text:
\section{Introduction}
The Large Hadron Collider (LHC) will hopefully soon discover the underlying theory for the TeV scale and might allow us to understand a more fundamental law of nature.
For more than three decades the idea of Supersymmetry has attracted the attention of many experts in the particle physics community
and the minimal supersymmetric extension of the Standard Model~\cite{MSSM1,MSSM2,MSSM3} (MSSM) is still considered one of the most appealing candidates for the theory of
particle physics at the TeV scale. It is well known that the MSSM provides an understanding of why the SM-like Higgs boson is light, contains a
cold dark matter candidate, allows for the unification of the gauge couplings and allows for the mechanism
of electroweak baryogenesis to explain the baryon asymmetry in the universe.

There are several open issues in the MSSM, one of them being the origin of the discrete symmetry R-parity~\cite{R1,R2}.
This symmetry plays a major role in the MSSM and it is defined as $R=(-1)^{3(B-L)+2S}$, where $B$, $L$ and $S$ stand for baryon number,
lepton number and spin, respectively. In many MSSM studies it is assumed that R-parity is conserved or explicitly broken
without understanding the origin of this symmetry. However, the fate of R-parity
is crucial for the discovery of supersymmetry since, as is well known, R-parity conservation give rise to channels with multi-jets, multi-leptons and missing
energy at the LHC, while signatures of broken R-parity are multi-leptons, multi-jets, and missing energy due to the SM neutrinos only.

The simplest and most elegant framework for the origin of R-parity is based on local B-L symmetry. This connection was explored for the first time in Ref.~\cite{Hayashi}, and in Ref.~\cite{Mohapatra} a simpler scenario was studied.
See also Ref.~\cite{Martin} for a complete discussion of how to gauge R-parity.\footnote{It is important to mention that the breaking of B-L in the context of the MSSM was studied for the first time in Ref.~\cite{AM}.
See also Refs.~\cite{Martin2,Masiero,Goran1,Goran2} for the study of R-parity in other models.}
Recently, we have investigated the simplest B-L models in
Refs.~\cite{Rp1,Rp2,Rp5,Rp6} and found the following main result

\begin{center}
{\it{The simplest theories based on local B-L make the following prediction:  \\
R-parity must be spontaneously broken at the TeV scale and  \\ one expects to observe lepton number violation at the LHC ! }}
\end{center}
In this letter we study in detail the theory proposed in Ref.~\cite{Rp2} which can be considered as the simplest gauge theory for R-parity violation.
In this context the only way to break local B-L and obtain the MSSM after symmetry breaking is to give a vacuum expectation value to one of
the right-handed sneutrino required by anomaly cancellation. One of the most
important features of this theory is that the B-L and R-parity breaking scales are determined by the soft supersymmetric breaking scale.
This idea was studied for the first time in Ref.~\cite{Rp1} which defined the simplest left-right symmetric model.

We review the theory and symmetry breaking mechanism in Sections~II and III. The full spectrum of the theory~\cite{Rp2} is discussed in Section~IV and the constraints coming from neutrino masses in Section~VI. We discuss the proton decay issue and the possible dark matter candidates in Section~V. In order to understand the testability of the theory
we study the properties of the new gauge boson, the neutralino decays in Section~VII and in Section~VIII the main production channels for the charged sleptons at the Large Hadron Collider. We find that the channels with four charged leptons (three with the same electric charge) and four jets give us the most striking signals
for the testability of lepton number violation at the LHC.

\section{The Minimal Gauge Theory for Spontaneous R-parity Violation}
The simplest gauge theory for spontaneous R-parity breaking was proposed in Ref.~\cite{Rp2}. In this context one can understand dynamically the origin of the
R-parity violating terms in the MSSM. Here we discuss the structure of the theory and the full spectrum.
\begin{itemize}
\item {\it \underline{Gauge Symmetry and Matter Fields}}: This theory is based on the gauge group $$SU(3)_C \otimes SU(2)_L \otimes U(1)_Y \otimes U(1)_{B-L},$$
and the different matter chiral superfields are given by
\begin{equation}
\hat{Q} = \left(
\begin{array} {c}
\hat{u} \\ \hat{d}
\end{array}
\right) \ \sim \ (2,1/3,1/3),
\ \
\hat{L} = \left(
\begin{array} {c}
 \hat{\nu} \\ \hat{e}
\end{array}
\right) \ \sim \ (2,-1,-1),
\end{equation}
\begin{equation}
\hat{u}^c \ \sim \ (1,-4/3,-1/3),
\ \
\hat{d}^c \ \sim \ (1,2/3,-1/3),
\ \
\hat{e}^c \ \sim \ (1,2,1).
\end{equation}
In order to cancel the $B-L$ anomalies one introduces three chiral superfields for the right-handed neutrinos:
\begin{equation}
\hat{\nu}^c \ \sim \ (1,0,1).
\end{equation}
\item {\it \underline{Higgs Sector}}: The Higgs sector is composed of two Higgs chiral superfields as in the MSSM
\begin{equation}
\hat{H}_u = \left(
\begin{array} {c}
	\hat{H}_u^+
\\
	\hat{H}_u^0
\end{array}
\right) \ \sim \ (2, 1, 0),
\ \ \
\hat{H}_d = \left(
\begin{array} {c}
	\hat{H}_d^0
\\
	\hat{H}_d^-
\end{array}
\right) \ \sim \ (2, -1, 0).
\end{equation}

\item {\it \underline{Interactions}:} With this field content the superpotential reads as
\begin{equation}
{\cal W}_{BL}={\cal W}_{MSSM} \ + \ Y_\nu \ \hat{L}^T \ i \sigma_2 \ \hat{H}_u \ \hat{\nu}^c,
\end{equation}
where
\begin{eqnarray}
	{\cal W}_{MSSM} &=& Y_u \ \hat{Q}^T \ i \sigma_2 \ \hat{H}_u \ \hat{u}^c
\ + \ Y_d \ \hat{Q}^T \ i \sigma_2 \ \hat{H}_d \ \hat{d}^c
\ + \ Y_e \ \hat{L}^T \ i \sigma_2 \ \hat{H}_d \  \hat{e}^c
\ + \ \mu \ \hat{H}_u^T \ i \sigma_2 \  \hat{H}_d. \nonumber \\
\end{eqnarray}
In addition to the superpotential, the model is also specified by the soft terms:
\begin{eqnarray}
\nonumber
V_{soft} & = & m_{\tilde \nu^c}^2 \abs{\tilde{\nu}^c}^2 \ + \ m_{\tilde L}^2 \ \abs{\tilde L}^2 \ + \ m_{\tilde e^c}^2 \ \abs{\tilde e^c}^2
\ + \ m_{H_u}^2 \abs{H_u}^2 + m_{H_d}^2 \abs{H_d}^2 \ + \ \left( \frac{1}{2} M_{BL} \tilde{B^{'}} \tilde{B^{'}} \right.
  	\nonumber
	\\
	& + &  \left. A_\nu \ \tilde{L}^T \ i \sigma_2 \ H_u \ \tilde{\nu}^c  \  + \  B\mu \ H_u^T \ i \sigma_2 \ H_d
		\ + \  \mathrm{h.c.} \right) \ + \ V_{soft}^{MSSM},
\label{soft}
\end{eqnarray}
where the terms not shown here correspond to terms in the soft MSSM potential.

Since we have a new gauge symmetry in the theory we need to modify the kinetic terms for all MSSM matter superfields,
and include the kinetic term for right-handed neutrino superfields
\begin{equation}
{\cal{L}}_{Kin} (\nu^c) = \int d^2 \theta d^2 \bar{\theta}  \   (\hat{\nu}^c)^\dagger  e^{g_{BL} \hat{V}_{BL}}  \hat{\nu}^c.
\end{equation}
Here $\hat{V}_{BL}$ is the B-L vector superfield. Using these interactions we can study the full spectrum of the theory.

\end{itemize}

\section{Electroweak and B-L Symmetry Breaking}
As in the MSSM, electroweak symmetry is broken by the vevs of $H_u^0$ and $H_d^0$, while $U(1)_{B-L}$
is broken due to the vev of right-handed sneutrinos. Notice that this is the only field which can break local $B-L$ and
give mass to the new neutral gauge boson in the theory. Therefore, the theory predicts spontaneous R-parity violation.
It is important to mention that the $B-L$ and R-parity breaking scales are determined by the soft supersymmetric breaking scale, and one must
expect lepton number violation at the LHC.

The neutral fields are defined as
\begin{eqnarray}
H_u^0 &=& \frac{1}{\sqrt{2}} \left(  v_u \ + \ h_u \right) \ + \ \frac{i}{\sqrt{2}} A_u, \\
H_d^0 &=& \frac{1}{\sqrt{2}} \left(  v_d \ + \ h_d \right) \ + \ \frac{i}{\sqrt{2}} A_d, \\
\tilde{\nu}^i &=& \frac{1}{\sqrt{2}} \left(  v_{L}^{i} \ + \  h_L^{i}   \right) \ + \ \frac{i}{\sqrt{2}} A_L^{i} , \\
\tilde{\nu}^c_i &=& \frac{1}{\sqrt{2}} \left(  v_{R}^{i} \ + \  h_R^{i}   \right) \ + \ \frac{i} {\sqrt{2}} A_R^{i},
\end{eqnarray}
and the relevant scalar potential reads as
\begin{eqnarray}
V &=& V_F \ + \ V_D \ + \ V_{soft}, \\
V_F &=&  |\mu|^2 |H_u^0|^2 \ + \  | - \mu H_d^0 + \tilde{\nu}_i Y_\nu^{ij} \tilde{\nu}^c_j |^2 \ + \ \sum_{i}  | Y_\nu^{ij} \tilde{\nu}^c_j|^2 |H_u^0|^2  \ + \ \sum_{j}  |\tilde{\nu}_i   Y_\nu^{ij}|^2 |H_u^0|^2, \\
V_D &=& \frac{(g_1^2 + g_2^2)}{8} \left(  |H_u^0|^2 - |H_d^0|^2 - \sum_{i} |\tilde{\nu}_i|^2 \right)^2 \ + \ \frac{g_{BL}^2}{8} \left( \sum_{i} ( |\tilde{\nu}^c_i|^2 - |\tilde{\nu}_i|^2 ) \right)^2, \\
V_{soft} &=& (\tilde{\nu}^c_i)^\dagger m_{\tilde{\nu}^c_{ij}}^2 \tilde{\nu}^c_j \ + \  \tilde{\nu}_i^\dagger m_{\tilde{L}_{ij}}^2 \tilde{\nu}_j \ + \  m_{H_u}^2 |H_u^0|^2 \ + \ m_{H_d}^2 |H_d^0|^2
\ + \ \left( \tilde{\nu}_i  a_\nu^{ij} \tilde{\nu}_j^c H_u^0 -  B \mu H_u^0 H_d^0 \ + \  \rm{h.c.}\right). \nonumber \\
\end{eqnarray}
Using the above scalar potential and assuming that all parameters are real we can find the minimization conditions
\begin{align}
	& v_u \left[  \mu^2 \ + \  \frac{1}{2} Y_\nu^{ij}  v_R^j Y_\nu^{ik} v_R^k \ + \  \frac{1}{2} v_L^i Y_\nu^{ij} v_L^k Y_\nu^{kj}
	\ + \ \frac{g_1^2 + g_2^2}{8} \left( v_u^2 - v_d^2 - v_L^i v_L^i \right) \ + \ m_{H_u}^2  \right]
\nonumber
\\
	\label{Eq1}
	& +  \frac{1}{\sqrt{2}} v_L^i a_\nu^{ij} v_R^j \ - \ B \mu v_d =0,
\\
\label{Eq2}
	& v_d \left[  \mu^2 \ - \ \frac{(g_1^2 + g_2^2)}{8} \left( v_u^2 - v_d^2 - v_L^i v_L^i \right) + m_{H_d}^2 \right]
	\ - \ \frac{1}{\sqrt{2}} \mu v_L^i Y_\nu^{ij} v_R^j \ - \ B \mu v_u =0,
\end{align}
\begin{align}
	& \frac{1}{2} v_L^i Y_\nu^{ij} v_R^j v_L^m Y_\nu^{mk} \ - \ \frac{1}{\sqrt{2}} \mu v_d v_L^i Y_\nu^{ik} \ + \ \frac{1}{2}
	v_u^2  Y_\nu^{ij} v_R^j Y_\nu^{ik} \ + \ \frac{g_{BL}^2}{8} \left( v_R^i v_R^i - v_L^i v_L^i \right) v_R^k
\nonumber
\\
	\label{Eq3}
	& \frac{1}{2} v_R^i \left[ (m_{\tilde{\nu}^c}^2)_{ki} + (m_{\tilde{\nu}^c}^2)_{ik} \right] \ + \
	\frac{1}{\sqrt{2}} v_L^i a_\nu^{ik} v_u =0,
\\
	& \frac{1}{2} v_L^i Y_\nu^{ij} v_R^j Y_\nu^{km} v_R^m \ - \ \frac{1}{\sqrt{2}} \mu v_d Y_\nu^{kj} v_R^j  \ + \ \frac{1}{2}
	v_u^2 v_L^i Y_\nu^{ij} Y_\nu^{kj} - \frac{(g_1^2 + g_2^2)}{8} \left( v_u^2-v_d^2 - v_L^i v_L^i \right) v_L^k
\nonumber
\\
	\label{Eq4}
	& -  \frac{g_{BL}^2}{8} \left( v_R^i v_R^i - v_L^i v_L^i \right) v_L^k \ + \ \frac{1}{2} v_L^i \left[ (m_{\tilde{L}}^2)_{ki}
	+ (m_{\tilde{L}}^2)_{ik} \right] \ + \ \frac{1}{\sqrt{2}} a_\nu^{kj} v_R^j v_u=0.
\end{align}
In order to have phenomenological allowed solutions the $v_L^i$ have to be small, and the $v_R^i$ have
to be much larger than $v_u, v_d$ and $v_L^i$. Up to negligibly small terms\footnote{The size would go as $\frac{Y_\nu \mu v_d v_L}{m_{\tilde \nu^c}^2} < 10^{-10}$. The maximum values for $Y_\nu$ and $v_L$ are about $10^{-6}$ and $10^{-2}$ GeV, respectively, see Fig.~\ref{vy}.} the right-handed sneutrino acquire a vev in only one family. A possible solution and the one used throughout this paper is $v_R^i=(0,0,v_R)$. In this case:
\begin{eqnarray}
\label{vR.sln}
v_R^2 &\approx& -\frac{8 (m_{\tilde{\nu}^c}^2)_{33}}{g_{BL}^2}, \\
v_L^k &\approx & \frac{v_R}{\sqrt{2}} \frac{\left( \mu v_d Y_\nu^{k3} - a_\nu^{k3} v_u \right)}{\left[ (m_{\tilde{L}}^2)_{kk} - \frac{(g_1^2 + g_2^2)}{8} (v_u^2 - v_d^2) - \frac{g_{BL}^2}{8} v_R^2\right]}.
\end{eqnarray}
Notice that the minimization conditions for $v_u$, Eq.~(\ref{Eq1}), and $v_d$, Eq.~(\ref{Eq2}) are not greatly altered from their MSSM equivalents since the extra terms are very small.

