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Overview of the Physics:-StandardModel package

 

Description

List of StandardModel Package Commands

References

Compatibility

Description

• 

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, Weinberg's angle, the Higgs boson, the fields and field strengths after breaking symmetries and most of the fields before that, the structure constants FSU3a,band Gell-Mann matrices λa, the electroweak charges and isospins, and also a command, Lagrangian, to retrieve the Lagrangian of the different sectors of the Standard Model like QED, QCD or Electro-Weak, or of all of it.

• 

Loading the package sets things to proceed computing with the model, as the fields and their computational properties, the commutation rules, covariant derivatives, the values of all the constants and matrices involved, the letters representing different kinds of tensor indices etc. Some of these settings, for instance the kinds of letters used to represent different kinds of indices, can be changed using the Physics Setup command.

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: uj,A , cj,A , tj,A , dj,A , sj,A , bj,A

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)
• 

A key observation in this implementation is that, with the exception of Lagrangian, FSU3 representing the structure constants FSU3a,b and Glambda representing the Gell-Mann matrices λa, the package's commands are not actually routines that perform tasks but representations of the physical fields of the model with the mathematical properties set and understood by the Maple system. Once the representations are in place, the actual computations are performed by the Physics package commands, e.g. scattering amplitudes are computed using FeynmanDiagrams, covariant and non-covariant derivatives are computed using D_ and d_, commutators and anticommutators using Commutator and AntiCommutator, and the simplification of tensor indices is performed by the Physics Simplify command. The Lagrangian command is used to retrieve all or part of the Lagrangian of the Standard Model, as shown in the Examples.

List of StandardModel Package Commands

• 

The following is a list of available commands.

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

 

%I__d

%I__e

%I__n

%I__u

%q__d

%q__e

%q__u

 

 

• 

The electric part of the electroweak charges and isospins of the model are all represented in inert form following the Maple convention: prefixed with the % symbol displayed in gray as in (1). If any of these constants is entered without that prefix you get its actual value; for the electric part of the electroweak charges of the model, in units of g__e, that is q__e = −1, q__u = 23, q__d = 13 respectively for the 1) the electron, muon and tauon, 2) the up, charm and top quarks and 3) the down strange and bottom quarks. For the isospins that is I__e = 12, I__u = 12, I__d = 12, I__n = 12, respectively for 1) the electron, muon and tauon, 2) the up, charm and top, 3) the down, strange and bottom, and 4) all the neutrinos.

• 

The electromagnetic coupling constant g__e is negative, equal to the charge of the electron, and g__s and g__w are respectively the the strong and weak coupling constants. These are the constants that enter as a factor in the gauge term of covariant derivatives, and so in the definition of the gauge field strengths 𝔾μ,ν,a and 𝕎μ,ν,J and in the interaction terms of the Lagrangian.

• 

The CKM  command is a representation for the Cabibbo Kobayashi Maskawa matrix.

• 

Further information relevant to the use of these commands is found under Conventions used in the Physics package (spacetime tensors, not commutative variables and functions, quantum states and Dirac notation).

• 

The typesetting of the package's representations for fields and constant follows as much as possible the one in textbooks and works under copy and paste - so that expressions from the output can be reused as input. To transform Maple worksheets using this package into LaTeX files see LaTeX.

Examples

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:-Leptons,StandardModel:-Quarks,StandardModel:-GaugeFields, 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)

The convention in (7) for the signs in the definitions of Aμ (using +sin) and Zμ(using -sin)  also follows ref [1] (page 702) and [3] (page 66), and the presentation of the Standard Model in Wikipedia, but this convention is not uniform in the literature, and ref [2] (vol II, page 307) uses the opposite convention for these signs.

