I only skimmed it. I don't understand it now, but I guess that in a week I could get a good enough understanding if I skip the most difficult parts. So my possible wrong ELI25:

The vacuum can transform spontaneously in another type of vacuum (like liquid water transform into ice at low temperature). Lucky this is very improbable because otherwise we would be all dead, and the transition is due to quantum tunneling.

To estimate the probability of the transition it is necessary to calculate the contribution of the virtual particles in the loops in the Feynman diagrams. A naïve calculation has too many infinite values to be useful.

In this article they calculate the contribution of multiple virtual particles, in particular when the Feynman diagrams has many loops. They have a nice pattern and somehow they cancel (most?) of the infinite values.

With this new calculation they can estimate how long we should wait until this tunneling effect between the two vacuum destroy the universe and luckily it's a long time. (Perhaps other thing can kill all of us before :).)

For this calculation it is particular important the mass of the heavy particles.

(Linkbait title: "Scientists say that God's particle may open portal to universe doom at any moment". Sorry. I couldn't resist.)

As I said earlier, this short version may be wrong. Any corrections or missing important points are welcome.