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# Shapes of magnetic monopoles in effective $SU(2)$ models

Petr Beneš<EMAIL_ADDRESS>Institute of Experimental and Applied
Physics,
Czech Technical University in Prague, Husova 240/5, 110 00 Prague 1, Czech
Republic  Filip Blaschke<EMAIL_ADDRESS>Institute of Experimental
and Applied Physics,
Czech Technical University in Prague, Husova 240/5, 110 00 Prague 1, Czech
Republic  Research Centre for Theoretical Physics and Astrophysics, Institute
of Physics,
Silesian University in Opava, Bezručovo náměstí 1150/13, 746 01 Opava, Czech
Republic

###### Abstract

We present a systematic exploration of a general family of effective $SU(2)$
models with an adjoint scalar. First, we discuss a redundancy in this class of
models and use it to identify seemingly different, yet physically equivalent
models. Next, we construct the Bogomol’nyi–Prasad–Sommerfield (BPS) limit and
derive analytic monopole solutions. In contrast to the ’t Hooft–Polyakov
monopole, included here as a special case, these solutions tend to exhibit
more complex energy density profiles. Typically, we obtain monopoles with a
hollow cavity at their core where virtually no energy is concentrated;
instead, most of the monopole’s energy is stored in a spherical shell around
its core. Moreover, the shell itself can be structured, with several “sub-
shells”. A recipe for the construction of these analytic solutions is
presented.

_For Jiří Hošek on his 80th birthday._

Magnetic monopole; electroweak model; exact solutions; BPS limit

##  I Introduction

Magnetic monopoles are amongst the most important hypothetical particles in
contemporary physics. Although they still await experimental detection, there
are very strong theoretical reasons for their existence. Monopoles play a
major rôle in such disparate subjects as cosmology, particle physics, and
condensed matter physics. Their detection would constitute a fundamental
breakthrough as they would provide a completely new window to high-energy
physics.

In Blaschke:2022pwq  we have introduced a generalization of Lee–Weinberg’s
conceptual scheme Lee:1994sk  and presented a landscape of $U(1)$ gauge
theories with a scalar field $\phi$ and massive, complex vector fields
$W_{\mu}$, namely:

$\displaystyle\mathcal{L}_{U(1)}$ $\displaystyle=$
$\displaystyle-\frac{f_{1}^{2}(\phi/v)}{4g^{2}}F_{\mu\nu}^{2}-\frac{\eta(\phi/v)}{2}d_{\mu\nu}F^{\mu\nu}-\frac{\chi(\phi/v)g^{2}}{4}d_{\mu\nu}^{2}$
(1)
$\displaystyle{}-\frac{f_{2}^{2}(\phi/v)}{2}\big{|}D_{\mu}W_{\nu}-D_{\nu}W_{\mu}\big{|}^{2}+m^{2}(\phi/v)\big{|}W_{\mu}\big{|}^{2}$
$\displaystyle{}+\frac{1}{2}(\partial_{\mu}\phi)^{2}-\frac{\lambda}{2}(\phi^{2}-v^{2})^{2}\,,$

where
$d_{\mu\nu}=\mathrm{i}(W_{\mu}^{\dagger}W_{\nu}-W_{\mu}W_{\nu}^{\dagger})$ is
a dipole-moment tensor. These models represent the most general effective
field-theoretical descriptions of a Dirac monopole possessing a finite mass.
The rôle of form-functions $f_{1,2}(\phi/v)$, $\eta(\phi/v)$, $\chi(\phi/v)$
and $m(\phi/v)$ is to provide control over the properties of the “substrate”
in which is the monopole is placed ($\phi/v$ being the collective coordinate
of the coherent state). In particular:

  * •

$f_{1}$ is as field-dependent magnetic susceptibility,

  * •

$f_{2}$ describes field-dependent non-linear elastic properties of $W_{\mu}$
fields,

  * •

$\eta$ and $\chi$ control the dipole-moment interactions of $W_{\mu}$,

  * •

$m^{2}$ is a field-dependent mass.

Alternatively, we can think about (1) as an effective action of a more
fundamental theory with partially resumed loop diagrams that are embodied in
these form-functions. In short, we believe that Eq. (1) is something that one
could typically encounter down the stream of the renormalization group river
starting from an unknown, more fundamental theory of magnetic
monopoles.111This viewpoint is strengthened by appearance of four-derivative
terms in the canonical formulation of $SU(2)$ landscape in this paper.

The utility of considering such a vast landscape is the following: It is
perhaps intuitively clear that properties of classical solutions would not be
qualitatively different for all choices of form-functions. Instead, they would
fall into a small number of universality classes in which a modification of
various field-dependent couplings would result in only quantitative
difference, but not in a qualitative change in their properties and behavior.
By identifying and studying such universality classes, we can understand
something about monopoles even though we do not know the ultimate, correct
theory, provided, of course, that the “real” monopole in Nature (if ever we
found one) falls into one of these universality classes.

In this paper, we make a first step toward this goal and provide a systematic
classification of spherically-symmetric solutions of a particular sub-
landscape of (1) that is endowed with $SU(2)$ gauge symmetry. The advantages
of this restriction are both practical and utilitarian. Practical, as we can
focus our efforts more narrowly and utilitarian, as we can compare our
solutions with the canonical ’t Hooft–Polyakov monopole tHooft:1974kcl ;
Polyakov:1974ek , which itself is a particular point of the general $SU(2)$
sub-landscape. Furthermore, we can avoid the troubles with residual
singularities that are permanent in $U(1)$ models. However, as we will see,
the discussion of whether the solutions must be always singularity-free will
be one of the corollaries of this paper.

Magnetic monopoles in extended, exotic, and variously generalized models have
been studied before Ramadhan:2015qku ; Bazeia:2018eta ; Casana:2012un ;
Bazeia:2018fhg . One of the characteristics of spherically symmetric single
monopole solutions that seems to be linked to form-functions is the shape of
their energy density profiles. We will pay a special attention to them. As
already reported by Bazeia et al. Bazeia:2018fhg , monopoles can have
unexpected shapes (which they dubbed “small” and “hollow”), in comparison with
the ’t Hooft–Polyakov’s solution. In particular, the appearance of a hollow
cavity at the core of the monopole, where virtually no energy is stored, seems
to be more typical than not. In this paper, we provide explicit analytic
solutions for these hollow monopoles and we discuss the conditions under which
the cavity appears and what controls its extent.

The paper is organized as follows. First, in Sec. II, we introduce a general
class of $SU(2)$ models and discuss its form-invariance. Next, in Sec. III, we
construct the Bogomol’nyi–Prasad–Sommerfield (BPS) limit and formulate the
corresponding BPS equation of motion and energy density. In Sec. IV we
specialize to spherical symmetry, i.e., to single-monopole solutions, and
present a “recipe” for finding analytic solutions. In the subsequent Sec. V,
we show explicit examples. Finally, in Sec. VI, we summarize our results and
give an outlook to future directions of research.

##  II General (non-BPS) model

###  II.1 The Lagrangian

We consider the most general $SU(2)$ gauge theory that is quadratic in
$\boldsymbol{F}^{\mu\nu}$ and with scalar $\boldsymbol{\phi}$ in adjoint (real
triplet) representation with at most two derivatives. Accordingly, the
Lagrangian must be a linear combination of four algebraically independent
terms $(D^{\mu}\boldsymbol{\phi})^{2}$,
$(\boldsymbol{\phi}\\!\cdot\\!D^{\mu}\boldsymbol{\phi})^{2}$,
$(\boldsymbol{F}^{\mu\nu})^{2}$ and
$(\boldsymbol{\phi}\\!\cdot\\!\boldsymbol{F}^{\mu\nu})^{2}$, with the
coefficients being dimensionless and gauge-invariant functions of
$\boldsymbol{\phi}$. We find most convenient to write these terms as

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\frac{v^{2}}{2}\bigg{[}f_{1}^{2}\,\bigg{(}\frac{(D^{\mu}\boldsymbol{\phi})^{2}}{\boldsymbol{\phi}^{2}}-\frac{(\boldsymbol{\phi}\\!\cdot\\!D^{\mu}\boldsymbol{\phi})^{2}}{\boldsymbol{\phi}^{4}}\bigg{)}+f_{3}^{2}\,\frac{(\boldsymbol{\phi}\\!\cdot\\!D^{\mu}\boldsymbol{\phi})^{2}}{\boldsymbol{\phi}^{4}}\bigg{]}$
(2)
$\displaystyle{}-\frac{1}{4g^{2}}\bigg{[}f_{2}^{2}\,\bigg{(}(\boldsymbol{F}^{\mu\nu})^{2}-\frac{(\boldsymbol{\phi}\\!\cdot\\!\boldsymbol{F}^{\mu\nu})^{2}}{\boldsymbol{\phi}^{2}}\bigg{)}+f_{4}^{2}\frac{(\boldsymbol{\phi}\\!\cdot\\!\boldsymbol{F}^{\mu\nu})^{2}}{\boldsymbol{\phi}^{2}}\bigg{]}$
$\displaystyle{}-V(\boldsymbol{\phi}^{2})\,,$

where $f_{1,2,3,4}^{2}$ are the “form-functions” and where

$\displaystyle D^{\mu}\boldsymbol{\phi}$ $\displaystyle=$
$\displaystyle\partial^{\mu}\boldsymbol{\phi}+\boldsymbol{A}^{\mu}\times\boldsymbol{\phi}\,,$
(3) $\displaystyle\boldsymbol{F}^{\mu\nu}$ $\displaystyle=$
$\displaystyle\partial^{\mu}\boldsymbol{A}^{\nu}-\partial^{\nu}\boldsymbol{A}^{\mu}+\boldsymbol{A}^{\mu}\times\boldsymbol{A}^{\nu}\,,$
(4)

with the boldface denoting a three-vector in the gauge space. The potential
$V(\boldsymbol{\phi}^{2})\geq 0$ need not be specified, besides the assumption
that for $\boldsymbol{\phi}^{2}=v^{2}$ (where $v^{2}>0$) it vanishes,
triggering the spontaneous symmetry breakdown $SU(2)\to U(1)$.

In the bulk of this paper, we utilize the decomposition

$\displaystyle\boldsymbol{\phi}$ $\displaystyle=$ $\displaystyle
vH\boldsymbol{n}\,,$ (5)

where the isovector $\boldsymbol{n}$ is normalized as $\boldsymbol{n}^{2}=1$
and $H$ is a dimensionless gauge-invariant scalar function, that serves as an
argument of all form-functions:

$\displaystyle f_{i}^{2}$ $\displaystyle\equiv$ $\displaystyle
f_{i}^{2}(H)\,.\hskip 28.45274pt(i=1,2,3,4)$ (6)

The link of the $SU(2)$ Lagrangian (2) to the $U(1)$ Lagrangian (1) is made by
rewriting the former in the unitary gauge and matching the respective form-
functions (keeping in mind that $f^{2}_{1,2}$ in both Lagrangians can be
different). At this point, there is no need to make this link explicit and we
do not display it here.

Note that the general model (2) includes the usual renormalizable
Georgi–Glashow model as the special case $f_{1}^{2}=f_{3}^{2}=H^{2}$ and
$f_{2}^{2}=f_{4}^{2}=1$.

