# Changes between Version 7 and Version 8 of anomalyfreeZprime

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Timestamp:
06/05/20 17:18:12 (4 months ago)
Comment:

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 v7 {{{#!latex $\mbox{Var}[\tau(X_p,X_d)]=\mbox{Var}[E(\tau(X_p,X_d)|X_p)]+E[\mbox{Var}(\tau(X_p,X_d)|X_p)]$ \begin{align*} \mathcal{L}_\text{gauge} = -\frac{1}{4} \begin{pmatrix} \hat{B}_{\mu \nu} & \hat{Z}'_{\mu \nu} \end{pmatrix} \begin{pmatrix} 1 & s_{Z'} \\ s_{Z'} & 1 \end{pmatrix} \begin{pmatrix} \hat{B}_{\mu \nu} \\ \hat{Z}'_{\mu \nu} \end{pmatrix} \; , \end{align*} }}} and afternormalizing the kinetic terms and rotating to the mass eigenbasis, the masses of the vector bosons are given by {{{#!latex \begin{align*} m_\gamma &= 0 \notag \\ m_{Z}^2&= \dfrac{v^2}{4}(g^2+{g'}^2) \; \left(1-\dfrac{v^2}{v_S^2} \; \dfrac{s_{Z'}^2{g'}^2}{8g_{Z'}^2 q_S^2}\right) + \mathcal{O}\left( \dfrac{v^6}{v_{S}^4} \right) \\ m_{Z'}^2&= \dfrac{g_{Z'}^2q_S^2v_S^2}{2 c_{Z'}^2} + \dfrac{v^2}{4}{g'}^2 t_{Z'}^2 % + \dfrac{v^4}{v_S^2} \; \dfrac{s_{Z'}^2{g'}^2}{8g_{Z'}^2 q_S^2}(g^2+{g'}^2) + \mathcal{O}\left( \dfrac{v^4}{v_S^2} \right)  \;. \end{align*} }}} As a second structural ingredient we give mass to the new gauge boson by introducing a complex scalar $S$ with the potential {{{#!latex \begin{align*} \mathcal{L}_\text{scalar} = \frac{1}{2}\, ( D_\mu S) (D^\mu S)^\dagger + \mu_S^2 \, S^\dagger S + \frac{\lambda_S}{2} (S^\dagger S)^2 + \lambda_{HS} \, H^\dagger H \, S^\dagger S\; . \end{align*} }}} In this case the covariant derivative introduces the charge $q_S$ of the heavy scalar under the new gauge group.[[BR]] The couplings of the mass eigenstates to fermions and scalars play an important role in the following analysis and we find {{{#!latex \begin{align*} \mathcal{L_\text{fermion}}&= ej_\text{em} A \notag\\ &\phantom{=}- c_w s_3 t_{Z'} ej_\text{em} Z  +(c_3+s_ws_3t_{Z'})\frac{e}{s_wc_w}j_Z Z  + \frac{s_3}{c_{Z'}}g_{Z'}j_{Z'} Z\notag\\ &\phantom{=}- c_w c_3 t_{Z'} ej_\text{em} Z' +(s_wc_3t_{Z'}-s_3)\frac{e}{s_wc_w}j_Z Z' + \frac{c_3}{c_{Z'}}g_{Z'}j_{Z'} Z' \end{align*} }}} and {{{#!latex \begin{align*} \mathcal{L_\text{scalar}}&\ni \frac{v}{8}(g^2+g^{\prime 2}) (c_\alpha H-s_\alpha S) Z_{\mu}Z^\mu \\ &\phantom{\ni}  +\frac{v}{4}s_wt_{Z'}(g^2+g^{\prime 2}) (c_\alpha H-s_\alpha S) Z_\mu Z^{\prime \mu}\notag\\ &\phantom{\ni}  +\frac{v}{8}  s_w^2 t_{Z'}^2\bigg[c_\alpha \bigg(g^2\!+\!g^{\prime 2}\!+\!\frac{4g_{Z'}^2 q_S^2 t_\alpha}{s_w^2s_{Z'}^2}\frac{v_S}{v} \bigg) H- s_\alpha  \bigg(g^2\!+\!g^{\prime 2}\!-\!\frac{4g_{Z'}^2 q_S^2 t_\alpha}{s_w^2 s_{Z'}^2}\frac{v_S}{v} \bigg)S\bigg] Z'_\mu Z^{\prime \mu}\notag\,. \end{align*} }}} The phenomenology of anomaly-free $U(1)$-extensions can thus be described by a small number of model parameters. The Lagrangian features the most relevant new parameters {{{#!latex \begin{align*} \{ \; m_\chi, \, g_{Z'}, m_{Z'}, s_{Z'}, \, m_S, \lambda_{HS}\; \} \; . \end{align*} }}} The charges under the new $U(1)$-symmetry we assume to be of order one. As long as we focus on a heavy dark matter mediator with on-shell decays, $m_{Z'} > 2 m_\chi$, the dark matter mass mainly enters the computation of the mediator widths $\Gamma_{S,Z'}$.\bigskip The vector and scalar mediator masses are typically related. A hierarchy with a comparably light scalar $\lambda_S \ll g_{Z'}$ is possible, but not the focus of our paper. Alternatively, the scalar can be heavier than the vector, $g_{Z'}\ll \lambda_S< 4\pi$. In this case, the small gauge coupling suppresses the interaction of the new gauge boson with the Standard Model. This does not only affect the LHC production cross section, it also reduces the annihilation cross section in the early universe to the point where an efficient annihilation is only possible around the pole condition $m_{Z'} = 2 m_\chi$. The phenomenology of the vector mediator is determined by its couplings to the Standard Model and by its mass $m_{Z'}$.  In Eq.\eqref{eq:all_mixings} we see that couplings to SM fermions can arise through kinetic mixing ($\tchi$), through mixing with the $Z$-boson ($s_3$), or through the $U(1)$ charges of the fermions ($g_{Z'}$). The properties of the new scalar $S$ are largely independent of the dark matter properties. All couplings to a pair of SM particles proceed through the Higgs portal ($s_\alpha$), with the possible exception of a the coupling to right-handed neutrinos in the case of $U(1)_{B-L}$. Interesting features only arise in couplings linking both mediators, like the $Z'$-$S$-$Z$ coupling.