Forward backward CP asymmetry in $\tau^- \to K \pi \nu_{\tau}$ in the Left-Right Inverse seesaw model

This paper investigates the Left-Right Inverse Seesaw model's ability to explain CP violation in $\tau^- \to K \pi \nu_{\tau}$ decays, finding that while it cannot account for the integrated CP asymmetry observed by BaBar, it predicts a distinctive and pronounced signal in the differential forward-backward CP asymmetry driven by a non-decoupling scalar operator.

The Gist

Imagine the universe as a giant, complex machine where tiny particles called tau leptons (heavy cousins of the electron) sometimes break apart. When a tau particle decays, it can turn into a kaon (a type of particle containing a strange quark), a pion (a lighter particle), and a neutrino (a ghostly particle that barely interacts with anything).

Scientists have been watching this specific breakup closely because, according to our current "rulebook" of physics (the Standard Model), this event should happen in a perfectly symmetrical way. However, a previous experiment called BaBar noticed a tiny, puzzling glitch: the decay seemed to happen slightly differently depending on the direction of the particles, suggesting a violation of a fundamental symmetry called CP symmetry (which basically asks: "Does physics look the same if we swap matter for antimatter and flip left for right?").

This paper is like a team of detectives trying to solve that glitch using a new, more complex rulebook called the Left-Right Inverse Seesaw (LRIS) model. Here is what they found, explained simply:

1. The "Total Score" Didn't Change Much

The researchers first looked at the total number of these decays. They asked: "If we count every single tau decay that happens, does the new LRIS model explain the glitch BaBar saw?"

The Answer: No.
Even with their new, fancy model, the total difference between matter and antimatter decays remains incredibly tiny—so small that it's practically invisible. The new model is actually too strict. It has to obey other rules (like how other particles mix and interact) that force this total difference to stay near zero. So, if you are looking for a big change in the total count, this model doesn't provide it.

2. The "Directional Clue" is the Real Treasure

However, the detectives found something much more exciting. Instead of looking at the total count, they looked at the direction the particles fly in.

Imagine throwing a ball at a wall. In a normal world, it bounces back straight. But in this specific particle decay, the new model predicts that the particles will prefer to bounce slightly to the left or right depending on whether they are matter or antimatter.

This is called the Forward-Backward CP Asymmetry.

• The Analogy: Think of a spinning top. If you spin it one way, it might lean left; spin it the other way, and it leans right. The "total spin" might look the same, but the lean tells you the secret.
• The Discovery: The LRIS model predicts a very strong "lean" (a large asymmetry) in this directional signal, specifically when the particles have a certain energy level.

3. The "Magic Box" and the "Heavy Neutrino"

How does this model create such a strong directional signal?

• The Old Way (Tree Level): Imagine a direct path where a heavy "Charged Higgs" particle (a new kind of particle) tries to mediate the decay. But this path is blocked by strict traffic rules (flavor constraints) that make the effect tiny.
• The New Way (The Loop): The paper suggests a more complex path. Imagine a box diagram (a loop in the particle's path) where the tau particle briefly turns into a top quark (the heaviest known quark) and a heavy neutrino before turning back.
• The "Non-Decoupling" Trick: Usually, if a particle is very heavy, its effect on low-energy physics disappears (like a heavy elephant leaving no footprint on a trampoline). But in this specific "Inverse Seesaw" model, the heavy neutrino has a special property: its heaviness actually cancels out in the math. Instead of disappearing, its effect stays strong. It's like the elephant stepping on the trampoline, but the trampoline somehow remembers the weight perfectly, no matter how heavy the elephant gets.

4. The "Resonance Amplifier"

The paper points out that this directional signal gets supercharged at a specific energy level, around 1.4 GeV.

• The Analogy: Imagine pushing a child on a swing. If you push at the wrong time, nothing happens. But if you push exactly when the swing is at the top of its arc (the resonance), the swing goes much higher.
• The Reality: At this specific energy, a particle called the $K^*_0(1430)$ (a scalar resonance) acts like that perfect timing. It amplifies the signal from the heavy neutrino loop, making the "lean" (the asymmetry) huge and easy to spot.

