Model selection for pion photoproduction

Partial-wave analysis of meson and photon-induced reactions is needed to enable the comparison of many theoretical approaches to data. In both energy-dependent and independent parametrizations of partial waves, the selection of the model amplitude is crucial. Principles of the $S$ matrix are implemented to a different degree in different approaches; but a many times overlooked aspect concerns the selection of undetermined coefficients and functional forms for fitting, leading to a minimal yet sufficient parametrization. We present an analysis of low-energy neutral pion photoproduction using the least absolute shrinkage and selection operator (LASSO) in combination with criteria from information theory and $K$-fold cross validation. These methods are not yet widely known in the analysis of excited hadrons but will become relevant in the era of precision spectroscopy. The principle is first illustrated with synthetic data; then, its feasibility for real data is demonstrated by analyzing the latest available measurements of differential cross sections ($d\sigma /d\mathrm{\Omega }$), photon-beam asymmetries ($\mathrm{\Sigma }$), and target asymmetry differential cross sections ($d{\sigma }_T/d\equiv Td\sigma /d\mathrm{\Omega }$) in the low-energy regime.

Figure 1

The AIC, AICc, and BIC criteria for a simple ${\chi }^2$ fit to 22 data (see text). All three methods exhibit the minimum at the same value of $\lambda$ indicating the optimal, i.e., simplest model. The correction term for a finite number of data in the AICc indeed produces a more pronounced minimum than for the AIC. The discontinuities occur every time a fit parameter is set to zero by LASSO.

Figure 2

LASSO, information criteria, and cross validation for the $\mathcal{B}$ model. (a) Total ${\chi }_T^2\left(\lambda \right)={\chi }^2\left(\lambda \right)+P\left(\lambda \right)$ with penalty $P$ (orange points) and ${\chi }^2\left(\lambda \right)$ from data alone (blue points). (b) The ${\chi }^2$/d.o.f. with the number of parameters dynamically updated for each $\lambda$. (c) Absolute value of the parameters $a_i$ as a function of $\lambda$ in a logarithmic scale. The red lines indicate the finally chosen parameters; the gray lines show the unnecessary parameters. (d) AICc, AIC, and BIC. (e) The cross-validation ${\chi }_V^2$. See text for further explanations.

Figure 3

$S$- and $P$-wave partial waves from the fits to the $\mathcal{B}$ model ($D$ waves are not shown because the result obtained was identically zero as expected). The blue lines show the $\mathcal{B}$ model used to generate synthetic data (benchmark solution). The orange lines and bands show the unconstrained 46-parameter fit with zero penalty $\lambda =0$. The brown lines show the fit at optimal $\lambda =3$. Removing the parameters that are effectively zero and refitting produces the final result indicated with red lines and uncertainty bands. The vertical lines at $W=1120$ MeV indicate the upper limit of fitted data. Note that for $\text{Im}{P}_i$, all but the unconstrained (orange) results for $\lambda =0$, are identically zero. Also in other cases the curves lie on top of each other such that only the blue line is visible. Uncertainties are computed using bootstrap.

Figure 4

Different goodness-of-fit criteria as a function of the penalty parameter $\lambda$.

Figure 5

The $\lambda$ dependence of the same quantities as discussed in Fig. 2 but for the situation with actual data from experiment.

Figure 6

$S$- and $P$-wave partial waves from the analysis of real data. Labeling of the curves and uncertainty bands as in Fig. 3. The simplest model is indicated with the red lines and bands. The blue circles show the single-energy solution of [23] for the real parts of $S$- and $P$-wave partial waves. The green triangles indicate the single-energy extraction of $\text{Im}{E}_{0+}$ from [36]. Uncertainties are computed using bootstrap.

Figure 7

Differential cross section. Data from [23]. The labels in the plot indicate the scattering energy $W$ in MeV. The red lines show the central value of the simplest model as determined in this study (indicated with the red lines in Fig. 6).

Figure 8

Beam asymmetry. Data from [23]. See Fig. 7 for further description.

Figure 9

Polarized differential cross section ${\sigma }_T=d\sigma /d\mathrm{\Omega }T$. Data from [36]. See Fig. 7 for further description.