SUSY meets SMEFT: Complete one-loop matching of the general MSSM

This paper presents the complete one-loop matching of the general 124-parameter MSSM with conserved R-parity onto both the SMEFT and the two-Higgs-doublet-model EFT using the Matchete package, providing a comprehensive framework for systematic global studies of supersymmetric parameter space while retaining full flavor structure and non-degenerate superpartner masses.

The Gist

Imagine the universe as a giant, intricate machine. For decades, physicists have been trying to understand how this machine works using a blueprint called the Standard Model. It's a great blueprint, but we know it's incomplete. It doesn't explain gravity, dark matter, or why the machine seems to have a few missing gears.

Enter Supersymmetry (SUSY). This is a popular theory suggesting that for every known particle (like an electron or a quark), there is a hidden "super-partner" particle (a selectron or a squark) that is much heavier. These super-partners are like the heavy-duty, industrial-grade gears hidden deep inside the machine's casing, too heavy to see directly with our current tools.

This paper is essentially a translation manual for physicists. Here is the story of what they did, explained simply:

1. The Problem: Too Many Gears to Count

The specific version of Supersymmetry they are looking at is called the MSSM (Minimal Supersymmetric Standard Model). It's the most "minimal" version, but it's still incredibly complex. It has 124 free parameters.

• The Analogy: Imagine trying to predict the weather, but you have 124 different dials you can turn (temperature, humidity, wind speed, etc.), and you don't know the settings for any of them. If you try to test every single combination, you'd need more time than the universe has existed.

2. The Solution: The "Effective" Blueprint

Since we can't see these heavy super-partners directly, physicists use a trick called Effective Field Theory (EFT).

• The Analogy: Imagine you are looking at a car from 100 feet away. You can't see the individual pistons, spark plugs, or the ECU chip. But you can see the car moving, the exhaust fumes, and the noise it makes. You can write a "simplified rulebook" that describes how the car behaves without needing to know the exact engineering of every internal part.
• In physics, this rulebook is called SMEFT (Standard Model Effective Field Theory). It keeps the known particles but adds "correction terms" (like little notes in the margin) that account for the invisible heavy gears pushing and pulling from the background.

3. The Challenge: The "Translation" is Hard

To use this simplified rulebook, you need to know exactly how the heavy gears (the MSSM) translate into those correction notes (the SMEFT).

• The Analogy: It's like trying to translate a 1,000-page technical manual written in a complex language (MSSM) into a simple summary (SMEFT).
• The Catch: Previous attempts at this translation were like summarizing a book by only reading the first chapter or skipping the footnotes. They missed subtle connections and correlations between different parts of the theory. Also, the math is so heavy that doing it by hand is impossible; you need a super-computer.

4. The Breakthrough: The "Matchete" Package

The authors of this paper used a powerful software tool called Matchete. Think of Matchete as a super-automated translator that can digest the entire 124-parameter MSSM blueprint and spit out the complete, one-loop (highly precise) translation into the SMEFT rulebook.

They didn't just do a rough draft; they did the complete job:

• They included all the super-partners (squarks, sleptons, gauginos, higgsinos).
• They kept the masses of these particles different (non-degenerate), meaning they didn't assume all the heavy gears were the same size, which is a more realistic scenario.
• They preserved the flavor structure, ensuring that the "flavor" (like the difference between an electron and a muon) is handled correctly.

5. The Higgs Twist: The "Heavy" and "Light" Cousins

A major part of the paper deals with the Higgs boson (the particle that gives mass to others). In the MSSM, there are actually two Higgs doublets (like a pair of twins), but in our low-energy world, we only see one.

• The Analogy: Imagine the MSSM has two Higgs twins. One is light and stays with us (the one we found at 125 GeV). The other is very heavy and hides away. The paper explains exactly how to "integrate out" (remove) the heavy twin and figure out how its mere existence changes the behavior of the light twin. They proved that their method of removing the heavy twin matches perfectly with other ways of looking at the problem.

