⚛️ phenomenology

Four-loop electron EDM matching and timelike $\tau$ electric dipole form factor

This paper computes the four-loop matching of a CP-odd four- $\tau$ interaction to the electron electric dipole moment and the leading timelike $\tau$ dipole form factor, demonstrating that while exact cancellation of the static electron EDM is possible, the resulting timelike $\tau$ signal provides a unique, complementary probe for constraining such interactions at colliders.

Original authors: Chia-Wei Liu, Zi-Yue Zou

Published 2026-10-21

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Chia-Wei Liu, Zi-Yue Zou

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 universe is built on a set of fundamental rules that govern how particles interact, but hidden within those rules is a subtle asymmetry known as CP violation. This phenomenon is a tiny crack in the mirror of nature, where the laws of physics behave slightly differently if time were reversed or if particles were swapped with their antimatter counterparts. Physicists hunt for this asymmetry because it is one of the few clues we have to explain why the universe is made of matter rather than being an empty void of equal parts matter and antimatter. One of the most sensitive tools for detecting this hidden imbalance is the electric dipole moment, a property that would cause a particle like an electron to act like a tiny bar magnet with a positive and negative end, even though it has no electric charge. If such a moment exists, it signals a profound violation of symmetry. While scientists have measured the electron with incredible precision, finding no such moment yet, the heavier cousin of the electron, the tau particle, remains largely unexplored. The tau is unstable and short-lived, making it difficult to study, but its behavior could hold the key to new physics that the electron cannot reveal.

In a recent study, researchers performed a complex calculation to connect the behavior of these heavy tau particles to the electron, exploring a scenario where a new, unseen force interacts with tau particles in a way that violates symmetry. They focused on a specific type of interaction where four tau particles meet at a single point, a process that cannot happen directly in nature but can be modeled mathematically to understand what might happen if such a force existed. The team calculated how this interaction would ripple through the quantum world, eventually influencing the electron's electric dipole moment. They also looked at how this same interaction would affect the tau particle itself when it is created in high-energy collisions, specifically examining a property called the electric dipole form factor. This form factor describes how the tau responds to electromagnetic forces when it is moving at high speeds, and unlike the static electron, the tau's response can have both a real part and an imaginary part, the latter representing a specific kind of quantum absorption that occurs when the particle briefly splits into other forms.

The researchers discovered that even if the electron's electric dipole moment were perfectly zero—meaning the effects of this new force cancel out exactly for the electron—the tau particle would still show a clear signal. This is because the interaction creates a unique signature in the tau's behavior that cannot be erased by the same cancellation that hides the effect in the electron. Specifically, the tau particle would exhibit a distinct "absorptive" signal, a quantum effect that arises from the temporary creation of particle pairs, which a simple, static magnetic-like property cannot mimic. This finding is crucial because it means that experiments searching for the electron's dipole moment and experiments colliding particles to create tau pairs are not just looking at the same thing in different ways; they are looking at two different sides of the same coin. If the electron shows no sign of this new physics, it does not mean the physics doesn't exist; it simply means the effects are arranged in a way that hides from the electron but leaves a fingerprint on the tau.

By calculating the relationship between these two particles up to a very high level of precision, the team showed that the tau particle's signal is directly tied to the strength of the four-particle interaction, independent of the electron's result. This allows scientists to use data from particle colliders, which measure the real and imaginary parts of the tau's form factor, to constrain the possible ways this new force could be hiding. The study confirms that the imaginary part of the tau's signal, which comes from the quantum absorption of energy, cannot be faked or canceled out by a simple static property. This provides a powerful new way to test theories of new physics: if the electron remains silent, the tau particle can still speak, revealing the presence of new forces through its unique, time-dependent behavior. The work bridges the gap between the quiet, static world of electron measurements and the dynamic, high-energy world of particle collisions, showing that the search for the universe's deepest secrets requires listening to both the silence and the noise.

Technical Summary: Four-loop electron EDM matching and timelike $\tau$ electric dipole form factor

Problem Statement
Electric dipole moments (EDMs) serve as precision probes for CP violation in lepton interactions. While static electron EDM searches constrain heavy-lepton sources via radiative effects, direct measurements of the $\tau$ electric dipole form factor in $e^+e^- \to \tau^+\tau^-$ collisions provide access to both real and imaginary (absorptive) parts of the form factor. A consistent interpretation of these signals requires a unified framework connecting the static electron EDM to the timelike $\tau$ dipole form factor, particularly when new physics involves a CP-violating scalar–pseudoscalar mediator.

In scenarios where a mediator is heavier than the collision energy but lighter than the electroweak scale, its effects are described by a low-energy effective field theory (EFT). This EFT contains a CP-odd four- $\tau$ operator (dimension eight) and a $\tau$ dipole operator (dimension six). While the four- $\tau$ operator is generated at tree level by the mediator, the $\tau$ dipole arises only at one loop. Consequently, the four- $\tau$ interaction induces a $\tau$ -pair cut with a non-trivial momentum dependence and an absorptive part that a real, local $\tau$ dipole cannot reproduce. To fully constrain such models, one must compute the four-loop matching of the four- $\tau$ interaction onto the electron EDM, alongside the leading timelike $\tau$ form factor, ensuring that contributions from both the local dipole and the four- $\tau$ source are treated consistently at the same radiative order.

