# Kink-antikink collisions in the $\phi^{8}$ model: short-range to long-range
journey

Dionisio Bazeia
Departamento de Física, Universidade Federal da Paraíba,
João Pessoa - PB - 58051-970, Brazil
<EMAIL_ADDRESS>João G. F. Campos
Departamento de Física, Universidade Federal da Paraíba,
João Pessoa - PB - 58051-970, Brazil
<EMAIL_ADDRESS>Azadeh Mohammadi
Departamento de Física, Universidade Federal de Pernambuco,
Av. Prof. Moraes Rego, 1235, Recife - PE - 50670-901, Brazil
<EMAIL_ADDRESS>
###### Abstract

We studied kink-antikink collisions in (1+1)-dimensional spacetime for all
$Z_{2}$ symmetric $\phi^{8}$ models with four degenerate minima. Such a
polynomial model has only one free parameter, allowing us to conduct an
exhaustive analysis. We performed detailed simulations in all three sectors of
the model. We observed resonance windows from both localized and delocalized
modes, as well as a sector change with the formation of additional kink-
antikink pairs. Furthermore, we were able to show how collisions are modified
when two quadratic minima merge into a quartic one, causing the kinks to
acquire a long-range character. We demonstrated that when the tail not facing
the opposing kink is long-range, incoming kinks and antikinks decay directly
into radiation, as suggested in [1], by forming a large number of small kink-
antikink pairs. Finally, we briefly discussed whether our analysis could be
generalized to other polynomial models.

##  1 Introduction

Physics seeks to understand wave equations, which describe phenomena such as
waves in fluids, light propagation, and strings under tension. With access to
the quantum realm, wave equations have also been found to describe particles
and fields. In particular, some nonlinear wave equations have fascinating
structures which are localized and propagate without losing their shape [2,
3]. These can be topological or non-topological and sometimes are called
solitary waves or solitons.

In (1+1)-dimensional spacetime, localized structures known as kinks exist,
which connect different scalar field vacua. They appear in many systems that
are effectively unidimensional in space and have degenerate minimum energy
configurations. Examples include electron displacement in polyacetylene [4,
5], DNA properties [6], Josephson junctions [7], domain walls in ferromamgnets
[8], gold dislocations [9], properties of Rydberg atoms [10], graphene
deformations [11, 12], and branes [13, 14].

The sine-Gordon model is the prototype of an integrable model that features
elastic kink collisions, while the $\phi^{4}$ model is the prototype of a non-
integrable model that exhibits inelastic collisions. Interestingly, the
$\phi^{4}$ model exhibits a resonance phenomenon, as is now well known after a
series of pioneering works [15, 16, 17, 18, 19]. It consists of a kink and an
antikink temporarily trapped after an initial bounce and separating in
subsequent bounces. In short, it occurs because there is an exchange between
the kink’s translational and vibrational degrees of freedom. This picture is
now better understood in terms of a collective coordinate model [20, 21].

Recently, resonance and other exciting phenomena in kink-antikink collisions
have been investigated by many authors. Examples consist of polynomial models
[22, 23, 24, 25], hyperbolic models [26, 27, 28, 29], multikink collisions
[30, 31, 32, 33], multifield collisions [34, 35, 36, 37], the double sine-
Gordon model [38, 39, 40, 41], long-range [42, 43, 44, 45, 1] and compact [46,
47] kinks, and interactions with boundaries [48, 49, 50]. Interestingly, the
exchange mechanism can occur via the interaction of other types of oscillating
modes besides the kink’s shape mode. These can be delocalized modes [22, 45,
47], sphalerons [51], quasinormal modes [52, 53] and fermion bound states
[54]. In particular, there have been great advances in the collective
coordinate description of the resonance mechanism of the $\phi^{6}$ model via
delocalized modes [55].

A unique property of scalar fields in (1+1)-dimensional spacetime is their
dimensionless nature, which allows polynomials of any order in the scalar
field potential when constructing the Lagrangian. We are interested in
potentials with $\phi\to-\phi$ symmetry111See Refs. [56, 57] as examples of
how this property can be relaxed.. The $\phi^{4}$ model with symmetry breaking
has no free parameters after rescaling and has been studied extensively. The
$\phi^{6}$, on the other hand, has a single free parameter and is typically
analyzed at the first-order phase transition. At that point, it has also been
extensively studied. From an experimental point of view, some parameters need
to be tuned to maintain the system there. Thus, one also needs to study the
system’s behavior away from the first-order phase transition. Luckily, it can
be inferred by the study of a model proposed in Ref. [58], which also exhibits
a first-order phase transition. There, the authors studied the possibility of
moving away from it. Therefore, it is possible to say that the resonance
mechanism of the $\phi^{4}$ and $\phi^{6}$ models are generally better
understood compared with the higher-order polynomial models.

One may wonder what new phenomena will arise in kink-antikink collisions as we
increase the order of the polynomial and whether it is worthwhile to do so.
Specifically, we aim to investigate the $\phi^{8}$ model to understand what
new phenomena emerge. Kink-antikink interactions in $\phi^{8}$ models with
long-range tails reveal highly nontrivial new phenomena [43, 44, 45, 1]. See
also Refs. [59, 60, 61] for works where kinks with long-range tails are taken
into account and Ref. [62] for a recent review. However, we focus here on how
the system approaches the long-range regime and other phenomena unrelated to
the long-range property. While a few other works, such as [24, 33], have
explored kink-antikink collisions in the $\phi^{8}$ model, an exhaustive
analysis of the possible behaviors is still lacking in the literature. This
gap is what we aim to address in the present work.

Polynomial field potentials of orders 8, 10, and 12, along with their kink
solutions, have been listed in [63], while quantum corrections to $\phi^{8}$
kinks were computed in [64]. In Ref. [63], the authors show that the
$\phi^{8}$ potential has two free parameters. This means that an exhaustive
analysis is impractical. However, there is only one free parameter at the
first-order phase transition, with four degenerate minima, making the analysis
doable. We will pursue this goal in the present work. To obtain a $\phi^{8}$
model with four degenerate minima, a single parameter has to be tuned.
Therefore, at least in theory, it should not be more difficult to deal with
such a system at the first-order phase transition than a $\phi^{6}$ system in
general. In section 2, we discuss some properties of the $\phi^{8}$ model with
four minima. In section 3, we show the results of several types of kink-
antikink collisions, and we conclude our work in section 4.

##  2 Model

We aim to study the following scalar field model

$\mathcal{L}=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-V_{b}(\phi).$ (1)

The current work focuses on the $\phi^{8}$ model with four degenerate minima.
After rescaling, they can be described by the following family of potentials
with one free parameter

$V_{b}(\phi)=\frac{1}{2}(\phi^{2}-1)^{2}(\phi^{2}-b)^{2}.$ (2)

We will consider the case where $b>0$. Then, the minima are at $\phi=\pm 1$
and $\phi=\pm\sqrt{b}$. Moreover, the regions $0<b<1$ and $b>1$ are physically
equivalent by scaling arguments, so we will take $b\in(0,1)$.

The equations of motion are given by

$\phi_{tt}-\phi_{xx}+V_{b}^{\prime}(\phi)=0.$ (3)

The Bogomol’nyi–Prasad–Sommerfield (BPS) equation can be used to obtain the
kink and antikink solutions

$\phi_{x}=\pm(\phi^{2}-1)(\phi^{2}-b).$ (4)

Implicit analytical solutions to the BPS equation can be obtained as shown in
Ref. [63]. For our purposes, it is enough to compute the kink and antikink
solutions numerically, which is a simple task.

Figure 1: Potential and kink solutions for several values of $b$.

In the top row of Fig. 1, we show two potentials with $b$ in the range $0<b<1$
and their corresponding kinks. They possess two inner minima at
$\phi=\pm\sqrt{b}$ and two outer minima at $\phi=\pm 1$. Thus, there are two
different sectors. The inner sector corresponds to solutions that interpolate
between the two inner minima. It contains a symmetric inner kink and its
corresponding antikink. The outer sectors correspond to solutions that
interpolate between an inner minimum and an outer minimum. They contain two
asymmetric outer kinks and their corresponding antikinks. The kinks’ profiles
are denoted by $\phi_{K}$. To specify the sector, we use a superscript $(i)$
for the inner one and $(o)$ for the outer one. For antikinks, we put a bar
over the subscript $K$.

