CP violation in two meson tau decays

This paper applies an effective field theory formalism to two-meson tau decays, concluding that future experiments with 5% precision on $K^\pm K_S$ modes can test the maximum allowed CP rate asymmetry to either validate or refute the BaBar anomaly in $\tau\to K_S \pi\nu_\tau$ decays.

The Gist

Here is an explanation of the paper, translated into everyday language with some creative analogies.

The Big Mystery: Why Does the Universe Prefer Matter?

Imagine the Big Bang as a giant bakery that baked equal amounts of "matter cookies" and "antimatter cookies." If they were truly identical, they would have annihilated each other instantly, leaving nothing but empty space. But we are here, which means somehow, the universe ended up with a few extra matter cookies.

To explain this, physicists need a rule called CP Violation. Think of CP Violation as a tiny, subtle "glitch" in the universe's recipe book that makes matter and antimatter behave slightly differently. Without this glitch, the universe would be empty.

The Suspect: The Tau Particle

Physicists have been looking for this glitch in various places. Recently, they focused on a heavy particle called the Tau lepton (let's call it "Tau"). When Tau decays (breaks apart), it sometimes spits out a pair of particles: a Kaon and a Pion.

• The BaBar Anomaly: A few years ago, an experiment called BaBar looked at a specific decay (Tau $\to$ Kaon + Pion + Neutrino) and found something weird. The rate at which positive Taus decayed was slightly different from negative Taus. It was like flipping a coin and getting "Heads" 51% of the time and "Tails" 49% of the time, instead of a perfect 50/50 split.
• The Problem: The Standard Model (our current best recipe book) predicts this difference should be tiny. The BaBar result was much bigger than expected. Some scientists thought, "Aha! This must be New Physics!" (a new, undiscovered rule of the universe).

The Detective Work: Checking the Other Channels

This paper, written by Daniel Arturo López Aguilar, acts like a forensic audit. The author asks: "If the BaBar result is caused by New Physics, does that same New Physics show up in the other ways a Tau can decay?"

Specifically, the author looked at two other decay channels:

1. $\tau \to K^\pm K_S \nu_\tau$ (Two Kaons)
2. $\tau \to K^\pm \pi^0 \nu_\tau$ (Kaon and neutral Pion)

He used a mathematical tool called Effective Field Theory (EFT).

• The Analogy: Imagine you are trying to figure out what's inside a sealed box by shaking it. You can't see inside, but you can hear the rattle. EFT is like a set of rules that tells you how heavy objects inside the box should rattle based on their weight and shape, without needing to know exactly what they are made of.

The Findings: The "Smoking Gun" is in a Different Room

The author ran the numbers and found some very interesting results:

1. The "Two Kaon" Channel ($K K$) is the Gold Mine
In the channel where the Tau decays into two Kaons, the author found that if New Physics exists (to explain the BaBar anomaly), it should create a huge signal here.

• The Analogy: If the BaBar anomaly is a whisper in a quiet room, the "Two Kaon" channel is a shout in a stadium. The author predicts that if New Physics is real, we should see a 5% difference in how often positive and negative Taus decay here.
• Why? In this specific decay, the "rules" that usually hide New Physics are much looser. It's like a back door that is wide open, whereas the BaBar channel was a locked front door.

2. The "Kaon + Pion" Channel ($K \pi$) is a Dead End
For the channel with a Kaon and a neutral Pion, the predicted signal from New Physics is incredibly tiny.

• The Analogy: This is like trying to hear a pin drop in a hurricane. Even if New Physics exists, it's too quiet to be heard in this specific channel right now.

3. The "Two Pions" Channel ($\pi \pi$) is Also Quiet
Similar to the Kaon-Pion channel, the signal here is too small to be detected by current experiments.

The Verdict: What Should We Do Next?

The paper concludes with a clear roadmap for future experiments (like Belle-II and the future Super Tau-Charm Factory):

• Don't just stare at the BaBar anomaly. It's confusing and hard to explain.
• Look at the "Two Kaon" decay. If the BaBar anomaly is real and caused by New Physics, the "Two Kaon" channel should light up like a Christmas tree with a 5% asymmetry.
• The Test: If future experiments measure a 5% difference in the Two Kaon channel, it confirms the New Physics theory. If they measure zero, it suggests the BaBar result was just a fluke or a mistake, and we need to look elsewhere for the universe's missing matter.

Summary in One Sentence

This paper tells us that if the mysterious "glitch" found in one type of Tau decay is real, we will definitely find it in a different, easier-to-measure decay (Two Kaons), and if we don't find it there, the original mystery might just be a false alarm.

Technical Summary

Here is a detailed technical summary of the paper "CP violation in two meson tau decays" by Daniel Arturo López Aguilar.

1. Problem Statement

The paper addresses the BaBar anomaly regarding the charge-parity (CP) violation rate asymmetry in the decay $\tau \to K_S \pi \nu_\tau$.

• The Anomaly: BaBar measured a rate asymmetry of $-(0.36 \pm 0.23 \pm 0.11)\%$, which has the opposite sign to the Standard Model (SM) prediction of $(0.36 \pm 0.01)\%$. The discrepancy is at the $2.8\sigma$ level.
• The Theoretical Challenge: Previous attempts to explain this anomaly using New Physics (NP) coupling to an antisymmetric tensor current faced a significant hurdle. In the elastic region (dominant for $K_S \pi$), the phases of the vector and tensor form factors are equal due to Watson's theorem. This equality forces the CP-violating contribution to vanish unless "extreme fine-tuning" is applied, making a simple NP explanation difficult.
• The Goal: The author investigates whether other two-meson tau decay channels ($\pi\pi$, $K\bar{K}$, and $K\pi^0$) can provide alternative avenues to test this anomaly or reveal New Physics, using an Effective Field Theory (EFT) framework that avoids the constraints of the elastic region.

