text:
\section{Introduction}
\label{Introduction}

 With the discovery of the Higgs boson at the Large Hadron Collider
(LHC)~\cite{Aad:2012tfa,Chatrchyan:2012xdj}, all the particles predicted
by the Standard Model (SM) have now been observed, and the search for
new physics is well underway. Among the many puzzles this program may
shed light on is the large hierarchy between the mass of the Higgs and
the Planck scale. If a symmetry protects the Higgs mass from short
distance quantum effects, then we expect new particles with masses of
order the weak scale that are related to the SM particles by the
symmetry. The symmetry partners of the top quark, the top partners, are
expected to be particularly light. Despite increasingly sophisticated
efforts, however, such particles have yet to be discovered at the LHC.

The paradigm of neutral
naturalness~\cite{Chacko:2005pe,Barbieri:2005ri,Chacko:2005vw,Burdman:2006tz,Poland:2008ev,Cai:2008au,Craig:2014aea,Craig:2014roa,Batell:2015aha,Csaki:2017jby,Serra:2017poj,Cohen:2018mgv,Cheng:2018gvu,Dillon:2018wye,Xu:2018ofw}
addresses the hierarchy problem by incorporating color neutral top
partners. Since color neutral particles are much more difficult for the
LHC to discover, the bounds on this class of models remain relatively
weak. In particular, the Mirror Twin Higgs (MTH)
framework~\cite{Chacko:2005pe} protects the Higgs mass by employing
fermionic top partners that are singlets under all the SM gauge
groups.\footnote{Recently, SM-singlet \emph{scalar} top partner models
have also been constructed~\cite{Cohen:2018mgv,Cheng:2018gvu} .} This
protection results from a global symmetry in the Higgs sector combined
with a discrete $\mathbb{Z}_2$ symmetry that exchanges each SM field
with a mirror (``twin") copy that is charged under its own gauge groups.
The electroweak (EW) gauge symmetries of both the SM and twin sectors
are contained in the approximate global symmetry, which in the simplest
realization of the model is $SU(4)\times U(1)$. This global symmetry is
spontaneously broken at a scale $f$ down to $SU(3)\times U(1)$. The
longitudinal modes of the $W$ and $Z$ vector bosons and the physical
Higgs boson are among the resulting pseudo-Nambu-Goldsone bosons
(pNGBs). The mass of the Higgs is protected against large radiative
corrections by a combination of the non-linearly realized global
symmetry and the discrete $\mathbb{Z}_2$ twin symmetry.

After electroweak symmetry breaking the Higgs and its twin counterpart
mix. If the $\mathbb{Z}_2$ symmetry were exact, the scales of
electroweak symmetry breaking in the SM sector and the twin sector would
be identical, $v= v'$, where $v = 246$ GeV is the Higgs vacuum
expectation value (VEV) in the SM, and we employ primes to denote the twin
sector. Then, as a consequence of the mixing, the couplings of the Higgs
boson to SM states would be suppressed and it would have equal couplings
to both visible and twin states. This, however, conflicts with existing
experimental results on the couplings of the Higgs. Introducing soft
breaking of the discrete $\mathbb{Z}_2$ symmetry allows the Higgs VEV
$v$ to be small compared to $v'$, so that the couplings of the physical
Higgs boson to visible sector particles are close to their SM values,
while its couplings to twin particles are suppressed. In this framework
the ratio $v/f \equiv v/\sqrt{v^2 + {v'}^2}$ determines many
observables, such as the ratio of the masses of the SM particles and
their twin partners, as well as deviations in the couplings of the Higgs
boson away from their SM values. The soft-$\mathbb{Z}_2$ breaking also
introduces a tuning in the model, which scales as $(v/f)^2$.
Experimental bounds on the Higgs couplings constrain $f/v\gtrsim 3$,
while requiring the model be less than 10\% tuned indicates $f/v\lesssim
6$~\cite{Burdman:2014zta}. This provides a definite window for the twin
particle masses: between 3 and 6 times the mass of their SM
counterparts.

The MTH framework has been explored and expanded on in recent years. It
has been shown that the breaking of the discrete $\mathbb{Z}_2$ symmetry
can be realized
spontaneously~\cite{Beauchesne:2015lva,Yu:2016bku,Yu:2016swa,Jung:2019fsp,Batell:2019ptb}.
Ultraviolet completions have been constructed based on
supersymmetry~\cite{Falkowski:2006qq,Chang:2006ra,Craig:2013fga,Katz:2016wtw,
Badziak:2017syq,Badziak:2017kjk,Badziak:2017wxn} (see
also~\cite{Berezhiani:2005ek}), compositeness of the
Higgs~\cite{Geller:2014kta,Barbieri:2015lqa,Low:2015nqa}, and most
recently using a ``turtle" construction~\cite{Asadi:2018abu}. Composite
MTH models have been shown to be consistent with precision electroweak
constraints~\cite{Contino:2017moj} and flavor
bounds~\cite{Csaki:2015gfd}, and their collider signals
studied~\cite{Cheng:2015buv,Cheng:2016uqk}.

The cosmology of the MTH, in its original incarnation, is problematic.
In the early universe, Higgs-mediated interactions keep the mirror
sector in thermal equilibrium with the SM down to temperatures of order
a few GeV~\cite{Barbieri:2005ri}. Then the twin photon and twin
neutrinos give an overly large contribution to the total energy density
in radiation, leading to conflict with the bounds on dark radiation from
the cosmic microwave background and Big Bang nucleosynthesis. This
problem can be solved if the model is extended to incorporate an
asymmetric reheating process that contributes to the energy density in
the SM degrees of freedom, but not to the mirror
sector~\cite{Berezhiani:1995yi,Berezhiani:1995am}. To solve the problem,
this mechanism must operate at late times, after the two sectors have
decoupled. This can be realized without requiring additional breaking of
the discrete $Z_2$ symmetry that relates the two
sectors~\cite{Chacko:2016hvu,Craig:2016lyx}. An alternative approach is
to introduce hard breaking of the $Z_2$ into the twin sector Yukawa
couplings, thereby altering the spectrum of mirror
states~\cite{Farina:2015uea,Barbieri:2016zxn,Csaki:2017spo,Barbieri:2017opf}.
This affects the number of degrees of freedom in the two sectors at the
time of decoupling, allowing the cosmological bounds to be satisfied.
Once this problem has been resolved, questions such as the nature of
dark matter~\cite{Farina:2015uea,Cheng:2018vaj}, the origin of the
baryon asymmetry~\cite{Farina:2016ndq,Earl:2019wjw}, the order of the
electroweak phase transition~\cite{Fujikura:2018duw}, and the
implications of the MTH framework for large scale
structure~\cite{Chacko:2018vss}, can be addressed.

The cosmological challenges can also be solved by making the twin sector
vector-like~\cite{Craig:2016kue}. An even more radical approach is to
simply remove from the theory the two lighter generations of mirror
fermions and the twin photon, which do not play a role in solving the
little hierarchy problem. This construction, known as the Fraternal Twin
Higgs (FTH)~\cite{Craig:2015pha}, gives rise to distinctive signatures at the LHC involving
displaced vertices~\cite{Curtin:2015fna,Csaki:2015fba}, and admits
several promising dark matter
candidates~\cite{Craig:2015xla,Garcia:2015loa,Garcia:2015toa,Hochberg:2018vdo,Terning:2019hgj}.

In the MTH framework the discrete $\mathbb{Z}_2$ symmetry and the
resulting gauge invariance under [SU(2) $\times$ U(1)]$^2$ largely
sequester the SM fields from their twin counterparts. The Higgs boson
itself is the only low-energy portal between the two sectors that is
guaranteed by construction. However, if the UV completion is weakly
coupled, the radial mode associated with the breaking of the symmetry
can be light enough to probe at current and future
colliders~\cite{Ahmed:2017psb,Chacko:2017xpd,Buttazzo:2018qqp,Kilic:2018sew,Alipour-fard:2018mre}.
In the absence of additional new fields~\cite{Bishara:2018sgl}, there is
only one other renormalizable operator consistent with the symmetries
that can link the two sectors; a kinetic mixing of the SM hypercharge
gauge boson $B_\mu$ with its twin counterpart $B_\mu'$,
 \begin{equation}
\frac{\epsilon}{2}B_{\mu\nu}'B^{\mu\nu}.
\label{mixepsilon}
 \end{equation}
 Here $B_{\mu\nu}$ is the usual Abelian field strength. This operator
has the effect of giving the SM quarks and leptons charges proportional
to $\epsilon$ under the twin photon and twin $Z$, which are denoted $A'$
and $Z'$ respectively.

Apart from some discussion in the context of dark
matter~\cite{Craig:2015xla,Garcia:2015toa}, previous analyses have
largely neglected this mixing, since it is not radiatively generated in
the low-energy theory to at least three-loop
order~\cite{Chacko:2005pe,Low:2015nqa}. An $\epsilon$ generated at the
four-loop level is small enough to remain consistent with the very
strong bounds on the photon mixing with another massless
vector~\cite{Davidson:2000hf,Vogel:2013raa}; for a twin electron of mass
of order MeV these bounds require $\epsilon\lesssim 10^{-9}$. However,
ultraviolet completions of the MTH based on compositeness generically
introduce states charged under both SM and twin gauge groups, with the
result that $\epsilon$ is expected to receive one-loop corrections of
order $10^{-2}$\textendash$10^{-3}$. Such large values of $\epsilon$ can
only be accommodated if the twin photon $A'$ is massive.

 In this paper we consider the collider signals arising from kinetic
mixing between the hypercharge gauge boson of the SM and its twin
counterpart in the MTH model, in the framework in which the twin photon
is massive. We focus on the framework in which the discrete
$\mathbb{Z}_2$ symmetry is only softly broken. We determine the current
experimental bounds on this scenario, and explore the possibility of
discovering both the twin photon and the twin $Z$ at the LHC, or at a
future 100 TeV hadron collider such as the Future Circular Collider in
hadron mode (FCC-hh). Although the phenomenology of new vector bosons
that mix with the U(1$)_{\rm Y}$ of the SM has been thoroughly
explored~\cite{Holdom:1985ag,Babu:1997st,ArkaniHamed:2008qp,Abel:2008ai,Goodsell:2009xc,Goodsell:2011wn,Essig:2013lka,Hoenig:2014dsa,Curtin:2014cca},
the MTH framework introduces several novel features.