The cases with $b=7-4\sqrt{3}$ and $b=1/4$ were studied in Refs. [24] and
[33], respectively. In the bottom row of figure 1, we show the two limiting
cases where $b=0$ and $b=1$. Note that some of the minima are much flatter
than the ones in the previous graphs, leading to kinks with long-range
character. There are two asymmetric kinks and their corresponding antikinks
for $b=0$. They all possess a short-range tail on one side and a long-range
tail on the other side. This case was studied in detail in [43, 44, 45]. The
authors showed that the model exhibits resonance windows when the long-range
tails face each other. The exchange mechanism occurs because the kink-antikink
pair possesses vibrational modes. For $b=1$, there is a single symmetric kink
with a double long-range tail and its corresponding antikink. This case was
studied in detail in [1]. The authors showed that there are no resonance
windows in kink-antikink collisions and that the critical velocity is ultra-
relativistic. The explanation for this phenomenon comes from the long-range
character of the tail that is not facing the opposing kink, as we will show
shortly. From the discussion above, we see that by varying the parameter $b$,
we can explore the scattering behavior of kinks that gradually acquire a long-
range character. Therefore, studying such systems may shed more light on the
scattering behavior of long-range kinks in general.

Although the kinks will be computed numerically, their masses can be computed
analytically. The potential in (2) can be written as

$V_{b}(\phi)=\frac{1}{2}\left(\frac{dW_{b}}{d\phi}\right)^{2},$ (5)

where $W_{b}=W_{b}(\phi)$ is a function of the scalar field with the form

$W_{b}(\phi)=b\,\phi-\frac{1}{3}(1+b)\,\phi^{3}+\frac{1}{5}\phi^{5}.$ (6)

For static configurations, the corresponding energy density $\rho_{b}(x)$ can
be written in the following form

$\rho_{b}(x)=\frac{1}{2}\phi_{x}^{2}+\frac{1}{2}\left(\frac{dW_{b}}{d\phi}\right)^{2}=\pm\frac{dW_{b}}{dx}+\frac{1}{2}\left(\phi_{x}\mp\frac{dW_{b}}{d\phi}\right)^{2}.$
(7)

Since the kink solutions obey the first-order equations in (4), their masses
are given by the energies of the static solutions at the respective sectors,
i.e., $M_{b}=|W_{b}(\phi(\infty))-W_{b}(\phi(-\infty))|$. Thus, they can be
computed as $M_{b}^{(i)}=|W_{b}(\sqrt{b})-W_{b}(-\sqrt{b})|$ and
$M_{b}^{(o)}=|W_{b}(1)-W_{b}(\sqrt{b})|$ resulting in

$M_{b}^{(i)}=\frac{4}{15}b^{3/2}(5-b),\quad
M_{b}^{(o)}=\frac{2}{15}(1-\sqrt{b})^{3}(1+3\sqrt{b}+b),$ (8)

reproducing the results obtained in Ref. [63]. Fig. 2 represents them
graphically.

Figure 2: Masses of inner and outer kinks as a function of $b$.

The scattering behavior of kink-antikink systems depends on the presence of
vibrational modes. Therefore, studying the kink’s stability equation, which
describes such modes, is essential. Substituting the ansatz
$\phi(x,t)=\phi_{K}(x,t)+\eta(x)\cos(\omega t)$ in the equation of motion, we
obtain a Schrödinger-like equation

$-\eta^{\prime\prime}(x)+U(x)\eta(x)=\omega^{2}\eta(x).$ (9)

The effective potential is defined as $U(x)\equiv
V_{b}^{\prime\prime}(\phi_{K}(x))$. It is well known that this equation has no
negative eigenvalues, and thus, the kink is stable. Due to the translational
invariance of the model, the kink possesses a zero mode which is proportional
to the derivative of the solution itself, given by

$\eta_{0}(x)\propto\phi_{K}^{\prime}(x).$ (10)

Figure 3: (Left) Effective potential for several values of $b$. (Right) The
corresponding discrete spectra as a function of $b$. The top and bottom rows
correspond to the inner and outer kinks, respectively. The zero mode frequency
is shown in magenta and the first excited state is shown in red. The continuum
spectrum starts at the black dashed line.

The potential $U(x)$ and its spectrum for several values of $b$ are shown in
Fig. 3. The top row describes the effective potential for the symmetric inner
kink $\phi_{K}^{(i)}$ with parameter $0<b<1$. For small $b$, the potential is
a well, and it acquires a volcano shape as $b$ increases, consistent with the
long-range character. The threshold energy which separates the discrete from
the continuum modes is shown as a dashed line in the right panels. It is equal
to the mass of the excitations around the vacua (mesons). For the inner kink,
we find that $U_{\text{th}}(b)=4b(1-b)^{2}$. It is interesting to take a look
at two limiting values. We have that $U_{\text{th}}(1)=0$ because the tail
becomes long-range. Also $U_{\text{th}}(0)=0$, but for a different reason.
This time, it occurs because the inner kink becomes increasingly small as
$b\to 0$. Finally, we find that the spectrum contains only the zero mode,
except for a small range where there is a single vibrational mode with a
frequency close to the threshold value.

In the lower panels, we show the potential $U(x)$ for an asymmetric outer
(anti)kink, $\phi_{K}^{(o)}$. The two asymptotic values of the potential are

$U(x\to-\infty)=4(1-b)^{2},\quad U(x\to\infty)=4b(1-b)^{2}.$ (11)

The discrete spectrum is separated from the continuum at the smaller value,
$U_{\text{th}}=4b(1-b)^{2}$. It is the same expression as in the previous
case. Therefore, we obtain a vanishing threshold value again at the two
limiting cases, that is, $U_{\text{th}}(0)=U_{\text{th}}(1)=0$. However, the
picture is inverted for the outer kink, meaning that $b=0$ corresponds to the
outer kink acquiring a long-range tail and $b=1$ to it becoming increasingly
small. Again, the spectrum contains no vibrational modes or a single
vibrational mode with a frequency close to the threshold of the continuous
spectrum.

##  3 Results

We performed numerical evaluations of kink-antikink collisions. It is achieved
by integrating the equations of motions starting with the following initial
conditions

$\phi(x,t)=\phi_{K}(\gamma(x+X_{0}-v_{i}t))+\phi_{\bar{K}}(\gamma(x-X_{0}+v_{i}t))-C,$
(12)

where $X_{0}=12.5$ is the kink’s initial position and $v_{i}$ is the absolute
value of the initial velocity. This is known as the additive ansatz, and the
constant $C$ is needed to adjust its limiting values. This expression
approximates the system’s dynamics well when the kinks are far away. Of
course, evolution does not obey this superposition anymore as the kinks start
to overlap.

We will discuss all three existing types of kink-antikink collision. The first
one is between inner kinks. Without loss of generality, we can consider only
kink-antikink collisions. In the outer sector, one can have either kink-
antikink or antikink-kink collisions because the kinks are asymmetric. To
avoid ambiguities when referring to them, we will always start our simulations
with outer kinks which interpolate between vacua with positive values of
$\phi$.

The integration methods are standard. We discretize space in the interval
$-400.0<x<400.0$ separated by 8192 equally spaced grid points and with
periodic boundary conditions. The spatial derivatives are approximated by a
Fourier spectral method, and the resulting set of ordinary differential
equations are integrated using a fifth-order Runge-Kutta method with adaptive
time step and error control. To avoid returning radiation from the boundaries,
we include damping in the regions $x<-380.0$ and $x>380.0$. The damping is
multiplied by a bump function 222The bump function was obtained by shifting
the center of the following expression to the boundaries
$B(x)=\begin{cases}5\exp\left(1-\frac{20^{2}}{20^{2}-x^{2}}\right),&-20<x<20,\\\
0,&\text{otherwise}.\end{cases}.$ with a maximum value at the boundaries
$x=\pm 400.0$. We measure the error in our numerical method by calculating the
total energy conservation when the damping is turned off. The maximum relative
error in our simulations is of order $10^{-5}$.

Now, care must be taken when simulating kinks with long-range tails because
the additive ansatz is not a good approximation anymore. The same is true when
the kinks become very small. As described above, the two phenomena appear in
the limits $b\to 0$ and $b\to 1$. Here, we employ the method developed in [1],
adding an extra minimization level on the one developed in [44] to initialize
such systems. In short, the method consists of minimizing the configuration
obtained by the additive ansatz in order for the field $\phi$ to obey the
static kink equation as closely as possible and the velocity field
$\dot{\phi}$ to obey the zero mode equation as close as possible. This is
performed via a nonlinear least-square minimization. In order to the
minimization procedure be computationally viable we reduced the box to
$-100.0<x<100.0$ separated by 2048 grid points, while the damping was included
in the regions $x<-80.0$ and $x>80.0$.

In Ref. [62], the authors raised doubt on the efficiency of the minimization
method in [1]. Here, we employed the method extensively, showing that it is
viable for large-scale computer simulations. Therefore, we give further
evidence that the method is highly computationally efficient.

