 Overview - Maple Help

Overview of the Physics:-StandardModel package Description

 • StandardModel is a Physics's package that implements computational representations for the mathematical objects formulating the Standard Model in particle physics. The package includes field representations for the leptons and quarks, Weinberg's angle, the Higgs boson, the fields and field strengths after breaking symmetries and most of the fields before that, the structure constants ${\mathrm{FSU3}}_{a,b}$and Gell-Mann matrices ${{\mathrm{\lambda }}}_{a}$, the electroweak charges and isospins, and also a command, Lagrangian, to retrieve the Lagrangian of the different sectors of the Standard Model like QED, QCD or Electro-Weak, or of all of it.
 • Loading the package sets things to proceed computing with the model, as the fields and their computational properties, the commutation rules, covariant derivatives, the values of all the constants and matrices involved, the letters representing different kinds of tensor indices etc. Some of these settings, for instance the kinds of letters used to represent different kinds of indices, can be changed using the Physics Setup command.
 > $\mathrm{with}\left(\mathrm{Physics}\right):\mathrm{with}\left(\mathrm{StandardModel}\right)$
 ${}\mathrm{_______________________________________________________}$
 $\mathrm{Setting}\mathrm{lowercaselatin_is}\mathrm{letters to represent}\mathrm{Dirac spinor}\mathrm{indices}$
 $\mathrm{Setting}\mathrm{lowercaselatin_ah}\mathrm{letters to represent}\mathrm{SU\left(3\right) adjoint representation, \left(1..8\right)}\mathrm{indices}$
 $\mathrm{Setting}\mathrm{uppercaselatin_ah}\mathrm{letters to represent}\mathrm{SU\left(3\right) fundamental representation, \left(1..3\right)}\mathrm{indices}$
 $\mathrm{Setting}\mathrm{uppercaselatin_is}\mathrm{letters to represent}\mathrm{SU\left(2\right) adjoint representation, \left(1..3\right)}\mathrm{indices}$
 $\mathrm{Setting}\mathrm{uppercasegreek}\mathrm{letters to represent}\mathrm{SU\left(2\right) fundamental representation, \left(1..2\right)}\mathrm{indices}$
 ${}\mathrm{_______________________________________________________}$
 $\mathrm{Defined as the electron, muon and tau leptons and corresponding neutrinos:}{{\mathrm{e}}}_{{j}}{,}{{\mathrm{\mu }}}_{{j}}{,}{{\mathrm{\tau }}}_{{j}}{,}{{\mathrm{ElectronNeutrino}}}_{{j}}{,}{{\mathrm{MuonNeutrino}}}_{{j}}{,}{{\mathrm{TauonNeutrino}}}_{{j}}$
 $\mathrm{Defined as the up, charm, top, down, strange and bottom quarks:}{{\mathrm{u}}}_{{j}{,}{A}}{,}{{\mathrm{c}}}_{{j}{,}{A}}{,}{{\mathrm{t}}}_{{j}{,}{A}}{,}{{\mathrm{d}}}_{{j}{,}{A}}{,}{{\mathrm{s}}}_{{j}{,}{A}}{,}{{\mathrm{b}}}_{{j}{,}{A}}$
 $\mathrm{Defined as gauge tensors:}{{\mathrm{B}}}_{{\mathrm{\mu }}}{,}{{\mathrm{𝔹}}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}{,}{{\mathrm{A}}}_{{\mathrm{\mu }}}{,}{{\mathrm{𝔽}}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}{,}{{\mathrm{W}}}_{{\mathrm{\mu }}{,}{J}}{,}{{\mathrm{𝕎}}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}{,}{J}}{,}{{\mathrm{WPlusField}}}_{{\mathrm{μ}}}{,}{{\mathrm{WPlusFieldStrength}}}_{{\mathrm{μ}}{,}{\mathrm{ν}}}{,}{{\mathrm{WMinusField}}}_{{\mathrm{μ}}}{,}{{\mathrm{WMinusFieldStrength}}}_{{\mathrm{μ}}{,}{\mathrm{ν}}}{,}{{\mathrm{Z}}}_{{\mathrm{\mu }}}{,}{{\mathrm{ℤ}}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}{,}{{\mathrm{G}}}_{{\mathrm{\mu }}{,}{a}}{,}{{\mathrm{𝔾}}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}{,}{a}}$
 $\mathrm{Defined as Gell-Mann \left(Glambda\right), Pauli \left(Psigma\right) and Dirac \left(Dgamma\right) matrices:}{{\mathrm{\lambda }}}_{{a}}{,}{{\mathrm{\sigma }}}_{{J}}{,}{{\mathrm{\gamma }}}_{{\mathrm{\mu }}}$
 $\mathrm{Defined as the electric, weak and strong coupling constants:}\mathrm{g__e}{,}\mathrm{g__w}{,}\mathrm{g__s}$
 $\mathrm{Defined as the charge in units of |g__e| for 1\right) the electron, muon and tauon, 2\right) the up, charm and top, and 3\right) the down, strange and bottom:}\mathrm{%q__e}{=}{-1}{,}\mathrm{%q__u}{=}\frac{{2}}{{3}}{,}\mathrm{%q__d}{=}{-}\frac{{1}}{{3}}$
 $\mathrm{Defined as the weak isospin for 1\right) the electron, muon and tauon, 2\right) the up, charm and top, 3\right) the down, strange and bottom, and 4\right) all the neutrinos:}\mathrm{%I__e}{=}{-}\frac{{1}}{{2}}{,}\mathrm{%I__u}{=}\frac{{1}}{{2}}{,}\mathrm{%I__d}{=}{-}\frac{{1}}{{2}}{,}\mathrm{%I__n}{=}\frac{{1}}{{2}}{,}$
 $\mathrm{You can use the active form without the % prefix, or the \text{'}value\text{'} command to give the corresponding value to any of the inert representations}\mathrm{%q__e}{,}\mathrm{%q__u}{,}\mathrm{%q__d}{,}\mathrm{%I__e}{,}\mathrm{%I__u}{,}\mathrm{%I__d}{,}\mathrm{%I__n}{,}$
 ${}\mathrm{_______________________________________________________}$
 ${}\mathrm{Default differentiation variables for d_, D_ and dAlembertian are:}{}\left\{X=\left(x{,}y{,}z{,}t\right)\right\}$
 $\mathrm{Minkowski spacetime with signatre}\left(\mathrm{- - - +}\right)$
 ${}\mathrm{_______________________________________________________}$
 $\left[\mathrm{%I__d}{,}\mathrm{%I__e}{,}\mathrm{%I__n}{,}\mathrm{%I__u}{,}\mathrm{%q__d}{,}\mathrm{%q__e}{,}\mathrm{%q__u}{,}{\mathrm{BField}}{,}{\mathrm{BFieldStrength}}{,}{\mathrm{Bottom}}{,}{\mathrm{CKM}}{,}{\mathrm{Charm}}{,}{\mathrm{Down}}{,}{\mathrm{ElectromagneticField}}{,}{\mathrm{ElectromagneticFieldStrength}}{,}{\mathrm{Electron}}{,}{\mathrm{ElectronNeutrino}}{,}{\mathrm{FSU3}}{,}{\mathrm{Glambda}}{,}{\mathrm{GluonField}}{,}{\mathrm{GluonFieldStrength}}{,}{\mathrm{HiggsBoson}}{,}{\mathrm{Lagrangian}}{,}{\mathrm{Muon}}{,}{\mathrm{MuonNeutrino}}{,}{\mathrm{Strange}}{,}{\mathrm{Tauon}}{,}{\mathrm{TauonNeutrino}}{,}{\mathrm{Top}}{,}{\mathrm{Up}}{,}{\mathrm{WField}}{,}{\mathrm{WFieldStrength}}{,}{\mathrm{WMinusField}}{,}{\mathrm{WMinusFieldStrength}}{,}{\mathrm{WPlusField}}{,}{\mathrm{WPlusFieldStrength}}{,}{\mathrm{WeinbergAngle}}{,}{\mathrm{ZField}}{,}{\mathrm{ZFieldStrength}}{,}\mathrm{g__e}{,}\mathrm{g__s}{,}\mathrm{g__w}\right]$ (1)
 • A key observation in this implementation is that, with the exception of representing the structure constants ${\mathrm{FSU3}}_{a,b}$ and $\mathrm{Glambda}$ representing the Gell-Mann matrices ${{\mathrm{\lambda }}}_{a}$, the package's commands are not actually routines that perform tasks but representations of the physical fields of the model with the mathematical properties set and understood by the Maple system. Once the representations are in place, the actual computations are performed by the Physics package commands, e.g. scattering amplitudes are computed using FeynmanDiagrams, covariant and non-covariant derivatives are computed using D_ and d_, commutators and anticommutators using Commutator and AntiCommutator, and the simplification of tensor indices is performed by the Physics Simplify command. The $\mathrm{Lagrangian}$ command is used to retrieve all or part of the Lagrangian of the Standard Model, as shown in the Examples. List of StandardModel Package Commands

 • The following is a list of available commands.

 $\mathrm{BField}$ $\mathrm{BFieldStrength}$ $\mathrm{Bottom}$ $\mathrm{CKM}$ $\mathrm{Charm}$ $\mathrm{Down}$ $\mathrm{ElectromagneticField}$ $\mathrm{ElectromagneticFieldStrength}$ $\mathrm{Electron}$ $\mathrm{ElectronNeutrino}$ $\mathrm{FSU3}$ $\mathrm{Glambda}$ $\mathrm{GluonField}$ $\mathrm{GluonFieldStrength}$ $\mathrm{HiggsBoson}$ $\mathrm{Muon}$ $\mathrm{MuonNeutrino}$ $\mathrm{Strange}$ $\mathrm{Tauon}$ $\mathrm{TauonNeutrino}$ $\mathrm{Top}$ $\mathrm{Up}$ $\mathrm{WField}$ $\mathrm{WFieldStrength}$ $\mathrm{WMinusField}$ $\mathrm{WMinusFieldStrength}$ $\mathrm{WPlusField}$ $\mathrm{WPlusFieldStrength}$ $\mathrm{WeinbergAngle}$ $\mathrm{ZField}$ $\mathrm{ZFieldStrength}$ $\mathrm{g__e}$ $\mathrm{g__s}$ $\mathrm{g__w}$ $\mathrm{%I__d}$ $\mathrm{%I__e}$ $\mathrm{%I__n}$ $\mathrm{%I__u}$ $\mathrm{%q__d}$ $\mathrm{%q__e}$ $\mathrm{%q__u}$

 • The electric part of the electroweak charges and isospins of the model are all represented in inert form following the Maple convention: prefixed with the % symbol displayed in gray as in (1). If any of these constants is entered without that prefix you get its actual value; for the electric part of the electroweak charges of the model, in units of $\mathrm{g__e}$, that is  respectively for the 1) the electron, muon and tauon, 2) the up, charm and top quarks and 3) the down strange and bottom quarks. For the isospins that is , respectively for 1) the electron, muon and tauon, 2) the up, charm and top, 3) the down, strange and bottom, and 4) all the neutrinos.
 • The electromagnetic coupling constant $\mathrm{g__e}$ is negative, equal to the charge of the electron, and $\mathrm{g__s}$ and $\mathrm{g__w}$ are respectively the the strong and weak coupling constants. These are the constants that enter as a factor in the gauge term of covariant derivatives, and so in the definition of the gauge field strengths ${{\mathbf{𝔾}}}_{\mathrm{\mu },\mathrm{\nu },a}$ and ${{\mathbf{𝕎}}}_{\mathrm{\mu },\mathrm{\nu },J}$ and in the interaction terms of the Lagrangian.
 • The  command is a representation for the Cabibbo Kobayashi Maskawa matrix.
 • Further information relevant to the use of these commands is found under Conventions used in the Physics package (spacetime tensors, not commutative variables and functions, quantum states and Dirac notation).
 • The typesetting of the package's representations for fields and constant follows as much as possible the one in textbooks and works under copy and paste - so that expressions from the output can be reused as input. To transform Maple worksheets using this package into LaTeX files see LaTeX. Examples

The Leptons, Quarks, Gauge Fields and structure constants of the model

The massless fields of the model are the electromagnetic field ${\mathbf{A}}$, the gluons ${\mathbf{G}}$ and neutrinos $\mathrm{MuonNeutrino},\mathrm{TauonNeutrino}$and $\mathrm{ElectronNeutrino}$

 > $\mathrm{Setup}\left(\mathrm{massless}\right)$
 $\mathrm{* Partial match of \text{'}}\mathrm{massless}\mathrm{\text{'} against keyword \text{'}}\mathrm{masslessfields}\text{'}$
 $\mathrm{_______________________________________________________}$
 $\left[{\mathrm{masslessfields}}{=}\left\{{\mathbf{G}}{,}{\mathrm{MuonNeutrino}}{,}{\mathrm{TauonNeutrino}}{,}{\mathbf{A}}{,}{\mathrm{ElectronNeutrino}}\right\}\right]$ (2)

The Leptons and Quarks of the model are

 > $\mathrm{StandardModel}:-\mathrm{Leptons}$
 $\left[{\mathbf{e}}{,}{\mathbf{\mu }}{,}{\mathbf{\tau }}{,}{\mathrm{ElectronNeutrino}}{,}{\mathrm{MuonNeutrino}}{,}{\mathrm{TauonNeutrino}}\right]$ (3)
 >
 $\left[{\mathbf{u}}{,}{\mathbf{c}}{,}{\mathbf{t}}{,}{\mathbf{d}}{,}{\mathbf{s}}{,}{\mathbf{b}}\right]$ (4)

