# Instability of electroweak homogeneous vacua in strong magnetic fields

Adam Gardner∗ and Israel Michael Sigal111Dept. of Math., U. of Toronto,
Toronto, Canada<EMAIL_ADDRESS><EMAIL_ADDRESS>
()

###### Abstract

We consider the classical vacua of the Weinberg-Salam (WS) model of
electroweak forces. These are no-particle, static solutions to the WS
equations minimizing the WS energy locally.

We study the WS vacuum solutions exhibiting a non-vanishing average magnetic
field, $\vec{b}$, and prove that (i) there is a magnetic field threshold
$b_{*}$ such that for $|\vec{b}|<b_{*}$, the vacua are translationally
invariant (and the magnetic field is constant), while, for $|\vec{b}|>b_{*}$,
they are not, (ii) for $|\vec{b}|>b_{*}$, there are non-translationally
invariant solutions with lower energy per unit volume and with the discrete
translational symmetry of a 2D lattice in the plane transversal to $\vec{b}$,
and (iii) the lattice minimizing the energy per unit volume approaches the
hexagonal one as the magnetic field strength approaches the threshold $b_{*}$.

In the absence of particles, the Weinberg-Salam model reduces to the Yang-
Mills-Higgs (YMH) equations for the gauge group $U(2)$. Thus our results can
be rephrased as the corresponding statements about the $U(2)$-YMH equations

##  1 Introduction

The Weinberg-Salam (WS) model of electroweak interactions was the first
triumph of the program to unify the four fundamental forces of nature. It is a
key part of the standard model of elementary particles. It unifies
electromagnetic and weak interactions, two of the three forces dealt with in
the standard model. It involves particle, gauge and the Higgs fields.

While the gauge fields describe the electroweak interactions, the role of the
Higgs field is to convert the original massless fields (zero masses are
required by the relativistic invariance) to massive ones. This phenomenon is
called the Higgs mechanism. This mechanism, together with the Goldstone
theorem, leads to all gauge particles but one acquiring mass, resulting in two
massive bosons – denoted W and Z – and a massless one – the photon. The W and
Z particles where discovered experimentally 16 years after their theoretical
prediction.

In this paper, we consider the vacuum solutions of the classical WS model with
a non-vanishing average magnetic field ($\vec{b}$). These are static, no-
particle solutions minimizing the WS energy locally for a fixed $\vec{b}$. We
prove that (i) there is a magnetic field threshold $b_{*}$ such that for
$|\vec{b}|<b_{*}$, the vacua are translationally invariant, while, for
$|\vec{b}|>b_{*}$, they are not, (ii) for $|\vec{b}|>b_{*}$, there are non-
translationally invariant solutions with lower energy per unit volume and with
the discrete translational symmetry of a 2D lattice in the plane transversal
to the magnetic field, and (iii) the lattice minimizing the energy of the
latter solutions per unit volume approaches the hexagonal one as the magnetic
field strength approaches the threshold $b_{*}$. We expect that these
solutions are stable under field fluctuations and, in fact, minimize the
energy locally.

The phenomenon above was investigated extensively in the physics literature
(see e.g. [12, 19, 25, 26] and the references therein). It is similar to the
one occurring in superconductivity and the solutions whose existence we
establish are analogous to the superconducting Abrikosov vortex lattices ([1],
see e.g. [34], for a review). It is estimated in [25] that the spontaneous
symmetry breaking take place at the critical average magnetic field of
approximately $10^{24}$ Gauss $=10^{20}$ Tesla. By comparison, the strongest
magnetic field produced on Earth is $10^{14}$ Tesla.

Note that, in the absence of particles, the WS system reduces to the Yang-
Mills-Higgs (YMH) one with the gauge group $U(2)$. So ultimately, these are
the equations we deal with.

The only rigorous result ([40, 41]) on the classical WS model deals with the
vortices in the self-dual regime, where the WS (or corresponding YMH)
equations are equivalent to the first order equations, and it uses this
equivalence in an essential way. (The self-dual regime in this context was
discovered in [6, 7, 8], see also [38, 39].)

Open problems and further directions:

(a) Stability of the emerging solutions.

(b) Quantum corrections to the values of the classical critical magnetic field
$b_{*}$ and the optimal lattice shape parameter $\tau_{*}$.

(c) Existence of vortex lattices at $|\vec{b}|\gg b_{*}$.

For the stability and existence problems, (a) and (c), see e.g. [35, 37] and
[36], respectively. The last problem brings up the regime of ‘sparse’ vortex
lattices as opposite to the case of $|\vec{b}|$ close to (and $>$) $b_{*}$
resulting in densely packed vortices: the lattice step $\rightarrow 0$ as
$|\vec{b}|\rightarrow b_{*}$ and $\rightarrow\infty$ as
$|\vec{b}|\rightarrow\infty$. Hence the existence of vortex lattices at
$|\vec{b}|\gg b_{*}$ is closely related to the problem of existence of
vortices (elementary excitations).

For the quantum corrections, problem (b), a natural approach would involve a
‘mean-field’/quasi-free approximation and a Bogoliubov-type expansion of a
regularized (say, lattice) WS model around it, see e.g. [14, 15].

The paper is organized as follows. In Section 2, we formulate the problem and
describe results. In Sections 3 \- 4, we fix the gauge and pass from the
original Yang-Mills fields to the W and Z (massive boson) and A (photon)
fields and rescale the resulting equations. The proofs of the main results are
given in Section 5 (Theorem 2.1), Sections 6 \- 10 (Theorem 2.2) and Section
11 (Theorem 2.3). In Appendix A, we discuss various covariant derivatives used
in the main text and in Appendix B, we review the time-dependent YMH equations
and derive the expression for the conserved energy as well as the YMH
equations used in the main text. Furthermore, there, we write the YMH
equations in the coordinate form and derive a convenient expression for the
energy functional. In Appendices D.1 \- D.2, we derive the WS equations in 3D
and 2D, respectively, in terms of the fields $W$, $Z$, $A$ and $\varphi$. In
the remaining appendices, we carry out technical computations.

Throughout the paper, we use the Einstein convention of summing over repeated
indices.

#### Acknowledgements

The second author is grateful to Nicholas Ercolani for many instructive and
stimulating discussions of the YMH equations.

The research on this paper is supported in part by NSERC Grant No. NA7901.

##  2 No-particle and vacuum sectors of the Weinberg-Salam model

The no-particle sector of the Weinberg-Salam (WS) model involves the
interacting Higgs and $SU(2)$ and $U(1)$ gauge fields, $\Phi$ and $V$ and $X$,
while the particle fields are set to zero. The field $\Phi$ is a vector-
function defined on the Minkowski space-time $\mathbb{R}^{3+1}$ with values in
$\mathbb{C}^{2}$, and the fields $V$ and $X$ are one-forms on
$\mathbb{R}^{3+1}$ with values in the algebras $\mathfrak{su}(2)$ and
$\mathfrak{u}(1)$, respectively. We write

$Q=gV+g^{\prime}X,$

where $g$ and $g^{\prime}$ are coupling constants, which is a one-form with
values in $\mathfrak{u}(2)$. We consider $SU(2)$ as a matrix group and $U(1)$
as multiples of the identity matrix ${{\bf 1}}$ acting on $\mathbb{C}^{2}$.

These fields satisfy the WS equations, which are the Euler-Lagrange equations
for the action functional

$\mathcal{S}(\Psi,Q)=\int_{M}\big{(}\langle\nabla_{Q}\Psi,\nabla_{Q}\Psi\rangle_{\Omega^{1}_{V}}^{\eta}-\frac{1}{2}\lambda(\|\Phi\|_{\mathbb{C}^{2}}^{2}-\varphi_{0}^{2})^{2}+\langle
F_{Q},F_{Q}\rangle_{\Omega^{2}_{\mathfrak{g}}}^{\eta}\big{)},$ (2.1)

where $M$ is a bounded domain in spacetime $\mathbb{R}^{d+1}$ equipped with
the Minkowski metric $\eta$ of signature $(+,-,...,-)$, $\lambda$ and
$\varphi_{0}$ are positive parameters, and the remaining symbols are defined
as follows:

$\nabla_{Q}$ is the covariant derivative mapping $\mathbb{C}^{2}$-valued
functions (sections) into $\mathbb{C}^{2}$-valued one-forms defined as

$\nabla_{Q}=d+Q,$ (2.2)

with $d$, the exterior derivative;

$F_{Q}$ is the curvature $2$-form of the connection one-form $Q$, given by

$F_{Q}=dQ+\frac{1}{2g}[Q,Q],$ (2.3)

where $[A,B]$ is defined in local coordinates $\\{x^{i}\\}$ as

$\displaystyle[A,B]:=[A_{i},B_{j}]dx^{i}\wedge dx^{j}=[B,A],$ (2.4)

with $A=A_{i}dx^{i}$ and $B=B_{i}dx^{i}$;

$\Omega^{p}_{U}\equiv U\otimes\Omega^{p}$ denotes the space of $U$-valued
$p$-forms with the inner product,

$\langle A,B\rangle_{\Omega^{p}_{U}}^{\eta}:=\langle
A_{\alpha},B^{\alpha}\rangle_{U},$ (2.5)

where $A=A_{\alpha}dx^{\alpha}$ and $B=B_{\alpha}dx^{\alpha}$ are $U$-valued
$p$-forms, $\alpha$ is a $p$-form index and $\langle\cdot,\cdot\rangle_{U}$ is
the inner product on $U$ 222For $\mathfrak{u}(2)$, the inner product is given
by $\langle
g,h\rangle_{\mathfrak{u}(2)}:=2\operatorname{Tr}(g^{*}h)=-2\operatorname{Tr}(gh)$.
with the indices raised and lowered with help of the metric $\eta$ on $M$.

Solutions of the no-particle WS equations solve also the full WS system as
well as that for the standard model of the particle physics.

The vacuum sector of the Weinberg-Salam (WS) model consists of static, no-
particle solutions. The static Higgs and $SU(2)$ and $U(1)$ gauge fields
$\Phi$, $V$ and $X$ are now defined on the physical space $\mathbb{R}^{3}$
with the same respective values as in the time-dependent case. Geometrically,
$V,X$ and $Q$ can be thought of as connection one-forms on the trivial bundles
$\mathbb{R}^{3}\times SU(2),\mathbb{R}^{3}\times U(1)$ and
$\mathbb{R}^{3}\times U(2)$.

The fields $\Phi$, $V$ and $X$ satisfy the static no-particle WS equations,
which are the Euler-Lagrange equations for the static WS energy functional
originating in (2.1)333For a discussion of the the time-dependent theory and a
derivation of the energy functional (2.6) see [23], [26], [32], [33] and
Appendix B.

$E_{N}^{WS}(\Phi,Q):=\int_{N}\big{(}\|\nabla_{Q}\Phi\|_{\Omega^{1}_{\mathbb{C}^{2}}}^{2}+\frac{1}{2}\lambda(\|\Phi\|_{\mathbb{C}^{2}}^{2}-\varphi_{0}^{2})^{2}+\frac{1}{2}\|F_{Q}\|_{\Omega^{2}_{\mathfrak{u}(2)}}^{2}\big{)},$
(2.6)

where $N$ is a bounded domain in $\mathbb{R}^{3}$ with appropriate boundary
conditions (specified in (2.22) below) and $\|\cdot\|_{\Omega^{p}_{U}}$ is the
norm on the space $\Omega^{p}_{U}:=U\otimes\Omega^{p}$ of $U$-valued $p$-forms
at $x\in N$ (e.g. for $B=B_{i}dx^{i}\in\Omega^{1}_{U}$, we have
$\|B(x)\|_{\Omega^{1}_{U}}:=(\sum_{i}\|B_{i}(x)\|^{2}_{U})^{1/2}$ with the
usual Euclidean metric and with the indices running through $1,2,3$). The
symbols $\nabla_{Q}$ and $F_{Q}$ are as defined above but without the time
component.

Since $Q=gV+g^{\prime}X$ and $X$ has the values in the centre, $u(1)$, of the
algebra $u(2)$, we have $F_{Q}=F_{V}+F_{X}$, where

$F_{V}:=dV+\frac{g}{2}[V,V]\ \text{ and }\ F_{X}:=dX$ (2.7)

are the curvatures of the connections $V$ and $X$444For more discussion of
covariant derivatives and their curvatures, see Appendices A for the general
case, or Appendix (C), for the case of the gauge group $G=U(2)$. and
$\|F_{Q}\|^{2}_{\Omega^{2}_{\mathfrak{u}(2)}}=\|F_{V}\|^{2}_{\Omega^{2}_{\mathfrak{u}(2)}}+\|F_{X}\|^{2}_{\Omega^{2}_{\mathfrak{u}(1)}}$.

We introduce the covariant derivative $d_{Q}$ mapping $\mathfrak{u}(2)$-valued
one-forms into $\mathfrak{u}(2)$-valued two-forms as555This formula originates
in the equation $(\delta_{Q}F_{Q})(B)=d_{Q}B$, where $\delta_{Q}$ is the
Gâteaux derivative with respect to $Q$.

$d_{Q}B:=dB+[Q,B]=d_{V}B:=dB+g[V,B].$ (2.8)

The Euler-Lagrange equations for energy functional (2.6) are given by (see
Appendix B666These equations could be converted formally back into the time-
dependent ones by taking the adjoints in the Minkowski metric instead of the
Euclidian one, see (B.4)-(B.5), Appendix B.)

$\displaystyle\nabla_{Q}^{*}\nabla_{Q}\Phi=\lambda(\varphi_{0}^{2}-\|\Phi\|^{2})\Phi,$
(2.9) $\displaystyle d_{Q}^{*}F_{Q}=J(\Phi,Q),$ (2.10)

where $\nabla_{Q}^{*}$ is the adjoint of $\nabla_{Q}$ and maps
$\mathbb{C}^{2}$-valued one-forms into $\mathbb{C}^{2}$-valued functions,
$d_{Q}^{*}$ is the adjoint of $d_{Q}$ and maps $\mathfrak{u}(2)$-valued two-
forms into $\mathfrak{u}(2)$-valued one-forms, and $J(\Phi,Q)$ is the
electroweak current, which is the $\mathfrak{u}(2)$-valued one-form given by

$\displaystyle J(\Phi,Q)$
$\displaystyle:=-\frac{ig}{2}\tau_{a}\operatorname{Im}\langle\tau_{a}\Phi,\nabla_{Q}\Phi\rangle-\frac{ig^{\prime}}{2}\tau_{0}\operatorname{Im}\langle\tau_{0}\Phi,\nabla_{Q}\Phi\rangle,$
(2.11)

where summing over repeated indices is understood, $\tau_{0}:=\mathbf{1}$ and
$\tau_{a},a=1,2,3,$ are the Pauli matrices,

$\displaystyle\tau_{1}:=\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right),\ \tau_{2}:=\left(\begin{array}[]{cc}0&-i\\\
i&0\end{array}\right),\ \tau_{3}:=\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right).$ (2.18)

(The Pauli matrices, multiplied by $-i/2$, form an orthonormal basis in
$\mathfrak{su}(2)$ with the inner product $\langle
g,h\rangle_{\mathfrak{su}(2)}:=2\operatorname{Tr}(g^{*}h)=-2\operatorname{Tr}(gh)$.)
We call system (2.9)-(2.10) the (static) WS equations.

The energy functional (2.6) and Euler-Lagrange equations (2.9) - (2.10) are
invariant under the group of rigid motions and the gauge transformations
(gauge symmetry)

$\displaystyle(\Phi(x),V(x),X(x))$
$\displaystyle\mapsto(h_{1}(x)h_{2}(x)\Phi(x),h_{1}(x)V(x)h_{1}^{-1}(x)$
$\displaystyle-$ $\displaystyle
i\frac{2}{g}h_{1}(x)dh_{1}^{-1}(x),X(x)-i\frac{2}{g^{\prime}}h_{2}(x)dh_{2}^{-1}(x)),$
(2.19) $\displaystyle\forall h_{1}(x)\in SU(2),\ h_{2}(x)\in U(1).$ (2.20)

For each $b\in\mathbb{R}$, Eqs. (2.9) - (2.10) have the constant magnetic
field solution given (up to a gauge symmetry) by

$U^{b}_{*}:=(\Phi_{0},-\frac{i}{2}\tau_{3}A^{b}\sin\theta,-\frac{i}{2}\tau_{0}A^{b}\cos\theta),$
(2.21)

where $\Phi_{0}:=(0,\varphi_{0})$, $A^{b}(x)$ is a magnetic potential of the
constant magnetic field of strength $b$ and $\theta$ is _Weinberg’s angle_ ,
given by $\tan\theta=g^{\prime}/g$.777Indeed,
$d_{Q}\Phi_{0}=(gV+g^{\prime}X)\Phi_{0}=(gA^{b}\sin\theta\tau_{3}+g^{\prime}A^{b}\cos\theta\tau_{0})\Phi_{0}=g^{\prime}A^{b}\cos\theta(\tau_{3}+\tau_{0})\Phi_{0}$.
Since $(\tau_{3}+\tau_{0})\Phi_{0}=0$, this implies $d_{Q}\Phi_{0}=0$. From
$d_{Q}\Phi_{0}=0$, it is easy to see that (2.21) solves (2.9) - (2.10). This
solution is gauge-translationally invariant, i.e. invariant under translations
up to a gauge symmetry. It corresponds to the ‘total vacuum’ with the constant
magnetic field $dA^{b}(x)=\sum_{(ijk)}b_{i}dx^{j}\wedge dx^{k}$.

In what follows, we identify the magnetic field 2-form,
$\sum_{(ijk)}b_{i}dx^{j}\wedge dx^{k}$, in $\mathbb{R}^{3}$ with the 3-vector,
$\vec{b}=(b_{1},b_{2},b_{3})$, and look for solutions with a fixed average
magnetic field $\vec{b}$. This breaks the rotational symmetry and therefore it
is natural to consider solutions which do not depend on the coordinate along
$\vec{b}$.

With the notation $b=|\vec{b}|$, we show that for appropriate perturbations,
(i) (2.21) is linearly stable for $b<b_{*}$ and unstable for $b>b_{*}$, where
$b_{*}:=g^{2}\varphi_{0}^{2}/2e$, (ii) at $b=b_{*}$, a new solution breaking
the gauge-translational invariance bifurcates. This solution has the discrete
translational symmetry of a lattice in the plane orthogonal to $\vec{b}$ and
has lower energy per unit area, (iii) the lattice shape minimizing the energy
per unit area approaches the hexagonal lattice as $b$ approaches $b_{*}$.

To formulate these results precisely, we introduce some definitions. Since we
consider solutions which do not depend on the coordinate along $\vec{b}$, we
can restrict our analysis to the plane $\perp\vec{b}$. We choose the
$x^{3}$-axis along $\vec{b}$ and identify the plane $\perp\vec{b}$ with
$\mathbb{R}^{2}$.

We fix a lattice $\mathcal{L}$ in $\mathbb{R}^{2}$ and say a triple
$(\Phi(x),V(x),X(x))$ is $\mathcal{L}$-_gauge-periodic_ , or,
$\mathcal{L}$_-equivariant_ , if and only if it satisfies the equation

$(T^{gauge}_{\gamma_{s}})^{-1}T^{trans}_{s}(\Phi,V,X)=(\Phi,V,X),\
\quad\forall s\in\mathcal{L},$ (2.22)

for some $\gamma_{s}\in C^{1}(\mathbb{R}^{2},SU(2)\times U(1))$. Here
$T^{gauge}_{\gamma}$ is given by the right-hand side of (2), with
$h_{1}(x)h_{2}(x)=\gamma(x)$, and $T^{trans}_{s}$ is the group of
translations, $T^{trans}_{s}f(x)=f(x+s)$. (When $\mathcal{L}$ is clear, we
omit it from the definition above.)

We denote by $\mathcal{H}^{s}_{\mathcal{L}}$ the space of locally Sobolev
space $\mathcal{L}$-equivariant triples $(\Phi,V,X)$ on $\mathbb{R}^{2}$ with
the inner product given by the standard Sobolev inner product restricted to an
arbitrary fundamental domain $\Omega$ of $\mathcal{L}$, and let
$L^{2}_{\mathcal{L}}=\mathcal{H}^{0}_{\mathcal{L}}$.

We say a solution $U_{*}:=(\Phi_{*},V_{*},X_{*})$ of the WS system (2.9) -
(2.10) is _energeticaly stable_ if and only if it is a local minimum of the WS
energy $E_{N}^{WS}$, in the sense that the spectrum of the _$L^{2}$ -Hessian_
of $E_{N}^{WS}$ at $U_{*}$ on $L^{2}_{\mathcal{L}}$ (which is real) is non-
negative. $U_{*}$ is said to be _unstable_ if it is a saddle point of
$E_{N}^{WS}$ (so that the spectrum of its hessian has a negative part).

For an $\mathcal{L}$-equivariant triple $U$ and a fundamental domain $\Omega$
of $\mathcal{L}$, we define the energy per fundamental cell

$E^{\mathcal{L}}(U):=\frac{1}{|\Omega|}E_{\Omega}^{WS}(U),$ (2.23)

where $|\Omega|$ denotes the area of $\Omega$. This energy is independent the
choice of $\Omega$.

In what follows, $\Omega$ denotes an arbitrary (but fixed throughout)
fundamental domain of $\mathcal{L}$, and $|\mathcal{L}|$, the area of a
fundamental cell of $\mathcal{L}$, which is independent of the choice of the
cell (and in particular, $|\mathcal{L}|=|\Omega|$).

Finally, let $M_{W}:=\frac{1}{\sqrt{2}}g\varphi_{0}$,
$M_{Z}:=\frac{1}{\sqrt{2}\cos\theta}g\varphi_{0}$ and
$M_{H}:=\sqrt{2}\lambda\varphi_{0}$. These are the masses of the W, Z and
Higgs bosons, respectively.888This nomenclature will be explained in the
discussion following equation (3). Let

$b_{*}:=\frac{g^{2}\varphi_{0}^{2}}{2e}=\frac{M_{W}^{2}}{e},\ \quad
e:=g\sin\theta.$ (2.24)

With the above definitions, we will prove the following:

######  Theorem 2.1.

The homogeneous vacuum solution (2.21) is energeticallly stable for $b<b_{*}$
and unstable for $b>b_{*}$.

######  Theorem 2.2.

Let $\mathcal{L}$ be a lattice satisfying
$0<1-\frac{M_{W}^{2}}{2\pi}|\mathcal{L}|\ll 1$ and assume that
$M_{Z}<M_{H}$.999This assumption is justified experimentally since
$M_{Z}=91.1876\pm 0.0021GeV/c^{2}$ [17] and $M_{H}=125.09\pm 0.31GeV/c^{2}$
[13] Then the following holds:

  1. (a)

Equations (2.9) - (2.10) have a non-trivial solution
$U_{\mathcal{L}}\in\mathcal{H}^{2}_{\mathcal{L}}$ in a neighbourhood of vacuum
solution (2.21);

  2. (b)

$U_{\mathcal{L}}$ is unique, up to gauge symmetry transformation, in a
neighbourhood ofvacuum solution (2.21);

  3. (c)

$U_{\mathcal{L}}$ has energy per unit area less than vacuum solution (2.21):
$E^{\mathcal{L}}(U_{\mathcal{L}})<E^{\mathcal{L}}(U^{b}_{*})$.

