text:
\section{Introduction}
Although the literature on dark matter models is vast and diverse, the organizational structure of many of these models
is similar.   The visible sector includes all the fields normally associated with the minimal standard model;  the dark sector
consists of a collection of fields that communicate very weakly with the visible sector; the messenger or portal sector
consists of those fields that allow for a weak coupling between the visible and dark sectors.   In this paper, we are
interested in a possible portal for non-Abelian dark matter models, specifically ones in which fermionic dark matter is
charged under a non-Abelian dark gauge group.    Examples of non-Abelian dark matter models can be found in
Refs.~\cite{Choquette:2015mca,Cheung:2014tha,Cline:2014kaa,Cline:2010kv,Carone:2010ha,Chen:2009ab,Zhang:2009dd}
and~\cite{Davoudiasl:2013jma,Chiang:2013kqa},  though we will not focus on models like those in
Refs.~\cite{Davoudiasl:2013jma,Chiang:2013kqa} where the dark gauge boson is itself the dark matter.   We are interested
here in a mechanism that allows the dark gauge boson to develop a small coupling to the visible sector, adequate enough to
facilitate the annihilation of the dark matter for a successful thermal freeze-out, without running afoul of direct detection bounds.
We focus on a model in which the effective coupling between the dark and visible sectors does not appear at the same order in
the dark matter annihilation and the dark-matter-nucleon elastic scattering cross sections.

One approach is to arrange for couplings between the non-Abelian dark gauge bosons and standard model leptons, but
not quarks; in this case, dark matter annihilation would proceed via tree-level diagrams, while diagrams involving quarks
would be higher-order.  However, this simple approach can lead to significant model building complications.   For
example,  if one tries to couple the dark gauge bosons to the standard model leptons directly, then the dark gauge
bosons are potentially no longer ``dark,'' unless their gauge coupling is taken to be very small.   However, this choice
suppresses the coupling of the dark gauge bosons to both the dark and visible sectors, making it ineffective as a
channel for dark matter annihilation.  Moreover, such direct couplings lead generically to chiral
anomalies, which must be cancelled by additional states that are charged under both the dark and standard model gauge
groups.    There is no guarantee that the simplest Higgs field content of the dark and visible sectors will have the correct
quantum numbers to provide Yukawa couplings for these additional states, so that
additional Higgs representations may be required.   Another potential problem is that charging standard model leptons
under the new non-Abelian group may either restrict the form of the standard model lepton Yukawa matrices in unwanted
ways, or forbid them entirely, unless a Higgs field charged under both the dark and standard model gauge groups is introduced.
While the proliferation of fields implied by these considerations does not rise to the level of a no-go theorem, it does make the approach described a lot less appealing.

Fortunately, there is a simple way to avoid the complications described above:  the non-Abelian dark gauge boson may
couple to a vector-like state $\chi$ that can mix with standard model leptons {\em after} the gauge symmetries of the
theory (both dark and visible) are spontaneously broken.  We will refer to the $\chi$ states as heavy, vector-like leptons.
If the dark gauge boson's coupling to dark matter is $g_D$, which may be substantial, then the induced coupling to the
standard model lepton in the mass eigenstate basis will be proportional to $\theta g_D$ where $\theta$ is a small mixing angle. Since
the gauge boson couples directly to a vector-like state, anomalies are cancelled, and a mass term
$-M_\chi \overline{\chi} \, \chi$ can be written down at tree-level.  The range of $M_\chi$ is then determined by
phenomenological requirements, for example, that the mixing angle $\theta$ is large enough to produce the desired value
of the dark matter relic density via dark matter annihilation to a standard model lepton-anti-lepton pair.  In this paper,
we will present an explicit and renormalizable model that illustrates this proposal.  Our focus differs from that of
Refs.~\cite{Choquette:2015mca,Cheung:2014tha,Cline:2014kaa,Cline:2010kv,Carone:2010ha,Chen:2009ab,Zhang:2009dd}, where
the origin of higher-dimension operators connecting the dark and visible sectors was either unspecified, or assumed to arise from a
sector whose flavor structure and phenomenology was not explicitly investigated.

To obtain a sufficient dark matter annihilation cross section to standard model particles, the mixing angle $\theta$ cannot
be too small; this implies that the heavy, vector-like leptons $\chi$ cannot be arbitrarily heavy.   As we will see later, a
$\chi$ mass of order $100$~TeV would not naturally lead to the desired annihilation cross section.  At face value, this seems
problematic, since the literature includes bounds on vector-like heavy leptons that exceed $100$~TeV~\cite{Ishiwata:2013gma}.  Such
stringent bounds, however, come from consideration of lepton-flavor-violating processes that emerge when the vector-like
states mix with all three standard model lepton flavors.   One expects such mixing to be present generically, and this
would doom the approach that we have just outlined.   In this paper, we show how a more favorable outcome can be
achieved via discrete symmetries that allow us to suppress the unwanted mass mixing arbitrarily.  In our model,
vector-like leptons mix only with a single flavor of the standard model leptons, which in turn
does not mix substantially with the remaining two flavors, thus avoiding problems with lepton flavor violation.   We refer to
this restricted pattern of mass mixing as ``flavor sequestering."  We will show that the discrete symmetry used to achieve this result does
not adversely affect the remaining flavor structure of the charged leptons or neutrino mass matrices.
Phenomenological considerations place constraints on the mass spectrum of the flavor-sequestered vector-like lepton
states that can be tested in direct collider searches.

Our paper is organized as follows.   In the next section, we define the simplest model that illustrates a portal involving
vector-like leptons and flavor sequestering.  In Sec.~3, we show how the flavor structure of the theory can be achieved using
a discrete symmetry, so that exclusive mixing with one standard model lepton generation is obtained and lepton-flavor-
violating effects avoided.  In Sec.~4 we discuss phenomenological constraints on the model parameters, including the
requirement that the correct dark matter relic density is obtained, and the constraints from dark matter-nucleon elastic
scattering, which follows from kinetic mixing that is induced after the non-Abelian gauge group is spontaneously broken.
In the final section, we summarize our conclusions.

\section{The Model} \label{sec:model}

We consider the simplest non-Abelian dark gauge group, SU(2)$_D$.   As stated earlier, we denote the heavy, vector-like leptons
$\chi$, and assume the quantum numbers
\begin{equation}
\chi_L \sim \chi_R \sim ({\bf 2},{\bf 1},{\bf 1},-1) \,\,\, ,
\end{equation}
where we indicate the representations of SU(2)$_D \times$ SU(3)$_C\times$ SU(2)$_W\times$U(1)$_Y$, in that order.  In other words, these states
are SU(2)$_D$ doublets, but have the same electroweak charges as right-handed leptons.   We further assume the simplest assignment for the
dark matter, {\em i.e.}, that it is a doublet under SU(2)$_D$.  However, to avoid a Witten anomaly~\cite{Witten:1982fp} there must be an even number of
SU(2) fermion doublets,
so we take
\begin{equation}
\psi_L \sim \psi_R \sim ({\bf 2},{\bf 1},{\bf 1},0)\,\,\, .
\end{equation}
Since the $\psi$ fields are charged only under SU(2)$_D$, we can construct Dirac or Majorana mass terms, or both.  We will assume
Dirac mass terms, for simplicity, though it is easy to make this the only possibility by imposing additional discrete symmetries.  For
example, an unbroken $Z_3$ symmetry can forbid Majorana masses for $\psi$, and also serve as the symmetry which stabilizes the dark
matter, which we identify henceforth as the lightest component of the $\psi$ doublet.

We assume that the dark gauge symmetry is spontaneously broken by two SU(2)$_D$ Higgs field representations,
\begin{equation}
H_D \sim ({\bf 2},{\bf 1},{\bf 1},0) \,\,\,\,\, \mbox{  and  } \,\,\,\,\, H_T \sim ({\bf 3},{\bf 1},{\bf 1},0) \,\,\,\, .
\end{equation}
We show at the end of this section that the Higgs potential has local minima consistent with the pattern of vacuum expectation values (vevs):
\begin{equation}
\langle H_D \rangle = \left(\begin{array}{c} v_{D1} \\ v_{D2} \end{array}\right) \,\,\,\,\, \mbox{ and } \,\,\,\,\, \langle H_T \rangle = \left(\begin{array}{cc}
v_T/2 & 0 \\ 0 & -v_T/2 \end{array}\right) \,\,\, . \label{eq:hdhtv}
\end{equation}
If we decompose $H_T = H_T^a \,(\sigma^a/2)$, where the $\sigma^a$ are Pauli matrices, then the $H_T$ vev above corresponds to $\langle H_T^3 \rangle = v_T$ and
$\langle H_T^a \rangle = 0$ for $a=1, 2$.   In fact, an arbitrary vev for $H_T$ can always be rotated into the $H_T^3$ direction by an SU(2)$_D$ transformation.
With this choice, vevs in both components of $H_D$ are expected, and one of those can be made real by a further SU(2)$_D$ phase rotation.   The fact that the remaining
$H_D$ vev in Eq.~(\ref{eq:hdhtv}) is assumed real will be shown to be consistent with the minimization of a potential later.

