Changes between Version 7 and Version 8 of anomalyfreeZprime


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

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  • anomalyfreeZprime

    v7 v8  
    7272
    7373{{{#!latex
    74 $\mbox{Var}[\tau(X_p,X_d)]=\mbox{Var}[E(\tau(X_p,X_d)|X_p)]+E[\mbox{Var}(\tau(X_p,X_d)|X_p)]$
     74\begin{align*}
     75\mathcal{L}_\text{gauge}
     76= -\frac{1}{4}
     77\begin{pmatrix} \hat{B}_{\mu \nu} & \hat{Z}'_{\mu \nu} \end{pmatrix}
     78\begin{pmatrix} 1 & s_{Z'} \\ s_{Z'} & 1 \end{pmatrix}
     79\begin{pmatrix} \hat{B}_{\mu \nu} \\ \hat{Z}'_{\mu \nu} \end{pmatrix} \; ,
     80\end{align*}
    7581}}}
    7682
     83and afternormalizing the kinetic terms and rotating to the mass eigenbasis, the masses of the vector bosons are given by
     84
     85{{{#!latex
     86\begin{align*}
     87m_\gamma &= 0 \notag \\
     88m_{Z}^2&=
     89 \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)
     90 + \mathcal{O}\left( \dfrac{v^6}{v_{S}^4} \right) \\
     91 m_{Z'}^2&=
     92 \dfrac{g_{Z'}^2q_S^2v_S^2}{2 c_{Z'}^2} + \dfrac{v^2}{4}{g'}^2 t_{Z'}^2
     93% + \dfrac{v^4}{v_S^2} \; \dfrac{s_{Z'}^2{g'}^2}{8g_{Z'}^2 q_S^2}(g^2+{g'}^2)
     94 + \mathcal{O}\left( \dfrac{v^4}{v_S^2} \right)  \;.
     95\end{align*}
     96}}}
     97
     98As a second structural ingredient we give mass to the new gauge boson
     99by introducing a complex scalar $S$ with the potential
     100
     101{{{#!latex
     102\begin{align*}
     103\mathcal{L}_\text{scalar}
     104= \frac{1}{2}\, ( D_\mu S) (D^\mu S)^\dagger
     105 + \mu_S^2 \, S^\dagger S
     106 + \frac{\lambda_S}{2} (S^\dagger S)^2
     107 + \lambda_{HS} \, H^\dagger H \, S^\dagger S\; .
     108\end{align*}
     109}}}
     110
     111In this case the covariant derivative introduces the charge $q_S$ of
     112the heavy scalar under the new gauge group.[[BR]]
     113
     114The couplings of the mass eigenstates to fermions and scalars play an important role in the following analysis and we find
     115
     116{{{#!latex
     117\begin{align*}
     118\mathcal{L_\text{fermion}}&= ej_\text{em} A \notag\\
     119&\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\\
     120&\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'
     121  + \frac{c_3}{c_{Z'}}g_{Z'}j_{Z'} Z'
     122\end{align*}
     123}}}
     124
     125and
     126
     127{{{#!latex
     128\begin{align*}
     129\mathcal{L_\text{scalar}}&\ni \frac{v}{8}(g^2+g^{\prime 2}) (c_\alpha H-s_\alpha S) Z_{\mu}Z^\mu \\
     130&\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\\
     131&\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\,.
     132\end{align*}
     133}}}
     134
     135The phenomenology of
     136anomaly-free $U(1)$-extensions can thus be described by a small number of
     137model parameters. The Lagrangian features the most relevant new
     138parameters
     139
     140{{{#!latex
     141\begin{align*}
     142\{ \; m_\chi, \, g_{Z'}, m_{Z'}, s_{Z'}, \, m_S, \lambda_{HS}\; \} \; .
     143\end{align*}
     144}}}
     145
     146The charges under the new $U(1)$-symmetry we assume to be of order
     147one. As long as we focus on a heavy dark matter mediator with
     148on-shell decays, $m_{Z'} > 2 m_\chi$, the dark matter mass mainly
     149enters the computation of the mediator widths $\Gamma_{S,Z'}$.\bigskip
     150
     151The vector and scalar mediator masses are typically related.
     152A hierarchy with a comparably light scalar
     153$\lambda_S \ll g_{Z'}$ is possible, but not the focus of our
     154paper. Alternatively, the scalar can be heavier than the
     155vector, $g_{Z'}\ll \lambda_S< 4\pi$. In this case, the small gauge
     156coupling suppresses the interaction of the new gauge boson with the
     157Standard Model. This does not only affect the LHC production cross section, it
     158also reduces the annihilation cross section in the early universe to
     159the point where an efficient annihilation is only possible around the
     160pole condition $m_{Z'} = 2 m_\chi$.
     161
     162The phenomenology of the vector mediator is determined by its
     163couplings to the Standard Model and by its mass $m_{Z'}$.  In
     164Eq.\eqref{eq:all_mixings} we see that couplings to SM fermions can
     165arise through kinetic mixing ($\tchi$), through mixing with the
     166$Z$-boson ($s_3$), or through the $U(1)$ charges of the fermions
     167($g_{Z'}$).
     168
     169The properties of the new scalar $S$ are largely independent of the
     170dark matter properties. All couplings to a pair of SM particles
     171proceed through the Higgs portal ($s_\alpha$), with the possible
     172exception of a the coupling to right-handed neutrinos in the case of
     173$U(1)_{B-L}$. Interesting features only arise in couplings linking
     174both mediators, like the $Z'$-$S$-$Z$ coupling.
     175
     176