# Scattering of kinks in scalar-field models
with higher-order self-interactions

Aliakbar Moradi Marjaneh1, Fabiano C. Simas2,3 and D. Bazeia4
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>1Department
of Physics, Quchan Branch, Islamic Azad University, Quchan, Iran
2Programa de Pós-Graduação em Física, Universidade Federal do Maranhão (UFMA),
Campus Universitário do Bacanga, 65085-580, São Luís, Maranhão, Brazil
3Centro de Ciências de Chapadinha (CCCh), Universidade Federal do Maranhão
(UFMA),
65000-000 Chapadinha, MA, Brazil
4Departamento de Física, Universidade Federal da Paraíba (UFPB), 58051-970
João Pessoa, PB, Brazil

###### Abstract

Higher-order scalar field models in two dimensions, including the $\phi^{8}$
model, have been researched. It has been shown that for some special cases of
the minima positions of the potential, the explicit kink solutions can be
found. However, in physical applications, it is very important to know all the
explicit solutions of a model for any minima position. In the present study,
with the help of some deformation functions, we have shown that higher-order
scalar field theories can be obtained with explicit kinks. In particular, we
introduced two deformation functions that, when applied to the well known
$\phi^{4}$ and $\phi^{6}$ models, produce modified $\phi^{8}$ and $\phi^{10}$
models, respectively, with all their explicit kink-like solutions which depend
on a single parameter. Since this parameter controls the position of the
minima of the potential, we have found interesting new solutions in many
distinct cases. We have also studied the kink mass, the behavior of the
excitation spectra and several kink-antikink collisions for these two new
modified models. The collision outcome is determined by the initial
configuration, specifically the sequence in which the kink-antikink and
antikink-kink pairings emerge. Another interesting finding is the suppression
of resonance windows, which may be explained by the presence of a set of
internal modes in the model.

##  I Introduction

Topological configurations such as stable particle-like objects with smooth
structure and finite mass, may play important role in physics. They are
solutions of field-theoretical models that have been used for at least sixty
years on different scales Ginzburg.ZhETF.1950 ; Bishop.PhysD.1980 ;
Rajaraman.book.1982 ; Coleman.book.1985 ; Vilenkin.book.2000 ;
Manton.book.2004 ; Dauxois.book.2006 ; Vachaspati.book.2010 ;
Kevrekidis.book.2019 ; Optics1 , and are of interest to cosmology, high energy
physics, condensed matter physics and optics. Scientists have always tried to
introduce models that can describe the physical behavior of real examples, and
they usually start with simple models in $(1+1)$ dimensional space-time and
gradually add new degrees of freedom and dimensions to achieve more complex
situations. Among the several possibilities, famous models such as the
integrable sine-Gordon system Rajaraman.book.1982 ; Coleman.book.1985 ;
Vachaspati.book.2010 ; Javidan.cc.2010 ; Moradi.EPJB.2018 ;
Mukhopadhyay.JHEP.2022 and non-integrable models such as $\phi^{4}$
Kevrekidis.book.2019 ; mainak2023 ; Campbell.PhysD.1983 ; anninos.1991 ;
Belova.PhysD.1988 ; Goodman.SIAM_JADS.2005 ; Moradi.CNSNS.2017 ;
Dorey.JHEP.2017 ; Dorey.PLB.2018 ; Weigel.JPCS.2014 ; Weigel.PRD.2016 ;
Askari.CSF.2020 and $\phi^{6}$ Weigel.JPCS.2014 ; Weigel.PRD.2016 ;
Dorey.PRL.2011 ; Gani.PRD.2014 ; Romanczukiewicz.PLB.2017 ; Weigel.PLB.2017 ;
Weigel.AHEP.2017 ; Moradi.JHEP.2017 ; Demirkaya.JHEP.2017 ; Lima.JHEP.2019 ;
Moradi.EPJB.2022 ; Saadatmand.2023 have been investigated in the field of
nonlinear physics. By deforming these models or creating different conditions,
more complex models are obtained, which are usually closer to reality. The
models obtained by changing the sine-Gordon model, such as the double sine-
Gordon model, belong to this category Peyrard.PhysD.1983.msG ;
Campbell.PhysD.1986.dsG ; Bazeia.EPJC.2011 ; Bazeia.EPJC.2013 ; Gani.EPJC.2018
; Belendryasova.JPCS.2019 ; Gani.EPJC.2019 ; riazi . Different models, whether
periodic or polynomial, are obtained by deformation procedure, which have
applications in different fields Bazeia.PRD.2002 ; Bazeia.PRD.2004 ;
Bazeia:2005hu ; Bazeia.PRD.2006 ; Bazeia.EPJC.2018 ; Moradi.CSF.2022 . For
example, black holes, tachyon matter cosmology and quintessential inflation
can be studied with hyperbolic models QiangWen.PRD.2015 ;
Pourhassan.IJMPD.2017 ; Agarwal.PLB.2017 . Although more complex models are
richer in explaining the physical behavior of various examples, it becomes
more difficult to obtain analytical solutions for them. For example, in
solving the nonlinear Klein-Gordon equation for potentials of order eight and
higher, there are analytical solutions just for some cases where the minima of
the potential are located in specific positions; see, e.g., Refs.
Bazeia.PRD.2006 ; Gani.PRD.2020 ; Khare.2021 .

Kink scattering in many models has revealed the complexities of the outcomes.
In particular, in Ref. Campbell.PhysD.1983 the study shows the kink-antikink
scattering in the $\phi^{4}$ model. The explanation for the appearance of the
two-bounce windows is related to the presence of the shape mode. In another
important work Dorey.PRL.2011 , the formation of resonant windows was
analyzed, however, the explanation was based on the presence of vibrational
states due to a perturbation potential for the antikink-kink pair. Another
counterexample to the mechanism of energy exchange between translational and
vibrational modes can be seen in Ref. simas.2016 . Notably, two-bounce windows
are suppressed even when internal modes are present. Other works that address
the collision of kinks and depict various types of outcomes can also be cited.
For instance, the scattering between wobbling kinks alonso.2021 ; joao.2021 ,
investigation of models with more scalar fields alonso.2020 ; alonso.2018 ;
halava.2012 and a kink-antikink scattering scenario with an additional scalar
field located in its quantum vacuum vacha.2023 . In addition, it is worth
noting that kink scattering may appear under the presence of a spectral wall,
giving rise to interesting new results adam.2019 ; joao.2023 ; adam.2022 ;
adam.2021 . Moreover, a recent investigation has examined how a deformation
function in scalar field models can change the solution, mass, internal
structure and collision process of kinks, also leading to new results of
current interest Moradi.CSF.2022 .

Motivated by the above new results, in this work we study properties of two
polynomial potentials, the deformed $\phi^{8}$ and $\phi^{10}$ models,
described by deformations of the standard $\phi^{4}$ and $\phi^{6}$ models. To
do this, we organize the paper as follows. In section II, we review the
general statement of the deformation procedure. Then, in the next two sections
III and IV, we define two deformation functions dependent on real parameter
$a$, and introduce modified deformed $\phi^{8}$ and $\phi^{10}$ models. There,
we also explore the scattering of kinks. The impact of the internal modes and
the distinctions between kink-antikink and antikink-kink collisions are also
studied in these two Sections. We also investigate the explicit kinks, their
masses, the stability potentials and the values of the internal modes. These
two new deformed models, are obtained via a deformation function that depends
on a single real parameter, which is used to control the positions of the
minima of potentials and can affect the main features of the models and guide
us to investigate the two models. The work is closed in Section V, where we
add our conclusions and comment on issues concerning future directions of new
research on the subject.

##  II Deformation procedure

Let us start reviewing the methodology related to the deformation procedure.
The dynamics of the real scalar field for a field theoretical model in $(1+1)$
space-time dimensions is described by the Lagrangian density

$\displaystyle\mathcal{L}=\frac{1}{2}\left(\frac{\partial\phi}{\partial
t}\right)^{2}-\frac{1}{2}\left(\frac{\partial\phi}{\partial
x}\right)^{2}-V(\phi).$ (1)

It yields the Klein-Gordon equation of motion

$\frac{\partial^{2}\phi}{\partial t^{2}}-\frac{\partial^{2}\phi}{\partial
x^{2}}+\frac{dV(\phi)}{d\phi}=0.$ (2)

In the static case, $\phi=\phi(x)$, this equation can be reduced to the
following first-order equations

$\frac{d\phi}{dx}=\pm\sqrt{2V(\phi)}.$ (3)

We suppose we can solve these equations to find static solutions analytically.
In this case, by applying a deformation function, $f(\phi)$, on the potential
${V}(\phi)$ Bazeia.PRD.2002 ; Bazeia.PRD.2004 ; Bazeia.PRD.2006 ;
Bazeia:2005hu ; Bazeia.EPJC.2018 , a new deformed model with the deformed
potential, $\tilde{V}(\phi)$, is formed

$\tilde{V}(\phi)=\frac{V[\phi\to f(\phi)]}{[f^{\prime}(\phi)]^{2}},$ (4)

such that, in the new model the new static kinks are given by

$\tilde{\phi}_{K}(x)=f^{-1}[\phi_{K}(x)].$ (5)

For solutions that solve the above first-order equations, we can obtain the
kink energy or kink mass as follows

$\displaystyle mass$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{+\infty}\bigg{(}\frac{\partial\phi}{\partial
x}\bigg{)}^{2}dx.$ (6)

Also, after adding a small perturbation to the static kink, in the form
$\phi(x,t)=\phi_{K}(x)+\eta(x)\cos(\omega t)$, one can find the Schrödinger-
like equation which allows studying linear stability and kink excitation
spectra

$-\frac{d^{2}\eta}{dx^{2}}+U\eta=\omega^{2}\eta,$ (7)

where $U=U(x)$ is the kink stability potential, which is obtained from the
second derivative of the potential $V(\phi)$ (or $\tilde{V}(\phi)$ for
modified model), with respect to the kink $\phi_{K}(x)$ (or
$\tilde{\phi}_{K}(x)$)

$\displaystyle
U(x)=\frac{d^{2}V(\phi)}{d\phi^{2}}\bigg{|}_{\phi=\phi_{K}(x)}.$ (8)

The above Eq. (7) and the corresponding stability potential are important for
the scattering process, since they provide the number of internal modes, which
directly influence the output of the collision.

