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A continuum limit of a quantum field theory sends the ultraviolet cutoff ${\Lambda }_0$ to infinity while holding chosen physical masses and amplitudes fixed. In cutoff units this requires a diverging correlation length, so the bare couplings must be tuned onto the critical surface of an ultraviolet renormalization-group fixed point. Each relevant direction of a fixed point requires one coordinate fixed by measurement; predictivity requires only finitely many such directions, and irrelevant microscopic details disappear.

Several behaviors are possible. In an asymptotically free theory the trajectory approaches a Gaussian fixed point in the ultraviolet and interactions vanish logarithmically. In an asymptotically safe quantum field theory it approaches a non-Gaussian fixed point with finitely many relevant directions. A theory with a Landau pole may possess only a trivial quantum field theory as its continuum limit; retaining a nonzero interaction then requires a finite cutoff. Finally, if no ultraviolet-complete trajectory exists, the model remains an effective field theory valid only below a physical cutoff.