Amplitude Analysis of Singly Cabibbo-Suppressed Decay $Λ^{+}_{c}\to p K^{+} K^{-}$

Using 4.4 fb⁻¹ of $e^{+}e^{-}$ annihilation data collected by the BESIII detector, this study performs an amplitude analysis of the singly Cabibbo-suppressed decay $\Lambda^{+}_{c}\to p K^{+} K^{-}$, resulting in the first observation of $\Lambda(1405)K^{+}$ and $\Lambda(1670)K^{+}$ modes, the measurement of branching fractions for several intermediate resonances, and an updated, more precise branching fraction for the overall decay that supersedes previous BESIII results.

The Gist

Imagine the subatomic world as a bustling, chaotic construction site where tiny particles are constantly being built, broken apart, and rebuilt. In this paper, scientists from the BESIII Collaboration (a team of physicists working at a massive particle accelerator in China) act like forensic investigators trying to solve a specific mystery: How does a particle called the $\Lambda_c^+$ (Lambda-c-plus) break apart?

Here is the story of their investigation, explained simply.

1. The Mystery: A Rare Breakup

The star of the show is the $\Lambda_c^+$, a heavy particle made of three smaller "quarks" (like a tiny Lego brick made of three different colored blocks). Usually, when this particle decays (breaks apart), it follows a very common, predictable path.

But sometimes, it takes a "rare route." This specific decay, turning into a proton ($p$), a positive kaon ($K^+$), and a negative kaon ($K^-$), is called a "Singly Cabibbo-Suppressed" decay.

• The Analogy: Imagine you have a deck of cards. Usually, you draw an Ace (a common event). But sometimes, you manage to draw a specific, rare card combination that the rules say should happen very rarely. This paper is about studying that rare hand.

2. The Crime Scene: The BESIII Detector

To catch this rare event, the team used the BESIII detector, which is like a giant, ultra-high-speed 3D camera surrounding a collision point.

• They smashed electrons and positrons (matter and antimatter) together at incredibly high speeds.
• This created a shower of new particles, including our suspect, the $\Lambda_c^+$.
• They collected data equivalent to 4.4 billion collisions (4.4 inverse femtobarns) to find just enough of these rare breakups to study them.

3. The Investigation: Amplitude Analysis

The scientists didn't just count how many times the particle broke; they wanted to know how it broke.

• The Problem: When the $\Lambda_c^+$ breaks into a proton and two kaons, it doesn't always happen in one single step. It often goes through "middlemen" or intermediate resonances.
• The Analogy: Imagine a glass vase shattering on the floor. Did it hit the floor and break into three pieces at once? Or did it hit a table first, break into two pieces, and then one of those pieces hit the floor and broke again?
• The Method: The team used a technique called Amplitude Analysis. Think of this as a sophisticated audio mixer. They listened to the "sound" of the particles' masses and energies to separate the different "tracks" (the different ways the particle could have broken).

4. The Discoveries: Finding the "Middlemen"

By analyzing the data, they found that the decay happens through specific intermediate steps. They identified four main "middlemen" (resonances):

1. $\phi(1020)$: The most common path. The particle briefly turns into a $\phi$ meson before splitting. (Like the vase hitting a table first).
2. $f_0(980)$: Another common path.
3. $\Lambda(1405)K^+$: A new discovery! This is the first time scientists have clearly seen this specific path. It's like finding a hidden door in the house that no one knew existed.
4. $\Lambda(1670)K^+$: Another new discovery! Another hidden path revealed for the first time.

They calculated exactly how often each path happens (the "Branching Fraction"). For example, about 57% of the time, it goes through the $\phi$ path, while the new paths happen much less frequently but are still significant.

5. The Result: A More Precise Measurement

Before this study, scientists had a rough guess about how often this rare decay happened.

• Old Measurement: A bit fuzzy, like looking at a photo in the fog.
• New Measurement: The team improved the precision by 1.5 times. They now know the probability of this event is roughly 0.001% (or $9.94 \times 10^{-4}$).

This new number is very close to what the world average predicted, which is great news. It means our current theories about how the "strong force" (the glue holding quarks together) works are holding up well.

Why Does This Matter?

