From the EFT to the UV: the complete SMEFT one-loop dictionary

This paper presents a complete one-loop dictionary connecting the Standard Model Effective Field Theory to arbitrary ultraviolet completions involving heavy fermions and scalars, implemented in the updated SOLD package to facilitate systematic phenomenological studies such as explaining the $B\rightarrow K \overline{\nu}\nu$ measurement tension.

The Gist

Imagine the Standard Model of particle physics as a massive, incredibly detailed instruction manual for how the universe works. For decades, scientists have been following this manual perfectly. But recently, they've noticed a few pages where the instructions seem slightly off—experiments are showing results that don't quite match what the manual predicts.

This paper is about a new, powerful tool designed to help scientists figure out what's missing from the manual without having to rewrite the whole book from scratch.

Here is the breakdown of the paper using simple analogies:

1. The Problem: The "Bottom-Up" vs. "Top-Down" Puzzle

Scientists approach these mysteries in two ways:

• Bottom-Up (The Detective): They look at the weird data (the "symptoms") and say, "Okay, something is wrong here. Let's write down a list of rules that could explain this, without guessing what the culprit is yet." This is called Effective Field Theory (EFT). It's like a detective listing all possible suspects based on the crime scene.
• Top-Down (The Architect): They start with a specific theory of new physics (a "UV model") and try to see if it fits the data. This is like a detective picking a specific suspect and checking their alibi.

The Gap: The problem is that there are thousands of possible suspects (theories). Trying to check every single one against the data is like trying to find a needle in a haystack by checking every single piece of hay one by one. It's slow, messy, and inefficient.

2. The Solution: The "Dictionary" (SOLD)

The authors of this paper created a tool called SOLD. Think of SOLD as a universal translator or a dictionary that connects the "Detective's list" (EFT) directly to the "Architect's blueprints" (UV models).

• What it does: If you tell SOLD, "I need a theory that explains this specific weird signal," it instantly spits out a list of every possible particle combination that could cause it.
• The Upgrade: Previous versions of this dictionary only worked for simple, direct connections (like a tree branch). This new version works for complex, loop-level connections.
  • Analogy: Imagine a tree. The old dictionary could only tell you how a leaf grows directly from a branch. The new dictionary can explain how a leaf grows from a branch through a complex network of roots and soil interactions (loops). This is crucial because some new physics is too heavy to show up directly; it only reveals itself through these complex "loops."

3. How It Works: The "Recipe Book"

Instead of scientists having to write out complex math equations for every new theory they want to test, they just use SOLD.

• Input: You tell the computer, "I have a new particle with these specific properties (like its electric charge or spin)."
• Process: SOLD uses a pre-calculated "recipe book" (the dictionary) to instantly tell you: "If you add this particle, here is exactly how it changes the rules of the universe."
• Output: It gives you a list of consequences. "If you add this particle, you will see this change in the B-meson decay, but this change in the muon's magnetic moment."

4. The Real-World Test: The "B-Meson Mystery"

To prove their tool works, the authors used it to investigate a specific mystery: B-mesons decaying into Kaons and neutrinos ($B \to K\nu\nu$).

• The Mystery: Recent experiments (by Belle II) showed this process happens about 3 times more often than the Standard Model predicts. It's a huge discrepancy.
• The Search: Scientists wanted to know: "What new heavy particles could cause this?"
• The Result:
  • They first looked for simple solutions (one new particle). SOLD said, "Nope, that doesn't work without breaking other rules."
  • They looked for two-particle solutions. SOLD said, "Still no, the math doesn't add up without causing other problems."
  • They looked for three-particle solutions. Here, they found a "magic trick." They discovered a specific combination of three new particles where the "bad" effects canceled each other out (like noise-canceling headphones), allowing the "good" effect (explaining the B-meson mystery) to shine through.

5. Why This Matters

This paper isn't just about solving one specific puzzle; it's about giving scientists a GPS for the future.

• Efficiency: Before, checking a theory took days of manual calculation. Now, with SOLD, it takes seconds.
• Completeness: It ensures scientists don't miss the "hidden" solutions that only appear in complex loops.
• Guidance: If future experiments find more "glitches" in the universe, scientists can use this dictionary to instantly know which theories are worth investigating and which are dead ends.

In a nutshell: The authors built a super-smart search engine that translates "weird experimental data" into "possible new physics theories," helping scientists navigate the vast ocean of possibilities to find the one that explains our universe's secrets.

Technical Summary

1. Problem Statement

Effective Field Theories (EFTs), specifically the Standard Model Effective Field Theory (SMEFT), serve as a bridge between experimental data and Beyond the Standard Model (BSM) physics. This process involves two complementary approaches:

• Bottom-up: Extracting Wilson Coefficients (WCs) from experimental data.
• Top-down: Matching specific UV-complete models to these WCs.

While tree-level dictionaries connecting UV models to SMEFT operators at dimension six are well-established, a complete one-loop dictionary has been lacking. This gap is critical because:

1. Many observables (e.g., the muon anomalous magnetic moment) receive leading contributions only at the loop level.
2. Renormalization group mixing causes tree-level and loop-level operators to mix.
3. Models with specific symmetries (e.g., $Z_2$ symmetry for Dark Matter candidates) often involve quadratic couplings (involving two BSM fields) that contribute to SMEFT only at the loop level.
4. Existing tools (like MatchmakerEFT) require users to implement full Lagrangians for every specific model, making systematic scans over the vast landscape of possible UV completions computationally cumbersome and inefficient.

2. Methodology

The authors present SOLD (Standard Model One-loop Dictionary), a Mathematica package that automates the construction of the complete one-loop dictionary for SMEFT at dimension six.

