Lagrangian - Maple Help
For the best experience, we recommend viewing online help using Google Chrome or Microsoft Edge.

Online Help

All Products    Maple    MapleSim


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 Lterm 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 LK,LN,LC,LH,LHV,LWWV,LWWVV,LY, 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:

• 

LK is the kinetic part, including the dynamic and mass (quadratic) terms;

• 

LN is the neutral current;

• 

LC is the charged current;

• 

LH has the Higgs three and four point self interaction terms;

• 

LHV contains the Higgs interactions with the gauge vector bosons W+, W and Z;

• 

LWWV includes the gauge three-point self interactions between the fields A,W+,W and Z;

• 

LWWVV contains the gauge four-point self interactions between the fields A,W+,W and Z;

• 

LY contains the Yukawa interactions between the fermions and the Higgs field.

Examples

withPhysics:

withStandardModel

_______________________________________________________

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: ej , μj , τj , ElectronNeutrinoj , MuonNeutrinoj , TauonNeutrinoj

Defined as the up, charm, top, down, strange and bottom quarks: uA,j , cA,j , tA,j , dA,j , sA,j , bA,j

Defined as gauge tensors: Bμ , 𝔹μ,ν , Aμ , 𝔽μ,ν , Wμ,J , 𝕎μ,ν,J , WPlusFieldμ , WPlusFieldStrengthμ,ν , WMinusFieldμ , WMinusFieldStrengthμ,ν , Zμ , μ,ν , Gμ,a , 𝔾μ,ν,a

Defined as Gell-Mann (Glambda), Pauli (Psigma) and Dirac (Dgamma) matrices: λa , σJ , γμ

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 = 23, %q__d = 13

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 = 12, %I__u = 12, %I__d = 12, %I__n = 12

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

Setupmassless

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

_______________________________________________________

masslessfields=G,MuonNeutrino,TauonNeutrino,A,ElectronNeutrino

(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

StandardModel:-Leptons

e,μ,τ,ElectronNeutrino,MuonNeutrino,TauonNeutrino

(3)

StandardModel:-Quarks

u,c,t,d,s,b

(4)

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

StandardModel:-GaugeFields

A,𝔽,B,𝔹,W,𝕎,G,𝔾,WMinusField,WMinusFieldStrength,WPlusField,WPlusFieldStrength,Z,

(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

Setupanticommutativeprefix=f__L,f__Q,f__U,f__D

anticommutativeprefix=f__D,f__L,f__Q,f__U

(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

CompactDisplayStandardModel:-Leptons,StandardModel:-Quarks,StandardModel:-GaugeFields,HiggsBoson,f__L,f__Q,f__U,f__DX,quiet:

interfaceimaginaryunit=i:

