BRST and Superfield Formalism—A Review

This article, which is a review with substantial original material, is meant to offer a comprehensive description of the superfield representations of BRST and anti-BRST algebras and their applications to some field-theoretic topics. After a review of the superfield formalism for gauge theories, we present the same formalism for gerbes and diffeomorphism invariant theories. The application to diffeomorphisms leads, in particular, to a horizontal Riemannian geometry in the superspace. We then illustrate the application to the description of consistent gauge anomalies and Wess–Zumino terms for which the formalism seems to be particularly tailor-made. The next subject covered is the higher spin YM-like theories and their anomalies. Finally, we show that the BRST superfield formalism applies as well to the N=1 super-YM theories formulated in the supersymmetric superspace, for the two formalisms go along with each other very well.

Anomalous Z′ bosons for anomalous B decays

Motivated by the intriguing discrepancies in b → sℓℓ transitions, the fermion mass problem, and a desire to preserve the accidental symmetries of the Standard Model (SM), we extend the SM by an anomalous U(1)X gauge symmetry where X = Y3 + a(Lμ− Lτ)/6. The heavy Z′ boson associated with spontaneously breaking U(1)X at the TeV scale mediates the b → sℓℓ anomalies via $$ {\mathcal{O}}_9^{\mu}\sim \frac{1}{\Lambda^2}\left(\overline{s}{\gamma}_{\rho }{P}_Lb\right)\left(\overline{\mu}{\gamma}^{\rho}\mu \right) $$ O 9 μ ~ 1 Λ 2 s ¯ γ ρ P L b μ ¯ γ ρ μ . We show that this model, which features mixed gauge anomalies involving U(1)X and hypercharge, can be made anomaly-free for any a ∈ ℤ by integrating in a pair of charged fermions whose masses naturally reside somewhere between 1 and 30 TeV. The gauge symmetry permits only the third family Yukawas at the renormalisable level, and so the light quark masses and mixings are controlled by accidental U(2)3 flavour symmetries which we assume are minimally broken alongside U(1)X. The lepton sector is not governed by U(2) symmetries, but rather one expects a nearly diagonal charged lepton Yukawa with me,μ « mτ. The model does not explain the hierarchy me « mμ, but it does possess high quality lepton flavour symmetries that are robust to the heavy physics responsible for generating me,μ. We establish the viability of these models by checking agreement with the most important experimental constraints. We comment on how the model could also explain neutrino masses and the muon g − 2.

A new anomaly-free flipped 341 model

A new flipped [Formula: see text] model without exotic electric charges is proposed. All the quark families are arranged in the same representation while lepton generations are in different representations leading to a tree level FCNC. Moreover, it is shown that the cancellation of the triangle gauge anomalies requires new additional leptons a 10-plet and a quadruplet. All fermion masses have been also discussed. Furthermore, using the most recent experimental data of the branching ratios of [Formula: see text] and [Formula: see text] rare decay modes, stringent bounds on the heavy neutral bosons masses and the muon-electron mixing matrix element are obtained.

Complete unification in 10d E8

A grand unification scenario is presented that is based on quantum field theory, and where a single [Formula: see text] gauge superfield in 10 dimensions is used to obtain all the particle content of the Standard Model at low energies. The key feature of the formulation lies in the dimensional reduction used to break the gauge symmetry and to determine the low energy spectrum. It is shown that, through the orbifold [Formula: see text], and its corresponding Wilson lines, the symmetry is broken to the Standard Model one, generating a particular model that includes the Minimal Supersymmetric Standard Model spectrum. Furthermore it is also shown that the model is free of gauge anomalies at all levels by itself, i.e. without the need to include any additional representations of fields. Thus a complete unification of the Standard Model into a single gauge superfield is shown to be formally plausible. Although this paper does not include a phenomenological study of the specific model (currently being investigated), some interesting questions and observations are included as motivation for the scenario.

Omega vs. pi, and 6d anomaly cancellation

In this note we review the role of homotopy groups in determining non-perturbative (henceforth ‘global’) gauge anomalies, in light of recent progress understanding global anomalies using bordism. We explain why non-vanishing of πd(G) is neither a necessary nor a sufficient condition for there being a possible global anomaly in a d-dimensional chiral gauge theory with gauge group G. To showcase the failure of sufficiency, we revisit ‘global anomalies’ that have been previously studied in 6d gauge theories with G = SU(2), SU(3), or G2. Even though π6(G) ≠ 0, the bordism groups $$ {\Omega}_7^{\mathrm{Spin}}(BG) $$ Ω 7 Spin BG vanish in all three cases, implying there are no global anomalies. In the case of G = SU(2) we carefully scrutinize the role of homotopy, and explain why any 7-dimensional mapping torus must be trivial from the bordism perspective. In all these 6d examples, the conditions previously thought to be necessary for global anomaly cancellation are in fact necessary conditions for the local anomalies to vanish.

Comments on gauge anomalies at dimension-six in the Standard Model Effective Field Theory

We study whether higher-dimensional operators in effective field theories, in particular in the Standard Model Effective Field Theory (SMEFT), can source gauge anomalies via the modification of the interactions involved in triangle diagrams. We find no evidence of such gauge anomalies at the level of dimension-6 operators that can therefore be chosen independently to each others without spoiling the consistency of SMEFT, at variance with recent claims. The underlying reason is that gauge-invariant combinations of Goldstone bosons and massive gauge fields are allowed to couple to matter currents which are not conserved. We show this in a toy model by computing the relevant triangle diagrams, as well as by working out Wess-Zumino terms in the bosonic EFT below all fermion masses. The same approach applies directly to the Standard Model both at the renormalisable level, providing a convenient and unusual way to check that the SM is anomaly free, as well as at the non-renormalisable level in SMEFT.

A note on gauge anomaly cancellation in effective field theories

The conditions for the absence of gauge anomalies in effective field theories (EFT) are rivisited. General results from the cohomology of the BRST operator do not prevent potential anomalies arising from the non-renormalizable sector, when the gauge group is not semi-simple, like in the Standard Model EFT (SMEFT). By considering a simple explicit model that mimics the SMEFT properties, we compute the anomaly in the regularized theory, including a complete set of dimension six operators. We show that the dependence of the anomaly on the non-renormalizable part can be removed by adding a local counterterm to the theory. As a result the condition for gauge anomaly cancellation is completely controlled by the charge assignment of the fermion sector, as in the renormalizable theory.

Electric dipole moments, new forces and dark matter

New sources of CP violation beyond the Standard Model are crucial to explain the baryon asymmetry in the Universe. We discuss the impact of new CP violating interactions in theories where a dark matter candidate is predicted by the cancellation of gauge anomalies. In these theories, the constraint on the dark matter relic density implies an upper bound on the new symmetry breaking scale from which all new states acquire their masses. We investigate in detail the predictions for electric dipole moments and show that if the relevant CP-violating phase is large, experiments such as the ACME collaboration will be able to fully probe the theory.

Some comments on 6d global gauge anomalies

Global gauge anomalies in 6d associated with non-trivial homotopy groups π6(G) for G = SU(2), SU(3), and G2 were computed and utilized in the past. In the modern bordism point of view of anomalies, however, they come from the bordism groups Ω7spin (BG), which are in fact trivial and therefore preclude their existence. Instead, it was noticed that a proper treatment of the 6d Green-Schwarz mechanism reproduces the same anomaly cancellation conditions derived from π6(G). In this paper, we revisit and clarify the relation between these two different approaches.