dynamics

dynamics#

import ampform.dynamics

Lineshape functions that describe the dynamics of an interaction.

See also

Dynamics and Phase space factor definitions.

class SimpleBreitWigner(s, mass, width, *args, evaluate: bool = False, **kwargs)[source]#

Bases: Expr

Simple Breit–Wigner with \(m_0 \Gamma_0\) in the numerator.

(1)#\[\begin{split} \begin{aligned} \mathcal{R}^\mathrm{BW}\left(s; m_{0}, \Gamma_{0}\right) \;&=\; \frac{\Gamma_{0} m_{0}}{- i \Gamma_{0} m_{0} + m_{0}^{2} - s} \\ \end{aligned}\end{split}\]
class BreitWigner(s, mass, width, m1, m2, angular_momentum, meson_radius, phsp_factor, *args, evaluate: bool = False, **kwargs)[source]#

Bases: Expr

Relativistic Breit–Wigner with \(m_0 \Gamma_0\) in the numerator.

Uses an EnergyDependentWidth in the denominator (see Equations (2) and (3)).

(2)#\[\begin{split} \begin{aligned} \mathcal{R}^\mathrm{BW}_{L}\left(s; m_{0}, \Gamma_{0}\right) \;&=\; \frac{\Gamma_{0} m_{0}}{m_{0}^{2} - i m_{0} \Gamma_{0}\left(s\right) - s} \\ \end{aligned}\end{split}\]
phsp_factor: PhaseSpaceFactorProtocol[source]#
energy_dependent_width() → EnergyDependentWidth | Basic[source]#
class EnergyDependentWidth(s, mass0, gamma0, m_a, m_b, angular_momentum, meson_radius, phsp_factor, name, *args, evaluate: bool = False, **kwargs)[source]#

Bases: Expr

Mass-dependent width, coupled to the pole position of the resonance.

See Equation (50.28) in PDG2021, §Resonances, p.9 and [Asner and Hanhart, 2012], equation (6). Default value for phsp_factor is PhaseSpaceFactor.

Note that the FormFactor of AmpForm is normalized in the sense that equal powers of \(z\) appear in the nominator and the denominator, while the definition in the PDG (as well as some other sources), always have \(1\) in the nominator of the Blatt–Weisskopf. In that case, one needs an additional factor \(\left(q/q_0\right)^{2L}\) in the definition for \(\Gamma(m)\).

With that in mind, the “mass-dependent” width in a relativistic_breit_wigner_with_ff becomes:

(3)#\[\begin{split} \begin{aligned} \Gamma_{0}\left(s\right) \;&=\; \frac{\Gamma_{0} \mathcal{F}_{L}\left(s, m_{a}, m_{b}\right)^{2} \rho\left(s\right)}{\mathcal{F}_{L}\left(m_{0}^{2}, m_{a}, m_{b}\right)^{2} \rho_{0}\left(m_{0}^{2}\right)} \\ \end{aligned}\end{split}\]

where \(F_L\) is defined by (1), \(q\) is defined by (5), and \(\rho\) is (by default) defined by (1).

phsp_factor: PhaseSpaceFactorProtocol[source]#
class MultichannelBreitWigner(s, mass, channels, *args, evaluate: bool = False, **kwargs)[source]#

Bases: Expr

Breit–Wigner with a running width summed over several decay channels.

Each channel is a ChannelArguments term \(\Gamma_i^\text{ch}(s)\), giving

\[\frac{1}{m_0^2 - s - i \sum_i g_i^2 \rho_i(s) F_{L_i}^2(s)},\]

where \(g_i^2\) is the coupling squared, \(\rho_i\) is a PhaseSpaceFactor, and \(F_{L_i}\) is a FormFactor. Unlike an EnergyDependentWidth, a channel term is not normalized at the pole position.

(4)#\[\begin{split} \begin{aligned} \mathcal{R}^\mathrm{BW}_\mathrm{multi}\left(s; g1sq, g2sq\right) \;&=\; \frac{1}{m_{0}^{2} - i m_{0} \left(\Gamma^\text{ch}\left(s; m_{0}, g1sq\right) + \Gamma^\text{ch}\left(s; m_{0}, g2sq\right)\right) - s} \\ \end{aligned}\end{split}\]
class ChannelArguments(s, mass, coupling_squared, m1, m2, angular_momentum, meson_radius, *args, evaluate: bool = False, **kwargs)[source]#

Bases: Expr

One channel term \(\Gamma_i^\text{ch}(s)\).

\[\Gamma_i^\text{ch}(s) = \frac{g_i^2}{m_0} \rho_i(s) F_{L_i}^2(s)\]
relativistic_breit_wigner(s, mass0, gamma0) → Expr[source]#

Relativistic Breit–Wigner lineshape.

See Without form factor and [Asner and Hanhart, 2012].

Deprecated since version 0.17.0: Use SimpleBreitWigner instead.

relativistic_breit_wigner_with_ff(s, mass0, gamma0, m_a, m_b, angular_momentum, meson_radius, phsp_factor: PhaseSpaceFactorProtocol = <class 'ampform.dynamics.phasespace.PhaseSpaceFactor'>) → Expr[source]#

Relativistic Breit–Wigner with FormFactor.

See With form factor and PDG2021, §Resonances, p.9.

The general form of a relativistic Breit–Wigner with Blatt–Weisskopf form factor is:

(5)#\[\mathcal{R}^\mathrm{BW}_{L}\left(s; m_{0}, \Gamma_{0}\right) \mathcal{F}_{L}\left(s, m_{a}, m_{b}\right)\]

where \(\Gamma(s)\) is defined by (3), \(B_L^2\) is defined by (2), and \(q^2\) is defined by (5).

formulate_form_factor(s, m_a, m_b, angular_momentum, meson_radius) → Expr[source]#

Formulate a Blatt–Weisskopf form factor.

Deprecated since version 0.16.0: Use FormFactor instead.

Submodules and Subpackages