Methodology
The authors perform calculations within low-energy quantum electrodynamics (QED) using pole masses for the electron ( $m_e$ ) and $\tau$ ( $m_\tau$ ) and the on-shell (OS) charge $e$ . The analysis adopts a specific radiative counting scheme where the local $\tau$ dipole $d_\tau$ is of order $O(e m_\tau C_{SP} / 8\pi^2)$ , ensuring that the local dipole and the four- $\tau$ loop contribute at the same source order.

Renormalization Group Evolution: The authors derive the one-loop renormalization group equations for the Wilson coefficients $d_\tau(\mu)$ and $C_{SP}(\mu)$ . They solve these equations to express the coefficients at a reference scale $\mu_0 = m_\tau$ , neglecting higher-order source terms.
Timelike Form Factor Calculation: The $\tau$ electromagnetic vertex is decomposed into form factors. The authors calculate the contribution of the renormalized four- $\tau$ loop to the electric dipole form factor $F_3(s)$ for $\gamma^*(q) \to \tau^-(p_-) + \tau^+(p_+)$ with $s > 4m_\tau^2$ . This involves evaluating the bubble diagram and determining the absorptive part arising from the $\tau$ -pair cut.
Four-loop Electron EDM Matching: The core calculation involves the four-loop matching of the CP-odd four- $\tau$ $τ$ interaction onto the electron EDM. This is performed using dimensional regularization ( $D = 4 - 2\epsilon$ $D = 4 - 2 ϵ$ ) and the Breitenlohner–Maison–'t Hooft–Veltman (BMHV) prescription for $\gamma_5$ $γ_{5}$ .
- Evanescent Operators: To handle the $\gamma_5$ algebra and operator mixing correctly, the authors introduce evanescent operators ( $E_1, E_2, E_3$ ) and determine their finite parts. These operators are defined such that their renormalized matrix elements do not contribute to the electron EDM coefficient $d_e$ .
- Diagrammatic Sources: The calculation includes all relevant four-loop source topologies, specifically external- and internal-photon partitions (1+3 and 2+2) and mixing counterterms.
- Numerical Evaluation: The finite, dimensionless integrals ( $I_{13}, I_{22}$ ) are evaluated numerically using convergent parameter integrals. The calculation utilizes tools such as FeynCalc for Dirac algebra, LiteRed2 for integration-by-parts (IBP) reduction, and custom Julia codes for numerical integration. The results were cross-checked using FIRE6, pySecDec, and high-precision Python/mpmath integration.

Key Contributions and Results

Four-loop Matching Coefficient ( $B_{SP}$ ): The authors compute the leading four-loop contribution of the four- $\tau$ interaction to the electron EDM. The result is expressed as a coefficient $B_{SP}$ in the expression for the induced electron EDM $\delta d_e$ . The value is determined to be:
$B_{SP} = I_{13} + I_{22} - \frac{19775}{41472} + \frac{\pi^4}{960} - \frac{11}{96}\zeta(3) \approx 6.104$
This calculation reproduces the known three-loop transfer coefficient for the local dipole ( $19\zeta(3)/4 - 71/24$ ) and extends the result to include the four-loop four- $\tau$ source.
Timelike $\tau$ Form Factor: The paper provides the explicit expression for the timelike $\tau$ electric dipole form factor $F_3(s)$ , which includes a term proportional to $C_{SP}(m_\tau)$ containing a logarithmic dependence and an imaginary part ( $i\pi\beta$ ). This imaginary part represents the absorptive signal from the $\tau$ -pair cut.
Scale Independence: Both the four-loop matching result and the leading timelike form factor are shown to be independent of the subtraction scale $\mu$ at this order, provided the coefficients are evaluated at $\mu = m_\tau$ .
Phenomenological Constraints: The authors analyze the constraints on the real source coefficients $d_\tau(m_\tau)$ and $C_{SP}(m_\tau)$ in the plane of these parameters.
- The electron EDM bound (e.g., from HfF+) constrains a linear combination of these coefficients, effectively selecting a narrow band around the cancellation line where $\delta d_e = 0$ .
- However, this cancellation does not eliminate the timelike $\tau$ signal. The absorptive part of the $\tau$ form factor is proportional to $C_{SP}$ and cannot be canceled by a real local dipole.
- Consequently, while the electron EDM search allows for a specific direction of cancellation between the two sources, the timelike $\tau$ measurements (specifically the absorptive part) constrain this otherwise open direction.

Significance
The paper establishes a consistent EFT framework linking static electron EDM searches and timelike $\tau$ dipole measurements in the presence of CP-violating scalar mediators. The primary significance lies in demonstrating that exact cancellation of the electron EDM does not imply the absence of new physics signals in $\tau$ pair production. Specifically, the momentum-dependent and absorptive components of the timelike $\tau$ form factor, generated by the four- $\tau$ operator, remain observable even when the static electron EDM vanishes. This allows collider experiments (such as Belle II and STCF) to complement static EDM searches by constraining the "cancellation direction" in the parameter space of CP-violating sources, thereby providing a more complete picture of potential CP violation in the third generation of leptons.

More like this