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Standard Model
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The Standard Model of particle physics is a theory of three of the four known fundamental interactions and the elementary particles that take part in these interactions. These particles make up all visible matter in the universe. The standard model is a gauge theory of the electroweak and strong interactions with the gauge group SU(3)×SU(2)×U(1).
Numerous experiments carried out since the mid-20th century have yielded findings consistent with the Standard Model.
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Encyclopedia
The Standard Model of particle physics is a theory of three of the four known fundamental interactions and the elementary particles that take part in these interactions. These particles make up all visible matter in the universe. The standard model is a gauge theory of the electroweak and strong interactions with the gauge group SU(3)×SU(2)×U(1).
Numerous experiments carried out since the mid-20th century have yielded findings consistent with the Standard Model. The Standard Model falls short of being a complete theory of fundamental interactions because it does not include gravity and because it is incompatible with the recent observation of neutrino oscillations.
Historical background
The first step towards the Standard Model was Sheldon Glashow's discovery, in 1963, of a way to combine the electromagnetic and weak interactions. In 1967, Steven Weinberg and Abdus Salam incorporated the Higgs mechanism into Glashow's electroweak theory, giving it its modern form. The Higgs mechanism is also believed to give rise to the rest masses of all the elementary particles the Standard Model accounts for, the W and Z bosons, and the fermions, the latter broken down into quarks and leptons.
After the discovery at CERN of neutral weak currents, caused by boson exchange, the electroweak theory became widely accepted. Glashow, Salam, and Weinberg shared the 1979 Nobel Prize in Physics for discovering the electroweak theory. The W and Z bosons were discovered experimentally in 1981, and their masses were found to be as the Standard Model predicted.
The theory of the strong interaction, to which many contributed, acquired its modern form around 1973-74, when experiments confirmed that the hadrons were composed of fractionally charged quarks.
Overview
At present, matter and energy are best understood in terms of the kinematics and interactions of elementary particles. To date, physics has reduced the laws governing the behavior and interaction of all known forms of matter and energy, to a small set of fundamental laws and theories. A major goal of physics is to find the "common ground" that would unite all of these theories into one integrated theory of everything, of which all the other known laws would be special cases, and from which the behavior of all matter and energy could be derived (at least in principle). "Details can be worked out if the situation is simple enough for us to make an approximation, which is almost never, but often we can understand more or less what is happening." (Feynman's lectures on Physics, Vol 1. 2–7)
The Standard Model groups two major extant theories — quantum electroweak and quantum chromodynamics — into an internally consistent theory describing the interactions between all experimentally observed particles. The Standard Model describes each type of particle in terms of a mathematical field, via quantum field theory. For a technical description of these fields and their interactions, see Standard Model (mathematical formulation).
Particle content
Elementary particles: fermions
In the Standard Model, fermions are defined as elementary particles having spin-, and that respect the Pauli Exclusion Principle in accordance with the spin-statistics theorem. There are 12 known fermions, each with a corresponding antiparticle. They are classified according to how they interact (or equivalently, by what charges they carry). There are six quarks (up, down, charm, strange, top, bottom), and six leptons (electron, muon, tauon, and their corresponding neutrinos).
Pairs from each classification are grouped together to form a generation, with corresponding particles exhibiting similar physical behavior (see table at right).
The defining property of the quarks is that they carry color charge, and hence, interact via the strong force. The infrared confining behavior of the strong force results in quarks being perpetually (or at least since very soon after the start of the big bang) bound to one another, forming color-neutral composite particles (hadrons) containing either a quark and an antiquark (mesons) or three quarks (baryons). The familiar proton and the neutron are the two baryons having the smallest mass. Quarks also carry electric charge and weak isospin. Hence they interact with other fermions both electromagnetically and via the weak nuclear interaction.
The remaining six fermions do not carry color charge and are called leptons. The three neutrinos do not carry electric charge either, so their motion is directly influenced only by the weak nuclear force, which makes them notoriously difficult to detect. However, by virtue of carrying an electric charge, the electron, muon and the tauon interact electromagnetically.
