\section{Introduction}
\label{Introduction}

In relativistic quantum field theory,
the nature of the field modes associated
with the spontaneous breaking of an internal symmetry
is now standard lore.
When a global internal symmetry is spontaneously broken,
one or more massless modes called Nambu-Goldstone (NG) modes appear
\cite{ng}.
If instead the symmetry is local,
the Higgs mechanism can occur:
the massless NG modes play the role
of additional components of the gauge fields,
which then propagate as massive modes
\cite{hm}.
In either case,
the spontaneous symmetry breaking
is typically driven by a potential term $V$
in the Lagrange density.
The vacuum field configuration lies
in a minimum $V_0$ of $V$.
The massless NG modes can be understood
as field excitations about the vacuum
that preserve the value $V_0$,
and they are associated with the broken generators of the symmetry.
For many potentials,
there are also additional excitations
involving other values of $V$.
These excitations,
often called Higgs modes,
correspond to additional massive modes
that are distinct from any massive gauge fields.

This standard picture changes when the spontaneous breaking
involves a spacetime symmetry rather than an internal one.
In this work,
we focus on spontaneous breaking
of Lorentz and diffeomorphism symmetries,
for which the corresponding Higgs mechanisms
exhibit some unique features
\cite{ks}.
Spontaneous Lorentz violation occurs
when one or more Lorentz-nonsinglet field configurations
acquire nonzero vacuum expectation values.
The field configurations of interest
can include fundamental vectors or tensors,
derivatives of scalars and other fields,
and Lorentz-nonsinglet composites.
The nonzero vacuum values are manifest
both on the spacetime manifold and in local frames
\cite{akgrav}.
Their origin in spontaneous violation
implies both local Lorentz violation and diffeomorphism violation,
along with the existence of NG modes
\cite{bk}.

At the level of an underlying Planck-scale theory,
numerous proposals exist that involve
spontaneous Lorentz violation,
including ones based on
string theory
\cite{ksp},
noncommutative field theories
\cite{ncqed},
spacetime-varying fields
\cite{spacetimevarying},
quantum gravity
\cite{qg},
random-dynamics models
\cite{fn},
multiverses
\cite{bj},
brane-world scenarios
\cite{brane},
supersymmetry
\cite{susy},
and massive gravity
\cite{modgrav1,modgrav2,modgrav3}.
At experimentally accessible scales,
the observable signals resulting from Lorentz breaking
can be described using effective field theory
\cite{kp}.
The general realistic effective field theory
containing the Lagrange densities
for both general relativity and the Standard Model
along with all scalar terms
involving operators for Lorentz violation
is called the Standard-Model Extension (SME)
\cite{ck,akgrav}.
Searches for low-energy signals of Lorentz violation
represent a promising avenue of investigation involving
the phenomenology of quantum gravity
\cite{cpt07}.
Numerous experimental measurements of SME coefficients
for Lorentz violation have already been performed \cite{kr},
including ones with
photons \cite{photonexpt},
electrons \cite{eexpt,eexpt2,eexpt3},
protons and neutrons \cite{ccexpt,spaceexpt,bnsyn},
mesons \cite{hadronexpt},
muons \cite{muexpt},
neutrinos \cite{nuexpt},
the Higgs \cite{higgs},
and gravity \cite{gravexpt,bak06}.

For spontaneous Lorentz and diffeomorphism breaking,
a general analysis
of the nature of the NG modes and Higgs mechanisms
is provided in Ref.\ \cite{bk}.
One result
is that the spontaneous breaking
of local Lorentz symmetry
implies spontaneous breaking of diffeomorphism symmetry
and vice versa.
Since six local Lorentz transformations and four diffeomorphisms
can be broken,
up to ten NG modes can appear.
To characterize these,
it is natural to adopt the vierbein formalism
\cite{uk},
in which the roles of local Lorentz transformations
and diffeomorphisms are cleanly distinguished.
It turns out that the vierbein itself
naturally incorporates all ten modes.
In an appropriate gauge,
the six Lorentz NG modes appear
in the antisymmetric components of the vierbein,
while the four diffeomorphism NG modes
appear along with the usual gravitational modes
in the symmetric components.

The dynamical behavior of the various NG modes is determined
by the structure of the action
\cite{bk}.
In a Lagrange density formed from tensor quantities
and with diffeomorphism-covariant kinetic terms,
the diffeomorphism NG modes are nonpropagating.
This feature, unique to diffeomorphism symmetry,
can be viewed as arising
because the diffeomorphism NG field excitations
that preserve $V_0$ include metric excitations,
and the combined excitations cancel
at propagation order in covariant derivatives and curvature.
In contrast,
the number of propagating Lorentz NG modes
is strongly model dependent.
For example,
choosing the kinetic terms in the Lagrange density
of the original field theory
to eliminate possible ghost modes
can also prevent the propagation of one or more Lorentz NG modes,
leaving them instead as auxiliary fields.

Several types of possible Higgs mechanisms can be distinguished
when spacetime symmetries are spontaneously broken.
These have features distinct from the
conventional Higgs mechanism of gauge field theories
\cite{hm}
and Higgs mechanisms involving gravity without
Lorentz violation
\cite{hmnonlv}.
The analysis of Higgs mechanisms can be performed
either using the vierbein formalism,
which permits tracking of Lorentz and diffeomorphism properties,
or by working directly with fields on the spacetime manifold.
The results of both approaches are equivalent.

For local Lorentz symmetry,
the role of the gauge fields in the vierbein formalism
is played by the spin connection.
The \it Lorentz Higgs mechanism \rm
occurs when the Lorentz NG modes
play the role of extra components of the spin connection
\cite{bk}.
Some components of the spin connection then acquire mass
via the covariant derivatives in the kinetic
part of the Lagrange density.
Explicit models displaying the Lorentz Higgs mechanism are known.
For this mechanism to occur,
the components of the spin connection
must propagate as independent degrees of freedom,
which requires a theory based on Riemann-Cartan geometry.
If instead the theory is based on Riemann geometry,
like General Relativity,
the spin connection is fixed nonlinearly
in terms of the vierbein and its derivatives.
The presence of these derivatives ensures
that no mass terms emerge
from the kinetic part of the Lagrange density,
although the vierbein propagator is modified.

In the context of diffeomorphism symmetry,
the role of the gauge fields is played by the metric.
For spontaneous diffeomorphism breaking in Riemann spacetime,
a conventional
\it diffeomorphism Higgs mechanism \rm
cannot generate a mass for the graviton
because the connection
and hence the analogue of the usual mass term
involve derivatives of the metric
\cite{ks}.
Also,
since diffeomorphism NG modes are nonpropagating
in a Lagrange density with covariant kinetic terms
for reasons mentioned above,
the propagating NG degrees of freedom
required to generate massive fields via a
conventional diffeomorphism Higgs mechanism are lacking.

In a conventional gauge theory with a nonderivative potential $V$,
the gauge fields are absent from $V$
and so the potential cannot directly contribute to the gauge masses.
However,
in spontaneous Lorentz and diffeomorphism violation,
massive Higgs-type modes involving the vierbein can
arise via the \it alternative Higgs mechanism, \rm
which involves the potential $V$
\cite{ks}.
The key point is that $V$ contains both tensor and metric fields,
so field fluctuations about the vacuum value $V_0$
can contain quadratic mass terms involving the metric.

In this work,
we study the nature and properties
of the additional massive Higgs-type modes
arising from this alternative Higgs mechanism.
A general treatment is provided
for a variety of types of potentials $V$
in gravitationally coupled theories with Riemann geometry.
We investigate the effects of the massive modes
on the physical properties of gravity.
In certain theories with spontaneous Lorentz violation,
the NG modes can play the role of photons
\cite{bk},
and we also examine the effects of the massive modes
on electrodynamics in this context.

The next section of this work provides a general discussion
of the origin and basic properties of the massive Higgs-type modes.
Section
\ref{Bumblebee Models}
analyzes the role of these modes
in vector theories with spontaneous Lorentz violation,
known as bumblebee models.
In Sec.\ \ref{Examples},
the massive modes in
a special class of bumblebee models are studied in more detail
for several choices of potential in the Lagrange density,
and their effects
on both the gravitational and electromagnetic interactions
are explored.
Section \ref{Summary} summarizes our results.
Some details about transformation laws are provided in the
Appendix.
Throughout this work,
the conventions and notations of Refs.\ \cite{akgrav,bk}
are used.

\section{Massive Modes}
\label{Massive Modes}

The characteristics of the massive Higgs-type modes
that can arise from the alternative Higgs mechanism
depend on several factors.
Among them are
the type of field configuration
acquiring the vacuum expectation value
and the form of the Lagrange density,
including the choice of potential $V$
inducing spontaneous breaking
of Lorentz and diffeomorphism symmetries.
In this section,
we outline some generic features associated with
the alternative Higgs mechanism
in a theory of a general tensor field $T_{\lambda\mu\nu\cdots}$
in a Riemann spacetime with metric $g_{\mu\nu}$.
We consider first consequences
of the choice of potential $V$,
then discuss properties of vacuum excitations,
and finally offer comments on the massive modes
arising from the alternative Higgs mechanism.

\subsection{Potentials}
\label{Potentials}

The potential $V$ in the Lagrange density is taken
to trigger a nonzero vacuum expectation value
\begin{equation}
\vev{T_{\lambda\mu\nu\cdots}}=
t_{\lambda\mu\nu\cdots}
\label{Tvev}
\end{equation}
for the tensor field,
thereby spontaneously breaking
local Lorentz and diffeomorphism symmetries.
In general,
$V$ varies with $T_{\lambda\mu\nu\cdots}$,
its covariant derivatives,
the metric $g_{\mu\nu}$,
and possibly other fields.
However,
for simplicity we suppose here that
$V$ has no derivative couplings
and involves only $T_{\lambda\mu\nu\cdots}$
and $g_{\mu\nu}$.
We also suppose that $V$ is everywhere positive
except at its degenerate minima,
which have $V = 0$ and are continuously connected
via the broken Lorentz and diffeomorphism generators.
The vacuum is chosen to be the particular minimum
in which $T_{\lambda\mu\nu\cdots}$ attains
its nonzero value \rf{Tvev}.

Since the Lagrange density is an observer scalar,
$V$ must depend on fully contracted combinations
of $T_{\lambda\mu\nu\cdots}$ and $g_{\mu\nu}$.
Provided $T_{\lambda\mu\nu\cdots}$ has finite rank,
the number of independent scalar combinations is limited.
For example,
for a symmetric two-tensor field $C_{\mu\nu}$
there are four independent possibilities,
which can be given explicitly
in terms of traces of powers of $C_{\mu\nu}$ and $g_{\mu\nu}$
\cite{kpgr}.
It is convenient to denote generically
these scalar combinations as $X_m$,
where $m= 1,2,\ldots, M$ ranges over the number $M$
of independent combinations.
The functional form of the potential therefore takes the form
\begin{equation}
V = V(X_1,X_2,\ldots,X_M)
\end{equation}
in terms of the scalars $X_m$.

The definition of the scalars $X_m$
can be chosen so that $X_m = 0$ in the vacuum for all $m$.
For example,
a choice involving a quadratic combination of the tensor
with zero vacuum value is
\begin{equation}
X =
T_{\lambda\mu\nu\cdots}
g^{\lambda\alpha} g^{\mu\beta} g^{\nu\gamma} \cdots
T_{\alpha\beta\gamma\cdots} \pm t^2,
\label{X}
\end{equation}
with $t^2$ the norm
\begin{equation}
t^2 =
\mp t_{\lambda\mu\nu\cdots}
\vev{g^{\lambda\alpha}} \vev{g^{\mu\beta}} \vev{g^{\nu\gamma}} \cdots
t_{\alpha\beta\gamma\cdots} ,
\label{norm}
\end{equation}
where $\vev{g^{\lambda\alpha}}$
is the vacuum value of the inverse metric.
The $\mp$ sign is introduced for convenience
so that $t^2$ can be chosen nonnegative.
In principle,
$t^2$ could vary with spacetime position,
which would introduce explicit spacetime-symmetry breaking,
but it suffices for present purposes
to take $t^2$ as a real nonnegative constant.

The $M$ conditions $X_m = 0$ fix the vacuum value
$t_{\lambda\mu\nu\cdots}$.
If only a subset of $N$ of these $M$ conditions
is generated in a given theory,
then the value of $t_{\lambda\mu\nu\cdots}$ is specified
up to $(M-N)$ coset transformations,
and so the vacuum is degenerate under $(M-N)$
additional gauge symmetries.
Note that these $(M-N)$ freedoms are distinct
from Lorentz and diffeomorphism transformations.

It is useful to distinguish two classes of potentials $V$.
The first consists of smooth functionals $V$ of $X_m$
that are minimized by the conditions $X_m=0$
for at least some $m$.
These potentials therefore satisfy $V = V_m^\prime = 0$
in the vacuum,
where $V_m^\prime$ denotes the derivative with respect to $X_m$,
and they fix the vacuum value of $T_{\lambda\mu\nu\cdots}$
to $t_{\lambda\mu\nu\cdots}$ modulo possible gauge freedoms.
A simple example with the quantity $X$ in Eq.\ \rf{X} is
\begin{equation}
V_S(X) = {\textstyle{1\over 2}} \kappa X^2,
\label{VS}
\end{equation}
where $\kappa$ is a constant.
In this case,
the vacuum value $t_{\lambda\mu\nu\cdots}$
is a solution of $V = V^\prime = 0$,
where $V^\prime$ denotes a derivative with respect to $X$.
If the matrix $V_{mn}^{\prime\prime}$
of second derivatives has nonzero eigenvalues,
the smooth functionals $V$ can give rise
to quadratic mass terms in the Lagrange density
involving the tensor and metric fields.
Potentials $V$ of this type are therefore associated
with the alternative Higgs mechanism,
and they are the primary focus of our attention.

A second class of potentials
introduces Lagrange-multiplier fields $\lambda_m$
for at least some $m$,
to impose directly the conditions $X_m=0$
as constraint terms in the Lagrange density.
We consider here both linear and quadratic functional
forms for these constraints.
An explicit linear example is
\begin{equation}
V_L(\lambda, X) = \lambda X,
\label{VL}
\end{equation}
while a quadratic one is
\begin{equation}
V_Q(\lambda, X) = {\textstyle{1\over 2}} \lambda X^2.
\label{VQ}
\end{equation}
In each example,
$\lambda$ is a Lagrange-multiplier field
and has equation of motion with solution $X=0$,
so the value $t_{\lambda\mu\nu\cdots}$
is a vacuum solution for the tensor.
Potentials in the Lagrange-multiplier class are unlikely
to be physical in detail
because they enforce singular slicings
in the phase space for the fields.
However,
when used with care they can be useful
as limiting approximations to potentials in the smooth class,
including those inducing the alternative Higgs mechanism
\cite{ks}.
Note that positivity of the potentials
can constrain the range of the Lagrange multiplier field.
For example,
the off-shell value of
$\lambda$ in $V_L$ must have the same sign as $X$,
while that of $\lambda$ in $V_Q$ must be non-negative.

\subsection{Excitations}
\label{Excitations}

Field excitations about the vacuum solution \rf{Tvev}
can be classified in five types:
gauge modes,
NG modes,
massive modes,
Lagrange-multiplier modes,
or spectator modes.
Gauge modes arise if the potential $V$ fixes only
$(M-N)$ of the $M$ conditions $X_m = 0$,
so that the vacuum is unspecified up to $N$ conditions.
These modes can be disregarded for most purposes here
because they can be eliminated via gauge fixing
without affecting the physics.
The NG modes are generated by the virtual action
of the broken Lorentz and diffeomorphism generators
on the symmetry-breaking vacuum.
They preserve the vacuum condition $V=0$.
Massive modes are excitations for which the symmetry breaking
generates quadratic mass terms.
Lagrange-multiplier modes are excitations
of the Lagrange multiplier field.
Finally,
spectator modes are the remaining modes in the theory.

For smooth potentials,
field excitations preserving the conditions $X_m = 0$ for all $m$
have potential $V = 0$.
They also satisfy $V_m^\prime = 0$.
The NG modes are of this type.
Excitations with $X_m \neq 0$
having nonzero potential $V \neq 0$
and $V_m^\prime \neq 0$
are massive modes,
with mass matrix determined by
the second derivatives $V_{mn}^{\prime\prime}$.
Since the smooth potentials depend
on the tensor $T_{\lambda\mu\nu\cdots}$
and the metric $g_{\mu\nu}$,
the corresponding massive modes
involve combinations of excitations of these fields.

In contrast,
for Lagrange-multiplier potentials
the conditions $X_m = 0$ always hold on shell,
which implies $V=0$ for all on-shell excitations.
If also $V_m^\prime = 0$,
then the excitations remain in the potential minimum
and include the NG modes.
For linear functional forms of $V$,
it follows that $V_m^\prime = \lambda_m$,
so $V_m^\prime$ is nonzero
only when the Lagrange-multiplier field $\lambda_m$ is excited.
Any excitations of $T_{\lambda\mu\nu\cdots}$ and $g_{\mu\nu}$
must have $V = V_m^\prime = 0$.
For quadratic functional forms of $V$,
one finds $V_m^\prime = 0$ for all excitations,
including the $\lambda$ field.
We therefore can conclude
that the combinations of $T_{\lambda\mu\nu\cdots}$ and $g_{\mu\nu}$
playing the role of massive modes for smooth potentials
are constrained to zero for Lagrange-multiplier potentials.
Evidently,
studies of the alternative Higgs mechanism
must be approached with care
when the Lagrange-multiplier approximation to a smooth potential
is adopted.

The tensor excitations about the vacuum can be expressed
by expanding $T_{\lambda\mu\nu\cdots}$ as
\begin{equation}
T_{\lambda\mu\nu\cdots} =
t_{\lambda\mu\nu\cdots} + \tau_{\lambda\mu\nu\cdots},
\label{tau}
\end{equation}
where the excitation
$\tau_{\lambda\mu\nu\cdots}$
is defined as the difference
$\tau_{\lambda\mu\nu\cdots} \equiv
\delta T_{\lambda\mu\nu\cdots}
= T_{\lambda\mu\nu\cdots} - t_{\lambda\mu\nu\cdots}$
between the tensor and its vacuum value.
We also expand the metric
\begin{equation}
g_{\mu\nu} = \vev{g_{\mu\nu}} + h_{\mu\nu}
\label{ghmunu}
\end{equation}
in terms of the metric excitations $h_{\mu\nu}$
about the metric background value
$\vev{g_{\mu\nu}}$.
For simplicity and definiteness,
in much of what follows we take
the background metric to be that of Minkowski spacetime,
$\vev{g_{\mu\nu}} = \eta_{\mu\nu}$.
We also suppose $t_{\lambda\mu\nu\cdots}$ is constant
in this background,
so that
$\partial_\alpha t_{\lambda\mu\nu\cdots} = 0$
in cartesian coordinates.

Other choices can be made for the expansion of the tensor
about its vacuum value.
One alternative
is to expand the contravariant version of the tensor as
\begin{equation}
T^{\lambda\mu\nu\cdots} =
\overline t^{\lambda\mu\nu\cdots} + \widetilde T^{\lambda\mu\nu\cdots}.
\label{olt}
\end{equation}
In a Minkowski background,
the vacuum values in the two expansions \rf{tau} and \rf{olt}
are related simply by
\begin{equation}
\overline t^{\lambda\mu\nu\cdots}
= \eta^{\lambda\alpha}\eta^{\mu\beta}\eta^{\nu\gamma}\cdots
t_{\alpha\beta\gamma\cdots}.
\end{equation}
However,
the relationship between the two tensor excitations
at leading order involves also the metric excitation:
\begin{equation}
\widetilde T^{\lambda\mu\nu\cdots}
=
\tau^{\lambda\mu\nu\cdots}
- h^{\lambda\alpha}t_\alpha^{\pt{\alpha}\mu\nu\cdots}
- h^{\mu\alpha}t^{\lambda\pt{\alpha}\nu\cdots}_{\pt{\lambda}\alpha}
- h^{\nu\alpha}t^{\lambda\mu\pt{\alpha}\cdots}_{\pt{\lambda\mu}\alpha}
- \ldots.
\end{equation}
In this expression,
indices have been raised using the Minkowski metric.

