dynamics

dynamics#

import ampform.dynamics

Lineshape functions that describe the dynamics of an interaction.

See also

Dynamics and Analytic continuation

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

Bases: Expr

Simple, non-relativistic Breit-Wigner with \(1\) in the nominator.

(1)#\[\begin{split} \begin{array}{rcl} \mathcal{R}^\mathrm{BW}\left(s; m_{0}, \Gamma_{0}\right) &=& \frac{1}{- i \Gamma_{0} m_{0} + m_{0}^{2} - s} \\ \end{array}\end{split}\]

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 \(1\) in the nominator.

SimpleBreitWigner with EnergyDependentWidth as width (see Equations (1) and (3)).

(2)#\[\begin{split} \begin{array}{rcl} \mathcal{R}^\mathrm{BW}_{L}\left(s; m_{0}, \Gamma_{0}\right) &=& \mathcal{R}^\mathrm{BW}\left(s; m_{0}, \Gamma_{0}\left(s\right)\right) \\ \end{array}\end{split}\]

phsp_factor: PhaseSpaceFactorProtocol[source]#

energy_dependent_width() → Expr[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 et al., 2006], 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)#\[\begin{split} \begin{array}{rcl} \Gamma_{0}\left(s\right) &=& \frac{\Gamma_{0} \mathcal{F}_{L}\left(s, m_{a}, m_{b}\right)^{2} \rho\left(s\right)}{\mathcal{F}_{L}\left(m_{0}^{2}, m_{a}, m_{b}\right)^{2} \rho_{0}\left(m_{0}^{2}\right)} \\ \end{array}\end{split}\]

where \(F_L\) is defined by (1), \(q\) is defined by (1), and \(\rho\) is (by default) defined by (2).

phsp_factor: PhaseSpaceFactorProtocol[source]#

name: str | None[source]#

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

Bases: Expr

BreitWigner for multiple channels.

(4)#\[\begin{split} \begin{array}{rcl} \mathcal{R}^\mathrm{BW}_\mathrm{multi}\left(s; \Gamma_{1}, \Gamma_{2}\right) &=& \mathcal{R}^\mathrm{BW}\left(s; m_{0}, \frac{\Gamma_{1} m_{0} \mathcal{F}_{L_{1}}\left(s, m_{a1}, m_{b1}\right)^{2}}{\sqrt{s}} + \frac{\Gamma_{2} m_{0} \mathcal{F}_{L_{2}}\left(s, m_{a2}, m_{b2}\right)^{2}}{\sqrt{s}}\right) \\ \end{array}\end{split}\]

with \(\mathcal{R}^\mathrm{BW}\left(s; m_{0}, \Gamma_{0}\right)\) defined by Equation (1), a SimpleBreitWigner.

channels: list[ChannelArguments][source]#

class ChannelArguments(width: Any, m1: Any = 0, m2: Any = 0, angular_momentum: Any = 0, meson_radius: Any = 1)[source]#

Bases: object

Arguments for a channel in a MultichannelBreitWigner.

formulate_width(s: Any, m0: Any) → Expr[source]#: \[\frac{\Gamma_{1} m_{0} \mathcal{F}_{L_{1}}\left(s, m_{a1}, m_{b1}\right)^{2}}{\sqrt{s}}\]

relativistic_breit_wigner(s, mass0, gamma0) → Expr[source]#

Relativistic Breit-Wigner lineshape.

See Without form factor and [Asner et al., 2006].

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

relativistic_breit_wigner_with_ff(s, mass0, gamma0, m_a, m_b, angular_momentum, meson_radius, phsp_factor: ~ampform.dynamics.phasespace.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:

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

where \(\Gamma(s)\) is defined by (3), \(B_L^2\) is defined by (2), and \(q^2\) is defined by (1).

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.

Submodules and Subpackages

dynamics

Contents

dynamics#