Lagrangian - Maple Help

Physics[StandardModel][Lagrangian] - retrieve the Lagrangian of the different sectors of the Standard Model like QED, QCD or Electro-Weak, or of all of it.

 Calling Sequence Lagrangian() Lagrangian(sector) Lagrangian(sector, options)

Parameters

 sector - can be QED, QCD, electroweak or all (default value)

Options

 • applied : (default = false), to return applied the products of differential operators times field functions
 • expanded : (default = false), to return expanded the sums over leptons and quarks
 • interaction : (default = false), to return only the interaction Lagrangian terms
 • showterms : (default = false), related to the electroweak part, to return the corresponding Lagrangian terms as an explicit sum of different ${L}_{\mathrm{term}}$ labels, with equations showing what is the contents of each label
 • term = ... : related to the electroweak part, the right-hand side can be any of the labels ${L}_{K},{L}_{N},{L}_{C},{L}_{H},{L}_{\mathrm{HV}},{L}_{\mathrm{WWV}},{L}_{\mathrm{WWVV}},{L}_{Y}$, to return only the corresponding Lagrangian term as shown when using the showterms option

Description

 • One of the distinctive aspects of the Standard Model is the complexity of its Lagrangian. In this context, Lagrangian returns the Lagrangian of the model after symmetry breaking, optionally restricted to only the interaction terms, or only one of its QED, QCD and electroweak sectors, or only one of the different sub-terms involved in the electroweak part; all of that with the sums over leptons and quarks optionally expanded.
 • All the algebraic expressions returned by Lagrangian are fully computable; so you can use them as starting point to construct other Lagrangians (add or subtract terms), or the Action and related field equations (see d_, D_ for covariant derivatives, diff and Fundiff for functional differentiation), or to compute scattering amplitudes (see FeynmanDiagrams and FeynmanIntegral). NOTE: the output of Lagrangian explicitly includes all the tensor indices of different kinds, like spacetime, spinor, su3 and su2 adjoint and fundamental representations.
 • If called with no arguments Lagrangian returns the whole Lagrangian for the Standard Model (free fields and interaction terms), i.e. the QCD part plus the electroweak part. The sums over leptons and quarks entering the Lagrangian are returned not expanded, using the %add command, an inert representation of add. For QED and QCD, the free fields part is expressed using products of covariant derivative differential operators (D_) times the field functions. That resembles the usual way we represent these terms using paper and pencil and is useful to see the whole structure in its most compact form.
 • The default output of Lagrangian can be restricted or tailored in several ways:
 – You can indicate the sector, QED, QCD or electroweak (synonym: ElectroWeak) you are interested in, so that only the related terms are returned.
 – For QED and QCD, you can use the keyword applied to have the covariant derivative differential operators D_ applied, not just multiplied by the field functions; or use the Library:-ApplyProductsOfDifferentialOperators command on the output of Lagrangian to get the same result.
 – You can use the keyword expand to get the sums over leptons and quarks expanded; or use the value command on the output of Lagrangian to get the same result.
 – Use the keyword interaction to get only the interaction terms; this is relevant when computing scattering amplitudes (see FeynmanDiagrams) where only the interaction part of the Lagrangian is used.
 – The electroweak part of the Lagrangian is particularly complicated. It has, however, an algebraic structure of physically recognizable terms. Use the showterms keyword to see that structure, and to see only one of those terms use term = ... where the right-hand side is any of the following L[sector] (synonym L__sector) labelled according to the Wikipedia electroweak page as:
 • ${L}_{K}$ is the kinetic part, including the dynamic and mass (quadratic) terms;
 • ${L}_{N}$ is the neutral current;
 • ${L}_{C}$ is the charged current;
 • ${L}_{H}$ has the Higgs three and four point self interaction terms;
 • ${L}_{\mathrm{HV}}$ contains the Higgs interactions with the gauge vector bosons ${\mathbf{W}}^{+}$, ${\mathbf{W}}^{-}$ and $\mathbf{Z}$;
 • ${L}_{\mathrm{WWV}}$ includes the gauge three-point self interactions between the fields $\mathbf{A},{\mathbf{W}}^{+},{\mathbf{W}}^{-}$ and $\mathbf{Z}$;
 • ${L}_{\mathrm{WWVV}}$ contains the gauge four-point self interactions between the fields $\mathbf{A},{\mathbf{W}}^{+},{\mathbf{W}}^{-}$ and $\mathbf{Z}$;
 • ${L}_{Y}$ contains the Yukawa interactions between the fermions and the Higgs field.

Examples

 > $\mathrm{with}\left(\mathrm{Physics}\right):$
 > $\mathrm{with}\left(\mathrm{StandardModel}\right)$
 ${}\mathrm{_______________________________________________________}$
 $\mathrm{Setting}\mathrm{lowercaselatin_is}\mathrm{letters to represent}\mathrm{Dirac spinor}\mathrm{indices}$
 $\mathrm{Setting}\mathrm{lowercaselatin_ah}\mathrm{letters to represent}\mathrm{SU\left(3\right) adjoint representation, \left(1..8\right)}\mathrm{indices}$
 $\mathrm{Setting}\mathrm{uppercaselatin_ah}\mathrm{letters to represent}\mathrm{SU\left(3\right) fundamental representation, \left(1..3\right)}\mathrm{indices}$
 $\mathrm{Setting}\mathrm{uppercaselatin_is}\mathrm{letters to represent}\mathrm{SU\left(2\right) adjoint representation, \left(1..3\right)}\mathrm{indices}$
 $\mathrm{Setting}\mathrm{uppercasegreek}\mathrm{letters to represent}\mathrm{SU\left(2\right) fundamental representation, \left(1..2\right)}\mathrm{indices}$
 ${}\mathrm{_______________________________________________________}$
 $\mathrm{Defined as the electron, muon and tau leptons and corresponding neutrinos:}{{\mathrm{e}}}_{{j}},{{\mathrm{\mu }}}_{{j}},{{\mathrm{\tau }}}_{{j}},{{\mathrm{ElectronNeutrino}}}_{{j}},{{\mathrm{MuonNeutrino}}}_{{j}},{{\mathrm{TauonNeutrino}}}_{{j}}$
 $\mathrm{Defined as the up, charm, top, down, strange and bottom quarks:}{{\mathrm{u}}}_{{A}{,}{j}},{{\mathrm{c}}}_{{A}{,}{j}},{{\mathrm{t}}}_{{A}{,}{j}},{{\mathrm{d}}}_{{A}{,}{j}},{{\mathrm{s}}}_{{A}{,}{j}},{{\mathrm{b}}}_{{A}{,}{j}}$
 $\mathrm{Defined as gauge tensors:}{{\mathrm{B}}}_{{\mathrm{\mu }}},{{\mathrm{𝔹}}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}},{{\mathrm{A}}}_{{\mathrm{\mu }}},{{\mathrm{𝔽}}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}},{{\mathrm{W}}}_{{\mathrm{\mu }}{,}{J}},{{\mathrm{𝕎}}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}{,}{J}},{{\mathrm{WPlusField}}}_{{\mathrm{μ}}},{{\mathrm{WPlusFieldStrength}}}_{{\mathrm{μ}}{,}{\mathrm{ν}}},{{\mathrm{WMinusField}}}_{{\mathrm{μ}}},{{\mathrm{WMinusFieldStrength}}}_{{\mathrm{μ}}{,}{\mathrm{ν}}},{{\mathrm{Z}}}_{{\mathrm{\mu }}},{{\mathrm{ℤ}}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}},{{\mathrm{G}}}_{{\mathrm{\mu }}{,}{a}},{{\mathrm{𝔾}}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}{,}{a}}$
 $\mathrm{Defined as Gell-Mann \left(Glambda\right), Pauli \left(Psigma\right) and Dirac \left(Dgamma\right) matrices:}{{\mathrm{\lambda }}}_{{a}},{{\mathrm{\sigma }}}_{{J}},{{\mathrm{\gamma }}}_{{\mathrm{\mu }}}$
 $\mathrm{Defined as the electric, weak and strong coupling constants:}\mathrm{g__e},\mathrm{g__w},\mathrm{g__s}$
 $\mathrm{Defined as the charge in units of |}\mathrm{g__e}\mathrm{| for 1\right) the electron, muon and tauon, 2\right) the up, charm and top, and 3\right) the down, strange and bottom:}\mathrm{%q__e}={-1},\mathrm{%q__u}=\frac{{2}}{{3}},\mathrm{%q__d}={-}\frac{{1}}{{3}}$
 $\mathrm{Defined as the weak isospin for 1\right) the electron, muon and tauon, 2\right) the up, charm and top, 3\right) the down, strange and bottom, and 4\right) all the neutrinos:}\mathrm{%I__e}={-}\frac{{1}}{{2}},\mathrm{%I__u}=\frac{{1}}{{2}},\mathrm{%I__d}={-}\frac{{1}}{{2}},\mathrm{%I__n}=\frac{{1}}{{2}}$
 $\mathrm{You can use the active form without the % prefix, or the \text{'}value\text{'} command to give the corresponding value to any of the inert representations}\mathrm{%q__e},\mathrm{%q__u},\mathrm{%q__d},\mathrm{%I__e},\mathrm{%I__u},\mathrm{%I__d},\mathrm{%I__n}$
 ${}\mathrm{_______________________________________________________}$
 ${}\mathrm{Default differentiation variables for d_, D_ and dAlembertian are:}{}\left\{X=\left(x{,}y{,}z{,}t\right)\right\}$
 $\mathrm{Minkowski spacetime with signatre}\left(\mathrm{- - - +}\right)$
 ${}\mathrm{_______________________________________________________}$
 $\left[\mathrm{%I__d}{,}\mathrm{%I__e}{,}\mathrm{%I__n}{,}\mathrm{%I__u}{,}\mathrm{%q__d}{,}\mathrm{%q__e}{,}\mathrm{%q__u}{,}{\mathrm{BField}}{,}{\mathrm{BFieldStrength}}{,}{\mathrm{Bottom}}{,}{\mathrm{CKM}}{,}{\mathrm{Charm}}{,}{\mathrm{Down}}{,}{\mathrm{ElectromagneticField}}{,}{\mathrm{ElectromagneticFieldStrength}}{,}{\mathrm{Electron}}{,}{\mathrm{ElectronNeutrino}}{,}{\mathrm{FSU3}}{,}{\mathrm{Glambda}}{,}{\mathrm{GluonField}}{,}{\mathrm{GluonFieldStrength}}{,}{\mathrm{HiggsBoson}}{,}{\mathrm{Lagrangian}}{,}{\mathrm{Muon}}{,}{\mathrm{MuonNeutrino}}{,}{\mathrm{Strange}}{,}{\mathrm{Tauon}}{,}{\mathrm{TauonNeutrino}}{,}{\mathrm{Top}}{,}{\mathrm{Up}}{,}{\mathrm{WField}}{,}{\mathrm{WFieldStrength}}{,}{\mathrm{WMinusField}}{,}{\mathrm{WMinusFieldStrength}}{,}{\mathrm{WPlusField}}{,}{\mathrm{WPlusFieldStrength}}{,}{\mathrm{WeinbergAngle}}{,}{\mathrm{ZField}}{,}{\mathrm{ZFieldStrength}}{,}\mathrm{g__e}{,}\mathrm{g__s}{,}\mathrm{g__w}\right]$ (1)

The massless fields of the model are the electromagnetic and gluon fields and the three neutrinos

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

Note that using Physics noncommutative and anticommutative fields are displayed in different colors. You change these colors using Setup.

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)
 > $\mathrm{StandardModel}:-\mathrm{Quarks}$
 $\left[{\mathbf{u}}{,}{\mathbf{c}}{,}{\mathbf{t}}{,}{\mathbf{d}}{,}{\mathbf{s}}{,}{\mathbf{b}}\right]$ (4)

The Gauge fields, and their related field strengths displayed with Open Face type fonts

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

To represent the interaction Lagrangians for the QCD and electroweak sectors as sums over leptons and quarks, all of them fermions, it is useful to introduce four anticommutative prefixes, used below as summation indices in the formulas

 > $\mathrm{Setup}\left(\mathrm{anticommutativeprefix}=\left\{\mathit{f__L},\mathit{f__Q},\mathit{f__U},\mathit{f__D}\right\}\right)$
 $\left[{\mathrm{anticommutativeprefix}}{=}\left\{\mathrm{f__D}{,}\mathrm{f__L}{,}\mathrm{f__Q}{,}\mathrm{f__U}\right\}\right]$ (6)

For readability, omit from the display of formulas the functionality of all the fields entering the Standard Model (see CompactDisplay) and use the lowercase i instead of the uppercase I to represent the imaginary unit

 > $\mathrm{CompactDisplay}\left(\left(\mathrm{StandardModel}:-\mathrm{Leptons},\mathrm{StandardModel}:-\mathrm{Quarks},\mathrm{StandardModel}:-\mathrm{GaugeFields},\mathrm{HiggsBoson},\mathit{f__L},\mathit{f__Q},\mathit{f__U},\mathit{f__D}\right)\left(X\right),\mathrm{quiet}\right):$
 > $\mathrm{interface}\left(\mathrm{imaginaryunit}=i\right):$

The Lagrangian of the whole Standard Model after symmetry breaking, in its most compact form

