Parametrisation and dictionary for CP violating Higgs boson interactions

This paper establishes a unified framework and explicit dictionaries to connect various theoretical parametrisations of CP-violating Higgs boson interactions, thereby resolving translation inconsistencies and facilitating robust comparisons between theoretical studies and experimental LHC analyses.

The Gist

Imagine the universe as a giant, complex machine built by a master architect (the Standard Model of physics). For a long time, we thought this machine was perfectly symmetrical, like a snowflake. But we know the universe isn't perfectly symmetrical; there is more matter than antimatter, and we need to find out why. One of the biggest clues might be hidden in the "Higgs boson," a particle discovered a few years ago that gives other particles their mass.

This paper is essentially a translation guide for scientists trying to find out if the Higgs boson breaks the rules of symmetry (specifically, a rule called "CP symmetry").

Here is the breakdown of what the paper does, using simple analogies:

1. The Problem: Everyone Speaks a Different Dialect

Imagine a group of detectives (experimentalists at the LHC) trying to solve a crime. They are gathering clues about the Higgs boson. However, the theoretical physicists who help them interpret the clues are all speaking different languages.

• One group speaks "Higgs Basis" (a direct description of the particle's properties).
• Another group speaks "κ's and Angles" (using ratios and rotation angles, like describing a door by how far it's open).
• A third group speaks "CP Fractions" (describing how much of the particle is "good" vs. "bad" symmetry).
• A fourth group speaks "SMEFT" and "HEFT" (complex mathematical frameworks that act like blueprints for the entire machine, including parts we haven't seen yet).

The problem is that when a detective says, "The door is open 30 degrees," a theorist using the "Blueprint" language might interpret that as a completely different number. This leads to confusion, mistakes, and missed clues.

2. The Solution: The "Dictionary"

The authors of this paper have built a universal dictionary. Their goal is to create a clear map that translates between all these different languages.

• If an experiment measures a specific "angle," the dictionary tells you exactly what that means in the "Blueprint" language.
• If a theorist calculates a result using "CP Fractions," the dictionary tells you how to convert that into the "Higgs Basis" numbers that experiments actually measure.

This ensures that when a scientist in New York says "X," a scientist in Madrid knows exactly what "X" means, without guessing or making assumptions.

3. The "Blueprints" vs. The "Real House"

The paper distinguishes between two types of blueprints:

• SMEFT (The Strict Blueprint): This assumes the Higgs boson is part of a specific, rigid family structure (a doublet). It's like assuming every house must have a specific number of windows. If the Higgs behaves differently, this blueprint might be missing some rooms.
• HEFT (The Flexible Blueprint): This is a more general approach. It treats the Higgs as a unique individual that doesn't have to follow the strict family rules. This allows for "extra rooms" or weird features that the strict blueprint doesn't allow.

The authors point out that some features allowed in the "Flexible Blueprint" (HEFT) simply don't exist in the "Strict Blueprint" (SMEFT). Their dictionary helps scientists realize when they are looking at a feature that only the Flexible Blueprint can explain, preventing them from trying to force a square peg into a round hole.

4. Why This Matters

The paper argues that to find the "smoking gun" of new physics (which could explain why the universe exists as it does), we need to stop talking past each other.

• Current Status: Experiments at the Large Hadron Collider (LHC) are already looking for these symmetry-breaking effects.
• Future: As we get more data (like the High-Luminosity LHC), the measurements will get sharper.
• The Paper's Role: By providing this unified framework, the paper ensures that when we finally find a deviation from the Standard Model, we can immediately understand what it is, rather than getting stuck arguing about how to describe it.

Summary

Think of this paper as the Rosetta Stone for Higgs physics. It doesn't discover a new particle or prove a new theory. Instead, it organizes the chaos of different mathematical descriptions into a single, coherent system. This allows the global scientific community to compare notes accurately, ensuring that the search for the secrets of the universe's asymmetry is conducted with a single, clear voice.

Technical Summary

1. Problem Statement

The search for Charge-Parity (CP) violating interactions in the Standard Model (SM) Higgs sector is a primary objective of the LHC and future collider programs. Such interactions are crucial for explaining the Baryon Asymmetry of the Universe (BAU) via electroweak baryogenesis. However, a significant barrier to interpreting experimental results exists:

• Fragmented Theoretical Frameworks: Experimental analyses (ATLAS, CMS) and theoretical studies utilize diverse parametrisations, including the Higgs basis, coupling modifiers ($\kappa$'s and angles), CP fractions, and Effective Field Theories (SMEFT and HEFT).
• Inconsistencies: Differing conventions, assumptions, and definitions across these frameworks make translating results between them nontrivial. This leads to potential inconsistencies when comparing theoretical predictions with experimental constraints or when performing global fits.
• Missing Links: While mappings exist between some frameworks (e.g., SMEFT to Higgs basis), comprehensive dictionaries connecting all major approaches—particularly including the more general Higgs Effective Field Theory (HEFT)—were previously lacking or incomplete.

