κ-Deformed Exponentials, Minimal Length and Renormalization Group Flow

The $\kappa$ -deformed exponential ${\exp }_{\kappa }\left(x\right)=\exp \left(\frac{1}{\kappa }\text{arcsinh}\left(\kappa x\right)\right)$ studied by Kaniadakis [1]-[3] allows to construct “deformed” Lorentz transformations associated with the ordinary velocity boost rapidity parameter $\xi$ and which can be recast in terms of ordinary Lorentz transformations (involving the ordinary exponential) but associated with a $\kappa$ -deformed (modified) rapidity parameter ${\xi }_{\kappa }=\xi f\left(\kappa \xi \right)$ given by a $\xi$ -dependent scaling of the original $\xi$ rapidity parameter. It is shown that when both the $\kappa$ parameter and $\xi \to \infty$ , and the double scaling limit $\xi \frac{\ln \left(2\kappa \xi \right)}{\kappa \xi }=\infty \times 0$ is finite and nonzero, it leads to a finite value for the $\kappa$ -deformed boost rapidity parameter ${\xi }_{\kappa =\infty }$