Fully Geant4 compatible package for the simulation of Dark Matter in fixed target experiments

A package for the simulation of Dark Matter (DM) particles is presented. It is intended for fixed target experiments. We chose the Geant4 toolkit framework as a standard in order to be compatible with particle tracking programs used in many High Energy Physics (HEP) experiments. One part of the package is a collection of codes that return the cross sections of DM particles’ production in such processes as electron and muon bremsstrahlung off nuclei, annihilation in flight and Primakoff processes. In most cases they are calculated at runtime. In the case of bremsstrahlung processes the total cross sections are either calculated numerically at exact tree level (ETL) or calculated at runtime using approximations validated by comparing with results of the ETL calculations. Other parts contain codes for the simulation of the above mentioned processes and DM particle classes.


Introduction
One of the popular directions of searches for physics beyond the Standard Model (SM) deals with light Dark Matter (LDM) models. They predict LDM particles χ with masses below a few GeV/c 2 interacting with SM particles via a light mediator. Initially it was assumed that the mediator is a 1 − vector boson called "dark photon" [1], but different quantum numbers for mediator are also possible. Such models are well motivated because they are compatible the hypothesis of DM thermal origin. The example of a relevant review is the 2019 CERN Physics Beyond Colliders report [2].
Due to comparably small masses of new LDM particles and very small production cross sections, experiments at accelerators with beam energies in the range ∼ 10−100 GeV suit very well to probe the LDM hypothesis. In many cases these are beam-dump experiments, with active or inactive thick target [3,4,5,6]. In contrast with colliders, the DM particles in such experiments can be produced by SM particles with a wide spectrum of energies, accompanied by other particles in a variety of topologies. Consequently, the optimal way to simulate the LDM production processes is to perform it inside the program for the full simulation of the experimental setup.
We created a Geant4 [7] compatible package DMG4 for the simulation of a number of LDM production processes. This package was initially presented and described in the paper [8]. The current version is 1.01. Here we remind the main features of the package and present important new developments.

DMG4 package structure
The DMG4 package is a set of DM process classes and DM particle definition classes. On top of this, there is a customizable DM physics class that assembles everything together. Historically, there is also a subpackage DarkMatter with a collection of routines for the cross section calculation. The DMG4 structure is shown in Figure 1. The new particles introduced in the current version of the package are listed in Table 1. The PDG codes are ascribed according to the rules in [9] (slightly extended).   Although such mediators as a dark photon decay into DM particles χ, we don't implement the corresponding decay tables. This is not important as long as the result is the same: the energy transferred to the produced mediator is carried away. For this reason in the following sections we call "dark matter" all produced dark sector particles. There are models with partly visible DM particles, where some decay products can be observed through cascade decays. The implementation of such models and corresponding particles is in our future plans.
The current version of DMG4 package includes the following DM production processes: • Bremsstrahlung-like process of the type pN → pN X, where p is a projectile (e − , e + , µ − , µ + ), and X is a DM mediator • Primakoff process of photon conversion γN → aN , where a is an axion-like particle (ALP) [10] • Resonant in-flight positron annihilation on atomic electrons e + e − → X → χχ, where χ is a dark matter mediator decay product [11].
In the case of annihilation the decay width of the mediator X into χ particles is important. This involves additional model parameters.
The simulation run physics is configured in the method DarkMatterPhysicsConfigure called from the constructor of the factory class DarkMatterPhysics. The factory then creates an instance of one of the process classes of the subpackage DarkMatter, instantiates and registers the needed particles and processes provided by the DMG4 package. The list of DarkMatter process classes and required parameters are shown in Figure 2. Here, "Threshold" is the cut-off minimal energy of particles that can initiate the processes of DM production (needed to avoid simulation of very soft DM particles that are anyway undetectable), and "BiasSigmaFactor" is a parameter that can bias the production cross section, i.e. increase it in such a way that the fraction of events with DM production is not too small. We note that the fraction of events with DM production should still be significantly smaller than 1 to avoid distortion of the energy and coordinate distributions.
The classes of the subpackage DarkMatter package contain methods that calculate the differential and total cross sections. There are also methods that sample events of DM particle production. We explain this in more detail in the next section.

