EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2022-287 LHCb-PAPER-2022-044

${\mathchar 28931\relax}^{+}_{c}$ polarimetry using the dominant hadronic mode

LHCb collaboration

The polarimeter vector field for multibody decays of a spin-half baryon is
introduced as a generalisation of the baryon asymmetry parameters. Using a
recent amplitude analysis of the ${{\mathchar 28931\relax}^{+}_{c}}\rightarrow
p{{K}^{-}}{{\pi}^{+}}$ decay performed at the LHCb experiment, we compute the
distribution of the kinematic-dependent polarimeter vector for this process in
the space of Mandelstam variables to express the polarised decay rate in a
model-agnostic form. The obtained representation can facilitate polarisation
measurements of the ${{\mathchar 28931\relax}^{+}_{c}}$ baryon and eases
inclusion of the ${{\mathchar 28931\relax}^{+}_{c}}\rightarrow
p{{K}^{-}}{{\pi}^{+}}$ decay mode in hadronic amplitude analyses.

submitted to JHEP

© 2024 CERN for the benefit of the LHCb collaboration. CC BY 4.0 licence.

##  1 Introduction

Polarisation characterises a preferred orientation of the particle spin in
space, and it is conserved for a freely moving particle. Therefore, the
strength and direction of the polarisation of particles produced in collisions
or decays gives access to details of the underlying fundamental theory.
Polarisation is widely exploited in spectroscopy studies of conventional and
exotic QCD states. Particularly, studies of pentaquarks in ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow
P_{c}^{+}(\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}p){{K}^{-}}$ decays in
the LHCb experiment take advantage of angular correlations of the muon pairs
appearing in the ${{J\mskip-3.0mu/\mskip-2.0mu\psi}}\rightarrow\mu^{+}\mu^{-}$
transition, which improves the sensitivity to the pentaquark spin [1].111The
charge-conjugate is implied for reactions and particles throughout the text,
unless stated otherwise. Angular analysis of the quantum numbers of
${\mathchar 28938\relax}_{c}^{**0}$ baryon states,222The double-star
superscript is used throughout the paper to indicate all excited states of the
hadron families. appearing as ${{\mathchar 28932\relax}^{+}_{c}}{{K}^{-}}$
resonances in ${{\mathchar 28938\relax}^{-}_{b}}\rightarrow{{\mathchar
28932\relax}^{+}_{c}}\pi^{-}{{K}^{-}}$ decays, is found to be sensitive to the
${\mathchar 28938\relax}_{c}^{**0}$ baryon parity only when the angular
distribution of the ${{\mathchar
28932\relax}^{+}_{c}}\rightarrow{p}{{K}^{-}}{{\pi}^{+}}$ transition is
considered [2]. This, in turn, requires detailed knowledge of the ${{\mathchar
28932\relax}^{+}_{c}}\rightarrow{p}{{K}^{-}}{{\pi}^{+}}$ decay matrix
elements. The polarisation of promptly produced mesons and baryons is an
essential observable that is sensitive to the mechanisms behind quark
hadronisation [3, 4, 5]. Many proposed searches for physics beyond the
Standard Model are based on semileptonic decays of the ${{\mathchar
28931\relax}^{0}_{b}}$ baryon, such as ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{\ell^{-}}{{\overline{\nu}}_{\ell}}$ transitions [6, 7,
8, 9, 10, 11, 12, 13, 14], where certain properties of the $b\rightarrow c$
transition current are reflected in the ${{\mathchar 28931\relax}^{+}_{c}}$
polarisation. For instance, the sign of the longitudinal polarisation of the
${{\mathchar 28931\relax}^{+}_{c}}$ baryon in ${{\mathchar
28931\relax}^{0}_{b}}$ decays provides a test for the left-handedness of the
$b\rightarrow c$ current [6]. The transition amplitude of the ${{\mathchar
28931\relax}^{+}_{c}}\rightarrow{p}{{K}^{-}}{{\pi}^{+}}$ decay recently
determined by LHCb [15] enables such measurement of the polarisation. A new
opportunity to test the Standard Model by measuring the electric and magnetic
dipole moment of charm baryons using spin rotation in crystals at the LHC
experiments is discussed in Refs. [16, 17, 18]. Recent feasibility studies
[19, 20] show that the measurement of the electric dipole moment of the
${{\mathchar 28931\relax}^{+}_{c}}$ and ${{\mathchar 28932\relax}^{+}_{c}}$
baryons can be performed in the near future.

In general, the measurement of the polarisation in baryon decays deserves
special consideration, since the final state of the decay contains a baryon,
e.g. a proton, whose spin state is usually not observed. Due to the averaging
over the baryon spin state, the influence of the initial-state polarisation on
the measurable observables is diminished. For example, angular distributions
in the decay of a fermion to a fermion and a scalar are insensitive to the
initial polarisation of the fermion when the decay conserves parity. However,
the forward-backward asymmetry in the distribution of the helicity angle of
the decay products is non-zero in weak decays, where both the parity-
conserving and parity-violating currents contribute to the transition
amplitude. In that case, the differential decay rate reads [21]

$\displaystyle\frac{2}{\Gamma}\frac{\mathrm{d}\Gamma}{\mathrm{d}\cos\theta}=1+P\alpha\cos\theta\,,$
(1)

where $\Gamma$ is the partial width for the considered decay, $\theta$ is the
helicity angle of the decay-product fermion, $P$ is the longitudinal
polarisation of the initial state, and $\alpha$ is the parity-violating
asymmetry parameter of the reaction. This parameter serves two roles. First,
it is a fundamental hadronic observable that characterises parity violation in
the weak transition. Second, it is a practical quantity, since the value of
$\alpha$ does not depend on the production mechanism of the decaying fermion.
The measurement of the angular distributions in the decay is a universal tool
to study the physics behind the production polarisation once its asymmetry
parameter is known.

The observables for polarisation measurements have been extensively explored
for the decays of the ${\tau^{-}}$ lepton [22, 23, 24]. For the polarised
decay, the differential decay rate reads

$\displaystyle\frac{\Phi}{\Gamma}\frac{\mathrm{d}\Gamma}{\mathrm{d}\Phi}\propto
1+\vec{P}\cdot\vec{h}\,,$ (2)

where $\Phi$ is the Lorentz-invariant phase space for the decay, $\vec{P}$ is
the polarisation vector of the ${\tau^{-}}$ lepton in its rest frame, and
$\vec{h}$ is a polarimeter vector. In semileptonic decays, the polarimeter
vector is a unit vector whose direction depends on the final state [25, 26,
27, 28, 24].

For the decay of a spin-half hadron, Eq. (2) still holds, but the length of
the polarimeter vector can be less than $1$. The direction of $\vec{h}$ is
related to the orientation of the final state particles, but does not have to
coincide with any of the momentum vectors. Instead, the length and orientation
of the polarimeter vector depends on the kinematic variables of the decay. As
shown in this paper, the dependence on the decay-plane orientation can be
factored out as

$\displaystyle\vec{h}=R(\phi,\theta,\chi)\vec{\alpha}\,,$ (3)

where $R$ is a three-dimensional rotation matrix with $\phi$, $\theta$, and
$\chi$ being the Euler angles that describe the orientation of the decay
products in space. The aligned polarimeter vector, denoted by $\vec{\alpha}$,
describes the direction of the polarimeter vector with respect to the decay-
product vectors. The dependence of $\vec{\alpha}$ on the kinematic variables
is specific to the decay reaction and needs to be determined experimentally.
For two-body decays with a baryon and a pseudoscalar in the final state, the
vector $\vec{\alpha}$ only has a longitudinal component, which matches the
asymmetry parameter $\alpha$ in Eq. (1). For multibody decays, the
$\vec{\alpha}$ vector forms a continuous vector field in the space of the
kinematic var