A massive twin photon requires contributions to the masses of the mirror
gauge bosons beyond those from electroweak symmetry breaking. This can
be accomplished either by simply introducing a mass term for the twin
hypercharge gauge boson, or by extending the Higgs sector. In our
analysis, we consider both cases. For the first case, we simply include
an explicit Proca mass term for the twin hypercharge boson,
 \begin{equation}
\frac{m_{B'}^2}{2}B_\mu'B'^\mu,
\label{eq:Bpmass}
 \end{equation}
 where $m_{B'}$ is a free parameter in the Lagrangian. Since a Proca
mass term can be obtained from the St\"{u}ckelberg construction, which
is unitary and renormalizable, after gauge fixing, this theory is
ultraviolet complete. The Proca mass can also be thought of as arising
from the VEV of a new Higgs field that carries charge under twin
hypercharge, but not under twin SU(2)$_{\rm L}$~\cite{Batell:2019ptb}. If this Higgs field is
heavy, it will decouple from the low energy spectrum, but if it carries
only a small charge under U(1$)_{\rm Y'}$, the twin photon, though no
longer massless, will still be light.

Alternatively, rather than include an explicit mass term for $B_{\mu}'$,
the twin photon can acquire a mass from the VEV of an additional Higgs
field in the twin sector. For concreteness, we consider a scenario in
which the Higgs content of the theory is extended to include two Higgs
doublets in each of the SM and twin
sectors~\cite{Chacko:2005vw,Beauchesne:2015lva,Yu:2016bku,Yu:2016swa}.
The VEVs of the two doublets in the visible sector must be aligned to
leave the photon massless, but the twin sector doublets are not so
constrained. In particular, the VEVs of the two Higgs doublets in the
twin sector must be somewhat misaligned to ensure that the twin photon
has a mass.

As a consequence of the coupling in Eq.~(\ref{mixepsilon}), the massive
$Z'$ and $A'$ mix with the hypercharge gauge boson. Therefore the SM
quarks and leptons acquire charges under the twin photon and twin $Z$,
which can therefore be searched for at colliders. At hadron colliders,
production through light quarks followed by decay into dileptons is a
particularly promising channel. We find that values of the mixing
parameter $\epsilon$ as small as $10^{-2}$ can potentially be probed at
the LHC and FCC-hh.

In the softly-broken $\mathbb{Z}_2$ scenario, a few physical parameters
determine all the couplings in the model that are relevant for LHC
searches. In particular, in the models we consider, the collider signals
depend on only three parameters beyond those in the SM. Consequently, if
more than three independent measurements can be performed, the
parameters of the model are overdetermined, allowing these
theories to be tested. Measuring the masses of the $A'$ and $Z'$, along with
both their dilepton resonance production rates, achieves this goal.

In the following section

 we outline the interactions of the neutral vector bosons in the MTH
framework, and study their production rates and branching fractions.
This is done for two scenarios; a model in which the twin hypercharge
gauge boson is given an explicit Proca mass (THPM), and a twin two Higgs
doublet model (T2HDM). In Sec.~\ref{sec:constraints} we determine the
existing bounds on these two models from direct searches and from
precision measurements. Then, in Sec.~\ref{s.discovery}, we explore the
discovery reach of the high luminosity LHC (HL-LHC) and a future 100 TeV
hadron collider, and consider the prospects for testing the THPM and the
T2HDM as the underlying origin of these resonances. We conclude in
Sec.~\ref{s.Con}.

\section{Neutral Vectors in the Mirror Twin Higgs}
\label{Mirror Twin Higgs Models}

In this section we study the effects of kinetic mixing on the masses and
couplings of the neutral vector bosons in the MTH model, in the scenario
in which the twin photon is massive. We then explore the implications of
mixing for the production cross sections and decay widths of these
particles at colliders. We focus on the two case studies of a Proca mass
for the twin hypercharge gauge boson and a two Higgs doublet version of
the MTH model. For each case we obtain the production cross sections and
branching fractions of the electrically neutral twin gauge bosons $A'$
and $Z'$ at the LHC and at a future 100 TeV collider.

\subsection{Proca Mass for Twin Hypercharge}

In this first case, we begin by considering the ${\rm SU(2)}_{\rm L} \times
{\rm U}(1)_{\rm Y} \times {\rm SU(2)}_{\rm L'}\times {\rm U(1)}_{\rm Y'}$
gauge sector,
 \begin{align}
\mathcal{L}\supset&-\frac{1}{2}{\rm Tr}\left[{W_{\mu\nu}}{W}^{\mu\nu}\right]-\frac{1}{2}{\rm Tr}\left[{W_{\mu\nu}^{\p}}{W^{\p}}^{\mu\nu}\right]-\frac{1}{4}{B_{\mu\nu}}{B}^{\mu\nu}-\frac{1}{4}{B_{\mu\nu}^{\p}}{B^{\p}}^{\mu\nu}\nonumber\\
&+\frac{\epsilon}{2 }{B_{\mu\nu}^{\p}}B^{\mu\nu}+\frac{m^2_{B'}}{2}B^{\p}_{\mu}B^{\p\mu},
\label{Lagrangian}
 \end{align}
 where primed fields belong to the twin sector and $\epsilon$ is the
kinetic mixing parameter. For any spin-1 field $X_{\mu}$, we employ the
notation
 \beq
X_{\mu\nu}=\partial_\mu X_\nu-\partial_\nu X_\mu,
 \eeq
 to denote the field strength, with the usual generalization to the
non-Abelian case.

Rather than work directly with $B'$ and $W'^{3}$, it is convenient to go
over to the analogue of the familiar electroweak basis of the photon and
$Z$ in the twin sector. Writing $s_W \equiv \sin\theta_W$ and $c_W
\equiv \cos\theta_W$, where $\theta_W$ is the weak mixing angle, we
define
 \begin{equation}
\overline{Z}_{\mu}^{\p} \equiv c_WW_\mu^{\p 3}-s_WB_{\mu}^\p,
\ \ \ \
\overline{A}_{\mu}^\p \equiv s_WW_\mu^{\p 3}+c_WB_{\mu}^\p \;.
 \end{equation}
 We find that, as expected, the Proca mass induces mass mixing between
the two vector bosons,
 \beq
\frac12 \left(\overline{A}_{\mu}^\p\;\; \overline{Z}_{\mu}^{\p} \right)\left( \begin{array}{cc}
c_W^2m^2_{B'} & -s_W c_W m^2_{B'}\\
-s_W c_W m^2_{B'} &m^2_{\overline{Z}^\p}+ s_W^2m^2_{B'}
\end{array}\right)\left(\begin{array}{c}
\overline{A}_{\mu}^\p\\
\overline{Z}_{\mu}^{\p}
\end{array} \right).\label{e.PMTHMassMat}
 \eeq
 Here $m^2_{\overline{Z}^\p} \equiv (v'/v)^2 m^2_{Z_0}$, where
$m_{Z_0}$ is the mass of the $Z$ boson in the SM. This mass matrix for
the neutral twin sector vector bosons can be diagonalized by a simple
rotation of the fields, with the mixing angle given by
 \begin{align}
 \sin2\phi=\frac{m^2_{B'}s_{2W}}{\sqrt{\left( m^2_{B'}-m^2_{\overline{Z}^\p}\right)^2+4s_W^2m^2_{B'}m^2_{\overline{Z}^\p}}}\,.
\label{mixangle1}
 \end{align}
 We label the lighter of the two mass eigenstates as $A_{\mu}'$, and the
heavier one as $Z_{\mu}'$. Note that in the limit $m^2_{B'} \gg
m^2_{\overline{Z}^\p}$, Eq.~(\ref{mixangle1}) implies that
$\phi=\theta_W$. Essentially, this undoes the weak mixing, meaning that
for large $m_{B'}$, the mass eigenstates are, to a good approximation,
simply the $W_3'$ and the $B'$ with masses $m_{\overline{Z}^\p} c_{W}$
and $m_{B'}$ respectively. In this limit the $Z'$, which has its mass
set by $m_{B'}$, couples more strongly to the visible sector than the
lighter $A'$, which decouples from the SM as $m_{B'}$ increases.
Additional details are given in Appendix~\ref{a.Diag}.

In the opposite limit, $m^2_{B'} \ll m^2_{\overline{Z}^\p}$, the mixing angle
$\phi$ tends to zero. The mass eigenstates are then, to a good
approximation, simply $\overline{A}_{\mu}^{\p}$ and $\overline{Z}_{\mu}^{\p}$, with masses
$m_{B'} c_W$ and $m_{\overline{Z}^\p}$, respectively. In this limit it is the
lighter eigenstate, the $A'$, which couples more strongly to the visible
sector, while the couplings of the heavier $Z'$ are suppressed by a
relative factor of $\tan\theta_W$.

\begin{figure}
\centering
\includegraphics[trim={8mm 11mm 8mm 8mm},clip,width=0.6\textwidth]{fig_mbp_mv_fov35_eps01.pdf}
\caption{The mass eigenvalues for the twin neutral vector bosons as a function of the Proca mass $m_{B'}$ in the THPM model for $\epsilon=0.1$ and $f/v=3$ and 5.}
\label{fig:masseigenvalues}
\end{figure}

To analyze the experimental signals, we must transform from
Eq.~\eqref{Lagrangian} to a basis diagonal in both mass and kinetic
terms. This is accomplished by first performing a shift of $B$ and then
rotating into the mass basis. The details of this procedure are given in
Appendix~\ref{a.Diag}, where we obtain the field transformation to
leading order in $\epsilon$. The couplings of the diagonalized fields to
SM and twin particles are provided in Appendix~\ref{a.VFcouplings},
again to leading order in $\epsilon$. However, this perturbative
analysis breaks down near mass degeneracies, where one of the twin
sector gauge bosons becomes close in mass to the SM $Z$ boson. Therefore
the results we present have been obtained by diagonalizing the system
numerically. For small $\epsilon$, away from mass degeneracies, the mass
eigenvalues are close to the values obtained by diagonalizing the matrix
Eq.~(\ref{e.PMTHMassMat}). The mass eigenvalues as a function of
$m_{B'}$, for the benchmark values $f/v=3$ and 5 and $\epsilon=0.1$, are
shown in Fig.~\ref{fig:masseigenvalues}. We see that
for large $m_{B'}$, the mass of the heavier $Z'$ is close to $m_{B'}$,
while that of the lighter $A'$ asymptotes to $(v'/v) m_W$.
\begin{figure}
\centering
\includegraphics[trim={8mm 11mm 8mm 8mm},clip,width=0.49\textwidth]{fig_map_sigzp_fov35_eps01.pdf}
\includegraphics[trim={8mm 11mm 8mm 8mm},clip,width=0.49\textwidth]{fig_map_sigap_fov35_eps01.pdf}
\caption{Production cross section of the $Z'$ (left) and $A'$ (right) in the THPM model at the LHC and at a future 100~TeV hadron collider for $\epsilon=0.1$ and $f/v=3$ and 5.}
\label{fig:productionxsec}
\end{figure}

