Consider the beta functions of eqs. (52.15) and (52.16).
a) Let ρ≡λ/g2, and compute dρ/dlnμ. Express your answer in terms of g and ρ. Explain why it is better to work with g and ρ rather than g and λ. Hint: the answer is mathematical, not physical.
b) Show that there are two fixed points, ρ+∗ and ρ−∗, where dρ/dlnμ=0, and find their values.
c) Suppose that, for some particular value of the renormalization scale μ, we have ρ=0 and g⋘1. What happens to ρ at much higher values of μ (but still low enough to keep g≪1)? At much lower values of μ?
d) Same question, but with an initial value of ρ=5.
e) Same question, but with an initial value of ρ=−5.
f) Find the trajectory in the (ρ,g) plane that is followed for each of the three starting points as μ is varied up and down. Hint: you should find that the trajectories take the form
g=g0ρ−ρ−∗ρ−ρ+∗ν
for some particular exponent ν. Put arrows on the trajectories that point in the direction of increasing μ.
g) Explain why ρ−∗ is called an ultraviolet stable fixed point, and why ρ+∗ is called an infrared stable fixed point.