The major appeal of the more traditional approaches to beyond the standard model building such as supersymmetry
(as well as technicolor, extra dimensions, etc.) is that they address the hierarchy
problem, and that they shed light on the apparent unification of couplings, both within the
context of local effective field theory (EFT). However, this apparent theoretical appeal of supersymmetry does not exclude approaches
that do not necessarily follow the local EFT paradigm. For example, in the Spectral Standard Model approach of Connes et al. [147, 148, 149, 152–156, 189] the hierarchy problem can be addressed
in a completely different fashion [154]. The crucial noncommutative geometric (and thus in some sense non-local)
aspect of the SM is found in the Higgs sector, which in principle comes with an extra (second)
scale, to be distinguished from the usual UV scale of local EFT. The hierarchy between the
Higgs and the UV (Planck) scale can be associated (as shown by Chamseddine and Connes
in Ref. [154]) with the natural exponential factor that comes from the dynamical discrete
geometry of the Higgs sector. Similarly, the apparent gauge unification (in the guise of an
effective SO(10) relation between the gauge couplings) is also incorporated into the Spectral
SM. These aspects of the NCG approach to the SM are almost completely unknown in
the particle physics community, and at the moment, almost completely undeveloped from a
phenomenological viewpoint.
One of our aims in our upcoming review of the Spectral SM [162] is to clarify these interesting
features of the NCG approach to the SM and make them palatable to the wider
phenomenological community. We are also motivated by a deeper need to understand the
limitations of the local EFT paradigm from the point of view of the physics of quantum
gravity, which is usually, rather naively, ignored at the currently interesting particle physics
scales, by invoking the concept of decoupling, which represents another central feature of
the local EFT and which is also challenged by the NCG approach to the SM. Finally, as we
discuss in the next concluding subsection of this paper, the usual RG analysis of the local
EFT should be re-examined in the new light of the non-commutative/non-local structure of
the SM, and the apparent existence of two natural (and naturally related) physics scales.
One of the most
interesting aspects of the NCG of the SM and its Pati-Salam-like completion is the existence
of the GUT scale which can be found in the close proximity to the Planck scale, i.e., the scale
of quantum gravity. Given this fact as well as the presence of a hidden fundamental noncommutative
structure in this approach, this suggests that the hierarchy problem should get
a quantum gravitational rather than an effective field theory treatment. The more convincing
physical meaning of this GUT scale also comes after one realizes that Connes’ approach also
produces a gravity sector in parallel with the standard model (and its Pati-Salam completion)
and thus the GUT scale should be viewed as being close to the natural scale of gravity, i.e.,
the Planck scale, and indeed the two scales are not that far apart in the non-commutative
approach...
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