The three-body decay of ${{\mathchar
28931\relax}^{+}_{c}}\rightarrow{p}{{K}^{-}}{{\pi}^{+}}$ is the golden
hadronic mode for reconstructing this charm baryon experimentally. Firstly,
this decay mode has one of the largest branching fraction among all hadronic
${{\mathchar 28931\relax}^{+}_{c}}$ decays. Secondly, the final state of this
decay only consists of charged particles, which have a high reconstruction
efficiency. The transition amplitude of the ${{\mathchar
28931\relax}^{+}_{c}}\rightarrow{p}{{K}^{-}}{{\pi}^{+}}$ decays was studied
previously by the ACCMOR collaboration [29, 30] and the E791 collaboration
[31, 32]. These studies indicated a large polarisation in the production of
the ${{\mathchar 28931\relax}^{+}_{c}}$ baryon, but a detailed amplitude
description could not be obtained due to the limited statistics and
difficulties in formulating the amplitude model. A new result on the amplitude
analysis has been published by the LHCb collaboration recently, based on a
sample of $400\,000$ decays with an equal amount of ${{\mathchar
28931\relax}^{+}_{c}}$ and ${{\kern
1.79993pt\overline{\kern-1.79993pt\mathchar 28931\relax}}{}^{-}_{c}}$ baryons
[15]. The candidates are selected from semileptonic decays of beauty hadrons
produced in $pp$ collisions at a centre-of-mass energy of
$13\text{\,Te\kern-1.00006ptV}$, corresponding to an integrated luminosity of
$1.7\text{\,fb}^{-1}$.333Natural units with $c=1$ are used throughout this
paper. The study in Ref. [15] reveals the presence of twelve different decay
chains with resonances in all two-body subsystems: six ${\mathchar
28931\relax}^{**}$ states, three ${\mathchar 28929\relax}^{**++}$ states, and
three $K^{**0}$ states. The average polarisation of the ${{\mathchar
28931\relax}^{+}_{c}}$ baryon in semileptonic decays of beauty hadrons is also
measured. The asymmetry parameter $\alpha$ is reported for quasi-two-body
decays of the ${{\mathchar 28931\relax}^{+}_{c}}$ state to a baryon and a
pseudoscalar.

The utilisation of the ${{\mathchar 28931\relax}^{+}_{c}}\rightarrow
p{{K}^{-}}{{\pi}^{+}}$ decay parametrisation is highly non-trivial given the
complexity and model-dependence of the amplitude analysis. In this paper, we
simplify the usage of these results by converting the transition amplitude for
the ${{\mathchar 28931\relax}^{+}_{c}}\rightarrow{p}{{K}^{-}}{{\pi}^{+}}$
decay to a model-agnostic representation using the distribution of the aligned
polarimeter vector $\vec{\alpha}$. The components of the vector are computed
on a two-dimensional grid of the Dalitz-plot variables. Special attention is
devoted to the propagation of uncertainties. The theoretical aspects of the
present paper are in line with the original consideration of the ${{\mathchar
28931\relax}^{+}_{c}}$ polarimetry in three-body decays done by Bjorken [33].
The recent work of Wei et al. [34] discusses the ${{\mathchar
28931\relax}^{+}_{c}}\rightarrow p{{K}^{-}}{{\pi}^{+}}$ decay from a similar
angle, by suggesting measuring the model-independent parameters referred to as
the calibration parameters. We find that these parameters are equal to the
averaged polarimeter components. The values are reported in this document for
completeness. We find a significant reduction of the uncertainty on the
polarisation measurement if the complete polarimetry field is used instead of
the averaged values.

The paper is structured as follows. In Section 2, we discuss general features
of the polarisation measurement in multibody decays and introduce the
polarimeter vector in general form by factoring out the overall rotation.
Section 3 is dedicated to the computation of the polarimeter distribution for
${{\mathchar 28931\relax}^{+}_{c}}\rightarrow{p}{{K}^{-}}{{\pi}^{+}}$ decays,
and we conclude in Section 4. Appendix A presents the amplitude model. The
relation between the model parameters used in this analysis and in the
previous analysis [15] are given in Appendix B. The $C\\!P$ violation effects
in the ${{\mathchar 28931\relax}^{+}_{c}}$ decays are discussed in Appendix C.
Appendix D illustrates how to use the polarimetry fields in spectroscopy
studies by writing the reaction amplitude for $B^{+}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{\kern 1.79993pt\overline{\kern-1.79993pt\mathchar
28931\relax}}{}^{-}_{c}}K^{+}$ decays with all 12 kinematic variables.

##  2 Polarisation sensitivity

We consider a multibody decay of a spin-half fermion decaying to three or more
particles. The total differential decay rate $\mathrm{d}\Gamma/\mathrm{d}\Phi$
for the polarised decay of this fermion is computed as
$\left|\mathcal{M}\right|^{2}/(2m_{0})$, where $m_{0}$ is the mass of the
decaying particle, and $\left|\mathcal{M}\right|^{2}$ is the averaged squared
matrix element. The latter is obtained by combining the transition amplitudes
with their conjugates through the spin-density matrix and summing over all
spin indices,

$\displaystyle\left|\mathcal{M}\right|^{2}=\sum_{\nu_{0},\nu_{0}^{\prime},\\{\lambda\\}}\mathfrak{R}_{\nu_{0}^{\prime},\nu_{0}}T_{\nu_{0}^{\prime},\\{\lambda\\}}^{*}T_{\nu_{0},\\{\lambda\\}}\,,$
(4)

where $T_{\nu_{0}^{(\prime)},\\{\lambda\\}}$ is the transition amplitude,
$\nu_{0}^{(\prime)}$ is the spin projection of the decaying fermion to the $z$
axis of the production plane, and
$\\{\lambda\\}=\\{\lambda_{1},\lambda_{2},\dots,\lambda_{n}\\}$ is the
combined index for the helicities of the final state particles numbered by the
low index $i$. The spin-density matrix $\mathfrak{R}$ is a two-dimensional
matrix in the spin-projection basis computed in the rest frame of the decaying
particle, with three polarisation degrees of freedom,
$\mathfrak{R}_{\nu_{0}^{\prime},\nu_{0}}=1+\vec{P}\cdot\vec{\sigma}^{P}_{\nu_{0}^{\prime},\nu_{0}}$,
where $\vec{P}=(P_{x},P_{y},P_{z})$ is a polarisation vector and
$\vec{\sigma}^{P}=(\sigma^{P}_{x},\sigma^{P}_{y},\sigma^{P}_{z})$ are the
Pauli matrices. The coordinate system for the polarisation vector and the spin
quantisation axis are fixed by the production process. The $z$ axis is given
by the momentum of the decay particle, the $y$ axis is normal to the
production plane, and the $x$ axis completes the right-handed coordinate
system.

The rotation properties of the transition amplitude
$T_{\nu_{0},\\{\lambda\\}}$ are defined by the total spin of the system, which
is $1/2$ for a fermion decay. For every point in the decay phase space, the
configuration of the final-state particle momenta forms a rigid body that can
be rotated to the common aligned configuration [35], where one of the momenta
is chosen as reference for the $z$ axis and the second noncollinear vector is
used to establish the direction perpendicular to the decay plane, the $y$
axis. The entire angular dependence of the decay amplitude is factored out
explicitly,

$\displaystyle
T_{\nu_{0},\\{\lambda\\}}=\sum_{\nu}D_{\nu_{0},\nu}^{1/2*}(\phi,\theta,\chi)A_{\nu,\\{\lambda\\}}(\kappa)\,,$
(5)

where the index $\nu$ is the spin projection of the decaying fermion onto the
$z$ axis of the aligned configuration, $D_{\nu_{0},\nu}^{1/2}$ is the Wigner
$D$-function, $A_{\nu,\\{\lambda\\}}$ is the transition amplitude in the
aligned configuration, and $\kappa$ denotes the kinematic variables which the
aligned reaction amplitude depends on. The number of variables is counted as
$3n-7$, with $n$ being the number of the decay products, and the seven
constraints originate from the four conservation laws and the three overall
rotations of the system. Notably, the Euler angles in the $Z$-$Y$-$Z$
convention shown in Fig. 1 coincide with the regular helicity angles used in
the construction of the decay amplitude as a sequence of two-body transitions.
Namely, for a process $P+Q\rightarrow A(\rightarrow B+C+D)\,+X$, the angles
$(\theta,\phi)$, and $\chi$ are the spherical angles of the momentum sum
$\vec{p}_{B}+\vec{p}_{C}$ in the helicity frame of $A$ obtained from the
centre-of-momentum frame of the reaction, and the polar angle of particle $B$
in the helicity frame of $BC$ obtained from the $A$ rest frame, respectively.

