Large-N expansion (large-$N$ expansion)

The large-N expansion organizes an $O(N)$-symmetric theory in powers of $1/N$. At leading order, an auxiliary constraint field is determined by a self-consistent saddle-point or gap equation.

A gap equation determines a physical mass or excitation gap self-consistently from an interaction and a regulated fluctuation integral.

A critical coupling is a coupling value at which a gap equation first admits a massless solution. Tuning through it can separate symmetric gapped and symmetry-broken phases.

At an upper critical dimension, a nominal power-law critical scaling acquires a multiplicative logarithmic correction because the coupling is marginal.