\subsection{Radiative Symmetry Breaking}
In the MSSM, the large top Yukawa coupling drives the up-type soft Higgs mass squared parameter to negative values for generic boundary conditions leading to radiative electroweak symmetry breaking~\cite{Ibanez:1982fr}; a celebrated success of the MSSM. A valid question is then if the same success is possible in achieving a tachyonic right-handed sneutrino mass in this $B-L$ model as required by Eq.~(\ref{vR.sln}).  Unfortunately, this is not possible through a large Yukawa coupling since the Yukawa couplings of the right-handed neutrino are all dictated to be small by neutrino masses.  However, there is an alternate possibility whereby a positive mass squared parameter for the right-handed sneutrino at the high scale will run to a tachyonic value at the low scale. This is due to the presence of the so-called $S$-term (due to $D$-term contributions to the RGE) in the soft mass RGE, as discussed for this $B-L$ model in~\cite{Ambroso:2009jd, Ambroso:2009sc, Ambroso:2010pe}. A short outline of the mechanism follows.

The RGE for the right-handed sneutrino soft mass squared parameter is
\begin{equation}
	16 \pi^2 \frac{d m_{\tilde \nu^c}^2}{d t} = - 3 g_{BL}^2 \left|M_{BL}\right|^2 + \frac{3}{4} g_{BL}^2 S_{BL},
\end{equation}
with
\begin{equation}
	S_{BL} = \text{Tr}\left(2 m_{\tilde Q}^2 - m_{\tilde u^c}^2 - m_{\tilde d^c}^2 - 2 m_{\tilde L}^2 + m_{\tilde e^c}^2 + m_{\tilde \nu^c}^2\right),
\end{equation}
where the trace is over the three generations of the fermions and the soft mass parameters in the trace are for the squark doublet, right-handed up squark, right-handed down squark, slepton doublet, right-handed charged slepton and right-handed sneutrino, respectively. The gaugino mass term always drives the sneutrino mass parameter positive at the low scale but if the overall sign of the $S$-term is positive, it could lead to the opposite effect. Such an effect would require a non-zero $S$-term at the high scale, which is not possible if the soft masses are universal across the generations of each flavor. An example of a suitable boundary condition with minimal variation from the popular MSUGRA Ansatz is universal boundary conditions for all sfermions except for the right-handed sneutrinos, which might have the boundary conditions
\begin{equation}
	m_{\tilde \nu^c_1}^2 = m_{\tilde \nu^c_2}^2 = P \, m_0^2,
	\quad \quad m_{\tilde \nu^c_3}^2 = Q \, m_0^2,
\end{equation}
where $m_0$ is the universal mass and $P > 1$ and $Q<1$. The boundary condition for $S_{BL}$ is then
\begin{equation}
	S_{BL} = (2 P + Q - 1)m_0^2,
\end{equation}
having the necessary sign to contribute negatively to the sneutrino soft mass parameter as it is evolved from the high scale down. The necessary sizes of $P$ and $Q$ depend on the size of the gaugino mass parameter which has the opposite effect, see~\cite{Ambroso:2009jd, Ambroso:2009sc, Ambroso:2010pe} for more details. So while, the traditional radiative symmetry breaking from universal boundary conditions is not possible in these models, it is possible to radiatively break $B-L$ through this $S$-term starting from a positive value. For the implementation of the radiative mechanism in the non-minimal model see Ref.~\cite{Fate}.

\section{Mass Spectrum and Lepton Number Violation}

\subsection{ R-Parity Violating Interactions}
After symmetry breaking lepton number is spontaneously broken in the form of bilinear R-parity violating interactions. There are no trilinear R-parity violating interactions at the renormalizable level. These bilinear interactions mix the leptons with the Higgsinos and gauginos:
	$$\frac{1}{2} g_{BL} v_{R} ( \nu_3^c \tilde B^{'} ), \quad
	\frac{1}{2} g_{2} v_L^i ( \nu_i \tilde W^0 ), \quad
	\frac{1}{\sqrt{2}} g_{2} v_L^i (e_i \tilde W^+ ),$$
	$$\frac{1}{2} g_{1} v_L ^i( \nu_i \tilde B ), \quad
	\frac{1}{\sqrt{2}} Y_\nu^{i3} v_R ( L^T_i i\sigma_2 \ \tilde{H}_u ), \quad
	\frac{1}{\sqrt{2}} Y_\nu^{i3} v_L^i ( \tilde{H}_u^0 \ \nu^c_3  ), \quad
	\frac{1}{\sqrt{2}} Y_e^i v_L^i ( \tilde{H}_d^- \ e^c_i ).$$
The first term is new and is the only term not suppressed by neutrino
masses. The fifth term corresponds to the so-called $\epsilon$ term,
and second, third and fourth terms are small but important for the decay
of neutralinos and charginos. See Section~V for the discussion of the baryon number violating operators.

There are also lepton number violating interactions coming from the soft terms and the B-L D-term.
From $V_{soft}$ one gets
$$ A_\nu^{i3} \frac{v_R}{\sqrt{2}} \tilde{L}_i^T \ i \sigma_2 \ H_u,$$
while from the D-term one finds
$$g_{BL}^2 v_R \ \tilde{\nu}^c \left( \tilde{q}^\dagger \frac{1}{6} \tilde{q} \ - \  \tilde{l}^\dagger \frac{1}{2} \tilde{l} \right).$$
As one can expect these terms are important to understand the scalar sector of the theory.

\subsection{Mass Spectrum}

\underline{Gauge Boson:}
The neutral gauge boson associated to the $B-L$ gauge group is $Z_{BL}$.
Using the covariant derivate for the right-handed sneutrinos, $D_\mu \tilde{\nu}^c = \partial_\mu \tilde{\nu}^c + i \frac{g_{BL}}{2} B_\mu^{'} \tilde{\nu}^c$,
the mass term for $Z_{BL}$ is:
\begin{equation}
	M_{Z_{BL}}=\frac{g_{BL}}{2} v_R.
\end{equation}
Now, using the experimental collider constraint~\cite{Carena:2004xs}:
\begin{equation}
\label{ZBL.const}
	\frac{M_{Z_{BL}}}{g_{BL}} \geq 3 \text{ TeV},
\end{equation}
and Eq. (21) one finds the condition
\begin{equation}
	| (m_{\tilde{\nu}^c)_{33}} | > 2.12 \ g_{BL} \ \rm{TeV}.
\end{equation}
Then, if $g_{BL} =0.1$ the soft mass above has to be larger than 200 GeV.
This condition can be easily satisfied without assuming a very heavy spectrum for the supersymmetric particles.

\underline{Neutralinos and Neutrinos:}
As in any supersymmetric theory where $R$-parity is broken all the fermions with the same quantum numbers mix and
form physical states which are linear combinations of the original fields. The neutralinos in this theory are a linear combination
of the  fields, $\left(\nu_i, \ \nu^c_j, \ \tilde B', \ \tilde B, \ \tilde W^0, \ \tilde H_d^0, \ \tilde H_u^0\right)$. Then, the neutralino mass matrix is given by
\begin{equation}
	{\cal M}_{N} =
	\begin{pmatrix}

			0
		&

			\frac{1}{\sqrt{2}} \ Y_\nu^{ij} v_u
		&

			-\frac{1}{2} g_{BL} \ v_L^i
		&

			-\frac{1}{2} g_1 \ v_L^i
		&

			\frac{1}{2} g_2 \ v_L^i
		&

			0
		&

			\frac{1}{\sqrt{2}} \ Y_\nu^{ij} \ v_R^j
	\\

			\frac{1}{\sqrt{2}} \ Y_\nu^{ij} \ v_u
		&

			0
		&

			\frac{1}{2} g_{BL} \ v_R^j
		&

			0
		&

			0
		&

			0
		&

			\frac{1}{\sqrt{2}} \ Y_\nu^{ij} \ v_L^i
	\\

			-\frac{1}{2} g_{BL} \ v_L^i
		&

			\frac{1}{2} g_{BL} \ v_R^j
		&

			M_{BL}
		&

			0
		&

			0
		&

			0
		&

			0
	\\

			-\frac{1}{2} g_1 \ v_L^i
		&

			0
		&

			0
		&

			M_1
		&

			0
		&

			-\frac{1}{2} g_1 v_d
		&

			\frac{1}{2} g_1 v_u
	\\

			\frac{1}{2} g_2 \ v_L^i
		&

			0
		&

			0
		&

			0
		&

			M_2
		&

			\frac{1}{2} g_2 v_d
		&

			-\frac{1}{2} g_2 v_u
	\\

			0
		&

			0
		&

			0
		&

			-\frac{1}{2} g_1 v_d
		&

			\frac{1}{2} g_2 v_d
		&

			0
		&

			-\mu
	\\

			\frac{1}{\sqrt{2}} \ Y_\nu^{ij} \ v_R^j
		&

			\frac{1}{\sqrt{2}} \ Y_\nu^{ij} \ v_L^i
		&

			0
		&

			\frac{1}{2} g_1 v_u
		&

			-\frac{1}{2} g_2 v_u
		&

			-\mu
		&

			0
	\end{pmatrix}.
\label{neutralino}
\end{equation}
We have discussed above that only one right-handed sneutrinos get a vev, $v_R^i=(0,0,v_R)$.
Now, integrating out the neutralinos one can find the mass matrix for the light neutrinos. In this case
one has three active neutrinos and two sterile neutrinos, and the mass matrix in the basis
$(\nu_e, \nu_\mu, \nu_\tau, \nu^c_e, \nu^c_\mu)$ is given by
\begin{widetext}
\begin{equation}
\label{Mnu}
M_\nu =
\begin{pmatrix}
	A \ v_{L}^i v_{L}^j
	+ B \ \left[Y_\nu^{i3} v_{L}^j + Y_\nu^{j3} v_{L}^i \right]
	+ C \ Y_\nu^{i3} Y_\nu^{j3}
	&
	\frac{1}{\sqrt{2}} v_u Y_\nu^{i \beta}
\\
	\frac{1}{\sqrt{2}} v_u Y_\nu^{\alpha j}
	&
	O_{2\times2}
\end{pmatrix},
\end{equation}
where
\begin{align}
	A & = \frac{2 \mu^2}{\tilde m^3}, \ \ \
	B = \left(\frac{v_u}{\sqrt{2} v_R}  + \frac{\sqrt{2} \mu v_d v_R}{\tilde m^3}\right),
	\ \ \
	C = \left(\frac{2 M_{BL} v_u^2}{g_{BL}^2 v_R^2} + \frac{v_d^2 v_R^2}{\tilde m^3}\right),
\end{align}
\begin{align}
	\tilde m^{3} & = \frac
		{
			4
			\left[
				\mu v_u v_d \left(g_1^2 M_2 + g_2^2 M_1 \right)
				- 2 M_1 M_2 \mu^2
			\right]
		}
		{g_1^2 M_2 + g_2^2 M_1}
		.
\end{align}
\end{widetext}
Here $\alpha$ and $\beta$ take only the values 1 and 2. From the experimental limits on active neutrino masses we obtain $(Y_\nu)_{i \alpha} \lesssim 10^{-12}$.
This can be compared to $(Y_\nu)_{i 3} \lesssim 10^{-5}$, which is less constrained because of the TeV scale
seesaw suppression. It has been pointed out
in Ref.~\cite{Barger:2010iv, Ghosh:2010hy} (and earlier in a different context~\cite{Mohapatra}) that this theory predicts the existence of two light sterile neutrinos which
are degenerate or lighter than the active neutrinos, a so-called $3+2$ neutrino model. A sample possible spectrum is displayed in Fig.~\ref{nu.spec}.
\begin{figure}[h]
	\includegraphics[scale=0.8]{imgNuSpec}
	\caption{Sample spectra for neutrino masses in the normal and inverted hierarchies.}
\label{nu.spec}
\end{figure}

Recently, it has been shown in~\cite{Hamann:2010bk} that precision cosmology and big-bang nucleosynthesis mildly favor extra radiation in the universe beyond photons and ordinary neutrinos, lending support to the existence of sub-eV sterile neutrinos.

