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.

Answer

Running the RGEs in reverse should be valid so long as you don't integrate over a scale where degrees of freedom enter/leave the theory. If you integrated out the electrons in QED, you'd have irrevocably lost that information in your low energy description of interacting photons. You'd see some non-renormalizable theory with interacting corrections to pure EM but RG evolving to the UV wouldn't tell you what that would be. Just like RG evolving QED to the UV keeps you unaware of the strong or the weak sector physics.

On the other hand, so long as you've not crossed any characteristic scale in your theory, the theory at the scales you've integrated out should be the same as the theory at the scale you're currently at. So you should be able to go back to where you came from.

To summarize, so long as you don't integrate out some characteristic scale, you can keep going back and forth.