Guest post: Alejandro Rivero, “sBootstrap”

Alejandro Rivero is a theoretical physics PhD from Zaragoza University who, to his own regret, was too good with computers and too stubborn about research paths. As there was a lot more of work for computer scientists than postdocts for non commutative geometry with the orthodox Connes approach, he went into the former. But he still has more confidence on NCG than in strings. Currently he works in the BIFI, where he is involved in the project of a national infrastructure for volunteer computing.

CHEWISH PHYSICS

Coming back to the RadLab from his exile, Chew developed in the early sixties a new approach to strong interactions: the bootstrap. He observed that there was nothing in a Feynman diagram allowing to know which particles were composite and which ones were fundamental ones. Raising the flag of “nuclear democracy”, and feeling himself backed by observations of Feynman and Heisenberg, he launched a S-matrix-based program aiming to find an unique self-consistent theory of the nuclear strong force. “Nuclear democracy” was to mean that a particle can be interpreted as composite or elementary depending of its role in a particular diagram, so there were no elementary particles at all, or all the particles were equally elementary. The pursuit of further pieces of matter, subcomponents and preons and prepreons and so on, was to come to an end by asking some general properties to the scattering matrix, the particles coming ready from its pole structure.

During the first years the approach enjoyed some attention from labs everywhere, because even if you did not agree with Chew’s goals, the S-matrix was one of the very few tools available to approach to the study of strong interactions. But for the late sixties it started to be clear that the answer for strong force was more of the same: the hadrons were equal between them… because they were all composite, made of quarks. QCD became the new game, and S-matrix theory was abandoned. At the same time, one of the arguments of Chew theory, s-t duality, evolved by itself coming to beget “dual models” and then “string theory”, and then no a democracy but an unique ruler, the Planck-scale superstring.

Now, open bosonic strings, the ones you can make with two quarks, or a quark and an antiquark, lie in “Regge trajectories”: a dependence between energy and angular momentum, in a way such that all the strings of a same trajectory have a same composition: they are just excitations of its fundamental, spin 0 state. In this way, the number of strings depends of the number of different flavours of quarks you have
available to combine. And these strings are bosons. Assume each spin 0 state is not degenerate (choose the lowest energy one). Let’s postulate that for each spin 0 string there is a fermionic degree of freedom with the same electric and colour charge, but elementary. If it is so, we have saved not the “nuclear democracy” nor the bootstrap, but part of its philosophy: each elementary particle is not composite, but it is supersymmetric to a composite.

COULD THE sBOOTSTRAP WORK ? PREDICTIONS

A first test is to check for matching in degrees of freedom. For n generations of quarks and leptons, we have n charged leptons, and then 2 n degrees of freedom of charge +1 and 2n d.o.f. of charge -1. On the other side, we can combine quarks U and D and their antiquarks to form n^2 strings of charge +1 and the same of charge -1. Then 2n=n^2 implies n=2. This is already bad, because we know there are three generations. But in the neutral side it is worse: the n neutrinos give us 4 n degrees of charge 0, while in the composite side we have either 2 n^2 neutral combinations of quarks and antiquarks that should reduce to 2 n^2 -1 after taking care of the U(1) singlet (as in SU(3) you get an octet instead of a nonet).

Even if we are willing to accept the result n=2 and the extra U(1), we still need to confront quarks. And here the bartering definitely breaks. On one side, the UD sector has n^2 combinations to form charge +1/3, for which there are 2n degrees of freedom in the elementary side. Well, n^2=2n still implies n=2. But on the other side, when combining D quarks to get charge -2/3 you get n(n+1)/2 of these, while you need 2n. So from the DD sector we obtain n=3. The idea fails in the quark sector.

Thus the sBootstrap conjecture predicts 2 generations and can only work assuming an extra U(1) in the combinations for neutral strings and neglecting the possibility of partners for down quarks. Kind of failure for our idea.