 > $\mathrm{Lagrangian}\left(\right)$
 ${\mathrm{%add}}{}\left({\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({\mathrm{f__Q}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{i}{}{{{\mathrm{Dgamma}}}_{{\mathrm{μ}}}}_{{j}{,}{k}}{}{{\mathrm{D_}}}_{{\mathrm{μ}}}{-}{{m}}_{\mathrm{f__Q}}{}{{\mathrm{KroneckerDelta}}}_{{j}{,}{k}}{,}{\mathrm{f__Q}}_{{A}{,}{k}}{}\left({X}\right)\right){,}\mathrm{f__Q}{=}\left[{\mathrm{Up}}{,}{\mathrm{Charm}}{,}{\mathrm{Top}}{,}{\mathrm{Down}}{,}{\mathrm{Strange}}{,}{\mathrm{Bottom}}\right]\right){-}\frac{{{\mathbf{𝔾}}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}{,}{a}}^{{2}}}{{4}}{-}\frac{{{\mathbf{𝔽}}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}^{{2}}}{{4}}{-}\frac{{{\mathrm{WPlusFieldStrength}}}_{{\mathrm{μ}}{,}{\mathrm{ν}}}{}{{\mathrm{WMinusFieldStrength}}}_{{\mathrm{μ}}{,}{\mathrm{ν}}}}{{2}}{+}{{m}}_{{\mathbf{W}}}^{{2}}{}{{\mathrm{WPlusField}}}_{{\mathrm{μ}}}{}{{\mathrm{WMinusField}}}_{{\mathrm{μ}}}{-}\frac{{{\mathbf{ℤ}}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}^{{2}}}{{4}}{+}\frac{{{m}}_{{\mathbf{Z}}}^{{2}}{}{{\mathbf{Z}}}_{{\mathrm{\mu }}}^{{2}}}{{2}}{+}\frac{{{{\partial }}_{{\mathrm{\mu }}}{}\left({\mathrm{HiggsBoson}}{}\left({X}\right)\right)}^{{2}}}{{2}}{-}\frac{{{m}}_{{\mathbf{\Phi }}}^{{2}}{}{{\mathrm{HiggsBoson}}{}\left({X}\right)}^{{2}}}{{2}}{+}{\mathrm{%add}}{}\left({\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({\mathrm{f__L}}_{{j}}{}\left({X}\right)\right){,}{i}{}{{{\mathrm{Dgamma}}}_{{\mathrm{μ}}}}_{{j}{,}{k}}{}{{\mathrm{d_}}}_{{\mathrm{μ}}}{}\left({\mathrm{f__L}}_{{k}}{}\left({X}\right){,}\left[{X}\right]\right){-}{{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{Dgamma}}}_{{\mathrm{μ}}}}_{{j}{,}{k}}{}{\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{f__L}{=}\left[{\mathrm{ElectronNeutrino}}{,}{\mathrm{MuonNeutrino}}{,}{\mathrm{TauonNeutrino}}\right]\right){+}{\mathrm{%add}}{}\left({\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({\mathrm{f__Q}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{i}{}{{{\mathrm{Dgamma}}}_{{\mathrm{μ}}}}_{{j}{,}{k}}{}{{\mathrm{d_}}}_{{\mathrm{μ}}}{}\left({\mathrm{f__Q}}_{{A}{,}{k}}{}\left({X}\right){,}\left[{X}\right]\right){-}{{m}}_{\mathrm{f__Q}}{}{\mathrm{f__Q}}_{{A}{,}{j}}{}\left({X}\right)\right){,}\mathrm{f__Q}{=}\left[{\mathrm{Up}}{,}{\mathrm{Charm}}{,}{\mathrm{Top}}{,}{\mathrm{Down}}{,}{\mathrm{Strange}}{,}{\mathrm{Bottom}}\right]\right){+}\mathrm{g__e}{}{\left({{\mathrm{\gamma }}}_{{\mathrm{\mu }}}\right)}_{{j}{,}{k}}{}\left(\mathrm{%q__e}{}{\mathrm{%add}}{}\left({\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({\mathrm{f__L}}_{{j}}{}\left({X}\right)\right){,}{\mathrm{f__L}}_{{k}}{}\left({X}\right)\right){,}\mathrm{f__L}{=}\left[{\mathrm{Electron}}{,}{\mathrm{Muon}}{,}{\mathrm{Tauon}}\right]\right){+}\mathrm{%q__u}{}{\mathrm{%add}}{}\left({\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({\mathrm{f__Q}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{\mathrm{f__Q}}_{{A}{,}{k}}{}\left({X}\right)\right){,}\mathrm{f__Q}{=}\left[{\mathrm{Up}}{,}{\mathrm{Charm}}{,}{\mathrm{Top}}\right]\right){+}\mathrm{%q__d}{}{\mathrm{%add}}{}\left({\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({\mathrm{f__Q}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{\mathrm{f__Q}}_{{A}{,}{k}}{}\left({X}\right)\right){,}\mathrm{f__Q}{=}\left[{\mathrm{Down}}{,}{\mathrm{Strange}}{,}{\mathrm{Bottom}}\right]\right)\right){}{{\mathrm{ElectromagneticField}}}_{{\mathrm{μ}}}{}\left({X}\right){+}\frac{\mathrm{g__w}{}\left({\left({{\mathrm{\gamma }}}_{{\mathrm{\mu }}}\right)}_{{j}{,}{k}}{}\left({{\mathrm{\delta }}}_{{k}{,}{l}}{+}{\left({{\mathrm{\gamma }}}_{{5}}\right)}_{{k}{,}{l}}\right){}\left(\mathrm{%I__e}{}{\mathrm{%add}}{}\left({\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({\mathrm{f__L}}_{{j}}{}\left({X}\right)\right){,}{\mathrm{f__L}}_{{l}}{}\left({X}\right)\right){,}\mathrm{f__L}{=}\left[{\mathrm{Electron}}{,}{\mathrm{Muon}}{,}{\mathrm{Tauon}}\right]\right){+}\mathrm{%I__n}{}{\mathrm{%add}}{}\left({\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({\mathrm{f__L}}_{{j}}{}\left({X}\right)\right){,}{\mathrm{f__L}}_{{l}}{}\left({X}\right)\right){,}\mathrm{f__L}{=}\left[{\mathrm{ElectronNeutrino}}{,}{\mathrm{MuonNeutrino}}{,}{\mathrm{TauonNeutrino}}\right]\right){+}\mathrm{%I__u}{}{\mathrm{%add}}{}\left({\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({\mathrm{f__Q}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{\mathrm{f__Q}}_{{A}{,}{l}}{}\left({X}\right)\right){,}\mathrm{f__Q}{=}\left[{\mathrm{Up}}{,}{\mathrm{Charm}}{,}{\mathrm{Top}}\right]\right){+}\mathrm{%I__d}{}{\mathrm{%add}}{}\left({\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({\mathrm{f__Q}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{\mathrm{f__Q}}_{{A}{,}{l}}{}\left({X}\right)\right){,}\mathrm{f__Q}{=}\left[{\mathrm{Down}}{,}{\mathrm{Strange}}{,}{\mathrm{Bottom}}\right]\right)\right){-}{{\mathrm{sin}}{}\left({\mathrm{WeinbergAngle}}\right)}^{{2}}{}{\left({{\mathrm{\gamma }}}_{{\mathrm{\mu }}}\right)}_{{j}{,}{k}}{}\left(\mathrm{%q__e}{}{\mathrm{%add}}{}\left({\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({\mathrm{f__L}}_{{j}}{}\left({X}\right)\right){,}{\mathrm{f__L}}_{{k}}{}\left({X}\right)\right){,}\mathrm{f__L}{=}\left[{\mathrm{Electron}}{,}{\mathrm{Muon}}{,}{\mathrm{Tauon}}\right]\right){+}\mathrm{%q__u}{}{\mathrm{%add}}{}\left({\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({\mathrm{f__Q}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{\mathrm{f__Q}}_{{A}{,}{k}}{}\left({X}\right)\right){,}\mathrm{f__Q}{=}\left[{\mathrm{Up}}{,}{\mathrm{Charm}}{,}{\mathrm{Top}}\right]\right){+}\mathrm{%q__d}{}{\mathrm{%add}}{}\left({\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({\mathrm{f__Q}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{\mathrm{f__Q}}_{{A}{,}{k}}{}\left({X}\right)\right){,}\mathrm{f__Q}{=}\left[{\mathrm{Down}}{,}{\mathrm{Strange}}{,}{\mathrm{Bottom}}\right]\right)\right)\right){}{{\mathrm{ZField}}}_{{\mathrm{μ}}}{}\left({X}\right)}{{\mathrm{cos}}{}\left({\mathrm{WeinbergAngle}}\right)}{-}\frac{\mathrm{g__w}{}\sqrt{{2}}{}{\left({{\mathrm{\gamma }}}_{{\mathrm{\mu }}}\right)}_{{j}{,}{k}}{}\left({{\mathrm{\delta }}}_{{k}{,}{l}}{+}{\left({{\mathrm{\gamma }}}_{{5}}\right)}_{{k}{,}{l}}\right){}\left(\left({\mathrm{%add}}{}\left({\mathrm{%add}}{}\left({{\mathrm{CKM}}}_{\mathrm{f__U}{,}\mathrm{f__D}}{}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({\mathrm{f__U}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{\mathrm{f__D}}_{{A}{,}{l}}{}\left({X}\right)\right){,}\mathrm{f__U}{=}\left[{\mathrm{Up}}{,}{\mathrm{Charm}}{,}{\mathrm{Top}}\right]\right){,}\mathrm{f__D}{=}\left[{\mathrm{Down}}{,}{\mathrm{Strange}}{,}{\mathrm{Bottom}}\right]\right){+}{\mathrm{%add}}{}\left({\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{f__L}}_{{1}}}_{{j}}{}\left({X}\right)\right){,}{{\mathrm{f__L}}_{{2}}}_{{l}}{}\left({X}\right)\right){,}\mathrm{f__L}{=}\left[\left[{\mathrm{ElectronNeutrino}}{,}{\mathrm{Electron}}\right]{,}\left[{\mathrm{MuonNeutrino}}{,}{\mathrm{Muon}}\right]{,}\left[{\mathrm{TauonNeutrino}}{,}{\mathrm{Tauon}}\right]\right]\right)\right){}{{\mathrm{WPlusField}}}_{{\mathrm{μ}}}{}\left({X}\right){+}\left({\mathrm{%add}}{}\left({\mathrm{%add}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{CKM}}}_{\mathrm{f__U}{,}\mathrm{f__D}}\right){}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({\mathrm{f__D}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{\mathrm{f__U}}_{{A}{,}{l}}{}\left({X}\right)\right){,}\mathrm{f__U}{=}\left[{\mathrm{Up}}{,}{\mathrm{Charm}}{,}{\mathrm{Top}}\right]\right){,}\mathrm{f__D}{=}\left[{\mathrm{Down}}{,}{\mathrm{Strange}}{,}{\mathrm{Bottom}}\right]\right){+}{\mathrm{%add}}{}\left({\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{f__L}}_{{2}}}_{{j}}{}\left({X}\right)\right){,}{{\mathrm{f__L}}_{{1}}}_{{l}}{}\left({X}\right)\right){,}\mathrm{f__L}{=}\left[\left[{\mathrm{ElectronNeutrino}}{,}{\mathrm{Electron}}\right]{,}\left[{\mathrm{MuonNeutrino}}{,}{\mathrm{Muon}}\right]{,}\left[{\mathrm{TauonNeutrino}}{,}{\mathrm{Tauon}}\right]\right]\right)\right){}{{\mathrm{WMinusField}}}_{{\mathrm{μ}}}{}\left({X}\right)\right)}{{2}}{-}\frac{\mathrm{g__w}{}{{m}}_{{\mathbf{\Phi }}}^{{2}}{}\left({{\mathrm{HiggsBoson}}{}\left({X}\right)}^{{3}}{+}\frac{{{\mathrm{HiggsBoson}}{}\left({X}\right)}^{{4}}}{{8}{}{{m}}_{{\mathbf{W}}}}\right)}{{4}{}{{m}}_{{\mathbf{W}}}}{+}\left(\frac{\mathrm{g__w}{}{\mathrm{HiggsBoson}}{}\left({X}\right)}{{{m}}_{{\mathbf{W}}}}{+}\frac{{\mathrm{g__w}}^{{2}}{}{{\mathrm{HiggsBoson}}{}\left({X}\right)}^{{2}}}{{4}{}{{m}}_{{\mathbf{W}}}^{{2}}}\right){}\left({{m}}_{{\mathbf{W}}}^{{2}}{}{{\mathrm{WPlusField}}}_{{\mathrm{μ}}}{}\left({X}\right){}{{\mathrm{WMinusField}}}_{{\mathrm{μ}}}{}\left({X}\right){+}\frac{{{{\mathrm{ZField}}}_{{\mathrm{μ}}}{}\left({X}\right)}^{{2}}{}{{m}}_{{\mathbf{Z}}}^{{2}}}{{2}}\right){-}{i}{}\mathrm{g__w}{}\left(\left({{\mathrm{WPlusFieldStrength}}}_{{\mathrm{μ}}{,}{\mathrm{ν}}}{}\left({X}\right){}{{\mathrm{WMinusField}}}_{{\mathrm{μ}}}{}\left({X}\right){-}{{\mathrm{WPlusField}}}_{{\mathrm{μ}}}{}\left({X}\right){}{{\mathrm{WMinusFieldStrength}}}_{{\mathrm{μ}}{,}{\mathrm{ν}}}{}\left({X}\right)\right){}\left({{\mathrm{ElectromagneticField}}}_{{\mathrm{ν}}}{}\left({X}\right){}{\mathrm{sin}}{}\left({\mathrm{WeinbergAngle}}\right){-}{{\mathrm{ZField}}}_{{\mathrm{ν}}}{}\left({X}\right){}{\mathrm{cos}}{}\left({\mathrm{WeinbergAngle}}\right)\right){+}{{\mathrm{WMinusField}}}_{{\mathrm{ν}}}{}\left({X}\right){}{{\mathrm{WPlusField}}}_{{\mathrm{μ}}}{}\left({X}\right){}\left({{\mathrm{ElectromagneticFieldStrength}}}_{{\mathrm{μ}}{,}{\mathrm{ν}}}{}\left({X}\right){}{\mathrm{sin}}{}\left({\mathrm{WeinbergAngle}}\right){-}{{\mathrm{ZFieldStrength}}}_{{\mathrm{μ}}{,}{\mathrm{ν}}}{}\left({X}\right){}{\mathrm{cos}}{}\left({\mathrm{WeinbergAngle}}\right)\right)\right){-}\frac{{\mathrm{g__w}}^{{2}}{}\left(\left({2}{}{{\mathrm{WPlusField}}}_{{\mathrm{μ}}}{}\left({X}\right){}{{\mathrm{WMinusField}}}_{{\mathrm{μ}}}{}\left({X}\right){+}{\left({{\mathrm{ElectromagneticField}}}_{{\mathrm{μ}}}{}\left({X}\right){}{\mathrm{sin}}{}\left({\mathrm{WeinbergAngle}}\right){-}{{\mathrm{ZField}}}_{{\mathrm{μ}}}{}\left({X}\right){}{\mathrm{cos}}{}\left({\mathrm{WeinbergAngle}}\right)\right)}^{{2}}\right){}\left({2}{}{{\mathrm{WPlusField}}}_{{\mathrm{ν}}}{}\left({X}\right){}{{\mathrm{WMinusField}}}_{{\mathrm{ν}}}{}\left({X}\right){+}{\left({{\mathrm{ElectromagneticField}}}_{{\mathrm{ν}}}{}\left({X}\right){}{\mathrm{sin}}{}\left({\mathrm{WeinbergAngle}}\right){-}{{\mathrm{ZField}}}_{{\mathrm{ν}}}{}\left({X}\right){}{\mathrm{cos}}{}\left({\mathrm{WeinbergAngle}}\right)\right)}^{{2}}\right){+}{\left({{\mathrm{WPlusField}}}_{{\mathrm{μ}}}{}\left({X}\right){}{{\mathrm{WMinusField}}}_{{\mathrm{ν}}}{}\left({X}\right){+}{{\mathrm{WPlusField}}}_{{\mathrm{ν}}}{}\left({X}\right){}{{\mathrm{WMinusField}}}_{{\mathrm{μ}}}{}\left({X}\right){+}\left({{\mathrm{ElectromagneticField}}}_{{\mathrm{μ}}}{}\left({X}\right){}{\mathrm{sin}}{}\left({\mathrm{WeinbergAngle}}\right){-}{{\mathrm{ZField}}}_{{\mathrm{μ}}}{}\left({X}\right){}{\mathrm{cos}}{}\left({\mathrm{WeinbergAngle}}\right)\right){}\left({{\mathrm{ElectromagneticField}}}_{{\mathrm{ν}}}{}\left({X}\right){}{\mathrm{sin}}{}\left({\mathrm{WeinbergAngle}}\right){-}{{\mathrm{ZField}}}_{{\mathrm{ν}}}{}\left({X}\right){}{\mathrm{cos}}{}\left({\mathrm{WeinbergAngle}}\right)\right)\right)}^{{2}}\right)}{{4}}{-}\frac{\mathrm{g__w}{}\left({\mathrm{%add}}{}\left({{m}}_{\mathrm{f__L}}{}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({\mathrm{f__L}}_{{j}}{}\left({X}\right)\right){,}{\mathrm{f__L}}_{{j}}{}\left({X}\right)\right){,}\mathrm{f__L}{=}\left[{\mathrm{Electron}}{,}{\mathrm{Muon}}{,}{\mathrm{Tauon}}{,}{\mathrm{ElectronNeutrino}}{,}{\mathrm{MuonNeutrino}}{,}{\mathrm{TauonNeutrino}}\right]\right){+}{\mathrm{%add}}{}\left({{m}}_{\mathrm{f__Q}}{}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({\mathrm{f__Q}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{\mathrm{f__Q}}_{{A}{,}{j}}{}\left({X}\right)\right){,}\mathrm{f__Q}{=}\left[{\mathrm{Up}}{,}{\mathrm{Charm}}{,}{\mathrm{Top}}{,}{\mathrm{Down}}{,}{\mathrm{Strange}}{,}{\mathrm{Bottom}}\right]\right)\right){}{\mathrm{HiggsBoson}}{}\left({X}\right)}{{2}{}{{m}}_{{\mathbf{W}}}}$ (7)

