A Mathematical Preon Foundation for the Standard Model


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Further, a simple charge function determines the charge of every object. Further still, the only free assignable parameters for the entire model are four mass constants for each fermion generation. This is essentially a summary of my book: Articles hosted may not yet have been verified by peer-review and should be treated as preliminary. In particular, anything that appears to include financial or legal advice or proposed medical treatments should be treated with due caution.

Add your own feedback and questions here: You are equally welcome to be positive or negative about any paper but please be polite. In an Abelian commutative group such as the U 1 we use here , since the generators t a all commute with each other, the structure constants vanish. The two-index objects are the field strengths derived from W and G the vector fields. There are also two extra hidden parameters: Note that we have to redefine a new U 1 symmetry of weak hypercharge , different from QED, in order to achieve the unification with the weak force.

Foundations of the notion of elementary particle in the Standard Model - Physics Stack Exchange

The first convention, used in this article, is equivalent to the earlier Gell-Mann—Nishijima formula. It makes the hypercharge be twice the average charge of a given isomultiplet. One may then define the conserved current for weak isospin as. As explained above , these currents mix to create the physically observed bosons, which also leads to testable relations between the coupling constants.

To explain in a simpler way , we can see the effect of the electroweak interaction by picking out terms from the Lagrangian.

Standard Model

The U 1 symmetry, on the other hand, is similar to electromagnetism, but acts on all " weak hypercharged " fermions both left- and right-handed via the neutral Z 0 , as well as the charged fermions via the photon. The quantum chromodynamics QCD sector defines the interactions between quarks and gluons , with SU 3 symmetry, generated by T a. Since leptons do not interact with gluons, they are not affected by this sector. The Dirac Lagrangian of the quarks coupled to the gluon fields is given by.

We see that the mass-generating interaction is achieved by constant flipping of particle chirality. Additionally, we know experimentally that the W and Z bosons are massive, but a boson mass term contains the combination e. Therefore, none of the standard model fermions or bosons can "begin" with mass, but must acquire it by some other mechanism. The solution to both these problems comes from the Higgs mechanism , which involves scalar fields the number of which depend on the exact form of Higgs mechanism which to give the briefest possible description are "absorbed" by the massive bosons as degrees of freedom, and which couple to the fermions via Yukawa coupling to create what looks like mass terms.

The weak hypercharge Y W of both components is 1. There is a parameter here, at first hidden within the shape of the potential, that is very important. It is also much smaller than the Planck scale; it is approximately equal to the Higgs mass, and sets the scale for the mass of everything else.

This is the only real fine-tuning to a small nonzero value in the Standard Model, and it is called the Hierarchy problem. The Yukawa interaction terms are.

As previously mentioned, evidence shows neutrinos must have mass. But within the standard model, the right-handed neutrino does not exist, so even with a Yukawa coupling neutrinos remain massless. Current status is that experimental evidence for sterile neutrinos is not convincing. Another possibility to consider is that the neutrino satisfies the Majorana equation , which at first seems possible due to its zero electric charge.

In this case the mass term is. Here we are essentially flipping between left-handed neutrinos and right-handed anti-neutrinos it is furthermore possible but not necessary that neutrinos are their own antiparticle, so these particles are the same. For both left and right chirality cases, Majorana terms violate lepton number , but possibly at a level beyond the current sensitivity of experiments to detect such violations.

Since in any case new fields must be postulated to explain the experimental results, neutrinos are an obvious gateway to searching physics beyond the Standard Model.

The Standard Model has the following fields. These describe one generation of leptons and quarks, and there are three generations, so there are three copies of each field. By CPT symmetry, there is a set of right-handed fermions with the opposite quantum numbers.

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The column " representation " indicates under which representations of the gauge groups that each field transforms, in the order SU 3 , SU 2 , U 1. Symbols used are common but not universal; superscript C denotes an antiparticle; and for the U 1 group, the value of the weak hypercharge is listed. Note that there are twice as many left-handed lepton field components as left-handed antilepton field components in each generation, but an equal number of left-handed quark and antiquark fields.