You might ask, "Who cares about a particle breaking into three other particles?"

• Understanding the Rules: This helps us understand the "Strong Force," which is one of the four fundamental forces of nature but is the hardest to understand mathematically.
• Searching for New Physics: These decays are sensitive to CP Violation (a subtle difference between matter and antimatter). If we find a tiny glitch in how these particles decay, it could explain why the universe is made of matter instead of antimatter. By mapping out the "normal" behavior so precisely, scientists can spot the tiny "glitches" that might point to new, undiscovered physics.

Summary

In short, the BESIII team acted like master detectives. They caught a rare particle decay, used advanced math to figure out exactly which "middlemen" were involved in the breakup (finding two new ones in the process), and measured the frequency of the event with much higher precision than before. It's a victory for understanding the fundamental building blocks of our universe.

Technical Summary

Here is a detailed technical summary of the paper "Amplitude Analysis of Singly Cabibbo-Suppressed Decay $\Lambda_c^+ \to pK^+K^-$":

1. Problem Statement and Motivation

The decay of the charmed baryon $\Lambda_c^+$ into a proton and two kaons ($\Lambda_c^+ \to pK^+K^-$) is a singly Cabibbo-suppressed (SCS) process. Such decays are critical for:

• CP Violation Searches: SCS decays can carry significant weak phases, making them sensitive probes for CP violation beyond the Standard Model.
• Understanding Decay Dynamics: The decay involves complex strong interaction dynamics where perturbative and non-perturbative QCD regimes intersect.
• Intermediate Resonances: The decay proceeds through various intermediate resonant states (e.g., $\phi(1020)$, $f_0(980)$, $\Lambda(1405)$, $\Lambda(1670)$). Previous measurements of the total branching fraction lacked the precision and model dependence required to disentangle these contributions accurately.
• Theoretical Discrepancies: Theoretical predictions for the branching fraction of the dominant channel $\Lambda_c^+ \to p\phi(1020)$ vary significantly across different models (e.g., Pole models vs. Covariant Confined Quark Models).

The primary goal of this work is to perform the first amplitude analysis of this decay to determine the contributions of intermediate resonances and to update the total branching fraction with improved precision.

2. Methodology

Data Sample and Detector

• Data Source: The analysis utilizes $e^+e^-$ annihilation data collected by the BESIII detector at the BEPCII collider.
• Integrated Luminosity: $4.4 \text{ fb}^{-1}$.
• Center-of-Mass Energies: Data was collected at six energy points ranging from $4600$to$4698$ MeV.
• Event Selection:
  • Charged tracks are identified as protons ($p$) and kaons ($K^\pm$) using ionization energy loss ($dE/dx$) and time-of-flight (TOF) measurements.
  • $\Lambda_c^+$ candidates are reconstructed from $pK^+K^-$ combinations.
  • Kinematic variables used for selection include the energy difference ($\Delta E$) and the beam-energy-constrained mass ($M_{BC}$).
  • Background Suppression: An unbinned maximum-likelihood fit to the $M_{BC}$ distribution is performed. The sPlot technique is employed to extract a background-subtracted, weighted signal sample for the amplitude analysis.

Amplitude Analysis Framework

• Formalism: The analysis uses the helicity amplitude formalism implemented in the open-source framework TF-PWA.
• Decay Chain Modeling: The three-body decay is modeled as a series of quasi-two-body sequential decays (e.g., $\Lambda_c^+ \to R K^+$, followed by $R \to pK^-$).
• Resonance Models:
  • $\phi(1020)$ and $\Lambda(1670)$: Described using relativistic Breit-Wigner functions. Detector resolution effects are convolved with the line shape.
  • $\Lambda(1405)$ and $f_0(980)$: Described using a two-coupled-channels Flatté model to account for their proximity to threshold and strong coupling to multiple channels ($\pi\pi/K\bar{K}$ for $f_0$; $p\bar{K}/\Sigma\pi$ for $\Lambda(1405)$).
• Fit Strategy: A simultaneous fit is performed across all six energy points. The fit parameters include the complex amplitudes (magnitudes and phases) of the resonant components.
• Validation: Input-Output (IO) tests using 500 pseudo-experiments ("toy" samples) are conducted to validate the stability of the fit and correct for biases.

