Amplitude analysis of $B^0 \rightarrow η_c(1S) K^+ π^-$ decays

Using LHCb data, this paper performs an amplitude analysis of $B^0 \rightarrow \eta_c(1S) K^+ \pi^-$ decays to measure their branching fraction and finds no evidence for the previously reported exotic resonance in the $\eta_c(1S) \pi^-$ system, with the data being well-described by known $K^{0*}$ resonances.

The Gist

The Big Picture: A Cosmic Detective Story

Imagine the LHCb detector at CERN as a giant, ultra-high-speed camera taking pictures of billions of tiny, invisible collisions. In this specific study, the scientists are looking at a very rare event: a heavy particle called a $B^0$ meson decaying (falling apart) into three smaller particles: a proton-antiproton pair (which comes from an $\eta_c$ meson), a kaon ($K^+$), and a pion ($\pi^-$).

Think of the $B^0$ meson as a heavy, unstable suitcase that instantly bursts open. The scientists want to know exactly how it bursts. Does it break apart all at once? Or does it go through a specific "middleman" step?

The Mystery: Are There "Exotic" Particles?

For decades, physicists have been hunting for "exotic" particles. Standard particles are like simple Lego bricks (made of two or three smaller pieces). Exotic particles are like complex Lego structures made of four or five pieces stuck together in weird ways.

In a previous study (using less data), the LHCb team thought they saw a ghost in the machine: a new, exotic particle they called $T_{c\bar{c}}(4100)^-$. They saw a "bump" in the data suggesting this particle existed, acting as a middleman that briefly formed before the final particles flew apart.

The Goal of This Paper:
The scientists returned with a much larger dataset (about double the size of the previous one) to see if that "ghost" was real or just a trick of the light. They wanted to confirm if this exotic particle exists or if the data can be explained by known, standard particles.

The Investigation: Sorting the Clues

To solve this, the scientists used a technique called Amplitude Analysis.

The Analogy: The Orchestra
Imagine the decay of the $B^0$ meson is a piece of music played by an orchestra.

• The known particles (called $K^*$ resonances) are the standard instruments (violins, drums, flutes) we know how to play.
• The exotic particle would be a brand new, strange instrument we've never heard before.

The scientists recorded the "music" (the data) and tried to figure out which instruments were playing.

1. The Baseline Model: First, they tried to explain the music using only the standard instruments they already knew about.
2. The Extended Model: Then, they tried adding the "strange new instrument" (the exotic $T_{c\bar{c}}(4100)^-$) to see if it made the music sound better.

The Findings: The Ghost Disappears

Here is what they discovered:

1. The Known Instruments Were Enough: When they used only the known standard particles (the $K^*$ resonances), the model fit the data very well. The "music" was explained perfectly without needing a new instrument.
2. The Exotic Candidate Faded: When they added the exotic particle to the model, it did make the fit look slightly better mathematically. However, when they accounted for all the possible "noise" and errors in their equipment (systematic uncertainties), the evidence for this new particle disappeared.
3. The Verdict: The "bump" they saw in the previous study was likely just a statistical fluke or a misunderstanding of the background noise. With more data, the case for the exotic $T_{c\bar{c}}(4100)^-$ particle is not confirmed.

The Analogy:
Imagine you hear a strange noise in your attic. You think it's a ghost. You call a detective (the first study), and they say, "Yeah, that sounds like a ghost."
You wait a year, get better recording equipment, and record the noise again (this study). This time, the detective listens closely and says, "Actually, that's just the wind blowing through a loose window. The ghost isn't there."

The Other Result: Measuring the "Frequency"

While they didn't find the ghost, they did measure something very important: How often does this decay happen?

They calculated the branching fraction.

• Analogy: If you have a bag of 10,000 $B^0$ mesons, how many of them will break apart into this specific trio of particles?
• The Result: They found that about 582 out of every 1 million $B^0$ mesons decay this way.
• They reported this number with high precision, giving physicists a solid reference point for future theories.

Summary

• What they did: They analyzed a massive amount of collision data to study how a specific particle breaks apart.
• What they looked for: Evidence of a new, exotic particle made of four quarks.
• What they found: The data is perfectly explained by known, standard particles. The evidence for the exotic particle seen in an earlier, smaller study is not confirmed with this larger dataset.
• What they measured: They precisely measured the probability of this decay occurring, providing a new standard number for the scientific community.

In short: The scientists looked hard for a new type of particle, but the universe told them, "Nope, just the usual suspects this time." They also took a very accurate census of how often this event happens.

Technical Summary

Technical Summary: Amplitude Analysis of $B^0 \to \eta_c(1S)K^+\pi^-$ Decays

Problem and Context
The paper addresses the search for exotic hadronic states, specifically those with quark compositions beyond the conventional $q\bar{q}$ mesons and $qqq$ baryons, such as tetraquarks ($qq\bar{q}\bar{q}$). Previous theoretical models and experimental observations, including the $T_{cc}(3900)^-$ and various charmonium-like states, suggest the existence of such particles. A specific candidate, the $T_{c\bar{c}}(4100)^-$, was previously reported by the LHCb collaboration in a 2018 analysis of the $B^0 \to \eta_c K^+ \pi^-$ decay channel. This state was hypothesized to be an isovector resonance decaying to $\eta_c \pi^-$, potentially related to the $T_{cc}(3900)^-$ via heavy-quark spin symmetry. However, the initial evidence was based on a dataset of 4.7 fb$^{-1}$. The current study aims to re-evaluate the existence and properties of this exotic candidate using a significantly larger dataset (9 fb$^{-1}$) to determine if the previous observation holds under increased statistical power and improved systematic control.

