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Standard model (basic details)
 
 
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For a basic description, see the article on the Standard Model
Standard Model

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....
.


This is a detailed description of the standard model
Standard Model

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....
 (SM) of particle physics
Particle physics

Particle physics is a branch of physics that studies the elementary particle constituents of matter and radiation, and the interactions between them....
. It describes how the lepton
Lepton

Leptons are a family of elementary particles, alongside quarks and gauge bosons . Like quarks, leptons are fermions and are subject to the electromagnetic force, the gravitational force, and weak interaction....
s, quark
Quark

Quarks are a type of elementary particle and major constituents of matter. They are the only particles in the Standard Model to experience all four fundamental interaction, which are also known as fundamental interactions....
s, gauge boson
Gauge boson

In particle physics, gauge bosons are bosonic particles that act as carriers of the fundamental interactions of nature. More specifically, elementary particles whose interactions are described by gauge theory exert forces on each other by the exchange of gauge bosons, usually as virtual particles....
s and the Higgs particle
Higgs boson

In particle physics, the Higgs boson is a massive Scalar field theory elementary particle predicted to exist by the Standard Model.The Higgs boson is the only Standard Model particle that has not yet been observed....
 fit together. It gives an outline of the main physics that the theory describes, and new directions in which it is moving.

It might be helpful to read this article along with the companion overview of the standard model
Standard Model

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....
.

article uses the Dirac basis
Gamma matrices

In mathematical physics, the gamma matrices, , also known as the Dirac matrices, form a matrix-valued representation of a set of orthogonal basis vectors for contravariant vectors in space time, from which can be constructed a Clifford algebra....
 instead of the more appropriate Weyl basis for describing spinors.






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For a basic description, see the article on the Standard Model
Standard Model

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....
.


This is a detailed description of the standard model
Standard Model

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....
 (SM) of particle physics
Particle physics

Particle physics is a branch of physics that studies the elementary particle constituents of matter and radiation, and the interactions between them....
. It describes how the lepton
Lepton

Leptons are a family of elementary particles, alongside quarks and gauge bosons . Like quarks, leptons are fermions and are subject to the electromagnetic force, the gravitational force, and weak interaction....
s, quark
Quark

Quarks are a type of elementary particle and major constituents of matter. They are the only particles in the Standard Model to experience all four fundamental interaction, which are also known as fundamental interactions....
s, gauge boson
Gauge boson

In particle physics, gauge bosons are bosonic particles that act as carriers of the fundamental interactions of nature. More specifically, elementary particles whose interactions are described by gauge theory exert forces on each other by the exchange of gauge bosons, usually as virtual particles....
s and the Higgs particle
Higgs boson

In particle physics, the Higgs boson is a massive Scalar field theory elementary particle predicted to exist by the Standard Model.The Higgs boson is the only Standard Model particle that has not yet been observed....
 fit together. It gives an outline of the main physics that the theory describes, and new directions in which it is moving.

It might be helpful to read this article along with the companion overview of the standard model
Standard Model

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....
.

A chiral gauge theory

This article uses the Dirac basis
Gamma matrices

In mathematical physics, the gamma matrices, , also known as the Dirac matrices, form a matrix-valued representation of a set of orthogonal basis vectors for contravariant vectors in space time, from which can be constructed a Clifford algebra....
 instead of the more appropriate Weyl basis for describing spinors. The Weyl basis
Gamma matrices

In mathematical physics, the gamma matrices, , also known as the Dirac matrices, form a matrix-valued representation of a set of orthogonal basis vectors for contravariant vectors in space time, from which can be constructed a Clifford algebra....
 is more convenient because there is no natural correspondence between the left-handed and right-handed fermion fields other than that generated dynamically through the Yukawa couplings after the Higgs field has acquired a vacuum expectation value (VEV
Vacuum expectation value

In quantum field theory the vacuum expectation value of an Operator is its average, expected value in the Vacuum#The quantum-mechanical vacuum....
).

The chirality
Chirality (physics)

A phenomenon is said to be chiral if it is not identical to its mirror image . The Spin of a particle may be used to define a handedness for that particle....
 projections of a Dirac field ? are "Left" chirality:   "Right" chirality:  

where is the the fifth gamma matrix
Gamma matrices

In mathematical physics, the gamma matrices, , also known as the Dirac matrices, form a matrix-valued representation of a set of orthogonal basis vectors for contravariant vectors in space time, from which can be constructed a Clifford algebra....
.

These are needed because the SM is a chiral gauge theory, ie, the two helicities are treated differently.

