# Semi-BPS sphaleron and its dynamics

A. Alonso-Izquierdo  Department of Applied Mathematics, University of
Salamanca, Casas del Parque 2 and
Institute of Fundamental Physics and Mathematics, University of Salamanca,
Plaza de la Merced 1, 37008 - Salamanca, Spain  S. Navarro-Obregón
Departamento de Física Teórica, Atómica y Óptica, Universidad de Valladolid,
47011 Valladolid, Spain.  K. Oles  Institute of Theoretical Physics,
Jagiellonian University, Lojasiewicza 11, Kraków, Poland  J. Queiruga
Department of Applied Mathematics, University of Salamanca, Casas del Parque 2
and
Institute of Fundamental Physics and Mathematics, University of Salamanca,
Plaza de la Merced 1, 37008 - Salamanca, Spain  T. Romanczukiewicz  Institute
of Theoretical Physics, Jagiellonian University, Lojasiewicza 11, Kraków,
Poland  A. Wereszczynski  Institute of Theoretical Physics, Jagiellonian
University, Lojasiewicza 11, Kraków, Poland

###### Abstract

We construct a simple field theory in which a sphaleron, i.e., a saddle-point
particle-like solution, forms a semi-BPS state with a background defect that
is an impurity. This means that there is no static force between the sphaleron
and the impurity. Therefore such a sphaleron-impurity system is very much like
usual BPS multi-solitons, however, still possessing an unstable direction
allowing for its decay. We study dynamics of the sphaleron in such a system.

##  I Motivation

Sphalerons are highly nonperturbative static solutions, existing in various
nonlinear theories, and to some extent, resembling topological solitons [1].
They are localized, typically particle-like field configurations obeying
vacuum boundary conditions, which, however, do not have to necessarily carry
any topological charge. What is important, they are saddle point solutions
i.e., posses an unstable direction along which they decay. Thus their small
perturbation spectrum possesses a negative mode.

Sphalerons have been found in many theories with or without topological
solitons, see e.g., [2, 3, 4, 5, 6, 7, 8, 9, 10], and for a review see [11].
Therefore, they are more fundamental and generic objects than topological
solitons. Of course, the most prominent example is given by the electroweak
sphaleron [12], see also [13, 14, 15]. On the contrary to stable topological
solitons, which are important for the static or vacuum sectors of a theory
(see e.g., instantons and their importance for the structure of the vacuum),
sphalerons are crucial also for real time dynamics, e.g., real-time
transitions between vacua or scattering processes. In the electroweak theory
the sphaleron is responsible for baryon charge violation effects.

Although sphalerons are known to be very important for time evolution of
nonlinear theories, with and without topological solitons, they are much less
understood than their stable solitonic counterparts. Specifically, the study
of their dynamics or their impact on dynamics of topological solitons is in a
rather initial stage. One reason for that is the complexity of interactions.
Indeed, similarly as in the case of solitons, a sphaleron interacts in the
three main ways:

First of all, it can be attracted or repelled by sphalerons, solitons or other
localized excitations of matter fields. It means that there is a static force
acting on a sphaleron due to the presence of defects. This amounts to a
kinetic motion.

Secondly, sphaleron may possess internal degrees of freedom (DoF) i.e.,
normal, or even quasi-normal modes, which may be excited during scattering
processes and which temporary can store some fraction of energy. In the case
of topological solitons internal DoF amount to an appearance of fractal
structures in the final state formation via the resonant energy transfer
mechanism, [16, 17]. In fact, it has recently been shown that sphaleron can
provide DoF which trigger resonant transfer phenomenon in kink-antikink
collisions [18]. Hence, they can actively modify solitonic dynamics even
though they do not appear in the initial or finite state.

Thirdly, sphaleron typically meets radiation which also can modify its
dynamics.

Another reason which increases the complexity of the sphaleron dynamics is
that it is not an ultimately stable solution and therefore it has a negative,
unstable mode. Hence its decay should be also taken into account.
Interestingly, this mode is extremely important for the recently discovered
sphaleron-oscillon correspondence [19] which states that properties of
sphaleron and its oscillon (i.e., an oscillon to which this sphaleron decays)
are intimately related. This unexpected correspondence partially unifies these
two, at the first glance independent, phenomena further underlying the
importance of sphalerons in nonlinear dynamics.

In the case of topological soliton, there exists a very special limit, called
Bogomol’nyi-Prasad-Sommerfield (BPS) limit [20, 21], which significantly
simplifies dynamics and allows for a very nontrivial insight into the static
as well as dynamical properties of solitons. In this limit there is no static
force between the solitons, which means that the energy of a class of
solutions in a topologically fixed sector is degenerated. For example, the
constituent solitons, in a multi-soliton sector, can be placed in any spatial
point and all such solutions possess exactly the same energy. Here the most
famous examples are the abelian Higgs vortices at the critical coupling, the
t’Hooft-Polyakov monopoles or the $SU(2)$ Yang-Mills instantons. Importantly,
the lowest energy dynamics of solitons in such BPS models finds an elegant
formulation in terms of geodesic motion on a pertinent moduli space, which is
a space of coordinates parameterizing the energetically equivalent solutions
[22]. No such a BPS-type construction for sphalerons has been studied yet.

The aim of the present work is to propose a framework in which dynamics of a
sphaleron does enjoy a sort of BPS limit. We will call such objects semi-BPS
sphalerons to underline their similarities as well as differences in
comparison with the typical BPS solitons. Specifically, we introduce probably
the simplest family of theories where a sphaleron remains to be a BPS-like
state even in a presence of other localized field configurations, which in our
case will be a background, non-dynamical defect i.e., an impurity. This choice
is dictated by an obvious reason. Namely, although providing a nontrivial
background, which may equivalently be treated as a nontrivial medium, the
impurity itself does not introduce new kinetic DoF. This, together with the
BPS-like property, may simplify the complexity of the problem allowing for a
new insight into the problem of dynamics and interactions of localized
solutions.

However, it should be underlined that, in the case of the sphaleron, being a
semi-BPS object has a slightly different, probably even more nontrivial,
meaning. As always, it denotes that:

  1. 1.

There is no static force between such a sphaleron and other localized object
(impurity/solitons etc.).

  2. 2.

The solution possesses a zero mode (although the impurity explicitly breaks
the translational invariance).

On the other hand, a semi-BPS sphaleron does not saturate any topological
bound but has a negative mode which can lower the energy of this semi-BPS
solution due to a decay along the unstable direction, exactly as required for
a genuine sphaleron. Due to that, even in its BPS-like limit the sphaleron
reveals much more involved pattern of interactions than its BPS solitonic
counterparts.

We remark that the notion of semi-BPS sphalerons agrees with the concept of
semi-BPS solutions, that is solutions which solve a Bogomolny equation on a
double-cover of a complex $\phi$-plane [23].

##  II Sphaleron in a scalar field theory

Let us consider the standard real field theory in (1+1) dimensions

$L=\int_{-\infty}^{\infty}\left(\frac{1}{2}\phi_{t}^{2}-\frac{1}{2}\phi_{x}^{2}-U(\phi)\right)dx.$
(1)

If the potential has two vacua as for example in the $\phi^{4}$ theory,
$U_{\phi^{4}}=\frac{1}{2}(1-\phi^{2})^{2}$, we have topological solitons,
usually called kink (or antikink) interpolating between the vacua,
$\phi_{+}>\phi_{-}$. An important feature of a kink (or antikink) is that it
saturates the pertinent topological energy bound in a given topological
sector. Indeed, consider the static energy integral

$\displaystyle E$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}\left(\frac{1}{2}\phi_{x}^{2}+U(\phi)\right)dx$
$\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}\left(\frac{1}{\sqrt{2}}\phi_{x}\mp\sqrt{U}\right)^{2}dx\mp\int_{-\infty}^{\infty}\sqrt{2U}\phi_{x}dx$
$\displaystyle\geq$
$\displaystyle\left|\int_{\phi(-\infty)}^{\phi(\infty)}\sqrt{2U}d\phi\right|=\int_{\phi_{-}}^{\phi_{+}}\sqrt{2U}d\phi\,|Q|,$

where the field at infinities must have one of vacuum values. $Q$ is the
topological charge, $\pm 1$ for the kink and antikink respectively. This
inequality is saturated for solutions of the so-called Bogomolny equations

$\frac{1}{\sqrt{2}}\phi_{x}=\pm\sqrt{U}.$ (3)

The kink obeys the equation with plus sign, while the antikink with minus
sign.

Figure 1: The deformed $\phi^{4}$ potential $U_{s}(\phi)$ for different $s$.
Figure 2: Sphalerons for different values of s in the model $U_{s}(\phi)$.

To find a genuine, spacially localized sphaleron we need a potential where one
of the global minima is transformed into a local minimum. Hence, in such
models, true vacuum $\phi_{v}$ is supplemented by a false vacuum $\phi_{f}$.
Without losing generality we assume that $\phi_{v}>\phi_{f}$. In addition, we
chose that at the false vacuum the potential takes the zero value,
$U(\phi_{f})=0$. Since, $U(\phi_{v})<U(\phi_{f})$, there exists a field value
$\phi_{0}$ at which the potential also vanishes, $U(\phi_{0})$. We call this
point a return point.