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

Lagrangian

%add`*`conjugatef__QA,jX,Dgammaμj,kD_μmf__QKroneckerDeltaj,k,f__QA,kX,f__Q=Up,Charm,Top,Down,Strange,Bottom𝔾μ,ν,a24𝔽μ,ν24WPlusFieldStrengthμ,νWMinusFieldStrengthμ,ν2+mW2WPlusFieldμWMinusFieldμμ,ν24+mZ2Zμ22+μHiggsBosonX22mΦ2HiggsBosonX22+%add`*`conjugatef__LjX,Dgammaμj,kd_μf__LkX,Xmf__Lf__LjX,f__L=Electron,Muon,Tauon+%addDgammaμj,k`*`conjugatef__LjX,d_μf__LkX,X,f__L=ElectronNeutrino,MuonNeutrino,TauonNeutrino+%add`*`conjugatef__QA,jX,Dgammaμj,kd_μf__QA,kX,Xmf__Qf__QA,jX,f__Q=Up,Charm,Top,Down,Strange,Bottom+g__eγμj,k%q__e%add`*`conjugatef__LjX,f__LkX,f__L=Electron,Muon,Tauon+%q__u%add`*`conjugatef__QA,jX,f__QA,kX,f__Q=Up,Charm,Top+%q__d%add`*`conjugatef__QA,jX,f__QA,kX,f__Q=Down,Strange,BottomElectromagneticFieldμX+g__wγμj,kδk,l+γ5k,l%I__e%add`*`conjugatef__LjX,f__LlX,f__L=Electron,Muon,Tauon+%I__n%add`*`conjugatef__LjX,f__LlX,f__L=ElectronNeutrino,MuonNeutrino,TauonNeutrino+%I__u%add`*`conjugatef__QA,jX,f__QA,lX,f__Q=Up,Charm,Top+%I__d%add`*`conjugatef__QA,jX,f__QA,lX,f__Q=Down,Strange,BottomsinWeinbergAngle2γμj,k%q__e%add`*`conjugatef__LjX,f__LkX,f__L=Electron,Muon,Tauon+%q__u%add`*`conjugatef__QA,jX,f__QA,kX,f__Q=Up,Charm,Top+%q__d%add`*`conjugatef__QA,jX,f__QA,kX,f__Q=Down,Strange,BottomZFieldμXcosWeinbergAngleg__w2γμj,kδk,l+γ5k,l%add%addCKMf__U,f__D`*`conjugatef__UA,jX,f__DA,lX,f__U=Up,Charm,Top,f__D=Down,Strange,Bottom+%add`*`conjugatef__L1jX,f__L2lX,f__L=ElectronNeutrino,Electron,MuonNeutrino,Muon,TauonNeutrino,TauonWPlusFieldμX+%add%addconjugateCKMf__U,f__D`*`conjugatef__DA,jX,f__UA,lX,f__U=Up,Charm,Top,f__D=Down,Strange,Bottom+%add`*`conjugatef__L2jX,f__L1lX,f__L=ElectronNeutrino,Electron,MuonNeutrino,Muon,TauonNeutrino,TauonWMinusFieldμX2g__wmΦ2HiggsBosonX3+HiggsBosonX48mW4mW+g__wHiggsBosonXmW+g__w2HiggsBosonX24mW2mW2WPlusFieldμXWMinusFieldμX+ZFieldμX2mZ22g__wWPlusFieldStrengthμ,νXWMinusFieldμXWPlusFieldμXWMinusFieldStrengthμ,νXElectromagneticFieldνXsinWeinbergAngleZFieldνXcosWeinbergAngle+WMinusFieldνXWPlusFieldμXElectromagneticFieldStrengthμ,νXsinWeinbergAngleZFieldStrengthμ,νXcosWeinbergAngleg__w22WPlusFieldμXWMinusFieldμX+ElectromagneticFieldμXsinWeinbergAngleZFieldμXcosWeinbergAngle22WPlusFieldνXWMinusFieldνX+ElectromagneticFieldνXsinWeinbergAngleZFieldνXcosWeinbergAngle2+WPlusFieldμXWMinusFieldνX+WPlusFieldνXWMinusFieldμX+ElectromagneticFieldμXsinWeinbergAngleZFieldμXcosWeinbergAngleElectromagneticFieldνXsinWeinbergAngleZFieldνXcosWeinbergAngle24g__w%addmf__L`*`conjugatef__LjX,f__LjX,f__L=Electron,Muon,Tauon,ElectronNeutrino,MuonNeutrino,TauonNeutrino+%addmf__Q`*`conjugatef__QA,jX,f__QA,jX,f__Q=Up,Charm,Top,Down,Strange,BottomHiggsBosonX2mW

(7)

In the output above we see, among other things, the γ5 Dirac matrix, and the Cabibbo - Kobayashi - Maskawa matrix 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 γ5, 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

LagrangianQED

`*`conjugateElectronjX,Dgammaμj,kD_~mumElectronKroneckerDeltaj,k,ElectronkX14`*`ElectromagneticFieldStrengthμ,ν,ElectromagneticFieldStrength~mu,~nu

(8)

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

LagrangianQED,applied

`*`conjugateElectronjX,D_μElectronkX,XDgamma~muj,k`*`mElectron,KroneckerDeltaj,k,ElectronkX14`*`ElectromagneticFieldStrengthμ,ν,ElectromagneticFieldStrength~mu,~nu

(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

expand

Dgamma~muj,k`*`conjugateElectronjX,d_μElectronkX,XDgamma~muj,kg__e`*`conjugateElectronjX,ElectronkX,ElectromagneticFieldμXmElectronKroneckerDeltaj,k`*`conjugateElectronjX,ElectronkX14`*`ElectromagneticFieldStrengthμ,ν,ElectromagneticFieldStrength~mu,~nu