In the output above we see, among other things, the ${\mathrm{\gamma }}_{5}$ Dirac matrix, and the Cabibbo - Kobayashi - Maskawa matrix $\mathbf{𝕄}$, and the tensor indices of different kinds all explicit. See StandardModel for the notational conventions used, which are standard in the literature but for a few things, like a sign in the definition of ${\mathrm{\gamma }}_{5}$, that depend on the reference. Although this result is the complete Standard Model Lagrangian, it contains not expanded sums over the leptons and quarks, and in the dynamic part (free fields) the covariant derivative operator D_ does not apply but multiply the field functions, all this allowing for a representation that is both computable and as in textbooks. Passing the optional argument applied makes the covariant derivative operator be applied instead of multiplied, and passing the optional argument expanded makes all the sums be expanded (performed).

The Quantum Electrodynamics (QED) Lagrangian

The simplest sector of this Lagrangian (8) is the QED one

 > $\mathrm{Lagrangian}\left(\mathrm{QED}\right)$
 ${\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Electron}}}_{{j}}{}\left({X}\right)\right){,}{i}{}{{{\mathrm{Dgamma}}}_{{\mathrm{μ}}}}_{{j}{,}{k}}{}{{\mathrm{D_}}}_{{\mathrm{~mu}}}{-}{{m}}_{{\mathrm{Electron}}}{}{{\mathrm{KroneckerDelta}}}_{{j}{,}{k}}{,}{{\mathrm{Electron}}}_{{k}}{}\left({X}\right)\right){-}\frac{{1}}{{4}}{}{\mathrm{*}}{}\left({{\mathrm{ElectromagneticFieldStrength}}}_{{\mathrm{μ}}{,}{\mathrm{ν}}}{,}{{\mathrm{ElectromagneticFieldStrength}}}_{{\mathrm{~mu}}{,}{\mathrm{~nu}}}\right)$ (8)

The applied form can be obtained using the Library command ApplyProductsOfDifferentialOperators over the output (9) or passing the optional argument applied

 > $\mathrm{Lagrangian}\left(\mathrm{QED},\mathrm{applied}\right)$
 ${\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Electron}}}_{{j}}{}\left({X}\right)\right){,}{i}{}{{\mathrm{D_}}}_{{\mathrm{μ}}}{}\left({{\mathrm{Electron}}}_{{k}}{}\left({X}\right){,}\left[{X}\right]\right){}{{{\mathrm{Dgamma}}}_{{\mathrm{~mu}}}}_{{j}{,}{k}}{-}{\mathrm{*}}{}\left({{m}}_{{\mathrm{Electron}}}{,}{{\mathrm{KroneckerDelta}}}_{{j}{,}{k}}{,}{{\mathrm{Electron}}}_{{k}}{}\left({X}\right)\right)\right){-}\frac{{1}}{{4}}{}{\mathrm{*}}{}\left({{\mathrm{ElectromagneticFieldStrength}}}_{{\mathrm{μ}}{,}{\mathrm{ν}}}{,}{{\mathrm{ElectromagneticFieldStrength}}}_{{\mathrm{~mu}}{,}{\mathrm{~nu}}}\right)$ (9)

Only the interaction part of this Lagrangian is relevant when computing scattering amplitudes. To get that part, you can either expand the covariant derivative operator

 > $\mathrm{expand}\left(\right)$
 ${i}{}{{{\mathrm{Dgamma}}}_{{\mathrm{~mu}}}}_{{j}{,}{k}}{}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Electron}}}_{{j}}{}\left({X}\right)\right){,}{{\mathrm{d_}}}_{{\mathrm{μ}}}{}\left({{\mathrm{Electron}}}_{{k}}{}\left({X}\right){,}\left[{X}\right]\right)\right){-}{{{\mathrm{Dgamma}}}_{{\mathrm{~mu}}}}_{{j}{,}{k}}{}\mathrm{g__e}{}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Electron}}}_{{j}}{}\left({X}\right)\right){,}{{\mathrm{Electron}}}_{{k}}{}\left({X}\right){,}{{\mathrm{ElectromagneticField}}}_{{\mathrm{μ}}}{}\left({X}\right)\right){-}{{m}}_{{\mathrm{Electron}}}{}{{\mathrm{KroneckerDelta}}}_{{j}{,}{k}}{}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Electron}}}_{{j}}{}\left({X}\right)\right){,}{{\mathrm{Electron}}}_{{k}}{}\left({X}\right)\right){-}\frac{{1}}{{4}}{}{\mathrm{*}}{}\left({{\mathrm{ElectromagneticFieldStrength}}}_{{\mathrm{μ}}{,}{\mathrm{ν}}}{,}{{\mathrm{ElectromagneticFieldStrength}}}_{{\mathrm{~mu}}{,}{\mathrm{~nu}}}\right)$ (10)

or pass the optional keyword expanded, in which case also the trace of $\mathrm{𝔽__μ,ν}$ gets expanded

 > $\mathrm{Lagrangian}\left(\mathrm{QED},\mathrm{expanded}\right)$
 ${i}{}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Electron}}}_{{j}}{}\left({X}\right)\right){,}{{\mathrm{d_}}}_{{\mathrm{μ}}}{}\left({{\mathrm{Electron}}}_{{k}}{}\left({X}\right){,}\left[{X}\right]\right)\right){}{{{\mathrm{Dgamma}}}_{{\mathrm{~mu}}}}_{{j}{,}{k}}{-}\mathrm{g__e}{}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Electron}}}_{{j}}{}\left({X}\right)\right){,}{{\mathrm{Electron}}}_{{k}}{}\left({X}\right){,}{{\mathrm{ElectromagneticField}}}_{{\mathrm{μ}}}{}\left({X}\right)\right){}{{{\mathrm{Dgamma}}}_{{\mathrm{~mu}}}}_{{j}{,}{k}}{-}{{m}}_{{\mathrm{Electron}}}{}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Electron}}}_{{j}}{}\left({X}\right)\right){,}{{\mathrm{Electron}}}_{{j}}{}\left({X}\right)\right){-}\frac{{1}}{{4}}{}{\mathrm{*}}{}\left({{\mathrm{d_}}}_{{\mathrm{μ}}}{}\left({{\mathrm{ElectromagneticField}}}_{{\mathrm{ν}}}{}\left({X}\right){,}\left[{X}\right]\right){-}{{\mathrm{d_}}}_{{\mathrm{ν}}}{}\left({{\mathrm{ElectromagneticField}}}_{{\mathrm{μ}}}{}\left({X}\right){,}\left[{X}\right]\right){,}{{\mathrm{d_}}}_{{\mathrm{~mu}}}{}\left({{\mathrm{ElectromagneticField}}}_{{\mathrm{~nu}}}{}\left({X}\right){,}\left[{X}\right]\right){-}{{\mathrm{d_}}}_{{\mathrm{~nu}}}{}\left({{\mathrm{ElectromagneticField}}}_{{\mathrm{~mu}}}{}\left({X}\right){,}\left[{X}\right]\right)\right)$ (11)

 > $\mathrm{remove}\left(\mathrm{has},,\left[\mathrm{d_},m\right]\right)$
 ${-}{{{\mathrm{Dgamma}}}_{{\mathrm{~mu}}}}_{{j}{,}{k}}{}\mathrm{g__e}{}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Electron}}}_{{j}}{}\left({X}\right)\right){,}{{\mathrm{Electron}}}_{{k}}{}\left({X}\right){,}{{\mathrm{ElectromagneticField}}}_{{\mathrm{μ}}}{}\left({X}\right)\right)$ (12)

or simpler: pass the keyword interaction

 > $\mathrm{Lagrangian}\left(\mathrm{QED},\mathrm{interaction}\right)$
 ${-}{{{\mathrm{Dgamma}}}_{{\mathrm{~mu}}}}_{{j}{,}{k}}{}\mathrm{g__e}{}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Electron}}}_{{j}}{}\left({X}\right)\right){,}{{\mathrm{Electron}}}_{{k}}{}\left({X}\right){,}{{\mathrm{ElectromagneticField}}}_{{\mathrm{μ}}}{}\left({X}\right)\right)$ (13)

All the algebraic expressions returned by Lagrangian are fully computable in that further calculations can proceed starting from them. For example (see FeynmanDiagrams), this is the self-energy of the electron

 > $\mathrm{FeynmanDiagrams}\left(,\mathrm{incoming}=\left[\mathrm{Electron}\right],\mathrm{outgoing}=\left[\mathrm{Electron}\right],\mathrm{numberofloops}=1,\mathrm{diagrams}\right)$

 ${-}{\mathrm{%FeynmanIntegral}}{}\left({-}\frac{{1}}{{8}}{}\frac{{{{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{Uspinor}}}_{{\mathrm{Electron}}}}_{{l}}{}\left(\mathrm{P__1_}\right){}{\mathrm{conjugate}}{}\left({{{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{Uspinor}}}_{{\mathrm{Electron}}}}_{{m}}{}\left(\mathrm{P__2_}\right)\right){}{\mathrm{g__e}}^{{2}}{}{{{\mathrm{Dgamma}}}_{{\mathrm{~alpha}}}}_{{m}{,}{n}}{}{{{\mathrm{Dgamma}}}_{{\mathrm{~nu}}}}_{{p}{,}{l}}{}\left(\left({\mathrm{P__1}}_{{\mathrm{β}}}{+}{\mathrm{p__2}}_{{\mathrm{β}}}\right){}{{{\mathrm{Dgamma}}}_{{\mathrm{~beta}}}}_{{n}{,}{p}}{+}{{m}}_{{\mathrm{Electron}}}{}{{\mathrm{KroneckerDelta}}}_{{n}{,}{p}}\right){}{{\mathrm{g_}}}_{{\mathrm{α}}{,}{\mathrm{ν}}}{}{\mathrm{Dirac}}{}\left({-}\mathrm{P__2}{+}\mathrm{P__1}\right)}{{{\mathrm{π}}}^{{3}}{}\left({\left(\mathrm{P__1}{+}\mathrm{p__2}\right)}^{{2}}{-}{{m}}_{{\mathrm{Electron}}}^{{2}}{+}{i}{}{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{ε}}\right){}\left({\mathrm{p__2}}^{{2}}{+}{i}{}{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{ε}}\right)}{,}\left[\left[\mathrm{p__2}\right]\right]\right)$ (14)

