Global anomalies in the Standard Model(s) and beyond

Abstract

We analyse global anomalies and related constraints in the Standard Model (SM) and various Beyond the Standard Model (BSM) theories. We begin by considering four distinct, but equally valid, versions of the SM, in which the gauge group is taken to be G = G SM /Γ n , with G SM = SU(3) × SU(2) × U(1) and Γ n isomorphic to Z/n where n ∈ {1, 2, 3, 6}. In addition to deriving constraints on the hypercharges of fields transforming in arbitrary representations of the SU(3) × SU(2) factor, we study the possibility of global anomalies in theories with these gauge groups by computing the bordism groups Ω Spin 5 (BG) using the Atiyah-Hirzebruch spectral sequence. In two cases we show that there are no global anomalies beyond the Witten anomaly, while in the other cases we show that there are no global anomalies at all, illustrating the subtle interplay between local and global anomalies. While freedom from global anomalies has been previously shown for the specific fermion content of the SM by embedding the SM in an anomaly-free SU(5) GUT, our results here remain true when the SM fermion content is extended arbitrarily. Going beyond the SM gauge groups, we show that there are no new global anomalies in extensions of the (usual) SM gauge group by U(1) m for any integer m, which correspond to phenomenologically well-motivated BSM theories featuring multiple Z bosons. Nor do we find any new global anomalies in various grand unified theories, including Pati-Salam and trinification models. We also consider global anomalies in a family of theories with gauge group SU(N) × Sp(M) × U(1), which share the phase structure of the SM for certain (N, M). Lastly, we discuss a BSM theory in which the SM fermions are defined using a spin c structure, for example by gauging B − L. Such a theory may be extended to all orientable four-manifolds, and we find no global anomalies.

The algebra of anomaly interplay

“…For our next example, we move up two dimensions and revisit the anomaly interplay between U(2) and SU(2) gauge theories in d = 4, each defined using a spin structure [16] (see also [47]). The map π : SU(2) → U(2) will denote the usual embedding of SU(2) ⊂ U(2) as the subgroup of 2-by-2 unitary matrices that have determinant one.…”

Topological superconductors on superstring worldsheets

“…There we have used a slightly more general notation, with Ω X d (Y ) representing the bordism group of d-dimensional manifolds with X structure equipped with a map to Y . Then for example a structure X consisting of spin structure and a 2 gauge field can equivalently be thought of as having (X , Y ) = (spin structure, B 2 ), 3 For other recent applications of bordism groups to high-energy theory, see [18][19][20][21][22][23][24][25][26]. 4 More precisely, d X as defined here classifies invertible phases whose partition functions do not depend continuously on the background fields.…”

“…This review will not be comprehensive -for more thorough introductions to the AHSS, the reader can consult e.g. [18,21,25,82]. The basic ingredients in the AHSS are the E 2 page and a set of differentials.…”

Some torsion classes in the Chow ring and cohomology of BPGLn

“…This suggests that the could be of use in the study of the (algebraic) period‐index problem due to Colliot‐Thélène [7]. The cohomology ring also appears in the study of anomalies in particle physics such as [8, 10, 15]. Antieau [3] and Kameko [24] considered the cohomology of for some finite cover of to construct counterexamples for the integral Tate conjecture.…”