After diagonalization the SM fermions acquire charges under the $A'$ and
the $Z'$. This allows the twin vector bosons to be produced and detected
at colliders. The production cross sections of the $A'$ and $Z'$ at the
13 TeV LHC and at a 100~TeV future hadron collider are shown in
Fig.~\ref{fig:productionxsec}, for $\epsilon=0.1$ and $f/v = 3$ and 5.
We see that for small values of its mass the $A'$ cross section is
large, but drops off quickly as the mass increases. This occurs because,
for large $m_{B'}$, the $A'$ is almost entirely composed of $W_3'$,
which does not mix directly with SM hypercharge. Consequently, as
$m_{B'}$ increases, the cross section plummets. In contrast, the
production cross section of the $Z'$ remains sizable even for masses in
the TeV range.
\begin{figure}
\centering
\includegraphics[trim={8mm 10mm 8mm 8mm},clip,width=0.49\textwidth]{fig_mzp_BRs_fov3_eps01.pdf}
\includegraphics[trim={8mm 10mm 8mm 8mm},clip,width=0.49\textwidth]{fig_map_BRs_fov3_eps01.pdf}\\
\includegraphics[trim={8mm 11mm 8mm 8mm},clip,width=0.49\textwidth]{fig_mzp_BRs_fov5_eps01.pdf}
\includegraphics[trim={8mm 11mm 8mm 8mm},clip,width=0.49\textwidth]{fig_map_BRs_fov5_eps01.pdf}
\caption{Branching ratios of the $Z'$ (left) and $A'$ (right) for the THPM model for $\epsilon=0.1$ and $f/v=3$(5) for the top (bottom) row. The light blue curve is for a single SM lepton flavor, while the dark blue curve corresponds to all six quark flavors.}
\label{fig:BRs}
\end{figure}

The corresponding branching fractions are shown in Fig.~\ref{fig:BRs}.
As expected, decays to twin fermions dominate the width. Nevertheless,
the branching fractions into SM fermions can be large enough for
discovery in a clean resonant channel such as dileptons. Note that for
large values of $m_{B'}$, the branching fraction of $A'$ into dileptons
is much smaller than the corresponding $Z'$ branching fraction. These
hierarchies in the production cross sections and dilepton branching
ratios translate into a much greater discovery potential for the $Z'$ at
colliders in most of parameter space, as we show below.

\subsection{Twin Two Higgs Doublet Model\label{T4HDM}}

We now turn to the twin two Higgs doublet model. We label the two
visible sector Higgs doublets as $H_1$ and $H_2$, and their twin
counterparts as $H_1'$ and $H_2'$. Since the photon is massless, the
VEVs of $H_1$ and $H_2$ must be aligned. However, the VEVs of the
doublets in the twin sector need to be misaligned if they are to give
mass to both the neutral gauge bosons in the twin sector.

The alignment of the two twin doublet VEVs can be quantified in many ways.
For instance, the quantity
 \begin{equation}
\frac{|H^{'\dag}_{1}H^{'}_{2}|^{2}}  {(H^{'\dag}_{1}H^{'}_{1})(H^{'\dag}_{2}H^{'}_{2})},
 \end{equation}
 tends to one in the limit of perfect alignment, and to zero in the
limit of perfect anti-alignment. The limit of perfect alignment does not
serve our purposes since it leaves one vector boson massless, but any
other configuration will result in both the neutral gauge bosons in the
twin sector acquiring masses. For concreteness, we restrict our
attention to the limit of perfect anti-alignment in the twin sector. We
further assume that of the two visible sector doublets, only $H_1$
acquires a VEV, while $\langle H_2 \rangle = 0$. It is straightforward
to construct a potential for the scalar sector that gives rise to these
features. While the potential for $H_2$ and $H_2'$ must respect the
discrete $\mathbb{Z}_2$ twin symmetry, it need not obey the global SU(4)
$\times$ U(1) symmetry. Therefore the scalar states in $H_2$, the second
visible sector doublet, are not required to be pNGBs and can
therefore naturally be heavy.

The fact that both $H_1'$ and $H_2'$ acquire VEVs implies that there are
two additional pNGB states in the mirror sector. If these
states are to acquire a mass, there must be interactions that couple
$H_1'$ and $H_2'$ in the potential for the scalar fields. We can lift
these additional pNGBs from the low energy theory by
including the term
 \begin{equation}
V= 	\lambda \left(\left| {H_1^\dagger  H_2}\right|^2+\left| {H'^\dagger_1  H'_2}\right|^2\right),\label{e.scalarMasses}
 \end{equation}
 in the scalar potential. These terms respect the discrete
$\mathbb{Z}_2$ symmetry, and therefore do not generate a large mass for
the light Higgs. For $\lambda \gtrsim 1$, these states acquire masses
above those of the twin gauge bosons, and we can neglect their dynamics
at low energies.

We can then write the VEVs of the Higgs fields as,
 \begin{equation}
H_1=\left( \begin{array}{c}
0\\
v/\sqrt{2}
\end{array}
\right), \ \ \ \
H'_1=\left( \begin{array}{c}
0\\
 f_1\cos\vartheta/\sqrt{2}
\end{array}
\right),\ \ \ \
H_2=\left( \begin{array}{c}
0\\
0
\end{array}
\right),
 \ \ \ \
H_2'=\left( \begin{array}{c}
\displaystyle f_2/\sqrt{2} \\
0
\end{array}
\right),
\label{e.linearTwinFourHiggsDoubletsAsector}
 \end{equation}
where $\sin\vartheta=v/f_1$. The mass terms for the gauge bosons arise from the kinetic terms for the Higgs bosons,
 \begin{align}
{\cal L}& \supset\left| {D}_{\mu} {H_1}\right|^2+\left| {D}_{\mu} H_2\right|^2+\left|D'_{\mu} H'_1\right|^2+\left| D'_{\mu} H'_2\right|^2~.\label{Lagrangiantwin}
 \end{align}
 From this we obtain the usual mass terms for the visible sector gauge bosons,
 \beq
\frac{g^2v^2}{4}W^+_\mu W^{-\mu}+\frac{g^2v^2}{8c_W^2}Z_{\mu} Z^\mu~.\label{e.AvecBosons}
 \eeq
 The mass matrix for the twin sector is more complicated. In this case we
find
 \begin{align}
&\frac{g^2}{4}\left(f_1^2\cos^2\vartheta+f_2^2 \right)W'^+_\mu W'^{-\mu}+\frac{g^2}{2}s_W^2f_2^2\overline{A}'_{\mu} \overline{A}'^\mu+\frac{g^2s_W}{2c_W}f_2^2c_{2W}\overline{A}'_{\mu} \overline{Z}'^\mu\nonumber\\
&+\frac{g^2}{8c_W^2}\left[f_1^2\cos^2\vartheta+f_2^2c_{2W}^2 \right]\overline{Z}'_{\mu} \overline{Z}'^\mu\, .
\end{align}
 where $\cos n\theta_W\equiv c_{nW}$. It is convenient to define
$m_{\overline{Z}^\p}=m_{Z_0}\cot\vartheta$ where $m_{Z_0}$ is the $Z$
mass in the SM and $m_{\overline{A}'}=gf_2s_W$. Expressed in terms of
these variables, the mass matrix is given by ($t_{2W}\equiv\tan 2\theta_W$)
 \beq
\frac12 \left(\overline{A}_{\mu}^\p\;\; \overline{Z}_{\mu}^{\p} \right)\left( \begin{array}{cc}
m_{\overline{A}'}^2 & m_{\overline{A}'}^2/t_{2W}\\
m_{\overline{A}'}^2/t_{2W} &m^2_{\overline{Z}^\p}+ m_{\overline{A}'}^2/t_{2W}^2
\end{array}\right)\left(\begin{array}{c}
\overline{A}_{\mu}^\p\\
\overline{Z}_{\mu}^{\p}
\end{array} \right),\label{e.T2HDMMassMat}
 \eeq
 which admits no massless state for $m_{\overline{A}'}^2>0$.

\begin{figure}
\centering
\includegraphics[trim={8mm 11mm 8mm 8mm},clip,width=0.6\textwidth]{fig_fp_mv_fov35_eps01.pdf}\vspace{-0.25cm}
\caption{The mass eigenvalues for the twin neutral vector bosons as a function of the twin photon mass $m_{\overline{A}'}$ in the T2HDM for $\epsilon=0.1$ and $f_{1}/v=3$ and 5.}
\label{fig:masseigenvalues2}
\end{figure}
From this point, the determination of the physical states and couplings
in this model proceeds just as in the THPM model, with an example of how the physical masses depend on $m_{\overline{A}^\p} $ given in Fig.~\ref{fig:masseigenvalues2}. The mixing angle that
diagonalizes the mass matrix in Eq.~(\ref{e.T2HDMMassMat}) is now given
by
 \begin{equation}
 \sin2\phi= -\frac{m_{\overline{A}'}^2s_{4W}}{\sqrt{\left(m^2_{\overline{Z}^\p}s_{2W}^2+c_{4W}m^2_{\overline{A}^\p} \right)^2+s_{4W}^2m^4_{\overline{A}^\p}}}\,.
 \end{equation}
 In the limit $m^2_{\overline{A}'} \ll m^2_{\overline{Z}^\p}$, the
mixing angle $\phi$ tends to $-\pi/2$, and the mass eigenstates become, to a
good approximation, just $\overline{A}_{\mu}^{\p}$ and
$\overline{Z}_{\mu}^{\p}$, with masses $m_{\overline{A}'}$ and
$m_{\overline{Z}^\p}$ respectively. In this limit the lighter
eigenstate, the $A'$, couples more strongly to the visible sector, while
the couplings of the heavier $Z'$ are suppressed by a relative factor of
$\tan\theta_W$. In the opposite limit, $m^2_{\overline{A}'} \gg
m^2_{\overline{Z}^\p}$, the mass eigenvalues are given by $m_{\overline{Z}^\p}
\sin 2\theta_W$ and $m_{\overline{A}^\p}/\sin 2\theta_W$. In this limit
it is again the lighter eigenstate, the $A'$, that couples more strongly
to the visible sector, while the couplings of the heavier $Z'$ are
suppressed by the same relative factor of $\tan\theta_W$. This is very
different from the THPM model, in which the couplings of the $A'$ to the
visible sector vanish in the limit that the $Z'$ is heavy.

