Symmetries from outer automorphisms and unorthodox group extensions

TL;DR

The paper addresses how to systematically construct symmetries in quantum field theories by extending a minimal seed symmetry to larger groups. It formalizes two extension classes: normal extensions realized by outer automorphisms and unorthodox extensions, and demonstrates their application in the 2HDM and 3HDM to reproduce all exact global symmetry groups observed in these models. It shows that, for the 2HDM, all discrete and continuous symmetries (including CP-type transformations) can be obtained by consecutive outer automorphism extensions of the seed symmetry CP1, with SU(2) arising from either CP2 or as an unorthodox extension of CP1; similar logic applies to the 3HDM’s finite Higgs-flavor groups. The work highlights the practical utility of this framework for narrowing the space of possible symmetries in model-building, guiding brute-force scans, and informing machine-learning searches, while also noting that unorthodox extensions are sometimes necessary, particularly when simple groups are involved. It suggests a complementary top-down approach based on covariant transforming basis invariants to identify realizable symmetry groups directly from basis changes, potentially constraining the landscape of allowed couplings without exhaustive testing.”

Abstract

Symmetries play an essential role in the construction and phenomenology of quantum field theories (QFTs). We discuss how to construct symmetries of QFTs by extending minimal "seed" symmetry groups to larger groups that contain the seed(s) as subgroup(s). On the one hand, there are so-called "normal" extensions, which are given by outer automorphisms of the original symmetry group (including the trivial one) and contain the seed as a normal subgroup. On the other hand, there can be "unorthodox extensions" which do not have this property. We demonstrate our logic on the most general scalar potentials of the two- and three-Higgs-doublet models (2HDM and 3HDM). For the 2HDM, we show that all symmetry groups, including the different possible classes of CP and continuous symmetry groups, can be obtained from extensions of the smallest possible symmetry CP1 by consecutive outer automorphisms. Scanning over normal and unorthodox group extensions might be the easiest way to "machine learn" the possible symmetries of a QFT. However, many of the groups constructible in this way may not be realizable in a concrete model, in the sense that they lead to additional accidental symmetries. Hence, we also comment on a different, "top-down" way to obtain the possible realizable symmetry groups of a QFT based on the covariant transformation of couplings under the most general basis changes.

Figures (3)

• Figure 1: Cartoon of split and non-split normal group extensions of a group $G$ by an outer automorphism $\mathrm{Out}(G)$ to a larger group $\Gamma$ (left and middle panel). We also show unorthodox extensions by a new generator $\mathsf{h}$ (right panel). Unorthodox extensions can also be either split (not shown) or non-split, and they can be finite or infinite.
• Figure 2: The symmetry map of the 2HDM scalar sector, adapted from Bento:2020jei. Horizontal steps correspond to enforcing non-trivial relations among the existing basis invariants of a ring, which is one-to-one with enforcing specific alignments of covariants. Vertical steps correspond to entirely removing the indicated covariant objects, which coincides with collapsing the ring of invariants to a smaller subring. Here we show that symmetry enhancements along all arrows are given by extensions of smaller symmetry groups. Solid arrows correspond to normal extensions by outer automorphisms (which are all trivial outer automorphisms in this case). Dashed arrows correspond to unorthodox extensions.
• Figure 3: ("Igor's tree") The flavor-type finite realizable symmetry groups of the 3HDM, adopted from Ivanov:2012fpIvanov:2014doa and modified for our purpose. All arrows denote subgroup relationships. Solid arrows indicate normal extension (outer automorphism), with thick(thin) solid arrows corresponding to non-trivial(trivial) outer automorphisms. Dashed arrows indicate unorthodox extensions (see classification in Sec. \ref{['sec:extensions_generalities']}). The underlined Abelian group $\mathbbm{Z}_3\times\mathbbm{Z}_3$ is not realizable but included to explicitly show that all groups can be obtained through normal extensions. On the r.h.s. we show possible choices of explicit matrix generators of the respective groups, see Eq. \ref{['eq:3HDM_generators']}. Dotted arrows here indicate that the explicitly stated generators need to be basis-transformed to make the subgroup relation manifest.