There are two reasons for writing the Lagrangian (2) with the basis of
projector-like structures. The first one is that the condition for the energy
density to be bounded from below is very simple: Each of the corresponding
functions $f_{i}^{2}$ must be non-negative,

$f_{i}^{2}\geq 0\,,$ (7)

as indicated by the square notation.

The second reason is that the form of the Lagrangian (2) is well suited for
the formalism to be used later in this paper. E.g., using the decomposition
(5) the first two terms in (2) (proportional to $f_{1}^{2}$ and $f_{3}^{2}$)
can be respectively rewritten very simply as

$\displaystyle\frac{(D^{\mu}\boldsymbol{\phi})^{2}}{\boldsymbol{\phi}^{2}}-\frac{(\boldsymbol{\phi}\\!\cdot\\!D^{\mu}\boldsymbol{\phi})^{2}}{\boldsymbol{\phi}^{4}}$
$\displaystyle=$ $\displaystyle(D^{\mu}\boldsymbol{n})^{2}\,,$ (8a)
$\displaystyle\frac{(\boldsymbol{\phi}\\!\cdot\\!D^{\mu}\boldsymbol{\phi})^{2}}{\boldsymbol{\phi}^{4}}$
$\displaystyle=$ $\displaystyle\frac{(\partial^{\mu}H)^{2}}{H^{2}}\,.$ (8b)

After introducing spherically symmetric Ansatz later on, similar
simplification will occur also for the gauge part.

Finally, we will for convenience assume, without loss of generality,222This
normalization can be always enforced by redefining
$\boldsymbol{\phi}\to\boldsymbol{\phi}/f_{1}(1)$, $v\to v/|f_{1}(1)|$ and
$g\to g\,|f_{2}(1)|$, which is equivalent to $f_{1,2}\to f_{1,2}/f_{1,2}(1)$.
the normalization

$\displaystyle f_{1}^{2}(1)\ =\ f_{2}^{2}(1)$ $\displaystyle=$ $\displaystyle
1\,.$ (9)

In this way, the kinetic terms are properly normalized in the vacuum
$\boldsymbol{\phi}^{2}=v^{2}$.

The non-canonical model (2) has necessarily the same spectrum of fluctuations
near the $SU(2)$-breaking vacuum as does the Georgi–Glashow model. Namely,
there is a massless gauge field (photon) corresponding to the unbroken $U(1)$
subgroup, a pair of massive charged vector fields ($W^{\pm}$), and a massive
real scalar field (Higgs). The latter non-vanishing masses are given, due to
the normalization (9), by the standard formulas $M_{W}=vg$ and
$M_{\mathrm{Higgs}}=2v\sqrt{V^{\prime\prime}(v^{2})}$.

###  II.2 Form-invariance and redundancy

Let us consider a transformation (field redefinition) of the scalar triplet
$\boldsymbol{\phi}$ to a new $\tilde{\boldsymbol{\phi}}$ that only changes its
length, namely

$\boldsymbol{\phi}=vH\boldsymbol{n}\hskip 22.76219pt\longrightarrow\hskip
22.76219pt\tilde{\boldsymbol{\phi}}=v\tilde{H}\boldsymbol{n}\,,$ (10a) where
$\tilde{H}$ is related to $H$ as $H=\alpha(\tilde{H})\,.$ (10b)

Here $\alpha(\tilde{H})$ is an arbitrary invertible and differentiable
function that satisfies $\alpha^{2}(\pm 1)=1$ to preserve the vacuum value.
Under this transformation, the original Lagrangian $\mathcal{L}$ of Eq. (2)
transforms into a new Lagrangian $\tilde{\mathcal{L}}$ of the form

$\displaystyle\tilde{\mathcal{L}}$ $\displaystyle=$
$\displaystyle\frac{v^{2}}{2}\bigg{[}\tilde{f}_{1}^{2}\bigg{(}\frac{(D^{\mu}\tilde{\boldsymbol{\phi}})^{2}}{\tilde{\boldsymbol{\phi}}^{2}}-\frac{(\tilde{\boldsymbol{\phi}}\\!\cdot\\!D^{\mu}\tilde{\boldsymbol{\phi}})^{2}}{\tilde{\boldsymbol{\phi}}^{4}}\bigg{)}+\tilde{f}_{3}^{2}\frac{(\tilde{\boldsymbol{\phi}}\\!\cdot\\!D^{\mu}\tilde{\boldsymbol{\phi}})^{2}}{\tilde{\boldsymbol{\phi}}^{4}}\bigg{]}$
(11)
$\displaystyle{}-\frac{1}{4g^{2}}\bigg{[}\tilde{f}_{2}^{2}\bigg{(}(\boldsymbol{F}^{\mu\nu})^{2}-\frac{(\tilde{\boldsymbol{\phi}}\\!\cdot\\!\boldsymbol{F}^{\mu\nu})^{2}}{\tilde{\boldsymbol{\phi}}^{2}}\bigg{)}+\tilde{f}_{4}^{2}\frac{(\tilde{\boldsymbol{\phi}}\\!\cdot\\!\boldsymbol{F}^{\mu\nu})^{2}}{\tilde{\boldsymbol{\phi}}^{2}}\bigg{]}$
$\displaystyle{}-\tilde{V}(\tilde{\boldsymbol{\phi}}^{2})\,,$

where $\tilde{V}(\tilde{\boldsymbol{\phi}}^{2})=V(\boldsymbol{\phi}^{2})$ and
where the new functions $\tilde{f}_{i}^{2}$ are given in terms of the old
functions $f_{i}^{2}$ as

$\displaystyle\tilde{f}_{i}^{2}(\tilde{H})$ $\displaystyle=$
$\displaystyle\begin{cases}f_{i}^{2}\big{(}\alpha(\tilde{H})\big{)}\,,&(i=1,2,4)\\\\[2.0pt]
\displaystyle
f_{i}^{2}(\alpha(\tilde{H})\big{)}\bigg{(}\tilde{H}\frac{\alpha^{\prime}(\tilde{H})}{\alpha(\tilde{H})}\bigg{)}^{2}\,,&(i=3)\end{cases}\hskip
28.45274pt$ (12)

The non-trivial transformation law for $f_{3}^{2}$ is due to the fact that
only this term contains a derivative of $H$, as can be seen from (8).

There is a special worth noting case of the transformation (10):
$H=1/\tilde{H}$. Obviously, when this transformation is applied twice, we
obtain the original theory, hence we can call it a _duality_. Moreover, since
$\tilde{H}\alpha^{\prime}/\alpha=-1$, all four $f_{i}^{2}$ transform uniformly
as $\tilde{f}_{i}^{2}(\tilde{H})=f_{i}^{2}(1/\tilde{H})$.

Notably, the transformed Lagrangian $\tilde{\mathcal{L}}$ is _of the same
form_ as $\mathcal{L}$. We can thus say that $\mathcal{L}$ is _form-invariant_
under the transformation (10) in the sense that (10) does not introduce a new
_kind_ of term that was absent beforehand. Moreover, both $\mathcal{L}$ and
$\tilde{\mathcal{L}}$ describe the same physics. This means, in particular,
that while the corresponding solutions can differ, they must both yield the
same energy density. Consequently, of the four seemingly independent functions
$f_{i}^{2}$ only three are (physically) independent, since the other one can
be fixed by (12).

The form-invariance of the theory implies that a singularity in $H$ is not
necessarily an issue, as it may be cured by a suitable transformation (10). We
will showcase a particular example of this in Sec. V.1.

###  II.3 Dressed vs. canonical fields

As mentioned, we can utilize the transformation (10) to make one of the four
functions $\tilde{f}_{i}^{2}$ to attain a desired form. Although we shall not
adopt any single one, let us point out a few choices of interest.

If, for instance, we want the kinetic term of the scalar triplet
$\boldsymbol{\phi}$ to be canonically normalized, we should demand
$f_{1}^{2}=H^{2}$. Another, slightly less obvious choice, is to demand
$f_{3}^{2}=H^{2}$, which makes the kinetic term for the scalar singlet $vH$
canonical.

To achieve canonical normalization of gauge fields, however, the
transformation (10) is not enough, as it cannot arrange $f_{2}^{2}=1$. In
fact, to make $f_{2}^{2}=1$ we need to perform a _field-redefinition of the
gauge fields_ as333We present a generalization of this transformation, also in
combination with the transformation (10), in Appendix A.

$\displaystyle\boldsymbol{A}^{\mu}$ $\displaystyle=$
$\displaystyle\tilde{\boldsymbol{A}}^{\mu}+\bigg{(}\frac{1}{f_{2}}-1\bigg{)}\frac{\boldsymbol{\phi}\times\tilde{D}^{\mu}\boldsymbol{\phi}}{\boldsymbol{\phi}^{2}}\,,$
(13)

where
$\tilde{D}^{\mu}\boldsymbol{\phi}=\partial^{\mu}\boldsymbol{\phi}+\tilde{\boldsymbol{A}}^{\mu}\times\boldsymbol{\phi}$
is a covariant derivative with respect to the transformed gauge field
$\tilde{\boldsymbol{A}}^{\mu}$. Accordingly, the Lagrangian (2) turns into

$\displaystyle\tilde{\mathcal{L}}$ $\displaystyle=$
$\displaystyle\frac{v^{2}}{2}\bigg{[}\frac{f_{1}^{2}}{f_{2}^{2}}\bigg{(}\frac{(\tilde{D}^{\mu}\boldsymbol{\phi})^{2}}{\boldsymbol{\phi}^{2}}-\frac{(\boldsymbol{\phi}\\!\cdot\\!\tilde{D}^{\mu}\boldsymbol{\phi})^{2}}{\boldsymbol{\phi}^{4}}\bigg{)}+f_{3}^{2}\frac{(\boldsymbol{\phi}\\!\cdot\\!\tilde{D}^{\mu}\boldsymbol{\phi})^{2}}{\boldsymbol{\phi}^{4}}\bigg{]}$
$\displaystyle-\frac{1}{4g^{2}}\bigg{[}(\boldsymbol{F}^{\mu\nu})^{2}+(f_{4}^{2}-1)\frac{(\boldsymbol{\phi}\\!\cdot\\!\boldsymbol{F}^{\mu\nu})^{2}}{\boldsymbol{\phi}^{2}}\bigg{]}-V(\boldsymbol{\phi}^{2})$
$\displaystyle-\frac{1}{2g^{2}}\boldsymbol{d}^{\mu\nu}\\!\cdot\\!\Bigg{\\{}f_{2}\bigg{[}\tilde{\boldsymbol{F}}_{\mu\nu}-\frac{(\boldsymbol{\phi}\\!\cdot\\!\tilde{\boldsymbol{F}}_{\mu\nu})}{\boldsymbol{\phi}^{2}}\boldsymbol{\phi}\bigg{]}+f_{4}^{2}\frac{(\boldsymbol{\phi}\\!\cdot\\!\tilde{\boldsymbol{F}}_{\mu\nu})}{\boldsymbol{\phi}^{2}}\boldsymbol{\phi}\Bigg{\\}}$
$\displaystyle-\frac{1}{4g^{2}}\bigg{[}f_{2}^{2}\bigg{(}(\boldsymbol{d}_{\mu\nu})^{2}-\frac{(\boldsymbol{\phi}\\!\cdot\\!\boldsymbol{d}_{\mu\nu})^{2}}{\boldsymbol{\phi}^{2}}\bigg{)}+f_{4}^{2}\frac{(\boldsymbol{\phi}\\!\cdot\\!\boldsymbol{d}_{\mu\nu})^{2}}{\boldsymbol{\phi}^{2}}\bigg{]}\,,$

where we denoted (here, the prime represents derivative with respect to $H$)

