Models for the Electric Dipole Moment and Anomalous Magnetic Moment of the Tau Lepton

This paper investigates two benchmark models featuring radiative tau mass generation that predict sizable anomalous magnetic moments and electric dipole moments for the tau lepton, with one model yielding particularly large EDM signals within the reach of future Belle II measurements.

The Gist

Imagine the Tau lepton as a heavy, short-lived cousin of the electron and the muon. In the world of particle physics, these particles are like tiny spinning tops. Usually, they spin perfectly symmetrically. However, if they have a slight "lopsidedness" in their electric charge (an Electric Dipole Moment or EDM) or if their magnetic spin is slightly stronger than expected (an Anomalous Magnetic Moment or $g-2$), it's a huge red flag. It suggests that invisible, unknown forces are messing with them.

This paper is like a detective story where the authors build two different "crime scenes" (theoretical models) to explain how these Tau particles might be getting lopsided.

The Big Idea: The "Radiative" Mass

In the standard story of the universe (the Standard Model), particles get their mass from interacting with a field called the Higgs, kind of like wading through thick molasses. But the authors propose a different idea for the Tau: Radiative Mass Generation.

Imagine the Tau doesn't get its mass directly from the molasses. Instead, it gets its mass by borrowing energy from a loop of invisible, exotic particles that pop in and out of existence. It's like the Tau is a child who doesn't have an allowance, so it has to earn money by doing chores (interacting with these new particles) to buy its own mass.

Because this "earning process" happens in a loop, it naturally creates the lopsidedness (EDM and $g-2$) the authors are looking for.

The Two Suspects (The Models)

The authors test two different scenarios, depending on what kind of "exotic" particles are doing the chores.

1. The "Majorana Fermion" Model (The Ghostly Neutrino Suspect)

• The Cast: This model introduces neutral fermions (particles that are their own antiparticles, like ghosts) and charged scalars (heavy, charged cousins of the Higgs).
• The Result: This setup is very effective at creating a "lopsided" Tau.
  • It predicts a magnetic anomaly ($g-2$) that is about 100,000 times larger than the standard prediction.
  • It predicts an electric dipole moment (EDM) that is huge for particle physics standards ($10^{-19}$ e cm).
• The Catch: To make this work, the new particles need to be relatively light (around the mass of a proton or slightly heavier, roughly 100 GeV) and the interactions between them must be quite strong.

2. The "Real Scalar" Model (The Heavy Higgs Suspect)

• The Cast: This model swaps the roles. Now we have a charged fermion (a heavy, charged particle) and neutral scalars (heavy, neutral cousins of the Higgs).
• The Result:
  • It still predicts a large magnetic anomaly ($g-2$), similar to the first model.
  • However, the electric dipole moment (EDM) is much smaller—about 10 times smaller than in the first model.
• Why the difference? The authors explain that in this model, the new particles tend to have very similar masses (they are "degenerate"). It's like two runners on a track; if they run at the exact same speed, their effects cancel each other out, leaving a smaller net result.

The "Smoking Gun" Test

How can we tell which model is right? The authors point out a simple sign flip:

• In the Majorana model, the magnetic anomaly is positive.
• In the Real Scalar model, the magnetic anomaly is negative.

It's like checking if a coin landed on heads or tails. Future experiments will measure the Tau's magnetic spin to see which sign it has, effectively ruling out one of the suspects.

The Constraints (The Rules of the Game)

The authors didn't just dream up these models; they had to make sure they didn't break the known laws of physics. They checked their models against:

1. The Higgs Boson: The new particles interact with the Higgs. If they interact too much, the Higgs would decay into Tau particles too often, which we haven't seen yet. Their models stay just within the safe limits.
2. Old Experiments (LEP): Experiments from the 1990s set a minimum weight for new charged particles. The authors ensure their new particles are heavy enough to have escaped detection back then.
3. Symmetry: They checked that the new particles don't mess up the balance between electrons, muons, and Taus in a way that contradicts current data.

The Conclusion

The paper concludes that if we find a large electric dipole moment or a specific magnetic anomaly in the Tau lepton, it could be the first sign of these "radiative mass" models.

• If the EDM is large (around $10^{-19}$ e cm), it points strongly toward the Majorana Fermion model.
• If the EDM is smaller but the magnetic anomaly is still huge, it might point toward the Real Scalar model.

The authors are essentially saying: "We have built two blueprints for new physics that fit all the current rules. If the next generation of experiments (like Belle II) finds these specific signals, we will know exactly which blueprint describes our universe."

Note: The paper focuses entirely on theoretical particle physics and does not discuss any medical, clinical, or immediate technological applications.