(10)

or pass the optional keyword expanded, in which case also the trace of 𝔽__μ,ν gets expanded

LagrangianQED,expanded

`*`conjugateElectronjX,d_μElectronkX,XDgamma~muj,kg__e`*`conjugateElectronjX,ElectronkX,ElectromagneticFieldμXDgamma~muj,kmElectron`*`conjugateElectronjX,ElectronjX14`*`d_μElectromagneticFieldνX,Xd_νElectromagneticFieldμX,X,d_~muElectromagneticField~nuX,Xd_~nuElectromagneticField~muX,X

(11)

then discard the non-interaction terms

removehas,,d_,m

Dgamma~muj,kg__e`*`conjugateElectronjX,ElectronkX,ElectromagneticFieldμX

(12)

or simpler: pass the keyword interaction

LagrangianQED,interaction

Dgamma~muj,kg__e`*`conjugateElectronjX,ElectronkX,ElectromagneticFieldμX

(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

FeynmanDiagrams,incoming=Electron,outgoing=Electron,numberofloops=1,diagrams

%FeynmanIntegral18Physics:-FeynmanDiagrams:-UspinorElectronlP__1_conjugatePhysics:-FeynmanDiagrams:-UspinorElectronmP__2_g__e2Dgamma~alpham,nDgamma~nup,lP__1β+p__2βDgamma~betan,p+mElectronKroneckerDeltan,pg_α,νDiracP__2+P__1π3P__1+p__22mElectron2+Physics:-FeynmanDiagrams:-εp__22+Physics:-FeynmanDiagrams:-ε,p__2

(14)

The Quantum Chromodynamics (QCD) Lagrangian

Next in complexity is the QCD Lagrangian

LagrangianQCD

%add`*`conjugatef__QA,jX,Dgammaμj,kD_~mumf__QKroneckerDeltaj,k,f__QA,kX,f__Q=Up,Charm,Top,Down,Strange,Bottom14`*`GluonFieldStrengthμ,ν,a,GluonFieldStrength~mu,~nu,a

(15)

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

value

`*`conjugateUpA,jX,Dgammaμj,kD_~mumUpKroneckerDeltaj,k,UpA,kX+`*`conjugateCharmA,jX,Dgammaμj,kD_~mumCharmKroneckerDeltaj,k,CharmA,kX+`*`conjugateTopA,jX,Dgammaμj,kD_~mumTopKroneckerDeltaj,k,TopA,kX+`*`conjugateDownA,jX,Dgammaμj,kD_~mumDownKroneckerDeltaj,k,DownA,kX+`*`conjugateStrangeA,jX,Dgammaμj,kD_~mumStrangeKroneckerDeltaj,k,StrangeA,kX+`*`conjugateBottomA,jX,Dgammaμj,kD_~mumBottomKroneckerDeltaj,k,BottomA,kX14`*`GluonFieldStrengthμ,ν,a,GluonFieldStrength~mu,~nu,a

(16)

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

LagrangianQCD,expanded

`*`conjugateUpA,jX,d_μUpA,kX,X12g__sGlambdaaA,B`*`UpB,kX,GluonFieldμ,aXDgamma~muj,kmUpKroneckerDeltaj,kUpA,kX+`*`conjugateCharmA,jX,d_μCharmA,kX,X12g__sGlambdaaA,B`*`CharmB,kX,GluonFieldμ,aXDgamma~muj,kmCharmKroneckerDeltaj,kCharmA,kX+`*`conjugateTopA,jX,d_μTopA,kX,X12g__sGlambdaaA,B`*`TopB,kX,GluonFieldμ,aXDgamma~muj,kmTopKroneckerDeltaj,kTopA,kX+`*`conjugateDownA,jX,d_μDownA,kX,X12g__sGlambdaaA,B`*`DownB,kX,GluonFieldμ,aXDgamma~muj,kmDownKroneckerDeltaj,kDownA,kX+`*`conjugateStrangeA,jX,d_μStrangeA,kX,X12g__sGlambdaaA,B`*`StrangeB,kX,GluonFieldμ,aXDgamma~muj,kmStrangeKroneckerDeltaj,kStrangeA,kX+`*`conjugateBottomA,jX,d_μBottomA,kX,X12g__sGlambdaaA,B`*`BottomB,kX,GluonFieldμ,aXDgamma~muj,kmBottomKroneckerDeltaj,kBottomA,kX14`*`d_μGluonFieldν,aX,Xd_νGluonFieldμ,aX,X+g__sFSU3a,b,c`*`GluonFieldμ,bX,GluonFieldν,cX,d_~muGluonField~nu,aX,Xd_~nuGluonField~mu,aX,X+g__sFSU3a,d,e`*`GluonField~mu,dX,GluonField~nu,eX

