Light new physics and the $\tau$ lepton dipole moments: prospects at Belle II

This paper demonstrates that measurements of asymmetries in $e^+e^- \to \tau^+\tau^-$ at Belle II, particularly those arising from the imaginary parts of light new physics contributions even without electron polarization, can be interpreted as model-dependent constraints on the $\tau$ lepton's dipole moments, offering a novel avenue for new physics searches using existing data.

The Gist

Imagine the universe as a giant, complex machine built according to a specific instruction manual called the Standard Model. Physicists have been checking this manual for decades, looking for typos or missing pages that might hint at a "New Physics" (NP) operating behind the scenes.

One of the best ways to find these hidden instructions is by looking at how particles spin and wobble. This "wobble" is called a dipole moment. Think of it like a tiny bar magnet inside a particle. If the magnet is stronger or weaker than the manual predicts, it means there's a secret force or particle messing with it.

The Problem: The "Ghost" Particle

Scientists have already measured these wobbles for the electron and the muon (a heavier cousin of the electron) with incredible precision. They found some strange hints that the manual might be wrong.

However, there is a third, even heavier cousin called the tau ($\tau$) lepton. It's like a super-heavy, super-fast version of the electron. The problem? The tau is so unstable that it dies almost instantly after being created. It's like trying to measure the weight of a firework while it's exploding; you barely have time to look at it before it's gone. Because of this, measuring the tau's "wobble" is notoriously difficult, and we haven't been able to check the manual for this particle as well as the others.

The Proposed Solution: The "Asymmetry" Trick

The paper suggests a clever way to catch the tau's wobble at the Belle II experiment in Japan. Instead of trying to weigh the firework directly, they propose watching how the fireworks fly apart when two beams of particles smash together.

Specifically, they look at a process where an electron and a positron (anti-electron) collide to create a pair of taus. By measuring the angles at which these taus fly out, scientists can spot an asymmetry.

• The Analogy: Imagine spinning a top. If the top is perfectly balanced, it spins straight. If it's slightly unbalanced (has a "dipole moment"), it wobbles and leans to one side. The paper proposes that by watching which way the taus lean (the asymmetry), we can calculate how unbalanced they are.

Usually, to see this lean clearly, you need to spin the incoming electron beam like a gyroscope (polarization). The paper notes that if the "New Physics" is heavy (like a heavy boulder hidden in the machine), this method works perfectly and tells us exactly how the tau is wobbling.

The Twist: "Light" New Physics

Here is where the paper gets interesting. What if the "New Physics" isn't a heavy boulder, but a light, ghostly particle (like a light scalar or vector boson)?

If the new particle is light, it doesn't just sit there; it zips around inside the collision, creating a "loop" of activity.

1. The Imaginary Part: In the world of quantum mechanics, these light particles can create something called an "imaginary part" in the math.
2. The Analogy: Think of a heavy boulder (heavy NP) as a rock that just sits in the road, slowing traffic down (a real effect). A light ghost (light NP) is like a ghost that passes through the cars, making them phase in and out of reality. This "phasing" creates a new kind of signal that doesn't require the electron beam to be spinning (polarized) to be seen.

The Key Discovery: The authors show that even without the fancy spinning electron beam, we can still detect these light ghosts by looking at a specific type of asymmetry. The "ghosts" leave a unique fingerprint (an imaginary part) that we can measure right now with the data Belle II is already collecting.

The Results: How Heavy is the Ghost?

The team ran simulations to see how well this method works for different "weights" of these new particles:

• Heavy Particles: As the new particle gets heavier, the signal fades away, and we eventually just see the standard "wobble" predicted by the old manual. This is expected.
• Light Particles: If the new particle is light, the signal stays strong.
• The Spin Difference: They found that spin-0 particles (like axions) leave a lingering signal for much longer as they get heavier compared to spin-1 particles (like light vector bosons). It's as if the spin-0 ghosts are "stickier" and harder to ignore, even when they get a bit heavier.

The Bottom Line

This paper is a roadmap for how to use the Belle II collider to hunt for new, light particles that might be messing with the tau lepton.

• The Good News: We don't necessarily need to wait for a massive upgrade to the machine (like a polarized electron beam) to find these light particles. We can use the "imaginary" signals from the light ghosts that are already accessible with current data.
• The Goal: If we can measure these signals, we can finally put a number on the tau's "wobble" and see if it matches the Standard Model or if it reveals a hidden layer of the universe.

In short: The authors are saying, "We have a new way to look at the tau lepton's wobble. Even if the new physics is light and ghostly, we can catch it without needing the most expensive equipment upgrades, simply by looking at the angles of the particles flying apart."

Technical Summary

Technical Summary: Light New Physics and the $\tau$ Lepton Dipole Moments: Prospects at Belle II

Problem Statement
Lepton dipole moments (electric $d_\ell$ and magnetic $a_\ell$) are precision probes for physics beyond the Standard Model (SM). While electron and muon moments are measured with extreme precision, the $\tau$ lepton's short lifetime makes accessing its dipole moments experimentally challenging. Current techniques struggle to reach the $\sim 10^{-5}$ precision required to test realistic New Physics (NP) scenarios.

