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Physics[FeynmanDiagrams] - compute an expansion of the Scattering matrix for a model in coordinates representation, or the Scattering matrix element in momentum representation, and the corresponding Feynman Diagrams

Calling Sequence

FeynmanDiagrams(InteractionLagrangian, ...(optional arguments)...)

Parameters

InteractionLagrangian

-

the interaction Lagrangian depending on quantum fields

Description

• 

A scattering matrix S relates the initial and final states, ⁢i⁢ and ⁢f⁢, of an interacting system. In an 4-dimensional spacetime with coordinates X, S can be written as:

S=T⁡ⅇi⁢∫L⁡XⅆX4

  

where i is the imaginary unit and L is the interaction Lagrangian, written in terms of quantum fields depending on the spacetime coordinates X. The T symbol means time-ordered. For the terminology used in this page, see for instance chapter IV, "The Scattering Matrix", in ref.[1]. This exponential can be expanded as

S=1+S1+S2+S3+...

  

where

Sn=⁡inn!⁢∫…⁢∫T⁡L⁡X__1,…,L⁡X__n⁢ⅆX__14⁢…⁢ⅆX__n4

  

and T⁡L⁡X1,...,L⁡Xn is the time-ordered product of n interaction Lagrangians evaluated at different points. Note the factor inn! included in the definition of Sn used in this page.

Coordinates representation

• 

The default output of FeynmanDiagrams(L(X)) is thus S=1+S1+S2+S3 in coordinates representation, S truncated by default at order = 3. The expansion of the time-ordered product entering the integrand of each Sn (herein called the expansion of Sn) is computed using Wick's theorems, and Sn itself, including the factor inn!, is expressed using the FeynmanIntegral command.

• 

The expansion of Sn consists of a sum of terms, each of which is a product of certain pairings of the operator field functions entering L times the normal product of the remaining unpaired operators of free fields. The pairings between fields are the propagators, the Green functions of the associated free fields.

• 

Each term of the expansion of Sn corresponds to a Feynman diagram with n vertices (the number n of interaction Lagrangians in Sn). Within the diagram, each of the pairings of fields (propagators) has a corresponding internal line, and each field entering the normal product of unpaired field operators has a corresponding external leg. Each of those terms has a different configuration of external legs (normal products of unpaired fields), and the expanded Sn being returned includes all the possible configurations of external legs.

Momentum representation

• 

The transition probability for a process with initial and final states ⁢i⁢ and ⁢f⁢ in momentum representation is given by the square of the scattering matrix elements ⁢f⁢|⁢S⁢|⁢i⁢. The states ⁢i⁢ and ⁢f⁢, with respectively s and r free particles, are normalized such that the average number per unit volume of each of the s and r particles is equal to one (see sec. 21.2 of ref.[1], "Normalization of the State amplitude"). In this sense, each of the s and r fields respectively indicated in incomingparticles and outgoingparticles represent a 1-particle per unit volume with precisely defined average momentum. All the s particles in ⁢i⁢ have different sets of quantum numbers, and the r particles in ⁢f⁢ are only those whose momentum has been altered by the scattering process.

• 

The output of FeynmanDiagrams(L(X), incomingparticles = [...], outgoingparticles = [...], numberofloops = n) is thus the S-matrix element ⁢f⁢|⁢S⁢|⁢i⁢ in momentum representation (see formula (2) of sec. 21.1 in ref.[1]), of the form

⁢f⁢|⁢S⁢|⁢i⁢=F⁡pi,pf⁢δ⁡∑i=1s⁡pi−∑f=1r⁡pf

  

The δ (Dirac) is the expected singular factor due to the conservation of the energy-momentum of the s initial (incoming) and r final (outgoing) particles, and F⁡pi,pf, also called the amplitude, is the sum of terms, in momentum representation, corresponding to Feynman diagrams with the number of loops indicated and s+r external legs.

• 

When the option output = probabilitydensity is given, the output is the transition probability density, that is, the average number of scattered particles per unit time and per unit volume having momenta within the infinitesimal volumes d 3⁢p1⁢ ...⁢d 3⁢pr. That probability density is constructed using the square of the scattering matrix element ⁢f⁢|⁢S⁢|⁢i⁢2, as shown in formula (13) of sec. 21.1 of ref.[1]:

dw=2⁢π3⁢s−4⁢n__1⁢...⁢n__s⁢F⁡pi,pf2⁢δ⁡∑i=1s⁡pi−∑f=1r⁡pf⁢ d 3⁢p1⁢ ...⁢d 3⁢pr

  

where each n1⁢...⁢ns represents the particle density of each of the s particles in the initial state ⁢i⁢, and for an interpretation of the square of the δ factor that appears in ⁢f⁢|⁢S⁢|⁢i⁢2 see sec. 64 of ref.[3]. In order to get the number of scattered particles per unit of time and volume that happen in the final momentum region Δ⁢p1 .. Δ⁢pr, the probability density dw needs to be integrated over that particular region.

• 

The truncation order of S in ⁢f⁢|⁢S⁢|⁢i⁢, if not specified, is some finite n automatically derived from the interaction Lagrangian, the indicated number of loops, and the number and kind of incoming and outgoing particles. Terms Sm with m>n don't contribute to the indicated process.

• 

Although ⁢f⁢|⁢S⁢|⁢i⁢ in momentum representation can be computed departing from the expansion of S in coordinates representation, it is simpler to compute it directly using the Feynman rules in momentum representation (for the conventions used, see chapter V of ref.[1], "The Feynman rules and Diagrams") together with a program for the generation of the Feynman diagrams at any order, implemented in the Maple language following the presentation in ref.[4].

The input representing the interaction Lagrangian

• 

Given an interaction Lagrangian, FeynmanDiagrams considers a field of the problem any function of spacetime variables (set using Coordinates or Setup). It is recommended that you also set the field operators as quantum operator using Setup, and fields that were not set as quantum operators are automatically set as such when you call FeynmanDiagrams.

• 

To represent the Dagger of a complex field, say C⁡X, use Dagger, as in Dagger(C(X)) or Dagger(C[mu](X)). To represent spinor fields, use anticommutative functions. For that purpose, you can set an anticommutative prefix using Setup. To represent the Dirac conjugate of a spinor field, say Q(X), use DiracConjugate, as in DiracConjugate(Q(X)).

