Observation of $B_c^+ \to D h^+ h^-$ decays

Using LHCb data corresponding to an integrated luminosity of $9~\text{fb}^{-1}$, this paper reports the first observation of the $B_c^+ \to D^+ K^+ \pi^-$, $B_c^+ \to D^{*+} K^+ \pi^-$, and $B_c^+ \to D_s^+ K^+ K^-$ decays and determines their branching fractions relative to the $B_c^+ \to B_s^0 \pi^+$ decay, opening new avenues for studying CP violation in beauty mesons.

The Gist

Imagine the universe as a giant, high-speed particle collider, like a cosmic racetrack where tiny subatomic particles zoom around at nearly the speed of light. In this race, the LHCb experiment at CERN acts like a super-sophisticated camera and detective team, trying to catch rare, fleeting moments where these particles crash and transform into something new.

This paper is about the team catching a very specific, rare type of "traffic accident" involving a particle called the $B_c^+$ meson.

The Main Character: The $B_c^+$ Meson

Think of the $B_c^+$ meson as a unique, heavy-duty delivery truck in the particle world. Unlike most trucks that carry just one type of heavy cargo, this one carries two different heavy "quarks" (the building blocks of matter) at the same time. Because it's so heavy and unstable, it doesn't last long; it quickly breaks apart into smaller, lighter particles.

Scientists have been studying these trucks for years, but they are like rare, elusive ghosts. They are much harder to find than other common particles because they are produced so rarely in the collisions.

The Discovery: Three New Routes

In this paper, the LHCb team announces they have spotted three new ways this heavy truck breaks apart. Before this, these specific routes were just theories or guesses. Now, they are confirmed observations.

The team looked at data from 9 years of particle collisions (imagine reviewing 9 years of security footage from the racetrack). They found that the $B_c^+$ meson was decaying (breaking apart) into three specific combinations of particles:

1. A charm meson ($D^+$) plus a pion and a kaon.
2. An excited charm meson ($D^{*+}$) plus a pion and a kaon.
3. A strange charm meson ($D_s^+$) plus two kaons.

The Analogy: Imagine you have a mysterious, complex machine (the $B_c^+$) that you know breaks down. You've seen it break into a few common ways. But suddenly, you discover three new ways it can fall apart, and you've never seen these specific combinations before. This paper says, "Yes, we saw them! Here is the proof."

How They Did It: The Detective Work

Finding these particles is like looking for a specific needle in a haystack the size of a mountain. The collisions produce billions of particles, but the $B_c^+$ decays they wanted are incredibly rare.

1. The Filter (The BDT): The team used a smart computer program (a "Boosted Decision Tree") trained to act like a seasoned detective. It was taught to ignore the "noise" (common, boring particles) and flag anything that looked like the rare decay they were hunting.
2. The Comparison (The Normalization): To measure exactly how often these rare events happened, they compared them to a "standard candle"—a known, common decay ($B_c^+ \to B_s^0 \pi^+$). It's like trying to count how many rare blue cars pass a street. Instead of counting every single car, you count how many blue cars pass compared to how many red cars (which you know the exact number of) pass. This helps cancel out errors in the counting.
3. The Mass Spectra (The Fingerprint): When they plotted the data, they saw clear "bumps" or peaks in the graphs (Figure 2). These bumps are the fingerprints of the new particles. The peaks were high enough to be statistically significant (more than 5 "standard deviations"), which in the world of physics means, "We are 99.9999% sure this isn't a fluke."

What's Inside the Decay?

The paper also looked at how these particles broke apart. It turns out they didn't just fall apart randomly. They went through intermediate steps, like a relay race.

• Some went through a $K^*$ resonance (a short-lived particle that acts like a stepping stone).
• Others went through a $\phi$ meson (another stepping stone made of strange quarks).

The Metaphor: Imagine the $B_c^+$ meson is a parent dropping off kids at school. Instead of just driving them straight there, the parent stops at a gas station ($K^*$) or a coffee shop ($\phi$) first. By studying these "stops," scientists can understand the rules of the road (the laws of physics) that govern how these particles interact.

Why Does This Matter?

You might ask, "So what? Who cares about a rare particle breaking apart?"

