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# Mechanization of scalar field theory in 1+1 dimensions

Filip Blaschke  filip.blaschke(at)fpf.slu.cz Research Centre for Theoretical
Physics and Astrophysics, Institute of Physics, Silesian University in Opava,
Bezručovo nám. 1150/13, 746 01 Opava, Czech Republic.
and Institute of Experimental and Applied Physics, Czech Technical University
in Prague, Husova 240/5, 110 00 Prague 1, Czech Republic  Ondřej Nicolas
Karpíšek  karponius(at)gmail.com Faculty of Philosophy and Science in Opava,
Silesian University in Opava, Bezručovo nám. 1150/13, 746 01 Opava, Czech
Republic,

###### Abstract

The ‘mechanization’ is a procedure of replacing a scalar field in 1+1
dimensions with a piece-wise linear function, i.e. a finite graph consisting
of $N$ joints (vertices) and straight segments (edges). As a result, the field
theory is approximated by a sequence of algebraically tractable, general-
purpose collective coordinate mechanical models. We observe the step-by-step
emergence of dynamical objects and associated phenomena as the $N$ increases.
Mech-kinks and mech-oscillons – mechanical analogs of kinks and oscillons
(bions) – appear in the simplest models, while more intricate dynamical
patterns, such as bouncing phenomenon and bion pair-production, emerge
gradually as decay states of high $N$ mech-oscillons.

kink, oscillon, collective coordinate model

##  I Introduction

Scattering of solitons provides a unique window to the inner workings of non-
linear field theories. Even in the case of the simplest solitons – kinks – the
rich phenomenology that we see has not been completely understood despite four
decades of theoretical research Campbell:1983xu ; Moshir:1981ja ;
Belova:1985fg ; Anninos:1991un ; Halavanau:2012dv ; Dorey:2011yw ;
Weigel:2013kwa ; Haberichter:2015xga ; Ashcroft:2016tgj ; Manton:2020onl ;
Manton:2021ipk ; Adam:2019prh ; Adam:2021gat ; 2019arXiv190903128K .

The prototypical non-linear field theory is the $\phi^{4}$ model with a single
scalar field and double-well potential
$V(\phi)=\tfrac{1}{2}\bigl{(}1-\phi^{2}\bigr{)}^{2}$, where the kink solution
has an especially simple form: $\phi_{K}(x)=\tanh(x)$. Numerical
investigations of kink ($K$) anti-kink ($\bar{K}$) collisions reveal an
intricate pattern of possible outcomes Campbell:1983xu ; Moshir:1981ja ;
Belova:1985fg ; Anninos:1991un ; Adam:2021gat ; Manton:2021ipk : i) Above the
critical velocity $v_{\rm crit}\approx 0.26$, the colliding pair is
immediately reflected and escapes to infinity. ii) Below this threshold, the
solitons sometimes form a ‘bound state’ (bion) that slowly decays to the
vacuum via the emission of radiation. iii) Before final reflection and escape
to infinity, the kink and anti-kink bounce off each other several times. These
bounces occur in narrow windows of initial velocities that are nested and
fractal-like punctuated by ‘bion chimneys’ and are observed only for initial
velocities larger than $v_{\rm min}\approx 0.18$.

To understand these phenomena is, at its core, a problem of complexity. During
the $K\bar{K}$ collisions, the field enters a deeply non-linear regime, where
our analytic and ‘weak-field’ perturbative tools fail. It could be the case
that a very large number of degrees of freedom participate with equal
importance, and we may never untangle the web of inner-dependencies. That
would be a scenario of _computational irreducibility_ that is characteristic
for some discrete dynamical systems, such as cellular automata Wolf .
Fortunately, for solitons it has been an overwhelming experience that the
opposite is true – there is a surprisingly large _reducibility_. Namely, a
deep understanding of the dynamics seems to be possible through tracking only
a few effective degrees of freedom, called _collective coordinates_ (CCs).

This is most evident for multi-soliton configurations that are BPS
Bogomolny:1975de ; Manton:2004tk . The small-velocity scattering of BPS
solitons can be accurately described using the moduli (geodetic)
approximation, where the CCs are time-dependent parameters (moduli) of the
static solution, such as mutual separations, etc. Integrating the field’s
kinetic energy for this background we get a metric of the moduli space – a
curved manifold of BPS solutions – and the scattering of solitons is
equivalent to the geodetic motion Manton:1981mp .

For a non-BPS case, the canonical moduli space does not exist and we are
forced to guess. However, a general strategy, which can be perhaps called an
_adiabatic_ method, uses the same principle, namely it introduces an ad hoc
moduli space that aspires to capture the dynamics under investigation. The
coordinates of this space become dynamical variables in the resulting
Collective Coordinate Model (CCM), usually resembling classical mechanics of
interacting point masses.

In the case of $\phi^{4}$ kinks, there are no static multi-kink solutions
(however, one can stabilize them by adding impurities Adam:2019prh ). The use
of CCM for $\phi^{4}$ kinks has a long, fruitful, and somewhat complicated
history (see 2019arXiv190903128K  for a recent overview). For our purposes,
let us highlight two important results.

First, for a single kink, it has been noticed a long-time ago RiceMJ  that a
correct CCM (in the sense of both qualitative and quantitative agreement) must
include not only the position modulus but also a scaling modulus. If we plug
the ansatz

$\phi_{\rm bkg}=\tanh\bigl{(}b(t)(x-a(t))\bigr{)}$ (1)

into the Lagrangian and integrate over $x$, the resulting effective theory
(Eq. (48)), while non-linear, can be exactly integrated and contain both
Lorentz-boosted kink solution and exact harmonic motion, intimately connected
with the so-called _Derrick mode_ of the kink Manton:2020onl . This mode has
been recognized as a key tool for recovering Lorentz covariance that is
typically lost in CCMs.

The second result is the recent identification of quantitatively correct CCM
for $K\bar{K}$ scattering Manton:2020onl ; Manton:2021ipk ; Adam:2021gat  that
is based on the observation that a simple $K\bar{K}$ superposition ansatz

$\phi_{\rm
bkg}=\tanh\bigl{(}b(t)(x+a(t))\bigr{)}-\tanh\bigl{(}b(t)(x-a(t))\bigr{)}-1\,,$
(2)

leads to moduli space with unremovable singularity at the point $a=0$.
Therefore, in Adam:2021gat  the authors proposed a perturbative approach for
incorporating the scaling modulus order by order. The resulting CCMs do not
suffer from the singularity at $a=0$ and seem to qualitatively and
quantitatively match observed $K\bar{K}$ dynamics.

The use of CCMs is, of course, not limited just to $K\bar{K}$ scattering, but
it can be also applied also on other problems, such as the lifetime of an
oscillon. A field-theoretical oscillon is a meta-stable state that is observed
in the collapse of bubbles – usually taken as Gaussian peaks placed on top of
a vacuum. Compared with models with only a single vacuum, when the scalar
field theory has two or more vacua, the lifetime of such bubbles is typically
much longer than one would naively expect. Further, certain universal features
are shared across various initial conditions and models. At the onset, the
initial profile quickly settles to a certain shape around which it oscillates
quasi-periodically for a long time slowly radiating away energy, until it
suddenly decays. The oscillons enjoy a long history of investigation, although
typically in spatial dimensions higher than one Gleiser:1993pt ;
Copeland:1995fq ; Andersen:2012wg ; Salmi:2012ta ; Saffin:2006yk ;
Fodor:2006zs ; Gleiser:2009ys ; Olle:2020qqy .

It is easy to construct a simple CCM for an oscillon that captures some of its
key properties. In $\phi^{4}$ theory, if we use the background

$\phi_{\rm bkg}=-1+A(t)\mathrm{e}^{-x^{2}/R(t)^{2}}\,,$ (3)

we get an effective Lagrangian

$\sqrt{\frac{2}{\pi}}L_{\rm
osc}=\frac{1}{2}R{\dot{A}}^{2}+\frac{1}{2}A\dot{A}\dot{R}+\frac{3{\dot{R}}^{2}A^{2}}{8R}\\\
-\frac{A^{2}}{2R}-2A^{2}R+2\sqrt{\frac{2}{3}}A^{3}R-\frac{A^{4}R}{2\sqrt{2}}\,.$
(4)

Numerical investigations of this CCM111See Sec. IV where we study essentially
the same model. reveal that for most initial values there exists, after a
short initial phase, long quasi-periodic regime after which the ‘bubble’
collapses exponentially fast ($A\to 0$ and $R\to\infty$). In that regard, the
above CCM is qualitatively faithful to what is going on in field theory, at
least for small energies. For energies large enough, we can expect that the
oscillon decays into a $K\bar{K}$ pair (that can also undergo a few bounces
before separation) as Fig. 1 illustrates.

Figure 1: The plot of the energy density and field profiles for several
timestamps in the $\phi^{4}$ model showcasing a disintegration of a Gaussian
peak into kink-anti-kink pair.

This behavior is, of course, impossible to predict using (4), which is
inadequate for any situations where the field significantly departs from the
Gaussian shape. Of course, one can attempt to remedy this by introducing more
complicated ansatzes.

Concrete CCMs will have limited applicability specific for a given situation.
However, there are also general limitations. The most apparent one is
pragmatical. Typically, when the number of collective coordinates exceeds
$N=2$, the resulting formulae algebraically explode. Indeed, it is hard to
come up with an $N$-dimensional moduli space for which the effective
Lagrangian can be written down in a closed (and managable) form. We can see
that for $K\bar{K}$ scattering CCMs which includes either normal modes
2019arXiv190903128K  or higher Derrick’s modes Adam:2021gat . Simply put,
$N$-point CCMs are typically _algebraically intractable_.

This leads to a closely related limitation, namely that CCMs are usually not
introduced as members of a perturbative schema that approaches full
configuration space in a limit. In other words, CCMs are not usually
_exhaustive_. Having constructed one CCM, however successful, it is not
apriori clear how to make a next step that would further deepen our
understanding. In other words, typical CCMs are isolated islands of limited
applicability that have no relation with one another.

Of course, this aspect might be both strength and weakness of the adiabatic
approach. We can use it to zoom in on a particular aspect of the theory, such
as $K\bar{K}$ scattering, and ignore everything else. By construction, a CCM
reflects our _apriori_ knowledge about the solution space. Thus, it is
difficult to be surprised and we cannot use CCMs to discover completely new
phenomena.222Let us, however, point out that the spectral wall phenomenon
Adam:2019xuc  might be a good counterexample for this generalizing statement.

In contrast, in this paper, we present a general-purpose CCM which attempts to
overcome the aforementioned limitations. The three criteria that we demand are
i) _algebraic tractability_ for an $N$-point CCM, ii) _exhaustiveness_ , e.g.
that the CCM approaches the field theory as $N\to\infty$, and iii)
_qualitative_ agreement between dynamical phenomena. Note that we do not (yet)
demand _quantitative_ agreement but rather we regard it as an ultimate test.

In particular, in this paper, we present _simplest_ (that we can imagine)
general-purpose CCM that passes these criteria. However, we do not claim that
it is also an optimal approach; indeed, there are deficiencies that we will
point out. We regard our construction as a proof of concept.