\underline{Charginos and Charged Leptons:}
In this theory the chargino mass matrix, in the basis $\left(e^c_j, \ \tilde W^+, \ \tilde H_u^+\right)$ and $\left(e_i, \ \tilde W^-, \ \tilde H_d^-\right)$, is given by
\begin{equation}
\label{chargino}
{\cal M}_{{\tilde \chi}^{\pm}}	=
	\left(
	\begin{array}{cc}
		0
	&
		M_C
	\\
		M_C^T
	&	0
	\end{array}
	\right),
\end{equation}
with
\begin{equation}
	M_C =
	\begin{pmatrix}

			-\frac{1}{\sqrt{2}} Y_e^{ij} v_d
		&

			0
		&

			\frac{1}{\sqrt{2}} Y_e^{ij}  v_L^j
	\\

			\frac{1}{\sqrt{2}} \ g_2  v_L^i
		&

			M_2
		&

			\frac{1}{\sqrt{2}} g_2 v_d
	\\

			-\frac{1}{\sqrt{2}} Y_\nu^{ij} v_R^j
		&

			\frac{1}{\sqrt{2}} g_2 v_u
		&

			\mu
	\end{pmatrix}.
\end{equation}

\underline{Squarks and Sleptons:}
In the sfermion sector, the mass matrices $\mathcal{M}_{\tilde u}^2$, and $\mathcal{M}_{\tilde d}^2$ for squarks,
and $\mathcal{M}_{\tilde e}^2$ for charged sleptons, in the basis $\left(\tilde f, \  {\tilde f}^{c*} \right)$, are given by
\begin{eqnarray}
\mathcal{M}_{\tilde u}^2&=&\left(\begin{array}{cc}
		m_{\tilde Q}^2
		\ + \ m_{u}^2
		\ + \  \left(\frac{1}{2} \ - \  \frac{2}{3} s_W^2 \right) \ M_Z^2 \ c_{2\beta}
		\ + \ \frac{1}{3} D_{BL}
		&
		\frac{1}{\sqrt 2} \left(a_u \ v_u - Y_u \ \mu \ v_d\right)
	\\
		\frac{1}{\sqrt 2} \left(a_u \ v_u - Y_u \ \mu \ v_d\right)
		&
		m_{\tilde u^c}^2
		\ + \ m_{u}^2
		\ + \ \frac{2}{3} M_Z^2 \ c_{2\beta}\ s_W^2
		\ - \ \frac{1}{3} D_{BL}
	\end{array}\right),
	\nonumber \\
\end{eqnarray}
\begin{eqnarray}
\mathcal{M}_{\tilde d}^2 &=& \left(\begin{array}{cc}
		m_{\tilde Q}^2
		\ + \ m_{d}^2
		\ - \  \left(\frac{1}{2}  \ - \ \frac{1}{3} \  s^2_W \right) M_Z^2 \ c_{2 \beta}
		\ + \ \frac{1}{3} D_{BL}
		&
		\frac{1}{\sqrt 2} \left(Y_d \ \mu \ v_u - a_d \ v_d\right)
	\\
		\frac{1}{\sqrt 2} \left(Y_d \ \mu \ v_u - a_d \ v_d\right)
		&
		m_{\tilde d^c}^2
		\ + \ m_{d}^2
		\ - \  \frac{1}{3} \ M_Z^2  \ c_{2\beta} \ s^2_W
		\ - \ \frac{1}{3} D_{BL}
	\end{array}\right),
	\nonumber \\
	\\
\mathcal{M}_{\tilde e}^2 &=& \left(\begin{array}{cc}
\label{Selectron.Mass}
		m_{\tilde L}^2
		\ + \ m_{e}^2
		\ - \ \left( \frac{1}{2} \ - s_W^2 \right) M_Z^2 \ c_{2\beta}
		\ - \ D_{BL}
		&
		\frac{1}{\sqrt 2} \left(Y_e \ \mu \ v_u - a_e \ v_d\right)
	\\
		\frac{1}{\sqrt 2} \left(Y_e \ \mu \ v_u - a_e \ v_d\right)
		&
		m_{\tilde e^c}^2
		\ + \ m_{e}^2
		\ - \  M_Z^2 \ c_{2\beta}\ s_W^2
		\ + \ D_{BL}
	\end{array}\right), \nonumber \\
\end{eqnarray}
where $c_{2\beta} = \cos 2\beta$, $s_W = \sin\theta_W$ and
\begin{equation}
D_{BL} \equiv \frac{1}{8} \ g_{BL}^2  v^2_R= \frac{1}{2} M_{Z_{BL}}^2.
\end{equation}
$m_u, \ m_d$ and $m_e$ are the respective fermion masses and $a_u, \ a_d$ and $a_e$
are the trilinear $a$-terms corresponding to the Yukawa couplings $Y_u, \ Y_d$ and $Y_e$.
Typically, it is assumed that substantial left-right mixing occurs only in the third generation.
Regardless, the physical states are related to the gauge states by
\begin{align}
\label{eq_squarkmixing}
	\begin{pmatrix}
		\tilde f_1
		\\
		\tilde f_2
	\end{pmatrix}
	& =
	\begin{pmatrix}
		\cos \theta_{\tilde f}
		&
		\sin \theta_{\tilde f}
		\\
		- \sin \theta_{\tilde f}
		&
		\cos \theta_{\tilde f}
	\end{pmatrix}
	\begin{pmatrix}
		\tilde f
		\\
		\tilde f^{c*}
	\end{pmatrix}.
\end{align}
The masses in the sneutrino sector are given by
\begin{eqnarray}
M_{\tilde{\nu}_i}^2 &=& m_{\tilde{L}_i}^2 \ + \ \frac{1}{2} M_{Z}^2 \cos 2 \beta - \frac{1}{2} M_{Z_{BL}}^2, \\
M_{\tilde{\nu}_3^c}^2 &=& M_{Z_{BL}}^2, \\
M_{\tilde{\nu}_\alpha^c}^2 &=& m_{\tilde{\nu}^c_\alpha}^2 \ + \ D_{BL},
\end{eqnarray}
and $\alpha=1..2$. For simplicity we listed the mass matrices in
the limit $v_L^i, a_\nu, Y_\nu \to 0$. For the most general expressions see Appendix~A.
It is important to mention that all sfermion masses are modified due to the existence of the B-L D-term.

In order to understand the properties of the spectrum we assume a simplified spectrum for the superpartners.
In the case of the sfermions we will assume for simplicity the same value for all soft masses. In this case if we neglect the left-right
mixing the full spectrum of sfermions will be defined by $M_{SUSY}$ (the universal soft supersymmetry breaking mass), $\tan \beta$ and the mass of the B-L
 gauge boson. Using this simplified spectrum we show in Fig.1 the values for the sfermion masses for different values of $M_{Z_{BL}}$.
 Notice that the condition that left-handed slepton masses have to be positive
 impose a bound on the $M_{Z_{BL}}$ for a given value of $M_{SUSY}$, i.e. $M_{Z_{BL}} < \sqrt{2} M_{\tilde{L}}$. In this way, we can appreciate that the spectrum
 can be very constrained. As it is well-known, in the general case one cannot predict the spectrum since the soft masses are unknown.

\begin{figure}[h]
\includegraphics[scale=0.6]{Spectrum}
\caption{Spectrum for sfermion masses assuming the same value for all soft masses, $M_{SUSY}=1$ TeV and $\tan \beta=5$.}
\label{spectrum}
\end{figure}

\section{Nucleon Stability and Dark Matter}
It is well-known that when R-parity is broken in a supersymmetric theory one has to understand the possible constraints coming from proton decay~\cite{proton}.
In the MSSM one has several interactions which could mediate proton decay at tree level and one-loop level. At the renormalizable level one has the lepton and baryon
number violating interactions
\begin{equation}
{\cal{W}}_{RpV}=\epsilon \hat{L} \hat{H}_u \ + \  \lambda \hat{L} \hat{L} \hat{e}^c \ + \   \lambda^{'} \hat{Q} \hat{L} \hat{d}^c \ + \  \lambda^{''} \hat{u}^c \hat{d}^c \hat{d}^c,
\end{equation}
which are not allowed in our theory before B-L breaking, and in general one has the dimension five operators
\begin{equation}
{\cal{W}}_{RpC}^5= \frac{\lambda_\nu}{\Lambda} \hat{L} \hat{L} \hat{H}_u \hat{H}_u \ + \  \frac{\lambda_L}{\Lambda} \hat{Q} \hat{Q} \hat{Q} \hat{L} \ + \  \frac{\lambda_R}{\Lambda} \hat{u}^c \hat{d}^c \hat{u}^c \hat{e}^c
\ + \  \frac{\lambda_{\nu^c}}{\Lambda} \hat{u}^c \hat{d}^c \hat{d}^c \hat{\nu}^c.
\end{equation}
Notice that the first term in the above equation is not allowed in our theory, but the last terms can mediate proton decay. The operators $QQQL$ and $u^c d^c u^c e^c$ mediate proton decay
at one-loop level and typically the scale $\Lambda$ should be larger than $10^{17}$ GeV in order to satisfy the experimental bounds on proton decay. For a detailed discussion
see Ref.~\cite{proton}. Once B-L is broken by the vev of the right-handed sneutrinos one finds new contributions to proton decay at tree level.
From the Yukawa coupling $Y_\nu \hat{L} \hat{H}_u \hat{\nu}^c$ one gets the lepton number violating interaction $L \tilde{H}_u$ and from the last term,
$\hat{u}^c \hat{d}^c \hat{d}^c \hat{\nu}^c$, in the above equation one gets the interaction $\tilde{u}^c d^c s^c$. Using these interactions and integrating
out the neutralinos and squarks we find the following constraint
\begin{equation}
\frac{\lambda^{1123}_{\nu^c}}{\Lambda} \frac{Y_u Y_\nu^{i3}}{M_{\tilde{u}}^2} \frac{v_R^2}{2 M_{\tilde{\chi}^0}} \ < \ 10^{-30} \ \rm{GeV}^{-2},
\end{equation}
from the channel $p \to K^+ \nu$. Then, assuming that $\lambda^{1123}_{\nu^c} \sim 1$, and $Y_{\nu}^{i3} \sim 10^{-6}$ (see Fig.~\ref{vy})
one gets $\Lambda > 10^{17}$ GeV. This constrain is similar to the one we get from the dimension five operators.
Therefore, one can say that if the above couplings are order one the cutoff of the theory has to be large. Also one can think about
possible scenarios where the couplings are small, see for example~\cite{Yuval}.

At first glance, finding a dark matter in $R$-parity violating theories seems hopeless.  But while the traditional neutralino LSP case is no longer valid, the situation is not lost. As first discussed in~\cite{Takayama:2000uz}, such models can have an unstable LSP gravitino, with a lifetime longer than the age of the universe.  The strong suppression on its lifetime is due to both the planck mass ($M_P$) suppression associated with its interaction strength and bilinear $R$-parity violation which is small due to neutrino masses and must facilitate the decay of the LSP. In the mass insertion approximation, this can be understood as the gravitino going into a photon and neutralino which then has some small mixing with the neutrino due to $R$-parity violation ($m_{\chi \, \nu}$), thereby allowing $\tilde G \to \gamma \nu$ as in Fig.~3. Adopting approximations made in~\cite{Takayama:2000uz}, the lifetime for the gravitino decaying into a photon and neutrino (in years) is about
\begin{figure}[h]
	\includegraphics[scale=0.4]{imgGravitinoDK}
	\put(-30,20){$\tilde G$}
	\put(-15,25){$\gamma$}
	\put(-12.3, 8){\Large+}
	\put(-18, 8.5){$\chi$}
	\put(-8, 0){$\nu$}
	\put(-9.8, 10.4){$m_{\chi \, \nu}$}
\caption{Gravitino decay into a photon and a neutrino. }
\label{gravitino}
\end{figure}
\begin{equation}
	\tau(\tilde G \to \gamma \nu) \sim 2 \times 10^{10}
	\left(\frac{m_{3/2}}{100 \text{ GeV}} \right)^{-3}
	\left(\frac{m_{\chi \nu}/m_\chi}{10^{-6}} \right)^{-2}
	\text{ years},
\end{equation}
which for appropriate values of the gravitino mass leads to a long enough life time. Unlike in $R$-parity conserving models with a gravitino LSP, there are no issues with big bang nucleosynthesis from slow NLSP decay since the NLSP decays more promptly through $R$-parity violating interactions. Several interesting studies have been conducted on the signatures and constraints of unstable gravitino dark matter, see for example~\cite{Takayama:2000uz,Buchmuller:2007ui}.

\section{Experimental Constraints}
In order to understand the different lepton number violating decays in the theory we need to understand which are the main constraints from neutrino experiments.
Today, we know well the numerical values for two of the neutrino mixings and the mass squared differences. The neutrino mixing matrix $V_{PMNS}$ is defined as
\begin{equation}
	V_{PMNS} =
	\begin{pmatrix}
		c_{12} \, c_{13}
		&
		c_{13} \, s_{12}
		&
		s_{13}
		\\
		-c_{23} \, s_{12}  -   c_{12} \, s_{13} \, s_{23}
		&
		c_{12} \, c_{23} - s_{12} \, s_{13} \, s_{23}
		&
		c_{13} \, s_{23}
		\\
		s_{12} \, s_{23}   -    c_{12} \, c_{23} \, s_{13}
		&
		-c_{12} \, s_{23} - c_{23} \, s_{12} \, s_{13}
		&
		c_{13} \, c_{23}
	\end{pmatrix},
\end{equation}
where $c_{ij} = \cos \theta_{ij}$ and $s_{ij} = \sin \theta_{ij}$ with $0 \leq \theta_{ij} \leq \pi/2$.
The physical neutrino masses are contained in $m_\nu=diag(m_{\nu_1},m_{\nu_2},m_{\nu_3})$.
As it is well-known, there are two possible neutrino spectra:
\begin{align}
\begin{split}
	\text{Normal Hierarchy (NH):} & \quad
		m_{\nu_1}, \ \
		m_{\nu_2} = \sqrt{ m_{\nu_1}^2 + \Delta m_{21}^2}, \ \
		m_{\nu_3} = \sqrt{ m_{\nu_1}^2 + |\Delta m_{31}^2| },
	\\
	\text{Inverted Hierarchy (IH):} & \quad
		m_{\nu_1} = \sqrt{m_{\nu_3}^2 + |\Delta m_{31}^2|}, \ \
		m_{\nu_2} = \sqrt{ m_{\nu_1}^2 + \Delta m_{21}^2}, \ \
		m_{\nu_3},
\end{split}
\end{align}
where~\cite{arXiv:1103.0734}
\begin{align}
	7.27 \times 10^{-5} \text{eV}^2 \leq & \, \Delta m_{21}^2 \, \leq 8.03 \times 10^{-5} \text{ eV}^2,
\\
	2.17 \times 10^{-3} \text{ eV}^2 < & \, |\Delta m_{31}^2| \, < 2.54 \times 10^{-3} \text{ eV}^2,
\end{align}
are the solar and atmospheric mass squared differences, respectively.

