Crossing Symmetric Spinning S-matrix Bootstrap: EFT bounds

Abstract

We develop crossing symmetric dispersion relations for describing 2-2 scattering of identical external particles carrying spin. This enables us to import techniques from Geometric Function Theory and study two sided bounds on low energy Wilson coefficients. We consider scattering of photons, gravitons in weakly coupled effective field theories. We provide general expressions for the locality/null constraints. Consideration of the positivity of the absorptive part leads to an interesting connection with the recently conjectured weak low spin dominance. We also construct the crossing symmetric amplitudes and locality constraints for the massive neutral Majorana fermions and parity violating photon and graviton theories. The techniques developed in this paper will be useful for considering numerical S-matrix bootstrap in the future.

Snowmass White Paper: UV Constraints on IR Physics

“…This conjecture was confirmed by the beautiful results of [93] where they used the power of the null-constraints together with the scattering amplitude in impact parameter space, or equivalently at −Λ 2 < t < 0 to define a set of positive integrals which could be used to constrain the leading s 2 terms in the scattering amplitudes even in the presence of the massless spin-2 pole. These bounds have been put on the leading order corrections to Einstein gravity in [94], complementing the results of [88][89][90]. An interesting observation is that strong bounds can be obtained by assuming that the lower spin partial waves dominate in the dispersion relation, as shown in explicit examples, via geometric function theory or via SDP numerically [88][89][90]94].…”

“…These bounds have been put on the leading order corrections to Einstein gravity in [94], complementing the results of [88][89][90]. An interesting observation is that strong bounds can be obtained by assuming that the lower spin partial waves dominate in the dispersion relation, as shown in explicit examples, via geometric function theory or via SDP numerically [88][89][90]94].…”

“…The second problem is that even at tree level the presence of a spin-2 massless pole undermines the assumption of a dispersion relation with two subtractions for t ≥ 0 and yet the positivity properties of the Legendre/Gegenbauer polynomials only apply for t ≥ 0. There is however no problem working with more than 2 subtractions, and a powerful set of bounds using the EFThedron and null-constraints can be derived for tree level EFT graviton scattering amplitudes [88][89][90]. The validity of the leading positivity bounds in the presence of a spin-2 pole was considered in [91,92] where it was argued that the leading positivity bounds could be mildly violated in gravitational effective field theories by an M 2 Pl suppressed amount.…”

“…Possible uses of dispersion relations/sum rules to constrain dimension-6 operators, with extra assumptions, have also been discussed in [56,[151][152][153][154]. Various positivity bounds have also been applied to constrain chiral perturbation theory [40,42,[155][156][157][158][159][160][161][162][163][164] and the Euler-Heisenberg EFT [89,165] In the context of the SMEFT, the most relevant positivity bounds are the leading forward limit bounds, which already come in at the order of dimension-8. For identical particle scattering, this means that the coefficient in front of s 2 is positive [38]; In the presence of multiple fields such as in the SMEFT, generalized elastic bounds are essentially the same after superimposing different fields, which still uses the standard optical theorem.…”

Celestial insights into the S-matrix bootstrap

“…As we will see, these Feynman blocks in the Celestial variables have remarkable properties. One of the main advantages of working with the CSDR is that it leads to a fascinating connection with an area of mathematics called Geometric Function Theory (GFT) [24][25][26]. The origin of two sided bounds on Wilson coefficients gets related to the famous Bieberbach conjecture (de Branges' theorem).…”