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Quarks And Leptons. An Introductory Course In Modern Particle Physics

Francis Halzen, Alan D. Martin

Chapter 10

Quantum Chromodynamics - all with Video Answers

Educators

Chapter Questions

03:59

Problem 1

Derive (10.3)-(10.5). It is easiest to work in the laboratory frame; the variables are given in Section 6.8. In the deep inelastic limit, the electron beam energy $E \gg E^{\prime}$. Then, as $\nu^{2} \sim E^{2}$ and $Q^{2} \simeq$ $4 E E^{\prime} \sin ^{2}(\theta / 2)$, we have $\nu^{2} \gg Q^{2}$. We also have introduced the $\gamma^{*}$-proton center-of-mass energy squared:
$$
s=(q+p)^{2}=M^{2}+2 M v-Q^{2} \simeq 2 M K,
$$

Ajay Singhal
Ajay Singhal
Numerade Educator
11:06

Problem 2

We outline the various stages of the calculation. First, show that for $\gamma^{*}(q) q(p) \rightarrow q\left(p^{\prime}\right)$,
$$
\overline{|\mathscr{M}|^{2}}=2 e_{i}^{2} e^{2} p \cdot q
$$
where we have averaged over transverse polarization states of the incoming $\gamma^{*}$. From Section $4.3$, we have
$$
F d \dot{\sigma}_{T}=\overline{\left.|\mathscr{}|\right|^{2}}(2 \pi)^{4} \delta^{(4)}\left(p^{\prime}-p-q\right) \frac{d^{3} p^{\prime}}{2 p_{0}^{\prime}(2 \pi)^{3}}
$$
where $F$ is the $\gamma^{*} \mathrm{q}$ flux factor. Calculate $F \hat{\sigma}_{T}$ by making use of $(6.47) .$ Use $F \hat{\sigma}_{0}=8 \pi^{2} \alpha$, see (10.5).

To determine the parton model prediction for $F_{2} / x$ of (10.4), we input in (10.8) the following cross section ratio for $\gamma^{*} \mathrm{q} \rightarrow \mathrm{q}$ :
$$
\frac{1}{\partial_{0}}\left(\hat{\sigma}_{T}+\hat{\sigma}_{L}\right)=e_{i}^{2} \delta(1-z),
$$
see $(10.10)$ and (10.11). After substitution, we obtain
$$
\frac{F_{2}\left(x, Q^{2}\right)}{x}=\sum_{i} e_{i}^{2} \int_{x}^{1} \frac{d y}{y} f_{i}(y) \delta\left(1-\frac{x}{y}\right)=\sum_{i} e_{i}^{2} f_{i}(x)
$$
An identical expression is found for $2 F_{1}$. The parton model results of (9.13) and (9.14) are indeed reproduced.Show that the parton model diagram, Fig. 10.1, gives
$$
\begin{aligned}
&\frac{\hat{\sigma}_{T}\left(z, Q^{2}\right)}{\hat{\sigma}_{0}}=e_{i}^{2} \delta(1-z) \\
&\hat{\sigma}_{L}\left(z, Q^{2}\right)=0
\end{aligned}
$$