The Gauge fields

 > $\mathrm{StandardModel}:-\mathrm{GaugeFields}$
 $\left[{\mathbf{A}}{,}{\mathbf{𝔽}}{,}{\mathbf{B}}{,}{\mathbf{𝔹}}{,}{\mathbf{W}}{,}{\mathbf{𝕎}}{,}{\mathbf{G}}{,}{\mathbf{𝔾}}{,}{\mathrm{WMinusField}}{,}{\mathrm{WMinusFieldStrength}}{,}{\mathrm{WPlusField}}{,}{\mathrm{WPlusFieldStrength}}{,}{\mathbf{Z}}{,}{\mathbf{ℤ}}\right]$ (5)

For readability, omit the functionality of all these fields from the display of formulas that follows (see CompactDisplay) and use the lowercase $i$ instead of the uppercase $\mathrm{I}$ to represent the imaginary unit

 >
 >

The definitions of the gauge fields can be seen as with any other tensor of the Physics package using the keyword $\mathrm{definition}$

 > $\mathrm{ElectromagneticField}\left[\mathrm{definition}\right]$
 ${{\mathbf{A}}}_{{\mathrm{\mu }}}{=}{\mathrm{sin}}{}\left({\mathrm{WeinbergAngle}}\right){}{{\mathbf{W}}}_{{\mathrm{\mu }}\phantom{{3}}}^{\phantom{{\mathrm{\mu }}}{3}}{+}{\mathrm{cos}}{}\left({\mathrm{WeinbergAngle}}\right){}{{\mathbf{B}}}_{{\mathrm{\mu }}}$ (6)
 >
 $\left[{{\mathrm{ElectromagneticField}}}_{{\mathrm{μ}}}{=}{\mathrm{sin}}{}\left({\mathrm{WeinbergAngle}}\right){}{{\mathrm{WField}}}_{{\mathrm{μ}}{,}{\mathrm{~3}}}{+}{\mathrm{cos}}{}\left({\mathrm{WeinbergAngle}}\right){}{{\mathrm{BField}}}_{{\mathrm{μ}}}{,}{{\mathrm{ElectromagneticFieldStrength}}}_{{\mathrm{μ}}{,}{\mathrm{ν}}}{=}{{\mathrm{d_}}}_{{\mathrm{μ}}}{}\left({{\mathrm{ElectromagneticField}}}_{{\mathrm{ν}}}{}\left({X}\right){,}\left[{X}\right]\right){-}{{\mathrm{d_}}}_{{\mathrm{ν}}}{}\left({{\mathrm{ElectromagneticField}}}_{{\mathrm{μ}}}{}\left({X}\right){,}\left[{X}\right]\right){,}{{\mathrm{BField}}}_{{\mathrm{μ}}}{=}\left(\left[\begin{array}{cccc}{\mathrm{BField}}_{1}& {\mathrm{BField}}_{2}& {\mathrm{BField}}_{3}& {\mathrm{BField}}_{4}\end{array}\right]\right){,}{{\mathrm{BFieldStrength}}}_{{\mathrm{μ}}{,}{\mathrm{ν}}}{=}{{\mathrm{d_}}}_{{\mathrm{μ}}}{}\left({{\mathrm{BField}}}_{{\mathrm{ν}}}{}\left({X}\right){,}\left[{X}\right]\right){-}{{\mathrm{d_}}}_{{\mathrm{ν}}}{}\left({{\mathrm{BField}}}_{{\mathrm{μ}}}{}\left({X}\right){,}\left[{X}\right]\right){,}{{\mathrm{WField}}}_{{\mathrm{μ}}{,}{J}}{=}\left(\left[\begin{array}{ccc}{\mathrm{WField}}_{1,1}& {\mathrm{WField}}_{1,2}& {\mathrm{WField}}_{1,3}\\ {\mathrm{WField}}_{2,1}& {\mathrm{WField}}_{2,2}& {\mathrm{WField}}_{2,3}\\ {\mathrm{WField}}_{3,1}& {\mathrm{WField}}_{3,2}& {\mathrm{WField}}_{3,3}\\ {\mathrm{WField}}_{4,1}& {\mathrm{WField}}_{4,2}& {\mathrm{WField}}_{4,3}\end{array}\right]\right){,}{{\mathrm{WFieldStrength}}}_{{\mathrm{μ}}{,}{\mathrm{ν}}{,}{J}}{=}{{\mathrm{d_}}}_{{\mathrm{μ}}}{}\left({{\mathrm{WField}}}_{{\mathrm{ν}}{,}{J}}{}\left({X}\right){,}\left[{X}\right]\right){-}{{\mathrm{d_}}}_{{\mathrm{ν}}}{}\left({{\mathrm{WField}}}_{{\mathrm{μ}}{,}{J}}{}\left({X}\right){,}\left[{X}\right]\right){+}\mathrm{g__w}{}{{\mathrm{LeviCivita}}}_{{J}{,}{K}{,}{L}}{}{\mathrm{*}}{}\left({{\mathrm{WField}}}_{{\mathrm{μ}}{,}{K}}{}\left({X}\right){,}{{\mathrm{WField}}}_{{\mathrm{ν}}{,}{L}}{}\left({X}\right)\right){,}{{\mathrm{GluonField}}}_{{\mathrm{μ}}{,}{a}}{=}\left(\left[\begin{array}{cccccccc}{\mathrm{GluonField}}_{1,1}& {\mathrm{GluonField}}_{1,2}& {\mathrm{GluonField}}_{1,3}& {\mathrm{GluonField}}_{1,4}& {\mathrm{GluonField}}_{1,5}& {\mathrm{GluonField}}_{1,6}& {\mathrm{GluonField}}_{1,7}& {\mathrm{GluonField}}_{1,8}\\ {\mathrm{GluonField}}_{2,1}& {\mathrm{GluonField}}_{2,2}& {\mathrm{GluonField}}_{2,3}& {\mathrm{GluonField}}_{2,4}& {\mathrm{GluonField}}_{2,5}& {\mathrm{GluonField}}_{2,6}& {\mathrm{GluonField}}_{2,7}& {\mathrm{GluonField}}_{2,8}\\ {\mathrm{GluonField}}_{3,1}& {\mathrm{GluonField}}_{3,2}& {\mathrm{GluonField}}_{3,3}& {\mathrm{GluonField}}_{3,4}& {\mathrm{GluonField}}_{3,5}& {\mathrm{GluonField}}_{3,6}& {\mathrm{GluonField}}_{3,7}& {\mathrm{GluonField}}_{3,8}\\ {\mathrm{GluonField}}_{4,1}& {\mathrm{GluonField}}_{4,2}& {\mathrm{GluonField}}_{4,3}& {\mathrm{GluonField}}_{4,4}& {\mathrm{GluonField}}_{4,5}& {\mathrm{GluonField}}_{4,6}& {\mathrm{GluonField}}_{4,7}& {\mathrm{GluonField}}_{4,8}\end{array}\right]\right){,}{{\mathrm{GluonFieldStrength}}}_{{\mathrm{μ}}{,}{\mathrm{ν}}{,}{a}}{=}{{\mathrm{d_}}}_{{\mathrm{μ}}}{}\left({{\mathrm{GluonField}}}_{{\mathrm{ν}}{,}{a}}{}\left({X}\right){,}\left[{X}\right]\right){-}{{\mathrm{d_}}}_{{\mathrm{ν}}}{}\left({{\mathrm{GluonField}}}_{{\mathrm{μ}}{,}{a}}{}\left({X}\right){,}\left[{X}\right]\right){+}\mathrm{g__s}{}{{\mathrm{FSU3}}}_{{a}{,}{b}{,}{c}}{}{\mathrm{*}}{}\left({{\mathrm{GluonField}}}_{{\mathrm{μ}}{,}{b}}{}\left({X}\right){,}{{\mathrm{GluonField}}}_{{\mathrm{ν}}{,}{c}}{}\left({X}\right)\right){,}{{\mathrm{WMinusField}}}_{{\mathrm{μ}}}{=}\frac{{1}}{{2}}{}\left({{\mathrm{WField}}}_{{\mathrm{μ}}{,}{\mathrm{~1}}}{+}{i}{}{{\mathrm{WField}}}_{{\mathrm{μ}}{,}{\mathrm{~2}}}\right){}\sqrt{{2}}{,}{{\mathrm{WMinusFieldStrength}}}_{{\mathrm{μ}}{,}{\mathrm{ν}}}{=}{{\mathrm{d_}}}_{{\mathrm{μ}}}{}\left({{\mathrm{WMinusField}}}_{{\mathrm{ν}}}{}\left({X}\right){,}\left[{X}\right]\right){-}{{\mathrm{d_}}}_{{\mathrm{ν}}}{}\left({{\mathrm{WMinusField}}}_{{\mathrm{μ}}}{}\left({X}\right){,}\left[{X}\right]\right){,}{{\mathrm{WPlusField}}}_{{\mathrm{μ}}}{=}\frac{{1}}{{2}}{}\left({{\mathrm{WField}}}_{{\mathrm{μ}}{,}{\mathrm{~1}}}{-}{i}{}{{\mathrm{WField}}}_{{\mathrm{μ}}{,}{\mathrm{~2}}}\right){}\sqrt{{2}}{,}{{\mathrm{WPlusFieldStrength}}}_{{\mathrm{μ}}{,}{\mathrm{ν}}}{=}{{\mathrm{d_}}}_{{\mathrm{μ}}}{}\left({{\mathrm{WPlusField}}}_{{\mathrm{ν}}}{}\left({X}\right){,}\left[{X}\right]\right){-}{{\mathrm{d_}}}_{{\mathrm{ν}}}{}\left({{\mathrm{WPlusField}}}_{{\mathrm{μ}}}{}\left({X}\right){,}\left[{X}\right]\right){,}{{\mathrm{ZField}}}_{{\mathrm{μ}}}{=}{\mathrm{cos}}{}\left({\mathrm{WeinbergAngle}}\right){}{{\mathrm{WField}}}_{{\mathrm{μ}}{,}{\mathrm{~3}}}{-}{\mathrm{sin}}{}\left({\mathrm{WeinbergAngle}}\right){}{{\mathrm{BField}}}_{{\mathrm{μ}}}{,}{{\mathrm{ZFieldStrength}}}_{{\mathrm{μ}}{,}{\mathrm{ν}}}{=}{{\mathrm{d_}}}_{{\mathrm{μ}}}{}\left({{\mathrm{ZField}}}_{{\mathrm{ν}}}{}\left({X}\right){,}\left[{X}\right]\right){-}{{\mathrm{d_}}}_{{\mathrm{ν}}}{}\left({{\mathrm{ZField}}}_{{\mathrm{μ}}}{}\left({X}\right){,}\left[{X}\right]\right)\right]$ (7)

The convention in (7) for the signs in the definitions of ${{\mathbf{A}}}_{\mathrm{\mu }}$ (using +sin) and ${{\mathbf{Z}}}_{\mathrm{\mu }}$(using -sin)  also follows ref  (page 702) and  (page 66), and the presentation of the Standard Model in Wikipedia, but this convention is not uniform in the literature, and ref  (vol II, page 307) uses the opposite convention for these signs.

The definitions of covariant derivatives (not shown above) follow references  (pages 690 and 802) and  (pages 355 of vol I and 155 of vol II), also the Wikipedia (webpages for QED and QCD) . E.g. for the Electron ${{\mathbf{e}}}_{j}$

 >
 ${{\mathrm{D_}}}_{{\mathrm{μ}}}{}\left({{\mathrm{Electron}}}_{{j}}{}\left({X}\right){,}\left[{X}\right]\right){=}{{\mathrm{d_}}}_{{\mathrm{μ}}}{}\left({{\mathrm{Electron}}}_{{j}}{}\left({X}\right){,}\left[{X}\right]\right){+}{i}{}\mathrm{g__e}{}{\mathrm{*}}{}\left({{\mathrm{Electron}}}_{{j}}{}\left({X}\right){,}{{\mathrm{ElectromagneticField}}}_{{\mathrm{μ}}}{}\left({X}\right)\right)$ (8)

where $\mathrm{g__e}<0$ is equal to the electric charge of the electron, and for the Top,

 >
 ${{\mathrm{D_}}}_{{\mathrm{μ}}}{}\left({{\mathrm{Up}}}_{{j}{,}{A}}{}\left({X}\right){,}\left[{X}\right]\right){=}{{\mathrm{d_}}}_{{\mathrm{μ}}}{}\left({{\mathrm{Up}}}_{{j}{,}{A}}{}\left({X}\right){,}\left[{X}\right]\right){-}\frac{{1}}{{2}}{}{i}{}\mathrm{g__s}{}{\mathrm{*}}{}\left({{{\mathrm{Glambda}}}_{{a}}}_{{A}{,}{B}}{,}{{\mathrm{Up}}}_{{j}{,}{B}}{}\left({X}\right){,}{{\mathrm{GluonField}}}_{{\mathrm{μ}}{,}{a}}{}\left({X}\right)\right)$ (9)

where $\mathrm{g__s}>0$ is the strong coupling constant. The Gell-Mann matrices, that enter gauge terms in the interaction Lagrangian of the StandardModel are represented by $\mathrm{Glambda}$, implemented as a tensor with an SU(3) adjoint representation index, all of whose components are matrices