$\displaystyle\boldsymbol{d}_{\mu\nu}$ $\displaystyle\equiv$
$\displaystyle\phantom{+\,\,}H\bigg{(}\frac{1}{f_{2}}\bigg{)}^{\prime}\frac{(\boldsymbol{\phi}\\!\cdot\\!\tilde{D}_{\mu}\boldsymbol{\phi})(\boldsymbol{\phi}\times\tilde{D}_{\nu}\boldsymbol{\phi})-(\mu\leftrightarrow\nu)}{\boldsymbol{\phi}^{4}}$
(15)
$\displaystyle{}-2\bigg{(}\frac{1}{f_{2}}-1\bigg{)}\frac{(\boldsymbol{\phi}\\!\cdot\\!\tilde{D}_{\mu}\boldsymbol{\phi})(\boldsymbol{\phi}\times\tilde{D}_{\nu}\boldsymbol{\phi})-(\mu\leftrightarrow\nu)}{\boldsymbol{\phi}^{4}}$
$\displaystyle{}+2\bigg{(}\frac{1}{f_{2}}-1\bigg{)}\frac{\tilde{D}_{\mu}\boldsymbol{\phi}\times\tilde{D}_{\nu}\boldsymbol{\phi}}{\boldsymbol{\phi}^{2}}$
$\displaystyle{}+\bigg{(}\frac{1}{f_{2}}-1\bigg{)}^{2}\frac{\boldsymbol{\phi}\\!\cdot\\!(\tilde{D}_{\mu}\boldsymbol{\phi}\times\tilde{D}_{\nu}\boldsymbol{\phi})}{\boldsymbol{\phi}^{4}}\boldsymbol{\phi}\,.$

Indeed, the transformation (13) did the trick: The gauge kinetic term in
(II.3) is canonically normalized. Notice, however, that (13) is not form-
invariant, and we pay a price of introducing additional interaction terms
(those with $\boldsymbol{d}_{\mu\nu}$) that are of order
$\sim\mathcal{O}((D\boldsymbol{\phi})^{4})$.

However, the appearance of these new terms in (II.3) has a clear physical
interpretation. In the original Lagrangian (2), the form-function $f_{2}^{2}$
models a non-linear response of the substrate (described by the scalar triplet
$\boldsymbol{\phi}$) to the presence of $SU(2)$ fields. By transforming from
the _dressed_ fields $\boldsymbol{A}_{\mu}$ to new _canonical_ fields
$\tilde{\boldsymbol{A}}_{\mu}$, this effect does not disappear, but rather
manifests itself through the appearance of the dipole-moment tensor
interactions.

These interactions, being higher-order in derivatives and non-renormalizable,
are normally omitted from the classical action of Yang–Mill–Higgs theory.
However, we can view them as effective, partial resummation of quantum
corrections at all orders that are modelled here classically via the function
$f_{2}^{2}$. In other words, the Lagrangian (II.3) represents a semi-classical
description of theory with charged bosons that have non-trivial electric and
magnetic moments.

Since the _canonical_ Lagrangian (II.3) is far more complicated than the
original _dressed_ Lagrangian (2), we will use the “dressed” formalism
throughout the paper unless stated otherwise. However, the exact
correspondence between both Lagrangians shows that the original theory (2) is
nothing but canonical $SU(2)$ theory with explicit dipole-moment interactions.

##  III BPS model

###  III.1 Derivation

Let us consider the energy density of a static configuration
($\partial^{0}=\boldsymbol{A}^{0}=0$) of (2):

$\displaystyle\mathcal{E}$ $\displaystyle=$
$\displaystyle\frac{v^{2}}{2}\bigg{[}f_{1}^{2}\frac{(D^{i}\boldsymbol{\phi})^{2}}{\boldsymbol{\phi}^{2}}+(f_{3}^{2}-f_{1}^{2})\frac{(\boldsymbol{\phi}\\!\cdot\\!D^{i}\boldsymbol{\phi})^{2}}{\boldsymbol{\phi}^{2}}\bigg{]}$
$\displaystyle{}+\frac{1}{2g^{2}}\bigg{[}f_{2}^{2}(\boldsymbol{B}^{i})^{2}+(f_{4}^{2}-f_{2}^{2})\frac{(\boldsymbol{\phi}\\!\cdot\\!\boldsymbol{B}^{i})^{2}}{\boldsymbol{\phi}^{2}}\bigg{]}+V(\boldsymbol{\phi}^{2})\,,$

where we defined
$\boldsymbol{B}^{i}\equiv-\frac{1}{2}\epsilon^{ijk}\boldsymbol{F}^{jk}$. This
can be rewritten identically as

$\displaystyle\mathcal{E}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\bigg{[}\frac{f_{1}}{H}D^{i}\boldsymbol{\phi}-\frac{f_{2}}{g}\boldsymbol{B}^{i}+x\frac{(\boldsymbol{\phi}\\!\cdot\\!\boldsymbol{B}^{i})}{gv^{2}H^{2}}\boldsymbol{\phi}-y\frac{(\boldsymbol{\phi}\\!\cdot\\!D^{i}\boldsymbol{\phi})}{v^{2}H^{3}}\boldsymbol{\phi}\bigg{]}^{2}$
(17)
$\displaystyle{}+\frac{f_{1}f_{2}}{gH}\partial^{i}(\boldsymbol{\phi}\\!\cdot\\!\boldsymbol{B}^{i})-\frac{xf_{1}+yf_{2}-xy}{gH^{2}}(\partial^{i}H)(\boldsymbol{\phi}\\!\cdot\\!\boldsymbol{B}^{i})$
$\displaystyle{}+\frac{f_{4}^{2}-(x-f_{2})^{2}}{2g^{2}v^{2}H^{2}}(\boldsymbol{\phi}\\!\cdot\\!\boldsymbol{B}^{i})^{2}+\frac{f_{3}^{2}-(y-f_{1})^{2}}{2v^{2}H^{4}}(\boldsymbol{\phi}\\!\cdot\\!D^{i}\boldsymbol{\phi})^{2}$
$\displaystyle{}+V(\boldsymbol{\phi}^{2})\,,$

where $x$, $y$ are arbitrary functions of $H$. We used
$D^{i}\boldsymbol{B}^{i}=0$ in order to write
$(D^{i}\boldsymbol{\phi})\\!\cdot\\!\boldsymbol{B}^{i}=\partial^{i}(\boldsymbol{\phi}\\!\cdot\\!\boldsymbol{B}^{i})$.

We can now derive the BPS theory in three steps. First, we discard the
potential, setting $V=0$, but keeping the boundary condition
$\boldsymbol{\phi}^{2}=v^{2}$. Next, we demand

$x\ =\ f_{2}+f_{4}\,,\hskip 28.45274pty\ =\ f_{1}+f_{3}\,,$ (18)

so that the terms on the third line in Eq. (17) vanish. Finally, we require

$\displaystyle\bigg{(}\frac{f_{1}f_{2}}{gH}\bigg{)}^{\prime}$ $\displaystyle=$
$\displaystyle-\frac{xf_{1}+yf_{2}-xy}{gH^{2}}\,,$ (19)

as then the terms on the second line in (17) can be combined into a total
derivative:

$\displaystyle\frac{f_{1}f_{2}}{gH}\partial^{i}(\boldsymbol{\phi}\\!\cdot\\!\boldsymbol{B}^{i})-\frac{xf_{1}+yf_{2}-xy}{gH^{2}}(\partial^{i}H)(\boldsymbol{\phi}\\!\cdot\\!\boldsymbol{B}^{i})$
(20) $\displaystyle\ =\
\partial^{i}\bigg{(}\frac{f_{1}f_{2}}{gH}\boldsymbol{\phi}\\!\cdot\\!\boldsymbol{B}^{i}\bigg{)}\,.$

Substituting the expressions (18) for $x$, $y$ into (19), we obtain the
equation

$\displaystyle f_{3}f_{4}$ $\displaystyle=$ $\displaystyle
H\big{(}f_{1}f_{2}\big{)}^{\prime}\,,$ (21)

i.e., a condition to be satisfied by the functions $f_{i}$ to allow for a BPS
theory in the limit of a vanishing potential.

###  III.2 Formulation

The coveted BPS condition (21) suggests that, for a BPS theory, it may be more
convenient to use another parametrization. Namely, let us consider the set of
functions $F_{i}(H)$ defined as

$\displaystyle F_{1}$ $\displaystyle=f_{1}f_{2}\,,$ $\displaystyle F_{2}$
$\displaystyle=f_{1}/f_{2}\,,$ (22a) $\displaystyle HF_{3}^{\prime}$
$\displaystyle=f_{3}f_{4}\,,$ $\displaystyle HF_{4}^{\prime}$
$\displaystyle=f_{3}/f_{4}\,,$ (22b)

with the inverse relations

$\displaystyle f_{1}^{2}$ $\displaystyle=F_{1}F_{2}\,,$ $\displaystyle
f_{2}^{2}$ $\displaystyle=F_{1}/F_{2}\,,$ (23a) $\displaystyle f_{3}^{2}$
$\displaystyle=H^{2}F_{3}^{\prime}F_{4}^{\prime}\,,$ $\displaystyle f_{4}^{2}$
$\displaystyle=F_{3}^{\prime}/F_{4}^{\prime}\,.$ (23b)

The functions $F_{i}$ are defined in terms of $f_{i}$ uniquely up to a sign,
therefore the conditions $f_{i}^{2}\geq 0$ for the stability of the theory
become

$\mathop{\rm sgn}\nolimits F_{1}=\mathop{\rm sgn}\nolimits F_{2}\,,\hskip
28.45274pt\mathop{\rm sgn}\nolimits F_{3}^{\prime}=\mathop{\rm sgn}\nolimits
F_{4}^{\prime}\,.$ (24)

Furthermore, we impose the normalization

$\big{|}F_{i}(1)\big{|}=1\,,\hskip 28.45274pt(i=1,2,3,4)$ (25)

which, for $F_{1,2}$, is equivalent to $f_{1,2}^{2}(1)=1$, while for $F_{3,4}$
we utilized the freedom to choose the constant of integration in $F_{3,4}=\int
F^{\prime}_{3,4}\,\mathrm{d}H$ any way we like. Finally, let us note that in
terms of $F_{i}$’s the transformation (10) attains a simpler form (compared
with (12)):

$\displaystyle\tilde{F}_{i}(\tilde{H})$ $\displaystyle=$ $\displaystyle
F_{i}\big{(}\alpha(\tilde{H})\big{)}\,.\hskip 22.76219pt(i=1,2,3,4)$ (26)

The Lagrangian (2) (with $V=0$) can be equivalently expressed as

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\frac{v^{2}}{2}\bigg{[}F_{1}F_{2}\frac{(D^{\mu}\boldsymbol{\phi})^{2}}{\boldsymbol{\phi}^{2}}+\big{(}H^{2}F_{3}^{\prime}F_{4}^{\prime}-F_{1}F_{2}\big{)}\frac{(\boldsymbol{\phi}\\!\cdot\\!D^{\mu}\boldsymbol{\phi})^{2}}{\boldsymbol{\phi}^{4}}\bigg{]}$
(27)
$\displaystyle{}-\frac{1}{4g^{2}}\bigg{[}\frac{F_{1}}{F_{2}}(\boldsymbol{F}^{\mu\nu})^{2}+\bigg{(}\frac{F_{3}^{\prime}}{F_{4}^{\prime}}-\frac{F_{1}}{F_{2}}\bigg{)}\frac{(\boldsymbol{\phi}\\!\cdot\\!\boldsymbol{F}^{\mu\nu})^{2}}{\boldsymbol{\phi}^{2}}\bigg{]}\,.$

Notice that now the renormalizable Georgi–Glashow model corresponds to a
special (and rather symmetric) case $F_{1}=F_{2}=F_{3}=F_{4}=H$.