We can now say something more concrete about the mass spectrum of the model.  The relevant Lagrangian terms are ${\cal L} \supset {\cal L}_\psi + {\cal L}_{\chi e}$, where
\begin{equation}
{\cal L}_\psi = - M_\psi\, \overline{\psi}_L\, \psi_R + \lambda_s \, \overline{\psi}_L\, H_T \,\psi_R + \mbox{ h.c. }  \,\,\,\, , \label{eq:lpsi}
\end{equation}
and
\begin{equation}
{\cal L}_{\chi e} = - M_\chi\, \overline{\chi}_L\, \chi_R + \lambda_s' \, \overline{\chi}_L\, H_T \,\chi_R - y_1 \, \overline{\chi}_L \,H_D\, e_R
- y_2  \, \overline{\chi}_L \,\widetilde{H}_D\, e_R- y_e \,  \overline{L}_L\, H \,e_R + \mbox{ h.c. }   \,\,\, ,  \label{eq:lchi}
\end{equation}
where $\widetilde{H}_D \equiv i \sigma^2 H_D^*$, and the final term is the usual standard model Yukawa coupling for a single lepton flavor.  Eq.~(\ref{eq:lchi}) assumes
the existence of a symmetry that leads to exclusive mixing between any one standard model, right-handed charged lepton flavor (called $e_R$ above) and the
vector-like $\chi$ fields.  We show how this flavor sequestering can be arranged by a discrete symmetry in Sec.~\ref{sec:zn}.    The first terms in Eqs.~(\ref{eq:lpsi}) and (\ref{eq:lchi}) provide a common mass for each component of the given doublet,
while the second terms lead to mass splittings proportional to the vev $v_T$. The third
and fourth terms in Eq.~(\ref{eq:lchi}) allow mixing between the standard model lepton $e_R$ and the $\chi$ fields, since the coupling
to the dark doublet Higgs field $H_D$ allows for the formation of an SU(2)$_D$ singlet.  The final term leads to an $e$ mass when the standard model Higgs field develops a
vacuum expectation value $\langle H \rangle = (0 , v/\sqrt{2})$, with $v = 246$~GeV.  Defining the column vector  $\Upsilon \equiv (e, \,\chi^{(1)},\, \chi^{(2)})^T$, which displays the two components of the $\chi$ doublet,  we may write the mass matrix that is produced after spontaneous symmetry breaking by
\begin{equation}
{\cal L}_{mass}^{\chi e} = - \overline{\Upsilon}_L  M \,\Upsilon_R + \mbox{ h.c. }  \,\,\, ,
\end{equation}
where
\begin{equation}
M = \left(\begin{array}{ccc}  \frac{h_e v}{\sqrt{2}}  & 0 & 0 \\
\frac{(y_1 v_{1D} + y_2 v_{2D})}{\sqrt{2}} & M_\chi - \frac{\lambda_s'v_T}{2} & 0 \\
\frac{(y_1 v_{2D} - y_2 v_{1D})}{\sqrt{2} }   & 0 &  M_\chi + \frac{\lambda_s'v_T}{2}  \end{array}\right) \equiv \left(\begin{array}{ccc} m_0 & 0 & 0 \\ m_1 & M_1& 0 \\ m_2 & 0 & M_2
\end{array}\right) \,\,\, , \label{eq:bigM}
\end{equation}
where the second form is a convenient parametrization.  This matrix can be diagonalized by a bi-unitary transformation,  $M = U_L \, M^{diag} \, U_R^\dagger$.   While this diagonalization can be done numerically, there are certain limits that are relevant to us in which simple
results can be obtained.   In particular, when  $M_1$,  $M_2$ $>>$  $m_1$, $m_2$ $>>m_0$, we find that the largest mixing angles, which occur in $U_R$, are given by
\begin{equation}
U_R = \left(\begin{array}{ccc}  1 -\frac{1}{2}\left(\frac{m_1^2}{M_1^2}+\frac{m_2^2}{M_2^2}\right)
& m_1/M_1 & m_2/M_2 \\  -m_1/M_1 & 1-\frac{1}{2} \frac{m_1^2}{M_1^2}  & -\frac{M_1}{M_2}\frac{m_1 m_2}{M_1^2-M_2^2} \\ -m_2/M_2
& \frac{M_2}{M_1}\frac{m_1 m_2}{M_1^2-M_2^2} & 1-\frac{1}{2} \frac{m_2^2}{M_2^2} \end{array}\right) + \cdots \,\,\, , \label{eq:ur}
\end{equation}
where the $\cdots$ represent terms that are cubic order or higher in $m_i/M_j$.   For this case, we can now find the leading coupling of the dark gauge fields $A_{D\mu}^a$ to the mass eigenstate fields.  In the gauge basis, the coupling to $\Upsilon_R$ can be written
\begin{equation}
{\cal L} = i \overline{\Upsilon}_R  \gamma^\mu (\partial_\mu - i g_D A^a_{D\mu} {\cal T}^a) \Upsilon_R + \cdots  \,\,\, ,
\end{equation}
where
\begin{equation}
{\cal T}^a = \left(\begin{array}{c|c} 0 & 0\\  \hline
0 & \, T^a \end{array} \right)  \,\,\, ,
\end{equation}
and $T^a = \sigma^a/2$, $a=1, \ldots, 3$,  are the generators of SU(2).  The zero in the $1$-$1$ element reflects the fact that the standard model lepton is not charged
under the dark gauge group.  In the mass eigenstate basis, the couplings of the $a^{th}$ dark gauge boson are therefore proportional to $U_R^\dagger {\cal T}^a U_R$.  In the same approximation as Eq.~(\ref{eq:ur}), these matrices are given by
\begin{eqnarray}
&&U_R^\dagger {\cal T}^a U_R = \nonumber \\
  &&\left[ \left(\begin{array}{ccc} \frac{m_1 \, m_2}{M_1 \, M_2} & -\frac{m_2}{2 \, M_2} & -\frac{m_1}{2\, M_1} \\
 -\frac{m_2}{2 \, M_2} & 0 & \frac{1}{2} \\
 -\frac{m_1}{2\, M_1} & \frac{1}{2} & 0 \end{array} \right) \,\,\, ,\,\,\,
 \left(\begin{array}{ccc} 0 & -\frac{i \, m_2}{2 \, M_2} & \frac{i \, m_1}{2\, M_1} \\
 \frac{i \, m_2}{2 \, M_2} & 0 & -\frac{i}{2} \\
 -\frac{i\, m_1}{2\, M_1} & \frac{i}{2} & 0 \end{array} \right)  \,\,\, , \,\,\,
 \left(\begin{array}{ccc} \frac{m_1^2}{2\, M_1^2} - \frac{m_2^2}{2\, M_2^2} & -\frac{m_1}{2 \, M_1} & \frac{m_2}{2\, M_2} \\
 -\frac{m_1}{2 \, M_1} & \frac{1}{2} & 0 \\
 \frac{m_2}{2\, M_2} & 0 & -\frac{1}{2} \end{array} \right) \right] \,\,\,, \nonumber \\ \label{eq:utu}
 \end{eqnarray}
where we only show results to linear order in $m_i/M_j$, with the exception of the $1$-$1$ entries, because of their relevance to our subsequent
discussion.  For example, for the lightest dark gauge boson, $A^3_D$, the coupling to  $e^+\, e^-$ is given by
\begin{equation}
g_D \overline{\Upsilon}_R  \gamma^\mu  A^3_{D\mu} (U_R^\dagger{\cal T}^3 U_R) \Upsilon_R  = \frac{g_D}{2} \left(\frac{m_1^2}{M_1^2}-\frac{m_2^2}{M_2^2} \right) \overline{e}_R \gamma^\mu  A^3_{D\mu} e_R + \cdots
\end{equation}
which provides the $A_D^3$ gauge boson with a decay channel (since we assume its mass is greater that $2 \,m_e$) and provides the dominant portal for dark matter
annihilation into standard model particles.   For later convenience, we define
\begin{equation}
\theta^2 \equiv g_D\left(\frac{m_1^2}{M_1^2}-\frac{m_2^2}{M_2^2} \right) \,\,\, . \label{eq:ourtheta}
\end{equation}
We illustrate the qualitative idea in Fig.~\ref{fig:one}  that the dark matter annihilation process of interest emerges from mixing that affects
two of the external legs.
\begin{figure}[t]
  \begin{center}
    \includegraphics[width=0.3\textwidth]{annihilation.pdf}
    \caption{ Qualitative picture of dark matter annihilation to a charged lepton-anti-lepton pair, as discussed in the text. The insertions labelled by $\theta$ represent mass mixing. \label{fig:one}}
      \end{center}
\end{figure}

We note that in the case where $m_0$ is comparable to $m_1$ and $m_2$ we find via numerical diagonalization that our expression $U_R$
in Eq.~(\ref{eq:ur}) still provides an accurate approximation.  Moreover, we can prove that $m_0$ appears only as a higher-order correction to
$\theta$, as defined in Eq.~(\ref{eq:ourtheta}), the quantity that is most relevant to our phenomenological discussion later.   The argument is as follows:  if $m_1$ or $m_2$ where to vanish, then $U_R$ must become the identity.  This implies that any corrections to the
$1$-$2$, $1$-$3$, $2$-$1$ and $3$-$1$ entries of $U_R$ that are proportional to $m_0$ must come at no lower order than $m_0 m_{1,2} / M_{1,2}^2$. This potential contribution is nonetheless higher-order than the values  shown for these entries in Eq.~(\ref{eq:ur}).  It is also the case that the $1$-$1$ entry of $U_R^\dagger {\cal T}^3 U_R$, from which $\theta$ is extracted, depends only on these four entries.  Hence,
the value of $\theta$, which controls the induced coupling of $A^3_D$ to the chosen standard model lepton flavor, remains unaffected at leading order.

Eq.~(\ref{eq:utu}) indicates that all states other than the lightest $\psi$ mass eigenstate have available decay channels that ultimately lead to standard model particles.  Let us give
some examples to establish this point, assuming the dark sector mass scale is significantly larger than the electron mass scale (we assume $>2$~GeV in Sec.~\ref{sec:pheno}), and that the
dark gauge bosons are heavier than the dark matter fermion.   The coupling matrices $U_R^\dagger {\cal T}^a U_R$, for $a=1$ and $a=3$ allow decays of $A^1_D$ and $A_D^3$ directly
to $e^+\, e^-$; the same is not true for $a=2$, but the $A^2_D$ boson does couple to the two different $\psi$ mass eigenstates, which we will call $\psi^{(1)}$ (the lighter, dark matter component)
and $\psi^{(2)}$ (the heavier).  The eigenstate $\psi^{(2)}$ can decay to dark matter $\psi^{(1)}$ plus $e^+ \, e^-$ via $A^1_D$ exchange.   Hence $A^2_D$ can decay to two dark matter particles
and an $e^+ \, e^-$ pair, whether or not $\psi^{(1)}$ is on shell.  Due to the $\chi A_D^{3} e$ couplings in $U_R^\dagger {\cal T}^3 U_R$,  both $\chi$ mass eigenstates can decay to a
same-sign $e$ plus an $e^+\, e^-$ pair via $A_D^3$ exchange. Finally, the exotic Higgs fields $H_D$ and $H_T$ couple to fermion pairs via their Yukawa couplings.  Since we have already established that those fermions couple ultimately to either $e$'s or $\psi^{(1)}$'s, our claim is established.

Since the $\chi$ and $e_R$ have identical electroweak quantum numbers, there is no effect on the coupling of the $Z$ boson to $e_R$ in the mass eigenstate basis.  However, $\chi$
and $e_L$ couple differently to the electroweak gauge bosons, and diagonalization of Eq.~(\ref{eq:bigM}) also involves a left-handed rotation matrix $U_L$ which differs from the identity.
Fortunately, the left-handed mixing angles are much smaller than those in Eq.~(\ref{eq:ur}) so that this does not present any phenomenological difficulties.  For example, the fractional shift in
the standard model $Z e_L e_L$ vertex is of ${\cal O}(\frac{m_0 m_1}{M_1^2}\frac{m_0 m_2}{M_2^2})$, which is negligible given the spectrum we assume in Sec.~\ref{sec:pheno}.  We also may take the
mostly $\chi$ mass eigenstates to be heavy enough so that rare $Z$ decays to $\chi \, e$ are kinematically forbidden.

Finally, let us return to the issue of the spontaneous breaking of the dark gauge symmetry.  In the effective theory well below the electroweak scale, the most general renormalizable potential
involving the dark Higgs fields is given by
\begin{equation}
\begin{split}
\label{eq:higgspotential}
V(H_D, H_T) &= -m_D^2 H_D^\dagger H_D -m_T^2 \tr(H_T H_T)
+\lambda_1(H_D^\dagger H_D)^2
+ \lambda_2 \big[\tr(H_T H_T)\big]^2
\\ &+ \lambda_3 H_D^\dagger H_T H_T H_D
+\mu_1 H_D^\dagger H_T H_D+ \left(\mu_2 H_D^\dagger H_T \widetilde H_D + \text{h.c.}\right),
\end{split}
\end{equation}
where we have used the fact that $H_T^\dagger = H_T.$ We assume the potential does not violate CP, so that all the couplings are real. Further, we require at least one of
($-m_D^2$, $-m_T^2$) to be negative so that the $H_D$ and $H_T$ fields may develop non-zero vevs. It should be noted that there are other terms involving the Higgs fields that could be added to the potential, such as $\widetilde H_D^\dagger \widetilde H_D, \, \tr( H_T^4), \,
\widetilde H_D^\dagger H_T \widetilde H_D, \, H_D^\dagger H_D \tr(H_T H_T)$, but these are not linearly independent of the terms included in Eq.~(\ref{eq:higgspotential}) and so have been omitted.

The Higgs doublet assumes the standard real-field parametrization,
\begin{equation}
\label{eq:hd}
H_D = \frac{1}{\sqrt2}\begin{pmatrix} \phi_1 + i\phi_2 \\ \phi_3 + i\phi_4\end{pmatrix},
\end{equation}
while the Higgs triplet can be represented by a $2\x2$ matrix of real fields $H_1, H_2$ and $H_3$,
\begin{equation}
\label{eq:ht}
H_T = H^a \frac{\sigma^a}{2} = \frac{1}{2}\begin{pmatrix} H_3 & H_1 - iH_2 \\ H_1+iH_2 & -H_3 \end{pmatrix}.
\end{equation}
The normalization assures canonical kinetic terms. We proceed to show that there exists a stable, local minimum of the potential for the pattern of vacuum expectation values described in Eq.~(\ref{eq:hdhtv}). One approach to studying the potential is to fix all the parameters and search for minima, using standard steepest descent algorithms. However the downside to this approach is that one may then have to repeatedly discard local minima that do not provide the pattern of vevs desired for the model. So instead, we will fix the vevs and work backwards, showing that an extremum exists that is also a local minimum for a fixed set of parameters.

