Revisiting the $\Lambda_c^+ \to n\pi^+\eta$ decay in light of the BESIII measurement

Technical Summary: Revisiting the $\Lambda^+_c \to n\pi^+\eta$ decay in light of the BESIII measurement

Problem Statement
The internal structure of light scalar mesons, particularly the $a_0(980)$, remains a subject of controversy within Quantum Chromodynamics (QCD), with interpretations ranging from conventional $q\bar{q}$ states to multi-quark configurations or dynamically generated resonances. While the Cabibbo-favored decay $\Lambda^+_c \to \Lambda\pi^+\eta$ has been extensively studied to probe $a_0(980)$ production mechanisms, the singly Cabibbo-suppressed decay $\Lambda^+_c \to n\pi^+\eta$ offers a complementary channel where the final-state $\Lambda$ is replaced by a nucleon.

Recent BESIII measurements of the $\Lambda^+_c \to n\pi^+\eta$ branching fraction yielded a value roughly half of previous theoretical predictions that neglected intermediate resonances, suggesting significant destructive interference. Furthermore, while the BESIII collaboration observed a tentative bump structure around 1160 MeV in the $\pi^+\eta$ invariant mass spectrum, the limited event statistics and coarse bin size (33 MeV) prevented a clear identification of the $a_0(980)$ signal or the precise nature of the excited nucleon contributions. This paper aims to systematically investigate the decay dynamics of $\Lambda^+_c \to n\pi^+\eta$ to clarify the production mechanism of the $a_0(980)$ and the roles of excited nucleon states ($N(1440)$, $N(1535)$) in light of these experimental constraints.

Methodology
The authors employ the chiral unitary approach to construct a theoretical framework for the decay. The study incorporates three distinct theoretical models to isolate the contributions of various mechanisms:

1. Fundamental Mechanisms:

   • Tree-level Production: The process is initiated via the external $W$-emission mechanism, mapped onto hadronic levels using SU(3) flavor symmetry. This yields direct $n\pi^+\eta$ production and intermediate meson-baryon pairs ($\pi N, \eta N, K\Lambda$) and meson-meson pairs ($K\bar{K}, \pi\eta$).
   • Dynamically Generated Resonances:
     • The $N(1535)$ resonance is dynamically generated through $S$-wave pseudoscalar meson–octet baryon interactions ($\pi N, \eta N, K\Lambda, K\Sigma$).
     • The $a_0(980)$ resonance is dynamically generated through $S$-wave pseudoscalar meson–pseudoscalar meson interactions ($\pi\eta, K\bar{K}$).
   • Final State Interactions (FSI): The scattering amplitudes for these interactions are solved using the Bethe-Salpeter equation with a cutoff regularization scheme.

2. Additional Resonance Contributions:

   • To address the bump structure near 1160 MeV and improve spectral fitting, the authors include contributions from the $P$-wave resonance $N(1440)$ (in the $\pi^+n$ channel) and the $D$-wave resonance $a_2(1320)$ (in the $\pi^+\eta$ channel).

3. Model Construction and Fitting:

   • Model A: Includes only the tree-level amplitude and the dynamically generated $N(1535)$ and $a_0(980)$ contributions.
   • Model B: Adds the $a_2(1320)$ contribution to Model A.
   • Model C: Adds the $N(1440)$ contribution to Model A.
   • The models are fitted to the BESIII experimental data on the $\pi^+\eta$ invariant mass spectrum (19 data points). Free parameters include production strengths ($V_p, V_{N(1440)}, V_{a_2(1320)}$) and relative phase angles ($\phi_1, \phi_2$).

Key Results

• Model Performance: Model A (without excited nucleons) fails to satisfactorily reproduce the spectral behavior in the low-energy region. Model B, which includes the $a_2(1320)$, shows only a marginal improvement in the high-mass region and does not significantly enhance the description of the low-mass spectrum. In contrast, Model C, which includes the $N(1440)$ resonance, yields a substantially better fit quality ($\chi^2/\text{d.o.f.} = 0.34$), indicating that the excited nucleon state $N(1440)$ plays a crucial role in describing the experimental data.
• Role of $a_0(980)$: The numerical results indicate that the $a_0(980)$ provides a significant contribution to the decay process. However, destructive interference among the tree-level, $N(1535)$, and $a_0(980)$ amplitudes results in a small cusp structure around 1 GeV in the $\pi^+\eta$ invariant mass distribution.
• Experimental Limitations: The authors note that with the current BESIII bin size of 33 MeV, the cusp structure associated with the $a_0(980)$ is effectively washed out, explaining the lack of a statistically significant signal in the existing data. Consequently, the precise contribution fraction of the $a_0(980)$ cannot be reliably extracted from the current dataset.
• Spectral Features:
  • The $\eta n$ invariant mass spectrum shows a prominent near-threshold enhancement around 1535 MeV, corresponding to the $N(1535)$.
  • The $\pi^+ n$ invariant mass spectrum exhibits a distinct peak associated with the $N(1440)$.
  • The $a_0(980)$ contributes primarily as a smooth background in these spectra.
• Dalitz Plots: Theoretical predictions for Dalitz plots ($M^2_{\pi^+\eta}$ vs. $M^2_{\eta n}$ and $M^2_{\pi^+\eta}$ vs. $M^2_{\pi^+ n}$) reveal a distinct signal structure for $a_0(980)$ near $M^2_{\pi^+\eta} = 0.96 \text{ GeV}^2$ and a clear band for $N(1535)$ near $M^2_{\eta n} = 2.35 \text{ GeV}^2$.

Significance and Claims
The paper claims that its systematic theoretical investigation clarifies the decay dynamics of $\Lambda^+_c \to n\pi^+\eta$ by demonstrating that:

1. Excited nucleon states, specifically $N(1440)$, are essential for reproducing the experimental $\pi^+\eta$ invariant mass spectrum in both low and high energy regions.
2. The $a_0(980)$ resonance makes a non-negligible contribution to the process, but its signal is suppressed by destructive interference, which accounts for the absence of a clear peak in current BESIII measurements.
3. The inability to extract the precise $a_0(980)$ contribution fraction is a direct consequence of insufficient experimental statistics and coarse binning.

The authors conclude that future high-precision and high-statistics experimental measurements of the $\Lambda^+_c \to n\pi^+\eta$ decay, particularly with finer binning and detailed Dalitz plot analyses, are necessary to reveal the intrinsic nature of the $a_0(980)$ and to quantify the specific roles of different excited nucleon states in this decay channel.