Custom dynamics¶

import graphviz
import qrules
import sympy as sp

We start by generating allowed transitions for a simple decay channel, just like in Formulate amplitude model:

reaction = qrules.generate_transitions(
    initial_state=("J/psi(1S)", [+1]),
    final_state=[("gamma", [+1]), "pi0", "pi0"],
    allowed_intermediate_particles=["f(0)(980)", "f(0)(1500)"],
    allowed_interaction_types=["strong", "EM"],
    formalism="helicity",
)

dot = qrules.io.asdot(reaction, collapse_graphs=True)
graphviz.Source(dot)

Next, create a HelicityAmplitudeBuilder using get_builder():

from ampform import get_builder

model_builder = get_builder(reaction)

In Formulate amplitude model, we used set_dynamics() with some standard lineshape builders from the builder module. These builders have a signature that follows the ResonanceDynamicsBuilder Protocol:

import inspect

from ampform.dynamics.builder import (
    ResonanceDynamicsBuilder,
    create_relativistic_breit_wigner,
)

print(inspect.getsource(ResonanceDynamicsBuilder))
print(inspect.getsource(create_relativistic_breit_wigner))

class ResonanceDynamicsBuilder(Protocol):
    """Protocol that is used by `.set_dynamics`.

    Follow this `~typing.Protocol` when defining a builder function that is to
    be used by `.set_dynamics`. For an example, see the source code
    `.create_relativistic_breit_wigner`, which creates a
    `.relativistic_breit_wigner`.

    .. seealso:: :doc:`/usage/dynamics/custom`
    """

    def __call__(
        self, resonance: Particle, variable_pool: TwoBodyKinematicVariableSet
    ) -> BuilderReturnType:
        ...

def create_relativistic_breit_wigner(
    resonance: Particle, variable_pool: TwoBodyKinematicVariableSet
) -> BuilderReturnType:
    """Create a `.relativistic_breit_wigner` for a two-body decay."""
    inv_mass = variable_pool.incoming_state_mass
    res_mass = sp.Symbol(f"m_{resonance.name}")
    res_width = sp.Symbol(f"Gamma_{resonance.name}")
    expression = relativistic_breit_wigner(
        s=inv_mass ** 2,
        mass0=res_mass,
        gamma0=res_width,
    )
    parameter_defaults = {
        res_mass: resonance.mass,
        res_width: resonance.width,
    }
    return expression, parameter_defaults

A function that behaves like a ResonanceDynamicsBuilder should return a tuple of some Expr (which formulates your lineshape) and a dict of Symbols to some suggested initial values. This signature is required so that set_dynamics() knows how to extract the correct symbol names and their suggested initial values from a StateTransition.

The Expr you use for the lineshape can be anything. Here, we use a Gaussian function and wrap it in a function. As you can see, this function stands on its own, independent of ampform:

def my_dynamics(x: sp.Symbol, mu: sp.Symbol, sigma: sp.Symbol) -> sp.Expr:
    return sp.exp(-((x - mu) ** 2) / sigma ** 2 / 2) / (
        sigma * sp.sqrt(2 * sp.pi)
    )

x, mu, sigma = sp.symbols("x mu sigma")
sp.plot(my_dynamics(x, 0, 1), (x, -3, 3), axis_center=(0, 0))
my_dynamics(x, mu, sigma)

\[\displaystyle \frac{\sqrt{2} e^{- \frac{\left(- \mu + x\right)^{2}}{2 \sigma^{2}}}}{2 \sqrt{\pi} \sigma}\]

We can now follow the example of the create_relativistic_breit_wigner() to create a builder for this custom lineshape:

from typing import Dict, Tuple

from qrules.particle import Particle

from ampform.dynamics.builder import TwoBodyKinematicVariableSet


def create_my_dynamics(
    resonance: Particle, variable_pool: TwoBodyKinematicVariableSet
) -> Tuple[sp.Expr, Dict[sp.Symbol, float]]:
    res_mass = sp.Symbol(f"m_{resonance.name}")
    res_width = sp.Symbol(f"sigma_{resonance.name}")
    expression = my_dynamics(
        x=variable_pool.incoming_state_mass,
        mu=res_mass,
        sigma=res_width,
    )
    parameter_defaults = {
        res_mass: resonance.mass,
        res_width: resonance.width,
    }
    return expression, parameter_defaults

Now, just like in Build SymPy expression, it’s simply a matter of plugging this builder into set_dynamics() and we can formulate() a model with this custom lineshape:

for name in reaction.get_intermediate_particles().names:
    model_builder.set_dynamics(name, create_my_dynamics)
model = model_builder.formulate()

As can be seen, the HelicityModel.parameter_defaults section has been updated with the some additional parameters for the custom parameter and there corresponding suggested initial values:

display(*sorted(model.parameter_defaults, key=lambda p: p.name))

\[\displaystyle C_{J/\psi(1S) \to f_{0}(1500)_{0} \gamma_{+1}; f_{0}(1500) \to \pi^{0}_{0} \pi^{0}_{0}}\]

\[\displaystyle C_{J/\psi(1S) \to f_{0}(980)_{0} \gamma_{+1}; f_{0}(980) \to \pi^{0}_{0} \pi^{0}_{0}}\]

\[\displaystyle m_{f(0)(1500)}\]

\[\displaystyle m_{f(0)(980)}\]

\[\displaystyle \sigma_{f(0)(1500)}\]

\[\displaystyle \sigma_{f(0)(980)}\]

Let’s quickly have a look what this lineshape looks like. First, check which Symbols remain once we replace the parameters with their suggested initial values. These are the kinematic variables of the model:

expr = model.expression.doit().subs(model.parameter_defaults)
free_symbols = tuple(sorted(expr.free_symbols, key=lambda s: s.name))
free_symbols

(m_12, theta_1+2)

To create an invariant mass distribution, we should integrate out the \(\theta\) angle. This can be done with integrate():

m, theta = free_symbols
integrated_expr = sp.integrate(
    expr,
    (theta, 0, sp.pi),
    meijerg=True,
    conds="piecewise",
    risch=None,
    heurisch=None,
    manual=None,
)
integrated_expr.n(1)

\[\displaystyle 5.0 \cdot 10^{1} e^{- 277.777777777778 \left(m_{12} - 0.99\right)^{2}} + 2.0 \cdot 10^{1} e^{- 180.806441326531 \left(0.664010624169987 m_{12} - 1\right)^{2}} + 6.0 \cdot 10^{1} e^{- 90.4032206632653 \left(0.664010624169987 m_{12} - 1\right)^{2}} e^{- 138.888888888889 \left(m_{12} - 0.99\right)^{2}}\]

Finally, here is the resulting expression as a function of the invariant mass, with custom dynamics!

x1, x2 = 0.6, 1.9
sp.plot(integrated_expr, (m, x1, x2), axis_center=(x1, 0));

AmpForm

Custom dynamics¶