We call our approach _mechanization_ ; we replace a smooth field $\phi(x,t)$
by piece-wise linear segments – i.e. a linear graph with a given number of
‘joints’ (vertices) connected with straight ‘segments’ (edges). The positions
of joints and the associated field values (populations on the graph) are the
time-dependent CCs. In other words, the continuous field – upon mechanization
– becomes a jagged line resembling an axle connecting wheels on a steam train
(see Fig. 2).

Despite conceptual simplicity, the mechanization is surprisingly successful in
capturing essential phenomena of scalar field theories, such as bouncing and
bion pair-production in $K\bar{K}$ scattering. As it does not require any
prior knowledge, we can in fact _discover_ these phenomena by gradual
exploration of ‘mech-models’. Indeed, the first two smallest values of $N$
reveals the mech-kink solution (a mechanical analog of a field-theoretical
kink) and the mech-oscillon (an analog of oscillon) with the same CCMs as
those in RiceMJ  and Eq. (4) without any prior insight. Furthermore, bouncing
phenomenon, bion pair-production as well as more intricate patterns can be
easily spotted in decay states of higher $N$ mech-oscillons. As these ‘mech-
phenomena’ gradually appear in an ordered sequence in complexity, we can use
mechanization as a tool for _systematic_ exploration of a scalar field theory.

The requirement for algebraic tractability is essential to this feature. As we
will show, mechanization yields compact formulae for an arbitrary number of
joints. In particular, we can write down the effective Lagrangian, the
equations of motion and the metric for $N$ joints explicitly.

In Sec. II we describe the mechanization method, the effective Lagrangian, its
symmetries, and the metric. In Sec. III we investigate mech-models with
topological boundary conditions – mech-kinks, while in Sec. IV we turn our
attention to topologically trivial configurations – mech-oscillons. Lastly, in
Sec. V we summarize our findings and discuss limitations and possible
improvements.

##  II Mechanization

###  II.1 Mech-field

Figure 2: A depiction of a mech-field $\phi_{M}(x,t)$ as a sequence of $N$
straight segments connected via free-moving joints.

_Mechanization_ is simply a replacement of a continuous field $\phi(x,t)$ with
a piece-wise linear function which we dub a _mech-field_ $\phi_{M}(x,t)$. We
express it mathematically as333We use the symbol $\xrightarrow[M]{}$ to
indicate the mechanization of a continuous variable.

$\phi(x,t)\xrightarrow[M]{}\phi_{M}(x,t)\equiv\sum\limits_{a=-1}^{N}\bigl{(}k_{a+1}(t)(x-x_{a}(t))+\phi_{a}(t)\bigr{)}\chi_{a}\,,$
(5)

where $x_{a}(t)$ are positions of $N+1$ joints,
$\phi_{a}(t)\equiv\phi(x_{a}(t))$ are the field values and

$k_{a+1}\equiv\frac{\Delta\phi_{a}}{\Delta
x_{a}}=\frac{\phi_{a+1}-\phi_{a}}{x_{a+1}-x_{a}}=\tan\bigl{(}\theta_{a}(t)\bigr{)}$
(6)

are the slopes of $a$-th segments (see Fig. 2). For our definition of mech-
field to work, we formally assign joints to both spatial infinities, namely
$x_{-1}=-\infty$ and $x_{N+1}=+\infty$. Furthermore, we always assume that the
outermost segments are horizontal and lie in some vacua, i.e.
$k_{0}=k_{N+1}=0$ and $\phi_{-1}=\phi_{0}=v_{\rm L}$,
$\phi_{N}=\phi_{N+1}=v_{\rm R}$, where $v_{\rm L(R)}$ are left (right) vacuum
values. Also, the positions of segments form an ordered sequence $x_{0}(t)\leq
x_{1}(t)\leq\ldots\leq x_{N}(t)$.

The $\chi_{a}$’s are the indicator functions for each segment, namely
$\chi_{a}=1$ inside the interval $(x_{a},x_{a+1})$ and zero outside:

$\chi_{a}\equiv\theta(x-x_{a})-\theta(x-x_{a+1})=-\Delta\theta(x-x_{a})\,,$
(7)

where the Heaviside step function is defined by

$\theta(x)=\lim_{\varepsilon\to
0^{+}}\int\limits_{-\infty}^{\infty}\frac{\mathrm{d}\omega}{2\pi\mathrm{i}}\frac{\mathrm{e}^{\mathrm{i}\omega
x}}{\omega-\mathrm{i}\varepsilon}=\left\\{\begin{matrix}1&\mbox{ if }x>0\\\
0&\mbox{ if }x<0\end{matrix}\right.$ (8)

The value $\theta(0)$ is left unspecified, but it has no impact on physics.

In certain sense, it might seem that there are two many variables. A standard
way of discretization of space would be to replace a continuum variable $x$
with a lattice or in general with a fixed grid. In contrast, in our approach,
the grid is itself dynamic. A dynamic grid is often employed to improve the
precision of numerical integration of differential equations. There, however,
the choice of grid spacing is dictated by optimization, while in our case the
dynamics originate by construction.

###  II.2 From mech-field to field

It is easy to imagine a formal limit that give us back the original continuous
field variable $\phi(x,t)$. If we send $N\to\infty$ in such a way that the
joints become dense on the entire $x$-axis, i.e. $x_{a}(t)\to\xi$, the
summation becomes an integration:

$\displaystyle\phi_{M}(x,t)=\sum\limits_{a=-1}^{N}\bigl{(}k_{a+1}(t)(x-x_{a}(t))+\phi_{a}(t)\bigr{)}\chi_{a}$
$\displaystyle\xrightarrow[N\to\infty]{}\int\limits_{-\infty}^{\infty}\mathrm{d}\xi\,\bigl{(}\phi^{\prime}(\xi)(x-\xi)+\phi(\xi)\bigr{)}\delta(x-\xi)=\phi(x)\,,$
(9)

where we used the following correspondence: $k_{a+1}\to\phi^{\prime}(\xi)$ and
$\chi_{a}\to\delta(x-\xi)\mathrm{d}\xi$ as $N\to\infty$.

Even without taking the continuous limit, we can directly relate the discrete
variables $\\{x_{a}(t),\phi_{a}(t)\\}$ and the time-dependent Fourier
coefficients of $\phi(x,t)$ through Eq. (8). In particular, after some
algebra, we have

$\phi(x,t)\xrightarrow[M]{}\frac{1}{2}\int\limits_{-\infty}^{\infty}\frac{\mathrm{d}\omega}{2\pi}\bigl{(}Y(\omega)\mathrm{e}^{\mathrm{i}\omega
x}+Y^{*}(\omega)\mathrm{e}^{-\mathrm{i}\omega x}\bigr{)}+\frac{v_{\rm
L}+v_{\rm R}}{2}\,,$ (10)

where we denoted

$Y(\omega)\equiv-\frac{1}{\omega^{2}}\sum\limits_{a}^{N}\Delta
k_{a}(t)\mathrm{e}^{-\mathrm{i}\omega x_{a}(t)}\,.$ (11)

Notice that $Y(\omega)$ are nothing but Fourier coefficients of the mech-field
$\phi_{M}(x,t)-v_{\rm L}$, while $Y^{*}(\omega)$ are Fourier coefficients of
$\phi_{M}(x,t)-v_{\rm R}$.

It is worth mentioning that one can assign a set of $2N$ discrete variables
$\\{x_{a}(t),\phi_{a}(t)\\}$ to a given function $\phi(x,t)$. This can be
achieved by expanding $Y(\omega)$ into a Taylor series in $\omega$ up to
$2N$-th power and matching the result with equivalent series for the Fourier
image of $\phi(x,t)-v_{\rm L}$. From this point of view, mechanization can be
roughly understood as an attempt to faithfully represent small frequencies
(long wavelengths) using the most simple functions available – piece-wise
linear functions.

###  II.3 Effective Lagrangian

We derive the effective Lagrangian by plugging the formula (5) into a
canonical Lagrangian for a single scalar field

${\mathcal{L}}=\frac{1}{2}{\dot{\phi}}^{2}-\frac{1}{2}\phi^{\prime\,2}-V(\phi)\,,$
(12)

where the potential has two degenerate minima $v_{\rm L}$ and $v_{\rm R}$,
i.e. $V(v_{\rm L,R})=V^{\prime}(v_{\rm L,R})=0$.

The integration procedure is straightforward and yields compact formulas. Let
us illustrate it on $\phi_{M}^{\prime\,2}/2$. If we adopt the derivative
$\phi_{M}^{\prime}$ as a differential consequence of (5)

$\displaystyle\phi^{\prime}(x,t)\xrightarrow[M]{}\sum\limits_{a=-1}^{N}k_{a+1}\chi_{a}\equiv\phi_{M}^{\prime}(x,t)\,,$
(13)

it follows that

$\displaystyle\frac{1}{2}\int\limits_{-\infty}^{\infty}\mathrm{d}x\,\phi^{\prime\,2}$
$\displaystyle\xrightarrow[M]{}\frac{1}{2}\sum\limits_{a,b}k_{a+1}k_{b+1}\int\limits_{-\infty}^{\infty}\mathrm{d}x\,\chi_{a}\chi_{b}$
$\displaystyle=\frac{1}{2}\sum\limits_{a=0}^{N-1}k_{a+1}^{2}(x_{a+1}-x_{a})\,,$
(14)

where we have used an obvious identity $\chi_{a}\chi_{b}=\chi_{a}\delta_{ab}$.
Notice that the resulting formula only involves the coupling of neighboring
joints, and it is also manifestly positive (as $x_{a+1}\geq x_{a}$ by
definition).