 > $\mathrm{Glambda}\left[\right]$
 ${{\mathrm{Glambda}}}_{{a}}{=}\left(\left[\begin{array}{cccccccc}{\mathrm{Glambda}}_{1}& {\mathrm{Glambda}}_{2}& {\mathrm{Glambda}}_{3}& {\mathrm{Glambda}}_{4}& {\mathrm{Glambda}}_{5}& {\mathrm{Glambda}}_{6}& {\mathrm{Glambda}}_{7}& {\mathrm{Glambda}}_{8}\end{array}\right]\right)$ (10)
 >
 ${{\mathrm{\lambda }}}_{{1}}{=}\left[\begin{array}{ccc}{0}& {1}& {0}\\ {1}& {0}& {0}\\ {0}& {0}& {0}\end{array}\right]{,}{{\mathrm{\lambda }}}_{{2}}{=}\left[\begin{array}{ccc}{0}& {-i}& {0}\\ {i}& {0}& {0}\\ {0}& {0}& {0}\end{array}\right]{,}{{\mathrm{\lambda }}}_{{3}}{=}\left[\begin{array}{ccc}{1}& {0}& {0}\\ {0}& {-1}& {0}\\ {0}& {0}& {0}\end{array}\right]{,}{{\mathrm{\lambda }}}_{{4}}{=}\left[\begin{array}{ccc}{0}& {0}& {1}\\ {0}& {0}& {0}\\ {1}& {0}& {0}\end{array}\right]{,}{{\mathrm{\lambda }}}_{{5}}{=}\left[\begin{array}{ccc}{0}& {0}& {-i}\\ {0}& {0}& {0}\\ {i}& {0}& {0}\end{array}\right]{,}{{\mathrm{\lambda }}}_{{6}}{=}\left[\begin{array}{ccc}{0}& {0}& {0}\\ {0}& {0}& {1}\\ {0}& {1}& {0}\end{array}\right]{,}{{\mathrm{\lambda }}}_{{7}}{=}\left[\begin{array}{ccc}{0}& {0}& {0}\\ {0}& {0}& {-i}\\ {0}& {i}& {0}\end{array}\right]{,}{{\mathrm{\lambda }}}_{{8}}{=}\left[\begin{array}{ccc}\frac{\sqrt{{3}}}{{3}}& {0}& {0}\\ {0}& \frac{\sqrt{{3}}}{{3}}& {0}\\ {0}& {0}& {-}\frac{{2}{}\sqrt{{3}}}{{3}}\end{array}\right]$ (11)

These matrices satisfy a SU(3) algebra

 > $\mathrm{Library}:-\mathrm{DefaultAlgebraRules}\left(\mathrm{Glambda}\right)$
 ${\mathrm{%Commutator}}{}\left({{\mathrm{Glambda}}}_{{b}}{,}{{\mathrm{Glambda}}}_{{c}}\right){=}{2}{}{i}{}{{\mathrm{FSU3}}}_{{a}{,}{b}{,}{c}}{}{{\mathrm{Glambda}}}_{{a}}$ (12)

The structure constants ${\mathrm{FSU3}}_{a,b,c}$entering (12) and interaction Lagrangian terms of the StandardModel form a 3-dimensional array of 8 x 8 matrices represented by the command $\mathrm{FSU3}$, implemented as a tensor with three SU(3) adjoint representation indices. As with any other tensor of the Physics package, to see its components you can use the keyword $\mathrm{matrix}$, e.g.

 > $\mathrm{FSU3}\left[1,b,c,\mathrm{matrix}\right]$
 ${{\mathrm{FSU3}}}_{{1}{,}{b}{,}{c}}{=}\left(\left[\begin{array}{cccccccc}0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0& 0\\ 0& -1& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& \frac{1}{2}& 0\\ 0& 0& 0& 0& 0& -\frac{1}{2}& 0& 0\\ 0& 0& 0& 0& \frac{1}{2}& 0& 0& 0\\ 0& 0& 0& -\frac{1}{2}& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right]\right)$ (13)

or, for a more general exploration of the components of  ${\mathrm{FSU3}}_{a,b,c}$you can use the command TensorArray with the option $\mathrm{explore}$

 > $\mathrm{TensorArray}\left(\mathrm{FSU3}\left[a,b,c\right],\mathrm{explore}\right)$
 ${}{{\mathrm{FSU3}}}_{{a}{,}{b}{,}{c}}\left(\mathrm{ordering of free indices}=\left[a{,}b{,}c\right]\right)$ (14)

 $\mathbf{Index 1}$ abc

$\mathbf{Value of Index 1}$  The tensorial equation for the Gell-Mann matrices

 >
 ${\mathrm{%Commutator}}{}\left({{\mathrm{Glambda}}}_{{b}}{,}{{\mathrm{Glambda}}}_{{c}}\right){=}{2}{}{i}{}{{\mathrm{FSU3}}}_{{a}{,}{b}{,}{c}}{}{{\mathrm{Glambda}}}_{{a}}$ (15)

is computable for each value of its tensor indices, e.g.

 > $\mathrm{SumOverRepeatedIndices}\left(\right)$
 ${\mathrm{%Commutator}}{}\left({{\mathrm{Glambda}}}_{{b}}{,}{{\mathrm{Glambda}}}_{{c}}\right){=}{2}{}{i}{}\left({{\mathrm{FSU3}}}_{{1}{,}{b}{,}{c}}{}{{\mathrm{Glambda}}}_{{1}}{+}{{\mathrm{FSU3}}}_{{2}{,}{b}{,}{c}}{}{{\mathrm{Glambda}}}_{{2}}{+}{{\mathrm{FSU3}}}_{{3}{,}{b}{,}{c}}{}{{\mathrm{Glambda}}}_{{3}}{+}{{\mathrm{FSU3}}}_{{4}{,}{b}{,}{c}}{}{{\mathrm{Glambda}}}_{{4}}{+}{{\mathrm{FSU3}}}_{{5}{,}{b}{,}{c}}{}{{\mathrm{Glambda}}}_{{5}}{+}{{\mathrm{FSU3}}}_{{6}{,}{b}{,}{c}}{}{{\mathrm{Glambda}}}_{{6}}{+}{{\mathrm{FSU3}}}_{{7}{,}{b}{,}{c}}{}{{\mathrm{Glambda}}}_{{7}}{+}{{\mathrm{FSU3}}}_{{8}{,}{b}{,}{c}}{}{{\mathrm{Glambda}}}_{{8}}\right)$ (16)
 > $\mathrm{eval}\left(,\left[b=4,c=5\right]\right)$
 ${\mathrm{%Commutator}}{}\left({{\mathrm{Glambda}}}_{{4}}{,}{{\mathrm{Glambda}}}_{{5}}\right){=}{2}{}{i}{}\left(\frac{{1}}{{2}}{}{{\mathrm{Glambda}}}_{{3}}{+}\frac{{1}}{{2}}{}\sqrt{{3}}{}{{\mathrm{Glambda}}}_{{8}}\right)$ (17)

Activating the left-hand side,

 > $\mathrm{value}\left(\right)$
 ${2}{}{i}{}{{\mathrm{FSU3}}}_{{4}{,}{5}{,}{a}}{}{{\mathrm{\lambda }}}_{{a}}{=}{2}{}{i}{}\left(\frac{{{\mathrm{\lambda }}}_{{3}}}{{2}}{+}\frac{\sqrt{{3}}{}{{\mathrm{\lambda }}}_{{8}}}{{2}}\right)$ (18)
 > $\mathrm{expand}\left(\mathrm{SumOverRepeatedIndices}\left(\right)\right)$
 ${i}{}{{\mathrm{\lambda }}}_{{3}}{+}{i}{}\sqrt{{3}}{}{{\mathrm{\lambda }}}_{{8}}{=}{i}{}{{\mathrm{\lambda }}}_{{3}}{+}{i}{}\sqrt{{3}}{}{{\mathrm{\lambda }}}_{{8}}$ (19)

To see all the components of (12) ≡ $\mathrm{%Commutator}\left({\mathrm{Glambda}}_{b},{\mathrm{Glambda}}_{c}\right)=2I{\mathrm{FSU3}}_{a,b,c}{{\mathrm{\lambda }}}_{a}$ at once you can use TensorArray

 > $\mathrm{TensorArray}\left(\right)$
 $\left[\begin{array}{cccccccc}\mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{1},{\mathrm{Glambda}}_{1}\right)=0& \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{1},{\mathrm{Glambda}}_{2}\right)=2{}I{}{\mathrm{Glambda}}_{3}& \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{1},{\mathrm{Glambda}}_{3}\right)=-2{}I{}{\mathrm{Glambda}}_{2}& \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{1},{\mathrm{Glambda}}_{4}\right)=I{}{\mathrm{Glambda}}_{7}& \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{1},{\mathrm{Glambda}}_{5}\right)=-I{}{\mathrm{Glambda}}_{6}& \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{1},{\mathrm{Glambda}}_{6}\right)=I{}{\mathrm{Glambda}}_{5}& \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{1},{\mathrm{Glambda}}_{7}\right)=-I{}{\mathrm{Glambda}}_{4}& \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{1},{\mathrm{Glambda}}_{8}\right)=0\\ \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{2},{\mathrm{Glambda}}_{1}\right)=-2{}I{}{\mathrm{Glambda}}_{3}& \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{2},{\mathrm{Glambda}}_{2}\right)=0& \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{2},{\mathrm{Glambda}}_{3}\right)=2{}I{}{\mathrm{Glambda}}_{1}& \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{2},{\mathrm{Glambda}}_{4}\right)=I{}{\mathrm{Glambda}}_{6}& \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{2},{\mathrm{Glambda}}_{5}\right)=I{}{\mathrm{Glambda}}_{7}& \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{2},{\mathrm{Glambda}}_{6}\right)=-I{}{\mathrm{Glambda}}_{4}& \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{2},{\mathrm{Glambda}}_{7}\right)=-I{}{\mathrm{Glambda}}_{5}& \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{2},{\mathrm{Glambda}}_{8}\right)=0\\ \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{3},{\mathrm{Glambda}}_{1}\right)=2{}I{}{\mathrm{Glambda}}_{2}& \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{3},{\mathrm{Glambda}}_{2}\right)=-2{}I{}{\mathrm{Glambda}}_{1}& \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{3},{\mathrm{Glambda}}_{3}\right)=0& \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{3},{\mathrm{Glambda}}_{4}\right)=I{}{\mathrm{Glambda}}_{5}& \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{3},{\mathrm{Glambda}}_{5}\right)=-I{}{\mathrm{Glambda}}_{4}& \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{3},{\mathrm{Glambda}}_{6}\right)=-I{}{\mathrm{Glambda}}_{7}& \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{3},{\mathrm{Glambda}}_{7}\right)=I{}{\mathrm{Glambda}}_{6}& \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{3},{\mathrm{Glambda}}_{8}\right)=0\\ \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{4},{\mathrm{Glambda}}_{1}\right)=-I{}{\mathrm{Glambda}}_{7}& \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{4},{\mathrm{Glambda}}_{2}\right)=-I{}{\mathrm{Glambda}}_{6}& \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{4},{\mathrm{Glambda}}_{3}\right)=-I{}{\mathrm{Glambda}}_{5}& \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{4},{\mathrm{Glambda}}_{4}\right)=0& \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{4},{\mathrm{Glambda}}_{5}\right)=I{}\left(\sqrt{3}{}{\mathrm{Glambda}}_{8}+{\mathrm{Glambda}}_{3}\right)& \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{4},{\mathrm{Glambda}}_{6}\right)=I{}{\mathrm{Glambda}}_{2}& \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{4},{\mathrm{Glambda}}_{7}\right)=I{}{\mathrm{Glambda}}_{1}& \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{4},{\mathrm{Glambda}}_{8}\right)=-I{}\sqrt{3}{}{\mathrm{Glambda}}_{5}\\ \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{5},{\mathrm{Glambda}}_{1}\right)=I{}{\mathrm{Glambda}}_{6}& \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{5},{\mathrm{Glambda}}_{2}\right)=-I{}{\mathrm{Glambda}}_{7}& \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{5},{\mathrm{Glambda}}_{3}\right)=I{}{\mathrm{Glambda}}_{4}& \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{5},{\mathrm{Glambda}}_{4}\right)=-I{}\left(\sqrt{3}{}{\mathrm{Glambda}}_{8}+{\mathrm{Glambda}}_{3}\right)& \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{5},{\mathrm{Glambda}}_{5}\right)=0& \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{5},{\mathrm{Glambda}}_{6}\right)=-I{}{\mathrm{Glambda}}_{1}& \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{5},{\mathrm{Glambda}}_{7}\right)=I{}{\mathrm{Glambda}}_{2}& \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{5},{\mathrm{Glambda}}_{8}\right)=I{}\sqrt{3}{}{\mathrm{Glambda}}_{4}\\ \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{6},{\mathrm{Glambda}}_{1}\right)=-I{}{\mathrm{Glambda}}_{5}& \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{6},{\mathrm{Glambda}}_{2}\right)=I{}{\mathrm{Glambda}}_{4}& \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{6},{\mathrm{Glambda}}_{3}\right)=I{}{\mathrm{Glambda}}_{7}& \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{6},{\mathrm{Glambda}}_{4}\right)=-I{}{\mathrm{Glambda}}_{2}& \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{6},{\mathrm{Glambda}}_{5}\right)=I{}{\mathrm{Glambda}}_{1}& \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{6},{\mathrm{Glambda}}_{6}\right)=0& \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{6},{\mathrm{Glambda}}_{7}\right)=I{}\left(-{\mathrm{Glambda}}_{3}+\sqrt{3}{}{\mathrm{Glambda}}_{8}\right)& \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{6},{\mathrm{Glambda}}_{8}\right)=-I{}\sqrt{3}{}{\mathrm{Glambda}}_{7}\\ \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{7},{\mathrm{Glambda}}_{1}\right)=I{}{\mathrm{Glambda}}_{4}& \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{7},{\mathrm{Glambda}}_{2}\right)=I{}{\mathrm{Glambda}}_{5}& \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{7},{\mathrm{Glambda}}_{3}\right)=-I{}{\mathrm{Glambda}}_{6}& \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{7},{\mathrm{Glambda}}_{4}\right)=-I{}{\mathrm{Glambda}}_{1}& \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{7},{\mathrm{Glambda}}_{5}\right)=-I{}{\mathrm{Glambda}}_{2}& \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{7},{\mathrm{Glambda}}_{6}\right)=-I{}\left(-{\mathrm{Glambda}}_{3}+\sqrt{3}{}{\mathrm{Glambda}}_{8}\right)& \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{7},{\mathrm{Glambda}}_{7}\right)=0& \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{7},{\mathrm{Glambda}}_{8}\right)=I{}\sqrt{3}{}{\mathrm{Glambda}}_{6}\\ \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{8},{\mathrm{Glambda}}_{1}\right)=0& \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{8},{\mathrm{Glambda}}_{2}\right)=0& \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{8},{\mathrm{Glambda}}_{3}\right)=0& \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{8},{\mathrm{Glambda}}_{4}\right)=I{}\sqrt{3}{}{\mathrm{Glambda}}_{5}& \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{8},{\mathrm{Glambda}}_{5}\right)=-I{}\sqrt{3}{}{\mathrm{Glambda}}_{4}& \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{8},{\mathrm{Glambda}}_{6}\right)=I{}\sqrt{3}{}{\mathrm{Glambda}}_{7}& \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{8},{\mathrm{Glambda}}_{7}\right)=-I{}\sqrt{3}{}{\mathrm{Glambda}}_{6}& \mathrm{%Commutator}{}\left({\mathrm{Glambda}}_{8},{\mathrm{Glambda}}_{8}\right)=0\end{array}\right]$ (20)