With the above methodology, we can now discuss the modified $\phi^{8}$ and
$\phi^{10}$ models, as well as their respective collision results.

##  III modified $\phi^{8}$ model

First, we introduce the deformation function $f_{1}[\phi]$, see Fig. 1(a)

$f_{1}[\phi]=\pm\tanh\left(\frac{a}{a-1}\left(\tanh^{-1}(\phi)-\tanh^{-1}(a\phi)\right)-c_{1}\right).$
(9)

where $a$ is a real parameter and $c_{1}$ is an arbitrary constant. Now, by
using of Eq. (4), this deformation function can change the $\phi^{4}$ model
with the potential $V^{(4)}=\frac{1}{2}(1-\phi^{2})^{2}$ to the following
modified $\phi^{8}$ model

$\displaystyle\tilde{V}^{(8)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\frac{\left(1-\phi^{2}\right)^{2}\left(\frac{1}{a^{2}}-\phi^{2}\right)^{2}}{\left(\frac{1}{a}+\phi^{2}\right)^{2}}.$
(10)

This potential has four minima at $\phi=\pm 1,\pm\frac{1}{a}$. We consider the
parameter equal or greater than one, $a\geq 1$, without changing the
generality of the problem. The potential profiles for $a=2$ and $a=3$ are
shown in Fig. 1(b). We notice that the point of maximum potential in the
central region decreases with increasing $a$. From Eq. (5), one can find all
explicit kinks solution of the model

$\displaystyle\tilde{\phi}^{(8)}\to\begin{cases}\tilde{\phi}^{(8)}_{AS}(\frac{1}{a},1)=\frac{(1-a)}{2a}\tanh\left(\frac{(1-a)}{a}x\right)+\sqrt{\left(\frac{1-a}{2a}\right)^{2}\tanh^{2}\left(\frac{(1-a)}{a}x\right)+\frac{1}{a}},\\\
\tilde{\phi}^{(8)}_{S}(-\frac{1}{a},\frac{1}{a})=\frac{1}{2\left(e^{\frac{(2(a-1))x}{a}}-1\right)}\Bigg{(}\left(\frac{1}{a}-1\right)\left(e^{\frac{(2(a-1))x}{a}}+1\right)\\\
\qquad\qquad\qquad+\sqrt{\frac{4\left(e^{\frac{(2(a-1))x}{a}}-1\right)^{2}}{a}+\left(\left(1-\frac{1}{a}\right)\left(e^{\frac{(2(a-1))x}{a}}+1\right)\right)^{2}}\Bigg{)},\\\
\tilde{\phi}^{(8)}_{AS}(-1,-\frac{1}{a})=\frac{(1-a)}{2a}\tanh\left(\frac{(1-a)}{a}x\right)-\sqrt{\left(\frac{1-a}{2a}\right)^{2}\tanh^{2}\left(\frac{(1-a)}{a}x\right)+\frac{1}{a}}.\end{cases}$
(11)

These kinks include a symmetric kink, $\tilde{\phi}^{(8)}_{S}$, which is in
between the minima $-\frac{1}{a}$ and $\frac{1}{a}$, and two similar
asymmetric kinks, $\tilde{\phi}^{(8)}_{AS}$, which are located in the sectors
$(-1,-\frac{1}{a})$ and $(\frac{1}{a},1)$. As a result of this asymmetry, the
behavior of the collision will depend on the initial configuration, since the
kink-antikink case differs from the antikink-kink one, yielding different
effects. The symmetric and asymmetric solutions are depicted for $a=2$ and
$a=3$ in Figs. 1(c) and 1(d), respectively.

(a) Deformation function (plus sign, $c_{1}=0$)

(b) Potentials

(c) Modified $\phi^{8}$ kinks for $a=2$

(d) Modified $\phi^{8}$ kinks for $a=3$

Figure 1: (a) The deformation function $f_{1}[\phi]$ as a function of $\phi$,
(b) modified $\phi^{8}$ potential and modified $\phi^{8}$ kinks for (c) $a=2$
and (d) $a=3$. The $\phi^{4}$ model is represented by the dashed red line.

In the case of $a=1$, the expressions for the potential and kink solution are
given by

$\displaystyle\begin{cases}\tilde{V}^{(8)}(a=1)=\frac{1}{2}\frac{\left(\phi^{2}-1\right)^{4}}{\left(\phi^{2}+1\right)^{2}},\\\
\tilde{\phi}^{(8)}_{S}(-1,1)=\frac{\sqrt{4x^{2}+1}-1}{2x}.\end{cases}$ (12)

(a) Potential

(b) Kink

Figure 2: Deformed $\phi^{8}$ potential and kink in the case of $a=1$ compared
well known $\phi^{4}$ model.

In Fig. 2 one depicts the potential and kink solution for $a=1$. It should be
emphasized that the potential contains only two minimum at $\pm 1$.
Furthermore, only symmetric kink remain after the loss of asymmetric kinks. It
is interesting to see that the minima are flatter. This behavior results in
defects with long-range tails. In Refs. ivan ; diojoaza , the authors studied
in detail the scattering of kinks with long- and short-range tails. In Ref.
ivan , the results revealed the existence of resonant windows due to the
presence of the bound state. In Ref. diojoaza , the output showed the decay of
the pair in radiation, favoring the appearance of kink-antikink pairs. A novel
technique is required for the development of collisions with long-range tails,
as described in Refs. chidede ; joazalong . We shall not handle kink-antikink
collisions for the scenario $a=1$. This work proposes to investigate
scattering in the presence of both symmetric and asymmetric kinks.

When $a\to\infty$, the modified $\phi^{8}$ potential (Eq. (10)) tends to well
known $\phi^{4}$ potential, with the symmetric kink solution
$\phi^{(4)}=\tanh(x)$, the mass of ${4}/{3}$ and the only shape mode with
frequency $\omega_{1}^{2}=3$. In this case, the potential has just two minima,
and the asymmetric kinks disappear.

Different values of parameter $a$ provide the examination of the changes of
the kinks relative to each other from the point of view of mass, internal mode
and stability potential, as well as the examination of different scatterings.
In this general case, the kink masses are obtained in terms of parameter $a$.
They are

$\displaystyle mass$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{+\infty}\bigg{(}\frac{\partial\phi}{\partial
x}\bigg{)}^{2}dx\to\begin{cases}\tilde{M}^{(8)}_{S}=\frac{6\sqrt{a}(a+1)^{2}\tan^{-1}\left(\frac{a}{\sqrt{a^{3}}}\right)-6a(a+1)-4}{3a^{3}},\\\
\tilde{M}^{(8)}_{AS}=-\frac{(a+1)^{2}\tan^{-1}\left(\frac{a-1}{2\sqrt{a}}\right)}{a^{5/2}}-\frac{2}{3a^{3}}+\frac{2}{3},\end{cases}$
(13)

where $\tilde{M}^{(8)}_{S}$ is the mass of symmetric kink in sector
$(-\frac{1}{a},\frac{1}{a})$ and $\tilde{M}^{(8)}_{AS}$ is the mass of
asymmetric kinks in the sectors $(-1,-\frac{1}{a})$ and $(\frac{1}{a},1)$. As
Fig. 3(a) shows, at $a=3.243327$ all kinks of this model have the same mass,
$\tilde{M}=0.117684$, and for values greater than this value of $a$, the mass
of asymmetric kinks is greater than the mass of symmetric kinks. In fact, the
parameter $a$ controls the mass ratio of symmetric and asymmetric kinks, which
provides different cases in the scattering of kinks. In the limit of $a\to 1$,
the asymmetric kink mass tends to zero. In the case of $a=1$, only the
(symmetric) kink in the model has a mass of
$\tilde{M}_{S}^{(8)}=\frac{2}{3}(3\pi-8)$.

(a)

(b)

(c)

(d)

Figure 3: Deformed $\phi^{8}$ model: (a) kink mass, (b) and (c) quantum
mechanical potential for symmetric and asymmetric kinks, respectively, as a
function of $x$ and (d) the squared frequencies $\omega^{2}_{1}$ of the
vibrational states as a function of the parameter $a$.

Apart from the mass, another quantity that is affected by the parameter $a$ is
the stability potential. Considering small fluctuations around the static
solution and substituting in the equation of motion, we arrive at the
Schrödinger-like equation with the following effective potential

$\displaystyle\tilde{U}^{(8)}=\frac{d^{2}\tilde{V}^{(8)}(\phi)}{d\phi^{2}}\bigg{|}_{\phi=\tilde{\phi}^{(8)}(x)}.$
(14)

Figs. 3(b) and 3(c) show the stability potential for different values of $a$
for symmetric and asymmetric kinks, respectively. The value of the potential
at the limits of $x\to\pm\infty$ in both cases is $\frac{4(a-1)^{2}}{a^{2}}$.
The Fig. 3(b) depicts a volcano-shape Schrödinger-like potential with a deep
minimum for small $a$ values. In particular, the potential with this behavior
allows the formation of resonances, making it difficult for internal modes to
be present. On the other hand, an increase in the parameter causes a change in
the shape of the potential. We can observe an increase in the asymptotic
maximum and a reduction in the potential minimum. Now, the linear perturbation
analysis for the asymmetric kink is not symmetrical relative to the reflection
$x\to-x$, see Fig. 3(c). It is worth noting that increasing $a$ promotes the
development of a maximum point in the potential as well as a rise in depth,
permitting the presence of vibrational states.

