Helicity versus canonical

Helicity versus canonical

import logging

import graphviz
import matplotlib as mpl
import numpy as np
import qrules
import sympy as sp
from IPython.display import HTML, Math
from matplotlib import cm
from rich.table import Table

import ampform

LOGGER = logging.getLogger()
LOGGER.setLevel(logging.ERROR)

In this notebook, we have a look at the decay

\[D_1(2420)^0 \to a_1(1260)^+ K^- \to (K^+K^0)K^-\]

in order to see the difference between a HelicityModel formulated in the canonical basis and one formulated in the helicity basis. To simplify things, we only look at spin projection \(+1\) for \(D_1(2420)^0\), because the intensities for each of the spin projections of \(D_1(2420)^0\) are incoherent, no matter which spin formalism we choose.

Tip

For more information about the helicity formalism, see [1], [2], and [3].

First, we use qrules.generate_transitions() to generate a ReactionInfo instance for both formalisms:

def generate_transitions(formalism: str):
    reaction = qrules.generate_transitions(
        initial_state=("D(1)(2420)0", [+1]),
        final_state=["K+", "K-", "K~0"],
        allowed_intermediate_particles=["a(1)(1260)+"],
        formalism=formalism,
    )
    builder = ampform.get_builder(reaction)
    return builder.formulate()


cano_model = generate_transitions("canonical-helicity")
heli_model = generate_transitions("helicity")

From components and parameter_defaults, we can see that the canonical formalism has a larger number of amplitudes.

table = Table(show_edge=False)
table.add_column("Formalism")
table.add_column("Coefficients", justify="right")
table.add_column("Amplitudes", justify="right")
table.add_row(
    "Canonical",
    str(len(cano_model.parameter_defaults)),
    str(len(cano_model.components) - 1),
)
table.add_row(
    "Helicity",
    str(len(heli_model.parameter_defaults)),
    str(len(heli_model.components) - 1),
)
table

 Formalism ┃ Coefficients ┃ Amplitudes 
━━━━━━━━━━━╇━━━━━━━━━━━━━━╇━━━━━━━━━━━━
 Canonical │            3 │          8 
 Helicity  │            3 │          3

The reason for this is that canonical basis distinguishes amplitudes over their \(LS\)-combinations. This becomes clear if we define \(a\) to be the amplitude without coefficient (\(A = C a\)), and consider what the full, coherent intensity looks like.

If we write the full intensity as \(I = \left|\sum_i A_i\right|^2\), then we have, in the case of the canonical basis:

def extract_amplitude_substitutions(model, colorize=False):
    amplitude_to_symbol = {}
    amplitude_names = sorted(c for c in model.components if c.startswith("A"))
    n_colors = len(amplitude_names)
    color_map = cm.brg(np.linspace(0, 1, num=n_colors + 1)[:-1])
    color_iter = (mpl.colors.to_hex(color) for color in color_map)
    for name in amplitude_names:
        expr = model.components[name]
        for par in model.parameter_defaults:
            if par in expr.args:
                expr /= par
        name = "a" + name[1:]
        if colorize:
            color = next(color_iter)
            name = Rf"\color{{{color}}}{{{name}}}"
        symbol = sp.Symbol(name)
        amplitude_to_symbol[expr] = symbol
    return amplitude_to_symbol


cano_amplitude_to_symbol = extract_amplitude_substitutions(cano_model)
heli_amplitude_to_symbol = extract_amplitude_substitutions(heli_model)


def render_amplitude_summation(model, colorize=False):
    amplitude_to_symbol = extract_amplitude_substitutions(model, colorize)
    collected_expr = sp.collect(
        model.expression.subs(amplitude_to_symbol).args[0].args[0],
        tuple(model.parameter_defaults),
    )
    terms = collected_expr.args
    latex = ""
    latex += R"\begin{align}"
    latex += Rf"\sum_i A_i & = {sp.latex(terms[0])}\\"
    for term in terms[1:]:
        latex += Rf"& + {sp.latex(term)} \\"
    latex += R"\end{align}"
    return Math(latex)


