# mathematics of particle physics

of the gauge field also be coupled to a current that lies in the triplet of SU(2). 3 μ {\displaystyle m_{e}={\frac {y_{e}}{\sqrt {2}}}v} The two-index objects are the field strengths derived from W and G the vector fields. The Standard Model of particle physics, which classifies elementary particles into several groups, is at the core of modern physics. For energy much less than the mass of the W-boson, the effective theory becomes the current–current contact interaction of the Fermi theory, v ≡ The free field model can be solved exactly, and then the solutions to the full model can be expressed as perturbations of the free field solutions, for example using the Dyson series. = p is the electric current and Similarly, the muons and their neutrinos are assigned a muon number of +1 and the tau leptons are assigned a tau lepton number of +1. In the 20th century, physicists began exploring the goings on at the smallest levels of matter, and among their most startling modern discoveries was the amount of different particles in the universe. (  The neutrino parameter values are still uncertain. To retain gauge invariance, the underlying fields must be massless, but the observable states can gain masses in the process. H {\displaystyle {\bar {\psi }}_{L}\psi _{L}} As an aside, the right-handed neutrino originally did not exist in the standard model – but the discovery of neutrino oscillation implies that neutrinos must have mass, and since chirality can change during the propagation of a massive particle, right-handed neutrinos must exist in reality. , Another possibility to consider is that the neutrino satisfies the Majorana equation, which at first seems possible due to its zero electric charge. , which is approximately 0.129, can be chosen as a free parameter. R 2 μ This has been proven for the CKM matrix, and is expected for the PMNS matrix. As an aside, if a complex phase term exists within either of these matrices, it will give rise to direct CP violation, which could explain the dominance of matter over antimatter in our current universe. where ti are the generators of the group. 4 + ¯ There are six distinct types of quark: Leptons are a type of fundamental particle that do not experience strong interaction. The interaction picture constitutes an intermediate between the two, where some time dependence is placed in the operators (the quantum fields) and some in the state vector. R For example, electron mass depends on the Yukawa coupling of electron to Higgs field, and its value is e GeV. μ g ¯ In an Abelian (commutative) group (such as the U(1) we use here), since the generators ta all commute with each other, the structure constants vanish. fields. v A similar argument in the quark sector also gives the same result for the electroweak theory. An obvious solution is to simply add a right-handed neutrino νR resulting in a Dirac mass term as usual. e However, gauge invariance now requires that the component A quick derivation is indeed presented at the article on Feynman diagrams. ) The Lagrangian can also be derived without using creation and annihilation operators (the "canonical" formalism), by using a "path integral" approach, pioneered by Feynman building on the earlier work of Dirac. 2 The unfortunate difference is that the terms tend to sound similar to each other. , where Conservation of baryon number implies that the number of quarks minus the number of antiquarks is a constant. ( W 2 {\displaystyle a_{r}^{\dagger }(\mathbf {p} )} 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. If a left-handed fermion spans some representation its antiparticle (right-handed antifermion) spans the dual representation (note that p The idea is that the state vector should only change when particles interact, meaning a free particle is one whose quantum state is constant. 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. here. In a unitarity gauge one can set 2 The local SU(3) × SU(2) × U(1) gauge symmetry is the internal symmetry. p ( A free particle can be represented by a mass term, and a kinetic term which relates to the "motion" of the fields. e ¯ {\displaystyle {\mathcal {L}}={\mathcal {L}}_{0}+{\mathcal {L}}_{\mathrm {I} }} Gauging of the lepton number is ruled out by experiment, leaving only the possible gauge group SU(2)L × U(1)Y. L r † {\displaystyle \psi _{\mathrm {e} }^{R}} Molecules are complex structures composed of multiple atoms bonded together. † and   e If ψ is thought of as an n × 1 matrix then † ψ μ Quarks always exist in combination to form subatomic particles known as hadrons. 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). μ For the spin-1 fields, first define the field strength tensor, for a given gauge field (here we use A), with gauge coupling constant g. The quantity  f abc is the structure constant of the particular gauge group, defined by the commutator. ), In addition to the accidental (but exact) symmetries described above, the Standard Model exhibits several approximate symmetries. is the non-vanishing vacuum expectation value of the Higgs field. The next step is to "couple" the gauge fields to the fermions, allowing for interactions. As previously mentioned, evidence shows neutrinos must have mass. can for example be seen to add one particle, because it will add 1 to the eigenvalue of the a-particle number operator, and the momentum of that particle ought to be p since the eigenvalue of the vector-valued momentum operator increases by that much.