For a real scalar field ϕ after performing all the 1-loop renormalization for dimensional regulator d=4−ϵ, ϵ→0+, I have found that the renormalized coupling λ can be related to the bare one by
λ(1+3(4π)2ϵλp)=λ0
I'm stuck trying to get the beta function from that equation. We call beta function to
β(λ)=μ∂λ∂μ
Taking into account that
λ=λpμϵ,[λp]=0,[μ]=1
My problem is that any way I use to get β(λ) from Eq. (1) gives a dependence on ϵ, but in Peskin it is used a different way to solve this and the solution is
β(λ)=3λ2(4π)2
How can I get the beta function via Eq. (1)?
Answer
The idea is that the bare quantities explicitly do not depend on μ, thus one has the equation 0=μdλ0dμ=(∂∂μ+β(λp)∂∂λp)(μϵλpZλ)=ϵμϵλpZλ+μϵβ(λp)d(λpZλ)dλp.
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