2.4 The Standard Model in Particle Physics
2.4.2 The Fundamental Particles in Nature
Basic ideas of the Standard Model.The Standard Model in elementary particle physics concerns the strong, weak, and electromagnetic force. The main idea of the Standard Model is the following.
(a) Fundamental particles.There exist precisely 12 fundamental particles in nature, namely, the 6 quarksu, d, c, s, t, band the 6 leptons
e−(electron), à−(muon), τ− (tauon), νe, νà, ντ(3 neutrinos).
These 12 fundamental particles are fermions (i.e., they have half-integer spin, 12.) For the six quarks u, d, c, s, t, b, physicists invented fancy names. They call them up, down, charm, strange, top, and bottom quarks, respectively. The 12 fundamental particles are divided into three generations:
First generation:u, d, e−, νe. Second generation:c, s, à−, νà. Third generation:t, b, τ−, ντ.
(b) Messenger particles.The 12 fundamental particles experience the three fundamental forces (strong, electromagnetic, and weak) by the exchange of 12 messenger particles, namely,
γ(the photon), W+, W−, Z0(3 weak gauge bosons), and 8 gluons.
These 12 messenger particles are bosons (integer spin, 0,). The 8 gluons provide the “glue” for keeping the quarks together.
Table 2.3. Fundamental particles in the Standard Model
leptons quarks
particle mass MeV/c2
electric charge
spin particle mass MeV/c2
electric charge
spin electron-
neutrino νe
<15ã10−6 0 12 upu 2−8 23e 12 electron
e− 0.511 −e 12 downd 5−15 −13e 12
myon- neutrino
νà
<0.17 0 12 charmc 1000−1600 23e 12 myon
à− 105.7 −e 12 stranges 100−300 −13e 12 tauon-
neutrino ντ
<24 0 12 topt 180 000
±12 000
2 3e 12 tauonτ− 1 777 −e 12 bottomb 4100−4500 −13e 12
(c) Antiparticles. To each fundamental particle, there exists an antiparticle of same mass and opposite electric charge. The 12 antiparticles are de- noted in the following way.
First generation: u, d(antiquarks),e+ (positron),νe (anti-electron neu- trino).
Second generation:c, s(antiquarks),à+ (antimuon),νà(anti-muon neu- trino).
Third generation: t, b (antiquarks), τ+ (antitauon), ντ (anti-tau neu- trino).
It is a typical property of our universe that matter highly dominates antimat- ter. In 1928 Dirac used his relativistic equation for the electron in order to predict the existence of the antiparticlee+ to the electrone−. This particle (called positron) was experimentally discovered in a Wilson cloud chamber in 1932 by Anderson (Nobel prize in physics in 1936). The antiprotonpwas dis- covered by Chamberlain and Segr´e in 1955 (Nobel prize in physics in 1959), and the antineutron was found by Cook in 1956.
Let us now discuss some more details of the Standard Model in particle physics. A thorough investigation of the physics of the Standard Model can be found in Volume V.
Strong force and the eight gluons.As a typical example, the proton pconsists of two uquarks and onedquark. Symbolically,p=uud. The rest massm0of a proton is equal to
Table 2.4. Messenger particles in the Standard Model electroweak force
particle mass GeV/c2 electric charge spin
photon 0 0
W− 80.3 −e
W+ 80.3 e
Z0 91.2 0
strong force
particle mass electric charge spin
8 gluons 0 0 0
m0= 1.67ã10−27kg.
This corresponds to the rest energyE=m0c2= 1.5ã10−18J. Physicists like to measure energies in eV (electron volt). In this energy scale, the rest energy of the proton is equal to
E= 0.938ã109eV.
As a rule of thumb, the rest energy of a proton is equal to 1 GeV (giga electron volt), and this equals the rest energy of 1836 electrons. Physi- cists use the prefixes giga, mega, nano, femto for the corresponding factors 109,106,10−9,10−15.For more information, see Table A.1 on page 935.
In 2008, the particle accelerator at CERN (Geneva, Switzerland)28 will reach particle energies which equal the rest energy of 7 000 protons (7TeV).
