# Complex BPS Skyrmions with real energy

Francisco Correa∘, Andreas Fring∙ and Takanobu Taira∙
$\bullet$ Department of Mathematics, City, University of London,
Northampton Square, London EC1V 0HB, UK
$\circ$ Instituto de Ciencias Físicas y Matemáticas, Universidad Austral de
Chile,
Casilla 567, Valdivia, Chile
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract:

We propose and investigate several complex versions of extensions and
restrictions of the Skyrme model with a well-defined Bogomolny-Prasad-
Sommerfield (BPS) limit. The models studied possess complex kink, anti-kink,
semi-kink, massless and purely imaginary compacton BPS solutions that all have
real energies. The reality of the energies for a particular solution is
guaranteed when a modified antilinear $\mathcal{CPT}$-symmetry maps the
Hamiltonian functional to its parity time-reversed complex conjugate and the
solution field to itself or a new field with degenerate energy. In addition to
the known BPS Skyrmion configurations we find new types that we refer to as
step, cusp, shell, and purely imaginary compacton solutions.

††conference: Complex BPS Skyrmions with real energy

##  1 Introduction

The Skyrme model [1] has been introduced as a potential candidate for a low
energy effective field theoretical description of a strongly interacting
matter theory, i.e. Quantum Chromodynamics, more than fifty years ago. It took
a fairly long time to demonstrate that the model could indeed arise as such
type of low energy effective theory in a limit for which the number of quark
colours is taken to be very large [2, 3]. The Skyrme model is perfectly
tailored to the nonperturbative nature of that energy regime and successfully
describes various key characteristics of atomic nuclei. The topological
soliton solutions of the model, the Skyrmions, are identified as Baryons with
integer topological charges $B$ being elements in the third homotopy group for
the $SU(2)$-group valued fields, $B\in\mathbb{Z\simeq\pi}_{3}(SU(2))$. The
field excitations around a trivial vacuum are identified as pions [4, 5, 6]
and there exist also variants of the model that include $\rho$, $\omega$ and
$A_{1}$ vector mesons [7, 8]. Early on in the exploration of the model it was
also noticed that Skyrmions allow for a fermionic interpretation [9] and that
one may formulate gauge theoretical versions of them [10]. Atiyah and Manton
established the remarkable fact that static Skyrmion solutions in
$\mathbb{R}^{3}$ can be approximated well by holonomies of $SU(2)$ Yang-Mills
instantons in $\mathbb{R}^{4}$ [11, 12].

Despite the success of the model on qualitative and conceptual aspects, it is
still way off on a quantitative level when comparing numerical solutions to
experimental measurements [13], as most quantities differ by a fair amount,
such as for instance the magnetic moments for the protons and neutrons which
are too small by about 30%. The Euler-Lagrange equation associated to the
Skyrme model is a complicated nonlinear wave equation for which various
solutions have been obtained numerically for small and large Baryon numbers
[14, 15, 16, 5, 17, 18, 19]. The energies for all these solutions show that
the binding energy, that is the energy required to separate a multi-Skyrmion
solution into single Skyrmions normalized by the Baryon number, is far too
large when compared to what is expected from experiments. Motivated by trying
to address this discrepancy, different variants of the original Skyrme model
have been explored. Especially promising are versions of the model with a
well-defined Bogomolny-Prasad-Sommerfield (BPS) [20, 21] limit as originally
proposed in [22].

These models exhibit a number of very appealing features: Firstly, they allow
for the construction of elegant exact analytical solutions in form of
topological solitons that satisfy the Bogomolny bounds. Secondly they
reproduce the linear relation between the binding energies and the baryon
number for small and large values. Thirdly, and most importantly, they resolve
the issue of the discrepancy of the large binding energies in the original
Skyrme model. In fact, in the BPS versions of the model the binding energies
are zero and one may adopt the view that quantum corrections will only
introduce small variations, hence producing the expected smaller values for
the binding energies. Taking corrections from collective coordinate
quantization of spin and isospin, the electrostatic Coulomb energies, and
small explicit breaking of the isospin symmetry into account lead to a very
good agreement between theory and experimental values for the binding energy
as shown in [23, 24, 25]. For a recent review on these type of BPS Skyrme
models see [26].

Motivated by the success of the BPS versions of the original Skyrme model, we
explore here further possible variants that include complex non-Hermitian
versions of these models. We demonstrate that some of their static solutions
have real topological energies despite being complex and thus these solutions
may also be associated to well-defined physical objects.

In order to overcome the so-called auxiliary field problem and emergence of
fourth-order time derivatives when introducing supersymmetry, complex versions
of the Skyrme model were previously studied [37, 38], by taking the fields to
be valued in a complexification of $SU(N)$, i.e. $SL(N,\mathbb{C})$. Besides
trying to overcome the above mentioned problems, these studies were guided by
the fact that the introduction of supersymmetry into the Skyrme model is
almost inevitably forcing the introduction of complex structures as the
underlying manifolds need to be of Kähler type for this purpose. However,
complex solutions and their reality conditions were not considered previously.

As argued in [27] the reality of the energy for some scalar field solutions
$\mathcal{\phi}_{i}$, $i=1,2,\ldots$, to the BPS equations or the equations of
motion is guaranteed when the following three conditions are met:

(i)

There exists a modified $\mathcal{CPT}$-symmetry that maps the Hamiltonian
functional to its parity time-reversed complex conjugate

$\mathcal{CPT}:\mathcal{H[}\phi(x_{\mu})\mathcal{]\rightarrow
H}^{\dagger}\mathcal{[}\phi(-x_{\mu})\mathcal{]},$ (1)

with $\mathcal{CPT}^{2}=\mathbb{I}$. Here $\mathcal{CPT}$-symmetry is not to
be taken literally as a simultaneous parity, time reversal and charge
conjugation, but be understood simply as an antilinear map of any kind in the
sense described by Wigner in [28].

(ii)

Two solutions $\mathcal{\phi}_{i}$ and $\mathcal{\phi}_{j}$, not necessarily
distinct, are related to each other by the modified $\mathcal{CPT}$-symmetry
as

$\mathcal{CPT}:\mathcal{\phi}_{i}(x_{\mu})\mathcal{\rightarrow\phi}_{j}(-x_{\mu}).$
(2)

(iii)

The energies $E[\mathcal{\phi}]$ of the two solutions $\mathcal{\phi}_{i}$ and
$\mathcal{\phi}_{j}$ are degenerate

$E[\mathcal{\phi}_{i}]=E[\mathcal{\phi}_{j}].$ (3)

Evidently when $\mathcal{\phi}_{i}=\mathcal{\phi}_{j}$ this condition holds
trivially and the energy is automatically guaranteed to be real. When
$\mathcal{\phi}_{i}\neq\mathcal{\phi}_{j}$ we must ensure that the energies
are degenerate to reach the same conclusion. As we shall see below this is
often a consequence of some symmetries in some coupling or integration
constants or by the fact that energies for solutions of the self-dual and
anti-self-dual BPS equations are identical as argued in [27].

For a more detailed reasoning on why these conditions and further examples we
refer the reader to [27] and references therein. We will present examples
below for models with solutions satisfying all three conditions so that their
energies are real, but we shall also explore the broken $\mathcal{CPT}$-regime
by presenting counter examples for solutions with complex energies for which
either or both conditions (ii) and (iii) do not hold.

Our manuscript is organized as follows: In section 2 we recall a general
Lagrangian density that encompasses a whole set of extensions and restrictions
of the standard version of the Skyrme model. In sections 3 we discuss a
complex, albeit pseudo Hermitian, version of the Skyrme model that possess new
types of solutions that satisfy all three conditions (1)-(3) and have
therefore real energies. In section 4 we discuss a version of the BPS Skyrme
model with a potential that leads to solutions that behave asymptotically as
kinks but with finite values a zero and also massless solutions with zero
energy. In section 5 we discuss a version of the model involving a whole ray
of Bender-Boettcher type potentials that possess fractional compacton and
semi-kink solutions with real energies. In section 6 we explore the broken
$\mathcal{CPT}$-regime by discussing a model for which either condition (ii)
and/or condition (iii) are not satisfied. Section 7 contains a discussion of a
Skyrmion submodel with complex semi-kink and soliton-like solutions. Our
conclusions are stated in section 8.

