# Scattering of fermionic isodoublets on the sine-Gordon kink

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

###### Abstract

The scattering of Dirac fermions on the sine-Gordon kink is studied both
analytically and numerically. To achieve invariance with respect to a discrete
symmetry, the sine-Gordon model is treated as a nonlinear $\sigma$-model with
a circular target space that interacts with fermionic isodublets through the
Yukawa interaction. It is shown that the diagonal and antidiagonal parts of
the fermionic wave function interact independently with the external field of
the sine-Gordon kink. The wave functions of the fermionic scattering states
are expressed in terms of the Heun functions. General expressions for the
transmission and reflection coefficients are derived, and their dependences on
the fermion momentum and mass are studied numerically. The existence condition
is found for two fermionic zero modes, and their analytical expressions are
obtained. It is shown that the zero modes do not lead to fragmentation of the
fermionic charge, but can lead to polarization of the fermionic vacuum. The
scattering of the diagonal and antidiagonal fermionic states is found to be
significantly different; this difference is shown to be due to the different
dependences of the energy levels of these bound states on the fermion mass,
and is in accordance with Levinson’s theorem.

##  I  Introduction

Many field theory models with spontaneously broken symmetry possess stable
localized solutions known as topological solitons [1]. One example of a
topological soliton is a kink [2], which is a one-dimensional static solution
of a $(1+1)$-dimensional field model. The best known are the kinks of the
$\phi^{4}$ [3, 4, 5] and the sine-Gordon models [6, 7]. The sine-Gordon model
has a number of remarkable properties; in particular, it possesses an infinite
number of conserved quantities at both the classical and quantum levels.
Furthermore, all time-dependent solutions of the sine-Gordon model are known,
and can be written down explicitly in closed form. The reason for this is that
the sine-Gordon model in $(1+1)$ dimensions is an integrable system. Another
special property of the sine-Gordon model is its equivalence to the massive
Thirring model, and the quantum soliton of the sine-Gordon model can be
identified with the fermion of the massive Thirring model [8].

The sine-Gordon model and its modifications are used in the study of a wide
range of phenomena, including QCD [9, 10, 11], condensed matter physics [12],
and solid state physics in relation to the Josephson junctions and their
associated magnetism and topological excitations [13, 14, 15, 16]. Models of
the sine-Gordon type are also used to describe vortex dynamics in
superconducting systems [17] and to investigate the Josephson current in some
systems [18].

The interaction between fermions and the background fields of topological
solitons leads to a number of interesting physical effects, such as
fragmentation of the fermionic number and polarization of the fermionic vacuum
[19, 20, 21, 22], superconductivity [23], and monopole catalysis of the proton
decay [24, 25, 26, 27]. In the case of a kink, fragmentation of the fermionic
number and polarization of the fermionic vacuum are possible [19]. In
addition, the kink’s field distorts the energy levels of the fermionic vacuum;
bound states can arise and continuum states can change compared to a free
fermion. These lead to a change in the zero-point fermion energy, and
consequently to the Casimir effect, in the presence of the kink [28].

The fermion-kink interactions, kink-antikink configurations, and domain walls
were also considered in Refs. [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39,
40]. The main purpose of the present work is to study the scattering and bound
states of fermions in the external field of the sine-Gordon kink. A
characteristic property of the sine-Gordon model is its invariance with
respect to discrete equidistant shifts of a scalar field. The usual Yukawa
interaction of a fermion with the scalar field will break this discrete
$\mathbb{Z}$-symmetry of the sine-Gordon model, and in order to avoid this,
the sine-Gordon model is treated as a nonlinear $\sigma$-model whose target
space is a circle, and the scalar field is treated as an angular variable.
Under these conditions, the Yukawa interaction of the $\sigma$-model’s two-
component nonlinear scalar field with a fermionic isodoublet will not break
the discrete $\mathbb{Z}$-symmetry.

This paper is structured as follows. In Sec. II, we describe briefly the
Lagrangian, symmetries, field equations, and kink solution of the sine-Gordon
model. In Sec. III, we consider fermion-kink scattering, and derive analytical
expressions for the fermionic scattering states and general expressions for
the transmission and reflection coefficients. In Sec. IV, we study the
fermionic bound states: the general properties of the fermionic bound states
are described, and their symmetries under the charge conjugation and parity
transformations are established. A condition for the existence of the
fermionic zero modes is found, and the properties of these modes are studied.
In Sec. V, we present numerical results relating to the fermion-kink
interaction. In the final section, we briefly summarize the results obtained
in this work. Appendix A contains some necessary information on the plane-wave
states of free fermions, and Appendix B gives an explanation of the dependence
of the energy of the fermionic bound state on the fermion mass at a
qualitative level.

Throughout the paper, the natural units $c=1$ and $\hbar=1$ are used.

##  II  Lagrangian, symmetries, and field equations of the model

The Lagrangian density of the $(1+1)$-dimensional sine-Gordon model has the
form

$\mathcal{L}=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-m^{4}\lambda^{-1}\bigl{(}1-\cos\bigl{(}m^{-1}\lambda^{1/2}\phi\bigr{)}\bigr{)},$
(1)

where $\phi(t,x)$ is a real scalar field, $m$ is the mass of the $\phi$-meson,
and $\lambda$ is the self-interaction coupling constant of the real scalar
field $\phi$. The Lagrangian $L=\int\mathcal{L}dx$ possesses symmetries under
the two discrete transformations:

$\bar{\phi}\left(t,x\right)\rightarrow-\bar{\phi}\left(t,x\right),$ (2)

and

$\bar{\phi}\left(t,x\right)\rightarrow\bar{\phi}\left(t,x\right)+2\pi
N,\;N\in\mathbb{Z},$ (3)

where the rescaled scalar field $\bar{\phi}=m^{-1}\lambda^{1/2}\phi$.
According to these symmetries, the classical vacua of model (1) are located at
the points

$\bar{\phi}\left(t,x\right)=2\pi N,\;N\in\mathbb{Z},$ (4)

which correspond to the zero minima of the potential
$U\left(\bar{\phi}\right)=1-\cos\left(\bar{\phi}\right)$. It follows from Eqs.
(3) and (4) that the field $\bar{\phi}=m^{-1}\lambda^{1/2}\phi$ may be
interpreted as an angular variable, defined modulo $2\pi$. We want the
Lagrangian to remain invariant under discrete transformations (3) when
fermions are included in the sine-Gordon model, and it is obvious that the
usual version of the Yukawa interaction $\bar{\psi}\phi\psi$ does not satisfy
this requirement.

The interpretation of $\bar{\phi}$ as an angular variable makes it possible to
formulate the sine-Gordon model as a nonlinear $\sigma$ model whose target
space is a circle [1]. To do this, we introduce the two-component isovector
scalar field

$\bm{\phi}=\left(\phi_{1},\phi_{2}\right)=\bigl{(}m\lambda^{-1/2}\cos\left(\bar{\phi}\right),\,m\lambda^{-1/2}\sin\left(\bar{\phi}\right)\bigr{)},$
(5)

which is invariant under discrete transformations (3). We want the interaction
between the Dirac fermionic field and isovector scalar field (5) to also be
invariant under transformations (3). This is most naturally achieved through
the Yukawa interaction
$\bm{\phi}\\!\bm{\cdot}\\!\bar{\psi}\bm{\tau}_{\perp}\psi$, where
$\bm{\tau}_{\perp}=(\tau_{1},\tau_{2})$ and $\tau_{1,2}$ are the corresponding
Pauli matrices. Indeed, the $SO(2)$ isovector
$\bar{\psi}\bm{\tau}_{\perp}\psi$ is also invariant under the $2\pi$ rotation
about the third isotopic axis, as is the $SO(2)$ isovector $\bm{\phi}$.
Moreover, the term $\bm{\phi}\\!\bm{\cdot}\\!\bar{\psi}\bm{\tau}\psi$ is
invariant under the full $SU(2)$ isotopic group, and hence the term
$\bm{\phi}\\!\bm{\cdot}\\!\bar{\psi}\bm{\tau}_{\perp}\psi$ is invariant under
the $U(1)$ subgroup of the $SU(2)$ isotopic group that corresponds to isotopic
rotations about the third axis.

In terms of the isovector field $\bm{\phi}$, the Lagrangian density of the
sine-Gordon model including an isodublet of the Dirac fermions takes the form

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\partial_{\mu}\bm{\phi\cdot}\partial^{\mu}\bm{\phi}-m^{4}\lambda^{-1}\bigl{(}1-m^{-1}\lambda^{1/2}\phi_{1}\bigr{)}$
$\displaystyle+\frac{\nu}{2}\left(\bm{\phi\cdot\phi}-m^{2}\lambda^{-1}\right)+i\bar{\psi}\gamma^{\mu}\partial_{\mu}\psi-g\bm{\phi\cdot}\bar{\psi}\bm{\tau}_{\perp}\psi,$

where the Lagrange multiplier $\nu$ is introduced to constrain $\bm{\phi}$ to
lie on the circle $\bm{\phi\cdot\phi}=m^{2}\lambda^{-1}$. It is understood
that the two types of indices of the Dirac field $\psi_{ia}$ correspond to its
spinor-isospinor structure. In $(1+1)$ dimensions, we use the following Dirac
matrices:

$\gamma^{0}=\sigma_{1},\;\gamma^{1}=-i\sigma_{2},\;\gamma_{5}=\gamma^{0}\gamma^{1}=\sigma_{3},$
(7)

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

It is readily seen that the Lagrangian density (II) is invariant under
discrete $\mathbb{Z}$-transformations (3). It is also invariant under discrete
$\mathbb{Z}_{2}$-transformation (2) provided that the Dirac fermionic field
$\psi_{ia}$ is transformed as $\psi\rightarrow\mathbb{I}\otimes\tau_{3}\psi$.
By varying the action $S=\int\mathcal{L}dxdt$ in the corresponding fields, we
obtain the field equations:

$\partial_{\mu}\partial^{\mu}\bm{\phi}-m^{3}\lambda^{-1/2}\mathbf{n}_{1}-\nu\bm{\phi}+g\bar{\psi}\bm{\tau}_{\perp}\psi=0,$
(8)

and

$i\gamma^{\mu}\partial_{\mu}\psi-g\bm{\phi}\bm{\cdot}\bm{\tau}_{\perp}\psi=0,$
(9)

where the Lagrange multiplier

$\nu=m^{-2}\lambda\bigl{(}\bm{\phi}\\!\bm{\cdot}\\!\partial_{\mu}\partial^{\mu}\bm{\phi}-m^{3}\lambda^{-1/2}\bm{\phi}\\!\bm{\cdot}\\!\mathbf{n}_{1}+g\bm{\phi}\\!\bm{\cdot}\\!\bar{\psi}\bm{\tau}_{\perp}\psi\bigr{)}$
(10)

and the unit isovector $\mathbf{n}_{1}=(1,0)$.

It is known that that among the numerous soliton solutions, the sine-Gordon
model possesses the static kink solution

$\phi_{\text{k}}(x)=4m\lambda^{-1/2}\arctan\left[\exp\left(mx\right)\right]$
(11)

or in terms of the isovector field

$\displaystyle\bm{\phi}_{\text{k}}(x)$ $\displaystyle=$ $\displaystyle
m\lambda^{-1/2}\left(1-2\text{sech}^{2}\left(mx\right),\right.$ (12)
$\displaystyle\left.-2\text{sech}\left(mx\right)\tanh\left(mx\right)\right).$

The sine-Gordon kink has mass

$M_{\text{k}}=8m^{3}\lambda^{-1}.$ (13)

Note that the mass of the kink is inversely proportional to the self-
interaction coupling constant, which is a characteristic property of soliton
solutions [41].

We shall now establish the symmetry properties of the Dirac equation (9) under
the discrete $C$, $P$, and $T$ transformations in the case of external field
of kink solution (12). It is easy to see that the kink solution has the
symmetry property

$\bm{\phi}_{\text{k}}\left(x\right)\\!\bm{\cdot}\\!\bm{\tau}_{\perp}=\bm{\phi}_{\text{k}}\left(-x\right)\\!\bm{\cdot}\\!\bm{\tau}_{\perp}^{\ast}.$
(14)

Using this symmetry property, it can easily be shown that if $\psi(t,x)$ is a
solution of the Dirac equation (9) in the external field of kink (12), then

$\displaystyle\psi^{C}\left(t,x\right)$ $\displaystyle=$
$\displaystyle\eta_{C}\gamma_{5}\otimes\tau_{1}\psi^{\ast}\left(t,x\right),$
(15a) $\displaystyle\psi^{P}\left(t,x\right)$ $\displaystyle=$
$\displaystyle\eta_{P}\gamma^{0}\otimes\tau_{1}\psi\left(t,-x\right),$ (15b)
$\displaystyle\psi^{T}\left(t,x\right)$ $\displaystyle=$
$\displaystyle\eta_{T}\gamma^{0}\otimes\tau_{1}\psi^{\ast}\left(-t,x\right),$
(15c)

where $\eta_{C}$, $\eta_{P}$, and $\eta_{T}$ are phase multipliers, are also
solutions of the Dirac equation in the external field of the kink. Of course,
the Lagrangian density (II) remains invariant under these $C$, $P$, and $T$
transformations.