Any Lagrange multipliers $\lambda_m$ in the theory
can also be expanded about their vacuum values
$\overline\lambda_m$
as
\begin{equation}
\lambda_m = \overline\lambda_m + \widetilde\lambda_m .
\end{equation}
For linear Lagrange-multiplier potentials,
the equations of motion for
$T_{\lambda\mu\nu\cdots}$ and $g_{\mu\nu}$
provide constraints on the vacuum values $\overline\lambda_m$.
In a Minkowski background
and a potential yielding $X_m = 0$ for all $m$,
the equations of motion for $T_{\lambda\mu\nu\cdots}$
can be solved to give
\begin{equation}
\overline\lambda_m = 0.
\end{equation}
For quadratic Lagrange-multiplier potentials,
the $\lambda_m$ are absent from the equations of motion.
Their vacuum values are therefore physically irrelevant,
and $\overline\lambda_m = 0$ can also be adopted in this case.
For the remainder of this work
we take $\overline\lambda_m = 0$,
and for notational simplicity
we write $\lambda_m$ for both
the full field $\lambda_m$ and the excitation $\widetilde\lambda_m$.

\subsection{Massive modes}
\label{Massive modes}

For a smooth potential $V$,
the massive excitations typically
involve a mixture of tensor and gravitational fields.
As an example,
consider the expression for $V_S$ in Eq.\ \rf{VS}.
This can be expanded in terms of the excitations
$\tau_{\lambda\mu\nu\cdots}$ and $h_{\mu\nu}$.
Retaining only terms up to quadratic order
in a Minkowski background gives
\begin{eqnarray}
V_S &\approx& 2 \kappa [ t^{\lambda\mu\nu\cdots} (\tau_{\lambda\mu\nu\cdots}
- {\textstyle{1\over 2}} h_{\lambda\alpha} t^\alpha_{\pt{\alpha}\mu\nu\cdots}
- {\textstyle{1\over 2}} h_{\mu\beta} t_{\lambda\pt{\beta}\nu\cdots}^{\pt{\lambda}\beta}
\nonumber\\
&&
\qquad
\qquad
\qquad
\quad
- {\textstyle{1\over 2}} h_{\nu\gamma} t_{\lambda\mu\pt{\gamma}\cdots}^{\pt{\lambda\mu}\gamma}
- \ldots)]^2 ,
\label{Vexpans}
\end{eqnarray}
where index contractions are performed
with the Minkowski metric $\eta_{\mu\nu}$.
Evidently,
the massive excitations in this example
involve linear combinations of $\tau_{\lambda\mu\nu\cdots}$
with contractions of $h_{\mu\nu}$ and $t_{\lambda\mu\nu\cdots}$.

The explicit expressions for the massive modes
can be modified by local Lorentz and diffeomorphism gauge fixing.
The action is symmetric
under 10 local Lorentz and diffeomorphism symmetries,
so there are 10 possible gauge conditions.
For an unbroken symmetry generator,
the effects of a gauge choice are conventional.
For a broken symmetry generator,
a gauge choice fixes the location in field space
of the corresponding NG mode.
For example,
suitable gauge choices can place all the NG modes
in the vierbein
\cite{bk}.
These gauge choices also affect the form of the massive modes.
They can be used to isolate some or all of the massive modes
as components of either the gravitational or tensor fields
by gauging other components to zero.
Alternatively,
the gauge freedom can be used to
simplify the equations of motion,
while the massive modes remain a mixture
of the excitations $\tau_{\lambda\mu\nu\cdots}$ and $h_{\mu\nu}$.

The behavior of the massive modes depends
on the form of the kinetic terms in the Lagrange density
as well as the form of the potential $V$.
A gravitational theory with a dynamical tensor field
$T_{\lambda\mu\nu\cdots}$
has kinetic terms for both
$g_{\mu\nu}$ and $T_{\lambda\mu\nu\cdots}$
and hence for both
$h_{\mu\nu}$ and $\tau_{\lambda\mu\nu\cdots}$.
Since in the alternative Higgs mechanism
the potential $V$ acts only as a source of mass,
the issue of whether the massive modes propagate dynamically
depends on the structure of these kinetic terms.
In particular,
propagating massive modes can be expected
only if the theory
\it without \rm
the potential $V$
contains the corresponding propagating massless modes.

It is desirable that any propagating modes
be unitary and ghost free.
To avoid unitarity issues with higher-derivative propagation,
the kinetic term for the metric excitation $h_{\mu\nu}$
can be taken to emerge as usual
from the Einstein-Hilbert action.
In the absence of the potential $V$,
this is also a ghost-free choice.
For the tensor field,
higher-derivative propagation can be avoided by writing
the general kinetic term ${\cal L}_{\rm K}$
as a weighted sum of all scalar densities
formed from quadratic expressions in $T_{\lambda\mu\nu\cdots}$
that involve two covariant derivatives.
For example,
one such scalar density is
\begin{equation}
{\cal L}_{\rm K} \sim
e T_{\lambda\mu\nu\cdots} D_\alpha D^\alpha T^{\lambda\mu\nu\cdots},
\label{kinetic}
\end{equation}
where
$e = \sqrt{-g}$ is the vierbein determinant.
In the absence of the potential $V$,
the ghost-free requirement places strong constraints
on allowed forms of ${\cal L}_{\rm K}$
and typically involves gauge symmetry for the tensor.
For a vector field,
for example,
the Maxwell action is the unique ghost-free combination
in the Minkowski spacetime limit.
In any case,
since the massive modes are combinations of
$h_{\mu\nu}$ and $\tau_{\lambda\mu\nu\cdots}$,
it follows that ghost-free propagation is possible
only if the kinetic terms
for these combinations are ghost free.
Note that the potential $V$
may explicitly break the tensor gauge symmetry,
so requiring ghost-free kinetic terms
is by itself insufficient to ensure ghost-free massive modes.

Under the assumption of Lorentz invariance,
the Fierz-Pauli action involving quadratic terms for $h_{\mu\nu}$
is the unique ghost-free choice for a free massive spin-2 field
\cite{fp}.
However,
when coupled to a covariantly conserved energy-momentum tensor,
the small-mass limit of a massive spin-2 field
includes modes that modify the gravitational bending of light
in disagreement with observation
\cite{vdvz}.
This presents an obstacle to constructing
a viable theory of massive gravity.
One avenue of investigation that might permit evading
this obstacle is to relax the assumption of Lorentz invariance.
Spontaneous Lorentz violation in closed string theory
has been proposed
\cite{modgrav1}
as a mechanism that might lead to graviton mass terms.
Models involving infrared modifications of gravity
have also been proposed
that involve spontaneous Lorentz violation with ghosts
\cite{modgrav2}
and that have explicit Lorentz violations
\cite{modgrav3}.
Explicit Lorentz violation
is incompatible with Riemann and Riemann-Cartan geometries
but may be compatible with Finsler or other geometries
\cite{akgrav,gyb}
or may be viewed as an approximation to spontaneous violation.
In the present context,
the possibility exists
that the massive modes from the alternative Higgs mechanism
could evade the Veltman-van Dam-Zakharov constraint
via their origin in spontaneous Lorentz violation
and their nature as mixtures of gravitational and tensor modes.
For example,
the expansion \rf{Vexpans}
includes quadratic terms involving
contractions of $h_{\mu\nu}$ with the vacuum value $t_{\lambda\mu\nu}$,
so a model with suitable vacuum values
and incorporating also a ghost-free propagator
for the corresponding modes
would describe a type of propagating massive gravity
without conventional Fierz-Pauli terms.

In the alternative Higgs mechanism,
the existence of propagating massive modes involving the metric
can be expected to affect gravitational physics.
Effects can arise directly from the modified graviton propagator
and also from the massive modes acting as sources
for gravitational interactions.
The latter can be understood by considering
the energy-momentum tensor $T^{\mu\nu}$ of the full theory,
which can be found by variation of the action
with respect to the metric $g_{\mu\nu}$.
The contribution $T_V^{\mu\nu}$ to $T^{\mu\nu}$
arising from the potential $V$ is
\begin{equation}
T_V^{\mu\nu} =
- g^{\mu\nu} V + 2 V_m^\prime \fr {\delta X_m} {\delta g_{\mu\nu}} ,
\label{TV}
\end{equation}
where a sum on $m$ is understood.
For the vacuum solution $X_m = 0$,
which satisfies $V = V_m^\prime = 0$,
$T_V^{\mu\nu}$ remains zero
and the gravitational sourcing is unaffected.
The same is true for excitations
for which $V$ and $V_m^\prime$ both vanish,
such as the NG modes.
However,
the massive modes arising from a smooth potential
have nonzero $V$ and $V_m^\prime$
and can therefore act as additional sources
for gravity.
These contributions can
lead to a variety of gravitational effects including,
for example,
modifications of the Newton gravitational potential
in the weak-field limit,
which could have relevance for dark matter,
or cosmological features such as dark energy
\cite{akgrav}.

Note that nonpropagating modes
can also have similar significant physical effects
on gravitational properties.
This possibility holds for any excitations appearing in $V$,
whether they are physical, ghost, or Lagrange-multiplier modes.
For example,
any Lagrange-multiplier fields are auxiliary by construction
and so the $\lambda_m$ excitations are nonpropagating.
Nonetheless,
for linear Lagrange-multiplier potentials
these excitations can contribute to $T_V^{\mu\nu}$
even though $V = 0$
because $V_m^\prime$ is nonvanishing.
However,
a theory with a quadratic Lagrange-multiplier potential
has $T_V^{\mu\nu}=0$
and therefore leaves the gravitational sourcing unaffected.

\section{Bumblebee Models}
\label{Bumblebee Models}

In this section,
we focus attention on the special class of theories
in which the role of the tensor $T_{\lambda\mu\nu\cdots}$
is played by a vector $B_\mu$
that acquires a nonzero vacuum expectation value $b_\mu$.
These theories,
called bumblebee models,
are among the simplest examples of field theories
with spontaneous Lorentz and diffeomorphism breaking.
In what follows,
bumblebee models are defined,
their properties under local Lorentz and
diffeomorphism transformations are presented,
their mode content is analyzed,
and issues involving
gauge fixing and alternative mode expansions are considered.

\subsection{Basics}
\label{Basics}

The action $S_B$ for a single bumblebee field $B_\mu$
coupled to gravity and matter can be written as
\begin{eqnarray}
S_B &=&
\int d^4 x~ {\cal L}_B
\nonumber\\
&=&
\int d^4 x~
({\cal L}_g + {\cal L}_{gB} + {\cal L}_{\rm K} + {\cal L}_V + {\cal L}_{\rm M}).
\end{eqnarray}
In Riemann spacetime,
the pure gravitational piece ${\cal L}_g$
of the Lagrange density
is usually taken to be the Einstein-Hilbert term
supplemented by the cosmological constant $\Lambda$.
The gravity-bumblebee couplings are described by ${\cal L}_{gB}$,
while ${\cal L}_{\rm K}$ contains the bumblebee kinetic terms
and any self-interaction terms.
The component ${\cal L}_V$ consists of the potential $V(B_\mu)$,
including terms triggering the spontaneous Lorentz violation.
Finally,
${\cal L}_{\rm M}$ involves the bumblebee coupling
to the matter or other sectors in the model.

The forms of ${\cal L}_{gB}$ and ${\cal L}_K$ are complicated
in the general case.
However,
if attention is limited to terms quadratic in $B_\mu$
involving no more than two derivatives,
then only five possibilities exist.
The Lagrange density ${\cal L}_B$ can then be written as
\begin{eqnarray}
{\cal L}_B &=&
\fr 1 {16\pi G} e (R - 2 \Lambda)
+ \si_1 eB^\mu B^\nu R_{\mu\nu} + \si_2 e B^\mu B_\mu R
\nonumber\\
&&
- \frac 14 \ta_1 e B^{\mu\nu} B_{\mu\nu}
+ \frac 12 \ta_2 e D_\mu B_\nu D^\mu B^\nu
\nonumber\\
&&
+ \frac 12 \ta_3 e D_\mu B^\mu D_\nu B^\nu
- eV
+ {\cal L}_{\rm M} ,
\label{bb}
\end{eqnarray}
where $G$ is the Newton gravitational constant
and the field-strength tensor $B_{\mu\nu}$
is defined as
\begin{equation}
B_{\mu\nu} = \partial_\mu B_\nu-\partial_\nu B_\mu .
\label{Bfield}
\end{equation}

The five real parameters
$\si_1$, $\si_2$, $\ta_1$, $\ta_2$, $\ta_3$
in Eq.\ \rf{bb}
are not all independent.
Up to surface terms,
which leave unaffected the equations of motion
from the action,
the condition
\begin{eqnarray}
\int d^4x~
(
eB^\mu B^\nu R_{\mu\nu}
- \frac 12 e B^{\mu\nu} B_{\mu\nu}
\hskip 50pt
\nonumber\\
+ e D_\mu B_\nu D^\mu B^\nu
- e D_\mu B^\mu D_\nu B^\nu
) = 0
\label{cond}
\end{eqnarray}
is identically satisfied,
so only four of the corresponding five terms in ${\cal L}_B$
are independent.
No term of the form $e B^\mu B_\mu R$
appears in the condition \rf{cond},
so the parameter $\si_2$ remains unaffected
while the four parameters $\si_1$, $\ta_1$, $\ta_2$, $\ta_3$
become linked.
For example,
the identity \rf{cond} implies that
the action for the Lagrange density \rf{bb}
with five nonzero parameters
$\si_1$, $\si_2$, $\ta_1$, $\ta_2$, $\ta_3$
is equivalent to an action of the same form
but with only four nonzero parameters
$\si_1^\prime = \si_1 + \ta_3$,
$\si_2^\prime = \si_2$,
$\ta_1^\prime = \ta_1 - 2 \ta_3$,
$\ta_2^\prime = \ta_2 - 2 \ta_3$,
while $\ta_3^\prime = 0$.
Moreover,
other factors may also constrain some of the five parameters.
For example,
certain models of the form ${\cal L}_B$
yield equations of motion that imply
additional relationships among the parameters.
Also,
certain physical limits
such as the restriction to Minkowski spacetime
can limit the applicability of the condition \rf{cond}.
Some cases with specific parameter values
may be of particular interest for
reasons of physics, geometry, or simplicity,
such as the models with
$\ta_1 = 1$, $\ta_2 = \ta_3 = 0$
considered below,
or the model with $\si_2 = - \si_1/2$
for which the bumblebee-curvature coupling involves
the Einstein tensor,
${\cal L}_B \supset \si_1 eB^\mu B^\nu G_{\mu\nu}$.
Since the most convenient choice of parameters
depends on the specific model
and on the physics being addressed,
it is useful to maintain the five-parameter form \rf{bb}
for generality.

Following the discussion in
Sec.\ \ref{Potentials},
we suppose the potential $V$ in Eq.\ \rf{bb}
has no derivative couplings
and is formed from scalar combinations
of the bumblebee field $B_\mu$
and the metric $g_{\mu\nu}$.
Only one independent scalar exists.
It can be taken as the bumblebee version of $X$ in Eq.\ \rf{X}:
\begin{equation}
X = B_\mu g^{\mu\nu} B_{\nu} \pm b^2,
\label{XB}
\end{equation}
where $b^2$ is a real nonnegative constant.
The potential $V(X)$ itself can be smooth in $X$,
like the form \rf{VS},
or it can involve Lagrange multipliers
like the form \rf{VL} or \rf{VQ}.
In any case,
the vacuum is determined by the single condition
\begin{equation}
X = B_\mu g^{\mu\nu} B_\nu \pm b^2 = 0  .
\label{Bsquare}
\end{equation}
In the vacuum,
the potential vanishes, $V(X)=0$,
and the fields $B_\mu$, $g_{\mu\nu}$
acquire vacuum values
\begin{equation}
B_\mu \to \vev{B_\mu} = b_\mu , \quad
g_{\mu\nu} \to \vev{g_{\mu\nu}}.
\label{vevs}
\end{equation}
The nonzero value $b_\mu$,
which obeys $b_\mu \vev{g^{\mu\nu}}b_\nu = \mp b^2$,
spontaneously breaks
both Lorentz and diffeomorphism symmetry.
Note that the choice of the potential $V$
can also have implications for the parameters in Eq.\ \rf{bb}.
For example,
if the potential takes
the quadratic Lagrange-multiplier form \rf{VQ},
then $\si_2$ can be taken as zero
because a nonzero value merely acts to rescale $G$ and $\Lambda$.
However,
a nonzero value of $\si_2$ can have nontrivial consequences
for models with other potentials,
such as the smooth quadratic form \rf{VS}.

For generic choices of parameters,
the Lagrange density \rf{bb}
is unitary because no more than two derivatives appear.
However,
as discussed in Sec.\ \ref{Massive modes},
the indefinite metric and generic absence of gauge invariance
typically imply the presence of ghosts
and corresponding negative-energy problems,
which can tightly constrain the viability of various models
\cite{ems}.
If the gravitational couplings and the potential $V$
are disregarded,
there is a unique set of parameters
ensuring the absence of ghosts:
$\ta_1 = 1$, $\ta_2 = \ta_3 = 0$.
With this choice,
the kinetic term for the bumblebee becomes
the Maxwell action,
in which the usual U(1) gauge invariance excludes ghosts.
When the gravitational terms and couplings are included,
this gauge invariance is maintained
for the parameter choice
$\ta_1 = 1$, $\ta_2 = \ta_3 = \si_1 = \si_2  = 0$.
The further inclusion of the potential $V$
breaks the U(1) gauge symmetry,
but the form of the kinetic term
ensures a remnant constraint
on the equations of motion
arising from the identity
\begin{equation}
D_\mu D_\nu B^{\mu\nu} \equiv 0.
\label{remconst}
\end{equation}
The action for these bumblebee models,
introduced in Ref.\ \cite{ks},
is therefore of particular interest.
The corresponding Lagrange density
\begin{eqnarray}
{\cal L}_{\rm KS} &=&
\fr 1 {16\pi G} e (R - 2 \Lambda)
- \frac 14 e B^{\mu\nu} B_{\mu\nu}
- eV + {\cal L}_{\rm M} ,
\qquad
\label{ksbb}
\end{eqnarray}
is investigated in more detail in Sec.\ \ref{Examples}.
The reader is warned
that some confusion about the relationship between
these models and ones with nonzero $\ta_2$ and $\ta_3$
exists in the literature.
In particular,
results for the models \rf{ksbb}
can differ from those obtained
in models with nonzero $\ta_2$, $\ta_3$
via straightforward adoption
of the limit $\ta_2, \ta_3 \to 0$,
due to the emergence of the remnant constraint \rf{remconst}.

