It may be useful in higher N versions of supersymmetry (SUSY) and supergravity (SUGRA) as a foundation for "within the Standard Model" theory that demonstrates deeper relationships between the components of the Standard Model, particularly because these theories are exceptions to an important "no-go" theorem in theoretical physics called the Coleman-Mandula no go theorem.

These goals are more worthwhile than exploring SUSY's best known crude N=1 form which is contrary to experiment, is baroque, and has been explored so heavily due to mathematical laziness, in order to explain the hierarchy non-problem, and in the interest of "naturalness."

**Background: What is the Coleman-Mandula no-go theorem?**

Basically, the Coleman-Mandula no-go theorem says that any theory that attempts to describe nature in a manner:

(1) consistent with the foundational principles of quantum mechanics, and

(2) also consistent with special relativity,

(3) that has massive fundamental particles which are consistent with those observed in real life at low energies:

(4) must have particle interactions that can be described in terms of a Lie Group, and

(5) can't have laws governing particle interactions that depend upon the laws of special relativity in a manner different from the way that they do in the Standard Model.

Since almost any realistic beyond the Standard Model theory must meet all three of the conditions for the no-go theorem to apply in order to meet rigorously tested experimental constraints, the conclusion of the theorem requires all such theories to have a single kind of core structure. This largely turns the process of inventing beyond the Standard Model theories of physics from an open ended inquiry into an elaborate multi-choice question. For example, while the theorem does not prescribe the conservation laws that are allowed in such a theory, all of its conservation laws must follow a very particular mathematical form.

More technically, this no-go theorem can be summed up as follows:

Every quantum field theory satisfying the assumptions,

1. Below any mass M, there are only finite number of particle types

2. Any two-particle state undergoes some reaction at almost all energies

3. The amplitude for elastic two body scattering are analytic functions of scattering angle at almost all energies.

and that has non-trivial interactions can only have a Lie group symmetry which is always a direct product of the PoincarĂ© group and an internal group if there is a mass gap: no mixing between these two is possible. As the authors say in the introduction to the 1967 publication, "We prove a new theorem on the impossibility of combining space-time and internal symmetries in any but a trivial way. . . .

Since "realistic" theories contain a mass gap, the only conserved quantities, apart from the generators of the PoincarĂ© group, must be Lorentz scalars.

The

**PoincarĂ© group**is a mathematical structure the defines the geometry of Minkowski space, which is the most basic space in which physical theories that are consistent with Einstein's theory of special relativity must follow.
A

**Lorentz scalar**is "is a scalar which is invariant under a Lorentz transformation. A Lorentz scalar may be generated from multiplication of vectors or tensors. While the components of vectors and tensors are in general altered by Lorentz transformations, scalars remain unchanged. A Lorentz scalar is not necessarily a scalar in the strict sense of being a (0,0)-tensor, that is, invariant under any base transformation. For example, the determinant of the matrix of base vectors is a number that is invariant under Lorentz transformations, but it is not invariant under any base transformation."
Notable Lorentz scalars include the "length" of a position vector, the "length" of a velocity vector, the inner product of acceleration and the velocity vector, the 4-momentum of a particle, the energy of a particle, the rest mass of a particle, the 3-momentum of a particle, and the 3-speed of a particle.

But, SUSY and SUGRA are important exceptions to the Coleman-Mandula no-go theorem. (There are also a few other exceptions to this no-go theorem which are beyond the scope of this post which have quite different applications.)

So, there are a variety of interesting ideas that one might want to try to implement in a beyond the Standard Model theory that it has been proved can only be implemented within the context of a SUSY or SUGRA model.