• 

When working in momentum representation beyond tree-level, the propagators of vector field that appear within Feynman integrals include their mass. To indicate that a field is massless use Setup, for instance as in Setup⁡massless=A.

• 

When matrix multiplication (e.g. of spinors) enters the interaction Lagrangian, you can enter the expression representing it with the matrix (spinorial) indices explicit. For example, for an interaction Lagrangian in quantum electrodynamics (QED) involving Dirac matrices, you can use DiracConjugate(Q[j](X)) * Dgamma[mu][j,k] * Q[k](X) * A[mu](X) where Q[j](X) and DiracConjugate(Q[j](X)) respectively represent a spinor field and its Dirac conjugate. Note that, depending on the case, the spinor and even spacetime indices can be omitted entirely without ambiguity (see the Examples section).

• 

The summation convention for spacetime, gauge, and spinor repeated indices in products entering the interaction Lagrangian is automatically used by FeynmanDiagrams.

The options for changing the default output

• 

Several options are available to change the default output of FeynmanDiagrams in coordinates and in momentum representation. See the Parameters section above. These options can be used alone or together and are illustrated in the Examples section. Briefly, to compute, in coordinates representation, only the terms that involve diagrams:

– 

with j external legs use the option numberofexternallegs = j;

– 

with some specific external legs specifying a particular physical process use the option normalproducts = ...;

  

and for both coordinates and momentum representation,

– 

to truncate the expansion of S or of ⁢f⁢|⁢S⁢|⁢i⁢ at an order different than the default use the option order = m;

– 

to compute only one term Sn of the expansion of S or of ⁢f⁢|⁢S⁢|⁢i⁢, use the option numberofvertices = n;

– 

to compute only the terms that involve diagrams with k loops (tree-level is k=0) use the option numberofloops = k;

– 

to additionally display the diagrams (drawings) corresponding to the terms appearing in the returned output use the option diagrams;

  

NOTE: the vertices of the drawings are displayed in red and encircled in black, and the current implementation of FeynmanDiagrams can only draw the diagrams than contain up to 6 vertices.

• 

When computing in coordinates representation up to order = m, it is convenient to previously set m coordinate systems using Setup or Coordinates to avoid having FeynmanDiagrams introducing labels to represent the m spacetime coordinates that enter the FeynmanIntegrals (each Sn). When additional labels are necessary, they are all of the form Xn, expressed using literal subscripts X__n, where n is an integer.

• 

When using the normalproducts, excludepropagators or crossedpropagators options, the fields entering the right-hand sides can be referenced by just their names, or also with their functionality. This can be used, for example, to filter out the propagator f⁡X,g⁡Y only, or all the propagators of the form f,g, which would include as well f⁡X,g⁡X and f⁡Y,g⁡Y.

Examples

Load the package, set three coordinate systems and set φ to represent a quantum operator

> 

with⁡Physics:

> 

Setup⁡mathematicalnotation=true,coordinates=X,Y,Z,quantumoperators=φ

⁢Systems of spacetime coordinates are:⁢X=x1,x2,x3,x4,Y=y1,y2,y3,y4,Z=z1,z2,z3,z4

_______________________________________________________

coordinatesystems=X,Y,Z,mathematicalnotation=true,quantumoperators=φ

(1)

Let L be the interaction Lagrangian

> 

L≔λ⁢φ⁡X4

L≔λ⁢φ⁡X4

(2)

The expansion of S in coordinates representation up to order = 3 is given by

> 

S=FeynmanDiagrams⁡L

S=1+I∫λ⁢φ⁡X4ⅆX 4+I22!∫∫16⁢λ2⁢φ⁡X3⁢φ⁡Y3⁢φ⁡X,φ⁡Y+72⁢λ2⁢φ⁡X2⁢φ⁡Y2⁢φ⁡X,φ⁡Y2+96⁢λ2⁢φ⁡X⁢φ⁡Y⁢φ⁡X,φ⁡Y3ⅆX 4ⅆY 4+I33!∫∫∫1728⁢λ3⁢φ⁡X2⁢φ⁡Y2⁢φ⁡Z2⁢φ⁡X,φ⁡Y⁢φ⁡X,φ⁡Z⁢φ⁡Y,φ⁡Z+10368⁢λ3⁢φ⁡X2⁢φ⁡Y⁢φ⁡Z⁢φ⁡X,φ⁡Y⁢φ⁡X,φ⁡Z⁢φ⁡Y,φ⁡Z2+2592⁢λ3⁢φ⁡X2⁢φ⁡Y2⁢φ⁡X,φ⁡Z2⁢φ⁡Y,φ⁡Z2+3456⁢λ3⁢φ⁡X2⁢φ⁡X,φ⁡Y⁢φ⁡X,φ⁡Z⁢φ⁡Y,φ⁡Z3+10368⁢λ3⁢φ⁡X⁢φ⁡Y⁢φ⁡X,φ⁡Y⁢φ⁡X,φ⁡Z2⁢φ⁡Y,φ⁡Z2+576⁢λ3⁢φ⁡X3⁢φ⁡Y3⁢φ⁡Z2⁢φ⁡X,φ⁡Z⁢φ⁡Y,φ⁡ZⅆX 4ⅆY 4ⅆZ 4

(3)

To compute only the tree-level terms of this result use numberofloops = 0. To produce the diagrams together with the algebraic expression of Streelevel, pass also the optional argument diagrams

> 

S__treelevel=FeynmanDiagrams⁡L,numberofloops=0,diagrams

S__treelevel=1+I∫λ⁢φ⁡X4ⅆX 4+I22!∫∫16⁢λ2⁢φ⁡X3⁢φ⁡Y3⁢φ⁡X,φ⁡YⅆX 4ⅆY 4+I33!∫∫∫576⁢λ3⁢φ⁡X3⁢φ⁡Y3⁢φ⁡Z2⁢φ⁡X,φ⁡Z⁢φ⁡Y,φ⁡ZⅆX 4ⅆY 4ⅆZ 4

(4)

The expansion of S  in coordinates representation to a specific order shows in a compact way the topology of the underlying Feynman diagrams. In the result above there are three terms, with 4, 6 and 8 external legs respectively. A ⁢f⁢|⁢S⁢|⁢i⁢ element in momentum representation corresponding to the term with 4 external legs is one where there are 2 incoming and 2 outgoing particles