Similarly, we calculate the kinetic energy

$\frac{1}{2}\int\limits_{-\infty}^{\infty}\mathrm{d}x\,\dot{\phi}^{2}\xrightarrow[M]{}\frac{1}{6}\sum\limits_{a=-1}^{N}\Delta
x_{a}\Bigl{(}\Phi_{a+1,a}^{2}+\Phi_{a+1,a}\Phi_{a,a}+\Phi_{a,a}^{2}\Bigr{)}\,,$

where we denoted $\Phi_{a,b}\equiv\dot{\phi}_{a}-k_{b+1}\dot{x}_{a}$ for
convenience. The above expression can be recast in manifestly positive form as

$\frac{1}{2}\int\limits_{-\infty}^{\infty}\mathrm{d}x\,\dot{\phi}^{2}\xrightarrow[M]{}\frac{1}{12}\sum\limits_{a=0}^{N-1}\Delta
x_{a}\Bigl{(}\bigl{(}\Phi_{a+1,a}+\Phi_{a,a}\bigr{)}^{2}\\\
+\Phi_{a+1,a}^{2}+\Phi_{a,a}^{2}\Bigr{)}\,,$ (15)

while most compact explicit formula reads

$\frac{1}{2}\int\limits_{-\infty}^{\infty}\mathrm{d}x\,\dot{\phi}^{2}\xrightarrow[M]{}\frac{1}{6}\sum\limits_{a=0}^{N-1}\dot{k}_{a+1}^{2}\bigl{(}\Delta
x_{a}\bigr{)}^{3}\\\ +\frac{1}{2}\sum\limits_{a=0}^{N-1}\Delta
x_{a}\bigl{(}\dot{\phi}_{a+1}-k_{a+1}\dot{x}_{a+1}\bigr{)}\bigl{(}\dot{\phi}_{a}-k_{a+1}\dot{x}_{a}\bigr{)}\,.$
(16)

Lastly, the potential energy is calculated as:

$\displaystyle\int\limits_{-\infty}^{\infty}\mathrm{d}x\,V(\phi)$
$\displaystyle\xrightarrow[M]{}\int\limits_{-\infty}^{\infty}\mathrm{d}x\,V\Bigl{(}\sum\limits_{a=-1}^{N}\bigl{(}k_{a+1}(x-x_{a})+\phi_{a}\bigr{)}\chi_{a}\Bigr{)}$
$\displaystyle=\sum\limits_{a=-1}^{N}\int\limits_{x_{a}}^{x_{a+1}}\mathrm{d}x\,V\Bigl{(}k_{a+1}(x-x_{a})+\phi_{a}\Bigr{)}$
$\displaystyle=\sum\limits_{a=-1}^{N}\frac{1}{k_{a+1}}\int\limits_{\phi_{a}}^{\phi_{a+1}}\mathrm{d}\xi\,V(\xi)$
$\displaystyle=\sum\limits_{a=0}^{N-1}\Delta
x_{a}\frac{{\mathcal{V}}\bigl{(}\phi_{a+1}\bigr{)}-{\mathcal{V}}\bigl{(}\phi_{a}\bigr{)}}{\phi_{a+1}-\phi_{a}}\,,$
(17)

where ${\mathcal{V}}$ is the primitive function of $V$. Notice that this is,
indeed, positive: ${\mathcal{V}}$ is an increasing function and hence
${\mathcal{V}}\bigl{(}\phi_{a+1}\bigr{)}-{\mathcal{V}}\bigl{(}\phi_{a}\bigr{)}$
will be either positive or negative if $\phi_{a+1}-\phi_{a}$ is positive or
negative.

Combining all our results, we can write the ‘mech-Lagrangian’ compactly as

$\displaystyle\int\limits_{-\infty}^{\infty}\mathrm{d}x\,{\mathcal{L}}\xrightarrow[M]{}\
L_{M}\,.$ $\displaystyle L_{M}=\sum\limits_{a=0}^{N-1}\Delta
x_{a}\biggl{[}\frac{1}{6}\Bigl{(}\Delta\dot{\phi}_{a}-\frac{\Delta\dot{x}_{a}}{\Delta
x_{a}}\Delta\phi_{a}\Bigr{)}^{2}-\frac{\Delta\phi_{a}^{2}}{2\Delta x_{a}^{2}}$
$\displaystyle+\frac{1}{2}\Bigl{(}\dot{\phi}_{a+1}-\frac{\Delta\phi_{a}}{\Delta
x_{a}}\dot{x}_{a+1}\Bigr{)}\Bigl{(}\dot{\phi}_{a}-\frac{\Delta\phi_{a}}{\Delta
x_{a}}\dot{x}_{a}\Bigr{)}-\frac{\Delta{\mathcal{V}}\bigl{(}\phi_{a}\bigr{)}}{\Delta\phi_{a}}\biggr{]}\,.$
(18)

###  II.4 Symmetries

Let us consider the relation between symmetries of the field theory (12) and
the mech-model (18).

First, the conserved quantities related to translational invariance of
spacetime, i.e. $x^{\mu}\to x^{\mu}+a^{\mu}$ – energy and momentum – are
preserved under mechanization. If we mechanize the continuous formulae we
obtain the discrete momentum

$\displaystyle P$
$\displaystyle=-\int\limits_{-\infty}^{\infty}\mathrm{d}x\,\dot{\phi}\phi^{\prime}\xrightarrow[M]{}\sum\limits_{a=0}^{N-1}\Delta\phi_{a}\bar{U}_{a}\equiv\sum\limits_{a=0}^{N-1}P_{a}\equiv
P_{M}$ (19)

and the discrete energy as

$\displaystyle E$
$\displaystyle=\int\limits_{-\infty}^{\infty}\mathrm{d}x\,\Bigl{(}\frac{1}{2}\dot{\phi}^{2}+\frac{1}{2}\phi^{\prime\,2}+V(\phi)\Bigr{)}$
$\displaystyle\xrightarrow[M]{}\sum\limits_{a=0}^{N-1}\biggl{(}\frac{1}{24}\dot{k}_{a+1}^{2}\bigl{(}\Delta
x_{a}\bigr{)}^{3}+\frac{1}{2}\bar{U}_{a}^{2}\Delta
x_{a}+\frac{1}{2}k_{a+1}^{2}\Delta x_{a}$ $\displaystyle+\Delta
x_{a}\frac{{\mathcal{V}}(\phi_{a+1})-{\mathcal{V}}(\phi_{a})}{\phi_{a+1}-\phi_{a}}\biggr{)}\equiv
E_{M}\,,$ (20)

where we introduced a quantity

$\displaystyle\bar{U}_{a}$ $\displaystyle\equiv-\frac{1}{\Delta
x_{a}}\int\limits_{x_{a}}^{x_{a+1}}\mathrm{d}x\,\dot{\phi}=-\frac{1}{2}\Bigl{(}\Phi_{a+1,a}+\Phi_{a,a}\Bigr{)}\,,$
(21)

which we loosely interpret as the average velocity of a segment. For future
reference, let us also introduce the notion of segment’s mass, namely twice of
the static free energy:

$M_{a}\equiv\int\limits_{x_{a}}^{x_{a+1}}\mathrm{d}x\,\phi^{\prime\,2}=k_{a+1}^{2}\Delta
x_{a}\,.$ (22)

The total mass of the mech-field is thus

$M\equiv\sum\limits_{a=1}^{N-1}M_{a}=\sum\limits_{a=1}^{N-1}k_{a+1}^{2}\Delta
x_{a}\,.$ (23)

As stated above, both momentum $P_{M}$ and energy $E_{M}$ are conserved
quantities. Indeed, the Lagrangian $L_{M}$ (18) has a translational symmetry
$x_{a}(t)\to x_{a}(t)+\alpha$ and it is also invariant under the time shift
$t\to t+\alpha$. Invoking Nöther’s theorem, we can compute the same
expressions $P_{M}$ and $E_{M}$ directly from $L_{M}$.

Unsurprisingly, the Lorentz invariance is generally lost. It is well known
that invariance under boosts leads to a conserved quantity444This follows from
the conservation law $\partial_{\mu}{\mathcal{M}}^{\mu\nu\rho}=0$, where
${\mathcal{M}}^{\mu\nu\rho}={\mathcal{T}}^{\mu\nu}x^{\rho}-{\mathcal{T}}^{\mu\rho}x^{\nu}\,,$
with ${\mathcal{T}}^{\mu\nu}$ being the canonical energy-momentum tensor.

$J^{01}=-tP+\int\limits_{-\infty}^{\infty}\mathrm{d}x\,{\mathcal{E}}x\,,$ (24)

that represents the uniform motion of the center of mass. A mechanized analog
of this would generally not be a constant of motion.

One may expect that the Galilean boost should still be a good symmetry of
$L_{M}$. However, if we make the transformation $x_{a}(t)\to x_{a}(t)-Vt$ we
find that the Lagrangian changes:

$L_{M}\to L_{M}-P_{M}V+\frac{1}{2}MV^{2}\,.$ (25)

The problem is that the total mass $M$ is generally not a constant of motion,
hence Galilean relativity is lost.

However, we can still relate solutions with different momenta. To do this, let
us redefine the variables as

$x_{a}(t)=\tilde{x}_{a}(t)+r(t)\,,$ (26)

where $r(t)$ is a site-independent variable proportional to the average
position (imposing the condition $\sum_{a}\tilde{x}_{a}=0$). It follows that

$\displaystyle L_{M}(x)$
$\displaystyle=L_{M}(\tilde{x})+P_{M}(\tilde{x})\dot{r}+\frac{1}{2}\dot{r}^{2}M(\tilde{x})\,,$
(27) $\displaystyle P_{M}(x)$
$\displaystyle=P_{M}(\tilde{x})+M(\tilde{x})\dot{r}\,,$ (28)

where we shown explicit dependence on $x=\\{x_{a}\\}$ or
$\tilde{x}=\\{\tilde{x}_{a}\\}$ variables. We may easily identify equation of
motion for $r(t)$ to be

$\frac{\mathrm{d}}{\mathrm{d}t}\Bigl{(}M(\tilde{x})\dot{r}+P_{M}(\tilde{x})\Bigr{)}=0\,.$
(29)

The expression in the brackets is a constant of motion, namely $P_{M}(x)$.
Thus, if we find a solution $\tilde{x}_{a}(t)$ with momentum
$P_{M}(\tilde{x})\equiv\tilde{P}$, we can switch to a different solution with
momentum $P_{M}(x)\equiv P$ via

$x_{a}(t)=\tilde{x}_{a}(t)+(P-\tilde{P})\int\frac{dt}{M(\tilde{x})}\,,$ (30)

where the second term is simply $r(t)$ integrated from its equation of motion.

Lastly, for completeness, let us display the equations of motion:

$\displaystyle\Delta\biggl{[}\frac{\mathrm{d}}{\mathrm{d}t}\biggl{(}\frac{1}{6}\bigr{(}\Delta
x_{a}\bigr{)}^{2}{\dot{k}_{a+1}}^{2}-\frac{1}{2}\Delta\phi_{a}\Phi_{a+1,a}\biggr{)}+\frac{1}{6}\frac{\Delta\dot{\phi}_{a}^{3}}{\Delta\dot{\phi}_{a}}-\frac{1}{6}k_{a+1}^{2}\Bigl{(}\frac{\Delta\dot{x}_{a}^{3}}{\Delta\dot{x}_{a}}-3\Bigr{)}-\frac{\Delta{\mathcal{V}}(\phi_{a})}{\Delta\phi_{a}}\biggr{]}=-\frac{\mathrm{d}}{\mathrm{d}t}P_{a}\,,$
(31)
$\displaystyle\Delta\biggl{[}\frac{\mathrm{d}}{\mathrm{d}t}\biggl{(}-\frac{1}{3}\bigr{(}\Delta
x_{a}\bigr{)}^{2}\dot{k}_{a+1}+\frac{1}{2}\Delta
x_{a}\Phi_{a+1,a}\biggr{)}-\frac{1}{6}\bigl{(}2\Delta\dot{\phi}_{a}\Delta\dot{x}_{a}+3\dot{x}_{a+1}\dot{\phi}_{a}+3\dot{x}_{a}\dot{\phi}_{a+1}\bigr{)}+\frac{1}{3}k_{a+1}\Bigl{(}\frac{\Delta\dot{x}_{a}^{3}}{\Delta\dot{x}_{a}}-3\Bigr{)}+\frac{\Delta
x_{a}}{\Delta\phi_{a}}\frac{\Delta{\mathcal{V}}(\phi_{a})}{\Delta\phi_{a}}\biggr{]}$
$\displaystyle=\frac{\mathrm{d}}{\mathrm{d}t}\Bigl{(}\Delta
x_{a}\bar{U}_{a}\Bigr{)}+V(\phi_{a+1})\Delta\frac{\Delta
x_{a}}{\Delta\phi_{a}}\,.$ (32)

Note that summation of the first line implies conservation of momentum, i.e.
$\dot{P}=\sum\dot{P}_{a}=0$.

