Mass dimension ($[X]$)

In units with $\hslash =c=1$, the mass dimension $[X]$ is the exponent with which a quantity scales under a common rescaling of energies and momenta. In $d$ spacetime dimensions a Lagrangian density has mass dimension $d$.

With a canonical kinetic term, a real scalar field has mass dimension $(D-2)/2$. A coupling constant multiplying ${\varphi }^n$ therefore has mass dimension $D-n(D-2)/2$. For $n>2$, it is dimensionless precisely at $D_c=2n/(n-2)$, giving six for a cubic interaction and four for a quartic interaction. This is a power counting statement about perturbation theory, not a claim that the corresponding quantum beta function vanishes.

Canonical free scalar and gauge-vector kinetic terms contain two derivatives and two fields, whereas the Dirac kinetic term contains one derivative and two spinor fields. Matching their mass dimension to $d$ in $d$ spacetime dimensions gives the displayed engineering dimensions. These are the dimensions used in power counting; interacting scaling dimensions can additionally contain anomalous contributions.