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Encyclopedia > Standard model (basic details)

This is a detailed description of the standard model (SM) of particle physics. It describes how the leptons, quarks, gauge bosons and the Higgs particle fit together. It gives an outline of the main physics that the theory describes, and new directions in which it is moving. The Standard Model of Fundamental Particles and Interactions For the Standard Model in Cryptography, see Standard Model (cryptography). ... Thousands of particles explode from the collision point of two relativistic (100 GeV per nucleon) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ... In physics, a lepton is a particle with spin-1/2 (a fermion) that does not experience the strong interaction (that is, the strong nuclear force). ... The six flavours of quarks and their most likely decay modes. ... Gauge bosons are bosonic particles which act as carriers of the fundamental forces of Nature. ... The Higgs boson is a hypothetical massive scalar elementary particle predicted to exist by the Standard Model of particle physics. ...


It might be helpful to read this article along with the companion overview of the standard model. The Standard Model of Fundamental Particles and Interactions For the Standard Model in Cryptography, see Standard Model (cryptography). ...

Contents

The classical theory

A chiral gauge theory

This article uses the Dirac basis instead of the more appropriate Weyl basis for describing spinors. The Weyl basis 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). In quantum field theory the vacuum expectation value (also called condensate) of an operator is its average, expected value in the vacuum. ...


The helicity projections of a Dirac field ψ are In particle physics, helicity is the projection of the angular momentum to the direction of motion: Because the angular momentum with respect to an axis has discrete values, helicity is discrete, too. ...

left helicity:  psi_L = frac{1}{2}(1-gamma_5)psi   and the right helicity:  psi_R = frac{1}{2}(1+gamma_5)psi

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 SU(2) the left handed and right handed helicities have different charges. 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 theory contains The weak isospin in theoretical physics parallels the idea of the isospin under the strong interaction, but applied under the weak interaction. ... In mathematics, the special unitary group of degree n, denoted SU(n), is the group of n×n unitary matrices with unit determinant. ... In physics, a coupling constant, usually denoted g, is a number that determines the strength of an interaction. ...

the left handed doublet of quarks QL = (uL,dL) and leptons EL = (νL,eL)
the right handed singlets of quarks uR and dR and the electron eR.

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. Now, however, a series of experiments including Super-Kamiokande 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 frac{(Hl)^2}{Lambda} also leads to neutrino oscillations. Super-Kamiokande, or Super-K for short, is a neutrino observatory in Japan. ...


This pattern is replicated in the next generations. We introduce a generation label i = 1,2,3 and write ui 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, QiL, as does the lepton doublet, EiL.


Why this?

What dictates this form of the weak isospin charges? The coupling of a right handed neutrino to matter in weak interactions was ruled out by experiment long ago. Benjamin Lee and J. Zinn-Justin, and Gerardus 't Hooft and Martinus Veltman in 1972 suggested the inclusion of left and right handed fields into the same multiplet. This possibility has been ruled out by experiment. This leaves the construction given above. Neutrinos are elementary particles denoted by the symbol ν. Travelling close to the speed of light, lacking electric charge and able to pass through ordinary matter almost undisturbed, they are extremely difficult to detect. ... The weak nuclear force or weak interaction is one of the four fundamental forces of nature. ... Category: ... Gerard t Hooft at Harvard University Gerardus (Gerard) t Hooft [ut-hooft] (The prefix ’t is pronounced as ‘ut’ and stands for ‘het’) (born July 5, 1946) is a professor in theoretical physics at Utrecht University, The Netherlands. ... Martinus J.G. Veltman (Tini for short) (born June 27, 1931, Waalwijk) is a 1999 Nobel Prize in Physics laureate for elucidating the quantum structure of electroweak interactions in physics, work done at Utrecht University, The Netherlands. ... In theoretical physics, a multiplet is formally a group representation of an algebra. ...