The solutions described in this theorem can be reinterpreted geometrically as
representing sections ($\Phi(x)$) and connections ($(V(x),X(x))$) on a $U(2)$
vector bundle over a torus (cf. [20]). However, a vector bundles over a torus
is topologically equivalent to a direct sum of line bundles. In our case, this
equivalence follows from equations (3.5) - (3.7) below.

For the next result, we introduce the standard parameterization of lattices in
$\mathbb{R}^{2}$. Identifying $\mathbb{R}^{2}$ with $\mathbb{C}$ via
$(x_{1},x_{2})\leftrightarrow x_{1}+ix_{2}$, we can view a lattice
$\mathcal{L}\subset\mathbb{R}^{2}$ as a subset of $\mathbb{C}$. It is a well-
known fact (see e.g. [4]) that any lattice $\mathcal{L}\subset\mathbb{C}$ can
be given a basis $r,r^{\prime}$ such that the ratio
$\tau=\frac{r^{\prime}}{r}$ belongs to the set

$\displaystyle\\{\tau\in\mathbb{C}:\operatorname{Im}\tau>0,\ |\tau|\geq 1,\
-\frac{1}{2}<\operatorname{Re}\tau\leq\frac{1}{2}\\},$ (2.25)

which is the fundamental domain, $\mathbb{H}/SL(2,\mathbb{Z})$, of the modular
group $SL(2,\mathbb{Z})$ acting on the Poincaré half-plane $\mathbb{H}$. For a
given $\mathcal{L}$, the parameter $\tau$ is unique and is used as a
parameterization (up to scaling) of the lattices. This gives the space of
(normalized) lattices a topology.

######  Theorem 2.3.

For $M_{Z}<M_{H}$, the lattice $\mathcal{L}_{*}$ minimizing the average
energy, $E^{\mathcal{L}}(U_{\mathcal{L}})$, approaches the hexagonal lattice
$\mathcal{L}_{\rm hex}$ as $b\to b_{*}$ in the sense that the shape parameter
$\tau_{*}$ of the lattice $\mathcal{L}_{*}$ approaches $\tau_{\rm
hex}=e^{i\pi/3}$ in $\mathbb{C}$.

Our approached is based on a careful examination of the linearization of the
WS equations on the homogeneous vacuum. The spectrum of the linearized problem
determines the domains of the linear, or energetic, stability and the
transition threshold. In the instability domain, we apply an equivariant
bifurcation theory. This gives Theorem 2.2(a) and (b). For Theorems 2.2(c) and
2.3, we carefully study the asymptotic behaviour of the energy functions for
small values of the bifurcation parameter.

##  3 Gauge fixing and $W$ and $Z$ bosons

In this section, we choose a particular gauge and pass from the fields (one-
forms) $V$ and $X$ to more suitable gauge fields. We eliminate a part of the
gauge freedom by assuming that the Higgs field $\Phi$ is of the form

$\Phi=(0,\varphi),$ (3.1)

with $\varphi$ real (this can be done using only the $SU(2)$ part of the gauge
group). Then

$\displaystyle\tau_{a}\Phi\neq 0,\ a=0,1,2,3,$ (3.2)

where, recall, $\tau_{a},\ a=1,2,3,$ are the Pauli matrices generating the Lie
algebra $su(2)$, and $\tau_{0}=\mathbf{1}$. However, there is one linear
combination of $\tau_{a}$’s (unique up to a scalar multiple) which annihilates
$\Phi$:

$\displaystyle(\tau_{3}+\tau_{0})\Phi=0.$ (3.3)

Thus, for the gauge $\Phi=(0,\varphi)$ the symmetries generated by
$\tau_{1},\tau_{2},\tau_{3}-\tau_{0}$ are broken and the $U(1)$ symmetry
generated by $\tau_{3}+\tau_{0}$ remains unbroken. The unbroken gauge symmetry
is given by transformations (2) with

$\displaystyle h_{1}(x):=$ $\displaystyle e^{-\frac{i}{2}\gamma(x)\tau_{3}}\in
SU(2),\ h_{2}(x):=e^{-\frac{i}{2}\gamma(x)\tau_{0}}\in U(1),$ (3.4)

where $\gamma\in C^{1}(\mathbb{R}^{3},\mathbb{R})$.

Continuing in the gauge $\Phi=(0,\varphi)$ and writing
$V=-\frac{i}{2}\tau_{a}V^{a}$101010Note that the lower indices $i,j,k$, as in
$A=A_{i}dx^{i}$, refer to vectorial components and run through $1,2$, while
the upper indices $a,b,c$, as in $V=-\frac{i}{2}\tau_{a}V^{a}$, refer to
$U(2)$-algebra components. and $X=-\frac{i}{2}\tau_{0}X^{0}$, where $X^{0}$
and $V^{a},a=1,2,3,$ are real fields (since $V$ takes values in $su(2)$ and
therefore $V^{*}=V$), we pass to the new fields corresponding to the broken
and unbroken generators, $\tau_{3}-\tau_{0}$ and $\tau_{3}+\tau_{0}$,
respectively:

$\displaystyle Z=V^{3}\cos\theta-X^{0}\sin\theta{\hbox{\quad and
\quad}}A=V^{3}\sin\theta+X^{0}\cos\theta,$ (3.5)

where, recall, $\theta$ is Weinberg’s angle, defined by
$\tan\theta=g^{\prime}/g$. Note that $Z$ and $A$ are real fields Moreover, it
is convenient to pass from the remaining two components, $V^{1},V^{2}$, of $V$
to a single complex field

$\displaystyle W=\frac{1}{\sqrt{2}}(V^{1}-iV^{2}).$ (3.6)

The gauge invariance of the original field equations with the unbroken gauge
symmetry given by transformations (2) with (3.4) leads to the invariance under
following gauge transformations:

$\displaystyle\tilde{T}^{gauge}_{\gamma}:(W,A,Z,\varphi)\mapsto(e^{i\gamma}W,A-\frac{1}{e}d\gamma,Z,\varphi),$
(3.7)

for $\gamma\in C^{1}(\mathbb{R}^{3},\mathbb{R})$, where, as usual,
$e^{i\gamma}W=\sum e^{i\gamma}W_{i}dx^{i}$ for $W=\sum W_{i}dx^{i}$,
$e:=g\sin\theta\
(=g^{\prime}\cos\theta=\frac{gg^{\prime}}{\sqrt{g^{2}+{g^{\prime}}^{2}}})$ is
the electron charge. Here, we replaced $\Phi:=(0,\varphi)$ by $\varphi$.

The WS energy in terms of $W,Z,A$ and $\varphi$ fields in 3D is given in
(D.1), Appendix D.1. The WS equations in terms of $W,Z,A$ and $\varphi$ in 3D
can be found by taking variational derivatives of this energy w.r.to different
fields.

In terms of $W,Z,A$ and $\varphi$ fields, the vacua (2.21) of the Weinberg-
Salam model become (up to a gauge symmetry):

$(0,A^{b}(x),0,\varphi_{0}),$ (3.8)

where, recall, $A^{b}(x)$ is a magnetic potential for the constant magnetic
field of strength $b$ in the $x^{3}$-direction, $dA^{b}(x)=bdx_{1}\wedge
dx_{2}$, and $\varphi_{0}$ is a positive constant from (2.6). We choose the
gauge so that $A^{b}(x)$ is of the form

$A^{b}(x)=\frac{b}{2}(-x_{2}dx_{1}+x_{1}dx_{2}).$ (3.9)

We will show that for a large magnetic field $b$, these homogeneous vacua
become unstable and new, inhomogeneous vacua emerge from them. This is a
bifurcation problem from the branch of gauge-translationally invariant
(homogeneous) solutions, (3.8).

Since we consider the WS system with the fields independent of the third
dimension $x^{3}$, i.e. in $\mathbb{R}^{2}$, we can choose the gauge with
$V_{3}=X_{3}=0$ (and hence $W_{3}=A_{3}=Z_{3}=0$).

Also, we will work in a fixed coordinate system, $\\{x^{i}\\}_{i=1}^{2}$ and
write the fields as $W=W_{i}dx^{i},$ $Z=Z_{i}dx^{i}$ and $A=A_{i}dx^{i}$. For
ease of comparing our arguments with earlier results, and given that we use
the standard Euclidean metric in $\mathbb{R}^{2}$, we identify (complex) one-
forms $W,Z$ and $A$ with the (complex) vector fields
$(W_{1},W_{2}),(Z_{1},Z_{2})$ and $(A_{1},A_{2})$. With this, we show in
Appendix D.2 that in this case, WS energy functional (2.6) can be written as

$\displaystyle E_{\Omega}^{WS}$
$\displaystyle(W,A,Z,\varphi)=\int_{\Omega}\big{[}|\operatorname{curl}_{gV^{3}}W|^{2}+\frac{1}{2}|\operatorname{curl}Z|^{2}+\frac{1}{2}|\operatorname{curl}A|^{2}$
$\displaystyle+\frac{1}{2}g^{2}\varphi^{2}|W|^{2}+\frac{1}{2}\kappa
g^{2}\varphi^{2}|Z|^{2}+\frac{g^{2}}{2}|\overline{W}\times W|^{2}$
$\displaystyle+ig(\operatorname{curl}V^{3})\overline{W}\times
W+|\nabla\varphi|^{2}+\frac{1}{2}\lambda(\varphi^{2}-\varphi_{0}^{2})^{2}\big{]},$
(3.10)

where $\kappa:=\frac{g^{2}}{2\cos^{2}\theta}$,
$\operatorname{curl}_{U}W:=(\nabla_{U})_{1}W_{2}-(\nabla_{U})_{2}W_{1}$,
$(\nabla_{U})_{i}:=\partial_{i}-iU_{i}$, $\partial_{i}\equiv\partial_{x^{i}}$
(for a $\mathfrak{u}(1)-$valued vector-field $U$),
$\xi\times\eta:=\xi_{1}\eta_{2}-\xi_{2}\eta_{1}$ and
$\operatorname{curl}V^{3}:=\partial_{1}V^{3}_{2}-\partial_{2}V^{3}_{1}$. It
follows from (3.5) that $V^{3}=Z\cos\theta+A\sin\theta$.

Expanding (3) in $\varphi$ around $\varphi_{0}$, we see that the $W$, $Z$ and
$\phi$ (Higgs) fields have the masses $M_{W}:=\frac{1}{\sqrt{2}}g\varphi_{0}$,
$M_{Z}:=\frac{1}{\sqrt{2}\cos\theta}g\varphi_{0}$ and
$M_{H}=\sqrt{2}\lambda\varphi_{0}$, respectively.

Using the relation $\xi\times\eta=J\xi\cdot\eta$, where $J$ is the symplectic
matrix,

$\displaystyle J:=\begin{pmatrix}0&-1\\\ 1&0\end{pmatrix},$ (3.11)

we find the Euler-Lagrange equations for (3), which give the WS system (2.9) -
(2.10) in 2D in terms of the fields $W$, $A$, $Z$ and $\varphi$

$\displaystyle[\operatorname{curl}_{gV^{3}}^{*}\operatorname{curl}_{gV^{3}}+\frac{g^{2}}{2}\varphi^{2}-ig(\operatorname{curl}V^{3})J+g^{2}(\overline{W}\times
W)J]W=0,$ (3.12)
$\displaystyle\operatorname{curl}^{*}\operatorname{curl}A+2e\operatorname{Im}[(\operatorname{curl}_{gV^{3}}W)J\overline{W}-\operatorname{curl}^{*}(\overline{W}_{1}W_{2})]=0,$
(3.13)
$\displaystyle[\operatorname{curl}^{*}\operatorname{curl}\\!+\kappa\varphi^{2}]Z+2g\cos\theta\operatorname{Im}[(\operatorname{curl}_{gV^{3}}W)J\overline{W}-\operatorname{curl}^{*}(\overline{W}_{1}W_{2})]=0,$
(3.14)
$\displaystyle[-\Delta+\lambda(\varphi^{2}-\varphi_{0}^{2})+\frac{g^{2}}{2}|W|^{2}+\frac{1}{2}\kappa|Z|^{2}]\varphi=0,$
(3.15)

where, recall, $\kappa=\frac{g^{2}}{2\cos^{2}\theta}$,
$V^{3}=Z\cos\theta+A\sin\theta$ and $\Delta$ is the standard Laplacian. (For a
derivation of (3.12) - (3.15) from (3), see Appendix D.2 and also [25, 40].)
Of course, (3.12) - (3.15) can also be derived directly from WS system (2.9) -
(2.10).

In terms of the $(W,A,Z,\varphi)$ fields, the lattice gauge - periodicity
(2.22) is expressed as

$(\tilde{T}^{gauge}_{\gamma_{s}})^{-1}T^{trans}_{s}(W,A,Z,\varphi)=(W,A,Z,\varphi),$
(3.16)

for all $s\in\mathcal{L},$ where $\gamma_{s}\in
C^{1}(\mathbb{R}^{2},\mathbb{R})$ for all $s\in\mathcal{L}$,
$\tilde{T}^{gauge}_{\gamma}$ given in (3.7) and $T^{trans}_{s}$ is the group
of translations, $T^{trans}_{s}f(x)=f(x+s)$. We say that $(W,A,Z,\varphi)$
satisfying (3.16) is an $\mathcal{L}$_-equivariant_ state. By evaluating the
effect of translation by $s+t$ in two different ways, we see that the family
of functions $\gamma_{s}$ has the co-cycle property111111A function
$\gamma_{s}:\mathcal{L}\times\mathbb{R}^{2}\rightarrow G$ satisfying the co-
cycle property (3.17) is called the automorphy exponent and $e^{i\gamma_{s}}$,
the automorphy factor.

$\displaystyle\gamma_{s+t}(x)-\gamma_{s}(x+t)-\gamma_{t}(x)\in
2\pi\mathbb{Z},\ \quad\forall s,t\in\mathcal{L}.$ (3.17)

Since $T^{trans}_{s}$ is an Abelian group, the co-cycle condition (3.17)
implies that, for any basis $\\{j_{1},j_{2}\\}$ in $\mathcal{L}$, the quantity

$c(\gamma_{s})=\frac{1}{2\pi}(\gamma_{j_{2}}(x+j_{1})+\gamma_{j_{1}}(x)-\gamma_{j_{1}}(x+j_{2})-\gamma_{j_{2}}(x))$
(3.18)

is independent of $x$ and of the choice of the basis $\\{j_{1},j_{2}\\}$, and
is an integer. This topological invariant is equal to the degree of the
corresponding line bundle.

One can show using Stokes’ Theorem, for any $A$ satisfying (3.16) - (3.18),
the magnetic flux through any fundamental domain $\Omega$ of the lattice
$\mathcal{L}$ is quantized:

$\frac{e}{2\pi}\int_{\Omega}dA=n,$ (3.19)

where $e$ is defined after (3.7) and $n=c(\gamma_{s})\in\mathbb{Z}$ defined in
(3.18). The left-hand side of (3.19) is called the _Chern number_ of the line
bundle corresponding to $\gamma_{s}$. (We note that $n$ is independent of the
choice of $\Omega$.)

The vacuum state (3.8) is $\mathcal{L}$-equivariant if and only if the
magnetic field $b$ is given by the relation

$b=\frac{2\pi}{e|\mathcal{L}|}n,$ (3.20)

where, by definition, $|\mathcal{L}|=|\Omega|$ for any fundamental cell
$\Omega$. In particular, $b$ is quantized. For such $b$, the vector field
$\frac{1}{e}A^{b}$ satisfies (3.19).

Furthermore, due to the reflection symmetry of the problem, we may assume that
$b\geq 0$. Clearly, we have:

######  Lemma 3.1.

Equations (2.9) - (2.10) for $\mathcal{L}$_-equivariant_ fields (2.22) in the
gauge $\Phi=(0,\varphi)$ are equivalent to Equations (3.12) - (3.15) for
$\mathcal{L}$_-equivariant_ fields (3.16), with the equivalence realized by
the transformation (3.5) - (3.6).

Finally, we use the invariance of (3.12) - (3.15) under the gauge
transformation (3.7) to choose a convenient gauge for the fields $W(x)$ and
$A(x)$. We say that the fields $(W,A,Z,\varphi)$ and
$(W^{\prime},A^{\prime},Z^{\prime},\varphi^{\prime})$ are _gauge-equivalent_
if there is $\gamma\in C^{1}(\mathbb{R}^{2},\mathbb{R})$ such that

$(W^{\prime},A^{\prime},Z^{\prime},\varphi^{\prime})=\tilde{T}^{gauge}_{\gamma}(W,A,Z,\varphi).$

Clearly, if $(W,A,Z,\varphi)$ and
$(W^{\prime},A^{\prime},Z^{\prime},\varphi^{\prime})$ are gauge-equivalent,
then $(W,A,Z,\varphi)$ solves (3.12) - (3.15) if and only if
$(W^{\prime},A^{\prime},Z^{\prime},\varphi^{\prime})$ solves (3.12) - (3.15).
The following proposition was first used in [29] and proven in [42] (an
alternate proof is given in Appendix A of [43]):

######  Proposition 3.2.

Let $(W^{\prime},A^{\prime},Z^{\prime},\varphi^{\prime})$ be an
$\mathcal{L}$-equivariant state and let $b$ be given by (3.20). Then there is
a $\mathcal{L}$-equivariant state $(W,A,Z,\varphi)$, gauge-equivalent to
$(W^{\prime},A^{\prime},Z^{\prime},\varphi^{\prime})$, which satisfies (3.16),
with $\chi_{s}(x)=\frac{eb}{2}s\wedge x+k_{s}$, i.e. such that

$\displaystyle W(x+s)=e^{i(\frac{eb}{2}s\wedge x+k_{s})}W(x),$ (3.21)
$\displaystyle A(x+s)=A(x)+\frac{b}{2}Js\ \forall s\in\mathcal{L},$ (3.22)
$\displaystyle\operatorname{div}A=0.$ (3.23)

Here $k_{s}$ satisfies the condition $k_{s+t}-k_{s}-k_{t}-\frac{eb}{2}s\wedge
t\in 2\pi\mathbb{Z}$, for all $s,t\in\mathcal{L}$, the matrix $J$ is given in
(3.11).

This gauge is consistent with the gauge chosen for the homogeneous vacua
(3.9).

Our goal is to prove the instability of the vacuum state (3.8) and the
existence of $\mathcal{L}-$equivariant (in the sense of (3.16)) solutions to
transformed WS system (3.12) - (3.15) having the properties described in
Theorems 2.2 and 2.3.

##  4 Rescaling

In this section, we rescale transformed WS system (3.12) - (3.15) to keep the
lattice size fixed. Specifically, we define the rescaled fields $(w,z,a,\phi)$
to be

$\displaystyle(w(x),a(x),z(x),\phi(x))$
$\displaystyle:=(rW(rx),rA(rx),rZ(rx),r\varphi(rx)),$ (4.1) $\displaystyle r$
$\displaystyle:=\sqrt{\frac{n}{eb}}=\sqrt{\frac{|\Omega|}{2\pi}}.$ (4.2)

where in the second equality (4.2), we used (3.20). Clearly,
$(W(x),A(x),Z(x),\varphi(x))$ is $\mathcal{L}$-equivariant if and only if
$(w(x),a(x),z(x),\phi(x))$ is $\mathcal{L}^{\prime}$-equivariant, where
$\mathcal{L}^{\prime}:=\frac{1}{r}\mathcal{L}$. Now, the size of a fundamental
domain, $\Omega^{\prime}$, of the rescaled lattice $\mathcal{L}^{\prime}$ is
fixed as $|\Omega^{\prime}|=2\pi$.

Plugging the rescaled fields into (3.12) - (3.15) gives the rescaled Weinberg-
Salem equations:

$\displaystyle[\operatorname{curl}_{\nu}^{*}\operatorname{curl}_{\nu}+\frac{g^{2}}{2}\phi^{2}-i(\operatorname{curl}\nu)J+g^{2}(\overline{w}\times
w)J]w=0,$ (4.3)
$\displaystyle\operatorname{curl}^{*}\operatorname{curl}a+2e\operatorname{Im}[(\operatorname{curl}_{\nu}w)J\overline{w}-\operatorname{curl}^{*}(\overline{w}_{1}w_{2})]=0,$
(4.4)
$\displaystyle[\operatorname{curl}^{*}\operatorname{curl}+\kappa\phi^{2}]z+2g\cos\theta\operatorname{Im}[(\operatorname{curl}_{\nu}w)J\overline{w}-\operatorname{curl}^{*}(\overline{w}_{1}w_{2})]=0,$
(4.5)
$\displaystyle[-\Delta+\lambda(\phi^{2}-\xi^{2})+\frac{g^{2}}{2}|w|^{2}+\frac{1}{2}\kappa|z|^{2}]\phi=0,$
(4.6)

where $\xi:=r\varphi_{0}$ (with $r$ given in (4.2)),
$\nu:=g(a\sin\theta+z\cos\theta)$ and, recall,
$\operatorname{curl}_{q}w=\nabla_{1}w_{2}-\nabla_{2}w_{1}$,
$\nabla_{i}:=\partial_{i}-iq_{i}$, $\partial_{i}\equiv\partial_{x^{i}}$ (for a
$\mathfrak{u}(1)-$valued vector-field $iq$) and, recall, $\overline{w}\times
w:=\overline{w}_{1}w_{2}-\overline{w}_{2}w_{1}$. We define the rescaled energy
by

$\mathcal{E}_{\Omega^{\prime}}(w,a,z,\phi;r):=r^{2}E_{\Omega}^{WS}(W,A,Z,\varphi).$
(4.7)

with $(W,A,Z,\varphi)$ related to $(w,a,z,\phi)$ by (4.1) and
$E_{\Omega}^{WS}(W,A,Z,\varphi)$ given in (3). Explicitly, we have

$\displaystyle\mathcal{E}_{\Omega^{\prime}}(w,a,z,\phi;r)$
$\displaystyle=\int_{\Omega^{\prime}}\big{(}|\operatorname{curl}_{\nu}w|^{2}+\frac{1}{2}|\operatorname{curl}a|^{2}+\frac{1}{2}|\operatorname{curl}z|^{2}$
$\displaystyle+\frac{1}{2}g^{2}\phi^{2}|w|^{2}+\frac{1}{2}\kappa\phi^{2}|z|^{2}+\frac{g^{2}}{2}|\overline{w}\times
w|^{2}$ $\displaystyle+i(\operatorname{curl}\nu)\overline{w}\times
w+|\nabla\phi|^{2}+\frac{1}{2}\lambda(\phi^{2}-\xi^{2})^{2}\big{)}.$ (4.8)

We note that after rescaling, the average magnetic flux per fundamental domain
becomes $n/e$ and the vacuum solution (3.8),

$\displaystyle m^{n,r}:=(0,\frac{1}{e}a^{n},0,\xi),$ (4.9)

where $a^{n}(x)\equiv A^{n}(x)=\frac{n}{2}Jx$, . Furthermore, (3.16) and
Proposition 3.2 imply that $(w,a,z,\phi)$ satisfy

$\displaystyle w(x+s)=e^{i(\frac{n}{2}s\times x+c_{s})}w(x)\text{ for all
}s\in\mathcal{L}^{\prime},$ (4.10) $\displaystyle
a(x+s)=a(x)+\frac{n}{2e}Js\text{ for all }s\in\mathcal{L}^{\prime},$ (4.11)
$\displaystyle\operatorname{div}a=0,$ (4.12) $\displaystyle
z(x+s)=z(x),\quad\phi(x+s)=\phi(x)\text{ for all }s\in\mathcal{L}^{\prime},$
(4.13)

where $c_{s}$ satisfies the condition $c_{s+t}-c_{s}-c_{t}-\frac{n}{2}s\times
t\in 2\pi\mathbb{Z}$, for all $s,t\in\mathcal{L}^{\prime}$.

