Angular distributions of differential cross sections from the latest CLAS data sets ~\cite{bradford}, for the reaction ${\gamma}+p{\rightarrow} K^{+} + {\Lambda}$ have been analyzed using associated Legendre polynomials. This analysis is based upon theoretical calculations in Ref.~\cite{fasano} where all sixteen observables in kaon photoproduction can be classified into four Legendre classes. Each observable can be described by an expansion of associated Legendre polynomial functions. One of the questions to be addressed is how many associated Legendre polynomials are required to describe the data. In this preliminary analysis, we used data models with different numbers of associated Legendre polynomials. We then compared these models by calculating posterior probabilities of the models. We found that the CLAS data set needs no more than four associated Legendre polynomials to describe the differential cross section data. In addition, we also show the extracted coefficients of the best model.
Introduction
Significant information on the structure of the nucleon can be obtained by studying its excitation spectrum. Over the last few decades, a large amount information about the spectrum of the nucleon has been collected. Most of this information has been extracted from pion-induced and pion photoproduction reactions. However, pionic reactions may have biased the information on the existence of certain resonances. Constituent quark model calculations predict a much richer resonance spectrum than has been observed in pion production experiments~\cite{capstick}. Predicted resonances which have not been observed are called "missing" resonances. Instead, the constituent quark model also predicts that these "missing" resonances may couple strongly to K$\Lambda$ and K$\Sigma$ channels or other final states involving vector mesons~\cite{capstick,mart1,mart2}. Since performing kaon-hyperon, kaon-nucleon or hyperon-nucleon scattering experiments is a daunting task, kaon photoproduction on the nucleon appears to be a good alternative solution~\cite{mart1,mart2}.
Experiments on kaon photoproduction and electroproduction started in the 1960s. However the old experimental data are often inconsistent and have large error bars. In recent years a large amount of data for kaon photoproduction has been collected. High statistics data from CLAS, for differential cross sections, recoil polarization, $C_{x}$ and $C_{z}$ double polarizations for the reaction $\gamma + p \rightarrow K^{+} + \Lambda$ have been published~\cite{bradford,bradfor2}. Additional experimental data have also been measured by SAPHIR~\cite{glander,tran,glander2}, LEPS~\cite{sumihama,zegers} and GRAAL~\cite{leres}.
Several previous analyses have been applied to the results of these experiments, such as Isobar models~\cite{mart1,mart2,ireland,janssen,janssen2} and Coupled channel models~\cite{shyklar,usov,penner}. However different theoretical model calculations often produce very different predictions. In Ref.\cite{fasano} all sixteen observables in kaon photoproduction were shown to be classified into the classes ${\cal L}_0(\hat{{\bf I}};\hat{{\bf E}};\hat{{\bf C_{z'}}};\hat{{\bf
L_{z'}}})$, ${\cal L}_{1a}(\hat{{\bf P}}; \hat{{\bf H}}; \hat{{\bf C_{x'}}}; \hat{{\bfL_{x'}}})$, ${\cal L}_{1b}(\hat{{\bf T}}; \hat{{\bf F}}; \hat{{\bf O_{x'}}};\hat{{\bf T_{z'}}})$ and ${\cal L}_2(\hat{{\bf {\Sigma}}}; \hat{{\bf G}}; \hat{{\bf O_{z'}}}; \hat{{\bf T_{x'}}})$, where each class is an expansion in a different set of associated Legendre polynomials. What is not apparent is how many terms in each expansion are required. This work attempts to address the issue by examining data models with different numbers of terms, and calculating which one has the greatest posterior probability. In this article we only focus on the differential cross section observables, which are described by the associated Legendre class ${\cal L}_0$. (PTPH)
Introduction
Significant information on the structure of the nucleon can be obtained by studying its excitation spectrum. Over the last few decades, a large amount information about the spectrum of the nucleon has been collected. Most of this information has been extracted from pion-induced and pion photoproduction reactions. However, pionic reactions may have biased the information on the existence of certain resonances. Constituent quark model calculations predict a much richer resonance spectrum than has been observed in pion production experiments~\cite{capstick}. Predicted resonances which have not been observed are called "missing" resonances. Instead, the constituent quark model also predicts that these "missing" resonances may couple strongly to K$\Lambda$ and K$\Sigma$ channels or other final states involving vector mesons~\cite{capstick,mart1,mart2}. Since performing kaon-hyperon, kaon-nucleon or hyperon-nucleon scattering experiments is a daunting task, kaon photoproduction on the nucleon appears to be a good alternative solution~\cite{mart1,mart2}.
Experiments on kaon photoproduction and electroproduction started in the 1960s. However the old experimental data are often inconsistent and have large error bars. In recent years a large amount of data for kaon photoproduction has been collected. High statistics data from CLAS, for differential cross sections, recoil polarization, $C_{x}$ and $C_{z}$ double polarizations for the reaction $\gamma + p \rightarrow K^{+} + \Lambda$ have been published~\cite{bradford,bradfor2}. Additional experimental data have also been measured by SAPHIR~\cite{glander,tran,glander2}, LEPS~\cite{sumihama,zegers} and GRAAL~\cite{leres}.
Several previous analyses have been applied to the results of these experiments, such as Isobar models~\cite{mart1,mart2,ireland,janssen,janssen2} and Coupled channel models~\cite{shyklar,usov,penner}. However different theoretical model calculations often produce very different predictions. In Ref.\cite{fasano} all sixteen observables in kaon photoproduction were shown to be classified into the classes ${\cal L}_0(\hat{{\bf I}};\hat{{\bf E}};\hat{{\bf C_{z'}}};\hat{{\bf
L_{z'}}})$, ${\cal L}_{1a}(\hat{{\bf P}}; \hat{{\bf H}}; \hat{{\bf C_{x'}}}; \hat{{\bfL_{x'}}})$, ${\cal L}_{1b}(\hat{{\bf T}}; \hat{{\bf F}}; \hat{{\bf O_{x'}}};\hat{{\bf T_{z'}}})$ and ${\cal L}_2(\hat{{\bf {\Sigma}}}; \hat{{\bf G}}; \hat{{\bf O_{z'}}}; \hat{{\bf T_{x'}}})$, where each class is an expansion in a different set of associated Legendre polynomials. What is not apparent is how many terms in each expansion are required. This work attempts to address the issue by examining data models with different numbers of terms, and calculating which one has the greatest posterior probability. In this article we only focus on the differential cross section observables, which are described by the associated Legendre class ${\cal L}_0$. (PTPH)
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