Quarks And Leptons. An Introductory Course In Modern Particle Physics

Francis Halzen, Alan D. Martin

Chapter 6 Electrodynamics of Spin- $\frac{1}{2}$ Particles - all with Video Answers

Educators

Chapter Questions

Problem 1

A "spinless" electron can interact with $A^{\mu}$ only via its charge; the coupling involves $\left(p_{f}+p_{i}\right)^{\mu}$. Show that
$$
\bar{u}_{f} \gamma^{\mu} u_{i}=\frac{1}{2 m} \bar{u}_{f}\left(\left(p_{f}+p_{i}\right)^{\mu}+i \sigma^{\mu \nu}\left(p_{f}-p_{i}\right)_{\nu}\right) u_{i},
$$
from which it is possible to establish that the physical spin- $\frac{1}{2}$ electron interacts via both its charge and its magnetic moment; see also Exercise 6.2. Equation (6.7) is known as the Gordon decomposition of the current.

Mayukh Banik

Numerade Educator

01:09

Problem 2

Show that in the nonrelativistic limit, the Gordon decomposition, (6.7), of the electron current, (6.6), separates the electron interaction with an electromagnetic field $A_{\mu}$ into a part arising from its charge, $-e$, and a part due to its magnetic moment, $-e / 2 m$. Assume that $A_{\mu}$ is independent of $t$, so that (6.4) becomes
$$
T_{f i}=-i 2 \pi \delta\left(E_{f}-E_{i}\right) \int j_{\mu}^{f i} A^{\mu} d^{3} x
$$
To identify the magnetic moment interaction $(-\mu \cdot \mathbf{B})$, it suffices to show that
$$
\int\left[-\frac{e}{2 m} \bar{\psi}_{f} i \sigma_{\mu \nu}\left(p_{f}-p_{i}\right)^{\nu} \psi_{i}\right] A^{\mu} d^{3} x=\int \psi_{A}^{f \dagger}\left(\frac{e}{2 m} \mathbf{\sigma} \cdot \mathbf{B}\right) \psi_{A}^{i} d^{3} x
$$
where $\psi_{A}$ denotes the upper two (or "large") components of $\psi$; compare with eqs. (5.31) and (5.32).

Raj Bala

Numerade Educator

01:44

Problem 3

Making use of (6.21), prove the trace theorems and the identities (6.24).

Harshita Goel

Numerade Educator

01:22

Problem 4

Assuming a vector-axial vector form of the weak interaction, explain why the electron emitted in the $\mu^{-}$-decay process, $\mu^{-} \rightarrow \mathrm{e}^{-} \bar{\nu}_{e} \nu_{\mu}$, must be left-handed. What is the helicity of $\mathrm{e}^{+}$from $\mu^{+}$decay?

Lazar Cvijovic

Numerade Educator

07:24

Problem 5

Use rotation matrix arguments to show that for "spinless" electrons and muons
$$
\mathscr{M}\left(\mathrm{e}^{-} \mathrm{e}^{+} \rightarrow \mu^{-} \mu^{+}\right) \propto \frac{t-u}{s}
$$
Compare the $s$-channel photon contribution of (4.47).

Abid Hussain

Numerade Educator

01:39

Problem 6

Show that the spin-averaged interference term between the two Feynman diagrams for electron-electron scattering is that shown in the table.

Daniel Gosser

Numerade Educator

03:53

Problem 7

Justify the following relations:
$$
\begin{aligned}
\int \frac{d^{3} p^{\prime}}{2 p_{0}^{\prime}} \delta^{(4)}\left(p+q-p^{\prime}\right) &=\int d^{3} p^{\prime} d p_{0}^{\prime} \delta^{(4)}\left(p+q-p^{\prime}\right) \theta\left(p_{0}^{\prime}\right) \delta\left(p^{\prime 2}-M^{2}\right) \\
&=\frac{1}{2 M} \delta\left(\nu+\frac{q^{2}}{2 M}\right) \\
&=\frac{1}{2 M A} \delta\left(E^{\prime}-E / A\right)
\end{aligned}
$$
where $A=1+(2 E / M) \sin ^{2} \frac{\theta}{2}$, and the step function $\theta(x)$ is 1 if $x>0$ and 0 otherwise.

