# Dimensional Transmutation

Summary:
I'd like to make sure I understand the term dimensional transmutation.
Hi,

I'd like to make sure my understanding of dimensional transmutation is correct.

The key insight of QFT is that introducing quantum fluctuation can break the scale invariance of a classical theory that is naively scale invariant. Equivalently, one says that the beta function can flow beyond the naive guess of 0 from dimensional analysis for dimensionless parameters.

From there one "extrapolates" the coupling by integrating the Callan-Symanzik equation and finds out that it blows up either in the IR (marginally relevant) of the UV (marginally irrelevant) at finite energy scale. The scale at which the coupling blows up is a new dimension-full scale. This value is determined by the lab energy scale and the dimensionless coupling strength we measured at that scale (the boundary condition of the callan symanzik extrapolation scheme).

People then use the term "dimensional transmutation" by saying that there is a mapping between the 2: the divergent scale where the coupling blows up and the lab measurement scale, so we can specify either one to fully fix the theory. This is just a statement that the callan-symanzik equation in those specific cases does not converge multiple boundary conditions (different lab measurements defining different theories) to the same blow up scale. If that happened, then the map would not be bijective preventing so called "dimensional-transmutation".

It is clear though that the blow up scale could be physically meaningless (one is extrapolating the running coupling outside of the regime where perturbation theory is valid).

Is that correct understanding of the term? I want to make sure i'm not oversimplifying.

Last edited:
ohwilleke and vanhees71

## Answers and Replies

Having read a bit more the literature, it seems the term dimensional transmutation is correctly defined above but is used in more general terms: it is the occurence of a energy scale in a theory that is classically scale invariant (does not have a preferred energy scale). The energy scale introduced is due to the non-vanishing of the beta function.

One could say that it is the breaking of classical scale invariance.

ohwilleke and vanhees71