##  2 The Skyrme model - extensions and restrictions

To establish our notations and conventions we briefly recall some key aspects
and definitions of the Skyrme model. Largely following [22, 26], we consider
an extended version of the standard Skyrme model described by variants of a
Lagrangian density of the general form

$\mathcal{L=\tilde{L}}_{0}+\mathcal{L}_{2}+\mathcal{L}_{4}+\mathcal{L}_{6}+\mathcal{L}_{0},$
(4)

where the different terms are defined as

$\mathcal{L}_{2}:=-\frac{f_{\pi}^{2}}{2}\mathop{\mathrm{T}r}\left(L_{\mu}L^{\mu}\right),~{}~{}\mathcal{L}_{4}:=\frac{1}{16e^{2}}\mathop{\mathrm{T}r}\left([L_{\mu},L_{\nu}]^{2}\right),~{}~{}\mathcal{L}_{6}:=-\lambda^{2}N_{0}^{2}B_{\mu}B^{\mu},~{}~{}\mathcal{L}_{0}:=-\mu^{2}V,$
(5)

with Lie algebraic currents in form of right Maurer Cartan forms, topological
current and $SU(2)$-group valued Skyrme fields

$L_{\mu}:=U^{\dagger}\partial_{\mu}U,~{}~{}~{}~{}~{}B^{\mu}:=\frac{1}{N_{0}}\varepsilon^{\mu\nu\rho\tau}\mathop{\mathrm{T}r}\left(L_{\nu}L_{\rho}L_{\tau}\right),~{}~{}~{}~{}~{}U:=e^{i\zeta(\sigma\cdot\vec{n})},$
(6)

respectively. Here $f_{\pi}$ can be interpreted as the pion decay constant and
the dimensionless constant $e$ is referred to as the Skyrme parameter. As is
well known, these parameters can be scaled away, so that we may set them both
to $1$ in what follows. Moreover, we denote by $\sigma$ the standard Pauli
matrices and take the three component unit vector to be of the form
$\vec{n}=(\sin\Theta\cos\Phi,\sin\Theta\sin\Phi,\cos\Theta)$ rather than the
rational map or stereographic projection often used instead in this context,
see e.g. [29]. Our space-time metric $g$ is taken to be
$\mathop{\mathrm{d}iag}g=(1,-1,-1,-1)$. The normalization constant $N_{0}$ is
chosen in such a way that the Baryon number $B=\int B_{0}d^{3}x\in\mathbb{Z}$
becomes an integer as it should be for a two flavour theory to guarantee that
Baryons with an even and odd number of quarks are Bosons and Fermions,
respectively. See for instance [3] for a more detailed reasoning on this
issue. For a standard static compacton solution the normalization constant is
usually taken to be $N_{0}=24\pi^{2}$.

Dropping and decomposing terms or further specifying the potential in the
general Lagrangian $\mathcal{L}$ gives rise to different versions of the
model. The original Skyrme model [1] is comprised of the sum of the sigma
model term $\mathcal{L}_{2}$ and the Skyrme term $\mathcal{L}_{4}$ with
occasionally the potential term $\mathcal{\tilde{L}}_{0}$ added which is of
the same functional form as $\mathcal{L}_{0}$. The BPS version of the model
introduced in [22] consists of the sum of $\mathcal{L}_{6}$, that mimics the
interactions generated by the vector mesons, and the potential term
$\mathcal{L}_{0}$.

Consistent submodels may be obtained by further decomposing terms in
$\mathcal{L}$. With our choice of the parameterization for the $SU(2)$-group
valued element $U$ the various parts of the Lagrangian take on the following
forms: For reasons that will become clear below, we decompose the sigma model
and the Skyrme term as

$\mathcal{L}_{2}=\mathcal{L}_{2}^{(1)}+\mathcal{L}_{2}^{(2)},~{}~{}\ \
\text{and~{}~{}\ \
~{}}\mathcal{L}_{4}=\mathcal{L}_{4}^{(1)}+\mathcal{L}_{4}^{(2)},$ (7)

with

$\displaystyle\mathcal{L}_{2}^{(1)}$ $\displaystyle=$
$\displaystyle\sin^{2}\zeta\left(\Theta_{\mu}\Theta^{\mu}+\Phi_{\mu}\Phi^{\mu}\sin^{2}\Theta\right),~{}~{}\
$ (8) $\displaystyle\mathcal{L}_{2}^{(2)}$ $\displaystyle=$
$\displaystyle\zeta_{\mu}\zeta^{\mu},$ (9)
$\displaystyle\mathcal{L}_{4}^{(1)}$ $\displaystyle=$
$\displaystyle\sin^{2}\zeta\left[\Theta_{\mu}\zeta^{\mu}\Theta_{\nu}\zeta^{\nu}-\Theta_{\mu}\Theta^{\mu}\zeta_{\nu}\zeta^{\nu}+\sin^{2}\Theta\left(\Phi_{\mu}\zeta^{\mu}\Phi_{\nu}\zeta^{\nu}-\Phi_{\mu}\Phi^{\mu}\zeta_{\nu}\zeta^{\nu}\right)\right],$
(10) $\displaystyle\mathcal{L}_{4}^{(2)}$ $\displaystyle=$
$\displaystyle\sin^{4}\zeta\sin^{2}\Theta\left(\Theta_{\mu}\Phi^{\mu}\Theta_{\nu}\Phi^{\nu}-\Theta_{\mu}\Theta^{\mu}\Phi_{\nu}\Phi^{\nu}\right).$
(11)

The extended part computes with

$B^{\mu}=\frac{1}{2N_{0}}\sin^{2}\zeta\sin\Theta\,\mathcal{B}^{\mu},~{}~{}~{}\
\ \
\mathcal{B}^{\mu}:=\varepsilon^{\mu\nu\rho\tau}\zeta_{\nu}\Theta_{\rho}\Phi_{\tau}$
(12)

to

$\mathcal{L}_{6}=-\frac{\lambda^{2}}{4}\sin^{4}\zeta\sin^{2}\Theta\mathcal{B}_{\mu}\mathcal{B}^{\mu}=\frac{\lambda^{2}}{4}\sin^{4}\zeta\sin^{2}\Theta\left[\varphi_{0}^{a}\mathcal{Q}_{a}^{i}\varphi_{0}^{b}\mathcal{Q}_{b}^{i}-\mathcal{B}_{0}\mathcal{B}_{0}\right],$
(13)

where
$\mathcal{Q}_{a}^{i}:=\frac{1}{2}\varepsilon_{abc}\varepsilon^{ijk}\varphi_{j}^{b}\varphi_{k}^{c}$,
$\varphi:=(\zeta,\Theta,\Phi)$ and $a,b,c,i,j,k\in\\{1,2,3\\}$.

Finally, the pion mass term in the standard BPS version of the model (BPSS)
$\mathcal{L}_{0}^{BPSS}=-\mu^{2}V$ is taken to involve the potential
$V=\frac{1}{2}\mathop{\mathrm{T}r}\left(\mathbb{I-}U\right)$ $=1-\cos\zeta$,
but we will allow here other forms of the potential as well. Further
extensions, including for instance a sextic derivative term [30] or
multiplying the terms with field dependent coupling constants [31] have also
been studied.

In what follows we shall investigate different combinations of various complex
extended or deformed versions of different parts of this model related to the
form of $\mathcal{L}$ in (4).

##  3 Pseudo Hermitian variants of Skyrme models

In this section our first guiding principle is to identify a
$\mathcal{CPT}$-symmetry in a Hermitian Hamiltonian and extend the model by
deforming or adding complex terms to convert it into a non-Hermitian
Hamiltonian that still respects this symmetry. Subsequently we try to identify
a pseudo Hermitian counterpart in a similar fashion as what is by now standard
for non-Hermitian quantum mechanical systems [32, 33]. For BPS systems in 1+1
dimensions this approach was recently applied successfully in [27]. We shall
now demonstrate that it can also be applied to 3+1 dimensional theories with
complex topological solutions.

