Chapter 14, Gauge Symmetries Video Solutions, Quarks and leptons: introductory course in modern particle physics

Problem 1

If you are unfamiliar with this formalism, consult, for example, Goldstein (1977) or work through the example given in Sakurai (1967), page 3.

Instead of writing down a relativistic wave equation, we simply choose a Lagrangian $\mathcal{L}$. Provided our choice is a Lorentz scalar, the equation of motion resulting from (14.4) will be covariant. For example, substituting the Lagrangian
$$
\mathcal{L}=\frac{1}{2}\left(\partial_\mu \phi\right)\left(\partial^\mu \phi\right)-\frac{1}{2} m^2 \phi^2
$$
into (14.4) gives the Klein-Gordon equation
$$
\partial_\mu \partial^\mu \phi+m^2 \phi=\left(\square^2+m^2\right) \phi=0 .
$$

There is no mystery here. The choice of $\mathcal{L}$ was specifically designed to reproduce (14.7).

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Problem 2

Verify that the Dirac equation follows from
$$
\mathcal{L}=i \bar{\psi} \gamma_\mu \partial^\mu \psi-m \bar{\psi} \psi,
$$
where each of the four components of $\psi$ and $\bar{\psi}$ is regarded as an independent field variable.

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Problem 3

Show that the substitution of the Lagrangian
$$
\mathcal{L}=-{ }_4^1 F_{\mu v} F^{\mu \nu}-j^\mu A_\mu
$$
into the Euler-Lagrange equation for $A_\mu$ gives the Maxwell equations, $(6.57)$,
$$
\partial_\mu F^{\mu \nu}=j^\nu,
$$
where $F^{\mu v} \equiv \partial^\mu A^v-\partial^v A^\mu$. Hence, show that the current is conserved, that is, $\partial_v j^v=0$.

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01:53

Problem 4

With the addition of a term $\frac{1}{2} m^2 A_\mu A^\mu$, show that the Lagrangian of (14.9) leads to an equation of motion
$$
\left(\square^2+m^2\right) A^\mu=j^\mu .
$$

The new term is therefore a photon mass contribution. We shall see in Section 14.3 that it is forbidden by gauge invariance. The photon is massless.

Manik Pulyani

Numerade Educator

04:47

Problem 5

Show that $d Q / d t=0$.

Scott Stetson

Numerade Educator

Problem 6

Show that $U(1)$ phase invariance of the Lagrangian
$$
\mathcal{L}=\left(\partial_\mu \phi\right)^*\left(\partial^\mu \phi\right)-m^2 \phi^* \phi
$$
of a complex scalar field implies the existence of a conserved current
$$
j^\mu=-i e\left(\phi^* \partial^\mu \phi-\phi \partial^\mu \phi^*\right),
$$
and compare with (3.25). Note that the Lagrangian, (14.19), for a complex field $\phi=\left(\phi_1+i \phi_2\right) / \sqrt{2}$ is normalized to ensure that
$$
\mathcal{L}(\phi)=\mathcal{L}\left(\phi_1\right)+\mathcal{L}\left(\phi_2\right),
$$
where $\mathcal{L}\left(\phi_i\right)$, the Lagrangian for a real field $\phi_i$, is given by $(14.6)$.

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01:21

Problem 7

Read about the Bohm-Aharonov effect. Suggested references are the Feynman Lectures on Physics, Volume 2, or Wu, T. T, and Yang, C. N. (1975) Phys. Rev. D12, 3845.

Suzanne W.

Numerade Educator

02:32

Problem 8

Show that det $U=e^{i \phi}$, where $\phi$ is real. We separate off such an overall phase by restricting the group transformations to those with det $U=+1$, see Section 2.3. We called this the group of special unitary $3 \times 3$ matrices $S U(3)$. Show that the requirement $\operatorname{det} U=+1$ implies $\operatorname{Tr}\left(T_a\right)=0$. Verify that $U^{\dagger}=U^{-1}$ requires
$$
\alpha_a T_a=\alpha_a^* T_a^{\dagger},
$$
so that choosing $T_a$ to be hermitian means the group parameters $\alpha_a$ are real.

Victor Salazar

Numerade Educator

01:10

Problem 9

Show that the structure constants $f_{a b c}$ are antisymmetric under interchange of any pair of indices.

Manik Pulyani

Numerade Educator

00:49

Problem 10

Due to the additional term in (14.38), $G_{\mu \nu}^a$ has a more complicated form than its counterpart in QED, (14.27). In order that the kinetic energy be invariant under (14.38), show that
$$
G_{\mu \nu}^a=\partial_\mu G_\nu^a-\partial_\nu G_\mu^a-g f_{a b c} G_\mu^b G_\nu^c .
$$

Alan Ghazarians

Numerade Educator

Problem 11

Using the prescription for obtaining the Feynman rules from the Lagrangian that we mentioned in Section 14.1, show that the vertex factors for the quark-gluon and triple gluon vertices of Fig. 14.1 are, respectively,
$$
\begin{gathered}
-i g \gamma_\mu\left(T_a\right)_{i j}, \\
-g f_{a b c}\left[g_{\mu \nu}\left(p_1-p_2\right)_\lambda+g_{\nu \lambda}\left(p_2-p_3\right)_\mu+g_{\lambda \mu}\left(p_3-p_1\right)_\nu\right] .
\end{gathered}
$$

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Problem 12

The Lagrangian for three interacting real fields $\phi_1, \phi_2, \phi_3$ is
$$
e=\frac{1}{2}\left(\partial_\mu \phi_i\right)^2-\frac{1}{2} \mu^2 \phi_i^2-\frac{1}{4} \lambda\left(\phi_i^2\right)^2
$$
with $\mu^2<0$ and $\lambda>0$, and where a summation of $\phi_i^2$ over $i$ is implied. Show that it describes a massive field of mass $\sqrt{-2 \mu^2}$ and two massless Goldstone bosons.

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