The discussion above defines bumblebee models
on the spacetime manifold,
without introducing a local Lorentz basis.
In this approach,
the Lorentz NG modes are hidden
within the bumblebee field $B_\mu$.
Adopting instead a vierbein formulation
reveals explicitly the local Lorentz properties of the models,
and it also provides a natural way to incorporate
spinor fields in the matter Lagrange density ${\cal L}_{\rm M}$.
The vierbein $\vb \mu a$
converts tensors expressed in a local basis
to ones on the spacetime manifold.
The spacetime metric $g_{\mu\nu}$ is related
to the Minkowski metric $\eta_{ab}$ in the local frame as
\begin{equation}
g_{\mu\nu} = \vb \mu a \vb \nu b  \eta_{ab} ,
\label{gvb}
\end{equation}
while the bumblebee spacetime vector $B_\mu$
is related to the local bumblebee vector $B_a$ as
\begin{equation}
B_\mu = \vb \mu a B_a .
\label{Bvb}
\end{equation}
A complete treatment in the vierbein formalism
involves also introducing the spin connection $\nsc \mu a b$,
which appears in covariant derivatives acting on local quantities.
In Riemann-Cartan geometry,
where the spacetime has both curvature and torsion,
the spin connection represents degrees of freedom
independent of the vierbein.
However,
experimental constraints on torsion are tight
\cite{torsion}.
In this work,
we restrict attention to Riemann geometry,
for which the torsion vanishes
and the spin connection is fixed in terms of the vierbein.
It therefore suffices for our purposes to
consider the vierbein degrees of freedom
and their role relative to the gravitational and NG modes.
Bumblebee models in the more general context
of Riemann-Cartan spacetime with nonzero torsion
are investigated in Refs.\ \cite{akgrav,bk}.

There is a substantial literature concerning
theories of vacuum-valued vectors coupled to gravity.
The five-parameter Lagrange density \rf{bb}
excluding the potential $V$ and the cosmological constant
was investigated by Will and Nordtvedt
in the context of vector-tensor models of gravity
\cite{wn}.
Kosteleck\'y and Samuel (KS)
\cite{ks}
introduced the potential $V$
triggering spontaneous Lorentz violation
and studied both the smooth quadratic potential
\rf{VS}
and the linear Lagrange-multiplier case
\rf{VL}
for the class of models given by Eq.\ \rf{ksbb}.
The presence of the potential
introduces a variety of qualitatively new effects,
including
the necessary breaking of U(1) gauge invariance
\cite{akgrav},
the existence of massless NG modes and massive modes
\cite{bk},
and potentially observable novel effects
for post-newtonian physics
\cite{bak06}
and for the matter sector
\cite{kleh}.
The potential $V$ also leads
to a candidate alternative description of the photon
\cite{bk}
and the graviton
\cite{akgrav,kpgr}.
In Minkowski spacetime,
more general potentials $V$ of hypergeometric form
are known to satisfy the one-loop exact renormalization group
\cite{baak}.
Models of the form \rf{bb}
with $\si_1 = \si_2 = 0$,
a unit timelike $b_\mu$,
a linear Lagrange-multiplier potential $V_L$,
and an additional fourth-order interaction for $B_\mu$
have been studied in some detail
as possible unconventional theories of gravity
\cite{bb1,bb2}.
Other works involving bumblebee models include
Refs.\ \cite{ems,cli,bmg,gjw,cfn,bb3}.

An aspect of bumblebee models of particular interest
is the appearance of massless propagating vector modes.
This feature suggests the prospect of an alternative
to the usual description of photons via U(1) gauge theory.
The central idea is to identify the photon modes
with the NG modes arising from spontaneous Lorentz violation.
Early discussions along these lines
centered on reinterpretating the photon or electron
in the context of special relativity
without physical Lorentz violation
\cite{dhfb,yn}.
More recently,
the Lorentz NG modes in certain bumblebee models
with physical Lorentz violation
have been shown to obey the Einstein-Maxwell equations
in Riemann spacetime in axial gauge
\cite{bk}.
These models are further considered below,
with the discussion initiated
in Sec.\ \ref{Bumblebee electrodynamics}.
One motivation of the present work
is to investigate the role of massive modes
and Lagrange-multiplier fields in this context.
In particular,
it is of interest to investigate possible modifications
to electrodynamics and gravity,
with an eye to novel phenomenological applications
of spontaneous Lorentz and diffeomorphism breaking.

\subsection{Transformations}
\label{Transformations}

In considering theories
with violations of Lorentz and diffeomorphism symmetry,
it is important to distinguish
between {\it observer} and {\it particle} transformations
\cite{ck}.
Under either an observer general coordinate transformation
or an observer local Lorentz transformation,
geometric quantities
such as scalars, vectors, tensors, and their derivatives
remain unchanged,
but the coordinate basis defining their components transforms.
In contrast,
particle transformations can change geometric quantities,
independently of any coordinate system and basis.

In theories without spacetime-symmetry breaking,
the component forms of the transformation laws
for particle and observer transformations are inversely related.
For example,
under infinitesimal particle diffeomorphisms
described by four infinitesimal dispacements $\xi^\mu$,
the components of the bumblebee field transform
according to the Lie derivative as
\begin{eqnarray}
B_\mu
&\rightarrow &
 B_\mu - (\partial_\mu \xi^\nu) B_\nu
- \xi^\nu \partial_\nu B_\mu ,
\nonumber\\
B^\mu
&\rightarrow &
B^\mu + (\partial_\nu \xi^\mu) B^\nu
- \xi^\nu \partial_\nu B^\mu,
\label{Bdiff}
\end{eqnarray}
while the metric transforms as
\begin{eqnarray}
g_{\mu\nu}
&\rightarrow &
g_{\mu\nu} - (\partial_\mu \xi^\alpha) g_{\alpha\nu}
- (\partial_\nu \xi^\alpha) g_{\mu\alpha} - \xi^\alpha \partial_\alpha g_{\mu\nu} ,
\nonumber\\
g^{\mu\nu}
&\rightarrow &
g^{\mu\nu} + (\partial_\alpha \xi^\mu) g^{\alpha\nu}
+ (\partial_\alpha \xi^\nu) g^{\mu\alpha} - \xi^\alpha \partial_\alpha g^{\mu\nu} .
\qquad
\label{gdiff}
\end{eqnarray}
Under infinitesimal observer general coordinate transformations,
which are the observer equivalent of diffeomorphisms,
the formulae for the transformations
of the bumblebee and metric components
take the same mathematical form
up to a possible sign change
in the arbitrary parameter $\xi^\mu$,
even though these transformations
are only the result of a change of coordinates.
Similarly,
under infinitesimal particle local Lorentz transformations
with six parameters $\epsilon_{ab} = -\epsilon_{ba}$
related to the local Lorentz group element by
$\Lambda_a^{\pt{a} b} \approx \delta_a^{\pt{a} b} + \epsilon_a^{\pt{a} b}$,
the local components of the bumblebee field transform as
\begin{equation}
B_a \rightarrow B_a + \epsilon_a^{\pt{a} b} B_b ,
\label{BLT}
\end{equation}
with the formula for $B^a$ following
from this by raising indices with $\eta^{ab}$.
Under observer local Lorentz transformations,
the corresponding transformation formulae
again take the same mathematical form.
However,
any form of physical Lorentz and diffeomorphism breaking
destroys the details of these equivalences.

A fundamental premise in classical physics
is that the properties of a physical system
are independent of the presence of a noninteracting observer.
The observer is free to select a coordinate system
to describe the physics of the system,
but the physics cannot depend on this choice.
In particular,
this must remain true
even when Lorentz and diffeomorphism symmetry are broken,
whether explicitly or spontaneously.
Viable candidate theories with spacetime-symmetry breaking
must therefore be invariant
under observer transformations,
which are merely changes of coordinate system.
For example,
the SME is formulated as a general effective field theory
that is invariant
under observer general coordinate transformations
and under observer local Lorentz transformations
\cite{akgrav,kp,ck}.

In contrast,
a theory with physical Lorentz and diffeomorphism breaking
cannot by definition remain fully invariant
under the corresponding particle transformations.
For example,
if the breaking is explicit,
the action of the theory changes under particle transformations.
If instead the breaking is spontaneous,
the action remains invariant
and the equations of motion transform covariantly,
as usual.
However,
the vacuum solution to the equations of motion
necessarily contains quantities with spacetime indices
that are unchanged by particle transformations.
These vacuum values and the excitations around them
can lead to physical effects revealing the symmetry breaking.

In bumblebee models,
the relevant spacetime vacuum values are those
of the bumblebee and metric fields,
denoted $\vev{B_\mu}$ and $\vev{g_{\mu\nu}}$.
These are unaffected by particle diffeomorphisms,
and their nonzero components thereby reveal
the broken particle diffeomorphisms.
In contrast,
$\vev{B_\mu}$ and $\vev{g_{\mu\nu}}$
both transform as usual
under observer general coordinate transformations,
which therefore are unbroken.
Analogous results hold for the local-frame vacuum value $\vev{B_a}$
of the local bumblebee field.
The components $\vev{B_a}$ remain unaffected
by particle local Lorentz transformations,
with the invariance of the nonzero components
resulting from the breaking of local Lorentz generators,
while the components $\vev{B_a}$ transform
under observer local Lorentz transformations
in the usual way.
Similarly,
the vacuum value $\vev{\vb \mu a}$ of the vierbein
is unchanged by both particle diffeomorphisms
and particle Lorentz transformations,
but it transforms as a vector under the
corresponding observer transformations.

By virtue of the relation \rf{Bvb}
between the spacetime and local bumblebee fields,
it follows that spontaneous local Lorentz violation
is necessarily accompanied by spontaneous diffeomorphism violation
and vice versa
\cite{bk}.
The point is that the vacuum value of the vierbein is nonzero,
so the existence of a nonzero $\vev{B_a}$
spontaneously breaking local Lorentz symmetry
also implies the existence of a nonzero $\vev{B_\mu}$
spontaneously breaking diffeomorphism symmetry,
and vice versa.
Note, however,
that this result fails for explicit violation,
where the analogue of the relation \rf{Bvb} is absent.
In general,
explicit local Lorentz violation occurs
when a nonzero quantity
$t_{abc\ldots}$
is externally prescribed in the local frame,
but the corresponding spacetime quantity
$t_{\lambda\mu\nu\ldots}\equiv
\vb \lambda a \vb \mu b \vb \nu c \cdots
t_{abc\ldots}$
is defined using the full vierbein
and hence remains invariant under diffeomorphisms.
Similarly,
explicit diffeomorphism violation occurs
when a nonzero quantity $t_{\lambda\mu\nu\ldots}$
is externally prescribed
on the spacetime manifold,
but the corresponding local quantity
defined via the inverse vierbein
remains invariant under local Lorentz transformations.

\subsection{Mode expansions}
\label{Mode expansions}

To study the content and behavior of the modes,
the fields can be expanded as infinitesimal excitations
about their vacuum values.
At the level of the Lagrange density,
it suffices for most purposes to keep only terms to second order
in the field excitations,
which linearizes the equations of motion.

Assuming for simplicity a Minkowski background
$\vev{g_{\mu\nu}} = \eta_{\mu\nu}$,
we expand the metric and its inverse as
\begin{equation}
g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu},
\quad
g^{\mu\nu} \approx \eta^{\mu\nu} - h^{\mu\nu} .
\label{gh}
\end{equation}
For the local bumblebee vector,
we write
\begin{equation}
B_a = b_a + \beta_a ,
\label{Bbe}
\end{equation}
where the vacuum value $\vev{B_a}$ is denoted $b_a$
and assumed constant,
and
the infinitesimal excitations are denoted $\beta_a$.
The vacuum condition \rf{Bsquare} implies
\begin{equation}
b_a \eta^{ab} b_b = \mp b^2.
\end{equation}

In a Minkowski background,
the vierbein vacuum value can be chosen to be
$\vev{\vb \mu a} = \delta_\mu^{\pt{\mu}a}$
in cartesian coordinates.
The expansions of the vierbein and its inverse are
\begin{eqnarray}
\vb \mu a
&\approx &
\delta_\mu^{\pt{\mu}a} + {\textstyle{1\over 2}} h_\mu^{\pt{\mu}a}
+ \chi_\mu^{\pt{\mu}a} ,
\nonumber\\
\ivb \mu a
&\approx &
\delta^\mu_{\pt{\mu}a} - {\textstyle{1\over 2}} h^\mu_{\pt{\mu}a}
+ \chi^\mu_{\pt{\mu}a} ,
\label{vhch}
\end{eqnarray}
where $h_{\mu a}$ and $\chi_{\mu a}$ are,
respectively,
the symmetric and antisymmetric components of the vierbein.
The covariant and contravariant components of the bumblebee field
follow from Eq.\ \rf{Bvb} as
\begin{eqnarray}
B_\mu
&\approx &
\delta_\mu^{\pt{\mu}a} (b_a + \beta_a)
+ ( {\textstyle{1\over 2}} h_\mu^{\pt{\mu}a} + \chi_\mu^{\pt{\mu}a}) b_a ,
\nonumber\\
B^\mu
&\approx &
\delta^\mu_{\pt{\mu}a} (b^a + \beta^a)
+ (-{\textstyle{1\over 2}} h^\mu_{\pt{\mu}a} + \chi^\mu_{\pt{\mu}a}) b^a .
\label{Bvbup}
\end{eqnarray}

Since the local and spacetime background metrics
are both Minkowski
and since the excitations are infinitesimal,
the distinction between the Latin local indices
and the Greek spacetime indices can be dropped.
We adopt Greek indices for most purposes that follow,
raising and lowering indices on purely first-order quantities
with $\eta^{\mu\nu}$ and $\eta_{\mu\nu}$.
The vierbein expansions \rf{vhch} can then be rewritten as
\begin{eqnarray}
\lvb \mu \nu
&\approx &
\eta_{\nu\sigma} \vb \mu \sigma
\approx ~
\eta_{\mu\nu} + {\textstyle{1\over 2}} h_{\mu\nu} + \chi_{\mu\nu} ,
\nonumber\\
\uvb \mu \nu
&\approx &
\eta^{\nu\sigma} \ivb \mu \sigma
\approx ~
\eta^{\mu\nu} - {\textstyle{1\over 2}} h^{\mu\nu} + \chi^{\mu\nu} .
\label{vhch2}
\end{eqnarray}
The vacuum value for the bumblebee vector becomes
\begin{equation}
\vev{B_\mu}
= \vev{\vb \mu a}  b_a
= \delta_\mu^{\pt{\mu}a} b_a \equiv b_\mu.
\label{bbvac}
\end{equation}

It is convenient to decompose the local vector
excitations $\beta_\mu \equiv \delta_\mu^{\pt{\mu}a} \beta_a$
into transverse
and longitudinal pieces with respect to $b_\mu$.
Excluding for simplicity
the case of lightlight $b_\mu$,
we write
\begin{equation}
\beta_\mu = \beta^{\rm t}_\mu + \beta \hat b_\mu ,
\quad \beta^{\rm t}_\mu b^\mu = 0,
\label{beta}
\end{equation}
where $\hat b_\mu = b_\mu/\sqrt{b^2}$
is a vector along the direction of $b_\mu$ obeying
$\hat b^\mu \hat b_\mu = \mp 1$.
Using this decomposition,
the bumblebee mode expansions \rf{Bvbup} become
\begin{eqnarray}
B_\mu
&\approx &
b_\mu
+ ({\textstyle{1\over 2}} h_{\mu\nu} + \chi_{\mu\nu}) b^\nu
+ \beta^{\rm t}_\mu + \beta \hat b_\mu ,
\nonumber\\
B^\mu
&\approx &
b^\mu
+ (- {\textstyle{1\over 2}} h^{\mu\nu} + \chi^{\mu\nu}) b_\nu
+ \beta^{{\rm t}\mu} + \beta \hat b^\mu .
\label{Bvb2}
\end{eqnarray}

It is instructive to count degrees of freedom
in the expressions \rf{vhch2} and \rf{Bvb2}.
On the left-hand side,
the vierbein has 16 components
and the bumblebee field 4, for a total of 20.
On the right-hand side,
the symmetric metric component $h_{\mu\nu}$ has 10,
the antisymmetric component $\chi_{\mu\nu}$ has 6,
the transverse bumblebee excitation $\beta^{\rm t}_\mu$ has 3,
and the longitudinal excitation $\beta$ has 1,
producing the required matching total of 20.
Of these 20 degrees of freedom,
6 are metric modes,
4 are bumblebee modes,
while 6 are associated with local Lorentz transformations
and 4 with diffeomorphisms.
The explicit transformations of all the field components
under particle diffeomorphisms
and local Lorentz transformations
can be deduced from the full-field expressions
and from the invariance of the vacuum values.
A list of formulae
for both particle and observer transformations
is provided in the Appendix.

Among all the excitations,
only the longitudinal component $\beta$
of the bumblebee field is invariant
under both particle diffeomorphisms and local Lorentz transformations.
It is therefore a physical degree of freedom
in any gauge.
Moreover,
using Eqs.\ \rf{Bsquare} and \rf{Bvb2}
reveals that exciting the $\beta$ mode alone
produces a nonzero value of $X$,
given at first order by
\begin{equation}
X = B_\mu g^{\mu\nu} B_\nu \pm b^2 \approx 2 (b^\mu \hat b_\mu) \beta .
\label{Bsquare2}
\end{equation}
As a result,
the excitation $\beta$ is associated with a
nonminimal value of the potential,
and it therefore cannot be an NG mode.
In fact,
for the case of a smooth quadratic potential,
$\beta$ is a massive mode.
In contrast,
in a theory with a Lagrange-multiplier potential
where the constraint $X = 0$ is enforced
as an equation of motion,
the massive mode $\beta$ identically vanishes.

\subsection{Gauge fixing and NG modes}
\label{Gauge fixing and NG modes}

Since the spacetime-symmetry breaking in bumblebee models
is spontaneous,
the four diffeomorphisms parametrized by $\xi_\mu$
and the six local Lorentz transformations
parametrized by $\epsilon_{\mu\nu}$
leave invariant the bumblebee action \rf{bb}
and transform covariantly the equations of motion.
Fixing the gauge freedom therefore requires
10 gauge conditions.

For the diffeomorphism freedom,
a choice common in the literature is the harmonic gauge
\begin{equation}
\partial_\mu \ol h^{\mu\nu}  = 0 ,
\label{harm}
\end{equation}
where
$\ol h^{\mu\nu} =  h^{\mu\nu} - {\textstyle{1\over 2}} \eta_{\mu\nu} h$
and $\ol h = - h \equiv - h^\mu_{\pt{\mu}\mu}$.
In the harmonic gauge,
the Einstein tensor becomes
$G_{\mu\nu} \approx -{\textstyle{1\over 2}} \mathchoice\sqr66\sqr66\sqr{2.1}3\sqr{1.5}3 \, \ol h_{\mu\nu}$ at linear order.
An alternative choice for the diffeomorphism degrees of freedom
is the axial gravitational gauge
\begin{equation}
h_{\mu\nu} b^\nu = 0.
\label{haxial}
\end{equation}
Both these gauge choices represent four conditions.
To fix the local Lorentz freedom,
it is common to eliminate
all six antisymmetric vierbein components
by imposing the six conditions
\begin{equation}
\chi_{\mu\nu} = 0.
\label{chigauge}
\end{equation}
Other choices are possible here too.
Consider,
for example,
the decomposition of $\chi_{\mu\nu}$
in terms of projections along $b_\mu$,
\begin{eqnarray}
\chi_{\mu\nu} &=&
\chi^{tt}_{\mu\nu}
\mp \chi_\mu^t \hat b_\nu
\pm \chi_\nu^t \hat b_\mu ,
\nonumber\\
\chi^{tt}_{\mu\nu} b^\nu
&=&
\chi^t_\nu b^\nu = 0,
\label{chidec}
\end{eqnarray}
which is the analogue of Eq.\ \rf{beta} for $\beta_\mu$.
The components $\chi^{tt}_{\mu\nu}$
and $\chi_\mu^t \equiv \chi_{\mu\nu} \hat b^\nu$
each contain three degrees of freedom.
Inspection of the transformation laws shows that
an alternative to fixing the local Lorentz gauge
via Eq.\ \rf{chigauge}
is the set of six conditions
\begin{equation}
\chi^{tt}_{\mu\nu} = \beta^t_\mu = 0.
\label{btgauge}
\end{equation}
Note,
however,
that the combination
$(\chi_{\mu\nu} b^\nu + \beta^{\rm t}_\mu)$
appearing in $B_\mu$
in Eq.\ \rf{Bvb2}
is invariant
and therefore cannot be gauged to zero.
Evidently,
the associated local Lorentz degrees of freedom
must remain somewhere in the theory.

