# Fermion scattering on topological solitons in the nonlinear $O(3)$
$\sigma$-model

A. Yu. Loginov<EMAIL_ADDRESS>Tomsk State University of Control
Systems and Radioelectronics, 634050 Tomsk, Russia

###### Abstract

The scattering of Dirac fermions in the background fields of topological
solitons of the $(2+1)$-dimensional nonlinear $O(3)$ $\sigma$-model is studied
using both analytical and numerical methods. General formulae describing
fermion scattering are obtained and the symmetry properties of the partial
scattering amplitudes and elements of the $S$-matrix are determined. Within
the framework of the Born approximation, the scattering amplitudes,
differential cross-sections, and total cross-sections of fermion-soliton
scattering are obtained in analytical forms, and their symmetry properties and
asymptotic behavior are investigated. The dependences of the first several
partial elements of the $S$-matrix on the momentum of the fermion are obtained
using numerical methods, and some properties of these dependences are
ascertained and discussed.

##  I  Introduction

Topological solitons of $(2+1)$-dimensional field models [1, 2, 3] play an
important role in field theory, high-energy physics, condensed matter physics,
cosmology, and hydrodynamics. Of these, the vortex solutions of the effective
theory of superconductivity [4] and those of the
$\left(2+1\right)$-dimensional Abelian Higgs model [5] are notable. Another
important example is provided by the soliton solutions of the
$(2+1)$-dimensional nonlinear $O\left(3\right)$ $\sigma$-model [6]. The
invariance of the static energy functional of the nonlinear $O\left(3\right)$
$\sigma$-model under scale transformations results in a corresponding zero
mode of the quadratic fluctuation operator in the functional neighborhood of a
given soliton solution. As a consequence, the $\sigma$-model possesses a one-
parameter family of soliton solutions with the same energy but different
sizes, rather than a soliton solution of fixed size.

Derrick’s theorem [7] offers a number of ways to fix the size of the soliton.
One of these is the addition of a potential term and a fourth-order term in
the field derivatives to the Lagrangian of the nonlinear $O\left(3\right)$
$\sigma$-model. The modified nonlinear $O\left(3\right)$ $\sigma$-model is
known as the baby Skyrme model, and its topological solitons [8, 9, 10] (known
as baby Skyrmions) have a fixed size. The baby Skyrme model and other
modifications to the nonlinear $O\left(3\right)$ $\sigma$-model have
applications in condensed matter physics [11, 12, 13, 14]. This model also has
applications in cosmology; it was shown in Refs. [15, 16, 17] that in the six-
dimensional Einstein-Skyrme model, the baby Skyrmion solutions realize warped
compactification of the extra dimensions and gravity localization on the four-
dimensional brane for the negative bulk cosmological constant.

The baby Skyrme model is a planar analogue of the original $(3+1)$-dimensional
Skyrme model [18], which can be regarded as an approximate, low-energy
effective theory of QCD that becomes exact as the number of quark colours
becomes large [19, 20]. It was shown in Refs. [21, 22] that nucleons and their
low-lying excitations, which are three-quark bound states from the viewpoint
of QCD, arise as a topological soliton of the Skyrme model (known as the
Skyrmion) and its low-lying excitations, respectively, whereas pions
correspond to linearized fluctuations in the model’s chiral scalar field.

Topological solitons formed from scalar fields can interact with fermion
fields via the Yukawa interaction. This fermion-soliton interaction may have a
significant impact on both the scalar field of the soliton and the fermion
field. In particular, it was shown in Refs. [23, 24, 25] that the Yukawa
interaction between the fermion and scalar fields of the $(3+1)$-dimensional
chiral-invariant linear $\sigma$-model results in the existence of the chiral
soliton, a stable configuration of the interacting scalar and fermion fields
that possesses bound fermion states. It was also shown in Refs. [26, 27, 28]
that an external chiral field can polarize the fermion vacuum; this
polarization can be interpreted as a contribution to the baryon charge of a
chiral soliton [24, 25]. Fermion bound states also exist in the background
field of a Skyrmion, and the properties of these bound states were
investigated in Refs. [29, 30]. The main feature of the fermion-soliton
interaction found in Refs. [23, 24, 25, 29, 30] is that the spectrum of the
Dirac operator shows a spectral flow of the eigenvalues, where the number of
zero-crossing normalized bound modes is equal to the winding number of the
soliton. The planar baby Skyrmions inherit this feature of the fermion-soliton
interaction. It was shown in Refs. [15, 16, 17] that in the background field
of a baby Skyrmion, the Dirac operator also shows a spectral flow of the
eigenvalues, and the number of zero-crossing normalized bounded modes turns
out to be equal to the winding number of the baby Skyrmion. This property of
the Dirac operator is preserved when the backreaction of the fermion field
coupled with the baby Skyrmion is taken into account [31].

Except for the Dirac sea fermions, only fermion bounded states belonging to
the discrete spectrum of the Dirac operator were considered in the works cited
above. One main aim of the present paper is to consider the fermion scattering
states belonging to the continuous spectrum of the Dirac operator. The
topological solitons of the nonlinear $O(3)$ $\sigma$-model are chosen as
background fields for the Dirac operator, since the nonlinear $O(3)$
$\sigma$-model is one of the few models for which exact analytical soliton
solutions are known. Having obtained the analytical soliton solution as a
background field of the Dirac operator makes it possible to obtain some
analytical results relating to fermion scattering. In particular, the
scattering amplitudes, differential cross-sections, and total cross-sections
can be obtained in an analytical form within the framework of the first Born
approximation.

This paper is structured as follows. In Sec. II, we describe briefly the
Lagrangian, symmetries, field equations, and topological solitons of the
nonlinear $O(3)$ $\sigma$-model. In Sec. III, general properties of fermion
scattering are considered, such as the forms of the fermion scattering wave
functions, their asymptotic behavior, and the symmetry properties of the
partial elements of the $S$-matrix. In Sec. IV, we give an analytical
description of fermion scattering within the framework of the first Born
approximation. In Sec. V, we present numerical results for the first several
partial elements of the $S$-matrix. In the final section, we briefly summarize
the results obtained in this work. Appendix A contains some necessary
information about the plane-wave and cylinder-wave states of free fermions. A
resonance behavior of some partial elements of the $S$-matrix is explained in
Appendix B.

Throughout this paper, we use the natural units $c=1$, $\hbar=1$.

##  II  Lagrangian, field equations, and topological solitons of the model

The model we are interested in is the $(2+1)$-dimensional nonlinear $O(3)$
$\sigma$-model, which includes a fermion field. The scalar isovector field
$\bm{\phi}$ of the model interacts with the spinor-isospinor fermion field
$\psi$ via the Yukawa coupling, leading to the Lagrangian

$\mathcal{L}=\frac{1}{2}\partial_{\mu}\bm{\phi}\bm{\cdot}\\!\partial^{\mu}\bm{\phi}+\frac{\lambda}{2}\left(\bm{\phi}\bm{\cdot}\\!\bm{\phi}-H^{2}\right)+i\bar{\psi}\gamma^{\mu}\partial_{\mu}\psi+h\mathbf{\phi}\bm{\cdot}\bar{\psi}\bm{\tau}\psi,$
(1)

where $h$ is the Yukawa coupling constant and $\lambda$ is the Lagrange
multiplier, which imposes the constraint
$\bm{\phi}\bm{\cdot}\\!\bm{\phi}=H^{2}$ on the scalar isovector field
$\bm{\phi}$. In $(2+1)$ dimensions, we shall use the following Dirac matrices:

$\gamma^{0}=\sigma_{3},\;\gamma^{1}=-i\sigma_{1},\;\gamma^{2}=-i\sigma_{2},$
(2)

where $\sigma_{i}$ are the Pauli matrices. To distinguish the Pauli matrices
$\sigma_{k}$ acting on the spinor index $i$ of the fermion field $\psi_{i,a}$
from those acting on the isospinor index $a$, we denote the latter as
$\tau_{k}$.

In natural units, the $(2+1)$-dimensional scalar field $\bm{\phi}$ has
dimension of $\text{mass}^{1/2}$. We adopt the squared parameter $H^{2}$ as
the energy (mass) unit, and use dimensionless variables:

$\displaystyle\mathbf{\phi}$ $\displaystyle\rightarrow$ $\displaystyle
H\mathbf{\phi},\;\psi\rightarrow H^{2}\psi,\;\mathbf{x}\rightarrow
H^{-2}\mathbf{x},$ $\displaystyle t$ $\displaystyle\rightarrow$ $\displaystyle
H^{-2}t,\;h\rightarrow Hh,\;\lambda\rightarrow H^{4}\lambda.$ (3)

In these new variables, the Lagrangian (1) takes the form

$\mathcal{L}=\frac{1}{2}\partial_{\mu}\bm{\phi}\bm{\cdot}\partial^{\mu}\bm{\phi}+\frac{\lambda}{2}\left(\bm{\phi}\bm{\cdot}\\!\bm{\phi}-1\right)+i\bar{\psi}\gamma^{\mu}\partial_{\mu}\psi+h\bm{\phi}\bm{\cdot}\bar{\psi}\bm{\tau}\psi.$
(4)

By varying the action $S=\int\mathcal{L}d^{2}xdt$ in $\bm{\phi}$ and
$\bar{\psi}$, we obtain the field equations of the model:

$\displaystyle\\!\\!\\!\partial_{\mu}\partial^{\mu}\bm{\phi}-\bm{\phi}\left(\bm{\phi}\\!\bm{\cdot}\\!\partial_{\mu}\partial^{\mu}\bm{\phi}\right)-h\bm{\phi}\left(\bm{\phi}\bm{\cdot}\bar{\psi}\bm{\tau}\psi\right)\\!+\\!h\bar{\psi}\bm{\tau}\psi$
$\displaystyle=0,$ (5) $\displaystyle
i\gamma^{\mu}\partial_{\mu}\psi+h\bm{\phi}\\!\bm{\cdot}\\!\bm{\tau}\psi$
$\displaystyle=0,$ (6)

where we use the constraint $\bm{\phi}\bm{\cdot}\\!\bm{\phi}=1$ to express the
Lagrange multiplier $\lambda$ in terms of the fields $\bm{\phi}$ and $\psi$.

Model (1) is invariant under global $SU(2)$ transformations of the isovector
field $\bm{\phi}$ and the isospinor field $\psi$. Its classical bosonic vacuum
is an arbitrary constant scalar field $\bm{\phi}(x)=\bm{\phi}_{\text{vac}}$,
which can be taken as $\bm{\phi}_{\text{vac}}=(0,0,-1)$. We see that the
vacuum $\bm{\phi}_{\text{vac}}$ breaks the original $SU(2)$ symmetry group of
model (1) to its $U(1)$ subgroup. Any finite energy field configuration
$\bm{\phi}(x)$ must tend to $\bm{\phi}_{\text{vac}}$ at spatial infinity. It
follows that the finite energy field configurations of model (1) are split
into topological classes that are elements of the homotopic group
$\pi_{2}\left(S^{2}\right)=\mathbb{Z}$, and are consequently characterized by
an integer winding number $n$.

It is well known [6] that the nonlinear $O(3)$ $\sigma$-model possesses a
variety of topological soliton solutions. These soliton solutions exist in
each topological sector of the model, and are absolute minima of the energy
functional. In the topological sector with a given nonzero winding number $n$,
the maximally symmetric soliton solution has the form

$\bm{\phi}\left(r,\theta\right)=\begin{pmatrix}\sin\left(f\left(r\right)\right)\cos\left(n\theta\right)\\\
\sin\left(f\left(r\right)\right)\sin\left(n\theta\right)\\\
\cos\left(f\left(r\right)\right)\end{pmatrix}\\!,$ (7)

where $r$ and $\theta$ are the polar coordinates, and the profile function is

$f\left(r\right)=2\arctan\left[\left(r/r_{0}\right)^{\left|n\right|}\right]\\!.$
(8)

Soliton solution (7) is invariant under the cyclic subgroup $\mathbb{Z}_{n}$
$(\theta\rightarrow\theta+2\pi k/n,\;k=0,\ldots,n-1)$ of the spatial rotation
group $SO(2)$. It is also invariant under the simultaneous action of spatial
rotation through an angle $\delta$ and internal rotation about the third
isotopic axis $\mathbf{u}_{3}=(0,0,1)$ through an angle $-n\delta$

$\bm{\phi}\left(r,\theta\right)=R_{\mathbf{u}_{3}}\left(-n\delta\right)\bm{\phi}\left(r,\theta+\delta\right),$
(9)

where the rotation matrix is

$R_{\mathbf{u}_{3}}\left(\varphi\right)=\begin{pmatrix}\cos\left(\varphi\right)&-\sin\left(\varphi\right)&0\\\
\sin\left(\varphi\right)&\cos\left(\varphi\right)&0\\\ 0&0&1\end{pmatrix}\\!.$
(10)

Note that soliton solution (7) depends on the positive parameter $r_{0}$,
which can be interpreted as the spatial size of the soliton. However, the
energy of the soliton solution

$\displaystyle E$ $\displaystyle=$ $\displaystyle
2\pi\int\limits_{0}^{\infty}\mathcal{E}\left(r\right)rdr=2\pi\int\limits_{0}^{\infty}\biggl{[}\frac{4n^{2}}{r^{2}}\left(\frac{r}{r_{0}}\right)^{2\left|n\right|}\biggr{.}$
(11)
$\displaystyle\biggl{.}\times\biggl{(}1+\left(\frac{r}{r_{0}}\right)^{2\left|n\right|}\biggr{)}^{-2}\biggr{]}rdr=4\pi\left|n\right|$

does not depend on $r_{0}$. This is because the bosonic part of the action
$S=\int\mathcal{L}d^{3}x$ of model (1) is invariant under scale
transformations $\mathbf{r}\rightarrow a\mathbf{r}$.

##  III  Fermions in the background field of the soliton

We consider fermion scattering on the topological soliton of the nonlinear
$O(3)$ $\sigma$-model in the external field approximation. In this
approximation, the backreaction of a fermion on the soliton’s field is
neglected, and the fermion-soliton scattering is described solely by the Dirac
equation (6). We can rewrite the Dirac equation in the Hamiltonian form

$i\frac{\partial\psi_{i,a}}{\partial t}=H_{i,a;\,j,b}\psi_{j,b},$ (12)

where the Hamiltonian

$H_{i,a;\,j,b}=\alpha_{ij}^{k}\otimes\mathbb{I}_{ab}\left(-i\partial_{k}\right)-h\beta_{ij}\otimes\phi^{c}\tau_{ab}^{c},$
(13)

$\mathbb{I}$ is the $2\times 2$ identity matrix,
$\alpha^{1}=\gamma^{0}\gamma^{1}=\sigma_{2}$,
$\alpha^{2}=\gamma^{0}\gamma^{2}=-\sigma_{1}$, and
$\beta=\gamma^{0}=\sigma_{3}$. In Eqs. (12) and (13), all spin and isospin
indices are explicitly shown, and the summation convention over repeated
indices is implied.

