= Renormalization-group flow
A renormalization-group flow tracks how a theory's effective couplings change as the observation scale changes. A coarse-graining step averages short-distance degrees of freedom and rescales the remaining variables, producing a new description for longer-distance measurements.
For dimensionless couplings $g_i$ and a logarithmic length scale $\ell$, write $dg_i/d\ell=\beta_i(g)$. A <renormalization-group fixed point> has all beta functions zero. Near it, some perturbations grow and others shrink, giving <relevant operators> and <irrelevant operators>. This explains how distinct microscopic models can approach the same long-distance behavior. The sign convention matters: a beta function defined using increasing energy has the opposite scale direction from one defined using increasing length.
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