Asymptotically (un)safe scattering amplitudes from scratch: a deep dive into the IR jungle

Abstract

We compute leading order quantum gravity contributions to a simple scalar scattering amplitude in Asymptotic Safety. Our model admits an analytic treatment so that several subtleties can be analysed. We find that (i) the existence of an asymptotically safe renormalisation group fixed point alone does not imply the boundedness of scattering amplitudes, (ii) gravitational logarithms can dominate the infrared regime of massless theories, (iii) a derivative expansion of the effective action fails quantitatively to predict the correct Wilson coefficients in massless theories, and (iv) standard renormalisation group improvement techniques fail qualitatively to describe the momentum dependence of correlation functions. Only momentum-dependent computations can resolve these issues. For theories that include massive fields, the derivative expansion can work effectively in most cases, but it can still fail for classically marginal couplings, and purely gravitational couplings. We also speculate about an effective realisation of the no-global-symmetries conjecture in Asymptotic Safety.

Figures (9)

• Figure 1: Diagrams that contribute to the $s$-channel process $\phi(p_1)\phi(p_2)\to\chi(p_3)\chi(p_4)$: gravity-mediated diagram (left) and contact term (right). The wavy double line in the left diagram indicates a fully dressed graviton propagator, and the dotted vertices are fully dressed as well. All momenta are ingoing, as indicated by the arrows.
• Figure 2: Double seagull diagram that we have to evaluate for the flow of the form factors $F_{i,k}^E(P^2)$. The wavy double lines represent graviton propagators, whereas the full and dashed lines correspond to the scalar fields. A regulator insertion $k\partial_k \mathfrak R_k$ is understood on one of the propagator lines.
• Figure 3: Fixed point solution $f_\ast^E(z)$ in units of the fixed point value of Newton's coupling, as a function of the dimensionless Euclidean squared momentum $z=p^2/k^2$. The function is finite at zero argument, and falls off like $1/z$ for large $z$. It is negative over the entire domain.
• Figure 4: Euclidean form factor $F^E$ in Planck units, as a function of Euclidean squared momenta in terms of the quantum gravity scale $G_N/2g_\ast$. Since we have a completely analytical expression, we can evaluate the form factor also for negative arguments. At vanishing argument, there is a logarithmic divergence, see \ref{['eq:LOFFexpanded']}.
• Figure 5: Momentum beta function in Planck units, as a function of Euclidean momenta in terms of the quantum gravity scale. Since we have a completely analytical expression, we can evaluate the form factor also for negative arguments. At vanishing argument, the function is continuous but not differentiable due to the logarithm in the form factor, cf. \ref{['eq:LOFFexpanded']} and \ref{['eq:LOmombeta']}.