Beyond Jarlskog: 699 invariants for CP violation in SMEFT

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 $\le 4$ and thus cannot appear in any physical observable at $\order{1/Λ^2}$. 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 $\le 4$. 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\(1/Λ^2\)$.

25 pages + 22 pages of appendices, 13 tables. v2: additional references, some extended discussions and fixed typos, matches JHEP version

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• 23, 381, 6242, 103268, 1743183, ...: Hilbert series for CP-violating operators in SMEFT
• Spelling Out Leptonic CP Violation in the Language of Invariant Theory
• Exploring the Flavor Symmetry Landscape