Methodology
The analysis utilizes proton-proton collision data collected by the LHCb detector at center-of-mass energies of $\sqrt{s} = 7, 8,$ and $13$ TeV, corresponding to an integrated luminosity of 9 fb$^{-1}$. The study focuses on the decay chain $B^0 \to \eta_c(1S) K^+ \pi^-$, with the $\eta_c(1S)$ reconstructed via its decay to a proton-antiproton pair ($\eta_c \to p\bar{p}$). This specific reconstruction mode was chosen to avoid the systematic uncertainties associated with distinguishing kaons and pions in the final state, which is necessary when using mesonic $\eta_c$ decay modes.

The analysis proceeds through the following steps:

1. Candidate Selection: $B^0$ candidates are reconstructed in the $p\bar{p}K^+\pi^-$ final state. A kinematic fit constrains the $B^0$ mass and origin vertex. A Boosted Decision Tree (BDT) algorithm is employed to suppress combinatorial background, trained separately for Run 1 and Run 2 data.
2. Yield Extraction: A two-dimensional extended maximum-likelihood fit is performed on the $m_{p\bar{p}K^+\pi^-}$ and $m_{p\bar{p}}$ mass distributions to extract the signal yields for both the $B^0 \to \eta_c K^+ \pi^-$ signal and the normalization channel $B^0 \to J/\psi K^+ \pi^-$. The $\eta_c$ and $J/\psi$ signals are isolated within specific mass windows.
3. Dalitz Plot (DP) Analysis: The dynamics of the three-body decay are analyzed using an unbinned maximum-likelihood fit on the Dalitz plot, defined by the squared invariant masses $m^2_{K^+\pi^-}$ and $m^2_{\eta_c\pi^-}$. The analysis accounts for the finite natural width of the $\eta_c$ meson by using four-momenta rather than fixed mass values.
4. Amplitude Modeling:
   • Baseline Model: Includes only known $K^*_0$ resonances decaying to $K^+\pi^-$ (specifically $K^*(892)^0$, $K^*(1410)^0$, $K^*_0(1430)^0$, $K^*_2(1430)^0$, $K^*(1680)^0$, and $K^*_0(1950)^0$) and a non-resonant S-wave component modeled with the LASS function.
   • Extended Model: Adds an exotic amplitude corresponding to the $T_{c\bar{c}}(4100)^-$ candidate decaying to $\eta_c \pi^-$. This candidate is tested under two quantum number hypotheses ($J^P = 0^+$ and $J^P = 1^-$).
5. Systematic Uncertainties: Extensive studies are conducted to evaluate systematic effects, including background parametrization, efficiency modeling, Dalitz plot boundary vetoes, and variations in resonance lineshape parameters.

Key Results

• Exotic Resonance Search: When an exotic $T_{c\bar{c}}(4100)^-$ amplitude is added to the baseline model, the fit quality improves slightly. The statistical significance of this contribution is found to be 3.6$\sigma$ when systematic uncertainties are neglected. However, when systematic uncertainties (particularly those related to background parametrization and efficiency modeling) are included, the significance drops to 2.5$\sigma$. Consequently, the paper concludes that the evidence for the $T_{c\bar{c}}(4100)^-$ state is not confirmed with the current dataset. The $J^P = 1^-$ hypothesis is preferred over $0^+$, but the discrimination is not statistically significant once systematics are considered.
• Amplitude Analysis: The data are well-described by the baseline model containing only known $K^*_0$ resonances. The dominant contributions are from $B^0 \to \eta_c K^*(892)^0$ (fit fraction $\approx 49\%$) and $B^0 \to \eta_c K^*_0(1430)^0$ (fit fraction $\approx 31\%$). The fit fractions, magnitudes, and phases for all intermediate $K^*_0$ states are reported with statistical and systematic uncertainties.
• Branching Fraction Measurement: The inclusive branching fraction for $B^0 \to \eta_c(1S) K^+ \pi^-$ is measured relative to the $B^0 \to J/\psi K^+ \pi^-$ normalization channel. The result is:
  $$\mathcal{B}(B^0 \to \eta_c(1S) K^+ \pi^-) = (5.82 \pm 0.20 \text{ (stat)} \pm 0.23 \text{ (syst)} \pm 0.55 \text{ (ext)}) \times 10^{-4}$$
  This value is consistent with the world average and the previous LHCb measurement. Product branching fractions for the intermediate $K^*_0$ resonances are also provided.

Significance
The paper claims that this analysis supersedes the previous LHCb study (Ref. [20]) by utilizing a dataset approximately twice the size. The primary significance lies in the rigorous re-evaluation of the $T_{c\bar{c}}(4100)^-$ candidate. While the previous analysis reported evidence for this exotic state, the current, larger dataset does not confirm its existence when systematic uncertainties are properly accounted for. The results suggest that the observed structures in the $\eta_c \pi^-$ system can be adequately described by known $K^*_0$ resonances and their interferences, without the need for an exotic tetraquark interpretation. Additionally, the paper provides the most precise measurement to date of the inclusive branching fraction for this decay mode and a detailed characterization of the $K^*_0$ contributions, serving as a benchmark for future searches for exotic hadrons in similar decay channels.