Physics for Maple 2022

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Physics

Maple provides a state-of-the-art environment for algebraic computations in Physics, with emphasis on ensuring that the computational experience is as natural as possible. The theme of the Physics project for Maple 2022 has been the consolidation of the functionality introduced in previous releases, including a significant speed-up across the package and significant enhancements in the areas of Particle Physics, Functional Differentiation in general relativity, and Integral Vector Calculus.

As part of its commitment to providing the best possible computational environment in Physics, Maplesoft launched a Maple Physics: Research and Development website in 2014, which enabled users to download research versions of the package, ask questions, and provide feedback. The results from this accelerated exchange have been incorporated into the Physics package in Maple 2022. The presentation below illustrates both the novelties and the kind of mathematical formulations that can now be performed.

The StandardModel package

Feynman Diagrams

FeynmanIntegral module with 9 new commands

Integral Vector Calculus and Parametrization of curves, surfaces and volumes

Functional Differentiation in General Relativity

CompactDisplay and Typesetting:-Suppress unified

Documentation advanced examples

See Also

The StandardModel package

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 of the model, as well as for Weinberg's angle, the Higgs boson, and the fields and field strengths after breaking symmetries and most of the fields before that. Loading the package sets things to proceed computing with the model.

>	$with (Physics) &colon; with (StandardModel)$

$_______________________________________________________$

$Setting lowercaselatin_is letters to represent Dirac spinor indices$

$Setting lowercaselatin_ah letters to represent SU(3) adjoint representation, (1..8) indices$

$Setting uppercaselatin_ah letters to represent SU(3) fundamental representation, (1..3) indices$

$Setting uppercaselatin_is letters to represent SU(2) adjoint representation, (1..3) indices$

$Setting uppercasegreek letters to represent SU(2) fundamental representation, (1..2) indices$

$_______________________________________________________$

$Defined as the electron, muon and tau leptons and corresponding neutrinos: e_{j}, μ_{j}, τ_{j}, {ElectronNeutrino}_{j}, {MuonNeutrino}_{j}, {TauonNeutrino}_{j}$

$Defined as the up, charm, top, down, strange and bottom quarks: u_{A, j}, c_{A, j}, t_{A, j}, d_{A, j}, s_{A, j}, b_{A, j}$

$Defined as gauge tensors: B_{μ}, 𝔹_{μ, ν}, A_{μ}, 𝔽_{μ, ν}, W_{μ, J}, 𝕎_{μ, ν, J}, {WPlusField}_{μ}, {WPlusFieldStrength}_{μ, ν}, {WMinusField}_{μ}, {WMinusFieldStrength}_{μ, ν}, Z_{μ}, ℤ_{μ, ν}, G_{μ, a}, 𝔾_{μ, ν, a}$

$Defined as Gell-Mann (Glambda), Pauli (Psigma) and Dirac (Dgamma) matrices: λ_{a}, σ_{J}, γ_{μ}$

$Defined as the electric, weak and strong coupling constants: g__e, g__w, g__s$

$Defined as the charge in units of | g__e | for 1) the electron, muon and tauon, 2) the up, charm and top, and 3) the down, strange and bottom: %q__e = −1, %q__u = \frac{2}{3}, %q__d = - \frac{1}{3}$

$Defined as the weak isospin for 1) the electron, muon and tauon, 2) the up, charm and top, 3) the down, strange and bottom, and 4) all the neutrinos: %I__e = - \frac{1}{2}, %I__u = \frac{1}{2}, %I__d = - \frac{1}{2}, %I__n = \frac{1}{2}$

$You can use the active form without the % prefix, or the 'value' command to give the corresponding value to any of the inert representations %q__e, %q__u, %q__d, %I__e, %I__u, %I__d, %I__n$

$_______________________________________________________$

$Default differentiation variables for d_, D_ and dAlembertian are: \{X = (x, y, z, t)\}$

$Minkowski spacetime with signatre (- - - +)$

$_______________________________________________________$

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

(1)

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

The massless fields of the model are the electromagnetic field $A$ , the gluons $G$ and neutrinos $MuonNeutrino, TauonNeutrino$ and $ElectronNeutrino$

>	$Setup (massless)$

$* Partial match of ' massless' against keyword ' masslessfields'$

$_______________________________________________________$

$[masslessfields = \{G, MuonNeutrino, TauonNeutrino, A, ElectronNeutrino\}]$

(2)

The Leptons and Quarks of the model are

>	$StandardModel :- Leptons$

$[e, μ, τ, ElectronNeutrino, MuonNeutrino, TauonNeutrino]$

(3)

>	$StandardModel :- Quarks$

$[u, c, t, d, s, b]$

(4)

The Gauge fields

>	$StandardModel :- GaugeFields$

$[A, 𝔽, B, 𝔹, W, 𝕎, G, 𝔾, WMinusField, WMinusFieldStrength, WPlusField, WPlusFieldStrength, Z, ℤ]$

(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 $I$ to represent the imaginary unit

>	$CompactDisplay (StandardModel :- Leptons (X), StandardModel :- Quarks (X), StandardModel :- GaugeFields (X), HiggsBoson (X), quiet)$

>	$interface (imaginaryunit = i) &colon;$

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

>	$ElectromagneticField [definition]$

$A_{μ} = \sin (WeinbergAngle) W_{μ}^{3} + \cos (WeinbergAngle) B_{μ}$

(6)