Existing proposals to measure $\tau$ dipole moments via asymmetries in $e^+e^- \to \tau^+\tau^-$ rely on an Effective Field Theory (EFT) argument. This framework assumes the NP scale ($m_{NP}$) is much larger than the center-of-mass energy ($\sqrt{s}$), allowing NP effects to be parameterized as local, real contributions to form factors $F_2$ and $F_3$. However, this interpretation fails if NP states are "light" ($m_{NP}^2 \lesssim s$), as they propagate at collider energies, introducing model-dependent momentum dependencies and potentially generating imaginary parts in the form factors. The paper addresses the gap in interpreting asymmetry measurements when light NP mediators (spin-0 and spin-1) are present.

Methodology
The authors analyze the $e^+e^- \to \tau^+\tau^-$ process at the Belle II collider ($\sqrt{s} \approx 10.58$ GeV) using a model-dependent approach for light NP.

1. Formalism: They parameterize the electromagnetic $\tau\tau\gamma$ vertex using form factors $F_1, F_2, F_3, F_A$. The dipole moments correspond to the zero-momentum limits of $F_2$ and $F_3$.
2. Asymmetries: They utilize specific asymmetries constructed from scattering angles, decay angles of $\tau \to h\nu$, and beam polarization.
   • Real Parts ($Re F_{2,3}$): Accessible via asymmetries ($A_T^\pm, A_L^\pm, A_{N,F3}^\pm$) requiring a polarized electron beam.
   • Imaginary Parts ($Im F_{2,3}$): Accessible via normal ($A_N^\pm$) and forward-backward ($A_{T,F3}^\pm$) asymmetries that do not require electron polarization, provided $s > (m_i + m_j)^2$.
3. NP Models: The study focuses on two classes of light mediators:
   • Spin-0 ($\phi$): Axions, axion-like particles (ALPs), and scalars, interacting via Yukawa-like couplings and dimension-5 operators ($\phi F_{\mu\nu}F^{\mu\nu}$).
   • Spin-1 ($X$): Light vector bosons, interacting via vector/axial-vector couplings and a Chern-Simons term.
4. Decoupling Analysis: The authors compute the contributions of these mediators to $F_{2,3}(s)$ and $a_\tau, d_\tau$. They specifically investigate the transition from the light NP regime to the heavy EFT limit as the mediator mass increases, checking for logarithmic dependencies and decoupling behaviors.

Key Contributions and Results

• Imaginary Parts without Polarization: A primary finding is that light NP generates non-vanishing imaginary parts in the form factors ($Im F_2, Im F_3$) above the $\tau^+\tau^-$ threshold. Crucially, these can be extracted via normal asymmetries ($A_N^\pm$) without a polarized electron beam. This allows for immediate NP searches using existing Belle II data, interpreting the results as constraints on the anomalous magnetic moment $a_\tau$ in a model-dependent manner.
• Decoupling Behavior:
  • Vector Mediators: As the mediator mass $M_X$ increases, the sensitivity to $a_\tau$ decouples rapidly, approaching the constant EFT limit.
  • Scalar Mediators: Scalar mediators exhibit a slower decoupling, displaying a residual logarithmic dependence on the mass ($\log M_\phi$). This behavior mirrors the Higgs boson contribution to $a_\tau$ in the SM. The authors derive explicit expressions showing that while the EFT limit is approached, the transition is governed by logarithmic corrections rather than a simple step function.
• Sensitivity Estimates: Assuming a future sensitivity of $10^{-6}$ on the effective form factors, the authors project constraints on $a_\tau$ and $d_\tau$.
  • For light mediators, the sensitivity remains competitive (typically below $10^{-5}$) across a wide mass range, only deteriorating significantly if accidental cancellations occur.
  • The sensitivity via $Im F_2$ (accessible without polarization) follows a similar decoupling pattern, remaining effective for scalar masses significantly higher than for vector masses.
• Threshold Enhancement: The paper notes that operating near the $\tau^+\tau^-$ threshold ($\sqrt{s} \approx 2m_\tau$) could enhance sensitivity by up to an order of magnitude. While this requires energy tuning or radiative return techniques (which reduce statistics), it represents a potential avenue for future improvements.

Significance and Claims
The paper claims to provide a necessary bridge between EFT interpretations and realistic light NP scenarios for $\tau$ dipole moment searches. Its significance lies in:

1. Reinterpreting Existing Data: Demonstrating that asymmetry measurements at Belle II can constrain light NP and $a_\tau$ even without the proposed electron beam polarization upgrade, specifically by utilizing the imaginary parts of form factors.
2. Model-Dependent Interpretation: Establishing that while EFT arguments fail for light mediators, the results can still be mapped to dipole moments if the specific momentum dependence of the loop is controlled.
3. Novel Search Channel: Highlighting the unique opportunity to probe light spin-0 and spin-1 mediators via normal asymmetries, a channel that is experimentally simpler (no polarization required) and theoretically distinct from the heavy NP case.

The authors conclude that a measurement of the normal asymmetry at Belle II would not only validate the methodology but also provide immediate physics insights into light NP, motivating such a measurement with current data. They suggest that similar programs could be pursued at other $e^+e^-$ machines like BESIII or a future Super Tau-Charm Factory.