Here we make a standard assumption that at the local minimum, $\phi_{f}$, the
potential behaves quadratically while the additional zero $\phi_{0}$ is
achieved linearly i.e.,

$U(\phi)=\frac{m_{f}}{2}(\phi_{f}-\phi)^{2}+O(\phi_{f}-\phi)^{3}\;\;\mbox{at}\;\;\phi\to\phi_{f}$
(4)

and

$U(\phi)=\beta(\phi_{0}-\phi)+O(\phi_{0}-\phi)^{2}\;\;\mbox{at}\;\;\phi\to\phi_{0}$
(5)

where $m_{f}$ and $\beta$ are positive constants. A consequence of this is
that the false vacuum is approached exponentially at $x\to\pm\infty$ but the
zero $\phi_{0}$ can be approached at a finite point. The extension to a higher
order false vacuum is straightforward.

As an example, which we would like to exploit in the paper, can serve a
modified $\phi^{4}$ potential (e.g., [26, 24])

$U_{s}(\phi)=2\phi^{2}\left(\phi-\tanh(s)\right)\left(\phi-\frac{1}{\tanh(s)}\right)$
(6)

where $s>0$ is a free parameter. For $s\to\infty$ we recover the $\phi^{4}$
theory with two true vacua at $\phi_{v}=0$ and $\phi_{v}=1$. Otherwise, there
is only one true vacuum at

$\phi_{v}=\frac{3}{4}\coth(2s)+\frac{1}{4}\sqrt{9\coth^{2}(2s)-8}$ (7)

and one false vacuum (local minimum) at $\phi_{f}=0$ at which
$U_{s}(\phi_{f})=0$, see Fig. 1. In the limit $s\to 0$ we effectively recover
the $\phi^{3}$ model where the true vacuum is sent to infinity [19]. This
potential has a quadratic zero at the false vacuum and a linear zero at the
return point $\phi_{0}=\tanh(s)$. Another linear zero at $\phi=\coth(s)$ is
not important for our considerations.

This model admits a sphaleron which interpolates between the false vacuum and
passes through the additional zero of the potential [24]. Its exact form reads

$\Phi_{sph}(x)=\frac{1}{2}\left(\tanh(x+s)-\tanh(x-s)\right).$ (8)

It may be viewed as a superposition of the $\phi^{4}$ kink and antikink
located at $\mp s$ respectively, see Fig. 2. For sufficiently large $s$ we
easily recognize both $\phi^{4}$ kink and antikink in the sphaleron profile.
For $s\to 0$ the solution is approximately $1/\cosh^{2}(x)$ which, up to some
rescalings, is the $\phi^{3}$ sphaleron [19, 25, 24].

In fact all potentials with a false vacuum amount to the appearance of such a
sphaleron. Interestingly, it may be treated as a solution which piece-wisely
obeys the Bogomolny equation. Namely,

$\frac{1}{\sqrt{2}}\phi_{x}=\left\\{\begin{array}[]{cc}+\sqrt{U}&x\leq 0\\\
&\\\ -\sqrt{U}&x\geq 0.\end{array}\right.$ (9)

At the gluing point (which here is set at the origin) the field takes
$\phi_{0}$ value. Therefore these two branches are connected forming a
$\mathcal{C}^{1}$ solution. In fact, one can easily show that all derivatives
are continuous at this point, which results in a smooth $\mathcal{C}^{\infty}$
sphaleron. This can be approached in a very elegant way in therms of
integration theory for kinks and sphalerons [23].

The fact that sphalerons in scalar field theories fulfill piece-wisely the
pertinent Bogomolny equation is a guideline which will be exploited in the
next part of the paper.

Of course, to prove that this is a sphaleron one has to compute the linear
perturbation spectrum and identify a negative, unstable mode. This is indeed
the case for our choice of the potential. The easiest way to verify this is to
compute the zero mode and show that it possesses one node. Therefore there
must be a lower (negative) energy mode.

Needless to say that, since the theory is translational invariant, the center
of the sphaleron can be moved from the origin to any spatially point $x_{0}$.
One may view this sphaleron as a trivial BPS-like object. There is a
(translation) zero mode but its action on the solution is trivial. For
example, the spectral structure is $x_{0}$ independent.

##  III Non-chiral BPS-impurity models

Typically, in single scalar field theory both kink or sphaleron, if placed in
the presence of other (antik)kink or sphaleron, are subject to an attraction
or repulsion. This also means that such a multi-particle state usually does
not have any zero mode. As a consequence, it does not obey any Bogomolny-type
equation. Hence, it is not a BPS solution.

To find multi-kink BPS solutions one needs to add, in a very special manner,
at least one new scalar field [27, 28], which obviously, makes investigations
more complicated both from analytical as well as numerical point of view.
However, it was recently shown that it is possible to couple the scalar field
$\phi$ to a background, non-dynamical field $\sigma(x)$ in such a way that BPS
property is preserved [29, 30]. It means that one obtains the simplest BPS-
type theory, still with only one dynamical field. Indeed, here a soliton does
not statically interact with the other localized object, i.e., the impurity
and therefore they can be located at any distance from each other without
change in the energy.

Figure 3: The BPS sphaleron-impurity solution for different values of the
modulus $a$. Upper panels: $\alpha=1$, lower panels $\alpha=-0.9$. Left
panels: $s=1$, right panels: $s=3$.

The usual BPS-impurity modification changes one of the Bogomolny equations (3)
into the following equation [29, 31]

$\frac{1}{\sqrt{2}}\phi_{x}=\sigma(x)W(\phi),$ (10)

where $W$ is a well-behaved function of $\phi$ and $W^{2}=U$.

Such a modification of the Bogomolny equation can be achieved if the original
field theory is deformed as [29]

$L=\int_{-\infty}^{\infty}\left(\frac{1}{2}\phi_{t}^{2}-\frac{1}{2}\left(\phi_{x}-\sigma(x)W(\phi)\right)^{2}\right)dx.$
(11)

Again, the existence of a first order Bogomolny equation implies that there is
a family of energetically degenerate kink-impurity solutions which are
parameterized by a continuous parameter $a$, called modulus and which may be
related to a distance between the kink and the impurity. In comparison to the
no impurity case, (3), the BPS sector is nontrivial. For example, the linear
mode structure of the soliton-impurity BPS solution depends on the distance
between the soliton and impurity. This renders the BPS-impurity model an
ideal, simplified laboratory for studying the role of the internal modes in
kink dynamics in the limit of vanishing static force. In fact, it allowed for
the discovery of the spectral wall phenomenon [32] which only very recently
has been observed in various solitonic models without any impurity [33, 34].
The BPS-impurity framework also allows for the computation of one-loop
corrections to kink-antikink processes, see [35].

Importantly, there is only one Bogomolny equation in this deformed model.
Hence, the resulting soliton can be treated as a chiral kink. Based on the
previous analysis, where we underlined the importance of the existence of two
Bogomolny equations, we may conclude that, while there are BPS kink-impurity
or even BPS kink-antikink-impurity solutions, no BPS sphaleron-impurity states
are possible in this set-up.

To construct a model which possesses a BPS sphaleron-impurity solution we have
to modify BPS-impurity Lagrangian in such a way that two Bogomolny equations
show up. It means that we need a non-chiral BPS-impurity theory.

For that reason let us consider the following background field deformed
Lagrangian

$L=\int_{-\infty}^{\infty}\left(\frac{1}{2}\phi_{t}^{2}-\frac{1}{2\sigma}\phi_{x}^{2}-\sigma
U(\phi)\right)dx,$ (12)

where again $\sigma$ is a given impurity. But now the background field
modifies the gradient term, too. To find Bogomolny equations we again analyze
the static energy integral

$\displaystyle E$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}\left(\frac{1}{2\sigma}\phi_{x}^{2}+\sigma
U(\phi)\right)dx$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}\left(\frac{1}{\sqrt{2\sigma}}\phi_{x}\mp\sqrt{\sigma
U}\right)^{2}dx\mp\int_{-\infty}^{\infty}\sqrt{2U}\phi_{x}dx$
$\displaystyle\geq$
$\displaystyle\left|\int_{\phi(-\infty)}^{\phi(\infty)}\sqrt{2U}d\phi\right|=\int_{\phi_{-}}^{\phi_{+}}\sqrt{2U}d\phi\,|Q|.$

The inequality is saturated if and only if

$\frac{1}{\sqrt{2\sigma}}\phi_{x}=\pm\sqrt{\sigma U}.$ (14)

This proves that such a simple impurity model has two Bogomolny equations and
therefore does not treat kink or antikink differently. Note, that the impurity
cannot be a function with negative value but otherwise it is not restricted by
any fine-tuned condition. Similarly as before a sphaleron-impurity BPS
solution is a solution which pice-wisely solves the impurity modified
Bogomolny equations (14). This requires a potential with false vacuum, e.g.,
(6).