The antikink solution is obtained from kink solution (12) through the
inversion $x\rightarrow-x$. Using the symmetry property (14), it can easily be
shown that if $\psi_{\text{k}}(t,x)$ is a solution of the Dirac equation (9)
in the external field of kink (12), then

$\psi_{\text{ak}}\left(t,x\right)=\mathbb{I}\otimes\tau_{1}\psi_{\text{k}}\left(t,x\right)$
(16)

is a solution of the Dirac equation (9) in the external field of the antikink
$\bm{\phi}_{\text{ak}}(x)=\bm{\phi}_{\text{k}}(-x)$.

As $x\rightarrow\pm\infty$, isovector kink field (12) tends to the vacuum
value

$\underset{x\rightarrow\pm\infty}{\lim}\bm{\phi}_{\text{k}}\left(x\right)=\bm{\phi}_{\text{vac}}=\bigl{(}m\lambda^{-1/2},0\bigr{)}.$
(17)

It follows that in the distant spatial regions $\left|x\right|\gg m^{-1}$, the
Dirac equation (9) describes free fermions (antifermions) with mass
$M=mg\lambda^{-1/2}$. The properties of the corresponding plane-wave solutions
of the Dirac equation and their explicit forms are given in Appendix A.

##  III  Fermion-kink scattering

We shall consider fermion scattering on the sine-Gordon kink (12) in the
external field approximation. In this approximation, the backreaction of a
fermion on the field of the kink is neglected, and kink-soliton scattering is
described solely by the Dirac equation (9). To neglect the fermion
backreaction, the contribution of the fermionic terms in Eqs. (8) and (10)
should be much less than that of the corresponding bosonic terms. Using
normalization condition (113c) to estimate the contribution of the fermionic
terms, it can be shown that the condition of their smallness has the form

$g^{2}\ll m^{2}\varepsilon L,$ (18)

where $\varepsilon$ is the fermion energy and $L$ is the normalized length. In
the nonrelativistic case, condition (18) becomes more stringent:

$g^{2}\ll m^{2}ML=gm^{2}\lambda^{-1/2}mL,$ (19)

where the fermion mass $M=mg\lambda^{-1/2}$.

We see that conditions (18) and (19) can always be fulfilled if the normalized
length $L$ is sufficiently large, corresponding to a low linear density of the
incident fermions. From the viewpoint of QFT, however, we are talking about
the scattering of a fermion of mass $M=mg\lambda^{-1/2}$ on a sine-Gordon kink
of mass $M_{\text{k}}=8m^{3}\lambda^{-1}$. To neglect the recoil of the kink
in fermion scattering, the mass of the kink $M_{\text{k}}$ must be much larger
than the mass of the fermion $M$, which leads to the condition

$g\ll 8m^{2}\lambda^{-1/2}.$ (20)

By comparing Eqs. (19) and (20), we see that they are equivalent if the
normalization length $L\approx 8m^{-1}$. In turn, it follows from Eq. (11)
that the value of $8m^{-1}$ can be interpreted as the spatial size of the
kink. Hence, Eqs. (19) and (20) become equivalent when there is on average one
fermion in a region whose length is equal to the spatial size of the kink.

Substituting the ansatz $\psi_{ia}\left(t,x\right)=\exp\left(-i\varepsilon
t\right)\psi_{ia}\left(x\right)$ and background kink field (12) into the Dirac
equation (9), we obtain the two independent systems of differential equations:

$\begin{pmatrix}\psi_{11}^{\prime}\\\
\psi_{22}^{\prime}\end{pmatrix}=\begin{pmatrix}i\varepsilon&-iMF\left(x\right)\\\
iMF^{\ast}\\!\\!\left(x\right)&-i\varepsilon\end{pmatrix}\begin{pmatrix}\psi_{11}\\\
\psi_{22}\end{pmatrix}$ (21)

and

$\begin{pmatrix}\psi_{12}^{\prime}\\\
\psi_{21}^{\prime}\end{pmatrix}=\begin{pmatrix}i\varepsilon&-iMF^{\ast}\\!\\!\left(x\right)\\\
iMF\left(x\right)&-i\varepsilon\end{pmatrix}\begin{pmatrix}\psi_{12}\\\
\psi_{21}\end{pmatrix},$ (22)

where the unitary function

$\displaystyle F(x)$ $\displaystyle=$
$\displaystyle\left(\frac{e^{mx}+i}{e^{mx}-i}\right)^{2}$ $\displaystyle=$
$\displaystyle
1-2\,\text{sech}^{2}\left(mx\right)+2i\,\text{sech}(mx)\tanh(mx).$

The Dirac equation (9) can be split into the two independent subsystems (21)
and (22) since the Dirac Hamiltonian

$H_{D}=\alpha\\!\otimes\\!\mathbb{I}\left(-i\partial_{x}\right)+g\beta\\!\otimes\\!\bm{\phi}_{\text{k}}\\!\bm{\cdot}\\!\bm{\tau}_{\perp}$
(24)

commutes with the spin-isospin operator
$T_{3}=\gamma_{5}\\!\otimes\\!\tau_{3}$:

$\left[H_{D},T_{3}\right]=0.$ (25)

Here, Eq. (21) contains only the diagonal elements of the matrix $\psi_{ia}$,
whereas Eq. (22) contains only the antidiagonal ones. Let us denote the
diagonal and antidiagonal parts of the matrix $\psi_{ia}$ as $\psi_{\text{d}}$
and $\psi_{\text{a}}$, respectively:

$\psi_{\text{d}}=\begin{pmatrix}\psi_{11}&0\\\
0&\psi_{22}\end{pmatrix},\quad\psi_{\text{a}}=\begin{pmatrix}0&\psi_{12}\\\
\psi_{21}&0\end{pmatrix}.$ (26)

It can then be easily shown that $\psi_{\text{d}}$ and $\psi_{\text{a}}$ are
the eigenmatrices of the operator $T_{3}$:

$T_{3}\psi_{\text{d}}=\psi_{\text{d}},\quad
T_{3}\psi_{\text{a}}=-\psi_{\text{a}}.$ (27)

It follows from Eqs. (25) and (27) that $\psi_{\text{d}}$ and
$\psi_{\text{a}}$ are independent from each other, and this is reflected in
Eqs. (21) and (22).

It can be shown that system (21) is equivalent to the second order
differential equation

$\displaystyle\psi_{11}^{\prime\prime}(x)+2im\,\text{sech}(mx)\psi_{11}^{\prime}(x)$
(28) $\displaystyle+\left(k^{2}+2\varepsilon
m\,\text{sech}(mx)\right)\psi_{11}(x)$ $\displaystyle=0$

together with the relation

$\psi_{22}(x)=M^{-1}F^{\ast}\\!\\!\left(x\right)\left(\varepsilon\psi_{11}(x)+i\psi_{11}^{\prime}(x)\right),$
(29)

where $k^{2}=\varepsilon^{2}-M^{2}$. Specifically, any solution to system (21)
satisfies Eqs. (28) and (29). Conversely, the function $\psi_{11}(x)$ that
satisfies Eq. (28) together with the function $\psi_{22}(x)$ obtained from Eq.
(29) form a solution to system (21). Indeed, in the latter case, the
derivative $\psi_{22}^{\prime}$ satisfies the relation

$\psi_{22}^{\prime}\left(x\right)=M^{-1}F^{\ast}\left(x\right)\left(\varepsilon\psi_{11}^{\prime}\left(x\right)-ik^{2}\psi_{11}\left(x\right)\right),$
(30)

where we use Eq. (28) to eliminate $\psi_{11}^{\prime\prime}$. It is easy to
see that by substituting Eqs. (29) and (30) into system (21), the latter is
identically satisfied. Similarly, system (22) is equivalent to the second-
order differential equation

$\displaystyle\psi_{12}^{\prime\prime}(x)-2im\,\text{sech}(mx)\psi_{12}^{\prime}(x)$
(31) $\displaystyle+\left(k^{2}-2\varepsilon
m\,\text{sech}(mx)\right)\psi_{12}(x)$ $\displaystyle=0$

together with the relation

$\psi_{21}(x)=M^{-1}F\left(x\right)\left(\varepsilon\psi_{12}(x)+i\psi_{12}^{\prime}(x)\right).$
(32)

By applying a change of variable $\xi=\tanh\left(mx\right)$, the second-order
differential equations (28) and (31) are reduced to the forms

$\displaystyle\psi_{11}^{\prime\prime}(\xi)-\frac{2\left(\xi-i\sqrt{1-\xi^{2}}\right)}{1-\xi^{2}}\psi_{11}^{\prime}(\xi)$
$\displaystyle+\frac{\left(k^{2}+2\varepsilon
m\sqrt{1-\xi^{2}}\right)}{m^{2}\left(1-\xi^{2}\right)^{2}}\psi_{11}(\xi)=0$
(33)

and

$\displaystyle\psi_{12}^{\prime\prime}(\xi)-\frac{2\left(\xi+i\sqrt{1-\xi^{2}}\right)}{1-\xi^{2}}\psi_{12}^{\prime}(\xi)$
$\displaystyle+\frac{\left(k^{2}-2\varepsilon
m\sqrt{1-\xi^{2}}\right)}{m^{2}\left(1-\xi^{2}\right)^{2}}\psi_{12}(\xi)=0.$
(34)

The solutions to differential equations (33) and (34) can be expressed in
terms of the local Heun functions [42, 43]. Let the fermionic wave fall on the
kink from the left. Then, as $x\rightarrow+\infty$, the solutions to Eqs. (33)
and (34) must approximate the transmitted plane wave, which is
$\propto\exp(ikx)$. In terms of the initial variable $x$, the corresponding
solutions to Eqs. (33) and (34) are

$\psi_{11}\left(x\right)=e^{ikx}Hl\left[\frac{1}{2},-i\frac{2\left(\varepsilon-k\right)}{m};-1,0,1-i\frac{2k}{m},1+i\frac{2k}{m};\frac{1}{1+ie^{mx}}\right]$
(35)

and

$\displaystyle\psi_{12}\left(x\right)$ $\displaystyle=$ $\displaystyle
e^{-\frac{\pi}{2}\frac{k}{m}}\left(e^{-i\frac{\pi}{4}}+e^{i\frac{\pi}{4}}e^{-mx}\right)^{2i\frac{k}{m}}e^{ikx}\times$
(36) $\displaystyle
Hl\left[\frac{1}{2},-2\frac{k^{2}}{m^{2}}-i\frac{2\varepsilon-k}{m};-1-i\frac{2k}{m},-i\frac{2k}{m},1-i\frac{2k}{m},1-i\frac{2k}{m};\frac{1}{1-ie^{mx}}\right],$

where we use the notation $Hl\left(a,q;\alpha,\beta,\gamma,\delta;z\right)$
for the six-parameter local Heun function [42, 43].

The local Heun function $Hl\left(a,q;\alpha,\beta,\gamma,\delta;z\right)$
represents the solution to the second-order differential Heun’s equation [42,
43]. Heun’s equation possesses four regular singular points located at $z=0$,
$z=1$, $z=a$, and $z=\infty$. In Eqs. (35) and (36), the arguments of the
local Heun functions tend to zero (one of the regular singular points) as
$x\rightarrow+\infty$. The local Heun function
$Hl\left(a,q;\alpha,\beta,\gamma,\delta;z\right)$ is analytic, and is equal to
one at the regular singular point $z=0$, where it can be expanded in a Taylor
series. In the complex $z$-plane, the radius of convergence of this series is
equal to the smallest of $\left|a\right|$ and one. It follows from Eqs. (35)
and (36) that in our case, it is equal to $1/2$.

Despite the finite radius of convergence of the series, the local Heun
function $Hl\left(1/2,q;\alpha,\beta,\gamma,\delta;z\right)$ can be
analytically continued to the whole complex plane with the cut
$\left[1/2,\infty\right)$. It will be defined at all finite points of the
complex plane except for the regular singular points $z=1/2$ and $z=1$. At the
same time, in Eqs. (35) and (36), the arguments of the local Heun functions
tend to one as $x\rightarrow-\infty$, and hence local solutions (35) and (36)
cannot be used as $x\rightarrow-\infty$.