Key Technical Components:

• General Theory Framework: Instead of defining specific models, the authors start with a generic renormalizable theory containing arbitrary heavy fermions ($\Psi$) and scalars ($\Phi$) arranged in multiplets. The Lagrangian includes all possible renormalizable interactions (Yukawa and scalar couplings) with unspecified gauge contractions represented by general Clebsch-Gordan (CG) tensors.
• Diagrammatic Matching: The package performs matching to the SMEFT Green's basis using a diagrammatic off-shell approach. It computes 1-particle-irreducible (1PI) diagrams.
• Basis Reduction: Results are projected from the redundant Green's basis to the minimal physical Warsaw basis using field redefinitions. The package handles the reduction of evanescent operators (structures that vanish in 4D but contribute in $d$-dimensions) and accounts for wavefunction renormalization effects on tree-level generated WCs.
• Group Theory Automation: Using the GroupMath package, SOLD calculates the specific CG tensors and group theory factors once the quantum numbers (SU(3), SU(2), U(1)) of the heavy fields are specified by the user.
• Two-Level Output:
  1. Class Level: Identifies the necessary restrictions on field representations (e.g., product of representations) to generate a specific operator.
  2. Model Level: Lists specific group representations that satisfy these restrictions.

3. Key Contributions

• Completeness: Unlike the previous iteration of SOLD (which only covered operators with field-strength tensors), this version covers all dimension-six SMEFT operators, including those that can receive tree-level contributions.
• Reverse Dictionary Capability: SOLD uniquely answers the inverse question: "Given a specific SMEFT operator, what are all possible scalar and fermion multi-field extensions that can generate it?" This is achieved without requiring the user to input a full Lagrangian.
• New Functionality:
  • ListModelsWarsaw: Generates lists of UV completions for a given operator.
  • MatchSchematic: Provides a schematic view of the matching result (diagram topology and coupling structure) without full numerical computation, allowing for rapid phenomenological assessment.
  • noSMcouplings Option: A novel feature that filters results to show only contributions generated purely by new physics interactions (excluding small SM couplings like $g_1, g_2, y_{light}$), isolating the "pure" BSM effects.
  • Parallelized Matching: Functions like Match2WarsawPar significantly reduce execution time for large scans.
• Integration: The package includes pre-computed tree-level results (matching Ref. [12]) and interfaces with MatchmakerEFT for full one-loop matching of specific promising models.

4. Results and Case Study

The authors demonstrate the utility of SOLD by investigating the tension in the branching ratio measurement of $B \to K \nu \bar{\nu}$ (specifically $R_K^{\nu\nu}$), which shows a $2.9\sigma$ deviation from the SM.

The Goal: Explain the required large Wilson Coefficients ($C_{\ell q}^{(1,3)}$ and $C_{\ell d}$) via UV completions.

Systematic Scan Results:

1. One-Field Extensions:

   • Scanned for single heavy particles generating the required pattern without SM couplings.
   • Result: Found that one-field extensions (e.g., a scalar doublet $\Phi \sim (1, 2, 1/2)$) generate the correct operator pattern but are suppressed by loop factors and linear couplings, making it impossible to reach the required magnitude ($O(10^{-2})$) without violating perturbative unitarity or other constraints.

2. Two-Field Extensions:

   • Explored combinations of scalars and fermions (e.g., $S_3 + Q_5$ or $S_3 + T_2$).
   • Result: While quadratic couplings allow for larger values, these models generically induce large contributions to other flavor-changing neutral currents (FCNCs), specifically $B_s - \bar{B}_s$ mixing and $B_s \to \mu\mu$, which are tightly constrained by experiment. No viable two-field solution was found.

3. Three-Field Extensions:

   • Explored models with three heavy fields (e.g., $\Pi_1 + N + D$).
   • Key Finding: Discovered a model involving a leptoquark $\Pi_1 \sim (3, 2, 1/6)$, a singlet fermion $N \sim (1, 1, 0)$, and a fermion $D \sim (3, 1, 2/3)$.
   • Mechanism: This model exhibits an accidental cancellation (reminiscent of the "magic zero" in dipole operators) in the Green's basis contributions to the $O_{qq}^{(1,3)}$ operators. When projected to the Warsaw basis, the dangerous flavor-off-diagonal contributions ($[2,3,2,3]$) vanish.
   • Outcome: This cancellation allows the model to generate the required $C_{\ell q}$ and $C_{\ell d}$ via loop-level effects while evading constraints from $B_s - \bar{B}_s$ mixing. The model can accommodate the pattern of the tension (qualitative agreement) but struggles to reach the large central values of the WCs implied by current data without pushing couplings to the perturbative unitarity limit.

5. Significance

• Systematic Phenomenology: SOLD transforms the search for UV completions from a model-by-model manual effort into a systematic, database-driven process. It allows physicists to quickly map the entire landscape of possible UV theories for a given EFT signature.
• Guiding Principle: The tool serves as a "pocket guide" for model builders, helping to identify viable UV scenarios and, crucially, to rule out large classes of models that cannot explain specific data patterns.
• Loop-Level Insights: The study highlights that while loop-level solutions are often suppressed, they can be viable if they involve quadratic couplings and specific symmetries that avoid tree-level constraints.
• Future Outlook: The authors note that if future experimental updates reduce the central value of the $B \to K \nu \bar{\nu}$ tension, loop-level solutions (like the three-field model found) could become the primary explanation. The package is designed to be continuously updated to include heavy vectors and further optimizations.

In summary, this work provides the first complete, automated, and user-friendly one-loop dictionary for SMEFT, significantly lowering the barrier for connecting low-energy experimental anomalies to high-energy theoretical models.