>	$map (u \to u [definition], StandardModel :- GaugeFields)$

$[{ElectromagneticField}_{μ} = \sin (WeinbergAngle) {WField}_{μ, ~3} + \cos (WeinbergAngle) {BField}_{μ}, {ElectromagneticFieldStrength}_{μ, ν} = {d_}_{μ} ({ElectromagneticField}_{ν} (X), [X]) - {d_}_{ν} ({ElectromagneticField}_{μ} (X), [X]), {BField}_{μ} = ([\begin{array}{c} {BField}_{1} & {BField}_{2} & {BField}_{3} & {BField}_{4} \end{array}]), {BFieldStrength}_{μ, ν} = {d_}_{μ} ({BField}_{ν} (X), [X]) - {d_}_{ν} ({BField}_{μ} (X), [X]), {WField}_{μ, J} = ([\begin{array}{c} {WField}_{1, 1} & {WField}_{1, 2} & {WField}_{1, 3} \\ {WField}_{2, 1} & {WField}_{2, 2} & {WField}_{2, 3} \\ {WField}_{3, 1} & {WField}_{3, 2} & {WField}_{3, 3} \\ {WField}_{4, 1} & {WField}_{4, 2} & {WField}_{4, 3} \end{array}]), {WFieldStrength}_{μ, ν, J} = {d_}_{μ} ({WField}_{ν, J} (X), [X]) - {d_}_{ν} ({WField}_{μ, J} (X), [X]) + g__w {LeviCivita}_{J, K, L} `*` ({WField}_{μ, K} (X), {WField}_{ν, L} (X)), {GluonField}_{μ, a} = ([\begin{array}{c} {GluonField}_{1, 1} & {GluonField}_{1, 2} & {GluonField}_{1, 3} & {GluonField}_{1, 4} & {GluonField}_{1, 5} & {GluonField}_{1, 6} & {GluonField}_{1, 7} & {GluonField}_{1, 8} \\ {GluonField}_{2, 1} & {GluonField}_{2, 2} & {GluonField}_{2, 3} & {GluonField}_{2, 4} & {GluonField}_{2, 5} & {GluonField}_{2, 6} & {GluonField}_{2, 7} & {GluonField}_{2, 8} \\ {GluonField}_{3, 1} & {GluonField}_{3, 2} & {GluonField}_{3, 3} & {GluonField}_{3, 4} & {GluonField}_{3, 5} & {GluonField}_{3, 6} & {GluonField}_{3, 7} & {GluonField}_{3, 8} \\ {GluonField}_{4, 1} & {GluonField}_{4, 2} & {GluonField}_{4, 3} & {GluonField}_{4, 4} & {GluonField}_{4, 5} & {GluonField}_{4, 6} & {GluonField}_{4, 7} & {GluonField}_{4, 8} \end{array}]), {GluonFieldStrength}_{μ, ν, a} = {d_}_{μ} ({GluonField}_{ν, a} (X), [X]) - {d_}_{ν} ({GluonField}_{μ, a} (X), [X]) + g__s {FSU3}_{a, b, c} `*` ({GluonField}_{μ, b} (X), {GluonField}_{ν, c} (X)), {WMinusField}_{μ} = \frac{1}{2} ({WField}_{μ, ~1} + ⅈ {WField}_{μ, ~2}) \sqrt{2}, {WMinusFieldStrength}_{μ, ν} = {d_}_{μ} ({WMinusField}_{ν} (X), [X]) - {d_}_{ν} ({WMinusField}_{μ} (X), [X]), {WPlusField}_{μ} = \frac{1}{2} ({WField}_{μ, ~1} - ⅈ {WField}_{μ, ~2}) \sqrt{2}, {WPlusFieldStrength}_{μ, ν} = {d_}_{μ} ({WPlusField}_{ν} (X), [X]) - {d_}_{ν} ({WPlusField}_{μ} (X), [X]), {ZField}_{μ} = \cos (WeinbergAngle) {WField}_{μ, ~3} - \sin (WeinbergAngle) {BField}_{μ}, {ZFieldStrength}_{μ, ν} = {d_}_{μ} ({ZField}_{ν} (X), [X]) - {d_}_{ν} ({ZField}_{μ} (X), [X])]$

(7)

Note that the conventions used in the definitions of covariant derivatives (not shown above) and field strength tensors, follow Peskin, S. "An Introduction to Quantum Field Theory", also the Wikipedia, and are not uniform in the literature: the gauge term involving the gluon in the covariant derivative of the quarks, e.g. the Top, $u_{j, A}$ , has a minus sign and the third term in the gluon field strength definition (shown above) has a plus sign:

>	$D_[mu] (Up [A, j] (X)) &colon; % = expand (%)$

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

(8)

>	$GluonFieldStrength [definition]$

${GluonFieldStrength}_{μ, ν, a} = {d_}_{μ} ({GluonField}_{ν, a} (X), [X]) - {d_}_{ν} ({GluonField}_{μ, a} (X), [X]) + g__s {FSU3}_{a, b, c} `*` ({GluonField}_{μ, b} (X), {GluonField}_{ν, c} (X))$

(9)

The convention for the signs in the definitions of $A_{μ}$ and $Z_{μ}$ in (7) also follow Peskin's book and the presentation of the Standard Model in Wikipedia.

The Gell-Mann matrices, that enter gauge terms in the interaction Lagrangian of the StandardModel are represented by $Glambda$ , implemented as a tensor with an SU(3) adjoint representation index, all of whose components are matrices

>	$Glambda []$

${Glambda}_{a} = ([\begin{array}{c} {Glambda}_{1} & {Glambda}_{2} & {Glambda}_{3} & {Glambda}_{4} & {Glambda}_{5} & {Glambda}_{6} & {Glambda}_{7} & {Glambda}_{8} \end{array}])$

(10)

>	$seq (Glambda [a, matrix], a = 1 .. 8)$

$λ_{1} = [\begin{array}{c} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{array}], λ_{2} = [\begin{array}{c} 0 & −ⅈ & 0 \\ ⅈ & 0 & 0 \\ 0 & 0 & 0 \end{array}], λ_{3} = [\begin{array}{c} 1 & 0 & 0 \\ 0 & −1 & 0 \\ 0 & 0 & 0 \end{array}], λ_{4} = [\begin{array}{c} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{array}], λ_{5} = [\begin{array}{c} 0 & 0 & −ⅈ \\ 0 & 0 & 0 \\ ⅈ & 0 & 0 \end{array}], λ_{6} = [\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{array}], λ_{7} = [\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & −ⅈ \\ 0 & ⅈ & 0 \end{array}], λ_{8} = [\begin{array}{c} \frac{\sqrt{3}}{3} & 0 & 0 \\ 0 & \frac{\sqrt{3}}{3} & 0 \\ 0 & 0 & - \frac{2 \sqrt{3}}{3} \end{array}]$

(11)

These matrices satisfy a SU(3) algebra

>	$Library :- DefaultAlgebraRules (Glambda)$

$%Commutator ({Glambda}_{b}, {Glambda}_{c}) = 2 ⅈ {FSU3}_{a, b, c} {Glambda}_{a}$

(12)

The structure constants ${FSU3}_{a, b, c}$ entering (12) and interaction Lagrangian terms of the StandardModel form a three-dimensional array of 8 x 8 matrices represented by the command $FSU3$ . 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 $matrix$ , e.g.

