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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.

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 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,TauonNeutrinoand ElectronNeutrino

Setupmassless

* 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,

(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

CompactDisplayStandardModel:-LeptonsX,StandardModel:-QuarksX,StandardModel:-GaugeFieldsX, HiggsBosonX,quiet 

interfaceimaginaryunit = i:

 

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

ElectromagneticFielddefinition

Aμ=sinWeinbergAngleWμ3μ3+cosWeinbergAngleBμ

(6)

mapu  udefinition,StandardModel:-GaugeFields 

ElectromagneticFieldμ=sinWeinbergAngleWFieldμ,~3+cosWeinbergAngleBFieldμ,ElectromagneticFieldStrengthμ,ν=d_μElectromagneticFieldνX,Xd_νElectromagneticFieldμX,X,BFieldμ=BField1BField2BField3BField4,BFieldStrengthμ,ν=d_μBFieldνX,Xd_νBFieldμX,X,WFieldμ,J=WField1,1WField1,2WField1,3WField2,1WField2,2WField2,3WField3,1WField3,2WField3,3WField4,1WField4,2WField4,3,WFieldStrengthμ,ν,J=d_μWFieldν,JX,Xd_νWFieldμ,JX,X+g__wLeviCivitaJ,K,L`*`WFieldμ,KX,WFieldν,LX,GluonFieldμ,a=GluonField1,1GluonField1,2GluonField1,3GluonField1,4GluonField1,5GluonField1,6GluonField1,7GluonField1,8GluonField2,1GluonField2,2GluonField2,3GluonField2,4GluonField2,5GluonField2,6GluonField2,7GluonField2,8GluonField3,1GluonField3,2GluonField3,3GluonField3,4GluonField3,5GluonField3,6GluonField3,7GluonField3,8GluonField4,1GluonField4,2GluonField4,3GluonField4,4GluonField4,5GluonField4,6GluonField4,7GluonField4,8,GluonFieldStrengthμ,ν,a=d_μGluonFieldν,aX,Xd_νGluonFieldμ,aX,X+g__sFSU3a,b,c`*`GluonFieldμ,bX,GluonFieldν,cX,WMinusFieldμ=12WFieldμ,~1+WFieldμ,~22,WMinusFieldStrengthμ,ν=d_μWMinusFieldνX,Xd_νWMinusFieldμX,X,WPlusFieldμ=12WFieldμ,~1WFieldμ,~22,WPlusFieldStrengthμ,ν=d_μWPlusFieldνX,Xd_νWPlusFieldμX,X,ZFieldμ=cosWeinbergAngleWFieldμ,~3sinWeinbergAngleBFieldμ,ZFieldStrengthμ,ν=d_μZFieldνX,Xd_νZFieldμX,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, uj,A , has a minus sign and the third term in the gluon field strength definition (shown above) has a plus sign:

D_muUpA,jX: % = expand%

D_μUpA,jX,X=d_μUpA,jX,X12g__s`*`GlambdaaA,B,UpB,jX,GluonFieldμ,aX

(8)

GluonFieldStrengthdefinition

GluonFieldStrengthμ,ν,a=d_μGluonFieldν,aX,Xd_νGluonFieldμ,aX,X+g__sFSU3a,b,c`*`GluonFieldμ,bX,GluonFieldν,cX

(9)

The convention for the signs in the definitions of Aμ and Zμin (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

Glambdaa=Glambda1Glambda2Glambda3Glambda4Glambda5Glambda6Glambda7Glambda8

(10)

seqGlambdaa,matrix, a=1..8

λ1=010100000,λ2=0−ⅈ000000,λ3=1000−10000,λ4=001000100,λ5=00−ⅈ00000,λ6=000001010,λ7=00000−ⅈ00,λ8=3300033000233

(11)

These matrices satisfy a SU(3) algebra

Library:-DefaultAlgebraRulesGlambda

%CommutatorGlambdab,Glambdac=2FSU3a,b,cGlambdaa

(12)

The structure constants FSU3a,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.