For the leptons, the gauge group can then be SU(2)_l times U(1)_L times U(1)_R. The two U(1) factors can be combined into U(1)_Y times U(1)_l where l is the lepton number. Gauging of the lepton number is ruled out by experiment, leaving only the possible gauge group SU(2)_L times U(1)_Y. 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 in 1961 and extended independently by Steven Weinberg and Abdus Salam in 1967 (and in rudimentary form by Julian Schwinger in 1957). In physics, a lepton is a particle with spin-1/2 (a fermion) that does not experience the strong interaction (that is, the strong nuclear force). ... In high energy physics, the lepton number is the number of leptons minus the number of antileptons. ... Sheldon Glashow at Harvard University Professor Sheldon Lee Glashow (born December 5, 1932) is an American physicist. ... Steven Weinberg (born May 3, 1933) is an American physicist. ... For other uses, see Abdus Salam (disambiguation). ... Julian Seymour Schwinger (February 12, 1918 -- July 16, 1994) was an American theoretical physicist. ...


The gauge field part

The gauge group has already been described. Now one needs the fields. The non-Abelian gauge field strength tensor The magnitude of an electric field surrounding two equally charged (repelling) particles. ...

F^a_{ munu}=partial_{mu}A^{a}_{ nu} - partial_{nu}A^{a}_{ mu} + g f^{abc}A^{b}_{mu}A^{c}_{nu}

in terms of the gauge field A^{a}_{mu}, where the subscript μ runs over spacetime dimensions (0 to 3) and the superscript a over the elements of the adjoint representation of the gauge group, and g is the gauge coupling constant. The quantity fabc is the structure constant of the gauge group, defined by the commutator [ta,tb] = fabctc. In an Abelian group, since the generators ta all commute with each other, the structure constants vanish, and the field tensor takes its usual Abelian form. In mathematics, the adjoint representation (or adjoint action) of a Lie group G is the natural representation of G on its own Lie algebra. ... In physics, a coupling constant, usually denoted g, is a number that determines the strength of an interaction. ... In mathematics, an algebra over a field K, or a K-algebra, is a vector space A over K equipped with a compatible notion of multiplication of elements of A. A straightforward generalisation allows K to be any commutative ring. ... Abelian, in mathematics, is used in many different definitions: In group theory: 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 morphisms Metabelian group is a group where the commutator subgroup is contained in... Abelian, in mathematics, is used in many different definitions: In group theory: 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 morphisms Metabelian group is a group where the commutator subgroup is contained in...


We need to introduce three gauge fields corresponding to each of the subgroups SU(3) times SU(2) times U(1)

  • The gluon field tensor will be denoted by G^{a}_{munu}, where the index a labels elements of the 8 representation of colour SU(3). The strong coupling constant will be labelled gs or g, the former where there is any ambiguity. The observations leading to the discovery of this part of the SM is discussed in the article in quantum chromodynamics.
  • The notation W^{a}_{munu} will be used for the gauge field tensor of SU(2) where a runs over the 3 of this group. The coupling will always be denoted by g. The gauge field will be denoted by W^{a}_{mu}.
  • The gauge field tensor for the U(1) of weak hypercharge will be denoted by Bμν, the coupling by g', and the gauge field by Bμ.

In mathematics, the special unitary group of degree n, denoted SU(n), is the group of n×n unitary matrices with unit determinant. ... Quantum chromodynamics (abbreviated as QCD) is the theory of the strong interaction (color force), a fundamental force describing the interactions of the quarks and gluons found in hadrons (such as the proton, neutron or pion). ... In mathematics, the special unitary group of degree n, denoted SU(n), is the group of n×n unitary matrices with unit determinant. ... In mathematics, the circle group, denoted by T (or in blackboard bold by ), is the multiplicative group of all complex numbers with absolute value 1. ... Weak hypercharge is twice the difference between the electrical charge and the weak isospin. ...

The gauge field Lagrangian

The gauge part of the electroweak Lagrangian is A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a function of the dynamical variables and concisely describes the equations of motion of the system. ...

mathcal{L}_{g} = -frac{1}{4} (W^{a}_{munu} W^{a,munu} + B_{munu}B^{munu})

The standard model Lagrangian consists of another similar term constructed using the gluon field tensor.


The W, Z and photon

The charged W bosons are the linear combinations In physics, the W and Z bosons are the elementary particles that mediate the weak nuclear force. ...

W^{pm}_{mu} = W^{1}_{mu} pm i W^{2}_{mu}

Z bosons (Zμ) and photons (Aμ) are mixtures of W3 and B. The precise mixture is determined by the Weinberg angle θW: In physics, the W and Z bosons are the elementary particles that mediate the weak nuclear force. ... In modern physics the photon is the elementary particle responsible for electromagnetic phenomena. ... The Weinberg angle or weak mixing angle is a parameter in the Abdus-Salam theory of the electroweak force. ...