>	$FSU3 [1, b, c, matrix]$

${FSU3}_{1, b, c} = ([\begin{array}{c} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & - 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{2} & 0 \\ 0 & 0 & 0 & 0 & 0 & - \frac{1}{2} & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{1}{2} & 0 & 0 & 0 \\ 0 & 0 & 0 & - \frac{1}{2} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array}])$

(13)

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

>	$TensorArray (FSU3 [a, b, c], explore)$

${FSU3}_{a, b, c} (ordering of free indices = [a, b, c])$

(14)

$Index 1$

Value of Index 1

The tensorial equation for the Gell-Mann matrices

$%Commutator ({Glambda}_{b}, {Glambda}_{c}) = 2 ⅈ {FSU3}_{a, b, c} {Glambda}_{a}$

(15)

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

>	$SumOverRepeatedIndices ()$

$%Commutator ({Glambda}_{b}, {Glambda}_{c}) = 2 ⅈ ({FSU3}_{1, b, c} {Glambda}_{1} + {FSU3}_{2, b, c} {Glambda}_{2} + {FSU3}_{3, b, c} {Glambda}_{3} + {FSU3}_{4, b, c} {Glambda}_{4} + {FSU3}_{5, b, c} {Glambda}_{5} + {FSU3}_{6, b, c} {Glambda}_{6} + {FSU3}_{7, b, c} {Glambda}_{7} + {FSU3}_{8, b, c} {Glambda}_{8})$

(16)

>	$eval (, [b = 4, c = 5])$

$%Commutator ({Glambda}_{4}, {Glambda}_{5}) = 2 ⅈ (\frac{1}{2} {Glambda}_{3} + \frac{1}{2} \sqrt{3} {Glambda}_{8})$

(17)

Activating the left-hand side,

>	$value ()$

$2 ⅈ {FSU3}_{4, 5, a} λ_{a} = 2 ⅈ (\frac{λ_{3}}{2} + \frac{\sqrt{3} λ_{8}}{2})$

(18)

>	$expand (SumOverRepeatedIndices ())$

$ⅈ λ_{3} + ⅈ \sqrt{3} λ_{8} = ⅈ λ_{3} + ⅈ \sqrt{3} λ_{8}$

(19)

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

>	$TensorArray ()$

$[\begin{array}{c} %Commutator ({Glambda}_{1}, {Glambda}_{1}) = 0 & %Commutator ({Glambda}_{1}, {Glambda}_{2}) = 2 I {Glambda}_{3} & %Commutator ({Glambda}_{1}, {Glambda}_{3}) = - 2 I {Glambda}_{2} & %Commutator ({Glambda}_{1}, {Glambda}_{4}) = I {Glambda}_{7} & %Commutator ({Glambda}_{1}, {Glambda}_{5}) = - I {Glambda}_{6} & %Commutator ({Glambda}_{1}, {Glambda}_{6}) = I {Glambda}_{5} & %Commutator ({Glambda}_{1}, {Glambda}_{7}) = - I {Glambda}_{4} & %Commutator ({Glambda}_{1}, {Glambda}_{8}) = 0 \\ %Commutator ({Glambda}_{2}, {Glambda}_{1}) = - 2 I {Glambda}_{3} & %Commutator ({Glambda}_{2}, {Glambda}_{2}) = 0 & %Commutator ({Glambda}_{2}, {Glambda}_{3}) = 2 I {Glambda}_{1} & %Commutator ({Glambda}_{2}, {Glambda}_{4}) = I {Glambda}_{6} & %Commutator ({Glambda}_{2}, {Glambda}_{5}) = I {Glambda}_{7} & %Commutator ({Glambda}_{2}, {Glambda}_{6}) = - I {Glambda}_{4} & %Commutator ({Glambda}_{2}, {Glambda}_{7}) = - I {Glambda}_{5} & %Commutator ({Glambda}_{2}, {Glambda}_{8}) = 0 \\ %Commutator ({Glambda}_{3}, {Glambda}_{1}) = 2 I {Glambda}_{2} & %Commutator ({Glambda}_{3}, {Glambda}_{2}) = - 2 I {Glambda}_{1} & %Commutator ({Glambda}_{3}, {Glambda}_{3}) = 0 & %Commutator ({Glambda}_{3}, {Glambda}_{4}) = I {Glambda}_{5} & %Commutator ({Glambda}_{3}, {Glambda}_{5}) = - I {Glambda}_{4} & %Commutator ({Glambda}_{3}, {Glambda}_{6}) = - I {Glambda}_{7} & %Commutator ({Glambda}_{3}, {Glambda}_{7}) = I {Glambda}_{6} & %Commutator ({Glambda}_{3}, {Glambda}_{8}) = 0 \\ %Commutator ({Glambda}_{4}, {Glambda}_{1}) = - I {Glambda}_{7} & %Commutator ({Glambda}_{4}, {Glambda}_{2}) = - I {Glambda}_{6} & %Commutator ({Glambda}_{4}, {Glambda}_{3}) = - I {Glambda}_{5} & %Commutator ({Glambda}_{4}, {Glambda}_{4}) = 0 & %Commutator ({Glambda}_{4}, {Glambda}_{5}) = I (\sqrt{3} {Glambda}_{8} + {Glambda}_{3}) & %Commutator ({Glambda}_{4}, {Glambda}_{6}) = I {Glambda}_{2} & %Commutator ({Glambda}_{4}, {Glambda}_{7}) = I {Glambda}_{1} & %Commutator ({Glambda}_{4}, {Glambda}_{8}) = - I \sqrt{3} {Glambda}_{5} \\ %Commutator ({Glambda}_{5}, {Glambda}_{1}) = I {Glambda}_{6} & %Commutator ({Glambda}_{5}, {Glambda}_{2}) = - I {Glambda}_{7} & %Commutator ({Glambda}_{5}, {Glambda}_{3}) = I {Glambda}_{4} & %Commutator ({Glambda}_{5}, {Glambda}_{4}) = - I (\sqrt{3} {Glambda}_{8} + {Glambda}_{3}) & %Commutator ({Glambda}_{5}, {Glambda}_{5}) = 0 & %Commutator ({Glambda}_{5}, {Glambda}_{6}) = - I {Glambda}_{1} & %Commutator ({Glambda}_{5}, {Glambda}_{7}) = I {Glambda}_{2} & %Commutator ({Glambda}_{5}, {Glambda}_{8}) = I \sqrt{3} {Glambda}_{4} \\ %Commutator ({Glambda}_{6}, {Glambda}_{1}) = - I {Glambda}_{5} & %Commutator ({Glambda}_{6}, {Glambda}_{2}) = I {Glambda}_{4} & %Commutator ({Glambda}_{6}, {Glambda}_{3}) = I {Glambda}_{7} & %Commutator ({Glambda}_{6}, {Glambda}_{4}) = - I {Glambda}_{2} & %Commutator ({Glambda}_{6}, {Glambda}_{5}) = I {Glambda}_{1} & %Commutator ({Glambda}_{6}, {Glambda}_{6}) = 0 & %Commutator ({Glambda}_{6}, {Glambda}_{7}) = I (- {Glambda}_{3} + \sqrt{3} {Glambda}_{8}) & %Commutator ({Glambda}_{6}, {Glambda}_{8}) = - I \sqrt{3} {Glambda}_{7} \\ %Commutator ({Glambda}_{7}, {Glambda}_{1}) = I {Glambda}_{4} & %Commutator ({Glambda}_{7}, {Glambda}_{2}) = I {Glambda}_{5} & %Commutator ({Glambda}_{7}, {Glambda}_{3}) = - I {Glambda}_{6} & %Commutator ({Glambda}_{7}, {Glambda}_{4}) = - I {Glambda}_{1} & %Commutator ({Glambda}_{7}, {Glambda}_{5}) = - I {Glambda}_{2} & %Commutator ({Glambda}_{7}, {Glambda}_{6}) = - I (- {Glambda}_{3} + \sqrt{3} {Glambda}_{8}) & %Commutator ({Glambda}_{7}, {Glambda}_{7}) = 0 & %Commutator ({Glambda}_{7}, {Glambda}_{8}) = I \sqrt{3} {Glambda}_{6} \\ %Commutator ({Glambda}_{8}, {Glambda}_{1}) = 0 & %Commutator ({Glambda}_{8}, {Glambda}_{2}) = 0 & %Commutator ({Glambda}_{8}, {Glambda}_{3}) = 0 & %Commutator ({Glambda}_{8}, {Glambda}_{4}) = I \sqrt{3} {Glambda}_{5} & %Commutator ({Glambda}_{8}, {Glambda}_{5}) = - I \sqrt{3} {Glambda}_{4} & %Commutator ({Glambda}_{8}, {Glambda}_{6}) = I \sqrt{3} {Glambda}_{7} & %Commutator ({Glambda}_{8}, {Glambda}_{7}) = - I \sqrt{3} {Glambda}_{6} & %Commutator ({Glambda}_{8}, {Glambda}_{8}) = 0 \end{array}]$

