How Does Dimensional Transmutation Impact Perturbation Theory in QCD?

[ShayanJ]

Here there is some explanation about dimensional transmutation that answered some of the questions I had about it.Those explanations are all I understand about the subject. But I still have some questions.

1) Is it correct to say that dimensional transmutation means you can replace the coupling constant in the Lagrangian with the energy scale using the RG flow equation? i.e. Does dimensional transmutation mean doing ## L[g]\to L[\Lambda] ##?

2) How is it that we still can do perturbation theory using the coupling constant in the weakly coupled (high energy) regime of QCD? How is it that dimensional transmutation doesn't affect that regime?

Thanks

 

[AndyW]

for your explanation.

Thank you for your interest in dimensional transmutation and for sharing your questions with us. I am happy to provide some additional information and clarification on this topic.

1) Yes, it is correct to say that dimensional transmutation allows for the replacement of the coupling constant in the Lagrangian with the energy scale using the renormalization group (RG) flow equation. This is because dimensional transmutation is a phenomenon that occurs when a theory has a dimensionless coupling constant, but the theory itself is not scale-invariant. In this case, the coupling constant becomes a function of the energy scale, and the RG flow equation describes how it changes with energy. So, in a sense, we can say that dimensional transmutation transforms the Lagrangian from being expressed in terms of a coupling constant to being expressed in terms of an energy scale.

2) This is a great question. In perturbation theory, we use the coupling constant to calculate the strength of interactions between particles. In the weakly coupled regime of QCD, where the energy scale is high, the coupling constant becomes small, and we can use perturbation theory to make accurate predictions. However, even though the coupling constant becomes small, it still has a non-zero value, and therefore, the effects of dimensional transmutation are still present. However, at high energies, these effects become negligible compared to the perturbative contributions, and we can still use perturbation theory to make accurate predictions.

I hope this helps to clarify your understanding of dimensional transmutation. If you have any further questions, please feel free to ask. Thank you for your curiosity and interest in this fascinating subject.