To represent, in what follows, the interaction Lagrangians for QCD and the Electro-Weak sector as sums over leptons and quarks, all of them fermions, it is useful to introduce two anticommutative prefixes to be used as summation indices

 >
 $\left[{\mathrm{anticommutativeprefix}}{=}\left\{\mathrm{f__L}{,}\mathrm{f__Q}\right\}\right]$ (21)
 > $\mathrm{CompactDisplay}\left(\left(\mathrm{f__L},\mathrm{f__Q}\right)\left(X\right)\right)$
 $\mathrm{f__L}{}\left({X}\right){}{\mathrm{will now be displayed as}}{}\mathrm{f__L}$
 $\mathrm{f__Q}{}\left({X}\right){}{\mathrm{will now be displayed as}}{}\mathrm{f__Q}$ (22)

The Quantum Electrodynamics (QED) sector of the Standard Model and its interaction Lagrangian

QED is about the interaction between electrons and the photon. The electron field is implemented as a Dirac spinor $\mathrm{Electron}\left[i\right]$, displayed ${{\mathbf{e}}}_{i}$, and the electromagnetic field as a spacetime tensor $\mathrm{ElectromagneticField}\left[\mathrm{mu}\right]$ displayed ${{\mathbf{A}}}_{\mathrm{\mu }}$. The interaction Lagrangian, used to compute scattering amplitudes, can be input as

 >
 ${-}\mathrm{g__e}{}{{{\mathrm{Dgamma}}}_{{\mathrm{~mu}}}}_{{i}{,}{j}}{}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Electron}}}_{{i}}{}\left({X}\right)\right){,}{{\mathrm{Electron}}}_{{j}}{}\left({X}\right){,}{{\mathrm{ElectromagneticField}}}_{{\mathrm{μ}}}{}\left({X}\right)\right)$ (23)

For example, the self-energies of the electron and the photon are given by

 > ${-}{\mathrm{%FeynmanIntegral}}{}\left({-}\frac{{1}}{{8}}{}\frac{{{{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{Uspinor}}}_{{\mathrm{Electron}}}}_{{k}}{}\left(\mathrm{P__1_}\right){}{\mathrm{conjugate}}{}\left({{{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{Uspinor}}}_{{\mathrm{Electron}}}}_{{l}}{}\left(\mathrm{P__2_}\right)\right){}{\mathrm{g__e}}^{{2}}{}{{{\mathrm{Dgamma}}}_{{\mathrm{~alpha}}}}_{{l}{,}{m}}{}{{{\mathrm{Dgamma}}}_{{\mathrm{~nu}}}}_{{n}{,}{k}}{}\left(\left({\mathrm{P__1}}_{{\mathrm{β}}}{+}{\mathrm{p__2}}_{{\mathrm{β}}}\right){}{{{\mathrm{Dgamma}}}_{{\mathrm{~beta}}}}_{{m}{,}{n}}{+}{{m}}_{{\mathrm{Electron}}}{}{{\mathrm{KroneckerDelta}}}_{{m}{,}{n}}\right){}{{\mathrm{g_}}}_{{\mathrm{α}}{,}{\mathrm{ν}}}{}{\mathrm{Dirac}}{}\left({-}\mathrm{P__2}{+}\mathrm{P__1}\right)}{{{\mathrm{π}}}^{{3}}{}\left({\left(\mathrm{P__1}{+}\mathrm{p__2}\right)}^{{2}}{-}{{m}}_{{\mathrm{Electron}}}^{{2}}{+}{i}{}{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{ε}}\right){}\left({\mathrm{p__2}}^{{2}}{+}{i}{}{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{ε}}\right)}{,}\left[\left[\mathrm{p__2}\right]\right]\right)$ (24)
 > ${-}{\mathrm{%FeynmanIntegral}}{}\left(\frac{{1}}{{16}}{}\frac{{{{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{PolarizationVector}}}_{{\mathrm{ElectromagneticField}}}}_{{\mathrm{ν}}}{}\left(\mathrm{P__1_}\right){}{\mathrm{conjugate}}{}\left({{{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{PolarizationVector}}}_{{\mathrm{ElectromagneticField}}}}_{{\mathrm{α}}}{}\left(\mathrm{P__2_}\right)\right){}{\mathrm{g__e}}^{{2}}{}{{{\mathrm{Dgamma}}}_{{\mathrm{~alpha}}}}_{{n}{,}{k}}{}{{{\mathrm{Dgamma}}}_{{\mathrm{~nu}}}}_{{m}{,}{l}}{}\left(\left({\mathrm{P__1}}_{{\mathrm{β}}}{+}{\mathrm{p__2}}_{{\mathrm{β}}}\right){}{{{\mathrm{Dgamma}}}_{{\mathrm{~beta}}}}_{{k}{,}{m}}{+}{{m}}_{{\mathrm{Electron}}}{}{{\mathrm{KroneckerDelta}}}_{{k}{,}{m}}\right){}\left({{m}}_{{\mathrm{Electron}}}{}{{\mathrm{KroneckerDelta}}}_{{l}{,}{n}}{+}{\mathrm{p__2}}_{{\mathrm{κ}}}{}{{{\mathrm{Dgamma}}}_{{\mathrm{~kappa}}}}_{{l}{,}{n}}\right){}{\mathrm{Dirac}}{}\left({-}\mathrm{P__2}{+}\mathrm{P__1}\right)}{{{\mathrm{π}}}^{{3}}{}\sqrt{\mathrm{E__1}{}\mathrm{E__2}}{}\left({\left(\mathrm{P__1}{+}\mathrm{p__2}\right)}^{{2}}{-}{{m}}_{{\mathrm{Electron}}}^{{2}}{+}{i}{}{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{ε}}\right){}\left({\mathrm{p__2}}^{{2}}{-}{{m}}_{{\mathrm{Electron}}}^{{2}}{+}{i}{}{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{ε}}\right)}{,}\left[\left[\mathrm{p__2}\right]\right]\right)$ (25)

To perform different manipulations of Feynman integrals, or their full evaluation when possible, see the FeynmanIntegral package.

The Lagrangian of QED can also be retrieved using the $\mathrm{Lagrangian}$ command

 > $\mathrm{Lagrangian}\left(\mathrm{QED}\right)$
 ${\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Electron}}}_{{j}}{}\left({X}\right)\right){,}{i}{}{{{\mathrm{Dgamma}}}_{{\mathrm{μ}}}}_{{j}{,}{k}}{}{{\mathrm{D_}}}_{{\mathrm{~mu}}}{-}{{m}}_{{\mathrm{Electron}}}{}{{\mathrm{KroneckerDelta}}}_{{j}{,}{k}}{,}{{\mathrm{Electron}}}_{{k}}{}\left({X}\right)\right){-}\frac{{1}}{{4}}{}{\mathrm{*}}{}\left({{\mathrm{ElectromagneticFieldStrength}}}_{{\mathrm{μ}}{,}{\mathrm{ν}}}{,}{{\mathrm{ElectromagneticFieldStrength}}}_{{\mathrm{~mu}}{,}{\mathrm{~nu}}}\right)$ (26)

Note the above is the full Lagrangian, not just the interaction part. It includes the trace of the electromagnetic field strength (last term) and is expressed using the covariant derivative operator ${▿}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}$. The notation used is product notation to represent the application of the differential operator. To transform this product notation into the application of the operator you can use

 > $\mathrm{Library}:-\mathrm{ApplyProductsOfDifferentialOperators}\left(\right)$
 ${\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Electron}}}_{{j}}{}\left({X}\right)\right){,}{i}{}{{{\mathrm{Dgamma}}}_{{\mathrm{μ}}}}_{{j}{,}{k}}{}{{\mathrm{D_}}}_{{\mathrm{~mu}}}{}\left({{\mathrm{Electron}}}_{{k}}{}\left({X}\right){,}\left[{X}\right]\right){-}{\mathrm{*}}{}\left({{m}}_{{\mathrm{Electron}}}{,}{{\mathrm{KroneckerDelta}}}_{{j}{,}{k}}{,}{{\mathrm{Electron}}}_{{k}}{}\left({X}\right)\right)\right){-}\frac{{1}}{{4}}{}{\mathrm{*}}{}\left({{\mathrm{ElectromagneticFieldStrength}}}_{{\mathrm{μ}}{,}{\mathrm{ν}}}{,}{{\mathrm{ElectromagneticFieldStrength}}}_{{\mathrm{~mu}}{,}{\mathrm{~nu}}}\right)$ (27)

or optionally request to $\mathrm{Lagrangian}$ for the operator to appear applied

 >
 ${\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Electron}}}_{{j}}{}\left({X}\right)\right){,}{i}{}{{\mathrm{D_}}}_{{\mathrm{μ}}}{}\left({{\mathrm{Electron}}}_{{k}}{}\left({X}\right){,}\left[{X}\right]\right){}{{{\mathrm{Dgamma}}}_{{\mathrm{~mu}}}}_{{j}{,}{k}}{-}{\mathrm{*}}{}\left({{m}}_{{\mathrm{Electron}}}{,}{{\mathrm{KroneckerDelta}}}_{{j}{,}{k}}{,}{{\mathrm{Electron}}}_{{k}}{}\left({X}\right)\right)\right){-}\frac{{1}}{{4}}{}{\mathrm{*}}{}\left({{\mathrm{ElectromagneticFieldStrength}}}_{{\mathrm{μ}}{,}{\mathrm{ν}}}{,}{{\mathrm{ElectromagneticFieldStrength}}}_{{\mathrm{~mu}}{,}{\mathrm{~nu}}}\right)$ (28)

To expand the covariant derivative, use expand

 >
 ${{\mathrm{D_}}}_{{\mathrm{μ}}}{}\left({{\mathrm{Electron}}}_{{k}}{}\left({X}\right){,}\left[{X}\right]\right){=}{{\mathrm{d_}}}_{{\mathrm{μ}}}{}\left({{\mathrm{Electron}}}_{{k}}{}\left({X}\right){,}\left[{X}\right]\right){+}{i}{}\mathrm{g__e}{}{\mathrm{*}}{}\left({{\mathrm{Electron}}}_{{k}}{}\left({X}\right){,}{{\mathrm{ElectromagneticField}}}_{{\mathrm{μ}}}{}\left({X}\right)\right)$ (29)

from where

 > $\mathrm{expand}\left(\right)$
 ${i}{}{{{\mathrm{Dgamma}}}_{{\mathrm{~mu}}}}_{{j}{,}{k}}{}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Electron}}}_{{j}}{}\left({X}\right)\right){,}{{\mathrm{d_}}}_{{\mathrm{μ}}}{}\left({{\mathrm{Electron}}}_{{k}}{}\left({X}\right){,}\left[{X}\right]\right)\right){-}{{{\mathrm{Dgamma}}}_{{\mathrm{~mu}}}}_{{j}{,}{k}}{}\mathrm{g__e}{}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Electron}}}_{{j}}{}\left({X}\right)\right){,}{{\mathrm{Electron}}}_{{k}}{}\left({X}\right){,}{{\mathrm{ElectromagneticField}}}_{{\mathrm{μ}}}{}\left({X}\right)\right){-}{{m}}_{{\mathrm{Electron}}}{}{{\mathrm{KroneckerDelta}}}_{{j}{,}{k}}{}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Electron}}}_{{j}}{}\left({X}\right)\right){,}{{\mathrm{Electron}}}_{{k}}{}\left({X}\right)\right){-}\frac{{1}}{{4}}{}{\mathrm{*}}{}\left({{\mathrm{ElectromagneticFieldStrength}}}_{{\mathrm{μ}}{,}{\mathrm{ν}}}{,}{{\mathrm{ElectromagneticFieldStrength}}}_{{\mathrm{~mu}}{,}{\mathrm{~nu}}}\right)$ (30)