##  5 The linearized problem

In this section we prove Theorem 2.1, describing the stability properties of
the vacuum (3.8). Equivalently, we will investigate the energetic stability of
the rescaled vacuum solution (4.9) of the rescaled WS equations (4.3) - (4.6).

Let $m:=(w,a,z,\phi)$ and denote by $G(b,m)\equiv G(m)$ the map given by the
left-hand side of (4.3) - (4.6), written explicitly as

$\displaystyle G(b,m)\equiv G(m)=(G_{1}(m),\dots,G_{4}(m)),$ (5.1)
$\displaystyle
G_{1}(m):=[\operatorname{curl}_{\nu}^{*}\operatorname{curl}_{\nu}+\frac{g^{2}}{2}\phi^{2}-i(\operatorname{curl}\nu)J+g^{2}(\overline{w}\times
w)J]w,$ (5.2) $\displaystyle
G_{2}(m):=\operatorname{curl}^{*}\operatorname{curl}a+2e\operatorname{Im}[(\operatorname{curl}_{\nu}w)J\overline{w}-\operatorname{curl}^{*}(\overline{w}_{1}w_{2})],$
(5.3) $\displaystyle
G_{3}(m):=[\operatorname{curl}^{*}\operatorname{curl}+\kappa\phi^{2}]z+2g\cos\theta\operatorname{Im}[(\operatorname{curl}_{\nu}w)J\overline{w}-\operatorname{curl}^{*}(\overline{w}_{1}w_{2})],$
(5.4) $\displaystyle
G_{4}(m):=[-\Delta+\lambda(\phi^{2}-\xi^{2})+\frac{g^{2}}{2}|w|^{2}+\frac{1}{2}\kappa|z|^{2}]\phi,$
(5.5)

where, recall, $J$ is the symplectic matrix given in (3.11),
$\xi:=r\varphi_{0}$ (with $r$ given in (4.2)),
$\nu:=g(a\sin\theta+z\cos\theta)$, $\Delta$ is the standard Laplacian and the
parameter $b$ enters through periodicity conditions (4.10) - (4.13). Now, the
WS system can be written as

$\displaystyle G(m)=0.$ (5.6)

Recall the definition of stability given above Eq. (2.23). To apply it to the
rescaled WS equations (4.3) - (4.6), we observe that the map $G$ is the
$L^{2}$-gradient, $\operatorname{grad}_{L^{2}}\mathcal{E}_{\Omega^{\prime}}$,
of the energy $\mathcal{E}_{\Omega^{\prime}}$, see (4), considered as a
functional of $u=(w,a,z,\phi)$. Hence the $L^{2}$-Hessian for
$\mathcal{E}_{\Omega^{\prime}}$ and $m$ is the formally symmetric operator

$\mathcal{E}_{\Omega^{\prime}}^{\prime\prime}(m):=\delta\operatorname{grad}_{L^{2}}\mathcal{E}_{\Omega^{\prime}}(m)=\delta
G(m),$

where $\delta$ is the Gâteau derivative, defined on the space of variations
$\mathcal{Y}$ (see below).

Denote the $L^{2}$-Hessian at the vacuum solution $m^{n,r}$ (see (4.9)) by

$L_{n,\mu}:=\delta G(m^{n,r}).$

As seen from its explicit form given below, the operator $L_{n,\mu}$, acting
on the space $\mathcal{Y}$, is self-adjoint and therefore its spectrum is
real.

Thus, applied to the rescaled WS equations (4.3) - (4.6), the definition of
stability can be rephrased as:

the vacuum solution $m^{n,r}$ is energetically stable (respectively, unstable)
if and only if $\inf\operatorname{spec}(L_{n,\mu})\geq 0$ (respectively,
$\inf\operatorname{spec}(L_{n,\mu})<0$).

We consider the operator $L_{n,\mu}$ on the space
$\mathcal{Y}:=L_{n}^{2}\times L^{2}_{0}\times L^{2}_{0}\times L^{2}$ tangent
to the space of $L^{2}_{\rm loc}$ functions of the form $(w,a,z,\phi)$
satisfying the gauge - periodicity conditions (4.10) - (4.13), with the domain

$\mathcal{X}:=\mathcal{H}_{n}^{2}\times\mathcal{H}^{2}_{0}\times\mathcal{H}^{2}_{0}\times\mathcal{H}^{2},$
(5.7)

where $\mathcal{H}_{n}^{s}$, $\mathcal{H}_{0}^{s}$ and $\mathcal{H}^{s}$ are
the respective Sobolev spaces for the $L^{2}$-spaces

$\displaystyle L_{n}^{2}:=\\{w\in
L^{2}_{loc}(\mathbb{R}^{2},\mathbb{C}^{2}):w(x+s)=e^{i(\frac{n}{2}s\times
x+c_{s})}w(x)\ \forall s\in\mathcal{L}^{\prime}\\},$ (5.8) $\displaystyle
L_{0}^{2}:=\\{\alpha\in
L^{2}_{loc}(\mathbb{R}^{2},\mathbb{R}^{2}):\alpha(x+s)=\alpha(x)\ \forall
s\in\mathcal{L}^{\prime},\;\operatorname{div}\alpha=0\\},$ (5.9)
$\displaystyle L^{2}:=\\{\psi\in
L^{2}_{loc}(\mathbb{R}^{2},\mathbb{R}):\psi(x+s)=\psi(x)\ \forall
s\in\mathcal{L}^{\prime}\\},$ (5.10)

(see (4.10) - (4.12)), with inner products given (for $s\in\mathbb{Z}_{\geq
0}$) by

$\displaystyle\langle
w,w^{\prime}\rangle_{\mathcal{H}_{n}^{s}}:=\frac{1}{|\Omega^{\prime}|}\sum\limits_{i=1}^{2}\sum\limits_{|\gamma|\leq
s}\int_{\Omega^{\prime}}\overline{(\nabla_{a^{n}})^{\gamma}w_{i}}(\nabla_{a^{n}})^{\gamma}w^{\prime}_{i},$
(5.11) $\displaystyle\langle
a,a^{\prime}\rangle_{\mathcal{H}^{s}_{0}}:=\frac{1}{|\Omega^{\prime}|}\sum\limits_{i=1}^{2}\sum\limits_{|\gamma|\leq
s}\int_{\Omega^{\prime}}\partial^{\gamma}a_{i}\partial^{\gamma}a^{\prime}_{i},$
(5.12)
$\displaystyle\langle\psi,\psi^{\prime}\rangle_{\mathcal{H}^{s}}:=\frac{1}{|\Omega^{\prime}|}\sum\limits_{|\gamma|\leq
s}\int_{\Omega^{\prime}}\partial^{\gamma}\psi\partial^{\gamma}\psi^{\prime},$
(5.13)

where $w^{\\#}=(w_{1}^{\\#},w_{2}^{\\#}),a^{\\#}=(a_{1}^{\\#},a_{2}^{\\#})$,
$\Omega^{\prime}$ is an arbitrary fundamental domain of the lattice
$\mathcal{L}^{\prime}$ and $\gamma$ is a multi-index. The
$\mathcal{L}^{\prime}$-equivariance of the above functions implies that these
inner products do not depend on the choice of fundamental domain
$\Omega^{\prime}$.

We compute the linear operator $L_{n,\mu}$ explicitly. In what follows we use
the notation $\oplus_{j}A_{j}$ for diagonal operator-matrices with the
operators $A_{j}$ on the diagonal. Passing from the parameter
$\xi=r\varphi_{0}$, or $r$, to the parameter $\mu:=g^{2}\xi^{2}/2$ and using
that $\nu\big{|}_{a=a^{n}/e,z=0}=\frac{1}{e}a^{n}g\sin\theta=a^{n}$, we find

$\displaystyle L_{n,\mu}=\oplus_{j=1}^{4}H_{j},$ (5.14) $\displaystyle
H_{1}(\mu):=\operatorname{curl}_{a^{n}}^{*}\operatorname{curl}_{a^{n}}+\mu-
niJ,$ (5.15) $\displaystyle
H_{2}(\mu):=\operatorname{curl}^{*}\operatorname{curl},$ (5.16) $\displaystyle
H_{3}(\mu):=\operatorname{curl}^{*}\operatorname{curl}+\frac{\mu}{\cos^{2}\theta},$
(5.17) $\displaystyle H_{4}(\mu):=-\Delta+\frac{4\lambda\mu}{g^{2}},$ (5.18)

where, recall,
$\operatorname{curl}_{q}w=(\nabla_{q})_{1}w_{2}-(\nabla_{q})_{2}w_{1}$,
$(\nabla_{q})_{i}:=\partial_{i}-iq_{i}$, $\partial_{i}\equiv\partial_{x^{i}}$.
(Note that the matrix $iJ$ is self-adjoint.)

The gauge invariance of Eq. (5.6) and the partial symmetry breaking of vacuum
solution (4.9) imply that $L_{n,\mu=n}$ has the gauge zero mode:

$\displaystyle L_{n,\mu=n}G_{f}=0,\ G_{f}:=(0,\nabla f,0,0).$ (5.19)

For a null vector $G_{f}$ defined in (5.19) to be in $\mathcal{X}$, $f$ must
satisfy $\operatorname{div}(\nabla f)=-\Delta f=0$. This implies that $f$ is a
linear function, $f(x)=c\cdot x+d$ for some $c\in\mathbb{R}^{2}$ and
$d\in\mathbb{R}$, and so

$\displaystyle G_{f}\in\mathcal{X}\Longrightarrow G_{f}=(0,c,0,0).$ (5.20)

In this section we shall prove the following result implying Theorem 2.1:

######  Theorem 5.1.

The operator $L_{n,\mu}$ on the space $\mathcal{X}$ has purely discrete
spectrum. For $\mu\neq n$, $L_{\mu,n}$ has the multiplicity $2$ eigenvalue $0$
with the eigenfuctions $(0,e_{i},0,0)$, $i=1,2,e_{1}=(1,0),e_{2}=(0,1)$ (see
(5.20)).

Furthermore, the smallest non-zero eigenvalue given by $\mu-n$, having
multiplicity $n$. For $\mu=n$, the eigenvalue $0$ has the multiplicity $n+2$.

Theorem 5.1 follows from Propositions 5.2 and 5.3 given below. $\Box$

######  Proposition 5.2.

The operators $H_{2}(\mu)$, $H_{3}(\mu)$ and $H_{4}(\mu)$ have purely discrete
spectra. Furthermore, $H_{3}(\mu)$ and $H_{4}(\mu)$ are strictly positive and
$H_{2}(\mu)$ is non-negative and has the null space
$\\{(0,c,0,0):c\in\mathbb{R}^{2}\\}$ of dimension $2$.

###### Proof.

The strict positivity of $H_{3}(\mu)$ and $H_{4}(\mu)$ and the non-negativity
of $H_{2}(\mu)$ are obvious. The discreteness of the spectra and the form of
the null space of $H_{2}(\mu)$ follow from the discreteness of the spectrum of
the Laplacian on compact domains and the identity
$\operatorname{curl}^{*}\operatorname{curl}v=-\Delta v$ when
$\operatorname{div}(v)=0$. To compute the null space of $H_{2}(\mu)$, we
observe that the solutions of the equations $\Delta v=0$ and
$\operatorname{div}(v)=0$ are constant vectors in $\mathbb{R}^{2}$. ∎

Let $\nabla_{q}:=\nabla-iq=((\nabla_{q})_{1},(\nabla_{q})_{2})$,
$(\nabla_{q})_{j}:=\partial_{j}-iq_{j}$, and
$\Delta_{q}:=\nabla_{q}^{2}=-\nabla_{q}^{*}\nabla_{q}$. We also introduce the
complexified covariant derivative
$\bar{\partial}_{q}:=(\nabla_{q})_{1}+i(\nabla_{q})_{2}.$ We have

######  Proposition 5.3.

(i) $H_{1}(\mu)$ is a self-adjoint operator on $\mathcal{H}^{2}_{n}$ and its
spectrum is given by

$\sigma(H_{1}(\mu))=\\{(m-1)n+\mu:m\in\mathbb{Z}_{\geq 0}\\}\cup\\{\mu\\},$
(5.21)

where $n:=eb|\mathcal{L}|/2\pi$.

(ii) The eigenspace of the eigenvalue $-n+\mu$ is $n$-dimensional and is
spanned by functions of the form

$\chi=(\beta,i\beta),\qquad\operatorname{curl}_{a^{n}}\chi=i\bar{\partial}_{a^{n}}\beta=0,$
(5.22)

and the eigenspace of the eigenvalue $\mu$ is of the form

$\operatorname{Null}{(H_{1}(\mu)-\mu)}=\\{\nabla_{a^{n}}f:f\in\mathcal{H}^{3}_{n}\\}.$
(5.23)

In the proof of this proposition, we use the following standard result whose
proof, for reader’s convenience, is given in Appendix H:

######  Proposition 5.4.

The operator $-\Delta_{a^{n}}$ is self-adjoint on its natural domain and its
spectrum is given by

$\sigma(-\Delta_{a^{n}})=\\{\,(m+1)n:m\in\mathbb{Z}_{\geq 0}\,\\},$ (5.24)

with each eigenvalue is of the multiplicity $n$. Moreover,

$\operatorname{Null}(-\Delta_{a^{n}}-n)=\operatorname{Null}\bar{\partial}_{a^{n}}.$
(5.25)

In more detail, we have

$\displaystyle\operatorname{Null}(-\Delta_{a^{n}}-n)=e^{\frac{in}{2}x^{2}(x^{1}+ix^{2})}V_{n},$
(5.26)

where $V_{n}$ is spanned by functions of the form (below
$z=(x^{1}+ix^{2})/\sqrt{\frac{2\pi}{\operatorname{Im}\tau}}$)

$\displaystyle\theta(z,\tau):=\sum_{m=-\infty}^{\infty}c_{m}e^{i2\pi mz},\
c_{m+n}=e^{-in\pi z}e^{i2m\pi\tau}c_{m}.$ (5.27)

Such functions are determined entirely by the values of $c_{0},\ldots,c_{n-1}$
and therefore form an $n$-dimensional vector space.

###### Proof of Proposition 5.3.

First, we will show that $\mathcal{H}^{2}_{n}=\mathcal{Y}\oplus\mathcal{Z}$
(the Hodge decomposition), where

$\displaystyle\mathcal{Y}:=$
$\displaystyle\\{w\in\mathcal{H}^{2}_{n}:\operatorname{div}_{a^{n}}w=0\\},$
(5.28) $\displaystyle\mathcal{Z}:=$
$\displaystyle\\{w\in\mathcal{H}^{2}_{n}:w=\nabla_{a^{n}}f\text{ for some
}f\in\mathcal{H}^{3}_{n}\\},$ (5.29)

with
$\operatorname{div}_{a^{n}}w:=(\nabla_{a^{n}})_{1}w_{1}+(\nabla_{a^{n}})_{2}w_{2}=-\nabla_{a^{n}}^{*}$.
We write any $w\in\mathcal{H}^{2}_{n}$ as $w=w_{0}+\nabla_{a^{n}}f$, where $f$
solves the equation $\Delta_{a^{n}}f=\operatorname{div}_{a^{n}}w$ and $w_{0}$
is defined by this relation. By Proposition 5.4, $0$ is not in the spectrum of
$\Delta_{a^{n}}$ and therefore the equation
$\Delta_{a^{n}}f=\operatorname{div}_{a^{n}}w$ has the unique solution
$f\in\mathcal{H}^{3}_{n}$. Then, since
$\Delta_{a^{n}}:=\operatorname{div}_{a^{n}}\nabla_{a^{n}}$, we have
$\operatorname{div}_{a^{n}}w_{0}=0$. This proves
$\mathcal{H}^{2}_{n}=\mathcal{Y}\oplus\mathcal{Z}$.

Now, recall that the operator $H_{1}(\mu)$ acts on complex vectors
$w=(w_{1},w_{2})$. The definition
$H_{1}(\mu):=\operatorname{curl}_{a^{n}}^{*}\operatorname{curl}_{a^{n}}-niJ+\mu$
and the relations $\operatorname{curl}_{a^{n}}^{*}=-J\nabla_{a^{n}}$ and

$\operatorname{curl}_{a^{n}}\nabla_{a^{n}}=[(\nabla_{a^{n}})_{1},(\nabla_{a^{n}})_{2}]=-in$

yield that $(H_{1}(\mu)-\mu)\nabla_{a^{n}}f=0$, which proves that the
$\mu$-eigenspace of $H_{1}(\mu)$ is of the form (5.23) giving the second part
of (ii).

By the above the subspace $\mathcal{Y}$ is invariant under $H_{1}(\mu)$. To
compute the spectrum of the operator $H_{1}(\mu)$ on the subspace
$\mathcal{Y}$, we use the definitions of $\operatorname{curl}_{a^{n}}$ and
$\operatorname{curl}_{a^{n}}^{*}$ and recall the relation
$[(\nabla_{a^{n}})_{1},(\nabla_{a^{n}})_{2}]=-in$ to compute

$\operatorname{curl}_{a^{n}}^{*}\operatorname{curl}_{a^{n}}=-\Delta_{a^{n}}-niJ+\nabla_{a^{n}}\operatorname{div}_{a^{n}}.$

By above, we have $H_{1}(\mu)w_{0}=(-\Delta_{a^{n}}-2niJ-\mu)w_{0}$, for any
$w_{0}\in\mathcal{Y}$. (We check using
$\operatorname{div}_{a^{n}}(-\Delta_{a^{n}}-2niJ)w_{0}=(-\Delta_{a^{n}})\operatorname{div}_{a^{n}}w_{0}=0$,
that $H_{1}(\mu)$ sends $\mathcal{Y}$ to $\mathcal{Y}$ and hence,
$\mathcal{Y}$ is invariant under $H_{1}(\mu)$.) Thus, we conclude that

$\displaystyle H_{1}(\mu)$ $\displaystyle(w_{0}\oplus
0)=(h_{1}-\mu)w_{0}\oplus 0,$ (5.30) $\displaystyle
h_{1}:=-\Delta_{a^{n}}-2niJ.$ (5.31)

Identifying one-forms with vector-fields, we compute

$U^{*}(iJ)U=\begin{pmatrix}-1&0\\\
0&1\end{pmatrix},\;\;U:=\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\\
-i&i\end{pmatrix},$ (5.32)

which gives

$U^{*}h_{1}U=\left(\begin{array}[]{cc}-\Delta_{a^{n}}+2n&0\\\
0&-\Delta_{a^{n}}-2n\end{array}\right).$ (5.33)

By Proposition 5.4, we know that

$\sigma(-\Delta_{a^{n}})=\\{\,(m+1)n:m\in\mathbb{Z}_{\geq 0}\\}$ (5.34)

and so the spectrum of $H_{1}(\mu)$ on $\mathcal{Y}$ is given by the first set
on the r.h.s. of (5.21). Hence, by
$\mathcal{H}^{2}_{n}=\mathcal{Y}\oplus\mathcal{Z}$, (5.21) follows, giving
(i).

Furthermore, by (5.33) and (5.34), any eigenvector $\chi$ of $h_{1}$
corresponding to the eigenvalue $-n$ must be of the form

$\chi=U(0,\beta)=\frac{1}{\sqrt{2}}(\beta,i\beta),$ (5.35)

where $\beta$ satisfies

$-\Delta_{a^{n}}\beta=n\omega.$ (5.36)

This relation, together with the equation
$\operatorname{Null}(-\Delta_{a^{n}}-n)=\operatorname{Null}\bar{\partial}_{a^{n}}$
(see (5.25)), implies $\overline{\partial}_{a^{n}}\beta=0$. Since
$\operatorname{curl}_{a^{n}}\chi=i\overline{\partial}_{a^{n}}\beta$, this
gives

$\operatorname{curl}_{a^{n}}\chi=i\overline{\partial}_{a^{n}}\omega=0.$ (5.37)

Furthermore, by Proposition 5.4, the space of such functions is
$n$-dimensional. Thus (after rescaling $\omega$ by a factor of $\sqrt{2}$)
$\chi$ is of the form (5.22). This gives also the first part of (ii)
completing the proof of the proposition. ∎

We see that the operator $H_{1}(\mu)$ is non-negative for the magnetic fields
satisfying $b<b_{*}:=g^{2}\varphi_{0}^{2}/2e=M_{W}^{2}/e$ and acquires a
negative eigenvalue $\mu-n=(b_{*}/b-1)\,n$ of multiplicity $n$ as the magnetic
field increases to $b>b_{*}$. Theorem 2.1 follows by undoing the rescaling
(4.1) - (4.2).

##  6 Setup of the bifurcation problem

We substitute $a=\frac{1}{e}a^{n}+\alpha$ (with
$\operatorname{div}(\alpha)=0$), $\phi=\xi+\psi$, $\nu=a^{n}+\tilde{\nu}$ and
$\xi=\sqrt{2\mu}/g$ into (4.3) - (4.6) and relabel the unknowns
$w,\alpha,z,\psi$ as $u_{1},u_{2},u_{3},u_{4}$ to obtain the system

$\displaystyle H_{i}u_{i}$ $\displaystyle=-J_{i}(\mu,u),\ i=1,\dots,4,$ (6.1)

where $u=(u_{1},u_{2},u_{3},u_{4})\equiv(w,\alpha,z,\psi)$, the operators
$H_{i}$ on the left-hand side are defined in (5.15) - (5.18), and

$\displaystyle J_{1}(\mu,u)$
$\displaystyle:=Mw+\frac{g^{2}}{2}\psi^{2}w+g\sqrt{2\mu}\psi
w-i(\operatorname{curl}\tilde{\nu})Jw+g^{2}(\overline{w}\times w)Jw,$ (6.2)
$\displaystyle J_{2}(\mu,u)$
$\displaystyle:=2e\operatorname{Im}[(\operatorname{curl}_{\nu}w)J\overline{w}-\operatorname{curl}^{*}(\overline{w}_{1}w_{2})],$
(6.3) $\displaystyle J_{3}(\mu,u)$
$\displaystyle:=2g\cos\theta\operatorname{Im}[(\operatorname{curl}_{\nu}w)J\overline{w}-\operatorname{curl}^{*}(\overline{w}_{1}w_{2})]+\kappa\frac{2\sqrt{2\mu}}{g}\psi
z+\kappa\psi^{2}z,$ (6.4) $\displaystyle J_{4}(\mu,u)$
$\displaystyle:=3\lambda\frac{\sqrt{2\mu}}{g}\psi^{2}+\lambda\psi^{3}+\frac{g^{2}}{2}|w|^{2}(\frac{\sqrt{2\mu}}{g}+\psi)+\frac{1}{2}\kappa|z|^{2}(\frac{\sqrt{2\mu}}{g}+\psi),$
(6.5)

with $\tilde{\nu}:=g(\alpha\sin\theta+z\cos\theta)$,
$\xi\times\eta:=\xi_{1}\eta_{2}-\xi_{2}\eta_{1}$, recall,
$\operatorname{curl}_{q}w=\nabla_{1}w_{2}-\nabla_{2}w_{1}$,
$\nabla_{i}:=\partial_{i}-iq_{i}$ and, recalling that
$w:\mathbb{R}^{2}\rightarrow\mathbb{C}^{2}$,

$\displaystyle
M:=\operatorname{curl}_{\nu}^{*}\operatorname{curl}_{\nu}-\operatorname{curl}_{a^{n}}^{*}\operatorname{curl}_{a^{n}}=$
$\displaystyle\begin{pmatrix}M_{22}&-M_{21}\\\ -M_{12}&M_{11}\end{pmatrix},$
(6.6)

with
$M_{ij}:=i\tilde{\nu}_{i}(\nabla_{a^{n}})_{j}+i\tilde{\nu}_{j}(\nabla_{a^{n}})_{i}+i\partial_{i}\tilde{\nu}_{j}+\tilde{\nu}_{i}\nu_{j}$.

