Weak corrections to Minimal Dark Matter annihilations

Abstract: We compute the one-loop weak corrections to the annihilation cross sections of fermionic Minimal Dark Matter multiplets. Infrared divergences cancel in the dominant $s$-wave combination relevant for the thermal relic abundance. Instead, infrared-enhanced corrections affect velocity-suppressed rates, through a Sudakov/Sommerfeld interplay. The corrections grow with the multiplet size and are at the $5\%$ level in the most motivated cases: the Higgsino-like doublet, the wino-like triplet, the stable quintuplet.

• The paper presents a detailed calculation of one-loop electroweak corrections to Minimal Dark Matter annihilation rates, achieving per-mille accuracy in relic density predictions.
• It decomposes the corrections into real and virtual components, addressing the interplay of Sommerfeld enhancement and non-cancelling Sudakov logarithms in velocity‐suppressed channels.
• Numerical analyses for Higgsino, wino, and quintuplet cases reveal corrections up to 10%, which significantly impact indirect detection signals and collider phenomenology.

One-Loop Weak Corrections to Minimal Dark Matter Annihilation

Overview and Motivation

The paper "Weak corrections to Minimal Dark Matter annihilations" (2606.18335) presents a systematic calculation of one-loop electroweak (EW) corrections to the DM annihilation cross sections within the framework of Minimal Dark Matter (MDM). The emphasis is on fermionic electroweak $n$-plets, specifically cases motivated by Higgsino ($n=2$), wino ($n=3$), and quintuplet ($n=5$) DM, with a focus on achieving an accurate relic density prediction at the per-mille precision level.

The relic abundance of MDM candidates is highly sensitive to the annihilation cross section, which motivates the inclusion of higher-order corrections. One-loop QCD corrections have been previously considered, but the EW contributions, including their interplay with non-perturbative Sommerfeld and infrared (IR) Sudakov effects, require systematic assessment, especially for large representations where these corrections grow with $n$.

Analytic Structure and Cancellation of Infrared Divergences

A detailed calculation of one-loop EW corrections to the dominant $s$-wave annihilation rates in the SU(2)$_L$-symmetric limit shows that all IR divergences cancel in the physically relevant inclusive rates for the relic abundance. This cancellation persists even when Sommerfeld enhancement factors, which depend on the total isospin $I$ of the initial DM pair, are included. The analysis demonstrates that the non-relativistic limit suppresses initial-state soft radiation at leading order: thus, the KLN cancellation applies channel-by-channel even in the presence of large non-perturbative corrections.

The authors explicitly separate the corrections into real and virtual components, categorize them via isospin, and demonstrate that only velocity-suppressed rates receive non-trivial IR-enhanced corrections from Sudakov and Sommerfeld interplay. The decomposition of cross section corrections into multiplet components is provided, including the impact of mass splittings and symmetry breaking below the EW transition.

Sudakov/Sommerfeld Interplay in $p$-Wave and Subleading Rates

In contrast to the $s$-wave, velocity-suppressed rates (genuine $n=2$0-wave, velocity-suppressed $n=2$1-wave) are affected by IR-enhanced logarithms due to a Sudakov/Sommerfeld interplay. The key finding is that, for these channels, the emission and absorption of soft EW gauge bosons can induce transitions between initial states of different total isospin, resulting in residual Sudakov logarithms which do not cancel, even after summing over real and virtual corrections. This effect is only relevant when the emission energy is large compared to the scale of Sommerfeld enhancement ($n=2$2).

These findings clarify that precision predictions for indirect detection signals, which can include velocity-suppressed channels, must account for these non-cancelling EW corrections, particularly in models with large representations.

Numerical Results: Higgsino, Wino, and Quintuplet Multiplets

The authors provide explicit numerical predictions for the most phenomenologically motivated cases:

• Higgsino-like Doublet ($n=2$3, $n=2$4): One-loop EW corrections increase the required DM mass for thermal relic abundance by approximately $n=2$5, with the total annihilation cross section receiving a $n=2$6 upward correction. Mass effects from SM particles, as well as additional $n=2$7 and top Yukawa couplings, are included at one-loop. Sommerfeld enhancement and bound-state effects are negligible for the thermal masses considered.
• Wino-like Triplet ($n=2$8, $n=2$9): EW loop corrections are numerically suppressed by accidental cancellations, with the required mass for the observed relic density in the range $n=3$0 TeV (including bound state formation). QCD corrections slightly increase, and EW corrections slightly decrease, the thermal mass.
• Quintuplet ($n=3$1, $n=3$2): EW corrections are larger, enhancing the annihilation cross section and reducing the thermal mass by up to $n=3$3. However, bound state formation (not included at one-loop EW accuracy) dominates for the TeV-scale quintuplet, and the full NLO prediction awaits a dedicated calculation of EW corrections to bound state formation.

The paper claims that for all cases, the corrections to the dominant $n=3$4-wave are IR finite, while sub-leading velocity-suppressed components require careful treatment of Sudakov-enhanced contributions. The corrections grow with the multiplet size $n=3$5, reaching $n=3$6 for the quintuplet.

Practical and Theoretical Implications

The results enable a more precise mapping of MDM parameter space to relic abundance and indirect detection rates. For large electroweak representations, the $n=3$7 loop corrections become non-negligible compared to experimental and astrophysical uncertainties. They also impact predictions for DM production and detection at colliders, where resonances and threshold effects linked to precise DM masses are relevant.

The non-trivial interplay of Sudakov and Sommerfeld effects in subleading channels suggests that incorporating both effects is mandatory for percent-level predictions. The cancellation of IR divergences in $n=3$8-wave channels, even after including Sommerfeld resummations, sets a robust standard for perturbative calculations in non-relativistic DM annihilation.

Outlook and Future Directions

Several avenues follow from these findings:

• The computation of electroweak corrections to bound state formation and decay is outstanding, especially for $n=3$9 MDM, and will be necessary for complete NLO relic predictions.
• The treatment of subleading velocity-suppressed rates should be extended to indirect detection signals, where they can be relevant for line searches.
• Extensions to include the effect of additional interactions or non-minimal scenarios (e.g., presence of a bino or scalar quartic couplings) would generalize the precision achieved here.

Conclusion

This work rigorously establishes the structure and size of one-loop weak corrections to the annihilation rates for fermionic Minimal Dark Matter candidates, demonstrating that the leading corrections are IR safe and that subleading channels are only affected by non-cancelling electroweak Sudakov logarithms when velocity-suppressed. The results provide per-mille-to-percent level accuracy in relic density calculations for the most important minimal DM scenarios, notably the Higgsino, wino, and quintuplet. These predictions inform both indirect detection and collider prospects, and emphasize the necessity of full NLO computations for robust DM phenomenology.