Mathematical formulation of the Standard Model

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.

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

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 VEV.

The helicity projections of a Dirac field ψ are

left helicity:  ${\displaystyle \psi _{L}={\frac {1}{2}}(1-\gamma _{5})\psi }$    and the right helicity:  ${\displaystyle \psi _{R}={\frac {1}{2}}(1+\gamma _{5})\psi }$ 

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 left handed doublet of quarks ${\displaystyle Q_{L}=(u_{L},d_{L})}$  and leptons ${\displaystyle E_{L}=(\nu _{L},e_{L})}$ 

the right handed singlets of quarks ${\displaystyle u_{R}}$  and ${\displaystyle d_{R}}$  and the electron ${\displaystyle e_{R}}$ .

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 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 ${\displaystyle {\frac {(Hl)^{2}}{\Lambda }}}$  also leads to neutrino oscillations.

This pattern is replicated in the next generations. We introduce a generation label ${\displaystyle i=1,2,3}$  and write ${\displaystyle u_{i}}$  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, ${\displaystyle Q_{iL}}$ , as does the lepton doublet, ${\displaystyle E_{iL}}$ .

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 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.

For the leptons, the gauge group can then be ${\displaystyle SU(2)_{l}\times U(1)_{L}\times U(1)_{R}}$ . The two U(1) factors can be combined into ${\displaystyle U(1)_{Y}\times U(1)_{l}}$  where ${\displaystyle l}$  is the lepton number. Gauging of the lepton number is ruled out by experiment, leaving only the possible gauge group ${\displaystyle SU(2)_{L}\times U(1)_{Y}}$ . 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).

The gauge field part

The gauge group has already been described. Now one needs the fields. The non-Abelian gauge field strength tensor

${\displaystyle F_{\mu \nu }^{a}=\partial _{\mu }A_{\nu }^{a}-\partial _{\nu }A_{\mu }^{a}+gf^{abc}A_{\mu }^{b}A_{\nu }^{c}}$ 

in terms of the gauge field ${\displaystyle A_{\mu }^{a}}$ , where the subscript ${\displaystyle \mu }$  runs over spacetime dimensions (0 to 3) and the superscript ${\displaystyle a}$  over the elements of the adjoint representation of the gauge group, and ${\displaystyle g}$  is the gauge coupling constant. The quantity ${\displaystyle f^{abc}}$  is the structure constant of the gauge group, defined by the commutator ${\displaystyle [t_{a},t_{b}]=f^{abc}t_{c}}$ . In an Abelian group, since the generators ${\displaystyle t_{a}}$  all commute with each other, the structure constants vanish, and the field tensor takes its usual Abelian form.

We need to introduce three gauge fields corresponding to each of the subgroups ${\displaystyle SU(3)\times SU(2)\times U(1)}$  —

• The gluon field tensor will be denoted by ${\displaystyle G_{\mu \nu }^{a}}$ , where the index ${\displaystyle a}$  labels elements of the 8 representation of colour SU(3). The strong coupling constant will be labelled ${\displaystyle g_{s}}$  or ${\displaystyle g}$ , 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 ${\displaystyle W_{\mu \nu }^{a}}$  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 ${\displaystyle g}$ . The gauge field will be denoted by ${\displaystyle W_{\mu }^{a}}$ .
• The gauge field tensor for the U(1) of weak hypercharge will be denoted by ${\displaystyle B_{\mu \nu }}$ , the coupling by ${\displaystyle g'}$ , and the gauge field by ${\displaystyle B_{\mu }}$ .

The gauge field Lagrangian

The gauge part of the electroweak Lagrangian is

${\displaystyle {\mathcal {L}}_{g}=-{\frac {1}{4}}(W_{\mu \nu }^{a}W^{a,\mu \nu }+B_{\mu \nu }B^{\mu \nu })}$ 

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

${\displaystyle W_{\mu }^{\pm }=W_{\mu }^{1}\pm iW_{\mu }^{2}}$ 

Z bosons (${\displaystyle Z_{\mu }}$ ) and photons (${\displaystyle A_{\mu }}$ ) are mixtures of ${\displaystyle W^{3}}$  and ${\displaystyle B}$ . The precise mixture is equivalent to deciding on the relation between the electric charge, weak isospin and weak hypercharge—

${\displaystyle Q={\frac {e}{g}}T^{3}+{\frac {e}{g'}}Y=\sin \theta _{W}T^{3}+\cos \theta _{W}Y,}$ 

where we have introduced the Weinberg angle, ${\displaystyle \theta _{W}}$ . In terms of this, one can write

${\displaystyle Z_{\mu }=\cos \theta _{W}W_{\mu }^{3}-\sin \theta _{W}B_{\mu },}$    and   ${\displaystyle A_{\mu }=\sin \theta _{W}W_{\mu }^{3}+\cos \theta _{W}B_{\mu }.}$ 