From translational invariance, we may infer the existence of the zero mode
($\omega_{0}=0$), which corresponds to the derivative of the solution, as
follows $\eta_{0}(x)={d\phi}/{dx}$. We numerically investigated the occurrence
of internal modes for both potentials. The Fig. 3(d) depicts the spectrum for
some $a$ values. We observe the occurrence of vibrational modes in asymmetric
kink, as indicated by the red line. Moreover, the dashed black line represents
the beginning of the continuous spectrum. It is worth noting that when $a$
increases, the frequency squared approaches the continuous mode. However, for
the symmetric case, no bound states are observed for small values of $a$. Only
values greater than $a\sim 4.6$ exhibit these states. In addition to the
analysis of the modes for an individual kink or antikink, we carried out the
investigation for the collective kink-antikink and antikink-kink pairs for the
asymmetric case. However, we discovered no additional states in any scenario.

In the following, we will discuss the kink-antikink and antikink-kink
scattering process of the asymmetric and symmetric solution. Due to the non
linearity of the model and solution behavior, the results are particularly
complex. In addition, the modification of the parameter $a$ and the initial
velocity produce intriguing effects. In this sense, we solved the equation of
motion with $4^{th}$ order finite-difference method with a spatial step
$\delta x=0.05$. For the time dependence we used a $6^{th}$ order symplectic
integrator method with a time step $\delta t=0.02$. We fixed $x_{0}=\pm 10$
for the initial position of the pair.

###  III.1 Asymmetric kink scattering

In this section, we initially investigate the kink-antikink collision process
of the solutions obtained from the deformation of the $\phi^{4}$ model. We
first investigate the asymmetric kink collision
$\tilde{\phi}^{(8)}_{AS}(\frac{1}{a},1)$. For numerical solutions, we used the
following initial conditions

$\displaystyle\tilde{\phi}(x,x_{0},v,0)$ $\displaystyle=$
$\displaystyle\tilde{\phi}_{AS}^{(8)}(x+x_{0},v,0)-\tilde{\phi}_{AS}^{(8)}(x-x_{0},-v,0)-1,$
(15) $\displaystyle\dot{\tilde{\phi}}(x,x_{0},v,0)$ $\displaystyle=$
$\displaystyle\dot{\tilde{\phi}}_{AS}^{(8)}(x+x_{0},v,0)-\dot{\tilde{\phi}}^{(8)}_{AS}(x-x_{0},-v,0).$
(16)

The time evolution in included with a boost for the static solutions, with
$\gamma=(1-v^{2})^{-1/2}$ and $v$ standing for the velocity.

(a)

(b)

Figure 4: Kink-antikink - Evolution of scalar field
$\tilde{\phi}^{(8)}_{AS}(\frac{1}{a},1)$ at the center of mass as a function
of time and initial velocity $v$ for (a) $a=2$ and (b) $a=3$.

The structure of scattering for some values of $a$ is depicted in Fig. 4.
Figures 4(a) and 4(b) correspond to $a=2$ and $a=3$, respectively. It is worth
noting that both have a single vibrational mode. For $a=2$, we observe the
formation of two-bounce windows. This figure shows the evolution of scalar
field at the center of mass as a function of time and initial velocity. The
white lines correspond to the interactions between the kinks. Resonance
windows are visible when we have only two white lines horizontally followed by
a vertical blue range. When the second horizontal line diverges, the critical
velocity is reached. For $a=3$, however, we do not notice the presence of the
resonant windows. In this case, for $v<v_{c}\approx 0.1203$, only bion states
are achieved and for $v>v_{c}$, the output corresponds to inelastic scattering
between the pair. Some kink-antikink scattering results for the case where
$a=2$ are shown in Fig. 5. We can observe the two-bounce behavior for the
first $(v=0.1930)$ and second $(v=0.2240)$ resonance window, respectively, in
Figs. 5(a) and 5(b). Additionally, we observe the development of the bion,
where the field oscillates and emits radiation (see Fig. 5(c)), as well as the
emergence of an inelastic scattering, as we can see in Fig. 5(d).

(a)

(b)

(c)

(d)

Figure 5: Kink-antikink - Evolution of scalar field
$\tilde{\phi}^{(8)}_{AS}(\frac{1}{a},1)$ in spacetime for $a=2$ with (a)
$v=0.1930$, (b) $v=0.2240$, (c) $v=0.1660$ and (d) $v=0.2598$.

An unusual pattern appears as the value of $a$ increases. The resonant windows
are completely suppressed even when the internal mode is present, and the
kink-antikink scattering results in an antikink-kink pair. In this sense, our
numerical results for some values of $a$ are depicted in Fig. 6. We see that
when $a=4$, the kink-antikink pair approaches, collides once and then
separates on a new vacuum. We can realize the result of the asymmetric kink-
antikink collision produces an antikink-kink pair that visits the vacuum of
the symmetric kink. These behaviors are related to the massive character of
the solutions. In the small $a$ region, the asymmetric kink is lighter than
the symmetric kink, which makes it impossible to change sectors. However, as
the parameter increases, the asymmetric kink becomes more massive, so sector
switching becomes possible diojoaza .

Notice that the increase in $a$ causes the formation of new antikink-kink
pairs. Consequently, the parameter $a$ has a bigger influence on the
development of new pairings, compared to the initial velocity. In particular,
compare Fig. 6(b) and Fig. 6(c). Although the latter has a small velocity, the
outcome reveals the development of two pairs. The generation of new pairs was
also seen in Ref. simas.2020 , where large kink scattering produces new pairs.
The presence of simply the zero mode, followed by the continuum mode, is
reported by the authors. On the other side, the model considered in this
paper, has a zero mode and an internal mode for the asymmetric kink.
Additionally, we further observe the appearance of an oscillating pulse after
the collision at $x=0$ \- see Fig. 6(d). The presence of this oscillation
causes the development of a new pair as a result of the disturbance it causes.
This phenomenon of oscillations at the collision center causing the
development of new kink pairs was also described in Ref. roman.2010 .

(a)

(b)

(c)

(d)

Figure 6: Kink-antikink - Evolution of scalar field
$\tilde{\phi}^{(8)}_{AS}(\frac{1}{a},1)$ in spacetime for (a) $a=4$ with
$v=0.20$, (b) $a=5$ with $v=0.50$, (c) $a=6$ with $v=0.18$ and (d) $a=7$ with
$v=0.30$.

In the following, we shall now consider antikink-kink scattering, due of the
asymmetric nature of the solution $\tilde{\phi}^{(8)}_{AS}(\frac{1}{a},1)$. We
used the following initial conditions

$\displaystyle\tilde{\phi}(x,x_{0},v,0)$ $\displaystyle=$
$\displaystyle\tilde{\phi}_{AS}^{(8)}(x-x_{0},-v,0)-\tilde{\phi}_{AS}^{(8)}(x+x_{0},v,0)-\frac{1}{a},$
(17) $\displaystyle\dot{\tilde{\phi}}(x,x_{0},v,0)$ $\displaystyle=$
$\displaystyle\dot{\tilde{\phi}}_{AS}^{(8)}(x-x_{0},-v,0)-\dot{\tilde{\phi}}^{(8)}_{AS}(x+x_{0},v,0),$
(18)

where $\tilde{\phi}(x,x_{0},v,t)=\tilde{\phi}(\gamma(x-vt))$ means a boost for
the static solution with $\gamma=(1-v^{2})^{-1/2}$.

We present a summary of the antikink-kink interaction findings for certain
values of $a$ in Fig. 7. In this figure we show the evolution of the scalar
field as a function of initial velocity and time. The collisions are
illustrated by the horizontal blue lines and the two-bounce windows are
distinguished by their diverging bands. For small values of $a$, we can
observe the presence of resonant windows. However, as the value of this
parameter increases, the number of windows decreases until they are completely
suppressed - see Fig. 7(a)-7(d). Additionally, the disappearance of the first
two-bounce windows leads to the emergence of false resonance windows. The peak
at $v\approx 0.28$ in Fig. 7(b) represents this false window behavior. In this
case, the antikink-kink pair approaches, collides twice and separates.
However, after one period, it collides again and forms an oscillatory state.
The appearance of false two-bounce windows was reported in Ref. Dorey.PRL.2011
. Furthermore, increasing this parameter contributes to an increase in the
critical velocity, which leads to a larger region with bion states.

(a)

(b)

(c)

(d)

Figure 7: Antikink-kink - Evolution of scalar field
$\tilde{\phi}^{(8)}_{AS}(\frac{1}{a},1)$ at the center of mass as a function
of time and initial velocity for (a) $a=2$, (b) $a=3$, (c) $a=4$ and (d)
$a=10$.

Unlike the asymmetric kink-antikink collision, the antikink-kink configuration
does not allow the pair to visit the symmetric vacuum, as we can seen in the
Fig. 8(a) and Fig. 8(b). We show the occurrence of two-bounce for $a=2$ and
$a=3$, respectively and the one-bounce behavior for $a=10$ (Fig. 8(c)).
Furthermore, we report the production of oscillating pulses for $a=10$ \- see
Fig. 8(d). Similar solutions were discovered in Ref. roman.2010 . As a result
of the collision of the antikink-kink pair, we observe two waves that scatter
almost harmonically.

The reason why kink-antikink and antikink-kink scattering yield different
results has to do with how the solutions are placed on the line that defines
the initial configuration. For instance, in the kink-antikink collision, the
region connecting the kink to the antikink does not vary with the change in
$a$. However, in the antikink-kink case, changes in $a$ alter the tails of the
antikink and kink. Moreover, the suppression of the resonance windows for the
collision of the asymmetric solution can be explained by the presence of the
set of internal modes of the model. For small values of $a$, the asymmetric
kink has just one internal mode. However, when the value increases, it favors
the occurrence of a vibrational state for the symmetric kink. As a result, the
internal mode of the symmetric solution influences the vibrational state of
the asymmetric kink. Consequently, the mechanism for exchanging resonant
energy between translational and vibrational modes become frustrated.

