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ENH: formulate Blatt–Weisskopf with spherical Hankel (#418)
* FEAT: simplify Hankel polynomial internally * MAINT: create separate `form_factor` submodule
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@@ -244,6 +244,7 @@ | |
| "hankel", | ||
| "helicities", | ||
| "helicity", | ||
| "Hippel", | ||
| "htmlcov", | ||
| "imap", | ||
| "ipynb", | ||
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| """Implementations of the form factor, or barrier factor.""" | ||
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| from __future__ import annotations | ||
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| from functools import lru_cache | ||
| from typing import Any | ||
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| import sympy as sp | ||
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| from ampform.sympy import unevaluated | ||
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| @unevaluated | ||
| class BlattWeisskopfSquared(sp.Expr): | ||
| r"""Normalized Blatt-Weisskopf function :math:`B_L^2(z)`, with :math:`B_L^2(1)=1`. | ||
| Args: | ||
| z: Argument of the Blatt-Weisskopf function :math:`B_L^2(z)`. A usual | ||
| choice is :math:`z = (d q)^2` with :math:`d` the impact parameter and | ||
| :math:`q` the breakup-momentum (see `.BreakupMomentumSquared`). | ||
| angular_momentum: Angular momentum :math:`L` of the decaying particle. | ||
| Note that equal powers of :math:`z` appear in the nominator and the denominator, | ||
| while some sources define an *non-normalized* form factor :math:`F_L` with :math:`1` | ||
| in the nominator, instead of :math:`z^L`. See for instance Equation (50.27) in | ||
| :pdg-review:`2021; Resonances; p.9`. We normalize the form factor such that | ||
| :math:`B_L^2(1)=1` and that :math:`B_L^2` is unitless no matter what :math:`z` is. | ||
| .. seealso:: :ref:`usage/dynamics:Form factor`, :doc:`TR-029<compwa:report/029>`, | ||
| and :cite:`chungFormulasAngularMomentumBarrier2015`. | ||
| With this, the implementation becomes | ||
| """ | ||
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| z: Any | ||
| angular_momentum: Any | ||
| _latex_repr_ = R"B_{{{angular_momentum}}}^2\left({z}\right)" | ||
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| def evaluate(self) -> sp.Expr: | ||
| ell = self.angular_momentum | ||
| z = sp.Dummy("z", nonnegative=True, real=True) | ||
| expr = ( | ||
| sp.Abs(SphericalHankel1(ell, 1)) ** 2 | ||
| / sp.Abs(SphericalHankel1(ell, sp.sqrt(z))) ** 2 | ||
| / z | ||
| ) | ||
| if not ell.free_symbols: | ||
| expr = expr.doit().simplify() | ||
| return expr.xreplace({z: self.z}) | ||
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| @unevaluated | ||
| class SphericalHankel1(sp.Expr): | ||
| r"""Spherical Hankel function of the first kind for real-valued :math:`z`. | ||
| See :cite:`VonHippel:1972fg`, Equation (A12), and :doc:`TR-029<compwa:report/029>` | ||
| for more info. `This page | ||
| <https://mathworld.wolfram.com/SphericalHankelFunctionoftheFirstKind.html>`_ | ||
| explains the difference with the *general* Hankel function of the first kind, | ||
| :math:`H_\ell^{(1)}`. | ||
| This expression class assumes that :math:`z` is real and evaluates to the following | ||
| series: | ||
| """ | ||
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| l: Any # noqa: E741 | ||
| z: Any | ||
| _latex_repr_ = R"h_{{{l}}}^{{(1)}}\left({z}\right)" | ||
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| def evaluate(self) -> sp.Expr: | ||
| l, z = self.args # noqa: E741 | ||
| k = sp.Dummy("k", integer=True, nonnegative=True) | ||
| return ( | ||
| (-sp.I) ** (1 + l) # type:ignore[operator] | ||
| * (sp.exp(z * sp.I) / z) | ||
| * _SymbolicSum( | ||
| sp.factorial(l + k) | ||
| / (sp.factorial(l - k) * sp.factorial(k)) | ||
| * (sp.I / (2 * z)) ** k, # type:ignore[operator] | ||
| (k, 0, l), | ||
| ) | ||
| ) | ||
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| class _SymbolicSum(sp.Sum): | ||
| """See [TR-029](https://compwa.github.io/report/029.html) for why this class is needed.""" | ||
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| def doit(self, deep: bool = True, **kwargs) -> sp.Expr: | ||
| if _get_indices(self): | ||
| expression = self.args[0] | ||
| indices = self.args[1:] | ||
| return _SymbolicSum(expression.doit(deep=deep, **kwargs), *indices) | ||
| return super().doit(deep=deep, **kwargs) | ||
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| @lru_cache(maxsize=None) | ||
| def _get_indices(expr: sp.Sum) -> set[sp.Basic]: | ||
| free_symbols = set() | ||
| for index in expr.args[1:]: | ||
| free_symbols.update(index.free_symbols) | ||
| return {s for s in free_symbols if not isinstance(s, sp.Dummy)} |
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