(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

>	$Setup (anticommutativeprefix = \{f__L, f__Q\})$

$[anticommutativeprefix = \{f__L, f__Q\}]$

(21)

>	$CompactDisplay ((f__L, f__Q) (X))$

$f__L (x, y, z, t) will now be displayed as f__L$

$f__Q (x, y, z, t) will now be displayed as f__Q$

(22)

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 $lowercaselatin_is$ and $uppercaselatin_ah$ letters. Gluons are tensors with one spacetime and one SU(3) adjoint representation index (1..8), respectively represented by $greek$ and $lowercaselatin_ah$ letters, and $g__s$ is the QCD coupling constant.

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

>	$L__QCD ≔ L__QG + L__GG$

$L__QCD ≔ L__QG + L__GG$

(23)

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

>	$L__QG ≔ \frac{g__s}{2} \cdot Dgamma [mu] [k, j] \cdot GluonField [mu, a] (X) \cdot Glambda [a] [A, B] \cdot %add (conjugate (f__Q [A, k] (X)) \cdot f__Q [B, j] (X), f__Q = StandardModel :- Quarks)$

(24)

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

$L__GG ≔ - g__s \cdot FSU3 [a, b, c] \cdot (d_[mu] (GluonField [nu, a] (X), [X]) \cdot GluonField [~ mu, b] (X) \cdot GluonField [~ nu, c] (X) + \frac{g__s}{4} \cdot FSU3 [e, d, c] \cdot GluonField [mu, a] (X) \cdot GluonField [lambda, b] (X) \cdot GluonField [~ mu, e] (X) \cdot GluonField [~ lambda, d] (X))$

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

(25)

From where

$L__QCD$

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

(26)

>	$L__QCD ≔ value (L__QCD)$

$\frac{1}{2} g__s `*` (`*` (conjugate ({Up}_{A, k} (X)), {Up}_{B, j} (X)) + `*` (conjugate ({Charm}_{A, k} (X)), {Charm}_{B, j} (X)) + `*` (conjugate ({Top}_{A, k} (X)), {Top}_{B, j} (X)) + `*` (conjugate ({Down}_{A, k} (X)), {Down}_{B, j} (X)) + `*` (conjugate ({Strange}_{A, k} (X)), {Strange}_{B, j} (X)) + `*` (conjugate ({Bottom}_{A, k} (X)), {Bottom}_{B, j} (X)), {GluonField}_{μ, a} (X), {Glambda}_{a}_{A, B}) {Dgamma}_{~mu}_{k, j} - g__s {FSU3}_{a, b, c} (`*` ({d_}_{μ} ({GluonField}_{ν, a} (X), [X]), {GluonField}_{~mu, b} (X), {GluonField}_{~nu, c} (X)) - \frac{1}{4} g__s {FSU3}_{c, d, e} `*` ({GluonField}_{μ, a} (X), {GluonField}_{λ, b} (X), {GluonField}_{~mu, e} (X), {GluonField}_{~lambda, d} (X)))$

(27)

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

>	$L__uG ≔ op (1, expand (value (L__QCD)))$

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

(28)

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

>	$FeynmanDiagrams (L__uG, incomingparticles = [Up, conjugate (Up)], outgoingparticles = [Up, conjugate (Up)], numberofloops = 0, diagrams)$