(a)

(b)

(c)

(d)

Figure 8: Antikink-kink - Evolution of scalar field
$\tilde{\phi}^{(8)}_{AS}(\frac{1}{a},1)$ in spacetime for (a) $a=2$ with
$v=0.25$, (b) $a=3$ with $v=0.3320$ and $a=10$ with (c) $v=0.70$ and (d)
$v=0.64$.

###  III.2 Symmetric kink scattering

Now, we investigate the kink-antikink collision process of the symmetric
solutions $\tilde{\phi}^{(8)}_{S}(-\frac{1}{a},\frac{1}{a})$. The kink-
antikink and antikink-kink scattering both generate the same outcomes. For
this reason, we shall exclusively study the kink-antikink collision. We used
the following initial conditions

$\displaystyle\tilde{\phi}(x,x_{0},v,0)$ $\displaystyle=$
$\displaystyle\tilde{\phi}_{S}^{(8)}(x+x_{0},v,0)-\tilde{\phi}_{S}^{(8)}(x-x_{0},-v,0)-\frac{1}{a},$
(19) $\displaystyle\dot{\tilde{\phi}}(x,x_{0},v,0)$ $\displaystyle=$
$\displaystyle\dot{\tilde{\phi}}_{S}^{(8)}(x+x_{0},v,0)-\dot{\tilde{\phi}}^{(8)}_{S}(x-x_{0},-v,0),$
(20)

where $\tilde{\phi}(x,x_{0},v,t)=\tilde{\phi}(\gamma(x-vt))$ means a boost for
the static solution with $\gamma=(1-v^{2})^{-1/2}$. In the first moment, we
examine the scattering by altering the parameter $a$ while keeping the initial
velocity constant. Some of these examples are illustrated in Fig. 9. For
$a=2$, we see the production of two antikink-kink pairs after collision.
Importantly, the scattered pairs are asymmetric kinks in the topological
sector $(-\frac{1}{a},-1)$. In addition, the increase in $a$ results in the
development of a single antikink-kink pair, as reported in Fig. 9(b) for
$a=3$. It is worth noting that in this region, the symmetrical kink is more
massive and becomes lighter as $a$ increases. There are no two-bounce windows
or bion development in this region of $a$ values. On the other hand, larger
values of $a$ modify this pattern and reveal the presence of a bion at the
center of the collision, as seen in Fig. 9(c) for $a=5$. The emergence of the
vibrational state for symmetric kink is associated with the alteration in
behavior.

(a)

(b)

(c)

Figure 9: Kink-antikink - Evolution of scalar field
$\tilde{\phi}^{(8)}_{S}(-\frac{1}{a},\frac{1}{a})$ in spacetime for (a) $a=2$,
(b) $a=3$ and (c) $a=5$ with $v=0.20$.

In order to better understand the impact of the parameter $a$, we developed a
outline for identifying the formation of the two-bounce windows by looking at
the number of bounces as a function of the initial velocity. This resonant
structure, as we know, may be constructed by counting the bounces of the
scalar field at the center of mass $\phi(0,t)$. We only include the structure
corresponding to one and two collisions in Fig. 10. The absence of any regions
suggests either the production bion or higher order collisions. In regions
with high velocities, only inelastic collisions are observed $(Nb=1)$.
Furthermore, for intermediate velocities, peaks can be seen in the pictures
depicting the region corresponding to two collisions $(Nb=2)$. In our
investigation, we see that when the parameter increases, the number of two-
bounce windows increases. Specifically, with $a=5$, we observe the formation
of only a few windows. However, for $a=12$, the number of windows is higher,
and these become wider. Clearly, the model contains two vibrational modes for
both symmetric and asymmetric kink at values around $a\sim 4.6$. The presence
of this set of internal modes in the model difficulties the energy exchange
mechanism during the collision. Consequently, creating resonant windows is
more challenging. On the other hand, increasing $a$ shows that
$\omega^{2}_{1}$, for the asymmetric solution, tends to approach the
continuous mode even more. Thus, for larger values of the parameter, there is
a greater contribution from the vibrational mode of the symmetric kink.
Therefore, the energy exchange mechanism between the translational and
vibrational modes is more effective, and there is an increase in the number of
two-bounce windows. In addition, with $a\rightarrow\infty$, the deformed
$\phi^{8}$ tends towards the $\phi^{4}$ model. One effect of this outcome is
decrease in critical velocity. The critical velocity approaches $v_{c}\approx
0.2598$ as $a$ increases.

(a)

(b)

(c)

Figure 10: Kink-antikink - Number of bounces versus initial velocity of scalar
field $\tilde{\phi}^{(8)}_{S}(-\frac{1}{a},\frac{1}{a})$ for (a) $a=6$, (b)
$a=7$, (c) $a=10$ and (d) $a=12$.

##  IV modified $\phi^{10}$ model

Second, we introduce the deformation function $f_{2}[\phi]$, as we can see in
Fig. 11(a)

$f_{2}[\phi]=\frac{1-(1+e^{2a})\phi^{2}}{1-(1-e^{2a})\phi^{2}}.$ (21)

Here, $a$ is a real parameter. By using of Eq. (4), this deformation function
can change the $\phi^{4}$ model with the potential
$V^{(4)}=\frac{1}{2}(1-\phi^{2})^{2}$ to the $\phi^{6}$ model with the
potential $V^{(6)}=\frac{1}{2}\phi^{2}(1-\phi^{2})^{2}$. We can use again
$f_{2}[\phi]$ in $\phi^{6}$ model and find the following modified $\phi^{10}$
model,

$\displaystyle\tilde{V}^{(10)}$ $\displaystyle=$
$\displaystyle\frac{\phi^{2}\left(\phi^{2}-1\right)^{2}\left(\left(e^{2a}+1\right)\phi^{2}-1\right)^{2}}{2\left(\left(e^{2a}-1\right)\phi^{2}+1\right)^{2}}.$
(22)

This potential has five minima, at $0,\pm\frac{1}{\sqrt{1+e^{2a}}},$ and $\pm
1$. The behavior of the potential is shown in Fig. 11(b). We consider positive
values for the parameter $a$.

For this model, the kink solutions are written as

$\displaystyle\tilde{\phi}^{(10)}$ $\displaystyle=$
$\displaystyle\begin{cases}\tilde{\phi}_{(0,\frac{1}{\sqrt{e^{2a}+1}})}^{(10)}=\sqrt{\frac{2e^{2x}}{2\left(e^{2a}+1\right)e^{2x}+\sqrt{4e^{2(a+x)}+1}+1}},\\\
\tilde{\phi}_{(\frac{1}{\sqrt{e^{2a}+1}},1)}^{(10)}=\frac{1}{\sqrt{2}}\sqrt{\frac{\sqrt{4e^{2(a-x)}+1}+2e^{2(a-x)}+2e^{-2x}+1}{2e^{2(a-x)}+e^{4a-2x}+e^{-2x}+1}}.\end{cases}$
(23)

In Figs. 11(c) and 11(d) one shows these kinks for two different values of
parameter $a$. The configuration of the solution reveals a kink that varies
between the minima $\frac{1}{\sqrt{e^{2a}+1}}$ and $1$, which we refer to as
up kink. In addition to this, we can also see another kink between $0$ and
$\frac{1}{\sqrt{e^{2a}+1}}$, which will be referred to here as down kink. The
behavior of collision will also depend on the initial configuration,
therefore, kink-antikink scattering differs from antikink-kink collision.

In particular, at $a\to\infty$, the modified $\phi^{10}$ potential tends to
well known $\phi^{6}$ potential, $V^{(6)}=\frac{1}{2}\phi^{2}(1-\phi^{2})^{2}$
with the asymmetric kink solution $\phi^{(6)}=\sqrt{\frac{1+\tanh(x)}{2}}$,
the mass of $\frac{1}{4}$ and without any shape mode for a single kink or
antikink Dorey.PRL.2011 . As the parameter $a$ increases, only up kink
$\phi_{(\frac{1}{\sqrt{e^{2a}+1}},1)}^{(10)}$ remains while down kink
$\phi_{(0,\frac{1}{\sqrt{e^{2a}+1}})}^{(10)}$ is eliminated.

(a) Deformation function

(b) Potentials

(c) Modified $\phi^{10}$ kinks for $a=0.5$

(d) Modified $\phi^{10}$ kinks for $a=1.0$

Figure 11: (a) The deformation function $f_{2}[\phi]$ as a function of $\phi$,
(b) modified $\phi^{10}$ potential and modified $\phi^{10}$ kinks for (c)
$a=0.5$ and (d) $a=1.0$. The $\phi^{6}$ model is represented by the dashed red
line.

The mass of these kinks are as the following, which is plotted in Fig. 12(a).

$\displaystyle mass$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{+\infty}(\frac{\partial\phi}{\partial x})^{2}dx$
(24) $\displaystyle=$
$\displaystyle\begin{cases}\tilde{M}_{(0,\frac{1}{\sqrt{e^{2a}+1}})}^{(10)}=\frac{-4e^{2a}+e^{4a}+2e^{6a}-4e^{4a}\left(e^{2a}+1\right)\log\left(\frac{2e^{2a}}{e^{2a}+1}\right)+1}{4\left(\left(1-e^{2a}\right)^{3}\left(e^{2a}+1\right)\right)},\\\
\tilde{M}_{(\frac{1}{\sqrt{e^{2a}+1}},1)}^{(10)}=\frac{1}{16}e^{a}\text{csch}^{2}(a)\left(\sinh(a)\\!+\\!\cosh(a)\\!+\\!\text{sech}(a)\\!+\\!2\text{csch}(a)\ln\big{(}\frac{\tanh(a)+1}{e^{2a}}\big{)}\right)\;\;\end{cases}$

We can check that at $a\approx 0.471354$, modified $\phi^{10}$ kinks have the
same mass, $mass\approx 0.0562126$.

