The Sill distribution for all partial waves

Technical Summary: Extension of the Sill Distribution to Arbitrary Partial Waves

Problem Statement
In high-energy physics, short-lived resonances are typically described by energy-dependent spectral functions. The standard relativistic Breit–Wigner (rBW) distribution is widely used due to its simplicity; however, it lacks a physical lower energy threshold. This omission becomes problematic when the resonance peak lies close to the decay threshold, leading to significant deviations in line shape that the rBW cannot capture. While advanced parametrizations incorporating energy-dependent widths and threshold effects exist, they often sacrifice exact normalization or introduce complex analytic structures that are difficult to implement in data fitting.

The "Sill distribution," proposed in Ref. [19], offers a solution by explicitly incorporating the physical threshold ($s_{th}$) while maintaining exact normalization and a simple analytic form. However, the original formulation is inherently restricted to $s$-wave decays ($\ell=0$), as its threshold behavior scales as $\sqrt{s - s_{th}} \propto k_{CM}$. This limits its applicability to resonances decaying via higher partial waves, where the threshold behavior follows the Wigner threshold law ($\delta(s) \propto k_{CM}^{2\ell+1}$).

Methodology
The authors extend the Sill distribution to arbitrary partial waves ($\ell$) by modifying the self-energy function within the framework of the propagator for an unstable state. The derivation proceeds as follows:

1. Self-Energy Modeling: The imaginary part of the self-energy, $\text{Im}[\Pi_\ell(s)]$, is constructed to satisfy two conditions:

   • Near the threshold ($s \to s_{th}$), it must scale as $(s - s_{th})^{\ell + 1/2}$ to respect the Wigner threshold law for the $\ell$-th partial wave.
   • At high energies, it must recover the asymptotic behavior of the standard Breit–Wigner form, scaling as $\sqrt{s}$.

   This is achieved by introducing a sub-threshold scale parameter $s_0$ (where $s_0 < s_{th}$) and defining:
   $$\text{Im}[\Pi_\ell(s)] = \frac{\tilde{\Gamma} (s - s_{th})^{\ell + 1/2}}{(s - s_0)^\ell} \theta(s - s_{th})$$
   where $\tilde{\Gamma}$ controls the asymptotic width.

2. Dispersion Relations: The real part of the self-energy, $\text{Re}[\Pi_\ell(s)]$, is derived using dispersion (Kramers–Krönig) relations. The authors perform the integration using contour methods, resulting in a finite, algebraic expression for $\text{Re}[\Pi_\ell(s)]$ that depends on $\ell$, $s_{th}$, and $s_0$.

3. Mass Renormalization: A physical mass parameter $M_{Sill}$ is defined by the condition that the real part of the inverse propagator vanishes at $s = M_{Sill}^2$. This ensures the pole of the propagator is correctly positioned relative to the physical mass.

4. Final Formulation: The generalized $\ell$-wave Sill distribution, $\rho^{(\ell)}_{Sill}(s)$, is constructed by substituting the modeled real and imaginary parts of the self-energy into the spectral function definition $\rho(s) = -\frac{1}{\pi} \text{Im}[G(s)]$. The resulting expression retains exact normalization ($\int_{s_{th}}^\infty \rho^{(\ell)}_{Sill}(s) ds = 1$) for any choice of parameters.

Key Contributions

• Generalization of the Sill Distribution: The paper provides the first analytic extension of the Sill distribution to arbitrary orbital angular momentum $\ell$.
• Preservation of Properties: The new formulation maintains the key advantages of the original Sill: exact normalization, explicit threshold behavior, and a simple algebraic form suitable for numerical implementation.
• Analytic Expressions: The authors derive explicit formulas for the real part of the self-energy for any $\ell$, with a specific focus on the $p$-wave ($\ell=1$) case, which yields a relatively compact expression.
• Decay Width Definition: A generalized energy-dependent decay width $\Gamma(s)$ is defined that correctly interpolates between the threshold suppression and the asymptotic limit.

Results: Application to $\Delta(1232)^{++}$
The authors benchmark the $p$-wave Sill distribution using the baryonic resonance $\Delta(1232)^{++} \to p\pi^+$. This system is chosen because the resonance mass is close to the $p\pi^+$ threshold ($\Gamma/M \approx 0.73$), making threshold effects significant.

• Data Fitting: The $p$-wave Sill was fitted to SAID proton-pion scattering data in the $P_{33}$ channel. The fit included no explicit background terms; the parameter $s_0$ was allowed to vary to absorb smooth non-resonant contributions.
• Performance: The model describes the phase angles and the real and imaginary parts of the scattering amplitude very well up to $\sqrt{s} \approx 1.35$ GeV. The authors note that using an $s$-wave Sill or a standard rBW distribution results in significantly poorer fits, confirming the necessity of the correct partial-wave threshold behavior.
• Parameter Extraction:
  • The fitted on-shell mass is $M_{Sill} = 1231.90 \pm 0.14$ MeV.
  • The fitted on-shell width is $\Gamma_{Sill} = 130.3$ MeV.
  • The complex pole position was extracted as $\text{Re}[\sqrt{s_{pole}}] = 1213.12 \pm 0.22$ MeV and $-2\text{Im}[\sqrt{s_{pole}}] = 98.39 \pm 0.42$ MeV.
• Comparison: The extracted pole position, mass, and width are in agreement with independent determinations compiled by the Particle Data Group (PDG).

Significance and Claims
The paper claims that the extended Sill distribution provides a robust, simple, and exactly normalized tool for describing resonances near threshold, particularly for higher partial waves. The authors emphasize that the $p$-wave Sill successfully reproduces the $\Delta(1232)^{++}$ line shape and its fundamental properties (mass, width, and pole position) without the need for complex QFT-derived spectral functions or ad-hoc normalization factors.

The authors state that this formalism is ready for application to other hadronic resonances, particularly those relevant to femtoscopy studies. They also suggest that a straightforward extension to multiple decay channels could allow for more realistic applications involving simultaneous contributions from several thresholds. The work does not claim to replace full QFT calculations but offers a phenomenological alternative that balances physical correctness with computational simplicity.