From Lagrangian to Higgs physics constraints for SUSY and non-SUSY models: interfacing FlexibleSUSY with HiggsTools and Lilith

This paper presents an automated interface between the FlexibleSUSY framework and HiggsTools/Lilith, enabling the direct validation of global Higgs sector agreement with experimental data for both supersymmetric and non-supersymmetric BSM models, including those with CP-violation and invisible decays.

The Gist

Imagine you are a detective trying to solve a mystery about the universe. You have a theory about how things work (a "Beyond the Standard Model" or BSM theory), but you need to check if your theory matches the clues left behind by nature. The biggest clue we have right now is the Higgs boson, a particle discovered in 2012 that gives other particles their mass.

This paper is about building a super-powered, automated detective tool that helps physicists check if their new theories are compatible with the real-world data we have about the Higgs boson.

Here is the breakdown of how this tool works, using simple analogies:

1. The Problem: Too Many Theories, Too Much Math

Physicists have created hundreds of new theories (like the "Supersymmetric" models or "Two Higgs Doublet" models) to explain things the current Standard Model can't.

• The Old Way: To test a theory, a physicist had to write complex equations by hand, calculate how the Higgs boson should behave in that theory, and then manually compare those numbers to experimental data from the Large Hadron Collider (LHC). It was like trying to solve a Sudoku puzzle by hand while someone else is shouting the rules at you. It was slow, prone to errors, and hard to do for many different theories at once.
• The New Tool (FlexibleSUSY): Think of FlexibleSUSY as a "Universal Translator" or a "Recipe Generator." You give it the ingredients (the fields and laws of your new theory), and it automatically cooks up the predictions (masses, decay rates, etc.).

2. The New Feature: The "Higgs Inspector"

The authors of this paper added a new module to FlexibleSUSY. They connected it to two existing expert systems called HiggsTools and Lilith.

• The Analogy: Imagine FlexibleSUSY is a chef who bakes a cake based on a secret recipe. HiggsTools and Lilith are the official taste-testers who have a strict checklist of what a "perfect" cake (the Standard Model Higgs) should taste like.
• The Connection: Before this paper, the chef had to take the cake out, measure it, write down the numbers, and then walk over to the taste-testers to ask, "Does this match?"
• The Innovation: Now, the chef and the taste-testers are in the same room. As soon as the cake is baked, the system automatically measures it, compares it to the official checklist, and instantly tells the chef: "You're good!" or "You're missing a pinch of salt (or in this case, your theory is ruled out)."

3. What Does It Actually Check?

The tool checks three main things, which are like checking the quality of a cake:

• The Flavor (Couplings): Does the Higgs interact with other particles (like top quarks or electrons) with the right strength? If your theory says the Higgs should be twice as strong with electrons as the Standard Model predicts, the tool flags it.
• The Hidden Ingredients (Invisible Decays): Sometimes, a particle might decay into things we can't see, like Dark Matter. It's like a cake that mysteriously loses a slice without anyone seeing it. The tool calculates how much "missing slice" your theory predicts and checks if that amount is allowed by current experiments.
• The CP-Violation (The Twist): Some theories suggest the Higgs has a "handedness" or a twist (CP-violation). The tool checks if the Higgs behaves like a mirror image or a twisted version of itself, which is a very subtle detail that older tools might miss.

4. The Three Test Cases

To prove their new tool works, the authors tested it on three different "recipes":

1. The Two-Higgs Model (2HDM): A theory where there are two Higgs bosons instead of one. The tool scanned thousands of variations to see which ones were still allowed.
2. The CP-Violating NMSSM: A complex Supersymmetric model where the Higgs has that "twist" mentioned above. The tool successfully identified which versions of this theory were still possible.
3. The MRSSM (Invisible Decay): A model where the Higgs decays into invisible Dark Matter particles. The tool showed that if the "missing slice" is too big, the theory is ruled out.