The extremization of Eq.~(\ref{eq:higgspotential}) with the fields set to the vevs shown in Eq.~(\ref{eq:hdhtv})
provides the following nontrivial, linearly independent constraints:
\begin{equation}
\begin{split}
-m_D^2 v_{D1}+\lambda_1 v_{D1}^3+\lambda_1 v_{D1} v_{D2}^2+\mu_2 v_{D2} v_T+\frac{1}{4} v_{D1} v_T (\lambda_3 v_T+2 \mu_1) &= 0
\\ \frac{1}{2} \left(-\mu_2 v_{D1}^2+\mu_1 v_{D1} v_{D2}+\mu_2 v_{D2}^2\right) &= 0
\\ -m_T^2 v_T+\frac{1}{4} \lambda_3 v_T \left(v_{D1}^2+v_{D2}^2\right)+\frac{1}{4} \mu_1 (v_{D1}^2-v_{D2}^2) +\mu_2 v_{D1} v_{D2}+\lambda_2 v_T^3 &= 0.
\end{split}
\end{equation}
For the purpose of numerical evaluation we work here in units where $\mu_1=1$. For fixed choices of the vevs and the couplings
$\lambda_{1,2,3}$, we may then determine $m_D, \, m_T$ and $\mu_2$.  To determine whether the extremum is a minimum, maximum
or saddle point, we need to examine the eigenvalues of the mass squared matrix (the second derivative matrix with all fields set to their
vevs and with the solutions for $m_D$, $m_T$ and $\mu_2$ corresponding to the extremum).  Since $SU(2)_D$ is spontaneously broken
to nothing, we expect three Goldstone bosons, one for each broken SU(2) generator, according to Goldstone's theorem.  Thus we would
expect three of the eigenvalues to be zero, corresponding to the massless degrees of freedom that are ``eaten" by the dark gauge bosons.
The remaining eigenvalues must be positive for the extremum to be a local minimum. For example, let us set
$v_T=v_{D1} = v_{D2}/2 = \lambda_{1,2,3}= \mu_1$ (here we require $v_{D1} \neq v_{D2}$ for a solution to exist). Then we find $m_D^2 = 53/12, \, m_T^2 = 1/6$ and $\mu_2 = -2/3$. The corresponding mass squared eigenvalues are
$\{0, \, 0, \, 0, \, 3.75, \, 3.75, \, 4, \, 10\},$ in units of $\mu_1^2$, thus confirming that we are at a local minimum of the potential.   This provides an existence proof that local minima exist in which the pattern of vevs shown in Eq.~(\ref{eq:hdhtv}) is obtained.  It is not difficult to find similar solutions for other choices of $v_{D1}$, $v_{D2}$ and $v_T$.

The SU(2)$_D$ breaking vevs affect the $\chi$-$e$ mass spectrum via Eq.~(\ref{eq:bigM}); the triplet vev also splits the $\psi$ mass
eigenstates
\begin{equation}
m_{\psi^{(1)}} = M_\psi - \frac{1}{2}\lambda_s v_T, \quad m_{\psi^{(2)}} = M_\psi + \frac{1}{2}\lambda_s v_T
\end{equation}
for
$\psi_{L,R} = \left( \psi^{(1)}, \; \psi^{(2)}\right)_{L,R}^T$.  The gauge field spectrum is obtained from the kinetic terms for $H_D$ and $H_T$,
\begin{equation}
 \mathcal L_{kin}(H_D, H_T) = (D_\mu H_D)^\dagger (D^\mu H_D) + \tr \big[ (D_\mu H_T)^\dagger (D^\mu H_T)\big],
\end{equation}
where $D_\mu H_D = \partial_\mu H_D - ig_D A_{D\mu}^a \tfrac{\sigma^a}{2} H_D$
and $D_\mu H_T = \partial_\mu H_T - i g_D \tfrac{\sigma^a}{2} A_{D\mu}^a H_T + ig_D A_{D\mu}^a H_T \tfrac{\sigma^a}{2}$. Following symmetry breaking the gauge bosons develop masses
\begin{equation}
 m_{A_D^1}^2 = m_{A_D^2}^2 = \frac{g_D^2}{4}(v_{D1}^2+v_{D_2}^2+4\, v_T^2), \quad m_{A_D^3}^2 = \frac{g_D^2}{4}(v_{D1}^2+v_{D_2}^2).
\end{equation}
In splitting the $\psi$ and $A_D$ multiplet masses, the triplet vev leads to a simple low-energy effective theory consisting of the dark matter
$\psi^{(1)}$ (we assume $\lambda_s>0$) and the mediator $A_D^3$, which has small induced couplings to a right-handed standard model lepton flavor.  This effective theory is relevant below the masses of the heavy vector-like leptons, $\psi^{(2)}$  and the $A_D^{1,2}$ bosons, which we will associate with a common scale, for simplicity.  In addition, we will see that the triplet vev leads to induced couplings of the
dark matter to quarks via kinetic mixing, which will lead to avenues for direct detection.   We discuss the phenomenology of this scenario in Sec.~\ref{sec:pheno}.

\section{Flavor sequestering} \label{sec:zn}

In this section, we show that it is possible to allow for non-negligible mixing between one flavor of the standard model leptons and the heavier,
vector-like leptons, while suppressing the mixing with the other standard-model flavors, so that bounds on lepton-flavor-violating processes
become irrelevant.   In the discussion below, we refer to that one flavor as the electron $e$, though the approach described applies equally
well if the chosen flavor were $\mu$ or $\tau$. Let us consider the structure of the standard model Yukawa matrices first, and then
introduce couplings to the vector-like states.

We represent the three generation of standard model lepton doublets by ${L_i}_L$ and the right-handed charged leptons by ${E_i}_R$, for $i=1, \ldots,  3$,
We imagine that the Yukawa couplings are determined by a flavor symmetry of the form $Z_N \times G_F$.   Our interest is in the effect of the $Z_N$ factor,
while we do not commit to any specific $G_F$.  We aim to show that the restrictions that follow from the $Z_N$ symmetry are sufficient to suppress the flavor
mixing effects that we would like to avoid, while remaining compatible with a variety of possible flavor models that may determine the remaining, detailed
structure of the Yukawa matrices.

We represent an element of $Z_N$ by $\omega^j$, for $j=1,\ldots N$, where $\omega^N \equiv 1$.   We assign the following transformation properties to
the $L$ and $E$ fields, representing them here as column vectors:
\begin{equation}
L_L \rightarrow \Omega\, L_L \,\,\,\,\, \mbox{ and } \,\,\,\,\, E_R \rightarrow \Omega\, E_R \,\,\, ,
\end{equation}
where
\begin{equation}
\Omega = \left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & \,\, \omega^{-n} & 0 \\ 0 & 0 & \,\, \omega^{n} \end{array}\right) \,\,\, .
\end{equation}
Note that $\omega^{-n} \equiv \omega^{N-n}$.  Assuming that the standard model Higgs doublet is unaffected by the $Z_N$ symmetry, the transformation properties of the charged-lepton
Yukawa matrix entries that lead to invariant couplings are summarized by
\begin{equation}
Y_E \sim \left(\begin{array}{c|cc} 1 & \omega^n & \omega^{-n} \\ \hline \omega^{-n} & 1 & \omega^{-2n} \\ \omega^n & \omega^{2n} & 1 \end{array}\right) \,\,\, ,
\end{equation}
where the transformation property of, for example, the $1$-$2$ entry is understood to be $Y_E^{12} \rightarrow   \omega^n \, Y_E^{12}$, and so on.   We will choose $N=2 \, n$ so that the entire two-by-two block on the lower right is
unconstrained by the $Z_N$ symmetry, the least restrictive possibility that meets our needs\footnote{This choice is also compatible with $G_F$ having a non-Abelian
component in which two flavors of standard model leptons transform as a doublet.  However, it is sufficient (and simplest) for present purposes to imagine that $
G_F$ has only Abelian factors.}.  The amount by which the electron mass eigenstate is affected by the second and third generation fields, however, is entirely controlled by the size of $n$, once $Z_N$ breaking fields are introduced, as we discuss later.

A symmetry affecting the left-handed charged leptons also affects the left-handed neutrinos, so we must verify that neutrino phenomenology is not adversely
affected.  For example, if we had imposed a $Z_2$ symmetry, with $n=1$, and required it to remain exactly unbroken, we can also completely eliminate mixing
between the first generation charged leptons and those of the second and third generations.  However, if we then introduce three generations of right-handed
neutrinos $N_i$, for $i=1, \ldots, 3$, one can show that there are no $Z_2$ charge assignments for the $N$ fields that leads to the correct neutrino mass squared
differences and mixing angles, assuming the light mass eigenstates follow from the see-saw mechanism.   However, more favorable results may be obtained when the $Z_N$ symmetry is spontaneously broken.  Here, we assume the same
transformation for all three right-handed
neutrino fields:
\begin{equation}
N_R \rightarrow \omega^p \, N_R \,\,\, ,
\end{equation}
where $p$ is an integer.  Defining the Dirac neutrino mass via ${\cal L} \supset \overline{L}_L \widetilde{H} Y_{LR} N_R + \mbox{h.c.}$, the transformation properties of the Yukawa coupling is given by
\begin{equation}
Y_{LR} \sim \left(\begin{array}{c|cc} \omega^{-p} & \omega^{-p} & \omega^{-p} \\ \hline \omega^{-n-p} & \omega^{-n-p} & \omega^{-n-p} \\ \omega^{n-p} & \omega^{n-p} & \omega^{n-p} \end{array}\right) \,\,\, .
\end{equation}
For the choice $n = 2 \, p$, or equivalently $N = 2 \, n = 4 \, p$, we may use the fact that $\omega^{-n-p} \equiv \omega^p$ and $\omega^{n-p} \equiv \omega^p$ to write
\begin{equation}
Y_{LR} \sim \left(\begin{array}{c|cc} \omega^{-p} & \omega^{-p} & \omega^{-p} \\ \hline \omega^{p} & \omega^{p} & \omega^{p} \\ \omega^{p} & \omega^{p} & \omega^{p} \end{array}\right) \,\,\, .
\end{equation}
The significance of this form is clear if we assume that there is a flavon field $\rho$ with the
$Z_N$ transformation property
\begin{equation}
\rho \rightarrow \omega \, \rho \,\,\, ,
\end{equation}
and a vacuum expectation value such that $\langle \rho \rangle/M \equiv \epsilon $ is a small parameter. Here $M$ is the flavor scale, which is the ultraviolet cut off of the effective theory. Then all the entries of $Y_{LR}$ are
non-vanishing, and proportional to either $(\rho/M)^p$ or to $(\rho^*/M)^p$.   Hence, we may write
\begin{equation}
Y_{LR} = \epsilon^p \, \widetilde{Y}_{LR} \,\,\, ,
\label{eq:lrt}
\end{equation}
where $\widetilde{Y}_{LR}$ is a three-by-three matrix that is thus far arbitrary.  Following a similar argument, we define the right-handed neutrino Majorana mass
matrix by the Lagrangian term $\overline{N^c}_R M_{RR} N_R$, and see immediately that
\begin{equation}
M_{RR} \rightarrow \omega^{-2p} M_{RR} \,\,\, .
\end{equation}
Again, this is consistent with the transformation property of $(\rho^*/M)^{2p}$, so we may write
\begin{equation}
M_{RR} = \epsilon^{2p} \widetilde{M}_{RR} \,\,\, ,
\label{eq:rrt}
\end{equation}
where $\widetilde{M}_{RR}$ is a three-by-three Majorana mass matrix that is also arbitrary thus far.  With $\widetilde{Y}_{LR}$ and
$\widetilde{M}_{RR}$ arbitrary, it is possible to obtain any desired neutrino phenomenology, which demonstrates that the $Z_N$ symmetry
does not lead to unwanted phenomenological restrictions.  Theories that predict the detailed structure of $\widetilde{Y}_{LR}$ and
$\widetilde{M}_{RR}$ by the breaking of an additional symmetry $G_F$ are compatible with this framework. Note that the overall powers of $\epsilon$
in Eqs.~(\ref{eq:lrt}) and (\ref{eq:rrt}) scale out of the see-saw formula which determines the Majorana
mass matrix for the three light neutrino mass eigenstates
\begin{equation}
M_{LL} = M_{LR} \, M_{RR}^{-1} \, M_{LR}^T ,\,\,\,
\end{equation}
where $M_{LR} = (v/\sqrt{2}) \, Y_{LR}$.
The effect of the $Z_N$ symmetry on the form of the charged lepton Yukawa matrix is to
impose the form
\begin{equation}
Y_E \sim \left(\begin{array}{c|cc} y_{11} & \epsilon^n \tilde{y}_{12} & \epsilon^n \tilde{y}_{13} \\ \hline \epsilon^n \tilde{y}_{21} & y_{22} & y_{23} \\ \epsilon^n \tilde{y}_{31} & y_{32} & y_{33} \end{array}\right) \,\,\, .
\end{equation}
For $\epsilon$ sufficiently small, or $n$ sufficiently large, or both, we can make $Y_E$ as close to block diagonal as we like.