Right handed singlets, left handed doublets


Under the weak isospin
Weak isospin

The weak isospin in particle physics is a quantum number relating to the weak interaction, and parallels the idea of isospin under the strong interaction....
 SU(2)
Special unitary group

In mathematics, the special unitary group of degree n, denoted SU, is the group of n×n unitary matrix Matrix with determinant 1....
 the left-handed and right-handed chiralities
Chirality (physics)

A phenomenon is said to be chiral if it is not identical to its mirror image . The Spin of a particle may be used to define a handedness for that particle....
 have different charges
Coupling constant

In physics, a coupling constant, usually denoted g, is a number that determines the strength of an interaction. Usually the Lagrangian or the Hamiltonian mechanics of a system can be separated into a kinetic part and an interaction part....
. The left-handed particles are weak-isospin doublets (2), whereas the right-handed are singlets (1). The right-handed neutrino does not exist in the standard model. (However, in some extensions of the standard model they do.) The up-type quarks are charge 2/3 quarks: u, c, t. The charge -1/3 quarks (d, s, b) are called down-type quarks. The charged leptons (e, µ, t) are denoted by l, and their corresponding neutrinos by ?. The theory contains the left handed doublet of quarks and leptons the right handed singlets of quarks and and the charged leptons . There is no right-handed neutrino in the SM. This is essentially by definition. When the Standard Model was written down, there was no evidence for neutrino mass
Neutrino

Neutrinos are elementary particles that travel close to the speed of light, lack an electric charge, are able to pass through ordinary matter almost undisturbed and are thus extremely difficult to detect....
. Now, however, a series of experiments including Super-Kamiokande
Super-Kamiokande

Super-Kamiokande, or Super-K for short, is a Neutrino detector in the city of Hida, Gifu, Gifu Prefecture, Japan. The observatory was designed to search for proton decay, study solar neutrino and Neutrino#Atmospheric neutrinoss, and keep watch for supernovas in the Milky Way Galaxy....
 have indicated that neutrinos indeed have a tiny mass. This fact can be simply accommodated in the Standard Model by adding a right-handed neutrino. This, however, is not strictly necessary. For example, the dimension 5 operator also leads to neutrino oscillation
Neutrino oscillation

Neutrino oscillation is a quantum mechanics phenomenon predicted by Bruno Pontecorvo whereby a neutrino created with a specific lepton flavor can later be Quantum measurement to have a different flavor....
s.

This pattern is replicated in the next generations. We introduce a generation label and write to denote the three generations of up-type quarks, and similarly for the down type quarks. The left-handed quark doublet also carries a generation index, , as does the lepton doublet, .

Why this?


What dictates this form of the weak isospin charges? The coupling of a right-handed neutrino
Neutrino

Neutrinos are elementary particles that travel close to the speed of light, lack an electric charge, are able to pass through ordinary matter almost undisturbed and are thus extremely difficult to detect....
 to matter in weak interactions was ruled out by experiment long ago. Benjamin Lee
Benjamin W. Lee

Benjamin Whiso Lee or Ben Lee, was a Korean-American theoretical physicist. His work in theorical particle physics exerted great influence on the development of the standard model in the late 20th century especially on the renormalization and the charm quark....
 and J. Zinn-Justin, and Gerardus 't Hooft
Gerardus 't Hooft

Gerardus 't Hooft is a professor in theoretical physics at Utrecht University, the Netherlands. He shared the 1999 Nobel Prize in Physics with Martinus J....
 and Martinus Veltman
Martinus J. G. Veltman

Martinus Justinus Godefriedus Veltman is a Netherlands theoretical physicist. He shared the 1999 Nobel Prize in physics with his former student Gerardus 't Hooft for their work on particle theory....
 in 1972 suggested the inclusion of left and right-handed fields into the same multiplet
Multiplet

In theoretical physics, a multiplet is formally a group representation of an algebra.The best known example is a spin multiplet, which is a group representation of an SU subgroup of the Lorentz group....
. This possibility has been ruled out by experiment. This leaves the construction given above.

