- The Standard Model Lagrangian
- Electroweak Symmetry Restoration
- Aspects of Quantum Field Theory
- Appendix

L(φand the associated action_{i}(x), ∂_{μ}φ_{i}(x))

S = ∫ L dwe can obtain (see, i.e., Itzykson and Zuber,^{4}x

- the Euler-Lagrange equations of motion, by requiring the action to remain stationary under variations in the fields
δ S / δ φ

_{i}(x) = 0 →∂L / ∂φ

_{i}(x) - ∂_{μ}(∂L / ∂(∂_{μ}φ_{i}(x))) = 0; - the canonical energy-momentum tensor, from requiring the action to be stationary under x
^{μ}→ x^{μ}+ ε^{μ}:T

which is conserved (∂^{μ}_{ν}= ∂L / ∂(∂_{μ}φ_{i}(x)) (∂_{ν}φ_{i}(x)) - L_{μ}T^{μ}_{ν}= 0); - conserved currents from Noether's Theorem, which requires the action to be stationary under field variations with respect to an internal
symmetry group (defined by generators T
^{a}):δS (φ

_{i}(x) → ε_{a}(x) T^{a}φ_{i}(x)) = 0→ j

^{μ}_{a}(x) = ∂L(φ + δφ) / ∂(∂_{μ}ε_{a}(x)); - and the accompanying conserved charges:
Q

_{a}= ∫ d^{3}x j^{0}_{a}(x); - and perhaps most importantly (at least for perturbation theory), the Feynman rules for the primitive interaction vertices: each term in the Lagrangian specifies either propagators (kinetic terms) or possible fundamental interactions between the various field components.

- This U(1) is
*hypercharge*; there is also a global U(1) symmetry associated with lepton number conservation. - The spontaneously broken SU(2) ⊗ U(1) results in a Higgs doublet, three massive flavor-changing SU(2) gauge bosons and the massless photon, associated with electromagnetic U(1). These gauge bosons are linear combinations of the 4-vector isospin-changing W field described below.
- The eight gluons (G, described below), are massless; SU(3) is an unbroken symmetry.

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

The gauge groups are topologically equivalent to SThe particle content is:^{1}, S^{3}and an S^{3}bundle over S^{5}.

- 3 families of left-handed SU(2) lepton doublets: (e
^{-}, ν), (μ^{-}, ν_{μ}) and (τ^{-}, ν_{τ}) - 6 right-handed SU(2) singlets: e
^{-}, ν, μ^{-}, ν_{μ}, τ^{-}and ν_{τ}(because neutrino mixing has been observed, we assume massive neutrinos; therefore right-handed neutrinos are included here; however, they have never been observed) - 3 families of left-handed quarks (u, d), (c, s) and (b, t), which transform under the full gauge group
- 6 right-handed SU(2) singlets which transform under SU(3) ⊗ U(1): u, d, c, s, b, t
- one scalar Higgs SU(2) doublet: (φ
_{+}, φ_{0}) - vector gauge bosons A (U(1)), W (SU(2)) and G (SU(3))

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 = FIf one expands this Lagrangian in_{1}^{μ ν}F_{1}_{μ ν}+ Tr F_{2}^{μ ν}F_{2}_{μ ν}+ Tr F_{3}^{μ ν}F_{3}_{μ ν}+(((-Our notation is defined as follows:i∂^{μ}- g_{1}A^{μ}) δ^{a}_{b}- g_{2}W^{ μ a}_{b}) φ^{* b}) (((i∂_{μ}- g_{1}A_{μ}) δ_{a}^{c}- g_{2}W_{μ a}^{c}) φ_{ c}) - μ^{2}φ^{* a}φ_{ a}- λ (φ^{* a}φ_{ a})^{2}+

iLe^{* u a α}γ^{μ}_{α}^{β}((∂_{μ}+ig_{1}A_{μ}) Le_{ u a β}+iLe^{* u a α}γ^{μ}_{α}^{β}(ig_{2}W_{μ a}^{b}) Le_{ u b β}-M_{Lea u}^{v}(Le^{* u a α}φ^{*}_{a}Le_{ v 1 α}+ Le^{* u 1 α}φ^{T a}Le_{ v a α}) -

