Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 303 / 2 / a / ii

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- Solution ii

Solution

g_{i} (ℓ)

under coarse-graining scale

b = e^{ℓ}

, their beta function is

β_{i} (g) = d g_{i} / d ℓ

. At a renormalization-group fixed point

g_{*}

all beta functions vanish. Linearization gives

\frac{d}{d ℓ} δ g_{i} = M_{ij} δ g_{j}, M_{ij} = \frac{\partial β _{i}}{\partial g _{j}}_{g_{*}} .

The eigenvalues

y_{a}

of the stability matrix of a renormalization-group fixed point are the scaling dimensions of the associated coupling directions:

δ g_{a} (ℓ) = e^{y_{a} ℓ} δ g_{a} (0)

.

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