The Quantum Chromodynamics (QCD) Lagrangian

Next in complexity is the QCD Lagrangian

 > $\mathrm{Lagrangian}\left(\mathrm{QCD}\right)$
 ${\mathrm{%add}}{}\left({\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({\mathrm{f__Q}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{i}{}{{{\mathrm{Dgamma}}}_{{\mathrm{μ}}}}_{{j}{,}{k}}{}{{\mathrm{D_}}}_{{\mathrm{~mu}}}{-}{{m}}_{\mathrm{f__Q}}{}{{\mathrm{KroneckerDelta}}}_{{j}{,}{k}}{,}{\mathrm{f__Q}}_{{A}{,}{k}}{}\left({X}\right)\right){,}\mathrm{f__Q}{=}\left[{\mathrm{Up}}{,}{\mathrm{Charm}}{,}{\mathrm{Top}}{,}{\mathrm{Down}}{,}{\mathrm{Strange}}{,}{\mathrm{Bottom}}\right]\right){-}\frac{{1}}{{4}}{}{\mathrm{*}}{}\left({{\mathrm{GluonFieldStrength}}}_{{\mathrm{μ}}{,}{\mathrm{ν}}{,}{a}}{,}{{\mathrm{GluonFieldStrength}}}_{{\mathrm{~mu}}{,}{\mathrm{~nu}}{,}{a}}\right)$ (15)

To activate only the sum over quarks, without expanding or applying the covariant derivatives, you can use the value command

 > $\mathrm{value}\left(\right)$
 ${\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Up}}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{i}{}{{{\mathrm{Dgamma}}}_{{\mathrm{μ}}}}_{{j}{,}{k}}{}{{\mathrm{D_}}}_{{\mathrm{~mu}}}{-}{{m}}_{{\mathrm{Up}}}{}{{\mathrm{KroneckerDelta}}}_{{j}{,}{k}}{,}{{\mathrm{Up}}}_{{A}{,}{k}}{}\left({X}\right)\right){+}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Charm}}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{i}{}{{{\mathrm{Dgamma}}}_{{\mathrm{μ}}}}_{{j}{,}{k}}{}{{\mathrm{D_}}}_{{\mathrm{~mu}}}{-}{{m}}_{{\mathrm{Charm}}}{}{{\mathrm{KroneckerDelta}}}_{{j}{,}{k}}{,}{{\mathrm{Charm}}}_{{A}{,}{k}}{}\left({X}\right)\right){+}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Top}}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{i}{}{{{\mathrm{Dgamma}}}_{{\mathrm{μ}}}}_{{j}{,}{k}}{}{{\mathrm{D_}}}_{{\mathrm{~mu}}}{-}{{m}}_{{\mathrm{Top}}}{}{{\mathrm{KroneckerDelta}}}_{{j}{,}{k}}{,}{{\mathrm{Top}}}_{{A}{,}{k}}{}\left({X}\right)\right){+}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Down}}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{i}{}{{{\mathrm{Dgamma}}}_{{\mathrm{μ}}}}_{{j}{,}{k}}{}{{\mathrm{D_}}}_{{\mathrm{~mu}}}{-}{{m}}_{{\mathrm{Down}}}{}{{\mathrm{KroneckerDelta}}}_{{j}{,}{k}}{,}{{\mathrm{Down}}}_{{A}{,}{k}}{}\left({X}\right)\right){+}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Strange}}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{i}{}{{{\mathrm{Dgamma}}}_{{\mathrm{μ}}}}_{{j}{,}{k}}{}{{\mathrm{D_}}}_{{\mathrm{~mu}}}{-}{{m}}_{{\mathrm{Strange}}}{}{{\mathrm{KroneckerDelta}}}_{{j}{,}{k}}{,}{{\mathrm{Strange}}}_{{A}{,}{k}}{}\left({X}\right)\right){+}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Bottom}}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{i}{}{{{\mathrm{Dgamma}}}_{{\mathrm{μ}}}}_{{j}{,}{k}}{}{{\mathrm{D_}}}_{{\mathrm{~mu}}}{-}{{m}}_{{\mathrm{Bottom}}}{}{{\mathrm{KroneckerDelta}}}_{{j}{,}{k}}{,}{{\mathrm{Bottom}}}_{{A}{,}{k}}{}\left({X}\right)\right){-}\frac{{1}}{{4}}{}{\mathrm{*}}{}\left({{\mathrm{GluonFieldStrength}}}_{{\mathrm{μ}}{,}{\mathrm{ν}}{,}{a}}{,}{{\mathrm{GluonFieldStrength}}}_{{\mathrm{~mu}}{,}{\mathrm{~nu}}{,}{a}}\right)$ (16)

To expand all of the QCD Lagrangian, that is the sum, covariant derivatives and trace of the gluon field strength $\mathrm{𝔾__μ,ν,a}$, pass expanded

 > $\mathrm{Lagrangian}\left(\mathrm{QCD},\mathrm{expanded}\right)$
 ${\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Up}}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{i}{}\left({{\mathrm{d_}}}_{{\mathrm{μ}}}{}\left({{\mathrm{Up}}}_{{A}{,}{k}}{}\left({X}\right){,}\left[{X}\right]\right){-}\frac{{1}}{{2}}{}{i}{}\mathrm{g__s}{}{{{\mathrm{Glambda}}}_{{a}}}_{{A}{,}{B}}{}{\mathrm{*}}{}\left({{\mathrm{Up}}}_{{B}{,}{k}}{}\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}}}_{{A}{,}{k}}{}\left({X}\right)\right){+}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Charm}}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{i}{}\left({{\mathrm{d_}}}_{{\mathrm{μ}}}{}\left({{\mathrm{Charm}}}_{{A}{,}{k}}{}\left({X}\right){,}\left[{X}\right]\right){-}\frac{{1}}{{2}}{}{i}{}\mathrm{g__s}{}{{{\mathrm{Glambda}}}_{{a}}}_{{A}{,}{B}}{}{\mathrm{*}}{}\left({{\mathrm{Charm}}}_{{B}{,}{k}}{}\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}}}_{{A}{,}{k}}{}\left({X}\right)\right){+}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Top}}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{i}{}\left({{\mathrm{d_}}}_{{\mathrm{μ}}}{}\left({{\mathrm{Top}}}_{{A}{,}{k}}{}\left({X}\right){,}\left[{X}\right]\right){-}\frac{{1}}{{2}}{}{i}{}\mathrm{g__s}{}{{{\mathrm{Glambda}}}_{{a}}}_{{A}{,}{B}}{}{\mathrm{*}}{}\left({{\mathrm{Top}}}_{{B}{,}{k}}{}\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}}}_{{A}{,}{k}}{}\left({X}\right)\right){+}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Down}}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{i}{}\left({{\mathrm{d_}}}_{{\mathrm{μ}}}{}\left({{\mathrm{Down}}}_{{A}{,}{k}}{}\left({X}\right){,}\left[{X}\right]\right){-}\frac{{1}}{{2}}{}{i}{}\mathrm{g__s}{}{{{\mathrm{Glambda}}}_{{a}}}_{{A}{,}{B}}{}{\mathrm{*}}{}\left({{\mathrm{Down}}}_{{B}{,}{k}}{}\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}}}_{{A}{,}{k}}{}\left({X}\right)\right){+}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Strange}}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{i}{}\left({{\mathrm{d_}}}_{{\mathrm{μ}}}{}\left({{\mathrm{Strange}}}_{{A}{,}{k}}{}\left({X}\right){,}\left[{X}\right]\right){-}\frac{{1}}{{2}}{}{i}{}\mathrm{g__s}{}{{{\mathrm{Glambda}}}_{{a}}}_{{A}{,}{B}}{}{\mathrm{*}}{}\left({{\mathrm{Strange}}}_{{B}{,}{k}}{}\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}}}_{{A}{,}{k}}{}\left({X}\right)\right){+}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Bottom}}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{i}{}\left({{\mathrm{d_}}}_{{\mathrm{μ}}}{}\left({{\mathrm{Bottom}}}_{{A}{,}{k}}{}\left({X}\right){,}\left[{X}\right]\right){-}\frac{{1}}{{2}}{}{i}{}\mathrm{g__s}{}{{{\mathrm{Glambda}}}_{{a}}}_{{A}{,}{B}}{}{\mathrm{*}}{}\left({{\mathrm{Bottom}}}_{{B}{,}{k}}{}\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}}}_{{A}{,}{k}}{}\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)$ (17)

For computing scattering amplitudes, only the interaction part of this Lagrangian is relevant. Although one can extract that part from the output above by removing terms, as done in (13), it is simpler to pass the keyword interaction

 > $\mathrm{Lagrangian}\left(\mathrm{QCD},\mathrm{interaction}\right)$
 $\frac{{1}}{{2}}{}\mathrm{g__s}{}{{{\mathrm{Glambda}}}_{{a}}}_{{A}{,}{B}}{}{\mathrm{*}}{}\left({\mathrm{%add}}{}\left({\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({\mathrm{f__Q}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{\mathrm{f__Q}}_{{B}{,}{k}}{}\left({X}\right)\right){,}\mathrm{f__Q}{=}\left[{\mathrm{Up}}{,}{\mathrm{Charm}}{,}{\mathrm{Top}}{,}{\mathrm{Down}}{,}{\mathrm{Strange}}{,}{\mathrm{Bottom}}\right]\right){,}{{\mathrm{GluonField}}}_{{\mathrm{μ}}{,}{a}}{}\left({X}\right)\right){}{{{\mathrm{Dgamma}}}_{{\mathrm{~mu}}}}_{{j}{,}{k}}{-}\mathrm{g__s}{}{{\mathrm{FSU3}}}_{{a}{,}{b}{,}{c}}{}\left({\mathrm{*}}{}\left({{\mathrm{d_}}}_{{\mathrm{μ}}}{}\left({{\mathrm{GluonField}}}_{{\mathrm{ν}}{,}{a}}{}\left({X}\right){,}\left[{X}\right]\right){,}{{\mathrm{GluonField}}}_{{\mathrm{~mu}}{,}{b}}{}\left({X}\right){,}{{\mathrm{GluonField}}}_{{\mathrm{~nu}}{,}{c}}{}\left({X}\right)\right){-}\frac{{1}}{{4}}{}\mathrm{g__s}{}{{\mathrm{FSU3}}}_{{c}{,}{d}{,}{e}}{}{\mathrm{*}}{}\left({{\mathrm{GluonField}}}_{{\mathrm{μ}}{,}{a}}{}\left({X}\right){,}{{\mathrm{GluonField}}}_{{\mathrm{α}}{,}{b}}{}\left({X}\right){,}{{\mathrm{GluonField}}}_{{\mathrm{~mu}}{,}{e}}{}\left({X}\right){,}{{\mathrm{GluonField}}}_{{\mathrm{~alpha}}{,}{d}}{}\left({X}\right)\right)\right)$ (18)

and to have also the sum expanded pass also expanded

 > $\mathrm{Lagrangian}\left(\mathrm{QCD},\mathrm{interaction},\mathrm{expanded}\right)$
 $\frac{{1}}{{2}}{}\mathrm{g__s}{}{{{\mathrm{Glambda}}}_{{a}}}_{{A}{,}{B}}{}{\mathrm{*}}{}\left({\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Up}}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{{\mathrm{Up}}}_{{B}{,}{k}}{}\left({X}\right)\right){+}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Charm}}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{{\mathrm{Charm}}}_{{B}{,}{k}}{}\left({X}\right)\right){+}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Top}}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{{\mathrm{Top}}}_{{B}{,}{k}}{}\left({X}\right)\right){+}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Down}}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{{\mathrm{Down}}}_{{B}{,}{k}}{}\left({X}\right)\right){+}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Strange}}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{{\mathrm{Strange}}}_{{B}{,}{k}}{}\left({X}\right)\right){+}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Bottom}}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{{\mathrm{Bottom}}}_{{B}{,}{k}}{}\left({X}\right)\right){,}{{\mathrm{GluonField}}}_{{\mathrm{μ}}{,}{a}}{}\left({X}\right)\right){}{{{\mathrm{Dgamma}}}_{{\mathrm{~mu}}}}_{{j}{,}{k}}{-}\mathrm{g__s}{}{{\mathrm{FSU3}}}_{{a}{,}{b}{,}{c}}{}\left({\mathrm{*}}{}\left({{\mathrm{d_}}}_{{\mathrm{μ}}}{}\left({{\mathrm{GluonField}}}_{{\mathrm{ν}}{,}{a}}{}\left({X}\right){,}\left[{X}\right]\right){,}{{\mathrm{GluonField}}}_{{\mathrm{~mu}}{,}{b}}{}\left({X}\right){,}{{\mathrm{GluonField}}}_{{\mathrm{~nu}}{,}{c}}{}\left({X}\right)\right){-}\frac{{1}}{{4}}{}\mathrm{g__s}{}{{\mathrm{FSU3}}}_{{c}{,}{d}{,}{e}}{}{\mathrm{*}}{}\left({{\mathrm{GluonField}}}_{{\mathrm{μ}}{,}{a}}{}\left({X}\right){,}{{\mathrm{GluonField}}}_{{\mathrm{α}}{,}{b}}{}\left({X}\right){,}{{\mathrm{GluonField}}}_{{\mathrm{~mu}}{,}{e}}{}\left({X}\right){,}{{\mathrm{GluonField}}}_{{\mathrm{~alpha}}{,}{d}}{}\left({X}\right)\right)\right)$ (19)

The amplitude at tree level for the process with two incoming and two outgoing Up quarks (particle and antiparticle) exchanging a gluon

 > $\mathrm{FeynmanDiagrams}\left(,\mathrm{incomingparticles}=\left[\mathrm{Up},\stackrel{&conjugate0;}{\mathrm{Up}}\right],\mathrm{outgoingparticles}=\left[\mathrm{Up},\stackrel{&conjugate0;}{\mathrm{Up}}\right],\mathrm{numberofloops}=0,\mathrm{diagrams}\right)$