5. Why This Matters

• Speed & Automation: It turns a process that used to take days of manual calculation into a process that takes minutes.
• Universality: It works for any theory you can write down, whether it involves Supersymmetry (SUSY) or not. You don't need to be a math wizard to use it; you just need to define your model.
• Future-Proofing: As new data comes in from the LHC, this tool can instantly re-evaluate all existing theories to see if they still hold up.

The Bottom Line

This paper describes a bridge between theoretical imagination and experimental reality. It automates the process of asking, "Does my new idea about the universe fit with what we actually see in the lab?" By doing so, it helps physicists quickly discard bad ideas and focus their energy on the theories that might actually be true. It's like upgrading from a manual map to a GPS that instantly reroutes you if you're driving toward a dead end.

Technical Summary

Here is a detailed technical summary of the paper "From Lagrangian to Higgs physics constraints for SUSY and non-SUSY models: interfacing FlexibleSUSY with HiggsTools and Lilith."

1. Problem Statement

The discovery of the 125 GeV Higgs boson has shifted the focus of Beyond the Standard Model (BSM) physics toward precision measurements of Higgs properties and searches for additional scalars. Future colliders (HL-LHC, FCC, CLIC, etc.) aim to measure these properties with high precision. However, current experimental data already places stringent constraints on BSM theories.

The primary challenge is the lack of a fully automated workflow that connects a user-defined BSM model (defined by its Lagrangian and field content) directly to state-of-the-art experimental constraints on the Higgs sector. While tools like FlexibleSUSY can calculate mass spectra and decay widths, and tools like HiggsTools and Lilith can compare these against experimental data, they previously operated in isolation. Users had to manually extract partial widths, calculate effective couplings, and format inputs for HiggsTools/Lilith, making the process cumbersome and prone to error, especially for complex models involving CP-violation or invisible decays.

2. Methodology

The authors present a new interface within FlexibleSUSY (version 2.9.0) that seamlessly connects the spectrum generator to HiggsTools and Lilith. The methodology involves three main technical components:

A. Construction of Normalized Effective Couplings

To interface with HiggsTools and Lilith, the BSM Higgs properties must be translated into a "reduced coupling" ($\kappa$) framework relative to the Standard Model (SM).

• Calculation: FlexibleSUSY calculates loop-corrected partial widths for BSM Higgs bosons ($\phi$).
• Normalization: It constructs a corresponding SM model with the same Higgs mass as the BSM state. The effective couplings ($c_i$) are extracted from the partial widths and normalized to the SM values ($c_i^{SM}$) to obtain $\kappa_i = c_i / c_i^{SM}$.
• CP-Structure: The framework distinguishes between CP-even ($c$) and CP-odd ($\tilde{c}$) couplings. For CP-violating models, both real and imaginary parts of the couplings are calculated.
• Special Cases: For kinematically forbidden decays (e.g., $h \to t\bar{t}$ for a 125 GeV Higgs), couplings are extracted from tree-level vertices.

B. Treatment of Invisible and Undetected Widths

A critical feature of the interface is the automated handling of non-SM decay modes:

• Invisible Decays: Identified by checking decays into stable, neutral, R-parity odd (or equivalent) particles (e.g., Dark Matter candidates).
• Undetected Decays: Calculated as the residual width: $\Gamma_{undet} = \Gamma_{tot} - \Gamma_{inv} - \sum \Gamma_{SM}$.
• Implementation: These widths are passed to HiggsTools (as partial widths) and Lilith (as branching ratios) to ensure the total width is correctly rescaled, which affects the predicted signal strengths.