Below we will use colors to describe the field values. To make the comparison
between graphs easier, we will use the same color scheme in all graphs in the
same subsection. However, the color schemes in different subsections may be
different.

###  3.1 Inner sector kink-antikink collision

The field evolution in this sector corresponds to a collision between an inner
kink and an inner antikink. An important phenomenon that we will observe is
kinks changing sector. It means that when the kinks get in contact, they do
not bounce. Instead, they continue moving and form a pair of kinks in the
neighboring sector, the outer kinks. This is a well-known phenomenon that
happens, for instance, in the sine-Gordon kink-antikink collisions.

Figure 4: Final value of $\phi$ at the collision center as a function of $b$
and $v_{i}$. We are considering kink-antikink collisions between inner kinks.

The system’s behavior as a function of the parameter $b$ and the initial
velocity $v_{i}$ can be summarized in Fig. 4. The color corresponds to the
field’s value at the collision center at a time $t_{f}=60.0/v_{i}$. It is
possible to distinguish several different regions. Some of them are listed
below:

  * •

Region $A$: The kinks form a slowly decaying bound state after colliding. Such
a state is called a bion, which is illustrated in Fig. 5(a).

  * •

Region $B$: The kinks collide and then reflect as illustrated in Fig. 5(b).

  * •

Region $C$: The kinks collide, change sectors, and then separate. This case is
illustrated in Fig. 5(c).

  * •

Region $D$: Same as $C$, but now an oscillating pulse is formed in the center.
Fig. 5(d) illustrates this case.

  * •

Region $E$: The kinks collide and change sectors. Moreover, an extra pair of
outer kink is formed. This leads to four outer kinks in the final state. Fig.
5(e) illustrates this case. In the case presented in the figure, the four
outer kinks form two bions. A significant amount of radiation is also present.

  * •

Region $F$: The kinks change sector and create an extra pair of outer kinks.
Moreover, an oscillating pulse is formed at the center. Fig. 5(f) illustrates
this case.

  * •

Region $G$: The kinks collide, forming three pairs of outer kinks in the final
state. An example is given in Fig. 5(g).

  * •

Region $H$: The kinks collide and form seven pairs of outer kinks in the final
state. An example is given in Fig. 5(h).

  * •

Region $I$: In the limit the outer kinks are very light, many kink-antikink
pairs are formed. As the kinks are also very small, they form a pattern
similar to the one in radiation emission. Fig. 5(i) illustrates this case.

(a) $b=0.05$, $v_{i}=0.2$.

(b) $b=0.1$, $v_{i}=0.4$.

(c) $b=0.17$, $v_{i}=0.2$.

(d) $b=0.23$, $v_{i}=0.2$.

(e) $b=0.26$, $v_{i}=0.2$.

(f) $b=0.358$, $v_{i}=0.2$.

(g) $b=0.37$, $v_{i}=0.2$.

(h) $b=0.55$, $v_{i}=0.2$.

(i) $b=0.82$, $v_{i}=0.2$.

Figure 5: Evolution of $\phi(x,t)$ in spacetime for kink-antikink collisions
between inner kinks.

A similar behavior observed in regions $C$ to $H$ was previously reported in
Ref. [40] when analyzing the double sine-Gordon model. We see that the outer
kink mass decreases with $b$, creating more pairs as the mass vanishes for
$b=1$. The evidence is $\phi(0,t_{f})$ changing color from white to yellow. So
every time a new kink-antikink pair is created, the vacuum at the center needs
to be adjusted accordingly, indicated by the color change. Moreover, Our
numerical simulations show that when the creation of a pair is almost
energetically favored, oscillating pulses appear. Increasing further the
initial energy, the oscillating pulse turns into a kink-antikink pair. This
result is in agreement with Ref. [40]. Moreover, the extra kink-antikink pairs
may form bound states among them. However, in the present case, the bound
structures are less stable than in Ref. [40], meaning that they are more
likely to decay into radiation.

As $b$ approaches one, all the extra kink pairs create a radiation pattern.
Furthermore, we see in Fig. 4 that this pattern repeats at least up to the
largest exhibited value, $v_{i}=0.5$. Hence, to overcome this effect, a large
initial velocity is needed. For $b=1$, the outer kinks cease to exist, and the
inner kink acquires long-range tails at both sides. This case was studied in
Ref. [1], and the same pattern, as well as an ultrarelativistic critical
velocity, was observed. Therefore, we found how the annihilation into
radiation is achieved as two quadratic minima merge to form a quartic one. It
is through the creation of many small kink-antikink pairs.

In Ref. [62], the authors also raised doubt on the claims presented in Ref.
[1]. The claims were that the kinks decay into radiation without bion
formation and possess an ultrarelativistic critical velocity for the kinks
with long-range tails on both sides. The key ingredient for the new behavior
is a long-range tail in the direction that is not facing the opposing kink.
Above, we presented a more detailed physical description of the phenomena,
corroborating such claims. Interestingly, it has recently been reported a
different scenario where complete kink-antikink annihilation can occur, which
is the scattering of long-range kinks by a half-BPS preserving impurity [65].

In regions $A$ and $B$, the outer kinks are more massive than the inner ones.
Hence, it is not possible to change sectors, and the kinks bounce back when
they collide, resembling the $\phi^{4}$ model. There are also resonance
windows at the boundary between the two regions, which is consistent with the
presence of a vibrational mode. Moreover, the inner kinks become increasingly
small as $b$ vanishes. In such a case, we observe that the critical velocity
separating regions $A$ and $B$ sharply increases near this point. This pattern
will also occur in the other cases considered below.

The simulations for $b\leq 0.6$ were performed after minimizing the initial
condition as described previously. Such a procedure was crucial to obtain
correct results. In the limit $b\to 1$, the minimization procedure gives
correct results for $v>0.25$. However, there is no visible difference compared
to non-minimized simulations. For $v<0.25$, our minimized algorithm runs into
trouble because the box is too small and $t_{f}$ is too large. Note that the
collision creates a large amount of radiation which travels at speed near
$c=1$, see Fig. 5(i). This creates a spurious effect from returning radiation
that cannot be eliminated by the damping term at the boundaries. Therefore, we
did not use minimized initial conditions in this region, but we ran several
tests to ensure that the result is still physically correct.

###  3.2 Outer sector antikink-kink collision

In this section, we will discuss the field evolution of antikink-kink
collisions in the outer sector. In such case, the field in the region between
the kinks is at the inner vacuum $\phi=\sqrt{b}$. Among all cases, this is the
simplest one. In this sector, the kink-antikink configuration allows
delocalized modes due to the asymmetry of the isolated kinks. Such modes are
excitations of the scalar field, which appear due to the potential well
created by the antikink-kink pair. Moreover, there is also a localized
vibrational mode. We will see shortly how these affect the resonance
structure.

Figure 6: Final value of $\phi$ at the collision center as a function of $b$
and $v_{i}$. We are considering antikink-kink collisions between outer kinks.

The system’s behavior as a function of the parameter $b$ and the initial
velocity $v_{i}$ is summarized in Fig. 6. It shows the field’s value at the
collision center at a time $t_{f}=60.0/v_{i}$. It is possible to distinguish
between three different regions, which are listed below:

  * •

Region $A$: In the large smooth region above $B$ and $C$, the kinks collide
and then reflect. The behavior is illustrated in Fig. 7(a).

  * •

Region $B$: The kinks form a bion, a state where the kink and the antikink are
bound. The bion slowly decays to a trivial vacuum after a large number of
collisions. Fig. 7(b) illustrates this case.

  * •

Region $C$: The kinks collide and then depending on the initial velocity form
a bion or reflect after more than a single bounce. The latter are resonance
windows. The two-, three- and four-bounce windows are illustrated in Fig.
7(c), Figs. 7(d) and (e), and Fig. 7(f), respectively.

(a) $b=0.45$, $v_{i}=0.3$.

(b) $b=0.07$, $v_{i}=0.06$.

(c) $b=0.15$, $v_{i}=0.09$.

(d) $b=0.12$, $v_{i}=0.089$.

(e) $b=0.66$, $v_{i}=0.074$.

(f) $b=0.125$, $v_{i}=0.109$.

Figure 7: Evolution of $\phi(x,t)$ in spacetime for antikink-kink collisions
between outer kinks.

Interestingly, nothing special happens as we approach the limit $b=0$, where
the tail facing the opposing kink is long-range. In other words, the system’s
resonance structure at $b=0$ is qualitatively the same as that of systems with
exponential tails, which also possess delocalized modes. This is expected from
the analysis in [45].