Figure 1: Definition of the decay-plane orientation angles $\phi$, $\theta$,
and $\chi$ related to polarisation of the particle $A$, produced in the
process $P+Q\rightarrow A+X$, and observed in the decay, $A\rightarrow B+C+D$.
The left yellow plane contains the momenta of the production process, while
the right green plane contains the momenta of the decay reaction. The central
blue plane is defined by the momenta $\vec{p}_{X}$ and $\vec{p}_{D}$ in the
rest frame of the decay particle $A$. The white arrows show the orientation of
the two different quantisation axes of the particle $A$ spin in its rest
frame.

By substituting Eq. (5) into Eq. (4), we get an expression for the polarised
decay rate in terms of the kinematic variables and the rotational degrees of
freedom,

$\displaystyle\left|\mathcal{M}\right|^{2}=\sum_{\nu_{0},\nu_{0}^{\prime},\nu,\nu^{\prime}}\mathfrak{R}_{\nu_{0}^{\prime},\nu_{0}}\,\,D_{\nu_{0}^{\prime},\nu^{\prime}}^{1/2*}(\phi,\theta,\chi)\,D_{\nu_{0},\nu}^{1/2}(\phi,\theta,\chi)\,\,\mathfrak{X}_{\nu^{\prime},\nu}(\kappa)\,,$
(6)

where $\mathfrak{X}$ is a hermitian $2\times 2$ matrix that encapsulates the
entire dependence on the decay dynamics. It can be expanded in the basis of
the Pauli matrices and the identity matrix,

$\displaystyle\mathfrak{X}_{\nu^{\prime},\nu}(\kappa)$
$\displaystyle=\sum_{\\{\lambda\\}}A_{\nu^{\prime},\\{\lambda\\}}^{*}(\kappa)A_{\nu,\\{\lambda\\}}(\kappa)\,,$
(7)
$\displaystyle=\frac{I_{0}(\kappa)}{2}\left(1+\vec{\alpha}(\kappa)\cdot\vec{\sigma}^{P}\right)_{\nu^{\prime},\nu}\,.$

Here, $I_{0}(\kappa)$ is the total differential decay rate,
$I_{0}=\sum_{\nu,\\{\lambda\\}}\left|A_{\nu,\\{\lambda\\}}\right|^{2}$. The
aligned polarimeter vector $\vec{\alpha}$ is computed by expanding the squared
decay amplitude in the basis of the Pauli matrices,

$\displaystyle\vec{\alpha}(\kappa)=\sum_{\nu^{\prime},\nu,\\{\lambda\\}}A^{*}_{\nu^{\prime},\\{\lambda\\}}\vec{\sigma}_{\nu^{\prime},\nu}A_{\nu,\\{\lambda\\}}\,\big{/}\,I_{0}(\kappa)\,.$
(8)

Equation (6) is simplified using properties of the rotation group and the
Pauli matrices. The Wigner $D$-matrix in Eq. (5) has $4\pi$ invariance, since
it belongs to the spin-$1/2$ representation of the rotation group. However,
dependence of the physical observables on the rotation must correspond to the
physical, $2\pi$-folded representations. Particularly, the Cornwell theorem
from group theory (see for example Section 3, Chapter 5 of Ref. [36]) gives
the relation between the rotation of the transition amplitude (spin-$1/2$
representation of the SU(2) group) and the rotation of the three-dimensional
vector (spin-1 representation of the SU(2) group). With this, the expression
for the differential decay rate gets a simple form,

$\displaystyle\left|\mathcal{M}(\phi,\theta,\chi,\kappa)\right|^{2}=I_{0}(\kappa)\,\bigg{(}1+\sum_{i,j}P_{i}R_{ij}(\phi,\theta,\chi)\alpha_{j}(\kappa)\bigg{)}\,,$
(9)

where $R_{ij}(\phi,\theta,\chi)$ is a three-dimensional rotation matrix
implementing the Euler transformation to a physical vector. It matches Eq. (2)
with the matrix product, $R\,\vec{\alpha}$, being the polarimeter vector
$\vec{h}$. The quantities $\vec{\alpha}(\kappa)$ give a model-agnostic
representation for polarisation dependence of the decay rate. To incorporate
the decay variables to a more complex reaction, Eq. (7) should be used. For
example, an amplitude analysis of ${{{B}^{+}}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{\kern 1.79993pt\overline{\kern-1.79993pt\mathchar
28931\relax}}{}^{-}_{c}}{{K}^{+}}$, where ${\mathchar 28931\relax}^{+}_{c}$
and ${\kern 1.79993pt\overline{\kern-1.79993pt\mathchar
28931\relax}}{}^{-}_{c}$ decays to $p{{K}^{-}}{{\pi}^{+}}$ and
$\bar{p}{{K}^{+}}{{\pi}^{-}}$ final states, respectively, would greatly
benefit from the polarimeter field $\vec{\alpha}$. The decay rate is expressed
using the $\mathfrak{X}^{{\mathchar 28931\relax}^{+}_{c}}$ and
$\mathfrak{X}^{{\kern 1.25995pt\overline{\kern-1.25995pt\mathchar
28931\relax}}{}^{-}_{c}}$ matrices in Appendix D.

The differential decay rate remains sensitive to the initial polarisation even
after integration over the kinematic variables $\kappa$. In that case, the
rate is only a function of the three production angles,

$\displaystyle\frac{8\pi^{2}}{\Gamma}\frac{\mathrm{d}^{3}\Gamma}{\mathrm{d}\phi\,\mathrm{d}\cos\theta\,\mathrm{d}\chi}=1+\sum_{i,j}P_{i}R_{ij}(\phi,\theta,\chi)\overline{\alpha}_{j}\,,$
(10)

where the components of the averaged aligned polarimeter vector
$\vec{\overline{\alpha}}$ are defined as,

$\displaystyle\overline{\alpha}_{j}=\int
I_{0}\alpha_{j}\mathrm{d}^{n}\kappa\,\big{/}\,\int
I_{0}\,\mathrm{d}^{n}\kappa\,.$ (11)

The integration over the kinematic variables simplifies the analysis, but
leads to an increased uncertainty on the results. As discussed below, the
method proposed in Ref. [24] can be used to quantify this effect.

The polarisation measurement for ${\tau^{-}}$ lepton decays is formulated in
general terms in Refs. [37, 24]. The authors introduce a scalar polarisation
sensitivity $S=1/(\sigma\sqrt{N})$, where $N$ is the number of events in the
sample analysed using the unbinned likelihood method and $\sigma$ is the
expected uncertainty for the value of the polarisation. The method has been
applied to evaluate the scalar polarisation sensitivity of the ${{\mathchar
28931\relax}^{+}_{c}}\rightarrow p{{K}^{-}}{{\pi}^{+}}$ decays in Refs. [17,
15]. If the probability density function for the decay rate reads as $W=f+pg$,
with $f$ and $g$ being the normalised polarisation-independent and
polarisation-dependent terms of the decay rate, respectively, and $p$ being a
small component of the polarisation vector, then the squared polarisation
sensitivity $S^{2}$, computed as the second derivative of the negative log-
likelihood, is equal to the variance of the ratio, $g/f$ [24]. Using Eq. (9),
it is straightforward to identify the $f$ and $g$ functions:
$f=I_{0}(\kappa)/f_{0}$, and
$g=\sum_{j}I_{0}(\kappa)R_{3j}(\phi,\theta,\chi)\alpha_{j}(\kappa)/f_{0}$,
where $f_{0}$ is the normalisation constant. For small polarisation values,
the variance can be evaluated for $p=0$, so that the three integrals over the
Euler angles are analytic.