Package DarkMatter and ETL cross sections
The cross sections of the DM production processes are calculated starting from the Lagrangians, an example of which is shown below. Four different quantum numbers of DM mediator particles are considered [12,13]. For simplicity, which in most cases does not limit the possibilities of the package, we assume that all other DM particles (χ) are fermions. More information can be found in [8]. For the e + e − annihilation and bremsstrahlung, in the case of a vector mediator, we have: where V is the mixing (or coupling) parameters, m V is the mass of mediator. For the electron and positron beams the package DarkMatter has the analytical formulas for the differential and total cross sections obtained using the improved Weizsacker-Williams (IWW) approximation [3]. It was shown that they can be significantly inexact in some regions of parameter space [14], so the package includes also the tabulated correction factors (K-factors). They are defined as the ratio of the total cross section values calculated at ETL [12,13,14] to the IWW analytical values. The ETL calculations were performed using the symbolic computation software Mathematica [15]. In some regions of the parameter space, where the difference is important, the tabulated differential cross sections calculated at ETL are used instead of analytical formulas.
For the muon bremsstrahlung process we recently implemented the Weizsacker-Williams (WW) approach [16]. This leads to more sophisticated formulas, but gives cross sections much closer to ETL calculations (within 2 -3%), so there is no need of K -factors. The difference of the experiments with muon beams from the ones with electron beams is that the recoil muon track should be reconstructed by tracker. This makes the accurate sampling of the recoil muon angle Ψ very important, in addition to the usual sampling of variable X = E A /E 0 , where E 0 is the initial particle energy. Such sampling is to be performed using the double differential cross section dσ/dXdΨ. The corresponding formula was obtained by rather complicated analytical integration of the initial cross section over the A angle Θ. The integration method was thoroughly checked by comparing total cross sections calculated by numerical integration of the differential cross sections dσ/dXdΨ and dσ/dXdΘ.
The implementation of the WW differential cross sections for muon beams is one of the most important developments of the DMG4 package as compared to the version described in [8].
The package DarkMatter contains also the methods that return widths of DM mediators calculated at tree-level. For example for the vector particle DMParticleXBoson decaying into e + e − we have:

Cross-checks and usage in active beam-dump experiments
After the implementation of the WW approximation for the muon beams, in order to check the sophisticated formulas, we performed a cross-check of the package. We compared the A yield in the limit M A =0 with the muon bremsstrahlung process implemented in Geant4. The absolute yield and the variable X distributions agreed within 2%.
Other cross-checks were performed by comparing some distributions with the ones obtained from the ETL calculations.
The DMG4 package was used to calculate the sensitivity to Dark Matter of an experiment that uses a missing energy signature in the electron beam. The sensitivity is defined as the expected 90% C.L. upper limit on the parameter in the case of no signal and very small background. The calculations are performed for the typical energy of the electron beam at the CERN SPS of 100 GeV and a lead/plastic electromagnetic calorimeter ECAL [17] as an active target.
A signal event is defined as an event with energy deposition in the ECAL smaller than 0.5E beam . We require no significant activity in the hadron calorimeter and several veto detectors. The sensitivity is calculated for the statistics corresponding to 5×10 12 EOT (electrons on target) assuming the background-free conditions and 100% efficiency of the experiment. The result for the vector mediator A is shown in Figure 3. The contribution from the annihilation processes is shown. It is significant for A masses above 100 MeV, but is more model-dependent.
Another example of usage is the simulation of the DM search experiment in the muon beam. The corresponding mediator can decay into SM particles or only invisibly into DM particles [18]. In the former case the new vector particle is usually called Z , and we can have the missing energy signature due to decays of Z into neutrinos. Both options are available in DMG4. The emission occurs in the ECAL similar to the one used in the electron beam-dump. The distribution of the variable X for different mediator masses is shown in Figure 4.

Conclusion
The package DMG4 for the simulation of light dark matter production in fixed target experiments has been created. It can be used in full simulation programs based on the Geant4 framework.
The The next development plans include semivisible decay modes (cascade decays), WW approximation for electrons, new processes such as µ → τ conversion and millicharged particle production. We plan to work on convenience and flexibility.