The critical velocity near $b=0.4$ becomes less than $0.025$ and is not
visible in Fig. 6. Due to the kink’s asymmetry, delocalized vibrational modes
create resonance windows for $b<0.3$. This is the region where the kinks are
mostly asymmetrical. However, not all values of $b$ in the interval $[0,1]$
exhibit resonance windows. It is possible to see that the windows are
suppressed near $b=0.3$. By increasing the parameter $b$, a vibrational mode
appears, and the resonance structure is recovered. Increasing even further,
the vibrational mode and the resonance windows cease to exist again.

In region $C$, the set of resonance windows form a quasi-fractal structure,
meaning that $(n+1)$-bounce windows accumulate near the edge of $n$-bounce
windows. To illustrate this phenomenon, examples of three and four-bounce
resonance windows are given in Fig. 7. Moreover, we also found higher-bounce
windows in kink-antikink collisions in the inner and outer sectors, evidencing
that they also form a quasi-fractal structure.

Another effect observed in Fig. 6 is that the kinks become increasingly small
and the critical velocity increasingly large as $b$ approaches one. Moreover,
we should mention that the simulations for $b<0.12$ and $b\geq 0.88$ were
performed after minimizing the initial condition as described previously. Such
a procedure was crucial to obtain correct results.

###  3.3 Outer sector kink-antikink collision

The field evolution in this sector corresponds to a kink-antikink collision
between outer kinks. The kinks are initially separated by the vacuum at
$\phi=1$. The main differences with the previous case are that the kink’s
asymmetry character does not allow delocalized modes’ presence, and the outer
kink-antikink pair can change sector after a collision and turn into an
antikink-kink pair in the inner sector.

Figure 8: Final value of $\phi$ at the collision center as a function of $b$
and $v_{i}$. We are considering kink-antikink collisions between outer kinks.

Fig. 8 summarizes the system’s behavior as a function of the parameter $b$ and
the initial velocity $v_{i}$. It shows the field’s value at the collision
center at a time $t_{f}=60.0/v_{i}$. It is possible to distinguish six
different regions, which will be listed below:

  * •

Region $A$: The kinks change sector, and an extra pair of inner kinks is
created. It is illustrated in Fig. 9(a).

  * •

Region $B$: The kinks change sector after the collision and separate. It is
illustrated in Fig. 9(b).

  * •

Region $C$: The kinks annihilate forming a bion as illustrated in Fig. 9(c).

  * •

Region $D$: The kinks either form a bion or bounce more than once and then
separate. The latter are resonance windows. The two-bounce case is illustrated
in Fig. 9(d).

  * •

Region $E$: The kinks collide and then reflect as illustrated in Fig. 9(e).

  * •

Region $F$: The kinks change sectors twice before separating as illustrated in
Fig. 9(f).

(a) $b=0.045$, $v_{i}=0.38$.

(b) $b=0.08$, $v_{i}=0.38$.

(c) $b=0.4$, $v_{i}=0.3$.

(d) $b=0.67$, $v_{i}=0.34$.

(e) $b=0.7$, $v_{i}=0.55$.

(f) $b=0.16$, $v_{i}=0.64$.

Figure 9: Evolution of $\phi(x,t)$ in spacetime for kink-antikink collisions
between outer kinks.

As $b\to 0$, we end up with an interesting situation. The tail that is not
facing the opposite kink becomes long-range. As we move towards this limit,
the formation of extra pairs of kinks from region $B$ to $A$ is completely
analogous to the behavior of kink-antikink collisions in the inner sector. It
occurs because the initial energy is sufficient to create more pairs as the
inner kinks become lighter. Similarly, it eventually culminates in
annihilation directly into radiation and an ultrarelativistic critical
velocity. This is in agreement with the results obtained in Ref. [1].

The inner kink is more massive in regions $D$ and $E$. Hence, it is not
possible to change sectors anymore. Instead, we observe the standard resonance
phenomenon, which is consistent with the presence of a vibrational mode. The
region $F$ is at the boundary between the cases where it is possible to change
sectors and where it is not. For the observed behavior to occur, the
parameters need to be tuned. First, a sector change occurs, but the energy of
the kinks in the new sector is very small. Thus, they do not separate and
bounce back. Still, the energy is enough for the kinks to return to the
original sector and separate. As a last observation, we see that the kinks
become extremely small as $b$ approaches one, and we observe a sharp increase
in the critical velocity again.

The simulations for $b\geq 0.76$ were performed after minimizing the initial
condition as described previously. Again, such a procedure was crucial to
obtain correct results. In the limit $b\to 0$, the tails become long-range,
but the minimization procedure is unnecessary because they do not face the
opposing kink.

##  4 Conclusion

We considered a class of $\phi^{8}$ scalar field models in (1+1)-dimensions.
The scalar potentials which describe the theory contain four degenerate minima
whose positions depend on one free parameter, $b$. In this scenario, a
detailed analysis of kink-antikink collisions was made. This model exhibits
two inner kinks related by parity or equivalently $\phi\to-\phi$
transformations. There are also four outer kinks related by parity and
$\phi\to-\phi$ transformations. We computed the kink profiles and spectra
numerically while the masses were computed analytically. Then, we analyzed
kink-antikink collisions. Three types of kink-antikink collisions exist in the
present model. We performed detailed numerical simulations of all cases and
found different regions in parameter space, which were carefully analyzed.
Here, we also used the minimization method proposed in Ref. [1] on a large
scale, showing that it is highly computationally efficient.

We conclude that the $\phi^{8}$ model exhibits new phenomena with respect to
lower-order polynomial models. It was already known that long-range tails
occur. The case with four degenerate minima exhibits other phenomena and
approaches the long-range case when $b=0$ and $b=1$. In particular, we found a
rich fractal structure of resonance windows coming from both localized and
delocalized modes. We also found sector change with the formation of
oscillating pulses and extra kink-antikink pairs. The critical velocity
exhibits a sharp increase in collisions between very small kinks. A
fascinating case was obtained in Fig. 9(f), where the kinks separate after
changing sectors twice. More importantly, we provided a path from short-range
to long-range behavior. When two quadratic minima merge, a large number of
small kink-antikink pairs may form, creating a radiation pattern. This occurs
when the tail not facing the opposing kink acquires long-range character.
Therefore, our results confirm that, in such case, incoming kink and antikink
form a large number of small kink-antikink pairs or decay directly into
radiation.

One perspective for future works is analyzing in detail the rich structure of
resonance windows that we found. For instance, it would be interesting to
measure the resonant frequency of different sets of windows numerically and
study their fractality. One could also study the $\phi^{8}$ model away from
the first-order phase transition. As there are two parameters, one should be
fixed to allow a similar analysis to the one performed here. We firmly believe
that new and exciting phenomena will be encountered. Furthermore, one could
move on to the $\phi^{10}$ and $\phi^{12}$ models, which have three and four
free parameters, respectively. The $\phi^{10}$ model with five degenerate
minima has only one free parameter, and a similar analysis can be performed.
However, two parameters need to be tuned to put the system in such a state. On
the other hand, to fully comprehend the $\phi^{12}$ model is much more
challenging. The case with six degenerate minima has two free parameters, and
the remaining two must be tuned to put the system in such a state.
Understanding these systems would be a fascinating challenge where new
phenomena will be unveiled, and an even richer fractal structure may be found.
It would also contribute to a better understanding of the value of having
polynomial potentials with increasing power in the non-linearity of the field
self-interaction.

## Acknowledgments

We acknowledge financial support from the National Council for Scientific and
Technological Development - CNPq, Grant No. 303469/2019-6 (DB), No.
150166/2022-2 (JGFC), and No. 309368/2020-0 (AM). DB thanks Paraíba State
Research Foundation, FAPESQ-PB, Grant N. 0015/2019. AM thanks financial
support from the Brazilian agency CAPES and Universidade Federal de Pernambuco
Edital Qualis A. The simulations presented here were conducted on the SDumont
cluster at the Brazilian laboratory LNCC (Laboratório Nacional de Computação
Científica).

