  To investigate the possible geometric structure of the proton, an improved
stringy proton model is constructed beyond the smallest distance approximation,
where the constituent quarks are connected by gluon tubes which merge at the
Fermat point of the quark triangle. The exclusive diffractive vector meson
production process in electron-proton deep inelastic scattering is used to test
the stringy structure of the proton. We calculate the coherent and incoherent
differential cross sections of the exclusive diffractive $J/\Psi$
photoproduction in the framework of Color Glass Condensate. The results show
that our calculations are in good agreement with HERA data. Especially, our
results give a better description of the HERA data at small $t$ as compared to
the ones from the hot spot model where the constituent quarks are uncorrelated
distributed in the proton. Meanwhile, the radius of the proton resulting from
the improved stringy proton model is coincident with the one from fitting to
the data from GlueX Collaboration at Jefferson Lab, which indicates that the
predictive power of the stringy proton model is significantly improved once it
goes beyond the smallest distance approximation. Moreover, we assume that the
transverse shape of gluon tube satisfies Gaussian distribution, and explore the
distribution width of the individual gluon tubes. We find an interesting result
that the up quark induced gluon tube seems to have larger distribution width
than the down quark induced gluon tube, which is favored by the HERA data.
