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.

good stuff

Media:HommezPhDThesis.ps

The polarized cross sections, which are related to the polarized structure function, are not measured directly. However the asymmetry related to them can be easily determined from the measurements:

[math]A_1 = \frac{\sigma_{1/2} - \sigma_{3/2}}{\sigma_{1/2} + \sigma_{3/2}}[/math]

where [math]\sigma_{1/2}[/math] ([math]\sigma_{3/2}[/math]) is the photon absorption cross section ones whose spin is antiparallel(parallel) to the target polarization. The photon of spin 1 is absorbed by only quarks whose spin is in the opposite direction from the photon's spin. Therefore, the polarized structure function can be related to the difference of these cross sections. However, the unpolarized structure function is the some of those two cross sections. The structure functions described above is based on quark parton model, where the contribution from the sea quarks is minimized. The double spin asymmetry can be written in terms of valence quark distribution functions for proton and neutron targets in this kinematic range:

[math]A_{1, p}^{I} = \frac {4\Delta u_v (x) + \Delta d_v (x)} {4 u_v (x) + d_v (x)} [/math] (1)

[math]A_{1, n}^{I} = \frac {\Delta u_v (x) + 4\Delta d_v (x)} {u_v (x) + 4d_v (x)} [/math] (2)

In inclusive DIS the asymmetry [math]A_{||}[/math] of cross sections, which is related to the beam and target polarization, only two measurable and controlled quantities, can be written in the following way:

[math]A_{meas} = P_B P_T f_D A_{||}[/math]

where [math]P_B[/math] and [math]P_T[/math] are the beam and target polarization respectively and [math]f_D[/math] - target dilution factor, the cross section factor caused by polarizable nucleons in the target.

In inclusive DIS the asymmetry of the photon absorption in the nucleon can be expressed in terms of the asymmetry [math]A_{||}[/math] of cross sections:

[math]A_{||} = D(1 + \eta \gamma )A_1[/math]

where [math]\eta = \frac{\epsilon \gamma y}{[1- (1-y) \epsilon ]}[/math] is a kinematic factor. The degree of polarization transfer from the electron to the virtual photon, so called depolarization factor D is given below:

[math]D = \frac{1- (1-y) \epsilon}{1 + \epsilon R}[/math]

where [math]\epsilon[/math] is the polarization parameter of the virtual photon:

[math]\epsilon = [1 + \frac{2 \vec{q}^2}{Q^2} tan^2 \frac{\theta}{2}]^{-1} = \frac{1 - y -\frac{1}{4} \gamma^2 y^2}{1-y+\frac{1}{4}y^2(\gamma^2+2)}[/math]

In inclusive DIS the polarization of the individual quarks(antiquarks) can be extrapolated from the fits to the data, relying on assumptions such as the Bjorken sum rule and SU(3) symmetry for the sea quark.

There is an alternative approach getting the information about the contributions from the individual quarks(antiquarks) using semi-inclusive Deep Inelastic Scattering instead of inclusive DIS, where in the final state in coincidence with the scattered lepton(in our case electron) hadron is detected too. In SIDIS after lepton scattering off the target emits an energetic virtual photon, which interacts with the quarks inside the nucleon. The energy and momentum transfer from the virtual photon to the quarks is so large that the struck quark is expelled from the nucleon forming the jet of hadrons. Hadrons created from the fragmentation of the expelled quark can be used as an one more method to tag its flavor. "This exploitation of hadrons in SIDIS measurements requires knowledge of the probabilities of the various types h of hadrons emerging from a struck quark of a given flavor q. These probabilities are embodied in the fragmentation function [math]D_q^h(z,Q^2)[/math], where [math]z=E_h/ \nu[/math] and [math]\nu[/math] and [math]E_h[/math] are the energies in the target rest frame of the absorbed virtual photon(th struck quark) and the detected hadron."[ref: flavor decomposition of the sea quark helicity dist. in the nucleon from SIDIS]. The correlation between the flavor of the probed quark and the detected hadron in the final state can be obtained from the fragmentation function, which is the probability that a quark of flavor q will fragment into a h type hadron. For instance, the detection of a [math]\pi^-[/math] meson in the final state, which is the bound state of [math]\bar{u}[/math](up anti) and d(down) quarks, indicates that either a [math]\bar{u}[/math](up antiquark) or a d(down) quark was probed in the nucleon. In addition, it is known that at high values of [math]X_B[/math] the Bjorken Scaling variable the sea quarks(like [math]\bar{u}[/math](up antiquark)) do not have a major contribution in the nucleon.

The semi-inclusive pion electro-production asymmetries can be written in terms of the cross sections of produced hadrons in the final state([math]\pi^-[/math] and [math]\pi^+[/math]):

[math]A[/math]^{[math]\pi^+ - \pi^-[/math]} =[math]\frac {\sigma^{\pi^+ - \pi^-}_{1/2} - \sigma^{\pi^+ - \pi^-}_{3/2}} {\sigma^{\pi^+ - \pi^-}_{1/2} + \sigma^{\pi^+ - \pi^-}_{3/2}} [/math]

• Extracting [math]\frac{\Delta d(u)}{d(u)}[/math]

In the semi inclusive deep inelastic scattering [math]\pi^+[/math] and [math]\pi^-[/math] production cross section can be written in the following way:

[math]\sigma = \frac{x(P+l)^2}{4 \pi \alpha^2} (\frac{2y^2}{1+(1-y)^2})\frac{d^3\sigma}{dxdydz}[/math]

and

[math]\Delta \sigma = \frac{x(P+l)^2}{4 \pi \alpha^2}(\frac{y}{2-y})[\frac{d^3\sigma_{+-}}{dxdydz} - \frac{d^3\sigma_{++}}{dxdydz}][/math]

P and l are the four momenta of the target and electron. [math]\sigma_{nm}[/math] refer to a nucleon of n helicity and a scattering lepton(electron) of helicity m. Rest of the variables are described in Table.1.

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