It is known [42, 43], however, that Heun’s equation has another local solution
such that the local Heun functions entering it will be analytic in the
neighborhood of the singular point $z=1$. The same is also true for the
solutions to Eqs. (33) and (34). Specifically, it can be shown that the
solutions $\psi_{11}$ and $\psi_{12}$ can be represented in the alternative
forms

$\displaystyle\psi_{11}$ $\displaystyle=$ $\displaystyle
c_{1}e^{ikx}Hl\left[\frac{1}{2},i\frac{2\left(\varepsilon-k\right)}{m};-1,0,1+i\frac{2k}{m},1-i\frac{2k}{m};\frac{1}{1-ie^{-mx}}\right]+$
(37) $\displaystyle
c_{2}e^{-ikx}\left(-i+e^{mx}\right)^{2i\frac{k}{m}}Hl\left[\frac{1}{2},-2\frac{k^{2}}{m^{2}}+i\frac{2\varepsilon+k}{m};-1-i\frac{2k}{m},-i\frac{2k}{m},1-i\frac{2k}{m},1-i\frac{2k}{m};\frac{1}{1-ie^{-mx}}\right]$

and

$\displaystyle\psi_{12}$ $\displaystyle=$ $\displaystyle
d_{1}e^{-\pi\frac{k}{m}}e^{ikx}\left(i+e^{mx}\right)^{-2i\frac{k}{m}}Hl\left[\frac{1}{2},-2\frac{k^{2}}{m^{2}}+i\frac{2\varepsilon-k}{m};-1+i\frac{2k}{m},i\frac{2k}{m},1+i\frac{2k}{m},1+i\frac{2k}{m};\frac{1}{1+ie^{-mx}}\right]+$
(38) $\displaystyle
d_{2}e^{-\pi\frac{k}{m}}e^{-ikx}Hl\left[\frac{1}{2},i\frac{2\left(k+\varepsilon\right)}{m};-1,0,1-i\frac{2k}{m},1+i\frac{2k}{m};\frac{1}{1+ie^{-mx}}\right],$

where $c_{1}$, $c_{2}$, $d_{1}$, and $d_{2}$ are constant coefficients. Now
the arguments of the local Heun functions tend to zero as
$x\rightarrow-\infty$, meaning that the local Heun functions remain analytic
in this limit. Forms (35) and (37) of the solution $\psi_{11}$ have
overlapping domains of analyticity in the variable $x$. The same is also true
for forms (36) and (38) of the solution $\psi_{12}$. It follows that by
equating the alternative expressions for $\psi_{11}(x)$ and its derivative
$d\psi_{11}(x)/dx$, we can find the coefficients $c_{1}$ and $c_{2}$ in Eq.
(37), and in Eq. (38), the coefficients $d_{1}$ and $d_{2}$ can be found in a
similar way. The $x$-coordinate of the matching point does not matter, and can
lie in the interval $(-\infty,\infty)$. For reasons of symmetry, we choose the
coordinate of the matching point as $x=0$. In this case, the coefficients
$c_{1}$ and $c_{2}$ can be written as

$\displaystyle c_{1}$ $\displaystyle=$
$\displaystyle\frac{4sk\chi_{\text{tr}}\left(s^{\ast}\right)\chi_{\text{rf}}\left(s\right)-m\left[\chi_{\text{tr}}^{\prime}\left(s^{\ast}\right)\chi_{\text{rf}}\left(s\right)+\chi_{\text{tr}}\left(s^{\ast}\right)\chi_{\text{rf}}^{\prime}\left(s\right)\right]}{4sk\chi_{\text{rf}}\left(s\right)\chi_{\text{in}}\left(s\right)+mW[\chi_{\text{rf}}\left(s\right),\chi_{\text{in}}\left(s\right)]},$
(39) $\displaystyle c_{2}$ $\displaystyle=$ $\displaystyle
2^{-i\frac{k}{m}}e^{-\frac{\pi}{2}\frac{k}{m}}\frac{m\left[\chi_{\text{tr}}^{\prime}\left(s^{\ast}\right)\chi_{\text{in}}\left(s\right)+\chi_{\text{tr}}\left(s^{\ast}\right)\chi_{\text{in}}^{\prime}\left(s\right)\right]}{4sk\chi_{\text{rf}}\left(s\right)\chi_{\text{
in}}\left(s\right)+mW[\chi_{\text{rf}}\left(s\right),\chi_{\text{in}}\left(s\right)]},$
(40)

where the functions

$\displaystyle\chi_{\text{in}}\left(z\right)$ $\displaystyle=$ $\displaystyle
Hl\left[\frac{1}{2},i\frac{2\left(\varepsilon-k\right)}{m};-1,0,1+i\frac{2k}{m},1-i\frac{2k}{m};z\right],$
(41) $\displaystyle\chi_{\text{rf}}\left(z\right)$ $\displaystyle=$
$\displaystyle
Hl\left[\frac{1}{2},-2\frac{k^{2}}{m^{2}}+i\frac{2\varepsilon+k}{m};-1-i\frac{2k}{m},-i\frac{2k}{m},1-i\frac{2k}{m},1-i\frac{2k}{m};z\right],$
(42) $\displaystyle\chi_{\text{tr}}\left(z\right)$ $\displaystyle=$
$\displaystyle
Hl\left[\frac{1}{2},-i\frac{2\left(\varepsilon-k\right)}{m};-1,0,1-i\frac{2k}{m},1+i\frac{2k}{m};z\right],$
(43)

the Wronskian

$W[\chi_{\text{rf}}\left(s\right),\chi_{\text{in}}\left(s\right)]=\left[\chi_{\text{rf}}\left(z\right)\chi_{\text{in}}^{\prime}\left(z\right)-\chi_{\text{rf}}^{\prime}\left(z\right)\chi_{\text{in}}\left(z\right)\right]_{z=s},$
(44)

and the variable $s=\left(1+i\right)/2$. Similarly, the coefficients $d_{1}$
and $d_{2}$ are

$\displaystyle d_{1}$ $\displaystyle=$
$\displaystyle-2^{2i\frac{k}{m}}\frac{4sk\eta_{\text{tr}}\left(s\right)\eta_{\text{rf}}\left(s^{\ast}\right)-m\left[\eta_{\text{tr}}^{\prime}\left(s\right)\eta_{\text{rf}}\left(s^{\ast}\right)+\eta_{\text{tr}}\left(s\right)\eta_{\text{rf}}^{\prime}\left(s^{\ast}\right)\right]}{4s^{\ast}k\eta_{\text{rf}}\left(s^{\ast}\right)\eta_{\text{in}}\left(s^{\ast}\right)-mW[\eta_{\text{rf}}\left(s^{\ast}\right),\eta_{\text{
in}}\left(s^{\ast}\right)]},$ (45) $\displaystyle d_{2}$ $\displaystyle=$
$\displaystyle
2^{i\frac{k}{m}}e^{\frac{\pi}{2}\frac{k}{m}}\frac{4k\eta_{\text{tr
}}\left(s\right)\eta_{\text{in}}\left(s^{\ast}\right)-m\left[\eta_{\text{tr}}^{\prime}\left(s\right)\eta_{\text{in}}\left(s^{\ast}\right)+\eta_{\text{tr}}\left(s\right)\eta_{\text{in}}^{\prime}\left(s^{\ast}\right)\right]}{4s^{\ast}k\eta_{\text{rf}}\left(s^{\ast}\right)\eta_{\text{in}}\left(s^{\ast}\right)-mW[\eta_{\text{rf}}\left(s^{\ast}\right),\eta_{\text{in}}\left(s^{\ast}\right)]},$
(46)

where the functions

$\displaystyle\eta_{\text{in}}\left(z\right)$ $\displaystyle=$ $\displaystyle
Hl\left[\frac{1}{2},-2\frac{k^{2}}{m^{2}}+i\frac{2\varepsilon-k}{m};-1+i\frac{2k}{m},i\frac{2k}{m},1+i\frac{2k}{m},1+i\frac{2k}{m};z\right],$
(47) $\displaystyle\eta_{\text{rf}}\left(z\right)$ $\displaystyle=$
$\displaystyle
Hl\left[\frac{1}{2},i\frac{2\left(k+\varepsilon\right)}{m};-1,0,1-i\frac{2k}{m},1+i\frac{2k}{m};z\right],$
(48) $\displaystyle\eta_{\text{tr}}\left(z\right)$ $\displaystyle=$
$\displaystyle
Hl\left[\frac{1}{2},-2\frac{k^{2}}{m^{2}}-i\frac{2\varepsilon-k}{m};-1-i\frac{2k}{m},-i\frac{2k}{m},1-i\frac{2k}{m},1-i\frac{2k}{m};z\right],$
(49)

the Wronskian

$W[\eta_{\text{rf}}\left(s^{\ast}\right),\eta_{\text{in}}\left(s^{\ast}\right)]=\left[\eta_{\text{rf}}\left(z\right)\eta_{\text{in}}^{\prime}\left(z\right)-\eta_{\text{rf}}^{\prime}\left(z\right)\eta_{\text{in}}\left(z\right)\right]_{z=s^{\ast}},$
(50)

and the variable $s^{\ast}=\left(1-i\right)/2$. In Eqs. (39)–(50), the
subscripts “in”, “rf”, and “tr” refer to the incident, transmitted, and
reflected fermionic wave, respectively.

The coefficients $c_{1}$ and $c_{2}$ ($d_{1}$ and $d_{2}$) contain all the
information about the scattering of the diagonal (antidiagonal) component
$\psi_{\text{d}}$ ($\psi_{\text{a}}$) of the fermionic wave on the sine-Gordon
kink. It follows from Eqs. (35) and (37) that the asymptotics of scattering
for the $\psi_{11}$ component can be schematically represented as

$c_{1}e^{ikx}\rightarrow e^{ikx}+c_{2}e^{\pi\frac{k}{m}}e^{-ikx},$ (51)

which corresponds to the splitting of the incident wave into the transmitted
and reflected waves. Next, using Eq. (29) for the $\psi_{22}$ component and
the expression
$j^{\mu}=\bar{\psi}\gamma^{\mu}\\!\otimes\\!\mathbb{I}\psi=\left(\psi^{\ast}\psi,\psi^{\ast}\sigma_{3}\\!\otimes\\!\mathbb{I}\psi\right)$
for the fermionic current, we obtain expressions for the incident,
transmitted, and reflected currents of the diagonal component
$\psi_{\text{d}}$:

$\displaystyle j_{\text{in}}$ $\displaystyle=$ $\displaystyle
c_{1}c_{1}^{\ast}\bigl{(}1-M^{-2}\left(\varepsilon-k\right)^{2}\bigr{)},$ (52)
$\displaystyle j_{\text{rf}}$ $\displaystyle=$ $\displaystyle
c_{2}c_{2}^{\ast}e^{2\pi\frac{k}{m}}\bigl{(}1-M^{-2}\left(\varepsilon+k\right)^{2}\bigr{)},$
(53) $\displaystyle j_{\text{tr}}$ $\displaystyle=$ $\displaystyle
1-M^{-2}\left(\varepsilon-k\right)^{2}.$ (54)

Using Eqs. (52)–(53), we obtain expressions for the transmission and
reflection coefficients:

$\displaystyle T$ $\displaystyle=$
$\displaystyle\frac{j_{\text{tr}}}{j_{\text{in}}}=\left|c_{1}\right|^{-2},$
(55) $\displaystyle R$ $\displaystyle=$
$\displaystyle\frac{\left|j_{\text{rf}}\right|}{j_{\text{in}}}=\frac{\left|c_{2}\right|^{2}}{\left|c_{1}\right|^{2}}e^{2\pi\frac{k}{m}}\left(1+2M^{-2}k\left(\varepsilon+k\right)\right),$
(56)

which correspond to the scattering of the diagonal component $\psi_{\text{d}}$
on the sine-Gordon kink. In the process of scattering, the transmitted
diagonal fermionic wave acquires a phase shift $\delta_{\text{d}}$ with
respect to the incident diagonal fermionic wave. Eq. (51) tells us that this
phase shift

$\delta_{\text{d}}(k)=-\text{arg}[c_{1}(k)].$ (57)

In a similar way to Eq. (51), the scattering of the $\psi_{12}$ component can
be schematically written as

$d_{1}e^{ikx}\rightarrow e^{ikx}+d_{2}e^{-\pi\frac{k}{m}}e^{-ikx}.$ (58)

In the same way as above, we can sequentially obtain expressions for the
fermionic currents:

$\displaystyle j_{\text{in}}$ $\displaystyle=$ $\displaystyle
d_{1}d_{1}^{\ast}\bigl{(}1-M^{-2}\left(\varepsilon-k\right)^{2}\bigr{)},$ (59)
$\displaystyle j_{\text{rf}}$ $\displaystyle=$ $\displaystyle
d_{2}d_{2}^{\ast}e^{-2\pi\frac{k}{m}}\bigl{(}1-M^{-2}\left(\varepsilon+k\right)^{2}\bigr{)},$
(60) $\displaystyle j_{\text{tr}}$ $\displaystyle=$ $\displaystyle
1-M^{-2}\left(\varepsilon-k\right)^{2},$ (61)

and the corresponding transmission and reflection coefficients:

$\displaystyle T$ $\displaystyle=$
$\displaystyle\frac{j_{\text{tr}}}{j_{\text{in}}}=\left|d_{1}\right|^{-2},$
(62) $\displaystyle R$ $\displaystyle=$
$\displaystyle\frac{\left|j_{\text{rf}}\right|}{j_{\text{in}}}=\frac{\left|d_{2}\right|^{2}}{\left|d_{1}\right|^{2}}e^{-2\pi\frac{k}{m}}\left(1+2M^{-2}k\left(\varepsilon+k\right)\right).$
(63)

Eqs. (59)–(63) correspond to the scattering of the antidiagonal component
$\psi_{\text{a}}$ on the sine-Gordon kink. From Eq. (58), it follows that when
the antidiagonal component $\psi_{\text{a}}$ is scattered, the phase shift

$\delta_{\text{a}}(k)=-\text{arg}[d_{1}(k)].$ (64)

Let us now define the mean value of the isospin $x$-projection $I_{1}$ for a
plane-wave fermionic state as

