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## Feynman Diagrams - the scattering matrix in coordinates and momentum representation

Maple

Feynman Diagrams
The scattering matrix in coordinates and momentum representation

 Mathematical methods for particle physics was one of the weak spots in the Physics package. There existed a FeynmanDiagrams command, but its capabilities were too minimal. People working in the area asked for more functionality. These diagrams are the cornerstone of calculations in particle physics (collisions involving from the electron to the Higgs boson), for example at the CERN. As an introduction for people curious, not working in the area, see "Why Feynman Diagrams are so important".
 This post is thus about a new development in Physics: a full rewriting of the FeynmanDiagrams command, now including a myriad of new capabilities (mainly a. b. and c. in the Introduction), reversing the previous status of things entirely. This is work in collaboration with Davide Polvara from Durham University, Centre for Particle Theory.
 The complexity of this material is high, so the introduction to the presentation below is as brief as it can get, emphasizing the examples instead. This material is reproducible in Maple 2019.2 after installing the Physics Updates, v.598 or higher.
 At the end they are attached the worksheet corresponding to this presentation and a PDF version of it, as well as the new FeynmanDiagrams help page with all the explanatory details.

Introduction

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

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

 where

 and  is the time-ordered product of n interaction Lagrangians evaluated at different points. The S matrix formulation is at the core of perturbative approaches in relativistic Quantum Field Theory.
 In connection, the FeynmanDiagrams  command has been rewritten entirely for Maple 2020. In brief, the new functionality includes computing:
 a. The expansion  in coordinates representation up to arbitrary order (the limitation is now only your hardware)
 b. The S-matrix element  in momentum representation up to arbitrary order for given number of loops and initial and final particles (the contents of the  and  states); optionally, also the transition probability density, constructed using the square of the scattering matrix element , as shown in formula (13) of sec. 21.1 of ref.[1].
 c. The Feynman diagrams (drawings) related to the different terms of the expansion of S or of its matrix elements .
 Interaction Lagrangians involving derivatives of fields, typically appearing in non-Abelian gauge theories, are also handled, and several options are provided enabling restricting the outcome in different ways, regarding the incoming and outgoing particles, the number of loops, vertices or external legs, the propagators and normal products, or whether to compute tadpoles and 1-particle reducible terms.

Examples

For illustration purposes set three coordinate systems , and set  to represent a quantum operator

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 >
 (1.1)

Let  be the interaction Lagrangian

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 (1.2)

The expansion of  in coordinates representation, computed by default up to order = 3 (you can change that using the option order = n), by definition containing all possible configurations of external legs, displaying the related Feynman Diagrams, is given by

 >

 (1.3)

The expansion of   in coordinates representation to a specific order shows in a compact way the topology of the underlying Feynman diagrams. Each integral is represented with a new command, FeynmanIntegral , that works both in coordinates and momentum representation. To each term of the integrands corresponds a diagram, and the correspondence is always clear from the symmetry factors.

In a typical situation, one wants to compute a specific term, or scattering process, instead of the S matrix up to some order with all possible configurations of external legs. For example, to compute only the terms of this result that correspond to diagrams with 1 loop use numberofloops = 1 (for tree-level, use numerofloops = 0)

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 (1.4)

In the result above there are two terms, with 4 and 6 external legs respectively.

A scattering process with matrix element  in momentum representation, corresponding to the term with 4 external legs (symmetry factor = 72), could be any process where the total number of incoming + outgoing parties is equal to 4. For example, one with 2 incoming and 2 outgoing particles. The transition probability for that process is given by

 >

 (1.5)

When computing in momentum representation, only the topology of the corresponding Feynman diagrams is shown (i.e. the diagrams associated to the corresponding Feynman integral in coordinates representation).

The transition matrix element  is related to the transition probability density  (formula (13) of sec. 21.1 of ref.[1]) by

where  represent the particle densities of each of the s particles in the initial state , the  (Dirac) is the expected singular factor due to the conservation of the energy-momentum and the amplitude is related to  via

To directly get the probability density  instead ofuse the option output = probabilitydensity

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 (1.6)

In practice, the most common computations involve processes with 2 or 4 external legs. To restrict the expansion of the scattering matrix in coordinates representation to that kind of processes use the numberofexternallegs option. For example, the following computes the expansion of  up to order = 3, restricting the outcome to the terms corresponding to diagrams with only 2 external legs

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 (1.7)

This result shows two Feynman integrals, with respectively 2 and 3 loops, the second integral with two terms. The transition probability density in momentum representation for a process related to the first integral (1 term with symmetry factor = 96) is then

 >

 (1.8)

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

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 (1.9)

This computation can also be performed to higher orders. For example, with 3 loops, in coordinates and momentum representations, corresponding to the other two terms and diagrams in (1.7)

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 (1.10)

A corresponding S-matrix element in momentum representation:

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 (1.11)

Consider the interaction Lagrangian of Quantum Electrodynamics (QED). To formulate this problem on the worksheet, start defining the vector field .

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 (1.12)

Set lowercase Latin letters from i to s to represent spinor indices (you can change this setting according to your preference, see Setup ), also the (anticommutative) spinor field will be represented by , so set  as an anticommutativeprefix, and set  and  as quantum operators

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 (1.13)

The matrix indices of the Dirac matrices  are written explicitly and use conjugate  to represent the Dirac conjugate

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 (1.14)

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

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 (1.15)

The same computation but with only 2 external legs results in the diagrams with 1 loop that correspond to the self-energy of the electron and the photon (page 218 of ref.[1])

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 (1.16)

where the diagram with two spinor legs is the electron self-energy. To restrict the output furthermore, for example getting only the self-energy of the photon, you can specify the normal products you want:

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 (1.17)

The corresponding S-matrix elements in momentum representation

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 (1.18)

In this result we see  spinor (see ref.[2]), and the propagator of the field  with a mass . To indicate that this field is massless use

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 (1.19)

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

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 (1.20)

The self-energy of the photon:

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 (1.21)

where  is the corresponding polarization vector.

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  where a is a SU2 index (see eq.(12) of sec. 19.4 of ref.[1]). The interaction Lagrangian can be entered as follows

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 (1.22)
 >
 >
 (1.23)
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 (1.24)

The transition probability density at tree-level for a process with two incoming and two outgoing B particles is given by

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 (1.25)

 (1.26)

To simplify the repeated indices, us the option simplifytensorindices. To check the indices entering a result like this one use Check ; there are no free indices, and regarding the repeated indices:

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 (1.27)

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

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 (1.28)
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 (1.29)
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 (1.30)
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 (1.31)
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 (1.32)
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 (1.33)
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 (1.34)

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 shown in the help page for FeynmanDiagrams .

 >
 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.

FeynmanDiagrams_and_the_Scattering_Matrix.PDF

FeynmanDiagrams_and_the_Scattering_Matrix.mw

FeynmanDiagrams_-_help_page.mw

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

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