda\right)-\phi_{n}^{\ast}\left(\bar{\lambda}\psi_{nL}\right)\right\\}.$
(1)

In Eq. (1), the matrix $\epsilon=-i\gamma_{0}\gamma_{2}\gamma_{5}$, the Latin
indices $n$ and $m$ run over the set $[-1,0,1]$, and the covariant derivatives

$\displaystyle D_{\mu}\phi_{n}$
$\displaystyle=\partial_{\mu}\phi_{n}-iq_{n}A_{\mu}\phi_{n},$ (2a)
$\displaystyle D_{\mu}\psi_{n}$
$\displaystyle=\partial_{\mu}\psi_{n}-iq_{n}A_{\mu}\psi_{n},$ (2b)

where $q_{n}=ne$ are the Abelian charges of the left-chiral superfield
$\Phi_{n}$. To avoid $U(1)$-$U(1)$-$U(1)$ and $U(1)$-graviton-graviton
anomalies, the sum of the $U(1)$ quantum numbers of all left-chiral
superfields and the sum of their cubes should vanish, which is obviously true
in our case. The field-dependent coefficients $f_{nm}$ and the interaction
potential $V\left(\phi\right)$ are expressed in terms of the superpotential

$f\left(\phi\right)=m\phi_{-1}\phi_{+1}+g\phi_{-1}\phi_{0}\phi_{+1},$ (3)

where $m$ is a mass parameter and $g$ is a coupling constant. The coefficients
$f_{nm}=\partial^{2}f/\partial\phi_{n}\partial\phi_{m}$, and the interaction
potential

$\displaystyle V\left(\phi\right)$
$\displaystyle=\sum\limits_{n}\left|\partial
f/\partial\phi_{n}\right|^{2}+\frac{1}{2}\Bigl{(}\sum\limits_{n}q_{n}\phi_{n}^{\ast}\phi_{n}\Bigr{)}^{2}$
$\displaystyle=\left|m+g\phi_{0}\right|^{2}\left(\left|\phi_{-1}\right|^{2}+\left|\phi_{+1}\right|^{2}\right)$
$\displaystyle+g^{2}\left|\phi_{-1}\right|^{2}\left|\phi_{+1}\right|^{2}$
$\displaystyle+\frac{e^{2}}{2}\left(\left|\phi_{+1}\right|^{2}-\left|\phi_{-1}\right|^{2}\right)^{2}.$
(4)

The field equations of model (1) have the form

$\partial_{\mu}F^{\mu\nu}=j^{\nu},$ (5) $\displaystyle
D_{\mu}D^{\mu}\phi_{n}-\frac{\partial
V}{\partial\phi_{n}^{\ast}}-\frac{1}{2}\sum\limits_{k^{\prime}m^{\prime}}f_{k^{\prime}m^{\prime}n}^{\ast}\left(\psi_{k^{\prime}L}^{\text{T}}\epsilon\psi_{m^{\prime}L}\right)^{\ast}$
$\displaystyle-i\sqrt{2}q_{n}\left(\overline{\lambda}\psi_{nL}\right)=0,$ (6)
$\not{D}\psi_{nL}-\sum\limits_{m^{\prime}}f_{nm^{\prime}}^{\ast}\epsilon\left(\overline{\psi_{m^{\prime}L}}\right)^{\text{T}}-i\sqrt{2}q_{n}\phi_{n}\lambda_{R}=0,$
(7)
$\not{\partial}\lambda+i\sqrt{2}\sum\limits_{m^{\prime}}q_{m^{\prime}}\left\\{\phi_{m^{\prime}}\epsilon\left(\overline{\psi_{m^{\prime}L}}\right)^{\text{T}}+\phi_{m^{\prime}}^{\ast}\psi_{m^{\prime}L}\right\\}=0,$
(8)

where the coefficients
$f_{kmn}=\partial^{3}f/\partial\phi_{k}\partial\phi_{m}\partial\phi_{n}$ and
the electromagnetic current

$j^{\nu}=i\sum\limits_{n}q_{n}\phi_{n}^{\ast}\overset{\longleftrightarrow}{D^{\nu}}\phi_{n}-i\sum\limits_{n}q_{n}\overline{\psi_{nL}}\gamma^{\nu}\psi_{nL}.$
(9)

Later on, we shall also need the expression for the energy density of an
electrically charged bosonic field configuration of the model

$\displaystyle\mathcal{E}$
$\displaystyle=\frac{1}{2}E_{i}E_{i}+\sum\limits_{n}\left\\{\left(D_{t}\phi_{n}\right)^{\ast}D_{t}\phi_{n}\right.$
$\displaystyle\left.+\left(D_{i}\phi_{n}\right)^{\ast}D_{i}\phi_{n}\right\\}+V\left(\phi\right),$
(10)

where $E_{i}=F_{i0}$ are the components of the electric field strength.

##  3 Ansatz and some properties of the nontopological soliton

The model (1) can be viewed as the Abelian gauge version of a model of the
Wess-Zumino type [25]. In Ref. [26], it was shown that for superpotentials of
the type in Eq. (3), these models admit the existence of nontopological
solitons. It follows from continuity considerations that nontopological
solitons can also exist in gauge model (1), at least for sufficiently small
values of the gauge coupling constant $e$.

