Generate transitions¶

Warning

The PWA Expert System has been split up into QRules and AmpForm. Please use these packages instead!

Note

The formulas and figures on this page have been generated with the lineshapes in the lineshape module, so as to glue them back in to the API of that module.

A Partial Wave Analysis starts by defining an amplitude model that describes the reaction process that is to be analyzed. Such a model is generally very complex and requires a fair amount of effort by the analyst (you). This gives a lot of room for mistakes.

The ‘expert system’ is responsible to give you advice on the form of an amplitude model, based on the problem set you define (initial state, final state, allowed interactions, intermediate states, etc.). Internally, the system propagates the quantum numbers through the reaction graph while satisfying the specified conservation rules. How to control this procedure is explained in more detail below.

Afterwards, the amplitude model of the expert system can be exported into Tensorwaves. The model can for instance be used to generate a data set (toy Monte Carlo) for this reaction and to optimize its parameters to resemble an actual data set as good as possible. For more info on that see Formulate amplitude model.

Note

Simple channels can be treated with the generate_transitions() façade function. This notebook shows how to treat more complicated cases with the StateTransitionManager.

1. Define the problem set¶

We first define the boundary conditions of our physics problem, such as initial state, final state, formalism type, etc. and pass all of that information to the StateTransitionManager. This is the main user interface class of the expertsystem.

By default, the StateTransitionManager loads all particles from the PDG. The expertsystem would take a long time to check the quantum numbers of all these particles, so in this notebook, we use a smaller subset of relatively common particles.

from expertsystem.reaction import InteractionTypes, StateTransitionManager

stm = StateTransitionManager(
    initial_state=["J/psi(1S)"],
    final_state=["gamma", "pi0", "pi0"],
    formalism_type="helicity",
)

StateTransitionManager

The StateTransitionManager (STM) is the main user interface class of the {mod}`expertsystem`. The boundary conditions of your physics problem, such as the initial state, final state, formalism type, etc., are defined through this interface.

create_problem_sets of the STM creates all problem sets ― using the boundary conditions of the StateTransitionManager instance. In total 3 steps are performed. The creation of reaction topologies. The creation of InitialFacts, based on a topology and the initial and final state information. And finally the solving settings such as the conservation laws and quantum number domains to use at which point of the topology.
By default, all three interaction types (EM, STRONG, and WEAK) are used in the preparation stage. However, it is also possible to choose the allowed interaction types globally via set_allowed_interaction_types.

After the preparation step, you can modify the problem sets returned by create_problem_sets to your liking. Since the output of this function contains quite a lot of information, the expertsystem aids in the configuration (especially the STM).

A subset of particles that are allowed as intermediate states can also be specified: either through the STM's constructor or by setting the instance allowed_intermediate_particles.

2. Prepare Problem Sets¶

Create all ProblemSet’s using the boundary conditions of the StateTransitionManager instance. By default it uses the isobar model (tree of two-body decays) to build Topology’s. Various InitialFacts are created for each topology based on the initial and final state. Lastly some reasonable default settings for the solving process are chosen. Remember that each interaction node defines its own set of conservation laws.

The StateTransitionManager (STM) defines three interaction types:

Interaction	Strength
strong	\(60\)
electromagnetic (EM)	\(1\)
weak	\(10^{-4}\)

By default, all three are used in the preparation stage. The create_problem_sets() method of the STM generates graphs with all possible combinations of interaction nodes. An overall interaction strength is assigned to each graph and they are grouped according to this strength.

problem_sets = stm.create_problem_sets()

3. Find solutions¶

If you are happy with the default settings generated by the StateTransitionManager, just start with solving directly!

This step takes about 23 sec on an Intel(R) Core(TM) i7-6820HQ CPU of 2.70GHz running, multi-threaded.

result = stm.find_solutions(problem_sets)

The find_solutions() method returns a Result object from which you can extract the transitions. Now, you can use get_intermediate_particles() to print the names of the intermediate states that the StateTransitionManager found:

print("found", len(result.transitions), "solutions!")
result.get_intermediate_particles().names

found 414 solutions!

{'a(0)(1450)0',
 'a(0)(980)0',
 'a(1)(1260)0',
 'a(1)(1640)0',
 'a(2)(1320)0',
 'a(2)(1700)0',
 'b(1)(1235)0',
 'f(0)(1370)',
 'f(0)(1500)',
 'f(0)(1710)',
 'f(0)(500)',
 'f(0)(980)',
 'f(1)(1285)',
 'f(1)(1420)',
 "f(2)'(1525)",
 'f(2)(1270)',
 'f(2)(1950)',
 'f(2)(2010)',
 'f(2)(2300)',
 'f(2)(2340)',
 'h(1)(1170)',
 'omega(1420)',
 'omega(1650)',
 'omega(782)',
 'phi(1020)',
 'phi(1680)',
 'rho(1450)0',
 'rho(1700)0',
 'rho(770)0'}

About the number of solutions

The “number of transitions” is the total number of allowed StateTransitionGraph instances that the StateTransitionManager has found. This also includes all allowed spin projection combinations. In this channel, we for example consider a \(J/\psi\) with spin projection \(\pm1\) that decays into a \(\gamma\) with spin projection \(\pm1\), which already gives us four possibilities.

On the other hand, the intermediate state names that was extracted with Result.get_intermediate_particles(), is just a set of the state names on the intermediate edges of the list of transitions, regardless of spin projection.

Now we have a lot of solutions that are actually heavily suppressed (they involve two weak decays).

In general, you can modify the ProblemSets returned by create_problem_sets() directly, but the STM also comes with functionality to globally choose the allowed interaction types. So, go ahead and disable the EM and InteractionTypes.WEAK interactions:

stm.set_allowed_interaction_types([InteractionTypes.STRONG])
problem_sets = stm.create_problem_sets()
result = stm.find_solutions(problem_sets)

print("found", len(result.transitions), "solutions!")
result.get_intermediate_particles().names

found 192 solutions!

