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DOC: manual formulation of helicity amplitudes for pgamma->LambdaKPi #104

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DOC: manual formulation of helicity amplitudes for pgamma->LambdaKPi #104

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d20d12c
initial commit
shenvitor Sep 2, 2024
c229304
show A^2 and A^3 expressions
shenvitor Sep 2, 2024
a9874c1
starting draft
shenvitor Sep 2, 2024
1c8990c
symbolic intensity
shenvitor Sep 3, 2024
6307f25
draft of function to formulate A
shenvitor Sep 5, 2024
514907e
Amplitudes explicit form
shenvitor Sep 6, 2024
a360fcf
gitignore .pixi/
shenvitor Nov 8, 2024
83d2bb7
address to comment https://github.com/ComPWA/gluex-nstar/pull/104#pul…
shenvitor Nov 8, 2024
3991946
https://github.com/ComPWA/gluex-nstar/pull/104#discussion_r1747309082
shenvitor Nov 8, 2024
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154 changes: 146 additions & 8 deletions docs/lambda-k-pi/manual-dpd.ipynb
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Original file line number Diff line number Diff line change
Expand Up @@ -54,9 +54,11 @@
},
{
"cell_type": "markdown",
"metadata": {},
"metadata": {
"jp-MarkdownHeadingCollapsed": true
},
"source": [
"# Reaction "
"## Reaction "
]
},
{
Expand Down Expand Up @@ -206,7 +208,17 @@
"outputs": [],
"source": [
"half = sp.Rational(1, 2)\n",
"\n",
"three_half = sp.Rational(3, 2)\n",
"five_half = sp.Rational(5, 2)\n",
"seven_half = sp.Rational(7, 2)"
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]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"A1 = sp.IndexedBase(\"A^1\")\n",
"A2 = sp.IndexedBase(\"A^2\")\n",
"A3 = sp.IndexedBase(\"A^3\")\n",
Expand Down Expand Up @@ -303,16 +315,41 @@
" m_k = sp.Symbol(rf\"m_{{{resonance_name}}}\")\n",
" gamma_k = sp.Symbol(rf\"\\Gamma_{{{resonance_name}}}\")\n",
" theta23, phi23 = sp.symbols(r\"theta_23 phi_23\") # noqa: F841\n",
" s23, lambda_R = sp.symbols(r\"s_{23} \\lambda_K^*\")\n",
" s23, lambda_R = sp.symbols(r\"s_{23} \\lambda_{K^*}\")\n",
" delta = sp.KroneckerDelta(-half, lambda_R + half)\n",
" d_k = Wigner.d(1, lambda_R, 0, theta23)\n",
" return sp.Sum(\n",
" (gamma_k * m_k * delta * c_k * d_k / (-sp.I * gamma_k * m_k + m_k**2 - s23)),\n",
" (lambda_R, -lambda_r_value, lambda_r_value),\n",
" )\n",
"\n",
"\n",
"formualte_A1(\"K^{*}(1410)\", 1)"
" )"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"A1 = (\n",
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Be careful not to overwrite the previous A1 definition. This should be the A1_expr, or even something like a mapping of sp.Symbol to sp.Expr as dictionary. Something like:

amplitudes = {}
amplitudes[A1[-half, +half]] = formulate_A1(...) + ...

" formualte_A1(\"K^{*}(1410)^+\", 1)\n",
" + formualte_A1(\"K^{*}(1680)^+\", 1)\n",
" + formualte_A1(\"K^{*}(892)^+\", 1)\n",
" + formualte_A1(\"K^{*}_0(1430)^+\", 0)\n",
" + formualte_A1(\"K^{*}_0(700)\", 0)\n",
" + formualte_A1(\"K^{*}_2(1430)\", 2)\n",
" + formualte_A1(\"K^{*}_3(1780)\", 3)\n",
" + formualte_A1(\"K^{*}_4(2045)\", 4)\n",
")\n",
"A1"
Comment on lines +338 to +348
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This will need to be automated, i.e., some loop over the $K^*$ resonances found by QRules, and computation of the helicity. Actually, your formulate_A1() is more something like formulate_chain_amplitude_a1() and formulate_a1() creates the coherent sum of those chain amplitudes.

