Show that deep inelastic electron electromagnetic scattering...

Show that deep inelastic electron electromagnetic scattering on an isoscalar target gives $$ \frac{d \sigma(\mathrm{eN} \rightarrow \mathrm{eX})}{d x d y}=\frac{2 \pi \alpha^{2}}{Q^{4}} x s\left[1+(1-y)^{2}\right] \frac{5}{18}[Q(x)+\bar{Q}(x)] $$ per nucleon, see Exercise 9.5. Note that, in contrast to $\nu \mathrm{N} \rightarrow \mu \mathrm{X}$, (12.78) embodies parity conservation so $Q$ and $\bar{Q}$ appear symmetrically.

Show that deep inelastic electron electromagnetic scattering on an isoscalar target gives
$$
\frac{d \sigma(\mathrm{eN} \rightarrow \mathrm{eX})}{d x d y}=\frac{2 \pi \alpha^{2}}{Q^{4}} x s\left[1+(1-y)^{2}\right] \frac{5}{18}[Q(x)+\bar{Q}(x)]
$$
per nucleon, see Exercise 9.5. Note that, in contrast to $\nu \mathrm{N} \rightarrow \mu \mathrm{X}$, (12.78) embodies parity conservation so $Q$ and $\bar{Q}$ appear symmetrically.

Key Concepts

Sum over Quark Charges

The factor that appears in the cross section, such as the 5/18 in the given formula, originates from summing over the squared electric charges of the quarks involved in the scattering process. In an isoscalar nucleon, this sum reflects the contributions from the up and down quarks weighted by their electric charge squared. This weighting is crucial for calculating the overall cross section in electromagnetic interactions.

Kinematic Variables in DIS

In deep inelastic scattering, several kinematic variables are central to the description of the process. Key variables include Q² (the negative square of the four-momentum transfer), x (the Bjorken scaling variable), y (the inelasticity or fractional energy loss), and s (the center-of-mass energy squared). These variables determine the kinematical region of the scattering and are essential for defining the differential cross section.

Quark Distribution Functions (Structure Functions)

The quark distribution functions describe the probability of finding a quark or antiquark carrying a certain fraction of the nucleon’s momentum. In the context of deep inelastic scattering, these functions appear in the description of the nucleon’s structure and are integrated over in the cross section formulas. They encapsulate the dynamics of the quark constituents as probed by the high-energy electron.

Isoscalar Target

An isoscalar target is one in which the numbers of protons and neutrons are equal, leading to an averaged distribution of up and down quarks. This symmetry simplifies the analysis of scattering cross sections, because the contributions from up and down quarks are balanced when summed over the nucleons, allowing for a more straightforward interpretation of the quark distribution functions.

Deep Inelastic Scattering (DIS)

DIS is a high-energy scattering process where leptons (such as electrons) probe the internal structure of hadrons (like nucleons) by exchanging a virtual photon. The process is fundamental for revealing the point-like constituents (quarks and gluons) inside nucleons, and it involves kinematic variables such as the momentum transfer squared (Q²) and the Bjorken scaling variable (x), which measures the fraction of the nucleon’s momentum carried by the struck parton.

Electromagnetic Interaction and Parity Conservation

The electromagnetic interaction, mediated by the photon, intrinsically conserves parity. This conservation means that the interaction treats left-handed and right-handed components symmetrically, implying that quark and antiquark contributions appear on equal footing in the scattering cross section. This parity symmetry contrasts with weak interactions, which are parity-violating.

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