Probing $\tau$ lepton dipole moments at future Lepton Colliders

This study demonstrates that future lepton colliders, specifically the Future Circular Collider and a multi-TeV muon collider, can significantly improve constraints on $\tau$ lepton dipole moments by several orders of magnitude through various production and decay channels, thereby offering a powerful probe for new physics beyond the Standard Model.

The Gist

The Hunt for the Tau's "Secret Superpowers"

Imagine the universe is a giant, high-speed race track. For decades, physicists have been watching the fastest runners: the electron and the muon. They have studied these particles so closely that they know their every move, down to the tiniest wobble. These wobbles, called "dipole moments," are like the particles' unique fingerprints. If a fingerprint doesn't match the official ID card (the Standard Model of physics), it means there's a secret agent (New Physics) hiding in the crowd.

But there's a third runner on the track: the tau lepton. The tau is the heavyweight champion—it's heavy, but it's also incredibly short-lived. It exists for only a fraction of a second before vanishing. Because it dies so fast, we can't catch it in a trap to measure its wobbles directly like we do with the electron. For a long time, the tau's "fingerprint" has been blurry and poorly understood.

This paper is a proposal for how to finally get a crystal-clear look at the tau's secrets using two futuristic, super-powered racetracks: the FCC-ee (a giant electron-positron collider) and a Muon Collider (a machine that smashes muons together at mind-boggling speeds).

Here is how they plan to do it, using some everyday analogies:

1. The Problem: The Tau is Too Fast to Catch

Think of the tau lepton like a ghost that flashes by in a dark room. You can't grab it to measure its "magnetic personality" (magnetic dipole moment) or its "electric personality" (electric dipole moment). Instead, physicists have to guess its properties by watching how it behaves when it's created in a collision. It's like trying to figure out the weight of a hummingbird by watching how the wind blows when it flies past, rather than putting it on a scale.

2. The Plan: Two Different Kinds of Microscopes

The authors suggest using two different "microscopes" to zoom in on the tau, because each machine sees the world differently.

Microscope A: The FCC-ee (The Precision Photographer)

• What it is: A massive ring where electrons and positrons smash together. It's designed to be incredibly clean and precise, like a high-end studio camera.
• The Strategy: It will produce billions of taus. Because the environment is so clean, it can spot tiny, subtle deviations in how taus are created.
• The Analogy: Imagine trying to hear a whisper in a library. The FCC-ee is the library. It's so quiet that even the faintest whisper (a tiny deviation in the tau's behavior) can be heard. This machine is best at measuring the tau's magnetic quirks because it can count so many events with such high precision.

Microscope B: The Muon Collider (The Heavyweight Hammer)

• What it is: A machine that smashes muons together at energies far higher than anything we have today. It's like a sledgehammer compared to the FCC-ee's scalpel.
• The Strategy: Instead of looking for tiny whispers, it looks for big, violent crashes. At these high energies, the laws of physics change slightly if "New Physics" exists.
• The Analogy: Imagine trying to find a hidden flaw in a piece of glass. The FCC-ee looks at the glass under a microscope. The Muon Collider hits the glass with a hammer. If there's a hidden crack (New Physics), the glass will shatter in a specific, weird way that only happens at high speeds. This machine is best at measuring the tau's electric quirks because the high energy amplifies those specific signals.

3. The Special Moves: How They Catch the Ghost

The paper outlines four specific "moves" (processes) the colliders will use to catch the tau:

• The Direct Smash ($\ell^+\ell^- \to \tau^+\tau^-$): Smashing particles together to create tau pairs directly. This is the standard way to study them.
• The Photon Flash ($\gamma\gamma \to \tau^+\tau^-$): Using the light (photons) emitted by the colliding particles to create taus. It's like using a camera flash to illuminate a dark corner.
• The Higgs Party ($\mu^+\mu^- \to \tau^+\tau^-H$): Creating a Higgs boson (the "God particle") along with the taus. The Higgs acts like a spotlight. If the tau has a secret superpower, it will interact with the Higgs in a weird way, making the party look different than expected.
• The Vector-Boson Scatter: A complex interaction where particles bounce off each other like billiard balls, exchanging force carriers. This is a very sensitive test for new forces.

