Renormalization and the renormalization group Rachel Wooten December 5, 2008 Quantum field theory

What is renormalization? Describe some system by a function F(x) and bare coupling constant g0. Could be Hamiltonian, correlation function, cross- section, differential equation, etc… Divergences occur in perturbation expansion terms. How do we deal with them? Renormalization: a procedure to reparametrize a theory in terms of finite, known values instead of infinite or undefined ones.

Renormalization Expand F(x) perturbatively in g0 and functions of x *This expansion is poorly defined. There are infinities. *There is only 1 coupling constant, so 1 experimental measurement can fix F(x) it at some point x = . Call this value *This is the renormalization prescription. *Renormalization hypothesis: reparametrization of F in terms of a physical quantity is enough to turn perturbation expansion into a well-defined expansion.

Renormalization But, cannot just plug gR into expansion because expansion is ill-defined. First, we must regularize: make all Fi(x) into new finite functions, Fi(x) that turn into Fi (x) as -> ∞ E.g. ---> Here’s a schematic of what we do: So define

Renormalization Renormalization performed recursively, at each order. Order g0: -----> Order g02: only freedom to eliminate divergence is to redefine g0

Renormalization As we take the limit  --> ∞, F(x) is finite for all x at this order if and only if the divergent part of F1(x) is exactly cancelled by the divergent part of F1( ) For this to occur, F1(x) must be x-independent! Renormalization procedure = adding a divergent term 2 to F to remove divergence. Divergences move to higher orders and are part of g0, not part of the Fi(x) The g0 are infinite in  --> ∞ limit.

The form of the divergences The case of g0 dimensionless, x not dimensionless. Dimensional analysis and renormalizability determine form of divergences. Separate F1(x) into a singular and a regular part, Since g0 is dimensionless, so are F(x) and all the Fi(x). Redefining regular part of F1, we can get singular part x-independent Also, F1 dimensionless, so must be function of x/ ( is the only other quantity with dimension)

The renormalization group Selection of x =  not unique. Could parametrize in terms of , ’, or ’’ Equivalence class of parametrizations of same theory. Choice of parametrization should not matter.

The renormalization group Eliminating g0 between two equations, Group law verified perturbatively, However,  very large. To get gR’(gR, /’), had to invert to get Not legal mathematically because of increasing powers of  We need to construct f so that

The renormalization group If f defined such that 1. Its expansion at order n given by nth order perturbation theory 2. The group law is exactly verified, Then f is a “self-similar approximation” at order n of the exact relationship between gR and gR’ Cannot transform between two very different scales perturbatively, but can use perturbative approach to move between two similar scales. Describe the curve between scales with calculus,  Gives infinitessimal evolution of coupling constants with scale

So what? Dimensionless coupling constant theories behave similarly (natural log dependence) for VERY different systems. (e.g. critical phenomena, QED  = scale beyond which new physics required to describe system For critical phenomena,  = atomic spacing scale where field theory description no longer sufficient. Renormalization not magic, and NOT fundamentally a quantum phenomenon!

References “A hint of renormalization”, B. Delamotte, Am. J. Phys. 72(2004)170. Peskin & Schroeder, Chapter 12