thresholds 0.4.0

Find two-body hadronic thresholds compatible with given J^P quantum numbers

ThresholdFinder

Finds n-body hadronic thresholds compatible with given J^P quantum numbers. Given a mass range and a target J^P, it scans all combinations of PDG hadrons whose combined mass falls in that range and checks whether they can couple — via some total orbital angular momentum L — to produce the desired quantum numbers. The default is two-body; use --n-body (or n_body= in the API) to search three-body or higher final states.

Requirements

• Python >= 3.11
• particle >= 0.24

pip install thresholds

Usage

Command line

threshold-finder mass_min mass_max [J P] [options]

Positional arguments:

Argument	Description
mass_min	Lower bound of the threshold search range (MeV)
mass_max	Upper bound of the threshold search range (MeV)
J	Target total angular momentum (integer or half-integer, e.g. 1, 0.5). Optional if --particles is given.
P	Target parity: +1 or -1. Optional if --particles is given.

J and P must either both be given or both be omitted. When omitted, --particles is required and the 3 lowest J^P combinations the pair can produce are used automatically.

Optional arguments:

Flag	Default	Description
--max-L L	auto	Maximum orbital angular momentum to consider. Without this flag, L is capped automatically at J + J₁ + J₂ + 4 for each pair.
--charge CHARGE	0	Required total electric charge of the two-particle system.
--status S [S ...]	0	PDG status codes to include: 0 = well-established, 1 = evidence but unconfirmed, 2 = omitted from summary tables.
--unique-pairs	off	Show each particle combination only once (keeping the lowest L), instead of one entry per valid L.
--n-body N	2	Number of particles in the final state (must be >= 2). Default is 2 (two-body). Use 3 for three-body, etc.

Flavor conservation flags (all optional, independent):

Flag	Description
--u N	Required net u-quark number of the pair (#u − #ū)
--d N	Required net d-quark number of the pair (#d − #d̄)
--s N	Required net s-quark number of the pair (#s − #s̄)
--c N	Required net charm of the pair (#c − #c̄)
--b N	Required net bottomness of the pair (#b − #b̄)
--particles P1 P2	Derive all flavor numbers automatically from two PDG particle names

Only the flags you provide are enforced. Omit a flag to leave that flavor unconstrained. Pairs involving particles with undefined quark content (e.g. η, ω — mixed states like uū+dd̄) are excluded when any flavor flag is set.

--particles does two things:

1. Derives flavor conservation automatically — no need to compute net quark numbers by hand. Explicit --u/--d/... flags override the derived values.
2. Checks feasibility — if J and P are given, the tool first verifies that the specified pair can actually produce that J^P at some L. If not, a warning is printed and no results are shown for that J^P.
3. Auto-detects J^P — if J and P are omitted, the 3 lowest J^P combinations the pair can produce (ordered by minimum L) are determined and each is searched separately.

Examples

Find all 1⁻ channels with threshold between 250 and 300 MeV (the ρ region):

$ python3 -m threshold_finder.cli 250 300 1 -1

Thresholds for J^P = 1^-  in [250.0, 300.0] MeV  (max L = ∞)
Found 1 combination(s):
  pi+ + pi-  threshold=279.1 MeV  L=1  J^P=1^-

Find 1⁻ channels near 1 GeV with full flavor conservation (all net quark numbers = 0):

$ python3 -m threshold_finder.cli 900 1100 1 -1 --u 0 --d 0 --s 0 --c 0 --b 0 --unique-pairs

Thresholds for J^P = 1^-  in [900.0, 1100.0] MeV  (max L = ∞)  flavor: u=+0, d=+0, s=+0, c=+0, b=+0
Found 4 combination(s):
  pi+ + rho(770)-  threshold=914.7 MeV  L=1  J^P=1^-
  pi- + rho(770)+  threshold=914.7 MeV  L=1  J^P=1^-
  K+ + K-  threshold=987.4 MeV  L=1  J^P=1^-
  K0 + K~0  threshold=995.2 MeV  L=1  J^P=1^-

Constrain only strangeness (leave u/d free) to find kaonic channels:

$ python3 -m threshold_finder.cli 600 700 0 -1 --s -1 --unique-pairs

Thresholds for J^P = 0^-  in [600.0, 700.0] MeV  (max L = ∞)  flavor: s=-1
Found 4 combination(s):
  pi0 + K(L)0  threshold=632.6 MeV  L=0  J^P=0^-
  ...

