well as virtual contributions from the ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}_{v}^{*}(2007)^{-}$ and $B_{v}^{*0}$
states. In the absence of sufficiently detailed models for the
${{D}^{-}}{{\pi}^{+}}$ S-wave, a quasi-model-independent description based on
spline interpolation is used. Projections of the results of the fit onto the
data are shown in Fig. 9.

Figure 9:  Projections of the data and fit results onto
$m({{D}^{-}}{{\pi}^{+}})_{\rm min}$ for
${{{B}^{+}}}\rightarrow{{D}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ candidates observed by
the LHCb collaboration [94] on (top left) linear and (bottom right)
logarithmic $y$-axis scales. Here, $m({{D}^{-}}{{\pi}^{+}})_{\rm min}$ is the
smaller of the two values of $m({{D}^{-}}{{\pi}^{+}})$ for each
${{{B}^{+}}}\rightarrow{{D}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ candidate. A legend
describing the various contributions is also given. The (bottom right) Argand
diagram of the ${{D}^{-}}{{\pi}^{+}}$ S-wave amplitude shows the expected
phase motion corresponding to the ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}_{0}^{*}(2400)^{0}$ resonance. The
numbered points correspond to the spline knots.

###  8.3 Speed

It is essential for the Laura++ amplitude analysis package to run quickly,
since otherwise the large data samples available in modern experiments can
lead to unmanageably long execution time. In this subsection some performance
benchmarks are provided. More specifically, a selection of the examples that
are provided with the package (several of which are based on analyses
presented in the previous subsection) are run out of the box on the same
machine (an Intel Core i5-3570 3.4${\rm\,GHz}$ quad-core CPU with
8${\rm\,Gbytes}$ of RAM). In each case, timings for both generation of 50 toy
datasets and for fitting those same 50 datasets are provided in Table 3. The
fitting times are averaged over 20 fits with randomised starting parameters.
The scenario demonstrated in each example is as follows:

  * •

GenFit3pi.cc
Example analysis of the symmetric final state
${{{B}^{+}}}\rightarrow{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$, using
LauSimpleFitModel (i.e. not including effects of $C\\!P$ violation). By
default there are 1500 signal events per experiment and the signal isobar
model contains five components: $\rho(770)^{0}$, $f_{0}(980)$, $f_{2}(1270)$,
$\rho(1450)^{0}$, and a nonresonant component. All of the resonance parameters
are fixed in the fit. There is also a background component, which defaults to
being uniformly distributed in the Dalitz plot and consists of 1250 background
events. A version of this example implemented in python, GenFit3pi.py, is also
included in the Laura++ package.

  * •

GenFit3K.cc
Example analysis of the symmetric final state
${{{B}^{+}}}\rightarrow{{K}^{+}}{{K}^{+}}{{K}^{-}}$, using LauSimpleFitModel
(i.e. not including effects of $C\\!P$ violation). By default there are 5000
signal events per experiment and the signal isobar model contains three
components: $\phi(1020)$, $f_{2}^{\prime}(1525)$ and a nonresonant component.
The mass and width of both the $\phi(1020)$ and the $f_{2}^{\prime}(1525)$ are
floating parameters by default and a two-stage fit is employed. No background
contributions are included. To demonstrate further the impact on the
performance of floating resonance parameters, this example is run with:

    * –

no floating resonance parameters,

    * –

only the mass of the $\phi(1020)$ floating,

    * –

the mass and width of the $\phi(1020)$ floating,

    * –

the mass and width of both the $\phi(1020)$ and $f^{\prime}_{2}(1525)$
floating,

and the timings for each of those scenarios are provided in Table 4.

  * •

GenFit3KS.cc
Example analysis of the fully-symmetric final state
${{B}^{0}}\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{K}^{0}_{\rm\scriptscriptstyle S}}{{K}^{0}_{\rm\scriptscriptstyle S}}$,
using LauSimpleFitModel (i.e. not including effects of $C\\!P$ violation). By
default there are 10000 signal events per experiment and the signal isobar
model contains four components: $f_{0}(980)$, $f_{0}(1710)$, $f_{2}(2010)$ and
${\chi_{{c}0}}$. All of the resonance parameters are fixed in the fit. No
background contributions are included.

  * •

GenFitDs2KKpi.cc
Example analysis of the decay
${{D}^{+}_{s}}\rightarrow{{\pi}^{+}}{{K}^{+}}{{K}^{-}}$, using
LauSimpleFitModel (i.e. not including effects of $C\\!P$ violation). By
default there are 10000 signal events and the signal isobar model contains
three components: $\phi(1020)$, ${{K}^{*}}(892)^{0}$ and a nonresonant
component. All of the resonance parameters are fixed in the fit. No background
contributions are included.

  * •

GenFitEFKLLM.cc
Example analysis of the decay ${{B}^{0}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{K}^{+}}{{\pi}^{-}}$, using
LauSimpleFitModel (i.e. not including effects of $C\\!P$ violation). By
default there are 5000 signal events per experiment and the signal isobar
model contains two components, which are the two parts of the EFKLLM model for
the ${{K}^{+}}{{\pi}^{-}}$ S-wave (see Eq. (52) in App. A). This example
mainly serves to demonstrate how to use this particular model.

  * •

GenFitBelleCPKpipi.cc
Example analysis of the decay
${{{B}^{+}}}\rightarrow{{K}^{+}}{{\pi}^{+}}{{\pi}^{-}}$, using LauCPFitModel,
which includes effects from $C\\!P$ violation. By default there are 5000
signal events per experiment and the signal isobar model contains seven
components: ${{K}^{*}}(892)^{0}$, ${K}^{*}_{0}(1430)^{0}$, $\rho(770)^{0}$,
$f_{0}(980)$, $\chi_{{c}0}$, and two nonresonant components. The slope of the
${{\pi}^{+}}{{\pi}^{-}}$ exponential nonresonant model and the effective range
and scattering length of the ${{K}^{+}}{{\pi}^{-}}$ LASS nonresonant component
are floating parameters by default. The isobar coefficients use the
LauBelleCPCoeffSet form to parameterise the $C\\!P$ violation (see Table 1)
and a two-stage fit is employed. No background contributions are included.

  * •

GenFitKpipi.cc
Example analysis of the decay
${{{B}^{+}}}\rightarrow{{K}^{+}}{{\pi}^{+}}{{\pi}^{-}}$, using LauCPFitModel,
which includes effects from $C\\!P$ violation. This example is based closely
on the BaBar analysis from Ref. [36]. By default there are 4585 signal events
per experiment and the signal isobar model contains seven components:
${{K}^{*}}(892)^{0}$, ${K}^{*}_{0}(1430)^{0}$, ${K}^{*}_{2}(1430)^{0}$,
$\rho(770)^{0}$, $\omega(782)$, $f_{0}(980)$, $f_{2}(1270)$,
$f^{\prime}_{0}(1300)$, $\chi_{{c}0}$, and a nonresonant component. All of the
resonance parameters are fixed in the fit. The isobar coefficients use the
LauCartesianCPCoeffSet form to parameterise the $C\\!P$ violation (see Table
1) and a two-stage fit is employed. Background contributions for combinatorial
candidates and those from other $B$ decays are included.

  * •

KMatrixExample.cc and KMatrixDto3pi.cc
These examples demonstrate how to use the K-matrix description of the S-wave.
The first scenario is for the
${{B}^{0}}\rightarrow{{\pi}^{+}}{{\pi}^{-}}{{K}^{0}_{\rm\scriptscriptstyle
S}}$ DP but the only terms included in the signal model are $\rho(770)^{0}$,
${{K}^{*}}(892)^{+}$ and a K-matrix component for the ${{\pi}^{+}}{{\pi}^{-}}$
S-wave. By default there are 5000 signal events per experiment. The second
scenario is for the symmetric
${{D}^{+}}\rightarrow{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ DP and includes in the
signal model the $\rho(770)^{0}$, $f_{2}(1270)$, and the K-matrix. By default
there are 20000 signal events per experiment. In both scenarios there are no
background contributions and all resonance parameters are fixed in the fit.

  * •

GenFitNoDP.cc and GenFitNoDPMultiDim.cc
These examples demonstrate how to use the package to perform fits to variables
other than the DP. The first case fits only a single variable (the invariant
mass of the ${{B}^{+}}$ candidate), while the second performs a 2D fit. In
both cases there are 5000 signal events. In the first case there are two
background components included, which have yields of 7000 and 3000 events. In
the second case there is a single background component that has a yield of
7000 events. All yield parameters are floating, along with some of the shape
parameters of the PDFs. Asymmetric uncertainties are evaluated in the first
case.

  * •

runMasterSlave.sh, Master.cc and Slave.cc
This example demonstrates how to perform a simultaneous fit to two categories
of data in the decay channel
${{B}^{0}}\rightarrow{{\pi}^{0}}{{\pi}^{0}}{{K}^{0}_{\rm\scriptscriptstyle
S}}$. The data are split based on the reconstruction category of the
${K}^{0}_{\rm\scriptscriptstyle S}$ candidate. In one category there are 500
signal and 1200 background events, while in the second there are 750 signal
and 2500 background events. The common signal DP model contains contributions
from the $f_{0}(980)$, $f_{2}(1270)$, ${{K}^{*}}(892)^{0}$, and
${K}^{*}_{0}(1430)^{0}$ resonances. The mass of the ${{K}^{*}}(892)^{0}$
resonance is a floating parameter of the fit and a two-stage fit is employed.
The background component is distributed uniformly in the DP by default. To
demonstrate further the impact on the performance of simultaneous fitting,
this example is run with:

    * –

the Master and the two Slave processes all running on the same host,

    * –

the Master and the two Slave processes running on three separate hosts,

and the timings for each of these scenarios are provided in Table 5. To run
this particular example, the hosts have dual Intel Xeon E5-2620v2
2.1${\rm\,GHz}$ 6-core CPUs and 64${\rm\,Gbytes}$ of RAM and are connected via
10${\rm\,Gbits/s}$ ethernet.

Table 3:  Speed of execution of the examples provided with the Laura++ package. The times given are the total to generate/fit 50 toy experiments. The examples differ, as explained in detail in the text, not only through the complexity of the model but also through the number of signal and background events in each pseudoexperiment.  Example | Execution time of | Execution
---|---|---
| toy generation | time of fit
GenFit3pi.cc | $7\rm{\,s}$ | $9\rm{\,s}$
GenFit3K.cc | $291\rm{\,s}$ | $2513\rm{\,s}$
GenFit3KS.cc | $78\rm{\,s}$ | $20\rm{\,s}$
GenFitDs2KKpi.cc | $47\rm{\,s}$ | $9\rm{\,s}$
GenFitEFKLLM.cc | $106\rm{\,s}$ | $3\rm{\,s}$
GenFitBelleCPKpipi.cc | $14\rm{\,s}$ | $18909\rm{\,s}$
GenFitKpipi.cc | $36\rm{\,s}$ | $1073\rm{\,s}$
KMatrixExample.cc | $194\rm{\,s}$ | $217\rm{\,s}$
KMatrixDto3pi.cc | $46\rm{\,s}$ | $251\rm{\,s}$
GenFitNoDP.cc | $2\rm{\,s}$ | $11\rm{\,s}$
GenFitNoDPMultiDim.cc | $3\rm{\,s}$ | $6\rm{\,s}$
Table 4:  Speed of execution of the GenFit3K.cc example provided with the Laura++ package with different sets of floating resonance parameters. The times given are the total to fit 50 toy experiments.  Floated parameters | Execution time of fit
---|---
None | $6\rm{\,s}$
Mass of $\phi(1020)$ | $702\rm{\,s}$
Mass and width of $\phi(1020)$ | $1090\rm{\,s}$
Mass and width of $\phi(1020)$ and $f^{\prime}_{2}(1525)$ | $2513\rm{\,s}$
Table 5:  Speed of execution of the fitting portion of the runMasterSlave.sh example provided with the Laura++ package with the Master and two Slave processes running either on the same or separate hosts. The times given are the total to fit 50 toy experiments.  Host setup | Execution time of fit
---|---
Master and two Slave processes on single host | $1972\rm{\,s}$
Master and two Slave processes on three separate hosts | $2151\rm{\,s}$

##  9 Future developments

There are several features that would help to improve and extend the
functionality of Laura++ in the future. Some of these potential future
developments are described below. In addition, it is anticipated that the
Laura++ code will be continually updated to take advantage of features in the
latest C++ standards.

###  9.1 Plotting the amplitude

Currently the user can easily make plots of the DP distribution or its
projections from the result of the fit (see Sec. 7.2). It can also be of
interest to draw amplitude-level quantities, for example to show the phase
variation with two-body invariant mass of a particular partial wave. While
this is currently possible in Laura++ (see Fig. 9 for an example), it would be
desirable to provide an interface to simplify matters for the user.