M Hassan Anwar

Numerade Educator

12:45

Problem 8

Show that the cross section for elastic scattering of unpolarized electrons from spinless point-like particles is
$$
\left.\frac{d \sigma}{d \Omega}\right|_{\mathrm{lab} .}=\left(\frac{\alpha^{2}}{4 E^{2} \sin ^{4} \frac{\theta}{2}}\right) \frac{E^{\prime}}{E} \cos ^{2} \frac{\theta}{2}
$$
where as before we neglect the mass of the electron. Justify using (6.18) with $L_{\mu \nu}^{\text {muon }}$ replaced by $\left(p+p^{\prime}\right)_{\mu}\left(p+p^{\prime}\right)_{\nu}$. Comparing the cross section with that for $\mathrm{e}^{-} \mu^{-} \rightarrow \mathrm{e}^{-} \mu^{-}$, we see that the $\sin ^{2}(\theta / 2)$ in $(6.50)$ is due to scattering from the magnetic moment of the muon.

Ren Jie Tuieng

Numerade Educator

09:33

Problem 9

Maxwell's equations of classical electrodynamics are, in vacuo,
$$
\begin{array}{ll}
\nabla \cdot \mathbf{E}=\rho, & \nabla \times \mathbf{E}+\frac{\partial \mathbf{B}}{\partial t}=0 \\
\nabla \cdot \mathbf{B}=0, & \nabla \times \mathbf{B}-\frac{\partial \mathbf{E}}{\partial t}=\mathbf{j}
\end{array}
$$
(where we are using Heaviside-Lorentz rationalized units, see Appendix C of Aitchison and Hey). Show that these equations are equivalent to the following covariant equation for $A^{\mu}$ :
$$
\square^{2} A^{\mu}-\partial^{\mu}\left(\partial_{\nu} A^{\nu}\right)=j^{\mu},
$$
with $j^{\mu}=(\rho, \mathbf{j})$, and where $A^{\mu}=(\phi, \mathbf{A})$, the four-vector potential, is related to the electric and magnetic fields by
$$
\mathbf{E}=-\frac{\partial \mathbf{A}}{\partial t}-\nabla \phi, \quad \mathbf{B}=\nabla \times \mathbf{A}
$$

Carson Merrill

Numerade Educator

01:24

Problem 10

Verify that $\mathbf{E}$ and $\mathbf{B}$ in (6.55) are unchanged by the gauge transformation
$$
A_{\mu} \rightarrow A_{\mu}^{\prime}=A_{\mu}+\partial_{\mu} \chi
$$
where $\chi$ can be any function of $x$. Use this freedom to write Maxwell's equations in the form
$$
\square^{2} A^{\mu}=j^{\mu} \quad \text { with } \partial_{\mu} A^{\mu}=0 .
$$

Varsha Aggarwal

Numerade Educator

02:42

Problem 11

Determine how the linear combinations
$$
\begin{aligned}
&\varepsilon_{R}=-\sqrt{\frac{1}{2}}\left(\varepsilon_{1}+i \varepsilon_{2}\right) \\
&\varepsilon_{L}=\sqrt{\frac{1}{2}}\left(\varepsilon_{1}-i \varepsilon_{2}\right)
\end{aligned}
$$
transform under a rotation $\theta$ about the $z$ axis. Hence, show that $\varepsilon_{R}$ and $\varepsilon_{L}$ describe a photon of helicity $+1$ and $-1$, respectively; $\varepsilon_{R, L}$ are called circular polarization vectors.

Mahendra K

Numerade Educator

02:27

Problem 12

Show that (in the transverse gauge) the completeness relation is
$$
\sum_{\lambda=R, L}\left(\varepsilon_{\lambda}\right)_{i}^{*}\left(\varepsilon_{\lambda}\right)_{j}=\delta_{i j}-\hat{q}_{i} \hat{q}_{j}
$$
If $\varepsilon$ were along $\mathbf{q}$, it would be associated with a helicity-zero photon. This state is missing because of the transversality condition, $\mathbf{q} \cdot \boldsymbol{\varepsilon}=0$. It can only be absent because the photon is massless. We return to a further discussion of photon polarization vectors in Section 6.13.

Keshav Singh

Numerade Educator

01:39

Problem 13

Verify that the inverse of the "momentum space operator" of (6.78) does not exist.

