Four-fermion operators, $Z$-boson exchange, and $\tau$ lepton dipole moments

This paper investigates how $Z$-boson exchange and four-fermion operators influence $\tau$ lepton dipole moment constraints derived from $e^+e^-\to\tau^+\tau^-$ asymmetry measurements, demonstrating that while these effects are small, they are crucial for precision studies and that specific loop-level observables could enable the determination of the anomalous magnetic moment $a_\tau$ even without a polarized electron beam.

The Gist

Imagine you are a detective trying to solve a mystery about a tiny, elusive particle called the tau lepton (let's call him "Tau"). Tau is a heavy cousin of the electron, and physicists want to know exactly how he spins and how he reacts to magnetic fields. These reactions are called his "dipole moments."

Think of Tau's magnetic personality like a tiny bar magnet. If he spins in a way that doesn't quite match the rules of the Standard Model (our current rulebook for physics), it's a sign that there are "ghosts" or new, unknown particles interacting with him.

This paper is a guidebook for the Belle II experiment (a giant particle collider in Japan) on how to catch these ghosts without getting tricked by "false alarms."

Here is the breakdown of their investigation using simple analogies:

1. The Goal: Measuring the Spin

The scientists want to measure Tau's "magnetic personality" (the anomalous magnetic moment, $a_\tau$) and his "electric personality" (the electric dipole moment, $d_\tau$).

• The Method: They smash electrons and positrons together to create pairs of Tau particles.
• The Clue: They look at how these Taus fly apart. If they fly apart in a specific, lopsided way (an "asymmetry"), it tells them about Tau's magnetic personality.
• The Upgrade: Usually, you need a special "polarized" electron beam (like a flashlight shining only one way) to see these clues clearly. The paper discusses what happens if we get that upgrade, but also what we can do without it.

2. The False Alarms (The Noise)

The problem is that the universe is noisy. When you try to measure Tau's magnetic personality, other things happen that look like the signal you want. The paper identifies two main sources of "noise" that could mess up the measurement.

A. The Z-Boson "Ghost"

Imagine you are trying to hear a whisper (the magnetic signal) in a room. Suddenly, a heavy door slams shut nearby. That slam is the Z-boson.

• The Z-boson is a heavy particle that mediates the weak nuclear force. It interacts with Tau, but it's not the magnetic signal you are looking for.
• The Finding: The authors calculated that this "door slam" creates a tiny ripple in the data, about 3 millionths ($3 \times 10^{-6}$) of the total signal.
• Why it matters: If you want to measure the magnetic personality with extreme precision (to see if it matches the "Schwinger term," a famous theoretical prediction), you must subtract this door slam. If you don't, you might think you found new physics when you just heard a door close.

B. The Four-Fermion "Crowd"

Now imagine the room is so crowded that people are bumping into each other. These are four-fermion operators.

• In physics, this means four particles interacting at once, rather than just two. It's like a chaotic mosh pit instead of a quiet conversation.
• The Finding: These interactions are usually very weak. The authors estimate they create a noise level of about 10 millionths ($10^{-5}$), depending on how heavy the new physics is.
• The Twist: While these interactions are mostly a nuisance for the main measurement, they have a secret superpower.

3. The Secret Superpower: Seeing the Invisible

Here is the most creative part of the paper.

Usually, to see the "magnetic signal," you need that special polarized electron beam (the flashlight). But the authors realized something clever:

• The Loop Trick: If you take those "mosh pit" interactions (four-fermion operators) and let them happen inside a tiny loop (a quantum loop), they generate a special kind of "imaginary" signal.
• The Analogy: Think of it like a shadow. You can't see the object directly, but if you shine a light just right, the shadow reveals the shape.
• The Result: This "shadow" (an imaginary part of the signal) can be detected using a specific measurement called the Normal Asymmetry ($A_N$).
• The Benefit: You don't need the polarized electron beam to see this shadow! This means we can hunt for these new physics effects right now at Belle II, even before the polarization upgrade is finished.

