Measurement of inclusive production of charmonium states in $b$-hadron decays via their decay into $\phi \phi$

Using LHCb Run 2 data, this study measures the inclusive production branching fractions of $\chi_{c0}$, $\chi_{c1}$, and $\chi_{c2}$ charmonium states in $b$-hadron decays via their $\phi\phi$ decay mode, reports the production rate of $\eta_c(2S)$, and achieves the most precise measurement to date of the $\eta_c(1S)$ mass.

The Gist

Imagine the Large Hadron Collider (LHC) at CERN as the world's most powerful "particle smasher." It fires protons at each other at nearly the speed of light, creating a chaotic explosion of new particles. Among the debris, heavy particles called b-hadrons are born. These are like unstable, heavy-duty delivery trucks that carry a "charm" particle inside them.

This paper is about the LHCb team acting as high-speed traffic cops and forensic investigators. They are trying to figure out exactly what happens when these heavy "delivery trucks" (b-hadrons) break down. Specifically, they are looking for a very specific, rare type of cargo: charmonium.

The Mystery Cargo: Charmonium

Think of charmonium as a "charm couple." It's a tiny particle made of a charm quark and an anti-charm quark holding hands. There are different "families" of these couples, named after Greek letters like $\eta_c$ (eta-c) and $\chi_c$ (chi-c).

Usually, these couples are hard to spot because they decay (fall apart) into invisible or messy particles. But in this study, the scientists are looking for a very specific "signature" of decay:

1. The heavy b-hadron breaks apart.
2. It releases a charmonium couple.
3. That couple immediately splits into two $\phi$ (phi) mesons.
4. Those $\phi$ mesons then split into pairs of kaons (which are like heavy cousins of pions).

So, the team is hunting for a chain reaction: b-hadron $\rightarrow$ Charmonium $\rightarrow$ Two $\phi$ mesons $\rightarrow$ Four Kaons.

The Detective Work: How They Did It

The team used data from 2015 to 2018 (known as "Run 2"), which is like having a massive library of 5.9 billion "pages" of collision data.

1. The Filter: They built a digital sieve to catch only the events where four kaons were found.
2. The Reconstruction: They checked if those four kaons could be grouped into two pairs, where each pair looked like a $\phi$ meson. If they did, they checked if those two $\phi$ mesons looked like they came from a charmonium parent.
3. The "Fingerprint" (Mass): Every particle has a specific "weight" (mass). The scientists looked at the total weight of the four kaons. If they saw a spike (a bump) in the data at a specific weight, it meant a specific charmonium particle had been found.
   • They found bumps for the $\eta_c(1S)$, $\eta_c(2S)$, and the three $\chi_c$ states ($\chi_{c0}, \chi_{c1}, \chi_{c2}$).

The Big Discoveries

1. How Often Does This Happen? (Branching Fractions)

The scientists wanted to know the odds: "If a b-hadron breaks, what are the chances it turns into a specific charmonium type?"

• They found that the $\chi_{c1}$ and $\chi_{c0}$ states are produced quite frequently (about 1.5 times out of every 1,000 b-hadron decays).
• The $\chi_{c2}$ is a bit rarer (about 0.5 times out of 1,000).
• They also measured the $\eta_c(2S)$, but since they don't know exactly how often that specific particle decays into $\phi$ mesons, they could only measure the combined probability of it being made and decaying that way.

Why does this matter?
Physicists have theories (like NRQCD) that predict these numbers. It's like having a recipe book for the universe. If the recipe says "add 1 cup of flour" but the cake comes out with 2 cups, the recipe is wrong. These measurements are a strict test of our "recipe" for how matter is created.

2. The Weighing Scale (Mass Measurements)

Just like you can weigh a fruit to see if it's ripe, physicists weigh particles to understand their nature.

• They measured the mass of the $\eta_c(1S)$ with incredible precision: 2984.1 MeV.
• This is the most precise measurement of this particle's weight ever made. It's like weighing a grain of sand and knowing its weight to within a fraction of a microgram. This helps refine our understanding of the strong force that holds quarks together.

3. The "Interference" Puzzle

One of the coolest parts of the paper is dealing with interference.
Imagine two waves in a pond meeting. If they peak at the same time, they make a huge wave (constructive interference). If one peaks while the other troughs, they cancel out (destructive interference).
In quantum physics, particles act like waves. The scientists realized that the signals for some of these particles were "talking" to each other, making the data look messy. They had to write complex math to untangle the waves and figure out the true signal. It was like trying to hear a single singer in a choir where everyone is singing slightly out of tune, but they managed to isolate the soloist perfectly.

