Regularized Lednicky-Lyuboshitz formula for higher partial waves in femtoscopy

TL;DR

This paper addresses the inclusion of higher partial waves in femtoscopy by regularizing the generalized Lednicky–Lyuboshitz formula, which previously exhibited unphysical behavior near the origin. It introduces three regularized LL schemes (rLL(cut), rLL(delta), rLL(well)) and demonstrates that rLL(cut) offers stable, accurate results when compared to the Koonin–Pratt reference across Gaussian potentials and resonant scenarios. A key result is the linkage between the regularization cutoff and the traditional effective-range correction, $r_c \approx r_0/2$, providing a consistent way to extend ERC ideas to $l\ge1$. The work further reveals how partial-wave contributions depend on source size and interaction parameters, and presents heatmaps illustrating the interaction dependence, including resonance ridges, highlighting the practical utility of the regularized framework for constraining hadron–hadron interactions in high-energy collisions.

Abstract

Femtoscopy is one of the promising experimental approaches to put constraints on interactions between various species of hadrons from the momentum correlation functions measured in high-energy nuclear collision experiments. The Koonin-Pratt and Lednicky-Lyuboshitz formulae provide useful expressions of the correlation functions and have been widely used to analyze the experimental data based on the assumption that the effect of higher partial waves is negligible. Those formulae can be generalized for higher partial waves, but the generalized Lednicky-Lyuboshitz formula produces wrong results due to a singular behavior of the asymptotic wave function at the origin. In this study, we attempt to solve the problem by regularizing the generalized Lednicky-Lyuboshitz formula with a cutoff and validate it using the Koonin-Pratt formula as a reference. We also show the relationship between the cutoff in the regularized Lednicky-Lyuboshitz and the effective-range correction in the original Lednicky-Lyuboshitz formula. Using the obtained formula, we investigate the source-size dependence, the validity of the effective range expansion, and the cutoff dependence of the correlation function. We also discuss the interaction dependence using the heatmap as a function of the scattering-length parameter $1/a_l$ and the momentum $q$.

Figures (12)

• Figure 1: $y = x\partial_x \ln j_l(x)$ for different angular-momentum quantum numbers $l$. The solid red, blue dashed, green dotted, and purple dash-dotted lines correspond to $l = 0$, $1$, $2$, and $3$, respectively.
• Figure 2: Magnitudes of partial-wave contributions to the correlation functions. Panels (a)--(c) correspond to different potentials, $V_\mathrm{2G}(r)$, $V_\mathrm{1GA}(r)$, and $V_\mathrm{1GR}(r)$, respectively. The line styles for different $l$'s are the same as Fig. \ref{['fig:xdlnj']}. The source size is fixed to $R = 2.0~\mathrm{fm}$, and the reduced mass is $\mu = 469~\mathrm{MeV}$.
• Figure 3: Corrections to correlation functions with different asymptotic forms (rLL) with different potentials. The reduced mass is $\mu = 469~\mathrm{MeV}$, the source size is $R = 2~\mathrm{fm}$, and the cutoff is $r_c=1.2~\mathrm{fm}$.
• Figure 4: Source-size $R$ dependence of partial-wave contributions to the correlation function. We also compare the results with and without the effective range expansion. The solid black line shows the exact integration by the KP formula, and the red dashed lines show the regularized LL formula with the exact $f_l(q)$. The blue dotted lines show the regularized LL formula with $f_l(q)$ approximated by the effective range expansion up to $\mathcal{O}(q^2)$: $f_l(q) = q^{2l}/(-1/a_l + r_lq^2/2 - iq^{2l+1})$. The potential is $V_\mathrm{2G}(r)$, the reduced mass is $\mu = 469~\mathrm{MeV}$, and the cutoff is $r_c=1.2~\mathrm{fm}$.
• Figure 5: Partial-wave amplitudes $f_l(q)$ as functions of the momentum $q$. The solid lines show the exact $f_l(q)$ for the potential, and the dashed lines show the effective range expansion up to $\mathcal{O}(q^2)$: $f_l(q) = q^{2l}/(-1/a_l + r_lq^2/2 - iq^{2l+1})$. The bold red lines correspond to the real parts of $f_l(q)$, and the thin blue lines to the imaginary parts. The potential is $V_\mathrm{2G}$, and the reduced mass is $\mu = 469~\mathrm{MeV}$.