You can also optionally request the Lagrangian to appear expanded, in which case ${{\mathbf{𝔽}}}_{\mathrm{\mu },\mathrm{\nu }}$ is also expanded using its definition

 > $\mathrm{ElectromagneticFieldStrength}\left[\mathrm{definition}\right]$
 ${{\mathrm{ElectromagneticFieldStrength}}}_{{\mathrm{μ}}{,}{\mathrm{ν}}}{=}{{\mathrm{d_}}}_{{\mathrm{μ}}}{}\left({{\mathrm{ElectromagneticField}}}_{{\mathrm{ν}}}{}\left({X}\right){,}\left[{X}\right]\right){-}{{\mathrm{d_}}}_{{\mathrm{ν}}}{}\left({{\mathrm{ElectromagneticField}}}_{{\mathrm{μ}}}{}\left({X}\right){,}\left[{X}\right]\right)$ (31)
 >
 ${i}{}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Electron}}}_{{j}}{}\left({X}\right)\right){,}{{\mathrm{d_}}}_{{\mathrm{μ}}}{}\left({{\mathrm{Electron}}}_{{k}}{}\left({X}\right){,}\left[{X}\right]\right)\right){}{{{\mathrm{Dgamma}}}_{{\mathrm{~mu}}}}_{{j}{,}{k}}{-}\mathrm{g__e}{}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Electron}}}_{{j}}{}\left({X}\right)\right){,}{{\mathrm{Electron}}}_{{k}}{}\left({X}\right){,}{{\mathrm{ElectromagneticField}}}_{{\mathrm{μ}}}{}\left({X}\right)\right){}{{{\mathrm{Dgamma}}}_{{\mathrm{~mu}}}}_{{j}{,}{k}}{-}{{m}}_{{\mathrm{Electron}}}{}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Electron}}}_{{j}}{}\left({X}\right)\right){,}{{\mathrm{Electron}}}_{{j}}{}\left({X}\right)\right){-}\frac{{1}}{{4}}{}{\mathrm{*}}{}\left({{\mathrm{d_}}}_{{\mathrm{μ}}}{}\left({{\mathrm{ElectromagneticField}}}_{{\mathrm{ν}}}{}\left({X}\right){,}\left[{X}\right]\right){-}{{\mathrm{d_}}}_{{\mathrm{ν}}}{}\left({{\mathrm{ElectromagneticField}}}_{{\mathrm{μ}}}{}\left({X}\right){,}\left[{X}\right]\right){,}{{\mathrm{d_}}}_{{\mathrm{~mu}}}{}\left({{\mathrm{ElectromagneticField}}}_{{\mathrm{~nu}}}{}\left({X}\right){,}\left[{X}\right]\right){-}{{\mathrm{d_}}}_{{\mathrm{~nu}}}{}\left({{\mathrm{ElectromagneticField}}}_{{\mathrm{~mu}}}{}\left({X}\right){,}\left[{X}\right]\right)\right)$ (32)

To get only the interaction Lagrangian part of this expression you can use the keyword $\mathrm{interaction}$

 > $\mathrm{Lagrangian}\left(\mathrm{QED},\mathrm{interaction}\right)$
 ${-}{{{\mathrm{Dgamma}}}_{{\mathrm{~mu}}}}_{{j}{,}{k}}{}\mathrm{g__e}{}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Electron}}}_{{j}}{}\left({X}\right)\right){,}{{\mathrm{Electron}}}_{{k}}{}\left({X}\right){,}{{\mathrm{ElectromagneticField}}}_{{\mathrm{μ}}}{}\left({X}\right)\right)$ (33)

The Quantum Chromodynamics (QCD) sector of the Standard Model and its interaction Lagrangian

QCD is about the interaction between quarks and gluons and the self-interaction of the latter. Quarks are implemented as tensors with one spinor and one SU(3) fundamental representation (1..3) indices. Unless set otherwise, according to the starting message these indices are represented by $\mathrm{lowercaselatin_is}$ and $\mathrm{uppercaselatin_ah}$ letters. Gluons are tensors with one spacetime and one SU(3) adjoint representation index (1..8), respectively represented by $\mathrm{greek}$ and $\mathrm{lowercaselatin_ah}$ letters, and   $\mathrm{g__s}$ is the QCD coupling constant.

The interaction Lagrangian for the QCD can then be introduced as the sum of two terms

 >
 $\mathrm{L__QCD}{≔}\mathrm{L__QG}{+}\mathrm{L__GG}$ (34)

where $\mathrm{L__QG}$ represents the part involving the interaction between quarks and gluons, and $\mathrm{L__GG}$ the part related to the self-interaction between gluons. $\mathrm{L__QG}$ is given by

 >
 $\frac{{1}}{{2}}{}\mathrm{g__s}{}{\mathrm{*}}{}\left({\mathrm{%add}}{}\left({\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({\mathrm{f__Q}}_{{k}{,}{A}}{}\left({X}\right)\right){,}{\mathrm{f__Q}}_{{j}{,}{B}}{}\left({X}\right)\right){,}\mathrm{f__Q}{=}\left[{\mathrm{Up}}{,}{\mathrm{Charm}}{,}{\mathrm{Top}}{,}{\mathrm{Down}}{,}{\mathrm{Strange}}{,}{\mathrm{Bottom}}\right]\right){,}{{\mathrm{GluonField}}}_{{\mathrm{μ}}{,}{a}}{}\left({X}\right){,}{{{\mathrm{Glambda}}}_{{a}}}_{{A}{,}{B}}\right){}{{{\mathrm{Dgamma}}}_{{\mathrm{~mu}}}}_{{k}{,}{j}}$ (35)

The self-interactions of the gluons $\mathrm{L__GG}$ can be written using the structure constants ${\mathrm{FSU3}}_{d,a,b}$ and the Gell-Mann matrices ${{\mathrm{\lambda }}}_{a}$

 >
 ${-}\mathrm{g__s}{}{{\mathrm{FSU3}}}_{{a}{,}{b}{,}{c}}{}\left({\mathrm{*}}{}\left({{\mathrm{d_}}}_{{\mathrm{μ}}}{}\left({{\mathrm{GluonField}}}_{{\mathrm{ν}}{,}{a}}{}\left({X}\right){,}\left[{X}\right]\right){,}{{\mathrm{GluonField}}}_{{\mathrm{~mu}}{,}{b}}{}\left({X}\right){,}{{\mathrm{GluonField}}}_{{\mathrm{~nu}}{,}{c}}{}\left({X}\right)\right){-}\frac{{1}}{{4}}{}\mathrm{g__s}{}{{\mathrm{FSU3}}}_{{c}{,}{d}{,}{e}}{}{\mathrm{*}}{}\left({{\mathrm{GluonField}}}_{{\mathrm{μ}}{,}{a}}{}\left({X}\right){,}{{\mathrm{GluonField}}}_{{\mathrm{λ}}{,}{b}}{}\left({X}\right){,}{{\mathrm{GluonField}}}_{{\mathrm{~mu}}{,}{e}}{}\left({X}\right){,}{{\mathrm{GluonField}}}_{{\mathrm{~lambda}}{,}{d}}{}\left({X}\right)\right)\right)$ (36)

From where

 > $\mathrm{L__QCD}$
 $\frac{{1}}{{2}}{}\mathrm{g__s}{}{\mathrm{*}}{}\left({\mathrm{%add}}{}\left({\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({\mathrm{f__Q}}_{{k}{,}{A}}{}\left({X}\right)\right){,}{\mathrm{f__Q}}_{{j}{,}{B}}{}\left({X}\right)\right){,}\mathrm{f__Q}{=}\left[{\mathrm{Up}}{,}{\mathrm{Charm}}{,}{\mathrm{Top}}{,}{\mathrm{Down}}{,}{\mathrm{Strange}}{,}{\mathrm{Bottom}}\right]\right){,}{{\mathrm{GluonField}}}_{{\mathrm{μ}}{,}{a}}{}\left({X}\right){,}{{{\mathrm{Glambda}}}_{{a}}}_{{A}{,}{B}}\right){}{{{\mathrm{Dgamma}}}_{{\mathrm{~mu}}}}_{{k}{,}{j}}{-}\mathrm{g__s}{}{{\mathrm{FSU3}}}_{{a}{,}{b}{,}{c}}{}\left({\mathrm{*}}{}\left({{\mathrm{d_}}}_{{\mathrm{μ}}}{}\left({{\mathrm{GluonField}}}_{{\mathrm{ν}}{,}{a}}{}\left({X}\right){,}\left[{X}\right]\right){,}{{\mathrm{GluonField}}}_{{\mathrm{~mu}}{,}{b}}{}\left({X}\right){,}{{\mathrm{GluonField}}}_{{\mathrm{~nu}}{,}{c}}{}\left({X}\right)\right){-}\frac{{1}}{{4}}{}\mathrm{g__s}{}{{\mathrm{FSU3}}}_{{c}{,}{d}{,}{e}}{}{\mathrm{*}}{}\left({{\mathrm{GluonField}}}_{{\mathrm{μ}}{,}{a}}{}\left({X}\right){,}{{\mathrm{GluonField}}}_{{\mathrm{λ}}{,}{b}}{}\left({X}\right){,}{{\mathrm{GluonField}}}_{{\mathrm{~mu}}{,}{e}}{}\left({X}\right){,}{{\mathrm{GluonField}}}_{{\mathrm{~lambda}}{,}{d}}{}\left({X}\right)\right)\right)$ (37)