Find open-charm 1⁻ thresholds near ψ(3770) with net zero flavor:

$ python3 -m threshold_finder.cli 3700 3900 1 -1 --u 0 --d 0 --s 0 --c 0 --b 0 --unique-pairs

...
  D0 + D~0  threshold=3729.7 MeV  L=1  J^P=1^-
  D+ + D-   threshold=3739.3 MeV  L=1  J^P=1^-
  ...
  D0 + D*(2007)~0  threshold=3871.7 MeV  L=1  J^P=1^-
  ...

Omit J and P to auto-detect the 3 lowest J^P combinations D0 + Lambda can produce:

$ threshold-finder 2800 3000 --particles 'D0' 'Lambda' --unique-pairs

Flavor conservation derived from 'D0' + 'Lambda':
  u = +0
  d = +1
  s = +1
  c = +1

No J^P given — using the 3 lowest combinations 'D0' + 'Lambda' can produce:
  J^P = 1/2^-  (lowest at L=0)
  J^P = 1/2^+  (lowest at L=1)
  J^P = 3/2^+  (lowest at L=1)

Thresholds for J^P = 1/2^-  in [2800.0, 3000.0] MeV  ...
...
Thresholds for J^P = 1/2^+  in [2800.0, 3000.0] MeV  ...
...
Thresholds for J^P = 3/2^+  in [2800.0, 3000.0] MeV  ...
...

If the requested J^P cannot be produced by the given pair at any L, a warning is shown and that J^P is skipped:

$ threshold-finder 2800 3000 0.5 +1 --particles 'pi+' 'pi-'

WARNING: 'pi+' + 'pi-' cannot produce J^P = 1/2^+ at any L

Derive flavor numbers from reference particles (D0 + Lambda defines the channel):

$ threshold-finder 2800 3000 0.5 -1 --particles 'D0' 'Lambda' --unique-pairs

Flavor conservation derived from 'D0' + 'Lambda':
  u = +0
  d = +1
  s = +1
  c = +1

Thresholds for J^P = 1/2^-  in [2800.0, 3000.0] MeV  (max L = ∞)  flavor: u=+0, d=+1, s=+1, c=+1
Found 5 combination(s):
  pi- + Xi(c)(2790)+  threshold=2931.5 MeV  L=1  J^P=1/2^-
  K- + Sigma(c)(2455)+  threshold=2946.3 MeV  L=0  J^P=1/2^-
  ...

If the reference threshold is outside the search range, a warning is printed but the search still runs:

$ threshold-finder 2500 2800 0.5 -1 --particles 'D0' 'Lambda' --unique-pairs

WARNING: threshold of D0 + Lambda = 2980.5 MeV is above mass_max = 2800.0 MeV
Flavor conservation derived from 'D0' + 'Lambda':
  ...

If a particle has ambiguous quark content (mixed state), the tool reports what could be determined and prints a ready-to-edit command with ??? placeholders for the unknown flavors:

$ threshold-finder 2800 3000 0.5 -1 --particles 'eta' 'Lambda'

ERROR: Quark content is ambiguous (mixed/superposition state) for:
  eta  (PDG quarks string: 'x(uU+dD)+y(sS)')
Determined from ['Lambda']: d=+1, s=+1, u=+1

Set the remaining flavor flags manually. Example command:
  threshold-finder 2800.0 3000.0 0.5 -1 --u 1 --d 1 --s 1 --c ??? --b ???

If a particle name is not found, 5 suggestions are printed as ready-to-run commands:

$ threshold-finder 2800 3000 0.5 -1 --particles 'D0' 'Lmabda'

ERROR: Unknown particle 'Lmabda'
Did you mean one of these?
  threshold-finder 2800.0 3000.0 0.5 -1 --particles 'D0' 'Lambda'
  threshold-finder 2800.0 3000.0 0.5 -1 --particles 'D0' 'Lambda~'
  threshold-finder 2800.0 3000.0 0.5 -1 --particles 'D0' 'Lambda(c)+'
  threshold-finder 2800.0 3000.0 0.5 -1 --particles 'D0' 'Lambda(b)0'
  threshold-finder 2800.0 3000.0 0.5 -1 --particles 'D0' 'Lambda(1520)'