 ${-}\frac{{i}{}{\left({{\mathbit{u}}}_{{\mathbf{u}}}\right)}_{{C}{,}{l}}{}\left(\mathrm{P__1_}\right){}\stackrel{{&conjugate0;}}{{\left({{\mathbit{v}}}_{{\mathbf{u}}}\right)}_{{E}{,}{m}}{}\left(\mathrm{P__2_}\right)}{}\stackrel{{&conjugate0;}}{{\left({{\mathbit{u}}}_{{\mathbf{u}}}\right)}_{{F}{,}{n}}{}\left(\mathrm{P__3_}\right)}{}{\left({{\mathbit{v}}}_{{\mathbf{u}}}\right)}_{{G}{,}{p}}{}\left(\mathrm{P__4_}\right){}{\mathrm{g__s}}^{{2}}{}{\left({{\mathrm{\lambda }}}_{{g}}\right)}_{{F}{,}{G}}{}{\left({{\mathrm{\gamma }}}_{\phantom{{}}\phantom{{\mathrm{\kappa }}}}^{\phantom{{}}{\mathrm{\kappa }}}\right)}_{{n}{,}{p}}{}{\left({{\mathrm{\lambda }}}_{{f}}\right)}_{{E}{,}{C}}{}{\left({{\mathrm{\gamma }}}_{\phantom{{}}\phantom{{\mathrm{\beta }}}}^{\phantom{{}}{\mathrm{\beta }}}\right)}_{{m}{,}{l}}{}{{g}}_{{\mathrm{\beta }}{,}{\mathrm{\kappa }}}{}{{\mathrm{\delta }}}_{{f}{,}{g}}{}{\mathrm{\delta }}{}\left({-}{\mathrm{P__3}}_{\phantom{{}}\phantom{{\mathrm{\lambda }}}}^{\phantom{{}}{\mathrm{\lambda }}}{-}{\mathrm{P__4}}_{\phantom{{}}\phantom{{\mathrm{\lambda }}}}^{\phantom{{}}{\mathrm{\lambda }}}{+}{\mathrm{P__1}}_{\phantom{{}}\phantom{{\mathrm{\lambda }}}}^{\phantom{{}}{\mathrm{\lambda }}}{+}{\mathrm{P__2}}_{\phantom{{}}\phantom{{\mathrm{\lambda }}}}^{\phantom{{}}{\mathrm{\lambda }}}\right)}{{16}{}{{\mathrm{\pi }}}^{{2}}{}\left(\left({\mathrm{P__1}}_{{\mathrm{\sigma }}}{+}{\mathrm{P__2}}_{{\mathrm{\sigma }}}\right){}\left({\mathrm{P__1}}_{\phantom{{}}\phantom{{\mathrm{\sigma }}}}^{\phantom{{}}{\mathrm{\sigma }}}{+}{\mathrm{P__2}}_{\phantom{{}}\phantom{{\mathrm{\sigma }}}}^{\phantom{{}}{\mathrm{\sigma }}}\right){+}{i}{}{\mathbf{\epsilon }}\right)}{+}\frac{{i}{}{\left({{\mathbit{u}}}_{{\mathbf{u}}}\right)}_{{C}{,}{l}}{}\left(\mathrm{P__1_}\right){}\stackrel{{&conjugate0;}}{{\left({{\mathbit{v}}}_{{\mathbf{u}}}\right)}_{{E}{,}{m}}{}\left(\mathrm{P__2_}\right)}{}\stackrel{{&conjugate0;}}{{\left({{\mathbit{u}}}_{{\mathbf{u}}}\right)}_{{F}{,}{n}}{}\left(\mathrm{P__3_}\right)}{}{\left({{\mathbit{v}}}_{{\mathbf{u}}}\right)}_{{G}{,}{p}}{}\left(\mathrm{P__4_}\right){}{\mathrm{g__s}}^{{2}}{}{\left({{\mathrm{\lambda }}}_{{g}}\right)}_{{E}{,}{G}}{}{\left({{\mathrm{\gamma }}}_{\phantom{{}}\phantom{{\mathrm{\kappa }}}}^{\phantom{{}}{\mathrm{\kappa }}}\right)}_{{m}{,}{p}}{}{\left({{\mathrm{\lambda }}}_{{f}}\right)}_{{F}{,}{C}}{}{\left({{\mathrm{\gamma }}}_{\phantom{{}}\phantom{{\mathrm{\beta }}}}^{\phantom{{}}{\mathrm{\beta }}}\right)}_{{n}{,}{l}}{}{{g}}_{{\mathrm{\beta }}{,}{\mathrm{\kappa }}}{}{{\mathrm{\delta }}}_{{f}{,}{g}}{}{\mathrm{\delta }}{}\left({-}{\mathrm{P__3}}_{\phantom{{}}\phantom{{\mathrm{\lambda }}}}^{\phantom{{}}{\mathrm{\lambda }}}{-}{\mathrm{P__4}}_{\phantom{{}}\phantom{{\mathrm{\lambda }}}}^{\phantom{{}}{\mathrm{\lambda }}}{+}{\mathrm{P__1}}_{\phantom{{}}\phantom{{\mathrm{\lambda }}}}^{\phantom{{}}{\mathrm{\lambda }}}{+}{\mathrm{P__2}}_{\phantom{{}}\phantom{{\mathrm{\lambda }}}}^{\phantom{{}}{\mathrm{\lambda }}}\right)}{{16}{}{{\mathrm{\pi }}}^{{2}}{}\left(\left({\mathrm{P__1}}_{{\mathrm{\sigma }}}{-}{\mathrm{P__3}}_{{\mathrm{\sigma }}}\right){}\left({\mathrm{P__1}}_{\phantom{{}}\phantom{{\mathrm{\sigma }}}}^{\phantom{{}}{\mathrm{\sigma }}}{-}{\mathrm{P__3}}_{\phantom{{}}\phantom{{\mathrm{\sigma }}}}^{\phantom{{}}{\mathrm{\sigma }}}\right){+}{i}{}{\mathbf{\epsilon }}\right)}$ (20)

The probability density of the same process at 1 loop

 > $\mathrm{FeynmanDiagrams}\left(,\mathrm{incomingparticles}=\left[\mathrm{Up},\stackrel{&conjugate0;}{\mathrm{Up}}\right],\mathrm{outgoingparticles}=\left[\mathrm{Up},\stackrel{&conjugate0;}{\mathrm{Up}}\right],\mathrm{numberofloops}=1,\mathrm{diagrams},\mathrm{output}=\mathrm{probabilitydensity}\right)$