> 

FeynmanDiagrams⁡L,incomingparticles=φ,φ,outgoingparticles=φ,φ,numberofloops=0,diagrams

3⁢I2⁢λ⁢δ⁡P__1⁢μ⁢μ+P__2⁢μ⁢μ−P__3⁢μ⁢μ−P__4⁢μ⁢μπ2⁢E__1⁢E__2⁢E__3⁢E__4

(5)

To compute the S-matrix element in momentum representation corresponding to the term with 6 external legs, change in the input above φ,φ by φ,φ,φ, and to get the element corresponding to 8 external legs use φ,φ,φ,φ. The following computes the expansion of S up to order = 3, only the terms corresponding to diagrams with two external legs

> 

FeynmanDiagrams⁡L,numberofexternallegs=2,diagrams

I22!∫∫96⁢λ2⁢φ⁡X⁢φ⁡Y⁢φ⁡X,φ⁡Y3ⅆX 4ⅆY 4+I33!∫∫∫3456⁢λ3⁢φ⁡X2⁢φ⁡X,φ⁡Y⁢φ⁡X,φ⁡Z⁢φ⁡Y,φ⁡Z3+10368⁢λ3⁢φ⁡X⁢φ⁡Y⁢φ⁡X,φ⁡Y⁢φ⁡X,φ⁡Z2⁢φ⁡Y,φ⁡Z2ⅆX 4ⅆY 4ⅆZ 4

(6)

Already from the diagrams, in this result there are terms with either 2 or 3 loops. The transition probability density for a process with two external legs - say one incoming and one outgoing particle - so corresponding to the first term with 2 loops, is then

> 

FeynmanDiagrams⁡L,incomingparticles=φ,outgoingparticles=φ,numberofloops=2,diagrams,output=probabilitydensity

∏i=11⁡ni⁢F2⁢δ⁡−P__2+P__1⁢∏f=11⁡dP→f32⁢πwhereF=∫∫3⁢I8⁢λ2π7⁢E__1⁢E__2⁢P__1+p__2+p__32−m__φ2+I⁢ε⁢p__22−m__φ2+I⁢ε⁢p__32−m__φ2+I⁢εⅆp__2 4ⅆp__3 4

(7)

In the above, the spacetime indices in the square of momenta (denominators of propagators) entering the amplitude F are implicit. To make those indices explicit, use the option putindicesinsquareofmomentum

> 

F=FeynmanDiagrams⁡L,incoming=φ,outgoing=φ,numberofloops=2,putindices

F=∫∫3⁢I8⁢λ2⁢δ⁡−P__2⁢β⁢β+P__1⁢β⁢βπ7⁢E__1⁢E__2⁢P__1μ+p__2μ+p__3μ⁢P__1⁢μ⁢μ+p__2⁢μ⁢μ+p__3⁢μ⁢μ−m__φ2+I⁢ε⁢p__2ν⁢p__2⁢ν⁢ν−m__φ2+I⁢ε⁢p__3α⁢p__3⁢α⁢α−m__φ2+I⁢εⅆp__2 4ⅆp__3 4

(8)

Up to order = 3, all the terms of the expansion of S that contain three loops (see (6)) actually have only 2 external legs, and the corresponding Feynman Diagrams only 3 vertices

> 

FeynmanDiagrams⁡L,numberofloops=3

I33!∫∫∫3456⁢λ3⁢φ⁡X2⁢φ⁡X,φ⁡Y⁢φ⁡X,φ⁡Z⁢φ⁡Y,φ⁡Z3+10368⁢λ3⁢φ⁡X⁢φ⁡Y⁢φ⁡X,φ⁡Y⁢φ⁡X,φ⁡Z2⁢φ⁡Y,φ⁡Z2ⅆX 4ⅆY 4ⅆZ 4

(9)

A corresponding S-matrix element in momentum representation:

> 

FeynmanDiagrams⁡L,incomingparticles=φ,outgoingparticles=φ,numberofloops=3

∫∫∫9⁢λ3⁢δ⁡−P__2+P__132⁢π11⁢E__1⁢E__2⁢p__3+p__4+p__52−m__φ2+I⁢ε⁢−P__1+P__2−p__3−p__4−p__52−m__φ2+I⁢ε⁢p__32−m__φ2+I⁢ε⁢p__42−m__φ2+I⁢ε⁢p__52−m__φ2+I⁢εⅆp__3 4ⅆp__4 4ⅆp__5 4+2⁢∫∫∫9⁢λ3⁢δ⁡−P__2+P__132⁢π11⁢E__1⁢E__2⁢−P__2+p__4+p__52−m__φ2+I⁢ε⁢−P__1+P__2−p__3−p__4−p__52−m__φ2+I⁢ε⁢p__32−m__φ2+I⁢ε⁢p__42−m__φ2+I⁢ε⁢p__52−m__φ2+I⁢εⅆp__3 4ⅆp__4 4ⅆp__5 4+∫∫∫9⁢λ3⁢δ⁡−P__2+P__132⁢π11⁢E__1⁢E__2⁢−P__1−p__2+P__2−p__4−p__52−m__φ2+I⁢ε⁢p__22−m__φ2+I⁢ε⁢−P__2+p__4+p__52−m__φ2+I⁢ε⁢p__42−m__φ2+I⁢ε⁢p__52−m__φ2+I⁢εⅆp__2 4ⅆp__4 4ⅆp__5 4

(10)

To compute only one term, say S2, and for instance including terms with tadpoles use the options numberofvertices = 2 and includetadpoles

> 

S2=FeynmanDiagrams⁡L,numberofvertices=2,includetadpoles

S2=I22!∫∫72⁢λ2⁢φ⁡X2⁢φ⁡Y2⁢φ⁡X,φ⁡Y2+16⁢λ2⁢φ⁡X3⁢φ⁡Y3⁢φ⁡X,φ⁡Y+144⁢λ2⁢φ⁡X2⁢φ⁡X,φ⁡Y2⁢φ⁡Y,φ⁡Y+96⁢λ2⁢φ⁡X⁢φ⁡Y⁢φ⁡X,φ⁡Y3ⅆX 4ⅆY 4

(11)

Consider an interaction Lagrangian involving two fields. To indicate that η also represents a quantum field operator (φ got set as quantum operator at the beginning), use Setup.

