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

To continue, we will need ComputableDAGs.jl. Since ComputableDAGs.jl uses RuntimeGeneratedFunctions as the return type of ComputableDAGs.get_compute_function, we need to initialize it in our current module.

using ComputableDAGs
using RuntimeGeneratedFunctions
RuntimeGeneratedFunctions.init(@__MODULE__)

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: [0.70775116924521, 0.14068121467573746, 0.3713022323562346, 0.1538118131635875]
     -> incoming photon: [0.7303299825618491, 0.18231826441601218, 0.2143475990135847, 0.19766212561352836]
    outgoing particles:
     -> outgoing electron: [0.6277362747221342, 0.6064755512096093, 0.45018356992197417, 0.5650617101523023]
     -> outgoing electron: [0.27843908479735036, 0.20585958678221983, 0.8158791982170412, 0.49028571949646327]
     -> outgoing electron: [0.5335758159826485, 0.5906354856927607, 0.7221970636872301, 0.07682903174626199]
     -> outgoing positron: [0.3157717324197383, 0.9432233886968963, 0.23323980212024276, 0.706855243342187]
     -> outgoing positron: [0.29977260296541564, 0.23694871417269514, 0.10616842207618615, 0.015014064268333072]

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

func = get_compute_function(
    dag, proc, cpu_st(), @__MODULE__; concrete_input_type = typeof(psp)
);

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

func(psp)

0.010133907867323565

We can benchmark the execution speed too:

using BenchmarkTools
@benchmark func($psp)

BenchmarkTools.Trial: 10000 samples with 1 evaluation per sample.
 Range (min … max):  28.042 μs … 104.456 μs  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     30.617 μs               ┊ GC (median):    0.00%
 Time  (mean ± σ):   31.145 μs ±   2.538 μs  ┊ GC (mean ± σ):  0.00% ± 0.00%

           ▂██▄▂▂                                     ▁▁▁      ▂
  ▅▄▁▁▁▁▃▁▁███████▇▇▇▆▆▆▆▇▆▄▅▄▄▅▁▄▁▅▄▄▅▄▅▁▄▄▃▄▃▁▄▃▁▄▆████▇▆▆▆▄ █
  28 μs         Histogram: log(frequency) by time      41.9 μs <

 Memory estimate: 256 bytes, allocs estimate: 2.

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