# kocolosk/thesis

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 \section{The Simple Parton Model} Over the past century studies of spin in elementary particle physics have proven their worth time and again, exposing weaknesses in theories that were otherwise able to explain the measurements of the day. The first indication that the proton was itself a composite particle came from a spin experiment, namely Stern's discovery that the magnetic moment of the proton is incompatible with the Dirac prediction for spin-$$\frac{1}{2}$$ particles. As a result, the \textit{structure function} was introduced in scattering cross sections to codify our lack of knowledge about the true internal structure of the nucleon. In 1964, Gell-Mann and Zweig independently proposed models \cite{GellMann:1964nj, Zweig:1964jf} in which hadrons are composed of a set of point-like elementary particles. These models provided a convenient taxonomy for the zoo of particles which had been identified in experiments, but it was unclear whether the quarks'', to use Gell-Mann's term, represented actual physical entities. Five years later, Feynman and Bjorken and Paschos postulated that the quarks -- they called them partons -- would behave quasi-free at high energies \cite{Feynman:1969ej, Bjorken:1969ja}. A consequence of this model is that in the high energy limit the structure functions of the proton measured in deep inelastic scattering depend only on the (dimensionless) ratio of the momentum transfer of the virtual photon and the energy loss of the scattered electron. This Bjorken scaling'' behavior was soon observed at SLAC by Friedman, Kendall, and Taylor \cite{Breidenbach:1969kd}. Physicists were initially reluctant to identify the partons implied by the SLAC experiment with the quarks in the models by Gell-Mann and Zweig, but eventually it became clear that they were one and the same. In the simple parton model Bjorken's $F_1$ structure function is expressed in terms of the number densities $q(x)$ of quarks and $\bar q(x)$ of antiquarks as % \begin{equation} F_1(x, Q^2) = \frac{1}{2}\sum_{j}{e_j^2[q_j(x) + \bar{q}_j(x)]} \end{equation} % where the sum is taken over quark flavors $j$ and $e_j$ is the electromagnetic charge of flavor $j$. In longitudinally polarized DIS we define an analogous polarization density $\Delta q(x) \equiv q_+(x) - q_-(x)$ as the difference in number density between quarks whose spins are aligned with the (longitudinal) spin of the proton and quarks whose spins are anti-aligned; the polarized analogue to $F_1$ is then % \begin{equation} g_1(x, Q^2) = \frac{1}{2}\sum_{j}{e_j^2[\Delta q_j(x) + \Delta \bar{q}_j(x)]}. \label{eqn:simple-g1} \end{equation} Typically one assumes $$SU(3)_F$$ flavor symmetry and thus it is useful to express the integral of $$g_1$$ in terms of quantities which have specific $$SU(3)_F$$ transformation properties: % \begin{equation} \Gamma_1^p = \int_0^1 dx~g_1^p(x) = \frac{1}{9}\left[a_0 + \frac{3}{4}a_3 + \frac{1}{4}a_8\right]. \label{eqn:g1} \end{equation} % The $$a_j$$ are the hadronic matrix elements of an octet of quark $SU(3)_F$ axial-vector currents $J_{5\mu}^i$ and a flavor singlet axial current $J_{5\mu}^0$, and are related to the polarized quark densities in the proton as % \begin{eqnarray} a_0 & = & (\Delta u + \Delta \bar{u}) + (\Delta d + \Delta \bar{d}) + (\Delta s + \Delta \bar{s}) \nonumber \\ a_3 & = & (\Delta u + \Delta \bar{u}) - (\Delta d + \Delta \bar{d}) \nonumber \\ a_8 & = & (\Delta u + \Delta \bar{u}) + (\Delta d + \Delta \bar{d}) - 2(\Delta s + \Delta \bar{s}) \label{eqn:su3-dis} \end{eqnarray} % In the limit of massless partons the non-singlet currents are scale-independent quantities, and are known from $\beta$-decay measurements \cite{Amsler:2008zzb}: % Stiegler says a_8 = 3F+D, but Anselmino and Ashman agree on the (-) \begin{eqnarray} a_3 & = & g_A = F+D = 1.2670 \pm 0.0035 \nonumber \\ a_8 & = & 3F-D = 0.585 \pm 0.025. \label{eqn:beta-decay} \end{eqnarray} % Hence a measurement of $$\Gamma_1^p$$ allows the extraction of the flavor singlet $$a_0$$, the quark spin contribution to the spin of the proton. If one assumes that the strange quark distribution does not contribute to the spin of the proton, as Ellis and Jaffe did in 1974 \cite{Ellis:1973kp}, Equations \ref{eqn:su3-dis} and \ref{eqn:beta-decay} allow a \textit{prediction} of the quark spin contribution to the spin of the proton, namely $$a_0 = a_8 \approx 0.6$$.