Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 305 / 4 / a

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- Solution a

Solution

Since

α_{i} = g_{i}^{2} / (4 π)

, the stated renormalization-group beta function implies

\frac{d α _{i}}{d l o g μ} = - \frac{β _{i}}{2 π} α_{i}^{2}, \frac{d α _{i}^{- 1}}{d l o g μ} = \frac{β _{i}}{2 π} .

For

β_{i} > 0

, the running coupling decreases at high energy: the theory is asymptotically free and becomes strong in the infrared, as in Quantum chromodynamics. For

β_{i} < 0

, it is infrared free but grows toward an ultraviolet Landau pole.

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