Show that σ(e^-) ≈(Eν in GeV) ×10^-41 cm^2, where Ev is the laboratory energy of the neutrino.

step 1 For this exercise we have that P sub sigma is equal to 1 .602 times 10 to the power of negative

Show that σ(e^-) ≈(Eν in GeV) ×10^-41 cm^2, where Ev is...

Show that $$ \sigma\left(\nu_e \mathrm{e}^{-}\right) \approx\left(E_\nu \text { in } \mathrm{GeV}\right) \times 10^{-41} \mathrm{~cm}^2, $$ where $E_v$ is the laboratory energy of the neutrino. The Feynman diagram for $\tilde{\nu} \mathrm{e}^{-} \rightarrow \mathrm{e}^{-} \tilde{\nu}_e$ is shown in Fig. 12.8b. We see that it can be obtained by crossing the neutrinos in $\nu_e \mathrm{e}^{-} \rightarrow \mathrm{e}^{-} \nu_e$ of diagram (a); see Sections 4.6 and 4.7. We therefore simply replace $s$ by $t$ in (12.57): $$ \begin{aligned} \frac{1}{2} \sum_{\text {spins }}|\mathscr{T}|^2 & =16 G^2 t^2 \\ & =4 G^2 s^2(1-\cos \theta)^2, \end{aligned} $$ where $\theta$ is the angle between the incoming $\bar{\nu}_e$ and the outgoing $\mathrm{e}^{-}$(see Fig. 12.9), and $$ t=-\frac{s}{2}(1-\cos \theta), $$ see (4.45). From (12.61), we obtain $$ \frac{d \sigma}{d \Omega}\left(\bar{\nu}_e \mathrm{e}^{-}\right)=\frac{G^2 s}{16 \pi^2}(1-\cos \theta)^2, $$ and integrating over angles yields $$ \sigma\left(\bar{v}_e \mathrm{e}^{-}\right)=\frac{G^2 s}{3 \pi} . $$ Comparing with (12.60) gives $$ \sigma\left(\bar{v}_e \mathrm{e}^{-}\right)=\frac{1}{3} \sigma\left(\nu_e \mathrm{e}^{-}\right) . $$ Results (12.59), (12.62), and (12.64) expose the $\gamma^\mu\left(1-\gamma^5\right)$ structure of the weak current in a way that can be experimentally checked. We can convince ourselves of this important statement by comparing the results with those obtained for the $\gamma^\mu$ vertex in the electromagnetic process $\mathrm{e} \mu \rightarrow \mathrm{e} \mu$ or by performing the following exercise.

Show that
$$
\sigma\left(\nu_e \mathrm{e}^{-}\right) \approx\left(E_\nu \text { in } \mathrm{GeV}\right) \times 10^{-41} \mathrm{~cm}^2,
$$
where $E_v$ is the laboratory energy of the neutrino.
The Feynman diagram for $\tilde{\nu} \mathrm{e}^{-} \rightarrow \mathrm{e}^{-} \tilde{\nu}_e$ is shown in Fig. 12.8b. We see that it can be obtained by crossing the neutrinos in $\nu_e \mathrm{e}^{-} \rightarrow \mathrm{e}^{-} \nu_e$ of diagram (a); see Sections 4.6 and 4.7. We therefore simply replace $s$ by $t$ in (12.57):
$$
\begin{aligned}
\frac{1}{2} \sum_{\text {spins }}|\mathscr{T}|^2 & =16 G^2 t^2 \\
& =4 G^2 s^2(1-\cos \theta)^2,
\end{aligned}
$$
where $\theta$ is the angle between the incoming $\bar{\nu}_e$ and the outgoing $\mathrm{e}^{-}$(see Fig. 12.9), and
$$
t=-\frac{s}{2}(1-\cos \theta),
$$
see (4.45). From (12.61), we obtain
$$
\frac{d \sigma}{d \Omega}\left(\bar{\nu}_e \mathrm{e}^{-}\right)=\frac{G^2 s}{16 \pi^2}(1-\cos \theta)^2,
$$
and integrating over angles yields
$$
\sigma\left(\bar{v}_e \mathrm{e}^{-}\right)=\frac{G^2 s}{3 \pi} .
$$

Comparing with (12.60) gives
$$
\sigma\left(\bar{v}_e \mathrm{e}^{-}\right)=\frac{1}{3} \sigma\left(\nu_e \mathrm{e}^{-}\right) .
$$

Results (12.59), (12.62), and (12.64) expose the $\gamma^\mu\left(1-\gamma^5\right)$ structure of the weak current in a way that can be experimentally checked. We can convince ourselves of this important statement by comparing the results with those obtained for the $\gamma^\mu$ vertex in the electromagnetic process $\mathrm{e} \mu \rightarrow \mathrm{e} \mu$ or by performing the following exercise.

Key Concepts

Weak Interaction

The weak interaction is one of the four fundamental forces in nature, responsible for processes like beta decay and neutrino scattering. It is mediated by massive vector bosons (W and Z bosons) at high energies or can be approximated by an effective four-fermion interaction (characterized by Fermi's constant) at low energies. This interaction exhibits unique properties, such as parity violation, due to its chiral V-A (vector minus axial vector) structure.

Feynman Diagrams

Feynman diagrams provide a graphical way to represent and calculate particle interactions. In processes involving weak interactions, they depict the exchange of heavy bosons or effective vertices that summarize the interaction. By illustrating the incoming and outgoing particles and their interactions, Feynman diagrams help in setting up the mathematical expressions for scattering amplitudes.

Crossing Symmetry

Crossing symmetry is a fundamental principle in quantum field theory that relates different scattering processes by interchanging particles between the initial and final states. By 'crossing' a particle from one side of the interaction to the other (or replacing it with its antiparticle), one can derive relationships between the amplitudes (and thus cross sections) of related processes, simplifying calculations and providing deeper insight into the underlying symmetries.

V-A (Vector minus Axial Vector) Structure

The V-A structure refers to the specific combination of vector and axial vector currents (typically represented as gamma^?(1-?^5) in the interaction vertex) that characterizes weak interactions. This structure is essential in explaining phenomena like parity violation and has observable consequences in scattering processes. The distinctive V-A behavior affects the angular distribution and energy dependence of scattering cross sections.

Differential Cross Section and Angular Integration

The differential cross section describes how the probability of a scattering event is distributed over various angles. It is derived from the squared scattering amplitude summed over spins and often involves factors arising from the underlying interaction (such as the weak interaction’s V-A structure). Integrating the differential cross section over all angles provides the total cross section, which is directly comparable to experimental measurements.

Please give Ace some feedback