Technical Summary

Technical Summary: Models for the Electric Dipole Moment and Anomalous Magnetic Moment of the Tau Lepton

Problem Statement
The electric dipole moment (EDM) and anomalous magnetic moment ($g-2$) of charged leptons serve as precision probes for physics beyond the Standard Model (SM). While the SM predicts extremely suppressed values for charged lepton EDMs (requiring CP violation) and precise values for $g-2$, current experimental limits on the tau lepton ($\tau$) remain orders of magnitude weaker than those for the electron and muon due to the $\tau$'s short lifetime. However, upcoming facilities like Belle II, BEPCII, and CEPC are projected to improve sensitivity to $|a_\tau| \sim 10^{-5}$ and $|d_\tau| \sim 10^{-19}$ e cm. The central problem addressed is identifying new physics models capable of generating sizable $\tau$ EDM and $g-2$ signals within these projected reach, specifically leveraging mechanisms where the $\tau$ mass is radiatively generated.

Methodology
The authors investigate a class of models where the tau lepton mass is generated radiatively via loop diagrams involving new exotic particles. This mechanism avoids the typical loop suppression of dipole operators because the loop factor is effectively absorbed into the expression for the radiatively generated mass.

The study focuses on two benchmark models distinguished by the hypercharge assignment ($Y_\psi$) of a new fermion $\psi$:

1. Majorana Fermion (MF) Model ($Y_\psi = 0$): Contains neutral Majorana fermions ($\psi$) and charged scalars ($\phi, \eta$). The model includes a softly broken $Z_2$ symmetry to forbid tree-level Yukawa couplings and ensure the mass is radiative.
2. Real Scalar (RS) Model ($Y_\psi = -1$): Contains a charged fermion ($\psi$) and neutral scalars ($\phi, \eta$).

Key Methodological Steps:

• Lagrangian Construction: Defined Yukawa interactions ($y_\phi, y_\eta$) and scalar mixing terms ($a H \eta^\dagger \phi$) that generate the $\tau$ mass at the one-loop level.
• CP Violation: Identified physical CP-violating phases arising from complex mass parameters or couplings that cannot be removed by field redefinitions. These phases are crucial for generating non-zero EDMs.
• Form Factor Calculation: Computed the dipole form factors $F_2(0)$ (related to $a_\tau$) and $F_3(0)$ (related to $d_\tau$) using Passarino-Veltman loop integrals ($B_0, C_0, C_1, C_{00}$).
• Constraints: Applied rigorous experimental and theoretical constraints, including:
  • Higgs decay branching ratios ($h \to \tau^+\tau^-$) and invisible decays.
  • LEP bounds on charged scalar and fermion masses.
  • Lepton Flavor Universality (LFU) in $Z \to \tau^+\tau^-$ decays.
  • Electroweak precision observables (S, T, U parameters) and the $W$ boson mass.
  • Perturbative unitarity bounds on quartic couplings.

Key Contributions and Results
The paper presents numerical predictions for $a_\tau$ and $d_\tau$ within the viable parameter spaces of the MF and RS models, assuming exotic particle masses in the $\mathcal{O}(100)$ GeV range.

• MF Model Results:

  • Predicts a large anomalous magnetic moment $a_\tau \sim \mathcal{O}(10^{-5})$ with a positive sign.
  • Predicts an EDM $|d_\tau| \sim \mathcal{O}(10^{-19})$ e cm.
  • The viable parameter space is constrained primarily by the LEP bound on charged scalars and the requirement that the product of Yukawa couplings $y_\phi y_\eta < 4\pi$.
  • The predicted $|d_\tau|$ approaches the indirect limit derived from electron EDM constraints via three-loop light-by-light mechanisms.

• RS Model Results:

  • Predicts $|a_\tau| \sim \mathcal{O}(10^{-5})$ but with a negative sign, offering a potential discriminator between the two models.
  • Predicts a significantly smaller EDM, $|d_\tau|$ roughly one order of magnitude lower than the MF model. This suppression arises from the degeneracy of the exotic scalar masses in the loop functions.
  • The constraint on $y_\phi y_\eta$ is milder compared to the MF model.

• Phenomenological Distinctions:

  • The sign of $a_\tau$ is identified as a model-dependent signal: positive for MF and negative for RS.
  • Both models naturally accommodate a dark matter candidate (the lightest exotic particle) stabilized by an unbroken $Z_2$ symmetry, though detailed dark matter phenomenology is reserved for future work.

Significance and Claims
The authors claim that their models provide concrete benchmarks for new physics that can generate sizable $\tau$ EDM and $g-2$ signals.

• Testability: The predicted signals ($a_\tau \sim 10^{-5}$ and $d_\tau \sim 10^{-19}$ e cm) are within the reach of future updates to measurements at Belle II and other lepton facilities.
• Model Discrimination: The distinct signs of $a_\tau$ and the relative magnitudes of $d_\tau$ between the MF and RS models offer a pathway to distinguish between different hypercharge assignments and particle content in future data.
• Radiative Mass Mechanism: The study reinforces the viability of radiative mass generation as a mechanism to enhance dipole moments without violating perturbativity, provided the new physics scale is near the electroweak scale.

The paper concludes that precise measurements of the tau dipole moments offer a promising avenue for uncovering new sources of CP violation and probing physics beyond the Standard Model, specifically testing the hypothesis of radiatively generated lepton masses.