The radius of the proton is equal to 10−15m = 1 fermi. Experiments show that electrons and quarks have a radius which is less than 0.001 fermi. In fact, nowadays the electron and the six quarks are considered to be point-like particles. Observe that most space of the proton is filled with massless gluons.
Each quark has three charges called red, green, and blue by physicists. The gluons see the color charge. The action of gluons onto the color charge causes the strong force. There are 8 gluons. As we will see later on, this depends on the fact that the dimension of the Lie algebra su(3) is equal to 8. Since the strong force is based on the color charge, the theory of the strong force is called quantum chromodynamics. An atom has a radius of 10−10m = 100 000 fermi. Therefore, the strong force does not play any role for the interaction between the electrons and the nucleus of an atom.
28CERN stands forConseil Europ´een pour la Recherche Nucl´eaire(European Or- ganization for Nuclear Research at Geneva); this was founded in 1953.
The total rest energy of the three quarks of the proton is equal to 30MeV.
Consider the rough form of Heisenberg’s uncertainty relation
∆xã∆p∼.
The quark moves in a proton of radius∆x= 10−15m. Therefore, the quark has the momentum∆p=/∆x, and the total energy
E=
m20c4+c2(∆p)2= 200 MeV.
Hence the total energy of the three quarks of the proton is equal to 0.6 GeV.
The remaining energy 0.4 GeV of the proton corresponds to the motion of gluons. This means that the proton has an extremely large binding energy of quarks which cannot be computed with the usual methods of perturbation theory. Physicists use highly specialized supercomputers in order to compute the binding energy of the proton, the neutron, and mesons. Mathematically, the computations are based on grid models in the framework of gauge lattice theory. The results are in good agreement with experiments.
Baryons and mesons as composite particles. An elementary parti- cle is called a hadron iff it experiences the strong force. Quarks are elemen- tary hadrons. Concerning composite hadrons, one has to distinguish between baryons and mesons.
• Each baryon consists of three quarks (e.g., the proton, the neutron, the lambda, and the sigma). Baryons have half-integer spin.
• Each meson consists of quark-antiquark pairs (e.g., the three mesons π+, π0, π− or the kaonK0). Mesons have integer spin.
The existence of mesons, which have about 250 electron masses (or 1/7 of the proton mass), was predicted by Yukawa in 1935 (Nobel prize in physics in 1949). Experimentally, mesons were discovered in cosmic rays by Powell in 1947 (Nobel prize in physics in 1950). Baryons and mesons are white, that is, the color charges red, green, and blue neutralize each other so that we cannot see the color charges. Only the gluons can see the color charges.
The existence of gluons was experimentally verified at the DESY accelerator PETRA (Hamburg, Germany) in 1979.
Electroweak force, the photon, and the three weak gauge bosons.
Maxwell’s classical theory of electromagnetism from 1864 unified the electric force with the magnetic force into the electromagnetic force. In the Standard Model of particle physics, the electromagnetic force is unified with the weak force. This yields the so-called electroweak force. The corresponding messen- ger particles are the photonγ and the three weak gauge bosons W−, W+, and Z0. As we will show later on, the appearance of 4 bosons depends on the fact that the dimension of the Lie algebra u(1)×su(2) is equal to 4. The process
d→u+e−+νe (2.53)
e− νe
d u
W−
^
^
Fig. 2.12.Beta decay of thedquark
describes the decay of a d quark into a u quark, an electron, and an anti- electron neutrino. This decay is caused by the exchange of aW−-boson (Fig.
2.12). Since we have n = ddu and p = uud for the neutron and proton, respectively, the process (2.53) is responsible for the crucial beta decay of the neutronn→p+e−+νe.
Lifetime of elementary particles.As a rule of thumb, physicists use the formula
∆t=
m0c2
for the lifetime, ∆t, of an elementary particle. This is a consequence of the energy-time uncertainty mentioned on page 142. Here, h, c, and m0 denote Planck’s constant, the velocity of light in a vacuum, and the rest mass of the particle, respectively. Recall that:=h/2π.In particular, massless particles like the photon, the gluon, and the graviton have an infinite lifetime. If a messenger particle has the lifetime∆t, then it can move the distancerduring its lifetime. This tells us that the range of the corresponding force is equal to
r=cã∆t= m0c.