$- \frac{ⅈ {(u_{u})}_{C, l} (P__1_) \overset{&conjugate0;}{{(v_{u})}_{E, m} (P__2_)} \overset{&conjugate0;}{{(u_{u})}_{F, n} (P__3_)} {(v_{u})}_{G, p} (P__4_) {g__s}^{2} {(γ_{}^{α})}_{n, p} {(γ_{}^{ν})}_{m, l} g_{α, ν} δ_{b, c} δ (- {P__3}_{~beta} - {P__4}_{~beta} + {P__1}_{~beta} + {P__2}_{~beta}) {(λ_{c})}_{F, G} {(λ_{b})}_{E, C}}{16 π^{2} (({P__1}_{κ} + {P__2}_{κ}) ({P__1}_{~kappa} + {P__2}_{~kappa}) + ⅈ ε)} + \frac{ⅈ {(u_{u})}_{C, l} (P__1_) \overset{&conjugate0;}{{(v_{u})}_{E, m} (P__2_)} \overset{&conjugate0;}{{(u_{u})}_{F, n} (P__3_)} {(v_{u})}_{G, p} (P__4_) {g__s}^{2} {(γ_{}^{α})}_{m, p} {(γ_{}^{ν})}_{n, l} g_{α, ν} δ_{b, c} δ (- {P__3}_{~beta} - {P__4}_{~beta} + {P__1}_{~beta} + {P__2}_{~beta}) {(λ_{c})}_{E, G} {(λ_{b})}_{F, C}}{16 π^{2} (({P__1}_{~kappa} - {P__3}_{~kappa}) ({P__1}_{κ} - {P__3}_{κ}) + ⅈ ε)}$

(29)

>	$L__GGGG ≔ op (- 1, expand (L__QCD))$

$\frac{1}{4} {g__s}^{2} {FSU3}_{a, b, c} {FSU3}_{c, d, e} `*` ({GluonField}_{λ, b} (X), {GluonField}_{μ, a} (X), {GluonField}_{~lambda, d} (X), {GluonField}_{~mu, e} (X))$

(30)

The amplitude for the process with two incoming and two outgoing gluons

>	$FeynmanDiagrams (L__GGGG, incomingparticles = [GluonField, GluonField], outgoingparticles = [GluonField, GluonField], numberofloops = 0, diagrams)$

$\frac{\frac{ⅈ}{16} {g__s}^{2} δ (- {P__3}_{~sigma} - {P__4}_{~sigma} + {P__1}_{~sigma} + {P__2}_{~sigma}) {(ϵ_{G})}_{ν, f} (P__1_) {(ϵ_{G})}_{α, g} (P__2_) \overset{&conjugate0;}{{(ϵ_{G})}_{β, h} (P__3_)} \overset{&conjugate0;}{{(ϵ_{G})}_{κ, a1} (P__4_)} (({FSU3}_{c, g, h} {FSU3}_{a1, c, f} - {FSU3}_{a1, c, h} {FSU3}_{c, f, g}) g_{}^{β, ν} g_{}^{α, κ} + g_{}^{κ, ν} ({FSU3}_{c, f, h} {FSU3}_{a1, c, g} + {FSU3}_{a1, c, h} {FSU3}_{c, f, g}) g_{}^{α, β} + (- {FSU3}_{c, g, h} {FSU3}_{a1, c, f} - {FSU3}_{c, f, h} {FSU3}_{a1, c, g}) g_{}^{α, ν} g_{}^{β, κ})}{π^{2} \sqrt{E__1 E__2 E__3 E__4}}$

(31)

The Electroweak sector of the Standard Model and its interaction Lagrangian

The computation of scattering amplitudes is performed with the model after symmetry breaking. The electro-weak interaction before symmetry breaking, 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

>	$L__EW ≔ L__g + L__f + L__h + L__y$

$L__EW ≔ L__g + L__f + L__h + L__y$

(32)

Out of these four, in the Maple 2022.0 implementation of $StandardModel$ it is possible to represent the first term, $L__g$ , the kinetic term for the $W_{μ, J}$ and $B_{μ}$ vector bosons

>	$L__g ≔ - \frac{1}{4} \cdot (WFieldStrength {[μ, ν, J]}^{2} + BFieldStrength {[μ, ν]}^{2})$

$L__g ≔ - \frac{𝕎_{μ, ν, J} 𝕎_{J}^{μ, ν}}{4} - \frac{𝔹_{μ, ν} 𝔹_{}^{μ, ν}}{4}$

(33)

Introducing the definitions of these tensors we have

>	$BFieldStrength [definition], WFieldStrength [definition]$

$𝔹_{μ, ν} = \partial_{μ} ({BField}_{ν} (X)) - \partial_{ν} ({BField}_{μ} (X)), 𝕎_{μ, ν, J} = \partial_{μ} ({WField}_{ν, J} (X)) - \partial_{ν} ({WField}_{μ, J} (X)) + g__w ε_{J, K, L} {WField}_{μ, K} (X) {WField}_{ν, L} (X)$

(34)

>	$L__g ≔ SubstituteTensor (, L__g)$

$- \frac{1}{4} `*` ({d_}_{μ} ({WField}_{ν, J} (X), [X]) - {d_}_{ν} ({WField}_{μ, J} (X), [X]) + g__w {LeviCivita}_{J, K, L} `*` ({WField}_{μ, K} (X), {WField}_{ν, L} (X)), {d_}_{~mu} ({WField}_{~nu, J} (X), [X]) - {d_}_{~nu} ({WField}_{~mu, J} (X), [X]) + g__w {LeviCivita}_{J, M, N} `*` ({WField}_{~mu, M} (X), {WField}_{~nu, N} (X))) - \frac{1}{4} `*` ({d_}_{μ} ({BField}_{ν} (X), [X]) - {d_}_{ν} ({BField}_{μ} (X), [X]), {d_}_{~mu} ({BField}_{~nu} (X), [X]) - {d_}_{~nu} ({BField}_{~mu} (X), [X]))$

(35)

The $L__f$ term is the kinetic term for the fermions of the model before symmetry breaking, and their interaction with the gauge bosons $W_{μ, K}$ and $B_{μ}$ is through the covariant derivative. Note that the electron field $e_{j}$ , 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. So the terms you get expanding the covariant derivatives of the leptons and quarks, e.g.

>	$D_[mu] (Electron [j] (X)) &colon; % = expand (%)$

${D_}_{μ} ({Electron}_{j} (X), [X]) = {d_}_{μ} ({Electron}_{j} (X), [X]) + ⅈ g__e `*` ({Electron}_{j} (X), {ElectromagneticField}_{μ} (X))$

(36)

>	$D_[mu] (Up [A, j] (X)) &colon; % = expand (%)$

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

(37)

are of no use for constructing the Lagrangian before symmetry breaking. The $L__h$ term involves the Higgs boson before symmetry breaking (here too, the HiggsBoson field implemented in the StandardModel in Maple 2022 is the Higgs after symmetry breaking) and the $L__y$ formulates the Yukawa interaction with the fermions.