The bumblebee vacuum value $\vev{B_\mu} = b_\mu$
breaks one of the four diffeomorphism symmetries.
The broken generator is associated with
the projected component $\xi_\nu b^\nu$
of the parameter $\xi_\mu$.
Analogously,
the vacuum value $\vev{B_a} = b_a$ breaks
three of the six local Lorentz symmetries.
The broken generators are associated with the components
$\epsilon_{ab} b^b \approx \epsilon_{\mu\nu} b^\nu$
of $\epsilon_{\mu\nu}$ projected along $b^\nu$.
In each case,
the unbroken generators are associated with the
orthogonal complements to the projections.

Since the vacuum breaks one particle diffeomorphism
and three local Lorentz transformations,
four NG modes appear.
A useful general procedure to identify these modes
is first to make virtual particle transformations
using the broken generators
acting on the appropriate vacuum values
for the fields in $V$,
and then to promote the corresponding parameters
$\epsilon_{\mu\nu}$ and $\xi_\mu$ to field excitations
\cite{bk}:
\begin{equation}
\xi_\mu \to \Xi_\mu,
\quad
\epsilon_{\mu\nu} \to {\cal E}_{\mu\nu} = - {\cal E}_{\nu\mu}.
\label{promotion}
\end{equation}
The properties of $\Xi_\mu$ and ${\cal E}_{\mu\nu}$
under various particle and observer transformations
are given in the Appendix.
The projections
$\xi_\nu b^\nu$ and $\epsilon_{\mu\nu} b^\nu$
associated with the broken generators
determine the NG modes,
which are therefore
$\Xi_\nu b^\nu$ and ${\cal E}_{\mu\nu} b^\nu$.

We can follow this procedure to elucidate
the relationship between
the NG modes and the component fields
in the decomposition \rf{Bvb2} of $B_\mu$.
Consider first the diffeomorphism NG mode,
which is generated by a virtual transformation
acting on the spacetime vacuum value $\vev{B_\mu}$
and involving the broken diffeomorphism generator.
The relevant transformation is given in Eq.\ \rf{pdiffs},
and it yields
\begin{equation}
\vev {B_\mu} \to b_\mu - (\partial_\mu \Xi_\nu) b^\nu .
\end{equation}
Comparison of this result to the form of Eq.\ \rf{Bvb2}
reveals that the four vierbein combinations
$({\textstyle{1\over 2}} h_{\mu\nu} + \chi_{\mu\nu})b^\nu$
contain the diffeomorphism NG mode.
This agrees with the result obtained by combining
virtual diffeomorphisms on the vacuum values
of the component fields in Eq.\ \rf{Bvb2}:
\begin{eqnarray}
\vev{h_{\mu\nu} b^\nu}
&\to &
-(\partial_\mu \Xi_\nu + \partial_\nu
\Xi_\mu) b^\nu ,
\nonumber\\
\vev{\chi_{\mu\nu} b^\nu}
&\to &
- {\textstyle{1\over 2}} (\partial_\mu \Xi_\nu - \partial_\nu \Xi_\mu) b^\nu ,
\nonumber\\
\vev{\beta^t_\mu}
&\to &
0,
\quad
\vev{\beta} \to 0.
\label{XiEpredef}
\end{eqnarray}
Note also that
the combinations $({\textstyle{1\over 2}} h_{\mu\nu} + \chi_{\mu\nu})b^\nu$
maintain the potential minimum $V=0$,
as is expected for an NG mode.

In contrast,
the three Lorentz NG modes ${\cal E}_{\mu\nu} b^\nu$
are generated by virtual transformations
acting on the local-frame vacuum value $\vev{B_a}$
and involving the broken local Lorentz generators:
\begin{equation}
\vev {B_a} \to b_a + {\cal E}_{ab} b^b ,
\label{LLvirttr}
\end{equation}
Comparison with Eqs.\ \rf{Bbe} and \rf{beta}
shows that the transverse field $\beta^t_\mu$
contains the three Lorentz NG modes ${\cal E}_{\mu\nu} b^\nu$.
Note that there are exactly three degrees of freedom
in $\beta^t_\mu$,
all of which maintain the potential minimum $V=0$.
This result can be used to connect
the component fields in the decomposition of $B_\mu$
with the three Lorentz NG modes ${\cal E}_{\mu\nu} b^\nu$.
The vacuum value $\vev{B_\mu}$ itself is invariant
under local Lorentz transformations,
so another component field must also contain
the Lorentz NG modes.
Performing virtual local Lorentz transformations
on the vacuum values of the component fields gives
\begin{eqnarray}
\vev{h_{\mu\nu} b^\nu}
&\to &
0,
\quad \vev{\chi_{\mu\nu} b^\nu}
\to
-{\cal E}_{\mu\nu} b^\nu ,
\nonumber\\
\vev{\beta^t_\mu}
&\to &
{\cal E}_{\mu\nu} b^\nu ,
\quad
\vev{\beta} \to 0,
\label{vllt}
\end{eqnarray}
which shows that the three combinations
$\vev{\chi_{\mu\nu} b^\nu}$
also contain the three Lorentz NG modes.

Combining the results \rf{XiEpredef} and \rf{vllt}
reveals the following NG mode content
for the component field combinations
in the decomposition \rf{Bvb2} of $B_\mu$:
\begin{eqnarray}
(-{\textstyle{1\over 2}} h_{\mu\nu} + \chi_{\mu\nu}) b^\nu
&= &
- (\partial_\mu \Xi_\nu) b^\nu - {\cal E}_{\mu\nu} b^\nu ,
\nonumber\\
\beta^t_\mu
&= &
{\cal E}_{\mu\nu} b^\nu ,
\nonumber\\
\beta &\supset& 0.
\label{combined}
\end{eqnarray}
We see that the four combinations
$({\textstyle{1\over 2}} h_{\mu\nu} + \chi_{\mu\nu})b^\nu$
contain a mixture of the diffeomorphism NG mode
and the three Lorentz NG modes,
while the three combinations $\beta^t_\mu$
contain only the three Lorentz NG modes.

By fixing the 10 gauge freedoms in the theory,
the above results can be used to determine
the physical content of the field $B_\mu$.
Suppose first we adopt
the 10 conditions \rf{haxial} and \rf{chigauge}.
Then,
the bumblebee field becomes
\begin{eqnarray}
B_\mu
&\approx &
b_\mu + \beta^t_\mu + \beta \hat b_\mu
\nonumber\\
&=&
b_\mu + {\cal E}_{\mu\nu} b^\nu + \beta \hat b_\mu ,
\label{bgauge1}
\end{eqnarray}
where the result \rf{combined} has been used.
Note that the fixed gauge means that fields
with different transformation properties
can appear on the left- and right-hand sides.
In this gauge,
the four components of $B_\mu$
are decomposed into
three Lorentz NG modes associated with $\beta^t_\mu$
and the one massive mode $\beta$.
The diffeomorphism NG mode is absent.
It obeys
\begin{equation}
(\partial_\mu \Xi_\nu)b^\nu = -(\partial_\nu \Xi_\mu) b^\nu
\label{diffeocond1}
\end{equation}
and hence
$b^\mu(\partial_\mu \Xi_\nu)b^\nu = 0$,
and it is locked to the Lorentz NG modes via the equation
\begin{equation}
(\partial_\mu \Xi_\nu)b^\nu = - {\cal E}_{\mu\nu} b^\nu.
\label{diffeocond2}
\end{equation}
For the diffeomorphism NG mode,
this gauge is evidently analogous to the unitary gauge
in a nonabelian gauge theory.

An alternative gauge choice could be to impose
the 10 conditions \rf{haxial} and \rf{btgauge} instead.
This gives
\begin{eqnarray}
B_\mu
&\approx &
b_\mu + \chi^t_\mu + \beta \hat b_\mu
\nonumber\\
&=&
b_\mu + {\cal E}_{\mu\nu} b^\nu + \beta \hat b_\mu ,
\label{bgauge2}
\end{eqnarray}
where in this gauge
the Lorentz NG modes ${\cal E}_{\mu\nu} b^\nu$
are identified with $\chi^t_\mu$
instead of $\beta^t_\mu$.
One way to understand this identification
is to perform a local Lorentz transformation
with parameter $\epsilon_{\mu\nu} = -{\cal E}_{\mu\nu}$
on the first result in Eq.\ \rf{bgauge1}.
This changes the value of $\beta^t_\mu$
from ${\cal E}_{\mu\nu} b^\nu$ to zero,
while simultaneously converting
$\chi^t_\mu$ from zero
to $\chi^t_\mu ={\cal E}_{\mu\nu} b^\nu$.
In this gauge,
the explicit decomposition \rf{bgauge2} of $B_\mu$
in terms of NG modes is the same as that in Eq.\ \rf{bgauge1},
but the three Lorentz NG modes
are now associated with the components $\chi^t_\mu$
of the vierbein.
The diffeomorphism mode remains absent.
It still obeys the condition \rf{diffeocond1}
and is locked to the Lorentz NG modes
by Eq.\ \rf{diffeocond2}.

Partial gauge conditions can also be imposed.
For example,
suppose the choice \rf{btgauge}
is made for the local Lorentz gauge,
while the diffeomorphism gauge remains unfixed.
Then,
the four degrees of freedom in
the combination $({\textstyle{1\over 2}} h_{\mu\nu} + \chi_{\mu\nu})b^\nu$
consist of the three Lorentz NG modes
and the diffeomorphism NG mode,
and all the NG modes are contained in the vierbein
\cite{bk}.
The bumblebee field can be written as
\begin{equation}
B_\mu
\approx
b_\mu
-(\partial_\mu \Xi_\nu) b^\nu + {\cal E}_{\mu\nu} b^\nu
+ \beta \hat b_\mu .
\label{Bvbredef}
\end{equation}
Note that only one projection of $\Xi_\nu$ appears,
even though four diffeomorphism choices remain.
The corresponding expression for $B^\mu$
includes additional metric contributions
and is given by
\begin{equation}
B^\mu \approx b^\mu + (\partial_\nu \Xi^\mu) b^\nu
+ {\cal E}^{\mu\nu} b_\nu
+ \beta \hat b^\mu .
\label{upBvbredef}
\end{equation}
This equation contains contributions
from all four fields $\Xi_\mu$.
However,
if the diffeomorphism excitations
are restricted only to the one for the broken generator,
for which $\Xi_\mu$ obeys $b_\mu \Xi_\nu = b_\nu \Xi_\mu$,
then $B^\mu$ also reduces to an expression
depending only on the diffeomorphism NG mode $\Xi_\nu b^\nu$.
Related results are obtained in Ref.\ \cite{bk},
which investigates the fate of the NG modes
using a decomposition of the vierbein
into transverse and longitudinal components along $b_\mu$.
This decomposition leads to the same relations as
Eqs.\ \rf{Bvbredef} and \rf{upBvbredef}
when the condition $b_\mu \Xi_\nu = b_\nu \Xi_\mu$ is applied.

Even without gauge fixing,
the fields $\Xi_\mu$ cancel at linear order
in both $G_{\mu\nu}$ and $B_{\mu\nu}$.
As a result,
propagating diffeomorphism NG modes cannot appear.
This is a special case of a more general result.
By virtue of their origin as virtual particle transformations,
the diffeomorphism NG modes appear as certain components
of representation-irreducible fields with nonzero vacuum values.
However,
diffeomorphism invariance ensures
these modes enter in the metric and bumblebee fields
in combinations that cancel in a diffeomorphism-invariant action,
including the general bumblebee action \rf{bb}.
In contrast,
the Lorentz NG modes do play a role in the bumblebee action.
They can be identified with a massless vector field
$A_\mu \equiv {\cal E}_{\mu\nu} b^\nu$
in the fixed axial gauge $A_\mu b^\mu \approx 0$.
The fully gauge-fixed expression \rf{bgauge1}
for $B_\mu$ then becomes
\begin{equation}
B_\mu \approx b_\mu + A_\mu + \beta \hat b_\mu ,
\label{bba}
\end{equation}
where the transverse components
have the form of photon fields
in the axial gauge,
and the longitudinal mode is the massive mode $\beta$.

\subsection{Alternative expansions}
\label{Alternative expansions}

The analysis in the previous subsections
is based on the local-frame expansion \rf{Bbe}
of the bumblebee field $B_a$.
However,
other mode expansions are possible,
including ones in which the vierbein makes no explicit appearance
and the local Lorentz transformations are no longer manifest.
This subsection offers a few comments
on two alternative expansions used in some of the literature.

The first alternative mode expansion
is specified using the covariant spacetime components
of the bumblebee field
\cite{bk,kp},
\begin{equation}
B_\mu = b_\mu + {\cal E}_\mu ,
\label{epdown}
\end{equation}
where ${\cal E}_\mu$ represents the excitations
of the spacetime bumblebee field
around the vacuum value $b_\mu$.
It follows that the contravariant components are
\begin{equation}
B^\mu \approx b^\mu + {\cal E}^\mu - h^{\mu\nu} b_\nu .
\label{epup}
\end{equation}
These fields are linked to the vierbein
and the local-frame fluctuations $\beta_\mu$ via
\begin{equation}
{\cal E}_\mu =
({\textstyle{1\over 2}} h_{\mu\nu} + \chi_{\mu\nu}) b^\nu
+ \beta^{\rm t}_\mu + \beta \hat b_\mu .
\label{redef1}
\end{equation}
The fields ${\cal E}_\mu$ are
scalars under particle local Lorentz transformations,
but transform under particle diffeomorphisms as
\begin{eqnarray}
{\cal E}_\mu &\rightarrow& {\cal E}_\mu - (\partial_\mu \xi_\alpha) b^\alpha ,
\nonumber\\
{\cal E}^\mu \equiv \eta^{\mu\nu} {\cal E}_\nu &\rightarrow& {\cal E}^\mu -
\eta^{\mu\nu} (\partial_\nu \xi_\alpha) b^\alpha .
\label{epdiff}
\end{eqnarray}
Note that this usage of ${\cal E}^\mu$
differs from that in Ref.\ \cite{bk}.

The second alternative expansion
\cite{bak06,bb1,cli,bmg,gjw}
starts instead with the contravariant bumblebee components
\begin{equation}
B^\mu = b^\mu + {\cal E}^{\prime\mu} ,
\label{epprimeup}
\end{equation}
where a prime is used to distinguish the excitations
${\cal E}^{\prime\mu}$ from the previous case.
The corresponding covariant components are then
\begin{equation}
B_\mu \approx b_\mu + {\cal E}^\prime_\mu + h_{\mu\nu} b^\nu .
\label{epprimedown}
\end{equation}
The field redefinitions connecting these to the vierbein
and previous case are
\begin{eqnarray}
{\cal E}^{\prime\mu} &=&
(- {\textstyle{1\over 2}} h^{\mu\nu} + \chi^{\mu\nu}) b_\nu
+ \beta_{\rm t}^\mu + \beta \hat b^\mu
\nonumber\\
&=& \eta^{\mu\nu} {\cal E}_\mu - h^{\mu\nu} b_\nu .
\label{redef2}
\end{eqnarray}
The fields ${\cal E}^{\prime\mu}$
are scalars under particle local Lorentz transformations
but transform under particle diffeomorphisms as
\begin{eqnarray}
{\cal E}^{\prime\mu}
&\rightarrow& {\cal E}^{\prime\mu} + (\partial_\alpha \xi^\mu) b^\alpha ,
\nonumber\\
{\cal E}^\prime_\mu \equiv \eta_{\mu\nu} {\cal E}^{\prime\mu}
&\rightarrow& {\cal E}^\prime_\mu + (\partial_\alpha \xi_\mu) b^\alpha .
\label{epprimediff}
\end{eqnarray}
Note that the fields ${\cal E}_\mu$ and ${\cal E}^\prime_\mu$
have different transformation properties
and contain different mixes of the bumblebee and metric excitations.
However,
${\cal E}_\mu$ and ${\cal E}^\prime_\mu$ take the same gauge-fixed form
in the gauge \rf{haxial}.

The two alternative expansions are useful because
the excitations ${\cal E}_\mu$ and ${\cal E}^\prime_\mu$ are invariant
under infinitesimal local particle Lorentz transformations.
In these variables,
the local Lorentz symmetry is hidden
and only the diffeomorphism symmetry is manifest.
The Lorentz NG modes lie in the $b_\mu$-tranverse components
of ${\cal E}_\mu$ or ${\cal E}^\prime_\mu$.
The massive mode $\beta$ joins the diffeomorphism mode
in the longitudinal component of ${\cal E}_\mu$.
It can be identified from Eq.\ \rf{redef1} as the combination
\begin{equation}
\beta =  \fr
{b^\mu ({\cal E}_\mu - {\textstyle{1\over 2}} h_{\mu\nu} b^\nu)}
{b^\alpha\hat b_\alpha}
= \mp \hat b^\mu ({\cal E}_\mu - {\textstyle{1\over 2}} h_{\mu\nu} b^\nu) .
\label{massep}
\end{equation}
Notice that $\beta$ appears here as a
diffeomorphism-invariant combination of field components
that transform nontrivially.

The choice of diffeomorphism gauge has interesting
consequences for ${\cal E}_\mu$ and ${\cal E}^\prime_\mu$.
First,
note that ${\cal E}_\mu$ cannot be gauged to zero,
since only one degree of freedom $\xi_\alpha b^\alpha$ appears in
its transformation law.
However,
it is possible to gauge ${\cal E}^\prime_\mu$ to zero.
The corresponding gauge condition for ${\cal E}_\mu$
is ${\cal E}_\mu = h_{\mu\nu} b^\nu$.
In either of these gauges,
the massive mode becomes
\begin{equation}
\beta = \mp {\textstyle{1\over 2}} \hat b^\mu h_{\mu\nu} \hat b^\nu ,
\label{massh}
\end{equation}
thereby becoming part of the metric.
The bumblebee field strength reduces to
\begin{equation}
B_{\mu\nu} =
(\partial_\mu h_{\nu\sigma} - \partial_\nu h_{\mu\sigma}) b^\sigma ,
\label{Bmunuh}
\end{equation}
and so is also given by the metric.
Evidently,
in these gauges the theory is defined entirely
in terms of the metric excitations $h_{\mu\nu}$.
Requiring ${\cal E}_\mu$ to vanish
or dropping the term
$h_{\mu\nu} b^\nu$ in Eq.\ \rf{epprimedown}
therefore improperly sets to zero
the bumblebee field strength $B_{\mu\nu}$,
which alters the equations of motion governing
the Lorentz NG and massive modes
\cite{bmg}.
It follows from Eq.\ \rf{Bmunuh}
that the effective Lagrange density
can contain additional kinetic terms for $h_{\mu\nu}$,
beyond those arising from the gravitational action,
that originate from the bumblebee kinetic terms ${\cal L}_{\rm K}$.
These additional terms propagate the Lorentz NG modes,
disguised as massless modes contained in the metric.