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$\displaystyle\quad+\alpha_{0}P(\alpha_{0})\pi_{\alpha_{0}}({\bf
V}_{2,\alpha_{0}}+{\bf
K}_{2,\alpha_{0}})\pi_{\alpha_{0}}\mathcal{M}_{\partial_{\alpha}\log\rho_{\alpha_{0}}}\pi_{\alpha_{0}}P(\alpha_{0})$
$\displaystyle=P(\alpha_{0})\pi_{\alpha_{0}}({\bf V}_{2,\alpha_{0}}+{\bf
K}_{2,\alpha_{0}})\pi_{\alpha_{0}}P(\alpha_{0})$
$\displaystyle\quad+\alpha_{0}P(\alpha_{0})\pi_{\alpha_{0}}({\bf
V}_{2,\alpha_{0}}+{\bf
K}_{2,\alpha_{0}})\pi_{\alpha_{0}}P(\alpha_{0})\mathcal{M}_{\partial_{\alpha}\log\rho_{\alpha_{0}}}P(\alpha_{0})$
$\displaystyle=P(\alpha_{0})\pi_{\alpha_{0}}({\bf V}_{2,\alpha_{0}}+{\bf
K}_{2,\alpha_{0}})\pi_{\alpha_{0}}P(\alpha_{0})\left(I+\alpha_{0}\mathcal{M}_{\partial_{\alpha}\log\rho_{\alpha_{0}}}\right)P(\alpha_{0})$
$\displaystyle=-\frac{1}{\alpha_{0}}P(\alpha_{0})A_{0}P(\alpha_{0})\left(I+\alpha_{0}\mathcal{M}_{\partial_{\alpha}\log\rho_{\alpha_{0}}}\right)P(\alpha_{0}).$

Hence,

$\displaystyle
P(\alpha_{0})\nabla\Phi(0,\alpha_{0})P(\alpha_{0})+P(\alpha_{0})A_{1}P(\alpha_{0})(\alpha-\alpha_{0})$
$\displaystyle\qquad=P(\alpha_{0})(I-A_{0})P(\alpha_{0})-\frac{\alpha-\alpha_{0}}{\alpha_{0}}P(\alpha_{0})A_{0}P(\alpha_{0})\left(I+\alpha_{0}\mathcal{M}_{\partial_{\alpha}\log\rho_{\alpha_{0}}}\right)P(\alpha_{0}).$

Combining this with [22, DEFINITION, THEOREM 9.2] and
$(I_{\mathcal{H}_{0}}+\tilde{M}_{0})$ is invertible on $\mathcal{H}_{0}$, we
arrive at

$\displaystyle{\rm
det}_{2,\mathcal{H}_{0}}\left(P(\alpha_{0})\nabla\Phi(0,\alpha_{0})P(\alpha_{0})+P(\alpha_{0})A_{1}P(\alpha_{0})(\alpha-\alpha_{0})\right)$
$\displaystyle\qquad=e^{\mathrm{tr}{\left(\tilde{A}_{0}+\frac{\alpha-\alpha_{0}}{\alpha_{0}}\tilde{A}_{0}(I_{\mathcal{H}_{0}}+\tilde{M}_{0})\right)}}{\rm
det}_{\mathcal{H}_{0}}\left(I_{\mathcal{H}_{0}}-\tilde{A}_{0}-\frac{\alpha-\alpha_{0}}{\alpha_{0}}\tilde{A}_{0}(I_{\mathcal{H}_{0}}+\tilde{M}_{0})\right)$
$\displaystyle\qquad=e^{\frac{\alpha}{\alpha_{0}}\mathrm{tr}{\left(\tilde{A}_{0}\right)}+\frac{\alpha-\alpha_{0}}{\alpha_{0}}\mathrm{tr}{\left(\tilde{A}_{0}\tilde{M}_{0}\right)}}{\rm
det}_{\mathcal{H}_{0}}\left(\tilde{A}_{0}(I_{\mathcal{H}_{0}}+\tilde{M}_{0})\right)$
$\displaystyle\qquad\quad\times{\rm
det}_{\mathcal{H}_{0}}\left((I_{\mathcal{H}_{0}}+\tilde{M}_{0})^{-1}\left(\tilde{A}_{0}^{-1}-I_{\mathcal{H}_{0}}\right)-\frac{\alpha-\alpha_{0}}{\alpha_{0}}\right).$

Since the algebraic multiplicity of the eigenvalue $0$ of
$(I_{\mathcal{H}_{0}}+\tilde{M}_{0})^{-1}\left(\tilde{A}_{0}^{-1}-I_{\mathcal{H}_{0}}\right)$
is odd, we find that ${\rm
det}_{\mathcal{H}_{0}}\left((I_{\mathcal{H}_{0}}+\tilde{M}_{0})^{-1}\left(\tilde{A}_{0}^{-1}-I_{\mathcal{H}_{0}}\right)-\frac{\alpha-\alpha_{0}}{\alpha_{0}}\right)$
changes sign. This implies that

${\rm
det}_{2,\mathcal{H}_{0}}\left(P(\alpha_{0})\nabla\Phi(0,\alpha_{0})P(\alpha_{0})+P(\alpha_{0})A_{1}P(\alpha_{0})(\alpha-\alpha_{0})+o(\alpha-\alpha_{0})\right)$

changes sign at $\alpha=\alpha_{0}$. Therefore, ${\rm
det}_{2}(I+\alpha\pi_{\alpha}({\bf V}_{2,\alpha}+{\bf
K}_{2,\alpha})\pi_{\alpha})$ changes sign at $\alpha=\alpha_{0}$.

∎

###### Proof of Corollary 3.6.

Existence of solutions for (3.16) follows from Corollary 2.4. For
$\sigma_{0}$, we can choose $0<\hat{\sigma}<\sigma_{0}<\check{\sigma}$ and
$\bar{V}(x)=C_{\sigma_{0}}(1+|x|)^{\gamma_{1}}$ for some $C_{\sigma_{0}}>0$
such that (A1) holds. Let $\alpha=\frac{\beta}{2\sigma^{2}}$. Then

$\displaystyle\mathcal{T}(x;\rho,\alpha)=\frac{\exp\left\\{-\frac{2\alpha}{\beta}V_{0}(x)+\alpha\int_{\mathbb{R}^{d}}H(x-y)\rho(y)\bar{\mu}(\mathrm{d}y)+\bar{V}(x)\right\\}}{\int_{\mathbb{R}}\exp\left\\{-\frac{1}{\sigma^{2}}\left(V_{0}(x)+\frac{\beta}{2}\int_{\mathbb{R}^{d}}H(x-y)\rho(y)\mathrm{d}y\right)\right\\}\bar{\mu}(\mathrm{d}x)}.$

Since for all $f\in L^{2}(\mu_{\alpha_{0}})$

$\displaystyle\mu_{\alpha_{0}}(f\pi_{\alpha_{0}}{\bf
V}_{2,\alpha_{0}}\pi_{\alpha_{0}}f)$
$\displaystyle=\sum_{i,j=1}^{l}G_{ij}\mu_{\alpha_{0}}((v_{i}-\mu_{\alpha_{0}}(v_{i}))f)\mu_{\alpha_{0}}(v_{j}(f-\mu_{\alpha_{0}}(f))$
$\displaystyle=\sum_{i,j=1}^{l}G_{ij}\mu_{\alpha_{0}}((v_{i}-\mu_{\alpha_{0}}(v_{i}))f)\mu_{\alpha_{0}}((v_{j}-\mu_{\alpha_{0}}(v_{j}))f)$
$\displaystyle\geq 0,$

we find that $I+\alpha_{0}\pi_{\alpha_{0}}V_{2,\alpha_{0}}\pi_{\alpha_{0}}$ is
invertible on $L^{2}(\mu_{\alpha_{0}})$. Hence, the first assertion of this
corollary can follow from Lemma 3.1 directly, and we focus on the bifurcation
point in the following discussion.

We prove that $0$ is the eigenvalue of $I+\alpha_{0}\pi_{\alpha_{0}}({\bf
V}_{2,\alpha_{0}}+{\bf K}_{2,\alpha_{0}})\pi_{\alpha_{0}}$ with odd algebraic
multiplicity. It is clear that ${\bf K}_{2,\alpha_{0}}$ and ${\bf
V}_{2,\alpha_{0}}$ are self-adjoint operators on $L^{2}(\mu_{\alpha_{0}})$.
For all $f\in L^{2}(\mu_{\alpha_{0}})$, due to that $K_{2}(x,\cdot)$ is anti-
symmetric, $V_{2}(\cdot,y)$ and $\mu_{\alpha_{0}}$ are symmetric, we have that
${\bf V}_{2,\alpha_{0}}\pi_{\alpha}{\bf K}_{2,\alpha_{0}}=0$ and

$\displaystyle{\bf K}_{2,\alpha_{0}}\pi_{\alpha_{0}}{\bf V}_{2,\alpha_{0}}f$
$\displaystyle=\int_{\mathbb{R}^{d}}K_{2}(x,z)\mu_{\alpha_{0}}(\mathrm{d}z)\int_{\mathbb{R}^{d}}(V_{2}(z,y)-\mu_{\alpha_{0}}(V_{2}(\cdot,y))f(y)\mu_{\alpha_{0}}(\mathrm{d}y)$
$\displaystyle=\int_{\mathbb{R}^{d}}\left(\int_{\mathbb{R}^{d}}K_{2}(x,z)(V_{2}(z,y)-\mu_{\alpha_{0}}(V_{2}(\cdot,y))\mu_{\alpha_{0}}(\mathrm{d}z)\right)f(y)\mu_{\alpha_{0}}(\mathrm{d}y)$
$\displaystyle=0.$