In order to understand the allowed values for the vevs of the left-handed sneutrinos and the Dirac Yukawa couplings we assume for simplicity that the off-diagonal block matrices in Eq.~(\ref{Mnu})
are zero, hence decoupling the two light sterile neutrinos from the active ones. In this case, all neutrino masses and mixing originate from the upper-left block matrix in Eq.~(\ref{Mnu}), which we label $m_\nu$.  The flavor pattern, and hence the rank of this matrix, dictates that one neutrino will be massless.  This matrix is diagonalized by the PMNS matrix
\begin{equation}
	m_\nu = V_\text{PMNS}^T \  M_\nu \ V_\text{PMNS},
\end{equation}
where $m_\nu = \text{diag}(0, \sqrt{\Delta m_{21}^2}, \sqrt{|\Delta m_{31}^2|})$ in the Normal Hierarchy
and $m_\nu = \text{diag}(\sqrt{|\Delta m_{31}^2|}, \sqrt{|\Delta m_{31}^2| + \Delta m_{21}^2}, 0)$ in the Inverted Hierarchy.
This yields a system of six equations quadratic in the vevs of the right-handed sneutrinos and Yukawa couplings, although solving
for these is not the most efficient way to proceed.  Instead, notice that the product above yields the following six terms
\begin{align}
\begin{split}
	\label{VvRelation}
	V^j & \equiv v_{L}^i \, {V_\text{PMNS}}^{ij},
	\\
	Y^j & \equiv  Y_\nu^{i3} \, V_\text{PMNS}^{ij}.
\end{split}
\end{align}
\begin{itemize}
\item{Normal Hierarchy}

In the Normal Hierarchy one obtains the following six equations
\begin{align}
	A V_1^2 + 2 B V_1 Y_1 + C Y_1^2 = & \, 0,
\\
	A V_1 V_2 + B \left(V_1 Y_2 + V_2 Y_1 \right)+ C Y_1 Y_2 = & \, 0,
\\
	A V_1 V_3 + B \left(V_1 Y_3 + V_3 Y_1 \right)+ C Y_1 Y_3 = & \, 0,
\\
	A V_2 V_3 + B \left(V_2 Y_3 + V_3 Y_2 \right) + C Y_2 Y_3 = & \, 0,
\\
	A V_2^2 + 2 B V_2 Y_2 + C Y_2^2 = &\,  m_2,
\\
	A V_3^2 + 2 B V_3 Y_3 + C Y_3^2 = & \ m_3.
\end{align}
In order for these equations to be consistent, $V_1 = Y_1 = 0$.  While this condition represents some fine-tuning between parameters, it is a result of
the simplifying assumption that the sterile and active light states decouple.  In a more general scenario, this condition would not be necessary.

The remaining system of equations, the last three, is underdetermined with three equations and four unknowns. Solving with respect to $Y_3$ yields
\begin{align}
	Y_2 & = \epsilon_1
		\sqrt{\frac{m_2}{m_3}}
		\sqrt
		{
			\frac
			{
				- Y_3^2 R - A m_3
			}
			{
				R
			}
		},
\\
	V_3 & = \frac
		{
			-B Y_3 + \epsilon_3   \sqrt{Y_3^2 R + A m_3}
		}
		{
			A
		},
\\
V_2 & = \frac
		{
			-B Y_2 + \epsilon_2   \sqrt{Y_2^2 R + A m_2}
		}
		{
			A
		},
\end{align}
with
\begin{equation}
	R \equiv B^2 - A C, \ \
	\epsilon_1  = \pm 1, \  \
	\epsilon_2  = \pm 1, \  \   \epsilon_3= \frac{\epsilon_1}{\epsilon_2} \frac{R Y_3}{\sqrt{R^2 Y_3^2}}.
\end{equation}
Inverting Eq.~(\ref{VvRelation}) one translates these solutions to the variables of interest.
The result being that specifying the SUSY spectrum and B-L parameters ($M_{Z_{BL}}$ and $g_{BL}$)
as well as $Y_3$ specifies all the values for $v_{L}^{i}$ and $Y_\nu^{i3}$.

\item Inverted Hierarchy

In the case of the inverted spectrum one can use the same procedure. However, one needs to make the following replacements:
$m_1 \leftrightarrow m_3$, $Y_1 \leftrightarrow Y_3$, $V_1 \leftrightarrow V_3$. In this way when we solve the equations for $V_i$ and $Y_i$, one obtains
$V_3=0$, $Y_3=0$ and the solutions above where we have made the previous substitutions.

\end{itemize}
\begin{table}[htdp]
\begin{center}
\begin{tabular}{|c|c|}
\hline
$\quad$ Parameter $\quad$ & $\quad \quad$ Range $\quad \quad$
\\
\hline
$M_1$ &	100 - 1200 GeV
\\
$M_2$ &	100 - 1200 GeV
\\
$|\mu|$ &	100 - 1200 GeV
\\
$\tan \beta$ & 3 - 50
\\
$|Y_3|$ & $10^{-7} - 10^{-5}$
\\
$M_{BL}$ & 100 - 1200 GeV
\\
$M_{Z_{BL}}$	&	1000 GeV
\\
$g_{BL}$	&	0.33
\\
\hline
\end{tabular}
\end{center}
\caption{Parameters and ranges scanned for the plots in Figs.~\ref{vy}, \ref{LSP.DL.Bino}--~\ref{LSP.BR.Higgsino}.}
\label{tab:scan}
\end{table}%
In Fig.~\ref{vy} we show the allowed values for the vevs of the left-handed sneutrinos and the Dirac Yukawa couplings for a scan over the parameters listed in Table~\ref{tab:scan}. As one can appreciate the allowed values for $v_L^i$ are in the range $(10^{-2}-10)$ MeV while the Yukawa couplings change between $10^{-7}$ and $10^{-5}$.
Now, using these results we are ready to discuss all R-parity violating decays.

\begin{figure}[h!]
	\includegraphics[scale=0.75]{v1Y1plotNH}
	\put(-40,-4){(a)}
	\includegraphics[scale=0.75]{v1Y1plotIH}
	\put(-40,-4){(b)}
\\
	\includegraphics[scale=0.75]{v2Y2plotNH}
	\put(-40,-4){(c)}
	\includegraphics[scale=0.75]{v2Y2plotIH}
	\put(-40,-4){(d)}
\\
	\includegraphics[scale=0.75]{v3Y3plotNH}
	\put(-40,-4){(e)}
	\includegraphics[scale=0.75]{v3Y3plotIH}
	\put(-40,-4){(f)}
\caption
{
	Allowed values for the $v_L^i$ versus $(Y_\nu)_{i3}$ in agreement with the neutrino masses and mixings constraints in the NH (IH) in a,c and e (b,d and f) for a scan over the parameters listed in Table~\ref{tab:scan}.
}
\label{vy}
\end{figure}

\section{Lepton Number Violation and Decays}
At this point, the relevant pieces of this model have be laid out and the question of interesting signals can now be tackled. Since lepton number is violated, same-sign dileptons and multijet are possible final states. Such signatures are interesting since they have no SM background. However, the final states depend critically on the nature of the LSP and since R-parity is violated the possibilities are more numerous than normal, \textit{i.e.} colored and charged fields.

These possibilities are briefly discuss in Appendix~\ref{LSPs}. We find that the most clear single of lepton number violation (and therefore the most interesting for us) results from the decays of a neutralino LSP through the process (see Fig~\ref{Signal})
\footnote{Throughout this paper, shorthand such as $pp \to \tilde e^* \tilde e$ represents the process $pp \to \tilde e^* \tilde e + X$, where the activity associated with $X$ has low transverse momentum and is not associated with the relevant physics of interest. Alternatively, this notation represents all possible production methods of $\tilde e^* \tilde e$ from the partons inside the proton taking their respective parton distribution functions  into account.}:
$$pp \  \to \gamma^*, Z^*, Z_{BL}^* \ \to \ \tilde{e}^*_i \tilde{e}_i \  \to  \ e^{\pm}_i \ e^{\mp}_i \ e^{\mp}_j \ e^{\mp}_k \ 4j.$$
In order to quantify this signal we will continue by investigating the decays of the $Z_{BL}$ gauge boson, the charged slepton and the neutralino and in the next section, the production mechanism for the charged sleptons at the LHC.
\begin{figure}[h!]
	\includegraphics[scale=1.25]{imgSignal}
	\put(-67,20){\Large $\tilde e^-_i$}
\caption
{
Topology of the signals with multi-leptons.
}
\label{Signal}
\end{figure}
\subsection{B-L Gauge Boson Decays}
The $Z_{BL}$ boson can decay into a pair of charged fermions, light neutrinos, and into two
sfermions. The partial widths for the decay into particles $P_1, P_2$ of masses $m_1, m_2$ are given by
\begin{equation}
	\Gamma(Z_{BL} \to P_1 P_2) =
	\frac{1}{16 \pi M_{Z_{BL}}} \, \left| \overline{\mathcal{M}} (Z_{BL} \to P_1 P_2) \right|^2 \,
	\sqrt{ \left( 1 - \frac{(m_1 +m_2)^2}{M_{Z_{BL}}^2} \right)
		 \left( 1 - \frac{(m_1 -m_2)^2}{M_{Z_{BL}^2}} \right)
	},
\end{equation}
where the squared matrix elements for specific final states are in Appendix~\ref{ZPDecay}.
The $Z_{BL}$ branching ratios are plotted in Fig.~\ref{ZBL.BR} for a fixed soft universal mass for all sfermions, $ M_{SUSY}=1$ TeV, versus the $Z_{BL}$ mass.
Decay channels into SUSY particles only open up for a $Z_{BL}$ mass of around $1.2$ TeV and at much 1.4 TeV, the sleptons become tachyonic.
As can be appreciated from Fig.~\ref{spectrum}, tachyonic sleptons are reached before decay channels into the squarks can open.
The branching ratios for the sleptons are divided up into sneutrinos, smuons plus selectrons and staus in anticipation of the associated signals.
However, each individual slepton pair has the same branching ratio, about $2.5\%$ at $m_{Z_{BL}} = 1.4$ TeV.
\begin{figure}[h!]
	\includegraphics[scale=0.65]{BrZBL1}
\caption
{
$Z_{BL}$ branching ratios versus $Z_{BL}$ mass for a universal soft mass for all sfermions, $M_{SUSY}=1$ TeV.
	Here the subscript one refers to the lightest eigenstate in each family and this case corresponds to the purely left-handed
	slepton (zero mixing angle is assumed). Given the universal soft mass for the sfermions, only the left-handed slepton
	channels can be open. Right-handed squark channels can open for larger values of $M_{Z_{BL}}$ but only at the
	cost of unphysical tachyonic slepton masses.
	}
\label{ZBL.BR}
\end{figure}
In Fig. 6 we show the predictions for the total decay width and the invisible decays of the $Z_{BL}$ gauge boson.
Since the new gauge boson can decay into five light neutrinos the invisible decay can be large, a few GeV when
the mass is above 1 TeV. These properties of the $Z_{BL}$ are very important in order to discover this theory at the LHC.
In summary, one can say that the new neutral gauge boson is B-L like with branching ratios
$$ \rm{Br}( Z_{BL} \to e^+_i e^-_i) \sim 40 \%, \  \rm{Br} (Z_{BL}  \to \nu \nu) \sim 35 \% ,  \rm{Br}(Z_{BL} \to jj) \sim 20 \%, \rm{and} \ \rm{Br} (Z_{BL} \to \bar{t} t) \sim 5 \%,$$
since the branching ratios for the SUSY decays are small  and the invisible decay can be large. For example when
$M_{Z_{BL}} = 1.2 \ \rm{TeV}$ the invisible decay width is $\Gamma_{Z_{BL}}(\rm{invisible}) \sim 3$  GeV.
\\
\begin{figure}[h!]
	\includegraphics[scale=0.7]{GZBL}
\caption
{The total decay width, $\Gamma(Z_{BL})$, versus $Z_{BL}$ mass for a universal soft mass for all sfermions, $M_{SUSY}=1$ TeV.
}
\label{GZBL}
\end{figure}
\subsection{Charged Slepton Decays}
\label{slep.dk}
The leading decay channels for the charged left-handed sleptons, $\tilde e_i^\pm$, are the decays into neutralinos and charginos
\begin{equation*}
	\tilde e_i^\pm \to e_i^\pm \tilde \chi^0_a,
	\ \ \ \tilde e_i^\pm \to \nu \tilde \chi^\pm_A,
	\ \ \ \tilde e_i^\pm \to \tilde \nu_j W^\pm,
\end{equation*}
where $i,j$ are lepton generational indices, $a$ labels the neutralinos from lightest to heaviest and $A$ labels the charginos from lightest to heaviest.  In addition to these decay modes there are various $R$-parity violating decays which only dominate when the slepton is the LSP. The last channel above usually involves an off-shell product particle (a three-body decay) and is therefore suppressed.  The decay widths for the remaining two relevant channels are given by
\begin{align}
\label{slepton.neutralino}
	\Gamma(\tilde e_i^\pm \to e_i^\pm \tilde \chi^0_a) & = \frac{m_{\tilde e_i}}{ 32 \pi}
	\left(
		\left| g_{BL} N_{2a} + g_1 N_{3a} + g_2 N_{4a} \right|^2
		+ \left| \frac{2 \sqrt{2} m_{e_i}}{v_d} N_{5a}\right|^2
	\right)
		\left(1 - \frac{m_{\tilde \chi^0_a}^2} {m_{\tilde e_i}^2}\right)^2,
	\\
\label{slepton.chargino}
	\Gamma(\tilde e_i^\pm \to \nu_j \tilde \chi^\pm_A) & = \frac{m_{\tilde e_i}}{ 16 \pi}
			\left| g_2 V^-_{1A} \right|^2
		\left(1 - \frac{m_{\tilde \chi^-_A}^2} {m_{\tilde e_i}^2}\right)^2,
\end{align}
where $N_{ab}$ diagonalizes the neutralino mass matrix and in the chargino sector one has $V^- X V^+  = \text{diag} \left(m_{\tilde \chi_1}, m_{\tilde \chi_2}\right).$ Due to the presence of the right-handed neutrino and $\tilde B'$, position 2, 3, 4 and 5 in the $N$ refer to $\tilde B', \tilde B, \tilde W  \ \text{and}, \tilde{H}_d$, respectively.  While it is hard to make predictions for the branching ratios of the charged sleptons decaying into a charged lepton and LSP without knowing the details of the spectrum, we briefly outline the best case scenario. Our final results will be given both with this best case scenario in mind and with arbitrary $\text{Br}(\tilde e_i \to e_i \tilde \chi_1^0)$.

For a mostly bino LSP, it is possible that charged selectrons decay one hundred percent into the LSP since the charginos and other neutralinos could be heavier.  For a wino LSP however, the lightest chargino channel is very likely to be open.  Because of the factor of 2 difference between Eq.~(\ref{slepton.neutralino}), and Eq.(\ref{slepton.chargino}) if all other neutralinos and charginos are kinematically disallowed, one can expect
\begin{equation}
\frac{\Gamma(\tilde e_i^\pm \to e_i^\pm \tilde \chi^0_1)}{\Gamma(\tilde e_i^\pm \to \nu_j \tilde \chi^\pm_1)} \sim \frac{1}{2},
\end{equation}
meaning a $33 \%$ branching ratio for a charged slepton into a wino LSP. Meanwhile, the left-handed selectron does not couple to the charged Higgsino.
Therefore, in the same limit where all other neutralinos and charginos are out of kinematic range, the charged sleptons decays one hundred percent into the LSP for a Higgsino LSP.