\section{Examples}
\label{Examples}

In this section,
some simple examples are developed
to illustrate and enrich the general results obtained
in the discussions above.
We choose to work within the class of KS\ bumblebee models
with Lagrange density given by
Eq.\ \rf{ksbb},
which avoids
\it a priori \rm
propagating ghost fields
and the complications of nonminimal gravitational couplings.
For definiteness,
we set $\Lambda=0$ and
choose the matter-bumblebee coupling to be
\begin{equation}
{\cal L}_{\rm M} =
- e B_\mu J_{\rm M}^\mu ,
\end{equation}
where $J_{\rm M}^\mu$ is understood to be a current
formed from matter fields in the theory.
In principle,
this current could be formed from dynamical fields
in the theory or could be prescribed externally.
For the latter case,
diffeomorphism invariance is explicitly broken
unless the current satisfies a suitable differential constraint.

The section begins by
presenting some general results for this class of models,
including ones related to the equations of motion,
conservation laws,
and the connection to Einstein-Maxwell electrodynamics.
We then turn to a more detailed discussion
of three specific cases
with different explicit potentials $V(X)$,
where the bumblebee field combination $X$
is defined as in Eq.\ \rf{XB}.
The three cases involve
the smooth quadratic potential
$V_S(X)$ of Eq.\ \rf{VS},
the linear Lagrange-multiplier potential
$V_L(\lambda, X)$ of Eq.\ \rf{VL},
and the quadratic Lagrange-multiplier potential
$V_Q(\lambda, X)$ of Eq.\ \rf{VQ}.
The massive modes are studied for each potential,
and their effects on gravity and electromagnetism are explored.

\subsection{General considerations}
\label{General considerations}

\subsubsection{Equations of motion and conservation laws}
\label{Equations of motion and conservation laws}

Varying the Lagrange density \rf{ksbb}
with respect to the metric
yields the gravitational equations of motion
\begin{eqnarray}
G^{\mu\nu} &=& 8 \pi G T^{\mu\nu} .
\label {geq}
\end{eqnarray}
In this expression,
the total energy-momentum tensor
$T^{\mu\nu}$ can be written as a sum of two terms,
\begin{equation}
T^{\mu\nu} = T^{\mu\nu}_{\rm M }+ T^{\mu\nu}_B .
\label{Ttotal}
\end{equation}
The first is the energy-momentum tensor
$T^{\mu\nu}_{\rm M}$
arising from the matter sector.
The second is the energy-momentum tensor
$T^{\mu\nu}_B$
arising from the bumblebee kinetic and potential terms,
\begin{equation}
T^{\mu\nu}_B =
T^{\mu\nu}_{\rm K}
+ T_V^{\mu\nu} ,
\label{TBmunu}
\end{equation}
where $T^{\mu\nu}_{\rm K}$ and $T_V^{\mu\nu}$
are given by
\begin{eqnarray}
T^{\mu\nu}_{\rm K} &=&
B^{\mu\alpha} B^\nu_{\pt{\nu}\alpha}
- \frac 1 4 g^{\mu\nu} B_{\alpha\beta} B^{\alpha\beta}
\label{tk}
\end{eqnarray}
and
\begin{eqnarray}
T_V^{\mu\nu} &=&
- V g^{\mu\nu} + 2 V^\prime B^\mu B^\nu .
\label{tv}
\end{eqnarray}

Varying instead with respect to the bumblebee field
generates the remaining equations of motion,
\begin{eqnarray}
D_\nu B^{\mu\nu} &=& J^\mu.
\label{Beq}
\end{eqnarray}
Like the total energy-momentum tensor,
the total current $J^\mu$ can be written as the sum of two terms
\begin{equation}
J^\mu = J_{\rm M}^\mu + J_B^\mu .
\end{equation}
The partial current $J_{\rm M}^\mu$
is associated with the matter sector
and acts as an external source for the bumblebee field.
The partial current $J_B^\mu$
arises from the bumblebee self-interaction,
and it is given explicitly as
\begin{equation}
J_B^\mu = - 2 V^\prime B^\mu .
\label{JB}
\end{equation}

The contracted Bianchi identities for $G_{\mu\nu}$
lead to conservation of the total energy-momentum tensor,
\begin{equation}
D_\mu T^{\mu\nu} \equiv
D_\mu (T^{\mu\nu}_{\rm M} + T^{\mu\nu}_B) = 0 .
\label{Tconsv}
\end{equation}
The antisymmetry of the bumblebee field strength $B_{\mu\nu}$
implies the remnant constraint \rf{remconst}
and hence a second conservation law,
\begin{equation}
D_\mu J^\mu \equiv
 D_\mu (J_{\rm M}^\mu + J_B^\mu) = 0 .
\label{Jconsv}
\end{equation}
Note that this second conservation law is a
special feature of KS\ bumblebee models.
It is a direct consequence of choosing
the \it a priori \rm ghost-free action \rf{ksbb}.
Note also that this conservation law holds
even though the potential term $V$
excludes any local U(1) gauge symmetry
in these models.

\subsubsection{Bumblebee currents}
\label{Bumblebee currents}

The bumblebee current $J_B^\mu$ defined in Eq.\ \rf{JB}
vanishes when $V^\prime = 0$.
This situation holds for the vacuum solution and for NG modes,
and it then follows from Eq.\ \rf{Jconsv}
that the matter current $J_{\rm M}^\mu$
is covariantly conserved.
However,
$V^\prime \ne 0$ in the presence of a nonzero massive mode
or a nonzero linear Lagrange multiplier,
whereupon the bumblebee current $J_B^\mu$
acts as an additional source
for the bumblebee field equation \rf{Beq}.
A nonzero massive mode or nonzero linear Lagrange multiplier
contributes to the bumblebee component $T^{\mu\nu}_B$
of the energy-momentum tensor too,
and it therefore also acts as an additional source
for the gravitational field equations \rf{geq}.

The contributions to the energy-momentum tensor $T^{\mu\nu}_B$
stemming from $V^\prime \ne 0$ can be positive or negative.
This is a generic feature,
as can be seen from the expression
for $T_V^{\mu\nu}$ in Eq.\ \rf{TV}
and the general bumblebee Lagrange density \rf{bb}.
The full energy-momentum tensor $T^{\mu\nu}$ is conserved,
but the prospect of negative values for $T^{\mu\nu}_B$
implies stability issues for the models.
Under suitable circumstances,
for example,
unbounded negative values of $T^{\mu\nu}_B$
can act as unlimited sources of energy
for the matter sector.
Stability is also a potential issue
in the absence of matter.
For example,
for the case where $J_{\rm M}^\mu$ is disregarded
and the potential involves a linear Lagrange multiplier
producing $\lambda \ne 0$ on shell,
at least one set of initial values is known
that yields a negative-energy solution
\cite{cj}.

Whether instabilities occur in practice depends on
the form of the Lagrange density and on the initial conditions.
The situation is comparatively favorable for KS\ bumblebee models
because the extra conservation law \rf{Jconsv}
can play a role.
In principle,
negative-energy contributions in the initial state
can be eliminated by choosing initial conditions
such that $V^\prime = 0$,
while the conservation law can prevent the
eventual development of a destabilizing mode.
One way to see this is to expand
the conservation law Eq.\ \rf{Jconsv}
at leading order in a Minkowski background.
For nonzero $\vev{B_0}$,
we obtain
\begin{equation}
\partial_0 V^\prime \approx
\fr 1 {2 B^0} ( \partial_\mu J_{\rm M}^\mu
- 2 V^\prime \partial_\mu B^\mu - 2 B^j \partial_j V^\prime ) .
\label{d0Vprime}
\end{equation}
Suppose the matter current is independently conserved,
$D_\mu J_{\rm M}^\mu \approx \partial_\mu J_{\rm M}^\mu = 0$.
The initial condition $V^\prime = 0$
then implies $\partial_0 V^\prime = 0$ initially.
Taking further derivatives shows that all time derivatives
of $V^\prime$ vanish initially and
so $V^\prime$ remains zero for all time,
indicating stability is maintained.

It is interesting to note that
the matter-current conservation law $D^\mu J^{\rm M}_\mu = 0$
emerges naturally in the limit
of vanishing NG and massive modes,
where the bumblebee field reduces to
its vacuum value $B_\mu \rightarrow \vev{B_\mu} = b_\mu$.
This agrees with a general conjecture
made in Ref.\ \cite{akgrav}.
The line of reasoning is as follows.
In the limit,
the bumblebee theory with spontaneous Lorentz violation
takes the form of a theory with explicit Lorentz violation
with couplings only to $b_\mu$.
However,
theories with explicit Lorentz violation
contain an incompatibility between the Bianchi identities
and the energy-momentum conservation law.
The incompatibility is avoided for spontaneous Lorentz violation
by the vanishing of a particular variation in the action,
which reduces in the limit to the requirement of current
conservation $D^\mu J^{\rm M}_\mu = 0$.
Nonetheless,
if $B_\mu$ is excited away from this limit,
current conservation in the matter sector may fail to hold.

In practice,
the potential instability may be irrelevant for physics.
For example,
if a bumblebee model is viewed
as an effective field theory emerging
from an underlying stable quantum theory of gravity,
any apparent instabilities may reflect incomplete
knowledge of constraints on the massive modes
or of countering effects that come into play
at energy scales above those of the effective theory.
Under these circumstances,
in KS\ bumblebee models
it may suffice for practical purposes simply to postulate
that the matter and bumblebee currents do not mix
and hence obey separate conservation laws,
\begin{equation}
D_\mu J_{\rm M}^\mu = 0 ,
\qquad
D_\mu J_B^\mu = 0 .
\label{JJB0}
\end{equation}
For instance,
one can impose that only matter terms ${\cal L}_M$
with a global U(1) symmetry are allowed,
as is done for the case of Minkowski spacetime
in Ref.\ \cite{cfn}.
Another option might be to disregard $J^{\rm M}_\mu$ altogether
and allow couplings between the matter
and bumblebee fields only through gravity.
In what follows,
we adopt Eqs.\ \rf{JJB0} as needed
to investigate modifications of gravity
arising from massive modes
and to study the conditions
under which Einstein-Maxwell solutions
emerge in bumblebee models.

\subsubsection{Bumblebee electrodynamics}
\label{Bumblebee electrodynamics}

An interesting aspect of the KS\ bumblebee models
is their prospective interpretation as alternative theories
of electrodynamics with physical signatures of Lorentz violation
\cite{bk}.
In this approach,
basic electrodynamics properties
such as the masslessness of photons are viewed as consequences
of spontaneous local Lorentz and diffeomorphism breaking
rather than of exact local U(1) symmetry.
A key point is that KS\ bumblebee models have no dependence
on second time derivatives of $B_0$.
They therefore feature an additional primary constraint
relative to the bumblebee models \rf{bb}
with more general kinetic terms.
The primary constraint generates a secondary constraint
in the form of a modified Gauss law,
which permits a physical interpretation of the theory
in parallel to electrodynamics.
This modified Gauss law is unavailable to
other bumblebee models.

In a suitable limit,
the equations of motion \rf{geq} and \rf{Beq}
for the KS\ bumblebee models
reduce to the Einstein-Maxwell equations
in a fixed gauge for the metric and photon fields.
To demonstrate this,
we work in asymptotically Minkowski spacetime
and choose the gauge in which the
bumblebee field is given by Eq.\ \rf{bba}.
The limit of interest is $\beta \to 0$,
corresponding to zero massive mode.
For the case with the Lagrange-multiplier potential \rf{VL},
where $\beta$ is constrained to zero,
an equivalent result also follows
in the limit $\lambda \to 0$.
Since $T_V^{\mu\nu}$ acquires nonzero contributions
only from the massive mode or linear Lagrange multiplier,
it follows that as $\beta \to 0$ or $\lambda \to 0$
the bumblebee energy-momentum tensor $T^{\mu\nu}_B$
reduces to that of Einstein-Maxwell electrodynamics,
\begin{eqnarray}
T^{\mu\nu}_B &\to&
T^{\mu\nu}_{\rm K}
\equiv T^{\mu\nu}_{\rm EM}
= F^{\mu\alpha} F_{\alpha}^{\pt{\alpha}\nu}
- \frac 1 4 g_{\mu\nu} F_{\alpha\beta} F^{\alpha\beta} .
\qquad
\label{TBBEM}
\end{eqnarray}
In this equation,
the bumblebee field strength is
reinterpreted as the electromagnetic field strength
\begin{equation}
B_{\mu\nu} \equiv F_{\mu\nu} =
\partial_\mu A_\nu - \partial_\nu A_\mu ,
\end{equation}
via the mode expansion \rf{bba},
and the gravitational field equations \rf{geq}
reduce to their Einstein-Maxwell counterparts.
Similarly,
the bumblebee current $J_B^\mu$ in Eq.\ \rf{JB}
vanishes as $\beta \to 0$ or $\lambda \to 0$
because $V^\prime \to 0$.
This leaves only the conventional matter current $J_{\rm M}^\mu$,
which by virtue of Eq.\ \rf{Jconsv} obeys covariant conservation.
The bumblebee field equations \rf{Beq}
therefore also reduce to their Einstein-Maxwell counterparts.

Even when $\beta$ vanishes,
the interpretation of the model as bumblebee electrodynamics
can in principle be distinguished
from conventional Einstein-Maxwell electrodynamics
through nontrivial SME-type couplings
involving the vacuum value $b_\mu$
in the matter equations of motion.
Depending on the types of couplings appearing in
the matter-sector Lagrange density ${\cal L}_{\rm M}$,
a variety of effects could arise.
At the level of the minimal SME,
for example,
axial vector couplings involving $b_\mu$
can be sought in numerous experiments,
including ones with
electrons \cite{eexpt,eexpt2},
protons and neutrons \cite{ccexpt,spaceexpt},
and muons \cite{muexpt}.
Other types of SME coefficients can also be generated
from the vacuum value $b_\mu$.
For example,
the nonzero symmetric traceless SME coefficients $c_{\mu\nu}$
might emerge via the symmetric traceless product
$b_\mu b_\nu - b^\alpha b_\alpha \eta_{\mu\nu}/4$,
and experimental searches for these include ones with
electrons \cite{eexpt,eexpt3},
protons, and neutrons \cite{ccexpt,spaceexpt}.

If the model conditions permit the field $\beta$ to be nonzero,
further deviations from Einstein-Maxwell electrodynamics can arise.
For example,
in the weak-field regime
where $\beta\hat b_\mu$ is small compared to $b_\mu$,
the bumblebee current $J_B^\mu$ is linear in $\beta$
and acts as an additional source in the equation
of motion \rf{Beq} for the field strength
$B_{\mu\nu}$.
Effects of this type are investigated
in the next few subsections.
For large fields,
nonlinear effects in the
bumblebee current and energy-momentum tensor
may also play a role.
However,
the field value of $\beta$ is typically suppressed
when the mass of $\beta$ is large,
so if the mass is set by the Planck scale
then the linear approximation is likely to suffice
for most realistic applications.

In the limit of zero $\beta$,
the physical fields are the field strengths $B_{\mu\nu}$ and
the gravitational field.
Excitations of the bumblebee field $B_\mu$
in the classical theory are then unobservable,
in parallel with the potential $A_\mu$ in classical electromagnetism.
This may have consequences for leading-order effects
in the weak-field limit.
For example,
the weak static limit
in general bumblebee models \rf{bb}
requires $B_\mu$ itself to be time independent.
However,
at leading order in the KS\ bumblebee model,
it can suffice to require that only the field strength $B_{\mu\nu}$
be time independent
while $B_\mu$ itself has time dependence,
in analogy with Maxwell electrodynamics in Minkowski spacetime.
For example,
the Coulomb electric field $\vec E = - \partial_0 \vec A$
for a static point charge $q$
emerges in temporal gauge $A_0 = 0$
from a time-dependent $A_\mu = (0, t \partial_j \Phi_q)$,
where $\Phi_q$ is the Coulomb potential.
Likewise,
in certain leading-order weak static solutions
for which $\beta$ acts as a source of charge,
the bumblebee field $B_\mu$ may naturally
exhibit potential lines converging at the source of charge,
similar to the physical configurations
with singular derivatives of $A_\mu$
that occur in classical electrodynamics.

Another interesting issue is the role of quantum effects.
Since bumblebee electrodynamics involves gravity,
it faces the same quantization challenges
as other theories of gravity and electrodynamics,
including Einstein-Maxwell theory.
Some work on the renormalization structure at one loop
has been performed
\cite{baak,renorm},
and in the limit of zero massive mode and in Minkowski spacetime
the usual properties of quantum electrodynamics
are expected to hold.
Addressing this issue in detail
is an interesting open problem
that lies beyond the scope of this work.

\subsection{Smooth quadratic potential}
\label{Smooth quadratic potential}

This subsection considers the specific KS\ bumblebee model \rf{ksbb}
having smooth quadratic potential \rf{VS}
with the definition \rf{XB}:
\begin{equation}
V = V_S(X) = {\textstyle{1\over 2}} \kappa (B_\mu g^{\mu\nu} B_{\nu} \pm b^2)^2.
\label{smoothlagv}
\end{equation}
For definiteness,
we adopt the mode expansion
$B_\mu = b_\mu + {\cal E}_\mu$ of Eq.\ \rf{epdown}
in a Minkowski background
and assume weak fields $h_{\mu\nu}$ and ${\cal E}_\mu$
so that the gravitational and bumblebee equations of motion
\rf{geq} and \rf{Beq} can be linearized.

One goal is to investigate deviations from
Einstein-Maxwell theory arising
from the presence of a weak nonzero massive mode $\beta$.
We therefore focus on dominant corrections
to the linearized Einstein-Maxwell equations
arising from $\beta$.
In this approximation,
the bumblebee component of the energy-momentum tensor becomes
\begin{equation}
T_B^{\mu\nu} \approx
T_{\rm EM}^{\mu\nu} + 4 \kappa (b^\alpha \hat b_\alpha) b^\mu b^\nu \beta ,
\label{linTV}
\end{equation}
where $T_{\rm EM}^{\mu\nu}$ is
the zero-$\beta$ limit of $T_{\rm K}^{\mu\nu}$
given in Eq.\ \rf{TBBEM}
and the other term arises from
$T_V^{\mu\nu}$.
The bumblebee current reduces to
\begin{equation}
J_B^{\mu} \approx - 4 \kappa (b^\alpha \hat b_\alpha) b^\mu \beta .
\label{Jbe}
\end{equation}
The constraint obtained by assuming
current conservation in the matter sector
becomes
\begin{equation}
b^\mu \partial_\mu \beta \approx 0.
\label{linbeconstraint}
\end{equation}
These expressions reveal that at leading order
the massive mode $\beta$
acts as a source for gravitation and electrodynamics,
subject to the constraint \rf{linbeconstraint}.

Note that the linearization procedure
can alter the time behavior and dynamics of the fields
in the presence of a nonzero massive mode $\beta$.
Suppose,
for example,
that $b_\mu$ is timelike
and we adopt the global observer frame in which
$b_\mu = (b,0,0,0)$.
Nonlinearities in the current $J_B^\mu$
associated with nonzero $\beta$
generate time dependence for most solutions
because the spatial current $\vec J_B$
does work on the field strength $B_{\mu\nu}$.
However,
the linearization \rf{Jbe}
implies $J_B^\mu$ is nonzero only along the direction of $b^\mu$,
reducing to a pure charge density
$J_B^\mu \approx (\rho_B,0,0,0)$
with $\rho_B \approx - 4 \kappa b^2 \beta$,
and the current-conservation law \rf{linbeconstraint}
then requires $\rho_B$ and hence $\beta$ to be static.
Nonetheless,
the linearization procedure captures
the dominant effects of nonzero $\beta$.

