Search for new physics in $B \to K \pi \pi \gamma$ with Belle II data

This paper discusses the amplitude analysis of $B \to K_{res} \gamma \to K \pi^+ \pi^- \gamma$ decays using Belle II data, which is essential for distinguishing CP eigenstates from non-CP eigenstates to constrain new physics through time-dependent CP-asymmetry measurements.

The Gist

The Big Picture: Hunting for Ghosts in the Machine

Imagine the Standard Model of physics as a very strict, well-written rulebook for how the universe works. Scientists have checked this rulebook for decades, and it usually fits perfectly. However, they suspect there might be "ghosts" (new, unknown particles) hiding in the shadows, breaking the rules in subtle ways.

This paper is about a specific experiment at the Belle II lab in Japan (run by the Belle II collaboration) trying to catch these ghosts. They are looking at a specific type of particle decay: a heavy particle called a B-meson breaking apart into a Kaon, two pions, and a photon (light).

The Mystery: The "Chirality" of Light

In the Standard Model, when a B-meson decays into a photon, the photon is almost always "left-handed" (like a left-handed screw). If scientists find a significant number of "right-handed" photons, it would be a smoking gun for new physics.

To measure this, they look at CP-asymmetry. Think of this like a dance between a particle and its mirror-image twin (antiparticle).

• If the dance is perfectly symmetrical, the rules are standard.
• If the dance is lopsided, something new is pushing the dancers.

However, there is a problem. The final result of the decay (Kaon + two pions) can be reached through many different "paths" or "routes." Some of these routes are "CP-eigenstates" (perfectly symmetrical dances), while others are "non-CP eigenstates" (messy, asymmetrical dances).

The Analogy: Imagine trying to hear a specific violin solo (the signal) in a crowded room. But the room is full of people talking, singing, and clapping (background noise and different decay paths). If you just listen to the whole room, the violin solo gets drowned out. You need to separate the solo from the noise to know how loud the violin actually is.

The Solution: The "Amplitude Analysis"

The paper explains that to find the new physics, they must perform an Amplitude Analysis. This is like being a super-sound engineer who can isolate every single instrument in the orchestra to see exactly how they are playing together.

1. The Orchestra: The decay doesn't happen in one straight line. The B-meson turns into a "resonance" (a temporary, heavy particle) which then breaks apart. There are many possible resonances (like $K_1$, $K^*$, etc.), each with different spins and properties.
2. The Interference: These different paths don't just happen one after another; they happen at the same time and "interfere" with each other, like waves in a pond crashing into one another. Sometimes they boost the signal; sometimes they cancel it out.
3. The Goal: The scientists built a complex mathematical model (a "decay model") that describes every possible path and how they interfere. They use this model to calculate a "dilution factor."
   • Analogy: If the messy dances (non-CP eigenstates) are 90% of the crowd, they "dilute" the signal of the symmetrical dances. The dilution factor tells them exactly how much the signal is being watered down so they can correct for it.

How They Did It (The Lab Work)

• The Data: They used data from the SuperKEKB collider, which smashes electrons and positrons together to create billions of B-mesons.
• The Filter: They used a statistical trick called sPlot to separate the real B-meson decays from the background noise (random collisions that look similar but aren't).
• The Simulation: The standard computer programs used to simulate these events weren't good enough because they didn't understand the complex "interference" between the different paths. So, the team used a new tool called AmpGen to create a realistic simulation of how these particles should behave if their new model is correct.

The Results So Far

The paper presents preliminary work.

• They have successfully built the mathematical model that describes all the possible ways the B-meson can decay into a Kaon and two pions.
• They have tested this model on simulated data and shown that it can successfully "fit" the data, meaning it can figure out the strength and phase of each different path.
• The Next Step: Now that the "engine" is built, they need to tune it (test its robustness) and then apply it to the real data collected by Belle II.

Why This Matters

Once they apply this model to the real data, they will be able to calculate the true CP-asymmetry without the "dilution" caused by the messy decay paths. This will give them a precise measurement of the "left-handedness" vs. "right-handedness" of the photon.

If the result deviates from the Standard Model's prediction, it won't just be a small error; it will be evidence that a new, heavy particle is lurking in the quantum loop, changing the rules of the universe.

In short: The paper is about building a sophisticated mathematical filter to separate the "signal" from the "noise" in a complex particle decay, so scientists can finally see if the universe is breaking its own rules.

Technical Summary

Based on the provided preprint, here is a detailed technical summary of the paper "Search for new physics in B →Kππγ with Belle II data" by Sahil Saha on behalf of the Belle II collaboration.

1. Problem Statement

The paper addresses the search for physics beyond the Standard Model (BSM) through the measurement of time-dependent CP asymmetry in the radiative decay $B^0 \to K_S^0 \pi^+ \pi^- \gamma$.

• Theoretical Context: In the Standard Model (SM), the flavor-changing neutral current (FCNC) transition $b \to s\gamma$ is forbidden at tree level and occurs via a penguin diagram. It is highly suppressed by the Glashow–Iliopoulos–Maiani (GIM) mechanism. New heavy particles in electroweak loops could alter the Wilson coefficients ($C_7$ and $C'_7$), leading to deviations in the CP asymmetry.
• The Challenge: The measured CP asymmetry ($S_{B^0 \to K_S^0 \pi^+ \pi^- \gamma}$) is an "effective" parameter diluted by the presence of non-CP eigenstates in the final state. The final state $K_S^0 \pi^+ \pi^-$ is a superposition of various intermediate resonances (kaonic resonances, $K_{res}$). Some of these resonances are CP eigenstates (e.g., $\rho^0$), while others are not.
• The Goal: To constrain New Physics (NP) without assuming specific field content, one must extract the "pure" CP asymmetry ($S_{CP}$) from the measured effective asymmetry. This requires calculating a dilution factor ($D$), which depends on the interference between different decay amplitudes. Currently, this factor cannot be determined without a full amplitude analysis of the decay dynamics.