]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"Math(aslatex(A1, terms_per_line=1))"
]
},
{
Expand Down Expand Up @@ -345,6 +382,57 @@
"A2_expr"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"def formualte_A2(resonance_name, lambda_r_value): # noqa: N802\n",
" c_k = sp.Symbol(rf\"C_{{{resonance_name}}}\")\n",
" m_k = sp.Symbol(rf\"m_{{{resonance_name}}}\")\n",
" gamma_k = sp.Symbol(rf\"\\Gamma_{{{resonance_name}}}\")\n",
" theta31, phi31 = sp.symbols(r\"theta_31 phi_31\") # noqa: F841\n",
" s31, lambda_R = sp.symbols(r\"s_{31} \\lambda_{\\Sigma^*}\")\n",
" delta = sp.KroneckerDelta(-half, lambda_R + half)\n",
" d_k = Wigner.d(1, lambda_R, 0, theta31)\n",
" return sp.Sum(\n",
" (gamma_k * m_k * delta * c_k * d_k / (-sp.I * gamma_k * m_k + m_k**2 - s31)),\n",
" (lambda_R, -lambda_r_value, lambda_r_value),\n",
" )\n",
"\n",
"\n",
"formualte_A2(r\"\\Sigma(1385)^0\", three_half)"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"A2 = (\n",
" formualte_A2(r\"\\Sigma(1385)^0\", three_half)\n",
" + formualte_A2(r\"\\Sigma(1660)^0\", half)\n",
" + formualte_A2(r\"\\Sigma(1670)^0\", three_half)\n",
" + formualte_A2(r\"\\Sigma(1750)^0\", half)\n",
" + formualte_A2(r\"\\Sigma(1775)^0\", five_half)\n",
" + formualte_A2(r\"\\Sigma(1915)^0\", five_half)\n",
" + formualte_A2(r\"\\Sigma(1940)^0\", three_half)\n",
" + formualte_A2(r\"\\Sigma(2030)^0\", seven_half)\n",
")\n",
"A2"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"Math(aslatex(A2, terms_per_line=1))"
]
},
{
"cell_type": "markdown",
"metadata": {},
Expand All @@ -367,6 +455,56 @@
")\n",
"A3_expr"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"def formualte_A3(resonance_name, lambda_r_value): # noqa: N802\n",
" c_k = sp.Symbol(rf\"C_{{{resonance_name}}}\")\n",
" m_k = sp.Symbol(rf\"m_{{{resonance_name}}}\")\n",
" gamma_k = sp.Symbol(rf\"\\Gamma_{{{resonance_name}}}\")\n",
" theta12, phi12 = sp.symbols(r\"theta_12 phi_12\") # noqa: F841\n",
" s12, lambda_R = sp.symbols(r\"s_{12} \\lambda_{N^*}\")\n",
" delta = sp.KroneckerDelta(half, lambda_R + half)\n",
" d_k = Wigner.d(1, lambda_R, 0, theta12)\n",
" return sp.Sum(\n",
" (gamma_k * m_k * delta * c_k * d_k / (-sp.I * gamma_k * m_k + m_k**2 - s12)),\n",
" (lambda_R, -lambda_r_value, lambda_r_value),\n",
" )\n",
"\n",
"\n",
"formualte_A3(\"N(1650)^0\", half)"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"A3 = (\n",
" formualte_A3(\"N(1650)^+\", half)\n",
" + formualte_A3(\"N(1675)^+\", five_half)\n",
" + formualte_A3(\"N(1680)^+\", five_half)\n",
" + formualte_A3(\"N(1700)^+\", three_half)\n",
" + formualte_A3(\"N(1710)^+\", half)\n",
" + formualte_A3(\"N(1720)^+\", three_half)\n",
" + formualte_A3(\"N(2190)^+\", seven_half)\n",
")\n",
"A3"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"Math(aslatex(A3, terms_per_line=1))"
]
}
],
"metadata": {
Expand Down
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