micrOMEGAs5.0 : freeze-in

Geneviève Bélanger; Fawzi Boudjema; Andreas Goudelis; Alexander Pukhov; Bryan Zaldivar

Source-linked AI summary

micrOMEGAs5.0 : freeze-in

Geneviève Bélanger, Fawzi Boudjema, Andreas Goudelis, Alexander Pukhov, Bryan Zaldivar

arXiv:1801.03509v3Jan 10, 2018hep-ph

TL;DR

The paper addresses complications in computing freeze-in dark matter abundances for general Standard Model extensions. It extends micrOMEGAs with a formalism and code for multiple freeze-in scenarios, finding that proper quantum statistics can change predicted relic densities from a few percent to about a factor of two or more.

Problem
General freeze-in calculations involve complications from high-temperature loss of equilibrium and non-Maxwell-Boltzmann phase-space distributions in dark matter production.
Method
The authors develop freeze-in Boltzmann formalism and extend micrOMEGAs to compute production from bath scatterings and mediator or dark-sector decays in general models.
Results
Proper Bose-Einstein and Fermi-Dirac statistics can change the predicted relic density by a few percent up to a factor of two or more, depending on the scenario.
Takeaways & Limitations
The upgraded code enables freeze-in abundance calculations across simple and complicated new-physics scenarios while quantifying when the Maxwell-Boltzmann approximation is inaccurate.
Takeaways & Limitations
The routines assume all odd FIMPs decay into the lightest one, so they overestimate the dark matter yield when dark-sector FIMPs predominantly decay into bath particles.

Abstract

from arXiv · show

We present a major upgrade of the micrOMEGAs dark matter code to compute the abundance of feebly interacting dark matter candidates through the freeze-in mechanism in generic extensions of the Standard Model of particle physics. We develop the necessary formalism in order to solve the freeze-in Boltzmann equations while making as few simplifying assumptions as possible concerning the phase-space distributions of the particles involved in the dark matter production process. We further show that this formalism allows us to treat different freeze-in scenarios and discuss the way it is implemented in the code. We find that, depending on the New Physics scenario under consideration, the effect of a proper treatment of statistics on the predicted dark matter abundance can range from a few percent up to a factor of two, or more. We moreover illustrate the underlying physics, as well as the various novel functionalities of micrOMEGAs, by presenting several example results obtained for different dark matter models.

1 Univ. Grenoble Alpes, USMB, CNRS, LAPTh, F-74940 Annecy, France

The supplied affiliations identify Sorbonne Université and Moscow State University affiliations.

Sorbonne Université affiliation includes CNRS and LPTHE in Paris, France.
Moscow State University affiliation is associated with the Skobeltsyn Institute of Nuclear Physics in Moscow, Russia.

1 Introduction

The paper addresses the lack of a public tool for computing FIMP freeze-in relic densities by extending micrOMEGAs with general freeze-in formalism and functionality. It emphasizes proper quantum statistics, which can substantially change predicted abundances.

A public computational tool for solving FIMP freeze-in relic-density calculations was previously unavailable.
Freeze-in produces feebly coupled dark matter through bath-particle scattering or mediator and dark-sector-particle decays.
micrOMEGAs now computes freeze-in abundances across extensions of the Standard Model, including scenarios with multiple mediators or dark-sector particles.
The code treats bath-particle phase-space distributions with Bose-Einstein or Fermi-Dirac statistics instead of assuming Maxwell-Boltzmann distributions.
Quantum-statistics effects can change predicted relic densities by a factor of two, especially for annihilation processes initiated by bosons.
Users must declare FIMPs and specify the reheating temperature; freeze-in calculations support up to two dark matter candidates.
The release adds process-level channel selection, freeze-out calculations for selected dark-sector particles, and three freeze-in sample models.
FIMP detection signatures in direct detection, indirect detection, and collider searches are left for future work.

2 Relic density of FIMPs

Freeze-in begins with a negligible dark-matter abundance and produces FIMPs through decays or annihilations while neglecting their inverse annihilations into bath particles. The simplified picture can involve intermediate long-lived states and has model-dependent thermal behavior.

Freeze-in assumes a very small initial dark-matter number density and production through decays or annihilations of other particles.
FIMP annihilations into bath particles can be neglected because both their couplings and number densities are small.
Production typically continues until the parent particles become Boltzmann suppressed, although intermediate potentially long-lived particles can complicate this picture.
Unlike freeze-out, a larger FIMP coupling to the visible sector implies a larger dark-matter density.