Let us define the shifted field
$\varphi_{0}\left(\mathbf{x},t\right)=mg^{-1}+\phi_{0}\left(\mathbf{x},t\right)$.
To find a nontopological soliton solution, we shall use the spherically
symmetrical ansatz:

$\displaystyle\phi_{+1}\left(\mathbf{x},t\right)$
$\displaystyle=2^{-\frac{1}{2}}\exp\left(-i\omega
t\right)f_{+1}\left(r\right),$ (11a)
$\displaystyle\phi_{-1}\left(\mathbf{x},t\right)$
$\displaystyle=2^{-\frac{1}{2}}\exp\left(i\omega
t\right)f_{-1}\left(r\right),$ (11b)
$\displaystyle\varphi_{0}\left(\mathbf{x},t\right)$
$\displaystyle=2^{-\frac{1}{2}}(\chi_{1}\left(r\right)+i\chi_{2}\left(r\right)),$
(11c) $\displaystyle A^{\mu}\left(\mathbf{x},t\right)$
$\displaystyle=\left(\Phi\left(r\right),\,0\right).$ (11d)

The energy density (10), written in terms of the ansatz functions (11), takes
the form

$\displaystyle\mathcal{E}$
$\displaystyle=\frac{1}{2}\Omega^{2}\left(f_{-1}^{2}+f_{+1}^{2}\right)+\frac{1}{2}\Phi^{\prime
2}$ $\displaystyle+\frac{1}{2}\left(f_{-1}^{\prime 2}+f_{+1}^{\prime
2}+\chi_{1}^{\prime 2}+\chi_{2}^{\prime 2}\right)+V,$ (12)

where the interaction potential

$\displaystyle V$
$\displaystyle=\frac{g^{2}}{4}\left(f_{-1}^{2}+f_{+1}^{2}\right)\left(\chi_{1}^{2}+\chi_{2}^{2}\right)$
$\displaystyle+\frac{g^{2}}{4}f_{-1}^{2}f_{+1}^{2}+\frac{e^{2}}{8}\left(f_{+1}^{2}-f_{-1}^{2}\right)^{2},$
(13)

the function $\Omega\left(r\right)=\omega-e\Phi\left(r\right)$, and the prime
indicates the derivative with respect to $r$. The Lagrangian density
$\mathcal{L}$ differs from the energy density $\mathcal{E}$ only in regard to
the sign of the terms in the second line of Eq. (12). The electromagnetic
current of spherically symmetrical field configuration (11) is

$j^{\nu}=\left(e\Omega\left(f_{-1}^{2}+f_{+1}^{2}\right),0,0,0\right).$ (14)

Substituting ansatz (11) into the bosonic parts of field equations (5) and
(6), we obtain a system of nonlinear differential equations for the ansatz
functions:

$\Omega^{\prime\prime}+\frac{2}{r}\Omega^{\prime}-e^{2}\left(f_{-1}^{2}+f_{+1}^{2}\right)\Omega=0,$
(15) $f_{\pm 1}^{\prime\prime}+\frac{2}{r}f_{\pm 1}^{\prime}+\frac{\partial
U}{\partial f_{\pm 1}}=0,$ (16)
$\chi_{1,2}^{\prime\prime}+\frac{2}{r}\chi_{1,2}^{\prime}+\frac{\partial
U}{\partial\chi_{1,2}}=0,$ (17)

where the effective potential

$U=\frac{1}{2}\Omega^{2}\left(f_{-1}^{2}+f_{+1}^{2}\right)-V.$ (18)

The regularity of the soliton field configuration and the finiteness of the
soliton energy lead to the following boundary conditions:

$\displaystyle f_{\pm 1}^{\prime}\left(0\right)$ $\displaystyle=0,$
$\displaystyle f_{\pm 1}\left(r\right)$
$\displaystyle\underset{r\rightarrow\infty}{\longrightarrow}0,$ (19a)
$\displaystyle\chi_{1,2}^{\prime}\left(0\right)$ $\displaystyle=0,$
$\displaystyle\chi_{1,2}\left(r\right)$
$\displaystyle\underset{r\rightarrow\infty}{\longrightarrow}\chi_{1,2\,\text{vac}},$
(19b) $\displaystyle\Omega^{\prime}\left(0\right)$ $\displaystyle=0,$
$\displaystyle\Omega\left(r\right)$
$\displaystyle\underset{r\rightarrow\infty}{\longrightarrow}\omega.$ (19c)

The boundary conditions in Eqs. (19b) and (19c) need some explanation. From
Eqs. (1) and (4), it follows that the classical vacuum of model (1) is

$F_{\mu\nu}=0,\quad\phi_{\pm 1}=0,\quad\phi_{0}=\phi_{0\,\text{vac}},$ (20)

where $\phi_{0\,\text{vac}}$ is an arbitrary complex constant. From Eq. (20),
it follows that model (1) has an infinite number of vacua at the classical
level, as reflected in the boundary condition in Eq. (19b). All of these vacua
are invariant under both the $U(1)$ gauge and $\mathcal{N}=1$ supersymmetry
transformations. According to the non-renormalisation theorems [27, 28], this
will also be true when perturbative quantum corrections are taken into
account.

