288. A Pyramid Scheme for particle physics
Save This Post as PDFThis is a guest post by Jean-Francois Fortin from Rutgers. Dmitry.
First of all I would like to thank Dmitry for requesting a guest post of the recent paper “A Pyramid Scheme for Particle Physics” (arXiv:0901.3578) that I coauthored with Tom Banks. The paper discusses a model of dynamical supersymmetry (SUSY) breaking which is not plagued by an accidental unbroken R-symmetry or Landau poles and which contains dark matter candidates possibly relevant for the cosmological positron excess measured by ATIC, PAMELA and PPB-BETS.
The Pyramid Scheme is based on the Intriligator-Seiberg-Shih (ISS) mechanism of dynamical SUSY breaking in metastable vacua. The ISS mechanism relies on Seiberg duality in the free magnetic range, where SQCD (which is UV-complete and strongly-coupled at low energies in this range) flows in the infrared to the same fixed point than an UV-incomplete IR-free magnetic dual. The low energy magnetic dual can be studied reliably to investigate possible metastable SUSY breaking minima and ISS showed their existence. One interesting feature of the metastable vacua ISS found is the large unbroken flavor group. It can be weakly-gauged in order to embed the standard model (SM) and thus mediate SUSY breaking directly. There are also drawbacks of the ISS vacua. The large unbroken flavor group usually leads to the blow-up of the SM gauge couplings well before the unification scale. Moreover, the ISS vacua has an unbroken R-symmetry which forbids Majorana gaugino masses, clearly excluded by experiments.
A lot of research has been done where the ISS framework is modified in such a way that the SUSY breaking sector is more phenomenologically viable. The R-symmetry can be easily broken but the Landau poles are difficult to avoid, which partly undermines the motivations for SUSY as physics beyond the SM since gauge coupling unification might be lost. The Pyramid Scheme takes a different approach, looking instead at SQCD with the same number of flavors and colors. In this case there is no magnetic gauge group, the dual theory consisting of baryons and mesons, and the existence of a metastable SUSY breaking minimum is yet to be proved. However in the Pyramid Scheme electroweak symmetry breaking is tied to SUSY breaking, thus if one assumes the existence of such a metastable SUSY breaking minimum, the requirement of electroweak symmetry breaking leads to R-symmetry breaking, giving masses to the SM gauginos.
This is nice, but what about gauge coupling unification? To solve the Landau pole problem, one needs to have a theory with a large enough flavor group to embed the SM but without too many extra fields charged under the SM. This is where the idea of trinification enters. The standard strategy to ensure gauge coupling unification is to use messenger fields in multiplets of grand unified theory gauge groups like 
or 
. The Pyramid Scheme takes a different approach with an 
gauge group which can be used to embed the SM. The 
– symmetry rotates the 
factors into each other, forcing them to have the same gauge coupling at the unification scale. Another 
factor is annexed to the model to act as the strongly-coupled gauge group. This forces on us three types of trianons (the electric quarks) which are all charged under the gauge group. The semidirect product of and 
looks like something that might naturally come from D-branes in Type II string theory.
Fig. 1. Quiver diagram of the Pyramid Scheme. SM particles are in broken multiplets running around the base.
Notice that each type of trianon transforms under one flavor only. This therefore solves our problem; the SM, which is embedded in the subgroup of 
, has only three extra flavor messenger fields per gauge group, and Landau poles are avoided successfully. But now, if the electric theory is effectively , we have nine flavors and three colors, which is six flavors too many. Thus two types of trianons have to be fairly massive such that only with three flavors is effectively left at energies below their masses. With that prescription we recover the Pyramid Scheme. Notice that we could have given mass to one type of trianon only, which would have led us back to ISS (with its unbroken R-symmetry), but this does not fit well with the idea of cosmological SUSY breaking, pioneered by Tom Banks, which is the true underlying mechanism at work here.
Cosmological SUSY breaking (CSB) conjectures a relation between the cosmological constant 
and the scale of SUSY breaking 
. The idea comes from M-theory in asymptotically flat spacetime, which is conjectured to be supersymmetric, thus leading to the aforementionned relation. In CSB the cosmological constant is seen as an input parameter determining the total number of quantum states of asymptotically flat de Sitter space, which is represented by the logarithm of the Bekenstein-Hawking entropy. The cosmological constant/SUSY breaking scale relation leads to another relation between the cosmological constant and the gravitino mass 
, of the form 
. Achieving gauge coupling unification with an appropriate gravitino mass is thus now possible, which was not the case for the Pentagon model, the predecessor of the Pyramid Scheme. The Pyramid Scheme is thus compatible with the greater principle underlying CSB.
