⚛️ phenomenology

Full Next-To Leading-Order Electroweak and QCD Corrections to the Relic Density in the CxSM

This paper presents the first calculation of next-to-leading-order QCD and electroweak corrections to the Dark Matter relic density within the complex singlet extended Standard Model (CxSM), demonstrating that these corrections can significantly alter exclusion limits and are implemented in the publicly available code RelExt@NLO.

Original authors: Pavao Brica, Karim Elyaouti, Pedro Gabriel, Margarete Mühlleitner, Rui Santos

Published 2026-07-22

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CC BY 4.0

Original authors: Pavao Brica, Karim Elyaouti, Pedro Gabriel, Margarete Mühlleitner, Rui Santos

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, bustling cosmic party that started with a massive explosion. Most of the guests at this party are the stuff we know and love: stars, planets, and us. But there's a mysterious, invisible crowd that makes up about 85% of the guest list. We can't see them, we can't touch them, and they don't talk to us, but we know they are there because their gravity holds the galaxy together. Scientists call this invisible crowd "Dark Matter." It's one of the biggest mysteries in science today. We know it exists, but we have no idea what it's made of. Is it a heavy ghost? A tiny, invisible particle? To solve this, physicists build "models"—like detailed blueprints for a new kind of particle—that try to explain how Dark Matter behaves.

One of the most popular ideas is that Dark Matter particles were created in the hot, dense early universe and then slowly "froze out" as the cosmos cooled down, leaving behind just the right amount to fill the universe today. To figure out if a specific blueprint is correct, scientists calculate how many of these particles should be left over. If the blueprint predicts too many or too few, it's probably wrong. However, doing these calculations is like trying to bake a cake with a recipe that only lists "a pinch of salt" and "some flour." It's a rough estimate. To get a truly delicious, perfect cake, you need to measure every single grain of sugar and every drop of milk with extreme precision. This is where the work of Pavao Brica and his team comes in. They are the master bakers who decided to stop guessing and start measuring the invisible ingredients of the Dark Matter recipe with the highest precision possible.

In this paper, the authors tackle a specific blueprint called the "Complex Singlet Extended Standard Model" (or CxSM for short). Think of the Standard Model as the current, best-selling cookbook of particle physics, but it's missing a chapter on Dark Matter. The CxSM adds a new, invisible ingredient—a complex scalar field—to the mix to create a Dark Matter candidate. The team's goal was to take the calculation of how much Dark Matter remains in the universe and upgrade it from a rough sketch to a high-definition, 3D rendering. They did this by calculating "Next-to-Leading Order" (NLO) corrections. If the first calculation (Leading Order) is like estimating the size of a shadow, the NLO corrections are like measuring the shadow while accounting for the wind, the angle of the sun, and the texture of the ground. They looked at two main forces that could tweak the numbers: the strong nuclear force (QCD) and the electroweak force (which includes electricity and magnetism).

The authors found that these "fine-tuning" corrections are not just tiny, boring details; they are actually game-changers. For many of the scenarios they tested, the corrections were small, changing the predicted amount of Dark Matter by just a few percent. However, in some specific situations—particularly when the Dark Matter particles were heavy and annihilated into pairs of Higgs bosons—the corrections were massive, shifting the prediction by up to 166%! This is like realizing that a recipe you thought would make a small cupcake actually makes a giant cake, or vice versa.

Crucially, the paper shows that these corrections can flip the verdict on whether a theory is right or wrong. There were specific points in their blueprint that looked perfect when using the rough, old calculation (Leading Order), but once the high-precision NLO corrections were applied, those points were ruled out because they predicted too much or too little Dark Matter. Conversely, some points that looked impossible at first glance turned out to be valid once the precise corrections were included. The team also discovered that the way they chose to handle certain mathematical "adjustments" (called renormalization schemes) mattered a lot. If they chose a scheme that involved large gaps between particle masses, the corrections became huge and the math got messy. But by choosing a smarter scheme, they could keep the corrections manageable and the predictions reliable, with remaining uncertainties as low as 1%.

To make these complex calculations accessible to other scientists, the authors didn't just write down the math; they built a new, public computer program called "RelExt@NLO." This tool allows anyone to run these high-precision tests on their own theories. The paper concludes that while the Standard Model is a great start, understanding the invisible universe requires us to stop guessing and start calculating with extreme precision. By including these next-level corrections, we can finally tell the difference between a theory that is just "close enough" and one that is truly the key to unlocking the mystery of Dark Matter.

Technical Summary: Full Next-To Leading-Order Electroweak and QCD Corrections to the Relic Density in the CxSM

Problem Statement
While the Standard Model (SM) successfully predicts many experimental observables, it fails to account for Dark Matter (DM). Theoretical extensions, such as the Complex Singlet Extended Standard Model (CxSM), introduce DM candidates stabilized by discrete $Z_2$ symmetries. A critical challenge in these models is the tension between satisfying the observed DM relic density ( $\Omega_{DM}h^2 \approx 0.12$ ) and direct detection limits. Current computational tools often calculate the thermally averaged cross section (TAC) and the resulting relic density only at Leading Order (LO). The lack of higher-order corrections introduces significant theoretical uncertainty, potentially leading to the incorrect inclusion or exclusion of viable parameter points. This work addresses the need for precise Next-to-Leading Order (NLO) predictions for DM annihilation in the CxSM.