Eqs. (13), (17), and (18) tell us that $\chi_{1}$ and $\chi_{2}$ satisfy the
same linear homogeneous differential equation, while Eq. (19b) tells us that
$\chi_{1}$ and $\chi_{2}$ satisfy the same homogeneous boundary condition at
$r=0$. It follows that the ratio $\chi_{2}(r)/\chi_{1}(r)$ does not depend on
$r$, and is equal to $\chi_{2\,\text{vac}}/\chi_{1\,\text{vac}}$. The phase of
the ansatz function $\varphi_{0}(r)=2^{-1/2}(\chi_{1}(r)+i\chi_{2}(r))$ is
therefore a constant. However, from Eqs. (12) and (13), it follows that in
this case, the energy density and the Lagrangian density do not depend on the
phase of $\varphi_{0}\left(r\right)$. Without loss of generality, we can set
this phase (and hence $\chi_{2}(r)$) equal to zero.

The field configurations of model (1) are determined up to gauge
transformations. In particular, the choice of ansatz (11) is equivalent to the
choice of the radial gauge. However, this gauge does not fix the soliton field
configuration completely; to do this, we need to impose an additional
condition $\Phi(\infty)=0$, which is equivalent to Eq. (19c).

The basic property of any non-topological soliton is that it is an extremum of
the energy functional $E$ at a fixed value of some Noether charge $Q_{N}$ (in
our case $E=4\pi\int\nolimits_{0}^{\infty}\mathcal{E}(r)r^{2}dr$ and
$Q_{N}=4\pi e^{-1}\int\nolimits_{0}^{\infty}j^{0}(r)r^{2}dr$). This property
results in the differential relation

$dE/dQ_{N}=\Omega_{\infty},$ (21)

where $\Omega_{\infty}\equiv\Omega(\infty)=\omega-e\Phi(\infty)=\omega$. Note
that a similar relation also holds for the electrically charged magnetic
monopoles [29].

Eqs. (13) and (18) tell us that the potentials $V$ and $U$ are invariant under
the permutation $f_{-1}\leftrightarrow f_{+1}$. It follows that if
$f_{-1}(r)$, $f_{+1}(r)$, $\chi_{1}(r)$, and $\Omega(r)$ is a solution of
system (15) – (17), then $f_{+1}(r)$, $f_{-1}(r)$, $\chi_{1}(r)$, and
$\Omega(r)$ is also a solution. Using qualitative research methods for
differential equations, it can be shown that the solutions $f_{-1}(r)$ and
$f_{+1}(r)$ coincide when the gauge coupling constant $e=0$. In the following,
we define the function
$\delta\left(r,e^{2}\right)=f_{+1}\left(r,e^{2}\right)-f_{-1}\left(r,e^{2}\right)$,
where the dependence on the gauge coupling constant is explicitly indicated
and we use the fact that the potential $V$ depends on $e$ only through
$e^{2}$. The function $\delta\left(r,e^{2}\right)$ satisfies the nonlinear
differential equation

$\displaystyle\delta^{\prime\prime}+\frac{2}{r}\delta^{\prime}+\left[\Omega^{2}+2^{-1}g^{2}\left(f_{-1}^{2}-\chi_{1}^{2}\right)-2e^{2}f_{-1}^{2}\right]\delta$
$\displaystyle+2^{-1}\left(g^{2}-4e^{2}\right)f_{-1}\delta^{2}-2^{-1}e^{2}\delta^{3}=0,$
(22)

where the dependence of $\delta$, $\Omega$, $f_{-1}$, and $\chi_{1}$ on $r$
and $e^{2}$ is omitted. From Eq. (19a), it follows that
$\delta\left(r,e^{2}\right)$ satisfies the boundary conditions

$\delta^{\prime}\left(0,e\right)=0,\quad\delta\left(\infty,e\right)=0.$ (23)

Our goal is to find the derivatives
$\delta^{(n)}\equiv\partial^{n}\delta/\partial e^{n}$ at $e=0$. To do this, we
differentiate Eq. (22) with respect to $e$, and then set $e=0$. As a result,
we obtain the trivial linear equation
$\delta^{\left(1\right)\prime\prime}+2r^{-1}\delta^{\left(1\right)\prime}=0$.
Its solution must satisfy the boundary conditions in Eq. (23) differentiated
with respect to $e$, and it is therefore easy to see that the solution is
$\delta^{\left(1\right)}(r,0)=0$. Thus, we have established that
$\delta(r,0)=0$ and $\delta^{\left(1\right)}(r,0)=0$. By continuing to
differentiate Eq. (22) with respect to $e$, setting $e=0$, and taking into
account the previous results at each step, it can be shown that
$\delta^{(n)}(r,0)=0$ for any $n\geq 0$. It follows that
$\delta\left(r,e^{2}\right)$ vanishes, and hence
$f_{+1}\left(r,e^{2}\right)=f_{-1}\left(r,e^{2}\right)\equiv
f\left(r,e^{2}\right)$.

We now examine the asymptotics of the soliton fields for large $r$. Suppose
that $f(r)$ tends to zero exponentially as $r\rightarrow\infty$. In this case,
we can neglect the nonlinear terms in Eqs. (15) and (17), and obtain the
asymptotic forms of $\Omega(r)$ and $\chi_{1}(r)$ as $r\rightarrow\infty$:

$\Omega\sim\omega-\frac{e}{4\pi}\frac{Q}{r},$ (24)
$\chi_{1}\sim\chi_{1\,\text{vac}}-\frac{1}{4\pi}\frac{Q_{\text{s}}}{r},$ (25)

where $Q=4\pi\int\nolimits_{0}^{\infty}j^{0}(r)r^{2}dr$ is the electric charge
of the soliton, and $Q_{s}$ is the scalar charge defined by analogy with the
large-distance asymptotics $\Phi\sim Q/(4\pi r)$ for the electric potential.
We see that both $\Omega=\omega-e\Phi$ and $\chi_{1}$ tend rather slowly
($\propto r^{-1}$) to their limiting values as $r\rightarrow\infty$. It should
be noted that nontopological solitons with a long-range scalar field were
studied in Refs. [26, 30, 31]. Furthermore, electrically charged
nontopological solitons with a long-range scalar field were studied in Refs.
[21, 23, 32].

