A tentative noncommutative geometric superconnection approach for beyond Standard Model phenomenology
...we have investigated the possibility of reviving the superconnection formalism ﬁrst discussed in 1979 by Ne’eman , Fairlie [5, 6], and others [7,8,9]. The original observation of Ne’eman was that the SU(2)L×U(1)Y gauge ﬁelds and the Higgs doublet in the SM could be embedded into a single su(2/1) superconnection [10, 11] with the SU(2)L×U(1)Y gauge ﬁelds constituting the even part of the superconnection and the Higgs doublet φ constituting the odd part... This embedding predicts sin2 θW = 1/4 as well as the Higgs quartic coupling, the latter leading to a prediction of the Higgs mass [12,13,14]. The leptons and quarks could also be embedded into irreducible representations of su(2/1) [15,16,17,18], thereby ﬁxing their electroweak quantum numbers in a natural fashion. Subsequently, suggestions have been made to incorporate QCD into the formalism by extending the superalgebra to su(5/1) [20,21,22]. Though the appearance of the su(2/1) superconnection suggested an underlying ‘internal’ SU(2/1) supersymmetry, gauging this supersymmetry to obtain the superconnection proved problematic as discussed in Refs. [23, 24]... Due to these, and various other problems, interest in the approach waned.
(Submitted on 26 Sep 2014)
It was subsequently recognized, however, that the appearance of a superconnection does not necessarily require the involvement of the familiar boson↔fermion supersymmetry. This development follows the 1990 paper of Connes and Lott  who constructed a new description of the SM using the framework of noncommutative geometry (NCG) in which the Higgs doublet appears as part of the Yang-Mills ﬁeld (i.e. connection) in a space-time with a modiﬁed geometry. The full Yang-Mills ﬁeld in this approach was described by a superconnection, the off-diagonal elements of which were required to be bosonic. The NGC-superconnection approach to the SM was studied by many authors and a vast literature on the subject exists, e.g. Refs. [30–55] to give just a representative list. Though these works differ from each other in detail, the basic premise is the same. The models are all of the Kaluza-Klein type in which the extra dimension is discrete and consists of only two points. In other words, the model spacetime consists of two 3 + 1 dimensional ‘branes.’ In such a setup, the connection must be generalized to connect not just points within each brane, but also to bridge the gap between the two. If the left-handed fermions live on one brane and the right-handed fermions on the other, then the connections within each brane, i.e. the even part of the superconnection, will involve the usual SM gauge ﬁelds which couple to fermions of that chirality. In contrast, the connection across the gap, i.e. the odd part of the superconnection, connects fermions of opposite chirality and can be identiﬁed with the Higgs doublet. In this approach, both the even and odd parts of the superconnection are bosonic, the Z2-grading of the super- algebra resulting not from fermionic degrees of freedom but from the existence of the two ‘branes’ (on which the chirality γ5 provides the Z2 grading operator), and the deﬁnition of the generalized exterior derivative d in the discrete direction. That is, the superconnection emerges from the ‘geometry’ of the discrete extra dimension...
The extra-discrete-dimension interpretation of the superconnection model also solves the problem that the prediction sin2θW=1/4 is not stable under renormalization group running and can only be imposed at one scale [64, 65]. That scale can be interpreted as the scale at which the SM with sin2θW=1/4 emerges from the underlying discrete extra dimension model... Given the current experimental knowledge of the SM, this scale turns out to be ∼ 4 TeV , suggesting a phenomenology that could potentially be explored at the LHC, as well as the existence of a new fundamental scale of nature at those energies... These developments notwithstanding, a deﬁnitive recipe for constructing a NCG Kaluza-Klein model for a given algebra still seems to be in the works. Different authors use different deﬁnitions of the exterior derivative d, which, naturally, lead to different Higgs sectors and different predictions. In the Spectral SM of Connes et al. [59–63], for instance, the prediction for the U(1)×SU(2)×SU(3) gauge couplings are of the SO(10) GUT type, pushing up the scale of emergence to the GUT scale. The Spectral SM is not particularly predictive either: the fermionic masses and mixings must all be put in by hand into the operator D. Thus, the NCG- superconnection approach still has much to be desired and further development is called for.
The su(2/1) superconnection also predicts the Higgs quartic coupling at that scale, from which in turn one can predict the Higgs boson mass to be ∼170 GeV. As discussed in Ref. , lowering this prediction down to ∼126 GeV requires the introduction of extra scalar degrees of freedom which modify the renormalization group equations (RGE) of the Higgs couplings. Those degrees of freedom would be available, for instance, if the su(2/1) superconnection were extended to su(2/2). The extra-discrete-dimensional su(2/2) model shares the same prediction for sin2θW as the su(2/1) version, and therefore the same scale (∼4 TeV) at which an effective SU(2)L×SU(2)R×U(1)B−L gauge theory can be expected to emerge. Thus, explaining the Higgs mass within the NCG-superconnection formalism seems to demand an extension of the SM gauge group.