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
04:55

Problem 3

In the center-of-mass frame of the parton process $\gamma^{*} \mathrm{q}_{1} \rightarrow \mathrm{q}_{2} \mathrm{~g}$ of Fig. $10.7$, show that
$$
\begin{aligned}
&\hat{s}=2 k^{2}+2 k q_{0}-Q^{2}-4 k^{\prime 2}, \\
&\hat{t}=-Q^{2}-2 k^{\prime} q_{0}+2 k k^{\prime} \cos \theta=-2 k k^{\prime}(1-\cos \theta), \\
&\hat{u}=-2 k k^{\prime}(1+\cos \theta),
\end{aligned}
$$
where $k, k^{\prime}$ are the magnitudes of the center-of-mass momenta $\mathbf{k}, \mathbf{k}^{\prime}$. Note that for the virtual photon, $q_{0}^{2}-k^{2}-Q^{2}$. A useful result is
$$
4 k k^{\prime}=-\hat{t}-\hat{u}=\hat{s}+Q^{2}
$$
The interesting quantity is the transverse momentum of the outgoing quark, $p_{T}=k^{\prime} \sin \theta$. Show that
$$
p_{T}^{2}=\frac{\operatorname{sit} 2}{\left(s+Q^{2}\right)^{2}}
$$
or, in the limit of small-angle scattering. $-\hat{t} \approx \hat{s}$, that
$$
p_{T}^{2}=\frac{s(-\hat{\imath})}{s+Q^{2}} .
$$
Further, show that for small scattering angles $(\cos \theta \simeq 1)$,
$$
d \Omega=\frac{4 \pi}{s} d p_{T}^{2}
$$
$\underset{\rightarrow q_{2} g .}{\text { Figy } 10.7 \text { Center-of-mass frame for } \gamma^{*} q_{1}}$
10.4 The Gluon Emission Cross Section
213
It is clear from $(10.17)$ that at high energy ( $\vec{s}$ large), the $\gamma^{*} \mathrm{q} \rightarrow \mathrm{qg}$ cross section peaks as $-\hat{i} \rightarrow 0$. Referring back to Fig. $4.8$ and the related discussion, we see that this is due to quark exchange in the $t$ channel. We can therefore approximate the cross section by its forward peak. For forward scattering, we find, from (10.26) and (4.35), that
$$
\frac{d \dot{\sigma}}{d p_{T}^{2}}=\frac{1}{16 \pi s^{2}} \overline{|S M|^{2}}
$$

Abid Hussain
Abid Hussain
Numerade Educator
01:50

Problem 4

Derive (10.27), Make use of (10.26) and $(10.20)$, together with (4.34). Show that the $\gamma^{*} \mathrm{q}$ flux factor is given by $2 s$ using the convention of (8.48).

Mukesh Devi
Mukesh Devi
Numerade Educator
01:45

Problem 5

Show that the maximal transverse momentum for the two-body interaction $\gamma^{*} \mathrm{q} \rightarrow$ qg is given by
$$
\left(p_{T}^{2}\right)_{\max }=\frac{\vec{s}}{4}=Q^{2} \frac{1-z}{4 z}
$$

Suzanne W.
Suzanne W.
Numerade Educator
01:06

Problem 6

Study the origin of the $\log Q^{2}$ term. Recall that the $\gamma^{*} \mathrm{q} \rightarrow \mathrm{qg}$ cross section, $d \dot{\sigma} / d p_{T}^{2}$, is dominated by the forward peak. The $t$-channel quark propagator leads to a factor $1 / p_{T}^{4}$. Show that helicity conservation at the gluon vertex weakens this singularity by introducing a factor $p_{T}^{2}$ in the numerator.

Carlos Henrique De Lima
Carlos Henrique De Lima
Numerade Educator
07:40

Problem 7

The origin of the scaling violation of $q\left(x, Q^{2}\right)$ given by (10.37) can be traced back to (10.32). There, we assumed that $\alpha_{s}$ is a constant. Show that (10.37) is also obtained for a running coupling constant. Assume that $\alpha_{s}$ in (10.32) is $\alpha_{s}\left(p_{T}^{2}\right)$ as given by (7.65).

Nathan Prins
Nathan Prins
Numerade Educator
02:19

Problem 8

Show that the color factor for $\gamma^{*} \mathrm{~g} \rightarrow \mathrm{q} \overline{\mathrm{q}}$ is $\frac{1}{2}$.

Narayan Hari
Narayan Hari
Numerade Educator
15:01

Problem 9

Verify that for $\gamma^{*} \mathrm{~g} \rightarrow \mathrm{q} \overline{\mathrm{q}}$,
$$
\overline{|\mathscr{I}|^{2}}=32 \pi^{2}\left(e_{q}^{2} \alpha \alpha_{s}\right) \frac{1}{2}\left(\frac{\hat{u}}{\hat{t}}+\frac{\hat{t}}{\hat{u}}-\frac{2 \hat{s} Q^{2}}{\hat{t} \hat{u}}\right),
$$
using $\sum \varepsilon_{\mu}^{*} \varepsilon_{\nu}=-g_{\mu \nu}$ for the $\gamma^{*}$-polarization sum. Hence, show that (10.34) for the proton structure function contains the additional contribution
$(10.40)$
where $g(y)$ is the gluon density in the proton and where
$$
P_{q g}(z)=\frac{1}{2}\left(z^{2}+(1-z)^{2}\right)
$$
represents the probability that a gluon annihilates into a q $\bar{q}$ pair such that the quark has a fraction $z$ of its momentum. Detailed measurements of the scaling violations of $F_{2}\left(x, Q^{2}\right)$ probe the gluon distribution inside the proton through (10.40).

