# Radiative Corrections

## Phase Space Diagrams

Shown below are <math>Q^2</math> vs.<math>W</math> plots for the 4- and 5-pass data sets. Displayed on each is a phase space boundary (the red curve) inside which cross section spectra are needed to carry out the radiative corrections to our data (shown in black). Also shown are the data for the E01-012 and E94-010 experiments in blue and green, respectively. The vertical line indicates the pion production threshold. The upper boundary indicates a line for which <math>E_s</math> (incoming electron energy) is a constant. The lower bound is calculated by varying the quantity <math>E_{s}^{min} </math> (see Mo and Tsai, Rev. Mod. Phys. 41, 205 1969), defined by:

<math>E_s^{min}(E_p) = \frac{m_{\pi}^2 + 2Mm_{\pi} + 2ME_p}{2M - 2E_p(1 - \cos \theta)}</math>,

where <math>m_{\pi}</math> is the pion mass, <math>M</math> is the nucleon mass, <math>E_p</math> is the scattered electron energy and <math>\theta</math> is the scattering angle of the electron. <math>E_s^{min}</math> is varied from the above value (evaluated using the smallest <math>E_p</math> value in the data set) up to the <math>E_s</math> value for the data set which needs the radiative corrections in steps of 1 MeV.

We see in the plots below that the E01-012 experiment's data falls more or less right in the middle of the boundary for both of our data sets, while the E94-010 data set falls just barely inside the boundary, closer to the edge than inside. Based on this, the E94-010 data set will be a good limiting case for our QFS model. A reasonable fit to not only our data sets but also to the other experiments' data will allow us to trust our QFS model when generating spectra to 'fill in the gaps' between the spectra shown.