\subsection{Neutralino Decays}
\label{chi.decay}
The leading decay channels for the lightest neutralino, $\tilde{\chi}_1^0$, include
\begin{align}
\label{chi.decays}
	\tilde{\chi}_1^0 \to e^{\pm}_i W^{\mp},
	\  \
	\tilde{\chi}_1^0 \to \nu_i Z,
	\  \
	\tilde{\chi}_1^0  \to  \nu_i h_k,
	\ \
	\tilde{\chi}_1^0 \to e_i^\pm H^\mp.
\end{align}
The amplitude for the two first channels are proportional to the mixing between the leptons and neutralinos, while the last one is proportional to the Dirac-like Yukawa terms. While decays to all the MSSM
Higgses are possible, typically, only the lightest MSSM Higgs, $h$ ($k = 1$), is light enough for the scenario we consider
here and so we will only take it into account.

A naive estimation of the decay width yields
\begin{equation}
\Gamma (\tilde{\chi}_1^0) \sim \frac{g_2^2}{32 \pi} |V_{\nu \chi }|^2 M_{\chi},
\end{equation}
where $V_{\chi \nu}$ is the mixing between the neutralino and neutrino which is proportional to $\sqrt{m_\nu/M_{\chi}}$.
Assuming that $m_\nu < 0.1$ eV the decay length one finds $L(\tilde{\chi}_1^0) \gg 0.6$ mm.
Therefore, even without making a detailed analysis of the decays of the lightest neutralino one expects signals with lepton
number violation and displaced vertices in part of the parameter space. For a recent analysis of the neutralino decays in R-parity violating models see Ref.~\cite{Porod, Bobrovskyi:2011vx}.

The specific decay width expressions are
\begin{eqnarray}
\label{lWL}
\Gamma^{e_i W_L}&\equiv&
\Gamma(\tilde{\chi}_a \to e_i^\pm W_L^\mp)= \frac{g_2^2}{64\pi
M_W^2}|V_{i a}|^2 m_{\tilde{\chi}_a}^3 \left(1-  \frac{m_W^2}{m_{\tilde{\chi}_a}^2} \right)^2,
\\
\label{lWT}
\Gamma^{e_i W_T}&\equiv&\Gamma(\tilde{\chi}_a \to e_i^\pm W_T^\mp)=
	\frac{g_2^2}{32\pi  }|V_{i a}|^2 m_{\tilde{\chi}_a}    \left(1-  \frac{m_W^2}{m_{\tilde{\chi}_a}^2} \right)^2,
\\
\label{nuZL}
\Gamma^{\nu_i Z_L}&\equiv&\Gamma(\tilde{\chi}_a\to \nu_i Z_L)=\frac{g_2^2}{
64\pi M_W^2}|V_{i a}|^2 m_{\tilde{\chi}_a}^3 \left(1-  \frac{m_Z^2}{m_{\tilde{\chi}_a}^2} \right)^2,
\\
\label{nuZT}
\Gamma^{\nu_i Z_T}&\equiv&\Gamma(\tilde{\chi}_a \to \nu_i Z_T)=\frac{g_2^2}{
32\pi c_W^2}|V_{i a}|^2 m_{\tilde{\chi}_a} \left(1-  \frac{m_Z^2}{m_{\tilde{\chi}_a}^2} \right)^2,
\\
\label{nuh}
\Gamma^{\nu_i h}&\equiv&\Gamma(\tilde{\chi}_a \to \nu_i h) =
	\frac{g_2^2}{64\pi M_W^2}
	|V_{i a}|^2 \cos^2 \alpha \;
	m_{\tilde{\chi}_a}^3 \left(1-  \frac{m_h^2}{m_{\tilde{\chi}_a}^2} \right)^2.
\end{eqnarray}
Here $\alpha$ is the mixing angle in the Higgs sector and in the decoupling limit, $M_A^2 \gg M_Z^2$, which we assume: $\cos \alpha = \sin \beta$. The index $i$ indicates the generation of lepton and $a$ the neutralino with $a = 6$ is the heaviest and $a=1$ is the lightest.  These expressions depend on the mixing between the light neutrinos and the neutralinos, $V_{i a}$, which is derived in Appendix~\ref{Vla}.  Of course, only the decays of the LSP are relevant since the decays of the other neutralinos will be dominated by $R$-parity conserving decays, hence $a=1$ for our purposes.
\begin{table}[htdp]
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
Hierarchy & LSP Personality & Bino & Wino & Higgsino
\\
\hline
&Decay Length &	1.1 mm & 0.03 mm & $1 \times 10^{-4}$ mm
\\
&2Br$(\chi_1^0 \to e^- W^+)$  & 4 $\%$	& 2 $\%$ & 13 $\%$
\\
&2Br$(\chi_1^0 \to \mu^- W^+)$  & 26 $\%$	& 12 $\%$ & 27 $\%$
\\
NH &2Br$(\chi_1^0 \to \tau^- W^+)$  & 61 $\%$	& 54 $\%$ & 30 $\%$
\\
&Br$(\chi_1^0 \to \nu Z^0)$  & 10 $\%$	& 29 $\%$ & 28 $\%$
\\
&Br$(\chi_1^0 \to \nu h)$  & 0 $\%$	& 3 $\%$ & 1 $\%$
\\
\hline
\hline
&Decay Length					& 0.6 mm	& 0.01 mm	& $1 \times 10^{-5}$ mm
\\
&2Br$(\chi_1^0 \to e^- W^+)$ 		& 17 $\%$	& 3 $\%$			& 25 $\%$
\\
&2Br$(\chi_1^0 \to \mu^- W^+)$	& 36 $\%$	& 32 $\%$		& 19 $\%$
\\
IH &2Br$(\chi_1^0 \to \tau^- W^+)$   & 38 $\%$		& 34 $\%$      & 26 $\%$
\\
&Br$(\chi_1^0 \to \nu Z^0)$		& 10 $\%$		& 29 $\%$	& 28 $\%$
\\
&Br$(\chi_1^0 \to \nu h)$			& 0 $\%$			& 3 $\%$		& 1 $\%$
\\
\hline
\end{tabular}
\end{center}
\caption
{
Values of interest for a sample point in parameter space: $\epsilon_1 = \epsilon_2 = 1, Y_3 = Y_1 = 10^{-6}, M_{Z_{BL}} = M_{\tilde{B}'} = 1 \text{ TeV}, \tan \beta =5 \text{ and } m_h = 125 \text{ GeV}$.  Here $M_1, M_2 \text{ and } \mu$ are 100 GeV, 500 GeV and 500 GeV for the bino LSP case, 500 GeV, 150 GeV and 500 GeV for the wino LSP case and 500 GeV, 500 GeV and 100 GeV for the Higgsino LSP case respectively.
}
\label{tab:test.points}
\end{table}
Table \ref{tab:scan}  displays the values of interest for a specific point in parameter space to gain an appreciation for possible values. Notice that in these scenarios the branching ratios for the channels with charged leptons can be large.

In Figs.~\ref{LSP.DL.Bino}-\ref{LSP.DL.Higgsino} are the decay lengths versus LSP mass resulting from a scan over all the possible values of $\epsilon_1$ and $\epsilon_2$ and over the parameters and ranges specified in Table~\ref{tab:scan}.  The points are divided according to the largest component of the LSP and the neutrino hierarchy with a dominantly bino, wino and Higgsino LSP in the NH shown in (a) and for an IH in (b), respectively. The relevant decay lengths can be understood by studying the mixings in Eq.~(\ref{neutralino}).  Since the higgsino-neutrino decay strength is the largest, $\sim Y_\nu v_R$, the Higgsino LSP has the shortest decay length.  It is followed by the wino LSP with mixing $\sim g_2 v_L$ and finally the bino with coupling $\sim g_1 v_L$ and therefore the largest possible decay lengths.  Displaced vertices associated with the lifetime of the LSP will only be discernible in a very limited part of the parameter space.

\begin{figure}[h!]
	\includegraphics[scale=0.85]{imgDLNHBino}
	\put(-40,-4){(a)}
	\includegraphics[scale=0.85]{imgDLIHBino}
	\put(-40,-4){(b)}
\caption{
Decay length in millimeters versus LSP mass for a dominantly bino LSP in (a) for a NH and in (b) for an IH.
Parameters are scanned according to the ranges specified in Table~\ref{tab:scan} and over all values of $\epsilon_1$ and $\epsilon_2$.}
\label{LSP.DL.Bino}
\end{figure}
\begin{figure}
	\includegraphics[scale=0.85]{imgDLNHWino}
	\put(-40,-4){(a)}
	\includegraphics[scale=0.85]{imgDLIHWino}
	\put(-40,-4){(b)}
\caption{
Decay length in millimeters versus LSP mass for a dominantly wino LSP in (a) for a NH and in (b) for an IH.
Parameters are scanned according to the ranges specified in Table~\ref{tab:scan} and over all values of $\epsilon_1$ and $\epsilon_2$.}
\label{LSP.DL.Wino}
\end{figure}
\begin{figure}
	\includegraphics[scale=0.85]{imgDLNHHiggsino}
	\put(-40,-4){(a)}
	\includegraphics[scale=0.885]{imgDLIHHiggsino}
	\put(-40,-4){(b)}
\caption
{Decay length in millimeters versus LSP mass for a dominantly Higgsino LSP in (a) for a NH and in (b) for an IH.
Parameters are scanned according to the ranges specified in Table~\ref{tab:scan} and over all values of $\epsilon_1$ and $\epsilon_2$.}
\label{LSP.DL.Higgsino}
\end{figure}

The LSP branching ratios into the various possible channels versus the LSP mass are displayed in Figs.~\ref{LSP.BR.Bino}-\ref{LSP.BR.Higgsino} scanning over the parameters in Table~\ref{tab:scan} and plotting the dominantly bino, wino and Higgsino LSP in (a) for a NH and in (b) for an IH.  The lack of variance with scanned parameters displayed in the $\nu \; Z$ and $\nu \; h$ channels are due to the sum over all three flavors of neutrinos and  also exists for the sum over the three charged lepton plus $W^\pm$ channels which total about $50 \%$ (or more if the other channels hadn't fully turned on yet).  Although it is not obvious from the Figs.~\ref{LSP.BR.Bino}-\ref{LSP.BR.Higgsino}, the branching ratio to the electron $W^\pm$ channel is always smaller then either the $\mu^\mp W^\pm$ or the $\tau^\mp W^\pm$ in the NH. Since we now know the properties of the neutralinos and selectrons decays we are ready to study the production channels.

\begin{figure}[t!]
	\includegraphics[scale=0.85]{imgBrNHBino}
	\put(-40,-4){(a)}
	\includegraphics[scale=0.85]{imgBrIHBino}
	\put(-40,-4){(b)}
\caption{
LSP branching ratios versus LSP mass for a dominantly bino LSP in (a) for a NH and in (b) for an IH.
	Parameters are scanned according to the ranges specified in Table~\ref{tab:scan} and over all values of $\epsilon_1$ and $\epsilon_2$.
	}
\label{LSP.BR.Bino}
\end{figure}
\begin{figure}
	\includegraphics[scale=0.85]{imgBrNHWino}
	\put(-40,-4){(a)}
	\includegraphics[scale=0.85]{imgBrIHWino}
	\put(-40,-4){(b)}
\caption{LSP branching ratios versus LSP mass for a dominantly wino  LSP in (a) for a NH and in (b) for an IH.
Parameters are scanned according to the ranges specified in Table~\ref{tab:scan} and over all values of $\epsilon_1$ and $\epsilon_2$.}
\label{LSP.BR.Wino}
\end{figure}
\begin{figure}
	\includegraphics[scale=0.85]{imgBrNHHiggsino}
	\put(-40,-4){(a)}
	\includegraphics[scale=0.85]{imgBrIHHiggsino}
	\put(-40,-4){(b)}
\caption
{
	LSP branching ratios versus LSP mass for a dominantly Higgsino LSP in (a) for a NH and in (b) for an IH.
	Parameters are scanned according to the ranges specified in Table~\ref{tab:scan} and over all values of $\epsilon_1$ and $\epsilon_2$.
}
\label{LSP.BR.Higgsino}
\end{figure}

\section{Production Mechanisms and Signals}
The lepton number violating signal discussed in the beginning of the previous section proceeds from the pair production of charged sleptons. While the MSSM contributions to this production drop rapidly with charged slepton mass (since that mass must be above the $Z$ threshold), in this model further contributions due to the $Z'$ resonance can significantly increase the cross section. These enhancement are discussed first followed by a study of the expected number of events at the LHC.