Linda Winkler
Linda Winkler
Numerade Educator
11:08

Problem 10

How would you set about verifying that
$$
\begin{aligned}
&P_{g q}(z)=\frac{4}{3} \frac{1+(1-z)^{2}}{z}, \\
&P_{g g}(z)=6\left(\frac{1-z}{z}+\frac{z}{1-z}+z(1-z)\right) ?
\end{aligned}
$$

Matthew Winsor
Matthew Winsor
Numerade Educator
00:45

Problem 11

Express (10.42) in symbolic form.

Julie Silva
Julie Silva
Numerade Educator
02:23

Problem 12

EXERCISE $10.12$ Obtain the evolution equations for the combinations
$$
\begin{aligned}
q_{N S} & \equiv q_{i}-q_{j} \\
q_{S} & \equiv \sum_{i} q_{i} .
\end{aligned}
$$
The subscripts are conventional and, in fact, are used to indicate that the combinations refer to nonsinglet and singlet combinations of the quark flavor group.

Daniel Sneed
Daniel Sneed
Numerade Educator
01:22

Problem 13

Use $(10.51)$ and $(10.52)$ to show that
$$
P_{q q}(z)=\frac{4}{3} \frac{1+z^{2}}{(1-z)_{+}}+2 \delta(1-z)
$$

AG
Ankit Gupta
Numerade Educator
05:50

Problem 14

Use momentum conservation to justify
$$
\int_{0}^{1} d z z\left(q_{S}\left(z, Q^{2}\right)+g\left(z, Q^{2}\right)\right)=1
$$
Hence, determine the $\delta(1-z)$ term in $P_{g g}$ and verify
$$
P_{g g}(z)=6\left(\frac{1-z}{z}+\frac{z}{(1-z)_{+}}+z(1-z)\right)+\left(\frac{11}{2}-\frac{n_{f}}{3}\right) \delta(1-z),
$$
where $n_{f}$ is the number of quark flavors.

Mahnoor Amin
Mahnoor Amin
Numerade Educator
01:45

Problem 15

Show that momentum conservation at the QCD vertex requires (for $z<1$ )
$$
\begin{aligned}
&P_{q q}(z)=P_{g q}(1-z) \\
&P_{q g}(z)=P_{q g}(1-z) \\
&P_{g g}(z)=P_{g g}(1-z)
\end{aligned}
$$
Check that the explicit formulas for the "splitting" functions satisfy these relations.

Suzanne W.
Suzanne W.
Numerade Educator
02:59

Problem 16

If $\alpha_{s}\left(Q^{2}\right)=c / \log Q^{2}$, show that (10.37) leads to
$$
\frac{\int x^{n-1} q\left(x, Q^{2}\right) d x}{\int x^{n-1} q\left(x, Q_{0}^{2}\right) d x}=\left(\frac{\log Q^{2}}{\log Q_{0}^{2}}\right)^{A_{n}}
$$
where
$$
\begin{aligned}
A_{n} &=\frac{c}{2 \pi} \int_{0}^{1} x^{n-1} P_{q q}(x) d x \\
&=\frac{c}{2 \pi} \frac{4}{3}\left(-\frac{1}{2}+\frac{1}{n(n+1)}-2 \sum_{j=2}^{n} \frac{1}{j}\right)
\end{aligned}
$$
That is, in QCD, the moments $(n \geq 1)$ of the quark structure functions decrease as calculable powers of $\log Q^{2} \cdot c$ is given by (7.65).
The observant reader may have noticed what appears to be a contradiction in our interpretation of the $P$ functions. For example, we regard the $P_{q q}$ term as the correction factor to the quark density that arises from allowing for gluon emission. However, there are two diagrams: one with the gluon emitted from the initial quark line and the other with the gluon radiated from the final quark line; see Fig. 10.2. Our picture is only valid if the first diagram dominates. Then, the emitted gluon can be considered as part of the proton structure. It is a "partonlike" diagram. It turns out that both diagrams are required to ensure gauge invariance of the amplitude, but that the second only plays the role of canceling the contributions from the unphysical polarization states of the gluon. Adopting a physical gauge, in which we sum only over transverse gluons, only the first diagram remains.

Hafiz Shahzaib
Hafiz Shahzaib
Numerade Educator