# Supertrace sum rule

1. Oct 26, 2015

### mitchell porter

Two years ago, this paper appeared:

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

$m_H^2 + m_W^2 +m_Z^2 + \Sigma_{f=q,l} m_f^2 = v^2$

... 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 before that by 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.,

$\Sigma_{bosons} m_b^2 = \Sigma_{fermions} m_f^2$

As it turns out, this second relation does have 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 to rule out spontaneously 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 lurking inside the 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.

2. Oct 26, 2015

### mitchell porter

I should have called this thread "Supertrace sum rule and the standard model"...

3. Oct 27, 2015

### fzero

The supertrace is actually weighted by the number of spin degrees of freedom and minus sign for fermions, so
$$\text{STr} (M^2) = \sum_{\{s\}} (-1)^{2s} (2s+1) m_s^2$$
and it is hard to reconcile this with the sum appearing above.

I am usually skeptical of numerology, but I wasn't sure why Lopez Castro and Pestieau didn't actually give the numerical result for their expression, so I calculated
$$\frac{1}{\sqrt{2}G_F} - (m_H^2 + m_Z^2 + m_W^2 + m_t^2 + m_b^2) \sim 180 \pm 490 ~\text{GeV}^2,$$
where the uncertainty is dominated by the uncertainty in the top and Higgs masses. So actually, within experimental uncertainty, we can ignore the particles with masses $< 100~\text{GeV}$ and say that
$$\frac{1}{\sqrt{2}G_F} - (m_H^2 + m_t^2 ) \approx 0.$$
This is probably a simpler relationship to explain, since it doesn't involve every particle in the SM. In particular, there could be some relation to a unitarity constraint or some other relationship between the top Yukawa and Higgs potential.

Edit: My last comments are incorrect, since they ignored the fact that the squared masses appear. So just the Higgs and top mass squared are not enough to cancel the VEV squared, even within experimental uncertainty.

Last edited: Oct 28, 2015
4. Oct 30, 2015

### mitchell porter

fzero is right that the supertrace employed in standard susy theory is weighted by the spins. I was "misled" by the preprint on D3-brane model building - more about that in a moment.

This started last year when I learned about "Veltman conditions" and their relationship to unbroken supersymmetry. As I understand it, a Veltman condition is a type of cancellation among the parameters of a theory that will tame divergences, and sometimes it can be motivated as a low-energy consequence of a zero-supertrace property holding at the supersymmetry scale.

In particular, I noticed that Veltman's original condition was

$m_H^2 + 2m_W^2 + m_Z^2 - 4m_t^2 = 0$

whereas the second claim made by Lopez Castro and Pestieau is that

$m_H^2 + m_W^2 + m_Z^2 - m_t^2 - \Sigma_{other fermions} m_f^2$

is small - similar except for the coefficients.

So I wondered if this part of LC&P's sum rule might be produced by a "deformation" of the usual supertrace rule. Then I forgot about it, until this week.

Now please take a look at "D3-brane model building and the supertrace rule". First, they do state the supertrace rule as simply "sum of squares of boson masses equals sum of squares of fermion masses". You might think they're being careless and approximate and leaving out small factors, and maybe they are; but then, they go on to the substance of their paper, which is about the worldvolume theories of D3-branes, and they prove a tree-level relation (their equation 16) which does look like this "wrong" version of the supertrace rule - without the weights. So I am not sure what's going on.

5. Oct 30, 2015

### fzero

I think there's a common factor of 2 because they are dealing with chiral multiplets. So the fermions are Weyl spinors and the spin factor is 2, while the scalars are complex, so there is also factor of 1 for the spin, then a factor of 2 for the number of real species coming in the supertrace.