Each member of a generation has greater mass than the corresponding particles of lower generations. The first generation charged particles do not decay; hence all ordinary (baryonic) matter is made of such particles. Specifically, all atoms consist of electrons orbiting atomic nuclei ultimately constituted of up and down quarks. Second and third generations charged particles, on the other hand, decay with very short half lives, and are observed only in very high-energy environments. Neutrinos of all generations also do not decay and pervade the universe, but rarely interact with baryonic matter.
Force mediating particles
Forces in physics are the ways that particles interact and influence each other. At a macro level, the electromagnetic force allows particles to interact with one another via electric and magnetic fields, and the force of gravitation allows two particles with mass to attract one another in accordance with Newton's Law of Gravitation. The standard model explains such forces as resulting from matter particles exchanging other particles, known as force mediating particles. When a force mediating particle is exchanged, at a macro level the effect is equivalent to a force influencing both of them, and the particle is therefore said to have mediated (i.e., been the agent of) that force. Force mediating particles are believed to be the reason why the forces and interactions between particles observed in the laboratory and in the universe exist.
The known force mediating particles described by the Standard Model also all have spin (as do matter particles), but in their case, the value of the spin is 1, meaning that all force mediating particles are bosons. As a result, they do not follow the Pauli Exclusion Principle. The different types of force mediating particles are described below.
- Photons mediate the electromagnetic force between electrically charged particles. The photon is massless and is well-described by the theory of quantum electrodynamics.
- The , , and gauge bosons mediate the weak interactions between particles of different flavors (all quarks and leptons). They are massive, with the being more massive than the . The weak interactions involving the act on exclusively left-handed particles and right-handed antiparticles. Furthermore, the carry an electric charge of +1 and −1 and couple to the electromagnetic interactions. The electrically neutral boson interacts with both left-handed particles and antiparticles. These three gauge bosons along with the photons are grouped together which collectively mediate the electroweak interactions.
- The eight gluons mediate the strong interactions between color charged particles (the quarks). Gluons are massless. The eightfold multiplicity of gluons is labeled by a combination of color and an anticolor charge (e.g., red–antigreen). Because the gluon has an effective color charge, they can interact among themselves. The gluons and their interactions are described by the theory of quantum chromodynamics.
The interactions between all the particles described by the Standard Model are summarized by the diagram at the top of this section.
The Higgs boson
The Higgs particle is a massive scalar elementary particle predicted by the Standard Model. It has no intrinsic spin, and for that reason is classified as a boson (like the force mediating particles, which have integer spin). Because an exceptionally large amount of energy and beam luminosity are required to create a Higgs boson in high energy colliders, it is the only fundamental particle predicted by the Standard Model that has yet to be observed.
The Higgs boson plays a unique role in the Standard Model, by explaining why the other elementary particles, the photon and gluon excepted, are massive. In particular, the Higgs boson would explain why the photon has no mass, while the W and Z bosons are very heavy. Elementary particle masses, and the differences between electromagnetism (mediated by the photon) and the weak force (mediated by the W and Z bosons), are critical to many aspects of the structure of microscopic (and hence macroscopic) matter. In electroweak theory, the Higgs boson generates the masses of the leptons (electron, muon, and tauon) and quarks.
As yet, no experiment has directly detected the existence of the Higgs boson, but there is some indirect evidence for it. It is hoped that the Large Hadron Collider at CERN will confirm the existence of this particle.
Chien-Peng Yuan et al believe that the Higgs boson may have been produced but overlooked:
"...experimenters may have already overlooked a Higgs particle, argues theorist Chien-Peng Yuan of Michigan State University in East Lansing and his colleagues. They considered the simplest possible supersymmetric theory. Ordinarily, theorists assume that the lightest of theory's five Higgses is the one that drags on the W and Z. Those interactions then feed back on Higgs and push its mass above 121 times the mass of the proton, the highest mass searched for at CERN's Large Electron–Positron (LEP) collider, which ran from 1989 to 2000. But it's possible that the lightest Higgs weighs as little as 65 times the mass of a proton and has been missed, Yuan and colleagues argue in a paper to be published in Physical Review Letters."
