Study of the KL0→π+π-γ direct emission vertex

We have perfomed studies of the ${K}_{L}^{0}\\ensuremath{\\rightarrow}{\\ensuremath{\\pi}}^{+}{\\ensuremath{\\pi}}^{\\ensuremath{-}}\\ensuremath{\\gamma}$ direct emission ( $\\mathrm{DE}$) and inner Bremsstrahlung ( $\\mathrm{IB}$) vertices, based on data collected by KTeV during the 1996 Fermilab fixed target run. We find ${a}_{1}/{a}_{2}\\phantom{\\rule{0ex}{0ex}}=\\phantom{\\rule{0ex}{0ex}}\\ensuremath{-}0.737\\ifmmode\\pm\\else\\textpm\\fi{}0.034{\\mathrm{GeV}}^{2}$ for the $\\mathrm{DE}$ form-factor parameter in the $\\ensuremath{\\rho}$-propagator parametrization, and report on fits of the form factor to linear and quadratic functions as well. We concurrently measure $\\ensuremath{\\Gamma}({K}_{L}^{0}\\ensuremath{\\rightarrow}{\\ensuremath{\\pi}}^{+}{\\ensuremath{\\pi}}^{\\ensuremath{-}}\\ensuremath{\\gamma}{,E}_{\\ensuremath{\\gamma}}^{*}g20\\mathrm{MeV})/\\ensuremath{\\Gamma}({K}_{L}^{0}\\ensuremath{\\rightarrow}{\\ensuremath{\\pi}}^{+}{\\ensuremath{\\pi}}^{\\ensuremath{-}})\\phantom{\\rule{0ex}{0ex}}=\\phantom{\\rule{0ex}{0ex}}(20.8\\ifmmode\\pm\\else\\textpm\\fi{}0.3)\\ifmmode\\times\\else\\texttimes\\fi{}{10}^{\\ensuremath{-}3}$, and a ${K}_{L}^{0}\\ensuremath{\\rightarrow}{\\ensuremath{\\pi}}^{+}{\\ensuremath{\\pi}}^{\\ensuremath{-}}\\ensuremath{\\gamma}$ $\\mathrm{DE}/(\\mathrm{DE}+\\mathrm{IB})$ branching ratio of $0.683\\ifmmode\\pm\\else\\textpm\\fi{}0.011$.