1. Testing the Rulebook (The Standard Model): The Standard Model is the current "rulebook" of physics. It predicts how particles should behave. By measuring exactly how often these rare decays happen (the "branching fractions"), scientists can check if the rulebook is correct. If the numbers don't match the predictions, it could mean there's "New Physics" hiding in the shadows—something the current rulebook doesn't know about.
2. The Mystery of Matter vs. Antimatter (CP Violation): The universe is made of matter, but the Big Bang should have created equal amounts of matter and antimatter, which would have destroyed each other. Something must have tipped the scales. These decays are a perfect place to look for "CP violation"—a subtle difference in how matter and antimatter behave. If we find a difference here, it helps explain why we exist at all.
3. Opening the Door: This paper is just the beginning. Now that we know these three routes exist, the LHCb team (which is getting upgraded to be even more powerful) can study them in much greater detail. They can look for those subtle differences between matter and antimatter that could unlock the secrets of the universe.

The Bottom Line

The LHCb collaboration has successfully found three new, rare ways a heavy particle called the $B_c^+$ meson breaks apart. They used a massive amount of data and clever statistical tools to prove these events are real. This discovery gives physicists a new set of tools to test the fundamental laws of nature and perhaps one day explain why the universe is made of matter instead of nothingness.

It's a bit like finding three new, secret passages in a castle you thought you knew completely. Now, you can explore them and see what treasures (or new physics) lie inside.

Technical Summary

1. Problem and Motivation

The $B_c^+$ meson is unique among $B$ mesons as it contains two heavy quarks ($b$ and $\bar{c}$), allowing it to decay via three distinct weak interaction processes: $b$-quark transition, $c$-quark transition, and weak annihilation ($\bar{b}c \to W^+ \to \bar{q}q'$). While previous studies have focused on two-body decays or specific open-charm modes, the dynamics of three-body decays involving open charm ($B_c^+ \to D h^+ h^-$) remain largely unexplored experimentally.

These decays are theoretically significant because:

• They involve interfering amplitudes from $bc$ annihilation and $b \to u\bar{u}s(d)$ tree and penguin diagrams.
• They offer a rich environment for studying non-perturbative Quantum Chromodynamics (QCD) and resonance structures (e.g., excited $K^0$ or $D^0$ resonances, $\phi$ mesons).
• They provide a new avenue for investigating Charge-Parity (CP) violation in beauty mesons, a crucial test for the Standard Model (SM) and potential New Physics.

Prior to this work, only a few $B_c^+$ decays with open charm hadrons had been observed (e.g., $B_c^+ \to D^0 K^+$ and $B_c^+ \to D^+ K^{*0}$). This paper addresses the gap by searching for and observing three specific three-body decay channels.

2. Methodology

Data Sample and Detector:

• Experiment: LHCb (Large Hadron Collider beauty) experiment at CERN.
• Data: Proton-proton ($pp$) collision data collected at center-of-mass energies of 7, 8, and 13 TeV.
• Integrated Luminosity: $9 \text{ fb}^{-1}$.
• Detector: A single-arm forward spectrometer ($2 < \eta < 5$) optimized for heavy-flavor physics, featuring a vertex locator, tracking system, Ring Imaging Cherenkov (RICH) detectors for particle identification (PID), calorimeters, and muon chambers.

Reconstruction and Selection:

• Signal Channels:
  1. $B_c^+ \to D^+ K^+ \pi^-$
  2. $B_c^+ \to D^{*+} K^+ \pi^-$ (where $D^{*+} \to D^0 \pi^+$)
  3. $B_c^+ \to D_s^+ K^+ K^-$
• Normalization Channel: $B_c^+ \to B_s^0 \pi^+$ (with $B_s^0 \to D_s^- \pi^+$ and $D_s^- \to K^+ K^- \pi^-$). This channel is used to cancel systematic uncertainties related to production and detection efficiencies.
• Candidate Selection:
  • Trigger: Hardware trigger requiring high transverse energy ($E_T > 3.5$ GeV) in the calorimeter; software trigger requiring displaced vertices and high $p_T$ tracks.
  • Offline Cuts: Kinematic constraints, PID requirements, and vertex quality cuts.
  • Multivariate Analysis (MVA): A Boosted Decision Tree (BDT) classifier was trained to separate signal from combinatorial background. The training used simulated signal and data sidebands ($6450-6750$ MeV/$c^2$).
  • Mass Constraints: A kinematic fit was applied with constraints on the $D$ meson masses to improve resolution, particularly for the $D^{*+}$ channel.