2.1 Notations and useful formulas

The formalism defines phase-space, thermodynamic, and expansion quantities needed to evolve freeze-in production with temperature. It uses quantum-statistical distributions for particles in kinetic equilibrium and relates time evolution to temperature through entropy conservation.

The section introduces notation and formulas used before presenting the freeze-in relic-density equations and their solution.
A species i is described by a phase-space distribution f_i, with number and energy densities determined from its internal degrees of freedom, momentum, and energy.
Particles in kinetic equilibrium follow Fermi-Dirac or Bose-Einstein distributions, with η_s=1 for bosons and η_s=−1 for fermions.
The radiation-era energy and entropy densities are expressed using the effective degrees of freedom g_eff and h_eff.
The Hubble rate uses the Planck mass M_Pl=1.22·10^19 GeV and is roughly H(T)≈10^−18(T/GeV)^2 GeV.
Entropy conservation relates cosmic time and temperature during the evolution.

2.2 General freeze-in Boltzmann equation

The general freeze-in treatment evolves the dark matter yield from bath-particle collision terms while assuming feeble interactions and negligible initial dark matter abundance. It distinguishes production through decays and bath annihilations and accounts for reheating-temperature dependence.

General formulation: The Boltzmann framework describes the time-temperature evolution of dark matter through integrated collision terms for reactions between initial and final particle sets.The collision term uses distribution functions, transition amplitudes, phase-space integration, and combinatorial factors.
Freeze-in assumptions: Freeze-in assumes feeble dark matter couplings and negligible initial abundance, allowing the analysis to retain only dark matter production processes.A nonzero initial abundance could instead add an independent contribution when annihilations are negligible.
Yield calculation: The final yield is obtained by integrating production from the reheating temperature TR to the present temperature T0.The yield is the comoving number density Yχ = nχ/s.
Scope and dependence: The abundance can depend on TR when reheating occurs near mediator or dark matter masses or when production is dominated by high temperatures.Dark-sector FIMPs are assumed eventually to decay into dark matter and a visible-sector particle.
Production channels: The framework covers equilibrium and nonequilibrium mediator decays, including late decays, alongside 2 →2 annihilations of bath particles.The listed annihilation final states are either two dark matter particles or one dark matter and one bath particle.

2.3 1 →2 (2 →1) processes

The decay formalism computes dark matter production from 1 →2 and inverse processes with medium-dependent statistics, treating mediators both in and out of thermal equilibrium. It provides numerical and approximate yield calculations and quantifies the impact of quantum statistics.

2.3 1 →2 (2 →1) processes: The formalism treats decays of bath particles, mediators, and dark-sector particles that produce one or two dark matter particles.The decaying particle may be an even mediator producing two odd FIMPs or an odd dark-sector state producing dark matter and a bath particle.
Decay rates in a medium: Medium effects enter the decay rate through the distribution functions of the final particles, while Lorentz-invariant phase space permits evaluation in the mediator rest frame.The resulting rate is interpreted as the decay rate in a thermal medium.
Collision terms: The integrated collision term is expressed through the generalized function ˜K1, which reduces to the usual K1(mY/T) under Maxwell-Boltzmann statistics.The statistical factor is isolated in S, and the mediator-to-dark-matter case is obtained as a special form.
Equilibrium mediator: For a mediator in thermal equilibrium with the Standard Model bath, the abundance follows from its equilibrium distribution and its decay width into dark matter.The treatment assumes mediator couplings to bath states are large while dark-matter partial widths remain much smaller.
Approximate yield: The production kernel xY^2 e^-xY peaks at xY = 3 with width σ ≈1.7, enabling an approximate yield for sufficiently large TR.For a bosonic mediator, the approximation has about 1% precision when mY > 3 GeV, and TR dependence is below 1% when TR > 2mY.
Example: Z′ mediator: Figure 1 compares differential dark matter yields for a Z′ mediator with large SM-dominated width, large DM-dominated width, and very small SM-dominated width.The example uses mZ′ = 104 GeV and only the process u, ¯u →χ¯χ.
Statistics effects: Quantum statistics can shift the abundance substantially: Fermi-Dirac effects decrease it by about 25% for fermion-coupled mediators, while bosonic couplings can increase it by up to 80%.For equilibrium bosonic mediators, Bose-Einstein statistics alone produce a 3.5% increase; fermion mediators produce a 2.5% decrease relative to Maxwell-Boltzmann statistics.