By substituting Eqs. (24) and (25) into Eq. (16), retaining the terms linear
in $f(r)$, and solving the resulting differential equation, we obtain the
large-distance asymptotics of $f(r)$ as

$\displaystyle f\left(r\right)$ $\displaystyle\sim f_{\infty}e^{-\Delta
r}\left(\Delta r\right)^{\beta}$
$\displaystyle\times\left(1-\frac{a^{2}}{32\pi^{2}\Delta^{3}r}-\frac{b}{8\pi\Delta^{2}r}\right),$
(26)

where $f_{\infty}$ is a constant,

$\displaystyle\Delta$
$\displaystyle=\left(\omega_{\max}^{2}-\omega^{2}\right)^{1/2},$ (27a)
$\displaystyle a$
$\displaystyle=e\omega_{\max}\left|Q\right|-g\left|\omega\right|\left|Q_{\text{s}}\right|,$
(27b) $\displaystyle b$
$\displaystyle=e\left|\omega\right|\left|Q\right|-g\omega_{\max}\left|Q_{\text{s}}\right|,$
(27c) $\displaystyle\beta$ $\displaystyle=-1-b/\left(4\pi\Delta\right),$ (27d)

and the parameter $\omega_{\max}=2^{-1/2}g\left|\chi_{1\,\text{vac}}\right|$.
We see that our assumption about the exponential asymptotics of $f(r)$ turned
out to be correct; we also see that the long-range terms in the asymptotics of
$\Omega(r)$ and $\chi_{1}(r)$ modify the pre-exponential factor in the
asymptotics of $f(r)$. Furthermore, we can conclude that the nontopological
soliton cannot exist when $\left|\omega\right|>\omega_{\max}$, since in this
case asymptotics (26) shows oscillating behavior, leading to an infinite
energy and charge for the corresponding field configuration.

The presence of two long-range fields in Eqs. (24) and (25) leads to a
modification of the intersoliton interaction potential at large distances. It
can be shown that in the case of large distances and low velocities, the
leading term of the intersoliton interaction potential is

$V_{12}=\frac{Q^{\left(1\right)}Q^{\left(2\right)}-Q_{\text{s}}^{\left(1\right)}Q_{\text{s}}^{\left(2\right)}}{4\pi
r_{12}},$ (28)

where $Q^{\left(i\right)}$ ($Q_{\text{s}}^{\left(i\right)}$) is the electric
(scalar) charge of the $i$-th soliton. Eq. (28) tells us that the energy of
the intersoliton interaction is the sum of the energies of the Coulomb and
scalar interactions. Depending on the signs of $Q^{(1)}$ and $Q^{(2)}$, the
Coulomb energy may be both positive (repulsion) and negative (attraction). At
the same time, it follows from the inhomogeneity of the boundary condition in
Eq. (19b) that for the fixed vacuum in Eq. (20), the scalar charges
$Q_{\text{s}}^{\left(i\right)}$ of the solitons must have the same sign.
Hence, unlike the Coulomb field, the long-range scalar field always leads to
attraction between solitons.

##  4 Fermionic zero modes

The Lagrangian density (1) is written in the Wess-Zumino gauge, meaning that
the corresponding action $S=\int\mathcal{L}d^{4}x$ is not invariant under the
usual $\mathcal{N}=1$ supersymmetry transformations. However, it will be
invariant under the modified supersymmetry transformations [33]:

$\displaystyle\delta\phi_{n}$
$\displaystyle=\sqrt{2}\overline{\alpha_{R}}\psi_{nL},$ (29a)
$\displaystyle\delta\psi_{nL}$
$\displaystyle=\sqrt{2}\gamma^{\mu}\left(D_{\mu}\phi_{n}\right)\alpha_{R}+\sqrt{2}\mathcal{F}_{n}\alpha_{L},$
(29b) $\displaystyle\delta A_{\mu}$
$\displaystyle=\bar{\alpha}\gamma_{\mu}\lambda,$ (29c)
$\displaystyle\delta\lambda$
$\displaystyle=i\mathcal{D}\gamma_{5}\alpha-\frac{1}{4}F_{\mu\nu}\left[\gamma^{\mu},\gamma^{\nu}\right]\alpha,$
(29d)

where

$\alpha=-i\begin{pmatrix}\epsilon_{a}\\\
\sum\nolimits_{b}e_{ab}\epsilon_{b}^{\ast}\end{pmatrix},$ (30)
$\begin{pmatrix}\epsilon_{1}\\\
\epsilon_{2}\end{pmatrix}=\begin{pmatrix}\epsilon_{11}+i\epsilon_{12}\\\
\epsilon_{21}+i\epsilon_{22}\end{pmatrix},$ (31)
$\mathcal{F}_{n}=-\left(\partial f/\partial\phi_{n}\right)^{\ast},$ (32)

and

$\mathcal{D}=e\left(\phi_{+1}^{\ast}\phi_{+1}-\phi_{-1}^{\ast}\phi_{-1}\right).$
(33)

In Eq. (31), $\epsilon_{ij}$ are real infinitesimal anticommuting
transformation parameters and $e_{ab}$ is an antisymmetric $2\times 2$ matrix
with $e_{12}=+1$, from which it follows that $\alpha$ in Eq. (30) is the
Majorana spinor. In Eq. (32), the auxiliary fields $\mathcal{F}_{n}$ are
expressed in terms of superpotential (3), and it is assumed that all the
fields in Eqs. (29a)–(29d) satisfy field equations (5)–(8).