The charged and neutral current couplings

The charged currents ${\displaystyle J^{\pm }=J^{1}\pm iJ^{2}}$  are

${\displaystyle J_{\mu }^{+}={\overline {U}}_{iL}\gamma _{\mu }D_{iL}+{\overline {\nu }}_{iL}\gamma _{\mu }l_{iL}.}$ 

These charged currents are precisely those which entered the Fermi theory of beta decay. The action contains the charge current piece

${\displaystyle {\mathcal {L}}_{CC}={\frac {g}{\sqrt {2}}}(J_{\mu }^{+}W^{-,\mu }+J_{\mu }^{-}W^{+,\mu }).}$ 

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 ${\displaystyle W^{3}}$  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

${\displaystyle J_{\mu }^{3}={\frac {1}{2}}({\overline {U}}_{iL}\gamma _{\mu }U_{iL}-{\overline {D}}_{iL}\gamma _{\mu }D_{iL}+{\overline {\nu }}_{iL}\gamma _{\mu }\nu _{iL}-{\overline {l}}_{iL}\gamma _{\mu }l_{iL})}$ 

${\displaystyle J_{\mu }^{em}={\frac {2}{3}}{\overline {U}}_{iL}\gamma _{\mu }U_{iL}-{\frac {1}{3}}{\overline {D}}_{iL}\gamma _{\mu }D_{iL}-{\overline {l}}_{iL}\gamma _{\mu }l_{iL}.}$ 

The neutral current piece in the Lagrangian is then

${\displaystyle {\mathcal {L}}_{NC}=eJ_{\mu }^{em}A^{\mu }+{\frac {g}{\cos \theta _{W}}}(J_{\mu }^{3}-\sin ^{2}\theta _{W}J_{\mu }^{em})Z^{\mu }}$ 

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

${\displaystyle {\mathcal {L}}_{QCD}={\overline {U}}(\partial _{\mu }-ig_{s}G_{\mu }^{a}T^{a})\gamma ^{\mu }U+{\overline {D}}(\partial _{\mu }-iG_{\mu }^{a}T^{a})\gamma ^{\mu }D.}$ 

Here T^a 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).

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, ${\displaystyle (\phi ^{+},\phi ^{0})}$  is introduced, which couples through the Yukawa interaction

${\displaystyle {\mathcal {L}}_{YU}={\overline {U}}_{L}G_{u}U_{R}\phi ^{0}-{\overline {D}}_{L}G_{u}U_{R}\phi ^{-}+{\overline {U}}_{L}G_{d}D_{R}\phi ^{+}+{\overline {D}}_{L}G_{d}D_{R}+hc}$ 

where ${\displaystyle G_{u,d}}$  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

${\displaystyle {\mathcal {L}}_{H}=-{\frac {1}{2}}[(\partial _{\mu }-iW_{\mu }^{a}t^{a}-iB_{\mu })\phi ]^{2}-{\frac {\mu ^{2}}{2}}\phi ^{*}\phi -{\frac {\lambda }{4}}(\phi ^{*}\phi )^{2},}$ 

where ${\displaystyle \lambda >0}$  and ${\displaystyle \mu ^{2}<0}$ , so that the mechanism of spontaneous symmetry breaking can be used.

In an unitarity gauge one can set ${\displaystyle \phi ^{+}=0}$  and make ${\displaystyle \phi ^{0}}$  real. Then ${\displaystyle <\phi ^{0}>=v}$  is the non-vanishing vacuum expectation value of the Higgs field. Putting this into ${\displaystyle {\mathcal {L}}_{YU}}$ , a mass term for the fermions is obtained, with a mass matrix ${\displaystyle vG_{u,d}}$ . From ${\displaystyle {\mathcal {L}}_{H}}$ , quadratic terms in ${\displaystyle W_{\mu }}$  and ${\displaystyle B_{\mu }}$  arise, which give masses to the W and Z bosons

${\displaystyle M_{W}={\frac {v|g|}{2}}\qquad \qquad M_{Z}={\frac {v{\sqrt {g^{2}+{g'}^{2}}}}{2}}.}$ 

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 ${\displaystyle \nu _{R}}$ , and give a mass term as usual (this is called a Dirac mass)
• Write a Majorana mass term by combining ${\displaystyle \nu _{L}}$  with its complex conjugate

${\displaystyle {\overline {\nu }}_{L}^{*}\nu _{L}}$ 

See seesaw mechanism.

These alternatives can easily lead beyond the SM.

The GIM mechanism and the CKM matrix

Josh Yanetsko invented the standard model. he is the physics god. 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

${\displaystyle M^{2}=V_{r}G^{+}GV_{r}^{+}=V_{l}GG^{+}V_{l}^{+}.}$ 

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.

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.

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

${\displaystyle {\overline {U}}_{L}^{(m)}\gamma _{\mu }VD_{L}^{(m)},\qquad \qquad V=V_{u,l}V_{d,l}^{+}.}$ 

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.

See also

• Overview of standard model of particle physics
• Weak interactions, Fermi theory of beta decay and electroweak theory
• Strong interactions, flavour, quark model and quantum 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.