Scalar mediator

Overview

The scalar_mediator module contains three models for which dark matter interacts with the Standard Model through a scalar mediator. For energies \(\mu\gg 1~\mathrm{GeV}\), the interaction Lagrangian can for this theory is given by:

\[\begin{split}\mathcal{L}_{\mathrm{Int}(S)} & = -S \left( g_{S\chi} + g_{Sf} \sum_f \frac{y_f}{\sqrt{2}} \bar{f} f \right) \\ & \hspace{2cm} + \frac{S}{\Lambda} \left( g_{SG} \frac{\alpha_\mathrm{EM}}{4\pi} F_{\mu\nu} F^{\mu\nu} + g_{SF} \frac{\alpha_s}{4\pi} G_{\mu\nu}^a G^{a \mu\nu} \right)\end{split}\]

where the \(y_{f}\)’s are the Yukawa couplings for the Standard Model fermions and \(F_{\mu\nu}\) and \(G^{a}_{\mu\nu}\) are the field strength tensors for the photon and gluons (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 scalar mediator to the mesons, we use Chiral Perturbation theory. The interaction Lagrangian becomes:

\[\begin{split}\mathcal{L}_{\mathrm{Int}(S)} & = \frac{2 g_{SG}}{9 \Lambda} S \left[ (\partial_\mu \pi^0) (\partial^\mu \pi^0) + 2 (\partial_\mu \pi^+) (\partial^\mu \pi^-) \right]\\ & \hspace{1cm} + \frac{4 i e g_{SG}}{9 \Lambda} S A^\mu \left[ \pi^- (\partial_\mu \pi^+) - \pi^+ (\partial_\mu \pi^-) \right]\\ & \hspace{1cm} - \frac{B (m_u + m_d)}{6} \left( \frac{3 g_{Sf}}{v_h} + \frac{2 g_{SG}}{3 \Lambda} \right) S \left[ (\pi^0)^2 + 2 \pi^+ \pi^- \right]\\ & \hspace{1cm} + \frac{B (m_u + m_d) g_{SG}}{81 \Lambda} \left( \frac{2 g_{SG}}{\Lambda} - \frac{9 g_{Sf}}{v_h} \right) S^2 \left[ (\pi^0)^2 + 2 \pi^+ \pi^- \right]\\ & \hspace{1cm} + \frac{4 e^2 g_{SF}}{9\Lambda} S \pi^+ \pi^- A_\mu A^\mu\\ & \hspace{1cm} - g_{S \chi} S \bar{\chi} \chi - g_{Sf} S \sum_{\ell=e,\mu} \frac{y_\ell}{\sqrt{2}} \bar{\ell} \ell.\end{split}\]

where \(B\approx 2800~\mathrm{MeV}\), \(v_{h} = 246~\mathrm{GeV}\) and the model parameters are:

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

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

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

  4. \(g_{Sf}\): coupling of scalar mediator to standard model fermions,

  5. \(g_{SG}\): effective coupling of scalar mediator to gluons,

  6. \(g_{SF}\): effective coupling of scalar mediator to photons and

  7. \(\Lambda\): cut-off scale for the effective interactions.

In addition to the generic scalar mediator mode, hazma also contains specialized models for realizations of the Higgs-portal and Heavy-quark theories. In the Higgs-portal model, we assume that the scalar mediator doesn’t directly interact with the standard model particles aside from the Higgs. We assume the scalar mixes with the Higgs and inherits all its interactions to the Standard model through the mixing. The Higgs-portal model contains the following parameters:

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

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

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

  4. \(\sin\theta\): mixing angle between the scalar mediator and Higgs.

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

\[g_{Sf} = \sin\theta, g_{SG} = 3\sin\theta, g_{SF} = -\frac{5}{6}\sin\theta, \Lambda = v_{h}.\]

The Heavy-quark model assumes that there exists a new heavy quark and that the scalar mediator only couples the heavy quark. The parameters of this model are:

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

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

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

  4. \(g_{SQ}\): coupling of scalar mediator to the heavy quark,

  5. \(Q_{Q}\): charge of the heavy quark,

  6. \(m_{Q}\): mass of the heavy quark.

The relationships between these parameters and the generic parameters are:

\[g_{SG} = g_{SQ}, g_{SF} = 2Q_{Q}^2g_{SQ}, \Lambda = m_{Q}.\]

For details on how to uses these classes, see Basic Usage.

Classes