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Advanced quantum field theory for particle physics course of the University of Oxford

Past exam of the advanced quantum field theory for particle physics course of the University of Oxford

2025

{c}

Write $X\simeq Y$ when the two densities differ by a <total derivative>. Under the <boundary conditions> used in the <action>, they have the same integral. The following identities establish the <integration-by-parts reduction of scalar operators>:
$$
\begin{aligned}
\partial_\mu(\phi^3\partial^\mu\phi)&=3\phi^2(\partial\phi)^2+\phi^3\Box\phi,
&\mathcal O_2&\simeq-\frac13\mathcal O_3,\\
\partial_\mu(\phi\partial^\mu\Box\phi)&=\partial_\mu\phi\partial^\mu\Box\phi+\phi\Box^2\phi,
&\mathcal O_4&\simeq-\mathcal O_5,\\
\partial_\mu(\partial^\mu\phi\Box\phi)&=(\Box\phi)^2+\partial_\mu\phi\partial^\mu\Box\phi,
&\mathcal O_5&\simeq-\mathcal O_6,\\
\partial_\mu(\partial_\nu\phi\partial^\mu\partial^\nu\phi)&=
(\partial_\mu\partial_\nu\phi)^2+\partial_\nu\phi\partial^\nu\Box\phi,
&\mathcal O_7&\simeq-\mathcal O_5.
\end{aligned}
$$
Consequently
$$
\boxed{\{\phi^6,\ \phi^3\Box\phi,\ (\Box\phi)^2\}}
$$
spans the <operator basis in effective field theory> before <equation-of-motion reduction of scalar operators>. These four independent relations reduce the seven representatives to three; the three sectors have different field or derivative content, so <integration by parts> cannot identify them further.

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Past exam of the advanced quantum field theory for particle physics course of the University of Oxford / 2025 / 1 / e / Solution

Past exam of the advanced quantum field theory for particle physics course of the University of Oxford / 2025 / 1 / e / Solution

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