This table is based in part on data gathered by the Particle Data Group. Upon writing the most general Lagrangian with massless neutrinos, one finds that the dynamics depend on 19 parameters, whose numerical values are established by experiment. Straightforward extensions of Standard Model with massive neutrinos need 7 more parameters, 3 masses and 4 PMNS matrix parameters, for a total of 26 parameters. The 19 certain parameters are summarized here.


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The choice of free parameters is somewhat arbitrary. Instead of fermion masses, dimensionless Yukawa couplings can be chosen as free parameters. From the theoretical point of view, the Standard Model exhibits four additional global symmetries, not postulated at the outset of its construction, collectively denoted accidental symmetries , which are continuous U 1 global symmetries.

The transformations leaving the Lagrangian invariant are:. The first transformation rule is shorthand meaning that all quark fields for all generations must be rotated by an identical phase simultaneously. By Noether's theorem , each symmetry above has an associated conservation law: Conservation of baryon number implies that the number of quarks minus the number of antiquarks is a constant. Within experimental limits, no violation of this conservation law has been found. The Standard Model predicts that each of these three numbers should be conserved separately in a manner similar to the way baryon number is conserved.

These numbers are collectively known as lepton family numbers LF. This result depends on the assumption made in Standard Model that neutrinos are massless. Experimentally, neutrino oscillations demonstrate that individual electron, muon and tau numbers are not conserved. In addition to the accidental but exact symmetries described above, the Standard Model exhibits several approximate symmetries. A similar argument in the quark sector also gives the same result for the electroweak theory.

High Energy Particle Physics

These charged currents are precisely those that entered the Fermi theory of beta decay. The action contains the charge current piece. For energy much less than the mass of the W-boson, the effective theory becomes the current—current interaction of the Fermi theory. However, this mixes with the U 1 , and another current in that sector is needed. These currents must be uncharged in order to conserve charge. So we require the neutral currents. From Wikipedia, the free encyclopedia.

For a less mathematical description, see Standard Model. Quantum electrodynamics Electroweak interaction Quantum chromodynamics Higgs mechanism. For example, the mass of an electron is really a coupling between a left-handed electron and a right-handed electron, which is the antiparticle of a left-handed positron.


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Also neutrinos show large mixings in their mass coupling, so it's not accurate to talk about neutrino masses in the flavor basis or to suggest a left-handed electron antineutrino. However, several contemporary experiments prove that neutrinos oscillate between their flavour states, which could not happen if all were massless. It is straightforward to extend the model to fit these data but there are many possibilities, so the mass eigenstates are still open. Particle Data Group Neutrino mass, mixing, and flavor change" PDF. Journal of Physics G. Since quarks can't be isolated because of QCD confinement , the quantity here is supposed to be the mass of the quark at the renormalization scale of the QCD scale.

Landau pole , but the predictions extracted from the Standard Model by current methods are all self-consistent.


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For a further discussion see e. Journal of High Energy Physics. There are non-perturbative effects which do not conserve baryon number: String theory Loop quantum gravity Causal dynamical triangulation Canonical quantum gravity Superfluid vacuum theory Twistor theory. Up quark antiquark Down quark antiquark Charm quark antiquark Strange quark antiquark Top quark antiquark Bottom quark antiquark. Photon Gluon W and Z bosons. Glueball Theta meson T meson.

A Mathematical Preon Foundation for the Standard Model A Mathematical Preon Foundation for the Standard Model
A Mathematical Preon Foundation for the Standard Model A Mathematical Preon Foundation for the Standard Model
A Mathematical Preon Foundation for the Standard Model A Mathematical Preon Foundation for the Standard Model
A Mathematical Preon Foundation for the Standard Model A Mathematical Preon Foundation for the Standard Model
A Mathematical Preon Foundation for the Standard Model A Mathematical Preon Foundation for the Standard Model

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