Note that system (6.1) can be also written as
$G(m^{n,r}+u)|_{\xi=\sqrt{2\mu}/g}=0$, where $G$ is defined in (5.1) and
$m^{n,r}:=(0,\frac{1}{e}a^{n},0,\xi)$.

Applying $\operatorname{div}$ to the second equation in (6.1), we find that a
solution $(\mu,u)$ should satisfy $\operatorname{div}J_{2}(\mu,u)=0$. To prove
that a solution $(\mu,u)$ satisfies this constraint, we consider the following
auxiliary problem

$\displaystyle F(\mu,u)=0,\ \text{ where }\
F(\mu,u):=L_{n,\mu}u+P^{\prime}J(\mu,u),$ (6.7)

where $P^{\prime}=\mathbf{1}\otimes P_{0}\otimes\mathbf{1}\otimes\mathbf{1}$,
with $P_{0}$ the orthogonal projection onto the divergence-free vector fields
($P_{0}=\frac{1}{-\Delta}\operatorname{curl}^{*}\operatorname{curl}$), and,
recall, $L_{n,\mu}=\oplus H_{i}$ and $J(\mu,u)$ given in (5.14) and

$\displaystyle J(\mu,u)$ $\displaystyle:=(J_{1}(\mu,u),\dots,J_{4}(\mu,u)).$
(6.8)

We consider $F(\mu,u)$ as a map from the space
$\mathbb{R}_{>0}\times\mathcal{X}$, where
$\mathcal{X}:=\mathcal{H}_{n}^{2}\oplus\mathcal{H}^{2}_{0}\oplus\mathcal{H}^{2}_{0}\oplus\mathcal{H}^{2}$,
to the space $\mathcal{Y}:=L_{n}^{2}\oplus L_{0}^{2}\oplus L_{0}^{2}\oplus
L^{2}$, and let $F=(F_{1},\dots,F_{4})$, where

$F_{i}(\mu,u)=H_{i}u+\delta_{i,2}P_{0}J_{i}(\mu,u),\quad i=1,\dots,4.$ (6.9)

In what follows, we denote the partial (real) Gâteaux derivatives with respect
to $\\#$ by $\delta_{\\#}$.

######  Proposition 6.1.

Assume $(\mu,u)$ is a solution of the system (6.7) satisfying the gauge -
periodicity conditions (4.10) - (4.13). Then $\operatorname{div}J(\mu,u)=0$
and therefore $(\mu,u)$ solves the original system (6.1).

###### Proof.

We follow [43]. Assume $\chi\in H^{1}_{\rm loc}$ and is $\mathcal{L}-$periodic
(we say, $\chi\in H^{1}_{\rm per}$). The gauge invariance implies that

$E_{\Omega^{\prime}}(e^{is\chi}w,a+s\nabla\chi,z,\phi)=E_{\Omega^{\prime}}(w,a,z,\phi),$
(6.10)

where $E_{\Omega^{\prime}}(w,a,z,\phi)$ is given in (4). Differentiating this
equation with respect to $s$ at $s=0$ gives
$\delta_{w}E_{\Omega^{\prime}}(w,a,z,\phi)(i\chi
w)+\delta_{a}E_{\Omega^{\prime}}(w,a,z,\phi)(\nabla\chi)=0$. Now, we use the
fact that the partial Gâteaux derivative with respect to $w$ vanishes,
$\delta_{w}E_{\Omega^{\prime}}(w,a,z,\phi)=0$, and that
$\operatorname{curl}\nabla\chi=0$, and integrate by parts, to obtain

$\displaystyle\langle J(\mu,u),\nabla\chi\rangle=0.$ (6.11)

(Due to conditions (4.10) - (4.13) and the $\mathcal{L}-$periodicity of
$\chi$, there are no boundary terms.) Since the last equation holds for any
$\chi\in H^{1}_{\rm per}$, we conclude that $\operatorname{div}J(\mu,u)=0$. ∎

In Sections 7 \- 8 we solve equation (6.7), subject to conditions (4.10) -
(4.13).

In conclusion of this section, we investigate properties of the map
$F(\mu,u)$. For $f=(f_{1},f_{2},f_{3},f_{4})$ and $\delta\in\mathbb{R}$,
define the global transformation

$T_{\delta}f=(e^{i\delta}f_{1},f_{2},f_{3},f_{4}).$ (6.12)

######  Proposition 6.2.

$F(\mu,u)$ has the following properties:

  1. (i)

$F:\mathbb{R}_{>0}\times\mathcal{X}\to\mathcal{Y}$ is continuously
differentiable of all orders;

  2. (ii)

$F(\mu,0)=0$ for all $\mu\in\mathbb{R}_{>0}$;

  3. (iii)

$\delta_{u}F(\mu,0)=L_{n,\mu}$ for all $\mu\in\mathbb{R}_{>0}$;

  4. (iv)

$F(\mu,T_{\delta}u)=T_{\delta}F(\mu,u)$ for all $\delta\in\mathbb{R}$;

  5. (v)

$\langle u,F(\mu,u)\rangle_{\mathcal{Y}}\in\mathbb{R}$ (respectively $\langle
w,F_{1}(\mu,u)\rangle_{L_{n}^{2}}\in\mathbb{R}$) for all $u\in\mathcal{X}$
(respectively $w\in\mathcal{H}_{n}^{2}$).

###### Proof.

$(i)$ follows because $F$ is a polynomial in the components of $u$ and their
first- and second-order (covariant) derivatives. $(ii)$, $(iii)$ and $(iv)$
follow from an easy calculation (in fact, $u$ and $L_{n,\mu}$ were defined so
that $(ii)$ and $(iii)$ hold). For $(v)$, it suffices to show that $\langle
w,F_{1}(\mu,u)\rangle_{L_{n}^{2}}\in\mathbb{R}$. To simplify notation we
return to the coordinates
$(w,a,z,\phi)=(w,\frac{1}{e}a^{n}+\alpha,z,\frac{\sqrt{2\mu}}{g}+\psi)$. Then

$\displaystyle\langle w,F_{1}(\mu,u)\rangle_{L_{n}^{2}}$
$\displaystyle=\frac{1}{|\Omega^{\prime}|}\int_{\Omega^{\prime}}|\operatorname{curl}_{\nu}w|^{2}+\frac{1}{|\Omega^{\prime}|}\int_{\Omega^{\prime}}\frac{g^{2}}{2}\phi^{2}|w^{2}|$
$\displaystyle+$
$\displaystyle\frac{1}{|\Omega^{\prime}|}\int_{\Omega^{\prime}}i(\operatorname{curl}\nu)(\overline{w}\times
w)+\frac{1}{|\Omega^{\prime}|}\int_{\Omega^{\prime}}g^{2}|\overline{w}\times
w|^{2}.$ (6.13)

The first, second and fourth terms are clearly real, while the third term is
real because $\nu$ is real and $\overline{w}\times w$ is imaginary. ∎

##  7 Reduction to a Finite-Dimensional Problem

In this section we shall reduce solving equation (6.7), i.e. $F(\mu,u)=0$,
with $u=(u_{1},u_{2},u_{3},u_{4})$ $\equiv(w,\alpha,z,\psi)$ and
$F:\mathbb{R}_{>0}\times\mathcal{X}\rightarrow\mathcal{Y}$ defined in (6.7) -
(6.8)), to a finite-dimensional problem.

Recall that $L_{n,\mu}$ is defined in (5.14). Let $P$ be the orthogonal
projection onto $\mathcal{K}:=\operatorname{Null}(L_{n,\mu=n})$, which can be
written explicitly as

$\displaystyle P=P_{1}\oplus P_{2}\oplus 0\oplus 0,$ (7.1) $\displaystyle
P_{1}w:=-\frac{1}{2\pi i}\oint_{\gamma_{n}}(H_{1}(n)-z)^{-1}w\;dz,$ (7.2)
$\displaystyle P_{2}\alpha:=\langle\alpha\rangle,$ (7.3)

where $H_{1}(n)$ is defined in (5.15), $\gamma_{n}$ is any simple closed curve
in $\mathbb{C}$ containing the eigenvalue $0$ and no other eigenvalues of
$H_{1}(n)$ (see Proposition 5.3), and $\langle\alpha\rangle$ is the mean value
of $\alpha$ in $\Omega^{\prime}$,
$\langle\alpha\rangle:=\frac{1}{|\Omega^{\prime}|}\int_{\Omega^{\prime}}\alpha$.
$P_{1}$ is a projection onto $\operatorname{Null}(H_{1}(n))$ (spanned by
vectors of the form (5.22)). Since $H_{1}(n)$ is self-adjoint, $P_{1}$ is an
orthogonal projection (relative to the inner product of $L^{2}_{n}$). By
Theorem 5.1, $\mathcal{K}:=\operatorname{Null}(L_{n,\mu=n})$ is
$(n+1)$-dimensional.

Let $P^{\perp}=1-P$ be the projection onto the orthogonal complement of
$\mathcal{K}$. Then we may rewrite the equation $F(\mu,u)=0$ (see (6.7)) as

$\displaystyle PF(\mu,v+u^{\prime})=0,$ (7.4) $\displaystyle
P^{\perp}F(\mu,v+u^{\prime})=0,$ (7.5)

where $v:=Pu,\ u^{\prime}:=P^{\perp}u$.

Our next goal is to solve (7.5) for $u^{\prime}$ in terms of $\mu$ and $v$.
Recall that the partial (real) Gâteaux derivatives with respect to $\\#$ are
denoted by $\delta_{\\#}$. Let
$\mathcal{X}^{\perp}:=P^{\perp}\mathcal{X}=\mathcal{X}\ominus\mathcal{K}$ and
$\mathcal{Y}^{\perp}:=P^{\perp}\mathcal{Y}=\mathcal{Y}\ominus\mathcal{K}$, and
let $\partial_{i}\equiv\partial_{x_{i}}$.

######  Proposition 7.1.

There is a neighbourhood $U\subset\mathbb{R}_{>0}\times\mathcal{K}$ of $(n,0)$
such that for every $(\mu,v)\in U$, equation (7.5) for $u^{\prime}$ has a
unique solution $u^{\prime}=u^{\prime}(\mu,v)$. Furthermore, this solution
$u^{\prime}=(u^{\prime}_{1},u^{\prime}_{2},u^{\prime}_{3},u^{\prime}_{4})$ has
the following properties:

$\displaystyle
u^{\prime}:\mathbb{R}_{>0}\times\mathcal{K}\to\mathcal{X}^{\perp}\text{ is
continuously differentiable of all orders;}$ (7.6)
$\displaystyle\|(\nabla_{a^{n}})_{j}^{m}u_{1}^{\prime}\|_{\mathcal{H}^{2}_{n}}=\mathcal{O}(\|v\|_{\mathcal{X}}^{2}),$
(7.7)
$\displaystyle\|\partial_{j}^{m}u_{k}^{\prime}\|_{\mathcal{H}^{2}_{k}}=\mathcal{O}(\|v\|^{2}_{\mathcal{X}}),$
(7.8)
$\displaystyle||\delta_{v_{i}}(\nabla_{a^{n}})_{j}^{m}u^{\prime}_{1}(\mu,v_{\hat{i}})||_{\mathcal{H}^{2}_{n}}\lesssim||v_{\hat{i}}||_{\mathcal{X}},$
(7.9)
$\displaystyle||\delta_{v_{i}}\partial_{j}^{m}u^{\prime}_{k}(\mu,v_{\hat{i}})||_{\mathcal{H}^{2}_{k}}\lesssim||v_{\hat{i}}||_{\mathcal{X}},$
(7.10)
$\displaystyle||\partial_{\mu}u^{\prime}(\mu,v)||_{\mathcal{X}}\lesssim||v||^{2}_{\mathcal{X}};$
(7.11)

where $i=1,...4,\;m=0,1,\;j=1,2,\;k=2,3,4$, $v_{\hat{i}}\equiv v|_{v_{i}=0}$,
for $v=(v_{1},v_{2},v_{3},v_{4})$ and $i=1,...,4$, and
$\mathcal{H}^{2}_{k}=\mathcal{H}^{2}_{0}$, $\mathcal{H}^{2}_{0}$,
$\mathcal{H}^{2}$ for $k=2,3,4$.

###### Proof.

Define
$F^{\perp}:\mathbb{R}_{>0}\times\mathcal{K}\times\mathcal{X}^{\perp}\to\mathcal{Y}^{\perp}$
by

$F^{\perp}(\mu,v,u^{\prime}):=P^{\perp}F(\mu,v+u^{\prime}).$ (7.12)

By Proposition 6.2 $(i)$ and $(ii)$, $F^{\perp}$ is continuously
differentiable of all orders as a map between Banach spaces and
$F^{\perp}(\mu,0,0)=0$ for all $\mu\in\mathbb{R}_{>0}$. Furthermore,

$\displaystyle\delta_{u^{\prime}}F^{\perp}(\mu,0,0)=P^{\perp}L_{n,\mu}P^{\perp}|_{\mathcal{X}^{\perp}},$
(7.13)

which is invertible for $\mu=n$ because $P^{\perp}$ is the projection onto the
orthogonal complement of $\mathcal{K}=\operatorname{Null}(L_{n,\mu=n})$. By
the Implicit Function Theorem (see e.g. [16]), there exists a function
$u^{\prime}(\mu,v)$ with continuous derivatives of all orders such that for
$(\mu,v)$ in a sufficiently small neighbourhood
$U\subset\mathbb{R}_{>0}\times\mathcal{K}$ of $(n,0)$, $(\mu,v,u^{\prime})$
solves (7.5) if and only if $u^{\prime}=u^{\prime}(\mu,v)$. This proves the
first statement and property (7.6).

We define the operator

$\displaystyle
L_{n,\mu}^{\perp}:=P^{\perp}L_{n,\mu}P^{\perp}|_{\mathcal{X}^{\perp}}:\mathcal{X}^{\perp}\rightarrow\mathcal{Y}^{\perp}.$
(7.14)

Then by (6.7) and (7.13), we can write equation (7.5) as
$L_{n,\mu}^{\perp}u^{\prime}=-P^{\perp}P^{\prime}J(\mu,u)$. By Theorem 5.1 and
the relation
$\mathcal{K}:=\operatorname{Null}(L_{n,\mu=n})=\operatorname{Null}(L_{n,\mu}-\mu+n)$,
for $\mu$ in a neighbourhood of $n$, the operator $L_{n,\mu}^{\perp}$ has a
uniformly bounded inverse
$(L_{n,\mu}^{\perp})^{-1}:\mathcal{Y}^{\perp}\to\mathcal{X}^{\perp}$. Hence
equation $L_{n,\mu}^{\perp}u^{\prime}=-P^{\perp}P^{\prime}J(\mu,u)$, with
$(\mu,v)\in U$ (replacing $U$ with a smaller neighbourhood if necessary), is
equivalent to

$\displaystyle
u^{\prime}=-(L_{n,\mu}^{\perp})^{-1}P^{\perp}P^{\prime}J(\mu,u);$ (7.15)

hence

$\displaystyle\|u^{\prime}\|_{\mathcal{X}}\lesssim\|J(\mu,u)\|_{\mathcal{Y}},$
(7.16)

uniformly in $\mu$. Recall that
$\mathcal{X}=\mathcal{H}^{2}_{n}\oplus\mathcal{H}^{2}_{0}\oplus\mathcal{H}^{2}_{0}\oplus\mathcal{H}^{2}$
and $\mathcal{Y}=L^{2}_{n}\oplus L^{2}_{0}\oplus L^{2}_{0}\oplus L^{2}$.
$J(\mu,u)$ is a polynomial in the components of $u$ and their first-order
(covariant) derivatives consisting of terms of degree at least $2$, so the
left-hand side of (7.16) can be bounded above by a sum of products of one
$\mathcal{L}^{2}$-norm and at least one $\mathcal{L}^{\infty}$-norm of these
terms. $\mathcal{H}^{1}$ is trivially continously embedded in
$\mathcal{L}^{2}$, and by the Sobolev Embedding Theorem, $\mathcal{H}^{1}$ is
continuously embedded in $\mathcal{L}^{\infty}$. Therefore,

$\displaystyle\|J(\mu,u)\|_{\mathcal{Y}}\lesssim\|u\|_{\mathcal{X}}^{2}.$
(7.17)

Recalling that $u=v+u^{\prime}$, this proves (7.7) and (7.8) when $m=0$. The
other case is proven similarly.

For $v=(v_{1},\dots,v_{4})$, we let $v_{\widehat{i}}\equiv v|_{v_{i}=0}$,
$i=1,\dots,4$. By the Taylor theorem for Banach spaces (see e.g. [16]), we
have

$\displaystyle
u^{\prime}(\mu,v)=u^{\prime}(\mu,v_{\widehat{i}})+\delta_{v_{i}}u^{\prime}(\mu,v_{\widehat{i}})v_{i}+R_{2}(\mu,v_{\widehat{i}})(v_{i}),$
(7.18) $\displaystyle
R_{2}(\mu,v_{\widehat{i}})(v_{i}):=\int_{0}^{1}(1-t)\delta_{v_{i}}^{2}u^{\prime}(\mu,v_{\widehat{i}}+tv_{i})(v_{i},v_{i})dt.$
(7.19)

Let $(\mu,v)\in U$ with $||v_{\widehat{i}}||=||v_{i}||=1$, and let
$\epsilon>0$. Then

$\displaystyle||\delta_{v_{i}}u^{\prime}(\mu,\epsilon v_{\widehat{i}})$
$\displaystyle\epsilon v_{i}||_{\mathcal{X}}=||u^{\prime}(\mu,\epsilon
v)-u^{\prime}(\mu,\epsilon v_{\widehat{i}})-R_{2}(\mu,\epsilon
v_{\widehat{i}})(\epsilon v_{i})||_{\mathcal{X}}$
$\displaystyle\leq||u^{\prime}(\mu,\epsilon
v)||_{\mathcal{X}}+||u^{\prime}(\mu,\epsilon v_{\widehat{i}})||_{\mathcal{X}}$
$\displaystyle+\epsilon^{2}||v_{i}||^{2}\sup\limits_{0\leq t\leq
1}(1-t)||\delta_{v_{i}}^{2}u^{\prime}(\mu,\epsilon v_{\widehat{i}}+t\epsilon
v_{i})||^{2}_{\mathcal{X}^{*}\otimes\mathcal{X}^{*}\otimes\mathcal{X}}$
$\displaystyle\lesssim\epsilon^{2}.$ (7.20)

with the norm taken in the appropriate space for $v_{i}$. Taking the supremum
over all $v_{i}$ with $||v_{i}||=1$ gives

$\displaystyle||\delta_{v_{i}}u^{\prime}(\mu,\epsilon
v_{\widehat{i}})||_{\mathcal{X}}\lesssim\epsilon,\
\|v_{\widehat{i}}\|_{\mathcal{X}}=1,$ (7.21)

proving (7.9) - (7.10) for $m=0$. The other cases are proven in exactly the
same way.

Again by Taylor’s Theorem,

$\displaystyle\partial_{\mu}u^{\prime}(\mu,v)=\partial_{\mu}u^{\prime}(\mu,0)+\partial_{\mu}\delta_{v}u^{\prime}(\mu,0)v+\tilde{R}_{2}(\mu,0)(v),$
(7.22)
$\displaystyle\tilde{R}_{2}(\mu,0)(v):=\int_{0}^{1}(1-t)\partial_{\mu}\delta_{v}^{2}u^{\prime}(\mu,tv)(v,v)dt.$
(7.23)

By Equations (7.8) and (7.9) - (7.10) with $m=0$, we have
$u^{\prime}(\mu,0)=0$ and $\delta_{v}u^{\prime}(\mu,0)=0$, so

$\displaystyle||\partial_{\mu}u^{\prime}(\mu,v)||_{\mathcal{X}}$
$\displaystyle=||\tilde{R}_{2}(\mu,0)(v)||_{\mathcal{X}}$ (7.24)
$\displaystyle\leq||v||_{\mathcal{X}}^{2}\sup\limits_{0\leq t\leq
1}(1-t)||\partial_{\mu}\delta_{v}^{2}u^{\prime}(\mu,tv)||^{2}_{\mathcal{X}^{*}\otimes\mathcal{X}^{*}\otimes\mathcal{X}}$
(7.25) $\displaystyle\lesssim||v||^{2}_{\mathcal{X}},$ (7.26)

proving (7.11). ∎

We plug the solution $u^{\prime}=u^{\prime}(\mu,v)$ into equation (7.4) to get
the _bifurcation equation_

$\gamma(\mu,v):=PF(\mu,v+u^{\prime}(\mu,v))=0.$ (7.27)

######  Corollary 7.2.

In a neighbourhood of $(n,0)$ in $\mathbb{R}_{>0}\times\mathcal{X}$, the pair
$(\mu,u)$ solves (6.7) if and only if $(\mu,v)$ solves the finite-dimensional
equation (7.27). Moreover, a solution of (6.7) can be constructed from a
solution $(\mu,v)$ of (7.27) by setting $u=v+u^{\prime}(\mu,v)$, where
$u^{\prime}(\mu,v)$ is given by Proposition 7.1.

Since $F:\mathbb{R}_{>0}\times\mathcal{X}\to\mathcal{Y}$ and
$u^{\prime}:\mathbb{R}_{>0}\times\mathcal{K}\to\mathcal{Y}^{\perp}$ have been
shown to be continuously differentiable of all orders, we conclude:

######  Corollary 7.3.

$\gamma:\mathbb{R}\times\mathcal{K}\to\mathcal{K}$ is continuously
differentiable of all orders.