 

The definitions of covariant derivatives (not shown above) follow references [1] (pages 690 and 802) and [2] (pages 355 of vol I and 155 of vol II), also the Wikipedia (webpages for QED and QCD) . E.g. for the Electron ej

D_muElectronjX: % = expand% 

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

(8)

where g__e<0 is equal to the electric charge of the electron, and for the Top, uj,A 

D_muUpj&comma;AX&colon; % &equals; expand%

D_&mu;Upj&comma;AX&comma;X&equals;d_&mu;Upj&comma;AX&comma;X12g__s`*`GlambdaaA&comma;B&comma;Upj&comma;BX&comma;GluonField&mu;&comma;aX

(9)

 

where g__s&gt;0 is the strong coupling constant. 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&equals;Glambda1Glambda2Glambda3Glambda4Glambda5Glambda6Glambda7Glambda8

(10)

seqGlambdaa&comma;matrix&comma; a&equals;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&comma;Glambdac&equals;2FSU3a&comma;b&comma;cGlambdaa

(12)

The structure constants FSU3a&comma;b&comma;centering (12) and interaction Lagrangian terms of the StandardModel form a 3-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&comma;b&comma;c&comma;matrix

FSU31&comma;b&comma;c&equals;00000000001000000100000000000012000000120000001200000012000000000000

(13)

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

TensorArrayFSU3a&comma;b&comma;c&comma;explore

FSU3a&comma;b&comma;c ordering of free indices=a&comma;b&comma;c

(14)

Index 1

 

Value of Index 1

 

The tensorial equation for the Gell-Mann matrices

 

%CommutatorGlambdab&comma;Glambdac&equals;2FSU3a&comma;b&comma;cGlambdaa

(15)

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

SumOverRepeatedIndices

%CommutatorGlambdab&comma;Glambdac&equals;2FSU31&comma;b&comma;cGlambda1&plus;FSU32&comma;b&comma;cGlambda2&plus;FSU33&comma;b&comma;cGlambda3&plus;FSU34&comma;b&comma;cGlambda4&plus;FSU35&comma;b&comma;cGlambda5&plus;FSU36&comma;b&comma;cGlambda6&plus;FSU37&comma;b&comma;cGlambda7&plus;FSU38&comma;b&comma;cGlambda8

(16)

eval&comma;b&equals;4&comma;c&equals;5

%CommutatorGlambda4&comma;Glambda5&equals;212Glambda3&plus;123Glambda8

(17)

Activating the left-hand side,

value

2FSU34&comma;5&comma;aλa=2λ32+3λ82

(18)

expandSumOverRepeatedIndices

λ3+3λ8=λ3+3λ8

(19)