Branching Fraction Measurement

• The total branching fraction is calculated using the signal yield derived from a simultaneous fit to the $M_{BC}$ distributions at all energy points.
• The detection efficiency is determined using Monte Carlo (MC) simulations generated based on the amplitude analysis results (rather than a simple phase-space model) to ensure higher accuracy.

3. Key Contributions

1. First Amplitude Analysis: This is the first study to perform a full amplitude analysis of $\Lambda_c^+ \to pK^+K^-$, allowing for the separation of overlapping resonant structures and the measurement of their interference.
2. Discovery of New Modes: The analysis provides the first observation of the decays:
   • $\Lambda_c^+ \to \Lambda(1405)K^+$
   • $\Lambda_c^+ \to \Lambda(1670)K^+$
3. Precision Branching Fractions: The study updates the branching fraction of the parent decay and provides the first precise measurements of the branching fractions for the individual resonant sub-channels.
4. Improved Precision: The measurement of the total branching fraction improves the precision of the previous BESIII result by a factor of approximately 1.5.

4. Key Results

Branching Fractions

Using the PDG average for the total branching fraction to minimize overall uncertainty, the absolute branching fractions for the resonant components are measured as:

• $\Lambda_c^+ \to p\phi(1020)$: $(0.62 \pm 0.05_{\text{stat}} \pm 0.02_{\text{syst}} \pm 0.03_{\text{PDG}}) \times 10^{-3}$
  • This is the most precise determination to date and agrees well with Pole model and CCQM predictions.
• $\Lambda_c^+ \to p f_0(980)$: $(0.43 \pm 0.17 \pm 0.16 \pm 0.02) \times 10^{-3}$
• $\Lambda_c^+ \to \Lambda(1405)K^+$: $(0.23 \pm 0.10 \pm 0.06 \pm 0.01) \times 10^{-3}$ (First Observation)
• $\Lambda_c^+ \to \Lambda(1670)K^+$: $(0.13 \pm 0.11 \pm 0.09 \pm 0.01) \times 10^{-3}$ (First Observation)

Total Branching Fraction

The updated branching fraction for the inclusive decay is:
$$\mathcal{B}(\Lambda_c^+ \to pK^+K^-) = (9.94 \pm 0.65_{\text{stat}} \pm 0.50_{\text{syst}}) \times 10^{-4}$$

• This result is consistent with the current world average within one standard deviation.
• It deviates from the older CLEO measurement by approximately $2.0\sigma$.

Fit Fractions (FF)

The relative contributions (Fit Fractions) of the components in the baseline model are:

• $p\phi(1020)$: $57 \pm 5 \pm 2%$(Significance:$16.6\sigma$)
• $p f_0(980)$: $40 \pm 16 \pm 15%$(Significance:$3.6\sigma$)
• $\Lambda(1405)K^+$: $21 \pm 9 \pm 6%$(Significance:$4.6\sigma$)
• $\Lambda(1670)K^+$: $12 \pm 10 \pm 8%$(Significance:$5.0\sigma$)
• Note: The sum of FFs exceeds 100% (130%) due to constructive interference between amplitudes.

5. Significance

• Refining QCD Models: The precise measurement of the $\Lambda_c^+ \to p\phi(1020)$ branching fraction helps discriminate between competing theoretical models (Pole vs. CCQM), providing crucial constraints on the dynamics of charmed baryon decays.
• CP Violation Sensitivity: By establishing a robust amplitude model with known phases and magnitudes for the intermediate states, this work lays the necessary groundwork for future searches for CP violation in this channel. Accurate signal modeling is essential to distinguish CP-violating asymmetries from strong phase differences.
• Discovery of New Physics: The first observation of $\Lambda(1405)K^+$ and $\Lambda(1670)K^+$ modes expands the known decay landscape of the $\Lambda_c^+$ baryon, offering new insights into the spectroscopy of excited baryons.
• Methodological Benchmark: The successful application of the sPlot technique combined with a sophisticated amplitude analysis and Flatté line shapes sets a new standard for analyzing complex three-body decays in charmed baryon physics.