###  3.1 Complex boosted BPS Skyrme models

We start with the standard BPS Skyrme model consisting of
$\mathcal{L}_{6}+\mathcal{L}_{0}^{BPSS}$ by noting that it remains invariant
under the antilinear $\mathcal{CPT}$-transformation: $\zeta\rightarrow-\zeta$,
$\imath\rightarrow-\imath\,$. Thus we may introduce a complex shift in
$\zeta\rightarrow\zeta+\imath\kappa~{}$with $\kappa\in\mathbb{R}$ without
breaking that symmetry. We denote here and in what follows the imaginary unit
as $\imath:=\sqrt{-1}$ to distinguish it from indices $i$. Choosing $\kappa=$
$-\mathop{\mathrm{a}rctanh}\epsilon$ with $\epsilon\in\mathbb{R}$ and using
the identities
$\sqrt{1-\epsilon^{2}}\sin\left(\zeta-\imath\mathop{\mathrm{a}rctanh}\epsilon\right)=\sin\zeta-\imath\epsilon\cos\zeta$,
$\sqrt{1-\epsilon^{2}}\cos\left(\zeta-\imath\mathop{\mathrm{a}rctanh}\epsilon\right)=\cos\zeta+\imath\epsilon\sin\zeta$,
we obtain a $\mathcal{CPT}$-symmetrically extended BPS Skyrme model of the
form

$\mathcal{L}_{\text{b}}=-\frac{\lambda^{2}}{4}\left(\sin\zeta-\imath\epsilon\cos\zeta\right)^{4}\sin^{2}\Theta\mathcal{B}_{\mu}\mathcal{B}^{\mu}-\mu^{2}\left(\sqrt{1-\epsilon^{2}}-\cos\zeta-\imath\epsilon\sin\zeta\right),$
(14)

after re-scaling the coupling constants as
$\lambda\rightarrow\lambda(1-\epsilon^{2})$,
$\mu\rightarrow\mu(1-\epsilon^{2})^{1/4}$. By design, for vanishing $\epsilon$
the model reduces to the standard BPS Skyrme model $\lim_{\epsilon\rightarrow
0}\mathcal{L}_{\text{b}}=\mathcal{L}_{6}+\mathcal{L}_{0}^{BPSS}$ as introduced
and discussed in [22]. We shall now demonstrate that the energies for the
topological solutions to the equations of motion resulting from
$\mathcal{L}_{\text{b}}$ and its corresponding Hermitian counterpart are
identical and real.

####  3.1.1 Topological energies for the real solutions of the Hermitian
counterpart

At first we derive the Hamiltonian corresponding to $\mathcal{L}_{\text{b}}$
in the standard fashion by computing the conjugate canonical momenta

$\Pi^{a}=\frac{\delta\mathcal{L}_{\text{b}}}{\delta\varphi_{0}^{a}}=G_{ac}\varphi_{0}^{c},~{}~{}\
\text{\ with \
}G_{ac}=\frac{\lambda^{2}}{2}\left(\sin\zeta-\imath\epsilon\cos\zeta\right)^{4}\sin^{2}\Theta\mathcal{Q}_{a}^{i}\mathcal{Q}_{c}^{i},$
(15)

so that

$\mathcal{H}_{\text{b}}=\frac{1}{2}\Pi^{a}G_{ac}^{-1}\Pi^{c}-\mathcal{L}_{\text{b}},~{}~{}\
\text{\ with \
}G_{ac}^{-1}=\frac{2\varphi_{i}^{a}\varphi_{i}^{c}}{J^{2}\lambda^{2}\left(\sin\zeta-\imath\epsilon\cos\zeta\right)^{4}\sin^{2}\Theta},$
(16)

where $J:=$
$\frac{1}{2}\varepsilon_{abc}\varepsilon^{ijk}\varphi_{i}^{a}\varphi_{j}^{b}\varphi_{k}^{c}$.

While overall our considerations are mainly classical, we now briefly appeal
to the quantum field theoretic version of the model, by assuming the standard
canonical equal time commutation relation
$\left[\varphi^{a}(r,t),\Pi^{b}(r^{\prime},t)\right]=i\delta^{ab}\delta(r-r^{\prime})$
between the fields $\varphi^{a}(r,t)$ and their conjugate momentum operators
$\Pi^{a}(r,t)$. We then use a slightly modified version of the Dyson operator
as employed in [34, 27]

$\eta=\exp\left[-\mathop{\mathrm{a}rctanh}\epsilon\sum\nolimits_{a}\int
dx\Pi^{a}(r,t)\right],$ (17)

to map the non-Hermitian Hamiltonian functional $\mathcal{H}_{\text{b}}$ to a
Hermitian counterpart $\mathfrak{h}_{\text{b}}$ by means of the adjoint action
of $\eta$

$\mathfrak{h}_{\text{b}}=\eta\mathcal{H}_{\text{b}}\eta^{-1}=\frac{1}{2}\Pi^{a}G_{ac}^{-1}\Pi^{c}+\frac{\tilde{\lambda}^{2}}{4}\sin^{4}\zeta\sin^{2}\Theta\mathcal{B}_{\mu}\mathcal{B}^{\mu}+\tilde{\mu}^{2}(1-\cos\zeta).$
(18)

We notice that $\mathfrak{h}_{\text{b}}$ is in fact the standard BPS Skyrme
model with reversing the previous re-scaling of the coupling constants as
$\lambda\rightarrow\tilde{\lambda}=\lambda(1-\epsilon^{2})$,
$\mu\rightarrow\tilde{\mu}=\mu(1-\epsilon^{2})^{1/4}$.

In this case the static BPS solution that saturates the Bogomolny bound is
known to be computable exactly [22] when using spherical space-time
coordinates $(x,y,z)\rightarrow(r,\theta,\phi)$ with $r\in[0,\infty)$,
$\theta\in[0,\pi)$, $\phi\in[0,2\pi)$ and the identifications $\Theta=\theta$,
$\Phi=n\phi$ with $n\in\mathbb{Z}$ together with the assumption that $\zeta$
is a function of $r$ only. In this case one obtains a well-defined real
compacton solution, see e.g. [35] for what that entails in general,

$\zeta_{r}(r)=\left\\{\begin{array}[]{ll}2\arccos\left(\frac{1}{\sqrt{2}}\left|\frac{\tilde{\mu}}{n\tilde{\lambda}}\right|^{1/3}r\right){}{}&\text{for
}r\in\left[0,r_{c}=\sqrt{2}\left|\frac{n\tilde{\lambda}}{\tilde{\mu}}\right|^{1/3}\right]\\\
0&\text{otherwise}\end{array}\right.,$ (19)

with real energy

$E=8\pi\tilde{\mu}^{2}\int\nolimits_{0}^{r_{c}}r^{2}V\left[\zeta_{r}(r)\right]dr=\frac{64}{15}\sqrt{2}\left|n\right|\tilde{\mu}\tilde{\lambda}\pi(1-\epsilon^{2})^{5/4}.$
(20)

Next we show that there are in fact more solutions in this case and how the
same energy results from a direct computation for the complex solution of the
non-Hermitian system (14).

####  3.1.2 Energies for the complex solutions of the non-Hermitian system

We adopt here and below the approach proposed in [36], which slightly
reformulates the BPS theory and exploits the self-duality and anti-self-
duality between certain fields. For this purpose we first note that the
Hamiltonian density for static solutions may be expressed as

$\mathcal{H}_{\text{b}}=A^{2}+\tilde{A}^{2},$ (21)

with

$A:=\frac{\lambda}{2}\left(\sin\zeta-\imath\epsilon\cos\zeta\right)^{2}\sin\Theta\mathcal{B}_{0},~{}~{}\
\ \ \ \tilde{A}=\mu
V=\mu\left(\sqrt{1-\epsilon^{2}}-\cos\zeta-\imath\epsilon\sin\zeta\right)^{1/2}.$
(22)

The self-duality and anti-self-duality between the fields $A$ and $\tilde{A}$

$A=\pm\tilde{A},$ (23)

is then interpreted as being identical to the BPS equations [20, 21]. The
energy functional for the solutions of (23) therefore acquires the form

$E_{\text{b}}=\int d^{3}x\left(A^{2}+\tilde{A}^{2}\right)=\pm 2\int
d^{3}x~{}A\tilde{A}.$ (24)

Note that since the fields $A$ and $\tilde{A}$ are now complex, the energy of
the static BPS solutions $E_{\text{b}}$ may no longer saturate the Bogomolny
bound.

