Differentiable Multi-scale Effective Field Theory Likelihoods for Beyond the Standard Model Phenomenology

This paper introduces a differentiable, multi-scale Effective Field Theory framework that integrates renormalization-group evolution, matching, and experimental constraints to enable efficient gradient-based inference in large parameter spaces, as demonstrated through comprehensive 374-parameter SMEFT analyses.

The Gist

The Big Picture: Hunting Ghosts with a New Kind of Net

Imagine you are a detective trying to catch a ghost. You can't see the ghost directly, but you know it's there because it's messing with the furniture in your living room. In the world of particle physics, the "ghosts" are New Particles (heavy, undiscovered particles) that are too massive to be created directly by our current particle colliders (like the Large Hadron Collider, or LHC).

Instead of catching the ghost, physicists look for the "furniture mess"—tiny, subtle deviations in how known particles behave. To make sense of this mess, they use a mathematical tool called an Effective Field Theory (EFT). Think of EFT as a giant, complex spreadsheet that predicts how the furniture should move if the ghost is there, and how it moves if the ghost isn't.

The Problem: The Spreadsheet is Too Heavy and Too Slow

For years, physicists have been trying to fill out this spreadsheet. But there were two major problems:

1. Too Many Variables: The spreadsheet has hundreds of columns (parameters). Trying to solve it was like trying to find a specific grain of sand on a beach while blindfolded.
2. The "Scale" Issue: The ghost might be influencing the furniture from a very high energy level (like a mountain peak), but we are measuring the mess at a low level (the valley). Connecting the mountain to the valley requires complex math called "Renormalization Group Evolution." Doing this math was like trying to solve a Rubik's cube while running a marathon. It was so slow and computationally expensive that scientists had to simplify the problem, ignoring huge chunks of data or making risky guesses.

The Solution: The "Differentiable" Super-Tool

This paper introduces a revolutionary new way to handle this spreadsheet. The authors, Aleks Smolkovič and Peter Stangl, built a system where the entire mathematical model is differentiable.

What does "differentiable" mean in plain English?
Imagine you are hiking down a mountain in thick fog.

• Old Way: You take a step, check if you are lower, take another step, check again. If the mountain is huge, this takes forever. You might get stuck in a small dip and think you've reached the bottom.
• New Way (Differentiable): You have a magical compass that instantly tells you exactly which direction is "down" and how steep the slope is at your exact feet. You can slide down the mountain instantly, finding the lowest point (the best answer) without getting lost.

Because their new tool (called jelli) is differentiable, it can use "gradient-based" methods. This means it can instantly calculate the slope of the data in every direction at once. This allows them to:

• Optimize: Find the best fit for the data in seconds instead of weeks.
• Sample: Explore the entire "mountain range" of possibilities to see where the ghost is most likely hiding, using a technique called Hamiltonian Monte Carlo (which is like a super-smart random walk).

The "Multi-Scale" Magic

The paper also emphasizes Multi-scale analysis.

• Analogy: Imagine you are trying to understand a crime. You have evidence from the crime scene (low energy), the suspect's alibi (medium energy), and the suspect's bank records from a different country (high energy).
• Old Way: Scientists often looked at just one piece of evidence at a time or tried to force them to fit together with rigid rules.
• New Way: This new tool seamlessly connects all the evidence. It takes the "high energy" rules (the bank records) and mathematically "evolves" them down to the "low energy" crime scene, ensuring everything is consistent. It does this without breaking the math, allowing them to look at 374 different variables at the same time.

The Results: A Giant Leap Forward

The authors tested their new tool with two massive experiments:

1. The Small Test (6 Dimensions): They looked at a specific type of particle decay (b → sℓℓ). The new tool confirmed what they already knew but did it much faster and with more precision, proving the "compass" works.
2. The Big Test (374 Dimensions): This is the headline. They analyzed Drell-Yan data (particles smashing together to create new particles) combined with low-energy flavor data.
   • They varied 374 parameters simultaneously. This is the most complex analysis of its kind ever done.
   • They found that when you look at all 374 variables together, the "shape" of the solution is weird. It's not a simple circle; it's a long, stretched-out tube.
   • The "Volume" Effect: They discovered that in high dimensions, the "volume" of the solution space can trick you. Sometimes, the most likely answer (the peak of the mountain) isn't where the "average" answer lies because there is so much "empty space" (volume) further away that pulls the average. Their tool can see this geometry clearly, whereas older methods would get confused.