FSU31,b,c,matrix

FSU31,b,c=00000000001000000100000000000012000000120000001200000012000000000000

(13)

or, for a more general exploration of the components of  FSU3a,b,cyou can use the command TensorArray with the option explore

TensorArrayFSU3a,b,c,explore

FSU3a,b,c ordering of free indices=a,b,c

(14)

Index 1

 

Value of Index 1

 

The tensorial equation for the Gell-Mann matrices

 

%CommutatorGlambdab,Glambdac=2FSU3a,b,cGlambdaa

(15)

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

SumOverRepeatedIndices

%CommutatorGlambdab,Glambdac=2FSU31,b,cGlambda1+FSU32,b,cGlambda2+FSU33,b,cGlambda3+FSU34,b,cGlambda4+FSU35,b,cGlambda5+FSU36,b,cGlambda6+FSU37,b,cGlambda7+FSU38,b,cGlambda8

(16)

eval,b=4,c=5

%CommutatorGlambda4,Glambda5=212Glambda3+123Glambda8

(17)

Activating the left-hand side,

value

2FSU34,5,aλa=2λ32+3λ82

(18)

expandSumOverRepeatedIndices

λ3+3λ8=λ3+3λ8

(19)

To see all the components of (12) ≡ %CommutatorGlambdab,Glambdac=2IFSU3a,b,cλa at once you can use TensorArray

TensorArray

%CommutatorGlambda1,Glambda1=0%CommutatorGlambda1,Glambda2=2IGlambda3%CommutatorGlambda1,Glambda3=2IGlambda2%CommutatorGlambda1,Glambda4=IGlambda7%CommutatorGlambda1,Glambda5=IGlambda6%CommutatorGlambda1,Glambda6=IGlambda5%CommutatorGlambda1,Glambda7=IGlambda4%CommutatorGlambda1,Glambda8=0%CommutatorGlambda2,Glambda1=2IGlambda3%CommutatorGlambda2,Glambda2=0%CommutatorGlambda2,Glambda3=2IGlambda1%CommutatorGlambda2,Glambda4=IGlambda6%CommutatorGlambda2,Glambda5=IGlambda7%CommutatorGlambda2,Glambda6=IGlambda4%CommutatorGlambda2,Glambda7=IGlambda5%CommutatorGlambda2,Glambda8=0%CommutatorGlambda3,Glambda1=2IGlambda2%CommutatorGlambda3,Glambda2=2IGlambda1%CommutatorGlambda3,Glambda3=0%CommutatorGlambda3,Glambda4=IGlambda5%CommutatorGlambda3,Glambda5=IGlambda4%CommutatorGlambda3,Glambda6=IGlambda7%CommutatorGlambda3,Glambda7=IGlambda6%CommutatorGlambda3,Glambda8=0%CommutatorGlambda4,Glambda1=IGlambda7%CommutatorGlambda4,Glambda2=IGlambda6%CommutatorGlambda4,Glambda3=IGlambda5%CommutatorGlambda4,Glambda4=0%CommutatorGlambda4,Glambda5=I3Glambda8+Glambda3%CommutatorGlambda4,Glambda6=IGlambda2%CommutatorGlambda4,Glambda7=IGlambda1%CommutatorGlambda4,Glambda8=I3Glambda5%CommutatorGlambda5,Glambda1=IGlambda6%CommutatorGlambda5,Glambda2=IGlambda7%CommutatorGlambda5,Glambda3=IGlambda4%CommutatorGlambda5,Glambda4=I3Glambda8+Glambda3%CommutatorGlambda5,Glambda5=0%CommutatorGlambda5,Glambda6=IGlambda1%CommutatorGlambda5,Glambda7=IGlambda2%CommutatorGlambda5,Glambda8=I3Glambda4%CommutatorGlambda6,Glambda1=IGlambda5%CommutatorGlambda6,Glambda2=IGlambda4%CommutatorGlambda6,Glambda3=IGlambda7%CommutatorGlambda6,Glambda4=IGlambda2%CommutatorGlambda6,Glambda5=IGlambda1%CommutatorGlambda6,Glambda6=0%CommutatorGlambda6,Glambda7=IGlambda3+3Glambda8%CommutatorGlambda6,Glambda8=I3Glambda7%CommutatorGlambda7,Glambda1=IGlambda4%CommutatorGlambda7,Glambda2=IGlambda5%CommutatorGlambda7,Glambda3=IGlambda6%CommutatorGlambda7,Glambda4=IGlambda1%CommutatorGlambda7,Glambda5=IGlambda2%CommutatorGlambda7,Glambda6=IGlambda3+3Glambda8%CommutatorGlambda7,Glambda7=0%CommutatorGlambda7,Glambda8=I3Glambda6%CommutatorGlambda8,Glambda1=0%CommutatorGlambda8,Glambda2=0%CommutatorGlambda8,Glambda3=0%CommutatorGlambda8,Glambda4=I3Glambda5%CommutatorGlambda8,Glambda5=I3Glambda4%CommutatorGlambda8,Glambda6=I3Glambda7%CommutatorGlambda8,Glambda7=I3Glambda6%CommutatorGlambda8,Glambda8=0