 ${\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{ProbabilityDensity}}{}\left({4}{}{{\mathrm{π}}}^{{2}}{}{\mathrm{%mul}}{}\left({{n}}_{\mathrm{i__1}}{,}\mathrm{i__1}{=}{1}{..}{2}\right){}{{\mathrm{abs}}{}\left(\mathrm{F__1}\right)}^{{2}}{}{\mathrm{Dirac}}{}\left({-}\mathrm{P__3}{-}\mathrm{P__4}{+}\mathrm{P__1}{+}\mathrm{P__2}\right){}{\mathrm{%mul}}{}\left({{\mathrm{dP_}}}_{{f}}^{{3}}{,}{f}{=}{1}{..}{2}\right){,}\mathrm{F__1}{=}{-}{\mathrm{%FeynmanIntegral}}{}\left(\frac{{1}}{{1024}}{}\frac{{{{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{Uspinor}}}_{{\mathrm{Up}}}}_{{C}{,}{l}}{}\left(\mathrm{P__1_}\right){}{\mathrm{conjugate}}{}\left({{{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{Vspinor}}}_{{\mathrm{Up}}}}_{{E}{,}{m}}{}\left(\mathrm{P__2_}\right)\right){}{\mathrm{conjugate}}{}\left({{{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{Uspinor}}}_{{\mathrm{Up}}}}_{{F}{,}{n}}{}\left(\mathrm{P__3_}\right)\right){}{{{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{Vspinor}}}_{{\mathrm{Up}}}}_{{G}{,}{p}}{}\left(\mathrm{P__4_}\right){}{\mathrm{g__s}}^{{4}}{}{{{\mathrm{Glambda}}}_{{\mathrm{a1}}}}_{{\mathrm{D3}}{,}{G}}{}{{{\mathrm{Dgamma}}}_{{\mathrm{~sigma}}}}_{{\mathrm{j1}}{,}{p}}{}{{{\mathrm{Glambda}}}_{{h}}}_{{F}{,}{\mathrm{D1}}}{}{{{\mathrm{Dgamma}}}_{{\mathrm{~lambda}}}}_{{n}{,}{r}}{}{{{\mathrm{Glambda}}}_{{g}}}_{{E}{,}{H}}{}{{{\mathrm{Dgamma}}}_{{\mathrm{~kappa}}}}_{{m}{,}{q}}{}{{{\mathrm{Glambda}}}_{{f}}}_{{\mathrm{D2}}{,}{C}}{}{{{\mathrm{Dgamma}}}_{{\mathrm{~beta}}}}_{{s}{,}{l}}{}\left(\left({-}{\mathrm{P__3}}_{{\mathrm{τ}}}{+}{\mathrm{P__1}}_{{\mathrm{τ}}}{+}{\mathrm{p__4}}_{{\mathrm{τ}}}\right){}{{{\mathrm{Dgamma}}}_{{\mathrm{~tau}}}}_{{q}{,}{s}}{+}{{m}}_{{\mathrm{Up}}}{}{{\mathrm{KroneckerDelta}}}_{{q}{,}{s}}\right){}{{\mathrm{KroneckerDelta}}}_{{\mathrm{D2}}{,}{H}}{}{{\mathrm{g_}}}_{{\mathrm{β}}{,}{\mathrm{λ}}}{}{{\mathrm{KroneckerDelta}}}_{{f}{,}{h}}{}{{\mathrm{g_}}}_{{\mathrm{κ}}{,}{\mathrm{σ}}}{}{{\mathrm{KroneckerDelta}}}_{{\mathrm{a1}}{,}{g}}{}\left({{m}}_{{\mathrm{Up}}}{}{{\mathrm{KroneckerDelta}}}_{{\mathrm{j1}}{,}{r}}{+}{\mathrm{p__4}}_{{\mathrm{υ}}}{}{{{\mathrm{Dgamma}}}_{{\mathrm{~upsilon}}}}_{{r}{,}{\mathrm{j1}}}\right){}{{\mathrm{KroneckerDelta}}}_{{\mathrm{D1}}{,}{\mathrm{D3}}}}{{{\mathrm{π}}}^{{6}}{}\left({\left({-}\mathrm{P__3}{+}\mathrm{P__1}{+}\mathrm{p__4}\right)}^{{2}}{-}{{m}}_{{\mathrm{Up}}}^{{2}}{+}{i}{}{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{ε}}\right){}\left({\left({-}\mathrm{P__3}{+}\mathrm{p__4}\right)}^{{2}}{+}{i}{}{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{ε}}\right){}\left({\left(\mathrm{P__3}{-}\mathrm{P__2}{-}\mathrm{P__1}{-}\mathrm{p__4}\right)}^{{2}}{+}{i}{}{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{ε}}\right){}\left({\mathrm{p__4}}^{{2}}{-}{{m}}_{{\mathrm{Up}}}^{{2}}{+}{i}{}{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{ε}}\right)}{,}\left[\left[\mathrm{p__4}\right]\right]\right){-}{\mathrm{%FeynmanIntegral}}{}\left(\frac{{1}}{{1024}}{}\frac{{{{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{Uspinor}}}_{{\mathrm{Up}}}}_{{C}{,}{l}}{}\left(\mathrm{P__1_}\right){}{\mathrm{conjugate}}{}\left({{{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{Vspinor}}}_{{\mathrm{Up}}}}_{{E}{,}{m}}{}\left(\mathrm{P__2_}\right)\right){}{\mathrm{conjugate}}{}\left({{{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{Uspinor}}}_{{\mathrm{Up}}}}_{{F}{,}{n}}{}\left(\mathrm{P__3_}\right)\right){}{{{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{Vspinor}}}_{{\mathrm{Up}}}}_{{G}{,}{p}}{}\left(\mathrm{P__4_}\right){}{\mathrm{g__s}}^{{4}}{}{{{\mathrm{Glambda}}}_{{h}}}_{{\mathrm{D3}}{,}{G}}{}{{{\mathrm{Dgamma}}}_{{\mathrm{~lambda}}}}_{{\mathrm{j1}}{,}{p}}{}{{{\mathrm{Glambda}}}_{{\mathrm{a1}}}}_{{F}{,}{\mathrm{D1}}}{}{{{\mathrm{Dgamma}}}_{{\mathrm{~sigma}}}}_{{n}{,}{r}}{}{{{\mathrm{Glambda}}}_{{g}}}_{{E}{,}{H}}{}{{{\mathrm{Dgamma}}}_{{\mathrm{~kappa}}}}_{{m}{,}{q}}{}{{{\mathrm{Glambda}}}_{{f}}}_{{\mathrm{D2}}{,}{C}}{}{{{\mathrm{Dgamma}}}_{{\mathrm{~beta}}}}_{{s}{,}{l}}{}\left(\left({-}{\mathrm{P__2}}_{{\mathrm{τ}}}{+}{\mathrm{P__3}}_{{\mathrm{τ}}}{-}{\mathrm{p__4}}_{{\mathrm{τ}}}\right){}{{{\mathrm{Dgamma}}}_{{\mathrm{~tau}}}}_{{q}{,}{s}}{+}{{m}}_{{\mathrm{Up}}}{}{{\mathrm{KroneckerDelta}}}_{{q}{,}{s}}\right){}{{\mathrm{KroneckerDelta}}}_{{\mathrm{D2}}{,}{H}}{}{{\mathrm{g_}}}_{{\mathrm{β}}{,}{\mathrm{λ}}}{}{{\mathrm{KroneckerDelta}}}_{{f}{,}{h}}{}{{\mathrm{g_}}}_{{\mathrm{κ}}{,}{\mathrm{σ}}}{}{{\mathrm{KroneckerDelta}}}_{{\mathrm{a1}}{,}{g}}{}\left({{m}}_{{\mathrm{Up}}}{}{{\mathrm{KroneckerDelta}}}_{{\mathrm{j1}}{,}{r}}{+}{\mathrm{p__4}}_{{\mathrm{υ}}}{}{{{\mathrm{Dgamma}}}_{{\mathrm{~upsilon}}}}_{{r}{,}{\mathrm{j1}}}\right){}{{\mathrm{KroneckerDelta}}}_{{\mathrm{D1}}{,}{\mathrm{D3}}}}{{{\mathrm{π}}}^{{6}}{}\left({\left({-}\mathrm{P__2}{+}\mathrm{P__3}{-}\mathrm{p__4}\right)}^{{2}}{-}{{m}}_{{\mathrm{Up}}}^{{2}}{+}{i}{}{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{ε}}\right){}\left({\left(\mathrm{P__3}{-}\mathrm{P__2}{-}\mathrm{P__1}{-}\mathrm{p__4}\right)}^{{2}}{+}{i}{}{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{ε}}\right){}\left({\left({-}\mathrm{P__3}{+}\mathrm{p__4}\right)}^{{2}}{+}{i}{}{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{ε}}\right){}\left({\mathrm{p__4}}^{{2}}{-}{{m}}_{{\mathrm{Up}}}^{{2}}{+}{i}{}{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{ε}}\right)}{,}\left[\left[\mathrm{p__4}\right]\right]\right){+}{\mathrm{%FeynmanIntegral}}{}\left(\frac{{1}}{{1024}}{}\frac{{{{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{Uspinor}}}_{{\mathrm{Up}}}}_{{C}{,}{l}}{}\left(\mathrm{P__1_}\right){}{\mathrm{conjugate}}{}\left({{{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{Vspinor}}}_{{\mathrm{Up}}}}_{{E}{,}{m}}{}\left(\mathrm{P__2_}\right)\right){}{\mathrm{conjugate}}{}\left({{{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{Uspinor}}}_{{\mathrm{Up}}}}_{{F}{,}{n}}{}\left(\mathrm{P__3_}\right)\right){}{{{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{Vspinor}}}_{{\mathrm{Up}}}}_{{G}{,}{p}}{}\left(\mathrm{P__4_}\right){}{\mathrm{g__s}}^{{4}}{}{{{\mathrm{Glambda}}}_{{\mathrm{a1}}}}_{{\mathrm{D3}}{,}{G}}{}{{{\mathrm{Dgamma}}}_{{\mathrm{~sigma}}}}_{{\mathrm{j1}}{,}{p}}{}{{{\mathrm{Glambda}}}_{{g}}}_{{F}{,}{H}}{}{{{\mathrm{Dgamma}}}_{{\mathrm{~kappa}}}}_{{n}{,}{q}}{}{{{\mathrm{Glambda}}}_{{h}}}_{{E}{,}{\mathrm{D1}}}{}{{{\mathrm{Dgamma}}}_{{\mathrm{~lambda}}}}_{{m}{,}{r}}{}{{{\mathrm{Glambda}}}_{{f}}}_{{\mathrm{D2}}{,}{C}}{}{{{\mathrm{Dgamma}}}_{{\mathrm{~beta}}}}_{{s}{,}{l}}{}\left(\left({\mathrm{P__3}}_{{\mathrm{τ}}}{-}{\mathrm{p__4}}_{{\mathrm{τ}}}\right){}{{{\mathrm{Dgamma}}}_{{\mathrm{~tau}}}}_{{q}{,}{s}}{+}{{m}}_{{\mathrm{Up}}}{}{{\mathrm{KroneckerDelta}}}_{{q}{,}{s}}\right){}{{\mathrm{KroneckerDelta}}}_{{\mathrm{D2}}{,}{H}}{}{{\mathrm{g_}}}_{{\mathrm{β}}{,}{\mathrm{λ}}}{}{{\mathrm{KroneckerDelta}}}_{{f}{,}{h}}{}\left(\left({\mathrm{P__3}}_{{\mathrm{υ}}}{-}{\mathrm{P__2}}_{{\mathrm{υ}}}{-}{\mathrm{P__1}}_{{\mathrm{υ}}}{-}{\mathrm{p__4}}_{{\mathrm{υ}}}\right){}{{{\mathrm{Dgamma}}}_{{\mathrm{~upsilon}}}}_{{r}{,}{\mathrm{j1}}}{+}{{m}}_{{\mathrm{Up}}}{}{{\mathrm{KroneckerDelta}}}_{{\mathrm{j1}}{,}{r}}\right){}{{\mathrm{KroneckerDelta}}}_{{\mathrm{D1}}{,}{\mathrm{D3}}}{}{{\mathrm{g_}}}_{{\mathrm{κ}}{,}{\mathrm{σ}}}{}{{\mathrm{KroneckerDelta}}}_{{\mathrm{a1}}{,}{g}}}{{{\mathrm{π}}}^{{6}}{}\left({\left(\mathrm{P__3}{-}\mathrm{p__4}\right)}^{{2}}{-}{{m}}_{{\mathrm{Up}}}^{{2}}{+}{i}{}{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{ε}}\right){}\left({\left(\mathrm{P__3}{-}\mathrm{P__1}{-}\mathrm{p__4}\right)}^{{2}}{+}{i}{}{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{ε}}\right){}\left({\left(\mathrm{P__3}{-}\mathrm{P__2}{-}\mathrm{P__1}{-}\mathrm{p__4}\right)}^{{2}}{-}{{m}}_{{\mathrm{Up}}}^{{2}}{+}{i}{}{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{ε}}\right){}\left({\mathrm{p__4}}^{{2}}{+}{i}{}{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{ε}}\right)}{,}\left[\left[\mathrm{p__4}\right]\right]\right){+}{\mathrm{%FeynmanIntegral}}{}\left(\frac{{1}}{{1024}}{}\frac{{{{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{Uspinor}}}_{{\mathrm{Up}}}}_{{C}{,}{l}}{}\left(\mathrm{P__1_}\right){}{\mathrm{conjugate}}{}\left({{{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{Vspinor}}}_{{\mathrm{Up}}}}_{{E}{,}{m}}{}\left(\mathrm{P__2_}\right)\right){}{\mathrm{conjugate}}{}\left({{{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{Uspinor}}}_{{\mathrm{Up}}}}_{{F}{,}{n}}{}\left(\mathrm{P__3_}\right)\right){}{{{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{Vspinor}}}_{{\mathrm{Up}}}}_{{G}{,}{p}}{}\left(\mathrm{P__4_}\right){}{\mathrm{g__s}}^{{4}}{}{{{\mathrm{Glambda}}}_{{\mathrm{a1}}}}_{{F}{,}{H}}{}{{{\mathrm{Dgamma}}}_{{\mathrm{~sigma}}}}_{{n}{,}{q}}{}{{{\mathrm{Glambda}}}_{{g}}}_{{E}{,}{\mathrm{D1}}}{}{{{\mathrm{Dgamma}}}_{{\mathrm{~kappa}}}}_{{m}{,}{r}}{}{{{\mathrm{Glambda}}}_{{h}}}_{{\mathrm{D3}}{,}{G}}{}{{{\mathrm{Dgamma}}}_{{\mathrm{~lambda}}}}_{{\mathrm{j1}}{,}{p}}{}{{{\mathrm{Glambda}}}_{{f}}}_{{\mathrm{D2}}{,}{C}}{}{{{\mathrm{Dgamma}}}_{{\mathrm{~beta}}}}_{{s}{,}{l}}{}\left(\left({\mathrm{P__3}}_{{\mathrm{τ}}}{+}{\mathrm{p__4}}_{{\mathrm{τ}}}\right){}{{{\mathrm{Dgamma}}}_{{\mathrm{~tau}}}}_{{q}{,}{s}}{+}{{m}}_{{\mathrm{Up}}}{}{{\mathrm{KroneckerDelta}}}_{{q}{,}{s}}\right){}{{\mathrm{KroneckerDelta}}}_{{\mathrm{D2}}{,}{H}}{}{{\mathrm{g_}}}_{{\mathrm{β}}{,}{\mathrm{λ}}}{}{{\mathrm{KroneckerDelta}}}_{{f}{,}{h}}{}\left(\left({-}{\mathrm{P__3}}_{{\mathrm{υ}}}{+}{\mathrm{P__1}}_{{\mathrm{υ}}}{-}{\mathrm{P__4}}_{{\mathrm{υ}}}{-}{\mathrm{p__4}}_{{\mathrm{υ}}}\right){}{{{\mathrm{Dgamma}}}_{{\mathrm{~upsilon}}}}_{{r}{,}{\mathrm{j1}}}{+}{{m}}_{{\mathrm{Up}}}{}{{\mathrm{KroneckerDelta}}}_{{\mathrm{j1}}{,}{r}}\right){}{{\mathrm{KroneckerDelta}}}_{{\mathrm{D1}}{,}{\mathrm{D3}}}{}{{\mathrm{g_}}}_{{\mathrm{κ}}{,}{\mathrm{σ}}}{}{{\mathrm{KroneckerDelta}}}_{{\mathrm{a1}}{,}{g}}}{{{\mathrm{π}}}^{{6}}{}\left({\left(\mathrm{P__3}{+}\mathrm{p__4}\right)}^{{2}}{-}{{m}}_{{\mathrm{Up}}}^{{2}}{+}{i}{}{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{ε}}\right){}\left({\left(\mathrm{P__3}{-}\mathrm{P__1}{+}\mathrm{p__4}\right)}^{{2}}{+}{i}{}{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{ε}}\right){}\left({\left({-}\mathrm{P__3}{+}\mathrm{P__1}{-}\mathrm{P__4}{-}\mathrm{p__4}\right)}^{{2}}{-}{{m}}_{{\mathrm{Up}}}^{{2}}{+}{i}{}{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{ε}}\right){}\left({\mathrm{p__4}}^{{2}}{+}{i}{}{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{ε}}\right)}{,}\left[\left[\mathrm{p__4}\right]\right]\right)\right)$ (21)

The Electro-Weak Lagrangian

The electroweak sector of the Standard Model Lagrangian is significantly more complicated.

 > $\mathrm{Lagrangian}\left(\mathrm{electroweak}\right)$
  (22)

To decipher this result it is useful to see the structure of physically recognizable terms; click on the equal symbols $=$ after where to highlight the ${L}_{\mathrm{term}}$ label and the formula it represents