Analysis Strategy:

• Yield Extraction: An unbinned maximum-likelihood fit was performed simultaneously on the invariant mass spectra of all four channels ($D^+ K^+ \pi^-$, $D^{*+} K^+ \pi^-$, $D_s^+ K^+ K^-$, and $B_s^0 \pi^+$).
  • Signal Shape: Sum of a Gaussian and a Crystal Ball function (with power-law tails).
  • Background Shape: Exponential function.
  • Normalization Contamination: The $B_c^+ \to B_s^{*0} \pi^+$ component (where the photon is lost) was modeled using RapidSim.
• Efficiency Correction: Due to the non-uniform phase space distribution of three-body decays, efficiencies were calculated as a function of two-dimensional mass combinations ($m_{D^{(*)}K}$, $m_{D^{(*)}h}$). Corrections were applied for PID, trigger, and $p_T$ weighting (calibrated using $B_c^+ \to J/\psi \pi^+$).
• Systematic Uncertainties: Evaluated for fit models, simulation sample size, $p_T$ weighting, efficiency binning, trigger corrections, and PID calibration.

3. Key Contributions

1. First Observation: This paper reports the first observation of the following three decay modes:
   • $B_c^+ \to D^+ K^+ \pi^-$
   • $B_c^+ \to D^{*+} K^+ \pi^-$
   • $B_c^+ \to D_s^+ K^+ K^-$
2. Branching Fraction Ratios: The paper provides precise measurements of the branching fractions (BF) for these modes relative to the normalization channel $B_c^+ \to B_s^0 \pi^+$.
3. Resonance Structure Analysis: The study identifies the dominant intermediate resonant structures in these decays, providing insight into the decay dynamics.
4. Methodological Framework: It establishes a robust framework for measuring multi-body $B_c^+$ decays, including the application of 2D efficiency maps and simultaneous fitting strategies.

4. Results

Signal Yields and Significance:
All three signal channels were observed with statistical significances exceeding 5 standard deviations (after accounting for systematic uncertainties from the fit model).

• $B_c^+ \to D^+ K^+ \pi^-$: $230 \pm 23$ events.
• $B_c^+ \to D^{*+} K^+ \pi^-$: $87 \pm 12$ events.
• $B_c^+ \to D_s^+ K^+ K^-$: $68 \pm 13$ events.

Branching Fraction Ratios ($R$):
The ratios $R(B_c^+ \to D h^+ h^-) \equiv \mathcal{B}(B_c^+ \to D h^+ h^-) / \mathcal{B}(B_c^+ \to B_s^0 \pi^+)$ were determined as:

• $R(B_c^+ \to D^+ K^+ \pi^-) = (1.96 \pm 0.23_{\text{stat}} \pm 0.08_{\text{syst}} \pm 0.10_{\text{BF}}) \times 10^{-3}$
• $R(B_c^+ \to D^{*+} K^+ \pi^-) = (3.67 \pm 0.55_{\text{stat}} \pm 0.24_{\text{syst}} \pm 0.20_{\text{BF}}) \times 10^{-3}$
• $R(B_c^+ \to D_s^+ K^+ K^-) = (1.61 \pm 0.35_{\text{stat}} \pm 0.13_{\text{syst}} \pm 0.07_{\text{BF}}) \times 10^{-3}$

(Note: The third uncertainty is due to the limited precision of the $D$-meson branching fractions.)

Resonance Observations:

• $B_c^+ \to D^+ K^+ \pi^-$: A significant peak compatible with the $K^{*}(892)^0$ state and a smaller peak compatible with $K^{*}(1430)^0$ were observed in the $K^+ \pi^-$ mass spectrum.
• $B_c^+ \to D^{*+} K^+ \pi^-$: A peak compatible with $K^{*}(892)^0$ was visible in the $K^+ \pi^-$ spectrum.
• $B_c^+ \to D_s^+ K^+ K^-$: A significant contribution in the low-mass $K^+ K^-$ region was observed, originating largely from the $\phi(1020)$ resonance. A hint of the $f'_2(1525)$ state was also noted.

5. Significance

• Standard Model Tests: These measurements provide critical data to test theoretical models of $B_c^+$ decays, particularly those involving the interplay between annihilation and tree-level diagrams. The results favor a dominant annihilation contribution in the decay amplitude, consistent with previous findings but now extended to three-body modes.
• CP Violation Prospects: The observation of these channels opens a new frontier for searching for local CP violation. The rich interference patterns in three-body decays (Dalitz plot analyses) are highly sensitive to CP-violating phases, which are difficult to access in two-body decays.
• Future Physics: With the LHCb Upgrade I (completed recently) and the upcoming Run 3/4 data collection, the increased luminosity and improved trigger capabilities will allow for amplitude analyses of these modes. This will enable precise measurements of polarization fractions, resonance properties, and CP asymmetries, deepening the understanding of heavy-quark dynamics and potential New Physics beyond the Standard Model.

In conclusion, this paper marks a significant milestone in $B_c^+$ physics by expanding the known decay landscape to include three-body open-charm modes, providing the first quantitative branching fraction measurements, and laying the groundwork for future precision CP violation studies.