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DOC: show how to collect Wigner D-functions
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@redeboer
redeboer committed Jan 13, 2023
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2 changes: 2 additions & 0 deletions .cspell.json
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"maxdepth",
"meshgrid",
"mypy",
"mystnb",
"nansum",
"nbconvert",
"nbformat",
Expand All @@ -119,6 +120,7 @@
"tqdm",
"unsrt",
"venv",
"wigners",
"wspace",
"xlabel",
"xreplace",
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364 changes: 364 additions & 0 deletions docs/collect-wigners.ipynb
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{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Collect angular distribution symbols\n",
"\n",
"```{autolink-concat}\n",
"```"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"jupyter": {
"source_hidden": true
},
"mystnb": {
"code_prompt_show": "Import Python libraries"
},
"tags": [
"hide-cell",
"scroll-input"
]
},
"outputs": [],
"source": [
"from __future__ import annotations\n",
"\n",
"import itertools\n",
"from typing import Iterable\n",
"\n",
"import qrules\n",
"import sympy as sp\n",
"from IPython.display import Latex, Markdown\n",
"\n",
"from ampform_dpd import DalitzPlotDecompositionBuilder, simplify_latex_rendering\n",
"from ampform_dpd.decay import (\n",
" IsobarNode,\n",
" Particle,\n",
" ThreeBodyDecay,\n",
" ThreeBodyDecayChain,\n",
")\n",
"from ampform_dpd.io import as_markdown_table, aslatex, perform_cached_doit\n",
"from ampform_dpd.spin import filter_parity_violating_ls, generate_ls_couplings\n",
"\n",
"simplify_latex_rendering()"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"jupyter": {
"source_hidden": true
},
"mystnb": {
"code_prompt_show": "Define particles"
},
"tags": [
"hide-input"
]
},
"outputs": [],
"source": [
"PDG = qrules.load_pdg()\n",
"PARTICLE_DB = {\n",
" p.name: Particle(\n",
" name=p.name,\n",
" latex=p.latex,\n",
" spin=p.spin,\n",
" parity=int(p.parity),\n",
" mass=p.mass,\n",
" width=p.width,\n",
" )\n",
" for p in PDG\n",
" if p.parity is not None\n",
"}\n",
"PARTICLE_DB[\"Lambda(2000)\"] = Particle(\n",
" name=\"Lambda(2000)\",\n",
" latex=R\"\\Lambda(2000)\",\n",
" spin=0.5,\n",
" parity=-1,\n",
" mass=2.0,\n",
" width=(0.020 + 0.400) / 2,\n",
")\n",
"Λc = PARTICLE_DB[\"Lambda(c)+\"]\n",
"p = PARTICLE_DB[\"p\"]\n",
"π = PARTICLE_DB[\"pi+\"]\n",
"K = PARTICLE_DB[\"K-\"]\n",
"PARTICLE_TO_ID = {Λc: 0, p: 1, π: 2, K: 3}\n",
"Markdown(as_markdown_table(list(PARTICLE_TO_ID)))"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"jupyter": {
"source_hidden": true
},
"mystnb": {
"code_prompt_show": "Select resonances"
},
"tags": [
"hide-input"
]
},
"outputs": [],
"source": [
"resonance_names = [\n",
" \"Lambda(1405)\",\n",
" \"Lambda(1520)\",\n",
" \"Lambda(1600)\",\n",
" \"Lambda(1670)\",\n",
" \"Lambda(1690)\",\n",
" \"Lambda(2000)\",\n",
" \"Delta(1232)+\",\n",
" \"Delta(1600)+\",\n",
" \"Delta(1700)+\",\n",
" \"K(0)*(700)+\",\n",
" \"K*(892)0\",\n",
" \"K(2)*(1430)0\",\n",
"]\n",
"resonances = [PARTICLE_DB[name] for name in resonance_names]\n",
