The Dirac equation predicts 4 kinds of fundamental particle: A, B, C and D.

We understand that types A and B would appear as electrons in a cloud chamber, while C and D would appear as positrons.

We assume that B can behave as electrons that occupy the shells in normal matter atoms, while D can take that role in anti-matter atoms.

So what else is there to say about electrons?

The first thing is that in real-world experiments, what we may measure as an electron may not correspond to a single particle of type B in our model.

This is because it is possible to manufacture a range of composite particles that have overall characteristics (charge, spin, mass) similar to a single particle of type B.

For example, consider the composite particle with composition AB_{2}.

With a small amount of reasoning, we can deduce that such a particle should be temporarily stable. For example, the A could be stationary in the middle, with the Bs on opposite sides of a circular orbit. Yes, the Bs repel each other, but theyre both attracted to the A in the middle more than theyre repelled by the other B because the A is closer. The A is of course repelled by the Bs, but since theyre on either side, its kind of sandwiched in the middle.

Now, if we add up the spin, charge and mass of the constituents, we get the following answers:

Particle | Charge | Spin | Mass |
---|---|---|---|

AB_{2} | -1 | +3/2,+1/2,-1/2 or -3/2 | +1 |

Note that these are exactly the same answers as for a single particle of type B, if we assume a spin value of +1/2 or -1/2.

We can extend this logic to say that any composite particle with the
composition A_{n}B_{n+1} would behave in a similar way.

Particle | Charge | Spin | Mass |
---|---|---|---|

A_{n}B_{n+1} | -1 | +(2n+1)/2 ... -(2n+1)/2 | +1 |

Furthermore, we know that As and Bs have the same response to an external electrostatic field, so we can expect them to hang around in more-or-less the same configuration as if the field wasnt there. Therefore we can imagine situations where a bunch of As and Bs arrive and are measured as a single electron.

We can also imagine situations where any of the individual constituents may be measured as a single electron, since the Bs have the same properties anyway and the As can be mistaken for Bs in a cloud chamber.

We propose this as the "shotgun" model of the electron.

This could explain some of the weird quantum mechanical behaviour that the standard model of particle physics gives us. We have a model where an electron can be either a single fundamental particle, or a whole bunch of particles masquerading as a single composite particle.

Clearly an electron can be in many places at once, if its actually a composite particle and any of its constituents can be measured (or mistaken) as an electron.

If we place a particle of type 'A' at rest, next to a particle of type 'B', also at rest, an interesting thing happens.

The positive charge 'A' is attracted to the negative charge 'B' according to the Coulomb Law of electrostatics:

F = K_{e} q_{i} q_{j} ⁄ r^{2} where K_{e} = 1 ⁄ (4 π ε0)

However, because 'A' has a negative mass, it moves in the opposite direction to which it is pushed (i.e. away from 'B') according to Newton's second law of motion:

F = m a

Likewise, 'B' is attracted to 'A' by the Coulomb law, only this time it has a positive mass and so moves towards 'A'.

The net effect is that they accelerate together in a straight line.

A curious side-effect is that they always remain the same distance apart, depending only on their initial separation.

Note that this analysis applies to each of the 4 composite particles:

- 'AB'
- 'BC'
- 'CD'
- 'DA'

Hence why this behaviour was marked with the symbol γ in the interaction chart.

We can imagine placing one of each of the 4 particles 'A','B','C' and 'D' together in a square.

We might expect that they would chase each other round and round.

However, this analysis turns out to be false because of the following diagonal asymmetry:

- 'A' and 'C' repel
- 'B' and 'D' attract

Also we note that while 'A's and 'B's have the same response to an external electromagnetic field (as do 'C's and 'D's), 'B's and 'C's do not (and neither do 'D's and 'A's). In particular, we would expect 'AB' and 'CD' to be relatively stable, while 'BC' and 'DA' are relatively unstable.

The net effect is that the "Ring of 4" particle 'ABCD' decays very quickly into 'AB' and 'CD'. This theme turns out to be quite common across a range of interactions that we will look at later.

According to our stats, the 'AB' particle has mass 0, which we arrive at by adding up the individual masses +1 and -1.

But what does this really mean? Given that both A and B have the same response to an external electromagnetic field, ideally we need a measure that will tell us how they move.

The trouble is that we expect different behaviour depending on which direction the field is applied.

If applied from the side (i.e. at right angles to the direction of motion) we expect both 'A' and 'B' to be deflected by the same amount. Note that this does not imply that the photon changes its primary direction of motion, merely that it is deflected to the side for as long as the external field is applied.

If applied from the front, then we would not expect the external field to make any difference at all. This is because the photon is already travelling at close to the speed of light under its own internal acceleration.

If applied from behind, then it depends how strong the external field is. If it is not strong enough to counteract the photon's internal acceleration, then we would expect it to make little difference overall. However, if it is strong enough, then the photon may be brought to rest and even reversed.

Clearly there is no single scalar "mass" parameter that describes all of these scenarios. We therefore suggest that the traditional mass definition, arrived at by simply adding up the constituent masses, has no useful meaning.

Instead we propose 2 different "mass" parameters for the photon. These will be defined in the direction of motion and at right angles to the direction of motion. Therefore they will be mass vectors rather than scalars. We will return to this later.