(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

Setupanticommutativeprefix = f__L,f__Q

anticommutativeprefix=f__L,f__Q

(21)

CompactDisplayf__L,f__QX

f__Lx,y,z,twill now be displayed asf__L

f__Qx,y,z,twill now be displayed asf__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__QCDL__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  g__s2Dgammamuk, jGluonFieldmu, aXGlambdaaA, B %addconjugatef__QA,kXf__QB,jX,f__Q=StandardModel:-Quarks

12g__s`*`%add`*`conjugatef__QA,kX,f__QB,jX,f__Q=Up,Charm,Top,Down,Strange,Bottom,GluonFieldμ,aX,GlambdaaA,BDgamma~muk,j

(24)

The self-interactions of the gluons L__GG can be written using the structure constants FSU3d,a,b and the Gell-Mann matrices λa

L__GG  g__s FSU3a, b, cd_muGluonFieldnu, aX, X GluonField~mu, bX GluonField~nu, cX + g__s4FSU3e, d, cGluonFieldmu, aX GluonFieldlambda, bXGluonField~mu, eXGluonField~lambda, dX

g__sFSU3a,b,c`*`d_μGluonFieldν,aX,X,GluonField~mu,bX,GluonField~nu,cX14g__sFSU3c,d,e`*`GluonFieldμ,aX,GluonFieldλ,bX,GluonField~mu,eX,GluonField~lambda,dX

(25)

From where

L__QCD

12g__s`*`%add`*`conjugatef__QA,kX,f__QB,jX,f__Q=Up,Charm,Top,Down,Strange,Bottom,GluonFieldμ,aX,GlambdaaA,BDgamma~muk,jg__sFSU3a,b,c`*`d_μGluonFieldν,aX,X,GluonField~mu,bX,GluonField~nu,cX14g__sFSU3c,d,e`*`GluonFieldμ,aX,GluonFieldλ,bX,GluonField~mu,eX,GluonField~lambda,dX

(26)

L__QCD   valueL__QCD

12g__s`*``*`conjugateUpA,kX,UpB,jX+`*`conjugateCharmA,kX,CharmB,jX+`*`conjugateTopA,kX,TopB,jX+`*`conjugateDownA,kX,DownB,jX+`*`conjugateStrangeA,kX,StrangeB,jX+`*`conjugateBottomA,kX,BottomB,jX,GluonFieldμ,aX,GlambdaaA,BDgamma~muk,jg__sFSU3a,b,c`*`d_μGluonFieldν,aX,X,GluonField~mu,bX,GluonField~nu,cX14g__sFSU3c,d,e`*`GluonFieldμ,aX,GluonFieldλ,bX,GluonField~mu,eX,GluonField~lambda,dX

(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  op1, expandvalueL__QCD  

12g__sDgamma~muk,j`*`conjugateUpA,kX,UpB,jX,GluonFieldμ,aX,GlambdaaA,B

(28)

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

FeynmanDiagramsL__uG, incomingparticles = Up, conjugateUp, outgoingparticles = Up, conjugateUp, numberofloops = 0, diagrams 

uuC,lP__1_vuE,mP__2_&conjugate0;uuF,nP__3_&conjugate0;vuG,pP__4_g__s2γααn,pγννm,lgα,νδb,cδP__3~betaP__4~beta+P__1~beta+P__2~betaλcF,GλbE,C16π2P__1κ+P__2κP__1~kappa+P__2~kappa+ε+uuC,lP__1_vuE,mP__2_&conjugate0;uuF,nP__3_&conjugate0;vuG,pP__4_g__s2γααm,pγννn,lgα,νδb,cδP__3~betaP__4~beta+P__1~beta+P__2~betaλcE,GλbF,C16π2P__1~kappaP__3~kappaP__1κP__3κ+ε

(29)

L__GGGG  op1, expandL__QCD 

14g__s2FSU3a,b,cFSU3c,d,e`*`GluonFieldλ,bX,GluonFieldμ,aX,GluonField~lambda,dX,GluonField~mu,eX

(30)

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

FeynmanDiagramsL__GGGG,incomingparticles=GluonField, GluonField,outgoingparticles=GluonField,GluonField,numberofloops=0,diagrams  