Furthermore, $\gamma(\mu,v)$ inherits the following symmetry of $F(\mu,u)$,
which we will use to find a solution of (7.27):

######  Lemma 7.4.

Let $T_{\delta}$ be given by (6.12). For every $\delta\in\mathbb{R}$ and
$(\mu,v)$ in a neighbourhood of $(n,0)$, we have

$\displaystyle u^{\prime}(\mu,T_{\delta}v)$
$\displaystyle=T_{\delta}u^{\prime}(\mu,v),$ (7.28)
$\displaystyle\gamma(\mu,T_{\delta}v)$
$\displaystyle=T_{\delta}\gamma(\mu,v).$ (7.29)

###### Proof.

For equation (7.28), we note that by Proposition 6.2 $(iv)$

$\displaystyle P^{\perp}F(\mu,T_{\delta}v+T_{\delta}u^{\prime}(\mu,v))$
$\displaystyle=P^{\perp}T_{\delta}F(\mu,v+u^{\prime}(\mu,v))$ (7.30)
$\displaystyle=T_{\delta}P^{\perp}F(\mu,v+u^{\prime}(\mu,v))=0.$

(Here we used $P^{\perp}T_{\delta}=T_{\delta}P^{\perp}$, which follows because
$T_{\delta}=e^{i\delta}\oplus 1\oplus 1\oplus 1$ and $P^{\perp}=1-P$ where $P$
is defined in (7.1).) Since $u^{\prime}=u^{\prime}(\mu,T_{\delta}v)$ is the
unique solution to $P^{\perp}F(\mu,T_{\delta}v+u^{\prime})=0$ for $(\mu,v)$ in
a neighbourhood $U\subset\mathbb{R}\times\mathcal{K}$ of $(n,0)$, we conlcude
that $u^{\prime}(\mu,T_{\delta}v)=T_{\delta}u^{\prime}(\mu,v)$.

For equation (7.29), we note that by (7.28) and Proposition 6.2 $(iv)$,

$\displaystyle\gamma(\mu,T_{\delta}v)$
$\displaystyle=PF(\mu,T_{\delta}v+u^{\prime}(\mu,T_{\delta}v))=PF(\mu,T_{\delta}(v+u^{\prime}(\mu,v)))$
$\displaystyle=T_{\delta}PF(\mu,v+u^{\prime}(\mu,v))=T_{\delta}\gamma(\mu,v)$
(7.31)

(where again we used $PT_{\delta}=T_{\delta}P$). ∎

##  8 The bifurcation result when $n=1$

######  Theorem 8.1.

Assume that $n=1$ and $|1-b_{*}/b|\ll 1$, $b_{*}:=M_{W}^{2}/e$. Then there
exists $\epsilon>0$ and a branch
$(\mu_{s},u_{s}):=(\mu_{s},w_{s},\alpha_{s},z_{s},\psi_{s})$, with
$s\in[0,\sqrt{\epsilon})$, of non-trivial solutions of equation (6.1), unique
modulo a gauge symmetry in a sufficiently small neighbourhood of the rescaled
vacuum solution (4.9) in $\mathbb{R}_{>0}\times\mathcal{X}$, such that

$\displaystyle\begin{cases}w_{s}=s\chi+sg_{1}(s^{2}),\\\
\alpha_{s}=g_{2}(s^{2}),\\\ z_{s}=g_{3}(s^{2}),\\\
\alpha\alpha_{s}=g_{4}(s^{2}),\\\ \mu_{s}=n+g_{5}(s^{2}),\end{cases}$ (8.1)

where $\chi$ solves the eigenvalue problem $H_{1}(n)\chi=0$ (it is defined in
(5.22), see Proposition 5.3),
$\mu:=g^{2}\xi^{2}/2=g^{2}r^{2}\varphi_{0}^{2}/2$,
$g_{1}:[0,\epsilon)\to\mathcal{H}_{n}^{2}$ and is orthogonal to
$\operatorname{Null}(H_{1}(n))$, $g_{2}:[0,\epsilon)\to\mathcal{H}^{2}_{0}$,
$g_{3}:[0,\epsilon)\to\mathcal{H}^{2}_{0}$,
$g_{4}:[0,\epsilon)\to\mathcal{H}^{2}$,
$g_{5}:[0,\epsilon)\to\mathbb{R}_{>0}$, and $g_{j}$ for $j=1,\cdots,5$ are
functions, continuously differentiable of all orders in $s$, such that
$g_{j}(0)=0$.

###### Proof of Theorem 8.1.

For the proof below, recall that we denote the partial (real) Gâteaux
derivatives with respect to $\\#$ by $\delta_{\\#}$, and let
$\partial_{i}\equiv\partial_{x_{i}}$.

By Proposition 6.1, solving equation (6.1) is equivalent to solving (6.7). By
Corollary 7.2, solving (6.7) is equivalent to solving the bifurcation equation
(7.27). Hence, we address the latter equation.

Recall that $P$ is the projection onto
$\mathcal{K}=\operatorname{Null}L_{n,\mu=n}=\operatorname{Null}(H_{1}(n))\times\\{\text{constants}\\}\times\\{0\\}\times\\{0\\}$.
The projection onto constant vector fields in $\mathcal{H}_{0}^{2}$ can be
written as the mean value
$\langle\alpha\rangle:=\frac{1}{|\Omega^{\prime}|}\int_{\Omega^{\prime}}\alpha$.
Since $\text{dim}\operatorname{Null}(H_{1}(n))=1$ for $n=1$, we may choose
$\chi\in\operatorname{Null}(H_{1}(n))$ such that

$\displaystyle P(w,\alpha,z,\psi)=(s\chi,c,0,0),$ (8.2) $\displaystyle
s:=\langle\chi,w\rangle_{L_{n}^{2}}\in\mathbb{C},\
c:=\langle\alpha\rangle\in\mathbb{R}^{2},$ (8.3)

and $\chi$ satisfies $\|\chi\|_{L_{n}^{2}}^{2}=\langle|\chi|^{2}\rangle=1$
(see (5.11)), where, recall, $\chi$ is described in (5.22). Hence we may write
the $\gamma$ from the bifurcation equation (7.27) as
$\gamma=(\tilde{\gamma}_{1}\chi,\tilde{\gamma}_{2},0,0)$, where
$\tilde{\gamma}_{1},\tilde{\gamma}_{2}:\mathbb{R}_{>0}\times\mathbb{C}\times\mathbb{R}^{2}\to\mathbb{C}$
are given by

$\displaystyle\tilde{\gamma}_{1}(\mu,s,c)$
$\displaystyle:=\langle\chi,F_{1}(\mu,v(s,c)+u^{\prime}(\mu,v(s,c))\rangle_{L_{n}^{2}},$
(8.4) $\displaystyle\tilde{\gamma}_{2}(\mu,s,c)$ $\displaystyle:=\langle
F_{2}(\mu,v(s,c)+u^{\prime}(\mu,v(s,c))\rangle,$ (8.5)

where, recall, $F_{j}$, $j=1,\dots,4$ are defined by (6.9), $s\in\mathbb{C},\
c\in\mathbb{R}^{2}$ and (see (8.2))

$\displaystyle v(s,c)$ $\displaystyle:=(s\chi,c,0,0).$ (8.6)

Note that $\tilde{\gamma}_{1}$ and $\tilde{\gamma}_{2}$ are continuously
differentiable of all orders in $\mu$, $s$ and $c$ by Corollary 7.3.
($\tilde{\gamma}_{2}$ is independent of $\mu$.) The bifurcation equation
(7.27) is then equivalent to the equations

$\displaystyle\tilde{\gamma}_{1}(\mu,s,c)$ $\displaystyle=0,$ (8.7)
$\displaystyle\tilde{\gamma}_{2}(\mu,s,c)$ $\displaystyle=0.$ (8.8)

######  Lemma 8.2.

There exists a neighbourhood $U\subset\mathbb{R}_{>0}\times\mathbb{R}_{>0}$ of
$(n,0)$ and a unique function $c:U\to\mathbb{R}^{2}$ with continuous
derivatives of all orders such that

$\tilde{\gamma}_{2}(\mu,s,c(\mu,s^{2}))=0$ (8.9)

and

$||\partial^{l}_{\mu}c(\mu,s^{2})||_{\mathbb{R}^{2}}=\mathcal{O}(|s|^{2}),\
l=0,1.$ (8.10)

###### Proof.

Recall that $F_{2}(\mu,u)=H_{2}(\mu)\alpha+P_{0}J_{2}(\mu,u)$ (see Equation
(6.7)), with $P_{0}$ the projection onto the divergence-free vector fields and

$\displaystyle u=(w,\alpha,z,\psi)=v+u^{\prime},$ (8.11)

where $v=v(s,c)$ and $u^{\prime}=u^{\prime}(\mu,v)$ solves (7.5). By
definition, $(\mathbf{1}-P_{0})f=\Delta^{-1}\nabla\operatorname{div}f$ and
therefore $\langle(\mathbf{1}-P_{0})f\rangle=0$. Hence $\langle
P_{0}f\rangle=\langle f\rangle$. This and the relation $\langle
H_{2}(\mu)\alpha\rangle=\frac{1}{|\Omega^{\prime}|}\int_{\Omega^{\prime}}\operatorname{curl}^{*}\operatorname{curl}\alpha=0$
give

$\tilde{\gamma}_{2}(\mu,s,c)=\langle
J_{2}(\mu,v(s,c)+u^{\prime}(\mu,v(s,c)))\rangle.$ (8.12)

Using (6.3), $\nu=a^{n}+\tilde{\nu}$,
$\operatorname{curl}_{a^{n}}w=\operatorname{curl}_{a^{n}}w-i\tilde{\nu}\times
w$ and that the final term in (6.3) vanishes after taking the mean, we find

$\displaystyle\langle
J_{2}(\mu,u)\rangle=2e\operatorname{Im}\langle(\operatorname{curl}_{a^{n}}w-i\tilde{\nu}\times
w)J\overline{w}\rangle.$ (8.13)

Recall $u^{\prime}=(w^{\prime},\alpha^{\prime},z^{\prime},\psi^{\prime})$.
Then (8.6) and (8.11) give $w=s\chi+w^{\prime}$ and (using that
$e=g\sin\theta$) $\tilde{\nu}=ec+\nu^{\prime}$. Using these relations and
$\operatorname{curl}_{a^{n}}\chi=0$ (by (5.22)) and (8.12) and (8.13), we find
for
$\overline{\gamma}_{2}(\mu,s,c):=(2e)^{-1}|s|^{-2}\tilde{\gamma}_{2}(\mu,s,c)$

$\displaystyle\overline{\gamma}_{2}(\mu,s,c)$
$\displaystyle:=-e\langle\operatorname{Re}[(c\times\chi)J\overline{\chi}]\rangle+\operatorname{Im}s^{-1}\langle(\operatorname{curl}_{a^{n}}w^{\prime})J\overline{\chi}\rangle$
(8.14)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\operatorname{Im}\langle\bar{R}_{2}(\mu,s,c)\rangle,$
(8.15) $\displaystyle\bar{R}_{2}(\mu,s,c)$
$\displaystyle:=|s|^{-2}[-i(ec\times s\chi)J\overline{w^{\prime}}-i(ec\times
w^{\prime})J\overline{w^{\prime}}$ (8.16) $\displaystyle-i(ec\times
w^{\prime})J\overline{s\chi}-i(\nu^{\prime}\times
w^{\prime})J\overline{s\chi}-i(\nu^{\prime}\times
s\chi)J\overline{w^{\prime}}$ (8.17) $\displaystyle-i(\nu^{\prime}\times
s\chi)J\overline{s\chi}-i(\nu^{\prime}\times
w^{\prime})J\overline{w^{\prime}}+(\operatorname{curl}_{a^{n}}w^{\prime})J\overline{w^{\prime}}].$
(8.18)

Note that we expect (8.14) $=\mathcal{O}(|s|^{2})$ and (8.15)
$=\mathcal{O}(|s|^{4})$. We now simplify (8.14). For the first term on the
right-hand side, we use (5.22) and the condition $\langle|\chi|^{2}\rangle=1$
to compute

$\langle\operatorname{Re}[(c\times\chi)J\overline{\chi}]\rangle=-\frac{1}{2}c.$
(8.19)

For the second term on the right-hand side of (8.14), we use $\langle
fJ\overline{\chi}\rangle=\langle
f(i\overline{\eta},\overline{\eta})\rangle=\langle
f\bar{\eta}\rangle(i,1)=\langle\eta,f\rangle(i,1)$ and integrate by parts to
compute

$\displaystyle\langle(\operatorname{curl}_{a^{n}}w^{\prime})J\overline{\chi}\rangle$
$\displaystyle=\langle\eta,\operatorname{curl}_{a^{n}}w^{\prime}\rangle(i,1)=\langle\operatorname{curl}_{a^{n}}^{*}\eta,w^{\prime}\rangle(i,1).$
(8.20)

Abusing notation, we write in what follows $w(\mu,s,c)\equiv w(\mu,v(s,c))$.
Then (8.14) becomes

$\displaystyle\overline{\gamma}_{2}(\mu,s,c)=$
$\displaystyle\frac{1}{2}ec+\operatorname{Im}s^{-1}\langle\operatorname{curl}_{a^{n}}^{*}\eta,w^{\prime}(\mu,s,c)\rangle\rangle(i,1)+\operatorname{Im}\langle\tilde{R}_{2}(\mu,s,c)\rangle.$
(8.21)

Now, Equation (7.7), with $m=0$, implies that

$\displaystyle|\operatorname{Im}\langle\operatorname{curl}_{a^{n}}^{*}\eta,w^{\prime}(\mu,s,c)\rangle\rangle|=\mathcal{O}(|s|^{2}).$
(8.22)

Furthermore, we show below the following estimate on the remainder:

$\displaystyle||\operatorname{Im}\langle\partial_{c}^{l}\tilde{R}_{2}(\mu,s,c)\rangle||_{\mathbb{R}^{2}}=\mathcal{O}(|s|^{2-l}),\
l=0,1.$ (8.23)

Hence $\overline{\gamma}_{2}(\mu,0,0)=0$. To apply the Implicit Function
Theorem to solve for $c$ as a function of $\mu$ and $s$, we have to estimate
the derivative

$\displaystyle\partial_{c}\overline{\gamma}_{2}(\mu,s,c)=$
$\displaystyle\frac{1}{2}e{{\bf
1}}+\operatorname{Im}s^{-1}\langle\operatorname{curl}_{a^{n}}^{*}\eta,\partial_{c}w^{\prime}(\mu,s,c)\rangle(i,1)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\quad+\operatorname{Im}\langle\partial_{c}\tilde{R}_{2}(\mu,s,c)\rangle.$
(8.24)

at $(n,s,0)$. At the first step, we use the following

######  Lemma 8.3.

Using Dirac’s bra-ket notation, we have

$\displaystyle(\partial_{c}w^{\prime})(n,s,0)$
$\displaystyle=-n^{-1}es|\operatorname{curl}^{*}_{a^{n}}\eta\rangle\langle(1,i)|+\mathcal{O}(|s|^{2}).$
(8.25)

###### Proof of Lemma 8.3.

By definition (7.2), $P^{\perp}_{1}$ projects onto the orthogonal complement
of the eigenspace of $H_{1}(n)$ corresponding to the eigenvalue $0$ and
therefore the operator $H_{1}^{\perp}(n)$ is invertible on
$\operatorname{Ran}P^{\perp}_{1}$. Hence (6.1) with $i=1$ can be rewritten as
$w^{\prime}=-(H_{1}^{\perp}(n))^{-1}P^{\perp}_{1}J_{1}(n,u)$ (which is the
first component of (7.15)), which gives

$\displaystyle\partial_{c}w^{\prime}=-(H_{1}^{\perp}(n))^{-1}P^{\perp}_{1}\partial_{c}J_{1}(n,u),$
(8.26)

where $u\equiv u(s,c):=v(s,c)+u^{\prime}(\mu,v(s,c))$. By (6.2) and (6.6), we
have

$\displaystyle\partial_{c}J_{1}(n,u)=\partial_{c}[\operatorname{curl}_{\nu}^{*}\operatorname{curl}_{\nu}w].$
(8.27)

Using $w=s\chi+w^{\prime},$ $\nu=a^{n}+ec+\nu^{\prime}$ and
$\operatorname{curl}_{\nu}=\operatorname{curl}_{a^{n}}+iJ(ec+\nu^{\prime})\cdot$,
$\operatorname{curl}_{\nu}^{*}=\operatorname{curl}_{a^{n}}^{*}-iJ(ec+\nu^{\prime})$
and that $\nu^{\prime}=\mathcal{O}(|s|^{2})$, we compute

$\displaystyle\partial_{c}J_{1}(n,u)c^{\prime}$
$\displaystyle=s\partial_{c}[\operatorname{curl}_{\nu}^{*}\operatorname{curl}_{\nu}]\chi
c^{\prime}+\mathcal{O}(|s|^{2})$
$\displaystyle=sie[-Jc^{\prime}\operatorname{curl}_{\nu}+\operatorname{curl}_{\nu}^{*}Jc^{\prime}\cdot]\chi+\mathcal{O}(|s|^{2})$
(8.28)
$\displaystyle=sie[-Jc^{\prime}\operatorname{curl}_{a^{n}+ec}+\operatorname{curl}_{a^{n}+ec}^{*}Jc^{\prime}\cdot]\chi+\mathcal{O}(|s|^{2}).$
(8.29)

Since
$\operatorname{curl}_{a^{n}}\chi=\nabla_{1}i\beta-\nabla_{2}\beta=i\bar{\partial}_{a^{n}}\beta=0$
and
$Jc^{\prime}\cdot\chi=(-c_{2}^{\prime},c_{1}^{\prime})\cdot(\beta,i\beta)=-c_{2}^{\prime}\beta+c_{1}^{\prime}i\beta=i(c^{\prime}_{1}+ic_{2}^{\prime})\beta$
and therefore
$\operatorname{curl}_{a^{n}}^{*}Jc^{\prime}\cdot\chi=i\operatorname{curl}_{a^{n}}^{*}\beta(c^{\prime}_{1}+ic_{2}^{\prime})$,
this yields

$\displaystyle\partial_{c}J_{1}(n,u)c^{\prime}\big{|}_{c=0}$
$\displaystyle=-se\operatorname{curl}_{a^{n}}^{*}\beta(c^{\prime}_{1}+ic_{2}^{\prime})+\mathcal{O}(|s|^{2}).$
(8.30)

By Proposition 5.3(ii),
$\operatorname{Null}{(H_{1}(\mu)-\mu+n)}=\\{\chi=(\beta,i\beta):\operatorname{curl}_{a^{n}}\chi=i\bar{\partial}_{a^{n}}\eta=0\\}$.
The relation $\operatorname{curl}_{a^{n}}\chi=0$ implies also
$\langle\chi,\operatorname{curl}_{a^{n}}^{*}\chi\rangle=\langle\operatorname{curl}_{a^{n}}\chi,\chi\rangle=0$,
which, for $n=1$, gives that
$P^{\perp}_{1}\partial_{c}J_{1}(n,u)c^{\prime}=\partial_{c}J_{1}(n,u)c^{\prime}$
and therefore

$\displaystyle P^{\perp}_{1}\partial_{c}J_{1}(n,u)c^{\prime}$
$\displaystyle=-se\operatorname{curl}_{a^{n}}^{*}\beta(c^{\prime}_{1}+ic_{2}^{\prime})+\mathcal{O}(|s|^{2}).$
(8.31)

By (5.22), we have $\operatorname{curl}_{a^{n}}^{*}\eta=i\nabla_{a^{n}}\beta$,
and by (5.23), we have $H_{1}(n)\nabla_{a^{n}}\beta=n\nabla_{a^{n}}\beta$;
hence
$(H_{1}^{\perp}(n))^{-1}\operatorname{curl}_{a^{n}}^{*}\beta=n^{-1}\operatorname{curl}_{a^{n}}^{*}\beta$.
This relation, together with (8.26) and (8.31), yields

$\displaystyle\partial_{c}w^{\prime}c^{\prime}$
$\displaystyle=sen^{-1}\operatorname{curl}_{a^{n}}^{*}\beta(c^{\prime}_{1}+ic_{2}^{\prime})+\mathcal{O}(|s|^{2}),$
(8.32)

which gives (8.25). ∎

Using Equation (8.25), we calculate the second term on the right-hand side of
(8) at $(n,s,0)$:

$\displaystyle\operatorname{Im}s^{-1}\langle\operatorname{curl}_{a^{n}}^{*}\beta,\partial_{c}w^{\prime}(\mu,s,c)c^{\prime}\rangle(i,1)$
$\displaystyle=en^{-1}\operatorname{Im}\langle\operatorname{curl}_{a^{n}}^{*}\beta,\operatorname{curl}_{a^{n}}^{*}\beta\rangle(c^{\prime}_{1}+ic_{2}^{\prime})(i,1).$
(8.33)

The inner product term is real. Integrating it by parts and using that, by
Equation (5.36), $\beta$ satisfies
$\operatorname{curl}_{a^{n}}\operatorname{curl}_{a^{n}}^{*}\beta=-\Delta_{a^{n}}\beta=n\beta$
and using that
$\|\beta\|_{L^{2}_{n}}^{2}=\frac{1}{2}\|\chi\|_{L^{2}_{n}}^{2}=\frac{1}{2}$,
gives

$\displaystyle\langle\operatorname{curl}_{a^{n}}^{*}\beta,\operatorname{curl}_{a^{n}}^{*}\beta\rangle=$
$\displaystyle\langle\beta,-\Delta_{a^{n}}\beta\rangle_{L^{2}_{n}}=\frac{1}{2}n.$
(8.34)

The last two equations and the relation
$\operatorname{Im}(c^{\prime}_{1}+ic_{2}^{\prime})(i,1)=\operatorname{Im}\bigg{(}\begin{array}[]{cc}i&-1\\\
1&i\end{array}\bigg{)}c^{\prime}=\mathbf{1}c^{\prime}$ imply

$\displaystyle\operatorname{Im}s^{-1}\langle\operatorname{curl}_{a^{n}}^{*}\beta,\partial_{c}w^{\prime}(\mu,s,c)\rangle(i,1)=\frac{1}{2}e{{\bf
1}}.$ (8.35)

This, together with (8), gives

$\displaystyle\partial_{c}\overline{\gamma}_{2}(n,s,0)=$
$\displaystyle\frac{1}{2}e{{\bf 1}}+\frac{1}{2}e{{\bf
1}}+\operatorname{Im}\langle\partial_{c}\tilde{R}_{2}(n,s,0)\rangle.$ (8.36)

Therefore, (8.36) and (8.23) (with $l=1$) imply

$\displaystyle\partial_{c}\overline{\gamma}_{2}(n,0,0)=e{{\bf 1}},$ (8.37)

proving that $\partial_{c}\overline{\gamma}_{2}(n,0,0)$ is invertible, as
required.