We know from Quantum Electrodynamics that electrons can absorb or emit photons in different circumstances. Is it possible to model that behaviour using our new-found definitions of electrons and photons?

Starting with a single fundamental electron of type 'B', if it absorbs
a single photon 'AB' then it will form a composite particle of type
'AB_{2}', because the number of fundamental particles is always conserved
in our model.

According to our definition, any particle with the formula 'A_{n}B_{n+1}'
has the same overall properties as a single 'B' particle and is
therefore also considered an electron.

We suggest that there are a range of configurations where 'AB_{2}' is
stable, depending on the initial conditions (i.e. positions and speeds
of the various components). This much could be studied via computer
simulation.

Therefore, in our model, we expect electrons to absorb photons frequently.

Likewise, if we start with a particle of type 'AB_{2}', then we can
imagine a circumstance where one of the 'B' particles get a little too
close to the central 'A' and destabilises the overall configuration.
In this case, the newly formed 'AB' particle accelerates to the speed
of light, while the remaining 'B' particle becomes a separate electron
in its own right.

According to the model, there is no limit to how many photons can be
absorbed by an electron. If we start with a particle of type 'A_{n}B_{n+1}'
and absorb another photon, we simply get a particle of type 'A_{n+1}B_{n+2}'
which is also an electron.

Note that this model does imply that there is a "ground state" of the electron 'B', where it cannot emit any more 'AB' photons, simply because there are none remaining.

We speculate that this "ground state" has a significant effect when electrons go into orbit around atomic nuclei. However, until we have a satisfactory model of the Proton and Neutron to work with (which we don't right now), then this is as far as we will go.

This same model can explain Bremsstrahlung or "braking radiation". In this case,
a composite electron of type 'A_{n}B_{n+1}' emits photons of type 'AB'
in response to an external electromagnetic field. We can imagine this happening when
the disruption caused by the external field causes a 'B' to stray too close to an 'A',
after which the decomposition is inevitable.

Bremsstrahlung has a continuous spectrum, which again is a likely consequence of this model if the initial separations of the 'AB' particles are random.

Note that there is a key prediction here - namely that there is a limit to how much Bremsstrahlung a composite electron can emit before it reaches the ground state and becomes a single 'B'.

Note that combinations with the same number of 'A's and 'B's would be electrically neutral and so would not be expected to leave a vapour trail in a cloud chamber, unless the individual constituents were widely-spaced.

However, if we ignore interactions between the constituents, the composite particle would still follow the same path as an individual 'A' or 'B' particle, because all constituents have the same response to the external field.

The 'AB' particle is a special case and behaves much like a photon, as discussed earlier. In this case the interaction between the 'A' and 'B' is significant (if closely spaced) and would dominate any effect from the external field.

However, combinations such as 'A_{2}B_{2}' could be temporarily stable if the 'A's formed
an 'AA' pair in the middle with the 'B's orbiting outside. In these cases, the interactions between
the constituents would balance out and so the effects from the external field would dominate.

Particle | Charge | Spin | Mass |
---|---|---|---|

'A_{n}B_{n}' | 0 | +n, +(n-1)... 0 ... -(n-1), -n | 0 |

Likewise, we can extend this analysis to equal numbers of 'C' and 'D' particles:

Particle | Charge | Spin | Mass |
---|---|---|---|

'C_{n}D_{n}' | 0 | +n, +(n-1)... 0 ... -(n-1), -n | 0 |

Computer simulation would be required to get a handle on how stable such particles might be.
For example, as soon as one of the 'B's strays too close to the 'A's then the composite particle 'A_{n}B_{n}' would decay
into n 'AB's. We would therefore expect a finite lifetime. Likewise for a 'C_{n}D_{n}' particle.

So far we have largely considered particles of type 'A' and 'B'. In
particular, the particle 'A_{n}B_{n+1}' is considered to be an electron while 'AB' is a photon.

The analysis works exactly the same way if we substitute 'C's
for 'A's and 'D's for 'B's. The only difference is that the charge of
a particle 'C_{n}D_{n+1}' works out as +1 instead of -1.

Particle | Charge | Spin | Mass |
---|---|---|---|

'C_{n}D_{n+1}' | +1 | +(2n+1)/2 ... -(2n+1)/2 | +1 |

The natural position to take is that this is the formula for a positron (i.e. an anti-matter electron).

However, we now have a bit of a conundrum. According to our definitions, we would not expect a ground-state positron 'D' to absorb a photon 'AB' because while 'A' is attracted to 'D', 'D' is repelled by 'A' . Therefore the dynamics of the situation is completely different. In most cases we would expect 'D' to simply side-step an incoming 'AB' and so the photon would bypass the positron altogether (i.e. not interact at all).

Instead, we would expect a of 'CD' particle to be absorbed by a
'D' particle to form a positron of type 'CD_{2}'.

But what in the heck is a 'CD' particle? According to our definition, it is the anti-matter equivalent of a photon 'AB'. It has the following characteristics:

Particle | Charge | Spin | Mass |
---|---|---|---|

'CD' | 0 | +1, 0 or -1 | 0 |

and travels at the speed of light, in much the same way as a photon.

Furthermore, whereas a photon can be absorbed by an electron, a 'CD' particle would be expected to bypass an electron 'B' in most cases. This is because 'B' is repelled by 'C' and so would side-step it.