(a)

(b)

(c)

(d)

Figure 12: Deformed $\phi^{10}$ model: (a) kink mass, (b) and (c) quantum
mechanical potential for up and down kinks as a function of $x$, respectively,
and (d) the squared frequencies $\omega^{2}_{1}$ of the vibrational state as
function of parameter $a$ for down kink.

Finally, similar to the previous section, we deal with the stability potential
and we arrive at the Schrödinger-like equation

$\displaystyle-\eta_{xx}+\tilde{U}^{(10)}=\omega^{2}\eta$ (25)

where the effective potential is given by

$\displaystyle\tilde{U}^{(10)}=\frac{d^{2}\tilde{V}^{(10)}(\phi)}{d\phi^{2}}|_{\phi=\tilde{\phi}^{(10)}(x)}.$
(26)

These potentials are plotted in Figs. 12(b) and 12(c) for up and down kink,
respectively. It is important to note that the potential $\tilde{U}^{(10)}$ is
not symmetrical with respect to the reflection $x\to-x$. The potential
$\tilde{U}_{up}^{(10)}$ for the kink is similar to that obtained in Ref.
Dorey.PRL.2011 for the $\phi^{6}$ model for small values of $a$. The increase
in the parameter shows the appearance of a maximum in the potential. The
numerical analysis of this potential only favors the appearance of the zero
mode for the individual kink or antikink. However, while examining the
collective antikink-kink pair, some internal modes can be found. The potential
for the down kink $\tilde{U}_{down}^{(10)}$ is depicted in Fig. 12(c). Note
the presence of equal asymptotic limits and a minimum that moves away from the
origin with increasing $a$. Here, we also investigated the occurrence of bound
states. The results are depicted in Fig. 12(d) and show that the value of
$\omega^{2}_{1}$ approaches to the continuous mode as $a$ decreases.

Now let us investigate the kink scattering process for the $\phi^{10}$
modified model. We obtain several outcomes by varying the parameter $a$ and
the initial velocity. In this sense, we solved the equation of motion with
$4^{th}$ order finite-difference method with a spatial step $\delta x=0.05$.
For the time dependence we used a $6^{th}$ order symplectic integrator method
with a time step $\delta t=0.02$. We fixed $x_{0}=\pm 10$ for the initial
position of the pair.

###  IV.1 Down kink scattering

We will discuss here the kink-antikink collision of the down solution
$\tilde{\phi}_{(0,\frac{1}{\sqrt{e^{2a}+1}})}^{(10)}$. For the sake of
simplicity, the kink is expressed as $\tilde{\phi}_{d}^{K}$ and the antikink
as $\tilde{\phi}_{d}^{\bar{K}}$. For numerical solutions, we used the
following initial conditions

$\displaystyle\tilde{\phi}(x,x_{0},v,0)$ $\displaystyle=$
$\displaystyle\tilde{\phi}_{d}^{K}(x+x_{0},v,0)+\tilde{\phi}_{d}^{\bar{K}}(x-x_{0},-v,0)-\phi_{\mu},$
(27) $\displaystyle\dot{\tilde{\phi}}(x,x_{0},v,0)$ $\displaystyle=$
$\displaystyle\dot{\tilde{\phi}}_{d}^{K}(x+x_{0},v,0)+\dot{\tilde{\phi}}^{\bar{K}}_{d}(x-x_{0},-v,0),$
(28)

where $\tilde{\phi}(x,x_{0},v,t)=\tilde{\phi}(\gamma(x-vt))$ means a boost for
the static solution with $\gamma=(1-v^{2})^{-1/2}$ and
$\phi_{\mu}=\frac{1}{\sqrt{e^{2a}+1}}$ is one vacuum of the theory.

(a)

(b)

Figure 13: Kink-antikink - Evolution of scalar field $\tilde{\phi}_{d}$ in
spacetime for $a=1.0$ with (a) $v=0.05$ and (b) $v=0.10$.

We start the analysis of collisions by varying the initial velocity $v$. The
Fig. 13 depicts two forms of behavior when $a=1$. Fig. 13(a) corresponds to
the bion state, where the scalar field oscillates erratically at the center of
mass after the interaction. The other behavior for this scattering is marked
by topological sector changing. In this case, the pair approaches, collides
and the scalar field exchanges vacuum - see Fig. 13(b).

Additionally, we altered the value of the parameter and performed exhaustive
collisions for various initial velocity values. The scattering of this
configuration showed the appearance of few two-bounce behaviors, indicating
that the two-bounce windows for this case are very sensitive. Note the need
for a high value of the final time in order to observe scattering completely.
For instance, in Fig. 14 we note that the kink-antikink pair approaches,
collides, and after a long period, collides again and then entirely moves
away. Notice that the value of the field decreases as the value of $a$
increases, compare Fig. 14(a) for $a=3$ with Fig. 14(b) for $a=6$.

(a)

(b)

Figure 14: Kink-antikink - Evolution of scalar field $\tilde{\phi}_{d}$ in
spacetime for (a) $a=3.0$ with $v=0.0934$ and (b) $a=6.0$ with $v=0.0936$.

We shall now explore the antikink-kink scattering of the solution
$\tilde{\phi}^{K}_{d}$. The initial conditions used in this problem are given
by

$\displaystyle\tilde{\phi}(x,x_{0},v,0)$ $\displaystyle=$
$\displaystyle\tilde{\phi}_{d}^{\bar{K}}(x-x_{0},-v,0)+\tilde{\phi}_{d}^{K}(x+x_{0},v,0),$
(29) $\displaystyle\dot{\tilde{\phi}}(x,x_{0},v,0)$ $\displaystyle=$
$\displaystyle\dot{\tilde{\phi}}_{d}^{\bar{K}}(x-x_{0},-v,0)+\dot{\tilde{\phi}}^{K}_{d}(x+x_{0},v,0),$
(30)

where $\tilde{\phi}(x,x_{0},v,t)=\tilde{\phi}(\gamma(x-vt))$ means a boost for
the static solution with $\gamma=(1-v^{2})^{-1/2}$. In this scenario, two-
bounce windows are absent for small values of $a$. However, the development of
the resonant structure is facilitated by the increase of this parameter.

(a)

(b)

(c)

Figure 15: Antikink-kink - Number of bounces versus initial velocity of scalar
field $\tilde{\phi}_{(0,\frac{1}{\sqrt{e^{2a}+1}})}^{(10)}$ for (a) $a=1$, (b)
$a=3$ and (c) $a=6$.

The number of bounces relative to the initial velocity is computed in Fig. 15.
Take note that there is no two-bounce window for $a=1$ Figs 15(a). Only two
regions are created for this value of $a$, the region with inelastic
collisions for velocities greater than the critical velocity and the region
with bion states for $v<v_{c}=0.532$. In contrast, as illustrated in Figs.
15(b) and 15(c), double collisions happen for greater values of $a$. We can
see that at certain velocity levels, the kinks collide twice and then
separate. The behavior with two collisions is denoted by $Nb=2$, whereas the
behavior with only one collision is marked by $Nb=1$. The value of the
critical velocity increases as a result of increasing the parameter.

The proximity of the vibrational mode value to the continuous spectrum serves
as the basis for the explanation of why the two-bounce windows vanish for
small values of $a$. The continuous mode in this configuration is obtained for
frequencies higher than $\omega^{2}>1$. Notice that the value of
$\omega_{1}^{2}$ gets closer and closer to $1$ as $a$ decreases. As a result
of the elimination of the bound mode in the continuum, a quasinormal mode
forms. This is a necessary component of the phenomenon known as spectral walls
adam.2019 .

Again, the scattering results are affected by how the initial condition is
established. The kink-antikink configuration, for example, has its asymptotic
values invariant, however, the center of the collision shows the field value
decreasing as $a$ increases. In contrast, the antikink-kink scattering has its
center of mass invariant and the variation of $a$ causes the asymptotic values
of the defect to change.

###  IV.2 Up kink scattering

In this section we will study the kink-antikink and antikink-kink collision of
the up solution $\tilde{\phi}_{(\frac{1}{\sqrt{e^{2a}+1}},1)}^{(10)}$. For the
sake of simplicity, the kink is written as $\tilde{\phi}_{u}^{K}$ and the
antikink as $\tilde{\phi}_{u}^{\bar{K}}$. In this first part, the kink-
antikink scattering we will be examined. We used the following initial
conditions

$\displaystyle\tilde{\phi}(x,x_{0},v,0)$ $\displaystyle=$
$\displaystyle\tilde{\phi}_{u}^{K}(x+x_{0},v,0)+\tilde{\phi}_{u}^{\bar{K}}(x-x_{0},-v,0)-1,$
(31) $\displaystyle\dot{\tilde{\phi}}(x,x_{0},v,0)$ $\displaystyle=$
$\displaystyle\dot{\tilde{\phi}}_{u}^{K}(x+x_{0},v,0)+\dot{\tilde{\phi}}^{\bar{K}}_{u}(x-x_{0},-v,0),$
(32)

where $\tilde{\phi}(x,x_{0},v,t)=\tilde{\phi}(\gamma(x-vt))$ means a boost for
the static solution with $\gamma=(1-v^{2})^{-1/2}$. It is important to note
here that there is no vibrational mode in this configuration, even for the
kink-antikink pair. A center barrier is produced by the perturbation potential
for the collective kink-antikink pair, making the presence of bound states
impossible. This behavior is similar to the potential in Ref. Dorey.PRL.2011 .
As a consequence, the two-bounce windows are absent.

(a)

(b)

(c)

(d)

Figure 16: Kink-antikink - Evolution of scalar field $\tilde{\phi}_{u}$ in
spacetime for (a) $a=0.4$ with $v=0.45$ (b) $a=0.5$ with $v=0.2$, (c) $a=0.9$
with $v=0.16$ and (d) $a=1.0$ with $v=0.1$.