Recall that, by (8.21), (8.22) and (8.23) (with $l=0$), we have

$\displaystyle\overline{\gamma}_{2}(n,0,0)=0.$ (8.38)

Since $\partial_{c}\overline{\gamma}_{2}(n,0,0)$ is invertible, by the
Implicit Function Theorem there exists a unique function
$\tilde{c}:\mathbb{R}_{>0}\times\mathbb{C}\to\mathbb{R}^{2}$ with continuous
derivatives of all orders such that
$\overline{\gamma}_{2}(\mu,s,\tilde{c}(\mu,s))=0$ for $(\mu,s)$ in a
sufficiently small neighbourhood of $(n,0)$. Furthermore, the symmetry (7.29)
implies that
$\overline{\gamma}_{2}(\mu,|s|,\tilde{c}(\mu,s))=\overline{\gamma}_{2}(\mu,e^{i\arg
s}|s|,\tilde{c}(\mu,s))=\overline{\gamma}_{2}(\mu,s,\tilde{c}(\mu,s))=0$, so
by the uniqueness of the branch $\tilde{c}(\mu,s)$ we have

$\tilde{c}(\mu,s)=\tilde{c}(\mu,|s|).$ (8.39)

In particular, $\partial_{\mu}^{l}\tilde{c}(\mu,s)$, $l=0,1$, restricted to
$s\in\mathbb{R}$ are even functions with continuous derivatives of all orders;
thus $\partial_{s}\partial_{\mu}^{l}\tilde{c}(\mu,0)=0$ and hence
$\partial_{\mu}^{l}\tilde{c}(\mu,s)=\mathcal{O}(|s|^{2})$, since the first two
terms of the Taylor expansion are $0$. We define
$c:\mathbb{R}_{>0}\times\mathbb{R}_{>0}\to\mathbb{R}^{2}$ by
$c(\mu,s):=\tilde{c}(\mu,\sqrt{s})$, which is a function with continuous
derivatives of all orders satisfying
$||\partial_{\mu}^{l}c(\mu,s^{2})||_{\mathbb{R}^{2}}=\mathcal{O}(|s|^{2})$,
$l=0,1$, and
$\tilde{\gamma}_{2}(\mu,s,c(\mu,s^{2}))=|s|^{2}\overline{\gamma}_{2}(\mu,s,c(\mu,s^{2}))=0$,
as required. ∎

######  Lemma 8.4.

For $\epsilon>0$ sufficiently small, there exists a unique function
$\mu:[0,\epsilon)\to\mathbb{R}_{>0}$ with continuous derivatives of all orders
such that

$\tilde{\gamma}_{1}(\mu(s^{2}),s,c(\mu(s^{2}),s^{2}))=0.$ (8.40)

###### Proof.

To simplify notation for this lemma, we set $u=v_{s}+u^{\prime}_{s}$, with
$v\equiv v_{s}\equiv(s\chi,c(\mu,s^{2}),0,0),\ u^{\prime}\equiv
u^{\prime}_{s}\equiv u^{\prime}(\mu,v_{s}),\ c\equiv c(\mu,s^{2})$.

We first show that $\tilde{\gamma}_{1}(\mu,s,c)\in\mathbb{R}$ for
$s\in\mathbb{R}$. Since $u^{\prime}$ by definition solves
$P_{1}^{\perp}F_{1}(\mu,v+u^{\prime})=0$, where
$P_{1}^{\perp}w^{\prime}=w^{\prime}$ and $P_{1}^{\perp}$ is self-adjoint, we
have

$\displaystyle\langle
w^{\prime},F_{1}(\mu,v+u^{\prime})\rangle_{L^{2}_{n}}=\langle
w^{\prime},P_{1}^{\perp}F_{1}(\mu,v+u^{\prime})\rangle_{L^{2}_{n}}=0.$ (8.41)

Therefore, for $s\neq 0$, we find

$\displaystyle\tilde{\gamma}_{1}(\mu,s,c)$ $\displaystyle=s^{-1}\langle
s\chi,F_{1}(\mu,v+u^{\prime}\rangle_{L^{2}_{n}}$ $\displaystyle=s^{-1}\langle
s\chi+w^{\prime},F_{1}(\mu,v+u^{\prime})\rangle_{L^{2}_{n}},$ (8.42)

which is real by Proposition 6.2 $(v)$. Furthermore, by equations (7.29) and
(8.39), we have
$\tilde{\gamma}_{1}(\mu,s,c(\mu,s^{2}))=e^{i\text{arg}(s)}\tilde{\gamma}_{1}(\mu,|s|,c(\mu,|s|^{2}))$,
so we may restrict $s$ to be real.

Next, we show that

$\displaystyle\tilde{\gamma}_{1}(n,s,c(n,s^{2}))=\mathcal{O}(|s|^{2})$ (8.43)

Indeed,

$\displaystyle|\tilde{\gamma}_{1}(n,s,c(n,s^{2}))|$
$\displaystyle\leq||\chi||_{L^{2}_{n}}||F_{1}(n,v+u^{\prime})||_{L^{2}_{n}}$
$\displaystyle\leq||\chi||_{L^{2}_{n}}\big{[}||H_{1}(n)(s\chi+w^{\prime})||_{L^{2}_{n}}$
$\displaystyle\quad+||J_{1}(n,v+u^{\prime})||_{L^{2}_{n}})\big{]}.$ (8.44)

Recall that $H_{1}(n)\chi=0$, so that

$\displaystyle|\tilde{\gamma}_{1}(n,s,c(n,s^{2}))|$
$\displaystyle\leq||\chi||_{L^{2}_{n}}\big{[}||H_{1}(n)||_{L^{2}_{n}\otimes(L^{2}_{n})^{*}}||w^{\prime}||_{L^{2}_{n}}$
$\displaystyle\quad+||J_{1}(n,v+u^{\prime})||_{L^{2}_{n}}\big{]}.$ (8.45)

By the definition $v\equiv v_{s}\equiv(s\chi,c(\mu,s^{2}),0,0)$ and equation
(8.10), $||v||_{\mathcal{X}}=\mathcal{O}(|s|)$; hence by Proposition 7.1,

$\displaystyle||w^{\prime}||_{L^{2}_{n}}\leq||w^{\prime}||_{\mathcal{H}^{2}_{n}}=\mathcal{O}(|s|^{2}).$
(8.46)

Furthermore, by equation (7.17) and recalling that $H_{1}(n)\chi=0$,

$\displaystyle||J_{1}(n,v+u^{\prime})||_{L^{2}_{n}}\leq||J_{1}(n,v+u^{\prime})||_{\mathcal{H}^{2}_{n}}\lesssim||v+u^{\prime}||_{\mathcal{X}}^{2}=\mathcal{O}(|s|^{2}).$
(8.47)

This proves that (8.45) is $\mathcal{O}(|s|^{2})$, as required.

In light of equation (8.43), we can define a function
$\overline{\gamma}_{1}:\mathbb{R}_{>0}\times\mathbb{R}_{>0}\to\mathbb{R}$ with
continuous derivatives of all orders by

$\overline{\gamma}_{1}(\mu,s)\equiv\begin{cases}s^{-1}\tilde{\gamma}_{1}(\mu,s,c(\mu,s^{2})),\quad
s\neq 0,\\\ 0,\quad s=0.\end{cases}$ (8.48)

We now find a non-trivial branch of solutions $(\mu,s)=(\tilde{\mu}(s),s)$ by
applying the Implicit Function Theorem to $\overline{\gamma}_{1}$. First, we
prove the following proposition to bound the polynomials of functions
appearing below:

######  Lemma 8.5.

Let $X$ be one of the spaces $\mathcal{H}^{2}_{n}$, $\mathcal{H}_{0}$ or
$\mathcal{H}^{2}$ defined before equation (5.8). Let $p(x_{1},...,x_{n})$ be a
polynomial with positive coefficients and let $f_{1},...,f_{n}\in X$. Then
$||p(f_{1},...,f_{n})||_{X}\lesssim p(||f_{1}||_{X},...,||f_{n}||_{X})$.

###### Proof.

Write $p(x_{1},...,x_{n})=\sum_{|\alpha|\leq N}p_{\alpha}x^{\alpha}$, where
$\alpha=(\alpha_{1},...,\alpha_{n})$ is a multi-index,
$x^{\alpha}=\prod_{i=1}^{n}x_{i}^{\alpha_{i}}$ and $p_{\alpha}\geq 0$. Since
by the Sobolev Embedding Theorem (see e.g. [2]), $X$ is a Banach algebra, we
have

$\displaystyle||p(f_{1},...,f_{n})||_{X}$ $\displaystyle\leq\sum_{|\alpha|\leq
N}p_{\alpha}||f^{\alpha}||_{X}$ $\displaystyle\lesssim\sum_{|\alpha|\leq
N}p_{\alpha}\prod_{i=1}^{n}||f_{i}||_{X}^{\alpha_{i}}$
$\displaystyle=p(||f_{1}||_{X},...,||f_{n}||_{X}),$

which implies the desired result. ∎

######  Lemma 8.6.

There exists $\epsilon>0$ and a unique function
$\tilde{\mu}:(-\sqrt{\epsilon},\sqrt{\epsilon})\to\mathbb{R}_{>0}$ with
continuous derivatives of all orders such that $\tilde{\mu}(0)=n$ and
$\mu=\tilde{\mu}(s)$ solves $\overline{\gamma}_{1}(\mu,s)=0$ for
$s\in(-\sqrt{\epsilon},\sqrt{\epsilon})$. Moreover, $\tilde{\mu}$ is an even
function: $\tilde{\mu}(s)=\tilde{\mu}(-s)$.

###### Proof.

Recall that $F_{1}(\mu,u)=H_{1}(\mu)w+J_{1}(\mu,u)$ (where $H_{1}(\mu)$ and
$J_{1}(\mu,u)$ are defined in (5.15) and (6.2)). Using that
$\partial_{\mu}F_{1}(\mu,u)=(1+\frac{g}{2\sqrt{2\mu}}\psi)w$ and setting
$u=v_{s}+u^{\prime}_{s}$, with $v\equiv v_{s}\equiv(s\chi,c(\mu,s^{2}),0,0),\
u^{\prime}\equiv u^{\prime}_{s}\equiv u^{\prime}(\mu,v_{s}),\ c=c(\mu,s^{2})$,
we compute

$\displaystyle\partial_{\mu}$
$\displaystyle[s^{-1}F_{1}(\mu,v+u^{\prime})]=s^{-1}(1+\frac{g}{2\sqrt{2\mu}}\psi^{\prime})(s\chi+w^{\prime})$
$\displaystyle+s^{-1}\sum\limits_{i=1}^{4}\delta_{u_{i}}F_{1}w(\mu,v+u^{\prime})(\partial_{\mu}v_{i}+\partial_{\mu}u^{\prime}_{i})$
$\displaystyle=s^{-1}(1+\frac{g}{2\sqrt{2\mu}}\psi^{\prime})(s\chi+w^{\prime})+s^{-1}\delta_{\alpha}F_{1}(\mu,v+u^{\prime})\partial_{\mu}c$
$\displaystyle+s^{-1}\sum\limits_{i=1}^{5}\delta_{u_{i}}F_{1}(\mu,v+u^{\prime})\partial_{\mu}u^{\prime}_{i}.$
(8.49)

By Lemma 8.2, $\|\partial_{\mu}^{l}c\|_{\mathbb{R}^{2}}=\mathcal{O}(|s|^{2})$,
$l=0,1$. Since $||v||_{\mathcal{X}}$ is $\mathcal{O}(|s|)$, by Proposition 7.1
the terms $||\partial_{\mu}^{l}u^{\prime}_{i}||$ ($l=0,1$, $i=1,\dots,4$, with
the norms taken in the appropriate spaces), are $\mathcal{O}(|s|^{2})$. By
Lemma 8.5, this implies that all terms in (8) containing
$c,w^{\prime},\alpha^{\prime},z^{\prime},\psi^{\prime}$ or their
$\mu$-derivatives vanish at $(\mu,s)=(n,0)$. Therefore

$\partial_{\mu}[s^{-1}F_{1}(\mu,v+u^{\prime})]|_{(\mu,s)=(n,0)}=\chi$ (8.50)

and hence

$\partial_{\mu}\overline{\gamma}_{1}(n,0)=\langle\chi,\partial_{\mu}[s^{-1}F_{1}(\mu,s)]\,|_{(\mu,s)=(n,0)}\rangle_{L^{2}_{n}}=\|\chi\|_{L^{2}_{n}}^{2}\neq
0.$ (8.51)

Since $\overline{\gamma}_{1}(\mu,s)$ is continuously differentiable of all
orders in $\mu$ and $s$, by the Implicit Function Theorem, we obtain the first
statement of the lemma.

By the symmetry $\overline{\gamma}_{1}(\mu,-s)=-\overline{\gamma}_{1}(\mu,s)$
of $\overline{\gamma}_{1}$ and the uniqueness of the branch $\tilde{\mu}(s)$,
we have $\tilde{\mu}(s)=\tilde{\mu}(-s)$, which gives the second statement. ∎

We define $\mu(s)\equiv\tilde{\mu}(\sqrt{s})$, which is a function with
continuous derivatives of all orders for $s\in[0,\epsilon)$ for the same
reasons that $c(\mu,s):=\tilde{c}(\mu,\sqrt{s})$ was shown to be continuously
differentiable of all orders in Lemma 8.2. Furthermore, $\mu$ satisfies
$\tilde{\gamma}_{1}(\mu(s^{2}),s,c(\mu(s^{2}),s^{2}))=s\overline{\gamma}_{1}(\mu(s^{2}),s,c(\mu(s^{2}),s^{2}))=0$,
as required. ∎

We will now use the branch of solutions to (8.7) - (8.8), provided by Lemmas
8.2 and 8.4, and Corollary 7.2 to obtain the corresponding unique branch,
$(\mu_{s},u_{s})$, of solutions to (6.7), with

$\displaystyle\mu_{s}$ $\displaystyle\equiv\mu(s^{2}),\quad u_{s}\equiv
v_{s}+u^{\prime}_{s},$ (8.52) $\displaystyle v_{s}$
$\displaystyle\equiv(s\chi,c_{s},0,0),\quad c_{s}\equiv c(\mu_{s},s^{2}),$
(8.53) $\displaystyle u^{\prime}_{s}$ $\displaystyle\equiv
u^{\prime}(\mu,v_{s}).$ (8.54)

(8.52) - (8.54) have continuous $s$-derivatives of all orders because each
component function has continuous derivatives of all orders. Symmetry (7.28)
with $\delta=\pi$ and the relation
$T_{\pi}(f_{1},f_{2},f_{3},f_{4})=(-f_{1},f_{2},f_{3},f_{4})$ imply that
$(u_{s}^{\prime})_{1}$ is an odd function of $s$ and $(u_{s}^{\prime})_{2}$,
$(u_{s}^{\prime})_{3}$ and $(u_{s}^{\prime})_{4}$ are even functions of $s$.
Arguing as in the case of Lemma 8.2 above shows that the functions:

$\displaystyle g_{1}(s)$
$\displaystyle:=\begin{cases}\frac{1}{\sqrt{s}}(u^{\prime}_{\sqrt{s}})_{1},\quad&s\neq
0,\\\ 0,\quad&s=0,\end{cases}\quad
g_{2}(s):=c_{\sqrt{s}}+(u^{\prime}_{\sqrt{s}})_{2},$ (8.55) $\displaystyle
g_{3}(s)$ $\displaystyle:=(u^{\prime}_{\sqrt{s}})_{3},\quad
g_{4}(s):=(u^{\prime}_{\sqrt{s}})_{4},\quad g_{5}(s):=\mu_{\sqrt{s}}-n,$
(8.56)

are well-defined for $s\geq 0$ and have continuous derivatives of all orders.
By Proposition 7.1, these functions have the properties listed in Theorem 8.1.
The above definitions and equations (8.52) - (8.54) imply
$u_{s}=(s\chi,\frac{1}{e}a^{n},0,0)+(g_{1}(s),\dots,g_{4}(s))$. Hence, this
solution is of the form (8.1). Now, by Proposition 6.1, this also solves
system (4.3) - (4.6), completing the proof. ∎

##  9 Proof of Theorem 2.2(a), (b)

Recall that $M_{W}$, $M_{Z}$, $M_{H}$ are the masses of the $W$, $Z$ and Higgs
bosons, respectively, and that $\tau$ is the shape parameter of the lattice
$\mathcal{L}$ (see the paragraph before Theorem 2.3 of Section 2). We
introduce the notation

$\displaystyle\langle f\rangle:=\frac{1}{|\Omega|}\int_{\Omega}f\ \qquad\text{
and }\ \qquad\langle
f\rangle^{\prime}:=\frac{1}{|\Omega^{\prime}|}\int_{\Omega^{\prime}}f,$ (9.1)

the averages of $f$ over fundamental domains $\Omega$ and $\Omega^{\prime}$.
Furthermore, we will use the space (cf. (5.10))

$\displaystyle L^{2}_{\mathcal{L}}:=\\{\psi\in
L^{2}_{loc}(\mathbb{R}^{2},\mathbb{R}):\psi(x+s)=\psi(x)\ \forall
s\in\mathcal{L}\\}.$ (9.2)

We will need the following function:

$\displaystyle\alpha_{M_{Z},M_{H}}(\tau)$
$\displaystyle:=\langle|B|^{2}G_{M_{Z},M_{H}}(|B|^{2})\rangle/\langle|B|^{2}\rangle^{2},$
(9.3)

where $B$ is a differentiable complex function on $\mathbb{R}^{2}$ satisfying
$\bar{\partial}_{A^{b}}B=0$, where $\bar{\partial}_{q}$ is the complexified
covariant derivative given by
$\bar{\partial}_{q}:=(\nabla_{q})_{1}+i(\nabla_{q})_{2},$ and
$\chi(x+s)=e^{i(\frac{eb}{2}s\wedge x+k_{s})}\chi(x),\ \forall
s\in\mathcal{L}$ (cf. (3.21))121212$\chi$ can be expressed in terms of the
Jacobi theta function, see Appendix H, and $G_{M,M^{\prime}}$ is the operator-
family on the space $L^{2}_{\mathcal{L}}$ given by

$\displaystyle G_{M,M^{\prime}}:=G_{M^{\prime}}-G_{M},\ \quad\text{ where }\
\quad G_{M}:=(-\Delta+M^{2})^{-1}.$ (9.4)

Furthermore, we introduce the function (see [25])

$\displaystyle\eta_{M_{Z},M_{H}}(\tau)=[M_{W}^{2}\alpha_{M_{Z},M_{H}}(\tau)+\sin^{2}\theta]^{-1}.$
(9.5)

######  Proposition 9.1.

If $M_{Z}<M_{H}$, the parameter $s$ of the branch (8.1) is related to the
magnetic field strength by

$\displaystyle s^{2}$
$\displaystyle=\frac{eb}{g^{2}2\langle|B|^{2}\rangle}\eta_{M_{Z},M_{H}}(\tau)\omega+R_{s}(\omega),\
\quad\omega:=1-\frac{M_{W}^{2}}{eb},$ (9.6)

where $R_{s}(\omega)$ is a real, smooth function of $\omega$ satisfying

$\displaystyle R_{s}(\omega)=\mathcal{O}(|\omega|^{2}).$ (9.7)

Before proving Proposition 9.1, we shall see how it implies statements (a) and
(b) of Theorem 2.2.

###### Proof of Theorem 2.2(a), (b).

Since the operator $G_{M_{Z},M_{H}}$ is positivity preserving, when
$M_{Z}<M_{H}$, the function $G_{M_{Z},M_{H}}(|B|^{2})$ is positive; in
particular, $\alpha_{M_{Z},M_{H}}(\tau)$ and hence $\eta_{M_{Z},M_{H}}(\tau)$
are positive. Furthermore, when the right-hand side of (9.6) is positive, we
may take the square root, solving for $s$ as a function of $b$, $s=s(b)$,
having continuous derivatives of all orders. When $|1-\frac{M_{W}^{2}}{eb}|\ll
1$, the right-hand side of (9.6) is positive if and only if
$1-\frac{M_{W}^{2}}{eb}>0$.131313The condition $0<1-\frac{M_{W}^{2}}{eb}\ll 1$
is equivalent to the condition $0<1-\frac{M_{W}^{2}}{2\pi}|\mathcal{L}|\ll 1$
of Theorem 2.2. Plugging $s=s(b)$ into (8.1) (i.e. passing from the
bifurcation parameter $s$ to the physical parameter $b$), undoing the
rescaling (4.1), and recalling that $b_{*}=\frac{M_{W}^{2}}{e}$, we arrive at
the branch, $U_{\mathcal{L}}\equiv(W_{b},A_{b},Z_{b},\varphi_{b})$, of
solutions of (3.12) - (3.15), which has the properties listed in statements
(a) and (b) of Theorem 2.2. ∎

The following statement follows from the proof above:

######  Lemma 9.2.

$U_{\mathcal{L}}$ is continuously differentiable of all orders in $b$ for $b$
in an open right half-interval of $b_{*}$

###### Proof of Proposition 9.1.

Consider the solution branch $(\mu_{s},w_{s},a_{s},z_{s})$ given in equation
(8.1) and described in Theorem 8.1. Using Taylor’s Theorem for Banach spaces
(see e.g. [16]) and recalling the relation $\xi=\sqrt{2\mu}/g$, we may expand
this branch in $s$ as follows:

$\displaystyle\begin{cases}w_{s}=s\chi+s^{3}w^{\prime}+\mathcal{O}(|s|^{5}),\\\
a_{s}=\frac{1}{e}a^{n}+s^{2}a^{\prime}+s^{4}a^{\prime\prime}+\mathcal{O}(|s|^{6}),\\\
z_{s}=s^{2}z^{\prime}+\mathcal{O}(|s|^{4}),\\\
\psi_{s}:=\phi_{s}-\xi_{s}=s^{2}\psi^{\prime}+\mathcal{O}(|s|^{4}),\\\
\xi_{s}:=\sqrt{2\mu_{s}}/g=\sqrt{2n}/g+s^{2}\xi^{\prime}+\mathcal{O}(|s|^{4}),\end{cases}$
(9.8)

where $w^{\prime},a^{\prime},z^{\prime},\psi^{\prime},\xi^{\prime}$ and
$a^{\prime\prime}$ are the coefficients of $s^{2}$ and $s^{4}$, respectively,
in the Taylor expansion of $g_{j}(s^{2})$, $j=0,...,5$, in (8.1), and $\chi$
is defined in (5.22). Here $\mathcal{O}(|s|^{p})$ stand for various error
terms which, together with their (covariant) derivatives, have norms of order
$\mathcal{O}(|s|^{p})$ when taken in the appropriate spaces.

To rewrite the asymptotics in terms of the parameter $b$, we analyze how $s$
depends on $b$. For this, we use the definitions $\xi_{s}=\sqrt{2\mu_{s}}/g$
and $\mu:=\frac{1}{2}(g\xi)^{2}=\frac{1}{2}(gr\varphi_{0})^{2}$, with
$r:=\sqrt{\frac{n}{eb}}$ (see (4.2)) to find the following equation for
$s^{2}$:

$\displaystyle\xi_{s}=\sqrt{\frac{n}{eb}}\varphi_{0}.$ (9.9)

To solve this equation for $s^{2}$, we use the Implicit Function Theorem. By
(9.8), we can write $\xi_{s}=\sqrt{2n}/g+g_{\xi}(s^{2})$, where recall,
$g_{\xi}(0)=0$ and $g^{\prime}_{\xi}(0)=\xi^{\prime}$. Hence, we have to show
that $\xi^{\prime}\neq 0$.

######  Lemma 9.3.

We have $\xi^{\prime}\neq 0$.

