helicity

import ampform.helicity

Generate an amplitude model with the helicity formalism.

class CanonicalAmplitudeBuilder(reaction_result: ReactionInfo)[source]

Bases: ampform.helicity.HelicityAmplitudeBuilder

Amplitude model generator for the canonical helicity formalism.

This class defines a full amplitude in the canonical formalism, using the helicity formalism as a foundation. The key here is that we take the full helicity intensity as a template, and just exchange the helicity amplitudes \(F\) as a sum of canonical amplitudes \(A\):

\[F^J_{\lambda_1,\lambda_2} = \sum_{LS} \mathrm{norm}(A^J_{LS})C^2.\]

Here, \(C\) stands for Clebsch-Gordan factor.

class HelicityAmplitudeBuilder(reaction: ReactionInfo)[source]

Bases: object

Amplitude model generator for the helicity formalism.

formulate()HelicityModel[source]
set_dynamics(particle_name: str, dynamics_builder: ResonanceDynamicsBuilder)None[source]
class HelicityModel(expression: Expr, parameter_defaults: Dict[Symbol, Union[float, complex, int]], components: Dict[str, Expr], adapter: HelicityAdapter, particles: ParticleCollection)[source]

Bases: object

__eq__(other)

Method generated by attrs for class HelicityModel.

property adapter
property components
property expression
property parameter_defaults
particles: ParticleCollection
sum_components(components: Iterable[str])Expr[source]

Coherently or incoherently add components of a helicity model.

extract_particle_collection(transitions: Iterable[StateTransition])ParticleCollection[source]

Collect all particles from a collection of state transitions.

formulate_clebsch_gordan_coefficients(transition: StateTransition, node_id: int)Expr[source]

Compute the two Clebsch-Gordan coefficients for a state transition node.

In the canonical basis (also called partial wave basis), Clebsch-Gordan coefficients ensure that the projection of angular momentum is conserved (Kutschke 1996, p. 4). When calling generate_transitions() with formalism="canonical-helicity", AmpForm formulates the amplitude in the canonical basis from amplitudes in the helicity basis using the transformation in Chung 2014, Eq. (4.32). See also Kutschke 1996, Eq. (28).

This function produces the two Clebsch-Gordan coefficients in Chung 2014, Eq. (4.32). For a two-body decay \(1 \to 2, 3\), we get:

(1)\[C^{s_1,\lambda}_{L,0,S,\lambda} C^{S,\lambda}_{s_2,\lambda_2,s_3,-\lambda_3}\]

with:

  • \(s_i\) the intrinsic Spin.magnitude of each state \(i\),

  • \(\lambda_{2}, \lambda_{3}\) the helicities of the decay products (can be taken to be their spin_projection when following a constistent boosting procedure),

  • \(\lambda=\lambda_{2}-\lambda_{3}\),

  • \(L\) the total angular momentum of the final state pair (l_magnitude),

  • \(S\) the coupled spin magnitude of the final state pair (s_magnitude),

  • and \(C^{j_3,m_3}_{j_1,m_1,j_2,m_2} = \langle j1,m1;j2,m2|j3,m3\rangle\), as in Clebsch-Gordan Coefficients.

Example

>>> import qrules
>>> reaction = qrules.generate_transitions(
...     initial_state=[("J/psi(1S)", [+1])],
...     final_state=[("gamma", [-1]), "f(0)(980)"],
... )
>>> transition = reaction.transitions[1]  # angular momentum 2
>>> formulate_clebsch_gordan_coefficients(transition, node_id=0)
CG(1, -1, 0, 0, 1, -1)*CG(2, 0, 1, -1, 1, -1)
\[C^{s_1,\lambda}_{L,0,S,\lambda} C^{S,\lambda}_{s_2,\lambda_2,s_3,-\lambda_3} = C^{1,(-1-0)}_{2,0,1,(-1-0)} C^{1,(-1-0)}_{1,-1,0,0} = C^{1,-1}_{2,0,1,-1} C^{1,-1}_{1,-1,0,0}\]
../_images/graphviz_1.svg
formulate_wigner_d(transition: StateTransition, node_id: int)Expr[source]

Compute WignerD for a transition node.

Following Kutschke 1996, Eq. (10). For a two-body decay \(1 \to 2, 3\), we get

(2)\[D^{s_1}_{m_1,\lambda_2-\lambda_3}(-\phi,\theta,0)\]

with:

  • \(s_1\) the Spin.magnitude of the decaying state,

  • \(m_1\) the spin_projection of the decaying state,

  • \(\lambda_{2}, \lambda_{3}\) the helicities of the decay products in in the restframe of \(1\) (can be taken to be their intrinsic spin_projection when following a constistent boosting procedure),

  • and \(\phi\) and \(\theta\) the helicity angles (see also get_helicity_angle_label()).

Note that \(\lambda_2, \lambda_3\) are ordered by their number of children, then by their state ID (see TwoBodyDecay).

See Kutschke 1996, Eq. (30) for an example of Wigner-\(D\) functions in a sequential two-body decay.

Example

>>> import qrules
>>> reaction = qrules.generate_transitions(
...     initial_state=[("J/psi(1S)", [+1])],
...     final_state=[("gamma", [-1]), "f(0)(980)"],
... )
>>> transition = reaction.transitions[0]
>>> formulate_wigner_d(transition, node_id=0)
WignerD(1, 1, -1, -phi_0, theta_0, 0)
\[D^{s_1}_{m_1,\lambda_2-\lambda_3}\left(-\phi,\theta,0\right) = D^{1}_{+1,(-1-0)}\left(-\phi_0,\theta_0,0\right) = D^{1}_{1,-1}\left(-\phi_0,\theta_0,0\right)\]
../_images/graphviz_0.svg
get_prefactor(transition: StateTransition)float[source]

Calculate the product of all prefactors defined in this transition.

group_transitions(transitions: Iterable[StateTransition])List[List[StateTransition]][source]

Match final and initial states in groups.

Each StateTransition corresponds to a specific state transition amplitude. This function groups together transitions, which have the same initial and final state (including spin). This is needed to determine the coherency of the individual amplitude parts.

Submodules and Subpackages