Find three-body 0⁻ thresholds near 420 MeV (the 3π region):

$ threshold-finder 400 450 0 -1 --n-body 3 --max-L 0 --unique-pairs

Thresholds for J^P = 0^-  in [400.0, 450.0] MeV  (max L = 0)  (3-body)
Found 2 combination(s):
  pi- + pi0 + pi+  threshold=414.1 MeV  L=0  J^P=0^-
  pi0 + pi0 + pi0  threshold=404.9 MeV  L=0  J^P=0^-

Find 2⁺ channels with threshold 500–700 MeV, restricting to L ≤ 2:

$ python3 -m threshold_finder.cli 500 700 2 +1 --max-L 2 --unique-pairs

Thresholds for J^P = 2^+  in [500.0, 700.0] MeV  (max L = 2)
Found 7 combination(s):
  pi0 + K(L)0  threshold=632.6 MeV  L=2  J^P=2^+
  ...

Python API

from threshold_finder import ThresholdFinder, FlavorFilter

finder = ThresholdFinder(
    mass_min=900,
    mass_max=1100,
    J_target=1,
    P_target=-1,
    n_body=2,                                # 2 = two-body (default); set 3 for three-body, etc.
    max_L=None,                              # None = automatic
    total_charge=0.0,
    flavor_filter=FlavorFilter(u=0, d=0, s=0, c=0, b=0),  # all net quark numbers = 0
    status_filter=(0,),                      # established particles only
)
result = finder.run()

print(result)  # formatted summary

for c in result.combinations:
    print(" + ".join(c.particles), "  L =", c.L, "  threshold =", c.threshold, "MeV")

Three-body search:

finder = ThresholdFinder(
    mass_min=400,
    mass_max=450,
    J_target=0,
    P_target=-1,
    n_body=3,
    max_L=0,
)
result = finder.run()
for c in result.combinations:
    print(" + ".join(c.particles), "  threshold =", c.threshold, "MeV")

Constrain only specific flavors by omitting the rest:

# Only require net charm = 0; u, d, s, b are unconstrained
flavor_filter=FlavorFilter(c=0)

# Only require net strangeness = -1
flavor_filter=FlavorFilter(s=-1)

ThresholdResult has the fields J_target, P_target, mass_min, mass_max, max_L, n_body, flavor_filter, and combinations (a list of CombinationResult).

Each CombinationResult contains:

Field	Type	Description
particles	tuple[str, ...]	PDG names of all particles in the final state
masses	tuple[float, ...]	Masses of all particles (MeV)
charges	tuple[float, ...]	Charges of all particles
spins	tuple[float, ...]	J values of all particles
parities	tuple[int, ...]	Parities of all particles
threshold	float	Sum of all particle masses (MeV)
L	int	Total orbital angular momentum
J_total	float	Total angular momentum (= J_target)
P_total	int	Total parity (= P_target)

For two-body results the legacy per-particle properties (particle1, particle2, mass1, mass2, charge1, charge2, J1, J2, P1, P2, identical) are still available as convenience aliases.

Physics

The tool checks whether a set of n particles (each with Jᵢ^Pᵢ) in a state of total orbital angular momentum L can produce the target J^P.

Parity:

P_total = P₁ · P₂ · … · Pₙ · (-1)^L

Angular momentum: J_total must be reachable by sequentially coupling all particle spins J₁ ⊗ J₂ ⊗ … ⊗ Jₙ via the triangle rule, then adding L.

Identical bosons (two-body only): For two identical bosons (e.g. π⁰π⁰), the spatial wave function must be symmetric under exchange, which requires L to be even. For n > 2 this constraint is not enforced (exact symmetrisation of n-body wave functions depends on the full Bose/Fermi statistics of all permutations).

Flavor conservation: Net quark numbers are computed as #quark − #antiquark for each flavor (u, d, s, c, b). They are additive over all n particles in the final state. Setting a flavor to 0 requires the combination to have no net quark content in that flavor. Particles with mixed or superposition quark content (η, ω, φ, π⁰, …) have undefined quark numbers and are excluded from any result when a flavor constraint is active.

Particle data (masses, J, P, charge, quark content) are read from the PDG via the particle package. Only hadrons with known mass, J, and P are considered.