\subsubsection{Propagating modes}
\label{Propagating modes}

To study free propagation of the gravitational and bumblebee fields
in the absence of charge and matter,
set $J_{\rm M}^\mu = T_{\rm M}^{\mu\nu} = 0$
in the linearized equations of motion.
The gravitational equations \rf{geq} then become
\begin{eqnarray}
&
\mathchoice\sqr66\sqr66\sqr{2.1}3\sqr{1.5}3 \, \ol h_{\mu\nu}
+ \eta_{\mu\nu} \partial^\alpha\partial^\beta \ol h_{\alpha\beta}
- \partial^\alpha\partial_\mu \ol h_{\alpha\nu}
- \partial^\alpha\partial_\nu \ol h_{\alpha\mu}
\nonumber\\
&
\hskip 20pt
\approx
- 64 \pi G \kappa (b^\alpha \hat b_\alpha) b_\mu b_\nu \beta .
\label{grav3}
\end{eqnarray}
Contributions from $T_{\rm EM}^{\mu\nu}$
are second-order in ${\cal E}_\mu$ and can be neglected
in this context.
The bumblebee equations \rf{Beq} reduce to
\begin{equation}
\mathchoice\sqr66\sqr66\sqr{2.1}3\sqr{1.5}3 \, {\cal E}_\mu - \partial_\mu \partial^\nu {\cal E}_\nu
\approx 4 \kappa (b^\alpha \hat b_\alpha)  b_\mu \beta .
\label{epeq2}
\end{equation}
Note that the massive mode $\beta$
is defined by Eq.\ \rf{massep}
in terms of ${\cal E}_\mu$ and $h_{\mu\nu}$,
so it is not an independent field
in these equations.
Note also that
the linearized current-conservation law
\rf{linbeconstraint}
follows by taking a derivative of Eq.\ \rf{epeq2}.

To investigate the behavior of the massive mode $\beta$,
it is convenient to combine the above two equations
to obtain
\begin{equation}
\mathchoice\sqr66\sqr66\sqr{2.1}3\sqr{1.5}3 \, \beta
- 4 \kappa (b^\alpha b_\alpha) (1 + 4 \pi G b^\beta b_\beta ) \beta
\approx
\fr 1 {b^\gamma \hat b_\gamma} b^\lambda \partial_\lambda \partial^\mu
({\cal E}_\mu - \ol h_{\mu\nu} b^\nu).
\label{beeq}
\end{equation}
At first glance,
this equation might suggest
that $\beta$ can be a propagating massive field.
However,
$\beta$ depends on the fields appearing
on the right-hand side,
so the modes are still coupled and
care is required in determining the dispersion properties,
including the mass value.

A suitable choice of diffeomorphism gauge clarifies matters.
The modes remain entangled in the harmonic gauge \rf{harm}
and also in the barred axial gravitational gauge
$\ol h_{\mu\nu} b^\nu = 0$.
However,
a sufficient decoupling can be achieved
by adopting the axial gravitational gauge \rf{haxial}
and by decomposing ${\cal E}_\mu$ into pieces
parallel and perpendicular to $b_\mu$.
In this gauge,
the longitudinal component ${\cal E}$ of ${\cal E}_\mu$
reduces to ${\cal E} = \beta$,
so that
\begin{eqnarray}
{\cal E}_\mu &=& {\cal E}^t_\mu + {\cal E} \hat b_\mu
= A_\mu + \beta \hat b_\mu .
\end{eqnarray}
In this equation,
we write ${\cal E}^t_\mu \equiv A_\mu $
for the transverse components of ${\cal E}_\mu$
to make easier the task of tracking
$\beta$-dependent deviations
from conventional electrodynamics.
These components satisfy the axial condition
$A_\mu b^\mu= 0$.
We emphasize that the axial condition
is \it not \rm a gauge choice.
It is a consequence of projecting along $b_\mu$
and is independent of the gravitational gauge fixing.
It does, however, have the same mathematical form
as an axial-gauge condition in electrodynamics,
even though the bumblebee models have no U(1) symmetry.

With these choices,
the usual Einstein equations for
$\ol h_{\mu\nu}$
in axial gravitational gauge
are recovered from the result \rf{grav3}
when $\beta$ vanishes
\cite{bk}.
For nonzero $\beta$,
only one linearly independent combination
of the Einstein equations is changed.
It can be written
\begin{eqnarray}
\mathchoice\sqr66\sqr66\sqr{2.1}3\sqr{1.5}3 \, \ol h
+ 2 \partial^\mu\partial^\nu \ol h_{\mu\nu}
&\approx&
- 64 \pi G \kappa (b^\alpha \hat b_\alpha) (b^\beta b_\beta) \beta .
\label{grav4}
\end{eqnarray}
Since the usual propagation-transverse components of $\ol h_{\mu\nu}$
remain unaffected,
the physical graviton modes propagate normally.
The bumblebee equations \rf{epeq2} become
\begin{equation}
\mathchoice\sqr66\sqr66\sqr{2.1}3\sqr{1.5}3 \, A_\mu - \partial_\mu \partial^\nu A_\nu
+ \fr 1 {b^\beta  b_\beta} {b_\mu b^\alpha } \partial_\alpha \partial^\nu A_\nu
\approx 0
\end{equation}
and
\begin{equation}
\mathchoice\sqr66\sqr66\sqr{2.1}3\sqr{1.5}3 \, \beta
- 4 \kappa (b^\alpha b_\alpha) \beta
\approx
\fr 1 {b^\beta \hat b_\beta} b^\alpha \partial_\alpha \partial^\mu A_\mu .
\label{axbeq}
\end{equation}
When the massive mode $\beta$ vanishes,
these equations reduce to those of electrodynamics
in axial gauge
\cite{bk}.
A nonzero value of $\beta$ modifies the equations,
but the usual propagation-transverse components of $A_\mu$
and hence the physical photon modes remain unaffected.

Since $\beta$ and $A_\mu$ are independent fields,
the dispersion properties of the massive mode $\beta$
can be identified from Eq.\ \rf{axbeq}.
Any solutions of the theory
describing freely propagating modes
are required to obey the boundary conditions
that the spacetime be asymptotically flat
and that the bumblebee fields vanish at infinity.
At linear order,
these solutions are formed from harmonic plane waves
with energy-momentum vectors $p^\mu = (E,\vec p)$
obeying suitable dispersion relations.
The modes satisfying the asymptotic boundary conditions
are then constructed as Fourier superpositions
of the plane-wave solutions.
For the massive mode $\beta$,
the constraint equation \rf{linbeconstraint}
imposes the additional requirement
\begin{equation}
b^\mu p_\mu \approx 0 .
\label{bk}
\end{equation}
Any freely propagating massive mode
is therefore constrained to have an energy-momentum vector
orthogonal to the vacuum value $b_\mu$.
For harmonic-mode solutions to the equations of motion,
this requirement implies that the term on the right-hand side
of Eq.\ \rf{axbeq} vanishes in Fourier space.
The resulting dispersion law for the massive mode $\beta$
is then
\begin{equation}
p^\mu p_\mu
\approx - 4 \kappa b^\alpha b_\alpha ,
\label{kdisp}
\end{equation}
which involves the squared-mass parameter
\begin{equation}
M_\beta^2 = 4 \kappa b^\alpha b_\alpha .
\label{bemass}
\end{equation}
The sign of $M_\beta^2$
depends on whether $b_\mu$ is timelike or spacelike.
The scale of $M_\beta^2$ depends on both $\kappa$ and $b$.
Inspection of Eqs.\ \rf{grav4} or \rf{axbeq}
shows that the limits $|M_\beta|^2 \to \infty$
or $\kappa\to \infty$ with $b_\mu$ fixed
are equivalent to taking the limit of vanishing
massive mode, $\beta\to 0$.
The discussion in Sec.\ \ref{Bumblebee electrodynamics}
then implies that the limit of large $|M_\beta|^2$
approximates Einstein-Maxwell theory.

For the case of a timelike vacuum value $b_\mu$,
the parameter $M_\beta^2 = -4\kappa b^2$
has the wrong sign for a physical mass.
Adopting the global observer frame in which
$b_\mu = (b,0,0,0)$,
we see that the constraint \rf{bk}
forces the energy $E$ of this mode to vanish,
leaving the condition that the magnitude of the momentum
must remain fixed at
$|\vec p| = |M_\beta|$.
A time-independent mode with fixed spatial wavelength
cannot be Fourier superposed to form a physical wave packet
satisfying the asymptotic boundary conditions.
It follows that no physical propagating massive mode appears
when $b_\mu$ is timelike.

If instead the vacuum value $b_\mu$ is spacelike,
then $M_\beta^2 = 4\kappa b^2$ is positive
and is a candidate for a physical mass.
Choose the global observer coordinate system
such that the spacelike $b_\mu$
takes the form $b_\mu = (0,0,0,b)$.
The constraint \rf{bk} then requires the vanishing
of the momentum along the spatial direction $b_3$,
and the dispersion law \rf{kdisp} becomes
\begin{equation}
E^2 - (p_x^2 + p_y^2) \approx M_\beta^2 .
\label{spacelikedisp}
\end{equation}
This suggests that $\beta$ could propagate
as a harmonic plane wave
along spatial directions transverse to $b_\mu$.
However,
these harmonic plane waves are constant in $z$
and hence fail to satisfy the
asymptotic boundary conditions as $z\to \pm \infty$
unless their amplitude is zero.
Since all harmonics propagate under the same constraint \rf{bk},
no Fourier superposition can be constructed
to obey the boundary conditions.
It follows that
no physical propagating massive mode exists
for the case of a spacelike vector either.

The above results can also be confirmed
by obtaining the eigenvalues
of the Fourier-transformed equations of motion,
which form a 14$\times$14 matrix
determining the nature of the modes.
Consider,
for example,
the case of purely timelike $b_\mu$.
In the axial gravitational gauge,
four of the modes are zero,
corresponding to the four fixed degrees of freedom.
Another four eigenvalues describe modes propagating
via a massless dispersion law,
corresponding to the two photon and two graviton modes.
One eigenvalue has zero energy and sets
$|\vec p| = |M_\beta|$,
corresponding to the massive mode $\beta$.
The remaining 5 modes are auxiliary,
and all have zero energy.
Related results can be obtained
for the case of spacelike $b_\mu$.

\subsubsection{Weak static limit}
\label{Weak static limit}

Although the massive mode $\beta$ is nonpropagating,
it plays the role of an auxiliary field
and can thereby produce various effects.
For example,
its presence can affect the weak static limits
of the gravitational and bumblebee interactions.
In particular,
a nonzero $\beta$ can modify the forms of both
the Newton and Coulomb potentials.

To demonstrate this,
we consider the weak static limit of the field equations
with an external matter sector containing massive point charges.
The matter Lagrange density can be taken as
\begin{eqnarray}
{\cal L}_{\rm M} &=&
\sum_n m_n \int d\tau
\left[ g_{\mu\nu}
\fr {dx_n^\mu}{d\tau} \fr {dx_n^\nu}{d\tau}
\right]^{1/2}
\delta^4 (x - x_n(\tau))
\nonumber
\\
&&
+ \sum_n q_n \int d\tau ~ B_\mu(x)
\fr {dx_n^\mu} {d\tau} \delta^4 (x - x_n(\tau)),
\label{LM}
\end{eqnarray}
where $m_n$ and $q_n$ are the masses and charges
of the point particles,
respectively.
The energy-momentum tensor for the matter sector is
\begin{equation}
T_{\rm M}^{\mu\nu} = \fr 1 e \sum_n m_n
\int d\tau \fr {dx_n^\mu} {d \tau}  \fr {dx_n^\nu} {d \tau}
\delta^4 (x - x_n(\tau)) .
\label{TMm}
\end{equation}
Its combination with $T_B^{\mu\nu}$ given in Eq.\ \rf{TBmunu}
forms the total energy-momentum tensor, which is conserved.
The matter-sector current
$J_{\rm M}^\mu \equiv (\rho_q, \vec J)$ is
\begin{equation}
J_{\rm M}^\mu
= \fr 1 e \sum_n q_n \int d\tau
\fr {d x_n^\mu} {d \tau} \delta^4 (x - x_n(\tau)) .
\label{Jq}
\end{equation}
It is conserved, $D_\mu J_{\rm M}^\mu = 0$.

The standard route to adopting the weak static limit
is to linearize the equations of motion
in the fields
and then discard all time derivatives.
This meets no difficulty under suitable circumstances,
such as the vacuum value $b_\mu$ being purely spacelike,
and we follow this route in the present subsection.
However,
in certain cases additional care is required
because the $b_\mu$-transverse components of the bumblebee field
obey the axial condition $b^\mu {\cal E}^t_\mu = 0$,
which can imply time dependence even for static fields.
For example,
as discussed in Sec.\ \ref{Bumblebee electrodynamics},
the case of purely timelike $b_\mu$ and zero $\beta$
is equivalent to electrodynamics in temporal gauge,
for which the static Coulomb potential
involves a linear time dependence in $\vec A$.
The corresponding weak static limit with nonzero $\beta$
can therefore be expected to exhibit this feature.
We return to this issue in the next subsection.

To proceed,
it is useful to choose a diffeomorphism gauge
that simplifies the Einstein tensor
and allows direct comparison
between the geodesic equation and the Newton force law.
One possible choice is the harmonic gauge \rf{harm},
in which the static gravitational potential $\Phi_g$
is related to the metric by
$\Phi_g = -{\textstyle{1\over 2}} h_{00} = -{\textstyle{1\over 2}} \ol h_{00} + \frac 1 4 \ol h$.
The linearized gravitational equations become
\begin{equation}
-{\textstyle{1\over 2}} \mathchoice\sqr66\sqr66\sqr{2.1}3\sqr{1.5}3 \, \ol h^{\mu\nu}
\approx
8\pi G
[T_{\rm M}^{\mu\nu}
+ T_{\rm EM}^{\mu\nu}
+ 4 \kappa (b^\alpha \hat b_\alpha) b^\mu b^\nu \beta] ,
\label{lsharm}
\end{equation}
where $\beta$ is the massive mode
given in terms of ${\cal E}_\mu$ and $h_{\mu\nu}$
in Eq.\ \rf{massep},
and $T_{\rm EM}^{\mu\nu}$
is the zero-$\beta$ form of $T_{\rm K}^{\mu\nu}$
given in Eq.\ \rf{TBBEM}.

We can now extract the relevant equations
in the weak static limit.
The equation for the gravitational potential $\Phi_g$
is the 00 component of the trace-reversed form
of Eq.\ \rf{lsharm},
\begin{equation}
\nabla^2 \Phi_g
\approx
8\pi G
[\ol T{}_{\rm M}^{00}
+ \ol T{}_{\rm EM}^{00}
+ 2 \kappa (b^\alpha \hat b_\alpha) (b_0^2 + \vec b^{\, 2}) \beta] ,
\label{lsharm2}
\end{equation}
where the trace-reversed energy-momentum tensors
are defined by
$\ol T_{\mu\nu} \equiv T_{\mu\nu} - \eta_{\mu\nu} T^\alpha_{\pt{\alpha}\alpha}$
and are evaluated in the static limit.
The weak static bumblebee equations become
\begin{eqnarray}
\vec \nabla^2 {\cal E}_0
&\approx & \rho_q
+ 4 \kappa (b^\alpha \hat b_\alpha)  b_0 \beta ,
\nonumber\\
\vec \nabla^2 \vec {\cal E}
- \vec\nabla (\vec\nabla\cdot\vec {\cal E}) &\approx & \vec J
+ 4 \kappa (b^\alpha \hat b_\alpha)\vec b \beta ,
\label{stlinkappaeqs}
\end{eqnarray}
where $\rho_q$ and $\vec J$ are static charge and current sources.

The above equations show that
the massive mode $\beta$ can be viewed
in the weak static limit
as a simultaneous source of
energy density, charge density, and current density.
The relative weighting of the contributions
is controlled by the relative sizes of the components of $b_\mu$.
In fact,
the same is also true of contributions
to gravitomagnetic and higher-order gravitational effects
arising from other components of the gravitational equations.
In the interpretation as bumblebee electrodynamics,
the equations show that we can expect
deviations from the usual Newton and Coulomb force laws
when $\beta$ is nonzero.
Since the effects are controlled by components of $b_\mu$,
they are perceived differently from different observer frames,
much like charge and current in ordinary electrodynamics.
Also,
the unconventional source terms are linear in $\beta$,
so the solutions can be interpreted as superpositions
of conventional gravitational and electrodynamic fields
with effects due to $\beta$.
However,
since $\beta$ itself is formed as the combination \rf{massep}
of the metric and bumblebee fields,
the extra source terms in fact reflect
contributions from the gravitational and electrodynamic fields
that are absent in the Einstein-Maxwell limit.

\subsubsection{Timelike case}
\label{Timelike case}

In this subsection,
we consider in more detail the weak static limit
with timelike $b_\mu$.
We solve explicitly the weak static equations
in the special observer frame
for which $b_\mu = (b,0,0,0)$ is purely timelike,
and we provide some remarks about the more general case.
The special frame might be identified
with the natural frame of the cosmic microwave background.
However,
in laboratory searches for modifications
to weak static gravitation and electrodynamics,
this frame can at best be an approximation
because the Earth rotates and revolves about the Sun,
which in turn is moving
with respect to the cosmic microwave background.
Nonetheless,
the explicit solutions obtained here
offer useful insight into the modifications
to the Newton and Coulomb potentials
that are introduced by the massive mode.

For the purely timelike case,
both $h_{\mu\nu}$ and $B_{\mu\nu}$
are expected to be time independent in the weak static limit.
However,
time dependence must appear in $\vec {\cal E}$
even in the static limit
because as $\beta \to 0$
the match to electrodynamics
arises in the temporal gauge,
for which the standard Coulomb solution
has a linear time dependence.
For this reason,
in taking the weak static limit in what follows,
we keep time derivatives
acting on the vector components $\vec {\cal E}$.

The analysis for purely timelike $b_\mu$
could be performed in the harmonic gauge,
but it turns out that
choosing the transverse gauge instead
results in the complete decoupling from the massive mode
of nine of the 10 gravitational equations.
The transverse gauge is defined by
\begin{equation}
\partial^j h_{0j} = 0,
\qquad
\partial^j h_{jk} = \frac 1 3 \partial_k h^j_{\pt{j}j},
\label{htransverse}
\end{equation}
where $j$, $k$ range over the spatial directions.
In this gauge,
$G_{00} \approx - \nabla^2 h_{00} = 2 \nabla^ 2 \Phi_g$,
where $\Phi_g$ is the gravitational potential.

Taking the weak static limit of the gravitational and bumblebee
equations of motion in this gauge yields
\begin{eqnarray}
\nabla^2 \Phi_g
&\approx&
4\pi G
(T_{\rm M}^{00} + T_{\rm EM}^{00} - 4 \kappa b^3 \beta) ,
\nonumber\\
\vec \nabla^2 {\cal E}_0 - \vec \nabla \cdot \partial_0 \vec {\cal E}
&\approx & \rho_q
- 4 \kappa b^2 \beta ,
\nonumber\\
\mathchoice\sqr66\sqr66\sqr{2.1}3\sqr{1.5}3 \, \vec {\cal E}
- \vec\nabla (\vec\nabla\cdot\vec {\cal E}) &\approx & \vec J .
\label{trlinkappaeqs}
\end{eqnarray}
The other nine components of $h_{\mu\nu}$
obey equations that are identical to those
of general relativity in the weak static limit.
In Eq.\ \rf{trlinkappaeqs},
$T_{\rm M}^{00}$ is the static energy density
in the matter sector,
and $T_{\rm EM}^{00}$ is the static $00$ component
of the zero-$\beta$ limit of $T_{\rm K}^{\mu\nu}$
given in Eq.\ \rf{TBBEM}.
The charge density $\rho_q$ and current density $\vec J$
are also understood to be time-independent distributions.