For the lepton
Lepton

Leptons are a family of elementary particles, alongside quarks and gauge bosons . Like quarks, leptons are fermions and are subject to the electromagnetic force, the gravitational force, and weak interaction....
s, the gauge group can then be . The two U(1) factors can be combined into where is the lepton number
Lepton number

In high energy physics, the lepton number is the number of leptons minus the number of antileptons.In equation form,so all leptons have assigned a value of +1, antileptons −1, and non-leptonic particles 0....
. Gauging of the lepton number is ruled out by experiment, leaving only the possible gauge group . A similar argument in the quark sector also gives the same result for the electroweak theory. This form of the theory developed from a suggestion by Sheldon Glashow
Sheldon Lee Glashow

Sheldon Lee Glashow is an United States physics. He is the Metcalf Professor of Mathematics and Physics at Boston University....
 in 1961 and extended independently by Steven Weinberg
Steven Weinberg

Steven Weinberg is an United States physicist and Nobel Prize in Physics for his contributions with Abdus Salam and Sheldon Lee Glashow to the Electroweak interaction of the weak force and electromagnetism interaction between elementary particles....
 and Abdus Salam
Abdus Salam

Abdus Salam was a Demographics of Pakistan theoretical physicist, Astrophysicist and Nobel laureate in Physics for his work in electroweak theory....
 in 1967 (and in rudimentary form by Julian Schwinger
Julian Schwinger

Julian Seymour Schwinger was an United States theoretical physicist. He is best known for his work on the theory of quantum electrodynamics, in particular for developing a relativistically invariant perturbation theory, and for renormalizing QED to one loop order....
 in 1957).

The gauge field part


The gauge group has already been described. Now one needs the field
Field (physics)

In physics, a field is a physical quantity associated to each point of spacetime. A field can be classified as a scalar field, a vector field, or a tensor field, according to whether the value of the field at each point is a scalar , a vector , or, more generally, a tensor, respectively....
s. The non-Abelian gauge field strength tensor in terms of the gauge field , where the subscript runs over spacetime dimensions (0 to 3) and the superscript over the elements of the adjoint representation
Adjoint representation

In mathematics, the adjoint representation of a Lie group G is the natural group representation of G on its own Lie algebra. This representation is the linearized version of the group action of G on itself by conjugation ....
 of the gauge group, and is the gauge coupling constant
Coupling constant

In physics, a coupling constant, usually denoted g, is a number that determines the strength of an interaction. Usually the Lagrangian or the Hamiltonian mechanics of a system can be separated into a kinetic part and an interaction part....
. The quantity is the structure constant
Algebra over a field

In mathematics, an algebra over a field is an algebraic structure consisting of a vector space together with an Binary operation, usually called multiplication, that combines any two vectors to form a third vector....
 of the gauge group, defined by the commutator . In an Abelian
Abelian

Abelian may refer to:* Abelians, a 4th century Christian sect* Hovhannes Abelian, an Armenian actor* A number of different mathematic terms named after Niels Henrik Abel::*Abelian group, a group in which the binary operation is commutative:**Category of abelian groups Ab has abelian groups as objects and group homomorphisms as morph...
 group, since the generators all commute with each other, the structure constants vanish, and the field tensor takes its usual Abelian
Abelian

Abelian may refer to:* Abelians, a 4th century Christian sect* Hovhannes Abelian, an Armenian actor* A number of different mathematic terms named after Niels Henrik Abel::*Abelian group, a group in which the binary operation is commutative:**Category of abelian groups Ab has abelian groups as objects and group homomorphisms as morph...
 form.

We need to introduce three gauge fields corresponding to each of the subgroups —
  • The gluon field tensor will be denoted by , where the index labels elements of the 8 representation of colour SU(3)
    Special unitary group

    In mathematics, the special unitary group of degree n, denoted SU, is the group of n×n unitary matrix Matrix with determinant 1....
    . The strong coupling constant will be labelled or , the former where there is any ambiguity. The observations leading to the discovery of this part of the SM are discussed in the article in quantum chromodynamics
    Quantum chromodynamics

    Quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons ....
    .
  • The notation will be used for the gauge field tensor of SU(2)
    Special unitary group

    In mathematics, the special unitary group of degree n, denoted SU, is the group of n×n unitary matrix Matrix with determinant 1....
     where a runs over the 3 of this group. The coupling will always be denoted by . The gauge field will be denoted by .
  • The gauge field tensor for the U(1)
    Circle group

    In mathematics, the circle group, denoted by T , is the multiplicative group of all complex numbers with absolute value 1, i.e., the unit circle in the complex plane....
     of weak hypercharge
    Weak hypercharge

    The weak hypercharge in particle physics is a conserved quantum number relating the electrical charge and the third component of weak isospin, and is similar to the Gell-Mann?Nishijima formula for the hypercharge of strong interactions ....
     will be denoted by , the coupling by , and the gauge field by .


