  Using an approach based on Soft Collinear Effective Theory (SCET) and Heavy
Quark Effective Theory (HQET) we determine the $b$-quark fragmentation function
from electron-positron annihilation data at the $Z$-boson peak at
next-to-next-to leading order, with next-to-next-to leading log resummation of
DGLAP logarithms, and next-to-next-to-next-to leading log resummation of
endpoint logarithms. This analysis improves, by one order, the previous
extraction of the $b$-quark fragmentation function. We find that while the
addition of the next order in the calculation does not much shift the extracted
form of the fragmentation function, it does reduce theoretical errors
indicating that the expansion is converging. Using an approach based on
effective field theory allows us to systematically control theoretical errors.
While the fits of theory to data are generally good, the fits seem to be
hinting that higher order correction from HQET may be needed to explain the
$b$-quark fragmentation function at smaller values of momentum fraction.
