helicity
helicity¶
import ampform.helicity
Generate an amplitude model with the helicity formalism.
- class ParameterValues(parameters: Mapping[Symbol, Union[float, complex, int]])[source]¶
Bases:
Mapping
Ordered mapping to
ParameterValue
with convenient getter and setter.>>> a, b, c = sp.symbols("a b c") >>> parameters = ParameterValues({a: 0.0, b: 1+1j, c: -2}) >>> parameters[a] 0.0 >>> parameters["b"] (1+1j) >>> parameters["b"] = 3 >>> parameters[1] 3 >>> parameters[2] -2 >>> parameters[2] = 3.14 >>> parameters[c] 3.14
- class HelicityModel(intensity: Expr, amplitudes: Mapping[Indexed, Expr], parameter_defaults: Mapping[Symbol, Union[float, complex, int]], kinematic_variables: Mapping[Symbol, Expr], components: Mapping[str, Expr], reaction_info: ReactionInfo)[source]¶
Bases:
object
- intensity: Expr¶
Main expression describing the intensity over
kinematic_variables
.
- amplitudes: OrderedDict[Indexed, Expr]¶
Definitions for the amplitudes that appear in
intensity
.The main
intensity
is a sum over amplitudes for each initial and final state helicity combination. These amplitudes are indicated with assp.Indexed
instances and this attribute provides the definitions for each of these. See also TR-014.
- parameter_defaults: ParameterValues¶
A mapping of suggested parameter values.
Keys are
Symbol
instances from the mainexpression
that should be interpreted as parameters (as opposed tokinematic_variables
). The symbols are ordered alphabetically by name with natural sort order. Values have been extracted from the inputReactionInfo
.
- kinematic_variables: OrderedDict[Symbol, Expr]¶
Expressions for converting four-momenta to kinematic variables.
- components: OrderedDict[str, Expr]¶
A mapping for identifying main components in the
expression
.Keys are the component names (
str
), formatted as LaTeX, and values are sub-expressions in the mainexpression
. The mapping is anOrderedDict
that orders the component names alphabetically with natural sort order.
- reaction_info: ReactionInfo¶
- property expression: Expr¶
Expression for the
intensity
with all amplitudes fully expressed.Constructed from
intensity
by substituting its amplitude symbols with the definitions withamplitudes
.
- rename_symbols(renames: Union[Iterable[Tuple[str, str]], Mapping[str, str]]) HelicityModel [source]¶
Rename certain symbols in the model.
Renames all
Symbol
instance that appear inexpression
,parameter_defaults
,components
, andkinematic_variables
. This method can be used to couple parameters.- Parameters
renames – A mapping from old to new names.
- Returns
A new instance of a
HelicityModel
with symbols in all attributes renamed accordingly.
- class DynamicsSelector(transitions: Union[ReactionInfo, Iterable[StateTransition]])[source]¶
Bases:
Mapping
Configure which
ResonanceDynamicsBuilder
to use for each node.- assign(selection: Any, builder: ResonanceDynamicsBuilder) None [source]¶
Assign a
ResonanceDynamicsBuilder
to a selection of nodes.Currently, the following types of selections are implements:
str
: Select transition nodes by the name of theparent
Particle
.TwoBodyDecay
ortuple
of aStateTransition
with a node ID: set dynamics for one specific transition node.
- class HelicityAmplitudeBuilder(reaction: ReactionInfo, stable_final_state_ids: Optional[Iterable[int]] = None, scalar_initial_state_mass: bool = False)[source]¶
Bases:
object
Amplitude model generator for the helicity formalism.
- Parameters
reaction – The
ReactionInfo
from which toformulate()
an amplitude model.stable_final_state_ids – Put final state ‘invariant’ masses (\(m_0, m_1, \dots\)) under
HelicityModel.parameter_defaults
(with a scalar suggested value) instead ofkinematic_variables
(which are expressions to compute an event-wise array of invariant masses). This is useful if final state particles are stable.stable_final_state_ids –
Put the invariant of the initial state (\(m_{012\dots}\)) under
HelicityModel.parameter_defaults
(with a scalar suggested value) instead ofkinematic_variables
. This is useful if four-momenta were generated with or kinematically fit to a specific initial state energy.See also
- property adapter: HelicityAdapter¶
Converter for computing kinematic variables from four-momenta.
- property dynamics_choices: DynamicsSelector¶
- property stable_final_state_ids: Optional[Set[int]]¶
IDs of the final states that should be considered stable.
The ‘invariant’ mass symbols for these final states will be inserted as scalar values into the
parameter_defaults
.
- property scalar_initial_state_mass: bool¶
Add initial state mass as scalar value to
parameter_defaults
.See also
- set_dynamics(particle_name: str, dynamics_builder: ResonanceDynamicsBuilder) None [source]¶
- formulate() HelicityModel [source]¶
- class CanonicalAmplitudeBuilder(reaction_result: ReactionInfo)[source]¶
Bases:
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.
See also
- 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 ([3], p. 4). When calling
generate_transitions()
withformalism="canonical-helicity"
, AmpForm formulates the amplitude in the canonical basis from amplitudes in the helicity basis using the transformation in [1], Eq. (4.32). See also [3], Eq. (28).This function produces the two Clebsch-Gordan coefficients in [1], 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)
\[C^{s_1,\lambda}_{L,0,S,\lambda} C^{S,\lambda}_{s_2,\lambda_2,s_3,-\lambda_3} = C^{1,(-1-0)}_{2,0,1,(-1-0)} C^{1,(-1-0)}_{1,-1,0,0} = C^{1,-1}_{2,0,1,-1} C^{1,-1}_{1,-1,0,0}\]
- formulate_wigner_d(transition: StateTransition, node_id: int) Expr [source]¶
Compute
WignerD
for a transition node.Following [3], 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 [3], 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)
\[D^{s_1}_{m_1,\lambda_2-\lambda_3}\left(-\phi,\theta,0\right) = D^{1}_{+1,(-1-0)}\left(-\phi_0,\theta_0,0\right) = D^{1}_{1,-1}\left(-\phi_0,\theta_0,0\right)\]
- 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