\subsection{Sleptons Production Mechanisms}
The main production channel for the charged sleptons is through the photon, the Z gauge boson and the $Z_{BL}$ boson
\begin{displaymath}
q(p_1) \bar{q}(p_2) \  \to \ \gamma, Z^*, Z^*_{BL} \  \to \  \tilde{e}^* (p_3) \tilde{e}  (p_4).
\end{displaymath}
The hadronic cross section is given by
\begin{equation}
d \sigma_{pp \to \tilde{e}^* \tilde{e}} (s) = \sum_{q=u,d,c,s}  \int_{\tau_0}^{1} d \tau \ \frac{d {\cal{L}}^{pp}_{q \bar{q}}}{d \tau}  d \hat{\sigma}_{q \bar{q} \to \tilde{e}^* \tilde{e}} (\hat{s}),
\end{equation}
where $\tau_0=4 M_{\tilde{e}}^2 / s$ and the differential partonic cross section is
\begin{equation}
d \hat{\sigma}_{q \bar{q} \to \tilde{e}^* \tilde{e}} (\hat{s}) = | \overline{{\cal M}}_{q \bar{q} \to \tilde{e}^* \tilde{e}} (\hat{s})|^2 \frac{\rm{d PS}^{(2)}}{2 \hat{s}}.
\end{equation}
Here $\rm{d PS}^{(2)}=d \hat{t}/ 8 \pi \hat{s}$ is the two particle phase-space element and $\hat{s}= \tau s$, where $s$ is the hadronic center-of-mass energy squared.
As it is well-known the parton luminosities are given by
\begin{equation}
\frac{\rm{d} {\cal L}_{ab}^{AB} }{\rm{d} \tau}= \frac{1}{1 + \delta_{ab}} \int_{\tau}^1 \left(   f_{a/A} (x,\mu)  f_{b/B} (\frac{\tau}{x}, \mu) +  f_{a/B} (\frac{\tau}{x}, \mu) f_{b/A} (x, \mu) \right),
\end{equation}
where the functions $f_{a/A} (x,\mu)$ are the particle distribution functions (PDFs).
The amplitude squared for these processes can be written as
\begin{equation}
 | \overline{{\cal M}}_{q \bar{q} \to \tilde{e}^* \tilde{e}} (\hat{s})|^2 = \frac{2}{3} \left( \hat{u} \hat{t} - M_{\tilde{e}}^4 \right)   | {{\cal A}}_{q \bar{q} \to \tilde{e}^* \tilde{e}} (\hat{s})|^2,
\end{equation}
with $\hat{s}=(p_1 + p_2)^2$, $\hat{t}=(p_1-p_3)^2$, $\hat{u}=(p_1-p_4)^2$ and
\begin{equation}
{{\cal A}}_{q \bar{q} \to \tilde{e}^* \tilde{e}} (\hat{s})=\frac{C_{q\bar{q}\gamma} C_{\gamma \tilde{e}^* \tilde{e} }}{\hat{s}} \ + \ \frac{2C_{q\bar{q} Z} C_{Z \tilde{e}^* \tilde{e} }}{\hat{s}-M_Z^2 + i M_Z \Gamma_Z}
\ + \ \frac{C_{q\bar{q} Z_{BL}} C_{Z_{BL} \tilde{e}^* \tilde{e} }}{\hat{s}-M_{Z_{BL}}^2 + i M_{Z_{BL}} \Gamma_{Z_{BL}}},
\end{equation}
where
\begin{eqnarray}
C_{\bar{q} q \gamma}&=&e_q \ e, \  \  C_{\gamma \tilde{e}^*_L \tilde{e}_L}=e_l \ e, \ \ C_{Z \tilde{e}^*_L \tilde{e}_L} = \frac{e}{\sin 2 \theta_W} L_e, \\
C_{\bar{q}_L q_L Z} &=& \frac{e L_q}{\sin 2 \theta_W}, \  \   C_{\bar{q}_R q_R Z} = \frac{e R_q}{\sin 2 \theta_W}, \ \
C_{\bar{f} f Z_{BL}}= g_{BL} \frac{n_{BL}^f}{2}, \ \ C_{\tilde{f}^* \tilde{f} Z_{BL}}= g_{BL} \frac{n_{BL}^f}{2}.
\end{eqnarray}
Here $L_f=I_f^3 -  e_f \sin^2 \theta_W$ and $R_f=-e_f \sin^2 \theta_W$, where $I_f^3$ is the isospin of the fermion f.
Now, using the equations
\begin{eqnarray}
\hat{u}&=& 2 M_{\tilde{e}}^2 - \hat{t}-\hat{s}, \\
\hat{t} &=& M_{\tilde{e}}^2 - \frac{\hat{s}}{2} + y \sqrt{\hat{s} \left( \frac{\hat{s}}{4} - M_{\tilde{e}}^2 \right)} ,
\end{eqnarray}
we can compute the cross section
\begin{equation}
\sigma_{pp \to \tilde{e}^*_L \tilde{e}_L} (s)= \sum_{q=u,d,s} \ \int_{-1}^1 dy  \int_{\tau_0}^1 d \tau \frac{d {\cal L}^{pp}_{q \bar{q}}}{d \tau} \ \sigma (M_{\tilde{e}}, y, \tau, s).
\end{equation}

\begin{figure}[h!]
	\includegraphics[scale=0.8]{imgppToe1e1LHC7}
	\put(-40,-4){(a)}
	\includegraphics[scale=0.8]{imgppToe1e1LHC14}
	\put(-40,-4){(b)}
\caption
{Drell-Yan production cross sections for the charged sleptons in our model. The dashed line corresponds to the prediction in the MSSM and the solid lines show the results in our model for different values of the $Z_{BL}$ mass. The gauge coupling $g_{BL}$ is assumed to be at the maximum value allowed by the experimental constraints: Eq~(\ref{ZBL.const}).
}
\label{Slepton.sigma}
\end{figure}

The numerical results for the selectron production cross sections are shown in Fig.~\ref{Slepton.sigma} for different scenarios, with $g_{BL}$ assumed to be the maximum value allowed by the experimental constraints: Eq~(\ref{ZBL.const}).
We have compared our analytical results for the cross section with the results in Ref.~\cite{Dawson} and
found the same result in the case of the MSSM. In Fig.~\ref{Slepton.sigma} we can see that even in the MSSM the cross section can be large and when the $Z_{BL}$ is included
the cross section can be even larger due to the resonance enhancement. For example when the $M_{Z_{BL}}=1$ TeV one can have a cross section above
1 fb where the selectron mass is below 450 GeV and $\sqrt{s}=7$ TeV.
\subsection{Signals with Multi-Leptons}
In this paper we wish to investigate the most promising signals associated with lepton number violation, through the process
\begin{equation}
	q \bar{q} \  \to
	\ \gamma, Z^*, Z^*_{BL} \ \to
	\  \tilde{e}_i^* \tilde{e}_i \to e_i^+ e_i^- \tilde{\chi}^0_1 \tilde{\chi}^0_1 \  \to
	\  e_i^+ e_i^- e^{\pm}_j e^{\pm}_k \ 4j,
\end{equation}
where $i,j,k=1..3$ are generational indices. See Fig.~\ref{Signal} for the illustration of these signals. However, we will focus on the electron and muon channels since hadronic activity associated with the tau would blemish the signal of lepton number violation, namely three same-sign leptons,
one lepton with opposite sign, four jets and no missing energy. Taking a cue from Eq.~(\ref{Selectron.Mass}), which shows that the left-handed (right-handed) sleptons receive a negative (positive) contribution to their mass from the $B-L$ $D$-term, we assume that only the left-handed sleptons are producible through this process.

We begin by giving an estimate for the number of events in the limit of mostly bino, wino and Higgsino LSP.
We present results for a 7 TeV LHC with $10 \text{ fb}^{-1}$ of data. The combinatorics factor for the channels of interest are given by
\begin{align}
\begin{split}
	\mathcal F_{jk}  = &
	2 \left(2 - \delta_{jk} \right) \, \text{Br}\left(\tilde e^\pm_i \to e^\pm_i \tilde \chi_1^0\right)^2 \times
	\text{Br}\left(\tilde \chi_1^0 \to e^\pm_j W^\mp \right)
\\
	& \times \text{Br}\left(\tilde \chi_1^0 \to e^\pm_k W^\mp \right)
	\times \text{Br}\left(W^\pm \to jj\right)^2,
\end{split}
\end{align}
so that the final states are
$$
e^\pm e^\mp e^\mp e^\mp 4j, \quad e^\pm e^\mp e^\mp \mu^\mp 4j, \quad e^\pm e^\mp \mu^\mp \mu^\mp 4j,
$$
$$
\mu^\pm \mu^\mp e^\mp e^\mp 4j, \quad \mu^\pm \mu^\mp e^\mp \mu^\mp 4j, \quad \mu^\pm \mu^\mp \mu^\mp \mu^\mp 4j.
$$
\begin{table}[htdp]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Hierarchy& LSP & $\text{Br}(\tilde \chi_1^0 \to e^\pm W^\mp)$	& $\text{Br}(\tilde \chi_1^0 \to \mu^\pm W^\mp)$ &
$\quad \quad \ \mathcal{F}_{ee} \quad \quad \ $ & $\ \quad \quad \mathcal{F}_{e \mu} \quad \quad \ $ & $\quad \quad \ \mathcal{F}_{\mu \mu} \quad \quad \ $
\\
\hline
NH 	& Bino & 1-20\%	& 10-50\% & 0.0001-0.036 & 0.002-0.18 & 0.01-0.22
\\
NH	& Wino&	1-20 \%	& 10-50\% & 0.00001-0.004 & 0.0002-0.02 & 0.001-0.024
\\
NH	& $\quad$ Higgsino $\quad$&	2-25 \%	& 10-50\% & 0.0004-0.056 & 0.004-0.22 & 0.008-0.24
\\
\hline
\hline
IH 	& Bino & 10-60\%	& 10-30 \% & 0.009-0.32 & 0.018-0.32 & 0.009-0.08
\\
IH 	& Wino & 10-60\%	& 13-30\% & 0.001-0.035 & 0.003-0.034 & 0.002-0.009
\\
IH 	& Higgsino & 10-60\%	& 13-35\% & 0.008-0.32 & 0.024-0.37 & 0.016-0.11
\\
\hline
\end{tabular}
\end{center}
\caption{Ranges for the branching ratios of the LSP to charge lepton and $W$ boson taken from the corresponding dense regions in Figs.~\ref{LSP.BR.Bino}~-~\ref{LSP.BR.Higgsino}. These are used to calculate the overall combinatorics factor, $\mathcal{F}_{jk}$ for the final state $e_i^\pm e_i^\mp e^\pm_j e^\pm_k 4j$.  Values are separated by the composition of the LSP: mostly bino, wino and Higgsino and for both the normal and inverted hierarchies.}
\label{Comb}
\end{table}
We assume that $\text{Br}\left(\tilde{e}^\pm_i \to e^\pm_i \tilde{\chi}^0_1 \right) \sim 100 \%, 33 \%, 100 \%$ for a bino, wino and Higgsino LSP respectively, following the discussion in Section~\ref{slep.dk}.  The branching ratio of the $W$ boson into jets is about 67\%. For the RPV neutralino decays we pick the most prominent regions from Figs.~\ref{LSP.BR.Bino}~-~\ref{LSP.BR.Higgsino} and display these values along with $\mathcal{F}_{jk}$ in Table~\ref{Comb}.  Values are shown for both the normal and inverted hierarchies.

To calculate the number of events expected after 10 $\text{fb}^{-1}$ of data, we convolute the combinatorics in Table~\ref{Comb} with the cross sections for a 1 TeV $Z_{BL}$ shown in Fig.~\ref{Slepton.sigma} and multiply by ten; the results are displayed in Figs.~\ref{Num.Events.Bino}~-~\ref{Num.Events.Higgsino}. Notice that in most of the scenarios one can have several events which are basically background free.
The main backgrounds coming from $t\bar{t} W Z$ and $jjjj W^{\pm} W^{\pm} Z$ are very suppressed.

\begin{figure}[h!]
	\includegraphics[scale=0.85]{imgNumNHBino7}
	\put(-40,-4){(a)}
	\includegraphics[scale=0.85]{imgNumIHBino7}
	\put(-40,-4){(b)}
\caption{
Number of $e_i^\pm e_i^\mp e_j^\pm e_k^\pm 4j$ at a 7 TeV LHC for 10 $\text{fb}^{-1}$ of data for a bino LSP. Branching ratio values are shown in Table~\ref{Comb}, while cross section values are taken from Fig~\ref{Slepton.sigma}.
Data is divided into (a) for the NH, and (b) for the IN.}
\label{Num.Events.Bino}
\end{figure}

\begin{figure}
	\includegraphics[scale=0.85]{imgNumNHWino7}
	\put(-40,-4){(a)}
	\includegraphics[scale=0.85]{imgNumIHWino7}
	\put(-40,-4){(b)}
\caption{
Number of $e_i^\pm e_i^\mp e_j^\pm e_k^\pm 4j$ at a 7 TeV LHC for 10 $\text{fb}^{-1}$ of data for a wino LSP. Branching ratio values are shown in Table~\ref{Comb}, while cross section values are taken from Fig~\ref{Slepton.sigma}.
Data is divided into (a) for the NH, and (b) for the IN.
}
\label{Num.Events.Wino}
\end{figure}
\begin{figure}
	\includegraphics[scale=0.85]{imgNumNHHiggsino7}
	\put(-40,-4){(a)}
	\includegraphics[scale=0.85]{imgNumIHHiggsino7}
	\put(-40,-4){(b)}
\caption
{Number of $e_i^\pm e_i^\mp e_j^\pm e_k^\pm 4j$ at a 7 TeV LHC for 10 $\text{fb}^{-1}$ of data for a Higgsino LSP. Branching ratio values are shown in Table~\ref{Comb}, while cross section values are taken from Fig~\ref{Slepton.sigma}.
Data is divided into (a) for the NH, and (b) for the IN.
	}
\label{Num.Events.Higgsino}
\end{figure}
In order to understand the testability of the model at the LHC we show curves of constant number of $e_i^\pm e_j^\mp e_j^\pm e_j^\pm 4j$ events per 10 $\text{fb}^{-1}$ of data in Fig.~\ref{Const.Num.Events}
in the $\text{Br}(\tilde \chi^0_1 \to e_j^\pm W^\mp)-\text{Br}(\tilde e_i^\pm \to e_i^\pm \tilde \chi^0_1)$ plane. Values are shown for a seven TeV LHC, with a 1 TeV $Z_{BL}$
and $m_{\tilde e_i}=200$ GeV. In the case of the observation of such events a the LHC, the $Z_{BL}$ mass can be reconstructed from electron-electron and muon-muon events and the selectron mass may be reconstructible from its decay into two leptons and two jets. Therefore allowing the calculation of the cross section for charged slepton pair-production. A plot such as Fig.\ref{Const.Num.Events} can be used to get a better handle on the two unknown branching ratios and shed further light on the model.