The explicit form of the massive mode $\beta$
follows from Eq.\ \rf{massep}
and is found to be
\begin{equation}
\beta = {\cal E}_0 - b \Phi_g .
\label{betimelike}
\end{equation}
For purely timelike $b_\mu$,
the constraint \rf{linbeconstraint} reduces to
\begin{equation}
\partial_0 \beta \approx 0,
\label{timeindepbe}
\end{equation}
so $\beta$ is time independent at this order.
As described in the previous subsection,
$\beta$ can be interpreted as a source
for $\Phi_g$ and ${\cal E}_0$.
Here,
$\beta$ acts as a static source
of energy density and charge density with
\begin{eqnarray}
T_B^{\mu\nu} &=& b \rho_B \eta^{0\mu} \eta^{0\nu} ,
\nonumber\\
J_B^\mu &=& (\rho_B,0,0,0),
\label{betastsource}
\end{eqnarray}
where $\rho_B = - 4 \kappa b^2 \beta$.

To find the explicit modifications
to the gravitational and electromagnetic potentials
caused by $\beta$,
we focus on the case of a point particle
of mass $m$ and charge $q$
located at the origin.
The source terms in this case
have the form
\begin{eqnarray}
T_{\rm M}^{00} &=& \rho_m = m \delta^{(3)}(\vec x) ,
\nonumber\\
J_{\rm M}^0 &\equiv& \rho_q = q \delta^{(3)}(\vec x) ,
\label{pointsource}
\end{eqnarray}
and all other components of
$T_{\rm M}^{\mu\nu}$ and $J_{\rm M}^\mu$ are zero.
In the absence of Lorentz violation,
the weak static solution to the Einstein-Maxwell equations
with these source terms
consists of the linearized Reissner-Nordstr\"om metric
and corresponding electromagnetic fields.
Denote by $\Phi_m$ the associated gravitational potential.
It obeys
\begin{equation}
\vec \nabla^2 \Phi_m =
4 \pi G (\rho_m + T_{\rm EM}^{00} ).
\label{RN}
\end{equation}
In the limit $q \rightarrow 0$,
it reduces to the Newton gravitational potential and
obeys the usual Poisson equation
$\nabla^2 \Phi_m = 4 \pi G \rho_m$.
Similarly,
denote the conventional Coulomb potential by $\Phi_q$.
It obeys the Poisson equation
\begin{equation}
\nabla^2 \Phi_q = - \rho_q.
\label{fish}
\end{equation}

It is also convenient to express
the massive mode $\beta$ in terms of a potential $\Phi_B$.
We define
\begin{equation}
\beta \equiv
\fr 1 {| M_\beta |^2}
\vec \nabla^2 \Phi_B (\vec x) ,
\label{Phbe}
\end{equation}
where $| M_\beta |^2$ is the absolute value
of the squared mass in Eq.\ \rf{bemass}.
Note that $\Phi_B$
is time-independent due to the constraint
\rf{timeindepbe}.
Since $\beta$ acts as a source of static charge density
given by Eq.\ \rf{betastsource},
the definition of $\Phi_B$ can be interpreted
as a third Poisson equation,
\begin{equation}
\vec\nabla^2 \Phi_B = - \rho_B .
\end{equation}
The time independence of both $\beta$ and $\Phi_B$
means they are determined by their initial values.

For the point particle of mass $m$ and charge $q$,
the general solution for
the gravitational potential $\Phi_g$
and the bumblebee field ${\cal E}_\mu$
in the weak static limit
can be expressed in terms of the conventional
potentials $\Phi_m$ and $\Phi_q$
and the bumblebee potential $\Phi_B$.
We obtain
\begin{eqnarray}
\Phi_g &=&
\Phi_m - 4\pi G b \Phi_B ,
\nonumber\\
{\cal E}_0 &=&
b \Phi_m - 4\pi G b^2 \Phi_B
+ \fr 1 {| M_\beta |^2} \vec \nabla^2 \Phi_B ,
\nonumber \\
{\cal E}_j &=& t \partial_j [
\Phi_q + b \Phi_m
+(1 - 4 \pi G b^2 ) \Phi_B
+ \fr 1 {| M_\beta |^2} \vec \nabla^2 \Phi_B
].
\nonumber \\
\label{Phisol}
\end{eqnarray}
The corresponding
static gravitational field $\vec g$,
static electric field $\vec E$,
and static magnetic field $\vec B$
are found to be
\begin{eqnarray}
\vec g &=& - \vec \nabla \Phi_m + 4\pi G b \vec \nabla \Phi_B ,
\nonumber\\
\vec E &=& - \vec \nabla \Phi_q - \vec \nabla \Phi_B ,
\nonumber\\
\vec B &=& 0 .
\label{EB}
\end{eqnarray}
The static Maxwell equations include a modified Gauss law
\begin{eqnarray}
\vec \nabla \cdot \vec E &=&
- \vec \nabla^2 \Phi_q - \vec \nabla^2 \Phi_B
\nonumber \\
&=& \rho_q + \rho_B ,
\label{gauss}
\end{eqnarray}
together with the usual static laws
\begin{eqnarray}
\vec \nabla \cdot \vec B &=& 0,
\nonumber\\
\vec \nabla \times \vec E &=& 0,
\nonumber\\
\vec \nabla \times \vec B &=& 0.
\end{eqnarray}
These equations demonstrate for the purely timelike case
the ways in which a nonzero massive mode $\beta$ modifies
the conventional static gravitational and electrodynamic fields.

Compared to electrodynamics,
the purely timelike KS\ bumblebee model
introduces an additional degree of freedom
into the problem of determining
the static electric field $\vec E$.
This extra degree of freedom is the massive mode $\beta$,
and the extent of its effects depends
on the initial conditions for $\beta$.
In the absence of
a satisfactory underlying theory predicting $\beta$
or of direct experimental observation of $\beta$,
these initial conditions are undetermined.

One natural choice is to adopt $\Phi_B = 0$
as an initial condition,
which implies $\Phi_B (\vec x) \approx 0$ for all time
and hence corresponds to zero massive mode $\beta$.
The solution \rf{Phisol}
then reduces to the weak static limit of
Einstein-Maxwell theory,
as expected.
The gravitational potential becomes $\Phi_g = \Phi_m$.
The bumblebee field reduces to
${\cal E}_\mu = (b \Phi_m, t \partial_j (\Phi_q + b \Phi_m))$,
which despite the appearance of the gravitational potential
implies $\vec E = - \vec \nabla \Phi_q$
and the usual Gauss and Poisson equations
$\vec \nabla \cdot E = - \vec \nabla^2 \Phi_q = \rho_q$.
We note in passing that conventional electrodynamics
is also recovered in the limit $|M_\beta|^2 \rightarrow \infty$
because taking this limit in Eq.\ \rf{trlinkappaeqs}
implies $\beta \to 0$ and hence $\Phi_B \to 0$.
Thus,
even with a nonzero massive mode $\beta$,
a theory with a large mass parameter $|M_\beta|$
approximates Einstein-Maxwell theory in the low-energy regime,
in agreement with the discussion following Eq.\ \rf{bemass}.

Other choices of initial conditions on $\Phi_B$
might appear in the context of a more fundamental theory
for which a bumblebee model
provides an effective partial low-energy theory.
One set of examples with nonzero $\Phi_B$
consists of solutions with zero matter current,
$J_{\rm M}^\mu\equiv 0$.
Imposing $J_{\rm M}^\mu\equiv 0$ by hand
is a common approach in the literature.
It implies the bumblebee field has no couplings to matter
and hence is unrelated to electrodynamics,
being a field that interacts only gravitationally.
The massless Lorentz NG modes then propagate freely
as sterile fields that carry energy and momentum
but convey no direct forces on charged particles.
We emphasize here that a theory of this type
can nonetheless lead to modifications of the
gravitational interaction,
contrary to some claims in the literature.
This is readily illustrated
in the context of the purely timelike example
considered in this subsection.
With $J_{\rm M}^\mu\equiv 0$,
the linearized solutions for a static point particle of mass $m$
are still given by Eqs.\ \rf{Phisol},
but $\Phi_q = 0$ and the fields $\vec E$ and $\vec B$
are largely irrelevant.
However,
the gravitational potential $\Phi_g$ is modified
by the potential $\Phi_B$ for the massive mode $\beta$,
with the value of $\Phi_B$ fixed by initial conditions.

In the absence of additional
theoretical or experimental information,
the possibilities for model building are vast.
One could,
for example,
consider initial conditions with $\Phi_B$ proportional to $\Phi_m$,
\begin{equation}
\Phi_B = \fr \alpha {4 \pi G b} \Phi_m ,
\label{scalePh}
\end{equation}
with $\alpha$ a constant.
For this class of solutions,
$\Phi_g$ has the usual form $\Phi_g = -G^* m/r$
for a point mass
but with a scaled value of the Newton gravitational constant
\begin{equation}
G^* = ( 1 - \alpha ) G_N ,
\label{Gstar}
\end{equation}
where $G_N$ is the value of $G^*$ for $\Phi_B = 0$.
This rescaling has no observable effects
in the special observer frame
because laboratory measurements
would determine $G^*$ instead of $G_N$,
although sidereal, annual, and solar motions
might introduce detectable effects in realistic experiments.
One could equally well consider instead examples
in which $\Phi_B$ has
nonlinear functional dependence on $\Phi_m$
or has other coordinate dependence.
Various forms could be proposed as candidate models
to explain phenomena such as dark matter
or possibly dark energy.
With the choice of $\Phi_B$ being conjecture
at the level of the linearized equations,
this approach is purely phenomenological.
However,
if a bumblebee model appears
as part of the effective theory
in a complete theory of quantum gravity,
the form of $\Phi_B$ could well be fixed
and predictions for dark matter and perhaps dark energy
could become possible.

We conclude this subsection with some comments
about the case where $b_\mu$ takes a more general timelike
form in the given observer frame $O$.
To investigate this,
one can either study directly the weak static limit
of the gravitational and bumblebee equations
with a charged massive point particle at rest at the origin
in $O$ as before,
or one can boost the observer frame $O$
to another frame $O^\prime$
in which $b_\mu$ is purely timelike
but the point particle is moving.
The two pictures are related by observer Lorentz transformations
and are therefore physically equivalent.

In the frame $O$ in which the particle is at rest,
the matter energy-momentum tensor and current
are given by Eq.\ \rf{pointsource} as before,
but the bumblebee energy-momentum tensor $T_B^{\mu\nu}$
and current $J_B^\mu$
have additional nonzero components
due to the nonzero spacelike components of $b_\mu$.
The complete solution therefore involves components of $h_{\mu\nu}$
other than the gravitational potential $\Phi_g$.
However,
we can gain insight into the behavior of the massive mode
by considering the constraint \rf{linbeconstraint}
and performing an observer Lorentz boost
with velocity $\vec v_B = - \vec b/b_0$
to a frame $O^\prime$ in which $b_\mu$ becomes purely timelike.
Since this transformation must reduce
the constraint \rf{linbeconstraint}
to the purely timelike form \rf{timeindepbe},
it follows that in the original frame $O$
the constraint takes the form
\begin{equation}
\partial_0 \beta + \vec v_B \cdot \vec \nabla \beta = 0,
\end{equation}
which implies
\begin{equation}
\beta = \beta (\vec x - \vec v_B t).
\end{equation}
We see that the massive mode $\beta$ moves in the frame $O$
with a velocity $\vec v_B$ equal to the relative boost velocity
linking $b_\mu$ to its purely timelike form.
The gravitational and electrodynamic equations
are therefore modified by time-dependent source terms
moving with velocity $v_B$ relative to the point particle.
An equivalent result is obtained
by working directly in the frame $O^\prime$,
in which $b_\mu$ is purely timelike
and $\beta$ is therefore static,
but in which all components of
the matter energy-momentum tensor and current are nonzero
and represent a particle moving with velocity $-\vec v_B$
relative to $\beta$.
As before,
the form of the massive mode $\beta$
is set by initial conditions.
Since the equations are linear in either frame,
the solutions consist of a superposition of
conventional potentials and the massive-mode contributions
with relative motion.

\subsection{Linear Lagrange-multiplier potential}
\label{Linear Lagrange-multiplier potential}

In this subsection,
we discuss the KS\ bumblebee model \rf{ksbb}
with the linear Lagrange-multiplier potential \rf{VL}
and the definition \rf{XB}:
\begin{equation}
V = V_L(\lambda, X) = \lambda (B_\mu g^{\mu\nu} B_{\nu} \pm b^2).
\label{linlagv}
\end{equation}
Bumblebee models with this potential
have been widely studied at the linearized level
for about two decades
\cite{ks}.
Here,
we compare and contrast this theory
to the case with the smooth quadratic potential $V_S$.
We show the Lagrange multiplier $\lambda$
produces effects in the $V_L$ model
that are very similar to those
of the massive mode $\beta$ in the $V_S$ model.
Paralleling the treatment of the $V_S$ model
in Sec.\ \ref{Smooth quadratic potential},
the mode expansion
$B_\mu = b_\mu + {\cal E}_\mu$ of Eq.\ \rf{epdown}
in a Minkowski background is adopted,
and the fields $h_{\mu\nu}$ and ${\cal E}_\mu$ are assumed weak
so that linearization can be performed.

Variation of the action produces
the gravitational and bumblebee equations of motion
\rf{geq} and \rf{Beq},
together with the Lagrange-multiplier constraint
\begin{equation}
X = B_\mu g^{\mu\nu} B_\nu \pm b^2 = 0  .
\label{lmconst}
\end{equation}
This condition enforces the vanishing of the massive mode,
as discussed in Sec.\ \ref{Excitations},
which leaves only the Lorentz NG modes
as possible excitations of the bumblebee field $B_\mu$.
However,
there is also an additional degree of freedom,
the Lagrange multiplier $\lambda$ itself,
that appears in the equations of motion.

One might naively expect the $V_L$ model to yield solutions
identical to those obtained
in the infinite-mass limit $|M_\beta| \rightarrow \infty$
of the $V_S$ model.
For example,
one might reason that an infinite mass
would make energetically impossible
any field excitations away from the minimum of $V_S$,
leading to the constraint \rf{lmconst}.
However,
the potential $V_L$ is a function of two combinations of fields,
$V_L = V_L(\lambda, X)$,
whereas $V_S(X)$ involves only one.
The infinite-mass limit indeed suppresses $X$ excitations
away from $X=0$ in $V_S$,
but it contains no match for $\lambda$.
There is thus an extra field degree of freedom in $V_L$
relative to the infinite-mass limit of $V_S$.
For example,
$V_S^\prime \rightarrow 0$ in the infinite-mass limit
because no excitations of $X$ are allowed,
whereas $V_L^\prime = \lambda$,
which need not vanish.
We therefore conclude that
the correspondence between
the infinite-mass limit of the $V_S$ model
and the $V_L$ model can occur only when $\lambda = 0$.

To gain intuition about the effects
associated with the Lagrange-multiplier field $\lambda$,
consider its role in the equations of motion.
Since the constraint \rf{lmconst} ensures
no massive excitations can appear,
the bumblebee energy-momentum tensor \rf{TBmunu} reduces to
\begin{equation}
T^{\mu\nu}_B =
T^{\mu\nu}_{\rm EM}
+ T_V^{\mu\nu} .
\label{lmTBmunu}
\end{equation}
We find
\begin{eqnarray}
T_V^{\mu\nu} &=& 2 \lambda B^\mu B^\nu \approx 2 b^\mu b^\nu \lambda,
\end{eqnarray}
where the last form is the leading-order contribution
in the linearized limit.
Similarly,
the bumblebee current \rf{JB}
becomes
\begin{eqnarray}
J_B^\mu &=& - 2 \lambda B^\mu \approx -2 b^\mu \lambda.
\label{TJlm}
\end{eqnarray}
Conservation of the matter current $J_M^\mu$
implies the constraint
\begin{equation}
b^\mu \partial_\mu \lambda \approx 0.
\label{lmconstraint}
\end{equation}
When $\lambda \to 0$,
all these equations reduce
to those of Einstein-Maxwell theory,
in agreement with the discussion of bumblebee electrodynamics
in Sec.\ \ref{Bumblebee electrodynamics}.
Moreover,
by comparison with
Eqs.\ \rf{linTV}--\rf{linbeconstraint},
we see that the Lagrange multiplier $\lambda$ in the $V_L$ model
plays a role very comparable
to that of the massive mode $\beta$ in the $V_S$ model.
In effect,
$\lambda$ acts as an additional source of energy-momentum density
and current density in the equations of motion.

The propagating modes for the KS\ bumblebee model
with the linear Lagrange-multiplier potential $V_L$
have been investigated elsewhere
\cite{ks,bk}.
The usual two graviton modes
propagate as free transverse massless modes
independently of $\lambda$,
as do the usual two photon modes
emerging from the Lorentz NG modes.
Since $\lambda$ has no kinetic terms,
it is auxiliary and cannot propagate.

The weak static limit of the $V_L$ model
can also be studied,
following the lines of the discussion for the $V_S$ model
in Sec.\ \ref{Timelike case}.
Consider,
for example,
the case of purely timelike $b_\mu = (b,0,0,0)$.
The constraint \rf{lmconstraint} then implies
that $\lambda$ must be time independent at leading order,
\begin{equation}
\partial_0 \lambda \approx 0.
\label{lmfixing}
\end{equation}
A nonzero $\lambda$ therefore acts
as an additional static source of energy density
and charge density,
which can modify the static potentials.
We again adopt the transverse gauge \rf{htransverse},
for which the gravitational potential is
$\Phi_g = -h_{00}/2$.
Linearizing the constraint \rf{lmconst}
for the purely timelike case yields
\begin{equation}
{\cal E}_0 - b \Phi_g = 0 .
\label{lambdaconstraint}
\end{equation}
As expected,
this condition corresponds in the context of the $V_S$ model
to enforcing the vanishing of the massive mode $\beta$.
In the context of Einstein-Maxwell theory,
it corresponds as before to a gauge-fixing condition
that reduces to the usual temporal gauge
in the absence of gravity.

For a single point particle of charge $q$ and mass $m$
at rest at the origin,
the energy density $\rho_m$ and charge density $\rho_q$
are given by Eq.\ \rf{pointsource}.
The equations of motion in the weak static limit become
\begin{eqnarray}
\nabla^2 \Phi_g
&\approx&
4\pi G
(\rho_m + T_{\rm EM}^{00} + 2 b^2 \lambda ) ,
\nonumber\\
\vec \nabla^2 {\cal E}_0 - \vec \nabla \cdot \partial_0 \vec {\cal E}
&\approx & \rho_q
+ 2 b \lambda ,
\nonumber\\
\mathchoice\sqr66\sqr66\sqr{2.1}3\sqr{1.5}3 \, \vec {\cal E}
- \vec\nabla (\vec\nabla\cdot\vec {\cal E}) &\approx & 0 .
\label{linlambdaeqs}
\end{eqnarray}
Comparison of these expressions
with Eqs.\ \rf{trlinkappaeqs} for the $V_S$ model
again reveals the correspondence between the roles
of the Lagrange-multiplier field $\lambda$ and the massive mode $\beta$.