In fact, the impurity dependent Bogomolny equation can be rewritten in the
impurity free form after introducing a new base space coordinate $y$

$\frac{dy}{dx}=\sigma(x)\;\Rightarrow\;y=\int\sigma dx+a$ (15)

where $a$ is an integration constant. Then, we arrive at the usual (no
impurity) Bogomolny equations

$\phi_{y}=\pm\sqrt{2U}.$ (16)

Solving these equations and replacing the $y$ coordinate in terms of the
original coordinate $x$ leads to a family of BPS sphaleron-impurity solutions.
Note, that this solution is parameterized by a modulus $a$ reflecting the BPS
property of the solution.

Let us now turn to our potential (6) and for concreteness assume the following
form of the background field

$\sigma(x)=1+\frac{\alpha}{\cosh^{2}(x)},$ (17)

which is an exponentially-like localized impurity centered at the origin.
$\alpha>-1$ is a free parameter measuring its strength. For $\alpha=0$ the
impurity vanishes and we recover the usual theory (1). For our choice of the
impurity the sphaleron-impurity solution reads

$\displaystyle\Phi_{sph}(x;a)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\tanh(x+\alpha\tanh(x)+a+s)\right.$ (18)
$\displaystyle-$ $\displaystyle\left.\tanh(x+\alpha\tanh(x)+a-s)\right),$

where the modulus $a$ can be related to the distance between the sphaleron
(position of the maximum of the field) and the impurity (origin). Indeed, for
$|a|\gg 1$, we get the undeformed sphaleron far away from the origin
(impurity). However, as $a$ tends to 0, the sphaleron approaches the impurity
and deforms. For positive $\alpha$ the original sphaleron gets squeezed as
approaching the impurity, while for $\alpha\in(-1,0)$ it is widened. As we
approach the limiting value $\alpha\to-1$ derivative of the field at the
origin tends to 0. This behaviour is presented in Fig. 3.

We remark that the sphaleron-impurity solution enjoys a reflection symmetry

$\Phi_{sph}(-x;-a)=\Phi_{sph}(x;a).$ (19)

All these solutions possess exactly the same $a$-independent energy

$E[\Phi_{sph}(x;a)]=\frac{9\sinh(2s)+\sinh(6s)-24s\cosh(2s)}{6\sinh^{2}(2s)}$
(20)

and form a BPS-type family of sphalerons in the sense explained before. The
sphaleron can be placed at any distance from the localized impurity without
changing its energy. Thus, there is no static force between the sphaleron and
impurity.

In the subsequent part of the work we will analyze various aspects of dynamics
of such a semi-BPS sphaleron-impurity system.

##  IV The mode structure

Figure 4: The frequency of the bound mode of the impurity.

Although the energy is degenerated while we change the modulus $a$ it is not
the case for the linear mode structure. This is related to the fact that the
impurity is not just a uniform vacuum and therefore nontrivially modifies the
shape of the sphaleron. To see that we consider the static BPS solution with
an addition of a small perturbation $\eta(x,t)$

$\phi(x,t)=\Phi_{sph}(x;a)+\eta(x,t)$ (21)

and insert it into the full time-dependent field equation

$\phi_{tt}-\frac{1}{\sigma}\phi_{xx}+\frac{\sigma_{x}}{\sigma^{2}}\phi_{x}+\sigma
U_{\phi}=0.$ (22)

At linear order we get

$\displaystyle\omega^{2}(a)\eta(x;a)=$ (23)
$\displaystyle\left[-\frac{d}{dx}\left(\frac{1}{\sigma}\frac{d}{dx}\right)+\sigma
U^{\prime\prime}(\Phi_{sph}(x;a))\right]\eta(x;a),$

where we assume that $\eta(x,t;a)=e^{i\omega(a)t}\eta(x;a)$. Here, $\omega(a)$
is frequency of the pertinent mode $\eta(x;a)$. Dependence on the modulus $a$
is explicitly marked.

Asymptotically, for $a\to\pm\infty$, the semi-BPS sphaleron $\Phi_{sph}(x;a)$
is just an infinitely separated system of the sphaleron in the original (no
impurity) model $\Phi_{sph}(x)$ and the impurity. Thus, the mode structure is
just a superposition of the modes of the free sphaleron and modes which are
added due to the background field itself. The former ones are localized on the
sphaleron while the later ones are localized on the impurity (here at the
origin). In our example, the free sphaleron, at least for not too small $s$,
is just a pair of well separated kink and antikink and therefore it has two
positive discrete modes $\eta_{1,2}(x)$ originated by the shape modes of each
of the constituent solitons. Thus, as $s$ becomes larger their frequencies
tend to $\omega^{2}_{1,2}=3$. There is also a zero mode
$\eta_{0}(x)\sim\partial_{x}\Phi_{sph}(x)$. Finally, there is the negative
mode, $\eta_{-1}(x)$ whose frequency tends to 0 for $s\to\infty$. In other
words, the sphaleron becomes quasi-stable for large $s$.

As we already noted the impurity may add some additional modes. This is found
by linear perturbation around $\phi=0$ solution. In our example there is only
one such a mode localized at the origin and it arises for $\alpha<0$, see Fig.
4. This plot is independent of the value of the model parameter $s$. It
follows from the fact that $U^{\prime\prime}(\phi=0)=4$. This mode will
actively participate in dynamics of the sphaleron.

Figure 5: The dependence of the mode structure of the sphaleron-impurity
solution on the modulus $a$. Upper panel: $s=3$, $\alpha=3$. Lower panel:
$s=3$, $\alpha=-0.5$.

If the sphaleron approaches the impurity it gets deformed and the spectral
structure changes in a more and more significant manner. Of course, due to the
BPS nature of the solution the zero mode still exists even though the
translational symmetry is lost

$\eta_{0}(x;a)=N(s,\alpha)\;\partial_{a}\Phi_{sph}(x;a),$ (24)

where the normalization constant $N(s,\alpha)$ is fixed for a given potential
and background field., i.e., for given $s$ and $\alpha$. Importantly, the zero
mode has one node. This implies that the solution is a genuine sphaleron with
one negative energy, unstable mode. In general, however, the spectral
structure may be deformed in a rather drastic way, see Fig. 5.

First of all, the background field can have impact on the unstable mode.
Namely, as $a\to 0$, the frequency of the unstable mode frequency rises/
lowers for negative/positive strength $\alpha$, respectively. Thus,
approaching the origin the sphaleron becomes more or less stable depending on
the impurity.

Secondly, also the spectrum of the massive bound modes changes. For
sufficiently large positive impurity the bound modes of the free sphaleron can
cross the mass threshold at a certain $a=a_{sw}$. This fact is known to
trigger the spectral wall phenomenon which has a significant impact on kink
dynamics [32]. In section VI we will study this phenomenon closer.

Finally, we remark that there is an interesting possibility related to the
change of the frequency of the negative mode. By an appropriate choice of the
impurity a genuine, fast decaying sphaleron can be transformed into a quasi-
stable sphaleron with the negative mode very close to 0. This happens at a
certain distance from the impurity due to the sphaleron-impurity interaction.

##  V The geodesic dynamics

In the well known BPS soliton limit, BPS solutions saturate the pertinent
topological bound. This means that in a given topological sector there are no
solutions with lower energy. A natural consequence of that is that the
simplest, lowest energy dynamics occurs via transitions between energetically
equivalent BPS solutions, which reflects an excitation of the zero mode. This
found an elegant formulation as a geodesic motion on the corresponding moduli
space of the BPS states [22].

In the case of the BPS sphalerons we do have a family of energetically
degenerated solutions $\mathcal{M}[a]=\\{\Phi_{sph}(x;a)\\}$ and therefore the
concept of geodesic motion may still make a sense. However, the sphaleron is
not the global energy minimizer in its, here trivial, topological sector.
Hence, in the space of field configurations, there is an unstable direction
along the negative mode $\eta_{-1}(x;a)$. Therefore looking from this
perspective the BPS sphaleron solutions span a sort of unstable moduli space.

Figure 6: Collision of the sphaleron ($s=3$) for repulsive ($\alpha=1$) and
attractive ($\alpha=-0.5$) BPS impurity, with $v_{in}=0.195$.

As always, the metric on such a moduli space is defined as [1, 11]

$g(a)=\int_{-\infty}^{\infty}\left(\frac{d\Phi_{sph}(x;a)}{da}\right)^{2}dx$
(25)

This can be obtained by inserting the BPS sphaleron-impurity solutions into
the original field-theoretical Lagrangian and promoting the modulus $a$ to a
time dependent collective coordinate $a(t)$. Such an insertion gives rise to a
one-dimensional collective coordinate model (CCM)

$L[a]=\frac{1}{2}g(a)\dot{a}^{2},$ (26)

whose solution is just a geodesic flow on the moduli space $\mathcal{M}[a]$.
We omitted a dynamically unimportant constant potential term. Under particular
circumstances the unstable mode may remain unexcited also when the sphaleron
passes through the impurity. This frequently occurs when the sphaleron is a
quasi-stable solution which happens for sufficiently large $s$. Then, one can
expect that, for a long time, the evolution follows the geodesic flow on the
unstable moduli space $\mathcal{M}[a]$. This is precisely what we saw in our
numerical analysis.