4. The Big Reveal: Why This Matters

The authors ran the numbers (simulations) and found something exciting:

• The FCC-ee could improve our knowledge of the tau's magnetic personality by 10 to 100 times better than we know it today.
• The Muon Collider could improve our knowledge of the tau's electric personality by 100 times (two orders of magnitude) better than the FCC-ee!

The Takeaway:
Currently, the tau lepton is the "wild card" of the particle world. We know the electron and muon are playing by the rules, but the tau might be breaking them. If the tau is found to have a "dipole moment" that doesn't match the Standard Model, it would be a smoking gun for New Physics. It could explain why the universe has more matter than antimatter, or reveal hidden dimensions.

By building these two future colliders, we aren't just looking for a slightly different number; we are building a better camera and a stronger hammer to finally see the tau lepton clearly. If the tau is hiding a secret, these machines will be the ones to expose it.

Technical Summary

1. Problem Statement

The electric and magnetic dipole moments of the electron and muon are among the most stringent tests of the Standard Model (SM) and sensitive probes for New Physics (NP). However, the corresponding dipole moments of the $\tau$ lepton ($a_\tau$ and $d_\tau$) remain poorly constrained.

• Current Limitations: Direct measurement is impossible due to the $\tau$'s short lifetime. Current constraints rely on indirect measurements at LEP, the LHC, and Belle II.
  • LEP-II limits: $-0.052 < a_\tau < 0.013$ and $|d_\tau| < 3.7 \times 10^{-16} \text{ e cm}$.
  • LHC (ATLAS/CMS) limits: $a_\tau \in [-2.4, 4.7] \times 10^{-3}$ and $|d_\tau| < 2.0 \times 10^{-17} \text{ e cm}$.
  • Belle II projections: Targeting $|a_\tau| \lesssim 10^{-5}$ and $|d_\tau| \lesssim 10^{-19} \text{ e cm}$, but these rely on specific asymmetry assumptions.
• The Gap: There is a need to explore the potential of future high-energy lepton colliders to improve these bounds by several orders of magnitude, particularly in scenarios involving enhanced couplings to third-generation fermions.

2. Methodology

The authors employ an Effective Field Theory (EFT) approach within the Standard Model Effective Field Theory (SMEFT) framework to describe NP effects at a scale $\Lambda$ well above the electroweak scale.

• Theoretical Framework:

  • The analysis focuses on dimension-6 dipole operators involving the $\tau$ lepton, Higgs field ($H$), and gauge bosons ($B_{\mu\nu}, W^I_{\mu\nu}$).
  • The relevant Lagrangian terms generate deviations in the anomalous magnetic moment ($\Delta a_\tau$) and electric dipole moment ($d_\tau$), as well as their weak counterparts ($\Delta a^Z_\tau, d^Z_\tau$).
  • Perturbative unitarity bounds are applied to ensure the validity of the EFT expansion at high energies.

• Facilities Analyzed:

  1. FCC-ee (Future Circular Collider - electron-positron): Operating at the Z-pole (Tera-Z), WW threshold, Higgs factory (240 GeV), and top factory (365 GeV).
  2. Multi-TeV Muon Collider ($\mu$C): Operating at $\sqrt{s} = 3, 6, 10, 14$ TeV with high luminosity scaling ($L \propto s$).

• Processes Investigated:

  • Direct Pair Production: $e^+e^-/\mu^+\mu^- \to \tau^+\tau^-$.
  • Photon-Photon Collisions: $\gamma\gamma \to \tau^+\tau^-$ (via equivalent photon approximation).
  • Associated Higgs Production: $\ell^+\ell^- \to \tau^+\tau^-H$ (sensitive to the $H\tau\tau\gamma^*$ vertex).
  • Radiative Higgs Decays: $H \to \tau^+\tau^-\gamma$.
  • Vector Boson Scattering (VBS): $\ell^+\ell^- \to \ell^+\ell^-\tau^+\tau^-$ and $\mu^+\mu^- \to \bar{\nu}\nu\tau^+\tau^-$.

• Simulation Details:

  • Numerical analyses include Initial State Radiation (ISR) and kinematic cuts (e.g., invariant mass, angular separation, transverse momentum).
  • Detector acceptance and reconstruction efficiencies (e.g., 80% for $\tau$ and $b$-tagging) are incorporated.
  • Backgrounds from SM processes (e.g., $Z \to \tau\tau$, $ZH$) are suppressed via specific selection cuts (Z-veto, Higgs mass windows).