(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

LagrangianQCD,interaction

12g__sGlambdaaA,B`*`%add`*`conjugatef__QA,jX,f__QB,kX,f__Q=Up,Charm,Top,Down,Strange,Bottom,GluonFieldμ,aXDgamma~muj,kg__sFSU3a,b,c`*`d_μGluonFieldν,aX,X,GluonField~mu,bX,GluonField~nu,cX14g__sFSU3c,d,e`*`GluonFieldμ,aX,GluonFieldα,bX,GluonField~mu,eX,GluonField~alpha,dX

(18)

and to have also the sum expanded pass also expanded

LagrangianQCD,interaction,expanded

12g__sGlambdaaA,B`*``*`conjugateUpA,jX,UpB,kX+`*`conjugateCharmA,jX,CharmB,kX+`*`conjugateTopA,jX,TopB,kX+`*`conjugateDownA,jX,DownB,kX+`*`conjugateStrangeA,jX,StrangeB,kX+`*`conjugateBottomA,jX,BottomB,kX,GluonFieldμ,aXDgamma~muj,kg__sFSU3a,b,c`*`d_μGluonFieldν,aX,X,GluonField~mu,bX,GluonField~nu,cX14g__sFSU3c,d,e`*`GluonFieldμ,aX,GluonFieldα,bX,GluonField~mu,eX,GluonField~alpha,dX

(19)

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

FeynmanDiagrams,incomingparticles=Up,Up&conjugate0;,outgoingparticles=Up,Up&conjugate0;,numberofloops=0,diagrams

uuC,lP__1_vuE,mP__2_&conjugate0;uuF,nP__3_&conjugate0;vuG,pP__4_g__s2λgF,Gγκκn,pλfE,Cγββm,lgβ,κδf,gδP__3λλP__4λλ+P__1λλ+P__2λλ16π2P__1σ+P__2σP__1σσ+P__2σσ+ε+uuC,lP__1_vuE,mP__2_&conjugate0;uuF,nP__3_&conjugate0;vuG,pP__4_g__s2λgE,Gγκκm,pλfF,Cγββn,lgβ,κδf,gδP__3λλP__4λλ+P__1λλ+P__2λλ16π2P__1σP__3σP__1σσP__3σσ+ε

(20)

The probability density of the same process at 1 loop

FeynmanDiagrams,incomingparticles=Up,Up&conjugate0;,outgoingparticles=Up,Up&conjugate0;,numberofloops=1,diagrams,output=probabilitydensity

