  We give a new definition of higher arithmetic Chow groups for smooth
projective varieties defined over a number field, which is similar to Gillet
and Soul\'e's definition of arithmetic Chow groups. We also give a compact
description of the intersection theory of such groups. A consequence of this
theory is the definition of a height pairing between two higher algebraic
cycles, of complementary dimensions, whose real regulator class is zero. This
description agrees with Beilinson's height pairing for the classical arithmetic
Chow groups. We also give examples of the higher arithmetic intersection
pairing in dimension zero that, assuming a conjecture by Milnor on the
independence of the values of the dilogarithm, are non zero.