This is equal to the so-called reduced Compton wave lengthλCof the particle.
Since the weak gauge bosons W± and Z0 have a rest mass of about 100 GeV/c2, their lifetime is 10−23s, and the range of the weak force is less than 0.01 fermi = 10−17m.
The quark confinement.Note that there exists a fundamental differ- ence between the electromagnetic force and the strong force (Fig. 2.13).
• The electromagnetic force vanishes for large distances and it goes to infinity if the distance goes to zero.
• In contrast to this, the strong force vanishes if the distance between the quarks goes to zero and it becomes infinite if the distance between the quarks goes to infinity.
This implies the crucial fact that the quarks behave like free particles for small distances less than 0.2 fermi = 0.2ã10−15m (asymptotic freedom). For
(a) strength of electric forceFe
(rdistance) - 6
r Fe
(b) strength of color forceFc
- 6
r Fc
Fig. 2.13. The quark confinement
distances more than one fermi between the quarks, the strong force is very large. This property of the strong force is responsible for the fact that free quarks have never been observed. This is the so-called quark confinement. A complete theoretical understanding of the quark confinement is still missing.
History of the Standard Model of particle physics.The quark hy- pothesis was formulated by Gell-Mann in 1964 (Nobel prize in physics in 1969). A similar theory was independently proposed by Zweig in 1964. In Zweig’s approach the quarks were called aces. In the very beginning of his theory, Gell-Mann was not sure whether the quarks are merely mathematical constructions (based on the representation theory of the group SU(3)) or real physical objects. The breakthrough came from physical experiments. In 1968, deeply inelastic electron-proton scattering experiments were performed at SLAC of Stanford University (California, U.S.A.). These experiments es- tablished that the proton possesses an internal structure which corresponds to a decomposition of the proton into three quarks.
The first theory of the weak force (β-decay) dates back to Fermi in 1933.
This model worked successfully for fairly low energies. In particular, Fermi was able to compute the cross sections for neutrino reactions. Since the in- finities of Fermi’s quantum field theory could not be renormalized, physicists were looking for an improved theory. In 1967 and 1968, Weinberg and Salam, respectively, formulated independently a model which unified the weak and electromagnetic force. They based their models in part on work developed by Glashow in 1961. Therefore, this model is called the Glashow–Salam–
Weinberg model (in 1979 Glashow, Salam, and Weinberg were awarded the Nobel prize in physics). The sophisticated renormalization of this model was shown by ’t Hooft in 1971 (Nobel prize in physics together with Veltman in 1999). He used mathematical tools developed for Feynman integrals by Faddeev and Popov in the 1960s (cancellation of ghosts by factorizing with respect to gauge orbits). In 1974, the reaction
p+νà→n+π++νà
was observed at Argonne National Laboratory. This process is based on the exchange of an electrically neutral Z0 boson, predicted by the Glashow–
Salam–Weinberg model. This model was finally established experimentally by the discovery of the three weak gauge bosonsW± andZ0 at the CERN particle accelerator (Geneva, Switzerland) in 1983. This experiment needs very high energies, since the W and Z bosons have a rest mass of approxi- mately 100 proton masses. Rubbia and van der Meer were awarded the 1984 Nobel prize in physics for performing this fundamental experiment, together with a large group of experimentalists at the CERN proton-antiproton col- lider. The particles of the Standard Model were discovered in the following years:
• Free leptons: electron (1895), muon (1937), electron neutrino (1956), muon neutrino (1961), tauon (1975), tauon neutrino (1975).
• Bound quarks:u, d, s(1970),c(1974),b(1977),t(1994).
• Messenger particles: photon (1922), gluons (1979), three weak gauge bosons (1983).
• Composite particles: proton (1914), neutron (1932),π-meson (1947), and J/ϕ-meson (1974).
• Antiparticles: anti-electron (positron) (1932), antiproton (1955), antineu- tron (1956).
These particles have the following lifetimes: photon and gluon (∞), uquark and proton (> 1032 years), electron (> 1023 years), d quark and neutron (887s), muon (10−6s), squark (10−8s), c andb quark (10−12s), Z andW± bosons (10−25s),tquark (10−25s). Nowadays physicists know about 80 com- posite particles.
