Application of the 3-Loop FlexibleEFTHiggs Method to the MSSM and the NMSSM

This paper presents an extensive analysis of the light CP-even Higgs boson pole mass in the MSSM and NMSSM using the newly implemented 3-loop FlexibleEFTHiggs hybrid calculation in FlexibleSUSY, focusing on its robustness in scenarios with highly non-degenerate SUSY spectra and providing an improved, uncertainty-assessed Higgs mass prediction for the NMSSM.

The Gist

Imagine the universe as a giant, complex machine. For decades, physicists have been trying to understand how this machine works, specifically how it gives mass to the tiny particles that make up everything around us. In 2012, they found a crucial part of this machine: the Higgs boson. It's like finding the engine of a car; you know it's there, but you don't know exactly how powerful it is or how it was built.

This paper is about a team of physicists (Thomas Kwasnita, Dominik Stöckinger, Alexander Voigt, and Johannes Wünsche) who built a new, ultra-precise calculator to predict the weight (mass) of this Higgs engine. They tested this calculator on two different "blueprints" for the universe: the MSSM (a popular, slightly upgraded version of our current physics) and the NMSSM (an even more complex, upgraded version).

Here is a simple breakdown of what they did and what they found, using everyday analogies:

1. The Problem: Two Different Ways to Measure

Imagine you are trying to measure the height of a mountain.

• Method A (Fixed Order): You stand at the base and measure step-by-step. This works great if the mountain is small (low energy), but if the mountain is huge, your steps become too small to count accurately, and you miss the big picture.
• Method B (Effective Field Theory): You stand on a helicopter far away and look at the whole mountain. This works great for huge mountains, but if the mountain is small, you miss the tiny details at the base.

For a long time, physicists had to choose one method or the other. If the "new particles" in these blueprints were heavy (like a giant mountain), they used Method B. If they were light, they used Method A. But since we don't know how heavy these new particles are, picking the wrong method gives a wrong answer.

2. The Solution: The "Hybrid" Calculator

The authors used a hybrid method called FlexibleEFTHiggs. Think of this as a smart drone that can do both jobs at once.

• It can zoom in to see the tiny details at the base (like Method A).
• It can zoom out to see the massive scale of the whole mountain (like Method B).
• It stitches these two views together perfectly, so it works whether the new particles are light, heavy, or a mix of both.

They upgraded this drone to 3-loop precision. In physics, "loops" are like layers of detail. A 1-loop calculation is a rough sketch; a 2-loop is a detailed drawing; a 3-loop is a photorealistic, high-definition 3D model. This is the most detailed version of this calculator ever made for these specific blueprints.

3. Testing the Calculator: The "Stress Test"

The team didn't just build the calculator; they put it through a stress test to see if it breaks under weird conditions.

• The "Standard" Test: They first tested it on "standard" scenarios where all the new particles have similar weights (like a family of identical twins). The calculator worked perfectly.
• The "Chaos" Test: Then, they tested it on "non-degenerate" scenarios. Imagine a family where one twin is a giant, another is a dwarf, and the third is a normal-sized adult. This is a messy, uneven situation.
  • Result: The calculator remained robust. It didn't crash. It handled the messy, uneven weights of the particles and still gave a reliable prediction.
  • One Catch: They found that if the "gluino" (a specific heavy particle) gets extremely heavy compared to the others, the calculator gets a bit jittery and the uncertainty grows. It's like trying to balance a scale when one side has a feather and the other has a boulder; it's hard to get a perfect reading.

4. The NMSSM Upgrade: Adding a Secret Ingredient

The NMSSM is like the MSSM blueprint but with a secret ingredient (a new particle called a "singlet") added to the mix.

• Before this paper, no one had built a 3-loop calculator specifically for this secret ingredient.
• The authors added this new ingredient to their drone. They checked if the secret ingredient changed the weight of the Higgs engine.
• Result: Yes, it does! Depending on how strong the "secret ingredient" interacts with the rest of the machine, the predicted weight of the Higgs can go up or down. The calculator successfully tracked these changes.

