PWA Software Pages

Warning

These pages are under development.

The PWA Software Pages serve two purposes:

1. They aim to bring together the many Partial Wave Analysis frameworks out there on the market through interlinked documentation.

2. They provide a dynamic platform to collect and maintain knowledge on both PWA theory and software tools.

Introduction

Warning

These pages are under development.

While the Standard Model provides an incredibly accurate description of the most fundamental constituents of matter, it remains difficult to describe interactions at the level of bound states (hadrons). This is because the fundamental force that dominates at this range―the strong force―is characterized by an asymptotic freedom: it has a strong coupling constant at low momentum transfer.

One the one hand, this asymptotic behavior of the strong force forces quarks to bound together in colorless groups (color confinement), giving particle physicists an amazingly varied spectrum of quark combinations to study. On the other, the asymptotic running of the coupling constant makes it almost impossible to predict from first principles how bound states of quarks interact at lower energies.

Theoreticians have developed several descriptive models and numerical tools (lattice QCD most importantly), but these models often rest on several assumptions or fail to describe larger systems. It therefore remains important to study particle interactions at the lower energies.

Partial Wave Analysis is a collection of techniques that allows us to perform these studies.

Todo

• Importance of decomposing interactions

• Strong QCD coupling constant at low momentum transfer

• Cannot compute intermediate states from theory

• Need to collect properties: mass, width, spin, parity

Scattering theory

Tip

An extensive, chronological lists of publications on scattering theory can be found here.

Transition amplitude

Todo

• Amplitudes and intensities

• Scattering operator \(S\) (S-matrix)

• Lead-up to \(T\)-matrix (transition operator)

• Distinction dynamical part and angular part

Partial waves

Todo

Decomposition into partial waves [Peters, 2004], p.3:

Consider an incident wave \(\left|i\right> = Psi_i\). In an experimental setting, we can assume a vanishing potential at \(t\rightarrow\infty\), which allows us to expand the incoming wave in terms of Legendre polynomials \(P_l\), with \(l\) the angular orbital momentum:

\[\Psi_i(r,\theta) = \sum_{l=0}^\infty U_l(r) P_l(\theta) \]

The \(U_l\) factor can be parametrized further. At this stage, it is important to point that the angular orbital momentum of the incoming state is used to characterize specific systems. Here, it is common to use the labels used for electron orbitals:

\(l\)	Label	Origin
0	s	“sharp”
1	p	“principal”
2	d	“diffuse”
3	f	“fundamental”
4	g	rest is alphabetical
…	…	…

In PWA, it is therefore common to distinguish these wave contributions as \(S\)-wave, \(P\)-wave, etc.

Todo

• Separating out angular and radial wave functions using Legendre polynomials

• What we can see from this and how it relates to analysis techniques

• (?) Difference with amplitude analysis

• Possible initial states: 1, 2, and multi-body

• Suitable for propagation, not re-scattering

Isobar model

Mandelstam variables

Show code cell source Hide code cell source

import matplotlib.pyplot as plt
from feynman import Diagram

fig, ax = plt.subplots(1, 3, figsize=(9, 3), frameon=False, tight_layout=True)
fig.patch.set_visible(False)
for a in ax:
    a.axis("off")
    a.set_xlim(0, 1)
    a.set_ylim(0, 0.8)

s_channel = Diagram(ax[0])
# Vertices
v_p1 = s_channel.vertex(xy=(0.1, 0.7), marker="")
v_p2 = s_channel.vertex(tuple(v_p1.xy), dy=-0.6, marker="")
v_p3 = s_channel.vertex(tuple(v_p1.xy), dx=0.8, marker="")
v_p4 = s_channel.vertex(tuple(v_p3.xy), dy=-0.6, marker="")
v1 = s_channel.vertex(tuple(v_p1.xy), dx=0.2, dy=-0.3, marker="")
v2 = s_channel.vertex(tuple(v_p3.xy), dx=-0.2, dy=-0.3, marker="")
# Lines
p1 = s_channel.line(v_p1, v1, arrow=False)
p2 = s_channel.line(v1, v_p2, arrow=False)
interaction = s_channel.line(v1, v2, style="dotted", arrow=False)
e4 = s_channel.line(v2, v_p3, arrow=False)
e5 = s_channel.line(v_p4, v2, arrow=False)
# Labels
v_p1.text("$p_1$", x=-0.06, y=0)
v_p2.text("$p_2$", x=-0.06, y=0)
v_p3.text("$p_3$", x=+0.06, y=0)
v_p4.text("$p_4$", x=+0.06, y=0)
s_channel.text(0.5, 0.08, "$s$-channel", fontsize=15)
s_channel.text(0.5, 0, "$s = (p_1 + p_2)^2 = (p_3 + p_4)^2$", fontsize=12)
s_channel.plot()