This QCD Lagrangian can also be retrieved using $\mathrm{Lagrangian}$

 > $\mathrm{Lagrangian}\left(\mathrm{QCD},\mathrm{expanded}\right)$
 ${\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Up}}}_{{j}{,}{A}}{}\left({X}\right)\right){,}{i}{}\left({{\mathrm{d_}}}_{{\mathrm{μ}}}{}\left({{\mathrm{Up}}}_{{k}{,}{A}}{}\left({X}\right){,}\left[{X}\right]\right){-}\frac{{1}}{{2}}{}{i}{}\mathrm{g__s}{}{\mathrm{*}}{}\left({{{\mathrm{Glambda}}}_{{a}}}_{{A}{,}{B}}{,}{{\mathrm{Up}}}_{{k}{,}{B}}{}\left({X}\right){,}{{\mathrm{GluonField}}}_{{\mathrm{μ}}{,}{a}}{}\left({X}\right)\right)\right){}{{{\mathrm{Dgamma}}}_{{\mathrm{~mu}}}}_{{j}{,}{k}}{-}{{m}}_{{\mathrm{Up}}}{}{{\mathrm{KroneckerDelta}}}_{{j}{,}{k}}{}{{\mathrm{Up}}}_{{k}{,}{A}}{}\left({X}\right)\right){+}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Charm}}}_{{j}{,}{A}}{}\left({X}\right)\right){,}{i}{}\left({{\mathrm{d_}}}_{{\mathrm{μ}}}{}\left({{\mathrm{Charm}}}_{{k}{,}{A}}{}\left({X}\right){,}\left[{X}\right]\right){-}\frac{{1}}{{2}}{}{i}{}\mathrm{g__s}{}{\mathrm{*}}{}\left({{{\mathrm{Glambda}}}_{{a}}}_{{A}{,}{B}}{,}{{\mathrm{Charm}}}_{{k}{,}{B}}{}\left({X}\right){,}{{\mathrm{GluonField}}}_{{\mathrm{μ}}{,}{a}}{}\left({X}\right)\right)\right){}{{{\mathrm{Dgamma}}}_{{\mathrm{~mu}}}}_{{j}{,}{k}}{-}{{m}}_{{\mathrm{Charm}}}{}{{\mathrm{KroneckerDelta}}}_{{j}{,}{k}}{}{{\mathrm{Charm}}}_{{k}{,}{A}}{}\left({X}\right)\right){+}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Top}}}_{{j}{,}{A}}{}\left({X}\right)\right){,}{i}{}\left({{\mathrm{d_}}}_{{\mathrm{μ}}}{}\left({{\mathrm{Top}}}_{{k}{,}{A}}{}\left({X}\right){,}\left[{X}\right]\right){-}\frac{{1}}{{2}}{}{i}{}\mathrm{g__s}{}{\mathrm{*}}{}\left({{{\mathrm{Glambda}}}_{{a}}}_{{A}{,}{B}}{,}{{\mathrm{Top}}}_{{k}{,}{B}}{}\left({X}\right){,}{{\mathrm{GluonField}}}_{{\mathrm{μ}}{,}{a}}{}\left({X}\right)\right)\right){}{{{\mathrm{Dgamma}}}_{{\mathrm{~mu}}}}_{{j}{,}{k}}{-}{{m}}_{{\mathrm{Top}}}{}{{\mathrm{KroneckerDelta}}}_{{j}{,}{k}}{}{{\mathrm{Top}}}_{{k}{,}{A}}{}\left({X}\right)\right){+}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Down}}}_{{j}{,}{A}}{}\left({X}\right)\right){,}{i}{}\left({{\mathrm{d_}}}_{{\mathrm{μ}}}{}\left({{\mathrm{Down}}}_{{k}{,}{A}}{}\left({X}\right){,}\left[{X}\right]\right){-}\frac{{1}}{{2}}{}{i}{}\mathrm{g__s}{}{\mathrm{*}}{}\left({{{\mathrm{Glambda}}}_{{a}}}_{{A}{,}{B}}{,}{{\mathrm{Down}}}_{{k}{,}{B}}{}\left({X}\right){,}{{\mathrm{GluonField}}}_{{\mathrm{μ}}{,}{a}}{}\left({X}\right)\right)\right){}{{{\mathrm{Dgamma}}}_{{\mathrm{~mu}}}}_{{j}{,}{k}}{-}{{m}}_{{\mathrm{Down}}}{}{{\mathrm{KroneckerDelta}}}_{{j}{,}{k}}{}{{\mathrm{Down}}}_{{k}{,}{A}}{}\left({X}\right)\right){+}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Strange}}}_{{j}{,}{A}}{}\left({X}\right)\right){,}{i}{}\left({{\mathrm{d_}}}_{{\mathrm{μ}}}{}\left({{\mathrm{Strange}}}_{{k}{,}{A}}{}\left({X}\right){,}\left[{X}\right]\right){-}\frac{{1}}{{2}}{}{i}{}\mathrm{g__s}{}{\mathrm{*}}{}\left({{{\mathrm{Glambda}}}_{{a}}}_{{A}{,}{B}}{,}{{\mathrm{Strange}}}_{{k}{,}{B}}{}\left({X}\right){,}{{\mathrm{GluonField}}}_{{\mathrm{μ}}{,}{a}}{}\left({X}\right)\right)\right){}{{{\mathrm{Dgamma}}}_{{\mathrm{~mu}}}}_{{j}{,}{k}}{-}{{m}}_{{\mathrm{Strange}}}{}{{\mathrm{KroneckerDelta}}}_{{j}{,}{k}}{}{{\mathrm{Strange}}}_{{k}{,}{A}}{}\left({X}\right)\right){+}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Bottom}}}_{{j}{,}{A}}{}\left({X}\right)\right){,}{i}{}\left({{\mathrm{d_}}}_{{\mathrm{μ}}}{}\left({{\mathrm{Bottom}}}_{{k}{,}{A}}{}\left({X}\right){,}\left[{X}\right]\right){-}\frac{{1}}{{2}}{}{i}{}\mathrm{g__s}{}{\mathrm{*}}{}\left({{{\mathrm{Glambda}}}_{{a}}}_{{A}{,}{B}}{,}{{\mathrm{Bottom}}}_{{k}{,}{B}}{}\left({X}\right){,}{{\mathrm{GluonField}}}_{{\mathrm{μ}}{,}{a}}{}\left({X}\right)\right)\right){}{{{\mathrm{Dgamma}}}_{{\mathrm{~mu}}}}_{{j}{,}{k}}{-}{{m}}_{{\mathrm{Bottom}}}{}{{\mathrm{KroneckerDelta}}}_{{j}{,}{k}}{}{{\mathrm{Bottom}}}_{{k}{,}{A}}{}\left({X}\right)\right){-}\frac{{1}}{{4}}{}{\mathrm{*}}{}\left({{\mathrm{d_}}}_{{\mathrm{μ}}}{}\left({{\mathrm{GluonField}}}_{{\mathrm{ν}}{,}{a}}{}\left({X}\right){,}\left[{X}\right]\right){-}{{\mathrm{d_}}}_{{\mathrm{ν}}}{}\left({{\mathrm{GluonField}}}_{{\mathrm{μ}}{,}{a}}{}\left({X}\right){,}\left[{X}\right]\right){+}\mathrm{g__s}{}{{\mathrm{FSU3}}}_{{a}{,}{b}{,}{c}}{}{\mathrm{*}}{}\left({{\mathrm{GluonField}}}_{{\mathrm{μ}}{,}{b}}{}\left({X}\right){,}{{\mathrm{GluonField}}}_{{\mathrm{ν}}{,}{c}}{}\left({X}\right)\right){,}{{\mathrm{d_}}}_{{\mathrm{~mu}}}{}\left({{\mathrm{GluonField}}}_{{\mathrm{~nu}}{,}{a}}{}\left({X}\right){,}\left[{X}\right]\right){-}{{\mathrm{d_}}}_{{\mathrm{~nu}}}{}\left({{\mathrm{GluonField}}}_{{\mathrm{~mu}}{,}{a}}{}\left({X}\right){,}\left[{X}\right]\right){+}\mathrm{g__s}{}{{\mathrm{FSU3}}}_{{a}{,}{d}{,}{e}}{}{\mathrm{*}}{}\left({{\mathrm{GluonField}}}_{{\mathrm{~mu}}{,}{d}}{}\left({X}\right){,}{{\mathrm{GluonField}}}_{{\mathrm{~nu}}{,}{e}}{}\left({X}\right)\right)\right)$ (38)
 >
 $\frac{{1}}{{2}}{}\mathrm{g__s}{}{\mathrm{*}}{}\left({\mathrm{%add}}{}\left({\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({\mathrm{f__Q}}_{{j}{,}{A}}{}\left({X}\right)\right){,}{\mathrm{f__Q}}_{{k}{,}{B}}{}\left({X}\right)\right){,}\mathrm{f__Q}{=}\left[{\mathrm{Up}}{,}{\mathrm{Charm}}{,}{\mathrm{Top}}{,}{\mathrm{Down}}{,}{\mathrm{Strange}}{,}{\mathrm{Bottom}}\right]\right){,}{{{\mathrm{Glambda}}}_{{a}}}_{{A}{,}{B}}{,}{{\mathrm{GluonField}}}_{{\mathrm{μ}}{,}{a}}{}\left({X}\right)\right){}{{{\mathrm{Dgamma}}}_{{\mathrm{~mu}}}}_{{j}{,}{k}}{-}\mathrm{g__s}{}{{\mathrm{FSU3}}}_{{a}{,}{b}{,}{c}}{}\left({\mathrm{*}}{}\left({{\mathrm{d_}}}_{{\mathrm{μ}}}{}\left({{\mathrm{GluonField}}}_{{\mathrm{ν}}{,}{a}}{}\left({X}\right){,}\left[{X}\right]\right){,}{{\mathrm{GluonField}}}_{{\mathrm{~mu}}{,}{b}}{}\left({X}\right){,}{{\mathrm{GluonField}}}_{{\mathrm{~nu}}{,}{c}}{}\left({X}\right)\right){-}\frac{{1}}{{4}}{}\mathrm{g__s}{}{{\mathrm{FSU3}}}_{{c}{,}{d}{,}{e}}{}{\mathrm{*}}{}\left({{\mathrm{GluonField}}}_{{\mathrm{μ}}{,}{a}}{}\left({X}\right){,}{{\mathrm{GluonField}}}_{{\mathrm{α}}{,}{b}}{}\left({X}\right){,}{{\mathrm{GluonField}}}_{{\mathrm{~mu}}{,}{e}}{}\left({X}\right){,}{{\mathrm{GluonField}}}_{{\mathrm{~alpha}}{,}{d}}{}\left({X}\right)\right)\right)$ (39)
 >
 $\frac{{1}}{{2}}{}\mathrm{g__s}{}{\mathrm{*}}{}\left({\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Up}}}_{{j}{,}{A}}{}\left({X}\right)\right){,}{{\mathrm{Up}}}_{{k}{,}{B}}{}\left({X}\right)\right){+}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Charm}}}_{{j}{,}{A}}{}\left({X}\right)\right){,}{{\mathrm{Charm}}}_{{k}{,}{B}}{}\left({X}\right)\right){+}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Top}}}_{{j}{,}{A}}{}\left({X}\right)\right){,}{{\mathrm{Top}}}_{{k}{,}{B}}{}\left({X}\right)\right){+}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Down}}}_{{j}{,}{A}}{}\left({X}\right)\right){,}{{\mathrm{Down}}}_{{k}{,}{B}}{}\left({X}\right)\right){+}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Strange}}}_{{j}{,}{A}}{}\left({X}\right)\right){,}{{\mathrm{Strange}}}_{{k}{,}{B}}{}\left({X}\right)\right){+}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Bottom}}}_{{j}{,}{A}}{}\left({X}\right)\right){,}{{\mathrm{Bottom}}}_{{k}{,}{B}}{}\left({X}\right)\right){,}{{{\mathrm{Glambda}}}_{{a}}}_{{A}{,}{B}}{,}{{\mathrm{GluonField}}}_{{\mathrm{μ}}{,}{a}}{}\left({X}\right)\right){}{{{\mathrm{Dgamma}}}_{{\mathrm{~mu}}}}_{{j}{,}{k}}{-}\mathrm{g__s}{}{{\mathrm{FSU3}}}_{{a}{,}{b}{,}{c}}{}\left({\mathrm{*}}{}\left({{\mathrm{d_}}}_{{\mathrm{μ}}}{}\left({{\mathrm{GluonField}}}_{{\mathrm{ν}}{,}{a}}{}\left({X}\right){,}\left[{X}\right]\right){,}{{\mathrm{GluonField}}}_{{\mathrm{~mu}}{,}{b}}{}\left({X}\right){,}{{\mathrm{GluonField}}}_{{\mathrm{~nu}}{,}{c}}{}\left({X}\right)\right){-}\frac{{1}}{{4}}{}\mathrm{g__s}{}{{\mathrm{FSU3}}}_{{c}{,}{d}{,}{e}}{}{\mathrm{*}}{}\left({{\mathrm{GluonField}}}_{{\mathrm{μ}}{,}{a}}{}\left({X}\right){,}{{\mathrm{GluonField}}}_{{\mathrm{α}}{,}{b}}{}\left({X}\right){,}{{\mathrm{GluonField}}}_{{\mathrm{~mu}}{,}{e}}{}\left({X}\right){,}{{\mathrm{GluonField}}}_{{\mathrm{~alpha}}{,}{d}}{}\left({X}\right)\right)\right)$ (40)

Each of these terms has different contributions to a scattering amplitude. For example, take the first term with the interaction between $\mathrm{Up}$ quarks and gluons and last one with the self-interaction between four gluons.

 >
 $\frac{{1}}{{2}}{}\mathrm{g__s}{}{{{\mathrm{Dgamma}}}_{{\mathrm{~mu}}}}_{{j}{,}{k}}{}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Up}}}_{{j}{,}{A}}{}\left({X}\right)\right){,}{{\mathrm{Up}}}_{{k}{,}{B}}{}\left({X}\right){,}{{{\mathrm{Glambda}}}_{{a}}}_{{A}{,}{B}}{,}{{\mathrm{GluonField}}}_{{\mathrm{μ}}{,}{a}}{}\left({X}\right)\right)$ (41)

The amplitude for the process with two incoming and two outgoing $\mathrm{Up}$ quarks (particle and antiparticle)

