Show that in the rest frame of the target proton, ν=E-E^', y=(E-E^')/(E) where E and E^' are t

step 1 This question, to start the proof, we have that a hydrogen atom whose electron has initial orbit

Show that in the rest frame of the target proton, ν=E-E^', ...

Show that in the rest frame of the target proton, $$ \nu=E-E^{\prime}, \quad y=\frac{E-E^{\prime}}{E} $$ where $E$ and $E^{\prime}$ are the initial and final electron energies, respectively. Evaluation of the cross section for ep $\rightarrow \mathrm{eX}$ is a straightforward repetition of the same calculation for $\mathrm{e}^{-} \mu^{-} \rightarrow \mathrm{e}^{-} \mu^{-}$(or ep $\rightarrow$ ep) scattering with the substitution of $W_{\mu \nu}$, given by $(8.27)$, for $L_{\mu \nu}^{\text {muon }}$ (or $L_{\mu \nu}^{p}$ ). Using the expression (6.25) for $\left(L^{e}\right)^{\mu \nu}$ and noting (8.25), we find $$ \left(L^{e}\right)^{\mu \nu} W_{\mu \nu}=4 W_{1}\left(k \cdot k^{\prime}\right)+\frac{2 W_{2}}{M^{2}}\left[2(p \cdot k)\left(p \cdot k^{\prime}\right)-M^{2} k \cdot k^{\prime}\right] . $$ In the laboratory frame, this becomes $$ \left(L^{e}\right)^{\mu \nu} W_{\mu \nu}=4 E E^{\prime}\left\{\cos ^{2} \frac{\theta}{2} W_{2}\left(\nu, q^{2}\right)+\sin ^{2} \frac{\theta}{2} 2 W_{1}\left(\nu, q^{2}\right)\right\} $$ see (6.44). By including the flux factor, (4.32), and the phase space factor for the outgoing electron, (4.24), we can obtain the inclusive differential cross section for inelastic electron-proton scattering, ep $\rightarrow \mathrm{eX}$, $$ d \sigma=\frac{1}{4\left((k \cdot p)^{2}-m^{2} M^{2}\right)^{1 / 2}}\left\{\frac{e^{4}}{q^{4}}\left(L^{e}\right)^{\mu \nu} W_{\mu \nu} 4 \pi M\right\} \frac{d^{3} k^{\prime}}{2 E^{\prime}(2 \pi)^{3}} $$ where $\overline{|\mathscr{T}|^{2}}$ is given by the expression in the braces [recall (6.18)]. The extra factor of $4 \pi M$ arises because we have adopted the standard convention for the normalization of $W^{\mu \nu}$. Inserting (8.32) in (8.33) yields $$ \left.\frac{d \sigma}{d E^{\prime} d \Omega}\right|_{\mathrm{lab}}=\frac{\alpha^{2}}{4 E^{2} \sin ^{4} \frac{\theta}{2}}\left\{W_{2}\left(\nu, q^{2}\right) \cos ^{2} \frac{\theta}{2}+2 W_{1}\left(\nu, q^{2}\right) \sin ^{2} \frac{\theta}{2}\right\} $$

Show that in the rest frame of the target proton,
$$
\nu=E-E^{\prime}, \quad y=\frac{E-E^{\prime}}{E}
$$
where $E$ and $E^{\prime}$ are the initial and final electron energies, respectively.
Evaluation of the cross section for ep $\rightarrow \mathrm{eX}$ is a straightforward repetition of the same calculation for $\mathrm{e}^{-} \mu^{-} \rightarrow \mathrm{e}^{-} \mu^{-}$(or ep $\rightarrow$ ep) scattering with the substitution of $W_{\mu \nu}$, given by $(8.27)$, for $L_{\mu \nu}^{\text {muon }}$ (or $L_{\mu \nu}^{p}$ ). Using the expression (6.25) for $\left(L^{e}\right)^{\mu \nu}$ and noting (8.25), we find
$$
\left(L^{e}\right)^{\mu \nu} W_{\mu \nu}=4 W_{1}\left(k \cdot k^{\prime}\right)+\frac{2 W_{2}}{M^{2}}\left[2(p \cdot k)\left(p \cdot k^{\prime}\right)-M^{2} k \cdot k^{\prime}\right] .
$$
In the laboratory frame, this becomes
$$
\left(L^{e}\right)^{\mu \nu} W_{\mu \nu}=4 E E^{\prime}\left\{\cos ^{2} \frac{\theta}{2} W_{2}\left(\nu, q^{2}\right)+\sin ^{2} \frac{\theta}{2} 2 W_{1}\left(\nu, q^{2}\right)\right\}
$$
see (6.44). By including the flux factor, (4.32), and the phase space factor for the outgoing electron, (4.24), we can obtain the inclusive differential cross section for inelastic electron-proton scattering, ep $\rightarrow \mathrm{eX}$,
$$
d \sigma=\frac{1}{4\left((k \cdot p)^{2}-m^{2} M^{2}\right)^{1 / 2}}\left\{\frac{e^{4}}{q^{4}}\left(L^{e}\right)^{\mu \nu} W_{\mu \nu} 4 \pi M\right\} \frac{d^{3} k^{\prime}}{2 E^{\prime}(2 \pi)^{3}}
$$
where $\overline{|\mathscr{T}|^{2}}$ is given by the expression in the braces [recall (6.18)]. The extra factor of $4 \pi M$ arises because we have adopted the standard convention for the normalization of $W^{\mu \nu}$. Inserting (8.32) in (8.33) yields
$$
\left.\frac{d \sigma}{d E^{\prime} d \Omega}\right|_{\mathrm{lab}}=\frac{\alpha^{2}}{4 E^{2} \sin ^{4} \frac{\theta}{2}}\left\{W_{2}\left(\nu, q^{2}\right) \cos ^{2} \frac{\theta}{2}+2 W_{1}\left(\nu, q^{2}\right) \sin ^{2} \frac{\theta}{2}\right\}
$$

Key Concepts

Differential Cross Section Calculation

The calculation of the differential cross section involves combining the dynamics of the interaction, phase space factors, and flux normalization. It requires careful treatment of factors like the electromagnetic coupling, the propagator terms, and the integration over the final state momenta. This framework is central to predicting and analyzing scattering outcomes in particle physics experiments.

Structure Functions

Structure functions, typically denoted as W? and W? in the context of deep inelastic scattering, encode the distribution of charge and current within the proton. They are functions of the kinematic variables (like ? and q²) and carry essential information about the internal dynamics and partonic substructure. Their measurement provides insight into the fundamental constituents of the proton.

Kinematic Variables

These are the fundamental quantities that characterize the energy and momentum exchange in scattering events. In the context of electron-proton scattering, variables like the energy transfer (?), the inelasticity (y), and the momentum transfer (q²) quantify how much energy is transferred from the electron to the proton and how the momentum is redistributed during the collision. Evaluating these in specific frames, such as the rest frame of the target proton, simplifies the analysis.

Frame Transformations

Transformations between different reference frames, such as from the laboratory frame to the rest frame of the target proton, are crucial for analyzing scattering processes. The choice of frame affects the representation of kinematic variables and simplifies the calculation by exploiting symmetries, for instance, by setting the target proton at rest.

Leptonic and Hadronic Tensors

In scattering theory, the leptonic tensor (derived from the electron currents) and the hadronic tensor (which encapsulates the complex internal structure of the proton) are key components. Their contraction yields measurable quantities that directly relate to the cross section. This formalism separates the well-understood electromagnetic interaction of the electrons from the more complex structure of the proton.

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