Why This Matters

This isn't just about doing math faster. It changes how we search for new physics.

• No More Guessing: Previously, scientists had to assume certain things about the "flavor" of particles (like assuming the ghost only likes red furniture). Now, they can let the data speak for itself without those artificial limits.
• Connecting the Dots: It allows them to connect low-energy experiments (like those measuring rare decays) with high-energy collider data in a single, consistent framework.
• The Future: This tool is a stepping stone. Eventually, they want to use it to test specific theories about what the "ghost" actually is (UV models), not just describe the mess it's making.

In Summary:
The authors built a super-smart, mathematically connected map that allows physicists to navigate the incredibly complex landscape of particle physics. Instead of stumbling around in the dark, they now have a GPS that can handle hundreds of variables at once, connecting high-energy and low-energy clues to find the hidden "ghosts" of the universe.

Technical Summary

1. Problem Statement

The search for physics Beyond the Standard Model (BSM) has shifted from direct discovery of heavy particles to indirect searches via precision measurements. Effective Field Theories (EFTs), specifically the Standard Model Effective Field Theory (SMEFT) and the Weak Effective Theory (WET), provide the framework to describe these effects. However, current global EFT analyses face significant computational and conceptual bottlenecks:

• High Dimensionality: Full SMEFT analyses involve hundreds of Wilson coefficients (parameters). Traditional methods struggle to explore these high-dimensional spaces efficiently.
• Multi-scale Complexity: Data spans energy scales from low-energy flavor physics (GeV) to high-energy collider data (TeV). Connecting these requires Renormalization Group (RG) evolution and matching between EFTs, which are computationally expensive to re-evaluate during statistical sampling.
• Basis and Flavor Dependence: To make analyses tractable, researchers often impose flavor assumptions (e.g., flavor universality) or restrict the parameter space to a subset of coefficients. These assumptions introduce spurious basis dependence and obscure correlations between operators.
• Inference Limitations: Existing tools rely on computationally expensive sampling or profiling methods that cannot easily utilize gradient-based optimization or Hamiltonian Monte Carlo (HMC) in high dimensions, limiting the ability to perform fully global, basis-independent fits.

2. Methodology

The authors introduce a paradigm shift by formulating the entire EFT likelihood as a fully differentiable function of the Wilson coefficients using automatic differentiation (specifically via the JAX library).

Key Technical Components:

• Differentiable Framework: The likelihood $\mathcal{L}(\vec{C})$ is constructed as a smooth, differentiable function of the Wilson coefficients $\vec{C}$. This allows for the analytic computation of gradients ($\nabla \mathcal{L}$) and Hessians ($\nabla^2 \mathcal{L}$).
• Modular Construction:
  • RG Evolution & Matching: Implemented via pre-computed evolution matrices $U(\mu_i, \mu)$ that map Wilson coefficients from a reference scale $\mu$ to the characteristic scale $\mu_i$ of each observable. This avoids solving differential equations numerically during the fit.
  • Observable Predictions: Theoretical predictions are encoded using the Polynomial Observable Prediction exchange format (POPxf). Observables are expressed as polynomials in Wilson coefficients, allowing efficient evaluation and differentiation.
  • Likelihood Assembly: The global likelihood combines experimental data with theoretical uncertainties (which can depend on Wilson coefficients) into a single differentiable object.
• Inference Techniques Enabled:
  • Gradient-Based Optimization: Using L-BFGS to find likelihood modes and perform profiling.
  • Hessian-Based Preconditioning: Using the analytic Hessian at the Standard Model point to define coordinate transformations (Hessian-normalized and whitened coordinates). This removes local correlations and normalizes scales, making high-dimensional sampling stable.
  • Hamiltonian Monte Carlo (HMC): Using the No-U-Turn Sampler (NUTS) to sample the posterior distribution in high-dimensional spaces, leveraging gradients for efficient exploration.