###  II.5 Moduli space

Integrating the kinetic energy over the field where certain parameters
(moduli) are promoted to time-dependent variables, i.e.
$\phi\bigl{(}x,t;\\{{\bf X}(t)\\}\bigr{)}$, yields a metric on the moduli
space spanned by the coordinates $\\{{\bf X}(t)\\}$.

Our variables are $N+1$ positions $\\{x_{a}(t)\\}$ and $N-1$ field values
$\\{\phi_{a}(t)\\}$ (the two outermost ones are forced to lie on vacua), i.e.
$\\{{\bf X}(t)\\}\equiv\\{x_{a}(t),\phi_{a}(t)\\}$ giving rise to
$2N$-dimensional moduli space.

Studying the metric is especially useful for understanding the singularities
of the moduli space. Given our heuristic choice of coordinates, we have no
right to expect that the moduli space will be geodetically complete. Indeed,
it is not. However, unlike most $N$-point CCM’s, we can write down our metric
on the back of an envelope:555To avoid awkward formulae, we extend the
summation ranges for $\phi$ coordinates, although
$\dot{\phi}_{0}=\dot{\phi}_{N}=0$.

$\frac{1}{2}\int\limits_{-\infty}^{\infty}\mathrm{d}x\,\dot{\phi}_{M}^{2}=\frac{1}{2}\dot{\bf
X}^{T}{\bf g^{XX}}\dot{\bf X}\\\
=\frac{1}{2}\sum\limits_{a,b=0}^{N}\Bigl{(}g_{ab}^{xx}\dot{x}_{a}\dot{x}_{b}+2g_{ab}^{x\phi}\dot{x}_{a}\dot{\phi}_{b}+g_{ab}^{\phi\phi}\dot{\phi}_{a}\dot{\phi}_{b}\Bigr{)}\,,$
(33)

where

$\displaystyle g_{aa}^{xx}=\frac{1}{3}\frac{(\Delta\phi_{a})^{2}}{\Delta
x_{a}}+\frac{1}{3}\frac{(\Delta\phi_{a-1})^{2}}{\Delta x_{a-1}}\,,$ (34)
$\displaystyle g_{a\,a+1}^{xx}=\frac{1}{6}\frac{(\Delta\phi_{a})^{2}}{\Delta
x_{a}}\,,\hskip
8.53581ptg_{a\,a-1}^{xx}=\frac{1}{6}\frac{(\Delta\phi_{a-1})^{2}}{\Delta
x_{a-1}}\,,$ (35) $\displaystyle
g_{aa}^{x\phi}=-\frac{1}{3}\bigl{(}\phi_{a+1}-\phi_{a-1}\bigr{)}\,,\hskip
8.53581ptg_{a\,a+1}^{x\phi}=-\frac{1}{6}\Delta\phi_{a}\,,$ (36) $\displaystyle
g_{a\,a-1}^{x\phi}=-\frac{1}{6}\Delta\phi_{a-1}\,,\hskip
8.53581ptg_{aa}^{\phi\phi}=\frac{1}{3}\bigl{(}x_{a+1}-x_{a-1}\bigr{)}\,$ (37)
$\displaystyle g_{a\,a+1}^{\phi\phi}=\frac{1}{6}\Delta x_{a}\,,\hskip
8.53581ptg_{a\,a-1}^{\phi\phi}=\frac{1}{6}\Delta x_{a-1}\,.$ (38)

The components $g^{xx}$, $g^{x\phi}$ and $g^{\phi\phi}$ are tri-diagonal due
to only neighboring interactions.

There are many singularities in this metric, but it is quite straightforward
to appreciate why. Most apparently, when the $x$-distance of neighboring
joints becomes zero, i.e. $\Delta x_{a}=0$, the components of $g^{xx}$
diverge. Indeed, when $\Delta x_{a}=0$ and $\phi_{a}\not=\phi_{a+1}$ the mech-
field becomes multi-valued.666Let us note that when $\Delta x_{a}\to 0$ and
$\Delta\phi_{a}\to 0$, such that the ratio $(\Delta\phi_{a})^{2}/\Delta x_{a}$
is kept fixed, the metric remains completely regular. However, this is not
true for components of the Riemann tensor, which diverge. It is possible to
remove these singularities to infinity by a change of coordinates, e.g.

$x_{a}(t)=x_{0}(t)+\sum\limits_{c=0}^{a}\mathrm{e}^{b_{c}(t)}\,,$ (39)

so that $\Delta x_{a}=\mathrm{e}^{b_{a+1}}$. This also enforces the ordering
$x_{0}<x_{1}<\ldots x_{N}$. We have used these coordinates in all our
numerical calculations.

A more subtle issue arises when a segment becomes flat, i.e.
$\Delta\phi_{a}=0$. If the segment in question is not on the edges, then the
metric remains well-defined. In fact, $g^{xx}$ and $g^{x\phi}$ become block-
diagonal, suggesting dynamical decoupling of the left and right parts of the
mech-field, in agreement with intuition. If the segment is on the border, i.e.
$\Delta\phi_{0}=0$ or $\Delta\phi_{N}=0$, this represents a sudden uncoupling
of an edge joint. Both cases manifest an abrupt change in the mech-field,
which is confirmed by looking at the formula for the determinant:

$\bigl{|}{\bf
g^{XX}}\bigr{|}=\frac{1}{12^{N}}\prod\limits_{a=0}^{N-1}\frac{(\Delta\phi_{a})^{4}(\Delta\phi_{a}\Delta
x_{a+1}-\Delta\phi_{a+1}\Delta x_{a})^{2}}{(\Delta x_{a})^{2}}\,.$ (40)

As we see, the volume measure vanishes if any $\Delta\phi_{a}=0$.

We can also observe that when neighboring slopes become identical, i.e.
$k_{a}=k_{a+1}$, the determinant vanishes too. However, these instances seem
to be coordinate singularities. We checked that for the first few $N$ the
expression $\sqrt{{\bf g^{XX}}}{\bf R^{XX}}$, where ${\bf R^{XX}}$ is the
Ricci scalar, is well defined in the limit of equal subsequent slopes.

##  III Mech-kink

In this section, we shall begin a systematic investigation of solutions of
mech-model (18) for increasing value of $N$. Here we mostly focus on static
solutions of topological configurations, that we call mech-kinks. As we shall
see, the model for a simplest mech-kink is totally integrable and formally
equivalent to the relativistic CCM of a BPS kink Adam:2021gat .

###  III.1 $N=1$ mech-kink

Figure 3: Simplest mechanical model of a kink = ‘mech-kink’.

Let us consider $N=1$ case with topological boundary conditions, i.e.
$\phi_{0}=v_{\rm L}\not=\phi_{1}=v_{\rm R}$, which we call a _mech-kink_ (see
Fig. 3). The Lagrangian (18) for $N=1$ is explicitly given by

$L_{M}^{N=1}=\frac{\bigl{(}\Delta\phi_{0}\bigr{)}^{2}}{24\Delta
x_{0}}\Bigl{(}\bigl{(}\Delta\dot{x}_{0}\bigr{)}^{2}-12+12\,\dot{\bar{x}}^{2}\Bigr{)}-\kappa\Delta
x_{0}\,,$ (41)

where we have denoted the average position $\bar{x}\equiv(x_{0}+x_{1})/2$ and
the constant

$\kappa\equiv\frac{1}{v_{\rm R}-v_{\rm L}}\int\limits_{v_{\rm L}}^{v_{\rm
R}}\mathrm{d}\xi\,V(\xi)\,.$ (42)

The equations of motion reads

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\Bigl{(}\frac{(\Delta\phi_{0})^{2}\Delta\dot{x}_{0}}{12\Delta
x_{0}}\Bigr{)}$
$\displaystyle=-\frac{\bigl{(}\Delta\phi_{0}\bigr{)}^{2}}{24(\Delta
x_{0})^{2}}\Bigl{(}\bigl{(}\Delta\dot{x}_{0}\bigr{)}^{2}-12+12\,\dot{\bar{x}}^{2}\Bigr{)}-\kappa\,,$
(43)
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\Bigl{(}\frac{(\Delta\phi_{0})^{2}\dot{\bar{x}}}{\Delta
x_{0}}\Bigr{)}$ $\displaystyle=0\,.$ (44)

The two conserved quantities reads:

$\displaystyle E_{M}^{N=1}$
$\displaystyle=\frac{\bigl{(}\Delta\phi_{0}\bigr{)}^{2}}{24\Delta
x_{0}}\Bigl{(}\bigl{(}\Delta\dot{x}_{0}\bigr{)}^{2}+12+12\,\dot{\bar{x}}^{2}\Bigr{)}+\kappa\Delta
x_{0}\,,$ (45) $\displaystyle P_{M}^{N=1}$
$\displaystyle=\frac{\bigl{(}\Delta\phi_{0}\bigr{)}^{2}}{\Delta
x_{0}}\dot{\bar{x}}\,.$ (46)

Note that $\Delta\phi_{0}=v_{\rm R}-v_{\rm L}$ is a constant.

Let us first consider a static mech-kink: $\Delta\dot{x}_{0}=\dot{\bar{x}}=0$.
Solving the equations of motion (43)-(44) we find the energy $E_{M,\,{\rm
static}}^{N=1}\equiv m_{K}$ and the width $R_{K}\equiv\Delta x_{0}$ to be

$m_{K}=(v_{\rm R}-v_{\rm L})\sqrt{2\kappa}\,,\hskip
14.22636ptR_{K}=\frac{v_{\rm R}-v_{\rm L}}{\sqrt{2\kappa}}\,.$ (47)

For concreteness, if we consider $\phi^{4}$ potential
$V(\phi)=(1-\phi^{2})^{2}/2$ we have $v_{\rm R}=-v_{\rm L}=1$ and
$\kappa=4/15$. The corresponding numbers for a static mech-kink reads
$m_{K}=\sqrt{32/15}$ and $R_{K}=\sqrt{15/2}$. Notice that mech-kink’s mass is
equal to the total mass of the mech-field, i.e.
$m_{K}=M_{0}=(\Delta\phi_{0})^{2}/\Delta x_{0}$.

