# Under what conditions are the renormalization group equations “reversible”?

As I understand it, the renormalization group is only a semi-group because the coarse graining part of a renormalization step consisting of

1. Summing / integrating over the small scales (coarse graining)

2. Calculating the new effective Hamiltonian or Lagrangian

3. Rescaling of coupling constants, fields, etc.

is generally irreversible.

So when doing a renormalization flow analysis one usually starts from an initial action valid at an initial renormalization time $t_0$ (or scale $l_0$)

$$t = \ln(\frac{l}{l_0}) = -\ln(\frac{\Lambda}{\Lambda_0})$$

and integrates the renormalization group equations

$$\dot{S} = -\Lambda\frac{\partial S}{\partial \Lambda} \doteq \frac{\partial S}{\partial t}$$

forward in renormalization time towards the IR regime.

Under what conditions (if any) are the renormalization group transformations invertible such that the renormalization group equations are reversible in renormalization time and can be integrated "backwards" towards negative renormalization times and smaller scales (the UV regime)?

As an example where it obviously can be done, the calculation of coupling constant unification comes to my mind.

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