Fermionic zero modes are generated by the action of transformations (29b) and
(29d) on purely bosonic field configuration (11). To represent these in a
compact form, we introduce a column $\Psi$ consisting of four fermionic fields
included in the Lagrangian (1). The transposed form of $\Psi$ is

$\Psi^{\text{T}}=N\left(\psi_{+1L}^{\text{T}},\psi_{0L}^{\text{T}},\psi_{-1L}^{\text{T}},\lambda^{\text{T}}\right),$
(34)

where

$\psi_{\pm 1L}=\begin{pmatrix}A_{\pm 1}f+Cf^{\prime}\\\ B_{\pm
1}f+Df^{\prime}\\\ 0\\\ 0\end{pmatrix}e^{\mp i\omega t},$ (35)
$\psi_{0L}=\begin{pmatrix}i\epsilon_{1}2^{-\frac{1}{2}}gf^{2}+C\chi_{1}^{\prime}\\\
i\epsilon_{2}2^{-\frac{1}{2}}gf^{2}+D\chi_{1}^{\prime}\\\ 0\\\
0\end{pmatrix},$ (36)
$\lambda=i\Phi^{\prime}\begin{pmatrix}\epsilon_{1}c+\epsilon_{2}e^{-i\varphi}s\\\
-\epsilon_{2}c+\epsilon_{1}e^{i\varphi}s\\\
-\epsilon_{2}^{\ast}c+\epsilon_{1}^{\ast}e^{-i\varphi}s\\\
-\epsilon_{1}^{\ast}c-\epsilon_{2}^{\ast}e^{i\varphi}s\end{pmatrix},$ (37)

and $N$ is a normalisation factor. For brevity, in Eqs. (35)–(37), we use the
notation

$\displaystyle A_{\pm 1}$ $\displaystyle=\pm
i\epsilon_{2}^{\ast}\Omega+i2^{-\frac{1}{2}}\epsilon_{1}g\chi_{1},$ (38a)
$\displaystyle B_{\pm 1}$ $\displaystyle=\mp
i\epsilon_{1}^{\ast}\Omega+i2^{-\frac{1}{2}}\epsilon_{2}g\chi_{1},$ (38b)
$\displaystyle C$
$\displaystyle=-\epsilon_{2}^{\ast}c+\epsilon_{1}^{\ast}e^{-i\varphi}s,$ (38c)
$\displaystyle D$
$\displaystyle=-\epsilon_{1}^{\ast}c-\epsilon_{2}^{\ast}e^{i\varphi}s,$ (38d)

where $c=\cos(\theta)$, $s=\sin(\theta)$,
$\epsilon_{1}=\epsilon_{11}+i\epsilon_{12}$, and
$\epsilon_{2}=\epsilon_{21}+i\epsilon_{22}$.

Eqs. (35)–(37) depend linearly on the four anticommuting parameters
$\epsilon_{ij}$, and hence Eq. (34) can be written as
$\Psi=\sum\nolimits_{ij}\epsilon_{ij}\Psi_{ij}$. It follows that there are
four (according to the number of the $\mathcal{N}=1$ supersymmetry generators)
independent fermionic zero modes $\Psi_{ij}$ expressed in terms of ansatz
functions (11). It can be shown that the components of the fermionic zero
modes $\Psi_{ij}$ satisfy field equations (7) and (8), provided that the
ansatz functions $\Omega$, $f$, and $\chi_{1}$ satisfy Eqs. (15)–(17). The
fermionic zero modes satisfy the orthonormality condition

$\int\Psi_{ij}^{\dagger}\Psi_{i^{\prime}j^{\prime}}d^{3}x=\delta_{ii^{\prime}}\delta_{jj^{\prime}},$
(39)

provided that the normalisation factor

$\displaystyle N$
$\displaystyle=\left[2\pi\int\nolimits_{0}^{\infty}\left[4\left(\Phi^{\prime
2}+f^{\prime 2}\right)+2\chi_{1}^{\prime 2}\right.\right.$
$\displaystyle\biggl{.}\left.+g^{2}f^{4}+2f^{2}\left(2\Omega^{2}+g^{2}\chi_{1}^{2}\right)\right]r^{2}dr\biggr{]}^{-\frac{1}{2}}.$
(40)

From Eq. (37), it follows that the gaugino component $\lambda$ of the
fermionic zero mode $\Psi_{ij}$ is proportional to the electric field strength
$E_{r}=-\Phi^{\prime}$ of the soliton, and therefore decreases rather slowly
($\propto r^{-2}$) at large distances. Furthermore, Eqs. (25) and (36) tell us
that at large distances, the component $\psi_{0L}\propto\chi_{1}^{\prime}\sim
Q_{\text{s}}/(4\pi r^{2})$. We see that similarly to the $\lambda$ component,
the $\psi_{0L}$ component of $\Psi_{ij}$ decreases slowly ($\propto r^{-2}$)
at large distances. In contrast, Eqs. (26) and (35) tell us that the two
remaining components $\psi_{\pm 1L}$ of $\Psi_{ij}$, which correspond to the
short-range scalar fields $\phi_{\pm 1}$, decrease exponentially away from the
soliton.