$\left\langle
I_{1}\right\rangle=\frac{1}{2}\frac{\psi^{\ast}\mathbb{I}\otimes\tau_{1}\psi}{\psi^{\ast}\mathbb{I}\otimes\mathbb{I}\psi}.$
(65)

It follows from Eq. (65) that the mean value $\left\langle I_{1}\right\rangle$
of the isospin $x$-projection lies in the interval $[-1/2,1/2]$, and is
relativistically invariant, as it should be. It can easily be shown that the
mean value $\left\langle I_{1}\right\rangle$ vanishes identically for the both
the diagonal and antidiagonal fermionic states:

$\displaystyle\left\langle I_{1}\right\rangle_{\text{d, in}}$ $\displaystyle=$
$\displaystyle\left\langle I_{1}\right\rangle_{\text{d, rf}}=\left\langle
I_{1}\right\rangle_{\text{ d, tr}}=0,$ (66) $\displaystyle\left\langle
I_{1}\right\rangle_{\text{a, in}}$ $\displaystyle=$ $\displaystyle\left\langle
I_{1}\right\rangle_{\text{a, rf}}=\left\langle I_{1}\right\rangle_{\text{ a,
tr}}=0.$ (67)

However, for a linear combination
$\alpha\psi_{\text{d}}+\beta\psi_{\text{a}}$, the mean value $\left\langle
I_{1}\right\rangle$ is different from zero. In particular, it can be shown
that

$\displaystyle\left\langle I_{1}\right\rangle_{\text{in}}$ $\displaystyle=$
$\displaystyle\frac{\text{Re}\left[\alpha^{\ast}\beta\right]}{\left|\alpha\right|^{2}+\left|\beta\right|^{2}},$
(68) $\displaystyle\left\langle I_{1}\right\rangle_{\text{tr}}$
$\displaystyle=$ $\displaystyle\frac{\text{Re}\left[\alpha^{\ast}\beta
c_{1}d_{1}^{\ast}\right]}{\left|\alpha d_{1}\right|^{2}+\left|\beta
c_{1}\right|^{2}},$ (69) $\displaystyle\left\langle
I_{1}\right\rangle_{\text{rf}}$ $\displaystyle=$
$\displaystyle\frac{\text{Re}\left[\alpha^{\ast}\beta
c_{1}c_{2}^{\ast}d_{1}^{\ast}d_{2}\right]}{e^{-2\pi\frac{k}{m}}\left|\alpha
c_{2}d_{1}\right|^{2}+e^{2\pi\frac{k}{m}}\left|\beta c_{1}d_{2}\right|^{2}}$
(70)

for the incident, reflected, and transmitted waves of the scattering state
$\alpha\psi_{\text{d}}+\beta\psi_{\text{a}}$, respectively.

##  IV  Fermionic bound states

Let us investigate the presence of fermionic bound states in the external
field of the sine-Gordon kink. It is obvious that the energy $\varepsilon$ of
a fermionic bound state must be less than the fermion mass $M$. In this case,
the parameter $k^{2}=\varepsilon^{2}-M^{2}$ becomes negative:
$k^{2}\equiv-\kappa^{2}<0$. It then follows from Eqs. (28), (29), (31), and
(32) that the components of the wave function of a fermionic bound state are
$\propto\exp\left(-\kappa\left|x\right|\right)$ as
$\left|x\right|\rightarrow\infty$. Next, it is easily shown that under charge
conjugation (15a), the diagonal fermionic wave function becomes an
antidiagonal one, and vice versa. Since the charge conjugation reverses the
sign of the energy of the fermionic state, the bound states of the Dirac
Hamiltonian (24) can be split into pairs, each consisting of diagonal and
antidiagonal bound states with opposite energies and connected to each other
by charge conjugation (15a).

Further, it can be shown that the Dirac Hamiltonian (24), parity operator
(15b), and operator $T_{3}$ (25) commute with each other. It follows that the
parity transformation leaves the type (diagonal or antidiagonal) of a
fermionic state unchanged. At the same time, it is known that one-dimensional
bound states are nondegenerate [44]. By combining these facts, we can conclude
that diagonal and antidiagonal bound states should possess certain parities.

We first consider a diagonal fermionic bound state $\psi_{\text{d}}$. In Eq.
(35), the argument of the local Heun function tends to zero as
$x\rightarrow\infty$, and hence the local Heun function tends to unity. It
follows that under the replacement $k\rightarrow i\kappa$, the transmitted
fermionic wave will have the correct bound state asymptotics
$\propto\exp\left(-\kappa x\right)$ as $x\rightarrow\infty$. Under the
replacement $k\rightarrow i\kappa$ in Eq. (37), the reflected wave also will
have the correct asymptotics $\propto\exp\left(\kappa x\right)$ as
$x\rightarrow-\infty$; however, in Eq. (37), the incident wave will then be
$\propto\exp\left(-\kappa x\right)$ and will increase indefinitely as
$x\rightarrow-\infty$. To eliminate the incorrect asymptotic behavior of the
incident fermionic wave in Eq. (37), the coefficient
$c_{1}\left(\varepsilon,k\right)$ must vanish at
$\varepsilon=\varepsilon_{n},k=i\kappa_{n}=i\left(M^{2}-\varepsilon_{n}^{2}\right)^{1/2}$,
where $\varepsilon_{n}$ is the energy of a diagonal bound state. Note that the
coefficient $c_{1}\left(\varepsilon,k\right)$ is explicitly determined by Eq.
(39) and Eqs. (41)–(44). Thus, the energy levels $\varepsilon_{n}$ of the
diagonal fermionic bound states are determined by the solutions of the
transcendental equation

$c_{1}\bigl{(}\varepsilon,i\left(M^{2}-\varepsilon^{2}\right)^{1/2}\bigr{)}=0.$
(71)

Similarly, we can conclude that the energy levels of the antidiagonal
fermionic bound states are determined by the transcendental equation

$d_{1}\bigl{(}\varepsilon,i\left(M^{2}-\varepsilon^{2}\right)^{1/2}\bigr{)}=0,$
(72)

where the coefficient $d_{1}\left(\varepsilon,k\right)$ is explicitly
determined by Eq. (45) and Eqs. (47)–(50). Due to the symmetry of the Dirac
equation under charge conjugation (15a), the energy levels of the diagonal
(antidiagonal) antifermionic bound states are opposite in sign to those of the
antidiagonal (diagonal) fermionic bound states.

As in the case of fermion scattering, the wave functions of the bound states
described above are the result of matching two local analytic solutions of Eq.
(28) (or Eq. (31)) at some intermediate point. At the same time, there are
bound states whose wave functions are analytic in the entire complex plane,
excluding the point at infinity. To demonstrate this, we consider Eq. (35). As
mentioned above, after the replacement $k\rightarrow i\kappa$, Eq. (35) will
show the correct asymptotic behavior $\propto\exp\left(-\kappa x\right)$ as
$x\rightarrow\infty$. However, the factor $\exp\left(-\kappa x\right)$
increases indefinitely as $x\rightarrow-\infty$, whereas at large negative
$x$, the correct asymptotics of the bound state should be
$\propto\exp\left(\kappa x\right)$. Hence, to compensate for the unlimited
growth of the factor $\exp\left(-\kappa x\right)$ and to obtain the correct
(i.e., $\propto\exp\left(\kappa x\right)$) asymptotic behavior of the bound
state, the local Heun function in Eq. (35) must tend to zero as
$\exp\left(2\kappa x\right)$, as $x\rightarrow-\infty$. The argument of the
local Heun function entering Eq. (35) tends to unity as $x\rightarrow-\infty$.
It follows that the local Heun function should be analytic in the vicinity of
unity.

Recall that Heun’s equation has four regular singular points at $z=0$, $z=a$,
$z=1$, and $z=\infty$. The local Heun function
$Hl\left(a,q;\alpha,\beta,\gamma,\delta;z\right)$ is analytic in the vicinity
of the regular singular point $z=0$, but in the general case, it is not
defined in the vicinity of the regular singular points $z=a$ and $z=1$.
However, if a local Heun function is analytic in the vicinity of $z=0$ and
$z=1$, it must also be analytic in the vicinity of $z=a=1/2$, since the
regular singular point $z=a=1/2$ lies between the other two regular singular
points $z=0$ and $z=1$. The situation in which the local Heun function
$Hl\left(a,q;\alpha,\beta,\gamma,\delta;z\right)$ is analytic in a domain
containing the three adjacent singularities $z=0$, $z=a$, and $z=1$ is rather
specific. In this case, $Hl\left(a,q;\alpha,\beta,\gamma,\delta;z\right)$ is
also a solution around the fourth singularity $z=\infty$ and is reduced to the
Heun polynomial [42, 43]. A necessary condition for this is $\alpha=-n$, where
$n$ is a positive integer. Note that the local Heun functions in Eq. (35) have
the parameter $\alpha=-1$.

When $\alpha=-n$, where $n$ is a positive integer, the $(n+1)\text{-th}$
coefficient in the series expansion of
$Hl\left(a,q;\alpha,\beta,\gamma,\delta;z\right)$ is a polynomial in $q$ of
order $n+1$. If $q$ is a root of that polynomial, then the $(n+1)$-th
coefficient vanishes and with it all the following ones, so the series is
truncated and the local Heun function becomes the Heun polynomial. In our
case, the series expansion of the local Heun function in Eq. (35) has the form

$\displaystyle
Hl\left[\frac{1}{2},-\frac{2\left(\kappa+i\varepsilon\right)}{m};-1,0,1+\frac{2\kappa}{m},1-\frac{2\kappa}{m};z\right]$
$\displaystyle=$ $\displaystyle
1-\frac{4i\left(\varepsilon-i\kappa\right)}{m+2\kappa}z-\frac{4\varepsilon\left(\varepsilon-i\kappa\right)}{\left(m+\kappa\right)\left(m+2\kappa\right)}z^{2}+O\left(z^{3}\right),$

where the parameter $\kappa=\left(M^{2}-\varepsilon^{2}\right)^{1/2}$. We see
that in Eq. (IV), the coefficient of $z^{2}$ vanishes iff the energy
$\varepsilon=0$. In this case, the local Heun function (IV) is reduced to the
first-order Heun polynomial
$Hp_{1,1}\left[1/2,-2M/m;-1,0,1+2M/m,1-2M/m;z\right]=1-4zM/\left(m+2M\right)$.
The Heun polynomial vanishes at the regular singular point $z=1$ iff the
following condition holds

$m=2M,$ (74)

which is equivalent to the condition

$2g=\lambda^{1/2}.$ (75)

We conclude that in the external field of the sine-Gordon kink, the diagonal
bound state with zero energy (diagonal zero mode) exists iff condition (74)
holds. Similarly, it can be shown that this is also true for the antidiagonal
zero mode. In this case, the analytic wave function arises from the reflected
wave in Eq. (38), and the local Heun function entering the reflected wave also
has the parameter $\alpha=-1$.

Let us denote the diagonal and antidiagonal zero modes as $\psi_{0\text{d}}$
and $\psi_{0\text{a}}$, respectively. These zero modes can then be written in
the form

$\psi_{0\text{d}}=\sqrt{\frac{2}{\pi
i}}M^{1/2}\begin{pmatrix}-\dfrac{e^{Mx}}{1+ie^{2Mx}}&0\\\
0&\dfrac{e^{Mx}}{i+e^{2Mx}}\end{pmatrix},$ (76)
$\psi_{0\text{a}}=\sqrt{\frac{2}{\pi
i}}M^{1/2}\begin{pmatrix}0&\dfrac{e^{Mx}}{i+e^{2Mx}}\\\
\dfrac{e^{Mx}}{1+ie^{2Mx}}&0\end{pmatrix}.$ (77)

The zero modes (76) and (77) are normalized to unity and become
$\propto\exp\left(-M\left|x\right|\right)$ as
$\left|x\right|\rightarrow\infty$.

We now investigate the properties of the zero modes (76) and (77) under the
charge conjugation (15a) and parity transformation (15b). Choosing the phase
factors $\eta_{C}$ and $\eta_{P}$ to be equal to $1$ and $-1$, respectively,
we find that

$\displaystyle\psi_{0\text{d}}^{C}\left(t,x\right)$ $\displaystyle=$
$\displaystyle\psi_{0\text{a}}\left(t,x\right),$ (78)
$\displaystyle\psi_{0\text{a}}^{C}\left(t,x\right)$ $\displaystyle=$
$\displaystyle\psi_{0\text{d}}\left(t,x\right),$ (79)
$\displaystyle\psi_{0\text{d}}^{P}\left(t,x\right)$ $\displaystyle=$
$\displaystyle\psi_{0\text{d}}\left(t,x\right),$ (80)
$\displaystyle\psi_{0\text{a}}^{P}\left(t,x\right)$ $\displaystyle=$
$\displaystyle-\psi_{0\text{a}}\left(t,x\right).$ (81)

We see that the diagonal and antidiagonal zero modes turn into each other
under the $C$-conjugation. We also see that these zero modes are the
eigenstates of the parity operator $P$, and that their eigenvalues (i.e.,
parities) are opposite. Later, from our numerical results, we shall see that
the antidiagonal and diagonal zero modes should be regarded as fermionic and
antifermionic, respectively.

