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

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.

with(Physics):

with(StandardModel);

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{Setting}\mathrm{lowercaselatin\_is}\mathrm{letters\; to\; represent}\mathrm{Dirac\; spinor}\mathrm{indices}$

$\mathrm{Setting}\mathrm{lowercaselatin\_ah}\mathrm{letters\; to\; represent}\mathrm{SU(3)\; adjoint\; representation,\; (1..8)}\mathrm{indices}$

$\mathrm{Setting}\mathrm{uppercaselatin\_ah}\mathrm{letters\; to\; represent}\mathrm{SU(3)\; fundamental\; representation,\; (1..3)}\mathrm{indices}$

$\mathrm{Setting}\mathrm{uppercaselatin\_is}\mathrm{letters\; to\; represent}\mathrm{SU(2)\; adjoint\; representation,\; (1..3)}\mathrm{indices}$

$\mathrm{Setting}\mathrm{uppercasegreek}\mathrm{letters\; to\; represent}\mathrm{SU(2)\; fundamental\; representation,\; (1..2)}\mathrm{indices}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{Defined\; as\; the\; electron,\; muon\; and\; tau\; leptons\; and\; corresponding\; neutrinos:}{\mathrm{e}}_i,{\mathrm{\mu }}_i,{\mathrm{\tau }}_i,{{\mathrm{\nu }}^{\left(\mathrm{e}\right)}}_i,{{\mathrm{\nu }}^{\left(\mathrm{\mu }\right)}}_i,{{\mathrm{\nu }}^{\left(\mathrm{\tau }\right)}}_i$

$\mathrm{Defined\; as\; the\; up,\; charm,\; top,\; down,\; strange\; and\; bottom\; quarks:}{\mathrm{u}}_{A,i},{\mathrm{c}}_{A,i},{\mathrm{t}}_{A,i},{\mathrm{d}}_{A,i},{\mathrm{s}}_{A,i},{\mathrm{b}}_{A,i}$

$\mathrm{Defined\; as\; gauge\; tensors:}{\mathrm{B}}_{\mathrm{\mu }},{\mathrm{𝔹}}_{\mathrm{\mu },\mathrm{\nu }},{\mathrm{A}}_{\mathrm{\mu }},{\mathrm{𝔽}}_{\mathrm{\mu },\mathrm{\nu }},{\mathrm{W}}_{\mathrm{\mu },J},{\mathrm{𝕎}}_{\mathrm{\mu },\mathrm{\nu },J},{{\mathrm{W}}^{\mathrm{+}}}_{\mathrm{\mu }},{{\mathrm{𝕎}}^{\mathrm{+}}}_{\mathrm{\mu },\mathrm{\nu }},{{\mathrm{W}}^{\mathrm{-}}}_{\mathrm{\mu }},{{\mathrm{𝕎}}^{\mathrm{-}}}_{\mathrm{\mu },\mathrm{\nu }},{\mathrm{Z}}_{\mathrm{\mu }},{\mathrm{\mathbb{Z}}}_{\mathrm{\mu },\mathrm{\nu }},{\mathrm{G}}_{\mathrm{\mu },a},{\mathrm{𝔾}}_{\mathrm{\mu },\mathrm{\nu },a}$

$\mathrm{Defined\; as\; Gell-Mann\; (Glambda),\; Pauli\; (Psigma)\; and\; Dirac\; (Dgamma)\; matrices:}{\mathrm{\lambda }}_a,{\mathrm{\sigma }}_J,{\mathrm{\gamma }}_{\mathrm{\mu }}$

$\mathrm{Defined\; as\; the\; electric,\; weak\; and\; strong\; coupling\; constants:}\mathrm{g\_\_e},\mathrm{g\_\_w},\mathrm{g\_\_s}$

$\mathrm{Defined\; as\; the\; charge\; in\; units\; of\; |}\mathrm{g\_\_e}\mathrm{|\; for\; 1)\; the\; electron,\; muon\; and\; tauon,\; 2)\; the\; up,\; charm\; and\; top,\; and\; 3)\; the\; down,\; strange\; and\; bottom:}\mathrm{q\_\_e}=\mathrm{-1},\mathrm{q\_\_u}=\frac{2}{3},\mathrm{q\_\_d}=-\frac{1}{3}$