Some results are achieved when we vary the values of the initial velocity and
the parameter $a$. For instance, the kink-antikink pair approaches, collides,
and eventually separates from one another in Fig. 16(a) when $a=0.4$. This
result occurs for velocities above the critical velocity $v>v_{c}=0.4332$. On
the other hand, when the collision occurs for small values of the initial
velocity, we observe only a bion behavior. The behavior of the scattering
begins to change when $a>0.5$. In this case, we realize only the formation of
an antikink-kink down pair after the collision, as shown in Fig. 16(b). It
should be noted that the velocity after the collision is greater than the
approach velocity of the kink-antikink pair. This is due to the up kink being
more massive than the down kink. Consequently, there is kinetic energy to
accelerate the newly formed anti-kink-kink pair. Additionally, the increase in
$a$ causes an oscillation at the center of mass - see Fig. 16(c). This
behavior favors the perturbation of the pair and consequently, we note the
presence of a double antikink-kink Fig. 16(d). Notice that the kinks are now
part of the other topological sector. Another feature is that the new pair
scatters with less radiation. These behaviors are comparable to those observed
in reference simas.2020 .

Let us now examine the antikink-kink scattering process of the up solution
$\tilde{\phi}_{(\frac{1}{\sqrt{e^{2a}+1}},1)}^{(10)}$. The initial conditions
for this configuration are expressed as follows

$\displaystyle\tilde{\phi}(x,x_{0},v,0)$ $\displaystyle=$
$\displaystyle\tilde{\phi}_{u}^{\bar{K}}(x-x_{0},-v,0)+\tilde{\phi}_{u}^{K}(x+x_{0},v,0)-\phi_{\mu},$
(33) $\displaystyle\dot{\tilde{\phi}}(x,x_{0},v,0)$ $\displaystyle=$
$\displaystyle\dot{\tilde{\phi}}_{u}^{\bar{K}}(x-x_{0},-v,0)+\dot{\tilde{\phi}}^{K}_{u}(x+x_{0},v,0),$
(34)

where $\tilde{\phi}(x,x_{0},v,t)=\tilde{\phi}(\gamma(x-vt))$ means a boost for
the static solution with $\gamma=(1-v^{2})^{-1/2}$ and
$\phi_{\mu}=\frac{1}{\sqrt{e^{2a}+1}}$. Once again, an excited mode is not
revealed by the perturbation analysis for a single kink or antikink. However,
the Schrödinger potential ensures the existence of numerous bound states when
we analyze the collective antikink-kink pair. This is favored due to the
configuration of how the initial condition is written, generating a
perturbation potential with a central well and asymptotic to
$\tilde{U}^{(10)}_{up}\to 4$ with $x\to\pm\infty$. The Ref. Dorey.PRL.2011
also exhibits this feature.

(a)

(b)

Figure 17: Antikink-kink - Number of bounces versus initial velocity of scalar
field $\tilde{\phi}_{(\frac{1}{\sqrt{e^{2a}+1}},1)}^{(10)}$ for (a) $a=0.1$
and (b) $a=0.3$.

The presence of vibrational modes therefore denotes the emergence of two-
bounce windows. However, due to the complexity of the resonant energy exchange
between the extra bound modes, such resonance windows are extremely thin. The
Fig. 17 depicts the structure of antikink-kink scattering, which includes the
number of collisions relative to the initial velocity. The result for $a=0.1$
is shown in Fig. 17(a). For $v>v_{c}=0.0746$, it describes the region with
just one collision ($Nb=1$). For velocities smaller than the critical
velocity, we observe the bounces structure between the bion region and with
more collisions. One can see that the critical velocity decreases with
increasing parameter $a$. This can be seen in Fig. 17(b) for $a=0.3$, where
$v_{c}=0.0476$. The critical velocity continues decreasing as $a$ increases,
for instance, for $a=1.0$, we realize that $v_{c}=0.0379$. This illustrates
the zone of bion states contracting and the region of inelastic collision
expanding.

##  V Conclusion

In this work we have used the deformation procedure to introduce two new
models, the modified $\phi^{8}$ and $\phi^{10}$ models. These models are
respectively obtained by deforming models $\phi^{4}$ and $\phi^{6}$ with the
aid of deformation techniques. The models contain a real parameter $a$. We
have shown that this parameter controls the potentials, the kinklike
solutions, their masses, the stability potentials and the internal modes
associated to the stability of the systems.

The modified $\phi^{8}$ model contains both types of symmetric and asymmetric
kinks. The stability analysis showed that the asymmetric kink has a shape
mode, while the symmetric kink has an internal mode for $a\geq 4.6$. We
performed the kink collision for both solutions. We investigated the kink-
antikink and antikink-kink collision processes in the asymmetric case. For
small values of $a$, we observe two-bounce windows for kink-antikink. However,
increasing the parameter starts the process of suppressing the resonant
windows and the formation of new antikink-kink pairs. This behavior is related
to the massive character of the asymmetric kink, allowing sector exchange.
When we consider the antikink-kink interaction, we also see the presence of
two-bounce windows for small values of $a$. The increase in this parameter
shows the appearance of false two-bounce windows, until the complete
annihilation of the resonances. The significance of the parameter in the
initial configuration accounts for the difference in the results between kink-
antikink and antikink-kink scattering. The increase in $a$ favors the
appearance of a new vibrational mode for the symmetric kink, which in turn
influences the internal mode of the asymmetric kink. Consequently, the
resonant energy exchange mechanism between the modes is frustrated. Now, the
scattering analysis for the symmetrical kink revealed the absence of fractal
structures for small values of $a$. However, the formation of antikink-kink
pairs in the asymmetrical sector was noticeable. Furthermore, large values of
the parameter modify the results, indicating the presence of a resonant
structure. In addition, with $a\to\infty$, the deformed $\phi^{8}$ tends
towards the $\phi^{4}$ model.

The $\phi^{10}$ model contains five minima and two forms of asymmetric kinks
known as down and up kink. Considering small perturbations around the static
solutions, we arrive at a Schrödinger equation with different perturbation
potentials for the down and up kink. We numerically investigated the
occurrence of vibrational modes at both potentials. However, we observed the
presence of only one shape mode for the potential for an individual down kink.
Numerical analysis of kink-antikink and antikink-kink scattering showed
complete suppression of the resonant windows for small values of $a$. However,
as this parameter increased, some two-bounce behavior was observed. The
proximity of the square frequency to the continuous spectrum serves as the
basis for explaining why the two-bounce windows disappear for small values of
$a$. Furthermore, we investigated the kink-antikink collision for the up kink.
The results in this scenario reveal the emergence of a more accelerated
antikink-kink down pair after the collision. This is due to the up kink being
more massive than the down kink. Resonant windows are absent in this case
because the perturbation potential of the kink-antikink pair generates a
barrier, which inhibits the presence of the bound modes. Finally, antikink-
kink scattering indicates the appearance of two-bounce windows. We can notice
that the excited modes for this case are revealed by the perturbation analysis
for the antikink-kink pair.

## Acknowledgements

AMM would like to thank Islamic Azad University Quchan branch for the grant.
This study was financed in part by the Coordenação de Aperfeiçoamento de
Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001. It was also
financed in part by Conselho Nacional de Desenvolvimento Científico e
Tecnológico, Grant No. 303469/2019-6 (DB) and by Paraiba State Research
Foundation, Grant No. 0015/2019 (DB).

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# Calabi-Yau metrics and string compactification

Michael R. Douglas<EMAIL_ADDRESS>Simons Center for Geometry
and Physics, Stony Brook University, Stony Brook NY 11794

###### Abstract

Yau proved an existence theorem for Ricci-flat Kähler metrics in the 1970’s,
but we still have no closed form expressions for them. Nevertheless there are
several ways to get approximate expressions, both numerical and analytical. We
survey some of this work and explain how it can be used to obtain physical
predictions from superstring theory.

To appear in Nuclear Physics B special issue: 60 Years of Calabi Conjecture.

##  1 Introduction

Superstring theory is formulated in ten space-time dimensions, and to describe
our universe we must postulate that the six extra spatial dimensions form a
compact manifold ${\cal M}$ of diameter smaller than $10^{-17}\mbox{cm}$. To a
good approximation, the metric on ${\cal M}$ must satisfy the equations of
motion of general relativity, which set the Ricci tensor of ${\cal M}$ equal
to a tensor which describes the energy of the vacuum. In the simplest case, we
postulate that the vacuum has zero energy, and thus ${\cal M}$ must have a
Ricci-flat metric.

One’s first thought might be to take ${\cal M}$ to be the six-torus. However,
this does not work, because it leads to a four dimensional theory with too
much symmetry to describe our universe. In particular, every isometry of
${\cal M}$ leads to a particle in four dimensions (a “graviphoton”) which can
be excluded by observation. Thus, ${\cal M}$ must be a Ricci-flat manifold
without isometries.111 One might imagine fixing this problem by postulating
other fields or structures on ${\cal M}$ which are not invariant under the
isometry. While the idea is reasonable, in the end the torus still doesn’t
work.

Such manifolds are extremely rare. In six dimensions, there is only one known
class of examples, the complex Kähler manifolds of vanishing first Chern
class. These conditions imply that the Ricci tensor is globally a total
derivative, and that starting from a given Kähler potential, in principle one
can add a function to set the Ricci tensor to zero. Calabi conjectured in 1957
that this was so, but to show this one must show that a highly nonlinear
Monge-Ampere equation always has a solution. Yau famously proved Calabi’s
conjecture for Ricci flat Kähler manifolds in 1977. [1] Such manifolds ${\cal
M}$, which are universally known as Calabi-Yau manifolds, are thus the
simplest candidates for the extra dimensions in our universe.