After symmetry breaking

For the purpose of computing scattering amplitudes, the formulation of the interaction Lagrangian after symmetry breaking is more relevant; this one is given by

>	$L__EW ≔ L__K + L__N + L__C + L__H + L__HV + L__WWV + L__WWVV + L__Y;$

$L__EW ≔ L__K + L__N + L__C + L__H + L__HV + L__WWV + L__WWVV + L__Y;$

(38)

where we use the notation shown in the Wikipedia page for the weak interaction. As illustration, we compute here the $L__K$ and $L__N$ 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 $A_{μ}$ and $Z_{μ}$ .

Following the Wikipedia page mentioned, the kinetic term $L__K$ is given by

$L__K ≔ - \frac{1}{4} ElectromagneticFieldStrength {[mu, nu]}^{2} - \frac{1}{2} WPlusFieldStrength [mu, nu] \cdot WMinusFieldStrength [mu, nu] + \frac{1}{2} \cdot m {[WField]}^{2} \cdot WPlusField [mu] \cdot WMinusField [mu] - \frac{1}{4} ZFieldStrength {[mu, nu]}^{2} + \frac{1}{2} m {[ZField]}^{2} \cdot ZField {[mu]}^{2} + \frac{1}{2} d_[mu] {(HiggsBoson (X))}^{2} - \frac{m {[HiggsBoson]}^{2}}{2} \cdot HiggsBoson {(X)}^{2} + %add (conjugate (f__L [j] (X)) \cdot (Dgamma [mu] [j, k] \cdot i \cdot d_[mu] (f__L [k] (X)) - m [f__L] \cdot f__L [j] (X)), f__L = StandardModel :- Leptons [1 .. 3]) + %add (conjugate (f__L [j] (X)) \cdot (Dgamma [mu] [j, k] \cdot i \cdot d_[mu] (f__L [k] (X))), f__L = StandardModel :- Leptons [4 .. 6]) + %add (conjugate (f__Q [A, j] (X)) \cdot (Dgamma [mu] [j, k] \cdot i \cdot d_[mu] (f__Q [A, k] (X)) - m [f__Q] \cdot f__Q [A, j] (X)), f__Q = StandardModel :- Quarks)$

(39)

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

>	$value ()$

$- \frac{1}{4} `*` ({ElectromagneticFieldStrength}_{μ, ν}, {ElectromagneticFieldStrength}_{~mu, ~nu}) - \frac{1}{2} `*` ({WPlusFieldStrength}_{μ, ν}, {WMinusFieldStrength}_{~mu, ~nu}) + \frac{1}{2} m_{WField}^{2} `*` ({WPlusField}_{μ}, {WMinusField}_{~mu}) - \frac{1}{4} `*` ({ZFieldStrength}_{μ, ν}, {ZFieldStrength}_{~mu, ~nu}) + \frac{1}{2} m_{ZField}^{2} `*` ({ZField}_{μ}, {ZField}_{~mu}) + \frac{1}{2} `*` ({d_}_{μ} (HiggsBoson (X), [X]), {d_}_{~mu} (HiggsBoson (X), [X])) - \frac{1}{2} m_{HiggsBoson}^{2} `^` (HiggsBoson (X), 2) + `*` (conjugate ({Electron}_{j} (X)), ⅈ {d_}_{μ} ({Electron}_{k} (X), [X]) {Dgamma}_{~mu}_{j, k} - m_{Electron} {Electron}_{j} (X)) + `*` (conjugate ({Muon}_{j} (X)), ⅈ {d_}_{μ} ({Muon}_{k} (X), [X]) {Dgamma}_{~mu}_{j, k} - m_{Muon} {Muon}_{j} (X)) + `*` (conjugate ({Tauon}_{j} (X)), ⅈ {d_}_{μ} ({Tauon}_{k} (X), [X]) {Dgamma}_{~mu}_{j, k} - m_{Tauon} {Tauon}_{j} (X)) + ⅈ `*` (conjugate ({ElectronNeutrino}_{j} (X)), {d_}_{μ} ({ElectronNeutrino}_{k} (X), [X])) {Dgamma}_{~mu}_{j, k} + ⅈ `*` (conjugate ({MuonNeutrino}_{j} (X)), {d_}_{μ} ({MuonNeutrino}_{k} (X), [X])) {Dgamma}_{~mu}_{j, k} + ⅈ `*` (conjugate ({TauonNeutrino}_{j} (X)), {d_}_{μ} ({TauonNeutrino}_{k} (X), [X])) {Dgamma}_{~mu}_{j, k} + `*` (conjugate ({Up}_{A, j} (X)), ⅈ {d_}_{μ} ({Up}_{A, k} (X), [X]) {Dgamma}_{~mu}_{j, k} - m_{Up} {Up}_{A, j} (X)) + `*` (conjugate ({Charm}_{A, j} (X)), ⅈ {d_}_{μ} ({Charm}_{A, k} (X), [X]) {Dgamma}_{~mu}_{j, k} - m_{Charm} {Charm}_{A, j} (X)) + `*` (conjugate ({Top}_{A, j} (X)), ⅈ {d_}_{μ} ({Top}_{A, k} (X), [X]) {Dgamma}_{~mu}_{j, k} - m_{Top} {Top}_{A, j} (X)) + `*` (conjugate ({Down}_{A, j} (X)), ⅈ {d_}_{μ} ({Down}_{A, k} (X), [X]) {Dgamma}_{~mu}_{j, k} - m_{Down} {Down}_{A, j} (X)) + `*` (conjugate ({Strange}_{A, j} (X)), ⅈ {d_}_{μ} ({Strange}_{A, k} (X), [X]) {Dgamma}_{~mu}_{j, k} - m_{Strange} {Strange}_{A, j} (X)) + `*` (conjugate ({Bottom}_{A, j} (X)), ⅈ {d_}_{μ} ({Bottom}_{A, k} (X), [X]) {Dgamma}_{~mu}_{j, k} - m_{Bottom} {Bottom}_{A, j} (X))$

(40)

Introducing the definition of the field strengths $𝔽_{μ, ν}$ , ${WPlusFieldStrength}_{μ, ν}$ , ${WMinusFieldStrength}_{μ, ν}$ and $ℤ_{μ, ν}$