$\mathrm{Defined\; as\; the\; weak\; isospin\; for\; 1)\; the\; electron,\; muon\; and\; tauon,\; 2)\; the\; up,\; charm\; and\; top,\; 3)\; the\; down,\; strange\; and\; bottom,\; and\; 4)\; all\; the\; neutrinos:}\mathrm{I\_\_e}=-\frac{1}{2},\mathrm{I\_\_u}=\frac{1}{2},\mathrm{I\_\_d}=-\frac{1}{2},\mathrm{I\_\_n}=\frac{1}{2}$

$\mathrm{You\; can\; use\; the\; active\; form\; without\; the\; \%\; prefix,\; or\; the\; \text{'}value\text{'}\; command\; to\; give\; the\; corresponding\; value\; to\; any\; of\; the\; inert\; representations}\mathrm{q\_\_e},\mathrm{q\_\_u},\mathrm{q\_\_d},\mathrm{I\_\_e},\mathrm{I\_\_u},\mathrm{I\_\_d},\mathrm{I\_\_n}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{Default\; differentiation\; variables\; for\; d\_,\; D\_\; and\; dAlembertian\; are:}\left\{X=\left(x\,y\,z\,t\right)\right\}$

$\mathrm{Minkowski\; spacetime\; with\; signatre}\left(\mathrm{-\; -\; -\; +}\right)$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\left[\mathrm{I\_\_d}\,\mathrm{I\_\_e}\,\mathrm{I\_\_n}\,\mathrm{I\_\_u}\,\mathrm{q\_\_d}\,\mathrm{q\_\_e}\,\mathrm{q\_\_u}\,\mathrm{BField}\,\mathrm{BFieldStrength}\,\mathrm{Bottom}\,\mathrm{CKM}\,\mathrm{Charm}\,\mathrm{Down}\,\mathrm{ElectromagneticField}\,\mathrm{ElectromagneticFieldStrength}\,\mathrm{Electron}\,\mathrm{ElectronNeutrino}\,\mathrm{FSU3}\,\mathrm{Glambda}\,\mathrm{GluonField}\,\mathrm{GluonFieldStrength}\,\mathrm{HiggsBoson}\,\mathrm{Lagrangian}\,\mathrm{Muon}\,\mathrm{MuonNeutrino}\,\mathrm{Strange}\,\mathrm{Tauon}\,\mathrm{TauonNeutrino}\,\mathrm{Top}\,\mathrm{Up}\,\mathrm{WField}\,\mathrm{WFieldStrength}\,\mathrm{WMinusField}\,\mathrm{WMinusFieldStrength}\,\mathrm{WPlusField}\,\mathrm{WPlusFieldStrength}\,\mathrm{WeinbergAngle}\,\mathrm{ZField}\,\mathrm{ZFieldStrength}\,\mathrm{g\_\_e}\,\mathrm{g\_\_s}\,\mathrm{g\_\_w}\right]$

Setup(massless);

$\mathrm{*\; Partial\; match\; of\; \text{'}}\mathrm{massless}\mathrm{\text{'}\; against\; keyword\; \text{'}}\mathrm{masslessfields}\text{'}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\left[\mathrm{masslessfields}=\left\{\mathbf{G}\,{\mathbf{\nu }}^{\left(\mathrm{\mu }\right)}\,{\mathbf{\nu }}^{\left(\mathrm{\tau }\right)}\,\mathbf{A}\,{\mathbf{\nu }}^{\left(\mathrm{e}\right)}\right\}\right]$

StandardModel:-Leptons;

$\left[\mathbf{e}\,\mathbf{\mu }\,\mathbf{\tau }\,{\mathbf{\nu }}^{\left(\mathrm{e}\right)}\,{\mathbf{\nu }}^{\left(\mathrm{\mu }\right)}\,{\mathbf{\nu }}^{\left(\mathrm{\tau }\right)}\right]$

StandardModel:-Quarks;

$\left[\mathbf{u}\,\mathbf{c}\,\mathbf{t}\,\mathbf{d}\,\mathbf{s}\,\mathbf{b}\right]$

StandardModel:-GaugeFields;

$\left[\mathbf{A}\,\mathbf{𝔽}\,\mathbf{B}\,\mathbf{𝔹}\,\mathbf{W}\,\mathbf{𝕎}\,\mathbf{G}\,\mathbf{𝔾}\,{\mathbf{W}}^{\mathrm{-}}\,{\mathbf{𝕎}}^{\mathrm{-}}\,{\mathbf{W}}^{\mathrm{+}}\,{\mathbf{𝕎}}^{\mathrm{+}}\,\mathbf{Z}\,\mathbf{\mathbb{Z}}\right]$