###  9.2 Decay-time-dependent fits

As mentioned in Sec. 1, the evolution of DP structure with decay time of a $B$
or $D$ meson can be of interest to study $C\\!P$ violation effects. For
example, studies of ${{B}^{0}}\rightarrow{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{0}}$
and $D{{\pi}^{+}}{{\pi}^{-}}$ decays are of interest to measure the angles
$\alpha$ and $\beta$ of the CKM Unitarity Triangle with low theoretical
uncertainty. Studies of ${{B}^{0}}\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}$ and ${{K}^{0}_{\rm\scriptscriptstyle
S}}{{K}^{+}}{{K}^{-}}$ decays enable determinations of $\beta$ that are not as
clean, but are potentially sensitive to effects of physics beyond the Standard
Model. Also, the ${{D}^{0}}\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}$ channel appears the most sensitive to possible
$C\\!P$ violation effects associated with charm mixing. Many other channels
are potentially of interest.

The Laura++ package has been used for decay-time-dependent DP analysis, for
example, in Refs. [44, 50]. However, this implementation was experiment-
specific and therefore unsuitable for use in the more general case. Further
development is necessary to establish a model that can deal generically with
issues such as flavour tagging, decay time resolution and acceptance as well
as production and detection asymmetries.

###  9.3 Alternative handling of resolution effects

The treatment of resolution discussed in Sec. 3 and Sec. 5.2 is completely
general, but is likely to be inefficient in certain cases. For example, in the
${{{B}^{+}}}\rightarrow{{K}^{+}}{{K}^{-}}{{K}^{+}}$ decay, there are narrow
contributions from the $\phi(1020)$ and $\chi_{c0}$ resonances in specific
regions of the DP, for which resolution effects may be important. The rest of
the DP is populated with broad or nonresonant states so that resolution can be
safely neglected. An approach in which Gaussian (or non-Gaussian) smearing of
Dalitz plot position can be used in only selected regions of the phase space
may be useful. In such a case it will be necessary to take care to avoid
issues due to edge effects, including the possibility of an event being
smeared to positions outside the kinematic boundary of the DP.

###  9.4 Non-zero spins

There are many interesting three-body decays that include particles with non-
zero spin in the initial and final states, which cannot be fitted using the
current version of Laura++. Large samples of $b$-baryons are available in the
LHCb data samples, which have baryons in the initial and final states, for
example ${{\mathchar 28931\relax}^{0}_{b}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{p}{{\pi}^{-}}$ and ${{\mathchar
28932\relax}^{-}_{b}}\rightarrow{p}{{K}^{-}}{{K}^{-}}$ decays. Decays of
${B}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}hh^{\prime}$,
where $h$ and $h^{\prime}$ are charged pions or kaons, are also interesting
and contain the spin-1 ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$
particle. Another group of decay modes, ${B}\rightarrow{{D}^{*}}hh^{\prime}$,
which include the vector ${D}^{*}$ meson, are interesting for spectroscopy of
$D^{**}$ and $D_{s}^{**}$ states.

To enable the Laura++ package to cope with the decays above, several things
would require updating or changing. Firstly, the phase space of the problem is
expanded from two to five dimensions where additional degrees of freedom would
be some angular variables. The helicity of particles of non-zero spin, like
the ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$, must also be considered,
requiring a sum over the helicity states. For the initial and final state this
sum must be incoherent and for the intermediate states a coherent sum is
required. The Zemach spin terms currently implemented must also be changed for
the non-zero spin particles. One potential way to calculate the spin factors
would be to interface Laura++ with the qft++ package [95], which allows the
spin terms to be calculated for any process.

###  9.5 Genetic algorithms

To ensure that the global minimum in the NLL has been found, Laura++ allows
many fits to be performed with randomised starting values for the floated fit
parameters. This method works well, but can become time consuming if the
global minimum is not found in a high proportion of fit attempts. Genetic
algorithms could provide a method to find sensible starting values for the
floated parameters such that the global minimum is always found. This could be
achieved by interfacing to existing software packages with implementations of
genetic algorithms.

###  9.6 Interface to EvtGen

The EvtGen package [96] is designed to predominantly simulate the decays of
$b$\- and $c$-hadrons. It is important in experimental particle physics to
produce simulated data samples that are realistic and match the true data
samples. Typically in three-body decays the simulated events are produced flat
in the DP, without resonant contributions. It would be beneficial if Laura++
could be used directly to provide realistic DP distributions for simulated
samples of three-body decays.

##  10 Summary

The Laura++ package provides a flexible and optimised framework for Dalitz-
plot analysis. While it can be used for the decay of any stable spin-zero
particle to any final state containing three stable spin-zero particles, it
has until now been most widely used for decays of $B$ or $D$ mesons to three
pseudoscalars. Features included in Laura++ allow the physics of such decays
to be probed in detail, including studies of the resonances appearing in the
contributing partial waves, and investigations of $C\\!P$-violating effects.
Use of the Laura++ software has resulted in numerous publications to date,
with many more expected in future with the increasingly large data samples
available at LHCb, Belle II and other experiments.

##  11 Acknowledgements

The Laura++ package has been under development for many years, with support
principally from the Science and Technology Facilities Council (United
Kingdom) and by the European Research Council under FP7. Individual authors
have received support from Marie Skłodowska-Curie Actions and from the
University of Warwick. We thank the Laura++ user community for feedback, bug
reports, testing and other contributions to improve the package, in particular
Louis Henry, Adlene Hicheur, Patricia Magalhães, Jussara Miranda, Sian Morgan
and Charlotte Wallace. We also acknowledge productive discussions regarding
aspects of Dalitz plot analysis with many members of the BaBar, Belle and LHCb
collaborations, notably Eli Ben-Haim, Alex Bondar, Jeremy Dalseno, Bill
Dunwoodie, Brian Meadows and Anton Poluektov. Similarly, instructive
communications with members of the theory community have been of great
benefit; in particular we thank Vladimir Anisovich, David Bugg, Leonard
Lesniak, Benoit Loiseau, Mike Pennington, Alessandro Pilloni, Andrey Sarantsev
and Adam Szczepaniak. Furthermore, we thank Bertram Kopf and Matthias Steinke
for providing access to the PAWIAN software (PANDA collaboration) for cross-
checking the $K$-matrix implementation. Finally, we acknowledge useful input
on technical features of the package from René Brun and Bertrand Echenard.

Appendices

##  Appendix A Formulae for available lineshapes

This section presents the complete formulae for all resonance shapes
implemented in Laura++. Table A1 gives the list of shapes, together with the
corresponding LauResonanceModel enumeration integer that is required to
specify the resonance type for the LauIsobarDynamics addResonance function, as
well as the equation number(s) that provide the expression for the resonance
mass term $R(m)$ used in Eq. (4). The K-matrix shape is particularly
complicated and is therefore described in a dedicated subsection.

Table A1:  List of the allowed resonance shape types. The LauResonanceModel
case-sensitive enumeration in the LauAbsResonance abstract class specifies the
integer that selects the resonance type for the addResonance function. For
example, the simple Breit–Wigner integer type is LauAbsResonance::BW.

Shape name | Enumeration |  $R(m)$ Eq.
---|---|---
Simple Breit–Wigner | BW | (37)
Relativistic Breit–Wigner (RBW) | RelBW | (6)
Modified Breit–Wigner from Gounaris–Sakurai (GS) | GS | (39)
Flatté or coupled-channel Breit–Wigner | Flatte | (44)
$\sigma$ or $f_{0}(500)$ | Sigma | (47)
$\kappa$ or low-mass $K\pi$ scalar | Kappa | (47)
Low-mass $D\pi$ scalar | Dabba | (49)
LASS $K\pi$ S-wave | LASS | (50)
Resonant part of $K\pi$ LASS | LASS_BW | (50) (2nd term)
Non-resonant part of $K\pi$ LASS | LASS_NR | (50) (1st term)
Form-factor description of the $K\pi$ S-wave | EFKLLM | (52)
S-wave using $K$-matrix and $P$-vector | KMatrix | (70)–(76)
Uniform non-resonant (NR) | FlatNR | $R(m)\equiv 1$
Theoretical NR model | NRModel | (53)
Empirical NR exponential | BelleNR | (55)
Empirical NR power-law | PowerLawNR | (56)
Empirical NR exponential for symmetrised DPs | BelleSymNR | (57)
Empirical NR Taylor expansion for symmetrised DPs | TaylorNR | (58)
Empirical NR polynomial | PolNR | (59)
Model-independent partial wave (magnitude & phase) | MIPW_MagPhase | (61)
Model-independent partial wave (real & imaginary) | MIPW_RealImag | (61)
Incoherent Gaussian shape | GaussIncoh | (62)
$\rho-\omega$ mixing: GS for $\rho$, RBW for $\omega$ | RhoOmegaMix_GS | (60)
neglecting $\Delta^{2}$ denominator term | RhoOmegaMix_GS_1 | (60)
$\rho-\omega$ mixing: RBW for both $\rho$ and $\omega$ | RhoOmegaMix_RBW | (60)
neglecting $\Delta^{2}$ denominator term | RhoOmegaMix_RBW_1 | (60)

The simple Breit–Wigner lineshape is given by

$R(m)=\frac{1}{m-m_{0}-\frac{i}{2}\Gamma_{0}}\equiv\frac{(m-m_{0})+\frac{i}{2}\Gamma_{0}}{(m-m_{0})^{2}+\frac{\Gamma^{2}_{0}}{4}}\,,$
(37)

where $m_{0}$ is the pole mass and $\Gamma_{0}$ is the resonance width. The
more commonly used relativistic Breit–Wigner lineshape is described in Sec.
2.1. We note here that the relativistic Breit–Wigner lineshape can also
describe so-called virtual contributions, from resonances with masses outside
the kinematically accessible region of the Dalitz plot, with one modification:
in the calculation of the momenta, the mass $m_{0}$ is set to a value
$m_{0}^{\rm{eff}}$ within the kinematically allowed range. This is
accomplished with the ad-hoc formula

$m_{0}^{\rm{eff}}(m_{0})=m^{\rm{min}}+\frac{1}{2}(m^{\rm{max}}-m^{\rm{min}})\left[1+\tanh\left(\frac{m_{0}-\frac{m^{\rm{min}}+m^{\rm{max}}}{2}}{m^{\rm{max}}-m^{\rm{min}}}\right)\right]\,,$
(38)

where $m^{\rm{max}}$ and $m^{\rm{min}}$ are the upper and lower limits of the
kinematically allowed mass range. For virtual contributions, only the tail of
the RBW function enters the Dalitz plot.

The Gounaris–Sakurai form of the Breit–Wigner lineshape [97] is usually used
as an alternative model for the $\rho$ resonance. It is given by

$R(m)=\frac{1+D\cdot\Gamma_{0}/m_{0}}{(m_{0}^{2}-m^{2})+f(m)-i\,m_{0}\Gamma(m)}\,,$
(39)

where

$f(m)=\Gamma_{0}\,\frac{m_{0}^{2}}{q_{0}^{3}}\,\left[\;q^{2}\left[h(m)-h(m_{0})\right]+\left(\,m_{0}^{2}-m^{2}\,\right)\,q^{2}_{0}\,\frac{dh}{ds}\bigg{|}_{m_{0}}\;\right]\,,$
(40)

$q$ is the magnitude of the momentum of one of the daughter particles in the
resonance rest-frame,

$h(m)=\frac{2}{\pi}\,\frac{q}{m}\,\ln\left(\frac{m+2q}{2m_{\pi}}\right)\,,$
(41)

and

$\frac{dh}{ds}\bigg{|}_{m_{0}}=h(m_{0})\left[(8q_{0}^{2})^{-1}-(2m_{0}^{2})^{-1}\right]\,+\,(2\pi
m_{0}^{2})^{-1}\,.$ (42)

The constant parameter $D$ is given by [97]

$D=\frac{3}{\pi}\frac{m_{\pi}^{2}}{q_{0}^{2}}\,\ln\left(\frac{m_{0}+2q_{0}}{2m_{\pi}}\right)+\frac{m_{0}}{2\pi\,q_{0}}-\frac{m_{\pi}^{2}m_{0}}{\pi\,q_{0}^{3}}\,.$
(43)

The Flatté [98], or coupled two-channel Breit–Wigner, lineshape is usually
used to model $f_{0}(980)$, $K^{*}_{0}(1430)$ and $a_{0}(980)$ states. It was
originally introduced in the form

$R(m)=\frac{1}{(m_{0}^{2}-m^{2})-im_{0}[\Gamma_{1}(m)+\Gamma_{2}(m)]}\,.$ (44)

The decay widths in the two systems are usually represented by products of
couplings and dimensionless phase-space factors:

$\Gamma_{1}(m)=g_{1}f_{A}\left(\frac{1}{3}\sqrt{1-(m_{1,1}+m_{1,2})^{2}/m^{2}}+\frac{2}{3}\sqrt{1-(m_{1,3}+m_{1,4})^{2}/m^{2}}\right)\,,$
(45)
$\Gamma_{2}(m)=g_{2}f_{A}\left(\frac{1}{2}\sqrt{1-(m_{2,1}+m_{2,2})^{2}/m^{2}}+\frac{1}{2}\sqrt{1-(m_{2,3}+m_{2,4})^{2}/m^{2}}\right)\,.$
(46)

Here the fractional coefficients come from isospin conservation, $m_{i,j}$
denotes the invariant mass of the daughter particle $j$ (1–4) in channel $i$
(1–2), and $g_{1}$ and $g_{2}$ are coupling constants whose values are assumed
to be those provided in Table A2. The Clebsch-Gordan coefficients in Eqs. 45
and 46 are not guaranteed to be correct for every possible resonance that
could be modelled with the Flatté lineshape, but are appropriate for every
case considered in Table A2. The expressions for the widths are continued
analytically ($\Gamma\rightarrow i|\Gamma|$) when $m$ is below any of the
specific channel thresholds, contributing to the real part of the amplitude,
while the Adler-zero term $f_{A}=(m^{2}-s_{A})/(m^{2}_{0}-s_{A})$ can be used
to suppress false kinematic singularities when $m$ goes below threshold [99]
(otherwise $f_{A}$ is set to unity).