Field content
The standard model has the following fields:
Spin 1
- A U(1) gauge field with coupling (weak U(1) or weak hypercharge)
- An SU(2) gauge field with coupling (weak SU(2) or weak isospin)
- An SU(3) gauge field with coupling (strong SU(3) or color)
Spin 1/2
The spin 1/2 particles are in representations of the gauge groups. For the U(1) group, we list the value of the weak hypercharge instead.
The left-handed fermionic fields are:
- An SU(3) singlet, SU(2) doublet with U(1) weak hypercharge -1 (left-handed lepton)
- An SU(3) singlet, SU(2) singlet with U(1) weak hypercharge 2 (left-handed antilepton)
- An SU(3) triplet, SU(2) doublet, with U(1) weak hypercharge (left-handed quarks)
- An SU(3) triplet, SU(2) singlet, with U(1) weak hypercharge - (left-handed up-type antiquark)
- An SU(3) triplet, SU(2) singlet, with U(1) weak hypercharge (left-handed down-type antiquark)
By CPT symmetry, there is a set of right-handed fermions with the opposite quantum numbers.
This describes one generation of leptons and quarks, and there are three generations, so there are three copies of each field. 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.
Spin 0
- An SU(2) doublet H with U(1) hyper-charge -1 (Higgs field)
note that , summed over the two SU(2) components, is invariant under both SU(2) and under U(1), and so it can appear as a renormalizable term in the Lagrangian, as can its square.
This field acquires a vacuum expectation value, leaving a combination of the weak isospin and hypercharge unbroken. This is the electromagnetic gauge group, and the photon remains massless. The standard formula for the electric charge (which defines the normalization of the weak hypercharge, which would otherwise be somewhat arbitrary) is:
-
Beware that the normalization sometimes is also used.
Lagrangian
The Lagrangian for the spin 1 and spin 1/2 fields is the most general renormalizable gauge field Lagrangian with no fine tunings:
Spin 1:
where the traces are over the SU(2) and SU(3) indices hidden in W and G respectively. The two-index objects are the field strengths derived from W and G the vector fields. There are also two extra hidden parameters: the theta angles for SU(2) and SU(3).
Note that the spin 1/2 particles can have no mass terms, because there is no right/left helicity pair with the same SU(2) and SU(3) representation and the same weak hypercharge. This means that if the gauge charges were conserved in the vacuum, none of the spin 1/2 particles could ever swap helicity, and they would all be massless.
For a neutral fermion, for example, a hypothetical right-handed lepton N, or in relativistic two-spinor notation, with no SU(3),SU(2) representation and zero charge, it is possible to add the term:
-
and this term gives the neutral fermion a Majorana mass. Since the generic value for M will be of order 1, such a particle would generically be unacceptably heavy.
Note that the interactions are completely determined by the theory – the leptons introduce no extra parameters.
Higgs mechanism
The Lagrangian for the Higgs includes the most general renormalizable self interaction:
-
The parameter v2 has dimensions of mass squared, and it gives the location where the classical Lagrangian is at a minimum. In order for the Higgs mechanism to work, v2 must be a positive number. v has units of mass, and it is the only parameter in the standard model which is not dimensionless. 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.
It is traditional to choose the SU(2) gauge so that the Higgs doublet in the vacuum has expectation value (v,0).
Masses and CKM matrix
The rest of the interactions are the most general spin-0 spin- Yukawa interactions, and there are many of these. These constitute most of the free parameters in the model. The Yukawa couplings generate the masses and mixings once the Higgs gets its vacuum expectation value.
The terms generate a mass term for each of the three generations of leptons. There are 9 of these terms, but by relabeling L and R, the matrix can be diagonalized. Since only the upper component of H is nonzero, the upper SU(2) component of L mixes with R to make the electron, the muon, and the tauon, leaving over a lower massless component, the neutrino.