Setup(anticommutativeprefix = {f__L, f__Q, f__U, f__D});

$\left[\mathrm{anticommutativeprefix}=\left\{\mathrm{f\_\_D}\,\mathrm{f\_\_L}\,\mathrm{f\_\_Q}\,\mathrm{f\_\_U}\right\}\right]$

CompactDisplay((StandardModel:-Leptons, StandardModel:-Quarks, StandardModel:-GaugeFields, HiggsBoson, f__L, f__Q, f__U, f__D)(X), quiet):

interface(imaginaryunit = i):

Lagrangian();

$\left(\sum _{\mathrm{f\_\_Q}=\left[\mathbf{u}\,\mathbf{c}\,\mathbf{t}\,\mathbf{d}\,\mathbf{s}\,\mathbf{b}\right]}{\overline{\mathrm{f\_\_Q}}}_{A,j}\left(\mathrm{i}{\left({\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\right)}_{j,k}{▿}_{\mathrm{\mu }}-m_{\mathrm{f\_\_Q}}{\mathrm{\delta }}_{j,k}\right){\mathrm{f\_\_Q}}_{A,k}\right)-\frac{{\mathbf{𝔾}}_{\mathrm{\mu },\mathrm{\nu },a}{\mathbf{𝔾}}_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\nu }}a}^{\phantom{}\mathrm{\mu },\mathrm{\nu }\phantom{a}}}{4}-\frac{{\mathbf{𝔽}}_{\mathrm{\mu },\mathrm{\nu }}{\mathbf{𝔽}}_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\mu },\mathrm{\nu }}}{4}-\frac{{{\mathbf{𝕎}}^{\mathrm{+}}}_{\mathrm{\mu },\mathrm{\nu }}{{\mathbf{𝕎}}^{\mathrm{-}}}_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\mu },\mathrm{\nu }}}{2}+m_{\mathbf{W}}^2{{\mathbf{W}}^{\mathrm{+}}}_{\mathrm{\mu }}{{\mathbf{W}}^{\mathrm{-}}}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}-\frac{{\mathbf{\mathbb{Z}}}_{\mathrm{\mu },\mathrm{\nu }}{\mathbf{\mathbb{Z}}}_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\mu },\mathrm{\nu }}}{4}+\frac{m_{\mathbf{Z}}^2{\mathbf{Z}}_{\mathrm{\mu }}{\mathbf{Z}}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}}{2}+\frac{{\partial }_{\mathrm{\mu }}\left(\mathbf{\Phi }\right){\partial }_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\left(\mathbf{\Phi }\right)}{2}-\frac{m_{\mathbf{\Phi }}^2{\mathbf{\Phi }}^2}{2}+\left(\sum _{\mathrm{f\_\_L}=\left[\mathbf{e}\,\mathbf{\mu }\,\mathbf{\tau }\right]}{\overline{\mathrm{f\_\_L}}}_j\left(\mathrm{i}{\left({\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\right)}_{j,k}{\partial }_{\mathrm{\mu }}\left({\mathrm{f\_\_L}}_k\right)-m_{\mathrm{f\_\_L}}{\mathrm{f\_\_L}}_j\right)\right)+\left(\sum _{\mathrm{f\_\_L}=\left[{\mathbf{\nu }}^{\left(\mathrm{e}\right)}\,{\mathbf{\nu }}^{\left(\mathrm{\mu }\right)}\,{\mathbf{\nu }}^{\left(\mathrm{\tau }\right)}\right]}\mathrm{i}{\overline{\mathrm{f\_\_L}}}_j{\partial }_{\mathrm{\mu }}\left({\mathrm{f\_\_L}}_k\right){\left({\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\right)}_{j,k}\right)+\left(\sum _{\mathrm{f\_\_Q}=\left[\mathbf{u}\,\mathbf{c}\,\mathbf{t}\,\mathbf{d}\,\mathbf{s}\,\mathbf{b}\right]}{\overline{\mathrm{f\_\_Q}}}_{A,j}\left(\mathrm{i}{\left({\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\right)}_{j,k}{\partial }_{\mathrm{\mu }}\left({\mathrm{f\_\_Q}}_{A,k}\right)-m_{\mathrm{f\_\_Q}}{\mathrm{f\_\_Q}}_{A,j}\right)\right)+\mathrm{g\_\_e}\left(\mathrm{q\_\_e}\left(\sum _{\mathrm{f\_\_L}=\left[\mathbf{e}\,\mathbf{\mu }\,\mathbf{\tau }\right]}{\overline{\mathrm{f\_\_L}}}_j{\mathrm{f\_\_L}}_k\right)+\mathrm{q\_\_u}\left(\sum _{\mathrm{f\_\_Q}=\left[\mathbf{u}\,\mathbf{c}\,\mathbf{t}\right]}{\overline{\mathrm{f\_\_Q}}}_{A,j}{\mathrm{f\_\_Q}}_{A,k}\right)+\mathrm{q\_\_d}\left(\sum _{\mathrm{f\_\_Q}=\left[\mathbf{d}\,\mathbf{s}\,\mathbf{b}\right]}{\overline{\mathrm{f\_\_Q}}}_{A,j}{\mathrm{f\_\_Q}}_{A,k}\right)\right){\mathbf{A}}_{\mathrm{\mu }}{\left({\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\right)}_{j,k}+\frac{\mathrm{g\_\_w}\left({\left({\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\right)}_{j,k}\left({\mathrm{\delta }}_{k,l}+{\left({\mathrm{\gamma }}_5\right)}_{k,l}\right)\left(\mathrm{I\_\_e}\left(\sum _{\mathrm{f\_\_L}=\left[\mathbf{e}\,\mathbf{\mu }\,\mathbf{\tau }\right]}{\overline{\mathrm{f\_\_L}}}_j{\mathrm{f\_\_L}}_l\right)+\mathrm{I\_\_n}\left(\sum _{\mathrm{f\_\_L}=\left[{\mathbf{\nu }}^{\left(\mathrm{e}\right)}\,{\mathbf{\nu }}^{\left(\mathrm{\mu }\right)}\,{\mathbf{\nu }}^{\left(\mathrm{\tau }\right)}\right]}{\overline{\mathrm{f\_\_L}}}_j{\mathrm{f\_\_L}}_l\right)+\mathrm{I\_\_u}\left(\sum _{\mathrm{f\_\_Q}=\left[\mathbf{u}\,\mathbf{c}\,\mathbf{t}\right]}{\overline{\mathrm{f\_\_Q}}}_{A,j}{\mathrm{f\_\_Q}}_{A,l}\right)+\mathrm{I\_\_d}\left(\sum _{\mathrm{f\_\_Q}=\left[\mathbf{d}\,\mathbf{s}\,\mathbf{b}\right]}{\overline{\mathrm{f\_\_Q}}}_{A,j}{\mathrm{f\_\_Q}}_{A,l}\right)\right)-{\sin \left({\mathbf{\θ}}_{\mathbf{w}}\right)}^2{\left({\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\right)}_{j,k}\left(\mathrm{q\_\_e}\left(\sum _{\mathrm{f\_\_L}=\left[\mathbf{e}\,\mathbf{\mu }\,\mathbf{\tau }\right]}{\overline{\mathrm{f\_\_L}}}_j{\mathrm{f\_\_L}}_k\right)+\mathrm{q\_\_u}\left(\sum _{\mathrm{f\_\_Q}=\left[\mathbf{u}\,\mathbf{c}\,\mathbf{t}\right]}{\overline{\mathrm{f\_\_Q}}}_{A,j}{\mathrm{f\_\_Q}}_{A,k}\right)+\mathrm{q\_\_d}\left(\sum _{\mathrm{f\_\_Q}=\left[\mathbf{d}\,\mathbf{s}\,\mathbf{b}\right]}{\overline{\mathrm{f\_\_Q}}}_{A,j}{\mathrm{f\_\_Q}}_{A,k}\right)\right)\right){\mathbf{Z}}_{\mathrm{\mu }}}{\cos \left({\mathbf{\θ}}_{\mathbf{w}}\right)}-\frac{\mathrm{g\_\_w}\sqrt{2}\left({\mathrm{\delta }}_{k,l}+{\left({\mathrm{\gamma }}_5\right)}_{k,l}\right)\left(\left(\left(\sum _{\mathrm{f\_\_D}=\left[\mathbf{d}\,\mathbf{s}\,\mathbf{b}\right]}\sum _{\mathrm{f\_\_U}=\left[\mathbf{u}\,\mathbf{c}\,\mathbf{t}\right]}{\mathbf{𝕄}}_{\mathrm{f\_\_U},\mathrm{f\_\_D}}{\overline{\mathrm{f\_\_U}}}_{A,j}{\mathrm{f\_\_D}}_{A,l}\right)+\left(\sum _{\mathrm{f\_\_L}=\left[\left[{\mathbf{\nu }}^{\left(\mathrm{e}\right)}\,\mathbf{e}\right]\,\left[{\mathbf{\nu }}^{\left(\mathrm{\mu }\right)}\,\mathbf{\mu }\right]\,\left[{\mathbf{\nu }}^{\left(\mathrm{\tau }\right)}\,\mathbf{\tau }\right]\right]}{\overline{{\mathrm{f\_\_L}}_1}}_j{\left({\mathrm{f\_\_L}}_2\right)}_l\right)\right){{\mathbf{W}}^{\mathrm{+}}}_{\mathrm{\mu }}+\left(\left(\sum _{\mathrm{f\_\_D}=\left[\mathbf{d}\,\mathbf{s}\,\mathbf{b}\right]}\sum _{\mathrm{f\_\_U}=\left[\mathbf{u}\,\mathbf{c}\,\mathbf{t}\right]}{\overline{\mathrm{f\_\_D}}}_{A,j}{\mathrm{f\_\_U}}_{A,l}\stackrel{\&conjugate0;}{{\mathbf{𝕄}}_{\mathrm{f\_\_D},\mathrm{f\_\_U}}}\right)+\left(\sum _{\mathrm{f\_\_L}=\left[\left[{\mathbf{\nu }}^{\left(\mathrm{e}\right)}\,\mathbf{e}\right]\,\left[{\mathbf{\nu }}^{\left(\mathrm{\mu }\right)}\,\mathbf{\mu }\right]\,\left[{\mathbf{\nu }}^{\left(\mathrm{\tau }\right)}\,\mathbf{\tau }\right]\right]}{\overline{{\mathrm{f\_\_L}}_2}}_j{\left({\mathrm{f\_\_L}}_1\right)}_l\right)\right){{\mathbf{W}}^{\mathrm{-}}}_{\mathrm{\mu }}\right){\left({\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\right)}_{j,k}}{2}-\frac{\mathrm{g\_\_w}{m}_{\mathbf{\Phi }}^2\left({\mathbf{\Phi }}^3+\frac{{\mathbf{\Phi }}^4}{8m_{\mathbf{W}}}\right)}{4m_{\mathbf{W}}}+\left(\frac{\mathrm{g\_\_w}\mathbf{\Phi }}{m_{\mathbf{W}}}+\frac{{\mathrm{g\_\_w}}^2{\mathbf{\Phi }}^2}{4m_{\mathbf{W}}^2}\right)\left(m_{\mathbf{W}}^2{{\mathbf{W}}^{\mathrm{+}}}_{\mathrm{\mu }}{{\mathbf{W}}^{\mathrm{-}}}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}+\frac{m_{\mathbf{Z}}^2{\mathbf{Z}}_{\mathrm{\mu }}{\mathbf{Z}}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}}{2}\right)-\mathrm{i}\mathrm{g\_\_w}\left(\left({{\mathbf{𝕎}}^{\mathrm{+}}}_{\mathrm{\mu },\mathrm{\nu }}{{\mathbf{W}}^{\mathrm{-}}}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}+{{\mathbf{W}}^{\mathrm{+}}}_{\mathrm{\mu }}{{\mathbf{𝕎}}^{\mathrm{-}}}_{\mathrm{\nu }\phantom{\mathrm{\mu }}}^{\phantom{\mathrm{\nu }}\mathrm{\mu }}\right)\left({\mathbf{A}}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\sin \left({\mathbf{\θ}}_{\mathbf{w}}\right)-{\mathbf{Z}}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\cos \left({\mathbf{\θ}}_{\mathbf{w}}\right)\right)+{{\mathbf{W}}^{\mathrm{-}}}_{\mathrm{\nu }}{{\mathbf{W}}^{\mathrm{+}}}_{\mathrm{\mu }}\left({\mathbf{𝔽}}_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\mu },\mathrm{\nu }}\sin \left({\mathbf{\θ}}_{\mathbf{w}}\right)-{\mathbf{\mathbb{Z}}}_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\mu },\mathrm{\nu }}\cos \left({\mathbf{\θ}}_{\mathbf{w}}\right)\right)\right)-\frac{{\mathrm{g\_\_w}}^2\left(\left(2{{\mathbf{W}}^{\mathrm{+}}}_{\mathrm{\mu }}{{\mathbf{W}}^{\mathrm{-}}}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}+\left({\mathbf{A}}_{\mathrm{\mu }}\sin \left({\mathbf{\θ}}_{\mathbf{w}}\right)-{\mathbf{Z}}_{\mathrm{\mu }}\cos \left({\mathbf{\θ}}_{\mathbf{w}}\right)\right)\left({\mathbf{A}}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\sin \left({\mathbf{\θ}}_{\mathbf{w}}\right)-{\mathbf{Z}}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\cos \left({\mathbf{\θ}}_{\mathbf{w}}\right)\right)\right)\left(2{{\mathbf{W}}^{\mathrm{+}}}_{\mathrm{\nu }}{{\mathbf{W}}^{\mathrm{-}}}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}+\left({\mathbf{A}}_{\mathrm{\nu }}\sin \left({\mathbf{\θ}}_{\mathbf{w}}\right)-{\mathbf{Z}}_{\mathrm{\nu }}\cos \left({\mathbf{\θ}}_{\mathbf{w}}\right)\right)\left({\mathbf{A}}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\sin \left({\mathbf{\θ}}_{\mathbf{w}}\right)-{\mathbf{Z}}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\cos \left({\mathbf{\θ}}_{\mathbf{w}}\right)\right)\right)+\left({{\mathbf{W}}^{\mathrm{+}}}_{\mathrm{\mu }}{{\mathbf{W}}^{\mathrm{-}}}_{\mathrm{\nu }}+{{\mathbf{W}}^{\mathrm{+}}}_{\mathrm{\nu }}{{\mathbf{W}}^{\mathrm{-}}}_{\mathrm{\mu }}+\left({\mathbf{A}}_{\mathrm{\mu }}\sin \left({\mathbf{\θ}}_{\mathbf{w}}\right)-{\mathbf{Z}}_{\mathrm{\mu }}\cos \left({\mathbf{\θ}}_{\mathbf{w}}\right)\right)\left({\mathbf{A}}_{\mathrm{\nu }}\sin \left({\mathbf{\θ}}_{\mathbf{w}}\right)-{\mathbf{Z}}_{\mathrm{\nu }}\cos \left({\mathbf{\θ}}_{\mathbf{w}}\right)\right)\right)\left({{\mathbf{W}}^{\mathrm{+}}}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}{{\mathbf{W}}^{\mathrm{-}}}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}+{{\mathbf{W}}^{\mathrm{+}}}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}{{\mathbf{W}}^{\mathrm{-}}}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}+\left({\mathbf{A}}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\sin \left({\mathbf{\θ}}_{\mathbf{w}}\right)-{\mathbf{Z}}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\cos \left({\mathbf{\θ}}_{\mathbf{w}}\right)\right)\left({\mathbf{A}}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\sin \left({\mathbf{\θ}}_{\mathbf{w}}\right)-{\mathbf{Z}}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\cos \left({\mathbf{\θ}}_{\mathbf{w}}\right)\right)\right)\right)}{4}-\frac{\mathrm{g\_\_w}\left(\left(\sum _{\mathrm{f\_\_L}=\left[\mathbf{e}\,\mathbf{\mu }\,\mathbf{\tau }\,{\mathbf{\nu }}^{\left(\mathrm{e}\right)}\,{\mathbf{\nu }}^{\left(\mathrm{\mu }\right)}\,{\mathbf{\nu }}^{\left(\mathrm{\tau }\right)}\right]}{m}_{\mathrm{f\_\_L}}{\overline{\mathrm{f\_\_L}}}_j{\mathrm{f\_\_L}}_j\right)+\left(\sum _{\mathrm{f\_\_Q}=\left[\mathbf{u}\,\mathbf{c}\,\mathbf{t}\,\mathbf{d}\,\mathbf{s}\,\mathbf{b}\right]}{m}_{\mathrm{f\_\_Q}}{\overline{\mathrm{f\_\_Q}}}_{A,j}{\mathrm{f\_\_Q}}_{A,j}\right)\right)\mathbf{\Phi }}{2m_{\mathbf{W}}}$