⚛️ phenomenology

Dynamical resonances $\Lambda(1405)$ , $\Lambda(1670)$ , and $f_0(980)$ in the process $\Lambda_c^+ \to p K^+ K^-$

Motivated by new BESIII data, this paper investigates the $\Lambda_c^+ \to p K^+ K^-$ decay by incorporating dynamically generated resonances $\Lambda(1405)$ , $\Lambda(1670)$ , $f_0(980)$ , and $\phi(1020)$ , successfully reproducing the observed invariant mass distributions and offering insights into the intrinsic properties of these low-lying excited states.

Original authors: Sheng-Chao Zhang, Gui-Ye Guo, Guan-Ying Wang, Li-Sheng Geng, En Wang, Ju-Jun Xie

Published 2026-10-20

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CC BY 4.0

Original authors: Sheng-Chao Zhang, Gui-Ye Guo, Guan-Ying Wang, Li-Sheng Geng, En Wang, Ju-Jun Xie

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ 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

The universe is built from a small set of fundamental particles that stick together to form everything we see, from the atoms in our bodies to the stars in the sky. Among these building blocks are protons and neutrons, which reside in the heart of every atom. However, nature is not limited to just these two; it also creates heavier, short-lived cousins known as baryons. Some of these exotic particles are made of three quarks, while others appear to be temporary clusters of a meson and a baryon, held together by the strong nuclear force. Understanding how these particles form, interact, and decay is like reading the instruction manual for the universe's most powerful glue. One particularly interesting family of these particles involves the lambda baryon, a heavy cousin of the proton that carries a strange quark. Scientists have long been puzzled by the specific ways these lambda particles behave when they are excited, or given extra energy, because their internal structure often defies simple explanation.

Recently, researchers turned their attention to a specific decay process involving a charmed lambda baryon, a heavy particle containing a charm quark. When this particle breaks apart, it transforms into a proton and two kaons, which are particles containing strange quarks. This transformation is a rare event, governed by the weak nuclear force, but it offers a unique window into the strong force that governs how particles interact in the final moments of the decay. By studying the masses of the particles produced in this breakup, scientists can see if any temporary, hidden structures formed during the process. These structures, called resonances, are like fleeting echoes of a particle that exists for only a fraction of a second before vanishing. The goal is to determine whether these echoes are simple combinations of three quarks or if they are more complex, dynamic molecules formed by particles orbiting each other.

A team of physicists, led by researchers from institutions in China, decided to investigate this specific decay using a sophisticated theoretical framework. They focused on the process where a charmed lambda baryon decays into a proton, a positive kaon, and a negative kaon. The team constructed a detailed model that accounted for the various ways these particles could interact with one another as they flew apart. They considered the possibility that the proton and the negative kaon might briefly form a temporary bond, creating a resonance known as the lambda(1405) or the lambda(1670). These are not standard particles but rather dynamic states that emerge from the interaction of mesons and baryons. The researchers also looked at the interaction between the two kaons, checking if they could form a temporary bond to create a scalar meson like the f0(980), or if they simply passed through the well-known vector meson phi(1020).

To test their ideas, the team compared their theoretical calculations against real experimental data collected by the BESIII collaboration, a group of scientists who have been studying these decays with high precision. The researchers simulated the entire decay process, calculating how often particles of certain masses should appear based on their model of the interactions. They then overlaid their predictions onto the actual measurements of the particle masses recorded by the detectors. The results were striking. The team found that their model, which included the dynamic formation of the lambda(1405) and lambda(1670) resonances, could accurately reproduce the patterns seen in the data. Specifically, the model successfully explained a sharp rise in the number of events right at the threshold where the proton and negative kaon could first form a bond, a feature linked to the lambda(1405). It also reproduced a distinct dip or "cusp" structure around 1670 million electron volts, a signature associated with the lambda(1670) resonance.

In the distribution of the two kaons, the model clearly showed the presence of the f0(980) meson, which appeared as an enhancement near the threshold where the two kaons could form a pair. The contribution from the phi(1020) meson was also clearly visible as a prominent peak, confirming its role as a major pathway in this decay. The researchers tested two different versions of their model: one where the mass and width of the phi(1020) were fixed to known values, and another where these values were allowed to float and be determined by the data. The second version provided an even better match to the experimental results, with a statistical fit that indicated a high level of agreement. This suggests that the dynamic generation of these particles through interactions is a robust description of what is happening.

The study also calculated the likelihood, or branching fraction, of these specific decay paths. The team found that the probability of the charmed lambda decaying into a proton and a kaon pair via the lambda(1405) and lambda(1670) routes matched the experimental measurements within the margin of error. Similarly, the calculated probabilities for the f0(980) and phi(1020) pathways aligned well with the observed data. These findings reinforce the idea that the lambda(1405) and lambda(1670) are not just simple three-quark particles but are likely dynamic states formed by the interaction of mesons and baryons. The research provides a coherent picture of how these complex interactions shape the final outcome of the decay, offering a clearer view of the forces at play.

Looking ahead, the authors suggest that future experiments with even higher precision will be crucial. Facilities like the proposed Super Tau-Charm Facility, along with continued work by the BESIII and Belle II collaborations, will be able to measure these decays with greater detail. Such improvements will allow scientists to probe the intrinsic properties of these low-lying excited states and the light scalar mesons with much greater clarity. By refining our understanding of how these particles form and interact, we move closer to a complete picture of the strong nuclear force and the diverse ways matter can organize itself in the subatomic world. This work does not just confirm a theory; it provides a concrete, tested framework for understanding the fleeting, dynamic structures that exist in the brief moments between the creation and destruction of heavy particles.