Z_{mu}=cos theta_W W^{3}_{mu} - sin theta_W B_{mu},   and   A_{mu} = sin theta_W W^{3}_{mu} + cos theta_W B_{mu}.

The electric charge, weak isospin and weak hypercharge are related by Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. ... The weak isospin in theoretical physics parallels the idea of the isospin under the strong interaction, but applied under the weak interaction. ... Weak hypercharge is twice the difference between the electrical charge and the weak isospin. ...

 Q = T^3 + Y , .

The charged and neutral current couplings

The charged currents J^{pm}=J^{1}pm i J^{2} are

J^+_mu = overline U_{iL}gamma_mu D_{iL} +overline nu_{iL}gamma_mu l_{iL}.

These charged currents are precisely those which entered the Fermi theory of beta decay. The action contains the charge current piece In physics, Fermis interaction is an old explanation of the weak force, proposed by Enrico Fermi. ...

mathcal{L}_{CC} = frac g{sqrt2}(J_mu^+W^{-,mu}+J_mu^-W^{+,mu}).

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 W3 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

J_mu^3 = frac12(overline U_{iL}gamma_mu U_{iL} - overline D_{iL}gamma_mu D_{iL} + overline nu_{iL}gamma_mu nu_{iL} - overline l_{iL}gamma_mu l_{iL})
J_mu^{em} = frac23overline U_{iL}gamma_mu U_{iL} -frac13overline D_{iL}gamma_mu D_{iL} - overline l_{iL}gamma_mu l_{iL}.

The neutral current piece in the Lagrangian is then

mathcal{L}_{NC} = e J_mu^{em} A^mu + frac g{costheta_W}(J_mu^3-sin^2theta_WJ_mu^{em})Z^mu

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

mathcal{L}_{QCD} = overline U (partial_mu-ig_sG_mu^a T^a)gamma^mu U + overline D (partial_mu-iG_mu^a T^a)gamma^mu D.

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 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 masses could be neglected with impunity). Neutrinos are elementary particles denoted by the symbol ν. Travelling close to the speed of light, lacking electric charge and able to pass through ordinary matter almost undisturbed, they are extremely difficult to detect. ... For the band, see 1990s (band). ... Neutrinos are elementary particles denoted by the symbol ν. Travelling close to the speed of light, lacking electric charge and able to pass through ordinary matter almost undisturbed, they are extremely difficult to detect. ...


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, +0) is introduced, which couples through the Yukawa interaction In particle physics, Yukawa interaction, named after Hideki Yukawa, is an interaction between a scalar field and a Dirac field of the type . The Yukawa interaction can be used to describe the strong nuclear force between nucleons (which are fermions), mediated by pions (which are scalar mesons). ...

mathcal{L}_{YU} = overline U_L G_u U_R phi^0 - overline D_L G_u U_R phi^- + overline U_L G_d D_R phi^+ + overline D_L G_d D_R phi^0 + hc

where Gu,d 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

mathcal{L}_H = -frac12[(partial_mu -i W_mu^a t^a -i B_mu)phi]^2 - frac{mu^2}2 phi^*phi-fraclambda4 (phi^*phi)^2,

where λ > 0 and μ2 < 0, so that the mechanism of spontaneous symmetry breaking can be used. Spontaneous symmetry breaking in physics takes place when a system that is symmetric with respect to some symmetry group goes into a vacuum state that is not symmetric. ...


In a unitarity gauge one can set φ + = 0 and make φ0 real. Then < φ0 > = v is the non-vanishing vacuum expectation value of the Higgs field. Putting this into mathcal{L}_{YU}, a mass term for the fermions is obtained, with a mass matrix vGu,d. From mathcal{L}_H, quadratic terms in Wμ and Bμ arise, which give masses to the W and Z bosons In the physics of gauge theories, gauge fixing (also called choosing a gauge) denotes a mathematical procedure for coping with redundant degrees of freedom in field variables. ... In quantum field theory the vacuum expectation value (also called condensate) of an operator is its average, expected value in the vacuum. ...

M_W = frac{v|g|}2 qquadqquad M_Z=frac{vsqrt{g^2+{g'}^2}}2.