5. The Bottom Line: How Sure Are We?

Every measurement has a margin of error (uncertainty). The authors calculated how much their prediction could be off.

• For most normal scenarios, the uncertainty is very small (about 0.8 to 1 GeV, which is roughly the weight of a proton). This is excellent precision.
• For the "chaos" scenarios with very uneven particle weights, the uncertainty can get larger (up to 4 GeV in extreme cases).
• They compared their new calculator against other existing calculators (like FeynHiggs and NMSSMCalc). Their new 3-loop version agreed well with the others but offered better stability and precision in tricky situations.

Summary

This paper is about building and testing the most advanced ruler physicists have for measuring the Higgs boson's mass in complex, supersymmetric universes.

• The Tool: A hybrid calculator that works for both light and heavy new particles.
• The Upgrade: It now includes 3-loop precision (the highest detail level) for both the MSSM and the NMSSM.
• The Verdict: The tool is reliable and robust, even when the universe's new particles have very different weights. It confirms that we can trust these predictions to help us understand if these new particles exist and what they might look like.

They didn't discover new particles in this paper; they just built a better microscope to look for them.

Technical Summary

Technical Summary: Application of the 3-Loop FlexibleEFTHiggs Method to the MSSM and the NMSSM

Problem Statement
The discovery of the Higgs boson confirmed the Standard Model (SM) mechanism for mass generation but left the Higgs mass ($M_h$) itself as an unexplained free parameter. Supersymmetric (SUSY) extensions, such as the Minimal Supersymmetric Standard Model (MSSM) and the Next-to-Minimal Supersymmetric Standard Model (NMSSM), offer a solution by predicting $M_h$ within the correct order of magnitude. However, calculating $M_h$ with high precision is challenging because the mass scale of new SUSY particles ($M_S$) is unknown.

• If $M_S \sim M_Z$, Fixed-Order (FO) calculations are precise but fail if large logarithmic corrections are neglected.
• If $M_S \gg M_Z$, Effective Field Theory (EFT) calculations resum large logarithms ($\ln(M_S/M_Z)$) but neglect power-suppressed terms ($v^2/M_S^2$).
  A "hybrid" approach is required to combine the virtues of both methods. While the FlexibleEFTHiggs method was previously established at the 1-loop and 2-loop levels, and applied to degenerate SUSY spectra, there was a need to extend this to 3-loop precision and validate its robustness in non-degenerate (split) SUSY mass spectra for both the MSSM and the NMSSM.

Methodology
The authors utilize the FlexibleEFTHiggs method, implemented in the public code FlexibleSUSY (version 2.9.0). This hybrid approach operates through the following steps:

1. Matching: At a high matching scale $Q_m \sim M_S$, the full BSM model (MSSM/NMSSM) is matched to the SM. The quartic Higgs coupling $\hat{\lambda}_{SM}$ is determined by imposing a Higgs pole mass matching condition: $(M_h^{BSM})^2 = (M_h^{SM})^2$.
2. Parametrization: Crucially, the matching is performed using a full-model parametrization (expressed in terms of BSM parameters) rather than an EFT parametrization. This allows for the natural inclusion of terms suppressed by powers of $v^2/M_S^2$ and leads to an effective resummation of QCD-enhanced terms involving the stop-mixing parameter ($X_t$).
3. Running: The derived SM parameters are evolved down to the electroweak scale $Q_{low} \approx M_Z$ using SM Renormalization Group Equations (RGEs) up to 4-loop order.
4. Calculation: The Higgs pole mass is calculated in the SM at the scale $Q_{pole} = M_t$, including 1-loop, 2-loop, and 3-loop QCD contributions.

