Clifford-based spectral action and renormalization group analysis of the gauge couplings

Abstract

The Spectral Action Principle in noncommutative geometry derives the actions of the Standard Model and General Relativity (along with several other gravitational terms) by reconciling them in a geometric setting, and hence offers an explanation for their common origin. However, one of the requirements in the minimal formalism, unification of the gauge coupling constants, is not satisfied, since the basic construction does not introduce anything new that can change the renormalization group (RG) running of the Standard Model. On the other hand, it has been recently argued that incorporating structure of the Clifford algebra into the finite part of the spectral triple, the main object that encodes the complete information of a noncommutative space, gives rise to five additional scalar fields in the basic framework. We investigate whether these scalars can help to achieve unification. We perform a RG analysis at the one-loop level, allowing possible mass values of these scalars to float from the electroweak scale to the putative unification scale. We show that out of twenty configurations of mass hierarchy in total, there does not exist even a single case that can lead to unification. In consequence, we confirm that the spectral action formalism requires a model-construction scheme beyond the (modified) minimal framework.2 Unification in the Pati-Salam models based on NCG can be realized in a variety of ways, as displayed in Refs. [21][22][23][24].3 The hypercharge normalization adopted in this paper is given as Q em = I L 3 + Y . 4 Therefore, these fields, despite of the fact that they carry right quantum numbers to be called leptoquarks, are in fact by definition not leptoquarks in this formalism. Note that the quantum number assignment alone does not necessarily guarantee that the scalars couple to lepton-quark pairs as leptoquarks by definition should do. Thus, these scalars are not relevant to discussion of scalar leptoquarks in relation to the ongoing LHC searches, for instance in the context of reported B-decay anomalies [27][28][29][30][31][32][33][34][35][36], unlike the case for the scalar leptoquarks in the Pati-Salam models from NCG [16] which can indeed be relevant to these anomalies [24].

A survey of spectral models of gravity coupled to matter

“…Imposing the hermiticity Λ µ = Λ * µ implies Λ µ ∈ R, and also automatically yields Λ µ = −Λ µ . Furthermore, Q µ = Q * µ implies that Q µ is a real-linear combination of the Pauli matrices, which span i su (2). Finally, the condition that V µ be hermitian yields V µ ∈ i u(3), so V µ is a U (3) gauge field.…”

“…This gives rise to a model Beyond the Standard Model: an analysis of the phenomenological consequences is performed in [59,37]. In [2] it was then found that this model does not exhibit grand unification of the Standard Model couplings.…”

Spectral Noncommutative Geometry Standard Model and all that

“…The extra terms are compatible with known physics. A complete phenomenological analysis has not been yet performed, but their role for the renormalization flow is studied in [13].…”

“…where χ is a cutoff function, and Λ is the cutoff scale. The former is assumed to be an arbitrary function such that the trace is well defined 13 . A natural choice of this function is the characteristic function of the unit interval or its smooth approximation.…”

“…From now on Tr stands for the functional trace on L 2 13. Recall that here we assume spacetime, to be compact, Euclidean and without a boundary.…”