Two years ago, this paper appeared:(adsbygoogle = window.adsbygoogle || []).push({});

http://arxiv.org/abs/1305.4208

Relation between masses of particles and the Fermi constant in the electroweak Standard Model

G. Lopez Castro, J. Pestieau

(Submitted on 17 May 2013)

An empirical formula relating the physical masses of elementary particles and the Fermi constant is proposed. Although no mechanism or theoretical model behind this formula is advocated, we seek for a possible physical interpretation. If not a simple numerical coincidence, this formula may motivate theorists to search for relations among the different sectors of the Electroweak Standard model.

The relation in the title is simply

[itex] m_H^2 + m_W^2 +m_Z^2 + \Sigma_{f=q,l} m_f^2 = v^2 [/itex]

... the sum of the squares of the masses of the standard model particles, equals the square of the Higgs VEV.

Also, a very similar observation was made two years beforethatby A. Garces Doz, in a comment at Lubos Motl's blog. (That was before the true Higgs mass was established; Garces Doz used the value mH = 119 GeV.)

I am not aware of any theoretical framework capable of explaining that relation.

The authors also write: "Observe that in the left-hand side of Eq. (1) there are almost equal contributions of fermions and bosons", i.e.,

[itex] \Sigma_{bosons} m_b^2 = \Sigma_{fermions} m_f^2 [/itex]

As it turns out, this second relationdoeshave an explanation. It's called the supertrace sum rule, and it is, apparently, characteristic of spontaneously broken supersymmetry. However, ironically, this connection has traditionally been used torule outspontaneously broken supersymmetry.

See, for example, the very first page of today's preprint "D3-brane model building and the supertrace rule", where it is stated: "The main advantage of soft supersymmetry breaking compared to spontaneous breaking, is that the former can avoid the supertrace sum rule ... and hence avoid the existence of supersymmetric particles much lighter than the top quark, which is essentially ruled out by recent LHC results."

Or try 2011's "Visible Supersymmetry Breaking and an Invisible Higgs" ... "By the standard lore, visible sector supersymmetry breaking is phenomenologically excluded by the supertrace sum rule" ... or indeed, any number of standard accounts of supersymmetric phenomenology.

This standard reasoning - which is used to conclude that supersymmetry must first be broken first among presently unknown particles, and then transmitted to the standard model superfields by some mediating interaction - is employed within the framework of the supersymmetrized standard model. That is, the authors are thinking that along with a top quark there's a stop squark, along with gauge bosons there are gauginos, and so on through all the SM particles in the familiar fashion. They are assuming that the supertrace sum rule would pertain to the masses of the SM particles combined with the masses of their superpartners.

However, the supertrace-like relation discovered by Lopez Castro and Pestieau, is a purely intra-SM phenomenon. No new particles are called upon. Therefore, if it is to be explained by supersymmetry, that would seem to require that supersymmetry has somehow been lurkinginsidethe standard model all this time.

There are a handful of attempts in that direction that come to mind. Hermann Nicolai and collaborators have been trying to save an old idea of Gell-Mann's, for obtaining the standard model from N=8 supergravity (here is the most recent paper). Stephen Adler has constructed a grand unified theory which is not supersymmetric but which has "boson-fermion balance", also inspired by N=8 supergravity. And I have to mention Physics Forums stalwart Alejandro Rivero (arivero), whose combinatorial approach has been the subject of a long thread here.

Returning to orthodoxy for a moment, I read that the supertrace relation also has implications for the cosmological constant. If the relation is exact, the c.c. is zero; if it is merely close to exact, the c.c. will be nonzero but small. Long-vanished PF commenter "smoit" talked about this from a string-theory perspective here. It is tempting to think that there could be a connection to the full Lopez Castro & Pestieau sum rule, which involves the Higgs VEV as well.

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# Supertrace sum rule

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