To get a compactification which solves all of the superstring equations of
motion and leads to the Standard Model gauge group $SU(3)\times SU(2)\times
U(1)$ at low energies requires a few more ingredients. The most important is
that ${\cal M}$ must carry a holomorphic vector bundle ${\cal V}$ of rank 3, 4
or 5, with a connection which solves the Yang-Mills equations. This
requirement is deduced by identifying the Standard Model gauge group with the
commutant of the holonomy group of ${\cal V}$ in $E_{8}$ (one of the two
$E_{8}$’s of the ten-dimensional gauge group). For the holonomy groups
$SU(3)$, $SU(4)$ and $SU(5)$, the commutants are $E_{6}$, $SO(10)$ and $SU(5)$
respectively. These are the “grand unified gauge groups” and are in striking
distance of the Standard Model gauge group, in the sense that after tensoring
${\cal V}$ with a nontrivial flat bundle (so, with a finite holonomy group),
the commutant will indeed be $SU(3)\times SU(2)\times U(1)$. This construction
favors a non-simply connected ${\cal M}$, so that the flat bundle will be
easily obtained.

Much as for the Ricci flat metric, the simplifications of Kähler geometry
allow reducing the Yang-Mills equations to a simpler form – the hermitian
Yang-Mills equation – and lead to a testable necessary condition for a
solution, that ${\cal V}$ is $\mu$-stable. By theorems of Donaldson and
Uhlenbeck-Yau, this condition is sufficient. Anomaly cancellation in
superstring theory constrains the Chern classes of ${\cal V}$ to be
$c_{1}({\cal V})=0$ and $c_{2}({\cal V})=c_{2}({\cal M})$. While it is not
immediately obvious which ${\cal M}$ carry such bundles, for ${\cal M}$
Calabi-Yau, there is an evident candidate: ${\cal V}\cong T{\cal M}$, the
tangent bundle of ${\cal M}$.

These ingredients were first put together in 1985 in a famous paper of
Candelas, Horowitz, Strominger and Witten (CHSW). [4] This was the first
proposal ever for a derivation starting from a fundamental physical theory
which could lead all the way to the Standard Model. Since then, while other
proposals have been made, this remains the prototypical and arguably the best
proposal yet made for the fundamental structure which leads to the physical
laws of our universe. The study of Calabi-Yau compactification also led to the
discovery of mirror symmetry, and many other developments in the interface
between string theory and mathematics.

After the gauge group, the next real world observable to reproduce is the
number of generations of quarks and leptons, which is three in the Standard
Model. In CHSW this arises as the index of a Dirac operator acting on sections
of ${\cal V}\otimes\Lambda^{2}T{\cal M}$. Following the ansatz that ${\cal
V}\cong T{\cal M}$, this index can be shown to equal half the Euler character
of ${\cal M}$, and thus one wants a Calabi-Yau manifold with $\chi=6$. One
furthermore wants $\pi_{1}({\cal M})\cong\mathbb{Z}_{3}\times\mathbb{Z}_{3}$
to get a flat bundle breaking $E_{6}$ to the Standard Model gauge group. And
indeed, such a manifold was very quickly constructed [5, 8] and used for
string phenomenology. [9] After thirty years, although there are a few
competitors [23], this “three generation manifold” is still the leading
candidate within this class of model.

Over the years, several other classes of string compactifications have been
found that can reproduce the Standard Model. These include the heterotic
“$(0,2)$” models, type II orientifold compactifications with Dirichlet branes,
F theory on elliptically fibered Calabi-Yau fourfolds, and M theory on
manifolds of $G_{2}$ holonomy. If we assume that there is low energy
supersymmetry (meaning low compared to the compactification scale), this may
be the complete list.222 The main exception to this claim are the “non-
geometric” compactifications which rely for their existence on special
features of string theory, such as nonperturbative effects on the world-sheet
or in space-time, or non-geometric monodromies. On the other hand, there are
many cases in which geometric interpretations for such constructions were
found later, so the claim has not yet been disproven.

The geometric heterotic $(0,2)$ models, also called “superstrings with
torsion,” are obtained by taking ${\cal V}$ different from $T{\cal M}$, though
still satisfying the anomaly cancellation constraints. As an example, one
could take a deformation of $T{\cal M}\oplus\mathbb{C}^{K}$. The metric and
gauge connection now must satisfy a set of equations given by Strominger in
[7], involving an additional scalar field. In [15] Li and Yau proved that
solutions exist for small deformations of $T{\cal M}\oplus\mathbb{C}^{K}$.333
This was generalized to certain non-Kähler manifolds in [19]. The number of
generations in a $(0,2)$ model is half the third Chern class of ${\cal V}$,
opening a new way to get three generations. Many explicit examples are known,
such as [34] which obtains precisely the spectrum of the minimal
supersymmetric Standard Model.

##  2 Physical predictions from Calabi-Yau manifolds

Once one has matched the particle content of the Standard Model, one would
like to make more detailed physical predictions, such as the masses of quarks
and leptons. Needless to say, this is a long story, but for the well-
understood constructions it can be summarized as follows.

First, in the present state of the art, quantum corrections must be small at
the compactification scale. This is simply because we do not have a complete
formulation of string/M theory which could be used to make general
computations with quantum corrections. Certainly, the use of duality arguments
has led to major advances in this direction, but they do not yet reach the
goal. Thus, our calculations will be based on the geometry of metrics and
connections.

Related to this, we need to assume $N=1$ supersymmetry at low energies. There
are many arguments for this, but the simplest is that supersymmetry
drastically constrains the quantum corrections and forces many of them to be
zero. In this case, the masses of quarks and leptons are determined by the
“cubic Yukawa couplings.” These are the amplitudes for a given quark or lepton
to interact with the Higgs field, which after electroweak symmetry breaking
gives mass to the quarks and leptons. Given supersymmetry, the Higgs field has
a fermionic partner (the “Higgsino”) which is not fundamentally different from
the quarks and leptons; thus these couplings have complete permutation
symmetry. There are many further details: for example supersymmetry requires
there to be two Higgs superfields, there are logarithmic corrections from the
renormalization group, and so on, but the basic inputs are the cubic Yukawa
couplings.

Thus, a good approximation to the problem for the heterotic string is the
following. We start with ten-dimensional super Yang-Mills theory, and find
explicit normalized zero modes of the Dirac operator on ${\cal M}$ coupled to
the Yang-Mills connection on ${\cal V}$, label these $\psi_{a}$. We then
compute the triple overlap of these wave functions,

$C_{abc}=\int_{\cal M}\psi_{a}\cdot\psi_{b}\cdot\psi_{c}.$ (1)

The case with $a$ and $b$ corresponding to a particular quark, and $c$
corresponding to the Higgsino determines the ratio of the quark mass to the
Higgs expectation value. For example, for $a=b$ the top quark and $c$ the “up”
Higgs, matching the observed top quark mass requires $C=1$ to a fairly good
accuracy (!)

One can go some distance in this direction without knowing the metric on
${\cal M}$, because the zero modes have a topological interpretation. [6] The
best case is for ${\cal V}\cong T{\cal M}$, as then Kähler geometry relates
the zero modes of the Dirac operator to harmonic forms. On a Calabi-Yau
threefold, the nontrivial harmonic forms will be of Hodge type $(1,1)$ and
$(2,1)$, and Hodge duals of these. These correspond to “generations” and
“antigenerations” of quarks and leptons respectively, with gauge charges in
the fundamental and antifundamental of $E_{6}$. For a more general ${\cal V}$,
the relevant forms are the $(0,p)$-forms with values in ${\cal V}$ and its
various tensor power bundles.

One can even get exact formulas for some Yukawa couplings, such as those
between a triplet of $(2,1)$-forms. These are

$C_{abc}=\int_{\cal
M}\Omega^{3,0}\wedge\left[\psi_{a}\wedge\psi_{b}\wedge\psi_{c}\right],$ (2)

where $\Omega^{3,0}$ is the holomorphic three-form, and the brackets denote
contractions of the holomorphic indices of the $\psi$’s with further
$\Omega^{3,0}$’s. Importantly, this formula is “topological” in that it does
not depend on the specific $(2,1)$-forms we use, only their cohomology class,
and thus we do not need to know the metric on ${\cal M}$ to compute it. Even
better, it can be shown to receive no string world-sheet corrections. Many
examples have been worked out, e.g. see [30].

While Eq. (2) is very powerful, it is not the answer to the physical question
we asked unless we can compute it for normalized $(2,1)$-forms. The
straightforward way to do this is to do the integrals

$K_{a,\bar{b}}=\int_{\cal
M}\omega\wedge\omega\wedge\left[\psi_{a}\wedge\bar{\psi}_{\bar{b}}\right]$ (3)

(where $\omega$ is the Kähler form) and go to a basis in which
$K_{a,\bar{b}}=\delta_{a,\bar{b}}$. This formula does depend on the specific
representatives we take for the $\psi$’s, and finding the harmonic forms
requires knowing the Ricci-flat metric on ${\cal M}$.

In some cases, there are ways around this. In particular, when ${\cal V}\cong
T{\cal M}$, one can compute Eq. (3) using “special geometry.” Mathematically,
deformations of $T{\cal M}$ integrate to deformations of the complex moduli of
${\cal M}$, and the metric on this moduli space can be derived using Weil-
Petersson geometry. However, the generalization of this approach to other
${\cal V}$ is not known. Thus, the next step in obtaining physical predictions
is to find the Ricci-flat metric on ${\cal M}$, the hermitian Yang-Mills
connection on ${\cal V}$, and the normalized ${\cal V}$-valued $(0,p)$ forms.

##  3 Analytic descriptions of Ricci-flat metrics

At present there are no closed form expressions for the Ricci-flat metric on
any nontrivial compact Calabi-Yau. This includes the K3 manifold which is even
hyperkähler and (in principle) amenable to the twistor transform, as well as
moduli spaces of instantons on K3 which are (again in principle) amenable to
the Nahm transform. This is not for want of trying and thus we can confidently
state that this is a “hard problem.”

