Long term stability in hadron colliders in presence of synchrotron oscillations and tune ripple

The problem of long-term losses in hadron colliders such as the Large Hadron Collider (LHC) is considered. A previous formula that provides the reduction of dynamic aperture with the number of turns is generalized to include also the relevant cases of off-momentum and tune ripple. The dynamic aperture turns out to shrink with a power of the inverse logarithm of the number of turns. Long-term tracking data of the LHC are analysed in this framework. The formula proves to hold in all cases, and the possibility of using its extrapolation to predict long-term losses are explored.