Specifically, in Fig. 6 we show profiles of the field while the sphaleron
passes through the impurity. In the upper panel, for $s=3$ and $\alpha=2$, the
sphaleron is distorted by the impurity exactly as dictated by the moduli space
analysis. The initial velocity of the sphaleron is $v_{in}=0.15$. The profiles
at fixed time $t$ are consistent with the BPS solutions at some value of the
modulus $a$. Furthermore, actual full field theory dynamics is also very well
reproduced by the CCM geodesic flow, see Fig. 7.

However, the sphaleron by definition possesses an unstable mode and it may be
excited, due to higher order nonlinear effects, while the sphaleron goes
through the impurity.

In Fig. 6, lower panel, we present an example of such a process where after
passing the impurity the sphaleron destabilizes and starts to decay. Whenever
the unstable mode gets excited then the geodesic flow on the $\mathcal{M}[a]$
space breaks down and the sphaleron falls apart.

To conclude this section, the geodesic motion based on the CCM model may still
quite well describe the simplest dynamics of a semi-BPS sphaleron but, as one
might expect, it is a less powerful tool if compared with BPS solitons.

Figure 7: Comparison between full numerics and the moduli space flow dynamics
for a non-decaying boosted sphaleron. The parameter of the model are
$s=3,\alpha=2$. The initial velocity is $v=0.15$.

##  VI Spectral wall

Now we will go beyond the simplest geodesic dynamics and investigate how
excitation of massive bound modes changes the interaction between the semi-BPS
sphaleron and impurity. In particular, we are interested in the case where a
bound mode of the static sphaleron-impurity solution crosses the mass
threshold at some $a$. It is known that such a property, if existing in a BPS
kink theories, gives rise to the spectral wall phenomenon.

Speaking precisely, a spectral wall phenomenon is an obstacle in soliton
dynamics due to a transition of a mode into the continuum spectrum. This
happens for a certain value of a moduli space coordinate $a=a_{sw}$ which can
be translated to a distance $x_{sw}$ between the soliton and its collision
partner (another soliton or impurity). The effect strongly depends on the
amount of the mode exited on the soliton. If its amplitude $A$ is smaller than
a critical value $A<A_{crit}$ then the soliton passes $x_{sw}$ with a
distortion growing with increasing of the amplitude. We say that it passes the
spectral wall. If $A>A_{crit}$ then the soliton gets reflected by the spectral
wall where the reflection point occurs sooner for bigger $A$. Finally, if
$A=A_{crit}$, the soliton forms a very long living quasi-stationary state at a
fix position $x=x_{sw}$.

Interestingly, the spectral wall phenomenon has a very selective nature.
Namely, each mode which crosses the mass threshold has its own spectral wall.
Furthermore, position of the spectral wall can be at very large distance from
the interaction partner. Thus, it can trigger a long range interaction.

To study this issue in the context of sphaleron we take, as an initial
configuration, a boosted semi-BPS sphaleron with one of its massive modes
$\eta(x;a)$ excited. We also assume that at $t=0$ the sphaleron is at a large
distance $x_{0}$ from the impurity. Hence, at $t=0$

$\displaystyle\phi(x,t)$ $\displaystyle=$
$\displaystyle\Phi_{sph}(\gamma(x-vt+x_{0}))$ $\displaystyle+$ $\displaystyle
A\eta(\gamma(x-vt+x_{0}))\cos(\omega\gamma(t-vx))$

which provides Cauchy data for the numerical analysis. Of course, the included
mode must hit the mass threshold for some $a$ (i.e., for some distance between
the sphaleron and origin). This is the case for a sufficiently big strength
$\alpha$ of the impurity. Furthermore, for simplicity we will search for
spectral walls in the quasi-stable sphaleron limit, that is for a large $s$.
In particular, in our example, we choose $s=8$ and $\alpha=3$. Due to that,
the free sphaleron looks like a molecule of kink and antikink of $\phi^{4}$
theory with a large separation between the constituent solitons. Hence, its
massive modes are very well approximated by a symmetric and antisymmetric
superposition of the shape modes of the $\phi^{4}$ kinks

Figure 8: The collision of the sphaleron with a spectral wall. The lines show
positions $z_{1},z_{2}$ of the composite kink and antikink in the sphaleron.
The colors denote cases with different amplitude of the excited mode. Dashed
lines are the positions of the spectral wall.

$\eta_{\pm}(x)=N_{\pm}\left(\eta_{sh}(x+s)\pm\eta_{sh}(x-s)\right),$ (28)

where

$\eta_{sh}(x)\approx\frac{\sinh(x)}{\cosh^{2}(x)},$ (29)

is the usual shape mode of the $\phi^{4}$ kink and $N_{\pm}$ are normalization
constants. The initial velocity of the sphaleron is $v_{in}=0.01$. In Fig. 8
we present the dynamics of the sphaleron with the $\eta_{+}$ mode excited.
From the perspective of the composite anti-kink the corresponding spectral
wall is located at $x_{sw}=1.25$. Indeed, at this distance from the origin the
mode hits the mass threshold. We clearly see in our numerics the appearance of
the spectral wall. At the critical value of the amplitude of the mode the
composite antikink forms a stationary state at $x=x_{sw}$.

The same pattern is observed for smaller $s$. However, for $s<4$ addition of
the shape mode responsible for the spectral wall should be compensated by an
addition of unstable mode to prevent the sphaleron to collapse [19].

Obviosuly, such a spectral wall acts as another destabilizing factor which
contributes to the decay of the sphaleron.

##  VII Decay of the semi-BPS sphaleron

By definition, the sphaleron is an unstable solution, here, with two decay
channels. The first is a trivial one and describes a decay into the true
vacuum. The sphaleron splits into kink and antikink which continually
accelerate in opposite directions. The second possibility is much more
interesting. It corresponds to the decay into the false vacuum and creation of
an oscillon. However, this oscillon is not a BPS-type object. Furthermore, it
is not immersed in the trivial vacuum but in a rather complicated background
provided by the impurity. Hence, its patterns of interactions are much more
involved than in the usual non-impurity model.

For simplicity, we analyze the decay of sphaleron when it is located at the
origin. Then, the problem maintains the reflection symmetry. However, one
should be aware that decay of the semi-BPS sphaleron depends on its position
with respect to the impurity.

One interesting scenario occurs if the impurity possesses its own internal
mode. Then, the sequence of events is the following. First, the sphaleron
decays into its oscillon which oscillates in the background of the impurity.
As a consequence, the oscillon loses energy, reduces its amplitude and
increases its frequency. After quite a long time the oscillons settle down at
a mode of the impurity which finally slowly decays via emission of radiation.
So, we may conclude that in this case the oscillon is trapped by the impurity
and transformed into to a bound mode.

Figure 9: Decay of the sphaleron for $s=3$ and $\alpha=-0.5$.

In Fig. 9 we present an example of such a behavior for $s=3$ and
$\alpha=-0.5$. In the first phase, for $t<1050$, the large oscillon gradually
lowers its amplitude, see the upper panel. As time grows, we also see a
characteristic double oscillations of the amplitude. Suddenly at $t\approx
1050$ there is a rapid jump both in the amplitude and frequency, see the upper
and bottom panel. The reason for that is the radiation burst effect. It arises
due to the fact that the frequency of oscillations is half of the mass
threshold (m=2) and therefore the double harmonics may freely propagate. The
increase of radiation is clearly visible in the central panel, where we
present the field measured at $x=20$. After this moment of the evolution the
system tends to a configuration being the impurity with a mode excited. This
is a very slow process which we see particularly well if a dumping term is
added to the field equation.

Figure 10: Decay of the sphaleron and interaction of the emerging oscillon
with the impurity: $\phi(x,t)$ for $s=0.2$. Upper: $\alpha=0.0595$ \- ejection
of the oscillons; central: $\alpha=0.0575$ \- bouncing of oscillons; lower:
$\alpha=0.05925$ \- formation of a quasi-stationary state.

Several comments are in order. Firstly, initially the sphaleron reappears at
the turning points of the oscillon oscillations. We can also see that the
sphaleron carries some excitations of its internal mode, see upper panel and
small wiggles in Fig. 9 when $\phi(0)$ is close to $\phi=1$.

Secondly, one can notice that radiation in the large oscillon phase is much
lower than in the phase where the internal mode starts to participate in the
decay of the oscillon.

Thirdly, the existence of a mode of the impurity effectively looks as
appearance of an attractive force between the oscillon and impurity. In our
example, this is indeed the case for a negative $\alpha$.