Surprisingly, the Lagrangian $L_{M}^{N=1}$ is formally equivalent to a
relativistic CCM for a BPS kink (Eq. (II.17) of Adam:2021gat ):

$L[a,b]=\frac{1}{2}Mb\dot{a}^{2}+\frac{Q}{2b^{3}}\dot{b}^{2}-\frac{1}{2}M\Bigl{(}b+\frac{1}{b}\Bigr{)},$
(48)

which is obtained in $\phi^{4}$ theory using the anstaz
$\phi_{K}=\tanh\bigl{(}b(x-a)\bigr{)}$. Here, $M=4/3$ is the kink’s mass and
$Q$ is the second moment of static kink’s energy density:

$Q=\int\limits_{-\infty}^{\infty}\mathrm{d}x\,x^{2}\phi_{K}^{\prime\,2}(x)=\frac{\pi^{2}-6}{9}\,.$
(49)

The similarity of (41) and (48) becomes explicit if we set

$\Delta x_{0}=\frac{R_{K}}{b}\,,\hskip 14.22636pt\bar{x}=a$ (50)

The Lagrangian $L_{M}^{N=1}$ switches to a form:

$L_{M}[a,b]=\frac{1}{2}m_{k}b\dot{a}^{2}+\frac{q_{M}}{2b^{3}}\dot{b}^{2}-\frac{1}{2}m_{k}\Bigl{(}b+\frac{1}{b}\Bigr{)}\,,$
(51)

where

$q_{M}\equiv\int\limits_{-\infty}^{\infty}\mathrm{d}x\,x^{2}\phi_{M}^{\prime\,2}(x)=\frac{1}{12}R_{K}\Delta\phi_{0}^{2}\,.$
(52)

For $\phi^{4}$ model $m_{K}=\sqrt{32/15}\approx 1.46$ which is not far off
from the BPS value $4/3\approx 1.33$. However, $q_{M}=\sqrt{5/6}\approx 0.91$
is more than twice of the field-theoretical value $Q\approx 0.43$.

Let us stress that apriori we had no right to expect this formal
correspondence. The mech-field $\phi_{M}^{N=1}$ is not a BPS solution of the
field theory, yet we see _precisely_ the same effective Lagrangian as for the
BPS kink. There seems to be a certain universality of CCMs, regarding the
structure of terms. Indeed, it is easy to show that the same effective
Lagrangian as in (48) arises for any background $\phi=f(b(x-a))$ provided that
$f^{\prime\,2}$ has finite first three moments. Thus, the key ingredient is
not the shape of the solution but the inclusion of a scaling modulus. It is
somewhat of an accident that $N=1$ mech-field also falls into this class of
backgrounds, as the scaling modulus appears as the slope of the middle
segment.

Given the exact correspondence with relativistic CCM, it is not surprising
that we will find the same results, namely the Lorentz covariance and the
existence of a Derrick mode. However, let us stress that from the point of
view of the mech-model both results are very unexpected!

Indeed, if we consider a mech-kink moving with a uniform velocity $v$, i.e.
$\Delta\dot{x}=0$, $\dot{\bar{x}}=v$, the equations of motion (43)-(44) can be
easily solved. We find that energy $E_{M}^{N=1}\equiv E_{K}$ and width $\Delta
x_{0}$ follow formulae for a relativistic particle:

$E_{K}=\frac{m_{K}}{\sqrt{1-v^{2}}}\,,\hskip 14.22636pt\Delta
x_{0}=R_{K}\sqrt{1-v^{2}}$ (53)

Let us now fix the center position to the origin ${\bar{x}}=0$. If the energy
$E_{K}$ is above the static energy, i.e. $E_{K}>m_{K}$, the mech-kink
oscillates with angular frequency $\omega_{M}=\sqrt{12}/R_{K}$

$\Delta
x_{0}=\frac{E_{K}}{2\kappa}-\frac{\sqrt{E_{K}^{2}-m_{k}^{2}}}{2\kappa}\sin\bigl{(}\omega_{M}t\bigr{)}\,.$
(54)

This vibrational mode is independent on the shape of the potential. Thus, it
is not a shape-mode – a massive normal mode of the kink. Indeed, not all kinks
have the shape mode as is well known Dorey:2011yw . This mode is more
appropriately identified with the so-called Derrick mode, which arises due to
infinitesimal scaling of the static solution and which exists for all kinks
Adam:2021gat .

In the field theory, the frequency of the Derrick mode is
$\omega_{D}^{2}=M/Q$. In the mech-model, it is again formally the same
$\omega_{M}^{2}=m_{K}/q_{M}$. Specifically, in $\phi^{4}$ theory,
$\omega_{D}^{2}\approx 3.1$ and $\omega_{M}^{2}\approx 4.4$.

We can construct a general solution of (43)-(44) as we have the same number of
unknowns, namely $\Delta x_{0}$ and $\bar{x}$, and constants of motion, i.e.
$E_{K}$ and $P_{K}$. Indeed, Eq. (44) is equivalent to $\dot{P}_{K}=0$ which
implies the conservation of the momentum. Furthermore, Eq. (44) can be
linearized as

$\Delta\ddot{x}_{0}=\frac{12E_{K}}{(\Delta\phi_{0})^{2}}-\frac{12}{(\Delta\phi_{0})^{2}}\Delta
x_{0}\Bigl{(}\kappa+\frac{P_{K}^{2}}{(\Delta\phi_{0})^{2}}\Bigr{)}\,.$ (55)

The general solution thus reads

$\Delta x_{0}=R_{K}\frac{m_{K}}{P_{K}^{2}+m_{K}^{2}}\biggl{(}E_{K}\\\
-\sqrt{E_{K}^{2}-P_{K}^{2}-m_{K}^{2}}\sin\Bigl{(}\frac{\sqrt{12}t\sqrt{P_{K}^{2}+m_{K}^{2}}}{m_{K}R_{K}}\Bigr{)}\biggr{)}\,.$
(56)

###  III.2 $N>1$ static mech-kinks

Figure 4: An example of joint-ejection for $N=2$ mech-kink, where the third
joint escapes to $\infty$. Figure 5: A symmetric example where the outermost
joints of $N=4$ mech-kink get ejected, leaving behind slightly excited $N=2$
mech-kink.

All non-trivial static configurations are mech-kinks, by which we mean any
configuration for which the outer segments lie in different vacua.777We
exclude the possibility of an inner segment lying in a vacuum as that would
lead to decoupled mech-kink and anti-mech-kink solutions. These solutions
nevertheless exist and can be considered as exact free mech-kink gas
solutions. For instance, we depict $N=2$ mech-kink on Fig. 6.

Figure 6: Schematic depiction of $N=2$ mech-kink.

The static equations of motion translate to the following conditions for the
lengths of the segments

$\displaystyle\Delta x_{0}$ $\displaystyle=\sqrt{\frac{(\phi_{1}-v_{\rm
L})^{3}}{2\bigl{(}{\mathcal{V}}(\phi_{1})-{\mathcal{V}}(v_{\rm
L})\bigr{)}}}\,,$ (57) $\displaystyle\Delta x_{1}$
$\displaystyle=\sqrt{\frac{(v_{\rm
R}-\phi_{1})^{3}}{2\bigl{(}{\mathcal{V}}(v_{\rm
R})-{\mathcal{V}}(\phi_{1})\bigr{)}}}\,,$ (58)

while the field values satisfy an algebraic equation

$k_{0}-k_{1}+V(\phi_{1})\biggl{(}\frac{1}{k_{0}}-\frac{1}{k_{1}}\biggr{)}=0\,,$
(59)

where $k_{0}=(\phi_{1}-v_{\rm L})/\Delta x_{0}$ and $k_{1}=(v_{\rm
R}-\phi_{1})/\Delta x_{1}$ are the slopes of the segments. This condition can
be reduced to a simpler form, namely $V(\phi_{1})=k_{0}k_{1}$, provided that
$k_{0}\not=k_{1}$. If $k_{0}=k_{1}$ we would simply get back $N=1$ mech-kink
solution.888This is true also for trivial solutions $\phi_{1}=v_{\rm L}$ and
$\phi_{1}=v_{\rm R}$.

To be concrete, let us consider $\phi^{4}$ model. We found two minimum-energy
solutions that are $\Delta x_{0}\leftrightarrows\Delta x_{1}$,
$\phi_{1}\leftrightarrows-\phi_{1}$ reflections of each other (see Fig. 7):

$\displaystyle\phi_{1}$ $\displaystyle=\frac{\pm
1}{12}\sqrt{139-\sqrt{3865}}\approx\pm 0.73\,,$ (60) $\displaystyle\Delta
x_{0}$
$\displaystyle=\frac{1}{80}\Bigl{(}3\sqrt{1373+7\sqrt{3865}}\pm\sqrt{3\bigl{(}4519-59\sqrt{3865}\bigr{)}}\Bigr{)}$
$\displaystyle\approx 1.594\pm 0.631\,,$ (61) $\displaystyle\Delta x_{1}$
$\displaystyle=\frac{1}{80}\Bigl{(}3\sqrt{1373+7\sqrt{3865}}\mp\sqrt{3\bigl{(}4519-59\sqrt{3865}\bigr{)}}\Bigr{)}$
$\displaystyle\approx 1.594\mp 0.631\,.$ (62)

Their energies are the same

$E_{K,\pm}=\frac{1}{216}\sqrt{\frac{1}{10}(989543-773\sqrt{3865})}\approx
1.42\\\ <E_{K}=\sqrt{\frac{32}{15}}\approx 1.46$ (63)

and are roughly 3% smaller than the energy of $N=1$ mech-kink. This is
understandable as adding more segments should get us closer to the exact kink
mass $M=4/3\approx 1.33$.

Figure 7: Comparison between exact kink solution in the double-well model and
two static solutions for $N=2$ mech-kinks.

For $N=3$, there is only a single minimum-energy solution which we show in
Fig. 8. Its energy is $E_{K}^{N=3}\approx 1.37$.

Figure 8: Comparison between the kink solution in the double-well model and a
static $N=3$ mech-kink.

For $N=4$, we find again two mirror-image solutions with energies
$E_{K}^{N=4}\approx 1.36$. This pattern repeats. For odd $N$ we get a unique
solution, while for even $N$ we find two degenerate solutions.

The static equations of motion (see Eqs. (31)-(32)) for arbitrary $N$ reduces
to

$\displaystyle k_{a+1}\equiv\frac{\Delta\phi_{a}}{\Delta
x_{a}}=\sqrt{\frac{2\bigl{(}{\mathcal{V}}(\phi_{a+1})-{\mathcal{V}}(\phi_{a})\bigr{)}}{\phi_{a+1}-\phi_{a}}}\,,$
(64) $\displaystyle k_{a+1}k_{a}=2V(\phi_{a})\,.$ (65)

Notice that in the continuous limit, the first formula becomes the BPS
equation for a kink, i.e. $\phi^{\prime}=\sqrt{2V(\phi)}$. We can simplify the
above system to a set of $N-1$ algebraic equations for $\phi_{a}$’s, namely

$V(\phi_{a})=\sqrt{\frac{\bigl{(}{\mathcal{V}}(\phi_{a+1})-{\mathcal{V}}(\phi_{a})\bigr{)}\bigl{(}{\mathcal{V}}(\phi_{a})-{\mathcal{V}}(\phi_{a-1})\bigr{)}}{\bigl{(}\phi_{a+1}-\phi_{a}\bigr{)}\bigl{(}\phi_{a}-\phi_{a-1}\bigr{)}}}\,.$
(66)

It is also easy to see that every static solution has its boosted version,
meaning that the energy is equal to the static energy times the Lorentz
$\gamma$ factor, the field values are unchanged, while $\Delta x_{a}$’s are
contracted by $\sqrt{1-v^{2}}$.