The invariance of soliton solution (7) under combined transformation (9)
results in the existence of the conserved generalized angular momentum or
grand spin, which we denote by $K_{3}$:

$\left[H,K_{3}\right]=0,$ (14)

where

$K_{3\,i,a;\,j,b}\\!=\\!\mathbb{I}_{ij}\otimes\mathbb{I}_{ab}\epsilon_{kl}x_{k}(-i\partial_{l})\\!+\\!\frac{1}{2}\sigma_{3\,ij}\otimes\mathbb{I}_{ab}\\!+\\!\frac{n}{2}\mathbb{I}_{ij}\mathbf{\otimes}\tau_{3\,ab}$
(15)

and the $2\times 2$ antisymmetric matrix $\epsilon_{kl}=i\sigma_{2\,kl}$. It
follows from Eq. (15) that the grand spin $K_{3}$ is the sum of the orbital
part
$L_{3}=\mathbb{I}\otimes\mathbb{I}\epsilon_{kl}x_{k}\left(-i\partial_{l}\right)$,
the spin part $s_{3}=\left(\sigma_{3}/2\right)\otimes\mathbb{I}$, and the
isospin part $nI_{3}=n\mathbb{I}\otimes\tau_{3}/2$. None of the terms $L_{3}$,
$s_{3}$, and $I_{3}$ is conserved separately in the background field of
soliton solution (7). Using Eq. (15), we can obtain the eigenfunctions
$\psi^{\left(m\right)}$ of the operator $K_{3}$. These eigenfunctions satisfy
the relation

$K_{3}\psi_{ia}^{\left(m\right)}=m\psi_{ia}^{\left(m\right)}$ (16)

and can be written in the compact matrix form

$\psi_{ia}^{\left(m\right)}=\left(\begin{array}[]{cc}c_{11}\left(r\right)e^{il_{11}\theta}&c_{12}\left(r\right)e^{il_{12}\theta}\\\
c_{21}\left(r\right)e^{il_{21}\theta}&c_{22}\left(r\right)e^{il_{22}\theta}\end{array}\right)\\!,$
(17)

where the orbital quantum numbers

$\displaystyle l_{11}$ $\displaystyle=$ $\displaystyle\left(2m-n-1\right)/2,$
(18a) $\displaystyle l_{12}$ $\displaystyle=$
$\displaystyle\left(2m+n-1\right)/2,$ (18b) $\displaystyle l_{21}$
$\displaystyle=$ $\displaystyle\left(2m-n+1\right)/2,$ (18c) $\displaystyle
l_{22}$ $\displaystyle=$ $\displaystyle\left(2m+n+1\right)/2,$ (18d)

and $c_{ia}\left(r\right)$ are some radial wave functions. The eigenfunctions
$\psi_{ia}^{\left(m\right)}(r,\theta)$ should be single-valued functions of
the polar angle $\theta$. This fact and Eq. (18) tell us that the eigenvalues
$m$ are integers for odd winding numbers $n$, and half-integers for even
winding numbers $n$.

Note that, in general, the full system of field equations (5) and (6) cannot
be described in terms of the ansatz given by Eqs. (7) and (17). It can be
shown that this ansatz is compatible with Eqs. (5) and (6) only under the
condition $\text{Im}\left[c_{11}c_{12}^{\ast}-c_{21}c_{22}^{\ast}\right]=0$.
Obviously, the real (modulo a common constant phase factor) radial wave
functions $c_{ia}(r)$ will satisfy this condition. In particular, the radial
wave functions of bound fermionic states, if any, can always be chosen to be
real. This is because these wave functions satisfy a system of linear
differential equations with real $r$-dependent coefficients and real boundary
conditions $c_{ia}(r)\rightarrow 0$ as $r\rightarrow\infty$. In contrast, the
radial wave functions of the fermionic scattering states satisfy some complex
asymptotic boundary condition (i.e., superposition of incident and outgoing
waves), and therefore do not need to satisfy the reality condition
$\text{Im}\left[c_{11}c_{12}^{\ast}-c_{21}c_{22}^{\ast}\right]=0$. When the
backreaction terms are neglected in Eq. (5) (as in the external field
approximation), the ansatz given by Eqs. (7) and (17) becomes valid without
restriction.

We now discuss the symmetry properties of the Dirac equation (12) under
discrete transformations. Unlike the Dirac equation (70) that describes the
states of free fermions, Eq. (12) is not invariant under $C$-transformation
(73). Instead, the Dirac equation (12) transforms into one that describes
fermions in the background field of the topological soliton with the opposite
winding number $n$. The Dirac equation (12) is also not invariant under the
modified $C$-transformation

$\psi\left(t,\mathbf{x}\right)\rightarrow\psi^{C^{\prime}}\left(t,\mathbf{x}\right)=i\gamma^{1}\mathbb{\otimes}\tau_{2}\psi^{\ast}\left(t,\mathbf{x}\right),$
(19)

which acts on the both spin and isospin indices of the fermion field. It can
easily be shown that $C^{\prime}$-transformation (19) changes the sign of the
Yukawa coupling constant $h$ in Eq. (13). The reason for the noninvariance of
Eq. (12) under $C^{\prime}$-transformation (19) is that the fundamental
representation of the $SU(2)$ group is pseudoreal. The noninvariance of Eq.
(12) under any version of the $C$-transformation leads to substantial
differences between fermion and antifermion scattering in the background field
of the soliton.

Under the inversion $\mathbf{x}\rightarrow-\mathbf{x}$, the isospin component
$\phi_{3}$ of soliton solution (7) does not change, whereas both the
$\phi_{1}$ and $\phi_{2}$ components are unchanged for even $n$ and the sign
changes for odd $n$. It follows that the Dirac equation (12) is invariant
under the $P$-transformation

$\psi\left(t,\mathbf{x}\right)\rightarrow\psi^{P}\left(t,\mathbf{x}\right)=\gamma_{0}\otimes\tau_{3}^{n}\psi\left(t,-\mathbf{x}\right),$
(20)

where $n$ is the winding number of the soliton and we use the fact that
$\tau_{3}^{n}$ is $\mathbb{I}$ for even $n$ and $\tau_{3}$ for odd $n$. Note
that the eigenfunctions $\psi^{\left(m\right)}$ of the grand spin $K_{3}$ are
also eigenfunctions of the $P$-transformation:

$\psi^{\left(m\right)P}=\left(-1\right)^{m\pm\frac{n+1}{2}}\psi^{\left(m\right)},$
(21)

where the upper (lower) sign is used for odd (even) winding numbers $n$.
Hence, unlike the $(3+1)$-dimensional case, parity conservation does not lead
to new restrictions on fermion scattering. This is because, unlike the
$(3+1)$-dimensional case, the inversion is equivalent to a rotation through an
angle $\pi$ in $(2+1)$ dimensions. Let us denote by the symbol $\Pi_{2}$ the
operation of coordinate reflection about the $Ox_{1}$ axis:
$\left(x_{1},x_{2}\right)\rightarrow\left(x_{1},-x_{2}\right)$. It can then
easily be shown that the Dirac equation (12) is invariant under the combined
$\Pi_{2}T$ transformation

$\psi\left(t,\mathbf{x}\right)\rightarrow\psi^{\Pi_{2}T}\left(t,\mathbf{x}\right)=\psi^{\ast}\left(-t,x_{1},-x_{2}\right).$
(22)

Finally, the the Dirac equation (12) is invariant under the combined
transformation

$\psi\left(t,\mathbf{x}\right)\rightarrow\gamma^{2}\otimes\tau_{2}\psi\left(t,x_{1},-x_{2}\right),$
(23)

which changes the signs of the orbital, spin, and isospin parts of grand spin
(15). Note that the invariance of the Dirac equation (12) under
transformations (22) and (23) results from the specific property
$\bm{\tau}\\!\bm{\cdot}\\!\bm{\phi}\left(x_{1},-x_{2}\right)=\bm{\tau}^{\ast}\\!\\!\bm{\cdot}\\!\bm{\phi}\left(x_{1},x_{2}\right)$
of soliton field configuration (7).

We now turn to the radial dependence of the eigenfunctions of the grand spin.
By substituting Eqs. (7), (8), and (17) into the Dirac equation (12), we
obtain the system of linear differential equations for the radial wave
functions $c_{ia}(r)$

$\frac{dc_{ia}}{dr}=\Lambda_{ia;jb}c_{jb},$ (24)

where the matrix

$\Lambda_{ia;jb}=\left(\begin{array}[]{cccc}\dfrac{l_{11}}{r}&0&\varepsilon+h-\dfrac{2h}{1+\left(r/r_{0}\right){}^{2\left|n\right|}}&-\dfrac{2h\left(r/r_{0}\right){}^{\left|n\right|}}{1+\left(r/r_{0}\right){}^{2\left|n\right|}}\\\
0&\dfrac{l_{12}}{r}&-\dfrac{2h\left(r/r_{0}\right){}^{\left|n\right|}}{1+\left(r/r_{0}\right){}^{2\left|n\right|}}&\varepsilon-h+\dfrac{2h}{1+\left(r/r_{0}\right){}^{2\left|n\right|}}\\\
-\varepsilon+h-\dfrac{2h}{1+\left(r/r_{0}\right){}^{2\left|n\right|}}&-\dfrac{2h\left(r/r_{0}\right){}^{\left|n\right|}}{1+\left(r/r_{0}\right){}^{2\left|n\right|}}&-\dfrac{l_{21}}{r}&0\\\
-\dfrac{2h\left(r/r_{0}\right){}^{\left|n\right|}}{1+\left(r/r_{0}\right){}^{2\left|n\right|}}&-\varepsilon-h+\dfrac{2h}{1+\left(r/r_{0}\right){}^{2\left|n\right|}}&0&-\dfrac{l_{22}}{r}\end{array}\right)$
(25)

and $\varepsilon$ is the energy of the fermionic state.

Eqs. (24) and (25) define the system of four linear differential equations of
first order. In the general case, the solution to this system depends on four
independent parameters. However, fermionic states should be described by
regular solutions to system (24). It can be shown that the regularity
condition imposed at the origin reduces the number of the independent
parameters from four to two. In the neighborhood of the origin, a regular
solution to system (24) has the form

$\displaystyle c_{11}\left(r\right)$ $\displaystyle=$ $\displaystyle
r^{\left|l_{11}\right|}\left(a_{0}+O\left(r^{2}\right)\right),$ (26a)
$\displaystyle c_{12}\left(r\right)$ $\displaystyle=$ $\displaystyle
r^{\left|l_{12}\right|}\left(b_{0}+O\left(r^{2}\right)\right),$ (26b)
$\displaystyle c_{21}\left(r\right)$ $\displaystyle=$ $\displaystyle
r^{\left|l_{21}\right|}\left(c_{0}+O\left(r^{2}\right)\right),$ (26c)
$\displaystyle c_{22}\left(r\right)$ $\displaystyle=$ $\displaystyle
r^{\left|l_{22}\right|}\left(d_{0}+O\left(r^{2}\right)\right),$ (26d)

where only two of the four parameters $a_{0}$, $b_{0}$, $c_{0}$, and $d_{0}$
are independent. Note that each of the radial wave functions $c_{ia}(r)$ has a
definite parity under the substitution $r\rightarrow-r$. This is because in
Eq. (24), the differential operator $d/dr$ and each of the nonzero elements of
the matrix $\Lambda_{ia;jb}$ also have a definite parity under this
substitution.

We present the linear relations between the parameters $a_{0}$, $b_{0}$,
$c_{0}$, and $d_{0}$ for the cases of the first odd $(n=1)$ and the first even
$(n=2)$ soliton winding numbers. By choosing $b_{0}$ and $c_{0}$ as the
independent parameters, we obtain for $a_{0}$ and $d_{0}$:

$\displaystyle a_{0}$ $\displaystyle=$ $\displaystyle-
c_{0}\frac{2m}{\varepsilon+h}+\frac{c_{0}}{2}\left(\varepsilon-h\right)\delta_{m0},$
$\displaystyle d_{0}$ $\displaystyle=$ $\displaystyle-
b_{0}\frac{\varepsilon-h}{2(m+1)}+c_{0}\frac{2\left|h\right|}{r_{0}(\varepsilon+h)}\frac{m}{m+1}$
(27)

in the case where $n=1$ and $m\geq 0$, and

$\displaystyle a_{0}$ $\displaystyle=$ $\displaystyle-
c_{0}\frac{2\left(m-1/2\right)}{\varepsilon+h}+\frac{c_{0}}{2}\left(\varepsilon-h\right)\delta_{m\frac{1}{2}},$
$\displaystyle\;d_{0}$ $\displaystyle=$ $\displaystyle-
b_{0}\frac{\varepsilon-h}{3+2m}+c_{0}\frac{2\left|h\right|}{r_{0}^{2}\left(\varepsilon+h\right)}\frac{2m-1}{2m+3}$
(28)

in the case where $n=2$ and $m\geq 1/2$. The expressions obtained from Eqs.
(27) and (28) by the substitution

$a_{0}\rightarrow d_{0},\,d_{0}\rightarrow a_{0},\,c_{0}\rightarrow-
b_{0},\,b_{0}\rightarrow-c_{0},\,m\rightarrow-m.$ (29)

are valid for negative $m$. Finally, the expressions obtained from Eqs. (27)
and (28) by the substitution

$a_{0}\rightarrow b_{0},\,b_{0}\rightarrow a_{0},\,c_{0}\rightarrow
d_{0},\,d_{0}\rightarrow c_{0},\,h\rightarrow-h$ (30)

can be used in the cases $n=-1$ and $n=-2$, respectively.