Really? Let’s review… we have done an extra assumption that actually does not appear in Nature: we have assumed that all the quarks bind into strings. This is not true: the top meson has a mass greater than the W, and then it is disintegrated under electroweak force before being allowed to link into a QCD bind.

Back to the blackboard: let be n the number of generations, s the number of U quarks and r the number of D quarks. Obviously n equal or greater than r and s, and we ask both to be greater than zero. The rest of the history is section 3 of my e-print arxiv:0710.1526. Now the equations are
2 n = rs,
4n = r^2 + s^2 -1

for the lepton side, and

2n = rs,
2n = r (r+1)/2

for the quark side, and all of them solve uniquely to n=3, s=2, r=3. The sBootstrap conjecture predicts three generations AND predicts that while all the D quarks are light, only two U quarks are light.

PROSPECTIVE

There are some hints that the framework of superstring theory could be fit to keep developing on this conjecture. If we look to the colour side, the standard model with massive neutrinos has 24 fermionic degrees of freedom for each colour, including the neutral possibility. Generation-wise, it makes 8 degrees of freedom by generation; then we could have some hope of fitting them into the massless states of some sector of a superstring.

Furthermore, it is worthwhile to think of the quark-antiquark composites as oriented open strings, while the quark-quark and antiquark-antiquark are different sectors of unoriented open strings. To close an unoriented string you need to zip it against another unoriented string from the opposite sector, and then the resulting closed string carries a charge of the kind colour+anticolour, similar to a gluon. On the other side, an oriented string can close upon itself, giving a closed string uncoloured but perhaps still with electric charge… pretty much as the electroweak bosons, and it could explain the strange similarity in decay rates between the Z, W particles and the most stable mesons, of which Dorigo was kind enough to speak time ago in this blog.

A different question is, what will the LHC find, if all the superpartners have already being found? “And the prize for the experimental finding of supersymmetry goes to… hmm, Cecil Frank Powell again???“. Giving that we do not expect a QFT to have elementary fermions beyond spin 1/2, it can happen we will not find any superpartner more. And yet, we are telling that the mechanism giving mass to the stringy-bosons is the same that the mechanism giving mass to the elementary-fermions. So there should be definitely something to decrypt in the electroweak scale.

THE BALL BACK TO HEP ENERGIES

The lightness of the five quarks, and the heavy character of the top, is predicated with reference to the electroweak scale. Moreover, the superpartners are the particles bound by the QCD strong force. If we can work out some model based on superstrings, they will not be superstrings at the Planck scale but superstrings down under the TeV. It is hard to guess if a modern string theorist will be happy or sad about the sBootstrap idea.

A string-like binding fits well with other expectations of the amateur spectrologists in physicsforums.com, all of them from HEP energy data: the quotient between the mass of Z and W has been noticed, by Hans de Vries, to be very much as the one of the binding two relativistic particles with total angular momentum 1 and 1/2. Some breaking in the neutral vs charged pion was also related to the quotient between the mass of a lepton and those of a gauge boson, and some of the octet breaking or mixing was very near of relationships involving only lepton masses. Furthermore, Koide’s formula works better when mixing composite particles, and then it could be a formula to be met in the initial three supermultiplets, before further symmetry breaking and mixing in the strong sector. This fact has been another of the guiding principles towards our suggestion of a composite elementary symmetry. Of course all the amateur findings, without a backing theory, could be birthday coincidences: how many people must meet in a room to have a 50% probability of finding two persons with a common birthday? But note that some formulae are grouped in very nicely symmetric families.

In any case, the plot associated to the Z width mystery provides an approximation of the scale range: from the mass of the pion -and muon- to the point where the scaling of charged pion meets the scaling of neutral pion. The latter goes with the cube of the mass, the former with the quintic power. The lines meet at 2.6 TeV, and of course the effect of the breaking of electroweak symmetry starts to blur already before, as we climb beyond the 0.1 TeV of the Z mass.