 > ${-}\frac{{i}{}{\left({{\mathbit{u}}}_{{\mathbf{u}}}\right)}_{{l}{,}{C}}{}\left(\stackrel{{\to }}{\mathrm{P__1}}\right){}\stackrel{{&conjugate0;}}{{\left({{\mathbit{v}}}_{{\mathbf{u}}}\right)}_{{m}{,}{E}}{}\left(\stackrel{{\to }}{\mathrm{P__2}}\right)}{}\stackrel{{&conjugate0;}}{{\left({{\mathbit{u}}}_{{\mathbf{u}}}\right)}_{{n}{,}{F}}{}\left(\stackrel{{\to }}{\mathrm{P__3}}\right)}{}{\left({{\mathbit{v}}}_{{\mathbf{u}}}\right)}_{{p}{,}{G}}{}\left(\stackrel{{\to }}{\mathrm{P__4}}\right){}{\mathrm{g__s}}^{{2}}{}{\left({{\mathrm{\gamma }}}_{\phantom{{}}\phantom{{\mathrm{\alpha }}}}^{\phantom{{}}{\mathrm{\alpha }}}\right)}_{{n}{,}{p}}{}{\left({{\mathrm{\gamma }}}_{\phantom{{}}\phantom{{\mathrm{\nu }}}}^{\phantom{{}}{\mathrm{\nu }}}\right)}_{{m}{,}{l}}{}{{g}}_{{\mathrm{\alpha }}{,}{\mathrm{\nu }}}{}{{\mathrm{\delta }}}_{{b}{,}{c}}{}{\mathrm{\delta }}{}\left({-}{\mathrm{P__3}}_{\phantom{{}}\phantom{{\mathrm{\beta }}}}^{\phantom{{}}{\mathrm{\beta }}}{-}{\mathrm{P__4}}_{\phantom{{}}\phantom{{\mathrm{\beta }}}}^{\phantom{{}}{\mathrm{\beta }}}{+}{\mathrm{P__1}}_{\phantom{{}}\phantom{{\mathrm{\beta }}}}^{\phantom{{}}{\mathrm{\beta }}}{+}{\mathrm{P__2}}_{\phantom{{}}\phantom{{\mathrm{\beta }}}}^{\phantom{{}}{\mathrm{\beta }}}\right){}{\left({{\mathrm{\lambda }}}_{{c}}\right)}_{{F}{,}{G}}{}{\left({{\mathrm{\lambda }}}_{{b}}\right)}_{{E}{,}{C}}}{{16}{}{{\mathrm{\pi }}}^{{2}}{}\left(\left({\mathrm{P__1}}_{{\mathrm{\kappa }}}{+}{\mathrm{P__2}}_{{\mathrm{\kappa }}}\right){}\left({\mathrm{P__1}}_{\phantom{{}}\phantom{{\mathrm{\kappa }}}}^{\phantom{{}}{\mathrm{\kappa }}}{+}{\mathrm{P__2}}_{\phantom{{}}\phantom{{\mathrm{\kappa }}}}^{\phantom{{}}{\mathrm{\kappa }}}\right){+}{i}{}{\mathbf{\epsilon }}\right)}{+}\frac{{i}{}{\left({{\mathbit{u}}}_{{\mathbf{u}}}\right)}_{{l}{,}{C}}{}\left(\stackrel{{\to }}{\mathrm{P__1}}\right){}\stackrel{{&conjugate0;}}{{\left({{\mathbit{v}}}_{{\mathbf{u}}}\right)}_{{m}{,}{E}}{}\left(\stackrel{{\to }}{\mathrm{P__2}}\right)}{}\stackrel{{&conjugate0;}}{{\left({{\mathbit{u}}}_{{\mathbf{u}}}\right)}_{{n}{,}{F}}{}\left(\stackrel{{\to }}{\mathrm{P__3}}\right)}{}{\left({{\mathbit{v}}}_{{\mathbf{u}}}\right)}_{{p}{,}{G}}{}\left(\stackrel{{\to }}{\mathrm{P__4}}\right){}{\mathrm{g__s}}^{{2}}{}{\left({{\mathrm{\gamma }}}_{\phantom{{}}\phantom{{\mathrm{\alpha }}}}^{\phantom{{}}{\mathrm{\alpha }}}\right)}_{{m}{,}{p}}{}{\left({{\mathrm{\gamma }}}_{\phantom{{}}\phantom{{\mathrm{\nu }}}}^{\phantom{{}}{\mathrm{\nu }}}\right)}_{{n}{,}{l}}{}{{g}}_{{\mathrm{\alpha }}{,}{\mathrm{\nu }}}{}{{\mathrm{\delta }}}_{{b}{,}{c}}{}{\mathrm{\delta }}{}\left({-}{\mathrm{P__3}}_{\phantom{{}}\phantom{{\mathrm{\beta }}}}^{\phantom{{}}{\mathrm{\beta }}}{-}{\mathrm{P__4}}_{\phantom{{}}\phantom{{\mathrm{\beta }}}}^{\phantom{{}}{\mathrm{\beta }}}{+}{\mathrm{P__1}}_{\phantom{{}}\phantom{{\mathrm{\beta }}}}^{\phantom{{}}{\mathrm{\beta }}}{+}{\mathrm{P__2}}_{\phantom{{}}\phantom{{\mathrm{\beta }}}}^{\phantom{{}}{\mathrm{\beta }}}\right){}{\left({{\mathrm{\lambda }}}_{{c}}\right)}_{{E}{,}{G}}{}{\left({{\mathrm{\lambda }}}_{{b}}\right)}_{{F}{,}{C}}}{{16}{}{{\mathrm{\pi }}}^{{2}}{}\left(\left({\mathrm{P__1}}_{\phantom{{}}\phantom{{\mathrm{\kappa }}}}^{\phantom{{}}{\mathrm{\kappa }}}{-}{\mathrm{P__3}}_{\phantom{{}}\phantom{{\mathrm{\kappa }}}}^{\phantom{{}}{\mathrm{\kappa }}}\right){}\left({\mathrm{P__1}}_{{\mathrm{\kappa }}}{-}{\mathrm{P__3}}_{{\mathrm{\kappa }}}\right){+}{i}{}{\mathbf{\epsilon }}\right)}$ (42)
 >
 $\frac{{1}}{{4}}{}{\mathrm{g__s}}^{{2}}{}{{\mathrm{FSU3}}}_{{a}{,}{b}{,}{c}}{}{{\mathrm{FSU3}}}_{{c}{,}{d}{,}{e}}{}{\mathrm{*}}{}\left({{\mathrm{GluonField}}}_{{\mathrm{α}}{,}{b}}{}\left({X}\right){,}{{\mathrm{GluonField}}}_{{\mathrm{μ}}{,}{a}}{}\left({X}\right){,}{{\mathrm{GluonField}}}_{{\mathrm{~alpha}}{,}{d}}{}\left({X}\right){,}{{\mathrm{GluonField}}}_{{\mathrm{~mu}}{,}{e}}{}\left({X}\right)\right)$ (43)

The amplitude at tree level for the process with two incoming and two outgoing gluons

 > $\frac{\frac{{i}}{{16}}{}{\mathrm{g__s}}^{{2}}{}{\mathrm{\delta }}{}\left({-}{\mathrm{P__3}}_{\phantom{{}}\phantom{{\mathrm{\sigma }}}}^{\phantom{{}}{\mathrm{\sigma }}}{-}{\mathrm{P__4}}_{\phantom{{}}\phantom{{\mathrm{\sigma }}}}^{\phantom{{}}{\mathrm{\sigma }}}{+}{\mathrm{P__1}}_{\phantom{{}}\phantom{{\mathrm{\sigma }}}}^{\phantom{{}}{\mathrm{\sigma }}}{+}{\mathrm{P__2}}_{\phantom{{}}\phantom{{\mathrm{\sigma }}}}^{\phantom{{}}{\mathrm{\sigma }}}\right){}{\left({{\mathbf{ϵ}}}_{{\mathbf{G}}}\right)}_{{\mathrm{\nu }}{,}{f}}{}\left(\stackrel{{\to }}{\mathrm{P__1}}\right){}{\left({{\mathbf{ϵ}}}_{{\mathbf{G}}}\right)}_{{\mathrm{\beta }}{,}{g}}{}\left(\stackrel{{\to }}{\mathrm{P__2}}\right){}\stackrel{{&conjugate0;}}{{\left({{\mathbf{ϵ}}}_{{\mathbf{G}}}\right)}_{{\mathrm{\kappa }}{,}{h}}{}\left(\stackrel{{\to }}{\mathrm{P__3}}\right)}{}\stackrel{{&conjugate0;}}{{\left({{\mathbf{ϵ}}}_{{\mathbf{G}}}\right)}_{{\mathrm{\lambda }}{,}{\mathrm{a1}}}{}\left(\stackrel{{\to }}{\mathrm{P__4}}\right)}{}\left(\left({-}{{\mathrm{FSU3}}}_{{\mathrm{a1}}{,}{c}{,}{f}}{}{{\mathrm{FSU3}}}_{{c}{,}{g}{,}{h}}{-}{{\mathrm{FSU3}}}_{{\mathrm{a1}}{,}{c}{,}{g}}{}{{\mathrm{FSU3}}}_{{c}{,}{f}{,}{h}}\right){}{{g}}_{\phantom{{}}\phantom{{\mathrm{\beta }}{,}{\mathrm{\nu }}}}^{\phantom{{}}{\mathrm{\beta }}{,}{\mathrm{\nu }}}{}{{g}}_{\phantom{{}}\phantom{{\mathrm{\kappa }}{,}{\mathrm{\lambda }}}}^{\phantom{{}}{\mathrm{\kappa }}{,}{\mathrm{\lambda }}}{+}\left({{\mathrm{FSU3}}}_{{\mathrm{a1}}{,}{c}{,}{f}}{}{{\mathrm{FSU3}}}_{{c}{,}{g}{,}{h}}{-}{{\mathrm{FSU3}}}_{{\mathrm{a1}}{,}{c}{,}{h}}{}{{\mathrm{FSU3}}}_{{c}{,}{f}{,}{g}}\right){}{{g}}_{\phantom{{}}\phantom{{\mathrm{\kappa }}{,}{\mathrm{\nu }}}}^{\phantom{{}}{\mathrm{\kappa }}{,}{\mathrm{\nu }}}{}{{g}}_{\phantom{{}}\phantom{{\mathrm{\beta }}{,}{\mathrm{\lambda }}}}^{\phantom{{}}{\mathrm{\beta }}{,}{\mathrm{\lambda }}}{+}{{g}}_{\phantom{{}}\phantom{{\mathrm{\lambda }}{,}{\mathrm{\nu }}}}^{\phantom{{}}{\mathrm{\lambda }}{,}{\mathrm{\nu }}}{}{{g}}_{\phantom{{}}\phantom{{\mathrm{\beta }}{,}{\mathrm{\kappa }}}}^{\phantom{{}}{\mathrm{\beta }}{,}{\mathrm{\kappa }}}{}\left({{\mathrm{FSU3}}}_{{\mathrm{a1}}{,}{c}{,}{g}}{}{{\mathrm{FSU3}}}_{{c}{,}{f}{,}{h}}{+}{{\mathrm{FSU3}}}_{{\mathrm{a1}}{,}{c}{,}{h}}{}{{\mathrm{FSU3}}}_{{c}{,}{f}{,}{g}}\right)\right)}{{{\mathrm{\pi }}}^{{2}}{}\sqrt{\mathrm{E__1}{}\mathrm{E__2}{}\mathrm{E__3}{}\mathrm{E__4}}}$ (44)

The Electroweak sector of the Standard Model and its interaction Lagrangian

Before symmetry breaking

The electro-weak interaction before symmetry breaking, that are not used to compute observable scattering amplitudes, but from where the formulation after symmetry breaking is derived, can be expressed as  a sum of four terms mentioned in the Wikipedia page for the weak interaction

 >
 $\mathrm{L__EW}{≔}\mathrm{L__g}{+}\mathrm{L__f}{+}\mathrm{L__h}{+}\mathrm{L__y}$ (45)

Out of these four, in the Maple 2022.0 implementation of $\mathrm{StandardModel}$ it is possible to represent the first term, $\mathrm{L__g}$, the kinetic term for the ${{\mathbf{W}}}_{\mathrm{\mu },J}$ and ${{\mathbf{B}}}_{\mathrm{μ}}$ vector bosons

 >
 $\mathrm{L__g}{≔}{-}\frac{{{\mathbf{𝕎}}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}{,}{J}}{}{{\mathbf{𝕎}}}_{\phantom{{}}\phantom{{\mathrm{\mu }}{,}{\mathrm{\nu }}}{J}}^{\phantom{{}}{\mathrm{\mu }}{,}{\mathrm{\nu }}\phantom{{J}}}}{{4}}{-}\frac{{{\mathbf{𝔹}}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}{}{{\mathbf{𝔹}}}_{\phantom{{}}\phantom{{\mathrm{\mu }}{,}{\mathrm{\nu }}}}^{\phantom{{}}{\mathrm{\mu }}{,}{\mathrm{\nu }}}}{{4}}$ (46)

Introducing the definitions of these tensors we have

 > $\mathrm{BFieldStrength}\left[\mathrm{definition}\right],\mathrm{WFieldStrength}\left[\mathrm{definition}\right]$
 ${{\mathbf{𝔹}}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}{=}{{\partial }}_{{\mathrm{\mu }}}{}\left({{\mathrm{BField}}}_{{\mathrm{ν}}}{}\left({X}\right)\right){-}{{\partial }}_{{\mathrm{\nu }}}{}\left({{\mathrm{BField}}}_{{\mathrm{μ}}}{}\left({X}\right)\right){,}{{\mathbf{𝕎}}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}{,}{J}}{=}{{\partial }}_{{\mathrm{\mu }}}{}\left({{\mathrm{WField}}}_{{\mathrm{ν}}{,}{J}}{}\left({X}\right)\right){-}{{\partial }}_{{\mathrm{\nu }}}{}\left({{\mathrm{WField}}}_{{\mathrm{μ}}{,}{J}}{}\left({X}\right)\right){+}\mathrm{g__w}{}{{\mathrm{\epsilon }}}_{{J}{,}{K}{,}{L}}{}{{\mathrm{WField}}}_{{\mathrm{μ}}{,}{K}}{}\left({X}\right){}{{\mathrm{WField}}}_{{\mathrm{ν}}{,}{L}}{}\left({X}\right)$ (47)
 >
 ${-}\frac{{1}}{{4}}{}{\mathrm{*}}{}\left({{\mathrm{d_}}}_{{\mathrm{μ}}}{}\left({{\mathrm{WField}}}_{{\mathrm{ν}}{,}{J}}{}\left({X}\right){,}\left[{X}\right]\right){-}{{\mathrm{d_}}}_{{\mathrm{ν}}}{}\left({{\mathrm{WField}}}_{{\mathrm{μ}}{,}{J}}{}\left({X}\right){,}\left[{X}\right]\right){+}\mathrm{g__w}{}{{\mathrm{LeviCivita}}}_{{J}{,}{K}{,}{L}}{}{\mathrm{*}}{}\left({{\mathrm{WField}}}_{{\mathrm{μ}}{,}{K}}{}\left({X}\right){,}{{\mathrm{WField}}}_{{\mathrm{ν}}{,}{L}}{}\left({X}\right)\right){,}{{\mathrm{d_}}}_{{\mathrm{~mu}}}{}\left({{\mathrm{WField}}}_{{\mathrm{~nu}}{,}{J}}{}\left({X}\right){,}\left[{X}\right]\right){-}{{\mathrm{d_}}}_{{\mathrm{~nu}}}{}\left({{\mathrm{WField}}}_{{\mathrm{~mu}}{,}{J}}{}\left({X}\right){,}\left[{X}\right]\right){+}\mathrm{g__w}{}{{\mathrm{LeviCivita}}}_{{J}{,}{M}{,}{N}}{}{\mathrm{*}}{}\left({{\mathrm{WField}}}_{{\mathrm{~mu}}{,}{M}}{}\left({X}\right){,}{{\mathrm{WField}}}_{{\mathrm{~nu}}{,}{N}}{}\left({X}\right)\right)\right){-}\frac{{1}}{{4}}{}{\mathrm{*}}{}\left({{\mathrm{d_}}}_{{\mathrm{μ}}}{}\left({{\mathrm{BField}}}_{{\mathrm{ν}}}{}\left({X}\right){,}\left[{X}\right]\right){-}{{\mathrm{d_}}}_{{\mathrm{ν}}}{}\left({{\mathrm{BField}}}_{{\mathrm{μ}}}{}\left({X}\right){,}\left[{X}\right]\right){,}{{\mathrm{d_}}}_{{\mathrm{~mu}}}{}\left({{\mathrm{BField}}}_{{\mathrm{~nu}}}{}\left({X}\right){,}\left[{X}\right]\right){-}{{\mathrm{d_}}}_{{\mathrm{~nu}}}{}\left({{\mathrm{BField}}}_{{\mathrm{~mu}}}{}\left({X}\right){,}\left[{X}\right]\right)\right)$ (48)