The main benefit of the new parametrization is that the BPS condition (21)
acquires a very simple form

$\displaystyle F_{3}$ $\displaystyle=$ $\displaystyle F_{1}\,,$ (28)

as well as the BPS equation itself:444The BPS equation is obtained simply as a
condition for the square bracket in (17) to vanish (so that the energy density
is just a total derivative). However, as the resulting equation is a bit messy
and complicated, some of its consequences have to be used to obtain the
elegant form (29).

$\displaystyle D^{i}\boldsymbol{\phi}$ $\displaystyle=$
$\displaystyle\frac{H}{g}\bigg{[}\frac{1}{F_{2}}\bigg{(}\boldsymbol{B}^{i}-\frac{\boldsymbol{\phi}\\!\cdot\\!\boldsymbol{B}^{i}}{\boldsymbol{\phi}^{2}}\boldsymbol{\phi}\bigg{)}+\frac{1}{HF_{4}^{\prime}}\frac{\boldsymbol{\phi}\\!\cdot\\!\boldsymbol{B}^{i}}{\boldsymbol{\phi}^{2}}\boldsymbol{\phi}\bigg{]}\,.\hskip
28.45274pt$ (29)

When it is satisfied, the energy density is just a total derivative, as
required:

$\displaystyle\mathcal{E}$ $\displaystyle=$
$\displaystyle\partial^{i}\bigg{(}\frac{F_{1}}{gH}\boldsymbol{\phi}\\!\cdot\\!\boldsymbol{B}^{i}\bigg{)}\,,$
(30)

but it can be also rewritten in two other equivalent ways as

$\displaystyle\mathcal{E}$ $\displaystyle=$ $\displaystyle
v^{2}\bigg{[}F_{1}F_{2}\frac{(D^{i}\boldsymbol{\phi})^{2}}{\boldsymbol{\phi}^{2}}+\big{(}H^{2}F_{1}^{\prime}F_{4}^{\prime}-F_{1}F_{2}\big{)}\frac{(\boldsymbol{\phi}\\!\cdot\\!D^{i}\boldsymbol{\phi})^{2}}{\boldsymbol{\phi}^{4}}\bigg{]}$
(31b) $\displaystyle=$
$\displaystyle\frac{1}{g^{2}}\bigg{[}\frac{F_{1}}{F_{2}}(\boldsymbol{B}^{i})^{2}+\bigg{(}\frac{F_{1}^{\prime}}{F_{4}^{\prime}}-\frac{F_{1}}{F_{2}}\bigg{)}\frac{(\boldsymbol{\phi}\\!\cdot\\!\boldsymbol{B}^{i})^{2}}{\boldsymbol{\phi}^{2}}\bigg{]}\,.$

###  III.3 Magnetic field and mass of a static solution

Asymptotically, in the vacuum, the field-strength tensor corresponding to the
unbroken “electromagnetic” $U(1)$ subgroup is
$F^{\mu\nu}\equiv\frac{1}{v}\boldsymbol{\phi}_{\infty}\cdot\boldsymbol{F}^{\mu\nu}$.
Therefore, we can define and calculate the asymptotic magnetic field as

$\displaystyle B^{i}$ $\displaystyle\equiv$
$\displaystyle-\frac{1}{2}\epsilon^{ijk}F^{jk}$ (32a) $\displaystyle=$
$\displaystyle\frac{1}{v}\boldsymbol{\phi}_{\infty}\cdot\boldsymbol{B}^{i}$
(32b) $\displaystyle=$
$\displaystyle\frac{1}{2}H(\infty)\varepsilon^{ijk}(\partial^{j}\boldsymbol{n}\times\partial^{k}\boldsymbol{n})\\!\cdot\\!\boldsymbol{n}\,,$
(32c)

where $H(\infty)=\pm 1$ is value of $H$ at spatial infinity. The magnetic
charge of a static configuration follows as

$\displaystyle q_{\mathrm{m}}$ $\displaystyle=$
$\displaystyle\lim_{r\to\infty}\int_{S^{2}}\mathrm{d}S_{i}\,B_{i}$ (33a)
$\displaystyle=$
$\displaystyle\frac{1}{2}H(\infty)\underbrace{\lim_{r\to\infty}\int_{S^{2}}\mathrm{d}S_{i}\,\varepsilon^{ijk}(\partial^{j}\boldsymbol{n}\times\partial^{k}\boldsymbol{n})\\!\cdot\\!\boldsymbol{n}}_{\mathclap{\displaystyle
8\pi N}}\hskip 28.45274pt$ (33b) $\displaystyle=$ $\displaystyle H(\infty)4\pi
N\,,$ (33c)

where $N\in\mathbb{Z}$ is the degree of the mapping $\boldsymbol{n}$.

We calculate the mass of a static configuration using the Gauss–Ostrogradsky
theorem (valid for regular $\mathcal{E}$). Using the previous result (33), we
obtain

$\displaystyle M$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{3}}\mathrm{d}^{3}x\,\mathcal{E}$ (34a)
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{3}}\mathrm{d}^{3}x\,\partial^{i}\bigg{(}\frac{F_{1}}{gH}\boldsymbol{\phi}\\!\cdot\\!\boldsymbol{B}^{i}\bigg{)}$
(34b) $\displaystyle=$
$\displaystyle\lim_{r\to\infty}\int_{S^{2}}\mathrm{d}S_{i}\,\frac{F_{1}}{gH}\boldsymbol{\phi}\\!\cdot\\!\boldsymbol{B}^{i}$
(34c) $\displaystyle=$
$\displaystyle\frac{vF_{1}(1)}{2g}\underbrace{\lim_{r\to\infty}\int_{S^{2}}\mathrm{d}S_{i}\,\varepsilon^{ijk}(\partial^{j}\boldsymbol{n}\times\partial^{k}\boldsymbol{n})\\!\cdot\\!\boldsymbol{n}}_{\mathclap{\displaystyle
8\pi N=2H(\infty)q_{\mathrm{m}}}}\hskip 28.45274pt$ (34d) $\displaystyle=$
$\displaystyle F_{1}(1)\frac{4\pi v}{g}N\,,$ (34e)

where $F_{1}(1)=\pm 1$. It follows that for the mass to be positive, the two
signs must be related as

$\displaystyle F_{1}(1)\,\mathop{\rm sgn}\nolimits N$ $\displaystyle=$
$\displaystyle+1\,,$ (35)

so that $M=\frac{4\pi v}{g}|N|\geq 0$.

Moreover, the magnetic charge can be rewritten as
$q_{\mathrm{m}}=\sigma\,4\pi|N|$, where the sign $\sigma\equiv
F_{1}(1)H(\infty)=\pm 1$ informs us whether we have a monopole
($q_{\mathrm{m}}>0$) or antimonopole ($q_{\mathrm{m}}<0$).

##  IV Spherical symmetry

###  IV.1 The Ansatz

Let us now focus on a spherically symmetric $N=1$ solutions for which we adopt
the standard “hedgehog” Ansatz:

$\boldsymbol{\phi}\ =\ vH\frac{\boldsymbol{r}}{r}\,,\hskip
28.45274pt\boldsymbol{A}^{i}\ =\
-\frac{\boldsymbol{r}\times\partial^{i}\boldsymbol{r}}{r^{2}}(1-K)\,,$ (36)

or in components,

$\phi_{a}\ =\ vH\frac{x_{a}}{r}\,,\hskip 28.45274ptA^{i}_{a}\ =\
-\frac{1}{r^{2}}\varepsilon_{abi}x_{b}(1-K)\,.$ (37)

Both form factors $H$ and $K$ are functions of $r=|\boldsymbol{r}|$ and
satisfy the boundary conditions

$H(\infty)\ =\ 1\,,\hskip 28.45274ptK(\infty)\ =\ 0\,,$ (38)

that follow from the requirement of the finite total energy. Notice that we
have deliberately chosen $H(\infty)=+1$ instead of $-1$; this, together with

$\displaystyle F_{1}(1)\ =\ F_{2}(1)$ $\displaystyle=$ $\displaystyle+1\,,$
(39)

that follows from (35) and from $N=1$, means that we are considering, without
loss of generality, only monopoles and not anti-monopoles.

At this point it is customary to impose the conditions $H(0)=0$ and $K(0)=1$,
so that the fields $\boldsymbol{\phi}$, $\boldsymbol{A}^{i}$ are regular at
the origin. (Provided that $K-1$ and $H$ go to zero fast enough, i.e., at
least linearly and quadratically, respectively.) In case of ’t Hooft–Polakov
monopole in the Georgi–Glashow theory (i.e., $F_{i}=H$ for all $i$), these
conditions are critical for having a finite mass. However, in our enlarged
settings (i.e., $F_{i}\neq H$ for some $i$) we should carefully reconsider
their necessity and explore the implications of allowing certain types of
singularities in these fields.

Thus, in this paper, we shall adopt a philosophy that a divergence in _gauge-
dependent_ fields $\boldsymbol{A}^{i}$ and _form-dependent_ field $H$ (see our
comment at the end of Sec. II.2) is not – by itself – necessarily a problem.
What will bother us, however, will be regularity of the energy density in the
origin, which is without any doubts a physical quantity that must not diverge.

