  The proton form factors at large momentum transfer are dominated by two
contributions which are associated with the hard and soft rescattering
respectively. Motivated by a very active experimental form factor program at
intermediate values of momentum transfers, $Q^{2}\sim 5-15 \text{GeV}^{2}$,
where an understanding in terms of only a hard rescattering mechanism cannot
yet be expected, we investigate in this work the soft rescattering contribution
using soft collinear effective theory (SCET). Within such description, the form
factor is characterized, besides the hard scale $Q^2$, by a semi-hard scale $Q
\Lambda$, which arises due to presence of soft spectators, with virtuality
$\Lambda^2$ ($\Lambda \sim 0.5$ GeV), such that $Q^{2}\gg Q\Lambda\gg
\Lambda^{2}$. We show that in this case a two-step factorization can be
successfully carried out using the SCET approach. In a first step (SCET$_I$),
we perform the leading order matching of the QCD electromagnetic current onto
the relevant SCET$_I$ operators and perform a resummation of large logarithms
using renormalization group equations. We then discuss the further matching
onto a SCET$_{II}$ framework, and propose the complete factorization formula
for the Dirac form factor, accounting for both hard and soft contributions. We
also present a qualitative discussion of the phenomenological consequences of
this new framework.