 At the intermediate value $m_{\overline{A}'}=s_{2W}m_{\overline{Z}'}$,
corresponding to $\phi=\theta_W-\pi/2$, the $\overline{Z}'$ field
rotates into $W'^3$, which means it is orthogonal to $B'$. As we see in
Figs.~\ref{fig:productionxsec2HDM} and~\ref{fig:BRs2HDM}, this causes
the $Z'$ to completely decouple from the visible sector, suppressing the
production cross section and branching into visible states. At the same
point, of course, the $A'$ is perfectly aligned with twin hypercharge, so its coupling to the visible sector is enhanced.

\begin{figure}
\centering
\includegraphics[trim={8mm 11mm 8mm 8mm},clip,width=0.49\textwidth]{fig_mzp_sigzp_fov35_eps01_2d.pdf}
\includegraphics[trim={8mm 11mm 8mm 8mm},clip,width=0.49\textwidth]{fig_map_sigap_fov35_eps01_2d.pdf}
\caption{Production cross section of the $Z'$ (left) and $A'$ (right) at the LHC and at a future 100~TeV hadron collider in the T2HDM for $\epsilon=0.1$ and $f_1/v=3$ and 5. For $m_{\overline{A}'}=s_{2W}m_{\overline{Z}'}$ the alignment of $Z'$ is orthogonal to twin hypercharge. This closes the portal to the SM sector, reducing production.}
\label{fig:productionxsec2HDM}
\end{figure}

\begin{figure}
\centering
\includegraphics[trim={8mm 8mm 8mm 8mm},clip,width=0.49\textwidth]{fig_mzp_BRs_f3v_eps01.pdf}
\includegraphics[trim={8mm 8mm 8mm 8mm},clip,width=0.49\textwidth]{fig_map_BRs_f3v_eps01.pdf}\\
\includegraphics[trim={4mm 5mm 4mm 3mm},clip,width=0.49\textwidth]{fig_mzp_BRs_f5v_eps01.pdf}
\includegraphics[trim={3mm 5mm 3mm 3mm},clip,width=0.49\textwidth]{fig_map_BRs_f5v_eps01.pdf}
\caption{Branching ratios of the $Z'$ (left) and $A'$ (right) in the T2HDM for $\epsilon=0.1$ and $f_1/v=3$ and 5. The blue curve is for a single SM lepton flavor while the dark blue curve corresponds to all six quark flavors. For $m_{\overline{A}'}=s_{2W}m_{\overline{Z}'}$ the alignment of $Z'$ is orthogonal to twin hypercharge. This closes the portal to the SM sector, reducing branching to visible states.}
\label{fig:BRs2HDM}
\end{figure}

To analyze the experimental signals, we transform to a basis in which
the mass and kinetic terms of the $A'$ and $Z'$, as well as those of the
SM photon and $Z$, are diagonal. As in the THPM case, this is
accomplished by first performing a shift of the SM hypercharge gauge
boson $B$ and then rotating into the mass basis, as discussed in
Appendix~\ref{a.Diag}. Once the kinetic and mass terms of the vector
bosons have been diagonalized, it is straightforward to calculate their
couplings to fermions. The details can be found in
Appendix~\ref{a.VFcouplings}. In Figs.~\ref{fig:productionxsec2HDM}
and~\ref{fig:BRs2HDM} we plot the production cross sections and
branching fractions, respectively, of the $A'$ and $Z'$ in the T2HDM. We
see that the production cross section for the $A'$ is always larger than
for the $Z'$. The branching fraction of the $A'$ to SM final states is
also greater than for the $Z'$. These features, along with its lighter
mass, enhance the prospects for $A'$ discovery while inhibiting $Z'$
discovery. We also see a striking feature when
$m_{\overline{A}'}=s_{2W}m_{\overline{Z}'}$, where the $Z'$ is
orthogonal to twin hypercharge and its couplings to SM states vanish.

\section{Existing Constraints\label{sec:constraints}}

In this section we determine the constraints on the vector boson sector
of the MTH model from direct searches and from precision
electroweak observables. These are displayed in
Fig.~\ref{fig:constraints}, the left plot showing the THPM model and the
right plot the T2HDM. The top (bottom) row is for the benchmark of
$f_{(1)}/v = 3(5)$.

\begin{itemize}
 \item LEP analyses place strong limits on any new physics contributions that affect the properties of the $Z$ boson. Quantities that can be determined directly from measurements at the $Z$-pole include the total width of the $Z$ boson, as well as the cross sections into various SM final states through the $Z$ resonance. Given these measurements, theory can be employed to fit other parameters in a straightforward way. In particular, theory relates the resonant dilepton production cross section and the total width to the leptonic width of the $Z$ in a model-independent way. This can therefore be used to constrain models of new physics. A different combination of the direct $Z$-pole measurements allows the invisible width of the $Z$ boson to be determined, again in a model-independent way, which provides a powerful independent constraint on new physics. We therefore begin our study of existing constraints on the MTH model with these two LEP $Z$-pole measurements.

The measured invisible width of the $Z$-boson is $\Gamma_\text{Inv}^\text{LEP}=499.0\pm 1.5\,
\text{MeV}$~\cite{Patrignani:2016xqp}. The predicted value for the SM
contribution is $\Gamma_\text{Inv}^\text{SM}=501.3\pm 0.6$
MeV~\cite{Carena:2003aj}. Therefore, the preferred central value and
associated uncertainty of any potential new physics contribution are
given by
 \beq
\Delta\Gamma_\text{Inv}=-2.2\pm 1.6 \;\hbox{MeV}. \label{difference}
 \eeq
 In our models, there are two contributions to
$\Delta\Gamma_\text{Inv}$, a {\it reduction} due to the change in the
couplings to the neutrinos, and a strictly positive contribution due to
$Z$ decays to kinematically accessible twin states. We find  that
the reduction in the SM width dominates, so the MTH invisible $Z$
width is generically smaller than the SM prediction, bringing it closer to the LEP
measurement. Consequently, this does not significantly constrain either
model.

The modification of the $Z$-boson coupling to electrons in the MTH model, as well as to the twin sector states also constrains the parameter space. The LEP bound on the partial width of $Z\to e^{+} e^{-}$ is given by~\cite{Patrignani:2016xqp}
\beq
\Gamma\left(Z \to e^+ e^-\right) = 83.91\,\pm\, 0.12 ~\text{MeV}.
\eeq
We require this partial width to be within the 2$\sigma$ range from 83.67 MeV to 84.15 MeV in our model, which excludes the regions above the black dashed lines in Fig.~\ref{fig:constraints}.

\item
 Electroweak precision data as encoded in the oblique
parameters~\cite{Peskin:1990zt}, in particular the $T$ parameter, also
constrain the relevant parameter space. The mixing among the neutral
vector bosons alters the relation between the $Z$ boson mass and the $W$
boson mass in the SM. The leading order correction is given in
Eq.~\eqref{e.massesZ}. Using the notation of
ref.~\cite{Appelquist:2002mw} we have
 \beq
\frac{\alpha T}{1+\alpha T}=1-\frac{m_Z^2}{m_{Z_0}^2}.\label{e.leadingT}
\eeq
 Here $m_{Z_0}$ is the $Z$ boson mass in the SM, and $\alpha$ is the
fine structure constant evaluated at $m_{Z_0}$.

\hspace{18pt}While this effect is the largest, there are other
contributions to both the $S$ and $T$ parameters from the reduction of
the Higgs coupling to SM states. These effects have been calculated in
the general case~\cite{Falkowski:2013dza}. Applied to our models, we obtain
 \begin{equation}
T\approx-\frac{3v^2}{8\pi c_W^2 f^2_{(1)}}\ln\frac{\Lambda_\text{UV}}{m_{Z_0}}, \ \ \ \ S\approx\frac{v^2}{6\pi f^2_{(1)}}\ln\frac{\Lambda_\text{UV}}{m_{Z_0}}.
\label{S&T}
 \end{equation}
 While this contribution to $T$ is smaller than Eq.~\eqref{e.leadingT},
it also has the opposite sign, reducing the deviation. We use the
current bound on the $T$-parameter leaving $U$ a free
parameter~\cite{Baak:2014ora}. For $\Lambda_\text{UV}=5$ TeV the
contribution to $S$ in Eq.~(\ref{S&T}) varies from about 0.02 to 0.008,
while the contribution to $T$ varies from $-0.06$ to $-0.02$, as $f/v$
changes from 3 to 5. In this case, the 95\% exclusion contours require
$T<0.14$ and 0.13 respectively. The excluded region is shown as the area
above the solid black line in Fig.~\ref{fig:constraints}.

\hspace{18pt} We see from the figure that both the $Z\to e^+e^-$ partial
width and the $T$-parameter bounds are extremely restrictive in the
T2HDM case. This occurs because, in the T2HDM, the lighter state
continues to have sizable couplings to the SM even as $m_{\overline{A}'}$
gets large, so the bound remains almost unchanged even as the $Z'$
becomes heavy. In contrast, in the THPM model, the lighter $A'$
decouples from the SM sector in the limit that the mass of the $Z'$ is
large, so the bounds on this scenario fall off as $m_{Z'}$
increases.

\item
 Direct searches at the LHC place strong bounds on dilepton resonances
produced from a $\bar{q}$-$q$ initial state. We use the ATLAS
limits~\cite{Aaboud:2017buh,Aad:2019fac}, and employ
{\sc{MadGraph5}}~\cite{Alwall:2014hca} to simulate backgrounds. The $A'$
and $Z'$ production and decay at the LHC are simulated using the MSTW
PDFs~\cite{Martin:2009iq}. Our analysis is performed at the parton level
since we are only dealing with leptons in the final state. In
Fig.~\ref{fig:constraints} we show the 95\% exclusion region from the
resonant $Z'$ (blue shaded) and $A'$ (red shaded) production with
subsequent decays to electrons and muons as a function of $m_{B'}$
(left) or $m_{\overline{A}'}$ (right) and $\epsilon$ and
$f_{(1)}/v=3(5)$ on the top (bottom) row.

