The Paradox
du/dℓ = −y_u · u (y_u > 0)
u flows to 0 under RG — it is irrelevant. So naively free energy scales as f ~ |t|^(2-α) with MF exponents.
f = |t|2−α Φ(h/|t|βδ, u/|t|y_u/y_t)
But: u appears in denominator of free energy! As t→0, u/|t|^(y_u/y_t) → ∞ even though u→0. The scaling function Φ has a singularity at large argument — this changes the effective exponent.
ξ ~ |t|−ν but M ~ |t|β*
β* = β − (y_u/y_t) × (correction)
Classic example: φ⁴ theory above d=4, where u is irrelevant (d>4) but appears in the denominator of the free energy minimum.