Reviving Kaluza-Klein: SM Bosons & Fermions

[mitchell porter]

https://arxiv.org/abs/2105.02899
Higher-dimensional routes to the Standard Model bosons
Joao Baptista
[Submitted on 6 May 2021]

In the old spirit of Kaluza-Klein, we consider a spacetime of the form P = M₄×K, where K is the Lie group SU(3) equipped with a left-invariant metric that is not fully right-invariant. This metric has a U(1)×SU(3) isometry group, corresponding to the massless gauge bosons, and depends on a parameter ϕ with values in a subspace of su(3) isomorphic to C². It is shown that the classical Einstein-Hilbert Lagrangian density R_P−2Λ on the higher-dimensional manifold P, after integration over K, encodes not only the Yang-Mills terms of the Standard Model over M₄, as in the usual Kaluza-Klein calculation, but also a kinetic term |d^Aϕ|² identical to the covariant derivative of the Higgs field.

For Λ in an appropriate range, it also encodes a potential V(|ϕ|²) having absolute minima with |ϕ₀|² ≠ 0, thereby inducing mass terms for the remaining gauge bosons. The classical masses of the resulting Higgs-like and gauge bosons are explicitly calculated as functions of the vacuum value |ϕ₀|² in the simplest version of the model. In more general versions, the values of the strong and electroweak gauge coupling constants are given as functions of the parameters of the left-invariant metric on K.

https://arxiv.org/abs/2105.02901
Higher-dimensional routes to the Standard Model fermions
Joao Baptista
[Submitted on 6 May 2021]

In the old spirit of Kaluza-Klein, we consider a spacetime of the form P = M₄×K, where K is the Lie group SU(3) equipped with a left-invariant metric that is not fully right-invariant. We observe that a complete generation of fermionic fields can be encoded in the 64 components of a single spinor over the 12-dimensional spacetime. The behaviour of the spinorial function along the internal space K can be chosen so that, after pairing and fibre-integration over K, the resulting Dirac kinetic terms in four dimensions couple to the u(1)⊕su(2)⊕su(3) gauge fields in the exact chiral representations present in the Standard Model. Although we describe the action of the internal Dirac operator on the 12-dimensional spinor, the full calculation of the fermionic mass terms produced by the model is longer and is not carried out here. We calculate instead the action of the internal Laplace operator on the spinor components.

 

[MathematicalPhysicist]

It seems that people ran out of new ideas...

 

[mitchell porter]

It seems that people ran out of new ideas...

It's more that the "old" ideas are still far from completely explored.

The apex of theoretical orthodoxy is that we are living somewhere in the landscape of solutions to string theory. This idea makes a great deal of sense, and actually brings more order to post-1970s theory than one had reason to expect. Theorists went from quantum fields, to quantum fields in extra dimensions, to strings and branes in extra dimensions. One might have expected that this exploration of possible worlds was simply divergent, bringing to light increasingly diverse possibilities, but strings turned out to also imply a great convergence, in the sense that all the string theories turned out to be aspects of a single theory, whose potentialities encompass most of the ideas that try to generalize the successes of the standard model.

If we were being systematic in considering the theoretical vistas opened up since the 1970s, we might say: The main thing is to search the string landscape for the physics of our world. That's a very big task, akin to the classification of finite simple groups, but it is a logical goal. Similarly, and in parallel, one may wish to explore all the possibilities arising at earlier stages - all possible Kaluza-Klein theories, all possible grand unified theories (i.e. gauge theories with a simple symmetry group). And one may do this with an eye on constraints implied by string theory, or even in defiance of them, just in case string theory isn't the last word after all.

It's common in mathematical practice, that mathematicians will conceive of a class of objects, and then will be interested in whether a particular possible property is possessed by any member of that class. That's all I'm talking about, except that we're exploring classes of physical theories, and the property of interest is, matching experiment. And so what I'm saying is that there are theoretical ideas, that are by now many decades old, and which are part of that convergent synthesis, whose possible forms are still being discovered. They've been thoroughly studied, and certain useful rules of thumb distilled from that study, but they have not been exhaustively studied. There's still opportunities for people to come up with new variations on old themes.

These papers caught my eye, because I thought they might be a realization of one such idea due to @arivero, namely, to look for a Kaluza-Klein realization of the low-energy limit of the standard model in which one has Dirac fermions interacting via U(1) x SU(3). (The full standard model has Weyl fermions interacting via U(1) x SU(2) x SU(3), paired into Dirac fermions via Higgs yukawa interactions.) Unfortunately, the actual model here seems to be Dirac fermions interacting via SU(3) x SU(3), with one of the SU(3)s reduced to U(1) x SU(2) by an unexplained constraint on the metric of the extra dimensions. That makes it less interesting to me, but it still might be worthwhile to understand: what are the new ideas here, what is their relationship to the old wisdom regarding Kaluza-Klein, and what are the physical prospects of the model.

 

[MathematicalPhysicist]

Is there really an end to how much you can theorize in physics if the universe is infinite?
It's obviously a philosophical question, but every researcher is bound to think of it.

 

[MathematicalPhysicist]

I should have said a limit not an end, but the meaning isn't lost.