Explicitly the BPS equations (23) may be written as

$\frac{\lambda}{2}\frac{\left(\sin\zeta-\imath\epsilon\cos\zeta\right)^{2}}{\mu\sqrt{V}}\sin\Theta\varepsilon^{ijk}\partial_{i}\zeta\partial_{j}\Theta\partial_{k}\Phi=\pm
1.$ (25)

Since $\varepsilon_{ijk}\zeta_{i}\Theta_{j}\Phi_{k}$ is simply the Jacobian
for the variable transformation $(x,y,z)\rightarrow(\Theta,\Phi\,,\zeta)$ the
multiplication of (25) by the volume element $d^{3}x$ leads to

$\frac{\lambda}{2}\frac{\left(\sin\zeta-\imath\epsilon\cos\zeta\right)^{2}}{\mu\sqrt{V}}\sin\Theta
d\zeta d\Theta d\Phi=\pm r^{2}\sin\theta drd\theta d\phi,$ (26)

where we used spherical coordinates on the right hand side. With the same
identifications between $(r,\theta,\phi)$ and $(\zeta,\Theta,\Phi)$ as chosen
in the previous section and together with the aforementioned trigonometric
identities the relation (26) converts into

$\frac{n\tilde{\lambda}}{2r^{2}}\sin^{2}\left(\zeta-\imath\mathop{\mathrm{a}rctanh}\epsilon\right)\frac{d\zeta}{dr}=\pm\tilde{\mu}\sqrt{1-\cos\left(\zeta-\imath\mathop{\mathrm{a}rctanh}\epsilon\right)}.$
(27)

These equations is easily integrated out by separating variables.
Corresponding to the different branches we obtain different types of solutions

$\zeta_{i,m}^{\pm}(r)=\tilde{\zeta}_{i,m}^{\pm}(r)+\imath\mathop{\mathrm{a}rctanh}\epsilon=2\arccos\left[\omega^{i}\frac{(n\tilde{\lambda}c\mp\tilde{\mu}r^{3})^{1/3}}{\sqrt{2}n^{1/3}\tilde{\lambda}^{1/3}}\right]+2\pi
m+\imath\mathop{\mathrm{a}rctanh}\epsilon,~{}$ (28)

for $i=0,1,2$, $m\in\mathbb{Z}$ and $\omega=e^{2\pi\imath/3}$ denoting the
third root of unity. We analytically continue here the $\arccos$-function to
the entire complex plane by the well-known formula $\arccos
z=-\imath\ln\left(z\pm\sqrt{z^{2}-1}\right)$. Note that for the Hermitian
case, i.e. $\epsilon=0$, all these solutions also arise, but in that case one
simply discards the complex solutions or the parts of the solutions that
become complex after a certain value of $r$, by requiring solutions to be
real. In order to identify possible compacton solutions in the real part we
need to specify the critical values $r_{0}$ for which the solution vanish,
$\tilde{\zeta}_{i}^{\pm}(r_{0})=0$, and also those values $r_{\pi}$ for which
$\tilde{\zeta}_{i}^{\pm}(r_{\pi})=\pi$. We obtain

$r_{0,i}^{\pm}:=\omega^{i}\left[\frac{\pm
n\tilde{\lambda}\left(c-2^{3/2}\right)}{\tilde{\mu}}\right]^{1/3}~{},~{}\ \
\text{and}~{}~{}\ \ \ r_{\pi,i}^{\pm}:=\omega^{i}\left(\frac{\pm
n\tilde{\lambda}c}{\tilde{\mu}}\right)^{1/3}.$ (29)

These values are irrelevant when complex, whereas when real they may produce
different types of scenarios depending on their ordering and signs of the
constants. In figure 1 we depict some interesting possibilities.

Figure 1: The $\tilde{\zeta}$-part of the solutions of the BPS equation (27)
for different scenarios with $n=\tilde{\lambda}=1$, $\mu=3/2$ and different
choices for $c$. The different relative orderings are: panel (a) $r_{\pi}\leq
0\leq r_{0}$ with $c=1/2$, panel (b) $0\leq r_{\pi}\leq$ $r_{0}$ with
$c=-1/2$, panel (c) $r_{0}\leq$ $0\leq r_{\pi}$ with $c=1/2$ and panel (d)
$0\leq r_{0}\leq r_{\pi}$ with $c=7/2$. Real parts correspond to solid lines
and imaginary parts to dotted ones.

It is clear from figure 1 that we may construct compacton type solutions in
various ways. Obvious choices are

$\tilde{\zeta}_{\text{BPS}}(r):=\left\\{\begin{array}[]{ll}\tilde{\zeta}_{0,0}^{-}{}{}&\text{for
\ \ }0\leq r\leq r_{0}^{-}\\\ 0&\text{for \ \
}r_{0}^{-}<r\end{array}\right.,~{}~{}~{}\ \ \text{\
}\tilde{\zeta}_{\text{St}}(r):=\left\\{\begin{array}[]{ll}\tilde{\zeta}_{1,0}^{-}{}{}&\text{for
\ \ }0\leq r\leq r_{\pi}^{-}\\\ \tilde{\zeta}_{0,0}^{-}&\text{for \ \
}r_{\pi}^{-}\leq r\leq r_{0}^{-}\\\ 0&\text{for \ \
}r_{0}^{-}<r\end{array}\right..$ (30)

Noting that $r_{\pi,i}^{+}(c)=r_{\pi,i}^{-}(-c)$, we may also glue together
solution that are self-dual with those that are anti-self-dual as

$\tilde{\zeta}_{\text{Cusp}}(r):=\left\\{\begin{array}[]{ll}\tilde{\zeta}_{0,0}^{+}{}{}&\text{for
\ \ }0\leq r\leq r_{\pi}^{+}=r_{\pi}^{-}\\\ \tilde{\zeta}_{0,0}^{-}&\text{for
\ \ }r_{\pi}^{-}\leq r\leq r_{0}^{-}\\\ 0&\text{for \ \
}r_{0}^{-}<r\end{array}\right.,~{}~{}~{}\ \
\tilde{\zeta}_{\text{Shell}}(r):=\left\\{\begin{array}[]{ll}0&\text{for \ \
}r<r_{0}^{+}\\\ \tilde{\zeta}_{0,0}^{+}{}{}&\text{for \ \ }r_{0}^{+}\leq r\leq
r_{\pi}^{+}=r_{\pi}^{-}\\\ \tilde{\zeta}_{0,0}^{-}&\text{for \ \
}r_{\pi}^{-}\leq r\leq r_{0}^{-}\\\ 0&\text{for \ \
}r_{0}^{-}<r\end{array}\right..$ (31)

A purely imaginary compacton solution is obtained as
$\tilde{\zeta}_{\text{iBPS}}(r):=\tilde{\zeta}_{0,0}^{+}$ for $r<r_{0}^{+}$
and $0$ otherwise. We have dropped here the second subscript on
$r_{0,i}^{\pm}$ and $r_{\pi,i}^{\pm}$ as the branch that produces a real
values depends on the values of $\tilde{\lambda}$, $\tilde{\mu}$ and $c$. It
appears that in this way one is combining solutions from different equations.
However, noting that the equation of motion resulting from (14) is simply the
square of the BPS equations (27), see e.g. [22] for a derivation when
$\epsilon=0$, we adopt here the view that the latter is more fundamental.
Hence, we are combining solutions for one single equation with different
choices of integration constants in different domains. Whilst the first order
derivative are discontinuous at the ‘gluing points’ $r_{0}^{\pm}$ and
$r_{\pi}^{\pm}$ in the solutions in (30) and (31), we may argue here in a
similar way as in [22] to establish that the solutions are in fact well
defined solutions. The derivative $d\tilde{\zeta}/dr$ always occurs multiplied
with a $\sin^{2}\tilde{\zeta}$ in the BPS equations, so that the left and
right limits of this combination is always finite at the gluing points, but
might differ by a sign. Since this sign is irrelevant in the equations of
motion the solutions are well defined and lead to meaningful values for the
energy density and the Baryon number density. We depict the configurations
(31) - (30) in figure 2.

Figure 2: The BPS solution $\tilde{\zeta}_{\text{BPS}}$ with $n=1$, $c=1/2$,
$\tilde{\mu}=3/2$, $\tilde{\lambda}=1$, the step solution
$\tilde{\zeta}_{\text{St}}$ with $n=1$, $-c=1/2$, $\tilde{\mu}=3/2$,
$\tilde{\lambda}=1$, the cusp solution $\tilde{\zeta}_{\text{Cusp}}$ with
$n=1$, $c=3/2$, $\tilde{\mu}=3/2$, $\tilde{\lambda}=1$, the shell solution
$\tilde{\zeta}_{\text{Shell}}$ with $n=1$, $c=11/2$, $\tilde{\mu}=3/2$,
$\tilde{\lambda}=1$ and the purely imaginary solution
$\tilde{\zeta}_{\text{iBPS}}$ with $n=1$, $c=11/2$, $\tilde{\mu}=3/2$,
$\tilde{\lambda}=1$.

In figure 3 we present the Skyrmion solutions of compacton type (31) - (30) as
slices in form of level curves. We may compare with figure 2. In panel (a) we
have a standard real (fractional) Skyrmion $\tilde{\zeta}_{\text{BPS}}$
starting at a finite value at $r=0$ and then decaying to zero at some critical
value $r_{0}^{-}$. In panel (b) we depict the solution
$\tilde{\zeta}_{\text{St}}$ taking on the form of a step like function with an
inflection point at $r_{\pi}^{-}$. The solution $\tilde{\zeta}_{\text{Cusp}}$
shown in panel (c) has a discontinuous first order derivative at
$r=r_{\pi}^{+}=r_{\pi}^{-}$, which is usually referred to as peakons in the
context of 1+1 dimensional integrable systems. The most interesting
structure$\ \tilde{\zeta}_{\text{Shell}}$ is seen in panel (d), which
corresponds to a real shell with a peakon structure. We may even change this
solution in the region $r<r_{0}^{+},$ by defining it as $\
\tilde{\zeta}_{\text{Core}}(r)=\tilde{\zeta}_{0,0}^{+}$ for $r<r_{0}^{+}$,
hence adding a purely imaginary core to it. It turns out that this is
consistent as the core has also real energies despite the fact that it is
complex.