A drastically different scenario happens if the impurity does not host any
bound mode. The sphaleron again decays into its oscillon which evolves in the
background of the impurity. After a short time this oscillon is destabilized
by the impurity and splits into two oscillons which are ejected in opposite
directions, see Fig. 10 upper panel, where $s=0.2$ and $\alpha=0.0595$. Hence,
we observe a repulsive interaction between the oscillon and impurity. In our
example it occurs for positive $\alpha$.

However, a detailed analysis shows that the situation is much more involved.
The interaction between the ejected oscillons and the impurity possesses also
an attractive channel. An example of that evolution is presented Fig. 10
central panel. We see that both ejected oscillons turn back and bounce around
the origin. Here $s=0.2$ and $\alpha=0.0575$.

Interestingly, we discovered a limiting situation where these oscillons form a
long living quasi-stationary state. Indeed, for a very long time their
positions stay fixed at a certain distance from the impurity. This is clearly
visible in Fig. 10 lower panel, where $s=0.2$ and $\alpha=0.05925$.

##  VIII Semi-BPS sphaleron-impurity collisions

As we already pointed out, once the sphaleron decays, the system completely
loses its semi-BPS property and the resulting oscillon reveals a rather
nontrivial pattern of interaction with the impurity. This is precisely
observed if the semi-BPS sphaleron is scattered with the impurity. The main
outcome of such processes is the creation of the oscillon in the first
collisions and its subsequent interaction with the impurity.

In Fig. 11 we present an example of collisions of a quasi-stable sphaleron,
$s=2.5$, with an attractive impurity, $\alpha=-0.5$. For smaller velocities,
$v_{in}<v_{1}\approx 0.051$, the sphaleron is always destabilized by the
impurity and decays into an oscillon. Then the oscillon is trapped by this
attractive impurity exactly as described in the previous section i.e., it
excites a bound mode of the impurity. On the other hand, if the initial
velocity is larger than $v_{2}\approx 0.182$, then the sphaleron passes
through the impurity. This again destabilizes the sphaleron which eventually,
at a certain distance from the impurity, decays into a boosted oscillon

Figure 11: Examples of scenarios for the sphaleron impurity collisions for
$s=2.5$ and $\alpha=-0.5$ for increasing initial velocity: from 0.051 to 0.192

However, between these two cases there is a variety of different processes. We
define the following qualitatively distinct classes of behaviour:

  1. 1.

Bounces of the oscillon.
Sphaleron decays into oscillon which is then attracted by the impurity. As a
consequence, the oscillon performs oscillations around the impurity.

  2. 2.

Oscillon trapping or ejection.
Such oscillations release energy and after a few bounces we end up with
trapping of the oscillon (eventually leading to an excited impurity) or
ejecting the oscillon both in the forward and in the backward direction.

  3. 3.

Oscillon stationary state.
The oscillon can also form a stationary state where it stays at a constant
distance from the impurity. This case resembles very much the stationary state
found in decay of a semi-BPS sphaleron in the presence of repulsive impurity.

  4. 4.

Reappearance of sphaleron.
Sphaleron collapses but then it is created again. After the creation it can
decay into the oscillon or into a pair of kink and antikink which escape from
each other.

By varying the initial velocity of the sphaleron we found an evidence that the
formation of the final state reveals a structure of a probably chaotic nature.
Indeed, different scenarios follow one after the other in a rather random way,
see Fig. 12, where we plot the field at the large time after collision. One
can easily see that the trapping of the oscillon by the impurity dominates.
Other possibilities like backward and forward ejection of the oscillon as well
as the kink-antikink creation, are also clearly visible.

Figure 12: The spatial dependence of the field $\phi(x)$ at large time for
different initial velocity $v_{in}$.

##  IX Summary

The main achievement of the present work is establishing the existence of a
type of BPS sphaleron-impurity solutions. We call them semi-BPS sphalerons
because of the fact that they comprise two rather different features.

On the one hand, they resemble the usual BPS systems known from the theory of
topological solitons, which means the existence of a family of energetically
equivalent static solutions which obey a lower order differential equation.
This results in the appearance of a moduli space with a coordinate being a
continuous parameter (modulus) parameterising the solutions and in existence
of a zero mode which is not related with the trivial translation invariance of
the theory. Specifically, in our setup, the sphaleron can be put at any
distance from the impurity and the total energy of the system remains
unchanged. This means that there is no static force between sphaleron and
impurity.

On the other hand, there is a fundamental difference in comparison with BPS
solitons. Namely, the semi-BPS sphaleron does not saturate any topological
energy bound and therefore there is an unstable direction along which such a
static solution can decay. In other words, there is a negative mode in the
spectrum of small perturbations. This is, of course, an expected property of a
genuine sphaleron.

As pointed out by Manton [23] the semi-BPS property means that the solutions
obey the corresponding Bogomolny equation on a double-cover of the complex
plane of $\phi$.

After proving the existence of semi-BPS sphaleron-impurity solutions we
investigated their dynamical properties. The reason is obvious. As in BPS
multi-soliton cases, the absence of a static force between localized objects
which participate in scatterings (here the sphaleron and impurity) provides a
simplified environment and allows for a deeper insight into dynamical
properties of sphalerons. Indeed, we found that the semi-BPS sphaleron
impurity system reveals many properties known form the usual BPS multi-soliton
systems.

For example, in the simplest dynamics, i.e., where only the kinetic DoF is
excited, the semi-BPS sphaleron follows the geodesic motion on the pertinent
moduli space, which means that during such a time evolution the field passes
through the available BPS states.

Next, if one excites a positive bound mode, the sphaleron may be subject to
the spectral wall phenomenon. As always, the necessary condition is that such
a mode touches the mass threshold at certain value of the modulus, which
translates into a certain distance between the sphaleron and impurity.

We remark that there is a range of the model parameter, where the sphaleron
looks like an almost stable kink-antikink molecule. Here it occurs for
sufficiently large $s$. Hence, it is natural to expect that collisions of such
quasi-stable sphalerons may find an interpretation as a four-kink scattering.

This standard BPS-soliton like dynamics is strongly modified once the unstable
mode is excited. Then the field probes a direction in the configuration space
along which the sphaleron can decay forming e.g., an oscillon. Of course, due
to the saddle point nature of the sphaleron the unstable mode can be excited
by an arbitrarily small perturbation and therefore, after sufficient amount of
time, it will always be the main factor of the evolution. Importantly, the
oscillon is not a BPS object and therefore interacts with the impurity in a
rather sophisticated manner. In this sense the semi-BPS sphaleron is a less
powerful concept than the usual BPS limit.

Nonetheless, our framework still allows for the study of the sphaleron decay
in a simplified, BPS-like situation i.e., in the limit of the absence of the
sphaleron-impurity force. Simultaneously, the produced oscillon is immersed in
a very nontrivial environment which amounts to appearance of an intriguing
oscillon-impurity stationary state. Its origin, as well as understanding its
properties and importance requires further studies.

The current work can be in a natural way extended to models where the
background field is replaced by a dynamical field. The resulting field theory
will contain at least two coupled scalar fields. Some of such models indeed
host a family of energetically equivalent unstable solutions. See for instance
the Montonen-Sarker-Trullinger-Bishop (MSTB) model [36, 37], where a one-
parameter family of sphalerons (in this case, non-topological solutions
connecting and returning to the same point in a two-dimensional internal
space) exists [28]. They could decay into one of the vacua, possibly giving
rise to radiation emission and the presence of oscillons, but they could also
decay into two less energetic kinks. It would be interesting to check how the
dynamical properties of sphalerons and oscillon found in our work look in this
model. Sphalerons can also be found in the Bazeia-Nascimento-Ribeiro-Toledo
(BNRT) model [38] for certain values of the coupling constant. In this case,
they are unstable topological kinks that can decay into other less energetic
kinks [39]. From our perspective, the study of these processes is also
relevant. Further examples of theories with sphalerons can be found along the
line described in [40].

A more general question is whether any sphaleron can be understood as an
unstable multi-soliton state. Here it is indeed the case. For large $s$ we
clearly appreciate a kink-antikink inner structure. In fact, for all $s$
considered here, the sphaleron can be written as a superposition of a kink and
antikink. Interestingly, such a picture reproduce the unstable and the zero
mode of the sphaleron (and possible also the shape modes). Namely, the zero
mode is related to a shift of the positions of the constituent kink and
antikink in the same direction, $z_{1,2}\to z_{1,2}+b$ while the unstable mode
corresponds to a similar shift but in opposite direction, $z_{1}\to z_{1}+b$,
$z_{2}\to z_{2}-b$. It is clear that the second transformation changes the
distance between the constituents.

A related problem is how properties of sphalerons are affected by external
conditions. This includes, for example, the appearance of external magnetic
and electric fields or other localized objects (solitons or sphalerons). As we
saw, for some impurities, the frequency of the negative mode comes closer to
0, which may transform a rather unstable sphaleron to a quasi-stable version.
We note that impact of a nontrivial background (e.g., external magnetic field)
on properties of sphalerons has a great physical importance [41].