###  III.3 Dynamics of mech-kinks

There are several questions about mech-kinks that interest us which are,
however, outside the scope of this paper. For instance, we would like to know
how fast the static energy approaches the BPS bound as a function of $N$.

Regarding the dynamics, an important query for $N\geq 2$ mech-kinks is whether
there exist any exact periodic solutions. We have not been able to find the
answer analytically. Numerically, however, we have glimpsed a promising
candidate (see Fig. 16).

A related problem is the investigation of normal modes of the mech-kinks. In
particular, we would like to study how the spectrum of small fluctuations
varies with increasing $N$, how many spurious modes there are (compared with
field theory), etc. In Fig. 15, we show a numerical solution of a slightly
perturbed static $N=2$ mech-kink indicating the presence of at least one
normal mode.

Ultimately, we would like to categorize the dynamics of mech-kinks for a vast
set of initial conditions to obtain a robust understanding of Cauchy’s problem
for each $N$. This task, however, is too time-consuming for our purpose here.
At present, we have only sampled the evolution of a few $N\geq 2$ mech-kinks
for random initial conditions via numerical integration of equations of
motion.

One phenomenon that we found to be endemic for all $N\geq 2$ mech-fields is
the _joint-ejection_ , i.e. when one of the outer joints rapidly approaches
vacuum and flies either to the left or right infinity, leaving behind an
effective $N-1$ mech-field. We observe this already for $N=2$ mech-kinks,
where the left-over piece is an excited $N=1$ mech-kink (see Fig. 4).
Curiously, two joint-ejections can happen simultaneously for the boundary
joints, leaving behind $N-2$ mech-field (see Fig. 5).

The joint-ejection is indicative of a general tendency for a mech-field to
simplify itself as time increases. In fact, it is not unreasonable to think
that all initial configurations for mech-kinks eventually settle to an exited
$N=1$ mech-kink.

##  IV Mech-oscillon

Figure 9: Schematic depiction of $N=2$ mech-oscillon.

In this section, we investigate non-topological configurations that have
$v_{\rm L}=v_{\rm R}\equiv v$. We call them generically ‘mech-oscillons’. The
simplest mech-oscillon is shown in Fig. 9.

###  IV.1 $N=2$: mech-field

Figure 10: An evolution of a mech-oscillon with initial conditions $R(0)=5$
and $A(0)=2$. Figure 11: An evolution of a mech-oscillon with initial
conditions $R(0)=7.1$ and $A(0)=1.3$. Figure 12: An evolution of a mech-
oscillon with initial conditions $R(0)=10$ and $A(0)=2$. Figure 13: An
evolution of a mech-oscillon with initial conditions $R(0)=10.1$ and $A(0)=2$.

To make the analysis as simple as possible, let us investigate symmetric
configuration, i.e. a triangle of base length $R(t)$ placed on top of the
vacuum $v$ with height $v+A(t)$ centered at the origin. In other words, we set

$\displaystyle x_{0}(t)=-R(t)/2\,,\hskip 8.53581ptx_{1}(t)=0\,,\hskip
8.53581ptx_{2}(t)=R(t)/2\,,$ (67)
$\displaystyle\phi_{0}(t)=\phi_{2}(t)=v\,,\hskip
8.53581pt\phi_{1}(t)=v+A(t)\,.$ (68)

This gives us

$\displaystyle L_{M}^{N=2}=$
$\displaystyle\frac{1}{6}R\dot{A}^{2}+\frac{1}{6}A\dot{A}\dot{R}+\frac{A^{2}}{6R}\bigl{(}\dot{R}^{2}-12\bigr{)}$
$\displaystyle-\frac{R}{A}\bigl{(}{\mathcal{V}}(v+A)-{\mathcal{V}}(v)\bigr{)}\,.$
(69)

In particular, for $\phi^{4}$ potential
$V(\phi)=\frac{1}{2}\bigl{(}1-\phi^{2}\bigr{)}^{2}$we obtain (taking $v=-1$)

$\displaystyle 3L_{M}^{N=2}=$
$\displaystyle\frac{1}{2}R\dot{A}^{2}+\frac{1}{2}A\dot{A}\dot{R}+\frac{A^{2}}{2R}\dot{R}^{2}$
$\displaystyle-\frac{6A^{2}}{R}-2A^{2}R+\frac{3}{2}A^{3}R-\frac{3}{10}A^{4}R\,.$
(70)

Note that $L_{M}^{N=2}$ has the same structure as in Eq. (4). Again, the
similarity is due to the universality of CCMs for this type of background.
Indeed, it is easy to see that the same terms with varying coefficients appear
when using $\phi_{\rm bkg}=-1+Af\bigl{(}x/R\bigr{)}$, given some obvious
convergence properties of $f$. Again, $N=2$ mech-kink falls into this class
somewhat accidentally due to the imposed reflection symmetry. If we relax this
restriction, i.e. $\Delta x_{0}\not=\Delta x_{1}$, we no longer fit into this
class, but a larger universality class of CCMs with three variables. However,
we do not believe that this would yield qualitatively different dynamics.

We found it advantageous to use an exponential ansatz:999This ansatz removes
the contact singularities and avoids $\Delta\phi_{0}=0$ problem, making these
coordinates especially convenient for numerical calculations. Notice that
$A(t)\geq 0$ cannot go below zero. We numerically verified that even for
general ansatz, the line $A(t)=0$ acts as a reflective barrier if approached
either from above or below, hence our ansatz is not a loss of generality.

$R(t)=\mathrm{e}^{b(t)}\,,\hskip 14.22636ptA(t)=\mathrm{e}^{a(t)}\,.$ (71)

In these variables, the equations of motions in $\phi^{4}$ model reads:

$\displaystyle\ddot{a}=$
$\displaystyle\dot{b}^{2}-\dot{a}^{2}+5\mathrm{e}^{a}-\frac{7}{5}\mathrm{e}^{2a}-20\mathrm{e}^{-2b}-4\,,$
(72) $\displaystyle\ddot{b}=$
$\displaystyle-2\dot{a}\dot{b}-\dot{b}^{2}-\mathrm{e}^{a}+\frac{2}{5}\mathrm{e}^{2a}+16\mathrm{e}^{-2b}\,,$
(73) $\displaystyle E_{\rm osc}=$
$\displaystyle\frac{1}{6}\mathrm{e}^{2a+b}\Bigl{(}\dot{a}^{2}+\dot{a}\dot{b}+\dot{b}^{2}+4\Bigr{)}$
$\displaystyle+2\mathrm{e}^{2a-b}+\frac{1}{10}\mathrm{e}^{4a+b}-\frac{1}{2}\mathrm{e}^{3a+b}\,.$
(74)

where $E_{\rm osc}$ is the mech-oscillon’s energy. The momentum is $P_{\rm
osc}=0$ by construction.

We plot several solutions with varying initial conditions in Figs. 10-13 that
illustrate a typical behavior of a mech-oscillon. Namely, there is an initial
quasi-periodic, chaotic phase followed by a decaying phase, in which the mech-
oscillon rapidly collapses to vacuum. In other words, the mech-oscillon seems
to have a well-defined lifetime, whose duration is very sensitive to initial
conditions. Indeed, the lifetime of a mech-oscillon can range from very short
(Fig. 10) to extremely long (Fig. 13).101010 Despite our best efforts, we did
not observe the decay of this mech-oscillon. We only know that its lifetime
must be longer than $\sim 74000$ time units. In fact, there is a common
understanding that in 1+1 dimensions, oscillons can have infinite lifetimes.
In Fig. 14, we display a ‘map’ of lifetime’s dependence on the initial height
and length of the triangle. Each color corresponds to a particular value of a
$\log_{10}$ of the lifetime.

Lastly, we can understand the decaying phase by studying the asymptotic
properties. It is easy to show that as $t\to\infty$ we have

$a(t)\sim-\sqrt{\frac{1}{3}V^{\prime\prime}(v)}t\,,\hskip 5.69054ptb(t)\sim
2\sqrt{\frac{1}{3}V^{\prime\prime}(v)}t\,,$ (75)

In other words, the height of the mech-oscillon exponentially decays, while
its width exponentially grows. Also notice that $b+2a$ remains constant.

Figure 14: A ‘map’ of the lifetimes for $N=2$ mech-oscillons for initial
values for $A$ and $R$. The colors represent $\log_{10}$ of the lifetime,
while lifetimes beyond $10^{3}$ time units are all represented by the
brightest color.

###  IV.2 $N\geq 3$ mech-oscillons

Mech-oscillons display a range of behaviors for $N\geq 3$. Among others, the
most significant is mech-$K\bar{K}$ pair production.

We can observe it already for $N=3$, where the mech-$K\bar{K}$ pair
immediately fly apart and decouple from each other as the middle segment falls
onto the vacuum. More interestingly, for $N\geq 4$, the created
mech-$K\bar{K}$ pair remains connected via a mech-oscillon that facilitates
the bouncing phenomenon (see Figs. 17-20). An interesting possibility suggests
itself, namely a connection between the distribution of bouncing windows and
the lifetime of mech-oscillon. As far as the authors are aware, this
connection has not been explored in the field theory.

It would be an interesting future project to map out the structure of the
bouncing windows for $N=4$ mech-oscillons and compare it with Fig. 14. In that
way, we may confirm this connection in our mech-models. The question of
whether the same can be meaningfully established in field theory, however, is
subtle.

In general, a nice feature of mech-models is the gradual discovery of new
behaviors as $N$ increases. For instance, the sequential ejection of two
mech-$K\bar{K}$ pairs becomes possible only at $N=7$. For $N=8$, we can not
only observe the ejection of two mech-$K\bar{K}$ pairs, but the trailing pair
can also undergo a few bounces before escaping to infinity as Figs. 21-22
illustrate.

A different type of behaviour is the ejection of a pair of mech-oscillons,
which we show on Figs. 23-25. In some instances (Figs. 23-24) the ejected
mech-oscillons has large amplitudes that span the gap between the two vacua.
These could be viewed as mech-bions, tight bound states of mech-$K\bar{K}$
pairs. In other instances (Fig. 25), the mech-oscillons have small amplitudes.
More than anything else, they could represent mechanical analogs of a
radiative decay.

For higher $N$ still (and sufficient energies) the mech-osillon may display a
various combination of all these processes. In our analysis we investigated
mech-fields up to $N=11$.

##  V Summary

In this paper we have presented a general-purpose $N$-point CCM for a simple
scalar field theory in $1+1$ dimensions. It is based on the ‘mechanization’
procedure in which a continuous field is replaced by a pice-wise linear
function. The conceptual simplicity of our construction gave us algebraically
tractable CCMs. Our numerical investigations indicate qualitative agreement
between phenomena observed in mech-models and the field theory.