$\displaystyle S_{0}^{2}$
$\displaystyle=\int\left.\frac{g^{2}}{f+p\,g}\right|_{p=0}\mathrm{d}^{3}(\phi,\theta,\chi)\,\mathrm{d}^{n}\kappa$
$\displaystyle=\frac{1}{3}\int
I_{0}\left|\vec{\alpha}\right|^{2}\mathrm{d}^{n}\kappa\,\big{/}\,\int
I_{0}\,\mathrm{d}^{n}\kappa\,,$ (12)

where the factor $3$ reflects the vector properties of $\vec{\alpha}$.
Equation (12) has a simple interpretation: the polarisation sensitivity is
proportional to the averaged length of the polarimeter vector over the phase
space domain, weighted by the probability distribution for the unpolarised
decay. When an analysis only uses angular variables, as in Eq. (10), the
polarisation sensitivity can be computed from the length of the averaged
polarimeter vector with
$\overline{S}_{0}^{2}=(\overline{\alpha}_{x}^{2}+\overline{\alpha}_{y}^{2}+\overline{\alpha}_{z}^{2})/3$.
The ratio $S_{0}/\overline{S}_{0}$ gives the expected increase of the
statistical uncertainty on the polarisation value if the kinematic dependence
is ignored.

##  3 Polarisation sensitivity for $\boldsymbol{{{\mathchar
28931\relax}^{+}_{c}}\rightarrow p{{K}^{-}}{{\pi}^{+}}}$ decays

A three-body decay is the smallest system where the polarimetry field develops
a non-trivial behaviour. In this case, the aligned amplitude depends on two
kinematic variables. The natural choice for these degrees of freedom are the
Mandelstam variables, often presented in a Dalitz plot [38], in which the
aligned polarimeter vector forms a continuous vector field.

To compute the polarisation sensitivity for the ${{\mathchar
28931\relax}^{+}_{c}}\rightarrow p{{K}^{-}}{{\pi}^{+}}$ decays, the results of
the recent amplitude analysis of this decay in Ref. [15] are used. The model
of Ref. [15] is mapped to the conventions used in the present analysis in
Appendix B. There are two important differences in our setup. First, the decay
plane in the ${{\mathchar 28931\relax}^{+}_{c}}$ rest frame is oriented using
the $K^{**0}$ decay chains as a reference: the sum of the pion and kaon
momenta defines the positive $z$ direction and the frame is oriented such that
the $x$ component of the pion momentum is positive. For the previous analysis,
the proton is aligned with the $z$ axis and the $x$ component of the pion
momentum is negative, i.e. all momenta vectors are rotated by $\pi$ about the
$y$ axis. Second, the alignment of the decay chains is performed for the
helicity state of the proton by matching the rotations as worked out in Ref.
[35] and the alignment rotations are expressed via the Thomas angle (see
derivation on p. 215 of Ref. [39]) for the canonical state of the proton in
the model of Ref. [15]. The couplings are defined to match the two ways of
formulating the amplitude. The obtained model is validated against the
implementation of Ref. [15] up to differences in floating-point precision.

The distribution of the $\vec{\alpha}$ components is shown in Fig. 2. The
polarimeter vector at the phase-space point given in the Dalitz-plot
coordinates, $m^{2}({{K}^{-}}{{\pi}^{+}})$ and $m^{2}({{K}^{-}}p)$, is
indicated with an arrow projected onto the $xz$ plane.

Figure 2: Aligned polarimeter vector field in Dalitz-plot coordinates. The $z$
and $x$ components of the $\alpha$ vector are shown by the horizontal and
vertical projections of the arrow, respectively. The colour indicates the
length of the polarimeter vector. The sketch in the top right corner shows the
decay-plane orientation. The momentum arrows for the pion and the kaon are
shown in gray, since their orientation depends on the kinematic variables,
$m^{2}({{K}^{-}}{{\pi}^{+}})$ and $m^{2}(p{{K}^{-}})$. The shaded regions
indicate where the decay rate of each subsystem exceeds 70%. These decay rates
are computed by setting the helicity of all decay chains in the other
subsystems to zero.

The length of the polarimeter vector, shown by the colour, changes from point
to point. However, it is greater than $0.5$ for most of the kinematic domain,
indicating significant contributions of both parity-conserving and parity-
violating currents [40]. The structures in Fig. 2 are driven by resonances in
different subsystems and their interference. For the ${{\mathchar
28931\relax}^{+}_{c}}$ baryon decaying to a baryon and a (pseudo)scalar, the
aligned polarimeter vector points in the same direction as the momentum of one
of the two particles. The resonance then decays, but this does not influence
the direction of the polarimeter vector. If the $z$ axis is chosen parallel to
the momentum of the resonance of the decay chain, the vector map is
homogeneous, i.e. the polarimeter vector is the same at every point in the
phase space, as shown on the left panel of Fig. 3. For a combination of many
decay chains, $\vec{\alpha}$ is not aligned with any momenta. In fact,
interference between the decay chains might cause $\alpha_{y}\neq 0$, meaning
that the $\vec{\alpha}$ vector points out of the momentum plane.

If an alternative aligned configuration is obtained from the default one by
the rotation $R$, the $\alpha$ vector is transformed under the inverse
rotation,

$\displaystyle\alpha_{i}\xrightarrow[R]{}\alpha_{i}^{\prime}=\sum_{j}R^{-1}_{ij}\alpha_{j}\,,$
(13)

where $\vec{\alpha}^{\,\prime}$ is the aligned polarimeter vector for the
transformed configuration. The default alignment choice in this paper is
$\vec{z}\uparrow\uparrow-\vec{p}_{p}$, i.e. the proton momentum is opposite to
the $z$ axis. To update the convention for either
$\vec{z}\uparrow\uparrow-\vec{p}_{{{\pi}^{+}}}$ or
$\vec{z}\uparrow\uparrow-\vec{p}_{{{K}^{-}}}$, one needs to apply
$R_{y}^{-1}(\zeta^{0}_{i(1)})$ for every point of the $\vec{\alpha}$ field,
where $i=2$ or $i=3$, respectively. The angle $\zeta^{0}_{i(1)}$ is the Wigner
angle for spin alignment of the ${{\mathchar 28931\relax}^{+}_{c}}$ baryon, as
given in [35]. Figure 3 shows the $\vec{\alpha}$ field for the ${{\mathchar
28931\relax}^{+}_{c}}\rightarrow{\mathchar 28931\relax}(1520)\pi^{+}$ decay
chain with two different decay-plane alignments. For this decay chain, the
aligned polarimeter vector always points along the momentum of the ${\mathchar
28931\relax}(1520)$ resonance. When the proton momentum is used as a reference
for the $z$ axis, the polarimeter vector is tilted for every point on the map.

Figure 3: The aligned polarimeter vector field in Dalitz-plot coordinates for
the ${{\mathchar 28931\relax}^{+}_{c}}\rightarrow{\mathchar
28931\relax}(1520)(\rightarrow pK^{-})\pi^{+}$ decay chain only. The decay
plane in the rest frame of the ${{\mathchar 28931\relax}^{+}_{c}}$ baryon is
aligned to the $K^{**0}$ decay chains and ${\mathchar 28931\relax}^{**}$ decay
chain for the left and right panel, respectively. See Fig. 2 for notations.

The rotation $R$ in Eq. (13) that relates the alternative conventions for the
Euler angles is global, i.e. it does not depend on the kinematic variables.
For example, in Ref. [15] the $z$ axis in the aligned configuration is
determined by the momentum of the proton, rather than the opposite direction,
as done here. With this convention, we get $R=R_{y}(\pi)$, which flips the
sign of $\alpha_{x}$ and $\alpha_{z}$ for every point of the Dalitz plot. By
using the alternative particle ordering, $p(1)K^{-}(2)\pi^{+}(3)$, it can be
seen that the flip of the decay plane corresponds to the global $R_{z}(\pi)$
rotation. In the conventions of Ref. [34], the proton momentum determines the
$x$ axis with $xy$ being the decay plane, and the kaon momentum having the
positive $y$ projection. This aligned configuration is obtained with the
rotation $R_{W}=R_{z}(\pi/2)R_{y}(\pi/2)$.

