Long-range φ⁴ theories have RG beta functions that satisfy a gradient flow with A matching the sphere free energy F̃ at leading nontrivial order.
Irreversibility of the Flux of the Renormalization Group in a 2D Field Theory
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Matching $A$ with $F$ in long-range QFTs
Long-range φ⁴ theories have RG beta functions that satisfy a gradient flow with A matching the sphere free energy F̃ at leading nontrivial order.