# kocolosk/thesis

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 \section{First Experimental Tests} In polarized deep inelastic scattering, a longitudinally polarized lepton beam is scattered off of nucleon targets polarized parallel or perpendicular to the beam axis. Asymmetries are formed by comparing event rates for scattering in different spin configurations. For a spin $\frac{1}{2}$ target, the asymmetries of interest are % Stiegler 2.3 is useful here \begin{equation} A_{\parallel} = \frac{\sigma^{\uparrow \Downarrow} - \sigma^{\uparrow \Uparrow}}{\sigma^{\uparrow \Downarrow} + \sigma^{\uparrow \Uparrow}}, ~~~~~~~ A_{\perp} = \frac{\sigma^{\uparrow \perp} - \sigma^{\uparrow \top}}{\sigma^{\uparrow \perp} + \sigma^{\uparrow \top}} \end{equation} Spin-dependent cross sections can be calculated by contracting the elastic Compton amplitude $T_{\mu \nu}$ with the photon polarization vectors; in the presence of parity conservation and time reversal, four of these are independent \cite{Close:1979bt}: % \begin{eqnarray} \sigma_{1/2} & = & F_1 + g_1 - \gamma^2 g_2, \nonumber \\ \sigma_{3/2} & = & F_1 - g_1 + \gamma^2 g_2, \nonumber \\ \sigma_L & = & -F_1 + F_2(1+\gamma^2)/(2x), \nonumber \\ \sigma_{TL} & = & \sqrt{2}\gamma (g_1+g_2). \end{eqnarray} % Here $\gamma^2 = Q^2/v^2$. These four cross sections are commonly rearranged into a pair of virtual photon asymmetries $A_1$ and $A_2$: % \begin{equation} A_1 = \frac{\sigma_{1/2} - \sigma_{3/2}}{\sigma_{1/2} + \sigma_{3/2}}, ~~~~ A_2 = \frac{\sigma_{TL}}{\sigma_T} \end{equation} % The longitudinal and transverse DIS asymmetries can then be written in terms of these virtual photon asymmetries. In the case of $A_{\parallel}$ we have \begin{equation} A_{\parallel} = D(A_1 + \eta A_2), \end{equation} % where the coefficients $D$ and $\eta$ can be approximated to first order in $\gamma$ in terms of the usual DIS kinematic variables and $R = \frac{\sigma_{L}}{\sigma_T}$: % Detailed derivation of these results can be found in \cite{Anselmino:1994gn} \begin{equation} D \approx \frac{y(2-y)}{y^2 + 2(1-y)(1+R)}, ~~~~~~~~ \eta \approx \frac{2(1-y)}{y(2-y)} \frac{\sqrt{Q^2}}{E}. \end{equation} % Similar equations exist for $A_{\perp}$, such that a measurement of both asymmetries allows an extraction of both $A_1$ and $A_2$. $D$ can be thought of as a depolarization factor arising from the fact that the photon is not fully aligned with the lepton beam, and $\eta$ is a kinematic factor that is usually small. Finally, the polarized structure functions can be written in terms of $A_{1,2}$: \begin{equation} g_1 = \frac{F_2}{2x(1+R)}(A_1+\gamma A_2), ~~~~~ g_2 = \frac{F_2}{2x(1+R)}(A_2/\gamma - A_1). \end{equation} Thus, measurements of $A_{\parallel}$, $A_{\perp}$, $F_2$, and $R$ are sufficient to determine the polarized structure functions of the nucleon. The first DIS experiments to extract $g_1$ using this methodology were E80 and E130, conducted in the late 1970s and early 1980s at SLAC. These experiments scattered longitudinally polarized electron beams off of longitudinally polarized proton targets and measured $A_{\parallel}^p$ in the range $0.1 < x < 0.7$. Using the positivity limit $A_2 < \sqrt{R}$ they determined that $A_{\parallel}/D$ was a good approximation for $A_1$, and after exploiting that assumption their results \cite{Alguard:1976bm, Baum:1983ha} were consistent with expectations from the parton model. % there are 2-3 more result papers cited in Baum:1983ha if I want them In 1988, the European Muon Collaboration (EMC) published data on asymmetries of longitudinally polarized muon beams scattering off of longitudinally polarized proton targets. The EMC experiment boasted kinematic coverage down to $x = 0.01$, an order of magnitude lower than the earlier SLAC experiments, and the Collaboration extracted measurements of the proton's $g_1$ structure function using the same assumption that $A_1 \approx A_{\parallel}/D$. The EMC data on $A_1$ are consistent with the results from SLAC in their overlapping kinematic regime, but at low $x$ the EMC results deviate significantly from parton model predictions. As shown in Figure \ref{fig:emc-g1p}, the value of $\Gamma_1^p$ obtained from the EMC extraction is incompatible with the prediction from Ellis and Jaffe. Solving for the polarized parton densities using this result and the beta decay measurements in \ref{eqn:beta-decay}, one finds that the strange quarks possess a significant polarization antiparallel to the proton, and that the quark spin contribution to the spin of the proton is much smaller than expected. The EMC result sparked what was once termed a spin crisis'' in particle physics. Successive polarized DIS experiments at CERN, SLAC, and DESY confirmed and refined the EMC measurement of $$g_1^p$$ with improved precision over a wider kinematic range \cite{Adams:1994zd}, and measured both $$g_1^n$$ \cite{Anthony:1993uf} and $$g_1^d$$ \cite{Adeva:1993km} which allowed a verification of the critical Bjorken sum rule. A recent global analysis \cite{Leader:2006xc} of these data determined the integral quark polarization to be $$\Delta \Sigma = 0.24 \pm 0.04$$. The conclusion drawn from the EMC data still holds true --- the quark helicities alone cannot explain how the proton gets its spin. \begin{figure} \includegraphics[width=1.0\textwidth]{figures/emc-g1p} \caption{EMC extraction of $g^1_p$ and its integral compared to the prediction from Ellis-Jaffe \cite{Ashman:1987hv}} \label{fig:emc-g1p} \end{figure}