Electron EDM and $\Gamma(\mu \to e \gamma)$ in the 2HDM

This paper presents the first complete two-loop calculation of the electron electric dipole moment and lepton-flavor violating decay rates within the unconstrained two-Higgs doublet model, incorporating general Yukawa interactions and Higgs potential phases, with results made available through a public Python implementation.

The Gist

Imagine the universe as a giant, complex machine built from tiny, invisible Lego bricks called particles. For decades, scientists have had a very successful instruction manual for how these bricks interact, called the Standard Model. It explains almost everything we see in particle accelerators.

However, there's a missing piece in the manual. The manual can't explain how the universe got started with the right mix of matter and anti-matter. It's like a recipe that tells you how to bake a cake perfectly, but fails to explain why the cake rose in the first place. To fix this, scientists propose adding a new, secret ingredient to the recipe. In this paper, the authors explore a specific version of this secret ingredient called the Two-Higgs Doublet Model (2HDM). Think of the Standard Model as having one "flavor" of Higgs field (like vanilla), and this new model adds a second one (like chocolate), creating a whole new world of possibilities.

The Mystery of the "Wobbly" Electron

The authors are investigating two main mysteries using this new model:

1. The Electron's "Wobble" (Electric Dipole Moment):
   Imagine an electron as a tiny, spinning top. In a perfectly symmetrical world, this top spins evenly. But if the electron has an "electric dipole moment" (EDM), it's like the top is slightly lopsided or "wobbly." This wobble is a sign that the laws of physics treat "left" and "right" differently (a property called CP violation).

   • The Paper's Claim: The authors calculated, for the first time, exactly how big this wobble would be if the "Two-Higgs" model were true. They didn't just look at the simple interactions; they looked at the complex, messy interactions that happen when particles bounce off each other in loops (like a ball bouncing off a wall, then a ceiling, then back to the wall). They found that if this new model is real, the electron's wobble could be much larger than previously thought, depending on the "flavor" and "phase" (a kind of hidden angle) of the new particles.

2. The "Wrong-Color" Switch (Lepton Flavor Violation):
   Normally, particles are very loyal. A muon (a heavy cousin of the electron) should decay into an electron and a neutrino, but it should never suddenly turn into an electron and a flash of light (a photon) on its own. That would be like a red Lego brick spontaneously turning into a blue one while glowing.

   • The Paper's Claim: The authors calculated how often this "wrong-color switch" (specifically $\mu \to e\gamma$) would happen in their new model. They found that the new Higgs particles could act as a bridge, allowing the muon to cheat the rules and turn into an electron plus a photon much more easily than the old rules allowed.

How They Did It: The "Double-Loop" Detective Work

Calculating these effects is incredibly hard. It's like trying to predict the exact path of a pinball that bounces off hundreds of bumpers, where the bumpers themselves are moving and changing shape.

• One-Loop vs. Two-Loop: In physics, "loops" represent the complexity of the calculation. A "one-loop" calculation is like a ball bouncing once. A "two-loop" calculation is like the ball bouncing twice, interacting with more particles along the way.
• The Breakthrough: Previous studies often stopped at the simple "one-bounce" level or made simplifying assumptions (like ignoring certain angles or phases). This paper is the first to do a complete "two-bounce" (two-loop) calculation that includes every possible way the new Higgs particles can interact, including all the complex angles and "phases" (hidden directions) that could exist.

The "Universal Translator" (The Python Code)

One of the most practical parts of this paper is that the authors didn't just write down thousands of pages of math formulas. They realized that other scientists would need to use these results to test their own theories against real data.

So, they built a Python computer program (a digital translator) that takes the complex math and turns it into a tool anyone can use. If you have a specific set of numbers for your new physics model, you can plug them into their code, and it will instantly tell you: "If your model is right, here is exactly how much the electron should wobble, and here is how often the muon should turn into an electron."

The Bottom Line

This paper is a massive "checklist" for physicists. It says: "We have calculated the most detailed, complete prediction for how these new particles would affect the electron and muons. If you want to test if this 'Two-Higgs' model is real, you must compare your experimental data against these specific numbers, not the old, simplified ones."

They have provided the most accurate map yet for where to look for the "wobble" and the "wrong-color switch," ensuring that if we find these effects in the future, we can correctly identify if they are caused by this specific new model of the universe.

Technical Summary

1. Problem Statement

The Standard Model (SM) of particle physics fails to explain the observed matter-antimatter asymmetry of the universe (baryogenesis) because the amount of CP violation it contains is insufficient, and the electroweak phase transition is a smooth crossover (second-order) rather than a first-order transition.

The Two-Higgs-Doublet Model (2HDM) is a popular extension that introduces a second Higgs doublet, potentially providing:

• A strong first-order electroweak phase transition.
• Additional sources of CP violation via complex phases in the Higgs potential and Yukawa couplings.

However, these new CP-violating phases are tightly constrained by the non-observation of Electric Dipole Moments (EDMs) in elementary particles (like the electron) and atoms. A critical issue in previous studies is that in the most general 2HDM (unconstrained by specific flavor symmetries), contributions from different diagrams can cancel each other out, evading experimental bounds. To rigorously test the 2HDM, one must calculate all leading contributions to EDMs and Lepton Flavor Violating (LFV) decays (specifically $\mu \to e\gamma$ and $\tau \to e/\mu\gamma$) without assuming specific flavor structures or neglecting higher-order loop effects.

2. Methodology

The authors perform a complete, leading-order two-loop calculation within the unconstrained 2HDM.