 > $\mathrm{Lagrangian}\left(\mathrm{electroweak},\mathrm{showterms}\right)$
 $\mathrm{L__K}{+}\mathrm{L__N}{+}\mathrm{L__C}{+}\mathrm{L__H}{+}\mathrm{L__HV}{+}\mathrm{L__WWV}{+}\mathrm{L__WWVV}{+}\mathrm{L__Y}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{where}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left[\mathrm{L__K}{=}{-}\frac{{{\mathbf{𝔽}}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}^{{2}}}{{4}}{-}\frac{{{\mathrm{WPlusFieldStrength}}}_{{\mathrm{μ}}{,}{\mathrm{ν}}}{}{{\mathrm{WMinusFieldStrength}}}_{{\mathrm{μ}}{,}{\mathrm{ν}}}}{{2}}{+}{{m}}_{{\mathbf{W}}}^{{2}}{}{{\mathrm{WPlusField}}}_{{\mathrm{μ}}}{}{{\mathrm{WMinusField}}}_{{\mathrm{μ}}}{-}\frac{{{\mathbf{ℤ}}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}^{{2}}}{{4}}{+}\frac{{{m}}_{{\mathbf{Z}}}^{{2}}{}{{\mathbf{Z}}}_{{\mathrm{\mu }}}^{{2}}}{{2}}{+}\frac{{{{\partial }}_{{\mathrm{\mu }}}{}\left({\mathrm{HiggsBoson}}{}\left({X}\right)\right)}^{{2}}}{{2}}{-}\frac{{{m}}_{{\mathbf{\Phi }}}^{{2}}{}{{\mathrm{HiggsBoson}}{}\left({X}\right)}^{{2}}}{{2}}{+}{\mathrm{%add}}{}\left({\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({\mathrm{f__L}}_{{j}}{}\left({X}\right)\right){,}{i}{}{{{\mathrm{Dgamma}}}_{{\mathrm{μ}}}}_{{j}{,}{k}}{}{{\mathrm{d_}}}_{{\mathrm{μ}}}{}\left({\mathrm{f__L}}_{{k}}{}\left({X}\right){,}\left[{X}\right]\right){-}{{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{Dgamma}}}_{{\mathrm{μ}}}}_{{j}{,}{k}}{}{\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{f__L}{=}\left[{\mathrm{ElectronNeutrino}}{,}{\mathrm{MuonNeutrino}}{,}{\mathrm{TauonNeutrino}}\right]\right){+}{\mathrm{%add}}{}\left({\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({\mathrm{f__Q}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{i}{}{{{\mathrm{Dgamma}}}_{{\mathrm{μ}}}}_{{j}{,}{k}}{}{{\mathrm{d_}}}_{{\mathrm{μ}}}{}\left({\mathrm{f__Q}}_{{A}{,}{k}}{}\left({X}\right){,}\left[{X}\right]\right){-}{{m}}_{\mathrm{f__Q}}{}{\mathrm{f__Q}}_{{A}{,}{j}}{}\left({X}\right)\right){,}\mathrm{f__Q}{=}\left[{\mathrm{Up}}{,}{\mathrm{Charm}}{,}{\mathrm{Top}}{,}{\mathrm{Down}}{,}{\mathrm{Strange}}{,}{\mathrm{Bottom}}\right]\right){,}\mathrm{L__N}{=}\mathrm{g__e}{}{\left({{\mathrm{\gamma }}}_{{\mathrm{\mu }}}\right)}_{{j}{,}{k}}{}\left(\mathrm{%q__e}{}{\mathrm{%add}}{}\left({\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({\mathrm{f__L}}_{{j}}{}\left({X}\right)\right){,}{\mathrm{f__L}}_{{k}}{}\left({X}\right)\right){,}\mathrm{f__L}{=}\left[{\mathrm{Electron}}{,}{\mathrm{Muon}}{,}{\mathrm{Tauon}}\right]\right){+}\mathrm{%q__u}{}{\mathrm{%add}}{}\left({\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({\mathrm{f__Q}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{\mathrm{f__Q}}_{{A}{,}{k}}{}\left({X}\right)\right){,}\mathrm{f__Q}{=}\left[{\mathrm{Up}}{,}{\mathrm{Charm}}{,}{\mathrm{Top}}\right]\right){+}\mathrm{%q__d}{}{\mathrm{%add}}{}\left({\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({\mathrm{f__Q}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{\mathrm{f__Q}}_{{A}{,}{k}}{}\left({X}\right)\right){,}\mathrm{f__Q}{=}\left[{\mathrm{Down}}{,}{\mathrm{Strange}}{,}{\mathrm{Bottom}}\right]\right)\right){}{{\mathrm{ElectromagneticField}}}_{{\mathrm{μ}}}{}\left({X}\right){+}\frac{\mathrm{g__w}{}\left({\left({{\mathrm{\gamma }}}_{{\mathrm{\mu }}}\right)}_{{j}{,}{k}}{}\left({{\mathrm{\delta }}}_{{k}{,}{l}}{+}{\left({{\mathrm{\gamma }}}_{{5}}\right)}_{{k}{,}{l}}\right){}\left(\mathrm{%I__e}{}{\mathrm{%add}}{}\left({\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({\mathrm{f__L}}_{{j}}{}\left({X}\right)\right){,}{\mathrm{f__L}}_{{l}}{}\left({X}\right)\right){,}\mathrm{f__L}{=}\left[{\mathrm{Electron}}{,}{\mathrm{Muon}}{,}{\mathrm{Tauon}}\right]\right){+}\mathrm{%I__n}{}{\mathrm{%add}}{}\left({\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({\mathrm{f__L}}_{{j}}{}\left({X}\right)\right){,}{\mathrm{f__L}}_{{l}}{}\left({X}\right)\right){,}\mathrm{f__L}{=}\left[{\mathrm{ElectronNeutrino}}{,}{\mathrm{MuonNeutrino}}{,}{\mathrm{TauonNeutrino}}\right]\right){+}\mathrm{%I__u}{}{\mathrm{%add}}{}\left({\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({\mathrm{f__Q}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{\mathrm{f__Q}}_{{A}{,}{l}}{}\left({X}\right)\right){,}\mathrm{f__Q}{=}\left[{\mathrm{Up}}{,}{\mathrm{Charm}}{,}{\mathrm{Top}}\right]\right){+}\mathrm{%I__d}{}{\mathrm{%add}}{}\left({\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({\mathrm{f__Q}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{\mathrm{f__Q}}_{{A}{,}{l}}{}\left({X}\right)\right){,}\mathrm{f__Q}{=}\left[{\mathrm{Down}}{,}{\mathrm{Strange}}{,}{\mathrm{Bottom}}\right]\right)\right){-}{{\mathrm{sin}}{}\left({\mathrm{WeinbergAngle}}\right)}^{{2}}{}{\left({{\mathrm{\gamma }}}_{{\mathrm{\mu }}}\right)}_{{j}{,}{k}}{}\left(\mathrm{%q__e}{}{\mathrm{%add}}{}\left({\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({\mathrm{f__L}}_{{j}}{}\left({X}\right)\right){,}{\mathrm{f__L}}_{{k}}{}\left({X}\right)\right){,}\mathrm{f__L}{=}\left[{\mathrm{Electron}}{,}{\mathrm{Muon}}{,}{\mathrm{Tauon}}\right]\right){+}\mathrm{%q__u}{}{\mathrm{%add}}{}\left({\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({\mathrm{f__Q}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{\mathrm{f__Q}}_{{A}{,}{k}}{}\left({X}\right)\right){,}\mathrm{f__Q}{=}\left[{\mathrm{Up}}{,}{\mathrm{Charm}}{,}{\mathrm{Top}}\right]\right){+}\mathrm{%q__d}{}{\mathrm{%add}}{}\left({\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({\mathrm{f__Q}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{\mathrm{f__Q}}_{{A}{,}{k}}{}\left({X}\right)\right){,}\mathrm{f__Q}{=}\left[{\mathrm{Down}}{,}{\mathrm{Strange}}{,}{\mathrm{Bottom}}\right]\right)\right)\right){}{{\mathrm{ZField}}}_{{\mathrm{μ}}}{}\left({X}\right)}{{\mathrm{cos}}{}\left({\mathrm{WeinbergAngle}}\right)}{,}\mathrm{L__C}{=}{-}\frac{\mathrm{g__w}{}\sqrt{{2}}{}{\left({{\mathrm{\gamma }}}_{{\mathrm{\mu }}}\right)}_{{j}{,}{k}}{}\left({{\mathrm{\delta }}}_{{k}{,}{l}}{+}{\left({{\mathrm{\gamma }}}_{{5}}\right)}_{{k}{,}{l}}\right){}\left(\left({\mathrm{%add}}{}\left({\mathrm{%add}}{}\left({{\mathrm{CKM}}}_{\mathrm{f__U}{,}\mathrm{f__D}}{}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({\mathrm{f__U}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{\mathrm{f__D}}_{{A}{,}{l}}{}\left({X}\right)\right){,}\mathrm{f__U}{=}\left[{\mathrm{Up}}{,}{\mathrm{Charm}}{,}{\mathrm{Top}}\right]\right){,}\mathrm{f__D}{=}\left[{\mathrm{Down}}{,}{\mathrm{Strange}}{,}{\mathrm{Bottom}}\right]\right){+}{\mathrm{%add}}{}\left({\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{f__L}}_{{1}}}_{{j}}{}\left({X}\right)\right){,}{{\mathrm{f__L}}_{{2}}}_{{l}}{}\left({X}\right)\right){,}\mathrm{f__L}{=}\left[\left[{\mathrm{ElectronNeutrino}}{,}{\mathrm{Electron}}\right]{,}\left[{\mathrm{MuonNeutrino}}{,}{\mathrm{Muon}}\right]{,}\left[{\mathrm{TauonNeutrino}}{,}{\mathrm{Tauon}}\right]\right]\right)\right){}{{\mathrm{WPlusField}}}_{{\mathrm{μ}}}{}\left({X}\right){+}\left({\mathrm{%add}}{}\left({\mathrm{%add}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{CKM}}}_{\mathrm{f__U}{,}\mathrm{f__D}}\right){}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({\mathrm{f__D}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{\mathrm{f__U}}_{{A}{,}{l}}{}\left({X}\right)\right){,}\mathrm{f__U}{=}\left[{\mathrm{Up}}{,}{\mathrm{Charm}}{,}{\mathrm{Top}}\right]\right){,}\mathrm{f__D}{=}\left[{\mathrm{Down}}{,}{\mathrm{Strange}}{,}{\mathrm{Bottom}}\right]\right){+}{\mathrm{%add}}{}\left({\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{f__L}}_{{2}}}_{{j}}{}\left({X}\right)\right){,}{{\mathrm{f__L}}_{{1}}}_{{l}}{}\left({X}\right)\right){,}\mathrm{f__L}{=}\left[\left[{\mathrm{ElectronNeutrino}}{,}{\mathrm{Electron}}\right]{,}\left[{\mathrm{MuonNeutrino}}{,}{\mathrm{Muon}}\right]{,}\left[{\mathrm{TauonNeutrino}}{,}{\mathrm{Tauon}}\right]\right]\right)\right){}{{\mathrm{WMinusField}}}_{{\mathrm{μ}}}{}\left({X}\right)\right)}{{2}}{,}\mathrm{L__H}{=}{-}\frac{\mathrm{g__w}{}{{m}}_{{\mathbf{\Phi }}}^{{2}}{}\left({{\mathrm{HiggsBoson}}{}\left({X}\right)}^{{3}}{+}\frac{{{\mathrm{HiggsBoson}}{}\left({X}\right)}^{{4}}}{{8}{}{{m}}_{{\mathbf{W}}}}\right)}{{4}{}{{m}}_{{\mathbf{W}}}}{,}\mathrm{L__HV}{=}\left(\frac{\mathrm{g__w}{}{\mathrm{HiggsBoson}}{}\left({X}\right)}{{{m}}_{{\mathbf{W}}}}{+}\frac{{\mathrm{g__w}}^{{2}}{}{{\mathrm{HiggsBoson}}{}\left({X}\right)}^{{2}}}{{4}{}{{m}}_{{\mathbf{W}}}^{{2}}}\right){}\left({{m}}_{{\mathbf{W}}}^{{2}}{}{{\mathrm{WPlusField}}}_{{\mathrm{μ}}}{}\left({X}\right){}{{\mathrm{WMinusField}}}_{{\mathrm{μ}}}{}\left({X}\right){+}\frac{{{{\mathrm{ZField}}}_{{\mathrm{μ}}}{}\left({X}\right)}^{{2}}{}{{m}}_{{\mathbf{Z}}}^{{2}}}{{2}}\right){,}\mathrm{L__WWV}{=}{-i}{}\mathrm{g__w}{}\left(\left({{\mathrm{WPlusFieldStrength}}}_{{\mathrm{μ}}{,}{\mathrm{ν}}}{}\left({X}\right){}{{\mathrm{WMinusField}}}_{{\mathrm{μ}}}{}\left({X}\right){-}{{\mathrm{WPlusField}}}_{{\mathrm{μ}}}{}\left({X}\right){}{{\mathrm{WMinusFieldStrength}}}_{{\mathrm{μ}}{,}{\mathrm{ν}}}{}\left({X}\right)\right){}\left({{\mathrm{ElectromagneticField}}}_{{\mathrm{ν}}}{}\left({X}\right){}{\mathrm{sin}}{}\left({\mathrm{WeinbergAngle}}\right){-}{{\mathrm{ZField}}}_{{\mathrm{ν}}}{}\left({X}\right){}{\mathrm{cos}}{}\left({\mathrm{WeinbergAngle}}\right)\right){+}{{\mathrm{WMinusField}}}_{{\mathrm{ν}}}{}\left({X}\right){}{{\mathrm{WPlusField}}}_{{\mathrm{μ}}}{}\left({X}\right){}\left({{\mathrm{ElectromagneticFieldStrength}}}_{{\mathrm{μ}}{,}{\mathrm{ν}}}{}\left({X}\right){}{\mathrm{sin}}{}\left({\mathrm{WeinbergAngle}}\right){-}{{\mathrm{ZFieldStrength}}}_{{\mathrm{μ}}{,}{\mathrm{ν}}}{}\left({X}\right){}{\mathrm{cos}}{}\left({\mathrm{WeinbergAngle}}\right)\right)\right){,}\mathrm{L__WWVV}{=}{-}\frac{{\mathrm{g__w}}^{{2}}{}\left(\left({2}{}{{\mathrm{WPlusField}}}_{{\mathrm{μ}}}{}\left({X}\right){}{{\mathrm{WMinusField}}}_{{\mathrm{μ}}}{}\left({X}\right){+}{\left({{\mathrm{ElectromagneticField}}}_{{\mathrm{μ}}}{}\left({X}\right){}{\mathrm{sin}}{}\left({\mathrm{WeinbergAngle}}\right){-}{{\mathrm{ZField}}}_{{\mathrm{μ}}}{}\left({X}\right){}{\mathrm{cos}}{}\left({\mathrm{WeinbergAngle}}\right)\right)}^{{2}}\right){}\left({2}{}{{\mathrm{WPlusField}}}_{{\mathrm{ν}}}{}\left({X}\right){}{{\mathrm{WMinusField}}}_{{\mathrm{ν}}}{}\left({X}\right){+}{\left({{\mathrm{ElectromagneticField}}}_{{\mathrm{ν}}}{}\left({X}\right){}{\mathrm{sin}}{}\left({\mathrm{WeinbergAngle}}\right){-}{{\mathrm{ZField}}}_{{\mathrm{ν}}}{}\left({X}\right){}{\mathrm{cos}}{}\left({\mathrm{WeinbergAngle}}\right)\right)}^{{2}}\right){+}{\left({{\mathrm{WPlusField}}}_{{\mathrm{μ}}}{}\left({X}\right){}{{\mathrm{WMinusField}}}_{{\mathrm{ν}}}{}\left({X}\right){+}{{\mathrm{WPlusField}}}_{{\mathrm{ν}}}{}\left({X}\right){}{{\mathrm{WMinusField}}}_{{\mathrm{μ}}}{}\left({X}\right){+}\left({{\mathrm{ElectromagneticField}}}_{{\mathrm{μ}}}{}\left({X}\right){}{\mathrm{sin}}{}\left({\mathrm{WeinbergAngle}}\right){-}{{\mathrm{ZField}}}_{{\mathrm{μ}}}{}\left({X}\right){}{\mathrm{cos}}{}\left({\mathrm{WeinbergAngle}}\right)\right){}\left({{\mathrm{ElectromagneticField}}}_{{\mathrm{ν}}}{}\left({X}\right){}{\mathrm{sin}}{}\left({\mathrm{WeinbergAngle}}\right){-}{{\mathrm{ZField}}}_{{\mathrm{ν}}}{}\left({X}\right){}{\mathrm{cos}}{}\left({\mathrm{WeinbergAngle}}\right)\right)\right)}^{{2}}\right)}{{4}}{,}\mathrm{L__Y}{=}{-}\frac{\mathrm{g__w}{}\left({\mathrm{%add}}{}\left({{m}}_{\mathrm{f__L}}{}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({\mathrm{f__L}}_{{j}}{}\left({X}\right)\right){,}{\mathrm{f__L}}_{{j}}{}\left({X}\right)\right){,}\mathrm{f__L}{=}\left[{\mathrm{Electron}}{,}{\mathrm{Muon}}{,}{\mathrm{Tauon}}{,}{\mathrm{ElectronNeutrino}}{,}{\mathrm{MuonNeutrino}}{,}{\mathrm{TauonNeutrino}}\right]\right){+}{\mathrm{%add}}{}\left({{m}}_{\mathrm{f__Q}}{}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({\mathrm{f__Q}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{\mathrm{f__Q}}_{{A}{,}{j}}{}\left({X}\right)\right){,}\mathrm{f__Q}{=}\left[{\mathrm{Up}}{,}{\mathrm{Charm}}{,}{\mathrm{Top}}{,}{\mathrm{Down}}{,}{\mathrm{Strange}}{,}{\mathrm{Bottom}}\right]\right)\right){}{\mathrm{HiggsBoson}}{}\left({X}\right)}{{2}{}{{m}}_{{\mathbf{W}}}}\right]$ (23)

In this result we see a sum of ${L}_{\mathrm{terms}}$, and after where there is a list of equations with the formulas represented by each ${L}_{\mathrm{term}}$. Take from the above, for instance, only the charged current ${L}_{C}$ term that involves interaction between the leptons and the corresponding neutrinos: you can do that with the mouse, copy and paste, or using the term = ... option