###  IV.2 The BPS equations and the energy density

The BPS equation (29) under the Ansatz (36) decouples into a system of two
ordinary differential equations for $K$ and $H$:

$\displaystyle\partial_{\rho}(\log K)$ $\displaystyle=$ $\displaystyle-
F_{2}(H)\,,$ (40a) $\displaystyle\partial_{\rho}H$ $\displaystyle=$
$\displaystyle\frac{1-K^{2}}{\rho^{2}}\frac{1}{F_{4}^{\prime}(H)}\,,$ (40b)

where we introduced a dimensionless radius

$\displaystyle\rho$ $\displaystyle\equiv$ $\displaystyle vgr\,.$ (41)

From the first equation we can immediately read off (due to (38) and (39)) the
asymptotic behavior of $K$:

$\displaystyle K$ $\displaystyle\sim$ $\displaystyle\exp(-\rho)\hskip
14.22636pt\mbox{as}\hskip 14.22636pt\rho\to\infty\,.$ (42)

The energy density reads

$\displaystyle\frac{\mathcal{E}}{v^{4}g^{2}}$ $\displaystyle=$
$\displaystyle\frac{\partial_{\rho}\big{[}F_{1}(1-K^{2})\big{]}}{\rho^{2}}$
(43) $\displaystyle=$ $\displaystyle
2F_{1}F_{2}\frac{K^{2}}{\rho^{2}}+\frac{F_{1}^{\prime}}{F_{4}^{\prime}}\frac{(1-K^{2})^{2}}{\rho^{4}}\,,$
(44)

where, on the second line, we used the BPS equations to eliminate the
derivatives with respect to $\rho$. The first line can be immediately
integrated over 3-volume to obtain the mass of the monopole

$\displaystyle M$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{3}}\\!\mathrm{d}^{3}x\,\mathcal{E}\ =\
\frac{4\pi
v}{g}\int_{0}^{\infty}\\!\mathrm{d}\rho\,\rho^{2}\frac{\mathcal{E}}{v^{4}g^{2}}\
=\ \frac{4\pi v}{g}\,,\hskip 28.45274pt$ (45)

in agreement with the general formula (34) (using $N=1$ and (39)).

###  IV.3 Solving the BPS equations

We can recast the system of the two first-order equations (40) into a single
second-order equation by a straightforward substitution. First, let us for
convenience introduce the shorthand

$\displaystyle\kappa$ $\displaystyle\equiv$
$\displaystyle-\partial_{\rho}(\log K)\,.$ (46)

Also, from now on we assume $F_{2}$ to be invertible.555However, see the
discussion at the end of the paper in Sec. VI. Then we have

$\displaystyle H$ $\displaystyle=$ $\displaystyle F_{2}^{-1}(\kappa)\,.$ (47)

Next, we introduce two functions $F$, $G$ of $\kappa$ as

$\displaystyle F(\kappa)$ $\displaystyle\equiv$ $\displaystyle
F_{1}\big{(}F_{2}^{-1}(\kappa)\big{)}\,,$ (48a) $\displaystyle G(\kappa)$
$\displaystyle\equiv$ $\displaystyle F_{4}\big{(}F_{2}^{-1}(\kappa)\big{)}\,.$
(48b)

Their derivatives can be expressed as

$F^{\prime}(\kappa)\ =\ \frac{F_{1}^{\prime}}{F_{2}^{\prime}}\,,\hskip
28.45274ptG^{\prime}(\kappa)\ =\ \frac{F_{4}^{\prime}}{F_{2}^{\prime}}\,,$
(49)

where the primes on the right-hand sides are derivatives with respect to $H$,
which itself is understood as a function of $\kappa$ via (47).

Expressing $H$ from (40a) as (47) and substituting it into (40b), we obtain a
single second-order equation for the form factor $K$:

$\displaystyle\partial_{\rho}^{2}(\log K)$ $\displaystyle=$
$\displaystyle-\frac{1-K^{2}}{\rho^{2}}\frac{1}{G^{\prime}(\kappa)}\,,$ (50)

where we used also (49).

Using the definitions above we can rewrite the energy density (44) as

$\displaystyle\frac{\mathcal{E}}{v^{4}g^{2}}$ $\displaystyle=$ $\displaystyle
2F(\kappa)\kappa\frac{K^{2}}{\rho^{2}}+\frac{F^{\prime}(\kappa)}{G^{\prime}(\kappa)}\frac{(1-K^{2})^{2}}{\rho^{4}}\,.$
(51)

In this form, the energy density depends only on $K$ (and its derivative via
$\kappa$) and there is no explicit dependence on $H$. Therefore the invariance
of the energy density under the transformation $H\to\tilde{H}$, which we
showed in Sec. II.3, is now manifest. Also note that while the energy density
in the form (44) depended on three functions, now it depends only on two.
Recall that of the original four functions $F_{i}$ that define our theory, one
is removed by the BPS condition and another is removed by the form-invariance
of the Lagrangian. The two functions $F$ and $G$ are exactly the two remaining
_physical_ functions invariant under the transformation (10): $F\equiv
F_{1}\circ F_{2}^{-1}=F_{1}\circ\alpha\circ\alpha^{-1}\circ F_{2}^{-1}$ (and
analogously for $G$).

This suggests to switch from the parameterization of our BPS Lagrangian in
terms of three functions $F_{1}$, $F_{2}$, $F_{4}$ to another parameterization
in terms of $F$ and $G$ and one non-physical function that would merely
represent the freedom under the form-invariance (10). The most straightforward
way to do this is to trade $F_{1}$, $F_{4}$ for $F$, $G$ by defining

$\displaystyle F_{1}(H)$ $\displaystyle=$ $\displaystyle
F\big{(}F_{2}(H)\big{)}\,,$ (52a) $\displaystyle F_{4}(H)$ $\displaystyle=$
$\displaystyle G\big{(}F_{2}(H)\big{)}\,.$ (52b)

Here, we let $F_{2}$ play the rôle of the non-physical function. In this way
we can recast the Lagrangian (27) – yet again – as

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\frac{v^{2}}{2}\bigg{[}FF_{2}\frac{(D^{\mu}\boldsymbol{\phi})^{2}}{\boldsymbol{\phi}^{2}}+\big{[}(HF_{2}^{\prime})^{2}F^{\prime}G^{\prime}-FF_{2}\big{]}\frac{(\boldsymbol{\phi}\\!\cdot\\!D^{\mu}\boldsymbol{\phi})^{2}}{\boldsymbol{\phi}^{4}}\bigg{]}$
(53)
$\displaystyle{}-\frac{1}{4g^{2}}\bigg{[}\frac{F}{F_{2}}(\boldsymbol{F}^{\mu\nu})^{2}+\bigg{(}\frac{F^{\prime}}{G^{\prime}}-\frac{F}{F_{2}}\bigg{)}\frac{(\boldsymbol{\phi}\\!\cdot\\!\boldsymbol{F}^{\mu\nu})^{2}}{\boldsymbol{\phi}^{2}}\bigg{]}\,,$

where the primes at $F$, $G$ are derivatives with respect to
$\kappa=F_{2}(H)$. The virtue of this (final!) form is that it separates the
functions $F$, $G$ that are physical from the function $F_{2}$ that is
arbitrary and merely represents our freedom to perform form-invariant
transformation (10).

Lastly, let us comment on a special case of $F(\kappa)=1$. (It follows that
also $F_{1}$ must be then a constant function, $F_{1}(H)=1$.) There are two
consequences. First, the Lagrangian (53) reduces to

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\frac{v^{2}}{2}F_{2}(D^{\mu}\boldsymbol{n})^{2}-\frac{1}{4g^{2}}\frac{1}{F_{2}}\bigg{[}(\boldsymbol{F}^{\mu\nu})^{2}-\frac{(\boldsymbol{\phi}\\!\cdot\\!\boldsymbol{F}^{\mu\nu})^{2}}{\boldsymbol{\phi}^{2}}\bigg{]}\,,\hskip
22.76219pt$ (54)

so that the scalar singlet $H$, living now only in $F_{2}(H)$, becomes a non-
dynamical field. Second, the energy density cannot be simultaneously regular
and positive. This can be seen by assuming power function behavior of $K$ for
small $\rho$ as $K=a\rho^{n}+\ldots$ The energy density then reads
$\frac{\mathcal{E}}{v^{4}g^{2}}=-\frac{(K^{2})^{\prime}}{\rho^{2}}=-2na^{2}\rho^{2n-3}+\ldots$
Thus, either $n\geq 3/2$ (or $n\geq 1/2$, if we allow for integrable
singularity), so that $\mathcal{E}$ is in the origin regular (but negative),
or $n<0$, implying positive (but singular) $\mathcal{E}$. For these reasons,
we will not consider the case $F(\kappa)=1$.

###  IV.4 Recipe for finding analytic solutions of BPS monopoles

At this point, it is the right time to briefly recapitulate the previous
developments and formulate a recipe for finding a theory that allows an
analytic monopole solution.

  1. 1.

The first step is to find a function $G(\kappa)$ such that the equation (50)
can be solved in a closed form for $K(\rho)$.

Taking into account the definition (48b), we deduce that $G(\kappa)$ must be
defined and continuous on some neighborhood of $\kappa=1$. This is to allow
for the existence of a function $F_{2}(H)$ that would not only satisfy the
condition $F_{2}(1)=1$ (recall (39) and (47)), but to be also invertible for
all $H\in[0,\infty)$.

  2. 2.

Next step is to choose a function $F(\kappa)$. There are two requirements that
come from the desired physical properties of the energy density. (Notice that
in its expression (51) everything except $F$ is already fixed.) First, $F$
must satisfy

$\displaystyle\mathop{\rm sgn}\nolimits F(\kappa)$ $\displaystyle=$
$\displaystyle\mathop{\rm sgn}\nolimits\kappa\,,$ (55a)
$\displaystyle\mathop{\rm sgn}\nolimits F^{\prime}(\kappa)$ $\displaystyle=$
$\displaystyle\mathop{\rm sgn}\nolimits G^{\prime}(\kappa)\,,$ (55b)

for all $\kappa$ (i.e., for all $\rho\geq 0$), so that both terms in the
energy density (51) are non-negative. Second, $F$ must be such that the energy
density is regular.

Having $G^{\prime}$, $F$ and $K$, we have all physical ingredients that allow
us, e.g., to draw a plot of the energy density (51).

  3. 3.

The last step is choosing the function $F_{2}(H)$. This has a twofold meaning.
Primarily, it allows us to find the remaining form factor $H$ through the
equation (47). More generally, however, with $F_{2}$ at hand, we can write
down the full Lagrangian (53). Recall that, taking into account the form-
invariance (10), all invertible functions $F_{2}$ are equally good and lead to
physically equivalent Lagrangians.

###  IV.5 Solvable case

To the best of our knowledge, the only case when an analytic solution to the
equation (50) is known666This is not entirely correct. In fact, it _is_
possible to find many analytic solutions to (50) by turning the logic upside
down and using a kind of “reverse engineering”: Given a prescribed form
$K(\rho)$, we can then look for $G^{\prime}(\kappa)$ that solves (50).
However, as simple as it sounds, it is surprisingly difficult to find in this
way a $G^{\prime}$ that meets the conditions for its domain, as discussed at
the point 1. of Sec. IV.4. In particular, all $G^{\prime}$ we managed to find
using this method happened to be singular at $\kappa=1$. is when

$\displaystyle G(\kappa)$ $\displaystyle=$ $\displaystyle\kappa\,.$ (56)

This, by the way, corresponds to

$\displaystyle F_{4}$ $\displaystyle=$ $\displaystyle F_{2}\,,$ (57)

which is certainly satisfying from an aesthetic point of view as it sort of
symmetrizes the theory – compare it with the BPS condition $F_{3}=F_{1}$.

Upon substituting $G^{\prime}(\kappa)=1$ into the equation (50), we obtain the
general solution $K=c\rho/\sinh[c(\rho-\rho_{0})]$, where the constants of
integration $c$, $\rho_{0}$ are in principle arbitrary. After all, this
solution satisfies the boundary condition $K(\infty)=0$ for all values of $c$,
$\rho_{0}$.