Written in terms of the left-handed fermion fields (including the massless
“neutrino” $\psi_{0L}$), the Lagrangian (1) is not invariant under the $P$ and
$C$ transformations; it is, however, invariant under the combined $CP$
transformation. Under the latter transformation, the original soliton solution
($f(r)\exp(\mp i\omega t)$, $\chi_{1}(r)$, $\Phi(r)$, $\Omega(r)$) of the
energy $E$ and electric charge $Q$ is transformed into an antisoliton solution
($f(r)\exp(\pm i\omega t)$, $\chi_{1}(r)$, $-\Phi(r)$, $-\Omega(r)$) of the
energy $E$ and electric charge $-Q$. It can be shown that under the $CP$
transformation, the fermionic zero modes $\Psi_{ij}$ of the soliton turn into
those $\tilde{\Psi}_{ij}$ of the antisoliton:

$\displaystyle\left[\Psi_{11}(x)\right]^{CP}$ $\displaystyle=$
$\displaystyle-\tilde{\Psi}_{22}(x),$
$\displaystyle\left[\Psi_{12}(x)\right]^{CP}$ $\displaystyle=$
$\displaystyle-\tilde{\Psi}_{21}(x),$
$\displaystyle\left[\Psi_{21}(x)\right]^{CP}$ $\displaystyle=$
$\displaystyle\tilde{\Psi}_{12}(x),$
$\displaystyle\left[\Psi_{22}(x)\right]^{CP}$ $\displaystyle=$
$\displaystyle\tilde{\Psi}_{11}(x).$ (41)

This is because the $CP$ transformation is a discrete symmetry of the
Lagrangian (1), and hence must convert one fermion-soliton solution into
another.

##  5 Numerical results

The system of differential equations (15) – (17) with boundary conditions (19)
represents a mixed boundary value problem on the semi-infinite interval
$r\in\left[0,\infty\right)$. To solve this system, we use the numerical
methods provided in the Maple package [34].

Formally, the boundary value problem (15) – (19) depends on five parameters:
$\omega$, $m$, $g$, $e$, and $\chi_{1\,\text{vac}}$. However, it is easily
shown that the energy and Noether charge of the soliton depends nontrivially
on only three dimensionless parameters:

$\displaystyle
E\left(\omega,m,g,e,\chi_{1\,\text{vac}}\right)=mg^{-2}\tilde{E}\left(\tilde{\omega},\tilde{e},\tilde{\chi}_{1\,\text{vac}}\right),$
(42) $\displaystyle
Q_{N}\left(\omega,m,g,e,\chi_{1\,\text{vac}}\right)=g^{-2}\tilde{Q}_{N}\left(\tilde{\omega},\tilde{e},\tilde{\chi}_{1\,\text{vac}}\right),$
(43)

where $\tilde{\omega}=\omega/m$, $\tilde{e}=e/g$, and
$\tilde{\chi}_{1\,\text{vac}}=\chi_{1\,\text{vac}}/m$. Hence, without loss of
generality, we can set the parameters $m$ and $g$ equal to unity. In addition,
we set the dimensionless parameter $\tilde{\chi}_{1\,\text{vac}}=2\sqrt{2}$ in
these numerical calculations.

Figure 1:  Dependence of the soliton energy $\tilde{E}$ on the phase frequency
$\tilde{\omega}$ for several values of the gauge coupling constant
$\tilde{e}$.

Figure 1 shows the dependence of the soliton energy $\tilde{E}$ on the phase
frequency $\tilde{\omega}$ for several values of the gauge coupling constant
$\tilde{e}$. We see that for each $\tilde{e}$, the phase frequency
$\tilde{\omega}\in\left(\tilde{\omega}_{\min}(\tilde{e}),\tilde{\omega}_{\max}\right]$,
where $\tilde{\omega}_{\max}=2^{-1/2}\tilde{\chi}_{1\,\text{vac}}=2$. As
$\tilde{e}$ decreases, the minimum allowable frequency
$\tilde{\omega}_{\min}(\tilde{e})$ falls monotonically, reaching the limiting
value $\tilde{\omega}_{\min}(0)=0$. Using numerical methods, we can show that
as $\tilde{\omega}\rightarrow\tilde{\omega}_{\min}(\tilde{e})$, the soliton
energy

$\tilde{E}\left(\tilde{\omega},\tilde{e}\right)\sim
a(\tilde{e})(\tilde{\omega}-\tilde{\omega}_{\min}(\tilde{e}))^{-2},$ (44)

where $a(\tilde{e})$ is a function of $\tilde{e}$. It follows that the soliton
energy increases indefinitely as
$\tilde{\omega}\rightarrow\tilde{\omega}_{\min}(\tilde{e})$. On the other
hand, $\tilde{\omega}_{\min}(\tilde{e})$ monotonically increases with
$\tilde{e}$, meaning that there is a limiting value $\tilde{e}_{\max}$ for
which $\tilde{\omega}_{\min}(\tilde{e}_{\max})=\tilde{\omega}_{\max}$. It
follows that the nontopological soliton can exist only when
$\tilde{e}\in\left[0,\tilde{e}_{\max}\right)$.