Using the zero modes $\psi_{0\text{d}}$ and $\psi_{0\text{a}}$, we can form
even and odd linear combinations:

$\displaystyle\psi_{0\text{e}}$ $\displaystyle=$ $\displaystyle
2^{-1/2}\left(\psi_{0\text{d}}+\psi_{0\text{a}}\right),$ (82)
$\displaystyle\psi_{0\text{o}}$ $\displaystyle=$ $\displaystyle
i2^{-1/2}\left(\psi_{0\text{d}}-\psi_{0\text{a}}\right),$ (83)

which are the eigenstates of the $C$-conjugation operator

$\displaystyle\psi_{0\text{e}}^{C}\left(t,x\right)$ $\displaystyle=$
$\displaystyle\psi_{0\text{e}}\left(t,x\right),$ (84)
$\displaystyle\psi_{0\text{o}}^{C}\left(t,x\right)$ $\displaystyle=$
$\displaystyle\psi_{0\text{o}}\left(t,x\right).$ (85)

At the same time, the zero modes $\psi_{0\text{e}}$ and $\psi_{0\text{o}}$
turn into each other up to a phase factor under the parity transformation

$\displaystyle\psi_{0\text{e}}^{P}\left(t,x\right)$ $\displaystyle=$
$\displaystyle-i\psi_{0\text{o}}\left(t,x\right),$ (86)
$\displaystyle\psi_{0\text{o}}^{P}\left(t,x\right)$ $\displaystyle=$
$\displaystyle i\psi_{0\text{e}}\left(t,x\right).$ (87)

Note also that the mean value of the isospin $x$-projection vanishes for all
types of zero modes:

$\left\langle I_{1}\right\rangle_{\psi_{0\text{d}}}=\left\langle
I_{1}\right\rangle_{\psi_{0\text{a}}}=\left\langle
I_{1}\right\rangle_{\psi_{0\text{e}}}=\left\langle
I_{1}\right\rangle_{\psi_{0\text{o}}}=0.$ (88)

Eqs. (84) and (85) tell us that $\psi_{0\text{e}}$ and $\psi_{0\text{o}}$ are
the Majorana spinors. Using Eqs. (76), (77), (82), and (83), it can be shown
that $\psi_{0\text{e}}$ and $\psi_{0\text{o}}$ satisfy the relation

$\bar{\psi}_{0\text{e}}\bm{\tau}_{\perp}\psi_{0\text{e}}=\bar{\psi}_{0\text{o}}\bm{\tau}_{\perp}\psi_{0\text{o}}=0.$
(89)

It then follows from Eqs. (8) and (10) that the Majorana fermions
corresponding to $\psi_{0\text{e}}$ and $\psi_{0\text{o}}$ have no effect on
the field of the kink, meaning that the external field approximation becomes
exact in this case.

Eqs. (57) and (64) determine the phase shifts in the scattering of the
diagonal and antidiagonal fermionic component, respectively. These phase
shifts depend on the magnitude of the fermion’s momentum. The difference in
the phase shifts
$\Delta_{\text{d,\,a}}=\delta_{\text{d,\,a}}\left(0\right)-\delta_{\text{d,\,a}}\left(\infty\right)$
plays an important role in the theory of scattering [44, 45, 46]; in
particular, Levinson’s theorem [47] establishes a relation between this
difference and the number of bound states for a given scattering channel. For
a one-dimensional case, Levinson’s theorem has the form [48]:

$\Delta=\pi\left(n_{b}-1/2\right),$ (90)

where $\Delta$ and $n_{b}$ are the difference in the phase shifts and the
number of bound states in a given scattering channel, respectively. In our
case, Levinson’s theorem is written as

$\Delta_{\text{d,\,a}}=\pi\left(n_{\text{d,\,a}}-1/2\right),$ (91)

where $n_{\text{d}}$ $(n_{\text{a}})$ is the number of diagonal (antidiagonal)
fermionic bound states.

Another well-known field theory kink solution is the kink of the $\phi^{4}$
model [3, 4, 5]. Unlike the sine-Gordon kink, the $\phi^{4}$ kink has a single
Majorana zero mode that exists for all values of the Yukawa coupling constant.
The presence of this zero mode leads to fragmentation of the fermionic charge
and polarization of the fermionic vacuum [19]. We now consider the effect of
the presence of two zero modes (76) and (77) in the field of the sine-Gordon
kink. Recall that these zero modes exist iff condition (74) holds. First, we
note that one can take either $\psi_{0\text{a}}$ and $\psi_{0\text{ d}}$ or
$\psi_{0\text{e}}$ and $\psi_{0\text{o}}$ as the states of zero energy. In the
first case, the second quantized fermionic field is written as

$\displaystyle\Psi\left(t,x\right)$ $\displaystyle=$ $\displaystyle
b_{0}\psi_{0\text{a}}\left(x\right)+d_{0}^{{\dagger}}\psi_{0\text{d}}\left(x\right)$
$\displaystyle+\sum\limits_{r\geqslant
1}\left(b_{r}e^{-i\varepsilon_{r}t}\psi_{r\text{d}}\left(x\right)+d_{r}^{{\dagger}}e^{i\varepsilon_{r}t}\psi_{r\text{a}}\left(x\right)\right),$

where it is understood that the fermion-kink system is placed in an one-
dimensional box of large but finite length, so that all fermionic energy
levels are discrete. In Eq. (IV), the role of the diagonal and antidiagonal
zero modes is reversed compared to the corresponding nonzero modes. Our
numerical results will show that all nonzero diagonal (antidiagonal) modes are
fermionic (antifermionic) and that the situation is reversed for zero modes.
The annihilation and creation operators satisfy the anticommutation relations

$\bigl{\\{}b_{r},b_{r^{\prime}}^{{\dagger}}\bigr{\\}}=\bigl{\\{}d_{r},d_{r^{\prime}}^{{\dagger}}\bigr{\\}}=\delta_{rr^{\prime}},$
(93)

where $r,r^{\prime}=0,1,2,\ldots\,.$ The other anticommutators involving the
annihilation and creation operators vanish.

Next, to calculate the operator of the fermionic charge, we use the normally
ordered fermionic current

$j^{\mu}=2^{-1}\left[\Psi^{{\dagger}},\gamma^{0}\gamma^{\mu}\Psi\right].$ (94)

The corresponding expression for the fermionic charge operator is

$\displaystyle Q$ $\displaystyle=$ $\displaystyle
2^{-1}\int\left(\Psi^{{\dagger}}\Psi-\Psi\Psi^{{\dagger}}\right)dx$
$\displaystyle=$ $\displaystyle
b_{0}^{{\dagger}}b_{0}-d_{0}^{{\dagger}}d_{0}+\sum\limits_{r\geqslant
1}\left(b_{r}^{{\dagger}}b_{r}-d_{r}^{{\dagger}}d_{r}\right).$

It follows from Eq. (IV) that the contributions of the zero modes
$\psi_{0\text{a}}$ and $\psi_{0\text{d}}$ to the fermionic charge are
completely analogous to those of the nonzero modes. Let us denote the state
vectors in the subspace of zero modes by
$\left|n_{\text{a}},n_{\text{d}}\right\rangle_{(1)}$, where
$n_{\text{a}\,(\text{d})}=0,1$ is the occupation number of the antidiagonal
(diagonal) zero mode. It then follows from Eqs. (93) and (IV) that

$\displaystyle Q\left|0,0\right\rangle_{(1)}$ $\displaystyle=$ $\displaystyle
0,$ (96a) $\displaystyle Q\left|1,0\right\rangle_{(1)}$ $\displaystyle=$
$\displaystyle\left|1,0\right\rangle_{(1)},$ (96b) $\displaystyle
Q\left|0,1\right\rangle_{(1)}$ $\displaystyle=$
$\displaystyle-\left|0,1\right\rangle_{(1)},$ (96c) $\displaystyle
Q\left|1,1\right\rangle_{(1)}$ $\displaystyle=$ $\displaystyle 0.$ (96d)

We see that in the case of the zero modes $\psi_{0\text{a}}$ and
$\psi_{0\text{ d}}$, there is no fermionic charge fragmentation, since all
eigenvalues of $Q$ are integers. Furthermore, Eq. (96a) tells us that the
sine-Gordon kink does not polarize the fermionic vacuum
$\left|0,0\right\rangle_{(1)}$ because its fermionic charge vanishes.

Next, we consider the case where zero energy fermions are in the Majorana
states $\psi_{0\text{e}}$ and $\psi_{0\text{o}}$. The second quantized
fermionic field is then written as

$\displaystyle\Psi\left(t,x\right)$ $\displaystyle=$
$\displaystyle\alpha\psi_{0\text{e}}\left(x\right)+\beta\psi_{0\text{o}}\left(x\right)$
$\displaystyle+\sum\limits_{r\geqslant
1}\left(b_{r}e^{-i\varepsilon_{r}t}\psi_{r\text{d}}\left(x\right)+d_{r}^{{\dagger}}e^{i\varepsilon_{r}t}\psi_{r\text{a}}\left(x\right)\right),$

where the annihilation operators

$\displaystyle\alpha$ $\displaystyle=$ $\displaystyle
2^{-1/2}\bigl{(}b_{0}+d_{0}^{{\dagger}}\bigr{)},$ (98a) $\displaystyle\beta$
$\displaystyle=$ $\displaystyle
i2^{-1/2}\bigl{(}b_{0}-d_{0}^{{\dagger}}\bigr{)}$ (98b)

satisfy the anticommutation relations

$\left\\{\alpha,\alpha^{{\dagger}}\right\\}=\left\\{\beta,\beta^{{\dagger}}\right\\}=1.$
(99)

Note that Eq. (IV) contains only the annihilation operators of the zero modes
$\psi_{0\text{e}}$ and $\psi_{0\text{o}}$, and the creation operators of the
corresponding zero antimodes are absent. This is because the wave functions of
the Majorana zero modes $\psi_{0\text{e}}$ and $\psi_{0\text{o}}$ are
invariant under the charge conjugation, and hence the corresponding zero modes
and antimodes are the same.

In terms of $\alpha$ and $\beta$, the fermionic charge $Q$ has the form

$Q=\alpha^{{\dagger}}\alpha+\beta^{{\dagger}}\beta-1+\sum\limits_{r\geqslant
1}\left(b_{r}^{{\dagger}}b_{r}-d_{r}^{{\dagger}}d_{r}\right).$ (100)

We denote the state vectors in the subspace of the Majorana zero modes
$\psi_{0\text{e}}$ and $\psi_{0\text{o}}$ by
$\left|n_{\text{e}},n_{\text{o}}\right\rangle_{(2)}$. The state vectors
$\left|n_{\text{e}},n_{\text{o}}\right\rangle_{(2)}$ are linear combinations
of the state vectors $\left|n_{\text{a}},n_{\text{d}}\right\rangle_{(1)}$:

$\displaystyle\left|0,0\right\rangle_{(2)}$ $\displaystyle=$
$\displaystyle\left|0,1\right\rangle_{(1)},$ (101a)
$\displaystyle\left|1,0\right\rangle_{(2)}$ $\displaystyle=$ $\displaystyle
2^{-1/2}\bigl{(}\left|0,0\right\rangle_{(1)}+\left|1,1\right\rangle_{(1)}\bigr{)},$
(101b) $\displaystyle\left|0,1\right\rangle_{(2)}$ $\displaystyle=$
$\displaystyle
2^{-1/2}\bigl{(}\left|0,0\right\rangle_{(1)}-\left|1,1\right\rangle_{(1)}\bigr{)},$
(101c) $\displaystyle\left|1,1\right\rangle_{(2)}$ $\displaystyle=$
$\displaystyle\left|1,0\right\rangle_{(1)}.$ (101d)

From Eqs. (99) and (100), we obtain the relations:

$\displaystyle Q\left|0,0\right\rangle_{(2)}$ $\displaystyle=$
$\displaystyle-\left|0,0\right\rangle_{(2)},$ (102a) $\displaystyle
Q\left|1,0\right\rangle_{(2)}$ $\displaystyle=$ $\displaystyle 0,$ (102b)
$\displaystyle Q\left|0,1\right\rangle_{(2)}$ $\displaystyle=$ $\displaystyle
0,$ (102c) $\displaystyle Q\left|1,1\right\rangle_{(2)}$ $\displaystyle=$
$\displaystyle\left|1,1\right\rangle_{(2)}.$ (102d)

We see that as in the previous case, there is no fermionic charge
fragmentation. At the same time, it follows from Eq. (102a) that the fermionic
charge of the vacuum state is equal to minus one, and hence the sine-Gordon
kink polarizes the fermionic vacuum $\left|0,0\right\rangle_{(2)}$.

