Dynamics¶

By default, the dynamic terms in an amplitude model are set to \(1\) by the HelicityAmplitudeBuilder. The method set_dynamics() can then be used to set dynamics lineshapes for specific resonances. The dynamics.builder module provides some tools to set standard lineshapes (see below), but it is also possible to set custom dynamics.

The standard lineshapes provided by AmpForm are illustrated below. For more info, have a look at the following pages:

%matplotlib widget
import matplotlib.pyplot as plt
import mpl_interactions.ipyplot as iplt
import numpy as np
import sympy as sp
from IPython.display import Math

import symplot

Form factor¶

AmpForm uses Blatt-Weisskopf functions \(B_L\) as barrier factors (also called form factors, see BlattWeisskopfSquared):

from ampform.dynamics import BlattWeisskopfSquared

L = sp.Symbol("L", integer=True)
z = sp.Symbol("z", real=True)
ff2 = BlattWeisskopfSquared(L, z)

Math(f"{sp.latex(ff2)} = {sp.latex(ff2.doit())}")

\[\begin{split}\displaystyle B_{L}^2\left(z\right) = \begin{cases} 1 & \text{for}\: L = 0 \\\frac{2 z}{z + 1} & \text{for}\: L = 1 \\\frac{13 z^{2}}{9 z + \left(z - 3\right)^{2}} & \text{for}\: L = 2 \\\frac{277 z^{3}}{z \left(z - 15\right)^{2} + \left(2 z - 5\right) \left(18 z - 45\right)} & \text{for}\: L = 3 \\\frac{12746 z^{4}}{25 z \left(2 z - 21\right)^{2} + \left(z^{2} - 45 z + 105\right)^{2}} & \text{for}\: L = 4 \\\frac{998881 z^{5}}{z^{5} + 15 z^{4} + 315 z^{3} + 6300 z^{2} + 99225 z + 893025} & \text{for}\: L = 5 \\\frac{118394977 z^{6}}{z^{6} + 21 z^{5} + 630 z^{4} + 18900 z^{3} + 496125 z^{2} + 9823275 z + 108056025} & \text{for}\: L = 6 \\\frac{19727003738 z^{7}}{z^{7} + 28 z^{6} + 1134 z^{5} + 47250 z^{4} + 1819125 z^{3} + 58939650 z^{2} + 1404728325 z + 18261468225} & \text{for}\: L = 7 \\\frac{4392846440677 z^{8}}{z^{8} + 36 z^{7} + 1890 z^{6} + 103950 z^{5} + 5457375 z^{4} + 255405150 z^{3} + 9833098275 z^{2} + 273922023375 z + 4108830350625} & \text{for}\: L = 8 \end{cases}\end{split}\]

The Blatt-Weisskopf form factor is used to ‘dampen’ the breakup-momentum at threshold and when going to infinity. A usual choice for \(z\) is therefore \(z=q^2d^2\) with \(q^2\) the breakup_momentum_squared() and \(d\) the impact parameter (also called meson radius):

from ampform.dynamics import breakup_momentum_squared

m, m_a, m_b, d = sp.symbols("m, m_a, m_b, d")
s = m ** 2
q_squared = breakup_momentum_squared(s, m_a, m_b)
ff2 = BlattWeisskopfSquared(L, z=q_squared * d ** 2)

Relativistic Breit-Wigner¶

AmpForm has two types of relativistic Breit-Wigner functions. Both are compared below ― for more info, see the links to the API.

Without form factor¶

The ‘normal’ relativistic_breit_wigner() looks as follows:

from ampform.dynamics import relativistic_breit_wigner

m, m0, w0 = sp.symbols("m, m0, Gamma0")
rel_bw = relativistic_breit_wigner(s=m ** 2, mass0=m0, gamma0=w0)
rel_bw

\[\displaystyle \frac{\Gamma_{0} m_{0}}{- i \Gamma_{0} m_{0} - m^{2} + m_{0}^{2}}\]

With form factor¶

The relativistic Breit-Wigner can be adapted slightly, so that its amplitude goes to zero at threshold (\(m_0 = m_a + m_b\)) and that it becomes normalizable. This is done with form factors and can be obtained with the function relativistic_breit_wigner_with_ff():

from ampform.dynamics import phase_space_factor  # noqa: F401
from ampform.dynamics import ComplexSqrt, breakup_momentum_squared


def breakup_momentum(s: sp.Symbol, m_a: sp.Symbol, m_b: sp.Symbol) -> sp.Expr:
    q_squared = breakup_momentum_squared(s, m_a, m_b)
    return ComplexSqrt(q_squared)

from ampform.dynamics import (
    phase_space_factor_ac,
    relativistic_breit_wigner_with_ff,
)

rel_bw_with_ff = relativistic_breit_wigner_with_ff(
    s=s,
    mass0=m0,
    gamma0=w0,
    m_a=m_a,
    m_b=m_b,
    angular_momentum=L,
    meson_radius=d,
    phsp_factor=phase_space_factor_ac,
)