Let $R_{V}$ be closure of the range of $\pi_{\alpha_{0}}{\bf
V}_{2,\alpha_{0}}\pi_{\alpha_{0}}$ and $R_{K}$ be closure of the range of
$\pi_{\alpha_{0}}{\bf K}_{2,\alpha_{0}}\pi_{\alpha_{0}}$. Then $R_{V}\perp
R_{K}$, $R_{V}\subset{\rm Ker}(\pi_{\alpha_{0}}{\bf
K}_{2,\alpha_{0}}\pi_{\alpha_{0}})$ and $R_{K}\subset{\rm
Ker}(\pi_{\alpha_{0}}{\bf V}_{2,\alpha_{0}}\pi_{\alpha_{0}})$. Then there is
subspace $\mathcal{H}$ such that $L^{2}(\mu_{\alpha_{0}})=R_{V}\oplus
R_{K}\oplus\mathcal{H}$ and

$\mathcal{H}\subset{\rm Ker}(\pi_{\alpha}{\bf
V}_{2,\alpha}\pi_{\alpha})\cap{\rm Ker}(\pi_{\alpha}{\bf
K}_{2,\alpha}\pi_{\alpha}).$

For $0\neq f\in L^{2}(\mu_{\alpha_{0}})$ with
$f+\alpha_{0}\pi_{\alpha_{0}}({\bf V}_{2,{\alpha_{0}}}+{\bf
K}_{2,{\alpha_{0}}})\pi_{\alpha_{0}}f=0$, there is $f=f_{1}+f_{2}$ for some
$f_{1}\in R_{V}$ and $f_{2}\in R_{K}$. Then

$\left(f_{1}+\alpha_{0}\pi_{\alpha_{0}}{\bf
V}_{2,\alpha_{0}}\pi_{\alpha_{0}}f_{1}\right)+\left(f_{2}+\alpha_{0}\pi_{\alpha_{0}}{\bf
K}_{2,\alpha_{0}}\pi_{\alpha_{0}}f_{2}\right)=0.$

This yields that

$\left\\{\begin{array}[]{l}f_{1}+\alpha_{0}\pi_{\alpha_{0}}{\bf
V}_{2,\alpha_{0}}\pi_{\alpha_{0}}f_{1}=0,\\\
f_{2}+\alpha_{0}\pi_{\alpha_{0}}{\bf
K}_{2,\alpha_{0}}\pi_{\alpha_{0}}f_{2}=0.\end{array}\right.$

Since $I+\alpha_{0}\pi_{\alpha_{0}}V_{2,\alpha_{0}}\pi_{\alpha_{0}}$ is
invertible, $f_{1}=0$. Thus, $0$ is an eigenvalue of
$I+\alpha_{0}\pi_{\alpha_{0}}({\bf V}_{2,\alpha_{0}}+{\bf
K}_{2,\alpha_{0}})\pi_{\alpha_{0}}$ if and only if $0$ is an eigenvalue of
$I+\alpha_{0}\pi_{\alpha_{0}}{\bf
K}_{2,\alpha_{0}}\pi_{\alpha_{0}}=I+\alpha_{0}{\bf K}_{2,\alpha_{0}}$, and

${\rm Ker}(I+\alpha_{0}\pi_{\alpha_{0}}\left({\bf V}_{2,\alpha_{0}}+{\bf
K}_{2,\alpha_{0}}\right)\pi_{\alpha_{0}})={\rm Ker}(I+\alpha_{0}{\bf
K}_{2,\alpha_{0}}).$ (3.49)

Moreover,

$\displaystyle I+\alpha_{0}\pi_{\alpha_{0}}({\bf V}_{2,\alpha_{0}}+{\bf
K}_{2,\alpha_{0}})\pi_{\alpha_{0}}$
$\displaystyle=\left(I+\alpha_{0}\pi_{\alpha_{0}}{\bf
V}_{2,\alpha_{0}}\pi_{\alpha_{0}}\right)\left(I+\alpha_{0}{\bf
K}_{2,\alpha_{0}}\right)$ $\displaystyle=\left(I+\alpha_{0}{\bf
K}_{2,\alpha_{0}}\right)\left(I+\alpha_{0}\pi_{\alpha_{0}}{\bf
V}_{2,\alpha_{0}}\pi_{\alpha_{0}}\right).$

Combining this with
$I+\alpha_{0}\pi_{\alpha_{0}}V_{2,\alpha_{0}}\pi_{\alpha_{0}}$ is invertible
on $L^{2}(\mu_{\alpha_{0}})$ and [19, Theorem 21.2.6 and Theorem 25.2.2’], the
algebraic multiplicity of $0$ as eigenvalue of
$I+\alpha_{0}\pi_{\alpha_{0}}({\bf V}_{2,\alpha_{0}}+{\bf
K}_{2,\alpha_{0}})\pi_{\alpha_{0}}$ is the same as $0$ as eigenvalue of
$I+\alpha_{0}{\bf K}_{2,\alpha_{0}}$.
Since $R_{K}\subset\mathcal{H}_{K}$, the space $\mathcal{H}_{K}:={\rm
span}[k_{1},\cdots,k_{m}]$ is an invariant space of $I+\alpha_{0}{\bf
K}_{2,\alpha_{0}}$. Due to that $k_{1},\cdots,k_{m}$ are linearly independent,
$I+\alpha_{0}{\bf K}_{2,\alpha_{0}}$ can be represented under the basis
$[k_{1},\cdots,k_{m}]$ as the matrix $I+\alpha_{0}GG(\sigma_{0})$. Then $0$ is
an eigenvalue of $I+\alpha_{0}{\bf K}_{2,\alpha_{0}}$ if and only if $0$ is an
eigenvalue of the matrix $I+\alpha_{0}GG(\sigma_{0})$, and they have the same
algebraic multiplicities. By (1) in the assumption, we have that $0$ is an
eigenvalue of $I+\alpha_{0}\pi_{\alpha_{0}}({\bf V}_{2,\alpha_{0}}+{\bf
K}_{2,\alpha_{0}})\pi_{\alpha_{0}}$ with odd algebraic multiplicity.

Next, we give a representation of
$\mathcal{M}_{\partial_{\alpha}\log\rho_{\alpha_{0}}}\Big{|}_{\mathcal{H}_{K}}$.
Let $P_{K}$ be the orthogonal projection from $L^{2}(\mu_{\alpha_{0}})$ on to
$\mathcal{H}_{K}$. Then for all $f\in L^{2}(\mu_{\alpha_{0}})$

$P_{K}f=\sum_{i,j=1}^{m}G_{ij}(\sigma_{0})^{-1}\mu_{\alpha_{0}}(fk_{j})k_{i}.$

Since $v_{1},\cdots,v_{l}$ are linearly independent,
$\pi_{\alpha_{0}}(v_{1}),\cdots,\pi_{\alpha_{0}}(v_{l})$ are also linearly
independent. Let $\mathcal{H}_{V}(\alpha_{0})={\rm
span}[\pi_{\alpha_{0}}(v_{1}),\cdots,\pi_{\alpha_{0}}(v_{l})]$ and $P_{V}$ be
the orthogonal projection from $L^{2}(\mu_{\alpha_{0}})$ on to
$\mathcal{H}_{V}(\alpha_{0})$. Then

$\displaystyle(I-P_{V})\pi_{\alpha_{0}}{\bf V}_{2,\alpha_{0}}=0,$
$\displaystyle P_{V}f=$
$\displaystyle\sum_{i,j=1}^{l}J^{-1}_{ij}(\alpha_{0})\mu_{\alpha_{0}}(f\pi_{\alpha_{0}}(v_{j}))\pi_{\alpha_{0}}(v_{i}).$

According to (3.7), $\theta^{\prime}(\alpha)=\frac{2}{\beta}$, $V_{1}=0$ and
the representation of $P_{V}$, we have that

