Truncated partial-wave analysis for $\eta$-photoproduction observables via Bayesian statistics

A truncated partial-wave analysis is performed for $\eta$ photoproduction off the proton using the polarization observables ${\sigma }_0, \mathrm{\Sigma }, T, E, F$, and $G$. Through this approach, model-independent estimates of the electromagnetic multipole parameters are calculated. Based on these estimates, predictions are made for polarization observables that have not yet been measured. These predictions identify promising future measurements that could resolve the inherent mathematical ambiguities within the results. Bayesian inference is combined for the first time with truncated partial-wave analysis, analyzing different truncation orders for six energy bins near the $\eta p$-production threshold, i.e., $E_{\gamma }^{\mathrm{lab}}\in [750,1250]\mathrm{MeV}$.

Figure 1

Posterior predictive check for the profile functions ${\sigma }_0, \check{G}, \check{\mathrm{\Sigma }}, \check{E}, \check{T}$, and $\check{F}$ for truncation order ${\ell }_{\text{max}}=1$ and energy bins $E_{\gamma }^{\text{lab}}=[750,850,950,1050,1150,1250]\mathrm{MeV}$. The reproduced data distributions for the different solutions are shown together with the original data with statistical uncertainties as black points. Each solution group is drawn in a different color and each peak of a distribution corresponds to an accidental ambiguity. In addition, the corresponding values from EtaMAID2018 [48] (dashed line), BnGa-2019 [42] (dotted line), and JüBo-2022 [47] (dash-dotted line) are shown as well.

Figure 2

Posterior predictive check for the profile functions ${\sigma }_0, \check{G}, \check{\mathrm{\Sigma }}, \check{E}, \check{T}$, and $\check{F}$ for truncation order ${\ell }_{\text{max}}=2$ and energy bins $E_{\gamma }^{\text{lab}}=[750,850,950,1050,1150,1250]\mathrm{MeV}$. The reproduced data distributions for the different solutions are shown together with the original data with statistical uncertainties as black points. Each solution group is drawn in a different color and each peak of a distribution corresponds to an accidental ambiguity. In addition, the corresponding values from EtaMAID2018 [48] (dashed line), BnGa-2019 [42] (dotted line), and JüBo-2022 [47] (dash-dotted line) are shown as well.

Figure 3

Energy and angular coverage of the six observables ${\sigma }_0, \mathrm{\Sigma }, G, E, T$, and $F$ [38, 39, 40, 41, 42] which were used for the analysis. The energies used, $E_{\gamma }^{\text{lab}}=[750,850,950,1050,1150,1250]\mathrm{MeV}$, are determined by the observable $G$.

Figure 4

Example for a correlation matrix. The correlations between the data points of the unpolarized differential cross section ${\sigma }_0$ and the profile functions used, as well as the correlation between the profile functions themselves, is shown for $E_{\gamma }^{\text{lab}}=750\mathrm{MeV}$. Each square represents a certain data point. The color encodes the correlation strength ranging from $-1$ (darker colors) to $+1$ (lighter colors).

Figure 5

Unique correlation matrix element values as a function of the laboratory frame energy. The color encodes the correlation strength ranging from $-1$ (darker colors) to $+1$ (lighter colors).

Figure 6

MCMC convergence diagnostics for the truncation order ${\ell }_{\text{max}}=2$. Shown are the potential-scale-reduction statistic $\widehat{R}$ (the gray, dashed line indicates the value of 1.01) and the Monte Carlo standard error (MCSE) for the median divided by the median in percent (the gray, dashed line indicates the value of $1\%$). Each solution group is drawn in a different color.

Figure 7

Solutions of the multipole parameters $\text{Re}\left(E_{0+}\right)$ and $\text{Re}\left(M_{2+}\right)$ for a truncation order of ${\ell }_{\text{max}}=2$, for the energy bins $E_{\gamma }^{\text{lab}}=[750,850,950,1050,1150,1250]\mathrm{MeV}$. Each solution group is drawn in a different color and each peak of a distribution corresponds to an accidental ambiguity. The different parts of the tripartite plots are explained at the beginning of Sec. 6. The natural logarithm was used to calculate the log posterior density (lpd).

Figure 8

Solutions of the multipole parameter $\text{Im}\left(M_{2+}\right)$ for a truncation order of ${\ell }_{\text{max}}=2$, for the energy bins $E_{\gamma }^{\text{lab}}=[750,850,950,1050,1150,1250]\mathrm{MeV}$. In addition, solutions of systematic parameters for a truncation order of ${\ell }_{\text{max}}=2$ for the energy bin $E_{\gamma }^{\text{lab}}=950\mathrm{MeV}$ are shown. Each solution group is drawn in a different color and each peak of a distribution corresponds to an accidental ambiguity. The different parts of the tripartite plots are explained at the beginning of Sec. 6. The natural logarithm was used to calculate the log posterior density (lpd).

Figure 9

Predicted data distributions for the polarization observables $C_{z^{\prime }}, H, T_{x^{\prime }}, O_{z^{\prime }}, L_{x^{\prime }}, P, T_{z^{\prime }}, O_{x^{\prime }}, L_{z^{\prime }}$, and $C_{x^{\prime }}$ for the energy bins $E_{\gamma }^{\text{lab}}=[750,850,950,1050,1150,1250]\mathrm{MeV}$, using a truncation order of ${\ell }_{\text{max}}=2$. Each solution group is drawn in a different color and each peak of a distribution corresponds to an accidental ambiguity. In addition, the corresponding values from EtaMAID2018 [48] (dashed line), BnGa-2019 [42] (dotted line), and JüBo-2022 [47] (dash-dotted line) are shown as well.

Figure 10

MCMC convergence diagnostics for the truncation order ${\ell }_{\text{max}}=1$. Shown are the potential-scale-reduction statistic $\widehat{R}$ (the gray, dashed line indicates the value of 1.01) and the Monte Carlo standard error (MCSE) for the median divided by the median in percent (the gray, dashed line indicates the value of $1\%$). Each solution group is drawn in a different color.

Figure 11

Illustration of the first 1000 sampling points of a chain with initial value at 3.7 (the blue vertical line). The first sampling point is drawn in red. The chain converges from its starting point to a more likely solution, i.e., with higher log posterior density (lpd) value. The natural logarithm was used to calculate the lpd.

Figure 12

Marginal multipole solutions for the truncation order ${\ell }_{\text{max}}=2$ for the energy bins $E_{\gamma }^{\text{lab}}=[750,850,950,1050,1150,1250]\mathrm{MeV}$. In addition, the multipole parameter predictions from EtaMAID2018 [48] (dashed line), BnGa-2019 [42], (dotted line) and JüBo-2022 [47] (dash-dotted line) are shown as well. The relevance of a solution is represented by a transition from sienna (less relevant) to blue (more relevant) hues. However, for a detailed comparison between the solutions and their relevance to each other, the reader is advised to the tripartite multipole parameter figures in Figs. 7 and 8 and Ref. [49].

Figure 13

Adapted workflow to monitor MCMC convergence due to a multimodal posterior.