2. Methodology

The study employs a Low-Energy Effective Field Theory (LEFT) formalism derived from the Standard Model Effective Field Theory (SMEFT).

• Lagrangian Construction: The author utilizes the most general dimension-six effective Lagrangian for $\tau^- \to \bar{u}D\nu_\tau$ decays ($D=d, s$). This includes:
  • Standard Vector and Axial-vector currents.
  • Scalar ($S$), Pseudoscalar ($P$), and Tensor ($T$) operators parameterized by Wilson coefficients ($\hat{\epsilon}_S, \hat{\epsilon}_P, \hat{\epsilon}_T$).
• Amplitude Calculation: The decay amplitudes are computed for specific channels ($K^-\pi^0$, $K^-K_S$, etc.) by contracting lepton currents with hadronic matrix elements.
• Hadronic Inputs:
  • Form Factors: The calculation relies on dispersive form factors ($F_+, F_0, F_T$) for vector, scalar, and tensor currents.
  • Phase Shifts: A critical component is the phase difference between the vector ($\delta_+$) and tensor ($\delta_T$) form factors. The paper notes that in the inelastic region (above the $K\bar{K}$ threshold), these phases can differ, allowing for non-zero CP violation even without fine-tuning.
  • Constraints: The imaginary parts of the Wilson coefficients (which drive CP violation) are constrained by:
    • Neutron Electric Dipole Moment (EDM) limits.
    • Neutral $D$-meson mixing ($D^0-\bar{D}^0$).
    • Renormalization Group Evolution (RGE) linking tensor and scalar coefficients.
• Observables: The paper calculates:
  • Rate Asymmetry ($A_{CP}^{rate}$): The difference in decay rates between $\tau^+$ and $\tau^-$.
  • Forward-Backward Asymmetry ($A_{FB}$): An angular distribution observable sensitive to interference terms.

3. Key Contributions

• Extension to Other Channels: The paper extends the EFT analysis previously applied to $K_S \pi$ to the $\pi^\pm \pi^0$, $K^\pm K_S$, and $K^\pm \pi^0$ channels.
• Identification of the Inelastic Advantage: The author demonstrates that the $K^\pm K_S$ channel is fully inelastic. Unlike the $K_S \pi$ channel (dominated by the elastic region where phases align), the inelastic nature of $K K$ allows the vector and tensor form factor phases to differ significantly, removing the suppression of the NP contribution.
• Relaxed Bounds on Wilson Coefficients: The analysis shows that the constraints on the tensor Wilson coefficient $\hat{\epsilon}_T^d$ (relevant for $K K$) are roughly 20 times weaker than those on $\hat{\epsilon}_T^s$ (relevant for $K \pi$), due to different constraints from neutron EDM and $D$-meson mixing.

4. Key Results

The numerical analysis yields the following maximum possible CP-violating asymmetries ($A_{CP}^{rate}$) driven by New Physics:

Decay Channel	SM Contribution	Max NP Contribution	Total Max $A_{CP}$	Observability
$\tau \to \pi^\pm \pi^0 \nu_\tau$	Negligible	$\le 3 \times 10^{-5}$	$\sim 10^{-5}$	Too small for current/future experiments.
$\tau \to K^\pm \pi^0 \nu_\tau$	Negligible	$\le 6 \times 10^{-7}$	$\sim 10^{-7}$	Too small to probe soon.
$\tau \to K^\pm K_S \nu_\tau$	$\sim 3.8 \times 10^{-3}$	$\le 2.3 \times 10^{-4}$	$\sim 4.0 \times 10^{-3}$	Highly promising.

• The $K^\pm K_S$ Breakthrough:
  • The NP contribution in the $K K$ channel can reach up to $2.3 \times 10^{-4}$.
  • Crucially, a measurement precision of 5% on the total rate asymmetry would be sufficient to detect this maximum allowed NP contribution.
  • The Forward-Backward asymmetry ($A_{FB}$) in this channel is predicted to be of order $-0.1$ in a large portion of the spectrum, which is experimentally accessible.
• Comparison to $K_S \pi$: In the $K_S \pi$ channel, the NP contribution is orders of magnitude smaller than the SM contribution, making it difficult to isolate. In contrast, the $K K$ channel offers a "clean" signal where NP effects are relatively larger and experimentally distinguishable.

5. Significance and Conclusion

• Resolving the Conundrum: The paper suggests that if the BaBar anomaly is indeed due to New Physics, it might not be visible in the $K_S \pi$ channel due to theoretical suppression (phase alignment). Instead, the $K^\pm K_S$ channel offers a more favorable environment to test the same New Physics operators.
• Experimental Roadmap: The results provide a clear target for current and future experiments (Belle-II and the proposed Super Tau-Charm Factory). A 5% precision measurement of the $K^\pm K_S$ rate asymmetry can either:
  1. Support the BaBar anomaly by detecting a deviation consistent with the allowed NP bounds.
  2. Cast further doubt on the anomaly if no deviation is found, implying the BaBar result might be a statistical fluctuation or systematic error.
• Theoretical Validation: The work reinforces the utility of EFT in connecting high-energy New Physics to low-energy observables while rigorously accounting for hadronic uncertainties and symmetry constraints.

In summary, López Aguilar identifies the $\tau \to K^\pm K_S \nu_\tau$ decay as the most promising "clean" channel to probe the New Physics potentially responsible for the BaBar CP violation anomaly, offering a realistic path for experimental verification in the near future.