Physics:-FeynmanDiagrams:-ProbabilityDensity4π2%mulni__1,i__1=1..2absF__12DiracP__3P__4+P__1+P__2%muldP_f3,f=1..2,F__1=%FeynmanIntegral11024Physics:-FeynmanDiagrams:-UspinorUpC,lP__1_conjugatePhysics:-FeynmanDiagrams:-VspinorUpE,mP__2_conjugatePhysics:-FeynmanDiagrams:-UspinorUpF,nP__3_Physics:-FeynmanDiagrams:-VspinorUpG,pP__4_g__s4Glambdaa1D3,GDgamma~sigmaj1,pGlambdahF,D1Dgamma~lambdan,rGlambdagE,HDgamma~kappam,qGlambdafD2,CDgamma~betas,lP__3τ+P__1τ+p__4τDgamma~tauq,s+mUpKroneckerDeltaq,sKroneckerDeltaD2,Hg_β,λKroneckerDeltaf,hg_κ,σKroneckerDeltaa1,gmUpKroneckerDeltaj1,r+p__4υDgamma~upsilonr,j1KroneckerDeltaD1,D3π6P__3+P__1+p__42mUp2+Physics:-FeynmanDiagrams:-εP__3+p__42+Physics:-FeynmanDiagrams:-εP__3P__2P__1p__42+Physics:-FeynmanDiagrams:-εp__42mUp2+Physics:-FeynmanDiagrams:-ε,p__4%FeynmanIntegral11024Physics:-FeynmanDiagrams:-UspinorUpC,lP__1_conjugatePhysics:-FeynmanDiagrams:-VspinorUpE,mP__2_conjugatePhysics:-FeynmanDiagrams:-UspinorUpF,nP__3_Physics:-FeynmanDiagrams:-VspinorUpG,pP__4_g__s4GlambdahD3,GDgamma~lambdaj1,pGlambdaa1F,D1Dgamma~sigman,rGlambdagE,HDgamma~kappam,qGlambdafD2,CDgamma~betas,lP__2τ+P__3τp__4τDgamma~tauq,s+mUpKroneckerDeltaq,sKroneckerDeltaD2,Hg_β,λKroneckerDeltaf,hg_κ,σKroneckerDeltaa1,gmUpKroneckerDeltaj1,r+p__4υDgamma~upsilonr,j1KroneckerDeltaD1,D3π6P__2+P__3p__42mUp2+Physics:-FeynmanDiagrams:-εP__3P__2P__1p__42+Physics:-FeynmanDiagrams:-εP__3+p__42+Physics:-FeynmanDiagrams:-εp__42mUp2+Physics:-FeynmanDiagrams:-ε,p__4+%FeynmanIntegral11024Physics:-FeynmanDiagrams:-UspinorUpC,lP__1_conjugatePhysics:-FeynmanDiagrams:-VspinorUpE,mP__2_conjugatePhysics:-FeynmanDiagrams:-UspinorUpF,nP__3_Physics:-FeynmanDiagrams:-VspinorUpG,pP__4_g__s4Glambdaa1D3,GDgamma~sigmaj1,pGlambdagF,HDgamma~kappan,qGlambdahE,D1Dgamma~lambdam,rGlambdafD2,CDgamma~betas,lP__3τp__4τDgamma~tauq,s+mUpKroneckerDeltaq,sKroneckerDeltaD2,Hg_β,λKroneckerDeltaf,hP__3υP__2υP__1υp__4υDgamma~upsilonr,j1+mUpKroneckerDeltaj1,rKroneckerDeltaD1,D3g_κ,σKroneckerDeltaa1,gπ6P__3p__42mUp2+Physics:-FeynmanDiagrams:-εP__3P__1p__42+Physics:-FeynmanDiagrams:-εP__3P__2P__1p__42mUp2+Physics:-FeynmanDiagrams:-εp__42+Physics:-FeynmanDiagrams:-ε,p__4+%FeynmanIntegral11024Physics:-FeynmanDiagrams:-UspinorUpC,lP__1_conjugatePhysics:-FeynmanDiagrams:-VspinorUpE,mP__2_conjugatePhysics:-FeynmanDiagrams:-UspinorUpF,nP__3_Physics:-FeynmanDiagrams:-VspinorUpG,pP__4_g__s4Glambdaa1F,HDgamma~sigman,qGlambdagE,D1Dgamma~kappam,rGlambdahD3,GDgamma~lambdaj1,pGlambdafD2,CDgamma~betas,lP__3τ+p__4τDgamma~tauq,s+mUpKroneckerDeltaq,sKroneckerDeltaD2,Hg_β,λKroneckerDeltaf,hP__3υ+P__1υP__4υp__4υDgamma~upsilonr,j1+mUpKroneckerDeltaj1,rKroneckerDeltaD1,D3g_κ,σKroneckerDeltaa1,gπ6P__3+p__42mUp2+Physics:-FeynmanDiagrams:-εP__3P__1+p__42+Physics:-FeynmanDiagrams:-εP__3+P__1P__4p__42mUp2+Physics:-FeynmanDiagrams:-εp__42+Physics:-FeynmanDiagrams:-ε,p__4

(21)

The Electro-Weak Lagrangian

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

Lagrangianelectroweak

(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 Lterm label and the formula it represents

