# Black Hole Production in the Presence of a Maximal Momentum in Horizon Wave Function Formalism

## TL;DR

This paper explores how a maximal momentum affects black hole formation probabilities within the Horizon Wave Function framework, revealing that black hole likelihood depends on mass, the GUP parameter, and can approach unity for certain conditions.

## Contribution

It introduces a generalized uncertainty principle with a maximal momentum into the Horizon Wave Function formalism, analyzing its impact on black hole probabilities and properties.

## Key findings

- Probability of black hole formation varies with mass and GUP parameter β.
- For larger β, black hole probability increases and approaches unity for high mass.
- Existence of a minimum probability and conditions where particles are fundamentally quantum black holes.

## Abstract

We study the Horizon Wave Function (HWF) description of a generalized uncertainty principle (GUP) black hole in the presence of two natural cutoffs as a minimal length and a maximal momentum. This is motivated by a metric which allows the existence of sub-Planckian black holes, where the black hole mass $m$ is replaced by $M=m\Big(1+\frac{\beta^{2}}{2}\frac{M_{pl}^{2}}{m^{2}}-\beta\frac{M_{pl}}{m}\Big)$. Considering a wave-packet with a Gaussian profile, we evaluate the HWF and the probability that the source might be a (quantum) black hole. By decreasing the free parameter the general form of probability distribution, ${\mathcal{P}}_{BH}$, is preserved , but this resulted in reducing the probability for the particle to be a black hole accordingly. The probability for the particle to be a black hole grows when the mass is increasing slowly for larger positive $\beta$, and for a minimum mass value it reaches to $0$. In effect, for larger $\beta$ the magnitude of $M$ and $r_{H}$ increases, matching with our intuition that either the particle ought to be more localized or more massive to be a black hole. The scenario undergoes a change for some values of $\beta$ significantly, where there is a minimum in ${\mathcal{P}}_{BH}$ , so this expresses that every particle can have some probability of decaying to a black hole. In addition, for sufficiently large $\beta$ we find that every particle could be fundamentally a quantum black hole.