The terms QHU generate up masses, while QHD generate down masses. But since there is more than one right-handed singlet in each generation, it is not possible to diagonalize both with a good basis for the fields, and there is an extra CKM matrix.
Theoretical aspects
Construction of the Standard Model Lagrangian
Parameters of the Standard Model | Symbol | Description | Renormalization
scheme (point) | Value |
|---|
| | Electron mass | | 511 keV | | | Muon mass | | 106 MeV | | | Tauon mass | | 1.78 GeV | | | Up quark mass | | 1.9 MeV | | | Down quark mass | | 4.4 MeV | | | Strange quark mass | | 87 MeV | | | Charm quark mass | | 1.32 GeV | | | Bottom quark mass | | 4.24 GeV | | | Top quark mass | (on-shell scheme) | 172.7 GeV | | | CKM 12-mixing angle | | 0.229 | | | CKM 23-mixing angle | | 0.042 | | | CKM 13-mixing angle | | 0.004 | | | CKM CP-violating Phase | | 0.995 | | | U(1) gauge coupling | | 0.357 | | | SU(2) gauge coupling | | 0.652 | | | SU(3) gauge coupling | | 1.221 | | | QCD Vacuum Angle | | ~0 | | | Higgs quadratic coupling | | Unknown | | | Higgs self-coupling strength | | Unknown |
Technically, quantum field theory provides the mathematical framework for the standard model, in which a Lagrangian controls the dynamics and kinematics of the theory. Each kind of particle is described in terms of a dynamical field that pervades space-time. The construction of the standard model proceeds following the modern method of constructing most field theories: by first postulating a set of symmetries of the system, and then by writing down the most general renormalizable Lagrangian from its particle (field) content that observes these symmetries.
The global Poincaré symmetry is postulated for all relativistic quantum field theories. It consists of the familiar translational symmetry, rotational symmetry and the inertial reference frame invariance central to the theory of special relativity. The local SU(3)SU(2)U(1) gauge symmetry is an internal symmetry that essentially defines the standard model. Roughly, the three factors of the gauge symmetry give rise to the three fundamental interactions. The fields fall into different representations of the various symmetry groups of the Standard Model (see table). Upon writing the most general Lagrangian, one finds that the dynamics depend on 19 parameters, whose numerical values are established by experiment. The parameters are summarized in the table at right.
The QCD sector
The electroweak sector
The electroweak sector is a Yang–Mills gauge theory with the symmetry group ,
where is the gauge field; is the weak hypercharge — the generator of the U(1) group; is the
three-component SU(2) gauge field; are the Pauli matrices — infinitesimal generators of the SU(2) group, the subscript indicates that they only act on left fermions; and are coupling constants.
The Higgs sector
In the Standard Model, the Higgs field is a complex spinor of the group SU(2)L:
where the indexes and indicate the Q-charges of the components. The YW-charge of both components is 1.
Before symmetry breaking, the Higgs Lagrangian is:
for which you may also find the following abbreviation:
Additional symmetries of the Standard Model
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. The fields , and , are the 2nd (muon) and 3rd (tauon) generation analogs of and fields.
By Noether's theorem, each symmetry above has an associated conservation law: the conservation of baryon number, electron number, muon number, and tauon number. Each quark is assigned a baryon number of 1/3, while each antiquark is assigned a baryon number of -1/3. 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.
Similarly, each electron and its associated neutrino is assigned an electron number of +1, while the antielectron and the associated antineutrino carry -1 electron number. 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. 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). Symmetry works differently for quarks than for leptons, mainly because the Standard Model predicts that neutrinos are massless. However, it was recently found that neutrinos have small masses and oscillate between flavors, signaling that the conservation of lepton family number is violated.
In addition to the accidental (but exact) symmetries described above, the Standard Model exhibits several approximate symmetries. These are the "SU(2) custodial symmetry" and the "SU(2) or SU(3) quark flavor symmetry."