> 

Setup⁡quantumoperators=η

quantumoperators=η,φ

(12)
> 

L__2≔λ⁢φ⁡X4+σ⁢η⁡X3

L__2≔λ⁢φ⁡X4+σ⁢η⁡X3

(13)

In the expansion of S, there are no terms with one external leg φ and another η, that would be represented as a normal product of unpaired fields φ⁢η

> 

FeynmanDiagrams⁡L__2,normalproducts=_NP⁡φ,η

0

(14)

Compute, one at a time, the terms entering the expansion of S up to order 2 that have 0 (tree-level), 1, 2 and 3 loops

> 

S__treelevel=FeynmanDiagrams⁡L__2,order=2,numberofloops=0

S__treelevel=1+I∫λ⁢φ⁡X4+σ⁢η⁡X3ⅆX 4+I22!∫∫16⁢λ2⁢φ⁡X3⁢φ⁡Y3⁢φ⁡X,φ⁡Y+9⁢σ2⁢η⁡X2⁢η⁡Y2⁢η⁡X,η⁡YⅆX 4ⅆY 4

(15)
> 

FeynmanDiagrams⁡L__2,order=2,numberofloops=1

I22!∫∫72⁢λ2⁢φ⁡X2⁢φ⁡Y2⁢φ⁡X,φ⁡Y2+18⁢σ2⁢η⁡X⁢η⁡Y⁢η⁡X,η⁡Y2ⅆX 4ⅆY 4

(16)
> 

FeynmanDiagrams⁡L__2,order=2,numberofloops=2

I22!∫∫96⁢λ2⁢φ⁡X⁢φ⁡Y⁢φ⁡X,φ⁡Y3ⅆX 4ⅆY 4

(17)
> 

FeynmanDiagrams⁡L__2,order=2,numberofloops=3

0

(18)

Indeed, to see terms with three loops you'd need to consider the expansion of S up to order=3 at least

> 

FeynmanDiagrams⁡L__2,order=3,numberofloops=3,diagrams

I33!∫∫∫3456⁢λ3⁢φ⁡X2⁢φ⁡X,φ⁡Y⁢φ⁡X,φ⁡Z⁢φ⁡Y,φ⁡Z3+10368⁢λ3⁢φ⁡X⁢φ⁡Y⁢φ⁡X,φ⁡Y⁢φ⁡X,φ⁡Z2⁢φ⁡Y,φ⁡Z2ⅆX 4ⅆY 4ⅆZ 4

(19)

A corresponding S-matrix element in momentum representation with 1 incoming and 1 outgoing particles

> 

FeynmanDiagrams⁡L__2,incomingparticles=φ,outgoingparticles=φ,numberofloops=3

∫∫∫9⁢λ3⁢δ⁡−P__2+P__132⁢π11⁢E__1⁢E__2⁢p__3+p__4+p__52−m__φ2+I⁢ε⁢−P__1+P__2−p__3−p__4−p__52−m__φ2+I⁢ε⁢p__32−m__φ2+I⁢ε⁢p__42−m__φ2+I⁢ε⁢p__52−m__φ2+I⁢εⅆp__3 4ⅆp__4 4ⅆp__5 4+2⁢∫∫∫9⁢λ3⁢δ⁡−P__2+P__132⁢π11⁢E__1⁢E__2⁢−P__2+p__4+p__52−m__φ2+I⁢ε⁢−P__1+P__2−p__3−p__4−p__52−m__φ2+I⁢ε⁢p__32−m__φ2+I⁢ε⁢p__42−m__φ2+I⁢ε⁢p__52−m__φ2+I⁢εⅆp__3 4ⅆp__4 4ⅆp__5 4+∫∫∫9⁢λ3⁢δ⁡−P__2+P__132⁢π11⁢E__1⁢E__2⁢−P__1−p__2+P__2−p__4−p__52−m__φ2+I⁢ε⁢p__22−m__φ2+I⁢ε⁢−P__2+p__4+p__52−m__φ2+I⁢ε⁢p__42−m__φ2+I⁢ε⁢p__52−m__φ2+I⁢εⅆp__2 4ⅆp__4 4ⅆp__5 4

(20)

Compute the terms entering the integrand of S3, having four external legs (normal ordered products involving four fields). For the interaction Lagrangian L2, the normal products of unpaired fields in S3 only involve the φ field

> 

FeynmanDiagrams⁡L__2,numberofvertices=3,numberoflegs=4

I33!∫∫∫2592⁢λ3⁢φ⁡X2⁢φ⁡Y2⁢φ⁡X,φ⁡Z2⁢φ⁡Y,φ⁡Z2+10368⁢λ3⁢φ⁡X2⁢φ⁡Y⁢φ⁡Z⁢φ⁡X,φ⁡Y⁢φ⁡X,φ⁡Z⁢φ⁡Y,φ⁡Z2ⅆX 4ⅆY 4ⅆZ 4

(21)

To get the corresponding S-matrix element in momentum representation, replace numberoflegs=4 by a process with the same four legs, for example incoming=φ,outgoing=φ,φ,φ and numberofloops=2. Considering S2 instead, you also get normal products of η fields

> 

FeynmanDiagrams⁡L__2,numberofvertices=2,numberoflegs=4

I22!∫∫72⁢λ2⁢φ⁡X2⁢φ⁡Y2⁢φ⁡X,φ⁡Y2+9⁢σ2⁢η⁡X2⁢η⁡Y2⁢η⁡X,η⁡YⅆX 4ⅆY 4

(22)

For this Lagrangian and configuration of vertices and external legs, S2 includes a term (with symmetry factor = 96) to which corresponds a 1-particle reducible diagram that is also a tadpole:

> 

FeynmanDiagrams⁡L__2,numberofvertices=2,numberoflegs=4,reducible,tadpoles,diagrams

I22!∫∫96⁢λ2⁢φ⁡X3⁢φ⁡Y⁢φ⁡X,φ⁡Y⁢φ⁡Y,φ⁡Y+72⁢λ2⁢φ⁡X2⁢φ⁡Y2⁢φ⁡X,φ⁡Y2+9⁢σ2⁢η⁡X2⁢η⁡Y2⁢η⁡X,η⁡YⅆX 4ⅆY 4

(23)

Following is an example with the interaction Lagrangian of Quantum Electrodynamics (QED). First, define the vector field Aμ.