The uncertainty on the results of the amplitude analysis consists of three
classes: the statistical uncertainty, related to the size of the data sample,
the model uncertainty, inferred from a number of alternative setups, and the
systematic uncertainty, which originates from experimental unknowns. These
uncertainties are propagated to the computed polarimeter vector fields and
other inferred quantities.

The statistical uncertainties are computed using the corresponding uncertainty
on the parameters for the default model of Ref. [15]. The parameter values are
assumed to be Gaussian-distributed and uncorrelated. A sample of 100 parameter
sets is drawn from the model parameters to compute the distribution of the
inferred quantities. To preserve correlations of the polarimeter field in the
kinematic domain, the sampled $\vec{\alpha}$ distributions are provided as
supplementary material [41].

The model uncertainties are computed from the alternative models that are
described in Ref. [15]. The polarimeter field is computed for every
alternative model and presented in supplemental material for this paper. For
scalar inferred quantities, the model uncertainty is computed by the extrema
of the sample of alternative models. These model uncertainties dominate over
the statistical uncertainties for all the checked quantities. Once the
$\vec{\alpha}$ field is used for the polarisation measurements with Eq. (9),
or for a spectroscopy analysis (see Appendix D for an example), it becomes
straightforward to propagate all uncertainties by simply replacing the field
of the default model to the fields provided in the supplementary material.

The systematic uncertainty is found to be negligible compared to the
statistical and model uncertainties. The effect is accounted for by inflating
the statistical uncertainties used in the resampling, namely, the statistic
and systematic uncertainties on the sampled parameters are added in
quadrature. Figure 4 summarises the effect of the uncertainties on the
polarimetry field. The model uncertainty dominates over the other sources of
uncertainty. Areas with the largest uncertainties overlap with regions where
there is a small number of events and where tails of different resonances
interfere. All computations are performed with libraries from the ComPWA
project[42] and cross-checked with the ThreeBodyDecay.jl package[43].

Figure 4: Uncertainties on the length of the aligned polarimeter vector for
each phase-space point. The left panel shows combined statistical and
systematic uncertainties and the right panel shows the model uncertainties.

A distinct feature of multibody decays with respect to two-body processes is
that the averaged value of the polarimeter vector defined by Eq. (11) has
transverse components. The averaged asymmetry $\vec{\overline{\alpha}}$ is
computed with Eq. (10). The result for the averaged aligned polarimeter vector
is shown in Table 1.

Table 1: Averaged values of the polarimetry fields in Cartesian and spherical coordinates. The central value and the model uncertainties are shown in the second and fourth columns, respectively. The second column presents the standard deviation of the distribution in the parameter sampling that includes both statistical and systematic uncertainties. Observable | Central | Stat. and syst. | Model
---|---|---|---
$\overline{\alpha}_{x}$ | [$\%$] | $-6.26$ | $0.45$ | $1.48$
$\overline{\alpha}_{y}$ | [$\%$] | $0.89$ | $0.89$ | $1.27$
$\overline{\alpha}_{z}$ | [$\%$] | $-27.80$ | $2.37$ | $4.04$
$\overline{\alpha}$ | [$\%$] | $28.51$ | $2.40$ | $3.79$
$\overline{\theta}$ | [rad] | $2.92$ | $<0.01$ | $0.05$
$\overline{\phi}$ | [rad] | $3.00$ | $0.14$ | $0.21$

In addition to the Cartesian components, the table gives the averaged vector
in spherical coordinates, with $\overline{\alpha}$, $\overline{\theta}$, and
$\overline{\phi}$ being the length of the averaged polarimeter vector and its
polar and azimuthal angles, respectively. Correlation coefficients in the
spherical representation are expected to be lower. Indeed, we find the
correlation over the uncertainties on the
$(\overline{\alpha}_{z},\overline{\alpha}_{x})$ components to be about
$100\%$, and the correlation coefficient of
$(\overline{\alpha}_{y},\overline{\alpha}_{x})$,
$(\overline{\alpha}_{z},\overline{\alpha}_{y})$ to be both about $10\%$. The
correlation of the spherical components is found to be $70\%$, $50\%$, and
$10\%$ for $(\overline{\alpha},\overline{\theta})$,
$(\overline{\theta},\overline{\phi})$, and
$(\overline{\alpha},\overline{\phi})$, respectively. The non-zero value of
$\overline{\alpha}_{y}$ is caused by the interference of various decay chains
contributing to the ${{\mathchar 28931\relax}^{+}_{c}}$ decays, since it
vanishes for all chains considered individually. The polarisation sensitivity
$\sqrt{3}\;\overline{S}_{0}$ with the averaged polarimeter vector is found to
be $0.29\pm 0.02\pm 0.04$. It can be compared to $\sqrt{3}\,S_{0}=0.67\pm
0.01\pm 0.02$ when the polarimetry field is used. Hence, the polarisation can
also be measured using the integrated values from Eq. (10) at the cost of an
increased statistical uncertainty on the polarisation by, roughly, a factor
$2.35$. The averaged values correspond to the calibration parameters, $G_{0}$,
$G_{1}$, and $G_{2}$, discussed in Ref. [34], up to a convention on the
overall rotation, $R_{W}$, discussed above. Equation (13) suggests the
translation between these calibration parameters and the averaged polarimeter
components: $G_{0}=\overline{\alpha}_{y}$, $G_{1}=-\overline{\alpha}_{x}$, and
$G_{2}=-\overline{\alpha}_{z}$.

The polarimeter vector for the charge-conjugated mode is obtained using exact
$C\\!P$ symmetry (see Appendix C for the details),

$\displaystyle\alpha_{x}^{{{\kern 1.25995pt\overline{\kern-1.25995pt\mathchar
28931\relax}}{}^{-}_{c}}}$ $\displaystyle=-\alpha_{x}^{{{\mathchar
28931\relax}^{+}_{c}}}\,,$ $\displaystyle\alpha_{y}^{{{\kern
1.25995pt\overline{\kern-1.25995pt\mathchar 28931\relax}}{}^{-}_{c}}}$
$\displaystyle=\alpha_{y}^{{{\mathchar 28931\relax}^{+}_{c}}}\,,$
$\displaystyle\alpha_{z}^{{{\kern 1.25995pt\overline{\kern-1.25995pt\mathchar
28931\relax}}{}^{-}_{c}}}$ $\displaystyle=-\alpha_{z}^{{{\mathchar
28931\relax}^{+}_{c}}}\,.$ (14)

Any deviation from these relations would indicate $C\\!P$ violation. Given
that $C\\!P$ violation effects are expected to be negligible for polarisation
studies, the expression for the $\vec{\alpha}_{{{\kern
1.25995pt\overline{\kern-1.25995pt\mathchar 28931\relax}}{}^{-}_{c}}}$
obtained under exact $C\\!P$ symmetry can be used for the ${{\kern
1.79993pt\overline{\kern-1.79993pt\mathchar 28931\relax}}{}^{-}_{c}}$ decays
(see Appendix D for an example).

##  4 Summary

The distribution of the spin state of a fermion contains important information
on its production process. Polarisation can be considered as a tool for
enriching the list of observables sensitive to fundamental physics, ranging
from the search of new phenomena beyond the Standard Model to studies of the
fundamentals of QCD. To access the polarisation of a fermion in a decay, the
transition amplitude needs to be known. More precisely, one needs to
characterise how the kinematic distributions of the decay products reflect the
initial polarisation.

In this paper, a model-agnostic representation of the fermion decay rate on
the entire space of kinematic dimensions for a multibody decay of a fermion is
introduced. The polarised decay rate has been expressed in terms of Euler
angles of the decay-plane orientation and a kinematics-dependent, aligned
polarimeter vector. The distribution of the polarimeter vector has been
explored for the dominant ${{\mathchar 28931\relax}^{+}_{c}}\rightarrow
p{{K}^{-}}{{\pi}^{+}}$ hadronic decay. Using the recent amplitude analysis
results of Ref. [15] by LHCb, the polarimeter vector distribution has been
evaluated. The statistical and model uncertainties are carefully propagated to
the inferred quantities based on the original LHCb analysis. Knowledge of the
polarimeter distribution of the ${{\mathchar 28931\relax}^{+}_{c}}$ baryon in
its main hadronic decay mode opens many new avenues for testing the Standard
Model and determining the properties of strong-interacting systems. The
numerical results provided in this paper facilitate the use of the
${{\mathchar 28931\relax}^{+}_{c}}\rightarrow{p}{{K}^{-}}{{\pi}^{+}}$
transition in any practical case without the need to implement amplitude
models.

