On the Standard Model of Particle Physics

Prelude

Since our discussion of the Standard Model will center around its Lagrangian, it seems prudent to remember some of the reasons we are so interested in a Lagrangian formulation in the first place. From a Lagrangian function
L(φi(x), ∂μ φi(x))
and the associated action
S = ∫ L d4x
we can obtain (see, i.e., Itzykson and Zuber, Quantum Field Theory):

It is for this last reason that we will be particularly interested in examining the types of terms included in:

The Standard Model Lagrangian

The fields in the Standard Model transform under the gauge group SU(3) ⊗ SU(2)left ⊗ U(1):

(See Huang, Quarks, Leptons and Gauge Fields, for more detail.)

The gauge groups are topologically equivalent to S1, S3 and an S3 bundle over S5.
The particle content is:

In all cases, antiparticles have identical mass, but opposite electric charge and helicity.
Our version of the Lagrangian assumes Dirac spinors and minimal Higgs couplings, with neutrino masses:
L = F1μ ν F1μ ν + Tr F2μ ν F2μ ν + Tr F3μ ν F3μ ν +
(((-iμ - g1 Aμ) δab - g2 W μ ab) φ* b) (((iμ - g1 Aμ) δac - g2 Wμ ac) φ c) - μ2 φ* a φ a - λ (φ* a φ a)2 +

i Le* u a α γμαβ ((∂μ + i g1 Aμ) Le u a β + i Le* u a α γμαβ ( i g2 Wμ ab) Le u b β -

MLea uv (Le* u a α φ*a Le v 1 α + Le* u 1 α φT a Le v a α) -
MLeb uv (Le* u a α φa Le v 2 α + Le* u 2 α φ* T a Le v a α) +
i Qu* u r a α γμαβ ((∂μ + i g1 Aμ δrs) + i g3 Gμ rs) Qu u s a β + i Qu* u r a α γμαβ ( i g2 Wμ ab δrs) Qu u s b β -
MQua uv (Qu* u r a α φ*a Qu v r 1 α + Qu* u r 1 α φT a Qu v r a α) -
MQub uv (Qu* u r a α φa Qu v r 2 α + Qu* u r 2 α φ* T a Qu v r a α)
Our notation is defined as follows:

μ and ν are spacetime indices"Tr" denotes the trace over generator indices
α and β are spinor indices"*" denotes the complex conjugate
a, b and c are SU(2) isospin indicesa superscript "T" denotes the transpose
r and s are SU(3) color indicesblue denotes the left-hand projection of a fermion field
u and v are family indices (assuming 3 families each of leptons and quarks)red denotes the right-hand projection of a fermion field
Le represents the lepton isospin doubletλ (> 0) and μ (imaginary) parameterize the Higgs potential
Qu represents the (colored) quark isospin doubletMLea, MLeb, MQua and MQub are each 3 x 3 mass/isospin mixing matrices
φ represents the Higgs isospin doublet (φ+, φ0)g1, g2 and g3 are the coupling constants for U(1), SU(2) and SU(3), respectively
φ* represents the conjugate Higgs isospin doublet (φ0, - φ*+)
A is the U(1) hypercharge gauge bosonF1μ &nu = ∂μ Aν - ∂ν Aμ
W is the SU(2) isospin gauge bosonF2μ &nu = ∂μ Wν - ∂ν Wμ - g2 [Wμ, Wν]
G is the SU(3) color gauge bosonF3μ &nu = ∂μ Gν - ∂ν Gμ - g3 [Gμ, Gν]

Note that g1, g2 and g3 are functions of the distance scale; at large scales, g3 > g1 > g2; at small scales, g1 grows while g3 gets smaller. At very small scales, the three couplings seem to converge.
This Lagrangian is symmetric under the following local infinitesimal transformations:

If one expands this Lagrangian in every index, one obtains 12,793 terms (detailed in the
appendix). Of these,

It is probably not possible to overestimate the importance of the covariant derivative and how gauge invariance limits the types of terms which are possible. Just as in General Relativity the covariant derivative allows us to define a consistent idea of the tangent space, in the standard model it allows us to consistently define isospin and color space, and hypercharge. And it is responsible for:

Electroweak Symmetry Restoration

Above the electroweak symmetry breaking scale, the Higgs decouples since <φ> = 0 is then a stable classical minimum of the potential (Kolb & Turner, The Early Universe, p. 198).