There are a number of explicit expressions for Ricci-flat metrics on
noncompact Calabi-Yau manifolds, starting with the “self-dual gravitational
instantons” of Eguchi-Hanson and the Taub-NUT spaces. The best cases are the
resolutions of the canonical singularities $\mathbb{C}^{2}/\Gamma$ with
$\Gamma$ a discrete subgroup of $SU(2)$. These are hyperkähler and there are
explicit hyperkähler quotient constructions for the metrics. In particular, it
is easy to see that these metrics are asymptotically locally Euclidean (ALE).
One can also construct moduli spaces of Yang-Mills instantons on these
resolved quotients with their hyperkähler metrics. [10]

As was pointed out by Page [2], one can get a nice picture of certain Ricci-
flat K3 metrics by starting with the flat quotient $T^{4}/\mathbb{Z}_{2}$,
excising the neighborhood of the fixed points, and gluing in an Eguchi-Hanson
metric at each one. These are generally known as Kummer surfaces. This idea
has been much used in existence proofs, for example in Joyce’s construction of
metrics of $G_{2}$ holonomy. [12] Using these techniques, in principle one
could patch together harmonic forms to compute Eq. (3) for resolved orbifolds.

The other general construction of explicit Ricci-flat metrics is to assume so
much continuous symmetry that the Kähler potential becomes a function of a
single variable. The Monge-Ampère equation then reduces to an ODE. This was
done for $\mathbb{C}^{n}/\mathbb{Z}_{n}$ by Calabi. [3] There are many
examples in the physics literature, where they are known as “cohomogeneity
one” metrics.

Once one moves on to $\mathbb{C}^{3}/\Gamma$, or other Kähler manifolds, while
many cases still admit quotient constructions, these are symplectic quotients
which have no reason to be Ricci-flat. An amusing case for which one can get a
Ricci-flat threefold from a quotient is to realize
$\mathbb{C}^{3}/\mathbb{Z}_{3}$ using the McKay quiver, and boldly continue
the two moment map (Fayet-Iliopoulos) parameters to complex conjugate values.
This procedure reproduces Calabi’s metric, without the need to solve any
differential equations. [14] I wonder if this has some deep significance.

The Strominger-Yau-Zaslow approach to mirror symmetry [13] suggests the
following strategy. One starts with a torus fibration $T^{n}\rightarrow{\cal
M}\rightarrow B_{n}$ and a “semi-flat structure” on ${\cal M}$. This is a map
from $B_{n}$ to the moduli space of flat tori which can be lifted to a Ricci-
flat metric, which however is singular for nontrivial fibrations. One then
adds a series of instanton corrections which smooth out the singularities and
(again in principle) sum to a smooth Ricci-flat metric. An interesting and
related recent development is the work of Gaiotto, Moore and Neitzke [28] on
hyperkähler metrics and instanton corrections in four-dimensional
supersymmetric gauge theory.

##  4 Numerical descriptions of Calabi-Yau metrics

Even if analytic expressions for Calabi-Yau metrics are found someday, it
seems likely that they will be very complicated. This is a particularly safe
bet for realistic string compactification manifolds and bundles, which are not
so simple even from the algebraic geometric point of view.

For many purposes, not just our string compactification questions, it would be
more useful to have a simple rough description of the metric, which could be
used to compute its properties to order one (but controlled) accuracy.
Numerical methods would seem well suited for this goal.

The numerical study of Calabi-Yau metrics was initiated by Headrick and
Wiseman [17] who studied Kummer surfaces. Their method was to discretize the
Monge-Ampere equation on an explicit lattice obtained by introducing
coordinate patches, one on the $T^{4}/\mathbb{Z}_{2}$ quotient and one on the
Eguchi-Hanson patch (by symmetry one can force all the Eguchi-Hanson patches
to be isometric). They then applied Gauss-Seidel relaxation to solve this
equation. Besides exhibiting the corrections to the flat and Eguchi-Hanson
patch geometries, they went on to get a low-lying eigenvalue of the scalar
Laplacian.

While the lattice leads to fairly good results, accurate to around 1% with a
few days work on a desktop computer, it comes with some problems. In $D$
dimensions, a lattice with linear extent $N$ has $N^{D}$ lattice sites, so for
$D=6$ a computer memory will only allow $N\sim 20$ or so. Numerical accuracy
is generally proportional to $1/N$. While 5% global accuracy is better than we
might need, the curvature is usually concentrated in certain regions, and this
forces us to larger $N$. This sort of problem is usually dealt with using
adaptive or multiscale methods, in which the lattice spacing varies according
to the local gradients of the functions involved. An adaptive discretization
scheme on a manifold of complicated topology would be rather complicated to
implement.

Even using a simpler discretization, a good deal of work is required to find
explicit coordinate patches and overlap functions. It would be much better if
the discretization could be derived from the geometry of the manifold, either
intrinsically or using some simple embedding. This was done in the work of
Donaldson [18], which introduced several new ingredients.

First, a natural finite dimensional space of Kähler metrics can be obtained by
embedding ${\cal M}$ using sections of a line bundle ${\cal L}$, and using as
metric the curvature of a unitary connection on ${\cal L}$. As Donaldson
comments, “The potential utility of this point of view has been advocated over
many years by Yau,” and indeed it turns out to be quite useful here.

This is much more concrete than it may sound. Consider a quintic hypersurface
in $\mathbb{P}^{4}$; what we are doing is looking for the best approximation
to a Ricci-flat metric in the space ${\cal K}_{N}$ of Kähler potentials

$K_{N,h}=\log\sum_{I_{1}\ldots I_{N},J_{1}\ldots J_{N}}H_{I_{1}\ldots
I_{N},J_{1}\ldots J_{N}}Z^{I_{1}}\ldots
Z^{I_{N}}{\bar{Z}}^{J_{1}}\ldots{\bar{Z}}^{J_{N}}$ (4)

where $H$ is a constant hermitian matrix. While one might have thought that
this $N$ is no better than the linear lattice extent $N$, in fact the
potential accuracy is better than $N^{-\nu}$ for any $\nu$ ! This is because
the Ricci-flat metrics we are looking for are analytic, and this behavior is
analogous to the fast decay of Fourier coefficients of analytic functions.

Donaldson then replaced the problem of finding the Ricci-flat metric by that
of finding the “balanced metric.” We can define this in terms of the following
map from hermitian metrics $h$ on ${\cal L}$ to ${\cal K}_{N}$. Given a metric
$h$, we compute the integral over ${\cal M}$ of the inner product between
sections, to get a hermitian metric $H^{-1}$ on the space of sections. We then
invert this to get $H$ in Eq. (4). Applying this map to the hermitian metric
$h=e^{-K}$ corresponding to $K$, we get a map from ${\cal K}_{N}$ to itself.
The balanced metrics are then the fixed points of this map. A variant is to
assume a given volume form $\nu$ on ${\cal M}$; this type of balanced map
exists by results of Bourguignon, Li and Yau [11]. For a Calabi-Yau manifold,
one can take $\nu=\Omega\wedge\bar{\Omega}$.

Using the Tian-Yau-Zelditch-Lu expansion for the diagonal of the Szegö kernel,
one can then show that as $N\rightarrow\infty$, the balanced metric converges
to the Ricci-flat metric, with corrections of order $1/N^{2}$ (granting
$c_{1}({\cal M})=0$). While we don’t have space to repeat these arguments
here, let me mention the paper [29] which rederives this expansion using
physics techniques.

Donaldson went on to implement this procedure on K3, getting 1% accuracy now
with $N=9$. While much simpler, one difficult point remained – namely, the use
of explicit coordinates to do integrals over ${\cal M}$. Now multidimensional
numerical integrals are usually best done by Monte Carlo, and as it happens it
is easy to produce random points drawn from the restriction of a Fubini-Study
measure to a subvariety. This procedure was implemented in [21], leading to
metrics on quintic threefolds of comparable quality, but with minimal
programming effort and computer resources. The method was used to study
eigenfunctions of the scalar Laplacian in [27], and a more accurate adaptive
version of the method was developed in [33].

As another option, rather than find the balanced metric, one can instead
minimize an error term related to the Ricci curvature. Numerical optimization
is quite efficient and this procedure was successfully carried out for a
Calabi-Yau metric in [31]. See also [22] and [25], which studied Kähler-
Einstein metrics on toric surfaces using a variety of numerical techniques,

Donaldson’s approach can also be used to obtain approximate hermitian Yang-
Mills connections and normalized $(0,p)$-forms. The idea is to embed ${\cal
V}$ by sections into a higher dimensional Grassmannian manifold Gr, and pull
back a simple connection on Gr to get a connection on ${\cal V}$ which depends
on the parameters of the embedding. Actually, the anomaly cancellation
condition $c_{1}({\cal V})=0$ implies that ${\cal V}$ will have no global
sections, but this can be finessed by taking sections of ${\cal V}\otimes{\cal
L}^{N}$ for some positive line bundle ${\cal L}$.

The concept of balanced metric was defined in this context by Wang [16], who
proved that a series of balanced metrics will converge to the metric
associated to a hermitian Yang-Mills connection as
$N\rightarrow\infty$.444Strictly speaking, one gets a “weak hermitian Yang-
Mills connection” solving the equation $\omega\cdot F=R(\omega)$ [16]. The
Kähler form $\omega$ on ${\cal M}$, which appears in the hermitian Yang-Mills
equation, is the curvature of ${\cal L}$. Furthermore, one can adapt the same
iterative procedure to find the balanced metric. If it exists, this procedure
was proven to converge by Seyyedali [26]. These ideas were implemented and
numerical solutions found in [20, 32, 33].

Now one has a necessary condition for a hermitian Yang-Mills solution, namely
that ${\cal V}$ is $\mu$-stable. If it is not, then the theorem of Wang
implies that no balanced metric can exist, and therefore the iterative
procedure cannot converge. In [32, 33], this idea was developed into a
numerical method for checking stability.

To summarize, we now have all of the ingredients required to compute both Eq.
(2) and Eq. (3). Although we still do not know which Calabi-Yau manifold makes
up the extra dimensions of our universe, if and when we do, we will be ready
to derive masses and coupling constants from string theory.

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