The Bottom Line

This paper is a triumph of precision. The LHCb team didn't just find these particles; they counted them with high accuracy and weighed them with record-breaking precision.

• The Analogy: Imagine trying to find a specific, rare coin in a mountain of gravel. Not only did they find the coin, but they also counted exactly how many coins were in the mountain, weighed the coin to the nearest microgram, and figured out that the coins were sometimes "bumping" into each other in a way that made them harder to spot.
• The Impact: These results confirm that our current theories of particle physics are mostly on the right track, but they also provide the ultra-precise data needed to fix the tiny cracks in the theory. It's a step forward in understanding the fundamental building blocks of our universe.

Technical Summary

1. Problem and Motivation

The production of charmonium ($c\bar{c}$) states in $b$-hadron decays is a critical testing ground for Quantum Chromodynamics (QCD), specifically within the Non-Relativistic QCD (NRQCD) framework. Theoretical predictions rely on separating short-distance perturbative contributions from long-distance non-perturbative parameters (matrix elements).

• Theoretical Challenge: For $P$-wave charmonia ($\chi_{cJ}$), theoretical predictions for absolute production rates in $b$-decays are imprecise because the Color Singlet (CS) contribution becomes negligible or negative, making the dominant Color Octet (CO) contribution difficult to constrain. However, the relative production rates of $\chi_{cJ}$ states ($J=0,1,2$) offer a cleaner test, as both CS and CO contributions are expected to scale with $2J+1$.
• Experimental Gap: Previous measurements by LHCb (Run 1) using the $\phi\phi$ decay channel were statistically limited. Furthermore, precise measurements of charmonium masses and widths in $b$-decays provide stringent tests of quarkonium spectroscopy.
• Goal: This analysis aims to improve the precision of inclusive production branching fractions for $\eta_c(1S)$, $\eta_c(2S)$, and $\chi_{cJ}(1P)$ states in $b$-hadron decays, and to measure their masses and widths with higher precision using the larger Run 2 dataset.

2. Methodology

Dataset and Detector

• Data: Proton-proton collision data collected by the LHCb experiment during Run 2 (2015–2018), corresponding to an integrated luminosity of 5.9 fb$^{-1}$.
• Decay Chain: The analysis reconstructs charmonium states decaying into a pair of $\phi$ mesons, which subsequently decay into charged kaon pairs:
  $$b \to \mathcal{H}_c X \to (\phi\phi) X \to (K^+K^-)(K^+K^-) X$$
  where $\mathcal{H}_c$ represents $\eta_c(1S)$, $\eta_c(2S)$, $\chi_{c0}$, $\chi_{c1}$, or $\chi_{c2}$.
• Selection:
  • Four kaon candidates with $p > 5$ GeV and $p_T > 0.65$ GeV.
  • Kaon pairs forming $\phi$ candidates within the mass window $1002\text{--}1038$ MeV.
  • $\phi\phi$ pairs forming charmonium candidates within $2800\text{--}4000$ MeV.
  • Displacement Cuts: To isolate $b$-hadron decays from prompt production, tracks must have a significant impact parameter relative to the primary vertex, and a minimum pseudo-proper lifetime is required.

Analysis Techniques

1. Yield Extraction (2D Fit):

   • A two-dimensional unbinned maximum-likelihood fit is performed on the two $K^+K^-$ mass combinations in 5 MeV bins of the four-kaon mass.
   • The $\phi$ signal is modeled with a relativistic Breit-Wigner (RBW) convolved with a Gaussian.
   • Backgrounds include combinatorial $K^+K^-$ pairs and mis-reconstructed $\phi$ candidates.

2. Mass and Width Fit:

   • The $\phi\phi$ mass spectra are fitted simultaneously for all years using a binned $\chi^2$ approach.
   • Signal Model: Each charmonium state is modeled by an RBW function convolved with a sum of two Gaussians (to model detector resolution).
   • Interference: A critical innovation in this analysis is the inclusion of interference terms between the resonant RBW signal and a non-resonant $\phi\phi$ background (modeled as a plane wave). This is particularly important for states with orbital angular momentum $L=1$ ($\eta_c$) and $L=0$ ($\chi_{c0}, \chi_{c2}$).
   • Angular Momentum: Constraints are applied based on Bose symmetry (e.g., $L=1$ is forbidden for $\chi_{c1} \to \phi\phi$).

