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Classical color field modified minijet model for $pp$ and $\bar{p} p$ total cross section

In a recent paper, we have evaluated the $gg\rightarrow gg$ scattering amplitude in the presence of classical color field generated by the colliding protons in the leading order approximation within the pQCD. In this work, we show that this amplitude can be resumed to obtain the classical color field modified $gg \rightarrow gg$ elastic scattering amplitude. This modified amplitude is suppressed when the longitudinal momentum fraction, $x$, of the incident gluon is small. Minijet cross section is calculated using the modified amplitude. We show that the $pp$ and $\bar{p} p$ cross section from $\sqrt{s} = 5$ GeV to 30 TeV can be described as a sum of a hard component contributed by the modified minijet model and a soft component due to the exchange of the pomeron and of the I=0 exchange-degenerate $ω$ and $f$ trajectories. The predicted cross section has a $\ln^2 s$ asymptotic behavior which satisfies the Froissart bound.