Probing the imaginary parts and their $q^2$ dependences for the tau $g-2$ and EDM

This paper investigates the $q^2$ dependence and absorptive imaginary parts of the tau lepton's anomalous magnetic and electric dipole moments within the SMEFT and 2HDM frameworks, demonstrating that combining measurements from Belle II and the Super Tau-Charm Facility can significantly improve current bounds and uniquely probe the energy evolution of these form factors.

The Gist

Imagine the tau particle as a tiny, hyper-active spinning top. In the world of particle physics, these tops have two special "personality traits" that scientists love to measure:

1. The Magnetic Wiggle (g-2): How much the top wobbles when it's near a magnet.
2. The Electric Tilt (EDM): How much the top leans to one side if it's near an electric field. This leaning is a sign of a very rare symmetry breaking (CP violation).

For a long time, scientists have been able to measure these traits for the lighter "siblings" of the tau (the electron and the muon) with incredible precision. But the tau is much heavier and dies almost instantly, making it a "ghost" that is very hard to catch and study.

This paper is like a detective story where the authors propose a new way to catch the tau's secrets by looking at how its personality changes depending on how fast it's moving and what energy level it's at.

The Core Mystery: The "Ghost" Energy

Usually, when we measure these traits, we pretend the tau is sitting still (like a parked car). But in the real world, when we smash particles together in colliders (like Belle II or the STCF), the tau is created with a lot of energy.

The authors point out a fascinating twist:

• The Real Part: This is the "standard" measurement we are used to.
• The Imaginary Part: This is a weird, hidden component that only appears when the tau has enough energy to briefly turn into a pair of other particles and then snap back. Think of it like a magician's trick: the tau briefly "dissolves" into a pair of ghosts (a tau and an anti-tau) and then reappears. This process leaves a "shadow" or an imaginary number in the math.

Until now, no one has really tried to measure this "shadow" or "imaginary" part for the tau. The paper argues that this shadow is actually a golden ticket to finding new physics.

Two Ways to Look at the Problem

The authors use two different lenses to solve the mystery:

1. The "Black Box" Lens (Effective Field Theory):
Imagine you don't know what's inside a machine, but you can poke it and see how it reacts. The authors treat the tau's interactions as a "black box." They show that if there is new physics (like a hidden force) causing the tau to lean (EDM), that same force must also make it wobble (g-2). You can't have one without the other. They also prove that even if the "static" rules say the EDM should be tiny, the "dynamic" energy of the collision can create a much larger, measurable signal.

2. The "Blueprint" Lens (Two-Higgs-Doublet Model):
Here, they build a specific machine to test their theory. They imagine a universe with extra "Higgs" particles (like extra types of snowflakes). They calculate that if a light, new particle exists (around 2 GeV, which is light for a new particle), it would act like a magnifying glass.

• It would make the tau's "wobble" (g-2) and "lean" (EDM) much bigger than we expect.
• Crucially, it would generate a huge "imaginary" shadow that we can actually see.

The New Detective Tools

The paper proposes a clever new way to measure these traits at the Belle II and STCF colliders.

Instead of just counting how many taus are made, they suggest looking at the dance moves of the particles the tau decays into.

• When a tau dies, it shoots out other particles (like pions or rho mesons).
• The direction these particles fly depends on how the tau was spinning.
• By analyzing the angles and correlations of these flying particles, the scientists can mathematically separate the "Real" wobble from the "Imaginary" shadow.

It's like trying to figure out how a spinning top is wobbling by watching the ripples it makes in a pond, rather than trying to grab the top itself.

The Big Payoff

The authors calculate that with these new techniques:

• Belle II and STCF can improve our knowledge of the tau's "wobble" (g-2) by more than 10 times (an order of magnitude).
• They can finally measure the "Imaginary" part, which has never been done before.
• By comparing data from STCF (lower energy) and Belle II (higher energy), they can map out exactly how these traits change as the energy changes. This is like watching a movie of the tau's personality evolving, rather than just taking a single snapshot.

Summary

In simple terms, this paper says: "The tau particle is hiding a secret 'imaginary' side that only shows up at high energies. We have a new mathematical map and a new set of camera lenses (using particle angles) to catch it. If we use the Belle II and STCF colliders together, we can not only find this hidden side but also see how it changes with energy, potentially revealing new forces of nature that we've never seen before."

Technical Summary

Technical Summary: Probing the Imaginary Parts and $q^2$ Dependences for the $\tau$ $g-2$ and EDM

Problem Statement
The $\tau$ lepton's anomalous magnetic dipole moment (MDM), $a_\tau$, and electric dipole moment (EDM), $d_\tau$, serve as precision probes for electroweak dynamics and potential new physics (NP). However, experimental constraints on the $\tau$ sector remain significantly weaker than those for electrons and muons. A critical theoretical gap exists in understanding the behavior of these quantities when the interacting photon is off-shell ($q^2 \neq 0$). Specifically, in the timelike kinematic region ($q^2 = s > 4m_\tau^2$), the form factors $a_\tau(q^2)$ and $d_\tau(q^2)$ can develop absorptive imaginary parts. While the static limits ($q^2 \to 0$) are well-defined, the $q^2$ dependence and the generation of these imaginary components in the context of NP remain under-explored. Furthermore, current experimental bounds on $a_\tau$ are insufficient to test Standard Model (SM) one-loop predictions, and the $\tau$ EDM is highly suppressed in the SM, making it a sensitive probe for CP-violating NP.