t_channel = Diagram(ax[1])
# Vertices
v_p1 = t_channel.vertex(xy=(0.1, 0.7), marker="")
v_p2 = t_channel.vertex(tuple(v_p1.xy), dy=-0.6, marker="")
v_p3 = t_channel.vertex(tuple(v_p1.xy), dx=0.8, marker="")
v_p4 = t_channel.vertex(tuple(v_p3.xy), dy=-0.6, marker="")
v1 = t_channel.vertex(tuple(v_p1.xy), dx=0.4, dy=-0.15, marker="")
v2 = t_channel.vertex(tuple(v_p2.xy), dx=0.4, dy=+0.15, marker="")
# Lines
p1 = t_channel.line(v_p1, v1, arrow=False)
p2 = t_channel.line(v1, v_p3, arrow=False)
interaction = t_channel.line(v1, v2, style="dotted", arrow=False)
e4 = t_channel.line(v2, v_p2, arrow=False)
e5 = t_channel.line(v_p4, v2, arrow=False)
# Labels
v_p1.text("$p_1$", x=-0.06, y=0)
v_p2.text("$p_2$", x=-0.06, y=0)
v_p3.text("$p_3$", x=+0.06, y=0)
v_p4.text("$p_4$", x=+0.06, y=0)
t_channel.text(0.5, 0.08, "$t$-channel", fontsize=15)
t_channel.text(0.5, 0, "$t = (p_1 - p_3)^2 = (p_4 - p_2)^2$", fontsize=12)
t_channel.plot()

u_channel = Diagram(ax[2])
# Vertices
v_p1 = u_channel.vertex(xy=(0.1, 0.7), marker="")
v_p2 = u_channel.vertex(tuple(v_p1.xy), dy=-0.6, marker="")
v_p3 = u_channel.vertex(tuple(v_p1.xy), dx=0.8, marker="")
v_p4 = u_channel.vertex(tuple(v_p3.xy), dy=-0.6, marker="")
v1 = u_channel.vertex(tuple(v_p1.xy), dx=0.4, dy=-0.15, marker="")
v2 = u_channel.vertex(tuple(v_p2.xy), dx=0.4, dy=+0.15, marker="")
# Lines
p1 = u_channel.line(v_p1, v1, arrow=False)
p2 = u_channel.line(v1, v_p4, arrow=False)
interaction = u_channel.line(v1, v2, style="dotted", arrow=False)
e4 = u_channel.line(v2, v_p2, arrow=False)
e5 = u_channel.line(v_p3, v2, arrow=False)
# Labels
v_p1.text("$p_1$", x=-0.06, y=0)
v_p2.text("$p_2$", x=-0.06, y=0)
v_p3.text("$p_3$", x=+0.06, y=0)
v_p4.text("$p_4$", x=+0.06, y=0)
u_channel.text(0.5, 0.08, "$u$-channel", fontsize=15)
u_channel.text(0.5, 0, "$u = (p_1 - p_4)^2 = (p_3 - p_2)^2$", fontsize=12)
u_channel.plot();

Formalisms

Warning

These pages are under development.

Helicity formalism

Warning

These pages are under development.

