Over the past two decades scientists have achieved a significant improvement of our understanding of the transport of energetic particles across a mean magnetic field. Due to test-particle simulations as well as powerful non-linear analytical tools our understanding of this type of transport is almost complete. However, previously developed non-linear analytical theories do not always agree perfectly with simulations. Therefore, a correction factor $a^2$ was incorporated into such theories with the aim to balance out inaccuracies. In this paper a new analytical theory for perpendicular transport is presented. This theory contains the previously developed unified non-linear transport theory, the most advanced theory to date, in the limit of small Kubo number turbulence. For two-dimensional turbulence new results are obtained. In this case the new theory describes perpendicular diffusion as a process which is sub-diffusive while particles follow magnetic field lines. Diffusion is restored as soon as the turbulence transverse complexity becomes important. For long parallel mean free paths one finds that the perpendicular diffusion coefficient is a reduced field line random walk limit. For short parallel mean free paths, on the other hand, one gets a hybrid diffusion coefficient which is a mixture of collisionless Rechester & Rosenbluth and fluid limits. Overall the new analytical theory developed in the current paper is in agreement with heuristic arguments. Furthermore, the new theory agrees almost perfectly with previously performed test-particle simulations without the need of the aforementioned correction factor $a^2$ or any other free parameter.
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