Now we include the vector-like state $\chi$ with the same electroweak quantum numbers as a right-handed electron, but charged also under a dark gauge group. Yukawa couplings involving
$\overline{\chi}_L e_R$ and a dark Higgs field are unaffected by the $Z_N$ symmetry, while those involving $\overline{\chi}_L \mu_R$  or $\overline{\chi}_L \tau_R$ transform by $\omega^{\pm n}$.
These potential sources of unwanted mixing that may emerge after the dark gauge symmetry is spontaneously broken are therefore highly suppressed by the same factors of $\epsilon^n$ that
appear in the unwanted entries in $Y_E$.      We conclude that it is possible to make the $\chi$, $e$, $\mu$, $\tau$ mass matrix as block diagonal as desired,
by suitable choice of $\epsilon^n$,  such that $\chi$ mixes substantially only with $e$, or any one desired lepton flavor, by a similar construction.

The question of which lepton flavor is selected to mix with the heavier, vector-like states impacts the phenomenology of the dark gauge bosons.  For example, if the mixing only involves the
$\tau$ lepton, then bounds on the $A_D^a$ from searches for $s$-channel resonances in low-energy $e^+ e^-$ collisions, or from indirect processes like the electron or muon $g-2$
would be irrelevant.  The phenomenology in the case where the mixing involves either a first or second generation lepton would lead to more meaningful constraints, but one that would depend on
other assumptions about the spectrum, for example if $A_D^3$ decays visibly or invisibly, which depends on the dark matter mass.   In the following section, we will assume the least constrained possibility, that the $\chi$'s mix with the $\tau$, and consider the wide range of phenomenological issues associated with the other two possibilities in separate
work~\cite{shikha}.   This has the appealing aesthetic feature that the flavor symmetry distinguishes the third generation from the other two, an idea that has appeared in many other contexts in
the literature on the flavor structure of the standard model~\cite{some}.

\section{Phenomenology}  \label{sec:pheno}

\subsection{Relic Density}

The scattering amplitude for $s$-channel dark matter annihilation into standard model particles depicted in Fig.~\ref{fig:one}, with $e$ replaced by $\tau$,
is given by
\begin{equation}
\mathcal M( \psi^{(1)} \overline{\psi^{(1)}} \to \tau^+ \tau^-)
=\frac{ ig_D^2\theta^2}{4\big(q^2-m_{A_D^3}^2+im_{A_D^3}\Gamma^D\big)} \,\, \overline v(p') \gamma^\mu u(p) \,\, \overline u(k) \gamma_\mu v(k')
\end{equation}
where $p\; (p')$ is the momentum of the incoming dark matter fermion (anti-fermion), $k\; (k')$ is the the momentum of the outgoing $\tau^-$
($\tau^+$) and $q=p+p'$ is the momentum flowing through the $A_D^3$ propagator. As discussed in Sec.~\ref{sec:model},
the lightest gauge boson $A_D^3$ couples to the vector-like states $\chi^{(1)}$ and $\chi^{(2)}$, which then mix with a standard model
lepton flavor (chosen here as $\tau$) after spontaneous symmetry breaking. This results in a factor of $\theta^2$, defined in
Eq.~(\ref{eq:ourtheta}), in the scattering amplitude.

Our numerical results for dark matter annihilation depend on assumptions about the dark particle mass spectrum and couplings.  We assume the picture described earlier, where the lightest states consist of $\psi^{(1)}$ and $A^3_D$, and decays of  $A^3_D$
to any of the heavier exotic states are not kinematically allowed.  For the mass range studied in this section, $A_D^3$ can decay to $\tau^+\tau^-$, and possibly also
$\psi^{(1)}\overline{\psi^{(1)}}$, depending on the dark matter mass.  Consequently, the total decay width of the dark gauge boson
appearing in the propagator is given by
\begin{equation}
\Gamma^D = \Gamma\left(A_D^3 \to \tau^+\tau^-\right) + \Theta \left( m_{A_D^3} - 2 \, m_{\psi^{(1)}}\right)\, \Gamma\left(A_D^3 \to \psi^{(1)}\overline{\psi^{(1)}}\right)
\end{equation}
where $\Theta$ is a step function, {\em i.e.}, $\Theta(x)=1 \text{ if } x\geq0 \text{ and }  \Theta(x)=0 \text{ if } x<0$, and
\begin{equation}
\Gamma\left(A_D^3 \to \tau^+\tau^-\right) = \frac{1}{48\pi} g_D^2 m_{A_D^3} \theta^4 \left(1+\frac{2\, m_\tau^2}{m_{A_D^3}^2}\right)\left(1-\frac{4\, m_\tau^2}{m_{A_D^3}^2}\right)^{1/2} \,\,\, ,
\end{equation}
\begin{equation}
\Gamma\left(A_D^3 \to \psi^{(1)}\overline{\psi^{(1)}}\right) =  \frac{1}{48\pi} g_D^2 m_{A_D^3}  \left(1+\frac{2\, m_{\psi^{(1)}}^2}{m_{A_D^3}^2}\right)\left(1-\frac{4\, m_{\psi^{(1)}}^2}{m_{A_D^3}^2}\right)^{1/2} \,\,\, .
\end{equation}
Since the mean dark matter velocity is typically around 220 km/s~\cite{pdg}, we work in the non-relativistic limit where
$E_{\psi^{(1)}} \approx m_{\psi^{(1)}}$.  We then find the thermally averaged annihilation cross section times velocity
 \begin{equation}
 \label{eq:crosssection}
 \braket{\sigma_A v} =  \frac{g_D^4 \theta^4}{32\pi}\frac{2m_{\psi^{(1)}}^2+m_\tau^2}{(4m_{\psi^{(1)}}^2-m_{A_D^3}^2)^2+m_{A_D^3}^2 \Gamma_D^2}  \left(1-\frac{m_\tau^2}{m_{\psi^{(1)}}^2}\right)^{1/2}.
 \end{equation}
Using this we calculate the freeze-out temperature $T_F$ and the dark matter relic density by standard methods \cite{Kolb:1990vq}. Dark matter freeze out occurs when the interaction probability per unit time $\Gamma_{\psi^{(1)}}$, equals the
expansion rate of the universe, $H$, {\em i.e.},
\begin{equation}
\label{eq:freezeout}
 \left.\frac{\Gamma_{\psi^{(1)}}}{H} \right|_{T=T_F} = \left.\frac{n_{EQ}^{\psi^{(1)}} \braket{\sigma_A v}}{H}\right|_{T=T_F} \simeq 1.
 \end{equation}
 Here $n_{EQ}^{\psi^{(1)}}$ is the equilibrium number density of the dark matter particle, given by
 \begin{equation}
 n_{EQ}^{\psi^{(1)}} = 2 \left( \frac{m_{\psi^{(1)}} T}{2\pi}\right)^{3/2} e^{-m_{\psi^{(1)}}/T}.
 \end{equation}
Freeze-out occurs during the radiation-dominated epoch in which case
\begin{equation}
H = 1.66 \,g_*^{1/2} T^2 / M_{pl},
\end{equation}
where $M_{pl} = 1.22 \times 10^{19}$ GeV is the Planck mass and $g_*(T)$ the number of relativistic degrees of freedom at temperature $T$,
\begin{equation}
g_*(T) = \sum_{i = bosons} g_i \left(\frac{T_i}{T}\right)^4 + \frac{7}{8} \sum_{i=fermions} g_i \left(\frac{T_i}{T}\right)^4.
\end{equation}
Finally the dark matter relic density is given by
\begin{equation}
\label{eq:relicdensity}
\Omega_D h^2 = \frac{2\cdot(1.07 \times 10^9 \text{ GeV}^{-1})\, x_F}{\sqrt{g_*(T_F)} M_{Pl} \braket{\sigma_A v}}.
\end{equation}
We define $x_F \equiv m_{\psi^{(1)}}/ T_F$ where $T_F$ is obtained by solving Eq.~(\ref{eq:freezeout}). The factor of 2 is included because we are accounting for the density of dark matter particles and antiparticles. We require Eq.~(\ref{eq:relicdensity}) to
reproduce the WMAP result $0.1186 \pm 0.0020$ \cite{pdg} within two standard deviations.

To display our results, we fix $m_{A^3}$ and $\theta$ and find the regions of the $g_D$-$m_{\psi^{(1)}}$ plane in which the desired dark matter
relic density is obtained.  We assume that the mixing angle remains small ($\theta<1$) but not so small that a satisfactory dark matter annihilation cross section cannot be obtained.  So that the dark gauge coupling remains perturbative, we assume
$\alpha_D / (4\pi) < 1/3$ or equivalently $g_D < 4\pi/\sqrt3 \sim 7.25$; one-loop corrections become comparable to tree-level amplitudes when $\alpha / (4\pi) \approx 1$, so one-third of this value is a reasonable upper limit on the dark coupling constant. For the purposes of
determining $g_*$, we assume all exotic mass eigenstates other than $\psi^{(1)}$ and $A_D^3$, are at $m_Z= 91.1876$~GeV.   With this choice, the $Z$ boson cannot decay into $\chi \overline{\chi}$ or $\chi \tau$, which could lead to an unacceptable broadening of the precisely measured $Z$ boson width~\cite{pdg}.

\begin{figure}[h]
\centering
	\subfigure{%
		\includegraphics[scale=0.6]{md10tau}} \hspace{2em
	\subfigure{%
		\includegraphics[scale=0.6]{md100tau}}%
	\caption{Regions of the $g_D$-$m_{\psi^{(1)}}$ plane in which the dark matter relic density is within two standard deviations of the WMAP result $0.1186 \pm 0.0020$~\cite{pdg}, for fixed choices of $m_{A^3}$ and $\theta$.  The allowed bands are not
	perfectly smooth due to their dependence on $g_*$, which is not a continuous function.   The point of minimum $g_D$ corresponds to resonance annihilation, where $m_{\psi^{(1)}} = m_{A^3}/2$. Note that as $\theta$ decreases the range of $m_{\psi^{(1)}}$ in which $g_D$ remains
	perturbative moves towards the resonance region. \label{fig:plots}}
\end{figure}

Fig.~\ref{fig:plots} shows the regions of the $g_D$-$m_{\psi^{(1)}}$ plane in which the dark matter relic density is within two standard deviations
of the WMAP result $0.1186 \pm 0.0020$~\cite{pdg}, for fixed choices of $m_{A^3}$ and $\theta$.  We have intentionally centered the plots
around the point of resonance annihilation $m_{\psi^{(1)}} = m_{A^3}/2$ where the cross section is largest.  For small values of $g_D$ at fixed $\theta$, some tuning is required to achieve a large enough annihilation cross section.  However, Fig.~\ref{fig:plots}
indicates that we can have larger, perturbative values of $g_D$ without requiring that we sit unnaturally close to the resonance.  As $\theta$ is made progressively smaller, however, more tuning is required.  This is indicated by the narrowing
range in $m_{\psi^{(1)}}$ for each solution in which $g_D$ is also perturbative.