Variants of the Flatté lineshape have been used in the literature. In some
cases, e.g. Refs. [100, 101], the constant $m_{0}$ that multiplies the widths
in the denominator of Eq. (44) is absorbed into the couplings. As a
consequence the couplings have dimensions of mass-squared, and are sometimes
denoted as $g_{i}$ [100] and sometimes as $g_{i}^{2}$ [101]. In Table A2 all
values have been converted to be consistent with Eqs. (44)–(46). In Laura++ it
is only possible to use the Flatté lineshape for the systems specified in
Table A2. At construction time the resonance name is checked and the
corresponding parameter values are set; these can be modified by the user if
desired.

Table A2:  The four daughter particles used for each channel term $m_{ij}$, as
well as the coupling ($g_{1}$, $g_{2}$) and Adler-zero ($s_{A}$) constants for
the Flatté lineshapes. Units of $\mathrm{\,Ge\kern-1.00006ptV}$ for $g_{1,2}$
(or $\mathrm{\,Ge\kern-1.00006ptV}^{2}$ for $m_{0}g_{1,2}$ when taken from
Refs. [100, 101]) and ${\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ for
$s_{A}$ are implied.

Resonance | Channel 1 | Channel 2 |  $g_{1}$ or $m_{0}g_{1}$ |  $g_{2}$ or $m_{0}g_{2}$ | $s_{A}$ | Reference
---|---|---|---|---|---|---
$f_{0}(980)$ |  ${\pi}^{0}$,${\pi}^{0}$,${\pi}^{\pm}$,${\pi}^{\pm}$ |  ${K}^{\pm}$,${K}^{\pm}$,${K}^{0}$,${K}^{0}$ | 0.165 | $4.21g_{1}$ | — | [100]
$K^{*}_{0}(1430)^{0}$ |  ${K}^{0}$,${\pi}^{0}$,${K}^{\pm}$,${\pi}^{\pm}$ |  ${K}^{0}$,$\eta^{\prime}$,${K}^{0}$,$\eta^{\prime}$ | 0.304 | 0.380 | 0.234 | [99]
$K^{*}_{0}(1430)^{\pm}$ |  ${K}^{\pm}$,${\pi}^{0}$,${K}^{0}$,${\pi}^{\pm}$ |  ${K}^{\pm}$,$\eta^{\prime}$,${K}^{\pm}$,$\eta^{\prime}$ | 0.304 | 0.380 | 0.234 | [99]
$a_{0}(980)^{0}$ |  $\eta$,${\pi}^{0}$,$\eta$,${\pi}^{0}$ |  ${K}^{\pm}$,${K}^{\pm}$,${K}^{0}$,${K}^{0}$ | 0.105 | $1.03g_{1}$ | — | [101]
$a_{0}(980)^{\pm}$ |  $\eta$,${\pi}^{\pm}$,$\eta$,${\pi}^{\pm}$ |  ${K}^{\pm}$,${K}^{0}$,${K}^{\pm}$,${K}^{0}$ | 0.105 | $1.03g_{1}$ | — | [101]

The $\sigma$ or $f_{0}(500)\rightarrow\pi\pi$ and $\kappa$ or
$K^{*}_{0}(800)\rightarrow K\pi$ low-mass scalar resonances can be described
using the form

$R(m)=\frac{1}{M^{2}-s-iM\Gamma(s)}\,,$ (47)

where $M$ is the mass where the phase shift goes through 90∘ for real $s\equiv
m^{2}$, and the width

$\Gamma(s)=\sqrt{1-(m_{1}+m_{2})^{2}/s}\left(\frac{s-s_{A}}{M^{2}-s_{A}}\right)(b_{1}+b_{2}s)e^{-(s-M^{2})/A}\,,$
(48)

where the square-root term is the phase space factor, which requires the
invariant masses of the daughter particles $m_{1}$ and $m_{2}$, $s_{A}$ is the
Adler-zero constant, while $b_{1}$, $b_{2}$ and $A$ are additional constants
[99]. Table A3 gives the default values of the parameters.

Table A3: Default values of the parameters for the $\sigma$ and $\kappa$ lineshapes based on BES data [99]. Resonance |  $M$ (${\mathrm{Ge\kern-1.00006ptV\\!/}c^{2}}$) |  $b_{1}$ (${\mathrm{Ge\kern-1.00006ptV\\!/}c^{2}}$) |  $b_{2}$ (${\mathrm{Ge\kern-1.00006ptV\\!/}c^{2}}$) |  $A$ (${\mathrm{Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$) | $s_{A}$
---|---|---|---|---|---
$\sigma$ | 0.9264 | 0.5843 | 1.6663 | 1.082 | $0.5m^{2}_{\pi}$
$\kappa$ | 3.3 | 24.49 | 0.0 | 2.5 | $m^{2}_{K}-0.5m^{2}_{\pi}$

The $D\pi$ S-wave can be parameterised using the form provided by Bugg [102],
who labels the pole state as “dabba”:

$R(m)=\frac{1}{1-\beta(m^{2}-s_{0})-ib\rho(m^{2}-s_{A})e^{-\alpha(m^{2}-s_{0})}}\,,$
(49)

where $\rho$ is the Lorentz invariant phase space factor
$\sqrt{1-s_{0}/m^{2}}$, $s_{0}$ is the square of the sum of the invariant
masses of the $D$ ($m_{D}$) and $\pi$ ($m_{\pi}$) daughters, $s_{A}$ is the
Adler-zero term $m^{2}_{D}-0.5m^{2}_{\pi}$ that comes from chiral symmetry
breaking [103], while $b$ = 24.49, $\alpha$ = 0.1 and $\beta$ = 0.1.

The RBW function is a very good approximation for narrow resonances well
separated from any other resonant or nonresonant contribution in the same
partial wave. This approximation is known to be invalid in the $K\pi$ S-wave,
since the ${{\kern 1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*}_{0}}(1430)$
resonance interferes strongly with a slowly varying nonresonant term [104].
The so-called LASS lineshape [105] has been developed to combine these
amplitudes,

$\displaystyle R(m)$ $\displaystyle=$
$\displaystyle\frac{m}{q\cot{\delta_{B}}-iq}+e^{2i\delta_{B}}\frac{m_{0}\Gamma_{0}\frac{m_{0}}{q_{0}}}{(m_{0}^{2}-m^{2})-im_{0}\Gamma_{0}\frac{q}{m}\frac{m_{0}}{q_{0}}}\,,$
(50) $\displaystyle{\rm with}\ \cot{\delta_{B}}$ $\displaystyle=$
$\displaystyle\frac{1}{aq}+\frac{1}{2}rq\,,$ (51)

where $m_{0}$ and $\Gamma_{0}$ are now the pole mass and width of the ${{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*}_{0}}(1430)$, and $a$ and $r$ are
parameters that describe the shape. Most implementations of the LASS shape in
amplitude analyses of $B$ meson decays [106, 34] apply a cut-off to the slowly
varying part close to the charm hadron mass ($\sim
1.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$).

An alternative representation of the $K\pi$ S-wave amplitude can be made using
the EFKLLM model described in Ref. [107] (the acronym comes from the surnames
of the authors of that paper), which uses a tabulated form-factor
$f_{0}^{K\pi}(m^{2})$ that is interpolated using two splines (one each for the
magnitude and phase parts), multiplied by a scaling power-law mass-dependence
$m^{\ell}$, leading to

$R(m)=f_{0}^{K\pi}(m^{2})\cdot m^{\ell}\,,$ (52)

where suggested values for the exponent $\ell$ are zero for $\kappa$ (this is
also the default value) and $-2$ for ${{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*}_{0}}(1430)$.

Because of the large phase-space available in three-body $B$ meson decays, it
is possible to have nonresonant amplitudes (i.e. contributions that are not
associated with any known resonance, including virtual states) that are not
constant across the Dalitz plot. One possible parameterisation, based on
theoretical considerations of final-state interactions in
${{B}^{\pm}}\rightarrow{{K}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}$ decays [108], uses
the form

$R(m)=\left[m_{13}m_{23}f_{1}(m^{2}_{13})f_{2}(m^{2}_{23})e^{-d_{0}m^{4}_{13}m^{4}_{23}}\right]^{\frac{1}{2}}\,,$
(53)

where

$f_{j}(m^{2})=\frac{1}{1+e^{a_{j}(m^{2}-b_{j})}}\,,$ (54)

with the constant parameters $d_{0}=1.3232\times 10^{-3}$ GeV-8, $a_{1}=0.65$
GeV-2, $b_{1}=18$ GeV2, $a_{2}=0.55$ GeV-2 and $b_{2}=15$ GeV2 in natural
units.

There are several empirical methods that can be used to model the nonresonant
contributions. One is to use an exponential form factor [32]

$R(m)=e^{-\alpha m^{2}}\,,$ (55)

while another form is simply a power-law distribution

$R(m)=m^{-2\alpha}\,,$ (56)

where in both cases $\alpha$ is a parameter that must be determined from the
data. For symmetric DPs, the exponential form is modified to

$R(m)=e^{-\alpha m^{2}_{13}}+e^{-\alpha m^{2}_{23}}\,,$ (57)

while a Taylor expansion up to first order can also be used:

$R(m)=1+\frac{\alpha(m^{2}_{13}+m^{2}_{23})}{m^{2}_{P}}\,,$ (58)

where $m_{P}$ is the invariant mass of the parent $P$. Another possible
description for non-symmetric DPs is based on the polynomial function [109]

$R(m)=\left[m-\frac{1}{2}\left(m_{P}+\frac{1}{3}(m_{1}+m_{2}+m_{3})\right)\right]^{n}\,,$
(59)

where $m_{k}$ is the invariant mass of daughter particle $k$ and $n$ is the
integer order equal to 0, 1 or 2; a quadratic dependence in $m$ can be
constructed by using up to three polynomial $R(m)$ terms, one for each order
along with their individual $c_{j}$ amplitude coefficients.

We next come to the model that implements the $\rho-\omega$ mass mixing
amplitude described in Ref. [110]

$A_{\rho-\omega}=A_{\rho}\left[\frac{1+A_{\omega}\Delta|B|e^{i\phi_{B}}}{1-\Delta^{2}A_{\rho}A_{\omega}}\right],$
(60)

where $A_{\rho}$ is the $\rho$ lineshape, $A_{\omega}$ is the $\omega$
lineshape, $|B|$ and $\phi_{B}$ are the relative magnitude and phase of the
production amplitudes of $\rho$ and $\omega$, and
$\Delta\equiv\delta(m_{\rho}+m_{\omega})$, where $\delta$ governs the
electromagnetic mixing of $\rho$ and $\omega$ (with pole masses $m_{\rho}$ and
$m_{\omega}$). Here, the amplitude $A_{\omega}$ is always given by the RBW
form of Eq. (6), while the amplitude $A_{\rho}$ can either be represented
using the Gounaris–Sakurai formula given in Eq. (39) or the RBW form; the
required shape is selected using either the RhoOmegaMix_GS or RhoOmegaMix_RBW
enumeration integer labels given in Table A1. When ignoring the small
$\Delta^{2}$ term in the denominator of Eq. (60), this is equivalent to the
parameterisation described in Ref. [111]; this option can be chosen using
either the RhoOmegaMix_GS_1 or RhoOmegaMix_RBW_1 enumeration labels, depending
on what lineshape is needed for the $\rho$ resonance. From SU(3) symmetry, the
$\rho$ and $\omega$ are expected to be produced coherently, which gives the
prediction $|B|e^{i\phi_{B}}=1$. To avoid introducing any theoretical
assumptions, however, it is advisable that these parameters are left floating
in the fit. In general $\delta$ is complex, although the imaginary part is
small so this is neglected. The theory prediction for $\delta$ is around
$2\mathrm{\,Me\kern-1.00006ptV}$ [112], and previous analyses have found
$|\delta|$ to be $2.15\pm 0.35\mathrm{\,Me\kern-1.00006ptV}$ [110] and
$1.57\pm 0.16\mathrm{\,Me\kern-1.00006ptV}$, and $\rm{arg}~{}\delta$ to be
$0.22\pm 0.06$ [111]. These parameters can be also be floated in the fit.

If the dynamical structure of the DP cannot be described by any of the forms
given above, then the LauModIndPartWave class can be used to define a model-
independent partial wave component, using splines to produce an amplitude. It
requires a series of mass points called “‘knots”, in ascending order, which
sets the magnitude $r(m)$ and phase $\phi(m)$ at each point $m$ that can be
floated when fitted to data:

$R(m)=r(m)\left[\cos\phi(m)+i\sin\phi(m)\right]\,.$ (61)

The amplitude for points between knots is found using cubic spline
interpolation, and is fixed to zero at the kinematic boundary. There are two
implementations for representing the amplitudes: one uses magnitudes and
phases (MIPW_MagPhase), while the other uses real and imaginary parts
(MIPW_RealImag).