Todo

• Lorentz-invariance of helicity operator

• Types

  • Two-body decay amplitude

  • Sequential two-body decay amplitude

• Definition of angles

  • Production angle

  • Helicity angle

  • Treiman-Yang angle

  • Overview of kinematic variables

• Dynamics:

  • Wigner-D functions

  • Clebsch-Gordan coefficients

• Kinematic variables

• An example

Canonical formulation

Alignment problem

Recommended literature:

• General introductions to helicity angles:
  [Kutschke, 1996, Richman, 1984]

• Suggested solutions:
  [Chen and Ping, 2017, Marangotto, 2020, Wang et al., 2020, Mikhasenko et al., 2020]

• LHCb study that led to these solution papers:
  [Aaij et al., 2015]

Tensor formalisms

Rarita-Schwinger

Other spin formalisms

Covariant Tensor formalisms

[Anisovich et al., 2002]

Non-covariant tensor formalisms

[Zemach, 1965]

Spin-projection formalisms

Dynamics

Warning

These pages are under development.

K-matrix

Since scattering operator (\(S\)-matrix) formulates the transition amplitude from initial state \(\left|i\right>\) to final state \(\left|f\right>\) through \(\left<i\right|S\left|f\right>\), it is a unitary operator—probability is conserved, meaning \(SS^* = I\). Now, having defined the transition operator through \(S = I + iT\), we can introduce another operator: \(K^{-1} = T^{-1} + iI\) [Chung et al., 1995].

Todo

Explain why this new matrix interesting

Todo

Definition in terms of \(T\)-matrix

Special cases of the K-matrix

Breit-Wigner

Todo

Derive from \(K\)-matrix instead/as well.

A quantum mechanical state at rest with energy \(E_0\) can be described in terms of the wave function:

\[ \psi(t) = \psi_0 e^{-iE_0t} \]

Now, if we assume that the state has a decay width of \(\Gamma\), the probability density \(\psi^*\psi\) of this state can be described as:

\[ \psi^*\psi = \psi_0^*\psi_0 e^{-\Gamma t}. \]

The wave function itself then becomes:

\[ \psi(t) = \psi_0 e^{-iE_0t} e^{-\frac{\Gamma}{2} t} = \psi_0 e^{-i \left(E_0-\tfrac{i}{2}\Gamma\right) t}. \]

A particle with a finite decay width can therefore be described as a particle with complex energy:

\[ E' = E_0 - \frac{i}{2}\Gamma \]

Now, as an experimental physicist, one is interested in predicting the probability of observing the particle at energy \(E\) (we want to describe the observed invariant mass distributions). This can be achieved by applying a Fourier transform, so that \(\psi\) is described in terms of energy \(E\) (or frequency \(\omega\)) instead of time \(t\):

\[ \psi(E) \propto \psi_0 \int_0^\infty e^{i\left(E-E_0+\tfrac{i}{2}\Gamma\right)t}\,\mathrm{d}t \propto \frac{1}{\left(E-E_0\right) - \tfrac{i}{2}\Gamma} \]

The probability to observe the particle at energy \(E\) is therefore:

\[ \psi^*(E)\psi(E) \propto \frac{\frac{\Gamma^2}{4}}{\left(E-E_0\right)^2 + \frac{\Gamma^2}{4}} \]

Todo

Describe relation between \(\psi(E)\) and transition amplitude \(M\)

From this, one can see that the transition amplitude \(M\) is described by:

\[ M(E) \propto \frac{\frac{\Gamma}{2}}{\left(E-E_0\right) - i\frac{\Gamma}{2}} \]

because \(|\psi|^2\propto|M|\). This is called non-relativistic Breit-Wigner parametrization.

Todo

Describe how the relativistic Breit-Wigner formula:

\[ M(E) \propto \frac{m_0\Gamma}{m_0^2 - m_{ab}^2 - im_0\Gamma} \]

is derived and why this is important in case of \(\Gamma \gg m_0\).

Show code cell source Hide code cell source

from sympy import symbols
from sympy.plotting import plot

x, Gamma, E0 = symbols(R"x \Gamma E_0")
gamma2_4 = Gamma**2 / 4
breit_wigner = gamma2_4 / ((x - E0) ** 2 + gamma2_4)

plot(
    breit_wigner.subs({R"\Gamma": 0.25, "E_0": 0.75}),
    xlim=(0, 1.5),
    title="Non-relativistic Breit-Wigner",
    xlabel="$E$ (A.U.)",
);

Flatté

The importance of Unitarity

Difference between resonances and bound states?

