Role of AND operation in Side Channel Security

I was going through a side channel protection. It seems in the 3-shares (Appendix A) or 4-shares (Appendix B) has some computations that grows with the number of AND operations.

My question: Is there any standard metric by which I can measure the cost for side channel protection (under a given requirement, such as 3-shares) for an Exclusive-OR operation and an AND operation? Something like, 'For 3-shares, the cost for an Exclusive-OR operation is 3 Gate Equivalents and that of an AND operation is 5 Gate Equivalents. This grows to respectively 6 and 12 for 4-shares'.

For ISW masking scheme and its variants, for an AND on $$t$$ shares you would need $$O(t^2)$$ operations, while for XOR it's simply XOR of the shares componentwise, so $$t$$ XORs. Furthermore, for AND you need to refresh shares by using fresh randomness (there are works trying to reduce the necessary number of refreshes for a given circuit, and so this quantity depends on the circuit being protected).