Flavor Constraints from Unitarity and Analyticity

We use unitarity and analyticity of scattering amplitudes to constrain fermionic operators in the standard model effective field theory. For four-fermion operators at mass dimension 8, we scatter flavor superpositions in fixed standard model representations and find the Wilson coefficients to be constrained so that their contraction with any pair of pure density matrices is positive. These constraints imply that flavor-violating couplings are upper bounded by their flavor-conserving cousins. For instance, LEP data already appears to preclude certain operators in upcoming $\mu \to 3e$ measurements.

Figure 1

Positivity bounds from Eq. (9) on the parameter space of a two-flavor theory. In this example toy model, we enforce $CP$ symmetry so that all $c_{mnpq}^{e,1}\in \mathbb{R}$. For simplicity, in this illustration we further identify various Wilson coefficients, defining $c=c_{1111}=c_{2222}=c_{1221}$, $c_0=c_{1122}$, $c_1=c_{1112}=c_{1222}$, and $c_2=c_{1212}$, so that $c$, $c_0$ are flavor-conserving, while $c_1$ ($c_2$) violates flavor by 1 (respectively, 2) unit. Consistency demands that $c>0$ and, defining $x_i=c_i/c$, that $-2+4|x_1|<x_0+x_2<2$ and $2|x_0-x_2|<2-x_0-x_2+\sqrt{(x_0+x_2+2-4x_1)(x_0+x_2+2+4x_1)}$, for which projections are depicted above.