Investigations into the BFKL mechanism with a running QCD coupling

We present approximations of varying degree of sophistication to the integral equations for the (gluon) structure functions of a hadron (“the partonic flux factor”) in a model valid in the leading log approximation with a running coupling constant. The results are all of the BFKL type, i.e., a power in the Bjorken variable $x_B^{-\lambda }$ with the parameter $\lambda$ determined from the size ${\alpha }_0$ of the “effective” running coupling $\overline{\alpha }\equiv 3{\alpha }_s/\pi ={\alpha }_0/\ln {(k}_{\perp }^2)$ and varying depending upon the treatment of the transverse momentum pole. We also consider the implications for the transverse momentum ${(k}_{\perp })$ fluctuations along the emission chains and we obtain an exponential falloff in the relevant $\kappa \equiv \ln {(k}_{\perp }^2)$ variable, i.e., an inverse power ${(k}_{\perp }^2{)}^{-(2+\lambda )}$ with the same parameter $\lambda .$ This is different from the BFKL result for a fixed coupling, where the distributions are Gaussian in the $\kappa$ variable with a width as in a Brownian motion determined by “the length” of the emission chains, i.e., $\ln {(1/x}_B).\mathrm{}$ The results are verified by a realistic Monte Carlo simulation and we provide a simple physics motivation for the change.