Gravitational waves and the graviton.If we assume that all of the fundamental forces in nature are based on the exchange of messenger par- ticles, then we have to postulate the existence of an additional messenger particle called graviton which is responsible for the gravitational force. This is not a pure speculation. Let us discuss this. In 1974 Hulse and Taylor ob- served the pulsar PSR 1913+16 which has a distance of 20 000 light years from earth (Hulse and Taylor were awarded the Nobel prize in physics in 1993). This pulsar consists of two neutron stars. Each of them has 1.4 sun masses and a diameter of approximately 20 km. This means that the mass density is very large. The pulsation period of 0.0590299952709 seconds un- dergoes a periodic change because of the companion star. This is one of the stablest clocks in the universe. The two stars slowly approach each other because of a loss of gravitational energy due to gravitational radiation. On the basis of a post-Newtonian approximation to general relativity, computa- tions verify a number of predictions, including the formula for the energy loss from a binary system due to gravitational radiation.29 Observe that Hulse and Taylor established the existence of gravitational waves only in an indi- rect manner. In the near future, physicists will perform highly sensitive laser experiments in order to prove directly the existence of gravitational waves.
29This can be found in Straumann (2004).
(a) point particle
*
(b) string
Fig. 2.14.Motion of particles and strings
The idea is the following. If a gravitational wave hits a mirror, then the small change of the mirror position can be observed by the small deflection of a laser beam of length between 500 meters (in Germany) and some kilometers (the LIGO (Laser Interferometer Gravitational-Wave Observatory) project in the United States of America). In the future, it is planned to use spacelabs where the effective distance is about five million kilometers (the LISA (Laser Interferometer Space Antenna) project to be launched by NASA and ESA in about 2011).30 Right now extensive computer simulations are being per- formed in order to understand the pattern of the gravitational waves caused by supernova explosions or the collision of two black holes (resp. two col- lapsing binary neutron stars). Einstein’s theory of general relativity predicts that gravitational waves propagate with the speed of light and they have two different directions of polarization. Therefore, the hypothetical massless graviton of spin 2 should propagate with the speed of light.
String theory and the graviton. In the 1970s a true revolution took place in the thinking of theoretical physicists. Up to this time, it was assumed that the fundamental constituents of matter are particles. In contrast to this, modern string theory is based on the following fascinating hypothesis:
Elementary particles are not point-particles, but they are tiny strings living below the Planck lengthl= 10−35m.
The motion of a point-particle (resp. string) corresponds to a 1-dimensional world-line (resp. 2-dimensional world-sheet) (Fig. 2.14). There exists a very rich mathematical theory of 2-dimensional surfaces called Riemann surfaces.
It is typical for Riemann surfaces that they possess a conformal structure.
This explains why conformal field theory is closely related to string theory.
Note the following important fact:
The larger the symmetry of a physical system is, the more infor- mation about the structure of the system can be obtained from the mathematics of the relevant symmetry group.
In contrast to other dimensions, the two-dimensional (local) conformal group is huge. This is reflected by the richness of the classical theory of analytic functions on the complex plane. For example, in conformal field theory, the
30Details can be found in the monograph by Schutz (2003), Chap. 22.
structure of the fundamental Green’s function is mainly determined by the conformal symmetry. Surprisingly enough, each string theory contains a par- ticle of spin 2 which can be identified with the graviton. String theory is the most promising candidate for a unified theory of all four interactions in nature. However, one should also note that there is no experimental evidence for strings so far. It is expected that typical string effects can be only ob- served at extremely high energies. Therefore, one is looking for indirect effects which can be observed at much lower energy ranges as virtual particles or as a relic of the Big Bang (cosmic strings, magnetic monopoles, dark matter, dark energy, and so on). As an introduction to string theory, we recommend L¨ust and Theissen (1989) and Zwiebach (2004). Moreover, we refer to the standard textbooks by Green, Schwarz and Witten (1987), Vols. 1, 2, and Polchinski (1998), Vols. 1, 2. The history of string theory can be found in Greene (1999).