Experimental data

Warning

These pages are under development.

Types of experiments

Todo

• Formation vs production

PWA data

Todo

• Why are momentum tuples sufficient?

• How to determine initial 4-momentum?

• Observables and variables we want to compute

  • Luminosity

  • Cross-section

  • Branching fraction

  • (?) Fit fractions

Comparison with Measurements

Todo

Luminosity \(L\)

Measurements \(N\)

Cross section \(\sigma\)

\[ \frac{dN}{d\Phi_f} = L \cdot \frac{d\sigma}{d\Phi_f} \]

Number of events in a infinitesimal phase space element \(\Phi_f\) is proportional to the cross section of a initial state transitioning to the final state in the infinitesimal phase space element.

This section should clear up the phase space element problem we are having. I’m not sure its just a plotting problem. Since there one makes the transition from unbinned to binned data…

Miscellaneous topics

Warning

These pages are under development.

Todo

Other topics (re-categorize!)

• Phase-space

• Coherent vs incoherent

• \(P\)-vector

• (?) \(S\)-matrix

• (?) Spin density

• Constituent Quark Model

  • Asymptotic freedom and color confinement

  • \(J^CP(I^G)\)

  • Multiplets categorization

  • Importance of experimental data

• Overlapping resonances

• Coupled channels

The Standard Model

Warning

These pages are under development.

Leptons

Quarks

Constituent Quark Model

Warning

These pages are under development.

Asymptotic Freedom

Todo

• Confinement property of the strong force

• Constituent Quark Model: only colorless bound states

Mesons and Baryons

Todo

• Quantum numbers (\(J^{CP}(I^G)\))

  • baryon number \(B\)

  • hypercharge \(Y\)

  • charge \(Q\)

  • orbital angular momentum \(L\)

  • total spin \(J\)

  • parity \(P\)

  • charge \(Q\)

  • isospin \(I\)

  • general parity \(G\)

• multiplets

  • pseudoscalar mesons

  • vector mesons

Overview of latest insights

Warning

These pages are under development.

Todo

• Exotic states

• Glueballs

PWA Frameworks

Warning

These pages are under development.

Software inventory

Amplitude analysis projects

The following frameworks or projects are specifically designed to do amplitude analysis. The table is built from this YAML file and can be updated here.

Project	Collaboration	Since	Latest commit	C++	Python	Julia	Cuda
AmpGen	CLEO / LHCb	2018	01/2024	✓
AmpTools	BESIII / GlueX	2011	11/2023	✓			✓
BruFit	CLAS12	2020	01/2022	✓
ComPWA (C++)		2012	01/2024	✓
ComPWA project QRules AmpForm TensorWaves		2020	04/2024		✓
cFit		2012	05/2018	✓
FDC-PWA	BESIII	2000		✓
GPUPWA	BESIII	2011		✓
HAMMER		2016	08/2022	✓	✓
Ipanema		2017	03/2019	✓	✓
Laura++	LHCb	2013		✓
Mint2		2016	01/2020	✓
Pawian	BESIII / PANDA	2010	03/2024	✓
PyPWA	JLab	2014	05/2023		✓
Rio++	LHCb	2016		✓
ROOTPWA	COMPASS	2009	06/2020	✓	✓		✓
TARA	Crystal Barrel	1999		✓
TensorFlowAnalysis AmpliTF TFA2	LHCb	2016	10/2023		✓
TF-PWA	BESIII / LHCb	2019	03/2024		✓
ThreeBodyDecays.jl	LHCb	2019	03/2024			✓

General fitter packages

The following packages do not have functionality for amplitude building, but can do fits with high performance. The YAML database for this table can be updated here.

Project	Since	Latest commit	C++	Python	Cuda
GooFit	2013	06/2023	✓	✓	✓
Hydra	2016	12/2023	✓
RooFit	2000		✓
zfit	2018	04/2024		✓

Last update: 06 April 2024

Software development

Tip

Have a look at scikit-hep.org/developer and Towards a HEP Software Training curriculum! For development instructions for the ComPWA organization, see Help developing.

Analysis techniques

Warning

These pages are under development.