Finally, the incoherent Gaussian lineshape form is given by

$R(m)=e^{-(m-m_{0})^{2}/2G^{2}_{0}}\,,$ (62)

where $m_{0}$ is the mass peak and $G_{0}$ is the width. This can be used to
parameterise the amplitude for a very narrow resonance, where the measurement
of the width is dominated by experimental resolution effects, producing a
lineshape that is indistinguishable from a Gaussian distribution. The narrow
width ensures that the resonance will not interfere significantly with other
resonances in the DP, i.e. it will be incoherent. This form could also be used
to parameterise narrow background resonance contributions that would otherwise
be excluded with mass vetoes, such as the ${{D}^{0}}$ meson decay to
${{K}^{-}}{{\pi}^{+}}$ in the charmless mode
${{{B}^{-}}}\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}$, when used with the
addIncoherentResonance function of LauIsobarDynamics.

###  A.1 $K$-matrix

The isobar model, described earlier in Sec. 2, can be used to describe the
dynamics of three-body decays when the quasi two-body resonances are
relatively narrow and isolated. However, this model does not satisfy
scattering ($S$-matrix) unitarity, thereby violating the conservation of
quantum mechanical probability current, when there are broad, overlapping
resonances (with the same spin-parity), such as the intermediate S-wave states
$\sigma$ for $\pi\pi$ and $\kappa$ for $K\pi$ channels. Assuming that the
dynamics is dominated by two-body processes, meaning that the S-wave does not
interact with the rest of the products in the final state, then unitarity is
naturally conserved within the $K$-matrix approach [113], which was originally
developed for two-body scattering [114] and the study of resonances in nuclear
reactions [115, 116], but was extended to describe resonance production in a
more general way [117].

The scattering matrix $S$, describing the general transformation of an initial
state to a final state, can be defined as

$S\equiv I+2iT\,,$ (63)

where $I$ is the identity matrix, representing the case when the initial and
final states do not interact at all, and $T$ is the physical (observable)
transition matrix. Conservation of scattering probability means that the
$n\times n$ $S$ matrix, where $n$ is the number of channels, is unitary
($SS^{\dagger}$ = $S^{\dagger}S=I$). The factor $2i$ is introduced so that the
transition amplitude for a single resonance channel corresponds to a circle of
unit diameter centred at (0, $i/2$) in the complex plane; physically allowed
values of $T$ will be along the boundary (elastic scattering) or inside
(inelastic scattering) this unitarity circle. Using Eq. (63), it can be shown
that the $n\times n$ $K$ matrix operator, defined as

$K\equiv(T^{-1}+iI)^{-1}\,,$ (64)

is Hermitian ($K$ = $K^{\dagger}$) [113]. Furthermore, $K$ is real and
symmetric, owing to the time-reversal invariance of the $S$ and $T$ matrices.
Rearranging Eq. (64) gives the following expression for the scattering
transition operator in terms of the $K$ matrix:

$T=(I-iK)^{-1}K.$ (65)

The normalisation of the two-body wave functions requires the inclusion of
phase-space factors in both the initial and final states [118]. This then
leads to the following definition of the Lorentz-invariant transition
amplitude $\hat{T}$:

$T_{uv}\equiv\\{\rho_{u}^{\dagger}\\}^{\frac{1}{2}}\,\hat{T}_{uv}\\{\rho_{v}\\}^{\frac{1}{2}}\,,$
(66)

where $u$ and $v$ indicate the channel indices (from 1 to $n$) and $\rho$ is
the normalised diagonal $n\times n$ phase space matrix, whose elements are
equal to $2q/m$, where $q$ is the magnitude of the momentum of either daughter
in the rest-frame of the two-body state that has invariant mass $m=\sqrt{s}$.
In general, the phase space element of channel $u$ is given by

$\rho_{u}=\sqrt{\left(1-\frac{(m_{1u}+m_{2u})^{2}}{s}\right)\left(1-\frac{(m_{1u}-m_{2u})^{2}}{s}\right)}\,,$
(67)

where $m_{1u}$ and $m_{2u}$ are the rest masses of the two daughters [62], and
is continued analytically by setting $\rho_{u}$ to be $i|\rho_{u}|$ when the
channel is below its mass threshold, provided it does not cross into another
channel.

The Lorentz-invariant form of the $K$ matrix, which will also be real and
symmetric, can be written as

$\hat{K}^{-1}=\hat{T}^{-1}+i\rho\,,$ (68)

which then implies that the Lorentz-invariant transition amplitude is given by

$\hat{T}=(I-i\hat{K}\rho)^{-1}\cdot\hat{K}\,.$ (69)

We can now use this expression to give the general amplitude of the
_production_ of overlapping resonance states. This model or ansatz [117]
describes the amplitude of channel $u$ in terms of the initial
$\hat{P}$-vector preparation of channel states $v$, that has the same form as
$\hat{K}$, transforming (“scattering”) into the final state $u$ via the
propagator term $(I-i\hat{K}\rho)^{-1}$:

${\cal{F}}_{u}=\sum_{v=1}^{n}[I-i\hat{K}\rho]^{-1}_{uv}\cdot\hat{P}_{v}\,.$
(70)

The scattering $\hat{K}$ matrix can be parameterised as a combination of the
summation of $N$ poles with real bare masses $m_{\alpha}$, together with
nonresonant slowly-varying parts (SVPs), so-called since they essentially have
a $1/s$ dependence, with real (and symmetric) coupling constants
$f^{\rm{scatt}}_{uv}$ [119]:

$\hat{K}_{uv}(s)=\left(\sum_{\alpha=1}^{N}{\frac{g^{(\alpha)}_{u}g^{(\alpha)}_{v}}{m^{2}_{\alpha}-s}}+f^{\rm{scatt}}_{uv}\frac{m^{2}_{0}-s^{\rm{scatt}}_{0}}{s-s^{\rm{scatt}}_{0}}\right)f_{A0}(s)\,,$
(71)

where $g_{u}^{(\alpha)}$ denotes the real coupling constant of the pole
$m_{\alpha}$ to the channel $u$, the factor

$f_{A0}(s)=\left(\frac{1\,{\rm{{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}}}-s_{A0}}{s-s_{A0}}\right)\left(s-\frac{1}{2}s_{A}m_{\pi}^{2}\right)$
(72)

is the Adler zero term that suppresses the false kinematic singularity when
$s$ goes below the $\pi\pi$ production threshold [103], while $m_{0}^{2}$,
$s^{\rm{scatt}}_{0}$, $s_{A}$ and $s_{A0}$ are real constants of order unity;
typical values are
$m_{0}^{2}=1\,{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$,
$s^{\rm{scatt}}_{0}=-5\,{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$,
$s_{A}=1$, and $s_{A0}=0\,{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$. Note
that the real poles $m_{\alpha}$ are the masses of the so-called _bare_ states
of the system, which do not correspond to the masses and widths of
_resonances_ (mixtures of bare states) from the complex poles in the physical
$T$ matrix. The production vector $\hat{P}$ is parameterised in an analogous
form to the $\hat{K}$ matrix:

$\hat{P}_{v}(s)=\sum_{\alpha=1}^{N}{\frac{\beta_{\alpha}g^{(\alpha)}_{v}}{m^{2}_{\alpha}-s}+f^{\rm{prod}}_{v}\frac{m_{0}^{2}-s_{0}^{\rm{prod}}}{s-s_{0}^{\rm{prod}}}}\,,$
(73)

where $\beta_{\alpha}$ and $f^{\rm{prod}}_{v}$ (which both depend on the final
state channel $u$) are complex production constants for the poles and
nonresonant SVPs, respectively, and $s_{0}^{\rm{prod}}$ is of order unity and
is usually taken to be approximately equal to $s_{0}^{\rm{scatt}}$. It is
important that the production and scattering processes use the same poles
$m_{\alpha}$, otherwise the transition amplitude would vanish (diverge) at the
$\hat{K}$-matrix ($\hat{P}$-vector) poles; the singularities need to cancel
out for the total amplitude. Also note that the Adler zero suppression factor
given in Eq. (72) is generally not needed for $\hat{P}$, since its inclusion
does not improve the description of S-wave amplitudes found in experimental
data [22, 120, 121, 16].

In order to clarify what amplitudes are used, we can separate out the
production pole and SVP terms shown above. The amplitude of each production
pole $m_{\alpha}$ for the final state $u$ is given by

${\cal{A}}_{\alpha,u}(s)=\sum_{v=1}^{n}[I-i\hat{K}\rho]^{-1}_{uv}\frac{\beta_{\alpha}g^{(\alpha)}_{v}}{m^{2}_{\alpha}-s}\equiv\frac{\beta_{\alpha}}{m^{2}_{\alpha}-s}\sum_{v=1}^{n}[I-i\hat{K}\rho]^{-1}_{uv}g^{(\alpha)}_{v}\,,$
(74)

where we need to sum the propagator contributions over the channels $v$, while
the SVP production amplitudes are separated out for each individual
$v\rightarrow u$ channel as

${\cal{A}}_{{\rm{SVP}},uv}(s)=\frac{m_{0}^{2}-s_{0}^{\rm{prod}}}{s-s_{0}^{\rm{prod}}}[I-i\hat{K}\rho]^{-1}_{uv}f^{\rm{prod}}_{v}\,.$
(75)

We can then sum all of these contributions to give the total S-wave amplitude

${\cal{F}}_{u}=\sum_{\alpha=1}^{N}{\cal{A}}_{\alpha,u}+\sum_{v=1}^{n}{\cal{A}}_{{\rm{SVP}},uv}\,.$
(76)

The elements $\rho_{u}$ of the diagonal phase space matrix depend on the
channel type $u$. For $\pi\pi$ systems, the five available channels are
$\pi\pi$, $K\bar{K}$, $4\pi$, $\eta\eta$ and $\eta\eta^{\prime}$ multimeson
states [119]; note that $\eta^{\prime}\eta^{\prime}$ is above the open charm
threshold and is not considered. The phase space factor for $\pi\pi$ ($u=1$),
$K\bar{K}$ ($u=2$) and $\eta\eta$ ($u=4$) is given by Eq. (67), with $m_{1u}$
and $m_{2u}$ equal to the rest masses of the two pseudoscalars forming channel
$u$ ($m_{1u}=m_{2u}$). For $\eta\eta^{\prime}$ ($u=5$), the second
multiplicative term involving $m_{\eta}-m_{\eta^{\prime}}$ is ignored (set to
unity), since below threshold this crosses channels and we cannot continue
this analytically in the usual way. As given in Ref. [119], the phase space
term for the $4\pi$ multimeson state ($u=3$) is

$\rho_{3}(s)=\begin{cases}\rho_{31}(s)&\text{for
}s<1\,{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}\\\
\sqrt{1-(16m^{2}_{\pi}/s)}&\text{for }s\geq
1\,{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}\,;\end{cases}$ (77)
$\rho_{31}(s)=\rho_{0}\int\frac{ds_{1}}{\pi}\int\frac{ds_{2}}{\pi}\frac{M^{2}_{0}\Gamma(s_{1})\Gamma(s_{2})\sqrt{(s+s_{1}-s_{2})^{2}-4ss_{1}}}{s[(M^{2}_{0}-s_{1})^{2}+M^{2}_{0}\Gamma^{2}(s_{1})][(M^{2}_{0}-s_{2})^{2}+M^{2}_{0}\Gamma^{2}(s_{2})]}\,.$

Here, $s_{1}$ and $s_{2}$ refer to the invariant mass-squared of the two di-
pion states (which are simply considered as integration variables), $M_{0}$ is
the pole mass of the $\rho$ resonance (775 $\mathrm{\,Me\kern-1.00006ptV}$)
and $\Gamma(s)=\Gamma_{0}[1-(4m^{2}_{\pi}/s)]^{3/2}$ is the energy-dependent
width, where $\Gamma_{0}$ is taken to be 0.3 $\mathrm{\,Ge\kern-1.00006ptV}$,
which is approximately 75% of the total width of the $f_{0}(1370)\rightarrow
4\pi$ state [62]. The constant factor $\rho_{0}$ ensures continuity at
$s=1\,{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$, while the limits of
integration are $4m^{2}_{\pi}$ to $(\sqrt{s}-2m_{\pi})^{2}$ for $s_{1}$ and
$4m^{2}_{\pi}$ to $(\sqrt{s}-\sqrt{s_{1}})^{2}$ for $s_{2}$ in order to
satisfy kinematic constraints. The $\rho_{31}$ term needs to be evaluated
numerically and is approximated very well by a 6th order polynomial in $s$:

$\rho_{31}(s)=1.0789s^{6}+0.1366s^{5}-0.2974s^{4}-0.2084s^{3}+0.1385s^{2}-0.0193s+0.0005.$
(78)

For $K\pi$ systems, we have the three channels $K\pi$, $K\eta^{\prime}$ and
$K\pi\pi\pi$ multimeson states [122]. The phase space factors for the first
two channels ($u$ = 1,2) are again given by Eq. (67), while the multimeson
phase space element is given by

$\rho_{K\pi\pi\pi}(s)=\begin{cases}r_{0}\left[1-((m_{K}-3m_{\pi})/s)\right]^{5/2}&\;\text{for
}s<1.44\,{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}\\\ 1&\;\text{for }s\geq
1.44\,{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}\,,\end{cases}$ (79)

where $r_{0}$ is a constant of continuity at
$s=1.44\,{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$.