"Markdown(as_markdown_table(resonances))"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"jupyter": {
"source_hidden": true
},
"mystnb": {
"code_prompt_show": "Generate decay"
},
"tags": [
"hide-input"
]
},
"outputs": [],
"source": [
"def load_three_body_decay(\n",
" resonance_names: Iterable[str],\n",
" particle_definitions: dict[str, Particle],\n",
" min_ls: bool = True,\n",
") -> ThreeBodyDecay:\n",
" resonances = [particle_definitions[name] for name in resonance_names]\n",
" chains: list[ThreeBodyDecayChain] = []\n",
" for res in resonances:\n",
" chains.extend(_create_isobar(res, min_ls))\n",
" return ThreeBodyDecay(\n",
" states={state_id: particle for particle, state_id in PARTICLE_TO_ID.items()},\n",
" chains=tuple(chains),\n",
" )\n",
"\n",
"\n",
"def _create_isobar(resonance: Particle, min_ls: bool) -> list[ThreeBodyDecayChain]:\n",
" if resonance.name.startswith(\"K\"):\n",
" child1, child2, spectator = π, K, p\n",
" elif resonance.name.startswith(\"L\"):\n",
" child1, child2, spectator = K, p, π\n",
" elif resonance.name.startswith(\"D\"):\n",
" child1, child2, spectator = p, π, K\n",
" else:\n",
" raise NotImplementedError\n",
" prod_ls_couplings = _generate_ls(Λc, resonance, spectator, conserve_parity=False)\n",
" dec_ls_couplings = _generate_ls(resonance, child1, child2, conserve_parity=True)\n",
" if min_ls:\n",
" decay = IsobarNode(\n",
" parent=Λc,\n",
" child1=IsobarNode(\n",
" parent=resonance,\n",
" child1=child1,\n",
" child2=child2,\n",
" interaction=min(dec_ls_couplings),\n",
" ),\n",
" child2=spectator,\n",
" interaction=min(prod_ls_couplings),\n",
" )\n",
" return [ThreeBodyDecayChain(decay)]\n",
" chains = []\n",
" for dec_ls, prod_ls in itertools.product(dec_ls_couplings, prod_ls_couplings):\n",
" decay = IsobarNode(\n",
" parent=Λc,\n",
" child1=IsobarNode(\n",
" parent=resonance,\n",
" child1=child1,\n",
" child2=child2,\n",
" interaction=dec_ls,\n",
" ),\n",
" child2=spectator,\n",
" interaction=prod_ls,\n",
" )\n",
" chains.append(ThreeBodyDecayChain(decay))\n",
" return chains\n",
"\n",
"\n",
"def _generate_ls(\n",
" parent: Particle, child1: Particle, child2: Particle, conserve_parity: bool\n",
") -> list[tuple[int, sp.Rational]]:\n",
" ls = generate_ls_couplings(parent.spin, child1.spin, child2.spin)\n",
" if conserve_parity:\n",
" return filter_parity_violating_ls(\n",
" ls, parent.parity, child1.parity, child2.parity\n",
" )\n",
" return ls\n",
"\n",
"\n",
"DECAY = load_three_body_decay(\n",
" resonance_names,\n",
" particle_definitions=PARTICLE_DB,\n",
" min_ls=True,\n",
")\n",
"Latex(aslatex(DECAY, with_jp=True))"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"mystnb": {
"code_prompt_show": "Define dynamics builder that returns a symbol only"
},
"tags": [
"hide-input"
]
},
"outputs": [],
"source": [
"def formulate_dynamics_symbol(\n",
" decay_chain: ThreeBodyDecayChain,\n",
") -> tuple[sp.Symbol, dict[sp.Symbol, float]]:\n",
" s = sp.latex(_get_mandelstam_s(decay_chain))\n",
" l_dec = sp.Rational(decay_chain.outgoing_ls.L)\n",
" l_prod = sp.Rational(decay_chain.incoming_ls.L)\n",