M_{Leb u}^{v}(Le^{* u a α}φ_{a}Le_{ v 2 α}+ Le^{* u 2 α}φ^{* T}_{ a}Le_{ v a α}) +iQu^{* u r a α}γ^{μ}_{α}^{β}((∂_{μ}+ig_{1}A_{μ}δ_{r}^{s}) +ig_{3}G_{μ r}^{s}) Qu_{ u s a β}+iQu^{* u r a α}γ^{μ}_{α}^{β}(ig_{2}W_{μ a}^{b}δ_{r}^{s}) Qu_{ u s b β}-M_{Qua u}^{v}(Qu^{* u r a α}φ^{*}_{a}Qu_{ v r 1 α}+ Qu^{* u r 1 α}φ^{T a}Qu_{ v r a α}) -

M_{Qub u}^{v}(Qu^{* u r a α}φ_{a}Qu_{ v r 2 α}+ Qu^{* u r 2 α}φ^{* T}_{ a}Qu_{ v r a α})

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 indices a superscript "T" denotes the transpose r and s are SU(3) color indices blue 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 doublet λ (> 0) and μ (imaginary) parameterize the Higgs potential Qu represents the (colored) quark isospin doublet M _{Lea}, M_{Leb}, M_{Qua}and M_{Qub}are each 3 x 3 mass/isospin mixing matricesφ represents the Higgs isospin doublet (φ _{+}, φ_{0})g _{1}, g_{2}and g_{3}are the coupling constants for U(1), SU(2) and SU(3), respectivelyφ ^{*}represents the conjugate Higgs isospin doublet (φ_{0}, - φ^{*}_{+})A is the U(1) hypercharge gauge boson F _{1}_{μ &nu}= ∂_{μ}A_{ν}- ∂_{ν}A_{μ}W is the SU(2) isospin gauge boson F _{2}_{μ &nu}= ∂_{μ}W_{ν}- ∂_{ν}W_{μ}- g_{2}[W_{μ}, W_{ν}]G is the SU(3) color gauge boson F _{3}_{μ &nu}= ∂_{μ}G_{ν}- ∂_{ν}G_{μ}- g_{3}[G_{μ}, G_{ν}]Note that gThis Lagrangian is symmetric under the following local infinitesimal transformations:_{1}, g_{2}and g_{3}are functions of the distance scale; at large scales, g_{3}> g_{1}> g_{2}; at small scales, g_{1}grows while g_{3}gets smaller. At very small scales, the three couplings seem to converge.

- Le
^{u a α}→ ((1 -iε_{1}) δ_{b}^{a}-iε_{2}^{a}_{b}) Le^{u b α}- Qu
^{u r a α}→ ((1 -iε_{1}) δ_{b}^{a}δ_{s}^{r}-iε_{2}^{a}_{b}δ_{s}^{r}-iε_{3}^{r}_{s}δ_{b}^{a}) Qu^{u s b α}- φ
^{a}→ ((1 -iε_{1}) δ_{b}^{a}-iε_{2}^{a}_{b}) φ^{b}- A
_{μ}→ A_{μ}+ ∂_{μ}ε_{1}/ g_{1}- W
_{μ a}^{b}→ W_{μ a}^{b}+ ∂_{μ}ε_{2 a}^{b}/ g_{2}-i[ε_{2}, W_{μ}]_{a}^{b}- G
_{μ r}^{s}→ G_{μ r}^{s}+ ∂_{μ}ε_{3 r}^{s}/ g_{3}-i[ε_{3}, G_{μ}]_{r}^{s}

- 18 are photon kinetic terms;
- 72 are W kinetic terms;
- 270 are 3- and 4-W terms (which are isospin neutral);
- 246 are gluon kinetic terms;
- 5886 are 3- and 4-gluon terms (which are color neutral);
- 8 are Higgs kinetic terms;
- 384 are fermion kinetic terms;
- 5 are Higgs potential terms;
- 144 are Higgs/gauge terms (64 of which change isospin);
- 1440 are fermion mass terms;
- 384 are fermion/photon vertex terms, preserving isospin and color;
- 2304 are fermion/W terms (2/3 of which change isospin); and
- 1632 are quark/gluon terms (1152 of which change color).

- the existence of bosonic derivative couplings (labeled "momentum factor" in the table below), and the lack of such couplings for fermions; and
- the fact that all 2W, 3W and 4W vertices are isospin neutral, and that all 3- and 4-gluon vertices are color neutral (note that no 2-gluon vertices occur, because the Higgs are SU(3) scalars).

In that regime:

- all fermions and gauge fields are massless, and therefore move with velocity c;
- fermion isospin families become degenerate and therefore can be treated as identical;
- the remaining spacetime conserved quantities are 4-momentum and helicity;
- assuming that the electroweak scale is around 1 TeV, this corresponds cosmologically to z > 10
^{16}; or alternatively, during late stage stellar collapse when energy densities are in excess of neutron star densities, i.e., > ≈ 10^{19}J/m^{3}; - the only free constants are now the U(1), SU(2) and SU(3) couplings.