\hspace{18pt} As we saw in Fig.~\ref{fig:masseigenvalues}, the $A'$ mass
in the THPM model asymptotes to $(v'/v) m_{W}$ as $m_{B'}$ gets large,
with similar behavior in the T2HDM. However, in the THPM case both the
production cross section (see Fig.~\ref{fig:productionxsec}) and
dilepton branching ratio of the $A'$ (see Fig.~\ref{fig:BRs}) fall off
rapidly in this region, where $A'$ decouples from the SM. Therefore, the
$A'$ is unconstrained for larger values of $m_{B'}$. Conversely, in the
T2HDM the production (see Fig.~\ref{fig:productionxsec2HDM}) and
dilepton branching fraction of the $A'$ (see Fig.~\ref{fig:BRs2HDM})
remain sizable as $m_{\overline{A}'}$ increases, making the bound from
$A'$ searches strong over the entire mass range of interest. In the THPM
model the $Z'$ limits persist, dominating the limits at higher $m_{B'}$.
In the T2HDM, however, the $Z'$ decouples from the SM at
$m_{\overline{A}'}=s_{2W}m_{\overline{Z}'}$, causing the bound to vanish
at this point. Even away from this point, the couplings of the $Z'$ to
the SM are, in general, smaller than those of the $A'$, so the bounds
on the T2HDM are dominated by the limits on the $A'$.

\end{itemize}

\begin{figure}
\centering
\includegraphics[trim={9mm 8mm 9mm 9mm},clip,width=0.49\textwidth]{fig_bounds_mbpeps_fov3.pdf}
\includegraphics[trim= 9mm 7mm 9mm 6mm,clip,width=0.49\textwidth]{fig_bounds_fpeps_fov3.pdf}\\
\includegraphics[trim={9mm 10mm 9mm 8mm},clip,width=0.49\textwidth]{fig_bounds_mbpeps_fov5.pdf}
\includegraphics[trim={9mm 10mm 9mm 8mm},clip,width=0.49\textwidth]{fig_bounds_fpeps_fov5.pdf}
\caption{Compilation of constraints on the MTH model in terms of $\epsilon$ and either $m_{B^\p}$ (THPM model on the left) or $m_{\overline{A}'}$ (T2HDM on the right) with $f_{(1)}/v=3$(5) in the top (bottom) row. Indirect constraints are shown in black. The 13 TeV LHC dilepton resonance search bounds on the $A'$ ($Z'$) are shown in red (blue). }
\label{fig:constraints}
\end{figure}

\section{Discovery Prospects \label{s.discovery}}

Now that the constraints on this framework have been mapped out, we
determine the discovery and exclusion reach of the HL-LHC and a 100~TeV
hadron collider. We estimate the sensitivity of future dilepton searches
by a simple scaling up of existing results. As an example, if a run-II
dilepton resonance search at a given mass can exclude a signal cross
section of $\sigma_{S,13}$ at 95\% confidence level, then a similar
search at a 100~TeV collider will be sensitive to a cross section
$\sigma_{S,100}$ at the same confidence level, given by
 \beq
\sigma_{S,100}=\sigma_{S,13}\sqrt{\frac {\sigma_{\text{BG}100~{\rm TeV}}}{\sigma_{\text{BG}13~{\rm TeV}}}}\sqrt{\frac {\mathcal{L}_{13}}{\mathcal{L}_{100}}},
 \eeq
 where the square roots contain the ratios of the background cross
sections at the two colliders, as well as the ratio of the luminosities
of the two searches. We calculate the background cross sections at 13
and 100~TeV at parton level in {\sc MadGraph}. Our results for
$f_{(1)}/v=3$ and 5 are shown in Fig.~\ref{fig:futuresensitivityHLLHC}
for the HL-LHC and Fig.~\ref{fig:futuresensitivityFCC} for the FCC-hh
with a luminosity of $3000~{\rm fb}^{-1}$, the THPM model on the left
and the T2HDM on the right. As in the collider constraints of the
previous section, we see that in the THPM model the sensitivity to $A'$
falls off as $m_{B'}$ increases, while the sensitivity to $Z'$ persists.
Conversely, in the T2HDM the $A'$ sensitivity falls off only slowly with
$m_{\overline{A}'}$, while the sensitivity to the $Z'$ is much weaker
across the entire parameter range.

The result is that in the THPM model, it is typically the $Z'$ channel
that can be used to improve the limits on $\epsilon$ at the LHC and FCC,
gaining a factor of a few over the constraints shown in
Fig.~\ref{fig:constraints}, whereas in the T2HDM the $A'$ drives the
sensitivity. This must be considered quite impressive, since for these
resonances both the production cross section and the branching ratio
into SM states scale roughly as $\epsilon^{2}$.

\begin{figure}
\centering
\includegraphics[trim={9mm 8mm 9mm 9mm},clip,width=0.49\textwidth]{fig_mbp_eps_13fov3.pdf}
\includegraphics[trim={9mm 8mm 9mm 9mm},clip,width=0.49\textwidth]{fig_mA0p_eps_13fov3.pdf}\\
\includegraphics[trim={9mm 8mm 9mm 9mm},clip,width=0.49\textwidth]{fig_mbp_eps_13fov5.pdf}
\includegraphics[trim={9mm 8mm 9mm 9mm},clip,width=0.49\textwidth]{fig_mA0p_eps_13fov5.pdf}
\caption{Projected $\sigma$ contours of the $Z'$ (blue) and $A'$ (red) at the HL-LHC for a luminosity of 3000~fb$^{-1}$ for $f_{(1)}/v=3$(5) on the top (bottom) row. The THPM model is on the left and the T2HDM is on the right. The regions excluded by current LHC searches for the twin bosons are shaded in the corresponding color.}
\label{fig:futuresensitivityHLLHC}
\end{figure}

\begin{figure}
\centering
\includegraphics[trim={9mm 8mm 9mm 9mm},clip,width=0.49\textwidth]{fig_mbp_eps_100fov3.pdf}
\includegraphics[trim={9mm 8mm 9mm 9mm},clip,width=0.49\textwidth]{fig_mA0p_eps_100fov3.pdf}\\
\includegraphics[trim={9mm 8mm 9mm 9mm},clip,width=0.49\textwidth]{fig_mbp_eps_100fov5.pdf}
\includegraphics[trim={9mm 8mm 9mm 9mm},clip,width=0.49\textwidth]{fig_mA0p_eps_100fov5.pdf}
\caption{Projected $\sigma$ contours of the $Z'$ (blue) and $A'$ (red) at a FCC hadron machine for a luminosity of 3000~fb$^{-1}$ for $f_{(1)}/v=3$(5) on the top (bottom) row. The THPM model is on the left and the T2HDM is on the right. The regions excluded by current LHC searches for the twin bosons are shaded in the corresponding color.}
\label{fig:futuresensitivityFCC}
\end{figure}

In the models we consider, the vector boson sector can be specified by
three parameters: $f_{(1)}/v$, $\epsilon$, and either $m_{B'}$ or
$m_{\overline{A}'}$. Therefore, measuring four or more independent
observables can confirm the underlying structure. The observables in
question can be the masses of $A'$ and $Z'$ and their resonant dilepton
production rates, $\sigma_{\ell\ell,A'}$ and $\sigma_{\ell\ell, Z'}$.
This is a goal that could be pursued at the LHC alone. Both vectors
would appear as resonances in the dilepton spectrum. Then, measuring
both masses and any one event rate would completely specify the parameters of the model.
A measurement of the other rate then provides a test of the theory.

From Fig.~\ref{fig:futuresensitivityHLLHC}, we see that this is possible
at the LHC for the THPM model. Although some part of the parameter space
where both the $A'$ and the $Z'$ can be discovered is already ruled out by
the current bounds on the $Z'$, a sizable region remains. In the T2HDM,
however, this is not the case. Although the LHC has excellent
sensitivity to the $A'$, the regions where the $Z'$ can be discovered at
the HL-LHC are already ruled out by current bounds on the $A'$.

Before a FCC-hh machine begins taking data, it is very likely the tunnel
will be used for a lepton collider. The FCC-ee is projected to measure
some Higgs couplings to better than 1\%, an order of magnitude beyond
the HL-LHC~\cite{Dawson:2013bba}. In particular, the coupling between
the Higgs and $Z$ bosons may be measured to 0.3\% at 95\%
confidence~\cite{Gomez-Ceballos:2013zzn}. We therefore expect that
before the FCC-hh begins taking data, precise measurements of the Higgs
couplings will already have determined $v/f_{(1)}$ to high accuracy. In both the THPM model and the T2HDM, the hadron machine can then completely specify
the twin vector boson sector by measuring the mass and dilepton event
rate of just one of the vectors, either the $A'$ or the $Z'$. In each of these models, the mass
and dilepton event rate of the second vector are then predicted, and a
targeted search may be made to test these theories. From
Fig.~\ref{fig:futuresensitivityHLLHC} we see that in the THPM model,
there are regions of parameter space where both the $A'$ and $Z'$ can be
discovered at the FCC-hh. However, in the T2HDM the regions where the
$Z'$ can be discovered are already ruled out by the current LHC bounds
on the $A'$, so the model cannot be tested with this approach.

\section{Conclusions\label{s.Con}}

MTH models offer a simple solution to the little hierarchy problem
without introducing new states charged under the SM gauge groups. The
MTH framework predicts both a twin photon and a twin $Z$ boson. These
states can interact with the SM through kinetic mixing between the
hypercharge gauge boson of the SM and its mirror counterpart. If the
twin photon is massive, this mixing can be sizable without violating the
current experimental bounds. This portal can then be exploited by the
LHC and future colliders to discover the twin vector bosons.

We have determined the bounds on the $A'$ and $Z'$ vector boson masses
in a model in which the twin hypercharge gauge boson has a Proca mass,
and also in a twin version of the two-Higgs-doublet model. In most of
the parameter space, LHC searches for neutral gauge bosons constrain the
mixing parameter $\epsilon \lesssim 0.1$ for both the THPM and T2HDM.
The HL-LHC and a 100 TeV collider with 3000 $\text{fb}^{-1}$ of
luminosity can improve on the current bounds by a factor of a few in
most of the parameter space. In the THPM, in some regions of parameter space
the HL-LHC or a future 100 TeV machine can discover both the neutral
twin sector vector bosons. By measuring both the masses of these
particles and the event rates into dilepton final states these colliders
can test the THPM model. However, in the case of the T2HDM, the current
LHC bounds on the $A'$ already exclude the parameter space in which the
HL-LHC or FCC-hh would be expected to discover the $Z'$, and therefore we
cannot test the model with this approach.

