Deprecated since version 0.7.3: Use QRules and AmpForm instead

import expertsystem.amplitude.dynamics.lineshape

Lineshape functions that describe the dynamics.

class BlattWeisskopf(*args: Symbol, **hints: Any)[source]

Bases: expertsystem.amplitude.dynamics.lineshape.UnevaluatedExpression

Blatt-Weisskopf function \(B_L\), up to \(L \leq 4\).

  • 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 Chung et al. 1995, p. 415. For a good overview of where to use these Blatt-Weisskopf functions, see Asner 2006.

See also Form factor.

property angular_momentum
property d

Impact parameter, also called meson radius.

property q

Break-up momentum.

class UnevaluatedExpression(*args)[source]

Bases: sympy.core.expr.Expr

breakup_momentum(m_r: Symbol, m_a: Symbol, m_b: Symbol)Expr[source]
implement_expr(n_args: int)Callable[[Type[UnevaluatedExpression]], Expr][source]

Decorator for classes that derive from UnevaluatedExpression.

Implement a __new__ and doit method for a class that derives from Expr (via UnevaluatedExpression). It is important to derive from evaluate() method has to be implemented

relativistic_breit_wigner(mass: Symbol, mass0: Symbol, gamma0: Symbol)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 Asner 2006.

relativistic_breit_wigner_with_ff(mass: Symbol, mass0: Symbol, gamma0: Symbol, m_a: Symbol, m_b: Symbol, angular_momentum: Symbol, meson_radius: Symbol)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 Asner 2006.