3. Branching Fraction Calculation:

   • Relative Ratios: Ratios of efficiency-corrected yields ($N/\epsilon$) are calculated between states.
   • Absolute Normalization: Absolute branching fractions are derived using $b \to \eta_c(1S)X$ as a normalization channel, utilizing world-average values for $B(b \to \eta_c(1S)X)$ and $B(\eta_c(1S) \to \phi\phi)$.
   • Efficiency: Determined via simulation (Pythia/EvtGen/Geant4) and corrected for data-simulation differences in tracking and Particle Identification (PID).

3. Key Contributions

• First High-Precision Measurement of $\chi_{cJ}$ in $b$-decays via $\phi\phi$: This paper provides the most precise measurements to date of the inclusive production of $\chi_{c0}$, $\chi_{c1}$, and $\chi_{c2}$ in $b$-hadron decays using this specific decay channel.
• Interference Modeling: The analysis rigorously accounts for interference between resonant signals and non-resonant $\phi\phi$ backgrounds, a factor previously neglected or treated simplistically. This reduces systematic biases in yield extraction.
• Precision Spectroscopy: It delivers the most precise single measurement of the $\eta_c(1S)$ mass to date ($2984.1 \pm 0.5 \pm 0.5$ MeV) and provides competitive measurements for $\eta_c(2S)$ and $\chi_{cJ}$ masses and widths.
• $\eta_c(2S)$ Production: It sets the first constraints on the product of branching fractions $B(b \to \eta_c(2S)X) \times B(\eta_c(2S) \to \phi\phi)$, as the individual decay width $B(\eta_c(2S) \to \phi\phi)$ is currently unknown.

4. Results

Branching Fractions

The absolute branching fractions for $\chi_{cJ}$ production in $b$-hadron decays are measured as:

• $\mathcal{B}(b \to \chi_{c0}X) = (1.34 \pm 0.13_{\text{stat}} \pm 0.06_{\text{syst}} \pm 0.37_{\text{ext}}) \times 10^{-3}$
• $\mathcal{B}(b \to \chi_{c1}X) = (1.58 \pm 0.12 \pm 0.09 \pm 0.44) \times 10^{-3}$
• $\mathcal{B}(b \to \chi_{c2}X) = (0.55 \pm 0.08 \pm 0.05 \pm 0.15) \times 10^{-3}$

The ratio of production rates relative to $\chi_{c0}$ are:

• $\mathcal{B}(b \to \chi_{c1}X) / \mathcal{B}(b \to \chi_{c0}X) = 1.18 \pm 0.13 \pm 0.08 \pm 0.07$
• $\mathcal{B}(b \to \chi_{c2}X) / \mathcal{B}(b \to \chi_{c0}X) = 0.41 \pm 0.07 \pm 0.04 \pm 0.03$

The product of branching fractions for $\eta_c(2S)$ is:

• $\mathcal{B}(b \to \eta_c(2S)X) \times \mathcal{B}(\eta_c(2S) \to \phi\phi) = (4.0 \pm 0.6 \pm 0.6 \pm 1.1) \times 10^{-7}$

Note: The third uncertainty in all results stems from the limited precision of external normalization constants (specifically $B(\eta_c(1S) \to \phi\phi)$).

Masses and Widths

• $\eta_c(1S)$ Mass: $2984.1 \pm 0.5 \pm 0.5$ MeV. This matches the world average and is the most precise single measurement to date.
• $\eta_c(1S)$ Width: $34.0 \pm 1.1 \pm 0.7$ MeV.
• $\chi_{c0}$ Width: $16.3 \pm 2.9 \pm 0.4$ MeV.
• $\eta_c(2S)$ Width: $18.6 \pm 5.1 \pm 1.2$ MeV.
• All measured masses are consistent with world averages within $2\sigma$.

5. Significance

• Validation of NRQCD: The measured ratios $\chi_{c1}/\chi_{c0}$ and $\chi_{c2}/\chi_{c0}$ provide crucial data points to test NRQCD factorization assumptions. The results are consistent with the expectation that production scales roughly with $2J+1$, though with significant deviations that require further theoretical refinement.
• Spectroscopy Precision: The $\eta_c(1S)$ mass measurement sets a new benchmark for precision in heavy quarkonium spectroscopy, reducing uncertainties that affect lattice QCD comparisons.
• Methodological Advancement: The successful implementation of interference modeling in the $\phi\phi$ channel demonstrates a robust technique for future analyses of similar hadronic decays where non-resonant backgrounds are significant.
• Future Physics: These results constrain the long-distance matrix elements in NRQCD, improving the predictive power of QCD for heavy quarkonium production in both $b$-decays and prompt production.

In conclusion, this paper represents a significant step forward in understanding charmonium production mechanisms and provides the most precise spectroscopic parameters for several charmonium states derived from $b$-hadron decays.