• Model Framework: They utilize the most general 2HDM, allowing for:
  • Complex phases in the Higgs potential parameters ($m^2_{12}, \lambda_{5,6,7}$).
  • General complex, flavor-non-diagonal Yukawa couplings ($\rho_f$ matrices) for all fermion generations.
  • No assumptions of CP conservation or specific alignment limits (e.g., Type-I or Type-II are treated as specific limits, not assumptions).
• Effective Field Theory (EFT) Approach:
  • The calculation is performed by integrating out heavy particles (Higgs bosons, top quark, $W/Z$ bosons) at the electroweak scale.
  • They match the full theory to an effective leptonic Lagrangian defined below the electroweak scale.
  • QED running effects are neglected (suppressed by $\alpha$), and strong interaction effects are handled by evaluating quark masses at the weak scale.
• Calculation Tools & Techniques:
  • Software: Diagrams generated with qgraf, algebraic manipulation with FORM (via the MaRTIn package), and Feynman rules derived using FeynRules.
  • Gauge Choice: Calculations are performed in a generalized $R_\xi$ gauge using the Background Field Method. This ensures that the final physical results are explicitly gauge-parameter independent.
  • $\gamma_5$ Scheme: To handle closed fermion loops involving $\gamma_5$ (arising from pseudo-scalar couplings), the authors employ the 't Hooft-Veltman (HV) scheme. This is crucial because the commonly used NDR (Naive Dimensional Regularization) scheme is algebraically inconsistent for these specific diagrams. They include necessary counterterms to ensure consistency and agreement with known results.
• Observables:
  • Electron EDM ($d_e$): Defined via the effective operator $\mathcal{L}_{eff} \supset -\frac{d_e}{2} (\bar{e}\sigma_{\mu\nu}i\gamma_5 e)F^{\mu\nu}$.
  • LFV Decays: $\mu \to e\gamma$ and $\tau \to \ell\gamma$, parameterized by dipole coefficients $c_L$ and $c_R$.

3. Key Contributions

This work provides the first complete analytic calculation of these observables in the general 2HDM, including:

1. Full Two-Loop Contributions: They calculate all leading two-loop terms that are quadratic in Yukawa couplings. This includes:
   • Barr-Zee diagrams: With internal fermion loops (top, bottom, charm, tau) and neutral/charged Higgs exchanges.
   • Bosonic diagrams: Diagrams involving vertices from the Higgs kinetic terms and the Higgs potential (involving $W, Z, \gamma, H^\pm, h^0$).
2. General Flavor and CP Structure: Unlike previous works that often assumed real couplings or specific flavor textures, this calculation retains general complex phases and off-diagonal flavor entries in the Yukawa matrices ($\rho_f$).
3. New Diagrams: They identify and calculate contributions from diagrams involving charged Higgs bosons and the Higgs potential ($d^{H^\pm \gamma}_e, d^{H^\pm Z}_e, d^{H^\pm W}_e$) which were previously absent in the literature for the general case.
4. Gauge Invariance Verification: They explicitly demonstrate that the sum of all contributions is independent of the gauge parameter $\xi$, a non-trivial check given the complexity of the two-loop bosonic diagrams.
5. Public Implementation: The authors provide a Python code (via a public Git repository) that implements their analytic results, allowing users to compute Wilson coefficients and decay rates for arbitrary input parameters.

4. Key Results and Formulas

The paper presents explicit analytic expressions for the electron EDM ($d_e$) and the dipole coefficients ($c_{L/R}$) for $\mu \to e\gamma$.

• Decomposition: The results are split into one-loop and two-loop contributions.
  • One-loop: Dominated by neutral Higgs exchange.
  • Two-loop: Dominated by Barr-Zee type diagrams (fermion loops) and bosonic diagrams.
• Specific Terms:
  • Fermionic Barr-Zee: Contributions from top, bottom, charm, and tau loops with neutral ($h_k$) and charged ($H^\pm$) Higgs exchange.
  • Bosonic Contributions: Terms arising from the Higgs kinetic terms ($d^{kin.}_e$) and the Higgs potential ($d^{H^\pm \gamma}_e$, etc.).
  • Loop Functions: The authors provide explicit forms for complex loop functions ($f_1$ through $f_{11}$) involving dilogarithms ($Li_2$) and logarithmic terms, valid for arbitrary mass ratios.
• Limiting Cases: The results successfully reproduce known limits:
  • Real, diagonal Yukawa couplings match results from Ref. [10].
  • Small fermion mass limits match Ref. [44] for $\mu \to e\gamma$.
  • The one-loop results for $\mu \to e\gamma$ are consistent with existing literature.

5. Significance

• Completeness: This is the most comprehensive calculation of electron EDM and LFV decays in the 2HDM to date. It removes the ambiguity of "missing" contributions that could artificially suppress or enhance predictions.
• Phenomenological Impact: By providing the full set of contributions, the paper enables a rigorous global fit of the 2HDM parameter space against experimental data (ACME II for EDM, MEG for $\mu \to e\gamma$). It clarifies how cancellations between different diagrams can allow large CP phases to exist without violating current bounds, or conversely, how specific regions are excluded.
• Methodological Benchmark: The use of the HV scheme for $\gamma_5$ in Barr-Zee diagrams and the verification of gauge independence in the background field method sets a new standard for precision calculations in BSM physics.
• Tool for Future Research: The provided Python code allows the community to immediately apply these results to constrain 2HDM models, study baryogenesis scenarios, and interpret future experimental data from EDM and flavor factories.

In summary, this paper resolves the theoretical incompleteness of previous 2HDM predictions for EDMs and LFV decays, providing the necessary analytic tools to fully test the viability of the 2HDM as a solution to the baryogenesis problem.