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

TensorArray

%CommutatorGlambda1&comma;Glambda1&equals;0%CommutatorGlambda1&comma;Glambda2&equals;2IGlambda3%CommutatorGlambda1&comma;Glambda3&equals;2IGlambda2%CommutatorGlambda1&comma;Glambda4&equals;IGlambda7%CommutatorGlambda1&comma;Glambda5&equals;IGlambda6%CommutatorGlambda1&comma;Glambda6&equals;IGlambda5%CommutatorGlambda1&comma;Glambda7&equals;IGlambda4%CommutatorGlambda1&comma;Glambda8&equals;0%CommutatorGlambda2&comma;Glambda1&equals;2IGlambda3%CommutatorGlambda2&comma;Glambda2&equals;0%CommutatorGlambda2&comma;Glambda3&equals;2IGlambda1%CommutatorGlambda2&comma;Glambda4&equals;IGlambda6%CommutatorGlambda2&comma;Glambda5&equals;IGlambda7%CommutatorGlambda2&comma;Glambda6&equals;IGlambda4%CommutatorGlambda2&comma;Glambda7&equals;IGlambda5%CommutatorGlambda2&comma;Glambda8&equals;0%CommutatorGlambda3&comma;Glambda1&equals;2IGlambda2%CommutatorGlambda3&comma;Glambda2&equals;2IGlambda1%CommutatorGlambda3&comma;Glambda3&equals;0%CommutatorGlambda3&comma;Glambda4&equals;IGlambda5%CommutatorGlambda3&comma;Glambda5&equals;IGlambda4%CommutatorGlambda3&comma;Glambda6&equals;IGlambda7%CommutatorGlambda3&comma;Glambda7&equals;IGlambda6%CommutatorGlambda3&comma;Glambda8&equals;0%CommutatorGlambda4&comma;Glambda1&equals;IGlambda7%CommutatorGlambda4&comma;Glambda2&equals;IGlambda6%CommutatorGlambda4&comma;Glambda3&equals;IGlambda5%CommutatorGlambda4&comma;Glambda4&equals;0%CommutatorGlambda4&comma;Glambda5&equals;I3Glambda8&plus;Glambda3%CommutatorGlambda4&comma;Glambda6&equals;IGlambda2%CommutatorGlambda4&comma;Glambda7&equals;IGlambda1%CommutatorGlambda4&comma;Glambda8&equals;I3Glambda5%CommutatorGlambda5&comma;Glambda1&equals;IGlambda6%CommutatorGlambda5&comma;Glambda2&equals;IGlambda7%CommutatorGlambda5&comma;Glambda3&equals;IGlambda4%CommutatorGlambda5&comma;Glambda4&equals;I3Glambda8&plus;Glambda3%CommutatorGlambda5&comma;Glambda5&equals;0%CommutatorGlambda5&comma;Glambda6&equals;IGlambda1%CommutatorGlambda5&comma;Glambda7&equals;IGlambda2%CommutatorGlambda5&comma;Glambda8&equals;I3Glambda4%CommutatorGlambda6&comma;Glambda1&equals;IGlambda5%CommutatorGlambda6&comma;Glambda2&equals;IGlambda4%CommutatorGlambda6&comma;Glambda3&equals;IGlambda7%CommutatorGlambda6&comma;Glambda4&equals;IGlambda2%CommutatorGlambda6&comma;Glambda5&equals;IGlambda1%CommutatorGlambda6&comma;Glambda6&equals;0%CommutatorGlambda6&comma;Glambda7&equals;IGlambda3&plus;3Glambda8%CommutatorGlambda6&comma;Glambda8&equals;I3Glambda7%CommutatorGlambda7&comma;Glambda1&equals;IGlambda4%CommutatorGlambda7&comma;Glambda2&equals;IGlambda5%CommutatorGlambda7&comma;Glambda3&equals;IGlambda6%CommutatorGlambda7&comma;Glambda4&equals;IGlambda1%CommutatorGlambda7&comma;Glambda5&equals;IGlambda2%CommutatorGlambda7&comma;Glambda6&equals;IGlambda3&plus;3Glambda8%CommutatorGlambda7&comma;Glambda7&equals;0%CommutatorGlambda7&comma;Glambda8&equals;I3Glambda6%CommutatorGlambda8&comma;Glambda1&equals;0%CommutatorGlambda8&comma;Glambda2&equals;0%CommutatorGlambda8&comma;Glambda3&equals;0%CommutatorGlambda8&comma;Glambda4&equals;I3Glambda5%CommutatorGlambda8&comma;Glambda5&equals;I3Glambda4%CommutatorGlambda8&comma;Glambda6&equals;I3Glambda7%CommutatorGlambda8&comma;Glambda7&equals;I3Glambda6%CommutatorGlambda8&comma;Glambda8&equals;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 &equals; f__L&comma;f__Q

anticommutativeprefix=f__L&comma;f__Q

(21)

CompactDisplayf__L&comma;f__QX

f__LXwill now be displayed asf__L

f__QXwill now be displayed asf__Q

(22)

 

The Quantum Electrodynamics (QED) sector of the Standard Model and its interaction Lagrangian

 

QED is about the interaction between electrons and the photon. The electron field is implemented as a Dirac spinor Electroni, displayed ei, and the electromagnetic field as a spacetime tensor ElectromagneticFieldmu displayed Aμ. The interaction Lagrangian, used to compute scattering amplitudes, can be input as

L__QED  g__e conjugateElectroniXDgamma~mui&comma; jElectronjXElectromagneticFieldmuX

g__eDgamma~mu&comma;j`*`conjugateElectronX&comma;ElectronjX&comma;ElectromagneticField&mu;X

(23)

For example, the self-energies of the electron and the photon are given by

FeynmanDiagramsL__QED&comma;incomingparticles=Electron&comma;outgoing=Electron&comma;numberofloops=1&comma;diagrams  

%FeynmanIntegral18Physics:-FeynmanDiagrams:-UspinorElectronkP__1_conjugatePhysics:-FeynmanDiagrams:-UspinorElectronlP__2_g__e2Dgamma~alphal&comma;mDgamma~nun&comma;kP__1&beta;&plus;p__2&beta;Dgamma~betam&comma;n&plus;mElectronKroneckerDeltam&comma;ng_&alpha;&comma;&nu;DiracP__2&plus;P__1&pi;3P__1&plus;p__22mElectron2&plus;Physics:-FeynmanDiagrams:-&epsilon;p__22&plus;Physics:-FeynmanDiagrams:-&epsilon;&comma;p__2