Lagrangianelectroweak,showterms

L__K+L__N+L__C+L__H+L__HV+L__WWV+L__WWVV+L__YwhereL__K=𝔽μ,ν24WPlusFieldStrengthμ,νWMinusFieldStrengthμ,ν2+mW2WPlusFieldμWMinusFieldμμ,ν24+mZ2Zμ22+μHiggsBosonX22mΦ2HiggsBosonX22+%add`*`conjugatef__LjX,Dgammaμj,kd_μf__LkX,Xmf__Lf__LjX,f__L=Electron,Muon,Tauon+%addDgammaμj,k`*`conjugatef__LjX,d_μf__LkX,X,f__L=ElectronNeutrino,MuonNeutrino,TauonNeutrino+%add`*`conjugatef__QA,jX,Dgammaμj,kd_μf__QA,kX,Xmf__Qf__QA,jX,f__Q=Up,Charm,Top,Down,Strange,Bottom,L__N=g__eγμj,k%q__e%add`*`conjugatef__LjX,f__LkX,f__L=Electron,Muon,Tauon+%q__u%add`*`conjugatef__QA,jX,f__QA,kX,f__Q=Up,Charm,Top+%q__d%add`*`conjugatef__QA,jX,f__QA,kX,f__Q=Down,Strange,BottomElectromagneticFieldμX+g__wγμj,kδk,l+γ5k,l%I__e%add`*`conjugatef__LjX,f__LlX,f__L=Electron,Muon,Tauon+%I__n%add`*`conjugatef__LjX,f__LlX,f__L=ElectronNeutrino,MuonNeutrino,TauonNeutrino+%I__u%add`*`conjugatef__QA,jX,f__QA,lX,f__Q=Up,Charm,Top+%I__d%add`*`conjugatef__QA,jX,f__QA,lX,f__Q=Down,Strange,BottomsinWeinbergAngle2γμj,k%q__e%add`*`conjugatef__LjX,f__LkX,f__L=Electron,Muon,Tauon+%q__u%add`*`conjugatef__QA,jX,f__QA,kX,f__Q=Up,Charm,Top+%q__d%add`*`conjugatef__QA,jX,f__QA,kX,f__Q=Down,Strange,BottomZFieldμXcosWeinbergAngle,L__C=g__w2γμj,kδk,l+γ5k,l%add%addCKMf__U,f__D`*`conjugatef__UA,jX,f__DA,lX,f__U=Up,Charm,Top,f__D=Down,Strange,Bottom+%add`*`conjugatef__L1jX,f__L2lX,f__L=ElectronNeutrino,Electron,MuonNeutrino,Muon,TauonNeutrino,TauonWPlusFieldμX+%add%addconjugateCKMf__U,f__D`*`conjugatef__DA,jX,f__UA,lX,f__U=Up,Charm,Top,f__D=Down,Strange,Bottom+%add`*`conjugatef__L2jX,f__L1lX,f__L=ElectronNeutrino,Electron,MuonNeutrino,Muon,TauonNeutrino,TauonWMinusFieldμX2,L__H=g__wmΦ2HiggsBosonX3+HiggsBosonX48mW4mW,L__HV=g__wHiggsBosonXmW+g__w2HiggsBosonX24mW2mW2WPlusFieldμXWMinusFieldμX+ZFieldμX2mZ22,L__WWV=−ⅈg__wWPlusFieldStrengthμ,νXWMinusFieldμXWPlusFieldμXWMinusFieldStrengthμ,νXElectromagneticFieldνXsinWeinbergAngleZFieldνXcosWeinbergAngle+WMinusFieldνXWPlusFieldμXElectromagneticFieldStrengthμ,νXsinWeinbergAngleZFieldStrengthμ,νXcosWeinbergAngle,L__WWVV=g__w22WPlusFieldμXWMinusFieldμX+ElectromagneticFieldμXsinWeinbergAngleZFieldμXcosWeinbergAngle22WPlusFieldνXWMinusFieldνX+ElectromagneticFieldνXsinWeinbergAngleZFieldνXcosWeinbergAngle2+WPlusFieldμXWMinusFieldνX+WPlusFieldνXWMinusFieldμX+ElectromagneticFieldμXsinWeinbergAngleZFieldμXcosWeinbergAngleElectromagneticFieldνXsinWeinbergAngleZFieldνXcosWeinbergAngle24,L__Y=g__w%addmf__L`*`conjugatef__LjX,f__LjX,f__L=Electron,Muon,Tauon,ElectronNeutrino,MuonNeutrino,TauonNeutrino+%addmf__Q`*`conjugatef__QA,jX,f__QA,jX,f__Q=Up,Charm,Top,Down,Strange,BottomHiggsBosonX2mW