### Code Availability Statement

A software used for this paper is available at Zenodo repository [41]. The
analysis documentation is hosted at the webpage, lc2pkpi-
polarimetry.docs.cern.ch.

##  Appendix A Decay amplitude for the ${{\mathchar
28931\relax}^{+}_{c}}\rightarrow p{{K}^{-}}{{\pi}^{+}}$ decay in the helicity
formalism

The helicity amplitude for the ${{\mathchar
28931\relax}^{+}_{c}}\rightarrow{p}{{K}^{-}}{{\pi}^{+}}$ decay reads as a sum
of three partial wave series, incorporating $K^{**0}$, ${\mathchar
28929\relax}^{**++}$, and ${\mathchar 28931\relax}^{**}$ resonances. Numbering
the particles as ${{\mathchar
28931\relax}^{+}_{c}}(0)\rightarrow{p}(1){{\pi}^{+}}(2){{K}^{-}}(3)$, the
aligned amplitude reads

$\displaystyle\mathcal{A}_{\nu,\lambda}(m_{K\pi},m_{pK})$
$\displaystyle=\sum_{k=1}^{3}\sum_{\nu^{\prime},\lambda^{\prime}}d_{\nu,\nu^{\prime}}^{1/2}(\zeta_{k(1)}^{0})\,\mathcal{A}_{\nu^{\prime},\lambda^{\prime}}^{\xi_{k}}\,d_{\lambda^{\prime},\lambda}^{1/2}(\zeta_{k(1)}^{1})\,,$
(15)

where $d_{\nu,\nu^{\prime}}^{1/2}$ is the Wigner $d$-function,
$\zeta^{1}_{k(1)}$ and $\zeta^{0}_{k(1)}$ are the Wigner rotation angles for
the proton (particle $1$) and the ${{\mathchar 28931\relax}^{+}_{c}}$ baryon
(particle 0). The index $\xi_{k}$ labels the amplitudes for the subsystems,
with $k=1$, $k=2$, and $k=3$ representing the subsystem of the $K^{**0}$,
${\mathchar 28929\relax}^{**++}$, and ${\mathchar 28931\relax}^{**}$
resonances, respectively. These amplitudes read

$\displaystyle\mathcal{A}^{\xi_{k}}_{\nu,\lambda}$
$\displaystyle=\sum_{j,\tau}\delta_{\nu,\tau-\lambda_{k}}\mathcal{H}^{{{\mathchar
28931\relax}^{+}_{c}}\rightarrow\xi_{k},P_{k}}_{\tau,\lambda_{k}}(-1)^{j_{k}-\lambda_{k}}\,d^{j}_{\lambda,0}(\theta_{ij})\,\mathcal{H}^{\xi\rightarrow
P_{i},P_{j}}_{\lambda_{i},\lambda_{j}}(-1)^{j_{j}-\lambda_{j}}\,.$ (16)

The helicity couplings in the strong decay of subchannel resonances obey
simple properties with respect to the parity transformation, which reduces the
number of couplings in the strong decay of isobars. Moreover, the magnitude of
the couplings cannot be determined separately and is set to 1, so that

$\displaystyle\mathcal{H}^{{\mathchar 28931\relax}^{**}\rightarrow
Kp}_{0,1/2}$ $\displaystyle=1\,,$ $\displaystyle\mathcal{H}^{{\mathchar
28929\relax}^{**}\rightarrow p\pi}_{1/2,0}$ $\displaystyle=1\,,$
$\displaystyle\mathcal{H}^{K^{**}\rightarrow\pi K}_{0,0}$ $\displaystyle=1,$
$\displaystyle\mathcal{H}^{{\mathchar 28931\relax}^{**}\rightarrow
Kp}_{0,-1/2}$ $\displaystyle=-P_{\Lambda}(-1)^{j-1/2}\,,$
$\displaystyle\mathcal{H}^{{\mathchar 28929\relax}^{**}\rightarrow
p\pi}_{-1/2,0}$ $\displaystyle=-P_{\Delta}(-1)^{j-1/2}\,.$ (17)

Angles with repeated lower indices are zero. The other angles are computed
from the Mandelstam variables using Appendix A of Ref. [35]. Note that the
angles $\zeta^{0}$ here are denoted by $\hat{\theta}$ in Ref. [35].

##  Appendix B Matching to the LHCb model

The decay plane used for formulating the amplitude model in Ref. [15] is
defined by the momenta of the proton and kaon in the ${{\mathchar
28931\relax}^{+}_{c}}$ rest frame. The $z$ axis is set by the proton momentum,
while the $y$ axis is orthogonal to the kaon and proton momenta. The
transition amplitude from Ref. [15] reads

$\displaystyle\mathcal{B}_{\nu,\lambda_{p}}$
$\displaystyle=\sum_{\lambda}\mathcal{K}^{{{\mathchar
28931\relax}^{+}_{c}}\rightarrow
K^{**}p}_{\lambda,\lambda_{p}}\delta_{\nu,\lambda+\lambda_{p}}\,d^{J_{K}}_{\lambda,0}(\theta_{K})$
$\displaystyle\quad+\sum_{\lambda,\mu_{p}}\mathcal{K}^{{{\mathchar
28931\relax}^{+}_{c}}\rightarrow{\mathchar
28931\relax}^{**}\pi}_{\lambda}\mathcal{K}^{\Lambda\rightarrow
pK}_{\mu_{p}}d^{1/2}_{\nu,\lambda}(\theta_{\Lambda})\,d^{J_{\Lambda}}_{\lambda,\mu_{p}}(\theta_{p}^{\Lambda})\,d^{1/2}_{\mu_{p},\lambda_{p}}(-(\theta_{p}^{\Lambda}+\theta_{\Lambda}+\alpha^{\Lambda}_{W}))$
$\displaystyle\quad+\sum_{\lambda,\mu_{p}}\mathcal{K}^{{{\mathchar
28931\relax}^{+}_{c}}\rightarrow{\mathchar
28929\relax}^{**}K}_{\lambda}\mathcal{K}^{\Delta\rightarrow
p\pi}_{\mu_{p}}d^{1/2}_{\nu,\lambda}(\theta_{\Delta})\,d^{J_{\Delta}}_{\lambda,\mu_{p}}(\theta_{p}^{\Delta})\,d^{1/2}_{\mu_{p},\lambda_{p}}(-(\theta_{p}^{\Delta}+\theta_{\Delta}-\alpha^{\Delta}_{W}))\,,$
(18)

where $\mathcal{K}^{A\rightarrow BC}_{\lambda_{B},\lambda_{C}}$ represents the
helicity coupling constant for the two-body decay $A\rightarrow BC$ with
$\lambda_{B}$ and $\lambda_{c}$ being the helicities of particles $B$ and $C$,
respectively. The helicity index for the pseudoscalar particles is omitted. An
algorithm to compute the helicity angles, denoted by the letter $\theta$, can
be found in Ref. [15]. The spin-alignment rotation is expressed via the Thomas
angle (see Ref. [39]) denoted by $\alpha_{W}^{\Lambda}$ and
$\alpha_{W}^{\Delta}$. They are related to the positive angles in Eq. (15) and
Eq. (16) as follows:

