Physics theoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passverified 2026-11-05 (gpt-6.1-sol)

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Linear beta coefficient from the scaling dimension

Physical scope

Linearized Wilsonian cutoff scaling near a declared two-dimensional conformal fixed point for a finite family of scalar scaling eigenoperators; no nonlinear beta function or global flow is asserted.

Statement

Let $N < \infty$ , let $g = (g^{1}, \dots, g^{N})$ be dimensionless Wilsonian couplings near a declared two-dimensional conformal fixed point, and let $O_{i}$ be scalar scaling eigenoperators of total dimensions $Δ_{i}$ . Use cutoff length $a$ and IR time $t = \log (a / a_{0})$ . For fixed dimensionful coefficients $λ_{p h y s}^{i}$ , set $g^{i} (a) = λ_{p h y s}^{i} a^{2 - Δ_{i}}$ . The engineering contribution to the beta function is $β_{e n g}^{i} = (2 - Δ_{i}) g^{i}$ . If the renormalized beta map is differentiable and its remaining terms vanish to second order at the fixed point, then $β^{i} (g) = (2 - Δ_{i}) g^{i} + O (∥ g ∥^{2})$ . This is a linearized scaling statement; it does not assert that marginal operators are exactly marginal.

Facts & Assumptions

Given: A finite family of dimensionless couplings and scalar scaling eigenoperators at the fixed point, a cutoff length $a$ , and the IR time convention $t = \log (a / a_{0})$ .

[F1]

Regulated conformal perturbation and beta-function conventions: $g^{i} (a) = λ_{p h y s}^{i} a^{2 - Δ_{i}}$ , where $λ_{p h y s}^{i}$ has units $[l e n g t h]^{Δ_{i} - 2}$ and $a$ has units of length.

[A1]

The fixed-point scaling law is $O_{i} (e^{t} x) = e^{- Δ_{i} t} O_{i} (x)$ , and the linearized coupling chart is the declared scaling-eigenoperator basis.

[A2]

Write the beta map as $β (g) = ((2 - Δ_{i}) g^{i})_{i = 1}^{N} + R (g)$ . The remainder satisfies $R (0) = 0$ and, for some $K, δ > 0$ , $∥ R (g) ∥ \leq K ∥ g ∥^{2}$ whenever $∥ g ∥ < δ$ , in the finite-dimensional Euclidean norm. This bound implies $D R (0) = 0$ ; it is the stated second-order hypothesis.

Proof

1.1givenF1A1algebra

Under $a (t) = a_{0} e^{t}$ , the definition in [F1] and the scaling law in [A1] give $g^{i} (t) = λ_{p h y s}^{i} (a_{0} e^{t})^{2 - Δ_{i}} = g^{i} (0) e^{(2 - Δ_{i}) t}$ .

2.1givenstep 1.1algebra

Differentiating this scalar exponential at $t = 0$ gives ${\frac{d g^{i}}{d t} ∣}_{0} = (2 - Δ_{i}) g^{i} (0)$ , which is exactly the engineering contribution to the beta vector in the chosen IR convention.

3.1givenA2step 2.1∎

By [A2], the derivative of the remainder at $g = 0$ is zero, so it does not change the linear part computed in 2.1. The full beta therefore has the stated linearization. For $Δ_{i} = 2$ the linear coefficient is zero and imposes no conclusion about higher-order terms.

Depends on

Regulated conformal perturbation and beta-function conventions

Linear beta coefficient from the scaling dimension

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources