Positivity bounds on Minimal Flavor Violation

TL;DR

The paper investigates whether positivity bounds on SMEFT dimension-8 four-fermion operators with two derivatives are compatible with Minimal Flavor Violation. By expressing MFV coefficients through Yukawa spurions and performing analytic and numerical analyses across $N_f=2$ and $N_f=3$, it shows that CKM entries do not enter at leading order and that Yukawa couplings shape the allowed region of flavor-blind MFV factors. The results indicate that the natural MFV scenario with order-one coefficients satisfies all positivity constraints, while the bounds significantly reduce the MFV parameter space by about $2^{14}$ across the 14 operator types considered. The work also discusses top-quark Yukawa resummation, CKM-dependence at subleading order, and outlines extensions to include dimension-6 operators and more general scattering configurations.

Abstract

From general analyticity and unitarity requirements on the UV theory, positivity bounds on the Wilson coefficients of the dimension-8 operators composed of 4 fermions and two derivatives appearing in the Standard Model Effective Field Theory have been derived recently. We explore the fate of these bounds in the context of models endowed with a Minimal Flavor Violation (MFV) structure, models in which the flavor structure of higher dimensional operators is inherited from the one already contained in the Yukawa sector of the Standard Model Lagrangian. Our goal is to check whether the general positivity bounds translate onto bounds on the Yukawa coefficients and/or on elements of the CKM matrix. MFV fixes the coefficients of dimension-8 operators up to some multiplicative flavor-blind factors and we find that, in the most generic setup, the freedom left by those unspecified coefficients is enough as not to constrain the parameters of the renormalizable Yukawa sector. On the contrary, the latter shape the allowed region for the former. Requiring said overall coefficients to take natural $\mathcal{O}(1)$ values could give rise to bounds on the Yukawa couplings. Remarkably, at leading order in an expansion in powers of the Yukawa matrices, no bounds on the CKM entries can be retrieved.

Figures (4)

• Figure 1: Plot showing the allowed region (in color) obtained for the (4-Q) operators restricted to $N_f=2$ with generic $\xi$ values, as $y_c$ changes. Every region associated to a larger $y_c$ value is contained in the previous ones: for instance, blue and dark orange regions are allowed for any $y_c$ roughly smaller than 4, but forbidden for larger values of $y_c$. The redundant $\xi_4$ has been set to 0. In this case $\xi_3>0$, and using that the bounds are invariant under a full rescaling, we have set $\xi_3=1$, and plot the remaining two independent coefficients. As explained in the text, values of $y_c>1$ are unphysical and are only plotted for visual reasons. The black point represents the natural MFV benchmark point $\xi_{1,2,3}=1$. The red line contours the region corresponding to the threshold value of $y_c=\sqrt{2(1+\sqrt{2})}$: for bigger values of $y_c$, the natural benchmark point does not belong to the allowed region any more. The region $\xi_1<-1$ is excluded for any value of $y_c$. For the physical value $y_c\approx 10^{-2}$, almost all points $(\xi_1\geq -1,\xi_2)$ are allowed.
• Figure 2: Plot showing in yellow the allowed region obtained for the (4-Q) (or, equivalently, (4-u)) operators with generic $\xi$ values. Using the scaling invariance of the bounds and since $\xi_3>0$, we have set $\xi_3=1$, and plot the remaining three independent coefficients. The red dot indicates the natural MFV benchmark point, $\xi_{1,2,3,4}=1$, that can be seen being inside the allowed region.
• Figure 3: Plot showing in yellow the allowed parameter space for $\xi^{d,i}_1$ and $\xi^{d,i}_3$, $i=1,2$. The black dot indicates the natural MFV benchmark point, $\xi_{1,3}=1$, that can be seen being inside the allowed region.
• Figure 4: Plots showing in yellow the allowed region obtained for the (2-u)(2-Q) operators, with generic $\xi$ values. Here $\xi_1>0$, so we have rescaled it to 1, and we show the allowed region for the remaining three. $y_t$ is set to 1. The red dot indicates the naturak MFV benchmark point, $\xi_{1,2,3,4}=1$, which can be seen being inside the allowed region.