\section*{Acknowledgements}

We are grateful to Aqeel Ahmed and Anson Hook for helpful discussions.
Z.C. is supported in part by the National Science Foundation under Grant
Number PHY-1620074. Z.C. would like to thank the Fermilab Theory Group for
hospitality during the completion of this work. Z.C.'s stay at Fermilab
was supported by the Fermilab Intensity Frontier Fellowship and by the
Visiting Scholars Award \#17-S-02 from the Universities Research
Association. The research of C.K. is supported by the National Science
Foundation Grant Number PHY-1620610. S.N. is supported by Vrije
Universiteit Brussel through the Strategic Research Program ``High
Energy Physics" and also supported by FWO under the EOS-be.h project n.
30820817. C.B.V. is supported by Department of Energy Grant Number
DE-SC-0009999.

redpajama_set_name: RedPajamaArXiv


AS IT STANDS: The Saga of the Shrinking Dollar: Remember When a 6-pack of Beer Was .99 cents?
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Written in clear and simple Spanish, this book introduces the grammatical and phonetical systems of Spanish by means of a contrastive analysis of Spanish and English. Presents a systems analysis to be used as a basis for error analysis in the classroom. Briefly outlines a model of language acquisition and discusses the relationship between linguistic theory and teaching methodology. Gives overview of articulatory phonetics and contrasts Spanish phonetics with English; develops the system of phonological rules of Spanish; and treats the formation of the verb tense-mood system. Considers morphology of nouns and adjectives, some illustrative derivational processes and morphophonemic processes, form and syntax of pronouns, semantics and the function of the lexicon in language teaching, verb tenses and their semantic functions, and subjunctive and indicative moods. Also offers suggestions for the study of language and culture. Specific theoretical models are avoided. Includes both mechanical exercises and topics for discussion.


The Fleurieu Peninsula was named by French explorer Nicolas Baudin in 1802, as he mapped the coast of what is now South Australia. He named the peninsula after another French explorer, Charles Pierre Claret de Fleurieu.
The west coast of the peninsula stretches from McLaren Vale down to Cape Jervis while the east coast runs from Cape Jervis up to Goolwa and includes the northern shores of Lake Alexandrina, the Langhorne Creek area and Strathalbyn. Hindmarsh Island at the mouth of the Murray is also included in the region.
The Rapid Bay Campground is a popular campground with excellent facilities and a large grassy area right on the beach. You can launch a boat, or fish from the beach or the new jetty.
Set on absolute beachfront, NRMA Victor Harbor Beachfront Holiday Park is the perfect base to explore South Australia's Fleurieu Peninsula.
The McLaren Vale Lakeside caravan park is nestled in the heart of McLaren Vale wine region and offers the perfect spot to relax, while being a convenient base for touring the Fleurieu Peninsula.
The Goolwa Camping & Tourist Park offers family friendly holiday accommodation at Goolwa, on South Australia's Fleurieu Peninsula. It is ideally positioned within walking distance to town, and makes an excellent base to enjoy the Fleurieu Peninsula.
Beachside Holiday Park is a full facility caravan park situated along the beach of Normanville, South Australia.
The Jetty Caravan Park is located on the Fleurieu Peninsula, South Australia, and just 75 km south of Adelaide's GPO.
Cruise the Coorong operates day cruises into the Coorong National Park from September through to May on Tuesdays, Fridays and Saturdays.
Cape Jervis Station offers powered and unpowered camping and caravan sites at competitive rates. There are breathtaking views of the ocean from almost anywhere that you set up camp.
Christies Beach Tourist Park is readily accessible from the city and right on the doorstep of the McLaren Vale wine region as well as the many other attractions of the Fleurieu Peninsula.
Moana Beach Tourist Park is only 10 minutes from the McLaren Vale wine region. We strive to be always friendly, tidy and green, and offer an ideal environment to relax with large grassed sites and sparkling clean amenities.
Nestled on one of South Australia's premier beachfront locations, the Port Elliot Holiday Park offers unrivalled picture postcard views from every angle.
Mundoo Boat Ramp issituated on the Mundoo Channel at Hindmarsh Island, providing boating access to the Coorong and the mouth of the Murray River. There is a jetty beside thr boat ramp.
Mount Compass Wetland is located at the far edge of the large gravel parking lot behind the Mount Compass shops. There are accessible toilets, a picnic table and a attractive boardwalk.
Briston-Smith Park is a day rest area at Goolwa South on the bank near the mouth of the Murray River. You can enjoy fishing and there is a concrete boat ramp here.
The Currency Creek Lions Park is a day rest area with some interesting history. Take a walk upstream from the park to the waterfall. Marvel at the Canoe Trees... hollowed out gum trees intended for the construction of canoes by Aboriginal people.
Alfred Langhorne Park is a pleasant picnic spot in the renowned wine growing centre of Langhorne Creek. There is a sheltered BBQ and picnic table in a pretty garden setting. An information board provides interesting reading about the history of the area.
Frank Potts Reserve is a nice shady spot with a short walk to the Langhorne Creek General Store and the Hotel, which serves good meals. This is an excellent rest area in the heart of a popular wine district. Plenty of shade and clean toilets.
Trig Campground is located in an open grassy area, this centrally located campground offers sheltered and shady sites with easy access to walking trails.
Victor Harbor Holiday and Cabin Park offers spacious Villas, Chalets, Cabins, Units, Onsite Vans and powered and non-powered sites. Our holiday park has accommodation options for all sized groups and families.
Second Valley Caravan Park is a well shaded park within easy stroll of the jetty, kiosk, safe sandy swimming beach, rock pools and historic boat sheds. Accommodation includes en-suite cabins, park cabins, on-site caravans, powered and un-powered sites.
Aldinga Beach Holiday Park is just 45 mins south of Adelaide and the airport. Close to local beaches of Aldinga and Port Willunga. Ideal stopover enroute to and from Kangaroo Island.
Wirrina Cove Caravan Park is nestled in between Normanville and Second Valley, shortly after the Hobart Memorial site at Lady Bay. The tree studded hideaway has many local furry residents, so please keep your speed down and your eyes open.
Spirit of the Coorong Cruises offer 2 hour, 4 hour, and 6 hour cruises from Goolwa, 90 kilometres south of Adelaide, South Australia, to the River Murray Mouth and into the Coorong National Park.
Woodcroft Park is only a short drive south of Adelaide, central to the magnificent south coast beaches and gateway to the McLaren Vale Wine region and the Fleurieu Peninsula.
Since early last century Newman's Horseradish has been available in South Australia. The Original Red Label Horseradish is one of South Australia's food icons.
The Strathalbyn Caravan Park is a small and peaceful park that caters for all range of travellers and their vehicles. The park is run by the Strathalbyn Oval Controlling Committee and exists to support participating sporting and community organisations.
Located in Victor Harbor, the iconic Adare property provides a unique mix of dormitory and caravan park accommodation and facilities. Owned and operated by the Uniting Church SA, the centre aims to provide exceptional Christian hospitality.
Mount Compass caravan Park is quietly situated off the main Victor Harbor highway at the hub of the Fleurieu Peninsular, within 20 minutes drive of all Surf, Swimming and Fishing beaches.
The peninsula is a fertile agricultural area, from the rich river loams to the sandy hills that gently roll across the green landscape and encompassing vineyards, olive groves, almond orchards, fruit and vegetable farms, sheep and cattle properties and many other forms of primary production.
The peninsula is also a happy hunting ground for tourists, with its surf beaches that are popular with Adelaide residents, its excellent fishing grounds both in the Murray River, the lakes and out into the Southern Ocean. It is also a haven for wildlife, particularly birds, with its wetland breeding grounds around the shores of the lakes.
There are some game reserves which attract both naturalists and hunters in season. With the advent of more natural flows being provided for in the Murray River, some areas which were in danger of becoming toxic swamps are now recovering to a healthier state and the birdlife is returning.
Cape Jervis at the tip of the peninsula is the embarkation point for the Kangaroo Island Ferry.
Hindmarsh Island near Goolwa is at the mouth of the Murray River and is a popular destination for both fishers and birdwatchers.


West Pier's charge for the Division One title gathered pace after they beat Whitby Fishermen 4-1.
Pier weren't at their fluent best, but managed to take the lead when neat link-up play down their right between Billy Logan and Chris Nelson led to Logan's cross being met by Sam Garnett at the back post.
It was all square at the break though, Fishermen grabbing a leveller after a scramble in the Pier box from a corner.
The crucial moment came when Fishermen spurned a glorious chance to take the lead, Pier hit them on the counter-attack and after winning a free-kick on the edge of the box, a Fishermen man was sent off after receiving his second yellow card.
Billy Logan's drilled free-kick was cleared out to Shaun Dodson, who fired into the top corner with a left-footed effort from the edge of the box.
Pier took charge from there on in and Martin Cooper grabbed a brace to hand them a 4-1 win against a Fishermens side who they meet in a huge North Riding County FA Saturday Challenge Cup semi-final next week.
Boss Andy Spivey said: "We weren't at our best but we managed to get over the line. We're expecting a tough game in the semi-final next week."
Nelson was Pier's man of the match.
Edgehill suffered a 3-2 home defeat against Hunmanby United as striker Cameron Dobson smashed in a superb match-winning hat-trick.
Danny White's foul in the box led to United's opener after 20 minutes, Dobson netting from the spot.
Dobson's second handed United a 2-0 lead at the break, but Edgehill improved after the break and were back in the game when Chris Duley spectacularly fired into his own net.
Star man Dobson completed his hat-trick with a neat finish into the bottom corner before Edgehill replied with a penalty of their own through Danny Collins after Lloyd Henderson was felled in the area.
United held on for the win though and Edgehill boss Steve Clegg conceded the visitors were good value for their win.
He said: "They deserved to win the game, they just about edged it and Dobson was by far the best player on the pitch."
Newlands' bid to retain their title received a boost with a 5-1 win at Westover Wasps.
Ben Luntley grabbed two and goals from Dan Freer, Liam Cooper and a Sean Exley pen rounded off the win for Shaun Mancrief's side.
Michael Wilson shone in the Park midfield while Jamie Gallagher also put in a decent performance.
Seamer were held to a 3-3 draw on the road at Sleights.
Sleights led 2-0 at half-time before two penalties brought Seamer level. Their first was converted by Luke Delve before Dan Bradbury fired in their second.
The home side looked to have won it late on, but Luke Purvis smashed in a last-gasp leveller to seal a point for Paddy Parke's side.


