Families from USp(32)

The same trick to descent to three generations can be done starting from the 495 of Sp(32). We are on it, as usual, in physicsforums


As noted there, we still lack of a variant showing the L-R chirality of the SM.  I wonder if we could use some exotic models from E6 or E8 for this. Particularly I am thinking on some innovative usage of the 27-dim irrep of E6, perhaps decoloring it so that all the three generations can be put where colour usually is. It could be interesting for instance if it goes to 24+3 purging the +4/3 quarks.

Models based on non commutative geometry with octonions or Jordan Algebras could be interesting to look at, for this task. Perhaps https://arxiv.org/abs/1604.01247 or something with hints of composite higgses.

wrong turns

Woit (What the Hell is Going On?) and Motl (wrong-turn-basins-gut-critics) have sparked a new round of the discussion on the existence, or not, of a wrong turn in the development of theoretical particle physics. The excuse this time is some comments of Nima Arkani-Hamed in a couple of recent lectures, but the running joke is already old, that something broke down around 1973-74.

When this discussion happened in 2006, inspired by Rovelli’s hep-th/0310077, I proposed two examples of -arguable- wrong turns: one in the seventies with the different interpretations of the meaning of renormalizability, and one centuries ago in the Principia, with the decision, after some discarded drafts, of corfirming angular momentum as a fluent quantity, allowing for infinitesimal changes. The moral of the examples was that even after the long way we eventually return to the right route, and with a lot of mathematical and physical stuff actually useful, collected along the path.

Them in 2011 I opened more specifically «The wrong turn of string theory», suggesting that it had been in mid 1970 the neglecting of its use as a theory of mesons, and that other way could have been the pairing of such mesons with their components, in an emulated, and recursive, sort of supersymmetry. Well, 13 pages and some summers later on this path, I found myself playing with group representations having 496 components as the best way to impose the needed symmetry. Perhaps there are some deep valleys across and bridges need to be deployed, but at the end it looks as that the destination allows for a number of turns, and it is a mater of taste to claim which ones are wrong.

Minute 0:56:12 of the first talk is specially hillarious because Nima is directly naming -and dismissing as random- the only known case where a boson and a fermion of the same charge were found having the same approximate mass, all of it under a title «Where in the World are SUSY». I mean, pion-muon. Then he in the next phrase dismisses the next case, charmonium-tau. And it does not even pauses to notice the joke; I had expected at least a punchline «These are not the scalars you are looking for»

Families from SO(32)

{\bf (1,24,1^c) }&+{\bf [1,15,\bar 3^c]}&+{\bf [1, \bar {15}, 3^c]}&+\\
1,24,8^c&+[1,10,\bar 6^c]&+[1,\bar {10},6^c]&+\\
(1,1,8^c)&&&+\\&(2,5,3^c)&+(2,\bar 5,\bar 3^c)&+\\

(or from SO(30), or perhaps just O(10)xU(3) or U(5)xU(3))

Point is, the first three lines seem to contain three generations with
electric and colour charge. It is possible to break the 24 and 15
from su5 to su3 + su2, and then identify the electric charge.

& Q_1 & Q_2 & su3 + su2 & Q_3 & N \\
(1,24,1^c) &0&0& (8, 1) + (1, 3) +(1, 1)&0 & 12\\
& 0 & 0 & (3,2) & 5 & 6\\
& 0 & 0 & (\bar 3,2) & -5 &6\\
(1,15,\bar 3^c)&0&4&(\bar 6,1)&4&6\\
(1, \bar {15}, 3^c)&0&-4&\\
(1,10,\bar 6^c)&0&4&(\bar 3,1)&4\\
(1,\bar {10},6^c)&0&-4&\\
(2,5,3^c)&2&\pm 2&(3,1)&2\\
&2&\pm 2&(1,2)&-3\\
(2,\bar 5,\bar 3^c)&-2&\pm 2&\\

We can choose 4(Q2+Q3)=-2/3 and Q3=-1/5 or 4(Q2+Q3)=1/3 and Q3=1/5

Honest problem is, I do not know how to identify weak isospin, nor weak hypercharge,
given that we are looking, I guess, to scalars. Perhaps the repr is too big, doubled?

«No room for self-referring vague half-elementary, half-composite particles»

Comment to a post from Lubos in his blog. Copy paste for reference.


As here you could be stressing the criticism of a comment ten years ago, allow me just a reminder for the people who was not here at that time. We set V as a five, (u,d,s,c,b), of SU(5). Then we go for composite scalars: the 24 out of V x V* = 24 +1 has six of charge +1, twelve neutrals and six of charge -1. The 15 out of VxV=15+10 has six of charge +2/3, six of charge -1/3, and three of 4/3. So this is the scalar content of three generations of the Supersymetrical Standard Model, except for the chiral 4/3 beastie.

This «Nuclear superdemocracy» actually fixes the number of generations because you can not make the same with any, only with N=3 and one «top» quark out of the set. Not a thing that Chew could have attempted in the sixties, with only u,d,s and no argument to look for squarks nor sleptons.

A peculiar coincidence is that the «fake sleptons», this is, the actual mesons bound with SU(3) colour, have approximately the same mass that the actual fermions, which is amusing when one considers that the origin of the mass is completely different: yukawas + higgs on one side, pure QCD on the other.

bootstraping a space from its tensor square

Consider the tensor square r?r of an irreducible group representation r with itself, and decompose it as irreducible representations. What can we said about the circumstance of finding the initial representation in the list? Or perhaps about finding its conjugate, as for instance in E6:


What groups have representations having this «bootstraping»? Can the irrep appear in both parts, symmetric and alternating, of the tensor square? Does it appear in an unique way, or can it be extracted from different combinations of the roots?

Similarly, consider the tensor square A?2 of an algebra. Are there situations where the new algebra does contain the initial one as a subalgebra in a non trivial way? This seems to generalise the question of generating an algebra from a finite number of elements and its n-times product, call it A×n, but perhaps it is not more general… still I wonder what can be said generically about such action. What I am expecting is that some ideal J can be chosen in
A?2 such that the quotient recovers the initial algebra. Or some similar mechanism, anyway.

The motivation of the post to be in BSM is, of course, my old observation that by choosing five quarks, out of all the set of three particle generations of the standard model, and pairing them we seem to be able to recover the full three generations, and I wondering if this phenomena could be tracked to some peculiar property in mathematical representation. Thinking it also in algebraic terms is interesting because the attempts to get generations out of the exceptional jordan algebra h3(O) or its twin h3(C?O) have some extra matter in the diagonal, an issue that also happens in the naive pairing.