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, DataTask: 505, QEDFeynmanDiagrams.ComputeTask_CollectTriples: 1, 
         QEDFeynmanDiagrams.ComputeTask_SpinPolCumulation: 1, QEDFeynmanDiagrams.ComputeTask_PropagatePairs: 60, QEDFeynmanDiagrams.ComputeTask_PairNegated: 9, 
         QEDFeynmanDiagrams.ComputeTask_Propagator: 60, QEDFeynmanDiagrams.ComputeTask_CollectPairs: 60, QEDFeynmanDiagrams.ComputeTask_TripleNegated: 150, 
         QEDFeynmanDiagrams.ComputeTask_BaseState: 7, QEDFeynmanDiagrams.ComputeTask_Triple: 30, QEDFeynmanDiagrams.ComputeTask_Pair: 60
  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: [0.7727641434976086, 0.3927659949273814, 0.8267472195085925, 0.0719570296373776]
     -> incoming photon: [0.6826483454114322, 0.2990298501136065, 0.884026833442817, 0.2037957241387528]
    outgoing particles:
     -> outgoing electron: [0.10541261545564962, 0.5129571017314462, 0.05797340857583011, 0.9732378593581387]
     -> outgoing electron: [0.9795398614482983, 0.2009140924909294, 0.0009143780531152279, 0.8674161223969643]
     -> outgoing electron: [0.013104653001469568, 0.7712460197829101, 0.3470835567103885, 0.12091623848213306]
     -> outgoing positron: [0.25770847325237844, 0.7269062422357176, 0.9547842614279513, 0.40079150758937665]
     -> outgoing positron: [0.019860604456588438, 0.08549780267669971, 0.1267117592284892, 0.9167546017661887]

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)

0.00842969526263296

We can benchmark the execution speed too:

using BenchmarkTools
@benchmark func($psp)

BenchmarkTools.Trial: 10000 samples with 1 evaluation per sample.
 Range (min … max):  47.204 μs … 134.665 μs  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     47.903 μs               ┊ GC (median):    0.00%
 Time  (mean ± σ):   48.367 μs ±   2.278 μs  ┊ GC (mean ± σ):  0.00% ± 0.00%

   ▄▆███▇▆▄▃▂▁▁▁▁                                    ▁▁▂▁▁▁    ▂
  ██████████████████▇▇▆▇▆▆▅▄▅▁▃▃▃▃▁▁▁▁▁▁▁▃▃▁▃▃▃▄▆▆▇▇████████▇▇ █
  47.2 μs       Histogram: log(frequency) by time      55.5 μs <

 Memory estimate: 256 bytes, allocs estimate: 2.

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