>	$ElectromagneticFieldStrength [definition]$

${ElectromagneticFieldStrength}_{μ, ν} = {d_}_{μ} ({ElectromagneticField}_{ν} (X), [X]) - {d_}_{ν} ({ElectromagneticField}_{μ} (X), [X])$

(41)

>	$WPlusFieldStrength [definition]$

${WPlusFieldStrength}_{μ, ν} = {d_}_{μ} ({WPlusField}_{ν} (X), [X]) - {d_}_{ν} ({WPlusField}_{μ} (X), [X])$

(42)

>	$WMinusFieldStrength [definition]$

${WMinusFieldStrength}_{μ, ν} = {d_}_{μ} ({WMinusField}_{ν} (X), [X]) - {d_}_{ν} ({WMinusField}_{μ} (X), [X])$

(43)

>	$ZFieldStrength [definition]$

${ZFieldStrength}_{μ, ν} = {d_}_{μ} ({ZField}_{ν} (X), [X]) - {d_}_{ν} ({ZField}_{μ} (X), [X])$

(44)

>	$L__K ≔ SubstituteTensor (,,,,)$

$- \frac{1}{4} `*` ({d_}_{μ} ({ElectromagneticField}_{ν} (X), [X]) - {d_}_{ν} ({ElectromagneticField}_{μ} (X), [X]), {d_}_{~mu} ({ElectromagneticField}_{~nu} (X), [X]) - {d_}_{~nu} ({ElectromagneticField}_{~mu} (X), [X])) - \frac{1}{2} `*` ({d_}_{μ} ({WPlusField}_{ν} (X), [X]) - {d_}_{ν} ({WPlusField}_{μ} (X), [X]), {d_}_{~mu} ({WMinusField}_{~nu} (X), [X]) - {d_}_{~nu} ({WMinusField}_{~mu} (X), [X])) + \frac{1}{2} m_{WField}^{2} `*` ({WPlusField}_{μ}, {WMinusField}_{~mu}) - \frac{1}{4} `*` ({d_}_{μ} ({ZField}_{ν} (X), [X]) - {d_}_{ν} ({ZField}_{μ} (X), [X]), {d_}_{~mu} ({ZField}_{~nu} (X), [X]) - {d_}_{~nu} ({ZField}_{~mu} (X), [X])) + \frac{1}{2} m_{ZField}^{2} `*` ({ZField}_{μ}, {ZField}_{~mu}) + \frac{1}{2} `*` ({d_}_{μ} (HiggsBoson (X), [X]), {d_}_{~mu} (HiggsBoson (X), [X])) - \frac{1}{2} m_{HiggsBoson}^{2} `^` (HiggsBoson (X), 2) + `*` (conjugate ({Electron}_{j} (X)), ⅈ {d_}_{μ} ({Electron}_{k} (X), [X]) {Dgamma}_{~mu}_{j, k} - m_{Electron} {Electron}_{j} (X)) + `*` (conjugate ({Muon}_{j} (X)), ⅈ {d_}_{μ} ({Muon}_{k} (X), [X]) {Dgamma}_{~mu}_{j, k} - m_{Muon} {Muon}_{j} (X)) + `*` (conjugate ({Tauon}_{j} (X)), ⅈ {d_}_{μ} ({Tauon}_{k} (X), [X]) {Dgamma}_{~mu}_{j, k} - m_{Tauon} {Tauon}_{j} (X)) + ⅈ `*` (conjugate ({ElectronNeutrino}_{j} (X)), {d_}_{μ} ({ElectronNeutrino}_{k} (X), [X])) {Dgamma}_{~mu}_{j, k} + ⅈ `*` (conjugate ({MuonNeutrino}_{j} (X)), {d_}_{μ} ({MuonNeutrino}_{k} (X), [X])) {Dgamma}_{~mu}_{j, k} + ⅈ `*` (conjugate ({TauonNeutrino}_{j} (X)), {d_}_{μ} ({TauonNeutrino}_{k} (X), [X])) {Dgamma}_{~mu}_{j, k} + `*` (conjugate ({Up}_{A, j} (X)), ⅈ {d_}_{μ} ({Up}_{A, k} (X), [X]) {Dgamma}_{~mu}_{j, k} - m_{Up} {Up}_{A, j} (X)) + `*` (conjugate ({Charm}_{A, j} (X)), ⅈ {d_}_{μ} ({Charm}_{A, k} (X), [X]) {Dgamma}_{~mu}_{j, k} - m_{Charm} {Charm}_{A, j} (X)) + `*` (conjugate ({Top}_{A, j} (X)), ⅈ {d_}_{μ} ({Top}_{A, k} (X), [X]) {Dgamma}_{~mu}_{j, k} - m_{Top} {Top}_{A, j} (X)) + `*` (conjugate ({Down}_{A, j} (X)), ⅈ {d_}_{μ} ({Down}_{A, k} (X), [X]) {Dgamma}_{~mu}_{j, k} - m_{Down} {Down}_{A, j} (X)) + `*` (conjugate ({Strange}_{A, j} (X)), ⅈ {d_}_{μ} ({Strange}_{A, k} (X), [X]) {Dgamma}_{~mu}_{j, k} - m_{Strange} {Strange}_{A, j} (X)) + `*` (conjugate ({Bottom}_{A, j} (X)), ⅈ {d_}_{μ} ({Bottom}_{A, k} (X), [X]) {Dgamma}_{~mu}_{j, k} - m_{Bottom} {Bottom}_{A, j} (X))$

(45)

The neutral current Lagrangian containing the interactions between fermions and the gauge bosons $A_{μ}$ and $Z_{μ}$ is expressed in terms of the electromagnetic and weak currents $J__E, μ$ and $J__W,μ$ as

>	$L__N ≔ g__e \cdot J [E, μ] \cdot ElectromagneticField [μ] (X) + \frac{g__w}{\cos (WeinbergAngle)} \cdot (J [W, mu] - \sin {(WeinbergAngle)}^{2} \cdot J [E, mu]) \cdot ZField [mu] (X)$

$g__e J_{E, μ} {ElectromagneticField}_{μ} (X) + \frac{g__w (J_{W, μ} - {\sin (WeinbergAngle)}^{2} J_{E, μ}) {ZField}_{μ} (X)}{\cos (WeinbergAngle)}$

(46)