It follows from Appendix A that the third component $I_{3}$ of the isospin is
conserved in the case of free fermions. The fermion scattering on the soliton
can therefore be regarded as a transition from the free “in” state, with
spatial momentum $\mathbf{p}=(p,0)$ and isospin $I_{3}$, to the free “out”
state, with spatial momentum
$\mathbf{p}^{\prime}=(p\cos(\theta),p\sin(\theta))$ and isospin
$I_{3}^{\prime}$. A transition without (with) a change in the isospin can be
considered to be elastic (inelastic). According to the theory of scattering
[32, 33], the scattering state that corresponds to the “in” isospin $I_{3}$
and the “out” isospin $I_{3}^{\prime}$ is described asymptotically by the wave
function

$\Psi_{I_{3}^{\prime},I_{3}}\sim\psi_{\varepsilon,\mathbf{p},I_{3}}\delta_{I_{3}^{\prime},I_{3}}+\frac{1}{\sqrt{2\varepsilon}}u_{\varepsilon,\mathbf{p}^{\prime},I_{3}^{\prime}\,}f_{I_{3}^{\prime},I_{3}}\left(p,\theta\right)\frac{e^{ipr}}{\sqrt{-ir}},$
(31)

where $\delta_{I_{3}^{\prime},I_{3}}$ is the Kronecker delta,
$\psi_{\varepsilon,\mathbf{p},I_{3}}$ is wave function (76) of the “in” state
with momentum $\mathbf{p}=(p,0)$ and isospin $I_{3}$,
$u_{\varepsilon,\mathbf{p}^{\prime},I_{3}^{\prime}}$ is the spinor-isospinor
amplitude of wave function (76) of the “out” state with momentum
$\mathbf{p}^{\prime}=(p\cos(\theta),p\sin(\theta))$ and isospin
$I_{3}^{\prime}$, $f_{I_{3}^{\prime},I_{3}}\left(p,\theta\right)$ is the
scattering amplitude, and the factor $-i$ under the square root sign is
introduced for convenience. Using Eqs. (81), (85), (86), and standard methods
from the theory of scattering [32, 33], we obtain an expansion of the
scattering amplitude $f_{I_{3}^{\prime},I_{3}}\left(p,\theta\right)$ in terms
of the partial scattering amplitudes
$f_{I_{3}^{\prime},I_{3}}^{\left(m\right)}\left(p\right)$

$f_{I_{3}^{\prime},I_{3}}\left(p,\theta\right)=e^{-iI_{3}^{\prime}\left(n+1\right)\theta}\sum\limits_{m}f_{I_{3}^{\prime},I_{3}}^{\left(m\right)}\left(p\right)e^{im\theta}.$
(32)

In turn, the partial scattering amplitudes
$f_{I_{3}^{\prime},I_{3}}^{\left(m\right)}\left(p\right)$ are expressed in
terms of the partial elements of the $S$-matrix

$\displaystyle f_{1/2,1/2}^{\left(m\right)}\left(p\right)$ $\displaystyle=$
$\displaystyle\frac{1}{i\sqrt{2\pi
p}}\left(S_{1/2,1/2}^{\left(m\right)}\left(p\right)-1\right),$ (33a)
$\displaystyle f_{-1/2,1/2}^{\left(m\right)}\left(p\right)$ $\displaystyle=$
$\displaystyle\frac{1}{i\sqrt{2\pi
p}}S_{-1/2,1/2}^{\left(m\right)}\left(p\right),$ (33b) $\displaystyle
f_{1/2,-1/2}^{\left(m\right)}\left(p\right)$ $\displaystyle=$
$\displaystyle\frac{1}{i\sqrt{2\pi
p}}S_{1/2,-1/2}^{\left(m\right)}\left(p\right),$ (33c) $\displaystyle
f_{-1/2,-1/2}^{\left(m\right)}\left(p\right)$ $\displaystyle=$
$\displaystyle\frac{1}{i\sqrt{2\pi
p}}\left(S_{-1/2,-1/2}^{\left(m\right)}\left(p\right)-1\right)\\!.$ (33d)

The importance of the partial elements of the $S$-matrix is that the
asymptotic behavior of the radial wave functions $c_{ia}(r)$ can be expressed
in terms of these as $r\rightarrow\infty$:

$c_{ia}(r)\sim\frac{\left(-1\right)^{1/4}}{\sqrt{2\pi
pr}}\begin{pmatrix}-i\sqrt{\frac{\varepsilon+h}{2\varepsilon}}\left[i^{-n}\left(-1\right)^{m}e^{-ipr}+S_{1/2,1/2}^{\left(m\right)}\left(p\right)e^{ipr}\right]&-\sqrt{\frac{\varepsilon-h}{2\varepsilon}}S_{-1/2,1/2}^{\left(m\right)}\left(p\right)e^{ipr}\\\
\sqrt{\frac{\varepsilon-h}{2\varepsilon}}\left[i^{-n}\left(-1\right)^{m+1}e^{-ipr}+S_{1/2,1/2}^{\left(m\right)}\left(p\right)e^{ipr}\right]&-i\sqrt{\frac{\varepsilon+h}{2\varepsilon}}S_{-1/2,1/2}^{\left(m\right)}\left(p\right)e^{ipr}\end{pmatrix}$
(34)

for the “in” isospin $I_{3}=1/2$, and

$c_{ia}(r)\sim\frac{\left(-1\right)^{1/4}}{\sqrt{2\pi
pr}}\begin{pmatrix}-i\sqrt{\frac{\varepsilon+h}{2\varepsilon}}S_{1/2,-1/2}^{\left(m\right)}\left(p\right)e^{ipr}&-\sqrt{\frac{\varepsilon-h}{2\varepsilon}}\left[i^{n}\left(-1\right)^{m}e^{-ipr}+S_{-1/2,-1/2}^{\left(m\right)}\left(p\right)e^{ipr}\right]\\\
\sqrt{\frac{\varepsilon-h}{2\varepsilon}}S_{1/2,-1/2}^{\left(m\right)}\left(p\right)e^{ipr}&-i\sqrt{\frac{\varepsilon+h}{2\varepsilon}}\left[i^{n}\left(-1\right)^{m+1}e^{-ipr}+S_{-1/2,-1/2}^{\left(m\right)}\left(p\right)e^{ipr}\right]\end{pmatrix}$
(35)

for the “in” isospin $I_{3}=-1/2$.

Using standard methods from the theory of scattering [32, 33], we obtain an
expression for the differential scattering cross-section of the process
$I_{3}\rightarrow I_{3}^{\prime}$ in terms of the scattering amplitude
$f_{I_{3}^{\prime},I_{3}}$

$d\sigma_{I_{3}^{\prime},I_{3}}/d\theta=\left|f_{I_{3}^{\prime},I_{3}}\left(p,\theta\right)\right|^{2}.$
(36)

Similarly, the partial cross-sections of the scattering process
$I_{3}\rightarrow I_{3}^{\prime}$ are expressed in terms of the partial
scattering amplitudes $f_{I_{3}^{\prime},I_{3}}^{\left(m\right)}$

$\sigma_{I_{3}^{\prime},I_{3}}^{\left(m\right)}=2\pi\left|f_{I_{3}^{\prime},I_{3}}^{\left(m\right)}\left(p\right)\right|^{2}.$
(37)

Note that in $(2+1)$ dimensions, the cross-sections
$d\sigma_{I_{3}^{\prime},I_{3}}/d\theta$ and
$\sigma_{I_{3}^{\prime},I_{3}}^{\left(m\right)}$ have the dimension of length
[32] in the initial dimensional units.

The partial matrix elements $S_{I_{3}^{\prime},I_{3}}^{\left(m\right)}$ must
satisfy a unitarity condition; this follows from the unitarity of the
$S$-matrix, $SS^{\dagger}=S^{\dagger}S=\mathbb{I}$, which is a consequence of
the conservation of probability. At the same time, the Dirac equation results
in conservation of the fermion current $j^{\mu}=\bar{\psi}\gamma^{\mu}\psi$,
the time component of which is the probability density. Hence, to obtain the
unitarity condition for $S_{I_{3}^{\prime},I_{3}}^{\left(m\right)}$ we shall
use the conservation of the fermion current: $\partial_{\mu}j^{\mu}=0$. From
Eq. (17), we obtain the contravariant components of the partial fermion
current
$j^{\mu\left(m\right)}=\bar{\psi}^{\left(m\right)}\gamma^{\mu}\psi^{\left(m\right)}$
in polar coordinates:

$\displaystyle j^{0\left(m\right)}$ $\displaystyle=$
$\displaystyle\left|c_{11}\right|^{2}+\left|c_{12}\right|^{2}+\left|c_{21}\right|^{2}+\left|c_{22}\right|^{2},$
(38a) $\displaystyle j^{r\left(m\right)}$ $\displaystyle=$ $\displaystyle
2\,\text{Im}\left[c_{22}c_{12}^{\ast}-c_{11}c_{21}^{\ast}\right],$ (38b)
$\displaystyle j^{\theta\left(m\right)}$ $\displaystyle=$
$\displaystyle-2r^{-1}\text{Re}\left[c_{11}c_{21}^{\ast}+c_{22}c_{12}^{\ast}\right].$
(38c)

The conservation of $j^{\mu\left(m\right)}$ results in the constancy of the
radial component of the fermion current, i.e.
$\partial_{r}j^{r\left(m\right)}=0$. The regularity of the eigenfunctions
$\psi^{\left(m\right)}$ at the origin leads us to the conclusion that the
radial component $j^{r\left(m\right)}$ vanishes. From Eqs. (34), (35), and
(38b), we obtain the asymptotic form of the radial component of the partial
fermion current in terms of the partial matrix elements
$S_{I_{3}^{\prime},I_{3}}^{\left(m\right)}$

$j^{r\left(m\right)}\sim\frac{v}{2\pi
pr}\left(\left|S_{1/2,I_{3}}^{\left(m\right)}\right|^{2}+\left|S_{-1/2,I_{3}}^{\left(m\right)}\right|^{2}-1\right),$
(39)

where $I_{3}=\pm 1/2$ is the isospin of the corresponding “in” fermion state,
and $v=\left(1-h^{2}\varepsilon^{-2}\right)^{1/2}$ is the asymptotic fermion
velocity. The vanishing of the radial component $j^{r\left(m\right)}$ results
in the diagonal part of the unitarity condition for the partial matrix
elements $S_{I_{3}^{\prime},I_{3}}^{\left(m\right)}$

$\sum\limits_{I_{3}^{\prime\prime}}\left[S_{I_{3}^{\prime\prime},I_{3}}^{\left(m\right)}\right]^{\ast}S_{I_{3}^{\prime\prime},I_{3}^{\prime}}^{\left(m\right)}=\delta_{I_{3},I_{3}^{\prime}}.$
(40)

We now turn to the symmetry properties of the partial matrix elements
$S_{I_{3}^{\prime},I_{3}}^{\left(m\right)}$. The basic symmetry property of
$S_{I_{3}^{\prime},I_{3}}^{\left(m\right)}$ is

$S_{-I_{3}^{\prime},-I_{3}}^{\left(-m\right)}=S_{I_{3}^{\prime},I_{3}}^{\left(m\right)}.$
(41)

This property follows from the invariance of the Dirac equation (12) under
combined transformation (23), which in turn is a consequence of the specific
form of soliton field configuration (7). The sequential action of the
$\Pi_{2}$ and $T$ transformations leaves the eigenvalue $m$ of grand spin (15)
unchanged but permutes the isospin labels $I_{3}$ and $I_{3}^{\prime}$. The
invariance of the Dirac equation (12) under $\Pi_{2}T$ transformation (22)
then results in the symmetry relation

$S_{I_{3}^{\prime},I_{3}}^{\left(m\right)}=S_{I_{3},I_{3}^{\prime}}^{\left(m\right)}.$
(42)

When $I_{3}=I_{3}^{\prime}$, relation (42) is trivial, but for unequal (i.e.
opposite) $I_{3}$ and $I_{3}^{\prime}$ it can be rewritten as

$S_{I_{3}^{\prime},I_{3}}^{\left(m\right)}=S_{-I_{3}^{\prime},-I_{3}}^{\left(m\right)}\quad\text{if}\quad
I_{3}+I_{3}^{\prime}=0$ (43)

and thus imposes nontrivial restrictions on
$S_{I_{3}^{\prime},I_{3}}^{\left(m\right)}$. Finally, from Eqs. (41) and (43),
we obtain a further symmetry relation for
$S_{I_{3}^{\prime},I_{3}}^{\left(m\right)}$

$S_{I_{3}^{\prime},I_{3}}^{\left(m\right)}=S_{I_{3}^{\prime},I_{3}}^{\left(-m\right)}\quad\text{if}\quad
I_{3}+I_{3}^{\prime}=0.$ (44)

The partial elements $S^{\left(m\right)}_{I_{3}^{\prime},I_{3}}$ of the
$S$-matrix depend on the three variables: the fermion momentum $p$, the Yukawa
coupling constant $h$, and the size of the soliton $r_{0}$. Using Eq. (24),
the explicit form (25) of the $\Lambda$-matrix, and the asymptotic form (31)
of the scattering state, it can be shown that in reality,
$S^{\left(m\right)}_{I_{3}^{\prime},I_{3}}$ depend only on two combinations of
$p$, $h$, and $r_{0}$:

$S_{I_{3}^{\prime},I_{3}}^{\left(m\right)}=\xi_{I_{3}^{\prime},I_{3}}^{\left(m\right)}\left(h^{-1}p,hr_{0}\right),$
(45)

where $\xi_{I_{3}^{\prime},I_{3}}^{\left(m\right)}$ are some functions of the
two variables. In particular, the dependence of
$S_{I_{3}^{\prime},I_{3}}^{\left(m\right)}$ on the size of the soliton $r_{0}$
occurs only through the combination $hr_{0}$.

##  IV  The Born approximation for the scattering amplitudes

The rather complicated structure of the matrix in Eq. (25) makes it impossible
to split system of differential equations (24) into smaller subsystems. Hence,
the solution to the system of four linear differential equations in Eq. (24)
is reduced to the solution to a linear differential equation of fourth order.
Due to its rather complex form, this equation cannot be solved analytically.
Consequently, scattering amplitudes (32) also cannot be obtained in an
analytical form. In view of this, it is important to investigate the fermion
scattering in the Born approximation, which gives us a chance to obtain an
approximate analytical expression for the scattering amplitudes
$f_{I_{3}^{\prime},I_{3}}$.

Eq. (4) tells us that the fermion-soliton interaction is described by the
Yukawa term

$V_{\text{int}}=-h\bar{\psi}\delta\bm{\phi}\bm{\cdot}\bm{\tau}\psi,$ (46)

where $\delta\bm{\phi}=\bm{\phi}-\bm{\phi}_{\text{v}}$ is the difference
between soliton field (7) and the vacuum field
$\bm{\phi}_{\text{v}}=\left(0,0,-1\right)$. The known condition of
applicability of the Born approximation [32] has the form
$\left|V_{\text{int}}\right|\ll pa/\left(m_{\psi}a^{2}\right)$, where $p$ is
the momentum of the fermion, $m_{\psi}$ is its mass, and $a$ is the size of
the area in which the fermion-soliton interaction is markedly different from
zero. In our case, this condition can be written as

$h^{2}r_{0}\ll p,$ (47)

where we have taken into account that in dimensionless variables (3) adopted
here, the fermion mass $m_{\psi}=h$. Note that fulfilling condition (47)
guarantees only that the amplitude of the scattered outgoing wave is much
smaller than the amplitude of the incident plane wave. At the same time, for
the Born approximation to be valid, the second-order Born amplitude should be
much smaller than the first-order one. We shall see later that for elastic
($I_{3}^{\prime}=I_{3}$) fermion scattering, this last condition may not be
satisfied even if condition (47) is met.