##  V  Numerical results

It follows from Eqs. (21) and (22) that the Dirac equations for the diagonal
and antidiagonal components of the fermionic wave function contain the meson
mass $m$ and fermion mass $M=mg\lambda^{-1/2}$ as parameters. Passing to the
dimensionless variable $\tilde{x}=mx$, it is easy to show that the solutions
to the Dirac equation depend only on the dimensionless variables
$\tilde{x}=mx$, $\tilde{M}=M/m$, $\tilde{\varepsilon}=\varepsilon/m$, and
$\tilde{k}=k/m$, in accordance with Eqs. (35) – (38). Hence, the meson mass
$m$ can be taken equal to unity in numerical calculations, while the
calculated values can be presented as functions of the dimensionless fermion
momentum $\tilde{k}=k/m$ or mass $\tilde{M}=M/m$. In terms of these
dimensionless variables, condition (20) ensuring the applicability of the
external field approximation takes the form

$\tilde{g}\ll 8\tilde{\lambda}^{-1/2},$ (103)

where $\tilde{g}=g/m$ and $\tilde{\lambda}=\lambda/m^{2}$. From Eq. (103), it
follows that the dimensionless fermion mass
$\tilde{M}=\tilde{g}\tilde{\lambda}^{-1/2}$ must be much less than the value
of $8\tilde{\lambda}^{-1}=8\times 10^{2}$, since we used
$\tilde{\lambda}=10^{-2}$ in the numerical calculations.

Figure 1 presents the dependences of the transition coefficient $T$ for the
antidiagonal component of the fermionic wave function on the dimensionless
fermion momentum $\tilde{k}$. These dependences were obtained using the
analytical expressions in Sec. III, and are shown for different values of the
dimensionless fermion mass $\tilde{M}$. We see that the curves $T(\tilde{k})$
have a characteristic tanh-like shape. As the fermion mass $\tilde{M}$
increases, the curves monotonically shift to the region of larger fermion
momenta. Figure 2 shows the curves $R(\tilde{k})$ that correspond to the
curves $T(\tilde{k})$ in Fig. 1. It was found that for the same values of
$\tilde{M}$, the curves $T(\tilde{k})$ and $R(\tilde{k})$ satisfy the
unitarity condition

$T(\tilde{k})+R(\tilde{k})=1.$ (104)

Let us define the fermion momentim $\tilde{k}_{1/2}$ by means of the relation
$T(\tilde{k}_{1/2})=R(\tilde{k}_{1/2})=1/2$. For a given value of $\tilde{M}$,
the value of $\tilde{k}_{1/2}$ determines the position of the midpoints of the
curves $T(\tilde{k})$ and $R(\tilde{k})$ in Figs. 1 and 2, respectively.
Hence, a fermion with momentum $\tilde{k}_{1/2}$ passes through the kink
barrier with probability $1/2$. Figure 3 presents the dependence
$\tilde{k}_{1/2}(\tilde{M})$ corresponding to the curves shown in Figs. 1 and
2. It was found numerically that $\tilde{k}_{1/2}\approx 4.2\tilde{M}^{2}$ for
$\tilde{M}\lesssim 0.1$, and $\tilde{k}_{1/2}\approx 1.43\tilde{M}^{1/2}$ for
$\tilde{M}\gtrsim 1$. Hence, the fermion velocity
$v_{1/2}\approx\tilde{k}_{1/2}/\tilde{M}<0.1$ if $\tilde{M}<0.024$. We see
that in the antidiagonal case, the fermions that pass through the kink barrier
with probability $1/2$ are nonrelativistic when the fermion mass $\tilde{M}$
is small. The fermions also become nonrelativistic for large fermion masses
$\tilde{M}>205$, as in this case, the fermion velocity $v_{1/2}\approx
1.43\tilde{M}^{-1/2}<0.1$. In the intermediate mass region, the fermions are
moderately relativistic.

Next, we turn to the diagonal component of the fermionic wave function. In
Figs. 4 and 5 we can see the curves $T(\tilde{k})$ and $R(\tilde{k})$,
respectively. These curves carry information about the interaction of the
diagonal fermionic component with the sine-Gordon kink. As in the previous
case, the curves in Figs. 4 and 5 which correspond to the same $\tilde{M}$
satisfy unitarity condition (104). However, except for this aspect, the
behavior of the curves in Figs. 4 and 5 differs substantially from those in
Figs. 1 and 2, respectively. In the antidiagonal case (Figs. 1 and 2), the
curves $T(\tilde{k})$ and $R(\tilde{k})$ monotonically shift to larger fermion
momenta with an increase in the fermion mass $\tilde{M}$. In contrast, in the
diagonal case (Figs. 4 and 5), the shift of the curves $T(\tilde{k})$ and
$R(\tilde{k})$ has an oscillatory character as $\tilde{M}$ increases.

Figure 6 shows the dependence $\tilde{k}_{1/2}(\tilde{M})$ corresponding to
the curves in Figs. 4 and 5. We see that the dependences
$\tilde{k}_{1/2}(\tilde{M})$ are completely different in the antidiagonal and
diagonal cases. The curve $\tilde{k}_{1/2}(\tilde{M})$ monotonically increases
in the antidiagonal case in Fig. 3, whereas it has an oscillatory character in
the diagonal case in Fig. 6. The amplitude of the oscillations is
approximately $0.14$, and the oscillation period increases with the fermion
mass $\tilde{M}$. In particular, it was found that for $n\gtrsim 3$, the
position $\tilde{M}_{n}$ of the $n$-th minimum of the curve
$\tilde{k}_{1/2}(\tilde{M})$ satisfies the asymptotic quadratic relation

$\tilde{M}_{n}\sim an^{2},$ (105)

where the coefficient $a\approx 0.18$. It follows that the oscillation period
(i.e., the distance between adjacent minima) grows linearly with $i$:

$\Delta\tilde{M}_{n}=\tilde{M}_{n}-\tilde{M}_{n-1}\sim 2an.$ (106)

From Eqs. (57) and (64), we can find numerical values for the phase shifts for
the antidiagonal and diagonal fermionic scattering states. It was found that
in the antidiagonal case, the phase shift $\Delta_{\text{a}}$ does not depend
on $\tilde{M}$ and is equal to $\pi/2$. In contrast, in the diagonal case, the
phase shift $\Delta_{\text{d}}$ increases step-wise by $\pi$ when the fermion
mass $\tilde{M}$ passes the next minimum in the curve
$\tilde{k}_{1/2}(\tilde{M})$ shown in Fig. 6. Thus, the dependences of the
phase shifts on the fermion mass $\tilde{M}$ are described by the expressions:

$\displaystyle\Delta_{\text{a}}\bigl{(}\tilde{M}\bigr{)}$ $\displaystyle=$
$\displaystyle\frac{\pi}{2},$ (107)
$\displaystyle\Delta_{\text{d}}\bigl{(}\tilde{M}\bigr{)}$ $\displaystyle=$
$\displaystyle\pi\biggl{[}\frac{1}{2}+\sum_{n=1}^{\infty}\theta\bigl{(}\tilde{M}-\tilde{M}_{n}\bigr{)}\biggr{]},$
(108)

where $\theta$ is the Heaviside function and $\tilde{M}_{1}\approx 0.85$ is
the minimum of the function $\tilde{k}_{1/2}(\tilde{M})$ in Fig. 6 that is
nearest to $\tilde{M}=0$. Levinson’s theorem (91) and Eq. (107) tell us that
for all values of the fermion mass $\tilde{M}$, there is only one antidiagonal
fermionic bound state in the external field of the sine-Gordon kink. On the
other hand, according to Levinson’s theorem (91) and Eq. (108), the number of
diagonal fermionic bound states increases step-wise with an increase in
$\tilde{M}$, and there is always at least one diagonal fermionic bound state
in the external field of the sine-Gordon kink.

In order to verify the validity of these statements, we need to investigate
the spectrum of bound states of the Dirac Hamiltonian. It is sufficient to
study the spectrum of the antidiagonal component of the Dirac Hamiltonian,
since for a given value of $\tilde{M}$, the energy eigenvalues of the diagonal
bound states are opposite in sign to those of the antidiagonal bound states.
To calculate the energy eigenvalues of the antidiagonal bound states, we used
the numerical methods provided by the Mathematica software package [49].
Figure 7 presents the dependence of the ratio $\tilde{\varepsilon}/\tilde{M}$
for the antidiagonal bound states of the Dirac Hamiltonian on the fermion mass
$\tilde{M}$. The corresponding curves for the diagonal bound states are
obtained from those shown in Fig. 7 by reflection with respect to the
horizontal zero axis.

Let us examine the main features of the curves shown in Fig. 7. Firstly, we
see that there is only one antidiagonal fermionic bound state in the entire
interval of fermion masses $\tilde{M}$. When $\tilde{M}\equiv M/m=1/2$, the
energy of this bound state is zero, and thus the sine-Gordon kink possesses a
fermionic zero mode as described in Sec. IV. Conversely, we see that as the
fermion mass $\tilde{M}$ increases, new antidiagonal antifermionic bound
states arise in the external field of the sine-Gordon kink, and one
antidiagonal antifermionic bound state exists for all values of $\tilde{M}$.
The situation is reversed for the diagonal bound states. In this case, there
is only one diagonal antifermionic bound state, whereas new diagonal fermionic
bound states arise as $\tilde{M}$ increases. When $\tilde{M}\equiv M/m=1/2$,
the energy of the diagonal antifermionic bound state vanishes, and it turns
into the antifermionic zero mode. The coordinates $\tilde{M}_{n}$ of the
points at which new antidiagonal antifermionic (diagonal fermionic) bound
states arise from the continuum coincide with the coordinates of the
corresponding minima of the curve $\tilde{k}_{1/2}(\tilde{M})$ in Fig. 6. It
was found that in Fig. 7, the curves
$\left|\tilde{\varepsilon}_{n}\right|/\tilde{M}$ tend quadratically to unity
as $\tilde{M}$ tends to $\tilde{M}_{n}$ from the right:

$\frac{\left|\tilde{\varepsilon}_{n}\right|}{\tilde{M}}\approx
1-\beta_{n}\bigl{(}\tilde{M}-\tilde{M}_{n}\bigr{)}^{2},$ (109)

where $\beta_{n}$ are some positive constants and the index $n=0,1,2,\ldots$
enumerates the curves in order of increasing $\tilde{M}_{n}$. A qualitative
explanation of this behavior of the curves
$\left|\tilde{\varepsilon}_{n}\right|/\tilde{M}$ is given in the Appendix B.

It follows from the above that Eqs. (107), (108), and the the behavior of the
curves in Figs. 3, 6, and 7 are in agreement with Levinson’s theorem. Indeed,
it follows from Eq. (108) that in the scattering of the diagonal state
fermions, the phase shift $\Delta_{\text{d}}$ increases step-wise by $\pi$
when $\tilde{M}$ passes the point $\tilde{M}_{n}$ at which a new diagonal
fermionic bound state arises from the continuum. At the same time, in Fig. 6,
the curve $\tilde{k}_{1/2}(\tilde{M})$ reaches a local near-zero minimum at
$\tilde{M}=\tilde{M}_{n}$. This corresponds to the fact that diagonal state
fermions with $\tilde{M}\approx\tilde{M}_{n}$ and small momenta $\tilde{k}$
almost completely transmit through the kink barrier. This resonance behavior
is due to the presence of a virtual level at $\tilde{k}\approx 0$ when
$\tilde{M}\approx\tilde{M}_{n}$, and is in accordance with the general
principles of scattering theory [44, 45, 46].

In contrast, it follows from Eq. (107) that in the scattering of the
antidiagonal state fermions, the phase shift $\Delta_{\text{a}}$ does not
depend on $\tilde{M}$ and is equal to $\pi/2$. From Fig. 3, we see that the
corresponding curve $\tilde{k}_{1/2}(\tilde{M})$ increases monotonically with
an increase in $\tilde{M}$ and has no local minima at nonzero $\tilde{M}$.
Hence, the scattering of the antidiagonal state fermions on the sine-Gordon
kink does not have a resonance character. This corresponds to the fact that in
the external field of the sine-Gordon kink, there are no antidiagonal
fermionic states arising at nonzero $\tilde{M}$ and there is only one such
state arising at $\tilde{M}=0$. In addition, we note that the roles of the
diagonal and antidiagonal states are reversed when passing from fermions to
antifermions. The diagonal (antidiagonal) antifermionic states are scattered
on the sine-Gordon kink in the same way as the antidagonal (diagonal)
fermionic states.

##  VI  Conclusion

In the present paper, fermion scattering in the background field of the sine-
Gordon kink has been investigated both analytically and numerically. To
achieve symmetry of the fermion-kink interaction under discrete transformation
(3), we treat the sine-Gordon model as a nonlinear $\sigma$-model with a
circular target space, which interacts with a fermionic isodoublet through the
Yukawa interaction. It was found that with respect to its spin and isospin
indices, the fermionic isodoublet can be divided into diagonal and
antidiagonal parts that interact with the sine-Gordon kink independently of
each other.

Studying the fermion-kink scattering, we have found analytical expressions for
the wave functions of the diagonal and antidiagonal fermionic states, and have
shown that these wave functions can be expressed in terms of the Heun
functions. Using the expressions obtained in this way and the matching
conditions for the fermionic wave functions, we have derived general
expressions for the fermionic transmission and reflection coefficients. It was
found that the scattering of the diagonal fermionic states differs
significantly from that of the antidiagonal states. In particular, for the
diagonal fermionic states, the dependence of the transmission and reflection
coefficients on the fermion mass has an oscillatory resonance character. In
contrast, this dependence has a monotonic nonresonance character for the
antidiagonal fermionic states. For antifermions the situation is reversed: the
scattering of the antidiagonal (diagonal) antifermionic states has a resonance
(nonresonance) character.

