Un, deux voire trois diamants de plus dans la poche du Mendiant* / One, two or three more diamonds in the pocket of the Beggar
In our previous paper , we found a reformulation of non-commutative geometry (NCG) that simpliﬁed and uniﬁed the mathematical axioms while, at the same time, resolving a problem with the NCG construction of the standard model Lagrangian, by precisely eliminating 7 terms which had previously been problematic. In this paper, we show that this same reformulation leads to a new perspective on the gauge symmetries associated to a given NCG, uncovering some that were previously missed. In particular, when we apply our formalism to the NCG traditionally used to describe the standard model of particle physics, we ﬁnd a new U(1)B-L gauge symmetry (and, correspondingly, a new B−L gauge boson [Cµ ]). This, in turn, implies the existence of a new complex Higgs ﬁeld σ that is a singlet under SU(3)C×SU(2)L×U(1)Y but transforms with charge +2 under U(1)B-L, allowing it to form a majorana-like yukawa coupling σνRνR with two right-handed neutrinos (so that, if it obtains a large VEV, it induces see-saw masses for the neutrinos). It is striking that, on the one hand, this precise extension of the standard model has been previously considered in the literature [22, 23] on the basis of its cosmological advantages; and, on the other hand, that the new ﬁeld σ can resolve a previous discrepancy between the observed Higgs mass and the NCG prediction [18, 19, 21]. Note that in the previous works [18, 19, 21] which introduced the σ ﬁeld for this purpose, it was a real ﬁeld, and a gauge singlet. By contrast, from the perspective presented here, the fact that σ is complex, and transforms under U(1)B-L, is the key to its existence: had it been real, it would not have been induced by the covariance argument of the previous section
Beggar = Standard Model in the spectral and almost commutative framework
The ordinary standard model makes sense, with or without the right-handed neutrino. In particular, since the right-handed neutrino is a singlet under the standard model gauge group, it does not contribute to anomaly cancellation: in the standard model, the anomalies cancel whether or not the right-handed neutrino exists. The right-handed neutrino has recently been tacked on to the standard model in order to account for the observed neutrino oscillations – but not for any independent theoretical reason. It is important to emphasize that the situation is very different in a Minimal Dimensionless Standard Model (MDSM) [that would consist of a standard model plus 3 right-handed neutrinos with the additional constraint of classical conformal invariance to eliminate all dimensionful couplings, augmented by a new U(1)B-L gauge symmetry carried by a new gauge boson Cµ and a complex scalar field, singlet under SU(3)C×SU(2)L×U(1)Y that couples to the new symmetry with charge +2]...
In this case, the right-handed neutrino is charged under U(1)B-L, and thus plays an essential role in the anomaly cancellation [requirement for renormalizability]... Relative to the standard model, this [MDSM] contains two new ﬁelds: the complex scalar ﬁeld [σ] and the (B−L) gauge ﬁeld Cµ. In this subsection, we would like to stress the elegantly economical way in which [σ] simultaneously accomplishes several important phenomenological tasks in this model. On the one hand, its properties – i.e. its SU(3)C×SU(2)L×U(1)Y×U(1)B-L charges ... – were determined by requirement that it should be able to provide a Majorana-like Yukawa coupling to right-handed neutrinos, in order to give a see-saw mechanism for neutrino mass.
On the other hand, these same properties are precisely what is needed in order for [σ] to perform another crucial task: spontaneously break (B−L) symmetry, and give mass to the Cµ boson via the Higgs mechanism, without leaving any additional unwanted Goldstone bosons.
At the same time, we have seen that [σ] expands the scalar sector in a way that leads to successful electroweak symmetry breaking via the Coleman-Weinberg mechanism, an inﬂaton candidate, a dark matter candidate, and a possible explanation for the cosmological matter/anti-matter asymmetry.
Finally, this model contains 4 diﬀerent quantities that all must be rather large, for very diﬀerent phenomenological/cosmological reasons: (i) the mass of the Z′µ boson, (ii) the masses of the heavy neutrinos; (iii) the mass of the [σ] boson, and (iv) the inﬂaton VEV. And yet, in [MDSM], all 4 large dimensionful quantities have a common origin in the large VEV of the ﬁeld [σ].
(Submitted on 1 Nov 2011 (v1), last revised 12 Mar 2013 (this version, v2)).