3. Key Contributions

• Comprehensive Channel Analysis: The paper systematically evaluates multiple production and decay channels across two distinct future collider paradigms (high-luminosity low-energy $e^+e^-$ vs. high-energy $\mu^+\mu^-$).
• EFT Validity and Unitarity: The authors carefully analyze the energy dependence of linear vs. quadratic NP contributions. They demonstrate that while linear terms dominate at low energies (FCC-ee Z-pole), quadratic terms dominate at high energies ($\mu$C), and they identify the energy scales where EFT validity breaks down (perturbative unitarity violation).
• Complementarity: The study highlights how FCC-ee and the Muon Collider offer complementary sensitivities: FCC-ee excels in precision at the Z-pole and via photon-photon fusion, while the Muon Collider leverages high-energy growth in associated production and VBS.

4. Key Results

A. FCC-ee Sensitivity

• Dominant Channels:
  • $\gamma\gamma \to \tau^+\tau^-$: Provides the most stringent bound on the anomalous magnetic moment due to high luminosity at the Z-pole.
    • Projected sensitivity: $|\Delta a_\tau| \lesssim 3.0 \times 10^{-5}$.
  • $e^+e^- \to \tau^+\tau^-$: Provides the strongest constraints on the electric dipole moment and weak dipole moments.
    • Projected sensitivity: $|d_\tau| \lesssim 2.5 \times 10^{-18} \text{ e cm}$, $|\Delta a^Z_\tau| \lesssim 1.0 \times 10^{-5}$.
• Higgs Channels: The radiative decay $H \to \tau^+\tau^-\gamma$ is a unique probe, though less constraining than direct production, it offers a clean environment to improve upon current LHC bounds.

B. Multi-TeV Muon Collider Sensitivity

• Energy Scaling: The reach for $\Delta a_\tau$ and $d_\tau$ improves significantly with energy ($\propto 1/\sqrt{s}$ or $1/s$ depending on the channel) due to the $L \propto s$ scaling and the energy growth of NP effects in quadratic terms.
• Dominant Channels:
  • Associated Production ($\mu^+\mu^- \to \tau^+\tau^-H$): This is the most sensitive probe at the $\mu$C. Unlike pair production, it does not saturate unitarity bounds as quickly, allowing for indefinite improvement with energy.
    • At $\sqrt{s} = 14$ TeV, the sensitivity to $\Delta a_\tau$ improves by one order of magnitude compared to FCC-ee.
  • Electric Dipole Moment ($d_\tau$): The $\mu$C provides constraints on $d_\tau$ that are more than two orders of magnitude stronger than FCC-ee at 14 TeV, driven by quadratic NP contributions.
• VBS Channels: While sensitive, VBS channels are challenging due to large QCD dijet backgrounds, limiting their reach compared to associated production.

C. Comparative Summary

• FCC-ee: Best for $a_\tau$ (via $\gamma\gamma$) and $d_\tau$ (via $e^+e^-$) at the precision frontier.
• Muon Collider: Best for $a_\tau$ and $d_\tau$ at the energy frontier ($\sqrt{s} > 10$ TeV), particularly for $d_\tau$ where it surpasses FCC-ee by $>100\times$.

5. Significance

• Order of Magnitude Improvements: Future lepton colliders can improve current experimental bounds on $\tau$ dipole moments by several orders of magnitude (e.g., from $10^{-17}$ to $10^{-19}$ e cm for $d_\tau$).
• Probing New Physics: These sensitivities allow for the exploration of NP scales up to $\Lambda \sim 10-30$ TeV, particularly in models with enhanced couplings to the third generation (e.g., Composite Higgs, U(2) flavor symmetries).
• Strategic Complementarity: The results underscore the necessity of both a high-luminosity Higgs/Z factory (FCC-ee) and a high-energy Muon Collider to fully map the parameter space of $\tau$ dipole moments. The FCC-ee probes the "linear" regime with high statistics, while the Muon Collider probes the "quadratic" regime where NP effects grow with energy.
• Motivation for Systematics: The paper motivates dedicated studies of systematic uncertainties (detector effects, background modeling) which were treated conservatively in this study, to fully realize the potential of these facilities.

In conclusion, the paper establishes that the next generation of lepton colliders will transform the $\tau$ lepton from a poorly constrained particle into a precision probe for New Physics, potentially revealing deviations from the Standard Model that are invisible to current facilities.