With $\lambda$ acting as an effective source of charge and energy density,
we can introduce a potential $\Phi_B$,
\begin{equation}
\lambda \equiv - \fr 1 {2 b} \nabla^2 \Phi_B  ,
\label{lambdaPhi}
\end{equation}
which equivalently can be viewed as the solution
to the Poisson equation
\begin{equation}
\nabla^2 \Phi_B = - \rho_B ,
\qquad
\rho_B \equiv 2b\lambda.
\label{lafish}
\end{equation}
As before,
let $\Phi_m$ denote the usual gravitational potential
obeying Eq.\ \rf{RN},
and let $\Phi_q$ denote the usual Coulomb potential
obeying Eq.\ \rf{fish}.
In terms of these three potentials,
the weak-field static solutions
to the equations of motion \rf{linlambdaeqs}
can be written as
\begin{eqnarray}
\Phi_g &=&
\Phi_m - 4\pi G b \Phi_B ,
\nonumber\\
{\cal E}_0 &=&
b \Phi_m - 4\pi G b^2 \Phi_B ,
\nonumber \\
{\cal E}_j &=& t \partial_j [
\Phi_q + b \Phi_m
+(1 - 4 \pi G b^2 ) \Phi_B
].
\label{Philambdasol}
\end{eqnarray}
The associated
static gravitational field $\vec g$,
static electric field $\vec E$,
and static magnetic field $\vec B$
are all given by the same mathematical expressions
as Eqs.\ \rf{EB},
but the potential $\Phi_B$ is now defined in terms of $\lambda$
according to Eq.\ \rf{lambdaPhi}.
Similarly,
a nonzero Lagrange-multipler field $\lambda$ yields
the same modified mathematical form \rf{gauss} of the Gauss law,
but the potential $\Phi_B$ and charge $\rho_B$
are now defined in terms of $\lambda$.

The bumblebee potential $\Phi_B$
in Eqs.\ \rf{Philambdasol}
is undetermined by the equations of motion \rf{linlambdaeqs}
and must be specified as an initial value.
This specification also fixes
the initial value of the Lagrange multiplier $\lambda$,
and the condition \rf{lmfixing}
then ensures that $\lambda$ remains unchanged for all time.
The situation here parallels that of the $V_S$ model,
where initial conditions must be imposed that
subsequently fix the bumblebee potential
and the massive mode $\beta$ for all time.
These results are special cases of a more general fact
often overlooked in the literature:
to be well defined,
all bumblebee models require explicit initial conditions
on the massive modes and Lagrange-multiplier fields.
The subsequent development of the massive modes
and Lagrange multipliers can then be deduced
from the equations of motion or from derived constraints.
In the absence of specified initial conditions,
physical interpretations and predictions
cannot be reliable.
Moreover,
in the absence of direct experimental observation
or prediction from an underlying theory,
the choice of initial conditions is largely unrestricted
and can lead to widely differing effects.

One natural initial condition is $\Phi_B = 0$, $\lambda = 0$.
The above results then reduce to Einstein-Maxwell theory
in the weak static limit,
as expected.
This $V_L$ model has only field excitations maintaining
both $V_L=0$ and $V_L^\prime = 0$.
It corresponds to the infinite-mass limit of $V_S$,
which is itself equivalent to Einstein-Maxwell theory.

Another possibility is to consider models
without direct coupling to matter,
$J_{\rm M}^\mu\equiv 0$,
which implies $\rho_q \equiv 0$, $\Phi_q \equiv 0$
in the above equations.
Like their $V_S$ counterparts
discussed in Sec.\ \ref{Timelike case},
these models are purely gravitational
and are unrelated to electrodynamics.
As always,
initial conditions on $\Phi_B$ must be specified.
One simple possibility,
for example,
is to choose $\Phi_B = - b \Phi_g$.
This $V_L$ model is contained
in the analysis of Ref.\ \cite{cli}.
The spatial components of the bumblebee vanish, ${\cal E}_j = 0$,
so the bumblebee field $B_\mu$ is parallel
to a timelike Killing vector.
The modifications to $\Phi_g$ in this example
involve a rescaling of the Newton gravitational constant.

Examples with both $\Phi_B$ and $\Phi_q$ nonzero
can also be considered.
These incorporate direct charge couplings to matter,
but the static weak-field limit yields
modified gravitational and Coulomb potentials
given by Eq.\ \rf{Philambdasol}.
One simple example is the choice
$\Phi_B = - \Phi_q$.
Since $\Phi_g \ne \Phi_m$,
this $V_L$ model has a modified gravitational potential.
The solution for the bumblebee field
has the form of a total derivative,
${\cal E}_\mu = \partial_\mu (t b \Phi_g)$.
In the limit $q \to 0$ and $\lambda \to 0$,
it provides an example of a solution
that with $\lambda = 0$ and hence $\Phi_B = 0$
has been identified in Ref.\ \cite{bb1}
as potentially flawed
due to the formation of shock discontinuities in ${\cal E}_\mu$.
Whether or not $\lambda$ is zero,
this solution is unusual.
Both the field strength $F_{\mu\nu}$
and the energy-momentum tensor $T_{\rm EM}^{\mu\nu}$ vanish
because the effective charge density $\rho_B$
associated with the Lagrange-multiplier field
cancels the matter charge density $\rho_q$.
Although the field strengths are zero,
the bumblebee field ${\cal E}_\mu$ must be nonzero
because the constraint \rf{lambdaconstraint}
implies ${\cal E}_0 = b \Phi_g$,
which cannot vanish in the presence of gravity.
For a point charge the solution
${\cal E}_\mu$ does indeed contain a singularity,
but this is physically unremarkable
as it merely reflects the usual $1/r$ dependence in $\Phi_g$.
The same behavior arises in the standard solutions
of Einstein-Maxwell theory in a gauge fixed
by Eq.\ \rf{lambdaconstraint},
to which the $V_L$ model is equivalent.

\subsection{Quadratic Lagrange-multiplier potential}
\label{Quadratic Lagrange-multiplier potential}

As a final example,
we consider in this subsection
the specific KS\ bumblebee model \rf{ksbb}
with the quadratic Lagrange-multiplier potential \rf{VQ}
\begin{equation}
V = V_Q(\lambda, X) = {\textstyle{1\over 2}} \lambda (B_\mu g^{\mu\nu} B_{\nu} \pm b^2)^2,
\label{quadlagv}
\end{equation}
where $X$ is defined in Eq.\ \rf{XB}.
As before,
we adopt the mode expansion
$B_\mu = b_\mu + {\cal E}_\mu$ of Eq.\ \rf{epdown}
in a Minkowski background,
and where useful we assume weak fields $h_{\mu\nu}$ and ${\cal E}_\mu$.

Bumblebee models with a quadratic Lagrange-multiplier potential
have not previously been considered in detail in the literature.
We introduce them here partly as a foil for the $V_L$ case,
in which the Lagrange multiplier plays a key role
in the physics of the model.
In contrast,
the Lagrange multiplier for the potential $V_Q$
decouples entirely from the classical dynamics.
The point is that
variation with respect to $\lambda$
yields the constraint $X^2=0$,
which is equivalent to $X=0$
and forces the massive mode to vanish.
However,
the quadratic nature of $V_Q$
means that
$\lambda$ always appears multiplied by $X$
in the gravitational and bumblebee field equations,
so the on-shell condition $X=0$ forces
the field $\lambda$ to decouple.
Also,
the potential obeys both $V_Q =0$ and $V_Q^\prime = 0$ on shell,
so the bumblebee energy-momentum tensor and bumblebee current
reduce to the standard expressions for electrodynamics
in curved spacetime.
The equations of motion are therefore
equivalent to those of Einstein-Maxwell electrodynamics
in the gauge $X=0$.
This correspondence holds both in the linearized limit
and in the full nonlinear theory.

Since the equations of motion generated
by a theory with a quadratic Lagrange-multiplier potential
are the equations in the absence of the constraint
plus the constraint itself,
introducing this type of potential
is equivalent at the classical level
to imposing the constraint by hand on the equations of motion.
The only distinction between the two approaches is the presence
of the decoupled Lagrange-multiplier field.
Bumblebee models with the potential $V_Q$
therefore incorporate a variety of models
in which constraints are imposed by hand.
For example,
Will and Nordtvedt have considered solutions
involving a nonzero background value for a vector
in bumblebee-type models with Lagrange density of the form \rf{bb}
but without a potential term
\cite{wn}.
This approach produces equations of motion
identical to those of the corresponding $V_Q$ model.

The $V_Q$ models are also related
to theories in which the constraint is substituted
into the action before varying,
although the correspondence is inexact.
For example,
Nambu has investigated the Maxwell action in Minkowski spacetime
using the constraint $X=0$ for the purely timelike case
as a nonlinear gauge condition
substituted into the action prior to the variation
\cite{yn}.
This represents gauge fixing at the level of the action,
and it yields a total of four equations for the fields:
the original constraint and
three equations of motion from the variation.
However,
the corresponding $V_Q$ model
yields five equations of motion instead,
one of which is the constraint.
The extra equation is the Gauss law,
which in the Nambu approach
is imposed as a separate initial condition
that subsequently holds at all times
by virtue of the three equations of motion and the constraint.

For the specific KS\ bumblebee model \rf{ksbb}
with potential $V_Q$ in Eq.\ \rf{quadlagv},
the freely propagating modes can be found
by linearizing the gravitational and bumblebee equations of motion
\rf{geq} and \rf{Beq}
and the constraint $X=0$.
The linearization generates the same equations
as emerge for the $V_L$ model with potential \rf{linlagv}
in the limit $\lambda \to 0$.
The propagating modes
consist of the usual graviton and photon in an axial gauge
\cite{bk},
as expected.
Similarly,
the weak static limit of the $V_Q$ model
with potential \rf{quadlagv}
produces equations identical at linear level
to those of the $V_L$ model
with potential \rf{linlagv}
in the limit $\lambda \to 0$.
The usual gravitational and electromagnetic
Poisson equations therefore emerge,
and the correct Newton and Coulomb potentials hold
at the linearized level.

The exact correspondence between the static limit
of the $V_Q$ bumblebee model and
the Newton and Coulomb potentials
involves the nonlinear constraint \rf{lmconst}.
The explicit forms of the solutions
are therefore also nonlinear.
For example,
consider again the case of a point particle
of mass $m$ and charge $q$
at rest at the origin
in the presence of a purely timelike vacuum value
$b_\mu = (b,0,0,0)$.
The equation of motion for $\lambda$ is the constraint \rf{lmconst},
which represents a nonlinear condition
relating ${\cal E}_\mu$ and $h_{\mu\nu}$.
Expanding this constraint through quadratic order gives
\begin{equation}
{\cal E}_0^2 +2 b {\cal E}_0 - 2 b^2 \Phi_g - 4 b \Phi_m {\cal E}_0 - {\cal E}_j^2
\approx 0 ,
\label{minquadV}
\end{equation}
where $\Phi_m$ obeys the conventional Poisson equation \rf{RN}
for the point source with mass and charge density
given by Eq.\ \rf{pointsource}.
In the transverse gauge,
the gravitational potential $\Phi_g$ obeys
\begin{equation}
\vec \nabla^2 \Phi_g \approx 4\pi G (\rho_m + T_{\rm EM}^{00}) ,
\label{EMeq}
\end{equation}
where the energy-momentum tensor $T_{\rm EM}^{00}$
has the usual Einstein-Maxwell form \rf{TBBEM}.
The solution for $\Phi_g$ is therefore
the standard gravitational potential
for a static point charge
in a curved spacetime in the weak-field limit.
At quadratic order,
the solution ${\cal E}_\mu = ({\cal E}_0, {\cal E}_j)$
satisfying the constraint \rf{minquadV}
and the field equations is found to be
\begin{eqnarray}
{\cal E}_0 &\approx & b \Phi_g + \fr {3b} 2 \Phi_g^2 + \fr {t^2} {2b}
[\partial_j (\Phi_q + b \Phi_g)]^2 ,
\nonumber \\
{\cal E}_j &\approx & t \partial_j (\Phi_q + b \Phi_g)
+ 3bt \Phi_g \partial_j \Phi_g
\nonumber \\
&&
+ \fr {t^3} {3b} [\partial_k (\Phi_q + b \Phi_g)] \partial_j
\partial_k (\Phi_q + b \Phi_g) ,
\label{quadep}
\end{eqnarray}
where $\Phi_q$ is the conventional Coulomb potential
obeying the Poisson equation \rf{fish}.
These expressions are both time dependent and nonlinear,
but they nonetheless generate
the usual static electric field $\vec E$
and magnetic field $\vec B$ for a point charge.
Explicit calculation to quadratic order yields
\begin{equation}
\vec E \approx - \vec \nabla \Phi_q ,
\qquad
\vec B \approx 0 ,
\label{EBfields2}
\end{equation}
which implies the usual form
$\vec \nabla \cdot \vec E \approx \rho_q$
of the Gauss law.
From these equations,
we see again that Einstein-Maxwell theory is recovered
despite the absence of U(1) symmetry.
In effect,
the nonlinear condition \rf{minquadV}
plays the role of a nonlinear gauge-fixing condition
in a U(1)-invariant theory,
removing a degree of freedom and leaving only NG modes
that propagate as photons
and generate the usual Coulomb potential in the weak-field limit.

\section{Summary}
\label{Summary}

In this paper,
we have investigated the properties of the massive modes
that can emerge from
spontaneous local Lorentz and diffeomorphism breaking.
In Riemann spacetime,
no massive modes of the conventional Higgs type
can appear because covariant kinetic terms
involve connections with derivatives.
However,
an alternative form of the Higgs mechanism can occur instead,
in which massive modes originate from quadratic couplings
in the potential $V$ inducing the symmetry breaking.

Section \ref{Massive Modes}
provides an analysis of this alternative Higgs mechanism
in the general context of an arbitrary tensor field.
Both smooth potentials and Lagrange-multiplier potentials
are considered.
Massive modes appear for a smooth potential
when excitations with $V^\prime \ne 0$ exist,
and they are formed as combinations of field and metric fluctuations.
For Lagrange-multiplier potentials
the massive modes are constrained to vanish,
but the Lagrange multiplier fields can play a related role.
The propagation of the massive modes
depends on the nature of the kinetic terms in the theory,
and the requirements of unitarity and ghost-free propagation
constrain possible models.
Even if the massive modes are nonpropagating
they can influence gravitational phenomena through,
for example,
effects on the static gravitational potential.
These modifications are of potential interest
in alternative theories of gravity
and descriptions of phenomena
such as dark matter or dark energy.

Following the general treatment,
we investigate in Sec.\ \ref{Bumblebee Models}
a broad class of theories
called bumblebee models
that involve gravitationally coupled vector fields
with spontaneous local Lorentz and diffeomorphism breaking.
For arbitrary quadratic kinetic terms,
the Lagrange density is given in Eq.\ \rf{bb}.
Along with the symmetry-breaking potential $V$
and a matter sector,
this Lagrange density involves five parameters,
four of which can be linearly independent
in specific models.
A particularly attractive class of theories
are the KS\ bumblebee models,
which have kinetic term of the Maxwell form
and hence an additional constraint
that minimizes problems with unitarity and ghosts.
These models also offer candidate alternatives
to Einstein-Maxwell electrodynamics.

In a series of subsections and the associated appendix,
we provide some results valid for general bumblebee models.
The observer and particle forms of
local Lorentz transformations and diffeomorphisms
are presented.
Using the vierbein formalism,
some decompositions of the bumblebee field and metric
are given that are suitable for
the identification of Lorentz and diffeomorphism NG modes.
The effects of various choices of
Lorentz and diffeomorphism gauges are described.
Alternative decompositions used in some of the literature
are also discussed,
in which the Lorentz NG modes are hidden
and only spacetime variables are used.

To provide explicit examples
and to gain insight via a more detailed analysis,
we focus attention in Sec.\ \ref{Examples}
on the class of KS\ bumblebee models.
The basic equations of motion and conservation laws are obtained,
and some properties of the bumblebee currents are considered.
The interpretation of these bumblebee models
as theories of electromagnetism and gravity,
known as bumblebee electrodynamics,
is discussed.
They contain four transverse massless modes,
two of which are massless gravitons
and two of which are massless photons,
along with a massive mode or Lagrange-multiplier mode.
When the massive mode or Lagrange multipler vanishes,
or in the limit of infinite mass,
conventional Einstein-Maxwell theory in an axial gauge
is recovered.

Section \ref{Examples} also contains subsections
considering in more detail various types of potentials,
including the smooth potential $V_S$
in Eq.\ \rf{smoothlagv},
the linear Lagrange-multiplier potential $V_L$
in Eq.\ \rf{linlagv},
and the quadratic Lagrange-multiplier potential $V_Q$
in Eq.\ \rf{quadlagv}.
For the $V_S$ model,
the gravitational and bumblebee equations of motion
are investigated to determine whether a physical massive mode
can propagate as a free field.
The sign of the squared mass term depends
on whether the bumblebee vacuum value $b_\mu$
is timelike or spacelike.
In the timelike case,
the massive mode is a ghost,
while in the spacelike case the squared mass has the usual sign.
However,
in both cases the dispersion law is unconventional
and no localized physical solutions satisfying
suitable asymptotic boundary conditions exist,
so the $V_S$ model has no freely propagating massive modes.
Nonetheless,
the massive mode has a physical impact as an auxiliary field,
acting as an additional source of energy-momentum density
and current density.
In the weak static limit,
for example,
the solutions to the equations of motion
in the presence of mass and charge
describe modified Newton and Coulomb potentials
according to Eq.\ \rf{Phisol}.
These may be of phenomenological interest
in the context of dark matter
and perhaps also dark energy.
The effects are controlled by the massive mode,
but its form is dynamically undetermined
and must be imposed via initial conditions.

Many of the results obtained for the $V_S$ model
apply also to the $V_L$ and $V_Q$ models
that have Lagrange-multiplier fields.
For example,
all the models contain four massless propagating modes
behaving like gravitons and photons.
Although the $V_L$ and $V_Q$ models generate an additional constraint
that eliminates the massive mode on shell,
the Lagrange-multiplier field appears instead
as a extra degree of freedom
playing a similar role to that of the massive mode
in the $V_S$ model.
The form of the Lagrange multiplier
must be set by initial conditions,
and if it is chosen to vanish
then Einstein-Maxwell theory is recovered.
The key difference between the $V_L$ and $V_Q$ models
lies in the role of the Lagrange multiplier,
which can affect the physics as an auxiliary mode
in the $V_L$ model
but which decouples from the theory in the $V_Q$ model.

In the context of bumblebee electrodynamics,
the massive mode or Lagrange-multiplier field
acts as an additional degree of freedom
relative to Einstein-Maxwell theory.
The extra freedom arises because the bumblebee model
has no U(1) gauge symmetry,
and the structure of the kinetic terms
implies that the freedom must be fixed as an initial condition.
In the absence of experimental evidence for a massive mode
or of guidance from an underlying theory,
the choice of initial condition is largely arbitrary.
A natural choice sets the massive mode or Lagrange multiplier field
to zero,
reducing the theory to Einstein-Maxwell electrodynamics
up to possible SME matter-sector couplings.
Bumblebee electrodynamics therefore provides
a candidate alternative explanation
for the existence of massless photons,
based on the masslessness of NG modes
instead of the usual gauge symmetry.
In any case,
the possibility of spontaneous breaking
of local Lorentz and diffeomorphism symmetry
remains a promising avenue for exploring
physics emerging from the Planck scale.

\section*{Acknowledgments}

This work was supported in part
by DOE grant DE-FG02-91ER40661,
NASA grant NAG3-2914,
and NSF grant PHY-0554663.