It is helpful to work through an alternative derivation of...

It is helpful to work through an alternative derivation of the parton model result, (9.13)-(9.15), in terms of the invariant variables of $(6.29)$ : $$ s \simeq 2 k \cdot p, \quad u \simeq-2 k^{\prime} \cdot p, \quad t \equiv-Q^2=-2 k \cdot k^{\prime}, $$ that is, the ep $\rightarrow$ eX rate is simply the incoherent sum over all the contributing partons. Show that the invariant variables for the parton subprocess are given by $$ \hat{s}=x s, \quad \hat{u}=x u, \quad \hat{t}=t . $$ Use these relations, together with the $\mu$ scattering amplitude of (6.30), to show that $$ \left(\frac{d \sigma}{d t d u}\right)_{\mathrm{eq}_i \rightarrow \mathrm{eq}_1}=x \frac{d \sigma}{d \hat{t} d \hat{u}}=x \frac{2 \pi \alpha^2 e_i^2}{t^2}\left(\frac{s^2+u^2}{s^2}\right) \delta(t+x(s+u)) . $$ Now, also express the left-hand side of (9.17) in terms of $s, t$, and $u$. It is simplest to use (8.31). Verify that $$ \left(\frac{d \sigma}{d t d u}\right)_{\mathrm{ep} \rightarrow \mathrm{eX}}=\frac{4 \pi \alpha^2}{t^2 s^2} \frac{1}{s+u}\left[(s+u)^2 x F_1-u s F_2\right], $$ where $F_1 \equiv M W_1$ and $F_2 \equiv \nu W_2$. Insert (9.18) and (9.19) into (9.17). Compare coefficients of $u s$ and $s^2+u^2$ and so obtain the master formula of the parton model:

It is helpful to work through an alternative derivation of the parton model result, (9.13)-(9.15), in terms of the invariant variables of $(6.29)$ :
$$
s \simeq 2 k \cdot p, \quad u \simeq-2 k^{\prime} \cdot p, \quad t \equiv-Q^2=-2 k \cdot k^{\prime},
$$
that is, the ep $\rightarrow$ eX rate is simply the incoherent sum over all the contributing partons. Show that the invariant variables for the parton subprocess are given by
$$
\hat{s}=x s, \quad \hat{u}=x u, \quad \hat{t}=t .
$$

Use these relations, together with the $\mu$ scattering amplitude of (6.30), to show that
$$
\left(\frac{d \sigma}{d t d u}\right)_{\mathrm{eq}_i \rightarrow \mathrm{eq}_1}=x \frac{d \sigma}{d \hat{t} d \hat{u}}=x \frac{2 \pi \alpha^2 e_i^2}{t^2}\left(\frac{s^2+u^2}{s^2}\right) \delta(t+x(s+u)) .
$$

Now, also express the left-hand side of (9.17) in terms of $s, t$, and $u$. It is simplest to use (8.31). Verify that
$$
\left(\frac{d \sigma}{d t d u}\right)_{\mathrm{ep} \rightarrow \mathrm{eX}}=\frac{4 \pi \alpha^2}{t^2 s^2} \frac{1}{s+u}\left[(s+u)^2 x F_1-u s F_2\right],
$$
where $F_1 \equiv M W_1$ and $F_2 \equiv \nu W_2$. Insert (9.18) and (9.19) into (9.17). Compare coefficients of $u s$ and $s^2+u^2$ and so obtain the master formula of the parton model:

Key Concepts

Parton Model

The parton model is a framework used to describe the scattering of leptons from hadrons by assuming that the hadron is made up of quasi-free, point-like constituents (partons). In this model, the scattering process is treated as an incoherent sum over individual hard-scattering events with these constituents, with each event characterized by the momentum fraction carried by a parton.

Invariant Variables

Invariant variables such as s, t, and u are scalar quantities constructed from the four-momenta of particles in the scattering process. They remain unchanged under Lorentz transformations, making them powerful tools for expressing the kinematics of high-energy collisions in a frame-independent way. Their definitions allow the formulation of relations between the observed and subprocess events by scaling, for instance, using the parton's momentum fraction.

Differential Cross Section

The differential cross section represents the probability for a scattering event to occur within a specific range of final state variables. In high-energy physics, differential cross sections are often expressed in terms of invariant variables and are related to the matrix elements of the scattering process. They encapsulate factors such as coupling constants, symmetry considerations, and phase space factors, and allow for comparisons between theoretical predictions and experimental measurements.

Structure Functions

Structure functions (usually denoted F1 and F2) quantify the distribution of quarks and gluons inside hadrons and embody the nonperturbative aspects of the hadronic structure. These functions appear in the description of deep inelastic scattering, where they relate the measured cross sections to the underlying partonic dynamics. They provide a connection between the experimental observables and the fundamental properties of the constituents of the hadron.

Delta Functions and Energy–Momentum Conservation

Delta functions are used in the formulation of scattering cross sections to enforce conservation of energy and momentum. In the context of scattering processes, they constrain the kinematic variables such that only physically allowed transitions contribute to the cross section. This ensures that the derived relations among invariant variables accurately reflect the conservation laws inherent in the underlying interaction.

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