Four-fermion operators at dimension 6: Dispersion relations and UV completions

A major task in phenomenology today is constraining the parameter space of Standard Model effective field theory and constructing models of fundamental physics from which the Standard Model derives. To this effect, we report an exhaustive list of sum rules for 4-fermion operators of dimension 6, connecting low-energy Wilson coefficients to cross sections in the UV. Unlike their dimension-8 counterparts which are amenable to a positivity bound, the discussion here is more involved due to the weaker convergence and indefinite signs of the dispersion integrals. We illustrate this by providing examples with weakly coupled UV completions leading to opposite signs of the Wilson coefficients for both convergent and nonconvergent dispersion integrals. We further decompose dispersion integrals under weak isospin and color groups, which lead to a tighter relation between IR measurements and UV models. These sum rules can become an effective tool for constructing consistent UV completions for Standard Model effective field theory following the prospective measurement of these Wilson coefficients.