Chapter 11, $\mathrm{e}^{+} \mathrm{e}^{-}$Annihilation and $\mathrm{QCD}$ Video Solutions, Quarks And Leptons. An Introductory Course In Modern Particle Physics

Problem 1

Fragmentation functions are of ten parametrized by the form
$$
D_{q}^{h}(z)=N \frac{(1-z)^{n}}{z}
$$
where $n$ and $N$ are constants. Show that
$$
N=(n+1)\langle z\rangle,
$$
where $\langle z\rangle$ is the average fraction of the quark energy carried by hadrons of type $\mathrm{h}$ after fragmentation. Further, show that
$$
n_{h} \sim \log \left(\frac{Q}{2 m_{h}}\right)
$$
for the two-jet process of Fig. 11.4. That is, the multiplicity of hadrons $\mathrm{h}$ grows logarithmically with the annihilation energy.

Guilherme Barros

Numerade Educator

02:44

Problem 2

The fragmentation functions $D(z)$ describe properties of partons and are therefore the same, no matter how the partons are produced. Consider the inclusive leptoproduction cross section $\sigma(\mathrm{ep} \rightarrow \mathrm{hX})$ and show that
$$
\frac{1}{\sigma} \frac{d \sigma}{d z}(\mathrm{ep} \rightarrow \mathrm{hX})=\frac{\sum_{q} e_{q}^{2} f_{q}(x) D_{q}^{h}(z)}{\sum_{q} e_{q}^{2} f_{q}(x)}
$$
where $f_{q}(x)$ are the proton structure functions of Chapter 9, see Fig. 11.6. The sum runs over the quarks and antiquarks that can be a parent of $h$.

Suzanne W.

Numerade Educator

04:34

Problem 3

Using charge conjugation and isospin invariance, show that
$$
\begin{aligned}
&D_{u}^{\pi^{+}}=D_{\bar{u}}^{\pi^{-}}=D_{d}^{\pi^{-}}=D_{d}^{\frac{\pi^{+}}{}}{ }, \\
&D_{u}^{\pi^{-}}=D_{\bar{u}}^{\pi^{+}}=D_{d}^{\pi^{+}}=D_{d}^{\frac{\pi}{d}}, \\
&D_{s}^{\pi^{+}}=D_{s}^{\pi^{-}} .
\end{aligned}
$$
Fig. 11.6 Deep inelastic leptoproduction of hadron $\mathrm{h} ;$ ep $\rightarrow \mathrm{hX}$.

Eduard Sanchez

Numerade Educator

09:32

Problem 4

Using the notation
$$
N_{p}^{\pi}(z) \equiv \frac{1}{\sigma} \frac{d \sigma}{d z}(\mathrm{ep} \rightarrow \pi \mathrm{X})
$$
show that, in the valence quark approximation for $\mathrm{p}, \mathrm{n}$,
$$
\frac{\int d z\left[N_{n}^{\pi^{+}}-N_{n}^{\pi^{-}}\right]}{\int d z\left[N_{p}^{\pi^{+}}-N_{p}^{\pi^{-}}\right]}=\frac{2}{7}
$$

Samuel Smith

Numerade Educator

02:04

Problem 5

Derive (11.19). Also show that the angle $\theta$ between the q and $\overline{\mathrm{q}}$ directions in Fig. $11.7$ is determined by the relation
$$
x_{\bar{q}}=\frac{2\left(1-x_{q}\right)}{2-x_{q}-x_{q} \cos \theta}
$$

Kaci Long

Numerade Educator

01:38

Problem 6

The $x_{T}$ distribution (11.30) can be translated into an "acollinearity" distribution $d \sigma / d \theta$, where $\theta$ is the angle between the $\mathrm{q}$ and $\overline{\mathrm{q}}$ jet directions defined in (11.20) and Fig. 11.7. Show that for $\theta$ not too large,
$$
\frac{1}{\sigma} \frac{d \sigma}{d \theta} \simeq \frac{8 \alpha_{s}}{3 \pi} \frac{1}{\theta} \log \left(\frac{1}{\theta^{2}}\right) .
$$

Manik Pulyani

Numerade Educator

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Problem 7

List all parton processes, in addition to those shown in Fig. 11.13, that can contribute to reaction (c) to the same order of $\alpha_{s}$. Comment on the relative strengths of the subprocesses, comparing, in particular, pp- and p $\bar{p}$-initiated reactions.

Erica Bartos

Numerade Educator

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Problem 8

Express the sum in (11.57) in terms of the valence and sea quark structure functions introduced in Chapter 9. Do the same for lepton pair production in $\bar{p} p$ and $\pi{ }^{\pm} p$ interactions.

Suzanne W.

Numerade Educator

03:44

Problem 9

The data for an (isoscalar) carbon target indicate that the ratio
$$
\frac{\sigma\left(\pi^{+} \mathrm{C} \rightarrow \mu^{-} \mu^{+} \mathrm{X}\right)}{\sigma\left(\pi^{-} \mathrm{C} \rightarrow \mu^{-} \mu^{+} \mathrm{X}\right)}
$$
is approximately unity for small values of $Q^{2} / \mathrm{s}$ but decreases toward $\frac{1}{4}$ as $Q^{2} / s$ approaches 1 . Explain why this is in agreement with the Drell-Yan prediction. Note that $x y=Q^{2} / s$.

Amany Waheeb

Numerade Educator

02:42

Problem 10

Including diagrams with gluons introduces logarithmic scaling violations in (11.58). Draw the diagrams which give an $O\left(\alpha^{2} \alpha_{s}\right)$ contribution to the lepton pair cross section. Compute the cross section for the diagram shown in Fig. 11.13e.

Narayan Hari

Numerade Educator