To determine the value of $c$, we have to consider the other form factor, $H$,
that can be calculated from $F_{2}(H)=\kappa$, where
$\kappa=-\partial_{\rho}(\log K)=c\coth[c(\rho-\rho_{0})]-1/\rho$. From
$\lim_{\rho\to\infty}\kappa=c$ (together with the normalization condition
$F_{2}(1)=1$) we learn that only $c=1$ is consistent with the boundary
condition $H(\infty)=1$.

Figure 1: Behavior of the function $\kappa=\coth(\rho-\rho_{0})-1/\rho$ for
various $\rho_{0}$. For $\rho_{0}<0$ it smoothly interpolates between
$-\infty$ and $1$, while for $\rho_{0}=0$ between $0$ and $1$. For
$\rho_{0}>0$ there is a singularity at $\rho=\rho_{0}$.

Next, to determine $\rho_{0}$, we have to resort to the requirement that each
of the two terms in the energy density

$\displaystyle\frac{\mathcal{E}}{v^{4}g^{2}}$ $\displaystyle=$ $\displaystyle
F(\kappa)\kappa\frac{2K^{2}}{\rho^{2}}+F^{\prime}(\kappa)\frac{(1-K^{2})^{2}}{\rho^{4}}$
(58)

has to be separately both regular and non-negative. However, a simple
inspection reveals that these conditions cannot be met for any function
$F(\kappa)$ as long as $\rho_{0}\neq 0$, due to the behavior of $\kappa(\rho)$
in such a case (see Fig. 1). Accordingly, we conclude that only $\rho_{0}=0$
is viable.

We therefore have

$\displaystyle\kappa$ $\displaystyle=$ $\displaystyle\coth\rho-\frac{1}{\rho}$
(59)

and the general solution for the form factors read

$K\ =\ \frac{\rho}{\sinh\rho}\,,\hskip 28.45274ptH\ =\ F_{2}^{-1}(\kappa)\,.$
(60)

Notice that for $F_{2}$ being the identity function this is nothing but the
notorious ’t Hooft–Polyakov solution in the BPS limit. We also remark that
since $K=1-\frac{1}{6}\rho^{2}+\mathcal{O}(\rho^{4})$, the gauge fields are
regular at the origin:

$\displaystyle A^{i}_{a}$ $\displaystyle=$
$\displaystyle-\frac{1}{6}\varepsilon_{abi}x_{b}v^{2}g^{2}\Big{[}1+\mathcal{O}(\rho^{2})\Big{]}\,,$
(61)

as is customary to require.

For the energy density (51) to be non-negative, the function $F$ must satisfy

$F(\kappa)\ \geq\ 0\,,\hskip 28.45274ptF^{\prime}(\kappa)\ \geq\ 0\,,$ (62)

for all $\kappa\in[0,1]$, while the requirement of regularity of $\mathcal{E}$
at $\rho=0$ implies that $F(\kappa\to 0)$ must go to zero sufficiently fast.
For instance, if it behaves near the origin like a power function

$\displaystyle F(\kappa)$ $\displaystyle\sim$ $\displaystyle\kappa^{N}\,,$
(63)

the exponent $N$ must satisfy

$\displaystyle N$ $\displaystyle\geq$ $\displaystyle 1\,.$ (64)

We also see that, except in the special case $N=1$ when
$\mathcal{E}(0)/v^{4}g^{2}=F^{\prime}(0)/3$, the energy density is always
vanishing at $\rho=0$. In other words, most of the energy of the monopole is
concentrated not in its centre, but rather in a spherical shell around it.
This is very similar to the _hollow monopoles_ discussed in Bazeia:2018fhg .
However, while the authors of Bazeia:2018fhg  found and studied the hollow
monopoles only numerically, we are going to present analytic solutions for
them.

##  V Examples

By choosing $G(\kappa)=\kappa$ we have fulfilled the first point in our recipe
of Sec. IV.4, so let us move to the next two points and provide some
particular examples of functions $F(\kappa)$ (to be able to draw energy
density) and $F_{2}(H)$ (to be able to write down Lagrangian density.)

###  V.1 Power function

As the simplest example let us take

$\displaystyle F$ $\displaystyle=$ $\displaystyle\kappa^{N}\,.$ (65)

In order that the energy density (with $G(\kappa)=\kappa$ and,
correspondingly, $K$ and $\kappa$ given by (59) and (60))

$\displaystyle\frac{\mathcal{E}}{v^{4}g^{2}}$ $\displaystyle=$
$\displaystyle\kappa^{N}\bigg{[}2\kappa\frac{K^{2}}{\rho^{2}}+N\frac{1}{\kappa}\frac{(1-K^{2})^{2}}{\rho^{4}}\bigg{]}$
(66a) $\displaystyle\xrightarrow[\rho\to 0]{}$
$\displaystyle\rho^{N-1}\bigg{[}\frac{N+2}{3^{N+1}}+\mathcal{O}(\rho^{2})\bigg{]}$
(66b)

be regular in the origin, we must demand $N\geq 1$ in accordance with (64). In
Fig. 2 we plot $\mathcal{E}$ for various values of $N$. As advertised, for
$N>1$ most of the energy is concentrated away from the centre and the
monopoles are hollow.

Figure 2: Energy density for a single monopole solution of the power-function
theory (67) for various $N=n/m$.

In order to write down the corresponding Lagrangian, we choose, for
simplicity, the function $F_{2}(H)$ to be a power function, too. Thus, we
arrive at

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\frac{v^{2}}{2}H^{n+m}\bigg{[}\frac{(D^{\mu}\boldsymbol{\phi})^{2}}{\boldsymbol{\phi}^{2}}+(nm-1)\frac{(\boldsymbol{\phi}\\!\cdot\\!D^{\mu}\boldsymbol{\phi})^{2}}{\boldsymbol{\phi}^{4}}\bigg{]}$
(67)
$\displaystyle{}-\frac{1}{4g^{2}}H^{n-m}\bigg{[}(\boldsymbol{F}^{\mu\nu})^{2}+\bigg{(}\frac{n}{m}-1\bigg{)}\frac{(\boldsymbol{\phi}\\!\cdot\\!\boldsymbol{F}^{\mu\nu})^{2}}{\boldsymbol{\phi}^{2}}\bigg{]}\,,\hskip
22.76219pt$

which in the language of $F_{i}$’s corresponds to

$\displaystyle F_{1}=F_{3}$ $\displaystyle=$ $\displaystyle H^{n}\,,$ (68a)
$\displaystyle F_{2}=F_{4}$ $\displaystyle=$ $\displaystyle H^{m}\,,$ (68b)

provided that

$\displaystyle N$ $\displaystyle=$ $\displaystyle\frac{n}{m}\,.$ (69)

The solution follows immediately as

$K\ =\ \frac{\rho}{\sinh\rho}\,,\hskip 28.45274ptH\ =\
\bigg{(}\coth\rho-\frac{1}{\rho}\bigg{)}^{\frac{1}{m}}\,.$ (70)

Interestingly, while the Lagrangian (67) is form-invariant under general
transformations $H\to\tilde{H}$ of Eq. (10), there is also a special subclass
of these transformations that affects only the parameters $n$ and $m$. Namely,
upon transforming

$\displaystyle H$ $\displaystyle\to$ $\displaystyle H^{\sigma}\,,$ (71)

where $\sigma\neq 0$ is arbitrary, we obtain a model of the same form as (67),
up to the rescaling of parameters

$n\to\sigma n\,,\hskip 28.45274ptm\to\sigma m\,.$ (72)

Note that the physical (measurable) parameter $N=n/m$ is invariant under (72).

Obviously, for $n=m=1$ this is just the well known ’t Hooft–Polyakov monopole
in the renormalizable Georgi–Glashow model (in the BPS limit) tHooft:1974kcl ;
Polyakov:1974ek . However, any model with $n/m=1$ is equivalent to the
Georgi–Glashow model, even though the Lagrangian and the solution $H$ are
different.

The form factor $H$, Eq. (70), behaves in the origin like
$H=(\rho/3+\mathcal{O}(\rho^{3}))^{1/m}$ and, correspondingly,
$\boldsymbol{\phi}=vH\boldsymbol{x}/r=v^{2}g\boldsymbol{x}\rho^{1/m-1}(1/3+\mathcal{O}(\rho^{2}))^{1/m}$.
If $1/m\geq 1$, the field $\boldsymbol{\phi}$ is regular, as is customary to
require. If, on the other hand, $1/m<1$, the field $\boldsymbol{\phi}$ is
singular at the origin. However, this singularity is completely unphysical and
harmless, as it can be transformed away using (71) with, e.g., $\sigma=1/m$.

###  V.2 Power-exponential function

As a slightly more complicated example, let us consider

$\displaystyle F$ $\displaystyle=$
$\displaystyle\kappa^{N}\exp\Big{[}a(\kappa^{M}-1)^{k}\Big{]}\,.$ (73)

Here we assume $a\neq 0$ and $M\neq 0$, as otherwise we would obtain the
previous case.

We take again $G=\kappa$. From the requirements that $F(\kappa=1)=1$ and
$F^{\prime}=F[N+akM\kappa^{M}(\kappa^{M}-1)^{k-1}]/\kappa\geq 0$ we find that
$N\geq 0$ and $aM>0$, while $k$ must be a positive, odd integer. Now there are
two options with rather different properties:

  * •

$a>0$ and $M>0$: In this case the energy density behaves for small $\rho$ like
a power function $\mathcal{E}\sim\rho^{N-1}$, so that it must be $N\geq 1$.

  * •

$a<0$ and $M<0$: In this case we have $\mathcal{E}\sim\exp(-1/\rho^{k|M|})$
(times a power function) as $\rho\to 0$ and $N\geq 0$.

Figure 3: Energy density for a single monopole solution of the power-
exponential-function theory (74) with $N=M=2$, $a=100$ and various $k$.
Figure 4: Energy density for a single monopole solution of the power-
exponential-function theory (74) with $N=2$, $M=-1$, $k=101$ and various
$a\leq 0$.

As an example of the Lagrangian that corresponds to this $F$, we can take

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\mathrm{e}^{a(H^{n\ell}-1)^{k}}H^{n-m}$ (74)
$\displaystyle\times\Bigg{\\{}\frac{v^{2}}{2}H^{2m}\bigg{[}\frac{(D^{\mu}\boldsymbol{\phi})^{2}}{\boldsymbol{\phi}^{2}}+\big{(}mns-1\big{)}\frac{(\boldsymbol{\phi}\\!\cdot\\!D^{\mu}\boldsymbol{\phi})^{2}}{\boldsymbol{\phi}^{4}}\bigg{]}$
$\displaystyle{}\hskip
14.22636pt-\frac{1}{4g^{2}}\bigg{[}(\boldsymbol{F}^{\mu\nu})^{2}+\bigg{(}\frac{n}{m}s-1\bigg{)}\frac{(\boldsymbol{\phi}\\!\cdot\\!\boldsymbol{F}^{\mu\nu})^{2}}{\boldsymbol{\phi}^{2}}\bigg{]}\Bigg{\\}}\,,\hskip
22.76219pt$

where we denoted for brevity

$\displaystyle s$ $\displaystyle\equiv$ $\displaystyle 1+a\ell
k\big{(}H^{n\ell}-1\big{)}^{k-1}H^{n\ell}\,.$ (75)

In the language of $F_{i}$’s we would specify the theory by

$\displaystyle F_{1}=F_{3}$ $\displaystyle=$ $\displaystyle
H^{n}\exp\Big{[}a(H^{n\ell}-1)^{k}\Big{]}\,,$ (76a) $\displaystyle
F_{2}=F_{4}$ $\displaystyle=$ $\displaystyle H^{m}\,,$ (76b)

which is related to the $F$ by the identification

$N\ =\ \frac{n}{m}\,,\hskip 28.45274ptM\ =\ \ell\frac{n}{m}\,.$ (77)

The form factors $K$ and $H$ are given by the very same expressions (70) as in
the previous power-function model.

