Abstract
The Standard Model of Particle Physics is a gauge theory of the group \(U_{Y_\mathrm{{w}}}(1)\,\times \,SU_{L}(2)\,\times \,SU_{c}(3) \). The gauge group is a direct product of three groups. Therefore, we will first briefly analyse them individually and assign to each sub-group a different type of interaction. Afterwards, we will organise the tree-level interaction pattern of all elementary particles by combining the sub-groups. Gauge invariance under the full gauge group requires all elementary fields to be massless. Finally, we will explain how the observed masses are generated by the phenomenon of spontaneous symmetry breaking and the Higgs mechanism.
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- 1.
Antiparticles are denoted by a bar over the symbol, not to be confused with the Dirac conjugate. The irreducible representations of \(U_{Y_\mathrm{{w}}}(1)\) can be written as \(\rho [U_{Y_\mathrm{{w}}}(1)]=\text {exp}[i\,Y_\mathrm{{w}}\,\theta (x)]\) and are labelled by the eigenvalue \(Y_\mathrm{{w}}\) (with the conjugated \(-Y_\mathrm{{w}}\)). Thus, we just write the real number \(Y_\mathrm{{w}}\) for \(\text {dim}_{\rho }[U_{y_\mathrm{{w}}}(1)]\) instead of a bold symbol. In this notation the \(U_{Y_\mathrm{{w}}}(1)\) singlet corresponds to \(Y_\mathrm{{w}}=0\).
- 2.
The group parameters \(g_{i}\) controlling the group action of a rigid symmetry are constant with respect to space-time.
- 3.
This follows from the complex integration contour of the Feynman propagator in QED.
- 4.
As an empirical fact, all observed objects in nature carry an integer multiple of the electric charge. This leads to the requirement that quarks have to carry a non-integer electric charge \(Q\).
- 5.
In fact, \(g_\mathrm{{s}}(E)\) diverges at very low energy scales \(E\) and perturbation theory breaks down.
- 6.
In QCD, the phenomenon of confinement was responsible for the short range interaction and not a mass of the force carrier — the gluons.
- 7.
The notation \(ISO(1,\,3)\) stems from the fact that the Poincaré group is a ten parametric Lie group and the generators of the corresponding Lie algebra are the Killing vectors of Minkowski space-time. See the definition of a Killing vector (B. 13) and set \(g_{\mu \nu }=\eta _{\mu \nu }\).
- 8.
The anti-electron-neutrino can be thought of as the electron-neutrino moving backwards in time.
- 9.
Although, neutrino oscillations suggest that neutrinos could have a mass, we do not consider the possibility of right-handed neutrinos. A sterile (not participating in the gauge interaction) heavy (at the GUT scale) right-handed neutrino could generate a tiny mass for the left-handed neutrino by the so-called seesaw mechanism. Since neutrinos are electrically neutral, they are their own antiparticles and have to be described as Majorana particles. Whether there do exist Majorana particles or not can in principle be tested by neutrino-less double beta-decay experiments [5].
- 10.
This is nothing else but a boundary term reflecting the connection between unstable systems and their sensitivity to boundary conditions.
- 11.
In the example of the ferromagnet, the formation of domain walls can be experimentally observed. They represent the boundaries between domains of different magnetizations \(\vec {M}_{i}\), denoted Weiss domains.
- 12.
Since the Yukawa coupling constant is dimensionless, it also respects the criterion of renormalizability, to be discussed in the next chapter.
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Steinwachs, C.F. (2014). Standard Model. In: Non-minimal Higgs Inflation and Frame Dependence in Cosmology. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-01842-3_3
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DOI: https://doi.org/10.1007/978-3-319-01842-3_3
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