$\displaystyle\bar{\theta}_{K}$ $\displaystyle=\theta_{23}\,,$
$\displaystyle\theta_{\Lambda}$ $\displaystyle=\pi-\zeta^{0}_{1(2)}\,,$
$\displaystyle\theta_{p}^{\Lambda}$ $\displaystyle=-(\pi-\theta_{31})\,,$
$\displaystyle\theta_{p}^{\Lambda}+\theta_{\Lambda}+\alpha^{\Lambda}_{W}$
$\displaystyle=-\zeta_{2(1)}^{1}\,,$ $\displaystyle\theta_{\Delta}$
$\displaystyle=-(\pi-\zeta^{0}_{3(1)})\,,$ $\displaystyle\theta_{p}^{\Delta}$
$\displaystyle=\theta_{12}\,,$
$\displaystyle\theta_{p}^{\Delta}+\theta_{\Delta}-\alpha^{\Delta}_{W}$
$\displaystyle=\zeta_{1(3)}^{1}\,.$ (19)

Equation (B) can be simplified using the properties of the Wigner
$d$-functions. The relation of the full amplitudes from Eq. (B) and Eq. (15)
is obtained by isolating the phase factor that depends on the external
helicity indices, which gives

$\displaystyle\mathcal{B}_{\nu,\lambda_{p}}$
$\displaystyle=(-1)^{1/2+\lambda_{p}}\mathcal{A}_{\nu,-\lambda_{p}}\,.$ (20)

The remaining phase factors redefine the product of the helicity couplings, so
that the helicity couplings $\mathcal{K}$ from Ref. [15] are related to the
helicity couplings $\mathcal{H}$ from this paper as

$\displaystyle\mathcal{K}^{{{\mathchar 28931\relax}^{+}_{c}}\rightarrow
K^{**}p}_{\lambda\lambda_{p}}$ $\displaystyle=\mathcal{H}^{{{\mathchar
28931\relax}^{+}_{c}}\rightarrow
K^{**}p}_{\lambda-\lambda_{p}}(-1)^{J_{K}}\,,$
$\displaystyle\mathcal{K}^{{{\mathchar
28931\relax}^{+}_{c}}\rightarrow{\mathchar
28931\relax}^{**}\pi}_{-\lambda}\mathcal{K}^{\Lambda\rightarrow
pK}_{-\mu_{p}}$ $\displaystyle=\mathcal{H}^{{{\mathchar
28931\relax}^{+}_{c}}\rightarrow{\mathchar
28931\relax}^{**}\pi}_{\lambda}(-1)^{J_{\Lambda}-1/2}\mathcal{H}^{\Lambda\rightarrow
pK}_{\mu_{p}}\,,$ $\displaystyle\mathcal{K}^{{{\mathchar
28931\relax}^{+}_{c}}\rightarrow{\mathchar
28929\relax}^{**}K}_{-\lambda}\mathcal{K}^{\Delta\rightarrow p\pi}_{-\mu_{p}}$
$\displaystyle=\mathcal{H}^{{{\mathchar
28931\relax}^{+}_{c}}\rightarrow{\mathchar
28929\relax}^{**}K}_{\lambda}\mathcal{H}^{\Delta\rightarrow\pi
p}_{\mu_{p}}\,.$ (21)

The decay couplings are fixed by the parity relation in Eq. (A) and can be
eliminated from Eq. (B) leading to the final mapping of the model parameters,

$\displaystyle\mathcal{H}^{{{\mathchar 28931\relax}^{+}_{c}}\rightarrow
K^{**}p}_{\lambda,\lambda_{p}}$
$\displaystyle=(-1)^{J_{K}}\mathcal{K}^{{{\mathchar
28931\relax}^{+}_{c}}\rightarrow K^{**}p}_{\lambda,-\lambda_{p}}\,,$
$\displaystyle\mathcal{H}^{{{\mathchar
28931\relax}^{+}_{c}}\rightarrow{\mathchar 28931\relax}^{**}\pi}_{\lambda}$
$\displaystyle=-P_{\Lambda}\mathcal{K}^{{{\mathchar
28931\relax}^{+}_{c}}\rightarrow{\mathchar
28931\relax}^{**}\pi}_{-\lambda}\,,$ $\displaystyle\mathcal{H}^{{{\mathchar
28931\relax}^{+}_{c}}\rightarrow{\mathchar 28929\relax}^{**}K}_{\lambda}$
$\displaystyle=-P_{\Delta}(-1)^{J_{\Delta}-1/2}\mathcal{K}^{{{\mathchar
28931\relax}^{+}_{c}}\rightarrow{\mathchar 28929\relax}^{**}K}_{-\lambda}\,.$
(22)

##  Appendix C Charge conjugation

The transition amplitude for the ${{\kern
1.79993pt\overline{\kern-1.79993pt\mathchar
28931\relax}}{}^{-}_{c}}\rightarrow\bar{p}{{K}^{+}}{{\pi}^{-}}$ decay is
constructed by applying the $C$ and $P$ operators to Eq. (5). The particles
are replaced by anti-particles (charge conjugation), the three momenta of the
final-state particles are flipped, and the helicity values are flipped (parity
inversion in the ${\mathchar 28931\relax}^{+}_{c}$ baryon rest frame). The
amplitude reads

$\displaystyle T_{\nu_{0},\lambda}^{{{\kern
1.25995pt\overline{\kern-1.25995pt\mathchar
28931\relax}}{}^{-}_{c}}}(\vec{p}_{1},\vec{p}_{2},\vec{p}_{3})=T_{\nu_{0},-\lambda}^{{{\mathchar
28931\relax}^{+}_{c}}}(-\vec{p}_{1},-\vec{p}_{2},-\vec{p}_{3}),$ (23)

where $\nu_{0}$ is a spin projection of the ${\mathchar 28931\relax}^{+}_{c}$
baryon onto the $z$ axis of the production plane. The amplitude is not flipped
under parity inversion (see Eq. 3.9 in Ref. [44]).

When the momenta of the final-state particles are flipped in space, the
polarisation angles transform as

$\displaystyle\phi$
$\displaystyle\rightarrow\left\\{\begin{array}[]{cl}\pi+\phi,&\text{ for
}\phi<0,\\\ -\pi+\phi,&\text{ for }\phi>0\end{array}\right.$
$\displaystyle\theta$ $\displaystyle\rightarrow\pi-\theta$ $\displaystyle\chi$
$\displaystyle\rightarrow\left\\{\begin{array}[]{cl}-\pi-\chi,&\text{ for
}\chi<0\,,\\\ \pi-\chi,&\text{ for }\chi>0\,.\end{array}\right.$ (28)

Using properties of the Wigner D-function,

$\displaystyle D_{\nu_{0},\nu}^{1/2}(\phi,\theta,\chi)$
$\displaystyle\rightarrow
D_{\nu_{0},-\nu}^{1/2}(\phi,\theta,\chi)\,e^{i\nu_{0}\pi}(-1)^{1/2+\nu_{0}}e^{i\nu\pi},$
(29)

we find that the amplitude transforms as

$\displaystyle O^{\nu}_{\lambda}\rightarrow
e^{-i\nu\pi}O^{-\nu}_{-\lambda}\,.$ (30)

With the definition of $\vec{\alpha}$ in Eq. (8), one finds a sign flip of
$\alpha_{x}$ and $\alpha_{z}$, while $\alpha_{y}$ remains the same for both
charge modes, therefore,

$\vec{\alpha}^{{{\kern 1.25995pt\overline{\kern-1.25995pt\mathchar
28931\relax}}{}^{-}_{c}}}=R_{y}(\pi)\vec{\alpha}^{{{\mathchar
28931\relax}^{+}_{c}}}\,,$ (31)

which matches Eq. (14).

