  We present a method to delens the acoustic peaks of the CMB temperature and
polarization power spectra internally, using lensing maps reconstructed from
the CMB itself. We find that when delensing CMB acoustic peaks with a lensing
potential map derived from the same CMB sky, a large bias arises in the
delensed power spectrum. The cause of this bias is that the noise in the
reconstructed potential map is derived from, and hence correlated with, the CMB
map when delensing. This bias is more significant relative to the signal than
an analogous bias found when delensing CMB B modes. We calculate the leading
term of this bias, which is present even in the absence of lensing. We also
demonstrate one method to remove this bias, using reconstructions from CMB
angular scales within given ranges to delens CMB scales outside of those
ranges. Some details relevant for a realistic analysis are also discussed, such
as the importance of removing mask-induced effects for successful delensing,
and a useful null test, obtained from randomizing the phases of the
reconstructed potential. Our findings should help current and next-generation
CMB experiments obtain tighter parameter constraints via the internal removal
of lensing-induced smoothing from temperature and E-mode acoustic peaks.