Curiously, Connes et al.’s Spectral SM with a GUT emergence scale also predicts the Higgs mass to be ∼170 GeV. Lowering this to ∼126 GeV requires the introduction of extra scalar degrees of freedom as discussed above [62, 63]. See also Refs. [66, 67]. Here too, the Higgs mass seems to suggest that the SM gauge group needs to be extended to SU(2)L×SU(2)R×U(1)B−L, or including the QCD sector, to SU(2)L×SU(2)R×SU(4).
Thus, the NCG-superconnection formalism already requires the extension of the SM gauge group to that of the left-right symmetric model (LRSM), or that of Pati-Salam...new gauge bosons?
Based on ... conventional phenomenological analyses, we conclude that the TeV scale LRSM predicted by the su(2/2) superconnection formalism, possibly with an underlying NCG, provides a wealth of new particles and predictions within reach of LHC and other experiments. The fact that the current experimental bounds on the LRSM and the corresponding predictions of the superconnection formalism are suspiciously close may be a sign that LHC is on the brink of discovering something new and exciting. With the center of mass energy of √s = 13 TeV for its second run, the LHC is well capable of observing the new particles of the model among which the most important are the right-handed gauge bosons (W±R, Z0) whose masses are ﬁxed by the formalism and range within the TeV scale. With the scale of 4 TeV, selected by the formalism itself, these masses will be within reach of the LHC, provided that the right handed neutrinos (NR) are light enough to make the corresponding channels accessible [There is nothing in the model which constraints the right-handed neutrinos NR to be light. With NR heavier than W±R , the Drell-Yan interactions will be highly suppressed and thus, although theoretically the TeV scale LRSM could still be viable, it will be very diﬃcult for the LHC to detect its signature through the W±R channels].
A number of relevant and important observations could be delivered in the lepton flavor violation branch as well, especially in µ→e conversion in nuclei, which we brieﬂy discussed in an earlier section. With the next generation of machines, COMET  and Mu2e  collaborations target to increase their sensitivity for this process from 10-13 to 10-17, which will signiﬁcantly improve the limits on new physics including LRSM. Moreover, the next generation of super B factories aim to increase the limit on LFV τ decays to a level of 10-9 [164, 165], which will also provide useful information on the nature of new physics.
UV/IR mixing : a conjectural quantum (or noncommutative spectral geometry/ field theory and strings) syncretism (?)
... we would like to comment on the observation made in Ref.  regarding the violation of decoupling in the Higgs sector, and how this violation may point to the more fundamental possibility of mixing of UV and IR degrees of freedom, given our view that a NCG underlies the Higgs sector. Such UV/IR mixing is known in the simpler context of non-commutative ﬁeld theory [169a,169b]...
At the moment we are not aware of an explicit UV/IR relation in the context of the NCG of Connes that underlies the superconnection formalism and the new view- point on the SM and the physics beyond it, as advocated in this paper. However, there exists a very speciﬁc toy model of non-commutative ﬁeld theory in which such UV/IR mixing has been explicitly demonstrated. The nice feature of this toy model is that it can be realized in a fundamental short distance theory, such as string theory [169a,169b]. The UV/IR mixing, characteristic of this type of non-commutative ﬁeld theory leads to the question of the existence of the proper continuum limit for non-commutative ﬁeld theory. This question can be examined from the point of view of non-perturbative Renormalization Group (RG). The proper Wilsonian analysis of this type of non-commutative theory has been done in Ref. . The UV/IR mixing leads to a new kind of the RG ﬂow: a double RG ﬂow, in which one ﬂows from the UV to IR and the IR to the UV and ends up, generically, at a self-dual ﬁxed point. It would be tantalizing if the NCG set- up associated with the SM, and in particular, the LRSM generalization discussed in this paper, would lead to the phenomenon of the UV/IR mixing and the double RG ﬂow with a self-dual ﬁxed point. Finally, we remark that it has been argued in a recent work on quantum gravity and string theory that such UV/IR mixing and the double RG might be a generic feature of quantum gravity coupled to matter [171, 172].
... we might reasonably expect that the the Higgs scale is mixed with the UV cut-off deﬁned by some more fundamental theory. Needless to say, at the moment this is only an exciting conjecture.
If this conjecture is true, given the results presented in this paper one could expect that the appearance of the LRSM degrees of freedom (as well as the embedded SM degrees of freedom) at low energy is essentially a direct manifestation of some eﬀective UV/IR mixing, and thus that on one hand the remnants of the UV physics can be expected at a low energy scale of 4 TeV, and conversely that the LRSM structure point to some unique features of the high energy physics of quantum gravity. In this context we recall the observations made in Ref.  about the special nature of the Pati-Salam model, which uniﬁes the LRSM with QCD, in certain constructions of string vacua. Even though this observation is mainly based on “groupology” and it is not deeply understood, this observation might be indicative that the Pati-Salam model is the natural completion of the SM, as suggested in this paper, in which the infrared physics associated with the Higgs sector is mixed with the ultraviolet physics of some more fundamental physics, such as string theory.