2. Methodology

The authors construct a unified framework to bridge these gaps through the following steps:

• Standardization of Notation: The paper establishes a consistent notation for CP-violating (CPV) operators across different formalisms, focusing on operators with up to four fields.
• Review of Formalisms:
  • Higgs Basis: Defined using SM mass eigenstates, parameterizing interactions via coefficients like $\tilde{\kappa}$, $\tilde{\lambda}$, and complex matrices for fermion couplings.
  • $\kappa$, Angles, and CP Fractions: Analyzed as experimental observables (ratios of couplings, mixing angles, or fractional cross-sections) used to reduce degeneracies in data fitting.
  • SMEFT: Reviewed in the Warsaw basis ($d=6$), detailing bosonic, dipole, Yukawa-like, and gauge-fermion operators.
  • HEFT: Reviewed as the most general EFT where the Higgs is a singlet, allowing for independent operators with multiple Higgs insertions that do not exist in SMEFT at $d=6$.
• Derivation of Matching Relations: The core methodology involves deriving explicit analytical "dictionaries" (matching equations) that translate Wilson coefficients between:
  1. SMEFT $\leftrightarrow$ Higgs Basis
  2. HEFT $\leftrightarrow$ Higgs Basis
  3. Experimental parameters ($\kappa$, angles, fractions) $\leftrightarrow$ Theoretical coefficients.
• Flavor Considerations: The analysis accounts for flavor assumptions (e.g., Minimal Flavour Violation vs. general flavor structures) and identifies which operators vanish or become CP-conserving under specific symmetry constraints.

3. Key Contributions

The paper provides several critical technical contributions:

A. Unified Dictionaries

• SMEFT $\leftrightarrow$ Higgs Basis: The authors explicitly map the Warsaw basis SMEFT coefficients to the Higgs basis parameters (e.g., relating $C_{H\tilde{W}B}$ to $\tilde{\kappa}_\gamma$). They highlight constraints imposed by gauge invariance, such as relations between coefficients with one vs. two Higgs insertions ($e c^{(2)}_{vv} = e c_{vv}$).
• HEFT $\leftrightarrow$ Higgs Basis: This is a novel contribution. The paper provides the first explicit dictionary translating HEFT operators (including those with arbitrary Higgs field functions $F_i(h)$) to the Higgs basis.
• Experimental $\leftrightarrow$ Theoretical: The paper clarifies the mathematical equivalence between experimental parameters (like CP mixing angles $\alpha$ or CP fractions $f_{CP}$) and ratios of theoretical coupling constants ($c'/c_0$). It demonstrates how to invert these relations to extract coupling constraints from experimental data.

B. Identification of HEFT-Unique Operators

The authors identify a class of CP-violating operators present in HEFT that have no counterpart in SMEFT at dimension-6. These include:

• Triple gauge couplings with an additional Higgs boson (e.g., $h W^+ W^- \gamma$).
• Specific bosonic and fermionic operators involving higher powers of the Higgs field or specific chiral structures that only appear at $d=8$ in SMEFT.
• The paper explicitly lists these "blind directions" where HEFT allows for CP violation that SMEFT ($d=6$) cannot capture, necessitating specific observables to break degeneracies.

C. Clarification of Experimental Parameterizations

The paper demystifies the relationship between:

• Coupling Ratios ($r_f$): $\tilde{\kappa}_f / \kappa_f$
• Mixing Angles ($\alpha$): $\tan^{-1}(\tilde{\kappa}_f / \kappa_f)$
• CP Fractions ($f_{CP}$): Ratios of partial widths or cross-sections.
  It provides the algebraic conversion between these forms, ensuring that experimental limits reported in one format can be correctly interpreted in another without loss of information or introduction of bias.

4. Results

• Explicit Equations: The paper presents a comprehensive set of equations (Sections 3.5.1 and 3.5.2) allowing researchers to convert Wilson coefficients between SMEFT, HEFT, and the Higgs basis.
• Flavor Constraints: The analysis details how flavor symmetries (like $U(3)^5$ or MFV) reduce the number of independent CP-violating parameters, specifically noting that Hermitian coefficients in certain sectors become CP-conserving in the flavor-universal limit.
• Validity Ranges: The authors emphasize that while experimental constraints on coupling ratios are robust, their interpretation within EFT requires checking the validity of the expansion (i.e., ensuring quadratic terms in EFT operators are negligible).
• Operator Redundancy: The paper identifies "null operators" in HEFT that do not contribute to physical observables, streamlining the basis for phenomenological studies.

5. Significance

• Facilitating Global Fits: By providing a consistent dictionary, the paper enables the combination of results from different channels and experiments into global EFT fits, reducing the risk of inconsistent interpretations.
• Bridging Theory and Experiment: It allows experimentalists to directly translate their limits on $\kappa$'s or CP fractions into constraints on SMEFT/HEFT Wilson coefficients, and vice versa, facilitating more robust comparisons.
• Future Collider Readiness: As the HL-LHC and future colliders (FCC, ILC, Muon Collider) aim for higher precision, the ability to distinguish between SMEFT and HEFT scenarios (especially via HEFT-unique operators) becomes critical. This paper provides the necessary theoretical infrastructure for such analyses.
• Standardization: It serves as a reference document to standardize conventions in the community, reducing errors caused by mismatched definitions in the literature.

In summary, this paper acts as a foundational "Rosetta Stone" for CP violation in the Higgs sector, unifying disparate theoretical languages and experimental metrics to enable a more precise and coherent search for physics beyond the Standard Model.