text:
Leeds Half Marathon 2022
The Leeds Half Marathon is Yorkshire's biggest and probably loudest event of its kind. Jane herself was proud to start the race back in 2003.
The route starts in The Headrow alongside Victoria Gardens, heads out of the city centre, takes in a section of the Ring Road and returns via Abbey Road and Kirkstall Road before finishing in Cookridge Street alongside Millennium Square.
If you would like to run for Leeds Hospitals Charity you can do so by booking a guaranteed charity place for £25.
By purchasing one of our charity places, you agree to pay a £25 registration fee and fundraise a minimum of £150.
Alternatively, if you already have a place in the half, you can still fundraise for us, simply by choosing to raise as much as you can in support of us.
Start: 8th May, 2022 at 9:00am
End: 8th May, 2022 at 4:00pm
Updated: 21st January, 2022
Location: Millennium Square, Leeds
Millennium Square, Leeds
Yorkshire Three Peaks
Follow a challenging route across the heart of the Yorkshire Dales to take on the county's three highest peaks: Pen-y-Ghent, Whernside and Ingleborough
Leeds Parks Challenge
The Leeds Park Challenge is BACK!! Complete 10 rounds of 5k in your local parks
Leeds 10K
New route for 2022! The Leeds Half Marathon is Yorkshire's biggest and probably loudest event of its kind. Jane herself was proud to start the race back in 2003.

redpajama_set_name: RedPajamaCommonCrawl


text:
c1900 Photograph of Rose Stone & Jane Jones, Presumably Rose E. (Doble) Stone & Jane Amanda Jones of Milo, Maine
c1910 photograph of two women identified on the reverse as Rose Stone and Jane Jones.
Because the photograph was found with a collection of photographs, many of which were related to the Milo, Maine, area, these women may have been teachers Rose Ethel (Doble) Stone and Jane Jones of Milo, Maine.
Rose married Charles W. Stone, Jr., in 1907. The photograph may have been taken after that date, or it may have been taken before Rose was married, with the identification provided at a later date.
See other posts that feature Jane Amanda Jones's relatives:
a photograph of Jane Jones' sister Martha (Jones) Prescott, wife of Edward P. Prescott of Milo, Maine. She's in a photograph of 10 women with only 3 of them identified. The post also shows a photograph of George Jones, presumably the father of Jane and Martha, and a photograph of Mildred Billings, which was inscribed to Jane Jones
Real Photo Postcard of Rachel Alice Prescott, daughter of Martha (Jones) Prescott)
Photograph of a child, presumably the child of Martha (Jones) Prescott
Photograph of Aunt Ethel Brown of Farmington, Maine - presumably Ethel L. or Ethelyn (Jones) Brown, sister of George A. Jones, father of Jane Amanda Jones - pictured below
From brief online research, hopefully correct and hopefully with the correct identities of the two women [corrections, additions and alternative identities requested]
Rose Ethel (Doble) Stone, hopefully the woman in the photograph, was born March 10, 1886 at Milo, Maine, the daughter of Benjamin Willis Doble and Lois Belle (Danforth) Doble. On June 11, 1907 at Milo, Maine, Rose married Charles Wesley Stone, Jr., son of Charles Wesley Stone and Frances Augusta (Cross) Stone. Charles, Jr., was born born February 11, 1880 and had graduated from the University of Maine with a degree in chemistry in 1903. At the time of their marriage, Rose was teaching and Charles was working as a chemist. Rose lived a long life and died in Virginia in 1980. Charles had predeceased her in Virginia in 1940.
Jane Amanda Jones, hopefully the other woman in the photograph, was born at Bucksport, Maine, on November 26, 1875, the daughter of railway contractors George A. Jones and Rachel (Lawrence) Jones. Interestingly, two other photographs were found in the collection containing the photograph above. One shows a man identified on the reverse as George Jones, who may have been Martha's father George A. Jones, who was born at Pugwash, Nova Scotia, on March 16, 1853.
Another photograph shows Mildred Billings who wrote on the reverse that her photograph was intended for Miss Jones, perhaps for Jane or her sister Martha. Both Jane and her sister Martha taught school, so either would have been a plausible "Miss Jones".
I don't believe Jane ever married. She died in 1959 and is buried in the Everegreen Cemetery at Milo, Maine, with family members.
If you have corrections and/or additions to the information above or if you have an alternate theory for either or both of the women, please leave a comment or contact me directly.
Early 1900s Photograph Mary, Florence & Areline an...
c1900 Photograph of Marjorie May Mills at Age Two
Vintage Candid Photo of Aunt Ethel Brown and Frien...
c1900 Photograph of Mary Stevens, Age 4; by the Ed...
1904 Graduation Photograph of Clarence Garland Gou...
1911 Photograph of Brothers Frederick Webber Dooe ...
1902 Photograph of Benjamin Chase at age 84, Possi...
1924 Snapshot of "Mother" at, presumably, Saint Fr...
Photograph of Two Young Women at Aroostook Falls i...
c1900 Photograph of Rose Stone & Jane Jones, Presu...
c1900 Photograph of Two Young Girls, Perhaps Daugh...
c1909 Photograph of Brothers Louis Edmond White an...
1926 Photograph of Emma Caroline (Kent) Perkins of...
Two Photographs from the Kittredge Family of Milo,...
c1900 Photo: Edie Lou Dinsmore; Elsie Thompson & M...
Early 1900s Photograph of a Child, Presumably a Ch...
c1920 Photograph of Zenas Morse and Herman Morse, ...
c1895 Photograph of 10 Women with 3 IDed: Cora Bra...
Two Early 1900s Photographs of Young Edgar Ward, P...
1919 Photo of Frank G. Thompson & Katherine (Hansc...
1865 Notice from I.O.O.F. Ancient Brothers Lodge a...


Apply for 2019 awards today!
The IEEE Region 1 Young Professional Award recognizes a young professional (within 15 years from first degree). 2019 IEEE Region 1 Young Professional Award Nomination Form [docx].
This award is designed to recognize those substantive projects or achievement of a relatively short nature, but which have left an undeniable imprint on the fabric of Young Professionals (YP) operations within Region 1. The award will be based on a selection that recognizes individuals involved with YP activities who are recognized for singular achievement in the development and completion of a project(s) or activity(ies) which are directed to the fulfillment of one or more of the Region's goals and/or objectives. Not to exceed one award annually.
The individuals nominated must be YP members at the time of nomination. The award must be approved within five years of the initiation of the project or event and the Region 1 Awards and Recognition Committee shall have the prerogative of carrying forward nominations to a more suitable time frame.
Complete and signed Region 1 Young Professionals Award Nomination Form.
All submissions must be made using this form. Any submissions not using the proper form or are not fully completed will not be reviewed.
Endorsed nominations by the Section Chair will then go to the Region 1 Awards and Recognition Committee for review and possible recommendation to the Region Assembly who may or may not approve the nomination. All deliberations are held in executive session and no information or feedback is to be expected during the process.
If there are any questions, please contact the Chair of the Region 1 Awards and Recognition Committee.
The Section Chair's signature is considered as an endorsement of the nominee from the Section. Please make sure the date of the application is entered. Email received from Section Chair will be considered as an electronic signature.
The Region 1 Awards and Recognition Committee will make a determination as to which nominee should receive the award and then will submit a formal recommendation to the Region Assembly. The Assembly will meet in executive session to review and vote to approve the Region 1 Young Professionals member who will receive the award.
An award plaque will be given to a Section representative for distribution at their respective annual award ceremony. The formal presentation of the award shall be made by the Region 1 Director or by his/her designee, at an appropriate time and location.


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Dispels the myths about Wicca, and gives step by step instructions on Witchcraft. Excellent for the beginner or intermediate practitioner. Explore a little bit of Pagan history. Where does the knowledge come from? Why was it repressed? Discover WHY and HOW Witchcraft works, and why anyone with an open mind can do it. Unlock the secrets of the astral plane and discover your true psychic potential.


text:
7 states sue EPA, seek wood boiler emissions rules
The coalition, which includes four New England states, wants the federal government to adopt updated emissions standards.
ALBANY, N.Y. — Seven states are suing the Environmental Protection Agency over health-damaging air pollution from outdoor wood-fired boilers that have become popular for residential heating.
The lawsuit filed Wednesday asks a federal court to order EPA to review and adopt updated emissions limits for the boilers. The coalition includes New York, Connecticut, Maryland, Massachusetts, Oregon, Rhode Island and Vermont.
New York Attorney General Eric Schneiderman says the EPA's existing emissions limits haven't been updated in 25 years and leave out popular types of residential wood heaters. Schneiderman cited EPA data saying emissions from wood-burning devices account for 13 percent of all soot pollution in the nation.
Soot is linked to public health problems including asthma, heart attacks and premature death.
An EPA spokeswoman says the agency is reviewing the lawsuit.


text:
Katjuša () je ruska vojna pesem o dekletu, hrepenečem po svojem ljubem, ki je v vojski. Katjuša je okrajšanka od imena Katarina (Екатерина).

Zložil jo je Matvej Blantner leta 1938, besedilo je napisal Mihail Izakovski. Prva jo je izvedla Lidija Ruslanova. Zmotno je mnenje, da je melodija Katjuše izvirno Blantnerjevo delo. Prvi jo je uporabil že Stravinski v operi Mavra (1922) in so jo kasneje priredili za Chanson Russe (1937).

Po pesmi Katjuša so nadeli vzdevek sovjetskemu večcevnemu raketnemu orožju BM-8, BM-13 in BM-31 Katjuša.