Using standard methods from field theory [34], we obtain an expression for the
first-order Born amplitude of the scattering process $I_{3}\rightarrow
I_{3}^{\prime}$

$f_{I_{3}^{\prime},I_{3}}=-\left(8\pi
p\right)^{-1/2}h\bar{u}_{\varepsilon,\mathbf{p}^{\prime},I_{3}^{\prime}}\mathbb{I\otimes}\left[\delta\mathbf{\phi}\left(\mathbf{q}\right)\mathbf{\bm{\cdot}\tau}\right]u_{\varepsilon,\mathbf{p},I_{3}},$
(48)

where

$\delta\bm{\phi}\left(\mathbf{q}\right)=\int\delta\bm{\phi}\left(\mathbf{x}\right)e^{-i\mathbf{qx}}d^{2}x$
(49)

and $\mathbf{q}=\mathbf{p}^{\prime}-\mathbf{p}$ is the momentum transfer. The
Born amplitudes can be obtained in an analytical form. For winding numbers
satisfying the condition $\left|n\right|\geqslant 2$, the amplitudes are
expressed in terms of the Meijer $G$ functions. In the important case when the
solitons have the winding numbers $n=1$ and $n=-1$, the Born amplitudes are
expressed in terms of modified Bessel functions of the second kind:

$\displaystyle f_{1/2,1/2}$ $\displaystyle=$
$\displaystyle\sqrt{2\pi}hr_{0}^{2}p^{-1/2}(\varepsilon+h$ (50a)
$\displaystyle-e^{-i\left(\vartheta_{2}-\vartheta_{1}\right)}(\varepsilon-h))\text{K}_{0}\left(qr_{0}\right),$
$\displaystyle f_{-1/2,1/2}$ $\displaystyle=$
$\displaystyle(-1)^{\varkappa}\sqrt{2\pi}hr_{0}^{2}p^{-1/2}$ (50b)
$\displaystyle\times
e^{i\varkappa\left(\vartheta_{1}+\vartheta_{2}\right)}q\text{K}_{1}\left(qr_{0}\right),$
$\displaystyle f_{1/2,-1/2}$ $\displaystyle=$
$\displaystyle(-1)^{\varkappa}\sqrt{2\pi}hr_{0}^{2}p^{-1/2}$ (50c)
$\displaystyle\times
e^{-i\varkappa\left(\vartheta_{1}+\vartheta_{2}\right)}q\text{K}_{1}\left(qr_{0}\right),$
$\displaystyle f_{-1/2,-1/2}$ $\displaystyle=$
$\displaystyle\sqrt{2\pi}hr_{0}^{2}p^{-1/2}(\varepsilon+h$ (50d)
$\displaystyle-e^{i\left(\vartheta_{2}-\vartheta_{1}\right)}(\varepsilon-h))\text{K}_{0}\left(qr_{0}\right),$

where $\vartheta_{1}$ ($\vartheta_{2}$) is the polar angle that defines the
direction of motion of the “in” (“out”) fermion,
$q=2p\sin\left(\left|\vartheta_{2}-\vartheta_{1}\right|/2\right)$ is the
absolute value of the momentum transfer, and $\varkappa$ is equal to $1$ for
$n=1$ and $0$ for $n=-1$. For antifermion scattering, the Born amplitudes take
a similar form

$\displaystyle f_{1/2,1/2}$ $\displaystyle=$
$\displaystyle-\sqrt{2\pi}hr_{0}^{2}p^{-1/2}(\varepsilon+h$ (51a)
$\displaystyle-e^{i\left(\vartheta_{2}-\vartheta_{1}\right)}(\varepsilon-h))\text{K}_{0}\left(qr_{0}\right),$
$\displaystyle f_{-1/2,1/2}$ $\displaystyle=$
$\displaystyle(-1)^{1+\varkappa}\sqrt{2\pi}hr_{0}^{2}p^{-1/2}$ (51b)
$\displaystyle\times
e^{-i\left(1-\varkappa\right)\left(\vartheta_{1}+\vartheta_{2}\right)}q\text{K}_{1}\left(qr_{0}\right),$
$\displaystyle f_{1/2,-1/2}$ $\displaystyle=$
$\displaystyle(-1)^{1+\varkappa}\sqrt{2\pi}hr_{0}^{2}p^{-1/2}$ (51c)
$\displaystyle\times
e^{i\left(1-\varkappa\right)\left(\vartheta_{1}+\vartheta_{2}\right)}q\text{K}_{1}\left(qr_{0}\right),$
$\displaystyle f_{-1/2,-1/2}$ $\displaystyle=$
$\displaystyle-\sqrt{2\pi}hr_{0}^{2}p^{-1/2}(\varepsilon+h$ (51d)
$\displaystyle-e^{-i\left(\vartheta_{2}-\vartheta_{1}\right)}(\varepsilon-h))\text{K}_{0}\left(qr_{0}\right).$

Note that the partial elements of the $S$-matrix
$S_{I_{3}^{\prime},I_{3}}^{\left(m\right)}=i\sqrt{2\pi
p}f_{I_{3}^{\prime},I_{3}}^{\left(m\right)}+\delta_{I_{3}^{\prime},I_{3}}$
that correspond to the Born amplitudes (50) and (51) can be written in form
(45).

From Eqs. (50) and (51) it follows that the amplitudes are Hermitian with
respect to the permutation of the isospins and momenta of the fermionic states

$f_{I_{3}^{\prime},I_{3}}\left(\vartheta^{\prime},\vartheta\right)=f_{I_{3},I_{3}^{\prime}}^{\ast}\left(\vartheta,\vartheta^{\prime}\right)$
(52)

as required for the Born approximation [32, 33]. Next, the invariance of Eq.
(12) under $\Pi_{2}T$ transformation (22) leads to a symmetry relation for the
Born amplitudes

$f_{I_{3}^{\prime},I_{3}}\left(\vartheta^{\prime},\vartheta\right)=f_{I_{3},I_{3}^{\prime}}\left(\pi-\vartheta,\pi-\vartheta^{\prime}\right).$
(53)

Another symmetry relation

$f_{I_{3}^{\prime},I_{3}}\left(\vartheta^{\prime},\vartheta\right)=f_{-I_{3}^{\prime},-I_{3}}\left(-\vartheta^{\prime},-\vartheta\right)$
(54)

follows from the invariance of Eq. (12) under combined transformation (23).
Note that in Eqs. (53) and (54), the minus signs before the angular variables
are due to the fact that the $\Pi_{2}$ transformation changes the signs of the
$y$ components of the momenta of the fermions. Finally, Eqs. (50) and (51)
tell us that scattering of the fermion in the background field of the soliton
with winding number $n=\pm 1$ is equivalent to scattering of the antifermion
in the background field of the soliton with the opposite winding number. We
use the superscript $[n,+]$ ($[n,-]$) to indicate the scattering of the
fermion (antifermion) in the background field of the soliton with a given
winding number $n$. It then follows from Eqs. (50) and (51) that the Born
amplitudes satisfy the relations:

$\displaystyle f_{I_{3}^{\prime},I_{3}}^{\left[\pm
1,+\right]}\left(\vartheta_{2},\vartheta_{1}\right)$ $\displaystyle=$
$\displaystyle\left(-1\right)^{I_{3}^{\prime}+I_{3}}f_{-I_{3}^{\prime},-I_{3}}^{\left[\mp
1,-\right]}\left(\vartheta_{2},\vartheta_{1}\right),$ (55a) $\displaystyle
f_{I_{3}^{\prime},I_{3}}^{\left[\pm
1,+\right]}\left(\vartheta_{2},\vartheta_{1}\right)$ $\displaystyle=$
$\displaystyle f_{I_{3}^{\prime},I_{3}}^{\left[\pm
1,+\right]}\left(\vartheta_{2}+\pi,\vartheta_{1}+\pi\right),$ (55b)
$\displaystyle f_{I_{3}^{\prime},I_{3}}^{\left[\pm
1,-\right]}\left(\vartheta_{2},\vartheta_{1}\right)$ $\displaystyle=$
$\displaystyle f_{I_{3}^{\prime},I_{3}}^{\left[\pm
1,-\right]}\left(\vartheta_{2}+\pi,\vartheta_{1}+\pi\right).$ (55c)

Eq. (55a) is a consequence of the fact that the winding number $n$ in the
Dirac equation (12) changes the sign under $C$-transformation (73). Eqs. (55b)
and (55c) follow from the invariance of the Dirac equation (12) under
$P$-transformation (20).

We can now ascertain the behavior of the Born amplitudes (50) for large and
small values of the momentum transfer $q$. Using the known asymptotic forms of
the modified Bessel functions $\text{K}_{0}\left(qr_{0}\right)$ and
$\text{K}_{1}\left(qr_{0}\right)$, we obtain asymptotic forms of the Born
amplitudes (50) at large values of $q$:

$\displaystyle f_{\pm 1/2,\pm 1/2}$ $\displaystyle\sim$ $\displaystyle
2^{-1/2}\pi hr_{0}^{3/2}e^{-qr_{0}}\left(1-e^{\mp
i\left(\vartheta_{2}-\vartheta_{1}\right)}\right)$ (56a)
$\displaystyle\times\sin\left(\left|\vartheta_{2}-\vartheta_{1}\right|/2\right)^{-1/2},$
$\displaystyle f_{\mp 1/2,\pm 1/2}$ $\displaystyle\sim$
$\displaystyle(-1)^{\varkappa}\sqrt{2}\pi hr_{0}^{3/2}e^{-qr_{0}}e^{\pm
i\varkappa\left(\vartheta_{2}+\vartheta_{1}\right)}$ (56b)
$\displaystyle\times\sin\left(\left|\vartheta_{2}-\vartheta_{1}\right|/2\right)^{1/2},$

where the angles $\vartheta_{1}$ and $\vartheta_{2}$ are fixed and
$\vartheta_{1}\neq\vartheta_{2}$. We see that the Born amplitudes decrease
exponentially with an increase in both the momentum transfer $q$ and the
effective size of the soliton $r_{0}$.

Next we consider the case of low momentum transfer $q$ and high fixed fermion
momentum $p$. This situation involves small scattering angles
$\Delta\vartheta\equiv\left|\vartheta_{2}-\vartheta_{1}\right|=2\arcsin\left[q/\left(2p\right)\right]\approx
q/p$. In this case, the Born amplitudes take the form

$\displaystyle f_{\pm 1/2,\pm 1/2}$
$\displaystyle\sim-2\sqrt{2\pi}h^{2}r_{0}^{2}p^{-1/2}\left(\ln\left(qr_{0}/2\right)+\gamma\right),$
(57a) $\displaystyle f_{\mp 1/2,\pm 1/2}$
$\displaystyle\sim(-1)^{\varkappa}\sqrt{2\pi}hr_{0}p^{-3/2}\left(p\pm
i\varkappa q\right),$ (57b)

where $\gamma$ is the Euler-Mascheroni constant. It follows from Eq. (57a)
that in the elastic channel (with no change in isospin), the Born amplitudes
diverge logarithmically as $q\rightarrow 0$. Conversely, the inelastic Born
amplitudes tend to constant values in this limit. More importantly, the
elastic Born amplitudes $\propto h^{2}$, whereas the inelastic ones $\propto
h$, as expected for the usual first Born approximation in which amplitudes are
proportional to a coupling constant. The reason for this behavior of the
elastic Born amplitudes lies in the spin-isospin structure of plane-wave
fermionic states (76) entering the Born amplitudes (48), and in the mechanism
of generation of the fermion mass in model (1).

In model (1), fermions gain mass due to spontaneous breaking of the $SU(2)$
global symmetry. In our dimensionless notation (3), the fermion mass
$m_{\psi}$ is equal to the Yukawa coupling constant $h$. Next, relativistic
invariance results in the factors $(\varepsilon\pm
m_{\psi})^{1/2}=(\varepsilon\pm h)^{1/2}$ in fermion wave functions (76). In
turn, these factors and the spin-isospin structure of Eq. (48) result in the
characteristic factors $\left[\varepsilon+h-e^{\mp
i\left(\vartheta_{2}-\vartheta_{1}\right)}(\varepsilon-h)\right]$ in elastic
Born amplitudes (50a), (50d), (51a), and (51d), whereas such factors are
absent from inelastic Born amplitudes (50b), (50c), (51b), and (51c). For
small scattering angles $\Delta\vartheta=\vartheta_{2}-\vartheta_{1}$ and
large fermion momenta $p$, these factors take the form $2h\pm
ip\Delta\vartheta$. We can see that for scattering angles
$\Delta\vartheta<2hp^{-1}$, the term $2h$ becomes predominant. Hence, the
terms that $\propto h^{2}$ also become predominant in elastic Born amplitudes
(50a), (50d), (51a), and (51d) that were obtained within the framework of the
first Born approximation. It follows that elastic Born amplitudes (50a),
(50d), (51a), and (51d) become inapplicable when the scattering angle
$\Delta\vartheta\lesssim 2hp^{-1}$. This is because the contribution of the
second Born approximation to the scattering amplitudes is $\propto h^{2}$, and
is therefore of the same order of magnitude as the contribution of the $h^{2}$
terms of the first Born approximation. Note that for large fermion momenta
$p$, the domain of $\Delta\vartheta$ in which the scattering amplitudes are
markedly different from zero is of the order of $(pr_{0})^{-1}$. It follows
that under the condition $2hr_{0}\sim 1$, the area $\Delta\vartheta\lesssim
2hp^{-1}$ in which the first Born approximation is incorrect may cover a
substantial part of the area $\Delta\vartheta\lesssim(pr_{0})^{-1}$ in which
the scattering amplitudes are markedly different from zero, and may even
overlap it completely. It was found that in the case of solitons with higher
winding numbers $n$, the characteristic factor $\left[\varepsilon+h-e^{\mp
i\left(\vartheta_{2}-\vartheta_{1}\right)}(\varepsilon-h)\right]$ also arises
in the elastic first-order Born amplitudes, meaning that these are
inapplicable when the scattering angle $\Delta\vartheta\lesssim 2hp^{-1}$.

Eqs. (32), (50), and (51) make it possible to obtain analytical expressions
for the partial amplitudes
$f_{I_{3}^{\prime},I_{3}}^{\left(m\right)}\left(p\right)$ in terms of the
Meijer G functions. As $p\rightarrow\infty$, these expressions tend to the
asymptotic forms:

$\displaystyle f_{\pm 1/2,\pm 1/2}^{\left(m\right)}$ $\displaystyle\sim$
$\displaystyle\sqrt{2\pi}hr_{0}p^{-1/2}$
$\displaystyle\times\left[hp^{-1}+2^{-2}p^{-2}r_{0}^{-2}\left(1-2\left(\varkappa\mp
m\right)\right)\right],$ $\displaystyle f_{\pm 1/2,\mp 1/2}^{\left(m\right)}$
$\displaystyle\sim$
$\displaystyle\left(-1\right)^{\varkappa}\sqrt{\pi/2}hp^{-1/2}$
$\displaystyle\times\left[p^{-1}+2^{-3}\left(1-4m^{2}\right)p^{-3}r_{0}^{-2}\right]$

for fermion scattering, and

$\displaystyle f_{\pm 1/2,\pm 1/2}^{\left(m\right)}$ $\displaystyle\sim$
$\displaystyle-\sqrt{2\pi}hr_{0}p^{-1/2}$
$\displaystyle\times\left[hp^{-1}+2^{-2}p^{-2}r_{0}^{-2}\left(1-2\left(\varkappa\pm
m\right)\right)\right],$ $\displaystyle f_{\pm 1/2,\mp 1/2}^{\left(m\right)}$
$\displaystyle\sim$
$\displaystyle\left(-1\right)^{1+\varkappa}\sqrt{\pi/2}hp^{-1/2}$
$\displaystyle\times\left[p^{-1}+2^{-3}\left(1-4m^{2}\right)p^{-3}r_{0}^{-2}\right]$

for antifermion scattering. It follows from Eq. (33) that the partial
amplitudes $f_{I_{3}^{\prime},I_{3}}^{\left(m\right)}$ must satisfy the same
symmetry relations as the partial matrix elements
$S_{I_{3}^{\prime},I_{3}}^{\left(m\right)}$, and we can see that the Born
partial amplitudes (58) and (59) satisfy symmetry relations (41), (42), (43),
and (44). Eqs. (58) and (59) tell us that the leading terms of the asymptotic
forms of $f_{I_{3}^{\prime},I_{3}}^{\left(m\right)}$ do not depend on $m$.
Furthermore, the leading terms of the elastic ($I_{3}=I_{3}^{\prime}$) partial
amplitudes are $\propto h^{2}$, whereas those of the inelastic ($I_{3}\neq
I_{3}^{\prime}$) ones are $\propto h$. Finally, we recall that the Born
partial amplitudes and corresponding partial elements of the $S$-matrix do not
satisfy the unitarity condition [32, 33].