(23)

In this result we see a sum of Lterms, and after where there is a list of equations with the formulas represented by each Lterm. Take from the above, for instance, only the charged current LC 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

Lagrangianelectroweak,term=LC

L__C=12g__w2KroneckerDeltak,l+Dgamma5k,l`*`%add%addCKMf__U,f__D`*`conjugatef__UA,jX,f__DA,lX,f__U=Up,Charm,Top,f__D=Down,Strange,Bottom+%add`*`conjugatef__L1jX,f__L2lX,f__L=ElectronNeutrino,Electron,MuonNeutrino,Muon,TauonNeutrino,Tauon,WPlusFieldμX+`*`%add%addconjugateCKMf__U,f__D`*`conjugatef__DA,jX,f__UA,lX,f__U=Up,Charm,Top,f__D=Down,Strange,Bottom+%add`*`conjugatef__L2jX,f__L1lX,f__L=ElectronNeutrino,Electron,MuonNeutrino,Muon,TauonNeutrino,Tauon,WMinusFieldμXDgamma~muj,k

(24)

Lagrangianelectroweak,term=LC,expanded

L__C=12g__w2KroneckerDeltak,l+Dgamma5k,l`*`CKMUp,Down`*`conjugateUpA,jX,DownA,lX+CKMCharm,Down`*`conjugateCharmA,jX,DownA,lX+CKMTop,Down`*`conjugateTopA,jX,DownA,lX+CKMUp,Strange`*`conjugateUpA,jX,StrangeA,lX+CKMCharm,Strange`*`conjugateCharmA,jX,StrangeA,lX+CKMTop,Strange`*`conjugateTopA,jX,StrangeA,lX+CKMUp,Bottom`*`conjugateUpA,jX,BottomA,lX+CKMCharm,Bottom`*`conjugateCharmA,jX,BottomA,lX+CKMTop,Bottom`*`conjugateTopA,jX,BottomA,lX+`*`conjugateElectronNeutrinojX,ElectronlX+`*`conjugateMuonNeutrinojX,MuonlX+`*`conjugateTauonNeutrinojX,TauonlX,WPlusFieldμX+`*`conjugateCKMUp,Down`*`conjugateDownA,jX,UpA,lX+conjugateCKMCharm,Down`*`conjugateDownA,jX,CharmA,lX+conjugateCKMTop,Down`*`conjugateDownA,jX,TopA,lX+conjugateCKMUp,Strange`*`conjugateStrangeA,jX,UpA,lX+conjugateCKMCharm,Strange`*`conjugateStrangeA,jX,CharmA,lX+conjugateCKMTop,Strange`*`conjugateStrangeA,jX,TopA,lX+conjugateCKMUp,Bottom`*`conjugateBottomA,jX,UpA,lX+conjugateCKMCharm,Bottom`*`conjugateBottomA,jX,CharmA,lX+conjugateCKMTop,Bottom`*`conjugateBottomA,jX,TopA,lX+`*`conjugateElectronjX,ElectronNeutrinolX+`*`conjugateMuonjX,MuonNeutrinolX+`*`conjugateTauonjX,TauonNeutrinolX,WMinusFieldμXDgamma~muj,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

FeynmanDiagramsrhs,incoming=Electron&conjugate0;,ElectronNeutrino,outgoing=Muon&conjugate0;,MuonNeutrino,numberofloops=0,diagrams

4vemP__1_&conjugate0;uElectronNeutrinonP__2_vμpP__3_uMuonNeutrinoqP__4_&conjugate0;2g__wγααq,p22g__wγ5r,pγααq,r22g__wγννm,n22g__wγ5s,nγννm,s2gα,ν+P__1ν+P__2νP__1α+P__2αmWMinusField2δP__3ββP__4ββ+P__1ββ+P__2ββπ2P__1κ+P__2κP__1κκ+P__2κκmWMinusField2+ε

(26)

The term LHV of the electroweak Lagrangian contains the interaction between the Higgs and the Z and W bosons

Lagrangianelectroweak,term=LHV,interaction

L__HV=`*`g__wHiggsBosonXmWField+14g__w2`^`HiggsBosonX,2mWField2,mWField2`*`WPlusFieldμ