nLab beta function


This entry is about the concept if quantum field theory. For the Euler beta function, related to the Gamma function, see there.


Algebraic Quantum Field Theory

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)



field theory:

Lagrangian field theory


quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization



States and observables

Operator algebra

Local QFT

Perturbative QFT



In statistical field theory and in perturbative quantum field theory, what is called the beta function is the logarithmic derivative of the running of the coupling constants under renormalization group flow. See there for more.


For Yang-Mills theory

In (Metsaev-Tseytlin 88) the 1-loop beta function for pure Yang-Mills theory was obtained as the point-particle limit of the partition function of a bosonic open string in a Yang-Mills background field. This provided a theoretical explanation for the observation, made earlier in (Nepomechie 83) that when computed via dimensional regularization then this beta function coefficient of Yang-Mills theory vanishes in spacetime dimension 26. This of course is the critical dimension of the bosonic string.

For more on this see at worldline formalism


The original informal discussion of beta functions for scaling transformations is due to

  • Murray Gell-Mann and F. E. Low, Quantum Electrodynamics at Small Distances, Phys. Rev. 95 (5) (1954), 1300–1312 (pdf)

there denoted “ψ\psi”. The notation “β\beta” is due to

Formulation in the rigorous context of causal perturbation theory/pAQFT, via the main theorem of perturbative renormalization, is due to

reviewed in

Discussion for Yang-Mills theory includes

Last revised on February 10, 2018 at 13:54:20. See the history of this page for a list of all contributions to it.