Apart from the fact that a 'CD' particle has an integer spin, these are sounding a lot like the properties of a neutrino.

We therefore propose the 'CD' particle as a neutrino, with the implication that neutrinos are the anti-matter equivalent of photons.

This means that in our model, neutrinos are now "spin-1 bosons" rather than "spin-1/2 fermions", which is a profound difference when compared to the Standard Model.

The shotgun model of the electron (and positron) leads to some specific predictions surrounding Electron / Positron Annihilation.

In the first instance, if electromagnetism is the only force in our model then we would not expect ground-state electrons and positrons to annihilate at all. Instead, the single 'B' and 'D' particles would simply go into one of the Kepler orbits around each other, due to their mutual attraction.

This would merely result in the production of a 'BD' pair.

However, if we start with composite electrons and positrons, then we have an altogether situation.

If an 'A_{n}B_{n+1}' electron approaches a 'C_{n}D_{n+1}' positron,
then what would we expect to happen?

The first thing to note is that the positron effectively represents an external electromagnetic field as far as the electron is concerned and vice versa. Therefore we would expect Bremsstrahlung radiation to occur for both. In the case of the electron, this will result in the emission of 'AB' photons. In the case of the positron, it will emit 'CD' neutrinos.

Because the 'A' and 'C' particles repel each other, we can expect the photons and neutrinos to be emitted in diametically opposite directions. As the electron and positron circle around each other, this should result in a scattering of photons and neutrinos.

However, it is possible that the 'A_{n}B_{n+1}'includes sub-structure of one of the neutral fragments 'A_{2}B_{2}'.
In this case we would expect to see 2 photons emitted in diametrically opposite directions.

Note that in all cases, we expect the decay process to result in a 'BD' pair, once all the 'AB' photons and 'CD' neutrinos have been exhausted. This tallies with one of the predictions made by Donald Hotson, which he referred to as an "epo" or "electron-positron pair". It also provides a mechanism for the rigorous conservation of spin and mass (i.e. "energy") during the annihilation process.

Through a similar line of reasoning, we can come up with a model for the reverse process of Pair Production.

In this case, we start with a 'BD' pair and an incoming 'AB' photon or 'CD' neutrino.

Under the correct conditions, an 'AB' photon would be absorbed by the 'B' electron, forming an 'AB_{2}' electron and dissociating from the 'D' positron in the process.

Likewise for an incoming 'CD' neutrino.

Continuing on this theme and starting with the 'A_{2}B_{2}C_{2}D_{2}' particle,
we can generate further particles by taking away one of the 'A's or 'C's:

Particle | Charge | Spin | Mass |
---|---|---|---|

'AB_{2}C_{2}D_{2}' | -1 | +7/2 ... -7/2 | +1 |

'A_{2}B_{2}CD_{2}' | +1 | +7/2 ... -7/2 | +1 |

In this case, the overall effect of the 2 internal 'BD' pairs is to stablise the 'A' next to the 'CC' or the 'AA' next to the 'C' respectively. We can imagine such an arrangement being at least temporarily stable, with much the same reasoning used for the Neutral Pion.

Alternatively, we can think of them as:

- 2 'CD's, an 'AB' and a spare 'B'
- 2 'AB's, a 'CD' and a spare 'D'

This leads to overall properties similar to a single 'B' or a single 'D' respectively.

In these cases, the vector mass argument would suggest a mass of 3 x 35 = 105 MeV.

We therefore propose:

- 'AB
_{2}C_{2}D_{2}' as a Muon - 'A
_{2}B_{2}CD_{2}' as an Anti-Muon

If we examine the decay products, we see that our model predicts a difference compared to the Standard Model:

- 'AB
_{2}C_{2}D_{2}' -> 'AB_{2}' + 2x 'CD' - 'A
_{2}B_{2}CD_{2}' -> 'CD_{2}' + 2x 'AB'

Therefore we expect:

- The muon to decay into an electron plus 2 neutrinos (where the electron may emit a further photon)
- The anti-muon to decay into a positron plus 2 photons (where the positron may emit a further neutrino)

Note that the first of these scenarios is in accordance with the Standard Model, whereas the second is most definitely not.

We suggest that any attempt to build a composite particle with the formula 'ABCD' is doomed to fail. This is because the dynamics of the 4 particles together is inherently unstable. Exhaustive computer simulation would be required to prove this beyond doubt, but we can take a short cut with some simple thought experiments.

We know that 'B' would tend to chase 'A' and that 'C' would tend to chase 'D'. We might also think that 'C' would chase 'B' and that 'A' would chase 'D'. However, whereas 'A' and 'B' have the same response to an external field, as do 'C' and 'D', it is not the case for 'B' and 'C' or 'D' and 'A'. The net effect is that any initial arrangement of 'A','B','C' and 'D' will tend to dissociate into 'AB' and 'CD'.

We could attempt to contrive the situation by aiming 'AB' and 'CD' directly at each other. However, given that 'A' and 'C' repel each other. we would expect this to be inherently unstable, with the slightest mis-alignment (or presence of an external field) to result in them deflecting off each other and heading off in opposite directions.

However, if we used pair instead of individual particles then we have an altogether situation.