Statistics

Warning

These pages are under development.

Fit techniques

Warning

These pages are under development.

Maximum likelihood

Coupled analyses

Visualization

Warning

These pages are under development.

Dalitz plot

Histogram

pwa_pages

import pwa_pages

Helper tools used in the Jupyter notebooks of the PWA Pages.

Submodules and Subpackages

repo

import pwa_pages.repo

Get information about a GitHub and GitLab repositories.

class Repo(name: str, url: str, full_name: str, languages: dict[str, float], first_commit: datetime, latest_commit: datetime)

Bases: object

__eq__(other)

Method generated by attrs for class Repo.

filter_languages(min_percentage: float) → list[str]

first_commit: datetime

full_name: str

property languages: list[str]

latest_commit: datetime

name: str

url: str

get_repo(url: str) → Repo | None

project_inventory

import pwa_pages.project_inventory

Helper tools for writing tables.

class Project(*, name: str, url: str, collaboration: List[str] | str | None = None, languages: List[str] = [], sub_projects: List[SubProject] | None = None, since: int = 0)

Bases: BaseModel

collaboration: List[str] | str | None

languages: List[str]

model_computed_fields: ClassVar[dict[str, ComputedFieldInfo]] = {}

A dictionary of computed field names and their corresponding ComputedFieldInfo objects.

model_config: ClassVar[ConfigDict] = {}

Configuration for the model, should be a dictionary conforming to [ConfigDict][pydantic.config.ConfigDict].

model_fields: ClassVar[dict[str, FieldInfo]] = {'collaboration': FieldInfo(annotation=Union[List[str], str, NoneType], required=False), 'languages': FieldInfo(annotation=List[str], required=False, default=[]), 'name': FieldInfo(annotation=str, required=True), 'since': FieldInfo(annotation=int, required=False, default=0), 'sub_projects': FieldInfo(annotation=Union[List[pwa_pages.project_inventory.SubProject], NoneType], required=False), 'url': FieldInfo(annotation=str, required=True)}

Metadata about the fields defined on the model, mapping of field names to [FieldInfo][pydantic.fields.FieldInfo].

This replaces Model.__fields__ from Pydantic V1.

name: str

since: int

sub_projects: List[SubProject] | None

url: str

class ProjectInventory(*, projects: List[Project], collaborations: Dict[str, str] = {})

Bases: BaseModel

collaborations: Dict[str, str]

model_computed_fields: ClassVar[dict[str, ComputedFieldInfo]] = {}

A dictionary of computed field names and their corresponding ComputedFieldInfo objects.

model_config: ClassVar[ConfigDict] = {}

Configuration for the model, should be a dictionary conforming to [ConfigDict][pydantic.config.ConfigDict].

model_fields: ClassVar[dict[str, FieldInfo]] = {'collaborations': FieldInfo(annotation=Dict[str, str], required=False, default={}), 'projects': FieldInfo(annotation=List[pwa_pages.project_inventory.Project], required=True)}

Metadata about the fields defined on the model, mapping of field names to [FieldInfo][pydantic.fields.FieldInfo].

This replaces Model.__fields__ from Pydantic V1.

projects: List[Project]

class SubProject(*, name: str, url: str)

Bases: BaseModel

model_computed_fields: ClassVar[dict[str, ComputedFieldInfo]] = {}

A dictionary of computed field names and their corresponding ComputedFieldInfo objects.

model_config: ClassVar[ConfigDict] = {}

Configuration for the model, should be a dictionary conforming to [ConfigDict][pydantic.config.ConfigDict].

model_fields: ClassVar[dict[str, FieldInfo]] = {'name': FieldInfo(annotation=str, required=True), 'url': FieldInfo(annotation=str, required=True)}

Metadata about the fields defined on the model, mapping of field names to [FieldInfo][pydantic.fields.FieldInfo].

This replaces Model.__fields__ from Pydantic V1.

name: str

url: str

export_json_schema(argv: Sequence[str] | None = None) → int

fix_html_alignment(src: str) → str

load_yaml(path: Path | str) → dict

to_html_table(inventory: ProjectInventory, selected_languages: List[str], *, fetch: bool = False, min_percentage: float = 2.5, hide_columns: Iterable[str] | None = None) → str

Glossary

Warning

These pages are under development.