C. The Interface Workflow

1. Input: The user defines a model in FlexibleSUSY (via SARAH) and sets flags in the SLHA input file (e.g., FlexibleDecay[7]=1 for HiggsTools, FlexibleDecay[8]=1 for Lilith).
2. Execution: FlexibleSUSY computes the mass spectrum, decay widths, and the normalized effective couplings.
3. Data Transfer: The couplings and widths are written to specific SLHA blocks (NORMALIZEDEFFHIGGSCOUPLINGS, IMNORMALIZEDEFFHIGGSCOUPLINGS, HIGGSSIGNALS, LILITH).
4. Analysis: HiggsTools and Lilith ingest this data to compute:
   • $\chi^2_{BSM}$: The goodness of fit for the BSM point.
   • $\chi^2_{SM}$: The reference SM fit.
   • $p$-value: Calculated via $\Delta\chi^2 = \max(\chi^2_{BSM} - \chi^2_{SM}, 0)$, assuming a $\chi^2$ distribution with 2 degrees of freedom.
   • HiggsBounds: Checks for exclusion limits on non-SM Higgs states (heavy scalars, pseudoscalars, charged Higgses).

3. Key Contributions

• Full Automation: The paper introduces a "Lagrangian-to-Constraints" pipeline. Users can define a BSM model and immediately obtain statistical validation (p-values, exclusion limits) without manual intervention.
• CP-Violation Support: The interface correctly handles complex parameters and CP-mixing angles, allowing for the study of CP-violating Higgs sectors (e.g., in the NMSSM) by distinguishing between CP-even and CP-odd coupling components.
• Invisible Decay Handling: It provides a robust mechanism to automatically identify and include invisible and undetected decay widths, which are crucial for Dark Matter models but often overlooked in simplified analyses.
• Dual Interface: The tool supports both Command-Line (C++) and Wolfram Language (Mathematica) interfaces, offering flexibility for different user workflows.
• Universality: The framework is model-agnostic, applicable to both Supersymmetric (SUSY) and non-SUSY models.

4. Results and Demonstrations

The authors validated the interface using three distinct phenomenological examples:

1. Real Type-II Two Higgs Doublet Model (2HDM):

   • Scanned the $m_{12}$ vs. $\tan\beta$ plane.
   • Demonstrated that HiggsTools (with 3% mass uncertainty) and Lilith (with a hard mass cut of 123–128 GeV) yield slightly different allowed regions.
   • Showed how HiggsBounds excludes regions based on searches for heavy $H$ and $A$ bosons (e.g., $ZZ$ and $\tau\tau$ channels).

2. CP-Violating Next-to-Minimal Supersymmetric Standard Model (NMSSM):

   • Analyzed a high-scale, $Z_3$-symmetric NMSSM with complex phases.
   • Demonstrated that including CP-sensitive observables (specifically the $h \to \tau\tau$ angular analysis from CMS) significantly constrains the CP-mixing angles ($\phi_\tau$), ruling out parameter points that would otherwise be allowed by rate-only measurements.

3. Minimal R-symmetric Supersymmetric Standard Model (MRSSM):

   • Investigated a model with a light Dirac neutralino Dark Matter candidate.
   • Showed that a large invisible branching ratio ($\sim 42\%$) for the 125 GeV Higgs drastically reduces the $p$-value, excluding points that would appear valid if invisible decays were ignored.
   • Highlighted the interplay between the invisible width rescaling effect and direct search limits (HiggsBounds).

5. Significance

This work represents a significant step forward in BSM phenomenology by democratizing access to high-precision Higgs constraints.

• Efficiency: It eliminates the bottleneck of manual data formatting, allowing researchers to scan large parameter spaces of complex, user-defined models rapidly.
• Accuracy: By utilizing loop-corrected widths to derive effective couplings, the tool captures higher-order BSM effects that tree-level approximations miss.
• Comprehensiveness: It unifies the treatment of visible, invisible, and undetected decays, ensuring that Dark Matter scenarios are tested rigorously against Higgs data.
• Future-Proofing: The automated nature of the tool allows for immediate integration of new experimental datasets (e.g., from HL-LHC or future $e^+e^-$ colliders) as they become available, ensuring that theoretical models are constantly tested against the latest data.

The authors note current limitations (e.g., the inability to compute cross-sections directly, requiring normalization to SM values, and a mass limit of $\sim 650$ GeV for the normalization procedure) but outline plans to address these in future updates.