In that regime:

We would like to consider an "unbroken standard model" Lagrangian as an effective field theory obtained when the mean energy rises above μ4/λ (the local maximum at φ=0). Here is that Lagrangian:

L = F1μ ν F1μ ν + Tr F2μ ν F2μ ν + Tr F3μ ν F3μ ν +
(-iμ φ* a)(iμ φa) +

i Le* a α γμαβ (∂μ + i g1 Aμ) Le a β + i Le* a α γμαβ ( i g2 Wμ ab) Le b β +

i Qu* r a α γμαβ ((∂μ + i g1 Aμ δab δrs) + i g2 Wμ ab δrs + i g3 Gμ rs δab) Qu s b β

All Higgs terms have gone to zero except the kinetic terms, and all fermion terms are reduced in number by 2/3 due to family degeneracy.
Let us consider this effective theory as a purely classical field theory (albeit one with scalar, spinor and vector fields). The Euler-Lagrange equations of motion (obtained for the unbarred fields by varying with respect to the barred ones) are (before gauge fixing)

Note that gauge generators (τi) have been inserted explicitly since most generators have been traced over. Symmetry group structure constants are designated fi a b c.

Aspects of Quantum Field Theory

Several aspects of Quantum Field Theory are particularly relevant to the Standard Model:


Appendix

Gauge group generators and Higgs terms have been marked by function (i.e., UbarD changes isospin from "down" to "up", and RbarG changes color from green to red), and the resulting Lagrangian terms are summarized below:
Note that all index summation was carried out explicitly before counting terms (the Pauli and Gell-Mann matrices were used as SU(2) and SU(3) generators, respectively, and the Dirac representation was used for the γ matrices).
markernumber of termstype of termsnotes
Yphot218photon kinetic terms
Yphot384fermion/photon verticespreserves isospin, color
Higgs1440fermion mass terms
Higgs22Higgs potential terms (μ)
Higgs28Higgs kinetic terms
Higgs2 Yphot16Higgs/photon verticespreserves isospin, φ momentum factor
Higgs2 Yphot28Higgs/photon verticespreserves isospin
Higgs2 Yphot UbarU4Higgs/photon/W verticesisospin neutral
Higgs2 Yphot UbarD16Higgs/photon/W verticeschanges isospin
Higgs2 Yphot DbarU16Higgs/photon/W verticeschanges isospin
Higgs2 Yphot DbarD4Higgs/photon/W verticesisospin neutral
Higgs2 UbarU8Higgs/W verticesisospin neutral, φ momentum factor
Higgs2 UbarD16Higgs/W verticeschanges isospin, φ momentum factor
Higgs2 DbarU16Higgs/W verticeschanges isospin, φ momentum factor
Higgs2 DbarD8Higgs/W verticesisospin neutral, φ momentum factor
Higgs2 UbarU24Higgs/2 W verticesisospin neutral
Higgs2 UbarD212Higgs/2 W verticesisospin neutral
Higgs2 DbarU212Higgs/2 W verticesisospin neutral
Higgs2 DbarD24Higgs/2 W verticesisospin neutral
subtotal 144 terms
Higgs43Higgs potential terms (λ)
DbarU UbarD UbarU2364 W verticesisospin neutral
DbarD DbarU UbarD UbarU364 W verticesisospin neutral
DbarU2 UbarD2184 W verticesisospin neutral
DbarD2 DbarU UbarD364 W verticesisospin neutral
subtotal 126 terms
DbarU UbarD UbarU723 W verticesisospin neutral, W momentum factor
DbarD DbarU UbarD723 W verticesisospin neutral, W momentum factor
subtotal 144 terms