The fermion-kink system has a rather interesting structure of the bound
states. Their energy levels are determined by zeros of the coefficient of the
incident wave in the asymptotics of the fermionic scattering states. The
number of bound states increases as the Yukawa coupling $g$ constant
increases, which is equivalent to an increase in the fermion mass $M$. The
growth in the number of bound states is asymptotically $\propto
M^{1/2}=m^{1/2}g^{1/2}\lambda^{-1/4}$. At the same time, for any nonzero $M$,
there are at least four bound states, of which two are fermionic and two are
antifermionic. The diagonal fermionic (antifermionic) bound states are related
by the charge conjugation to the antidiagonal antifermionic (fermionic) bound
states, meaning that the energy levels of the fermion-kink system can be
divided into pairs of levels with opposite energies.

In addition, the fermion-kink system will possesses two zero modes when
condition (74) is satisfied. Of these, the antidiagonal zero mode is
fermionic, while the diagonal one is antifermionic, and these modes are
related by the charge conjugation. The two linear combinations of the
fermionic and antifermionic zero modes are the eigenstates of the charge
conjugation operator, and hence are the Majorana zero modes. The Majorana zero
modes have no effect on the field configuration of the sine-Gordon kink.

It should be noted that as the fermion mass $M$ increases, the number of the
antidiagonal fermionic (diagonal antifermionic) states remains equal to one,
while the number of the antidiagonal antifermionic (diagonal fermionic) states
increases asymptotically $\propto M^{1/2}$. This difference in the properties
of the bound states appears to be closely related through Levinson’s theorem
to the above-mentioned difference in the scattering of the diagonal and
antidiagonal states on the sine-Gordon kink.

In addition to the sine-Gordon kink, there is the well-known kink solution of
the $(1+1)$-dimensional $\phi^{4}$ model [3, 4, 5]. The $\phi^{4}$ kink
possesses a single Majorana zero mode, which exists for all nonzero values of
the Yukawa coupling constant. This is the main difference from the sine-Gordon
kink, which has two zero modes, but only if condition (74) is satisfied. As in
the previous case, the Majorana zero mode has no effect on the field
configuration of the $\phi^{4}$ kink. The presence of the single Majorana zero
mode leads to fragmentation of the fermionic charge and polarization of the
fermionic vacuum in the external field of the $\phi^{4}$ kink, for any nonzero
value of the Yukawa coupling constant. In contrast, the presence of the two
zero modes $\psi_{0\text{a}}$ and $\psi_{0\text{d}}$ in the external field of
the sine-Gordon kink makes it possible to form the two Majorana zero modes
$\psi_{0\text{e}}$ and $\psi_{0\text{o}}$ in the exceptional case (74). The
presence of two zero modes makes fragmentation of the fermionic charge
impossible in the external field of the sine-Gordon kink. At the same time,
the fermionic charge of the vacuum state $\left|0,0\right\rangle_{(2)}$ is
equal to minus one, which indicates polarization of the fermionic vacuum in
the subspace of the Majorana zero modes.

As in the case of the sine-Gordon kink, the number of fermionic bound states
of the $\phi^{4}$ kink increases with an increase in the fermionic mass $M$.
However, the number of bound fermionic states of the $\phi^{4}$ kink is
asymptotically $\propto M$, while that of the sine-Gordon kink is
asymptotically $\propto M^{1/2}$. Hence, we can say that the $\phi^{4}$ kink
holds fermions more efficiently than the sine-Gordon kink.

##  Appendix A Plane-wave solutions to the Dirac equation

When $\left|x\right|\gg m^{-1}$, the Dirac equation (9) describing fermions in
external kink field (12) turns into the free Dirac equation

$i\gamma^{\mu}\otimes\mathbb{I}\partial_{\mu}\psi-M\mathbb{I}\otimes\tau_{1}\psi=0,$
(110)

where the fermion mass $M=mg\lambda^{-1/2}$, and we explicitly write out the
matrices acting on the spinor and isospinor indices. The free Dirac
Hamiltonian corresponding to Eq. (110) is

$H_{0}=\alpha\mathbb{\otimes
I}\left(-i\partial_{x}\right)+M\beta\mathbf{\otimes}\tau_{1},$ (111)

where $\alpha=\gamma^{0}\gamma^{1}$ and $\beta=\gamma^{0}$. The Hamiltonian
(111) commutes with the operators $p_{x}=-i\partial_{x}$ and
$I_{1}=2^{-1}\tau_{1}$ corresponding to the momentum and isospin
$x$-projection of a free fermion, respectively. It follows that the states of
free fermions can be characterized by the momentum $p_{x}$ and isospin
$x$-projection $I_{1}$. Since in the one-dimensional case fermions have no
spin, there exist four fermionic states at a fixed fermion momentum $p_{x}$.
These states correspond to the combinations of the isospin $x$-projections
$I_{1}=\pm 1/2$ and the fermion energies
$p_{0}=\pm\varepsilon=\pm(p_{x}^{2}+M^{2})^{1/2}$ with both signs.

Let us denote the wave function of a fermion with momentum $p_{x}$, energy
$p_{0}=\epsilon\left(p_{x}^{2}+M^{2}\right)^{1/2}$, where $\epsilon=\pm 1$,
and isospin $x$-projection $I_{x}$ by $\psi_{p_{x},\epsilon,I_{1}}$. Then, the
wave functions of free fermions have the form

$\displaystyle\psi_{p_{x},1,1/2}$ $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{\varepsilon
L}}\begin{pmatrix}\sqrt{\varepsilon+p_{x}}&\sqrt{\varepsilon+p_{x}}\\\
\sqrt{\varepsilon-p_{x}}&\sqrt{\varepsilon-
p_{x}}\end{pmatrix}\exp\left[-i\left(\varepsilon t-p_{x}x\right)\right],$
(112a) $\displaystyle\psi_{p_{x},-1,1/2}$ $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{\varepsilon
L}}\begin{pmatrix}-\sqrt{\varepsilon-p_{x}}&-\sqrt{\varepsilon-p_{x}}\\\
\sqrt{\varepsilon+p_{x}}&\sqrt{\varepsilon+p_{x}}\end{pmatrix}\exp\left[i\left(\varepsilon
t+p_{x}x\right)\right],$ (112b) $\displaystyle\psi_{p_{x},1,-1/2}$
$\displaystyle=$ $\displaystyle\frac{1}{2\sqrt{\varepsilon
L}}\begin{pmatrix}-\sqrt{\varepsilon+p_{x}}&\sqrt{\varepsilon+p_{x}}\\\
\sqrt{\varepsilon-p_{x}}&-\sqrt{\varepsilon-
p_{x}}\end{pmatrix}\exp\left[-i\left(\varepsilon t-p_{x}x\right)\right],$
(112c) $\displaystyle\psi_{p_{x},-1,-1/2}$ $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{\varepsilon L}}\begin{pmatrix}\sqrt{\varepsilon-
p_{x}}&-\sqrt{\varepsilon-p_{x}}\\\
\sqrt{\varepsilon+p_{x}}&-\sqrt{\varepsilon+p_{x}}\end{pmatrix}\exp\left[i\left(\varepsilon
t+p_{x}x\right)\right],$ (112d)

where $L$ is the normalized length. In Eqs. (112a)–(112d), the first (second)
index of the matrices corresponds to the spinor (isospinor) structure of the
fermionic wave function. The wave functions $\psi_{p_{x},\epsilon,I_{1}}$
satisfy the normalized relations:

$\displaystyle\bar{\psi}_{p_{x},\epsilon,I_{1}}\gamma^{\mu}\psi_{p_{x},\epsilon,I_{1}}$
$\displaystyle=$ $\displaystyle\left(1,\epsilon
p_{x}\varepsilon^{-1}\right)=(1,\epsilon v_{x}),$ (113a)
$\displaystyle\bar{\psi}_{\epsilon
p_{x},\epsilon,I_{1}}\gamma^{\mu}\psi_{\epsilon p_{x},\epsilon,I_{1}}$
$\displaystyle=$
$\displaystyle\left(1,p_{x}\varepsilon^{-1}\right)=(1,v_{x}),$ (113b)
$\displaystyle\bar{\psi}_{p_{x},\epsilon,I_{1}}\psi_{p_{x},\epsilon,I_{1}}$
$\displaystyle=$ $\displaystyle\left(-1\right)^{1/2-I_{1}}\epsilon
M\varepsilon^{-1},$ (113c)

where the normalized length $L$ is taken to be equal to unity.

Let us define the spinor-isospinor amplitude $u_{p_{x},\epsilon,I_{1}}$ of the
fermionic wave function $\psi_{p_{x},\epsilon,I_{1}}$ by the relation
$\psi_{p_{x},\epsilon,I_{1}}\equiv\left(2\varepsilon
L\right)^{-1/2}u_{p_{x},\epsilon,I_{1}}\exp\left[-i\left(\epsilon\varepsilon
t-p_{x}x\right)\right]$. Then, the amplitudes $u_{p_{x},\epsilon,I_{1}}$
satisfy the orthogonality and completeness relations:

$\displaystyle
u_{p_{x},\epsilon,I_{1}}^{{\dagger}}u_{p_{x},\epsilon^{\prime},I_{1}^{\prime}}$
$\displaystyle=$ $\displaystyle
2\varepsilon\delta_{\epsilon,\epsilon^{\prime}}\delta_{I_{1},I_{1}^{\prime}},$
(114a)
$\displaystyle\sum\limits_{\epsilon,\,I_{1}}u_{p_{x},\epsilon,I_{1}\left[i,a\right]}u_{p_{x},\epsilon,I_{1}\left[j,b\right]}^{{\dagger}}$
$\displaystyle=$ $\displaystyle 2\varepsilon\delta_{i,j}\delta_{a,b}$ (114b)

for the Hermitian conjugate case, and

$\displaystyle\bar{u}_{\epsilon
p_{x},\epsilon,I_{1}}u_{\epsilon^{\prime}p_{x},\epsilon^{\prime},I_{1}^{\prime}}=2M\epsilon\left(-1\right)^{1/2-I_{1}}\delta_{\epsilon,\epsilon^{\prime}}\delta_{I_{1},I_{1}^{\prime}},$
(115a) $\displaystyle\sum\limits_{I_{1}}u_{\epsilon
p_{x},\epsilon,I_{1}\left[i,a\right]}\bar{u}_{\epsilon
p_{x},\epsilon,I_{1}\left[j,b\right]}=\left\\{\gamma^{\mu}p_{\mu}+\epsilon
M\right\\}_{i,j}\delta_{a,b},$ (115b)
$\displaystyle\sum\limits_{\epsilon,\,I_{1}}\epsilon u_{\epsilon
p_{x},\epsilon,I_{1}\left[i,a\right]}\bar{u}_{\epsilon
p_{x},\epsilon,I_{1}\left[j,b\right]}=2M\mathbb{\delta}_{i,j}\delta_{a,b}$
(115c)

for the Dirac conjugate case. In Eqs. (114b), (115b), and (115c), the first
(second) index in square brackets is the spinor (isospinor) one.

As for the Dirac equation (9), the free Dirac equation (110) is invariant
under the $C$, $P$, and $T$ transformations (15a)–(15c). However, it is also
invariant under the additional variants of these transformations:

$\displaystyle\psi^{C^{\prime}}\left(t,x\right)$ $\displaystyle=$
$\displaystyle\eta_{C^{\prime}}\gamma_{5}\otimes\mathbb{I}\psi^{\ast}\left(t,x\right),$
(116a) $\displaystyle\psi^{C^{\prime\prime}}\left(t,x\right)$ $\displaystyle=$
$\displaystyle\eta_{C^{\prime\prime}}\mathbb{I}\otimes\tau_{2}\psi^{\ast}\left(t,x\right),$
(116b) $\displaystyle\psi^{C^{\prime\prime\prime}}\left(t,x\right)$
$\displaystyle=$
$\displaystyle\eta_{C^{\prime\prime\prime}}\mathbb{I}\otimes\tau_{3}\psi^{\ast}\left(t,x\right),$
(116c) $\displaystyle\psi^{P^{\prime}}\left(t,x\right)$ $\displaystyle=$
$\displaystyle\eta_{P^{\prime}}\gamma^{0}\otimes\mathbb{I}\psi\left(t,-x\right),$
(116d) $\displaystyle\psi^{T^{\prime}}\left(t,x\right)$ $\displaystyle=$
$\displaystyle\eta_{T^{\prime}}\gamma^{0}\otimes\mathbb{I}\psi^{\ast}\left(-t,x\right).$
(116e)

In particular, it follows from Eqs. (116b) and (116c) that

$\displaystyle\psi_{p_{x},\epsilon,I_{1}}$
$\displaystyle\overset{C^{\prime\prime}}{\longrightarrow}$
$\displaystyle\left(-1\right)^{\left(I_{1}+1\right)/2}\psi_{-p_{x},-\epsilon,-I_{1}},$
(117a) $\displaystyle\psi_{p_{x},\epsilon,I_{1}}$
$\displaystyle\overset{C^{\prime\prime\prime}}{\longrightarrow}$
$\displaystyle\psi_{-p_{x},-\epsilon,-I_{1}},$ (117b)

where the phase factors $\eta_{C^{\prime\prime}}$ and
$\eta_{C^{\prime\prime\prime}}$ are taken equal to $-i$ and $1$, respectively.
It follows that in the second quantization formalism, the negative frequency
fermionic wave function $\psi_{-p_{x},-1,-I_{1}}$ can be used to describe the
antifermion having the momentum $p_{x}$ and isospin $x$-projection $I_{1}$.