5. What This Means for the Future

The paper concludes that while the "Total Score" (integrated asymmetry) remains too small to explain the BaBar glitch, the "Directional Lean" (differential forward-backward asymmetry) is a golden signal.

• The Prediction: The model predicts a distinct peak in the directional signal right at the energy of the $K^*_0(1430)$ particle.
• The Test: The Belle II experiment (a massive particle collider in Japan) is expected to collect enough data to see this specific "lean." If they see this peak, it would be a smoking gun for the Left-Right Inverse Seesaw model and the existence of these heavy neutrinos.

In Summary:
The paper says, "Don't look at the total number of broken tau particles; that won't show us the new physics. Instead, look at which way they fly when they break apart. If you look at the direction near a specific energy (1.4 GeV), our new model predicts a huge, clear signal that current experiments like Belle II might finally be able to catch."

Technical Summary

1. Problem Statement

The paper addresses a $2.8\sigma$ discrepancy reported by the BaBar collaboration regarding the integrated CP asymmetry ($A_{CP}$) in the semileptonic decay $\tau^- \to K\pi\nu_\tau$.

• The Anomaly: The experimental value is $A_{CP}^{\text{exp}} = -(0.36 \pm 0.23 \pm 0.11)\%$, whereas the Standard Model (SM) prediction for direct CP violation is negligible ($\sim 10^{-12}$) due to the lack of necessary weak and strong phase differences.
• The Challenge: Previous attempts to explain this anomaly using loop-induced tensor operators in New Physics (NP) models typically fail because they suffer from severe $1/16\pi^2$ loop suppression and constraints from Watson's theorem (which correlates strong phases of vector and tensor form factors, suppressing the CP-violating interference).
• The Goal: To investigate whether the Left-Right Inverse Seesaw (LRIS) model can generate observable CP violation in this channel, specifically looking beyond the integrated asymmetry to differential observables.

2. Methodology

The authors employ an effective field theory approach combined with specific model calculations within the LRIS framework.

• Effective Hamiltonian: They construct the $|\Delta S|=1$ effective Hamiltonian at the scale $\mu \sim m_\tau$, including vector, axial-vector, scalar, pseudoscalar, and tensor operators. The decay is described by hadronic form factors ($F_+, F_0, F_T$) parameterized by resonances ($K^*(892)$, $K^*_0(1430)$, etc.).
• LRIS Model Framework:
  • The model extends the SM gauge symmetry to $SU(3)_C \times SU(2)_L \times SU(2)_R \times U(1)_{B-L}$.
  • It incorporates the Inverse Seesaw mechanism to generate small neutrino masses via heavy neutrinos ($N_i$) at the TeV scale.
  • The scalar sector includes a bidoublet $\phi$ and a right-handed doublet $\chi_R$, yielding physical charged Higgs bosons ($H^\pm$) and neutral scalars.
• Calculation of Wilson Coefficients:
  • Tree-Level: They calculate the contribution from charged Higgs exchange ($g_S^{\text{tree}}$).
  • Loop-Level: They calculate one-loop box diagrams involving internal heavy neutrinos ($N_i$), top quarks ($t$), and charged Goldstone bosons ($G^\pm$) or neutral Higgs ($H^0$).
• Phenomenological Constraints: The analysis rigorously applies constraints from:
  • $K^0-\bar{K}^0$ and $D^0-\bar{D}^0$ mixing (limiting flavor-changing neutral currents).
  • Neutrino non-unitarity bounds from tau decays.
  • LHC limits on $W_R$ and charged Higgs masses.
• Observables: The authors compute both the integrated CP asymmetry ($A_{CP}$) and the differential forward-backward CP asymmetry ($A_{CP}^{FB}(s)$), analyzing the interference between SM vector currents and NP scalar/tensor operators.