4. The "Schwinger Term" Goal

There is a famous theoretical limit called the Schwinger term. It's like a "Goldilocks zone" for Tau's magnetic personality.

• If we can measure Tau's magnetism with a precision of 1 in 100,000 ($10^{-5}$), we can check if he matches this Goldilocks zone.
• The paper says: "Hey, if we can measure that 'Normal Asymmetry' with a precision of 1 in 100,000, we can actually test this fundamental theory without needing the fancy polarized beam."

Summary: What does this mean for us?

1. Clean Up the Data: If we want to be super precise, we have to account for the "Z-boson door slam" and the "four-fermion crowd noise," or we'll get the wrong answer.
2. New Hunting Grounds: Even if we can't measure the main magnetic signal perfectly yet, we can use a clever "loop trick" to hunt for new physics (like heavy, unknown particles) using the "Normal Asymmetry."
3. No Polarization Needed: This new method works even without the expensive polarization upgrade, making it a great "intermediate goal" for the Belle II experiment right now.

In short, the authors are saying: "We know the room is noisy, but if we listen to the right echo (the Normal Asymmetry), we can hear the ghosts of new physics without needing the special flashlight."

Technical Summary

1. Problem Statement

The paper addresses the precision limits in measuring the anomalous magnetic moment ($a_\tau$) and electric dipole moment ($d_\tau$) of the $\tau$ lepton via the process $e^+e^- \to \tau^+\tau^-$ at B-factories (specifically Belle II and a potential future SuperKEKB polarization upgrade).

While asymmetry measurements are a promising avenue to constrain these dipole moments, achieving a precision of $\mathcal{O}(10^{-6})$ to test the Standard Model (SM) Schwinger term requires rigorous control over subleading effects. The authors identify two primary sources of potential contamination that must be quantified to avoid misinterpreting experimental data:

1. Electroweak effects: Specifically, tree-level and loop-level contributions from $Z$-boson exchange.
2. Beyond-the-Standard-Model (BSM) operators: Specifically, four-fermion operators in the Low-Energy Effective Field Theory (LEFT) that could mimic or obscure dipole operator signals.

The core challenge is determining whether these non-dipole contributions are negligible or if they constitute a significant background that limits the sensitivity to $a_\tau$ and $d_\tau$.

2. Methodology

The authors employ an Effective Field Theory (EFT) approach combined with detailed perturbative calculations:

• Formalism: They utilize a general parameterization of the lepton-photon vertex involving six form factors ($F_1$ to $F_6$). The dipole moments are extracted from the low-energy limits of $F_2$ (magnetic) and $F_3$ (electric).
• Cross-Section and Asymmetries: They derive the differential cross-sections for unpolarized and polarized beams, including spin-dependent terms. They define specific asymmetries ($A_N, A_T, A_L$, and their CP-odd counterparts) designed to isolate specific form factor interferences (e.g., $F_1-F_2$ interference for $a_\tau$).
• Perturbative Expansion:
  • Z-Boson Exchange: They calculate tree-level $Z$-boson exchange contributions and their interference with QED amplitudes. They also estimate one-loop $Z$-boson effects (anapole moments).
  • Four-Fermion Operators: They analyze the impact of dimension-6 LEFT operators (vectorial and scalar/ tensor) at both tree level and one-loop level.
• Loop Calculations: A key methodological innovation is the calculation of loop-level insertions where four-fermion operators or dipole operators generate imaginary parts in the form factors. These imaginary parts are accessible via the normal asymmetry ($A_N$) without requiring polarized electron beams.

