Renormalization group and effective theories

A Wilsonian action describes modes below a resolution scale. Integrating out a thin shell of higher momenta generates every local operator allowed by the symmetries, even if its coefficient was initially zero. Locality of the expansion requires external momenta and light masses small compared with the eliminated heavy scale. The coefficient of an operator is therefore a memory of unresolved physics, not permission to use the expansion at arbitrarily high energy.

If an operator has scaling dimension $\Delta$ in $d$ spacetime dimensions, its coefficient has mass dimension $d-\Delta$. Define a dimensionless coefficient at scale $\mu$ by multiplying by $\mu^{\Delta-d}$. Near a Gaussian fixed point, operators with $\Delta<d$ are relevant toward the infrared, those with $\Delta>d$ are irrelevant, and equality demands a quantum calculation. At an interacting fixed point, anomalous dimensions replace naive engineering dimensions; the labels are statements about a specified fixed point.

For a real scalar with interaction $\lambda\phi^4/4!$, the four-dimensional one-loop beta function is $\beta(\lambda)=3\lambda^2/(16\pi^2)+O(\lambda^3)$ in a mass-independent scheme. Separate variables in $d\lambda/d\log\mu=b\lambda^2$. The inverse coupling is linear in the logarithm of scale. This is more useful than treating a large logarithm as one enormous first-order correction: running resums a particular hierarchy of logarithms.

For positive coupling, increasing the renormalization scale increases this one-loop coupling. The formal pole in the resulting expression marks failure of that approximation; it is not evidence that a measured cross section literally becomes infinite there. Thresholds, further operators, higher loops, and possible ultraviolet physics must be considered before drawing conclusions outside weak coupling. The subtraction scale is also distinct from the physical energy, although choosing comparable scales often avoids unnecessarily large logarithms.

Matching fixes effective coefficients so that low-energy amplitudes agree with a more resolved theory or with experiment. Running then evolves those coefficients between scales without counting the heavy degrees of freedom twice. A dimension-eight interaction such as $(\partial\phi)^4/M^4$ produces a typical dimensionless four-point contribution proportional to $(E/M)^4$ times its coefficient. This estimate presumes the coefficient is not unexpectedly enhanced and that the relevant kinematic invariants are all below the heavy scale.

Keep three questions separate when reporting precision: how accurately were coefficients matched, how many quantum loops were retained, and how many powers of $E/M$ were retained? A result can have tiny statistical uncertainty while having a large truncation error. Symmetries may forbid the apparently leading correction, while resonances or small denominators can invalidate a simple estimate. An effective theory succeeds when its stated hierarchy is tested, rather than when every omitted term is called negligible by habit.