(24)

FeynmanDiagramsL__QED&comma;incomingparticles=ElectromagneticField&comma;outgoing=ElectromagneticField&comma;numberofloops=1&comma;diagrams  

%FeynmanIntegral116Physics:-FeynmanDiagrams:-PolarizationVectorElectromagneticField&nu;P__1_conjugatePhysics:-FeynmanDiagrams:-PolarizationVectorElectromagneticField&alpha;P__2_g__e2Dgamma~alphan&comma;kDgamma~num&comma;lP__1&beta;&plus;p__2&beta;Dgamma~betak&comma;m&plus;mElectronKroneckerDeltak&comma;mmElectronKroneckerDeltal&comma;n&plus;p__2&kappa;Dgamma~kappal&comma;nDiracP__2&plus;P__1&pi;3E__1E__2P__1&plus;p__22mElectron2&plus;Physics:-FeynmanDiagrams:-&epsilon;p__22mElectron2&plus;Physics:-FeynmanDiagrams:-&epsilon;&comma;p__2

(25)

To perform different manipulations of Feynman integrals, or their full evaluation when possible, see the FeynmanIntegral package.

The Lagrangian of QED can also be retrieved using the Lagrangian command

LagrangianQED

`*`conjugateElectronjX&comma;Dgamma&mu;j&comma;kD_~mumElectronKroneckerDeltaj&comma;k&comma;ElectronkX14`*`ElectromagneticFieldStrength&mu;&comma;&nu;&comma;ElectromagneticFieldStrength~mu&comma;~nu

(26)

Note the above is the full Lagrangian, not just the interaction part. It includes the trace of the electromagnetic field strength (last term) and is expressed using the covariant derivative operator μμ. The notation used is product notation to represent the application of the differential operator. To transform this product notation into the application of the operator you can use

Library:-ApplyProductsOfDifferentialOperators

`*`conjugateElectronjX&comma;Dgamma&mu;j&comma;kD_~muElectronkX&comma;X`*`mElectron&comma;KroneckerDeltaj&comma;k&comma;ElectronkX14`*`ElectromagneticFieldStrength&mu;&comma;&nu;&comma;ElectromagneticFieldStrength~mu&comma;~nu

(27)

or optionally request to Lagrangian for the operator to appear applied

LagrangianQED&comma; applied

`*`conjugateElectronjX&comma;D_&mu;ElectronkX&comma;XDgamma~muj&comma;k`*`mElectron&comma;KroneckerDeltaj&comma;k&comma;ElectronkX14`*`ElectromagneticFieldStrength&mu;&comma;&nu;&comma;ElectromagneticFieldStrength~mu&comma;~nu

(28)

To expand the covariant derivative, use expand

D_muElectronkX&colon; % &equals;expand%

D_&mu;ElectronkX&comma;X&equals;d_&mu;ElectronkX&comma;X&plus;g__e`*`ElectronkX&comma;ElectromagneticField&mu;X

(29)

 

from where

expand

Dgamma~muj&comma;k`*`conjugateElectronjX&comma;d_&mu;ElectronkX&comma;XDgamma~muj&comma;kg__e`*`conjugateElectronjX&comma;ElectronkX&comma;ElectromagneticField&mu;XmElectronKroneckerDeltaj&comma;k`*`conjugateElectronjX&comma;ElectronkX14`*`ElectromagneticFieldStrength&mu;&comma;&nu;&comma;ElectromagneticFieldStrength~mu&comma;~nu

(30)

You can also optionally request the Lagrangian to appear expanded, in which case 𝔽μ,ν is also expanded using its definition

ElectromagneticFieldStrengthdefinition

ElectromagneticFieldStrength&mu;&comma;&nu;&equals;d_&mu;ElectromagneticField&nu;X&comma;Xd_&nu;ElectromagneticField&mu;X&comma;X

(31)

