Stuff From PhD Proposal

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Deep inelastic scattering (DIS) of electrons by hadrons is a powerful tool for understanding the structure of the nucleon. When the momentum transferred to the target hadron is larger than the hadron mass, the inelastic scattering can be considered as the incoherent sum of the elastic scattering off the hadron constituents.

The way to think of this is that the electron hits the target with a photon whose momentum is the amount of momentum transfered.  If the photons wavelength is a lot less than the size of the target (1 fermi or 200 Mev) then you see the constituents

In the quark parton model, a proton is described in terms of fractionally charged constituents called up (u) and down (d) quarks. QCD extends the list of constituents to include antiquarks with all constituents interacting through the exchange of neutrally charged gluons. The QCD picture of a nucleon in terms of three valence quarks and a sea of quark-antiquark pairs has been supported by several deep inelastic scattering experiments [ CERN, SLAC, HERMES]

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In deep inelastic electron scattering, the high energy electron with initial energy [math]E[/math] and four-momentum [math]\vec{q}[/math], exchanges a virtual photon or boson which is scattered by a hadron. The scattering angle of the electron is [math]\theta[/math], measured from the incoming electron beam. The final state of the system is described by the four-momentum of the scattered electron and of the nucleon with invariant mass W. The kinematics of DIS can be described with four-momentum transfer square, the energy transfer and the Bjorken scaling variable. The four-momentum transfer is a measure of the magnifying power of the lepton probe. The probed distance scale can be written in terms of Q:

[math]d = \frac{\hbar c}{Q} = 0.2 \frac{GeV fm}{Q}[/math]

Using Fragmentation function

The semi-inclusive cross sections can be expressed in terms of the quark distribution functions and fragmentation functions:

[math]\frac{d^3 \sigma^h_{1/2(3/2)}}{dxdQ^2 dz}\approx \Sigma_q e_q^2 q^{+(-)}(x,Q^2)D_q^h(z,Q^2)[/math]

Related measured asymmetry as we have for inclusive DIS [math]A_1[/math] is introduced for semi-inclusive asymmetry too, which is written in terms of fragmentation functions and quark helicity densities:

[math]A_1^h(x,Q^2,z) = \frac{\Sigma_q \Delta q(x,Q^2) D_q^h(z,Q^2)}{\Sigma_{q^'} q^'(x,Q^2) D_{q^'}^h(z,Q^2)} \frac{(1+R(x))}{(1+\gamma^2)}[/math]

In the last equation factoring out the ratio of polarized to unpolarized quark distribution functions and introducing new term, so called purity [math]P_q^h(x,Q^2,z)[/math], which is the probability that in the case when the beam and target are unpolarized after the scattering the created hadron type of h is detected in the final state is the result of probing a quark q in the nucleon. "The purities are extracted from the Monte Carlo simulation". So the above equation can be rewritten in the following manner:

[math]A_1^h(x,Q^2,z) = \Sigma_q P_q^h(x,Q^2,z)\frac{\Delta q(x,Q^2)}{q(x,Q^2)} \frac{(1+R(x))}{(1+\gamma^2)}[/math]

where purities are defined following way:

[math]P^h_q(x,Q^2,z) = \frac{e^2_q q(x, Q^2)\int_{0.2}^{0.8}D^h_q(z,Q^2)dz}{\Sigma_{q^'} e^2_{q^'} q^'(x, Q^2)\int_{0.2}^{0.8}D^h_{q^'}(z,Q^2)dz}[/math]

The double-spin asymmetry [math]A_1^h[/math] can be written in a matrix form:

[math]\vec{A}(x) = P(x) \cdot \vec{Q}(x)[/math]

where [math]\vec{A}(x)[/math] represents the measured asymmetries for different targets and the final states of the detected hadron: [math]\vec{A}(x) = (A_{1p},{A_{1p}}^{\pi^+},{A_{1p}}^{\pi^-}, A_{1d}, {A_{1d}}^{\pi^+}, {A_{1d}}^{\pi^-})[/math] and the polarization information for different flavors of quarks and antiquarks is in [math]\vec{Q}(x)[/math] vector: [math]\vec{Q}(x) = (\frac{\Delta u}{u},\frac{\Delta d}{d}, \frac{\Delta s}{s}, \frac{\Delta u^-}{u^-}, \frac{\Delta d^-}{d^-}, \frac{\Delta s^-}{s^-})[/math]. When the Bjorken scaling variable [math]x_B\gt 0.3[/math], region where the sea quark contribution is zero.

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