6. Why Does This Matter?

• Connecting the Dots: Because Supersymmetry is a rigid theory, the "correction notes" in the SMEFT aren't random. They are all connected. If you tweak one dial in the MSSM, it changes many notes in the SMEFT rulebook in a specific, predictable pattern.
• The Future: Now that the authors have provided this complete translation (and the code to do it), experimentalists at the Large Hadron Collider (LHC) can take their data, look for these specific patterns, and say: "If Supersymmetry is real, it must look like this."
• Validation: They also checked their work against known results and found they matched, proving their "translator" works.

Summary

This paper is a massive computational achievement. The authors built a bridge between a complex, high-energy theory (MSSM) and the practical, low-energy tools (SMEFT) used to analyze data from particle colliders. They didn't just build a small footbridge; they built a highway, complete with all the necessary rules and signs, allowing physicists to systematically search for the hidden "super-partners" of the universe without getting lost in the math.

In a nutshell: They took a 124-dial control panel, figured out exactly how every single dial affects the simple dashboard we can see, and wrote down the instructions so anyone can use them to find new physics.

Technical Summary

1. Problem Statement

The search for physics beyond the Standard Model (BSM) often relies on the Standard Model Effective Field Theory (SMEFT) framework to interpret experimental data from the LHC and future colliders. However, global fits of SMEFT Wilson coefficients face a significant challenge: the theory contains nearly 2,500 free parameters (at dimension six), making model-independent analyses difficult without simplifying assumptions that can obscure correlations between different sectors.

To address this, specific BSM theories, such as the Minimal Supersymmetric Standard Model (MSSM), are used as Ultraviolet (UV) completions to constrain the SMEFT parameter space. While previous studies have matched parts of the MSSM to the SMEFT, they relied on approximations, such as:

• Integrating out only specific subsets of superpartners (e.g., only stops).
• Considering only leading-order contributions.
• Assuming degenerate masses or specific flavor structures.

The Gap: There was no complete, automated, one-loop matching of the general MSSM (with all 124 free parameters, conserved R-parity, and non-degenerate masses) onto the dimension-six SMEFT. This paper aims to fill that gap, providing a rigorous foundation for systematic global studies of the MSSM using EFT methods.

2. Methodology

The authors employed the Matchete package, a tool based on functional methods (specifically the Universal One-Loop Effective Action formalism), to perform the matching. Key methodological steps included:

• Lagrangian Formulation: The MSSM Lagrangian was reformulated from the standard Weyl spinor notation (common in SUSY literature) into four-component Dirac and Majorana spinors to ensure compatibility with the Matchete code and the Warsaw basis of SMEFT.
• Mass Basis and Decoupling:
  • All superpartners (sfermions, gauginos, Higgsinos) and the second Higgs doublet were integrated out at a single scale $\bar{\mu} \gg v$ (electroweak scale).
  • The Higgs sector was treated carefully. The authors rotated the two Higgs doublets into a soft-SUSY mass basis (before Electroweak Symmetry Breaking, EWSB) to distinguish the light SM-like doublet ($H$) from the heavy doublet ($\Phi$).
  • The Higgsino sector was redefined into a vector-like fermion $\Sigma$ to facilitate integration.
• Scheme Conversion (DR $\leftrightarrow$ MS): Since the MSSM is naturally defined in the Dimensional Reduction (DR) scheme to preserve SUSY, while SMEFT calculations typically use Dimensional Regularization (MS), the authors implemented a consistent scheme change. They derived finite counterterms to convert DR couplings to MS couplings before matching.
• Operator Reduction: The functional method generates a large number of redundant operators. The authors utilized Matchete's automated reduction capabilities (integration-by-parts, equations of motion, Fierz identities) to project the results onto the minimal Warsaw basis.
• Evanescent Operators: Special attention was paid to evanescent operators (operators that vanish in $D=4$ but contribute at one-loop in $D=4-2\epsilon$). The authors implemented a consistent prescription to absorb these effects into finite counterterms, ensuring the results are valid in an evanescent-free MS scheme.
• Alternative Scenario: As a byproduct, the authors also performed the matching onto a Two-Higgs-Doublet Model EFT (2HDM-EFT), where the second Higgs doublet is kept light and dynamic in the infrared spectrum.