We can also determine the asymptotic behavior of the partial amplitudes
$f_{I_{3}^{\prime},I_{3}}^{\left(m\right)}\left(p\right)$ as
$\left|m\right|\rightarrow\infty$. Using known methods [35] for the
calculation of the Fourier coefficients in the limit of large $m$, we obtain
the corresponding asymptotic forms of the Born partial amplitudes:

$\displaystyle f_{\pm 1/2,\pm 1/2}^{\left(m\right)}$ $\displaystyle\sim$
$\displaystyle\sqrt{\pi/2}hr_{0}^{2}p^{-1/2}\left[\left(\varepsilon+h\right)\left|m\mp\varkappa\right|^{-1}\right.$
(60a) $\displaystyle\left.-\left(\varepsilon-h\right)\left|m\mp\varkappa\pm
1\right|^{-1}\right],$ $\displaystyle f_{\pm 1/2,\mp 1/2}^{\left(m\right)}$
$\displaystyle\sim$
$\displaystyle\left(-1\right)^{\varkappa}\sqrt{\pi/2}hr_{0}^{3}p^{3/2}\left|m\right|^{-3}$
(60b)

for fermion scattering, and

$\displaystyle f_{\pm 1/2,\pm 1/2}^{\left(m\right)}$ $\displaystyle\sim$
$\displaystyle-\sqrt{\pi/2}hr_{0}^{2}p^{-1/2}\left[\left(\varepsilon+h\right)\left|m\pm\varkappa\mp
1\right|^{-1}\right.$ (61a)
$\displaystyle\left.-\left(\varepsilon-h\right)\left|m\pm\varkappa\right|^{-1}\right],$
$\displaystyle f_{\pm 1/2,\mp 1/2}^{\left(m\right)}$ $\displaystyle\sim$
$\displaystyle\left(-1\right)^{1+\varkappa}\sqrt{\pi/2}hr_{0}^{3}p^{3/2}\left|m\right|^{-3}$
(61b)

for antifermion scattering.

From the analytical expressions for the Born amplitudes (50) and Eq. (36), we
obtain differential scattering cross-sections for the processes
$I_{3}\rightarrow I_{3}^{\prime}$ in the Born approximation

$\displaystyle d\sigma_{\pm 1/2,\pm 1/2}/d\vartheta$ $\displaystyle=$
$\displaystyle 8\pi
p^{-1}h^{2}r_{0}^{4}\text{K}_{0}\left(2pr_{0}\sin\left(\vartheta/2\right)\right)^{2}$
(62a)
$\displaystyle\times\left(h^{2}+p^{2}\sin\left(\vartheta/2\right)^{2}\right),$
$\displaystyle d\sigma_{\mp 1/2,\pm 1/2}/d\vartheta$ $\displaystyle=$
$\displaystyle 8\pi
ph^{2}r_{0}^{4}\text{K}_{1}\left(2pr_{0}\sin\left(\vartheta/2\right)\right)^{2}$
(62b) $\displaystyle\times\sin\left(\vartheta/2\right)^{2}.$

Since the corresponding Born amplitudes in Eqs. (50) and (51) differ only by a
phase factor, the Born differential cross-sections for antifermion scattering
are the same as those for fermion scattering. Note that in accordance with Eq.
(57), the elastic differential cross-sections diverge logarithmically
according to the leading term $8\pi
h^{4}r_{0}^{4}p^{-1}\left(\gamma+\ln\left(pr_{0}\vartheta/2\right)\right)^{2}$,
whereas the inelastic ones tend to a constant value of $2\pi
h^{2}r_{0}^{2}p^{-1}$ as $\vartheta\rightarrow 0$ for large but fixed $p$. The
total cross-sections of the processes $I_{3}\rightarrow I_{3}^{\prime}$ can
also be obtained in analytical form

$\displaystyle\sigma_{\pm 1/2,\pm 1/2}$ $\displaystyle=$ $\displaystyle
8\pi^{2}p^{-1}h^{2}r_{0}^{4}$ (63f)
$\displaystyle\times\left\\{p^{2}G_{2,4}^{3,1}\left(4p^{2}r_{0}^{2}\left|\begin{array}[]{c}-\frac{1}{2},\frac{1}{2}\\\
0,0,0,-1\end{array}\right.\right)\right.$
$\displaystyle\left.+h^{2}G_{2,4}^{3,1}\left(4p^{2}r_{0}^{2}\left|\begin{array}[]{c}\frac{1}{2},\frac{1}{2}\\\
0,0,0,0\end{array}\right.\right)\right\\},$ $\sigma_{\mp 1/2,\pm
1/2}=8\pi^{2}ph^{2}r_{0}^{4}G_{2,4}^{3,1}\left(4p^{2}r_{0}^{2}\left|\begin{array}[]{c}-\frac{1}{2},\frac{1}{2}\\\
-1,0,1,-1\end{array}\right.\right),$ (63g)

where $G^{m,n}_{p,q}$ are the Meijer $G$ functions, defined according to Ref.
[36]. Finally, using known asymptotic expansions for the Meijer $G$ functions,
we obtain the asymptotics of the total cross-sections (63) for large values of
the fermion momenta $p$

$\displaystyle\sigma_{\pm 1/2,\pm 1/2}$ $\displaystyle=$
$\displaystyle\frac{1}{8}\pi^{3}h^{2}r_{0}$
$\displaystyle\times\left(\left(1+32h^{2}r_{0}^{2}\right)p^{-2}+O\left(p^{-4}\right)\right),$
$\displaystyle\sigma_{\mp 1/2,\pm 1/2}$ $\displaystyle=$
$\displaystyle\frac{3}{8}\pi^{3}h^{2}r_{0}\left(p^{-2}+O\left(p^{-4}\right)\right).$
(64b)

Note that Eqs. (62a), (63f), and (64), which are related to the elastic
channel, include terms with an additional factor $h^{2}$ in comparison with
the usual first Born approximation.

It follows from Eqs. (63) and (64) that in the Born approximation, the total
cross-section of the fermion scattering, which is equal to the sum of the
elastic (63f) and inelastic (63g) parts, is finite. Next, it follows from Eq.
(50) that for small scattering angles, the imaginary parts of the elastic Born
amplitudes $f_{\pm 1/2,\pm 1/2}$ have the form $\text{Im}\left[f_{\pm 1/2,\pm
1/2}\right]\sim\mp\sqrt{2\pi}hr_{0}^{2}p^{-1/2}\left(\varepsilon-h\right)\left(\gamma+\ln\left[pr_{0}\vartheta/2\right]\right)\vartheta$,
and consequently vanish at zero scattering angle. At the same time, in our
case, the optical theorem of the scattering theory can be written as

$\text{Im}\left[f_{I_{3},I_{3}}\left(p,\vartheta=0\right)\right]=\frac{1}{2}\sqrt{\frac{p}{2\pi}}\left(\sigma_{1/2,I_{3}}\left(p\right)+\sigma_{-1/2,I_{3}}\left(p\right)\right).$
(65)

We see that the optical theorem is not valid in the Born approximation. This
is because the optical theorem is a consequence of the unitarity of the
$S$-matrix, which is violated in the Born approximation.

##  V  Numerical results

The radial wave functions of fermionic scattering states are solutions to
system (24) that satisfy regularity condition (26). As $r\rightarrow\infty$,
the radial wave functions $c_{ia}(r)$ tend to their asymptotic forms (34) and
(35), which are expressed in terms of the partial elements of the $S$-matrix
(33). The components of Eq. (34) that correspond to the “out” isospin
$I_{3}^{\prime}=-1/2$ and those of Eq. (35) that corresponds to the “out”
isospin $I_{3}^{\prime}=1/2$ contain only outgoing waves, and this should be
clear from the physical background.

Our goal is to find the partial matrix elements
$S^{(m)}_{I_{3}^{\prime},I_{3}}(p)$ for a range of fermion momenta $p$, as
this will give the most complete description of the fermion scattering. To do
this, we numerically solve the system in Eq. (24) under regularity condition
(26) for a set of $p$ within some finite range. The origin $r=0$ in the
neighborhood of which regular expansion (26) is valid is the regular singular
point of the system in Eq. (24). Hence, the initial value problem (IVP) for
system (24) cannot be posed at $r=0$. To work around this problem, we shift
the point at which the IVP is posed to a short distance from the origin. The
initial values of the radial wave functions $c_{ia}(r)$ at the shifted initial
point $r_{0}$ are calculated using regular power expansion (26).

A regular solution to the system in Eq. (24) depends on the two complex
parameters $b_{0}$ and $c_{0}$. The linearity of the system makes it possible
to set one of these parameters (for instance $b_{0}$) to one. In this way, we
stay with the two real parameters $\text{Re}[c_{0}]$ and $\text{Im}[c_{0}]$.
Consider fermion scattering that corresponds to the “in” fermionic state with
isospin $I_{3}=1/2$. To obtain the radial wave functions $c_{ia}(r)$ that
correspond to this scattering state, we must satisfy the condition that the
radial wave functions $c_{i2}(r)$ contain only the outgoing wave as it is in
Eq. (34). It follows from Eq. (34) that for large $r$, the real and imaginary
parts of the outgoing radial wave functions $c_{i2}(r)$ should satisfy the
asymptotic condition:
$\text{Im}[\sqrt{pr}c_{i2}(pr+\delta_{2}+\pi/2)]=\text{Re}[\sqrt{pr}c_{i2}(pr+\delta_{2})]$,
where $\delta_{2}$ is some phase shift. To satisfy this condition, we can use
the two parameters $\text{Re}[c_{0}]$ and $\text{Im}[c_{0}]$. By varying
$\text{Re}[c_{0}]$ and keeping $\text{Im}[c_{0}]$ fixed, we can achieve
coincidence between the zeros of
$\text{Im}[\sqrt{pr}c_{i2}(pr+\delta_{2}+\pi/2)]$ and
$\text{Re}[\sqrt{pr}c_{i2}(pr+\delta_{2})]$. Next, by varying
$\text{Im}[c_{0}]$ and keeping $\text{Re}[c_{0}]$ fixed, we can achieve
equality of $\text{Im}[\sqrt{pr}c_{i2}(pr+\delta_{2}+\pi/2)]$ and
$\text{Re}[\sqrt{pr}c_{i2}(pr+\delta_{2})]$. The positions of the zeros of
$\text{Im}[\sqrt{pr}c_{i2}(pr+\delta_{2}+\pi/2)]$ and
$\text{Re}[\sqrt{pr}c_{i2}(pr+\delta_{2})]$ do not change under this
variation, because linear system (24) contains only real coefficient functions
and thus the imaginary parts of $c_{ia}(r)$ are $\propto\text{Im }[c_{0}]$,
since $i\text{Im}[c_{0}]$ is the only imaginary parameter of the problem. In
turn, linear scaling cannot change the location of the zeros of
$\text{Im}[c_{ia}(r)]$. When the asymptotic condition is satisfied, we can
assume that the radial wave functions $c_{ia}(r)$ correspond to the “in”
fermionic state with isospin $I_{3}=1/2$. Note that the asymptotic condition
for the “in” fermionic state with isospin $I_{3}=-1/2$ has a similar form:
$\text{Im}[\sqrt{pr}c_{i1}(pr+\delta_{1}+\pi/2)]=\text{Re}[\sqrt{pr}c_{i1}(pr+\delta_{1})]$.
The value of $r$ at which the asymptotic condition is satisfied should be
chosen based on the accuracy of asymptotic form (34) and on the low influence
of the background field of the soliton.

To solve the IVP numerically, we use the IVP solver provided by the Maple
package [37]. This solver finds a numerical solution to the IVP using the
Fehlberg fourth-fifth order Runge-Kutta method with degree four interpolant.
Having obtained the solution to the IVP, we can determine the coefficients of
the outgoing ($\propto\exp(ipr)$) and ingoing ($\propto\exp(-ipr)$) parts of
the radial wave functions $c_{ia}(r)$. To do this, we use the orthonormality
property of the exponential functions $\int\nolimits_{0}^{2\pi
p^{-1}}e^{\kappa_{1}ipr}e^{\kappa_{2}ipr}dr=2\pi
p^{-1}\delta_{0,\kappa_{1}+\kappa_{2}}$, where $\kappa_{1,2}=\pm 1$. Taking
into account the phase and normalization factors in Eq. (34), we obtain the
values of the partial matrix elements
$S_{I_{3}^{\prime},I_{3}}^{\left(m\right)}$ for a given $p$. To control the
correctness of the numerical solution to the IVP, we use unitarity condition
(40) and the equality of the pairs of
$S_{I_{3}^{\prime},I_{3}}^{\left(m\right)}$ obtained from the up and down
components of the columns in Eq. (34).

We now turn to a discussion of the numerical results. We consider only fermion
scattering on the elementary ($n=1$) topological soliton of the nonlinear
$O(3)$ $\sigma$-model. The scattering corresponds to the “in” fermionic state
with isospin $I_{3}=1/2$; from Eqs. (41), it follows that scattering with “in”
isospin $I_{3}=-1/2$ is in essence equivalent to the case considered here. The
size parameter $r_{0}$ and the Yukawa coupling constant $h$ are the only
parameters of the IVP. In order for the external field approximation to be
valid, the mass of the fermion, which is equal to $h$ in dimensionless units
(3), should be much lower than the mass of the soliton $4\pi$. Hence, we set
the parameters $h$ and $r_{0}$ equal to $0.1$ and $1$, respectively. According
to Eq. (45), the numerical solution corresponding to these values $h$ and
$r_{0}$ will also give us information about the partial elements of the
$S$-matrix for which the parameters satisfy the condition $hr_{0}=0.1$ The
presence of the inelastic scattering channel $I_{3}\rightarrow I_{3}^{\prime}$
($I_{3}\neq I_{3}^{\prime}$) results in $S_{1/2,1/2}^{\left(m\right)}$ leaving
the unitary circle, and thus the description of the scattering in terms of a
phase shift loses its advantage. In view of this, we shall describe the
partial elements of the $S$-matrix in terms of their real and imaginary parts.

It was found that the $p$-dependences of the partial matrix elements
$S_{I_{3}^{\prime},I_{3}}^{\left(0\right)}$,
$S_{I_{3}^{\prime},I_{3}}^{\left(\pm 1\right)}$, and
$S_{I_{3}^{\prime},I_{3}}^{\left(\left|m\right|\geq 2\right)}$ are
substantially different, and we therefore consider them separately here. For
better visualization of the $p$-dependences, we show them on log-linear plots.
Figure 1 shows the $p$-dependences related to the partial matrix elements
$S_{\pm 1/2,1/2}^{\left(0\right)}$. We can see that all of the curves in Fig.
1 are regular functions of the fermion momentum $p$. In particular, they all
tend to nonzero limits as $p\rightarrow 0$, and
$|S_{1/2,1/2}^{\left(0\right)}-1|^{2}$, $|S_{-1/2,1/2}^{\left(0\right)}|^{2}$,
$|\text{Re}[S_{1/2,1/2}^{\left(0\right)}-1]|$, and
$|\text{Im}[S_{-1/2,1/2}^{\left(0\right)}]|$ reach their maximal values at
$p=0$.

Figures 2 – 7 show the real parts, imaginary parts, and squared magnitudes of
the partial matrix elements $S_{\pm 1/2,1/2}^{\left(m\right)}$ as functions of
the fermion momentum $p$. The $p$-dependences are shown for
$\left|m\right|=2,3,4,\text{ and}\,5$. We can see that for $\left|m\right|\geq
2$, the curves of $\text{Re }[S_{1/2,1/2}^{\left(m\right)}\left(p\right)-1]$,
$\text{Im}[S_{1/2,1/2}^{\left(m\right)}\left(p\right)]$,
$|S_{1/2,1/2}^{\left(m\right)}\left(p\right)-1|^{2}$,
$\text{Re}[S_{-1/2,1/2}^{\left(m\right)}\left(p\right)]$,
$\text{Im}[S_{-1/2,1/2}^{\left(m\right)}\left(p\right)]$, and
$|S_{-1/2,1/2}^{\left(m\right)}\left(p\right)|^{2}$ have similar forms. All of
them have wide extrema at moderate values of $p$, and tend to zero as
$p\rightarrow\infty$ and to constant values as $p\rightarrow 0$. The only
difference is that for $\left|m\right|=2,3,\text{and}\,4$, the
$\text{Re}[S_{-1/2,1/2}^{\left(m\right)}\left(p\right)]$ curves have a node
located to the right of the maximum at moderate values of $p$.