2. Methodology

The paper outlines the development of a formalism for a model-dependent amplitude analysis of the charged isospin partner decay, $B^+ \to K^+ \pi^+ \pi^- \gamma$, which is statistically more favorable and assumed to have no direct CP violation.

A. Decay Model Construction

The analysis models the decay as a coherent sum of intermediate paths involving three steps:

1. Production: $B^+ \to K_{res}^+ \gamma$.
2. Resonance Decay: $K_{res}^+ \to K^+ \pi^+ \pi^-$ via various intermediate states.
3. Final State: The $K^+ \pi^+ \pi^-$ state.

• Intermediate States: The model includes a comprehensive list of kaonic resonances ($K_{res}$) with spin-parity $J^P = 1^\pm, 2^\pm$. These include:
  • $K_1(1270)^+$, $K_1(1400)^+$
  • $K^*(1410)^+$, $K^*(1680)^+$
  • $K_2^*(1430)^+$, $K_2(1770)^+$
• Decay Chains: Each resonance decays into combinations such as $K^* \pi$, $\rho K$, $\omega K$, or $K_2^* \pi$.
• Amplitude Formalism: The total probability is defined as $P_{total} = |\sum c_i A_i|^2$, where $c_i$ are complex weights and $A_i$ are amplitudes.
  • The amplitude $A_i$ is factored into spin-dependent terms ($S$), lineshapes ($T$), and Blatt-Weisskopf barrier factors ($BL$).
  • Kinematics: The analysis utilizes five observables: three invariant masses ($m_{K\pi\pi}$, $m_{K\pi}$, etc.) and two angles ($\cos\theta$, $\phi$) defined in the $K_{res}$ rest frame.
  • Interference: The model explicitly accounts for quantum interference between all decay modes using a coherent sum.

B. Data and Simulation

• Data Source: Belle II data collected from $e^+e^-$ collisions at the SuperKEKB collider ($\sqrt{s} = 10.58$ GeV, $\Upsilon(4S)$ resonance).
• Background Subtraction: The sPlot technique is used to extract background-subtracted distributions of the fit variables. The discriminating variable is $\Delta E$ (energy difference).
• Monte Carlo Generation: The standard generator EvtGen is insufficient because it lacks proper angular structures and interference between $K_{res}$ modes. Therefore, the collaboration uses AmpGen to generate the $B^+ \to K^+ \pi^+ \pi^- \gamma$ decay, which is then processed through the global Belle II simulation and reconstruction software.

C. Fitting Strategy

• The fit determines the complex weights (magnitudes and phases) of each intermediate decay path.
• Resonance pole masses and widths are fixed to Particle Data Group (PDG) averages.
• The output of the fit provides the full amplitude structure required to compute the dilution factor $D$ via integration over the phase space (Eq. 1.5).

3. Key Contributions

• First Formalism Development: This work presents the first development of an amplitude analysis formalism specifically for $B^+ \to K^+ \pi^+ \pi^- \gamma$ decays at Belle II, covering all possible intermediate kaonic resonances.
• Tool Integration: It successfully integrates the AmpGen generator with the Belle II simulation chain to overcome the limitations of EvtGen regarding angular distributions and interference.
• Dilution Factor Calculation: The methodology establishes the necessary framework to calculate the dilution factor $D$, which is the critical missing link to translate the measured effective CP asymmetry into a constraint on the Wilson coefficients ($C_7, C'_7$).
• Comprehensive Model: The model includes a wide spectrum of resonances ($K_1, K^*, K_2$) and their various decay chains, ensuring a complete description of the $K\pi\pi$ final state.

4. Results

• Preliminary Status: The paper reports on the preliminary developments and the validation of the fit strategy on simulated data.
• Simulation Validation: Figure 3.2 demonstrates a successful fit to generator-level simulated data using the full combination of decay pathways. The fit successfully retrieves the input amplitudes and phases, confirming the model's ability to resolve the complex interference patterns.
• Data Application: While the formalism is ready, the analysis of the actual Belle II dataset (registered so far) is pending. The current work focuses on defining the unbiased fit strategy and testing robustness before applying it to real data.
• Current Measurement Context: The paper references a previous measurement of the effective CP parameter $S_{B^0 \to K_S^0 \pi^+ \pi^- \gamma} = -0.29 \pm 0.11 \pm 0.05$ using Belle and Belle II data, which awaits the dilution factor correction to be interpreted as a New Physics constraint.

5. Significance

• New Physics Sensitivity: The measurement of CP asymmetry in radiative $B$ decays is one of the most sensitive probes for New Physics. By enabling the calculation of the dilution factor, this work allows the Belle II collaboration to convert current experimental limits into rigorous constraints on the chiral structure of the $b \to s\gamma$ transition (specifically the ratio $C'_7/C_7$).
• Model Independence: The approach allows for constraints on New Physics without needing to specify the exact nature of the new heavy particles, relying instead on the effective field theory (EFT) framework.
• Future Prospects: This analysis is a prerequisite for the next generation of precision measurements at Belle II. Once applied to the full dataset, it will significantly reduce the uncertainty on the CP asymmetry parameters, potentially revealing deviations from the Standard Model that could point to new particles or interactions.