The reason to consider this complicated and rather artificial model is that it
showcases more interesting energy density profiles with richer structure
(admittedly, for “unnaturally” large values of parameters). While we again
obtain the hollow monopoles, this time they have new features. As depicted in
Fig. 3, for some range of parameters there can be not only one minimum of the
energy density, but two. In Fig. 4, we see that the peak of the energy density
can be much sharper. Moreover, since in the latter case the energy density
falls off exponentially for small $\rho$ (due to $a<0$), there is a well-
defined finite region in the centre of the monopole with virtually vanishing
energy density – accordingly we can dub these solutions the _truly hollow_
monopoles.

##  VI Summary and outlook

In this paper we have investigated spherically symmetric solutions of a family
of general $SU(2)$ gauge theories (2) with adjoint scalar in the BPS limit.

First, we discussed the redundancy and physical rôle of the four functions
$f_{i}^{2}$ that define our model. There is a form-invariance of our theory
(2) that exploits a general field redefinition (10) of the gauge-invariant
part of the adjoint scalar $\boldsymbol{\phi}$. This redefinition leaves the
structure of the model intact and allows to fix one of the four functions
(e.g., to have the kinetic term of $\boldsymbol{\phi}$ canonically
normalized). By itself, however, the transformation (10) is not enough to
eliminate non-canonical structure of the (dressed) gauge fields. To achieve
that, we have introduced a form non-invariant transformation (13) that defines
canonically normalized gauge fields and through which the starting Lagrangian
(2) is recast into (II.3). In this way, the presence of field-dependent
$SU(2)$ magnetic susceptibility (functions $f_{2}^{2}$ and $f_{4}^{2}$) in Eq.
(2) are found in Eq. (II.3) to control the strength of dipole-moment
interactions. This clearly illustrates the physical rôle of $f_{2}^{2}$ and
$f_{4}^{2}$ and opens up an avenue for further investigations of their impact
on physics that we plan to do in the future. (In App. A, we discuss a
generalization of both transformations and showcase its group-like properties
and the impact on the hedgehog Ansatz.)

Second, we have formulated a BPS version of the theory with the help of
reparametrizing the Lagrangian (2) in terms of new, more “BPS-friendly”
functions $F_{i}$ (rather than $f_{i}^{2}$). This has several advantages: i)
the key BPS condition is stated very simply as $F_{3}=F_{1}$, ii) the form-
invariance (10) becomes a simple function composition, and iii) the solvable
cases that are studied in this paper – all of which can be regarded as certain
generalizations of the ’t Hooft–Polyakov monopole – belong to a special subset
that is defined simply as $F_{4}=F_{2}$.

Third, in the BPS limit there are only two physically relevant form-functions
that we have labelled $F$ and $G$ (see Eq. (52)). We have then constructed an
especially convenient form of the Lagrangian, (53), that is written solely in
terms of form-functions $F$, $G$ and $F_{2}$, with the freedom of performing
(10) being entirely contained in $F_{2}$. Indeed, this function does not
appear in a general formula (51) for energy density of spherically symmetric
solutions, which is thus manifestly form-invariant.

Having clarified the rôle of various functions, the rest of the paper presents
in Sec. V some concrete examples of analytic BPS solutions that illustrate an
application of a general “recipe” described in Sec. IV.4. These examples are
not meant to be exhaustive, rather they expose a key point of this study: How
the shape of a monopole (i.e., distribution of its energy) depends on the
form-function $F$. (The other form-function, $G$, is being fixed for
simplicity, although we suspect its rôle to be qualitatively the same.) While
the general dependence might be glimpsed from the formula (51), in particular,
we have seen that the position and number of extrema can be controlled by the
choice of $F$ (see Fig. 3).

A universal feature seems to be tendency of energy to concentrate in a shell
rather than at the monopole’s centre. In other words, a generic monopole of
our family of models tends to be _hollow_. The only exception in the examples
presented here are the cases $n=m$ of the simplest power-function model (67)
that are actually nothing but the ’t Hooft–Polyakov monopole, up to a field
redefinition (10). These cases also stand out by the presence of canonically
normalized gauge kinetic term in the Lagrangian density.

On the other hand, in all our examples of hollow monopoles the gauge kinetic
terms are always modified by some power of the scalar singlet $H$, manifesting
a non-trivial $SU(2)$ magnetic dipole-moment of the adjoint triplet.
Intuitively, the presence of hollow monopoles therefore seems to be connected
with vanishing (or “freezing”) of the gauge kinetic term at the monopole’s
centre, namely that the dipole-moment interactions that _screen_ the bare
monopole charge become effectively infinite. At this point, we present it as
an observation, however, we are planing to examine this issue more thoroughly
by investigating the properties of homogeneous “phases” of the theory (2) in a
future work.

Lastly, in this paper, we have only expounded the case of _invertible_ $F_{2}$
that allowed us to condense the system of two first-order BPS equations (40)
into a single second-order equation (50). However, there is an entire branch
of analytic solutions that correspond to a _non-invertible_ choices of
$F_{2}$. For instance, taking $F_{2}=1$ yields a particularly simple form of
$K$:

$\displaystyle K$ $\displaystyle=$ $\displaystyle\xi\mathrm{e}^{-\rho}\,,$
(78)

where $\xi$ is a constant of integration. (The other form factor, $H$, depends
on $F_{4}$ and is typically much more complicated.) First of all, near origin
$K\sim\xi(1-\rho+\mathcal{O}(\rho^{2}))$ and, correspondingly,
$A_{a}^{i}\sim\varepsilon_{abi}x_{b}(\xi-1+\mathcal{O}(r))/r^{2}$ as $r\to 0$.
Notice that this singularity only becomes milder if $\xi=1$, but does not
disappear completely. As discussed at the beginning of this paper, by itself a
singular behavior might not be problematic, as it can be transformed away by a
general field redefinition of the type discussed in Appendix A. Interestingly,
however, the physical requirements (like regularity of the energy density) do
not constrain $\xi$ to have a single value, but rather $\xi\in[-1,1]$. In
other words, $\xi$ is a completely new moduli of the BPS solution and it is
physical in the sense that it controls the shape of the energy density. In
particular, it is related to the width of the hollow cavity.

We plan to investigate this class of solutions in a separate paper. There we
hope to elucidate the physical origin of this moduli and whether the
corresponding solutions are physically viable. In particular, we have to
analyze the stability of these solutions, as presence of $\xi$ might lead to
dynamical instabilities. Further, we need to study them in non-BPS cases and
identify what are the exact conditions that lead to their presence.

###### Acknowledgements.

The authors would like to express the gratitude for the institutional support
of the Institute of Experimental and Applied Physics, Czech Technical
University in Prague (P. B. and F. B.), and of the Research Centre for
Theoretical Physics and Astrophysics, Institute of Physics, Silesian
University in Opava (F. B.). P. B. is indebted to Aštar Šeran and FSM for
invaluable discussions.

##  Appendix A General canonical transformation for $SU(2)$ adjoint fields

Let us consider the transformation
$\\{\boldsymbol{A}_{\mu},\boldsymbol{\phi}\\}\to\\{\tilde{\boldsymbol{A}}_{\mu},\tilde{\boldsymbol{\phi}}\\}$
generated by the functions $\alpha$, $h$, $k$, $\ell$ as

$\displaystyle\boldsymbol{\phi}$ $\displaystyle=$
$\displaystyle\frac{\alpha(\tilde{H})}{\tilde{H}}\tilde{\boldsymbol{\phi}}\,,$
(79a) $\displaystyle\boldsymbol{A}_{\mu}$ $\displaystyle=$
$\displaystyle\tilde{\boldsymbol{A}}_{\mu}+h(\tilde{H})\smash{\overbrace{\bigg{[}\frac{\tilde{D}_{\mu}\tilde{\boldsymbol{\phi}}}{v\tilde{H}}-\frac{\tilde{\boldsymbol{\phi}}\\!\cdot\\!\tilde{D}_{\mu}\tilde{\boldsymbol{\phi}}}{(v\tilde{H})^{3}}\tilde{\boldsymbol{\phi}}\bigg{]}}^{\mathclap{\displaystyle\tilde{D}_{\mu}\boldsymbol{n}}}}+k(\tilde{H})\frac{\tilde{\boldsymbol{\phi}}\\!\cdot\\!\tilde{D}_{\mu}\tilde{\boldsymbol{\phi}}}{(v\tilde{H})^{3}}\tilde{\boldsymbol{\phi}}$
(79b)
$\displaystyle{}+\big{[}\ell(\tilde{H})-1\big{]}\frac{\tilde{\boldsymbol{\phi}}\times\tilde{D}_{\mu}\tilde{\boldsymbol{\phi}}}{(v\tilde{H})^{2}}\,,$

where $\tilde{\boldsymbol{\phi}}=v\tilde{H}\boldsymbol{n}$ and
$\tilde{D}_{\mu}\tilde{\boldsymbol{\phi}}\equiv\partial_{\mu}\tilde{\boldsymbol{\phi}}+\tilde{\boldsymbol{A}}_{\mu}\times\tilde{\boldsymbol{\phi}}$.
Equation (79a) is equivalent to $H=\alpha(\tilde{H})$.