UbarU218W kinetic terms
DbarU UbarD36W kinetic terms
DbarD218W kinetic terms
subtotal 72 terms
UbarU384fermion/W verticesisospin neutral, preserves color
UbarD768fermion/W verticeschanges isospin, preserves color
DbarU768fermion/W verticeschanges isospin, preserves color
DbarD384fermion/W verticesisospin neutral, preserves color
subtotal 2304 terms
BbarG GbarB, BbarB2364 gluon verticescolor neutral
BbarR RbarB, BbarB2364 gluon verticescolor neutral
BbarG GbarB, GbarG21204 gluon verticescolor neutral
GbarR RbarG, GbarG2 1204 gluon verticescolor neutral
BbarR RbarB, RbarR21204 gluon verticescolor neutral
GbarR RbarG, RbarR21204 gluon verticescolor neutral
BbarG GbarB, BbarB GbarG844 gluon verticescolor neutral
BbarR RbarB, BbarB RbarR844 gluon verticescolor neutral
GbarR RbarG, GbarG RbarR724 gluon verticescolor neutral
BbarG GbarR RbarB, BbarB2884 gluon verticescolor neutral
BbarR RbarG GbarB, BbarB2884 gluon verticescolor neutral
BbarG GbarR RbarB, GbarG5764 gluon verticescolor neutral
BbarR RbarG GbarB, GbarG5764 gluon verticescolor neutral
BbarG GbarR RbarB, RbarR5764 gluon verticescolor neutral
BbarR RbarG GbarB, RbarR5764 gluon verticescolor neutral
BbarG2 GbarB2184 gluon verticescolor neutral
BbarR2 RbarB2184 gluon verticescolor neutral
GbarR2 RbarG2184 gluon verticescolor neutral
BbarG GbarB BbarR RbarB964 gluon verticescolor neutral
BbarG GbarB GbarR RbarG964 gluon verticescolor neutral
BbarR RbarB GbarR RbarG964 gluon verticescolor neutral
subtotal 4014 terms
BbarG GbarB, BbarB723 gluon verticescolor neutral, G momentum factor
BbarR RbarB, BbarB723 gluon verticescolor neutral, G momentum factor
BbarG GbarB, GbarG1443 gluon verticescolor neutral, G momentum factor
GbarR RbarG, GbarG1443 gluon verticescolor neutral, G momentum factor
BbarR RbarB, RbarR1443 gluon verticescolor neutral, G momentum factor
GbarR RbarG, RbarR1443 gluon verticescolor neutral, G momentum factor
BbarG GbarR RbarB5763 gluon verticescolor neutral, G momentum factor
BbarR GbarB RbarG5763 gluon verticescolor neutral, G momentum factor
subtotal 1872 terms
BbarB218gluon kinetic terms
GbarG260gluon kinetic terms
RbarR260gluon kinetic terms
BbarG GbarB36gluon kinetic terms
BbarR RbarB36gluon kinetic terms
GbarR RbarG36gluon kinetic terms
subtotal 270 terms
BbarB96quark gluon verticespreserves isospin, color neutral
GbarG192quark gluon verticespreserves isospin, color neutral
RbarR192quark gluon verticespreserves isospin, color neutral
BbarG192quark gluon verticespreserves isospin, changes color
BbarR192quark gluon verticespreserves isospin, changes color
GbarB192quark gluon verticespreserves isospin, changes color
GbarR192quark gluon verticespreserves isospin, changes color
RbarB192quark gluon verticespreserves isospin, changes color
RbarG192quark gluon verticespreserves isospin, changes color
subtotal 1632 terms
(none)384fermion kinetic terms
12793total number of terms
(the "momentum factor" comes from the derivative of one gauge boson or one Higgs)


©2013, Kenneth R. Koehler. All Rights Reserved.