$\displaystyle\partial_{\alpha}\log\rho_{\alpha_{0}}$
$\displaystyle=-\frac{2}{\beta}(I+\alpha_{0}\pi_{\alpha_{0}}{\bf
V}_{2,\alpha_{0}}\pi_{\alpha_{0}})^{-1}\pi_{\alpha_{0}}\left(V_{0}+{\bf
V}_{2,\alpha_{0}}\mathds{1}\right)$
$\displaystyle=-\frac{2}{\beta}(I+\alpha_{0}\pi_{\alpha_{0}}{\bf
V}_{2,\alpha_{0}}\pi_{\alpha_{0}})^{-1}P_{V}\left(\pi_{\alpha_{0}}\left(V_{0}+{\bf
V}_{2,\alpha_{0}}\mathds{1}\right)\right)$
$\displaystyle\quad-\frac{2}{\beta}(I+\alpha_{0}\pi_{\alpha_{0}}{\bf
V}_{2,\alpha_{0}}\pi_{\alpha_{0}})^{-1}(I-P_{V})\left(\pi_{\alpha_{0}}\left(V_{0}+{\bf
V}_{2,\alpha_{0}}\mathds{1}\right)\right)$
$\displaystyle=-\frac{2}{\beta}(I+\alpha_{0}\pi_{\alpha_{0}}{\bf
V}_{2,\alpha_{0}}\pi_{\alpha_{0}})^{-1}\left(P_{V}\pi_{\alpha_{0}}V_{0}+\pi_{\alpha_{0}}{\bf
V}_{2,\alpha_{0}}\mathds{1}\right)$
$\displaystyle\quad-\frac{2}{\beta}(I+\alpha_{0}\pi_{\alpha_{0}}{\bf
V}_{2,\alpha_{0}}\pi_{\alpha_{0}})^{-1}(I-P_{V})\pi_{\alpha_{0}}V_{0}$
$\displaystyle=-\frac{2}{\beta}\sum_{i=1}^{l}\left[(I+\alpha_{0}JJ(\alpha_{0}))^{-1}(w+\tilde{w})\right]_{i}\pi_{\alpha_{0}}(v_{i})$
$\displaystyle\quad-\frac{2}{\beta}(\pi_{\alpha_{0}}V_{0}-\sum_{i=1}^{l}w_{i}\pi_{\alpha_{0}}(v_{i})),$

where $I$ is an identity operator, which maybe on different space from line to
line. Choose an orthonormal basis $[\tilde{k}_{1},\cdots,\tilde{k}_{m}]$ of
$\mathcal{H}_{K}$ with inner product induced by $L^{2}(\mu_{\alpha_{0}})$. Let
$S\in\mathbb{R}^{m}\otimes\mathbb{R}^{m}$ such that
$k_{i}=\sum_{j=1}^{m}S_{ij}\tilde{k}_{j}$. Then $G(\sigma_{0})=SS^{*}$. By the
definition of $M_{K}(\alpha_{0})$, under the basis
$[\tilde{k}_{1},\cdots,\tilde{k}_{m}]$, the operator
$P_{K}\mathcal{M}_{\partial_{\alpha}\log\rho_{\alpha_{0}}}\Big{|}_{\mathcal{H}_{K}}$
can be represented as a matrix on $\mathcal{H}_{K}$, saying
$S^{-1}M_{K}(\alpha_{0})(S^{*})^{-1}$.

Finally, we prove that $(I_{\mathcal{H}_{0}}+\tilde{M}_{0})$ is invertible on
$\mathcal{H}_{0}$. Since

$\displaystyle K_{2}(x,y)$
$\displaystyle=\sum_{i,j=1}^{m}G_{ij}k_{i}(x)k_{j}(y)=\sum_{i,j=1}^{m}G_{ij}\left(\sum_{r=1}^{m}S_{ir}\tilde{k}_{r}(x)\right)\left(\sum_{n=1}^{m}S_{jn}k_{n}(y)\right)$
$\displaystyle=\sum_{i,j,r,n=1}^{m}S_{ir}G_{ij}S_{jn}\tilde{k}_{r}(x)\tilde{k}_{n}(y),$

${\bf K}_{2,\alpha_{0}}$ can be represented as the matrix $S^{*}GS$ under the
basis $[\tilde{k}_{1},\cdots,\tilde{k}_{m}]$. Then $\mathcal{H}_{0}$ can be
represented as ${\rm Ker}(I+\alpha_{0}S^{*}GS)$. It is clear that
$P(\alpha_{0})\subset P_{K}$. We denote by $P_{K}(\alpha_{0})$ the
representation of $P(\alpha_{0})$ restricted on $\mathcal{H}_{K}$. Then

$P_{K}(\alpha_{0})\mathcal{H}_{K}={\rm Ker}(I+\alpha_{0}S^{*}GS),$ (3.50)

and
$P(\alpha_{0})\mathcal{M}_{\partial_{\alpha}\log\rho_{\alpha_{0}}}\Big{|}_{\mathcal{H}_{K}}$
can be represented as $P_{K}(\alpha_{0})S^{-1}M_{K}(\alpha_{0})(S^{*})^{-1}$.
By using these matrices, we prove that
$\bullet$ $(I_{\mathcal{H}_{0}}+\tilde{M}_{0})$ is not invertible on
$\mathcal{H}_{0}$ if and only if the following system has a solution
$(w_{1},w_{2})\in\mathbb{R}^{2m}$ with $w_{2}\neq 0$

$\displaystyle\left(I+\alpha_{0}S^{*}GS\right)w_{2}=0,$ (3.51)
$\displaystyle\left(I+\alpha_{0}S^{-1}M_{K}(\alpha_{0})(S^{*})^{-1}\right)w_{2}=\left(I+\alpha_{0}S^{*}GS\right)w_{1}.$
(3.52)

Indeed, $(I_{\mathcal{H}_{0}}+\tilde{M}_{0})$ is not invertible on
$\mathcal{H}_{0}$ if and only if there exists $w_{2}\in\mathbb{R}^{m}$ with
$w_{2}\neq 0$ such that (3.51) holds and
$P_{K}(\alpha_{0})\left(I+\alpha_{0}S^{-1}M_{K}(\alpha_{0})(S^{*})^{-1}\right)w_{2}=0$.
Taking into account ${\rm Ker}(I+\alpha_{0}S^{*}GS)\perp{\rm
Ran}(I+\alpha_{0}S^{*}GS)$ and (3.50), we have that

$\left(I+\alpha_{0}S^{-1}M_{K}(\alpha_{0})(S^{*})^{-1}\right)w_{2}\in{\rm
Ran}(I+\alpha_{0}S^{*}GS).$

Thus there exists $w_{1}$ such that (3.52) holds. Conversely, if there exists
$(w_{1},w_{2})$ with $w_{2}\neq 0$ such that (3.51) and (3.52) hold, then

$P_{K}(\alpha_{0})(I+\alpha_{0}S^{-1}M_{K}(\alpha_{0})(S^{*})^{-1})w_{2}=P_{K}(\alpha_{0})\left(I+\alpha_{0}S^{*}GS\right)w_{1}=0.$

Thus $(I_{\mathcal{H}_{0}}+\tilde{M}_{0})$ is not invertible on
$\mathcal{H}_{0}$.
Rewrite (3.51) and (3.52) in the following form:

$\displaystyle T\left(\begin{array}[]{c}w_{1}\\\
w_{2}\end{array}\right):=\left[\begin{array}[]{cc}0&I+\alpha_{0}S^{*}GS\\\
(I+\alpha_{0}S^{*}GS)&-(I+\alpha_{0}S^{-1}M_{K}(\alpha_{0})(S^{*})^{-1})\end{array}\right]\left(\begin{array}[]{c}w_{1}\\\
w_{2}\end{array}\right)=0.$

It is clear that $(w_{1},0)$ is a solution of this system if and only if
$w_{1}\in{\rm Ker}(I+\alpha_{0}S^{*}GS)$. Thus there exists $(w_{1},w_{2})$
with $w_{2}\neq 0$ such that (3.51) and (3.52) holds if and only if

${\rm dim}{\rm Ker}\left(T\right)>{\rm dim}{\rm Ker}(I+\alpha_{0}S^{*}GS).$
(3.53)

Since

$\displaystyle{\rm dim}{\rm Ker}(I+\alpha_{0}S^{*}GS)=m-{\rm
rank}(I+\alpha_{0}S^{*}GS),$ $\displaystyle{\rm dim}{\rm
Ker}\left(T\right)=2m-{\rm dim}{\rm Ran}(T^{*})=2m-{\rm rank}(T),$

we have that (3.53) holds if and only if

$m+{\rm rank}(I+\alpha_{0}S^{*}GS)>{\rm rank}(T).$

Hence, $(I_{\mathcal{H}_{0}}+\tilde{M}_{0})$ is invertible on
$\mathcal{H}_{0}$ if and only if

$m+{\rm rank}(I+\alpha_{0}S^{*}GS)\leq{\rm rank}(T).$ (3.54)