It follows from Fig. 3 that the imaginary parts of the elastic partial
elements of the $S$-matrix satisfy the approximate relation

$\text{Im}[S_{1/2,1/2}^{\left(m\right)}(p)]\approx\text{Im}[S_{1/2,1/2}^{\left(-m+1\right)}(p)].$
(66)

Since
$|\text{Im}[S_{1/2,1/2}^{\left(m\right)}]|\gg|\text{Re}[S_{1/2,1/2}^{\left(m\right)}-1]|$,
approximate equality (66) results in the approximate equality of the squared
magnitudes
$|S_{1/2,1/2}^{\left(m\right)}-1|^{2}\approx|S_{1/2,1/2}^{\left(-m+1\right)}-1|^{2}$,
as shown in Fig. 4. It follows from Eqs. (17) and (18) that for $n=1$, the
change $m\rightarrow-m+1$ of the grand spin $K_{3}$ leads only to the
interchange of the absolute values of the orbital angular momenta $|l_{11}|$
and $|l_{21}|$. It follows from Eq. (26) that the centrifugal barrier does not
change as a whole for the elastic channel of fermion scattering, and thus the
fermion-soliton interaction is of the same order for the partial elastic
channels with $K_{3}=m$ and $K_{3}=-m+1$. This may explain the close values of
the dominant imaginary parts in Eq. (66).

We also found that the position of the maximum in
$|S_{1/2,1/2}^{\left(m\right)}-1|^{2}$ is approximately determined by the
linear expression

$p_{\text{max}}\approx 0.64\left|m-1/2\right|,$ (67)

whereas the heights of the maxima decrease monotonically (approximately
$\propto\left|m\right|^{-2}$) with an increase in $\left|m\right|$. Note that
Eq. (67) is compatible with Eq. (66) since $\left|m-1/2\right|$ is equal to
$m-1/2$ for positive $m$ and $\left(-m+1\right)-1/2$ for negative $m$. Eq.
(67) can be explained as follows. The partial matrix elements
$S_{1/2,1/2}^{\left(m\right)}$ describe the elastic fermion scattering in the
state with grand spin $K_{3}=m$. Eq. (15) tells us that with an increase in
$\left|m\right|$, the main contribution to $K_{3}$ comes from the orbital
part. Next, it follows from Eqs. (17) and (18) that for $n=1$, the orbital
angular momenta of the elastic components $c_{11}$ and $c_{21}$ of the fermion
radial wave function are $l_{11}=m-1$ and $l_{21}=m$, respectively. We see
that both $l_{11}$ and $l_{21}$ depend linearly on $m$. In the classical
limit, the absolute value of the orbital angular momentum $\left|l\right|=bp$,
where $b$ is an impact parameter. In our case, the impact parameter $b$ should
be on the order of the soliton’s size $r_{0}$. In the momentum representation,
the fermion radial wave function $c_{ia}$ that corresponds to the state with
$K_{3}=m$ should have a maximum in the neighborhood of the classical value
$p=\left|l\right|/b$, in accordance with Eq. (67).

It follows from Figs. 5 – 7 that in accordance with Eq. (44), the inelastic
partial matrix elements $S_{-1/2,1/2}^{\left(m\right)}$ coincide when they
have opposite values of $m$. Furthermore, similarly to Eq. (67) and for the
same reasons, the position of the maximum of
$|S_{-1/2,1/2}^{\left(m\right)}|^{2}$ is also determined by the linear
expression

$p_{\text{max}}\approx 1.37\left|m\right|.$ (68)

From a comparison between the $p$-dependences related to $S_{\pm
1/2,1/2}^{\left(0\right)}$ and those related to $S_{\pm
1/2,1/2}^{\left(|m|\geq 2\right)}$, we can see two main differences. Firstly,
there are no pronounced maxima for
$|S_{I_{3}^{\prime},I_{3}}^{\left(0\right)}-\delta_{I_{3}^{\prime},I_{3}}|^{2}$
at nonzero $p$, since for $m=0$, the contribution of the orbital part to the
grand spin $K_{3}$ is not dominant in comparison with those of the spin and
isospin parts. Secondly, the limiting values of
$|S_{I_{3}^{\prime},I_{3}}^{\left(0\right)}-\delta_{I_{3}^{\prime},I_{3}}|^{2}$
are much greater than those of
$|S_{I_{3}^{\prime},I_{3}}^{\left(\left|m\right|>2\right)}-\delta_{I_{3}^{\prime},I_{3}}|^{2}$
as $p\rightarrow 0$. This is because according to Eq. (18), both $l_{21}$ and
$l_{12}$ vanish when $m=0$ and $n=1$. It follows that in this case, the
centrifugal barrier is absent for both elastic and inelastic fermion
scattering. This leads to intense interactions between the fermions and the
core of the soliton. In turn, this results in large squared magnitudes for
both the elastic and inelastic partial elements of the $S$-matrix.

Using numerical methods, we were also able to ascertain some other features of
the curves shown in Figs. 1 – 7. In particular, we ascertained the asymptotic
behavior of $S_{I_{3}^{\prime},I_{3}}^{\left(m\right)}(p)$ as
$p\rightarrow\infty$:

$\displaystyle\text{Re}\left[S_{1/2,1/2}^{\left(m\right)}-1\right]$
$\displaystyle\sim$ $\displaystyle\alpha\left(m\right)p^{-2},$ (69a)
$\displaystyle\text{Im}\left[S_{1/2,1/2}^{\left(m\right)}\right]$
$\displaystyle\sim$ $\displaystyle\beta\left(m\right)p^{-1},$ (69b)
$\displaystyle\text{Re}\left[S_{-1/2,1/2}^{\left(m\right)}\right]$
$\displaystyle\sim$ $\displaystyle\gamma\left(m\right)p^{-1},$ (69c)
$\displaystyle\text{Im}\left[S_{-1/2,1/2}^{\left(m\right)}\right]$
$\displaystyle\sim$ $\displaystyle-\pi hp^{-1},$ (69d)

where $\alpha\left(m\right)$, $\beta\left(m\right)$, and
$\gamma\left(m\right)$ are $m$-dependent constants. Note that for
$\text{Im}[S_{-1/2,1/2}^{\left(m\right)}]$, we were able to find the exact
form of the coefficient of the leading asymptotic term. This is because the
Born approximation (58) perfectly describes the behavior of
$\text{Im}[S_{-1/2,1/2}^{\left(m\right)}]$ for $p\gtrsim 10$. At the same
time, the Born approximation (58) gives only a qualitative description of
$\text{Im}[S_{1/2,1/2}^{\left(m\right)}]$: it gives the correct ($\propto
p^{-1}$) leading asymptotic behavior, but an incorrect factor before the
leading asymptotic term. Note that in Eq. (69a), the leading asymptotic term
is $\propto p^{-2}$, while in Eqs. (69b) – (69d), the leading asymptotic terms
are $\propto p^{-1}$. This difference is due to the fact that
$\text{Im}[S_{1/2,1/2}^{\left(m\right)}]$,
$\text{Re}[S_{-1/2,1/2}^{\left(m\right)}]$, and
$\text{Im}[S_{-1/2,1/2}^{\left(m\right)}]$ tend to zero as
$p\rightarrow\infty$, while $\text{Re}[S_{1/2,1/2}^{\left(m\right)}]$ tends to
one in this limit. It can then easily be shown that asymptotic behavior (69a)
follows from the fulfillment of the unitarity condition
$|S_{1/2,1/2}^{\left(m\right)}|^{2}+|S_{-1/2,1/2}^{\left(m\right)}|^{2}=1$ in
the leading order in inverse powers of $p$.

It follows from Figs. 1 – 7 that as $p$ tends to zero, the real and imaginary
parts of the difference
$S_{I_{3}^{\prime},I_{3}}^{\left(m\right)}-\delta_{I_{3}^{\prime},I_{3}}$ tend
to some constants whose absolute values decrease monotonically with an
increase in $\left|m\right|$. Then, Eqs. (33) and (37) tell us that both the
elastic and inelastic partial cross-sections
$\sigma_{I_{3}^{\prime},I_{3}}^{\left(m\right)}=p^{-1}|S_{I_{3}^{\prime},I_{3}}^{\left(m\right)}-\delta_{I_{3}^{\prime},I_{3}}|^{2}$
diverge as $p^{-1}$ when $p\rightarrow 0$.

Next we turn to the partial matrix elements $S_{\pm 1/2,1/2}^{\left(\pm
1\right)}$. The dependence of the real and imaginary parts of $S_{\pm
1/2,1/2}^{\left(+1\right)}$ on the fermion momentum $p$ is shown in Fig. 8. We
can see that the $p$-dependences in Fig. 8 are in sharp contrast to those in
Figs. 1 – 7. In particular, the $p$-dependence of the elastic partial element
$S_{1/2,1/2}^{\left(+1\right)}$ shows pronounced resonance behavior at
extremely low values of $p$. We were able to achieve extremely small values of
$p$ to reveal two resonance valleys of
$\text{Re}[S_{1/2,1/2}^{\left(+1\right)}]$. The positions of the extrema in
$\text{Re}[S_{1/2,1/2}^{\left(+1\right)}]$ coincide with those of the zeros of
$\text{Im}[S_{1/2,1/2}^{\left(+1\right)}]$, as expected for resonance
structures. Note that $\text{Re}[S_{1/2,1/2}^{\left(+1\right)}]$ (and hence
$|S_{1/2,1/2}^{\left(+1\right)}|$) reaches the unitary boundary of $1$ at its
maxima. Hence, the elastic partial cross-section $\sigma_{1/2,1/2}^{(+1)}$
vanishes at the maxima of $\text{Re}[S_{1/2,1/2}^{\left(+1\right)}]$ as well
as the inelastic partial cross-section $\sigma_{-1/2,1/2}^{(+1)}$. Note that
both $\text{Re}[S_{1/2,1/2}^{\left(+1\right)}]$ and
$\text{Im}[S_{1/2,1/2}^{\left(+1\right)}]$ do not reach the unitary boundary
of $-1$ at their minima, although these are in close proximity. This follows
from the fact that the inelastic scattering does not vanish at the minima of
$\text{Re}[S_{1/2,1/2}^{\left(+1\right)}]$ and
$\text{Im}[S_{1/2,1/2}^{\left(+1\right)}]$. For a similar reason,
$\text{Im}[S_{1/2,1/2}^{\left(+1\right)}]$ also does not quite reach the
unitary boundary of $1$ at its maxima.

The $p$-dependences of the real and imaginary parts of the inelastic partial
matrix element $S_{-1/2,1/2}^{\left(+1\right)}$ also have a resonance
structure at small values of $p$. Moreover, the positions of the minima
(maxima) of $\text{Re}[S_{-1/2,1/2}^{\left(+1\right)}]$ coincide with those of
the maxima (minima) of $\text{Re}[S_{1/2,1/2}^{\left(+1\right)}]$, and a
similar situation holds for the imaginary parts
$\text{Im}[S_{-1/2,1/2}^{\left(+1\right)}]$ and
$\text{Im}[S_{1/2,1/2}^{\left(+1\right)}]$. The positions of the zeros of
$\text{Re}[S_{-1/2,1/2}^{\left(+1\right)}]$,
$\text{Im}[S_{-1/2,1/2}^{\left(+1\right)}]$, and
$\text{Im}[S_{1/2,1/2}^{\left(+1\right)}]$ also coincide, since
$\text{Re}[S_{1/2,1/2}^{\left(+1\right)}]$ reaches the unitary boundary of $1$
at this point. Note that in Fig. 8, the behavior of the curves is in
accordance with the unitarity condition
$|S_{1/2,1/2}^{\left(+1\right)}|^{2}+|S_{-1/2,1/2}^{\left(+1\right)}|^{2}=1$.

Figure 9 presents the $p$-dependence of the real and imaginary parts of the
partial matrix elements $S_{\pm 1/2,1/2}^{\left(-1\right)}$. We see that in
accordance with Eq. (44), the curves of
$\text{Re}[S_{-1/2,1/2}^{\left(-1\right)}]$ and
$\text{Im}[S_{-1/2,1/2}^{\left(-1\right)}]$ coincide with the curves for
$\text{Re}[S_{-1/2,1/2}^{\left(+1\right)}]$ and
$\text{Im}[S_{-1/2,1/2}^{\left(+1\right)}]$ shown in Fig. 8. In particular,
the curves of $\text{Re}[S_{-1/2,1/2}^{\left(-1\right)}]$ and
$\text{Im}[S_{-1/2,1/2}^{\left(-1\right)}]$ have the same resonance structure
as those of $\text{Re}[S_{-1/2,1/2}^{\left(+1\right)}]$ and
$\text{Im}[S_{-1/2,1/2}^{\left(+1\right)}]$, respectively. In Fig. 9, the
curves that correspond to the real and imaginary parts of the elastic partial
matrix element $S_{1/2,1/2}^{\left(-1\right)}$ also show resonance-like
behavior. However, the behavior of these curves differs from those of the
other resonance curves in Figs. 8 and 9. In particular, the points of the
maxima in $|S_{1/2,1/2}^{\left(-1\right)}|^{2}$ and
$\text{Re}[S_{1/2,1/2}^{\left(-1\right)}]$ do not coincide, and
$\text{Im}[S_{1/2,1/2}^{\left(-1\right)}]$ does not vanish at these points.
Hence, it can be said that $S_{1/2,1/2}^{\left(-1\right)}$ does not show true
resonance behavior. Instead, the resonance-like behavior of
$S_{1/2,1/2}^{\left(-1\right)}$ is caused by the unitarity condition
$|S_{1/2,1/2}^{\left(-1\right)}|^{2}+|S_{-1/2,1/2}^{\left(-1\right)}|^{2}=1$
and the true resonance behavior of the inelastic partial element
$S_{-1/2,1/2}^{\left(-1\right)}$ of the $S$-matrix.

In Figs. 8 and 9, all $p$-dependencies are shown on a logarithmic scale. We
see that in the resonance region, the $p$-dependences of $|S_{\pm
1/2,1/2}^{\left(\pm 1\right)}-\delta_{\pm 1/2,1/2}|^{2}$, $\text{Re}[S_{\pm
1/2,1/2}^{\left(\pm 1\right)}-\delta_{\pm 1/2,1/2}]$, and $\text{Im}[S_{\pm
1/2,1/2}^{\left(\pm 1\right)}]$ have the Breit-Wigner form on this scale.
Also, the positions of resonance peaks are at extremely low values of the
fermion momenta. Another feature is a periodic structure of the resonances in
Figs. 8 and 9. In particular, the distance between the resonance peaks is
approximately the same on the logarithmic scale. We were able to identify the
two resonance peaks of $|S_{\pm 1/2,1/2}^{\left(\pm 1\right)}-\delta_{\pm
1/2,1/2}|^{2}$ and reach the beginning of the third one. Therefore, we may
suppose the existence of a sequence of resonances (possibly infinite)
condensing to the zero fermion momentum.

