Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 305 / 2 / ii

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- Solution ii

Solution

The strong-coupling scale

Λ

is defined as the scale at which the one-loop inverse coupling vanishes. Setting

g^{- 2} (Λ) = 0

gives

Λ = Λ_{U V} exp [- \frac{3 ( 4 π ) ^{2}}{2 b _{0} g _{0}^{2}}] .

Equivalently, in terms of the coupling measured at any scale

μ

,

Λ = μ exp [- \frac{3 ( 4 π ) ^{2}}{2 b _{0} g ^{2} ( μ )}] .

This dimensional transmutation defines

Λ_{QCD}

for Quantum chromodynamics.

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