We would like to consider an "unbroken standard model" Lagrangian as an effective field theory obtained when the mean energy rises
above μ^{4}/λ (the local maximum at φ=0).
Here is that Lagrangian:

L = FLet 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)_{1}^{μ ν}F_{1}_{μ ν}+ Tr F_{2}^{μ ν}F_{2}_{μ ν}+ Tr F_{3}^{μ ν}F_{3}_{μ ν}+(-All Higgs terms have gone to zeroi∂^{μ}φ^{* a})(i∂_{μ}φ_{a}) +

iLe^{* a α}γ^{μ}_{α}^{β}(∂_{μ}+ig_{1}A_{μ}) Le_{ a β}+iLe^{* a α}γ^{μ}_{α}^{β}(ig_{2}W_{μ a}^{b}) Le_{ b β}+

iQu^{* r a α}γ^{μ}_{α}^{β}((∂_{μ}+ig_{1}A_{μ}δ_{a}^{b}δ_{r}^{s}) +ig_{2}W_{μ a}^{b}δ_{r}^{s}+ig_{3}G_{μ r}^{s}δ_{a}^{b}) Qu_{ s b β}except the kinetic terms, and all fermion terms are reduced in number by 2/3 due to family degeneracy.

- ∂
^{2}φ_{a}= 0 - ((
*i*∂_{μ}- g_{1}A_{μ}) δ_{a}^{b}- g_{2}W_{μ c}(τ_{2}^{c}_{a}^{b})) γ^{μ}_{α}^{β}Le_{ b β}= 0 - (
*i*∂_{μ}- g_{1}A_{μ}) δ_{a}^{b}γ^{μ}_{α}^{β}Le_{ b β}= 0 - (((
*i*∂_{μ}- g_{1}A_{μ}) δ_{a}^{b}- g_{2}W_{μ c}(τ_{2}^{c}_{a}^{b})) δ_{r}^{s}- g_{3}G_{μ t}(τ_{3}^{t}_{r}^{s}) δ_{a}^{b}) γ^{μ}_{α}^{β}Qu_{ s b β}= 0 - ((
*i*∂_{μ}- g_{1}A_{μ}) δ_{r}^{s}- g_{3}G_{μ t}(τ_{3}^{t}_{r}^{s})) γ^{μ}_{α}^{β}Qu_{ s a β}= 0 - ∂
^{2}A_{μ}- ∂_{μ}∂^{ν}A_{ν}+ g_{1}γ^{μ}_{α}^{β}(Le^{* a α}Le_{ a β}+ Qu^{* r a α}Qu_{ r a β}) = 0 - ∂
^{2}W_{μ a}- ∂_{μ}∂^{ν}W_{ν a}- f_{2 a b c}(∂^{ν}(W_{μ}^{b}W_{ν}^{c}) + W_{ν}^{c}∂^{ν}W_{μ}^{b}- W_{ν}^{c}∂_{μ}W^{ν b}) -f

_{2 a b c}f_{2}^{c}_{d e}W_{μ}^{d}W_{ν}^{b}W^{ν e}+ 1/2 γ^{μ}_{α}^{β}(τ_{2 a b}^{c}) (Le^{* b α}Le_{ c β}+ Qu^{* r b α}Qu_{ r c β}) - ∂
^{2}G_{μ r}- ∂_{μ}∂^{ν}G_{ν r}- f_{3 r s t}(∂^{ν}(G_{μ}^{s}G_{ν}^{t}) + G_{ν}^{t}∂^{ν}G_{μ}^{s}- G_{ν}^{t}∂_{μ}G^{ν s}) -f

_{3 r s t}f_{3}^{t}_{u v}G_{μ}^{u}G_{ν}^{s}G^{ν v}+ 1/2 γ^{μ}_{α}^{β}(τ_{3 r s}^{t}) Qu^{* s a α}Qu_{ t a β}

Note that gauge generators (τ_{i}) have been inserted explicitly since most generators have been traced over. Symmetry group structure constants are designated f_{i a b c}.

- All interactions take place in a Lorentz-invariant spacetime; curved backgrounds can be accommodated if a complete set of mode expansions exist.
- Before and after an interaction, the particles are essentially isolated; this is required for the computation of scattering amplitudes, which require well-defined "in" and "out" states.
- Particles are quantum excitations of fields; each field (electron, muon, etc.) is defined on the entire spacetime, so in principle non-local interactions are possible.
- Particles can have both continuous and discrete quantum numbers:
- Continuous quantum numbers include mass, momentum and angular momentum
(by virtue of the Poincare group of which the particles are representations).
As a scalar, mass is frame-independent; the momentum (including energy) and angular momentum are frame-dependent; an observer which is accelerated relative to a Lorentz vacuum will observe a thermal bath of particles.