##  VI  Conclusion

In the present paper, fermion scattering in the background fields of the
topological solitons of the nonlinear $O(3)$ $\sigma$-model has been
investigated both analytically and numerically. In particular, we have
ascertained the asymptotic behavior of the fermion wave functions for both
large and small values of the radial variable $r$. The symmetry properties of
the partial elements of the $S$-matrix under discrete transformations of the
Dirac Hamiltonian have been determined. In the framework of the first Born
approximation, a complete analytical investigation of fermion scattering in
the background field of the elementary topological soliton with winding number
$n=1$ has been carried out. In particular, the Born amplitudes, differential
cross-sections, and total cross-sections of fermion scattering have been
obtained in analytical form. The asymptotic behavior of the partial Born
amplitudes has been investigated for extreme values of the fermion momentum,
momentum transfer, and grand spin.

We have also performed a numerical study of fermion scattering in the
background field of the elementary topological solition of the nonlinear
$O(3)$ $\sigma$-model. In particular, we have obtained the $p$-dependences of
the partial elements $S_{I_{3}^{\prime},I_{3}}^{\left(m\right)}$ of the
$S$-matrix for $\left|m\right|\leq 5$, and have ascertained their main
features. In particular, we found that the partial elements
$S_{I_{3}^{\prime},I_{3}}^{\left(\pm 1\right)}$ of the $S$-matrix show
resonance behavior at small values of the fermion momentum.

###### Acknowledgements.

This work was supported by the Russian Science Foundation, grant No
19-11-00005.

##  Appendix A Free fermions

Let the isovector scalar field $\bm{\phi}$ be a constant field that takes the
value $(0,0,-1)$. This situation corresponds to the vacuum state in the
topologically trivial sector $(n=0)$ or distant regions in the topologically
nontrivial sectors $(n\neq 0)$. In this case, the Dirac equation (6) is
written as

$i\gamma^{\mu}\otimes\mathbb{I}\partial_{\mu}\psi-h\mathbb{I}\otimes\tau_{3}\psi=0$
(70)

or in the Hamiltonian form:

$i\frac{\partial\psi}{\partial t}=H_{0}\psi,$ (71)

where the free Hamiltonian

$H_{0}=\alpha^{k}\otimes\mathbb{I}\left(-i\partial_{k}\right)+h\beta\otimes\tau_{3}.$
(72)

The Dirac equation (70) is invariant under the $C$, $P$ and $\Pi_{2}T$
transformations:

$\displaystyle\psi\left(t,\mathbf{x}\right)$ $\displaystyle\rightarrow$
$\displaystyle\psi^{C}\left(t,\mathbf{x}\right)=i\gamma^{1}\mathbb{\otimes
I\,}\psi^{\ast}\left(t,\mathbf{x}\right),$ (73)
$\displaystyle\psi\left(t,\mathbf{x}\right)$ $\displaystyle\rightarrow$
$\displaystyle\psi^{P}\left(t,\mathbf{x}\right)=\gamma^{0}\mathbb{\otimes
I}\psi\left(t,-\mathbf{x}\right),$ (74)
$\displaystyle\psi\left(t,\mathbf{x}\right)$ $\displaystyle\rightarrow$
$\displaystyle\psi^{\Pi_{2}T}\left(t,\mathbf{x}\right)=\psi^{\ast}\left(-t,x_{1},-x_{2}\right).$
(75)

The Hamiltonian (72) commutes with the operator of grand spin (15), which is
reduced to the operator of the usual angular momentum in the topologically
trivial background vacuum field $\bm{\phi}_{\text{ vac}}=(0,0,-1)$. It also
commutes with the isospin generator $I_{3}=\tau_{3}/2$ and the momentum
operator $\mathbf{\hat{p}}=-i\nabla$. It therefore follows that the free
fermionic states can be characterized either by the momentum $\mathbf{p}$ and
the third isospin component $I_{3}$ (plane waves) or by the grand spin $K_{3}$
and the third isospin component $I_{3}$ (cylindrical waves).

In the compact matrix form, the plane-wave fermionic states $\psi_{\pm
p,I_{3}}$ with positive and negative energies can be written as

$\psi_{p,1/2}=\frac{1}{\sqrt{2\varepsilon}}\begin{pmatrix}\sqrt{\varepsilon+h}&0\\\
i\sqrt{\varepsilon-h}e^{i\theta_{\mathbf{p}}}&0\end{pmatrix}e^{-ipx},$ (76a)
$\psi_{p,-1/2}=\frac{1}{\sqrt{2\varepsilon}}\begin{pmatrix}0&-i\sqrt{\varepsilon-h}e^{-i\theta_{\mathbf{p}}}\\\
0&\sqrt{\varepsilon+h}\end{pmatrix}e^{-ipx},$ (76b)
$\psi_{-p,1/2}=\frac{1}{\sqrt{2\varepsilon}}\begin{pmatrix}-i\sqrt{\varepsilon-h}e^{-i\theta_{\mathbf{p}}}&0\\\
\sqrt{\varepsilon+h}&0\end{pmatrix}e^{ipx},$ (76c)
$\psi_{-p,-1/2}=\frac{1}{\sqrt{2\varepsilon}}\begin{pmatrix}0&\sqrt{\varepsilon+h}\\\
0&i\sqrt{\varepsilon-h}e^{i\theta_{\mathbf{p}}}\end{pmatrix}e^{ipx},$ (76d)

where $p=\left(\varepsilon,\mathbf{p}\right)$, $px=\varepsilon
t-\mathbf{p}\\!\bm{\cdot}\\!\mathbf{x}$, and $\theta_{\mathbf{p}}$ is the
azimuthal angle of the momentum $\mathbf{p}$. Note that the negative energy
wave functions are $C$-conjugates of the positive energy wave functions,
$\psi_{-p,\,I_{3}}=\left(\psi_{p,\,I_{3}}\right)^{C}=\sigma_{1}\mathbb{\otimes
I\,}\psi_{p,\,I_{3}}^{\ast}$. The wave functions $\psi_{\pm p,I_{3}}$ and
their amplitudes $u_{\pm p,I_{3}}$ defined by the formula $\psi_{\pm
p,I_{3}}=\left(2\varepsilon\right)^{-1/2}u_{\pm p,I_{3}}e^{\mp ipx}$ satisfy
the normalization conditions:

$\bar{\psi}_{\pm p,\,I_{3}}\gamma^{\mu}\otimes\mathbb{I}\psi_{\pm
p,\,I_{3}}=\left(1,\frac{p_{x}}{\varepsilon},\frac{p_{y}}{\varepsilon}\right)=\left(1,\,\mathbf{v}\right)$
(77)

and

$\bar{u}_{\pm p,I_{3}^{\prime}}u_{\pm p,\,I_{3}}=(-1)^{1/2\mp
I_{3}}2h\delta_{I_{3}^{\prime},I_{3}},\quad\bar{u}_{p,I_{3}^{\prime}}u_{-p,\,I_{3}}=0.$
(78)

It follows that the wave functions $\psi_{\pm p,I_{3}}$ have the normalization

$\displaystyle\int\bar{\psi}_{\pm
p^{\prime},I_{3}^{\prime}}(\mathbf{x})\psi_{\pm
p,I_{3}}(\mathbf{x})d^{2}\mathbf{x}=$ $\displaystyle(-1)^{1/2\mp
I_{3}}\frac{h}{\varepsilon}\left(2\pi\right)^{2}$
$\displaystyle\times\delta_{I_{3}^{\prime},I_{3}}\delta^{\left(2\right)}\left(\mathbf{p}-\mathbf{p}^{\prime}\right),$
(79)

where it is understood that in Eqs. (77)–(79), the summation is performed over
the spin and isospin indices of the corresponding wave functions and
amplitudes.

Let us consider the cylinder-wave fermionic states $\psi_{p,m,I_{3}}$ that
possess definite values of the grand spin $K_{3}$ and the isospin $I_{3}$. For
free fermions, the matrix $\Lambda$ in Eq. (24) takes the form

$\Lambda_{ia;jb}=\left(\begin{array}[]{cccc}\dfrac{l_{11}}{r}&0&\varepsilon+h&0\\\
0&\dfrac{l_{12}}{r}&0&\varepsilon-h\\\
-\varepsilon+h&0&-\dfrac{l_{21}}{r}&0\\\
0&-\varepsilon-h&0&-\dfrac{l_{22}}{r}\end{array}\right),$ (80)

where the orbital quantum numbers $l_{ia}$ are defined in Eq. (18). We see
that system (24) can be split into two independent subsystems that correspond
to the two isospin states $\pm 1/2$. These two subsystems can be solved
analytically, and their positive energy regular solutions are

$\displaystyle\psi_{p,m,1/2}$
$\displaystyle=\begin{pmatrix}\sqrt{\frac{p\left(\varepsilon+h\right)}{2\varepsilon}}J_{l_{11}}\left(pr\right)e^{il_{11}\theta}&0\\\
-\sqrt{\frac{p\left(\varepsilon-h\right)}{2\varepsilon}}J_{l_{21}}\left(pr\right)e^{il_{21}\theta}&0\end{pmatrix}e^{-i\varepsilon
t}\sim\sqrt{\frac{2}{\pi
r}}\begin{pmatrix}\sqrt{\frac{\varepsilon+h}{2\varepsilon}}\sin\left[pr-\pi\left(2l_{11}-1\right)/4\right]e^{il_{11}\theta}&0\\\
\sqrt{\frac{\varepsilon-h}{2\varepsilon}}\sin\left[pr-\pi\left(2l_{21}+3\right)/4\right]e^{il_{21}\theta}&0\end{pmatrix}e^{-i\varepsilon
t},$ (81a) $\displaystyle\psi_{p,m,-1/2}$
$\displaystyle=\begin{pmatrix}0&-\sqrt{\frac{p\left(\varepsilon-h\right)}{2\varepsilon}}J_{l_{12}}\left(pr\right)e^{il_{12}\theta}\\\
0&\sqrt{\frac{p\left(\varepsilon+h\right)}{2\varepsilon}}J_{l_{22}}\left(pr\right)e^{il_{22}\theta}\end{pmatrix}e^{-i\varepsilon
t}\sim\sqrt{\frac{2}{\pi
r}}\begin{pmatrix}0&\sqrt{\frac{\varepsilon-h}{2\varepsilon}}\sin\left[pr-\pi\left(2l_{12}+3\right)/4\right]e^{il_{12}\theta}\\\
0&\sqrt{\frac{\varepsilon+h}{2\varepsilon}}\sin\left[pr-\pi\left(2l_{22}-1\right)/4\right]e^{il_{22}\theta}\end{pmatrix}e^{-i\varepsilon
t},$ (81b)

where $p=\left|\bf{p}\right|$ and $J_{\nu}(pr)$ are the Bessel functions of
the first kind. The negative energy regular solutions are obtained from Eqs.
(81) by means of $C$-conjugation (73). Wave functions (81) satisfy the
normalization condition

$\displaystyle\int\psi_{p^{\prime},m^{\prime},I_{3}^{\prime}}^{\ast}(r,\theta)\psi_{p,m,I_{3}}(r,\theta)rdrd\theta=$
$\displaystyle 2\pi\delta_{m^{\prime}\\!,m}\delta_{I_{3}^{\prime},I_{3}}$
$\displaystyle\times\delta\left(p\\!-\\!p^{\prime}\right)\\!.$ (82)

Unlike wave functions (76), wave functions (81) have definite parity under
$P$-transformation (74)

$\displaystyle\psi^{P}_{p,m,I_{3}}\left(t,\mathbf{x}\right)$
$\displaystyle=\gamma^{0}\\!\otimes\\!\mathbb{I}\psi_{p,m,I_{3}}\left(t,-\mathbf{x}\right)$
$\displaystyle=\left(-1\right)^{m-nI_{3}-\frac{1}{2}}\psi_{p,m,I_{3}}\left(t,\mathbf{x}\right)\\!$
(83)

and are invariant under $\Pi_{2}T$-transformation (75).

The plane-wave fermionic states $\psi_{p,I_{3}}$ can be expanded in terms of
the cylinder-wave fermionic states $\psi_{p,m,I_{3}}$. To do this, we use the
well-known expansion of the plane wave in terms of cylinder waves

$e^{ipx}=e^{ipr\cos\left(\theta\right)}=\sum_{i=-\infty}^{\infty}i^{m}J_{m}\left(pr\right)e^{im\theta}.$
(84)

Eqs. (76), (81), and (84) give us the expansion

$\psi_{\left(\varepsilon,p,0\right),\,I_{3}}=\sum_{m=-\infty}^{\infty}a_{p,m,I_{3}}\psi_{p,m,I_{3}},$
(85)

where the expansion coefficients

$a_{p,m,I_{3}}=i^{m-I_{3}\left(n+1\right)}p^{-1/2}$ (86)

and all components of the $(2+1)$-dimensional momentum of the plane-wave state
are explicitly shown on the left-hand side of Eq. (85). Note that Eq. (86) is
valid only for the positive energy fermionic states. To obtain the expansion
coefficients for the negative energy fermionic states, we must take the
complex conjugate of the right-hand side of Eq. (86).

##  Appendix B Resonances of partial amplitudes at small fermion momenta

Let us ascertain the cause of occurrence of the resonance peaks in the partial
channel with grand spin $m=1$. It follows from Eqs. (18), (26), and (34) that
the characteristic feature of this partial channel is the absence of both the
kinematic suppression factor $(\varepsilon-h)^{1/2}$ and the centrifugal
barrier for the $c_{11}$ component of the radial wave function $c_{ia}(r)$. As
a result, the $c_{11}$ component is much larger than the other three
components of $c_{ia}(r)$. Using this fact, we can find an approximate
solution to system (24) for small values of $p$. To do this, we perform two
iterations. At the first iteration, we suppose that $c_{11}(r)$ is a constant
$\alpha_{0}$ and find $c_{12}(r)$, $c_{21}(r)$, and $c_{22}(r)$ neglecting all
terms except centrifugal and those that proportional to $\alpha_{0}$. At the
second iteration, we substitute $c_{12}(r)$, $c_{21}(r)$, and $c_{22}(r)$ into
the differential equation for $c_{11}(r)$ and integrate this equation. As a
result, we obtain the approximate iterative solution for the radial wave
function $c_{ia}(r)$

$\displaystyle c_{11}\left(r\right)$ $\displaystyle=$
$\displaystyle\frac{\alpha_{0}}{12}\Bigl{\\{}12-3p^{2}r^{2}-hr_{0}^{2}\left(\varepsilon+h\right)\Bigr{.}$
(87a)
$\displaystyle\times\Bigl{(}\pi^{2}-3\ln\left[\left(r^{2}+r_{0}^{2}\right)r_{0}^{-2}\right]\Bigr{.}$
$\displaystyle\times\left.\left(2+\ln\left[r_{0}^{2}\left(r^{2}+r_{0}^{2}\right)r^{-4}\right]\right)\right.$
$\displaystyle\left.\left.-6\text{Li}_{2}\left[r_{0}^{2}\left(r^{2}+r_{0}^{2}\right)^{-1}\right]\right)\right\\},$
$\displaystyle c_{12}\left(r\right)$ $\displaystyle=$
$\displaystyle\beta_{0}r-2^{-1}\alpha_{0}hr_{0}^{3}\left(\varepsilon+h\right)r^{-1}$
(87b)
$\displaystyle\times\ln\left[\left(r^{2}+r_{0}^{2}\right)r_{0}^{-2}\right],$
$\displaystyle c_{21}\left(r\right)$ $\displaystyle=$
$\displaystyle-2^{-1}\alpha_{0}\left(\varepsilon-h\right)r-\alpha_{0}hr_{0}^{2}r^{-1}$
(87c)
$\displaystyle\times\ln\left[\left(r^{2}+r_{0}^{2}\right)r_{0}^{-2}\right],$
$\displaystyle c_{22}\left(r\right)$ $\displaystyle=$
$\displaystyle\alpha_{0}hr_{0}\left(r_{0}^{2}r^{-2}\ln\left[\left(r^{2}\\!+\\!r_{0}^{2}\right)r_{0}^{-2}\right]\\!-\\!1\right),$
(87d)

where $\alpha_{0}$ and $\beta_{0}$ are constants, and $\text{Li}_{2}$ is the
dilogarithm function.

