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Quarks and leptons: introductory course in modern particle physics

Francis Halzen, Alan D. Martin

Chapter 9

Partons - all with Video Answers

Educators

Chapter Questions

04:09

Problem 1

Prove that $0 \leq x \leq 1$, as it must be if $x$ represents a momentum fraction; recall Exercise 8.11.

João Gabriel Alencar Caribé
João Gabriel Alencar Caribé
Numerade Educator

Problem 2

Convince yourself that the interaction time is much shorter than the time scale over which the partons inside the target interact with one another. Read the explicit derivation in J. D. Bjorken and E. A. Paschos, Phys. Rev. 185, 1975 (1969); see also Perl (1974).

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Problem 3

It is helpful to work through an alternative derivation of the parton model result, (9.13)-(9.15), in terms of the invariant variables of $(6.29)$ :
$$
s \simeq 2 k \cdot p, \quad u \simeq-2 k^{\prime} \cdot p, \quad t \equiv-Q^2=-2 k \cdot k^{\prime},
$$
that is, the ep $\rightarrow$ eX rate is simply the incoherent sum over all the contributing partons. Show that the invariant variables for the parton subprocess are given by
$$
\hat{s}=x s, \quad \hat{u}=x u, \quad \hat{t}=t .
$$

Use these relations, together with the $\mu$ scattering amplitude of (6.30), to show that
$$
\left(\frac{d \sigma}{d t d u}\right)_{\mathrm{eq}_i \rightarrow \mathrm{eq}_1}=x \frac{d \sigma}{d \hat{t} d \hat{u}}=x \frac{2 \pi \alpha^2 e_i^2}{t^2}\left(\frac{s^2+u^2}{s^2}\right) \delta(t+x(s+u)) .
$$

Now, also express the left-hand side of (9.17) in terms of $s, t$, and $u$. It is simplest to use (8.31). Verify that
$$
\left(\frac{d \sigma}{d t d u}\right)_{\mathrm{ep} \rightarrow \mathrm{eX}}=\frac{4 \pi \alpha^2}{t^2 s^2} \frac{1}{s+u}\left[(s+u)^2 x F_1-u s F_2\right],
$$
where $F_1 \equiv M W_1$ and $F_2 \equiv \nu W_2$. Insert (9.18) and (9.19) into (9.17). Compare coefficients of $u s$ and $s^2+u^2$ and so obtain the master formula of the parton model:

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Problem 4

In Chapter 8, we evaluated the ep $\rightarrow$ eX cross section for the electron to be scattered into the $d E^{\prime} d \Omega$ element in the target proton rest frame (the laboratory frame). Show that
$$
d E^{\prime} d \Omega=\frac{\pi}{E E^{\prime}} d Q^2 d \nu=\frac{2 M E}{E^{\prime}} \pi y d x d y,
$$
where $x$ and $y$ are the dimensionless variables
$$
x=\frac{Q^2}{2 M \nu}, \quad y=\frac{p \cdot q}{p \cdot k_{(\mathrm{lab} .)}}=\frac{\nu}{E},
$$
see (8.30). The allowed kinematic region $(0 \leq x, y \leq 1)$ is shown in Fig. 9.3.

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Problem 5

Use (9.23) to show that the parton model predicts
$$
\left(\frac{d \sigma}{d x d y}\right)_{e p \rightarrow \mathrm{eX}}=\frac{2 \pi \alpha^2}{Q^4} s\left[1+(1-y)^2\right] \sum_i e_i^2 x f_i(x)
$$
if particle masses are neglected.

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Problem 6

Show that
$$
1-y=\frac{p \cdot k^{\prime}}{p \cdot k} \simeq \frac{1}{2}(1+\cos \theta),
$$
where $\theta$ is the scattering angle in the electron-quark center-of-mass frame.
Hence, identify the $(1-y)^2$ and 1 terms in (9.24) with scattering between an electron and quark with opposite helicities and with the same helicity, respectively.

The parton model result $2 x F_1=F_2$ is known as the Callan-Gross relation. It is a consequence of the quarks having spin $\frac{1}{2}$ and is well borne out by the data.

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Problem 7

In the limit $\nu, Q^2 \rightarrow \infty$, with $x$ fixed, show that the Callan-Gross relation implies that the virtual photon-quark cross sections of $(8.53),(8.54)$ satisfy
$$
\frac{\sigma_L}{\sigma_T} \rightarrow 0
$$

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02:21

Problem 8

Starting from (6.51), show that if quarks had spin 0, $F_2(x)$ would still be given by $(9.13)$ but that $F_1(x)=0$ and hence $\sigma_T=0$.

Thus, in contrast to (9.26), spin-0 quarks would yield $\sigma_T / \sigma_L=0$. We can understand this difference by glancing at Fig. 9.4, which shows the head-on collision between the quark and the virtual photon. By conservation of $J_z$ (with $z$ along p), we see that a spin- 0 quark cannot absorb a photon of helicity $\lambda= \pm 1$, so $\sigma_T=0$. Suppose now the quark has spin $\frac{1}{2}$. We recall that its helicity is conserved in a high-energy interaction (see Section 6.6). This can only be achieved by a $\lambda= \pm 1$ photon; hence, $\sigma_L \rightarrow 0$ in this case.

Ashwin Banarsee
Ashwin Banarsee
Numerade Educator
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Problem 9

Show that the above expressions lead to the bounds
$$
\frac{1}{4} \leq \frac{F_2^{e n}(x)}{F_2^{e p}(x)} \leq 4
$$
whatever the value of $x$. The lower (upper) limit would be realized if only u (d) quarks were present in the proton.

Suzanne W.
Suzanne W.
Numerade Educator
04:21

Problem 10

Assume that the virtual photon-proton total cross section of Section 8.5 behaves like a constant as $x \rightarrow 0, \nu \rightarrow \infty$ for fixed $Q^2$, and hence show that
$$
f_i(x) \underset{x \rightarrow 0}{\longrightarrow} \frac{1}{x} .
$$

Thus, we have a logarithmic growth of partons at small $x$.

Guilherme Barros
Guilherme Barros
Numerade Educator

Problem 11

Discuss, on physical grounds, the behavior of $f_i(x)$ in the limit as $x \rightarrow 1$ when parton $i$ carries all the momentum of the proton. Counting rules have been proposed which argue that
$$
f_i(x) \underset{x \rightarrow 1}{\longrightarrow}(1-x)^{2 n_s-1},
$$
where $n_s$ is the number of spectator valence quarks which share between them the residual, vanishingly small momentum of the proton. Contrast the $x \rightarrow 1$ behavior of $u^p(x)$ with that of $u^{\nabla}(x)$, the u-quark structure function of a $\pi^{+}$-meson.

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