## Acknowledgements

K. O., T. R., and A. W. were supported by the Polish National Science Centre
(Grant No. NCN 2019/35/B/ST2/00059). A.A.I., S.N. and J.Q. were supported by
the Spanish Ministerio de Ciencia e Innovación (MCIN) with funding from the
European Union NextGenerationEU (PRTRC17.I1) and the Consejería de Educación,
Junta de Castilla y León, through QCAYLE project, as well as MCIN Project No.
PID2020-113406GB-I00 MTM. S.N. thanks JCyL for the pre-doctoral contract
financed through ESF+.

K.O. and A. W. thank Nick Manton for inspiring discussions and for sharing an
early version of [23].

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# Exploring Consequential Robot Sound:
Should We Make Robots Quiet and Kawaii-et?

Brian J. Zhang1, Knut Peterson1, Christopher A. Sanchez2, Naomi T. Fitter1 1
Collaborative Robotics and Intelligent Systems (CoRIS) Institute, Oregon State
University, Corvallis, OR 97331, USA. {zhangbr<EMAIL_ADDRESS><EMAIL_ADDRESS>School of Psychological Science, Oregon State
University, Corvallis, OR 97331, USA<EMAIL_ADDRESS>
###### Abstract

All robots create consequential sound—sound produced as a result of the
robot’s mechanisms—yet little work has explored how sound impacts human-robot
interaction. Recent work shows that the sound of different robot mechanisms
affects perceived competence, trust, human-likeness, and discomfort. However,
the physical sound characteristics responsible for these perceptions have not
been clearly identified. In this paper, we aim to explore key characteristics
of robot sound that might influence perceptions. A pilot study from our past
work showed that quieter and higher-pitched robots may be perceived as more
competent and less discomforting. To better understand how variance in these
attributes affects perception, we performed audio manipulations on two sets of
industrial robot arm videos within a series of four new studies presented in
this paper. Results confirmed that quieter robots were perceived as less
discomforting. In addition, higher-pitched robots were perceived as more
energetic, happy, warm, and competent. Despite the robot’s industrial purpose
and appearance, participants seemed to prefer more “cute” (or “kawaii”) sound
profiles, which could have implications for the design of more acceptable and
fulfilling sound profiles for human-robot interactions with practical
collaborative robots.

##  I Introduction

Like all machines, robots invariably produce sound alongside their motion in
the form of _consequential sound_ [1]. Robot sound can affect human-robot
interaction (HRI) by reducing the perceived usability of prosthetics [2],
improving human-robot localization [3], or even impeding interactions with
robot companions [4]. In the most extreme case, excessively loud sound has
precluded the usage of a robot [5]. Despite these examples, however, little is
known about robot consequential sound and how it might impact successful
design for HRI. This paper aims to _identify important physical
characteristics of robot consequential sound_ and measure the strength and
consistency of these characteristics’ effects on the _perception of robot arm
recordings_.

Past researchers have encountered considerable challenges when attempting to
connect objective auditory characteristics to the subjective perception of
robot consequential sounds. A survey of servo motor sounds found that the
correlations between the spectral centroid, servo weight, and subjective audio
strength were the only significant relationships between twelve objective
measures and six subjective measures [6]. Conversely, a survey with high-end
and low-end robot arm sounds failed to find a significant difference between
the two [7]. These past efforts relied only on sound samples from existing
robot arms and mechanisms (rather than richer audio-edited sets of existing
and prospective robot sounds), which limited the work’s ability to acutely
understand how specific characteristics of sound impacted perception. Other
research has considered audio-edited versions of consequential robot sound.
Work in [3] showed that adding broadband and tonal sound to robot movement
makes a robot more localizable and noticeable, though also more annoying.
Recent work by the authors has pointed towards the benefit of quieter and
higher-pitched robots, though a lack of granularity in conditions limits the
understanding of this effect [8]. While these past results demonstrate how
altering the sound profile of robots can improve experiences working with
collaborative robots, _key questions remain as to how finer audio
manipulations on promising objective scales, such as those shown in Fig. 1,
will influence subjective robot perception_.

To understand the current state of robot sound research, we reviewed related
work in Section II. Based on our previous pilot study results in [8], we
designed and conducted follow-up studies as described in Section III. The
results of _manipulations in loudness_ are presented in Section IV, and the
results of _manipulations in pitch_ are presented in Section V. Section VI
discusses our findings and related design implications for robot sound
profiles.

Figure 1: An overview of the study manipulations.

##  II Related Work

To inform our efforts, we explored prior work in robot sound and related
fields; the nascent field of robot sound can benefit from drawing on more
established bodies of work.

_Psychoacoustics & Sound Design:_ Psychoacoustics defines the three subjective
qualities of sound as _loudness_ , _pitch_ , and _timbre_. While _loudness_
and _pitch_ are well-known to non-psychoacousticians, the _timbre_ of a sound,
which can be broken down into factors such as sharpness, pleasantness,
roughness, and tonalness, encompasses all remaining perceptual properties of a
sound [9]. However, sound design for products and interactions must also take
context-dependent factors (e.g., the input of other senses, personal knowledge
and associations of sound) into account [10]. For instance, modulation in
mechanical sounds (e.g., gearboxes) is generally undesirable while modulation
in musical sounds (e.g., vocoders) is often desirable [11]; loudness may be
perceived as unwanted noise, but products such as vacuums and motorcycles may
benefit from increased loudness [12]. As such, good sound design usually
relies on a combination of proper sound design methods and expertise in
related fields such as acoustics, engineering, and psychology [11, 1, 13].

Product sound design defines the two main categories of product sound as
_consequential sound_ (e.g., the meshing of servo motor gears, the hum of
cooling fans, or the rattle of drivetrain chains) and _intentional sound_
(e.g., non-linguistic utterances, back-up alarm beeps, or played-back music).
As their origins and design spaces differ greatly, consequential and
intentional sound require different design processes and expertise [1].
Characterizing and designing consequential sound, which relies on product
components’ designs, materials, and manufacturing, is especially difficult [1,
10]. As a first step in this challenging process, we aim to _identify key
perceptual trends related to loudness and pitch of consequential robot
sounds._

_Sound in Human-Robot Interaction:_ Sound in HRI also falls into intentional
and consequential sound categories. Recent work has explored consequential
robot sounds and subtypes of intentional robot sounds such as _vocable_ and
_transformative_ sound. Vocable sound, or non-linguistic utterances, can
independently convey affect and amplify the affective interpretation of
actions [14]. Transformative sound, or intentional sound designed to be
associated with robot motion, also changes affective and objective responses.
In a study on localization of a visually obscured robot, playing broadband and
tonal sounds increased the accuracy and inference speed for localizing the
robot as well as perceived noticeability, localizability, and annoyance [3].
Work on masking undesirable consequential sound with musical sounds showed
that participants rated masked sound more positively than just consequential
sound [15, 16].

Consequential robot sound research has evaluated sound at the component,
robot, and inter-robot level. For robot components, a study on servomotor
sounds found that although the sounds were separable on six subjective
attribute scales, only one of these attributes was correlated with objective
characteristics of the servomotors [6]. Yet, participants were able to
accurately identify the lowest-quality and most inexpensive servomotor as the
most inappropriate, untrustworthy, weak, imprecise, and inexpensive [17].
Another project found that consequential sounds of a humanoid robot may
communicate emotions separately from the corresponding robot motions,
potentially requiring masking through sonification of motion [18]. Lastly,
comparisons between the sounds of a high-end and a low-end robot arm found
that the effect of sound trended differently across contexts, but the presence
of sound detracted overall [7]. While these investigations do compare current
robot sounds, they unfortunately do not offer clear direction on how to design
_future_ robot sound. Our work builds upon this body of research by
_investigating how systematic manipulations of consequential robot sound might
affect perceptions of a robot._

##  III Methods: All Studies

We conducted a series of four studies via Amazon Mechanical Turk (MTurk) to
explore the effects of modifying loudness and pitch on participant perceptions
of robots. All study procedures were approved by Oregon State University under
IRB-2019-0172.

###  III-A Hypotheses

These hypotheses arose from our past results in [8]:

  1. H1:

Reducing the loudness of a robot will lead to perceptions of lower discomfort
and higher competence.

  2. H2:

Lowering the pitch of a robot will lead to perceptions of higher discomfort.

###  III-B Study Design

The studies employed a UR5e robot arm. The arm performed motions mimicking
_pick & place_ and _screwdriving_ tasks without any props, as shown in Fig. 2.
We performed manipulations across two video types to serve as repetition for
confirming trends across different robot use contexts.

Figure 2: Cropped keyframes from the screwdriving and pick & place motions.