##  Appendix D Example of extending amplitude models with the polarimeter
field

Knowledge of the ${{\mathchar 28931\relax}^{+}_{c}}\rightarrow
p{{K}^{-}}{{\pi}^{+}}$ transition can be incorporated to any decay process
involving the decay as a subprocess using the polarimeter vector. The
${{{B}^{+}}}\rightarrow{{\mathchar 28931\relax}^{+}_{c}}{{\kern
1.79993pt\overline{\kern-1.79993pt\mathchar 28931\relax}}{}^{-}_{c}}{{K}^{+}}$
decay is used as an example to demonstrate how to incorporate the polarimetry
fields into the decay amplitude. The decay amplitude is a function of 12
variables: two Mandelstam invariant variables for the $B$ meson decay, three
angles and two Dalitz plot variables for each of the ${{\mathchar
28931\relax}^{+}_{c}}$ and ${\kern 1.79993pt\overline{\kern-1.79993pt\mathchar
28931\relax}}{}^{-}_{c}$ decays,

$\displaystyle\mathcal{A}_{\lambda,\bar{\lambda}}(m_{{{\mathchar
28931\relax}^{+}_{c}}{{K}^{+}}}^{2},m_{{{\kern
1.25995pt\overline{\kern-1.25995pt\mathchar
28931\relax}}{}^{-}_{c}}{{K}^{+}}}^{2};\phi,\theta,\chi,m_{p{{K}^{-}}}^{2},m_{{{K}^{-}}{{\pi}^{+}}}^{2};\bar{\phi},\bar{\theta},\bar{\chi},m_{\bar{p}{{K}^{+}}}^{2},m_{{{K}^{+}}{{\pi}^{-}}}^{2})=$
$\displaystyle\quad\sum_{\nu_{0},\bar{\nu}_{0},\nu,\bar{\nu}}O_{\nu_{0},\bar{\nu}_{0}}^{B}(m_{{{\mathchar
28931\relax}^{+}_{c}}{{K}^{+}}}^{2},m_{{{\kern
1.25995pt\overline{\kern-1.25995pt\mathchar
28931\relax}}{}^{-}_{c}}{{K}^{+}}}^{2})$ $\displaystyle\qquad\qquad\times
D^{1/2}_{\nu_{0},\nu}(\phi,\theta,\chi)O_{\nu,\lambda}^{{\mathchar
28931\relax}^{+}_{c}}(m_{p{{K}^{-}}}^{2},m_{{{K}^{-}}{{\pi}^{+}}}^{2})$
$\displaystyle\qquad\qquad\times
D^{1/2}_{\bar{\nu}_{0},\bar{\nu}}(\bar{\phi},\bar{\theta},\bar{\chi})O_{\bar{\nu},\bar{\lambda}}^{{\kern
1.25995pt\overline{\kern-1.25995pt\mathchar
28931\relax}}{}^{-}_{c}}(m_{\bar{p}{{K}^{+}}}^{2},m_{{{K}^{+}}{{\pi}^{-}}}^{2})\,,$
(32)

where $\lambda$ and $\bar{\lambda}$ are the helicity of the proton and anti-
proton in the ${\mathchar 28931\relax}^{+}_{c}$ and ${\kern
1.79993pt\overline{\kern-1.79993pt\mathchar 28931\relax}}{}^{-}_{c}$ rest
frames, respectively.

Then, the decay rate reads

$\displaystyle I(m_{{{\mathchar 28931\relax}^{+}_{c}}{{K}^{+}}}^{2},m_{{{\kern
1.25995pt\overline{\kern-1.25995pt\mathchar
28931\relax}}{}^{-}_{c}}{{K}^{+}}}^{2};\phi,\theta,\chi,m_{p{{K}^{-}}}^{2},m_{{{K}^{-}}{{\pi}^{+}}}^{2};\bar{\phi},\bar{\theta},\bar{\chi},m_{\bar{p}{{K}^{+}}}^{2},m_{{{K}^{+}}{{\pi}^{-}}}^{2})=$
$\displaystyle\quad\sum_{\nu_{0},\bar{\nu}_{0},\nu,\bar{\nu}}\sum_{\nu_{0}^{\prime},\bar{\nu}_{0}^{\prime},\nu^{\prime},\bar{\nu}^{\prime}}O_{\nu_{0}^{\prime},\bar{\nu}_{0}^{\prime}}^{B*}(m_{{{\mathchar
28931\relax}^{+}_{c}}{{K}^{+}}}^{2},m_{{{\kern
1.25995pt\overline{\kern-1.25995pt\mathchar
28931\relax}}{}^{-}_{c}}{{K}^{+}}}^{2})O_{\nu_{0},\bar{\nu}_{0}}^{B}(m_{{{\mathchar
28931\relax}^{+}_{c}}{{K}^{+}}}^{2},m_{{{\kern
1.25995pt\overline{\kern-1.25995pt\mathchar
28931\relax}}{}^{-}_{c}}{{K}^{+}}}^{2})$ $\displaystyle\qquad\qquad\times
D^{1/2*}_{\nu_{0}^{\prime},\nu^{\prime}}(\phi,\theta,\chi)D^{1/2}_{\nu_{0},\nu}(\phi,\theta,\chi)\mathfrak{X}_{\nu^{\prime},\nu}^{{\mathchar
28931\relax}^{+}_{c}}(m_{p{{K}^{-}}}^{2},m_{{{K}^{-}}{{\pi}^{+}}}^{2})$
$\displaystyle\qquad\qquad\times
D^{1/2*}_{\bar{\nu}_{0}^{\prime},\bar{\nu}^{\prime}}(\bar{\phi},\bar{\theta},\bar{\chi})D^{1/2}_{\bar{\nu}_{0},\bar{\nu}}(\bar{\phi},\bar{\theta},\bar{\chi})\,\mathfrak{X}_{\bar{\nu}^{\prime},\bar{\nu}}^{{\kern
1.25995pt\overline{\kern-1.25995pt\mathchar
28931\relax}}{}^{-}_{c}}(m_{\bar{p}{{K}^{+}}}^{2},m_{{{K}^{+}}{{\pi}^{-}}}^{2})\,,$
(33)

where $\mathfrak{X}$ is the matrix given by Eq. (7). If we integrate over the
degrees of freedom in the ${\mathchar 28931\relax}^{+}_{c}$ and ${\kern
1.79993pt\overline{\kern-1.79993pt\mathchar 28931\relax}}{}^{-}_{c}$ decays,
the expression reduces to a simple Dalitz-plot analysis,

$\displaystyle I^{\text{(just Dalitz)}}(m_{{{\mathchar
28931\relax}^{+}_{c}}{{K}^{+}}}^{2},m_{{{\kern
1.25995pt\overline{\kern-1.25995pt\mathchar
28931\relax}}{}^{-}_{c}}{{K}^{+}}}^{2})=\sum_{\nu_{0},\bar{\nu}_{0}}\left|O_{\nu_{0},\bar{\nu}_{0}}^{B}(m_{{{\mathchar
28931\relax}^{+}_{c}}{{K}^{+}}}^{2},m_{{{\kern
1.25995pt\overline{\kern-1.25995pt\mathchar
28931\relax}}{}^{-}_{c}}{{K}^{+}}}^{2})\right|^{2}\,.$ (34)

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## Supplementary material for LHCb-PAPER-2022-044

Numerical values for $I_{0}$, $\alpha_{x}$, $\alpha_{y}$, and $\alpha_{z}$
have been computed over a two-dimensional grid of the Dalitz plot coordinates
and are provided in easy-readable JSON files. The grid dimension is chosen to
be $100\\!\times\\!100$ values as a compromise between the precision and the
size of the output set. This supplemental material can be downloaded at
https://lc2pkpi-polarimetry.docs.cern.ch/_static/export/polarimetry-
field.tar.gz. Each JSON file provides the following data entries:

  * •

metadata: information about the amplitude model with which the field and
intensity was computed, including model description, parameters, and reference
subsystem that was used to align the amplitude models with Dalitz-plot
decomposition.

  * •

m^2_Kpi: an array of 100 values for $\sigma_{1}=m^{2}(K^{-}\pi^{+})$ that span
the $x$ axis of the Dalitz grid.

  * •

m^2_pK: an array of 100 values for $\sigma_{2}=m^{2}(pK^{-})$ that span the
$y$ axis of the Dalitz grid.

  * •

alpha_x, alpha_y, alpha_z: computed values for $\alpha_{x}$, $\alpha_{y}$,
$\alpha_{z}$ over the $100\\!\times\\!100$ grid array.

  * •

intensity: computed unpolarised intensity $I_{0}$ over the
$100\\!\times\\!100$ grid array.

Values on the grid that lie outside the phase space are given as NaN.