3. Key Contributions and Results

A. Z-Boson Contributions

• Tree-Level Exchange: The interference between tree-level QED and $Z$-boson exchange is helicity-suppressed (scaling as $m_e^2$) and negligible. However, the quadratic $Z$-boson contribution (pure $Z$ exchange squared) is not suppressed by electron mass.
• Magnitude: The authors estimate the $Z$-boson contribution to the transverse-longitudinal asymmetry ($\Delta A_{TL}$) at Belle II energies ($\sqrt{s} \approx 10.58$ GeV) to be approximately $3 \times 10^{-6}$.
• Implication: This is a critical systematic effect. If the goal is to measure $a_\tau$ with $10^{-6}$ precision, this $Z$-boson background must be subtracted. Other asymmetries sensitive to $d_\tau$ vanish after symmetrization.

B. Tree-Level Four-Fermion Operators

• Helicity Suppression: Similar to the $Z$-boson case, the interference between four-fermion operators and the QED photon exchange is helicity-suppressed (scaling as $m_e m_\tau$).
• Magnitude: The largest possible effect from four-fermion operators is estimated as $\simeq 10^{-5} C v^2/\Lambda^2$, where $C$ is the Wilson coefficient and $\Lambda$ is the BSM scale.
• Conclusion: For heavy BSM scenarios ($\Lambda \gg v$), these contributions are subdominant to dipole operators. However, for light BSM degrees of freedom or large Wilson coefficients, they could become relevant.

C. Loop-Level Effects and New Probes

The paper highlights a novel mechanism where loop-level insertions generate observable effects that are otherwise inaccessible:

1. Four-Fermion Operators: Closing a four-fermion vertex and attaching it to a photon line generates an imaginary part in the magnetic form factor ($\text{Im } F_2$) if the photon energy exceeds the threshold of the virtual fermion pair ($q^2 > 4m_\psi^2$).

   • This imaginary part can be measured via the normal asymmetry ($A_N$) without needing polarized electron beams.
   • This allows constraints on operators like $4\tau$ ($\bar{\tau}\tau\bar{\tau}\tau$) and semileptonic operators ($\bar{q}q\bar{\tau}\tau$) that are difficult to probe directly.
   • Projected Bounds: Assuming a measurement precision of $10^{-6}$ for $\text{Im } F_2$, the authors derive bounds on the scale $\Lambda/\sqrt{C}$ ranging from 6 GeV to 1200 GeV depending on the flavor (e.g., $\Lambda \gtrsim 343$ GeV for $\tau$ loops, $\Lambda \gtrsim 1181$ GeV for charm loops).

2. Dipole Operator Loops: The SM dipole operator itself generates an imaginary part at one-loop order.

   • This allows the determination of $a_\tau$ via the normal asymmetry ($A_N$) even without a polarized electron beam.
   • Sensitivity: To probe the Schwinger term ($a_\tau \approx \alpha/2\pi$) via this loop-induced imaginary part, a measurement of $A_N$ with a precision of $\lesssim 10^{-5}$ is required.

4. Significance

• Precision Benchmarking: The paper establishes that $Z$-boson contributions ($\sim 3 \times 10^{-6}$) are a non-negligible systematic for future high-precision measurements of $a_\tau$ at Belle II. Ignoring this would limit the ability to test the SM Schwinger term.
• New Experimental Strategy: It proposes a strategy to measure $a_\tau$ and constrain four-fermion operators using the normal asymmetry ($A_N$) and loop-induced imaginary parts. This is significant because it removes the immediate dependency on the technically challenging polarized electron beam upgrade for certain measurements.
• EFT Constraints: The work provides the first quantitative estimates for how loop-level four-fermion operators constrain Wilson coefficients (particularly for $4\tau$ and flavor-violating semileptonic operators) that are currently unconstrained by direct searches.
• Future Outlook: The analysis suggests that while current settings at Belle II can target the Schwinger term with $A_N$ measurements, future lepton colliders with higher energies (above the top threshold) could extend these probes to top-quark related operators.

In summary, the paper provides the necessary theoretical groundwork to ensure that future high-precision measurements of $\tau$ dipole moments are not limited by unaccounted electroweak or non-dipole BSM effects, while simultaneously identifying new avenues to probe BSM physics through loop-induced asymmetries.