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.

Idea

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.

Examples

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

References

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 “$\psi$”. The notation “$\beta$” is due to

• Curtis Callan, Broken Scale Invariance in Scalar Field Theory, Phys. Rev. D 2, 1541, 1970 (doi:10.1103/PhysRevD.2.1541)

• Kurt Symanzik, Small distance behaviour in field theory and power counting, Communications in Mathematical Physics. 18 (3): 227–246 (doi:10.1007/BF01649434)

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

• Romeo Brunetti, Michael Dütsch, Klaus Fredenhagen, Perturbative Algebraic Quantum Field Theory and the Renormalization Groups, Adv. Theor. Math. Physics 13 (2009), 1541-1599 (arXiv:0901.2038)

reviewed in

• Michael Dütsch, section 3.5.3 of From classical field theory to perturbative quantum field theory, 2018

Discussion for Yang-Mills theory includes

• R.I. Nepomechie, Remarks on quantized Yang-Mills theory in 26 dimensions, Physics Letters B Volume 128, Issues 3–4, 25 August 1983, Pages 177-178 Phys. Lett. B128 (1983) 177 (doi:10.1016/0370-2693(83)90385-4)

• Ruslan Metsaev, Arkady Tseytlin, On loop corrections to string theory effective actions, Nuclear Physics B Volume 298, Issue 1, 29 February 1988, Pages 109-132 (doi:10.1016/0550-3213(88)90306-9)