Determine the value and sign of the leading quantum correction to the Newtonian potential

Determine the value and sign of the leading-order quantum correction to the Newtonian potential in order to establish a reliable constraint on the quadratic generalized uncertainty principle parameter \(\beta_0\).

Background

The scale-dependent quadratic GUP effective metric relates the dimensionless GUP parameter $\beta_0$ to the renormalization-group correction parameter $\tilde{\omega}$ through $\beta_0\zeta^2\equiv\tilde{\omega}$. The paper attempts to constrain $\beta_0$ using published calculations of the leading quantum correction to the Newtonian potential.

The authors explicitly note that published results disagree about both the magnitude and sign of this correction. Because the inferred value of $\beta_0$ depends directly on that correction, resolving its value and sign is necessary for obtaining a robust GUP-parameter constraint and for determining the associated effective-metric parameters.