{'b(1)(1235)0',
 'f(0)(1370)',
 'f(0)(1500)',
 'f(0)(1710)',
 'f(0)(500)',
 'f(0)(980)',
 "f(2)'(1525)",
 'f(2)(1270)',
 'f(2)(1950)',
 'f(2)(2010)',
 'f(2)(2300)',
 'f(2)(2340)',
 'rho(1450)0',
 'rho(1700)0',
 'rho(770)0'}

Now note that, since a \(\gamma\) particle appears in one of the interaction nodes, the expert system knows that this node must involve EM interactions! Because the node can be an effective interaction, the weak interaction cannot be excluded, as it contains only a subset of conservation laws.

Since only the strong interaction was supposed to be used, this results in a warning and the STM automatically corrects the mistake.

Once the EM interaction is included, this warning disappears. Be aware, however, that the EM interaction is now available globally. Hence, there now might be solutions in which both nodes are electromagnetic.

stm.set_allowed_interaction_types(
    [InteractionTypes.STRONG, InteractionTypes.EM]
)
problem_sets = stm.create_problem_sets()
result = stm.find_solutions(problem_sets)

print("found", len(result.transitions), "solutions!")
result.get_intermediate_particles().names

found 318 solutions!

{'a(0)(1450)0',
 'a(0)(980)0',
 'a(2)(1320)0',
 'a(2)(1700)0',
 'b(1)(1235)0',
 'f(0)(1370)',
 'f(0)(1500)',
 'f(0)(1710)',
 'f(0)(500)',
 'f(0)(980)',
 "f(2)'(1525)",
 'f(2)(1270)',
 'f(2)(1950)',
 'f(2)(2010)',
 'f(2)(2300)',
 'f(2)(2340)',
 'h(1)(1170)',
 'omega(1420)',
 'omega(1650)',
 'omega(782)',
 'phi(1020)',
 'phi(1680)',
 'rho(1450)0',
 'rho(1700)0',
 'rho(770)0'}

Great! Now we selected only the strongest contributions. Be aware, though, that there are more effects that can suppress certain decays, like small branching ratios. In this example, the initial state \(J/\Psi\) can decay into \(\pi^0 + \rho^0\) or \(\pi^0 + \omega\).

decay	branching ratio
\(\omega\to\gamma+\pi^0\)	0.0828
\(\rho^0\to\gamma+\pi^0\)	0.0006

Unfortunately, the \(\rho^0\) mainly decays into \(\pi^0+\pi^0\), not \(\gamma+\pi^0\) and is therefore suppressed. This information is currently not known to the expert system, but it is possible to hand the expert system a list of allowed intermediate states.

# particles are found by name comparison,
# i.e. f2 will find all f2's and f all f's independent of their spin
stm.set_allowed_intermediate_particles(["f(0)", "f(2)"])

result = stm.find_solutions(problem_sets)

print("found", len(result.transitions), "solutions!")
result.get_intermediate_particles().names

found 138 solutions!

{'f(0)(1370)',
 'f(0)(1500)',
 'f(0)(1710)',
 'f(0)(500)',
 'f(0)(980)',
 "f(2)'(1525)",
 'f(2)(1270)',
 'f(2)(1950)',
 'f(2)(2010)',
 'f(2)(2300)',
 'f(2)(2340)'}

Now we have selected all amplitudes that involve f states. The warnings appear only to notify the user that the list of solutions is not exhaustive: for certain edges in the graph, no suitable particle was found (since only f states were allowed).

import graphviz

from expertsystem import io

dot = io.asdot(result, collapse_graphs=True, render_node=False)
graphviz.Source(dot)

4. Export generated transitions¶

The Result, StateTransitionGraph, and Topology can be serialized to and from a dict with io.asdict() and io.fromdict():

from expertsystem import io

graph = result.transitions[0]
io.asdict(graph.topology)

{'nodes': [0, 1],
 'edges': {-1: {'ending_node_id': 0},
  0: {'originating_node_id': 0},
  1: {'originating_node_id': 1},
  2: {'originating_node_id': 1},
  3: {'originating_node_id': 0, 'ending_node_id': 1}}}

This also means that the Result can be written to JSON or YAML format with io.write() and loaded again with io.load():

io.write(result, "transitions.json")
imported_result = io.load("transitions.json")
assert imported_result == result

Handy if it takes a lot of computation time to generate the transitions!

Warning

It’s not possible to pickle a Result, because StateTransitionGraph makes use of Generic.

5. Generate an amplitude model¶

Now that we are satisfied with the intermediate resonances, we are all set convert the solutions that the STM found to an amplitude model! This can be done with get_builder(). This function deduces the amplitude formalism from the formalism_type that you specified when constructing the STM. Its output HelicityAmplitudeBuilder builder class can HelicityAmplitudeBuilder.generate() a HelicityModel that defines the mathematical formulation of an amplitude model in the helicity (or canonical) formalism.

import expertsystem as es

model_builder = es.amplitude.get_builder(result)
amplitude_model = model_builder.generate()
next(iter(amplitude_model.components.values()))

\[\displaystyle C[J/\psi(1S) \to f_{2}(1270)_{+2} \gamma_{+1};f_{2}(1270) \to \pi^{0}_{0} \pi^{0}_{0}] D^{1}_{-1,-1}\left(- \phi_{1+2},\theta_{1+2},0\right) D^{2}_{-2,0}\left(- \phi_{1,1+2},\theta_{1,1+2},0\right)\]

From here on, have a look at Formulate amplitude model to see how to work with the HelicityModel class.