Starting with 'AA' and 'CC' pairs, the challenge is to build a composite particle that is stable, given that the 'A's repel the 'C's and so would have a natural tendency to fly apart.

The obvious thing to do is to attempt to stablise them with 'BD'
pairs. Furthermore, if we use 2 'BD' pairs then we can arrive at an
overall composition 'A_{2}B_{2}C_{2}D_{2}' which has a neat symmetry.

But surely 'A_{2}B_{2}C_{2}D_{2}' is just equivalent to having 2 'ABCD' particles
which we've already said are unstable. Well, not quite ...

In the pairs situation, we can imagine the 'AA' pair being stationary next to the 'CC' pair as long as there is a net inwards acceleration keeping them there. We can achieve this if, on average, the 'B's are the far side of the 'A's and the 'D's are the far side the 'C's.

Note that this is not what we would automatically think of, given that the 'B's and 'D's are supposed to be in 'BD' pairs. However, we could imagine orbits in which the 'B's and 'D's periodically visit the center and swing around each other, on their way back to the outside.

Clearly computer simulation would be required to judge whether such an arrangement could ever be stable. But if we make a leap of faith and assume that it is (at least temporarily) stable then what next?

In the event of a mis-alignment, or the presence of an external field, the 'B's would tend to pair up with the 'A's and the 'D's would tend to pair up with the 'C's. This results in the following decay products:

- 'AB' x 2
- 'CD' x 2

From our earlier definitions, this is equivalent to 2 photons and 2 neutrinos. But is there a real-world particle that corresponds to this situation?

According to the Standard Model, the answer is "no". However, we observe that the Neutral Pion is understood to have its main decay mode into 2 photons, which is quite close.

Furthermore, if look at the stats for the 'A_{2}B_{2}C_{2}D_{2}' particle, we can
see some more similarities:

Particle | Charge | Spin | Mass |
---|---|---|---|

'A_{2}B_{2}C_{2}D_{2}' | 0 | +4 ... -4 | 0 |

We note that it has an equal composition of matter ('A's and 'B's) and anti-matter ('C's and 'D's) and is therefore its own anti-particle.

Clearly the mass of 0 is inconsistent with that of the Neutral Pion (135 MeV). However, this bring us back to our earlier discussion of vector mass in relation to the 'AB' particle.

We suggest that in this situation, because there are effectively 2 'AB's acting in opposition to 2 'CD's, that the vector mass in the direction of motion (referred to earlier) is more signfiicant than the perpendicular mass.

Furthermore, if we make a great speculative leap and suggest that the vector mass in the direction of motion is equivalent to the real-world mass quantum 35MeV, then we can arrive at an approximate mass for the Neutral Pion of 4 x 35 = 140MeV.

One consequence of having a model with 4 fundamental charged, spin-1/2 particles is that it is impossible to generate a composite charged particle with spin 0. Therefore we can say from the outset that if the Standard Model is right and the charged pion is a pseudo-scalar meson with spin 0 then we're stuffed.

But if we allow for the possibility that the charged pion is a spin-1/2 particle after all, then how would we model it?

If we reverse-assemble a charged pion from its decay products of a muon and a neutrino, then we get to the following formulae:

Particle | Charge | Spin | Mass |
---|---|---|---|

'AB_{2}C_{3}D_{3}' | -1 | +9/2 ... -9/2 | +1 |

'A_{3}B_{3}CD_{2}' | +1 | +9/2 ... -9/2 | +1 |

We can think of these as an 'A' coupled to a 'C_{3}' or an 'A_{3}' coupled to a 'C'. We would expect 2 'BD's to be
sufficient for the job, given the overall repulsion is no different than for the 'AA' and 'CC' pairs in a neutral pion.

The vector mass works out as 4 x 35 = 140MeV.

This leads to the following decays:

- 'AB
_{2}C_{3}D_{3}' -> 'AB_{2}C_{2}D_{2}' + 'CD' - 'A
_{3}B_{3}CD_{2}' -> 'A_{2}B_{2}CD_{2}' + 'AB'

a.k.a. "charged pion -> muon + neutrino" (or anti-muon + photon)

Or:

- 'AB
_{2}C_{3}D_{3}' -> 'AB_{2}' + 3x 'CD' - 'A
_{3}B_{3}CD_{2}' -> 'CD_{2}' + 3x 'AB'

a.k.a. "charged pion -> electron + 3 neutrinos" (or positron + 3 photons)

We can also get an extra 'AB' photon if the electron decays to 'B' at the same time (or an extra 'CD' if the positron decays to 'D'), which correspond to the rare radiative decays.

However, it would be impossible for a charged pion to beta-decay into a neutral pion.

One alternative is if the decay into a muon involves the emission of a different kind of neutral particle (e.g. a 'BD' pair). This could be a way of accounting for a difference in neutrino flavour, which has so far been absent from any of our models.