The gauge field Lagrangian


The gauge part of the electroweak Lagrangian
Lagrangian

The Lagrangian, , of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics known as Lagrangian mechanics....
 is The standard model Lagrangian consists of another similar term constructed using the gluon field tensor.

The W, Z and photon


The charged W boson
W and Z bosons

The W and Z bosons are the elementary particles that mediate the weak force. Their discovery has been heralded as a major success for the Standard Model of particle physics....
s are the linear combinations Z boson
W and Z bosons

The W and Z bosons are the elementary particles that mediate the weak force. Their discovery has been heralded as a major success for the Standard Model of particle physics....
s and photon
Photon

In physics, the photon is an elementary particle, the quantum of the electromagnetic field and the basic unit of light and all other forms of electromagnetic radiation....
s are mixtures of and . The precise mixture is determined by the Weinberg angle
Weinberg angle

The Weinberg angle or weak mixing angle is a parameter in the Steven Weinberg-Abdus Salam theory of the electroweak force. It gives a relationship between the masses of the W and Z bosons , as well as the ratio of Z boson mediated interaction which behaves like a photon, i.e....
  and  

with

The electric charge
Electric charge

Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. Electrically charged matter is influenced by, and produces, electromagnetic fields....
 Q, weak isospin
Weak isospin

The weak isospin in particle physics is a quantum number relating to the weak interaction, and parallels the idea of isospin under the strong interaction....
 T3 (aka Tz) and weak hypercharge
Weak hypercharge

The weak hypercharge in particle physics is a conserved quantum number relating the electrical charge and the third component of weak isospin, and is similar to the Gell-Mann?Nishijima formula for the hypercharge of strong interactions ....
 YW are related by

Note: usually weak hypercharge is scaled so that

which is a formally equivalent to Gell-Mann–Nishijima formula
Gell-Mann–Nishijima formula

The Gell-Mann?Nishijima formula relates the baryon number B, the strangeness S, the isospin Iz of hadrons to the charge Q....
.

The charged and neutral current couplings


The charged currents are
These charged currents are precisely those which entered the Fermi theory of beta decay. The action contains the charge current piece
It will be discussed later in this article that the W boson becomes massive, and for energy much less than this mass, the effective theory becomes the current-current interaction of the Fermi theory.

However, gauge invariance now requires that the component of the gauge field also be coupled to a current which lies in the triplet of SU(2). However, this mixes with the U(1), and another current in that sector is needed. These currents must be uncharged in order to conserve charge. So we require the neutral currents
The neutral current piece in the Lagrangian is then
There are no mass terms for the fermions. Everything else will come through the scalar (Higgs) sector.

Quantum chromodynamics


Leptons carry no colour charge; quarks do. Moreover, the quarks have only vector couplings to the gluons, ie, the two helicities are treated on par in this part of the standard model. So the coupling term is given by
Here Ta stands for the generators of SU(3) colour. The mass term in QCD arises from interactions in the Higgs sector.

The Higgs field


One requires masses for the W, Z, quarks and leptons. Recent experiments have also shown that the neutrino
Neutrino

Neutrinos are elementary particles that travel close to the speed of light, lack an electric charge, are able to pass through ordinary matter almost undisturbed and are thus extremely difficult to detect....
 has a mass. However, the details of the mechanism that give the neutrinos a mass are not yet clear. So this article deals with the classic version of the SM (circa 1990s, when neutrino mass
Neutrino

Neutrinos are elementary particles that travel close to the speed of light, lack an electric charge, are able to pass through ordinary matter almost undisturbed and are thus extremely difficult to detect....
es could be neglected with impunity).

The Yukawa terms


Giving a mass to a Dirac field requires a term in the Lagrangian which couples the left and right helicities. A complex scalar doublet (charge 2) Higgs field, is introduced, which couples through the Yukawa interaction
Yukawa interaction

In particle physics, Yukawa's interaction, named after Hideki Yukawa, is an interaction between a scalar field and a Dirac field of the type...
where are 3×3 matrices of Yukawa couplings, with the ij term giving the coupling of the generations i and j.

Symmetry breaking


The Higgs part of the Lagrangian is
where and , so that the mechanism of spontaneous symmetry breaking
Spontaneous symmetry breaking

In physics, spontaneous symmetry breaking occurs when a system that is symmetry in physics with respect to some symmetry group goes into a vacuum state that is not symmetric....
 can be used.

