{title2=$\eta$}
{wiki}

An anomalous dimension is the interaction-generated difference between a full <scaling dimension> and its <engineering dimension>. For a scalar field, momentum-dependent self-energy corrections require <wave-function renormalization> and produce $\Delta_\phi=(d-2+\eta)/2$.

Anomalous dimension

The free energy is dimensionless in units with $k_BT=1$, so the integrand has momentum dimension $d$. Since a <derivative> has dimension one, the kinetic term gives
$$
2+2[\phi]_{\rm eng}=d,
\qquad
\boxed{[\phi]_{\rm eng}=\frac{d-2}{2}}.
$$
The mass term then gives
$$
[\mu_0^2]+2[\phi]_{\rm eng}=d,
\qquad
\boxed{[\mu_0^2]=2}.
$$
These are <engineering dimensions>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 303 / 2 / b / Solution

The operator $\nabla^{2m}\phi^{2n}$ has <engineering dimension>
$$
2m+2n\frac{d-2}{2}=2m+n(d-2).
$$
The action integral is dimensionless, so
$$
\boxed{[\alpha]=d-2m-n(d-2)=2n-2m-(n-1)d}.
$$
Thus $\alpha$ is a <relevant coupling>, <marginal coupling>, or <irrelevant coupling> according as
$$
\boxed{
d<\frac{2(n-m)}{n-1},\qquad
d=\frac{2(n-m)}{n-1},\qquad
d>\frac{2(n-m)}{n-1}}
$$
respectively.

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 303 / 2 / d / Solution

Under $x'=x/\zeta$ and the Gaussian field rescaling $\phi'(x')=\zeta^{(d-2)/2}\phi(x)$, a mass squared has <engineering dimension> two and a quartic coupling has dimension $4-d$. Therefore
$$
\boxed{A=2,\qquad B=4-d}.
$$

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 303 / 3 / a / iv / Solution

{title2=$[X]_{\mathrm{eng}}$}

The engineering dimension of a field, parameter or operator is its dimension under the rescaling that leaves the quadratic theory invariant. Interactions can change it into a full <scaling dimension>.

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 Incoming links: Engineering dimension