- Standard Model (mathematical formulation)
:"For a basic description, see the article on the

Standard Model ".This is a detailed description of the standard model (SM) of

particle physics . It describes how thelepton s,quark s,gauge boson s 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.It might be helpful to read this article along with the companion overview of the standard model.

**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).

The chirality projections of a Dirac field ψ are::“Left” chirality: $psi\_L\; =\; frac\{1\}\{2\}(1-gamma\_5)psi$::“Right” chirality: $psi\_R\; =\; frac\{1\}\{2\}(1+gamma\_5)psi$

where $gamma\_5$ is the the fifth gamma matrix.

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 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 charged leptons (e, μ, τ) are denoted by l, and their corresponding neutrinos by ν. The theory contains::the left handed doublet of quarks $Q\_L=\; (u\_L,\; d\_L)$ and leptons $E\_L=\; (\; u\_\{lL\},\; l\_L)$::the right handed singlets of quarks $u\_R$ and $d\_R$ and the charged leptons $l\_R$. 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 experimentsincludingSuper-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.This pattern is replicated in the next generations. We introduce a generation label $i=1,2,3$ and write $u\_i$ 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, $Q\_\{iL\}$, as does the lepton doublet, $E\_\{iL\}$.

**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 andJ. Zinn-Justin , andGerardus 't Hooft and Martinus Veltman in 1972 suggested the inclusion of left and right handed fields into the samemultiplet . This possibility has been ruled out by experiment. This leaves the construction given above.For the

lepton s, the gauge group can then be $SU(2)\_l\; imes\; U(1)\_L\; imes\; U(1)\_R$. The two U(1) factors can be combined into $U(1)\_Y\; imes\; U(1)\_l$ where $l$ is thelepton number . Gauging of the lepton number is ruled out by experiment, leaving only the possible gauge group $SU(2)\_L\; imes\; U(1)\_Y$. 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 bySteven Weinberg andAbdus Salam in 1967 (and in rudimentary form byJulian Schwinger in 1957).**The gauge field part**The gauge group has already been described. Now one needs the fields. The non-Abelian gauge field strength tensor:$F^a\_\{\; mu\; u\}=partial\_\{mu\}A^\{a\}\_\{\; u\}\; -\; partial\_\{\; u\}A^\{a\}\_\{\; mu\}\; +\; g\; f^\{abc\}A^\{b\}\_\{mu\}A^\{c\}\_\{\; u\}$in terms of the gauge field $A^\{a\}\_\{mu\}$, where the subscript $mu$ 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 gaugecoupling constant . The quantity $f^\{abc\}$ is the structure constant of the gauge group, defined by the commutator $[t\_\{a\},\; t\_\{b\}]\; =\; f^\{abc\}\; t\_\{c\}$. In anAbelian group, since the generators $t\_\{a\}$ all commute with each other, the structure constants vanish, and the field tensor takes its usualAbelian form.We need to introduce three gauge fields corresponding to each of the subgroups $SU(3)\; imes\; SU(2)\; imes\; U(1)$ —

*The gluon field tensor will be denoted by $G^\{a\}\_\{mu\; u\}$, where the index $a$ labels elements of the**8**representation of colour SU(3). The strong coupling constant will be labelled $g\_\{s\}$ or $g$, the former where there is any ambiguity. The observations leading to the discovery of this part of the SM are discussed in the article inquantum chromodynamics .