In order to estimate the reach of the LHC we also present curves of constant number of $e_i^\pm e_i^\mp e_j^\pm e_j^\pm 4j$ events per 10 $\text{fb}^{-1}$ of data in the $\text{Br}\left(\tilde e_i^\pm \to e_i^\pm \tilde \chi^0_1 \right)-m_{\tilde e_i}$ plane in Fig.~\ref{Const.Num.Events2}. This is again for a seven TeV LHC, with a 1 TeV $Z_{BL}$ and we show two possible values for $\text{Br}\left(\tilde \chi^0_1 \to e_j^\pm W^\mp \right)$, representing the upper (lower) part of that range in blue (green). One can see that even if the slepton mass is around 450 GeV one could observe a few events with multileptons and four jets. It is important to mention we satisfy the recent bounds coming from ATLAS~\cite{Atlas1, Atlas2}.
\begin{figure}[h!]
	\includegraphics[scale=0.9]{imgNumEvBrBr7}
\caption
{
	Curves of constant number of events for the final state $e_i^\pm e_i^\mp e_j^\pm e_j^\pm 4j$ in the  $\text{Br}\left(\tilde \chi^0_1 \to e_j^\pm W^\mp \right)-\text{Br}\left(\tilde e_i^\pm \to e_i^\pm \tilde \chi^0_1 \right)$ plane.  Values are shown for a seven TeV LHC, $M_{Z_{BL}}=1$ TeV and $m_{\tilde e_i}=200$ GeV.
}
\label{Const.Num.Events}
\end{figure}

\begin{figure}[h!]
	\includegraphics[scale=0.9]{imgNumEvBrme7}
\caption
{
	Curves of constant number of events for the final state $e_i^\pm e_i^\mp e_j^\pm e_j^\pm 4j$ in the  $\text{Br}\left(\tilde e_i^\pm \to e_i^\pm \tilde \chi^0_1 \right)-m_{\tilde e_i}$ plane.
	Values are shown for a seven TeV LHC, $M_{Z_{BL}}=1$ TeV and for two different values of $\text{Br}\left(\tilde \chi^0_1 \to e_j^\pm W^\mp \right)$, representing the upper (lower) part of that range in blue (green).
}
\label{Const.Num.Events2}
\end{figure}

\section{Summary and Outlook}
In this article we have studied in detail the theory proposed in Ref.~\cite{Rp2} which we consider the simplest gauge theory for R-parity violation.
This theory makes a prediction for the LHC since in order to break the B-L gauge symmetry the right-handed sneutrino must get a vacuum
expectation value and one should observe lepton number violation at colliders. We have found the following results:

\begin{itemize}

\item In Fig. 2 we have illustrated in a simple way that one can have a realistic scenario for all sfermion masses even
if some of the sfermion masses have a negative and large contribution from the B-L D-term in the theory. We have shown
that in order to avoid tachyonic masses one should satisfy the condition $M_{Z_{BL}} < \sqrt{2} m_{\tilde{L}}$. This is a
simple result which helps us to understand the constraints on the spectrum.

\item The full spectrum of the theory and the constraints coming from neutrino masses were analyzed in detail.
The spectrum for neutrinos is interesting since it contains five light neutrinos: three active neutrinos and two sterile neutrinos.
Using the experimental constraints on the masses and mixing for the active neutrinos we show in Fig. 4 the allowed
values for the vacuum expectations of the left-handed neutrinos and the Yukawa couplings. As we have discussed in the text,
these results are crucial to understand the decays of the lightest supersymmetric particle in the theory.

\item In Figs. 5 and 6 we have shown the properties of the new neutral gauge boson in the theory, the B-L gauge boson.
Since one has two extra light neutrinos in the theory the invisible decay width is larger in this case.
The contributions of the supersymmetric particle to the decay width are small and so the $Z_{BL}$
is like the B-L gauge boson in the non-SUSY scenarios.

\item We have investigated the neutralinos decays in great detail. In Figs. 7-9 we have shown the results for the decay
length in the different cases. As one can appreciate in Figs. 7-9 there are some scenarios in the Bino limit where one could expects
displaced vertices. The branching ratios have been investigated in Figs.10-12 and we can summarize the results in the following way
$$\rm{Br}( \tilde{\chi}^0_1 \to \tau W), \rm{Br}(\tilde{\chi}^0_1 \to \mu W)> \rm{Br}(\tilde{\chi}^0_1 \to e W),$$  in the Normal Hierarchy
and
$$\rm{Br}( \tilde{\chi}^0_1 \to e W), \rm{Br}(\tilde{\chi}^0_1 \to \mu W)> \rm{Br}(\tilde{\chi}^0_1 \to \tau W),$$ in the Inverted Hierarchy,
in majority of the parameter space.

\item We have studied the main production channels for the charged sleptons at the LHC. In this case one can produce the charged sleptons
through the photon and the Z as in the MSSM, and through the new neutral gauge boson, $Z_{BL}$, in our model.
As we have shown in Fig. 13 the production cross section can be large and thanks to the presence of the $Z_{BL}$ one
can have even larger values for the cross section due to the resonance enhancement. We should point out that this production channel (throught the photon and $Z$)
is very important to understand the signals in any model for R-parity violation.

\item The most striking signals for lepton number violation in this context  are the channels with three leptons with the same electric
charge and four jets. In Fig.~\ref{Num.Events.Bino}-\ref{Const.Num.Events2} we have shown that one can have a large number of events at the LHC with only 10 $\rm{fb}^{-1}$.
The background for these channels is suppressed, therefore there is a hope to test or rule out this theory in the near future.

\end{itemize}

{\textit{Acknowledgments}}:
The work of P.F.P. is supported by the James Arthur Fellowship, CCPP-New York University. P. F. P. thanks A. Haas for a discussion about
the searches for multi-leptons at the LHC.

redpajama_set_name: RedPajamaArXiv


Well thread is good, and idea of donate button is also awesome.
There used to be some Strattons NE of Jasper....is that you? If so, grew up about 2 miles from ya.
Just click the button at the bottom of the page and it will walk you through.
When you make a donation, is there any confirmation that it was received other than a payment to PayPal shown on my credit card statement?
I just made a (small) donation. This forum is worth a TON, and thanks to Darin for his work on creating/maintaining it.
I just made a donation. This forum is worth a few bucks from time to time. I have learned to stay away from the politics section... much too depressing.
I am new to this site and I think this site is best to explore knowledge and good source of information.
There are expenses and a good deal of time invested in maintaining the forum. If you enjoy the forum or find it useful, a couple bucks now and then would be greatly appreciated.
That's a great idea... !!! i will also think of to donate it... and tell more people to do it..!!!
I had some error.. do you guys faced it?
I do so appreciate the AC Forum!!! The short time it was down made me realize how much I enjoy it.
I threw in 20 bucks. How do we know if you have enough? You'll let us know?
I still gots my A/C but it's clear out in the barn now.
Paypal was easy and painless.
Hello, as president of the Upper Midwest A-C Club, the board of directors would like to have an ad on your website. We are willing to pay the $125.00 per year.
PM Darin directly. If I post his email here it will open him up to spam.
Very informative site. I am working to restore a D-14. Everyone should donate some money to this site.
This site has helped me out over and over and I will definitely be donating at least a few dollars from time to time. Although, I could use a new hat! Thanks!
I am looking forward to making some new friends on here and getting some advice on AC equipment is well worth a few bucks. Shops usually wont tell you anything at all they just say bring it in we'll take a look at it.
It's been over a year since anyone posted and for me to make a donation to the best site on the internet. Making a donation today.
Thanks to all who post and donate!


text:
You are here: Home / Archive / Arts in Nature / Black History Month FEMMEnomenal Spotlight: Dr. Grace Marilynn James
February 17, 2021 by Amy Joseph Landon
Black History Month FEMMEnomenal Spotlight: Dr. Grace Marilynn James
In honor of the 40th anniversary of the Artist in Residence Program and Kentucky Women's Suffrage Centennial in 2020, Bernheim presents FEMMEnomenal Bluegrass Botanicals by Gabriella Boros. This exhibition celebrates the extraordinary achievements of 10 Kentucky women through the invocation of 10 native Kentucky plants.
Meet Dr. Grace Marilynn James
Jefferson County, KY, 1923-1989
Photo Credit: U.S. National Library of Medicine, National Institutes of Health
Dr. Grace Marilynn James began her pediatric practice in Louisville when city hospitals were legally segregated. Despite these obstacles, she was the first Black woman on the faculty at the University of Louisville School of Medicine. She dedicated her practice to unwed mothers and the poorest of children. Dr. James was a tireless advocate for preventative care and universal healthcare, often giving her impoverished patients food, clothing, and toys.
Grace James was born in Charleston, West Virginia in 1923. After spending her early years in the town of Institute, West Virginia, Grace left to attend West Virginia State College. She completed post-graduate work there and at the University of Chicago before entering Meharry Medical College in Nashville, Tennessee, graduating with an M.D. degree in 1950. Dr. James then moved to New York City, where she completed an internship and pediatric residency at Harlem Hospital. Around this time, she also became a clinical fellow at Babies Hospital and the Vanderbilt Clinic, both part of Columbia Presbyterian Hospital. She also broadened her training to include child psychiatry, studying at Creedmoor State Hospital in Queens Village and becoming a fellow at the Albert Einstein School of Medicine, where she cared for children with disabilities.
In a fellowship application addressed to the National Urban League before she entered medical school, she explained that she had decided to pursue a career in medicine because of "interest in human suffering" and the suffering of Black Americans in particular. She also noted that a visit to Lincoln Hospital in the Bronx had inspired her to help "the ones who needed to be taught, educated and given a chance to learn sound principles of health." Dr. James made good on these promises when she moved to Louisville in 1953, opening a private pediatrics practice and a walk-in clinic serving Louisville's impoverished West End residents. In 1974, Dr. James also helped lead the Council on Urban Education and helped establish the West Louisville Health Education Program.
Long before others, Grace was outspoken about Black infant mortality and the underserved Black community at large. Eventually, she became affiliated with eight Louisville-area hospitals, and became the first Black American woman on the staff of Louisville Children's Hospital and one of only two Black women on the faculty of a southern medical school in the early 1950s.
In a series of public lectures and talks, as she advocated for Black American youth to embrace the study of medicine, she described the obstacles she had to face in her career, including the "triple jeopardy" confronting Black women in particular—that of "racism, capitalism, and sexism"—that prevented many from pursuing or completing their studies. Dr. James was a tireless activist for her causes, including comprehensive, preventive, and universal care.
FEMMEnomenal Bluegrass Botanicals celebrates the collective value women have had on our history, and much like the native plants in Kentucky, despite all the challenges, are still here, standing tall and beautiful, each in their own way.
Dr. Grace Marilynn James is represented by Harbinger of Spring (Erigenia bulbosa), an early bloomer in dense forests, as Grace was an early spokesperson for Black American health.
While in residence at Bernheim, Boros created a series of woodblock prints that celebrate ten notable Kentucky women. The prints, enlarged into fabric hangings, are suspended from trees along Lake Nevin Loop.
*Additional information for this post courtesy of the U.S. National Library of Medicine at the National Institutes of Health
Filed Under: Arts in Nature, enews1, Homepage Blog Feed

redpajama_set_name: RedPajamaCommonCrawl


Logan USD #326 in Logan, KS has been named the number one school district in Kansas by an independent website. Schooldigger is a national website that has reviewed the data of 255 school districts and 653 Kansas high schools. Their results placed Logan as the third highest ranked high school and the overall number one school district in Kansas.
WWW.Schooldigger.com was founded in 2006 to serve one purpose: to empower parents to make informed choices about choosing a school for their child. Their database contains detailed profiles for over 136,000 schools in every state in the U.S., including 20 years of enrollment data, several years of test scores, crime data, and real estate data, to help parents make better school choices. The site calculates school rankings based on state assessments of Common Core English Language Arts and Mathematics test scores released by individual State Department of Educations. These rankings are updated yearly and are based on the most recent test scores available. Their sources of data include the National Center for Education Statistics, U.S. Department of Education and individual State Departments of Education. They calculate an Average Standard Score by normalizing and averaging each school's test scores across all tests and grades. That average score is then sorted. The school with the highest score is ranked #1, the second highest #2, and so on. Schools, districts, and cities in the top 10% of the ranking list earn a five-star rating. The next 10% get 4 stars, etc.
The data is compared based on these scores, not on school or community population. Large urban schools are compared to small rural schools. There is no differentiation between school size. If a school has too small of a population, they exempt them from ranking. If a district has a class size under a certain threshold, it would be difficult to have reliable data to compare to other schools. The data and statistical information would still count, but they are not ranked as an individual school.
The Logan students and staff were told of their award during a presentation from principal David Kirkendall on Monday, March 27. The students gathered for a picture, then were congratulated by the staff with High-5's on the way back to class.


As part of User Research class at Pilot year - CIID, we spent two days in two different elderly homes in Copenhagen (Absalonhus and Bethania) - most of the residents have dementia. We interviewed some elderly people, the staff and the manager of both places. We could see the everyday life of the residents and how the home is structured and organised.
They want to be as autonomous as possible in terms of mobility and date/time understanding; and they want to have an active social life but also keep their privacy as a value.
We thought about some tools that could be developed to address these needs such as underlining natural cycles to keep them in contact with reality and passing time. We discussed using nature to connect them with the space/context and help them to relax. We also thought about how existing devices can be improve some existing devices such as phones, gym interfaces, watches, TV and radios, to make them more accessible for elderly people.