3. Key Contributions

• Completeness: This is the first complete one-loop matching of the general R-parity conserving MSSM (124 parameters) to the SMEFT. It includes all subleading contributions from all superpartners and captures all supersymmetry-governed correlations between Wilson coefficients.
• Automation and Code: The work required significant performance enhancements to the Matchete package (v0.4.0), including optimized operator reduction, improved memory management for large interaction terms, and the ability to handle heavy flavored particles (sfermions) consistently.
• Public Availability: The full matching conditions, code, and auxiliary materials (Mathematica notebooks, PDFs, and C++ classes via OperatorToC++) are made available on GitHub.
• Validation: The results were validated against existing literature (e.g., Refs. [73], [80], [84], [87]) in various limits (e.g., decoupling of heavy stops, specific flavor alignments). The authors confirmed agreement with known results while correcting or extending previous approximations (e.g., identifying missing contributions to gauge-boson self-energies in prior works).

4. Key Results

• Wilson Coefficients: The paper provides analytical expressions for all dimension-six Wilson coefficients in the Warsaw basis generated by the MSSM.
  • Tree-level: Only the heavy Higgs doublet $\Phi$ contributes, generating operators like $O_H$, $O_{uH}$, $O_{dH}$, etc., with coefficients dependent on $\tan\beta$ and $m_\Phi$.
  • One-loop: All BSM states contribute. The authors derived explicit formulas for operators modifying Higgs-gauge couplings ($O_{HG}, O_{HW}, O_{HB}, O_{HWB}$) and four-fermion operators.
• Threshold Corrections: The paper details the threshold corrections to renormalizable parameters (gauge couplings, Yukawas, Higgs mass, and quartic coupling) arising from the matching.
• Phenomenological Example (Stop-Bino Scenario):
  • The authors analyzed a simplified scenario where only the right-handed stop ($\tilde{t}_R$) and bino ($\tilde{B}$) are light.
  • They calculated the resulting Wilson coefficients relevant for top-pair production and Higgs physics.
  • Finding: The generated Wilson coefficients are generally very small (orders of magnitude below current LHC sensitivity) because the couplings are fixed by gauge interactions ($y_{DM} \propto g_1$) rather than being free parameters. This highlights the predictive power of the MSSM compared to generic simplified models.
• Higgs Mass: The derived corrections to the Higgs mass parameter $\Delta m_h^2$ reproduce the well-known one-loop stop corrections, validating the consistency of the EFT approach.

5. Significance

• Bridging UV and IR: This work provides the essential "dictionary" to translate constraints on SMEFT Wilson coefficients (from LHC data) directly into constraints on the fundamental parameters of the MSSM.
• Global Fits: By providing the full set of correlations between Wilson coefficients dictated by SUSY, this work enables more robust and less ambiguous global fits of the MSSM parameter space, avoiding the pitfalls of model-independent analyses that ignore these correlations.
• Benchmark for Automation: The successful application of functional methods to such a complex theory (124 parameters, multiple scales, flavor structures) demonstrates the maturity of automated EFT matching tools and sets a precedent for future studies of other complex BSM scenarios.
• Future Directions: The authors outline paths for future work, including multi-scale matching (for Split SUSY scenarios where masses span orders of magnitude) and scenarios where some superpartners (like light Higgsinos) are retained in the EFT spectrum.

In summary, this paper represents a major step forward in the precision phenomenology of Supersymmetry, moving from approximate, partial matching to a complete, automated, and rigorous one-loop connection between the MSSM and the SMEFT.