LagrangianQED&comma; expanded

`*`conjugateElectronjX&comma;d_&mu;ElectronkX&comma;XDgamma~muj&comma;kg__e`*`conjugateElectronjX&comma;ElectronkX&comma;ElectromagneticField&mu;XDgamma~muj&comma;kmElectron`*`conjugateElectronjX&comma;ElectronjX14`*`d_&mu;ElectromagneticField&nu;X&comma;Xd_&nu;ElectromagneticField&mu;X&comma;X&comma;d_~muElectromagneticField~nuX&comma;Xd_~nuElectromagneticField~muX&comma;X

(32)

To get only the interaction Lagrangian part of this expression you can use the keyword interaction

LagrangianQED&comma;interaction

Dgamma~muj&comma;kg__e`*`conjugateElectronjX&comma;ElectronkX&comma;ElectromagneticField&mu;X

(33)

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&plus;L__GG

L__QCDL__QG+L__GG

(34)

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&comma; jGluonFieldmu&comma; aXGlambdaaA&comma; B %addconjugatef__Qk&comma; AXf__Qj&comma; BX&comma;f__Q&equals;StandardModel:-Quarks

12g__s`*`%add`*`conjugatef__Qk&comma;AX&comma;f__Qj&comma;BX&comma;f__Q&equals;Up&comma;Charm&comma;Top&comma;Down&comma;Strange&comma;Bottom&comma;GluonField&mu;&comma;aX&comma;GlambdaaA&comma;BDgamma~muk&comma;j

(35)

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

L__GG  g__sFSU3a&comma; b&comma; cd_muGluonFieldnu&comma; aX&comma; X GluonField~mu&comma; bX GluonField~nu&comma; cX&plus; g__s4FSU3e&comma; d&comma; cGluonFieldmu&comma; aX GluonFieldlambda&comma; bXGluonField~mu&comma; eXGluonField~lambda&comma; dX

g__sFSU3a&comma;b&comma;c`*`d_&mu;GluonField&nu;&comma;aX&comma;X&comma;GluonField~mu&comma;bX&comma;GluonField~nu&comma;cX14g__sFSU3c&comma;d&comma;e`*`GluonField&mu;&comma;aX&comma;GluonField&lambda;&comma;bX&comma;GluonField~mu&comma;eX&comma;GluonField~lambda&comma;dX

(36)

From where

L__QCD

12g__s`*`%add`*`conjugatef__Qk&comma;AX&comma;f__Qj&comma;BX&comma;f__Q&equals;Up&comma;Charm&comma;Top&comma;Down&comma;Strange&comma;Bottom&comma;GluonField&mu;&comma;aX&comma;GlambdaaA&comma;BDgamma~muk&comma;jg__sFSU3a&comma;b&comma;c`*`d_&mu;GluonField&nu;&comma;aX&comma;X&comma;GluonField~mu&comma;bX&comma;GluonField~nu&comma;cX14g__sFSU3c&comma;d&comma;e`*`GluonField&mu;&comma;aX&comma;GluonField&lambda;&comma;bX&comma;GluonField~mu&comma;eX&comma;GluonField~lambda&comma;dX

(37)