The glossary can be used to collect terms that require further explanation or linking to literature, but are too general or specific to deserve an own section on these pages.

asymptotic freedom

The fact that the coupling constant of the strong force increases in strength for lower momentum.

baryon

A particle that consists of three quarks.

color confinement

The fact that quarks only exist in colorless, bound states (quark groups of which the colors add up to ‘white’). See also asymptotic freedom.

coupling constant

Todo

Define coupling constant

damping factor

form factor

Often defined with the symbol \(B_{\alpha_i}\).

formation experiment

Experiments where particles \(A\) and \(B\) are collided, resulting in a reaction:

\[ A + B \to R \to C_1 + \dots + C_n, \]

Here, \(R\) is the resonance in which one is often interested in hadron spectroscopy.

See PDG2020, §Resonances [T13]. Related: production experiment.

hadron

Particles that consist of two quarks (meson) or three quarks (baryon).

hadron spectroscopy

Todo

Define hadron spectroscopy

hyperon

A baryon with at least one strange quark.

K-matrix

A real, symmetric and hermitian operator that is defined from the T-matrix through \(K^{-1} = T^{-1} + iI\). See [Chung et al., 1995, Badalyan et al., 1982] and PDG2020, §Resonances [T13].

lattice QCD

A numerical tool with which to study particle interactions mediated by the strong force at low momentum transfer

meson

A particle that consists of two quarks.

P-vector

First described in [Aitchison, 1972]. See also [Chung et al., 1995] and PDG2020, §Resonances [T13].

production experiment

spectator particle

Experiments of the kind:

\[ \begin{align}\begin{aligned}A + B \to R + S \to [C_1 + \dots + C_n] + S\\Z \to R + S \to [C_1 + \dots + C_n] + S\end{aligned}\end{align} \]

Here, \(R\) is the resonance in which one is often interested in hadron spectroscopy, while \(S\) is a spectator particle. The first case is comparable to a formation experiment, while the second represents a decay process of particle \(Z\).

See PDG2020, §Resonances [T13].

Q-vector

See [Chung et al., 1995] and PDG2020, §Resonances [T13].

Quantum Chromodynamics (QCD)

The theory that describes the strong force on the most fundamental level. See PDG2020, §QCD [T13].

quark

Elementary particle that constitutes hadronic matter.

Quark Constituent Model (QCM)

Model with which to describe and categorize matter constituted of quarks (i.e. hadrons). See [T13] (PDG2020, §Quark Model).

resonance

See [T13] (PDG2020, §Resonances).

S-matrix

scattering operator

See [Martin and Spearman, 1970], Ch.4.

Standard Model

Most fundamental description of matter and forces

strong force

One of the four fundamental forces of the Standard Model

T-matrix

transition operator

The scattering operator can split into a non-interacting component \(I\) (identity operator) and a matrix \(T\) that describes the actual interactions through \(S = I + iT\). See [Martin and Spearman, 1970], Ch.4, and [Chung et al., 1995].

Bibliography

Tip

Download this bibliography as BibTeX here.

References

[T1]

K. Peters. A Primer on Partial Wave Analysis. arXiv, December 2004. arXiv:hep-ph/0412069.

[T2]

R. Kutschke. An Angular Distribution Cookbook. January 1996. home.fnal.gov/~kutschke/Angdist/angdist.ps.

[T3]

J. D. Richman. An Experimenter's Guide to the Helicity Formalism. June 1984. inspirehep.net/literature/202987.

[T4]

H. Chen and R.-G. Ping. Coherent helicity amplitude for sequential decays. Physical Review D, 95(7):076010, April 2017. doi:10.1103/PhysRevD.95.076010.

[T5]

D. Marangotto. Helicity Amplitudes for Generic Multibody Particle Decays Featuring Multiple Decay Chains. Advances in High Energy Physics, 2020:1–15, December 2020. doi:10.1155/2020/6674595.

[T6]

M. Wang et al. A novel method to test particle ordering and final state alignment in helicity formalism. arXiv, December 2020. arXiv:2012.03699.

