quasi-primary fields in CFT Ward identities

Hi,

I'm in the process of learning conformal field theory from Di
Francesco, Mathieu, Senechal,
and need some help with something I perceive as a paradox:

There are three Ward identities associated with global conformal
invariance.
These Ward identities express expectation values of the form <T X>
through terms that
only involve <X>, where T is the energy momentum tensor (indices
suppressed)
and X is any product of n fields at different spacetime points. It is
stressed at
various points in the book that the form of the Ward identities used is
valid only
for the case that the n fields comprising X are primary fields.
However, I do not
see why they shouldn't be valid for quasi-primary fields as well. After
all, these
equations are first derived in chapter 4 for general number of
spacetime dimensions D,
where no local conformal transformations are available (thus there is
no distinction
between primary and quasi-primary fields at this point). It is stated
that D=2 is
more subtle, since one cannot proof in general that T can be made
traceless,
but the authors assure the reader that this is no problem in practice.
From this
I must assume that these Ward identities are valid also when X is made
up of
quasi-primary fields.

On the other hand, this cannot be true, because it is shown in chapter
5 that
the same ward identities imply that <T(z) X(w)> diverges only as
(z-w)^(-2).
But this is in fact not the case when I choose X=T, which is a
quasi-primary field.
For then <T(z)T(w)> diverges as c/2 (z-w)^(-4), which defines the
conformal charge c.
This is also used to make the case that T is not a primary field, but a
quasi-primary
one. But again, I do not see why the implications of the Ward
identities (i.e.
<T(z)X(w1,w2...)> behaves as (z-w_i)^(-2) for small |z-w_i|, eq. 5.41
in the book)
do not hold for quasi-primary fields X also.

I would greatly appreciate any help with this puzzle.

.