This QCD Lagrangian can also be retrieved using Lagrangian

LagrangianQCD&comma;expanded

`*`conjugateUpj&comma;AX&comma;d_&mu;Upk&comma;AX&comma;X12g__s`*`GlambdaaA&comma;B&comma;Upk&comma;BX&comma;GluonField&mu;&comma;aXDgamma~muj&comma;kmUpKroneckerDeltaj&comma;kUpk&comma;AX&plus;`*`conjugateCharmj&comma;AX&comma;d_&mu;Charmk&comma;AX&comma;X12g__s`*`GlambdaaA&comma;B&comma;Charmk&comma;BX&comma;GluonField&mu;&comma;aXDgamma~muj&comma;kmCharmKroneckerDeltaj&comma;kCharmk&comma;AX&plus;`*`conjugateTopj&comma;AX&comma;d_&mu;Topk&comma;AX&comma;X12g__s`*`GlambdaaA&comma;B&comma;Topk&comma;BX&comma;GluonField&mu;&comma;aXDgamma~muj&comma;kmTopKroneckerDeltaj&comma;kTopk&comma;AX&plus;`*`conjugateDownj&comma;AX&comma;d_&mu;Downk&comma;AX&comma;X12g__s`*`GlambdaaA&comma;B&comma;Downk&comma;BX&comma;GluonField&mu;&comma;aXDgamma~muj&comma;kmDownKroneckerDeltaj&comma;kDownk&comma;AX&plus;`*`conjugateStrangej&comma;AX&comma;d_&mu;Strangek&comma;AX&comma;X12g__s`*`GlambdaaA&comma;B&comma;Strangek&comma;BX&comma;GluonField&mu;&comma;aXDgamma~muj&comma;kmStrangeKroneckerDeltaj&comma;kStrangek&comma;AX&plus;`*`conjugateBottomj&comma;AX&comma;d_&mu;Bottomk&comma;AX&comma;X12g__s`*`GlambdaaA&comma;B&comma;Bottomk&comma;BX&comma;GluonField&mu;&comma;aXDgamma~muj&comma;kmBottomKroneckerDeltaj&comma;kBottomk&comma;AX14`*`d_&mu;GluonField&nu;&comma;aX&comma;Xd_&nu;GluonField&mu;&comma;aX&comma;X&plus;g__sFSU3a&comma;b&comma;c`*`GluonField&mu;&comma;bX&comma;GluonField&nu;&comma;cX&comma;d_~muGluonField~nu&comma;aX&comma;Xd_~nuGluonField~mu&comma;aX&comma;X&plus;g__sFSU3a&comma;d&comma;e`*`GluonField~mu&comma;dX&comma;GluonField~nu&comma;eX

(38)

LagrangianQCD&comma;interaction 

12g__s`*`%add`*`conjugatef__Qj&comma;AX&comma;f__Qk&comma;BX&comma;f__Q&equals;Up&comma;Charm&comma;Top&comma;Down&comma;Strange&comma;Bottom&comma;GlambdaaA&comma;B&comma;GluonField&mu;&comma;aXDgamma~muj&comma;kg__sFSU3a&comma;b&comma;c`*`d_&mu;GluonField&nu;&comma;aX&comma;X&comma;GluonField~mu&comma;bX&comma;GluonField~nu&comma;cX14g__sFSU3c&comma;d&comma;e`*`GluonField&mu;&comma;aX&comma;GluonField&alpha;&comma;bX&comma;GluonField~mu&comma;eX&comma;GluonField~alpha&comma;dX

(39)

L__QCD  LagrangianQCD&comma;interaction&comma;expanded  

12g__s`*``*`conjugateUpj&comma;AX&comma;Upk&comma;BX&plus;`*`conjugateCharmj&comma;AX&comma;Charmk&comma;BX&plus;`*`conjugateTopj&comma;AX&comma;Topk&comma;BX&plus;`*`conjugateDownj&comma;AX&comma;Downk&comma;BX&plus;`*`conjugateStrangej&comma;AX&comma;Strangek&comma;BX&plus;`*`conjugateBottomj&comma;AX&comma;Bottomk&comma;BX&comma;GlambdaaA&comma;B&comma;GluonField&mu;&comma;aXDgamma~muj&comma;kg__sFSU3a&comma;b&comma;c`*`d_&mu;GluonField&nu;&comma;aX&comma;X&comma;GluonField~mu&comma;bX&comma;GluonField~nu&comma;cX14g__sFSU3c&comma;d&comma;e`*`GluonField&mu;&comma;aX&comma;GluonField&alpha;&comma;bX&comma;GluonField~mu&comma;eX&comma;GluonField~alpha&comma;dX

(40)

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&comma; expandL__QCD   

12g__sDgamma~muj&comma;k`*`conjugateUpj&comma;AX&comma;Upk&comma;BX&comma;GlambdaaA&comma;B&comma;GluonField&mu;&comma;aX

(41)

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

FeynmanDiagramsL__UG&comma; incomingparticles &equals; Up&comma; conjugateUp&comma; outgoingparticles &equals; Up&comma; conjugateUp&comma; numberofloops &equals; 0&comma; diagrams 

uul,CP__1vum,EP__2&conjugate0;uun,FP__3&conjugate0;vup,GP__4g__s2γααn,pγννm,lgα,νδb,cδP__3ββP__4ββ+P__1ββ+P__2ββλcF,GλbE,C16π2P__1κ+P__2κP__1κκ+P__2κκ+ε+uul,CP__1vum,EP__2&conjugate0;uun,FP__3&conjugate0;vup,GP__4g__s2γααm,pγννn,lgα,νδb,cδP__3ββP__4ββ+P__1ββ+P__2ββλcE,GλbF,C16π2P__1κκP__3κκP__1κP__3κ+ε

