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"## The minimal order differential equation for the all equal mass multiloop sunset integrals in general dimensions"
]
},
{
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"id": "8a363466",
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"source": [
"\\begin{equation}\n",
" \\Omega^{(n-1)}_\\circleddash( t,\\epsilon)={\\Omega^{(n)}_0\\over\n",
" \\textbf{F}_{n-1}(t)}\\left(\\textbf{U}_{n-1}^n\\over \\textbf{F}_{n-1}(t)^{n-1}\\right)^\\epsilon\n",
"\\end{equation}\n",
"with\n",
"\\begin{equation}\n",
" \\textbf{ U}_{n-1}= x_1\\cdots x_n\\sum_{i=1}^n {1\\over x_i},\\qquad\n",
" \\textbf{ F}_{n-1}(t)= \\textbf{U}_{n-1} \\left(\\sum_{i=1}^n x_i\\right)-t x_1\\cdots x_n\n",
"\\end{equation}"
]
},
{
"cell_type": "markdown",
"id": "a240515b",
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"source": [
"<center><h2>The all equal mass case up to 20 loops</h2></center>\n",
" The Picard-Fuchs operator at loop L is in PF[L]\n",
" $$\\mathscr{L}^{(l),\\epsilon} I^{(l)}(t,\\epsilon)=-(l+1)! {\\Gamma(1+\\epsilon)^l\\over \\Gamma(1+l\\epsilon)}$$\n",
" with \n",
"$$ \\mathscr{L}^{(l),\\epsilon}=\\sum_{r=0}^l \\mathscr{L}^{(l),r} \\epsilon^r$$\n",
"where $ \\mathscr{L}^{(l),r}$ is differential operator in $t$ of order $l-r$."
]
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{
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"execution_count": 1,
"id": "a5eed4cc",
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"outputs": [],
"source": [
"epsilon=var('epsilon')"
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "6a8857a9",
"metadata": {},
"outputs": [
{
"ename": "ModuleNotFoundError",
"evalue": "No module named 'ore_algebra'",
"output_type": "error",
"traceback": [
"\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
"\u001b[0;31mModuleNotFoundError\u001b[0m Traceback (most recent call last)",
"Cell \u001b[0;32mIn[2], line 1\u001b[0m\n\u001b[0;32m----> 1\u001b[0m \u001b[38;5;28;01mfrom\u001b[39;00m \u001b[38;5;21;01more_algebra\u001b[39;00m \u001b[38;5;28;01mimport\u001b[39;00m \u001b[38;5;241m*\u001b[39m\n\u001b[1;32m 2\u001b[0m OA, t, Dt \u001b[38;5;241m=\u001b[39m DifferentialOperators(QQ, \u001b[38;5;124m'\u001b[39m\u001b[38;5;124mt\u001b[39m\u001b[38;5;124m'\u001b[39m)\n\u001b[1;32m 3\u001b[0m OAepsilon, t, Dt \u001b[38;5;241m=\u001b[39m DifferentialOperators(QQ[epsilon], \u001b[38;5;124m'\u001b[39m\u001b[38;5;124mt\u001b[39m\u001b[38;5;124m'\u001b[39m)\n",
"\u001b[0;31mModuleNotFoundError\u001b[0m: No module named 'ore_algebra'"
]
}
],
"source": [
"from ore_algebra import *\n",
"OA, t, Dt = DifferentialOperators(QQ, 't')\n",
"OAepsilon, t, Dt = DifferentialOperators(QQ[epsilon], 't')"
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "1529aabc",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"loading result at loop order 1\n",
"loading result at loop order 2\n",
"loading result at loop order 3\n",
"loading result at loop order 4\n",
"loading result at loop order 5\n",
"loading result at loop order 6\n",
"loading result at loop order 7\n",
"loading result at loop order 8\n",
"loading result at loop order 9\n",
"loading result at loop order 10\n",
"loading result at loop order 11\n",
"loading result at loop order 12\n",
"loading result at loop order 13\n",
"loading result at loop order 14\n",
"loading result at loop order 15\n",
"loading result at loop order 16\n",
"loading result at loop order 17\n",
"loading result at loop order 18\n",
"loading result at loop order 19\n",
"loading result at loop order 20\n"
]
}
],
"source": [
"# PFEqualMass[l] where l is the loop order l=1,..,20\n",
"PFEqualMass=dict()\n",
"with open(\"PFSunsetEqualMass-1to20.txt\") as f:\n",
" for e in f.read().replace(\"\\n\", \"\").split(\";\"):\n",
" if e:\n",
" name, expr = e.split(\":=\")\n",
" num=int(name.strip(\"PFEqualMass\"))\n",
" print(\"loading result at loop order \",num)\n",
" PFEqualMass[num] = SR(expr)"
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "d20be3c4-4b4c-49f5-92a5-4c203b3c656d",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"-(t^2 - 4*t)*Dt - epsilon*t - t + 2"
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"PFEqualMass[1] "
]
},
{
"cell_type": "code",
"execution_count": 7,
"id": "695720ea-a37f-4857-a8f5-8c39cc7d598e",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(t^3 - 10*t^2 + 9*t)*Dt^2 + 2*epsilon^2*t + (3*epsilon*t^2 - 10*epsilon*t + 3*t^2 - 9*epsilon - 20*t + 9)*Dt + 2*epsilon^2 + 3*epsilon*t - 5*epsilon + t - 3"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"PFEqualMass[2] "
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "4a15bfb1-ec27-46db-a23e-c821b7b67b16",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"-(t^4 - 20*t^3 + 64*t^2)*Dt^3 - 6*epsilon^3*t - 6*(epsilon*t^3 - 10*epsilon*t^2 + t^3 - 15*t^2 + 32*t)*Dt^2 - 12*epsilon^3 - 11*epsilon^2*t - (11*epsilon^2*t^2 - 28*epsilon^2*t + 18*epsilon*t^2 - 64*epsilon^2 - 120*epsilon*t + 7*t^2 - 68*t + 64)*Dt + 14*epsilon^2 - 6*epsilon*t + 18*epsilon - t + 4"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"PFEqualMass[3] "
]
},
{
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"execution_count": null,
"id": "c049edf6-9beb-46bc-8273-656b4b73b35e",
"metadata": {},
"outputs": [],
"source": []
}
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