Dominador Tan

Numerade Educator

03:09

Problem 14

The condition $\partial_{\lambda} A^{\lambda}=0$ does not fully define the propagator. We are at liberty to rewrite wave equation (6.78) as
$$
\left[g^{\nu \lambda} \square^{2}-\left(1-\frac{1}{\xi}\right) \partial^{\nu} \partial^{\lambda}\right] A_{\lambda}=j^{\nu}
$$
In this case, use (6.79) to show that the propagator is
$$
\frac{i}{q^{2}}\left(-g_{\mu \nu}+(1-\xi) \frac{q_{\mu} q_{\nu}}{q^{2}}\right)
$$

Keshav Singh

Numerade Educator

01:06

Problem 15

For a vector particle of mass $M$, energy $E$, and momentum $\mathbf{p}$ along the $z$ axis, show that states of helicity $\lambda$ can be described by polarization vectors
$$
\begin{gathered}
\varepsilon^{(\lambda=\pm 1)}=\mp(0,1, \pm i, 0) / \sqrt{2} \\
\varepsilon^{(\lambda=0)}=(|\mathbf{p}|, 0,0, E) / M
\end{gathered}
$$

Narayan Hari

Numerade Educator

09:49

Problem 16

Show that the completeness relation is
$$
\sum_{\lambda} \varepsilon_{\mu}^{(\lambda) *} \varepsilon_{\nu}^{(\lambda)}=-g_{\mu \nu}+\frac{p_{\mu} p_{\nu}}{M^{2}}
$$
where the sum is over the three polarization states of the massive vector particle.

Robert Zaballa

Numerade Educator

01:29

Problem 17

Verify (6.100) by making use of the Fourier transform
$$
\frac{1}{|\mathbf{q}|^{2}}=\int d^{3} x e^{i \mathbf{q} \cdot \mathbf{x}} \frac{1}{4 \pi|\mathbf{x}|}
$$

Hast Aggarwal

Numerade Educator

04:00

Problem 18

Show that, individually, the amplitudes $9 \mathbb{R}_{1}$ and $9 \mathbb{R}_{2}$ are not gauge invariant but that their sum indeed satisfies (6.108).

Khoobchandra Agrawal

Numerade Educator

00:49

Problem 19

Repeat the above calculation for an incident virtual photon of mass $k^{2} \equiv-Q^{2}$. Continue to use (6.111). Show that for $\gamma^{*} \mathrm{e}^{-} \rightarrow$ $\gamma \mathrm{e}^{-}$(where $\gamma^{*}$ denotes a virtual photon),
$$
\overline{|\mathscr{T}|^{2}}=2 e^{4}\left(-\frac{u}{s}-\frac{s}{u}+\frac{2 Q^{2} t}{s u}\right)
$$
We shall make use of this result in Chapter 10 .

Salamat Ali

Numerade Educator

02:29

Problem 20

Restore the mass $m$ of the electron and show that at high energy, $s \rightarrow \infty$, the integrated cross section for Compton scattering is
$$
\sigma=\frac{1}{64 \pi^{2} s} \int \overline{\left.|T|^{2}\right|^{2}} d \Omega \rightarrow \frac{2 \pi \alpha^{2}}{s} \log \left(\frac{s}{m^{2}}\right)
$$
Note that at high energy the dominant contribution comes from $9 \mathrm{R}_{2}$, via a glancing collision in which the $u$-channel electron is almost on mass shell.

Mayukh Banik

Numerade Educator

04:23

Problem 21

Show, by using particle helicities, that high-energy Compton scattering via the first diagram of Fig. $6.12$ is, in the center-of-mass frame, given by
$$
\begin{aligned}
\overline{\left|I_{1}\right|^{2}} & \propto\left|d_{++}^{1 / 2}(\theta)\right|^{2}+\left|d_{--}^{1 / 2}(\theta)\right|^{2} \\
&=(1+\cos \theta) \simeq-\frac{u}{2 s}
\end{aligned}
$$
in agreement with (6.112). An example of this type of calculation is described in Section 6.6.

Arpit Gupta

Numerade Educator

01:22

Problem 22

Draw the lowest-order Feynman diagrams for the pair annihilation process
$$
\mathrm{e}^{+}\left(p_{1}, s_{1}\right)+\mathrm{e}^{-}\left(p_{2}, s_{2}\right) \rightarrow \gamma\left(k_{1}, \varepsilon_{1}\right)+\gamma\left(k_{2}, \varepsilon_{2}\right)
$$

Suzanne W.

Numerade Educator

01:04

Problem 23

Use the Feynman rules to evaluate $9 \mathrm{R}\left(\gamma \mathrm{e}^{-} \rightarrow \gamma \mathrm{e}^{-}\right)$corresponding to the two Feynman diagrams of Fig. $6.12$ with the electron taken to have spin 0. Show that the result is not invariant under the gauge transformation (6.66). Demonstrate that gauge invariance is restored if diagram $6.16$ is included with a vertex factor $2 i e^{2} g^{\mu \nu}$.

Chai Santi

Numerade Educator