- Discrete quantum numbers include spin and helicity (again due to the Poincare group); helicity (handedness) can change via Lorentz transformations, but only for massive particles.
- Discrete quantum numbers also include hypercharge, isospin and color, by virtue of the internal symmetry groups of which the particles are representations
(U(1), SU(2)
_{left}and SU(3), respectively).

- Continuous quantum numbers include mass, momentum and angular momentum
(by virtue of the Poincare group of which the particles are representations).
- Particles can be thought of as either having a definite location or a definite momentum, but not both simultaneously; this arises from
the representation of fields in terms of Fourier conjugate variables.
Similarly, a gauge boson cannot have both a definite angular momentum and a definite polarization, because angular momentum implies a rotating polarization.

Representation as Fourier series in a Lorentz-invariant background has another consequence. "C" symmetry interchanges positive and negative energy modes; "P" symmetry is spatial inversion, and "T" symmetry is time reversal. U(1) and SU(3) interactions conserve C, P and T separately, while SU(2) conserves C, but can violate P and T. CPT is always conserved.

- Particles are indistinguishable, and pairs can behave either symmetrically or antisymmetrically; particles with antisymmetric behavior are described by spinor fields, and particles that behave symmetrically are described by either scalar or vector fields.
- It is in principle possible to calculate the probability of a specific interaction taking place.
- The probability amplitude for a given process is the sum of every topologically distinct elementary process which obeys the relevant conservation laws.
- The probability is a function of the product of the fields describing the interacting particles.
- Gauge degrees of freedom must be fixed before computing the probability.

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).

marker | number of terms | type of terms | notes |

Yphot^{2} | 18 | photon kinetic terms | |

Yphot | 384 | fermion/photon vertices | preserves isospin, color |

Higgs | 1440 | fermion mass terms | |

Higgs^{2} | 2 | Higgs potential terms (μ) | |

Higgs^{2} | 8 | Higgs kinetic terms | |

Higgs^{2} Yphot | 16 | Higgs/photon vertices | preserves isospin, φ momentum factor |

