Empirical Reconstruction of the JSNS$^2$ KDAR $ν_μ$-$^{12}$C Missing-Energy Spectrum with a Two-Ex-Gaussian and Generalized-Tail Model

TL;DR

The paper addresses the challenge of accurately describing the missing-energy spectrum in monoenergetic KDAR $\nu_\mu$ scattering on $^{12}$C at $E_\nu\approx235.5$ MeV, where existing generators struggle to reproduce both the peak width and the high-$E_m$ tail. It introduces a compact, physically motivated empirical model consisting of two ex-Gaussian bound-state components (for the $p$- and $s$-shells) plus a generalized power-exponential continuum term, fitted to the JSNS$^2$ data in visible energy. The fit achieves $\chi^2/\text{ndf}\approx1.3$, extracting parameters that reveal a narrow $p$-shell peak and a broader, mildly skewed $s$-shell shaped by continuum coupling and multinucleon dynamics, with a continuum tail quantified by $\lambda\approx21$ MeV and $\beta\approx1.24$. This empirical framework provides a physically transparent, data-driven reference for phenomenological comparisons and guides future theoretical refinements of binding energy treatments, 2p-2h dynamics, and FSI in the few-hundred-MeV neutrino regime.

Abstract

Recent analyses of the JSNS$^2$ monoenergetic $ν_μ$ scattering on $^{12}$C at 235.5~MeV have compared the measured missing-energy spectrum with several nuclear models, including \textsc{NuWro}, \textsc{GiBUU}, and RMF+Achilles. While these models reproduce the overall peak position, their respective $χ^2$ values of $35.5$, $176.8$, and $58.1$ indicate that none can simultaneously describe the spectral width and the high-energy tail, reflecting limitations in the treatment of binding energy, two-particle--two-hole (2p-2h) excitations, and final-state interactions (FSI). To address these discrepancies, we introduce an empirical yet physically motivated representation of the spectrum based on two exponentially modified Gaussian (ex-Gaussian) components for p- and s-shell knockout and a generalized power-exponential continuum term describing multinucleon and FSI-induced strength. The fit reproduces the JSNS$^2$ data within the fitted energy range with $χ^2=8.0$ for 6 degrees of freedom. yielding parameters that quantify asymmetric broadening of the s-shell while preserving a narrow quasielastic p-shell response. This compact model demonstrates that a minimal empirical framework can capture key features of the nuclear response and provides a useful reference for phenomenological comparisons and future studies of quasielastic and 2p-2h dynamics in the few-hundred-MeV regime.

Figures (5)

• Figure 1: Comparison of the JSNS$^2$ missing--energy spectrum with the NuWro (v21) prediction. Top: area--normalized differential distribution per 5 MeV bins; black bars show statistical errors and red bars indicate statistical$\oplus$systematic uncertainties. The dashed blue histogram is the NuWro result. The upper axis labels $E_{\rm vis}$, while the lower axis labels $E_m$. Middle: bin--by--bin residuals $\Delta(\text{Data}-\text{NuWro})$ on the experimental binning. Bottom: data--to--model ratio. NuWro reproduces the peak location but does not capture the observed peak width and underestimates the strength in the intermediate-$E_m$ region and in the high-$E_m$ tail; the corresponding global comparison yields $\chi^2=35.5$ for 16 degrees of freedom.
• Figure 2: As in Fig. 1 (same panel layout and area normalization), but for GiBUU (2021p1). The magenta histogram shows the GiBUU prediction. Relative to the data, GiBUU overshoots strength in the low-- and intermediate--$E_m$ regions and shifts yield toward higher $E_m$, resulting in a large global comparison value $\chi^2=176.8$ for 16 degrees of freedom.
• Figure 3: As in Fig. 1 (same panel layout and area normalization), but for RMF+Achilles (green histogram). The model reproduces the peak location and captures part of the high--$E_m$ tail, while discrepancies remain in the detailed shape around $E_m\!\sim\!20$--30 MeV and in the peak width; the global comparison yields $\chi^2=58.1$ for 16 degrees of freedom.
• Figure 4: Illustration of the ex-Gaussian line shape used to describe asymmetric broadening in the missing-energy spectrum. (Left) Comparison between the right- and left-skewed forms, $\mathrm{exG_R}$ (solid) and $\mathrm{exG_L}$ (dashed), which are mirror reflections about the mean value ($\mathrm{exG_L}(E_m)=\mathrm{exG_R}(2\mu-E_m)$). (Right) Dependence of $\mathrm{exG_R}$ on the asymmetry scale $\tau$, together with the Gaussian limit (solid blue curve). Smaller $\tau$ values lead to a nearly symmetric Gaussian-like shape, whereas larger $\tau$ produce an extended high-$E_m$ tail. This behavior enables the ex-Gaussian to model realistic one-sided broadening due to final-state interactions and continuum coupling.
• Figure 5: Comparison between the JSNS$^2$ missing-energy spectrum and the best-fit model. The black histogram shows the experimental data, with dark and light gray bands representing the statistical and total (statistical + systematic) uncertainties, respectively. The orange dashed line denotes the best-fit model averaged over each experimental bin width. Within the fitted region of $10~\mathrm{MeV} < E_m < 120~\mathrm{MeV}$, the model closely follows the central values of the measured spectrum, reproducing both the p-shell and s-shell features as well as the overall high-$E_m$ continuum within uncertainties.