Vector mediator

Overview

The vector_mediator module contains two models for which dark matter interacts with the Standard Model through a vector mediator. For energies \(\mu\gg 1~\mathrm{GeV}\), the interaction Lagrangian is:

\[\mathcal{L}_{\mathrm{Int}(V)}= V_\mu \left( g_{V\chi} \bar{\chi} \gamma^\mu \chi + \sum_f g_{Vf} \bar{f} \gamma^\mu f \right) - \frac{\epsilon}{2} V^{\mu\nu} F_{\mu\nu}.\]

where the \(\epsilon\) is the kinetic mixing couplings for the Standard Model fermions and \(F_{\mu\nu}\) is the field strength tensor for the photon (we will describe the remaining parameters below.) For energies \(\mu<1~\mathrm{GeV}\), the quarks and gluons confine into mesons and baryons. In order to describe the interactions of the vector mediator to the mesons, we use Chiral Perturbation theory. The interaction Lagrangian becomes:

\[\begin{split}\mathcal{L}_{\mathrm{Int}(V)} & = -i (g_{Vu} - g_{Vd}) V^\mu \left( \pi^+ \partial_\mu \pi^- - \pi^- \partial_\mu \pi^+ \right)\\ & \hspace{1cm} + ( g_{Vu} - g_{Vd} )^2 V_\mu V^\mu \pi^+ \pi^-\\ & \hspace{1cm} + 2 e (Q_u - Q_d) (g_{Vu} - g_{Vd}) A_\mu V^\mu \pi^+ \pi^-\\ & \hspace{1cm} + \frac{1}{8\pi^2 f_\pi} \epsilon^{\mu\nu\rho\sigma} (\partial_\mu \pi^0)\\ & \hspace{2cm} \times \left\{ e (2 g_{Vu} + g_{Vd}) \left[ (\partial_\nu A_\rho) V_\sigma + (\partial_\nu V_\rho) A_\sigma \right] \right. \\ & \hspace{4cm} \left. + 3 (g_{Vu}^2 - g_{Vd}^2) (\partial_\nu V_\rho) V_\sigma \right\}\\ & \hspace{1cm} + V_{\mu}\left(g_{Ve}\bar{e}\gamma^{\mu}e + g_{V\mu}\bar{\mu}\gamma^{\mu}\mu\right)\end{split}\]

where \(f_{\pi}\approx 93~\mathrm{MeV}\) and the model parameters are:

  1. \(m_{\chi}\): dark matter mass,

  2. \(m_{V}\): vector mediator mass,

  3. \(g_{V\chi}\): coupling of vector mediator to dark matter,

  4. \(g_{Vq}\): (\(q=u,d,s\)) coupling of vector mediator to standard model quarks,

  5. \(g_{V\ell}\): (\(\ell=e,\mu\)) coupling of vector mediator to standard model leptons,

In addition to the generic vector mediator model, hazma also contains specialized models for realization for a theory in which the vector mediator mixes with the Standard model photon. In the kinetic-mixing model, we assume that the vector mediator doesn’t directly interact with the Standard model particles aside from the mixing with the photon. The vector then inherits its coupling to the charged Standard model fermions from the photon. The kinetic-mixing model contains the following parameters:

  1. \(m_{\chi}\): dark matter mass,

  2. \(m_{V}\): scalar mediator mass,

  3. \(g_{V\chi}\): coupling of scalar mediator to dark matter,

  4. \(\epsilon\): kinetic mixing parameter between the vector mediator and SM photon.

The generic couplings are obtained from these parameters through the following relationships:

\[g_{Vf} = \epsilon e Q_{q}\]

where \(f=(u,d,s,e,\mu)\). For details on how to uses these classes, see Basic Usage.

Classes