Key Contributions
The paper presents two primary advancements:

1. MSSM Extension to 3-Loops and Non-Degenerate Spectra:

   • The method is extended to include 3-loop QCD contributions ($\mathcal{O}(g_3^4 y_t^4)$) in the matching, achieving a precision of N3LO+N3LL.
   • The analysis moves beyond "standard" degenerate spectra to generic scenarios with non-degenerate soft-breaking mass parameters. This includes extensive scans of the gluino mass ($M_3$), the CP-odd Higgs mass ($m_A$), the $\mu$-parameter, and sfermion masses.
   • The study incorporates the Himalaya module for 3-loop threshold corrections, which applies different approximations based on the hierarchy between gluino and squark masses.

2. NMSSM Implementation and Validation:

   • A new application of the 3-loop FlexibleEFTHiggs method to the NMSSM is presented. This includes 2-loop matching corrections of $\mathcal{O}(g_3^2(y_t^4+y_b^4) + \dots)$ and dominant 3-loop QCD corrections.
   • The authors provide a rigorous uncertainty estimation for the NMSSM, specifically analyzing the reliability of matching scale variation in estimating missing NMSSM-specific higher-order terms (dependent on couplings $\lambda$ and $\kappa$). They demonstrate that matching scale variation effectively simulates missing terms up to 3-loop order.
   • The calculation is validated against Fixed-Order (FO) results, EFT results, and the NMSSMCalc code, showing excellent agreement in the MSSM limit and stable convergence properties.

Results

• MSSM Robustness: In scenarios with degenerate spectra, the 3-loop calculation yields theoretical uncertainties of approximately 1 GeV, consistent with previous studies. However, in non-degenerate scenarios (e.g., $M_3 \gg M_S$ or highly split squark masses), the uncertainty can increase significantly (up to >4 GeV in extreme cases like benchmark point M5). This instability is attributed to hierarchy transitions in the Himalaya module and large renormalization group running effects driven by large stop mixing ($x_t$) or bottom mixing ($x_b$).
• Parameter Dependencies: The Higgs mass is found to be highly sensitive to the stop mixing parameter $x_t$ and the sfermion masses. The dependence on the gluino mass $M_3$ is mild for $M_3 \lesssim M_S$ but grows for $M_3 \gg M_S$ due to 2-loop corrections.
• NMSSM Specifics: The NMSSM-specific couplings $\lambda$ and $\kappa$ can significantly alter $M_h$. For small $\tan\beta$, increasing $\lambda$ increases $M_h$, while for large $\tan\beta$, the effect can be negative. The 3-loop calculation shows excellent convergence, with the difference between 2-loop and 3-loop results being remarkably small.
• Uncertainty Estimation: The study confirms that matching scale variation is a reliable proxy for estimating missing NMSSM-specific higher-order terms, even when $\lambda$ and $\kappa$ are of $\mathcal{O}(1)$.

Significance
The paper claims that the implementation of the 3-loop FlexibleEFTHiggs method in FlexibleSUSY (v2.9.0) represents a state-of-the-art tool for Higgs mass calculations in both the MSSM and NMSSM.

• It provides a full-model parametrization that avoids the expansion in $v^2/M_S^2$, ensuring accuracy even when SUSY particles are not extremely heavy.
• It offers a robust framework for non-degenerate spectra, though the authors modestly note that in cases of extreme mass splitting or specific hierarchy transitions, the 2-loop result may currently be more trustworthy than the 3-loop result due to discontinuities in the 3-loop threshold corrections.
• The work establishes a reliable method for quantifying theory uncertainty in the NMSSM, addressing a gap in previous hybrid calculations which often lacked a comprehensive uncertainty analysis for NMSSM-specific parameters.

The authors conclude that while the method is highly precise for standard and moderately split scenarios, further improvements in the treatment of hierarchy transitions and renormalization schemes (such as the MDR scheme) are necessary to stabilize calculations in extreme non-degenerate regimes.