###### Proof.

We find relations between $\psi^{\prime}$, $a^{\prime}$ and $z^{\prime}$
entering (9.8). Plugging (9.8) into Equations (4.4) - (4.6), we obtain at
order $s^{4}$

$\displaystyle\begin{cases}-\Delta
a^{\prime}-e\operatorname{curl}^{*}|\chi|^{2}=0\\\
(-\Delta+\frac{n}{\cos^{2}\theta})z^{\prime}-g\cos\theta\operatorname{curl}^{*}|\chi|^{2}=0\\\
(-\Delta+\frac{4\lambda
n}{g^{2}})\psi^{\prime}+\frac{g}{2}\sqrt{2n}|\chi|^{2}=0.\\\ \end{cases}$
(9.10)

We solve these equations, using that
$\operatorname{curl}^{*}|\chi|^{2}=\operatorname{curl}^{*}(|\chi|^{2}-\langle|\chi|^{2}\rangle^{\prime})$
and
$|\chi|^{2}-\langle|\chi|^{2}\rangle^{\prime}\in\operatorname{Ran}(\Delta)$,
for the first one, to find 141414To check the solutions, one may use that
$\operatorname{curl}\operatorname{curl}^{*}=-\Delta$.

$\displaystyle\begin{cases}a^{\prime}=e\operatorname{curl}^{*}G_{0}(|\chi|^{2}-\langle|\chi|^{2}\rangle^{\prime})\\\
z^{\prime}=g\cos\theta\operatorname{curl}^{*}G_{m_{z}}(|\chi|^{2})\\\
\psi^{\prime}=-\frac{g}{2}\sqrt{2n}\;G_{m_{h}}(|\chi|^{2}),\\\ \end{cases}$
(9.11)

where $G_{m}:=(-\Delta+m^{2})^{-1}$ acting on the space
$L^{2}_{\mathcal{L}^{\prime}}$ (cf. (9.4)), and
$m_{z}:=\frac{\sqrt{n}}{\cos\theta}$ and $m_{h}:=\frac{\sqrt{4\lambda n}}{g}$
are the masses of the rescaled Z and Higgs boson ($\Phi$) fields, $z$ and
$\phi$, respectively. Next, we use the following relation proven in Appendix
E:

$\displaystyle\int_{\Omega^{\prime}}\mkern-9.0mug\sqrt{2n}\xi^{\prime}|\chi|^{2}=\int_{\Omega^{\prime}}\mkern-9.0mu\big{(}-g\sqrt{2n}\psi^{\prime}|\chi|^{2}+\operatorname{curl}\nu^{\prime}|\chi|^{2}-g^{2}|\chi|^{4}\big{)},$
(9.12)

where, recall, $\nu^{\prime}:=g(a^{\prime}\sin\theta+z^{\prime}\cos\theta)$.
First, we evaluate $\operatorname{curl}\nu^{\prime}$. The relations
$(-\Delta+m^{2})G_{m}=\mathbf{1}$ and
$\operatorname{curl}\operatorname{curl}^{*}=-\Delta$ imply
$\operatorname{curl}a^{\prime}=e(|\chi|^{2}-\langle|\chi|^{2}\rangle^{\prime})$.
Next, the second relation in (9.11) and the relation
$\operatorname{curl}\operatorname{curl}^{*}=-\Delta$ yield
$\operatorname{curl}z^{\prime}=g\cos\theta(-\Delta)(-\Delta+m_{z}^{2})^{-1}|\chi|^{2}$,
which, together with $m_{z}:=\frac{\sqrt{n}}{\cos\theta}$, gives
$\operatorname{curl}z^{\prime}=g\cos\theta|\chi|^{2}-g\frac{n}{\cos\theta}G_{m_{z}}|\chi|^{2}$.
Finally, using that $e:=g\sin\theta$, we conclude that

$\displaystyle\operatorname{curl}\nu^{\prime}=g^{2}|\chi|^{2}-eg\sin\theta\langle|\chi|^{2}\rangle-g^{2}nG_{m_{z}}(|\chi|^{2}).$
(9.13)

Plugging the last relation and equation (9.11) into the relation (9.12), gives

$\displaystyle g\sqrt{2n}\xi^{\prime}\langle$
$\displaystyle|\chi|^{2}\rangle^{\prime}=-g^{2}[m_{w}^{2}\langle
G_{m_{z},m_{h}}(|\chi|^{2})|\chi|^{2}\rangle^{\prime}+\sin^{2}\theta(\langle|\chi|^{2}\rangle^{\prime})^{2}],$
(9.14)

where $m_{w}:=\sqrt{n}$ is the mass of the rescaled $W$ boson field $w$ and
the operator-family $G_{m,m^{\prime}}$ is defined by (9.4) but on the space
$L^{2}_{\mathcal{L}^{\prime}}(\mathbb{R}^{2})$, rather than on
$L^{2}_{\mathcal{L}}(\mathbb{R}^{2})$. We solve for $\xi^{\prime}$ and
simplify the solution as follows:

$\xi^{\prime}=-\frac{g}{\sqrt{2n}}\langle|\chi|^{2}\rangle^{\prime}\eta^{\prime-1},$
(9.15)

where

$\eta^{\prime}\equiv\eta^{\prime}_{m_{z},m_{h}}(\tau):=[m_{w}^{2}\alpha^{\prime}_{m_{z},m_{h}}(\tau)+\sin^{2}\theta]^{-1},$
(9.16)

with, recall, $m_{w}:=\sqrt{n}$, $m_{z}:=\frac{\sqrt{n}}{\cos\theta}$ and
$m_{h}:=\frac{\sqrt{4\lambda n}}{g}$ the masses of the rescaled W, Z and Higgs
boson fields, $w$, $z$ and $\phi$, respectively, and

$\displaystyle\alpha^{\prime}_{m_{z},m_{h}}(\tau):=\langle|\chi|^{2}G_{m_{z},m_{h}}(|\chi|^{2})\rangle^{\prime}/(\langle|\chi|^{2}\rangle^{\prime})^{2}.$
(9.17)

The operator $G_{m_{z},m_{h}}$ is positivity preserving and therefore the
function $\alpha^{\prime}_{m_{z},m_{h}}(\tau)$ (and hence $\eta^{\prime}$) is
positive, if and only if $m_{z}<m_{h}$ (equivalently, $M_{Z}<M_{H}$), in which
case $\xi^{\prime}<0$. ∎

We now derive the estimate (9.6) - (9.7) for $s^{2}$. Equations (9.8) and
(9.9) give $\xi_{s}$ as a function of $s$ and $b$ respectively, yielding

$\displaystyle\xi_{s}^{2}=[\frac{\sqrt{2n}}{g}+g_{\xi}(s^{2})]^{2}=\frac{n}{eb}\varphi_{0}^{2},$
(9.18)

which can be rearranged to give

$\displaystyle\frac{2\sqrt{2n}}{g}g_{\xi}(s^{2})+g_{\xi}(s^{2})^{2}=\frac{2n}{g^{2}}\omega,$
(9.19)

where, recall, $\omega=1-\frac{M_{W}^{2}}{eb}$, with
$M_{W}=\frac{1}{\sqrt{2}}g\varphi_{0}$. Recall that $g_{\xi}(0)=0$ and
$g_{\xi}^{\prime}(0)=\xi^{\prime}$. We have

$\displaystyle\frac{d}{ds^{2}}|_{s^{2}=0}[\frac{2\sqrt{2n}}{g}g_{\xi}(s^{2})+g_{\xi}(s^{2})^{2}]=\frac{2\sqrt{2n}}{g}\xi^{\prime}.$
(9.20)

Since $\xi^{\prime}\neq 0$ and $g_{\xi}(s^{2})$ is continuously differentiable
of all orders (see Theorem 8.1), by the Implicit Function Theorem, we may
solve (9.19) for $s^{2}$, with the solution, $s^{2}=s^{2}(\omega)$, with
continuous derivatives of all orders in $\omega$. Explicitly, (9.19) - (9.20)
give

$\displaystyle s^{2}=\frac{g}{2\sqrt{2n}}$
$\displaystyle\xi^{\prime-1}\frac{2n}{g^{2}}\omega+\mathcal{O}(|\omega|^{2}).$
(9.21)

Plugging (9.15) into (9.21) gives

$\displaystyle
s^{2}=\frac{n}{g^{2}}\frac{\omega}{\langle|\chi|^{2}\rangle^{\prime}}\eta^{\prime}+\tilde{R}_{s}(\omega),$
(9.22)

where $\tilde{R}_{s}(\omega)$ satisfies
$\tilde{R}_{s}(\omega)=\mathcal{O}(|\omega|^{2})$. Furthermore, since the
solution $s^{2}=s^{2}(\omega)$ is continuously differentiable of all orders in
$\omega$, so is the remainder term $\tilde{R}_{s}(\omega)$.

To derive (9.6) from (9.22), we use the following lemma, proven in Appendix F,
relating the rescaled $\eta$-function and the original one:

######  Lemma 9.4.

The functions $\eta_{M_{Z},M_{H}}(\tau)$and
$\eta^{\prime}_{m_{z},m_{h}}(\tau)$ are related as

$\displaystyle\eta^{\prime}_{m_{z},m_{h}}(\tau)$
$\displaystyle=\eta_{M_{Z},M_{H}}(\tau)+R_{\eta}(\omega),$ (9.23)

where $R_{\eta}(\omega)$ is a real, smooth function of $\omega$ satisfying

$\displaystyle R_{\eta}(\omega)=\mathcal{O}(|\omega|).$ (9.24)

Now, we rescale back from $\chi$ to the original fields $(B,iB)$:

$\chi(x):=(rB(rx),irB(rx)),$

and use that $|\Omega|=r^{2}|\Omega^{\prime}|$ to obtain
$\langle|\chi|^{2}\rangle^{\prime}=2r^{2}\langle|B|^{2}\rangle$. Plugging the
latter relation and Eq. (9.23) into (9.22), we arrive at Eq. (9.6). ∎

##  10 Asymptotics of the Weinberg-Salam energy near $b=M_{W}^{2}/e$

The main result of this section is the following:

######  Theorem 10.1.

If $M_{Z}<M_{H}$, then the WS energy (3) of the branch of solutions (8.1) has
the following expansion:

$\displaystyle\frac{1}{|\Omega|}E_{\Omega}^{WS}(W_{b},A_{b},Z_{b},\varphi_{b})=\frac{1}{2}b^{2}-\frac{1}{2}b^{2}\sin^{2}\theta\
$ $\displaystyle\eta_{M_{Z},M_{H}}(\tau)\omega^{2}+R_{E}(\omega),$ (10.1)

where, recall, $\omega=1-\frac{M_{W}^{2}}{eb}$, with
$M_{W}=\frac{1}{\sqrt{2}}g\varphi_{0}$, and $R_{E}(\omega)$ is a real function
with continuous derivatives of all orders satisfying

$\displaystyle R_{E}(\omega)=\mathcal{O}(|\omega|^{3}).$ (10.2)

Before proving Theorem 10.1, we derive from it Theorem 2.2 (c).

###### Proof of Theorem 2.2 (c).

Since $\eta_{M_{Z},M_{H}}(\tau)$ is positive,151515See the discussion
following Proposition 9.1 for details. the second term in Equation (10.1) is
negative, and so for $0<1-\frac{M_{W}^{2}}{eb}\ll 1$, $E_{\Omega}^{WS}$ is
less than the vacuum energy $\frac{1}{2}b^{2}|\Omega|$. This proves Theorem
2.2 (c). ∎

###### Proof of Theorem 10.1.

Let
$\mathcal{E}^{\prime}(w_{s},a_{s},z_{s},\psi_{s}+\xi_{s};r):=\frac{1}{|\Omega^{\prime}|}\mathcal{E}_{\Omega^{\prime}}(w_{s},a_{s},z_{s},\psi_{s}+\xi_{s};r)$,
where $\mathcal{E}_{\Omega^{\prime}}$ is the rescaled WS energy given in (4).
In Appendix G, we derive the following expansion of $\mathcal{E}^{\prime}$
evaluated at family (9.8) of solutions, up to order $s^{4}$:

$\displaystyle\mathcal{E}^{\prime}(w_{s},a_{s},z_{s},\psi_{s}+\xi_{s};r)$
$\displaystyle=\frac{1}{2}\frac{n^{2}}{e^{2}}+s^{4}\frac{1}{|\Omega^{\prime}|}\int_{\Omega^{\prime}}\frac{1}{2}|\operatorname{curl}z^{\prime}|^{2}$
$\displaystyle+$
$\displaystyle\frac{1}{2}|\operatorname{curl}a^{\prime}|^{2}+\
g\sqrt{2n}(\psi^{\prime}+\xi^{\prime})|\chi|^{2}+\frac{n}{2\cos^{2}\theta}|z^{\prime}|^{2}$
$\displaystyle+$ $\displaystyle|\nabla\psi^{\prime}|^{2}+\frac{4\lambda
n}{g^{2}}\psi^{\prime
2}-|\chi|^{2}\operatorname{curl}\nu^{\prime}+\frac{g^{2}}{2}|\chi|^{4}$
$\displaystyle+$ $\displaystyle R_{\varepsilon}(s),$ (10.3)

where $R_{\varepsilon}(s)=\mathcal{O}(|s|^{6})$ and has continuous derivatives
of all orders, $\nu^{\prime}:=g(a^{\prime}\sin\theta+z^{\prime}\cos\theta)$
and, recall, $\xi_{s}=\sqrt{2\mu_{s}}/g$.

To simplify notation, in what follows, we shall suppress the arguments
$(w_{s},a_{s},z_{s},\psi_{s}+\xi_{s};r)$ of $\mathcal{E}^{\prime}$. We claim
the following relation:

$\displaystyle\mathcal{E}^{\prime}$
$\displaystyle=\frac{1}{2}\frac{n^{2}}{e^{2}}-s^{4}\frac{g^{2}}{2}\langle|\chi|^{2}\rangle^{2}\eta^{\prime-1}+R_{\varepsilon}(s),$
(10.4)

where, recall,
$\eta^{\prime}=\eta^{\prime}_{m_{z},m_{h}}(\tau,r):=\big{[}m_{w}^{2}\alpha_{m_{z},m_{h}}(\tau)+\sin^{2}\theta\big{]}^{-1}$
and $\alpha^{\prime}_{m_{z},m_{h}}(\tau,r)$ is given in (9.17).

###### Proof of (10.4).

We simplify the integral at order $s^{4}$ in (10) by applying equations (9.10)
for $a^{\prime}$, $z^{\prime}$ and $\psi^{\prime}$ to convenient groupings of
terms.

First, we address the $z^{\prime}$ terms. Integrating by parts and factoring
out $z^{\prime}$ gives

$\displaystyle\frac{1}{|\Omega^{\prime}|}\int_{\Omega^{\prime}}\frac{1}{2}|\operatorname{curl}z^{\prime}|^{2}+\frac{n}{\cos^{2}\theta}|z^{\prime}|^{2}$
$\displaystyle=\frac{1}{|\Omega^{\prime}|}\int_{\Omega^{\prime}}\frac{1}{2}z^{\prime}\cdot(-\Delta+\frac{n}{\cos^{2}\theta})z^{\prime}.$
(10.5)

Applying (9.10) for $z^{\prime}$ gives

$\displaystyle\frac{1}{|\Omega^{\prime}|}\int_{\Omega^{\prime}}\frac{1}{2}|\operatorname{curl}z^{\prime}|^{2}+\frac{n}{\cos^{2}\theta}|z^{\prime}|^{2}$
$\displaystyle=\frac{1}{|\Omega^{\prime}|}\int_{\Omega^{\prime}}\frac{1}{2}z^{\prime}\cdot
g\cos\theta\operatorname{curl}^{*}|\chi|^{2}.$ (10.6)

Integrating by parts again gives

$\displaystyle\frac{1}{|\Omega^{\prime}|}\int_{\Omega^{\prime}}\frac{1}{2}|\operatorname{curl}z^{\prime}|^{2}+\frac{n}{\cos^{2}\theta}|z^{\prime}|^{2}$
$\displaystyle=\frac{1}{|\Omega^{\prime}|}\int_{\Omega^{\prime}}\frac{1}{2}g\cos\theta(\operatorname{curl}z^{\prime})|\chi|^{2}.$
(10.7)

Next, we address the $a^{\prime}$ term. Integrating by parts gives

$\displaystyle\frac{1}{|\Omega^{\prime}|}\int_{\Omega^{\prime}}\frac{1}{2}|\operatorname{curl}a^{\prime}|^{2}$
$\displaystyle=\frac{1}{|\Omega^{\prime}|}\int_{\Omega^{\prime}}\frac{1}{2}a^{\prime}\cdot(-\Delta)a^{\prime}.$
(10.8)

Applying (9.10) for $a^{\prime}$ gives

$\displaystyle\frac{1}{|\Omega^{\prime}|}\int_{\Omega^{\prime}}\frac{1}{2}|\operatorname{curl}a^{\prime}|^{2}$
$\displaystyle=\frac{1}{|\Omega^{\prime}|}\int_{\Omega^{\prime}}\frac{1}{2}a^{\prime}\cdot
e\operatorname{curl}^{*}|\chi|^{2}.$ (10.9)

Integrating by parts again gives

$\displaystyle\frac{1}{|\Omega^{\prime}|}\int_{\Omega^{\prime}}\frac{1}{2}|\operatorname{curl}a^{\prime}|^{2}$
$\displaystyle=\frac{1}{|\Omega^{\prime}|}\int_{\Omega^{\prime}}\frac{1}{2}g\sin\theta(\operatorname{curl}a^{\prime})|\chi|^{2}.$
(10.10)

Next, we address the $\psi^{\prime}$ terms. Integrating by parts and factoring
out $\psi^{\prime}$ gives

$\displaystyle\frac{1}{|\Omega^{\prime}|}\int_{\Omega^{\prime}}|\nabla\psi^{\prime}|^{2}+\frac{4\lambda
n}{g^{2}}$ $\displaystyle\psi^{\prime 2}+g\sqrt{2n}\psi^{\prime}|\chi|^{2}$
$\displaystyle=\frac{1}{|\Omega^{\prime}|}\int_{\Omega^{\prime}}\psi^{\prime}(-\Delta+\frac{4\lambda
n}{g^{2}}+g\sqrt{2n}|\chi|^{2})\psi^{\prime}.$ (10.11)

Applying (9.10) for $\psi^{\prime}$ gives

$\displaystyle\frac{1}{|\Omega^{\prime}|}\int_{\Omega^{\prime}}|\nabla\psi^{\prime}|^{2}+\frac{4\lambda
n}{g^{2}}$ $\displaystyle\psi^{\prime
2}+g\sqrt{2n}\psi^{\prime}|\chi|^{2}=\frac{1}{|\Omega^{\prime}|}\int_{\Omega^{\prime}}\frac{g}{2}\sqrt{2n}\psi^{\prime}|\chi|^{2}.$
(10.12)

The $\xi^{\prime}$ term is addressed by (9.12).

Finally, there are two remaining terms of the integral at order $s^{4}$ in
(10), which we will not presently simplify:

$\displaystyle\frac{1}{|\Omega^{\prime}|}\int_{\Omega^{\prime}}-|\chi|^{2}\operatorname{curl}\nu^{\prime}+\frac{g^{2}}{2}|\chi|^{4}.$
(10.13)

Adding equations (9.12), (10.7), (10.10), (10.12) and (10.13) gives

$\displaystyle\mathcal{E}^{\prime}=\frac{1}{2}\frac{n^{2}}{e^{2}}+s^{4}\frac{1}{|\Omega^{\prime}|}\int_{\Omega^{\prime}}\frac{1}{2}g\sqrt{2n}\psi^{\prime}|\chi|^{2}-\frac{1}{2}\operatorname{curl}\nu^{\prime}|\chi|^{2}+\frac{1}{2}g^{2}|\chi|^{4}.$
(10.14)

Recall that $e=g\sin\theta$ and $m_{w}^{2}=m_{z}^{2}\cos^{2}\theta=n$.
Plugging Equation (9.13) for $\operatorname{curl}\nu^{\prime}$ and (9.11) for
$\psi^{\prime}$ into (10.14) gives Equation (10.4), as required. ∎

Plugging (9.22) into (10.4) gives

$\displaystyle\mathcal{E}^{\prime}=\frac{1}{2}\frac{n^{2}}{e^{2}}-$
$\displaystyle\frac{1}{2}\frac{n^{2}}{g^{2}}\omega^{2}\eta^{\prime}_{m_{z},m_{h}}(\tau,r)+\tilde{R}_{\varepsilon}(\omega),$
(10.15)

where $\tilde{R}_{\varepsilon}(\omega)$ has continuous derivatives of all
orders and satisfies
$\tilde{R}_{\varepsilon}(\omega)=\mathcal{O}(|\omega|^{3})$.

To compute the WS energy (3), evaluated at $(W_{b},A_{b},Z_{b},\varphi_{b})$,
we recall that $E_{\Omega}^{WS}=\frac{1}{r^{2}}\mathcal{E}_{\Omega^{\prime}}$,
which implies

$\displaystyle\frac{1}{|\Omega|}E_{\Omega}^{WS}=\frac{|\Omega^{\prime}|}{r^{2}|\Omega|}\mathcal{E}^{\prime}=\frac{e^{2}b^{2}}{n^{2}}\mathcal{E}^{\prime},\quad
r=\sqrt{\frac{|\Omega|}{|\Omega^{\prime}|}}=\sqrt{\frac{n}{eb}}.$ (10.16)

Equation (10.1) follows by plugging (10.15) and (9.23) into (10.16). Since the
remainder terms $\tilde{R}_{s}$, $R_{\alpha}$ and $\tilde{R}_{\varepsilon}$
(of (9.22), (9.23) and (10.15), respectively) have continuous derivatives of
all orders, so does the remainder term $R_{E}$. ∎

##  11 Shape of lattice solutions

In this section we shall prove Theorem 2.3. Recall the shape parameter $\tau$
described in the paragraph preceding (2.25). We return briefly to working with
the rescaled fields to prove that $\mathcal{E}_{\Omega^{\prime}}(u;r)$ (and
hence $E_{\Omega}^{WS}(U)$), given in (4), is continuously differentiable of
all orders in the shape parameter $\tau$ (restricted to domain (2.25)), which
enters through $\Omega^{\prime}$ and $\Omega$, as well as the spaces
containing $u$ and $U$. Below, we write

$\displaystyle
u_{\tau,b}(x)\equiv(w_{\tau,b}(x),a_{\tau,b}(x),z_{\tau,b}(x),\phi_{\tau,b}(x)),$
(11.1)
$\displaystyle\mathcal{E}(\tau,b,u)\equiv\mathcal{E}_{\Omega^{\prime}}(u;r),$
(11.2) $\displaystyle
U_{\tau,b}(x)\equiv(W_{\tau,b}(x),A_{\tau,b}(x),Z_{\tau,b}(x),\varphi_{\tau,b}(x)),$
(11.3) $\displaystyle E(\tau,b,U)\equiv E_{\Omega}^{WS}(U),$ (11.4)
$\displaystyle\mathcal{X}_{\tau}\equiv\mathcal{X},$ (11.5)

to emphasize the dependence of the family of solutions (9.8), the
corresponding energy $\eqref{WS-energy-resc}$ (respectively (3)) and the space
(5.7) containing these solutions on the shape parameter $\tau$, the magnetic
field strength $b$ and the position in space $x\in\mathbb{R}^{2}$. Also,
recall the notation $r:=\sqrt{n/eb}$.