Taking into account that

$\displaystyle{\rm rank}(T)$ $\displaystyle\leq{\rm
rank}(I+\alpha_{0}S^{*}GS)$ $\displaystyle\quad+{\rm
rank}([I+\alpha_{0}S^{-1}M_{K}(\alpha_{0})(S^{*})^{-1},-(I+\alpha_{0}S^{*}GS)])$
$\displaystyle\leq{\rm rank}(I+\alpha_{0}S^{*}GS)+m,$

we find that (3.54) holds if and only if $m+{\rm
rank}(I+\alpha_{0}S^{*}GS)={\rm rank}(T)$. By using $SS^{*}=G(\sigma_{0})$, we
have that

$\displaystyle I+\alpha_{0}S^{-1}M_{K}(\alpha_{0})(S^{*})^{-1}$
$\displaystyle=I+\alpha_{0}(S^{*})(S^{*})^{-1}S^{-1}M_{K}(\alpha_{0})(S^{*})^{-1}$
$\displaystyle=S^{*}\left(I+\alpha_{0}G(\sigma_{0})^{-1}M_{K}(\alpha_{0})\right)(S^{*})^{-1},$
$\displaystyle I+\alpha_{0}S^{*}GS$
$\displaystyle=I+\alpha_{0}S^{*}GS(S^{*})(S^{*})^{-1}$
$\displaystyle=S^{*}\left(I+\alpha_{0}GG(\sigma_{0})\right)(S^{*})^{-1}.$

Then

$\displaystyle{\rm rank}(I+\alpha_{0}S^{*}GS)$ $\displaystyle={\rm
rank}(I+\alpha_{0}GG(\sigma_{0})),$ $\displaystyle{\rm rank}(T)$
$\displaystyle={\rm rank}\left(\left[\begin{array}[]{cc}(S^{*})^{-1}&0\\\
0&(S^{*})^{-1}\end{array}\right]T\left[\begin{array}[]{cc}S^{*}&0\\\
0&S^{*}\end{array}\right]\right)$ $\displaystyle={\rm
rank}\left(\left[\begin{array}[]{cc}0&I+\alpha_{0}GG(\sigma_{0})\\\
I+\alpha_{0}GG(\sigma_{0})&-(I+\alpha_{0}G(\sigma_{0})^{-1}M_{K}(\alpha_{0}))\end{array}\right]\right).$

Therefore, we have that $(I_{\mathcal{H}_{0}}+\tilde{M}_{0})$ is invertible on
$\mathcal{H}_{0}$ if and only if (3.17) holds.

∎

##  4 Appendix: proofs of auxiliary lemmas

The following lemma is devoted to the regularity of $\psi(\mu)$ in Section 2.

######  Lemma 4.1.

If $f\in W^{1,p}_{loc}$ for some $p>d$, then $e^{f}\in W^{1,p}_{loc}$ and
$\nabla e^{f}=e^{f}\nabla f$.

###### Proof.

Since for any $\zeta\in C_{0}$, there is $N>0$ such that
$\displaystyle{\mathrm{supp}}\\{\zeta\\}\subset B_{N}$, where $B_{N}$ is the
open ball with radius $N$ and centre at $0$. Then
$e^{f}\zeta=e^{f\zeta_{2N}}\zeta$ and $f\zeta_{2N}\in W^{1,p}$. Hence, we
first assume that $f\in W^{1,p}$. In this case, there is a sequence
$\\{f_{m}\\}\subset C^{\infty}_{0}$ such that

$\displaystyle\lim_{m\rightarrow+\infty}\|f_{m}-f\|_{W^{1,p}}=0.$

Since $p>d$, it follows from the Morrey embedding theorem ([2, Theorem 9.12])
that $W^{1,p}\subset L^{\infty}$ with continuous injection. Then

$\displaystyle\|f\|_{\infty}\vee\sup_{m\geq 1}\|f_{m}\|_{\infty}$
$\displaystyle\leq C\left(\|f\|_{W^{1,p}}\vee\sup_{m\geq
1}\|f_{m}\|_{W^{1,p}}\right)<\infty,$ (4.1)
$\displaystyle\lim_{m\rightarrow+\infty}\|f_{m}-f\|_{\infty}$
$\displaystyle\leq C\lim_{m\rightarrow+\infty}\|f-f_{m}\|_{W^{1,p}}=0.$ (4.2)

By using the following fundamental inequality

$\displaystyle|e^{x}-e^{y}|\leq(|x-y|\wedge 1)e^{x\vee
y},~{}x,y\in\mathbb{R},$ (4.3)

we have that

$\displaystyle\|e^{f_{m}}-e^{f}\|_{L^{p}}+\|e^{f_{m}}\nabla f_{m}-e^{f}\nabla
f\|_{L^{p}}$ $\displaystyle\leq
e^{\|f_{m}\|_{\infty}\vee\|f\|_{\infty}}\|f-f_{m}\|_{L^{p}}+\|(e^{f_{m}}-e^{f})\nabla
f_{m}\|_{L^{p}}+e^{\|f\|_{\infty}}\|\nabla f_{m}-\nabla f\|_{L^{p}}$
$\displaystyle\leq
e^{\|f_{m}\|_{\infty}\vee\|f\|_{\infty}}\left(\|f-f_{m}\|_{L^{p}}+\|f-f_{m}\|_{\infty}\|\nabla
f_{m}\|_{L^{p}}\right)+e^{\|f\|_{\infty}}\|\nabla f_{m}-\nabla f\|_{L^{p}}.$

This, together with (4.1) and (4.2), implies that $e^{f_{m}}$ converges to
$e^{f}$ in $W^{1,p}$ and $\nabla e^{f}=e^{f}\nabla f$. For $f\in
W^{1,p}_{loc}$, $e^{f\zeta_{2N}}\in W^{1,p}$. Then
$e^{f}\zeta=e^{f\zeta_{2N}}\zeta\in W^{1,p}$. Hence, $e^{f}\in W^{1,p}_{loc}$
and $\nabla e^{f}=e^{f}\nabla f$.

∎

The following lemma is devoted to the invariant probability measure of
$L_{\mu}$. It is fundamental and we give the proof for readers’ convenient.

######  Lemma 4.2.

Assume (H). Then for each $\mu\in\mathscr{P}_{W_{0}}$,
$\hat{\mathcal{T}}(x,\mu)\bar{\mu}(\mathrm{d}x)$ is an invariant probability
measure of $L_{\mu}$.

###### Proof.

For every $g\in C_{0}^{\infty}$, due to
$V_{0}\in\mathcal{W}^{1,p}_{q,\bar{\mu}}$ and (2.4),
$\nabla\log(\psi(\mu)e^{-\bar{V}})\in L^{q}(\bar{\mu})\cap L^{p}_{loc}$. Then
$\langle\nabla\log(\psi(\mu)e^{-\bar{V}}),\nabla g\rangle\in
L^{q}(\bar{\mu})\cap L^{p}$. Hence, for all $g\in
C^{\infty}_{0}(\mathbb{R}^{d})$, there is $L_{\mu}g\in L^{1}(\bar{\mu})\cap
L^{1}$. It follows from the integration by part formula that ($\xi_{n}$ is
defined in Lemma 4.1)

$\begin{split}&\left|\int_{\mathbb{R}^{d}}\zeta_{n}(x)(L_{\mu}g)(x)\psi(x,\mu)\bar{\mu}(\mathrm{d}x)\right|\\\
&\qquad=\left|\int_{\mathbb{R}^{d}}\zeta_{n}(x)\mathrm{div}(\psi(x,\mu)e^{-\bar{V}}\nabla
g)(x)e^{\bar{V}(x)}\bar{\mu}(\mathrm{d}x)\right|\\\
&\qquad=\left|\frac{1}{n}\int_{\mathbb{R}^{d}}\chi^{\prime}(|x|/n)\left\langle\frac{x}{|x|},\nabla
g(x)\right\rangle\psi(x,\mu)\bar{\mu}(\mathrm{d}x)\right|\\\
&\qquad\leq\frac{2}{n}\left|\int_{n\leq|x|\leq 2n}|\nabla
g(x)|\psi(x,\mu)\bar{\mu}(\mathrm{d}x)\right|\\\
&\qquad\leq\frac{2\|\psi(\mu)\|_{\infty}}{n}\|\nabla
g\|_{\infty}\left|\int_{n\leq|x|\leq
2n}\bar{\mu}(\mathrm{d}x)\right|,\end{split}$ (4.4)

It follows from the dominated convergence theorem that

$\int_{\mathbb{R}^{d}}(L_{\mu}g)(x)\psi(x,\mu)\bar{\mu}(\mathrm{d}x)=\lim_{n\rightarrow+\infty}\int_{\mathbb{R}^{d}}\zeta_{n}(x)(L_{\mu}g)(x)\psi(x,\mu)\bar{\mu}(\mathrm{d}x)=0.$

∎

Acknowledgements

The author was supported by the National Natural Science Foundation of China
(Grant No. 12371153).

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