In the subplot in Fig. 1, we can see the curves
$\tilde{E}(\tilde{\omega},\tilde{e})$ in the vicinity of the maximum allowable
phase frequency $\tilde{\omega}_{\max}$. All the curves
$\tilde{E}(\tilde{\omega},\tilde{e})$ in the subplot tend to zero as
$\tilde{\omega}\rightarrow\tilde{\omega}_{\max}$. It has been found
numerically that as
$\tilde{\omega}\rightarrow\tilde{\omega}_{\min}(\tilde{e})$, the soliton
energy

$\tilde{E}\left(\tilde{\omega},\tilde{e}\right)\approx
b(\tilde{e})\left(\tilde{\omega}_{\max}-\tilde{\omega}\right)^{1/2},$ (45)

where $b(\tilde{e})$ is an increasing function of $\tilde{e}$.

According to Eq. (21), the curves $\tilde{Q}_{N}(\tilde{\omega},\tilde{e})$
are related to the curves $\tilde{E}(\tilde{\omega},\tilde{e})$ by the
integral relation
$\tilde{Q}_{N}\left(\tilde{\omega},\tilde{e}\right)=-\int\nolimits_{\tilde{\omega}}^{\tilde{\omega}_{\max}}\tau^{-1}\partial_{\tau}\tilde{E}\left(\tau,\tilde{e}\right)d\tau$.
It follows that the curves $\tilde{Q}_{N}(\tilde{\omega},\tilde{e})$ will be
similar to the curves $\tilde{E}(\tilde{\omega},\tilde{e})$ shown in Fig. 1;
in particular, the behavior of the curves
$\tilde{Q}_{N}(\tilde{\omega},\tilde{e})$ in the neighborhoods of
$\tilde{\omega}_{\min}$ and $\tilde{\omega}_{\max}$ is the same as that of the
curves $\tilde{E}(\tilde{\omega},\tilde{e})$.

Figure 2:  Dependence of the soliton energy $\tilde{E}$ on the Noether charge
$\tilde{Q}_{N}$ for several values of the gauge coupling constant $\tilde{e}$.

Figure 2 shows the dependence of the soliton energy $\tilde{E}$ on the Noether
charge $\tilde{Q}_{N}$ for several values of the gauge coupling constant
$\tilde{e}$. In Fig. 2, the black dashed line
$\tilde{E}=\tilde{\omega}_{\max}\tilde{Q}_{N}$ corresponds to the energy of a
plane-wave configuration with a given Noether charge $\tilde{Q}_{N}$. We see
that for all values of $\tilde{e}$ considered here, the energies of the
solitons with a given $\tilde{Q}_{N}$ are lower than the energy of the
corresponding plane-wave configuration. It follows that these solitons are
stable against decay into massive charged $\phi$-mesons.

Figure 3:  Scaled dimensionless functions $10\times\tilde{f}(\tilde{r})$
(solid black), $\tilde{\Phi}(\tilde{r})$ (solid red),
$\tilde{\chi}_{1}(\tilde{r})$ (solid blue),
$10^{4}\times\tilde{\mathcal{E}}(\tilde{r})$ (dashed orange), and
$10^{3}\times\tilde{j}_{N}^{0}(\tilde{r})$ (dashed brown). The functions
correspond to the parameters $\tilde{e}=0.1$ and $\tilde{\omega}=0.32214$.

We have established that the energy $\tilde{E}(\tilde{\omega},\tilde{e})$ and
the Noether charge $\tilde{Q}_{N}(\tilde{\omega},\tilde{e})$ of the soliton
increase indefinitely as
$\tilde{\omega}\rightarrow\tilde{\omega}_{\min}(\tilde{e})$. In view of this,
it would be interesting to explore the behavior of the soliton fields in this
limit. To do this, we define the dimensionless profile functions
$\tilde{f}(\tilde{r})=m^{-1}gf(r)$,
$\tilde{\chi}_{1}(\tilde{r})=m^{-1}g\chi_{1}(r)$, and
$\tilde{\Phi}(\tilde{r})=m^{-1}g\Phi(r)$, where $\tilde{r}=m^{-1}r$. We also
define the dimensionless energy density
$\tilde{\mathcal{E}}(\tilde{r})=m^{-4}g^{2}\mathcal{E}(r)$ and the
dimensionless Noether charge density
$\tilde{j}_{N}^{0}(\tilde{r})=m^{-3}g^{2}j_{N}^{0}(r)$. Figure 3 shows these
dimensionless functions for parameter values $\tilde{e}=0.1$ and
$\tilde{\omega}=0.32214$. Note that $\tilde{\omega}=0.32214$ is the minimum
value of the phase frequency, which we were able to achieve by numerical
methods for $\tilde{e}=0.1$. We see that only $\tilde{f}(\tilde{r})$ and
$\tilde{j}^{0}_{N}(\tilde{r})$ are localised, whereas
$\tilde{\Phi}(\tilde{r})$, $\tilde{\chi}_{1}(\tilde{r})$, and
$\tilde{\mathcal{E}}(\tilde{r})$ are long-range, which is consistent with the
asymptotic forms in Eqs. (24), (25), and (26). We also see that
$\tilde{\chi}_{1}(\tilde{r})\approx 0$ in the interior of the soliton. The
long-range character ($\propto r^{-4}$) of the energy density
$\tilde{\mathcal{E}}$ arises from the gradient of the long-range electric
potential $\tilde{\Phi}$ and the gradient of the long-range neutral scalar
field $\tilde{\chi}_{1}$. According to Eq. (14), the local character of the
charge density $\tilde{j}_{N}^{0}$ is due to the local character of the
function $\tilde{f}$. Note that the electrostatic repulsion causes the
electric charge density to increase near the surface of the soliton.

