Tutorial - Part 3: Testing a Simple Dark Matter Model

Effective Dark Matter

A prominent toy model to estimate and compare the sensitivity of different dark matter detection methods is the "effective operator approach":

Let us assume the standard model plus an extra dark matter particle χ which we assume to be a Dirac fermion. By a simple additional bifold symmetry we only allow pairwise interaction of this new particle such that is is rendered stable. Let us furthermore assume that there is a mediator vector particle V_μ that couples weakly to pairs of χ but also to pairs of Standard Model quarks q, such that dark matter can weakly interact with our known matter. We therefore have two new vector-like interactions g χ Γ^μ χ V_μ and g q Γ^μ q V_μ. (For the sake of simplicity, we have assumed that both q and χ have identical couplings g to the mediator. We also only consider u and d quarks in this analysis) The interaction can be illustrated as follows:

If the mediator has a mass much larger than the energy of a given experient (let's say beyond the TeV range), it cannot be resolved and one only oberserves a 4-fermi interaction just like in electroweak interactions below the W/Z-boson mass scale. In the language of QFT, we have the following effective operator: 1/Λ² (q Γ^μ q)(χ Γ_μ χ) In diagrammatic form, it looks as follows:

The prefactor of this coupling is often denoted as 1/Λ² and we will follow this convention here. Note that Λ has mass dimention 1.

The effective approach has two main advantages. One is related to the number of parameters, one to the comparability of different processes.

In the following, we will test two possibilities for the interaction vertex, namely

• vector like interaction: Γ^μ = γ^μ and
• axial vector like interaction Γ^μ = γ^μγ⁵.

We want to test the constraints the LHC can set on these models. For that purpose, everyone again should consider a different mass m_χ of the dark matter candidate. Furthermore, as few data points are required to find the exclusion line, everyone only has to analyse one of the two above models. For a given point, the task is to find the respective 90% confidence limit on Λ (Note that this is a difference confidence level as the 95% from before, as 90% is the traditional limit for dark matter searches!).