Beyond Jarlskog: 699 invariants for CP violation in SMEFT

TL;DR

This paper develops a comprehensive set of 699 flavor-invariant CP-odd conditions in SMEFT, enhancing the understanding of CP violation by linking invariants to physical observables and providing a detailed classification of CP-violating effects.

Contribution

It introduces a complete list of linear CP-odd flavor invariants in SMEFT and establishes their relation to physical CP violation, extending the analysis beyond the Jarlskog invariant.

Findings

Identifies 699 conditions for CP conservation in SMEFT at leading order.

Establishes a one-to-one correspondence between CP-odd invariants and physical CP violation.

Provides explicit lists of invariants and their relationship to CP-breaking parameters.

Abstract

As SMEFT is a framework of growing importance to analyze high-energy data, understanding its parameter space is crucial. The latter is commonly split into CP-even and CP-odd parts, but this classification is obscured by the fact that CP violation is actually a collective effect that is best captured by considering flavor-invariant combinations of Lagrangian parameters. First we show that fermion rephasing invariance imposes that several coefficients associated to dimension-six operators can never interfere with operators of dimension $\leq4$ and thus cannot appear in any physical observable at $\order{1/\Lambda^2}$. For those that can, instead, we establish a one-to-one correspondence with CP-odd flavor invariants, all linear with respect to SMEFT coefficients. We explicitly present complete lists of such linear CP-odd invariants, and carefully examine their relationship to CP breaking throughout the parameter space of coefficients of dimension $\leq 4$. Requiring that these invariants all vanish, together with the Jarlskog invariant, the strong-CP phase, and the 6 CP-violating dimension-6 bosonic operators, provides $699(+1+1+6)$ conditions for CP conservation to hold in any observable at leading order, $\cO\(1/\Lambda^2\)$.