[T7]

M. Mikhasenko et al. Dalitz-plot decomposition for three-body decays. Physical Review D: Particles and Fields, 101(3):034033, February 2020. doi:10.1103/PhysRevD.101.034033.

[T8]

R. Aaij et al. Observation of 𝐽/𝜓 𝑝 Resonances Consistent with Pentaquark States in 𝛬𝑏⁰ → 𝐽/𝜓𝐾⁻𝑝 Decays. Physical Review Letters, 115(7):072001, August 2015. doi:10.1103/PhysRevLett.115.072001.

[T9]

A. V. Anisovich et al. Moment-operator expansion for the two-meson, two-photon and fermion–antifermion states. Journal of Physics G: Nuclear and Particle Physics, 28(1):15–32, January 2002. doi:10.1088/0954-3899/28/1/302.

[T10]

C. Zemach. Use of Angular-Momentum Tensors. Physical Review, 140(1B):B97–B108, October 1965. doi:10.1103/PhysRev.140.B97.

[T11]

S.-U. Chung et al. Partial wave analysis in 𝐾-matrix formalism. Annalen der Physik, 507(5):404–430, May 1995. doi:10.1002/andp.19955070504.

[T12]

A.M. Badalyan et al. Resonances in coupled channels in nuclear and particle physics. Physics Reports, 82(2):31–177, February 1982. doi:10.1016/0370-1573(82)90014-X.

[T13]

Particle Data Group et al. Review of Particle Physics. Progress of Theoretical and Experimental Physics, 2020(8):083C01, August 2020. doi:10.1093/ptep/ptaa104.

[T14]

I.J.R. Aitchison. The 𝐾-matrix formalism for overlapping resonances. Nuclear Physics A, 189(2):417–423, July 1972. doi:10.1016/0375-9474(72)90305-3.

[T15]

A. D. Martin and T. D. Spearman. Elementary Particle Theory. North-Holland Pub. Co, Amsterdam, 1970. ISBN:978-0-7204-0157-8.

Further reading

• C. Amsler. The Quark Structure of Hadrons: An Introduction to the Phenomenology and Spectroscopy. Number 949 in Lecture Notes in Physics. Springer International Publishing : Imprint: Springer, Cham, 1st ed. 2018 edition, October 2018. ISBN:978-3-319-98527-5. doi:10.1007/978-3-319-98527-5.

• D. M. Asner et al. Dalitz Plot Analysis Formalism. In Review of Particle Physics: Volume I Reviews. January 2006. doi:10.1093/ptep/ptaa104.

• E. Byckling and K. Kajantie. Particle Kinematics. Wiley, London, New York, 1973. ISBN:978-0-471-12885-4.

• S.-U. Chung. Formulas for Angular-Momentum Barrier Factors (Version II). Technical Report, Brookhaven National Laboratory, March 2015. physique.cuso.ch/fileadmin/physique/document/2015_chung_brfactor1.pdf.

• S.-U. Chung. Spin Formalisms (Updated Version). Technical Report, Brookhaven National Laboratory, July 2014. suchung.web.cern.ch/spinfm1.pdf.

• B. Ketzer, B. Grube, and D. Ryabchikov. Light-Meson Spectroscopy with COMPASS. Progress in Particle and Nuclear Physics, December 2019. doi:10.1016/j.ppnp.2020.103755.

• E. Leader. Spin in Particle Physics. Number 15 in Cambridge Monographs on Particle Physics, Nuclear Physics, and Cosmology. Cambridge University Press, Cambridge ; New York, 2001. ISBN:978-0-521-35281-9.

• X.-Y. Li, X.-K. Dong, and H.-J. Jing. Spin-orbit amplitudes for decays with arbitrary spin. Nuclear Physics A, 1040:122761, December 2023. doi:10.1016/j.nuclphysa.2023.122761.

• R. G. Newton. Scattering Theory of Waves and Particles. Springer, New York, 2nd edition edition, 1982. ISBN:978-3-540-10950-1.

• S. Weinberg. The Quantum Theory of Fields, Volume 1: Foundations. Cambridge University Press, Cambridge ; New York, 1995. ISBN:978-0-521-55001-7.