Figure 3: Different types of solutions to the equations of motion as defined
in (31) - (30) with parameters $n=\tilde{\lambda}=1$, $\tilde{\mu}=3/2$ and
$c=1/2$ in panel (a), $c=-1/2$ in panel (b), $c=3/2$ in panel (c), $c=11/2$ in
panel (d).

Next we demonstrate that all types of solutions depicted in figures 2 and 3
possess real energies. We compute these energies on some domain
$r\in[\tilde{r}_{c},r_{c}]$ by using the general expression (24)

$\displaystyle E$ $\displaystyle=$
$\displaystyle\pm\tilde{\lambda}\tilde{\mu}\int
d^{3}x\left(\sin\zeta-\imath\epsilon\cos\zeta\right)^{2}\sin\Theta\mathcal{B}_{0}\left(\sqrt{1-\epsilon^{2}}-\cos\zeta-\imath\epsilon\sin\zeta\right)^{1/2}$
(32) $\displaystyle=$ $\displaystyle\pm
4\pi\tilde{\lambda}\tilde{\mu}n\int\nolimits_{\tilde{r}_{c}}^{r_{c}}dr\sin^{2}\left(\zeta(r)-\imath\mathop{\mathrm{a}rctanh}\epsilon\right)\sqrt{1-\cos\left(\zeta(r)-\imath\mathop{\mathrm{a}rctanh}\epsilon\right)}\frac{d\zeta}{dr}~{}~{}~{}~{}~{}~{}~{}~{}$
(33) $\displaystyle=$ $\displaystyle
8\pi\tilde{\mu}^{2}\int\nolimits_{\tilde{r}_{c}}^{r_{c}}drr^{2}V\left[\zeta(r)\right].$
(34)

In the last step we used once more equation (27). For the solutions
$\tilde{\zeta}_{\text{BPS}}$, $\tilde{\zeta}_{\text{St}}$ and
$\tilde{\zeta}_{\text{Cusp}}$ we calculate

$E_{\text{BPS/St,Cusp}}=\frac{8}{15}n\tilde{\mu}\tilde{\lambda}\pi\left(8\sqrt{2}\mp
10c\pm 3c^{5/3}\right),$ (35)

for $c\geq 0$ on the domains as indicated in figure 1. The upper signs stand
here for BPS and lower signs for the step and cusp solutions, which have the
same energies. As expected, the expressions (35) reduce to the energy of the
standard real case (20) in the limit $c\rightarrow 0$, since in that case the
fractional BPS Skyrmions become full BPS Skyrmions with $\zeta(r=0)=\pi$. For
the shell solution $\ \tilde{\zeta}_{\text{Shell}}$ and the purely imaginary
core solution $\tilde{\zeta}_{\text{iBPS}}$ we obtain the real energies

$E_{\text{Shell}}=\frac{128}{15}\sqrt{2}n\tilde{\mu}\tilde{\lambda}\pi,~{}\ \
\text{and}~{}\ \ \text{ }E_{\text{iBPS}}=-E_{\text{BPS}},$ (36)

respectively. The reality of the solutions is ensured by verifying that the
respective solutions satisfy all three conditions (1)-(3) for a particular
$\mathcal{CPT}^{\prime}$-symmetry. With condition (1) we identify here the
symmetry to

$\mathcal{CPT}^{\prime}:\zeta(x_{\mu})\rightarrow\zeta^{\ast}(-x_{\mu})+2\imath\mathop{\mathrm{a}rctanh}\epsilon=\zeta(-x_{\mu}).$
(37)

We are considering static solutions in which the angle dependence has already
been eliminated, so that our solutions only depend on $r$. Hence the change in
the arguments of the fields $x_{\mu}\rightarrow-x_{\mu}$ is automatically
satisfied. The $\mathcal{CPT}^{\prime}$-symmetry condition (37) is then easily
verified for our solutions $\zeta_{i,m}^{\pm}(r)$ in (28):
$\zeta_{i,m}^{\pm}(r)\rightarrow\left[\zeta_{i,m}^{\pm}(r)\right]^{\ast}+2\imath\mathop{\mathrm{a}rctanh}\epsilon=\zeta_{i,m}^{\pm}(r)$.
Since the solutions are mapped to themselves, the condition (iii) is
automatically satisfied and energies for these solutions must be real. Notice
that the symmetry $\mathcal{CPT}^{\prime}$ differs from the symmetry
$\mathcal{CPT}$ we used for the construction of the model.

We conclude this section with a brief comment on the values for the Baryon
number, that in general is no longer integer valued. Taking the normalization
constant to be $N_{0}=24\pi^{2}$ we obtain

$B=\int
B_{0}d^{3}x=\frac{n}{\pi}\left[\zeta^{\pm}(r=0)-\frac{1}{2}\sin\left[2\zeta^{\pm}(r=0)\right]\right],$
(38)

which is no longer integer valued.

It is worth pointing out that we may reach similar conclusions as in the
boosted model discussed in this section for a model with complex rotated
fields. With a slight modification of the Dyson map used in [39], having the
effect on the fields is that they transform as $\varphi^{a}\rightarrow
e^{-i\theta_{a}}\varphi^{a}$ and $\Pi^{a}\rightarrow e^{i\theta_{a}}\Pi^{a}$,
we may construct a new complex models. The model obtained in this manner also
possess complex BPS solutions with real energies.

##  4 Skyrme model with semi-kink and massless solutions

While most Skyrmion solutions are of compacton type, there exist also
interesting variants of the model $\mathcal{L}_{0}+\mathcal{L}_{6}$ with
potentials that lead to solutions which are partly of kink type with real
energies. We consider here the potential

$V_{SK}(\zeta)=\sin^{2}\zeta(1+\cos\zeta)^{2}.\ $ (39)

The corresponding BPS equations

$\tan\left(\frac{\zeta}{2}\right)\frac{d\zeta}{dr}=\pm\frac{2\mu}{n\lambda}r^{2},$
(40)

are easily solved to

$\zeta_{s}^{\pm}(r)=2s~{}\arccos\left(e^{\mp\frac{\mu
r^{3}}{3n\lambda}-c}\right),$ (41)

with $s=\pm 1$ and $c$ denoting an integration constant. A similar solutions
to $s=1$ was found in [24]. Evidently we have $\zeta_{s}^{\pm}(r_{0}^{\pm})=0$
for $r_{0,i}^{\pm}=\omega^{i}(\mp 3n\lambda c/\mu)^{1/3}$ and asymptotically
$\zeta_{s}^{\pm}$ acquires a finite value
$\lim_{r\rightarrow\infty}\zeta_{s}^{\pm}(r)=s\pi$ for $\pm\mu/n\lambda>0$. We
depict some sample solutions in figure 4 panel (a). For $r<r_{0}$ we notice
the previously observed standard real or purely imaginary compacton solutions,
but for $r>r_{0}$ the solutions $\zeta_{\pm}^{+}$ exhibit the interesting
feature of being of compacton type at $r=r_{0}$ and of kink type when
$r\rightarrow\infty$.

Figure 4: Panel (a): Purely imaginary and real compacton and semi-kink
solutions (41) resulting from the potential $V_{SK}(\zeta)$ with parameter
choices $n=\lambda=1$, $\mu=2$ and $c=\pm 1$ for $\gamma=\mp 1$. Panel (b):
Complex solutions with zero energy resulting from the potential
$V_{m0}(\zeta)$ with parameter choices $n=\lambda=10$, $\mu=2$ and $c=\pm
25.15$ for $\gamma=\mp 1$. Real parts correspond to solid lines and imaginary
parts to dotted ones.

Crucially, it turns out that the energies of these solutions are all real and
finite. From the general expression (34) we compute

$\displaystyle E_{\text{semi-kink}}\left(\zeta_{s}^{-}\right)$
$\displaystyle=$ $\displaystyle\frac{16n\lambda\mu\pi}{3},$ (42)
$\displaystyle E_{\text{real compacton}}\left(\zeta_{s}^{-}\right)$
$\displaystyle=$ $\displaystyle-\left(4e^{-6c}-3e^{-8c}-1\right)E_{\text{semi-
kink}},$ (43) $\displaystyle E_{\text{purely imaginary
compacton}}\left(\zeta_{s}^{+}\right)$ $\displaystyle=$
$\displaystyle\left(4e^{6c}-3e^{8c}-1\right)E_{\text{semi-kink}}\text{,}$ (44)

for $c>0$ and $n\lambda\mu>0$.