In this case, we can reverse-assemble a charged pion to get the following formulae:

Particle | Charge | Spin | Mass |
---|---|---|---|

'AB_{3}C_{2}D_{3}' | -1 | +9/2 ... -9/2 | +3 |

'A_{2}B_{3}CD_{3}' | +1 | +9/2 ... -9/2 | +3 |

The decays now work out as:

- 'AB
_{3}C_{2}D_{3}' -> 'AB_{2}C_{2}D_{2}' + 'BD' - 'A
_{2}B_{3}CD_{3}' -> 'A_{2}B_{2}CD_{2}' + 'BD'

a.k.a. "charged pion -> muon + gluon" (or anti-muon + gluon)

Or:

- 'AB
_{3}C_{2}D_{3}' -> 'AB_{2}' + 2x 'CD' + 'BD' - 'A
_{2}B_{3}CD_{3}' -> 'CD_{2}' + 2x 'AB' + 'BD'

a.k.a. "charged pion -> electron + 2 neutrinos + gluon" (or positron + 2 photons + gluon)

However, it is not clear from a dynamics point of view why the gluon would separate from the charged pion, given that it has no internally-generated acceleration. Also it is still impossible for a charged pion to beta-decay into a neutral pion. Also the vector mass works out as 3 x 35 = 105MeV assuming the extra 'BD' pair is not worth 35MeV.

A third attempt to fix this problem is as follows:

Particle | Charge | Spin | Mass |
---|---|---|---|

'A_{2}B_{3}C_{2}D_{2}' | -1 | +9/2 ... -9/2 | +1 |

'A_{2}B_{2}C_{2}D_{3}' | +1 | +9/2 ... -9/2 | +1 |

The vector mass works out as 4 x 35 = 140MeV.

The decays are:

- 'A
_{2}B_{3}C_{2}D_{2}' -> 'AB_{2}C_{2}D_{2}' + 'AB' - 'A
_{2}B_{2}C_{2}D_{3}' -> 'A_{2}B_{2}CD_{2}' + 'CD'

a.k.a. "charged pion -> muon + photon" (or anti-muon + neutrino)

Or:

- 'A
_{2}B_{3}C_{2}D_{2}' -> 'AB_{2}' + 2x 'CD' + 'AB' - 'A
_{2}B_{2}C_{2}D_{3}' -> 'CD_{2}' + 2x 'AB' + 'CD'

a.k.a. "charged pion -> electron + 2 neutrinos + photon" (or positron + 2 photons + neutrino)

Or:

- 'A
_{2}B_{3}C_{2}D_{2}' -> 'A_{2}B_{2}C_{2}D_{2}' + 'B' - 'A
_{2}B_{2}C_{2}D_{3}' -> 'A_{2}B_{2}C_{2}D_{2}' + 'D'

a.k.a. "charged pion -> neutral pion + electron" (or neutral pion + positron)

This would seem to be a plausible model for the elusive pion beta decay, albeit without the emission of a neutrino required by the Standard Model. Whilst we could add an extra neutrino (or photon) into the formula, this would result in a vector mass of 5 x 35 = 175MeV.

Neverthless, in the absence of an extra neutrino (or photon) it is not clear what the dynamics of the electron separation would be. Also with a positive overall mass, we might expect the charged pion to behave in a similar way to an electron or muon, which is clearly not the final answer.

A final attempt is as follows:

Particle | Charge | Spin | Mass |
---|---|---|---|

'A_{2}B_{2}C_{3}D_{2}' | -1 | +9/2 ... -9/2 | -1 |

'A_{3}B_{3}C_{2}D_{2}' | +1 | +9/2 ... -9/2 | -1 |

Yielding:

- 'A
_{2}B_{2}C_{3}D_{2}' -> 'A_{2}B_{2}C_{2}D_{2}' + 'C' - 'A
_{3}B_{2}C_{2}D_{2}' -> 'A_{2}B_{2}C_{2}D_{2}' + 'A'

a.k.a. "charged pion -> neutral pion + mistaken electron" (or neutral pion + mistaken positron)

The dynamics of 'A' or 'C' emission is more believable in this case, because 3-body systems are known to be unstable. Also we can speculate that the net negative overall mass is what makes the charged pion a Hadron rather than a Lepton.

However, we now have much more of a problem modelling the decay into muons, unless the emission of 'AA' or 'CC' is what actually happens, and these have somehow been mistaken for neutrinos (which seems unlikely).

Alternatively, there could be both positive- and negative-mass muons which might be mistaken for each other in experiments. This would put muons on much the same footing as the fundamental particles.

The decays are:

- 'A
_{2}B_{2}C_{3}D_{2}' -> 'A_{2}B_{2}C_{2}D' + 'CD' - 'A
_{3}B_{2}C_{2}D_{2}' -> 'A_{2}BC_{2}D_{2}' + 'AB'

a.k.a. "charged pion -> mistaken muon + neutrino" (or mistaken anti-muon + photon)

Or:

- 'A
_{2}B_{2}C_{3}D_{2}' -> 'C_{3}D_{2}' + 2x 'AB' - 'A
_{3}B_{2}C_{2}D_{2}' -> 'A_{3}B_{2}' + 2x 'CD'
<

a.k.a. "charged pion -> mistaken positron + 2 photons" (or mistaken electron + 2 neutrinos)

Perhaps there is no single formula for the charged pion, and it depends on which formula you have as to what the decay products would be in any given situation. In this case, a superposition is the most likely scenario and we need to retro-fit superpositions into the muon and electron definitions as well.

