Towards Two-to-Two Scattering of Scalars in Asymptotically Safe Quantum Gravity

Abstract

We compute the graviton-mediated two-to-two scattering amplitude and cross section for scalar particles in asymptotically safe quantum gravity. Specifically, we compute the full momentum dependence of the scalar-graviton three-point scattering vertex for spacelike momenta with the functional renormalisation group. We also discuss the analytic continuation to the Minkowski branch, and in particular its angular dependence. Then, the timelike part of the vertex is reconstructed and used to compute the scattering amplitude and cross-section. We show that the cross-section reduces to that in General Relativity at small energies, and it respects unitarity in the UV.

Figures (14)

• Figure 1: We display the non-perturbative scattering amplitude for graviton-mediated $s$-channel $\phi\phi\to\phi\phi$ scattering, see \ref{['fig:scattering']}. The computed data is displayed in red alongside a smooth interpolation in blue for visualisation purposes. The amplitude is shown as a function of the centre of mass energy $\sqrt{s}$, and we display the classical Planck scale with the vertical dashed line. $\mathcal{A}_s$ is bounded for trans-Planckian energies, thus compatible with unitarity.
• Figure 2: Feynman diagrams for the scattering of two identical scalars into two identical scalars, see \ref{['eq:FullChannelsAmplitude']}. All vertices and propagators are full 1PI vertices, see \ref{['sec:FRG']}. In this work, we focus on the first three mediated diagrams, corresponding to $s,t,u$-channels.
• Figure 3: Full 1PI scalar-graviton vertex with $p_h= -p_1 - p_2$. All momenta are considered as incoming.
• Figure 4: Diagrammatic representation of the flow of the scalar-graviton vertex. The single black lines represent scalar propagators, while the double blue lines represent graviton propagators. The crossed circles stand for regulator insertions. Note the absence of closed matter loops, since we are neglecting scalar self-interactions, see \ref{['eq:ScalarAction']}.
• Figure 5: Flow function $F(x_1,x_2,z)$, \ref{['eq:FlowFunctionP1P2andZ']}, of the scalar-graviton vertex defined in \ref{['eq:ShorthandOfFlow']} for generic radial momenta $x_1=\|p_1\|/k\, , x_2=\|p_2\|/k\,$ and $z$, see \ref{['eq:Anglez']}. $F(x_1,x_2,z)$ is the diagrammatic (dynamical) part of the flow of the Newton coupling \ref{['eq:FlowEquationFinalFinal']}. Left: projection onto the $p/k=x_1=x_2$ plane. It is shown as a function of the dimensionless momentum $p/k$ and the angle $z$. Right: projection onto the momentum symmetric plane $z=-1/2$ for independent $x_1,x_2$. In both figures, the red dashed lines are the slices $x_1=x_2$ with $z=-1/2$.