*The notation $W^\{a\}\_\{mu\; u\}$ 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) ofweak hypercharge will be denoted by $B\_\{mu\; u\}$, the coupling by $g\text{'}$, and the gauge field by $B\_\{mu\}$.**The gauge field Lagrangian**The gauge part of the electroweak

Lagrangian is:$mathcal\{L\}\_\{g\}\; =\; -frac\{1\}\{4\}\; (W^\{a\}\_\{mu\; u\}\; W^\{a,mu\; u\}\; +\; B\_\{mu\; u\}B^\{mu\; u\})$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:$W^\{pm\}\_\{mu\}\; =\; frac1sqrt2left(W^\{1\}\_\{mu\}\; pm\; i\; W^\{2\}\_\{mu\}\; ight)$

Z bosons ($Z\_\{mu\}$) andphoton s ($A\_\{mu\}$) are mixtures of $W^\{3\}$ and $B$. Theprecise mixture is determined by theWeinberg angle $heta\_W$::$Z\_\{mu\}=cos\; heta\_W\; W^\{3\}\_\{mu\}\; -\; sin\; heta\_WB\_\{mu\},$ and $A\_\{mu\}\; =sin\; heta\_W\; W^\{3\}\_\{mu\}\; +\; cos\; heta\_W\; B\_\{mu\},$with $cos\; heta\_W\; =\; frac\; g\; sqrt\{g^\{2\}+g\text{'}^\{2$

The

electric charge ,weak isospin andweak hypercharge arerelated by:$Q\; =\; T^3\; +\; Y\; ,\; .$**The charged and neutral current couplings**The charged currents $J^\{pm\}=J^\{1\}pm\; i\; J^\{2\}$ are::$J^+\_mu\; =\; overline\; U\_\{iL\}gamma\_mu\; D\_\{iL\}\; +overline\; u\_\{iL\}gamma\_mu\; l\_\{iL\}.$These charged currents are precisely those which entered the

Fermi theory of beta decay . The action contains the charge current piece::$mathcal\{L\}\_\{CC\}\; =\; frac\; g\{sqrt2\}(J\_mu^+W^\{-,mu\}+J\_mu^-W^\{+,mu\}).$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 $W^\{3\}$ 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**::$J\_mu^3\; =\; frac12(overline\; U\_\{iL\}gamma\_mu\; U\_\{iL\}\; -\; overline\; D\_\{iL\}gamma\_mu\; D\_\{iL\}\; +\; overline\; u\_\{iL\}gamma\_mu\; u\_\{iL\}\; -\; overline\; l\_\{iL\}gamma\_mu\; l\_\{iL\})$::$J\_mu^\{em\}\; =\; frac23overline\; U\_igamma\_mu\; U\_i\; -frac13overline\; D\_igamma\_mu\; D\_i\; -\; overline\; l\_igamma\_mu\; l\_i.$The neutral current piece in the Lagrangian is then::$mathcal\{L\}\_\{NC\}\; =\; e\; J\_mu^\{em\}\; A^mu\; +\; frac\; g\{cos\; heta\_W\}(J\_mu^3-sin^2\; heta\_WJ\_mu^\{em\})Z^mu$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::$mathcal\{L\}\_\{QCD\}\; =\; overline\; U\; (partial\_mu-ig\_sG\_mu^a\; T^a)gamma^mu\; U\; +\; overline\; D\; (partial\_mu-i\; g\_s\; G\_mu^a\; T^a)gamma^mu\; D.$Here

**T**stands for the generators of SU(3) colour. The mass term in QCD arises from interactions in the Higgs sector.^{a}**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).**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, $(phi^\{+\},\; phi^\{0\})$ is introduced, which couples through theYukawa interaction ::$mathcal\{L\}\_\{YU\}\; =\; overline\; U\_L\; G\_u\; U\_R\; phi^0\; -\; overline\; D\_L\; G\_u\; U\_R\; phi^-\; +\; overline\; U\_L\; G\_d\; D\_R\; phi^+\; +\; overline\; D\_L\; G\_d\; D\_R\; phi^0\; +\; hc$where $G\_\{u,d\}$ are 3×3 matrices of Yukawa couplings, with the "ij" term giving the coupling of the generations "i" and "j".**ymmetry breaking**The Higgs part of the Lagrangian is::$mathcal\{L\}\_H\; =\; [(partial\_mu\; -ig\; W\_mu^a\; t^a\; -ig\text{'}Y\_\{phi\}\; B\_mu)phi]\; ^2\; +\; mu^2\; phi^daggerphi-lambda\; (phi^daggerphi)^2,$where $lambda>0$ and $mu^\{2\}>0$, so that the mechanism of