The free Hamiltonian (111) commutes with the operator
$T_{3}=\gamma_{5}\\!\otimes\\!\tau_{3}$, which determines the type (diagonal
or antidiagonal) of the state. At the same time, the operator $T_{3}$ does not
commute with the isospin operator
$I_{1}=2^{-1}\mathbb{I}\\!\otimes\\!\tau_{1}$. It follows that the eigenstates
of the operator $T_{3}$ are linear combinations of the eigenstates of the
isospin operator $I_{1}$:

$\displaystyle\psi_{p_{x},1,\text{d}}$ $\displaystyle=$ $\displaystyle
2^{-1/2}\left(\psi_{p_{x},1,1/2}-\psi_{p_{x},1,-1/2}\right),$ (118a)
$\displaystyle\psi_{p_{x},1,\text{a}}$ $\displaystyle=$ $\displaystyle
2^{-1/2}\left(\psi_{p_{x},1,1/2}+\psi_{p_{x},1,-1/2}\right),$ (118b)
$\displaystyle\psi_{p_{x},-1,\text{d}}$ $\displaystyle=$ $\displaystyle
2^{-1/2}\left(\psi_{p_{x},-1,1/2}-\psi_{p_{x},-1,-1/2}\right),$ (118c)
$\displaystyle\psi_{p_{x},-1,\text{a}}$ $\displaystyle=$ $\displaystyle
2^{-1/2}\left(\psi_{p_{x},-1,1/2}+\psi_{p_{x},-1,-1/2}\right).$ (118d)

##  Appendix B Energy of a bound fermionic (antifermionic) state in the
vicinity of the transition to the continuum

In this appendix, we explain the behavior of the curves
$\left|\varepsilon_{n}\right|/M$ expressed by Eq. (109). Since for the
diagonal and antidiagonal cases the curves
$\left|\varepsilon_{n}\left(M\right)\right|$ are the same, we shall consider
only the diagonal case. As $M$ tends to $M_{n}$ from the right, the curve
$\left|\varepsilon_{n}\left(M\right)\right|/M$ tends to unity. It follows that
in this case, the parameter $\kappa=\left(M^{2}-\varepsilon^{2}\right)^{1/2}$
tends to zero.

Let us replace the independent variable $x$ in the differential equation (28)
by $\xi=\kappa x$. After this replacement, the differential equation takes the
form

$\displaystyle\psi_{11}^{\prime\prime}(\xi)+2im\kappa^{-1}\text{sech}\left(m\kappa^{-1}\xi\right)\psi_{11}^{\prime}(\xi)$
$\displaystyle-\left(1-2\varepsilon
m\kappa^{-2}\text{sech}\left(m\kappa^{-1}\xi\right)\right)\psi_{11}(\xi)=0.$
(119)

When the dimensionless variable $\left|\xi\right|\gg\kappa m^{-1}$, the
function $\text{sech}\left(m\kappa^{-1}\xi\right)$ exponentially tends to
zero. In this case, we can neglect the corresponding terms in Eq. (119) and
find that

$\psi_{11}(\xi)\propto\exp\left(-\left|\xi\right|\right)$ (120)

when $\left|\xi\right|\gg\kappa m^{-1}$. Next we turn to the region of
$\left|\xi\right|\ll\kappa m^{-1}$, where the function
$\text{sech}\left(m\kappa^{-1}\xi\right)$ can be set equal to unity. Using
this fact, we find an approximate solution to Eq. (119) in the region of
$\left|\xi\right|\ll\kappa m^{-1}$:

$\psi_{11}(\xi)\propto\exp\left(-i\kappa^{-1}\tau\xi\right),$ (121)

where the parameter

$\tau=m+\epsilon\left(m^{2}+2m\varepsilon-\kappa^{2}\right)^{1/2}$ (122)

and $\epsilon=\pm 1$.

Now we need to match solutions (120) and (121) at $\xi=\kappa m^{-1}$. Since,
for $\left|\xi\right|\ll\kappa m^{-1}$, Eq. (119) contains the imaginary
coefficient $2im\kappa^{-1}$, solution (121) is essentially complex. The
matching condition must be satisfied for both the real and imaginary parts of
solution (121). Let us consider the real part of Eq. (121):

$\text{Re}\left[\psi_{11}(\xi)\right]\propto\cos\left(\kappa^{-1}\tau\xi\right).$
(123)

By equating the logarithmic derivatives of Eqs. (120) and (123) at $\xi=\kappa
m^{-1}$, we arrive at the transcendental equation

$\tau\tan\left(m^{-1}\tau\right)=\kappa.$ (124)

From Eq. (124) it follows that $\tan\left(m^{-1}\tau\right)$ must tend to zero
together with $\kappa$. This, in turn, implies that the combination
$m^{-1}\tau$ can be written as

$m^{-1}\tau=\pi n+\Delta,$ (125)

where $n$ is a nonnegative integer and $\Delta\rightarrow 0$ as
$\kappa\rightarrow 0$. Combining Eqs. (122) and (125), we find that

$\left|\varepsilon_{n}\right|\underset{\kappa\rightarrow
0}{\longrightarrow}M_{n}=m\pi n\left(n\pi/2-1\right),$ (126)

where $\varepsilon_{n}$ is the energy of the $n$-th fermionic (antifermionic)
bound state. We see that for large $n$, the value of the fermion mass at which
the $n$-th fermionic (antifermionic) bound state arises from the continuum
becomes approximately proportional to $n^{2}$. Note that this fact is in
accordance with Eqs. (105) and (106).

By substituting Eq. (125) into Eq. (124), we obtain the transcendental
equation in terms of the parameters $\Delta$ and $\kappa$:

$m\left(\pi n+\Delta\right)\tan\left(\Delta\right)=\kappa.$ (127)

We now study the behavior of the curve $\left|\varepsilon_{n}(M)\right|$ in
the neighborhood of the fermion mass $M_{n}$, where both $\Delta$ and $\kappa$
tend to zero. Expanding the right-hand side of Eq. (127) in terms of $\Delta$
and keeping the first expansion terms, we obtain the equation

$mn\pi\Delta\left(1-\delta_{n0}\right)+m\Delta^{2}\delta_{n0}=\kappa.$ (128)

In Eq. (128), we can express both $\Delta$ and $\kappa$ in terms of
$\varepsilon$ and $M$, using Eqs. (122), (125), and the definition
$\kappa=\left(M^{2}-\varepsilon^{2}\right)^{1/2}$. In the neighborhood of
$M_{n}$ (defined in Eq. (126)), the variables $\varepsilon$ and $M$ can be
written as

$\displaystyle\varepsilon$ $\displaystyle=$ $\displaystyle
M_{n}+\Delta\varepsilon,$ (129a) $\displaystyle M$ $\displaystyle=$
$\displaystyle M_{n}+\Delta M.$ (129b)

As a result, we obtain a cumbersome expression implicitly defining
$\Delta\varepsilon$ as a function of $\Delta M$ in the neighborhood of
$M_{n}$:

$F\left(m,n,\Delta M,\Delta\varepsilon\right)=0.$ (130)

The value of $\Delta\varepsilon$ should vanish along with that of $\Delta M$,
and therefore Eq. (130) must be satisfied identically when $\Delta\varepsilon$
and $\Delta M$ vanish. This can be used to determine the sign factor
$\epsilon$ in Eq. (122):

$\epsilon=1-2\delta_{n0},$ (131)

where it is understood that in Eq. (122), the principal value of the square
root is used.

Using expression (130), treating $\Delta\varepsilon$ as a function of $\Delta
M$, and applying the rules for differentiation of an implicit function, we
obtain sequentially:

$\displaystyle\Delta\varepsilon\left(0\right)$ $\displaystyle=$ $\displaystyle
0,$ (132a) $\displaystyle\Delta\varepsilon^{\prime}\left(0\right)$
$\displaystyle=$ $\displaystyle 1,$ (132b)
$\displaystyle\Delta\varepsilon^{\prime\prime}\left(0\right)$ $\displaystyle=$
$\displaystyle-\frac{2\pi n}{m\left(\pi n-2\right)\left(\pi n-1\right)^{2}},$
(132c) $\displaystyle\Delta\varepsilon^{\prime\prime\prime}\left(0\right)$
$\displaystyle=$ $\displaystyle\frac{6}{m^{2}}\frac{-2+4\pi
n-\pi^{2}n^{2}+\pi^{2}n^{3}}{\left(\pi n-2\right)^{2}\left(\pi
n-1\right)^{3}}.$ (132d)

for $n=1,2,3,\ldots,$ and

$\displaystyle\Delta\varepsilon\left(0\right)$ $\displaystyle=$ $\displaystyle
0,$ (133a) $\displaystyle\Delta\varepsilon^{\prime}\left(0\right)$
$\displaystyle=$ $\displaystyle\pm 1,$ (133b)
$\displaystyle\Delta\varepsilon^{\prime\prime}\left(0\right)$ $\displaystyle=$
$\displaystyle 0,$ (133c)
$\displaystyle\Delta\varepsilon^{\prime\prime\prime}\left(0\right)$
$\displaystyle=$ $\displaystyle\mp 3m^{-2}$ (133d)

for $n=0$. The two signs in Eqs. (133b) and (133d) correspond to the two
curves starting at $\tilde{M}=0$ in Fig. 7.

It follows from Eqs. (132) and (133) that
$\left|\Delta\varepsilon^{\prime}\left(0\right)\right|$ is exactly equal to
one and that $\Delta\varepsilon^{\prime\prime}\left(0\right)$ vanishes when
$n=0$. We can show that this behavior of the function
$\Delta\varepsilon\left(\Delta M\right)$ corresponds to Eq. (109). To do this,
we suppose that in the neighborhood of $M_{n}$, the function
$\left|\varepsilon_{n}\right|/M$ has the form

$\displaystyle\frac{\left|\varepsilon_{n}\right|}{M}$ $\displaystyle=$
$\displaystyle
1-\alpha_{n}\left(M-M_{n}\right)-\beta_{n}\left(M-M_{n}\right)^{2}$ (134)
$\displaystyle-\gamma_{n}\left(M-M_{n}\right)^{3}+O\left[\left(M-M_{n}\right)^{4}\right],$

where $\alpha_{n}$, $\beta_{n}$, and $\gamma_{n}$ are constant coefficients.
Rewriting Eq. (134) in terms of
$\Delta\varepsilon=\left|\varepsilon_{n}\right|-M_{n}$ and $\Delta M=M-M_{n}$,
we obtain the expression

$\displaystyle\Delta\varepsilon$ $\displaystyle=$
$\displaystyle\left(1-M_{n}\alpha_{n}\right)\Delta
M-\left(\alpha_{n}+M_{n}\beta_{n}\right)\Delta M^{2}$ (135)
$\displaystyle-\left(\beta_{n}+M_{n}\gamma_{n}\right)\Delta
M^{3}+O\left(\Delta M^{4}\right).$

From Eqs. (132) and (133) it follows that in Eq. (135), the coefficient
$\alpha_{n}$ vanishes, whereas the coefficient $\beta_{n}\neq 0$ and is
positive for all $n$. Under these conditions, Eqs. (109) and (134) become
equivalent. Note also that in Eq. (135), the coefficient at $\Delta M^{2}$
vanishes when $n=0$, in accordance with Eq. (133c).

We conclude that the use of the rather rough approximation allows us to
explain the behavior of the curves in Fig. 7 at a qualitative level.

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Figure 1:  Dependences of the transmission coefficient $T$ for the
antidiagonal component of the fermionic wave function on the fermion momentum
$\tilde{k}$ for different values of the fermion mass $\tilde{M}$. Figure 2:
Dependences of the reflection coefficient $R$ for the antidiagonal component
of the fermionic wave function on the fermion momentum $\tilde{k}$ for
different values of the fermion mass $\tilde{M}$. Figure 3:  Dependence of the
parameter $\tilde{k}_{1/2}$ on the fermion mass $\tilde{M}$ for the
antidiagonal case.

Figure 4:  Dependences of the transmission coefficient $T$ for the diagonal
component of the fermionic wave function on the fermion momentum $\tilde{k}$
for different values of the fermion mass $\tilde{M}$. Figure 5:  Dependences
of the reflection coefficient $R$ for the diagonal component of the fermionic
wave function on the fermion momentum $\tilde{k}$ for different values of the
fermion mass $\tilde{M}$. Figure 6:  Dependence of the parameter
$\tilde{k}_{1/2}$ on the fermion mass $\tilde{M}$ for the diagonal case.

Figure 7:  Dependence of the ratio $\tilde{\varepsilon}/\tilde{M}$ for the
antidiagonal bound states of the Dirac Hamiltonian on the fermion mass
$\tilde{M}$.