# Modern Physics: Standard Model of Particle Physics (Invited Lecture)

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1 Modern Physics: Standard Model of Particle Physics (Invited Lecture) Pichet Vanichchapongjaroen The Institute for Fundamental Study, Naresuan University 1 Informations Lecturer Pichet Vanichchapongjaroen The Institute for Fundamental Study Naresuan University Thailand Office: TA208B Lecture Time Sc5-214 Department of Physics, Faculty of Science, Naresuan University 17th March :00-10:00 Webpage of lecture Objectives To discuss some basic topics on elementary particles and interactions in the Standard Model. Recommended Books (only need to read a part of section 1) [1] I. Aitchison and A. Hey, Gauge Theories in Particle Physics: A Practical Introduction: From Relativistic Quantum Mechanics to QED, Fourth Edition, vol. 1. Taylor & Francis, [2] D. Griffiths, Introduction to Elementary Particles. Physics textbook. Wiley, Elementary particles One of the goals of science is to identify the most fundamental matters and how they interact. From the history, it was first found that all the known matters at the time are 1

2 made of atoms. Atoms are made of nuclei and electrons. Nuclei are made of protons and neutrons which are in turn made of quarks. In the process, there are many more particles discovered, and there has been the successful attempt to classify and describe them. This attempt is the theory called the Standard Model. The Standard Model of particle physics is a theory concerning elementary particles and their interactions. The elementary particles are classified as fermions, gauge bosons, and Higgs boson. Fermions are elementary particles which are components of all known matters. Each of them is a spin-1/2 particle, which obeys Pauli exclusion principle. Fermions in the Standard Model are further classified as quarks and leptons. Quarks are elementary particles which have to be bound together. They are not directly observed independently. There are six quark flavours: up (u), down (d), charm (c), strange (s), top (t), and bottom (b). The flavours are classified by mass, electric charge. Quarks can be combined to form particles. u c t d s b Table 1: Quark flavours. When two quarks are combined, they form a particle called a meson. But when three quarks are combined, they form a particle called a baryon. There are many types of mesons and baryons, but most notably, protons and neutrons are baryons. Leptons are the types of elementary fermions which can be observed independently. There are six lepton flavours: electron (e ), muon (µ ), tau (τ ), electron neutrino (ν e ), muon neutrino (ν µ ), tau neutrino (ν τ ). e µ τ ν e ν µ ν τ Table 2: Lepton flavours. Bosons are particles of integral spin. In the Standard Model, there are Higgs boson (spin 0), and gauge bosons (spin 1). They are not restricted by Pauli exclusion principle. So two bosons of the same quantum numbers can exist together. Higgs bosons (H) are responsible for mass of particles in the Standard Model. H Table 3: Higgs boson. 2

3 Gauge bosons are responsible for interactions. Fermions in the Standard Model interact by exchanging the gauge bosons. There are 3 families of gauge bosons characterised by interactions they mediate. Photons (γ) are carriers of electromagnetic interaction. W ± and Z bosons are carriers of weak interaction. Gluons (g) are carriers of strong interaction. γ W ±, Z 0 g Table 4: Gauge bosons. Detailed studies of the properties of these particles requires a whole course. So let us come back to this only if we have time. 3 Antiparticles In 1928, Dirac proposed a theory which explains relativistic electron with properties agreeing with that from the experiments. However, Dirac theory predicts that apart from positive-energy solutions with E = m 2 c 4 + p 2 c 2, there are also negative-energy solutions with E = m 2 c 4 + p 2 c 2. The presence of negative-energy particles suggests instability. Positive energy electrons can keep going down to more and more negative energy states and keep emitting photons in the process. E=0 Figure 1: Positive energy electrons can keep going down to more and more negative energy states and keep emitting photons in the process. Dirac argues that from Pauli s exclusion principle, there can be no more than two electrons per state. So all negative energy states are usually filled. This keeps positive 3

4 energy electrons from falling down to negative energy states. The completely filled negative energy states are called Dirac sea. If an electron is excited from the sea, a hole will be left. This hole is interpreted as a particle of positive charge. It was discovered in 1931 by Anderson that this hole is the positron which is the antiparticle of the electron. E=0 E=0 Figure 2: Left: Dirac sea keeps a positive energy electron from falling down to negative energy states. Right: a hole in the sea is interpreted as a particle of positive charge. Every particle has an antiparticle. For example, an antiparticle of an electron e is a positron e +, an antiparticle of a proton p is an antiproton p, an antiparticle of a neutron n is an antineutron n, an antiparticle of a muon µ is an antimuon µ +. Some particles are their own antiparticle. For example, photon γ is its own antiparticle. 4 Interactions There are four types of known interactions in the nature. Gravitational interaction. This is described by Newton s law of gravitation. When two particles of mass m and M are apart by a distance r, the gravitational force between them is given by F = G NMm r 2, (4.1) where the Newton s constant G N = N m 2 /kg 2. The new description of gravitational interaction is given by general relativity by Einstein. In this theory the curvature of spacetime provides the gravitational interaction. The Standard Model cannot be combined with the gravitational interaction. Electromagnetic interaction. This interaction is explained by Maxwell s equation. Weak nuclear force. This is the interaction which is responsible for beta decay n p + e + ν e. (4.2) Strong nuclear force. This is the interaction which binds the nucleons together. 4