Field content of the Standard Model Field
(1st generation) | Spin | Gauge group
Representation | Baryon
Number | Electron
Number |
|---|
| Left-handed quark | | | (, , ) | | | | Left-handed up antiquark | | | (, , ) | | | | Left-handed down antiquark | | | (, , ) | | | | Left-handed lepton | | | (, , ) | | | | Left-handed antielectron | | | (, , ) | | | | Hypercharge gauge field | | | (, , ) | | | | Isospin gauge field | | | (, , ) | | | | Gluon field | | | (, , ) | | | | Higgs field | | | (, , ) | | |
List of standard model fermions
This table is based in part on data gathered by the Particle Data Group .
Tests and predictions
The Standard Model (SM) predicted the existence of the W and Z bosons, gluon, and the top and charm quarks before these particles were observed. Their predicted properties were experimentally confirmed with good precision. To give an idea of the success of the SM, the following table compares the measured masses of the W and Z bosons with the masses predicted by the SM:
| Quantity | Measured (GeV) | SM prediction (GeV) |
|---|
| Mass of W boson | 80.398±0.025 | 80.3900±0.0180 | | Mass of Z boson | 91.1876±0.0021 | 91.1874±0.0021 |
The SM also makes several predictions about the decay of Z bosons, which have been experimentally confirmed by the Large Electron-Positron Collider at CERN.
Challenges to the standard model
There is some experimental evidence consistent with neutrinos having mass, which the Standard Model does not allow. To accommodate such findings, the Standard Model can be modified by adding a non-renormalizable interaction of lepton fields with the square of the Higgs field. This is natural in certain grand unified theories, and if new physics appears at about 1016 GeV, the neutrino masses are of the right order of magnitude.
Currently, there is one elementary particle predicted by the Standard Model that has yet to be observed: the Higgs boson. A major reason for building the Large Hadron Collider is that the high energies of which it is capable are expected to make the Higgs observable. However, as of August 2008, there is only indirect empirical evidence for the existence of the Higgs boson, so that its discovery cannot be claimed.
A fair amount of theoretical and experimental research has attempted to extend the Standard Model into a theory of everything, a complete theory explaining all physical phenomena. Inadequacies of the Standard Model that motivate such research 'Beyond the Standard Model' include:
- Does not attempt to explain gravity, and there is no known way of adapting the quantum field theory of the sort the Standard Model employs freely, with general relativity, the canonical theory of gravity. This means, among other things, that we have no good theory for the very early universe;
- Seems rather ad-hoc and inelegant, requiring 19 numerical constants whose values are unrelated and arbitrary. Although the Standard Model, as it now stands, cannot explain why neutrinos have masses (and the specifics of neutrino mass are still unclear), it is believed that explaining neutrino mass will require an additional 7 or 8 constants;
- Gives rise to the hierarchy problem, namely why the weak scale and Planck scale are so disparate;
- Should be modified so as to be consistent with the emerging "standard model of cosmology." Specifically, a truly satisfactory theory of the elementary particles and of the fundamental interactions must explain the initial conditions of the universe that gave rise to certain observed properties of the present-day universe, properties such as the predominance of matter over antimatter (matter/antimatter asymmetry), and its isotropy and homogeneity over large distances.
See also
Introductory textbooks
Advanced textbooks
- Highlights gauge theory aspects of the Standard Model.
- Highlights dynamical and phenomenological aspects of the Standard Model.
- :Highlights group-theoretical aspects of the Standard Model.
Journal articles
- S.F. Novaes, Standard Model: An Introduction,
- D.P. Roy, Basic Constituents of Matter and their Interactions — A Progress Report,
- Y. Hayato et al., Search for Proton Decay through p ? ?K+ in a Large Water Cherenkov Detector. Phys. Rev. Lett. 83, 1529 (1999).
- Ernest S. Abers and Benjamin W. Lee, Gauge theories. Physics Reports (Elsevier) C9, 1–141 (1973).
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