  We derive a novel formula for the derivative of operator product expansion
(OPE) coefficients with respect to a coupling constant. The formula only
involves the OPE coefficients themselves, and no further input, and is in this
sense self-consistent. Furthermore, unlike other formal identities of this
general nature in quantum field theory (such as the formal expression for the
Lagrangian perturbation of a correlation function), our formula is completely
well-defined from the start, i.e. requires no further UV-renormalization. This
feature is a result of a cancelation of UV-divergences between various terms in
our identity. Our proof, and an analysis of the features, of our identity is
given for the example of massive, Euclidean $\varphi^4$ theory in 4 dimensional
Euclidean space, and relies heavily on the framework of the renormalization
group flow equations. It is valid to arbitrary, but finite orders in
perturbation theory. The final formula, however, makes no explicit reference to
the renormalization group flow, nor to perturbation theory, and we conjecture
that it also holds non-perturbatively. The identity can be applied
constructively because it gives recursive algorithm for the computation of OPE
coefficients to arbitrary (finite) perturbation order in terms of the zeroth
order coefficients corresponding to the underlying free field theory, which in
turn are trivial to obtain. We briefly illustrate the relation of this method
to more standard methods for computing the OPE in some simple examples.