The $\mathrm{L__f}$ term is the kinetic term for the fermions of the model before symmetry breaking, and their interaction with the gauge bosons ${{\mathbf{W}}}_{\mathrm{\mu },K}$ and ${{\mathbf{B}}}_{\mathrm{\mu }}$is through the covariant derivative. The $\mathrm{L__h}$ term involves the Higgs boson before symmetry breaking (the $\mathrm{HiggsBoson}\equiv {\mathbf{\Phi }}$ field implemented in the StandardModel in Maple 2022 is the Higgs after symmetry breaking) and the $\mathrm{L__y}$ formulates the Yukawa interaction with the fermions. Note that the electron field ${{\mathbf{e}}}_{j}$, as well as all the leptons are Dirac spinors that result after symmetry breaking. The quarks are also particles that appear through the symmetry breaking mechanism.

After symmetry breaking

For the purpose of computing scattering amplitudes, the formulation of the interaction Lagrangian after symmetry breaking is more relevant. Following the presentation in Wikipedia, the interaction Lagrangian of this sector is given by

 >
 $\mathrm{L__EW}{≔}\mathrm{L__K}{+}\mathrm{L__N}{+}\mathrm{L__C}{+}\mathrm{L__H}{+}\mathrm{L__HV}{+}\mathrm{L__WWV}{+}\mathrm{L__WWVV}{+}\mathrm{L__Y;}$ (49)

where we use the notation shown in the Wikipedia page for the weak interaction. As illustration, we compute here the $\mathrm{L__K}$ and $\mathrm{L__N}$ terms, respectively containing the kinetic terms corresponding to the free fields and the interaction terms between the fermions - leptons and quarks - and the gauge bosons ${{\mathbf{A}}}_{\mathrm{\mu }}$and ${{\mathbf{Z}}}_{\mathrm{\mu }}$. At the end, we use the $\mathrm{Lagrangian}$ command to retrieve all the terms of (49).

The kinetic term $\mathrm{L__K}$ can be entered as

 >
 $\mathrm{L__K}{=}{-}\frac{{1}}{{4}}{}{\mathrm{*}}{}\left({{\mathrm{ElectromagneticFieldStrength}}}_{{\mathrm{μ}}{,}{\mathrm{ν}}}{,}{{\mathrm{ElectromagneticFieldStrength}}}_{{\mathrm{~mu}}{,}{\mathrm{~nu}}}\right){-}\frac{{1}}{{2}}{}{\mathrm{*}}{}\left({{\mathrm{WPlusFieldStrength}}}_{{\mathrm{μ}}{,}{\mathrm{ν}}}{,}{{\mathrm{WMinusFieldStrength}}}_{{\mathrm{~mu}}{,}{\mathrm{~nu}}}\right){+}\frac{{1}}{{2}}{}{{m}}_{{\mathrm{WField}}}^{{2}}{}{\mathrm{*}}{}\left({{\mathrm{WPlusField}}}_{{\mathrm{μ}}}{,}{{\mathrm{WMinusField}}}_{{\mathrm{~mu}}}\right){-}\frac{{1}}{{4}}{}{\mathrm{*}}{}\left({{\mathrm{ZFieldStrength}}}_{{\mathrm{μ}}{,}{\mathrm{ν}}}{,}{{\mathrm{ZFieldStrength}}}_{{\mathrm{~mu}}{,}{\mathrm{~nu}}}\right){+}\frac{{1}}{{2}}{}{{m}}_{{\mathrm{ZField}}}^{{2}}{}{\mathrm{*}}{}\left({{\mathrm{ZField}}}_{{\mathrm{μ}}}{,}{{\mathrm{ZField}}}_{{\mathrm{~mu}}}\right){+}\frac{{1}}{{2}}{}{\mathrm{*}}{}\left({{\mathrm{d_}}}_{{\mathrm{μ}}}{}\left({\mathrm{HiggsBoson}}{}\left({X}\right){,}\left[{X}\right]\right){,}{{\mathrm{d_}}}_{{\mathrm{~mu}}}{}\left({\mathrm{HiggsBoson}}{}\left({X}\right){,}\left[{X}\right]\right)\right){-}\frac{{1}}{{2}}{}{{m}}_{{\mathrm{HiggsBoson}}}^{{2}}{}{\mathrm{^}}{}\left({\mathrm{HiggsBoson}}{}\left({X}\right){,}{2}\right){+}{\mathrm{%add}}{}\left({\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({\mathrm{f__L}}_{{j}}{}\left({X}\right)\right){,}{i}{}{{\mathrm{d_}}}_{{\mathrm{μ}}}{}\left({\mathrm{f__L}}_{{k}}{}\left({X}\right){,}\left[{X}\right]\right){}{{{\mathrm{Dgamma}}}_{{\mathrm{~mu}}}}_{{j}{,}{k}}{-}{{m}}_{\mathrm{f__L}}{}{\mathrm{f__L}}_{{j}}{}\left({X}\right)\right){,}\mathrm{f__L}{=}\left[{\mathrm{Electron}}{,}{\mathrm{Muon}}{,}{\mathrm{Tauon}}\right]\right){+}{\mathrm{%add}}{}\left({i}{}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({\mathrm{f__L}}_{{j}}{}\left({X}\right)\right){,}{{\mathrm{d_}}}_{{\mathrm{μ}}}{}\left({\mathrm{f__L}}_{{k}}{}\left({X}\right){,}\left[{X}\right]\right)\right){}{{{\mathrm{Dgamma}}}_{{\mathrm{~mu}}}}_{{j}{,}{k}}{,}\mathrm{f__L}{=}\left[{\mathrm{ElectronNeutrino}}{,}{\mathrm{MuonNeutrino}}{,}{\mathrm{TauonNeutrino}}\right]\right){+}{\mathrm{%add}}{}\left({\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({\mathrm{f__Q}}_{{j}{,}{A}}{}\left({X}\right)\right){,}{i}{}{{\mathrm{d_}}}_{{\mathrm{μ}}}{}\left({\mathrm{f__Q}}_{{k}{,}{A}}{}\left({X}\right){,}\left[{X}\right]\right){}{{{\mathrm{Dgamma}}}_{{\mathrm{~mu}}}}_{{j}{,}{k}}{-}{{m}}_{\mathrm{f__Q}}{}{\mathrm{f__Q}}_{{j}{,}{A}}{}\left({X}\right)\right){,}\mathrm{f__Q}{=}\left[{\mathrm{Up}}{,}{\mathrm{Charm}}{,}{\mathrm{Top}}{,}{\mathrm{Down}}{,}{\mathrm{Strange}}{,}{\mathrm{Bottom}}\right]\right)$ (50)

The inert sums over the leptons and quarks can be activated using value

 > $\mathrm{value}\left(\right)$
 $\mathrm{L__K}{=}{-}\frac{{1}}{{4}}{}{\mathrm{*}}{}\left({{\mathrm{ElectromagneticFieldStrength}}}_{{\mathrm{μ}}{,}{\mathrm{ν}}}{,}{{\mathrm{ElectromagneticFieldStrength}}}_{{\mathrm{~mu}}{,}{\mathrm{~nu}}}\right){-}\frac{{1}}{{2}}{}{\mathrm{*}}{}\left({{\mathrm{WPlusFieldStrength}}}_{{\mathrm{μ}}{,}{\mathrm{ν}}}{,}{{\mathrm{WMinusFieldStrength}}}_{{\mathrm{~mu}}{,}{\mathrm{~nu}}}\right){+}\frac{{1}}{{2}}{}{{m}}_{{\mathrm{WField}}}^{{2}}{}{\mathrm{*}}{}\left({{\mathrm{WPlusField}}}_{{\mathrm{μ}}}{,}{{\mathrm{WMinusField}}}_{{\mathrm{~mu}}}\right){-}\frac{{1}}{{4}}{}{\mathrm{*}}{}\left({{\mathrm{ZFieldStrength}}}_{{\mathrm{μ}}{,}{\mathrm{ν}}}{,}{{\mathrm{ZFieldStrength}}}_{{\mathrm{~mu}}{,}{\mathrm{~nu}}}\right){+}\frac{{1}}{{2}}{}{{m}}_{{\mathrm{ZField}}}^{{2}}{}{\mathrm{*}}{}\left({{\mathrm{ZField}}}_{{\mathrm{μ}}}{,}{{\mathrm{ZField}}}_{{\mathrm{~mu}}}\right){+}\frac{{1}}{{2}}{}{\mathrm{*}}{}\left({{\mathrm{d_}}}_{{\mathrm{μ}}}{}\left({\mathrm{HiggsBoson}}{}\left({X}\right){,}\left[{X}\right]\right){,}{{\mathrm{d_}}}_{{\mathrm{~mu}}}{}\left({\mathrm{HiggsBoson}}{}\left({X}\right){,}\left[{X}\right]\right)\right){-}\frac{{1}}{{2}}{}{{m}}_{{\mathrm{HiggsBoson}}}^{{2}}{}{\mathrm{^}}{}\left({\mathrm{HiggsBoson}}{}\left({X}\right){,}{2}\right){+}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Electron}}}_{{j}}{}\left({X}\right)\right){,}{i}{}{{\mathrm{d_}}}_{{\mathrm{μ}}}{}\left({{\mathrm{Electron}}}_{{k}}{}\left({X}\right){,}\left[{X}\right]\right){}{{{\mathrm{Dgamma}}}_{{\mathrm{~mu}}}}_{{j}{,}{k}}{-}{{m}}_{{\mathrm{Electron}}}{}{{\mathrm{Electron}}}_{{j}}{}\left({X}\right)\right){+}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Muon}}}_{{j}}{}\left({X}\right)\right){,}{i}{}{{\mathrm{d_}}}_{{\mathrm{μ}}}{}\left({{\mathrm{Muon}}}_{{k}}{}\left({X}\right){,}\left[{X}\right]\right){}{{{\mathrm{Dgamma}}}_{{\mathrm{~mu}}}}_{{j}{,}{k}}{-}{{m}}_{{\mathrm{Muon}}}{}{{\mathrm{Muon}}}_{{j}}{}\left({X}\right)\right){+}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Tauon}}}_{{j}}{}\left({X}\right)\right){,}{i}{}{{\mathrm{d_}}}_{{\mathrm{μ}}}{}\left({{\mathrm{Tauon}}}_{{k}}{}\left({X}\right){,}\left[{X}\right]\right){}{{{\mathrm{Dgamma}}}_{{\mathrm{~mu}}}}_{{j}{,}{k}}{-}{{m}}_{{\mathrm{Tauon}}}{}{{\mathrm{Tauon}}}_{{j}}{}\left({X}\right)\right){+}{i}{}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{ElectronNeutrino}}}_{{j}}{}\left({X}\right)\right){,}{{\mathrm{d_}}}_{{\mathrm{μ}}}{}\left({{\mathrm{ElectronNeutrino}}}_{{k}}{}\left({X}\right){,}\left[{X}\right]\right)\right){}{{{\mathrm{Dgamma}}}_{{\mathrm{~mu}}}}_{{j}{,}{k}}{+}{i}{}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{MuonNeutrino}}}_{{j}}{}\left({X}\right)\right){,}{{\mathrm{d_}}}_{{\mathrm{μ}}}{}\left({{\mathrm{MuonNeutrino}}}_{{k}}{}\left({X}\right){,}\left[{X}\right]\right)\right){}{{{\mathrm{Dgamma}}}_{{\mathrm{~mu}}}}_{{j}{,}{k}}{+}{i}{}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{TauonNeutrino}}}_{{j}}{}\left({X}\right)\right){,}{{\mathrm{d_}}}_{}\right)$