 > $\mathrm{Lagrangian}\left(\mathrm{electroweak},\mathrm{term}={L}_{C}\right)$
 $\mathrm{L__C}{=}{-}\frac{{1}}{{2}}{}\mathrm{g__w}{}\sqrt{{2}}{}\left({{\mathrm{KroneckerDelta}}}_{{k}{,}{l}}{+}{{{\mathrm{Dgamma}}}_{{5}}}_{{k}{,}{l}}\right){}\left({\mathrm{*}}{}\left({\mathrm{%add}}{}\left({\mathrm{%add}}{}\left({{\mathrm{CKM}}}_{\mathrm{f__U}{,}\mathrm{f__D}}{}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({\mathrm{f__U}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{\mathrm{f__D}}_{{A}{,}{l}}{}\left({X}\right)\right){,}\mathrm{f__U}{=}\left[{\mathrm{Up}}{,}{\mathrm{Charm}}{,}{\mathrm{Top}}\right]\right){,}\mathrm{f__D}{=}\left[{\mathrm{Down}}{,}{\mathrm{Strange}}{,}{\mathrm{Bottom}}\right]\right){+}{\mathrm{%add}}{}\left({\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{f__L}}_{{1}}}_{{j}}{}\left({X}\right)\right){,}{{\mathrm{f__L}}_{{2}}}_{{l}}{}\left({X}\right)\right){,}\mathrm{f__L}{=}\left[\left[{\mathrm{ElectronNeutrino}}{,}{\mathrm{Electron}}\right]{,}\left[{\mathrm{MuonNeutrino}}{,}{\mathrm{Muon}}\right]{,}\left[{\mathrm{TauonNeutrino}}{,}{\mathrm{Tauon}}\right]\right]\right){,}{{\mathrm{WPlusField}}}_{{\mathrm{μ}}}{}\left({X}\right)\right){+}{\mathrm{*}}{}\left({\mathrm{%add}}{}\left({\mathrm{%add}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{CKM}}}_{\mathrm{f__U}{,}\mathrm{f__D}}\right){}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({\mathrm{f__D}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{\mathrm{f__U}}_{{A}{,}{l}}{}\left({X}\right)\right){,}\mathrm{f__U}{=}\left[{\mathrm{Up}}{,}{\mathrm{Charm}}{,}{\mathrm{Top}}\right]\right){,}\mathrm{f__D}{=}\left[{\mathrm{Down}}{,}{\mathrm{Strange}}{,}{\mathrm{Bottom}}\right]\right){+}{\mathrm{%add}}{}\left({\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{f__L}}_{{2}}}_{{j}}{}\left({X}\right)\right){,}{{\mathrm{f__L}}_{{1}}}_{{l}}{}\left({X}\right)\right){,}\mathrm{f__L}{=}\left[\left[{\mathrm{ElectronNeutrino}}{,}{\mathrm{Electron}}\right]{,}\left[{\mathrm{MuonNeutrino}}{,}{\mathrm{Muon}}\right]{,}\left[{\mathrm{TauonNeutrino}}{,}{\mathrm{Tauon}}\right]\right]\right){,}{{\mathrm{WMinusField}}}_{{\mathrm{μ}}}{}\left({X}\right)\right)\right){}{{{\mathrm{Dgamma}}}_{{\mathrm{~mu}}}}_{{j}{,}{k}}$ (24)
 > $\mathrm{Lagrangian}\left(\mathrm{electroweak},\mathrm{term}={L}_{C},\mathrm{expanded}\right)$
 $\mathrm{L__C}{=}{-}\frac{{1}}{{2}}{}\mathrm{g__w}{}\sqrt{{2}}{}\left({{\mathrm{KroneckerDelta}}}_{{k}{,}{l}}{+}{{{\mathrm{Dgamma}}}_{{5}}}_{{k}{,}{l}}\right){}\left({\mathrm{*}}{}\left({{\mathrm{CKM}}}_{{\mathrm{Up}}{,}{\mathrm{Down}}}{}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Up}}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{{\mathrm{Down}}}_{{A}{,}{l}}{}\left({X}\right)\right){+}{{\mathrm{CKM}}}_{{\mathrm{Charm}}{,}{\mathrm{Down}}}{}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Charm}}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{{\mathrm{Down}}}_{{A}{,}{l}}{}\left({X}\right)\right){+}{{\mathrm{CKM}}}_{{\mathrm{Top}}{,}{\mathrm{Down}}}{}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Top}}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{{\mathrm{Down}}}_{{A}{,}{l}}{}\left({X}\right)\right){+}{{\mathrm{CKM}}}_{{\mathrm{Up}}{,}{\mathrm{Strange}}}{}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Up}}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{{\mathrm{Strange}}}_{{A}{,}{l}}{}\left({X}\right)\right){+}{{\mathrm{CKM}}}_{{\mathrm{Charm}}{,}{\mathrm{Strange}}}{}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Charm}}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{{\mathrm{Strange}}}_{{A}{,}{l}}{}\left({X}\right)\right){+}{{\mathrm{CKM}}}_{{\mathrm{Top}}{,}{\mathrm{Strange}}}{}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Top}}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{{\mathrm{Strange}}}_{{A}{,}{l}}{}\left({X}\right)\right){+}{{\mathrm{CKM}}}_{{\mathrm{Up}}{,}{\mathrm{Bottom}}}{}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Up}}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{{\mathrm{Bottom}}}_{{A}{,}{l}}{}\left({X}\right)\right){+}{{\mathrm{CKM}}}_{{\mathrm{Charm}}{,}{\mathrm{Bottom}}}{}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Charm}}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{{\mathrm{Bottom}}}_{{A}{,}{l}}{}\left({X}\right)\right){+}{{\mathrm{CKM}}}_{{\mathrm{Top}}{,}{\mathrm{Bottom}}}{}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Top}}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{{\mathrm{Bottom}}}_{{A}{,}{l}}{}\left({X}\right)\right){+}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{ElectronNeutrino}}}_{{j}}{}\left({X}\right)\right){,}{{\mathrm{Electron}}}_{{l}}{}\left({X}\right)\right){+}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{MuonNeutrino}}}_{{j}}{}\left({X}\right)\right){,}{{\mathrm{Muon}}}_{{l}}{}\left({X}\right)\right){+}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{TauonNeutrino}}}_{{j}}{}\left({X}\right)\right){,}{{\mathrm{Tauon}}}_{{l}}{}\left({X}\right)\right){,}{{\mathrm{WPlusField}}}_{{\mathrm{μ}}}{}\left({X}\right)\right){+}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{CKM}}}_{{\mathrm{Up}}{,}{\mathrm{Down}}}\right){}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Down}}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{{\mathrm{Up}}}_{{A}{,}{l}}{}\left({X}\right)\right){+}{\mathrm{conjugate}}{}\left({{\mathrm{CKM}}}_{{\mathrm{Charm}}{,}{\mathrm{Down}}}\right){}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Down}}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{{\mathrm{Charm}}}_{{A}{,}{l}}{}\left({X}\right)\right){+}{\mathrm{conjugate}}{}\left({{\mathrm{CKM}}}_{{\mathrm{Top}}{,}{\mathrm{Down}}}\right){}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Down}}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{{\mathrm{Top}}}_{{A}{,}{l}}{}\left({X}\right)\right){+}{\mathrm{conjugate}}{}\left({{\mathrm{CKM}}}_{{\mathrm{Up}}{,}{\mathrm{Strange}}}\right){}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Strange}}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{{\mathrm{Up}}}_{{A}{,}{l}}{}\left({X}\right)\right){+}{\mathrm{conjugate}}{}\left({{\mathrm{CKM}}}_{{\mathrm{Charm}}{,}{\mathrm{Strange}}}\right){}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Strange}}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{{\mathrm{Charm}}}_{{A}{,}{l}}{}\left({X}\right)\right){+}{\mathrm{conjugate}}{}\left({{\mathrm{CKM}}}_{{\mathrm{Top}}{,}{\mathrm{Strange}}}\right){}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Strange}}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{{\mathrm{Top}}}_{{A}{,}{l}}{}\left({X}\right)\right){+}{\mathrm{conjugate}}{}\left({{\mathrm{CKM}}}_{{\mathrm{Up}}{,}{\mathrm{Bottom}}}\right){}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Bottom}}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{{\mathrm{Up}}}_{{A}{,}{l}}{}\left({X}\right)\right){+}{\mathrm{conjugate}}{}\left({{\mathrm{CKM}}}_{{\mathrm{Charm}}{,}{\mathrm{Bottom}}}\right){}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Bottom}}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{{\mathrm{Charm}}}_{{A}{,}{l}}{}\left({X}\right)\right){+}{\mathrm{conjugate}}{}\left({{\mathrm{CKM}}}_{{\mathrm{Top}}{,}{\mathrm{Bottom}}}\right){}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Bottom}}}_{{A}{,}{j}}{}\left({X}\right)\right){,}{{\mathrm{Top}}}_{{A}{,}{l}}{}\left({X}\right)\right){+}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Electron}}}_{{j}}{}\left({X}\right)\right){,}{{\mathrm{ElectronNeutrino}}}_{{l}}{}\left({X}\right)\right){+}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Muon}}}_{{j}}{}\left({X}\right)\right){,}{{\mathrm{MuonNeutrino}}}_{{l}}{}\left({X}\right)\right){+}{\mathrm{*}}{}\left({\mathrm{conjugate}}{}\left({{\mathrm{Tauon}}}_{{j}}{}\left({X}\right)\right){,}{{\mathrm{TauonNeutrino}}}_{{l}}{}\left({X}\right)\right){,}{{\mathrm{WMinusField}}}_{{\mathrm{μ}}}{}\left({X}\right)\right)\right){}{{{\mathrm{Dgamma}}}_{{\mathrm{~mu}}}}_{{j}{,}{k}}$ (25)

A process at tree level with a positron and electronic neutrino incoming and the antimuon (antiparticle of the muon) and the muon neutrino outgoing after exchanging a W boson

 > $\mathrm{FeynmanDiagrams}\left(\mathrm{rhs}\left(\right),\mathrm{incoming}=\left[\stackrel{&conjugate0;}{\mathrm{Electron}},\mathrm{ElectronNeutrino}\right],\mathrm{outgoing}=\left[\stackrel{&conjugate0;}{\mathrm{Muon}},\mathrm{MuonNeutrino}\right],\mathrm{numberofloops}=0,\mathrm{diagrams}\right)$

 $\frac{{-}\frac{{i}}{{4}}{}\stackrel{{&conjugate0;}}{{\left({{\mathbit{v}}}_{{\mathbf{e}}}\right)}_{{m}}{}\left(\mathrm{P__1_}\right)}{}{\left({{\mathbit{u}}}_{{\mathrm{ElectronNeutrino}}}\right)}_{{n}}{}\left(\mathrm{P__2_}\right){}{\left({{\mathbit{v}}}_{{\mathbf{\mu }}}\right)}_{{p}}{}\left(\mathrm{P__3_}\right){}\stackrel{{&conjugate0;}}{{\left({{\mathbit{u}}}_{{\mathrm{MuonNeutrino}}}\right)}_{{q}}{}\left(\mathrm{P__4_}\right)}{}\left({-}\frac{{i}{}\sqrt{{2}}{}\mathrm{g__w}{}{\left({{\mathrm{\gamma }}}_{\phantom{{}}\phantom{{\mathrm{\alpha }}}}^{\phantom{{}}{\mathrm{\alpha }}}\right)}_{{q}{,}{p}}}{{2}}{-}\frac{{i}{}\sqrt{{2}}{}\mathrm{g__w}{}{\left({{\mathrm{\gamma }}}_{{5}}\right)}_{{r}{,}{p}}{}{\left({{\mathrm{\gamma }}}_{\phantom{{}}\phantom{{\mathrm{\alpha }}}}^{\phantom{{}}{\mathrm{\alpha }}}\right)}_{{q}{,}{r}}}{{2}}\right){}\left({-}\frac{{i}{}\sqrt{{2}}{}\mathrm{g__w}{}{\left({{\mathrm{\gamma }}}_{\phantom{{}}\phantom{{\mathrm{\nu }}}}^{\phantom{{}}{\mathrm{\nu }}}\right)}_{{m}{,}{n}}}{{2}}{-}\frac{{i}{}\sqrt{{2}}{}\mathrm{g__w}{}{\left({{\mathrm{\gamma }}}_{{5}}\right)}_{{s}{,}{n}}{}{\left({{\mathrm{\gamma }}}_{\phantom{{}}\phantom{{\mathrm{\nu }}}}^{\phantom{{}}{\mathrm{\nu }}}\right)}_{{m}{,}{s}}}{{2}}\right){}\left({-}{{g}}_{{\mathrm{\alpha }}{,}{\mathrm{\nu }}}{+}\frac{\left({\mathrm{P__1}}_{{\mathrm{\nu }}}{+}{\mathrm{P__2}}_{{\mathrm{\nu }}}\right){}\left({\mathrm{P__1}}_{{\mathrm{\alpha }}}{+}{\mathrm{P__2}}_{{\mathrm{\alpha }}}\right)}{{{m}}_{{\mathrm{WMinusField}}}^{{2}}}\right){}{\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)}{{{\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){-}{{m}}_{{\mathrm{WMinusField}}}^{{2}}{+}{i}{}{\mathbf{\epsilon }}\right)}$ (26)

The term ${L}_{\mathrm{HV}}$ of the electroweak Lagrangian contains the interaction between the Higgs and the Z and W bosons

 > $\mathrm{Lagrangian}\left(\mathrm{electroweak},\mathrm{term}={L}_{\mathrm{HV}},\mathrm{interaction}\right)$
 $\mathrm{L__HV}{=}{\mathrm{*}}{}\left(\frac{\mathrm{g__w}{}{\mathrm{HiggsBoson}}{}\left({X}\right)}{{{m}}_{{\mathrm{WField}}}}{+}\frac{{1}}{{4}}{}\frac{{\mathrm{g__w}}^{{2}}{}{\mathrm{^}}{}\left({\mathrm{HiggsBoson}}{}\left({X}\right){,}{2}\right)}{{{m}}_{{\mathrm{WField}}}^{{2}}}{,}{{m}}_{{\mathrm{WField}}}^{{2}}{}{\mathrm{*}}{}\left({{\mathrm{WPlusField}}}_{{\mathrm{μ}}}\right)\right)$