(42)

L__GGGG  op1&comma; expandL__QCD 

14g__s2FSU3a&comma;b&comma;cFSU3c&comma;d&comma;e`*`GluonField&alpha;&comma;bX&comma;GluonField&mu;&comma;aX&comma;GluonField~alpha&comma;dX&comma;GluonField~mu&comma;eX

(43)

The amplitude at tree level for the process with two incoming and two outgoing gluons

FeynmanDiagramsL__GGGG&comma;incomingparticles&equals;GluonField&comma; GluonField&comma;outgoingparticles&equals;GluonField&comma;GluonField&comma;numberofloops&equals;0&comma;diagrams  

16g__s2δP__3σσP__4σσ+P__1σσ+P__2σσϵGν,fP__1ϵGβ,gP__2ϵGκ,hP__3&conjugate0;ϵGλ,a1P__4&conjugate0;FSU3a1&comma;c&comma;fFSU3c&comma;g&comma;hFSU3a1&comma;c&comma;gFSU3c&comma;f&comma;hgβ,νβ,νgκ,λκ,λ+FSU3a1&comma;c&comma;fFSU3c&comma;g&comma;hFSU3a1&comma;c&comma;hFSU3c&comma;f&comma;ggκ,νκ,νgβ,λβ,λ+gλ,νλ,νgβ,κβ,κFSU3a1&comma;c&comma;gFSU3c&comma;f&comma;h+FSU3a1&comma;c&comma;hFSU3c&comma;f&comma;gπ2E__1E__2E__3E__4

(44)

 

The Electroweak sector of the Standard Model and its interaction Lagrangian

 

Before symmetry breaking

 

The electro-weak interaction before symmetry breaking, that are not used to compute observable scattering amplitudes, but 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&plus;L__f&plus;L__h&plus;L__y

L__EWL__g+L__f+L__h+L__y

(45)

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 B&mu; vector bosons

L__g  14WFieldStrength&mu;&comma;&nu;&comma;J2&plus;BFieldStrength&mu;&comma;&nu;2

L__g𝕎μ,ν,J𝕎μ,νJμ,νJ4𝔹μ,ν𝔹μ,νμ,ν4

(46)

Introducing the definitions of these tensors we have

BFieldStrengthdefinition&comma;WFieldStrengthdefinition

𝔹μ,ν=μBField&nu;XνBField&mu;X,𝕎μ,ν,J=μWField&nu;&comma;JXνWField&mu;&comma;JX+g__wεJ,K,LWField&mu;&comma;KXWField&nu;&comma;LX

(47)

L__g  SubstituteTensor&comma;L__g

14`*`d_&mu;WField&nu;&comma;JX&comma;Xd_&nu;WField&mu;&comma;JX&comma;X&plus;g__wLeviCivitaJ&comma;K&comma;L`*`WField&mu;&comma;KX&comma;WField&nu;&comma;LX&comma;d_~muWField~nu&comma;JX&comma;Xd_~nuWField~mu&comma;JX&comma;X&plus;g__wLeviCivitaJ&comma;M&comma;N`*`WField~mu&comma;MX&comma;WField~nu&comma;NX14`*`d_&mu;BField&nu;X&comma;Xd_&nu;BField&mu;X&comma;X&comma;d_~muBField~nuX&comma;Xd_~nuBField~muX&comma;X

(48)

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. The L__h term involves the Higgs boson before symmetry breaking (the HiggsBosonΦ 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. 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.

 

After symmetry breaking

 

For the purpose of computing scattering amplitudes, the formulation of the interaction Lagrangian after symmetry breaking is more relevant. Following the presentation in Wikipedia, the interaction Lagrangian of this sector is given by

L__EW  L__K&plus;L__N&plus;L__C&plus;L__H&plus;L__HV&plus;L__WWV&plus;L__WWVV&plus;L__Y;

L__EWL__K+L__N+L__C+L__H+L__HV+L__WWV+L__WWVV+L__Y;

(49)

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. At the end, we use the Lagrangian command to retrieve all the terms of (49).