Eq. (25) tells us that the asymptotics of $\tilde{\chi}_{1}$ is characterised
by the scalar charge $\tilde{Q}_{\text{s}}=gQ_{\text{s}}$. Using numerical
methods, we find that similarly to the energy $\tilde{E}$ and the Noether
charge $\tilde{Q}_{N}$, the scalar charge

$\tilde{Q}_{\text{s}}\left(\tilde{\omega},\tilde{e}\right)\propto\left(\tilde{\omega}-\tilde{\omega}_{\min}\left(\tilde{e}\right)\right)^{-2}$
(46)

as $\tilde{\omega}\rightarrow\tilde{\omega}_{\min}$. However, unlike the
Noether (electric) charge $Q_{N}$ ($Q=eQ_{N}$), the scalar charge
$Q_{\text{s}}$ is simply a definition and is not related to any symmetry of
model (1).

##  6 Conclusion

In the present paper, we show that an electrically charged nontopological
soliton exists in a version of $\mathcal{N}=1$ supersymmetric scalar
electrodynamics. A characteristic feature of this soliton is the presence of
two long-range fields, which slowly ($\propto r^{-1}$) tend to limiting
values: these are the electrostatic Coulomb field, and the electrically
neutral massless scalar field. The presence of these two long-range fields
leads to a modification of the intersoliton interaction in comparison with the
purely Coulomb case. Another feature of the soliton is that its energy and
electric charge take arbitrarily large values when the modulus of the phase
frequency tends to the minimum possible value. In contrast, the energy and
electric charge of the soliton vanish when the modulus of the phase frequency
tends to the maximum possible value.

We note that in the general case, the energy and electric charge of a
nontopological soliton cannot be arbitrarily large due to Coulomb repulsion
[10, 20]. We avoid this restriction because the attraction due to the massless
scalar field compensates for the Coulomb repulsion. A similar situation also
arises in the massless limit of the gauged Fridberg-Lee-Sirlin model [21, 23].
It is also worth noting that the electric charge and energy of the dyon
(electrically charged magnetic monopole) also cannot be arbitrarily large in
the general non-BPS case [35]. Only in the BPS limit, when the scalar field of
the dyon becomes massless, can the energy and electric charge take arbitrarily
large values.

The $\mathcal{N}=1$ supersymmetry of the model makes it possible to obtain
expressions for the fermionic zero modes in terms of bosonic fields of the
soliton. The fermionic zero modes are bound states of the fermion-soliton
system, and their components that correspond to the long-range bosonic fields
are also long-range. In accordance with the number of $\mathcal{N}=1$
supersymmetry generators, the number of independent fermionic zero modes of
the soliton is four. The fermionic zero modes of two solitons with opposite
electric charges are related by the $CP$ transformation.

In this work, we have investigated a nontopological soliton of an
$\mathcal{N}=1$ supersymmetric Abelian gauge model. It is known [36, 37],
however, that nontopological solitons can also exist in non-Abelian gauge
models. In particular, it was shown in Ref. [38] that an electrically charged
nontopological soliton exists in the Weinberg-Salam model of electroweak
interactions. This model allows for $\mathcal{N}=1$ supersymmetric extension,
and its fermionic sector contains both massive ($e$, $\mu$, $\tau$) and
massless ($\nu_{e}$, $\nu_{\mu}$, $\nu_{\tau}$) fermions. The bosonic
superpartners of the neutrinos (sneutrinos) also have zero masses. We can
assume that, similarly to the nonsupersymmetric case [38], an electrically
charged nontopological soliton also exists in this model, meaning that some
properties of this soliton will be similar to those studied in this work. In
particular, in addition to the long-range Coulomb field, this soliton will
have long-range fields of massless sneutrinos. Furthermore, it will be
possible to express the fermionic zero modes of this soliton in terms of its
bosonic fields.

## Acknowledgements

This work was supported by the Russian Science Foundation, grant No
23-11-00002.

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## Figure captions

Fig. 1. Dependence of the soliton energy $\tilde{E}$ on the phase frequency
$\tilde{\omega}$ for several values of the gauge coupling constant
$\tilde{e}$.

Fig. 2. Dependence of the soliton energy $\tilde{E}$ on the Noether charge
$\tilde{Q}_{N}$ for several values of the gauge coupling constant $\tilde{e}$.

Fig. 3. Scaled dimensionless functions $10\times\tilde{f}(\tilde{r})$ (solid
black), $\tilde{\Phi}(\tilde{r})$ (solid red), $\tilde{\chi}_{1}(\tilde{r})$
(solid blue), $10^{4}\times\tilde{\mathcal{E}}(\tilde{r})$ (dashed orange),
and $10^{3}\times\tilde{j}_{N}^{0}(\tilde{r})$ (dashed brown). The functions
correspond to the parameters $\tilde{e}=0.1$ and $\tilde{\omega}=0.32214$.