The epsilon expansion and critical exponents

Take the Euclidean $O(N)$ model with interaction $\lambda(\phi_i\phi_i)^2/4!$ and a tuned mass. Use dimensional regularization near $d=4-\epsilon$ and define a dimensionless renormalized coupling $g$ by factoring $8\pi^2\mu^\epsilon$ from the quartic coefficient. Our beta function differentiates with respect to increasing momentum scale $\mu$. The engineering contribution is then negative epsilon times g; changing to increasing length would reverse the flow convention.

At one loop the positive quadratic term balances that engineering contribution and gives a nonzero fixed point proportional to epsilon. This is the Wilson–Fisher point for small positive epsilon and positive physical component number. The numerical value assigned to g depends on its normalization and subtraction scheme; a critical exponent computed consistently to its stated order does not acquire that arbitrary normalization dependence. Finding a zero of one truncated beta function is only the first step, not a complete construction of a fixed-point theory.

A small departure of the thermal or mass-like scaling variable grows under coarse graining with eigenvalue $y_t$. The correlation length diverges with exponent $\nu=1/y_t$. Evaluating the mass anomalous dimension at the perturbative fixed point gives $y_t=2-(N+2)\epsilon/(N+8)+O(\epsilon^2)$. The two comes from the Gaussian mass dimension, and the correction comes from interacting critical fluctuations. A field anomalous dimension starts only at second order in epsilon in this model.

Invert the series and then truncate at the same order: $\nu=1/2+(N+2)\epsilon/[4(N+8)]+O(\epsilon^2)$. Numerically taking the reciprocal of the truncated expression for $y_t$ without re-expanding includes an uncontrolled subset of higher powers. That can be a declared approximation, but it is not the unique first-order prediction. This distinction becomes noticeable at epsilon near one, where different partial rearrangements disagree by terms that the calculation has not determined.

For sufficiently small epsilon the fixed point is weakly coupled, so omitted loop terms are parametrically suppressed. In three dimensions epsilon is one, and small-epsilon control alone no longer establishes numerical accuracy. Higher orders, carefully justified resummation, lattice calculations, experiments, and bootstrap bounds provide complementary evidence. Agreement among independent methods is more informative than treating a single low-order decimal as an exact critical exponent.

Universality also has conditions. The symmetry, dimensionality, interaction range, and relevant perturbations must place the system in the claimed universality class. Long-range forces, anisotropies, disorder, or an additional massless mode can change the fixed-point problem. The epsilon expansion is powerful because it tracks operators and eigenvalues systematically; it does not license relabeling every continuous transition as the same model. Some candidate fixed points disappear or require genuinely nonperturbative information away from the controlled neighborhood.