High energy particle physics theorists are disappointed and even dismayed that the Large Hadron Collider has shown up nothing beyond the Standard Model Higgs. Their favorite "Beyond Standard Model" physics, based on an idea called supersymmetry (SUSY) has not shown even a tiny hint of existing. Over on "Not Even Wrong", Peter Woit asked "“Is there any forseeable experimental data that would cause you to decide that SUSY was an idea that should be abandoned?”

Urs Schreiber gave a logical answer as to why physicists might think SUSY is relevant to physics - not just in a technical sense where it can make some computations tractable - but as a part of reality. You can follow the link (or see below the fold).

So, I thought that Woit's question was answered. Nothing but a mathematical theorem with a proof will serve to end the SUSY quest. Namely, something like:

On a side note, Charles Darwin, around the time of spelling out his theory of evolution, also wrote: "It is mere rubbish thinking at present of the origin of life-- one might as well think of the origin of matter." This is because he knew that the problem of the origin of life was well beyond the reach of the science of his time. Particle physics theorists, however, have believed for about forty years that a complete description of fundamental physics is within their grasp. Nature has proven to be rather elusive.

*** Somebody pointed out on Woit's blog that the O(16)xO(16) heterotic string has worldsheet supersymmetry but not spacetime supersymmetry (and no tachyons that would weigh against it).

An O(16) x O(16) Heterotic String, Luis Alvarez-Gaume, Paul H. Ginsparg, Gregory W. Moore, C. Vafa (Harvard U.) Feb 1986 - 8 pages Phys.Lett. B171 (1986) 155-162 (1986)

Urs Schreiber gave a logical answer as to why physicists might think SUSY is relevant to physics - not just in a technical sense where it can make some computations tractable - but as a part of reality. You can follow the link (or see below the fold).

So, I thought that Woit's question was answered. Nothing but a mathematical theorem with a proof will serve to end the SUSY quest. Namely, something like:

"Nature chose to have an ordinary group act on the supermanifold" because:

1. Using a supergroup on the supermanifold implies a necessary feature in the low energy theory that our observed low energy world lacks; or

2. Our low energy world has an observed feature that the use of a supergroup on the supermanifold cannot reproduce; or else,

3. Using a supergroup on a supermanifold produces a high-energy theory that fails for some reason (without even considering the low energy world).

Since the question is: “Is there any foreseeable experimental data that would cause you to decide that SUSY was an idea that should be abandoned?”", (3.) above need not concern us here. The Higgs as detected by the LHC, with nothing super- accompanying it, does not quite fall into either 1. or 2. without additional assumptions.Woit didn't like that answer and deleted it.

Therefore the SUSY search will continue.

On a side note, Charles Darwin, around the time of spelling out his theory of evolution, also wrote: "It is mere rubbish thinking at present of the origin of life-- one might as well think of the origin of matter." This is because he knew that the problem of the origin of life was well beyond the reach of the science of his time. Particle physics theorists, however, have believed for about forty years that a complete description of fundamental physics is within their grasp. Nature has proven to be rather elusive.

The fact that the wordline theory of an ordinary spinning particle (such as the electron) is 1d supersymmetric (see here for details and references) is interesting mostly because it is a precursor of the fact that the worldsheet theory of the “spinning string” (as it was originally called) just so happens to be 2d supersymmetric (and called the “superstring” only once this was realized), which in turn is interesting because it miraculously implies that the effective spacetime theory of which these strings are quanta is locally 10d supersymmetric.*** This is interesting because, while the automatic worldline supersymmetry of the electron does not imply that the spacetime theory is supersymmetric, just assuming that the electron is the point particle limit of a string (and nothing else) does imply local spacetime supersymmetry. This is a key prediction of the assumption of strings:

strings & fermions => supergravity

This of course does not prove that local spacetime supersymmetry (supergravity) is realistic, but it is one of the theorems that miraculously imply local spacetime supersymmetry and thus make it a plausible possibility. Of course this particular argument does require the assumption that particles are limits of strings, which might be wrong, and if so then the conclusion wouldn’t follow, of course.

On the other hand Deligne’s theorem needs no assumption beyond established principles of quantum physics (that elementary particles are labeled by irreducible linear representations of the local spacetime symmetry group), but as a payoff it does not imply that the local spacetime symmetry group is the super-Poincare group, it only implies that among all imaginable modifications of local spacetime supersymmetry (e.g. non-commutative groups, aka quantum groups or more bizarre possibilities) all except precisely the class of super-symmetry groups are excluded on mathematical grounds.

Combined with the observavation made in 1922, that the world does contain fermions, and the realization, a little later, that hence the phase space of every realistic physical theory (such as the standard model) is a supermanifold (the fermions constituting the odd-graded coordinates) it makes it rather plausible that the local symmetry group acting on this supermanifold in general mixed bosons and fermions, hence be a supersymmetry group. Of course this is not logically implied and could well be wrong, but the point to notice is that given the above circumstances, it would almost seem to require _more_ explanation why it shouldn’t, than the other way around: experiment shows that phase space is a supermanifold, and Deligne’s theorem says that the most general local spacetime symmetry group may be a supergroup, so why on earth would Nature then choose to constrain this possibility and have an ordinary group act on the supermanifold, if it could choose a supergroup.

Again, this does of course not prove local spacetime supersymmetry, but it serves to show that there are good reasons to expect that it might be: Because conversely, if the world were not locally supersymmetric, then there would be reason to ask: why this constraint? Why, if provided with the possibility to choose from the class of all super-groups, and with phase space of verified physic already being a supermanifold, why would Nature choose a purely bosonic symmetry group? It’s possible, but it shows that it is not insane to suspect that fundamentally (i.e. at high energy) it might be different.

That all said, recall again that all this is about high energy local supersymmetry (supergravity). There is no comparable argument from first-principles which would force that a theory of supergavity should settle in a vacuum where at the weak scale there is a global supersymmetry left (which is the kind of supersymmetry that is invoked for those arguments about naturalness, gauge coupling unification, dark matter etc.). In contrast, a global supersymmetry (a global Killing spinor) in a solution to the equations of motion of supergravity is about is unlikely as a global translation symmetry (of which the supersymmetry is a “square root”). We don’t expect the universe to be globally translation invariant in any one direction, even though locally it is, and analogously it would be surprising to find a global supersymmetry, even if locally everything would be supersymmetric.

In conclusion, the main question of interest seems to be a question rarely discussed at the moment:

Theoretically, the real question is: given a theory of supergravity (or some UV-completion thereof) what are the compactification mechanisms, what can one say about its effective “low” energy (say weak scale) vacua. Is there any actual mechanism, besides prejudice, that would prefer Calabi-Yau or G2 compactifications? Or maybe better: at which scale would such these be effective?

Experimentally, a good question would be: what would be experimental checks not of low energy supersymmetry, but of supergravity. For instance these authors here argue that plateau inflation (such as Starobisnky-inflation) — which is the model of inflation presently preferred by experiemtal data — gives a still better fit to experimental data after embedding into supergravity. Now, that particular argument will have its loopholes, but generally more experimental arguments for or against high energy supersymmetry (supergravity) would be interesting to have.

*** Somebody pointed out on Woit's blog that the O(16)xO(16) heterotic string has worldsheet supersymmetry but not spacetime supersymmetry (and no tachyons that would weigh against it).

An O(16) x O(16) Heterotic String, Luis Alvarez-Gaume, Paul H. Ginsparg, Gregory W. Moore, C. Vafa (Harvard U.) Feb 1986 - 8 pages Phys.Lett. B171 (1986) 155-162 (1986)