from ampform.dynamics import coupled_width

q_squared = breakup_momentum_squared(s, m_a, m_b)
ff_sq = BlattWeisskopfSquared(L, z=q_squared * d ** 2)
mass_dependent_width = coupled_width(
    s, m0, w0, m_a, m_b, L, d, phsp_factor=phase_space_factor_ac
)
rel_bw_with_ff.subs(
    {
        2 * q_squared: 2 * sp.Symbol("q^{2}(m)"),
        ff_sq: sp.Symbol(R"B_{L}^{2}\left(q\right)"),
        mass_dependent_width: sp.Symbol(R"\Gamma(m)"),
    }
)

\[\begin{split}\displaystyle \frac{\Gamma_{0} m_{0} \left(\begin{cases} 0 & \text{for}\: m^{2} < \left(m_{a} + m_{b}\right)^{2} \wedge L \neq 0 \\\sqrt{B_{L}^{2}\left(q\right)} & \text{otherwise} \end{cases}\right)}{- i \Gamma(m) m_{0} - m^{2} + m_{0}^{2}}\end{split}\]

Here, \(\Gamma(m)\) is the coupled_width() (also called running width or mass-dependent width), defined as:

from ampform.dynamics import coupled_width

L = sp.Symbol("L", integer=True)
d = sp.Symbol("d")
s = m ** 2
running_width = coupled_width(
    s=s,
    mass0=m0,
    gamma0=w0,
    m_a=m_a,
    m_b=m_b,
    angular_momentum=L,
    meson_radius=d,
    phsp_factor=phase_space_factor_ac,
)

q0_squared = breakup_momentum_squared(m0 ** 2, m_a, m_b)
ff = BlattWeisskopfSquared(L, z=q_squared * d ** 2)
ff0_sq = BlattWeisskopfSquared(L, z=q0_squared * d ** 2)
rho = phase_space_factor_ac(s, m_a, m_b)
rho0 = phase_space_factor_ac(m0 ** 2, m_a, m_b)
running_width = running_width.subs(
    {
        rho / rho0: sp.Symbol(R"\rho(m)") / sp.Symbol(R"\rho(m_{0})"),
        ff: sp.Symbol("B_{L}(q)"),
        ff0_sq: sp.Symbol("B_{L}(q_{0})"),
    },
)
Math(fR"\Gamma(m) = {sp.latex(running_width)}")

\[\displaystyle \Gamma(m) = \frac{B_{L}(q) \Gamma_{0} \rho(m)}{B_{L}(q_{0}) \rho(m_{0})}\]

It is possible to choose different formulations for the phase space factor \(\rho\), see Analytic continuation.

Comparison¶

fig, axes = plt.subplots(2, 1, figsize=(8, 6), sharex=True)
ax_real, ax_imag = axes
ax_imag.set_xlabel("$m$")
ax_real.set_ylabel(R"$\left|A\right|^2$")
ax_imag.set_ylabel(R"Im$\left(A\right)$")

# Real
ylim = (0, 1.1)
controls = iplt.plot(
    plot_domain,
    lambda *args, **kwargs: np.abs(np_rel_bw_with_ff(*args, **kwargs)) ** 2,
    label="analytic continuation",
    **sliders,
    ylim=ylim,
    ax=ax_real,
    linestyle="dotted",
    c="C0",
)
iplt.plot(
    plot_domain.real,
    lambda *args, **kwargs: np.abs(np_rel_bw_with_ff(*args, **kwargs)) ** 2,
    label="$with$ form factor",
    controls=controls,
    ylim=ylim,
    ax=ax_real,
)
iplt.plot(
    plot_domain.real,
    lambda *args, **kwargs: np.abs(np_rel_bw(*args, **kwargs)) ** 2,
    label="non-relativistic Breit-Wigner",
    controls=controls,
    ylim=ylim,
    ax=ax_real,
)
iplt.axvline(controls["m0"], c="gray", linestyle="dotted")

# Imaginary
iplt.plot(
    plot_domain,
    lambda *args, **kwargs: np_rel_bw_with_ff(*args, **kwargs).imag,
    label="analytic continuation",
    controls=controls,
    ylim=ylim,
    ax=ax_imag,
    alpha=0.5,
    linestyle="dotted",
    c="C0",
)
iplt.plot(
    plot_domain.real,
    lambda *args, **kwargs: np_rel_bw_with_ff(*args, **kwargs).imag,
    label="$with$ form factor",
    controls=controls,
    ylim=ylim,
    ax=ax_imag,
)
iplt.plot(
    plot_domain.real,
    lambda *args, **kwargs: np_rel_bw(*args, **kwargs).imag,
    label="non-relativistic Breit-Wigner",
    controls=controls,
    ylim=ylim,
    ax=ax_imag,
)
iplt.axvline(controls["m0"], c="gray", linestyle="dotted")

ax_real.legend(loc="upper right")
iplt.title(
    R"$m_0={m0:.1f} \qquad \Gamma_0={Gamma0:.1f} \qquad L={L} \qquad m_a={m_a:.1f} \qquad m_b={m_b:.1f}$",
    controls=controls,
    ax=ax_real,
)
fig.tight_layout()
plt.show()