Software Tools:
The authors released two public Python packages:

1. jelli: A JAX-based package for constructing and evaluating differentiable EFT likelihoods.
2. rgevolve: A package providing pre-computed RG evolution and matching matrices for SMEFT and WET.

3. Key Contributions

• First Fully Differentiable Global EFT Likelihoods: The paper demonstrates the first practical implementation of a likelihood that includes RG evolution, matching, and observable predictions as a single differentiable chain.
• Scalability to 374 Parameters: The authors successfully performed a global SMEFT fit with 374 real parameters (all dimension-six two-quark-two-lepton operators), which is the highest-dimensional SMEFT fit performed to date.
• Basis Independence: By avoiding flavor assumptions and retaining the full flavor structure, the framework ensures that constraints are basis-independent. Results can be transformed between "up-aligned" and "down-aligned" flavor bases without re-running the fit.
• Resolution of Volume Effects: The study highlights and analytically explains "volume effects" in high-dimensional spaces where marginalized posteriors can be dominated by large-volume regions with slightly lower likelihoods, leading to spurious tensions with the Standard Model. The framework allows for a clear distinction between profile likelihoods (tracking the best fit) and marginalized posteriors.

4. Results

The authors validated their framework through three progressively complex scenarios:

A. Six-dimensional WET Fit ($b \to s \ell \ell$)

• Setup: 181 observables at $\mu = 4.8$ GeV.
• Outcome: Excellent agreement between profiling (L-BFGS), marginalized HMC sampling, and Gaussian approximations derived from the Hessian. This validated the differentiability and accuracy of the framework in a controlled setting.

B. 374-parameter SMEFT Fit (Drell-Yan + $b \to s \nu \bar{\nu}$)

• Setup: Combined Drell-Yan data (691 bins) with $b \to s \nu \bar{\nu}$ observables to constrain flat directions associated with left-handed quark doublets.
• Outcome:
  • Successfully sampled the 374-dimensional posterior using NUTS with Hessian-based preconditioning.
  • Volume Effects: The analysis revealed that for operators like $O^{(3)}_{lq}$, the marginalized posterior showed a significant tension with the Standard Model (shifting away from zero), while the profile likelihood remained consistent with the SM. This was attributed to the large volume of parameter space where quadratic terms compensate for linear interference terms.
  • Constraints: Effective scales ($\Lambda_{eff}$) for many eigendirections were constrained above 10 TeV.

C. Multi-scale Combination with Full Flavor Observables

• Setup: Added 954 total observables, including low-energy flavor data (e.g., $K \to \pi \nu \bar{\nu}$, $B \to K^{(*)} \ell \ell$).
• Outcome:
  • The inclusion of low-energy data significantly altered the likelihood geometry, lifting degeneracies and creating highly anisotropic curvature.
  • Despite the increased complexity, the Hessian-preconditioned sampling remained feasible in the full 374-dimensional space.
  • The constraints on specific Wilson coefficient combinations improved by orders of magnitude compared to the Drell-Yan-only fit.

5. Significance and Outlook

• New Standard for Precision Phenomenology: This work establishes a new framework where global inference, multi-scale consistency, and basis independence are treated within a single differentiable structure.
• Bridge to UV Models: The framework is designed to be extended to Ultraviolet (UV) complete models. By combining differentiable EFT likelihoods with differentiable matching tools (e.g., Matchete), one can perform direct inference on UV model parameters, which are typically fewer in number than EFT coefficients, allowing for more complete exploration.
• Information Geometry: The ability to compute the Fisher Information metric efficiently opens the door to studying EFTs as "sloppy models" and addressing questions of naturalness and fine-tuning.
• Future Applications: The tools (jelli, rgevolve) are publicly available and can be applied to other areas, such as hadronic form-factor inference and next-to-leading order (NLO) global analyses.

In summary, this paper removes the computational barriers that have historically prevented fully global, basis-independent, and multi-scale EFT analyses, enabling a new generation of precision tests for the Standard Model.