Amplitudes, resonances, and the ultraviolet completion of gravity

This paper constructs, making use of the on-shell spinor-helicity formalism, a possible ultraviolet completion of gravity following a “bottom-up” approach. The assumptions of locality, unitarity, and causality i) require an infinite tower of resonances with increasing spin and quantized mass, ii) introduce a duality relation among crossed scattering channels, and iii) dress all gravitational amplitudes in the Standard Model with a form factor that closely resembles either the Veneziano or the Virasoro-Shapiro amplitude in string theory. As a consequence of unitarity, the theory predicts leading order deviations from General Relativity in the coupling of gravity to fermions that could be explained if space-time has torsion in addition to curvature.

Figure 1

Four-point amplitude with ingoing initial-state and outgoing finale-state particles (top-left diagram) and with all particles taken incoming (bottom-left diagram). On the right side, we show the three physical scattering regions (r), (b), and (g) as a function of the kinematic invariants $s_{ij}$, corresponding to the substitutions in terms of the conventional Mandelstam variables in Eqs. (1)–(3). For massless external particles, the inner triangle collapses into a point.

Figure 2

Chew-Frautschi spin/mass plot of the spectrum of exchanged resonances in the four-graviton scattering $g^{\pm 2}{g}^{\pm 2}\to g^{\pm 2}{g}^{\pm 2}$ (left panel) and $g^{\pm 2}{g}^{\mp 2}\to g^{\pm 2}{g}^{\mp 2}$ (right panel). The mass is quantized according to the relation $M_n^2=(1+n)M^2$ with $n=0,1,\dots \in \mathbb{N}$ on the $x$ axes. The rainbow colors in the legend mark, for each resonance, the value of $\mathrm{\Gamma }/M$ in units of ${\alpha }_{\mathrm{Pl}}\equiv M^2/M_{\mathrm{Pl}}^2$ [see Eqs. (71) and (74)].

Figure 3

Unitarity bound (allowed region shaded in green) on the parameter $b$ in Table 1 obtained from the positivity of the coefficients in the expansion in Eq. (82). The left side of the bound is described by the simple analytical formula $b\ge (4+n)/2(3+n)$, which asymptotes to $1/2$ for $n\to \infty$.