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

The scalar field path integral integrates over real field histories with vacuum boundary conditions. With $S_0[\varphi ]=\int d^{4}x{\mathcal{L}}_0$, the partition function isThe constant $\mathcal{N}$ fixes the desired vacuum normalization. A generating functional adds $\int d^{4}xJ\varphi$ to the action; the required partition function is its zero-source value. Equivalently, a Wick rotation gives $Z_E={\mathcal{N}}_E\int \mathcal{D}\varphi e^{-S_E[\varphi ]}$, with the positive-sign kinetic and potential terms in the Euclidean action when $m^2\ge 0$ and $\lambda >0$.

Vary the action, taking the field variation to vanish at the boundary. Integration by parts givesThus the Euler-Lagrange field equation isFor the field redefinition in part f, define the functional derivative $E(\varphi )=\delta S_0/\delta \varphi =-(\square \varphi +m^2\varphi +\lambda {\varphi }^3/6)$. The sign does not affect the equation of motion, but it fixes the sign of an added redundant operator.

The action is dimensionless, so in four dimensions the Lagrangian density has mass dimension four. Its kinetic term fixes $[\varphi ]=1$ and $[{\partial }_{\mu }]=1$. An interaction $g_D{\mathcal{O}}_D$ with $[{\mathcal{O}}_D]=D$ has $[g_D]=4-D$. ThereforeIn an effective field theory with heavy scale $\Lambda$, a higher-dimensional local operator enters as $c_D{\mathcal{O}}_D/{\Lambda }^{D-4}$. For characteristic scales $E,m\ll \Lambda$ and coefficients compatible with this expansion, its effects are suppressed by powers of the small ratios. Omitting these terms defines the leading approximation and its truncation error in effective field theory. It is not a fundamental prohibition: such terms must be retained if that accuracy is required. A continuum renormalizable quantum field theory can instead be defined by the closed set of counterterms of dimension at most four.

Up to total derivatives, all Lorentz scalar terms of dimension at most four built from one real scalar field areThe term $\varphi \square \varphi$ is equivalent to the kinetic term by integration by parts; $\square \varphi$ and $\square ({\varphi }^2)$ contribute only at the boundary. The displayed Lagrangian density omits the linear and cubic interactions and a field-independent constant. The exact Z2 symmetry $\varphi \mapsto -\varphi$ forbids both odd interactions, and radiative corrections preserving this symmetry cannot generate them. The constant shifts the vacuum energy and the normalization of $Z$ and can be dropped when gravity and absolute vacuum energy are not being studied. A constant vacuum counterterm may still be used when renormalizing unnormalized vacuum diagrams. The kinetic coefficient has already been fixed by field normalization.

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.

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:Consequentlyspans 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.

Use the sign convention $E(\varphi )=\delta S_0/\delta \varphi$ from part b, and letIn the path integral, make the local field redefinition ${\varphi }_{\mathrm{old}}=\chi -\eta f(\chi )$. The functional Taylor expansion of the action givesThe two first-order terms cancel. Neglecting the functional Jacobian as permitted,Thus $fE$ is a redundant operator at first order, by the equivalence theorem for field redefinitions. The same argument works for a local derivative-dependent $f$, which is needed to remove $(\square \varphi {)}^2$. Insertions and sources must be transformed too: arbitrary untransformed off shell correlation functions need not be equal.

Sign qualification. If instead $E_{+}=\square \varphi +m^2\varphi +\lambda {\varphi }^3/6$ is used, then $E_{+}=-\delta S_0/\delta \varphi$. An added $+\eta f{E}_{+}$ is removed by ${\varphi }_{\mathrm{old}}=\chi +\eta f(\chi )$. The minus shift requested in the source therefore fixes which sign is meant by its otherwise unspecified $E$.

Let $a=\lambda /6$. Within terms already suppressed by ${\Lambda }^{-2}$, the leading equation of motion gives $\square \varphi =-m^2\varphi -a{\varphi }^3$. Equality here means equivalence under a perturbative field redefinition, together with integration by parts, rather than a pointwise identity inside arbitrary off shell correlation functions. HenceFor the second relation, factor the difference as $(\square \varphi -m^2\varphi -a{\varphi }^3)(\square \varphi +m^2\varphi +a{\varphi }^3)$, which is explicitly a redundant operator. Ifthe reduced correction isThe first two coefficients shift the already present mass and quartic coupling constant. After these shifts,This is the equation-of-motion reduction of scalar operators through $O({\Lambda }^{-2})$; terms of $O({\Lambda }^{-4})$ are outside that truncation. The lower-dimensional shifts should not be silently dropped when $m\ne 0$.