Supersymmetry.Physicists assume that there exists a perfect symme- try between fermions and bosons at extremely high energies. This means that, for each fermion there exists precisely one boson called the supersym- metric partner of the fermion. For example, the supersymmetric partners of electrons, quarks, photons, and gravitons are called electrinos, quarkinos, photinos, and gravitinos, respectively. Note that this so-called supersymme- try is not observed in our real world today. Physicists assume that perfect supersymmetry did exist only shortly after the Big Bang at extremely high energies. However, physicists expect that the particle accelerators of the next generation will be able to prove the existence of supersymmetric particles.
The relevant calculations have been already performed in the framework of the so-called minimal supersymmetric Standard Model. For the renormaliza- tion of the minimal supersymmetric Standard Model see Hollik et al. (2002).
As an introduction to supersymmetry, we recommend Martin (1997) (a super- symmetry primer), Bailin and Love (1997), Kalka (1997), and Kane (2000).
We also refer to Wess and Bagger (1991) and Weinberg (1995), Vol. 3.
The Higgs particles. In gauge theories, the messenger particles are massless for mathematical reasons. In sharp contrast to this, the gauge bosons W± and Z0 possess a large mass of about 100 proton masses which corre- sponds to a rest energy of 100GeV. In order to explain theoretically the par- ticle masses ofW± and Z0, physicists invented a mathematical trick called the Higgs mechanism, by using gauge invariance and adding appropriate mass terms to the Lagrangian. In terms of physics, this means that the Standard Model has to be supplemented by a number of hypothetical particles called Higgs particles. Computations show that the mass of the lightest Higgs par- ticle should be between 114 and 193 proton masses. In 2008, the energy of the new CERN collider LHC31will be large enough in order to establish the
31The letters LHC stand for Large Hadron Collider. A detailed discussion of the LHC can be found in the article by B. Mansouli´e, Physics at the large hadron collider. In: Duplantier and Rivasseau (Eds.) (2003), pp. 311–331.
existence of Higgs particles on a sound experimental basis. Note that the Standard Model of particle physics would break down if the Higgs particle did not exist.
Noncommutative geometry and the Standard Model in particle physics. It was discovered by Connes and Lott in 1990 that there is a new kind of geometry behind the Standard Model of particle physics called non- commutative geometry.32 As an introduction to noncommutative geometry, we recommend the two monographs by Connes (1994) and Gracia-Bondia, V`arilly, and Figueroa (2001).
Originally, the Higgs particle was inserted into the Standard Model by hand. Noncommutative geometry implies the appearance of the Higgs particle in a natural way. This will be studied in Volume V on the physics of the Standard Model. Noncommutative geometry is a new branch of mathematics which studies the generalization of geometric properties in terms of operator algebras.
Quantum gravity.Most physicists assume that below the Planck length lP = 10−35m and the Planck time tP = 10−44s, space and time lose their classical geometric properties, and there appear new physical effects com- bining gravitation and quantum physics in a strange manner. This has been coined as quantum gravity. Moreover, it is thinkable that space and time did not exist at the very beginning of the universe. They were created later on. In contrast to space and time, physical states always exist. They can be described mathematically by operator algebras.
In the setting of noncommutative geometry, physical states are pri- mary and space-time is secondary.
As an introduction to different approaches to quantum gravity, we recom- mend the collection of articles by Giulini, Kiefer, and L¨ammerzahl (2003) (from theory to experimental search), the survey article by Ashtekhar and Lewandowski (2004) (loop quantum gravity), and the monograph by Kiefer (2004).
Most physicists expect that the creation of the final theory of quantum gravity will dramatically change our knowledge about space and time.
The main tasks of quantum field theory.There exist two fundamen- tal kinds of quantum states, namely, scattering states and bound states. In terms of classical celestial mechanics, scattering states correspond to comets and bound states correspond to closed orbits of planets (Fig. 2.15). Physicists use quantum field theory in order to compute
• the cross section of scattering processes,
32A. Connes and J. Lott, Particle models and noncommutative geometry, Nucl.
Phys. B (Proc. Suppl.)18(1990), 29–47. See also the collection of survey articles edited by F. Scheck, W. Wend, and H. Upmeier, Noncommutative Geometry and the Standard Model of Elementary Particle Physics, Springer, Berlin, 2002.