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Quarks And Leptons. An 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
00:24

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).

Sarah Mccrumb
Sarah Mccrumb
Numerade Educator
01:22

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):

Raymond Matshanda
Raymond Matshanda
Numerade Educator
05:09

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).

Chai Santi
Chai Santi
Numerade Educator
01:02

Problem 5

Use (9.23) to show that the parton model predicts
$$
\left(\frac{d \sigma}{d x d y}\right)_{\mathrm{ep} \rightarrow \mathrm{eX}}=\frac{2 \pi \alpha^{2}}{Q^{4}} s\left[1+(1-y)^{2}\right] \sum_{i} e_{i}^{2} x f_{i}(x)
$$

Raj Bala
Raj Bala
Numerade Educator
06:02

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.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
02:27

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

Keshav Singh
Keshav Singh
Numerade Educator
02:50

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 $\sigma_{T}=0$.

Declan Nell
Declan Nell
Numerade Educator
View

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
02:19

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

Keshav Singh
Keshav Singh
Numerade Educator
03:22

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
$$
f_{i}(x) \underset{x \rightarrow 1}{\longrightarrow}(1-x)^{2 n_{x}-1}
$$
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 $u^{\pi}(x)$, the u-quark structure function of a $\pi^{+}$-meson.

Suzanne W.
Suzanne W.
Numerade Educator