By choosing one heavy trianon to be of the 
type, one discovers a plethora of phenomenologically exciting features. This is not forced on us, but the phenomenological consequences are so attractive that it is the obvious choice to make. First of all, the Pyramid Scheme has two kinds of messengers, the heavy type trianon and the lighter moduli of the gauge theory with three flavors, called the pyrmeson and pyrma-baryons. Since the light moduli are only charged under 
while the heavier trianons are charged under there is a suppression of the gluino/chargino mass ratio, which solves the fine-tuning problem of usual gauge mediation. The trianon mass must not be too large however, otherwise the gluino would be unacceptably light. The same happens for the squark/slepton mass ratio.
Also, as mentioned above, in the Pyramid Scheme electroweak symmetry breaking is tied to SUSY breaking through the superpotential, and it is plausible that one obtains acceptable values for the 
and 
parameters, though more work is required to verify this claim.
Finally, the Pyramid Scheme might have something to say about cosmology. Gauge mediated SUSY breaking models do not have standard WIMP dark matter candidates, the LSP being the gravitino, which is too light. However, it is possible that baryon-like states of the hidden sector play the role of cold dark matter (this is not the case in ISS since the hidden sector baryon number is spontaneously broken in the metastable vacua), although a primordial asymmetry is required if the reheating temperature is above the confinement scale of the hidden sector. It is also possible that the pseudo Nambu-Goldstone boson (PNGB) of the spontaneously broken hidden sector baryon number plays the role of dark matter, although again a primordial asymmetry is required. One has to be careful with the last statement, since the PNGB effective Yukawa coupling to electrons violates bounds from stelar cooling rates, which forces the PNGB to be heavier than about an MeV. This was a problem in the Pentagon model where the pyrmion mass was generated by a dimension seven operator and thus needed the introduction of a new scale. In the Pyramid Scheme on the other hand, the pyrmion mass is generated by a dimension five operator and is of the order of a few MeV, as required.
The Pyramid Scheme, which has two unbroken baryon-like symmetries and one spontaneously broken baryon-like symmetry, thus provides a wealth of possibilities for explaining the dark matter and the ordinary baryon number asymmetry (which is generated through couplings with hidden sector currents). Assuming negligible primordial asymmetries and a low reheat temperature, there is a small window where non-thermal production of the lightest (SM neutral) pyrma-baryons of the unbroken baryon-like symmetries could account for the observed dark matter density and thus give good dark matter candidates. These dark matter particles are strongly-coupled under , which has a confining scale of a few TeV, and by analogy with QCD their annihilation cross section is energy-independent. Pushing the analogy further, they decay with high multiplicity to the PNGB, called the pyrmion, which is the QCD equivalent of the pion. This last observation is very interesting since the pyrmion does not carry color and is very light, thus decaying primarily to electron-positron pairs, photons and neutrinos, in agreement with the positron excess observed by ATIC, PAMELA and PPB-BETS.
Wrapping it up, with very acceptable numbers, the Pyramid Scheme falls in the right ballpark to explain the observed positron excess while avoiding the numerous possible phenomenological problems of direct gauge mediation. More work has to be done, but the model looks promising.
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Dear Jeff, it’s very interesting & greetings to Tom.
Are you saying that CSB and ISS are compatible? I thought that CSB required a complete departure from low-energy effective field theory while ISS required to take it completely seriously.
The Pyramid Scheme quiver is very interesting. I wondered whether it has a natural embedding into E8 which has a SU(2)^4 subgroup. But it probably doesn’t because the trianons do not appear in the decomposition of 248, and even the “27″ appear thrice.
Or maybe one should use a non-fundamental rep of E8? Their dimensions are pretty high.
It’s interesting why you didn’t call it Tetranification instead of Pyramid Scheme.
Dear Lubos,
One can look at the implications of CSB for low-energy effective field theory and the picture that emerges fits well with Nf=Nc SUSY QCD. So as far as I know there is nothing wrong with low-energy CSB. Also, the main reasons why we chose SU(3)^3 are one-loop gauge coupling unification and Landau poles. The quick way to see that gauge coupling unification occurs is to consider SU(3)^3 as a subgroup of the GUT gauge group E6, thus leading to conventional gauge coupling unification. We did not spend time on that nor on E8, considering instead D-branes in Type II as the origin of the SU(3)^3 gauge group, but the extra SU(3) is strongly-coupled and does not unify with the three SU(3)s, hence trinification rather than tetranification.
Cheers.
By the way, no need to apologize, Jeff is the usual nickname for Jean-Francois.
SU(2)^4 should have been SU(3)^4.
I apologize for the “Jeff” which I copied from Dmitry’s subtitle. I guess it was a “pronounced” version of J.F unless I am missing something.