The transformations of the covariant derivative and field-strength tensor
follow as

$\displaystyle\frac{D_{\mu}\boldsymbol{\phi}}{vH}$ $\displaystyle=$
$\displaystyle\ell\frac{\tilde{D}_{\mu}\tilde{\boldsymbol{\phi}}}{v\tilde{H}}-h\frac{\tilde{\boldsymbol{\phi}}\times\tilde{D}_{\mu}\tilde{\boldsymbol{\phi}}}{(v\tilde{H})^{2}}-\bigg{(}\ell-\tilde{H}\frac{\alpha^{\prime}}{\alpha}\bigg{)}\frac{\tilde{\boldsymbol{\phi}}\\!\cdot\\!\tilde{D}_{\mu}\tilde{\boldsymbol{\phi}}}{(v\tilde{H})^{3}}\tilde{\boldsymbol{\phi}}\,,$
$\displaystyle\boldsymbol{F}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\ell\,\tilde{\boldsymbol{F}}_{\mu\nu}-h\frac{\tilde{\boldsymbol{\phi}}\times\tilde{\boldsymbol{F}}_{\mu\nu}}{v\tilde{H}}-\big{(}\ell-1\big{)}\frac{\tilde{\boldsymbol{\phi}}\cdot\tilde{\boldsymbol{F}}_{\mu\nu}}{(v\tilde{H})^{2}}\tilde{\boldsymbol{\phi}}+\boldsymbol{d}_{\mu\nu}\,,$

where we defined

$\displaystyle\boldsymbol{d}_{\mu\nu}$ $\displaystyle\equiv$
$\displaystyle\phantom{+\,\,}\big{(}\tilde{H}h^{\prime}-k\ell\big{)}\frac{(\tilde{\boldsymbol{\phi}}\\!\cdot\\!\tilde{D}_{\mu}\tilde{\boldsymbol{\phi}})(\tilde{D}_{\nu}\tilde{\boldsymbol{\phi}})-(\mu\leftrightarrow\nu)}{(v\tilde{H})^{3}}+\big{(}\tilde{H}\ell^{\prime}+kh\big{)}\frac{(\tilde{\boldsymbol{\phi}}\\!\cdot\\!\tilde{D}_{\mu}\tilde{\boldsymbol{\phi}})(\tilde{\boldsymbol{\phi}}\times\tilde{D}_{\nu}\tilde{\boldsymbol{\phi}})-(\mu\leftrightarrow\nu)}{(v\tilde{H})^{4}}$
$\displaystyle{}-\Big{[}(\ell^{2}+h^{2}-1)-(\ell-1)^{2}\Big{]}\bigg{[}\frac{(\tilde{\boldsymbol{\phi}}\\!\cdot\\!\tilde{D}_{\mu}\tilde{\boldsymbol{\phi}})(\tilde{\boldsymbol{\phi}}\times\tilde{D}_{\nu}\tilde{\boldsymbol{\phi}})-(\mu\leftrightarrow\nu)}{(v\tilde{H})^{4}}-\frac{\tilde{D}_{\mu}\tilde{\boldsymbol{\phi}}\times\tilde{D}_{\nu}\tilde{\boldsymbol{\phi}}}{(v\tilde{H})^{2}}\bigg{]}+(\ell-1)^{2}\frac{\tilde{\boldsymbol{\phi}}\\!\cdot\\!(\tilde{D}_{\mu}\tilde{\boldsymbol{\phi}}\times\tilde{D}_{\nu}\tilde{\boldsymbol{\phi}})}{(v\tilde{H})^{4}}\tilde{\boldsymbol{\phi}}\,,$

with the primes being differentiations with respect to $\tilde{H}$. Under the
transformation (79) the original Lagrangian $\mathcal{L}$, Eq. (2), transforms
to another Lagrangian $\tilde{\mathcal{L}}$ of the form

$\displaystyle\tilde{\mathcal{L}}$ $\displaystyle=$
$\displaystyle\frac{v^{2}}{2}\bigg{[}\tilde{f}_{1}^{2}\bigg{(}\frac{(\tilde{D}^{\mu}\tilde{\boldsymbol{\phi}})^{2}}{\tilde{\boldsymbol{\phi}}^{2}}-\frac{(\tilde{\boldsymbol{\phi}}\\!\cdot\\!\tilde{D}^{\mu}\tilde{\boldsymbol{\phi}})^{2}}{\tilde{\boldsymbol{\phi}}^{4}}\bigg{)}+\tilde{f}_{3}^{2}\frac{(\tilde{\boldsymbol{\phi}}\\!\cdot\\!\tilde{D}^{\mu}\tilde{\boldsymbol{\phi}})^{2}}{\tilde{\boldsymbol{\phi}}^{4}}\bigg{]}-\frac{1}{4g^{2}}\bigg{[}\tilde{f}_{2}^{2}\bigg{(}(\boldsymbol{F}^{\mu\nu})^{2}-\frac{(\tilde{\boldsymbol{\phi}}\\!\cdot\\!\boldsymbol{F}^{\mu\nu})^{2}}{\tilde{\boldsymbol{\phi}}^{2}}\bigg{)}+\tilde{f}_{4}^{2}\frac{(\tilde{\boldsymbol{\phi}}\\!\cdot\\!\boldsymbol{F}^{\mu\nu})^{2}}{\tilde{\boldsymbol{\phi}}^{2}}\bigg{]}$
(83)
$\displaystyle{}-\frac{1}{2g^{2}}\boldsymbol{d}^{\mu\nu}\\!\cdot\\!\Bigg{\\{}f_{2}^{2}\ell\bigg{[}\tilde{\boldsymbol{F}}_{\mu\nu}-\frac{(\tilde{\boldsymbol{\phi}}\\!\cdot\\!\tilde{\boldsymbol{F}}_{\mu\nu})}{\tilde{\boldsymbol{\phi}}^{2}}\tilde{\boldsymbol{\phi}}\bigg{]}+f_{2}^{2}h\frac{\tilde{\boldsymbol{F}}_{\mu\nu}\times\tilde{\boldsymbol{\phi}}}{v\tilde{H}}+f_{4}^{2}\frac{(\tilde{\boldsymbol{\phi}}\\!\cdot\\!\tilde{\boldsymbol{F}}_{\mu\nu})}{\tilde{\boldsymbol{\phi}}^{2}}\tilde{\boldsymbol{\phi}}\Bigg{\\}}$
$\displaystyle{}-\frac{1}{4g^{2}}\bigg{[}f_{2}^{2}\bigg{(}(\boldsymbol{d}_{\mu\nu})^{2}-\frac{(\tilde{\boldsymbol{\phi}}\\!\cdot\\!\boldsymbol{d}_{\mu\nu})^{2}}{\tilde{\boldsymbol{\phi}}^{2}}\bigg{)}+f_{4}^{2}\frac{(\tilde{\boldsymbol{\phi}}\\!\cdot\\!\boldsymbol{d}_{\mu\nu})^{2}}{\tilde{\boldsymbol{\phi}}^{2}}\bigg{]}-\tilde{V}(\tilde{\boldsymbol{\phi}}^{2})\,,$

where the new functions $\tilde{f}_{i}^{2}=\tilde{f}_{i}^{2}(\tilde{H})$ are
given as

$\displaystyle\tilde{f}_{i}^{2}$ $\displaystyle=$ $\displaystyle
f_{i}^{2}\big{(}\ell^{2}+h^{2}\big{)}\,,\hskip 17.07164pt(i=1,2)$ (84a)
$\displaystyle\tilde{f}_{3}^{2}$ $\displaystyle=$ $\displaystyle
f_{3}^{2}\bigg{(}\tilde{H}\frac{\alpha^{\prime}}{\alpha}\bigg{)}^{2}\,,$ (84b)
$\displaystyle\tilde{f}_{4}^{2}$ $\displaystyle=$ $\displaystyle f_{4}^{2}\,,$
(84c)

with $f_{i}^{2}=f_{i}^{2}(\alpha(\tilde{H}))$, and
$\tilde{V}(\tilde{\boldsymbol{\phi}}^{2})=V(\boldsymbol{\phi}^{2})$.

We also obtain

$\displaystyle\boldsymbol{\phi}\\!\cdot\\!\boldsymbol{A}_{\mu}$
$\displaystyle=$
$\displaystyle\frac{\alpha}{\tilde{H}}\bigg{[}\tilde{\boldsymbol{\phi}}\\!\cdot\\!\tilde{\boldsymbol{A}}_{\mu}+k\frac{\tilde{\boldsymbol{\phi}}\\!\cdot\\!\tilde{D}_{\mu}\tilde{\boldsymbol{\phi}}}{v\tilde{H}}\bigg{]}\,,\hskip
56.9055pt$ (85)
$\displaystyle\boldsymbol{\phi}\\!\cdot\\!\boldsymbol{F}_{\mu\nu}$
$\displaystyle=$
$\displaystyle\frac{\alpha}{\tilde{H}}\bigg{[}\tilde{\boldsymbol{\phi}}\\!\cdot\\!\tilde{\boldsymbol{F}}_{\mu\nu}+(\ell^{2}+h^{2}-1)\frac{\tilde{\boldsymbol{\phi}}\\!\cdot\\!(\tilde{D}_{\mu}\tilde{\boldsymbol{\phi}}\times\tilde{D}_{\nu}\tilde{\boldsymbol{\phi}})}{\tilde{\boldsymbol{\phi}}^{2}}\bigg{]}\,.$

By performing two consecutive transformations (79) we obtain again a
transformation of the form (79). For the scalar it is obvious: If we transform
first $H_{1}\to H_{2}$ and then $H_{2}\to H_{3}$ with
$H_{1}=\alpha_{2}(H_{2})$ and $H_{2}=\alpha_{3}(H_{3})$, respectively, then
the combined transformation $H_{1}\to H_{3}$ is done via
$H_{1}=\alpha_{2}(\alpha_{3}(H_{3}))$, i.e., by simple function composition.
For the gauge fields the combined transformation is more complicated. If the
first transformation is by $h_{2}(H_{2})$, $k_{2}(H_{2})$, $\ell_{2}(H_{2})$
and the second one by $h_{3}(H_{3})$, $k_{3}(H_{3})$, $\ell_{3}(H_{3})$, then
the combination of the two transformation is done via

$\displaystyle h(h_{3})$ $\displaystyle=$
$\displaystyle\ell_{2}(H_{2})\,h_{3}(H_{3})+h_{2}(H_{2})\,\ell_{3}(H_{3})\,,$
(87a) $\displaystyle\ell(h_{3})$ $\displaystyle=$
$\displaystyle\ell_{2}(H_{2})\,\ell_{3}(H_{3})-h_{2}(H_{2})\,h_{3}(H_{3})\,,$
(87b) $\displaystyle k(h_{3})$ $\displaystyle=$ $\displaystyle
k_{3}(H_{3})+k_{2}(H_{2})\,H_{3}\frac{\alpha_{3}^{\prime}(H_{3})}{\alpha_{3}(H_{3})}\,.\hskip
28.45274pt$ (87c)

Unless $h=k=0$, the general transformation (79) does _not_ protect the
spherically symmetric Ansatz (37), but rather leads to

$\displaystyle\tilde{\phi}_{a}$ $\displaystyle=$ $\displaystyle
v\tilde{H}\frac{x_{a}}{r}\,,$ (88a) $\displaystyle\tilde{A}^{i}_{a}$
$\displaystyle=$
$\displaystyle-\frac{\varepsilon_{abi}x_{b}}{r^{2}}(1-\tilde{K})+\frac{\delta_{ia}r^{2}-x_{i}x_{a}}{r^{3}}\tilde{L}+\frac{x_{i}x_{a}}{r^{3}}\tilde{M}\,,\hskip
31.29802pt$ (88b)

where $\tilde{H}$ is obtained by inverting $\alpha(\tilde{H})=H$, while

$\tilde{K}\ \equiv\ \frac{\ell}{\ell^{2}+h^{2}}K\,,\hskip 8.53581pt\tilde{L}\
\equiv\ \frac{-h}{\ell^{2}+h^{2}}K\,,\hskip 8.53581pt\tilde{M}\ \equiv\
-kr\frac{\tilde{H}^{\prime}}{\tilde{H}}\,.$ (89)

Nevertheless, it is still possible to require the transformation (79) to
protect, at least, the transversality of the gauge and scalar fields:
$\boldsymbol{\phi}\cdot\boldsymbol{A}^{i}=\tilde{\boldsymbol{\phi}}\cdot\tilde{\boldsymbol{A}}^{i}=0$.
In other words, one can demand $\tilde{M}=0$, implying $k=0$.

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