spontaneous symmetry breaking can be used.In a unitarity gauge one can set $phi^\{+\}=0$ and make $phi^0$ real. Then $^0>=v$ 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 $v\; G\_\{u,d\}$. From $mathcal\{L\}\_H$, quadratic terms in $W\_\{mu\}$ and $B\_\{mu\}$ arise, which give masses to the "W" and "Z" bosons::$M\_W\; =\; frac\{v|g2\; qquadqquad\; M\_Z=frac\{vsqrt\{g^2+\{g\text{'}\}^22.$**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 $u\_R$, and give a mass term as usual (this is called a**Dirac mass**)

*Write a**Majorana mass**term by combining $u\_L$ with its complex conjugate:::$overline\; u\_L^*\; u\_L$See

seesaw mechanism .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 s. However, they can be diagonalized by separate unitary matrices acting on the two sides (this process is called asingular value decomposition ). In other words one can find diagonal matrices:$M^\{2\}\; =\; V\_r\; G^\{+\}\; G\; V\_r^\{+\}\; =\; V\_l\; G\; G^\{+\}\; V\_l^\{+\}.$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 ofquantum chromodynamics .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 thecharm quark was found, and therefore predicted this new flavour.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-violation s in the SM.**ee also***Overview of standard model of

particle physics

*Weak interactions,Fermi theory of beta decay and electroweak theory

*Strong interaction s, flavour,quark model andquantum chromodynamics

*For open questions, see quark matter, CP violation and neutrino masses

*Beyond the Standard Model **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.

* [*http://nuclear.ucdavis.edu/~tgutierr/files/stmL1.html PDF, PostScript, and LaTeX version of the Standard Model Lagrangian with explicit Higgs terms*]

*Wikimedia Foundation.
2010.*

### Look at other dictionaries:

**Standard Model**— The Standard Model of particle physics is a theory that describes three of the four known fundamental interactions together with the elementary particles that take part in these interactions. These particles make up all matter in the universe… … Wikipedia**Noncommutative standard model**— In theoretical particle physics, the non commutative Standard Model, mainly due to the French mathematician Alain Connes, uses his noncommutative geometry to devise an extension of the Standard Model to include a modified form of general… … Wikipedia**Mathematical analysis**— Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and… … Wikipedia**Mathematical optimization**— For other uses, see Optimization (disambiguation). The maximum of a paraboloid (red dot) In mathematics, computational science, or management science, mathematical optimization (alternatively, optimization or mathematical programming) refers to… … Wikipedia**Potts model**— In statistical mechanics, the Potts model, a generalization of the Ising model, is a model of interacting spins on a crystalline lattice. By studying the Potts model, one may gain insight into the behaviour of ferromagnets and certain other… … Wikipedia**Okumura Model**— The Okumura model for Urban Areas is a Radio propagation model that was built using the data collected in the city of Tokyo, Japan. The model is ideal for using in cities with many urban structures but not many tall blocking structures. The model … Wikipedia**List of mathematical topics in quantum theory**— This is a list of mathematical topics in quantum theory, by Wikipedia page. See also list of functional analysis topics, list of Lie group topics, list of quantum mechanical systems with analytical solutions. Contents 1 Mathematical formulation… … Wikipedia**Nearly-free electron model**— Electronic structure methods Tight binding Nearly free electron model Hartree–Fock method Modern valence bond Generalized valence bond Møller–Plesset perturbat … Wikipedia**Milne model**— General relativity Introduction Mathematical formulation Resources Fundamental concepts … Wikipedia**Log-distance path loss model**— The log distance path loss model is a radio propagation model that predicts the path loss a encounters inside a building or densely populated areas over distance.Applicable to / Under conditionsThe model is applicable to indoor propagation… … Wikipedia