16g__s2δP__3~sigmaP__4~sigma+P__1~sigma+P__2~sigmaϵGν,fP__1_ϵGα,gP__2_ϵGβ,hP__3_&conjugate0;ϵGκ,a1P__4_&conjugate0;FSU3c,g,hFSU3a1,c,fFSU3a1,c,hFSU3c,f,ggβ,νβ,νgα,κα,κ+gκ,νκ,νFSU3c,f,hFSU3a1,c,g+FSU3a1,c,hFSU3c,f,ggα,βα,β+FSU3c,g,hFSU3a1,c,fFSU3c,f,hFSU3a1,c,ggα,να,νgβ,κβ,κπ2E__1E__2E__3E__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__EWL__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 Wμ,J and Bμ vector bosons

L__g  14WFieldStrengthμ,ν,J2+BFieldStrengthμ,ν2

L__g𝕎μ,ν,J𝕎μ,νJμ,νJ4𝔹μ,ν𝔹μ,νμ,ν4

(33)

Introducing the definitions of these tensors we have

BFieldStrengthdefinition,WFieldStrengthdefinition

𝔹μ,ν=μBFieldνXνBFieldμX,𝕎μ,ν,J=μWFieldν,JXνWFieldμ,JX+g__wεJ,K,LWFieldμ,KXWFieldν,LX

(34)

L__g  SubstituteTensor,L__g

14`*`d_μWFieldν,JX,Xd_νWFieldμ,JX,X+g__wLeviCivitaJ,K,L`*`WFieldμ,KX,WFieldν,LX,d_~muWField~nu,JX,Xd_~nuWField~mu,JX,X+g__wLeviCivitaJ,M,N`*`WField~mu,MX,WField~nu,NX14`*`d_μBFieldνX,Xd_νBFieldμX,X,d_~muBField~nuX,Xd_~nuBField~muX,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 Wμ,K and Bμis through the covariant derivative. Note that the electron field ej, 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_muElectronjX:%  = expand%

D_μElectronjX,X=d_μElectronjX,X+g__e`*`ElectronjX,ElectromagneticFieldμX

(36)

D_muUpA,jX:% = expand%

D_μUpA,jX,X=d_μUpA,jX,X12g__s`*`GlambdaaA,B,UpB,jX,GluonFieldμ,aX

(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__EWL__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 Aμand Zμ.

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

L__K  14ElectromagneticFieldStrengthmu,nu2  12WPlusFieldStrengthmu,nuWMinusFieldStrengthmu,nu+12mWField2WPlusFieldmuWMinusFieldmu 14ZFieldStrengthmu,nu2+12mZField2ZFieldmu2+12d_muHiggsBosonX2mHiggsBoson22HiggsBosonX2 +%addconjugatef__LjXDgammamu j,kid_muf__LkX  mf__Lf__LjX, f__L = StandardModel:-Leptons1..3 +%addconjugatef__LjXDgammamu j,kid_muf__LkX , f__L = StandardModel:-Leptons4..6 +%addconjugatef__QA,jXDgammamuj,kid_muf__QA,kX  mf__Qf__QA,jX, f__Q = StandardModel:-Quarks 

14`*`ElectromagneticFieldStrengthμ,ν,ElectromagneticFieldStrength~mu,~nu12`*`WPlusFieldStrengthμ,ν,WMinusFieldStrength~mu,~nu+12mWField2`*`WPlusFieldμ,WMinusField~mu14`*`ZFieldStrengthμ,ν,ZFieldStrength~mu,~nu+12mZField2`*`ZFieldμ,ZField~mu+12`*`d_μHiggsBosonX,X,d_~muHiggsBosonX,X12mHiggsBoson2`^`HiggsBosonX,2+%add`*`conjugatef__LjX,d_μf__LkX,XDgamma~muj,kmf__Lf__LjX,f__L=Electron,Muon,Tauon+%add`*`conjugatef__LjX,d_μf__LkX,XDgamma~muj,k,f__L=ElectronNeutrino,MuonNeutrino,TauonNeutrino+%add`*`conjugatef__QA,jX,d_μf__QA,kX,XDgamma~muj,kmf__Qf__QA,jX,f__Q=Up,Charm,Top,Down,Strange,Bottom

(39)