The $K$-matrix formalism is a way to describe the dynamics of a set of broad,
overlapping resonances with the same isospin $I_{s}$, spin $L$ and parity $P$.
Final states with different $I_{s}L^{P}$ values would require the appropriate
number of $K$-matrices. To avoid overcomplicating the Dalitz plot analysis,
the usual procedure is to parameterise only the S-wave ($L^{P}$ = 0+)
components with the $K$-matrix approach, and then combine the other (narrow)
resonances with the isobar model. This means that the total amplitude would be
given by

${\cal
A}\left(m^{2}_{13},m^{2}_{23}\right)=\sum_{I_{s}}{\cal{F}}_{u,I_{s}}(s)+\sum_{j=1,j\neq{\cal{F}}_{u}}^{N}c_{j}F_{j}\left(m^{2}_{13},m^{2}_{23}\right)\,,$
(80)

where ${\cal{F}}_{u,I_{s}}(s)$ is the $K$-matrix amplitude defined in Eq. (70)
for the channel $u$ and isospin state $I_{s}$. The recommended procedure would
then be to first use scattering data to completely define the $K$-matrix
elements in Eq. (71), such as using the values quoted in Ref. [121] which are
obtained from a global analysis of $\pi\pi$ scattering data [119].
Subsequently in the DP analysis the user can fit for the coefficients
$\beta_{\alpha}$ and $f^{\rm{prod}}_{v}$ of the $\hat{P}$-vector used in Eqs.
(73), (74) and (75).

####  A.1.1 Implementation details for $K$-matrix

Special commands are required in order to use the $K$-matrix amplitude defined
in Eqs. (70) and (76), which is combined automatically with the other isobar
resonances to produce the total dynamical amplitude given by Eq. (80).

First, the $(I-i\hat{K}\rho)^{-1}$ propagator term is formed using the
defineKMatrixPropagator function in LauIsobarDynamics, which requires a
descriptive name, a text file containing a keyword-defined list of the
scattering and Adler zero coefficients, as well as an integer to specify which
daughter is the bachelor particle. This function also requires the total
number $n$ of $K$-matrix scattering channels (sum over $v=1$ to $n$), the
number of bare poles $N$ (sum of $m_{\alpha}$ terms), as well as the final
channel index $u$. Note that the complete $\hat{K}$ matrix in Eq. (71), which
is real and symmetric, is found for all possible values of $u$ and $v$ in
order to find the propagator; the specific $u$ index is only needed for
finding the final ${\cal{F}}_{u}$ amplitude. For $\pi\pi$ S-wave, we normally
have five channels ($\pi\pi$, $K\bar{K}$, $4\pi$, $\eta\eta$ and
$\eta\eta^{\prime}$ multimeson states), five poles and the index $u$ is equal
to 1 ($\pi\pi$). The production vector $\hat{P}$ defined in Eq. (73) is then
formed using the addKMatrixProdPole and addKMatrixProdSVP functions of
LauIsobarDynamics that create the $\beta_{\alpha}$ pole term given by Eq. (74)
(which internally sums the propagator function over the initial channels $v$
owing to the $g_{v}$ coupling dependence) and the slowly-varying
$f_{v}^{\rm{prod}}$ term given by Eq. (75), respectively. They each require a
descriptive name, the name of the propagator term defined earlier and the pole
or channel integer number (starting from 1). These functions also accept a
boolean useProdAdler to specify if the Adler zero suppression factor given in
Eq. (72) is also used for the production vector $\hat{P}$; by default
useProdAdler is set to false. Additional $K$-matrix amplitudes (e.g. for
different isospin settings) can be included by simply defining additional
propagators with unique names together with their required production terms.

Internally, the $K$-matrix propagator is defined by the LauKMatrixPropagator
class, in which each unique propagator is created using an instance of the
LauKMatrixPropFactory factory method, while the LauKMatrixProdPole and
LauKMatrixProdSVP classes represent the production pole and slowly-varying
terms, respectively. Since Root does not implement complex matrices, the
$K$-matrix propagator is expanded into real and imaginary parts using the
following method. If $A$, $B$, $C$ and $D$ are real matrices (TMatrix
objects), then the propagator can be expressed as

$(I-i\hat{K}\rho)^{-1}\equiv C+iD\equiv(A+iB)^{-1}\,,$ (81)

where $A$ is equal to $I+\hat{K}$Im($\rho$) and $B$ is $-\hat{K}$Re($\rho$).
Both $A$ and $B$ are completely determined if we know the real, symmetric
$\hat{K}$ matrix and the diagonal phase space matrix $\rho$ (which can have
imaginary terms if the invariant mass is below threshold). The real and
imaginary propagator terms are then given by

$C=(A+BA^{-1}B)^{-1}\hskip 28.45274pt{\rm and}\hskip 28.45274ptD=-A^{-1}BC\,.$
(82)

####  A.1.2 Pedagogical $K$-matrix plots

Figure 10: Plots showing properties of the $\pi\pi\rightarrow\pi\pi$ $K$-matrix S-wave transition amplitude, corresponding to the $T_{11}$ matrix element: Argand diagram, intensity or amplitude squared (showing the location of various “resonance structures”), phase shift $\delta$ and inelasticity $\eta$. Table A4: $\hat{K}$-matrix parameters taken from Ref. [121], which are obtained from a global analysis of $\pi\pi$ scattering data by Anisovich and Sarantsev [119]. Only $f_{1v}$ parameters are listed here ($\pi\pi$ S-wave). Masses $m_{\alpha}$ and couplings $g_{u}^{(\alpha)}$ are given in ${\mathrm{\,Ge\kern-0.90005ptV\\!/}c^{2}}$, while units of ${\mathrm{\,Ge\kern-0.90005ptV^{2}\\!/}c^{4}}$ for $s$-related quantities are implied; $s^{\rm{prod}}_{0}$ is taken from Ref. [22]. $\alpha$ | $m_{\alpha}$ | $g^{(\alpha)}_{1}[\pi\pi]$ | $g^{(\alpha)}_{2}[K\bar{K}]$ | $g^{(\alpha)}_{3}[4\pi]$ | $g^{(\alpha)}_{4}[\eta\eta]$ | $g^{(\alpha)}_{5}[\eta\eta^{\prime}]$
---|---|---|---|---|---|---
$1$ | $0.65100$ | $0.22889$ | $-0.55377$ |  $-$$0.00000$ | $-0.39899$ | $-0.34639$
$2$ | $1.20360$ | $0.94128$ |  $-$$0.55095$ |  $-$$0.00000$ |  $-$$0.39065$ |  $-$$0.31503$
$3$ | $1.55817$ | $0.36856$ |  $-$$0.23888$ |  $-$$0.55639$ |  $-$$0.18340$ |  $-$$0.18681$
$4$ | $1.21000$ | $0.33650$ |  $-$$0.40907$ |  $-$$0.85679$ |  $-$$0.19906$ | $-0.00984$
$5$ | $1.82206$ | $0.18171$ | $-0.17558$ | $-0.79658$ | $-0.00355$ |  $-$$0.22358$
| $s_{0}^{\rm{scatt}}$ | $f_{11}^{\rm{scatt}}$ | $f_{12}^{\rm{scatt}}$ | $f_{13}^{\rm{scatt}}$ | $f_{14}^{\rm{scatt}}$ | $f_{15}^{\rm{scatt}}$
| $-3.92637$ | $0.23399$ |  $-$$0.15044$ | $-0.20545$ |  $-$$0.32825$ |  $-$$0.35412$
| $s_{0}^{\rm{prod}}$ | $m_{0}^{2}$ | $s_{A}$ | $s_{A0}$ |  |
| $-3.0$ | $1.0$ | $1.0$ | $-0.15$ |  |

In order to better understand the properties of the $K$-matrix description we
will now show a series of instructional plots. The first of these is Fig. 10
which shows the transition amplitude of the $\pi\pi\rightarrow\pi\pi$ S-wave,
corresponding to the first element $T_{11}$ of the $T$ matrix defined in Eq.
(66), using the parameters given in Table A4 and where we are not considering
the effect of the production vector $\hat{P}$. Figure 10a) shows the phase
motion of the amplitude, which lies within a circle of unit diameter centred
on $(0,i/2$), while Fig. 10b) is the equivalent intensity or amplitude
squared. First, we can see that the amplitude follows the unit circle
anticlockwise, corresponding to the very broad $\sigma$ or $f_{0}(500)$
resonance structure, until we reach an invariant mass near to the threshold of
the $f_{0}(980)$ resonance, where its interference with the $\sigma$ produces
a striking dip in the intensity; the amplitude is purely elastic until we
reach the $f_{0}(980)$. As we follow the phase motion counterclockwise, new
channels such as $K\bar{K}$ open up at higher energies, producing other
resonance structures that give extra interference terms and so the scattering
process remains inelastic. A more detailed discussion of these features is
given in Ref. [123], which has a slightly different amplitude intensity
distribution at high invariant mass owing to different scattering data being
considered. If we now imagine a vector $\ell$ starting from the centre of the
unitarity circle $(0,i/2)$ and ending on the position of the complex amplitude
$T_{11}$, then the phase shift $\delta$ is defined as half of the angle that
$\ell$ subtends with the imaginary axis (anticlockwise is positive):

$\delta\equiv\frac{1}{2}{\rm{tan}}^{-1}\left(\frac{{\rm{Im}}T_{11}-\frac{1}{2}}{|{\rm{Re}}T_{11}|}\right)+\frac{\pi}{4}\hskip
14.22636pt\rm{radians},$ (83)

while the inelasticity $\eta$ is defined as twice the length of $\ell$

$\eta\equiv
2\left|T_{11}-{\frac{i}{2}}\right|=|S|=2\sqrt{({\rm{Im}}T_{11}-{\scriptstyle{\frac{1}{2}}})^{2}+({\rm{Re}}T_{11})^{2}}.$
(84)

Figure 10c) shows the evolution of $\delta$ with invariant $\pi\pi$ mass,
while Fig. 10d) shows the variation of the inelasticity $\eta$, where a purely
elastic (inelastic) process has $\eta=1$ ($\eta=0$); rapid changes in $\delta$
and $\eta$ are observed at the thresholds of various resonance structures.

Figure 11: First row of the $\hat{K}$ matrix, where red (blue) lines show only
the pole (SVP) terms, while the black (magenta) lines show the full elements
with (without) the Adler zero factor.

We now move onto the $\hat{K}$ matrix itself, which is the main ingredient of
the scattering propagator. Figure 11 shows the first row of the $\hat{K}$
matrix ($\hat{K}_{1j}$), where we have split up the various components that
make up each matrix element. The red lines show only the bare pole
$m_{\alpha}$ contributions, given by the first summation term on the right
hand side of Eq. (71), where $u=1$, $v=1-5$ and we sum over all five poles
($\alpha=1-5$), and the Adler zero suppression factor $f_{A0}(s)$ is set to
unity. All of the plots show the strong effect of the bare pole singularities.
The blue lines show the rather small SVP contributions, corresponding to the
second term on the right hand side of Eq. (71) (with $f_{A0}(s)=1$). The
summation of these various pole and SVP contributions is given by the magenta
lines, while the black lines show the inclusion of the Adler zero suppression
factor of Eq. (72), which only significantly effects the overall shapes at low
invariant mass ($m\lesssim 0.5\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$).
The absence of singularities for $\hat{K}_{13}$ below
$1.2\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ is due to the fact that the
coupling of the first two bare poles in the $4\pi$ channel is zero. Similar
distributions are obtained for the other rows of the $\hat{K}$ matrix.

Figure 12: Complex amplitude components of the first row of the propagator
$[I-i\hat{K}\rho]^{-1}_{1v}$ ($v\rightarrow\pi\pi$ channels), where red (blue)
lines show the real (imaginary) components while the black curves show the
magnitudes. The dotted horizontal lines denote the zero amplitude level.
Channels are a) $\pi\pi$, b) $K\bar{K}$, c) $4\pi$, d) $\eta\eta$ and e)
$\eta\eta^{\prime}$.

Next, let us look at the $s$-dependence of the first row of the propagator
matrix $[I-i\hat{K}\rho]^{-1}_{1v}$, since this will effectively modulate the
individual production pole and SVP shapes that are combined to form the total
$\pi\pi$ S-wave amplitude ${\cal{F}}_{1}(s)$ using Eq. (70). Figure 12 shows
the real and imaginary components, as well as the magnitude, of the propagator
elements. The overall impression we get is that the propagator amplitudes have
non-trivial variations as a function of $\sqrt{s}$, owing to the matrix
inversion process mixing and transposing the superposition of the bare pole
states $m_{\alpha}$. These poles essentially produce the various cusps and
peaks in the propagator amplitude, where the channel couplings
$g_{u,v}^{(\alpha)}$ and phase space $\rho_{u,v}$ (mass threshold) weighting
factors shift and distort these features away from the original $m_{\alpha}$
values. The corresponding Argand diagrams show similar behaviour as the
transition amplitude $T_{11}$ shown in Fig. 10a), although in general they
exhibit distortions due to the channel-dependent couplings and mass
thresholds. In particular, the $\pi\pi\rightarrow\pi\pi$ propagator
($u=1,v=1$) exactly matches the phase motion of $T_{11}$ if we first rotate
$T_{11}$ by 90 degrees anticlockwise ($\delta$ = 45 degrees) around the centre
of the unitarity circle at $(0,\frac{i}{2})$ and then shift it by the
translation $(\frac{1}{2},-\frac{i}{2})$.

