n-Pair Trident Scattering Process

In this file, we set up an n-pair trident scattering process. A trident process looks like $k e^- \to e^- (e^- e^+)^n$.

You can download this file as a jupyter notebook.

using QEDFeynmanDiagrams

We need QEDcore of the QEDjl-project for base functionality and a process type, for which we can use the Mocks submodule from QEDbase for this tutorial. Downstream, a ScatteringProcess from QEDprocesses.jl could be used, for example.

using QEDcore
using QEDbase.Mocks

Let's decide how many pairs our trident should produce:

n = 2;

Now we setup the scattering process accordingly. We only consider a single spin/polarization combination here. For an example with many spin and polarization combinations, refer to the n-photon Compton example

proc = QEDbase.Mocks.MockProcessSP(
    (Electron(), Photon()),                                                         # incoming particles
    (Electron(), ntuple(_ -> Electron(), n)..., ntuple(_ -> Positron(), n)...),     # outgoing particles
    (SpinUp(), PolX()),                                                             # incoming particle spin/pols
    (SpinUp(), ntuple(_ -> SpinUp(), 2 * n)...),                                    # outgoing particle spin/pols
)

QEDbase.Mocks.MockProcessSP{Tuple{Electron, Photon}, Tuple{Electron, Electron, Electron, Positron, Positron}, Tuple{SpinUp, PolarizationX}, NTuple{5, SpinUp}}((electron, photon), (electron, electron, electron, positron, positron), (spin up, x-polarized), (spin up, spin up, spin up, spin up, spin up))

The number_of_diagrams function returns how many valid Feynman diagrams there are for a given process.

number_of_diagrams(proc)

252

Next, we can generate the DAG representing the computation for our scattering process' squared matrix element. This uses ComputableDAGs.jl.

dag = graph(proc)

Graph:
  Nodes: Total: 943, QEDFeynmanDiagrams.ComputeTask_Propagator: 60, QEDFeynmanDiagrams.ComputeTask_CollectPairs: 60, 
         QEDFeynmanDiagrams.ComputeTask_CollectTriples: 1, QEDFeynmanDiagrams.ComputeTask_BaseState: 7, QEDFeynmanDiagrams.ComputeTask_Triple: 30, 
         QEDFeynmanDiagrams.ComputeTask_PairNegated: 9, QEDFeynmanDiagrams.ComputeTask_PropagatePairs: 60, QEDFeynmanDiagrams.ComputeTask_TripleNegated: 150, 
         ComputableDAGs.DataTask: 505, QEDFeynmanDiagrams.ComputeTask_Pair: 60, QEDFeynmanDiagrams.ComputeTask_SpinPolCumulation: 1
  Edges: 1553
  Total Compute Effort: 0.0
  Total Data Transfer: 0.0
  Total Compute Intensity: 0.0

To continue, we will need ComputableDAGs.jl.

using ComputableDAGs

Now we need an input for the function, which is a QEDcore.PhaseSpacePoint. For now, we generate random momenta for every particle. In the future, QEDevents will be able to generate physical PhaseSpacePoints.

psp = PhaseSpacePoint(
    proc,
    MockModel(),
    FlatPhaseSpaceLayout(TwoBodyRestSystem()),
    tuple((rand(SFourMomentum) for _ in 1:number_incoming_particles(proc))...),
    tuple((rand(SFourMomentum) for _ in 1:number_outgoing_particles(proc))...),
)

PhaseSpacePoint:
    process: QEDbase.Mocks.MockProcessSP{Tuple{Electron, Photon}, Tuple{Electron, Electron, Electron, Positron, Positron}, Tuple{SpinUp, PolarizationX}, NTuple{5, SpinUp}}((electron, photon), (electron, electron, electron, positron, positron), (spin up, x-polarized), (spin up, spin up, spin up, spin up, spin up))
    model: mock model
    phase space layout: FlatPhaseSpaceLayout{TwoBodyTargetSystem{Energy{2}}}(TwoBodyTargetSystem{Energy{2}}(Energy{2}()))
    incoming particles:
     -> incoming electron: [0.046497460104267474, 0.7231872842126934, 0.8564535184316862, 0.7593243740505685]
     -> incoming photon: [0.7337632916099343, 0.1368027978056554, 0.8330314048213406, 0.2868426931578971]
    outgoing particles:
     -> outgoing electron: [0.1032993303871862, 0.1316595311828468, 0.4569626935057627, 0.4324047748563856]
     -> outgoing electron: [0.16363060458841117, 0.14079302692857, 0.05735869974431995, 0.8434009733741221]
     -> outgoing electron: [0.778302199299943, 0.7440927157102011, 0.7297663322829567, 0.7761959119099375]
     -> outgoing positron: [0.8338192459745578, 0.4268252361951489, 0.4911682937973807, 0.5434596700355626]
     -> outgoing positron: [0.15093621960209236, 0.4606526124463929, 0.3184177873742151, 0.5341560278079245]

With the DAG, the process, and an input type to use, we can now generate the actual computable function:

func = compute_function(dag, proc, cpu_st(), @__MODULE__);

Finally, we can test that the function actually runs and computes something by simply calling it on the PhaseSpacePoint:

func(psp)

0.004449133083094162

We can benchmark the execution speed too:

using BenchmarkTools
@benchmark func($psp)

BenchmarkTools.Trial: 10000 samples with 1 evaluation per sample.
 Range (min … max):  46.046 μs …   4.865 ms  ┊ GC (min … max): 0.00% … 93.86%
 Time  (median):     48.952 μs               ┊ GC (median):    0.00%
 Time  (mean ± σ):   53.610 μs ± 117.929 μs  ┊ GC (mean ± σ):  5.85% ±  2.65%

   ▂▅▆▇████▇▆▆▅▄▃▂▂▁                ▁▁▁▁▁▁ ▁▁                  ▃
  ▇██████████████████▇▇▆▆▆▆▆▄▂▆▅▆▇▇████████████████▇█▆▇▇▇▇▅▆▄▅ █
  46 μs         Histogram: log(frequency) by time      69.2 μs <

 Memory estimate: 68.98 KiB, allocs estimate: 661.

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