In turn, these currents are expressed as

$J [E, μ] ≔ %q__e \cdot Dgamma [mu] [k, j] \cdot %add (conjugate (f__L [k] (X)) \cdot f__L [j] (X), f__L = [Electron, Muon, Tauon]) + %q__u \cdot Dgamma [mu] [k, j] \cdot %add (conjugate (f__Q [A, k] (X)) \cdot f__Q [A, j] (X), f__Q = [Up, Charm, Top]) + %q__d \cdot Dgamma [mu] [k, j] \cdot %add (conjugate (f__Q [A, k] (X)) \cdot f__Q [A, j] (X), f__Q = [Down, Strange, Bottom])$

$%q__e {Dgamma}_{μ}_{k, j} %add (`*` (conjugate ({f__L}_{k} (X)), {f__L}_{j} (X)), f__L = [Electron, Muon, Tauon]) + %q__u {Dgamma}_{μ}_{k, j} %add (`*` (conjugate ({f__Q}_{A, k} (X)), {f__Q}_{A, j} (X)), f__Q = [Up, Charm, Top]) + %q__d {Dgamma}_{μ}_{k, j} %add (`*` (conjugate ({f__Q}_{A, k} (X)), {f__Q}_{A, j} (X)), f__Q = [Down, Strange, Bottom])$

(47)

To activate only the sum over the different kinds of fermions,

>	$J [E, mu] ≔ eval (, %add = add)$

$%q__e {Dgamma}_{μ}_{k, j} (`*` (conjugate ({Electron}_{k} (X)), {Electron}_{j} (X)) + `*` (conjugate ({Muon}_{k} (X)), {Muon}_{j} (X)) + `*` (conjugate ({Tauon}_{k} (X)), {Tauon}_{j} (X))) + %q__u {Dgamma}_{μ}_{k, j} (`*` (conjugate ({Up}_{A, k} (X)), {Up}_{A, j} (X)) + `*` (conjugate ({Charm}_{A, k} (X)), {Charm}_{A, j} (X)) + `*` (conjugate ({Top}_{A, k} (X)), {Top}_{A, j} (X))) + %q__d {Dgamma}_{μ}_{k, j} (`*` (conjugate ({Down}_{A, k} (X)), {Down}_{A, j} (X)) + `*` (conjugate ({Strange}_{A, k} (X)), {Strange}_{A, j} (X)) + `*` (conjugate ({Bottom}_{A, k} (X)), {Bottom}_{A, j} (X)))$

(48)

To activate the sums and also the inert representations of the different charges you can use the value command

>	$J [E, mu] ≔ value ()$

$- {Dgamma}_{μ}_{k, j} (`*` (conjugate ({Electron}_{k} (X)), {Electron}_{j} (X)) + `*` (conjugate ({Muon}_{k} (X)), {Muon}_{j} (X)) + `*` (conjugate ({Tauon}_{k} (X)), {Tauon}_{j} (X))) + \frac{2}{3} {Dgamma}_{μ}_{k, j} (`*` (conjugate ({Up}_{A, k} (X)), {Up}_{A, j} (X)) + `*` (conjugate ({Charm}_{A, k} (X)), {Charm}_{A, j} (X)) + `*` (conjugate ({Top}_{A, k} (X)), {Top}_{A, j} (X))) - \frac{1}{3} {Dgamma}_{μ}_{k, j} (`*` (conjugate ({Down}_{A, k} (X)), {Down}_{A, j} (X)) + `*` (conjugate ({Strange}_{A, k} (X)), {Strange}_{A, j} (X)) + `*` (conjugate ({Bottom}_{A, k} (X)), {Bottom}_{A, j} (X)))$

(49)

For the weak current, from the Wikipedia reference mentioned,

$J [W, μ] ≔ Dgamma [mu] [k, j] \cdot (KroneckerDelta [j, l] - Dgamma [5] [j, l]) \cdot (%I__e \cdot %add (conjugate (f__L [k] (X)) \cdot f__L [l] (X), f__L = StandardModel :- Leptons [1 .. 3]) + %I__n \cdot %add (conjugate (f__L [k] (X)) \cdot f__L [l] (X), f__L = StandardModel :- Leptons [4 .. 6]) + %I__u \cdot %add (conjugate (f__Q [A, k] (X)) \cdot f__Q [A, l] (X), f__Q = StandardModel :- Quarks [1 .. 3]) + %I__d \cdot %add (conjugate (f__Q [A, k] (X)) \cdot f__Q [A, l] (X), f__Q = StandardModel :- Quarks [4 .. 6]))$

${Dgamma}_{μ}_{k, j} ({KroneckerDelta}_{j, l} - {Dgamma}_{5}_{j, l}) (%I__e %add (`*` (conjugate ({f__L}_{k} (X)), {f__L}_{l} (X)), f__L = [Electron, Muon, Tauon]) + %I__n %add (`*` (conjugate ({f__L}_{k} (X)), {f__L}_{l} (X)), f__L = [ElectronNeutrino, MuonNeutrino, TauonNeutrino]) + %I__u %add (`*` (conjugate ({f__Q}_{A, k} (X)), {f__Q}_{A, l} (X)), f__Q = [Up, Charm, Top]) + %I__d %add (`*` (conjugate ({f__Q}_{A, k} (X)), {f__Q}_{A, l} (X)), f__Q = [Down, Strange, Bottom]))$

(50)

To activate only the sums,

>	$J [W, μ] ≔ eval (, %add = add)$

${Dgamma}_{μ}_{k, j} ({KroneckerDelta}_{j, l} - {Dgamma}_{5}_{j, l}) (%I__e (`*` (conjugate ({Electron}_{k} (X)), {Electron}_{l} (X)) + `*` (conjugate ({Muon}_{k} (X)), {Muon}_{l} (X)) + `*` (conjugate ({Tauon}_{k} (X)), {Tauon}_{l} (X))) + %I__n (`*` (conjugate ({ElectronNeutrino}_{k} (X)), {ElectronNeutrino}_{l} (X)) + `*` (conjugate ({MuonNeutrino}_{k} (X)), {MuonNeutrino}_{l} (X)) + `*` (conjugate ({TauonNeutrino}_{k} (X)), {TauonNeutrino}_{l} (X))) + %I__u (`*` (conjugate ({Up}_{A, k} (X)), {Up}_{A, l} (X)) + `*` (conjugate ({Charm}_{A, k} (X)), {Charm}_{A, l} (X)) + `*` (conjugate ({Top}_{A, k} (X)), {Top}_{A, l} (X))))$

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