Discussing the features in Fig. 12 in detail, the first pole ($m_{1}=0.651$
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$) produces the first cusps around
0.65–0.8 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ in all of the channels,
except for $4\pi$ ($v=3$) which has a coupling of zero. The second pole
($m_{2}=1.2036$ ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$) produces cusps at
1, 1.1 and 1.5 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ for the $K\bar{K}$
$(v=2)$, $\eta\eta$ $(v=4)$ and $\eta\eta^{\prime}$ $(v=5)$ channels,
respectively, and also generates a very broad dip centred near 1.2
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ for the $\pi\pi$ $(v=1)$ channel.
The third pole ($m_{3}=1.55817$ ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$)
generates broad peaks near 1.55 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$,
where the low mass end terminates in a cusp at the threshold of the given
channel, except for the $\eta\eta^{\prime}$ mode where a narrow cusp at 1.5
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ is generated against a smoothly
varying amplitude. The fourth pole ($m_{4}=1.21$
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$) creates structures very similar to
the second pole owing to their almost degenerate mass values, with an
additional broad peak around 1.1–1.3
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ present for the $4\pi$ channel,
while the fifth pole ($m_{5}=1.82206$
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$) generates the broad peaks near 1.8
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ for all channels, with additional
cusps at 1 ($K\bar{K}$), 1.1 ($\eta\eta$) and 1.5
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ ($\eta\eta^{\prime}$) that are very
similar to those found for the second and fourth poles.

Figure 13: Complex amplitude components for the production poles defined in
Eq. (74), where the red (blue) lines show the real (imaginary) parts, while
the black lines show the magnitude. The magenta lines show what happens to the
magnitude when it is scaled by the Adler zero suppression factor $f_{A0}(s)$.
The dotted horizontal lines denote the zero amplitude level.

Figure 13 shows the pole production amplitudes ${\cal{A}}_{\alpha,u=1}(s)$
defined in Eq. (74), which are formed by modulating the pole singularity term
$1/(m^{2}_{\alpha}-s)$ with a weighted sum of the $s$-dependent propagator
distributions for all channels $v=1$ to $5$ shown in Fig. 12, along with
$\beta_{\alpha}\equiv 1$. Concentrating on the magnitudes, we can see that the
first pole ($m_{1}=0.651$ ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$) has an
amplitude that begins rather flat from the $\pi\pi$ threshold until it starts
to peak near 1 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ before rapidly
falling at higher invariant mass. Even though the $\pi\pi$ propagator
($\pi\pi\rightarrow\pi\pi$) is slowly decreasing from unity at the $\pi\pi$
threshold down to zero near 0.8 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$,
the presence of both the rapid rise of the amplitude from the pole singularity
at 0.65 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ as well as the increasingly
influential $K\bar{K}$ propagator ($K\bar{K}\rightarrow\pi\pi$), owing to its
larger coupling constant, ensures that the amplitude remains fairly constant
in the region below 1 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. The
propagators for all of the channels (except $4\pi$) have a peak near 1
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, and these combine to give the same
local peak for the first production pole. As we increase the invariant mass,
the falling shape of the pole singularity starts to dominate the amplitude
modulation, and so we get a rapid reduction in the magnitude no matter what
shapes the propagators have. The amplitude for the second production pole
($m_{2}=1.2036$ ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$) strongly depends
upon the $\pi\pi$ and $K\bar{K}$ propagator shapes, where the former has a
coupling constant almost double that of the latter. As we decrease the
invariant mass from $1.2$ ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, the pole
amplitude would be very small at the strong $K\bar{K}$ dip at 1
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ if not for the compensating sharp
peak in the $\pi\pi$ propagator. Likewise, the zero $\pi\pi$ propagator
amplitude at 0.8 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ is nullified by
the non-zero $K\bar{K}$ contribution. These two effects conspire to shift the
location of the sharp dip in the production pole amplitude by 50
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ from 1 to 0.95
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. As we decrease the invariant mass,
the pole amplitude becomes more influenced by the rising $\pi\pi$ propagator
until it starts to fall as we move further away from the pole mass. Above 1.2
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ the production amplitude tends to
follow the undulations of the $\pi\pi$ and $K\bar{K}$ propagators, producing
broad local peaks centred on 1.3 and 1.9
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ as well as a more narrow one at 1.5
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. The modulation of the third pole
($m_{3}=1.55817$ ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$) amplitude is
dominated by the $4\pi$ channel, although the other channels give significant
contributions, varying from 33% ($\eta\eta$) up to 66% $(\pi\pi$). The $4\pi$
propagator has a very broad maximum centred very close to the pole mass, and
so we would expect the production pole peak to remain very close to 1.56
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ (with an asymmetric width that is
slightly narrower on the low side). However, as we decrease the invariant
mass, the rising contributions from the $\pi\pi$ and $K\bar{K}$ propagators
effectively shift the production peak by 60
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ down to 1.5
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. Below 1.2
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, the modulations from the $\pi\pi$
and $K\bar{K}$ propagators become washed out since they are too far away from
the pole position. Above 1.5 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, the
width of the production amplitude remains very wide owing to the dominant
$4\pi$ propagator. The fourth pole located at 1.21
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ is almost degenerate with the
second pole (1.2036 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$) and so we
would naively expect them to have essentially identical production shapes.
However, the coupling coefficients are completely different, where now the
$4\pi$ channel propagator dominates, with a factor of two or more reduction in
the other contributions, leading to the production of the two broad peaks
centred around 1.4 and 1.7 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. For
invariant masses at 1 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ and below,
the amplitude does indeed closely follow the shape of the second production
pole since the $4\pi$ propagator becomes negligible and the $\pi\pi$ and
$K\bar{K}$ contributions dominate. The fifth and last pole ($m_{2}=1.82206$
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$) has an amplitude that is strongly
influenced by the $4\pi$ channel, with much smaller contributions from the
others. The large, broad $4\pi$ propagator maximum near 1.6
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ shifts the production peak by
around 70 ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ down to 1.75
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, while the other smaller production
peaks near 1.4 and 1.1 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ match those
seen in the $4\pi$ shape. The magenta lines in Fig. 13 show the effect of
multiplying the Adler zero suppression factor $f_{A0}(s)$ to the production
pole amplitudes, where we can see that it only reduces the magnitudes for
invariant masses below 1 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$.

The mass distributions of the SVP production amplitudes
${\cal{A}}_{{\rm{SVP}},u=1,v}(s)$ defined in Eq. (75) follows very closely the
shape modulations of the propagator terms shown in Fig. 12 (when
$f_{v}^{\rm{prod}}\equiv 1$), since each SVP is simply equal to the propagator
multiplied by the common function $4/(s+3)$, obtained using $m_{0}^{2}=1.0$
${\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ and $s_{0}^{\rm{prod}}=-3.0$
${\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$, which enhances (suppresses)
features at low (high) invariant mass.

Figure 14: Mass projections of the individual production pole (left) and SVP
(right) amplitudes, with the $\rho(770)$ resonance shown for comparison.

The final set of pedagogical plots are given in Fig. 14, which show the
normalised mass projections of the individual production pole and SVP
amplitudes, as well as the $\rho(770)$ resonance for comparison, using events
generated uniformly across the Dalitz plot. In general, we see that the
peaking structures observed in Fig. 13 are replicated here, with a reduction
in the intensity as the invariant mass approaches the $\pi\pi$ threshold
(fewer events are generated at the kinematic boundary), with an almost flat
intensity seen for invariant masses above 2
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, which is the effective cut-off for
the $K$-matrix parameterisation owing to the fact that there are no bare poles
defined in this region. The first production pole generates the peak
corresponding to the $f_{0}(980)$ along with a broad shoulder on the low mass
side, which can be referred to as the $f_{0}(500)$ or $\sigma$ resonance. The
third pole generates the $f_{0}(1500)$ peak, the fifth pole is the main
contributor for the $f_{0}(1710)$, while a combination of the second and
fourth poles (which have almost degenerate bare masses) produces peaks in the
$f_{0}(1370)$ region. All of these peaks are generated dynamically by the
$K$-matrix amplitude. The first two SVPs have rather oscillatory shapes in the
region around 1 to 2 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$; when they are
combined we can approximately obtain a very broad bump between the
$f_{0}(980)$ and $f_{0}(1370)$ peak locations. The third SVP generates a large
peak very near 1.6 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, in between the
$f_{0}(1500)$ and $f_{0}(1710)$ regions. Lastly, the fourth and fifth SVPs are
essentially degenerate, peaking at the same position of the $f_{0}(980)$ from
the first production pole. This means that we can ignore these two
contributions, or at least remove the fifth SVP, since nothing is gained by
their inclusion in the total amplitude description.

##  Appendix B Formulae for available angular distributions

The angular distributions and Blatt–Weisskopf form factors set out in Sec. 2.2
are the default settings in Laura++. However, other formalisms to describe the
angular distributions are also implemented in the package and it is
straightforward to switch between them. This appendix details these
alternative formalisms and illustrates the few additional lines of code
required to use them.

The four spin-factor formalisms are defined in the enumeration
LauAbsResonance::LauSpinType, which can take the values Zemach_P (the default
setting), Zemach_Pstar, Covariant, and Legendre. The simplest description of
the spin factors is that of the Legendre formalism, where the spin factors are
simply the Legendre polynomials (with some additional numerical constants in
order to maintain consistency of the phase conventions among the various
formalisms)

$\displaystyle L=0\ :\ T(\vec{p},\vec{q})$ $\displaystyle=$
$\displaystyle\phantom{-}\,1\,,$ (85) $\displaystyle L=1\ :\
T(\vec{p},\vec{q})$ $\displaystyle=$ $\displaystyle-\,2\,\cos{\theta}\,,$ (86)
$\displaystyle L=2\ :\ T(\vec{p},\vec{q})$ $\displaystyle=$
$\displaystyle\phantom{-}\,\frac{4}{3}\left[3\cos^{2}{\theta}-1\right]\,,$
(87) $\displaystyle L=3\ :\ T(\vec{p},\vec{q})$ $\displaystyle=$
$\displaystyle-\,\frac{24}{15}\left[5\cos^{3}{\theta}-3\cos{\theta}\right]\,,$
(88) $\displaystyle L=4\ :\ T(\vec{p},\vec{q})$ $\displaystyle=$
$\displaystyle\phantom{-}\,\frac{16}{35}\left[35\cos^{4}{\theta}-30\cos^{2}{\theta}+3\right]\,,$
(89) $\displaystyle L=5\ :\ T(\vec{p},\vec{q})$ $\displaystyle=$
$\displaystyle-\,\frac{32}{63}\left[63\cos^{5}{\theta}-70\cos^{3}{\theta}+15\cos{\theta}\right]\,.$
(90)

The spin factors for Zemach_P are those given in Eqs. (9)–(14), which differ
from the expressions of Eqs. (85)–(90) by factors of $(p\,q)^{L}$. Similarly,
those for Zemach_Pstar are the same as those for Zemach_P but with the
bachelor momentum evaluated in the rest frame of the parent particle
($p^{\ast}$), rather than that of the resonance ($p$). The angular
distributions have been implemented in Laura++ up to $L=5$, which is two units
larger than the maximum spin of any resonance observed to be produced in any
Dalitz plot to date [87, 88, 124].

The angular distributions discussed above are based on a non-relativistic
assumption. For certain channels, this may not be sufficiently precise, and
therefore the Covariant formalism is also made available. This is given by

$\displaystyle L=0\ :\ T(\vec{p},\vec{q})$ $\displaystyle=$
$\displaystyle\phantom{-}\,1\,,$ (91) $\displaystyle L=1\ :\
T(\vec{p},\vec{q})$ $\displaystyle=$
$\displaystyle-\,2\,(p^{\ast}q)\sqrt{1+\frac{p^{2}}{m^{2}_{P}}}\,\cos{\theta}\,,$
(92) $\displaystyle L=2\ :\ T(\vec{p},\vec{q})$ $\displaystyle=$
$\displaystyle\phantom{-}\,\frac{4}{3}\,(p^{\ast}q)^{2}\left(\frac{3}{2}+\frac{p^{2}}{m^{2}_{P}}\right)\left[3\cos^{2}{\theta}-1\right]\,,$
(93) $\displaystyle L=3\ :\ T(\vec{p},\vec{q})$ $\displaystyle=$
$\displaystyle-\,\frac{24}{15}\,(p^{\ast}q)^{3}\sqrt{1+\frac{p^{2}}{m^{2}_{P}}}\left(\frac{5}{2}+\frac{p^{2}}{m^{2}_{P}}\right)\left[5\cos^{3}{\theta}-3\cos{\theta}\right]\,,$
(94) $\displaystyle L=4\ :\ T(\vec{p},\vec{q})$ $\displaystyle=$
$\displaystyle\phantom{-}\,\frac{16}{35}\,(p^{\ast}q)^{4}\left(\frac{8p^{4}}{m^{4}_{P}}+\frac{40p^{2}}{m^{2}_{P}}+35\right)\left[35\cos^{4}{\theta}-30\cos^{2}{\theta}+3\right]\,.$
(95)

The first three of these expressions are derived in Ref. [125] and, based on
that work, the last two were derived in Ref. [124]. As can be seen from the
expressions, the differences between formalisms are more significant for
higher spin resonances, and particularly affect tails of the distributions. To
give an idea of the effect, the lineshapes for the $f_{2}(1270)$ and
$\rho_{3}(1690)^{0}$ resonances decaying to ${{\pi}^{+}}{{\pi}^{-}}$ are shown
in Fig. 15.