> 

Define⁡Aμ

Defined objects with tensor properties

Aμ,γμ,σμ,∂μ,gμ,ν,εα,β,μ,ν,Xμ,Yμ,Zμ

(24)

Set the type of letter used to represent spinor indices (you can change this setting according to your preference using Setup) and set A and Q as quantum operators, with Q being anticommutative to represent the electron field

> 

Setup⁡spinorindices=lowercaselatin_is,anticommutativeprefix=Q,op=A,Q

* Partial match of 'op' against keyword 'quantumoperators'

_______________________________________________________

anticommutativeprefix=Q,quantumoperators=A,Q,η,φ,spinorindices=lowercaselatin_is

(25)

So, to represent spinor indices, use lowercase Latin letters from i to s, the spinor fields will be represented by any name prefixed by Q, and the matrix indices related to the matrix product involving Dirac matrices are written explicitly in the interaction Lagrangian. Use DiracConjugate to represent the Dirac conjugate Qj¯

> 

L__QED≔α⁢DiracConjugate⁡Qj⁡X⁢Dgammaμj,k⁢Qk⁡X⁢Aμ⁡X

L__QED≔α⁢Qj⁡X¯⁢Qk⁡X⁢Aμ⁡X⁢γ⁢μ⁢μj,k

(26)

Compute S2, only the terms with 4 external legs, so all the corresponding graphs have no loops, and display the diagrams

> 

FeynmanDiagrams⁡L__QED,numberofvertices=2,numberoflegs=4,diagrams

I22!∫∫−α2⁢γ⁢μ⁢μj,k⁢γ⁢α⁢αi,l⁢Aμ⁡X⁢Aα⁡Y⁢Qk⁡X⁢Qi⁡Y¯⁢Ql⁡Y,Qj⁡X¯−α2⁢γ⁢μ⁢μj,k⁢γ⁢α⁢αi,l⁢Aμ⁡X⁢Aα⁡Y⁢Ql⁡Y⁢Qj⁡X¯⁢Qk⁡X,Qi⁡Y¯+α2⁢γ⁢μ⁢μj,k⁢γ⁢α⁢αi,l⁢Qk⁡X⁢Ql⁡Y⁢Qi⁡Y¯⁢Qj⁡X¯⁢Aα⁡Y,Aμ⁡XⅆX 4ⅆY 4

(27)

As shown next, there are only two terms entering the integrand of S2 with two external legs. They both have one loop, and correspond to the self-energy of the photon and the electron (page 218 of ref.[1]) and are given by

> 

FeynmanDiagrams⁡L__QED,numberofvertices=2,numberoflegs=2,diagrams

I22!∫∫−2⁢α2⁢γ⁢μ⁢μj,k⁢γ⁢α⁢αi,l⁢Qk⁡X⁢Qi⁡Y¯⁢Ql⁡Y,Qj⁡X¯⁢Aα⁡Y,Aμ⁡X−α2⁢γ⁢μ⁢μj,k⁢γ⁢α⁢αi,l⁢Aμ⁡X⁢Aα⁡Y⁢Ql⁡Y,Qj⁡X¯⁢Qk⁡X,Qi⁡Y¯ⅆX 4ⅆY 4

(28)

where the diagram with two spinor legs is the electron self-energy. The corresponding S-matrix elements in momentum representation (numberofvertices = 2 can be given, but in this case is not necessary):

> 

FeynmanDiagrams⁡L__QED,incomingparticles=Q,outgoing=Q,numberofloops=1,diagrams

−∫uQi⁡P→1⁢uQl⁡P→2¯⁢α2⁢γ⁢α⁢αl,m⁢γ⁢ν⁢νn,i⁢P__1β+p__2β⁢γ⁢β⁢βm,n+m__Q⁢δm,n⁢−gα,ν+p__2ν⁢p__2αm__A2⁢δ⁡−P__2+P__18⁢π3⁢P__1+p__22−m__Q2+I⁢ε⁢p__22−m__A2+I⁢εⅆp__2 4

(29)

In this result we see the propagator of the field Aμ with a mass mA. To indicate that this field is massless use

> 

Setup⁡massless=A

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

_______________________________________________________

masslessfields=A

(30)

Now the propagator for Aμ is the one of a massless vector field:

> 

FeynmanDiagrams⁡L__QED,incomingparticles=Q,outgoing=Q,numberofloops=1

−∫−uQi⁡P→1⁢uQl⁡P→2¯⁢α2⁢γ⁢α⁢αl,m⁢γ⁢ν⁢νn,i⁢P__1β+p__2β⁢γ⁢β⁢βm,n+m__Q⁢δm,n⁢gα,ν⁢δ⁡−P__2+P__18⁢π3⁢P__1+p__22−m__Q2+I⁢ε⁢p__22+I⁢εⅆp__2 4

(31)
> 

FeynmanDiagrams⁡L__QED,incomingparticles=A,outgoing=A,numberofloops=1,diagrams

(32)

In the two self-energy expressions above we see the u-spinor uQ and the polarization vector ϵA.

Compute S3 for this model, keeping only the terms with 3 external legs

> 

FeynmanDiagrams⁡L__QED,numberofvertices=3,numberoflegs=3,diagrams

I33!∫∫∫−6⁢α3⁢γ⁢μ⁢μj,k⁢γ⁢α⁢αi,l⁢γ⁢β⁢βm,n⁢Aμ⁡X⁢Ql⁡Y⁢Qm⁡Z¯⁢Qn⁡Z,Qj⁡X¯⁢Qk⁡X,Qi⁡Y¯⁢Aα⁡Y,Aβ⁡Z−α3⁢γ⁢μ⁢μj,k⁢γ⁢α⁢αi,l⁢γ⁢β⁢βm,n⁢Aμ⁡X⁢Aα⁡Y⁢Aβ⁡Z⁢Ql⁡Y,Qj⁡X¯⁢Qk⁡X,Qm⁡Z¯⁢Qn⁡Z,Qi⁡Y¯−α3⁢γ⁢μ⁢μj,k⁢γ⁢α⁢αi,l⁢γ⁢β⁢βm,n⁢Aμ⁡X⁢Aα⁡Y⁢Aβ⁡Z⁢Qn⁡Z,Qj⁡X¯⁢Qk⁡X,Qi⁡Y¯⁢Ql⁡Y,Qm⁡Z¯ⅆX 4ⅆY 4ⅆZ 4

(33)

Note the introduction of spacetime indices (covariant as subscripts and contravariant as superscripts) and spinor indices (all presented as subscripts) by FeynmanDiagrams; this preserves the summation convention for repeated indices by avoiding occurrences of indices more than once in each product. To verify for correctness, the indices and noncommutative products use Check.