Besedilo

Pesmi leta 1938
Ruske vojaške pesmi

redpajama_set_name: RedPajamaWikipedia


Richard F. Thomas, Jan M. Ziolkowski (ed.), The Virgil Encyclopedia (3 vols.). Chichester; Malden, MA: Wiley-Blackwell, 2013. Pp. 1,632. ISBN 9781405154987. $595.00.
Some disclosures to start, and not merely for form's sake. As a potential contributor to this project who pleaded the burden of prior commitments, I find myself on the outside looking in on the work of friends, colleagues, and people I either know or wish I knew. My respect for their achievement is thus personal as well as professional. Second, as editor-in-chief of the fifth and entirely online edition of the Oxford Classical Dictionary now at the organizational stage, I am deeply immersed in a venture that inevitably overlaps this one and has compelled serious attention to the future of reference works in what purports to be a Digital Age. Third, this essay was written in western Oregon, where the nearest public copy of the Enciclopedia Virgiliana sits three hundred miles up the freeway. And doesn't circulate. In the end, the University of Arizona library kindly loaned me volume one, which has made possible the comparisons to come between that work (hereafter EV) and this one (hereafter VE) but limited them to entries falling between Abante and Dauno.
The present work, surely destined for many more shelves than the EV, offers much to like and to admire. Its creation was a substantial undertaking, and the result is a significant achievement. Over 350 contributors produced some 2,200 entries of varying length but uniformly high quality designed to provide Virgil's modern readers and prospective readers with, in the editors' words, 'background to his poems, but also a foreground to their reception' (lxv). Both 'background' and 'foreground' are broadly defined to cover a wide spectrum of contemporary needs and interests: simply running an eye over the entry list offers a valuable lesson in the extent and depth of Virgil's achievement and his influence in the western tradition. Thus the broad and rich field of reception is explored not only through entries on writers both obvious (e.g. "Eliot, T.S."; "Lewis, C. S."; "Milton, John") and less obvious (e.g. "Heaney, Seamus"; "Wordsworth, William", though not "Walcott, Derek"), but periods (e.g. "World War I", "World War II"), media (e.g. a long string of entries under the general rubric 'art'), and (coming closer to home) the scholars, commentators, and translators whose names, if not careers, have grown familiar to generations of students and teachers. Entries carry the story of reception up to the present in all but the last category, which for understandable reasons restricts itself to scholars ensconced on the far side of Styx. There are in addition forty-eight pages of illustrations tucked into the third volume that provide not only the more or less expected artistic representations but maps and tables, including a helpful guide to the principal Roman monuments (Map 8) and a remarkably clear précis of the Roman political system in late Republic and Principate (Chart 3). By casting so deliberately wide a net, the VE makes a significant statement not only about the Virgilian corpus and its history of interpretation, but perhaps less intentionally about the state of Virgilian studies in the early twenty-first century.
Though a contemporary encyclopedia aimed at the Anglophone world will necessarily differ significantly from its Italian predecessor, comparisons are inevitable and even useful. Indeed, the editors themselves invite them: the design for their project initially worked from a spreadsheet containing all the headwords from the EV (lxvi). The new emphasis on which they settled is obvious in the treatment of such distinctly Anglophone preoccupations as "Optimism and Pessimism" and "Winners and Losers". "Vietnam War" examines the effect and putative effects of that unhappy episode on American scholarship. The kind of trivia that is always so useful in the Latin classroom is well represented in entries like "dollar bill", "American" and "mottoes and seals of American states", which turn out to be replete with Virgilian echoes, while "schools and schooling" takes readers from the lectures of Caecilius Epirota to the American Advanced Placement Program. A serious effort has also been made to explain mysteries of scholarly form, e.g. "Virgil, spelling of". In addition, the project has been well conceived and well edited for ease of reference: what good is information if readers cannot find it? That task is facilitated here by a generous use of cross-references and blind references, together with a List of Entries set conspicuously at the front, and a General Index at the back.
With some notable exceptions (e.g. "commentaries"; "critical theory"; "Rome, topography and architecture of"; "Shakespeare, William"; "Virgil, portraits of") articles tend toward the succinct. Where EV dedicated twenty-four of its generous columns to aqua, "water" gets barely three much smaller ones in VE. Monuments (e.g. "Ara pacis", "Apollo Palatinus, temple of") are treated primarily as ideological symbols, not material objects. Stylistic features receiving significant attention in EV (e.g. five columns on allitterazione, four on amplificazione) are relegated to an appendix of 'Stylistic Terms,' though a few merit entries of their own (e.g. two columns for "archaism" to EV's eight). This truncation is understandable, perhaps even welcome, but behind it lie choices and priorities that raise an important question, since the editors claim to 'have set as our ambit everything of importance that enters into Virgil, that is in Virgil, and that comes from Virgil…' (lxv). What, we may then ask, is not of importance? Or more to the point, what is not as important as it used to be?
Latin is one such thing, and not only when it comes to 'stylistic terms'. The VE is not interested in lexical nuance: 'we refer interested readers to many dictionaries and online resources' (lxvi). Those anxious to distinguish among acer, acerbus, and acidus or aer, aether, and aethra must either identify the required dictionaries and online resources for themselves or find a copy of the EV, where the former set merits five columns and the latter six. An encyclopedia, or at least a print encyclopedia, must of course draw the line between inclusion and exclusion somewhere, but was this one drawn in just the right place? Nuances of Latin vocabulary are not as easily grasped as what even a Google search will quickly supply regarding "Accius" and "Alcuin", while a philological question that goes unanswered is all too likely in time to become a question that goes unasked. Other technical matters are not so fully ignored, but can be significantly compressed: details may then be difficult to locate and extract. Where EV foregrounded such matters as ablativo assoluto, accusativi plurali in –is, eīs ed es, and accusativo alla greca, VE relegates them to entries on "syntax" and "morphology", with other, briefer treatments in "Grecism" and "Hellenism, linguistic". These are, I hasten to add, very good and useful entries, but the compressed attention to linguistic form and structure is again indicative of a shift away from the tools of close reading and the basic philological information that modern readers increasingly require to read Virgil in Latin with a depth of understanding that the editors may too readily be taking for granted in their audience.
Bibliography is another matter often kept in the background. The representation of scholarship is erratic and inconsistent. This is again the consequence, perhaps unintended, of an editorial decision: 'we have not encouraged contributors to produce comprehensive bibliographies. Rather we have encouraged them to focus on providing the best means of access' (lxix). Many entries do this quite well, either by focusing a potentially enormous field on the specifically Virgilian context (e.g. "Bacchus"; "Bacchylides"), by guiding readers into largely unfamiliar terrain (e.g. "badges", "Caribbean colonial"; "ballad, Virgil in"), or simply by shrewdness and good sense ("critical theory"), but others are oddly inadequate. How does the solitary reference to an RE article provide Anglophone readers with 'the best means of access' to "Callimachus"? It will certainly do no such thing for readers curious about "Terence", who will find in the RE only the preoccupations of early twentieth-century scholars and Gunther Jachmann's faint damns of praise.
All such complaints are subject to one common rejoinder: there is not enough space for everything. Hard choices have to be made. Since Time's advance only adds to our store of information, the accommodation of new findings and new interests must inevitably come at the expense of old ones. That is an unanswerable argument. The problem is not with the editors who must work within those restrictions, but with a publisher that insists on serving the needs of twenty-first century readers with eighteenth-century technology. An inevitable result of its failure to invest in the future is nicely produced volumes like these, that will cost your obliging library some $92 per pound (or €153 per kilo) to put on a reference shelf, where readers diligent enough to come in and find them will have the opportunity to flip pages and juggle volumes in pursuit of its riches. They will be well rewarded for that effort, but the number of readers able and willing to do so is not growing. Students in particular have already found an inviting and increasingly popular alternative, though the VE's editors are not kind to it: 'a printed encyclopedia of this sort is also a world apart from the web. It offers material that does not have to be unearthed by sorting through a dung-heap in which pearls of truth are buried amid mistakes, exaggerations, and misunderstandings. The volumes have been vetted and edited for accuracy and clarity' (lxx). I share their dedication to vetting and editing. Virgil may have had to search for pearls in a dung-heap, but his modern readers should certainly be spared that experience. Yet the editors invoke what is, or ought to be, a false dichotomy: an encyclopedia of this sort is a world apart from the web largely because no effort has been made to unite the accuracy and clarity of the former with the flexibility and accessibility of the latter. It is clearly not (or not yet) in Wiley-Blackwell's interest to do so, but it is most certainly in our interest to have it done, and at some point the editors and contributors to projects like this one are going to have to demand that their labor, their expertise, and their sheer love of the enterprise be given a more progressive format. Their own dedication to the field, so richly displayed in these volumes, deserves nothing less.


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ForbesOpinion
The English Language Is Here to Stay
Lee Kuan Yew
Contributor Group
Jun 18, 2014, 06:00am EDT |
This story appears in the June 29, 2014 issue of Forbes. Subscribe
That English has become a dominant world language is a consequence of history. The British Empire spread English to many parts of the globe, first through coloniz
ation in the 17th and 18th centuries and then through its leadership in the Industrial Revolution. The U.S.' subsequent economic superiority and political leadership secured English as the first or second language of many countries today. For countries to shift to any other major language, such as French or Spanish, would be extremely complicated and problematic.
I can see no circumstances under which Mandarin (China's official language) would displace English as the world's most important language. Although Mandarin has the largest number of native speakers, virtually all are in China. The Chinese historically weren't colonizers, so there are no far-flung former Chinese colonies speaking Mandarin today, the way the former colonies of England, France and Spain still speak the languages of those countries.
In any event, Mandarin would be a very difficult language for the rest of the world to learn and master. Even if you put Chinese words into pinyin form (roman characters), there are four tones to each character (often monosyllabic) that clarify the meaning. Regardless, I don't see the Chinese discarding their Mandarin characters and converting entirely to pinyin, as they are proud of their language, which has survived more than 5,000 years.
Chinese totally differs from most other languages in use today, as it consists of pictographs and ideographs, without any spelling or symbols to indicate which of the four tones for each character is intended. There isn't much bilingualism among people from different regions in China because the dialects are so different. Bilingualism among Europeans is much more common because so many of Europe's languages have Latin roots.
A small bicultural elite has formed in China and Singapore made up of people who are fluent in their mother tongue and have acquired, to varying degrees, fluency in English. This has happened in large part to facilitate trade and business. However, it's difficult to find people who are equally competent in both languages.
China's announcement that it has placed an air defense identification zone (ADIZ) over most of the East China Sea is the result of its need to define its immediate regional environment. This ADIZ is controversial partly because when it was first promulgated it established additional requirements beyond those normally imposed. It also covers contested maritime areas, including the Senkaku/Diaoyu Islands, and overlaps with an ADIZ that Japan had already established. This has created the potential for conflicting instructions being issued from different control agencies when commercial aircraft fly through the overlapping zones. Fundamentally, ADIZs can't impinge on freedom of navigation across all oceans and air space—an accepted rule of international law and one that's in the interest of an increasingly interconnected world. I don't see that changing. However, I can't see a major power like China ever subordinating itself to decisions made by the International Court of Justice, even though arbitration or adjudication would be preferable to armed conflict should bilateral negotiations fail.
With China's population of 1.3 billion—versus the U.S.' 300 million—and its growing economy, it's inevitable that China's GDP will eclipse America's during this century. With such a shift in power there could be changes in where the dividing line of the U.S.' influence in the Pacific Ocean falls, somewhere between Guam and Hawaii.
The U.S. has many friends and allies in Asia, but it must keep in mind that China's interests and influence in the region are growing. China is the largest trading partner of many Asian countries that also want to remain friends with the U.S. As its companies internationalize, China is also investing abroad, because it needs access to minerals and other natural resources. China has also declared its intention to become a maritime power. Therefore, it is crucial that the U.S. strengthen its ties with Asia. The Obama Administration's "rebalancing" toward Asia is a step in the right direction, but it must be sustained over the long term.