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

value

14`*`ElectromagneticFieldStrengthμ,ν,ElectromagneticFieldStrength~mu,~nu12`*`WPlusFieldStrengthμ,ν,WMinusFieldStrength~mu,~nu+12mWField2`*`WPlusFieldμ,WMinusField~mu14`*`ZFieldStrengthμ,ν,ZFieldStrength~mu,~nu+12mZField2`*`ZFieldμ,ZField~mu+12`*`d_μHiggsBosonX,X,d_~muHiggsBosonX,X12mHiggsBoson2`^`HiggsBosonX,2+`*`conjugateElectronjX,d_μElectronkX,XDgamma~muj,kmElectronElectronjX+`*`conjugateMuonjX,d_μMuonkX,XDgamma~muj,kmMuonMuonjX+`*`conjugateTauonjX,d_μTauonkX,XDgamma~muj,kmTauonTauonjX+`*`conjugateElectronNeutrinojX,d_μElectronNeutrinokX,XDgamma~muj,k+`*`conjugateMuonNeutrinojX,d_μMuonNeutrinokX,XDgamma~muj,k+`*`conjugateTauonNeutrinojX,d_μTauonNeutrinokX,XDgamma~muj,k+`*`conjugateUpA,jX,d_μUpA,kX,XDgamma~muj,kmUpUpA,jX+`*`conjugateCharmA,jX,d_μCharmA,kX,XDgamma~muj,kmCharmCharmA,jX+`*`conjugateTopA,jX,d_μTopA,kX,XDgamma~muj,kmTopTopA,jX+`*`conjugateDownA,jX,d_μDownA,kX,XDgamma~muj,kmDownDownA,jX+`*`conjugateStrangeA,jX,d_μStrangeA,kX,XDgamma~muj,kmStrangeStrangeA,jX+`*`conjugateBottomA,jX,d_μBottomA,kX,XDgamma~muj,kmBottomBottomA,jX

(40)

Introducing the definition of the field strengths 𝔽μ,ν, WPlusFieldStrengthμ,ν, WMinusFieldStrengthμ,ν and μ,ν

ElectromagneticFieldStrengthdefinition

ElectromagneticFieldStrengthμ,ν=d_μElectromagneticFieldνX,Xd_νElectromagneticFieldμX,X

(41)

WPlusFieldStrengthdefinition

WPlusFieldStrengthμ,ν=d_μWPlusFieldνX,Xd_νWPlusFieldμX,X

(42)

WMinusFieldStrengthdefinition

WMinusFieldStrengthμ,ν=d_μWMinusFieldνX,Xd_νWMinusFieldμX,X

(43)

ZFieldStrengthdefinition 

ZFieldStrengthμ,ν=d_μZFieldνX,Xd_νZFieldμX,X

(44)

L__K  SubstituteTensor,,,,

14`*`d_μElectromagneticFieldνX,Xd_νElectromagneticFieldμX,X,d_~muElectromagneticField~nuX,Xd_~nuElectromagneticField~muX,X12`*`d_μWPlusFieldνX,Xd_νWPlusFieldμX,X,d_~muWMinusField~nuX,Xd_~nuWMinusField~muX,X+12mWField2`*`WPlusFieldμ,WMinusField~mu14`*`d_μZFieldνX,Xd_νZFieldμX,X,d_~muZField~nuX,Xd_~nuZField~muX,X+12mZField2`*`ZFieldμ,ZField~mu+12`*`d_μHiggsBosonX,X,d_~muHiggsBosonX,X12mHiggsBoson2`^`HiggsBosonX,2+`*`conjugateElectronjX,d_μElectronkX,XDgamma~muj,kmElectronElectronjX+`*`conjugateMuonjX,d_μMuonkX,XDgamma~muj,kmMuonMuonjX+`*`conjugateTauonjX,d_μTauonkX,XDgamma~muj,kmTauonTauonjX+`*`conjugateElectronNeutrinojX,d_μElectronNeutrinokX,XDgamma~muj,k+`*`conjugateMuonNeutrinojX,d_μMuonNeutrinokX,XDgamma~muj,k+`*`conjugateTauonNeutrinojX,d_μTauonNeutrinokX,XDgamma~muj,k+`*`conjugateUpA,jX,d_μUpA,kX,XDgamma~muj,kmUpUpA,jX+`*`conjugateCharmA,jX,d_μCharmA,kX,XDgamma~muj,kmCharmCharmA,jX+`*`conjugateTopA,jX,d_μTopA,kX,XDgamma~muj,kmTopTopA,jX+`*`conjugateDownA,jX,d_μDownA,kX,XDgamma~muj,kmDownDownA,jX+`*`conjugateStrangeA,jX,d_μStrangeA,kX,XDgamma~muj,kmStrangeStrangeA,jX+`*`conjugateBottomA,jX,d_μBottomA,kX,XDgamma~muj,kmBottomBottomA,jX