" s_dec = sp.Rational(decay_chain.outgoing_ls.S)\n",
" s_prod = sp.Rational(decay_chain.incoming_ls.S)\n",
" parameter_defaults = {}\n",
" dynamics = sp.Symbol(\n",
" Rf\"X_{{{decay_chain.resonance.spin}}}^{{{l_prod},{s_prod};{l_dec},{s_dec}}}\\left({s}\\right)\"\n",
" )\n",
" return dynamics, parameter_defaults\n",
"\n",
"\n",
"def _get_mandelstam_s(decay: ThreeBodyDecayChain) -> sp.Symbol:\n",
" s1, s2, s3 = sp.symbols(\"sigma1:4\", nonnegative=True)\n",
" m1, m2, m3 = map(_to_mass_symbol, [p, π, K])\n",
" decay_masses = {_to_mass_symbol(p) for p in decay.decay_products}\n",
" if decay_masses == {m2, m3}:\n",
" return s1\n",
" if decay_masses == {m1, m3}:\n",
" return s2\n",
" if decay_masses == {m1, m2}:\n",
" return s3\n",
" raise NotImplementedError(\n",
" f\"Cannot find Mandelstam variable for {''.join(decay_masses)}\"\n",
" )\n",
"\n",
"\n",
"def _to_mass_symbol(particle: Particle) -> sp.Symbol:\n",
" state_id = PARTICLE_TO_ID.get(particle)\n",
" if state_id is not None:\n",
" return sp.Symbol(f\"m{state_id}\", nonnegative=True)\n",
" return sp.Symbol(f\"m_{{{particle.latex}}}\", nonnegative=True)"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"tags": [
"full-width",
"hide-input"
]
},
"outputs": [],
"source": [
"model_builder = DalitzPlotDecompositionBuilder(DECAY, min_ls=(False, True))\n",
"for chain in model_builder.decay.chains:\n",
" model_builder.dynamics_choices.register_builder(chain, formulate_dynamics_symbol)\n",
"model = model_builder.formulate(reference_subsystem=1)\n",
"model.intensity"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"mystnb": {
"code_prompt_show": "Set all couplings to one"
},
"tags": [
"hide-input"
]
},
"outputs": [],
"source": [
"couplings_to_one = {s: 1 for s in model.parameter_defaults if \"mathcal\" in str(s)}\n",
"\n",
"full_amplitudes = {k: perform_cached_doit(v) for k, v in model.amplitudes.items()}\n",
"full_expression = perform_cached_doit(model.intensity).xreplace(full_amplitudes)\n",
"full_expression = full_expression.doit().xreplace(couplings_to_one)\n",
"dynamic_symbols = sorted(\n",
" (s for s in full_expression.free_symbols if str(s).startswith(\"X\")),\n",
" key=lambda s: (str(s)[-10:], str(s)),\n",
")"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"jupyter": {
"source_hidden": true
},
"mystnb": {
"code_prompt_show": "Collect Wigner-D functions"
},
"tags": [
"hide-input",
"full-width"
]
},
"outputs": [],
"source": [
"src = R\"\"\"\n",
"\\begin{eqnarray}\n",
"\"\"\"\n",
"for x in dynamic_symbols:\n",
" filter_substitutions = {s: 1 if s == x else 0 for s in dynamic_symbols}\n",
" factor = full_expression.xreplace(filter_substitutions)\n",
" src += Rf\"{sp.latex(x)} &:& {sp.latex(factor)} \\\\\" \"\\n\"\n",
"src += R\"\\end{eqnarray}\"\n",
"Latex(src)"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3 (ipykernel)",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.8.15"
}
},
"nbformat": 4,
"nbformat_minor": 4
}
1 change: 1 addition & 0 deletions docs/index.md
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---
lc2pkpi
jpsi2ksp
collect-wigners
```

```{toctree}
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