**The Post**

The latest CERN Courier has a long article by Hermann Nicolai, mostly about quantum gravity. Nicolai makes the following interesting comments about supersymmetry and unification:

To the great disappointment of many, experimental searches at the LHC so far have found no evidence for the superpartners predicted by N = 1 supersymmetry. However, there is no reason to give up on the idea of supersymmetry as such, since the refutation of low-energy supersymmetry would only mean that the most simple-minded way of implementing this idea does not work. Indeed, the initial excitement about supersymmetry in the 1970s had nothing to do with the hierarchy problem, but rather because it offered a way to circumvent the so-called Coleman–Mandula no-go theorem – a beautiful possibility that is precisely not realised by the models currently being tested at the LHC.In fact, the reduplication of internal quantum numbers predicted by N = 1 supersymmetry is avoided in theories with extended (N > 1) supersymmetry.Among all supersymmetric theories, maximal N = 8 supergravity stands out as the most symmetric. Its status with regard to perturbative finiteness is still unclear, although recent work has revealed amazing and unexpected cancellations. However, there is one very strange agreement between this theory and observation, first emphasised by Gell-Mann: the number of spin-1/2 fermions remaining after complete breaking of supersymmetry is 48 = 3 × 16, equal to the number of quarks and leptons (including right-handed neutrinos) in three generations(see “The many lives of supergravity”). To go beyond the partial matching of quantum numbers achieved so far will, however, require some completely new insights, especially concerning the emergence of chiral gauge interactions.

I think this is an interesting perspective on the main problem with supersymmetry, which I’d summarize as follows. In N=1 SUSY you can get a chiral theory like the SM, but if you get the SM this way, you predict for every SM particle a new particle with the exact same charges (behavior under internal symmetry transformation), but spin differing by 1/2. This is in radical disagreement with experiment.What you’d really like is to use SUSY to say something about internal symmetry, and this is what you can do in principle with higher values of N. The problem is that you don’t really know how to get a chiral theory this way.That may be a much more fruitful problem to focus on than the supposed hierarchy problem.

From Not Even Wrong (italics in original, boldface emphasis mine).

## 4 comments:

Hi,

in case you missed it, I just found the source : http://journals.aps.org/prd/abstract/10.1103/PhysRevD.91.065029

Thanks a lot. I appreciate it. The abstract of the paper you linked and the full citation are as follows:

"In a scheme originally proposed by Gell-Mann, and subsequently shown to be realized at the SU(3)×U(1) stationary point of maximal gauged SO(8) supergravity by Warner and one of the present authors, the 48 spin-

1/2 fermions of the theory remaining after the removal of eight Goldstinos can be identified with the 48 quarks and leptons (including right-chiral neutrinos) of the Standard model, provided one identifies the residual SU(3) with the diagonal subgroup of the color group SU(3)c and a family symmetry SU(3)f. However, there remained a systematic mismatch in the electric charges by a spurion charge of ± 1/6. We here identify the “missing” U(1) that rectifies this mismatch, and that takes a surprisingly simple, though unexpected form."

Krzysztof A. Meissner and Hermann Nicolai, "Standard model fermions and N=8 supergravity", Phys. Rev. D 91, 065029 (24 March 2015).

Preprint at https://arxiv.org/abs/1412.1715

From the conclusion:

"What is clear, however, is that in such a scheme the W± and Z vector bosons would have to be composite, in a partial realization of the conjecture already made in [7], that SU(8) becomes dynamical. We recall that the ‘composite’ chiral SU(8) symmetry does not suffer from anomalies [14], and the same should be true for any subgroup of SU(8) that becomes dynamical.

The results of this article lend further credence to the remarkable coincidence, already exhibited in [5] and [3],

between the fermionic sector of N = 8 supergravity and the observed 48 spin- 1/2 fermions of the Standard Model.

Evidently this agreement would be spoilt if any new fundamental spin- 1/2 degrees of freedom (as predicted by all models of N = 1 low energy supersymmetry) were to be found at LHC. While the numerology is thus very suggestive, there remain, of course, the thorny open problems already listed in [3] (huge negative cosmological constant, mass spectrum, etc.), whose resolution would demand some new, and as yet unknown, dynamics which would also have to account for the final breaking of N = 2 supersymmetry.

So the above coincidence between theory and observation may yet turn out to be a mirage. At any rate,

and in view of the complete absence so far of any ‘new physics’ at LHC, it appears worthwhile to search for unconventional alternatives, of the type considered here, to currently popular ideas. In particular, the actual realization of supersymmetry in particle physics may require a more sophisticated implementation of this beautiful concept than in the N = 1 models currently thought to be phenomenologically viable."

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