Another interesting variant emerges when considering the potential
$V_{m0}(\zeta)=-V_{SK}(\zeta)$. In this case the solutions become
$\check{\zeta}_{s}^{\pm}(r)=2s~{}\arccos(e^{\mp\imath\frac{\mu
r^{3}}{3n\lambda}-\imath c})$, which vanish for $r_{0,i}^{\pm}=\omega^{i}(\mp
3n\lambda(c+2\pi m)/\mu)^{1/3}$ with $m\in\mathbb{Z}$ and
$\check{\zeta}_{s}^{\pm}(r_{\pi,i}^{\pm})=2\pi s$ for
$r_{\pi,i}^{\pm}=\omega^{i}[\mp 3n\lambda(c+2\pi(m+1/2))/\mu]^{1/3}$. A sample
solution is depicted in figure 4 panel (b). We observe a re-occurring complex
periodic shell solution that becomes squeezed for increasing $r$.
Interestingly the energies for these type of shell solutions is vanishing

$8\pi\mu^{2}\int\nolimits_{r_{0}^{+}}^{r_{0}^{-}}drr^{2}V_{m0}\left[\check{\zeta}_{s}^{\pm}(r)\right]=0.$
(45)

This suggests that the shell solutions may be interpreted as massless
Skyrmions.

We observe from (3) that the energies of the solutions are ensured to be real
by the $\mathcal{CPT}_{\pm}$-symmetries:
$\zeta(r)\rightarrow\pm\zeta^{\ast}(r)$. For the same reasons as in the
previous subsection there is no effect on the arguments of the fields. For the
complex solution $\check{\zeta}_{s}^{\pm}$ this reads $\mathcal{CPT}_{\pm}$:
$\zeta_{s}^{\pm}(c)\rightarrow\left[\zeta_{\pm
s}^{\pm}(c)\right]^{\ast}=\zeta_{\pm s}^{\mp}(-c)$. Thus in this case this
$\mathcal{CPT}_{\pm}$-symmetries map solutions to different solutions.
However, invoking condition (3) and noting that the energies for
$\check{\zeta}_{s}^{\pm}(r)$ are the same for both BPS equations and
independent of $s,c$, they must be real.

##  5 Skyrme model with a Bender-Boettcher type potential

We will now investigate further variants of the model
$\mathcal{L}_{0}+\mathcal{L}_{6}$ by allowing for a wider range of potentials
in $\mathcal{L}_{0}$, including the possibilities of functions of
$\mathop{\mathrm{T}r}\left[U(\zeta)\right]$ and even
$\mathop{\mathrm{T}r}\left[U(\imath\zeta)\right]$. As a first example we
consider the potential

$V_{BB}(\zeta)=\left(\imath\zeta\right)^{\varepsilon}\sin^{4}\zeta,~{}~{}\
\varepsilon\in\mathbb{R}.\ $ (46)

This potential closely resembles the classical prototype potential studied in
$\mathcal{PT}$-symmetric quantum mechanics [40], remaining invariant under the
$\mathcal{CPT}$-transformation: $\zeta\rightarrow-\zeta$,
$\imath\rightarrow-\imath$. Using the same parameterization and reasonings as
in the previous sections, the BPS equations derived in analogy to (27) read

$\frac{n\lambda}{2\mu}\frac{\sin^{2}\left(\zeta\right)}{\sqrt{V_{BB}}}d\zeta=\pm
r^{2}dr~{}~{}~{}\
\Rightarrow~{}~{}~{}\frac{d\zeta}{dr}=\pm\frac{2\mu}{n\lambda}\left(\imath\zeta\right)^{\varepsilon/2}r^{2}~{}.$
(47)

These equations are easily integrated, acquiring the following Gaussian form

$\zeta_{m}^{\pm}(r)=\left[\left|\frac{n\lambda}{\mu(\varepsilon-2)}\right|\frac{1}{(c+r^{3}/3)}\right]^{\frac{2}{\varepsilon-2}}e^{\imath\pi\left(\frac{3s}{2-\varepsilon}-\frac{1}{2}\right)}e^{2\pi\imath\frac{2m}{\varepsilon-2}}\text{,~{}~{}~{}~{}\
}s=\pm 1,c\in\mathbb{R},m\in\mathbb{Z}\text{~{}}.$ (48)

In principle the integration constant $c~{}$could be complex, but we only
obtain real energies for $c\in\mathbb{R}$ so we ignore that possibility in
what follows. We have defined the constant $s:=\mathop{\mathrm{s}ign}[\pm
n\lambda/\mu(\varepsilon-2)]$ where as above $\mathop{\mathrm{s}ign}$ denotes
the signum function. The last factor accounts for all the branches of $\zeta$,
as can either be seen by inserting $1=e^{2\pi\imath m}$ into the square
bracket or by noting that $\zeta\rightarrow\zeta
e^{2\pi\imath\frac{2m}{\varepsilon-2}}$ is a symmetry of equation (47). The
BPS solutions $\zeta_{m}^{\pm}(r)$ exhibit two different types of qualitative
behaviour. When $c\in\mathbb{R}^{-}$, $\varepsilon<2$ we obtain compacton
solutions with finite values at $r=0$ and
$\zeta_{m}^{\pm}[(3\left|c\right|)^{1/3}]=0$. For $c\in\mathbb{R}^{+}$,
$\varepsilon>2$ the solutions are finite at $r=0$ and tend to zero only for
$r\rightarrow\infty$. We illustrate these types of behaviour in figure 5.

Figure 5: BPS solutions $\zeta_{m}^{\pm}(r)$ for a Skyrme model with a Bender-
Boettcher type potential for the parameter choices $\lambda=1$, $\mu=2$,
$n=1$. In panels (a), (b) we have taken $c=-2$ and in panels (c), (d) we have
$c=0.2$.

By the same reasoning as in the previous subsections the energies for these
solutions are computed to

$E_{\text{BB}}^{\pm}=8\pi\mu^{2}\int\nolimits_{0}^{r_{c}}drr^{2}V\left[\zeta_{m}^{\pm}(r)\right],$
(49)

where $r_{c}=(3\left|c\right|)^{1/3}$ for the compacton solutions and
$r_{c}\rightarrow\infty$ for the unbounded ones. As is evident from (46) these
energies can be real when $\zeta$ is either purely imaginary or real. Together
with (48) real energies are found when

$\displaystyle\zeta\in-\imath\mathbb{R}^{+}\text{:\quad}\varepsilon=\frac{4m+4\ell-3s}{2\ell}\text{,\quad}\ell,m\in\mathbb{N}\text{,}$
(50) $\displaystyle\zeta\in\imath\mathbb{R}^{+}\text{: \
\quad}\varepsilon=\frac{2+4m+4\ell-3s}{1+2\ell}\text{,\quad}m,\varepsilon\in\mathbb{N}\text{,}\ell\in\mathbb{N}_{0}\text{,}$
(52) $\displaystyle\zeta\in\mathbb{R}\text{: \
\quad}\varepsilon=\frac{2(1+4m+2\ell-3s)}{1+2\ell}\text{,\quad}\ell,m\in\mathbb{N}_{0}\text{,}\ell,\varepsilon\in
4\mathbb{N}\text{.}$

Examples for these solutions are depicted in figure 5. In panel (b) of that
figure we also displayed a two solution real solutions with $\varepsilon\notin
4\mathbb{N}$. Next we plot the corresponding energies for these cases as
functions of $\varepsilon$ in figure 6.

Figure 6: Real energies $E_{\text{BB}}^{\pm}$ of the compacton (panel a) and
unbounded (panel b) BPS solutions for the cases (50) - (52) with $\lambda=1$,
$\mu=2$, $n=1$ and $c=0.2$ for several values of $\ell,m$.

We observe from figure 6 that the energies are finite and follow distinct
curves for the different cases. Moreover, for the case
$\zeta\in-\imath\mathbb{R}^{+}$ the curve is fairly dense and becomes more
connected when including more values for $\ell$ and $m$, hence $\varepsilon$.
In the other cases this can not be achieved due to the additional restriction
on $\varepsilon$ so that the distribution is more sparse. The transition at
$\varepsilon=2$ is not smooth.

