The Diederich-Forn{\ae}ss index I: for domains of non-trivial index

We study bounded pseudoconvex domains in complex Euclidean space. We define an index associated to the boundary and show this new index is equivalent to the Diederich-Forn{\ae}ss index defined in 1977. This connects the Diederich-Forn{\ae}ss index to boundary conditions and refines the Levi pseudoconvexity. We also prove the $\beta$-worm domain is of index $\pi/{(2\beta)}$. It is the first time that a precise non-trivial Diederich-Forn{\ae}ss index in Euclidean spaces is obtained. This finding also indicates that the Diederich-Forn{\ae}ss index is a continuum in $(0,1]$, not a discrete set. The ideas of proof involve a new complex geometric analytic technique on the boundary and detailed estimates on differential equations.