How to make looped animation of particles in Particle_SOP ?
I am not sure this is possible as is. Looping particles is usually finding a trick to bypass the randomness of particles generation and position.
Maybe have a look here for a first approach : Looping Noise Part 1: Ending at the Beginning — Simon Alexander-Adams
It’s possible. Just turn off all the features of the particle SOP that introduce randomness (jitter, variances turbulence, etc) and then introduce your randomness with point order and normals since point order determines the order in which the points will be used to emit particles, and the normals determine initial velocity.
Then you can loop the noise that controls those attributes, and keep it in sync with the timing by coordinating number of points, life expectancy and birth rate so that it will end up in the same state at some number of cycles and that will be your loop.
It will take a lot of work on your part to get all of those dialed in, and probably a lot of math to make sure everything lines up. Not easy to quickly demonstrate or answer in a single forum post unfortunately.