g (t) = \frac{g}{1 + 2 b g ^{2} t} .

lo g f (t) = 2 c \int_{0}^{t} \frac{g ^{2}}{1 + 2 b g ^{2} u} d u = \frac{c}{b} lo g (1 + 2 b g^{2} t), f (t) = (1 + 2 b g^{2} t)^{c / b} \sim (2 b g^{2})^{c / b} t^{c / b} .

For a reference two-point factor regular and nonzero at the free coupling,

C (r, g (t)) \to C (r, 0)

, so the large-momentum correction is a power

c / b

lo g (p^{2} / μ^{2})

. These are asymptotic freedom and logarithmic two-point scaling from a cubic beta function. If the supplied beta and gamma expressions are leading small-coupling terms rather than exact functions, they fix the leading logarithmic exponent, while subleading corrections and the prefactor depend on higher orders.