  We calculate the deuteron charge, electric quadrupole, and magnetic form
factors up to next-to-next-to-leading order in chiral effective field theory,
treating subleading corrections, especially that of chiral nuclear forces, in
perturbation theory. We examine the power counting based on naive dimensional
analysis by investigating the ultraviolet cutoff variation of these form
factors. We find that the N${}^2$LO magnetic form factor shows significant
cutoff dependence, suggesting the contact current operator responsible be
enhanced. After promotion to N${}^2$LO, it indeed renormalizes the magnetic
form factor. This is in agreement with a previous work based on
renormalization-group analysis. For the charge and quadrupole moments,
perturbative calculation allows us to study how they scale with multiple
low-energy parameters such as the pion mass, deuteron binding momentum, and
momentum transfer.