3. Key Contributions

The paper makes several distinct theoretical and phenomenological contributions:

1. Identification of Non-Decoupling Behavior: The authors demonstrate that specific box diagrams in the LRIS model exhibit a non-decoupling property. In the limit of heavy neutrino mass ($M_{N_i} \gg v_R$), the $1/M_{N_i}^2$ suppression from the loop integral is exactly canceled by the $M_{N_i}$ dependence in the Yukawa coupling ($Y_{\tau N G} \propto M_{N_i}/v_R$). This allows the scalar Wilson coefficient $g_S$ to remain unsuppressed by heavy mass scales.
2. Top-Quark Dominance: Unlike tree-level contributions which are suppressed by light quark masses (via Minimal Flavor Violation constraints), the loop-induced contribution accesses the top-quark Yukawa coupling. This bypasses the severe suppression from $K^0-\bar{K}^0$ mixing constraints that affect light-quark transitions.
3. Differential vs. Integrated Asymmetry: The paper establishes a crucial distinction:
   • Integrated Asymmetry ($A_{CP}$): Remains heavily suppressed ($\ll 10^{-7}$) because the dominant tensor contributions are suppressed by loop factors and Watson's theorem constraints (vector and tensor form factors share the same strong phase).
   • Forward-Backward Asymmetry ($A_{CP}^{FB}$): Is significantly enhanced. The interference between the SM vector current and the NP scalar operator ($g_S$) involves the scalar form factor $F_0(s)$ (dominated by $K^*_0(1430)$). Since the scalar and vector resonances have different strong-phase structures, Watson's theorem does not suppress this interference.
4. Kinematic Amplification: The authors show that the $K^*_0(1430)$ resonance acts as a kinematic amplifier for the CP-violating signal, creating a distinct peak in the differential asymmetry.

4. Results

• Integrated Asymmetry: The predicted integrated CP asymmetry in the LRIS model is $A_{CP}^{\text{LRIS}} \ll 10^{-7}$. This confirms that the LRIS model cannot explain the BaBar anomaly via the integrated rate, consistent with the suppression mechanisms found in other NP models.
• Forward-Backward Asymmetry:
  • The non-decoupling box diagram generates a scalar Wilson coefficient $|g_S^{\text{box}}| \sim \mathcal{O}(10^{-4})$.
  • This leads to a predicted integrated forward-backward asymmetry of $A_{CP}^{FB, \text{LRIS}} \approx 2.08 \times 10^{-4}$.
  • This represents an enhancement of $\sim 10^{10}$ over the SM prediction.
• Differential Distribution: The differential asymmetry $A_{CP}^{FB}(s)$ exhibits a pronounced peak at $\sqrt{s} \approx 1.4$ GeV, coinciding exactly with the mass of the $K^*_0(1430)$ scalar resonance.
• Benchmark Parameters: Using a benchmark scenario ($v_R = 3.0$ TeV, Yukawa couplings $\approx 0.5$, maximal CP phases), the results are consistent with all current flavor and collider constraints.

5. Significance

• Experimental Accessibility: The predicted signal ($A_{CP}^{FB} \sim 10^{-4}$) is within the potential reach of the Belle II experiment, which aims to collect $50 \text{ ab}^{-1}$ of data. This makes the forward-backward asymmetry a "golden observable" for testing the LRIS model.
• Model Discrimination: The specific kinematic shape (peak at 1.4 GeV) allows experimentalists to distinguish the LRIS scalar sector from other New Physics scenarios (e.g., Type-II 2HDM or Leptoquarks) that might predict different kinematic distributions or rely on tensor operators.
• Theoretical Insight: The work highlights the importance of differential observables over integrated ones in searching for CP violation. It demonstrates that even if an integrated anomaly cannot be explained, differential angular distributions can reveal new physics through interference effects that evade standard suppression theorems (Watson's theorem).
• Inverse Seesaw Phenomenology: It provides a concrete low-energy signature for the Inverse Seesaw mechanism, linking heavy neutrino physics to precision flavor observables in tau decays, independent of direct heavy neutrino production at colliders.

In conclusion, while the LRIS model cannot resolve the BaBar integrated CP anomaly, it predicts a robust, observable signature in the forward-backward CP asymmetry of $\tau \to K\pi\nu_\tau$, driven by non-decoupling scalar box diagrams and enhanced by the $K^*_0(1430)$ resonance. This offers a promising avenue for discovering TeV-scale left-right symmetry and inverse seesaw physics at Belle II.