The kinetic term L__K can be entered as

L__K &equals; 14ElectromagneticFieldStrengthmu&comma;nu2  12WPlusFieldStrengthmu&comma;nuWMinusFieldStrengthmu&comma;nu&plus;12mWField2WPlusFieldmuWMinusFieldmu 14ZFieldStrengthmu&comma;nu2&plus;12mZField2ZFieldmu2&plus;12d_muHiggsBosonX2mHiggsBoson22HiggsBosonX2 &plus;%addconjugatef__LjXDgammamu j&comma;kid_muf__LkX  mf__Lf__LjX&comma; f__L &equals; StandardModel:-Leptons1..3 &plus;%addconjugatef__LjXDgammamu j&comma;kid_muf__LkX &comma; f__L &equals; StandardModel:-Leptons4..6 &plus;%addconjugatef__Qj&comma; AXDgammamuj&comma;kid_muf__Qk&comma;AX  mf__Qf__Qj&comma;AX&comma; f__Q &equals; StandardModel:-Quarks 

L__K&equals;14`*`ElectromagneticFieldStrength&mu;&comma;&nu;&comma;ElectromagneticFieldStrength~mu&comma;~nu12`*`WPlusFieldStrength&mu;&comma;&nu;&comma;WMinusFieldStrength~mu&comma;~nu&plus;12mWField2`*`WPlusField&mu;&comma;WMinusField~mu14`*`ZFieldStrength&mu;&comma;&nu;&comma;ZFieldStrength~mu&comma;~nu&plus;12mZField2`*`ZField&mu;&comma;ZField~mu&plus;12`*`d_&mu;HiggsBosonX&comma;X&comma;d_~muHiggsBosonX&comma;X12mHiggsBoson2`^`HiggsBosonX&comma;2&plus;%add`*`conjugatef__LjX&comma;d_&mu;f__LkX&comma;XDgamma~muj&comma;kmf__Lf__LjX&comma;f__L&equals;Electron&comma;Muon&comma;Tauon&plus;%add`*`conjugatef__LjX&comma;d_&mu;f__LkX&comma;XDgamma~muj&comma;k&comma;f__L&equals;ElectronNeutrino&comma;MuonNeutrino&comma;TauonNeutrino&plus;%add`*`conjugatef__Qj&comma;AX&comma;d_&mu;f__Qk&comma;AX&comma;XDgamma~muj&comma;kmf__Qf__Qj&comma;AX&comma;f__Q&equals;Up&comma;Charm&comma;Top&comma;Down&comma;Strange&comma;Bottom

(50)

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

value

L__K&equals;14`*`ElectromagneticFieldStrength&mu;&comma;&nu;&comma;ElectromagneticFieldStrength~mu&comma;~nu12`*`WPlusFieldStrength&mu;&comma;&nu;&comma;WMinusFieldStrength~mu&comma;~nu&plus;12mWField2`*`WPlusField&mu;&comma;WMinusField~mu14`*`ZFieldStrength&mu;&comma;&nu;&comma;ZFieldStrength~mu&comma;~nu&plus;12mZField2`*`ZField&mu;&comma;ZField~mu&plus;12`*`d_&mu;HiggsBosonX&comma;X&comma;d_~muHiggsBosonX&comma;X12mHiggsBoson2`^`HiggsBosonX&comma;2&plus;`*`conjugateElectronjX&comma;d_&mu;ElectronkX&comma;XDgamma~muj&comma;kmElectronElectronjX&plus;`*`conjugateMuonjX&comma;d_&mu;MuonkX&comma;XDgamma~muj&comma;kmMuonMuonjX&plus;`*`conjugateTauonjX&comma;d_&mu;TauonkX&comma;XDgamma~muj&comma;kmTauonTauonjX&plus;`*`conjugateElectronNeutrinojX&comma;d_&mu;ElectronNeutrinokX&comma;XDgamma~muj&comma;k&plus;`*`conjugateMuonNeutrinojX&comma;d_&mu;MuonNeutrinokX&comma;X</