Kinetic mixing, custodial symmetry, $Z$, $Z^{\prime }$ interactions and $Z^{\prime }$ production in hadron colliders

Abstract

In this work we study the interactions of the $Z$ and $Z^{\prime }$ generated by kinetic mixing in a class of theories for physics beyond the standard model motivated by the dark matter problem, containing a spontaneously broken extra $U(1{)}_d$ gauge factor group in a hidden scenario and a Higgs sector respecting custodial symmetry. It is shown that custodial symmetry allows us to write the $Z$ and $Z^{\prime }$ couplings to standard model fermions in terms of the measured values of $\hat{I}\pm$, $G_F$, $M_Z$ and $M_{Z^{\prime }}$. Working at the loop level, we calculate the ratio $\ddot{I}{}_0=M_W^2/{\hat{c}}_Z^2{M}_Z^2\hat{\ddot{I}}$ used in the fit to electroweak precision data (EWPD) and use its value to estimate possible effects of kinetic mixing at the electroweak scale. For the $Z$ sector, we calculate the oblique parameters $S$ and $T$, finding that for $M_{Z^{\prime }}\ge M_Z$ our results are in agreement with the values of the oblique parameters extracted from the global fit to EWPD at $1\ddot{I}ƒ$ level. As to the $Z^{\prime }$ sector, we calculate the $Z^{\prime }$ contributions to charged lepton pair production at the Large Hadron Colliderin the well motivated case of dark matter entering particle physics as the matter fields of the $U(1{)}_d$ gauge symmetry with perturbative couplings at the electroweak scale, finding that data reported by the Compact Muon Selenoid Collaboration impose a lower limit $M_{Z^{\prime }}\gtrsim 5.0TeV$.

14 pages, updated version focused on the dark matter problem. Notation changed to harmonize it with the PDG one. Loop level analysis explicit for SM particles. Comparison with updated data by CMS for $c_u$ and $c_d$