(45)

The neutral current Lagrangian containing the interactions between fermions and the gauge bosons Aμand Zμ is expressed in terms of the electromagnetic and weak currents J__E, μ and J__W,μ as

L__N g__e JE, μElectromagneticFieldμX + g__wcosWeinbergAngleJW,mu  sinWeinbergAngle2JE,muZFieldmuX

g__eJE,μElectromagneticFieldμX+g__wJW,μsinWeinbergAngle2JE,μZFieldμXcosWeinbergAngle

(46)

In turn, these currents are expressed as

JE,μ  %q__eDgammamuk, j%addconjugatef__LkXf__LjX,f__L=Electron, Muon, Tauon +%q__uDgammamuk, j%addconjugatef__QA,kXf__QA,jX,f__Q=Up,Charm,Top +%q__dDgammamuk, j%addconjugatef__QA,kXf__QA,jX,f__Q=Down,Strange,Bottom

%q__eDgammaμk,j%add`*`conjugatef__LkX,f__LjX,f__L=Electron,Muon,Tauon+%q__uDgammaμk,j%add`*`conjugatef__QA,kX,f__QA,jX,f__Q=Up,Charm,Top+%q__dDgammaμk,j%add`*`conjugatef__QA,kX,f__QA,jX,f__Q=Down,Strange,Bottom

(47)

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

JE,mu eval,%add = add

%q__eDgammaμk,j`*`conjugateElectronkX,ElectronjX+`*`conjugateMuonkX,MuonjX+`*`conjugateTauonkX,TauonjX+%q__uDgammaμk,j`*`conjugateUpA,kX,UpA,jX+`*`conjugateCharmA,kX,CharmA,jX+`*`conjugateTopA,kX,TopA,jX+%q__dDgammaμk,j`*`conjugateDownA,kX,DownA,jX+`*`conjugateStrangeA,kX,StrangeA,jX+`*`conjugateBottomA,kX,BottomA,jX

(48)

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

JE,mu  value

Dgammaμk,j`*`conjugateElectronkX,ElectronjX+`*`conjugateMuonkX,MuonjX+`*`conjugateTauonkX,TauonjX+23Dgammaμk,j`*`conjugateUpA,kX,UpA,jX+`*`conjugateCharmA,kX,CharmA,jX+`*`conjugateTopA,kX,TopA,jX13Dgammaμk,j`*`conjugateDownA,kX,DownA,jX+`*`conjugateStrangeA,kX,StrangeA,jX+`*`conjugateBottomA,kX,BottomA,jX

(49)

For the weak current, from the Wikipedia reference mentioned,

JW,μ  Dgammamuk, jKroneckerDeltaj,lDgamma5j,l%I__e%addconjugatef__LkXf__LlX,f__L=StandardModel:-Leptons1..3 +%I__n%addconjugatef__LkXf__LlX,f__L=StandardModel:-Leptons4..6 +%I__u%addconjugatef__QA,kXf__QA,lX,f__Q=StandardModel:-Quarks1..3  +%I__d%addconjugatef__QA,kXf__QA,lX,f__Q=StandardModel:-Quarks4..6 

Dgammaμk,jKroneckerDeltaj,lDgamma5j,l%I__e%add`*`conjugatef__LkX,f__LlX,f__L=Electron,Muon,Tauon+%I__n%add`*`conjugatef__LkX,f__LlX,f__L=ElectronNeutrino,MuonNeutrino,TauonNeutrino+%I__u%add`*`conjugatef__QA,kX,f__QA,lX,f__Q=Up,Charm,Top+%I__d%add`*`conjugatef__QA,kX,f__QA,lX,f__Q=Down,Strange,Bottom

(50)

To activate only the sums,

JW,μ  eval,%add = add

Dgammaμk,jKroneckerDeltaj,lDgamma5j,l%I__e`*`conjugateElectronkX,ElectronlX+`*`conjugateMuonkX,MuonlX+`*`conjugateTauonkX,TauonlX+%I__n`*`conjugateElectronNeutrinokX,ElectronNeutrinolX+`*`conjugateMuonNeutrinokX,MuonNeutrinolX+`*`conjugateTauonNeutrinokX,TauonNeutrinolX+%I__u`*`conjugateUpA,kX,UpA,lX+`*`conjugateCharmA,kX,CharmA,lX+`*`conjugateTopA,kX,TopA,lX+