== Anomaly Free Z prime Model

=== Autors

* Martin Bauer
  * Durham University
  * martin.m.bauer@...

* Sascha Diefenbacher
  * Universität Hamburg
  * sascha.daniel.diefenbacher@...

* Tilman Plehn
  * Universität Heidelberg
  * plehn@...

* Michael Russell


* Daniel A. Camargo

=== Model Description

We consider consistent dark matter models with a spin-1 mediator Z' and a dark matter
fermion χ, charged under the new gauge group. The available options are purely singlet
SM fermions, gauged lepton number differences, or the well-known anomaly-free difference
between the lepton and baryon numbers 

The $Z'$ couplings to currents of SM fermionsare given by:[[BR]]

{{{#!latex
\begin{alignat}{9}
\mathcal{L}_\text{fermion} = -g_{Z'} j'_\mu & {Z'}^\mu \notag \\
j'_\mu&= 0 \qquad && U(1)_X \notag  \\
j'_\mu&= \bar L_i \gamma_\mu L_i 
          + \bar \ell_i\gamma_\mu \ell_i 
          - \bar L_j \gamma_\mu L_j -\bar\ell_j\gamma_\mu \ell_j
            \qquad && U(1)_{L_i-L_j} \notag \\ 
j'_\mu&=  \frac{1}{3}\bar Q \gamma_\mu Q 
          + \frac{1}{3}\bar u_R\gamma_\mu u_R 
          + \frac{1}{3}\bar d_R\gamma_\mu d_R
          - \bar L \gamma_\mu L 
          + \bar \ell\gamma_\mu \ell
            \qquad && U(1)_{B-L} \; ,
\end{alignat}
}}}

where $g_{Z'}$ denotes the dark gauge coupling.  The different
coupling structures shown above can be understood in terms of a flavor
structure of a dark gauge coupling matrix. 

The fermion current structure can be
generalized to include the dark matter current. To couple to the gauge
mediator the dark matter fermion has to be a Dirac fermion. To avoid
new anomalies, the dark matter candidate cannot be chiral and its
charges under the new gauge group are $q_{\chi_L}=q_{\chi_R}$. This
defines a dark fermion Lagrangian with a vector mass term


{{{#!latex
\begin{align*}
\mathcal{L}_\text{DM}= i \bar \chi \not{D} \chi - m_\chi \bar \chi \chi \; ,
\end{align*}
}}}

with the covariant derivative of the SM-singlet fermion 
{{{#!latex
$D_\mu=\partial_\mu -ig_{Z'} q_\chi \hat Z'_\mu$.
}}}

In all cases, the kinetic term for the $U(1)$ gauge bosons is not canonically normalized

{{{#!latex
$\mbox{Var}[\tau(X_p,X_d)]=\mbox{Var}[E(\tau(X_p,X_d)|X_p)]+E[\mbox{Var}(\tau(X_p,X_d)|X_p)]$
}}}