The question of quantitative agreement is left as a future task. At this
point, we may only resort to a hand-waving statement that mech-field should
resemble continuous theory more and more as $N$ increases.

The most useful aspect of our approach is the natural ordering in the
complexity of behaviors. As we have seen, the exploration of mech-models with
an increasing number of joints gradually opens new dynamical modes of the
mech-field.

Starting at $N=1$, we have ‘discovered’ a mech-kink that behaves as a
relativistic particle. Furthermore, the corresponding mech-model (51) turns
out to be the same term-wise as relativistically covariant CCM based on
position and scaling modulus (48). Let us point out that this result was given
us _for free_ without any attempt to recover lost Lorentz covariance that
motivates its construction in the field theory Manton:2020onl .

At $N=2$, we have found a mech-oscillon and its tendency to suddenly decay
into a vacuum after a period of time that sensitively depends on initial
conditions (see Fig. 14). The corresponding CCM (70) is, again, structurally
the same as the field-theoretical one (4).

For $N=2$ mech-kinks, we observed a phenomenon of joint-ejection (Fig. 4) that
seems to be endemic for all configurations with $N\geq 2$ (Fig. 5). The joint-
ejection exemplifies a general tendency that is characteristic across our
numerical data. Namely, the proclivity of initially tightly bound mech-field
to disintegrate over time into most basic configurations, such as $N=1$ mech-
kinks and $N=2$ mech-oscillons (or their pairs), that separate and gradually
decouple from each other.

The production of mech-$K\bar{K}$ pairs can be seen already at $N=3$ level,
but it is for $N=4$ mech-oscillons that the phenomenon of bouncing starts to
manifest (Figs. 17, 19 and 20). As the mech-$K\bar{K}$ pair is bound together
via a mech-oscillon, a curious connection between phenomena of bouncing and
mech-oscillon’s lifetime suggests itself. We have not investigated this
possibility in detail, however, and it remains an interesting future work.

At $N=6$, yet another mode opens up, namely an ejection of a pair of mech-
oscillons (Fig. 23). For $N=8$, we observed that the peaks of ejected mech-
oscillons sometimes reach all the way to the second vacuum (Fig. 24). In such
situations, it seems reasonable to view the ejecta as mech-bions – tight
mech-$K\bar{K}$ bound states. Other times, however, the mech-oscillons have
very small amplitudes. These situations are perhaps analogs of field-
theoretical radiation decay.

Of course, arbitrary combinations of the above processes can occur in sequence
for sufficiently high $N$ (Figs. 18, 21 and 22).

Let us stress that mechanization should be viewed as a proof-of-concept rather
than a serious attempt for general-purpose CCMs. Indeed, there are issues with
our construction. The most glaring one is the geodetic incompleteness of the
moduli space, which is perhaps the largest source of quantitative
disagreement.

As we discussed, singularities arise whenever the distance between two joints
becomes zero, i.e. $\Delta x_{a}=0$, or a middle segment becomes flat, i.e.
$\Delta\phi_{a}=0$. The second type of singularities introduces an unexpected
practical problem: it is quite challenging to investigate mech-$K\bar{K}$
scattering directly. This is because the configuration of initially separated
mech-kink and anti-mech-kink with a flat segment in between is dynamically
decoupled. Indeed, such a configuration is an exact solution of the equations
of motion describing free particles. However, upon contact, we arrive at
$\Delta x_{a}=0$ type singularity, and the equations of motion break down. For
this reason, we have mostly investigated the evolution of large $N$ mech-
oscillons that provide an indirect way of studying the scattering of mech-
kinks.

The resolution of moduli space singularities would be a fundamental step
forward. At present, it is not clear to us how we should accomplish it. It is
telling, however, that the singularities appear whenever the mech-field
suddenly changes its (effective) number of joints. For example, one may
continue beyond a singular collision of a free-moving mech-$K\bar{K}$ pair,
which is a $N=3$ configuration, by replacing it with a $N=2$ mech-oscillon.
Although intuitive, it is difficult to realize this approach in practice.

In other words, we should figure out how to dynamically connect different
$N$-sectors. Making the number of particles vary would also provide a step
towards restoring explicit Lorentz covariance. There is an obvious way to
achieve this that we already know, i.e. taking $N\to\infty$ and reintroducing
back continuous field. Whether there exists a middle ground where $N$ would
remain finite and discrete dynamical variable remains a tantalizing
possibility.

###### Acknowledgements.

O. N. K. would like to thank Lukáš Rafaj for useful assistance. The authors
are also indebted to A. Wereszczynski, T. Romanczukiewicz and K. Oles for
discussions and useful feedback. Also we would like to thank T.
Romanczukiewicz for his help with our numerical code. F. B. would like to
express his acknowledgment for the institutional support of the Research
Centre for Theoretical Physics and Astrophysics, Institute of Physics,
Silesian University in Opava and to the Institute of Experimental and Applied
Physics, Czech Technical University in Prague. This work was supported by the
Student Grant Foundation of the Silesian University in Opava, Grant No.
SGF/3/2021, which was realized within the EU OPSRE project entitled ”Improving
the quality of the internal grand scheme of the Silesian University in Opava”,
reg. number: CZ.02.2.69/0.0/0.0/19_073/0016951.

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Figure 15: Small perturbation of static $N=2$ mech-kink leading to quasi-
periodic oscillations. Figure 16: Quasi-periodic oscillations in $N=4$ mech-
kink. Figure 17: A mech-$K\bar{K}$ ejection from initial $N=4$ mech-oscillon
leaving behind $N=2$ mech-oscillon. Figure 18: An initial ejection of
outermost joints followed by ejection of a highly-excited mech-$K\bar{K}$ pair
for $N=6$ mech-oscillon. Figure 19: Initial ‘bounces’ before a mech-$K\bar{K}$
ejection for $N=4$ mech-oscillon . Figure 20: Sequence of long bounces before
a mech-$K\bar{K}$ ejection for $N=4$ mech-oscillon. Figure 21: Ejection of two
mech-$K\bar{K}$ pairs with two intermediary bounces of the second pair for
$N=8$ mech-oscillon. Figure 22: Ejection of two mech-$K\bar{K}$ pairs with
three intermediary bounces of the second pair for $N=8$ mech-oscillon. Figure
23: Ejection of a pair of mech-oscillons for $N=6$ mech-field. Figure 24:
Ejection of pair of large-amplitude mech-oscillons in $N=8$ configuration,
illustrating production of mech-bions (tight mech-$K\bar{K}$ bound states).
Figure 25: Ejection of small-amplitude mech-oscillons for $N=8$ mech-field.
Analog of radiative decay?


# Joint Deep Learning For Improved Myocardial Scar Detection From Cardiac MRI

###### Abstract

Automated identification of myocardial scar from late gadolinium enhancement
cardiac magnetic resonance images (LGE-CMR) is limited by image noise and
artifacts such as those related to motion and partial volume effect. This
paper presents a novel joint deep learning (JDL) framework that improves such
tasks by utilizing simultaneously learned myocardium segmentations to
eliminate negative effects from non-region-of-interest areas. In contrast to
previous approaches treating scar detection and myocardium segmentation as
separate or parallel tasks, our proposed method introduces a message passing
module where the information of myocardium segmentation is directly passed to
guide scar detectors. This newly designed network will efficiently exploit
joint information from the two related tasks and use all available sources of
myocardium segmentation to benefit scar identification. We demonstrate the
effectiveness of JDL on LGE-CMR images for automated left ventricular (LV)
scar detection, with great potential to improve risk prediction in patients
with both ischemic and non-ischemic heart disease and to improve response
rates to cardiac resynchronization therapy (CRT) for heart failure patients.
Experimental results show that our proposed approach outperforms multiple
state-of-the-art methods, including commonly used two-step segmentation-
classification networks, and multitask learning schemes where subtasks are
indirectly interacted.

Index Terms—  One, two, three, four, five

##  1 Introduction

Myocardial scarring is an important factor that leads to cardiac dysfunction
and poor prognosis of patients with heart diseases [1, 2]. Identifying the
presence and extent of myocardial scars is important to many clinical
interventions. For instance, it can significantly increase the response rate
of cardiac resynchronization therapy (CRT), since contractile improvements are
often shown on scar-free sites [3]. Another example is that the quantification
of scar tissue can provide prognostic information about the risk for
arrhythmia in patients with nonischemic cardiomyopathy undergoing
cardioverter-defibrillator implantation [4]. With rapid development of
advanced imaging techniques, LGE-CMR has become one of the most promising and
noninvasive methods to determine myocardial scar burden and guide therapeutic
decisions [5]. However, the variable quality of LGE images due to image
noises, motion artifacts, and partial volume effect makes the task of
automated scar detection challenging.

Inspired by the great performance of deep learning methods, recent research
has made the first attempt to address this issue by employing deep neural
networks to segment scar directly from LGE-CMR images [6, 7]. Despite
relatively low Dice scores, these works demonstrated that the predicted scar
segmentations can be substantially improved with provided myocardium
segmentation labels. Later, two-step approaches were developed to achieve
better performance by cascading the task of myocardium and scar segmentation
sequentially in two separate networks [8]. While the scar segmentation
accuracy was improved, the performance of two-step methods heavily depends on
the quality of myocardium segmentation. Another mainstream of incorporating
myocardium segmentations is through multitask learning (MTL). In contrast to
two-step approaches, MTL performs both tasks simultaneously and allows mutual
benefit from each other [9, 10]. However, MTL methods are fully data-driven
and the learned relationship between tasks is unclear. With little to no
knowledge on how the task relationship is utilized, the performance of MTL can
be unstable even with fine-tuned parameters. In addition, MTL heavily depends
on the quality of training data. It may learn an inaccurate task relationship
that misleads the learning towards suboptimal models when the data is low
quality with high noises.

In this paper, we propose a novel joint deep learning (JDL) framework that
efficiently solicits the knowledge of spatial relationships between the
myocardium and myocardial scars from jointly learned networks. In contrast to
previous methods treating related tasks in a separate or parallel manner, our
model (i) explicitly enforces a mutual guidance between the myocardium and
scar segmentation tasks in training, and (ii) directly incorporates the prior
information provided by predicted myocardium segmentations to identify the
scars. We develop an interactive joint deep network that connects myocardium
segmentation and myocardial scar detection by a message-passing module. More
specifically, we use the predicted myocardium probability maps as weighting
masks on the LGE images to eliminate the negative effects of non-ROI regions
in scar detection. Meanwhile, the myocardium segmentation network receives
extra supervision from the scar maps, which captures the scar region more
accurately.

To the best of our knowledge, we are the first to develop JDL models for
myocardial scar detection with learned myocardium segmentations from LGE-CMR
images. Experimental results show that our model outperforms a direct
segmentation network on original LGE images [7], fully automatic two-step
approaches [8], and recent MTL schemes [10]. The outcome of this research has
great potential to impact various image-guided clinical systems, such as
locating optimal left-ventricular-lead sites that are scar-free for CRT [11],
providing more accurate prognostic information in patients with known or
suspected coronary artery disease [1], and making the best possible
therapeutic decisions for coronary revascularization [12].