Methodology
The authors perform a full NLO calculation of QCD and Electroweak (EW) corrections to the DM annihilation cross section within the CxSM framework. The model features a complex scalar singlet $S$ with a softly broken global $U(1)$ symmetry, where the CP-odd component $A$ serves as the stable DM candidate, and the CP-even components mix with the SM Higgs doublet.

Key methodological components include:

Renormalization Scheme: The authors implement a comprehensive renormalization procedure. For the singlet Vacuum Expectation Value (VEV) $v_S$ $v_{S}$ and the Higgs mixing angle $\alpha$ $α$ , they introduce and compare multiple schemes:
- Singlet VEV ( $v_S$ ): Process-dependent schemes using the decay $h_i \to AA$ (where $h_i$ are the mass eigenstates). This includes the ZEM scheme (Zero External Momentum, setting $p^2=0$ ) and the On-Shell (OS) scheme (using physical decay kinematics). Two variants are tested: $ZEM_{h1}$ and $ZEM_{h2}$ , depending on which Higgs state is used for renormalization.
- Mixing Angle ( $\alpha$ ): The OS tadpole-pinched scheme and the $p^*$ tadpole-pinched scheme, combined with the Alternative Tadpole Scheme (ATS) to ensure gauge independence.
Infrared (IR) Divergences: To handle IR divergences arising from massless gauge bosons in loops and real emissions, the authors employ the Catani-Seymour dipole subtraction formalism. This allows for the cancellation of divergences between virtual corrections and real $2 \to 3$ emission processes in four dimensions.
Implementation: The calculations are implemented in the C++ code RelExt@NLO, an extension of the existing LO code RelExt. The code utilizes the COLLIER library for Passarino-Veltman functions and the Cuba library for multi-dimensional phase space integration.
Parameter Scan: A scan is performed over the CxSM parameter space ( $m_S, m_A, v_S, \alpha$ ) constrained by theoretical bounds (boundedness, unitarity) and experimental data (LHC Higgs signal strengths, EW precision observables).

Key Contributions

First NLO Calculation in CxSM: This work presents the first calculation of full NLO QCD and EW corrections to the DM relic density in the CxSM with a single DM candidate.
Renormalization Framework: The paper establishes a robust framework for renormalizing the singlet VEV and mixing angle, explicitly demonstrating how different schemes (ZEM vs. OS) affect the counterterms and the final physical predictions.
Public Tool: The authors release RelExt@NLO, making these precision calculations accessible for future studies of Higgs portal models.
Uncertainty Estimation: The authors propose a method to estimate theoretical uncertainties due to missing higher-order corrections by varying the renormalization schemes consistently (converting input parameters between schemes).

Results

Magnitude of Corrections:
- QCD Corrections: For DM masses below the $WW$ threshold ( $m_A < 80$ GeV), where annihilation into $b\bar{b}$ dominates, QCD corrections are significant, increasing the relic density by up to $\sim 60\%$ . This is attributed to the large QCD corrections to the $h \to b\bar{b}$ vertex.
- Electroweak Corrections: Above the $WW$ threshold, EW corrections dominate. While typically moderate (ranging from $-25\%$ to $+25\%$ for the bulk of points), they can become very large (up to $+166\%$ or $-75\%$ ) in specific scenarios.
Scheme Dependence: The results show a strong dependence on the renormalization scheme for $v_S$ $v_{S}$ .
- The $ZEM_{h1}$ scheme (using the lighter Higgs $h_1$ for renormalization with $p^2=0$ ) yields large corrections and poor perturbative convergence when there is a large mass gap between the decaying Higgs and the DM pair.
- The $ZEM_{h2}$ and OS schemes (using the heavier Higgs $h_2$ or physical kinematics) demonstrate better convergence and smaller corrections, as the kinematic restrictions of the scan prevent large mass gaps in these configurations.
Phenomenological Impact: The inclusion of NLO corrections significantly alters the allowed parameter space.
- Some points allowed at LO are excluded at NLO.
- Conversely, some points excluded at LO become allowed at NLO.
- This demonstrates that LO calculations can lead to erroneous conclusions regarding the viability of specific model points.
Theoretical Uncertainty: By comparing results across different renormalization schemes (after consistent parameter conversion), the authors estimate the remaining theoretical uncertainty. For the preferred schemes ( $ZEM_{h2}$ and OS), the uncertainty is found to be small (percent level, e.g., $\sim 1.6 \times 10^{-4}$ for the mixing angle variation), whereas the $ZEM_{h1}$ scheme shows larger uncertainties (up to $\sim 68\%$ in high mass regions) due to large mass gaps.

Significance
The paper claims that this work marks a "first step" towards the automated calculation of NLO relic densities for arbitrary models with DM candidates stabilized by discrete $Z_2$ symmetries. The primary significance lies in the demonstration that NLO corrections are of phenomenological impact, capable of shifting the boundaries of the allowed parameter space. The authors emphasize that relying solely on LO calculations risks discarding viable models or including excluded ones. Furthermore, the analysis of renormalization schemes provides guidelines for choosing schemes with good perturbative convergence (favoring OS or $ZEM_{h2}$ over $ZEM_{h1}$ in this context) to minimize theoretical uncertainties. The release of RelExt@NLO enables high-precision exploration of Higgs portal models, bridging the gap between theoretical predictions and increasingly precise experimental data.

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