Figure 15:  Lineshapes for the (left) $f_{2}(1270)$ and (right)
$\rho_{3}(1690)^{0}$ resonances decaying to ${{\pi}^{+}}{{\pi}^{-}}$ (in the
${{{B}^{+}}}\rightarrow{{K}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ Dalitz plot) with the
(blue) Legendre, (red) Zemach_P, (green) Zemach_Pstar and (magenta) Covariant
spin formalisms. In all cases the relativistic Breit–Wigner description is
used, with mass and width parameters as given in App. C and the two
Blatt–Weisskopf factors set to unity.

It is possible to switch between these different formalisms via a function of
the LauResonanceMaker factory object. For example, to use the Covariant
formalism one would do:

⬇

LauResonanceMaker& resMaker = LauResonanceMaker::get();

resMaker.setSpinFormalism( LauAbsResonance::Covariant );

It is important to note that any such operation must be performed prior to
constructing any resonances, i.e. before calling
LauIsobarDynamics::addResonance or LauIsobarDynamics::addIncoherentResonance
for the first time.

As the angular and Blatt–Weisskopf factors are strongly coupled, it is also
possible to straightforwardly modify the form of the Blatt–Weisskopf factors.
In particular, the momentum value used for the factor that is related to the
decay of the parent particle into the resonance and the bachelor can be
selected from the following options (defined in the
LauBlattWeisskopfFactor::RestFrame enumeration):

  * •

LauBlattWeisskopfFactor::ResonanceFrame, the momentum of the bachelor in the
rest frame of the resonance, $p$ (the default setting),

  * •

LauBlattWeisskopfFactor::ParentFrame, the momentum of the bachelor in the rest
frame of the parent, $p^{\ast}$,

  * •

LauBlattWeisskopfFactor::Covariant, the product of the momentum of the
bachelor in the rest frame of the parent, $p^{\ast}$, and a function of the
ratio of the energy and mass of the resonance in the rest frame of the parent,
$\sqrt{1+p^{2}/m^{2}_{P}}$. More precisely, this function is the expression in
the middle term in Eqs. (91) to (95) raised to the power of $1/L$.

This setting is changed as follows:

⬇

LauResonanceMaker& resMaker = LauResonanceMaker::get();

resMaker.setBWBachelorRestFrame( LauBlattWeisskopfFactor::ParentFrame );

where in this example the momentum of the bachelor in the rest frame of the
parent ($p^{\ast}$) is to be used. Again, this operation must be performed
before constructing any resonances.

In addition, it is possible to change the form of the Blatt–Weisskopf factors,
with the different types being defined by the
LauBlattWeisskopfFactor::BarrierType enumeration. The default setting,
corresponding to Eqs. (15)–(20), is given by
LauBlattWeisskopfFactor::BWPrimeBarrier and is recommended when the angular
terms contain momentum factors. One possible alternative is to use the
LauAbsResonance::Legendre angular terms and the
LauBlattWeisskopfFactor::BWBarrier form for the Blatt–Weisskopf factors:

$\displaystyle L=0\ :\ X(z)$ $\displaystyle=$ $\displaystyle 1\,,$ (96)
$\displaystyle L=1\ :\ X(z)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2z^{2}}{1+z^{2}}}\,,$ (97) $\displaystyle L=2\ :\
X(z)$ $\displaystyle=$ $\displaystyle\sqrt{\frac{13z^{4}}{z^{4}+3z^{2}+9}}\,,$
(98) $\displaystyle L=3\ :\ X(z)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{277z^{6}}{z^{6}+6z^{4}+45z^{2}+225}}\,,$ (99)
$\displaystyle L=4\ :\ X(z)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{12746z^{8}}{z^{8}+10z^{6}+135z^{4}+1575z^{2}+11025}}\,,$
(100) $\displaystyle L=5\ :\ X(z)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{998881z^{10}}{z^{10}+15z^{8}+315z^{6}+6300z^{4}+99225z^{2}+893025}}\,.$
(101)

An exponential form for these factors, LauBlattWeisskopfFactor::ExpBarrier,
which has been used in some analyses for virtual contributions has also been
implemented,

$X(z)=e^{-z^{L}}\,.$ (102)

To change the form of the barrier factors for all resonances, the following
lines are required

⬇

LauResonanceMaker& resMaker = LauResonanceMaker::get();

resMaker.setBWType( LauBlattWeisskopfFactor::BWBarrier );

where in this example the forms in Eqs. (96)–(101) are to be used. Again, this
operation should be performed before constructing any resonances.

As for the $T(\vec{p},\vec{q})$ terms, the differences between Blatt–Weisskopf
form factor formalisms are more significant for higher spin resonances, and
far from the peak of the resonance. An illustrative comparison of the shapes
is given in Fig. 16.

Figure 16:  Lineshapes for the (left) $f_{2}(1270)$ and (right)
$\rho_{3}(1690)^{0}$ resonances decaying to ${{\pi}^{+}}{{\pi}^{-}}$ (in the
${{{B}^{+}}}\rightarrow{{K}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ Dalitz plot) with
(blue) no Blatt–Weisskopf factors, and with the (red) ResonanceFrame, (green)
ParentFrame and (magenta) Covariant settings for evaluating the momentum that
enters the Blatt–Weisskopf factor associated with the decay of the parent
particle. In all cases the relativistic Breit–Wigner description is used, with
mass and width parameters as given in App. C and the Zemach_P formalism for
the spin factors.

It is possible to make all of the changes discussed in this Appendix at the
level of individual resonances, using the functions
LauAbsResonance::setSpinType and LauAbsResonance::setBarrierRadii, but this
requires much care to be taken and is not generally recommended.

##  Appendix C Standard resonances

This section provides the complete set of available resonances, indicating the
name, mass $m_{0}$, width $\Gamma_{0}$, spin, charge and Blatt–Weisskopf
barrier radius $r^{R}_{\rm BW}$. Table C1 contains information for light meson
resonances, Table C2 for charm, charmonium, strange-charm, beauty and strange-
beauty resonances, Table C3 for $K^{*}$ resonances and Table C4 for
nonresonant terms. Most data are taken from Ref. [62]. The tables list the
information contained in the information records for both neutral and
positively-charged resonances. Negatively-charged resonance records are
implemented as charge-conjugates of the positively charged ones; the plus sign
in the name is replaced with a minus sign.

In case a user wishes to modify the values of the parameters from those given
in the tables, the LauAbsResonance::changeResonance function, which takes the
mass, width and spin as arguments, can be used. The Blatt–Weisskopf barrier
radius can be changed with the LauAbsResonance::changeBWBarrierRadii function,
and other parameters specific to particular lineshapes can be changed with the
LauAbsResonance::setResonanceParameter function. The same approach can be used
to include a resonance that is not available in these tables, by using any of
the existing states of appropriate charge and redefining its properties.

Table C1: Standard light meson resonances defined in Laura++. Name |  $m_{0}$ (${\mathrm{Ge\kern-1.00006ptV\\!/}c^{2}}$) |  $\Gamma_{0}$ (${\mathrm{Ge\kern-1.00006ptV\\!/}c^{2}}$) | spin | charge |  $r^{R}_{\rm BW}$ ($\mathrm{Ge\kern-1.00006ptV}^{-1}$)
---|---|---|---|---|---
rho0(770) | 0.77526 | 0.1478 | 1 | 0 | 5.3
rho+(770) | 0.77511 | 0.1491 | 1 | 1 | 5.3
rho0(1450) | 1.465 | 0.400 | 1 | 0 | 4.0
rho+(1450) | 1.465 | 0.400 | 1 | 1 | 4.0
rho0(1700) | 1.720 | 0.250 | 1 | 0 | 4.0
rho+(1700) | 1.720 | 0.250 | 1 | 1 | 4.0
rho0(1900) | 1.909 | 0.130 | 1 | 0 | 4.0
rho+(1900) | 1.909 | 0.130 | 1 | 1 | 4.0
rho0_3(1690) | 1.686 | 0.186 | 3 | 0 | 4.0
rho+_3(1690) | 1.686 | 0.186 | 3 | 1 | 4.0
rho0_3(1990) | 1.982 | 0.188 | 3 | 0 | 4.0
rho+_3(1990) | 1.982 | 0.188 | 3 | 1 | 4.0
phi(1020) | 1.019461 | 0.004266 | 1 | 0 | 4.0
phi(1680) | 1.680 | 0.150 | 1 | 0 | 4.0
f_0(980) | 0.990 | 0.070 | 0 | 0 | —
f_2(1270) | 1.2751 | 0.1851 | 2 | 0 | 4.0
f_0(1370) | 1.370 | 0.350 | 0 | 0 | —
f’_0(1300) | 1.449 | 0.126 | 0 | 0 | —
f_2(1430) | 1.430 | 0.150 | 2 | 0 | 4.0
f_0(1500) | 1.505 | 0.109 | 0 | 0 | —
f’_2(1525) | 1.525 | 0.073 | 2 | 0 | 4.0
f_2(1565) | 1.562 | 0.134 | 2 | 0 | 4.0
f_2(1640) | 1.639 | 0.099 | 2 | 0 | 4.0
f_0(1710) | 1.722 | 0.135 | 0 | 0 | —
f_2(1810) | 1.816 | 0.197 | 2 | 0 | 4.0
f_2(1910) | 1.903 | 0.196 | 2 | 0 | 4.0
f_2(1950) | 1.944 | 0.472 | 2 | 0 | 4.0
f_2(2010) | 2.011 | 0.202 | 2 | 0 | 4.0
f_0(2020) | 1.992 | 0.442 | 0 | 0 | —
f_4(2050) | 2.018 | 0.237 | 4 | 0 | 4.0
f_0(2100) | 2.101 | 0.224 | 0 | 0 | —
omega(782) | 0.78265 | 0.00849 | 1 | 0 | 4.0
a0_0(980) | 0.980 | 0.092 | 0 | 0 | —
a+_0(980) | 0.980 | 0.092 | 0 | 1 | —
a0_0(1450) | 1.474 | 0.265 | 0 | 0 | —
a+_0(1450) | 1.474 | 0.265 | 0 | 1 | —
a0_2(1320) | 1.3190 | 0.1050 | 2 | 0 | 4.0
a+_2(1320) | 1.3190 | 0.1050 | 2 | 1 | 4.0
sigma0 | 0.475 | 0.550 | 0 | 0 | —
sigma+ | 0.475 | 0.550 | 0 | 1 | —
Table C2: Standard charm, charmonium, strange-charm, beauty and strange-beauty resonances defined in Laura++. Name |  $m_{0}$ (${\mathrm{Ge\kern-1.00006ptV\\!/}c^{2}}$) |  $\Gamma_{0}$ (${\mathrm{Ge\kern-1.00006ptV\\!/}c^{2}}$) | spin | charge |  $r^{R}_{\rm BW}$ ($\mathrm{Ge\kern-1.00006ptV}^{-1}$)
---|---|---|---|---|---
chi_c0 | 3.41475 | 0.0105 | 0 | 0 | —
chi_c1 | 3.51066 | 0.00084 | 1 | 0 | 4.0
chi_c2 | 3.55620 | 0.00193 | 2 | 0 | 4.0
X(3872) | 3.87169 | 0.0012 | 1 | 0 | 4.0
dabba0 | 2.098 | 0.520 | 0 | 0 | —
dabba+ | 2.098 | 0.520 | 0 | 1 | —
D*0 | 2.00696 | 0.0021 | 1 | 0 | 4.0
D*+ | 2.01026 | $83.4\times 10^{-6}$ | 1 | 1 | 4.0
D*0_0 | 2.318 | 0.267 | 0 | 0 | —
D*+_0 | 2.403 | 0.283 | 0 | 1 | —
D*0_2 | 2.4626 | 0.049 | 2 | 0 | 4.0
D*+_2 | 2.4643 | 0.037 | 2 | 1 | 4.0
D0_1(2420) | 2.4214 | 0.0274 | 1 | 0 | 4.0
D+_1(2420) | 2.4232 | 0.025 | 1 | 1 | 4.0
D0(2600) | 2.612 | 0.093 | 0 | 0 | —
D+(2600) | 2.612 | 0.093 | 0 | 1 | —
D0(2760) | 2.761 | 0.063 | 1 | 0 | 4.0
D+(2760) | 2.761 | 0.063 | 1 | 1 | 4.0
D0(3000) | 3.0 | 0.15 | 0 | 0 | —
D0(3400) | 3.4 | 0.15 | 0 | 0 | —
Ds*+ | 2.1121 | 0.0019 | 1 | 1 | 4.0
Ds*+_0(2317) | 2.3177 | 0.0038 | 0 | 1 | —
Ds*+_2(2573) | 2.5719 | 0.017 | 2 | 1 | 4.0
Ds*+_1(2700) | 2.709 | 0.117 | 1 | 1 | 4.0
Ds*+_1(2860) | 2.862 | 0.180 | 1 | 1 | 4.0
Ds*+_3(2860) | 2.862 | 0.058 | 3 | 1 | 4.0
B*0 | 5.3252 | 0.00 | 1 | 0 | 6.0
B*+ | 5.3252 | 0.00 | 1 | 1 | 6.0
Bs*0 | 5.4154 | 0.00 | 1 | 0 | 6.0
Table C3: Standard $K^{*}$ resonances defined in Laura++. Name |  $m_{0}$ (${\mathrm{Ge\kern-1.00006ptV\\!/}c^{2}}$) |  $\Gamma_{0}$ (${\mathrm{Ge\kern-1.00006ptV\\!/}c^{2}}$) | spin | charge |  $r^{R}_{\rm BW}$ ($\mathrm{Ge\kern-1.00006ptV}^{-1}$)
---|---|---|---|---|---
K*0(892) | 0.89581 | 0.0474 | 1 | 0 | 3.0
K*+(892) | 0.89166 | 0.0508 | 1 | 1 | 3.0
K*0(1410) | 1.414 | 0.232 | 1 | 0 | 4.0
K*+(1410) | 1.414 | 0.232 | 1 | 1 | 4.0
K*0_0(1430) | 1.425 | 0.270 | 0 | 0 | —
K*+_0(1430) | 1.425 | 0.270 | 0 | 1 | —
K*0_2(1430) | 1.4324 | 0.109 | 2 | 0 | 4.0
K*+_2(1430) | 1.4256 | 0.0985 | 2 | 1 | 4.0
K*0(1680) | 1.717 | 0.322 | 1 | 0 | 4.0
K*+(1680) | 1.717 | 0.322 | 1 | 1 | 4.0
K*0_0(1950) | 1.945 | 0.201 | 0 | 0 | —
K*+_0(1950) | 1.945 | 0.201 | 0 | 1 | —
kappa0 | 0.682 | 0.547 | 0 | 0 | —
kappa+ | 0.682 | 0.547 | 0 | 1 | —
Table C4: Standard nonresonant terms defined in Laura++. Name |  $m_{0}$ (${\mathrm{Ge\kern-1.00006ptV\\!/}c^{2}}$) |  $\Gamma_{0}$ (${\mathrm{Ge\kern-1.00006ptV\\!/}c^{2}}$) | spin | charge
---|---|---|---|---
NonReson | 0.0 | 0.0 | 0 | 0
NRModel | 0.0 | 0.0 | 0 | 0
BelleSymNR | 0.0 | 0.0 | 0 | 0
BelleNR | 0.0 | 0.0 | 0 | 0
BelleNR+ | 0.0 | 0.0 | 0 | 1
BelleNR_Swave | 0.0 | 0.0 | 0 | 0
BelleNR_Swave+ | 0.0 | 0.0 | 0 | 1
BelleNR_Pwave | 0.0 | 0.0 | 1 | 0
BelleNR_Pwave+ | 0.0 | 0.0 | 1 | 1
BelleNR_Dwave | 0.0 | 0.0 | 2 | 0
BelleNR_Dwave+ | 0.0 | 0.0 | 2 | 1
BelleNR_Fwave | 0.0 | 0.0 | 3 | 0
BelleNR_Fwave+ | 0.0 | 0.0 | 3 | 1
NRTaylor | 0.0 | 0.0 | 0 | 0
PolNR_S0 | 0.0 | 0.0 | 0 | 0
PolNR_S1 | 0.0 | 0.0 | 0 | 0
PolNR_S2 | 0.0 | 0.0 | 0 | 0
PolNR_P0 | 0.0 | 0.0 | 1 | 0
PolNR_P1 | 0.0 | 0.0 | 1 | 0
PolNR_P2 | 0.0 | 0.0 | 1 | 0

