Elastic positivity vs extremal positivity bounds in SMEFT: a case study in transversal electroweak gauge-boson scatterings

Abstract

The positivity bounds, derived from the axiomatic principles of quantum field theory (QFT), constrain the signs of Wilson coefficients and their linear combinations in the Standard Model Effective Field Theory (SMEFT). The precise determination of these bounds, however, can become increasingly difficult as more and more SM modes and oper- ators are taken into account. We study two approaches that aim at obtaining the full set of bounds for a given set of SM fields: 1) the traditional elastic positivity approach, which exploits the elastic scattering amplitudes of states with arbitrarily superposed helicities as well as other quantum numbers, and 2) the newly proposed extremal positivity approach, which constructs the allowed coefficient space directly by using the extremal representation of convex cones. Considering the electroweak gauge-bosons as an example, we demonstrate how the best analytical and numerical positivity bounds can be obtained in several ways. We further compare the constraining power and the efficiency of various approaches, as well as their applicability to more complex problems. While the new extremal approach is more constraining by construction, we also find that it is analytically easier to use, nu- merically much faster than the elastic approach, and much more applicable when more SM particle states and operators are taken into account. As a byproduct, we provide the best positivity bounds on the transversal quartic-gauge-boson couplings, required by the axiomatic principles of QFT, and show that they exclude ≈ 99.3% of the parameter space currently being searched at the LHC.

New positivity bounds from full crossing symmetry

“…In particular, positivity bounds have been used to constrain Standard Model EFT [19][20][21][22][23][24][25][26]. For example, it has been shown that the vast majority of the parameter space of the dimension 8 anomalous quartic gauge couplings is excluded by the positivity bounds [19,20,23,25]. For another example, strong constraints can also be obtained for Horndeski theory when pairing together with cosmological parameter estimation analysis [36].…”

Positivity bounds on Minimal Flavor Violation

“…Using the Warsaw basis for the dimension-6 SMEFT [28], and considering the high-energy phase where all particles are massless and the electroweak symmetry unbroken, it is straightforward to check that there are no dimension-6 contributions to the four-fermion positivity bounds at tree level. This differs from the case of 2 → 2 gauge bosons scattering, where non-redundant trilinear couplings enter the bounds at tree-level and strenghthen the bounds on dimension-8 operators [10,11,14].…”

“…More generally, every setting where the flavor-blind parameters are degenerate is seen to be compatible with the positivity conditions. Still, the positivity constraints are such as to reduce the full parameter space by a factor of 2 14 .…”

“…Recently, efforts have been made to constrain the coefficients in the SMEFT expansion [7,8], particularly regarding vector boson scattering [9][10][11] (see also refs. [12][13][14][15][16][17][18][19] for recent developments).…”

Low-Energy Effective Field Theory below the Electroweak Scale: dimension-8 operators

“…Furthermore, many dimension-8 operators are subject to positivity bounds arising from fundamental principles of quantum field theory including unitarity and analyticity, see [20][21][22][23][24][25][26][27][28][29] for some recent applications. The LEFT is a valid EFT in its own right, independent of whatever its UV completion may be.…”