Using a Canon Vixia HF R800 camera and a Blue Snowball microphone, the arm was
recorded completing these motions with different background objects. After we
captured the recordings, we manipulated the audio track of the videos to
create different stimuli. All stimuli included noise reduction with a level of
20 dB, 3.50 sensitivity, and 2 frequency smoothing bands and then underwent
condition-specific manipulation before background noise was added back into
the stimuli. Manipulations in loudness were performed in MATLAB and
manipulations in pitch were performed in Audacity using the Change Pitch tool
with the high-quality setting. The _pick & place_ motion stimuli were 18
seconds long and the _screwdriving_ motion stimuli were 14 seconds long. The
accompanying video compares several select stimuli, while the full set of
stimuli are available in [19].

###  III-C Participants

Participants were MTurk workers from the United States with qualifications of
$>$97% prior task approval rate and $>$5000 previously approved tasks. Study
instructions required that participants have normal or corrected-to-normal
vision and hearing and also be native English speakers. Participants were not
allowed to participate more than once across all studies. We compensated
participants with 3.75 USD for finishing the estimated 15-minute task. Based
on the results from [8], we performed an _a priori_ power analysis using
G*Power for a repeated-measures ANOVA with $\alpha=0.05$ and $\beta=0.05$
[20]. As the effect sizes ranged from small to medium, a sample of $N=100$
participants was recommended for each study to ensure adequate statistical
power.

###  III-D Procedure

Upon enrolling in the study, participants first were asked to provide informed
consent. Participants then completed a 15-minute survey implemented via
Qualtrics, beginning with an introductory module common across all studies.
This module included a loudness calibration video, which requested that the
respondent wear headphones or earbuds and set their device volume so that the
speech in the video was at a “loud but comfortable volume.” The survey also
requested that the participant not change their device volume for the
remainder of the study. Next, the participant was introduced to the UR5e via
an image of the UR5e in the lab environment and the following explanation:
_“We would like to know what you think about several versions of a robot. Here
is a photo of the robot in a lab environment.”_ Lastly, the module presented a
video stimulus featuring the Cassie robot walking on a treadmill and a post-
stimulus questionnaire as described in Section III-E. We included this
introductory stimulus across all studies to have a common point of reference
for all participants.

After finishing the introductory module, participants completed the study-
specific module, which included eight levels of manipulation for either
_loudness_ or _pitch_ on either the _pick & place_ or _screwdriving_ stimuli.
Studying eight levels of manipulation is consistent with prior work in the
robot sound field [21, 6]. The order of stimuli was counterbalanced such that
stimuli of adjacent levels would not play directly after one another, and the
order of stimuli was presented as evenly as possible. For each stimulus, the
survey page hid the “Next” button for 40 seconds and requested the participant
watch the stimulus video at the top of the page. The participant was also
asked to answer the post-stimulus questionnaire as described in Section III-E.
After the fourth and eighth stimulus presentations, participants had to
successfully complete an attention check question to continue. This step was
to help ensure that participants were attending to the task and not selecting
random responses.

Lastly, participants completed the closing module. This module contained a
free-response question that asked participants to describe the parts of the
robot videos that affected their responses the most in a minimum of 200
characters. Next, the survey contained a manipulation check to determine
whether participants could differentiate between stimuli when shown the videos
side-by-side. Participants who reported that the stimuli were the same did not
advance. Participants who reported that the stimuli were different then
completed the Negative Attitudes Towards Robots Scale (NARS) and a demographic
survey, both described in Section III-E, before receiving a completion code
for compensation.

Participants who did not finish the survey, failed attention or manipulation
checks, or attempted to take the survey more than once were excluded from the
study.

###  III-E Measurement

We recorded these subjective and perceptual measures:

  * •

_Post-stimulus questionnaire_ : after each stimulus, the Robotic Social
Attributes Scale (RoSAS) captured participant perceptions of _warmth_ ,
_competence_ , and _discomfort_ by combining six component attributes for each
subscale [22]. Participants rated each attribute on a six-point Likert scale
from “definitely not associated” to “definitely associated.” _Valence_ and
_arousal_ from the circumplex model of affect were acquired via participant
association of the robot with “happy,” which also contributes to the _warmth_
subscale, and an additional attribute of “energetic.” After all stimuli, a
free-response question requested the most important factors in participant
responses.

  * •

_Attitudes questionnaire_ : after the post-stimulus questionnaires, the NARS
questions captured general attitudes towards robots for potential use in
subsequent covariate analyses. Using a seven-point Likert scale, participants
indicated their agreement level with fourteen statements that were combined
into subscales of negative attitudes towards _interactions with robots_ ,
_social influence of robots_ , and _emotions in robots_ as described in [23].

  * •

_Demographic questionnaire_ : at the end of the survey, a final questionnaire
recorded demographic and occupational information.

This work focuses on the analysis of post-stimulus responses. The
questionnaires and stimulus videos are available in [19].

Figure 3: Post-stimulus RoSAS responses for Studies 1-2. Boxplots include
boxes from the 25th to the 75th percentiles, center lines for medians,
asterisks for means, whiskers up to 1.5 times the inter-quartile range, and
“+” marks for outliers. Brackets above the boxplots indicate pairwise
differences.

###  III-F Analysis

Responses to the post-stimulus questionnaire were analyzed using repeated-
measures analysis of variance (rANOVA) tests with an $\alpha=0.05$
significance level and adjusted with Greenhouse-Geisser sphericity
corrections. Pairwise comparisons were conducted to unpack any significant
main effects using Tukey’s HSD test, which adjusts for Type 1 error inflation.
We report effect size $\eta^{2}$, where $\eta^{2}=0.010$ is considered a small
effect, $\eta^{2}=0.040$ a medium effect, and $\eta^{2}=0.090$ a large effect
[24]. All statistical analyses were conducted using jamovi [25, 26, 27, 28].

##  IV Results: Studies 1-2, on Loudness

These studies tested H1 to determine how loudness might impact perception of
robots. Study 1 included _pick & place_ stimuli with 35.5% to 179.6% of the N5
loudness compared to the loudness calibration video [9]. Study 2 included
_screwdriving_ stimuli with 26.7% to 124.9% relative loudness.

###  IV-A Participants

Study 1 was completed by $N=99$ participants who were adults between 21 and 69
years of age ($M=36.3$, $SD=10.9$), with 62.6% cisgender men, 35.4% cisgender
women, 1.0% transgender women, and 1.0% non-binary individuals. An additional
14 respondents failed the manipulation check, and their responses are not
considered in these results.

Study 2 was completed by $N=94$ participants who were adults between 22 and 69
years of age ($M=38.9$, $SD=11.7$), with 54.3% cisgender men, 43.6% cisgender
women, and 2.1% non-binary individuals. An additional 12 respondents failed
the manipulation check for this study and were excluded from the results.

Figure 4: Post-stimulus RoSAS responses for Studies 3-4. Boxplots include
boxes from the 25th to the 75th percentiles, center lines for medians,
asterisks for means, whiskers up to 1.5 times the inter-quartile range, and
“+” marks for outliers. Brackets above the boxplots indicate pairwise
differences.

Introductory stimulus and NARS results did not differ significantly between
Study 1 and Study 2, suggesting that participants across these studies were
well-matched on these initial robot attitudes.

###  IV-B Effects of Loudness

Responses to post-stimulus questionnaires along RoSAS subscales for Studies
1-2 appear in Fig. 3. rANOVAs showed that, as _loudness_ increased: perceived
_valence_ decreased in Study 1 ($p<0.001$, $F(5.26,\,515.82)=4.16$,
$\eta^{2}=0.007$) but not in Study 2; perceived _arousal_ increased in Study 2
($p=0.003$, $F(5.74,\,533.80)=3.38$, $\eta^{2}=0.008$) but not in Study 1;
perceived _warmth_ decreased in Study 1 ($p=0.011$, $F(5.65,\,553.17)=2.88$,
$\eta^{2}=0.029$) but not in Study 2; perceived _competence_ decreased in
Study 1 ($p<0.001$, $F(5.67,\,555.56)=6.86$, $\eta^{2}=0.011$) but not in
Study 2; and perceived _discomfort_ increased in both Study 1 ($p<0.001$,
$F(3.45,\,338.33)=37.74$, $\eta^{2}=0.077$) and Study 2 ($p<0.001$,
$F(4.48,\,416.48)=16.89$, $\eta^{2}=0.040$).

##  V Results: Studies 3-4, on Pitch

These studies tested H2 to determine the effect of pitch on perception of
robots. Study 3 included _screwdriving_ stimuli frequency shifted by -24, -18,
-12, -6, 0, +4, +8, +12 semitones. Study 4 included _pick & place_ stimuli
shifted by -12, -6, 0, +6, +12, +18, +24, +30 semitones to investigate whether
trends from Study 3 persisted at higher pitches.

###  V-A Participants

Study 3 was completed by $N=89$ participants who were adults between 22 and 69
years of age ($M=38.5$, $SD=11.8$), with 58.4% cisgender men and 41.6%
cisgender women. An additional 15 respondents failed the manipulation check
and were excluded from the results.

Study 4 was completed by $N=100$ participants who were adults between 23 and
69 years of age ($M=36.8$, $SD=11.1$), with 53.0% cisgender men, 45.0%
cisgender women, and 1.0% transgender men. An additional 6 respondents failed
the manipulation check, and their responses are not considered in these
results.