> 

Check((33), 'all');

The products in the given expression check ok.

The repeated indices per term are: ...,...,..., the free indices are: ...

α,β,i,j,k,l,m,μ,n,α,β,i,j,k,l,m,μ,n,α,β,i,j,k,l,m,μ,n,∅

(34)

The transition probability density corresponding to the term with three external vector-boson legs

> 

FeynmanDiagrams⁡L__QED,incomingparticles=A,outgoingparticles=A,A,numberofloops=1,diagrams,output=probabilitydensity

(35)

The processes of S3 with 5 external legs (you can indicate one particular 5 legs' process, among all the possible ones, by specifying the particles entering as external legs  using the option normalproducts):

> 

FeynmanDiagrams⁡L__QED,numberofvertices=3,numberoflegs=5,diagrams

I33!∫∫∫−α3⁢γ⁢μ⁢μj,k⁢γ⁢α⁢αi,l⁢γ⁢β⁢βm,n⁢Aμ⁡X⁢Aα⁡Y⁢Aβ⁡Z⁢Qk⁡X⁢Qi⁡Y¯⁢Qn⁡Z,Qj⁡X¯⁢Ql⁡Y,Qm⁡Z¯−α3⁢γ⁢μ⁢μj,k⁢γ⁢α⁢αi,l⁢γ⁢β⁢βm,n⁢Aμ⁡X⁢Aα⁡Y⁢Aβ⁡Z⁢Qk⁡X⁢Qm⁡Z¯⁢Ql⁡Y,Qj⁡X¯⁢Qn⁡Z,Qi⁡Y¯−α3⁢γ⁢μ⁢μj,k⁢γ⁢α⁢αi,l⁢γ⁢β⁢βm,n⁢Aμ⁡X⁢Aα⁡Y⁢Aβ⁡Z⁢Ql⁡Y⁢Qj⁡X¯⁢Qk⁡X,Qm⁡Z¯⁢Qn⁡Z,Qi⁡Y¯−α3⁢γ⁢μ⁢μj,k⁢γ⁢α⁢αi,l⁢γ⁢β⁢βm,n⁢Aμ⁡X⁢Aα⁡Y⁢Aβ⁡Z⁢Ql⁡Y⁢Qm⁡Z¯⁢Qn⁡Z,Qj⁡X¯⁢Qk⁡X,Qi⁡Y¯−α3⁢γ⁢μ⁢μj,k⁢γ⁢α⁢αi,l⁢γ⁢β⁢βm,n⁢Aμ⁡X⁢Aα⁡Y⁢Aβ⁡Z⁢Qn⁡Z⁢Qj⁡X¯⁢Qk⁡X,Qi⁡Y¯⁢Ql⁡Y,Qm⁡Z¯−α3⁢γ⁢μ⁢μj,k⁢γ⁢α⁢αi,l⁢γ⁢β⁢βm,n⁢Aμ⁡X⁢Aα⁡Y⁢Aβ⁡Z⁢Qn⁡Z⁢Qi⁡Y¯⁢Ql⁡Y,Qj⁡X¯⁢Qk⁡X,Qm⁡Z¯+6⁢α3⁢γ⁢μ⁢μj,k⁢γ⁢α⁢αi,l⁢γ⁢β⁢βm,n⁢Aμ⁡X⁢Qk⁡X⁢Ql⁡Y⁢Qi⁡Y¯⁢Qm⁡Z¯⁢Aα⁡Y,Aβ⁡Z⁢Qn⁡Z,Qj⁡X¯−3⁢α3⁢γ⁢μ⁢μj,k⁢γ⁢α⁢αi,l⁢γ⁢β⁢βm,n⁢Aμ⁡X⁢Ql⁡Y⁢Qn⁡Z⁢Qi⁡Y¯⁢Qj⁡X¯⁢Aα⁡Y,Aβ⁡Z⁢Qk⁡X,Qm⁡Z¯−3⁢α3⁢γ⁢μ⁢μj,k⁢γ⁢α⁢αi,l⁢γ⁢β⁢βm,n⁢Aμ⁡X⁢Ql⁡Y⁢Qn⁡Z⁢Qj⁡X¯⁢Qm⁡Z¯⁢Aα⁡Y,Aβ⁡Z⁢Qk⁡X,Qi⁡Y¯ⅆX 4ⅆY 4ⅆZ 4

(36)

Depending on the interaction Lagrangian, it is frequently possible to entirely omit spinor and spacetime indices without ambiguity resulting in less cluttered output. For example, for QED the interaction Lagrangian L introduced lines above can be entered as

> 

L__qed≔α⁢DiracConjugate⁡Q⁡X⁢Q⁡X⁢A⁡X

L__qed≔α⁢Q⁡X¯⁢Q⁡X⁢A⁡X

(37)

In this way, the expressions look more compact. For example, the first two computations performed above with LQED with spinor and spacetime indices now become

> 

FeynmanDiagrams⁡L__qed,numberofvertices=2,numberoflegs=4

I22!∫∫−2⁢α2⁢A⁡X⁢A⁡Y⁢Q⁡X⁢Q⁡Y¯⁢Q⁡Y,Q⁡X¯−α2⁢Q⁡X¯⁢Q⁡Y¯⁢Q⁡X⁢Q⁡Y⁢A⁡X,A⁡YⅆX 4ⅆY 4

(38)

The S-matrix element in momentum representation corresponding to the term and diagram with 2 fermion and 2 boson external legs: compare with and without the spinor and spacetime indices and factorize the tree-level terms