Of course, the values of $\theta$ that are indicated in Fig~\ref{fig:plots}  are related to choices for the masses and coupling in the model, such that $\theta^2 = g_D \left(\frac{m_1^2}{M_1^2}-\frac{m_2^2}{M_2^2} \right)$, where the $m_i$ and
$M_i$ were defined in Eq.~(\ref{eq:bigM}).   It is not hard to verify that the values of $\theta$ shown in Fig.~\ref{fig:plots} can be achieved given the assumptions that went into the making of
the plots. For example, in the $m_{A^3}=10$~GeV plot, consider the point where $g_D \approx 1$ and $m_{\psi^{(1)}} \approx 8.5$~GeV, on the $\theta=0.1$ band.   Given our earlier
assumption in computing $g_*$ that the heavier exotic states are at $m_Z$, one can check that this is consistent with, for example, $v_{D1} = v_{D2} \approx 14$~GeV,  $v_T \approx 49$~GeV, $\lambda_s \approx 0.85$, and $y_1=y_2 \approx 0.06$, where the Yukawa couplings $y_i$ were defined in Eq.~(\ref{eq:lchi}).  Similar statements can be made about other points on the allowed bands\footnote{The scenario that we have considered assumes that communication between dark and visible sectors occurs primarily through the portal that we have proposed, involving mixing with vector-like leptons.   It is of course possible to have scenarios in which communication is also significant through Higgs portal couplings or other mediators.  The results presented in this section demonstrate the effectiveness of the portal we have proposed taken in
isolation; this may nonetheless accurately represent a subset of the parameters space of a more complicated model with other dark matter annihilation channels.}.

\subsection{Direct Detection}

The interactions that we have discussed to this point have involved leptons exclusively, but couplings to quarks that are generated at the loop level also have significant consequences.
In this section, we consider direct detection of the dark matter in the model via dark-matter-nucleon elastic scattering.   The couplings to quarks arise after the SU(2)$_D$ symmetry is
spontaneously broken, since kinetic mixing between $A^3_D$ and hypercharge is then allowed, via an effective dimension-5 operator
\begin{equation}
\mathcal{L}_{eff} = X \tr\left(\langle H_{T}\rangle T^{a}A^{a}_{D \mu\nu}\right)Y^{\mu\nu} \,\,\, , \label{eq:Leff1}
\end{equation}
\begin{figure}[t]
\centering
\includegraphics[width=0.75 \textwidth]{kmd.pdf}
\caption{Self energies leading to kinetic mixing between the third dark gauge boson $A_{D}^{(3)}$ and hypercharge $Y$ after SU(2)$_D$ is spontaneously broken. \label{fig:vertex}}
\end{figure}
where we have set the triplet Higgs to its vev, as per Eq.~(\ref{eq:hdhtv}).   Here, $X$ is a constant with units of GeV$^{-1}$ which is found by integrating out the ``heavy" physics, {\em i.e.},  the $\chi$ fields, the only fields that are charged both under SU(2)$_D$ and hypercharge U(1)$_Y$.   To proceed, we study the self-energy shown in Fig.~\ref{fig:vertex}, where
$\chi^{(1)}$ and $\chi^{(2)}$ here represent the heavy mass eigenstates, whose mass eigenvalues are given approximately by $m_{\chi^{(1)}} = M_\chi -\delta$ and $m_{\chi^{(2)}} = M_\chi + \delta$
where $\delta \equiv \lambda_s' \, v_T/2$.  (For the purposes of this estimate, we ignore mass mixing with the standard model lepton, which is a subleading correction.)   The first diagram is
given by
\begin{equation}
		i\mathcal{M}_{1} =  - \frac{g_{D} g_{Y}}{2}  \int \frac{d^{4}k}{(2\pi)^{4}}
		\frac{
			\tr\left[ \gamma^{\mu} \left( \slashed{k} + m_{\chi^{(1)}} \right) \gamma^{\nu} \left( \slashed{k} + \slashed{p} +m_{\chi^{(1)}} \right) \right] }{ [ k^{2}-m_{\chi^{(1)}}^{2}+i\epsilon ] [ (k+p)^{2}-m_{\chi^{(1)}}^{2}+i\epsilon] } \, \, \, .
\end{equation}
After carrying out this loop integral using dimensional regularization in $D=4-\epsilon$ dimensions, the amplitude is
\begin{equation}
i\mathcal{M}_{1} =- \frac{g_{D}g_{Y}}{8\pi^{2}} \int_{0}^{1} dx \ x(1-x) \left(\frac{4}{\epsilon}-2\gamma +2\log(4\pi) -2 \log(\Delta_{1})\right) i (g^{\mu\nu}p^{2}-p^{\mu}p^{\nu}) \, \, \, ,
\end{equation}
where $\Delta_{1} = m_{\chi^{(1)}}^{2} -x(1-x)p^{2}$.   Since $A_D^3$ couples to the $\chi$ proportional to $\sigma^3/2$, the amplitude $i\mathcal{M}_{2}$ shown in
Fig.~\ref{fig:vertex} will differ from $i\mathcal{M}_{1}$ by a overall minus sign and the replacement of the $\chi^{(1)}$ by the $\chi^{(2)}$ mass.  Hence, $\Delta_{1}$ is replaced by $\Delta_{2} = m_{\chi^{(2)}}^{2} -x(1-x)p^{2}$. Then, when these two amplitudes are added together, all terms in the remaining Feynman parameter integral cancel, except for the terms that depend on the fermion masses:
\begin{equation}
i\mathcal{M}_{1} + i \mathcal{M}_{2} = i(g^{\mu\nu}p^{2} - p^{\mu}p^{\nu}) \frac{g_{D}g_{Y}}{4\pi^{2}} \int_{0}^{1}dx\ x(1-x) \log\left(\frac{\Delta_{1}}{\Delta_{2}}\right) \,\,\, .
\end{equation}
Assuming the mass splitting $\delta$ is small compared to the $\chi$ masses (which will turn out to be the case) the integrand can be expanded in $\delta$.  The leading order term can be found
using $x(1-x)\log(\Delta_{1}/\Delta_{2}) \approx -\frac{4mx(1-x)}{m^{2}-x(1-x)p^{2}} \delta$.  Moreover, we can also expand the result in powers of momentum, which can later be compared to a
derivative expansion in the low-energy effective theory.  We find
\begin{equation}
i\mathcal{M}_{1} + i \mathcal{M}_{2} = - i \frac{g_{D}g_{Y}\delta}{6\pi^{2} M_\chi}(g^{\mu\nu}p^{2} - p^{\mu}p^{\nu}) + \cdots \, \, \, , \label{eq:chiloop}
\end{equation}
where the $\cdots$ represents terms involving higher powers of $\delta$ and $p^2/M_\chi^2$.   The result in Eq.~(\ref{eq:chiloop}) must be matched to a similar amplitude in the low-energy effective theory in which the $\chi$ fields have been integrated out.  We identify this as the tree-level amplitude associated with the Eq.~(\ref{eq:Leff1}), treated as a two-point vertex,
\begin{equation}
i\mathcal{A}  = i X v_{T} \left( p^{2}g^{\mu\nu}-p^{\mu}p^{\nu} \right) \,\,\, , \label{eq:AmpEff}
\end{equation}
from which we conclude
\begin{equation}
X = - \frac{g_{D}g_{Y}\delta}{6\pi^{2} M_\chi v_{T}} \, \, \, . \label{eq:KineticMixing}
\end{equation}
\begin{figure}
\centering
\includegraphics{scattering_cross_section_02.pdf}
\caption{The Feynman diagram for the scattering of the dark matter particles, $\psi^{(1)}$, off of protons, $P$, through kinetic mixing of the dark matter boson $A_{D}^{3}$ and the photon, $\gamma$.  \label{fig:crosssection}}
\end{figure}

Using Eqs.~(\ref{eq:Leff1}) and (\ref{eq:KineticMixing}), we can now calculate the cross section for dark matter scattering off of nucleons. We will be working in the limit of low momentum transfer
$q \sim {\cal O}(100)$~MeV ($\ll M_\chi$), where the effective description is accurate and where scattering through the $Z$ boson is suppressed by $q^2/m_Z^2 \sim 10^{-6}$ compared to the photon.  Hence,
we will consider kinetic mixing involving the photon only from here on. First, we consider the dark matter, $\psi^{(1)}$, scattering off of a quark, $q_{f}$, as in the diagram in Fig.~\ref{fig:crosssection}, with the protons replaced by a quark of flavor $f$.   This can be described by the effective dimension-six operator
\begin{equation}
 \mathcal{L}_{eff,q} = C_{f} \, \overline{\psi^{(1)}} \gamma^{\mu}\psi^{(1)} \overline{q_{f}}\gamma_{\mu}q_{f} \, \, \, . \label{eq:effqop}
 \end{equation}
In the full theory, this quark-dark matter scattering amplitude is
\begin{equation}
i\mathcal{M}_{f} = i X v_{T} Q_{f} \frac{g_{D}}{2} e \frac{1}{(q^{2}-m_{A^{3}_{D}}^{2} +i\epsilon)(q^{2}+i\epsilon)} \overline{\psi^{(1)}}\gamma^{\mu}\psi^{(1)}\overline{q_{f}}\gamma_{\mu}q_{f}
\end{equation}
or, in the limit of $q^{2}\ll m_{A^{(3)}_{D}}^{2}$,
\begin{equation}
i\mathcal{M}_{f}  =-i \frac{Xv_{T}Q_{f}}{m_{A^{3}_{D}}^{2}} \frac{g_{D}}{2} e \overline{\psi^{(1)}}\gamma^{\mu}\psi^{(1)}\overline{q_{f}}\gamma_{\mu}q_{f} \, \, \, .
\end{equation}
From this, we conclude the coefficient $C_{f}$ for quarks is
\begin{equation}
C_{f} = - \frac{g_d e Xv_{T}Q_{f}}{2 m_{A_{D}^{3}}^{2}} = \frac{g^2_{D} e^2 \delta Q_{f}}{12 \pi^{2} M_\chi m_{A_{D}^{3}}^{2} } \,\,\, .
\end{equation}

Of interest, however, is the effective interactions involving nucleons rather than quarks, which can be written
\begin{equation}
\mathcal{L}_{eff,N}  = C_n \, \overline{\psi^{(1)}}\gamma^{\mu} \psi^{(1)} \overline{n} \gamma_{\mu}n + C_p \, \overline{\psi^{(1)}}\gamma^{\mu} \psi^{(1)} \overline{p} \gamma_{\mu} p  \,\,\,.
\label{eq:effNop}
\end{equation}
Using the fact that the quark vector currents are conserved, so that the spatial integral of the zeroth component is a quark number operator, one can match matrix elements of Eq.~(\ref{eq:effqop})
between nucleon states with the same for Eq.~(\ref{eq:effNop}), from which one concludes $C_{n} = C_{u}+ 2 C_{d}$ and $C_{p} = 2 C_{u} + C_{d}$, for the neutron and proton, respectively.  (There
are no form factors as there would be for scalar quark operators.)   Since the flavor dependence of the $C_f$ comes only from the electric charge, the coefficient $C_n$ and thus the scattering amplitude for $\psi^{(1)}$ off of neutrons are both zero. Therefore, the only relevant scattering is with the proton, for which
\begin{equation}
C_{p} = 2\, C_{u}+C_{d} = \frac{g^2_{D} e^2 \delta }{12 \pi^{2} M_\chi m_{A_{D}^{3}}^{2} }\, \, \, .
\end{equation}
Taking into account that the dark matter is non-relativistic and that momentum transfers are small, a straightforward calculation of the scattering cross section yields
\begin{equation}
\lara{\sigma_{\psi^{(1)}p\rightarrow\psi^{(1)}p}} = \frac{g_{D}^{4}e^{4}m_{p}^{2}m_{\psi^{(1)}}^{2} }{ 576 \pi^{5} \left( m_{p} + m_{\psi^{(1)}} \right)^{2} m_{A^{3}_{D}}^{4} }
\left( \frac{2\delta}{M_\chi}\right)^{2} \,\,\, , \label{eq:crosssection2}
\end{equation}
where we have separated out the dependence on $2\delta/M_\chi$, the fractional mass spitting of the vector-like leptons.  Since this splitting is a free parameter in our model, we can use the
experimental bounds on the dark-matter-nucleon elastic scattering cross section to say something about the vector-like lepton spectrum.