Methodology
The authors investigate the generation of $q^2$ dependence and imaginary parts from two complementary theoretical perspectives:

1. Standard Model Effective Field Theory (SMEFT):

   • The study utilizes a model-independent approach within the Low-Energy Effective Field Theory (LEFT), matched to SMEFT.
   • The analysis focuses on two types of operators: dipole operators (dimension-5) and four-fermion operators (dimension-6 and dimension-8).
   • The authors calculate one-loop corrections involving four-fermion operators and standard QED vertices. This calculation explicitly demonstrates how absorptive imaginary parts are dynamically generated when $s$ exceeds the kinematic threshold ($s > 4m_\tau^2$), allowing intermediate $\tau$ leptons to go on-shell.
   • A key aspect of this framework is the correlation between $a_\tau$ and $d_\tau$ via common effective operators, particularly highlighting that while tree-level dipole coefficients are constrained by electron EDM limits, the dynamically generated four-fermion contributions are not subject to the same stringent indirect bounds.

2. UV-Complete Two-Higgs-Doublet Model (2HDM):

   • To provide a rigorous UV completion, the authors analyze a specific 2HDM scenario featuring a light neutral scalar $A$ (mass $\sim$ GeV scale) with CP-violating couplings to the $\tau$ lepton.
   • The interaction Lagrangian includes both scalar ($a$) and pseudoscalar ($b$) couplings. The authors compute one-loop contributions to the form factors $a_\tau(s)$ and $d_\tau(s)$ involving this scalar mediator.
   • The analysis incorporates constraints from BESIII (via $\gamma^* \to a\gamma$), OPAL (via $e^+e^- \to \gamma\gamma(\gamma)$), and LHC Higgs signal strengths ($h \to \tau^+\tau^-$).
   • The study examines the interference between scalar and pseudoscalar loop functions, noting that destructive interference can occur if couplings are comparable, necessitating a scalar-dominant scenario to maximize the anomalous MDM.

3. Experimental Proposal:

   • The paper proposes experimental methods to extract both real and imaginary components of the dipole form factors at $e^+e^-$ colliders, specifically utilizing the $\psi(2S)$ resonance at the Super Tau-Charm Facility (STCF) and the $\Upsilon(4S)$ resonance at Belle II.
   • The methodology relies on analyzing angular distributions and spin correlations in the process $e^+e^- \to \tau^+\tau^-$ followed by hadronic decays ($\tau \to h\nu$).
   • Specific operators are constructed from the momenta of the beam, the $\tau$ pair, and the decay hadrons to isolate $\text{Re}(a_\tau)$, $\text{Im}(a_\tau)$, $\text{Re}(d_\tau)$, and $\text{Im}(d_\tau)$.

Key Results

• EFT Findings: The calculation reveals that the absorptive imaginary parts of the form factors are directly proportional to the kinematic function $\beta_\tau \Theta(s - 4m_\tau^2)$ and the Wilson coefficients of four-fermion operators. Crucially, the dynamically generated part of the EDM is immune to the severe indirect constraints imposed by electron EDM measurements on the tree-level dipole coefficient.
• 2HDM Predictions: Within the viable parameter space of the 2HDM (constrained by BESIII, OPAL, and LHC data), the model predicts sizable values for both $a_\tau$ and $d_\tau$ at collider energies.
  • At STCF ($\sqrt{s} = 3.686$ GeV), the model predicts $a_\tau^{NP} \approx (1.1 + 5.3i) \times 10^{-3}$ and $d_\tau^{NP} \approx (1.96 + 1.10i) \times 10^{-17}$ e cm.
  • At Belle II ($\sqrt{s} = 10.58$ GeV), the predictions are $a_\tau^{NP} \approx (-0.8 + 2.1i) \times 10^{-3}$ and $d_\tau^{NP} \approx (-1.68 + 3.16i) \times 10^{-18}$ e cm.
  • The study highlights that the real parts of the form factors can suffer from destructive interference between scalar and pseudoscalar contributions, making the imaginary parts a robust target for detection.
• Experimental Sensitivity:
  • The proposed analysis suggests that Belle II and STCF can improve current bounds on $a_\tau$ by more than one order of magnitude.
  • Projected sensitivities at STCF for $a_\tau$ are $\delta \text{Re}(a_\tau) \approx 2.17 \times 10^{-3}$ and $\delta \text{Im}(a_\tau) \approx 4.90 \times 10^{-3}$.
  • For $d_\tau$, the projected sensitivities at STCF are $\delta \text{Re}(d_\tau) \approx 2.8 \times 10^{-18}$ e cm and $\delta \text{Im}(d_\tau) \approx 0.7 \times 10^{-18}$ e cm.
  • The combination of measurements at the distinct center-of-mass energies of STCF and Belle II allows for the first direct probe of the $q^2$ evolution of these dipole form factors.

Significance
The paper claims that its work provides a unique avenue to explicitly obtain information about the $q^2$ evolution of $\tau$ dipole form factors, a feature previously unexplored. By demonstrating that imaginary parts and significant $q^2$ running can be generated at levels accessible by current and near-future colliders, the authors motivate the necessity of measuring both real and imaginary components. The study emphasizes that combining measurements across different energy scales (STCF and Belle II) is essential not only for improving bounds on $a_\tau$ to the level required to test SM loop predictions but also for explicitly mapping the dynamical behavior of these form factors, thereby offering a powerful tool to discover CP-violating new physics in the third-generation lepton sector.