To get rid of the dependency of the space $\mathcal{X}_{\tau}$ containing
$u_{\tau,b}$, on the shape parameter $\tau$, we make the change of coordinates
$y=m_{\tau}^{-1}x$ with
$m_{\tau}=\frac{1}{\sqrt{\operatorname{Im}(\tau)}}\left(\begin{array}[]{cc}1&\text{Re}(\tau)\\\
0&\text{Im}(\tau)\end{array}\right)$. This defines a function

$\displaystyle M_{\tau}:\mathcal{X}_{\tau}$ $\displaystyle\to\mathcal{X}_{1},$
$\displaystyle(M_{\tau}u)(x)$ $\displaystyle=u(m_{\tau}x),$ (11.6)

that is linear in $u$ (this change of coordinates transforms $\Omega^{\prime}$
into a square of area $2\pi$). This in turn allows us to define the following
functions on the fixed space $\mathcal{X}_{1}$:

$\displaystyle G^{\prime}:\mathbb{C}\times\mathbb{R}\times\mathcal{X}_{1}$
$\displaystyle\to\mathbb{C}\times\mathbb{R}\times\mathcal{Y}_{1}$
$\displaystyle G^{\prime}(\tau,b,v)$
$\displaystyle=M_{\tau}G(b,M_{\tau}^{-1}v),$ (11.7)

and

$\displaystyle\varSigma:\mathbb{C}\times\mathbb{R}\times\mathcal{X}_{1}$
$\displaystyle\to\mathbb{C}\times\mathbb{R}\times\mathcal{Y}_{1}$
$\displaystyle\varSigma(\tau,b,v)$
$\displaystyle=M_{\tau}\varepsilon_{WS}(b,M_{\tau}^{-1}v),$ (11.8)

where, recall, $G(b,v)$ is the map given by the left-hand side of (4.3) -
(4.6), given explicitly in (5.1), and
$\varepsilon_{WS}(b,u):=\varepsilon_{WS}(u;r)$ is the rescaled energy density
given by the integrand in (4) ($\varepsilon_{WS}$ depends on the magnetic
field strength $b$ but does not directly depend on the shape parameter
$\tau$).

######  Lemma 11.1.

$G^{\prime}(\tau,b,v)$ and $\varSigma(\tau,b,v)$ are continuously
differentiable of all orders in $\operatorname{Re}(\tau)$,
$\operatorname{Im}(\tau)$, $b$ and $v$.

###### Proof.

Since $G(b,u)$ and $\varepsilon_{WS}(b,u)$ have continuous $b$ and $u$
derivatives of all orders, and $M_{\tau}$ is a linear map independent of $b$
and $v$, it follows that $G^{\prime}(\tau,b,v)$ and $\varSigma(\tau,b,v)$ have
continuous $b$\- and $v$-derivatives of all orders.

For the $\tau$-derivatives, note that

$\displaystyle M_{\tau}\circ\partial_{x_{1}}\circ
M_{\tau}^{-1}(v_{j})(x)=\frac{1}{\sqrt{\operatorname{Im}(\tau)}}\partial_{x_{1}}v_{j}(x),\quad
j=1,...,4,$ (11.9) $\displaystyle M_{\tau}\circ\partial_{x_{2}}\circ
M_{\tau}^{-1}(v_{j})(x)$
$\displaystyle\qquad=\frac{1}{\sqrt{\operatorname{Im}(\tau)}}(\operatorname{Re}(\tau)\partial_{x_{1}}v_{j}(x)+\operatorname{Im}(\tau)\partial_{x_{2}}v_{j}(x)),\quad
j=1,...,4,$ (11.10)

are continuously differentiable of all orders in $\operatorname{Re}(\tau)$ and
$\operatorname{Im}(\tau)$. Since $G(b,u)$ and $\varepsilon_{WS}(b,u)$ are
polynomials in the components of $u$ and their (covariant) derivatives,
$G^{\prime}$ and $\varSigma$ are simply $G$ and $\varepsilon_{WS}$ with the
coefficients of the derivative-containing terms multiplied by smooth functions
of $\operatorname{Re}(\tau)$ and $\operatorname{Im}(\tau)$. Therefore
$G^{\prime}(\tau,b,v)$ and $\varSigma(\tau,b,v)$ have continuous
$\operatorname{Re}(\tau)$\- and $\operatorname{Im}(\tau)$-derivatives of all
orders. ∎

######  Lemma 11.2.

$v_{\tau,b}:=M_{\tau}u_{\tau,b}$ is continuously differentiable of all orders
in $\text{Re}(\tau)$ and $\text{Im}(\tau)$.

###### Proof.

Let $\tau_{0}$ be an arbitrary shape parameter, and recall that $\delta_{\\#}$
denotes the partial (real) Gâteaux derivative with respect to $\\#$. Then
$G^{\prime}(\tau_{0},b,v_{\tau_{0},b})=\\\ M_{\tau_{0}}G(b,u_{\tau_{0},b})=0$,
$\delta_{v}G(\tau_{0},b,v_{\tau_{0},b})=M_{\tau_{0}}\circ\delta_{u}G(b,u_{\tau_{0},b})\circ
M_{\tau_{0}}^{-1}$ is invertible, and by Lemma 11.1, $G^{\prime}$ is
continuously differentiable of all orders in $\tau$, $b$ and $v$. Therefore,
by the Implicit Function Theorem, the unique solution $v_{\tau,b}$ to the
equation $G(\tau,b,v)=0$ is continuously differentiable of all orders in
$\text{Re}(\tau)$ and $\text{Im}(\tau)$ near
$(\text{Re}(\tau),\text{Im}(\tau))=(\text{Re}(\tau_{0}),\text{Im}(\tau_{0}))$.
Since $\tau_{0}$ was arbitrary, this proves the result. ∎

######  Proposition 11.3.

$E(\tau,b,U_{\tau,b})$ is continuously differentiable of all orders in
$\text{Re}(\tau)$ and $\text{Im}(\tau)$.

###### Proof.

To get rid of the dependency of $\mathcal{E}(\tau,b,u_{\tau,b})$ on the domain
of integration $\Omega^{\prime}$, we again make the change of coordinates
$y=m_{\tau}^{-1}x$. Then

$\displaystyle\mathcal{E}(\tau,b,u_{\tau,b})=$
$\displaystyle\int_{\Omega^{\prime}}\varepsilon_{WS}(b,u_{\tau,b})(x)\;d^{2}x$
$\displaystyle=$
$\displaystyle\int_{0}^{\sqrt{2\pi}}\int_{0}^{\sqrt{2\pi}}\varSigma(\tau,b,v_{\tau,b})(y)d^{2}y.$
(11.11)

By Lemma 11.2, $v_{\tau,b}$ has continuous $\operatorname{Re}(\tau)$\- and
$\operatorname{Im}(\tau)$-derivatives of all orders, and by Lemma 11.1,
$\varSigma$ has continuous derivatives of all orders mapping
$\mathbb{C}\times\mathbb{R}\times\mathcal{X}_{1}$ to
$\mathbb{C}\times\mathbb{R}\times\mathcal{Y}_{1}$. In particular, the
$\operatorname{Re}(\tau)$\- and $\operatorname{Im}(\tau)$-derivatives of
$\varSigma(\tau,b,v_{\tau,b})$ remain integrable, so we conclude that
$\mathcal{E}(\tau,b,u_{\tau,b})$ (and hence $E(\tau,b,U_{\tau,b})$) is
continuously differentiable of all orders in $\text{Re}(\tau)$ and
$\text{Im}(\tau)$. ∎

######  Theorem 11.4.

When $M_{Z}<M_{H}$, the minimizers $\tau_{b}$ of $E_{WS}(\tau,b,U_{\tau,b})$
are related to the maximizers $\tau_{*}$ of $\eta(M_{Z},M_{H};\tau)$ as
$\tau_{b}-\tau_{*}=\mathcal{O}(|1-\frac{M_{W}^{2}}{eb}|^{\frac{1}{2}})$. In
particular, $\tau_{b}\to\tau_{*}$ as $b\to b_{*}=M_{W}^{2}/e$.

###### Proof.

By Theorem 10.1, the minimizers of $E(\tau,b,U_{\tau,b})$ are equivalent to
the minimizers of the energy functional

$\displaystyle\tilde{E}(\tau,U_{\tau,b})$
$\displaystyle:=\omega^{-2}(E(\tau,b,U_{\tau,b})-\frac{1}{2}b^{2})$
$\displaystyle=-\frac{1}{2}b^{2}\sin^{2}\theta\
\eta(M_{Z},M_{H};\tau)+\mathcal{O}(|\omega|),$

where, recall, $\omega=1-\frac{M_{W}^{2}}{eb}$. Since
$\partial_{\tau}\tilde{E}(\tau,U_{\tau,b})|_{\tau=\tau_{b}}=0$, we have the
expansion

$\displaystyle\tilde{E}(\tau_{*},U_{\tau_{*},b})-\tilde{E}(\tau_{b},U_{\tau_{b},b})$
$\displaystyle=\frac{1}{2}\partial_{\tau}^{2}\tilde{E}(\tau,U_{\tau,b})|_{\tau=\tau_{b}}[\tau_{*}-\tau_{b}]^{2}+\mathcal{O}([\tau_{*}-\tau_{b}]^{3})$
$\displaystyle=$ $\displaystyle-\frac{1}{4}b^{2}\sin^{2}\theta\
(\partial^{2}_{\tau}\eta)|_{(M_{Z},M_{H},\tau_{b})}\ [\tau_{*}-\tau_{b}]^{2}$
$\displaystyle+\mathcal{O}([\tau_{*}-\tau_{b}]^{3})+\mathcal{O}(|\omega|).$
(11.12)

For both expansions to hold, we must have
$\tau_{b}-\tau_{*}=\mathcal{O}(|\omega|^{\frac{1}{2}})$, as required. ∎

The maximizer of $\eta(M_{Z},M_{H};\tau)$, defined in (9.5), was found
numerically in [25] (cf. [28, 3]):

######  Theorem 11.5 ([25]).

When $M_{Z}<M_{H}$, $\eta(M_{Z},M_{H};\tau)$ has a maximum at
$\tau_{*}=e^{i\pi/3}$.

Theorem 2.3 follows from Theorems 11.4 and 11.5.

##  Appendix A Covariant derivatives and curvature

In this appendix, we briefly review some basic definitions from gauge theory.
Recall that we use the Einstein convention of summing over repeated indices.

Let $V$ be an inner product vector space, $G$ a Lie group acting transitively
on $V$ via a unitary representation $\rho:g\mapsto\rho_{g}$, and let
$\mathfrak{g}$ be the Lie algebra of $G$ acting on $V$ via the representation
$\tilde{\rho}:A\mapsto\tilde{\rho}_{A}$ induced by $\rho$.

To simplify notation below, we take $V=\mathbb{C}^{m}$ and $G$ a matrix group,
acting on $V$ by matrix rules (and similarly for $\mathfrak{g}$) and write
$\rho_{g}\Psi=g\Psi$ and $\tilde{\rho}_{A}\Psi=A\Psi$. Moreover, we assume
that $G$ is either $U(m)$ or a Lie subgroup of $U(m)$.

Let $M$ be an open subset in a finite-dimensional vector space, with a metric
$h$ and local coordinates $\\{x^{i}\\}$, and let
$\partial_{i}\equiv\partial_{x^{i}}$.

For a ${\mathfrak{g}}$-valued connection (one-form) $A\equiv A_{i}dx^{i}$ on
$M$, we define the covariant derivatives:

\- $\nabla_{A}$, mapping functions (sections), $\Psi:M\rightarrow V$, into
$\mathfrak{g}$-valued one-forms, as

$\nabla_{A}\Psi:=d\Psi+A\Psi\equiv(\partial_{i}\Psi+A_{i}\Psi)dx^{i};$ (A.1)

\- $d_{A}$, mapping $\mathfrak{g}$-valued functions ($0$-forms) $f$ into
$\mathfrak{g}$-valued one-forms

$d_{A}f:=df+[A,f]\equiv(\partial_{i}f+[A_{i},f])dx^{i};$ (A.2)

\- $d_{A}$, mapping $\mathfrak{g}$-valued one-forms into $\mathfrak{g}$-valued
two-forms

$d_{A}B:=dB+[A,B],$ (A.3)

with $[A,B]$ defined in local coordinates $\\{x^{i}\\}$ as

$\displaystyle[A,B]:=[A_{i},B_{j}]dx^{i}\wedge dx^{j}\equiv[B,A],$ (A.4)

for $A=A_{i}dx^{i}$ and $B=B_{j}dx^{j}$.161616More generally, if $A$ is a
$\mathfrak{g}-$valued $p-$form and $B$ is a $\mathfrak{g}-$valued $q-$form,
written as $A=A^{a}\otimes\gamma_{a}$ and $B=B^{b}\otimes\gamma_{b}$, where
$A^{a}$ and $B^{b}$ are $p-$ and $q-$forms and $\\{\gamma_{a}\\}$ is a basis
in $\mathfrak{g}$, then  $[A,B]:=(A^{a}\wedge
B^{b})\otimes[\gamma_{a},\gamma_{b}]=(-1)^{pq+1}[B,A].$ (A.5)

The curvature form of the connection $A$ is the $\mathfrak{g}$-valued two-form
given by the formula

$\displaystyle F_{A}=dA+\frac{1}{2}[A,A].$ (A.6)

It is related to the curvature operator (denoted by the same symbol)
$F_{A}:=d_{A}\circ d_{A}$. As a simple computation shows, this operator is a
matrix-multiplication operator given by the matrix-valued 2-form (A.6).

Let $U$ be a vector space ($V$ or $\mathfrak{g}$ in our case) and let
$\Omega^{p}_{U}\equiv U\otimes\Omega^{p}$ denote the space of $U$-valued
$p$-forms. On $\Omega^{p}_{U}$, one defined the inner product,
$\langle\cdot,\cdot\rangle_{\Omega^{p}_{U}}\equiv\langle\cdot,\cdot\rangle_{\Omega^{p}_{U}}^{h}$
as

$\langle A,B\rangle_{\Omega^{p}_{U}}\equiv\langle
A,B\rangle_{\Omega^{p}_{U}}^{h}:=\langle A_{\alpha},B^{\alpha}\rangle_{U},$
(A.7)

where $A=A_{\alpha}dx^{\alpha}$ and $B=B_{\alpha}dx^{\alpha}$ are $U$-valued
$p$-forms, $\alpha$ is a $p$-form index and $\langle\cdot,\cdot\rangle_{U}$ is
the inner product on $U$. Here the indices are raised and lowered with help of
the inner product $h$ on $M$.

Above, we did not display the coupling constants. Doing so would change the
covariant derivative to $d_{A}\Psi=(d+gA)\Psi$, if $G$ is simple. If $G$ is
not simple, then each simple component of $G$ gets its own coupling constant,
as was done in the main text for $G=SU(2)\times U(1)$ (see also (C.2)-(C.6)
below).

##  Appendix B The time-dependent Yang-Mills-Higgs system

In this appendix, we briefly review the Yang-Mills-Higgs theory, including the
derivation of the energy functional (2.6). In what follows, we use the
convention of raising or lowering an index by contracting a tensor $T$ with
the metric tensor:

$T^{\alpha}_{i,\beta}=\eta_{ij}T^{j,\alpha}_{\beta}$ (B.1)

where $\eta$ is the Minkowski metric of signature $(+,-,...,-)$ on
$M\subset\mathbb{R}^{d+1}$ and $\alpha,\beta$ are multi-indices. The same
equations could be reinterpreted as stationary equations by taking the
Euclidean metric $\delta_{ij}$, instead of $\eta_{ij}$, and letting the
indices range over $1,\dots,d$, rather than $1,\dots,d+1$. In this case,
$T^{\alpha}_{i,\beta}=T^{i,\alpha}_{\beta}$.

#### Lagrangian.

Let $\Omega$ be a bounded domain in $\mathbb{R}^{d}$ and
$M=\Omega\times[0,T]\subset\mathbb{R}^{d+1}$ be spacetime equipped with the
Minkowski metric $\eta$ of signature $(+,-,...,-)$ and $V$ and $G$ be as in
Appendix A. The theory involves a Higgs field $\Psi:M\rightarrow V$
interacting with the gauge field $A$, a connection (one-form) on $M$ with
values in the algebra ${\mathfrak{g}}$. The dynamics are given by the
Lagrangian

$\mathcal{L}(\Psi,A):=\int_{\Omega}\big{(}\langle\nabla_{A}\Psi,\nabla_{A}\Psi\rangle_{\Omega^{1}_{V}}^{\eta}-U(\Psi)+\langle
F_{A},F_{A}\rangle_{\Omega^{2}_{\mathfrak{g}}}^{\eta}\big{)},$ (B.2)

with corresponding action $\mathcal{S}:=\int_{0}^{T}\mathcal{L}(\Psi,A)dt$,
$T>0$, given explicitly by

$\mathcal{S}(\Psi,A)=\int_{M}\big{(}\langle\nabla_{A}\Psi,\nabla_{A}\Psi\rangle_{\Omega^{1}_{V}}^{\eta}-U(\Psi)+\langle
F_{A},F_{A}\rangle_{\Omega^{2}_{\mathfrak{g}}}^{\eta}\big{)},$ (B.3)

where $U:V\rightarrow\mathbb{R}^{+}$ is a self-interaction potential, which is
assumed to be gauge invariant: $U(\rho_{g}\Psi)=U(\Psi)$. Typical examples of
$G,V$ and $U(\Psi)$ are $U(m),\mathbb{C}^{m}$ and
$U(\Psi)=\frac{1}{2}\lambda(1-\|\Psi\|^{2}_{V})^{2}$.

#### Euler-Lagrange equations.

The Euler-Lagrange equations (called Yang-Mills-Higgs equations) for the
fields $\Psi$ and $A$ are

$\displaystyle\nabla_{A}^{*_{\eta}}\nabla_{A}\Psi=U^{\prime}(\Psi),$ (B.4)
$\displaystyle d_{A}^{*_{\eta}}F_{A}=J(\Psi,A),$ (B.5)

where $\nabla_{A}^{*_{\eta}}$ and $d_{A}^{*_{\eta}}$ are the adjoints of
$\nabla_{A}$ and $d_{A}$ in the appropriate inner products involving the
metric $\eta$ and $J(\Psi,A)$ is the YMH current given by

$\displaystyle J(\Psi,A)$
$\displaystyle:=\operatorname{Re}\langle{\gamma_{a}}\Psi,\nabla_{A}\Psi\rangle_{V}\gamma_{a}$
(B.6)
$\displaystyle=\operatorname{Re}\langle{\gamma_{a}}\Psi,\nabla_{i}\Psi\rangle_{V}\gamma_{a}\otimes
dx^{i},$ (B.7)

where $\gamma_{a}$ is an orthonormal basis of $\mathfrak{g}$ and
$\nabla_{i}:=\partial_{i}+{A_{i}}$, with $\partial_{i}\equiv\partial_{x^{i}}$,
so that $\nabla_{A}\Psi=\nabla_{i}\Psi dx^{i}$. (B.5) is the Yang-Mills
equation.

###### Proof of (B.4) - (B.5).

For convenience, we assume periodic or Dirichlet boundary conditions and that
$\Psi$ and $A$ are $T$-periodic in $t$ and calculate the Gâteaux derivatives
formally.

Recall that $\delta_{\\#}$ denotes the partial (real) Gâteaux derivative with
respect to $\\#$. First we calculate the (complex) Gâteaux derivative of (B.3)
in the $\Psi$-direction. Define
$\partial_{z}\equiv\frac{1}{2}(\partial_{\operatorname{Re}z}-i\partial_{\operatorname{Im}z})$
and
$\delta_{\Psi}\equiv\frac{1}{2}(\delta_{\operatorname{Re}\Psi}-i\delta_{\operatorname{Im}\Psi})$.
Then
$\delta_{\Psi}\mathcal{S}(\Psi,A)\Psi^{\prime}=\partial_{z}\mathcal{S}(\Psi_{z},A)|_{z=0}$,
where $\Psi_{z}=\Psi+z\Psi^{\prime}$, $z\in\mathbb{C}$. Using this, we find

$\displaystyle\delta_{\Psi}\mathcal{S}(\Psi,A)\Psi^{\prime}$
$\displaystyle=\int_{M}\big{(}\langle\nabla_{A}\Psi,\nabla_{A}\Psi^{\prime}\rangle_{\Omega^{1}_{V}}-\langle
U^{\prime}(\Psi),\Psi^{\prime}\rangle_{V}\big{)}.$ (B.8)

Integrating the first term by parts and factoring out $\Psi^{\prime}$ gives

$\displaystyle\delta_{\Psi}\mathcal{S}(\Psi,A)\Psi^{\prime}$
$\displaystyle=\int_{M}\langle\nabla^{*}_{A}\nabla_{A}\Psi-U^{\prime}(\Psi),\Psi^{\prime}\rangle_{V}.$
(B.9)

For this derivative to be zero for every variation $\Psi^{\prime}$, (B.4) must
hold.

Next we calculate the Gâteaux derivative of (B.3) in the $A$-direction. Using
the definition $\delta_{A}f(A)B=\partial_{s}f(A_{s})|_{s=0}$, where
$A_{s}=A+sA^{\prime}$, $s\in\mathbb{R}$, we find

$\displaystyle\delta_{A}\mathcal{S}(\Psi,A)B$
$\displaystyle=\int_{M}\big{(}\langle
B\Psi,\nabla_{A}\Psi\rangle_{\Omega^{1}_{V}}+c.c.+2\langle
d_{A}B,F_{A}\rangle_{\Omega^{2}_{\mathfrak{g}}}\big{)}$ (B.10)
$\displaystyle=I+II.$ (B.11)

Writing $B=B^{a}\gamma_{a}=B_{i}^{a}dx^{i}\otimes\gamma_{a}$ (with $B_{i}^{a}$
real) and $\nabla_{A}\Psi=\nabla_{i}\Psi dx^{i}$, so that

$\langle B\Psi,\nabla_{A}\Psi\rangle_{\Omega^{1}_{V}}=\langle
B^{a},\langle\gamma_{a}\Psi,\nabla_{i}\Psi\rangle_{V}dx^{i}\rangle_{\Omega^{1}},$
(B.12)

and using that
$B^{a}C^{a}=-\operatorname{Tr}[(B^{c}\gamma_{c})(C^{a}\gamma_{a})]$ (since
$\operatorname{Tr}(\gamma_{c}^{*}\gamma_{a})=-\operatorname{Tr}(\gamma_{c}\gamma_{a})=\delta_{ca}$),
gives

$\displaystyle I=-\int_{M}$ $\displaystyle\langle
B,\langle\gamma_{a}\Psi,\nabla_{i}\Psi\rangle_{V}\gamma_{a}\otimes
dx^{i}\rangle_{\Omega^{1}_{\mathfrak{g}}}+c.c..$ (B.13)

which gives $I=\int_{M}\langle