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.

two equations
            talking


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.


Some relevant references


Original papers


Kenneth G. Wilson and Michael G Fischer

Critical exponents in 3.99 dimensions

Phys. Rev. Lett. 28, 240 (1972)


Joseph Polchinski

Renormalization and Effective Lagrangians.

Nucl. Phys. B231, 269-295 (1984)


Christoph Wetterich

Exact evolution equation for the effective potential.

Phys. Lett. B 301, 90 (1993)


Tim R. Morris

Derivative expansion of the exact renormalization group.

Phys.Lett.B329:241-248 (1994)

e-Print:hep-ph/9403340


Tim R. Morris

On truncations of the exact renormalization group.

Phys. Lett. B334:355-362 (1994)

e-Print:hep-th/9405190


Tim R. Morris, Michael D. Turner

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

Nucl. Phys. B509, 637-661 (1998)

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

Nonperturbative renormalization flow in quantum field theory and statistical physics.

Phys. Rept. 363, 223 92002)

e-Print:hep-ph/0005122


Jan M. Pawlowski

Aspects of the functional renormalization group
Annals Phys. 322, 2831-2915
(2007)

e-Print:hep-th/0512261


Bertrand Delamotte
An introduction to the nonperturbative renormalization group
Lect.Notes Phys. 852, 49-132
(2012)
e-Print: cond-mat/0702365

Holger Gies
Introduction to the functional RG and applications to gauge theories
Lecture Notes in Phys. 852, 287-348 (2012)
e-Print: hep-ph/0611146

Oliver Rosten (2010)

Fundamentals of the functional renormalization group.

e-Print:1003.1366 [hep-th]


Last update 16/06/2014