Effective delta sources and Newtonian limit in nonlocal gravity

TL;DR

The paper analyzes the Newtonian limit of nonlocal gravity models with exponential form factors $f_s(\Box)=\exp [(-\Box/\mu_s^2)^{N_s}]$ across real $N_s>0$, using an effective-source formalism to derive the smeared delta sources $\rho_s$, mass functions $M_s$, and Newtonian potentials $\chi_s$. It shows that $\rho_s$ is strictly positive for $0<N_s\le 1$ and that the mass function remains positive for all $N_s>0$, while oscillations appear in the potentials for $N_s>1$; the Newtonian solutions are regular and free of curvature singularities, and the limit $N_s\to\infty$ yields analytic forms with interesting oscillatory behavior and a nonuniform approach to GR. The authors provide multiple representations (power series, integrals, Fox–Wright/H-functions, Meijer $G$, and generalized exponential functions) and derive a simple large-$N_s$ approximation, enabling efficient weak-field phenomenology. They also compute leading one-loop logarithmic quantum corrections to the Newtonian limit, express them in terms of master integrals and special functions, and analyze their IR behavior and sign dependence on the renormalization scale, showing these corrections preserve regularity. The results extend the study to fractional $N_s$ values, offering a versatile framework for exploring nonlocal gravity phenomenology and its quantum-corrected weak-field regime.

Abstract

We investigate the Newtonian limit of a class of nonlocal gravity models with exponential form factors $f_s (\Box) = \exp [(-\Box/μ_s^2)^{N_s}]$. Our main goal is to identify similarities and differences between models in this family in regard to weak-field solutions. To this end, we use the effective source formalism to compare the related effective delta sources, mass functions, and Newtonian potentials. We obtain a variety of representations for these quantities in terms of series, integrals, and special functions, as well as simple approximations that capture the relevant dependence on the parameters $N_s$ and $μ_s$ - which can be used to explore the weak-field phenomenology of nonlocal gravity. We explain why only for $N_s>1$ the Newtonian potential oscillates and prove that, despite the oscillations, the effective masses are positive. Moreover, we verify that these linearized solutions are regular (without curvature singularities). Finally, we also calculate the form of the leading logarithmic quantum correction to the Newtonian potential in these models. In all our considerations, we assume that $N_s$ is a positive real parameter. The cases of non-integer $N_s$ might be applied beyond nonlocal gravity, in effective approaches to implement quantum corrections in the weak field regime.

Figures (20)

• Figure 1: Graph of the source $\rho_s(r)/(M\mu_s^3)$ as a function of $\mu_s r$ for different values of $N_s$. The oscillations are present only for $N_s > 1$. The value of $\rho_s (0)$ grows very fast as $N_s \to 0$.
• Figure 2: Plot of the dimensionless mass function $M_s(r)/M$ as a function of $\mu_s r$ for different values of $N_s$. The oscillations are present only for $N_s > 1$. For any value of $N_s$ we can observe the asymptotic limits: For $\mu_s r \ll 1$, $M_s(r) \sim r^3$, while for $\mu_s r \gg 1$, $M_s (r) \sim M$.
• Figure 3: Plot of the dimensionless mass function (\ref{['meff-lw']}) for Lee--Wick gravity, $M_s^{\text{LW}}(r)/M$, as a function of $b_s r$ for different values of $q_s = b_s/a_s$. ${M_s^{\text{LW}}}(r)$ assumes negative values for $q_s>2.67$.
• Figure 4: Plot of $\chi_s(r)/(G M \mu_s)$ as a function of $\mu_s r$ for different values of $N_s$. The oscillations are present only for $N_s > 1$. Note that for any $N_s >0$ we have $\chi'_s(0)=0$.
• Figure 5: (a)$M_s^{\infty}(r)/M$ as a function of $\mu_s r$, the horizontal dashed lines corresponds the value $(1 \pm 2/\pi) = (0.363,1.637)$. (b) Plot of the gravitational field $g_s^{\infty}(r)/GM$ as a function of $\mu_s r$. The dashed lines correspond to the standard $-1/r^2$ gravitational field. Although the mass function $M_s^\infty(r)$ does not have a definite limit when $r \gg 1/\mu_s$, the gravitational field $g_s^\infty(r) \to 0$ owning to the $r^{-2}$ damping factor in Eq. (\ref{['gs_def']}). Furthermore, the typical Newtonian singularity at $r=0$ is absent.