5 5 Selected topics We have given a quick overview of the Standard Model. But there are actually many topics one needs to master in order to properly study the Standard Models. Most notably, one inevitably needs to study quantum field theory (QFT). A good place to study QFT is at IF, Naresuan University. Here, we will not discuss QFT, but will try to cover some topics of the Standard Model in a bit more details. 5.1 Electric charge of the fundamental fermions Let us present the electric charge of the fundamental fermions. The antiparticles have Fermions Charge q/e u, c, t 2/3 d, s, b -1/3 e, µ, τ -1 ν e, ν µ, ν τ 0 opposite charge from their counter part. Antifermions Charge q/e ū, c, t -2/3 d, s, b 1/3 e +, µ +, τ + 1 ν e, ν µ, ν τ 0 Each proton and neutron is composed of 3 quarks of types u and d. From considering charge alone, we can argue that a proton is composed of two up quarks and one down quarks, while a neutron is composed of one up quark and two down quarks. We may write this schematically as p = uud, n = udd. (5.1) 5.2 Yukawa s theory The strong interaction between neutron and proton decreases rapidly when the particles are r 2 fm apart. The potential should decay more quicker than the Coulomb-like potential 1/r. Yukawa proposed that the potential energy for neutron-proton interaction is given by e r/a V ( r) = g2, (5.2) 4π r where g is the coupling constant of the Yukawa interaction, and a is a range parameter. This potential satisfies ( 2 1a ) V ( r) = g 2 δ (3) ( r). (5.3) 2 5

6 e r r 1 r Figure 3: Yukawa-like potential (red) decays rapidly than that of Coulomb-like potential (blue). Yukawa then extended the potential to be relativistic. For r 0, the equation is given by ( c 2 t 1 ) V ( r, t) = 0. (5.4) 2 a 2 This has a solution of the form where Comparing this with V ( r, t) exp ( ) p r i iet, (5.5) E 2 = p 2 c 2 + c2 2 one can conclude that the field V has a mass m given by a 2. (5.6) E 2 = p 2 c 2 + m 2 c 4, (5.7) m = ac. (5.8) For a 2 fm, we have m 100 MeV/c 2, which is the mass of a particle predicted by Yukawa. Let us do this more explicitly. Start from writing m = c ac 2. (5.9) Note that ( ) c = ( m 2 kg s 1 )( m s 1 ev ) = ev m J (5.10) So it is easy to see that m ev m m 1 c MeV/c2. (5.11) The corresponding process is a neutron decaying to a proton and the force-mediating particle corresponding to the field V. This mediating particle is called a pion π. The process is n p + π. (5.12) 6

7 However, this is not a real process as the rest mass of the neutron is less than the combined rest mass of the proton and the pion. In particle physics, there exists many virtual processes which can be understood as intermediate steps. Strong processes do not change the types of quarks. So the types of quarks contained within π can be determined from udd uud + π. (5.13) So π = ūd. (5.14) 5.3 The mass of the photon Yukawa s idea can also be applied to electromagnetic and weak interaction. The range of electromagnetic interaction is infinite. So a. This matches with the fact that the potential of the electromagnetic interaction is given by where we used the fact that V ( r) = e2 1 4πɛ 0 r, (5.15) e r/a 1 as a. (5.16) As a result, the mass of photon, the carrier of the electromagnetic particle, is m = ac 0 as a. (5.17) But this argument does not say explicitly whether photons are exactly massless or having extremely small mass. It can in fact be shown by the requirement of gauge symmetry that the photons are exactly massless. 5.4 Weak interactions In the nuclear β decay, n p + e + ν e (5.18) two types of interactions are responsible. Both of them are in the virtual process. This starts from the strong process n p + π, (5.19) which we have discussed. Then followed by a weak process π e + ν e. (5.20) Both of them can be summarised in a diagram The weak process is mediated by a W boson. The range of this interaction is very small. So by following the method of Yukawa, the W boson should be very heavy. It was found that the mass of the W boson is 80 GeV/c 2. So the range of the force is m. 7

8 Figure 4: Left: nuclear β decay process. Right: the intermediate virtual weak process. 8

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