\begin{figure}
\centering
\includegraphics[width=.4\textwidth]{dd.pdf}
\caption{Upper bound on the fractional mass splitting of the $\chi^{(1)}$ and $\chi^{(2)}$ fermions as a function of the mass of the dark matter particle, $\psi^{(1)}$, assuming  $g_{D}=0.3$ and
$m_{A^{3}_{D}} = 10\, \mbox{GeV}$.  The discontinuity in the curve reflects that the bounds on the dark-matter-nucleon elastic scattering cross section originate from the CDMSlite experiment~\cite{CDMSlite}
below $m_{\psi^{(1)}} \approx 6$~GeV, where the otherwise tighter bounds from the XENON1T experiment~\cite{xenon1t} do not exist. \label{fig:directdetection}}
\end{figure}

Using experimental bounds on the cross section from XENON1T~\cite{xenon1t} and CDMSlite~\cite{CDMSlite}, we show bounds on the $\chi^{(1)}$-$\chi^{(2)}$
mass splitting for dark matter masses between $2$~GeV and $10$~GeV.  The results of this calculation are shown in Fig.~\ref{fig:directdetection}, where a dark coupling of $g_{D}=0.3$ and a dark
boson mass of $m_{A^{3}_{D}} = 10\, \mbox{GeV}$ have been used. For dark matter masses below approximately $6$~GeV, the cross section bounds from CDMSlite are used, since no data from XENON1T is
available in this region. Although there is CDMSlite data for dark matter masses above $6$~GeV, these bounds are superceded by the stricter ones from XENON1T.  For the
range of $\psi^{(1)}$ masses in Fig.~\ref{fig:directdetection} that are affected by the XENON1T bounds, the masses of the charged fermions $\chi^{(1)}$ and $\chi^{(2)}$ are
degenerate at the $1$-$10$\% level at minimum.  This feature could be observed in collider searches for the vector-like leptons and possibly correlated with a dark matter direct detection signal.

\section{Conclusions}
We have presented an explicit, renormalizable non-Abelian dark SU(2) model which contains two vector-like fermion doublets. One of them, $\psi$, includes a dark matter
candidate; the other doublet, $\chi$, has the same electroweak quantum numbers as a right-handed electron, so that communication with the visible sector can occur via mass
mixing. The $\psi$ and $\chi$ fields communicate with each other via the dark gauge group, so that the dark matter may annihilate to standard model leptons.  The dark SU(2)
symmetry is spontaneously broken via a Higgs sector involving doublet and triplet fields.  The doublet vacuum expectation value (vev) leads to mixing between the $\chi$ and standard model lepton fields, while the triplet vev splits the mass spectrum leaving a simple lower-energy theory consisting of the dark matter (the lightest $\psi$ mass eigenstate) and
the mediator (the third component of the SU(2) gauge multiplet).  We identify a discrete flavor
symmetry that allows mixing  between the vector-like leptons $\chi$ and a single standard model lepton flavor exclusively; the remaining standard model lepton flavors may
mix only with each other.  This flavor sequestering eliminates lepton-flavor-violating effects, relaxing bounds on the vector-like lepton mass scale.  As a consequence, mixing
between the chosen lepton flavor and the $\chi$ can be large enough so that the correct relic density can be obtained exclusively via dark matter annihilation to lepton-anti-lepton pairs, for
perturbative values of the dark gauge coupling.  This is true even if no other significant annihilation channels are available that originate from other portals.

The structure of our model avoids complications that would ensue if we tried to couple the dark gauge bosons directly to standard model fields, such as the necessity of
including extraneous fermions to cancel chiral anomalies, or special Higgs representations to allow for acceptable standard model Yukawa couplings.  Unlike some of the non-Abelian dark
matter models appearing in the literature, the portal we present is renormalizable and completely specified, including the discrete flavor symmetries that control the pattern of mixing between
exotic and standard model fermions states.   The portal we define for communication between the dark and visible sectors presents a well defined framework for answering
phenomenological questions.  In the present work, we showed that there are regions of the dark gauge coupling - dark matter mass plane where the correct relic density is
obtained, and where current direct detection bounds are satisfied.   The latter consideration also allowed us to conclude that the two heavy lepton mass eigenstates (roughly the
two components of the $\chi$ doublet) are notably degenerate in mass (to keep kinetic mixing effects small), a feature that could be tested in collider searches for these states.
This observation, together with the distinct lepton flavor structure of the $\chi$ decays, suggests that the collider signatures of the portal that we have proposed are
worthy of future detailed investigation.

\begin{acknowledgments}
This work was supported by the NSF under Grant PHY-1519644.
\end{acknowledgments}

redpajama_set_name: RedPajamaArXiv


text:
UK Snooker Championship 2020: Judd Trump through to quarter-finals
Judd Trump won the UK Championship in 2011
Dates: 23 November to 6 December Venue: Marshall Arena, Milton Keynes
Coverage: Watch live on BBC TV, BBC iPlayer, BBC Red Button, BBC Sport website and BBC Sport app from Saturday, 28 November.
World number one Judd Trump held off Ricky Walden's fightback to reach the quarter-finals of the UK Championship.
Trump severely punished errors from Walden with breaks of 85 and 65 to open up a commanding 4-0 advantage.
Walden rallied by taking three of the next four frames, including a 70 break, but Trump took the one required to comfortably advance with a 6-3 victory.
Trump faces rival Kyren Wilson or former world champion Graeme Dott next, with the pair meeting in the evening.
Englishman Jack Lisowski made a century and further breaks of 65 and 62 to beat China's Xiao Guodong 6-4 and reach the last eight where he will face Zhou Yuelong.
Trump extends winning streak
With 12 ranking title triumphs since the start of the 2018-19 season, including the English Open and Northern Ireland Open this term, Trump has elevated his game to a different level, becoming a formidable attacking force yet equalling adept in the tactical department.
Although world number 46 Walden improved after the interval, Trump maintained control and extended his winning run to 13 matches, last losing in the Champion of Champions semi-final on 7 November.
Trump told BBC Two: "It is never easy against the top players and Ricky is a lot better player than his ranking is, I knew it was going to be tough. When you go 4-0 up, the other player has nothing to lose so is relaxed and he put me under pressure a bit towards the end.
"I am still practicing three or four hours a day and trying to improve and keep that number one spot for as long as possible.
"People are appreciative of having tournaments on, it still feels like the UK Championship, still feels special and you know it is a big event.
"I still get the butterflies walking into the venues, I want to win every game and don't want this winning streak to end."
'Obscene' Trump potting – analysis
1991 world champion John Parrott on BBC Two: "Ricky Walden gave himself too much to do, it was a valiant effort but you cannot give a 4-0 lead to the world number one
"With his cue power, Trump takes on shots that other players can't and pulls off some screamers. One long red that he potted was obscene."
Six-time world champion Steve Davis: "I don't think Judd is in a hurry to win matches, he was not attacking too much and is waiting for big chances to open up.
"He is an opportunist and that makes him very difficult to play against. As an opponent, there are very few hiding places because he is so adept at pulling off shots."
Sign up to My Sport to follow snooker news on the BBC app.
Related Itemschampionshipfinalsquartersnookerthroughtrump
Newsexplored 3rd December 2020
Romain Grosjean: Charles Leclerc feared Haas driver had died in Bahrain crash
AC Milan 4-2 Celtic: Italian side come from two goals down to win

redpajama_set_name: RedPajamaCommonCrawl


Rowe Ridge Vineyard & Winery is a winery in Kansas City, Kansas. Some wineries or vineyards offer wine discounts on large orders of wine. Contact Rowe Ridge Vineyard & Winery for more information on their wine discounts and wine promotions.


text:
Pyotr Stepanovich Kotlyarevsky (23 June 1782 – 2 November  1852) was a Russian military hero of the early 19th century.

Biography

He was born in the village of Olkhovatka near Kharkiv into a cleric's family. Kotlyarevsky was brought up in an infantry regiment quartered near Mozdok. He was promoted officer for his exploits during Count Zubov's Persian Campaign in 1796.

His  leadership and boldness made him a national celebrity during the Russo-Turkish War (1806–12)#Caucasus Front and the Russo-Persian War (1804-1813). In 1810 he took hold of Meghri Citadel, withstood a siege by the Persian army and then routed them on the Araks River. In 1812, he defeated Abbas Mirza in the Battle of Aslanduz and stormed Lankaran with as little as around 2,500 soldiers. Thereupon the Persians sued for peace, and Kotlyarevsky was promoted General of Infantry (a rank equal to that of Full General in other armies). Suffering from wounds, he had to retire from active service and spent the rest of his life in Feodosiya, where he eventually died as well.

A local artist, Ivan Aivazovsky, devised Kotlyarevsky's mausoleum, while Prince Vorontsov ordered his statue to be erected in Ganja.

External links
Biography

1782 births
1852 deaths
People from Voronezh Oblast
Russian generals
People of the Russo-Persian Wars
Recipients of the Order of St. George of the Second Degree
Recipients of the Order of St. George of the Third Degree

redpajama_set_name: RedPajamaWikipedia


Experience two World Heritage listed locations in style and comfort on this two day luxury Great Barrier Reef and World Heritage Daintree rainforest package.
Your first day you will be heading to the Outer Great Barrier Reef onboard our fast catamaran "Reef Experience".
Enjoying a more personalised, private and relaxed day as you begin your day in style in the exclusive Captains' Wheelhouse Top Deck Lounge" to our first pristine Outer Great Barrier Reef Location.
From the moment you board until you wave goodbye, you can relax in the knowledge that all of your needs are taken care of.
On day two you will explore the oldest living tropical rainforest in the world on your second tour from Cairns.
Visiting locations such as Mossman Gorge, the Daintree Rainforest, Cape Tribulation, enjoying the view from spectacular look outs, spotting crocodiles on the Daintree River, and native wildlife throughout the day.
All aboard a luxury vehicle, in a small group with a friendly, very knowledgeable guide who will explain our unique flora and fauna.
Discover the Natural wonders of the Great Barrier Reef with your Personal Valet caring for you in and out of the water!
Visit the oldest living Rainforest in the world that offers an extraordinary diversity of flora and fauna.
Guided Rainforest walks discovering the complex functions of a tropical mangrove, shaded by it's canopy of enormous fan palms.
*Please note seasonal weather conditions and operational requirements may cause a variation in the tour itinerary.
*Children's prices are for ages 4-11 years.
This website acts as a booking agent only for the tours advertised herein.
Although we endeavour to provide accurate information we are not liable for inaccurate information concerning services.
We are not liable for changes to tours, pricing or itineraries; all of which are subject to change without notice.
Neither this site, nor any associated company or agent advertised through this site is responsible for costs or losses resulting from alterations, cancellations, loss of baggage, delay or personal injury.
We strongly recommend everyone has their own travel insurance, and scuba diving insurance where applicable.
We reserve the right to cancel or vary the service at any time without notice.
Queensland, Australia has some of the strictest scuba diving rules in the world but also the safest scuba diving record.
We have a passion at Reef Experience to show you Australia's Great Barrier Reef so please make sure you understand the scuba diving laws. We are regulated by the Queensland Government Code of Practice and comply with all rules including medical as is the law.
To avoid disappointment on your day of travel we strongly recommend that all of our passengers complete our Dive Medical Questionnaire prior to booking as some medical conditions, age and / or medications may prevent you from scuba diving. Age restrictions will also apply to any diver training courses.
You should not travel to high altitude (fly or drive above 8000 feet) within 18 hours of completing a single scuba dive or 24 hours after multiple dives OR after any 'deep' dive. (A deep dive is any dive over 18 meters/60 feet). We recommended leaving a minimum of 24 hours after your last scuba dive before flying. You can fly and then dive the same day, but not dive and then fly. On the last day of your tour your final dive will be finished by around 12 noon. Please consider this when arranging ongoing travel plans which include flying or driving to altitude. Flying any time after 12 noon the following day is recommended.
Return helicopter transfer: Our helicopter tours are low flying scenic excursions over the coral reef that are designed to be a safe activity after scuba diving. You may book a helicopter transfer back to Cairns on the final day of your stay with us, even if you have scuba dived that day.
Experience a World Heritage listed locations in style and comfort on this one day luxury Great Barrier Reef and Daintree Rainforest package.
Looking for a trip to the Great Barrier Reef with that "Celebrity Feel"? Then our Great Barrier Reef "Top Deck Club" VIP Day is just the thing for you.
Great day with our family on the Reef with outstanding VIP service. Thank you for a great family memory.
Cairnstravel.net.au is proudly part of the Hot Getaways Company Hot Getaways Pty Ltd proudly develops high quality, user friendly websites which deliver your perfect holiday experience, every time. We have partnered with operators who share our passion and all tours, accommodation and attractions are carefully selected to deliver outstanding service. We're passionate about making your holiday memories last a lifetime, and satisfied customers are the testament to our ongoing success.


