Changes between Version 7 and Version 8 of anomalyfreeZprime

Timestamp:

Jun 5, 2020, 5:18:12 PM (5 years ago)

Author:

MartinBauer

Comment:

--

anomalyfreeZprime

@@ -72 +72 @@
 
 {{{#!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.
+
+