Continuing on this theme and starting with the 'A_{2}B_{2}C_{2}D_{2}' particle,
we can generate further particles by taking away one of the 'A's or 'C's:

Particle | Charge | Spin | Mass |
---|---|---|---|

'AB_{2}C_{2}D_{2}' | -1 | +7/2 ... -7/2 | +1 |

'A_{2}B_{2}CD_{2}' | +1 | +7/2 ... -7/2 | +1 |

In this case, the overall effect of the 2 internal 'BD' pairs is to stablise the 'A' next to the 'CC' or the 'AA' next to the 'C' respectively. We can imagine such an arrangement being at least temporarily stable, with much the same reasoning used for the Neutral Pion.

Alternatively, we can think of them as:

- 2 'CD's, an 'AB' and a spare 'B'
- 2 'AB's, a 'CD' and a spare 'D'

This leads to overall properties similar to a single 'B' or a single 'D' respectively.

In these cases, the vector mass argument would suggest a mass of 3 x 35 = 105 MeV.

We therefore propose:

- 'AB
_{2}C_{2}D_{2}' as a Muon - 'A
_{2}B_{2}CD_{2}' as an Anti-Muon

If we examine the decay products, we see that our model predicts a difference compared to the Standard Model:

- 'AB
_{2}C_{2}D_{2}' -> 'AB_{2}' + 2x 'CD' - 'A
_{2}B_{2}CD_{2}' -> 'CD_{2}' + 2x 'AB'

Therefore we expect:

- The muon to decay into an electron plus 2 neutrinos (where the electron may emit a further photon)
- The anti-muon to decay into a positron plus 2 photons (where the positron may emit a further neutrino)

Note that the first of these scenarios is in accordance with the Standard Model, whereas the second is most definitely not.

We suggest that any attempt to build a composite particle with the formula 'ABCD' is doomed to fail. This is because the dynamics of the 4 particles together is inherently unstable. Exhaustive computer simulation would be required to prove this beyond doubt, but we can take a short cut with some simple thought experiments.

We know that 'B' would tend to chase 'A' and that 'C' would tend to chase 'D'. We might also think that 'C' would chase 'B' and that 'A' would chase 'D'. However, whereas 'A' and 'B' have the same response to an external field, as do 'C' and 'D', it is not the case for 'B' and 'C' or 'D' and 'A'. The net effect is that any initial arrangement of 'A','B','C' and 'D' will tend to dissociate into 'AB' and 'CD'.

We could attempt to contrive the situation by aiming 'AB' and 'CD' directly at each other. However, given that 'A' and 'C' repel each other. we would expect this to be inherently unstable, with the slightest mis-alignment (or presence of an external field) to result in them deflecting off each other and heading off in opposite directions.

However, if we used pair instead of individual particles then we have an altogether situation.

Starting with 'AA' and 'CC' pairs, the challenge is to build a composite particle that is stable, given that the 'A's repel the 'C's and so would have a natural tendency to fly apart.

The obvious thing to do is to attempt to stablise them with 'BD'
pairs. Furthermore, if we use 2 'BD' pairs then we can arrive at an
overall composition 'A_{2}B_{2}C_{2}D_{2}' which has a neat symmetry.

But surely 'A_{2}B_{2}C_{2}D_{2}' is just equivalent to having 2 'ABCD' particles
which we've already said are unstable. Well, not quite ...

In the pairs situation, we can imagine the 'AA' pair being stationary next to the 'CC' pair as long as there is a net inwards acceleration keeping them there. We can achieve this if, on average, the 'B's are the far side of the 'A's and the 'D's are the far side the 'C's.

Note that this is not what we would automatically think of, given that the 'B's and 'D's are supposed to be in 'BD' pairs. However, we could imagine orbits in which the 'B's and 'D's periodically visit the center and swing around each other, on their way back to the outside.

Clearly computer simulation would be required to judge whether such an arrangement could ever be stable. But if we make a leap of faith and assume that it is (at least temporarily) stable then what next?

In the event of a mis-alignment, or the presence of an external field, the 'B's would tend to pair up with the 'A's and the 'D's would tend to pair up with the 'C's. This results in the following decay products:

- 'AB' x 2
- 'CD' x 2

From our earlier definitions, this is equivalent to 2 photons and 2 neutrinos. But is there a real-world particle that corresponds to this situation?

According to the Standard Model, the answer is "no". However, we observe that the Neutral Pion is understood to have its main decay mode into 2 photons, which is quite close.

Furthermore, if look at the stats for the 'A_{2}B_{2}C_{2}D_{2}' particle, we can
see some more similarities:

Particle | Charge | Spin | Mass |
---|---|---|---|

'A_{2}B_{2}C_{2}D_{2}' | 0 | +4 ... -4 | 0 |

We note that it has an equal composition of matter ('A's and 'B's) and anti-matter ('C's and 'D's) and is therefore its own anti-particle.

Clearly the mass of 0 is inconsistent with that of the Neutral Pion (135 MeV). However, this bring us back to our earlier discussion of vector mass in relation to the 'AB' particle.

We suggest that in this situation, because there are effectively 2 'AB's acting in opposition to 2 'CD's, that the vector mass in the direction of motion (referred to earlier) is more signfiicant than the perpendicular mass.

Furthermore, if we make a great speculative leap and suggest that the vector mass in the direction of motion is equivalent to the real-world mass quantum 35MeV, then we can arrive at an approximate mass for the Neutral Pion of 4 x 35 = 140MeV.

