Multiple on-shell configurations and integration order

A diagram can have more than one resonance where not all of them can be on shell at once. The process g g > h > Z Z > e+ e- mu+ mu- is a standard example: the Higgs mass, 125 GeV, is below twice the Z mass, so the two Z propagators cannot both sit on their pole while the Higgs also sits on its pole. Topology.topologies enumerates the on-shell configurations this forces, each one a separate integration channel.

Building the diagram

The Higgs propagator p0 splits into the two Z propagators p1 and p2, which each decay to a lepton pair:

import madspace as ms

M_H, W_H = 125.0, 0.00407
M_Z, W_Z = 91.188, 2.4952

props = [
    ms.Propagator(mass=M_H, width=W_H, pdg_id=25),
    ms.Propagator(mass=M_Z, width=W_Z, pdg_id=23),
    ms.Propagator(mass=M_Z, width=W_Z, pdg_id=23),
]
diagram = ms.Diagram(
    [0.0, 0.0], [0.0, 0.0, 0.0, 0.0], props,
    [["i0", "i1", "p0"], ["p0", "p1", "p2"], ["p1", "o0", "o1"], ["p2", "o2", "o3"]],
)

Enumerating the channels

Topology.topologies returns one channel per combination of on-shell flags that is both kinematically possible and not already covered by another channel:

topologies = ms.Topology.topologies(diagram)
print(f"number of channels: {len(topologies)}")
for t in topologies:
    on_shell = [d.on_shell for d in t.decays if d.mass != 0]
    print("on-shell flags (H, Z, Z):", on_shell)

Three channels come out: the Higgs on its pole with either Z on shell and the other one floating, and the Higgs off its pole with both Z bosons on shell:

number of channels: 3
on-shell flags (H, Z, Z): [True, True, False]
on-shell flags (H, Z, Z): [True, False, True]
on-shell flags (H, Z, Z): [False, True, True]

Inspecting a channel

Printing a Topology shows its decay tree, with the on_shell flag and the order in which each invariant is sampled:

print(topologies[0])

├── incoming: index=0, mass=0
├── incoming: index=1, mass=0
└── decay: order=0, mass=125, width=0.00407, e_min=0, e_max=0, pdg_id=25, on_shell=true, on_shell_boundary=false
    ├── decay: order=1, mass=91.188, width=2.4952, e_min=0, e_max=0, pdg_id=23, on_shell=true, on_shell_boundary=false
    │   ├── outgoing: index=0, mass=0
    │   └── outgoing: index=1, mass=0
    └── decay: order=2, mass=91.188, width=2.4952, e_min=0, e_max=0, pdg_id=23, on_shell=false, on_shell_boundary=false
        ├── outgoing: index=2, mass=0
        └── outgoing: index=3, mass=0

The on-shell Higgs is sampled first, then the on-shell Z, then the off-shell Z last, since its window depends on what energy the first two leave behind.

Breaking a tie with integration_order

Propagator also takes an integration_order argument. A lower value is sampled first. It only matters between propagators the on-shell status alone does not already order, such as the two Z bosons in the channel where both are on shell:

props[1] = ms.Propagator(mass=M_Z, width=W_Z, pdg_id=23, integration_order=1)
props[2] = ms.Propagator(mass=M_Z, width=W_Z, pdg_id=23, integration_order=0)
diagram_reordered = ms.Diagram(
    [0.0, 0.0], [0.0, 0.0, 0.0, 0.0], props,
    [["i0", "i1", "p0"], ["p0", "p1", "p2"], ["p1", "o0", "o1"], ["p2", "o2", "o3"]],
)
both_on_shell = [
    t for t in ms.Topology.topologies(diagram_reordered) if not t.decays[0].on_shell
][0]
print(both_on_shell)

The second Z, now with the lower integration_order, is sampled first:

├── incoming: index=0, mass=0
├── incoming: index=1, mass=0
└── decay: order=1, mass=125, width=0.00407, e_min=0, e_max=0, pdg_id=25, on_shell=false, on_shell_boundary=false
    ├── decay: order=2, mass=91.188, width=2.4952, e_min=0, e_max=0, pdg_id=23, on_shell=true, on_shell_boundary=false
    │   ├── outgoing: index=0, mass=0
    │   └── outgoing: index=1, mass=0
    └── decay: order=0, mass=91.188, width=2.4952, e_min=0, e_max=0, pdg_id=23, on_shell=true, on_shell_boundary=false
        ├── outgoing: index=2, mass=0
        └── outgoing: index=3, mass=0