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Quarks And Leptons. An Introductory Course In Modern Particle Physics

Francis Halzen, Alan D. Martin

Chapter 4

Electrodynamics of Spinless Particles - all with Video Answers

Educators

Chapter Questions

12:18

Problem 1

Working in a box of volume $V=L^{3}$, show that the number of allowed states of momentum with $x$ component in the range $p_{x}$ to $p_{x}+d p_{x}$ is $(L / 2 \pi) d p_{x}$. Convince yourself that you need to impose periodic boundary conditions on the wavefunction and its derivative to ensure no net particle flow out of the volume.
Turning now to the initial flux, we find that it is easiest to calculate it in the laboratory frame. The number of beam particles passing through unit area per unit time is $\left|\mathbf{v}_{A}\right| 2 E_{A} / V$, and the number of target particles per unit volume is $2 E_{B} / V$. To obtain a normalization-independent measure of the ingoing "density,"EXERCISE 4.I Working in a box of volume $V=L^{3}$, show that the number of allowed states of momentum with $x$ component in the range $p_{x}$ to $p_{x}+d p_{x}$ is $(L / 2 \pi) d p_{x}$. Convince yourself that you need to impose periodic boundary conditions on the wavefunction and its derivative to ensure no net particle flow out of the volume.
Turning now to the initial flux, we find that it is easiest to calculate it in the laboratory frame. The number of beam particles passing through unit area per unit time is $\left|\mathbf{v}_{A}\right| 2 E_{A} / V$, and the number of target particles per unit volume is $2 E_{B} / V$. To obtain a normalization-independent measure of the ingoing "density,"

Robert Zaballa
Robert Zaballa
Numerade Educator
07:26

Problem 2

In the center-of-mass frame for the process AB $\rightarrow$ CD, show that
$$
\begin{gathered}
d Q=\frac{1}{4 \pi^{2}} \frac{p_{f}}{4 \sqrt{s}} d \Omega \\
F=4 p_{i} \sqrt{s}
\end{gathered}
$$
and hence that the differential cross section is
$$
\left.\frac{d \sigma}{d \Omega}\right|_{c m}=\frac{1}{64 \pi^{2} s} \frac{p_{f}}{p_{i}}|\Omega|^{2}
$$
where $d \Omega$ is the element of solid angle about $\mathbf{p}_{C}, s=\left(E_{A}+E_{B}\right)^{2},\left|\mathbf{p}_{A}\right|=$ $\left|\mathbf{p}_{B}\right|=p_{i}$ and $\left|\mathbf{p}_{C}\right|=\left|\mathbf{p}_{D}\right|=p_{f}$.

Michael Talbot
Michael Talbot
Numerade Educator
02:29

Problem 3

Use (4.18) to show that for very high-energy "spinless" electron-muon scattering,
$$
\left.\frac{d \sigma}{d \Omega}\right|_{c m}=\frac{\alpha^{2}}{4 s}\left(\frac{3+\cos \theta}{1-\cos \theta}\right)^{2}
$$
where $\theta$ is the scattering angle and $\alpha=e^{2} / 4 \pi .$ Neglect the particle masses.

Mayukh Banik
Mayukh Banik
Numerade Educator
02:29

Problem 4

Use (4.18) to show that for very high-energy "spinless" electron-muon scattering,
$$
\left.\frac{d \sigma}{d \Omega}\right|_{c m}=\frac{\alpha^{2}}{4 s}\left(\frac{3+\cos \theta}{1-\cos \theta}\right)^{2}
$$
where $\theta$ is the scattering angle and $\alpha=e^{2} / 4 \pi .$ Neglect the particle masses.

Mayukh Banik
Mayukh Banik
Numerade Educator
03:58

Problem 5

We return to the application of the Feynman rules to some sample processes. For electron-electron scattering, the new feature is that we have identical particles in the initial and the final states, and so the amplitude should be symmetric under interchange of particle labels $\mathrm{C} \leftrightarrow \mathrm{D}$ (and $\mathrm{A} \leftrightarrow \mathrm{B}$ ). Consequently, in addition to the Feynman diagram of Fig. 4.4a, we have a second diagram, Fig. 4.4b, which, to maintain the order of A, B, C, and D, is drawn as Fig. 4.4c.

There is no way to experimentally distinguish whether electron $\mathrm{C}$ came from $\mathrm{A}$ or B, so we must add amplitudes (rather than probabilities). Thus, the invariant amplitude for the scattering of spinless electrons is, to lowest order, the sum of the amplitudes for diagrams (a) and (c):

Amit Srivastava
Amit Srivastava
Numerade Educator
06:03

Problem 6

Again, here we have two possible Feynman diagrams, Figs. $4.5$ a and $4.5 \mathrm{c}$. We are working only in terms of particle (electron) states, and so we must use the antiparticle prescription, (3.28), to translate these to diagrams (b) and (d), respectively. We can use the Feynman rules (obtained above and collected in Section 6.17) to calculate the (lowest-order) $\mathrm{e}^{-} \mathrm{e}^{+} \rightarrow \mathrm{e}^{-} \mathrm{e}^{+}$amplitude

Ozenc Gungor
Ozenc Gungor
Numerade Educator
02:16

Problem 7

For the crossed reaction $\mathrm{A} \overline{\mathrm{D}} \rightarrow \mathrm{CB}\left(\mathrm{e}^{-} \mathrm{e}^{-} \rightarrow \mathrm{e}^{-} \mathrm{e}^{-}\right)$, show that $u$ becomes the square of the total center-of-mass energy and that this process would become physical in a different kinematic region: $u \geq 4 m^{2}$, $t \leq 0$, and $s \leq 0$. (Note that, for example, $-p_{D}=(E, \mathbf{p})$, where $E$ and $\mathbf{p}$ refer to the incoming $\overline{\mathrm{D}}$ ).

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
03:32

Problem 8

If the $s$ channel process is $\mathrm{e}^{-} \mu^{-} \rightarrow \mathrm{e}^{-} \mu^{-}$, show that the boundaries of the physical regions of this and the crossed channel reactions are given by
$$
t=0, \quad s u=\left(M^{2}-m^{2}\right)^{2}
$$

Ameer Said
Ameer Said
Numerade Educator
02:36

Problem 9

Verify that crossing relation (4.42) is of the form
$$
\mathscr{R}_{\mathrm{e}^{-} \mathrm{e}^{+}}(s, t, u)=\operatorname{M}_{\mathrm{e}^{-} \mathrm{e}^{-}}(u, t, s)
$$

Chris Trentman
Chris Trentman
Numerade Educator
07:24

Problem 10

Show that the invariant amplitude, (4.41), for "spinless" electron-positron scattering can be written as
$$
\text { IR }_{\mathrm{e}^{-} \mathrm{e}^{+}}(s, t, u)=e^{2}\left(\frac{s-u}{t}+\frac{t-u}{s}\right) .
$$

Abid Hussain
Abid Hussain
Numerade Educator