⚛️ phenomenology

Exploring charmonia $\eta_c(6S,7S)$ in the $\Lambda_c^+\bar{\Lambda}_c^-$ invariant mass spectrum of $B^0 \to K_S^0\Lambda_c^+\bar{\Lambda}_c^-$

This paper analyzes the $B^0 \to K_S^0 \Lambda_c^+ \bar{\Lambda}_c^-$ decay to propose that the observed 4.63 GeV enhancement in the $\Lambda_c^+\bar{\Lambda}_c^-$ spectrum corresponds to the charmonium state $\eta_c(6S)$ , while noting a significant discrepancy between the fitted production rate and theoretical predictions from factorization and hadron-loop mechanisms.

Original authors: Cheng-Xi Liu, Jun-Zhang Wang, Xiang Liu

Published 2026-08-18

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CC0 1.0

Original authors: Cheng-Xi Liu, Jun-Zhang Wang, Xiang Liu

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Deep within the subatomic world, particles known as charmonia act as a kind of Rosetta Stone for physicists. These are not fundamental particles themselves, but rather bound systems made of a charm quark and its antimatter twin, the anti-charm quark, held together by the strong nuclear force. Much like the hydrogen atom in chemistry, which helped scientists map out the rules of electrons, the family of charmonia provides a testing ground for the laws governing how quarks interact. However, while the lower-energy members of this family are well cataloged, the higher-energy, more excited states remain a mystery. Finding these missing pieces is crucial because they reveal how the strong force behaves when particles are pushed to their limits, potentially exposing gaps in our current understanding of the universe's building blocks.

Recently, a team of researchers turned their attention to a specific decay process observed by the LHCb collaboration at CERN. In this event, a particle called a neutral B meson transforms into a strange neutral kaon and a pair of heavy baryons known as a Lambda-c and an anti-Lambda-c. When scientists plotted the mass of the Lambda-c and anti-Lambda-c pair, they noticed something peculiar: a small but noticeable pile-up of events around a mass of 4.63 billion electron volts. This accumulation was not a random fluctuation; it appeared consistently across different datasets, suggesting the presence of a hidden intermediate particle that briefly formed and then decayed into the observed pair. The question was simple yet profound: what was this particle?

To solve this puzzle, the researchers built a detailed mathematical model of the decay process, treating the event as a complex wave interference pattern. They started with what was already known, accounting for the background noise and two previously discovered particles, the Xi-c(2923) and Xi-c(2939), which also appear in related measurements. When they compared their model to the actual data, a gap remained near the 4.63 billion electron volt mark. The researchers then hypothesized that this gap was filled by a high-energy charmonium state, specifically a version of the eta-c particle in its sixth excited state, known as eta-c(6S). They also tested for a seventh excited state, eta-c(7S), to see if it could explain features at even higher masses.

By fixing the properties of these hypothetical particles based on theoretical predictions from a modified version of the Godfrey-Isgur model—a framework used to calculate the masses of quark systems—the team re-ran their analysis. The results were striking. When they included the eta-c(6S) with a mass of 4629.00 MeV and a width of 24.20 MeV, the model fit the data significantly better in the region of the 4.63 billion electron volt accumulation. Adding the eta-c(7S) at 4718.84 MeV further improved the description of the higher-mass part of the spectrum. The data now aligned with the theory, suggesting that the mysterious pile-up of events is indeed the signature of these high-lying charmonium states.

However, the story did not end with a simple identification. The researchers then asked a deeper question: how likely is it for a B meson to produce such a heavy charmonium state, and how likely is that state to decay into the specific baryon pair observed? They performed two separate calculations to estimate these probabilities. The first method, known as standard factorization, is a standard way to estimate production rates in particle physics. The second method involved a complex mechanism where the particles temporarily form a loop of other hadrons before decaying. When they combined these theoretical estimates, the predicted rate was far lower than what the experimental data suggested. The data implied a production rate that was roughly one to three orders of magnitude higher than the standard theoretical predictions.