For these models the $\mathcal{CPT}^{\prime}$-symmetry identified from (3)
must act as $\zeta\rightarrow-\zeta^{\ast}$. For our solutions in (48) this
becomes
$\zeta_{m}^{\pm}\rightarrow-(\zeta_{m}^{\pm})^{\ast}=\zeta_{-m}^{\mp}$. Noting
now that $\varepsilon(m,\ell,s)=\varepsilon(-m,-\ell,-s)$ in (50) and
$\varepsilon(m,\ell,s)=\varepsilon(-m,-\ell-1,-s)$ in (52), (52), we simply
have to choose a new $\ell^{\prime}=-\ell$, $\ell^{\prime}=-\ell-1$,
respectively, to obtain the same value for $\varepsilon$. This establishes
that $E[\zeta_{m}^{\pm}]=E[\zeta_{-m}^{\mp}]$ so that condition (iii) in (3)
also holds and the energy must therefore be real. Notice once more that the
$\mathcal{CPT}^{\prime}$-symmetry that ensures the reality of the energies is
different from $\mathcal{CPT}$, that was observed initially for
$V_{BB}(\zeta)$.

##  6 Skyrme model with complex trigonometric potentials

Next we study a model for which the Hamiltonian respects again the
$\mathcal{CPT}$-symmetry: $\zeta\rightarrow-\zeta$,
$\imath\rightarrow-\imath$, but which has solutions transforming under a
$\mathcal{CPT}^{\prime}$-symmetry to satisfy (1) with conditions (2) and/or
(3) violated. Thus we are in the broken $\mathcal{CPT}^{\prime}$-regime. For
this purpose we consider the variant of the model
$\mathcal{L}_{0}+\mathcal{L}_{6}$ involving the trigonometric potential

$V_{T}(\zeta)=\sin^{4}\zeta\cos^{4}(\zeta+i\epsilon),~{}~{}\
\varepsilon\in\mathbb{R}.$ (53)

We notice that unlike as in the pseudo Hermitian model discussed in section 3
only one of the factors in the potential is shifted so that the potential is
not simply boosted and most likely not pseudo Hermitian. The BPS equations
take the form

$\frac{n\lambda}{2\mu}\frac{\sin^{2}\zeta}{\sqrt{V_{T}}}d\zeta=\pm
r^{2}dr\quad\Rightarrow\quad\frac{d\zeta}{dr}=\pm
3\alpha\cos^{2}(\zeta+i\epsilon)r^{2},$ (54)

where we abbreviated $\alpha:=\frac{2\mu}{3n\lambda}$. Integrating this
equation we find the solutions

$\zeta_{\alpha,\gamma}^{\pm}(r)=-i\epsilon\pm\mathrm{arctan}\,\alpha(r^{3}+\gamma)\,.$
(55)

with integration constant $\gamma\in\mathbb{C}$. The symmetry identified from
condition (i) in (1) acts as $\mathcal{CPT}^{\prime}$:
$\zeta_{\alpha,\gamma}^{\pm}\rightarrow-\left(\zeta_{\alpha,\gamma}^{\pm}\right)^{\ast}=$
$\zeta_{\alpha,\gamma}^{\mp}$. Thus the second condition (2) still holds.
However, the energies of the two solutions related in this manner are in
general not degenerate, i.e. $E[\zeta_{\alpha,\gamma}^{+}]\neq
E[\zeta_{\alpha,\gamma}^{-}]$. Depending on the nature of the integration
constant $\gamma$ we find two different types of behaviour and we can still
find discrete values for the two $\mathcal{CPT}^{\prime}$ related solutions
that have degenerate energies.

#### Real integration constants $\gamma\in\mathbb{R}$

Computing the energy $E_{\alpha,\gamma}^{\pm}$ as in the previous sections,
the real and imaginary part acquire the form

$\displaystyle\mathrm{Re}\,E_{\alpha,\gamma}^{\pm}$
$\displaystyle=\frac{1}{32}\left(\frac{\pi}{\alpha}{-}\frac{2\gamma}{\alpha^{2}\gamma^{2}{+}1}\right)+\frac{1}{48}\left(\frac{2\gamma\left(\alpha^{2}\gamma^{2}{+}3\right)}{\left(\alpha^{2}\gamma^{2}{+}1\right)^{2}}{-}\frac{\pi}{\alpha}\right)\cosh
2\epsilon+\frac{\gamma\left(\alpha^{2}\gamma^{2}{-}3\right)\cosh
4\epsilon}{72\left(\alpha^{2}\gamma^{2}{+}1\right)^{3}}$
$\displaystyle+\frac{\gamma(2\cosh
2\epsilon{-}3)}{48\alpha}\mathrm{arctan}\,\alpha\gamma\,,$ (56)
$\displaystyle\mathrm{Im}\,E_{\alpha,\gamma}^{\pm}$
$\displaystyle=\mp\frac{\sinh
2\epsilon\left([3\alpha^{2}\gamma^{2}{-}1]\cosh(2\epsilon){+}3{+}3\alpha^{2}\gamma^{2}\right)}{36\alpha\left(\alpha^{2}\gamma^{2}{+}1\right)^{3}}.$
(57)

This in general the energy is complex and we have
$E_{\alpha,\gamma}^{+}=\left(E_{\alpha,\gamma}^{-}\right)^{\ast}$ and the
model is in the broken $\mathcal{CPT}^{\prime}$-phase. However, we note that
the imaginary part vanishes when parameterizing the integration constant as

$\gamma_{\ell}(\alpha,\epsilon)=\ell\text{sech}\,\epsilon\frac{\sqrt{\cosh
2\epsilon{-}3}}{\sqrt{6}\alpha},\quad\quad\ell=\pm\,.$ (58)

In this case we have also satisfied condition (iii) in (3) with
$E[\zeta_{\alpha,\gamma}^{+}]=E[\zeta_{\alpha,\gamma}^{-}]$ and the
$\mathcal{CPT}^{\prime}$-symmetry is restored. In order to keep the condition
$\gamma\in\mathbb{R}$, we must restrict
$\left|\epsilon\right|\in[\frac{1}{2}\mathrm{arccosh}\,3,\infty)$.

#### Purely imaginary integration constants $\gamma\in i\mathbb{R}$

Taking now $\gamma$ to be purely imaginary the
$\mathcal{CPT}^{\prime}$-symmetry acts as $\mathcal{CPT}^{\prime}$:
$\zeta_{\alpha,\gamma}^{\pm}\rightarrow-\left(\zeta_{\alpha,\gamma}^{\pm}\right)^{\ast}=$
$\zeta_{\alpha,-\gamma}^{\mp}$. The real and imaginary parts of the energies
become now

$\displaystyle\mathrm{Re}\,E_{\alpha,\gamma}^{\pm}$
$\displaystyle=\frac{\pi}{32\alpha}\left(1-\frac{2}{3}\cosh 2\epsilon\right),$
(59) $\displaystyle\mathrm{Im}\,E_{\alpha,\gamma}^{\pm}$
$\displaystyle=\pm\frac{\sinh 2\epsilon\left([1{+}3\alpha^{2}\gamma^{2}]\cosh
2\epsilon{-}3{+}3\alpha^{2}\gamma^{2}\right)}{36\alpha\left(1{-}\alpha^{2}\gamma^{2}\right)^{3}}+\gamma\frac{(2\cosh
2\epsilon{-}3)}{48\alpha}\mathrm{arctan}\,\alpha\gamma$
$\displaystyle-\frac{\gamma}{72}\frac{\left(\alpha^{2}\gamma^{2}{+}3\right)\cosh
4\epsilon+\left(1{-}\alpha^{2}\gamma^{2}\right)\left(3{-}\alpha^{2}\gamma^{2}\right)\cosh
2\epsilon}{(1{-}\alpha^{2}\gamma^{2})^{3}}.$ (60)

Interestingly the real part becomes very simple and does not depend on the
integration constant $\gamma$. We may, however, still find values for $\gamma$
as function of $\alpha$ and $\epsilon$ for which the imaginary part (60)
vanishes, but not in a closed form as in (58). In this case condition (iii) in
(3) becomes $E[\zeta_{\alpha,\gamma}^{+}]=E[\zeta_{\alpha,-\gamma}^{-}]$ and
the $\mathcal{CPT}^{\prime}$-symmetry is also restored.