We are your SOURCE for Fresh, Fast, Fit Foods! We make fresh, healthy meals daily that are ready to eat! Grab 1 or several as they last 7 days!
Sign up today and enjoy fresh, fit, perfectly-portioned meals for 21 days. Our team will work with you to create a customized meal program to help you reach your fitness and lifestyle goals. Get started by scheduling time with one of our team members today!


The 1916 Arts and Crafts-style house and adjacent Duroe Building contain vintage clothing, antiques, fire truck and agricultural tools. Visit one-room school and general store. Research archives and library.
Art Center with exhibit gallery, gift shop featuring Midwestern artists, classes and workshops. Special art events throughout the year.


text:
The Most Interesting Thing Joey Chestnut Has Devoured
Steven Ferdman/Getty Images
By Anthony Mastroianni/April 14, 2021 4:25 pm EST
If the name Joey Chestnut means anything to you, it's likely because you've seen his insane feats of throwing back Nathan's Hot Dogs like they are popcorn. While competitive eating in general — let alone scarfing down 75 hot dogs in 10 minutes (via New York Times) — may not be everyone's cup of tea, champion Joey "Jaws" Chestnut has thrown down with far worse than just frankfurters. In fact, Nicki Swift details a recent zombie-themed event that Chestnut took part in where, instead of dogs or donuts, the delicacy was a bit more organ-ic.
In an interview with GQ, Chestnut described some of the weirdest items he's ever had to scarf down. Though the competitive eater mentions a kimchi eating contest in Korea that was a bit unusual, it wasn't the memory of fermented cabbage that has really stuck with him through the years. It was the time he had to eat cow's brain. Sesos, which are brains, are actually a common filling for tacos in Mexico, according to Atlas Obscura. The article goes into great detail about the method to extract the meat — usually from goats or cows — that is then braised or grilled.
There are actually some benefits to eating cow brain
During the zombie pub crawl in Minneapolis, Joey Chestnut managed to eat a whopping 54 brain tacos in an incredible eight minutes (via GQ). Though Chestnut thought the flavor wasn't bad, he also said it did leave a metallic taste in his mouth. In another interview with San Diego Union-Tribune, the competitive eater describes the sesos as being pretty good because of all the garlic seared in, yet in the same breath Chestnut used the term "veiny and gelatinous." Not exactly two words you want to hear when describing a dish. Let alone a dish you eat 54 times in a row.
Even if you're an adventurous eater, you may want to weigh the advantages and possible disadvantages of the old cow brain taco. Livestrong says that, on one hand, beef brain contains a type of omega-3 fatty acid that is not only good for possibly preventing cardiovascular disease, but may ironically even help brain function. On the other hand, there is the risk that this type of meat can be infected with mad cow disease, according to the article. Whatever the case may be for Joey Chestnut, even after choking down 54 of these tacos, at least we can be sure he hasn't turned into a real zombie — unless of course, he starts getting an insatiable appetite for brains.


From the best family day out to the most wonderful places to stay, the UK Heritage Awards 2019 has honoured Britain's finest heritage experiences.
This year's UK Heritage Awards winners are crowned across 12 categories, which also include hidden gems, best festival, best wedding venue and outstanding customer service.
The best family day out, declare the judges, is a visit to Kent's Hever Castle and Gardens - once the childhood home of Henry VIII's second wife, Anne Boleyn, and which harbours 700 years of history.
They noted that there's plenty to do for everyone, from exploring the splendid rooms to roaming the 125 acres of grounds and gardens.
For children, there is a playground as well as the challenge of finding their way through a water maze and a 100-year-old Yew Maze.
The panel are also impressed with the fascinating collection of miniature model houses, the boating lake and the warm welcome in the shops and restaurants.
According to the judges, eating at The Alnwick Garden is a wonderful experience, thanks to the Treehouse Restaurant - complete with wobbly rope bridges, wooden walkways and twinkling lights – and its superb mix of Northumberland fare from locally sourced ingredients.
The Alnwick Garden itself is praised for being one of the world's most ambitious new gardens.
A once forgotten plot, it was transformed by Jane Percy, Duchess of Northumberland, into a space the judges say is now a 21st-century experience that's full of imagination, inspiration and fun.
There are more than 4,000 plant varieties to discover, thousands of seasonal blooms and stunning water sculptures.
Set in the beautiful Lake District, Askham Hall is a secluded, luxurious place to visit and relax, say the judges.
It is best known for its 3AA Rosette restaurant, Allium, where the menu is dictated by the cycle of life in the gardens and fields.
The restaurant works with the seasons and grows, sources and preserves the produce from its gardens, surrounding farms and woodland.
Thirlestane is described as a magnificent 16th century castle set in a quiet meadow landscape in the Scottish Borders.
The judges say it is one of the finest castles in Scotland and a simply stunning place to stay, with a fairy-tale façade.
There are eight en-suite bedrooms in five apartments, and all are furnished with a beautifully eclectic mix of contemporary furniture and original items from the castle collections, including dramatic portraits and tactile soft furnishings.
The castle can be let on a self-catered basis or groups can choose to be catered for in the castle's Victorian kitchen or in one of the state rooms.
St Giles is a family house with a long and rich history spanning over 400 years.
Following the Second World War, St Giles House fell into a period of disrepair and appeared on the English Heritage 'Buildings at Risk' register. In 2010, a project to bring the house back to life began.
What started as a simple plan to move the resident Ashley-Cooper family back into the house developed into a full restoration project that is still ongoing.
The house is now open by arrangement and judges were impressed that it runs events to share the vibrancy of the house.
In the hidden gem category, judges jointly awarded the prize to Harvington Hall in Worcestershire and 78 Derngate House in Northamptonshire.
The panel was impressed with Harvington Hall, a moated manor house, thanks to its ingenious secret priest hides - it has more than any other building in the country - and a rare collection of original Elizabethan wall paintings.
Originally built in the 1300s and developed magnificently in the late 1500s, the hall brings to life the fascinating history of the survival of Roman Catholic families and clergymen during the Reformation of the late 16th century.
Meanwhile, 78 Derngate House is the only house in England to have been designed by Charles Rennie Mackintosh.
It is described as a small terraced house that is home to a wonderful history.
It was radically transformed at the hands of Mackintosh between 1916 and 1917 and sensitively restored between 2002 and 2003.
The house next door, No80, was remodeled into a visitor centre and exhibition space.
Castle Howard is one of Britain's finest stately homes in the heart of North Yorkshire and the winner of the best event award for its 12 Days of Christmas festival.
The Castle Howard Christmas event in 2018 highlighted historical traditions and, according to judges, the theme was expertly interwoven with the Howard family history, bringing to life stories such as the 12 children of Lady Georgiana, 6th Countess of Carlisle.
The house was professionally decorated by theatre designers Charlotte Lloyd Webber and Bretta Gerecke, with each room reflecting an element of the classic song. There was also a 26ft tall Christmas tree.
Over 2,000 children were able to meet Father Christmas and there were gourmet festive afternoon teas and a roaring fire.
The award panel said it was 'exceptional organisation' with 'all of this plus live music, magic shows and the Christmas market executed to the highest standards'.
Winner of the contribution to heritage award is the historic Blenheim Palace, which has recently set out a new 10 year plan for the evolution of the estate.
Judges were impressed with its £40million restoration plan and its proposal to acquire key items lost from its collection.
It is hoped that the establishment of an endowment fund for the palace will triple Blenheim's contribution to the local economy.
Described as one of the most beautiful country homes in England, Adlington Hall reflects both the history of English architecture and the 700-year story of a single family, the Leghs, say the judges.
They found that as a wedding venue, Adlington Hall is a totally unique place to get married thanks to two beautiful indoor venues - the magnificent Great Hall and welcoming Hunting Lodge.
There are also two outdoor areas in which to hold the ceremony – the historic walled courtyard or the English Rose Garden.
Judges were also impressed with the dedicated team of wedding coordinators that ensure that everything runs totally smoothly.
Broughton Hall dates back to 1597 and has belonged to the Tempest Baronet family for 900 years.
Judges say it is one of the most exquisite event venues in the UK and one of the only exclusive hire historic houses in Yorkshire.
They add that the historic home creates a uniquely memorable impression for any corporate brand as associating with a house of such legacy offers the perfect backdrop for product launches, celebrations, training or corporate retreats.
It was also praised for the recent addition of The Land Rover Experience to the estate, which gives an added extra for team building and incentives.
Blenheim Palace, the birthplace of Sir Winston Churchill and home to over 300 years of history, scoops a second award by winning the must visit category.
Judges say there is just oodles to discover, from the Palace State Rooms and a Winston Churchill exhibition inside, to formal gardens, pleasure gardens and a butterfly house outside.
They also praised the eating options and tempting shops.
Meanwhile coming highly commended in the must visit category is the Malvern Hills, which boasts exhilarating climbs with fantastic views as well as gentle strolls through colourful wildflower meadows.
Judges advise walking, cycling or horse riding through the area.
It is also described as a place where man's activities can combine with nature in a cherished cultural landscape.
The winner of the outstanding customer experience award is based on not only the judges' experiences with the venue but also from outstanding client feedback.
Adlington Hall came out on top across all areas including: organisation, food and service.
It adds the accolade to its other prize for the best wedding venue.
The Guild Chapel, winner of the judges award, is described as one of Stratford-upon-Avon's most iconic and important historic buildings.
Its rich history dates back to the 13th century when it was built by the Guild of the Holy Cross, a prominent social and religious organisation.
As the Guild grew and prospered, so did its chapel, becoming central to medieval life in Stratford-upon-Avon.
Today, almost half a millennia later, the Chapel's rare and special wall paintings offer a unique glimpse back to that medieval past.
The judges were impressed with the dedication of the trust in securing Heritage Lottery funding to renovate these rare and icon paintings and the efforts to encourage and educate the local volunteers to ensure visitors enjoy this vital piece of Shakespeare's history.
Winner of the individual award is Chris Ridgway, who has worked at Castle Howard since 1984.
Chris has published works on Castle Howard's collections, architecture, landscape and history and told its story through lectures both nationally and internationally.
The judges wanted to recognise this and his work as the chair of the Yorkshire Country House Partnership.
The Partnership has a core of 12 major country houses across the region and associations with others in relation to specific projects.
Spanning more than six centuries in origin and an impressive range of architectural styles, the houses are united by both their geography and their historical roles at the heart of rural society.
It also provides a practical support network for house curators, which has resulted in many coordinated exhibitions for visitors to enjoy in Yorkshire.
Britain has a gorgeous countryside.