Introductory stimulus and NARS results did not differ significantly between
Study 3 and Study 4 with the exception of perceived discomfort from the
introductory stimulus ($t(187)=2.02$, $p=.04$). Thus, participants appear
generally consistent on their base attitudes towards robots.

###  V-B Effects of Pitch

Responses to post-stimulus questionnaires along RoSAS subscales for Studies
3-4 appear in Fig. 4. rANOVAs showed that, as _pitch_ increased: perceived
_valence_ increased in both Study 3 ($p<0.001$, $F(5.13,\,451.67)=5.01$,
$\eta^{2}=0.013$) and Study 4 ($p<0.001$, $F(4.16,\,412.10)=10.32$,
$\eta^{2}=0.021$); perceived _arousal_ increased in both Study 3 ($p<0.001$,
$F(5.89,\,525.08)=6.69$, $\eta^{2}=0.022$) and Study 4 ($p<0.001$,
$F(6.11,\,605.16)=4.62$, $\eta^{2}=0.013$); perceived _warmth_ increased in
both Study 3 ($p<0.001$, $F(5.09,\,448.24)=5.73$, $\eta^{2}=0.007$) and Study
4 ($p<0.001$, $F(4.23,\,418.67)=7.83$, $\eta^{2}=0.008$); perceived
_competence_ increased in Study 3 ($p=0.004$, $F(5.59,\,491.88)=3.28$,
$\eta^{2}=0.006$) but not in Study 4; and perceived _discomfort_ decreased in
both Study 3 ($p<0.001$, $F(5.49,\,482.94)=5.44$, $\eta^{2}=0.011$) and Study
4 ($p<0.001$, $F(4.81,\,476.44)=5.00$, $\eta^{2}=0.011$).

##  VI Discussion

Results partially supported the affective responses expected in H1. In Study
1, more quiet stimuli led to a lower perception of discomfort and a higher
perception of competence as expected, as well as a higher perception of
warmth, but perceptions of competence and warmth did not change in Study 2.
Notably, Study 1 led to a significant ($p<0.001$) increase in perceived
competence for more quiet stimuli with a small to medium effect size, but
Study 2 did not indicate any difference on this scale. Increases in valence
and warmth for quieter stimuli in Study 1 and increases in arousal for louder
stimuli in Study 2 had less than small effect sizes, while decreases for
discomfort for quieter stimuli had medium to large effect sizes in both
studies. Generally, an _increase in discomfort as the loudness increased_ was
the strongest and most repeatable result related to H1.

Data from the free-response survey field can help to explain and support the
observed differences in loudness perception. One participant remarked that
_“when I think of robots I think of soft whirls, maybe some slight clunking,”_
and objected to anything _“overly noisy or off-putting.”_ On a related note,
other respondents noted that the robot seemed to be making _“growling or
roaring animal-like noise,” “a ferocious growl,”_ or sound _“like a monster or
dragon,”_ although the audio was sampled from the natural consequential sound
of the robot. Participants labeled quieter sounds as _“gentle,” “more safe,”_
or _“less dangerous.”_ Conversely, louder sounds were _“scary,” “disturbing,”
“horrifying,” “dangerous,” “mean,” “aggressive,”_ and _“harsh.”_ The uniform
overall perception seems to be that quieter robot arms are preferable.

Results fully supported the affective responses expected in H2 as well as
additional affective responses not anticipated. In Studies 3 and 4, _higher-
pitched stimuli led to higher arousal, valence, and warmth, as well as lower
discomfort_. In Study 3, higher-pitched stimuli also led to higher perception
of competence, though with less than small effect size. For arousal, valence,
and discomfort, effect sizes ranged from small to medium, while the effect
size for warmth was less than small in both studies.

Again, free-response data can reinforce our understanding of these trends. One
participant remarked that _“the sounds they made changed the robot’s
‘personality,’ ”_ going on to state that _“some sounded cute,” “some sounded
horrifying,”_ and _“a few just sounded machine-like.”_ Despite the results
from Studies 3 and 4, we suspect that at some point, the high pitch of a robot
may ultimately be too high. Although we extended the high range of sounds in
Study 4 (compared to Study 3), we have not yet encountered this limit on the
group level; however, at least one respondent noted that _“deeper sounds
leaned more into the negative perceptions […] Interestingly, though, very high
pitched noises tended to the negative as well.”_ For now, the consensus seems
to be that higher-pitched robots are favorable, leading to descriptors such as
_“lively,” “cute,” “bright,” “curious,” “sweet,” “happy,”_ and _“friendly.”_
The incidence of these labels was highest for Study 4, which included the
highest-pitched stimuli. On the other hand, participants described lower-
pitched robots as _“evil,” “ominous,” “scary,”_ and _“as if it was working off
a Darth Vader audio kit.”_ So far, it appears that the pitching up of robot
arm sounds leads to a sort of positive ”sci-fi” perception of robots, in which
robots seem like a friendly _“alien thing”_ or WALL-E-like entity. Pitching
sounds down, on the other hand, leads to the opposite. This suggests that
users may associate the higher-pitched sounds with with a cute (or kawaii in
Japanese [29]) robot.

###  VI-A Design Implications

Our results indicate two clear themes that robot designers should carefully
consider: 1) _quieter robot arms are less discomforting_ than loud robots and
2) _higher-pitched robot arms seem more pleasant, energetic, and warm, as well
as less discomforting_ than lower-pitched robots. After repeating a similar
study design across different robot arms and use contexts, we are reasonably
confident that these results would extend to additional robot arms and use
cases. Follow-up investigations in person, in addition to replication across
other robotic systems, would serve as a helpful final reinforcement of these
findings.

For robot arm designers, we recommend selecting parts to reduce loudness
emitted by the system. Particularly when the trade-off is nominal, selecting
mechanical elements that reduce sound intensity will improve the comfort of
users. This design principle aligns well with workplace safety standards
(e.g., regulations on workplace sound levels [30]); however, we acknowledge
that it is not always desirable or feasible to eliminate robot sound
altogether. For example, the emission of audible robot sound can help people
safely localize robots in their environment (e.g., in [3]). In such cases, we
suggest keeping intensity levels close to the minimum audible level in a given
environment. Furthermore, when it is impossible or impractical to fully
eliminate sound, tactics like the following pitch adjustment principles might
serve as another option for improving robot favorability.

In terms of pitch, reducing lower-pitched sounds in the design of robot arms
seems to be most crucial. Transforming the consequential sounds of a robot to
a higher-pitch can also enhance perceptions of the system. While changes in
pitch intensity may inherently be linked to part selection, alterations to
pitch intensity based on real-time sampling might be an option to increase
positive perceptions of robots. For example, future robotic systems might
include a microphone and hardware for fast audio sampling and processing.
Coupled with the additional capability to then emit this modified sound
through speakers would enable the transformation of the overall robot sound
profile. Using this type of pipeline, one could overlay or augment robot
sounds in real time to achieve desired robot attributes.

###  VI-B Strengths & Limitations

One key strength of this work is its rigorous evaluation of the relationship
between physical sound attributes and HRI. At least within the application
space of robot arms, we identified repeatable results related to modulations
of loudness and pitch of robot sound. While research on robot sound has
historically seemed crucial but full of mixed results, our initial results and
proposed methods for research in this space offer the potential to alter the
state of the field. Our proposed sound editing approaches offer an accessible
path for most roboticists without the need of proprietary strategies and tools
of audio engineering.

A major limitation of this work is the reliance on video-based surveys (rather
than in-person interactions with robots) to evaluate consequential sound
effects. We plan to conduct follow-up in-person studies to confirm that the
observed trends extend to in-lab and in-the-wild interactions with robots. As
with most studies, the participant population was not fully representative of
all robot users. Due to our MTurk eligibility criteria, most results of this
work are centered on the cultural context of the United States. We plan to
conduct follow-up work with more diverse participant groups, and we welcome
others to build on this work with their own stakeholder populations of
interest.

###  VI-C Conclusions

In this work, we performed four online studies to investigate the effects of
robot sound. This allowed us to compare more levels (i.e., eight levels of
loudness or pitch per study) and attain more repeatable results than past
research on robot sound. Within the space of robot arms, our results
demonstrated that people perceive quieter and higher-pitched robots more
positively. These pitch results surprised us somewhat; we expected that
eventually a high-pitched robot would reach the point of annoyance, but even
our highest-pitched stimuli were not yet perceived as such. Rather, the high-
pitched sound appeared to largely make the robot seem cute or _kawaii_ [29].
Thus, our overall recommendation is to make robots quiet and high-pitched (or
“ _kawaii_ -et”). This work can inform robot designers, researchers, and audio
engineers who wish to improve perceptions of robots.

## Acknowledgments

We thank Prof. Joe Davidson for access to his UR5e robot, Kevin Green and
Jeremy Dao for the Cassie robot recording, Ameer Helmi for help with figures,
and Dylan Moore for study design guidance.

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