2.4 2 →2 processes

The 2 →2 treatment covers bath annihilations producing FIMPs, using an empirical correction for single-FIMP final states and temperature-dependent resonance widths to match mediator-decay calculations.

2 →2 collision terms: For two-FIMP final states, the integrated collision term can be treated similarly to the standard thermal freeze-out calculation after neglecting final-state FIMP blocking or stimulation factors.The approximation is (1∓fa)(1∓fb) ≃1.
Single-FIMP production: Single-FIMP production with an accompanying bath particle requires the full 12-dimensional collision-term integration in principle.micrOMEGAs instead introduces an empirical correction factor to reduce the computational cost.
Approximation accuracy: 2% precision is achieved by the correction factor when compared with the full numerical calculation, assuming the associated X particle thermalises rapidly.The comparison is performed for the model discussed in Section 4.3.
Resonance matching: micrOMEGAs sums all 2 →2 production channels and modifies resonant cross sections so their contributions match mediator-decay results.The effective mediator width is tabulated as a function of temperature and substituted into the relevant matrix elements.
Thermal corrections: The effective width receives temperature-dependent corrections for the mediator chemical potential and the thermal-bath branching ratio.A global rescaling is avoided because it would also alter off-resonance contributions.
Late decays: For delayed mediator decays, the branching ratio is modified using the survival probability of a mediator produced at Tc and decaying at Td.The decay width is small enough that Td is below the production temperature.
Scope boundary: The implementation does not yet treat χX final states when X belongs to the bath.The stated discussion focuses on dark matter pair production or final states containing only FIMPs.

3 Functions of micrOMEGAs

micrOMEGAs provides auxiliary functions and freeze-in routines for computing dark-matter abundances from decays and scattering, while exposing channel contributions and model-dependent assumptions. The implementation also documents limitations affecting freeze-out calculations and decay-based freeze-in yields.

Constants and auxiliary functions: The code provides functions for radiation degrees of freedom, the Hubble rate, entropy derivatives, and statistical-distribution corrections.gEff includes only Standard Model particles, while Hubble applies during radiation domination for T ≳100 eV.
Freeze-in routines: darkOmegaFiDecay computes freeze-in production from a named particle’s decays, with a switch for whether the decaying particle is in kinetic equilibrium.The routine uses different equations depending on whether the decaying particle is feeble and whether KE=1 or KE=0.
Freeze-in routines: darkOmegaFi22 isolates production through a user-defined 2 →2 process, while darkOmegaFi sums relevant bath-initiated processes involving odd feebly interacting particles.The vegas switch enables direct collision-term integration for precision checks, and production-temperature profiles can differ between decay and scattering treatments.
Freeze-in routines: In models with two dark sectors, darkOmegaFi computes the total abundance and sort2FiDm separates the two contributions.Odd FIMPs in each sector are assumed to decay into the lightest dark-matter particle of the same class.
Freeze-in routines: Channel-reporting routines expose relative contributions of 2 →2 and 1 →2 processes to Ωh2, with optional percentage output and a user-set cutoff.printChannelsFi writes selected channels to a file, using omegaFiCh weights to represent relative contributions.
Freeze-out interactions and limitations: Freeze-out routines retain previous behavior when no feebly interacting particles are declared, but FIMP declarations remove listed odd particles from freeze-out relic-density calculations.This can affect coannihilation when feeble particles are nearly degenerate with the dark matter, and the code does not verify the decay-rate condition relative to H(TFO).

4 Sample models and numerical results

The sample models illustrate freeze-in production through mediator decays and bath scatterings across distinct mass regimes, while showing that exact bath statistics can materially change relic-density predictions.