In a unitarity gauge
Gauge fixing

In the physics of gauge theory, gauge fixing denotes a mathematical procedure for coping with redundant Degrees of freedom in field variables....
 one can set and make real. Then is the non-vanishing vacuum expectation value
Vacuum expectation value

In quantum field theory the vacuum expectation value of an Operator is its average, expected value in the Vacuum#The quantum-mechanical vacuum....
 of the Higgs field. Putting this into , a mass term for the fermions is obtained, with a mass matrix . From , quadratic terms in and arise, which give masses to the W and Z bosons


Including neutrino mass


As mentioned earlier, in the SM classic there are no right-handed neutrinos. The same mechanism as the quarks would then give masses to the electrons, but because of the missing right-handed neutrino the neutrinos remain massless. Small changes can also accommodate massive neutrinos. Two approaches are possible—
  • Add , and give a mass term as usual (this is called a Dirac mass)
  • Write a Majorana mass term by combining with its complex conjugate


See seesaw mechanism
Seesaw mechanism

In theoretical physics, in the area of quantum field theory, the seesaw mechanism is a mechanism to generate very small numbers from "reasonable numbers" and very large numbers....
.

These alternatives can easily lead beyond the SM.

The GIM mechanism and the CKM matrix


The Yukawa couplings for the quarks are not required to have any particular symmetry, so they cannot be diagonalized by unitary transformation
Unitary transformation

Informally, a unitary transformation is a transformation that respects the dot product: the dot product of two vectors before the transformation is equal to their dot product after the transformation....
s. However, they can be diagonalized by separate unitary matrices acting on the two sides (this process is called a singular value decomposition
Singular value decomposition

In linear algebra, the singular value decomposition is an important Matrix decomposition of a rectangular real number or complex number matrix , with several applications in signal processing and statistics....
). In other words one can find diagonal matrices
This matrix V is called the Cabibbo-Kobayashi-Maskawa (CKM) matrix
CKM Matrix

#REDIRECTCabibbo?Kobayashi?Maskawa matrix...
. The matrix is usually not diagonal, and therefore causes mixing of the quark flavours. It also gives rise to CP-violations in the SM.

See also

  • Overview of standard model
    Standard Model

    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....
     of particle physics
    Particle physics

    Particle physics is a branch of physics that studies the elementary particle constituents of matter and radiation, and the interactions between them....
  • Weak interactions, Fermi theory of beta decay and electroweak theory
  • Strong interaction
    Strong interaction

    In particle physics, the strong interaction, or strong force, or color force, holds quarks and gluons together to form protons, neutrons and other particles....
    s, flavour
    Flavour (particle physics)

    In particle physics, flavour or flavor is a quantum number of elementary particles. In quantum chromodynamics flavour is a global symmetry....
    , quark model
    Quark model

    In physics, the quark model is a classification scheme for hadrons in terms of their valence quarks, i.e., the quarks which give rise to the quantum numbers of the hadrons....
     and quantum chromodynamics
    Quantum chromodynamics

    Quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons ....
  • For open questions, see quark matter
    QCD matter

    Quark matter or QCD matter refers to any of a number of theorized phase of matter whose degrees of freedom include quarks and gluons. These theoretical phases would occur at extremely high temperatures and densities, billions of times higher than can be produced in equilibrium in laboratories....
    , CP violation and neutrino mass
    Neutrino

    Neutrinos are elementary particles that travel close to the speed of light, lack an electric charge, are able to pass through ordinary matter almost undisturbed and are thus extremely difficult to detect....
    es
  • Beyond the Standard Model
    Beyond the Standard Model

    In physics, the Standard Model of particle physics is currently the best description of all experimental data.Nevertheless, there are reasons to believe that there are phenomena that are not accurately described by this theory and...
  • Weinberg angle
    Weinberg angle

    The Weinberg angle or weak mixing angle is a parameter in the Steven Weinberg-Abdus Salam theory of the electroweak force. It gives a relationship between the masses of the W and Z bosons , as well as the ratio of Z boson mediated interaction which behaves like a photon, i.e....
  • Electroweak interaction
    Electroweak interaction

    In particle physics, the electroweak interaction is the unified description of two of the four fundamental interactions of nature: electromagnetism and the weak interaction....
  • Fundamental interaction
    Fundamental interaction

    In physics, a fundamental interaction or fundamental force is a process by which elementary particles interact with each other. An interaction is often described as a field , and is mediated by the exchange of gauge bosons between particles....
  • Particle physics
    Particle physics

    Particle physics is a branch of physics that studies the elementary particle constituents of matter and radiation, and the interactions between them....