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

For dimension-six operators of a real scalar field, let $n$ count fields and $r$ count partial derivatives. Mass dimension gives $n+r=6$. Lorentz invariance requires even $r$. Discard derivatives acting on the entire expression, but do not yet identify different expressions by integration by parts. A complete list of seven representatives isFor $(n,r)=(6,0)$ there is one choice; for $(4,2)$ the derivative distribution is either $2$ or $1+1$; for $(2,4)$ it is $4$, $3+1$, or $2+2$, the last allowing two distinct metric contractions. Commuting partial derivatives identifies alternative orderings. Contractions with a Levi-Civita symbol vanish because an antisymmetric index pair necessarily contracts symmetric derivatives. The remaining case $(0,6)$ gives no field-dependent expression. No independent odd-$r$ Lorentz scalar exists.

The source's seven means this unreduced list. If “ignoring total derivative operators” were interpreted as already taking the full quotient by total derivatives, seven would be incorrect: the next part gives three representatives of that quotient. Dimensionful factors multiplying lower-dimensional terms do not count as new canonical-dimension-six field monomials.