⚛️ phenomenology

Full Three-Loop Electroweak Multiplet Contributions to the Electron Electric Dipole Moment

This paper presents a full three-loop calculation of the electron electric dipole moment induced by CP-violating Yukawa interactions of SU(2) $_L$ multiplets, revealing that the total contribution is three times larger than the previously estimated value derived solely from the electroweak-Weinberg operator.

Original authors: Tatsuya Banno, Junji Hisano, Teppei Kitahara, Kiyoto Ogawa, Naohiro Osamura

Published 2026-03-02

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CC BY 4.0

Original authors: Tatsuya Banno, Junji Hisano, Teppei Kitahara, Kiyoto Ogawa, Naohiro Osamura

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: The Perfectly Round Ball vs. The Egg

Imagine an electron. In the Standard Model of physics (our current best rulebook for how the universe works), an electron is like a perfectly round, smooth marble. It has no "lopsidedness."

However, physicists are looking for a tiny imperfection: an Electric Dipole Moment (EDM). If an electron had an EDM, it would be like a marble that is slightly squashed into the shape of an egg. One side would be slightly positive, and the other slightly negative.

Why do we care? Because a perfectly round marble implies the universe treats "left" and "right" (and matter and antimatter) exactly the same. But we know the universe doesn't treat them the same (otherwise, we wouldn't exist!). Finding an "egg-shaped" electron is the smoking gun for New Physics—rules of the universe that we haven't discovered yet.

The Detective Work: Listening for a Whisper

For decades, scientists have been trying to measure this "egg shape."

The Old Detective: The ACME II experiment was like a very sensitive microphone, but it could only hear a shout.
The New Detective: The JILA experiment and the upcoming ACME III are like super-sensitive microphones that can hear a whisper. They are getting 30 times more sensitive.

The paper asks: "If there are new, heavy particles hiding in the universe, will our new super-microphones be able to hear them?"

The Suspects: The "Heavy Hitters"

The paper focuses on a specific type of hypothetical new particle. Imagine a secret society of heavy particles living at a high energy level (the "TeV scale").

They are like giant, invisible bouncers at a club.
They interact with each other in a very specific, "twisted" way (called CP-violation).
They don't talk directly to the electron (the VIP guest). Instead, they talk to the W-bosons (the bouncers' security guards), who then whisper to the electron.

Because they don't talk directly, the "whisper" is very faint. In physics terms, this whisper is generated at the three-loop level.

Analogy: Imagine a game of "Telephone."
- One-loop: You tell a friend, they tell you back. (Clear message).
- Two-loop: You tell a friend, they tell another, who tells you. (Slightly garbled).
- Three-loop: You tell a friend, who tells another, who tells a third, who tells you. (Very garbled, very hard to hear).

The Big Discovery: We Were Underestimating the Volume

In a previous study (by the same team), the scientists tried to predict how loud this whisper would be. They used a shortcut method (Effective Field Theory). They calculated the signal coming from one specific path: the "Electroweak-Weinberg operator."

Think of this like calculating the volume of a song by only listening to the drums. They found the drums were loud enough that the new microphones might hear them.

But here is the twist in this new paper:
The team decided to stop using the shortcut and calculate the entire song (the "Full Three-Loop" calculation). They listened to the drums, the bass, the guitar, and the vocals all at once.

The Result:
When they calculated the full picture, the signal was three times louder than they thought!

The Old Math: "The drums are loud enough to hear."
The New Math: "The drums are loud, but the whole band is playing, making it three times as loud."

Why Does This Matter?

The "Three Times" Factor: Because the signal is three times stronger, the new experiments (ACME III) have a much better chance of finding these heavy particles.
Dark Matter Connection: The paper suggests these heavy particles might be Dark Matter. If they are, they need to be heavy (around the size of a proton multiplied by 1,000).
The "Quintuplet" Case: The paper highlights a specific scenario where there are 5 types of these particles working together (a "quintuplet"). In this case, the signal is boosted even more because of how they are arranged (mathematically, it scales with the cube of the number of particles). This makes the "egg shape" of the electron much more likely to be found.

The Conclusion: The Net is Tightening

The authors conclude that:

We don't need to wait for a miracle; the math says the signal is strong enough to be detected soon.
If the next generation of experiments (ACME III) sees an "egg-shaped" electron, it won't just be a tiny hint; it will be a massive confirmation that these heavy, twisted particles exist.
The "shortcut" method they used before was okay, but it missed a huge chunk of the signal. By doing the hard, full calculation, they have shown that the universe is much more likely to reveal its secrets to us than we previously thought.

In short: The universe might be hiding a secret party of heavy particles. We thought the music was too quiet to hear, but this paper proves the band is actually playing at full volume. Our new microphones are ready to catch the beat.

1. Problem Statement

The electron electric dipole moment (EDM, $d_e$ ) is a highly sensitive probe for CP violation beyond the Standard Model (BSM). While current experimental bounds (e.g., JILA: $|d_e| < 4.1 \times 10^{-30} e \cdot \text{cm}$ ) are stringent, next-generation experiments (ACME III) aim to improve sensitivity by an order of magnitude, reaching $\mathcal{O}(10^{-31}) e \cdot \text{cm}$ .

The paper addresses a specific BSM scenario involving additional $SU(2)_L$ multiplets (fermions $\psi_{A/B}$ and a complex scalar $S$ ) with CP-violating Yukawa interactions.