render_amplitude_summation(cano_model, colorize=True)
\[\begin{split}\displaystyle \begin{align}\sum_i A_i & = C_{D_{1}(2420)^{0} \xrightarrow[S=1]{L=0} K^{-} a_{1}(1260)^{+}; a_{1}(1260)^{+} \xrightarrow[S=0]{L=1} K^{+} \overline{K}^{0}} \left(\color{#0000ff}{a_{D_{1}(2420)^{0}_{+1} \xrightarrow[S=1]{L=0} K^{-}_{0} a_{1}(1260)^{+}_{+1}; a_{1}(1260)^{+}_{+1} \xrightarrow[S=0]{L=1} K^{+}_{0} \overline{K}^{0}_{0}}} + \color{#4000bf}{a_{D_{1}(2420)^{0}_{+1} \xrightarrow[S=1]{L=0} K^{-}_{0} a_{1}(1260)^{+}_{-1}; a_{1}(1260)^{+}_{-1} \xrightarrow[S=0]{L=1} K^{+}_{0} \overline{K}^{0}_{0}}} + \color{#80007f}{a_{D_{1}(2420)^{0}_{+1} \xrightarrow[S=1]{L=0} K^{-}_{0} a_{1}(1260)^{+}_{0}; a_{1}(1260)^{+}_{0} \xrightarrow[S=0]{L=1} K^{+}_{0} \overline{K}^{0}_{0}}}\right)\\& + C_{D_{1}(2420)^{0} \xrightarrow[S=1]{L=1} K^{-} a_{1}(1260)^{+}; a_{1}(1260)^{+} \xrightarrow[S=0]{L=1} K^{+} \overline{K}^{0}} \left(\color{#c0003f}{a_{D_{1}(2420)^{0}_{+1} \xrightarrow[S=1]{L=1} K^{-}_{0} a_{1}(1260)^{+}_{+1}; a_{1}(1260)^{+}_{+1} \xrightarrow[S=0]{L=1} K^{+}_{0} \overline{K}^{0}_{0}}} + \color{#fe0100}{a_{D_{1}(2420)^{0}_{+1} \xrightarrow[S=1]{L=1} K^{-}_{0} a_{1}(1260)^{+}_{-1}; a_{1}(1260)^{+}_{-1} \xrightarrow[S=0]{L=1} K^{+}_{0} \overline{K}^{0}_{0}}}\right) \\& + C_{D_{1}(2420)^{0} \xrightarrow[S=1]{L=2} K^{-} a_{1}(1260)^{+}; a_{1}(1260)^{+} \xrightarrow[S=0]{L=1} K^{+} \overline{K}^{0}} \left(\color{#3ec100}{a_{D_{1}(2420)^{0}_{+1} \xrightarrow[S=1]{L=2} K^{-}_{0} a_{1}(1260)^{+}_{0}; a_{1}(1260)^{+}_{0} \xrightarrow[S=0]{L=1} K^{+}_{0} \overline{K}^{0}_{0}}} + \color{#7e8100}{a_{D_{1}(2420)^{0}_{+1} \xrightarrow[S=1]{L=2} K^{-}_{0} a_{1}(1260)^{+}_{-1}; a_{1}(1260)^{+}_{-1} \xrightarrow[S=0]{L=1} K^{+}_{0} \overline{K}^{0}_{0}}} + \color{#be4100}{a_{D_{1}(2420)^{0}_{+1} \xrightarrow[S=1]{L=2} K^{-}_{0} a_{1}(1260)^{+}_{+1}; a_{1}(1260)^{+}_{+1} \xrightarrow[S=0]{L=1} K^{+}_{0} \overline{K}^{0}_{0}}}\right) \\\end{align}\end{split}\]

In the helicity basis, the \(LS\)-combinations have been summed over already and we can only see an amplitude for each helicity:

