Pseudo and quasi quark PDF in the BFKL approximation

I examine the high-energy behavior of the Ioffe-time distribution for the quark bi-local space-like separated operator using the high-energy operator product expansion. These findings have significant implications for lattice calculations, which require extrapolation for large Ioffe-time values. I perform an explicit Fourier transform for both the pseudo-PDF and quasi-PDF, and investigate their behavior within the first two leading twist contributions. I show that the quark pseudo-PDF captures the BFKL resummation (resummation of all twists) and exhibits a rising behavior for small $x_B$ values, while the quasi-PDF presents a different behavior. I demonstrate that an appropriate small-$x_B$ behavior cannot be achieved solely through DGLAP dynamics, emphasizing the importance of all-twist resummation. This study provides valuable insights into quark non-local operators' high-energy behavior and the limitations of lattice calculations in this context.