Topological superconductors might provide particle physicists new ideas to provide mass to fermions

A Kitaev-Wen mechanism to give mass to mirror fermions in a force and family unified model?

It was recently argued by X. G. Wen that the Spin(10) unified theory can be regularized on a 3d spatial lattice with continuous time [6]. The low energy limit of lattice gauge theory is necessarily non-chiral [7], so the continuum fermion fields that emerge from a Spin(10) lattice gauge theory must transform as the reducible representation 16⁺⊕16^-. The conclusion of Wen’s paper implies that the 16^- mirror fermions must have somehow obtained mass and decoupled from the low energy theory without breaking Spin(10) and without giving mass to the gauge bosons... 

The same type of argument was independently proposed by A. Kitaev to show that the free-fermion classification of 3He-B reduces under interactions from Z to Z₁₆ [10]. This means that 16 copies of topological superconductor can be smoothly deformed into an ordinary superconductor without going through a bulk phase transition and without breaking time reversal invariance. (This conclusion was later verified by a different approach [13].) This means the protected gapless (2+1)-dimensional edge modes decouple from the low energy theory even though time reversal symmetry forbids all mass terms in the Lagrangian. This also means that the (3 + 1)-dimensional bulk theory can be tuned through the point m=0 without closing the bulk mass gap. Lattice simulations supporting these types of arguments have also appeared recently [14, 15].[After the first draft of our paper was posted, additional numerical work appeared to further support the idea that this transition can be described by a continuum field theory [16].] 

Therefore, from these recent developments in condensed matter theory, we learn that in very special cases, one of which serendipitously happens to be the Standard Model (SM), it is possible for the fermion single particle spectrum to obtain an interaction-induced energy gap without any explicit fermion mass term in the Lagrangian. We will refer to this argument for “mass without mass terms” as the Kitaev-Wen mechanism. We emphasize again that in this approach, in contrast to the Higgs mechanism, the electroweak gauge symmetry remains unbroken and the gauge bosons remain massless.

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 We have intentionally emphasized only the situation in which none of the extra fermions have any influence below the usual scale MGUT∼10¹⁶ GeV, but this was just the simplest choice. We invite the interested reader to re-evaluate the possible importance of the additional states for TeV-scale physics.

Given the phenomenological success of the Higgs mechanism in particle physics, one could ask whether the Kitaev-Wen mechanism could also do the job without spontaneous symmetry breaking. (We have already explained in Sec. II.2 that this mechanism will not give mass to the gauge bosons, so this question pertains only to the fermion mass.) The physical intuition from Sec. II.3 implies that a fermion whose mass comes from the Kitaev-Wen mechanism has a propagator of the form

below the multiparticle threshold. This expresses the physical distinction between a fermion mass obtained from the Higgs mechanism and one obtained from the Kitaev-Wen mechanism. It is still unclear what the full phenomenological implications would be for an alternative version of particle physics based on this mechanism for generating fermion masses.

The Origin of Families and SO(18) Grand Unification

Yoni BenTov, A. Zee

(Submitted on 16 May 2015 (v1), last revised 4 Feb 2016 (this version, v2))

A realistic group which unifies gravity with gauge interactions and contains the Standard Model is SO(1,13) in a fourteen dimensional tangent space. It corresponds to SO(10) grand unified theory concerning the gauge fields content, however, it has double the number of fermions in the form of 16⁺⊕16^-. It is not easy to decouple the mirror fermions by giving them very heavy masses via Brout-Englert-Higgs mechanism. Instead, this could be done by appealing to a mechanism used for topological superconductors, where the 16^- could be made very heavy. One can go further and unify the three families by considering SO(1,21) instead of SO(1,13) with SO(18) grand unification group as argued in [5]. Since the Dirac operator plays a fundamental role in this setting, it is natural to look for connections between this construction and that of noncommutative geometry. In addition, the need to add Higgs scalar fields suggests that a total unification of gravity, gauge and Higgs fields within one geometrical setting, should be possible by replacing the continuous four-dimensional manifold by a noncommutative space which has both discrete and continuous structures [9]. 

On Unification of Gravity and Gauge Interactions

Ali H. Chamseddine, Viatcheslav Mukhanov

(Submitted on 6 Feb 2016 (v1), last revised 9 May 2016 (this version, v3))