  We define the notions of weak amenability and the Cowling-Haagerup constant
for extremal finite index subfactors of type II_1. We prove that the
Cowling-Haagerup constant only depends on the standard invariant of the
subfactor. Hence, we define the Cowling-Haagerup constant for standard
invariants. We explicitly compute the constant for Bisch-Haagerup subfactors
and prove that it is equal to the constant of the group involved in the
construction. Given a finite family of amenable standard invariants in the
sense of Popa, we prove that their free product in the sense of Bisch-Jones is
weakly amenable with constant 1. We show that the Cowling-Haagerup constant of
the tensor product of a finite family of standard invariants is equal to the
product of their Cowling-Haagerup constants.
