A Measurement of Proton Structure Function $g_2$ at Low $Q^2$

Measurements of the nucleon spin-dependent structure functions at low $Q^2$ have been proven to be powerful tools in testing the validity of effective theories of Quantum Chromodynamics. The neutron spin structure functions, $g_1^n$ and $g_2^n$, and the proton spin structure function, $g_1^p$, have been measured over a wide kinematic range. However, the proton spin structure function, $g_2^p$, remains largely unmeasured. The recent Jefferson Lab Hall A experiment E08-027 is an inclusive measurement of the proton $g_2$ structure function in the low momentum transfer region of $0.02 < Q^2 < 0.20$ GeV${}^{\mathrm{2}}$. This experiment will allow us to extract the generalized longitudinal-transverse spin polarizability $\delta_{LT}$ and test the Burkhardt-Cottingham sum rule at low $Q^2$. Chiral Perturbation Theory ($\chi PT$) is expected to work in this kinematic region and this measurement of $\delta_{LT}$ will give a benchmark test to $\chi PT$ calculations. The details of the experiment will be presented in this talk, along with the preliminary results and an overview of the analysis progress.