##  7 A new Skyrme submodel with complex BPS solutions and real energy

By decomposing the sigma model and the Skyrme term, Adam, Sanchez-Guillen and
Wereszczynski noticed in [41] that one may define further consistent and
solvable submodels by combining terms from either decomposition as

$\mathcal{L}_{+}^{(1)}:=\mathcal{L}_{2}^{(1)}+\mathcal{L}_{4}^{(1)},~{}~{}~{}\
~{}\text{and~{}~{}~{}~{}~{}}\mathcal{L}_{+}^{(2)}:=\mathcal{L}_{2}^{(2)}+\mathcal{L}_{4}^{(2)}.$

Choosing the coupling constant in front of $\mathcal{L}_{4}$ to be negative
relative to $\mathcal{L}_{2}$, we consider now a slight modification of the
second submodel defined by the Lagrangian densities

$\mathcal{L}_{-}^{(2)}:=\lambda\left(\mathcal{L}_{2}^{(2)}-\mathcal{L}_{4}^{(2)}\right),~{}~{}~{}\
\ ~{}\ \ \ \lambda\in\mathbb{C}.$ (61)

The corresponding Hamiltonian density for static solutions may then be written
as

$\mathcal{H}_{-}^{(2)}=\lambda(\bigtriangledown\zeta)^{2}-\lambda\sin^{4}\zeta\sin^{2}\Theta\left(\bigtriangledown\Theta\times\bigtriangledown\Phi\right)^{2}=A^{2}+\tilde{A}^{2},$
(62)

where the dual fields are defined as

$A_{i}=\sqrt{\lambda}\zeta_{i},\qquad\text{and\qquad}\tilde{A}_{i}=\imath\sqrt{\lambda}\sin^{2}\zeta\sin\Theta\varepsilon_{ijk}\Theta_{j}\Phi_{k}.$
(63)

Thus, the Hamiltonian density is of the same generic form as for the class of
general BPS models discussed in [36]. Hence, following the same reasoning, the
imposition of a self-duality and anti-self-duality between $A_{i}$ and
$\tilde{A}_{i}$,

$A_{i}=\pm\tilde{A}_{i}$ (64)

selects out the BPS equations [20, 21]. Thus the energy functional
$E_{-}^{(2)}$ for the solutions of (64) therefore acquires the form as in
equation (24).

We now solve the BPS equations (64) and subsequently compute the energies
$E_{-}^{(2)}$ for the solutions obtained. Multiplying (64) by $\Theta_{i}$,
$\Phi_{i}\,$, $\zeta_{i}$ and summing over $i$ we obtain the respective
equations

$\zeta_{i}\Theta_{i}=0,~{}~{}\
\zeta_{i}\Phi_{i}=0,~{}~{}~{}\text{and~{}~{}~{}}\zeta_{i}\zeta_{i}=\pm\imath\sin^{2}\zeta\sin\Theta\varepsilon_{ijk}\zeta_{i}\Theta_{j}\Phi_{k}.$
(65)

The first two constraints are satisfied by a suitable choice of the space-time
dependence of $\Theta$, $\Phi\,$, $\zeta$. Since
$\varepsilon_{ijk}\zeta_{i}\Theta_{j}\Phi_{k}$ is simply the Jacobian for the
variable transformation $(x,y,z)\rightarrow(\Theta,\Phi\,,\zeta)$, the
multiplication of the last equation by the volume element $d^{3}x$ in (65)
leads to

$(\bigtriangledown\zeta)^{2}d^{3}x=\pm\imath\sin^{2}\zeta\sin\Theta d\Theta
d\Phi\,d\zeta.$ (66)

Similarly as above, we choose spherical space-time coordinates
$(x,y,z)\rightarrow(r,\theta,\phi)$ with $r\in[0,\infty)$, $\theta\in[0,\pi)$,
$\phi\in[0,2\pi)$, identify $\Theta=\theta$, $\Phi=n\phi$ with
$n\in\mathbb{Z}$ and assume $\zeta(r)\in\mathbb{C}$. These choices will
automatically solve the first two equations in (65), whereas the last one
reduces to

$\frac{d\zeta}{dr}=\pm\imath\frac{n}{r^{2}}\sin^{2}\zeta.$ (67)

Apart from the $\imath$, this equation coincides with equation (3.6) in [41]
derived for $\mathcal{L}_{+}^{(2)}$ by expressing the unit vector $\vec{n}$ by
means of a stereographic projection. We solve equation (67) to

$\zeta_{\pm}^{(m)}(r)=\imath\mathop{\mathrm{a}rccoth}\left(c\mp\frac{n}{r}\right)+m\pi,~{}~{}~{}\
\ ~{}\ \ \ c\in\mathbb{C}\text{,}m\in\mathbb{Z}.$ (68)

As seen in figure 7 the imaginary parts of these solutions tend to zero for
$r\rightarrow\infty$, whereas the real parts approach asymptotically the
constant value $m\pi+\tilde{c}/2$ when taking $c=\imath\cot(\tilde{c}/2)$,
$\tilde{c}\in\mathbb{R}\backslash\\{2n\pi\\}$ with $n\in\mathbb{Z}$. Moreover
$\lim_{r\rightarrow 0}\zeta_{\pm}^{(m)}(r)=m\pi$.

Figure 7: Complex BPS solutions $\zeta_{-}^{(0)}$ for different values of $n$
and the initial condition $c$ for $\mathcal{L}_{-}^{(2)}$. Real parts as solid
and imginary parts as dotted lines.

At first sight the solution (68) may seem to be unattractive due to its
complex nature. However, first of all it is continuous throughout the entire
range of $r$ and thus overcomes an issue of the real solutions
$\zeta_{r}^{(m)}=\mathop{\mathrm{a}rccot}\left(c-\frac{n}{r}\right)+m\pi$
found for $\mathcal{L}_{+}^{(2)}$ in [41], which are discontinuous at $r=$
$n/c$. Moreover the energies for these solutions are real. We compute

$\displaystyle E_{-}^{(2)}\left(\zeta\right)$ $\displaystyle=$
$\displaystyle\pm 2\imath\lambda\int
d^{3}x\sin^{2}\zeta\sin\Theta\varepsilon_{ijk}\zeta_{i}\Theta_{j}\Phi_{k}$
$\displaystyle=$ $\displaystyle\pm 2\imath n\lambda\int\sin^{2}\zeta\sin\theta
d\theta d\phi\,d\zeta$ $\displaystyle=$ $\displaystyle\pm 8\pi\imath
n\lambda\int_{0}^{\infty}\sin^{2}\zeta\frac{d\zeta}{dr}dr$ $\displaystyle=$
$\displaystyle\pm 8\pi\imath
n\lambda\int_{\zeta(0)}^{\zeta(\infty)}\sin^{2}\zeta d\zeta=\left.\pm
2\pi\imath
n\lambda\left[2\zeta-\sin(2\zeta)\right]\right|_{\zeta(0)}^{\zeta(\infty)}.$
(70)

Thus taking now the complex coupling constant to be of the form
$\lambda=\imath\tilde{\lambda}$, $\tilde{\lambda}\in\mathbb{R}$, we obtain for
solutions $\zeta_{\pm}^{(m)}$ the real energies

$E_{-}^{(2)}\left(\zeta_{\pm}^{(m)}\right)=\pm 2\pi
n\tilde{\lambda}\left[\sin(\tilde{c})-\tilde{c}\right].$ (71)

We identify the $\mathcal{CPT}$-symmetry from (62) as
$\mathcal{CPT}:\zeta\rightarrow\zeta^{\ast}$, which for our solution (68)
becomes
$\zeta_{\pm}^{(m)}(\tilde{c})\rightarrow\left[\zeta_{\pm}^{(m)}(\tilde{c})\right]^{\ast}=$
$\zeta_{\mp}^{(-m)}(-\tilde{c})$. Since
$E_{-}^{(2)}\left[\zeta_{\pm}^{(m)}(\tilde{c})\right]=E_{-}^{(2)}\left[\zeta_{\mp}^{(-m)}(-\tilde{c})\right],$
the energies are guaranteed to be real by the antilinear symmetry
$\mathcal{CPT}$.

##  8 Conclusions

We have studied several variants of the Skyrme Lagrangian density
$\mathcal{L}$ in equation (4). Our main focus has been on finding complex
solutions to the self-dual and anti-self-dual versions of the BPS-equation or
equation of motion. Identifying the $\mathcal{CPT}$-symmetries for these
models from the requirement in (1) allowed us to check the remaining
conditions (ii), (iii) in (2), (3) for the constructed solutions, which when
satisfied ensures the reality of the energy. The broken $\mathcal{CPT}$-regime
was also investigated by providing a sample model in section 7 that is not
pseudo Hermitian possessing generic solutions for which neither of the
conditions (ii) or (iii) holds. However, when taking the integration constant
to be real or purely imaginary and parameterizing it in terms of the coupling
constants of the model the reality of the energy could be restored, which is
also reflected in the restored $\mathcal{CPT}$-symmetry of the model.

We found a number of novel Skyrmion configurations. Besides the standard
compacton or fractional compacton shapes of the BPS-Skyrmions, we found new
types of configurations such as complex kink, anti-kink, semi-kink, massless
solutions and purely imaginary compactons.

Acknowledgments: FC was partially supported by Fondecyt grant 1171475.

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