One consequence of having a model with 4 fundamental charged, spin-1/2 particles is that it is impossible to generate a composite charged particle with spin 0. Therefore we can say from the outset that if the Standard Model is right and the charged pion is a pseudo-scalar meson with spin 0 then we're stuffed.

But if we allow for the possibility that the charged pion is a spin-1/2 particle after all, then how would we model it?

If we reverse-assemble a charged pion from its decay products of a muon and a neutrino, then we get to the following formulae:

Particle | Charge | Spin | Mass |
---|---|---|---|

'AB_{2}C_{3}D_{3}' | -1 | +9/2 ... -9/2 | +1 |

'A_{3}B_{3}CD_{2}' | +1 | +9/2 ... -9/2 | +1 |

We can think of these as an 'A' coupled to a 'C_{3}' or an 'A_{3}' coupled to a 'C'. We would expect 2 'BD's to be
sufficient for the job, given the overall repulsion is no different than for the 'AA' and 'CC' pairs in a neutral pion.

The vector mass works out as 4 x 35 = 140MeV.

This leads to the following decays:

- 'AB
_{2}C_{3}D_{3}' -> 'AB_{2}C_{2}D_{2}' + 'CD' - 'A
_{3}B_{3}CD_{2}' -> 'A_{2}B_{2}CD_{2}' + 'AB'

a.k.a. "charged pion -> muon + neutrino" (or anti-muon + photon)

Or:

- 'AB
_{2}C_{3}D_{3}' -> 'AB_{2}' + 3x 'CD' - 'A
_{3}B_{3}CD_{2}' -> 'CD_{2}' + 3x 'AB'

a.k.a. "charged pion -> electron + 3 neutrinos" (or positron + 3 photons)

We can also get an extra 'AB' photon if the electron decays to 'B' at the same time (or an extra 'CD' if the positron decays to 'D'), which correspond to the rare radiative decays.

However, it would be impossible for a charged pion to beta-decay into a neutral pion.

One alternative is if the decay into a muon involves the emission of a different kind of neutral particle (e.g. a 'BD' pair). This could be a way of accounting for a difference in neutrino flavour, which has so far been absent from any of our models.

In this case, we can reverse-assemble a charged pion to get the following formulae:

Particle | Charge | Spin | Mass |
---|---|---|---|

'AB_{3}C_{2}D_{3}' | -1 | +9/2 ... -9/2 | +3 |

'A_{2}B_{3}CD_{3}' | +1 | +9/2 ... -9/2 | +3 |

The decays now work out as:

- 'AB
_{3}C_{2}D_{3}' -> 'AB_{2}C_{2}D_{2}' + 'BD' - 'A
_{2}B_{3}CD_{3}' -> 'A_{2}B_{2}CD_{2}' + 'BD'

a.k.a. "charged pion -> muon + gluon" (or anti-muon + gluon)

Or:

- 'AB
_{3}C_{2}D_{3}' -> 'AB_{2}' + 2x 'CD' + 'BD' - 'A
_{2}B_{3}CD_{3}' -> 'CD_{2}' + 2x 'AB' + 'BD'

a.k.a. "charged pion -> electron + 2 neutrinos + gluon" (or positron + 2 photons + gluon)

However, it is not clear from a dynamics point of view why the gluon would separate from the charged pion, given that it has no internally-generated acceleration. Also it is still impossible for a charged pion to beta-decay into a neutral pion. Also the vector mass works out as 3 x 35 = 105MeV assuming the extra 'BD' pair is not worth 35MeV.

A third attempt to fix this problem is as follows:

Particle | Charge | Spin | Mass |
---|---|---|---|

'A_{2}B_{3}C_{2}D_{2}' | -1 | +9/2 ... -9/2 | +1 |

'A_{2}B_{2}C_{2}D_{3}' | +1 | +9/2 ... -9/2 | +1 |

The vector mass works out as 4 x 35 = 140MeV.

The decays are:

- 'A
_{2}B_{3}C_{2}D_{2}' -> 'AB_{2}C_{2}D_{2}' + 'AB' - 'A
_{2}B_{2}C_{2}D_{3}' -> 'A_{2}B_{2}CD_{2}' + 'CD'

a.k.a. "charged pion -> muon + photon" (or anti-muon + neutrino)

Or:

- 'A
_{2}B_{3}C_{2}D_{2}' -> 'AB_{2}' + 2x 'CD' + 'AB' - 'A
_{2}B_{2}C_{2}D_{3}' -> 'CD_{2}' + 2x 'AB' + 'CD'

a.k.a. "charged pion -> electron + 2 neutrinos + photon" (or positron + 2 photons + neutrino)

Or:

- 'A
_{2}B_{3}C_{2}D_{2}' -> 'A_{2}B_{2}C_{2}D_{2}' + 'B' - 'A
_{2}B_{2}C_{2}D_{3}' -> 'A_{2}B_{2}C_{2}D_{2}' + 'D'

a.k.a. "charged pion -> neutral pion + electron" (or neutral pion + positron)

This would seem to be a plausible model for the elusive pion beta decay, albeit without the emission of a neutrino required by the Standard Model. Whilst we could add an extra neutrino (or photon) into the formula, this would result in a vector mass of 5 x 35 = 175MeV.