The Gap: Previous work (Ref. [15]) calculated the electron EDM contribution mediated by the electroweak-Weinberg operator (a CP-odd trilinear $W$ -boson coupling) generated at the two-loop level. This contribution was matched to the electron EDM at the one-loop level.
The Issue: The matching between the electroweak-Weinberg operator and the electron EDM does not receive logarithmic enhancement from renormalization group (RG) evolution because the associated anomalous dimension is zero. Consequently, the contribution from the CP-odd electron dipole operator (generated directly at the three-loop level) is of the same order as the Weinberg operator contribution.
Objective: To perform a full three-loop calculation of the electron EDM in this model to determine the total contribution, comparing it against the partial calculation (Weinberg operator only) to assess the true reach of future experiments.

2. Methodology

The authors employ a combination of Effective Field Theory (EFT) and full theory calculations:

Model Setup:
- Introduces two fermion multiplets ( $\psi_A, \psi_B$ ) and a complex scalar $S$ with CP-violating Yukawa couplings: $g_{\bar{B}AS} = X_{\bar{B}AS}(s + \gamma_5 a)$ and $g_{\bar{A}B\bar{S}} = X_{\bar{A}B\bar{S}}(s^* - \gamma_5 a^*)$ .
- Focuses on the simplest case where the representation is $(A, B, S) = (r, r, 1)$ , meaning the fermions are in an $SU(2)_L$ representation of dimension $r$ , and $S$ is a singlet.
- The CP-violating phase is encoded in the rephasing-invariant quantity $\text{Im}(s a^*) m_A m_B$ .
Calculation Strategy:
1. Review of EFT Approach: Re-evaluates the contribution via the electroweak-Weinberg operator ( $\mathcal{L}_W \propto \epsilon^{abc} W^a W^b \tilde{W}^c$ ). The Wilson coefficient $C_W$ is generated at two loops, and the matching to $d_e$ occurs at one loop.
2. Full Three-Loop Calculation: Directly calculates the electron EDM in the full theory without integrating out the heavy multiplets first.
  - Identifies 12 distinct Feynman diagrams contributing to the process (involving $W$ bosons, photons, and the heavy multiplets).
  - Confirms that diagrams involving $ZZ$ or $ZZ\gamma$ vertices vanish due to symmetry cancellations (clockwise vs. anticlockwise loops).
3. Loop Integration:
  - Uses Integration-by-Parts (IBP) reduction (via the code Kira and Fermat) to reduce complex three-loop integrals to a set of master integrals.
  - Defines master integrals $J[n_1, \dots, n_6]$ and evaluates them analytically for degenerate mass cases ( $m_A = m_B = m_S$ ) and numerically for non-degenerate cases.
  - Ensures UV and IR finiteness of the final result.

3. Key Contributions

First Full Three-Loop Calculation: This is the first complete calculation of the electron EDM induced by CP-violating Yukawa interactions of $SU(2)_L$ multiplets at the full three-loop level.
Quantitative Correction Factor: The paper establishes that the full three-loop result is exactly three times larger than the contribution calculated solely through the electroweak-Weinberg operator (in the limit where Next-to-Leading Order terms are neglected).
- $d_e^{\text{Full}} \approx 3 \times d_e^{\text{Weinberg}}$ .
Analytic Results: Provides analytic expressions for the loop functions $B_0$ and $B_1$ in the degenerate mass limit and discusses the behavior in hierarchical mass limits ( $m_A \ll m_S$ or $m_S \ll m_A$ ).
Group Theory Factor: Explicitly derives the dependence on the $SU(2)_L$ representation dimension $r$ , showing the result scales as $r(r^2-1)/12$ .

4. Results

Magnitude Enhancement:
- In the degenerate mass case ( $m_A = m_B = m_S = m$ ), the full EDM is:
  $d_e^{\text{Full}} \approx \frac{\alpha_2^2}{(16\pi^2)^2} \frac{r(r^2-1)}{12} \frac{\text{Im}(sa^*)}{m^2} \times 0.41$
- The Weinberg-only contribution was previously found to be proportional to $0.14/m^2$ . The ratio is $0.41/0.14 \approx 3$ .
Representation Dependence: The EDM scales significantly with the representation dimension $r$ . For the fermionic quintuplet ( $r=5$ ), which is motivated by Minimal Dark Matter models, the enhancement is substantial ( $5(24)/12 = 10$ ).
Experimental Reach:
- Current Bounds: For $r=5$ and $\text{Im}(sa^*) = 0.25$ , masses below $\sim 350$ GeV are already excluded by current JILA bounds.
- Future Projections: With the anticipated sensitivity of ACME III ( $\sim 10^{-31} e \cdot \text{cm}$ ), the model can probe $SU(2)_L$ multiplet masses up to the TeV scale (specifically up to $\sim 1-2$ TeV for $r=5$ ).
- The full calculation shifts the exclusion limits and discovery potential significantly compared to the partial EFT calculation.

5. Significance

Theoretical Precision: The paper demonstrates that neglecting the direct three-loop contribution from the CP-odd dipole operator leads to a significant underestimation (factor of 3) of the electron EDM in this class of BSM models. This highlights the necessity of full theory calculations even when EFT matching seems sufficient.
Dark Matter Connection: The results strengthen the link between Dark Matter candidates (specifically Minimal Dark Matter models with high $SU(2)_L$ representations) and low-energy precision experiments. The enhanced EDM signal makes these heavy multiplets testable via EDM experiments even if they are too heavy to be produced directly at the LHC.
Future Physics: The findings suggest that future EDM experiments will be powerful tools for probing TeV-scale new physics that generates EDMs only at the three-loop level, a regime previously thought to be less accessible.

In summary, Banno et al. provide a rigorous correction to the predicted electron EDM in models with heavy $SU(2)_L$ multiplets, revealing a factor-of-three enhancement that significantly improves the prospects for discovering such new physics in upcoming experiments.