This large discrepancy does not mean the discovery is wrong; rather, it suggests that our current tools for calculating these processes are incomplete. The standard methods might be missing important long-distance effects or other mechanisms that allow these heavy particles to form more easily than expected. The researchers conclude that while the evidence strongly points to the eta-c(6S) and eta-c(7S) being the source of the observed structures, the exact nature of their production and decay requires more precise data. Future measurements from the LHCb and the Belle II experiments will be essential to map out the precise shape of these peaks and to determine whether the missing physics lies in how these particles are made or how they fall apart. For now, the 4.63 billion electron volt bump stands as a promising candidate for a missing piece of the charmonium family, waiting for the next generation of data to confirm its identity and reveal the secrets of the strong force at high energies.

Technical Summary: Exploring Charmonia $\eta_c(6S, 7S)$ in the $\Lambda_c^+ \bar{\Lambda}_c^-$ Invariant Mass Spectrum of $B^0 \to K_S^0 \Lambda_c^+ \bar{\Lambda}_c^-$

Problem Statement
The LHCb Collaboration recently reported the first observation of the decay $B^0 \to K_S^0 \Lambda_c^+ \bar{\Lambda}_c^-$ , identifying two $\Xi_c^+$ states ( $\Xi_c(2923)^+$ and $\Xi_c(2939)^+$ ) in the $\Lambda_c^+ K_S^0$ invariant mass spectrum. Upon inspection of the $\Lambda_c^+ \bar{\Lambda}_c^-$ invariant mass spectrum from this decay, an accumulation of events was observed near 4.63 GeV. Similar accumulations were noted in previous analyses of $\bar{B}^0 \to \bar{K}^0 \Lambda_c^+ \bar{\Lambda}_c^-$ and $B^- \to \Lambda_c^+ \bar{\Lambda}_c^- K^-$ , though no statistically significant structure was established. This enhancement lies well above the $\Lambda_c^+ \bar{\Lambda}_c^-$ threshold and suggests the potential contribution of high-lying hidden-charm states. The paper investigates whether this structure corresponds to the missing high-lying pseudoscalar charmonium states, specifically $\eta_c(6S)$ and $\eta_c(7S)$ .

Methodology
The authors employ a combined amplitude analysis of the $\Lambda_c^+ \bar{\Lambda}_c^-$ and $\Lambda_c^+ K_S^0$ invariant mass spectra. The methodology proceeds in three main stages:

Amplitude Modeling and Baseline Fit:
A coherent amplitude model is constructed including a non-resonant term and the two established $\Xi_c$ contributions. The model uses relativistic Breit-Wigner functions for resonances and integrates over the physical three-body phase space. A baseline fit is performed to the LHCb data to establish the background and $\Xi_c$ contributions. The fit quality is evaluated using $\chi^2$ minimization against the published spectra, accounting for asymmetric uncertainties.
Spectroscopic Constraints:
To test the hypothesis that the 4.63 GeV enhancement is an $\eta_c$ state, the authors calculate the masses and open-charm decay widths of $\eta_c(6S)$ and $\eta_c(7S)$ using a screened Godfrey-Isgur (MGI) potential model and the Quark-Pair-Creation (QPC or $^3P_0$ ) model.
- MGI Model: The linear confining potential is replaced by a screened form to account for string breaking. Parameters are varied ( $\mu = 0.11, 0.12, 0.13$ GeV) to define mass ranges.
- QPC Model: Open-charm decay widths are calculated by creating a light $q\bar{q}$ pair. The results provide total widths used as proxies for the resonance total widths in the amplitude fit.
- Representative masses and widths are selected from these theoretical ranges: $(M, \Gamma) = (4629.00, 24.20)$ MeV for $\eta_c(6S)$ and $(4718.84, 22.84)$ MeV for $\eta_c(7S)$ .
Spectroscopic Fit and Theoretical Estimates:
The amplitude model is extended to include fixed $\eta_c(6S)$ and $\eta_c(7S)$ terms. The fit quality is compared against the baseline. Subsequently, the authors calculate the branching fractions for the complete decay chain $B^0 \to K_S^0 \eta_c(nS) \to K_S^0 \Lambda_c^+ \bar{\Lambda}_c^-$ :
- Weak Production: Estimated using factorization for $B^0 \to K_S^0 \eta_c(nS)$ .
- Baryonic Decay: Estimated for $\eta_c(nS) \to \Lambda_c^+ \bar{\Lambda}_c^-$ using a hadron-loop mechanism involving intermediate open-charm meson pairs ( $D^{(*)}\bar{D}^{(*)}$ ) and ground-state baryon exchange.

