An ERGE primer

An “Exact Renormalization Group Equation” or “Functional Renormalization Group Equation” is an equation describing the dependence of some  effective action as a cutoff is varied. There are various forms of ERGE, depending on what functional one is considering and on the way the cutoff is imposed. Polchinski's equation refers to the Wilson action, Wetterich's equation to the generating functional of 1PI Green functions, also known as the "effective average action". This picture shows Wetterich and Polchinski in Corfu, september 2010.

The following files contain a very quick introduction to the Wetterich equation and its use to find fixed points. The context is scalar theory and the fixed point in question is the Wilson-Fisher fixed point. The applications of the ERGE to gravity are technically more complicated in several ways, but the basic principles are not very different.

• ·         Derivation of the ERGE (in .pdf format or in postscript format)
• ·         Application of the ERGE to scalar theory: the Wilson-Fisher fixed point in d=3 (in .pdf format or in postscript format)
• ·         A Mathematica notebook calculating the above (thanks to A. Codello)

Original papers

Kenneth G. Wilson and Michael G Fischer (1972)

Critical exponents in 3.99 dimensions

Phys. Rev. Lett. 28, 240

Joseph Polchinski (1984)

Renormalization and Effective Lagrangians.

Nucl. Phys. B231, 269-295

Christoph Wetterich (1993)

Exact evolution equation for the effective potential.

Phys. Lett. B 301, 90.

Derivative expansion of the exact renormalization group.

Phys.Lett.B329:241-248

e-Print:hep-ph/9403340

Tim R. Morris (1994b)

On truncations of the exact renormalization group.

Phys. Lett. B334:355-362

e-Print:hep-th/9405190

Tim R. Morris, Michael D. Turner (1998)

Derivative expansion of the renormalization group in O(N) scalar field theory.

Nucl. Phys. B509 637-661

e-Print:hep-th/9704202

Daniel F. Litim (2001)

Optimised renormalisation group flows.

Phys. Rev. D 64, 105007.

e-Print:hep-th/0103195

(This paper introduces a very useful type of cutoff that gives the beta functions in closed form, aside from other good properties.)

Reviews

Tim R. Morris (1998)

Elements of the continuous renormalization group

Prog. Theor. Phys. Suppl.131 395-414

e-Print: hep-th/9802039

C. Bagnuls and C. Bervillier (2001)

Exact renormalization group equations: An introductory review.

Phys. Rept. 348, 91.

e-Print:hep-th/0002034

J. Berges, N. Tetradis, and C. Wetterich (2002)

Nonperturbative renormalization flow in quantum field theory and statistical physics.

Phys. Rept. 363, 223.

e-Print:hep-ph/0005122

Jan M. Pawlowski (2005)

Aspects of the functional renormalization group.

e-Print:hep-th/0512261

Fundamentals of the functional renormalization group.