> 

FeynmanDiagrams⁡L__qed,numberofvertices=2,incomingparticles=Q,DiracConjugate⁡Q,outgoingparticles=A,A,numberofloops=0,factortreelevel,diagrams

I8⁢1P__1ν−P__3ν⁢P__1⁢ν⁢ν−P__3⁢ν⁢ν−m__Q2+I⁢ε+1P__1ν−P__4ν⁢P__1⁢ν⁢ν−P__4⁢ν⁢ν−m__Q2+I⁢ε⁢α2⁢δ⁡P__1⁢μ⁢μ+P__2⁢μ⁢μ−P__3⁢μ⁢μ−P__4⁢μ⁢μ⁢vQ⁡P→2¯⁢ϵA⁡P→3&conjugate0;⁢ϵA⁡P→4&conjugate0;⁢uQ⁡P→1π2⁢E__3⁢E__4

(39)

Compare with the related interaction Lagrangian that has the spinor and spacetime indices explicit

> 

FeynmanDiagrams⁡L__QED,numberofvertices=2,incomingparticles=Q,DiracConjugate⁡Q,outgoingparticles=A,A,numberofloops=0,factortreelevel,diagrams

I8⁢P__1β−P__3β⁢γ⁢β⁢βm,n+m__Q⁢δm,n⁢γ⁢α⁢αl,m⁢γ⁢ν⁢νn,iP__1λ−P__3λ⁢P__1⁢λ⁢λ−P__3⁢λ⁢λ−m__Q2+I⁢ε+P__1β−P__4β⁢γ⁢β⁢βm,n+m__Q⁢δm,n⁢γ⁢ν⁢νl,m⁢γ⁢α⁢αn,iP__1λ−P__4λ⁢P__1⁢λ⁢λ−P__4⁢λ⁢λ−m__Q2+I⁢ε⁢α2⁢δ⁡P__1⁢κ⁢κ+P__2⁢κ⁢κ−P__3⁢κ⁢κ−P__4⁢κ⁢κ⁢vQl⁡P→2¯⁢ϵAα⁡P→4&conjugate0;⁢ϵAν⁡P→3&conjugate0;⁢uQi⁡P→1π2⁢E__3⁢E__4

(40)

The self-energy terms, that only have 2 external legs and one loop:

> 

FeynmanDiagrams⁡L__qed,numberofvertices=2,numberoflegs=2,diagrams

I22!∫∫−2⁢α2⁢Q⁡X⁢Q⁡Y¯⁢Q⁡Y,Q⁡X¯⁢A⁡X,A⁡Y−α2⁢A⁡X⁢A⁡Y⁢Q⁡Y,Q⁡X¯⁢Q⁡X,Q⁡Y¯ⅆX 4ⅆY 4

(41)

When working with non-Abelian gauge fields, the interaction Lagrangian involves derivatives. FeynmanDiagrams can handle that kind of interaction in momentum representation. Consider for instance a Yang-Mills theory with a massless field Bμ,a where a is a SU2 index (see eq.(12) of sec. 19.4 of ref.[1]). The interaction Lagrangian can be entered as

> 

Setup⁡su2indices=lowercaselatin_ah,massless=B,op=B

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

* Partial match of 'op' against keyword 'quantumoperators'

_______________________________________________________

masslessfields=A,B,quantumoperators=A,B,Q,η,φ,su2indices=lowercaselatin_ah

(42)
> 

Define⁡Bμ,a,quiet:

> 

F__Bμ,ν,a≔d_μ⁡Bν,a⁡X−d_ν⁡Bμ,a⁡X

F__Bμ,ν,a≔∂μ⁡Bν,a⁡X−∂ν⁡Bμ,a⁡X

(43)
> 

L≔g2⁢LeviCivitaa,b,c⁢F__Bμ,ν,a⁢Bμ,b⁡X⁢Bν,c⁡X+g24⁢LeviCivitaa,b,c⁢LeviCivitaa,e,f⁢Bμ,b⁡X⁢Bν,c⁡X⁢Bμ,e⁡X⁢Bν,f⁡X

L≔g⁢εa,b,c⁢∂μ⁡Bν,a⁡X−∂ν⁡Bμ,a⁡X⁢B⁢μb⁢μb⁡X⁢B⁢νc⁢νc⁡X2+g2⁢εa,b,c⁢εa,e,f⁢Bμ,b⁡X⁢Bν,c⁡X⁢B⁢μe⁢μe⁡X⁢B⁢νf⁢νf⁡X4

(44)

The transition probability density at tree-level (number of loops = 0) for a process with two incoming and two outgoing B particles is given by

> 

FeynmanDiagrams⁡L,incomingparticles=B,B,outgoingparticles=B,B,numberofloops=0,output=probabilitydensity,diagrams

(45)

This process can be computed with 1 or more loops, in which case the number of terms increases significantly. As another interesting non-Abelian model, consider the interaction Lagrangian of the electro-weak part of the Standard Model

> 

Coordinates⁡clear,Z

Unaliasing Z previously defined as a system of spacetime coordinates

(46)
> 

Setup⁡quantumoperators=W,Z

quantumoperators=A,B,Q,W,Z,η,φ

(47)
> 

Define⁡Wμ,Zμ

Defined objects with tensor properties

Aμ,Bμ,a,γμ,σμ,Qj,Wμ,Zμ,∂μ,gμ,ν,εα,β,μ,ν,Xμ,Yμ

(48)
> 

F__Wμ,ν≔d_μ⁡Wν⁡X−d_ν⁡Wμ⁡X

F__Wμ,ν≔∂μ⁡Wν⁡X−∂ν⁡Wμ⁡X

(49)
> 

F__Zμ,ν≔d_μ⁡Zν⁡X−d_ν⁡Zμ⁡X

F__Zμ,ν≔∂μ⁡Zν⁡X−∂ν⁡Zμ⁡X

(50)
> 

L__WZ≔I⁢g⁢cos⁡θ__w⁢Dagger⁡F__Wμ,ν⁢Wμ⁡X−Dagger⁡Wμ⁡X⁢F__Wμ,ν⁢Zν⁡X+Wν⁡X⁢Dagger⁡Wμ⁡X⁢F__Zμ,ν

L__WZ≔I⁢g⁢cos⁡θ__w⁢∂μ⁡Wν⁡X†−∂ν⁡Wμ⁡X†⁢W⁢μ⁢μ⁡X−Wμ⁡X†⁢∂⁢μ⁢μ⁡Wν⁡X−∂ν⁡W⁢μ⁢μ⁡X⁢Z⁢ν⁢ν⁡X+Wν⁡X⁢Wμ⁡X†⁢∂⁢μ⁢μ⁡Z⁢ν⁢ν⁡X−∂⁢ν⁢ν⁡Z⁢μ⁢μ⁡X

(51)

This interaction Lagrangian contains six different terms. The S-matrix element for the tree-level process with two incoming and two outgoing W particles is given below. Due to the use of size simplification and the option factortreelevel, the result below is a product of only 10 factors, two of which are nested several times, totaling a computational length of 2750. Expanding that product the computational length is 325,761 and the number of terms is 480.

