Zamolodchikov C-Theorem

2D RG flow: c-function decreasing UV→IR; fixed points at Ising (c=½), free boson (c=1), tricritical (c=7/10)

Initial g₁ (mass deformation): 0.50 Initial g₂ (interaction): 0.20

UV cutoff scale Λ: 10.0 Flow steps shown: 80

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c (UV)

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c (IR)

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Δc = c_UV − c_IR

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IR fixed point

Zamolodchikov (1986): In 2D QFT, ∃ function C(g) on coupling space such that: (1) C is stationary ⟺ at a CFT fixed point, (2) dC/d(ln μ) ≤ 0 along RG flow (monotone decreasing UV→IR).
At fixed points: C = central charge c. Known values: Ising c=½, free boson c=1, free fermion c=½, tricritical Ising c=7/10, 3-state Potts c=4/5, WZW SU(2)_k: c=3k/(k+2).
Proof: C defined via 2-point functions of stress tensor: C ~ (r⁴/3)⟨T(r)T(0)⟩. Positivity of Källén-Lehmann spectral weight → dC/d(ln r) ≤ 0.