text:
DnaE2 Polymerase Contributes to In Vivo Survival and the Emergence of Drug Resistance in Mycobacterium tuberculosis
@article{Boshoff2003DnaE2PC,
title={DnaE2 Polymerase Contributes to In Vivo Survival and the Emergence of Drug Resistance in Mycobacterium tuberculosis},
author={Helena I. M. Boshoff and Michael B. Reed and Clifton E. Barry and Valerie Mizrahi},
journal={Cell},
volume={113},
pages={183-193}
H. Boshoff, M. Reed, +1 author V. Mizrahi
Medicine, Biology
The presence of multiple copies of the major replicative DNA polymerase (DnaE) in some organisms, including important pathogens and symbionts, has remained an unresolved enigma. We postulated that one copy might participate in error-prone DNA repair synthesis. We found that UV irradiation of Mycobacterium tuberculosis results in increased mutation frequency in the surviving fraction. We identified dnaE2 as a gene that is upregulated in vitro by several DNA damaging agents, as well as during…
View on Elsevier
cell.com
Mycobacterium tuberculosis
DNA-Directed DNA Polymerase
Genus Mycobacterium
DNA-dependent binding of nargenicin to DnaE1 inhibits replication in Mycobacterium tuberculosis
M. Chengalroyen, Mandy K. Mason, +13 authors V. Mizrahi
Assessment of the antimycobacterial activity of nargenicin revealed that it is a bactericidal genotoxin that induces a DNA damage response in Mycobacterium tuberculosis (Mtb) and inhibits growth by blocking the replicative DNA polymerase, DnaE1.
View 10 excerpts, cites methods and background
Distinct mechanisms of DNA repair in mycobacteria and their implications in attenuation of the pathogen growth
Krishna Kurthkoti, U. Varshney
Mechanisms of Ageing and Development
The findings, using Mycobacterium smegmatis model, suggest that deficiency of various DNA repair functions in single or in combinations severely compromises their DNA repair capacity and attenuates their growth under conditions typically encountered in macrophages.
Expression and function of the mutator DNA polymerasencoding umuC-like genes in mycobacteria
Robyn Brackin
Through gene expression studies, it was found that UV irradiation of Mycobacterium smegmatis resulted in the up-regulation of dinP3, which was determined to be a Y-family polymerase by sequence analysis, the first example of a Yfamily polymerases in mycobacteria forming part of the SOS regulon.
View 20 excerpts, cites background
DNA Replication Fidelity in the Mycobacterium tuberculosis Complex.
D. Warner, Jeremy M. Rock, S. Fortune, V. Mizrahi
This chapter focuses on mechanisms involved in maintaining DNA replication fidelity in M. tuberculosis, and considers the potential to target components of the DNA replication machinery as part of novel therapeutic regimens designed to curb the emerging threat of drug-resistance.
Important role of the nucleotide excision repair pathway in Mycobacterium smegmatis in conferring protection against commonly encountered DNA-damaging agents.
Krishna Kurthkoti, Pradeep Kumar, Ruchi Jain, U. Varshney
The strain deficient in NER was the most sensitive to commonly encountered DNA-damaging agents such as UV, low pH, reactive oxygen species, hypoxia, and was also sensitive to acidified nitrite, Taken together with previous observations on NER-deficient M. tuberculosis, these results suggest that NER is an important DNA repair pathway in mycobacteria.
DNA Replication in Mycobacterium tuberculosis.
Z. Ditse, M. Lamers, D. Warner
Microbiology spectrum
Current knowledge of the machinery responsible for DNA replication in M. tuberculosis is summarized, the potential contribution of the expanded complement of mycobacterial DNA polymerases to mutagenesis is discussed, and the possible role of DNA replication, in particular, its regulation and coordination with cell division is considered.
Evidence for the role of Mycobacterium tuberculosis RecG helicase in DNA repair and recombination
R. S. Thakur, S. Basavaraju, +4 authors G. Nagaraju
The FEBS journal
It is shown that M. tuberculosis RecG (MtRecG) expression was induced in response to different genotoxic agents, and results support that MtRecG plays an important role in processes related to DNA metabolism under normal as well as stress conditions.
Division of labor between SOS and PafBC in mycobacterial DNA repair and mutagenesis
Oyindamola O. Adefisayo, P. Dupuy, Astha Nautiyal, J. Bean, M. Glickman
The complex transcriptional regulatory network of the DDR in mycobacteria is defined and new insight is provided into the regulatory mechanisms controlling the genesis of antibiotic resistance in M. tuberculosis.
Specific mutations in the Mycobacterium tuberculosis rpoB gene are associated with increased dnaE2 expression.
I. Bergval, P. Klatser, A. Schuitema, L. Oskam, R. Anthony
The presence of specific mutations in rpoB is not only associated with impaired fitness but also results in a detectable, moderate yet persistent increase in the expression of dnaE2 but not recA, indicating subsequent adaptation may be critical to restore fitness.
DNA repair systems and the pathogenesis of Mycobacterium tuberculosis: varying activities at different stages of infection.
A. Gorna, R. Bowater, J. Dziadek
This review focuses on different DNA repair systems from mycobacteria and identifies questions that remain in the understanding of how these systems have an impact upon the infection processes of these important pathogens.
DNA metabolism in mycobacterium tuberculosis: implications for drug resistance and strain variability.
H. Boshoff, S. Durbach, V. Mizrahi
Scandinavian journal of infectious diseases
A model in which stress-induced DNA damage in M. tuberculosis plays a role in the evolution of chromosomally encoded drug resistance mutations by altering the global mutation rate by mechanisms akin to SOS mutagenesis is proposed.
Identification of Some DNA Damage-Inducible Genes of Mycobacterium tuberculosis: Apparent Lack of Correlation with LexA Binding
P. C. Brooks, F. Movahedzadeh, E. Davis
It appears most likely that an alternative mechanism of gene regulation in response to DNA damage exists in M. tuberculosis, and LexA is the repressor protein responsible for regulating expression of these SOS genes in E. coli.
Characterization of Escherichia coli DNA Lesions Generated within J774 Macrophages
Eliana Schlosser-Silverman, M. Elgrably-Weiss, I. Rosenshine, R. Kohen, S. Altuvia
It is found that under these conditions, early in the infection the majority of E. coli cells are viable but gene expression is inhibited, and protection against the macrophage-induced DNA damage is an important component of the bacterial defense mechanism within macrophages.
Error-prone repair DNA polymerases in prokaryotes and eukaryotes.
M. Goodman
Annual review of biochemistry
This review is devoted primarily to a discussion of Y-family polymerase members that exhibit error-prone behavior, which poses challenging questions about their roles in targeting and trafficking.
Highly mutagenic replication by DNA polymerase V (UmuC) provides a mechanistic basis for SOS untargeted mutagenesis.
A. Maor-Shoshani, N. Reuven, G. Tomer, Z. Livneh
DNA polymerase V (pol V; UmuC), which previously has been shown to be a lesion-bypass DNA polymerase, was highly mutagenic during in vitro gap-filling replication of a gapped plasmid carrying the cro reporter gene, suggesting that replication by pol V, in the presence of UmuD', RecA, and ssDNA-binding protein, is the basis of chromosomal SOS untargeted mutagenesis.
Regulation of the Mycobacterium tuberculosis hypoxic response gene encoding α-crystallin
D. Sherman, M. Voskuil, D. Schnappinger, R. Liao, M. Harrell, G. Schoolnik
The results suggest a possible role for Rv3132c/3133c/Rv3134c in mycobacterial latency, and an apparent operon that includes the putative two-component response regulator pair R v3133/3132/3134/3 134c is suggested.
Persistence of Mycobacterium tuberculosis in macrophages and mice requires the glyoxylate shunt enzyme isocitrate lyase
J. Mckinney, K. H. Z. Bentrup, +7 authors D. Russell
It is reported that persistence of M. tuberculosis in mice is facilitated by isocitrate lyase (ICL), an enzyme essential for the metabolism of fatty acids, an observation with important implications for the treatment of chronic tuberculosis.
Two Essential DNA Polymerases at the Bacterial Replication Fork
E. Dervyn, C. Suski, +5 authors S. Ehrlich
Bacillus subtilis genome analysis shows that dnaEBS is essential for cell viability and for the elongation step of DNA replication, as is polC, and concludes that there are two different essential DNA polymerases at the replication fork of B. subtil is, as was previously observed in eukaryotes.
Molecular genetic basis of antimicrobial agent resistance in Mycobacterium tuberculosis: 1998 update.
S. Ramaswamy, J. Musser
Tubercle and lung disease : the official journal of the International Union against Tuberculosis and Lung Disease
Although remarkable advances have been made, much remains to be learned about the molecular genetic basis of drug resistance in Mycobacterium tuberculosis, it is reasonable to believe that development of new therapeutics based on knowledge obtained from the study of the molecular mechanisms of resistance will occur.
Exploring drug-induced alterations in gene expression in Mycobacterium tuberculosis by microarray hybridization.
M. Wilson, J. Derisi, +4 authors G. Schoolnik
It is shown that isoniazid induced several genes that encode proteins physiologically relevant to the drug's mode of action, including an operonic cluster of five genes encoding type II fatty acid synthase enzymes and fbpC, which encodes trehalose dimycolyl transferase.


Life Lines is a publication brought to you by the University's Employee and Family Assistance Program Provider, Homewood Human Solutions. With the aim of "Improving Your Quality of Life – One Step at a Time", this month's edition focuses on "Talking about Mental Illness".
Read Life Lines May 2014 to learn more.


Eva Bakkeslett is an artist, filmmaker, curator and gentle activist, exploring the potential for social change through gentle actions and subtle mind-shifts. Her socially engaged practice often combines film, participatory events and workshops; She frequently collaborates with other artist, activists, scientist and engaged people in her work.
Eva creates spaces and experiences that challenge our thinking and unravel new narratives that inspire and engage us to make sense of and embody sustainable and thriving ways. By revealing and reclaiming forgotten or rejected practices, concepts and cultures her work directs our attention to the patterns that connect us to the earth as a living organism.
Eva is particularly inspired by the process of fermentation and explores how this can be a method for re-imagining sustainable human cultures. She looks at how microbes can inspire creative problem solving, collaboration and transformation to find new ways to deal with the many challenges we are now facing in the world.
Eva shows, lectures, curates and performs her work worldwide. She has an MA in Art & Ecology from Dartington College of Art in England and lives with her family on the beautiful island of Engeløya in North Norway. Here she tries to walk the talk, grow and harvest her own food, and learn as much as possible about life and sustainability from old folks, animals, birds, trees and the earth whilst enjoying the beauty of every day.
Get in touch to discuss my work or a new project.


text:
Quicksilver (software) - Wikipedia, the free encyclopedia
http://QSApp.com
1.0ß60 (Build 3850)[1] / June 5, 2011; 2 months ago
Preview release
Application Launcher (utility software)
Quicksilver is a computer utility software program for Mac OS X, originally developed by Blacktree Software and distributed freely. It is essentially a graphical shell for the Mac OS X operating system, allowing users to use the keyboard to rapidly perform tasks such as launching applications, manipulating files, or sending e-mail. It is similar to the Mac OS X applications LaunchBar and Alfred, but uses a different interaction paradigm. Although a complex application, Quicksilver is based on a simple three-panel interface.
1 Interface
2 Extensibility
2.1 Triggers
2.2 Plug-ins
2.3 Flexibility
3.1 Alchemy
[edit] Interface
Quicksilver is designed to run all the time in the background while the operating system is running. It maintains a configurable "catalog" of items including files and folders on disk, address book contacts, Internet bookmarks, scripts, clipboard contents, etc. Pressing a configurable hotkey or a combination of keys, usually Control-Space, invokes the Quicksilver command window, which contains three panes.
The panes roughly correspond to the object, predicate, and attributes of the desired task. The user first chooses the object of the command (an application, a file, an e-mail address, an IM contact, a bookmark, etc.), then an action (predicate) to be performed on that object, and finally additional details (attributes) for the action.
The Quicksilver user interface provides access to a wide range of objects by applying incremental search to input typed on the keyboard. Most objects require typing only a few letters into any of the panes, reducing the set of suggestions and displaying the most likely match. Quicksilver uses a priority system based on previous usage (see mnemonic) to determine the most likely match, apparently "learning" from the user's actions.
Once an object is selected in the first panel, pressing the tab key moves the focus to the second pane to search for available actions for that object (the "predicate"). Each type of object has certain actions which can be performed with it. For example, files on the hard drive can be moved to the trash, revealed in the Finder, or opened in the default application for that file type.
Some actions require an "attribute" that provides further details or the specifics of an action. In this case, the third panel is used to find such an object on which to perform the command. For instance, the "Move to folder…" command requires the selection of a folder in the third panel. Quicksilver also accepts a direct-text entry mode in its panels (by pressing the "." key on a QWERTY keyboard or "=" on an AZERTY keyboard), making possible quick shell commands (Terminal) and instant messaging software (such as iChat and Adium).
Quicksilver's icon is based on the alchemical symbol for mercury (quicksilver being an archaic name for the element mercury).
[edit] Extensibility
[edit] Triggers
Quicksilver allows users to define "triggers," which perform a specific command (direct object + action + indirect object) whenever a predefined keyboard shortcut is pressed. For instance, the <"Documents", "Reveal"> command could be bound to the F7 key. Then pressing the F7 key would open the Documents folder in the Finder and bring it to the front.
In addition to keyboard triggers, the "Mouse Triggers" plugin allows commands to be bound to simple mouse actions such as clicks or mouseovers in different parts of the screen. A plugin called "Abracadabra" supports triggers activated through more complicated mouse gestures.
[edit] Plug-ins
Quicksilver has a plug-in architecture, with each plug-in providing integration with a program, interface, or new feature. For example, plug-ins exist for sending email via Mail without opening the application or manipulating images via text commands.
[edit] Flexibility
Because shell scripts and AppleScripts can be stored in the catalog, any function which can be performed using a script can be tied to Quicksilver, either through the command window, or through triggers. Because most Apple-native applications have extensive scripting libraries, many common tasks can easily be performed from Quicksilver. For instance, iTunes can be told to play or pause, increase or decrease the rating on the current track, or skip to the previous or next track.
There are various visual interfaces for the Quicksilver, alternative Constellation Menus and also a Notification Hub, which supports Growl via a plugin.
[edit] Development
On November 6, 2007, the source code for Quicksilver was made available via Google Code.[2]
Quicksilver is now developed by a team of volunteers with work on the open source project increasing throughout 2010.[3]
In November 2009, development shifted to using GitHub.[4]
At the end of 2010, a new website QSApp.com was launched, with the aim of unifying and collating all of Quicksilver's fragmented builds, plugins and support groups. Since its launch, the site has included a new Plugins Repository, Wiki and Downloads section. After several months of development, Quicksilver version β59 was released; a marked point in the history of the application.
[edit] Alchemy
Experimental trunk builds of Quicksilver, known as Alchemy, have many major changes.
Triggers are moving to a separate product, called Catalyst
All the little frameworks are being joined into one big one called Crucible. This includes extensions and core functionality that most apps and plugins will use. This is currently called QSBase.framework
The preferences will be greatly simplified. There will be Extras-style advanced preferences for the fiddly options.
Plugins are going to be hidden from most users, they'll activate themselves automatically or be installable from the web
β5X Plugins are incompatible.
These builds have four major components: Crucible, a framework with extension to AppKit and tools common to all Alchemy apps; elements, a framework supporting the plugin architecture; quicksilver, a command window driven launcher; and catalyst, which triggers a preference pane.
[edit] See also
Comparison of application launchers
^ a b Latest Preview release of Quicksilver, 2011-06-05, https://github.com/quicksilver/Quicksilver/downloads, retrieved 2011-06-05
^ Lu, Mat (2007-11-06), Quicksilver goes Open Source, http://www.tuaw.com/2007/11/06/quicksilver-goes-open-source/, retrieved 2007-11-06
^ Quicksilver GitHub, https://github.com/quicksilver/Quicksilver/, retrieved 10 June 2011
^ "Bug fixes to latest - svn/branches/B5X B56a7 (3825)". blacktree-quicksilver mailing list. 10 November 2009. http://groups.google.com/group/blacktree-quicksilver/msg/f3ffca725a6d5d33. Retrieved 17 November 2009.
[edit] External links
Quicksilver development homepage
Google Tech Talks presentation by Nicholas Jitkoff
Retrieved from "http://en.wikipedia.org/wiki/Quicksilver_(software)"
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