4.1 Vector portal: The Z′ portal considers dark fermion production through mediator decay or 2 →2 bath production, with regimes classified by mZ′, mχ, and TR.The off-shell, on-shell, and EFT regimes correspond respectively to off-shell annihilation, resonant mediator production, and an effective point-like interaction.
4.1 Vector portal: The vector-portal calculation reproduces the expected EFT 1/mZ′^4 behavior, an off-shell plateau, and increased on-shell abundance as decreasing mZ′ suppresses the decay width.On-shell production eventually becomes kinematically impossible.
4.1 Vector portal: 4% lower relic density appears in the EFT regime and 25% lower density in other regimes when exact Fermi-Dirac statistics replace Maxwell-Boltzmann statistics.For an on-shell mediator coupled mostly to SM fermions, the exact treatment instead increases the relic density by 4%.
4.2 Scalar portal: The singlet scalar model produces dark matter mainly through Higgs decay when mS < mh/2 and λhs << 1, with relic density proportional to λhs^2mS.When on-shell Higgs production is unavailable, all SM particles contribute, including W, Z, and h bosons.
4.2 Scalar portal: Nearly a factor 2 increase in dark-matter density arises from Bose-Einstein statistics in the singlet scalar model, while statistics effects are mild for on-shell Higgs decay.For the same coupling, the relic-density constraint can be met for sub-GeV and weak-scale dark matter.
4.3 Extended dark sector: In the extended dark sector, the scalar is produced through heavy-lepton decay and electron-mediated scatterings, with the heavy lepton maintained in equilibrium by gauge interactions.For large mass splitting decays dominate; as the splitting decreases, scattering processes take over.

5 Installation and sample output

The distribution includes sample models and commands for compiling and running micrOMEGAs5.0 freeze-in calculations, with output listing the total abundance and contributing channels.

Installation: The code distribution contains model directories that users can select before compiling the main code.The installation workflow begins by entering the micromegas_5.0 directory and unpacking the distribution.
Sample calculation: The sample program computes relic density through decay or scattering processes using a supplied parameter file.Running ./main data1.par produces the sample output.
Sample output: Freeze-in output reports a total Ωh^2 = 1.220E+00 and breaks it down by production channels.The listed channels include B,b; G,G; L,l; C,c; A,A; W−,W+; S,s; Z,Z; and h,h initial states.
Sample output: The sample output also reports mediator decay widths and branching fractions, including h → ~x1,~x1.This supplements the channel-by-channel relic-density contributions with decay information.

6 Discussion and conclusions

The paper extends micrOMEGAs to compute freeze-in dark-matter abundances across several scenarios while addressing high-temperature production and non-Maxwell-Boltzmann statistics. It reports that these effects can change predictions substantially, while the code remains limited in intermediate freeze-in/freeze-out regimes.

Discussion and conclusions: Freeze-in can be complicated because dark-matter production may be dominated by high temperatures, requiring density evolution to be tracked back to the reheating era.This contrasts with freeze-out, where equilibrium erases memory of high-temperature processes.
Discussion and conclusions: The new micrOMEGAs version treats different freeze-in types and computes production from bath scatterings or mediator and dark-sector decays.The decaying particle may or may not be in thermal equilibrium with the Standard Model.
Discussion and conclusions: Dark-matter production through mediator decays in infrared-dominated freeze-in peaks around mY /3.This reproduces a well-known result from the literature.
Discussion and conclusions: Using Maxwell-Boltzmann statistics for bath particles can mispredict the dark-matter abundance by up to a factor of 2.The approximation is therefore not always reliable.
Discussion and conclusions: The code does not check whether dark-matter couplings satisfy the freeze-in assumption, and the intermediate freeze-in/freeze-out regime is deferred to a future upgrade.The authors identify this intermediate regime as especially complicated because it requires more general Boltzmann equations.
Discussion and conclusions: The authors intend the tool to support studies of feebly coupled dark matter alongside collider and intensity-frontier searches.They specifically mention long-lived-particle searches at the LHC and searches for feebly coupled particles.

A Treatment of t-channel propagators

The t-channel treatment stabilizes high-energy cross-section calculations by isolating the pole region, fitting its angular behavior, and integrating the approximation symbolically. The method has identifiable failure cases involving multiple propagators and extremely narrow bosonic poles.

Treatment of t-channel propagators: High-energy freeze-in cross sections can become numerically unstable for 2 →2 processes containing a t-channel propagator.The relevant energies are much larger than the particle masses, making the pole contribution difficult to integrate directly.
Treatment of t-channel propagators: For a t-channel pole, micrOMEGAs first integrates the cross section away from the forward region, with cos θ13 < 1 −10^-10.The forward-angle region is then treated separately.
Treatment of t-channel propagators: The code samples the differential cross section at small scattering angles to determine A and C, then integrates the approximate formula symbolically.The angular dependence includes the pole contribution and uses θ13, the center-of-mass angle between incoming and outgoing particles.
Treatment of t-channel propagators: The approximation may fail when the lightest t-channel mass sets δ13 but a heavier propagator dominates, or when a pole is negligible near the cutoff yet important overall.The latter issue occurs for l = 2 and δ13 < 10^-30, and both issues are slated for a future release.