Splitting formulas for the logarithmic double ramification cycle

The logarithmic double ramification cycle is roughly a logarithmic Gromov–Witten invariant of ${\mathbb{P}}^1$. For classical Gromov–Witten invariants, formulas for the pullback along the gluing maps have been invaluable to the theory. For logarithmic Gromov–Witten invariants, such formulas have not yet been found. One issue is the fact that log stable maps cannot be glued. In this paper, we use the framework from [HS23] for gluing pierced log curves (a refinement of classical log curves) to give formulas for the pullback of the (log) (twisted) double ramification cycle along the loop gluing map.