Key Results

Spectral Fit: The inclusion of a fixed $\eta_c(6S)$ amplitude significantly improves the description of the $\Lambda_c^+ \bar{\Lambda}_c^-$ spectrum near 4.63 GeV, reducing the $\chi^2$ for that projection from 24.90 to 20.37. The addition of a fixed $\eta_c(7S)$ amplitude further improves the description of the higher-mass region (around 4.7 GeV), reducing the $\chi^2$ to 16.51. The $\Lambda_c^+ K_S^0$ spectrum description remains consistent with the baseline in both cases.
Branching Fraction Product: Using the narrow-width approximation and the fitted fraction of the $\eta_c(6S)$ signal ( $2.37\%$ ), the product of branching fractions is extracted as:
$\mathcal{B}(B^0 \to K_S^0 \eta_c(6S)) \times \mathcal{B}(\eta_c(6S) \to \Lambda_c^+ \bar{\Lambda}_c^-) \approx 6.2 \times 10^{-6}$
Theoretical Comparison:
- Theoretical estimates for the production rate $B^0 \to K_S^0 \eta_c(6S)$ via factorization yield values in the range $(0.63 \sim 1.22) \times 10^{-4}$ .
- Theoretical estimates for the decay $\eta_c(6S) \to \Lambda_c^+ \bar{\Lambda}_c^-$ via the hadron-loop mechanism yield branching fractions in the range $(5.96 \times 10^{-5} \sim 1.07 \times 10^{-3})$ , depending on the cutoff parameter $\alpha$ .
- The product of these theoretical estimates is roughly $1.7$ to $3.2$ orders of magnitude lower than the value extracted from the experimental fit ( $6.2 \times 10^{-6}$ ). Even with optimistic inputs for the decay constant, the theoretical product remains about 1.4 orders of magnitude lower.
- If the fitted product is combined with the upper end of the hadron-loop decay estimate, the implied production branching fraction would be $\sim 0.58\% \sim 10.4\%$ . The authors note that the upper end of this range is physically unreasonable for a color-suppressed exclusive decay, suggesting that the hadron-loop calculation likely favors larger cutoff values ( $\alpha \sim 3$ or higher) or that long-distance production mechanisms are significant.

Significance and Claims
The paper claims that the $\eta_c(6S)$ state is a plausible conventional candidate for the observed enhancement near 4.63 GeV in the $B^0 \to K_S^0 \Lambda_c^+ \bar{\Lambda}_c^-$ decay, as its predicted mass and width align with the data when included in the amplitude model. The $\eta_c(7S)$ is proposed as a candidate for the higher-mass structure.

However, the authors are modest regarding the definitive nature of these findings. They highlight a significant discrepancy between the experimentally extracted branching fraction product and the product of factorization and hadron-loop theoretical estimates. This discrepancy suggests that:

Factorization may not adequately describe the production of such highly excited charmonium states in $B$ decays.
Long-distance rescattering mechanisms may play a dominant role in production.
The hadron-loop calculation for the baryonic decay has substantial model dependence (specifically regarding the cutoff parameter).

The authors conclude that while the data supports the existence of a structure consistent with $\eta_c(6S)$ , more precise measurements of the line shapes and searches for similar structures in $B$ decays to kaons and open-charm meson pairs are required to confirm the nature of the state and clarify the production and decay mechanisms. The work serves as a phenomenological reference for future searches for high-lying $\eta_c$ states in baryonic $B$ -meson decays by LHCb and Belle II.

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