##  2 A Joint Deep Network For Scar Detection

Consider a set of training images $I_{1},\cdots,I_{N}$ with ground truth
labels of myocardium and myocardial scars, where $N$ is the number of images.
Our JDL framework simultaneously learns the tasks of myocardium and scar
segmentation in an interactive manner. With a joint optimization scheme, our
model produces two different outputs: a myocardium segmentation probability
map and a binary scar map. The myocardium probability map is used as a ROI-
mask that is iteratively passed to the learning process of scar detection.
Next, we will introduce the network architecture and optimization schemes.

###  2.1 Joint Network Architecture

Our designed network is conceptually simple and modifies two-step approaches
[8] to jointly optimized networks through a message passing module. The first
myocardium segmentation network takes LGE-CMR as an input and generates a
probability map. A message passing module then combines the information of
probability maps with the original images to reproduce a masked input data for
the scar segmentation network. This effectively suppresses noises or artifacts
from non-ROI regions in scar detection. It is worthy to mention that our
framework is highly flexible to various types of segmentation networks for
both tasks. In this paper, we employ pre-trained TransUNet [13], which shows
superior performance on medical image segmentation tasks over other commonly
used networks such as U-Net [14] An overview of our proposed approach is
presented in Fig. 1.

Fig. 1: The overview of our proposed framework, where the myocardium
segmentation network communicates with the scar segmentation network via the
message-passing module, and the two networks are jointly trained. The
$\bigodot$ means element-wise multiplication and refers to masking operation,
and $\bigoplus$ means summation for the joint optimization.

###  2.2 Loss and Optimization

Consider a set of ground truth segmentation labels $Y$, we use a weighted sum
of cross entropy and Dice loss $L(\cdot,\cdot)$ for both segmentation
networks, i.e.,

$L(Y,\hat{Y}(\theta))=-\sum
Y\log\hat{Y}(\theta)-w\sum(1-\frac{2Y\hat{Y}(\theta)+\epsilon}{Y+\hat{Y}(\theta)+\epsilon}),$

where $\hat{Y}(\theta)$ represents network prediction with parameter $\theta$,
$w$ is a weighting parameter, and $\epsilon$ is a small smoothing constant. In
this paper, we set $\epsilon=1e-6$ in all our experiments.

Supervised by manually delineated myocardium $M$ and myocardial scars $S$, our
model jointly learns to predict myocardium segmentation probability maps
$\hat{M}(\theta_{M})$ and binary scar maps $\hat{S}(\theta_{S})$. A message-
passing module effectively utilizes the predicted myocardium segmentations to
mask out non-ROI regions from original images, which provides new input data
to the scar network. The total loss function is then defined by a weighted sum
of the regularized myocardium segmentation loss $L_{m}$ and the scar
segmentation loss $L_{s}$

$\displaystyle L(\theta_{M},\theta_{S})=$
$\displaystyle\lambda\sum_{i=1}^{N}L_{m}\left(M_{i},\hat{M}_{i}(\theta_{M});I_{i}\right)$
$\displaystyle+L_{s}\left(S_{i},\hat{S}_{i}(\theta_{S});\hat{M}_{i}(\theta_{M})\odot
I_{i}\right)$
$\displaystyle+\beta_{M}\text{Reg}(\theta_{M})+\beta_{S}\text{Reg}(\theta_{S}),$
(1)

where $\\{\lambda,\beta_{M},\beta_{S}\\}$ are weighting parameters, $\odot$
denotes an element-wise production), and $\text{Reg}(\cdot)$ represents
network regularization, which is set to $L_{2}$ weight decay in our
experiments.

Joint network optimization. We use a joint optimization scheme to minimize the
loss in Eq. (2.2). Our scar segmentation network is trained using the output
of the myocardium segmentation network, whereas the myocardium segmentation
network is trained by taking a sum of weighted gradients from both loss
functions $L_{m}$ and $L_{s}$. The optimization of Eq. (2.2) splits into two
parts:

$\displaystyle\hat{\theta}_{M}=$
$\displaystyle\operatorname*{arg\,min}_{\theta_{M}}\lambda\sum_{i=1}^{N}L_{m}\left(M_{i},\hat{M}_{i}(\theta_{M});I_{i}\right)$
$\displaystyle+L_{s}\left(S_{i},\hat{S}_{i}(\theta_{S});\hat{M}_{i}(\theta_{M})\odot
I_{i}\right)$ $\displaystyle+\beta_{M}\text{Reg}(\theta_{M}),$
$\displaystyle\hat{\theta}_{S}=$
$\displaystyle\operatorname*{arg\,min}_{\theta_{S}}L_{s}\left(S_{i},\hat{S}_{i}(\theta_{S});\hat{M}_{i}(\theta_{M})\odot
I_{i}\right)+\beta_{S}\text{Reg}(\theta_{S}).$

This shows that our model explicitly allows two tasks mutually guided through
the message-passing module. In practice, we train the two subnetworks
simultaneously. A detailed description of parameter settings will be
introduced in the experiment section.

##  3 Experimental Results

###  3.1 Experimental Settings

We test the proposed JDL framework on a newly collected dataset of LGE-CMR
images in our institute. To validate the performance of our model, we compare
it with three types of baseline algorithms: a direct scar segmentation network
on original image data, a two-step myocardium and scar segmentation network,
and a multitask learning network with hard-parameter sharing. For fair
comparison, the backbone network of all methods is TransUNet [15] with the
transformer part pre-trained on imageNet-21K [13]. To evaluate volume overlap
between the predicted segmentation A and the manual segmentation B, we compute
the Dice similarity coefficient
$DSC(A,B)=2\left(|A|\cap|B|\right)/\left(|A|+|B|\right)$, where $\cap$ denotes
an intersection of two regions.

Data. We have been collecting LGE-CMR images with the size of $224\times 224$
from $168$ heart failure patients who had CMR studies prior to undergoing the
treatment of CRT. All ground truth labels, including segmentation of
myocardium and scar maps involved in our experiment were manually delineated
by experienced medical doctors. $510$ LGE-CMR images from $140$ patients are
used for training, and $125$ images from $28$ different patients are used for
testing. All images are normalized to $[0,1]$ before sending to the network.
We generated random affine transforms, including scaling, rotating,
translating and shearing, for data augmentation. While the data used in this
paper is not publicly available online, upon the acceptance of this
manuscript, we commit to share all images along with myocardium and scar
labels if an institutional data use agreement is in place.

Network training and hyper-parameter tuning. The joint network is trained on
GPU units with Adam optimizer [16]. We set the batch size and the number of
epochs to $40$ and $500$ respectively. We have the learning rate $\eta$, drop-
out rate $p$, loss weight $\lambda$ and $L_{2}$ weight decay weights
$\beta_{M}$ and $\beta_{S}$ as hyper-parameters to be tuned using Bayesian
optimization with Gaussian processes implemented by wandb [17] package. The
empirical optimal hyper-parameters for JDL are $\hat{\eta}=4.73e-4$, $p=0.39$,
$\hat{\lambda}=1.02$ and $\beta_{M}=\beta_{S}=5.58e-6$.

Compared methods. We compared our proposed approach with 3 other approaches:
the direct segmentation approach, the two-step approach and the multi-task
learning approach. In the direct segmentation approach a segmentation network
is directly trained to predicted the scar segmentation mask given one LGE MR
image. In the two-step approach, we train a myocardium segmentation network,
and separately train a scar segmentation network that takes MR image masked by
manual labeled myocardium segmentation mask. In the testing stage, we first
predict the myocardium mask using the myocardium segmentation network and use
them to mask the LGE MR images. Then, the masked images are sent to the scar
segmentation network to generate scar segmentation mask. All the networks
mentioned above use the TransUNet [13] as backbone. For the multi-task
learning approach, we adopted the multi-head network structure, where the
network has on shared sub-network followed by two sub-branch networks, one of
which generates myocardium segmentation mask and the other generates scar
segmentation mask. In the implementation, we duplicated the decoder (i.e. the
upscaling convolutional layers) of the TransUNet [13] as two sub-branch
networks. The loss function of (sub-branch) networks is the same with our
method as Eq.2.2.

###  3.2 Results & Discussion

Fig. 2: Scar Segmentation Dice score of Different Methods Fig. 3: A comparison
of predicted scar segmentations between our method and other approaches. Left
to right: input images, manually delineated ground truth labels, direct
segmentation on original images, two-step approach, multitask learning (MTL),
and our joint deep learning (JDL) framework.

The comparison of Dice scores of all methods are reported in Fig. 2. It shows
that our proposed JDL framework outperforms all other approaches in terms of
both accuracy and robustness. Examples of segmentation results are shown in
Fig. 3, which also demonstrates that our method achieves the closest scar
segmentations to manual delineations (ground truth) provided by experts.

##  4 Discussion

###  4.1 Interpretation of JDL Framwork

To further understand why our JDL achieves better performance, we also
computed the precision and recall of the segmentation results. For each
segmentation result, they are computed as
$\text{Precision}(S,\hat{S})=(S\cap\hat{S})/\|\hat{S}\|$ and
$\text{Recall}(S,\hat{S})=(S\cap\hat{S})/\|S\|$ where $S$ and $\hat{s}$ are
the ground-truth and predicted masks, respectively. The segmentation model
with high precision tends to capture all possible areas and misses less, while
the model with high recall tends to avoid false positive prediction and tries
to ensure all the captured regions are indeed the target region. The scar
segmentation precision and recall of different methods are shown in Fig. 4.
While our JDL model has both the best precision and recall performance, its
recall is significantly higher than all other methods, which indicates that
our JDL model is less likely to wrongly report the existence of scar at a
scar-free region, and this also matches the examples given in Fig.3.

Fig. 4: Scar segmentation precision and recall of all methods

###  4.2 Effect on Myocardium Segmentation Task

While in general, when multiple tasks are trained jointly, the performance
improvement of all tasks is expected. However, the mean Dice scores of the
myocardium segmentation of the two-step approach, the MTL approach and our
approach are $0.894$, $0.900$ and $0.891$, respectively. In other words, the
joint training strategy has little effect on the myocardium segmentation task,
while it does improve the scar segmentation task. This raise further
interpretability question to those approaches, which is crucial to build
trustable algoroithms and could be the future work of the proposed JDL method.

##  5 Conclusion

In this paper, we proposed a novel joint deep neural network for an improved
performance of myocardial scar detection with learned myocardium
segmentations. In contrast to existing deep learning based approaches that
train these tasks in a separate or parallel manner, our methods incorporates a
message-passing module that effectively passes information within the
subnetworks. Therefore, our model is able to explicitly use the information of
learned myocardium segmentations to guide the scar detection. Meanwhile, the
myocardium segmentation network receives additional supervision from the scar
labels, which improves the predicted segmentation accuracy. Experimental
results show that our method achieved the state-of-the-art performance on
myocardial scar segmentation. Our work has great potential to impact various
image-guided clinical systems, such as identifying optimal pacing sites that
are scar-free for efficient treatment of CRT [11]. We commit to make code
produced in this work publicly available online upon the acceptance of this
manuscript.

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