Problem Statement
The paper investigates the singly Cabibbo-suppressed decay process $\Lambda_c^+ \to pK^+K^-$ . While recent measurements by the BESIII Collaboration have provided updated branching fractions and invariant mass distributions for this channel, a complete theoretical description of the underlying dynamics remains necessary. Specifically, the observed spectra exhibit structures that cannot be explained solely by the quasi-two-body decay through the $\phi(1020)$ resonance. The authors aim to determine whether the inclusion of dynamically generated low-lying excited baryons ( $\Lambda(1405)$ and $\Lambda(1670)$ ) and scalar mesons ( $f_0(980)$ and $a_0(980)$ ), alongside the vector meson $\phi(1020)$ , can reproduce the experimental data, particularly the near-threshold enhancements and cusp structures observed in the $pK^-$ and $K^+K^-$ invariant mass spectra.

Methodology
The authors employ a theoretical framework based on the chiral unitary approach to describe the final-state interactions (FSIs). The analysis considers the decay at the quark level via internal and external $W^+$ emission mechanisms, followed by hadronization into meson-baryon and meson-meson pairs.

Dynamical Generation of Resonances:
- Baryons: The $\Lambda(1405)$ and $\Lambda(1670)$ states are treated as dynamically generated resonances arising from S-wave pseudoscalar meson-octet baryon interactions (specifically $\bar{K}N$ , $\pi\Sigma$ , $\eta\Lambda$ , etc.). These are calculated by solving the Bethe-Salpeter equation with coupled channels.
- Scalar Mesons: The $f_0(980)$ and $a_0(980)$ are generated via S-wave pseudoscalar meson-pseudoscalar meson interactions ( $\pi\pi$ , $K\bar{K}$ , $\eta\eta$ , etc.), also utilizing the coupled-channel formalism.
- Vector Meson: The contribution of the $\phi(1020)$ is included as an intermediate state decaying to $K^+K^-$ via P-wave, described using a derivative coupling and a Breit-Wigner propagator with a running width.
Amplitude Construction:
The total decay amplitude is constructed as a coherent sum of tree-level contributions, meson-baryon ( $T_{MB}$ ) interactions, meson-meson ( $T_{MM}$ ) interactions, and the $\phi(1020)$ contribution ( $T_\phi$ ). The authors account for interference phases between these components.
Data Comparison and Fitting:
The theoretical invariant mass distributions are compared with BESIII data. To address experimental limitations, the authors convolve the theoretical truth-level distributions with a detector resolution function (modeled as a weighted sum of two Gaussians) to simulate the binning and smearing effects of the experiment. Two fitting models are tested:
- Model A: The mass and width of the $\phi(1020)$ are fixed to Particle Data Group (PDG) values.
- Model B: The mass and width of the $\phi(1020)$ are treated as free parameters to be extracted from the fit.

Key Contributions and Results

Reproduction of Experimental Spectra: The theoretical calculations successfully reproduce the invariant mass distributions of $K^-p$ and $K^+K^-$ measured by BESIII. The best fit is achieved with Model B, yielding a $\chi^2/\text{d.o.f.} = 1.06$ .
Identification of Resonance Structures:
- $\Lambda(1405)$ : A near-threshold enhancement in the $pK^-$ invariant mass spectrum is identified as the signature of the $\Lambda(1405)$ resonance.
- $\Lambda(1670)$ : A distinct cusp structure around 1670 MeV in the $pK^-$ spectrum is reproduced. The authors attribute this to the $\Lambda(1670)$ resonance, noting that the interference between partial amplitudes transforms a dip structure (characteristic of the $\eta\Lambda$ threshold effect) into a pronounced cusp.
- $f_0(980)$ and $\phi(1020)$ : In the $K^+K^-$ spectrum, a prominent peak at 1020 MeV is attributed to the $\phi(1020)$ , while a near-threshold enhancement is associated with the isoscalar $f_0(980)$ . The contribution from the isovector $a_0(980)$ is found to be negligible.
Branching Fractions: By integrating the differential decay widths, the authors calculate branching fractions for the subprocesses $\Lambda_c^+ \to \Lambda(1405)K^+$ , $\Lambda_c^+ \to \Lambda(1670)K^+$ , $\Lambda_c^+ \to p f_0(980)$ , and $\Lambda_c^+ \to p \phi(1020)$ . These calculated values are consistent with the experimental measurements reported by BESIII.
Dalitz Plot Analysis: The Dalitz plot of $M_{pK^-}^2$ versus $M_{K^+K^-}^2$ clearly displays the signals of the $\Lambda(1405)$ , $\Lambda(1670)$ , $f_0(980)$ , and $\phi(1020)$ resonances.

Significance
The paper claims that its analysis provides a comprehensive theoretical description of the $\Lambda_c^+ \to pK^+K^-$ decay, validating the mechanism of dynamical generation for both low-lying excited baryons and light scalar mesons within the chiral unitary approach. The successful reproduction of the cusp structure at 1670 MeV and the threshold enhancement offers further insight into the nature of the $\Lambda(1670)$ and the interplay of final-state interactions. The authors conclude that future high-precision measurements by BESIII, Belle II, and the proposed Super Tau-Charm Facility will be essential for deepening the understanding of the intrinsic properties of these resonances and the nonleptonic weak decay dynamics of charmed baryons.

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