The neighborhood is relatively clean and the room is unexpectedly spacious. And the best thing about this place is that they have NETFLIX free access!
The staff was very helpful whenever we had a question. We loved the location! If you go just a few blocks away you hit an entire strip of restaurants and stores along Chestnut. There are great breakfast places and stores like Walgreens all within a short walk. The room exceeded our expectations of cleanliness.
Mr. Ash and Ed were super kind and helpful, great customer service!
Basic but adequate accommodations. Good for the price.
My partner and I live in this city, we booked a room to escape our home which is being renovated for a couple nights. We loved it and booked an extra day. The rates are excellent and the neighborhood is great. We took advantage of nearby late night dining i recommend Brazenhead and Delarosa! The staff here is incredibly accommodating. Ash, the manager is very friendly, let us check in 2 hours early at no charge and really keeps an eye out for all guests. He greeted me both nights upon my return with good wishes, a big friendly smile and pleasant conversation. Next time we need a "staycation" away from home we will certainly return. Located a block from gate to Presidio, we walked over to the beautiful golf course where they will rent clubs for cheap and there's a great driving range. Also its close to walking or biking iconic Golden Gate bridge. Highly recommended! There is no better deal in SF!
Location was perfect. Beach and bars very close. Short drive to Lombard street.
The property is well located and quiet. Staff is excellent. Clean and good size rooms.
Knights Inn This rating is a reflection of how the property compares to the industry standard when it comes to price, facilities and services available. It's based on a self-evaluation by the property. Use this rating to help choose your stay!
Located just 1,000 feet away from Presidio National Park, this hotel offers guest rooms with free WiFi. Free guest parking is available. San Francisco's iconic Golden Gate Bridge is just 1.9 mi away.
Knights Inn features amenities such a coffee machine and a flat-screen cable satellite TV in each room. Guests can take advantage of the hotel's free local calls.
Points of interest such as the Palace of Fine Arts, Chinatown and Fisherman's Wharf are only moments from San Francisco Knights Inn. Guests can also explore the shops and restaurants of Union Square, 3 mi away.
When would you like to stay at Knights Inn?
This air-conditioned room is furnished a cable satellite TV and a coffee machine.
Area Info – This neighborhood is a great choice for travelers interested in harbors, tourist attractions and scenery – Check location Good location – show map Guests loved walking around the neighborhood!
House Rules Knights Inn takes special requests – add in the next step!
Knights Inn accepts these cards and reserves the right to temporarily hold an amount prior to arrival.
Please note: Knights Inn is located on Lombard Street between Baker and Broderick.
The parking spots are limited and the size is kind of narrow and small. Make sure you have a small vehicle.
After leaving towles on floor. They simply took and did not replace. Found the maid asleep in a closet.
Noisy, and I felt itchy the whole time - though I think this may have been psychological.
We really didn't come encounter with much we didn't like! My husband wishes he would've had more pillows on the bed, but I believe we were both well pleased with the room and accommodations.
Terrible towels. Didn't look clean. There was a wire/cord taped together under bed. The toilet was not clean. Poor flooring and bad remodel of bathroom.
More power outlets to charge phones etc.. room can be brighter with more lights?
Extremely friendly helpful staff. Fast response. Hotel room - clean sheets but wooden floor needs to be mopped properly. Toilet/ dressing area is spacious there is a small walking wardrobe. Good location. Presedio park gate (for presidio free shuttle) is just 3 mins walk from hotel.
Pretty good location, friendly staff.


text:
Office of the President /
GREAVES, Lorraine
Lorraine GREAVES
D. University 2007
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Lorraine Greaves, Director General, BC Centre of Excellence for Women's Health, has played a pioneering role in addressing neglected and emerging women's health issues – such as addictions, tobacco use, mental health, and violence against women – in a collaborative and multidisciplinary way. Her leadership in bridging biomedical and social sciences, and integrating sex and gender analysis into Canadian health research, is recognized internationally.
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text:
iPhone 3GS supplies running low
Will be discontinued in the autumn, reports claim
Ben Camm-Jones (Macworld U.K.) 03 August, 2011 01:41
The days of the iPhone 3GS look to be numbered with reports suggesting that supplies of the device are running low ahead of a planned discontinuation of the product in the autumn.
A number of suppliers and mobile networks around the world are reportedly seeing dwindling numbers of the iPhone 3GS, ahead of the rumoured launch of a next-generation iPhone in September.
A source at an Orange UK store told 9to5Mac that employees have been advised that the iPhone 3GS is no longer available. You can still order an iPhone 3GS through the Orange UK website though.
Indeed, O2 , Vodafone, Three and T-Mobile all still have the iPhone 3GS listed on their sites. You can also still buy an iPhone 3GS through the Apple UK Store SIM-free for £428.
The next-generation iPhone is expected to launch in September and though some reports suggested that Apple might keep the iPhone 3GS as its low-cost option when the iPhone 5 launched, evidence seems to suggest it will be scrapped.
It was reported last month that Apple would launch not one but two new iPhones in September, one of which would be a low-cost model, which if true would put the final nail in the iPhone 3GS' coffin. However, 9to5Mac speculates whether the reduced supply could hint at Apple giving the 3GS a makeover, upgrading the screen or processor.
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A wet day it was.
While the adventure started on a sunny beach just outside the luxurious resort it ended in a wet experience.
The Crew™ met Chief Pirate One Leg Kidman and got instructed on what they need to visit when NOT crossing the border.
We continued up North and eventually we landed at the Atlantic.
Or euhh was it wet.


Going through a divorce or separation can at times feel overwhelming. While divorce is never a simple process, working with an experienced attorney can help. I have been helping clients throughout North Carolina resolve divorce issues for over 25 years.
If you are going through a divorce or seeking separation, contact me today to schedule an appointment. Call The Law Office of Kevin R. Brackett for straightforward divorce help.
For a divorce to be granted in North Carolina, a couple must live separate and apart for at least a year, with one of the parties having lived in the state for at least six months prior to filing. I can help you navigate the legal processes involved with a divorce and make the process more efficient and affordable.
We are increasingly seeing individuals deciding to start families at a later date and subsequently divorcing later in life. Such individuals may be bringing into their marriages more finances, real estate and other assets. Also, couples are now marrying after having children from a previous marriage.
Under such situations, it is important to have a family law lawyer on your side with the experience to deal with the unique circumstances of your case. I am an attorney who regularly handles high-asset divorces and who understands the facets of North Carolina's divorce law. More questions and answers about divorce and family law matters are on the FAQ page.
If domestic violence is involved, there are legal remedies that enable a spouse to get immediate ex parte relief without a court hearing or notice to the other party. If you feel like your safety, or the safety of your children, is compromised then contact me as soon as possible.
You can contact my law office in Greensboro by calling 336-273-8595. I am available to speak to you by appointment 9 a.m. to 6 p.m. Monday through Friday. I am also sometimes available during evenings and weekends under advance arrangement.


London Film Museum or The Movieum of London for tourists to explore cinema history, film studios, 'How Films Are Made' feature, artefacts on Batman, a spear, Superman tunic etc.
London Film Museum otherwise known as The Movieum of London allows tourists to explore the history of cinema. Tourist can learn about the great film studios, from Pinewood to MGM; and the 'How Films Are Made' feature, where tourist can discover the step-by-step process of making a movie. It is also home to some terrific artefacts, including Jack Nicholson's costume from Batman, a spear from Zulu, and Christopher Reeves' Superman tunic.
The London Film Museum is the only film museum of its kind in Great Britain, supporting the Film Industry and the talent within it. It was founded and created by Jonathan Sands in Feb 2008 following the success of Star Wars The Exhibition. About 50% of the museum's collection is from his private archive.
The London Film Museum's highlights include the 'Made in Britain' exhibition, which is dedicated to the blockbuster films that have been made in the UK, including Star Wars, Indiana Jones, Superman, and Alien. There is a special exhibition on the James Bond Film.
Tickets can be purchased in advance online or at the venue box office daily. There is good access with entrance from street level and a lift to all areas of the museum for disabled guests. The carer will get free access. Personal photography is allowed but with no flash.


It may be deep mid-winter, but the arts and flora will be in full bloom at the Worcester Art Museum and Tower Hill Botanic Garden in Boylston as the 11th annual "Flora in Winter" runs from Jan. 31 through Feb. 3.
"Flora Euphoria" will get matters under way from 5:30 to 8 p.m. Jan. 31 at WAM with entertainment, music, hors d'ouevres and a cash bar. Admission is $50, $30 for members. Reservations recommended; register online at www.worcesterart.org or call (508) 799-4406 x3123.


T-Mobile has HotSpot @Home, Sprint has AIRAVE, and Verizon has its Wireless Network Extender, leaving just AT&T among the States' big four carriers without a launched WiFi or femtocell solution for extending signals into the home -- but it looks like that's finally getting close to changing. We know that they've been in the process of trialing some units recently, and tipsters have observed that there's now a pretty slick site launched on AT&T's domain for its 3G MicroCell, an indication that they might be looking to go retail eventually. There's quite a bit of detail here; from the picture, we can make out that the unit comes from Cisco (versus Samsung for Sprint and Verizon), and like its competitors, the MicroCell will require a broadband connection to operate. It'll cover up to 5,000 square feet, allow up to four simultaneous voice or data connections (locked down so that your neighbors can't pilfer the signal), and most interestingly, will only work with 3G phones. We've heard that femtocells are more difficult to manage in a 2G GSM environment than in CDMA and WCDMA -- hence the 3G requirement -- but the cells offered by Sprint and Samsung only offer 2G coverage, so AT&T's arguably got an advantage here. We still don't know exactly when this is coming or for how much dough, but the site makes mention of a "3G MicroCell service plan," so we'd count on a fee for the pleasure of extending AT&T's network on their behalf. Follow the break for AT&T's full rundown of the device.
What is an AT&T 3G MicroCell™?
AT&T 3G MicroCell acts like a mini cellular tower in your home or small business environment. It connects to AT&T's network via your existing broadband internet service (such as DSL or cable) and is designed to support up to 10 3G capable wireless phones in a home or small business setting. With AT&T 3G MicroCell, you receive improved cellular signal performance for both voice calls and cellular data applications, like picture messaging and surfing the web for up to 4 simultaneous users.
* Enhanced coverage indoors - supports both voice and data up to 5000 square feet.
* Available unlimited minute plans - Individual or Family Plan.
* 3G handset compatible - works with any AT&T 3G Phone.
* Up to 4 simultaneous voice or data users supported.
* Seamless call hand-over - start calls on your 3G MicroCell and continue uninterrupted even if you leave the building.
* Installing your device near a window is strongly recommended to ensure access to Global Positioning System (GPS). A GPS link is needed to verify the device location during the initial startup.
* The 3G MicroCell device is portable. The device may be moved, provided the new location is within the AT&T authorized service area and properly registered online.


text:
WHO'S WHO & WHO WAS WHO
Royalty, rulers, and aristocracy
Printed from WHO'S WHO & WHO WAS WHO. Under the terms of the licence agreement, an individual user may print out a single article for personal use.
Abercrombie, Lascelles
https://doi.org/10.1093/ww/9780199540884.013.U205045
FBA 1937; MA (Liverpool and Oxford); Hon. LittD (Camb. and Manchester); Hon. DLit (Belfast)
Born 9 Jan. 1881; 6th s of William Abercrombie, Ashton-on-Mersey, Cheshire; ...
Text of WHO'S WHO and WHO WAS WHO copyright © A & C Black, an imprint of Bloomsbury Publishing plc. All other web site content copyright © Oxford University Press 2020.