  Let $B=(B_1(t),\ldots,B_d(t))$ be a $d$-dimensional fractional Brownian
motion with Hurst index $\alpha<1/4$. Defining properly iterated integrals of
$B$ is a difficult task because of the low H\"older regularity index of its
paths. Yet rough path theory shows it is the key to the construction of a
stochastic calculus with respect to $B$, or to solving differential equations
driven by $B$. We show in this paper how to obtain second-order iterated
integrals as the limit when the ultra-violet cut-off goes to infinity of
iterated integrals of weakly interacting fields defined using the tools of
constructive field theory, in particular, cluster expansion and
renormalization. The construction extends to a large class of Gaussian fields
with the same short-distance behaviour, called multi-scale Gaussian fields.
Previous constructions \cite{Unt-Holder,Unt-fBm} were of algebraic nature and
did not provide such a limiting procedure.
