Perturbative expansion of Beta function - Renormalization

[nigelscott]

I am trying to understand the basics of Renormalization. I have read that β encodes the running coupling and can be expanded as a power series as:

β(g) = ∂g/(∂ln(μ)) = β₀g³ + β₁g⁵ + ...

However, I don't understand how this is derived.. I assume that the terms correspond to 1 loop, 2 loops etc. Can somebody help me out.

 

[AndyW]

Hi there,

Firstly, let's define some terms. Renormalization is a technique used in theoretical physics to remove infinities that arise in certain calculations in quantum field theory. These infinities arise due to the fact that particles and fields can interact at arbitrarily short distances, leading to infinite results in certain calculations.

Now, let's look at the equation for β(g) that you have mentioned. This is known as the beta function, and it encodes the running coupling constant g, which is a measure of the strength of the interaction between particles. The beta function tells us how the coupling constant changes as we change the energy scale μ at which we are probing the system.

The beta function can be expanded as a power series, as you have mentioned. This expansion is known as the perturbative expansion, and it is an approximation that is valid when the coupling constant is small. The terms in the expansion correspond to the contributions from different loops in the Feynman diagrams that are used to calculate the beta function.

The first term, β0g3, corresponds to the contribution from 1 loop in the Feynman diagram. This means that it takes into account the interactions between 3 particles. The second term, β1g5, corresponds to the contribution from 2 loops in the Feynman diagram, which takes into account the interactions between 5 particles.

In general, the perturbative expansion of the beta function includes an infinite number of terms, each corresponding to the contribution from a higher number of loops. However, in practice, we only consider a finite number of terms in the expansion, as including an infinite number of terms is not feasible.

I hope this helps to clarify the origin of the terms in the perturbative expansion of the beta function. If you have any further questions, please let me know.