> 

FeynmanDiagrams⁡L__WZ,incomingparticles=W,W,outgoingparticles=W,W,numberofloops=0,factor,diagrams

* Partial match of 'factor' against keyword 'factortreelevel'

I16⁢−I⁢g⁢cos⁡θ__w⁢g⁢λ,τ⁢λ,τ⁢P__4⁢β⁢β+I⁢g⁢cos⁡θ__w⁢g⁢β,λ⁢β,λ⁢P__4⁢τ⁢τ−I⁢g⁢cos⁡θ__w⁢P__2⁢λ⁢λ⁢g⁢β,τ⁢β,τ+I⁢g⁢cos⁡θ__w⁢g⁢β,λ⁢β,λ⁢P__2⁢τ⁢τ+I⁢cos⁡θ__w⁢g⁢g⁢β,τ⁢β,τ⁢P__1⁢λ⁢λ−P__3⁢λ⁢λ−I⁢cos⁡θ__w⁢g⁢g⁢λ,τ⁢λ,τ⁢P__1⁢β⁢β−P__3⁢β⁢β⁢−I⁢g⁢cos⁡θ__w⁢g⁢κ,σ⁢κ,σ⁢P__3⁢α⁢α+I⁢g⁢cos⁡θ__w⁢g⁢α,κ⁢α,κ⁢P__3⁢σ⁢σ−I⁢g⁢cos⁡θ__w⁢P__1⁢κ⁢κ⁢g⁢α,σ⁢α,σ+I⁢g⁢cos⁡θ__w⁢g⁢α,κ⁢α,κ⁢P__1⁢σ⁢σ−I⁢cos⁡θ__w⁢g⁢g⁢α,σ⁢α,σ⁢P__1⁢κ⁢κ−P__3⁢κ⁢κ+I⁢cos⁡θ__w⁢g⁢g⁢κ,σ⁢κ,σ⁢P__1⁢α⁢α−P__3⁢α⁢α⁢−gσ,τ+−P__1σ+P__3σ⁢−P__1τ+P__3τm__Z2P__1χ−P__3χ⁢P__1⁢χ⁢χ−P__3⁢χ⁢χ−m__Z2+I⁢ε+−I⁢g⁢cos⁡θ__w⁢g⁢κ,τ⁢κ,τ⁢P__3⁢β⁢β+I⁢g⁢cos⁡θ__w⁢g⁢β,κ⁢β,κ⁢P__3⁢τ⁢τ−I⁢g⁢cos⁡θ__w⁢P__2⁢κ⁢κ⁢g⁢β,τ⁢β,τ+I⁢g⁢cos⁡θ__w⁢g⁢β,κ⁢β,κ⁢P__2⁢τ⁢τ+I⁢cos⁡θ__w⁢g⁢g⁢β,τ⁢β,τ⁢P__1⁢κ⁢κ−P__4⁢κ⁢κ−I⁢cos⁡θ__w⁢g⁢g⁢κ,τ⁢κ,τ⁢P__1⁢β⁢β−P__4⁢β⁢β⁢−I⁢g⁢cos⁡θ__w⁢g⁢λ,σ⁢λ,σ⁢P__4⁢α⁢α+I⁢g⁢cos⁡θ__w⁢g⁢α,λ⁢α,λ⁢P__4⁢σ⁢σ−I⁢g⁢cos⁡θ__w⁢P__1⁢λ⁢λ⁢g⁢α,σ⁢α,σ+I⁢g⁢cos⁡θ__w⁢g⁢α,λ⁢α,λ⁢P__1⁢σ⁢σ−I⁢cos⁡θ__w⁢g⁢g⁢α,σ⁢α,σ⁢P__1⁢λ⁢λ−P__4⁢λ⁢λ+I⁢cos⁡θ__w⁢g⁢g⁢λ,σ⁢λ,σ⁢P__1⁢α⁢α−P__4⁢α⁢α⁢−gσ,τ+−P__1σ+P__4σ⁢−P__1τ+P__4τm__Z2P__1χ−P__4χ⁢P__1⁢χ⁢χ−P__4⁢χ⁢χ−m__Z2+I⁢ε⁢δ⁡P__1⁢υ⁢υ+P__2⁢υ⁢υ−P__3⁢υ⁢υ−P__4⁢υ⁢υ⁢ϵWκ⁡P→3&conjugate0;⁢ϵWλ⁡P→4&conjugate0;⁢ϵWα⁡P→1⁢ϵWβ⁡P→2π2⁢E__1⁢E__2⁢E__3⁢E__4

(52)
> 

See Also

conjugate, Dagger, Dgamma, DiracConjugate, FeynmanIntegral, length, Physics, Physics conventions, Physics examples, Physics Updates, Tensors - a complete guide, Mini-Course Computer Algebra for Physicists, Physics[`*`], Psigma, Setup, simplify/size, StandardModel

References

  

[1] Bogoliubov, N.N., and Shirkov, D.V. Quantum Fields. Benjamin Cummings, 1982.

  

[2] Weinberg, S., The Quantum Theory Of Fields. Cambridge University Press, 2005.

  

[3] Berestetskii, V.B., Lifshitz, E.M., Pitaevskii, L.P., Quantum Electrodynamics. Vol.4, Course of Theoretical Physics, 2nd edition, Pergamon Press, 1982.

  

[4] Xiao, B., Wang, H., Zhu, S., A simple algorithm for automatic Feynman diagrams generation. Computer Physics Communications, Volume 184, Issue 8, (2013).