lineshape¶
import expertsystem.amplitude.dynamics.lineshape
Lineshape functions that describe the dynamics.
See also
-
class
BlattWeisskopf
(*args: sympy.core.symbol.Symbol, **hints: Any)[source]¶ Bases:
expertsystem.amplitude.dynamics.lineshape.UnevaluatedExpression
Blatt-Weisskopf function \(B_L\), up to \(L \leq 4\).
- Parameters
q – Break-up momentum. Can be computed with
breakup_momentum
.d – impact parameter \(d\), also called meson radius. Usually of the order 1 fm.
angular_momentum – Angular momentum \(L\) of the decaying particle.
(1)¶\[\begin{split}\displaystyle \begin{cases} 1 & \text{for}\: L = 0 \\\frac{\sqrt{2} \left|{d}\right| \left|{q}\right|}{\sqrt{d^{2} q^{2} + 1}} & \text{for}\: L = 1 \\\frac{\sqrt{13} d^{2} q^{2}}{\sqrt{9 d^{2} q^{2} + \left(d^{2} q^{2} - 3\right)^{2}}} & \text{for}\: L = 2 \\\sqrt{277} d^{2} q^{2} \sqrt{\frac{1}{d^{2} q^{2} \left(d^{2} q^{2} - 15\right)^{2} + \left(2 d^{2} q^{2} - 5\right) \left(18 d^{2} q^{2} - 45\right)}} \left|{d}\right| \left|{q}\right| & \text{for}\: L = 3 \\\frac{\sqrt{12746} d^{4} q^{4}}{\sqrt{25 d^{2} q^{2} \left(2 d^{2} q^{2} - 21\right)^{2} + \left(d^{4} q^{4} - 45 d^{2} q^{2} + 105\right)^{2}}} & \text{for}\: L = 4 \end{cases}\end{split}\]Each of these cases has been taken from [1], p. 415. For a good overview of where to use these Blatt-Weisskopf functions, see [2].
See also Form factor.
-
property
angular_momentum
¶
-
property
d
¶ Impact parameter, also called meson radius.
-
default_assumptions
= {}¶
-
doit
(**hints: Any) → sympy.core.expr.Expr¶ Evaluate objects that are not evaluated by default like limits, integrals, sums and products. All objects of this kind will be evaluated recursively, unless some species were excluded via ‘hints’ or unless the ‘deep’ hint was set to ‘False’.
>>> from sympy import Integral >>> from sympy.abc import x
>>> 2*Integral(x, x) 2*Integral(x, x)
>>> (2*Integral(x, x)).doit() x**2
>>> (2*Integral(x, x)).doit(deep=False) 2*Integral(x, x)
-
evaluate
() → sympy.core.expr.Expr[source]¶
-
property
q
¶ Break-up momentum.
-
class
UnevaluatedExpression
(*args)[source]¶ Bases:
sympy.core.expr.Expr
-
default_assumptions
= {}¶
-
abstract
evaluate
() → sympy.core.expr.Expr[source]¶
-
-
breakup_momentum
(m_r: sympy.core.symbol.Symbol, m_a: sympy.core.symbol.Symbol, m_b: sympy.core.symbol.Symbol) → sympy.core.expr.Expr[source]¶
-
implement_expr
(n_args: int) → Callable[[Type[expertsystem.amplitude.dynamics.lineshape.UnevaluatedExpression]], sympy.core.expr.Expr][source]¶ Decorator for classes that derive from
UnevaluatedExpression
.Implement a
__new__
anddoit
method for a class that derives fromExpr
(viaUnevaluatedExpression
). It is important to derive fromevaluate
method has to be implemented
-
relativistic_breit_wigner
(mass: sympy.core.symbol.Symbol, mass0: sympy.core.symbol.Symbol, gamma0: sympy.core.symbol.Symbol) → sympy.core.expr.Expr[source]¶ Relativistic Breit-Wigner lineshape.
(2)¶\[\displaystyle \frac{\Gamma m_{0}}{- i \Gamma m_{0} - m^{2} + m_{0}^{2}}\]See Without form factor and [2].
-
relativistic_breit_wigner_with_ff
(mass: sympy.core.symbol.Symbol, mass0: sympy.core.symbol.Symbol, gamma0: sympy.core.symbol.Symbol, m_a: sympy.core.symbol.Symbol, m_b: sympy.core.symbol.Symbol, angular_momentum: sympy.core.symbol.Symbol, meson_radius: sympy.core.symbol.Symbol) → sympy.core.expr.Expr[source]¶ Relativistic Breit-Wigner with
BlattWeisskopf
factor.For \(L=0\), this lineshape has the following form:
(3)¶\[\displaystyle \frac{\Gamma m_{0}}{- \frac{i \Gamma m_{0}^{2} \sqrt{\frac{\left(m^{2} - \left(m_{a} - m_{b}\right)^{2}\right) \left(m^{2} - \left(m_{a} + m_{b}\right)^{2}\right)}{m^{2}}}}{m \sqrt{\frac{\left(m_{0}^{2} - \left(m_{a} - m_{b}\right)^{2}\right) \left(m_{0}^{2} - \left(m_{a} + m_{b}\right)^{2}\right)}{m_{0}^{2}}}} - m^{2} + m_{0}^{2}}\]See With form factor and [2].