Renormalization Group - Appendix: Exact Renormalization Group Equations

Appendix: Exact Renormalization Group Equations

An exact renormalization group equation (ERGE) is one that takes irrelevant couplings into account. There are several formulations.

The Wilson ERGE is the simplest conceptually, but is practically impossible to implement. Fourier transform into momentum space after Wick rotating into Euclidean space. Insist upon a hard momentum cutoff, so that the only degrees of freedom are those with momenta less than Λ. The partition function is

For any positive Λ′ less than Λ, define S_Λ′ (a functional over field configurations φ whose Fourier transform has momentum support within ) as

Obviously,

In fact, this transformation is transitive. If you compute S_Λ′ from S_Λ and then compute S_Λ″ from S_Λ′, this gives you the same Wilsonian action as computing S_Λ″ directly from S_Λ.

The Polchinski ERGE involves a smooth UV regulator cutoff. Basically, the idea is an improvement over the Wilson ERGE. Instead of a sharp momentum cutoff, it uses a smooth cutoff. Essentially, we suppress contributions from momenta greater than Λ heavily. The smoothness of the cutoff, however, allows us to derive a functional differential equation in the cutoff scale Λ. As in Wilson's approach, we have a different action functional for each cutoff energy scale Λ. Each of these actions are supposed to describe exactly the same model which means that their partition functionals have to match exactly.

In other words, (for a real scalar field; generalizations to other fields are obvious)

and Z_Λ is really independent of Λ! We have used the condensed deWitt notation here. We have also split the bare action S_Λ into a quadratic kinetic part and an interacting part S_{int Λ}. This split most certainly isn't clean. The "interacting" part can very well also contain quadratic kinetic terms. In fact, if there is any wave function renormalization, it most certainly will. This can be somewhat reduced by introducing field rescalings. R_Λ is a function of the momentum p and the second term in the exponent is

when expanded. When, R_Λ(p)/p^2 is essentially 1. When, R_Λ(p)/p^2 becomes very very huge and approaches infinity. R_Λ(p)/p^2 is always greater than or equal to 1 and is smooth. Basically, what this does is to leave the fluctuations with momenta less than the cutoff Λ unaffected but heavily suppresses contributions from fluctuations with momenta greater than the cutoff. This is obviously a huge improvement over Wilson.

The condition that

can be satisfied by (but not only by)

Jacques Distler claimed without proof that this ERGE isn't correct nonperturbatively.

The Effective average action ERGE involves a smooth IR regulator cutoff. The idea is to take all fluctuations right up to an IR scale k into account. The effective average action will be accurate for fluctuations with momenta larger than k. As the parameter k is lowered, the effective average action approaches the effective action which includes all quantum and classical fluctuations. In contrast, for large k the effective average action is close to the "bare action". So, the effective average action interpolates between the "bare action" and the effective action.

For a real scalar field, we add an IR cutoff

to the action S where R_k is a function of both k and p such that for, R_k(p) is very tiny and approaches 0 and for, . R_k is both smooth and nonnegative. Its large value for small momenta leads to a suppression of their contribution to the partition function which is effectively the same thing as neglecting large scale fluctuations. We will use the condensed deWitt notation

for this IR regulator.

So,

where J is the source field. The Legendre transform of W_k ordinarily gives the effective action. However, the action that we started off with is really S+1/2 φ⋅R_k⋅φ and so, to get the effective average action, we subtract off 1/2 φ⋅R_k⋅φ. In other words,

can be inverted to give J_k and we define the effective average action Γ_k as

Hence,

thus

is the ERGE which is also known as the Wetterich equation.

As there are infinitely many choices of R_k, there are also infinitely many different interpolating ERGEs. Generalization to other fields like spinorial fields is straightforward.

Although the Polchinski ERGE and the effective average action ERGE look similar, they are based upon very different philosophies. In the effective average action ERGE, the bare action is left unchanged (and the UV cutoff scale—if there is one—is also left unchanged) but we suppress the IR contributions to the effective action whereas in the Polchinski ERGE, we fix the QFT once and for all but vary the "bare action" at different energy scales to reproduce the prespecified model. Polchinski's version is certainly much closer to Wilson's idea in spirit. Note that one uses "bare actions" whereas the other uses effective (average) actions.