render_amplitude_summation(heli_model)
\[\begin{split}\displaystyle \begin{align}\sum_i A_i & = C_{D_{1}(2420)^{0} \to K^{-}_{0} a_{1}(1260)^{+}_{+1}; a_{1}(1260)^{+} \to K^{+}_{0} \overline{K}^{0}_{0}} a_{D_{1}(2420)^{0}_{+1} \to K^{-}_{0} a_{1}(1260)^{+}_{+1}; a_{1}(1260)^{+}_{+1} \to K^{+}_{0} \overline{K}^{0}_{0}}\\& + C_{D_{1}(2420)^{0} \to K^{-}_{0} a_{1}(1260)^{+}_{-1}; a_{1}(1260)^{+} \to K^{+}_{0} \overline{K}^{0}_{0}} a_{D_{1}(2420)^{0}_{+1} \to K^{-}_{0} a_{1}(1260)^{+}_{-1}; a_{1}(1260)^{+}_{-1} \to K^{+}_{0} \overline{K}^{0}_{0}} \\& + C_{D_{1}(2420)^{0} \to K^{-}_{0} a_{1}(1260)^{+}_{0}; a_{1}(1260)^{+} \to K^{+}_{0} \overline{K}^{0}_{0}} a_{D_{1}(2420)^{0}_{+1} \to K^{-}_{0} a_{1}(1260)^{+}_{0}; a_{1}(1260)^{+}_{0} \to K^{+}_{0} \overline{K}^{0}_{0}} \\\end{align}\end{split}\]

Amplitudes in the canonical basis are formulated with regard to their \(LS\)-couplings. As such, they contain additional Clebsch-Gordan coefficients that serve as expansion coefficients.

def extract_amplitudes(model):
    return {
        expr: sp.Symbol(name)
        for name, expr in model.components.items()
        if name.startswith("A")
    }


cano_amplitudes = extract_amplitudes(cano_model)
heli_amplitudes = extract_amplitudes(heli_model)

expression, symbol = next(iter(cano_amplitude_to_symbol.items()))
display(symbol, Math(Rf"\quad = {sp.latex(expression)}"))
\[\displaystyle a_{D_{1}(2420)^{0}_{+1} \xrightarrow[S=1]{L=0} K^{-}_{0} a_{1}(1260)^{+}_{+1}; a_{1}(1260)^{+}_{+1} \xrightarrow[S=0]{L=1} K^{+}_{0} \overline{K}^{0}_{0}}\]
\[\displaystyle \quad = C^{0,0}_{0,0,0,0} C^{1,1}_{0,0,1,1} C^{1,0}_{1,0,0,0} C^{1,1}_{1,1,0,0} D^{1}_{1,0}\left(- \phi_{0,0+2},\theta_{0,0+2},0\right) D^{1}_{1,1}\left(- \phi_{0+2},\theta_{0+2},0\right)\]

In the helicity basis, these Clebsch-Gordan coefficients and Wigner-\(D\) functions have been summed up, leaving only a Wigner-\(D\) for each node in the decay chain (two in this case):

expression, symbol = next(iter(heli_amplitude_to_symbol.items()))
display(symbol, Math(Rf"\quad = {sp.latex(expression)}"))
\[\displaystyle a_{D_{1}(2420)^{0}_{+1} \to K^{-}_{0} a_{1}(1260)^{+}_{+1}; a_{1}(1260)^{+}_{+1} \to K^{+}_{0} \overline{K}^{0}_{0}}\]
\[\displaystyle \quad = D^{1}_{1,0}\left(- \phi_{0,0+2},\theta_{0,0+2},0\right) D^{1}_{1,1}\left(- \phi_{0+2},\theta_{0+2},0\right)\]

See formulate_wigner_d() and formulate_clebsch_gordan_coefficients() for how these Wigner-\(D\) functions and Clebsch-Gordan coefficients are computed for each node on a StateTransition.

We can see this also from the original ReactionInfo objects. Let’s select only the transitions where the \(a_1(1260)^+\) resonance has spin projection \(-1\) (taken to be helicity \(-1\) in the helicity formalism). We then see just one StateTransition in the helicity basis and three transitions in the canonical basis:

def render_selection(model):
    transitions = model.reaction_info.transitions
    selection = filter(
        lambda s: s.states[3].spin_projection == -1, transitions
    )
    dot = qrules.io.asdot(
        selection, render_node=True, render_final_state_id=False
    )
    return graphviz.Source(dot)


display(
    HTML("<b>Helicity</b> basis:"),
    render_selection(heli_model),
    HTML("<b>Canonical</b> basis:"),
    render_selection(cano_model),
)
Helicity basis:
../_images/formalism_18_1.svg
Canonical basis:
../_images/formalism_18_3.svg

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