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 νR, and give a mass term as usual (this is called a Dirac mass)
  • Write a Majorana mass term by combining νL with its complex conjugate
overlinenu_L^*nu_L

See seesaw mechanism. In theoretical physics, 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 transformations. However, they can be diagonalized by separate unitary matrices acting on the two sides (this process is called a singular value decomposition). In other words one can find diagonal matrices A unitary transformation is an isomorphism between two Hilbert spaces. ... In linear algebra, the singular value decomposition (SVD) is an important factorization of a rectangular real or complex matrix, with several applications in signal processing and statistics. ...

 M^{2} = V_r G^{+} G V_r^{+} = V_l G G^{+} V_l^{+}.

Using these matrices, one can define linear combinations of the quark fields used till now to get a definition of the quark fields in the mass basis. These are the quark flavours of quantum chromodynamics. Flavour (or flavor) is a quantum number of elementary particles related to their weak interactions. ... Quantum chromodynamics (abbreviated as QCD) is the theory of the strong interaction (color force), a fundamental force describing the interactions of the quarks and gluons found in hadrons (such as the proton, neutron or pion). ...


On making these transformations in the neutral current, one finds no mixing of flavours, provided there is a doublet of quarks in each generation. This cancellation of flavour changing neutral currents is referred to as the Glashow-Iliopoulos-Maiani (GIM) mechanism. This mechanism was proposed before the charm quark was found, and therefore predicted this new flavour. In theoretical physics, flavor changing neutral currents (FCNCs) are expressions that change the flavor of a fermion current without altering its electric charge. ... The charm quark is a second-generation quark with a charge of +(2/3)e. ...


However, if these transformations are made in the charged current, then one finds that the current takes the form

overline U^{(m)}_Lgamma_mu VD^{(m)}_L,qquadqquad V=V_{u,l} V^+_{d,l}.

This matrix V is called the Cabibbo-Kobayashi-Maskawa (CKM) 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. In the standard model of particle physics the Cabibbo Kobayashi Maskawa matrix (CKM matrix, sometimes earlier called KM matrix) is a unitary matrix which contains information on the strength of flavour changing weak decays. ... CP-symmetry is a symmetry obtained by a combination of the C-symmetry and the P-symmetry. ...


See also

The Standard Model of Fundamental Particles and Interactions For the Standard Model in Cryptography, see Standard Model (cryptography). ... Thousands of particles explode from the collision point of two relativistic (100 GeV per nucleon) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ... The weak nuclear force or weak interaction is one of the four fundamental forces of nature. ... In physics, Fermis interaction is an old explanation of the weak force, proposed by Enrico Fermi. ... In physics, the electroweak theory presents a unified description of two of the four fundamental forces of nature: electromagnetism and the weak nuclear force. ... The strong interaction or strong force is today understood to represent the interactions between quarks and gluons as detailed by the theory of quantum chromodynamics (QCD). ... Flavour (or flavor) is a quantum number of elementary particles related to their weak interactions. ... In physics, the quark model is a classification scheme for hadrons in terms of their valence quarks, ie, the quarks (and antiquarks) which give rise to the quantum numbers of the hadrons. ... Quantum chromodynamics (abbreviated as QCD) is the theory of the strong interaction (color force), a fundamental force describing the interactions of the quarks and gluons found in hadrons (such as the proton, neutron or pion). ... Quark matter or QCD matter refers to any of a number of phases of matter whose degrees of freedom include quarks and gluons. ... CP-symmetry is a symmetry obtained by a combination of the C-symmetry and the P-symmetry. ... Neutrinos are elementary particles denoted by the symbol ν. Travelling close to the speed of light, lacking electric charge and able to pass through ordinary matter almost undisturbed, they are extremely difficult to detect. ... This article or section seems not to be written in the formal tone expected of an encyclopedia entry. ...

References and external links

  • The quantum theory of fields (vol 2), by S. Weinberg (Cambridge University Press, 1996) ISBN 0-521-55002-5.
  • Theory of elementary particles, by T.P. Cheng and L.F. Lee (Oxford University Press, 1982) ISBN 0-19-851961-3.
  • An introduction to quantum field theory, by M.E. Peskin and D.V. Schroeder (HarperCollins, 1995) ISBN 0-201-50397-2.

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