Power counting in quantum field theory

Power counting assigns mass dimensions to fields and couplings. An interaction is relevant, marginal or irrelevant when its coupling has positive, zero or negative mass dimension, respectively.

The truncation error in effective field theory is the uncertainty from omitted orders of a stated energy, derivative or loop expansion. It is commonly estimated from the first omitted terms using an explicit assumption about Wilson coefficients. Small energy alone does not justify ignoring an anomalously large coupling or a nearby omitted pole.

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.

The superficial degree of divergence $D$ is the ultraviolet power predicted by rescaling all independent loop momenta together. For a scalar graph with $L$ loops and $I$ propagators behaving as $p^{-2}$ in $d$ dimensions, $D=dL-2I$ before cancellations and subdivergences are considered.

For a connected amputated Feynman diagram with the quartic Yukawa interaction in three spacetime dimensions, scalar quartic and scalar sextic vertices, $b$ external bosonic legs, $f$ external fermionic legs and $n_4$ scalar quartic vertices, the superficial degree of divergence is the displayed expression. Indeed $\omega =3L-2I_b-I_f$, $2I_b+b=2n_Y+4n_4+6n_6$, $2I_f+f=2n_Y$ and $L=I_b+I_f-n_Y-n_4-n_6+1$. Substitution eliminates the counts of the two marginal vertices. The bound concerns overall ultraviolet divergence, before numerator cancellations and subtraction of ultraviolet subdivergences.

For a connected scalar graph with nonderivative interaction $-\lambda {\varphi }^n$, $E$ external legs and $V$ vertices, $nV=2I+E$ and $L=I-V+1$. Since $[\varphi ]=(d-2)/2$ and $[\lambda ]=d-n(d-2)/2$, the superficial degree of divergence isThe graph contributes at order ${\lambda }^V$. This is simultaneous large-loop-momentum counting for an amputated graph; it does not test subdivergences or guarantee that the predicted pole coefficient is nonzero.

// Target: perturbative-quantum-field-theory.bigb

In massive four-dimensional phi-fourth theory, a six-point triangle is one-particle irreducible and has superficial degree $-2$. Insert a tadpole diagram self-energy into one of its internal lines. The resulting graph is still one-particle irreducible and still has degree $-2$, but its tadpole loop diverges. Recursive local counterterms remove the subgraph divergence before overall power counting is applied. Convergence requires attention to all ultraviolet subgraphs, not merely to simultaneous scaling of all loop momenta.

For interaction $\lambda {\Phi }^n(\partial \Phi {)}^r$ with a two-derivative kinetic term, each vertex has $n+r$ legs and $r$ momentum powers. Since $[\Phi ]=(d-2)/2$, its coupling has dimension $[\lambda ]=d-(n+r)(d-2)/2-r$. The graph identities $2I+E=(n+r)V$ and $L=I-V+1$, inserted into $D=dL-2I+rV$, prove the displayed formula. It is an upper bound on ultraviolet scaling; momentum contractions and symmetry can cause cancellations.

For a connected scalar field theory graph without derivative vertices, each independent loop momentum supplies $d$ powers and each internal scalar propagator removes two. Thus $D=dL-2I$. The graph identities $L=I-{\sum }_n{V}_n+1$ and $2I+E={\sum }_{n}n{V}_n$ give the displayed formula. The mass dimension of the field is $(d-2)/2$ and that of $g_n$ is $n-d(n/2-1)$. Negative-dimensional couplings allow the superficial degree to grow with vertex count. This is overall power counting in quantum field theory; subgraphs can diverge even when the whole graph has negative degree.

A marginal coupling is dimensionless under classical power counting.

The marginal interaction coefficient is distinct from the operator it multiplies.

A relevant coupling has positive mass dimension and grows under scaling toward long distances.

An irrelevant coupling has negative mass dimension and is suppressed toward long distances.

A power-counting nonrenormalizable interaction has a coupling of negative mass dimension, so higher loop orders generally require counterterms of unbounded operator dimension. It is naturally interpreted within an effective field theory below a cutoff.