Quantum gravity constraints from unitarity and analyticity

Abstract

We derive rigorous bounds on corrections to Einstein gravity using unitarity and analyticity of graviton scattering amplitudes. In D ≥ 4 spacetime dimensions, these consistency conditions mandate positive coefficients for certain quartic curvature operators. We systematically enumerate all such positivity bounds in D ¼ 4 and D ¼ 5 before extending to D ≥ 6. Afterwards, we derive positivity bounds for supersymmetric operators and verify that all of our constraints are satisfied by weakly coupled string theories. Among quadratic curvature operators, we find that the Gauss-Bonnet term in D ≥ 5 is inconsistent unless new degrees of freedom enter at the natural cutoff scale defined by the effective theory. Our bounds apply to perturbative ultraviolet completions of gravity.

Softness and amplitudes’ positivity for spinning particles

“…Equivalently, we can pick a convenient frame where p is aligned with k r since the forward amplitude depends only on s. Hence, the action of crossing spin-1 massless particles in the forward limit is equivalent again to k σ a ↔ −k −σ a everywhere including the contribution from the polarizations up to gauge transformations, crossing massless spin-1: 12) implying the relations (3.9) and (3.10).…”

“…This is why they have found several applications ranging e.g. from the a-theorem [3] to the theory of pions [4][5][6], from WW-scattering [1,7,8] to composite Higgs models [8][9][10][11], from quantum gravity [12] to inflation [13,14], from Galileons [15] to massive gravity [16], from the weak gravity conjecture [17,18] to the OPE coefficients [19], from the conformal blocks expansion [20] to the Mellin amplitudes for CFTs at large-N [21]. Virtually all literature have focused on positivity bounds for amplitudes of bosons with spin-0, -1 or -2; see [22] for an interesting exception that studied dimension-6 4-fermi interactions, and e.g.…”

“…And in fact, apart from e.g. [12,16], most of the positivity bounds for massive spin-1 or spin-2 bosons have actually focused on the "eaten" scalar modes, i.e. the spin-0 Goldstone Bosons (GB) or the Galileon mode, respectively.…”

“…Effectively, this is mathematically equivalent to consider fully unpolarized half-integer particles. 12 Should one be interested in a tree-level inequality, the (4.13) can be turned into a neat expression similar to (4.14), even when IR branch-cut from the polarizations or other IR poles below threshold are present, thanks to a simple trick from complex analysis. Because the tree-level EFT has no UV branch cut, the left-hand side in (4.13) is nothing but (minus) the residue at infinity calculated with the tree-level EFT: M…”

“…12 Note that we could take even further subtractions, that is obtain positivity conditions for higher derivatives of the amplitude, as long as the gap is open and µ 2 = 0 in order to regulate possible IR divergences.…”