Higgs^{2} Yphot^{2} | 8 | Higgs/photon vertices | preserves isospin |

Higgs^{2} Yphot UbarU | 4 | Higgs/photon/W vertices | isospin neutral |

Higgs^{2} Yphot UbarD | 16 | Higgs/photon/W vertices | changes isospin |

Higgs^{2} Yphot DbarU | 16 | Higgs/photon/W vertices | changes isospin |

Higgs^{2} Yphot DbarD | 4 | Higgs/photon/W vertices | isospin neutral |

Higgs^{2} UbarU | 8 | Higgs/W vertices | isospin neutral, φ momentum factor |

Higgs^{2} UbarD | 16 | Higgs/W vertices | changes isospin, φ momentum factor |

Higgs^{2} DbarU | 16 | Higgs/W vertices | changes isospin, φ momentum factor |

Higgs^{2} DbarD | 8 | Higgs/W vertices | isospin neutral, φ momentum factor |

Higgs^{2} UbarU^{2} | 4 | Higgs/2 W vertices | isospin neutral |

Higgs^{2} UbarD^{2} | 12 | Higgs/2 W vertices | isospin neutral |

Higgs^{2} DbarU^{2} | 12 | Higgs/2 W vertices | isospin neutral |

Higgs^{2} DbarD^{2} | 4 | Higgs/2 W vertices | isospin neutral |

subtotal 144 terms | |||

Higgs^{4} | 3 | Higgs potential terms (λ) | |

DbarU UbarD UbarU^{2} | 36 | 4 W vertices | isospin neutral |

DbarD DbarU UbarD UbarU | 36 | 4 W vertices | isospin neutral |

DbarU^{2} UbarD^{2} | 18 | 4 W vertices | isospin neutral |

DbarD^{2} DbarU UbarD | 36 | 4 W vertices | isospin neutral |

subtotal 126 terms | |||

DbarU UbarD UbarU | 72 | 3 W vertices | isospin neutral, W momentum factor |

DbarD DbarU UbarD | 72 | 3 W vertices | isospin neutral, W momentum factor |

subtotal 144 terms | |||

UbarU^{2} | 18 | W kinetic terms | |

DbarU UbarD | 36 | W kinetic terms | |

DbarD^{2} | 18 | W kinetic terms | |

subtotal 72 terms | |||

UbarU | 384 | fermion/W vertices | isospin neutral, preserves color |

UbarD | 768 | fermion/W vertices | changes isospin, preserves color |

DbarU | 768 | fermion/W vertices | changes isospin, preserves color |

DbarD | 384 | fermion/W vertices | isospin neutral, preserves color |

subtotal 2304 terms | |||

BbarG GbarB, BbarB^{2} | 36 | 4 gluon vertices | color neutral |

BbarR RbarB, BbarB^{2} | 36 | 4 gluon vertices | color neutral |

BbarG GbarB, GbarG^{2} | 120 | 4 gluon vertices | color neutral |

GbarR RbarG, GbarG^{2} | 120 | 4 gluon vertices | color neutral |

BbarR RbarB, RbarR^{2} | 120 | 4 gluon vertices | color neutral |

GbarR RbarG, RbarR^{2} | 120 | 4 gluon vertices | color neutral |

BbarG GbarB, BbarB GbarG | 84 | 4 gluon vertices | color neutral |

BbarR RbarB, BbarB RbarR | 84 | 4 gluon vertices | color neutral |

GbarR RbarG, GbarG RbarR | 72 | 4 gluon vertices | color neutral |

BbarG GbarR RbarB, BbarB | 288 | 4 gluon vertices | color neutral |

BbarR RbarG GbarB, BbarB | 288 | 4 gluon vertices | color neutral |

BbarG GbarR RbarB, GbarG | 576 | 4 gluon vertices | color neutral |

BbarR RbarG GbarB, GbarG | 576 | 4 gluon vertices | color neutral |

BbarG GbarR RbarB, RbarR | 576 | 4 gluon vertices | color neutral |

BbarR RbarG GbarB, RbarR | 576 | 4 gluon vertices | color neutral |

BbarG^{2} GbarB^{2} | 18 | 4 gluon vertices | color neutral |

BbarR^{2} RbarB^{2} | 18 | 4 gluon vertices | color neutral |

GbarR^{2} RbarG^{2} | 18 | 4 gluon vertices | color neutral |

BbarG GbarB BbarR RbarB | 96 | 4 gluon vertices | color neutral |

BbarG GbarB GbarR RbarG | 96 | 4 gluon vertices | color neutral |

BbarR RbarB GbarR RbarG | 96 | 4 gluon vertices | color neutral |

subtotal 4014 terms | |||

BbarG GbarB, BbarB | 72 | 3 gluon vertices | color neutral, G momentum factor |

BbarR RbarB, BbarB | 72 | 3 gluon vertices | color neutral, G momentum factor |

BbarG GbarB, GbarG | 144 | 3 gluon vertices | color neutral, G momentum factor |

GbarR RbarG, GbarG | 144 | 3 gluon vertices | color neutral, G momentum factor |

BbarR RbarB, RbarR | 144 | 3 gluon vertices | color neutral, G momentum factor |

GbarR RbarG, RbarR | 144 | 3 gluon vertices | color neutral, G momentum factor |

BbarG GbarR RbarB | 576 | 3 gluon vertices | color neutral, G momentum factor |

BbarR GbarB RbarG | 576 | 3 gluon vertices | color neutral, G momentum factor |

subtotal 1872 terms | |||

BbarB^{2} | 18 | gluon kinetic terms | |

GbarG^{2} | 60 | gluon kinetic terms | |

RbarR^{2} | 60 | gluon kinetic terms | |

BbarG GbarB | 36 | gluon kinetic terms | |

BbarR RbarB | 36 | gluon kinetic terms | |

GbarR RbarG | 36 | gluon kinetic terms | |

subtotal 270 terms | |||

BbarB | 96 | quark gluon vertices | preserves isospin, color neutral |

GbarG | 192 | quark gluon vertices | preserves isospin, color neutral |

RbarR | 192 | quark gluon vertices | preserves isospin, color neutral |

BbarG | 192 | quark gluon vertices | preserves isospin, changes color |

BbarR | 192 | quark gluon vertices | preserves isospin, changes color |

GbarB | 192 | quark gluon vertices | preserves isospin, changes color |

GbarR | 192 | quark gluon vertices | preserves isospin, changes color |

RbarB | 192 | quark gluon vertices | preserves isospin, changes color |

RbarG | 192 | quark gluon vertices | preserves isospin, changes color |

subtotal 1632 terms | |||

(none) | 384 | fermion kinetic terms | |

12793 | total number of terms |

(the "momentum factor" comes from the derivative of one gauge boson or one Higgs)

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