##  Appendix D PDF classes

This section details the formulae within classes that can be used to
parameterise additional PDFs ${\cal P}(x;p_{1},p_{2},...,p_{n})$ for the
likelihood function. Here, $x$ denotes the dependent variable, while
$p_{1},p_{2},...,p_{n}$ is the list of parameters in the form of a vector of
LauParameter objects, each containing a descriptive name (which must contain
the case-sensitive word shown in quotes), the value of the parameter and
optionally its validity range, uncertainty and constantness. All PDFs used in
Laura++ are normalised to unity, although the normalisation factors are
omitted in many equations in this section for brevity.

###  D.1 LauArgusPdf

The ARGUS threshold function [126] can be used to parameterise the shape of
combinatorial or partially-reconstructed backgrounds of the invariant mass $m$
of parent candidates:

${\cal P}(x;m_{0},\xi)=x\sqrt{1-x^{2}}e^{-\xi(1-x^{2})}\theta(x)\,,$ (103)

where $x=m/m_{0}$, $m_{0}$ is the end-point of the curve (“m0”), $\xi$ is the
shape parameter (“xi”), while $\theta(x\leq m_{0})=1$ and $\theta(x>m_{0})=0$.

###  D.2 LauBifurcatedGaussPdf

This PDF is the bifurcation of two Gaussians having different widths,
$\sigma_{L}$ (“sigmaL”) or $\sigma_{R}$ (“sigmaR”), to the left or right of
the “mean” $\mu$:

${\cal
P}(x;\mu,\sigma_{L},\sigma_{R})=\begin{cases}e^{-(x-\mu)^{2}/(2\sigma_{L}^{2})}&\text{for
}x\leq\mu\\\ e^{-(x-\mu)^{2}/(2\sigma_{R}^{2})}&\text{for
}x>\mu\,.\end{cases}$ (104)

###  D.3 LauChebychevPdf

This class implements a sum of Chebyshev polynomials of the first kind $T_{i}$
(up to seventh order):

${\cal P}(x;c_{i})=1+\sum_{i=1}^{n\leq 7}c_{i}T_{i}(x)\,,$ (105)

where $c_{i}$ is the parameter coefficient (“c1”, “c2”, etc.) and

$T_{n}(y)=\frac{(y-\sqrt{y^{2}-1})^{n}+(y+\sqrt{y^{2}-1})^{n}}{2}\,,\hskip
14.22636pty\equiv-1+\frac{2(x-x_{\rm{min}})}{(x_{\rm{max}}-x_{\rm{min}})}\,.$
(106)

###  D.4 LauCruijffPdf

The Cruijff PDF is a bifurcated Gaussian, which has different widths
$\sigma_{L}$ (“sigmaL”) and $\sigma_{R}$ (“sigmaR”) to the left and right of
the “mean” $\mu$, along with asymmetric tails $\alpha_{L}$ (“alphaL”) and
$\alpha_{R}$ (“alphaR”):

${\cal
P}(x;\mu,\sigma_{L},\sigma_{R},\alpha_{L},\alpha_{R})=\begin{cases}e^{-(x-\mu)^{2}/(2\sigma_{L}^{2}+\alpha_{L}(x-\mu)^{2})}&\text{for
}x\leq\mu\\\
e^{-(x-\mu)^{2}/(2\sigma_{R}^{2}+\alpha_{R}(x-\mu)^{2})}&\text{for
}x>\mu\,.\end{cases}$ (107)

###  D.5 LauCrystalBallPdf

The Crystal Ball function [127] is a PDF that contains a Gaussian core with a
continuous power-law tail on one side:

${\cal P}(x;\mu,\sigma,\alpha,n)=\begin{cases}e^{-\frac{1}{2}t^{2}}&\text{for
}t\geq|\alpha|\\\
\left(\frac{n}{|\alpha|}\right)^{n}e^{-\frac{1}{2}|\alpha|^{2}}\left(\frac{n}{|\alpha|}-|\alpha|-t\right)^{-n}&\text{otherwise}\,,\end{cases}$
(108)

where $\mu$ and $\sigma$ are the Gaussian “mean” and width (“sigma”),
respectively, $\alpha$ (“alpha”) is the positive or negative distance from the
mean in which the Gaussian and the tail parts match up, $n$ (“order”) is the
power exponent for the tail, while $t$ is equal to $(x-\mu)/\sigma$, which
changes sign if $\alpha$ is negative.

###  D.6 LauExponentialPdf

This exponential function is simply given by

${\cal P}(x;\lambda)=e^{\lambda x}\,,$ (109)

where $\lambda$ is the “slope” parameter.

###  D.7 LauGaussPdf

The Gaussian PDF is defined by a “mean” $\mu$ and width $\sigma$ (“sigma”):

${\cal
P}(x;\mu,\sigma)=\frac{1}{\sigma\sqrt{2\pi}}e^{-(x-\mu)^{2}/(2\sigma^{2})}\,.$
(110)

###  D.8 LauLinearPdf

This linear function only needs the gradient (“slope”) $\lambda$:

${\cal P}(x;\lambda)=\lambda x+c\,,$ (111)

where $c$ is the intercept and is evaluated using the range of the abscissa
$x$.

###  D.9 LauNovosibirskPdf

The Novosibirsk PDF is a Gaussian with a logarithmic exponent [128]:

${\cal
P}(x;\mu,\sigma,\tau)=\text{exp}\left[-\frac{1}{2}\left(\frac{\text{ln}^{2}[1+\Lambda\tau(x-\mu)]}{\tau^{2}}+\tau^{2}\right)\right]\,,\hskip
7.11317pt\text{with }\Lambda\equiv\frac{\text{sinh }(\tau\sqrt{\text{ln
}4})}{\sigma\tau\sqrt{\text{ln }4}}\,,$ (112)

where $\mu$ and $\sigma$ are the usual “mean” and width (“sigma”) values,
respectively, and $\tau$ is the “tail” parameter; as $\tau\rightarrow 0$, the
PDF converges to a normal Gaussian with width $\sigma$.

###  D.10 LauParametricStepFuncPdf

This parametric step function is a binned distribution whose parameters are
the contents of each bin (except one), essentially representing a histogram
with variable bin content. The content of the remaining bin is determined from
that of the others and the requirement of normalisation. The constructor
requires two vectors of LauParameter objects; the first stores the bin
contents or weights (which are parameters that can be fitted), while the
second stores the lower edge abscissa limits of each bin (in ascending order)
as well as the upper edge limit of the last bin. It can also be specified
whether the normalisation bin is the first or last; it is advisable to use the
bin with the larger content.

###  D.11 LauSigmoidPdf

The sigmoid function follows an “S” shape and is defined as

${\cal P}(x;a,b)=\frac{1}{1+e^{b-ax}}\,,$ (113)

with parameters “a” and “b” defining the steepness of the slope and the shift
of the distribution, respectively. A negative value of $a$ will flip the
distribution around the ordinate axis.

###  D.12 LauSumPdf

This class implements the summation of two PDFs, ${\cal P}_{1}$ and ${\cal
P}_{2}$, with a relative fraction (“frac”) $f$:

${\cal P}(x)=f{\cal P}_{1}(x)+(1-f){\cal P}_{2}(x)\,.$ (114)

###  D.13 Dalitz-plot-dependent PDFs

What follows are descriptions of PDFs with parameters that can vary across the
Dalitz plot via power-law scaling with Laurent polynomials containing either
positive or negative exponents. Here, the parameterisation of a given PDF
parameter $g$ is given by

$g=\sum_{i=1}^{n}{c_{i}D^{i}}\hskip 28.45274pt\text{or }\hskip
28.45274ptg=\sum_{i=1}^{n}{c_{i}D^{-i}}\,,$ (115)

where $D$ is the invariant mass-squared variable representing the DP position
and $c_{i}$ is the coefficient of expansion for the power term $i$. The
variable $D$ can be either $m^{2}_{13}$, $m^{2}_{23}$, or $m^{2}_{12}$, the
minimum or maximum values of $m^{2}_{13}$ or $m^{2}_{23}$, or the distance
from the DP centre. The constructors of these PDFs require vectors of the
coefficients $c_{i}$ for each function parameter $g$, as well as the pointer
to the LauDaughters object in order to find $D$ from the kinematics. The
LauDPDepBifurGaussPdf, LauDPDepCruijffPdf and LauDPDepGaussPdf classes
represent bifurcated Gaussian, Cruijff and normal Gaussian PDFs, given in Eqs.
(104), (107) and (110) respectively, with parameters that can vary according
to Eq. (115).

The LauDPDepMapPdf class can be used to define a PDF that requires different
functions depending on the DP region. It uses a Root histogram that divides
the DP, or its projection onto one axis, into ascending, numbered regions. The
region number (starting from zero) for a given value of $D$ is then used to
choose the corresponding PDF from the ordered list of functions provided as a
vector in the constructor.

The LauDPDepSumPdf class implements the sum of two PDFs, defined by Eq. (114),
in which the fraction $f$ (“frac”) depends on the variable $D$, using either
the contents from a two-dimensional histogram directly or a vector of
coefficients $c_{i}$ for the positive-power Laurent polynomial shown in Eq.
(115).

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