On Robust $β$-Spectra Shape Parameter Extraction

TL;DR

This work addresses the challenge of robustly extracting β-shape-function coefficients from β-decay spectra, where traditional power-polynomial parameterizations yield highly correlated coefficients and order-dependent interpretations. It proposes a weighted orthogonal-polynomial framework, constructed via Gram-Schmidt with a weight function $B(W,Z)$, to obtain coefficients $b_i$ that are largely independent of the expansion order and less covariant. Demonstrated on 32P data, the orthogonal-coefficient extraction shows stability across higher-order terms and improved interpretability, enabling more precise quantification of deviations from predictions and easier theory-experiment comparisons. The approach has potential for broader impact in nuclear-physics analyses and reactor antineutrino spectra, where robust, interpretable shape factors are essential.

Abstract

Experimental extraction of $β$-shape functions, C(W), is challenging. Comparing different experimental $β$-shapes to each other and to those predicted by theory in a consistent manner is difficult. This difficulty is compounded when different parameterizations of the $β$-shape function are used. Usually some form of a power polynomial of the total electron energy is chosen for this parametrization. This choice results in extracted coefficients that are highly correlated, with their physical meaning and numerical value dependent on the order of polynomial chosen. This is true for both theoretical and experimental coefficients, and leads to challenges when comparing coefficients from polynomials of different orders. Accurately representing the highly correlated uncertainties is difficult and subtle. These issues impact the underlying physical interpretation of shape function parameters. We suggest an alternative approach based on orthogonal polynomials. Orthogonal polynomials offer more stable coefficient extraction which is less dependent on the order of the polynomial, allow for easier comparison between theory and experimental coefficients from polynomials of different orders, and offer some observations on simple physical meaning and on the statistical limits of the extracted coefficients.

Figures (5)

• Figure 1: The first few powers, $W^{i}$, used to identify a shape factor for the 32P $\beta$-decay energy spectrum.
• Figure 2: First few weighted power polynomials fit to 32P $\beta$ decay data. Shaded regions correspond to (uncorrelated) 1-$\sigma$ bands for each fit component. These would be what is usually reported as uncertainties in a shape-factor measurement. It is easy to see that there are large cancellations between the various powers of $W$. Below $\approx 300$ keV the parent decay and online threshold effects impact the spectrum therefore this region is not used in the fit. The fit range is from 300 to 1550 keV.
• Figure 3: First few weighted orthogonal polynomials generated for studying the 32P $\beta$ decay. The range of the $\Phi_{i}$ is between $-0.7 \lesssim \Phi_{i}(W) \lesssim 7.0$ instead of the 2.5 order of magnitude larger range for the power polynomials shown in Figure \ref{['p32_w']}.
• Figure 4: First few weighted orthogonal polynomials fit to online 32P $\beta$ decay data. As in Figure \ref{['p32_powerpoly_fit']}, fit components have shaded regions corresponding to 1-$\sigma$ uncertainties, however in this case they are mostly narrower than the line width. The fit range is from 300 to 1550 keV. Inset (b) shows that the higher order components ($i>1$) lie near the edge or well within the statistical uncertainties indicated by in the gray-shaded region. This is the same 32P data shown in Figure \ref{['p32_powerpoly_fit']}.
• Figure 5: First few weighted orthogonal polynomials including $W^{-1}$ generated for studying the 32P $\beta$ decay. (a) Ordering of the basis functions is $\lbrace 1, W, W^{-1}, W^2...\rbrace$, i.e. the first basis function with a $W^{-1}$ term is $\Phi_2$. (b) Ordering of the basis functions is $\lbrace 1, W^{-1}, W, W^2...\rbrace$, i.e. the first basis function with a $W^{-1}$ term is $\Phi_1$.