Neverthless, in the absence of an extra neutrino (or photon) it is not clear what the dynamics of the electron separation would be. Also with a positive overall mass, we might expect the charged pion to behave in a similar way to an electron or muon, which is clearly not the final answer.

A final attempt is as follows:

Particle | Charge | Spin | Mass |
---|---|---|---|

'A_{2}B_{2}C_{3}D_{2}' | -1 | +9/2 ... -9/2 | -1 |

'A_{3}B_{3}C_{2}D_{2}' | +1 | +9/2 ... -9/2 | -1 |

Yielding:

- 'A
_{2}B_{2}C_{3}D_{2}' -> 'A_{2}B_{2}C_{2}D_{2}' + 'C' - 'A
_{3}B_{2}C_{2}D_{2}' -> 'A_{2}B_{2}C_{2}D_{2}' + 'A'

a.k.a. "charged pion -> neutral pion + mistaken electron" (or neutral pion + mistaken positron)

The dynamics of 'A' or 'C' emission is more believable in this case, because 3-body systems are known to be unstable. Also we can speculate that the net negative overall mass is what makes the charged pion a Hadron rather than a Lepton.

However, we now have much more of a problem modelling the decay into muons, unless the emission of 'AA' or 'CC' is what actually happens, and these have somehow been mistaken for neutrinos (which seems unlikely).

Alternatively, there could be both positive- and negative-mass muons which might be mistaken for each other in experiments. This would put muons on much the same footing as the fundamental particles.

The decays are:

- 'A
_{2}B_{2}C_{3}D_{2}' -> 'A_{2}B_{2}C_{2}D' + 'CD' - 'A
_{3}B_{2}C_{2}D_{2}' -> 'A_{2}BC_{2}D_{2}' + 'AB'

a.k.a. "charged pion -> mistaken muon + neutrino" (or mistaken anti-muon + photon)

Or:

- 'A
_{2}B_{2}C_{3}D_{2}' -> 'C_{3}D_{2}' + 2x 'AB' - 'A
_{3}B_{2}C_{2}D_{2}' -> 'A_{3}B_{2}' + 2x 'CD'
<

a.k.a. "charged pion -> mistaken positron + 2 photons" (or mistaken electron + 2 neutrinos)

Perhaps there is no single formula for the charged pion, and it depends on which formula you have as to what the decay products would be in any given situation. In this case, a superposition is the most likely scenario and we need to retro-fit superpositions into the muon and electron definitions as well.

In our scheme, we arrive at the following definitions:

- 'A's, 'B's, 'C's and 'D's are fundamental charged spin-1/2 particles.
- Each of them may be measured (or mistaken) as electrons or positrons in experiements.
- Fermions are fundamental or composite particles consisting of an odd number of 'A's, 'B's, 'C's or 'D's.
- Bosons are composite particles consisting of an even number of 'A's, 'B's, 'C's or 'D's
- Leptons are fundamental or composite particles with a net positive mass (predominantly 'B's or 'D's).
- Hadrons are composite particles with a net negative mass (predominantly 'A's or 'C's).
- Quarks are not explitly represented, but could be semi-accounted for as combinations of 'A's or 'C's with integer rather than fractional charges)

Particle | Proposed Formula |
---|---|

Electron | 'A_{n}B_{n+1}' (or 'A_{n+1}B_{n}') |

Positron | 'C_{n}D_{n+1}' (or 'C_{n+1}D_{n}') |

Photon | 'AB' |

Neutrino | 'CD' |

Gluon | 'BD' |

Muon | 'AB_{2}C_{2}D_{2}' (or 'A_{2}BC_{2}D_{2}') |

Anti-Muon | 'A_{2}B_{2}CD_{2}' (or 'A_{2}B_{2}CD_{2}') |

Neutral Pion | 'A_{2}B_{2}C_{2}D_{2}' |

Charged Pion (-) | 'A_{3}B_{2}C_{2}D_{2}' (or 'A_{2}B_{3}C_{2}D_{2}') |

Charged Pion (+) | 'A_{2}B_{2}C_{3}D_{2}' (or 'A_{2}B_{2}C_{2}D_{3}') |

Where particles may be mistaken for each other, the alternative formulae are shown in brackets. In these cases, a superposition of both formulae may be appropriate, depending on the situation.

Clearly the complexity of composite particles increases with the number of constituents and we feel that there is little merit in speculating further.

Therefore we will not attempt to provide models for more massive particles (kaons, protons, neutrons, tauons etc) at this stage. These will be the subject of further investigation.

So what gives if our model is to be believed?

- Neutrinos are anti-photons
- Both photons and neutrinos have zero mass
- Gluons are a by-product of electron-position annihilation
- Charged pions are spin-1/2 fermions
- Anti-Muons decay with the emission of 2 photons
- Neutral pions decay with the emission of 2 neutrinos in addition to 2 photons

As a parting shot, we will suggest that based on the 35MeV mass quantum, we could generate the following:

- Kaons with 14 mass quanta via a face-centered cubic crystal structure
- Protons and Neutrons with 27 mass quanta via a 3x3x3 cubic crystal structure

Such a model would also go a long way to explaining the proton spin crisis, although we would have a hard job explaining why protons are infinitely stable. To be continued ...