At small $r$, the components of the radial wave function can be written as

$\displaystyle c_{11}\left(r\right)$ $\displaystyle=$
$\displaystyle\alpha_{0}-2^{-2}\alpha_{0}p^{2}r^{2}+O\left(r^{4}\right),$
(88a) $\displaystyle c_{12}\left(r\right)$ $\displaystyle=$
$\displaystyle\left(\beta_{0}-2^{-1}\alpha_{0}hr_{0}\left(\varepsilon+h\right)\right)r$
(88b) $\displaystyle+O\left(r^{3}\right)\\!,$ $\displaystyle
c_{21}\left(r\right)$ $\displaystyle=$
$\displaystyle-2^{-1}\alpha_{0}\left(\varepsilon+h\right)r+O\left(r^{3}\right),$
(88c) $\displaystyle c_{22}\left(r\right)$ $\displaystyle=$
$\displaystyle-2^{-1}\alpha_{0}hr_{0}^{-1}r^{2}+O\left(r^{4}\right)$ (88d)

in accordance with Eq. (26). It was found that in a wide area of $r$, the
components of the radial wave function satisfy the conditions

$\displaystyle\left|c_{11}\right|$ $\displaystyle\gtrsim$
$\displaystyle\alpha_{0}\left(1-13.6\tau^{2}\right),$ (89a)
$\displaystyle\left|c_{12}\right|$ $\displaystyle\lesssim$ $\displaystyle
0.8\alpha_{0}\tau^{2},$ (89b) $\displaystyle\left|c_{21}\right|$
$\displaystyle\lesssim$ $\displaystyle 0.8\alpha_{0}\tau,$ (89c)
$\displaystyle\left|c_{22}\right|$ $\displaystyle\lesssim$
$\displaystyle\alpha_{0}\tau,$ (89d)

where the parameter $\tau=hr_{0}$. We see that under the condition $\tau\ll
1$, the absolute value of the $c_{11}$ component is much larger than those of
the other three components of the radial wave function.

When the distance from the core of the soliton is sufficiently large, we can
neglect the fermion-soliton interaction. In this case, the general solution to
system (24) is written as

$\displaystyle c_{11}\left(r\right)$ $\displaystyle\sim$ $\displaystyle
C_{1}J_{0}\left(pr\right)+C_{2}Y_{0}\left(pr\right),$ (90a) $\displaystyle
c_{12}\left(r\right)$ $\displaystyle\sim$
$\displaystyle-\sqrt{\frac{\varepsilon-h}{\varepsilon+h}}\left(C_{3}J_{1}\left(pr\right)+C_{4}Y_{1}\left(pr\right)\right),$
(90b) $\displaystyle c_{21}\left(r\right)$ $\displaystyle\sim$
$\displaystyle-\sqrt{\frac{\varepsilon-h}{\varepsilon+h}}\left(C_{1}J_{1}\left(pr\right)+C_{2}Y_{1}\left(pr\right)\right),$
(90c) $\displaystyle c_{22}\left(r\right)$ $\displaystyle\sim$ $\displaystyle
C_{3}J_{2}\left(pr\right)+C_{4}Y_{2}\left(pr\right),$ (90d)

where $C_{1}$ – $C_{4}$ are constant coefficients, $J_{\nu}$ and $Y_{\nu}$ are
the Bessel and Neumann functions of corresponding orders, respectively. Using
standard methods from the theory of scattering [32, 33], we obtain the
expressions for the partial elements of the $S$-matrix in terms of the
coefficients $C_{1}$ – $C_{4}$

$\displaystyle S_{1/2,1/2}^{\left(+1\right)}$ $\displaystyle=$
$\displaystyle\frac{C_{1}-iC_{2}}{C_{1}+iC_{2}},$ (91a) $\displaystyle
S_{-1/2,1/2}^{\left(+1\right)}$ $\displaystyle=$
$\displaystyle\frac{2p}{\varepsilon-h}\frac{C_{3}}{C_{1}+iC_{2}}.$ (91b)

To express the coefficients $C_{1}$ – $C_{4}$ in terms of physical parameters,
we join solutions (87) and (90) at $r=\kappa r_{0}$, where $\kappa$ is a
positive coefficient greater than one. The resulting expressions for $C_{1}$ –
$C_{4}$ are too lengthy to be presented here. Expanding these expressions in
$p$, holding lower-order terms, and substituting the resulting expressions in
Eq. (91), we obtain the partial matrix elements:

$S_{1/2,1/2}^{\left(+1\right)}=\frac{A+\pi^{-1}\Gamma\ln\left[pr_{0}\right]+i\Gamma/2}{A+\pi^{-1}\Gamma\ln\left[pr_{0}\right]-i\Gamma/2},$
(92)

and

$S_{-1/2,1/2}^{\left(+1\right)}=\frac{-i\tau\Gamma}{A+\pi^{-1}\Gamma\ln\left[pr_{0}\right]-i\Gamma/2}$
(93)

where

$\displaystyle A$ $\displaystyle=$ $\displaystyle
1+\kappa^{2}-\frac{1}{6}\tau^{2}\left(\left(12\gamma+\pi^{2}\right)\kappa^{2}+\pi^{2}\right)$
$\displaystyle+\frac{1}{2}\tau^{2}\left\\{2\kappa^{2}\ln\left[4\left(\kappa^{2}+1\right)\kappa^{-2}\right]\right.$
$\displaystyle\left.+\left\\{4\left(\kappa^{2}+1\right)\left(\gamma+\ln\left[2^{-1}\left(\kappa^{2}+1\right)^{1/4}\right]\right)+2\right\\}\right.$
$\displaystyle\times\left.\ln\left[\kappa^{2}+1\right]\right\\}+\tau^{2}\left(\kappa^{2}+1\right)\text{Li}_{2}\left[\left(\kappa^{2}+1\right)^{-1}\right],$
$\Gamma=2\pi\tau^{2}\left(\left(\kappa^{2}+1\right)\ln\left[\kappa^{2}+1\right]-\kappa^{2}\right),$
(95)

and $\gamma$ is the Euler-Mascheroni constant.

Using Eqs. (92), (93), and (33), we obtain the expressions for the squared
magnitudes of the partial amplitudes $f_{1/2,1/2}^{\left(+1\right)}$ and
$f_{-1/2,1/2}^{\left(+1\right)}$:

$\left|f_{1/2,1/2}^{\left(+1\right)}\right|^{2}=\frac{2}{\pi
p}\frac{\left(\Gamma/2\right)^{2}}{\left(A+\pi^{-1}\Gamma\ln\left[pr_{0}\right]\right)^{2}+\left(\Gamma/2\right)^{2}},$
(96)

and

$\left|f_{-1/2,1/2}^{\left(+1\right)}\right|^{2}=\frac{1}{2\pi
p}\frac{\Gamma_{r}\Gamma}{\left(A+\pi^{-1}\Gamma\ln\left[pr_{0}\right]\right)^{2}+\left(\Gamma/2\right)^{2}},$
(97)

where

$\Gamma_{r}=\tau^{2}\Gamma.$ (98)

We can see that the squared magnitudes of the partial amplitudes
$f_{1/2,1/2}^{\left(+1\right)}$ and $f_{-1/2,1/2}^{\left(+1\right)}$,
considered as functions of the logarithmic variable
$\pi^{-1}\Gamma\ln\left[pr_{0}\right]$, show resonance behavior of the Breit-
Wigner type with total decay width equal to $\Gamma$. It follows that on the
logarithmic scale, the squared magnitudes
$|S_{I_{3}^{\prime},I_{3}}^{\left(+1\right)}-\delta_{I_{3}^{\prime},I_{3}}|^{2}=2\pi
p|f_{I_{3}^{\prime},I_{3}}^{\left(+1\right)}|^{2}$ are of the symmetric Breit-
Wigner form in the resonance region, as it is in Fig. 8. At the same time, the
resonance peaks of
$|S_{I_{3}^{\prime},I_{3}}^{\left(+1\right)}-\delta_{I_{3}^{\prime},I_{3}}|^{2}$
have a strongly asymmetric form in the linear scale. In particular, the
resonance peaks have maxima at $p_{\max}=r_{0}^{-1}\exp\left(-\pi
A/\Gamma\right)$ and reach half of the maxima at $p_{\pm
1/2}=r_{0}^{-1}\exp\left(-\pi A/\Gamma\pm\pi/2\right)$. The asymmetry of the
resonance peaks is reflected in the relations

$\frac{p_{+1/2}}{p_{\max}}=\frac{p_{\max}}{p_{-1/2}}=\exp\left(\pi/2\right)\approx
4.81048,$ (99a) and
$\frac{p_{\max}-p_{-1/2}}{p_{+1/2}-p_{\max}}=\exp\left(-\pi/2\right)\approx
0.20788.$ (99b)

The position of the maxima of the resonance peaks depends exponentially on the
ratio $A/\Gamma$, which, in turn, depends on the parameters $\tau$ and
$\kappa$. This results in extremely low values of $p_{\max}$ when
$\tau\lesssim 0.1$ and $1\lesssim\kappa\lesssim 10^{2}$ in accordance with
Figs. 8 and 9.

The existence of the resonance peaks and their logarithmic character are due
to the term $\ln\left[pr_{0}\right]$ in Eqs. (92) and (93). In turn, this term
results from the leading term of the expansion of the Neumann function
$Y_{0}(pr)$ in the neighborhood of zero:
$Y_{0}\left(pr\right)=2\pi^{-1}\left(\gamma-\ln\left(2\right)+\ln\left[pr\right]\right)+O(\left(pr\right)^{2}\ln\left[pr\right])$.
Only the Neumann function $Y_{0}(pr)$ has the leading asymptotic behavior
$\propto\ln[pr]$ as $pr\rightarrow 0$, whereas the other Neumann functions
$Y_{l>0}(pr)\propto\left(pr\right)^{-l}$ in this limit. Therefore, the Neumann
functions $Y_{l>0}(pr)$ cannot lead to resonance peaks of the logarithmic
type.

It follows from Eq. (92) that the squared magnitude of
$S_{1/2,1/2}^{\left(+1\right)}$ is equal to one. Therefore, the unitarity
condition
$|S_{1/2,1/2}^{\left(+1\right)}|^{2}+|S_{-1/2,1/2}^{\left(+1\right)}|^{2}=1$
is not satisfied in the used approach. Eq. (93) tells us that the squared
magnitude of $S_{-1/2,1/2}^{\left(+1\right)}$ does not exceed the value of
$4\tau^{2}=\left(2hr_{0}\right)^{2}$, which is equal to $0.04$ for the values
of $h$ and $r_{0}$ used in Sec. V. It follows that the violation of the
unitarity is small when the parameter $\tau=hr_{0}$ is much smaller than one.

The partial matrix elements (92) and (93) have a simple pole located at
$p_{\text{pol}}=ir_{0}^{-1}\exp\left(-\pi A/\Gamma\right)=ip_{\max}$.
According to the theory of scattering [32, 33], a pole of an elastic partial
element of the $S$-matrix located at a positive imaginary momentum corresponds
to a bound state. In our case, however, the presence of the pole is only an
artifact of the used approach. Indeed, this pole exists for arbitrary small
values of $h$ and $r_{0}$, which is impossible for bound states. We also
failed to find fermion bound states in the partial channels with $m=\pm 1$
using numerical methods.

The obvious drawback of the used approach is that it is unable to explain the
periodic structure of the resonance peaks in Figs. 8 and 9. This approach,
however, can explain the location of the resonance peaks at extremely low
fermion momenta and the Breit-Wigner form of these peaks on the logarithmic
scale.

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Figure 1:  Dependence of $\text{Re}[S_{1/2,1/2}^{\left(0\right)}-1]$,
$\text{Im}[S_{1/2,1/2}^{\left(0\right)}]$,
$|S_{1/2,1/2}^{\left(0\right)}|^{2}$,
$\text{Re}[S_{-1/2,1/2}^{\left(0\right)}]$,
$\text{Im}[S_{-1/2,1/2}^{\left(0\right)}]$, and
$|S_{-1/2,1/2}^{\left(0\right)}|^{2}$ on the fermion momentum $p$. Figure 2:
Dependence of $\text{Re}[S_{1/2,1/2}^{\left(m\right)}-1]$ on the fermion
momentum $p$ for $|m|=2,3,4,\,\text{and}\;5$.

Figure 3:  Dependence of $\text{Im}[S_{1/2,1/2}^{\left(m\right)}]$ on the
fermion momentum $p$ for $|m|=2,3,4,\,\text{and}\;5$. Figure 4:  Dependence of
$|S_{1/2,1/2}^{\left(m\right)}-1|^{2}$ on the fermion momentum $p$ for
$|m|=2,3,4,\,\text{and}\;5$.

Figure 5:  Dependence of $\text{Re}[S_{-1/2,1/2}^{\left(m\right)}]$ on the
fermion momentum $p$ for $|m|=2,3,4,\,\text{and}\;5$. Figure 6:  Dependence of
$\text{Im}[S_{-1/2,1/2}^{\left(m\right)}]$ on the fermion momentum $p$ for
$|m|=2,3,4,\,\text{and}\;5$.

Figure 7:  Dependence of $|S_{-1/2,1/2}^{\left(m\right)}|^{2}$ on the fermion
momentum $p$ for $|m|=2,3,4,\,\text{and}\;5$. Figure 8:  Dependence of
$\text{Re}[S_{1/2,1/2}^{\left(+1\right)}-1]$,
$\text{Im}[S_{1/2,1/2}^{\left(+1\right)}]$,
$|S_{1/2,1/2}^{\left(+1\right)}|^{2}$,
$\text{Re}[S_{-1/2,1/2}^{\left(+1\right)}]$,
$\text{Im}[S_{-1/2,1/2}^{\left(+1\right)}]$, and
$|S_{-1/2,1/2}^{\left(+1\right)}|^{2}$ on the fermion momentum $p$. Figure 9:
Dependence of $\text{Re}[S_{1/2,1/2}^{\left(-1\right)}-1]$,
$\text{Im}[S_{1/2,1/2}^{\left(-1\right)}]$,
$|S_{1/2,1/2}^{\left(-1\right)}|^{2}$,
$\text{Re}[S_{-1/2,1/2}^{\left(-1\right)}]$,
$\text{Im}[S_{-1/2,1/2}^{\left(-1\right)}]$, and
$|S_{-1/2,1/2}^{\left(-1\right)}|^{2}$ on the fermion momentum $p$.